From 894f54e60d528b4503007ff8373bd6bc9d121fe4 Mon Sep 17 00:00:00 2001 From: Henry Date: Sun, 5 Jul 2026 18:45:22 -0700 Subject: [PATCH 1/4] Update mathlib4 import graph data and fix dot parser for isolated nodes Regenerate release/data/import_graph.txt from the current mathlib4 (v4.32.0-rc1), growing from 3763 to 8244 modules. The newer `lake exe graph` export emits isolated-node declaration lines ("X" [shape=ellipse];) for sink nodes, which the old parser choked on; gen_graph.py now handles both line forms. Co-Authored-By: Claude Sonnet 5 --- release/data/import_graph.txt | 23800 ++++++++++++++++++++++---------- script/gen_graph.py | 10 +- 2 files changed, 16389 insertions(+), 7421 deletions(-) diff --git a/release/data/import_graph.txt b/release/data/import_graph.txt index ea441c6..b2024f8 100644 --- a/release/data/import_graph.txt +++ b/release/data/import_graph.txt @@ -1,7527 +1,16489 @@ -3763 -Mathlib.Tactic.ApplyWith #404080 0 105.92252882497733 0.32231330369876443 -Mathlib.Tactic.Attr.Register #404080 -1 70.10416666666667 1.1160191259243795 -Mathlib.Tactic.CasesM #404080 -2 99.87447830537839 0.6021038234758789 -Mathlib.Tactic.Classical #404080 -3 87.54532885640924 0.8131729690018252 -Mathlib.Tactic.Clean #404080 -4 137.33333333333331 0.22255722064128597 -Mathlib.Tactic.Clear! #404080 -5 89.8438632579097 0.5248023347563459 -Mathlib.Tactic.ClearExcept #404080 -6 105.92252882497733 0.32231330369876443 -Mathlib.Tactic.Clear_ #404080 -7 105.92252882497733 0.32231330369876443 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8230 8231 +8224 0 +8225 0 +8226 0 +8227 1 8232 +8228 1 8241 +8229 2 8233 8234 +8230 0 +8231 0 +8232 2 8235 8236 +8233 3 8238 8237 8239 +8234 3 8238 8239 8240 +8235 0 +8236 0 +8237 1 8240 +8238 0 +8239 1 8241 +8240 0 +8241 2 8242 8243 +8242 0 +8243 0 diff --git a/script/gen_graph.py b/script/gen_graph.py index 2faf27b..543005d 100644 --- a/script/gen_graph.py +++ b/script/gen_graph.py @@ -37,8 +37,14 @@ with open(graph_path, 'r') as f: for line in f: if line.startswith(' '): - src, dst = line.strip().split(' -> ') - G.add_edge(src[1:-1], dst[1:-2]) + stripped = line.strip() + if ' -> ' in stripped: + src, dst = stripped.split(' -> ') + G.add_edge(src[1:-1], dst[1:-2]) + else: + # isolated node declaration with no outgoing edges, e.g. + # "Mathlib.X.Y" [shape=ellipse]; + G.add_node(stripped.split('"')[1]) G.remove_node('Mathlib') From 0cb986510cf52613fb5bd03e988e0d04fe8c279b Mon Sep 17 00:00:00 2001 From: Henry Date: Sun, 5 Jul 2026 18:45:30 -0700 Subject: [PATCH 2/4] Add mathematical kingdom 3D map data generation module New, independent data pipeline (kingdom/regions.yaml, kingdom/tier_overrides.yaml, script/gen_kingdom.py) that maps mathlib4's import graph onto mountains (topic regions) with two elevation scales: a heuristic macro_tier approximating abstraction level within a region (unification breadth + reverse PageRank, quantile-binned, with a manual override file for later curation), and micro_elevation for local dependency depth. Category theory is modeled as a shared summit layer floating above the other mountains. Regions are laid out with a golden-angle spiral packing so mountains never overlap. Full format documented in kingdom/schema.md, including known heuristic limitations. Output data lives in kingdom/data/kingdom_data.json and is fully decoupled from any renderer. Co-Authored-By: Claude Sonnet 5 --- kingdom/data/kingdom_data.json | 1 + kingdom/regions.yaml | 191 ++++++++++++++++ kingdom/schema.md | 93 ++++++++ kingdom/tier_overrides.yaml | 37 ++++ script/gen_kingdom.py | 386 +++++++++++++++++++++++++++++++++ 5 files changed, 708 insertions(+) create mode 100644 kingdom/data/kingdom_data.json create mode 100644 kingdom/regions.yaml create mode 100644 kingdom/schema.md create mode 100644 kingdom/tier_overrides.yaml create mode 100644 script/gen_kingdom.py diff --git a/kingdom/data/kingdom_data.json b/kingdom/data/kingdom_data.json new file mode 100644 index 0000000..73b4387 --- /dev/null +++ b/kingdom/data/kingdom_data.json @@ -0,0 +1 @@ +{"meta": {"source": "mathlib4", "toolchain": "leanprover/lean4:v4.32.0-rc1", "generated_at": "2026-07-05T06:47:30.822555+00:00", "node_count": 8244, "edge_count": 20865}, "regions": [{"id": "algebra", "name": "代数山", "color": "#ffff00", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 0.0, "z": 0.0}, "footprint_radius": 141.14432330065563, "tier_count": 6, "node_count": 2947}, {"id": "category_theory", "name": "范畴天空", "color": "#80a0ff", "is_summit_layer": true, "is_meta": false, "map_center": null, "footprint_radius": 85.76059701284734, "tier_count": 4, "node_count": 1088}, {"id": "topology", "name": "拓扑山", "color": "#ff00ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": -104.8926672403429, "z": 191.68340658285135}, "footprint_radius": 67.34953600434082, "tier_count": 6, "node_count": 671}, {"id": "analysis", "name": "分析山", "color": "#00ffff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 79.9841957298199, "z": -209.53168837541938}, "footprint_radius": 73.12427777421121, "tier_count": 6, "node_count": 791}, {"id": "measure_theory", "name": "测度之泽", "color": "#8000ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": -180.72905593888726, "z": -78.61938908080222}, "footprint_radius": 45.92515650490481, "tier_count": 5, "node_count": 312}, {"id": "probability", "name": "概率之海", "color": "#0000ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": -187.33258043995951, "z": 119.19943081116658}, "footprint_radius": 30.43221976787103, "tier_count": 4, "node_count": 137}, {"id": "dynamics", "name": "动力学山", "color": "#008040", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 187.0331041928763, "z": 132.17646513648592}, "footprint_radius": 15.160474926597782, "tier_count": 4, "node_count": 34}, {"id": "combinatorics", "name": "组合山", "color": "#800000", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 63.15373116965944, "z": 176.48684438039675}, "footprint_radius": 36.21380952067871, "tier_count": 5, "node_count": 194}, {"id": "order", "name": "序理论丘陵", "color": "#804000", "is_summit_layer": false, "is_meta": false, "map_center": {"x": -194.801068108119, "z": 30.68458674865894}, "footprint_radius": 45.92515650490481, "tier_count": 4, "node_count": 312}, {"id": "logic_set_theory", "name": "逻辑与集合基石", "color": "#0080ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": -31.024762780286682, "z": -210.68569978626203}, "footprint_radius": 27.515813635071744, "tier_count": 4, "node_count": 112}, {"id": "foundations_data", "name": "基础数据平原", "color": "#404040", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 213.4587560719115, "z": 40.3900910649158}, "footprint_radius": 66.08297814112194, "tier_count": 3, "node_count": 646}, {"id": "algebraic_geometry", "name": "代数几何山", "color": "#6040ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 210.94774597020242, "z": -67.32791746438433}, "footprint_radius": 30.543084323623898, "tier_count": 5, "node_count": 138}, {"id": "algebraic_topology", "name": "代数拓扑山", "color": "#ff80ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 148.98044036668762, "z": -107.66070958408105}, "footprint_radius": 32.57790662396834, "tier_count": 5, "node_count": 157}, {"id": "geometry", "name": "几何山", "color": "#ff8080", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 126.24457599466353, "z": 131.66361316600586}, "footprint_radius": 30.98257574831376, "tier_count": 5, "node_count": 142}, {"id": "computability", "name": "可计算性山", "color": "#bfff00", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 123.46583984486607, "z": 190.99525227450516}, "footprint_radius": 15.600000000000001, "tier_count": 4, "node_count": 36}, {"id": "model_theory", "name": "模型论山", "color": "#4080ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 167.9076905803438, "z": -165.42069835415253}, "footprint_radius": 15.160474926597782, "tier_count": 4, "node_count": 34}, {"id": "condensed", "name": "凝聚层塔", "color": "#ff0000", "is_summit_layer": false, "is_meta": false, "map_center": {"x": -185.0961031014865, "z": -150.75620258100128}, "footprint_radius": 15.160474926597782, "tier_count": 3, "node_count": 34}, {"id": 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"edge_count": 1}, {"from_region": "model_theory", "to_region": "computability", "edge_count": 1}, {"from_region": "probability", "to_region": "foundations_data", "edge_count": 1}, {"from_region": "analysis", "to_region": "algebraic_topology", "edge_count": 1}]} \ No newline at end of file diff --git a/kingdom/regions.yaml b/kingdom/regions.yaml new file mode 100644 index 0000000..02c830a --- /dev/null +++ b/kingdom/regions.yaml @@ -0,0 +1,191 @@ +# 数学王国地图 —— 山脉(区域)定义表 +# +# 这是对 script/gen_graph.py 里 color_and_rest_v 表的粗化重组: +# 把原来"每个 mathlib 顶层命名空间一个主题"的 26 条细分,合并成更少的 +# "山脉",让每座山脉能装下一整条从具体到抽象的阶梯(大海拔 tier 0..tier_count-1)。 +# +# 字段说明: +# id 区域唯一标识,用作 kingdom_data.json 里 node.region_id 的取值 +# name 显示名称 +# color 十六进制颜色,沿用原表配色习惯 +# namespace_prefixes 一个节点只要以 "Mathlib.." 开头就归入这个区域 +# (与 gen_graph.py 里 get_group 的匹配方式一致); +# 列表里第一个匹配到的区域生效,自上而下优先级从高到低 +# is_summit_layer true 表示这是悬浮在其他山脉之上的"天空层"(目前只有范畴论), +# 天空层节点不参与 map_center 的自身定位,而是取其主要依赖方 +# 区域的加权质心,macro_tier 会被强制抬到所有地面山脉之上 +# tier_count 这座山脉的大海拔(macro_tier)层数,用于分位数分箱 +# is_meta 可选,默认 false。true 表示这不是数学内容本身,而是 +# 元编程/证明战术工具箱(目前只有 Tactic), +# 渲染器可以选择用不同的视觉风格(比如画成工坊而非山脉) +# +# 匹配不到下面任何前缀的节点,一律落入代码里硬编码的 fallback 区域 "frontier" +# (未知边境),不需要在这里单独列出。 + +regions: + - id: algebra + name: "代数山" + color: "#ffff00" + namespace_prefixes: + - Algebra + - GroupTheory + - RingTheory + - FieldTheory + - RepresentationTheory + - NumberTheory + - LinearAlgebra + is_summit_layer: false + tier_count: 6 + + - id: category_theory + name: "范畴天空" + color: "#80a0ff" + namespace_prefixes: + - CategoryTheory + is_summit_layer: true + tier_count: 4 + + - id: topology + name: "拓扑山" + color: "#ff00ff" + namespace_prefixes: + - Topology + is_summit_layer: false + tier_count: 6 + + - id: analysis + name: "分析山" + color: "#00ffff" + namespace_prefixes: + - Analysis + is_summit_layer: false + tier_count: 6 + + - id: measure_theory + name: "测度之泽" + color: "#8000ff" + namespace_prefixes: + - MeasureTheory + is_summit_layer: false + tier_count: 5 + + - id: probability + name: "概率之海" + color: "#0000ff" + namespace_prefixes: + - Probability + is_summit_layer: false + tier_count: 4 + + - id: dynamics + name: "动力学山" + color: "#008040" + namespace_prefixes: + - Dynamics + is_summit_layer: false + tier_count: 4 + + - id: combinatorics + name: "组合山" + color: "#800000" + namespace_prefixes: + - Combinatorics + is_summit_layer: false + tier_count: 5 + + - id: order + name: "序理论丘陵" + color: "#804000" + namespace_prefixes: + - Order + is_summit_layer: false + tier_count: 4 + + - id: logic_set_theory + name: "逻辑与集合基石" + color: "#0080ff" + namespace_prefixes: + - Logic + - SetTheory + is_summit_layer: false + tier_count: 4 + + - id: foundations_data + name: "基础数据平原" + color: "#404040" + namespace_prefixes: + - Data + - Init + - Deprecated + - Mathport + is_summit_layer: false + tier_count: 3 + + - id: algebraic_geometry + name: "代数几何山" + color: "#6040ff" + namespace_prefixes: + - AlgebraicGeometry + is_summit_layer: false + tier_count: 5 + + - id: algebraic_topology + name: "代数拓扑山" + color: "#ff80ff" + namespace_prefixes: + - AlgebraicTopology + is_summit_layer: false + tier_count: 5 + + - id: geometry + name: "几何山" + color: "#ff8080" + namespace_prefixes: + - Geometry + is_summit_layer: false + tier_count: 5 + + - id: computability + name: "可计算性山" + color: "#bfff00" + namespace_prefixes: + - Computability + is_summit_layer: false + tier_count: 4 + + - id: model_theory + name: "模型论山" + color: "#4080ff" + namespace_prefixes: + - ModelTheory + is_summit_layer: false + tier_count: 4 + + - id: condensed + name: "凝聚层塔" + color: "#ff0000" + namespace_prefixes: + - Condensed + is_summit_layer: false + tier_count: 3 + + - id: information_theory + name: "信息论山" + color: "#8000ff" + namespace_prefixes: + - InformationTheory + is_summit_layer: false + tier_count: 3 + + - id: tactic + name: "战术工坊" + color: "#404080" + namespace_prefixes: + - Tactic + is_summit_layer: false + tier_count: 2 + is_meta: true + +# fallback 区域,代码里硬编码使用,不在上面 regions 列表中出现: +# id: frontier, name: "未知边境", color: "#202020", +# is_summit_layer: false, tier_count: 3 diff --git a/kingdom/schema.md b/kingdom/schema.md new file mode 100644 index 0000000..6793732 --- /dev/null +++ b/kingdom/schema.md @@ -0,0 +1,93 @@ +# 数学王国地图 —— 数据格式(`kingdom_data.json`) + +这份文档是**数据生成模块**(`script/gen_kingdom.py`)和**渲染模块**(目前尚未实现,未来可以是任何语言/引擎)之间唯一的契约。渲染器只需要读这份 JSON,完全不需要知道 mathlib、Lean、networkx 的存在。 + +数据由 `script/gen_kingdom.py` 从 mathlib4 的 `lake exe graph` 产出的 `import_graph.dot` 生成,山脉(区域)划分规则在 `kingdom/regions.yaml`,人工层级覆盖在 `kingdom/tier_overrides.yaml`。 + +## 顶层结构 + +```json +{ + "meta": { ... }, + "regions": [ ... ], + "nodes": [ ... ], + "edges": [ ... ], + "bridge_summary": [ ... ] +} +``` + +### `meta` + +| 字段 | 类型 | 说明 | +|---|---|---| +| `source` | string | 固定为 `"mathlib4"` | +| `toolchain` | string | 生成数据时 mathlib4 的 `lean-toolchain` 版本,如 `"leanprover/lean4:v4.32.0-rc1"` | +| `generated_at` | string (ISO 8601) | 生成时间戳 | +| `node_count` / `edge_count` | int | 冗余字段,方便渲染器/校验脚本快速核对而不用数组长度 | + +### `regions[]` —— 山脉 + +| 字段 | 类型 | 说明 | +|---|---|---| +| `id` | string | 唯一标识,对应 `kingdom/regions.yaml` 里的 `id`,或 fallback 区域 `"frontier"` | +| `name` | string | 显示名称,如 "代数山" | +| `color` | string (hex) | 建议配色 | +| `is_summit_layer` | bool | true = 悬浮在其他山脉之上的"天空层"(目前只有 `category_theory`) | +| `is_meta` | bool | true = 非数学内容,元编程/证明工具箱(目前只有 `tactic`) | +| `map_center` | `{x, z}` \| `null` | 这座山在地图平面上的中心坐标。**天空层区域此字段为 `null`**——它没有自己的地面位置,坐标要看它下面挂的节点各自的 `x, z`(见下) | +| `footprint_radius` | float \| `null` | 这座山脉节点分布的精确边界半径(`= LOCAL_SPREAD_FACTOR * sqrt(node_count)`,天空层为 `null`)。**山脉之间保证 `distance(centerA, centerB) >= footprint_radiusA + footprint_radiusB + margin`**,渲染器画"领地范围"时应该用这个精确值,而不是自己估算,否则可能画得比实际节点分布小,让节点看起来"溢出" | +| `tier_count` | int | 这座山脉大海拔的层数 | +| `node_count` | int | 落在这个区域里的节点数(冗余,便于校验) | + +### `nodes[]` —— mathlib 模块 = 地图上的一个点 + +| 字段 | 类型 | 说明 | +|---|---|---| +| `id` | string | 完整 mathlib 模块名,如 `"Mathlib.Algebra.Group.Defs"` | +| `region_id` | string | 所属山脉,对应 `regions[].id` | +| `micro_elevation` | float, 0.0–1.0 | **小海拔**:区域内最长依赖路径深度,归一化到 0~1。若 a 依赖 b,则 `micro_elevation(b) < micro_elevation(a)`(在同一区域内严格保证;跨区域依赖不参与这个排序,由 `is_bridge` 边表达) | +| `macro_tier` | int | **大海拔**:离散层级,范围 `[0, tier_count)`;天空层节点固定为 `SKY_BASE_TIER + 区域内局部 tier`(`SKY_BASE_TIER = 100`,保证任何天空层节点的 `macro_tier` 都高于任何地面山脉),用来保证"范畴论永远悬浮在所有地面山脉之上" | +| `macro_tier_score` | float | 分箱之前的连续启发式分数(见下方"大海拔启发式"),保留下来是为了以后调整分箱边界或做手工校准时可以复查 | +| `macro_tier_override` | int \| `null` | 非 `null` 时表示这个层级来自 `kingdom/tier_overrides.yaml` 的人工指定,而不是启发式计算 | +| `x`, `z` | float | 节点在地图平面上的坐标。地面山脉节点:`map_center` + 一个半径不超过 `footprint_radius` 的局部极坐标偏移(半径 = 归一化的 `micro_elevation`,角度 = 节点名的稳定哈希),保证同一座山的所有节点都落在它自己的 `footprint_radius` 圆内,不会侵入相邻山脉。天空层节点:其**依赖方各区域 `map_center` 按引用次数加权的质心**——被代数大量引用的范畴论节点,坐标会漂移到代数山上空 | +| `size` | float | 节点大小,直接复用 `gen_graph.py` 里已经在算的全局反向 PageRank 半径公式 | + +### `edges[]` —— import 依赖 + +| 字段 | 类型 | 说明 | +|---|---|---| +| `from`, `to` | string | 模块名,`from` 依赖 `to`(`from` 的 `micro_elevation`/`macro_tier` 应该 ≥ `to`,同区域内严格保证) | +| `is_bridge` | bool | `region_id(from) != region_id(to)` 时为 true,即跨山脉的边——地图上画成连接两座山峰的桥梁 | + +### `bridge_summary[]` —— 粗粒度桥梁汇总 + +```json +{"from_region": "category_theory", "to_region": "algebra", "edge_count": 143} +``` + +按 `(from_region, to_region)` 聚合的跨区边数量,渲染器可以只画这张汇总表里排名靠前的"主桥",而不必渲染全部两万条边里的每一条跨区边。 + +## 山脉布局算法 —— 保证互不重叠 + +每座地面山脉的节点分布被约束在一个半径精确等于 `footprint_radius` 的圆盘内(节点局部坐标用极坐标生成:半径 = 归一化 `micro_elevation` × `footprint_radius`,角度 = 节点名的稳定哈希)。有了这个精确边界,山脉中心 `map_center` 就可以用一个简单的黄金角螺旋摆放算法逐个放置:按节点数从大到小排序,依次沿螺旋线寻找第一个满足 `distance(新山, 已放置的每一座山) >= 两者 footprint_radius 之和 + 10` 的位置。这保证了**任意两座地面山脉的节点分布圆盘一定不重叠**,代价是不再像早期版本那样用"跨区边数量"驱动布局去让联系紧密的山脉互相靠近——保证不重叠现在优先于这一点。跨区边的强弱关系仍然完整保留在 `bridge_summary` 里,只是不再影响山脉摆放位置。 + +## 大海拔(`macro_tier`)启发式 —— 已知局限性,请务必阅读 + +大海拔本应表达"这一层比上一层在做**推广(Generalization)/统一(Unification)/遗忘(Abstraction)**"当中至少两条,这是一个数学教学法/编辑判断,**不是能从 import 依赖图直接算出来的东西**。当前实现用两个图论信号近似: + +1. **统一广度**(近似 Unification):把区域内节点按第三级命名空间分成若干"子理论簇"(比如代数山里 `Group`/`Ring`/`Field`/`NumberTheory` 各是一簇),统计有多少个不同的子理论簇最终依赖于某节点。依赖它的子簇越多,越像是在"缝合"多条原本独立的理论。 +2. **奠基广度**(近似 Generalization/Abstraction):直接复用 `gen_graph.py` 里已经在算的反向 PageRank——有多少东西直接或间接建立在它之上。 + +`macro_tier_score = normalize(统一广度) + normalize(奠基广度)`,再用分位数分箱(`numpy.percentile`,箱数取 `regions.yaml` 里的 `tier_count`)映射成整数 `macro_tier`。 + +**这套启发式大概率不会精确复现"数 → 同余 → 群 → 环 → 域 → Galois"这种叙事**。典型的失配例子:mathlib 里 `Nat`/`Int` 这类具体数系的反向 PageRank 几乎是全库最高(几乎所有东西都直接或间接用到自然数),启发式会把它们排得比 `Group.Defs` 更"奠基",但这不代表 `Nat` 在数学抽象阶梯上应该排在 `Group` 之上——恰恰相反,用户设想的阶梯里 `Nat` 应该在最底层。 + +因此: + +- `macro_tier_score` 字段被完整保留在输出里,方便以后复查/重新分箱。 +- 任何时候想手工纠正,不需要改数据格式或重写生成逻辑——直接在 `kingdom/tier_overrides.yaml` 里给该模块指定一个层级,重新跑一次 `gen_kingdom.py` 即可,该节点的 `macro_tier` 会直接取覆盖值,`macro_tier_override` 字段也会同步记录这一点。 +- 目前 `tier_overrides.yaml` 是空的,这是本阶段有意为之的产物——数据管线已经打好地基,数学内容的人工校准是下一阶段的工作。 + +## 与现有 2D 数据管线的关系 + +`kingdom_data.json` 是一条完全独立的数据管线,和现有的 `script/gen_graph.py` / `release/data/import_graph.txt` / bgfx 渲染器**互不影响、互不依赖**。两者共享同一份上游数据来源(`lake exe graph` 生成的 `import_graph.dot`),但下游各自处理,互相之间没有代码或数据格式上的耦合。 diff --git a/kingdom/tier_overrides.yaml b/kingdom/tier_overrides.yaml new file mode 100644 index 0000000..519315f --- /dev/null +++ b/kingdom/tier_overrides.yaml @@ -0,0 +1,37 @@ +# 人工大海拔(macro_tier)覆盖表 —— 目前为空。 +# +# script/gen_kingdom.py 里的 macro_tier 是用图论启发式(统一广度 + 反向 PageRank) +# 自动估算的,并不保证精确复现"数 -> 同余 -> 群 -> 环 -> 域 -> Galois -> 范畴" +# 这类数学教学法叙事(细节和局限性见 kingdom/schema.md)。 +# +# 以后如果想手工钦定某个具体概念该在第几层,在这里加一行: +# +# Mathlib.Algebra.Group.Defs: 1 +# Mathlib.FieldTheory.Galois.Basic: 4 +# +# 键是完整的 mathlib 模块名,值是该模块所在区域里的目标 tier(从 0 开始, +# 不能超过 regions.yaml 里对应区域的 tier_count - 1)。 +# gen_kingdom.py 读取这个文件时,任何在这里出现的节点都会跳过启发式计算, +# 直接使用这里指定的值,并在输出的 kingdom_data.json 里把该节点的 +# macro_tier_override 字段设成这个值(而不是 null)。 + +overrides: + # 代数山阶梯的第一轮人工校准:把 Group/Ring/Field 的基础定义压到阶梯低位, + # 把 Galois 理论簇整体抬到它们之上,修正启发式把"被广泛依赖的 Defs 文件" + # 误判为"最奠基"从而排到 Galois 之上的问题(具体案例见 kingdom/schema.md)。 + # algebra 山脉 tier_count=6,合法取值 0..5。 + Mathlib.Algebra.Group.Defs: 1 + Mathlib.Algebra.Ring.Defs: 2 + Mathlib.Algebra.Field.Defs: 3 + Mathlib.FieldTheory.Galois.Basic: 5 + Mathlib.FieldTheory.Galois.Abelian: 5 + Mathlib.FieldTheory.Galois.GaloisClosure: 5 + Mathlib.FieldTheory.Galois.Infinite: 5 + Mathlib.FieldTheory.Galois.IsGaloisGroup: 5 + Mathlib.FieldTheory.Galois.NormalBasis: 5 + Mathlib.FieldTheory.Galois.Notation: 5 + Mathlib.FieldTheory.Galois.Profinite: 5 + Mathlib.FieldTheory.Finite.GaloisField: 5 + Mathlib.RingTheory.Invariant.Galois: 5 + Mathlib.NumberTheory.NumberField.Cyclotomic.Galois: 5 + Mathlib.NumberTheory.RamificationInertia.Galois: 5 diff --git a/script/gen_kingdom.py b/script/gen_kingdom.py new file mode 100644 index 0000000..8458083 --- /dev/null +++ b/script/gen_kingdom.py @@ -0,0 +1,386 @@ +""" +Generate kingdom/data/kingdom_data.json ("数学王国地图") from a mathlib4 +`lake exe graph` import_graph.dot export. + +This is a sibling pipeline to gen_graph.py, not a replacement: it shares the +same upstream `import_graph.dot`, but produces a completely independent +JSON artifact (schema documented in kingdom/schema.md). Rendering is +intentionally out of scope here -- see kingdom/schema.md for the contract. + +Usage (run with cwd = repo root, same convention as gen_graph.py): + python script/gen_kingdom.py +""" +import os +import re +import sys +import math +import json +import hashlib +import datetime +import networkx as nx +import numpy as np +import yaml + +if hasattr(sys.stdout, 'reconfigure'): + sys.stdout.reconfigure(encoding='utf-8') + +REPO_ROOT = os.path.abspath(os.path.dirname(os.path.dirname(__file__))) +mathlib_src_path = os.path.abspath(os.path.join(REPO_ROOT, '..', 'mathlib4')) +graph_path = os.path.join(mathlib_src_path, 'import_graph.dot') +toolchain_path = os.path.join(mathlib_src_path, 'lean-toolchain') +regions_config_path = os.path.join(REPO_ROOT, 'kingdom', 'regions.yaml') +overrides_config_path = os.path.join(REPO_ROOT, 'kingdom', 'tier_overrides.yaml') +output_path = os.path.join(REPO_ROOT, 'kingdom', 'data', 'kingdom_data.json') + +FRONTIER_REGION = { + 'id': 'frontier', 'name': '未知边境', 'color': '#202020', + 'is_summit_layer': False, 'is_meta': False, 'tier_count': 3, + 'namespace_prefixes': [], +} +SKY_BASE_TIER = 100 # any summit-layer macro_tier is offset above this +LOCAL_SPREAD_FACTOR = 2.6 # per-region footprint radius = this * sqrt(node_count) +MOUNTAIN_MARGIN = 10.0 # minimum gap kept between any two mountains' footprints +GOLDEN_ANGLE = math.pi * (3 - math.sqrt(5)) + + +# --------------------------------------------------------------------------- +# 1. Parse import_graph.dot +# (handles both "a" -> "b"; edges and the newer "x" [shape=ellipse]; +# isolated-node declaration lines that the current importGraph exporter +# emits for sink nodes with no outgoing edges.) +# --------------------------------------------------------------------------- +edge_re = re.compile(r'^"(.+)" -> "(.+)";?$') +node_re = re.compile(r'^"(.+)" \[shape=ellipse\];?$') + +G = nx.DiGraph() +with open(graph_path, 'r', encoding='utf-8') as f: + for line in f: + line = line.strip() + m = edge_re.match(line) + if m: + G.add_edge(m.group(1), m.group(2)) + continue + m = node_re.match(line) + if m: + G.add_node(m.group(1)) + +if 'Mathlib' in G: + G.remove_node('Mathlib') + +print(f'# of nodes: {G.number_of_nodes()}') +print(f'# of edges: {G.number_of_edges()}') + +# NOTE on edge direction: a dot edge "u" -> "v" means v imports u, i.e. +# v depends on u (u is upstream/foundational, v is downstream/built-on-top). +# This is the same convention gen_graph.py relies on (nx.ancestors(node) is +# used there as "the set of things node depends on"). + + +# --------------------------------------------------------------------------- +# 2. Load region config + tier overrides +# --------------------------------------------------------------------------- +with open(regions_config_path, 'r', encoding='utf-8') as f: + regions_cfg = yaml.safe_load(f)['regions'] +for r in regions_cfg: + r.setdefault('is_meta', False) + +with open(overrides_config_path, 'r', encoding='utf-8') as f: + tier_overrides = yaml.safe_load(f).get('overrides') or {} + + +def assign_region(node): + for r in regions_cfg: + for prefix in r['namespace_prefixes']: + if node.startswith(f'Mathlib.{prefix}.'): + return r['id'] + return FRONTIER_REGION['id'] + + +region_by_id = {r['id']: r for r in regions_cfg} +region_by_id[FRONTIER_REGION['id']] = FRONTIER_REGION + +node_region = {n: assign_region(n) for n in G.nodes} + +nodes_by_region = {} +for n, rid in node_region.items(): + nodes_by_region.setdefault(rid, []).append(n) + + +# --------------------------------------------------------------------------- +# 3. Global reverse-PageRank (reused both as node "size" and as one of the +# two macro_tier heuristic signals -- same computation gen_graph.py +# already relies on for node radius). +# --------------------------------------------------------------------------- +page_rank = nx.pagerank(G.reverse()) +pr_max, pr_min = max(page_rank.values()), min(page_rank.values()) + + +def stable_hash01(name): + """Deterministic hash -> [0, 1), stable across processes/platforms + (unlike builtin hash(), which is salted per-process).""" + h = hashlib.md5(name.encode('utf-8')).hexdigest() + return int(h[:8], 16) / 0xFFFFFFFF + + +def cluster_key(node, depth=3): + parts = node.split('.') + return '.'.join(parts[:min(len(parts), depth)]) + + +# --------------------------------------------------------------------------- +# 4. Per-region computation: micro_elevation (local dependency depth) and +# the macro_tier heuristic (unification breadth + reverse-pagerank). +# --------------------------------------------------------------------------- +node_data = {} # id -> dict of computed fields (filled in below) +region_footprint_radius = {} # rid -> exact bounding radius of its node spread + +for rid, members in nodes_by_region.items(): + region = region_by_id[rid] + sub = G.subgraph(members) + topo = list(nx.topological_sort(sub)) + + # --- micro_elevation: longest path depth from region-local roots --- + depth = {} + for n in topo: + preds = list(sub.predecessors(n)) + depth[n] = 0 if not preds else 1 + max(depth[p] for p in preds) + max_depth = max(depth.values()) if depth else 0 + micro_elevation = {n: (depth[n] / max_depth if max_depth > 0 else 0.0) for n in members} + + # --- unification breadth: distinct sub-theory clusters among a node's + # descendants, computed bottom-up in reverse topological order so + # each node's cluster-set is a single union of its direct + # successors' already-computed sets (O(V+E), no repeated traversal) + desc_clusters = {} + for n in reversed(topo): + s = set() + for succ in sub.successors(n): + s.add(cluster_key(succ)) + s |= desc_clusters[succ] + desc_clusters[n] = s + unification_breadth = {n: len(desc_clusters[n]) for n in members} + + # --- combine + normalize within region, then quantile-bin into tiers --- + ub_vals = np.array([unification_breadth[n] for n in members], dtype=float) + pr_vals = np.array([(page_rank[n] - pr_min) / (pr_max - pr_min) if pr_max > pr_min else 0.0 + for n in members], dtype=float) + ub_norm = (ub_vals - ub_vals.min()) / (ub_vals.max() - ub_vals.min()) if ub_vals.max() > ub_vals.min() else ub_vals * 0 + score = 0.5 * ub_norm + 0.5 * pr_vals + + tier_count = region['tier_count'] + boundaries = np.unique(np.percentile(score, np.linspace(0, 100, tier_count + 1))) + if len(boundaries) < 2: + tiers = np.zeros(len(score), dtype=int) + else: + tiers = np.clip(np.digitize(score, boundaries[1:-1], right=True), 0, len(boundaries) - 2) + + # --- local planar spread: a bounded polar disc (radius = normalized + # micro_elevation, angle = a stable hash) so every region's + # footprint has an exact, predictable radius -- this is what makes + # guaranteed non-overlapping mountain placement possible below. + # Scaling by sqrt(region size) keeps node density comparable + # across differently-sized mountains. --- + footprint_radius = LOCAL_SPREAD_FACTOR * max(1.0, len(members) ** 0.5) + region_footprint_radius[rid] = footprint_radius + + for i, n in enumerate(members): + angle = stable_hash01(n) * 2 * math.pi + local_r = micro_elevation[n] * footprint_radius + node_data[n] = { + 'region_id': rid, + 'micro_elevation': micro_elevation[n], + 'macro_tier_score': float(score[i]), + 'macro_tier_heuristic': int(tiers[i]), + 'local_depth': depth[n], + 'local_x': local_r * math.cos(angle), + 'local_z': local_r * math.sin(angle), + } + +# apply summit-layer tier offset +for rid, members in nodes_by_region.items(): + if region_by_id[rid]['is_summit_layer']: + for n in members: + node_data[n]['macro_tier_heuristic'] += SKY_BASE_TIER + +# apply manual overrides +for n, override_tier in tier_overrides.items(): + if n in node_data: + node_data[n]['macro_tier_override'] = int(override_tier) + else: + print(f'WARNING: tier_overrides.yaml references unknown node {n!r}, ignoring') +for n in node_data: + node_data[n].setdefault('macro_tier_override', None) + node_data[n]['macro_tier'] = ( + node_data[n]['macro_tier_override'] + if node_data[n]['macro_tier_override'] is not None + else node_data[n]['macro_tier_heuristic'] + ) + + +# --------------------------------------------------------------------------- +# 5. Region map layout: place every ground mountain with a guaranteed +# non-overlapping golden-angle spiral packing (each region's footprint +# is an exact circle of radius region_footprint_radius[rid], so two +# regions overlap iff the distance between their centers is less than +# the sum of their radii + margin -- this is checked explicitly below, +# not left to a force-directed layout that ignores footprint size). +# Regions are placed largest-first so the biggest mountain anchors the +# center of the map and smaller ones spiral out around it. +# --------------------------------------------------------------------------- +ground_region_ids = [rid for rid, r in region_by_id.items() if not r['is_summit_layer']] +ground_region_ids.sort(key=lambda rid: -len(nodes_by_region.get(rid, []))) + +placed_circles = [] # list of (x, z, radius) +map_center = {} +for rid in ground_region_ids: + r = region_footprint_radius.get(rid, LOCAL_SPREAD_FACTOR) + if not placed_circles: + map_center[rid] = (0.0, 0.0) + placed_circles.append((0.0, 0.0, r)) + continue + t = 0 + while True: + t += 1 + angle = t * GOLDEN_ANGLE + dist = 3.0 * math.sqrt(t) + x, z = dist * math.cos(angle), dist * math.sin(angle) + if all(math.hypot(x - px, z - pz) >= (pr + r + MOUNTAIN_MARGIN) for px, pz, pr in placed_circles): + map_center[rid] = (x, z) + placed_circles.append((x, z, r)) + break + +# cross-region edge counts are no longer used to drive placement (guaranteed +# separation now takes priority over connectivity-based proximity), but are +# still reported in bridge_summary further down from the raw edge list. + + +# --------------------------------------------------------------------------- +# 6. Summit-layer node placement: weighted centroid of the map_centers of +# the (non-summit) regions that depend on each summit node, computed in +# a single pass over all edges (rather than per-node successor scans). +# --------------------------------------------------------------------------- +summit_region_ids = {rid for rid in nodes_by_region if region_by_id[rid]['is_summit_layer']} +summit_dependent_weight = {} # node -> {region_id: count} +for u, v in G.edges(): + # v depends on u; if u is a summit node and v's region differs, v's region "uses" u + if node_region.get(u) in summit_region_ids and node_region[v] not in summit_region_ids: + summit_dependent_weight.setdefault(u, {}) + summit_dependent_weight[u][node_region[v]] = summit_dependent_weight[u].get(node_region[v], 0) + 1 + +fallback_center = ( + float(np.mean([c[0] for c in map_center.values()])) if map_center else 0.0, + float(np.mean([c[1] for c in map_center.values()])) if map_center else 0.0, +) + +for n in node_data: + if node_data[n]['region_id'] in summit_region_ids: + weights = summit_dependent_weight.get(n) + if weights: + total = sum(weights.values()) + cx = sum(map_center[r][0] * w for r, w in weights.items()) / total + cz = sum(map_center[r][1] * w for r, w in weights.items()) / total + node_data[n]['x'] = cx + node_data[n]['local_z'] + node_data[n]['z'] = cz + node_data[n]['local_depth'] + else: + node_data[n]['x'] = fallback_center[0] + node_data[n]['local_z'] + node_data[n]['z'] = fallback_center[1] + node_data[n]['local_depth'] + else: + cx, cz = map_center[node_data[n]['region_id']] + node_data[n]['x'] = cx + node_data[n]['local_x'] + node_data[n]['z'] = cz + node_data[n]['local_z'] + + +# --------------------------------------------------------------------------- +# 7. Assemble output +# --------------------------------------------------------------------------- +toolchain = '' +if os.path.exists(toolchain_path): + with open(toolchain_path, 'r', encoding='utf-8') as f: + toolchain = f.read().strip() + +regions_out = [] +for rid, r in region_by_id.items(): + regions_out.append({ + 'id': rid, + 'name': r['name'], + 'color': r['color'], + 'is_summit_layer': r['is_summit_layer'], + 'is_meta': r['is_meta'], + 'map_center': None if r['is_summit_layer'] else { + 'x': map_center[rid][0], 'z': map_center[rid][1], + }, + 'footprint_radius': region_footprint_radius.get(rid), + 'tier_count': r['tier_count'], + 'node_count': len(nodes_by_region.get(rid, [])), + }) + +nodes_out = [] +for n, d in node_data.items(): + rank = page_rank[n] + size = 0.2 + 3 * ((rank - pr_min) / (pr_max - pr_min)) ** 0.5 if pr_max > pr_min else 0.2 + nodes_out.append({ + 'id': n, + 'region_id': d['region_id'], + 'micro_elevation': round(d['micro_elevation'], 4), + 'macro_tier': d['macro_tier'], + 'macro_tier_score': round(d['macro_tier_score'], 4), + 'macro_tier_override': d['macro_tier_override'], + 'x': round(d['x'], 3), + 'z': round(d['z'], 3), + 'size': round(size, 4), + }) + +edges_out = [] +bridge_summary_counter = {} +for u, v in G.edges(): + from_n, to_n = v, u # v depends on u + from_region, to_region = node_region[v], node_region[u] + is_bridge = from_region != to_region + edges_out.append({'from': from_n, 'to': to_n, 'is_bridge': is_bridge}) + if is_bridge: + key = (from_region, to_region) + bridge_summary_counter[key] = bridge_summary_counter.get(key, 0) + 1 + +bridge_summary_out = [ + {'from_region': ra, 'to_region': rb, 'edge_count': cnt} + for (ra, rb), cnt in sorted(bridge_summary_counter.items(), key=lambda kv: -kv[1]) +] + +output = { + 'meta': { + 'source': 'mathlib4', + 'toolchain': toolchain, + 'generated_at': datetime.datetime.now(datetime.timezone.utc).isoformat(), + 'node_count': len(nodes_out), + 'edge_count': len(edges_out), + }, + 'regions': regions_out, + 'nodes': nodes_out, + 'edges': edges_out, + 'bridge_summary': bridge_summary_out, +} + +os.makedirs(os.path.dirname(output_path), exist_ok=True) +with open(output_path, 'w', encoding='utf-8') as f: + json.dump(output, f, ensure_ascii=False, indent=None) + +print(f'wrote {output_path}') + + +# --------------------------------------------------------------------------- +# 8. Sanity report +# --------------------------------------------------------------------------- +print('\n--- per-region tier histogram ---') +for r in regions_out: + rid = r['id'] + tiers = [n['macro_tier'] for n in nodes_out if n['region_id'] == rid] + if not tiers: + continue + hist = {} + for t in tiers: + hist[t] = hist.get(t, 0) + 1 + print(f"{r['name']:12s} ({rid:20s}) n={len(tiers):5d} tiers={dict(sorted(hist.items()))}") + +print('\n--- top bridge_summary pairs ---') +for b in bridge_summary_out[:10]: + print(f" {b['from_region']:20s} -> {b['to_region']:20s}: {b['edge_count']}") From 4d84634cf5249fb582405094afe23be69423367b Mon Sep 17 00:00:00 2001 From: Henry Date: Thu, 9 Jul 2026 10:27:40 -0700 Subject: [PATCH 3/4] Extract real concept summaries and doc links from mathlib4 source gen_kingdom.py now pulls each module's own /-! ... -/ doc-string straight out of the local mathlib4 checkout (title + intro paragraph, truncated), plus a deterministic mathlib4_docs URL. 96% of the 8244 nodes get a title, 83% get a summary; both are real source content, not generated. Documented in kingdom/schema.md. Co-Authored-By: Claude Sonnet 5 --- kingdom/data/kingdom_data.json | 2 +- kingdom/schema.md | 3 +++ script/gen_kingdom.py | 36 ++++++++++++++++++++++++++++++++++ 3 files changed, 40 insertions(+), 1 deletion(-) diff --git a/kingdom/data/kingdom_data.json b/kingdom/data/kingdom_data.json index 73b4387..b5f10cc 100644 --- a/kingdom/data/kingdom_data.json +++ b/kingdom/data/kingdom_data.json @@ -1 +1 @@ -{"meta": {"source": "mathlib4", "toolchain": "leanprover/lean4:v4.32.0-rc1", "generated_at": "2026-07-05T06:47:30.822555+00:00", "node_count": 8244, "edge_count": 20865}, "regions": [{"id": "algebra", "name": "代数山", "color": "#ffff00", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 0.0, "z": 0.0}, "footprint_radius": 141.14432330065563, "tier_count": 6, "node_count": 2947}, {"id": "category_theory", "name": "范畴天空", "color": "#80a0ff", "is_summit_layer": true, "is_meta": false, "map_center": null, "footprint_radius": 85.76059701284734, "tier_count": 4, "node_count": 1088}, {"id": "topology", "name": "拓扑山", "color": "#ff00ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": -104.8926672403429, "z": 191.68340658285135}, "footprint_radius": 67.34953600434082, "tier_count": 6, "node_count": 671}, {"id": "analysis", "name": "分析山", "color": "#00ffff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 79.9841957298199, "z": -209.53168837541938}, "footprint_radius": 73.12427777421121, "tier_count": 6, "node_count": 791}, {"id": "measure_theory", "name": "测度之泽", "color": "#8000ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": -180.72905593888726, "z": -78.61938908080222}, "footprint_radius": 45.92515650490481, "tier_count": 5, "node_count": 312}, {"id": "probability", "name": "概率之海", "color": "#0000ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": -187.33258043995951, "z": 119.19943081116658}, "footprint_radius": 30.43221976787103, "tier_count": 4, "node_count": 137}, {"id": "dynamics", "name": "动力学山", "color": "#008040", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 187.0331041928763, "z": 132.17646513648592}, "footprint_radius": 15.160474926597782, "tier_count": 4, "node_count": 34}, {"id": "combinatorics", "name": "组合山", "color": "#800000", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 63.15373116965944, "z": 176.48684438039675}, "footprint_radius": 36.21380952067871, "tier_count": 5, "node_count": 194}, {"id": "order", "name": "序理论丘陵", "color": "#804000", "is_summit_layer": false, "is_meta": false, "map_center": {"x": -194.801068108119, "z": 30.68458674865894}, "footprint_radius": 45.92515650490481, "tier_count": 4, "node_count": 312}, {"id": "logic_set_theory", "name": "逻辑与集合基石", "color": "#0080ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": -31.024762780286682, "z": -210.68569978626203}, "footprint_radius": 27.515813635071744, "tier_count": 4, "node_count": 112}, {"id": "foundations_data", "name": "基础数据平原", "color": "#404040", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 213.4587560719115, "z": 40.3900910649158}, "footprint_radius": 66.08297814112194, "tier_count": 3, "node_count": 646}, {"id": "algebraic_geometry", "name": "代数几何山", "color": "#6040ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 210.94774597020242, "z": -67.32791746438433}, "footprint_radius": 30.543084323623898, "tier_count": 5, "node_count": 138}, {"id": "algebraic_topology", "name": "代数拓扑山", "color": "#ff80ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 148.98044036668762, "z": -107.66070958408105}, "footprint_radius": 32.57790662396834, "tier_count": 5, "node_count": 157}, {"id": "geometry", "name": "几何山", "color": "#ff8080", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 126.24457599466353, "z": 131.66361316600586}, "footprint_radius": 30.98257574831376, "tier_count": 5, "node_count": 142}, {"id": "computability", "name": "可计算性山", "color": "#bfff00", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 123.46583984486607, "z": 190.99525227450516}, "footprint_radius": 15.600000000000001, "tier_count": 4, "node_count": 36}, {"id": "model_theory", "name": "模型论山", "color": "#4080ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 167.9076905803438, "z": -165.42069835415253}, "footprint_radius": 15.160474926597782, "tier_count": 4, "node_count": 34}, {"id": "condensed", "name": "凝聚层塔", "color": "#ff0000", "is_summit_layer": false, "is_meta": false, "map_center": {"x": -185.0961031014865, "z": -150.75620258100128}, "footprint_radius": 15.160474926597782, "tier_count": 3, "node_count": 34}, {"id": "information_theory", "name": "信息论山", "color": "#8000ff", "is_summit_layer": false, "is_meta": false, "map_center": {"x": 137.897132011464, "z": 76.14053442820632}, "footprint_radius": 6.368673331236263, "tier_count": 3, "node_count": 6}, {"id": "tactic", "name": "战术工坊", "color": "#404080", "is_summit_layer": false, "is_meta": true, "map_center": {"x": -108.79655362873866, "z": -168.10208183871188}, "footprint_radius": 49.05670188669434, "tier_count": 2, "node_count": 356}, {"id": "frontier", "name": "未知边境", "color": "#202020", "is_summit_layer": false, "is_meta": false, "map_center": {"x": -1.9646108139449154, "z": 209.9693794445984}, "footprint_radius": 25.60703028466987, "tier_count": 3, "node_count": 97}], "nodes": [{"id": "Mathlib.Algebra.Order.Module.Equiv", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -13.004, "z": -30.796, "size": 0.2298}, {"id": "Mathlib.Algebra.Module.Equiv.Basic", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 4, "macro_tier_score": 0.392, "macro_tier_override": null, "x": 30.376, "z": 8.606, "size": 0.5343}, {"id": "Mathlib.Algebra.Order.Group.Equiv", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 1, "macro_tier_score": 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Note that the `SMul` instances are already defined in `Mathlib/Algebra/Order/Group/Synonym.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Module/Synonym.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Determinant.Misc", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 9.161, "z": -62.475, "size": 0.2511, "title": "Miscellaneous results about determinant", "summary": "In this file, we collect various formulas about determinant of matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Determinant/Misc.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 3, "macro_tier_score": 0.2078, "macro_tier_override": null, "x": 24.281, "z": -56.271, "size": 0.4219, "title": "Determinant of a matrix", "summary": "This file defines the determinant of a matrix, `Matrix.det`, and its essential properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Determinant/Basic.html"}, {"id": "Mathlib.Algebra.Ring.NegOnePow", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0229, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.4286, "title": "Integer powers of (-1)", "summary": "This file defines the map `negOnePow : ℤ → ℤˣ` which sends `n` to `(-1 : ℤˣ) ^ n`. The definition of `negOnePow` and some lemmas first appeared in contributions by Johan Commelin to the Liquid Tensor Experiment.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/NegOnePow.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Stochastic", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 1.83, "z": 0.315, "size": 0.2765, "title": "Row- and Column-stochastic matrices", "summary": "A square matrix `M` is *row-stochastic* if all its entries are non-negative and `M *ᵥ 1 = 1`. Likewise, `M` is *column-stochastic* if all its entries are non-negative and `1 ᵥ* M = 1`. This file defines these concepts and provides basic API for them. Note that *doubly stochastic* matrices (i.e. matrices that are both row- and column-stochastic) are defined in `Mathlib/Analysis/Convex/DoublyStochasticMatrix.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Stochastic.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Permutation", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0019, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.3186, "title": "Permutation matrices", "summary": "This file defines the matrix associated with a permutation", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Permutation.html"}, {"id": "Mathlib.RingTheory.RingHom.FiniteType", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 2, "macro_tier_score": 0.0084, "macro_tier_override": null, "x": 20.891, "z": 82.836, "size": 0.3071, "title": "The meta properties of finite-type ring homomorphisms.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/FiniteType.html"}, {"id": "Mathlib.RingTheory.FiniteStability", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.025, "macro_tier_override": null, "x": -78.274, "z": 15.826, "size": 0.3088, "title": "Stability of finiteness conditions in commutative algebra", "summary": "In this file we show that `Algebra.FiniteType` and `Algebra.FinitePresentation` are stable under base change.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/FiniteStability.html"}, {"id": "Mathlib.RingTheory.Finiteness.FiniteTypeLocal", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 2, "macro_tier_score": 0.0083, "macro_tier_override": null, "x": -30.843, "z": -71.644, "size": 0.2988, "title": "Locality of `Algebra.FiniteType`", "summary": "In this file we show that finite-type is local on the source and the target.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/FiniteTypeLocal.html"}, {"id": "Mathlib.RingTheory.Localization.InvSubmonoid", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 2, "macro_tier_score": 0.0219, "macro_tier_override": null, "x": -75.897, "z": 17.994, "size": 0.3191, "title": "Submonoid of inverses", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/InvSubmonoid.html"}, {"id": "Mathlib.RingTheory.RingHom.Finite", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 2, "macro_tier_score": 0.0183, "macro_tier_override": null, "x": -83.411, "z": -5.192, "size": 0.297, "title": "The meta properties of finite ring homomorphisms.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/Finite.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4196, "macro_tier_override": null, "x": -7.986, "z": -16.767, "size": 0.3017, "title": "Pointwise operations of sets in a group with zero", "summary": "This file proves properties of pointwise operations of sets in a group with zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Pointwise/Set/Basic.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Basic", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4698, "macro_tier_override": null, "x": -0.671, "z": 9.262, "size": 0.5679, "title": "Groups with an adjoined zero element", "summary": "This file describes structures that are not usually studied on their own right in mathematics, namely a special sort of monoid: apart from a distinguished “zero element” they form a group, or in other words, they are groups with an adjoined zero element. Examples are: * division rings; * the value monoid of a multiplicative valuation; * in particular, the non-negative real numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Basic.html"}, {"id": "Mathlib.Algebra.Group.Pointwise.Set.Basic", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4567, "macro_tier_override": null, "x": -7.526, "z": 14.924, "size": 0.5982, "title": "Pointwise operations of sets", "summary": "This file defines pointwise algebraic operations on sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pointwise/Set/Basic.html"}, {"id": "Mathlib.Algebra.PEmptyInstances", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -5.54, "z": -0.596, "size": 0.241, "title": "Instances on pempty", "summary": "This file collects facts about algebraic structures on the (universe-polymorphic) empty type, e.g. that it is a semigroup.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/PEmptyInstances.html"}, {"id": "Mathlib.Algebra.Group.Defs", "region_id": "algebra", "micro_elevation": 0.0263, "macro_tier": 1, "macro_tier_score": 0.5653, "macro_tier_override": 1, "x": 0.941, "z": 3.593, "size": 1.2931, "title": "Typeclasses for (semi)groups and monoids", "summary": "In this file we define typeclasses for algebraic structures with one binary operation. The classes are named `(Add)?(Comm)?(Semigroup|Monoid|Group)`, where `Add` means that the class uses additive notation and `Comm` means that the class assumes that the binary operation is commutative. The file does not contain any lemmas except for * axioms of typeclasses restated in the root namespace; * lemmas required for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Defs.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.Finset.Basic", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4551, "macro_tier_override": null, "x": 7.873, "z": -12.6, "size": 0.679, "title": "Big operators", "summary": "In this file we prove theorems about products and sums indexed by a `Finset`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Finset/Basic.html"}, {"id": "Mathlib.Algebra.Order.AbsoluteValue.Basic", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 4, "macro_tier_score": 0.4027, "macro_tier_override": null, "x": 26.433, "z": 8.794, "size": 0.3571, "title": "Absolute values", "summary": "This file defines a bundled type of absolute values `AbsoluteValue R S`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/AbsoluteValue/Basic.html"}, {"id": "Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.2767, "macro_tier_override": null, "x": -17.806, "z": 5.276, "size": 0.3066, "title": "Big operators indexed by intervals", "summary": "This file proves lemmas about `∏ x ∈ Ixx a b, f x` and `∑ x ∈ Ixx a b, f x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/BigOperators/Group/LocallyFinite.html"}, {"id": "Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4044, "macro_tier_override": null, "x": 12.734, "z": -10.827, "size": 0.322, "title": "Big operators on a finset in groups with zero involving order", "summary": "This file contains the results concerning the interaction of finset big operators with groups with zero, where order is involved.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/BigOperators/GroupWithZero/Finset.html"}, {"id": "Mathlib.RingTheory.Etale.QuasiFinite", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 52.991, "z": 98.023, "size": 0.2332, "title": "Etale local structure of finite maps", "summary": "We construct etale neighborhoods that split fibers of finite algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Etale/QuasiFinite.html"}, {"id": "Mathlib.Algebra.Polynomial.Div", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.2112, "macro_tier_override": null, "x": -33.286, "z": -60.115, "size": 0.3033, "title": "Division of univariate polynomials", "summary": "The main defs are `divByMonic` and `modByMonic`. The compatibility between these is given by `modByMonic_add_div`. We also define `rootMultiplicity`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Div.html"}, {"id": "Mathlib.Algebra.Field.IsField", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.3348, "macro_tier_override": null, "x": -9.501, "z": 5.822, "size": 0.3981, "title": "`IsField` predicate", "summary": "Predicate on a (semi)ring that it is a (semi)field, i.e. that the multiplication is commutative, that it has more than one element and that all non-zero elements have a multiplicative inverse. In contrast to `Field`, which contains the data of a function associating to an element of the field its multiplicative inverse, this predicate only assumes the existence and can therefore more easily be used to e.g. transfer…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/IsField.html"}, {"id": "Mathlib.Algebra.Polynomial.Inductions", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.2143, "macro_tier_override": null, "x": 59.691, "z": -25.731, "size": 0.2787, "title": "Induction on polynomials", "summary": "This file contains lemmas dealing with different flavours of induction on polynomials.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Inductions.html"}, {"id": "Mathlib.Algebra.Polynomial.Monic", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.2698, "macro_tier_override": null, "x": 40.951, "z": 52.849, "size": 0.3642, "title": "Theory of monic polynomials", "summary": "We give several tools for proving that polynomials are monic, e.g. `Monic.mul`, `Monic.map`, `Monic.pow`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Monic.html"}, {"id": "Mathlib.RingTheory.Multiplicity", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 3, "macro_tier_score": 0.2286, "macro_tier_override": null, "x": 17.11, "z": 14.279, "size": 0.3556, "title": "Multiplicity of a divisor", "summary": "For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves several basic results on it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Multiplicity.html"}, {"id": "Mathlib.RingTheory.Adjoin.PowerBasis", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 2, "macro_tier_score": 0.0101, "macro_tier_override": null, "x": -94.214, "z": 9.734, "size": 0.2514, "title": "Power basis for `R[x]`", "summary": "This file defines the canonical power basis on `R[x]`, where `x` is an integral element over `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Adjoin/PowerBasis.html"}, {"id": "Mathlib.RingTheory.PowerBasis", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 3, "macro_tier_score": 0.1412, "macro_tier_override": null, "x": 86.126, "z": -34.713, "size": 0.4169, "title": "Power basis", "summary": "This file defines a structure `PowerBasis R S`, giving a basis of the `R`-algebra `S` as a finite list of powers `1, x, ..., x^n`. For example, if `x` is algebraic over a ring/field, adjoining `x` gives a `PowerBasis` structure generated by `x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerBasis.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Basis", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.1421, "macro_tier_override": null, "x": -32.778, "z": 56.131, "size": 0.4099, "title": "Bases and matrices", "summary": "This file defines the map `Basis.toMatrix` that sends a family of vectors to the matrix of their coordinates with respect to some basis.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Basis.html"}, {"id": "Mathlib.GroupTheory.Descent", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.709, "z": -37.104, "size": 0.2, "title": "Descent Theorem", "summary": "We provide a proof of the following result. Let `G` be a group and `f : G →* G` an endomorphism of `G` that maps every subgroup of `G` into itself (e.g., `f = fun g ↦ g ^ n` when `G` is commutative). If there is a finite subset `s : Set G` and there exists a \"height\" function `h : G → ℝ` and constants `a, b, c : ℝ` such that * `s` surjects onto the quotient `G ⧸ f(G)`, * for all `g ∈ s` and `x : G`, `h x ≤ a * h (g…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Descent.html"}, {"id": "Mathlib.GroupTheory.Finiteness", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 4, "macro_tier_score": 0.3287, "macro_tier_override": null, "x": 25.034, "z": 22.154, "size": 0.3738, "title": "Finitely generated monoids and groups", "summary": "We define finitely generated monoids and groups. See also `Submodule.FG` and `Module.Finite` for finitely-generated modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Finiteness.html"}, {"id": "Mathlib.GroupTheory.Index", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.2565, "macro_tier_override": null, "x": 20.46, "z": 28.749, "size": 0.4595, "title": "Index of a Subgroup", "summary": "In this file we define the index of a subgroup, and prove several divisibility properties. Several theorems proved in this file are known as Lagrange's theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Index.html"}, {"id": "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 4, "macro_tier_score": 0.3262, "macro_tier_override": null, "x": 19.817, "z": -39.924, "size": 0.4125, "title": "Non-unital Subalgebras over Commutative Semirings", "summary": "In this file we define `NonUnitalSubalgebra`s and the usual operations on them (`map`, `comap`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.html"}, {"id": "Mathlib.Algebra.Algebra.NonUnitalHom", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 4, "macro_tier_score": 0.3567, "macro_tier_override": null, "x": 39.813, "z": -15.476, "size": 0.4742, "title": "Morphisms of non-unital algebras", "summary": "This file defines morphisms between two types, each of which carries: * an addition, * an additive zero, * a multiplication, * a scalar action. The multiplications are not assumed to be associative or unital, or even to be compatible with the scalar actions. In a typical application, the operations will satisfy compatibility conditions making them into algebras (albeit possibly non-associative and/or non-unital) but…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/NonUnitalHom.html"}, {"id": "Mathlib.LinearAlgebra.Span.Basic", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 4, "macro_tier_score": 0.3974, "macro_tier_override": null, "x": 18.676, "z": 38.416, "size": 0.7187, "title": "The span of a set of vectors, as a submodule", "summary": "* `Submodule.span s` is defined to be the smallest submodule containing the set `s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Span/Basic.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.Basic", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.4281, "macro_tier_override": null, "x": 3.403, "z": -22.025, "size": 0.4876, "title": "Definitions of group actions", "summary": "This file defines a hierarchy of group action type-classes on top of the previously defined notation classes `SMul` and its additive version `VAdd`: * `MulAction M α` and its additive version `AddAction G P` are typeclasses used for actions of multiplicative and additive monoids and groups; they extend notation classes `SMul` and `VAdd` that are defined in `Algebra.Group.Defs`; * `DistribMulAction M A` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/Basic.html"}, {"id": "Mathlib.Algebra.Group.Action.End", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.4408, "macro_tier_override": null, "x": -20.429, "z": -0.005, "size": 0.3776, "title": "Interaction between actions and endomorphisms/automorphisms", "summary": "This file provides two things: * The tautological actions by endomorphisms/automorphisms on their base type. * An action by a monoid/group on a type is the same as a hom from the monoid/group to endomorphisms/automorphisms of the type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/End.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.Defs", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.465, "macro_tier_override": null, "x": 10.266, "z": 7.976, "size": 0.7556, "title": "Definitions of group actions", "summary": "This file defines a hierarchy of group action type-classes on top of the previously defined notation classes `SMul` and its additive version `VAdd`: * `SMulZeroClass` is a typeclass for an action that preserves zero * `DistribSMul M A` is a typeclass for an action on an additive monoid (`AddZeroClass`) that preserves addition and zero * `DistribMulAction M A` is a typeclass for an action of a multiplicative monoid…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/Defs.html"}, {"id": "Mathlib.Algebra.Group.Action.Prod", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4254, "macro_tier_override": null, "x": 15.751, "z": 5.593, "size": 0.3833, "title": "Prod instances for additive and multiplicative actions", "summary": "This file defines instances for binary product of additive and multiplicative actions and provides scalar multiplication as a homomorphism from `α × β` to `β`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Prod.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Prod", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4273, "macro_tier_override": null, "x": 16.4, "z": -3.227, "size": 0.3693, "title": "Products of monoids with zero, groups with zero", "summary": "In this file we define `MonoidWithZero`, `GroupWithZero`, etc... instances for `M₀ × N₀`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Prod.html"}, {"id": "Mathlib.Algebra.BigOperators.Ring.List", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4109, "macro_tier_override": null, "x": -16.467, "z": -8.587, "size": 0.3151, "title": "Big operators on a list in rings", "summary": "This file contains the results concerning the interaction of list big operators with rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Ring/List.html"}, {"id": "Mathlib.Algebra.Order.BigOperators.Group.List", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4421, "macro_tier_override": null, "x": -6.069, "z": 13.561, "size": 0.3267, "title": "Big operators on a list in ordered groups", "summary": "This file contains the results concerning the interaction of list big operators with ordered groups/monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/BigOperators/Group/List.html"}, {"id": "Mathlib.Algebra.Order.Group.Nat", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4213, "macro_tier_override": null, "x": 2.052, "z": 14.715, "size": 0.573, "title": "The naturals form a linear ordered monoid", "summary": "This file contains the linear ordered monoid instance on the natural numbers. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Nat.html"}, {"id": "Mathlib.Algebra.Order.Sub.Basic", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4074, "macro_tier_override": null, "x": -8.212, "z": 12.381, "size": 0.4638, "title": "Lemmas about subtraction in unbundled canonically ordered monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Sub/Basic.html"}, {"id": "Mathlib.Algebra.Ring.Nat", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 1, "macro_tier_score": 0.0032, "macro_tier_override": null, "x": 5.736, "z": 7.302, "size": 0.4409, "title": "The natural numbers form a semiring", "summary": "This file contains the commutative semiring instance on the natural numbers. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Nat.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.Noetherian", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 2, "macro_tier_score": 0.0248, "macro_tier_override": null, "x": -47.166, "z": -84.271, "size": 0.2827, "title": null, "summary": "This file proves additional properties of the prime spectrum a ring is Noetherian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/Noetherian.html"}, {"id": "Mathlib.RingTheory.Artinian.Ring", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.0282, "macro_tier_override": null, "x": -27.878, "z": -64.833, "size": 0.2908, "title": "Artinian rings", "summary": "A ring is said to be left (or right) Artinian if it is Artinian as a left (or right) module over itself, or simply Artinian if it is both left and right Artinian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Artinian/Ring.html"}, {"id": "Mathlib.RingTheory.Ideal.MinimalPrime.Noetherian", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 2, "macro_tier_score": 0.0309, "macro_tier_override": null, "x": 66.836, "z": 1.707, "size": 0.3289, "title": "Finiteness of minimal primes", "summary": "We prove finiteness of minimal primes above an ideal. This is proved without reference to `PrimeSpectrum` to avoid heavy imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/MinimalPrime/Noetherian.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.Topology", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 3, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": -35.781, "z": 87.697, "size": 0.5024, "title": "The Zariski topology on the prime spectrum of a commutative (semi)ring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/Topology.html"}, {"id": "Mathlib.GroupTheory.Coset.Defs", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.4096, "macro_tier_override": null, "x": -4.485, "z": -21.83, "size": 0.4221, "title": "Cosets", "summary": "This file develops the basic theory of left and right cosets. When `G` is a group and `a : G`, `s : Set G`, with `open scoped Pointwise` we can write: * the left coset of `s` by `a` as `a • s` * the right coset of `s` by `a` as `MulOpposite.op a • s` (or `op a • s` with `open MulOpposite`, or `s <• a` with `open scoped Pointwise RightActions`) If instead `G` is an additive group, we can write (with `open scoped…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Coset/Defs.html"}, {"id": "Mathlib.Algebra.Order.CauSeq.Basic", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 1, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": -3.16, "z": 29.546, "size": 0.3097, "title": "Cauchy sequences", "summary": "A basic theory of Cauchy sequences, used in the construction of the reals and p-adic numbers. Where applicable, lemmas that will be reused in other contexts have been stated in extra generality. There are other \"versions\" of Cauchyness in the library, in particular Cauchy filters in topology. This is a concrete implementation that is useful for simplicity and computability reasons.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/CauSeq/Basic.html"}, {"id": "Mathlib.Algebra.Group.Action.Pi", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4263, "macro_tier_override": null, "x": -6.946, "z": -10.989, "size": 0.4999, "title": "Pi instances for multiplicative actions", "summary": "This file defines instances for `MulAction` and related structures on `Pi` types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Pi.html"}, {"id": "Mathlib.Algebra.Order.Field.Basic", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 2, "macro_tier_score": 0.0182, "macro_tier_override": null, "x": -3.116, "z": -27.683, "size": 0.4211, "title": "Lemmas about (linear) ordered (semi)fields", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Field/Basic.html"}, {"id": "Mathlib.Algebra.Order.Group.MinMax", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 2, "macro_tier_score": 0.0109, "macro_tier_override": null, "x": -22.422, "z": -8.953, "size": 0.3321, "title": "`min` and `max` in linearly ordered groups.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/MinMax.html"}, {"id": "Mathlib.Algebra.Ring.Pi", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4173, "macro_tier_override": null, "x": 8.01, "z": -16.756, "size": 0.4615, "title": "Pi instances for ring", "summary": "This file defines instances for ring, semiring and related structures on Pi Types", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Pi.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Ring", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 3, "macro_tier_score": 0.2313, "macro_tier_override": null, "x": -13.121, "z": 6.97, "size": 0.43, "title": "Commutativity and associativity of action of integers on rings", "summary": "This file proves that `ℕ` and `ℤ` act commutatively and associatively on (semi)rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Ring.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 3, "macro_tier_score": 0.1836, "macro_tier_override": null, "x": 55.416, "z": -60.054, "size": 0.3127, "title": "Characteristic polynomials", "summary": "We give methods for computing coefficients of the characteristic polynomial.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.html"}, {"id": "Mathlib.Algebra.Polynomial.Expand", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.1927, "macro_tier_override": null, "x": -71.913, "z": 8.635, "size": 0.3277, "title": "Expand a polynomial by a factor of p, so `∑ aₙ xⁿ` becomes `∑ aₙ xⁿᵖ`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Expand.html"}, {"id": "Mathlib.Algebra.Polynomial.Laurent", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.1867, "macro_tier_override": null, "x": 61.476, "z": -26.281, "size": 0.2886, "title": "Laurent polynomials", "summary": "We introduce Laurent polynomials over a semiring `R`. Mathematically, they are expressions of the form $$ \\sum_{i \\in \\mathbb{Z}} a_i T ^ i $$ where the sum extends over a finite subset of `ℤ`. Thus, negative exponents are allowed. The coefficients come from the semiring `R` and the variable `T` commutes with everything. Since we are going to convert back and forth between polynomials and Laurent polynomials, we…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Laurent.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 3, "macro_tier_score": 0.1831, "macro_tier_override": null, "x": 5.192, "z": -79.689, "size": 0.2553, "title": "Characteristic polynomials and the Cayley-Hamilton theorem", "summary": "We define characteristic polynomials of matrices and prove the Cayley–Hamilton theorem over arbitrary commutative rings. See the file `Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean` for corollaries of this theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.SchurComplement", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 3, "macro_tier_score": 0.1831, "macro_tier_override": null, "x": -67.727, "z": 38.693, "size": 0.2553, "title": "2×2 block matrices and the Schur complement", "summary": "This file proves properties of 2×2 block matrices `[A B; C D]` that relate to the Schur complement `D - C*A⁻¹*B`. Some of the results here generalize to 2×2 matrices in a category, rather than just a ring. A few results in this direction can be found in `Mathlib/CategoryTheory/Preadditive/Biproducts.lean`, especially the declarations `CategoryTheory.Biprod.gaussian` and `CategoryTheory.Biprod.isoElim`. Compare with…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/SchurComplement.html"}, {"id": "Mathlib.RingTheory.Polynomial.Nilpotent", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.1884, "macro_tier_override": null, "x": -69.569, "z": 11.854, "size": 0.3348, "title": "Nilpotency in polynomial rings.", "summary": "This file is a place for results related to nilpotency in (single-variable) polynomial rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Nilpotent.html"}, {"id": "Mathlib.NumberTheory.LSeries.Deriv", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": 60.015, "z": -49.823, "size": 0.2466, "title": "Differentiability and derivatives of L-series", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/Deriv.html"}, {"id": "Mathlib.NumberTheory.LSeries.Convergence", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -34.006, "z": 68.128, "size": 0.2727, "title": "Convergence of L-series", "summary": "We define `LSeries.abscissaOfAbsConv f` (as an `EReal`) to be the infimum of all real numbers `x` such that the L-series of `f` converges for complex arguments with real part `x` and provide some results about it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/Convergence.html"}, {"id": "Mathlib.NumberTheory.Real.GoldenRatio", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 83.485, "z": 3.821, "size": 0.2565, "title": "The golden ratio and its conjugate", "summary": "This file defines the golden ratio `φ := (1 + √5)/2` and its conjugate `ψ := (1 - √5)/2`, which are the two real roots of `X² - X - 1`. Along with various computational facts about them, we prove their irrationality, and we link them to the Fibonacci sequence by proving Binet's formula.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Real/GoldenRatio.html"}, {"id": "Mathlib.Algebra.Lie.IdealOperations", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 1, "macro_tier_score": 0.0031, "macro_tier_override": null, "x": -76.064, "z": -3.484, "size": 0.3232, "title": "Ideal operations for Lie algebras", "summary": "Given a Lie module `M` over a Lie algebra `L`, there is a natural action of the Lie ideals of `L` on the Lie submodules of `M`. In the special case that `M = L` with the adjoint action, this provides a pairing of Lie ideals which is especially important. For example, it can be used to define solvability / nilpotency of a Lie algebra via the derived / lower-central series.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/IdealOperations.html"}, {"id": "Mathlib.Algebra.Lie.Ideal", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 2, "macro_tier_score": 0.0063, "macro_tier_override": null, "x": -47.557, "z": -57.069, "size": 0.3854, "title": "Lie Ideals", "summary": "This file defines Lie ideals, which are Lie submodules of a Lie algebra over itself. They are defined as a special case of `LieSubmodule`, and inherit much of their structure from it. We also prove some basic properties of Lie ideals, including how they behave under Lie algebra homomorphisms (`map`, `comap`) and how they relate to the lattice structure on Lie submodules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Ideal.html"}, {"id": "Mathlib.Algebra.Order.Group.Indicator", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.4018, "macro_tier_override": null, "x": -7.296, "z": -21.058, "size": 0.2883, "title": "Support of a function in an order", "summary": "This file relates the support of a function to order constructions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Indicator.html"}, {"id": "Mathlib.Algebra.Group.Indicator", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4035, "macro_tier_override": null, "x": 8.193, "z": 4.371, "size": 0.3978, "title": "Indicator function", "summary": "In this file, we prove basic results about the indicator of a set. - `Set.indicator (s : Set α) (f : α → β) (a : α)` is `f a` if `a ∈ s` and is `0` otherwise. - `Set.mulIndicator (s : Set α) (f : α → β) (a : α)` is `f a` if `a ∈ s` and is `1` otherwise.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Indicator.html"}, {"id": "Mathlib.Algebra.Order.Group.Synonym", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.4696, "macro_tier_override": null, "x": 4.364, "z": -3.464, "size": 0.4672, "title": "Group structure on the order type synonyms", "summary": "Transfer algebraic instances from `α` to `αᵒᵈ`, `Lex α`, and `Colex α`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Synonym.html"}, {"id": "Mathlib.Algebra.Order.Group.Unbundled.Abs", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.4246, "macro_tier_override": null, "x": 20.261, "z": 2.609, "size": 0.4824, "title": "Absolute values in ordered groups", "summary": "The absolute value of an element in a group which is also a lattice is its supremum with its negation. This generalizes the usual absolute value on real numbers (`|x| = max x (-x)`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Unbundled/Abs.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4548, "macro_tier_override": null, "x": -12.775, "z": 2.406, "size": 0.5806, "title": "Canonically ordered monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Canonical/Defs.html"}, {"id": "Mathlib.Algebra.TrivSqZeroExt.Basic", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.0183, "macro_tier_override": null, "x": -0.77, "z": 57.567, "size": 0.3747, "title": "Trivial Square-Zero Extension", "summary": "Given a ring `R` together with an `(R, R)`-bimodule `M`, the trivial square-zero extension of `M` over `R` is defined to be the `R`-algebra `R ⊕ M` with multiplication given by `(r₁ + m₁) * (r₂ + m₂) = r₁ r₂ + r₁ m₂ + m₁ r₂`. It is a square-zero extension because `M^2 = 0`. Note that expressing this requires bimodules; we write these in general for a not-necessarily-commutative `R` as: ```lean variable {R M : Type*}…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/TrivSqZeroExt/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Prod", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 4, "macro_tier_score": 0.3642, "macro_tier_override": null, "x": 46.28, "z": 3.715, "size": 0.5612, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Prod.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.Lattice", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 4, "macro_tier_score": 0.3276, "macro_tier_override": null, "x": 49.93, "z": -24.721, "size": 0.6359, "title": "Complete lattice structure of subalgebras", "summary": "In this file we define `Algebra.adjoin` and the complete lattice structure on subalgebras. More lemmas about `adjoin` can be found in `Mathlib/RingTheory/Adjoin/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/Lattice.html"}, {"id": "Mathlib.Algebra.Order.Nonneg.Lattice", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2628, "title": "Lattice structures on the type of nonnegative elements", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Nonneg/Lattice.html"}, {"id": "Mathlib.RingTheory.RootsOfUnity.Minpoly", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.016, "macro_tier_override": null, "x": 101.013, "z": -31.659, "size": 0.2968, "title": "Minimal polynomial of roots of unity", "summary": "We gather several results about minimal polynomial of root of unity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RootsOfUnity/Minpoly.html"}, {"id": "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 2, "macro_tier_score": 0.0157, "macro_tier_override": null, "x": 2.936, "z": 94.67, "size": 0.2499, "title": "GCD domains are integrally closed", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GCDMonoid/IntegrallyClosed.html"}, {"id": "Mathlib.FieldTheory.Finite.Basic", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 3, "macro_tier_score": 0.0568, "macro_tier_override": null, "x": 88.771, "z": 46.659, "size": 0.4527, "title": "Finite fields", "summary": "This file contains basic results about finite fields. Throughout most of this file, `K` denotes a finite field and `q` is notation for the cardinality of `K`. See `RingTheory.IntegralDomain` for the fact that the unit group of a finite field is a cyclic group, as well as the fact that every finite integral domain is a field (`Fintype.fieldOfDomain`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Finite/Basic.html"}, {"id": "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 3, "macro_tier_score": 0.045, "macro_tier_override": null, "x": 12.603, "z": 103.235, "size": 0.3045, "title": "Minimal polynomials over a GCD monoid", "summary": "This file specializes the theory of minpoly to the case of an algebra over a GCD monoid.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.html"}, {"id": "Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 2, "macro_tier_score": 0.0252, "macro_tier_override": null, "x": 21.986, "z": -74.838, "size": 0.3837, "title": "Primitive roots of unity", "summary": "We define a predicate `IsPrimitiveRoot` on commutative monoids, expressing that an element is a primitive root of unity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RootsOfUnity/PrimitiveRoots.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.Nat", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 2, "macro_tier_score": 0.0159, "macro_tier_override": null, "x": 31.301, "z": -4.123, "size": 0.2843, "title": "Unique factorization of natural numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/Nat.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.ExactFunctor", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 2, "macro_tier_score": 0.02, "macro_tier_override": null, "x": -14.823, "z": -1.007, "size": 0.344, "title": "Exact functors", "summary": "In this file, it is shown that additive functors which preserves homology also preserves finite limits and finite colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/ExactFunctor.html"}, {"id": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4298, "macro_tier_override": null, "x": -16.042, "z": -4.692, "size": 0.3819, "title": "Big operators on a finset in groups with zero", "summary": "This file contains the results concerning the interaction of finset big operators with groups with zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/GroupWithZero/Finset.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Units.Basic", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.467, "macro_tier_override": null, "x": 1.518, "z": 12.911, "size": 0.6062, "title": "Lemmas about units in a `MonoidWithZero` or a `GroupWithZero`.", "summary": "We also define `Ring.inverse`, a globally defined function on any ring (in fact any `MonoidWithZero`), which inverts units and sends non-units to zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Units/Basic.html"}, {"id": "Mathlib.Algebra.Notation.Indicator", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.4338, "macro_tier_override": null, "x": 4.112, "z": 6.187, "size": 0.4547, "title": "Indicator function", "summary": "This file defines the indicator function of a set. More lemmas can be found in `Mathlib/Algebra/Group/Indicator.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Notation/Indicator.html"}, {"id": "Mathlib.Algebra.Polynomial.CoeffList", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -57.453, "z": 30.402, "size": 0.2338, "title": "A list of coefficients of a polynomial", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/CoeffList.html"}, {"id": "Mathlib.Algebra.Polynomial.EraseLead", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.27, "macro_tier_override": null, "x": 38.531, "z": 50.024, "size": 0.3771, "title": "Erase the leading term of a univariate polynomial", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/EraseLead.html"}, {"id": "Mathlib.FieldTheory.PolynomialGaloisGroup", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 1, "macro_tier_score": 0.004, "macro_tier_override": null, "x": -87.277, "z": -75.106, "size": 0.3119, "title": "Galois Groups of Polynomials", "summary": "In this file, we introduce the Galois group of a polynomial `p` over a field `F`, defined as the automorphism group of its splitting field. We also provide some results about some extension `E` above `p.SplittingField`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/PolynomialGaloisGroup.html"}, {"id": "Mathlib.FieldTheory.Galois.Basic", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 5, "macro_tier_score": 0.0602, "macro_tier_override": 5, "x": -101.941, "z": 49.417, "size": 0.4584, "title": "Galois Extensions", "summary": "In this file we define Galois extensions as extensions which are both separable and normal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Galois/Basic.html"}, {"id": "Mathlib.Algebra.Order.Interval.Finset.SuccPred", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.3161, "macro_tier_override": null, "x": -16.71, "z": 0.374, "size": 0.3504, "title": "Finset intervals in an additive successor-predecessor order", "summary": "This file proves relations between the various finset intervals in an additive successor/predecessor order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Interval/Finset/SuccPred.html"}, {"id": "Mathlib.Algebra.Ring.Subring.Order", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -1.097, "z": -24.118, "size": 0.2342, "title": "Subrings of ordered rings", "summary": "We study subrings of ordered rings and prove their basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Subring/Order.html"}, {"id": "Mathlib.Algebra.Order.Hom.Ring", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": -6.951, "z": -21.174, "size": 0.3592, "title": "Ordered ring homomorphisms", "summary": "Homomorphisms between ordered (semi)rings that respect the ordering.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Hom/Ring.html"}, {"id": "Mathlib.Algebra.Order.Ring.InjSurj", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 1, "macro_tier_score": 0.0039, "macro_tier_override": null, "x": -16.615, "z": 11.886, "size": 0.3043, "title": "Pulling back ordered rings along injective maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/InjSurj.html"}, {"id": "Mathlib.Algebra.Ring.Subring.Defs", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.3836, "macro_tier_override": null, "x": -18.562, "z": 0.599, "size": 0.3992, "title": "Subrings", "summary": "Let `R` be a ring. This file defines the \"bundled\" subring type `Subring R`, a type whose terms correspond to subrings of `R`. This is the preferred way to talk about subrings in mathlib. Unbundled subrings (`s : Set R` and `IsSubring s`) are not in this file, and they will ultimately be deprecated. We prove that subrings are a complete lattice, and that you can `map` (pushforward) and `comap` (pull back) them along…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Subring/Defs.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.Saturation", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 4.55, "z": 12.178, "size": 0.2, "title": "Saturation of a submonoid", "summary": "We define a submonoid `s` to be saturated if `x * y ∈ s → x ∈ s ∧ y ∈ s`. The type of all saturated submonoids forms a complete lattice. For a given submonoid `s` we construct the saturation of `s` as the smallest saturated submonoid containing `s`, which when the underlying type is a commutative monoid, is given by the formula `{x : M | ∃ y : M, x * y ∈ s}`. Saturated submonoids are used in the context of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/Saturation.html"}, {"id": "Mathlib.Algebra.Divisibility.Basic", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.4626, "macro_tier_override": null, "x": 1.273, "z": 7.319, "size": 0.4535, "title": "Divisibility", "summary": "This file defines the basics of the divisibility relation in the context of `(Comm)` `Monoid`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Divisibility/Basic.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.Basic", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4636, "macro_tier_override": null, "x": 6.171, "z": -9.278, "size": 0.4512, "title": "Submonoids: `CompleteLattice` structure", "summary": "This file defines a `CompleteLattice` structure on `Submonoid`s, define the closure of a set as the minimal submonoid that includes this set, and prove a few results about extending properties from a dense set (i.e. a set with `closure s = ⊤`) to the whole monoid, see `Submonoid.dense_induction` and `MonoidHom.ofClosureEqTopLeft`/`MonoidHom.ofClosureEqTopRight`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/Basic.html"}, {"id": "Mathlib.Algebra.Polynomial.Coeff", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 4, "macro_tier_score": 0.2897, "macro_tier_override": null, "x": -4.199, "z": -49.967, "size": 0.4575, "title": "Theory of univariate polynomials", "summary": "The theorems include formulas for computing coefficients, such as `coeff_add`, `coeff_sum`, `coeff_mul`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Coeff.html"}, {"id": "Mathlib.Algebra.CharP.Defs", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.3243, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.4635, "title": "Characteristic of semirings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/Defs.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.Support", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 4, "macro_tier_score": 0.2905, "macro_tier_override": null, "x": -34.607, "z": -33.674, "size": 0.4008, "title": "Lemmas about the support of a finitely supported function", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/Support.html"}, {"id": "Mathlib.Algebra.Polynomial.Basic", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 4, "macro_tier_score": 0.3299, "macro_tier_override": null, "x": -48.286, "z": 0.123, "size": 0.5877, "title": "Theory of univariate polynomials", "summary": "This file defines `Polynomial R`, the type of univariate polynomials over the semiring `R`, builds a semiring structure on it, and gives basic definitions that are expanded in other files in this directory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Basic.html"}, {"id": "Mathlib.Algebra.Quotient", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.4074, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2965, "title": "Algebraic quotients", "summary": "This file defines notation for algebraic quotients, e.g. quotient groups `G ⧸ H`, quotient modules `M ⧸ N` and ideal quotients `R ⧸ I`. The actual quotient structures are defined in the following files: * Quotient Group: `Mathlib/GroupTheory/QuotientGroup/Defs.lean` * Quotient Module: `Mathlib/LinearAlgebra/Quotient/Defs.lean` * Quotient Ring: `Mathlib/RingTheory/Ideal/Quotient/Defs.lean`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Quotient.html"}, {"id": "Mathlib.Algebra.Group.Action.Opposite", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4698, "macro_tier_override": null, "x": 5.488, "z": 9.698, "size": 0.5078, "title": "Scalar actions on and by `Mᵐᵒᵖ`", "summary": "This file defines the actions on the opposite type `SMul R Mᵐᵒᵖ`, and actions by the opposite type, `SMul Rᵐᵒᵖ M`. Note that `MulOpposite.smul` is provided in an earlier file as it is needed to provide the `NSMul.nsmul` and `ZSMul.zsmul` fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Opposite.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.MulOpposite", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4387, "macro_tier_override": null, "x": -12.925, "z": -7.326, "size": 0.3209, "title": "Mul-opposite subgroups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/MulOpposite.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Defs", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.4479, "macro_tier_override": null, "x": -0.921, "z": -20.408, "size": 0.4835, "title": "Definition of `orbit`, `fixedPoints` and `stabilizer`", "summary": "This file defines orbits, stabilizers, and other objects defined in terms of actions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Defs.html"}, {"id": "Mathlib.RingTheory.Adjoin.Singleton", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.0944, "macro_tier_override": null, "x": 51.318, "z": -39.894, "size": 0.2609, "title": "Adjoin one single element", "summary": "This file contains basic results on `Algebra.adjoin`, specifically on adjoining singletons.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Adjoin/Singleton.html"}, {"id": "Mathlib.RingTheory.Adjoin.Polynomial.Basic", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.2585, "macro_tier_override": null, "x": 31.989, "z": 54.441, "size": 0.4065, "title": "Polynomials and adjoining roots", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Adjoin/Polynomial/Basic.html"}, {"id": "Mathlib.RingTheory.Polynomial.Tower", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.1872, "macro_tier_override": null, "x": -61.226, "z": 15.442, "size": 0.3308, "title": "Algebra towers for polynomial", "summary": "This file proves some basic results about the algebra tower structure for the type `R[X]`. This structure itself is provided elsewhere as `Polynomial.isScalarTower` When you update this file, you can also try to make a corresponding update in `RingTheory.MvPolynomial.Tower`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Tower.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.Equiv", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -72.651, "z": -33.152, "size": 0.2403, "title": "Equivalences related to power series rings", "summary": "This file establishes a number of equivalences related to power series rings. * `MvPowerSeries.toAdicCompletionAlgEquiv` : the canonical isomorphism from multivariate power series to the adic completion of multivariate polynomials with respect to the ideal spanned by all variables when the index is finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/Equiv.html"}, {"id": "Mathlib.Algebra.Lie.OfAssociative", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 2, "macro_tier_score": 0.0176, "macro_tier_override": null, "x": -14.259, "z": 72.905, "size": 0.4413, "title": "Lie algebras of associative algebras", "summary": "This file defines the Lie algebra structure that arises on an associative algebra via the ring commutator. Since the linear endomorphisms of a Lie algebra form an associative algebra, one can define the adjoint action as a morphism of Lie algebras from a Lie algebra to its linear endomorphisms. We make such a definition in this file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/OfAssociative.html"}, {"id": "Mathlib.RingTheory.AdicCompletion.Algebra", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 2, "macro_tier_score": 0.0109, "macro_tier_override": null, "x": -8.729, "z": -66.286, "size": 0.3255, "title": "Algebra instance on adic completion", "summary": "In this file we provide an algebra instance on the adic completion of a ring. Then the adic completion of any module is a module over the adic completion of the ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AdicCompletion/Algebra.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.Ideal", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -64.835, "z": -43.366, "size": 0.2344, "title": "Lemmas about ideals of `MvPolynomial`", "summary": "Notably this contains results about monomial ideals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/Ideal.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.Rename", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -69.043, "z": 21.889, "size": 0.2344, "title": "Renaming variables of power series", "summary": "This file establishes the `rename` operation on multivariate power series under a map with finite fibers, which modifies the set of variables. Unlike polynomials, renaming variables in power series requires a finiteness condition on the map `f : σ → τ` between the index types. Specifically, we require that `f` has finite fibers, which is formalized as `Filter.TendstoCofinite f`. To see why this is necessary,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/Rename.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Substitution", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 2, "macro_tier_score": 0.0118, "macro_tier_override": null, "x": 73.338, "z": -11.831, "size": 0.3105, "title": "Substitutions in power series", "summary": "A (possibly multivariate) power series can be substituted into a (univariate) power series if and only if its constant coefficient is nilpotent. This is a particularization of the substitution of multivariate power series to the case of univariate power series. Because of the special API for `PowerSeries`, some results for `MvPowerSeries` do not immediately apply and a “primed” version is provided here.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Substitution.html"}, {"id": "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.3891, "macro_tier_override": null, "x": -15.312, "z": 13.523, "size": 0.339, "title": "Type tags for right action on the domain of a function", "summary": "By default, `M` acts on `α → β` if it acts on `β`, and the action is given by `(c • f) a = c • (f a)`. In some cases, it is useful to consider another action: if `M` acts on `α` on the left, then it acts on `α → β` on the right so that `(c • f) a = f (c • a)`. E.g., this action is used to reformulate the Mean Ergodic Theorem in terms of an operator on `L²`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/DomAct/Basic.html"}, {"id": "Mathlib.Algebra.BigOperators.RingEquiv", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": 16.192, "z": -4.148, "size": 0.317, "title": "Results about mapping big operators across ring equivalences", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/RingEquiv.html"}, {"id": "Mathlib.Algebra.Ring.Equiv", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4354, "macro_tier_override": null, "x": -14.828, "z": 0.929, "size": 0.6619, "title": "(Semi)ring equivs", "summary": "In this file we define an extension of `Equiv` called `RingEquiv`, which is a datatype representing an isomorphism of `Semiring`s, `Ring`s, `DivisionRing`s, or `Field`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Equiv.html"}, {"id": "Mathlib.Algebra.Ring.Opposite", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4444, "macro_tier_override": null, "x": -1.966, "z": -14.727, "size": 0.4773, "title": "Ring structures on the multiplicative opposite", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Opposite.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.Finset.Defs", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4636, "macro_tier_override": null, "x": -0.792, "z": -12.976, "size": 0.6274, "title": "Big operators", "summary": "In this file we define products and sums indexed by finite sets (specifically, `Finset`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Finset/Defs.html"}, {"id": "Mathlib.RingTheory.Nakayama", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.065, "macro_tier_override": null, "x": 42.407, "z": -49.262, "size": 0.3143, "title": "Nakayama's lemma", "summary": "This file contains some alternative statements of Nakayama's Lemma as found in [Stacks: Nakayama's Lemma](https://stacks.math.columbia.edu/tag/00DV).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Nakayama.html"}, {"id": "Mathlib.RingTheory.Finiteness.Nakayama", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.0676, "macro_tier_override": null, "x": -35.91, "z": -45.0, "size": 0.3371, "title": "Nakayama's lemma", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Nakayama.html"}, {"id": "Mathlib.RingTheory.Jacobson.Ideal", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.1975, "macro_tier_override": null, "x": 32.963, "z": -53.857, "size": 0.4085, "title": "Jacobson radical", "summary": "The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`. This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`. We can extend the idea of the nilradical of `R` to ideals of `R`, by letting the nilradical of an ideal `I` be the intersection of prime ideals containing `I`. Under this extension, the original nilradical is the radical of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Jacobson/Ideal.html"}, {"id": "Mathlib.Algebra.DirectSum.Internal", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.0323, "macro_tier_override": null, "x": -41.51, "z": 39.893, "size": 0.3491, "title": "Internally graded rings and algebras", "summary": "This module provides `DirectSum.GSemiring` and `DirectSum.GCommSemiring` instances for a collection of subobjects `A` when a `SetLike.GradedMonoid` instance is available: * `SetLike.gnonUnitalNonAssocSemiring` * `SetLike.gsemiring` * `SetLike.gcommSemiring` With these instances in place, it provides the bundled canonical maps out of a direct sum of subobjects into their carrier type: * `DirectSum.coeRingHom` (a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/DirectSum/Internal.html"}, {"id": "Mathlib.Algebra.Algebra.Operations", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 4, "macro_tier_score": 0.35, "macro_tier_override": null, "x": 34.983, "z": -40.949, "size": 0.538, "title": "Multiplication and division of submodules of an algebra.", "summary": "An interface for multiplication and division of sub-R-modules of an R-algebra A is developed.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Operations.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.Basic", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 4, "macro_tier_score": 0.331, "macro_tier_override": null, "x": 34.827, "z": 41.082, "size": 0.5608, "title": "Subalgebras over Commutative Semiring", "summary": "In this file we define `Subalgebra`s and the usual operations on them (`map`, `comap`). The `Algebra.adjoin` operation and complete lattice structure can be found in `Mathlib/Algebra/Algebra/Subalgebra/Lattice.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/Basic.html"}, {"id": "Mathlib.Algebra.DirectSum.Algebra", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 3, "macro_tier_score": 0.0365, "macro_tier_override": null, "x": -52.453, "z": 18.783, "size": 0.3351, "title": "Additively-graded algebra structures on `⨁ i, A i`", "summary": "This file provides `R`-algebra structures on external direct sums of `R`-modules. Recall that if `A i` are a family of `AddCommMonoid`s indexed by an `AddMonoid`, then an instance of `DirectSum.GMonoid A` is a multiplication `A i → A j → A (i + j)` giving `⨁ i, A i` the structure of a semiring. In this file, we introduce the `DirectSum.GAlgebra R A` class for the case where all `A i` are `R`-modules. This is the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/DirectSum/Algebra.html"}, {"id": "Mathlib.Algebra.Order.Antidiag.Prod", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 3, "macro_tier_score": 0.0342, "macro_tier_override": null, "x": -14.448, "z": 3.462, "size": 0.3286, "title": "Antidiagonal with values in general types", "summary": "We define a type class `Finset.HasAntidiagonal A` which contains a function `antidiagonal : A → Finset (A × A)` such that `antidiagonal n` is the finset of all pairs adding to `n`, as witnessed by `mem_antidiagonal`. Analogously, the type class `Finset.HasMulAntidiagonal A` contains a function `mulAntidiagonal : A → Finset (A × A)` such that `mulAntidiagonal n` is the finset of all pairs multiplying to `n`, as…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Antidiag/Prod.html"}, {"id": "Mathlib.Algebra.Star.MonoidHom", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 2, "macro_tier_score": 0.0192, "macro_tier_override": null, "x": 5.28, "z": 21.652, "size": 0.2841, "title": "Morphisms of star monoids", "summary": "This file defines the type of morphisms `StarMonoidHom` between monoids `A` and `B` where both `A` and `B` are equipped with a `star` operation. These morphisms are star-preserving monoid homomorphisms and are equipped with the notation `A →⋆* B`. The primary motivation for these morphisms is to provide a target type for morphisms which induce a corresponding morphism between the unitary groups in a star monoid.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/MonoidHom.html"}, {"id": "Mathlib.Algebra.Star.Basic", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.2918, "macro_tier_override": null, "x": 15.136, "z": -13.72, "size": 0.522, "title": "Star monoids, rings, and modules", "summary": "We introduce the basic algebraic notions of star monoids, star rings, and star modules. A star algebra is simply a star ring that is also a star module. These are implemented as \"mixin\" typeclasses, so to summon a star ring (for example) one needs to write `(R : Type*) [Ring R] [StarRing R]`. This avoids difficulties with diamond inheritance. For now we simply do not introduce notations, as different users are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/Basic.html"}, {"id": "Mathlib.RingTheory.IntegralClosure.Algebra.Ideal", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 2, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -29.039, "z": 82.315, "size": 0.2438, "title": "Integrality over ideals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IntegralClosure/Algebra/Ideal.html"}, {"id": "Mathlib.RingTheory.IntegralClosure.Algebra.Basic", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 3, "macro_tier_score": 0.177, "macro_tier_override": null, "x": 84.677, "z": -11.316, "size": 0.3168, "title": "Integral closure of a subring.", "summary": "Let `A` be an `R`-algebra. We prove that integral elements form a sub-`R`-algebra of `A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IntegralClosure/Algebra/Basic.html"}, {"id": "Mathlib.RingTheory.Polynomial.Wronskian", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -14.077, "z": 61.554, "size": 0.2487, "title": "Wronskian of a pair of polynomial", "summary": "This file defines Wronskian of a pair of polynomials, which is `W(a, b) = ab' - a'b`. We also prove basic properties of it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Wronskian.html"}, {"id": "Mathlib.Algebra.Polynomial.AlgebraMap", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 4, "macro_tier_score": 0.289, "macro_tier_override": null, "x": 6.967, "z": -60.889, "size": 0.6161, "title": "Theory of univariate polynomials", "summary": "We show that `A[X]` is an R-algebra when `A` is an R-algebra. We promote `eval₂` to an algebra hom in `aeval`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/AlgebraMap.html"}, {"id": "Mathlib.Algebra.Polynomial.Derivative", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.2054, "macro_tier_override": null, "x": 49.184, "z": 29.924, "size": 0.4116, "title": "The derivative map on polynomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Derivative.html"}, {"id": "Mathlib.LinearAlgebra.SesquilinearForm.Basic", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 3, "macro_tier_score": 0.1818, "macro_tier_override": null, "x": -37.423, "z": -36.105, "size": 0.443, "title": "Sesquilinear maps", "summary": "This file provides properties about sesquilinear maps and forms. The maps considered are of the form `M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M`, where `I₁ : R₁ →+* R` and `I₂ : R₂ →+* R` are ring homomorphisms and `M₁` is a module over `R₁`, `M₂` is a module over `R₂` and `M` is a module over `R`. Sesquilinear forms are the special case that `M₁ = M₂`, `M = R₁ = R₂ = R`, and `I₁ = RingHom.id R`. Taking additionally `I₂ = RingHom.id…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/SesquilinearForm/Basic.html"}, {"id": "Mathlib.RingTheory.Coprime.Basic", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.3304, "macro_tier_override": null, "x": -5.501, "z": -21.596, "size": 0.3414, "title": "Coprime elements of a ring or monoid", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Coprime/Basic.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.MappingCone", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 2, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 22.855, "z": -7.781, "size": 0.3131, "title": "The mapping cone of a morphism of cochain complexes", "summary": "In this file, we study the homotopy cofiber `HomologicalComplex.homotopyCofiber` of a morphism `φ : F ⟶ G` of cochain complexes indexed by `ℤ`. In this case, we redefine it as `CochainComplex.mappingCone φ`. The API involves definitions - `mappingCone.inl φ : Cochain F (mappingCone φ) (-1)`, - `mappingCone.inr φ : G ⟶ mappingCone φ`, - `mappingCone.fst φ : Cocycle (mappingCone φ) F 1` and - `mappingCone.snd φ :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.HomComplex", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 2, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": 9.139, "z": -20.326, "size": 0.3274, "title": "The cochain complex of homomorphisms between cochain complexes", "summary": "If `F` and `G` are cochain complexes (indexed by `ℤ`) in a preadditive category, there is a cochain complex of abelian groups whose `0`-cocycles identify to morphisms `F ⟶ G`. Informally, in degree `n`, this complex shall consist of cochains of degree `n` from `F` to `G`, i.e. arbitrary families for morphisms `F.X p ⟶ G.X (p + n)`. This complex shall be denoted `HomComplex F G`. In order to avoid type-theoretic…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCofiber", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 2, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -13.391, "z": -10.003, "size": 0.3012, "title": "The homotopy cofiber of a morphism of homological complexes", "summary": "In this file, we construct the homotopy cofiber of a morphism `φ : F ⟶ G` between homological complexes in `HomologicalComplex C c`. In degree `i`, it is isomorphic to `(F.X j) ⊞ (G.X i)` if there is a `j` such that `c.Rel i j`, and `G.X i` otherwise. (This is also known as the mapping cone of `φ`. Under the name `CochainComplex.mappingCone`, a specific API shall be developed for the case of cochain complexes…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCofiber.html"}, {"id": "Mathlib.Algebra.Algebra.Defs", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 4, "macro_tier_score": 0.3987, "macro_tier_override": null, "x": -23.23, "z": -11.677, "size": 0.6157, "title": "Algebras over commutative semirings", "summary": "In this file we define associative unital `Algebra`s over commutative (semi)rings. * algebra homomorphisms `AlgHom` are defined in `Mathlib/Algebra/Algebra/Hom.lean`; * algebra equivalences `AlgEquiv` are defined in `Mathlib/Algebra/Algebra/Equiv.lean`; * `Subalgebra`s are defined in `Mathlib/Algebra/Algebra/Subalgebra/Basic.lean`; * The category `AlgCat R` of `R`-algebras is defined in the file…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Defs.html"}, {"id": "Mathlib.Algebra.Order.Archimedean.Real.Basic", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 2, "macro_tier_score": 0.0076, "macro_tier_override": null, "x": 19.526, "z": 14.2, "size": 0.3914, "title": "The real numbers are an Archimedean floor ring, and a conditionally complete linear order.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Archimedean/Real/Basic.html"}, {"id": "Mathlib.Algebra.Order.Nonneg.Module", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.3172, "macro_tier_override": null, "x": -19.423, "z": 10.928, "size": 0.4058, "title": "Modules over nonnegative elements", "summary": "For an ordered ring `R`, this file proves that any (ordered) `R`-module `M` is also an (ordered) `R≥0`-module. Among other things, these instances are useful for working with `ConvexCone`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Nonneg/Module.html"}, {"id": "Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -20.811, "z": -94.303, "size": 0.2755, "title": "The group cohomology of a `k`-linear `G`-representation", "summary": "Let `k` be a commutative ring and `G` a group. This file defines the group cohomology of `A : Rep k G` to be the cohomology of the complex $$0 \\to \\mathrm{Fun}(G^0, A) \\to \\mathrm{Fun}(G^1, A) \\to \\mathrm{Fun}(G^2, A) \\to \\dots$$ with differential $d^n$ sending $f: G^n \\to A$ to the function mapping $(g_0, \\dots, g_n)$ to $$\\rho(g_0)(f(g_1, \\dots, g_n))$$ $$+ \\sum_{i = 0}^{n - 1} (-1)^{i + 1}\\cdot f(g_0, \\dots,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.html"}, {"id": "Mathlib.Algebra.Homology.ConcreteCategory", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": -52.259, "z": -41.701, "size": 0.2915, "title": "Homology of complexes in concrete categories", "summary": "The homology of short complexes in concrete categories was studied in `Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean`. In this file, we introduce specific definitions and lemmas for the homology of homological complexes in concrete categories. In particular, we give a computation of the connecting homomorphism of the homology sequence in terms of (co)cycles.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ConcreteCategory.html"}, {"id": "Mathlib.RepresentationTheory.Homological.Resolution", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": -80.801, "z": 49.419, "size": 0.2915, "title": "The standard and bar resolutions of `k` as a trivial `k`-linear `G`-representation", "summary": "Given a commutative ring `k` and a group `G`, this file defines two projective resolutions of `k` as a trivial `k`-linear `G`-representation. The first one, the standard resolution, has objects `k[Gⁿ⁺¹]` equipped with the diagonal representation, and differential defined by `(g₀, ..., gₙ) ↦ ∑ (-1)ⁱ • (g₀, ..., ĝᵢ, ..., gₙ)`. We define this as the alternating face map complex associated to an appropriate simplicial…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/Resolution.html"}, {"id": "Mathlib.GroupTheory.IsSubnormal", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -24.115, "z": -23.151, "size": 0.2, "title": "Subnormal subgroups", "summary": "In this file, we define subnormal subgroups. We also show some basic results about the interaction of subnormality and simplicity of groups. These should cover most of the results needed in this case.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/IsSubnormal.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.Pointwise", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 4, "macro_tier_score": 0.4309, "macro_tier_override": null, "x": -2.938, "z": 31.435, "size": 0.4696, "title": "Pointwise instances on `Subgroup` and `AddSubgroup`s", "summary": "This file provides the actions * `Subgroup.pointwiseMulAction` * `AddSubgroup.pointwiseMulAction` which matches the action of `Set.mulActionSet`. These actions are available in the `Pointwise` locale.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/Pointwise.html"}, {"id": "Mathlib.GroupTheory.QuotientGroup.Defs", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.4092, "macro_tier_override": null, "x": 24.043, "z": -2.192, "size": 0.4874, "title": "Quotients of groups by normal subgroups", "summary": "This file defines the group structure on the quotient by a normal subgroup.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/QuotientGroup/Defs.html"}, {"id": "Mathlib.GroupTheory.Subgroup.Simple", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 3, "macro_tier_score": 0.1935, "macro_tier_override": null, "x": -21.072, "z": 15.231, "size": 0.3036, "title": "Simple groups", "summary": "This file defines `IsSimpleGroup G`, a class indicating that a group has exactly two normal subgroups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Subgroup/Simple.html"}, {"id": "Mathlib.FieldTheory.RatFunc.Defs", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.029, "macro_tier_override": null, "x": -36.341, "z": -44.652, "size": 0.2633, "title": "The field of rational functions", "summary": "Files in this folder define the field `K⟮X⟯` of rational functions over a field `K`, show it is the field of fractions of `K[X]` and provide the main results concerning it. This file contains the basic definition. For connections with Laurent Series, see `Mathlib/RingTheory/LaurentSeries.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/RatFunc/Defs.html"}, {"id": "Mathlib.RingTheory.Localization.FractionRing", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 3, "macro_tier_score": 0.2322, "macro_tier_override": null, "x": 42.718, "z": 35.768, "size": 0.4981, "title": "Fraction ring / fraction field Frac(R) as localization", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/FractionRing.html"}, {"id": "Mathlib.Algebra.Ring.Action.ConjAct", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 3, "macro_tier_score": 0.1406, "macro_tier_override": null, "x": -22.753, "z": 21.888, "size": 0.3297, "title": "Conjugation action of a ring on itself", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Action/ConjAct.html"}, {"id": "Mathlib.Algebra.Ring.Action.Basic", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.4357, "macro_tier_override": null, "x": 17.562, "z": 10.436, "size": 0.5164, "title": "Group action on rings", "summary": "This file defines the typeclass of monoid acting on semirings `MulSemiringAction M R`. An example of a `MulSemiringAction` is the action of the Galois group `Gal(L/K)` on the big field `L`. Note that `Algebra` does not in general satisfy the axioms of `MulSemiringAction`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Action/Basic.html"}, {"id": "Mathlib.GroupTheory.GroupAction.ConjAct", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 4, "macro_tier_score": 0.4297, "macro_tier_override": null, "x": 25.45, "z": 15.338, "size": 0.3762, "title": "Conjugation action of a group on itself", "summary": "This file defines the conjugation action of a group on itself. See also `MulAut.conj` for the definition of conjugation as a homomorphism into the automorphism group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/ConjAct.html"}, {"id": "Mathlib.Algebra.Group.Invertible.Basic", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4699, "macro_tier_override": null, "x": -6.628, "z": -15.344, "size": 0.4394, "title": "Theorems about invertible elements", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Invertible/Basic.html"}, {"id": "Mathlib.Algebra.Group.Commute.Units", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4739, "macro_tier_override": null, "x": -14.135, "z": 4.577, "size": 0.4235, "title": "Lemmas about commuting pairs of elements involving units.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Commute/Units.html"}, {"id": "Mathlib.Algebra.Group.Invertible.Defs", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.4695, "macro_tier_override": null, "x": -2.885, "z": -4.766, "size": 0.3748, "title": "Invertible elements", "summary": "This file defines a typeclass `Invertible a` for elements `a` with a two-sided multiplicative inverse. The intent of the typeclass is to provide a way to write e.g. `⅟2` in a ring like `ℤ[1/2]` where some inverses exist but there is no general `⁻¹` operator; or to specify that a field has characteristic `≠ 2`. It is the `Type`-valued analogue to the `Prop`-valued `IsUnit`. For constructions of the invertible element…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Invertible/Defs.html"}, {"id": "Mathlib.Algebra.Group.Hom.Defs", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.5186, "macro_tier_override": null, "x": 0.33, "z": -5.562, "size": 0.859, "title": "Monoid and group homomorphisms", "summary": "This file defines the bundled structures for monoid and group homomorphisms. Namely, we define `MonoidHom` (resp., `AddMonoidHom`) to be bundled homomorphisms between multiplicative (resp., additive) monoids or groups. We also define coercion to a function, and usual operations: composition, identity homomorphism, pointwise multiplication and pointwise inversion. This file also defines the lesser-used (and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Hom/Defs.html"}, {"id": "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 3, "macro_tier_score": 0.0636, "macro_tier_override": null, "x": 53.997, "z": 91.051, "size": 0.2901, "title": "Algebraic Closure", "summary": "In this file we construct the algebraic closure of a field", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.html"}, {"id": "Mathlib.FieldTheory.IsAlgClosed.Basic", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 3, "macro_tier_score": 0.0855, "macro_tier_override": null, "x": 71.982, "z": -75.065, "size": 0.4513, "title": "Algebraically Closed Field", "summary": "In this file we define the typeclass for algebraically closed fields and algebraic closures, and prove some of their properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/IsAlgClosed/Basic.html"}, {"id": "Mathlib.FieldTheory.SplittingField.Construction", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 3, "macro_tier_score": 0.08, "macro_tier_override": null, "x": 92.729, "z": 38.193, "size": 0.4089, "title": "Splitting fields", "summary": "In this file we prove the existence and uniqueness of splitting fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/SplittingField/Construction.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Differentials.Basic", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 75.793, "z": -39.416, "size": 0.2478, "title": "The differentials of a morphism in the category of commutative rings", "summary": "In this file, given a morphism `f : A ⟶ B` in the category `CommRingCat`, and `M : ModuleCat B`, we define the type `M.Derivation f` of derivations with values in `M` relative to `f`. We also construct the module of differentials `CommRingCat.KaehlerDifferential f : ModuleCat B` and the corresponding derivation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Differentials/Basic.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.ChangeOfRings", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -57.619, "z": -14.558, "size": 0.4098, "title": "Change Of Rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.html"}, {"id": "Mathlib.Algebra.Category.Ring.Basic", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 3, "macro_tier_score": 0.0755, "macro_tier_override": null, "x": -9.535, "z": 18.067, "size": 0.449, "title": "Category instances for `Semiring`, `Ring`, `CommSemiring`, and `CommRing`.", "summary": "We introduce the bundled categories: * `SemiRingCat` * `RingCat` * `CommSemiRingCat` * `CommRingCat` along with the relevant forgetful functors between them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/Basic.html"}, {"id": "Mathlib.RingTheory.Kaehler.Basic", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 2, "macro_tier_score": 0.0229, "macro_tier_override": null, "x": -35.134, "z": -75.828, "size": 0.3179, "title": "The module of Kähler differentials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Kaehler/Basic.html"}, {"id": "Mathlib.RingTheory.Finiteness.Cofinite", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 1, "macro_tier_score": 0.0017, "macro_tier_override": null, "x": 47.085, "z": 42.072, "size": 0.2997, "title": "Co-finitely generated submodules", "summary": "This files defines the notion of a co-finitely generated submodule. A submodule `S` of a module `M` is co-finitely generated (or CoFG for short) if the quotient of `M` by `S` is finitely generated (i.e. FG).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Cofinite.html"}, {"id": "Mathlib.RingTheory.Noetherian.Basic", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 4, "macro_tier_score": 0.294, "macro_tier_override": null, "x": 29.727, "z": -53.594, "size": 0.4509, "title": "Noetherian rings and modules", "summary": "The following are equivalent for a module M over a ring R: 1. Every increasing chain of submodules M₁ ⊆ M₂ ⊆ M₃ ⊆ ⋯ eventually stabilises. 2. Every submodule is finitely generated. A module satisfying these equivalent conditions is said to be a *Noetherian* R-module. A ring is a *Noetherian ring* if it is Noetherian as a module over itself. (Note that we do not assume yet that our rings are commutative, so perhaps…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Noetherian/Basic.html"}, {"id": "Mathlib.Algebra.Homology.EulerCharacteristic", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 4.597, "z": 53.661, "size": 0.2, "title": "Euler characteristic of homological complexes", "summary": "The Euler characteristic is defined using the `ComplexShape.EulerCharSigns` typeclass, which provides the alternating signs for each index. This allows the definition to work uniformly for chain complexes, cochain complexes, and complexes with other index types. The definitions work on graded objects, with the homological complex versions defined as abbreviations that apply the graded object versions to `C.X` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/EulerCharacteristic.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Basic", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 3, "macro_tier_score": 0.0613, "macro_tier_override": null, "x": 16.351, "z": -41.464, "size": 0.4918, "title": "The category of `R`-modules", "summary": "`ModuleCat.{v} R` is the category of bundled `R`-modules with carrier in the universe `v`. We show that it is preadditive and show that being an isomorphism, monomorphism and epimorphism is equivalent to being a linear equivalence, an injective linear map and a surjective linear map, respectively.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Basic.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 2, "macro_tier_score": 0.0096, "macro_tier_override": null, "x": -1.48, "z": -12.916, "size": 0.4705, "title": "The short complexes attached to homological complexes", "summary": "In this file, we define a functor `shortComplexFunctor C c i : HomologicalComplex C c ⥤ ShortComplex C`. By definition, the image of a homological complex `K` by this functor is the short complex `K.X (c.prev i) ⟶ K.X i ⟶ K.X (c.next i)`. The homology `K.homology i` of a homological complex `K` in degree `i` is defined as the homology of the short complex `(shortComplexFunctor C c i).obj K`, which can be abbreviated…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.Finrank", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 3, "macro_tier_score": 0.2578, "macro_tier_override": null, "x": -12.063, "z": 50.582, "size": 0.3056, "title": "Finite dimension of vector spaces", "summary": "Definition of the rank of a module, or dimension of a vector space, as a natural number.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/Finrank.html"}, {"id": "Mathlib.Algebra.Field.GeomSum", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 2, "macro_tier_score": 0.008, "macro_tier_override": null, "x": 25.957, "z": 1.507, "size": 0.3536, "title": "Partial sums of geometric series in a field", "summary": "This file determines the values of the geometric series $\\sum_{i=0}^{n-1} x^i$ and $\\sum_{i=0}^{n-1} x^i y^{n-1-i}$ and variants thereof.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/GeomSum.html"}, {"id": "Mathlib.Algebra.Field.Basic", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.3935, "macro_tier_override": null, "x": 13.07, "z": 18.051, "size": 0.5127, "title": "Lemmas about division (semi)rings and (semi)fields", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/Basic.html"}, {"id": "Mathlib.Algebra.Ring.GeomSum", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.2775, "macro_tier_override": null, "x": 10.484, "z": -21.748, "size": 0.4153, "title": "Partial sums of geometric series in a ring", "summary": "This file determines the values of the geometric series $\\sum_{i=0}^{n-1} x^i$ and $\\sum_{i=0}^{n-1} x^i y^{n-1-i}$ and variants thereof. Several variants are recorded, generalising in particular to the case of a noncommutative ring in which `x` and `y` commute. Even versions not using division or subtraction, valid in each semiring, are recorded.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/GeomSum.html"}, {"id": "Mathlib.RingTheory.NoetherNormalization", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 86.827, "z": 20.193, "size": 0.2, "title": "Noether normalization lemma", "summary": "This file contains a proof by Nagata of the Noether normalization lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/NoetherNormalization.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Monad", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.0715, "macro_tier_override": null, "x": 65.313, "z": -14.289, "size": 0.2981, "title": "Monad operations on `MvPolynomial`", "summary": "This file defines two monadic operations on `MvPolynomial`. Given `p : MvPolynomial σ R`, * `MvPolynomial.bind₁` and `MvPolynomial.join₁` operate on the variable type `σ`. * `MvPolynomial.bind₂` and `MvPolynomial.join₂` operate on the coefficient type `R`. - `MvPolynomial.bind₁ f φ` with `f : σ → MvPolynomial τ R` and `φ : MvPolynomial σ R`, is the polynomial `φ(f 1, ..., f i, ...) : MvPolynomial τ R`. -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Monad.html"}, {"id": "Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 3, "macro_tier_score": 0.1772, "macro_tier_override": null, "x": -87.276, "z": 1.337, "size": 0.3291, "title": "# Integral closure as a characteristic predicate", "summary": "We prove basic properties of `IsIntegralClosure`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.html"}, {"id": "Mathlib.RepresentationTheory.Homological.GroupHomology.Basic", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": -16.564, "z": 97.026, "size": 0.2879, "title": "The group homology of a `k`-linear `G`-representation", "summary": "Let `k` be a commutative ring and `G` a group. This file defines the group homology of `A : Rep k G` to be the homology of the complex $$\\dots \\to \\bigoplus_{G^2} A \\to \\bigoplus_{G^1} A \\to \\bigoplus_{G^0} A$$ with differential $d_n$ sending $a\\cdot (g_0, \\dots, g_n)$ to $$\\rho(g_0^{-1})(a)\\cdot (g_1, \\dots, g_n)$$ $$+ \\sum_{i = 0}^{n - 1}(-1)^{i + 1}a\\cdot (g_0, \\dots, g_ig_{i + 1}, \\dots, g_n)$$ $$+ (-1)^{n +…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.html"}, {"id": "Mathlib.RepresentationTheory.Coinvariants", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 1, "macro_tier_score": 0.0041, "macro_tier_override": null, "x": 57.521, "z": 77.573, "size": 0.3201, "title": "Coinvariants of a group representation", "summary": "Given a commutative ring `k` and a monoid `G`, this file introduces the coinvariants of a `k`-linear `G`-representation `(V, ρ)`. We first define `Representation.Coinvariants.ker`, the submodule of `V` generated by elements of the form `ρ g x - x` for `x : V`, `g : G`. Then the coinvariants of `(V, ρ)` are the quotient of `V` by this submodule. We show that the functor sending a representation to its coinvariants is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Coinvariants.html"}, {"id": "Mathlib.RingTheory.Invariant.Defs", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 2, "macro_tier_score": 0.0148, "macro_tier_override": null, "x": 1.454, "z": -27.819, "size": 0.2864, "title": "Invariant Extensions of Rings", "summary": "Given an extension of rings `B/A` and an action of `G` on `B`, we introduce a predicate `Algebra.IsInvariant A B G` which states that every fixed point of `B` lies in the image of `A`. The main application is in algebraic number theory, where `G := Gal(L/K)` is the Galois group of some finite Galois extension of number fields, and `A := 𝓞K` and `B := 𝓞L` are their rings of integers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Invariant/Defs.html"}, {"id": "Mathlib.Algebra.Polynomial.SpecificDegree", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 16.953, "z": -74.232, "size": 0.2, "title": "Polynomials of specific degree", "summary": "Facts about polynomials that have a specific integer degree.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/SpecificDegree.html"}, {"id": "Mathlib.Algebra.Polynomial.FieldDivision", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 3, "macro_tier_score": 0.1913, "macro_tier_override": null, "x": -22.734, "z": 70.722, "size": 0.3779, "title": "Theory of univariate polynomials", "summary": "This file starts looking like the ring theory of $R[X]$", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/FieldDivision.html"}, {"id": "Mathlib.GroupTheory.Subgroup.Center", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 4, "macro_tier_score": 0.4418, "macro_tier_override": null, "x": 24.601, "z": 8.416, "size": 0.3726, "title": "Centers of subgroups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Subgroup/Center.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.Basic", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.4485, "macro_tier_override": null, "x": -24.035, "z": 2.277, "size": 0.426, "title": "Basic results on subgroups", "summary": "We prove basic results on the definitions of subgroups. The bundled subgroups use bundled monoid homomorphisms. Special thanks goes to Amelia Livingston and Yury Kudryashov for their help and inspiration.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/Basic.html"}, {"id": "Mathlib.GroupTheory.Submonoid.Center", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.4448, "macro_tier_override": null, "x": -22.276, "z": -0.676, "size": 0.3547, "title": "Centers of monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Submonoid/Center.html"}, {"id": "Mathlib.Algebra.Central.Defs", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.0067, "macro_tier_override": null, "x": -40.027, "z": 41.381, "size": 0.347, "title": "Central Algebras", "summary": "In this file we define the predicate `Algebra.IsCentral K D` where `K` is a commutative ring and `D` is a (not necessarily commutative) `K`-algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Central/Defs.html"}, {"id": "Mathlib.Algebra.Group.Action.Pointwise.Set.Finite", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 14.192, "z": -14.695, "size": 0.2338, "title": "Finiteness lemmas for pointwise operations on sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Pointwise/Set/Finite.html"}, {"id": "Mathlib.Algebra.Group.Action.Basic", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4645, "macro_tier_override": null, "x": 5.203, "z": -17.828, "size": 0.5736, "title": "More lemmas about group actions", "summary": "This file contains lemmas about group actions that require more imports than `Mathlib/Algebra/Group/Action/Defs.lean` offers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Basic.html"}, {"id": "Mathlib.Algebra.Group.Pointwise.Set.Scalar", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.4548, "macro_tier_override": null, "x": 0.352, "z": 7.42, "size": 0.4909, "title": "Pointwise scalar operations of sets", "summary": "This file defines pointwise scalar-flavored algebraic operations on sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pointwise/Set/Scalar.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.WithZero", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 3, "macro_tier_score": 0.0678, "macro_tier_override": null, "x": 20.388, "z": 1.292, "size": 0.262, "title": "Covariant instances on `WithZero`", "summary": "Adding a zero to a type with a preorder and multiplication which satisfies some axiom, gives us a new type which satisfies some variant of the axiom.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/WithZero.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.Canonical", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 3, "macro_tier_score": 0.2533, "macro_tier_override": null, "x": 10.669, "z": 15.201, "size": 0.4213, "title": "Linearly ordered commutative groups and monoids with a zero element adjoined", "summary": "This file sets up a special class of linearly ordered commutative monoids that show up as the target of so-called “valuations” in algebraic number theory. Usually, in the informal literature, these objects are constructed by taking a linearly ordered commutative group Γ and formally adjoining a zero element: `Γ ∪ {0}`. The disadvantage is that a type such as `NNReal` is not of that form, whereas it is a very common…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/Canonical.html"}, {"id": "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "region_id": "algebra", "micro_elevation": 0.9342, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 56.566, "z": 119.109, "size": 0.268, "title": "The Jacobi Symbol", "summary": "We define the Jacobi symbol and prove its main properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.html"}, {"id": "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "region_id": "algebra", "micro_elevation": 0.9211, "macro_tier": 1, "macro_tier_score": 0.004, "macro_tier_override": null, "x": -123.615, "z": 40.245, "size": 0.3082, "title": "Quadratic reciprocity.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.html"}, {"id": "Mathlib.Algebra.Order.Archimedean.Basic", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 3, "macro_tier_score": 0.1128, "macro_tier_override": null, "x": -7.211, "z": 21.087, "size": 0.4747, "title": "Archimedean groups and fields", "summary": "This file proves several results connected to the notion of Archimedean groups. Being Archimedean means that for all elements `x` and `y > 0` there exists a natural number `n` such that `x ≤ n • y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Archimedean/Basic.html"}, {"id": "Mathlib.Algebra.Order.Group.Pointwise.CompleteLattice", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 2, "macro_tier_score": 0.0074, "macro_tier_override": null, "x": -20.28, "z": 2.459, "size": 0.3183, "title": "Infima/suprema in ordered monoids and groups", "summary": "In this file we prove a few facts like “The infimum of `-s` is `-` the supremum of `s`”.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Pointwise/CompleteLattice.html"}, {"id": "Mathlib.RingTheory.Localization.NormTrace", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 2, "macro_tier_score": 0.0142, "macro_tier_override": null, "x": -70.119, "z": -100.536, "size": 0.3291, "title": "Field/algebra norm / trace and localization", "summary": "This file contains results on the combination of `IsLocalization` and `Algebra.norm`, `Algebra.trace` and `Algebra.discr`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/NormTrace.html"}, {"id": "Mathlib.RingTheory.Discriminant", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 2, "macro_tier_score": 0.0154, "macro_tier_override": null, "x": -12.782, "z": -120.037, "size": 0.3364, "title": "Discriminant of a family of vectors", "summary": "Given an `A`-algebra `B` and `b`, an `ι`-indexed family of elements of `B`, we define the *discriminant* of `b` as the determinant of the matrix whose `(i j)`-th element is the trace of `b i * b j`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Discriminant.html"}, {"id": "Mathlib.Algebra.Order.BigOperators.Ring.List", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.4005, "macro_tier_override": null, "x": 11.685, "z": -18.977, "size": 0.2744, "title": "Big operators on a list in ordered rings", "summary": "This file contains the results concerning the interaction of list big operators with ordered rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/BigOperators/Ring/List.html"}, {"id": "Mathlib.Algebra.Order.Ring.Canonical", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.4028, "macro_tier_override": null, "x": 8.024, "z": -18.787, "size": 0.4159, "title": "Canonically ordered rings and semirings.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Canonical.html"}, {"id": "Mathlib.Algebra.Order.Group.OrderIso", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.424, "macro_tier_override": null, "x": 9.191, "z": -13.961, "size": 0.4225, "title": "Inverse and multiplication as order isomorphisms in ordered groups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/OrderIso.html"}, {"id": "Mathlib.Algebra.Ring.Int.Defs", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4341, "macro_tier_override": null, "x": -3.212, "z": 8.712, "size": 0.7015, "title": "The integers are a ring", "summary": "This file contains the commutative ring instance on `ℤ`. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Int/Defs.html"}, {"id": "Mathlib.Algebra.Order.Ring.Archimedean", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -40.886, "z": -45.655, "size": 0.2342, "title": "Archimedean classes of a linearly ordered ring", "summary": "The archimedean classes of a linearly ordered ring can be given the structure of an `AddCommMonoid`, by defining * `0 = mk 1` * `mk x + mk y = mk (x * y)` For a linearly ordered field, we can define a negative as * `-mk x = mk x⁻¹` which turns them into a `LinearOrderedAddCommGroupWithTop`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Archimedean.html"}, {"id": "Mathlib.Algebra.Order.Archimedean.Class", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": -5.735, "z": 23.452, "size": 0.2653, "title": "Archimedean classes of a linearly ordered group", "summary": "This file defines archimedean classes of a given linearly ordered group. Archimedean classes measure to what extent the group fails to be Archimedean. For additive group, elements `a` and `b` in the same class are \"equivalent\" in the sense that there exist two natural numbers `m` and `n` such that `|a| ≤ m • |b|` and `|b| ≤ n • |a|`. An element `a` in a higher class than `b` is \"infinitesimal\" to `b` in the sense…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Archimedean/Class.html"}, {"id": "Mathlib.Algebra.Order.Group.DenselyOrdered", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 15.372, "z": 6.564, "size": 0.2582, "title": "Lemmas about densely linearly ordered groups.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/DenselyOrdered.html"}, {"id": "Mathlib.RingTheory.Valuation.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.0699, "macro_tier_override": null, "x": 59.091, "z": -6.336, "size": 0.4006, "title": "The basics of valuation theory.", "summary": "The basic theory of valuations (non-archimedean norms) on a commutative ring, following T. Wedhorn's unpublished notes “Adic Spaces” ([wedhorn_adic]). The definition of a valuation we use here is Definition 1.22 of [wedhorn_adic]. A valuation on a ring `R` is a monoid homomorphism `v` to a linearly ordered commutative monoid with zero, that in addition satisfies the following two axioms: * `v 0 = 0` * `∀ x y, v (x +…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/Basic.html"}, {"id": "Mathlib.GroupTheory.Commutator.Basic", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 4, "macro_tier_score": 0.3583, "macro_tier_override": null, "x": 12.201, "z": 27.094, "size": 0.3536, "title": "Commutators of Subgroups", "summary": "If `G` is a group and `H₁ H₂ : Subgroup G` then the commutator `⁅H₁, H₂⁆ : Subgroup G` is the subgroup of `G` generated by the commutators `h₁ * h₂ * h₁⁻¹ * h₂⁻¹`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Commutator/Basic.html"}, {"id": "Mathlib.GroupTheory.Subgroup.Centralizer", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 4, "macro_tier_score": 0.4347, "macro_tier_override": null, "x": 15.855, "z": -22.906, "size": 0.3503, "title": "Centralizers of subgroups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Subgroup/Centralizer.html"}, {"id": "Mathlib.RingTheory.Ideal.Colon", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.1723, "macro_tier_override": null, "x": -37.318, "z": 46.251, "size": 0.3662, "title": "The colon ideal", "summary": "This file defines `Submodule.colon N P` as the ideal of all elements `r : R` such that `r • P ⊆ N`. The normal notation for this would be `N : P` which has already been taken by type theory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Colon.html"}, {"id": "Mathlib.Algebra.Ring.Action.Pointwise.Set", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 3, "macro_tier_score": 0.174, "macro_tier_override": null, "x": 25.919, "z": 2.06, "size": 0.3949, "title": "Pointwise operations of sets in a ring", "summary": "This file proves properties of pointwise operations of sets in a ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Action/Pointwise/Set.html"}, {"id": "Mathlib.LinearAlgebra.Quotient.Defs", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 4, "macro_tier_score": 0.3497, "macro_tier_override": null, "x": 26.101, "z": 20.886, "size": 0.559, "title": "Quotients by submodules", "summary": "* If `p` is a submodule of `M`, `M ⧸ p` is the quotient of `M` with respect to `p`: that is, elements of `M` are identified if their difference is in `p`. This is itself a module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Quotient/Defs.html"}, {"id": "Mathlib.RingTheory.Ideal.Maps", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 4, "macro_tier_score": 0.3313, "macro_tier_override": null, "x": 54.059, "z": 19.804, "size": 0.5952, "title": "Maps on modules and ideals", "summary": "Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator` and `Submodule.annihilator`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Maps.html"}, {"id": "Mathlib.RingTheory.PowerSeries.WellKnown", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": 65.644, "z": 20.312, "size": 0.2531, "title": "Definition of well-known power series", "summary": "In this file we define the following power series: * `PowerSeries.invUnitsSub`: given `u : Rˣ`, this is the series for `1 / (u - x)`. It is given by `∑ n, x ^ n /ₚ u ^ (n + 1)`. * `PowerSeries.invOneSubPow`: given a commutative ring `S` and a number `d : ℕ`, `PowerSeries.invOneSubPow S d` is the multiplicative inverse of `(1 - X) ^ d` in `S⟦X⟧ˣ`. When `d` is `0`, `PowerSeries.invOneSubPow S d` will just be `1`. When…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/WellKnown.html"}, {"id": "Mathlib.Algebra.Algebra.Rat", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 3, "macro_tier_score": 0.0346, "macro_tier_override": null, "x": 2.91, "z": 27.705, "size": 0.4086, "title": "Further basic results about `Algebra`'s over `ℚ`.", "summary": "This file could usefully be split further.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Rat.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Basic", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 2, "macro_tier_score": 0.0221, "macro_tier_override": null, "x": 30.984, "z": -59.245, "size": 0.3981, "title": "Formal power series (in one variable)", "summary": "This file defines (univariate) formal power series and develops the basic properties of these objects. A formal power series is to a polynomial like an infinite sum is to a finite sum. Formal power series in one variable are defined from multivariate power series as `PowerSeries R := MvPowerSeries Unit R`. The file sets up the (semi)ring structure on univariate power series. We provide the natural inclusion from…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Basic.html"}, {"id": "Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 1, "macro_tier_score": 0.0045, "macro_tier_override": null, "x": 114.144, "z": 58.242, "size": 0.3434, "title": "Ramification of infinite places of a number field", "summary": "This file studies the ramification of infinite places of a number field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/InfinitePlace/Ramification.html"}, {"id": "Mathlib.NumberTheory.NumberField.InfinitePlace.Basic", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -9.016, "z": -125.965, "size": 0.363, "title": "Infinite places of a number field", "summary": "This file defines the infinite places of a number field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.html"}, {"id": "Mathlib.Algebra.Algebra.Hom", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 4, "macro_tier_score": 0.3952, "macro_tier_override": null, "x": -23.495, "z": -33.426, "size": 0.7191, "title": "Homomorphisms of `R`-algebras", "summary": "This file defines bundled homomorphisms of `R`-algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Hom.html"}, {"id": "Mathlib.Algebra.Algebra.Basic", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 4, "macro_tier_score": 0.3951, "macro_tier_override": null, "x": 30.958, "z": -23.719, "size": 0.7166, "title": "Further basic results about `Algebra`.", "summary": "This file could usefully be split further.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Basic.html"}, {"id": "Mathlib.Algebra.Divisibility.Hom", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4329, "macro_tier_override": null, "x": 7.977, "z": -4.753, "size": 0.4237, "title": "Mapping divisibility across multiplication-preserving homomorphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Divisibility/Hom.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Variables", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.2728, "macro_tier_override": null, "x": 25.201, "z": -59.916, "size": 0.4863, "title": "Variables of polynomials", "summary": "This file establishes many results about the variable sets of a multivariate polynomial. The *variable set* of a polynomial $P \\in R[X]$ is a `Finset` containing each $x \\in X$ that appears in a monomial in $P$.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Variables.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Degrees", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 4, "macro_tier_score": 0.2747, "macro_tier_override": null, "x": 19.041, "z": 60.204, "size": 0.4373, "title": "Degrees of polynomials", "summary": "This file establishes many results about the degree of a multivariate polynomial. The *degree set* of a polynomial $P \\in R[X]$ is a `Multiset` containing, for each $x$ in the variable set, $n$ copies of $x$, where $n$ is the maximum number of copies of $x$ appearing in a monomial of $P$.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Degrees.html"}, {"id": "Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.19, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2881, "title": "Cyclic groups", "summary": "`IsCyclic` is a predicate on a group stating that the group is cyclic. For the concrete cyclic group of order `n`, see `Data.ZMod.Basic`. * `isCyclic_of_prime_card` proves that a finite group of prime order is cyclic. cyclic group", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/SpecificGroups/Cyclic/Basic.html"}, {"id": "Mathlib.LinearAlgebra.FreeModule.Determinant", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 2, "macro_tier_score": 0.0112, "macro_tier_override": null, "x": 26.959, "z": -81.064, "size": 0.25, "title": "Determinants in free (finite) modules", "summary": "Quite a lot of our results on determinants (that you might know in vector spaces) will work for all free (finite) modules over any commutative ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeModule/Determinant.html"}, {"id": "Mathlib.LinearAlgebra.Determinant", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 3, "macro_tier_score": 0.1356, "macro_tier_override": null, "x": -77.301, "z": -31.763, "size": 0.4937, "title": "Determinant of families of vectors", "summary": "This file defines the determinant of an endomorphism, and of a family of vectors with respect to some basis. For the determinant of a matrix, see the file `LinearAlgebra.Matrix.Determinant`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Determinant.html"}, {"id": "Mathlib.Algebra.Torsor.Basic", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": -7.173, "z": 19.128, "size": 0.354, "title": "Torsors of group actions", "summary": "Further results for torsors, that are not in `Mathlib/Algebra/AddTorsor/Defs.lean` to avoid increasing imports there.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Torsor/Basic.html"}, {"id": "Mathlib.Algebra.Torsor.Defs", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 2, "macro_tier_score": 0.0074, "macro_tier_override": null, "x": -5.582, "z": -9.644, "size": 0.3833, "title": "Torsors of group actions", "summary": "This file defines torsors of additive and multiplicative group actions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Torsor/Defs.html"}, {"id": "Mathlib.Algebra.Group.End", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4585, "macro_tier_override": null, "x": 12.649, "z": 10.926, "size": 0.5343, "title": "Monoids of endomorphisms, groups of automorphisms", "summary": "This file defines * the endomorphism monoid structure on `Function.End α := α → α` * the endomorphism monoid structure on `Monoid.End M := M →* M` and `AddMonoid.End M := M →+ M` * the automorphism group structure on `Equiv.Perm α := α ≃ α` * the automorphism group structure on `MulAut M := M ≃* M` and `AddAut M := M ≃+ M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/End.html"}, {"id": "Mathlib.RingTheory.Bialgebra.SymmetricAlgebra", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -63.968, "z": 19.444, "size": 0.2, "title": "Bialgebra structure on `SymmetricAlgebra R M`", "summary": "`SymmetricAlgebra R M` is the cocommutative commutative `R`-bialgebra on `M` in which each generator `ι x` is primitive: `Δ(ι x) = ι x ⊗ 1 + 1 ⊗ ι x` and `ε(ι x) = 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Bialgebra/SymmetricAlgebra.html"}, {"id": "Mathlib.LinearAlgebra.SymmetricAlgebra.Basic", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -59.43, "z": 14.972, "size": 0.2676, "title": "Symmetric Algebras", "summary": "Given a commutative semiring `R`, and an `R`-module `M`, we construct the symmetric algebra of `M`. This is the free commutative `R`-algebra generated (`R`-linearly) by the module `M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/SymmetricAlgebra/Basic.html"}, {"id": "Mathlib.RingTheory.Bialgebra.Basic", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 1, "macro_tier_score": 0.0045, "macro_tier_override": null, "x": 16.89, "z": 62.768, "size": 0.3462, "title": "Bialgebras", "summary": "In this file we define `Bialgebra`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Bialgebra/Basic.html"}, {"id": "Mathlib.Algebra.BigOperators.Finprod", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 4, "macro_tier_score": 0.4058, "macro_tier_override": null, "x": -7.361, "z": 30.702, "size": 0.5165, "title": "Finite products and sums over types and sets", "summary": "We define products and sums over types and subsets of types, with no finiteness hypotheses. All infinite products and sums are defined to be junk values (i.e. one or zero). This approach is sometimes easier to use than `Finset.sum`, when issues arise with `Finset` and `Fintype` being data.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Finprod.html"}, {"id": "Mathlib.Algebra.BigOperators.Pi", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.408, "macro_tier_override": null, "x": -13.526, "z": -15.31, "size": 0.4015, "title": "Big operators for Pi Types", "summary": "This file contains theorems relevant to big operators in binary and arbitrary products of monoids and groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Pi.html"}, {"id": "Mathlib.Algebra.Notation.FiniteSupport", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4012, "macro_tier_override": null, "x": -9.191, "z": -1.322, "size": 0.3342, "title": "Finiteness of support", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Notation/FiniteSupport.html"}, {"id": "Mathlib.Algebra.FiniteSupport.Basic", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.4012, "macro_tier_override": null, "x": -23.343, "z": 6.164, "size": 0.3342, "title": "Make `fun_prop` work for finite (multiplicative) support", "summary": "We provide API lemmas for the predicate `HasFiniteMulSupport` (and its additivized version `HasFiniteSupport`) on functions so that `fun_prop` can prove it for functions that are built from other functions with finite multiplicative support.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/FiniteSupport/Basic.html"}, {"id": "Mathlib.Algebra.Module.End", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4137, "macro_tier_override": null, "x": 10.441, "z": 15.359, "size": 0.4061, "title": "Module structure and endomorphisms", "summary": "In this file, we define `Module.toAddMonoidEnd`, which is `(•)` as a monoid homomorphism. We use this to prove some results on scalar multiplication by integers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/End.html"}, {"id": "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 4, "macro_tier_score": 0.4024, "macro_tier_override": null, "x": 29.286, "z": -5.032, "size": 0.3981, "title": "Big operators on a finset in ordered rings", "summary": "This file contains the results concerning the interaction of finset big operators with ordered rings. In particular, this file contains the standard form of the Cauchy-Schwarz inequality, as well as some of its immediate consequences.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/BigOperators/Ring/Finset.html"}, {"id": "Mathlib.Algebra.Field.ModEq", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 21.95, "z": -10.055, "size": 0.2302, "title": "Congruence modulo multiples of an element in a (semi)field", "summary": "In this file we prove a few theorems about the congruence relation `_ ≡ _ [PMOD _]` in a division semiring or a semifield.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/ModEq.html"}, {"id": "Mathlib.Algebra.Group.ModEq", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 1, "macro_tier_score": 0.0039, "macro_tier_override": null, "x": 8.562, "z": 3.595, "size": 0.3, "title": "Equality modulo an element", "summary": "This file defines equality modulo an element in an additive commutative monoid. In case of a group, `a` and `b` are congruent modulo `p` iff `b - a ∈ zmultiples p`. In case of a monoid, the definition is a bit more complicated, and it is given with the use case of natural numbers in mind.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/ModEq.html"}, {"id": "Mathlib.Algebra.Group.Nat.Defs", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.4717, "macro_tier_override": null, "x": -3.309, "z": 4.482, "size": 0.6114, "title": "The natural numbers form a monoid", "summary": "This file contains the additive and multiplicative monoid instances on the natural numbers. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Nat/Defs.html"}, {"id": "Mathlib.Algebra.BigOperators.ModEq", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 21.189, "z": -11.572, "size": 0.2, "title": "Congruence modulo natural and integer numbers for big operators", "summary": "In this file we prove various versions of the following theorem: if `f i ≡ g i [MOD n]` for all `i ∈ s`, then `∏ i ∈ s, f i ≡ ∏ i ∈ s, g i [MOD n]`, and similarly for sums. We prove it for lists, multisets, and finsets, as well as for natural and integer numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/ModEq.html"}, {"id": "Mathlib.Algebra.BigOperators.Ring.Finset", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.4165, "macro_tier_override": null, "x": -4.893, "z": 21.742, "size": 0.5369, "title": "Results about big operators with values in a (semi)ring", "summary": "We prove results about big operators that involve some interaction between multiplicative and additive structures on the values being combined.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Ring/Finset.html"}, {"id": "Mathlib.LinearAlgebra.Finsupp.VectorSpace", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 4, "macro_tier_score": 0.3041, "macro_tier_override": null, "x": -53.698, "z": -4.147, "size": 0.4121, "title": "Linear structures on function with finite support `ι →₀ M`", "summary": "This file contains results on the `R`-module structure on functions of finite support from a type `ι` to an `R`-module `M`, in particular in the case that `R` is a field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Finsupp/VectorSpace.html"}, {"id": "Mathlib.Algebra.Polynomial.PartialFractions", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 33.389, "z": -62.174, "size": 0.2, "title": "Partial fractions", "summary": "For `f, g : R[X]`, if `g` is expressed as a product `g₁ ^ n₁ * g₂ ^ n₂ * ... * gₙ ^ nₙ`, where the `gᵢ` are monic and pairwise coprime, then there is a quotient `q` and for each `i` from 1 to n and for each `0 ≤ j < nᵢ` there is a remainder `rᵢⱼ` with degree less than the degree of `gᵢ`, such that the fraction `f / g` decomposes as `q + ∑ i j, rᵢⱼ / gᵢ ^ (j + 1)`. Since polynomials do not have a division, the main…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/PartialFractions.html"}, {"id": "Mathlib.RingTheory.Coprime.Lemmas", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.3313, "macro_tier_override": null, "x": 19.519, "z": -14.209, "size": 0.3935, "title": "Additional lemmas about elements of a ring satisfying `IsCoprime`", "summary": "and elements of a monoid satisfying `IsRelPrime` These lemmas are in a separate file to the definition of `IsCoprime` or `IsRelPrime` as they require more imports. Notably, this includes lemmas about `Finset.prod` as this requires importing BigOperators, and lemmas about `Pow` since these are easiest to prove via `Finset.prod`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Coprime/Lemmas.html"}, {"id": "Mathlib.Algebra.Module.Equiv.Defs", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 4, "macro_tier_score": 0.4006, "macro_tier_override": null, "x": -11.732, "z": -23.203, "size": 0.5819, "title": "(Semi)linear equivalences", "summary": "In this file we define * `LinearEquiv σ M M₂`, `M ≃ₛₗ[σ] M₂`: an invertible semilinear map. Here, `σ` is a `RingHom` from `R` to `R₂` and an `e : M ≃ₛₗ[σ] M₂` satisfies `e (c • x) = (σ c) • (e x)`. The plain linear version, with `σ` being `RingHom.id R`, is denoted by `M ≃ₗ[R] M₂`, and the star-linear version (with `σ` being `starRingEnd`) is denoted by `M ≃ₗ⋆[R] M₂`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Equiv/Defs.html"}, {"id": "Mathlib.Algebra.Module.Submodule.Defs", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 4, "macro_tier_score": 0.3976, "macro_tier_override": null, "x": -27.748, "z": 15.061, "size": 0.5628, "title": "Submodules of a module", "summary": "In this file we define * `Submodule R M` : a subset of a `Module` `M` that contains zero and is closed with respect to addition and scalar multiplication. * `Subspace k M` : an abbreviation for `Submodule` assuming that `k` is a `Field`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/Defs.html"}, {"id": "Mathlib.RingTheory.Bialgebra.Convolution", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -5.995, "z": -72.181, "size": 0.2611, "title": "Convolution product on bialgebra homs", "summary": "This file constructs the ring structure on algebra homs `C → A` where `C` is a bialgebra and `A` an algebra, and also the ring structure on bialgebra homs `C → A` where `C` and `A` are bialgebras. Both multiplications are given by ``` | μ | | / \\ f * g = f g | | \\ / δ | ``` diagrammatically, where `μ` stands for multiplication and `δ` for comultiplication.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Bialgebra/Convolution.html"}, {"id": "Mathlib.RingTheory.Bialgebra.TensorProduct", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0038, "macro_tier_override": null, "x": -43.271, "z": -55.75, "size": 0.2934, "title": "Tensor products of bialgebras", "summary": "We define the data in the monoidal structure on the category of bialgebras - e.g. the bialgebra instance on a tensor product of bialgebras, and the tensor product of two `BialgHom`s as a `BialgHom`. This is done by combining the corresponding API for coalgebras and algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Bialgebra/TensorProduct.html"}, {"id": "Mathlib.RingTheory.Coalgebra.Convolution", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 2, "macro_tier_score": 0.0064, "macro_tier_override": null, "x": 26.666, "z": 57.237, "size": 0.3221, "title": "Convolution product on linear maps from a coalgebra to an algebra", "summary": "This file constructs the ring and algebra structure on linear maps `C → A` where `C` is a coalgebra and `A` an algebra, where multiplication is given by `(f * g)(x) = ∑ f x₍₁₎ * g x₍₂₎` in Sweedler notation or ``` | μ | | / \\ f * g = f g | | \\ / δ | ``` diagrammatically, where `μ` stands for multiplication and `δ` for comultiplication.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Coalgebra/Convolution.html"}, {"id": "Mathlib.Algebra.Category.Ring.Under.Basic", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": 28.155, "z": -66.733, "size": 0.2687, "title": "Under `CommRingCat`", "summary": "In this file we provide basic API for `Under R` when `R : CommRingCat`. `Under R` is (equivalent to) the category of commutative `R`-algebras. For not necessarily commutative algebras, use `AlgCat R` instead.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/Under/Basic.html"}, {"id": "Mathlib.Algebra.Category.Ring.Constructions", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0724, "macro_tier_override": null, "x": -70.321, "z": -5.954, "size": 0.414, "title": "Constructions of (co)limits in `CommRingCat`", "summary": "In this file we provide the explicit (co)cones for various (co)limits in `CommRingCat`, including * tensor product is the pushout * tensor product over `ℤ` is the binary coproduct * `ℤ` is the initial object * `0` is the strict terminal object * Cartesian product is the product * arbitrary direct product of a family of rings is the product object (Pi object) * `RingHom.eqLocus` is the equalizer", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/Constructions.html"}, {"id": "Mathlib.Algebra.Module.Card", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -19.56, "z": 5.895, "size": 0.2374, "title": "Cardinality of a module", "summary": "This file proves that the cardinality of a module without zero divisors is at least the cardinality of its base ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Card.html"}, {"id": "Mathlib.Algebra.Module.Torsion.Free", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4413, "macro_tier_override": null, "x": -11.469, "z": -14.607, "size": 0.5484, "title": "Torsion-free modules", "summary": "This files defines a torsion-free `R`-(semi)module `M` as a (semi)module where scalar multiplication by a regular element `r : R` is injective as a map `M → M`. In the case of a module (group over a ring), this is equivalent to saying that `r • m = 0` for some `r : R`, `m : M` implies that `r` is a zero-divisor. If furthermore the base ring is a domain, this is equivalent to the naïve `r • m = 0 ↔ r = 0 ∨ m = 0`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Torsion/Free.html"}, {"id": "Mathlib.LinearAlgebra.FiniteDimensional.Lemmas", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 3, "macro_tier_score": 0.1884, "macro_tier_override": null, "x": -2.028, "z": -77.974, "size": 0.443, "title": "Finite-dimensional vector spaces", "summary": "This file contains some further development of finite-dimensional vector spaces, their dimensions, and linear maps on such spaces. Definitions are in `Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean` and results that require fewer imports are in `Mathlib/LinearAlgebra/FiniteDimensional/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.DivisionRing", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 3, "macro_tier_score": 0.2254, "macro_tier_override": null, "x": -8.966, "z": 75.614, "size": 0.3634, "title": "Dimension of vector spaces", "summary": "In this file we provide results about `Module.rank` and `Module.finrank` of vector spaces over division rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/DivisionRing.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.2065, "macro_tier_override": null, "x": 67.987, "z": -24.977, "size": 0.4113, "title": "Some results on free modules over rings satisfying strong rank condition", "summary": "This file contains some results on free modules over rings satisfying strong rank condition. Most of them are generalized from the same result assuming the base ring being a division ring, and are moved from the files `Mathlib/LinearAlgebra/Dimension/DivisionRing.lean` and `Mathlib/LinearAlgebra/FiniteDimensional/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.html"}, {"id": "Mathlib.LinearAlgebra.FiniteDimensional.Basic", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 3, "macro_tier_score": 0.2316, "macro_tier_override": null, "x": 74.222, "z": 3.087, "size": 0.3968, "title": "Finite-dimensional vector spaces", "summary": "Basic properties of finite-dimensional vector spaces, of their dimensions, and of linear maps on such spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FiniteDimensional/Basic.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Equiv", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4525, "macro_tier_override": null, "x": -8.255, "z": 10.043, "size": 0.4082, "title": "Isomorphisms of monoids with zero", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Equiv.html"}, {"id": "Mathlib.Algebra.Group.Equiv.Defs", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.5072, "macro_tier_override": null, "x": -7.151, "z": 2.01, "size": 0.6993, "title": "Multiplicative and additive equivs", "summary": "In this file we define two extensions of `Equiv` called `AddEquiv` and `MulEquiv`, which are datatypes representing isomorphisms of `AddMonoid`s/`AddGroup`s and `Monoid`s/`Group`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Equiv/Defs.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Hom", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4732, "macro_tier_override": null, "x": -5.604, "z": -9.631, "size": 0.6654, "title": "Monoid with zero and group with zero homomorphisms", "summary": "This file defines homomorphisms of monoids with zero. We also define coercion to a function, and usual operations: composition, identity homomorphism, pointwise multiplication and pointwise inversion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Hom.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": -61.416, "z": -86.221, "size": 0.3118, "title": "Projective general linear group", "summary": "In this file we define `Matrix.ProjGenLinGroup n R` as the quotient of `GL n R` by its center. We introduce notation `PGL(n, R)` for this group, which works if `n` is either a finite type or a natural number. If `n` is a number, then `PGL(n, R)` is interpreted as `PGL(Fin n, R)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Projective.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Basic", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -13.508, "z": -80.591, "size": 0.2715, "title": "Basic lemmas about the general linear group $GL(n, R)$", "summary": "This file lists various basic lemmas about the general linear group $GL(n, R)$. For the definitions, see `Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.ProjectiveSpecialLinearGroup", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -62.46, "z": -49.76, "size": 0.2614, "title": "Projective Special Linear Group", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/ProjectiveSpecialLinearGroup.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Nat", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.2968, "macro_tier_override": null, "x": -1.926, "z": -7.175, "size": 0.3854, "title": "The natural numbers form a cancellative `CommMonoidWithZero`", "summary": "This file contains the `CommMonoidWithZero` and `IsCancelMulZero` instances on the natural numbers. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Nat.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Defs", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.4789, "macro_tier_override": null, "x": -1.175, "z": -5.446, "size": 0.728, "title": "Typeclasses for groups with an adjoined zero element", "summary": "This file provides just the typeclass definitions, and the projection lemmas that expose their members.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Defs.html"}, {"id": "Mathlib.FieldTheory.PurelyInseparable.Exponent", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -41.339, "z": 111.438, "size": 0.2, "title": "The exponent of purely inseparable extensions", "summary": "This file defines the exponent of a purely inseparable extension (if one exists) and some related results. Most results are stated using `ringExpChar K` rather than using `[ExpChar K p]` parameter because it gives cleaner API. To use the results in a context with `[ExpChar K p]`, consider using `ringExpChar.eq K p` for substitution.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/PurelyInseparable/Exponent.html"}, {"id": "Mathlib.FieldTheory.PurelyInseparable.Basic", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 3, "macro_tier_score": 0.0449, "macro_tier_override": null, "x": -114.244, "z": -25.252, "size": 0.3705, "title": "Basic results about purely inseparable extensions", "summary": "This file contains basic definitions and results about purely inseparable extensions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/PurelyInseparable/Basic.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Quotient", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 3, "macro_tier_score": 0.2564, "macro_tier_override": null, "x": 13.732, "z": -28.429, "size": 0.4142, "title": "Properties of group actions involving quotient groups", "summary": "This file proves properties of group actions which use the quotient group construction, notably * the orbit-stabilizer theorem `MulAction.card_orbit_mul_card_stabilizer_eq_card_group` * the class formula `MulAction.selfEquivSigmaOrbitsQuotientStabilizer'` * Burnside's lemma `MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group`, as well as their analogues for additive groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Quotient.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.Actions", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 4, "macro_tier_score": 0.4246, "macro_tier_override": null, "x": 6.976, "z": -26.97, "size": 0.4014, "title": "Actions by `Subgroup`s", "summary": "These are just copies of the definitions about `Submonoid` starting from `Submonoid.mulAction`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/Actions.html"}, {"id": "Mathlib.GroupTheory.Coset.Basic", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 4, "macro_tier_score": 0.3726, "macro_tier_override": null, "x": -20.559, "z": -15.917, "size": 0.4026, "title": "Cosets", "summary": "This file develops the basic theory of left and right cosets. When `G` is a group and `a : G`, `s : Set G`, with `open scoped Pointwise` we can write: * the left coset of `s` by `a` as `a • s` * the right coset of `s` by `a` as `MulOpposite.op a • s` (or `op a • s` with `open MulOpposite`, or `s <• a` with `open scoped Pointwise RightActions`) If instead `G` is an additive group, we can write (with `open scoped…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Coset/Basic.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Basic", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 3, "macro_tier_score": 0.2544, "macro_tier_override": null, "x": 12.889, "z": 18.181, "size": 0.2998, "title": "Basic properties of group actions", "summary": "This file primarily concerns itself with orbits, stabilizers, and other objects defined in terms of actions. Despite this file being called `basic`, low-level helper lemmas for algebraic manipulation of `•` belong elsewhere.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Basic.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Hom", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.438, "macro_tier_override": null, "x": 12.011, "z": 18.772, "size": 0.5759, "title": "Equivariant homomorphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Hom.html"}, {"id": "Mathlib.RingTheory.Unramified.Locus", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -52.458, "z": -91.946, "size": 0.3365, "title": "Unramified locus of an algebra", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Unramified/Locus.html"}, {"id": "Mathlib.RingTheory.Etale.Kaehler", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 2, "macro_tier_score": 0.0184, "macro_tier_override": null, "x": 13.97, "z": -103.059, "size": 0.3059, "title": "The differential module and étale algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Etale/Kaehler.html"}, {"id": "Mathlib.RingTheory.LocalRing.ResidueField.Fiber", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 2, "macro_tier_score": 0.0254, "macro_tier_override": null, "x": 90.886, "z": 37.791, "size": 0.3357, "title": "The fiber of a ring homomorphism at a prime ideal", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/ResidueField/Fiber.html"}, {"id": "Mathlib.RingTheory.Support", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 2, "macro_tier_score": 0.0284, "macro_tier_override": null, "x": -76.188, "z": 34.348, "size": 0.3114, "title": "Support of a module", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Support.html"}, {"id": "Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody", "region_id": "algebra", "micro_elevation": 0.9342, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -50.715, "z": -121.716, "size": 0.2683, "title": "Convex Bodies", "summary": "The file contains the definitions of several convex bodies lying in the mixed space `ℝ^r₁ × ℂ^r₂` associated to a number field of signature `K` and proves several existence theorems by applying *Minkowski Convex Body Theorem* to those.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.html"}, {"id": "Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic", "region_id": "algebra", "micro_elevation": 0.9211, "macro_tier": 1, "macro_tier_score": 0.003, "macro_tier_override": null, "x": -109.47, "z": 70.119, "size": 0.3185, "title": "Canonical embedding of a number field", "summary": "The canonical embedding of a number field `K` of degree `n` is the ring homomorphism `K →+* ℂ^n` that sends `x ∈ K` to `(φ_₁(x),...,φ_n(x))` where the `φ_i`'s are the complex embeddings of `K`. Note that we do not choose an ordering of the embeddings, but instead map `K` into the type `(K →+* ℂ) → ℂ` of `ℂ`-vectors indexed by the complex embeddings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Divisibility", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4257, "macro_tier_override": null, "x": -9.908, "z": 11.071, "size": 0.4421, "title": "Divisibility in groups with zero.", "summary": "Lemmas about divisibility in groups and monoids with zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Divisibility.html"}, {"id": "Mathlib.Algebra.Divisibility.Units", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4239, "macro_tier_override": null, "x": -12.931, "z": 1.341, "size": 0.3652, "title": "Divisibility and units", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Divisibility/Units.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -61.614, "z": 34.411, "size": 0.2276, "title": "Projective Dimension in ModuleCat", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/ProjectiveDimension.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Ext.DimensionShifting", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -64.774, "z": 22.937, "size": 0.2659, "title": "Dimension Shifting", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Ext/DimensionShifting.html"}, {"id": "Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -16.364, "z": -31.262, "size": 0.2443, "title": "Correctness of Terminating Continued Fraction Computations (`GenContFract.of`)", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.html"}, {"id": "Mathlib.Algebra.ContinuedFractions.Computation.Translations", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -32.133, "z": -9.216, "size": 0.2353, "title": "Basic Translation Lemmas Between Structures Defined for Computing Continued Fractions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/ContinuedFractions/Computation/Translations.html"}, {"id": "Mathlib.Algebra.ContinuedFractions.TerminatedStable", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 7.657, "z": 12.732, "size": 0.2654, "title": "Stabilisation of gcf Computations Under Termination", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/ContinuedFractions/TerminatedStable.html"}, {"id": "Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 8.413, "z": 12.246, "size": 0.2654, "title": "Recurrence Lemmas for the Continuants (`conts`) Function of Continued Fractions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/ContinuedFractions/ContinuantsRecurrence.html"}, {"id": "Mathlib.Algebra.Star.NonUnitalSubalgebra", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 2, "macro_tier_score": 0.0072, "macro_tier_override": null, "x": 44.205, "z": -23.67, "size": 0.3034, "title": "Non-unital Star Subalgebras", "summary": "In this file we define `NonUnitalStarSubalgebra`s and the usual operations on them (`map`, `comap`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/NonUnitalSubalgebra.html"}, {"id": "Mathlib.Algebra.Star.StarAlgHom", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 3, "macro_tier_score": 0.2608, "macro_tier_override": null, "x": 8.765, "z": -47.484, "size": 0.3553, "title": "Morphisms of star algebras", "summary": "This file defines morphisms between `R`-algebras (unital or non-unital) `A` and `B` where both `A` and `B` are equipped with a `star` operation. These morphisms, namely `StarAlgHom` and `NonUnitalStarAlgHom` are direct extensions of their non-`star`red counterparts with a field `map_star` which guarantees they preserve the star operation. We keep the type classes as generic as possible, in keeping with the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/StarAlgHom.html"}, {"id": "Mathlib.Algebra.Star.Center", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 2, "macro_tier_score": 0.007, "macro_tier_override": null, "x": -19.328, "z": -14.468, "size": 0.279, "title": "`Set.center`, `Set.centralizer` and the `star` operation", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/Center.html"}, {"id": "Mathlib.Algebra.Star.SelfAdjoint", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.2821, "macro_tier_override": null, "x": -11.73, "z": 21.102, "size": 0.3707, "title": "Self-adjoint, skew-adjoint and normal elements of a star additive group", "summary": "This file defines `selfAdjoint R` (resp. `skewAdjoint R`), where `R` is a star additive group, as the additive subgroup containing the elements that satisfy `star x = x` (resp. `star x = -x`). This includes, for instance, (skew-)Hermitian operators on Hilbert spaces. We also define `IsStarNormal R`, a `Prop` that states that an element `x` satisfies `star x * x = x * star x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/SelfAdjoint.html"}, {"id": "Mathlib.Algebra.Star.Prod", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 2, "macro_tier_score": 0.0082, "macro_tier_override": null, "x": 12.223, "z": 18.635, "size": 0.2838, "title": "Basic Results about Star on Product Type", "summary": "This file provides basic results about the star on product types defined in `Mathlib/Algebra/Notation/Prod.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/Prod.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 3, "macro_tier_score": 0.1768, "macro_tier_override": null, "x": 48.72, "z": 67.902, "size": 0.3015, "title": "Cayley-Hamilton theorem for f.g. modules.", "summary": "Given a fixed finite spanning set `b : ι → M` of an `R`-module `M`, we say that a matrix `M` represents an endomorphism `f : M →ₗ[R] M` if the matrix as an endomorphism of `ι → R` commutes with `f` via the projection `(ι → R) →ₗ[R] M` given by `b`. We show that every endomorphism has a matrix representation, and if `f.range ≤ I • ⊤` for some ideal `I`, we may furthermore obtain a matrix representation whose entries…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.html"}, {"id": "Mathlib.Algebra.Module.SpanRank", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.1779, "macro_tier_override": null, "x": -53.471, "z": -46.058, "size": 0.2999, "title": "Minimum Cardinality of generating set of a submodule", "summary": "In this file, we define the minimum cardinality of a generating set for a submodule, which is implemented as `spanFinrank` and `spanRank`. `spanFinrank` takes value in `ℕ` and equals `0` when no finite generating set exists. `spanRank` takes value as a cardinal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/SpanRank.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.Range", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.0683, "macro_tier_override": null, "x": -35.084, "z": -3.769, "size": 0.3133, "title": "The range of a MonoidWithZeroHom", "summary": "Given a `MonoidWithZeroHom` `f : A → B` whose codomain `B` is a `LinearOrderedCommGroupWithZero`, we provide some order properties of the `MonoidWithZeroHom.ValueGroup₀` as defined in `Mathlib.Algebra.GroupWithZero.Range`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/Range.html"}, {"id": "Mathlib.Algebra.Ring.Torsion", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 3, "macro_tier_score": 0.0686, "macro_tier_override": null, "x": -15.655, "z": 9.992, "size": 0.3354, "title": "Torsion-free rings", "summary": "A characteristic zero domain is torsion-free.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Torsion.html"}, {"id": "Mathlib.Algebra.Opposites", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.5043, "macro_tier_override": null, "x": 0.211, "z": -5.567, "size": 0.6448, "title": "Multiplicative opposite and algebraic operations on it", "summary": "In this file we define `MulOpposite α = αᵐᵒᵖ` to be the multiplicative opposite of `α`. It inherits all additive algebraic structures on `α` (in other files), and reverses the order of multipliers in multiplicative structures, i.e., `op (x * y) = op y * op x`, where `MulOpposite.op` is the canonical map from `α` to `αᵐᵒᵖ`. We also define `AddOpposite α = αᵃᵒᵖ` to be the additive opposite of `α`. It inherits all…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Opposites.html"}, {"id": "Mathlib.Algebra.Notation.Pi.Defs", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 4, "macro_tier_score": 0.5097, "macro_tier_override": null, "x": -1.095, "z": 1.5, "size": 0.7052, "title": "Notation for algebraic operators on pi types", "summary": "This file provides only the notation for (pointwise) `0`, `1`, `+`, `*`, `•`, `^`, `⁻¹` on pi types. See `Mathlib/Algebra/Group/Pi/Basic.lean` for the `Monoid` and `Group` instances. There is also an instance of the `Star` notation typeclass, but no default notation is included.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Notation/Pi/Defs.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Limits", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.03, "macro_tier_override": null, "x": 52.836, "z": 22.866, "size": 0.3443, "title": "The category of R-modules has all limits", "summary": "Further, these limits are preserved by the forgetful functor --- that is, the underlying types are just the limits in the category of types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Limits.html"}, {"id": "Mathlib.Algebra.Category.Grp.Limits", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 3, "macro_tier_score": 0.0902, "macro_tier_override": null, "x": -22.728, "z": -8.145, "size": 0.3659, "title": "The category of (commutative) (additive) groups has all limits", "summary": "Further, these limits are preserved by the forgetful functor --- that is, the underlying types are just the limits in the category of types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Limits.html"}, {"id": "Mathlib.Algebra.Colimit.Module", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 3, "macro_tier_score": 0.0773, "macro_tier_override": null, "x": -55.269, "z": -7.033, "size": 0.3211, "title": "Direct limit of modules and abelian groups", "summary": "See Atiyah-Macdonald PP.32-33, Matsumura PP.269-270 Generalizes the notion of \"union\", or \"gluing\", of incomparable modules over the same ring, or incomparable abelian groups. It is constructed as a quotient of the free module instead of a quotient of the disjoint union so as to make the operations (addition etc.) \"computable\".", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Colimit/Module.html"}, {"id": "Mathlib.Algebra.Module.Shrink", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 4, "macro_tier_score": 0.3606, "macro_tier_override": null, "x": 14.534, "z": 25.917, "size": 0.4135, "title": "Transfer module and algebra structures from `α` to `Shrink α`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Shrink.html"}, {"id": "Mathlib.RingTheory.WittVector.Frobenius", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 110.29, "z": 15.893, "size": 0.2668, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/Frobenius.html"}, {"id": "Mathlib.Algebra.Algebra.ZMod", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 2, "macro_tier_score": 0.0174, "macro_tier_override": null, "x": -26.885, "z": -7.295, "size": 0.3187, "title": "The `ZMod n`-algebra structure on rings whose characteristic divides `n`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/ZMod.html"}, {"id": "Mathlib.RingTheory.WittVector.IsPoly", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 1, "macro_tier_score": 0.0041, "macro_tier_override": null, "x": 77.296, "z": 77.663, "size": 0.3857, "title": "The `IsPoly` predicate", "summary": "`WittVector.IsPoly` is a (type-valued) predicate on functions `f : Π R, 𝕎 R → 𝕎 R`. It asserts that there is a family of polynomials `φ : ℕ → MvPolynomial ℕ ℤ`, such that the `n`th coefficient of `f x` is equal to `φ n` evaluated on the coefficients of `x`. Many operations on Witt vectors satisfy this predicate (or an analogue for higher arity functions). We say that such a function `f` is a *polynomial function*.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/IsPoly.html"}, {"id": "Mathlib.RingTheory.QuotSMulTop", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 2, "macro_tier_score": 0.0292, "macro_tier_override": null, "x": 52.099, "z": 41.901, "size": 0.2766, "title": "Reducing a module modulo an element of the ring", "summary": "Given a commutative ring `R` and an element `r : R`, the association `M ↦ M ⧸ rM` extends to a functor on the category of `R`-modules. This functor is isomorphic to the functor of tensoring by `R ⧸ (r)` on either side, but can be more convenient to work with since we can work with quotient types instead of fiddling with simple tensors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/QuotSMulTop.html"}, {"id": "Mathlib.LinearAlgebra.DFinsupp", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 4, "macro_tier_score": 0.3252, "macro_tier_override": null, "x": -29.704, "z": 42.681, "size": 0.5254, "title": "Properties of the module `Π₀ i, M i`", "summary": "Given an indexed collection of `R`-modules `M i`, the `R`-module structure on `Π₀ i, M i` is defined in `Mathlib/Data/DFinsupp/Module.lean`. In this file we define `LinearMap` versions of various maps: * `DFinsupp.lsingle a : M →ₗ[R] Π₀ i, M i`: `DFinsupp.single a` as a linear map; * `DFinsupp.lmk s : (Π i : (↑s : Set ι), M i) →ₗ[R] Π₀ i, M i`: `DFinsupp.mk` as a linear map; * `DFinsupp.lapply i : (Π₀ i, M i) →ₗ[R]…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/DFinsupp.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Quotient", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.1385, "macro_tier_override": null, "x": -53.217, "z": 26.453, "size": 0.3386, "title": "Interaction between Quotients and Tensor Products", "summary": "This file contains constructions that relate quotients and tensor products. This file is also a home for results whose proof depends on both tensor products and linear algebraic quotients. Let `M, N` be `R`-modules, `m ≤ M` and `n ≤ N` be an `R`-submodules and `I ≤ R` an ideal. We prove the following isomorphisms:", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Quotient.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.RightExactness", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.1504, "macro_tier_override": null, "x": 41.732, "z": 49.835, "size": 0.3839, "title": "Right-exactness properties of tensor product", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/RightExactness.html"}, {"id": "Mathlib.Algebra.Group.Pointwise.Finset.Scalar", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 3, "macro_tier_score": 0.1997, "macro_tier_override": null, "x": -20.204, "z": -3.024, "size": 0.3508, "title": "Pointwise operations of finsets", "summary": "This file defines pointwise algebraic operations on finsets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pointwise/Finset/Scalar.html"}, {"id": "Mathlib.Algebra.Group.Pointwise.Set.Finite", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.3935, "macro_tier_override": null, "x": 18.552, "z": -0.843, "size": 0.3989, "title": "Finiteness lemmas for pointwise operations on sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pointwise/Set/Finite.html"}, {"id": "Mathlib.Algebra.Category.CommAlgCat.FiniteType", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": -65.009, "z": -46.38, "size": 0.2907, "title": "The category of finitely generated `R`-algebras", "summary": "We define the category of finitely generated `R`-algebras and show it is essentially small.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/CommAlgCat/FiniteType.html"}, {"id": "Mathlib.Algebra.Category.CommAlgCat.Basic", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 1, "macro_tier_score": 0.0039, "macro_tier_override": null, "x": 73.833, "z": 8.191, "size": 0.2987, "title": "The category of commutative algebras over a commutative ring", "summary": "This file defines the bundled category `CommAlgCat` of commutative algebras over a fixed commutative ring `R` along with the forgetful functors to `CommRingCat` and `AlgCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/CommAlgCat/Basic.html"}, {"id": "Mathlib.RingTheory.FinitePresentation", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 3, "macro_tier_score": 0.139, "macro_tier_override": null, "x": -56.205, "z": -54.085, "size": 0.3635, "title": "Finiteness conditions in commutative algebra", "summary": "In this file we define several notions of finiteness that are common in commutative algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/FinitePresentation.html"}, {"id": "Mathlib.RingTheory.RingHomProperties", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.0693, "macro_tier_override": null, "x": 65.441, "z": -31.041, "size": 0.3716, "title": "Properties of ring homomorphisms", "summary": "We provide the basic framework for talking about properties of ring homomorphisms. The following meta-properties of predicates on ring homomorphisms are defined * `RingHom.RespectsIso`: `P` respects isomorphisms if `P f → P (e ≫ f)` and `P f → P (f ≫ e)`, where `e` is an isomorphism. * `RingHom.StableUnderComposition`: `P` is stable under composition if `P f → P g → P (f ≫ g)`. * `RingHom.IsStableUnderBaseChange`:…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHomProperties.html"}, {"id": "Mathlib.RingTheory.Unramified.Dedekind", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 64.018, "z": 102.342, "size": 0.2478, "title": "Unramified algebras over Dedekind domains", "summary": "We prove that a domain finite and unramified over a Dedekind domain is a Dedekind domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Unramified/Dedekind.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.Dvr", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 3, "macro_tier_score": 0.05, "macro_tier_override": null, "x": -58.954, "z": 81.128, "size": 0.3461, "title": "Dedekind domains", "summary": "This file defines an equivalent notion of a Dedekind domain (or Dedekind ring), namely a Noetherian integral domain where the localization at every nonzero prime ideal is a DVR.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/Dvr.html"}, {"id": "Mathlib.RingTheory.Finiteness.Quotient", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 2, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 63.749, "z": -44.946, "size": 0.3004, "title": "Finiteness of quotient modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Quotient.html"}, {"id": "Mathlib.RingTheory.Unramified.Field", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 2, "macro_tier_score": 0.0173, "macro_tier_override": null, "x": -72.797, "z": -93.957, "size": 0.3119, "title": "Unramified algebras over fields", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Unramified/Field.html"}, {"id": "Mathlib.RingTheory.Smooth.StandardSmoothCotangent", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 2, "macro_tier_score": 0.0092, "macro_tier_override": null, "x": -83.135, "z": 62.488, "size": 0.2788, "title": "Cotangent complex of a submersive presentation", "summary": "Let `P` be a submersive presentation of `S` as an `R`-algebra and denote by `I` the kernel `R[X] → S`. We show - `SubmersivePresentation.free_cotangent`: `I ⧸ I ^ 2` is `S`-free on the classes of `P.relation i`. - `SubmersivePresentation.subsingleton_h1Cotangent`: `H¹(L_{S/R}) = 0`. - `SubmersivePresentation.free_kaehlerDifferential`: `Ω[S⁄R]` is `S`-free on the images of `dxᵢ` where `i ∉ Set.range P.map`. -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/StandardSmoothCotangent.html"}, {"id": "Mathlib.LinearAlgebra.Basis.Exact", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 2, "macro_tier_score": 0.0092, "macro_tier_override": null, "x": 35.801, "z": -40.236, "size": 0.2735, "title": "Basis from a split exact sequence", "summary": "Let `0 → K → M → P → 0` be a split exact sequence of `R`-modules, let `s : M → K` be a retraction of `f` and `v` be a basis of `M` indexed by `κ ⊕ σ`. Then if `s vᵢ = 0` for `i : κ` and `(s vⱼ)ⱼ` is linear independent for `j : σ`, then the images of `vᵢ` for `i : κ` form a basis of `P`. We treat linear independence and the span condition separately. For convenience this is stated not for `κ ⊕ σ`, but for an…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Basis/Exact.html"}, {"id": "Mathlib.RingTheory.Smooth.StandardSmooth", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 2, "macro_tier_score": 0.0091, "macro_tier_override": null, "x": 37.978, "z": -82.697, "size": 0.2571, "title": "Standard smooth algebras", "summary": "A standard smooth algebra is an algebra that admits a `SubmersivePresentation`. A standard smooth algebra is of relative dimension `n` if it admits a submersive presentation of dimension `n`. While every standard smooth algebra is smooth, the converse does not hold. But if `S` is `R`-smooth, then `S` is `R`-standard smooth locally on `S`, i.e. there exists a set `{ t }` of `S` that generates the unit ideal, such…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/StandardSmooth.html"}, {"id": "Mathlib.RingTheory.Etale.Basic", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 2, "macro_tier_score": 0.0206, "macro_tier_override": null, "x": -101.039, "z": -14.984, "size": 0.3103, "title": "Étale morphisms", "summary": "An `R`-algebra `A` is formally etale if `Ω[A⁄R]` and `H¹(L_{A/R})` both vanish. This is equivalent to the standard definition that \"for every `R`-algebra `B`, every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`\". An `R`-algebra `A` is étale if it is formally étale and of finite presentation. We show that the property extends onto nilpotent ideals, and that these…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Etale/Basic.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.Real", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 79.149, "z": 44.906, "size": 0.2, "title": "Real quadratic forms", "summary": "Sylvester's law of inertia `equivalent_one_neg_one_weighted_sum_squared`: A real quadratic form is equivalent to a weighted sum of squares with the weights being ±1 or 0. When the real quadratic form is nondegenerate we can take the weights to be ±1, as in `QuadraticForm.equivalent_one_zero_neg_one_weighted_sum_squared`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/Real.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 2, "macro_tier_score": 0.0109, "macro_tier_override": null, "x": -87.38, "z": 17.643, "size": 0.3932, "title": "Isometric equivalences with respect to quadratic forms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/IsometryEquiv.html"}, {"id": "Mathlib.Algebra.CharP.Invertible", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 2, "macro_tier_score": 0.0221, "macro_tier_override": null, "x": -20.083, "z": 3.74, "size": 0.3963, "title": "Invertibility of elements given a characteristic", "summary": "This file includes some instances of `Invertible` for specific numbers in characteristic zero. Some more cases are given as a `def`, to be included only when needed. To construct instances for concrete numbers, `invertibleOfNonzero` is a useful definition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/Invertible.html"}, {"id": "Mathlib.RingTheory.FreeCommRing", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.073, "macro_tier_override": null, "x": 60.023, "z": 33.451, "size": 0.3285, "title": "Free commutative rings", "summary": "The theory of the free commutative ring generated by a type `α`. It is isomorphic to the polynomial ring over ℤ with variables in `α`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/FreeCommRing.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Equiv", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.2715, "macro_tier_override": null, "x": 57.155, "z": -30.959, "size": 0.4404, "title": "Equivalences between polynomial rings", "summary": "This file establishes a number of equivalences between polynomial rings, based on equivalences between the underlying types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Equiv.html"}, {"id": "Mathlib.Algebra.MvPolynomial.CommRing", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.2712, "macro_tier_override": null, "x": -5.476, "z": 66.633, "size": 0.4298, "title": "Multivariate polynomials over a ring", "summary": "Many results about polynomials hold when the coefficient ring is a commutative semiring. Some stronger results can be derived when we assume this semiring is a ring. This file does not define any new operations, but proves some of these stronger results.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/CommRing.html"}, {"id": "Mathlib.RingTheory.FreeRing", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.0734, "macro_tier_override": null, "x": -34.602, "z": 6.914, "size": 0.268, "title": "Free rings", "summary": "The theory of the free ring over a type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/FreeRing.html"}, {"id": "Mathlib.Algebra.Homology.QuasiIso", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 2, "macro_tier_score": 0.0074, "macro_tier_override": null, "x": -16.156, "z": -4.284, "size": 0.3821, "title": "Quasi-isomorphisms", "summary": "A chain map is a quasi-isomorphism if it induces isomorphisms on homology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/QuasiIso.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Invertible", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": 11.863, "z": -58.233, "size": 0.2808, "title": "Invertible polynomials", "summary": "This file is a stub containing some basic facts about invertible elements in the ring of polynomials.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Invertible.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Basic", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 4, "macro_tier_score": 0.2775, "macro_tier_override": null, "x": -43.193, "z": 38.064, "size": 0.4945, "title": "Multivariate polynomials", "summary": "This file defines polynomial rings over a base ring (or even semiring), with variables from a general type `σ` (which could be infinite).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Basic.html"}, {"id": "Mathlib.RingTheory.AlgebraTower", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 3, "macro_tier_score": 0.2686, "macro_tier_override": null, "x": 13.66, "z": -48.247, "size": 0.359, "title": "Towers of algebras", "summary": "We set up the basic theory of algebra towers. An algebra tower A/S/R is expressed by having instances of `Algebra A S`, `Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the later asserting the compatibility condition `(r • s) • a = r • (s • a)`. In `Mathlib/FieldTheory/Tower.lean` we use this to prove the tower law for finite extensions, that if `R` and `S` are both fields, then `[A:R] = [A:S] [S:A]`. In this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AlgebraTower.html"}, {"id": "Mathlib.RingTheory.Morita.Matrix", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 60.542, "z": -17.939, "size": 0.2, "title": "Morita Equivalence between `R` and `Mₙ(R)`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Morita/Matrix.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Module", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 37.12, "z": -1.326, "size": 0.2478, "title": "Mₙ(R)-module structure on `Mⁿ`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Module.html"}, {"id": "Mathlib.RingTheory.Morita.Basic", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 22.841, "z": 56.871, "size": 0.2478, "title": "Morita equivalence", "summary": "Two `R`-algebras `A` and `B` are Morita equivalent if the categories of modules over `A` and `B` are `R`-linearly equivalent. In this file, we prove that Morita equivalence is an equivalence relation and that isomorphic algebras are Morita equivalent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Morita/Basic.html"}, {"id": "Mathlib.Algebra.Star.LinearMap", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -67.448, "z": -46.132, "size": 0.2617, "title": "Intrinsic star operation on linear maps", "summary": "This file defines the star operation on linear maps: `(star f) x = star (f (star x))`. This corresponds to a map being star-preserving, i.e., a map is self-adjoint iff it is star-preserving.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/LinearMap.html"}, {"id": "Mathlib.RingTheory.RingHom.Integral", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 22.155, "z": -90.176, "size": 0.2593, "title": "The meta properties of integral ring homomorphisms.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/Integral.html"}, {"id": "Mathlib.NumberTheory.NumberField.Discriminant.Basic", "region_id": "algebra", "micro_elevation": 0.9474, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -89.716, "z": -99.151, "size": 0.2718, "title": "Number field discriminant", "summary": "This file defines the discriminant of a number field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Discriminant/Basic.html"}, {"id": "Mathlib.Algebra.Module.ZLattice.Covolume", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 84.866, "z": 20.413, "size": 0.2623, "title": "Covolume of ℤ-lattices", "summary": "Let `E` be a finite-dimensional real vector space. Let `L` be a `ℤ`-lattice `L` defined as a discrete `ℤ`-submodule of `E` that spans `E` over `ℝ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/ZLattice/Covolume.html"}, {"id": "Mathlib.NumberTheory.NumberField.Discriminant.Defs", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": 99.128, "z": -75.21, "size": 0.2673, "title": "Number field discriminant", "summary": "This file defines the discriminant of a number field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Discriminant/Defs.html"}, {"id": "Mathlib.NumberTheory.NumberField.EquivReindex", "region_id": "algebra", "micro_elevation": 0.9342, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -46.002, "z": 123.574, "size": 0.2596, "title": "Reindexed basis", "summary": "This file introduces an equivalence between the set of embeddings of `K` into `ℂ` and the index set of the chosen basis of the ring of integers of `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/EquivReindex.html"}, {"id": "Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex", "region_id": "algebra", "micro_elevation": 0.9211, "macro_tier": 1, "macro_tier_score": 0.0037, "macro_tier_override": null, "x": 120.639, "z": 48.443, "size": 0.278, "title": "Totally real and totally complex number fields", "summary": "This file defines the type of totally real and totally complex number fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/InfinitePlace/TotallyRealComplex.html"}, {"id": "Mathlib.FieldTheory.IsPerfectClosure", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 99.487, "z": -65.036, "size": 0.2, "title": "`IsPerfectClosure` predicate", "summary": "This file contains `IsPerfectClosure` which asserts that `L` is a perfect closure of `K` under a ring homomorphism `i : K →+* L`, as well as its basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/IsPerfectClosure.html"}, {"id": "Mathlib.FieldTheory.PerfectClosure", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -38.515, "z": 92.596, "size": 0.2478, "title": "The perfect closure of a characteristic `p` ring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/PerfectClosure.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.TruncLEHomology", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 2, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -19.244, "z": -22.641, "size": 0.3015, "title": "The homology of a canonical truncation", "summary": "Given an embedding of complex shapes `e : Embedding c c'`, we relate the homology of `K : HomologicalComplex C c'` and of `K.truncLE e : HomologicalComplex C c'`. The main result is that `K.ιTruncLE e : K.truncLE e ⟶ K` induces a quasi-isomorphism in degree `e.f i` for all `i`. (Note that the complex `K.truncLE e` is exact in degrees that are not in the image of `e.f`.) All the results are obtained by dualising the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/TruncLEHomology.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.TruncGEHomology", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": -25.17, "z": -11.938, "size": 0.2576, "title": "The homology of a canonical truncation", "summary": "Given an embedding of complex shapes `e : Embedding c c'`, we relate the homology of `K : HomologicalComplex C c'` and of `K.truncGE e : HomologicalComplex C c'`. The main result is that `K.πTruncGE e : K ⟶ K.truncGE e` induces a quasi-isomorphism in degree `e.f i` for all `i`. (Note that the complex `K.truncGE e` is exact in degrees that are not in the image of `e.f`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/TruncGEHomology.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.TruncLE", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": -27.857, "z": -0.217, "size": 0.2576, "title": "The canonical truncation", "summary": "Given an embedding `e : Embedding c c'` of complex shapes which satisfies `e.IsTruncLE` and `K : HomologicalComplex C c'`, we define `K.truncGE' e : HomologicalComplex C c` and `K.truncLE e : HomologicalComplex C c'` which are the canonical truncations of `K` relative to `e`. In order to achieve this, we dualize the constructions from the file `Embedding.TruncGE`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/TruncLE.html"}, {"id": "Mathlib.Algebra.Homology.HomologySequence", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 2, "macro_tier_score": 0.0064, "macro_tier_override": null, "x": -5.0, "z": -15.949, "size": 0.3212, "title": "The homology sequence", "summary": "If `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0` is a short exact sequence in a category of complexes `HomologicalComplex C c` in an abelian category (i.e. `S` is a short complex in that category and satisfies `hS : S.ShortExact`), then whenever `i` and `j` are degrees such that `hij : c.Rel i j`, then there is a long exact sequence : `... ⟶ S.X₁.homology i ⟶ S.X₂.homology i ⟶ S.X₃.homology i ⟶ S.X₁.homology j ⟶ ...`. The connecting…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomologySequence.html"}, {"id": "Mathlib.Algebra.Homology.HomologicalComplexAbelian", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 2, "macro_tier_score": 0.0062, "macro_tier_override": null, "x": -8.681, "z": 14.284, "size": 0.3113, "title": "THe category of homological complexes is abelian", "summary": "If `C` is an abelian category, then `HomologicalComplex C c` is an abelian category for any complex shape `c : ComplexShape ι`. We also obtain that a short complex in `HomologicalComplex C c` is exact (resp. short exact) iff degreewise it is so.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomologicalComplexAbelian.html"}, {"id": "Mathlib.NumberTheory.ArithmeticFunction.Misc", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 2, "macro_tier_score": 0.0293, "macro_tier_override": null, "x": 35.192, "z": -20.757, "size": 0.3702, "title": "Miscellaneous arithmetic Functions", "summary": "This file defines some simple examples of arithmetic functions (functions `ℕ → R` vanishing at `0`, considered as a ring under Dirichlet convolution). Note that the Von Mangoldt and Möbius functions are in separate files.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ArithmeticFunction/Misc.html"}, {"id": "Mathlib.NumberTheory.ArithmeticFunction.Zeta", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 2, "macro_tier_score": 0.0285, "macro_tier_override": null, "x": 13.036, "z": 36.757, "size": 0.3204, "title": "The arithmetic function `ζ`", "summary": "We define `ζ` to be the arithmetic function with `ζ n = 1` for `0 < n` (whose Dirichlet series is the Riemann zeta function).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ArithmeticFunction/Zeta.html"}, {"id": "Mathlib.NumberTheory.LSeries.ZMod", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 62.909, "z": 55.016, "size": 0.2653, "title": "L-series of functions on `ZMod N`", "summary": "We show that if `N` is a positive integer and `Φ : ZMod N → ℂ`, then the L-series of `Φ` has analytic continuation (away from a pole at `s = 1` if `∑ j, Φ j ≠ 0`) and satisfies a functional equation. We also define completed L-functions (given by multiplying the naive L-function by a Gamma-factor), and prove analytic continuation and functional equations for these too, assuming `Φ` is either even or odd. The most…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/ZMod.html"}, {"id": "Mathlib.NumberTheory.LSeries.RiemannZeta", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 2, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -13.843, "z": -80.534, "size": 0.3072, "title": "Definition of the Riemann zeta function", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/RiemannZeta.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Associator", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 4, "macro_tier_score": 0.2846, "macro_tier_override": null, "x": -50.839, "z": -10.93, "size": 0.4316, "title": "Associators and unitors for tensor products of modules over a commutative ring.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Associator.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Map", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 4, "macro_tier_score": 0.3521, "macro_tier_override": null, "x": -32.074, "z": -38.543, "size": 0.4674, "title": "Tensor products and linear maps", "summary": "This file defines `TensorProduct.map`, the `R`-linear map from `M ⊗ N` to `M₂ ⊗ N₂` defined by a pair of linear (or more generally semilinear) maps `f : M → M₂` and `g : N → N₂`. The notation `f ⊗ₘ g` is available for this map. We also define one-sided versions `lTensor` and `rTensor`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Map.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Semi", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 3, "macro_tier_score": 0.0574, "macro_tier_override": null, "x": 26.475, "z": 20.411, "size": 0.3235, "title": "The category of `R`-modules", "summary": "If `R` is a semiring, `SemimoduleCat.{v} R` is the category of bundled `R`-semimodules with carrier in the universe `v`. We show that it is preadditive and show that being an isomorphism and monomorphism are equivalent to being a linear equivalence and an injective linear map respectively.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Semi.html"}, {"id": "Mathlib.Algebra.Category.Grp.Preadditive", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 3, "macro_tier_score": 0.0952, "macro_tier_override": null, "x": 16.129, "z": -12.537, "size": 0.3939, "title": "The category of additive commutative groups is preadditive.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Preadditive.html"}, {"id": "Mathlib.LinearAlgebra.BilinearMap", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 4, "macro_tier_score": 0.3671, "macro_tier_override": null, "x": 26.174, "z": 33.756, "size": 0.4884, "title": "Basics on bilinear maps", "summary": "This file provides basics on bilinear maps. The most general form considered are maps that are semilinear in both arguments. They are of type `M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P`, where `M` and `N` are modules over `R` and `S` respectively, `P` is a module over both `R₂` and `S₂` with commuting actions, and `ρ₁₂ : R →+* R₂` and `σ₁₂ : S →+* S₂`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/BilinearMap.html"}, {"id": "Mathlib.LinearAlgebra.SModEq.Pointwise", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 2, "macro_tier_score": 0.0091, "macro_tier_override": null, "x": -5.782, "z": 55.414, "size": 0.2688, "title": "Pointwise lemmas for modular equivalence", "summary": "In this file, we record more lemmas about `SModEq` on elements of modules or rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/SModEq/Pointwise.html"}, {"id": "Mathlib.LinearAlgebra.SModEq.Basic", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 3, "macro_tier_score": 0.1589, "macro_tier_override": null, "x": 10.774, "z": 50.872, "size": 0.3611, "title": "modular equivalence for submodule", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/SModEq/Basic.html"}, {"id": "Mathlib.LinearAlgebra.BilinearForm.Orthogonal", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 1, "macro_tier_score": 0.0027, "macro_tier_override": null, "x": 61.164, "z": 56.95, "size": 0.2891, "title": "Bilinear form", "summary": "This file defines orthogonal bilinear forms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/BilinearForm/Orthogonal.html"}, {"id": "Mathlib.LinearAlgebra.BilinearForm.Properties", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 3, "macro_tier_score": 0.0433, "macro_tier_override": null, "x": -23.354, "z": -78.307, "size": 0.3421, "title": "Bilinear form", "summary": "This file defines various properties of bilinear forms, including reflexivity, symmetry, alternativity, adjoint, and non-degeneracy. For orthogonality, see `Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/BilinearForm/Properties.html"}, {"id": "Mathlib.LinearAlgebra.SesquilinearForm.Orthogonal", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 2, "macro_tier_score": 0.0136, "macro_tier_override": null, "x": -12.984, "z": 52.269, "size": 0.3573, "title": "Orthogonal complement", "summary": "This file defines the orthogonal submodule of a submodule with respect to a sesqui-blinear map.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/SesquilinearForm/Orthogonal.html"}, {"id": "Mathlib.GroupTheory.Perm.MaximalSubgroups", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": -38.681, "z": 40.099, "size": 0.2839, "title": "Maximal subgroups of the symmetric groups", "summary": "* `Equiv.Perm.isCoatom_stabilizer`: if neither `s : Set α` nor its complementary subset is empty, and the cardinality of `s` is not half of that of `α`, then `MulAction.stabilizer (Equiv.Perm α) s` is a maximal subgroup of the symmetric group `Equiv.Perm α`. This is the *intransitive case* of the O'Nan-Scott classification.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/MaximalSubgroups.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Jordan", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 1, "macro_tier_score": 0.0039, "macro_tier_override": null, "x": 21.723, "z": -49.282, "size": 0.3025, "title": "Theorems of Jordan", "summary": "A proof of theorems of Jordan regarding primitive permutation groups. This mostly follows the book [Wielandt, *Finite permutation groups*][Wielandt-1964]. - `MulAction.IsPreprimitive.is_two_pretransitive` and `MulAction.IsPreprimitive.is_two_preprimitive` are technical lemmas that prove 2-pretransitivity / 2-preprimitivity for some group primitive actions given the transitivity / primitivity of `ofFixingSubgroup G…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Jordan.html"}, {"id": "Mathlib.RingTheory.AdjoinRoot", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 3, "macro_tier_score": 0.1348, "macro_tier_override": null, "x": 89.972, "z": -29.598, "size": 0.4284, "title": "Adjoining roots of polynomials", "summary": "This file defines the commutative ring `AdjoinRoot f`, the ring R[X]/(f) obtained from a commutative ring `R` and a polynomial `f : R[X]`. If furthermore `R` is a field and `f` is irreducible, the field structure on `AdjoinRoot f` is constructed. We suggest stating results on `IsAdjoinRoot` instead of `AdjoinRoot` to achieve higher generality, since `IsAdjoinRoot` works for all different constructions of `R[α]`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AdjoinRoot.html"}, {"id": "Mathlib.RingTheory.Ideal.Quotient.Noetherian", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.1373, "macro_tier_override": null, "x": 50.167, "z": -38.345, "size": 0.3264, "title": "Noetherian quotient rings and quotient modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Quotient/Noetherian.html"}, {"id": "Mathlib.RingTheory.Polynomial.Quotient", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 3, "macro_tier_score": 0.1495, "macro_tier_override": null, "x": 38.341, "z": -63.628, "size": 0.332, "title": "Quotients of polynomial rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Quotient.html"}, {"id": "Mathlib.RingTheory.Nilpotent.Lemmas", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.2083, "macro_tier_override": null, "x": -61.963, "z": -19.637, "size": 0.3952, "title": "Nilpotent elements", "summary": "This file contains results about nilpotent elements that involve ring theory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Nilpotent/Lemmas.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.ToLin", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.2584, "macro_tier_override": null, "x": -0.396, "z": 63.142, "size": 0.4867, "title": "Linear maps and matrices", "summary": "This file defines the maps to send matrices to a linear map, and to send linear maps between modules with a finite bases to matrices. This defines a linear equivalence between linear maps between finite-dimensional vector spaces and matrices indexed by the respective bases.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/ToLin.html"}, {"id": "Mathlib.RingTheory.Nilpotent.Defs", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 3, "macro_tier_score": 0.2177, "macro_tier_override": null, "x": -11.202, "z": 9.76, "size": 0.363, "title": "Definition of nilpotent elements", "summary": "This file proves basic facts about nilpotent elements. For results that require further theory, see `Mathlib/RingTheory/Nilpotent/Basic.lean` and `Mathlib/RingTheory/Nilpotent/Lemmas.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Nilpotent/Defs.html"}, {"id": "Mathlib.GroupTheory.MonoidLocalization.Cardinality", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 2, "macro_tier_score": 0.0101, "macro_tier_override": null, "x": 2.388, "z": -20.289, "size": 0.2451, "title": "Cardinality of localizations of commutative monoids", "summary": "This file contains some results on cardinality of localizations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/MonoidLocalization/Cardinality.html"}, {"id": "Mathlib.GroupTheory.MonoidLocalization.Basic", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 3, "macro_tier_score": 0.238, "macro_tier_override": null, "x": -10.398, "z": -15.388, "size": 0.383, "title": "Localizations of commutative monoids", "summary": "Localizing a commutative ring at one of its submonoids does not rely on the ring's addition, so we can generalize localizations to commutative monoids. We characterize the localization of a commutative monoid `M` at a submonoid `S` up to isomorphism; that is, a commutative monoid `N` is the localization of `M` at `S` iff we can find a monoid homomorphism `f : M →* N` satisfying 3 properties: 1. For all `y ∈ S`, `f…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/MonoidLocalization/Basic.html"}, {"id": "Mathlib.GroupTheory.OreLocalization.Cardinality", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 2, "macro_tier_score": 0.0125, "macro_tier_override": null, "x": 14.707, "z": -2.107, "size": 0.2792, "title": "Cardinality of Ore localizations", "summary": "This file contains some results on cardinality of Ore localizations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/OreLocalization/Cardinality.html"}, {"id": "Mathlib.Algebra.Star.Unitary", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 2, "macro_tier_score": 0.0204, "macro_tier_override": null, "x": -58.415, "z": 18.54, "size": 0.3675, "title": "Unitary elements of a star monoid", "summary": "This file defines `unitary R`, where `R` is a star monoid, as the submonoid made of the elements that satisfy `star U * U = 1` and `U * star U = 1`, and these form a group. This includes, for instance, unitary operators on Hilbert spaces. See also `Matrix.UnitaryGroup` for specializations to `unitary (Matrix n n R)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/Unitary.html"}, {"id": "Mathlib.Algebra.Algebra.Spectrum.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 2, "macro_tier_score": 0.033, "macro_tier_override": null, "x": 31.656, "z": 50.296, "size": 0.3897, "title": "Spectrum of an element in an algebra", "summary": "This file develops the basic theory of the spectrum of an element of an algebra. This theory will serve as the foundation for spectral theory in Banach algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Spectrum/Basic.html"}, {"id": "Mathlib.Algebra.Star.StarProjection", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 3, "macro_tier_score": 0.0342, "macro_tier_override": null, "x": 24.364, "z": -9.077, "size": 0.3329, "title": "Star projections", "summary": "This file defines star projections, which are self-adjoint idempotents. In star-ordered rings, star projections are non-negative. (See `IsStarProjection.nonneg` in `Mathlib/Algebra/Order/Star/Basic.lean`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/StarProjection.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.BigOperators", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4308, "macro_tier_override": null, "x": 5.309, "z": 11.867, "size": 0.5011, "title": "Submonoids: membership criteria for products and sums", "summary": "In this file we prove various facts about membership in a submonoid: * `list_prod_mem`, `multiset_prod_mem`, `prod_mem`: if each element of a collection belongs to a multiplicative submonoid, then so does their product; * `list_sum_mem`, `multiset_sum_mem`, `sum_mem`: if each element of a collection belongs to an additive submonoid, then so does their sum;", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/BigOperators.html"}, {"id": "Mathlib.Algebra.Group.Support", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.433, "macro_tier_override": null, "x": -6.872, "z": 2.82, "size": 0.3774, "title": "Support of a function", "summary": "In this file we prove basic properties of `Function.support f = {x | f x ≠ 0}`, and similarly for `Function.mulSupport f = {x | f x ≠ 1}`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Support.html"}, {"id": "Mathlib.Algebra.Homology.TotalComplexShift", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 15.244, "z": -21.063, "size": 0.239, "title": "Behaviour of the total complex with respect to shifts", "summary": "There are two ways to shift objects in `HomologicalComplex₂ C (up ℤ) (up ℤ)`: * by shifting the first indices (and changing signs of horizontal differentials), which corresponds to the shift by `ℤ` on `CochainComplex (CochainComplex C ℤ) ℤ`. * by shifting the second indices (and changing signs of vertical differentials). These two sorts of shift functors shall be abbreviated as `HomologicalComplex₂.shiftFunctor₁ C…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/TotalComplexShift.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.Shift", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 2, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -16.309, "z": 8.884, "size": 0.3382, "title": "The shift on cochain complexes and on the homotopy category", "summary": "In this file, we show that for any preadditive category `C`, the categories `CochainComplex C ℤ` and `HomotopyCategory C (ComplexShape.up ℤ)` are equipped with a shift by `ℤ`. We also show that if `F : C ⥤ D` is an additive functor, then the functors `F.mapHomologicalComplex (ComplexShape.up ℤ)` and `F.mapHomotopyCategory (ComplexShape.up ℤ)` commute with the shift by `ℤ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/Shift.html"}, {"id": "Mathlib.Algebra.Homology.TotalComplex", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 1, "macro_tier_score": 0.0022, "macro_tier_override": null, "x": -22.831, "z": 7.852, "size": 0.3393, "title": "The total complex of a bicomplex", "summary": "Given a preadditive category `C`, two complex shapes `c₁ : ComplexShape I₁`, `c₂ : ComplexShape I₂`, a bicomplex `K : HomologicalComplex₂ C c₁ c₂`, and a third complex shape `c₁₂ : ComplexShape I₁₂` equipped with `[TotalComplexShape c₁ c₂ c₁₂]`, we construct the total complex `K.total c₁₂ : HomologicalComplex C c₁₂`. In particular, if `c := ComplexShape.up ℤ` and `K : HomologicalComplex₂ c c`, then for any `n : ℤ`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/TotalComplex.html"}, {"id": "Mathlib.Algebra.Category.Grp.Basic", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 3, "macro_tier_score": 0.0974, "macro_tier_override": null, "x": 0.805, "z": -18.554, "size": 0.4789, "title": "Category instances for Group, AddGroup, CommGroup, and AddCommGroup.", "summary": "We introduce the bundled categories: * `GrpCat` * `AddGrpCat` * `CommGrpCat` * `AddCommGrpCat` along with the relevant forgetful functors between them, and to the bundled monoid categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Basic.html"}, {"id": "Mathlib.RingTheory.Coalgebra.Hom", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.0084, "macro_tier_override": null, "x": -57.149, "z": -6.965, "size": 0.3765, "title": "Homomorphisms of `R`-coalgebras", "summary": "This file defines bundled homomorphisms of `R`-coalgebras. We largely mimic `Mathlib/Algebra/Algebra/Hom.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Coalgebra/Hom.html"}, {"id": "Mathlib.RingTheory.Coalgebra.Basic", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 2, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": -29.284, "z": 47.398, "size": 0.4407, "title": "Coalgebras", "summary": "In this file we define `Coalgebra`, and provide instances for: * Commutative semirings: `CommSemiring.toCoalgebra` * Binary products: `Prod.instCoalgebra` * Finitely supported functions: `DFinsupp.instCoalgebra`, `Finsupp.instCoalgebra` * Finite pi functions: `Pi.instCoalgebra`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Coalgebra/Basic.html"}, {"id": "Mathlib.Algebra.BigOperators.Field", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 2, "macro_tier_score": 0.0119, "macro_tier_override": null, "x": 16.912, "z": -17.23, "size": 0.3881, "title": "Results about big operators with values in a field", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Field.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Expand", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 2, "macro_tier_score": 0.0071, "macro_tier_override": null, "x": -64.275, "z": 44.191, "size": 0.289, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Expand.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Nilpotent", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 2, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": -28.318, "z": 70.682, "size": 0.2942, "title": "Nilpotents and units in multivariate polynomial rings", "summary": "We prove that - `MvPolynomial.isNilpotent_iff`: A multivariate polynomial is nilpotent iff all its coefficients are. - `MvPolynomial.isUnit_iff`: A multivariate polynomial is invertible iff its constant term is invertible and its other coefficients are nilpotent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Nilpotent.html"}, {"id": "Mathlib.Algebra.Order.Ring.Finset", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 2, "macro_tier_score": 0.0103, "macro_tier_override": null, "x": 17.722, "z": -13.513, "size": 0.2744, "title": "`Finset.sup` and ring operations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Finset.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.Operations", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4646, "macro_tier_override": null, "x": 14.522, "z": -8.276, "size": 0.4845, "title": "Operations on `Submonoid`s", "summary": "In this file we define various operations on `Submonoid`s and `MonoidHom`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/Operations.html"}, {"id": "Mathlib.Algebra.Regular.Basic", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.3823, "macro_tier_override": null, "x": 6.456, "z": 9.082, "size": 0.3893, "title": "Regular elements", "summary": "By definition, a regular element in a commutative ring is a non-zero divisor. Lemma `IsRegular.of_ne_zero` implies that every non-zero element of an integral domain is regular. Since it assumes that the ring is a cancellative `MonoidWithZero` it applies also, for instance, to `ℕ`. The lemmas in Section `MulZeroClass` show that the `0` element is (left/right-)regular if and only if the `MulZeroClass` is trivial. This…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Regular/Basic.html"}, {"id": "Mathlib.GroupTheory.Congruence.Hom", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4335, "macro_tier_override": null, "x": -2.372, "z": 12.782, "size": 0.447, "title": "Congruence relations and homomorphisms", "summary": "This file contains elementary definitions involving congruence relations and morphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Congruence/Hom.html"}, {"id": "Mathlib.GroupTheory.OreLocalization.Basic", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 3, "macro_tier_score": 0.2392, "macro_tier_override": null, "x": 5.715, "z": 11.676, "size": 0.326, "title": "Localization over left Ore sets.", "summary": "This file defines the localization of a monoid over a left Ore set and proves its universal mapping property.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/OreLocalization/Basic.html"}, {"id": "Mathlib.Algebra.Order.Ring.GeomSum", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 8.865, "z": -24.442, "size": 0.2764, "title": "Partial sums of geometric series in an ordered ring", "summary": "This file upper- and lower-bounds the values of the geometric series $\\sum_{i=0}^{n-1} x^i$ and $\\sum_{i=0}^{n-1} x^i y^{n-1-i}$ and variants thereof. We also provide some bounds on the \"geometric\" sum of `a/b^i` where `a b : ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/GeomSum.html"}, {"id": "Mathlib.Algebra.Lie.AdjointAction.Basic", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": 73.777, "z": 70.642, "size": 0.2982, "title": "Properties of the adjoint action", "summary": "Theorems about the adjoint action `LieAlgebra.ad` on associative algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/AdjointAction/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Semisimple", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 2, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -92.12, "z": 39.64, "size": 0.3242, "title": "Semisimple linear endomorphisms", "summary": "Given an `R`-module `M` together with an `R`-linear endomorphism `f : M → M`, the following two conditions are equivalent: 1. Every `f`-invariant submodule of `M` has an `f`-invariant complement. 2. `M` is a semisimple `R[X]`-module, where the action of the polynomial ring is induced by `f`. A linear endomorphism `f` satisfying these equivalent conditions is known as a *semisimple* endomorphism. We provide basic…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Semisimple.html"}, {"id": "Mathlib.LinearAlgebra.FreeModule.IdealQuotient", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 29.454, "z": 76.222, "size": 0.2813, "title": "Ideals in free modules over PIDs", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeModule/IdealQuotient.html"}, {"id": "Mathlib.LinearAlgebra.FreeModule.Finite.Quotient", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.0129, "macro_tier_override": null, "x": 78.608, "z": 14.075, "size": 0.3112, "title": "Quotient of submodules of full rank in free finite modules over PIDs", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeModule/Finite/Quotient.html"}, {"id": "Mathlib.GroupTheory.FreeGroup.Reduce", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -17.906, "z": -18.851, "size": 0.2735, "title": "The maximal reduction of a word in a free group", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FreeGroup/Reduce.html"}, {"id": "Mathlib.GroupTheory.FreeGroup.Basic", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.3439, "macro_tier_override": null, "x": 13.435, "z": 20.06, "size": 0.3527, "title": "Free groups", "summary": "This file defines free groups over a type. Furthermore, it is shown that the free group construction is an instance of a monad. For the result that `FreeGroup` is the left adjoint to the forgetful functor from groups to types, see `Mathlib/Algebra/Category/Grp/Adjunctions.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FreeGroup/Basic.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.Abelian", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 2, "macro_tier_score": 0.026, "macro_tier_override": null, "x": 1.544, "z": 9.157, "size": 0.3714, "title": "Abelian categories have homology", "summary": "In this file, it is shown that all short complexes `S` in abelian categories have terms of type `S.HomologyData`. The strategy of the proof is to study the morphism `kernel.ι S.g ≫ cokernel.π S.f`. We show that there is a `LeftHomologyData` for `S` for which the `H` field consists of the coimage of `kernel.ι S.g ≫ cokernel.π S.f`, while there is a `RightHomologyData` for which the `H` is the image of `kernel.ι S.g ≫…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/Abelian.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.Limits", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 2, "macro_tier_score": 0.0249, "macro_tier_override": null, "x": 1.821, "z": -0.364, "size": 0.2989, "title": "Limits and colimits in the category of short complexes", "summary": "In this file, it is shown if a category `C` with zero morphisms has limits of a certain shape `J`, then it is also the case of the category `ShortComplex C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/Limits.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.Preadditive", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 2, "macro_tier_score": 0.0251, "macro_tier_override": null, "x": -7.198, "z": -1.837, "size": 0.3098, "title": "Homology of preadditive categories", "summary": "In this file, it is shown that if `C` is a preadditive category, then `ShortComplex C` is a preadditive category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/Preadditive.html"}, {"id": "Mathlib.Algebra.Module.Presentation.Tensor", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -18.41, "z": -52.585, "size": 0.2, "title": "Presentation of the tensor product of two modules", "summary": "Given presentations of two `A`-modules `M₁` and `M₂`, we obtain a presentation of `M₁ ⊗[A] M₂`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Presentation/Tensor.html"}, {"id": "Mathlib.Algebra.Module.Presentation.Basic", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 2, "macro_tier_score": 0.006, "macro_tier_override": null, "x": 52.469, "z": -12.149, "size": 0.3652, "title": "Presentations of modules", "summary": "Given a ring `A`, we introduce a structure `Relations A` which contains the data that is necessary to define a module by generators and relations. A term `relations : Relations A` involves two index types: a type `G` for the generators and a type `R` for the relations. The relation attached to `r : R` is an element `G →₀ A` which expresses the coefficients of the expected linear relation. One may think of `relations…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Presentation/Basic.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Basic", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 4, "macro_tier_score": 0.3546, "macro_tier_override": null, "x": 46.85, "z": -11.69, "size": 0.4776, "title": "Universal property of the tensor product", "summary": "Given any bilinear map `f : M →ₛₗ[σ₁₂] N →ₛₗ[σ₁₂] P₂`, there is a unique semilinear map `TensorProduct.lift f : TensorProduct R M N →ₛₗ[σ₁₂] P₂` whose composition with the canonical bilinear map `TensorProduct.mk` is the given bilinear map `f`. Uniqueness is shown in the theorem `TensorProduct.lift.unique`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Basic.html"}, {"id": "Mathlib.RingTheory.KrullDimension.Zero", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 3, "macro_tier_score": 0.058, "macro_tier_override": null, "x": -27.381, "z": 92.609, "size": 0.3632, "title": "Zero-dimensional rings", "summary": "We provide further API for zero-dimensional rings. Basic definitions and lemmas are provided in `Mathlib/RingTheory/KrullDimension/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/KrullDimension/Zero.html"}, {"id": "Mathlib.RingTheory.Jacobson.Ring", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 3, "macro_tier_score": 0.0587, "macro_tier_override": null, "x": 79.122, "z": 52.064, "size": 0.3407, "title": "Jacobson Rings", "summary": "The following conditions are equivalent for a ring `R`: 1. Every radical ideal `I` is equal to its Jacobson radical 2. Every radical ideal `I` can be written as an intersection of maximal ideals 3. Every prime ideal `I` is equal to its Jacobson radical Any ring satisfying any of these equivalent conditions is said to be Jacobson. Some particular examples of Jacobson rings are also proven. - `isJacobsonRing_quotient`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Jacobson/Ring.html"}, {"id": "Mathlib.Algebra.Order.Monoid.ToMulBot", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -20.316, "z": -2.146, "size": 0.2, "title": null, "summary": "Making an additive monoid multiplicative then adding a zero is the same as adding a bottom element then making it multiplicative.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/ToMulBot.html"}, {"id": "Mathlib.Algebra.Category.CoalgCat.Basic", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -57.956, "z": -19.928, "size": 0.2748, "title": "The category of coalgebras over a commutative ring", "summary": "We introduce the bundled category `CoalgCat` of coalgebras over a fixed commutative ring `R` along with the forgetful functor to `ModuleCat`. This file mimics `Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/CoalgCat/Basic.html"}, {"id": "Mathlib.RingTheory.Coalgebra.Equiv", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 2, "macro_tier_score": 0.0078, "macro_tier_override": null, "x": -58.981, "z": -7.288, "size": 0.3422, "title": "Isomorphisms of `R`-coalgebras", "summary": "This file defines bundled isomorphisms of `R`-coalgebras. We largely mirror the basic API of `Mathlib/Algebra/Module/Equiv/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Coalgebra/Equiv.html"}, {"id": "Mathlib.Algebra.Group.Pi.Basic", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.4927, "macro_tier_override": null, "x": -4.598, "z": 3.147, "size": 0.6333, "title": "Instances and theorems on pi types", "summary": "This file provides instances for the typeclass defined in `Algebra.Group.Defs`. More sophisticated instances are defined in `Algebra.Group.Pi.Lemmas` files elsewhere.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pi/Basic.html"}, {"id": "Mathlib.Algebra.Notation.Pi.Basic", "region_id": "algebra", "micro_elevation": 0.0263, "macro_tier": 4, "macro_tier_score": 0.5066, "macro_tier_override": null, "x": -1.456, "z": -3.417, "size": 0.6881, "title": "Very basic algebraic operations on pi types", "summary": "This file provides very basic algebraic operations on functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Notation/Pi/Basic.html"}, {"id": "Mathlib.Algebra.Field.NegOnePow", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -1.857, "z": 0.021, "size": 0.2929, "title": "Integer powers of `-1` in a field", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/NegOnePow.html"}, {"id": "Mathlib.Algebra.Field.Periodic", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": 10.297, "z": 21.837, "size": 0.3063, "title": "Periodic functions", "summary": "This file proves facts about periodic and antiperiodic functions from and to a field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/Periodic.html"}, {"id": "Mathlib.Algebra.QuadraticDiscriminant", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.3236, "title": "Quadratic discriminants and roots of a quadratic", "summary": "This file defines the discriminant of a quadratic and gives the solution to a quadratic equation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/QuadraticDiscriminant.html"}, {"id": "Mathlib.Algebra.EuclideanDomain.Field", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.2933, "macro_tier_override": null, "x": 6.002, "z": -13.591, "size": 0.3093, "title": "Instances for Euclidean domains", "summary": "* `Field.toEuclideanDomain`: shows that any field is a Euclidean domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/EuclideanDomain/Field.html"}, {"id": "Mathlib.Algebra.EuclideanDomain.Defs", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.2959, "macro_tier_override": null, "x": 9.165, "z": 1.491, "size": 0.3362, "title": "Euclidean domains", "summary": "This file introduces Euclidean domains and provides the extended Euclidean algorithm. To be precise, a slightly more general version is provided which is sometimes called a transfinite Euclidean domain and differs in the fact that the degree function need not take values in `ℕ` but can take values in any well-ordered set. Transfinite Euclidean domains were introduced by Motzkin and examples which don't satisfy the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/EuclideanDomain/Defs.html"}, {"id": "Mathlib.Algebra.Field.Defs", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 3, "macro_tier_score": 0.4321, "macro_tier_override": 3, "x": -1.564, "z": 9.153, "size": 0.6633, "title": "Division (semi)rings and (semi)fields", "summary": "This file introduces fields and division rings (also known as skewfields) and proves some basic statements about them. For a more extensive theory of fields, see the `FieldTheory` folder.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/Defs.html"}, {"id": "Mathlib.Algebra.Order.Monoid.NatCast", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4196, "macro_tier_override": null, "x": -1.334, "z": 11.063, "size": 0.299, "title": "Order of numerals in an `AddMonoidWithOne`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/NatCast.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4948, "macro_tier_override": null, "x": 6.619, "z": 6.512, "size": 0.8818, "title": "Ordered monoids", "summary": "This file develops the basics of ordered monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Unbundled/Basic.html"}, {"id": "Mathlib.Algebra.GCDMonoid.Finset", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 4, "macro_tier_score": 0.2949, "macro_tier_override": null, "x": 0.187, "z": -27.857, "size": 0.3412, "title": "GCD and LCM operations on finsets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GCDMonoid/Finset.html"}, {"id": "Mathlib.Algebra.GCDMonoid.Multiset", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 4, "macro_tier_score": 0.2942, "macro_tier_override": null, "x": 5.757, "z": -25.355, "size": 0.2845, "title": "GCD and LCM operations on multisets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GCDMonoid/Multiset.html"}, {"id": "Mathlib.Algebra.GCDMonoid.Nat", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 4, "macro_tier_score": 0.2942, "macro_tier_override": null, "x": -20.395, "z": -16.127, "size": 0.2845, "title": "ℕ and ℤ are normalized GCD monoids.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GCDMonoid/Nat.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.Lift", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.3277, "macro_tier_override": null, "x": 2.796, "z": -18.36, "size": 0.306, "title": "Lifting monoid algebras", "summary": "This file defines `liftNC`. For the definition of `MonoidAlgebra.lift`, see `Mathlib/Algebra/MonoidAlgebra/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/Lift.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.Defs", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.3302, "macro_tier_override": null, "x": 12.957, "z": 10.559, "size": 0.4376, "title": "Monoid algebras", "summary": "When the domain of a `Finsupp` has a multiplicative or additive structure, we can define a convolution product. To mathematicians this structure is known as the \"monoid algebra\", i.e. the finite formal linear combinations over a given semiring of elements of a monoid `M`. The \"group ring\" `ℤ[G]` or the \"group algebra\" `k[G]` are typical uses. In fact the construction of the \"monoid algebra\" makes sense when `M` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/Defs.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Pointwise.Finset", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 15.401, "z": 16.108, "size": 0.239, "title": "Pointwise operations of finsets in a group with zero", "summary": "This file proves properties of pointwise operations of finsets in a group with zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Pointwise/Finset.html"}, {"id": "Mathlib.Algebra.Group.Pointwise.Finset.Basic", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.3678, "macro_tier_override": null, "x": -20.425, "z": 0.398, "size": 0.4732, "title": "Pointwise operations of finsets", "summary": "This file defines pointwise algebraic operations on finsets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pointwise/Finset/Basic.html"}, {"id": "Mathlib.Algebra.DirectSum.AddChar", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -25.91, "z": 2.17, "size": 0.2302, "title": "Direct sum of additive characters", "summary": "This file defines the direct sum of additive characters.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/DirectSum/AddChar.html"}, {"id": "Mathlib.Algebra.DirectSum.Basic", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.2872, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.4021, "title": "Direct sum", "summary": "This file defines the direct sum of abelian groups, indexed by a discrete type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/DirectSum/Basic.html"}, {"id": "Mathlib.Algebra.Group.AddChar", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.3236, "macro_tier_override": null, "x": 11.962, "z": 20.971, "size": 0.3937, "title": "Characters from additive to multiplicative monoids", "summary": "Let `A` be an additive monoid, and `M` a multiplicative one. An *additive character* of `A` with values in `M` is simply a map `A → M` which intertwines the addition operation on `A` with the multiplicative operation on `M`. We define these objects, using the namespace `AddChar`, and show that if `A` is a commutative group under addition, then the additive characters are also a group (written multiplicatively). Note…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/AddChar.html"}, {"id": "Mathlib.Algebra.Category.MonCat.Yoneda", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -13.708, "z": -5.729, "size": 0.2, "title": "Yoneda embeddings", "summary": "This file defines a few Yoneda embeddings for the category of commutative monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/MonCat/Yoneda.html"}, {"id": "Mathlib.Algebra.Category.MonCat.Basic", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 3, "macro_tier_score": 0.0954, "macro_tier_override": null, "x": -7.011, "z": 10.948, "size": 0.4029, "title": "Category instances for `Monoid`, `AddMonoid`, `CommMonoid`, and `AddCommMonoid`.", "summary": "We introduce the bundled categories: * `MonCat` * `AddMonCat` * `CommMonCat` * `AddCommMonCat` along with the relevant forgetful functors between them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/MonCat/Basic.html"}, {"id": "Mathlib.Algebra.Group.Pi.Lemmas", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4703, "macro_tier_override": null, "x": -7.028, "z": 8.647, "size": 0.4524, "title": "Extra lemmas about products of monoids and groups", "summary": "This file proves lemmas about the instances defined in `Algebra.Group.Pi.Basic` that require more imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pi/Lemmas.html"}, {"id": "Mathlib.Algebra.Group.Even", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4411, "macro_tier_override": null, "x": -2.802, "z": 12.695, "size": 0.5112, "title": "Squares and even elements", "summary": "This file defines square and even elements in a monoid.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Even.html"}, {"id": "Mathlib.Algebra.Group.Nat.Hom", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4457, "macro_tier_override": null, "x": 5.183, "z": 11.922, "size": 0.403, "title": "Extensionality of monoid homs from `ℕ`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Nat/Hom.html"}, {"id": "Mathlib.Algebra.Ring.Hom.Defs", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4598, "macro_tier_override": null, "x": 12.545, "z": -3.41, "size": 0.6178, "title": "Homomorphisms of semirings and rings", "summary": "This file defines bundled homomorphisms of (non-unital) semirings and rings. As with monoid and groups, we use the same structure `RingHom a β`, a.k.a. `α →+* β`, for both types of homomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Hom/Defs.html"}, {"id": "Mathlib.Algebra.Group.Finsupp", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.3298, "macro_tier_override": null, "x": 12.885, "z": -1.73, "size": 0.3706, "title": "Additive monoid structure on `ι →₀ M`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Finsupp.html"}, {"id": "Mathlib.Algebra.Module.RingHom", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4029, "macro_tier_override": null, "x": -16.802, "z": -7.912, "size": 0.496, "title": "Composing modules with a ring hom", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/RingHom.html"}, {"id": "Mathlib.Algebra.Order.Module.Defs", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.3225, "macro_tier_override": null, "x": -12.052, "z": -16.495, "size": 0.4402, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Module/Defs.html"}, {"id": "Mathlib.Algebra.Order.Nonneg.Basic", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.3166, "macro_tier_override": null, "x": 15.971, "z": 4.929, "size": 0.3766, "title": "The type of nonnegative elements", "summary": "This file defines instances and prove some properties about the nonnegative elements `{x : α // 0 ≤ x}` of an arbitrary type `α`. Currently we only state instances and states some `simp`/`norm_cast` lemmas. When `α` is `ℝ`, this will give us some properties about `ℝ≥0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Nonneg/Basic.html"}, {"id": "Mathlib.NumberTheory.LegendreSymbol.Complex", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -23.152, "z": 124.147, "size": 0.2, "title": "Additive characters on finite fields", "summary": "We construct a primitive additive character on a finite field `F` with values in `ℂ`. This file is kept separate from `Mathlib.NumberTheory.LegendreSymbol.AddCharacter` to avoid importing the fundamental theorem of algebra and Bochner integral into that file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LegendreSymbol/Complex.html"}, {"id": "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 2, "macro_tier_score": 0.0071, "macro_tier_override": null, "x": 14.314, "z": 123.604, "size": 0.2856, "title": "Additive characters of finite rings and fields", "summary": "This file collects some results on additive characters whose domain is (the additive group of) a finite ring or field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LegendreSymbol/AddCharacter.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.Pointwise", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.432, "macro_tier_override": null, "x": -22.134, "z": 2.597, "size": 0.4311, "title": "Pointwise instances on `Submonoid`s and `AddSubmonoid`s", "summary": "This file provides: * `Submonoid.inv` * `AddSubmonoid.neg` and the actions * `Submonoid.pointwiseMulAction` * `AddSubmonoid.pointwiseAddAction` which matches the action of `Set.mulActionSet`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/Pointwise.html"}, {"id": "Mathlib.Algebra.Group.Action.Pointwise.Set.Basic", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.4365, "macro_tier_override": null, "x": 11.254, "z": 17.049, "size": 0.4338, "title": "Pointwise actions on sets", "summary": "This file proves that several kinds of actions of a type `α` on another type `β` transfer to actions of `α`/`Set α` on `Set β`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Pointwise/Set/Basic.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.Membership", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.448, "macro_tier_override": null, "x": 3.558, "z": -18.228, "size": 0.485, "title": "Submonoids: membership criteria", "summary": "In this file we prove various facts about membership in a submonoid: * `mem_iSup_of_directed`, `coe_iSup_of_directed`, `mem_sSup_of_directedOn`, `coe_sSup_of_directedOn`: the supremum of a directed collection of submonoid is their union. * `sup_eq_range`, `mem_sup`: supremum of two submonoids `S`, `T` of a commutative monoid is the set of products; * `closure_singleton_eq`, `mem_closure_singleton`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/Membership.html"}, {"id": "Mathlib.RingTheory.Ideal.Operations", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 4, "macro_tier_score": 0.3294, "macro_tier_override": null, "x": -12.003, "z": -54.407, "size": 0.4861, "title": "More operations on modules and ideals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Operations.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.Polynomial", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 96.725, "z": 18.239, "size": 0.2414, "title": "Prime spectrum of (multivariate) polynomials", "summary": "Also see `AlgebraicGeometry/AffineSpace` for the affine space over arbitrary schemes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/Polynomial.html"}, {"id": "Mathlib.LinearAlgebra.Charpoly.BaseChange", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 2, "macro_tier_score": 0.0074, "macro_tier_override": null, "x": -28.411, "z": 92.299, "size": 0.3166, "title": "The characteristic polynomial of base change", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Charpoly/BaseChange.html"}, {"id": "Mathlib.LinearAlgebra.Eigenspace.Zero", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -95.117, "z": -16.701, "size": 0.2771, "title": "Results on the eigenvalue 0", "summary": "In this file we provide equivalent characterizations of properties related to the eigenvalue 0, such as being nilpotent, having determinant equal to 0, having a non-trivial kernel, etc...", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Eigenspace/Zero.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Units.Equiv", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 3, "macro_tier_score": 0.0403, "macro_tier_override": null, "x": 15.197, "z": 6.958, "size": 0.3635, "title": "Multiplication by a nonzero element in a `GroupWithZero` is a permutation.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Units/Equiv.html"}, {"id": "Mathlib.Algebra.CubicDiscriminant", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -27.719, "z": 72.909, "size": 0.3143, "title": "Cubics and discriminants", "summary": "This file defines cubic polynomials over a semiring and their discriminants over a splitting field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CubicDiscriminant.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.HomComplexCohomology", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 57.304, "z": 26.522, "size": 0.2767, "title": "Cohomology of the hom complex", "summary": "Given `ℤ`-indexed cochain complexes `K` and `L`, and `n : ℤ`, we introduce a type `HomComplex.CohomologyClass K L n` which is the quotient of `HomComplex.Cocycle K L n` which identifies cohomologous cocycles. We construct this type of cohomology classes instead of using the homology API because `Cochain K L` can be considered both as a complex of abelian groups or as a complex of `R`-modules when the category is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/HomComplexCohomology.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 2, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -12.338, "z": 31.069, "size": 0.3257, "title": "Shifting cochains", "summary": "Let `C` be a preadditive category. Given two cochain complexes (indexed by `ℤ`), the type of cochains `HomComplex.Cochain K L n` of degree `n` was introduced in `Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean`. In this file, we study how these cochains behave with respect to the shift on the complexes `K` and `L`. When `n`, `a`, `n'` are integers such that `h : n' + a = n`, we obtain `rightShiftAddEquiv K…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.html"}, {"id": "Mathlib.Algebra.Category.Grp.Abelian", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 2, "macro_tier_score": 0.0197, "macro_tier_override": null, "x": 8.33, "z": -60.718, "size": 0.3255, "title": "The category of abelian groups is abelian", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Abelian.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.4215, "macro_tier_override": null, "x": 17.728, "z": 16.389, "size": 0.4079, "title": "Pointwise operations of sets in a group with zero", "summary": "This file proves properties of pointwise operations of sets in a group with zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/Pointwise/Set.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Injective", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 2, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": 55.885, "z": -25.156, "size": 0.2646, "title": "Injective objects in the category of $R$-modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Injective.html"}, {"id": "Mathlib.LinearAlgebra.Projectivization.Independence", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 18.543, "z": -75.765, "size": 0.239, "title": "Independence in Projective Space", "summary": "In this file we define independence and dependence of families of elements in projective space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Projectivization/Independence.html"}, {"id": "Mathlib.LinearAlgebra.Projectivization.Basic", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 1, "macro_tier_score": 0.0033, "macro_tier_override": null, "x": 41.645, "z": -63.746, "size": 0.3374, "title": "Projective Spaces", "summary": "This file contains the definition of the projectivization of a vector space over a field, as well as the bijection between said projectivization and the collection of all one dimensional subspaces of the vector space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Projectivization/Basic.html"}, {"id": "Mathlib.Algebra.Homology.ExactSequence", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 2, "macro_tier_score": 0.0204, "macro_tier_override": null, "x": 8.381, "z": 9.938, "size": 0.367, "title": "Exact sequences", "summary": "A sequence of `n` composable arrows `S : ComposableArrows C` (i.e. a functor `S : Fin (n + 1) ⥤ C`) is said to be exact (`S.Exact`) if the composition of two consecutive arrows are zero (`S.IsComplex`) and the diagram is exact at each `i` for `1 ≤ i < n`. Together with the inductive construction of composable arrows `ComposableArrows.precomp`, this is useful in order to state that certain finite sequences of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ExactSequence.html"}, {"id": "Mathlib.Algebra.Group.ConjFinite", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": 7.327, "z": 19.07, "size": 0.2596, "title": "Conjugacy of elements of finite groups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/ConjFinite.html"}, {"id": "Mathlib.Algebra.Group.Conj", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4475, "macro_tier_override": null, "x": 8.389, "z": 16.569, "size": 0.3803, "title": "Conjugacy of group elements", "summary": "See also `MulAut.conj` and `Quandle.conj`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Conj.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Units.Fintype", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 2, "macro_tier_score": 0.025, "macro_tier_override": null, "x": -2.581, "z": -18.391, "size": 0.3746, "title": "Fintype instances relating to units", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Units/Fintype.html"}, {"id": "Mathlib.Algebra.Group.Torsion", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.4993, "macro_tier_override": null, "x": -6.901, "z": 2.749, "size": 0.5576, "title": "Torsion-free monoids and groups", "summary": "This file proves lemmas about torsion-free monoids. A monoid `M` is *torsion-free* if `n • · : M → M` is injective for all non-zero natural numbers `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Torsion.html"}, {"id": "Mathlib.Algebra.Group.Basic", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.5301, "macro_tier_override": null, "x": 5.454, "z": -1.139, "size": 0.9889, "title": "Basic lemmas about semigroups, monoids, and groups", "summary": "This file lists various basic lemmas about semigroups, monoids, and groups. Most proofs are one-liners from the corresponding axioms. For the definitions of semigroups, monoids and groups, see `Mathlib/Algebra/Group/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Basic.html"}, {"id": "Mathlib.LinearAlgebra.FreeModule.ModN", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 44.921, "z": 54.429, "size": 0.2, "title": "Quotienting out a free `ℤ`-module", "summary": "If `G` is a rank `d` free `ℤ`-module, then `G/nG` is a finite group of cardinality `n ^ d`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeModule/ModN.html"}, {"id": "Mathlib.Algebra.EuclideanDomain.Int", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 3, "macro_tier_score": 0.2615, "macro_tier_override": null, "x": 12.447, "z": 20.687, "size": 0.3945, "title": "Instances for Euclidean domains", "summary": "* `Int.euclideanDomain`: shows that `ℤ` is a Euclidean domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/EuclideanDomain/Int.html"}, {"id": "Mathlib.Algebra.Module.ZMod", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 3, "macro_tier_score": 0.0825, "macro_tier_override": null, "x": 43.296, "z": -25.295, "size": 0.292, "title": "The `ZMod n`-module structure on Abelian groups whose elements have order dividing `n`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/ZMod.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.Free", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.2516, "macro_tier_override": null, "x": -51.597, "z": -45.381, "size": 0.4416, "title": "Rank of free modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/Free.html"}, {"id": "Mathlib.Algebra.Order.UpperLower", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 10.463, "z": -19.677, "size": 0.2523, "title": "Algebraic operations on upper/lower sets", "summary": "Upper/lower sets are preserved under pointwise algebraic operations in ordered groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/UpperLower.html"}, {"id": "Mathlib.LinearAlgebra.FreeModule.Basic", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 4, "macro_tier_score": 0.3109, "macro_tier_override": null, "x": 40.536, "z": -29.517, "size": 0.4219, "title": "Free modules", "summary": "We introduce a class `Module.Free R M`, for `R` a `Semiring` and `M` an `R`-module and we provide several basic instances for this class. Use `Finsupp.linearCombination_id_surjective` to prove that any module is the quotient of a free module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeModule/Basic.html"}, {"id": "Mathlib.Algebra.Module.ULift", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 4, "macro_tier_score": 0.3825, "macro_tier_override": null, "x": 5.956, "z": -27.213, "size": 0.4, "title": "`ULift` instances for module and multiplicative actions", "summary": "This file defines instances for `Module`, `MulAction` and related structures on `ULift` types. (Recall `ULift α` is just a \"copy\" of a type `α` in a higher universe.) We also provide `ULift.moduleEquiv : ULift M ≃ₗ[R] M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/ULift.html"}, {"id": "Mathlib.LinearAlgebra.Basis.Basic", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 4, "macro_tier_score": 0.3226, "macro_tier_override": null, "x": -6.558, "z": -47.839, "size": 0.5158, "title": "Basic results on bases", "summary": "The main goal of this file is to show the equivalence between bases and families of vectors that are linearly independent and whose span is the whole space. There are also various lemmas on bases on specific spaces (such as empty or singletons).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Basis/Basic.html"}, {"id": "Mathlib.RingTheory.Extension.Cotangent.LocalizationAway", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.0081, "macro_tier_override": null, "x": 10.957, "z": -105.29, "size": 0.28, "title": "Cotangent and localization away", "summary": "Let `R → S → T` be algebras such that `T` is the localization of `S` away from one element, where `S` is generated over `R` by `P : R[X] → S` with kernel `I` and `Q : S[Y] → T` is the canonical `S`-presentation of `T` with kernel `K`. Denote by `J` the kernel of the composition `R[X,Y] → T`. This file proves `J/J² ≃ₗ[T] T ⊗[S] (I/I²) × K/K²`. For this we establish the exact sequence: ``` 0 → T ⊗[S] (I/I²) → J/J² →…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Extension/Cotangent/LocalizationAway.html"}, {"id": "Mathlib.RingTheory.Kaehler.JacobiZariski", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 2, "macro_tier_score": 0.0196, "macro_tier_override": null, "x": -84.123, "z": -34.705, "size": 0.3192, "title": "The Jacobi-Zariski exact sequence", "summary": "Given algebras $R \\to S \\to T$, the Jacobi-Zariski exact sequence is a long exact sequence relating the first homology of the naive cotangent complexes and the Kähler differentials of the respective algebras. It takes the form: $$ H_1(L_{T/R}) \\to H_1(L_{T/S}) \\to T \\otimes_S \\Omega_{S/R} \\to \\Omega_{T/R} \\to \\Omega_{T/S} \\to 0 $$ The maps in the sequence are - `Algebra.H1Cotangent.map` - `Algebra.H1Cotangent.δ` -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Kaehler/JacobiZariski.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.ShortExact", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 2, "macro_tier_score": 0.0277, "macro_tier_override": null, "x": 6.907, "z": -11.014, "size": 0.4442, "title": "Short exact short complexes", "summary": "A short complex `S : ShortComplex C` is short exact (`S.ShortExact`) when it is exact, `S.f` is a mono and `S.g` is an epi.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/ShortExact.html"}, {"id": "Mathlib.GroupTheory.MonoidLocalization.Order", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -20.424, "z": -0.42, "size": 0.2, "title": "Ordered structures on localizations of commutative monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/MonoidLocalization/Order.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Defs", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4582, "macro_tier_override": null, "x": -5.115, "z": -9.9, "size": 0.5804, "title": "Ordered monoids", "summary": "This file provides the definitions of ordered monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Defs.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.TensorProduct", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 2, "macro_tier_score": 0.0257, "macro_tier_override": null, "x": -89.517, "z": 36.235, "size": 0.3559, "title": "Lemmas regarding the prime spectrum of tensor products", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/TensorProduct.html"}, {"id": "Mathlib.Algebra.Ring.Invertible", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.4168, "macro_tier_override": null, "x": -20.133, "z": -3.466, "size": 0.3987, "title": "Theorems about additively and multiplicatively invertible elements in rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Invertible.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Invertible", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4163, "macro_tier_override": null, "x": 2.737, "z": 18.369, "size": 0.3752, "title": "Theorems about invertible elements in a `GroupWithZero`", "summary": "We intentionally keep imports minimal here as this file is used by `Mathlib/Tactic/NormNum.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Invertible.html"}, {"id": "Mathlib.Algebra.Ring.Defs", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 2, "macro_tier_score": 0.4911, "macro_tier_override": 2, "x": 3.452, "z": -6.578, "size": 0.9467, "title": "Semirings and rings", "summary": "This file defines semirings, rings and domains. This is analogous to `Mathlib/Algebra/Group/Defs.lean` and `Mathlib/Algebra/Group/Basic.lean`, the difference being that those are about `+` and `*` separately, while the present file is about their interaction. the present file is about their interaction.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Defs.html"}, {"id": "Mathlib.Algebra.Lie.LieTheorem", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -91.463, "z": -66.846, "size": 0.2553, "title": "Lie's theorem for Solvable Lie algebras.", "summary": "Lie's theorem asserts that Lie modules of solvable Lie algebras over fields of characteristic 0 have a common eigenvector for the action of all elements of the Lie algebra. This result is named `LieModule.exists_forall_lie_eq_smul_of_isSolvable`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/LieTheorem.html"}, {"id": "Mathlib.Algebra.Lie.Weights.Basic", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -42.99, "z": -102.803, "size": 0.3517, "title": "Weight spaces of Lie modules of nilpotent Lie algebras", "summary": "Just as a key tool when studying the behaviour of a linear operator is to decompose the space on which it acts into a sum of (generalised) eigenspaces, a key tool when studying a representation `M` of Lie algebra `L` is to decompose `M` into a sum of simultaneous eigenspaces of `x` as `x` ranges over `L`. These simultaneous generalised eigenspaces are known as the weight spaces of `M`. When `L` is nilpotent, it…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Weights/Basic.html"}, {"id": "Mathlib.RingTheory.Finiteness.Nilpotent", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 17.322, "z": -64.575, "size": 0.2686, "title": "Nilpotent maps on finite modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Nilpotent.html"}, {"id": "Mathlib.Algebra.Category.Grp.Adjunctions", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -31.807, "z": 15.279, "size": 0.2488, "title": "Adjunctions regarding the category of (abelian) groups", "summary": "This file contains construction of basic adjunctions concerning the category of groups and the category of abelian groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Adjunctions.html"}, {"id": "Mathlib.GroupTheory.FreeAbelianGroup", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 4, "macro_tier_score": 0.3089, "macro_tier_override": null, "x": -33.294, "z": 3.0, "size": 0.3095, "title": "Free abelian groups", "summary": "The free abelian group on a type `α`, defined as the abelianisation of the free group on `α`. The free abelian group on `α` can be abstractly defined as the left adjoint of the forgetful functor from abelian groups to types. Alternatively, one could define it as the functions `α → ℤ` which send all but finitely many `(a : α)` to `0`, under pointwise addition. In this file, it is defined as the abelianisation of the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FreeAbelianGroup.html"}, {"id": "Mathlib.Algebra.Category.CoalgCat.Monoidal", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 57.319, "z": -26.488, "size": 0.2382, "title": "The monoidal category structure on `R`-coalgebras", "summary": "In `Mathlib/RingTheory/Coalgebra/TensorProduct.lean`, given two `R`-coalgebras `M, N`, we define a coalgebra instance on `M ⊗[R] N`, as well as the tensor product of two `CoalgHom`s as a `CoalgHom`, and the associator and left/right unitors for coalgebras as `CoalgEquiv`s. In this file, we declare a `MonoidalCategory` instance on the category of coalgebras, with data fields given by the definitions in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/CoalgCat/Monoidal.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 2, "macro_tier_score": 0.0133, "macro_tier_override": null, "x": 42.948, "z": 32.498, "size": 0.3424, "title": "The monoidal category structure on R-modules", "summary": "Mostly this uses existing machinery in `LinearAlgebra.TensorProduct`. We just need to provide a few small missing pieces to build the `MonoidalCategory` instance. The `SymmetricCategory` instance is in `Algebra.Category.ModuleCat.Monoidal.Symmetric` to reduce imports. Note the universe level of the modules must be at least the universe level of the ring, so that we have a monoidal unit. For now, we simplify by…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.html"}, {"id": "Mathlib.RingTheory.Coalgebra.TensorProduct", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 2, "macro_tier_score": 0.0074, "macro_tier_override": null, "x": 29.104, "z": -53.935, "size": 0.3144, "title": "Tensor products of coalgebras", "summary": "Suppose `S` is an `R`-algebra. Given an `S`-coalgebra `A` and `R`-coalgebra `B`, we can define a natural comultiplication map `Δ : A ⊗[R] B → (A ⊗[R] B) ⊗[S] (A ⊗[R] B)` and counit map `ε : A ⊗[R] B → S` induced by the comultiplication and counit maps of `A` and `B`. In this file we show that `Δ, ε` satisfy the axioms of a coalgebra, and also define other data in the monoidal structure on `R`-coalgebras, like the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Coalgebra/TensorProduct.html"}, {"id": "Mathlib.LinearAlgebra.Eigenspace.ContinuousLinearMap", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 73.121, "z": -36.478, "size": 0.241, "title": "Eigenspaces of continuous linear maps", "summary": "This file provides some basic properties of eigenspaces of continuous linear maps. These results are in a separate file to avoid heavy topology imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Eigenspace/ContinuousLinearMap.html"}, {"id": "Mathlib.LinearAlgebra.Eigenspace.Basic", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.0155, "macro_tier_override": null, "x": 66.75, "z": 43.838, "size": 0.3993, "title": "Eigenvectors and eigenvalues", "summary": "This file defines eigenspaces, eigenvalues, and eigenvectors, as well as their generalized counterparts. We follow Axler's approach [axler2024] because it allows us to derive many properties without choosing a basis and without using matrices. An eigenspace of a linear map `f` for a scalar `μ` is the kernel of the map `(f - μ • id)`. The nonzero elements of an eigenspace are eigenvectors `x`. They have the property…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Eigenspace/Basic.html"}, {"id": "Mathlib.FieldTheory.PurelyInseparable.Tower", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 62.985, "z": -102.981, "size": 0.2, "title": "Tower law for purely inseparable extensions", "summary": "This file contains results related to `Field.sepDegree`, `Field.insepDegree` and the tower law.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/PurelyInseparable/Tower.html"}, {"id": "Mathlib.FieldTheory.LinearDisjoint", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -29.748, "z": -111.235, "size": 0.2758, "title": "Linearly disjoint fields", "summary": "This file contains basics about the linearly disjoint fields. We adapt the definitions in . See the file `Mathlib/LinearAlgebra/LinearDisjoint.lean` and `Mathlib/RingTheory/LinearDisjoint.lean` for details.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/LinearDisjoint.html"}, {"id": "Mathlib.FieldTheory.PurelyInseparable.PerfectClosure", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 16.807, "z": -117.664, "size": 0.2827, "title": "Basic results about relative perfect closure", "summary": "This file contains basic results about relative perfect closures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/PurelyInseparable/PerfectClosure.html"}, {"id": "Mathlib.Algebra.Group.Pointwise.Set.BigOperators", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.3443, "macro_tier_override": null, "x": -7.359, "z": -17.051, "size": 0.3064, "title": "Results about pointwise operations on sets and big operators.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pointwise/Set/BigOperators.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.Finset.Piecewise", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4132, "macro_tier_override": null, "x": -16.641, "z": -1.563, "size": 0.4334, "title": "Interaction of big operators with piecewise functions", "summary": "This file proves lemmas on the sum and product of piecewise functions, including `ite` and `dite`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Finset/Piecewise.html"}, {"id": "Mathlib.RingTheory.GradedAlgebra.Homogeneous.Subsemiring", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -24.97, "z": -55.969, "size": 0.2, "title": "Homogeneous subsemirings of a graded semiring", "summary": "This file defines homogeneous subsemirings of a graded semiring, as well as operations on them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/GradedAlgebra/Homogeneous/Subsemiring.html"}, {"id": "Mathlib.RingTheory.GradedAlgebra.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.0337, "macro_tier_override": null, "x": -10.093, "z": 58.566, "size": 0.4175, "title": "Internally-graded rings and algebras", "summary": "This file defines the typeclass `GradedAlgebra 𝒜`, for working with an algebra `A` that is internally graded by a collection of submodules `𝒜 : ι → Submodule R A`. See the docstring of that typeclass for more information.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/GradedAlgebra/Basic.html"}, {"id": "Mathlib.GroupTheory.Archimedean", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 2, "macro_tier_score": 0.006, "macro_tier_override": null, "x": 27.451, "z": -4.742, "size": 0.2919, "title": "Archimedean groups", "summary": "This file proves a few facts about ordered groups which satisfy the `Archimedean` property, that is: `class Archimedean (α) [OrderedAddCommMonoid α] : Prop :=` `(arch : ∀ (x : α) {y}, 0 < y → ∃ n : ℕ, x ≤ n • y)` They are placed here in a separate file (rather than incorporated as a continuation of `Algebra.Order.Archimedean`) because they rely on some imports from `GroupTheory` -- bundled subgroups in particular.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Archimedean.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.Order", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 2, "macro_tier_score": 0.0127, "macro_tier_override": null, "x": -10.909, "z": 23.601, "size": 0.2943, "title": "Facts about ordered structures and ordered instances on subgroups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/Order.html"}, {"id": "Mathlib.LinearAlgebra.PerfectPairing.Restrict", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -31.933, "z": 77.231, "size": 0.2761, "title": "Restriction to submodules and restriction of scalars for perfect pairings.", "summary": "We provide API for restricting perfect pairings to submodules and for restricting their scalars.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/PerfectPairing/Restrict.html"}, {"id": "Mathlib.LinearAlgebra.PerfectPairing.Basic", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 1, "macro_tier_score": 0.0039, "macro_tier_override": null, "x": 56.476, "z": -59.058, "size": 0.2989, "title": "Perfect pairings", "summary": "This file defines perfect pairings of modules. A perfect pairing of two (left) modules may be defined either as: 1. A bilinear map `M × N → R` such that the induced maps `M → Dual R N` and `N → Dual R M` are both bijective. It follows from this that both `M` and `N` are reflexive modules. 2. A linear equivalence `N ≃ Dual R M` for which `M` is reflexive. (It then follows that `N` is reflexive.) In this file we…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/PerfectPairing/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.BaseChange", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -77.322, "z": -10.27, "size": 0.2561, "title": "Matrices and base change", "summary": "This file is a home for results about base change for matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/BaseChange.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.FixedDetMatrices", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -0.178, "z": 79.858, "size": 0.2603, "title": "Matrices with fixed determinant", "summary": "This file defines the type of matrices with fixed determinant `m` and proves some basic results about them. We also prove that the subgroup of `SL(2,ℤ)` generated by `S` and `T` is the whole group. Note: Some of this was done originally in Lean 3 in the kbb (https://github.com/kim-em/kbb/tree/master) repository, so credit to those authors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/FixedDetMatrices.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 3, "macro_tier_score": 0.1385, "macro_tier_override": null, "x": 55.078, "z": -55.231, "size": 0.3367, "title": "The Special Linear group $SL(n, R)$", "summary": "This file defines the elements of the Special Linear group `SpecialLinearGroup n R`, consisting of all square `R`-matrices with determinant `1` on the fintype `n` by `n`. In addition, we define the group structure on `SpecialLinearGroup n R` and the embedding into the general linear group `GeneralLinearGroup R (n → R)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.html"}, {"id": "Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 1, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": 41.283, "z": -47.778, "size": 0.3083, "title": "Homogeneous ideals of a graded algebra", "summary": "This file defines homogeneous ideals of `GradedRing 𝒜` where `𝒜 : ι → Submodule R A` and operations on them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/GradedAlgebra/Homogeneous/Ideal.html"}, {"id": "Mathlib.Algebra.Star.Pi", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 2, "macro_tier_score": 0.0087, "macro_tier_override": null, "x": -18.749, "z": 12.047, "size": 0.3277, "title": "Basic Results about Star on Pi Types", "summary": "This file provides basic results about the star on product types defined in `Mathlib/Algebra/Notation/Pi/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/Pi.html"}, {"id": "Mathlib.Algebra.Star.Subalgebra", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 1, "macro_tier_score": 0.0017, "macro_tier_override": null, "x": 54.45, "z": -23.813, "size": 0.3023, "title": "Star subalgebras", "summary": "A \\*-subalgebra is a subalgebra of a \\*-algebra which is closed under `*`. The centralizer of a \\*-closed set is a \\*-subalgebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/Subalgebra.html"}, {"id": "Mathlib.LinearAlgebra.FreeModule.Norm", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 78.365, "z": 56.437, "size": 0.2694, "title": "Norms on free modules over principal ideal domains", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeModule/Norm.html"}, {"id": "Mathlib.RingTheory.Norm.Defs", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 3, "macro_tier_score": 0.1207, "macro_tier_override": null, "x": 40.222, "z": 75.368, "size": 0.3327, "title": "Norm for (finite) ring extensions", "summary": "Suppose we have an `R`-algebra `S` with a finite basis. For each `s : S`, the determinant of the linear map given by multiplying by `s` gives information about the roots of the minimal polynomial of `s` over `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Norm/Defs.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Unbundled.Defs", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.4767, "macro_tier_override": null, "x": -4.284, "z": 6.069, "size": 0.5732, "title": "Covariants and contravariants", "summary": "This file contains general lemmas and instances to work with the interactions between a relation and an action on a Type. The intended application is the splitting of the ordering from the algebraic assumptions on the operations in the `Ordered[...]` hierarchy. The strategy is to introduce two more flexible typeclasses, `CovariantClass` and `ContravariantClass`: * `CovariantClass` models the implication `a ≤ b → c *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Unbundled/Defs.html"}, {"id": "Mathlib.Algebra.Order.IsBotOne", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.4821, "macro_tier_override": null, "x": -5.562, "z": 0.317, "size": 0.6427, "title": "Typeclasses expressing `IsBot 1` and `IsBot 0`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/IsBotOne.html"}, {"id": "Mathlib.GroupTheory.ArchimedeanDensely", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 2, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 41.969, "z": 7.947, "size": 0.3077, "title": "Archimedean groups are either discrete or densely ordered", "summary": "This file proves a few additional facts about linearly ordered additive groups which satisfy the `Archimedean` property -- they are either order-isomorphic and additively isomorphic to the integers, or they are densely ordered. They are placed here in a separate file (rather than incorporated as a continuation of `GroupTheory.Archimedean`) because they rely on some imports from pointwise lemmas.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/ArchimedeanDensely.html"}, {"id": "Mathlib.Algebra.Order.Group.Pointwise.Interval", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 2, "macro_tier_score": 0.0123, "macro_tier_override": null, "x": 23.499, "z": 18.187, "size": 0.4486, "title": "(Pre)images of intervals", "summary": "In this file we prove a bunch of trivial lemmas like “if we add `a` to all points of `[b, c]`, then we get `[a + b, a + c]`”. For the functions `x ↦ x ± a`, `x ↦ a ± x`, and `x ↦ -x` we prove lemmas about preimages and images of all intervals. We also prove a few lemmas about images under `x ↦ a * x`, `x ↦ x * a` and `x ↦ x⁻¹`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Pointwise/Interval.html"}, {"id": "Mathlib.Algebra.Group.Action.Defs", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4961, "macro_tier_override": null, "x": -9.226, "z": 1.053, "size": 0.7556, "title": "Definitions of group actions", "summary": "This file defines a hierarchy of group action type-classes on top of the previously defined notation classes `SMul` and its additive version `VAdd`: * `MulAction M α` and its additive version `AddAction G P` are typeclasses used for actions of multiplicative and additive monoids and groups; they extend notation classes `SMul` and `VAdd` that are defined in `Algebra.Group.Defs`; * `DistribMulAction M A` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Defs.html"}, {"id": "Mathlib.Algebra.Order.Interval.Set.Monoid", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.2789, "macro_tier_override": null, "x": 11.76, "z": -5.541, "size": 0.3087, "title": "Images of intervals under `(+ d)`", "summary": "The lemmas in this file state that addition maps intervals bijectively. The typeclass `ExistsAddOfLE` is defined specifically to make them work when combined with `IsOrderedCancelAddMonoid`; the lemmas below therefore apply to all ordered groups, but also to `ℕ` and `ℝ≥0`, which are not groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Interval/Set/Monoid.html"}, {"id": "Mathlib.Algebra.Group.Action.TransferInstance", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.3657, "macro_tier_override": null, "x": -10.852, "z": 7.158, "size": 0.3937, "title": "Transfer algebraic structures across `Equiv`s", "summary": "This continues the pattern set in `Mathlib/Algebra/Group/TransferInstance.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/TransferInstance.html"}, {"id": "Mathlib.Algebra.Group.Action.Faithful", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4779, "macro_tier_override": null, "x": 0.283, "z": -11.139, "size": 0.5477, "title": "Faithful group actions", "summary": "This file provides typeclasses for faithful actions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Faithful.html"}, {"id": "Mathlib.Algebra.Group.TransferInstance", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.3913, "macro_tier_override": null, "x": 2.498, "z": -8.943, "size": 0.4442, "title": "Transfer algebraic structures across `Equiv`s", "summary": "In this file we prove lemmas of the following form: if `β` has a group structure and `α ≃ β` then `α` has a group structure, and similarly for monoids, semigroups and so on.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/TransferInstance.html"}, {"id": "Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": 95.193, "z": -71.174, "size": 0.2955, "title": "Dimension formula and Sturm bound for level 1 modular forms", "summary": "This file proves the dimension formula and the Sturm bound for the space of modular forms for `𝒮ℒ` (= `SL(2, ℤ)`) of even weight.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/LevelOne/DimensionFormula.html"}, {"id": "Mathlib.NumberTheory.ModularForms.CuspFormSubmodule", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -113.286, "z": 0.555, "size": 0.264, "title": "Cusp form submodule and IsCuspForm predicate", "summary": "This file defines the inclusion of cusp forms into modular forms as a linear map, the cusp form submodule of modular forms, and the `IsCuspForm` predicate. It also provides a direct constructor `ModularForm.toCuspForm` for building cusp forms from modular forms with vanishing constant q-expansion coefficient (for `𝒮ℒ`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/CuspFormSubmodule.html"}, {"id": "Mathlib.NumberTheory.ModularForms.Discriminant", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 116.998, "z": 0.908, "size": 0.264, "title": "The modular discriminant Δ", "summary": "This file defines the modular discriminant `Δ(z) = η(z) ^ 24`, where `η` is the Dedekind eta function, and proves its key properties including invariance under the generators of `SL(2, ℤ)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/Discriminant.html"}, {"id": "Mathlib.Algebra.Order.Floor.Semifield", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -28.73, "z": -13.091, "size": 0.2683, "title": "Lemmas on `Nat.floor` and `Nat.ceil` for semifields", "summary": "This file contains basic results on the natural-valued floor and ceiling functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Floor/Semifield.html"}, {"id": "Mathlib.NumberTheory.Padics.PadicNumbers", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 2, "macro_tier_score": 0.0117, "macro_tier_override": null, "x": -29.477, "z": -53.732, "size": 0.3082, "title": "p-adic numbers", "summary": "This file defines the `p`-adic numbers (rationals) `ℚ_[p]` as the completion of `ℚ` with respect to the `p`-adic norm. We show that the `p`-adic norm on `ℚ` extends to `ℚ_[p]`, that `ℚ` is embedded in `ℚ_[p]`, and that `ℚ_[p]` is Cauchy complete.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/PadicNumbers.html"}, {"id": "Mathlib.NumberTheory.Padics.PadicNorm", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 2, "macro_tier_score": 0.0124, "macro_tier_override": null, "x": -28.099, "z": 9.664, "size": 0.2635, "title": "p-adic norm", "summary": "This file defines the `p`-adic norm on `ℚ`. The `p`-adic valuation on `ℚ` is the difference of the multiplicities of `p` in the numerator and denominator of `q`. This function obeys the standard properties of a valuation, with the appropriate assumptions on `p`. The valuation induces a norm on `ℚ`. This norm is a nonarchimedean absolute value. It takes values in `{0} ∪ {1/p^k | k ∈ ℤ}`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/PadicNorm.html"}, {"id": "Mathlib.RingTheory.Polynomial.Chebyshev", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -71.853, "z": 18.857, "size": 0.2725, "title": "Chebyshev polynomials", "summary": "The Chebyshev polynomials are families of polynomials indexed by `ℤ`, with integral coefficients.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Chebyshev.html"}, {"id": "Mathlib.Algebra.Polynomial.Sequence", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": 66.535, "z": -28.621, "size": 0.2474, "title": "Polynomial sequences", "summary": "We define polynomial sequences – sequences of polynomials `a₀, a₁, ...` such that the polynomial `aᵢ` has degree `i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Sequence.html"}, {"id": "Mathlib.GroupTheory.Perm.ViaEmbedding", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 3, "macro_tier_score": 0.0711, "macro_tier_override": null, "x": 13.736, "z": 12.499, "size": 0.246, "title": "`Equiv.Perm.viaEmbedding`, a noncomputable analogue of `Equiv.Perm.viaFintypeEmbedding`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/ViaEmbedding.html"}, {"id": "Mathlib.RingTheory.Smooth.Fiber", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": 120.369, "z": 23.138, "size": 0.2678, "title": "Flat and smooth fibers imply smooth", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/Fiber.html"}, {"id": "Mathlib.RingTheory.Etale.Field", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 2, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": -62.849, "z": 103.064, "size": 0.2798, "title": "Étale algebras over fields", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Etale/Field.html"}, {"id": "Mathlib.RingTheory.Flat.Equalizer", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": 70.955, "z": 40.53, "size": 0.2566, "title": "Base change along flat modules preserves equalizers", "summary": "We show that base change along flat modules (resp. algebras) preserves kernels and equalizers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/Equalizer.html"}, {"id": "Mathlib.RingTheory.Kaehler.TensorProduct", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": 63.575, "z": -57.065, "size": 0.2664, "title": "Kähler differential module under base change", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Kaehler/TensorProduct.html"}, {"id": "Mathlib.RingTheory.Smooth.Local", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 1, "macro_tier_score": 0.0045, "macro_tier_override": null, "x": -53.002, "z": -87.317, "size": 0.2411, "title": "Formally smooth local algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/Local.html"}, {"id": "Mathlib.RingTheory.Etale.Locus", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 1, "macro_tier_score": 0.0045, "macro_tier_override": null, "x": 71.013, "z": 80.992, "size": 0.2411, "title": "Etale locus of an algebra", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Etale/Locus.html"}, {"id": "Mathlib.Algebra.Polynomial.Splits", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 3, "macro_tier_score": 0.1802, "macro_tier_override": null, "x": 14.318, "z": 74.785, "size": 0.3754, "title": "Split polynomials", "summary": "A polynomial `f : R[X]` splits if it is a product of constant and monic linear polynomials.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Splits.html"}, {"id": "Mathlib.RingTheory.LocalRing.Basic", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 3, "macro_tier_score": 0.2374, "macro_tier_override": null, "x": 14.685, "z": 47.945, "size": 0.4503, "title": "Local rings", "summary": "We prove basic properties of local rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/Basic.html"}, {"id": "Mathlib.Algebra.Lie.Basic", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 2, "macro_tier_score": 0.0203, "macro_tier_override": null, "x": 15.971, "z": -39.617, "size": 0.4128, "title": "Lie algebras", "summary": "This file defines Lie rings and Lie algebras over a commutative ring together with their modules, morphisms and equivalences, as well as various lemmas to make these definitions usable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Basic.html"}, {"id": "Mathlib.Algebra.Module.Submodule.Equiv", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 4, "macro_tier_score": 0.3947, "macro_tier_override": null, "x": 25.025, "z": 32.297, "size": 0.5997, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/Equiv.html"}, {"id": "Mathlib.Algebra.Module.Rat", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.2795, "macro_tier_override": null, "x": 22.487, "z": -8.788, "size": 0.4041, "title": "Basic results about modules over the rationals.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Rat.html"}, {"id": "Mathlib.Algebra.Order.Archimedean.Hom", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 24.082, "z": -1.718, "size": 0.2712, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Archimedean/Hom.html"}, {"id": "Mathlib.Algebra.Group.Nat.Even", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4167, "macro_tier_override": null, "x": -14.47, "z": 3.369, "size": 0.3314, "title": "`IsSquare` and `Even` for natural numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Nat/Even.html"}, {"id": "Mathlib.RingTheory.Ideal.Int", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -95.798, "z": -56.915, "size": 0.2572, "title": "Ideal of `ℤ`", "summary": "We prove results about ideals of `ℤ` or ideals of extensions of `ℤ`. In particular, for `I` an ideal of a ring `R` extending `ℤ`, we prove several results about `absNorm (under ℤ I)` which is the smallest positive integer contained in `I`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Int.html"}, {"id": "Mathlib.RingTheory.Ideal.Norm.AbsNorm", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 2, "macro_tier_score": 0.0123, "macro_tier_override": null, "x": -99.052, "z": -46.849, "size": 0.3457, "title": "Ideal norms", "summary": "This file defines the absolute ideal norm `Ideal.absNorm (I : Ideal R) : ℕ` as the cardinality of the quotient `R ⧸ I` (setting it to 0 if the cardinality is infinite).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Norm/AbsNorm.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Simple", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 79.443, "z": 8.129, "size": 0.2676, "title": "Simple objects in the category of `R`-modules", "summary": "We prove simple modules are exactly simple objects in the category of `R`-modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Simple.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Algebra", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -35.498, "z": 32.733, "size": 0.241, "title": "Additional typeclass for modules over an algebra", "summary": "For an object in `M : ModuleCat A`, where `A` is a `k`-algebra, we provide additional typeclasses on the underlying type `M`, namely `Module k M` and `IsScalarTower k A M`. These are not made into instances by default. We provide the `Linear k (ModuleCat A)` instance.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Algebra.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Subobject", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -2.785, "z": 55.645, "size": 0.263, "title": "Subobjects in the category of `R`-modules", "summary": "We construct an explicit order isomorphism between the categorical subobjects of an `R`-module `M` and its submodules. This immediately implies that the category of `R`-modules is well-powered.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Subobject.html"}, {"id": "Mathlib.RingTheory.SimpleModule.Basic", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 3, "macro_tier_score": 0.109, "macro_tier_override": null, "x": -34.018, "z": 50.978, "size": 0.3606, "title": "Simple Modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SimpleModule/Basic.html"}, {"id": "Mathlib.RingTheory.Ideal.Defs", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 4, "macro_tier_score": 0.3451, "macro_tier_override": null, "x": -18.472, "z": 27.862, "size": 0.4918, "title": "Ideals over a ring", "summary": "This file defines `Ideal R`, the type of (left) ideals over a ring `R`. Note that over commutative rings, left ideals and two-sided ideals are equivalent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Defs.html"}, {"id": "Mathlib.Algebra.Order.Monovary", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 14.136, "z": 17.229, "size": 0.2713, "title": "Monovarying functions and algebraic operations", "summary": "This file characterises the interaction of ordered algebraic structures with monovariance of functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monovary.html"}, {"id": "Mathlib.Algebra.Order.Monoid.OrderDual", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 3, "macro_tier_score": 0.2528, "macro_tier_override": null, "x": 5.911, "z": 11.579, "size": 0.3428, "title": "Ordered monoid structures on the order dual.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/OrderDual.html"}, {"id": "Mathlib.LinearAlgebra.PiTensorProduct.Finite", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -9.691, "z": -56.751, "size": 0.2, "title": "A multiple tensor product of finitely generated modules is finitely generated", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/PiTensorProduct/Finite.html"}, {"id": "Mathlib.RingTheory.Finiteness.Basic", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 4, "macro_tier_score": 0.3205, "macro_tier_override": null, "x": -0.562, "z": 50.14, "size": 0.5182, "title": "Basic results on finitely generated (sub)modules", "summary": "This file contains the basic results on `Submodule.FG` and `Module.Finite` that do not need heavy further imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Basic.html"}, {"id": "Mathlib.LinearAlgebra.PiTensorProduct.Generators", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 3.32, "z": 55.616, "size": 0.2478, "title": "Generators of multiple tensor products", "summary": "Given a finite family of `R`-modules `M i`, if we have, for each `i`, a family of generators of the module `M i`, then the tensor products of these elements generate `⨂[R] i, M i`. In `LinearAlgebra.PiTensorProduct.Finite`, we deduce that if the modules `M i` are finitely generated, then so is `⨂[R] i, M i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/PiTensorProduct/Generators.html"}, {"id": "Mathlib.Algebra.Order.Ring.Defs", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4265, "macro_tier_override": null, "x": -17.64, "z": -5.807, "size": 0.5913, "title": "Ordered rings and semirings", "summary": "This file develops the basics of ordered (semi)rings. Each typeclass here comprises * an algebraic class (`Semiring`, `CommSemiring`, `Ring`, `CommRing`) * an order class (`PartialOrder`, `LinearOrder`) * assumptions on how both interact ((strict) monotonicity, canonicity) For short, * \"`+` respects `≤`\" means \"monotonicity of addition\" * \"`+` respects `<`\" means \"strict monotonicity of addition\" * \"`*` respects…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Defs.html"}, {"id": "Mathlib.Algebra.Order.Ring.Unbundled.Basic", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4211, "macro_tier_override": null, "x": 13.702, "z": 9.572, "size": 0.3906, "title": "Basic facts for ordered rings and semirings", "summary": "This file develops the basics of ordered (semi)rings in an unbundled fashion for later use with the bundled classes from `Mathlib/Algebra/Order/Ring/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Unbundled/Basic.html"}, {"id": "Mathlib.Algebra.CharZero.Defs", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.4357, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.5451, "title": "Characteristic zero", "summary": "A ring `R` is called of characteristic zero if every natural number `n` is non-zero when considered as an element of `R`. Since this definition doesn't mention the multiplicative structure of `R` except for the existence of `1` in this file characteristic zero is defined for additive monoids with `1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharZero/Defs.html"}, {"id": "Mathlib.Algebra.Order.Group.Defs", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4218, "macro_tier_override": null, "x": -5.3, "z": 13.88, "size": 0.4646, "title": "Ordered groups", "summary": "This file defines bundled ordered groups and develops a few basic results.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Defs.html"}, {"id": "Mathlib.Algebra.Ring.GrindInstances", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4286, "macro_tier_override": null, "x": -3.423, "z": 8.632, "size": 0.4665, "title": "Instances for `grind`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/GrindInstances.html"}, {"id": "Mathlib.Algebra.Order.Group.Unbundled.Int", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.4195, "macro_tier_override": null, "x": -22.282, "z": -0.419, "size": 0.4175, "title": "Facts about `ℤ` as an (unbundled) ordered group", "summary": "See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Unbundled/Int.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Unbundled.ExistsOfLE", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4593, "macro_tier_override": null, "x": 10.913, "z": -2.254, "size": 0.5243, "title": "Unbundled and weaker forms of canonically ordered monoids", "summary": "This file provides a Prop-valued mixin for monoids satisfying a one-sided cancellativity property, namely that there is some `c` such that `b = a + c` if `a ≤ b`. This is particularly useful for generalising statements from groups/rings/fields that don't mention negation or subtraction to monoids/semirings/semifields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Unbundled/ExistsOfLE.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Sheaf.Limits", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 52.016, "z": 38.98, "size": 0.2719, "title": "Limits in categories of sheaves of modules", "summary": "In this file, it is shown that under suitable assumptions, limits exist in the category `SheafOfModules R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Sheaf/Limits.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.Limits", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 1, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": 56.593, "z": 28.006, "size": 0.3117, "title": "Limits in categories of presheaves of modules", "summary": "In this file, it is shown that under suitable assumptions, limits exist in the category `PresheafOfModules R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Sheaf", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 1, "macro_tier_score": 0.0019, "macro_tier_override": null, "x": 48.239, "z": -40.744, "size": 0.3174, "title": "Sheaves of modules over a sheaf of rings", "summary": "In this file, we define the category `SheafOfModules R` when `R : Sheaf J RingCat` is a sheaf of rings on a category `C` equipped with a Grothendieck topology `J`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Sheaf.html"}, {"id": "Mathlib.Algebra.BigOperators.Fin", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.3472, "macro_tier_override": null, "x": -19.409, "z": -14.359, "size": 0.5226, "title": "Big operators and `Fin`", "summary": "Some results about products and sums over the type `Fin`. The most important results are the induction formulas `Fin.prod_univ_castSucc` and `Fin.prod_univ_succ`, and the formula `Fin.prod_const` for the product of a constant function. These results have variants for sums instead of products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Fin.html"}, {"id": "Mathlib.NumberTheory.LSeries.Linearity", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 75.291, "z": 11.366, "size": 0.2516, "title": "Linearity of the L-series of `f` as a function of `f`", "summary": "We show that the `LSeries` of `f : ℕ → ℂ` is a linear function of `f` (assuming convergence of both L-series when adding two functions).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/Linearity.html"}, {"id": "Mathlib.NumberTheory.LSeries.Basic", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 2, "macro_tier_score": 0.0066, "macro_tier_override": null, "x": -39.718, "z": -62.777, "size": 0.3368, "title": "L-series", "summary": "Given a sequence `f: ℕ → ℂ`, we define the corresponding L-series.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/Basic.html"}, {"id": "Mathlib.FieldTheory.Finite.Valuation", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": -18.312, "z": 100.489, "size": 0.2827, "title": "Valuations on an algebra over a finite field.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Finite/Valuation.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Localization", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 67.019, "z": -15.174, "size": 0.2512, "title": "Localized Module in ModuleCat", "summary": "For a ring `R` satisfying `[Small.{v} R]` and a submonoid `S` of `R`, this file defines an exact functor `ModuleCat.{v} R ⥤ ModuleCat.{v} (Localization S)`, see `ModuleCat.localizedModuleFunctor`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Localization.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 56.727, "z": 50.793, "size": 0.2755, "title": "Quasicoherent sheaves", "summary": "A sheaf of modules is quasi-coherent if it admits locally a presentation as the cokernel of a morphism between coproducts of copies of the sheaf of rings. When these coproducts are finite, we say that the sheaf is of finite presentation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Sheaf/Quasicoherent.html"}, {"id": "Mathlib.Algebra.Module.LocalizedModule.Away", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 39.943, "z": 27.131, "size": 0.2338, "title": "API for localized modules away from an element", "summary": "We provide some specialized API for the localization of a module away from an element.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LocalizedModule/Away.html"}, {"id": "Mathlib.RingTheory.OrderOfVanishing.Basic", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 71.426, "z": -70.397, "size": 0.2541, "title": "Order of vanishing", "summary": "This file defines the order of vanishing of an element of a ring as the length of the quotient of the ring by the ideal generated by that element. We also define the extension of this notion to the field of fractions", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/OrderOfVanishing/Basic.html"}, {"id": "Mathlib.RingTheory.KrullDimension.NonZeroDivisors", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -76.394, "z": 33.887, "size": 0.2753, "title": "Krull dimension and non-zero-divisors", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/KrullDimension/NonZeroDivisors.html"}, {"id": "Mathlib.RingTheory.Length", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.0592, "macro_tier_override": null, "x": -61.493, "z": 38.27, "size": 0.2886, "title": "Length of modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Length.html"}, {"id": "Mathlib.RingTheory.HopkinsLevitzki", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 2, "macro_tier_score": 0.0241, "macro_tier_override": null, "x": -80.319, "z": 56.896, "size": 0.3245, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HopkinsLevitzki.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Unbundled.Pow", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4697, "macro_tier_override": null, "x": 11.099, "z": 0.994, "size": 0.5359, "title": "Lemmas about the interaction of power operations with order in terms of `CovariantClass`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Unbundled/Pow.html"}, {"id": "Mathlib.RingTheory.KrullDimension.Regular", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 45.281, "z": 93.626, "size": 0.2, "title": "Krull Dimension of quotient regular sequence", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/KrullDimension/Regular.html"}, {"id": "Mathlib.RingTheory.Flat.TorsionFree", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 2, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": -102.004, "z": -5.349, "size": 0.3066, "title": "Relationships between flatness and torsionfreeness.", "summary": "We show that flat implies torsion-free, and that they're the same concept for rings satisfying a certain property, including Dedekind domains and valuation rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/TorsionFree.html"}, {"id": "Mathlib.RingTheory.KrullDimension.Module", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 64.295, "z": -74.529, "size": 0.2338, "title": "Krull Dimension of Module", "summary": "In this file we define `Module.supportDim R M` for an `R`-module `M` as the krull dimension of its support. It is equal to the krull dimension of `R / Ann M` when `M` is finitely generated.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/KrullDimension/Module.html"}, {"id": "Mathlib.RingTheory.Regular.RegularSequence", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 2, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -44.069, "z": -71.009, "size": 0.274, "title": "Regular sequences and weakly regular sequences", "summary": "The notion of a regular sequence is fundamental in commutative algebra. Properties of regular sequences encode information about singularities of a ring and regularity of a sequence can be tested homologically. However the notion of a regular sequence is only really sensible for Noetherian local rings. TODO: Koszul regular sequences, `H_1`-regular sequences, quasi-regular sequences, depth.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Regular/RegularSequence.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.LTSeries", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -34.003, "z": -96.318, "size": 0.2338, "title": "Lemmas about `LTSeries` in the prime spectrum", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/LTSeries.html"}, {"id": "Mathlib.FieldTheory.Isaacs", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 84.709, "z": -66.535, "size": 0.2, "title": "Algebraic extensions are determined by their sets of minimal polynomials up to isomorphism", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Isaacs.html"}, {"id": "Mathlib.FieldTheory.Normal.Basic", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 3, "macro_tier_score": 0.0651, "macro_tier_override": null, "x": 88.779, "z": -54.17, "size": 0.3176, "title": "Normal field extensions", "summary": "In this file we prove that for a finite extension, being normal is the same as being a splitting field (`Normal.of_isSplittingField` and `Normal.exists_isSplittingField`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Normal/Basic.html"}, {"id": "Mathlib.FieldTheory.PrimitiveElement", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 3, "macro_tier_score": 0.0608, "macro_tier_override": null, "x": -94.628, "z": 47.451, "size": 0.3925, "title": "Primitive Element Theorem", "summary": "In this file we prove the primitive element theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/PrimitiveElement.html"}, {"id": "Mathlib.GroupTheory.CosetCover", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -42.764, "z": -35.713, "size": 0.239, "title": "Lemma of B. H. Neumann on coverings of a group by cosets.", "summary": "Let the group $G$ be the union of finitely many, let us say $n$, left cosets of subgroups $C₁$, $C₂$, ..., $Cₙ$: $$ G = ⋃_{i = 1}^n C_i g_i. $$ * `Subgroup.exists_finiteIndex_of_leftCoset_cover` at least one subgroup $C_i$ has finite index in $G$. * `Subgroup.leftCoset_cover_filter_FiniteIndex` the cosets of subgroups of infinite index may be omitted from the covering. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/CosetCover.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 1, "macro_tier_score": 0.0039, "macro_tier_override": null, "x": 37.011, "z": -3.134, "size": 0.3764, "title": "Affine spaces", "summary": "This file defines affine subspaces (over modules) and the affine span of a set of points.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Defs.html"}, {"id": "Mathlib.LinearAlgebra.Span.Defs", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 4, "macro_tier_score": 0.3901, "macro_tier_override": null, "x": -31.92, "z": 15.04, "size": 0.512, "title": "The span of a set of vectors, as a submodule", "summary": "* `Submodule.span s` is defined to be the smallest submodule containing the set `s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Span/Defs.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Defs", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 1, "macro_tier_score": 0.003, "macro_tier_override": null, "x": 4.91, "z": -12.037, "size": 0.3174, "title": "Affine space", "summary": "In this file we introduce the following notation: * `AffineSpace V P` is an alternative notation for `AddTorsor V P` introduced at the end of this file. We tried to use an `abbreviation` instead of a `notation` but this led to hard-to-debug elaboration errors. So, we introduce a localized notation instead. When this notation is enabled with `open Affine`, Lean will use `AffineSpace` instead of `AddTorsor` both in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Defs.html"}, {"id": "Mathlib.Algebra.FiveLemma", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 2, "macro_tier_score": 0.0126, "macro_tier_override": null, "x": -25.831, "z": 47.259, "size": 0.286, "title": "The five lemma in terms of modules", "summary": "The five lemma for all abelian categories is proven in `CategoryTheory.Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono`. But for universe generality and ease of application in the unbundled setting, we reprove them here.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/FiveLemma.html"}, {"id": "Mathlib.Algebra.Exact.Basic", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 4, "macro_tier_score": 0.3002, "macro_tier_override": null, "x": -48.688, "z": -18.262, "size": 0.4736, "title": "Exactness of a pair", "summary": "* For two maps `f : M → N` and `g : N → P`, with `Zero P`, `Function.Exact f g` says that `Set.range f = Set.preimage g {0}` * For two maps `f : M → N` and `g : N → P`, with `One P`, `Function.MulExact f g` says that `Set.range f = Set.preimage g {1}` * For additive maps `f : M →+ N` and `g : N →+ P`, `Exact f g` says that `range f = ker g` * For multiplicative maps `f : M →* N` and `g : N →* P`, `MulExact f g` says…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Exact/Basic.html"}, {"id": "Mathlib.LinearAlgebra.TensorAlgebra.Basis", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -8.229, "z": 81.3, "size": 0.2, "title": "A basis for `TensorAlgebra R M`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorAlgebra/Basis.html"}, {"id": "Mathlib.LinearAlgebra.TensorAlgebra.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 2, "macro_tier_score": 0.0129, "macro_tier_override": null, "x": 33.554, "z": -49.05, "size": 0.381, "title": "Tensor Algebras", "summary": "Given a commutative semiring `R`, and an `R`-module `M`, we construct the tensor algebra of `M`. This is the free `R`-algebra generated (`R`-linearly) by the module `M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorAlgebra/Basic.html"}, {"id": "Mathlib.LinearAlgebra.FreeAlgebra", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -61.415, "z": -51.044, "size": 0.2617, "title": "Linear algebra properties of `FreeAlgebra R X`", "summary": "This file provides a `FreeMonoid X` basis on the `FreeAlgebra R X`, and uses it to show the dimension of the algebra is the cardinality of `List X`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeAlgebra.html"}, {"id": "Mathlib.LinearAlgebra.PiTensorProduct.Dual", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 59.791, "z": 13.456, "size": 0.2, "title": "Tensor products of dual spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/PiTensorProduct/Dual.html"}, {"id": "Mathlib.LinearAlgebra.Dual.Basis", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 3, "macro_tier_score": 0.1785, "macro_tier_override": null, "x": -45.976, "z": 6.473, "size": 0.3414, "title": "Bases of dual vector spaces", "summary": "The dual space of an $R$-module $M$ is the $R$-module of $R$-linear maps $M \\to R$. This file concerns bases on dual vector spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dual/Basis.html"}, {"id": "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.2683, "macro_tier_override": null, "x": 52.627, "z": -27.608, "size": 0.3992, "title": "Finite and free modules", "summary": "We provide some instances for finite and free modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeModule/Finite/Basic.html"}, {"id": "Mathlib.LinearAlgebra.PiTensorProduct.Basis", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 6.477, "z": 59.075, "size": 0.239, "title": "Basis for `PiTensorProduct`", "summary": "This file constructs a basis for `PiTensorProduct` given bases on the component spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/PiTensorProduct/Basis.html"}, {"id": "Mathlib.Algebra.Module.BigOperators", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 4, "macro_tier_score": 0.3787, "macro_tier_override": null, "x": 6.25, "z": -34.728, "size": 0.4688, "title": "Finite sums over modules over a ring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/BigOperators.html"}, {"id": "Mathlib.RingTheory.Ideal.Basic", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 4, "macro_tier_score": 0.3279, "macro_tier_override": null, "x": 17.843, "z": 44.869, "size": 0.388, "title": "Ideals over a ring", "summary": "This file contains an assortment of definitions and results for `Ideal R`, the type of (left) ideals over a ring `R`. Note that over commutative rings, left ideals and two-sided ideals are equivalent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Basic.html"}, {"id": "Mathlib.NumberTheory.Harmonic.Bounds", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 11.648, "z": -14.464, "size": 0.2, "title": null, "summary": "This file proves $\\log(n + 1) \\le H_n \\le 1 + \\log(n)$ for all natural numbers $n$.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Harmonic/Bounds.html"}, {"id": "Mathlib.NumberTheory.Harmonic.Defs", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 15.995, "z": -4.852, "size": 0.2773, "title": null, "summary": "This file defines the harmonic numbers. * `Mathlib/NumberTheory/Harmonic/Int.lean` proves that the `n`th harmonic number is not an integer. * `Mathlib/NumberTheory/Harmonic/Bounds.lean` provides basic log bounds.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Harmonic/Defs.html"}, {"id": "Mathlib.Algebra.Ring.AddAut", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -17.429, "z": 16.706, "size": 0.2765, "title": "Multiplication on the left/right as additive automorphisms", "summary": "In this file we define `AddAut.mulLeft` and `AddAut.mulRight`. See also `AddMonoidHom.mulLeft`, `AddMonoidHom.mulRight`, `AddMonoid.End.mulLeft`, and `AddMonoid.End.mulRight` for multiplication by `R` as an endomorphism instead of multiplication by `Rˣ` as an automorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/AddAut.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.Units", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4426, "macro_tier_override": null, "x": 4.709, "z": 14.091, "size": 0.4893, "title": "Multiplicative actions with zero on and by `Mˣ`", "summary": "This file provides the multiplicative actions with zero of a unit on a type `α`, `SMul Mˣ α`, in the presence of `SMulWithZero M α`, with the obvious definition stated in `Units.smul_def`. Additionally, a `MulDistribMulAction G M` for some group `G` satisfying some additional properties admits a `MulDistribMulAction G Mˣ` structure, again with the obvious definition stated in `Units.coe_smul`. This instance uses a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/Units.html"}, {"id": "Mathlib.Algebra.Group.Units.Opposite", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 3, "macro_tier_score": 0.0903, "macro_tier_override": null, "x": 1.044, "z": 11.094, "size": 0.2992, "title": "Units in multiplicative and additive opposites", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Units/Opposite.html"}, {"id": "Mathlib.Algebra.Module.Opposite", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4372, "macro_tier_override": null, "x": -16.602, "z": -1.933, "size": 0.4173, "title": "Module operations on `Mᵐᵒᵖ`", "summary": "This file contains definitions that build on top of the group action definitions in `Mathlib/Algebra/GroupWithZero/Action/Opposite.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Opposite.html"}, {"id": "Mathlib.Algebra.Order.Kleene", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -17.077, "z": 11.211, "size": 0.2838, "title": "Kleene algebras", "summary": "This file defines idempotent semirings and Kleene algebras, which are used extensively in the theory of computation. An idempotent semiring is a semiring whose addition is idempotent. An idempotent semiring is naturally a semilattice by setting `a ≤ b` if `a + b = b`. A Kleene algebra is an idempotent semiring equipped with an additional unary operator `∗`, the Kleene star, such that (informally) `a∗ = 1 + a + a * a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Kleene.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.SelmerGroup", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 94.78, "z": 42.814, "size": 0.2, "title": "Selmer groups of fraction fields of Dedekind domains", "summary": "Let $K$ be the field of fractions of a Dedekind domain $R$. For any set $S$ of prime ideals in the height one spectrum of $R$, and for any natural number $n$, the Selmer group $K(S, n)$ is defined to be the subgroup of the unit group $K^\\times$ modulo $n$-th powers where each element has $v$-adic valuation divisible by $n$ for all prime ideals $v$ away from $S$. In other words, this is precisely $$ K(S, n) :=…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/SelmerGroup.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.AdicValuation", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 3, "macro_tier_score": 0.0372, "macro_tier_override": null, "x": -94.043, "z": 39.866, "size": 0.3773, "title": "Adic valuations on Dedekind domains", "summary": "Given a Dedekind domain `R` of Krull dimension 1 and a maximal ideal `v` of `R`, we define the `v`-adic valuation on `R` and its extension to the field of fractions `K` of `R`. We prove several properties of this valuation, including the existence of uniformizers. We define the completion of `K` with respect to the `v`-adic valuation, denoted `v.adicCompletion`, and its ring of integers, denoted…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/AdicValuation.html"}, {"id": "Mathlib.Algebra.Group.Int.TypeTags", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -11.117, "z": -0.766, "size": 0.2478, "title": "Lemmas about `Multiplicative ℤ`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Int/TypeTags.html"}, {"id": "Mathlib.LinearAlgebra.PiTensorProduct.Finsupp", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -39.404, "z": 41.974, "size": 0.2541, "title": "Results on finitely supported functions.", "summary": "* `ofFinsuppEquiv`, the tensor product of the family `κ i →₀ M i` indexed by `ι` is linearly equivalent to `∏ i, κ i →₀ ⨂[R] i, M i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/PiTensorProduct/Finsupp.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Units", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 3, "macro_tier_score": 0.2509, "macro_tier_override": null, "x": 10.37, "z": -4.077, "size": 0.2768, "title": "Units in ordered monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Units.html"}, {"id": "Mathlib.Algebra.Group.Units.Defs", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.5013, "macro_tier_override": null, "x": -7.148, "z": -5.927, "size": 0.6273, "title": "Units (i.e., invertible elements) of a monoid", "summary": "An element of a `Monoid` is a unit if it has a two-sided inverse.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Units/Defs.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Submonoid.Primal", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.2997, "macro_tier_override": null, "x": 12.387, "z": -11.222, "size": 0.2789, "title": "Submonoid of primal elements", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Submonoid/Primal.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.Defs", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4714, "macro_tier_override": null, "x": -8.965, "z": -2.42, "size": 0.5512, "title": "Submonoids: definition", "summary": "This file defines bundled multiplicative and additive submonoids. We also define a `CompleteLattice` structure on `Submonoid`s, define the closure of a set as the minimal submonoid that includes this set, and prove a few results about extending properties from a dense set (i.e. a set with `closure s = ⊤`) to the whole monoid, see `Submonoid.dense_induction` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/Defs.html"}, {"id": "Mathlib.RepresentationTheory.Tannaka", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -70.718, "z": -63.008, "size": 0.2, "title": "Tannaka duality for finite groups", "summary": "In this file we prove Tannaka duality for finite groups. The theorem can be formulated as follows: for any integral domain `k`, a finite group `G` can be recovered from `FDRep k G`, the monoidal category of finite-dimensional `k`-linear representations of `G`, and the monoidal forgetful functor `forget : FDRep k G ⥤ FGModuleCat k`. The main result is the isomorphism `equiv : G ≃* Aut (forget k G)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Tannaka.html"}, {"id": "Mathlib.RepresentationTheory.FDRep", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 2, "macro_tier_score": 0.0062, "macro_tier_override": null, "x": -89.658, "z": -24.166, "size": 0.3094, "title": "`FDRep k G` is the category of finite-dimensional `k`-linear representations of `G`.", "summary": "If `V : FDRep k G`, there is a coercion that allows you to treat `V` as a type, and this type comes equipped with `Module k V` and `FiniteDimensional k V` instances. Also `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`. Conversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`, you can construct the bundled representation as `Rep.of ρ`. We prove Schur's Lemma: the dimension of the `Hom`-space between two irreducible…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/FDRep.html"}, {"id": "Mathlib.Algebra.Category.AlgCat.FilteredColimits", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -48.368, "z": -37.637, "size": 0.2, "title": "Filtered colimits in the category of `R`-algebras", "summary": "In this file we show that the forgetful functor from `R`-algebras to rings creates filtered colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/AlgCat/FilteredColimits.html"}, {"id": "Mathlib.Algebra.Category.AlgCat.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 2, "macro_tier_score": 0.0065, "macro_tier_override": null, "x": -55.44, "z": 21.406, "size": 0.3282, "title": "Category instance for algebras over a commutative ring", "summary": "We introduce the bundled category `AlgCat` of algebras over a fixed commutative ring `R` along with the forgetful functors to `RingCat` and `ModuleCat`. We furthermore show that the functor associating to a type the free `R`-algebra on that type is left adjoint to the forgetful functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/AlgCat/Basic.html"}, {"id": "Mathlib.Algebra.Category.Ring.Colimits", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 3, "macro_tier_score": 0.0704, "macro_tier_override": null, "x": -7.11, "z": 21.121, "size": 0.2956, "title": "The category of commutative rings has all colimits.", "summary": "This file uses a \"pre-automated\" approach, just as for `Mathlib/Algebra/Category/MonCat/Colimits.lean`. It is a very uniform approach, that conceivably could be synthesised directly by a tactic that analyses the shape of `CommRing` and `RingHom`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/Colimits.html"}, {"id": "Mathlib.Algebra.Category.Ring.FilteredColimits", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 1, "macro_tier_score": 0.0017, "macro_tier_override": null, "x": -1.182, "z": 22.255, "size": 0.3026, "title": "The forgetful functor from (commutative) (semi-) rings preserves filtered colimits.", "summary": "Forgetful functors from algebraic categories usually don't preserve colimits. However, they tend to preserve _filtered_ colimits. In this file, we start with a small filtered category `J` and a functor `F : J ⥤ SemiRingCat`. We show that the colimit of `F ⋙ forget₂ SemiRingCat MonCat` (in `MonCat`) carries the structure of a semiring, thereby showing that the forgetful functor `forget₂ SemiRingCat MonCat` preserves…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/FilteredColimits.html"}, {"id": "Mathlib.Algebra.Group.Units.Basic", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4974, "macro_tier_override": null, "x": -9.481, "z": 5.855, "size": 0.6163, "title": "Units (i.e., invertible elements) of a monoid", "summary": "An element of a `Monoid` is a unit if it has a two-sided inverse. This file contains the basic lemmas on units in a monoid, especially focusing on singleton types and unique types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Units/Basic.html"}, {"id": "Mathlib.Algebra.FiniteSupport.Defs", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.4069, "macro_tier_override": null, "x": 2.287, "z": -7.068, "size": 0.4847, "title": "Make `fun_prop` work for finite (multiplicative) support", "summary": "We define a new predicate `HasFiniteMulSupport` (and its additivized version) on functions and provide the infrastructure so that `fun_prop` can prove it for functions that are built from other functions with finite multiplicative support. The relevant API lemmas are provided in [Mathlib.Algebra.FiniteSupport.Basic](Mathlib/Algebra/FiniteSupport/Basic.lean).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/FiniteSupport/Defs.html"}, {"id": "Mathlib.Algebra.Category.Grp.Yoneda", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": 16.207, "z": -12.436, "size": 0.3039, "title": "Yoneda embeddings", "summary": "This file defines a few Yoneda embeddings for the category of commutative groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Yoneda.html"}, {"id": "Mathlib.GroupTheory.Focal", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 5.926, "z": 51.662, "size": 0.2, "title": "Focal Subgroup Theorem", "summary": "This file defines the focal subgroup and proves the Focal Subgroup Theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Focal.html"}, {"id": "Mathlib.GroupTheory.Abelianization.Defs", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 4, "macro_tier_score": 0.3189, "macro_tier_override": null, "x": -1.476, "z": 31.537, "size": 0.3162, "title": "The abelianization of a group", "summary": "This file defines the commutator and the abelianization of a group. It furthermore prepares for the result that the abelianization is left adjoint to the forgetful functor from abelian groups to groups, which can be found in `Mathlib/Algebra/Category/Grp/Adjunctions.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Abelianization/Defs.html"}, {"id": "Mathlib.GroupTheory.Transfer", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -28.538, "z": -41.231, "size": 0.2936, "title": "The Transfer Homomorphism", "summary": "In this file we construct the transfer homomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Transfer.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.Tower", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 4, "macro_tier_score": 0.2848, "macro_tier_override": null, "x": 33.241, "z": -47.006, "size": 0.394, "title": "Subalgebras in towers of algebras", "summary": "In this file we prove facts about subalgebras in towers of algebras. An algebra tower A/S/R is expressed by having instances of `Algebra A S`, `Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the latter asserting the compatibility condition `(r • s) • a = r • (s • a)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/Tower.html"}, {"id": "Mathlib.Algebra.Module.Projective", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 3, "macro_tier_score": 0.2629, "macro_tier_override": null, "x": 61.286, "z": -0.038, "size": 0.4047, "title": "Projective modules", "summary": "This file contains a definition of a projective module, the proof that our definition is equivalent to a lifting property, and the proof that all free modules are projective.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Projective.html"}, {"id": "Mathlib.LinearAlgebra.Basis.Fin", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 3, "macro_tier_score": 0.2543, "macro_tier_override": null, "x": -28.667, "z": 41.141, "size": 0.2821, "title": "Bases indexed by `Fin`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Basis/Fin.html"}, {"id": "Mathlib.LinearAlgebra.Basis.Prod", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 3, "macro_tier_score": 0.267, "macro_tier_override": null, "x": -9.986, "z": 51.033, "size": 0.3908, "title": "Bases for the product of modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Basis/Prod.html"}, {"id": "Mathlib.LinearAlgebra.Basis.SMul", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 3, "macro_tier_score": 0.2545, "macro_tier_override": null, "x": 46.789, "z": 18.031, "size": 0.3083, "title": "Bases and scalar multiplication", "summary": "This file defines the scalar multiplication of bases by multiplying each basis vector.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Basis/SMul.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Notation", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 3, "macro_tier_score": 0.2617, "macro_tier_override": null, "x": -51.188, "z": 16.745, "size": 0.401, "title": "Matrix and vector notation", "summary": "This file includes `simp` lemmas for applying operations in `Data.Matrix.Basic` to values built out of the matrix notation `![a, b] = vecCons a (vecCons b vecEmpty)` defined in `Data.Fin.VecNotation`. This also provides the new notation `!![a, b; c, d] = Matrix.of ![![a, b], ![c, d]]`. This notation also works for empty matrices; `!![,,,] : Matrix (Fin 0) (Fin 3)` and `!![;;;] : Matrix (Fin 3) (Fin 0)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Notation.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.StdBasis", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 4, "macro_tier_score": 0.2753, "macro_tier_override": null, "x": -37.426, "z": -43.748, "size": 0.3607, "title": "Standard basis on matrices", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/StdBasis.html"}, {"id": "Mathlib.RingTheory.Ideal.Span", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 4, "macro_tier_score": 0.339, "macro_tier_override": null, "x": -12.919, "z": -42.659, "size": 0.4751, "title": "Ideals generated by a set of elements", "summary": "This file defines `Ideal.span s` as the ideal generated by the subset `s` of the ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Span.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Diagonal", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 1, "macro_tier_score": 0.0019, "macro_tier_override": null, "x": -72.82, "z": -32.781, "size": 0.3157, "title": "Diagonal matrices", "summary": "This file contains some results on the linear map corresponding to a diagonal matrix (`range`, `ker` and `rank`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Diagonal.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.LinearMap", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 1, "macro_tier_score": 0.002, "macro_tier_override": null, "x": -77.102, "z": 11.807, "size": 0.3259, "title": "The rank of a linear map", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/LinearMap.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.Exact", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 2, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -10.076, "z": 4.757, "size": 0.5528, "title": "Exact short complexes", "summary": "When `S : ShortComplex C`, this file defines a structure `S.Exact` which expresses the exactness of `S`, i.e. there exists a homology data `h : S.HomologyData` such that `h.left.H` is zero. When `[S.HasHomology]`, it is equivalent to the assertion `IsZero S.homology`. Almost by construction, this notion of exactness is self dual, see `Exact.op` and `Exact.unop`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/Exact.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.SnakeLemma", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 2, "macro_tier_score": 0.0198, "macro_tier_override": null, "x": 14.846, "z": -0.58, "size": 0.3275, "title": "The snake lemma", "summary": "The snake lemma is a standard tool in homological algebra. The basic situation is when we have a diagram as follows in an abelian category `C`, with exact rows: L₁.X₁ ⟶ L₁.X₂ ⟶ L₁.X₃ ⟶ 0 | | | |v₁₂.τ₁ |v₁₂.τ₂ |v₁₂.τ₃ v v v 0 ⟶ L₂.X₁ ⟶ L₂.X₂ ⟶ L₂.X₃ We shall think of this diagram as the datum of a morphism `v₁₂ : L₁ ⟶ L₂` in the category `ShortComplex C` such that both `L₁` and `L₂` are exact, and `L₁.g` is epi and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.html"}, {"id": "Mathlib.GroupTheory.Torsion", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 3, "macro_tier_score": 0.0814, "macro_tier_override": null, "x": -43.249, "z": -21.473, "size": 0.2946, "title": "Torsion groups", "summary": "This file defines torsion groups, i.e. groups where all elements have finite order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Torsion.html"}, {"id": "Mathlib.GroupTheory.PGroup", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 3, "macro_tier_score": 0.0959, "macro_tier_override": null, "x": 44.629, "z": -12.802, "size": 0.3003, "title": "p-groups", "summary": "This file contains a proof that if `G` is a `p`-group acting on a finite set `α`, then the number of fixed points of the action is congruent mod `p` to the cardinality of `α`. It also contains proofs of some corollaries of this lemma about existence of fixed points.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/PGroup.html"}, {"id": "Mathlib.GroupTheory.Rank", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.0914, "macro_tier_override": null, "x": -32.812, "z": -12.98, "size": 0.2904, "title": "Rank of a group", "summary": "This file defines the rank of a group, namely the minimum size of a generating set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Rank.html"}, {"id": "Mathlib.Algebra.Lie.Free", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -70.773, "z": -32.791, "size": 0.239, "title": "Free Lie algebras", "summary": "Given a commutative ring `R` and a type `X` we construct the free Lie algebra on `X` with coefficients in `R` together with its universal property.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Free.html"}, {"id": "Mathlib.Algebra.FreeNonUnitalNonAssocAlgebra", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -17.805, "z": 46.876, "size": 0.2382, "title": "Free algebras", "summary": "Given a semiring `R` and a type `X`, we construct the free non-unital, non-associative algebra on `X` with coefficients in `R`, together with its universal property. The construction is valuable because it can be used to build free algebras with more structure, e.g., free Lie algebras. Note that elsewhere we have a construction of the free unital, associative algebra. This is called `FreeAlgebra`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/FreeNonUnitalNonAssocAlgebra.html"}, {"id": "Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -44.552, "z": -1.322, "size": 0.2739, "title": "Lie algebras as non-unital, non-associative algebras", "summary": "The definition of Lie algebras uses the `Bracket` typeclass for multiplication whereas we have a separate `Mul` typeclass used for general algebras. It is useful to have a special typeclass for Lie algebras because: * it enables us to use the traditional notation `⁅x, y⁆` for the Lie multiplication, * associative algebras carry a natural Lie algebra structure via the ring commutator and so we need them to carry both…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/NonUnitalNonAssocAlgebra.html"}, {"id": "Mathlib.Algebra.Lie.UniversalEnveloping", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -73.557, "z": -19.68, "size": 0.2382, "title": "Universal enveloping algebra", "summary": "Given a commutative ring `R` and a Lie algebra `L` over `R`, we construct the universal enveloping algebra of `L`, together with its universal property.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/UniversalEnveloping.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.PreservesHomology", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 2, "macro_tier_score": 0.026, "macro_tier_override": null, "x": 9.133, "z": 1.679, "size": 0.3714, "title": "Functors which preserves homology", "summary": "If `F : C ⥤ D` is a functor between categories with zero morphisms, we shall say that `F` preserves homology when `F` preserves both kernels and cokernels. This typeclass is named `[F.PreservesHomology]`, and is automatically satisfied when `F` preserves both finite limits and finite colimits. If `S : ShortComplex C` and `[F.PreservesHomology]`, then there is an isomorphism `S.mapHomologyIso F : (S.map F).homology ≅…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.QuasiIso", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 2, "macro_tier_score": 0.0254, "macro_tier_override": null, "x": 7.427, "z": -0.169, "size": 0.3376, "title": "Quasi-isomorphisms of short complexes", "summary": "This file introduces the typeclass `QuasiIso φ` for a morphism `φ : S₁ ⟶ S₂` of short complexes (which have homology): the condition is that the induced morphism `homologyMap φ` in homology is an isomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.html"}, {"id": "Mathlib.NumberTheory.Primorial", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 1.645, "z": -0.862, "size": 0.2385, "title": "Primorial", "summary": "This file defines the primorial function (the product of primes less than or equal to some bound), and proves that `primorial n ≤ 4 ^ n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Primorial.html"}, {"id": "Mathlib.NumberTheory.PrimeCounting", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0103, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2805, "title": "The Prime Counting Function", "summary": "In this file we define the prime counting function: the function on natural numbers that returns the number of primes less than or equal to its input.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/PrimeCounting.html"}, {"id": "Mathlib.Algebra.Order.Ring.Pow", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.1098, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.3434, "title": "Bernoulli's inequality", "summary": "In this file we prove several versions of Bernoulli's inequality. Besides the standard version `1 + n * a ≤ (1 + a) ^ n`, we also prove `a ^ n + n * a ^ (n - 1) * b ≤ (a + b) ^ n`, which can be regarded as Bernoulli's inequality for `b / a` multiplied by `a ^ n`. Also, we prove versions for different typeclass assumptions on the (semi)ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Pow.html"}, {"id": "Mathlib.Algebra.GCDMonoid.FinsetLemmas", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 2, "macro_tier_score": 0.008, "macro_tier_override": null, "x": 12.125, "z": -27.128, "size": 0.2606, "title": "`Finset.lcm` lemmas", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GCDMonoid/FinsetLemmas.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Polynomial", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 2, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": -41.086, "z": -52.744, "size": 0.2746, "title": "Some lemmas relating polynomials and multivariable polynomials.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Polynomial.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Division", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 2, "macro_tier_score": 0.0069, "macro_tier_override": null, "x": -53.263, "z": 43.415, "size": 0.2684, "title": "Division of `MvPolynomial` by monomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Division.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.Division", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 3, "macro_tier_score": 0.2144, "macro_tier_override": null, "x": -7.073, "z": -17.172, "size": 0.288, "title": "Division of `AddMonoidAlgebra` by monomials", "summary": "This file is most important for when `G = ℕ` (polynomials) or `G = σ →₀ ℕ` (multivariate polynomials). In order to apply in maximal generality (such as for `LaurentPolynomial`s), this uses `∃ d, g' = g + d` in many places instead of `g ≤ g'`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/Division.html"}, {"id": "Mathlib.Algebra.Module.LinearMap.Defs", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.4169, "macro_tier_override": null, "x": 9.767, "z": -22.079, "size": 0.7829, "title": "(Semi)linear maps", "summary": "In this file we define * `LinearMap σ M M₂`, `M →ₛₗ[σ] M₂` : a semilinear map between two `Module`s. Here, `σ` is a `RingHom` from `R` to `R₂` and an `f : M →ₛₗ[σ] M₂` satisfies `f (c • x) = (σ c) • (f x)`. We recover plain linear maps by choosing `σ` to be `RingHom.id R`. This is denoted by `M →ₗ[R] M₂`. We also add the notation `M →ₗ⋆[R] M₂` for star-linear maps. * `IsLinearMap R f` : predicate saying that `f : M…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LinearMap/Defs.html"}, {"id": "Mathlib.Algebra.Module.Pi", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.3921, "macro_tier_override": null, "x": 10.929, "z": 17.26, "size": 0.5042, "title": "Pi instances for modules", "summary": "This file defines instances for module and related structures on Pi Types", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Pi.html"}, {"id": "Mathlib.Algebra.Field.Rat", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.2792, "macro_tier_override": null, "x": 4.183, "z": -10.328, "size": 0.3916, "title": "The rational numbers form a field", "summary": "This file contains the field instance on the rational numbers. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/Rat.html"}, {"id": "Mathlib.Algebra.Group.Commute.Basic", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 3, "macro_tier_score": 0.2641, "macro_tier_override": null, "x": 1.07, "z": 9.224, "size": 0.3546, "title": "Additional lemmas about commuting pairs of elements in monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Commute/Basic.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4125, "macro_tier_override": null, "x": -11.649, "z": -14.464, "size": 0.4494, "title": "Further lemmas about units in a `MonoidWithZero` or a `GroupWithZero`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Units/Lemmas.html"}, {"id": "Mathlib.RingTheory.WittVector.Compare", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -103.096, "z": 51.278, "size": 0.2, "title": "Comparison isomorphism between `WittVector p (ZMod p)` and `ℤ_[p]`", "summary": "We construct a ring isomorphism between `WittVector p (ZMod p)` and `ℤ_[p]`. This isomorphism follows from the fact that both satisfy the universal property of the inverse limit of `ZMod (p^n)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/Compare.html"}, {"id": "Mathlib.RingTheory.WittVector.Truncated", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": -105.025, "z": 42.469, "size": 0.2862, "title": "Truncated Witt vectors", "summary": "The ring of truncated Witt vectors (of length `n`) is a quotient of the ring of Witt vectors. It retains the first `n` coefficients of each Witt vector. In this file, we set up the basic quotient API for this ring. The ring of Witt vectors is the projective limit of all the rings of truncated Witt vectors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/Truncated.html"}, {"id": "Mathlib.RingTheory.WittVector.Identities", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": 93.802, "z": -63.522, "size": 0.3023, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/Identities.html"}, {"id": "Mathlib.NumberTheory.Padics.RingHoms", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": -54.584, "z": -55.719, "size": 0.2725, "title": "Relating `ℤ_[p]` to `ZMod (p ^ n)`, aka `ℤ/p^nℤ`.", "summary": "In this file we establish connections between the `p`-adic integers `ℤ_[p]` and the integers modulo powers of `p`, `ℤ/p^nℤ`, implemented as `ZMod (p^n)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/RingHoms.html"}, {"id": "Mathlib.Algebra.Quaternion", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 1, "macro_tier_score": 0.0037, "macro_tier_override": null, "x": -11.977, "z": -67.663, "size": 0.2855, "title": "Quaternions", "summary": "In this file we define quaternions `ℍ[R]` over a commutative ring `R`, and define some algebraic structures on `ℍ[R]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Quaternion.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.2584, "macro_tier_override": null, "x": 21.973, "z": 63.144, "size": 0.4048, "title": "Lemmas about rank and `finrank` in rings satisfying strong rank condition.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/StrongRankCondition.html"}, {"id": "Mathlib.Algebra.Module.Torsion.Prod", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": -22.285, "z": -0.196, "size": 0.2518, "title": "Product of torsion-free modules", "summary": "This file shows that the product of two torsion-free modules is torsion-free.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Torsion/Prod.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Subalgebra", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 2, "macro_tier_score": 0.0067, "macro_tier_override": null, "x": 30.255, "z": -57.53, "size": 0.2245, "title": "Some results on tensor product of subalgebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Subalgebra.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Submodule", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 2, "macro_tier_score": 0.0113, "macro_tier_override": null, "x": 58.956, "z": 7.487, "size": 0.2666, "title": "Some results on tensor product of submodules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Submodule.html"}, {"id": "Mathlib.RingTheory.TensorProduct.Maps", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.2482, "macro_tier_override": null, "x": 0.722, "z": 63.139, "size": 0.5125, "title": "Maps between tensor products of R-algebras", "summary": "This file provides results about maps between tensor products of `R`-algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/Maps.html"}, {"id": "Mathlib.Algebra.Algebra.Pi", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 4, "macro_tier_score": 0.3298, "macro_tier_override": null, "x": -38.473, "z": -25.991, "size": 0.6117, "title": "The R-algebra structure on families of R-algebras", "summary": "The R-algebra structure on `Π i : I, A i` when each `A i` is an R-algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Pi.html"}, {"id": "Mathlib.LinearAlgebra.Pi", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 4, "macro_tier_score": 0.3583, "macro_tier_override": null, "x": -48.195, "z": -2.965, "size": 0.6034, "title": "Pi types of modules", "summary": "This file defines constructors for linear maps whose domains or codomains are pi types. It contains theorems relating these to each other, as well as to `LinearMap.ker`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Pi.html"}, {"id": "Mathlib.GroupTheory.DedekindFinite", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": 3.59, "z": -12.494, "size": 0.317, "title": "Finite monoids are Dedekind-finite", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/DedekindFinite.html"}, {"id": "Mathlib.Algebra.Polynomial.Lifts", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.1791, "macro_tier_override": null, "x": 53.19, "z": 43.504, "size": 0.301, "title": "Polynomials that lift", "summary": "Given semirings `R` and `S` with a morphism `f : R →+* S`, we define a subsemiring `lifts` of `S[X]` by the image of `RingHom.of (map f)`. Then, we prove that a polynomial that lifts can always be lifted to a polynomial of the same degree and that a monic polynomial that lifts can be lifted to a monic polynomial (of the same degree).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Lifts.html"}, {"id": "Mathlib.Algebra.Polynomial.Eval.Subring", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.1787, "macro_tier_override": null, "x": 54.221, "z": -19.354, "size": 0.2663, "title": "Evaluation of polynomials in subrings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Eval/Subring.html"}, {"id": "Mathlib.RingTheory.PowerSeries.PiTopology", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 54.978, "z": -44.247, "size": 0.2842, "title": "Product topology on power series", "summary": "Let `R` be with `Semiring R` and `TopologicalSpace R` In this file we define the topology on `PowerSeries σ R` that corresponds to the simple convergence on its coefficients. It is the coarsest topology for which all coefficients maps are continuous. When `R` has `UniformSpace R`, we define the corresponding uniform structure. This topology can be included by writing `open scoped PowerSeries.WithPiTopology`. When…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/PiTopology.html"}, {"id": "Mathlib.GroupTheory.FreeGroup.CyclicallyReduced", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 27.591, "z": 3.847, "size": 0.2, "title": null, "summary": "This file defines some extra lemmas for free groups, in particular about cyclically reduced words. We show that free groups are (strongly) torsion-free in the sense of `IsMulTorsionFree`, i.e., taking powers by every non-zero element `n : ℕ` is injective.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FreeGroup/CyclicallyReduced.html"}, {"id": "Mathlib.Algebra.Module.Prod", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.3909, "macro_tier_override": null, "x": 9.003, "z": -18.338, "size": 0.4292, "title": "Prod instances for module and multiplicative actions", "summary": "This file defines instances for binary product of modules", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Prod.html"}, {"id": "Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 26.892, "z": -102.385, "size": 0.2, "title": "Lemmas of Gauss and Eisenstein", "summary": "This file contains the Lemmas of Gauss and Eisenstein on the Legendre symbol. The main results are `ZMod.gauss_lemma` and `ZMod.eisenstein_lemma`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.html"}, {"id": "Mathlib.NumberTheory.LegendreSymbol.Basic", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 2, "macro_tier_score": 0.0062, "macro_tier_override": null, "x": 54.173, "z": -88.778, "size": 0.3115, "title": "Legendre symbol", "summary": "This file contains results about Legendre symbols. We define the Legendre symbol $\\Bigl(\\frac{a}{p}\\Bigr)$ as `legendreSym p a`. Note the order of arguments! The advantage of this form is that then `legendreSym p` is a multiplicative map. The Legendre symbol is used to define the Jacobi symbol, `jacobiSym a b`, for integers `a` and (odd) natural numbers `b`, which extends the Legendre symbol.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LegendreSymbol/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.Basic", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 3, "macro_tier_score": 0.2588, "macro_tier_override": null, "x": -49.93, "z": 4.622, "size": 0.3694, "title": "Dimension of modules and vector spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/Basic.html"}, {"id": "Mathlib.RingTheory.RingHom.LocallyStandardSmooth", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -125.437, "z": -26.2, "size": 0.2473, "title": "Smooth is locally standard smooth", "summary": "In this file we show that a ring homomorphism is smooth if and only if it is locally standard smooth.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/LocallyStandardSmooth.html"}, {"id": "Mathlib.RingTheory.RingHom.Locally", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": -74.446, "z": 37.975, "size": 0.2956, "title": "Target local closure of ring homomorphism properties", "summary": "If `P` is a property of ring homomorphisms, we call `Locally P` the closure of `P` with respect to standard open coverings on the (algebraic) target (i.e. geometric source). Hence for `f : R →+* S`, the property `Locally P` holds if it holds locally on `S`, i.e. if there exists a subset `{ t }` of `S` generating the unit ideal, such that `P` holds for all compositions `R →+* Sₜ`. Assuming without further mention…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/Locally.html"}, {"id": "Mathlib.RingTheory.RingHom.StandardSmooth", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": 36.064, "z": -121.028, "size": 0.2714, "title": "Standard smooth ring homomorphisms", "summary": "In this file we define standard smooth ring homomorphisms and show their meta properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/StandardSmooth.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -16.275, "z": -57.157, "size": 0.2619, "title": "Kaplansky criterion for factoriality", "summary": "We prove Kaplansky criterion for factoriality: an integral domain is a UFD if and only if every nonzero prime ideal contains a prime element.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/Kaplansky.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.Basic", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 3, "macro_tier_score": 0.2156, "macro_tier_override": null, "x": -13.674, "z": -24.27, "size": 0.3709, "title": "Basic results on unique factorization monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/Basic.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.Ideal", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 4, "macro_tier_score": 0.2944, "macro_tier_override": null, "x": 56.553, "z": -10.784, "size": 0.3047, "title": "Unique factorization and ascending chain condition on ideals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/Ideal.html"}, {"id": "Mathlib.Algebra.Order.Floor.Semiring", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 2, "macro_tier_score": 0.0064, "macro_tier_override": null, "x": -0.999, "z": 29.698, "size": 0.3251, "title": "Lemmas on `Nat.floor` and `Nat.ceil` for semirings", "summary": "This file contains basic results on the natural-valued floor and ceiling functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Floor/Semiring.html"}, {"id": "Mathlib.Algebra.FreeMonoid.Symbols", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -9.343, "z": 9.039, "size": 0.2, "title": "The finite set of symbols in a FreeMonoid element", "summary": "This is separated from the main FreeMonoid file, as it imports the finiteness hierarchy", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/FreeMonoid/Symbols.html"}, {"id": "Mathlib.Algebra.FreeMonoid.Basic", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4482, "macro_tier_override": null, "x": -6.021, "z": -9.376, "size": 0.3596, "title": "Free monoid over a given alphabet", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/FreeMonoid/Basic.html"}, {"id": "Mathlib.Algebra.Group.Pointwise.Set.Lattice", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.428, "macro_tier_override": null, "x": -3.321, "z": -18.272, "size": 0.3436, "title": "Indexed unions and intersections of pointwise operations of sets", "summary": "This file contains lemmas on taking the union and intersection over pointwise algebraic operations on sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pointwise/Set/Lattice.html"}, {"id": "Mathlib.Algebra.Regular.Pi", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 6.718, "z": 15.305, "size": 0.2, "title": "Results about `IsRegular` and pi types", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Regular/Pi.html"}, {"id": "Mathlib.Algebra.Regular.SMul", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4441, "macro_tier_override": null, "x": 13.346, "z": 6.529, "size": 0.5025, "title": "Action of regular elements on a module", "summary": "We introduce `M`-regular elements, in the context of an `R`-module `M`. The corresponding predicate is called `IsSMulRegular`. There are very limited typeclass assumptions on `R` and `M`, but the \"mathematical\" case of interest is a commutative ring `R` acting on a module `M`. Since the properties are \"multiplicative\", there is no actual requirement of having an addition, but there is a zero in both `R` and `M`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Regular/SMul.html"}, {"id": "Mathlib.LinearAlgebra.PiTensorProduct.DFinsupp", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 34.126, "z": 44.04, "size": 0.2761, "title": "Tensor products of finitely supported functions", "summary": "This file shows that taking `PiTensorProduct`s commutes with taking `DFinsupp`s in all arguments.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/PiTensorProduct/DFinsupp.html"}, {"id": "Mathlib.LinearAlgebra.PiTensorProduct.Basic", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 2, "macro_tier_score": 0.0104, "macro_tier_override": null, "x": 3.67, "z": 53.733, "size": 0.3644, "title": "Tensor product of an indexed family of modules over commutative semirings", "summary": "We define the tensor product of an indexed family `s : ι → Type*` of modules over commutative semirings. We denote this space by `⨂[R] i, s i` and define it as `FreeAddMonoid (R × Π i, s i)` quotiented by the appropriate equivalence relation. The treatment follows very closely that of the binary tensor product in `Mathlib/LinearAlgebra/TensorProduct/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/PiTensorProduct/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Multilinear.DFinsupp", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 3, "macro_tier_score": 0.2094, "macro_tier_override": null, "x": -17.42, "z": -50.963, "size": 0.3319, "title": "Interactions between finitely-supported functions and multilinear maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Multilinear/DFinsupp.html"}, {"id": "Mathlib.Algebra.Group.Int.Defs", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.4718, "macro_tier_override": null, "x": -0.943, "z": 5.491, "size": 0.5889, "title": "The integers form a group", "summary": "This file contains the additive group and multiplicative monoid instances on the integers. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Int/Defs.html"}, {"id": "Mathlib.Algebra.Order.Group.Unbundled.Basic", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4284, "macro_tier_override": null, "x": -7.685, "z": 10.485, "size": 0.4964, "title": "Ordered groups", "summary": "This file develops the basics of unbundled ordered groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Unbundled/Basic.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 71.881, "z": 42.632, "size": 0.3545, "title": "Finite-dimensional subspaces of affine spaces.", "summary": "This file provides a few results relating to finite-dimensional subspaces of affine spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.html"}, {"id": "Mathlib.RingTheory.Finiteness.Finsupp", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 4, "macro_tier_score": 0.2926, "macro_tier_override": null, "x": 47.251, "z": 25.846, "size": 0.333, "title": "Finiteness of (sub)modules and finitely supported functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Finsupp.html"}, {"id": "Mathlib.Algebra.FreeAbelianGroup.Finsupp", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 4, "macro_tier_score": 0.3047, "macro_tier_override": null, "x": 14.011, "z": -32.385, "size": 0.327, "title": "Isomorphism between `FreeAbelianGroup X` and `X →₀ ℤ`", "summary": "In this file we construct the canonical isomorphism between `FreeAbelianGroup X` and `X →₀ ℤ`. We use this to transport the notion of `support` from `Finsupp` to `FreeAbelianGroup`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/FreeAbelianGroup/Finsupp.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.Module", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 4, "macro_tier_score": 0.3311, "macro_tier_override": null, "x": 18.206, "z": 42.71, "size": 0.4707, "title": "Module structure on monoid algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/Module.html"}, {"id": "Mathlib.Algebra.Order.Module.Field", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 1, "macro_tier_score": 0.0019, "macro_tier_override": null, "x": 19.197, "z": -14.641, "size": 0.3214, "title": "Ordered vector spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Module/Field.html"}, {"id": "Mathlib.Algebra.QuadraticAlgebra.Defs", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 17.01, "z": 66.576, "size": 0.2465, "title": "Quadratic Algebra", "summary": "In this file we define the quadratic algebra `QuadraticAlgebra R a b` over a commutative ring `R`, and define some algebraic structures on it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/QuadraticAlgebra/Defs.html"}, {"id": "Mathlib.Algebra.Colimit.TensorProduct", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.0513, "macro_tier_override": null, "x": 6.036, "z": -59.122, "size": 0.2726, "title": "Tensor product with direct limit of finitely generated submodules", "summary": "We show that if `M` and `P` are arbitrary modules and `N` is a finitely generated submodule of a module `P`, then two elements of `N ⊗ M` have the same image in `P ⊗ M` if and only if they already have the same image in `N' ⊗ M` for some finitely generated submodule `N' ≥ N`. This is the theorem `Submodule.FG.exists_rTensor_fg_inclusion_eq`. The key facts used are that every module is the direct limit of its…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Colimit/TensorProduct.html"}, {"id": "Mathlib.Algebra.Colimit.Finiteness", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.0524, "macro_tier_override": null, "x": 52.995, "z": -22.495, "size": 0.2671, "title": "Modules as direct limits of finitely generated submodules", "summary": "We show that every module is the direct limit of its finitely generated submodules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Colimit/Finiteness.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.DirectLimit", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.0536, "macro_tier_override": null, "x": -6.004, "z": 57.258, "size": 0.2797, "title": "Tensor product and direct limits commute with each other.", "summary": "Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic as `R`-modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/DirectLimit.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.DualNumber", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -4.959, "z": -61.085, "size": 0.2676, "title": "Matrices of dual numbers are isomorphic to dual numbers over matrices", "summary": "Showing this for the more general case of `TrivSqZeroExt R M` would require an action between `Matrix n n R` and `Matrix n n M`, which would risk causing diamonds.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/DualNumber.html"}, {"id": "Mathlib.RingTheory.Flat.CategoryTheory", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 23.912, "z": 78.138, "size": 0.2338, "title": "Tensoring with a flat module is an exact functor", "summary": "In this file we prove that tensoring with a flat module is an exact functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/CategoryTheory.html"}, {"id": "Mathlib.RingTheory.Flat.Basic", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 3, "macro_tier_score": 0.0535, "macro_tier_override": null, "x": 13.12, "z": -78.773, "size": 0.4111, "title": "Flat modules", "summary": "A module `M` over a commutative semiring `R` is *mono-flat* if for all monomorphisms of modules (i.e., injective linear maps) `N →ₗ[R] P`, the canonical map `N ⊗ M → P ⊗ M` is injective (cf. [Katsov2004], [KatsovNam2011]). To show a module is mono-flat, it suffices to check inclusions of finitely generated submodules `N` into finitely generated modules `P`, and `P` can be further assumed to lie in the same universe…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/Basic.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.ModuleCat", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 2, "macro_tier_score": 0.0199, "macro_tier_override": null, "x": -20.19, "z": -63.736, "size": 0.3991, "title": "Homology and exactness of short complexes of modules", "summary": "In this file, the homology of a short complex `S` of abelian groups is identified with the quotient of `LinearMap.ker S.g` by the image of the morphism `S.moduleCatToCycles : S.X₁ →ₗ[R] LinearMap.ker S.g` induced by `S.f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/ModuleCat.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.0093, "macro_tier_override": null, "x": 54.814, "z": -17.607, "size": 0.2891, "title": "The monoidal closed structure on `Module R`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Monoidal/Closed.html"}, {"id": "Mathlib.GroupTheory.Sylow", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 3, "macro_tier_score": 0.094, "macro_tier_override": null, "x": -3.203, "z": -48.18, "size": 0.3266, "title": "Sylow theorems", "summary": "The Sylow theorems are the following results for every finite group `G` and every prime number `p`. * There exists a Sylow `p`-subgroup of `G`. * All Sylow `p`-subgroups of `G` are conjugate to each other. * Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow `p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow `p`-subgroup in `G`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Sylow.html"}, {"id": "Mathlib.RingTheory.SimpleRing.Field", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 51.542, "z": 15.623, "size": 0.2417, "title": "Simple ring and fields", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SimpleRing/Field.html"}, {"id": "Mathlib.RingTheory.SimpleRing.Basic", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 4, "macro_tier_score": 0.3318, "macro_tier_override": null, "x": 37.205, "z": 36.33, "size": 0.4913, "title": "Basic Properties of Simple rings", "summary": "A ring `R` is **simple** if it has only two two-sided ideals, namely `⊥` and `⊤`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SimpleRing/Basic.html"}, {"id": "Mathlib.Algebra.Field.Equiv", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 3, "macro_tier_score": 0.2298, "macro_tier_override": null, "x": 16.422, "z": -3.112, "size": 0.3583, "title": "If a semiring is a field, any isomorphic semiring is also a field.", "summary": "This is in a separate file to avoid needing to import `Field` in `Mathlib/Algebra/Ring/Equiv.lean`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/Equiv.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basis", "region_id": "algebra", "micro_elevation": 0.9211, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -50.234, "z": -119.904, "size": 0.2, "title": "The basis obtained from Geck's construction of Lie algebras from root systems", "summary": "The Geck construction of a Lie algebra associated to a root system, `RootPairing.GeckConstruction.lieAlgebra`, yields a simple Lie algebra with distinguished basis which we construct here.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/GeckConstruction/Basis.html"}, {"id": "Mathlib.Algebra.Lie.Basis", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -123.955, "z": 24.157, "size": 0.2338, "title": "Bases of semisimple Lie algebras", "summary": "In this file we define bases of semisimple Lie algebras. Given an indexing type `ι`, a basis of a Lie algebra consists of a non-degenerate matrix of integers `A` indexed by `ι × ι` and generators `h i`, `e i`, `f i` indexed by `ι`, each forming an `sl₂` triple, and satisfying the Chevalley-Serre relations: * `⁅h i, h j⁆ = 0` * `⁅h j, e i⁆ = A i j • e i` * `⁅h j, f i⁆ = -A i j • f i` * `⁅e i, f j⁆ = 0` (for `i ≠ j`)…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Basis.html"}, {"id": "Mathlib.Algebra.Lie.CartanCriterion", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -50.372, "z": 107.657, "size": 0.2516, "title": "Cartan's criteria", "summary": "The two **Cartan criteria** characterise solvability and semisimplicity of finite-dimensional Lie algebras over fields of characteristic zero in terms of the Killing form: solvability via its vanishing on `L × ⁅L, L⁆`, semisimplicity via its non-degeneracy.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/CartanCriterion.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -128.116, "z": 2.709, "size": 0.2338, "title": "Geck's construction of a Lie algebra associated to a root system yields semisimple algebras", "summary": "This file contains a proof that `RootPairing.GeckConstruction.lieAlgebra` yields semisimple Lie algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/GeckConstruction/Semisimple.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -85.652, "z": -95.313, "size": 0.2338, "title": "Relations in Geck's construction of a Lie algebra associated to a root system", "summary": "This file contains proofs that `RootPairing.GeckConstruction.lieAlgebra` contains `sl₂` triples satisfying relations associated to the Cartan matrix of the input root system.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/GeckConstruction/Relations.html"}, {"id": "Mathlib.NumberTheory.NumberField.InfiniteAdeleRing", "region_id": "algebra", "micro_elevation": 0.9342, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -101.81, "z": 83.793, "size": 0.2478, "title": "The infinite adele ring of a number field", "summary": "This file contains the formalisation of the infinite adele ring of a number field as the finite product of completions over its infinite places.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/InfiniteAdeleRing.html"}, {"id": "Mathlib.Algebra.Group.Pi.Units", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 2, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -4.584, "z": 10.156, "size": 0.3114, "title": "Units in pi types", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pi/Units.html"}, {"id": "Mathlib.NumberTheory.NumberField.Completion.InfinitePlace", "region_id": "algebra", "micro_elevation": 0.9211, "macro_tier": 1, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": -108.827, "z": -71.113, "size": 0.315, "title": "The completion of a number field at an infinite place", "summary": "This file contains the completion of a number field at an infinite place. This is ultimately achieved by applying the `UniformSpace.Completion` functor, however each infinite place induces its own `UniformSpace` instance on the number field, so the inference system cannot automatically infer these. A common approach to handle the ambiguity that arises from having multiple sources of instances is through the use of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Completion/InfinitePlace.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Reindex", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.1982, "macro_tier_override": null, "x": 38.141, "z": -50.323, "size": 0.3276, "title": "Changing the index type of a matrix", "summary": "This file concerns the map `Matrix.reindex`, mapping a `m` by `n` matrix to an `m'` by `n'` matrix, as long as `m ≃ m'` and `n ≃ n'`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Reindex.html"}, {"id": "Mathlib.Algebra.SkewPolynomial.Basic", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -25.518, "z": 49.528, "size": 0.2, "title": "Univariate skew polynomials", "summary": "Given a ring `R` and an endomorphism `φ` on `R` the skew polynomials over `R` are polynomials $$\\sum_{i= 0}^n a_iX^n, n\\geq 0, a_i\\in R$$ where the addition is the usual addition of polynomials $$\\sum_{i= 0}^n a_iX^n + \\sum_{i= 0}^n b_iX^n= \\sum_{i= 0}^n (a_i + b_i)X^n.$$ The multiplication, however, is determined by $$Xa = \\varphi (a)X$$ by extending it to all polynomials in the obvious way. Skew polynomials are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/SkewPolynomial/Basic.html"}, {"id": "Mathlib.Algebra.SkewMonoidAlgebra.Single", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -52.033, "z": -13.901, "size": 0.2478, "title": "Modifying skew monoid algebra at exactly one point", "summary": "This file contains basic results on updating/erasing an element of a skew monoid algebra using one point of the domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/SkewMonoidAlgebra/Single.html"}, {"id": "Mathlib.Algebra.SkewMonoidAlgebra.Support", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -53.842, "z": 1.308, "size": 0.2478, "title": "Lemmas about the support of an element of a skew monoid algebra", "summary": "For `f : SkewMonoidAlgebra k G`, `f.support` is the set of all `a ∈ G` such that `f.coeff a ≠ 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/SkewMonoidAlgebra/Support.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.Ker", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.458, "macro_tier_override": null, "x": -16.38, "z": -15.111, "size": 0.55, "title": "Kernel and range of group homomorphisms", "summary": "We define and prove results about the kernel and range of group homomorphisms. Special thanks goes to Amelia Livingston and Yury Kudryashov for their help and inspiration.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/Ker.html"}, {"id": "Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 27.726, "z": -96.378, "size": 0.2442, "title": "Functoriality of group cohomology", "summary": "Given a commutative ring `k`, a group homomorphism `f : G →* H`, a `k`-linear `H`-representation `A`, a `k`-linear `G`-representation `B`, and a representation morphism `Res(f)(A) ⟶ B`, we get a cochain map `inhomogeneousCochains A ⟶ inhomogeneousCochains B` and hence maps on cohomology `Hⁿ(H, A) ⟶ Hⁿ(G, B)`. We also provide extra API for these maps in degrees 0, 1, 2.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.html"}, {"id": "Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": 11.47, "z": 97.759, "size": 0.2972, "title": "The low-degree cohomology of a `k`-linear `G`-representation", "summary": "Let `k` be a commutative ring and `G` a group. This file contains specialised API for the cocycles and group cohomology of a `k`-linear `G`-representation `A` in degrees 0, 1 and 2. In `Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean`, we define the `n`th group cohomology of `A` to be the cohomology of a complex `inhomogeneousCochains A`, whose objects are `(Fin n → G) → A`; this is unnecessarily…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.html"}, {"id": "Mathlib.NumberTheory.Transcendental.Liouville.Residual", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 84.945, "z": -9.088, "size": 0.2338, "title": "Density of Liouville numbers", "summary": "In this file we prove that the set of Liouville numbers form a dense `Gδ` set. We also prove a similar statement about irrational numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Transcendental/Liouville/Residual.html"}, {"id": "Mathlib.NumberTheory.Transcendental.Liouville.Basic", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": 31.491, "z": -77.412, "size": 0.2908, "title": "Liouville's theorem", "summary": "This file contains a proof of Liouville's theorem stating that all Liouville numbers are transcendental. To obtain this result, there is first a proof that Liouville numbers are irrational and two technical lemmas. These lemmas exploit the fact that a polynomial with integer coefficients takes integer values at integers. When evaluating at a rational number, we can clear denominators and obtain precise inequalities…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Transcendental/Liouville/Basic.html"}, {"id": "Mathlib.Algebra.Polynomial.Mirror", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -66.835, "z": 1.73, "size": 0.2493, "title": "\"Mirror\" of a univariate polynomial", "summary": "In this file we define `Polynomial.mirror`, a variant of `Polynomial.reverse`. The difference between `reverse` and `mirror` is that `reverse` will decrease the degree if the polynomial is divisible by `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Mirror.html"}, {"id": "Mathlib.Algebra.Polynomial.Reverse", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.2703, "macro_tier_override": null, "x": -59.896, "z": 25.249, "size": 0.3919, "title": "Reverse of a univariate polynomial", "summary": "The main definition is `reverse`. Applying `reverse` to a polynomial `f : R[X]` produces the polynomial with a reversed list of coefficients, equivalent to `X^f.natDegree * f(1/X)`. The main result is that `reverse (f * g) = reverse f * reverse g`, provided the leading coefficients of `f` and `g` do not multiply to zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Reverse.html"}, {"id": "Mathlib.Algebra.NeZero", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.4785, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.5348, "title": "`NeZero` typeclass", "summary": "We give basic facts about the `NeZero n` typeclass.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/NeZero.html"}, {"id": "Mathlib.Algebra.Category.Grp.Kernels", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 2, "macro_tier_score": 0.0191, "macro_tier_override": null, "x": 27.777, "z": 10.553, "size": 0.2634, "title": "The concrete (co)kernels in the category of abelian groups are categorical (co)kernels.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Kernels.html"}, {"id": "Mathlib.Algebra.Category.Grp.EpiMono", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 2, "macro_tier_score": 0.0195, "macro_tier_override": null, "x": -6.361, "z": -27.121, "size": 0.3072, "title": "Monomorphisms and epimorphisms in `Group`", "summary": "In this file, we prove monomorphisms in the category of groups are injective homomorphisms and epimorphisms are surjective homomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/EpiMono.html"}, {"id": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2874, "title": "Summability of Eisenstein series", "summary": "We gather results about the summability of Eisenstein series, particularly the summability of the Eisenstein series summands, which are used in the proof of the boundedness of Eisenstein series at infinity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/EisensteinSeries/Summable.html"}, {"id": "Mathlib.NumberTheory.Divisors", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 3, "macro_tier_score": 0.2443, "macro_tier_override": null, "x": -23.528, "z": -11.066, "size": 0.3635, "title": "Divisor Finsets", "summary": "This file defines sets of divisors of a natural number. This is particularly useful as background for defining Dirichlet convolution.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Divisors.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Invertible", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 3, "macro_tier_score": 0.2005, "macro_tier_override": null, "x": 12.979, "z": 50.355, "size": 0.3269, "title": "Extra lemmas about invertible matrices", "summary": "A few of the `Invertible` lemmas generalize to multiplication of rectangular matrices. For lemmas about the matrix inverse in terms of the determinant and adjugate, see `Matrix.inv` in `Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Invertible.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.ConjTranspose", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 3, "macro_tier_score": 0.2627, "macro_tier_override": null, "x": 9.824, "z": -49.172, "size": 0.4431, "title": "Matrices over star rings.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/ConjTranspose.html"}, {"id": "Mathlib.RingTheory.Perfectoid.FontaineTheta", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -28.837, "z": 117.221, "size": 0.2676, "title": "Fontaine's θ map", "summary": "In this file, we define Fontaine's `θ` map, which is a ring homomorphism from the Witt vector `𝕎 R♭` of the tilt of a perfectoid ring `R` to `R` itself. Our definition of `θ` does not require that `R` is perfectoid in the first place. We only need `R` to be `p`-adically complete.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Perfectoid/FontaineTheta.html"}, {"id": "Mathlib.RingTheory.AdicCompletion.Functoriality", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 2, "macro_tier_score": 0.0092, "macro_tier_override": null, "x": 55.453, "z": 40.58, "size": 0.2762, "title": "Functoriality of adic completions", "summary": "In this file we establish functorial properties of the adic completion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AdicCompletion/Functoriality.html"}, {"id": "Mathlib.RingTheory.AdicCompletion.RingHom", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -16.864, "z": 66.613, "size": 0.2459, "title": "Lift of ring homomorphisms to adic completions", "summary": "Let `R`, `S` be rings, `I` be an ideal of `S`. In this file we prove that a compatible family of ring homomorphisms from a ring `R` to `S ⧸ I ^ n` can be lifted to a ring homomorphism `R →+* AdicCompletion I S`. If `S` is `I`-adically complete, then this compatible family of ring homomorphisms can be lifted to a ring homomorphism `R →+* S`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AdicCompletion/RingHom.html"}, {"id": "Mathlib.RingTheory.Perfectoid.Untilt", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 47.133, "z": -92.708, "size": 0.2459, "title": "Untilt Function", "summary": "In this file, we define the untilt function from the pretilt of a `p`-adically complete ring to the ring itself. Note that this is not the untilt *functor*.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Perfectoid/Untilt.html"}, {"id": "Mathlib.RingTheory.WittVector.TeichmullerSeries", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 42.823, "z": 110.876, "size": 0.2459, "title": "Teichmuller Series", "summary": "Let `R` be a characteristic `p` perfect ring. In this file, we show that every element `x` of the Witt vectors `𝕎 R` can be written as the (`p`-adic) summation of Teichmuller series, namely `∑ i, (teichmuller p (((frobeniusEquiv R p).symm ^ i) (x.coeff i)) * p ^ i)`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/TeichmullerSeries.html"}, {"id": "Mathlib.RingTheory.Unramified.Pi", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": 82.672, "z": -56.769, "size": 0.2468, "title": "Formal-unramification of finite products of rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Unramified/Pi.html"}, {"id": "Mathlib.RingTheory.Unramified.Basic", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 2, "macro_tier_score": 0.0213, "macro_tier_override": null, "x": 78.668, "z": -59.159, "size": 0.3537, "title": "Unramified morphisms", "summary": "An `R`-algebra `A` is formally unramified if `Ω[A⁄R]` is trivial. This is equivalent to the standard definition \"for every `R`-algebra, every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`\". It is unramified if it is formally unramified and of finite type. Note that there are multiple definitions in the literature. The definition we give is equivalent to the one in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Unramified/Basic.html"}, {"id": "Mathlib.RingTheory.FiniteLength", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.067, "macro_tier_override": null, "x": 66.819, "z": -22.707, "size": 0.2942, "title": "Modules of finite length", "summary": "We define modules of finite length (`IsFiniteLength`) to be finite iterated extensions of simple modules, and show that a module is of finite length iff it is both Noetherian and Artinian, iff it admits a composition series. We do not make `IsFiniteLength` a class, instead we use `[IsNoetherian R M] [IsArtinian R M]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/FiniteLength.html"}, {"id": "Mathlib.RingTheory.Artinian.Module", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.0923, "macro_tier_override": null, "x": 41.521, "z": -54.751, "size": 0.4555, "title": "Artinian rings and modules", "summary": "A module satisfying these equivalent conditions is said to be an *Artinian* R-module if every decreasing chain of submodules is eventually constant, or equivalently, if the relation `<` on submodules is well founded. A ring is said to be left (or right) Artinian if it is Artinian as a left (or right) module over itself, or simply Artinian if it is both left and right Artinian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Artinian/Module.html"}, {"id": "Mathlib.Algebra.Order.Sub.Defs", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4359, "macro_tier_override": null, "x": 10.739, "z": -2.975, "size": 0.6275, "title": "Ordered Subtraction", "summary": "This file proves lemmas relating (truncated) subtraction with an order. We provide a class `OrderedSub` stating that `a - b ≤ c ↔ a ≤ c + b`. The subtraction discussed here could both be normal subtraction in an additive group or truncated subtraction on a canonically ordered monoid (`ℕ`, `Multiset`, `ENNReal`, ...)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Sub/Defs.html"}, {"id": "Mathlib.Algebra.Ring.Shrink", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 3, "macro_tier_score": 0.0727, "macro_tier_override": null, "x": 18.57, "z": 0.279, "size": 0.3087, "title": "Transfer ring structures from `α` to `Shrink α`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Shrink.html"}, {"id": "Mathlib.Algebra.Group.Shrink", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.3615, "macro_tier_override": null, "x": -5.628, "z": 13.75, "size": 0.4013, "title": "Transfer group structures from `α` to `Shrink α`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Shrink.html"}, {"id": "Mathlib.Algebra.Ring.TransferInstance", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 3, "macro_tier_score": 0.0804, "macro_tier_override": null, "x": 16.456, "z": -2.929, "size": 0.3753, "title": "Transfer algebraic structures across `Equiv`s", "summary": "This continues the pattern set in `Mathlib/Algebra/Group/TransferInstance.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/TransferInstance.html"}, {"id": "Mathlib.RingTheory.Localization.Pi", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -19.619, "z": 96.454, "size": 0.2, "title": "Localizing a product of commutative rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Pi.html"}, {"id": "Mathlib.Algebra.Divisibility.Prod", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": 13.014, "z": -10.489, "size": 0.2583, "title": "Lemmas about the divisibility relation in product (semi)groups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Divisibility/Prod.html"}, {"id": "Mathlib.Algebra.Order.Star.Basic", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 2, "macro_tier_score": 0.0263, "macro_tier_override": null, "x": -8.996, "z": 26.365, "size": 0.4326, "title": "Star ordered rings", "summary": "We define the class `StarOrderedRing R`, which says that the order on `R` respects the star operation, i.e. an element `r` is nonnegative iff it is in the `AddSubmonoid` generated by elements of the form `star s * s`. In many cases, including all C⋆-algebras, this can be reduced to `0 ≤ r ↔ ∃ s, r = star s * s`. However, this generality is slightly more convenient (e.g., it allows us to register a `StarOrderedRing`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Star/Basic.html"}, {"id": "Mathlib.Algebra.Order.Group.Opposite", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 2, "macro_tier_score": 0.0238, "macro_tier_override": null, "x": -6.74, "z": -11.117, "size": 0.2933, "title": "Order instances for `MulOpposite`/`AddOpposite`", "summary": "This file transfers order instances and ordered monoid/group instances from `α` to `αᵐᵒᵖ` and `αᵃᵒᵖ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Opposite.html"}, {"id": "Mathlib.Algebra.Star.StarRingHom", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 3, "macro_tier_score": 0.2693, "macro_tier_override": null, "x": 21.539, "z": 5.722, "size": 0.3371, "title": "Morphisms of star rings", "summary": "This file defines a new type of morphism between (non-unital) rings `A` and `B` where both `A` and `B` are equipped with a `star` operation. This morphism, namely `NonUnitalStarRingHom`, is a direct extension of its non-`star`red counterpart with a field `map_star` which guarantees it preserves the star operation. As with `NonUnitalRingHom`, the multiplications are not assumed to be associative or unital.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/StarRingHom.html"}, {"id": "Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -12.654, "z": -10.92, "size": 0.2403, "title": "Equivalence of Recursive and Direct Computations of Convergents of Generalized Continued Fractions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.html"}, {"id": "Mathlib.NumberTheory.Height.Northcott", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 22.239, "z": -1.451, "size": 0.2302, "title": "Results on the Northcott property for heights", "summary": "Assume that `K` is a field with a family of admissible absolute values that satisfies the Northcott property for `mulHeight₁`. We provide instances showing that `K` also satisfies the Northcott property * for `logHeight₁`, * (TODO) for `Projectivization.mulHeight`, * (TODO) for `Projectivization.logHeight`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Height/Northcott.html"}, {"id": "Mathlib.NumberTheory.Height.Basic", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": 11.682, "z": -16.759, "size": 0.2926, "title": "Basic theory of heights", "summary": "This is an attempt at formalizing some basic properties of height functions. We aim at a level of generality that allows to apply the theory to algebraic number fields and to function fields (and possibly beyond). The general set-up for heights is the following. Let `K` be a field. * We have a `Multiset` of archimedean absolute values on `K` (with values in `ℝ`). * We also have a `Set` of non-archimedean (i.e.,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Height/Basic.html"}, {"id": "Mathlib.Algebra.Ring.Parity", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4261, "macro_tier_override": null, "x": -2.889, "z": -18.346, "size": 0.6312, "title": "Even and odd elements in rings", "summary": "This file defines odd elements and proves some general facts about even and odd elements of rings. As opposed to `Even`, `Odd` does not have a multiplicative counterpart.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Parity.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Block", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 3, "macro_tier_score": 0.1859, "macro_tier_override": null, "x": -43.787, "z": 64.551, "size": 0.3152, "title": "Block matrices and their determinant", "summary": "This file defines a predicate `Matrix.BlockTriangular` saying a matrix is block triangular, and proves the value of the determinant for various matrices built out of blocks.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Block.html"}, {"id": "Mathlib.Algebra.Module.Torsion.Field", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.3844, "macro_tier_override": null, "x": -6.867, "z": -19.24, "size": 0.3852, "title": "Vector spaces are torsion-free", "summary": "In this file, we show that any module over a division semiring is torsion-free. Note that more generally any reflexive module is torsion-free.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Torsion/Field.html"}, {"id": "Mathlib.Algebra.Category.Grp.LeftExactFunctor", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 29.132, "z": 5.855, "size": 0.2462, "title": "The forgetful functor `(C ⥤ₗ AddCommGroup) ⥤ (C ⥤ₗ Type v)` is an equivalence", "summary": "This is true as long as `C` is additive. Here, `C ⥤ₗ D` is the category of finite-limits-preserving functors from `C` to `D`. To construct a functor from `C ⥤ₗ Type v` to `C ⥤ₗ AddCommGrpCat.{v}`, notice that a left-exact functor `F : C ⥤ Type v` induces a functor `CommGrp C ⥤ CommGrp (Type v)`. But `CommGrp C` is equivalent to `C`, and `CommGrp (Type v)` is equivalent to `AddCommGrpCat.{v}`, so we turn this into a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/LeftExactFunctor.html"}, {"id": "Mathlib.RingTheory.SimpleModule.IsAlgClosed", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -73.653, "z": 76.034, "size": 0.2, "title": "Wedderburn–Artin Theorem over an algebraically closed field", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SimpleModule/IsAlgClosed.html"}, {"id": "Mathlib.RingTheory.SimpleModule.WedderburnArtin", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 24.679, "z": 68.095, "size": 0.2478, "title": "Wedderburn–Artin Theorem", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SimpleModule/WedderburnArtin.html"}, {"id": "Mathlib.RingTheory.Flat.Tensor", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 2, "macro_tier_score": 0.0282, "macro_tier_override": null, "x": -33.419, "z": -74.569, "size": 0.2909, "title": "Flat modules", "summary": "`M` is flat if `· ⊗ M` preserves finite limits (equivalently, pullbacks, or equalizers). If `R` is a ring, an `R`-module `M` is flat if and only if it is mono-flat, and to show a module is flat, it suffices to check inclusions of finitely generated ideals into `R`. See .", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/Tensor.html"}, {"id": "Mathlib.Algebra.Module.CharacterModule", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 2, "macro_tier_score": 0.0306, "macro_tier_override": null, "x": -17.865, "z": 62.497, "size": 0.31, "title": "Character module of a module", "summary": "For commutative ring `R` and an `R`-module `M` and an injective module `D`, its character module `M⋆` is defined to be `R`-linear maps `M ⟶ D`. `M⋆` also has an `R`-module structure given by `(r • f) m = f (r • m)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/CharacterModule.html"}, {"id": "Mathlib.RingTheory.Finiteness.ModuleFinitePresentation", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -60.277, "z": -75.451, "size": 0.2276, "title": "Finitely presented algebras and finitely presented modules", "summary": "In this file we establish relations between finitely presented as an algebra and finitely presented as a module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/ModuleFinitePresentation.html"}, {"id": "Mathlib.Algebra.Module.FinitePresentation", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.0359, "macro_tier_override": null, "x": -14.422, "z": 67.184, "size": 0.3661, "title": "Finitely Presented Modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/FinitePresentation.html"}, {"id": "Mathlib.Algebra.Star.Conjneg", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 8.464, "z": 22.611, "size": 0.239, "title": "Conjugation-negation operator", "summary": "This file defines the conjugation-negation operator, useful in Fourier analysis. The way this operator enters the picture is that the adjoint of convolution with a function `f` is convolution with `conjneg f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/Conjneg.html"}, {"id": "Mathlib.RingTheory.Extension.Presentation.Core", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": 18.385, "z": 89.124, "size": 0.2483, "title": "Presentations on subrings", "summary": "In this file we establish the API for realising a presentation over a subring of `R`. We define a property `HasCoeffs R₀` for a presentation `P` to mean the (sub)ring `R₀` contains the coefficients of the relations of `P`. subring `R₀` of `R` that contains the coefficients of the relations In this case there exists a model of `S` over `R₀`, i.e., there exists an `R₀`-algebra `S₀` such that `S` is isomorphic to `R…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Extension/Presentation/Core.html"}, {"id": "Mathlib.RingTheory.Extension.Presentation.Submersive", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 2, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": -52.4, "z": 72.117, "size": 0.3343, "title": "Submersive presentations", "summary": "In this file we define `PreSubmersivePresentation`. This is a presentation `P` that has fewer relations than generators. More precisely there exists an injective map from `σ` to `ι`. To such a presentation we may associate a Jacobian. `P` is then a submersive presentation, if its Jacobian is invertible. Algebras that admit such a presentation are called standard smooth. See `Mathlib.RingTheory.Smooth.StandardSmooth`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Extension/Presentation/Submersive.html"}, {"id": "Mathlib.RepresentationTheory.Homological.GroupCohomology.Shapiro", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 77.292, "z": 63.902, "size": 0.2, "title": "Shapiro's lemma for group cohomology", "summary": "Given a commutative ring `k` and a subgroup `S ≤ G`, the file `Mathlib/RepresentationTheory/Coinduced.lean` proves that the functor `Coind_S^G : Rep k S ⥤ Rep k G` preserves epimorphisms. Since `Res(S) : Rep k G ⥤ Rep k S` is left adjoint to `Coind_S^G`, this means `Res(S)` preserves projective objects. Since `Res(S)` is also exact, given a projective resolution `P` of `k` as a trivial `k`-linear `G`-representation,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/GroupCohomology/Shapiro.html"}, {"id": "Mathlib.RepresentationTheory.Coinduced", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 3.272, "z": 96.517, "size": 0.2676, "title": "Coinduced representations", "summary": "Given a commutative ring `k`, a monoid homomorphism `φ : G →* H`, and a `k`-linear `G`-representation `A`, this file introduces the coinduced representation $Coind_G^H(A)$ of `A` as an `H`-representation. By `coind φ A` we mean the submodule of functions `H → A` such that for all `g : G`, `h : H`, `f (φ g * h) = ρ g (f h)`. We define a representation of `H` on this submodule by sending `h : H` and `f : coind φ A` to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Coinduced.html"}, {"id": "Mathlib.RepresentationTheory.Induced", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -54.368, "z": -82.052, "size": 0.2676, "title": "Induced representations", "summary": "Given a commutative ring `k`, a group homomorphism `φ : G →* H`, and a `k`-linear `G`-representation `A`, this file introduces the induced representation $Ind_G^H(A)$ of `A` as an `H`-representation. By `ind φ A` we mean the `(k[H] ⊗[k] A)_G` with the `G`-representation on `k[H]` defined by `φ`. We define a representation of `H` on this submodule by sending `h : H` and `⟦h₁ ⊗ₜ a⟧` to `⟦h₁h⁻¹ ⊗ₜ a⟧`. We also prove…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Induced.html"}, {"id": "Mathlib.Algebra.Order.Field.Rat", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 13.795, "z": -24.202, "size": 0.3508, "title": "The rational numbers form a linear ordered field", "summary": "This file used to contain the linear ordered field instance on the rational numbers. TODO: rename this file to `Mathlib/Algebra/Order/GroupWithZero/NNRat.lean` See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Field/Rat.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": -71.841, "z": 94.69, "size": 0.3, "title": "Structural lemmas about finite crystallographic root pairings", "summary": "In this file we gather basic lemmas necessary for the classification of finite crystallographic root pairings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/Finite/Lemmas.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.Reduced", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0027, "macro_tier_override": null, "x": 69.939, "z": -79.463, "size": 0.2947, "title": "Reduced root pairings", "summary": "This file contains basic definitions and results related to reduced root pairings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/Reduced.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.Irreducible", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": 18.246, "z": 115.57, "size": 0.2806, "title": "Irreducible root pairings", "summary": "This file contains basic definitions and results about irreducible root systems.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/Irreducible.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.Retract", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 2, "macro_tier_score": 0.0064, "macro_tier_override": null, "x": 9.223, "z": -1.075, "size": 0.3276, "title": "Quasi-isomorphisms of short complexes are stable under retracts", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/Retract.html"}, {"id": "Mathlib.Algebra.Field.Subfield.Defs", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 3, "macro_tier_score": 0.2323, "macro_tier_override": null, "x": 21.251, "z": 14.98, "size": 0.4241, "title": "Subfields", "summary": "Let `K` be a division ring, for example a field. This file defines the \"bundled\" subfield type `Subfield K`, a type whose terms correspond to subfields of `K`. Note we do not require the \"subfields\" to be commutative, so they are really sub-division rings / skew fields. This is the preferred way to talk about subfields in mathlib. Unbundled subfields (`s : Set K` and `IsSubfield s`) are not in this file, and they…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/Subfield/Defs.html"}, {"id": "Mathlib.LinearAlgebra.LinearIndependent.Lemmas", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 4, "macro_tier_score": 0.3276, "macro_tier_override": null, "x": -30.828, "z": -39.547, "size": 0.4638, "title": "Linear independence", "summary": "This file collects consequences of linear (in)dependence and includes specialized tests for specific families of vectors, requiring more theory to state.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/LinearIndependent/Lemmas.html"}, {"id": "Mathlib.LinearAlgebra.Dual.Defs", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 4, "macro_tier_score": 0.3248, "macro_tier_override": null, "x": 30.312, "z": 32.678, "size": 0.3369, "title": "Dual vector spaces", "summary": "The dual space of an $R$-module $M$ is the $R$-module of $R$-linear maps $M \\to R$.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dual/Defs.html"}, {"id": "Mathlib.LinearAlgebra.Finsupp.SumProd", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 4, "macro_tier_score": 0.3248, "macro_tier_override": null, "x": -9.255, "z": 26.275, "size": 0.3408, "title": "`Finsupp`s and sum/product types", "summary": "This file contains results about modules involving `Finsupp` and sum/product/sigma types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Finsupp/SumProd.html"}, {"id": "Mathlib.LinearAlgebra.LinearIndependent.Basic", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 4, "macro_tier_score": 0.3315, "macro_tier_override": null, "x": -33.912, "z": 31.711, "size": 0.4014, "title": "Linear independence", "summary": "This file collects basic consequences of linear (in)dependence and includes specialized tests for specific families of vectors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/LinearIndependent/Basic.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 16.015, "z": -64.911, "size": 0.2719, "title": "The associated sheaf of a presheaf of modules", "summary": "In this file, given a presheaf of modules `M₀` over a presheaf of rings `R₀`, we construct the associated sheaf of `M₀`. More precisely, if `R` is a sheaf of rings and `α : R₀ ⟶ R.val` is locally bijective, and `A` is the sheafification of the underlying presheaf of abelian groups of `M₀`, i.e. we have a locally bijective map `φ : M₀.presheaf ⟶ A.val`, then we endow `A` with the structure of a sheaf of modules over…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Sheaf.ChangeOfRings", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": 62.495, "z": -17.875, "size": 0.2944, "title": "Change of sheaf of rings", "summary": "In this file, we define the restriction of scalars functor `restrictScalars α : SheafOfModules.{v} R' ⥤ SheafOfModules.{v} R` attached to a morphism of sheaves of rings `α : R ⟶ R'`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Sheaf/ChangeOfRings.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.OrderIso", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 2, "macro_tier_score": 0.0175, "macro_tier_override": null, "x": -17.21, "z": 6.981, "size": 0.3278, "title": "Multiplication by a positive element as an order isomorphism", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/OrderIso.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.Basic", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4216, "macro_tier_override": null, "x": -14.698, "z": 2.172, "size": 0.4121, "title": "Lemmas on the monotone multiplication typeclasses", "summary": "This file builds on `Mathlib/Algebra/Order/GroupWithZero/Unbundled/Defs.lean` by proving several lemmas that do not immediately follow from the typeclass specifications.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/Basic.html"}, {"id": "Mathlib.Algebra.BigOperators.Finsupp.Fin", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 3, "macro_tier_score": 0.269, "macro_tier_override": null, "x": -25.948, "z": -1.647, "size": 0.3155, "title": "`Finsupp.sum` and `Finsupp.prod` over `Fin`", "summary": "This file contains theorems relevant to big operators on finitely supported functions over `Fin`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Finsupp/Fin.html"}, {"id": "Mathlib.Algebra.BigOperators.Finsupp.Basic", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.3303, "macro_tier_override": null, "x": -16.199, "z": -17.902, "size": 0.4789, "title": "Big operators for finsupps", "summary": "This file contains theorems relevant to big operators in finitely supported functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Finsupp/Basic.html"}, {"id": "Mathlib.Algebra.Order.CauSeq.Completion", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 0, "macro_tier_score": 0.001, "macro_tier_override": null, "x": 8.29, "z": 30.464, "size": 0.3363, "title": "Cauchy completion", "summary": "This file generalizes the Cauchy completion of `(ℚ, abs)` to the completion of a ring with absolute value.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/CauSeq/Completion.html"}, {"id": "Mathlib.Algebra.Order.Ring.Rat", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 2, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": 23.989, "z": -10.028, "size": 0.3888, "title": "The rational numbers form a linear ordered field", "summary": "This file constructs the order on `ℚ` and proves that `ℚ` is a discrete, linearly ordered commutative ring. `ℚ` is in fact a linearly ordered field, but this fact is located in `Data.Rat.Field` instead of here because we need the order on `ℚ` to define `ℚ≥0`, which we itself need to define `Field`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Rat.html"}, {"id": "Mathlib.Algebra.Group.Subsemigroup.Membership", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.3848, "macro_tier_override": null, "x": 0.843, "z": 11.111, "size": 0.2354, "title": "Subsemigroups: membership criteria", "summary": "In this file we prove various facts about membership in a subsemigroup. The intent is to mimic `GroupTheory/Submonoid/Membership`, but currently this file is mostly a stub and only provides rudimentary support. * `mem_iSup_of_directed`, `coe_iSup_of_directed`, `mem_sSup_of_directed_on`, `coe_sSup_of_directed_on`: the supremum of a directed collection of subsemigroup is their union.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subsemigroup/Membership.html"}, {"id": "Mathlib.Algebra.Group.Subsemigroup.Basic", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4636, "macro_tier_override": null, "x": 3.819, "z": 8.464, "size": 0.4074, "title": "Subsemigroups: `CompleteLattice` structure", "summary": "This file defines a `CompleteLattice` structure on `Subsemigroup`s, and define the closure of a set as the minimal subsemigroup that includes this set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subsemigroup/Basic.html"}, {"id": "Mathlib.RingTheory.OrzechProperty", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 4, "macro_tier_score": 0.2754, "macro_tier_override": null, "x": -32.408, "z": 49.815, "size": 0.2944, "title": "Orzech property of rings", "summary": "In this file we define the following property of rings: - `OrzechProperty R` is a type class stating that `R` satisfies the following property: for any finitely generated `R`-module `M`, any surjective homomorphism `f : N → M` from a submodule `N` of `M` to `M` is injective. It was introduced in papers by Orzech [orzech1971], Djoković [djokovic1973] and Ribenboim [ribenboim1971], under the names `Π`-ring or…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/OrzechProperty.html"}, {"id": "Mathlib.RingTheory.Finiteness.Cardinality", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 4, "macro_tier_score": 0.2971, "macro_tier_override": null, "x": 57.014, "z": 7.999, "size": 0.4431, "title": "Finite modules and types with finitely many elements", "summary": "This file relates `Module.Finite` and `_root_.Finite`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Cardinality.html"}, {"id": "Mathlib.RingTheory.RingHom.OpenImmersion", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -77.717, "z": 30.73, "size": 0.2704, "title": "Standard Open Immersion", "summary": "We define the property `RingHom.IsStandardOpenImmersion` on ring homomorphisms: it means that the morphism is a localization map away from some element. We also define the equivalent `Algebra.IsStandardOpenImmersion`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/OpenImmersion.html"}, {"id": "Mathlib.RingTheory.LocalProperties.Basic", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 3, "macro_tier_score": 0.0723, "macro_tier_override": null, "x": -12.123, "z": -80.811, "size": 0.4896, "title": "Local properties of commutative rings", "summary": "In this file, we define local properties in general.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalProperties/Basic.html"}, {"id": "Mathlib.RingTheory.Finiteness.Projective", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.1848, "macro_tier_override": null, "x": 57.551, "z": 25.98, "size": 0.3146, "title": "Finite and projective modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Projective.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.TruncGE", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 2, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 25.381, "z": -5.641, "size": 0.2991, "title": "The canonical truncation", "summary": "Given an embedding `e : Embedding c c'` of complex shapes which satisfies `e.IsTruncGE` and `K : HomologicalComplex C c'`, we define `K.truncGE' e : HomologicalComplex C c` and `K.truncGE e : HomologicalComplex C c'` which are the canonical truncations of `K` relative to `e`. For example, if `e` is the embedding `embeddingUpIntGE p` of `ComplexShape.up ℕ` in `ComplexShape.up ℤ` which sends `n : ℕ` to `p + n` and `K…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/TruncGE.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -56.261, "z": 42.605, "size": 0.268, "title": "Colimits in categories of sheaves of modules", "summary": "In this file, we show that colimits of shape `J` exist in a category of sheaves of modules if it exists in the corresponding category of presheaves of modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Sheaf/Colimits.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 1, "macro_tier_score": 0.0022, "macro_tier_override": null, "x": 57.983, "z": 36.874, "size": 0.3381, "title": "The sheafification functor for presheaves of modules", "summary": "In this file, we construct a functor `PresheafOfModules.sheafification α : PresheafOfModules R₀ ⥤ SheafOfModules R` for a locally bijective morphism `α : R₀ ⟶ R.val` where `R₀` is a presheaf of rings and `R` a sheaf of rings. In particular, if `α` is the identity of `R.val`, we obtain the sheafification functor `PresheafOfModules R.val ⥤ SheafOfModules R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafification.html"}, {"id": "Mathlib.LinearAlgebra.Complex.Orientation", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -59.177, "z": 64.164, "size": 0.2404, "title": "The standard orientation on `ℂ`.", "summary": "This had previously been in `LinearAlgebra.Orientation`, but keeping it separate results in a significant import reduction.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Complex/Orientation.html"}, {"id": "Mathlib.LinearAlgebra.Complex.Module", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 2, "macro_tier_score": 0.0108, "macro_tier_override": null, "x": 64.993, "z": 0.973, "size": 0.3853, "title": "Complex number as a vector space over `ℝ`", "summary": "This file contains the following instances: * Any `•`-structure (`SMul`, `MulAction`, `DistribMulAction`, `Module`, `Algebra`) on `ℝ` imbues a corresponding structure on `ℂ`. This includes the statement that `ℂ` is an `ℝ` algebra. * any complex vector space is a real vector space; * any finite-dimensional complex vector space is a finite-dimensional real vector space; * the space of `ℝ`-linear maps from a real…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Complex/Module.html"}, {"id": "Mathlib.LinearAlgebra.Orientation", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": -24.945, "z": -81.706, "size": 0.2913, "title": "Orientations of modules", "summary": "This file defines orientations of modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Orientation.html"}, {"id": "Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": 48.227, "z": 100.453, "size": 0.2732, "title": "Rings with finite quotients", "summary": "A commutative ring is said to have finite quotients if, for any nonzero ideal `I` of `R`, the quotient `R ⧸ I` is finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Quotient/HasFiniteQuotients.html"}, {"id": "Mathlib.LinearAlgebra.Isomorphisms", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 4, "macro_tier_score": 0.2986, "macro_tier_override": null, "x": 51.968, "z": -1.844, "size": 0.4157, "title": "Isomorphism theorems for modules.", "summary": "* The Noether's first, second, and third isomorphism theorems for modules are proved as `LinearMap.quotKerEquivRange`, `LinearMap.quotientInfEquivSupQuotient` and `Submodule.quotientQuotientEquivQuotient`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Isomorphisms.html"}, {"id": "Mathlib.LinearAlgebra.Quotient.Basic", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 4, "macro_tier_score": 0.3324, "macro_tier_override": null, "x": -49.595, "z": 7.396, "size": 0.6206, "title": "Quotients by submodules", "summary": "* If `p` is a submodule of `M`, `M ⧸ p` is the quotient of `M` with respect to `p`: that is, elements of `M` are identified if their difference is in `p`. This is itself a module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Quotient/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Quotient.Card", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 4, "macro_tier_score": 0.2983, "macro_tier_override": null, "x": -2.044, "z": -35.227, "size": 0.3484, "title": null, "summary": "Results about the cardinality of a quotient module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Quotient/Card.html"}, {"id": "Mathlib.Algebra.Polynomial.Eval.Irreducible", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -31.122, "z": 52.796, "size": 0.2478, "title": "Mapping irreducible polynomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Eval/Irreducible.html"}, {"id": "Mathlib.Algebra.Polynomial.Eval.Degree", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 4, "macro_tier_score": 0.2883, "macro_tier_override": null, "x": 40.741, "z": 43.267, "size": 0.4829, "title": "Evaluation of polynomials and degrees", "summary": "This file contains results on the interaction of `Polynomial.eval` and `Polynomial.degree`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Eval/Degree.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -70.001, "z": -8.963, "size": 0.2, "title": "The category of sheaves of modules as a localization of presheaves of modules", "summary": "Similarly as the category of sheaves with values in a category identify to a localization of the category of presheaves with respect to those morphisms which become isomorphisms after sheafification (see the file `Mathlib/CategoryTheory/Sites/Localization.lean`), we show that the sheafification functor from presheaves of modules to sheaves of modules is a localization functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Sheaf/Localization.html"}, {"id": "Mathlib.RingTheory.Regular.IsSMulRegular", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": -19.828, "z": -79.273, "size": 0.2619, "title": "Lemmas about the `IsSMulRegular` Predicate", "summary": "For modules over a ring the proposition `IsSMulRegular r M` is equivalent to `r` being a *non-zero-divisor*, i.e. `r • x = 0` only if `x = 0` for `x ∈ M`. This specific result is `isSMulRegular_iff_smul_eq_zero_imp_eq_zero`. Lots of results starting from this, especially ones about quotients (which don't make sense without some algebraic assumptions), are in this file. We don't pollute the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Regular/IsSMulRegular.html"}, {"id": "Mathlib.Algebra.Module.Torsion.Basic", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.0829, "macro_tier_override": null, "x": -62.378, "z": -9.8, "size": 0.3891, "title": "Torsion submodules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Torsion/Basic.html"}, {"id": "Mathlib.RingTheory.Ideal.AssociatedPrime.Basic", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.0093, "macro_tier_override": null, "x": 54.156, "z": -45.25, "size": 0.2871, "title": "Associated primes of a module", "summary": "We provide the definition and related lemmas about associated primes of modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/AssociatedPrime/Basic.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.AreComplementary", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -26.744, "z": -16.78, "size": 0.2826, "title": "Complementary embeddings", "summary": "Given two embeddings `e₁ : c₁.Embedding c` and `e₂ : c₂.Embedding c` of complex shapes, we introduce a property `e₁.AreComplementary e₂` saying that the image subsets of the indices of `c₁` and `c₂` form a partition of the indices of `c`. If `e₁.IsTruncLE` and `e₂.IsTruncGE`, and `K : HomologicalComplex C c`, we construct a quasi-isomorphism `shortComplexTruncLEX₃ToTruncGE` between the cokernel of `K.ιTruncLE e₁ :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/AreComplementary.html"}, {"id": "Mathlib.Algebra.Group.Units.Equiv", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4791, "macro_tier_override": null, "x": 13.23, "z": -6.76, "size": 0.4872, "title": "Multiplicative and additive equivalence acting on units.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Units/Equiv.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Expand", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 81.192, "z": 19.802, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Expand.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.Expand", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 35.599, "z": 73.553, "size": 0.2478, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/Expand.html"}, {"id": "Mathlib.RepresentationTheory.Homological.GroupHomology.Shapiro", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -30.754, "z": 95.455, "size": 0.2, "title": "Shapiro's lemma for group homology", "summary": "Given a commutative ring `k` and a subgroup `S ≤ G`, the file `Mathlib/RepresentationTheory/Coinduced.lean` proves that the functor `Coind_S^G : Rep k S ⥤ Rep k G` preserves epimorphisms. Since `Res(S) : Rep k G ⥤ Rep k S` is left adjoint to `Coind_S^G`, this means `Res(S)` preserves projective objects. Since `Res(S)` is also exact, given a projective resolution `P` of `k` as a trivial `k`-linear `G`-representation,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/GroupHomology/Shapiro.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Commute", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.417, "macro_tier_override": null, "x": 7.704, "z": -14.833, "size": 0.4517, "title": "Lemmas about commuting elements in a `MonoidWithZero` or a `GroupWithZero`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Commute.html"}, {"id": "Mathlib.Algebra.Ring.Divisibility.Basic", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4235, "macro_tier_override": null, "x": -10.238, "z": -4.398, "size": 0.399, "title": "Lemmas about divisibility in rings", "summary": "Note that this file is imported by basic tactics like `linarith` and so must have only minimal imports. Further results about divisibility in rings may be found in `Mathlib/Algebra/Ring/Divisibility/Lemmas.lean` which is not subject to this import constraint.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Divisibility/Basic.html"}, {"id": "Mathlib.Algebra.Ring.Commute", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4215, "macro_tier_override": null, "x": 13.262, "z": 10.174, "size": 0.452, "title": "Semirings and rings", "summary": "This file gives lemmas about semirings, rings and domains. This is analogous to `Mathlib/Algebra/Group/Basic.lean`, the difference being that the former is about `+` and `*` separately, while the present file is about their interaction. For the definitions of semirings and rings see `Mathlib/Algebra/Ring/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Commute.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.List.Basic", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.452, "macro_tier_override": null, "x": 3.232, "z": 8.705, "size": 0.3865, "title": "Sums and products from lists", "summary": "This file provides basic results about `List.prod`, `List.sum`, which calculate the product and sum of elements of a list and `List.alternatingProd`, `List.alternatingSum`, their alternating counterparts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/List/Basic.html"}, {"id": "Mathlib.Algebra.QuadraticAlgebra.Basic", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 42.003, "z": 56.711, "size": 0.2478, "title": "Quadratic algebras: involution and norm.", "summary": "Let `R` be a commutative ring. We define: * `QuadraticAlgebra.star`: the quadratic involution * `QuadraticAlgebra.norm`: the norm We prove: * `QuadraticAlgebra.isUnit_iff_norm_isUnit`: `w : QuadraticAlgebra R a b` is a unit iff `w.norm` is a unit in `R`. * `QuadraticAlgebra.norm_mem_nonZero_divisors_iff`: `w : QuadraticAlgebra R a b` isn't a zero divisor iff `w.norm` isn't a zero divisor in `R`. * If `K` is a field,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/QuadraticAlgebra/Basic.html"}, {"id": "Mathlib.Algebra.Lie.SemiDirect", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 60.421, "z": -57.737, "size": 0.2, "title": "Semi-direct products", "summary": "This file defines the semi-direct sum of Lie algebras. These are the infinitesimal counterpart of semidirect products of (Lie) groups. Given two Lie algebras `K` and `L` over `R` as well as a Lie algebra homomorphism `ψ : L → LieDerivation R K K`, the underlying set of the semidirect sum is `K × L`, however the bracket is twisted by `ψ`. In this file we show that `SemiDirectSum K L ψ` is itself a Lie algebra and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/SemiDirect.html"}, {"id": "Mathlib.Algebra.Lie.Derivation.Basic", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 1, "macro_tier_score": 0.0027, "macro_tier_override": null, "x": -56.94, "z": 50.554, "size": 0.2938, "title": "Lie derivations", "summary": "This file defines *Lie derivations* and establishes some basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Derivation/Basic.html"}, {"id": "Mathlib.Algebra.Lie.Extension", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 55.069, "z": 60.372, "size": 0.239, "title": "Extensions of Lie algebras", "summary": "This file defines extensions of Lie algebras, given by short exact sequences of Lie algebra homomorphisms. They are implemented in two ways: `IsExtension` is a `Prop`-valued class taking two homomorphisms as parameters, and `Extension` is a structure that includes the middle Lie algebra. Because our sign convention for differentials is opposite that of Chevalley-Eilenberg, there is a change of signs in the \"action\"…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Extension.html"}, {"id": "Mathlib.Algebra.Lie.Prod", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 1, "macro_tier_score": 0.0037, "macro_tier_override": null, "x": -76.033, "z": -4.1, "size": 0.2805, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Prod.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 2, "macro_tier_score": 0.0063, "macro_tier_override": null, "x": 24.869, "z": 25.033, "size": 0.3838, "title": "The pretriangulated structure on the homotopy category of complexes", "summary": "In this file, we define the pretriangulated structure on the homotopy category `HomotopyCategory C (ComplexShape.up ℤ)` of an additive category `C`. The distinguished triangles are the triangles that are isomorphic to the image in the homotopy category of the standard triangle `K ⟶ L ⟶ mappingCone φ ⟶ K⟦(1 : ℤ)⟧` for some morphism of cochain complexes `φ : K ⟶ L`. This result first appeared in the Liquid Tensor…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/Pretriangulated.html"}, {"id": "Mathlib.RingTheory.RegularLocalRing.Defs", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -63.883, "z": -79.701, "size": 0.2, "title": "Regular local rings", "summary": "For a Noetherian local ring `R`, we define `IsRegularLocalRing` as `(maximalIdeal R).spanFinrank = ringKrullDim R`. This definition is equivalent to the dimension of the cotangent space over the residue field being equal to `ringKrullDim R`, (see `IsRegularLocalRing.iff_finrank_cotangentSpace`). For the next section, we define regular rings as Noetherian rings whose localization at every prime are regular local…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RegularLocalRing/Defs.html"}, {"id": "Mathlib.Algebra.Module.SpanRankOperations", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -87.076, "z": 6.059, "size": 0.2565, "title": "Span rank under operations", "summary": "In this file we show how operations on submodules interact with `Submodule.spanRank`. # Main Results * `Submodule.spanRank_baseChange_le`: Base change doesn't increase the span rank. * `TensorProduct.spanFinrank_top_eq_of_residueField`: For a finitely generated module over a local ring, the dimension of the base change to the residue field is equal to its span rank. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/SpanRankOperations.html"}, {"id": "Mathlib.RingTheory.Ideal.KrullsHeightTheorem", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 98.789, "z": -17.27, "size": 0.2954, "title": "Krull's Height Theorem", "summary": "In this file, we prove **Krull's principal ideal theorem** (also known as **Krullscher Hauptidealsatz**), and **Krull's height theorem** (also known as **Krullscher Höhensatz**).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.html"}, {"id": "Mathlib.RingTheory.KrullDimension.Field", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": 0.448, "z": -70.571, "size": 0.2687, "title": "The Krull dimension of a field", "summary": "This file proves that the Krull dimension of a field is zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/KrullDimension/Field.html"}, {"id": "Mathlib.RingTheory.KrullDimension.PID", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": -3.045, "z": -98.382, "size": 0.2658, "title": "The Krull dimension of a principal ideal domain", "summary": "In this file, we proved some results about the dimension of a principal ideal domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/KrullDimension/PID.html"}, {"id": "Mathlib.NumberTheory.LucasLehmer", "region_id": "algebra", "micro_elevation": 0.9474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 131.099, "z": -26.325, "size": 0.2, "title": "The Lucas-Lehmer test for Mersenne primes", "summary": "We define `lucasLehmerResidue : Π p : ℕ, ZMod (2^p - 1)`, and prove `lucasLehmerResidue p = 0 ↔ Prime (mersenne p)`. We construct a `norm_num` extension to calculate this residue to certify primality of Mersenne primes using `lucas_lehmer_sufficiency`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LucasLehmer.html"}, {"id": "Mathlib.NumberTheory.Fermat", "region_id": "algebra", "micro_elevation": 0.9342, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 120.246, "z": 54.107, "size": 0.2478, "title": "Fermat numbers", "summary": "The Fermat numbers are a sequence of natural numbers defined as `Nat.fermatNumber n = 2^(2^n) + 1`, for all natural numbers `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Fermat.html"}, {"id": "Mathlib.RingTheory.Fintype", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2642, "title": "Some facts about finite rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Fintype.html"}, {"id": "Mathlib.Algebra.Quandle", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -18.176, "z": 3.814, "size": 0.2, "title": "Racks and Quandles", "summary": "This file defines racks and quandles, algebraic structures for sets that bijectively act on themselves with a self-distributivity property. If `R` is a rack and `act : R → (R ≃ R)` is the self-action, then the self-distributivity is, equivalently, that ``` act (act x y) = act x * act y * (act x)⁻¹ ``` where multiplication is composition in `R ≃ R` as a group. Quandles are racks such that `act x x = x` for all `x`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Quandle.html"}, {"id": "Mathlib.Algebra.Algebra.Tower", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 4, "macro_tier_score": 0.3393, "macro_tier_override": null, "x": -35.97, "z": -26.322, "size": 0.7293, "title": "Towers of algebras", "summary": "In this file we prove basic facts about towers of algebras. An algebra tower A/S/R is expressed by having instances of `Algebra A S`, `Algebra R S`, `Algebra R A` and `IsScalarTower R S A`, the latter asserting the compatibility condition `(r • s) • a = r • (s • a)`. An important definition is `toAlgHom R S A`, the canonical `R`-algebra homomorphism `S →ₐ[R] A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Tower.html"}, {"id": "Mathlib.Algebra.Lie.Engel", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": 88.887, "z": 53.994, "size": 0.3058, "title": "Engel's theorem", "summary": "This file contains a proof of Engel's theorem providing necessary and sufficient conditions for Lie algebras and Lie modules to be nilpotent. The key result `LieModule.isNilpotent_iff_forall` says that if `M` is a Lie module of a Noetherian Lie algebra `L`, then `M` is nilpotent iff the image of `L → End(M)` consists of nilpotent elements. In the special case that we have the adjoint representation `M = L`, this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Engel.html"}, {"id": "Mathlib.Algebra.Lie.Nilpotent", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 1, "macro_tier_score": 0.0031, "macro_tier_override": null, "x": -35.111, "z": -79.913, "size": 0.3273, "title": "Nilpotent Lie algebras", "summary": "Like groups, Lie algebras admit a natural concept of nilpotency. More generally, any Lie module carries a natural concept of nilpotency. We define these here via the lower central series.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Nilpotent.html"}, {"id": "Mathlib.Algebra.Ring.Idempotent", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.3431, "macro_tier_override": null, "x": -6.176, "z": -13.513, "size": 0.4225, "title": "Idempotent elements of a ring", "summary": "This file proves result about idempotent elements of a ring, like: * `IsIdempotentElem.one_sub_iff`: In a (non-associative) ring, `a` is an idempotent if and only if `1 - a` is an idempotent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Idempotent.html"}, {"id": "Mathlib.Algebra.Field.Subfield.Basic", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 3, "macro_tier_score": 0.2288, "macro_tier_override": null, "x": -52.927, "z": 9.968, "size": 0.3682, "title": "Subfields", "summary": "Let `K` be a division ring, for example a field. This file concerns the \"bundled\" subfield type `Subfield K`, a type whose terms correspond to subfields of `K`. Note we do not require the \"subfields\" to be commutative, so they are really sub-division rings / skew fields. This is the preferred way to talk about subfields in mathlib. Unbundled subfields (`s : Set K` and `IsSubfield s`) are not in this file, and they…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/Subfield/Basic.html"}, {"id": "Mathlib.Algebra.Order.CompleteField", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 19.881, "z": 16.756, "size": 0.2814, "title": "Conditionally complete linear ordered fields", "summary": "This file shows that the reals are unique, or, more formally, given a type satisfying the common axioms of the reals (field, conditionally complete, linearly ordered) that there is an isomorphism preserving these properties to the reals. This is `ConditionallyCompleteLinearOrderedField.inducedOrderRingIso`. Moreover this isomorphism is unique. We show all conditionally complete linear ordered fields are archimedean.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/CompleteField.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.Defs", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4198, "macro_tier_override": null, "x": -0.416, "z": -9.277, "size": 0.3166, "title": "(Strict) monotonicity of multiplication by nonnegative (positive) elements", "summary": "This file defines eight typeclasses expressing monotonicity (strict monotonicity) of multiplication on the left or right by nonnegative (positive) elements in a preorder. For left multiplication (`a ↦ b * a`) we define the following typeclasses: * `PosMulMono`: If `b ≥ 0`, then `a₁ ≤ a₂ → b * a₁ ≤ b * a₂`. * `PosMulStrictMono`: If `b > 0`, then `a₁ < a₂ → b * a₁ < b * a₂`. * `PosMulReflectLT`: If `b ≥ 0`, then `b *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/Defs.html"}, {"id": "Mathlib.RingTheory.Ideal.Maximal", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 4, "macro_tier_score": 0.3308, "macro_tier_override": null, "x": 24.868, "z": -39.208, "size": 0.4967, "title": "Ideals over a ring", "summary": "This file contains an assortment of definitions and results for `Ideal R`, the type of (left) ideals over a ring `R`. Note that over commutative rings, left ideals and two-sided ideals are equivalent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Maximal.html"}, {"id": "Mathlib.RingTheory.Ideal.Prime", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 4, "macro_tier_score": 0.3298, "macro_tier_override": null, "x": 3.633, "z": 36.965, "size": 0.464, "title": "Prime ideals", "summary": "This file contains the definition of `Ideal.IsPrime` for prime ideals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Prime.html"}, {"id": "Mathlib.Algebra.ContinuedFractions.Determinant", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -2.944, "z": 16.453, "size": 0.2443, "title": "Determinant Formula for Generalized Continued Fraction", "summary": "We derive the so-called *determinant formula* for `GenContFract`: `Aₙ * Bₙ₊₁ - Bₙ * Aₙ₊₁ = (-a₀) * (-a₁) * .. * (-aₙ)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/ContinuedFractions/Determinant.html"}, {"id": "Mathlib.RingTheory.IntegralClosure.IntegrallyClosed", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 3, "macro_tier_score": 0.0748, "macro_tier_override": null, "x": -62.07, "z": -69.065, "size": 0.3709, "title": "Integrally closed rings", "summary": "An integrally closed ring `R` contains all the elements of `Frac(R)` that are integral over `R`. A special case of integrally closed rings are the Dedekind domains.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IntegralClosure/IntegrallyClosed.html"}, {"id": "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.0181, "macro_tier_override": null, "x": -32.127, "z": 73.11, "size": 0.2775, "title": "Eisenstein polynomials", "summary": "Given an ideal `𝓟` of a commutative semiring `R`, we say that a polynomial `f : R[X]` is *Eisenstein at `𝓟`* if `f.leadingCoeff ∉ 𝓟`, `∀ n, n < f.natDegree → f.coeff n ∈ 𝓟` and `f.coeff 0 ∉ 𝓟 ^ 2`. In this file we gather miscellaneous results about Eisenstein polynomials.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Eisenstein/Basic.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.HomologicalFunctor", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -22.964, "z": -31.523, "size": 0.2856, "title": "The homological functor", "summary": "In this file, it is shown that if `C` is an abelian category, then `homologyFunctor C (ComplexShape.up ℤ) n` is a homological functor `HomotopyCategory C (ComplexShape.up ℤ) ⥤ C`. As distinguished triangles in the homotopy category can be characterized in terms of degreewise split short exact sequences of cochain complexes, this follows from the homology sequence associated to a short exact sequence of homological…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/HomologicalFunctor.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 14.857, "z": -34.043, "size": 0.2895, "title": "Degreewise split exact sequences of cochain complexes", "summary": "The main result of this file is the lemma `HomotopyCategory.distinguished_iff_iso_trianglehOfDegreewiseSplit` which asserts that a triangle in `HomotopyCategory C (ComplexShape.up ℤ)` is distinguished iff it is isomorphic to the triangle attached to a degreewise split short exact sequence of cochain complexes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/DegreewiseSplit.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -2.173, "z": -61.248, "size": 0.2, "title": "Shifting an affine subspace towards a point", "summary": "This file introduces a \"shift\" transformation of affine subspace, where the subspace is translated relatively to a point `c`. This is equivalent to `AffineSubspace.map (AffineEquiv.constVAdd ..)`, but hides the detail of arbitrarily choosing a point in the subspace. Shifting is controlled by a parameter `r`, indicating how far the output space is to `c`. We set `r = 0` to mean the output space passes through `c`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Shift.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": 28.99, "z": -51.879, "size": 0.3746, "title": "Simplex in affine space", "summary": "This file defines n-dimensional simplices in affine space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Simplex/Basic.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 2, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -4.795, "z": 16.012, "size": 0.3356, "title": "The homotopy category", "summary": "`HomotopyCategory V c` gives the category of chain complexes of shape `c` in `V`, with chain maps identified when they are homotopic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory.html"}, {"id": "Mathlib.Algebra.Order.Ring.Star", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": -29.467, "z": 3.829, "size": 0.2595, "title": "Commutative star-ordered rings are ordered rings", "summary": "A noncommutative star-ordered ring is generally not an ordered ring. Indeed, in a star-ordered ring, nonnegative elements are self-adjoint, but the product of self-adjoint elements is self-adjoint if and only if they commute. Therefore, a necessary condition for a star-ordered ring to be an ordered ring is that all nonnegative elements commute. Consequently, if a star-ordered ring is spanned by it nonnegative…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Star.html"}, {"id": "Mathlib.Algebra.Polynomial.Identities", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.1876, "macro_tier_override": null, "x": 18.881, "z": 56.35, "size": 0.265, "title": "Theory of univariate polynomials", "summary": "The main def is `Polynomial.binomExpansion`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Identities.html"}, {"id": "Mathlib.Algebra.Polynomial.DenomsClearable", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -25.53, "z": 59.777, "size": 0.2438, "title": "Denominators of evaluation of polynomials at ratios", "summary": "Let `i : R → K` be a homomorphism of semirings. Assume that `K` is commutative. If `a` and `b` are elements of `R` such that `i b ∈ K` is invertible, then for any polynomial `f ∈ R[X]` the \"mathematical\" expression `b ^ f.natDegree * f (a / b) ∈ K` is in the image of the homomorphism `i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/DenomsClearable.html"}, {"id": "Mathlib.NumberTheory.Real.Irrational", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 2, "macro_tier_score": 0.0076, "macro_tier_override": null, "x": -41.878, "z": -70.169, "size": 0.3306, "title": "Irrational real numbers", "summary": "In this file we define a predicate `Irrational` on `ℝ`, prove that the `n`-th root of an integer number is irrational if it is not integer, and that `√(q : ℚ)` is irrational if and only if `¬IsSquare q ∧ 0 ≤ q`. We also provide dot-style constructors like `Irrational.add_ratCast`, `Irrational.ratCast_sub` etc. With the `Decidable` instances in this file, is possible to prove `Irrational √n` using `decide`, when `n`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Real/Irrational.html"}, {"id": "Mathlib.Algebra.Order.Interval.Set.Group", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": -19.926, "z": 13.633, "size": 0.3041, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Interval/Set/Group.html"}, {"id": "Mathlib.Algebra.GroupWithZero.ProdHom", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -15.175, "z": -13.677, "size": 0.2338, "title": "Homomorphisms for products of groups with zero", "summary": "This file defines homomorphisms for products of groups with zero, which is identified with the `WithZero` of the product of the units of the groups. The product of groups with zero `WithZero (αˣ × βˣ)` is a group with zero itself with natural inclusions. TODO: Give `GrpWithZero` instances of `HasBinaryProducts` and `HasBinaryCoproducts`, as well as a terminal object.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/ProdHom.html"}, {"id": "Mathlib.Algebra.Group.Prod", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4758, "macro_tier_override": null, "x": -14.857, "z": 0.125, "size": 0.5785, "title": "Monoid, group etc. structures on `M × N`", "summary": "In this file we define one-binop (`Monoid`, `Group` etc) structures on `M × N`. We also prove trivial `simp` lemmas, and define the following operations on `MonoidHom`s: * `fst M N : M × N →* M`, `snd M N : M × N →* N`: projections `Prod.fst` and `Prod.snd` as `MonoidHom`s; * `inl M N : M →* M × N`, `inr M N : N →* M × N`: inclusions of first/second monoid into the product; * `f.prod g` : `M →* N × P`: sends `x` to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Prod.html"}, {"id": "Mathlib.Algebra.GroupWithZero.WithZero", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4328, "macro_tier_override": null, "x": -14.631, "z": 2.581, "size": 0.3637, "title": "Adjoining a zero to a group", "summary": "This file proves that one can adjoin a new zero element to a group and get a group with zero. In valuation theory, valuations have codomain `{0} ∪ {c ^ n | n : ℤ}` for some `c > 1`, which we can formalise as `ℤᵐ⁰ := WithZero (Multiplicative ℤ)`. It is important to be able to talk about the maps `n ↦ c ^ n` and `c ^ n ↦ n`. We define these as `exp : ℤ → ℤᵐ⁰` and `log : ℤᵐ⁰ → ℤ` with junk value `log 0 = 0`. Junkless…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/WithZero.html"}, {"id": "Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 3, "macro_tier_score": 0.1799, "macro_tier_override": null, "x": -10.858, "z": -52.752, "size": 0.357, "title": "Integral closure of a subring.", "summary": "If A is an R-algebra then `a : A` is integral over R if it is a root of a monic polynomial with coefficients in R.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IntegralClosure/IsIntegral/Defs.html"}, {"id": "Mathlib.Algebra.Polynomial.Degree.Defs", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 4, "macro_tier_score": 0.2895, "macro_tier_override": null, "x": 50.578, "z": 12.08, "size": 0.4514, "title": "Degree of univariate polynomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Degree/Defs.html"}, {"id": "Mathlib.Algebra.Polynomial.Eval.Defs", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 4, "macro_tier_score": 0.2914, "macro_tier_override": null, "x": 43.6, "z": 24.768, "size": 0.4773, "title": "Evaluating a polynomial", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Eval/Defs.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.Isometries", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 84.553, "z": 33.644, "size": 0.2465, "title": "Linear equivalences of tensor products as isometries", "summary": "These results are separate from the definition of `QuadraticForm.tmul` as that file is very slow.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.TensorProduct", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -72.825, "z": -48.119, "size": 0.2687, "title": "The quadratic form on a tensor product", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.html"}, {"id": "Mathlib.RingTheory.TensorProduct.MonoidAlgebra", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.1901, "macro_tier_override": null, "x": 2.118, "z": -66.824, "size": 0.2998, "title": "Monoid algebras commute with base change", "summary": "In this file we show that monoid algebras are stable under pushout.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/MonoidAlgebra.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.Basic", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 4, "macro_tier_score": 0.2949, "macro_tier_override": null, "x": -25.406, "z": 41.062, "size": 0.4453, "title": "Algebra structure on monoid algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/Basic.html"}, {"id": "Mathlib.LinearAlgebra.DirectSum.Finsupp", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 4, "macro_tier_score": 0.2746, "macro_tier_override": null, "x": 39.107, "z": -42.252, "size": 0.4307, "title": "Results on finitely supported functions.", "summary": "* `TensorProduct.finsuppLeft`, the tensor product of `ι →₀ M` and `N` is linearly equivalent to `ι →₀ M ⊗[R] N` * `TensorProduct.finsuppScalarLeft`, the tensor product of `ι →₀ R` and `N` is linearly equivalent to `ι →₀ N` * `TensorProduct.finsuppRight`, the tensor product of `M` and `ι →₀ N` is linearly equivalent to `ι →₀ M ⊗[R] N` * `TensorProduct.finsuppScalarRight`, the tensor product of `M` and `ι →₀ R` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/DirectSum/Finsupp.html"}, {"id": "Mathlib.RingTheory.IsTensorProduct", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.2147, "macro_tier_override": null, "x": 60.871, "z": -22.798, "size": 0.384, "title": "The characteristic predicate of tensor product", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IsTensorProduct.html"}, {"id": "Mathlib.RingTheory.Polynomial.Hermite.Gaussian", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 24.638, "z": 56.116, "size": 0.2, "title": "Hermite polynomials and Gaussians", "summary": "This file shows that the Hermite polynomial `hermite n` is (up to sign) the polynomial factor occurring in the `n`th derivative of a Gaussian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Hermite/Gaussian.html"}, {"id": "Mathlib.RingTheory.Polynomial.Hermite.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -9.903, "z": -58.598, "size": 0.239, "title": "Hermite polynomials", "summary": "This file defines `Polynomial.hermite n`, the `n`th probabilists' Hermite polynomial.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Hermite/Basic.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.Evaluation", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 2, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": -65.687, "z": 20.174, "size": 0.2839, "title": "Evaluation of multivariate power series", "summary": "Let `σ`, `R`, `S` be types, with `CommRing R`, `CommRing S`. One assumes that `IsTopologicalRing R` and `IsUniformAddGroup R`, and that `S` is a complete and separated topological `R`-algebra, with `IsLinearTopology S S`, which means there is a basis of neighborhoods of 0 consisting of ideals. Given `φ : R →+* S`, `a : σ → S`, and `f : MvPowerSeries σ R`, `MvPowerSeries.eval₂ f φ a` is the evaluation of the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/Evaluation.html"}, {"id": "Mathlib.RingTheory.Ideal.BigOperators", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.1806, "macro_tier_override": null, "x": 32.87, "z": 12.832, "size": 0.3928, "title": "Big operators for ideals", "summary": "This contains some results on the big operators `∑` and `∏` interacting with the `Ideal` type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/BigOperators.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.PiTopology", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 2, "macro_tier_score": 0.0116, "macro_tier_override": null, "x": 44.523, "z": 49.876, "size": 0.2915, "title": "Product topology on multivariate power series", "summary": "Let `R` be with `Semiring R` and `TopologicalSpace R` In this file we define the topology on `MvPowerSeries σ R` that corresponds to the simple convergence on its coefficients. It is the coarsest topology for which all coefficient maps are continuous. When `R` has `UniformSpace R`, we define the corresponding uniform structure. This topology can be included by writing `open scoped MvPowerSeries.WithPiTopology`. When…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/PiTopology.html"}, {"id": "Mathlib.RingTheory.Coalgebra.MonoidAlgebra", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -11.042, "z": -67.822, "size": 0.257, "title": "The coalgebra structure on monoid algebras", "summary": "Given a type `X`, a commutative semiring `R` and a semiring `A` which is also an `R`-coalgebra, this file collects results about the `R`-coalgebra instance on `A[X]` inherited from the corresponding structure on its coefficients, defined in `Mathlib/RingTheory/Coalgebra/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Coalgebra/MonoidAlgebra.html"}, {"id": "Mathlib.NumberTheory.Harmonic.EulerMascheroni", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 18.096, "z": -4.174, "size": 0.271, "title": "The Euler-Mascheroni constant `γ`", "summary": "We define the constant `γ`, and give upper and lower bounds for it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Harmonic/EulerMascheroni.html"}, {"id": "Mathlib.RingTheory.Lasker", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -58.846, "z": -42.228, "size": 0.2, "title": "Lasker ring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Lasker.html"}, {"id": "Mathlib.Algebra.Module.LocalizedModule.Submodule", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.184, "macro_tier_override": null, "x": 57.836, "z": 13.669, "size": 0.3385, "title": "Localization of Submodules", "summary": "Results about localizations of submodules and quotient modules are provided in this file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LocalizedModule/Submodule.html"}, {"id": "Mathlib.Algebra.Ring.CentroidHom", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -13.381, "z": 24.433, "size": 0.239, "title": "Centroid homomorphisms", "summary": "Let `A` be a (nonunital, non-associative) algebra. The centroid of `A` is the set of linear maps `T` on `A` such that `T` commutes with left and right multiplication, that is to say, for all `a` and `b` in `A`, $$ T(ab) = (Ta)b, T(ab) = a(Tb). $$ In mathlib we call elements of the centroid \"centroid homomorphisms\" (`CentroidHom`) in keeping with `AddMonoidHom` etc. We use the `DFunLike` design, so each type of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/CentroidHom.html"}, {"id": "Mathlib.Algebra.Module.Hom", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.3868, "macro_tier_override": null, "x": 19.507, "z": -10.777, "size": 0.3341, "title": "Bundled Hom instances for module and multiplicative actions", "summary": "This file defines instances for `Module` on bundled `Hom` types. These are analogous to the instances in `Algebra.Module.Pi`, but for bundled instead of unbundled functions. We also define a bundled versions of `(· • ·)` as `AddMonoidHom.smul`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Hom.html"}, {"id": "Mathlib.Algebra.Ring.Subsemiring.Basic", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 4, "macro_tier_score": 0.3848, "macro_tier_override": null, "x": -21.992, "z": -13.869, "size": 0.403, "title": "Bundled subsemirings", "summary": "We define some standard constructions on bundled subsemirings: `CompleteLattice` structure, subsemiring `map`, `comap` and range (`rangeS`) of a `RingHom` etc.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Subsemiring/Basic.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.End", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4354, "macro_tier_override": null, "x": 11.393, "z": 12.23, "size": 0.4743, "title": "Group actions and (endo)morphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/End.html"}, {"id": "Mathlib.GroupTheory.OreLocalization.OreSet", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 3, "macro_tier_score": 0.2388, "macro_tier_override": null, "x": -0.536, "z": 11.13, "size": 0.2877, "title": "(Left) Ore sets", "summary": "This defines left Ore sets on arbitrary monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/OreLocalization/OreSet.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.MulAction", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4637, "macro_tier_override": null, "x": 4.047, "z": -10.382, "size": 0.4131, "title": "Actions by `Submonoid`s", "summary": "These instances transfer the action by an element `m : M` of a monoid `M` written as `m • a` onto the action by an element `s : S` of a submonoid `S : Submonoid M` such that `s • a = (s : M) • a`. These instances work particularly well in conjunction with `Monoid.toMulAction`, enabling `s • m` as an alias for `↑s * m`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/MulAction.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.Cardinal", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 2, "macro_tier_score": 0.016, "macro_tier_override": null, "x": 18.571, "z": -0.099, "size": 0.2895, "title": "Cardinality of monoid algebras", "summary": "This file computes the cardinality of `R[M]` in terms of `#R` and `#M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/Cardinal.html"}, {"id": "Mathlib.Algebra.Lie.Semisimple.Lemmas", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 56.215, "z": 100.489, "size": 0.2429, "title": "Lemmas about semisimple Lie algebras", "summary": "This file is a home for lemmas about semisimple and reductive Lie algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Semisimple/Lemmas.html"}, {"id": "Mathlib.Algebra.Lie.Semisimple.Basic", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": -67.488, "z": 58.241, "size": 0.2831, "title": "Semisimple Lie algebras", "summary": "The famous Cartan-Dynkin-Killing classification of semisimple Lie algebras renders them one of the most important classes of Lie algebras. In this file we prove basic results about simple and semisimple Lie algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Semisimple/Basic.html"}, {"id": "Mathlib.Algebra.Homology.ExactSequenceFour", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -11.664, "z": 9.203, "size": 0.2487, "title": "Exact sequences with four terms", "summary": "The main definition in this file is `ComposableArrows.Exact.cokerIsoKer`: given an exact sequence `S` (involving at least four objects), this is the isomorphism from the cokernel of `S.map' k (k + 1)` to the kernel of `S.map' (k + 2) (k + 3)`. This is intended to be used for exact sequences in abelian categories, but the construction works for preadditive balanced categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ExactSequenceFour.html"}, {"id": "Mathlib.Algebra.Polynomial.Module.AEval", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.0767, "macro_tier_override": null, "x": 21.541, "z": 59.356, "size": 0.3523, "title": "Action of the polynomial ring on module induced by an algebra element.", "summary": "Given an element `a` in an `R`-algebra `A` and an `A`-module `M` we define an `R[X]`-module `Module.AEval R M a`, which is a type synonym of `M` with the action of a polynomial `f` given by `f • m = Polynomial.aeval a f • m`. In particular `X • m = a • m`. In the special case that `A = M →ₗ[R] M` and `φ : M →ₗ[R] M`, the module `Module.AEval R M a` is abbreviated `Module.AEval' φ`. In this module we have `X • m = ↑φ…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Module/AEval.html"}, {"id": "Mathlib.Algebra.Module.Submodule.Invariant", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 4, "macro_tier_score": 0.3094, "macro_tier_override": null, "x": -24.329, "z": -28.067, "size": 0.3473, "title": "The lattice of invariant submodules", "summary": "In this file we defined the type `Module.End.invtSubmodule`, associated to a linear endomorphism of a module. Its utility stems primarily from those occasions on which we wish to take advantage of the lattice structure of invariant submodules. See also `Mathlib/Algebra/Polynomial/Module/AEval.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/Invariant.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 2, "macro_tier_score": 0.0218, "macro_tier_override": null, "x": -47.874, "z": 35.212, "size": 0.3815, "title": "Formal (multivariate) power series", "summary": "This file defines multivariate formal power series and develops the basic properties of these objects. A formal power series is to a polynomial like an infinite sum is to a finite sum. We provide the natural inclusion from multivariate polynomials to multivariate formal power series.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/Basic.html"}, {"id": "Mathlib.Algebra.Group.TypeTags.Basic", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.5006, "macro_tier_override": null, "x": -5.204, "z": 7.69, "size": 0.6596, "title": "Type tags that turn additive structures into multiplicative, and vice versa", "summary": "We define two type tags: * `Additive α`: turns any multiplicative structure on `α` into the corresponding additive structure on `Additive α`; * `Multiplicative α`: turns any additive structure on `α` into the corresponding multiplicative structure on `Multiplicative α`. We also define instances `Additive.*` and `Multiplicative.*` that actually transfer the structures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/TypeTags/Basic.html"}, {"id": "Mathlib.RingTheory.Valuation.DiscreteValuativeRel", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 46.651, "z": -45.263, "size": 0.2338, "title": "Discrete Valuative Relations", "summary": "Discrete valuative relations have a maximal element less than one in the value group. In the rank-one case, this is equivalent to the value group being isomorphic to `ℤᵐ⁰`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/DiscreteValuativeRel.html"}, {"id": "Mathlib.RingTheory.Valuation.RankOne", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 2, "macro_tier_score": 0.0072, "macro_tier_override": null, "x": 3.848, "z": 63.026, "size": 0.2991, "title": "Rank one valuations", "summary": "We define rank one valuations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/RankOne.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.KProjective", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -20.713, "z": 39.467, "size": 0.2429, "title": "K-projective cochain complexes", "summary": "We define the notion of K-projective cochain complex in an abelian category, and show that bounded above complexes of projective objects are K-projective.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/KProjective.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.KInjective", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -41.135, "z": 11.508, "size": 0.2782, "title": "K-injective cochain complexes", "summary": "We define the notion of K-injective cochain complex in an abelian category, and show that bounded below complexes of injective objects are K-injective.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/KInjective.html"}, {"id": "Mathlib.Algebra.Homology.CochainComplexOpposite", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 17.56, "z": -10.44, "size": 0.2553, "title": "Opposite categories of cochain complexes", "summary": "We construct an equivalence of categories `CochainComplex.opEquivalence C` between `(CochainComplex C ℤ)ᵒᵖ` and `CochainComplex Cᵒᵖ ℤ`, and we show that two morphisms in `CochainComplex C ℤ` are homotopic iff they are homotopic as morphisms in `CochainComplex Cᵒᵖ ℤ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/CochainComplexOpposite.html"}, {"id": "Mathlib.Algebra.Algebra.RestrictScalars", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 3, "macro_tier_score": 0.2517, "macro_tier_override": null, "x": -38.668, "z": 25.699, "size": 0.5463, "title": "The `RestrictScalars` type alias", "summary": "See the documentation attached to the `RestrictScalars` definition for advice on how and when to use this type alias. As described there, it is often a better choice to use the `IsScalarTower` typeclass instead.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/RestrictScalars.html"}, {"id": "Mathlib.RingTheory.TensorProduct.Basic", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 3, "macro_tier_score": 0.2566, "macro_tier_override": null, "x": -0.196, "z": 61.286, "size": 0.4622, "title": "The tensor product of R-algebras", "summary": "This file provides results about the multiplicative structure on `A ⊗[R] B` when `R` is a commutative (semi)ring and `A` and `B` are both `R`-algebras. On these tensor products, multiplication is characterized by `(a₁ ⊗ₜ b₁) * (a₂ ⊗ₜ b₂) = (a₁ * a₂) ⊗ₜ (b₁ * b₂)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/Basic.html"}, {"id": "Mathlib.GroupTheory.Order.Min", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2478, "title": "Minimum order of an element", "summary": "This file defines the minimum order of an element of a monoid.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Order/Min.html"}, {"id": "Mathlib.Algebra.Order.Group.Bounds", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 8.816, "z": -11.959, "size": 0.2, "title": "Least upper bound and the greatest lower bound in linear ordered additive commutative groups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Bounds.html"}, {"id": "Mathlib.RepresentationTheory.Rep.Res", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 2, "macro_tier_score": 0.01, "macro_tier_override": null, "x": 54.563, "z": 77.42, "size": 0.3414, "title": "Restriction of representations", "summary": "Given a group homomorphism `f : H →* G`, we have the restriction functor `resFunctor f : Rep k G ⥤ Rep k H` which sends a `G`-representation `ρ` to the `H`-representation `ρ.comp f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Rep/Res.html"}, {"id": "Mathlib.RepresentationTheory.Rep.Iso", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 2, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 86.294, "z": 34.292, "size": 0.3141, "title": "Equivalence between `Rep k G` and `ModuleCat k[G]`", "summary": "In this file we show that the category of `k`-linear representations of a monoid `G` is equivalent to the category of modules over the monoid algebra `k[G]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Rep/Iso.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 2, "macro_tier_score": 0.0127, "macro_tier_override": null, "x": -27.036, "z": -48.715, "size": 0.2998, "title": "The symmetric monoidal structure on `Module R`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -94.427, "z": -33.779, "size": 0.2, "title": "Isomorphisms with the even subalgebra of a Clifford algebra", "summary": "This file provides some notable isomorphisms regarding the even subalgebra, `CliffordAlgebra.even`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.Even", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 9.136, "z": -98.005, "size": 0.2565, "title": "The universal property of the even subalgebra", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/Even.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.Prod", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 49.877, "z": 76.115, "size": 0.2862, "title": "Quadratic form on product and pi types", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/Prod.html"}, {"id": "Mathlib.Algebra.Star.CHSH", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 27.187, "z": -11.993, "size": 0.2, "title": "The Clauser-Horne-Shimony-Holt inequality and Tsirelson's inequality.", "summary": "We establish a version of the Clauser-Horne-Shimony-Holt (CHSH) inequality (which is a generalization of Bell's inequality). This is a foundational result which implies that quantum mechanics is not a local hidden variable theory. As usually stated the CHSH inequality requires substantial language from physics and probability, but it is possible to give a statement that is purely about ordered \\*-algebras. We do…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/CHSH.html"}, {"id": "Mathlib.NumberTheory.FLT.Basic", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -57.28, "z": -26.573, "size": 0.2647, "title": "Statement of Fermat's Last Theorem", "summary": "This file states Fermat's Last Theorem. We provide a statement over a general semiring with specific exponent, along with the usual statement over the naturals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/FLT/Basic.html"}, {"id": "Mathlib.RingTheory.PrincipalIdealDomain", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 4, "macro_tier_score": 0.2957, "macro_tier_override": null, "x": 43.15, "z": 43.522, "size": 0.4735, "title": "Principal ideal rings, principal ideal domains, and Bézout rings", "summary": "A principal ideal ring (PIR) is a ring in which all left ideals are principal. A principal ideal domain (PID) is an integral domain which is a principal ideal ring. The definition of `IsPrincipalIdealRing` can be found in `Mathlib/RingTheory/Ideal/Span.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PrincipalIdealDomain.html"}, {"id": "Mathlib.NumberTheory.NumberField.DedekindZeta", "region_id": "algebra", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -113.705, "z": 83.623, "size": 0.2, "title": "The Dedekind zeta function of a number field", "summary": "In this file, we define and prove results about the Dedekind zeta function of a number field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/DedekindZeta.html"}, {"id": "Mathlib.Algebra.BigOperators.Ring.Nat", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 1.691, "z": 18.494, "size": 0.2695, "title": "Big operators on a finset in the natural numbers", "summary": "This file contains the results concerning the interaction of finset big operators with natural numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Ring/Nat.html"}, {"id": "Mathlib.NumberTheory.LSeries.SumCoeff", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -16.663, "z": -74.298, "size": 0.239, "title": "Partial sums of coefficients of L-series", "summary": "We prove several results involving partial sums of coefficients (or norm of coefficients) of L-series.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/SumCoeff.html"}, {"id": "Mathlib.NumberTheory.NumberField.Ideal.Asymptotics", "region_id": "algebra", "micro_elevation": 0.9868, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -34.241, "z": 135.013, "size": 0.239, "title": "Asymptotics on integral ideals of a number field", "summary": "We prove several asymptotics involving integral ideals of a number field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Ideal/Asymptotics.html"}, {"id": "Mathlib.RingTheory.LocalRing.RingHom.Basic", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.0941, "macro_tier_override": null, "x": -38.479, "z": -54.675, "size": 0.3334, "title": "Local rings homomorphisms", "summary": "We prove basic properties of local rings homomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/RingHom/Basic.html"}, {"id": "Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.1216, "macro_tier_override": null, "x": -59.181, "z": 26.884, "size": 0.4329, "title": "Maximal ideal of local rings", "summary": "We prove basic properties of the maximal ideal of a local ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/MaximalIdeal/Basic.html"}, {"id": "Mathlib.LinearAlgebra.LinearPMap", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 3, "macro_tier_score": 0.2496, "macro_tier_override": null, "x": -4.551, "z": 48.071, "size": 0.3528, "title": "Partially defined linear maps", "summary": "A `LinearPMap σ E F` or `E →ₛₗ.[σ] F` is a semilinear map from a submodule of `E` to `F` with a ring homomorphism `σ` between the scalars. This reduces to a linear map when `σ` is the identity. We define a `SemilatticeInf` with `OrderBot` instance on this, and define three operations: * `mkSpanSingleton` defines a partial linear map defined on the span of a singleton. * `sup` takes two partial linear maps `f`, `g`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/LinearPMap.html"}, {"id": "Mathlib.LinearAlgebra.Finsupp.Pi", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 4, "macro_tier_score": 0.294, "macro_tier_override": null, "x": -47.694, "z": -15.48, "size": 0.4071, "title": "Properties of the module `α →₀ M`", "summary": "* `Finsupp.linearEquivFunOnFinite`: `α →₀ β` and `a → β` are equivalent if `α` is finite * `FunOnFinite.map`: the map `(X → M) → (Y → M)` induced by a map `f : X ⟶ Y` when `X` and `Y` are finite. * `FunOnFinite.linearMmap`: the linear map `(X → M) →ₗ[R] (Y → M)` induced by a map `f : X ⟶ Y` when `X` and `Y` are finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Finsupp/Pi.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Finiteness", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 3, "macro_tier_score": 0.0568, "macro_tier_override": null, "x": 6.074, "z": 53.514, "size": 0.3583, "title": "Some finiteness results of tensor product", "summary": "This file contains some finiteness results of tensor product. - `TensorProduct.exists_multiset`, `TensorProduct.exists_finsupp_left`, `TensorProduct.exists_finsupp_right`, `TensorProduct.exists_finset`: any element of `M ⊗[R] N` can be written as a finite sum of pure tensors. See also `TensorProduct.span_tmul_eq_top`. - `TensorProduct.exists_finite_submodule_left_of_setFinite`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Finiteness.html"}, {"id": "Mathlib.Algebra.BigOperators.Associated", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 3, "macro_tier_score": 0.2148, "macro_tier_override": null, "x": -11.735, "z": -23.201, "size": 0.3189, "title": "Products of associated, prime, and irreducible elements.", "summary": "This file contains some theorems relating definitions in `Algebra.Associated` and products of multisets, finsets, and finsupps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Associated.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Associated", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.3954, "macro_tier_override": null, "x": -1.438, "z": 20.378, "size": 0.4683, "title": "Associated elements.", "summary": "In this file we define an equivalence relation `Associated` saying that two elements of a monoid differ by a multiplication by a unit. Then we show that the quotient type `Associates` is a monoid and prove basic properties of this quotient.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Associated.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.SemiringInverse", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 3, "macro_tier_score": 0.2304, "macro_tier_override": null, "x": -11.747, "z": -18.939, "size": 0.3294, "title": "Nonsingular inverses over semirings", "summary": "This file proves `A * B = 1 ↔ B * A = 1` for square matrices over a commutative semiring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/SemiringInverse.html"}, {"id": "Mathlib.Algebra.Group.Embedding", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.292, "macro_tier_override": null, "x": -4.402, "z": -3.415, "size": 0.46, "title": "The embedding of a cancellative semigroup into itself by multiplication by a fixed element.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Embedding.html"}, {"id": "Mathlib.GroupTheory.Perm.Sign", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 3, "macro_tier_score": 0.2531, "macro_tier_override": null, "x": 7.869, "z": 18.852, "size": 0.3606, "title": "Sign of a permutation", "summary": "The main definition of this file is `Equiv.Perm.sign`, associating a `ℤˣ` sign with a permutation. Other lemmas have been moved to `Mathlib/GroupTheory/Perm/Finite.lean`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Sign.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Slope", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 1, "macro_tier_score": 0.002, "macro_tier_override": null, "x": 43.245, "z": 28.877, "size": 0.3248, "title": "Slope of a function", "summary": "In this file we define the slope of a function `f : k → PE` taking values in an affine space over `k` and prove some basic theorems about `slope`. The `slope` function naturally appears in the Mean Value Theorem, and in the proof of the fact that a function with nonnegative second derivative on an interval is convex on this interval.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Slope.html"}, {"id": "Mathlib.Algebra.Polynomial.BigOperators", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.2693, "macro_tier_override": null, "x": -67.026, "z": 15.14, "size": 0.3315, "title": "Lemmas for the interaction between polynomials and `∑` and `∏`.", "summary": "Recall that `∑` and `∏` are notation for `Finset.sum` and `Finset.prod` respectively.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/BigOperators.html"}, {"id": "Mathlib.LinearAlgebra.LinearIndependent.Defs", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 4, "macro_tier_score": 0.3354, "macro_tier_override": null, "x": -28.686, "z": -34.114, "size": 0.3751, "title": "Linear independence", "summary": "This file defines linear independence in a module or vector space. It is inspired by Isabelle/HOL's linear algebra, and hence indirectly by HOL Light. We define `LinearIndependent R v` as `Function.Injective (Finsupp.linearCombination R v)`. Here `Finsupp.linearCombination` is the linear map sending a function `f : ι →₀ R` with finite support to the linear combination of vectors from `v` with these coefficients. The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/LinearIndependent/Defs.html"}, {"id": "Mathlib.RingTheory.Flat.IsBaseChange", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 80.903, "z": -11.491, "size": 0.2, "title": "Lemmas about `IsBaseChange` under exact sequences", "summary": "In this file, we show that if `S` is a flat `R`-algebra, taking kernels commutes with base change of modules from `R` to `S`. # Main Results For `S` an `R`-algebra, consider the following commutative diagram with exact rows, `M₁` `M₂` `M₃` `R`-modules, `N₁` `N₂` `N₃` `S`-modules, `R`-linear maps `f₁` `f₂` `i₁` `i₂` `i₃` and `S`-linear maps `g₁` `g₂`. M₁ --f₁--> M₂ --f₂--> M₃ | | | i₁ i₂ i₃ | | | v v v N₁ --g₁--> N₂…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/IsBaseChange.html"}, {"id": "Mathlib.Algebra.DirectSum.Idempotents", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -63.054, "z": 15.79, "size": 0.2, "title": "Decomposition of the identity of a semiring into orthogonal idempotents", "summary": "In this file we show that if a semiring `R` can be decomposed into a direct sum of (left) ideals `R = V₁ ⊕ V₂ ⊕ ⋯ ⊕ Vₙ` then in the corresponding decomposition `1 = e₁ + e₂ + ⋯ + eₙ` with `eᵢ ∈ Vᵢ`, each `eᵢ` is an idempotent and the `eᵢ`'s form a family of complete orthogonal idempotents.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/DirectSum/Idempotents.html"}, {"id": "Mathlib.RingTheory.Idempotents", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 2, "macro_tier_score": 0.0214, "macro_tier_override": null, "x": 15.958, "z": -61.094, "size": 0.2818, "title": "Idempotents in rings", "summary": "The predicate `IsIdempotentElem` is defined for general monoids in `Mathlib/Algebra/Group/Idempotent.lean`; ring-specific lemmas are in `Mathlib/Algebra/Ring/Idempotent.lean`. In this file we provide various results regarding idempotent elements in rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Idempotents.html"}, {"id": "Mathlib.Algebra.DirectSum.Decomposition", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 3, "macro_tier_score": 0.0366, "macro_tier_override": null, "x": 27.179, "z": 48.636, "size": 0.4022, "title": "Decompositions of additive monoids, groups, and modules into direct sums", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/DirectSum/Decomposition.html"}, {"id": "Mathlib.RingTheory.OreLocalization.NonZeroDivisors", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 3, "macro_tier_score": 0.2353, "macro_tier_override": null, "x": 3.943, "z": -23.819, "size": 0.2732, "title": "Ore Localization over nonZeroDivisors in monoids with zeros.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/OreLocalization/NonZeroDivisors.html"}, {"id": "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.3902, "macro_tier_override": null, "x": -20.892, "z": -7.758, "size": 0.4797, "title": "Non-zero divisors and smul-divisors", "summary": "In this file we define the submonoid `nonZeroDivisors` and `nonZeroSMulDivisors` of a `MonoidWithZero`. We also define `nonZeroDivisorsLeft` and `nonZeroDivisorsRight` for non-commutative monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/NonZeroDivisors.html"}, {"id": "Mathlib.RingTheory.OreLocalization.Basic", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 3, "macro_tier_score": 0.2366, "macro_tier_override": null, "x": -14.168, "z": 8.868, "size": 0.2929, "title": "Localization over left Ore sets.", "summary": "This file proves results on the localization of rings (monoids with zeros) over a left Ore set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/OreLocalization/Basic.html"}, {"id": "Mathlib.Algebra.Order.Positive.Ring", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 1, "macro_tier_score": 0.002, "macro_tier_override": null, "x": -11.543, "z": -16.855, "size": 0.3258, "title": "Algebraic structures on the set of positive numbers", "summary": "In this file we define various instances (`AddSemigroup`, `IsOrderedMonoid` etc) on the type `{x : R // 0 < x}`. In each case we try to require the weakest possible typeclass assumptions on `R` but possibly, there is a room for improvements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Positive/Ring.html"}, {"id": "Mathlib.Algebra.Ring.InjSurj", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4351, "macro_tier_override": null, "x": 9.175, "z": -6.323, "size": 0.4969, "title": "Pulling back rings along injective maps, and pushing them forward along surjective maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/InjSurj.html"}, {"id": "Mathlib.RingTheory.Unramified.Finite", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 2, "macro_tier_score": 0.0168, "macro_tier_override": null, "x": 97.88, "z": -21.839, "size": 0.2615, "title": "Various results about unramified algebras", "summary": "We prove various theorems about unramified algebras. In fact we work in the more general setting of formally unramified algebras which are essentially of finite type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Unramified/Finite.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Ideal", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -61.795, "z": 47.598, "size": 0.2, "title": "Ideals in power series.", "summary": "We gather miscellaneous results about prime ideals in `R⟦X⟧`. More precisely, we prove that, given a prime ideal `I` of `R⟦X⟧`, if the ideal generated by the constant coefficients of the `f ∈ I` is generated by `n` elements, then `I` is generated by at most `n + 1` elements (actually it is generated by `n` elements if `X ∉ I` and `n + 1` elements otherwise). This implies immediately that `R⟦X⟧` is noetherian if `R`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Ideal.html"}, {"id": "Mathlib.RingTheory.Noetherian.OfPrime", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -52.399, "z": 35.234, "size": 0.2302, "title": "Noetherian rings and prime ideals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Noetherian/OfPrime.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Inverse", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 2, "macro_tier_score": 0.0081, "macro_tier_override": null, "x": -68.299, "z": 33.662, "size": 0.2753, "title": "Formal power series - Inverses", "summary": "If the constant coefficient of a formal (univariate) power series is invertible, then this formal power series is invertible. (See the discussion in `Mathlib/RingTheory/MvPowerSeries/Inverse.lean` for the construction.) Formal (univariate) power series over a local ring form a local ring. Formal (univariate) power series over a field form a discrete valuation ring, and a normalization monoid. The definition…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Inverse.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Trunc", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 2, "macro_tier_score": 0.0125, "macro_tier_override": null, "x": 13.951, "z": 67.284, "size": 0.2715, "title": "Formal power series in one variable - Truncation", "summary": "`PowerSeries.trunc n φ` truncates a (univariate) formal power series to the polynomial that has the same coefficients as `φ`, for all `m < n`, and `0` otherwise.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Trunc.html"}, {"id": "Mathlib.RingTheory.Jacobson.Semiprimary", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.0891, "macro_tier_override": null, "x": -39.568, "z": 49.208, "size": 0.2864, "title": "Semiprimary rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Jacobson/Semiprimary.html"}, {"id": "Mathlib.RingTheory.Jacobson.Radical", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.2034, "macro_tier_override": null, "x": 2.182, "z": -59.389, "size": 0.3699, "title": "Jacobson radical of modules and rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Jacobson/Radical.html"}, {"id": "Mathlib.RingTheory.Localization.Free", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": 79.468, "z": 7.88, "size": 0.2471, "title": "Free modules and localization", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Free.html"}, {"id": "Mathlib.RingTheory.Localization.Finiteness", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0364, "macro_tier_override": null, "x": 33.648, "z": 62.034, "size": 0.3321, "title": "Finiteness properties under localization", "summary": "In this file we establish behaviour of `Module.Finite` under localizations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Finiteness.html"}, {"id": "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 3, "macro_tier_score": 0.2, "macro_tier_override": null, "x": 15.85, "z": 76.373, "size": 0.4178, "title": "Strong rank condition for commutative rings", "summary": "We provide a shortcut instance for the fact that any nontrivial commutative ring satisfies `StrongRankCondition`, meaning that if there is an injective linear map `(Fin n → R) →ₗ[R] Fin m → R`, then `n ≤ m`. This implies that any commutative ring satisfies `InvariantBasisNumber`: the rank of a finitely generated free module is well defined.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeModule/StrongRankCondition.html"}, {"id": "Mathlib.Algebra.Homology.HomologicalComplexBiprod", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -10.221, "z": 8.033, "size": 0.2664, "title": "Binary biproducts of homological complexes", "summary": "In this file, it is shown that if two homological complex `K` and `L` in a preadditive category are such that for all `i : ι`, the binary biproduct `K.X i ⊞ L.X i` exists, then `K ⊞ L` exists, and there is an isomorphism `biprodXIso K L i : (K ⊞ L).X i ≅ (K.X i) ⊞ (L.X i)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomologicalComplexBiprod.html"}, {"id": "Mathlib.Algebra.Homology.HomologicalComplexLimits", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 2, "macro_tier_score": 0.0065, "macro_tier_override": null, "x": 9.52, "z": 5.791, "size": 0.3309, "title": "Limits and colimits in the category of homological complexes", "summary": "In this file, it is shown that if a category `C` has (co)limits of shape `J`, then it is also the case of the categories `HomologicalComplex C c`, and the evaluation functors `eval C c i : HomologicalComplex C c ⥤ C` commute to these.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomologicalComplexLimits.html"}, {"id": "Mathlib.Algebra.Homology.Additive", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 2, "macro_tier_score": 0.0088, "macro_tier_override": null, "x": 9.018, "z": -6.545, "size": 0.443, "title": "Homology is an additive functor", "summary": "When `V` is preadditive, `HomologicalComplex V c` is also preadditive, and `homologyFunctor` is additive.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Additive.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.Multiset.Defs", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.4544, "macro_tier_override": null, "x": -6.03, "z": -4.338, "size": 0.3917, "title": "Sums and products over multisets", "summary": "In this file we define products and sums indexed by multisets. This is later used to define products and sums indexed by finite sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Multiset/Defs.html"}, {"id": "Mathlib.Algebra.Group.Idempotent", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4444, "macro_tier_override": null, "x": 4.836, "z": -10.039, "size": 0.3272, "title": "Idempotents", "summary": "This file defines idempotents for an arbitrary multiplication and proves some basic results, including: * `IsIdempotentElem.mul_of_commute`: In a semigroup, the product of two commuting idempotents is an idempotent; * `IsIdempotentElem.pow_succ_eq`: In a monoid `a ^ (n+1) = a` for `a` an idempotent and `n` a natural number.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Idempotent.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.MulOpposite", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4544, "macro_tier_override": null, "x": -7.829, "z": -10.378, "size": 0.3297, "title": "Submonoid of opposite monoids", "summary": "For every monoid `M`, we construct an equivalence between submonoids of `M` and that of `Mᵐᵒᵖ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/MulOpposite.html"}, {"id": "Mathlib.NumberTheory.ClassNumber.FunctionField", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 87.701, "z": -88.269, "size": 0.2, "title": "Class numbers of function fields", "summary": "This file defines the class number of a function field as the (finite) cardinality of the class group of its ring of integers. It also proves some elementary results on the class number.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ClassNumber/FunctionField.html"}, {"id": "Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 72.965, "z": -27.573, "size": 0.239, "title": "Admissible absolute values on polynomials", "summary": "This file defines an admissible absolute value `Polynomial.cardPowDegreeIsAdmissible` which we use to show the class number of the ring of integers of a function field is finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.html"}, {"id": "Mathlib.NumberTheory.ClassNumber.Finite", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -104.907, "z": 63.392, "size": 0.2621, "title": "Class numbers of global fields", "summary": "In this file, we use the notion of \"admissible absolute value\" to prove finiteness of the class group for number fields and function fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ClassNumber/Finite.html"}, {"id": "Mathlib.NumberTheory.FunctionField", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 70.815, "z": 83.614, "size": 0.2617, "title": "Function fields", "summary": "This file defines a function field and the ring of integers corresponding to it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/FunctionField.html"}, {"id": "Mathlib.Algebra.Homology.FullSubcategory", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 8.612, "z": -3.473, "size": 0.2324, "title": "Homological complexes in full subcategories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/FullSubcategory.html"}, {"id": "Mathlib.Algebra.Homology.HomologicalComplex", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 2, "macro_tier_score": 0.0093, "macro_tier_override": null, "x": 6.938, "z": 2.655, "size": 0.4589, "title": "Homological complexes.", "summary": "A `HomologicalComplex V c` with a \"shape\" controlled by `c : ComplexShape ι` has chain groups `X i` (objects in `V`) indexed by `i : ι`, and a differential `d i j` whenever `c.Rel i j`. We in fact ask for differentials `d i j` for all `i j : ι`, but have a field `shape` requiring that these are zero when not allowed by `c`. This avoids a lot of dependent type theory hell! The composite of any two differentials `d i…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomologicalComplex.html"}, {"id": "Mathlib.Algebra.Order.Sub.Unbundled.Basic", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4068, "macro_tier_override": null, "x": 10.633, "z": -7.48, "size": 0.4421, "title": "Lemmas about subtraction in an unbundled canonically ordered monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Sub/Unbundled/Basic.html"}, {"id": "Mathlib.RingTheory.GradedAlgebra.FiniteType", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -56.246, "z": 54.042, "size": 0.2345, "title": "Graded rings of finite type", "summary": "We show that graded rings of finite type (over the 0-th component) are generated by homogeneous elements of positive degree.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/GradedAlgebra/FiniteType.html"}, {"id": "Mathlib.RingTheory.FiniteType", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 3, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": -18.214, "z": 73.933, "size": 0.4758, "title": "Finiteness conditions in commutative algebra", "summary": "In this file we define a notion of finiteness that is common in commutative algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/FiniteType.html"}, {"id": "Mathlib.LinearAlgebra.Basis.MulOpposite", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -22.838, "z": 68.734, "size": 0.2478, "title": "Basis of an opposite space", "summary": "This file defines the basis of an opposite space and shows that the opposite space is finite-dimensional and free when the original space is.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Basis/MulOpposite.html"}, {"id": "Mathlib.LinearAlgebra.FiniteDimensional.Defs", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.2348, "macro_tier_override": null, "x": -50.943, "z": 48.839, "size": 0.3906, "title": "Finite-dimensional vector spaces", "summary": "This file defines finite-dimensional vector spaces and shows our definition is equivalent to alternative definitions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FiniteDimensional/Defs.html"}, {"id": "Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -15.296, "z": 115.997, "size": 0.2482, "title": "Eisenstein polynomials", "summary": "In this file we gather more miscellaneous results about Eisenstein polynomials", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.html"}, {"id": "Mathlib.RingTheory.Norm.Transitivity", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 2, "macro_tier_score": 0.0267, "macro_tier_override": null, "x": 114.514, "z": 12.025, "size": 0.3441, "title": "Transitivity of algebra norm", "summary": "Suppose we have an `R`-algebra `S` with a finite basis. For each `s : S`, the determinant of the linear map given by multiplying by `s` gives information about the roots of the minimal polynomial of `s` over `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Norm/Transitivity.html"}, {"id": "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 2, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": -109.373, "z": -6.604, "size": 0.2694, "title": "Cyclotomic polynomials and `expand`.", "summary": "We gather results relating cyclotomic polynomials and `expand`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Unbundled.Units", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 7.783, "z": -10.413, "size": 0.2, "title": "Lemmas for units in an ordered monoid", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Unbundled/Units.html"}, {"id": "Mathlib.Algebra.Category.Ring.Epi", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 63.442, "z": 26.399, "size": 0.2756, "title": "Epimorphisms in `CommRingCat`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/Epi.html"}, {"id": "Mathlib.Algebra.Algebra.Epi", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -66.749, "z": 3.816, "size": 0.2646, "title": "Algebras which are commutative ring epimorphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Epi.html"}, {"id": "Mathlib.Algebra.NoZeroSMulDivisors.Basic", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 3, "macro_tier_score": 0.2573, "macro_tier_override": null, "x": 22.206, "z": -1.884, "size": 0.4037, "title": "`NoZeroSMulDivisors`", "summary": "This file proves more lemmas about the `NoZeroSMulDivisors` class, which is deprecated in favor of `Module.IsTorsionFree`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/NoZeroSMulDivisors/Basic.html"}, {"id": "Mathlib.NumberTheory.Padics.Hensel", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -65.626, "z": -42.158, "size": 0.2, "title": "Hensel's lemma on `ℤ_p`", "summary": "This file proves Hensel's lemma on `ℤ_p`, roughly following Keith Conrad's writeup: Hensel's lemma gives a simple condition for the existence of a root of a polynomial. The proof and motivation are described in the paper [R. Y. Lewis, *A formal proof of Hensel's lemma over the p-adic integers*][lewis2019].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/Hensel.html"}, {"id": "Mathlib.NumberTheory.Padics.PadicIntegers", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 68.5, "z": -33.252, "size": 0.286, "title": "p-adic integers", "summary": "This file defines the `p`-adic integers `ℤ_[p]` as the subtype of `ℚ_[p]` with norm `≤ 1`. We show that `ℤ_[p]` * is complete, * is nonarchimedean, * is a normed ring, * is a local ring, and * is a discrete valuation ring. The relation between `ℤ_[p]` and `ZMod p` is established in another file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/PadicIntegers.html"}, {"id": "Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -6.558, "z": 60.934, "size": 0.2676, "title": "Tensor algebras as direct sums of tensor powers", "summary": "In this file we show that `TensorAlgebra R M` is isomorphic to a direct sum of tensor powers, as `TensorAlgebra.equivDirectSum`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.html"}, {"id": "Mathlib.LinearAlgebra.TensorPower.Basic", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.0073, "macro_tier_override": null, "x": 57.538, "z": 1.972, "size": 0.3074, "title": "Tensor power of a semimodule over a commutative semiring", "summary": "We define the `n`th tensor power of `M` as the n-ary tensor product indexed by `Fin n` of `M`, `⨂[R] (i : Fin n), M`. This is a special case of `PiTensorProduct`. This file introduces the notation `⨂[R]^n M` for `TensorPower R n M`, which in turn is an abbreviation for `⨂[R] i : Fin n, M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorPower/Basic.html"}, {"id": "Mathlib.Algebra.Azumaya.Matrix", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 23.185, "z": -60.725, "size": 0.2, "title": "Matrix algebra is an Azumaya algebra over R", "summary": "In this file we prove that finite-dimensional matrix algebra `Matrix n n R` over `R` is an Azumaya algebra where `R` is a commutative ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Azumaya/Matrix.html"}, {"id": "Mathlib.Algebra.Azumaya.Defs", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -62.109, "z": 11.385, "size": 0.2617, "title": "Azumaya Algebras", "summary": "An Azumaya algebra over a commutative ring `R` is a finitely generated, projective and faithful R-algebra where the tensor product `A ⊗[R] Aᵐᵒᵖ` is isomorphic to the `R`-endomorphisms of A via the map `f : a ⊗ b ↦ (x ↦ a * x * b.unop)`. TODO : Add the three more definitions and prove they are equivalent: · There exists an `R`-algebra `B` such that `B ⊗[R] A` is Morita equivalent to `R`; · `Aᵐᵒᵖ ⊗[R] A` is Morita…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Azumaya/Defs.html"}, {"id": "Mathlib.GroupTheory.Complement", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 2, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 8.365, "z": 36.189, "size": 0.3766, "title": "Complements", "summary": "In this file we define the complement of a subgroup.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Complement.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Bilinear", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -31.513, "z": -43.676, "size": 0.2676, "title": "Bundled versions of multiplication for matrices", "summary": "This file provides versions of `LinearMap.mulLeft` and `LinearMap.mulRight` which work for the heterogeneous multiplication of matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Bilinear.html"}, {"id": "Mathlib.RingTheory.Finiteness.Ideal", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.1864, "macro_tier_override": null, "x": 5.013, "z": 59.217, "size": 0.3522, "title": "Finitely generated ideals", "summary": "Lemmas about finiteness of ideal operations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Ideal.html"}, {"id": "Mathlib.GroupTheory.Perm.ConjAct", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": 4.213, "z": 44.372, "size": 0.2652, "title": "Some lemmas pertaining to the action of `ConjAct (Perm α)` on `Perm α`", "summary": "We prove some lemmas related to the action of `ConjAct (Perm α)` on `Perm α`: Let `α` be a decidable fintype. * `conj_support_eq` relates the support of `k • g` with that of `g` * `cycleFactorsFinset_conj_eq`, `mem_cycleFactorsFinset_conj'` and `cycleFactorsFinset_conj` relate the set of cycles of `g`, `g.cycleFactorsFinset`, with that for `k • g`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/ConjAct.html"}, {"id": "Mathlib.Algebra.Group.Action.Pointwise.Finset", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 2, "macro_tier_score": 0.0123, "macro_tier_override": null, "x": -22.137, "z": 2.575, "size": 0.3464, "title": "Pointwise actions of finsets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Pointwise/Finset.html"}, {"id": "Mathlib.GroupTheory.Perm.Cycle.Factors", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 3, "macro_tier_score": 0.2289, "macro_tier_override": null, "x": 5.885, "z": 42.307, "size": 0.3012, "title": "Cycle factors of a permutation", "summary": "Let `β` be a `Fintype` and `f : Equiv.Perm β`. * `Equiv.Perm.cycleOf`: `f.cycleOf x` is the cycle of `f` that `x` belongs to. * `Equiv.Perm.cycleFactors`: `f.cycleFactors` is a list of disjoint cyclic permutations that multiply to `f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Cycle/Factors.html"}, {"id": "Mathlib.NumberTheory.DirichletCharacter.Orthogonality", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -76.487, "z": 81.033, "size": 0.2338, "title": "Orthogonality relations for Dirichlet characters", "summary": "Let `n` be a positive natural number. The main result of this file is `DirichletCharacter.sum_char_inv_mul_char_eq`, which says that when `a : ZMod n` is a unit and `b : ZMod n`, then the sum `∑ χ : DirichletCharacter R n, χ a⁻¹ * χ b` vanishes when `a ≠ b` and has the value `n.totient` otherwise. This requires `R` to have enough roots of unity (e.g., `R` could be an algebraically closed field of characteristic…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/DirichletCharacter/Orthogonality.html"}, {"id": "Mathlib.NumberTheory.DirichletCharacter.Basic", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 2, "macro_tier_score": 0.0071, "macro_tier_override": null, "x": -40.858, "z": -0.009, "size": 0.2895, "title": "Dirichlet Characters", "summary": "Let `R` be a commutative monoid with zero. A Dirichlet character `χ` of level `n` over `R` is a multiplicative character from `ZMod n` to `R` sending non-units to 0. We then obtain some properties of `toUnitHom χ`, the restriction of `χ` to a group homomorphism `(ZMod n)ˣ →* Rˣ`. Main definitions: - `DirichletCharacter`: The type representing a Dirichlet character. - `changeLevel`: Extend the Dirichlet character χ…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/DirichletCharacter/Basic.html"}, {"id": "Mathlib.NumberTheory.MulChar.Duality", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -57.956, "z": -92.99, "size": 0.251, "title": "Duality for multiplicative characters", "summary": "Let `M` be a finite commutative monoid and `R` a ring that has enough `n`th roots of unity, where `n` is the exponent of `M`. Then the main results of this file are as follows.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/MulChar/Duality.html"}, {"id": "Mathlib.Algebra.Homology.SpectralSequence.ComplexShape", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -4.19, "z": -6.134, "size": 0.2721, "title": "Complex shapes for pages of spectral sequences", "summary": "In this file, we define complex shapes which correspond to pages of spectral sequences: * `ComplexShape.spectralSequenceNat`: for any `u : ℤ × ℤ`, this is the complex shape on `ℕ × ℕ` corresponding to differentials of `ComplexShape.up' u : ComplexShape (ℤ × ℤ)` with source and target in `ℕ × ℕ`. (With `u := (r, 1 - r)`, this will apply to the `r`th-page of first quadrant `E₂` cohomological spectral sequence). *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/SpectralSequence/ComplexShape.html"}, {"id": "Mathlib.Algebra.Homology.ComplexShape", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 2, "macro_tier_score": 0.0076, "macro_tier_override": null, "x": -0.733, "z": -5.523, "size": 0.3936, "title": "Shapes of homological complexes", "summary": "We define a structure `ComplexShape ι` for describing the shapes of homological complexes indexed by a type `ι`. This is intended to capture chain complexes and cochain complexes, indexed by either `ℕ` or `ℤ`, as well as more exotic examples. Rather than insisting that the indexing type has a `succ` function specifying where differentials should go, inside `c : ComplexShape` we have `c.Rel : ι → ι → Prop`, and when…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ComplexShape.html"}, {"id": "Mathlib.FieldTheory.KummerPolynomial", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 3, "macro_tier_score": 0.1152, "macro_tier_override": null, "x": -8.866, "z": 96.165, "size": 0.3351, "title": "Irreducibility of X ^ p - a", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/KummerPolynomial.html"}, {"id": "Mathlib.Algebra.Group.Nat.Units", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4146, "macro_tier_override": null, "x": 6.021, "z": -9.376, "size": 0.397, "title": "The unit of the natural numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Nat/Units.html"}, {"id": "Mathlib.Algebra.Order.Ring.Nat", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 3, "macro_tier_score": 0.2499, "macro_tier_override": null, "x": -20.311, "z": 2.188, "size": 0.4615, "title": "The natural numbers form an ordered semiring", "summary": "This file contains the commutative linear ordered semiring instance on the natural numbers. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Nat.html"}, {"id": "Mathlib.Algebra.Order.Ring.WithTop", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.29, "macro_tier_override": null, "x": -18.971, "z": 11.695, "size": 0.4258, "title": "Structures involving `*` and `0` on `WithTop` and `WithBot`", "summary": "The main results of this section are `WithTop.instOrderedCommSemiring` and `WithBot.instOrderedCommSemiring`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/WithTop.html"}, {"id": "Mathlib.Algebra.Order.Sub.WithTop", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 11.71, "z": 9.144, "size": 0.3667, "title": "Lemma about subtraction in ordered monoids with a top element adjoined.", "summary": "This file introduces a subtraction on `WithTop α` when `α` has a subtraction and a bottom element, given by `x - ⊤ = ⊥` and `⊤ - x = ⊤`. This will be instantiated mostly for `ℕ∞` and `ℝ≥0∞`, where the bottom element is zero. Note that there is another subtraction on objects of the form `WithTop α` in the file `Mathlib/Algebra/Order/AddGroupWithTop.lean`, setting `-⊤ = ⊤` as this corresponds to the additivization of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Sub/WithTop.html"}, {"id": "Mathlib.FieldTheory.ChevalleyWarning", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 98.279, "z": -27.833, "size": 0.2478, "title": "The Chevalley–Warning theorem", "summary": "This file contains a proof of the Chevalley–Warning theorem. Throughout most of this file, `K` denotes a finite field and `q` is notation for the cardinality of `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/ChevalleyWarning.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.RingHom", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 3, "macro_tier_score": 0.0751, "macro_tier_override": null, "x": 81.026, "z": -10.593, "size": 0.3201, "title": "Functoriality of the prime spectrum", "summary": "In this file we define the induced map on prime spectra induced by a ring homomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/RingHom.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.Basic", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.1125, "macro_tier_override": null, "x": -25.046, "z": -61.989, "size": 0.3714, "title": "Prime spectrum of a commutative (semi)ring", "summary": "For the Zariski topology, see `Mathlib/RingTheory/Spectrum/Prime/Topology.lean`. (It is also naturally endowed with a sheaf of rings, which is constructed in `AlgebraicGeometry.StructureSheaf`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/Basic.html"}, {"id": "Mathlib.RingTheory.LocalRing.ResidueField.Ideal", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 3, "macro_tier_score": 0.0786, "macro_tier_override": null, "x": -79.431, "z": -8.246, "size": 0.3308, "title": "The residue field of a prime ideal", "summary": "We define `Ideal.ResidueField I` to be the residue field of the local ring `Localization.Prime I`, and provide an `IsFractionRing (R ⧸ I) I.ResidueField` instance.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/ResidueField/Ideal.html"}, {"id": "Mathlib.RingTheory.Extension.Cotangent.Basic", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 2, "macro_tier_score": 0.0226, "macro_tier_override": null, "x": -86.745, "z": -20.542, "size": 0.3653, "title": "Naive cotangent complex associated to a presentation.", "summary": "Given a presentation `0 → I → R[x₁,...,xₙ] → S → 0` (or equivalently a closed embedding `S ↪ Aⁿ` defined by `I`), we may define the (naive) cotangent complex `I/I² → ⨁ᵢ S dxᵢ → Ω[S/R] → 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Extension/Cotangent/Basic.html"}, {"id": "Mathlib.RingTheory.Kaehler.Polynomial", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 2, "macro_tier_score": 0.0216, "macro_tier_override": null, "x": 84.87, "z": 9.761, "size": 0.2959, "title": "The Kähler differential module of polynomial algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Kaehler/Polynomial.html"}, {"id": "Mathlib.RingTheory.Extension.Presentation.Basic", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 2, "macro_tier_score": 0.0219, "macro_tier_override": null, "x": -8.356, "z": -86.886, "size": 0.3257, "title": "Presentations of algebras", "summary": "A presentation of an `R`-algebra `S` is a distinguished family of generators and relations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Extension/Presentation/Basic.html"}, {"id": "Mathlib.Algebra.BigOperators.NatAntidiagonal", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0009, "macro_tier_override": null, "x": 6.863, "z": 15.24, "size": 0.3289, "title": "Big operators for `NatAntidiagonal`", "summary": "This file contains theorems relevant to big operators over `Finset.NatAntidiagonal`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/NatAntidiagonal.html"}, {"id": "Mathlib.Algebra.Order.Nonneg.Field", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 1, "macro_tier_score": 0.002, "macro_tier_override": null, "x": 16.14, "z": -17.955, "size": 0.3237, "title": "Semifield structure on the type of nonnegative elements", "summary": "This file defines instances and prove some properties about the nonnegative elements `{x : α // 0 ≤ x}` of an arbitrary type `α`. This is used to derive algebraic structures on `ℝ≥0` and `ℚ≥0` automatically.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Nonneg/Field.html"}, {"id": "Mathlib.Algebra.Order.Field.Canonical", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 20.443, "z": -8.874, "size": 0.2593, "title": "Canonically ordered semifields", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Field/Canonical.html"}, {"id": "Mathlib.Algebra.Order.Nonneg.Ring", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 1, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": 19.32, "z": -11.109, "size": 0.3149, "title": "Bundled ordered algebra instance on the type of nonnegative elements", "summary": "This file defines instances and prove some properties about the nonnegative elements `{x : α // 0 ≤ x}` of an arbitrary type `α`. Currently we only state instances and states some `simp`/`norm_cast` lemmas. When `α` is `ℝ`, this will give us some properties about `ℝ≥0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Nonneg/Ring.html"}, {"id": "Mathlib.Algebra.Order.Field.GeomSum", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 23.181, "z": -15.45, "size": 0.2667, "title": "Partial sums of geometric series in an ordered field", "summary": "This file upper- and lower-bounds the values of the geometric series $\\sum_{i=0}^{n-1} x^i$ and $\\sum_{i=0}^{n-1} x^i y^{n-1-i}$ and variants thereof.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Field/GeomSum.html"}, {"id": "Mathlib.LinearAlgebra.Basis.Cardinality", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 4, "macro_tier_score": 0.2944, "macro_tier_override": null, "x": -3.836, "z": -46.27, "size": 0.3047, "title": "Results relating bases and cardinality.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Basis/Cardinality.html"}, {"id": "Mathlib.LinearAlgebra.Basis.Defs", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 4, "macro_tier_score": 0.3467, "macro_tier_override": null, "x": -43.298, "z": -10.578, "size": 0.539, "title": "Bases", "summary": "This file defines bases in a module or vector space. It is inspired by Isabelle/HOL's linear algebra, and hence indirectly by HOL Light.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Basis/Defs.html"}, {"id": "Mathlib.Algebra.Field.ZMod", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 3, "macro_tier_score": 0.0586, "macro_tier_override": null, "x": 19.718, "z": -13.932, "size": 0.3304, "title": "`ZMod p` is a field", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/ZMod.html"}, {"id": "Mathlib.RingTheory.LocalRing.ResidueField.Defs", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.0894, "macro_tier_override": null, "x": 35.035, "z": -56.943, "size": 0.3107, "title": "Residue Field of local rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/ResidueField/Defs.html"}, {"id": "Mathlib.RingTheory.ZMod", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": -42.042, "z": -51.985, "size": 0.2701, "title": "Ring-theoretic facts about `ZMod n`", "summary": "We collect a few facts about `ZMod n` that need some ring theory to be proved/stated.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/ZMod.html"}, {"id": "Mathlib.RingTheory.RingHom.Surjective", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": -80.127, "z": -23.75, "size": 0.2798, "title": "The meta properties of surjective ring homomorphisms.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/Surjective.html"}, {"id": "Mathlib.Algebra.Order.Group.Pointwise.Bounds", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 2, "macro_tier_score": 0.0071, "macro_tier_override": null, "x": -4.148, "z": 18.102, "size": 0.2905, "title": "Upper/lower bounds in ordered monoids and groups", "summary": "In this file we prove a few facts like “`-s` is bounded above iff `s` is bounded below” (`bddAbove_neg`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Pointwise/Bounds.html"}, {"id": "Mathlib.Algebra.Ring.CharZero", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4104, "macro_tier_override": null, "x": 15.943, "z": 5.021, "size": 0.4082, "title": "Characteristic zero rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/CharZero.html"}, {"id": "Mathlib.Algebra.Notation.Support", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.4515, "macro_tier_override": null, "x": 2.822, "z": 4.804, "size": 0.5244, "title": "Support of a function", "summary": "In this file we define `Function.support f = {x | f x ≠ 0}` and prove its basic properties. We also define `Function.mulSupport f = {x | f x ≠ 1}`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Notation/Support.html"}, {"id": "Mathlib.Algebra.Ring.Units", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4309, "macro_tier_override": null, "x": 0.812, "z": 14.835, "size": 0.4296, "title": "Units in semirings and rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Units.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.Dual", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 64.338, "z": 66.957, "size": 0.2319, "title": "Quadratic form structures related to `Module.Dual`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/Dual.html"}, {"id": "Mathlib.LinearAlgebra.Dual.Lemmas", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 3, "macro_tier_score": 0.1804, "macro_tier_override": null, "x": 67.099, "z": 43.302, "size": 0.5079, "title": "Dual vector spaces", "summary": "The dual space of an $R$-module $M$ is the $R$-module of $R$-linear maps $M \\to R$. This file contains basic results on dual vector spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dual/Lemmas.html"}, {"id": "Mathlib.Algebra.ContinuedFractions.Basic", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 1, "macro_tier_score": 0.0029, "macro_tier_override": null, "x": 10.089, "z": 4.73, "size": 0.3142, "title": "Basic Definitions/Theorems for Continued Fractions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/ContinuedFractions/Basic.html"}, {"id": "Mathlib.RingTheory.Localization.BaseChange", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.1912, "macro_tier_override": null, "x": 62.257, "z": -24.373, "size": 0.4208, "title": "Localized Module", "summary": "Given a commutative semiring `R`, a multiplicative subset `S ⊆ R` and an `R`-module `M`, we can localize `M` by `S`. This gives us a `Localization S`-module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/BaseChange.html"}, {"id": "Mathlib.RingTheory.WittVector.MulCoeff", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -54.11, "z": -101.638, "size": 0.252, "title": "Leading terms of Witt vector multiplication", "summary": "The goal of this file is to study the leading terms of the formula for the `n+1`st coefficient of a product of Witt vectors `x` and `y` over a ring of characteristic `p`. We aim to isolate the `n+1`st coefficients of `x` and `y`, and express the rest of the product in terms of a function of the lower coefficients. For most of this file we work with terms of type `MvPolynomial (Fin 2 × ℕ) ℤ`. We will eventually…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/MulCoeff.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Supported", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.0775, "macro_tier_override": null, "x": 3.228, "z": 66.78, "size": 0.3329, "title": "Polynomials supported by a set of variables", "summary": "This file contains the definition and lemmas about `MvPolynomial.supported`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Supported.html"}, {"id": "Mathlib.Algebra.Ring.Action.Pointwise.Finset", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -27.183, "z": -6.093, "size": 0.2, "title": "Pointwise actions on sets in a ring", "summary": "This file proves properties of pointwise actions on sets in a ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Action/Pointwise/Finset.html"}, {"id": "Mathlib.FieldTheory.Normal.Closure", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 3, "macro_tier_score": 0.0626, "macro_tier_override": null, "x": 94.686, "z": 47.335, "size": 0.2956, "title": "Normal closures", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Normal/Closure.html"}, {"id": "Mathlib.RingTheory.Polynomial.Vieta", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 1, "macro_tier_score": 0.0017, "macro_tier_override": null, "x": 76.347, "z": -15.979, "size": 0.2996, "title": "Vieta's Formula", "summary": "The main result is `Multiset.prod_X_add_C_eq_sum_esymm`, which shows that the product of linear terms `X + λ` with `λ` in a `Multiset s` is equal to a linear combination of the symmetric functions `esymm s`. From this, we deduce `MvPolynomial.prod_X_add_C_eq_sum_esymm` which is the equivalent formula for the product of linear terms `X + X i` with `i` in a `Fintype σ` as a linear combination of the symmetric…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Vieta.html"}, {"id": "Mathlib.NumberTheory.NumberField.Cyclotomic.Galois", "region_id": "algebra", "micro_elevation": 1.0, "macro_tier": 5, "macro_tier_score": 0.0, "macro_tier_override": 5, "x": -3.961, "z": -141.089, "size": 0.2, "title": "Galois theory for cyclotomic fields", "summary": "In this file, we study the Galois theory of cyclotomic extensions of `ℚ`. Let `n` be an integer. There is an isomorphism between `Gal(ℚ(ζₙ)/ℚ)` and `(ℤ/nℤ)ˣ` that sends `σ` to `a_σ` such that `σ (ζₙ) = ζₙ ^ a_σ`. Following [Washington][washington.cyclotomic], we define the bijection between subfields of `ℚ(ζₙ)` and subgroups of the group `Xₙ` of Dirichlet characters of level `n` such that `F` corresponds to `Y` if…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Cyclotomic/Galois.html"}, {"id": "Mathlib.FieldTheory.Finite.Extension", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 52.613, "z": -106.579, "size": 0.2276, "title": "Extensions of finite fields", "summary": "In this file we develop the theory of extensions of finite fields. If `k` is a finite field (of cardinality `q = p ^ m`), then there is a unique (up to in general non-unique isomorphism) extension `l` of `k` of any given degree `n > 0`. This extension is Galois with cyclic Galois group of degree `n`, and the (arithmetic) Frobenius map `x ↦ x ^ q` is a generator.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Finite/Extension.html"}, {"id": "Mathlib.NumberTheory.Cyclotomic.Gal", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 76.369, "z": 98.237, "size": 0.2276, "title": "Galois group of cyclotomic extensions", "summary": "In this file, we show the relationship between the Galois group of `K(ζₙ)` and `(ZMod n)ˣ`; it is always a subgroup, and if the `n`th cyclotomic polynomial is irreducible, they are isomorphic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Cyclotomic/Gal.html"}, {"id": "Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal", "region_id": "algebra", "micro_elevation": 0.9868, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -102.093, "z": 94.751, "size": 0.2276, "title": "Ideals in cyclotomic fields", "summary": "In this file, we prove results about ideals in cyclotomic extensions of `ℚ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Cyclotomic/Ideal.html"}, {"id": "Mathlib.NumberTheory.NumberField.Ideal.Basic", "region_id": "algebra", "micro_elevation": 0.9868, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 61.455, "z": 124.997, "size": 0.2276, "title": "Basic results on integral ideals of a number field", "summary": "We study results about integral ideals of a number field `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Ideal/Basic.html"}, {"id": "Mathlib.Algebra.Polynomial.Degree.CardPowDegree", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 46.681, "z": -60.156, "size": 0.2382, "title": "Absolute value on polynomials over a finite field.", "summary": "Let `𝔽_q` be a finite field of cardinality `q`, then the map sending a polynomial `p` to `q ^ degree p` (where `q ^ degree 0 = 0`) is an absolute value.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Degree/CardPowDegree.html"}, {"id": "Mathlib.Algebra.Order.AbsoluteValue.Euclidean", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 26.057, "z": 14.283, "size": 0.2762, "title": "Euclidean absolute values", "summary": "This file defines a predicate `AbsoluteValue.IsEuclidean abv` stating the absolute value is compatible with the Euclidean domain structure on its domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/AbsoluteValue/Euclidean.html"}, {"id": "Mathlib.RingTheory.Adjoin.Polynomial", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -54.738, "z": -35.055, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Adjoin/Polynomial.html"}, {"id": "Mathlib.RingTheory.Valuation.ExtendToLocalization", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 3, "macro_tier_score": 0.0357, "macro_tier_override": null, "x": 49.752, "z": -35.787, "size": 0.2666, "title": "Extending valuations to a localization", "summary": "We show that, given a valuation `v` taking values in a linearly ordered commutative *group* with zero `Γ`, and a submonoid `S` of `v.supp.primeCompl`, the valuation `v` can be naturally extended to the localization `S⁻¹A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/ExtendToLocalization.html"}, {"id": "Mathlib.RingTheory.Localization.Defs", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 3, "macro_tier_score": 0.2371, "macro_tier_override": null, "x": 18.933, "z": 22.902, "size": 0.4402, "title": "Localizations of commutative rings", "summary": "We characterize the localization of a commutative ring `R` at a submonoid `M` up to isomorphism; that is, a commutative ring `S` is the localization of `R` at `M` iff we can find a ring homomorphism `f : R →+* S` satisfying 3 properties: 1. For all `y ∈ M`, `f y` is a unit; 2. For all `z : S`, there exists `(x, y) : R × M` such that `z * f y = f x`; 3. For all `x, y : R` such that `f x = f y`, there exists `c ∈ M`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Defs.html"}, {"id": "Mathlib.NumberTheory.NumberField.ExistsRamified", "region_id": "algebra", "micro_elevation": 0.9737, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -134.67, "z": -27.405, "size": 0.2, "title": "Every number field has a ramified prime over `ℚ`", "summary": "...except `ℚ` itself. This is a trivial corollary of `NumberField.not_dvd_discr_iff_forall_mem` and `NumberField.abs_discr_gt_two` but is placed in a separate file to avoid large imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/ExistsRamified.html"}, {"id": "Mathlib.NumberTheory.NumberField.Discriminant.Different", "region_id": "algebra", "micro_elevation": 0.9605, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 134.774, "z": -14.692, "size": 0.2679, "title": "(Absolute) Discriminant and Different Ideal", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Discriminant/Different.html"}, {"id": "Mathlib.FieldTheory.Minpoly.Field", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 3, "macro_tier_score": 0.1469, "macro_tier_override": null, "x": -45.244, "z": -78.956, "size": 0.3712, "title": "Minimal polynomials on an algebra over a field", "summary": "This file specializes the theory of minpoly to the setting of field extensions and derives some well-known properties, amongst which the fact that minimal polynomials are irreducible, and uniquely determined by their defining property.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Minpoly/Field.html"}, {"id": "Mathlib.Algebra.Order.Sub.Unbundled.Hom", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -0.676, "z": -14.842, "size": 0.2221, "title": "Lemmas about subtraction in unbundled canonically ordered monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Sub/Unbundled/Hom.html"}, {"id": "Mathlib.Algebra.Ring.Basic", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4552, "macro_tier_override": null, "x": 11.047, "z": 1.462, "size": 0.5022, "title": "Semirings and rings", "summary": "This file gives lemmas about semirings, rings and domains. This is analogous to `Mathlib/Algebra/Group/Basic.lean`, the difference being that the former is about `+` and `*` separately, while the present file is about their interaction. For the definitions of semirings and rings see `Mathlib/Algebra/Ring/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Basic.html"}, {"id": "Mathlib.Algebra.Group.Equiv.Basic", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4992, "macro_tier_override": null, "x": -9.272, "z": -0.504, "size": 0.6302, "title": "Multiplicative and additive equivs", "summary": "This file contains basic results on `MulEquiv` and `AddEquiv`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Equiv/Basic.html"}, {"id": "Mathlib.Algebra.Group.Units.Hom", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4888, "macro_tier_override": null, "x": 12.723, "z": 2.671, "size": 0.5421, "title": "Monoid homomorphisms and units", "summary": "This file allows to lift monoid homomorphisms to group homomorphisms of their units subgroups. It also contains unrelated results about `Units` that depend on `MonoidHom`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Units/Hom.html"}, {"id": "Mathlib.RingTheory.LocalRing.Quotient", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": 20.679, "z": -77.134, "size": 0.2658, "title": null, "summary": "We gather results about the quotients of local rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/Quotient.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.OrzechProperty", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 2, "macro_tier_score": 0.0131, "macro_tier_override": null, "x": -61.282, "z": 41.988, "size": 0.3243, "title": "Bases of modules and the Orzech property", "summary": "It is shown in this file that any spanning set of a module over a ring satisfying the Orzech property of cardinality not exceeding the rank of the module must be linearly independent, and therefore is a basis.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/OrzechProperty.html"}, {"id": "Mathlib.LinearAlgebra.FreeModule.PID", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 68.58, "z": 4.301, "size": 0.3721, "title": "Free modules over PID", "summary": "A free `R`-module `M` is a module with a basis over `R`, equivalently it is an `R`-module linearly equivalent to `ι →₀ R` for some `ι`. This file proves a submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank, if `R` is a principal ideal domain (PID), i.e. we have instances `[IsDomain R] [IsPrincipalIdealRing R]`. We express \"free `R`-module of finite rank\" as a module `M` which has…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeModule/PID.html"}, {"id": "Mathlib.RingTheory.Ideal.Quotient.Index", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 2, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": 65.211, "z": -14.747, "size": 0.2321, "title": "Indices of ideals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Quotient/Index.html"}, {"id": "Mathlib.Algebra.Ring.Submonoid.Pointwise", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.3444, "macro_tier_override": null, "x": -21.936, "z": 10.084, "size": 0.3166, "title": "Elementwise monoid structure of additive submonoids", "summary": "These definitions are a cut-down versions of the ones around `Submodule.mul`, as that API is usually more useful. `SMul (AddSubmonoid R) (AddSubmonoid A)` is also provided given `DistribSMul R A`, and when `R = A` it is definitionally equal to the multiplication on `AddSubmonoid R`. These are all available in the `Pointwise` locale. Additionally, it provides various degrees of monoid structure: * `AddSubmonoid.one`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Submonoid/Pointwise.html"}, {"id": "Mathlib.Algebra.Module.Defs", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4598, "macro_tier_override": null, "x": 13.184, "z": -6.85, "size": 0.724, "title": "Modules over a ring", "summary": "In this file we define * `Module R M` : an additive commutative monoid `M` is a `Module` over a `Semiring R` if for `r : R` and `x : M` their \"scalar multiplication\" `r • x : M` is defined, and the operation `•` satisfies some natural associativity and distributivity axioms similar to those on a ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Defs.html"}, {"id": "Mathlib.Algebra.ModEq", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/ModEq.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -27.202, "z": 42.124, "size": 0.3529, "title": "Affine maps", "summary": "This file defines affine maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/AffineMap.html"}, {"id": "Mathlib.GroupTheory.Frattini", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 10.999, "z": -50.824, "size": 0.2, "title": "The Frattini subgroup", "summary": "We give the definition of the Frattini subgroup of a group, and three elementary results: * The Frattini subgroup is characteristic. * If every subgroup of a group is contained in a maximal subgroup, then the Frattini subgroup consists of the non-generating elements of the group. * The Frattini subgroup of a finite group is nilpotent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Frattini.html"}, {"id": "Mathlib.GroupTheory.Nilpotent", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -41.569, "z": 28.043, "size": 0.2951, "title": "Nilpotent groups", "summary": "An API for nilpotent groups, that is, groups for which the upper central series reaches `⊤`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Nilpotent.html"}, {"id": "Mathlib.GroupTheory.Commensurable", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 5.28, "z": 36.766, "size": 0.2649, "title": "Commensurability for subgroups", "summary": "Two subgroups `H` and `K` of a group `G` are commensurable if `H ∩ K` has finite index in both `H` and `K`. This file defines commensurability for subgroups of a group `G`. It goes on to prove that commensurability defines an equivalence relation on subgroups of `G` and finally defines the commensurator of a subgroup `H` of `G`, which is the elements `g` of `G` such that `gHg⁻¹` is commensurable with `H`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Commensurable.html"}, {"id": "Mathlib.Algebra.Group.Subsemigroup.Operations", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.3924, "macro_tier_override": null, "x": -11.889, "z": 11.748, "size": 0.3352, "title": "Operations on `Subsemigroup`s", "summary": "In this file we define various operations on `Subsemigroup`s and `MulHom`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subsemigroup/Operations.html"}, {"id": "Mathlib.NumberTheory.LSeries.Injectivity", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -76.527, "z": 15.093, "size": 0.2, "title": "A converging L-series determines its coefficients", "summary": "We show that two functions `f` and `g : ℕ → ℂ` whose L-series agree and both converge somewhere must agree on all nonzero arguments. See `LSeries_eq_iff_of_abscissaOfAbsConv_lt_top` and `LSeries_injOn`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/Injectivity.html"}, {"id": "Mathlib.Algebra.QuaternionBasis", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 70.068, "z": 8.422, "size": 0.2302, "title": "Basis on a quaternion-like algebra", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/QuaternionBasis.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Ceva", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -33.312, "z": -51.443, "size": 0.2478, "title": "Ceva's theorem.", "summary": "This file proves various versions of Ceva's theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Ceva.html"}, {"id": "Mathlib.Algebra.Homology.Linear", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 2, "macro_tier_score": 0.0067, "macro_tier_override": null, "x": -12.856, "z": -1.931, "size": 0.3426, "title": "The category of homological complexes is linear", "summary": "In this file, we define the instance `Linear R (HomologicalComplex C c)` when the category `C` is `R`-linear.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Linear.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.AB", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -63.021, "z": 22.324, "size": 0.2595, "title": "AB axioms in module categories", "summary": "This file proves that the category of modules over a ring satisfies Grothendieck's axioms AB5, AB4, and AB4\\*. Further, it proves that `R` is a separator in the category of modules over `R`, and concludes that this category is Grothendieck abelian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/AB.html"}, {"id": "Mathlib.Algebra.Category.Grp.AB", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 42.146, "z": 49.485, "size": 0.2797, "title": "AB axioms for the category of abelian groups", "summary": "This file proves that the category of abelian groups satisfies Grothendieck's axioms AB5, AB4, and AB4\\*.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/AB.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Colimits", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 2, "macro_tier_score": 0.0311, "macro_tier_override": null, "x": -52.862, "z": -10.31, "size": 0.4033, "title": "The category of R-modules has all colimits.", "summary": "From the existence of colimits in `AddCommGrpCat`, we deduce the existence of colimits in `ModuleCat R`. This way, we get for free that the functor `forget₂ (ModuleCat R) AddCommGrpCat` commutes with colimits. Note that finite colimits can already be obtained from the instance `Abelian (Module R)`. TODO: In fact, in `ModuleCat R` there is a much nicer model of colimits as quotients of finitely supported functions,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Colimits.html"}, {"id": "Mathlib.LinearAlgebra.SymmetricAlgebra.Basis", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -81.398, "z": -7.193, "size": 0.2, "title": "A basis for `SymmetricAlgebra R M`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/SymmetricAlgebra/Basis.html"}, {"id": "Mathlib.RingTheory.MvPolynomial", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.0091, "macro_tier_override": null, "x": -62.289, "z": -49.974, "size": 0.2586, "title": "Multivariate polynomials over fields", "summary": "This file contains basic facts about multivariate polynomials over fields, for example that the dimension of the space of multivariate polynomials over a field is equal to the cardinality of finitely supported functions from the indexing set to `ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial.html"}, {"id": "Mathlib.Algebra.Ring.PUnit", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.3909, "macro_tier_override": null, "x": 1.687, "z": 9.131, "size": 0.3784, "title": "`PUnit` is a commutative ring", "summary": "This file collects facts about algebraic structures on the one-element type, e.g. that it is a commutative ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/PUnit.html"}, {"id": "Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 33.11, "z": 119.944, "size": 0.287, "title": "Embeddings of number fields", "summary": "This file defines the embeddings of a number field and, in particular, the embeddings into the field of complex numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/InfinitePlace/Embeddings.html"}, {"id": "Mathlib.Algebra.Algebra.Hom.Rat", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": 38.958, "z": 17.517, "size": 0.2604, "title": "Homomorphisms of `ℚ`-algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Hom/Rat.html"}, {"id": "Mathlib.NumberTheory.NumberField.Basic", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 2, "macro_tier_score": 0.0145, "macro_tier_override": null, "x": -52.503, "z": 110.759, "size": 0.3455, "title": "Number fields", "summary": "This file defines a number field and the ring of integers corresponding to it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Basic.html"}, {"id": "Mathlib.Algebra.Module.LocalizedModule.Basic", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 3, "macro_tier_score": 0.1934, "macro_tier_override": null, "x": 24.254, "z": -39.591, "size": 0.3686, "title": "Localized Module", "summary": "Given a commutative semiring `R`, a multiplicative subset `S ⊆ R` and an `R`-module `M`, we can localize `M` by `S`. This gives us a `Localization S`-module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LocalizedModule/Basic.html"}, {"id": "Mathlib.Algebra.Homology.Factorizations.Basic", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 8.465, "z": 3.816, "size": 0.2421, "title": "Basic definitions for factorization lemmas", "summary": "We define the class of morphisms `degreewiseEpiWithInjectiveKernel : MorphismProperty (CochainComplex C ℤ)` in the category of cochain complexes in an abelian category `C`. When restricted to the full subcategory of bounded below cochain complexes in an abelian category `C` that has enough injectives, this is the class of fibrations for a model category structure on the bounded below category of cochain complexes in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Factorizations/Basic.html"}, {"id": "Mathlib.Algebra.Group.Subsemigroup.MulOpposite", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4538, "macro_tier_override": null, "x": 8.057, "z": -7.697, "size": 0.2796, "title": "Subsemigroup of opposite semigroups", "summary": "For every semigroup `M`, we construct an equivalence between subsemigroups of `M` and that of `Mᵐᵒᵖ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subsemigroup/MulOpposite.html"}, {"id": "Mathlib.Algebra.Group.Opposite", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.5031, "macro_tier_override": null, "x": -0.139, "z": 9.285, "size": 0.6423, "title": "Group structures on the multiplicative and additive opposites", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Opposite.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.Localization", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.0214, "macro_tier_override": null, "x": 6.51, "z": 70.271, "size": 0.2772, "title": "Localization and multivariate polynomial rings", "summary": "In this file we show some results connecting multivariate polynomial rings and localization.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/Localization.html"}, {"id": "Mathlib.RingTheory.Ideal.Quotient.Operations", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 4, "macro_tier_score": 0.2749, "macro_tier_override": null, "x": 8.979, "z": 60.625, "size": 0.5154, "title": "More operations on modules and ideals related to quotients", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Quotient/Operations.html"}, {"id": "Mathlib.RingTheory.TensorProduct.MvPolynomial", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.1803, "macro_tier_override": null, "x": 63.472, "z": -26.326, "size": 0.311, "title": "Tensor Product of (multivariate) polynomial rings", "summary": "Let `Semiring R`, `Algebra R S` and `Module R N`. * `MvPolynomial.rTensor` gives the linear equivalence `MvPolynomial σ S ⊗[R] N ≃ₗ[R] (σ →₀ ℕ) →₀ (S ⊗[R] N)` characterized, for `p : MvPolynomial σ S`, `n : N` and `d : σ →₀ ℕ`, by `rTensor (p ⊗ₜ[R] n) d = (coeff d p) ⊗ₜ[R] n` * `MvPolynomial.scalarRTensor` gives the linear equivalence `MvPolynomial σ R ⊗[R] N ≃ₗ[R] (σ →₀ ℕ) →₀ N` such that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/MvPolynomial.html"}, {"id": "Mathlib.LinearAlgebra.Complex.FiniteDimensional", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 2, "macro_tier_score": 0.0081, "macro_tier_override": null, "x": -16.986, "z": 70.409, "size": 0.4129, "title": "Complex number as a finite-dimensional vector space over `ℝ`", "summary": "This file contains the `FiniteDimensional ℝ ℂ` instance, as well as some results about the rank (`finrank` and `Module.rank`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Complex/FiniteDimensional.html"}, {"id": "Mathlib.GroupTheory.Perm.ClosureSwap", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 17.247, "z": -19.456, "size": 0.2338, "title": "Subgroups generated by transpositions", "summary": "This file studies subgroups generated by transpositions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/ClosureSwap.html"}, {"id": "Mathlib.GroupTheory.GroupAction.FixedPoints", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 3, "macro_tier_score": 0.065, "macro_tier_override": null, "x": 5.483, "z": -23.512, "size": 0.3132, "title": "Properties of `fixedPoints` and `fixedBy`", "summary": "This module contains some useful properties of `MulAction.fixedPoints` and `MulAction.fixedBy` that don't directly belong to `Mathlib/GroupTheory/GroupAction/Basic.lean`, as well as their interaction with `MulActionHom`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/FixedPoints.html"}, {"id": "Mathlib.GroupTheory.Perm.Support", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 3, "macro_tier_score": 0.2522, "macro_tier_override": null, "x": 17.873, "z": -5.046, "size": 0.2973, "title": "support of a permutation", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Support.html"}, {"id": "Mathlib.RingTheory.ClassGroup", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -79.82, "z": 63.735, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/ClassGroup.html"}, {"id": "Mathlib.RingTheory.ClassGroup.Basic", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 2, "macro_tier_score": 0.0089, "macro_tier_override": null, "x": 38.015, "z": 92.802, "size": 0.3426, "title": "The ideal class group", "summary": "This file defines the ideal class group `ClassGroup R` of fractional ideals of `R` inside its field of fractions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/ClassGroup/Basic.html"}, {"id": "Mathlib.Algebra.Ring.Action.End", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 3, "macro_tier_score": 0.1656, "macro_tier_override": null, "x": 22.546, "z": 12.949, "size": 0.2893, "title": "Ring automorphisms", "summary": "This file defines the automorphism group structure on `RingAut R := RingEquiv R R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Action/End.html"}, {"id": "Mathlib.Algebra.Ring.Action.Group", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.3866, "macro_tier_override": null, "x": -12.337, "z": -20.753, "size": 0.6163, "title": "If a group acts multiplicatively on a semiring, each group element acts by a ring automorphism.", "summary": "This result is split out from `Mathlib/Algebra/Ring/Action/Basic.lean` to avoid needing the import of `Mathlib/Algebra/GroupWithZero/Action/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Action/Group.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.Finset.Pi", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.411, "macro_tier_override": null, "x": 6.214, "z": 13.496, "size": 0.3201, "title": "Products over `univ.pi`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Finset/Pi.html"}, {"id": "Mathlib.Algebra.BigOperators.Ring.Multiset", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.411, "macro_tier_override": null, "x": -9.258, "z": -18.211, "size": 0.3201, "title": "Lemmas about `Multiset.sum` and `Multiset.prod` requiring extra algebra imports", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Ring/Multiset.html"}, {"id": "Mathlib.Algebra.Order.Ring.IsNonarchimedean", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 3, "macro_tier_score": 0.0491, "macro_tier_override": null, "x": 10.541, "z": 15.29, "size": 0.3592, "title": "Nonarchimedean functions", "summary": "A function `f : α → R` is nonarchimedean if it satisfies the strong triangle inequality `f (a + b) ≤ max (f a) (f b)` for all `a b : α`. This file proves basic properties of nonarchimedean functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/IsNonarchimedean.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 3, "macro_tier_score": 0.0502, "macro_tier_override": null, "x": 71.855, "z": 69.959, "size": 0.3623, "title": "Dedekind domains and ideals", "summary": "In this file, we prove some results on the unique factorization monoid structure of the ideals. The unique factorization of ideals and invertibility of fractional ideals can be found in `Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.html"}, {"id": "Mathlib.RingTheory.Valuation.Discrete.Basic", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 3, "macro_tier_score": 0.0359, "macro_tier_override": null, "x": -59.87, "z": 78.128, "size": 0.2824, "title": "Discrete Valuations", "summary": "Given a linearly ordered commutative group with zero `Γ`, a valuation `v : A → Γ` on a ring `A` is *discrete*, if there is an element `γ : Γˣ` that is `< 1` and generated the range of `v`, implemented as `MonoidWithZeroHom.valueGroup v`. When `Γ := ℤₘ₀` (defined in `Multiplicative.termℤₘ₀`), `γ = ofAdd (-1)` and the condition of being discrete is equivalent to asking that `ofAdd (-1 : ℤ)` belongs to the image, in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/Discrete/Basic.html"}, {"id": "Mathlib.RingTheory.Polynomial.Subring", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 3, "macro_tier_score": 0.1764, "macro_tier_override": null, "x": 53.656, "z": -4.66, "size": 0.2517, "title": "Polynomials over subrings", "summary": "Given a ring `K` with a subring `R`, we construct a map from polynomials in `K[X]` with coefficients in `R` to `R[X]`. We provide several lemmas to deal with coefficients, degree, and evaluation of `Polynomial.toSubring`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Subring.html"}, {"id": "Mathlib.Algebra.Polynomial.Eval.Coeff", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 4, "macro_tier_score": 0.286, "macro_tier_override": null, "x": 51.735, "z": -5.25, "size": 0.3995, "title": "Evaluation of polynomials", "summary": "This file contains results on the interaction of `Polynomial.eval` and `Polynomial.coeff`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Eval/Coeff.html"}, {"id": "Mathlib.Algebra.WithConv", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 2, "macro_tier_score": 0.0069, "macro_tier_override": null, "x": -26.148, "z": -14.116, "size": 0.2654, "title": "Type synonym for linear map convolutive ring and intrinsic star", "summary": "This files provides the type synonym `WithConv` which we will use in later files to put the convolutive product on linear maps instance and the intrinsic star instance. This is to ensure that we only have one multiplication, one unit, and one star. This is given for any type `A` so that we can have `WithConv (A →ₗ[R] B)`, `WithConv (A →L[R] B)`, `WithConv (Matrix m n R)`, etc.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/WithConv.html"}, {"id": "Mathlib.Algebra.Module.TransferInstance", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 4, "macro_tier_score": 0.3664, "macro_tier_override": null, "x": -19.426, "z": 19.966, "size": 0.4649, "title": "Transfer algebraic structures across `Equiv`s", "summary": "This continues the pattern set in `Mathlib/Algebra/Group/TransferInstance.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/TransferInstance.html"}, {"id": "Mathlib.Algebra.SkewMonoidAlgebra.Basic", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": -29.098, "z": 43.097, "size": 0.3061, "title": "Skew Monoid Algebras", "summary": "Given a monoid `G` acting on a ring `k`, the skew monoid algebra of `G` over `k` is the set of finitely supported functions `f : G → k` for which addition is defined pointwise and multiplication of two elements `f` and `g` is given by the finitely supported function whose value at `a` is the sum of `f x * (x • g y)` over all pairs `x, y` such that `x * y = a`, where `•` denotes the action of `G` on `k`. When this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/SkewMonoidAlgebra/Basic.html"}, {"id": "Mathlib.RingTheory.Algebraic.Basic", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 3, "macro_tier_score": 0.1858, "macro_tier_override": null, "x": 37.912, "z": 70.285, "size": 0.379, "title": "Algebraic elements and algebraic extensions", "summary": "An element of an R-algebra is algebraic over R if it is the root of a nonzero polynomial. An R-algebra is algebraic over R if and only if all its elements are algebraic over R. The main result in this file proves transitivity of algebraicity: a tower of algebraic field extensions is algebraic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Algebraic/Basic.html"}, {"id": "Mathlib.Algebra.Algebra.Equiv", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 4, "macro_tier_score": 0.3976, "macro_tier_override": null, "x": -42.327, "z": -5.743, "size": 0.8092, "title": "Isomorphisms of `R`-algebras", "summary": "This file defines bundled isomorphisms of `R`-algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Equiv.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.Finite", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.2424, "macro_tier_override": null, "x": 24.558, "z": -68.139, "size": 0.4696, "title": "Conditions for rank to be finite", "summary": "Also contains characterization for when rank equals zero or rank equals one.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/Finite.html"}, {"id": "Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -71.249, "z": 13.023, "size": 0.2, "title": "Analytic part of the Lindemann-Weierstrass theorem", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Transcendental/Lindemann/AnalyticalPart.html"}, {"id": "Mathlib.Algebra.Polynomial.SumIteratedDerivative", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -0.833, "z": -70.567, "size": 0.2276, "title": "Sum of iterated derivatives", "summary": "This file introduces `Polynomial.sumIDeriv`, the sum of the iterated derivatives of a polynomial, as a linear map. This is used in particular in the proof of the Lindemann-Weierstrass theorem (see https://github.com/leanprover-community/mathlib4/pull/6718).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/SumIteratedDerivative.html"}, {"id": "Mathlib.RingTheory.Int.Basic", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 2, "macro_tier_score": 0.0276, "macro_tier_override": null, "x": 15.171, "z": 61.294, "size": 0.3353, "title": "Divisibility over ℤ", "summary": "This file collects results for the integers that use ring theory in their proofs or cases of ℤ being examples of structures in ring theory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Int/Basic.html"}, {"id": "Mathlib.Algebra.Algebra.Spectrum.Quasispectrum", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 2, "macro_tier_score": 0.0069, "macro_tier_override": null, "x": -47.305, "z": -38.964, "size": 0.3584, "title": "Quasiregularity and quasispectrum", "summary": "For a non-unital ring `R`, an element `r : R` is *quasiregular* if it is invertible in the monoid `(R, ∘)` where `x ∘ y := y + x + x * y` with identity `0 : R`. We implement this both as a type synonym `PreQuasiregular` which has an associated `Monoid` instance (note: *not* an `AddMonoid` instance despite the fact that `0 : R` is the identity in this monoid) so that one may access the quasiregular elements of `R` as…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Spectrum/Quasispectrum.html"}, {"id": "Mathlib.Algebra.Algebra.Unitization", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 2, "macro_tier_score": 0.0066, "macro_tier_override": null, "x": 49.323, "z": -16.472, "size": 0.3369, "title": "Unitization of a non-unital algebra", "summary": "Given a non-unital `R`-algebra `A` (given via the type classes `[NonUnitalRing A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A]`) we construct the minimal unital `R`-algebra containing `A` as an ideal. This object `Unitization R A` is a type synonym for `R × A` on which we place a different multiplicative structure, namely, `(r₁, a₁) * (r₂, a₂) = (r₁ * r₂, r₁ • a₂ + r₂ • a₁ + a₁ * a₂)` where the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Unitization.html"}, {"id": "Mathlib.Algebra.Order.Ring.Ordering.Basic", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -38.362, "z": -14.062, "size": 0.2, "title": "Ring orderings", "summary": "We prove basic properties of (pre)orderings on rings and their supports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Ordering/Basic.html"}, {"id": "Mathlib.Algebra.Order.Ring.Ordering.Defs", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -34.138, "z": 18.858, "size": 0.2276, "title": "Ring orderings", "summary": "Let `R` be a commutative ring. A preordering on `R` is a subset closed under addition and multiplication that contains all squares, but not `-1`. The support of a preordering `P` is the set of elements `x` such that both `x` and `-x` lie in `P`. An ordering `O` on `R` is a preordering such that 1. `O` contains either `x` or `-x` for each `x` in `R` and 2. the support of `O` is a prime ideal. We define preorderings,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Ordering/Defs.html"}, {"id": "Mathlib.Algebra.Ring.SumsOfSquares", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 2, "macro_tier_score": 0.0063, "macro_tier_override": null, "x": -27.715, "z": -2.818, "size": 0.3133, "title": "Sums of squares", "summary": "We introduce a predicate for sums of squares in a ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/SumsOfSquares.html"}, {"id": "Mathlib.Algebra.Order.Antidiag.Finsupp", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 2, "macro_tier_score": 0.0221, "macro_tier_override": null, "x": 23.738, "z": 10.607, "size": 0.3385, "title": "Antidiagonal of finitely supported functions as finsets", "summary": "This file defines the finset of finitely functions summing to a specific value on a finset. Such finsets should be thought of as the \"antidiagonals\" in the space of finitely supported functions. Precisely, for a commutative monoid `μ` with antidiagonals (see `Finset.HasAntidiagonal`), `Finset.finsuppAntidiag s n` is the finset of all finitely supported functions `f : ι →₀ μ` with support contained in `s` and such…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Antidiag/Finsupp.html"}, {"id": "Mathlib.Algebra.Order.Antidiag.Pi", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 2, "macro_tier_score": 0.0218, "macro_tier_override": null, "x": -21.021, "z": 7.4, "size": 0.3188, "title": "Antidiagonal of functions as finsets", "summary": "This file provides the finset of functions summing to a specific value on a finset. Such finsets should be thought of as the \"antidiagonals\" in the space of functions. Precisely, for a commutative monoid `μ` with antidiagonals (see `Finset.HasAntidiagonal`), `Finset.piAntidiag s n` is the finset of all functions `f : ι → μ` with support contained in `s` and such that the sum of its values equals `n : μ`. We define…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Antidiag/Pi.html"}, {"id": "Mathlib.Algebra.CharP.Lemmas", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 3, "macro_tier_score": 0.2086, "macro_tier_override": null, "x": -0.247, "z": -1.841, "size": 0.354, "title": "Characteristic of semirings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/Lemmas.html"}, {"id": "Mathlib.Algebra.Category.HopfAlgCat.Monoidal", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -51.228, "z": -53.797, "size": 0.2, "title": "The monoidal structure on the category of Hopf algebras", "summary": "In `Mathlib/RingTheory/HopfAlgebra/TensorProduct.lean`, given two Hopf `R`-algebras `A, B`, we define a Hopf `R`-algebra instance on `A ⊗[R] B`. Here, we use this to declare a `MonoidalCategory` instance on the category of Hopf algebras, via the existing monoidal structure on `BialgCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/HopfAlgCat/Monoidal.html"}, {"id": "Mathlib.Algebra.Category.BialgCat.Monoidal", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 26.62, "z": -67.36, "size": 0.239, "title": "The monoidal structure on the category of bialgebras", "summary": "In `Mathlib/RingTheory/Bialgebra/TensorProduct.lean`, given two `R`-bialgebras `A, B`, we define a bialgebra instance on `A ⊗[R] B` as well as the tensor product of two `BialgHom`s as a `BialgHom`, and the associator and left/right unitors for bialgebras as `BialgEquiv`s. In this file, we declare a `MonoidalCategory` instance on the category of bialgebras, with data fields given by the definitions in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/BialgCat/Monoidal.html"}, {"id": "Mathlib.Algebra.Category.HopfAlgCat.Basic", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -24.733, "z": -68.076, "size": 0.239, "title": "The category of Hopf algebras over a commutative ring", "summary": "We introduce the bundled category `HopfAlgCat` of Hopf algebras over a fixed commutative ring `R` along with the forgetful functor to `BialgCat`. This file mimics `Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/HopfAlgCat/Basic.html"}, {"id": "Mathlib.RingTheory.HopfAlgebra.TensorProduct", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 38.348, "z": -61.445, "size": 0.2516, "title": "Tensor products of Hopf algebras", "summary": "We define the Hopf algebra instance on the tensor product of two Hopf algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HopfAlgebra/TensorProduct.html"}, {"id": "Mathlib.LinearAlgebra.Contraction", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 3, "macro_tier_score": 0.0744, "macro_tier_override": null, "x": 45.821, "z": -67.659, "size": 0.4068, "title": "Contractions", "summary": "Given modules $M, N$ over a commutative ring $R$, this file defines the natural linear maps: $M^* \\otimes M \\to R$, $M \\otimes M^* \\to R$, and $M^* \\otimes N → Hom(M, N)$, as well as proving some basic properties of these maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Contraction.html"}, {"id": "Mathlib.RingTheory.RingHom.Unramified", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 36.103, "z": -101.485, "size": 0.2854, "title": "The meta properties of unramified ring homomorphisms.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/Unramified.html"}, {"id": "Mathlib.Algebra.Polynomial.GroupRingAction", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.091, "macro_tier_override": null, "x": 42.212, "z": -54.221, "size": 0.2411, "title": "Group action on rings applied to polynomials", "summary": "This file contains instances and definitions relating `MulSemiringAction` to `Polynomial`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/GroupRingAction.html"}, {"id": "Mathlib.GroupTheory.Coset.Card", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 3, "macro_tier_score": 0.2559, "macro_tier_override": null, "x": -8.285, "z": 26.597, "size": 0.3284, "title": "Lagrange's theorem: the order of a subgroup divides the order of the group.", "summary": "* `Subgroup.card_subgroup_dvd_card`: Lagrange's theorem (for multiplicative groups); there is an analogous version for additive groups", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Coset/Card.html"}, {"id": "Mathlib.Algebra.Group.NatPowAssoc", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": 14.045, "z": 9.061, "size": 0.2593, "title": "Typeclasses for power-associative structures", "summary": "In this file we define power-associativity for algebraic structures with a multiplication operation. The class is a Prop-valued mixin named `NatPowAssoc`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/NatPowAssoc.html"}, {"id": "Mathlib.Algebra.Group.PNatPowAssoc", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2416, "title": "Typeclasses for power-associative structures", "summary": "In this file we define power-associativity for algebraic structures with a multiplication operation. The class is a Prop-valued mixin named `PNatPowAssoc`, where `PNat` means only strictly positive powers are considered.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/PNatPowAssoc.html"}, {"id": "Mathlib.Algebra.Squarefree.Basic", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.1423, "macro_tier_override": null, "x": 9.53, "z": -33.975, "size": 0.3656, "title": "Squarefree elements of monoids", "summary": "An element of a monoid is squarefree when it is not divisible by any squares except the squares of units. Results about squarefree natural numbers are proved in `Data.Nat.Squarefree`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Squarefree/Basic.html"}, {"id": "Mathlib.Algebra.Order.Field.Pi", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 11.556, "z": -5.955, "size": 0.2329, "title": "Lemmas about (finite domain) functions into fields.", "summary": "We split this from `Algebra.Order.Field.Basic` to avoid importing the finiteness hierarchy there.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Field/Pi.html"}, {"id": "Mathlib.Algebra.Order.Pi", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 1, "macro_tier_score": 0.0033, "macro_tier_override": null, "x": -3.06, "z": 20.198, "size": 0.3981, "title": "Pi instances for ordered groups and monoids", "summary": "This file defines instances for ordered group, monoid, and related structures on Pi types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Pi.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Counit", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": -43.712, "z": 42.957, "size": 0.2808, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Counit.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Eval", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 4, "macro_tier_score": 0.2744, "macro_tier_override": null, "x": -7.237, "z": 58.987, "size": 0.423, "title": "Multivariate polynomials", "summary": "This file defines functions for evaluating multivariate polynomials. These include generically evaluating a polynomial given a valuation of all its variables, and more advanced evaluations that allow one to map the coefficients to different rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Eval.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.ExtendHomology", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": -20.554, "z": -12.666, "size": 0.2495, "title": "Homology of the extension of a homological complex", "summary": "Given an embedding `e : c.Embedding c'` and `K : HomologicalComplex C c`, we shall compute the homology of `K.extend e`. In degrees that are not in the image of `e.f`, the homology is obviously zero. When `e.f j = j`, we construct an isomorphism `(K.extend e).homology j' ≅ K.homology j`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/ExtendHomology.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.Extend", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 2, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -21.91, "z": 4.076, "size": 0.3342, "title": "The extension of a homological complex by an embedding of complex shapes", "summary": "Given an embedding `e : Embedding c c'` of complex shapes, and `K : HomologicalComplex C c`, we define `K.extend e : HomologicalComplex C c'`, and this leads to a functor `e.extendFunctor C : HomologicalComplex C c ⥤ HomologicalComplex C c'`. This construction first appeared in the Liquid Tensor Experiment.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/Extend.html"}, {"id": "Mathlib.Algebra.Module.Basic", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.4062, "macro_tier_override": null, "x": 21.582, "z": -5.558, "size": 0.4171, "title": "Further basic results about modules.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Multilinear.Finsupp", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 3, "macro_tier_score": 0.2077, "macro_tier_override": null, "x": -55.511, "z": 4.756, "size": 0.2828, "title": "Interactions between finitely-supported functions and multilinear maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Multilinear/Finsupp.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.Different", "region_id": "algebra", "micro_elevation": 0.9211, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -125.952, "z": 32.192, "size": 0.2589, "title": "The different ideal", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/Different.html"}, {"id": "Mathlib.NumberTheory.RamificationInertia.Unramified", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": 110.208, "z": 65.385, "size": 0.2433, "title": "Unramified and ramification index", "summary": "We connect `Ideal.ramificationIdx` to the commutative algebra notion predicate of `IsUnramifiedAt`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/RamificationInertia/Unramified.html"}, {"id": "Mathlib.RingTheory.Conductor", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 2, "macro_tier_score": 0.0092, "macro_tier_override": null, "x": -58.849, "z": -74.214, "size": 0.2731, "title": "The conductor ideal", "summary": "This file defines the conductor ideal of an element `x` of `R`-algebra `S`. This is the ideal of `S` consisting of all elements `a` such that for all `b` in `S`, the product `a * b` lies in the `R`-subalgebra of `S` generated by `x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Conductor.html"}, {"id": "Mathlib.RingTheory.FractionalIdeal.Extended", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -100.519, "z": 18.146, "size": 0.2645, "title": "Extension of fractional ideals", "summary": "This file defines the extension of a fractional ideal along a ring homomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/FractionalIdeal/Extended.html"}, {"id": "Mathlib.RingTheory.Trace.Quotient", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": -58.77, "z": 111.779, "size": 0.2433, "title": null, "summary": "We gather results about the relations between the trace map on `B → A` and the trace map on quotients and localizations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Trace/Quotient.html"}, {"id": "Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -29.023, "z": -26.052, "size": 0.2, "title": "Termination of Continued Fraction Computations (`GenContFract.of`)", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.html"}, {"id": "Mathlib.Algebra.ContinuedFractions.Computation.Approximations", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 0.971, "z": 37.131, "size": 0.2625, "title": "Approximations for Continued Fraction Computations (`GenContFract.of`)", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/ContinuedFractions/Computation/Approximations.html"}, {"id": "Mathlib.RingTheory.IntegralClosure.Algebra.Defs", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 3, "macro_tier_score": 0.1789, "macro_tier_override": null, "x": 55.464, "z": -5.276, "size": 0.2865, "title": "Integral algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IntegralClosure/Algebra/Defs.html"}, {"id": "Mathlib.NumberTheory.NumberField.Cyclotomic.Basic", "region_id": "algebra", "micro_elevation": 0.9737, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": 122.153, "z": 62.974, "size": 0.3054, "title": "Ring of integers of cyclotomic fields", "summary": "We gather results about cyclotomic extensions of `ℚ`. In particular, we compute the ring of integers of a cyclotomic extension of `ℚ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Cyclotomic/Basic.html"}, {"id": "Mathlib.RingTheory.WittVector.Verschiebung", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 103.899, "z": -40.269, "size": 0.2668, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/Verschiebung.html"}, {"id": "Mathlib.RingTheory.WittVector.MulP", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 104.987, "z": 37.341, "size": 0.2668, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/MulP.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.IsValuedIn", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 1, "macro_tier_score": 0.0038, "macro_tier_override": null, "x": 74.652, "z": -72.411, "size": 0.3698, "title": "Root pairings taking values in a subring", "summary": "This file lays out the basic theory of root pairings over a commutative ring `R`, where `R` is an `S`-algebra, and the pairing between roots and coroots takes values in `S`. The main application of this theory is the theory of crystallographic root systems, where `S = ℤ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/IsValuedIn.html"}, {"id": "Mathlib.NumberTheory.LSeries.HurwitzZeta", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -73.453, "z": -31.336, "size": 0.2844, "title": "The Hurwitz zeta function", "summary": "This file gives the definition and properties of the following two functions: * The **Hurwitz zeta function**, which is the meromorphic continuation to all `s ∈ ℂ` of the function defined for `1 < re s` by the series `∑' n, 1 / (n + a) ^ s` for a parameter `a ∈ ℝ`, with the sum taken over all `n` such that `n + a > 0`; * the related sum, which we call the \"**exponential zeta function**\" (does it have a standard…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/HurwitzZeta.html"}, {"id": "Mathlib.NumberTheory.LSeries.HurwitzZetaEven", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -66.926, "z": -40.063, "size": 0.2727, "title": "Even Hurwitz zeta functions", "summary": "In this file we study the functions on `ℂ` which are the meromorphic continuation of the following series (convergent for `1 < re s`), where `a ∈ ℝ` is a parameter: `hurwitzZetaEven a s = 1 / 2 * ∑' n : ℤ, 1 / |n + a| ^ s` and `cosZeta a s = ∑' n : ℕ, cos (2 * π * a * n) / |n| ^ s`. Note that the term for `n = -a` in the first sum is omitted if `a` is an integer, and the term for `n = 0` is omitted in the second sum…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/HurwitzZetaEven.html"}, {"id": "Mathlib.NumberTheory.LSeries.HurwitzZetaOdd", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -24.194, "z": -74.154, "size": 0.2727, "title": "Odd Hurwitz zeta functions", "summary": "In this file we study the functions on `ℂ` which are the analytic continuation of the following series (convergent for `1 < re s`), where `a ∈ ℝ` is a parameter: `hurwitzZetaOdd a s = 1 / 2 * ∑' n : ℤ, sgn (n + a) / |n + a| ^ s` and `sinZeta a s = ∑' n : ℕ, sin (2 * π * a * n) / n ^ s`. The term for `n = -a` in the first sum is understood as 0 if `a` is an integer, as is the term for `n = 0` in the second sum (for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.Basic", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0268, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.4077, "title": "Short complexes", "summary": "This file defines the category `ShortComplex C` of diagrams `X₁ ⟶ X₂ ⟶ X₃` such that the composition is zero. Note: This structure `ShortComplex C` was first introduced in the Liquid Tensor Experiment.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/Basic.html"}, {"id": "Mathlib.Algebra.Order.ZeroLEOne", "region_id": "algebra", "micro_elevation": 0.0263, "macro_tier": 4, "macro_tier_score": 0.4764, "macro_tier_override": null, "x": 0.612, "z": 3.664, "size": 0.5065, "title": "Typeclass expressing `0 ≤ 1`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/ZeroLEOne.html"}, {"id": "Mathlib.Algebra.Order.BigOperators.Group.Finset", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.4154, "macro_tier_override": null, "x": 13.569, "z": 19.969, "size": 0.4713, "title": "Big operators on a finset in ordered groups", "summary": "This file contains the results concerning the interaction of finset big operators with ordered groups/monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/BigOperators/Group/Finset.html"}, {"id": "Mathlib.LinearAlgebra.Charpoly.ToMatrix", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 2, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": 94.689, "z": -2.21, "size": 0.3081, "title": "Characteristic polynomial", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Charpoly/ToMatrix.html"}, {"id": "Mathlib.LinearAlgebra.Eigenspace.Minpoly", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 2, "macro_tier_score": 0.0082, "macro_tier_override": null, "x": -1.001, "z": -87.281, "size": 0.2863, "title": "Eigenvalues are the roots of the minimal polynomial.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Eigenspace/Minpoly.html"}, {"id": "Mathlib.FieldTheory.IsRealClosed.Basic", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 56.061, "z": -13.105, "size": 0.2, "title": "Real Closed Field", "summary": "A field `R` is real closed if all of the following hold: 1. `R` is real (that is, `-1` is not a sum of squares in `R`). 2. for every `x` in `R`, one of `x` or `-x` is a square. 3. every odd-degree polynomial over `R` has a root in `R`. A real closed field is an algebraic generalisation of the real numbers. In this file we define real closed fields and prove some of their properties. TODO (Artie Khovanov) :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/IsRealClosed/Basic.html"}, {"id": "Mathlib.Algebra.Polynomial.Degree.Domain", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 4, "macro_tier_score": 0.2863, "macro_tier_override": null, "x": -45.937, "z": 31.526, "size": 0.3559, "title": "Univariate polynomials form a domain", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Degree/Domain.html"}, {"id": "Mathlib.Algebra.Ring.Semireal.Defs", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -17.624, "z": -23.924, "size": 0.2338, "title": "Semireal rings", "summary": "A semireal ring is a commutative ring (with unit) in which `-1` is *not* a sum of squares. For instance, linearly ordered rings are semireal, because sums of squares are positive and `-1` is not.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Semireal/Defs.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.ChangeOfRings", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 1, "macro_tier_score": 0.002, "macro_tier_override": null, "x": 61.897, "z": -12.483, "size": 0.329, "title": "Change of presheaf of rings", "summary": "In this file, we define the restriction of scalars functor `restrictScalars α : PresheafOfModules.{v} R' ⥤ PresheafOfModules.{v} R` attached to a morphism of presheaves of rings `α : R ⟶ R'`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/ChangeOfRings.html"}, {"id": "Mathlib.RingTheory.Adjoin.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.2655, "macro_tier_override": null, "x": 55.719, "z": -20.669, "size": 0.3735, "title": "Adjoining elements to form subalgebras", "summary": "This file contains basic results on `Algebra.adjoin`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Adjoin/Basic.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.Prod", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.2643, "macro_tier_override": null, "x": 45.582, "z": -35.169, "size": 0.2869, "title": "Products of subalgebras", "summary": "In this file we define the product of two subalgebras as a subalgebra of the product algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/Prod.html"}, {"id": "Mathlib.RingTheory.Jacobson.Polynomial", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 3, "macro_tier_score": 0.058, "macro_tier_override": null, "x": 21.368, "z": -73.084, "size": 0.2842, "title": "Jacobson radical of polynomial ring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Jacobson/Polynomial.html"}, {"id": "Mathlib.RingTheory.PrincipalIdealDomainOfPrime", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.038, "macro_tier_override": null, "x": -61.882, "z": 12.561, "size": 0.2726, "title": "Principal ideal domains and prime ideals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PrincipalIdealDomainOfPrime.html"}, {"id": "Mathlib.RingTheory.Ideal.Oka", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 3, "macro_tier_score": 0.0414, "macro_tier_override": null, "x": 59.167, "z": 15.979, "size": 0.2793, "title": "Oka predicates", "summary": "This file introduces the notion of Oka predicates and standard results about them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Oka.html"}, {"id": "Mathlib.LinearAlgebra.Finsupp.Span", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 4, "macro_tier_score": 0.3043, "macro_tier_override": null, "x": 42.566, "z": -13.222, "size": 0.2978, "title": "Finitely supported functions and spans", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Finsupp/Span.html"}, {"id": "Mathlib.LinearAlgebra.Projection", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 4, "macro_tier_score": 0.3112, "macro_tier_override": null, "x": 49.523, "z": -15.861, "size": 0.4718, "title": "Projection to a subspace", "summary": "In this file we define * `Submodule.projectionOnto (p q : Submodule R E) (h : IsCompl p q)`: the projection of a module `E` to a submodule `p` along its complement `q`; it is the unique linear map `f : E → p` such that `f x = x` for `x ∈ p` and `f x = 0` for `x ∈ q`. * `Submodule.projection` (p q : Submodule R E) (h : IsCompl p q)`: the projection `Submodule.projectionOnto` as a linear map from `E` to `E`. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Projection.html"}, {"id": "Mathlib.Algebra.Module.MinimalAxioms", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -16.714, "z": -0.041, "size": 0.2813, "title": "Minimal Axioms for a Module", "summary": "This file defines a constructor to define a `Module` structure on a Type with an `AddCommGroup`, while proving a minimum number of equalities.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/MinimalAxioms.html"}, {"id": "Mathlib.LinearAlgebra.Finsupp.Supported", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 4, "macro_tier_score": 0.3713, "macro_tier_override": null, "x": 35.285, "z": -20.598, "size": 0.4398, "title": "`Finsupp`s supported on a given submodule", "summary": "* `Finsupp.restrictDom`: `Finsupp.filter` as a linear map to `Finsupp.supported s`; `Finsupp.supported R R s` and codomain `Submodule.span R (v '' s)`; * `Finsupp.supportedEquivFinsupp`: a linear equivalence between the functions `α →₀ M` supported on `s` and the functions `s →₀ M`; * `Finsupp.domLCongr`: a `LinearEquiv` version of `Finsupp.domCongr`; * `Finsupp.congr`: if the sets `s` and `t` are equivalent, then…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Finsupp/Supported.html"}, {"id": "Mathlib.Algebra.Module.Submodule.Range", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 4, "macro_tier_score": 0.3965, "macro_tier_override": null, "x": -38.054, "z": 8.542, "size": 0.6391, "title": "Range of linear maps", "summary": "The range `LinearMap.range` of a (semi)linear map `f : M → M₂` is a submodule of `M₂`. More specifically, `LinearMap.range` applies to any `SemilinearMapClass` over a `RingHomSurjective` ring homomorphism. Note that this also means that dot notation (i.e. `f.range` for a linear map `f`) does not work.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/Range.html"}, {"id": "Mathlib.LinearAlgebra.Finsupp.LSum", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 4, "macro_tier_score": 0.3707, "macro_tier_override": null, "x": -34.655, "z": -6.644, "size": 0.4187, "title": "Sums as a linear map", "summary": "Given an `R`-module `M`, the `R`-module structure on `α →₀ M` is defined in `Data.Finsupp.Basic`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Finsupp/LSum.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.Prod", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 86.307, "z": 39.013, "size": 0.2, "title": "Clifford algebras of a direct sum of two vector spaces", "summary": "We show that the Clifford algebra of a direct sum is the graded tensor product of the Clifford algebras, as `CliffordAlgebra.equivProd`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/Prod.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.Grading", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 2, "macro_tier_score": 0.0071, "macro_tier_override": null, "x": 29.304, "z": -88.113, "size": 0.3666, "title": "Results about the grading structure of the clifford algebra", "summary": "The main result is `CliffordAlgebra.gradedAlgebra`, which says that the clifford algebra is a ℤ₂-graded algebra (or \"superalgebra\").", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/Grading.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Graded.Internal", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -45.513, "z": 46.407, "size": 0.239, "title": "Graded tensor products over graded algebras", "summary": "The graded tensor product $A \\hat\\otimes_R B$ is imbued with a multiplication defined on homogeneous tensors by: $$(a \\otimes b) \\cdot (a' \\otimes b') = (-1)^{\\deg a' \\deg b} (a \\cdot a') \\otimes (b \\cdot b')$$ where $A$ and $B$ are algebras graded by `ℕ`, `ℤ`, or `ι` (or more generally, any index that satisfies `Module ι (Additive ℤˣ)`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.html"}, {"id": "Mathlib.RingTheory.Localization.Away.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.1982, "macro_tier_override": null, "x": -37.005, "z": -46.502, "size": 0.3867, "title": "Localizations away from an element", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Away/Basic.html"}, {"id": "Mathlib.RingTheory.Localization.Module", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.1909, "macro_tier_override": null, "x": 21.308, "z": 53.484, "size": 0.3526, "title": "Modules / vector spaces over localizations / fraction fields", "summary": "This file contains some results about vector spaces over the field of fractions of a ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Module.html"}, {"id": "Mathlib.Algebra.Order.Ring.Opposite", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -15.512, "z": -13.293, "size": 0.2, "title": "Ordered ring instances for `MulOpposite`/`AddOpposite`", "summary": "This file transfers ordered (semi)ring instances from `R` to `Rᵐᵒᵖ` and `Rᵃᵒᵖ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Opposite.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.Symmetric.Defs", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 1, "macro_tier_score": 0.0029, "macro_tier_override": null, "x": -67.683, "z": -11.863, "size": 0.3104, "title": "Symmetric Polynomials and Elementary Symmetric Polynomials", "summary": "This file defines symmetric `MvPolynomial`s and the bases of elementary, complete homogeneous, power sum, and monomial symmetric `MvPolynomial`s. We also prove some basic facts about them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/Symmetric/Defs.html"}, {"id": "Mathlib.Algebra.BigOperators.Module", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -22.281, "z": -0.488, "size": 0.2754, "title": "Summation by parts", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Module.html"}, {"id": "Mathlib.Algebra.Order.Field.Power", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 6.323, "z": -27.13, "size": 0.2754, "title": "Lemmas about powers in ordered fields.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Field/Power.html"}, {"id": "Mathlib.RingTheory.Polynomial.Pochhammer", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.0427, "macro_tier_override": null, "x": -0.268, "z": 63.143, "size": 0.3022, "title": "The Pochhammer polynomials", "summary": "We define and prove some basic relations about `ascPochhammer S n : S[X] := X * (X + 1) * ... * (X + n - 1)` which is also known as the rising factorial and about `descPochhammer R n : R[X] := X * (X - 1) * ... * (X - n + 1)` which is also known as the falling factorial. Versions of this definition that are focused on `Nat` can be found in `Data.Nat.Factorial` as `Nat.ascFactorial` and `Nat.descFactorial`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Pochhammer.html"}, {"id": "Mathlib.Algebra.Order.Floor.Defs", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 2, "macro_tier_score": 0.007, "macro_tier_override": null, "x": 5.428, "z": 27.324, "size": 0.364, "title": "Floor and ceil", "summary": "We define the natural- and integer-valued floor and ceil functions on linearly ordered rings. We also provide `positivity` extensions to handle floor and ceil.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Floor/Defs.html"}, {"id": "Mathlib.Algebra.Order.Ring.Cast", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 2, "macro_tier_score": 0.0094, "macro_tier_override": null, "x": -25.357, "z": 5.747, "size": 0.4618, "title": "Order properties of cast of integers", "summary": "This file proves additional properties about the *canonical* homomorphism from the integers into an additive group with a one (`Int.cast`), particularly results involving algebraic homomorphisms or the order structure on `ℤ` which were not available in the import dependencies of `Mathlib/Data/Int/Cast/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Cast.html"}, {"id": "Mathlib.RingTheory.AdicCompletion.Completeness", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -76.278, "z": 34.147, "size": 0.2478, "title": "Completeness of the Adic Completion for Finitely Generated Ideals", "summary": "This file establishes that `AdicCompletion I M` is itself `I`-adically complete when the ideal `I` is finitely generated.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AdicCompletion/Completeness.html"}, {"id": "Mathlib.RingTheory.AdicCompletion.Exactness", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": 81.562, "z": -5.004, "size": 0.2581, "title": "Exactness of adic completion", "summary": "In this file we establish exactness properties of adic completions. In particular we show:", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AdicCompletion/Exactness.html"}, {"id": "Mathlib.RingTheory.AdicCompletion.Topology", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 66.333, "z": -8.363, "size": 0.2403, "title": "Connection between adic properties and topological properties", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AdicCompletion/Topology.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Integer", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 8.055, "z": 28.602, "size": 0.2725, "title": "Lemmas on integer matrices", "summary": "Here we collect some results about matrices over `ℚ` and `ℤ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Integer.html"}, {"id": "Mathlib.Algebra.Category.MonCat.Limits", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 3, "macro_tier_score": 0.0892, "macro_tier_override": null, "x": -12.635, "z": -10.943, "size": 0.2999, "title": "The category of (commutative) (additive) monoids has all limits", "summary": "Further, these limits are preserved by the forgetful functor --- that is, the underlying types are just the limits in the category of types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/MonCat/Limits.html"}, {"id": "Mathlib.NumberTheory.LSeries.DirichletContinuation", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -107.633, "z": 4.215, "size": 0.2685, "title": "Analytic continuation of Dirichlet L-functions", "summary": "We show that if `χ` is a Dirichlet character `ZMod N → ℂ`, for a positive integer `N`, then the L-series of `χ` has analytic continuation (away from a pole at `s = 1` if `χ` is trivial), and similarly for completed L-functions. All definitions and theorems are in the `DirichletCharacter` namespace.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/DirichletContinuation.html"}, {"id": "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -66.005, "z": -82.76, "size": 0.2653, "title": "The Euler Product for the Riemann Zeta Function and Dirichlet L-Series", "summary": "The first main result of this file is the Euler Product formula for the Riemann ζ function $$\\prod_p \\frac{1}{1 - p^{-s}} = \\lim_{n \\to \\infty} \\prod_{p < n} \\frac{1}{1 - p^{-s}} = \\zeta(s)$$ for $s$ with real part $> 1$ ($p$ runs through the primes). `riemannZeta_eulerProduct` is the second equality above. There are versions `riemannZeta_eulerProduct_hasProd` and `riemannZeta_eulerProduct_tprod` in terms of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/EulerProduct/DirichletLSeries.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.Bounds", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -7.048, "z": -15.156, "size": 0.2342, "title": "Lemmas about `BddAbove`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/Bounds.html"}, {"id": "Mathlib.RingTheory.Polynomial.ShiftedLegendre", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 56.779, "z": -27.627, "size": 0.2, "title": "shifted Legendre Polynomials", "summary": "In this file, we define the shifted Legendre polynomials `shiftedLegendre n` for `n : ℕ` as a polynomial in `ℤ[X]`. We prove some basic properties of the Legendre polynomials. * `factorial_mul_shiftedLegendre_eq`: The analogue of Rodrigues' formula for the shifted Legendre polynomials; * `shiftedLegendre_eval_symm`: The values of the shifted Legendre polynomial at `x` and `1 - x` differ by a factor `(-1)ⁿ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/ShiftedLegendre.html"}, {"id": "Mathlib.Algebra.Homology.SpectralObject.Homology", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 1.385, "z": -24.103, "size": 0.253, "title": "The homology of the differentials of a spectral object", "summary": "Let `X` be a spectral object indexed by a category `ι` in an abelian category `C`. Assume we have seven composable arrows `f₁`, `f₂`, `f₃`, `f₄`, `f₅`, `f₆`, `f₇` in `ι`. In this file, we compute the homology of the differentials, i.e. the homology of the short complex `E^{n - 1}(f₅, f₆, f₇) ⟶ E^n(f₃, f₄, f₅) ⟶ E^{n + 1}(f₁, f₂, f₃)`. The main definition for this is `dHomologyData` which is a homology data for this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/SpectralObject/Homology.html"}, {"id": "Mathlib.Algebra.Homology.SpectralObject.EpiMono", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 21.592, "z": -5.518, "size": 0.2833, "title": "Induced morphisms that are epi or mono", "summary": "Given a spectral object in an abelian category, we show that certain morphisms `E^n(f₁, f₂, f₃) ⟶ E^n(f₁', f₂', f₃')` are monomorphisms, epimorphisms or isomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/SpectralObject/EpiMono.html"}, {"id": "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 2, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 78.305, "z": -94.3, "size": 0.3155, "title": "Primitive roots in cyclotomic fields", "summary": "If `IsCyclotomicExtension {n} A B`, we define an element `zeta n A B : B` that is a primitive `n`th-root of unity in `B` and we study its properties. We also prove related theorems under the more general assumption of just being a primitive root, for reasons described in the implementation details section.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.html"}, {"id": "Mathlib.Algebra.Group.Action.Hom", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.454, "macro_tier_override": null, "x": -10.559, "z": 10.452, "size": 0.4226, "title": "Homomorphisms and group actions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Hom.html"}, {"id": "Mathlib.RingTheory.OreLocalization.Cardinality", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 2, "macro_tier_score": 0.0101, "macro_tier_override": null, "x": 27.444, "z": -11.392, "size": 0.2451, "title": "Cardinality of Ore localizations of rings", "summary": "This file contains some results on cardinality of Ore localizations of rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/OreLocalization/Cardinality.html"}, {"id": "Mathlib.RingTheory.OreLocalization.Ring", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 3, "macro_tier_score": 0.2358, "macro_tier_override": null, "x": -27.784, "z": 2.018, "size": 0.3165, "title": "Module and Ring instances of Ore Localizations", "summary": "The `Monoid` and `DistribMulAction` instances and additive versions are provided in `Mathlib/RingTheory/OreLocalization/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/OreLocalization/Ring.html"}, {"id": "Mathlib.Algebra.Category.Grp.Zero", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 2, "macro_tier_score": 0.006, "macro_tier_override": null, "x": 16.628, "z": 11.868, "size": 0.2867, "title": "The category of (commutative) (additive) groups has a zero object.", "summary": "`AddCommGroup` also has zero morphisms. For definitional reasons, we infer this from preadditivity rather than from the existence of a zero object.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Zero.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughProjectives", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 42.351, "z": -36.202, "size": 0.2909, "title": "Smallness of Ext-groups from the existence of enough projectives", "summary": "Let `C : Type u` be an abelian category (`Category.{v} C`) that has enough projectives. If `C` is locally `w`-small, i.e. the type of morphisms in `C` are `Small.{w}`, then we show that the condition `HasExt.{w}` holds, which means that for `X` and `Y` in `C`, and `n : ℕ`, we may define `Ext X Y n : Type w`. In particular, this holds for `w = v`. However, the main lemma `hasExt_of_enoughProjectives` is not made an…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/Ext/EnoughProjectives.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": 18.16, "z": 50.704, "size": 0.3567, "title": "Long exact sequences of `Ext`-groups", "summary": "In this file, we obtain the covariant long exact sequence of `Ext` when `n₀ + 1 = n₁`: `Ext X S.X₁ n₀ → Ext X S.X₂ n₀ → Ext X S.X₃ n₀ → Ext X S.X₁ n₁ → Ext X S.X₂ n₁ → Ext X S.X₃ n₁` when `S` is a short exact short complex in an abelian category `C`, `n₀ + 1 = n₁` and `X : C`. Similarly, if `Y : C`, there is a contravariant long exact sequence : `Ext S.X₃ Y n₀ → Ext S.X₂ Y n₀ → Ext S.X₁ Y n₀ → Ext S.X₃ Y n₁ → Ext…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/Ext/ExactSequences.html"}, {"id": "Mathlib.LinearAlgebra.Basis.Flag", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -17.028, "z": -51.095, "size": 0.2, "title": "Flag of submodules defined by a basis", "summary": "In this file we define `Basis.flag b k`, where `b : Basis (Fin n) R M`, `k : Fin (n + 1)`, to be the subspace spanned by the first `k` vectors of the basis `b`. We also prove some lemmas about this definition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Basis/Flag.html"}, {"id": "Mathlib.RepresentationTheory.Homological.GroupCohomology.Hilbert90", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 93.366, "z": 85.036, "size": 0.2, "title": "Hilbert's Theorem 90", "summary": "Let `L/K` be a finite extension of fields. Then this file proves Noether's generalization of Hilbert's Theorem 90: that the 1st group cohomology $H^1(Aut_K(L), L^\\times)$ is trivial. We state it both in terms of $H^1$ and in terms of cocycles being coboundaries. Hilbert's original statement was that if $L/K$ is Galois, and $Gal(L/K)$ is cyclic, generated by an element `σ`, then for every `x : L` such that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/GroupCohomology/Hilbert90.html"}, {"id": "Mathlib.RepresentationTheory.Homological.GroupCohomology.FiniteCyclic", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -34.692, "z": 94.095, "size": 0.2478, "title": "Group cohomology of a finite cyclic group", "summary": "Let `k` be a commutative ring, `G` a group and `A` a `k`-linear `G`-representation. Given endomorphisms `φ, ψ : A ⟶ A` such that `φ ∘ ψ = ψ ∘ φ = 0`, denote by `Chains(A, φ, ψ)` the periodic chain complex `... ⟶ A --φ--> A --ψ--> A --φ--> A --ψ--> A ⟶ 0` and by `Cochains(A, φ, ψ)` the periodic cochain complex `0 ⟶ A --ψ--> A --φ--> A --ψ--> A --φ--> A ⟶ ...`. When `G` is finite and generated by `g : G`, then `P :=…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/GroupCohomology/FiniteCyclic.html"}, {"id": "Mathlib.RingTheory.IntegralClosure.IntegralRestrict", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 2, "macro_tier_score": 0.0093, "macro_tier_override": null, "x": 77.065, "z": -97.692, "size": 0.293, "title": "Restriction of various maps between fields to integrally closed subrings.", "summary": "In this file, we assume `A` is an integrally closed domain; `K` is the fraction ring of `A`; `L` is a finite extension of `K`; `B` is the integral closure of `A` in `L`. We call this the AKLB setup.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.html"}, {"id": "Mathlib.RingTheory.LocalRing.Module", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 2, "macro_tier_score": 0.0251, "macro_tier_override": null, "x": 18.787, "z": -83.338, "size": 0.3141, "title": "Finite modules over local rings", "summary": "This file gathers various results about finite modules over a local ring `(R, 𝔪, k)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/Module.html"}, {"id": "Mathlib.RingTheory.Flat.EquationalCriterion", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 2, "macro_tier_score": 0.0245, "macro_tier_override": null, "x": -80.781, "z": -21.42, "size": 0.2441, "title": "The equational criterion for flatness", "summary": "Let $M$ be a module over a commutative ring $R$. Let us say that a relation $\\sum_{i \\in \\iota} f_i x_i = 0$ in $M$ is *trivial* (`Module.IsTrivialRelation`) if there exist a finite index type $\\kappa$ = `Fin k`, elements $(y_j)_{j \\in \\kappa}$ of $M$, and elements $(a_{ij})_{i \\in \\iota, j \\in \\kappa}$ of $R$ such that for all $i$, $$x_i = \\sum_j a_{ij} y_j$$ and for all $j$, $$\\sum_i f_i a_{ij} = 0.$$ The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/EquationalCriterion.html"}, {"id": "Mathlib.RingTheory.Ideal.Quotient.ChineseRemainder", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 2, "macro_tier_score": 0.0245, "macro_tier_override": null, "x": 54.615, "z": 38.564, "size": 0.2441, "title": "Module version of Chinese remainder theorem", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Quotient/ChineseRemainder.html"}, {"id": "Mathlib.RingTheory.LocalProperties.Exactness", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.0386, "macro_tier_override": null, "x": -54.127, "z": 48.127, "size": 0.3293, "title": "Local properties about linear maps", "summary": "In this file, we show that injectivity, surjectivity, bijectivity and exactness of linear maps are local properties. More precisely, we show that these can be checked at maximal ideals and on standard covers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalProperties/Exactness.html"}, {"id": "Mathlib.FieldTheory.Galois.Abelian", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 5, "macro_tier_score": 0.0113, "macro_tier_override": 5, "x": -52.529, "z": 106.621, "size": 0.2682, "title": "Abelian extensions", "summary": "In this file, we define the typeclass of abelian extensions and provide some basic API.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Galois/Abelian.html"}, {"id": "Mathlib.FieldTheory.Galois.Infinite", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 5, "macro_tier_score": 0.0206, "macro_tier_override": 5, "x": -102.875, "z": 55.731, "size": 0.3081, "title": "The Fundamental Theorem of Infinite Galois Theory", "summary": "In this file, we prove the fundamental theorem of infinite Galois theory and the special case for open subgroups and normal subgroups. We first verify that `IntermediateField.fixingSubgroup` and `IntermediateField.fixedField` are inverses of each other between intermediate fields and closed subgroups of the Galois group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Galois/Infinite.html"}, {"id": "Mathlib.NumberTheory.Padics.Complex", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -42.181, "z": 46.988, "size": 0.2, "title": "The field `ℂ_[p]` of `p`-adic complex numbers.", "summary": "In this file we define the field `ℂ_[p]` of `p`-adic complex numbers as the `p`-adic completion of an algebraic closure of `ℚ_[p]`. We endow `ℂ_[p]` with both a normed field and a valued field structure, induced by the unique extension of the `p`-adic norm to `ℂ_[p]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/Complex.html"}, {"id": "Mathlib.RingTheory.OreLocalization.OreSet", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 3, "macro_tier_score": 0.1765, "macro_tier_override": null, "x": 5.099, "z": -11.958, "size": 0.2647, "title": "(Left) Ore sets and rings", "summary": "This file contains results on left Ore sets for rings and monoids with zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/OreLocalization/OreSet.html"}, {"id": "Mathlib.Algebra.Ring.Regular", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 3, "macro_tier_score": 0.1779, "macro_tier_override": null, "x": -8.473, "z": -3.799, "size": 0.2985, "title": "Lemmas about regular elements in rings.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Regular.html"}, {"id": "Mathlib.Algebra.Colimit.DirectLimit", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 3, "macro_tier_score": 0.0769, "macro_tier_override": null, "x": 35.53, "z": 32.699, "size": 0.2824, "title": "Direct limit of algebraic structures", "summary": "We introduce all kinds of algebraic instances on `DirectLimit`, and specialize to the cases of modules and rings, showing that they are indeed colimits in the respective categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Colimit/DirectLimit.html"}, {"id": "Mathlib.GroupTheory.Perm.Option", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 3, "macro_tier_score": 0.2088, "macro_tier_override": null, "x": 21.007, "z": 7.442, "size": 0.2871, "title": "Permutations of `Option α`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Option.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 3, "macro_tier_score": 0.2023, "macro_tier_override": null, "x": -24.001, "z": -72.262, "size": 0.4219, "title": "Nonsingular inverses", "summary": "In this file, we define an inverse for square matrices of invertible determinant. For matrices that are not square or not of full rank, there is a more general notion of pseudoinverses which we do not consider here. The definition of inverse used in this file is the adjugate divided by the determinant. We show that dividing the adjugate by `det A` (if possible), giving a matrix `A⁻¹` (`nonsing_inv`), will result in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.html"}, {"id": "Mathlib.NumberTheory.LSeries.Nonvanishing", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 24.796, "z": -106.73, "size": 0.2755, "title": "The L-function of a Dirichlet character does not vanish on Re(s) ≥ 1", "summary": "The main result in this file is `DirichletCharacter.LFunction_ne_zero_of_one_le_re`: if `χ` is a Dirichlet character, `s ∈ ℂ` with `1 ≤ s.re`, and either `χ` is nontrivial or `s ≠ 1`, then the L-function of `χ` does not vanish at `s`. As a consequence, we have the corresponding statement for the Riemann ζ function: `riemannZeta_ne_zero_of_one_le_re` (which does not require `s ≠ 1`, since the junk value at `s = 1`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/Nonvanishing.html"}, {"id": "Mathlib.NumberTheory.Harmonic.ZetaAsymp", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 105.397, "z": 9.868, "size": 0.2685, "title": "Asymptotics of `ζ s` as `s → 1` or `s → 0`", "summary": "The goal of this file is to evaluate the limit of `ζ s - 1 / (s - 1)` as `s → 1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Harmonic/ZetaAsymp.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.Directed", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.0903, "macro_tier_override": null, "x": 27.167, "z": -50.759, "size": 0.2987, "title": "Subalgebras and directed Unions of sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/Directed.html"}, {"id": "Mathlib.Algebra.Star.Module", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 4, "macro_tier_score": 0.2758, "macro_tier_override": null, "x": 43.104, "z": 21.762, "size": 0.3866, "title": "The star operation, bundled as a star-linear equiv", "summary": "We define `starLinearEquiv`, which is the star operation bundled as a star-linear map. It is defined on a star algebra `A` over the base ring `R`. This file also provides some lemmas that need `Algebra.Module.Basic` imported to prove.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/Module.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.MonomialOrder.DegLex", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": 4.55, "z": -77.868, "size": 0.2727, "title": "Some lemmas about the degree lexicographic monomial order on multivariate polynomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/MonomialOrder/DegLex.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.MonomialOrder", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 2, "macro_tier_score": 0.0073, "macro_tier_override": null, "x": 33.143, "z": -68.552, "size": 0.3062, "title": "Degree, leading coefficient and leading term of polynomials with respect to a monomial order", "summary": "We consider a type `σ` of indeterminates and a commutative semiring `R` and a monomial order `m : MonomialOrder σ`. * `m.degree f` is the degree of `f` for the monomial ordering `m`. * `m.leadingCoeff f` is the leading coefficient of `f` for the monomial ordering `m`. * `m.Monic f` asserts that the leading coefficient of `f` is `1`. * `m.leadingTerm f` is the leading term of `f` for the monomial ordering `m`. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/MonomialOrder.html"}, {"id": "Mathlib.Algebra.Order.Module.HahnEmbedding", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -47.878, "z": -38.258, "size": 0.2302, "title": "Hahn embedding theorem on ordered modules", "summary": "This file proves a variant of the Hahn embedding theorem: For a linearly ordered module `M` over an Archimedean division ring `K`, there exists a strictly monotone linear map to lexicographically ordered `R⟦FiniteArchimedeanClass M⟧` with an archimedean `K`-module `R`, as long as there are embeddings from a certain family of Archimedean submodules to `R`. The family of Archimedean submodules…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Module/HahnEmbedding.html"}, {"id": "Mathlib.Algebra.Module.Submodule.Order", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -14.054, "z": 30.331, "size": 0.2298, "title": "Ordered instances on submodules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/Order.html"}, {"id": "Mathlib.Algebra.Order.Module.Archimedean", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": 34.73, "z": 6.238, "size": 0.2442, "title": "Archimedean classes for ordered module", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Module/Archimedean.html"}, {"id": "Mathlib.LinearAlgebra.Basis.VectorSpace", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 3, "macro_tier_score": 0.2528, "macro_tier_override": null, "x": 6.029, "z": -53.519, "size": 0.4849, "title": "Bases in a vector space", "summary": "This file provides results for bases of a vector space. Some of these results should be merged with the results on free modules. We state these results in a separate file to the results on modules to avoid an import cycle.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Basis/VectorSpace.html"}, {"id": "Mathlib.RingTheory.HahnSeries.Lex", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -57.401, "z": -15.393, "size": 0.2298, "title": "Lexicographical order on Hahn series", "summary": "In this file, we define lexicographical ordered `Lex R⟦Γ⟧`, and show this is a `LinearOrder` when `Γ` and `R` themselves are linearly ordered. Additionally, it is an ordered group or ring whenever `R` is.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HahnSeries/Lex.html"}, {"id": "Mathlib.Algebra.Group.Commute.Defs", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.5118, "macro_tier_override": null, "x": -5.799, "z": -4.643, "size": 0.759, "title": "Commuting pairs of elements in monoids", "summary": "We define the predicate `Commute a b := a * b = b * a` and provide some operations on terms `(h : Commute a b)`. E.g., if `a`, `b`, and c are elements of a semiring, and that `hb : Commute a b` and `hc : Commute a c`. Then `hb.pow_left 5` proves `Commute (a ^ 5) b` and `(hb.pow_right 2).add_right (hb.mul_right hc)` proves `Commute a (b ^ 2 + b * c)`. Lean does not immediately recognise these terms as equations, so…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Commute/Defs.html"}, {"id": "Mathlib.Algebra.Group.InjSurj", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.5042, "macro_tier_override": null, "x": 3.711, "z": -4.156, "size": 0.6418, "title": "Lifting algebraic data classes along injective/surjective maps", "summary": "This file provides definitions that are meant to deal with situations such as the following: Suppose that `G` is a group, and `H` is a type endowed with `One H`, `Mul H`, and `Inv H`. Suppose furthermore, that `f : G → H` is a surjective map that respects the multiplication, and the unit elements. Then `H` satisfies the group axioms. The relevant definition in this case is `Function.Surjective.group`. Dually, there…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/InjSurj.html"}, {"id": "Mathlib.NumberTheory.Height.MvPolynomial", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -60.703, "z": 48.982, "size": 0.2565, "title": "Height bounds for linear and polynomial maps", "summary": "We prove an upper bound for the height of the image of a tuple under a linear map. We also prove upper and lower bounds for the height of `fun i ↦ eval P i x`, where `P` is a family of homogeneous polynomials over the field `K` of the same degree `N` and `x : ι → K` with `ι` finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Height/MvPolynomial.html"}, {"id": "Mathlib.Algebra.Polynomial.Homogenize", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 69.746, "z": -30.55, "size": 0.2603, "title": "Homogenize a univariate polynomial", "summary": "In this file we define a function `Polynomial.homogenize p n` that takes a polynomial `p` and a natural number `n` and returns a homogeneous bivariate polynomial of degree `n`. If `n` is at least the degree of `p`, then `(homogenize p n).eval ![x, 1] = p.eval x`. We use `MvPolynomial (Fin 2) R` to represent bivariate polynomials instead of `R[X][Y]` (i.e., `Polynomial (Polynomial R)`), because Mathlib has a theory…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Homogenize.html"}, {"id": "Mathlib.Algebra.Order.Invertible", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 1, "macro_tier_score": 0.0022, "macro_tier_override": null, "x": -17.326, "z": -14.017, "size": 0.4003, "title": "Lemmas about `invOf` in ordered (semi)rings.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Invertible.html"}, {"id": "Mathlib.GroupTheory.FiniteAbelian.Duality", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": -93.225, "z": 53.96, "size": 0.2581, "title": "Duality for finite abelian groups", "summary": "Let `G` be a finite abelian group. For `M` a commutative monoid that has enough `n`th roots of unity, where `n` is the exponent of `G`, the main results in this file are: * `CommGroup.exists_apply_ne_one_of_hasEnoughRootsOfUnity`: Homomorphisms `G →* Mˣ` separate elements of `G`. * `CommGroup.monoidHom_mulEquiv_self_of_hasEnoughRootsOfUnity`: `G` is isomorphic to `G →* Mˣ`. * `CommGroup.monoidHomMonoidHomEquiv`: `G`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FiniteAbelian/Duality.html"}, {"id": "Mathlib.NumberTheory.MulChar.Lemmas", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 2, "macro_tier_score": 0.0104, "macro_tier_override": null, "x": -47.622, "z": -66.404, "size": 0.2825, "title": "Further Results on multiplicative characters", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/MulChar/Lemmas.html"}, {"id": "Mathlib.Algebra.Group.Ext", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 1, "macro_tier_score": 0.0019, "macro_tier_override": null, "x": 4.522, "z": -5.893, "size": 0.3175, "title": "Extensionality lemmas for monoid and group structures", "summary": "In this file we prove extensionality lemmas for `Monoid` and higher algebraic structures with one binary operation. Extensionality lemmas for structures that are lower in the hierarchy can be found in `Algebra.Group.Defs`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Ext.html"}, {"id": "Mathlib.LinearAlgebra.PiTensorProduct", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 46.52, "z": 30.66, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/PiTensorProduct.html"}, {"id": "Mathlib.NumberTheory.EllipticDivisibilitySequence", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2649, "title": "Elliptic divisibility sequences", "summary": "This file defines the type of an elliptic divisibility sequence (EDS) and a few examples.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/EllipticDivisibilitySequence.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.Localization", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -64.901, "z": -16.057, "size": 0.239, "title": "Localization of a UFD", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/Localization.html"}, {"id": "Mathlib.RingTheory.Localization.Ideal", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.1374, "macro_tier_override": null, "x": 40.821, "z": -50.584, "size": 0.3977, "title": "Ideals in localizations of commutative rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Ideal.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.ZPowers.Basic", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.4293, "macro_tier_override": null, "x": 20.792, "z": -8.021, "size": 0.3545, "title": "Subgroups generated by an element", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/ZPowers/Basic.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Semiconj", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4154, "macro_tier_override": null, "x": 9.625, "z": -11.318, "size": 0.384, "title": "Lemmas about semiconjugate elements in a `GroupWithZero`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Semiconj.html"}, {"id": "Mathlib.GroupTheory.SpecificGroups.KleinFour", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -42.005, "z": -7.754, "size": 0.2708, "title": "Klein Four Group", "summary": "The Klein (Vierergruppe) four-group is a non-cyclic abelian group with four elements, in which each element is self-inverse and in which composing any two of the three non-identity elements produces the third one.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/SpecificGroups/KleinFour.html"}, {"id": "Mathlib.GroupTheory.SpecificGroups.Cyclic", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 3, "macro_tier_score": 0.1914, "macro_tier_override": null, "x": 40.857, "z": 0.202, "size": 0.3797, "title": "Further properties of cyclic groups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/SpecificGroups/Cyclic.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Pi", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.415, "macro_tier_override": null, "x": 2.809, "z": 6.877, "size": 0.3641, "title": "Pi instances for groups with zero", "summary": "This file defines monoid with zero, group with zero, and related structure instances for pi types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Pi.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 23.992, "z": 2.7, "size": 0.2573, "title": "Subgroups generated by an element", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/ZPowers/Lemmas.html"}, {"id": "Mathlib.Algebra.Module.Submodule.Lattice", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 4, "macro_tier_score": 0.3955, "macro_tier_override": null, "x": 2.641, "z": -33.324, "size": 0.538, "title": "The lattice structure on `Submodule`s", "summary": "This file defines the lattice structure on submodules, `Submodule.CompleteLattice`, with `⊥` defined as `{0}` and `⊓` defined as intersection of the underlying carrier. If `p` and `q` are submodules of a module, `p ≤ q` means that `p ⊆ q`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/Lattice.html"}, {"id": "Mathlib.GroupTheory.MonoidLocalization.Away", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 3, "macro_tier_score": 0.1981, "macro_tier_override": null, "x": -12.865, "z": 18.198, "size": 0.3198, "title": "Localizing commutative monoids away from an element", "summary": "We treat the special case of localizing away from an element in the sections `AwayMap` and `Away`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/MonoidLocalization/Away.html"}, {"id": "Mathlib.RingTheory.Ideal.IsPrimary", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.1565, "macro_tier_override": null, "x": -11.34, "z": -62.117, "size": 0.3502, "title": "Primary ideals", "summary": "A proper ideal `I` is primary iff `xy ∈ I` implies `x ∈ I` or `y ∈ radical I`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/IsPrimary.html"}, {"id": "Mathlib.RingTheory.Ideal.Over", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.1675, "macro_tier_override": null, "x": -21.707, "z": 59.295, "size": 0.4047, "title": "Ideals over/under ideals", "summary": "This file concerns ideals lying over other ideals. Let `f : R →+* S` be a ring homomorphism (typically a ring extension), `I` an ideal of `R` and `J` an ideal of `S`. We say `J` lies over `I` (and `I` under `J`) if `I` is the `f`-preimage of `J`. This is expressed here by writing `I = J.comap f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Over.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.Defs", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 3, "macro_tier_score": 0.1553, "macro_tier_override": null, "x": -24.206, "z": -30.58, "size": 0.3477, "title": "Prime spectrum of a commutative (semi)ring as a type", "summary": "The prime spectrum of a commutative (semi)ring is the type of all prime ideals. For the Zariski topology, see `Mathlib/RingTheory/Spectrum/Prime/Topology.lean`. (It is also naturally endowed with a sheaf of rings, which is constructed in `AlgebraicGeometry.StructureSheaf`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/Defs.html"}, {"id": "Mathlib.RingTheory.Ideal.Quotient.PowTransition", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.0668, "macro_tier_override": null, "x": -37.473, "z": 46.126, "size": 0.2665, "title": "The quotient map from `R ⧸ I ^ m` to `R ⧸ I ^ n` where `m ≥ n`", "summary": "In this file we define the canonical quotient linear map from `M ⧸ I ^ m • ⊤` to `M ⧸ I ^ n • ⊤` and canonical quotient ring map from `R ⧸ I ^ m` to `R ⧸ I ^ n`. These definitions will be used in theorems related to `IsAdicComplete` to find a lift element from compatible sequences in the quotients. We also include results about the relation between quotients of submodules and quotients of ideals here.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Quotient/PowTransition.html"}, {"id": "Mathlib.RingTheory.Ideal.Quotient.Defs", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 4, "macro_tier_score": 0.3057, "macro_tier_override": null, "x": 29.68, "z": -19.084, "size": 0.4719, "title": "Ideal quotients", "summary": "This file defines ideal quotients as a special case of submodule quotients and proves some basic results about these quotients. See `RingCon.Quotient` for quotients of (possibly non-commutative) semirings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Quotient/Defs.html"}, {"id": "Mathlib.NumberTheory.Basic", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -21.707, "z": -41.042, "size": 0.2659, "title": "Basic results in number theory", "summary": "This file should contain basic results in number theory. So far, it only contains the essential lemma in the construction of the ring of Witt vectors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Basic.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.Basic", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 2, "macro_tier_score": 0.0067, "macro_tier_override": null, "x": -64.813, "z": -63.879, "size": 0.3428, "title": "Clifford Algebras", "summary": "We construct the Clifford algebra of a module `M` over a commutative ring `R`, equipped with a quadratic form `Q`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/Basic.html"}, {"id": "Mathlib.RingTheory.LocalProperties.Semilocal", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 70.992, "z": -76.002, "size": 0.2, "title": "Local properties for semilocal rings", "summary": "This file proves some local properties for a semilocal ring `R` (a ring with finitely many maximal ideals).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalProperties/Semilocal.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.PID", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": 67.085, "z": -77.026, "size": 0.2648, "title": "Criteria under which a Dedekind domain is a PID", "summary": "This file contains some results that we can use to test whether all ideals in a Dedekind domain are principal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/PID.html"}, {"id": "Mathlib.Algebra.CharP.MixedCharZero", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.785, "z": 70.55, "size": 0.2, "title": "Equal and mixed characteristic", "summary": "In commutative algebra, some statements are simpler when working over a `ℚ`-algebra `R`, in which case one also says that the ring has \"equal characteristic zero\". A ring that is not a `ℚ`-algebra has either positive characteristic or there exists a prime ideal `I ⊂ R` such that the quotient `R ⧸ I` has positive characteristic `p > 0`. In this case one speaks of \"mixed characteristic `(0, p)`\", where `p` is only…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/MixedCharZero.html"}, {"id": "Mathlib.Algebra.CharP.LocalRing", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -67.427, "z": -13.241, "size": 0.2676, "title": "Characteristics of local rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/LocalRing.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.PointwiseSMul", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -15.962, "z": -9.493, "size": 0.2676, "title": "Scalar multiplication by (additive) monoid rings on formal functions.", "summary": "Given sets `G` and `P`, with a left-cancellative scalar-multiplication (or vector-addition) of `G` on `P`, together with a module `V` over a semiring `R`, we define a convolution action of the monoid algebra `R[G]` on the set of functions `P → V`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/PointwiseSMul.html"}, {"id": "Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 2, "macro_tier_score": 0.0081, "macro_tier_override": null, "x": 7.018, "z": -40.25, "size": 0.2834, "title": "SubMulActions on complements of invariant subsets", "summary": "Given a `MulAction` of `G` on `α` and `s : Set α`, - `SubMulAction.ofFixingSubgroup` is the action of `FixingSubgroup G s` on the complement `sᶜ` of `s`. - We define equivariant maps that relate various of these `SubMulAction`s and permit to manipulate them in a relatively smooth way: * `SubMulAction.ofFixingSubgroup_equivariantMap`: the identity map from `sᶜ` to `α`, as an equivariant map relative to the injection…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/SubMulAction/OfFixingSubgroup.html"}, {"id": "Mathlib.Algebra.Group.Pointwise.Set.Card", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 3, "macro_tier_score": 0.0983, "macro_tier_override": null, "x": -20.336, "z": 1.949, "size": 0.3185, "title": "Cardinalities of pointwise operations on sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pointwise/Set/Card.html"}, {"id": "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 3, "macro_tier_score": 0.0655, "macro_tier_override": null, "x": 8.862, "z": -24.443, "size": 0.3495, "title": "Fixing submonoid, fixing subgroup of an action", "summary": "In the presence of an action of a monoid or a group, this file defines the fixing submonoid or the fixing subgroup, and relates it to the set of fixed points via a Galois connection.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/FixingSubgroup.html"}, {"id": "Mathlib.GroupTheory.GroupAction.SubMulAction.OfStabilizer", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": -28.124, "z": 14.347, "size": 0.2457, "title": "The SubMulAction of the stabilizer of a point on the complement of that point", "summary": "When a group `G` acts on a type `α`, the stabilizer of a point `a : α` acts naturally on the complement of that point. Such actions (as the similar one, `SubMulAction.ofFixingSubgroup`, for the fixing subgroup of a set acting on the complement of that set) are useful to study the multiple transitivity of the group `G`, since `n`-transitivity of `G` on `α` is equivalent to `n - 1`-transitivity of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/SubMulAction/OfStabilizer.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Primitive", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 2, "macro_tier_score": 0.0081, "macro_tier_override": null, "x": -21.76, "z": -32.366, "size": 0.2752, "title": "Primitive actions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Primitive.html"}, {"id": "Mathlib.NumberTheory.Height.EllipticCurve", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 61.667, "z": 50.739, "size": 0.2, "title": "The naïve height and the approximate parallelogram law", "summary": "This file defines the *naïve height* on an elliptic curve (over a field `K` with a theory of heights, i.e., satisfying `[Height.AdmissibleAbsValues K]`). The final goal of this file is to prove the *approximate parallelogram law* for (affine) points on elliptic curves, ``` |h(P+Q) + h(P-Q) - 2*(h(P) + h(Q))| ≤ C ``` where `h` is the naïve height, `P` and `Q` are affine points on a `WeierstrassCurve` and `C` is some…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Height/EllipticCurve.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.Contraction", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -91.477, "z": -36.337, "size": 0.273, "title": "Contraction in Clifford Algebras", "summary": "This file contains some of the results from [grinberg_clifford_2016]. The key result is `CliffordAlgebra.equivExterior`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.Fold", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 2, "macro_tier_score": 0.0064, "macro_tier_override": null, "x": -19.368, "z": -94.61, "size": 0.3266, "title": "Recursive computation rules for the Clifford algebra", "summary": "This file provides API for a special case `CliffordAlgebra.foldr` of the universal property `CliffordAlgebra.lift` with `A = Module.End R N` for some arbitrary module `N`. This specialization resembles the `list.foldr` operation, allowing a bilinear map to be \"folded\" along the generators. For convenience, this file also provides `CliffordAlgebra.foldl`, implemented via `CliffordAlgebra.reverse`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/Fold.html"}, {"id": "Mathlib.LinearAlgebra.ExteriorAlgebra.Basic", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 2, "macro_tier_score": 0.0062, "macro_tier_override": null, "x": -85.85, "z": -35.388, "size": 0.3096, "title": "Exterior Algebras", "summary": "We construct the exterior algebra of a module `M` over a commutative semiring `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.html"}, {"id": "Mathlib.NumberTheory.Padics.PadicVal.Defs", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": -18.64, "z": 15.343, "size": 0.3059, "title": "`p`-adic Valuation", "summary": "This file defines the `p`-adic valuation on `ℕ`, `ℤ`, and `ℚ`. The `p`-adic valuation on `ℚ` is the difference of the multiplicities of `p` in the numerator and denominator of `q`. This function obeys the standard properties of a valuation, with the appropriate assumptions on `p`. The `p`-adic valuations on `ℕ` and `ℤ` agree with that on `ℚ`. The valuation induces a norm on `ℚ`. This norm is defined in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/PadicVal/Defs.html"}, {"id": "Mathlib.RingTheory.Flat.Stability", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 3, "macro_tier_score": 0.0414, "macro_tier_override": null, "x": 19.524, "z": 79.348, "size": 0.3644, "title": "Flatness is stable under composition and base change", "summary": "We show that flatness is stable under composition and base change.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/Stability.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Ext.Finite", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 3.21, "z": -70.499, "size": 0.2276, "title": "`Ext`-modules between finitely generated modules over Noetherian rings are finitely generated", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Ext/Finite.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -51.551, "z": 6.821, "size": 0.2691, "title": "Ext-modules in linear categories", "summary": "In this file, we show that if `C` is an `R`-linear abelian category, then there is an `R`-module structure on the groups `Ext X Y n` for `X` and `Y` in `C` and `n : ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/Ext/Linear.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.Groebner", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 44.09, "z": -64.344, "size": 0.239, "title": "Division algorithm with respect to monomial orders", "summary": "We provide a division algorithm with respect to monomial orders in polynomial rings. Let `R` be a commutative ring, `σ` a type of indeterminates and `m : MonomialOrder σ` a monomial ordering on `σ →₀ ℕ`. Consider a family of polynomials `b : ι → MvPolynomial σ R` with invertible leading coefficients (with respect to `m`): we assume `hb : ∀ i, IsUnit (m.leadingCoeff (b i))`. * `MonomialOrder.div hb f` furnishes - a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/Groebner.html"}, {"id": "Mathlib.Algebra.Group.PUnit", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.399, "macro_tier_override": null, "x": -4.027, "z": -3.85, "size": 0.4386, "title": "`PUnit` is a commutative group", "summary": "This file collects facts about algebraic structures on the one-element type, e.g. that it is a commutative ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/PUnit.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Funext", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 2, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -73.242, "z": -12.412, "size": 0.32, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Funext.html"}, {"id": "Mathlib.Algebra.Polynomial.Roots", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.2091, "macro_tier_override": null, "x": 40.663, "z": -59.938, "size": 0.3786, "title": "Theory of univariate polynomials", "summary": "We define the multiset of roots of a polynomial, and prove basic results about it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Roots.html"}, {"id": "Mathlib.RingTheory.Ideal.MinimalPrime.Basic", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.1314, "macro_tier_override": null, "x": 30.575, "z": -57.361, "size": 0.3736, "title": "Minimal primes", "summary": "We provide various results concerning the minimal primes above an ideal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.html"}, {"id": "Mathlib.RingTheory.Noetherian.Defs", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 4, "macro_tier_score": 0.307, "macro_tier_override": null, "x": -27.11, "z": -44.374, "size": 0.3956, "title": "Noetherian rings and modules", "summary": "The following are equivalent for a module M over a ring R: 1. Every increasing chain of submodules M₁ ⊆ M₂ ⊆ M₃ ⊆ ⋯ eventually stabilises. 2. Every submodule is finitely generated. A module satisfying these equivalent conditions is said to be a *Noetherian* R-module. A ring is a *Noetherian ring* if it is Noetherian as a module over itself. (Note that we do not assume yet that our rings are commutative, so perhaps…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Noetherian/Defs.html"}, {"id": "Mathlib.Algebra.GroupWithZero.InjSurj", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4654, "macro_tier_override": null, "x": -1.257, "z": -9.2, "size": 0.5409, "title": "Lifting groups with zero along injective/surjective maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/InjSurj.html"}, {"id": "Mathlib.Algebra.GroupWithZero.NeZero", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.4678, "macro_tier_override": null, "x": 7.397, "z": -0.682, "size": 0.5152, "title": "`NeZero 1` in a nontrivial `MulZeroOneClass`.", "summary": "This file exists to minimize the dependencies of `Mathlib/Algebra/GroupWithZero/Defs.lean`, which is a part of the algebraic hierarchy used by basic tactics.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/NeZero.html"}, {"id": "Mathlib.LinearAlgebra.Finsupp.LinearCombination", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 4, "macro_tier_score": 0.3743, "macro_tier_override": null, "x": -14.022, "z": 40.348, "size": 0.6137, "title": "`Finsupp.linearCombination`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Finsupp/LinearCombination.html"}, {"id": "Mathlib.Algebra.Group.TypeTags.Pointwise", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 8.623, "z": 16.449, "size": 0.293, "title": "Lemmas about pointwise operations in the presence of `Multiplicative` and `Additive`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/TypeTags/Pointwise.html"}, {"id": "Mathlib.Algebra.Order.Floor.Div", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 21.805, "z": -4.606, "size": 0.2478, "title": "Flooring, ceiling division", "summary": "This file defines division rounded up and down. The setup is an ordered monoid `α` acting on an ordered monoid `β`. If `a : α`, `b : β`, we would like to be able to \"divide\" `b` by `a`, namely find `c : β` such that `a • c = b`. This is of course not always possible, but in some cases at least there is a least `c` such that `b ≤ a • c` and a greatest `c` such that `a • c ≤ b`. We call the first one the \"ceiling…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Floor/Div.html"}, {"id": "Mathlib.RingTheory.Smooth.Basic", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 2, "macro_tier_score": 0.0214, "macro_tier_override": null, "x": -43.251, "z": -90.481, "size": 0.362, "title": "Smooth morphisms", "summary": "An `R`-algebra `A` is formally smooth if `Ω[A⁄R]` is `A`-projective and `H¹(L_{A/R}) = 0`. This is equivalent to the standard definition that \"for every `R`-algebra `B`, every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`\". An `R`-algebra `A` is smooth if it is formally smooth and of finite presentation. We show that the property of being formally smooth extends…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/Basic.html"}, {"id": "Mathlib.RingTheory.Smooth.Kaehler", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 2, "macro_tier_score": 0.0207, "macro_tier_override": null, "x": 61.461, "z": 67.11, "size": 0.3155, "title": "Relation of smoothness and `Ω[S⁄R]`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/Kaehler.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 1, "macro_tier_score": 0.0045, "macro_tier_override": null, "x": -59.358, "z": -15.253, "size": 0.4465, "title": "Presheaves of modules over a presheaf of rings.", "summary": "Given a presheaf of rings `R : Cᵒᵖ ⥤ RingCat`, we define the category `PresheafOfModules R`. An object `M : PresheafOfModules R` consists of a family of modules `M.obj X : ModuleCat (R.obj X)` for all `X : Cᵒᵖ`, together with the data, for all `f : X ⟶ Y`, of a functorial linear map `M.map f` from `M.obj X` to the restriction of scalars of `M.obj Y` via `R.map f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.Map", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.4538, "macro_tier_override": null, "x": -18.587, "z": 8.477, "size": 0.4149, "title": "`map` and `comap` for subgroups", "summary": "We prove results about images and preimages of subgroups under group homomorphisms. The bundled subgroups use bundled monoid homomorphisms. Special thanks goes to Amelia Livingston and Yury Kudryashov for their help and inspiration.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/Map.html"}, {"id": "Mathlib.RingTheory.Spectrum.Maximal.Topology", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -89.463, "z": -36.367, "size": 0.2, "title": "The Zariski topology on the maximal spectrum of a commutative (semi)ring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Maximal/Topology.html"}, {"id": "Mathlib.LinearAlgebra.Projectivization.Action", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": -99.85, "z": -40.405, "size": 0.286, "title": "Group actions on projectivization", "summary": "Show that (among other groups), the general linear group and the special linear groups of `V` act on `ℙ K V`. Prove that these actions are 2-transitive.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Projectivization/Action.html"}, {"id": "Mathlib.GroupTheory.GroupAction.MultipleTransitivity", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 2, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 46.834, "z": -17.914, "size": 0.3329, "title": "Multiple transitivity", "summary": "* `MulAction.IsMultiplyPretransitive`: A multiplicative action of a group `G` on a type `α` is n-transitive if the action of `G` on `Fin n ↪ α` is pretransitive. * `MulAction.is_zero_pretransitive` : any action is 0-pretransitive * `MulAction.is_one_pretransitive_iff` : An action is 1-pretransitive iff it is pretransitive * `MulAction.is_two_pretransitive_iff` : An action is 2-pretransitive if for any `a`, `b`, `c`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/MultipleTransitivity.html"}, {"id": "Mathlib.LinearAlgebra.SpecialLinearGroup", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 17.346, "z": -96.889, "size": 0.2424, "title": "The special linear group of a module", "summary": "If `R` is a commutative ring and `V` is an `R`-module, we define `SpecialLinearGroup R V` as the subtype of linear equivalences `V ≃ₗ[R] V` with determinant 1. When `V` doesn't have a finite basis, the determinant is defined to be 1 and the definition gives `V ≃ₗ[R] V`. The interest of this definition is that `SpecialLinearGroup R V` has a group structure. (Starting from linear maps wouldn't have worked.) The file…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/SpecialLinearGroup.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.IsDiag", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 54.395, "z": -35.585, "size": 0.2424, "title": "Diagonal matrices", "summary": "This file contains the definition and basic results about diagonal matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/IsDiag.html"}, {"id": "Mathlib.LinearAlgebra.Center", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -98.348, "z": -19.625, "size": 0.2424, "title": "Center of the algebra of linear endomorphisms", "summary": "If `V` is an `R`-module, we say that an endomorphism `f : Module.End R V` is a *homothety* with central ratio if there exists `a ∈ Set.center R` such that `f x = a • x` for all `x`. By `Module.End.mem_subsemiringCenter_iff`, these linear maps constitute the center of `Module.End R V`. (When `R` is commutative, we can write `f = a • LinearMap.id`.) In what follows, `V` is assumed to be a free `R`-module. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Center.html"}, {"id": "Mathlib.Algebra.Algebra.Shrink", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 2, "macro_tier_score": 0.0225, "macro_tier_override": null, "x": 42.806, "z": 17.979, "size": 0.2778, "title": "Transfer module and algebra structures from `α` to `Shrink α`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Shrink.html"}, {"id": "Mathlib.Algebra.Algebra.TransferInstance", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 2, "macro_tier_score": 0.0287, "macro_tier_override": null, "x": 37.433, "z": -24.196, "size": 0.3309, "title": "Transfer algebraic structures across `Equiv`s", "summary": "This continues the pattern set in `Mathlib/Algebra/Group/TransferInstance.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/TransferInstance.html"}, {"id": "Mathlib.RingTheory.Nilpotent.Basic", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 3, "macro_tier_score": 0.1943, "macro_tier_override": null, "x": -24.788, "z": 22.429, "size": 0.3579, "title": "Nilpotent elements", "summary": "This file develops the basic theory of nilpotent elements. In particular it shows that the nilpotent elements are closed under many operations. For the definition of `nilradical`, see `Mathlib/RingTheory/Nilpotent/Lemmas.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Nilpotent/Basic.html"}, {"id": "Mathlib.RingTheory.Nullstellensatz", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 105.503, "z": -8.665, "size": 0.2403, "title": "Nullstellensatz", "summary": "This file establishes a version of Hilbert's classical Nullstellensatz for `MvPolynomial`s. The main statement of the theorem is `MvPolynomial.vanishingIdeal_zeroLocus_eq_radical`. The statement is in terms of new definitions `vanishingIdeal` and `zeroLocus`. Mathlib already has versions of these in terms of the prime spectrum of a ring, but those are not well-suited for expressing this result. Suggestions for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Nullstellensatz.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.Tower", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 3, "macro_tier_score": 0.1403, "macro_tier_override": null, "x": -43.375, "z": 43.297, "size": 0.3017, "title": "Algebra towers for multivariate polynomial", "summary": "This file proves some basic results about the algebra tower structure for the type `MvPolynomial σ R`. This structure itself is provided elsewhere as `MvPolynomial.isScalarTower` When you update this file, you can also try to make a corresponding update in `RingTheory.Polynomial.Tower`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/Tower.html"}, {"id": "Mathlib.RingTheory.Flat.Localization", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 3, "macro_tier_score": 0.0344, "macro_tier_override": null, "x": -83.318, "z": -6.509, "size": 0.3458, "title": "Flatness and localization", "summary": "In this file we show that localizations are flat, and flatness is a local property.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/Localization.html"}, {"id": "Mathlib.RingTheory.Ideal.IsPrincipal", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 2, "macro_tier_score": 0.0147, "macro_tier_override": null, "x": 62.956, "z": 4.867, "size": 0.2756, "title": "Principal Ideals", "summary": "This file deals with the set of principal ideals of a `CommRing R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/IsPrincipal.html"}, {"id": "Mathlib.Algebra.Homology.HomologicalComplexLimitsEventuallyConstant", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -12.519, "z": 13.717, "size": 0.2298, "title": "Limits of degreewise eventually constant systems", "summary": "Let `F : J ⥤ HomologicalComplex C c` be a functor from a cofiltered category such that for any degree `q`, the functor `F ⋙ eval C c q : J ⥤ C` is eventually constant. Let `cF` be a limit cone for `F`. For a given degree `q`, we show that for suitable `j : J`, the map `(cF.π.app j).f q` is an isomorphism, and that `cf.π.app j` is a quasi-isomorphism in degree `q`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomologicalComplexLimitsEventuallyConstant.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Projective", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 2, "macro_tier_score": 0.0163, "macro_tier_override": null, "x": -6.082, "z": -62.85, "size": 0.3191, "title": "The category of `R`-modules has enough projectives.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Projective.html"}, {"id": "Mathlib.RepresentationTheory.Rep.Basic", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 2, "macro_tier_score": 0.0128, "macro_tier_override": null, "x": 17.605, "z": 89.282, "size": 0.3034, "title": "`Rep k G` is the category of `k`-linear representations of `G`.", "summary": "Given a `G`-representation `ρ` on a module `V`, you can construct the bundled representation as `Rep.of ρ`. Conversely, given a bundled representation `A : Rep k G`, you can get the underlying module as `A.V` and the representation on it as `A.ρ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Rep/Basic.html"}, {"id": "Mathlib.RingTheory.LocalRing.Defs", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 3, "macro_tier_score": 0.2356, "macro_tier_override": null, "x": 7.617, "z": 8.133, "size": 0.3743, "title": "Local rings", "summary": "Define the notion of a local ring for non-commutative semirings. In the commutative case, this is shown to be equivalent to the familiar definition that there exists a unique maximal ideal in `IsLocalRing.of_unique_max_ideal` and `IsLocalRing.maximal_ideal_unique`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/Defs.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 2, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -37.018, "z": -36.52, "size": 0.3197, "title": "The Ext class of a short exact sequence", "summary": "In this file, given a short exact short complex `S : ShortComplex C` in an abelian category, we construct the associated class in `Ext S.X₃ S.X₁ 1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/Ext/ExtClass.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 2, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": 50.083, "z": -2.465, "size": 0.3201, "title": "Ext groups in abelian categories", "summary": "Let `C` be an abelian category (with `C : Type u` and `Category.{v} C`). In this file, we introduce the assumption `HasExt.{w} C` which asserts that morphisms between single complexes in arbitrary degrees in the derived category of `C` are `w`-small. Under this assumption, we define `Ext.{w} X Y n : Type w` as shrunk versions of suitable types of morphisms in the derived category. In particular, when `C` has enough…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/Ext/Basic.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.SingleTriangle", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 5.891, "z": -46.054, "size": 0.2912, "title": "The distinguished triangle of a short exact sequence in an abelian category", "summary": "Given a short exact short complex `S` in an abelian category, we construct the associated distinguished triangle in the derived category: `(singleFunctor C 0).obj S.X₁ ⟶ (singleFunctor C 0).obj S.X₂ ⟶ (singleFunctor C 0).obj S.X₃ ⟶ ...`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/SingleTriangle.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.Finite", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.3285, "macro_tier_override": null, "x": -15.681, "z": 9.951, "size": 0.2746, "title": "Submonoids", "summary": "This file provides some results on multiplicative and additive submonoids in the finite context.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/Finite.html"}, {"id": "Mathlib.NumberTheory.DirichletCharacter.GaussSum", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -122.941, "z": -36.144, "size": 0.2478, "title": "Gauss sums for Dirichlet characters", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/DirichletCharacter/GaussSum.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.TStructure", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 35.352, "z": -32.89, "size": 0.2723, "title": "The canonical t-structure on the derived category", "summary": "In this file, we introduce the canonical t-structure on the derived category of an abelian category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/TStructure.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.Fractions", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 44.902, "z": 11.811, "size": 0.2938, "title": "Calculus of fractions in the derived category", "summary": "We obtain various consequences of the calculus of left and right fractions for `HomotopyCategory.quasiIso C (ComplexShape.up ℤ)` as lemmas about factorizations of morphisms `f : Q.obj X ⟶ Q.obj Y` (where `X` and `Y` are cochain complexes). These `f` can be factored as a right fraction `inv (Q.map s) ≫ Q.map g` or as a left fraction `Q.map g ≫ inv (Q.map s)`, with `s` a quasi-isomorphism (to `X` or from `Y`). When…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/Fractions.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.ShortExact", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 2, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -36.659, "z": -25.353, "size": 0.3206, "title": "The distinguished triangle attached to a short exact sequence of cochain complexes", "summary": "Given a short exact short complex `S` in the category `CochainComplex C ℤ`, we construct a distinguished triangle `Q.obj S.X₁ ⟶ Q.obj S.X₂ ⟶ Q.obj S.X₃ ⟶ (Q.obj S.X₃)⟦1⟧` in the derived category of `C`. (See `triangleOfSES` and `triangleOfSES_distinguished`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/ShortExact.html"}, {"id": "Mathlib.Algebra.Category.Ring.Topology", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 57.118, "z": 44.537, "size": 0.2, "title": "Topology on `Hom(R, S)`", "summary": "In this file, we define topology on `Hom(A, R)` for a topological ring `R`, given by the coarsest topology that makes `f ↦ f x` continuous for all `x : A`. Alternatively, given a presentation `A = ℤ[xᵢ]/I`, this is the subspace topology `Hom(A, R) ↪ Hom(ℤ[xᵢ], R) = Rᶥ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/Topology.html"}, {"id": "Mathlib.RingTheory.TensorProduct.DirectLimitFG", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 70.9, "z": -22.174, "size": 0.2338, "title": "Tensor products and finitely generated submodules", "summary": "Various results about how tensor products of arbitrary modules are direct limits of tensor products of finitely-generated modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/DirectLimitFG.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Tower", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 4, "macro_tier_score": 0.2853, "macro_tier_override": null, "x": 25.041, "z": 47.682, "size": 0.4575, "title": "The `A`-module structure on `M ⊗[R] N`", "summary": "When `M` is both an `R`-module and an `A`-module, and `Algebra R A`, then many of the morphisms preserve the actions by `A`. The `Module` instance itself is provided elsewhere as `TensorProduct.leftModule`. This file provides more general versions of the definitions already in `LinearAlgebra/TensorProduct`. In this file, we use the convention that `M`, `N`, `P`, `Q` are all `R`-modules, but only `M` and `P` are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Tower.html"}, {"id": "Mathlib.RingTheory.Adjoin.FG", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.2484, "macro_tier_override": null, "x": -8.694, "z": -71.906, "size": 0.3452, "title": "Adjoining elements to form subalgebras", "summary": "This file develops the basic theory of finitely-generated subalgebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Adjoin/FG.html"}, {"id": "Mathlib.Algebra.Order.Quantale", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 3.625, "z": 10.537, "size": 0.2, "title": "Theory of quantales", "summary": "Quantales are the non-commutative generalization of locales/frames and as such are linked to point-free topology and order theory. Applications are found throughout logic, quantum mechanics, and computer science (see e.g. [Vickers1989] and [Mulvey1986]). The most general definition of quantale occurring in literature, is that a quantale is a semigroup distributing over a complete sup-semilattice. In our definition…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Quantale.html"}, {"id": "Mathlib.Algebra.Lie.Sl2", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -95.421, "z": 61.063, "size": 0.2563, "title": "The Lie algebra `sl₂` and its representations", "summary": "The Lie algebra `sl₂` is the unique simple Lie algebra of minimal rank, 1, and as such occupies a distinguished position in the general theory. This file provides some basic definitions and results about `sl₂`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Sl2.html"}, {"id": "Mathlib.Algebra.Order.SuccPred", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.3408, "macro_tier_override": null, "x": 14.204, "z": 4.358, "size": 0.4178, "title": "Interaction between successors and arithmetic", "summary": "We define the `SuccAddOrder` and `PredSubOrder` typeclasses, for orders satisfying `succ x = x + 1` and `pred x = x - 1` respectively. This allows us to transfer the API for successors and predecessors into these common arithmetical forms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/SuccPred.html"}, {"id": "Mathlib.Algebra.Category.CommHopfAlgCat", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -74.914, "z": 27.663, "size": 0.2, "title": "The category of commutative Hopf algebras over a commutative ring", "summary": "This file defines the bundled category `CommHopfAlgCat` of commutative Hopf algebras over a fixed commutative ring `R` along with the forgetful functor to `CommBialgCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/CommHopfAlgCat.html"}, {"id": "Mathlib.Algebra.Category.CommBialgCat", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -42.482, "z": 65.417, "size": 0.2338, "title": "The category of commutative bialgebras over a commutative ring", "summary": "This file defines the bundled category `CommBialgCat R` of commutative bialgebras over a fixed commutative ring `R` along with the forgetful functor to `CommAlgCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/CommBialgCat.html"}, {"id": "Mathlib.RingTheory.HopfAlgebra.Convolution", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 65.16, "z": -35.674, "size": 0.2585, "title": "Convolution product on Hopf algebra maps", "summary": "This file constructs the ring structure on bialgebra homs `C → A` where `C` and `A` are Hopf algebras and multiplication is given by ``` | μ | | / \\ f * g = f g | | \\ / δ | ``` diagrammatically, where `μ` stands for multiplication and `δ` for comultiplication.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HopfAlgebra/Convolution.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Regular", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.4448, "macro_tier_override": null, "x": -6.482, "z": 3.628, "size": 0.4105, "title": "Results about `IsRegular` and `0`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Regular.html"}, {"id": "Mathlib.RepresentationTheory.Homological.ContCohomology.Basic", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": 87.863, "z": 30.044, "size": 0.2955, "title": "Continuous cohomology", "summary": "We define continuous cohomology as the homology of the homogeneous cochain complex.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/ContCohomology/Basic.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Topology.Homology", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -41.063, "z": -45.496, "size": 0.264, "title": "`TopModuleCat` is a `CategoryWithHomology`", "summary": "`TopModuleCat R`, the category of topological `R`-modules, is not an abelian category. But since the topology on subquotients is well-defined, we can still talk about homology in this category. See the `CategoryWithHomology (TopModuleCat R)` instance in this file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Topology/Homology.html"}, {"id": "Mathlib.RepresentationTheory.Continuous.TopRep", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 21.328, "z": 88.466, "size": 0.264, "title": "Topological representations", "summary": "This file defines the category `TopRep k G` of topological representations of a monoid `G` over a topological ring `k`, and shows that it is equivalent to the category `Action (TopModuleCat k) G`. For a topological group `G` we define the invariants functor `TopRep.invariantsFunctor`, the coinduction functor `TopRep.coind₁Functor`, the restriction functor `TopRep.resFunctor` along a group homomorphism `φ : H →* G`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Continuous/TopRep.html"}, {"id": "Mathlib.RingTheory.Henselian", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -59.763, "z": -44.124, "size": 0.2, "title": "Henselian rings", "summary": "In this file we set up the basic theory of Henselian (local) rings. A ring `R` is *Henselian* at an ideal `I` if the following conditions hold: * `I` is contained in the Jacobson radical of `R` * for every polynomial `f` over `R`, with a *simple* root `a₀` over the quotient ring `R/I`, there exists a lift `a : R` of `a₀` that is a root of `f`. (Here, saying that a root `b` of a polynomial `g` is *simple* means that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Henselian.html"}, {"id": "Mathlib.Algebra.Polynomial.Taylor", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.1805, "macro_tier_override": null, "x": 32.67, "z": 64.643, "size": 0.3248, "title": "Taylor expansions of polynomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Taylor.html"}, {"id": "Mathlib.RingTheory.LocalRing.ResidueField.Basic", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.0831, "macro_tier_override": null, "x": -56.213, "z": 39.52, "size": 0.3408, "title": "Residue Field of local rings", "summary": "We prove basic properties of the residue field of a local ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/ResidueField/Basic.html"}, {"id": "Mathlib.RingTheory.AdicCompletion.Basic", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.0677, "macro_tier_override": null, "x": 25.486, "z": -59.796, "size": 0.3444, "title": "Completion of a module with respect to an ideal.", "summary": "In this file we define the notions of Hausdorff, precomplete, and complete for an `R`-module `M` with respect to an ideal `I`:", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AdicCompletion/Basic.html"}, {"id": "Mathlib.NumberTheory.FermatPsp", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 7.046, "z": 101.901, "size": 0.2, "title": "Fermat Pseudoprimes", "summary": "In this file we define Fermat pseudoprimes: composite numbers that pass the Fermat primality test. A natural number `n` passes the Fermat primality test to base `b` (and is therefore deemed a \"probable prime\") if `n` divides `b ^ (n - 1) - 1`. `n` is a Fermat pseudoprime to base `b` if `n` is a composite number that passes the Fermat primality test to base `b` and is coprime with `b`. Fermat pseudoprimes can also be…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/FermatPsp.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Stalk", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -27.017, "z": -57.072, "size": 0.2555, "title": "Module structure on stalks", "summary": "Let `M` be a presheaf of `R`-modules on a topological space. We endow `M.presheaf.stalk x` with an `R.stalk x`-module structure. The key characterizing lemma is `PresheafOfModules.germ_smul`, which describes the compatibility of the scalar action and `TopCat.Presheaf.germ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Stalk.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Prod", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 2, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": 52.475, "z": -18.721, "size": 0.2769, "title": "Tensor products of products", "summary": "This file shows that taking `TensorProduct`s commutes with taking `Prod`s in both arguments.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Prod.html"}, {"id": "Mathlib.RingTheory.LocalRing.ResidueField.Instances", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 2, "macro_tier_score": 0.0124, "macro_tier_override": null, "x": -44.204, "z": 85.862, "size": 0.2595, "title": "Instances on residue fields", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/ResidueField/Instances.html"}, {"id": "Mathlib.FieldTheory.Separable", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 3, "macro_tier_score": 0.1284, "macro_tier_override": null, "x": -9.78, "z": 94.209, "size": 0.3904, "title": "Separable polynomials", "summary": "We define a polynomial to be separable if it is coprime with its derivative. We prove basic properties about separable polynomials here.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Separable.html"}, {"id": "Mathlib.Algebra.Field.Opposite", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.2823, "macro_tier_override": null, "x": -10.78, "z": 12.774, "size": 0.3096, "title": "Field structure on the multiplicative/additive opposite", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/Opposite.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -27.777, "z": -68.898, "size": 0.2, "title": "Pullbacks of free sheaves of modules", "summary": "Let `S` (resp.`R`) be a sheaf of rings on a category `C` (resp. `D`) equipped with a Grothendieck topology `J` (resp. `K`). Let `F : C ⥤ D` be a continuous functor. Let `φ` be a morphism from `S` to the direct image of `R`. We introduce `unitToPushforwardObjUnit φ` which is the morphism in the category `SheafOfModules S` which corresponds to `φ`, and show that the adjoint morphism `pullbackObjUnitToUnit φ :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Sheaf/PullbackFree.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Sheaf.Free", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -45.501, "z": 56.353, "size": 0.2626, "title": "Free sheaves of modules", "summary": "In this file, we construct the functor `SheafOfModules.freeFunctor : Type u ⥤ SheafOfModules.{u} R` which sends a type `I` to the coproduct of copies indexed by `I` of `unit R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Sheaf/Free.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackContinuous", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 70.454, "z": 4.087, "size": 0.247, "title": "Pullback of sheaves of modules", "summary": "Let `S` and `R` be sheaves of rings over sites `(C, J)` and `(D, K)` respectively. Let `F : C ⥤ D` be a continuous functor between these sites, and let `φ : S ⟶ (F.sheafPushforwardContinuous RingCat.{u} J K).obj R` be a morphism of sheaves of rings. In this file, we define the pullback functor for sheaves of modules `pullback.{v} φ : SheafOfModules.{v} S ⥤ SheafOfModules.{v} R` that is left adjoint to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Sheaf/PullbackContinuous.html"}, {"id": "Mathlib.Algebra.Lie.TraceForm", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 102.623, "z": -52.218, "size": 0.262, "title": "The trace and Killing forms of a Lie algebra.", "summary": "Let `L` be a Lie algebra with coefficients in a commutative ring `R`. Suppose `M` is a finite, free `R`-module and we have a representation `φ : L → End M`. This data induces a natural bilinear form `B` on `L`, called the trace form associated to `M`; it is defined as `B(x, y) = Tr (φ x) (φ y)`. In the special case that `M` is `L` itself and `φ` is the adjoint representation, the trace form is known as the Killing…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/TraceForm.html"}, {"id": "Mathlib.Algebra.DirectSum.LinearMap", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 79.653, "z": 30.88, "size": 0.2475, "title": "Linear maps between direct sums", "summary": "This file contains results about linear maps which respect direct sum decompositions of their domain and codomain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/DirectSum/LinearMap.html"}, {"id": "Mathlib.Algebra.Lie.InvariantForm", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 60.152, "z": -65.79, "size": 0.2475, "title": "Lie algebras with non-degenerate invariant bilinear forms are semisimple", "summary": "In this file we prove that a finite-dimensional Lie algebra over a field is semisimple if it does not have non-trivial abelian ideals and it admits a non-degenerate reflexive invariant bilinear form. Here a form is *invariant* if it is invariant under the Lie bracket in the sense that `⁅x, Φ⁆ = 0` for all `x` or equivalently, `Φ ⁅x, y⁆ z = Φ x ⁅y, z⁆`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/InvariantForm.html"}, {"id": "Mathlib.Algebra.Lie.Weights.Cartan", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -50.365, "z": -101.475, "size": 0.2475, "title": "Weights and roots of Lie modules and Lie algebras with respect to Cartan subalgebras", "summary": "Given a Lie algebra `L` which is not necessarily nilpotent, it may be useful to study its representations by restricting them to a nilpotent subalgebra (e.g., a Cartan subalgebra). In the particular case when we view `L` as a module over itself via the adjoint action, the weight spaces of `L` restricted to a nilpotent subalgebra are known as root spaces. Basic definitions and properties of the above ideas are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Weights/Cartan.html"}, {"id": "Mathlib.Algebra.Lie.Weights.Linear", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -95.684, "z": -60.651, "size": 0.2544, "title": "Lie modules with linear weights", "summary": "Given a Lie module `M` over a nilpotent Lie algebra `L` with coefficients in `R`, one frequently studies `M` via its weights. These are functions `χ : L → R` whose corresponding weight space `LieModule.genWeightSpace M χ`, is non-trivial. If `L` is Abelian or if `R` has characteristic zero (and `M` is finite-dimensional) then such `χ` are necessarily `R`-linear. However in general non-linear weights do exist. For…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Weights/Linear.html"}, {"id": "Mathlib.LinearAlgebra.BilinearForm.TensorProduct", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": 57.838, "z": 57.724, "size": 0.263, "title": "The bilinear form on a tensor product", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/BilinearForm/TensorProduct.html"}, {"id": "Mathlib.LinearAlgebra.PID", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -41.561, "z": 74.638, "size": 0.2334, "title": "Linear maps of modules with coefficients in a principal ideal domain", "summary": "Since a submodule of a free module over a PID is free, certain constructions which are often developed only for vector spaces may be generalised to any module with coefficients in a PID. This file is a location for such results and exists to avoid making large parts of the linear algebra import hierarchy have to depend on the theory of PIDs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/PID.html"}, {"id": "Mathlib.Algebra.ContinuedFractions.Translations", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 1, "macro_tier_score": 0.0032, "macro_tier_override": null, "x": 12.966, "z": -0.943, "size": 0.335, "title": "Basic Translation Lemmas Between Functions Defined for Continued Fractions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/ContinuedFractions/Translations.html"}, {"id": "Mathlib.RingTheory.LocalProperties.ProjectiveDimension", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 6.691, "z": 81.441, "size": 0.2, "title": "The Projective Dimension Equal to Supremum over Localizations", "summary": "In this file, we proved that projective dimension equal to supremum over localizations", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalProperties/ProjectiveDimension.html"}, {"id": "Mathlib.RingTheory.LocalProperties.Projective", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.0212, "macro_tier_override": null, "x": 74.558, "z": 28.608, "size": 0.256, "title": "Being projective is a local property", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalProperties/Projective.html"}, {"id": "Mathlib.Algebra.Polynomial.RingDivision", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.2113, "macro_tier_override": null, "x": 42.381, "z": 56.429, "size": 0.3127, "title": "Theory of univariate polynomials", "summary": "We prove basic results about univariate polynomials.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/RingDivision.html"}, {"id": "Mathlib.RingTheory.AlgebraicIndependent.Transcendental", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 3, "macro_tier_score": 0.0626, "macro_tier_override": null, "x": -78.362, "z": 29.048, "size": 0.3008, "title": "Algebraic Independence", "summary": "This file relates algebraic independence and transcendence (or algebraicity) of elements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AlgebraicIndependent/Transcendental.html"}, {"id": "Mathlib.RingTheory.Algebraic.MvPolynomial", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 3, "macro_tier_score": 0.0636, "macro_tier_override": null, "x": 49.203, "z": 65.241, "size": 0.288, "title": "Transcendental elements in `MvPolynomial`", "summary": "This file lists some results on some elements in `MvPolynomial σ R` being transcendental over the base ring `R` and subrings `MvPolynomial.supported` of `MvPolynomial σ R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Algebraic/MvPolynomial.html"}, {"id": "Mathlib.RingTheory.AlgebraicIndependent.Basic", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0624, "macro_tier_override": null, "x": 19.782, "z": 67.743, "size": 0.2811, "title": "Algebraic Independence", "summary": "This file contains basic results on algebraic independence of a family of elements of an `R`-algebra", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AlgebraicIndependent/Basic.html"}, {"id": "Mathlib.NumberTheory.FLT.Polynomial", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -58.134, "z": -79.428, "size": 0.2, "title": "Fermat's Last Theorem for polynomials over a field", "summary": "This file states and proves the Fermat's Last Theorem for polynomials over a field. For `n ≥ 3` not divisible by the characteristic of the coefficient field `k` and (pairwise) nonzero coprime polynomials `a, b, c` (over a field) with `a ^ n + b ^ n = c ^ n`, all polynomials must be constants. More generally, we can prove non-solvability of the Fermat-Catalan equation: there are no non-constant polynomial solutions…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/FLT/Polynomial.html"}, {"id": "Mathlib.NumberTheory.FLT.MasonStothers", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -77.094, "z": -11.859, "size": 0.239, "title": "Mason-Stothers theorem", "summary": "This file states and proves the Mason-Stothers theorem, which is a polynomial version of the ABC conjecture. For (pairwise) coprime polynomials `a, b, c` (over a field) with `a + b + c = 0`, we have `max {deg(a), deg(b), deg(c)} + 1 ≤ deg(rad(abc))` or `a' = b' = c' = 0`. Proof is based on this online note by Franz Lemmermeyer http://www.fen.bilkent.edu.tr/~franz/ag05/ag-02.pdf, which is essentially based on Noah…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/FLT/MasonStothers.html"}, {"id": "Mathlib.RingTheory.Polynomial.IsIntegral", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 2, "macro_tier_score": 0.0069, "macro_tier_override": null, "x": -74.599, "z": -61.329, "size": 0.2645, "title": "Results about coefficients of polynomials being integral", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/IsIntegral.html"}, {"id": "Mathlib.NumberTheory.ArithmeticFunction.Liouville", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 18.734, "z": -38.387, "size": 0.2, "title": "The Liouville Function", "summary": "This file defines the Liouville function `λ(n)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ArithmeticFunction/Liouville.html"}, {"id": "Mathlib.RingTheory.Smooth.IntegralClosure", "region_id": "algebra", "micro_elevation": 0.9211, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -107.112, "z": -73.671, "size": 0.2371, "title": "Smooth base change commutes with integral closure", "summary": "In this file we aim to prove that smooth base change commutes with integral closure. We define the map `TensorProduct.toIntegralClosure : S ⊗[R] integralClosure R B →ₐ[S] integralClosure S (S ⊗[R] B)` and show that it is bijective when `S` is `R`-smooth.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/IntegralClosure.html"}, {"id": "Mathlib.RingTheory.Unramified.LocalStructure", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -128.114, "z": 2.791, "size": 0.2757, "title": "Local structure of unramified algebras", "summary": "In this file, we will prove that if `S` is a finite type `R`-algebra unramified at `Q`, then there exists `f ∉ Q` and a standard etale algebra `A` over `R` that surjects onto `S[1/f]`. Geometrically, this says that unramified morphisms locally are closed subsets of etale covers. As a corollary, we also obtain results about the local structure of etale and smooth algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Unramified/LocalStructure.html"}, {"id": "Mathlib.GroupTheory.Congruence.Defs", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4329, "macro_tier_override": null, "x": 11.136, "z": -0.4, "size": 0.4234, "title": "Congruence relations", "summary": "This file defines congruence relations: equivalence relations that preserve a binary operation, which in this case is multiplication or addition. The principal definition is a `structure` extending a `Setoid` (an equivalence relation), and the inductive definition of the smallest congruence relation containing a binary relation is also given (see `ConGen`). The file also proves basic properties of the quotient of a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Congruence/Defs.html"}, {"id": "Mathlib.NumberTheory.ModularForms.SlashInvariantForms", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 1, "macro_tier_score": 0.0017, "macro_tier_override": null, "x": -1.439, "z": 1.174, "size": 0.3033, "title": "Slash invariant forms", "summary": "This file defines functions that are invariant under a `SlashAction` which forms the basis for defining `ModularForm` and `CuspForm`. We prove several instances for such spaces, in particular that they form a module over `ℝ`, and over `ℂ` if the group is contained in `SL(2, ℝ)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/SlashInvariantForms.html"}, {"id": "Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0019, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.3164, "title": "Arithmetic subgroups of `GL(2, ℝ)`", "summary": "We define a subgroup of `GL (Fin 2) ℝ` to be *arithmetic* if it is commensurable with the image of `SL(2, ℤ)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/ArithmeticSubgroups.html"}, {"id": "Mathlib.NumberTheory.ModularForms.SlashActions", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2933, "title": "Slash actions", "summary": "This file defines a class of slash actions, which are families of right actions of a group on an a additive monoid, parametrized by some index type. This is modeled on the slash action of `GL (Fin 2) ℝ` on the space of modular forms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/SlashActions.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 125.437, "z": 14.629, "size": 0.2607, "title": "Geck's construction of a Lie algebra associated to a root system", "summary": "This file contains an implementation of Geck's construction of a semisimple Lie algebra from a reduced crystallographic root system. It follows [Geck](Geck2017) quite closely.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/GeckConstruction/Basic.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -121.457, "z": -34.594, "size": 0.2429, "title": "Supporting lemmas for Geck's construction of a Lie algebra associated to a root system", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/GeckConstruction/Lemmas.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.Submodule", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -61.72, "z": 25.702, "size": 0.2, "title": "Submodules of presheaves of modules", "summary": "Given a presheaf of modules `M` over a presheaf of rings `R` and a family of submodules `N X` of `M.obj X` that is stable under the restriction maps of `M`, we construct the corresponding subobject of `M` in the category `PresheafOfModules R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/Submodule.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.EpiMono", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 49.779, "z": -41.799, "size": 0.2758, "title": "Epimorphisms and monomorphisms in the category of presheaves of modules", "summary": "In this file, we give characterizations of epimorphisms and monomorphisms in the category of presheaves of modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/EpiMono.html"}, {"id": "Mathlib.NumberTheory.NumberField.FinitePlaces", "region_id": "algebra", "micro_elevation": 0.9342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 64.012, "z": 115.279, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/FinitePlaces.html"}, {"id": "Mathlib.NumberTheory.NumberField.Completion.FinitePlace", "region_id": "algebra", "micro_elevation": 0.9211, "macro_tier": 1, "macro_tier_score": 0.0027, "macro_tier_override": null, "x": -123.944, "z": -39.22, "size": 0.2894, "title": "Finite places of number fields", "summary": "This file defines finite places of a number field `K` as absolute values coming from an embedding into a completion of `K` associated to a non-zero prime ideal of `𝓞 K`. Many of the results in this file are expressed in the generality of: `R` is a Dedekind domain with field of fractions `K` such that `Module.Finite ℤ R` and `Module.Free ℤ R`. If `K` is a number field, then this characterises `R` as being isomorphic…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Completion/FinitePlace.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.ShortExact", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 26.932, "z": -30.725, "size": 0.2917, "title": "The mapping cone of a monomorphism, up to a quasi-isomorphism", "summary": "If `S` is a short exact short complex of cochain complexes in an abelian category, we construct a quasi-isomorphism `descShortComplex S : mappingCone S.f ⟶ S.X₃`. We obtain this by comparing the homology sequence of `S` and the homology sequence of the homology functor on the homotopy category, applied to the distinguished triangle attached to the mapping cone of `S.f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.Basic", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 2, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": 12.936, "z": -40.709, "size": 0.3634, "title": "The derived category of an abelian category", "summary": "In this file, we construct the derived category `DerivedCategory C` of an abelian category `C`. It is equipped with a triangulated structure. The derived category is defined here as the localization of cochain complexes indexed by `ℤ` with respect to quasi-isomorphisms: it is a type synonym of `HomologicalComplexUpToQuasiIso C (ComplexShape.up ℤ)`. Then, we have a localization functor `DerivedCategory.Q :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Dual.BaseChange", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": -37.728, "z": 78.712, "size": 0.2672, "title": "Base change for the dual of a module", "summary": "* `Module.Dual.congr` : equivalent modules have equivalent duals. If `f : Module.Dual R V` and `Algebra R A`, then * `Module.Dual.baseChange A f` is the element of `Module.Dual A (A ⊗[R] V)` deduced by base change. * `Module.Dual.baseChangeHom` is the `R`-linear map given by `Module.Dual.baseChange`. * `IsBaseChange.dual` : for finite free modules, taking dual commutes with base change.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dual/BaseChange.html"}, {"id": "Mathlib.RingTheory.TensorProduct.IsBaseChangeHom", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -60.028, "z": -60.786, "size": 0.2915, "title": "Base change properties for modules of linear maps", "summary": "* `IsBaseChange.linearMapRight`: If `M` is finite free and `P` is a base change of `N` to `S`, then `M →ₗ[R] P` is a base change of `M →ₗ[R] N` to `S`. * `IsBaseChange.linearMapLeftRight`: If `M` is finite free and `P` is a base change of `M` to `S`, if `Q` is a base change of `N` to `S`, then `P →ₗ[S] Q` is a base change of `M →ₗ[R] N` to `S`. * `IsBaseChange.end`: If `M` is finite free and `P` is a base change of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/IsBaseChangeHom.html"}, {"id": "Mathlib.Algebra.Group.Int.Units", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4136, "macro_tier_override": null, "x": 8.553, "z": 9.791, "size": 0.345, "title": "Units in the integers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Int/Units.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Restrict", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 1, "macro_tier_score": 0.0019, "macro_tier_override": null, "x": 35.336, "z": 43.076, "size": 0.3189, "title": "Affine map restrictions", "summary": "This file defines restrictions of affine maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Restrict.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 1, "macro_tier_score": 0.0045, "macro_tier_override": null, "x": -53.836, "z": -1.511, "size": 0.4045, "title": "Affine spaces", "summary": "This file gives further properties of affine subspaces (over modules) and the affine span of a set of points.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Basic.html"}, {"id": "Mathlib.Algebra.Order.SuccPred.TypeTags", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -14.343, "z": 3.874, "size": 0.2, "title": "Successor and predecessor on type tags", "summary": "This file declares successor and predecessor orders on type tags.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/SuccPred/TypeTags.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Unbundled.TypeTags", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 3, "macro_tier_score": 0.2511, "macro_tier_override": null, "x": 12.999, "z": -0.201, "size": 0.2947, "title": "Ordered monoid structures on `Multiplicative α` and `Additive α`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Unbundled/TypeTags.html"}, {"id": "Mathlib.RingTheory.Ideal.Nonunits", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 4, "macro_tier_score": 0.3016, "macro_tier_override": null, "x": 34.98, "z": -33.286, "size": 0.4446, "title": "The set of non-invertible elements of a monoid", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Nonunits.html"}, {"id": "Mathlib.GroupTheory.Perm.Centralizer", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -43.214, "z": 25.435, "size": 0.2557, "title": "Centralizer of a permutation and cardinality of conjugacy classes in the symmetric groups", "summary": "Let `α : Type` with `Fintype α` (and `DecidableEq α`). The main goal of this file is to compute the cardinality of conjugacy classes in `Equiv.Perm α`. Every `g : Equiv.Perm α` has a `g.cycleType : Multiset ℕ`. By `Equiv.Perm.isConj_iff_cycleType_eq`, two permutations are conjugate in `Equiv.Perm α` iff their cycle types are equal. To compute the cardinality of the conjugacy classes, we could use a purely…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Centralizer.html"}, {"id": "Mathlib.GroupTheory.NoncommCoprod", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 23.972, "z": 2.868, "size": 0.2379, "title": "Canonical homomorphism from a pair of monoids", "summary": "This file defines the construction of the canonical homomorphism from a pair of monoids. Given two morphisms of monoids `f : M →* P` and `g : N →* P` where elements in the images of the two morphisms commute, we obtain a canonical morphism `MonoidHom.noncommCoprod : M × N →* P` whose composition with `inl M N` coincides with `f` and whose composition with `inr M N` coincides with `g`. There is an analogue…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/NoncommCoprod.html"}, {"id": "Mathlib.GroupTheory.Perm.Cycle.PossibleTypes", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 45.537, "z": 16.06, "size": 0.2379, "title": "Possible cycle types of permutations", "summary": "* For `m : Multiset ℕ`, `Equiv.Perm.exists_with_cycleType_iff m` proves that there are permutations with cycleType `m` if and only if its sum is at most `Fintype.card α` and its members are at least 2.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Cycle/PossibleTypes.html"}, {"id": "Mathlib.GroupTheory.Perm.DomMulAct", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -15.213, "z": 16.286, "size": 0.2379, "title": "Subgroup of `Equiv.Perm α` preserving a function", "summary": "Let `α` and `ι` by types and let `f : α → ι` * `DomMulAct.mem_stabilizer_iff` proves that the stabilizer of `f : α → ι` in `(Equiv.Perm α)ᵈᵐᵃ` is the set of `g : (Equiv.Perm α)ᵈᵐᵃ` such that `f ∘ (mk.symm g) = f`. The natural equivalence from `stabilizer (Perm α)ᵈᵐᵃ f` to `{ g : Perm α // p ∘ g = f }` can be obtained as `subtypeEquiv mk.symm (fun _ => mem_stabilizer_iff)` * `DomMulAct.stabilizerMulEquiv` is the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/DomMulAct.html"}, {"id": "Mathlib.Algebra.Order.Hom.MonoidWithZero", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 3, "macro_tier_score": 0.0709, "macro_tier_override": null, "x": 20.313, "z": 2.173, "size": 0.3379, "title": "Ordered monoid and group homomorphisms", "summary": "This file defines morphisms between (additive) ordered monoids with zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Hom/MonoidWithZero.html"}, {"id": "Mathlib.Algebra.Order.Hom.Monoid", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 3, "macro_tier_score": 0.1967, "macro_tier_override": null, "x": -12.031, "z": -8.717, "size": 0.3699, "title": "Ordered monoid and group homomorphisms", "summary": "This file defines morphisms between (additive) ordered monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Hom/Monoid.html"}, {"id": "Mathlib.Algebra.Group.Pointwise.Set.Small", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 4.963, "z": -17.896, "size": 0.2, "title": "Small instances for pointwise operations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pointwise/Set/Small.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Pi", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 3, "macro_tier_score": 0.0373, "macro_tier_override": null, "x": -22.008, "z": -51.184, "size": 0.3096, "title": "Tensor product and products", "summary": "In this file we examine the behaviour of the tensor product with arbitrary and finite products. Let `S` be an `R`-algebra, `N` an `S`-module, `ι` an index type and `Mᵢ` a family of `R`-modules. We then have a natural map `TensorProduct.piRightHom`: `N ⊗[R] (∀ i, M i) →ₗ[S] ∀ i, N ⊗[R] M i` In general, this is not an isomorphism, but if `ι` is finite, then it is and it is packaged as `TensorProduct.piRight`. Also a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Pi.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.FreeLocus", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 2, "macro_tier_score": 0.0197, "macro_tier_override": null, "x": 81.25, "z": 58.787, "size": 0.3252, "title": "The free locus of a module", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/FreeLocus.html"}, {"id": "Mathlib.RingTheory.TensorProduct.Free", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.0559, "macro_tier_override": null, "x": 27.814, "z": 58.749, "size": 0.2907, "title": "Results on bases of tensor products", "summary": "In the file we construct a basis for the base change of a module to an algebra, and deduce that `Module.Free` is stable under base change.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/Free.html"}, {"id": "Mathlib.RingTheory.TensorProduct.IsBaseChangePi", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 2, "macro_tier_score": 0.0282, "macro_tier_override": null, "x": 17.84, "z": 66.359, "size": 0.2884, "title": "Base change properties", "summary": "This file proves that several constructions in linear algebra commute with base change, as expressed by `IsBaseChange`. * `IsBaseChange.prodMap`, `IsBaseChange.pi`: binary and finite products. In particular, localization of modules commutes with binary and finite products. * `IsBaseChange.directSum`: base change for direct sums * Homomorphism modules", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/IsBaseChangePi.html"}, {"id": "Mathlib.NumberTheory.EulerProduct.ExpLog", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -40.877, "z": 62.028, "size": 0.2639, "title": "Logarithms of Euler Products", "summary": "We consider `f : ℕ →*₀ ℂ` and show that `exp (∑ p in Primes, log (1 - f p)⁻¹) = ∑ n : ℕ, f n` under suitable conditions on `f`. This can be seen as a logarithmic version of the Euler product for `f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/EulerProduct/ExpLog.html"}, {"id": "Mathlib.NumberTheory.LSeries.Dirichlet", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 2, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -37.763, "z": -96.903, "size": 0.3051, "title": "L-series of Dirichlet characters and arithmetic functions", "summary": "We collect some results on L-series of specific (arithmetic) functions, for example, the Möbius function `μ` or the von Mangoldt function `Λ`. In particular, we show that `L ↗Λ` is the negative of the logarithmic derivative of the Riemann zeta function on `re s > 1`; see `LSeries_vonMangoldt_eq_deriv_riemannZeta_div`. We also prove some general results on L-series associated to Dirichlet characters (i.e., Dirichlet…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/Dirichlet.html"}, {"id": "Mathlib.NumberTheory.ModularForms.NormTrace", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -73.832, "z": 39.155, "size": 0.2, "title": "Norm and trace maps", "summary": "Given two subgroups `𝒢, ℋ` of `GL(2, ℝ)` with `𝒢.relindex ℋ ≠ 0` (i.e. `𝒢 ⊓ ℋ` has finite index in `ℋ`), we define a trace map from `ModularForm (𝒢 ⊓ ℋ) k` to `ModularForm ℋ k`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/NormTrace.html"}, {"id": "Mathlib.NumberTheory.ModularForms.LevelOne.Basic", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 1, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": -46.526, "z": 67.176, "size": 0.3088, "title": "Level one modular forms", "summary": "This file contains results specific to modular forms of level one, i.e. modular forms for `SL(2, ℤ)`. TODO: Add finite-dimensionality of these spaces of modular forms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/LevelOne/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.InvariantBasisNumber", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.1757, "macro_tier_override": null, "x": 59.33, "z": 30.822, "size": 0.3001, "title": "Invertible matrices over a ring with invariant basis number are square.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/InvariantBasisNumber.html"}, {"id": "Mathlib.LinearAlgebra.InvariantBasisNumber", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.2593, "macro_tier_override": null, "x": 39.723, "z": -51.451, "size": 0.3964, "title": "Invariant basis number property", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/InvariantBasisNumber.html"}, {"id": "Mathlib.Algebra.Lie.BaseChange", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 1, "macro_tier_score": 0.0029, "macro_tier_override": null, "x": 14.34, "z": 80.447, "size": 0.3121, "title": "Extension and restriction of scalars for Lie algebras and Lie modules", "summary": "Lie algebras and their representations have a well-behaved theory of extension and restriction of scalars.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/BaseChange.html"}, {"id": "Mathlib.Algebra.Lie.TensorProduct", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": -66.054, "z": 44.88, "size": 0.287, "title": "Tensor products of Lie modules", "summary": "Tensor products of Lie modules carry natural Lie module structures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/TensorProduct.html"}, {"id": "Mathlib.Algebra.Category.AlgCat.TensorAlgebra", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -5.528, "z": -61.037, "size": 0.2734, "title": "`TensorAlgebra` as a functor `ModuleCat R ⥤ AlgCat R`", "summary": "In this file we define the functor `AlgCat.tensorAlgebra : ModuleCat R ⥤ AlgCat R` sending an `R`-module `M` to the tensor algebra `TensorAlgebra R M`. We show that this functor is a left-adjoint to the forgetful functor `AlgCat R ⥤ ModuleCat R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/AlgCat/TensorAlgebra.html"}, {"id": "Mathlib.NumberTheory.DirichletCharacter.Bounds", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": -30.458, "z": 97.497, "size": 0.2481, "title": "Bounds for values of Dirichlet characters", "summary": "We consider Dirichlet characters `χ` with values in a normed field `F`. We show that `‖χ a‖ = 1` if `a` is a unit and `‖χ a‖ ≤ 1` in general.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/DirichletCharacter/Bounds.html"}, {"id": "Mathlib.NumberTheory.LSeries.Convolution", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": 77.187, "z": -11.241, "size": 0.2481, "title": "Dirichlet convolution of sequences and products of L-series", "summary": "We define the *Dirichlet convolution* `f ⍟ g` of two sequences `f g : ℕ → R` with values in a semiring `R` by `(f ⍟ g) n = ∑ (k * m = n), f k * g m` when `n ≠ 0` and `(f ⍟ g) 0 = 0`. Technically, this is done by transporting the existing definition for `ArithmeticFunction R`; see `LSeries.convolution`. We show that these definitions agree (`LSeries.convolution_def`). We then consider the case `R = ℂ` and show that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/Convolution.html"}, {"id": "Mathlib.NumberTheory.LSeries.Positivity", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": 5.949, "z": -79.636, "size": 0.2481, "title": "Positivity of values of L-series", "summary": "The main results of this file are as follows. * If `a : ℕ → ℂ` takes nonnegative real values and `a 1 > 0`, then `L a x > 0` when `x : ℝ` is in the open half-plane of absolute convergence; see `LSeries.positive` and `ArithmeticFunction.LSeries_positive`. * If in addition the L-series of `a` agrees on some open right half-plane where it converges with an entire function `f`, then `f` is positive on the real axis; see…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/Positivity.html"}, {"id": "Mathlib.NumberTheory.SumPrimeReciprocals", "region_id": "algebra", "micro_elevation": 0.0263, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": -2.19, "z": 3.0, "size": 0.2481, "title": "The sum of the reciprocals of the primes diverges", "summary": "We show that the sum of `1/p`, where `p` runs through the prime numbers, diverges. We follow the elementary proof by Erdős that is reproduced in \"Proofs from THE BOOK\". There are two versions of the main result: `not_summable_one_div_on_primes`, which expresses the sum as a sub-sum of the harmonic series, and `Nat.Primes.not_summable_one_div`, which writes it as a sum over `Nat.Primes`. We also show that the sum of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/SumPrimeReciprocals.html"}, {"id": "Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -19.443, "z": -40.108, "size": 0.2537, "title": "The von Mangoldt Function", "summary": "In this file we define the von Mangoldt function: the function on natural numbers that returns `log p` if the input can be expressed as `p^k` for a prime `p`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ArithmeticFunction/VonMangoldt.html"}, {"id": "Mathlib.Algebra.DirectSum.Module", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 4, "macro_tier_score": 0.2913, "macro_tier_override": null, "x": 21.44, "z": -49.406, "size": 0.538, "title": "Direct sum of modules", "summary": "The first part of the file provides constructors for direct sums of modules. It provides a construction of the direct sum using the universal property and proves its uniqueness (`DirectSum.toModule.unique`). The second part of the file covers the special case of direct sums of submodules of a fixed module `M`. There is a canonical linear map from this direct sum to `M` (`DirectSum.coeLinearMap`), and the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/DirectSum/Module.html"}, {"id": "Mathlib.Algebra.Order.Ring.Basic", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.4112, "macro_tier_override": null, "x": 15.945, "z": 12.771, "size": 0.3357, "title": "Basic lemmas about ordered rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Basic.html"}, {"id": "Mathlib.Algebra.Order.Ring.Unbundled.Rat", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 3, "macro_tier_score": 0.233, "macro_tier_override": null, "x": -4.155, "z": -23.783, "size": 0.4068, "title": "The rational numbers possess a linear order", "summary": "This file constructs the order on `ℚ` and proves various facts relating the order to ring structure on `ℚ`. This only uses unbundled type classes, e.g. `CovariantClass`, relating the order structure and algebra structure on `ℚ`. For the bundled `LinearOrderedCommRing` instance on `ℚ`, see `Algebra.Order.Ring.Rat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Unbundled/Rat.html"}, {"id": "Mathlib.Algebra.Ring.Rat", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 2, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -2.081, "z": -14.711, "size": 0.3846, "title": "The rational numbers are a commutative ring", "summary": "This file contains the commutative ring instance on the rational numbers. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Rat.html"}, {"id": "Mathlib.Algebra.FreeAlgebra", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.2565, "macro_tier_override": null, "x": 39.374, "z": -42.003, "size": 0.4158, "title": "Free Algebras", "summary": "Given a commutative semiring `R`, and a type `X`, we construct the free unital, associative `R`-algebra on `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/FreeAlgebra.html"}, {"id": "Mathlib.Algebra.Homology.Homotopy", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": -2.875, "z": -14.577, "size": 0.4067, "title": "Chain homotopies", "summary": "We define chain homotopies, and prove that homotopic chain maps induce the same map on homology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Homotopy.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.Torsion.Basic", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 62.792, "z": -36.1, "size": 0.2582, "title": "Rank and torsion", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/Torsion/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.Constructions", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.2487, "macro_tier_override": null, "x": 35.182, "z": 61.177, "size": 0.5557, "title": "Rank of various constructions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/Constructions.html"}, {"id": "Mathlib.Algebra.Module.Submodule.Ker", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 4, "macro_tier_score": 0.3963, "macro_tier_override": null, "x": -8.801, "z": 36.085, "size": 0.586, "title": "Kernel of a linear map", "summary": "This file defines the kernel of a linear map.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/Ker.html"}, {"id": "Mathlib.Algebra.Homology.Square", "region_id": "algebra", "micro_elevation": 0.0263, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -1.967, "z": -3.151, "size": 0.2333, "title": "Relation between pullback/pushout squares and kernel/cokernel sequences", "summary": "This file is the bundled counterpart of `Mathlib/Algebra/Homology/CommSq.lean`. The same results are obtained here for squares `sq : Square C` where `C` is an additive category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Square.html"}, {"id": "Mathlib.Algebra.Star.Rat", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.281, "macro_tier_override": null, "x": 21.956, "z": -3.818, "size": 0.2989, "title": "\\*-ring structure on `ℚ` and `ℚ≥0`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/Rat.html"}, {"id": "Mathlib.RingTheory.Regular.Flat", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.814, "z": 56.798, "size": 0.2, "title": "`RingTheory.Sequence.IsWeaklyRegular` is stable under flat base change", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Regular/Flat.html"}, {"id": "Mathlib.RingTheory.Flat.FaithfullyFlat.Basic", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 3, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 83.167, "z": 8.225, "size": 0.3704, "title": "Faithfully flat modules", "summary": "A module `M` over a commutative ring `R` is *faithfully flat* if it is flat and `IM ≠ M` whenever `I` is a maximal ideal of `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/FaithfullyFlat/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Quotient.Pi", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 2, "macro_tier_score": 0.0124, "macro_tier_override": null, "x": -16.315, "z": -49.375, "size": 0.2547, "title": "Submodule quotients and direct sums", "summary": "This file contains some results on the quotient of a module by a direct sum of submodules, and the direct sum of quotients of modules by submodules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Quotient/Pi.html"}, {"id": "Mathlib.Algebra.Module.Injective", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 2, "macro_tier_score": 0.0325, "macro_tier_override": null, "x": 26.587, "z": -53.151, "size": 0.2757, "title": "Injective modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Injective.html"}, {"id": "Mathlib.Algebra.Order.ToIntervalMod", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 4.392, "z": -27.509, "size": 0.2456, "title": "Reducing to an interval modulo its length", "summary": "This file defines operations that reduce a number (in an archimedean linearly ordered abelian group) to a number in a given interval, modulo the length of that interval.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/ToIntervalMod.html"}, {"id": "Mathlib.GroupTheory.Divisible", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 2, "macro_tier_score": 0.0301, "macro_tier_override": null, "x": -28.86, "z": 20.303, "size": 0.2618, "title": "Divisible Group and rootable group", "summary": "In this file, we define a divisible additive monoid and a rootable monoid with some basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Divisible.html"}, {"id": "Mathlib.RingTheory.HahnSeries.Valuation", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -22.462, "z": -57.022, "size": 0.2, "title": "Valuations on Hahn Series rings", "summary": "If `Γ` is a linearly ordered cancellative monoid and `R` is a domain, then the domain `R⟦Γ⟧` admits an additive valuation given by `orderTop`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HahnSeries/Valuation.html"}, {"id": "Mathlib.RingTheory.HahnSeries.Multiplication", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -47.749, "z": -32.164, "size": 0.3197, "title": "Multiplicative properties of Hahn series", "summary": "If `Γ` is ordered and `R` has zero, then `R⟦Γ⟧` consists of formal series over `Γ` with coefficients in `R`, whose supports are partially well-ordered. This module introduces multiplication and scalar multiplication on Hahn series. If `Γ` is an ordered cancellative commutative additive monoid and `R` is a semiring, then we get a semiring structure on `R⟦Γ⟧`. If `Γ` has an ordered vector-addition on `Γ'` and `R` has…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HahnSeries/Multiplication.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.Base", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 1, "macro_tier_score": 0.0029, "macro_tier_override": null, "x": -122.5, "z": 4.218, "size": 0.3097, "title": "Bases for root pairings / systems", "summary": "This file contains a theory of bases for root pairings / systems.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/Base.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.Chain", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 1, "macro_tier_score": 0.003, "macro_tier_override": null, "x": 103.255, "z": -62.534, "size": 0.3213, "title": "Chains of roots", "summary": "Given roots `α` and `β`, the `α`-chain through `β` is the set of roots of the form `α + z • β` for an integer `z`. This is known as a \"root chain\" and also a \"root string\". For linearly independent roots in finite crystallographic root pairings, these chains are always unbroken, i.e., of the form: `β - q • α, ..., β - α, β, β + α, ..., β + p • α` for natural numbers `p`, `q`, and the length, `p + q` is at most 3.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/Chain.html"}, {"id": "Mathlib.RepresentationTheory.Homological.ContCohomology.LowDegree", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 88.877, "z": -32.74, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/ContCohomology/LowDegree.html"}, {"id": "Mathlib.Algebra.NoZeroSMulDivisors.Defs", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.3683, "macro_tier_override": null, "x": -13.625, "z": -15.221, "size": 0.4088, "title": "`NoZeroSMulDivisors`", "summary": "This file defines the `NoZeroSMulDivisors` class, and includes some tests for the vanishing of elements (especially in modules over division rings).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/NoZeroSMulDivisors/Defs.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.EpiMono", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 3, "macro_tier_score": 0.0537, "macro_tier_override": null, "x": 44.05, "z": 27.635, "size": 0.3675, "title": "Monomorphisms in `Module R`", "summary": "This file shows that an `R`-linear map is a monomorphism in the category of `R`-modules if and only if it is injective, and similarly an epimorphism if and only if it is surjective.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/EpiMono.html"}, {"id": "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 2, "macro_tier_score": 0.0127, "macro_tier_override": null, "x": -40.858, "z": 101.67, "size": 0.2934, "title": "Evaluating cyclotomic polynomials", "summary": "This file states some results about evaluating cyclotomic polynomials in various different ways.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.html"}, {"id": "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 2, "macro_tier_score": 0.0163, "macro_tier_override": null, "x": 18.153, "z": -106.175, "size": 0.3188, "title": "Roots of cyclotomic polynomials.", "summary": "We gather results about roots of cyclotomic polynomials. In particular we show in `Polynomial.cyclotomic_eq_minpoly` that `cyclotomic n R` is the minimal polynomial of a primitive root of unity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.html"}, {"id": "Mathlib.NumberTheory.Padics.PadicVal.Basic", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 2, "macro_tier_score": 0.0271, "macro_tier_override": null, "x": 26.727, "z": -7.857, "size": 0.2911, "title": "`p`-adic Valuation", "summary": "This file defines the `p`-adic valuation on `ℕ`, `ℤ`, and `ℚ`. The `p`-adic valuation on `ℚ` is the difference of the multiplicities of `p` in the numerator and denominator of `q`. This function obeys the standard properties of a valuation, with the appropriate assumptions on `p`. The `p`-adic valuations on `ℕ` and `ℤ` agree with that on `ℚ`. The valuation induces a norm on `ℚ`. This norm is defined in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/PadicVal/Basic.html"}, {"id": "Mathlib.GroupTheory.Coxeter.Basic", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 39.348, "z": 42.027, "size": 0.2442, "title": "Coxeter groups and Coxeter systems", "summary": "This file defines Coxeter groups and Coxeter systems. Let `B` be a (possibly infinite) type, and let $M = (M_{i,i'})_{i, i' \\in B}$ be a matrix of natural numbers. Further assume that $M$ is a *Coxeter matrix* (`CoxeterMatrix`); that is, $M$ is symmetric and $M_{i,i'} = 1$ if and only if $i = i'$. The *Coxeter group* associated to $M$ (`CoxeterMatrix.Group`) has the presentation $$\\langle \\{s_i\\}_{i \\in B} \\vert…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Coxeter/Basic.html"}, {"id": "Mathlib.GroupTheory.Coxeter.Matrix", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -37.464, "z": 41.238, "size": 0.2455, "title": "Coxeter matrices", "summary": "Let us say that a matrix (possibly an infinite matrix) is a *Coxeter matrix* (`CoxeterMatrix`) if its entries are natural numbers, it is symmetric, its diagonal entries are equal to 1, and its off-diagonal entries are not equal to 1. In this file, we define Coxeter matrices and provide some ways of constructing them. We also define the Coxeter matrices `CoxeterMatrix.Aₙ` (`n : ℕ`), `CoxeterMatrix.Bₙ` (`n : ℕ`),…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Coxeter/Matrix.html"}, {"id": "Mathlib.GroupTheory.PresentedGroup", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -25.445, "z": -5.347, "size": 0.2567, "title": "Defining a group given by generators and relations", "summary": "Given a subset `rels` of relations of the free group on a type `α`, this file constructs the group given by generators `x : α` and relations `r ∈ rels`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/PresentedGroup.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 2, "macro_tier_score": 0.0125, "macro_tier_override": null, "x": -62.828, "z": -6.305, "size": 0.278, "title": "ZeroDivisors in a MvPowerSeries ring", "summary": "- `mem_nonZeroDivisors_of_constantCoeff` proves that a multivariate power series whose constant coefficient is not a zero divisor is itself not a zero divisor - `MvPowerSeries.order_mul` : multiplicativity of `MvPowerSeries.order` if the semiring `R` has no zero divisors", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/NoZeroDivisors.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.LexOrder", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 2, "macro_tier_score": 0.0125, "macro_tier_override": null, "x": 27.131, "z": -54.954, "size": 0.2696, "title": "LexOrder of multivariate power series", "summary": "Given an ordering of `σ` such that `WellFoundedGT σ`, the lexicographic order on `σ →₀ ℕ` is a well ordering, which can be used to define a natural valuation `lexOrder` on the ring `MvPowerSeries σ R`: the smallest exponent in the support.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/LexOrder.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.Order", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 2, "macro_tier_score": 0.0184, "macro_tier_override": null, "x": 24.485, "z": 56.183, "size": 0.3104, "title": "Order of multivariate power series", "summary": "We work with `MvPowerSeries σ R`, for `Semiring R`, and `w : σ → ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/Order.html"}, {"id": "Mathlib.RingTheory.Ideal.UFD", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -102.109, "z": 2.675, "size": 0.2, "title": "UFD criteria via height `1` prime ideals and localization", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/UFD.html"}, {"id": "Mathlib.RingTheory.Localization.Away.Lemmas", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": 40.891, "z": 70.748, "size": 0.2875, "title": "More lemmas on localization away", "summary": "This file contains lemmas on localization away from an element requiring more imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Away/Lemmas.html"}, {"id": "Mathlib.RingTheory.Binomial", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -59.058, "z": 27.153, "size": 0.2662, "title": "Binomial rings", "summary": "In this file we introduce the binomial property as a mixin, and define the `multichoose` and `choose` functions generalizing binomial coefficients. According to our main reference [elliott2006binomial] (which lists many equivalent conditions), a binomial ring is a torsion-free commutative ring `R` such that for any `x ∈ R` and any `k ∈ ℕ`, the product `x(x-1)⋯(x-k+1)` is divisible by `k!`. The torsion-free condition…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Binomial.html"}, {"id": "Mathlib.Algebra.Lie.Weights.Chain", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 43.643, "z": -106.552, "size": 0.2365, "title": "Chains of roots and weights", "summary": "Given roots `α` and `β` of a Lie algebra, together with elements `x` in the `α`-root space and `y` in the `β`-root space, it follows from the Leibniz identity that `⁅x, y⁆` is either zero or belongs to the `α + β`-root space. Iterating this operation leads to the study of families of roots of the form `k • α + β`. Such a family is known as the `α`-chain through `β` (or sometimes, the `α`-string through `β`) and the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Weights/Chain.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.Substitution", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.0116, "macro_tier_override": null, "x": -18.782, "z": 68.027, "size": 0.2987, "title": "Substitutions in multivariate power series", "summary": "Here we define the substitution of power series into other power series. We follow [Bourbaki, Algebra II, chap. 4, §4, n° 3][bourbaki1981] who present substitution of power series as an application of evaluation. For an `R`-algebra `S`, `f : MvPowerSeries σ R` and `a : σ → MvPowerSeries τ S`, `MvPowerSeries.subst a f` is the substitution of `X s` by `a s` in `f`. It is only well defined under one of the two…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/Substitution.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.LinearTopology", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 2, "macro_tier_score": 0.0113, "macro_tier_override": null, "x": 52.16, "z": -44.734, "size": 0.2565, "title": "Linear topology on the ring of multivariate power series", "summary": "- `MvPowerSeries.LinearTopology.basis`: the ideals of the ring of multivariate power series all coefficients the exponent of which is smaller than some bound vanish. - `MvPowerSeries.LinearTopology.hasBasis_nhds_zero` : the two-sided ideals from `MvPowerSeries.LinearTopology.basis` form a basis of neighborhoods of `0` if the topology of `R` is (left and right) linear.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/LinearTopology.html"}, {"id": "Mathlib.GroupTheory.Perm.Cycle.Concrete", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 46.234, "z": 4.251, "size": 0.2949, "title": "Properties of cyclic permutations constructed from lists/cycles", "summary": "In the following, `{α : Type*} [Fintype α] [DecidableEq α]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Cycle/Concrete.html"}, {"id": "Mathlib.RingTheory.Etale.Descent", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 70.813, "z": 104.566, "size": 0.2442, "title": "Etale descends along faithfully flat ring maps", "summary": "In this file we show that smooth, unramified and étale algebras descend along faithfully flat base change.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Etale/Descent.html"}, {"id": "Mathlib.Algebra.Group.Action.Option", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -11.974, "z": 5.062, "size": 0.2, "title": "Option instances for additive and multiplicative actions", "summary": "This file defines instances for additive and multiplicative actions on `Option` type. Scalar multiplication is defined by `a • some b = some (a • b)` and `a • none = none`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Option.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 3, "macro_tier_score": 0.1389, "macro_tier_override": null, "x": 33.301, "z": -74.622, "size": 0.3591, "title": "Matrices and linear equivalences", "summary": "This file gives the map `Matrix.toLinearEquiv` from matrices with invertible determinant, to linear equivs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 3, "macro_tier_score": 0.1386, "macro_tier_override": null, "x": -58.612, "z": 54.24, "size": 0.3408, "title": "The General Linear group $GL(n, R)$", "summary": "This file defines the elements of the General Linear group `Matrix.GeneralLinearGroup n R`, consisting of all invertible `n` by `n` `R`-matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Defs.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 3, "macro_tier_score": 0.1403, "macro_tier_override": null, "x": 72.636, "z": 15.571, "size": 0.302, "title": "Matrices associated with non-degenerate bilinear forms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Nondegenerate.html"}, {"id": "Mathlib.RingTheory.Localization.Integer", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 3, "macro_tier_score": 0.2043, "macro_tier_override": null, "x": 13.464, "z": -28.557, "size": 0.36, "title": "Integer elements of a localization", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Integer.html"}, {"id": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.IsBoundedAtImInfty", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -60.258, "z": 18.871, "size": 0.2477, "title": "Boundedness of Eisenstein series", "summary": "We show that Eisenstein series of weight `k` and level `Γ(N)` with congruence condition `a : Fin 2 → ZMod N` are bounded at infinity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/EisensteinSeries/IsBoundedAtImInfty.html"}, {"id": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 12.542, "z": -30.987, "size": 0.2824, "title": "Eisenstein Series", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/EisensteinSeries/Defs.html"}, {"id": "Mathlib.NumberTheory.ModularForms.Identities", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 5.407, "z": 61.047, "size": 0.2651, "title": "Identities of ModularForms and SlashInvariantForms", "summary": "Collection of useful identities of modular forms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/Identities.html"}, {"id": "Mathlib.Algebra.Group.Action.Sigma", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -4.044, "z": 12.355, "size": 0.2, "title": "Sigma instances for additive and multiplicative actions", "summary": "This file defines instances for arbitrary sum of additive and multiplicative actions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Sigma.html"}, {"id": "Mathlib.Algebra.AffineMonoid.Irreducible", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -31.589, "z": -15.724, "size": 0.2, "title": "An affine monoid with no non-trivial unit is generated by its irreducible elements", "summary": "This file proves that an additive cancellative monoid with no non-trivial unit unit is generated by its irreducible elements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/AffineMonoid/Irreducible.html"}, {"id": "Mathlib.Algebra.Group.Irreducible.Defs", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.3944, "macro_tier_override": null, "x": 9.41, "z": -5.968, "size": 0.3253, "title": "Irreducible elements in a monoid", "summary": "This file defines irreducible elements of a monoid (`Irreducible`), as non-units that can't be written as the product of two non-units. This generalises irreducible elements of a ring. We also define the additive variant (`AddIrreducible`). In decomposition monoids (e.g., `ℕ`, `ℤ`), this predicate is equivalent to `Prime` (see `irreducible_iff_prime`), however this is not true in general.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Irreducible/Defs.html"}, {"id": "Mathlib.RingTheory.Complex", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -75.854, "z": -46.826, "size": 0.2347, "title": "Lemmas about `Algebra.trace` and `Algebra.norm` on `ℂ`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Complex.html"}, {"id": "Mathlib.RingTheory.Trace.Defs", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 3, "macro_tier_score": 0.0371, "macro_tier_override": null, "x": 27.825, "z": 82.733, "size": 0.2981, "title": "Trace for (finite) ring extensions.", "summary": "Suppose we have an `R`-algebra `S` with a finite basis. For each `s : S`, the trace of the linear map given by multiplying by `s` gives information about the roots of the minimal polynomial of `s` over `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Trace/Defs.html"}, {"id": "Mathlib.Algebra.AffineMonoid.Basic", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -34.597, "z": -6.939, "size": 0.2, "title": "Affine monoids", "summary": "This file defines affine monoids as finitely generated cancellative torsion-free commutative monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/AffineMonoid/Basic.html"}, {"id": "Mathlib.Algebra.Group.Equiv.Opposite", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.471, "macro_tier_override": null, "x": 0.142, "z": 11.142, "size": 0.5408, "title": "Group isomorphism between a group and its opposite", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Equiv/Opposite.html"}, {"id": "Mathlib.RingTheory.Ideal.GoingUp", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 3, "macro_tier_score": 0.0838, "macro_tier_override": null, "x": 78.457, "z": 49.67, "size": 0.3794, "title": "Ideals over/under ideals in integral extensions", "summary": "This file proves some going-up results for integral algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/GoingUp.html"}, {"id": "Mathlib.NumberTheory.SelbergSieve", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 41.65, "z": 15.873, "size": 0.2, "title": "The Selberg Sieve", "summary": "We set up the working assumptions of the Selberg sieve, define the notion of an upper bound sieve and show that every upper bound sieve yields an upper bound on the size of the sifted set. We also define the Λ² sieve and prove that Λ² sieves are upper bound sieves. We then diagonalise the main term of the Λ² sieve. We mostly follow the treatment outlined by Heath-Brown in the notes to an old graduate course. One…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/SelbergSieve.html"}, {"id": "Mathlib.NumberTheory.ArithmeticFunction.Moebius", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 2, "macro_tier_score": 0.0251, "macro_tier_override": null, "x": -15.251, "z": 39.899, "size": 0.3107, "title": "The Möbius function and Möbius inversion", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ArithmeticFunction/Moebius.html"}, {"id": "Mathlib.Algebra.Module.DedekindDomain", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": 33.187, "z": -96.602, "size": 0.2545, "title": "Modules over a Dedekind domain", "summary": "Over a Dedekind domain, an `I`-torsion module is the internal direct sum of its `p i ^ e i`-torsion submodules, where `I = ∏ i, p i ^ e i` is its unique decomposition in prime ideals. Therefore, as any finitely generated torsion module is `I`-torsion for some `I`, it is an internal direct sum of its `p i ^ e i`-torsion submodules for some prime ideals `p i` and numbers `e i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/DedekindDomain.html"}, {"id": "Mathlib.Algebra.Order.Hom.Submonoid", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -5.183, "z": 17.834, "size": 0.2, "title": "Isomorphism of submonoids of ordered monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Hom/Submonoid.html"}, {"id": "Mathlib.RepresentationTheory.Homological.TateCohomology.Basic", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 105.158, "z": 12.155, "size": 0.2, "title": "Tate Cohomology", "summary": "This file defines the Tate cohomology of finite groups by taking homology of the Tate complex. We define the Tate complex by connecting the inhomogeneous chain complex with the inhomogeneous cochain complex using the norm map.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/TateCohomology/Basic.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.Connect", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -9.907, "z": -13.462, "size": 0.239, "title": "Connecting a chain complex and a cochain complex", "summary": "Given a chain complex `K`: `... ⟶ K.X 2 ⟶ K.X 1 ⟶ K.X 0`, a cochain complex `L`: `L.X 0 ⟶ L.X 1 ⟶ L.X 2 ⟶ ...`, a morphism `d₀ : K.X 0 ⟶ L.X 0` satisfying the identifies `K.d 1 0 ≫ d₀ = 0` and `d₀ ≫ L.d 0 1 = 0`, we construct a cochain complex indexed by `ℤ` of the form `... ⟶ K.X 2 ⟶ K.X 1 ⟶ K.X 0 ⟶ L.X 0 ⟶ L.X 1 ⟶ L.X 2 ⟶ ...`, where `K.X 0` lies in degree `-1` and `L.X 0` in degree `0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/Connect.html"}, {"id": "Mathlib.RepresentationTheory.Homological.GroupCohomology.LongExactSequence", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -93.494, "z": 41.137, "size": 0.239, "title": "Long exact sequence in group cohomology", "summary": "Given a commutative ring `k` and a group `G`, this file shows that a short exact sequence of `k`-linear `G`-representations `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0` induces a short exact sequence of complexes `0 ⟶ inhomogeneousCochains X₁ ⟶ inhomogeneousCochains X₂ ⟶ inhomogeneousCochains X₃ ⟶ 0`. Since the cohomology of `inhomogeneousCochains Xᵢ` is the group cohomology of `Xᵢ`, this allows us to specialize API about long exact…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/GroupCohomology/LongExactSequence.html"}, {"id": "Mathlib.RepresentationTheory.Homological.GroupHomology.LongExactSequence", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 63.971, "z": 82.0, "size": 0.239, "title": "Long exact sequence in group homology", "summary": "Given a commutative ring `k` and a group `G`, this file shows that a short exact sequence of `k`-linear `G`-representations `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0` induces a short exact sequence of complexes `0 ⟶ inhomogeneousChains X₁ ⟶ inhomogeneousChains X₂ ⟶ inhomogeneousChains X₃ ⟶ 0`. Since the homology of `inhomogeneousChains Xᵢ` is the group homology of `Xᵢ`, this allows us to specialize API about long exact sequences to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/GroupHomology/LongExactSequence.html"}, {"id": "Mathlib.NumberTheory.LSeries.AbstractFuncEq", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2637, "title": "Abstract functional equations for Mellin transforms", "summary": "This file formalises a general version of an argument used to prove functional equations for zeta and L-functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/AbstractFuncEq.html"}, {"id": "Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": 65.379, "z": -39.031, "size": 0.2637, "title": "Asymptotic bounds for Jacobi theta functions", "summary": "The goal of this file is to establish some technical lemmas about the asymptotics of the sums `F_nat k a t = ∑' (n : ℕ), (n + a) ^ k * exp (-π * (n + a) ^ 2 * t)` and `F_int k a t = ∑' (n : ℤ), |n + a| ^ k * exp (-π * (n + a) ^ 2 * t).` Here `k : ℕ` and `a : ℝ` (resp `a : UnitAddCircle`) are fixed, and we are interested in asymptotics as `t → ∞`. These results are needed for the theory of Hurwitz zeta functions (and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.html"}, {"id": "Mathlib.NumberTheory.LSeries.MellinEqDirichlet", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2637, "title": "Dirichlet series as Mellin transforms", "summary": "Here we prove general results of the form \"the Mellin transform of a power series in exp (-t) is a Dirichlet series\".", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/MellinEqDirichlet.html"}, {"id": "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 4, "macro_tier_score": 0.4036, "macro_tier_override": null, "x": -32.581, "z": -7.481, "size": 0.5462, "title": "Lemmas about group actions on big operators", "summary": "This file contains results about two kinds of actions: * sums over `DistribSMul`: `r • ∑ x ∈ s, f x = ∑ x ∈ s, r • f x` * products over `MulDistribMulAction` (with primed name): `r • ∏ x ∈ s, f x = ∏ x ∈ s, r • f x` * products over `SMulCommClass` (with unprimed name): `b ^ s.card • ∏ x ∈ s, f x = ∏ x ∈ s, b • f x` Note that analogous lemmas for `Module`s like `Finset.sum_smul` appear in other files.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/GroupWithZero/Action.html"}, {"id": "Mathlib.Algebra.Field.MinimalAxioms", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -11.113, "z": -0.82, "size": 0.2456, "title": "Minimal Axioms for a Field", "summary": "This file defines constructors to define a `Field` structure on a Type, while proving a minimum number of equalities.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/MinimalAxioms.html"}, {"id": "Mathlib.RingTheory.Ideal.Quotient.Over", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -63.783, "z": -12.524, "size": 0.2428, "title": "Lemmas about `primesOver` in quotient rings.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Quotient/Over.html"}, {"id": "Mathlib.RingTheory.Polynomial.IrreducibleRing", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 58.865, "z": 42.201, "size": 0.2, "title": "Polynomials over an irreducible ring", "summary": "This file contains results about the polynomials over an irreducible ring (i.e. a ring with only one minimal prime ideal, equivalently, whose spectrum is an irreducible topological space).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/IrreducibleRing.html"}, {"id": "Mathlib.NumberTheory.ModularForms.DedekindEta", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 1, "macro_tier_score": 0.0017, "macro_tier_override": null, "x": -95.473, "z": -64.367, "size": 0.3001, "title": "Dedekind eta function", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/DedekindEta.html"}, {"id": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 1, "macro_tier_score": 0.002, "macro_tier_override": null, "x": -101.193, "z": -50.93, "size": 0.3264, "title": "Summability of E2", "summary": "We collect here lemmas about the summability of the Eisenstein series `E2` that will be used to prove how it transforms under the slash action.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/EisensteinSeries/E2/Summable.html"}, {"id": "Mathlib.RingTheory.MatrixAlgebra", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.1842, "macro_tier_override": null, "x": -6.886, "z": 64.635, "size": 0.2573, "title": "Algebra isomorphisms between tensor products and matrices", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MatrixAlgebra.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Kronecker", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.2042, "macro_tier_override": null, "x": -62.357, "z": 9.937, "size": 0.3553, "title": "Kronecker product of matrices", "summary": "This defines the [Kronecker product](https://en.wikipedia.org/wiki/Kronecker_product).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Kronecker.html"}, {"id": "Mathlib.Algebra.Category.MonCat.FilteredColimits", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 9.865, "z": 11.109, "size": 0.2845, "title": "The forgetful functor from (commutative) (additive) monoids preserves filtered colimits.", "summary": "Forgetful functors from algebraic categories usually don't preserve colimits. However, they tend to preserve _filtered_ colimits. In this file, we start with a small filtered category `J` and a functor `F : J ⥤ MonCat`. We then construct a monoid structure on the colimit of `F ⋙ forget MonCat` (in `Type`), thereby showing that the forgetful functor `forget MonCat` preserves filtered colimits. Similarly for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/MonCat/FilteredColimits.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.Finset.Sigma", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4213, "macro_tier_override": null, "x": 6.431, "z": -15.428, "size": 0.3997, "title": "Product and sums indexed by finite sets in sigma types.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Finset/Sigma.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.FilteredColimits", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 25.065, "z": -49.759, "size": 0.2812, "title": "The forgetful functor from `R`-modules preserves filtered colimits.", "summary": "Forgetful functors from algebraic categories usually don't preserve colimits. However, they tend to preserve _filtered_ colimits. In this file, we start with a ring `R`, a small filtered category `J` and a functor `F : J ⥤ ModuleCat R`. We show that the colimit of `F ⋙ forget₂ (ModuleCat R) AddCommGrpCat` (in `AddCommGrpCat`) carries the structure of an `R`-module, thereby showing that the forgetful functor `forget₂…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/FilteredColimits.html"}, {"id": "Mathlib.Algebra.Category.Grp.FilteredColimits", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 1, "macro_tier_score": 0.0017, "macro_tier_override": null, "x": 19.933, "z": 4.476, "size": 0.3074, "title": "The forgetful functor from (commutative) (additive) groups preserves filtered colimits.", "summary": "Forgetful functors from algebraic categories usually don't preserve colimits. However, they tend to preserve _filtered_ colimits. In this file, we start with a small filtered category `J` and a functor `F : J ⥤ GrpCat`. We show that the colimit of `F ⋙ forget₂ GrpCat MonCat` (in `MonCat`) carries the structure of a group, thereby showing that the forgetful functor `forget₂ GrpCat MonCat` preserves filtered colimits.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/FilteredColimits.html"}, {"id": "Mathlib.RingTheory.Localization.AtPrime.Basic", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.105, "macro_tier_override": null, "x": 29.157, "z": -62.222, "size": 0.4728, "title": "Localizations of commutative rings at the complement of a prime ideal", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/AtPrime/Basic.html"}, {"id": "Mathlib.RingTheory.Localization.Submodule", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 3, "macro_tier_score": 0.0785, "macro_tier_override": null, "x": -27.77, "z": 74.874, "size": 0.388, "title": "Submodules in localizations of commutative rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Submodule.html"}, {"id": "Mathlib.RingTheory.Ideal.Finsupp", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 48.045, "z": -31.722, "size": 0.2302, "title": "Lemmas for action of ideals on submodules of `Finsupp`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Finsupp.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.IsOpenComapC", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 19.641, "z": 94.554, "size": 0.2, "title": null, "summary": "The morphism `Spec R[x] --> Spec R` induced by the natural inclusion `R --> R[x]` is an open map. The main result is the first part of the statement of Lemma 00FB in the Stacks Project. https://stacks.math.columbia.edu/tag/00FB", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/IsOpenComapC.html"}, {"id": "Mathlib.RingTheory.SimpleModule.Isotypic", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 63.076, "z": 2.92, "size": 0.2403, "title": "Isotypic modules and isotypic components", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SimpleModule/Isotypic.html"}, {"id": "Mathlib.RingTheory.Perfection", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 84.032, "z": -54.737, "size": 0.2575, "title": "Ring Perfection and Tilt", "summary": "In this file we define the perfection of a ring of characteristic p, and the tilt of a field given a valuation to `ℝ≥0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Perfection.html"}, {"id": "Mathlib.Algebra.CharP.Pi", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": -16.157, "z": 12.501, "size": 0.235, "title": "Characteristic of semirings of functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/Pi.html"}, {"id": "Mathlib.Algebra.CharP.Quotient", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 2, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": -15.094, "z": 57.48, "size": 0.261, "title": "Characteristic of quotient rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/Quotient.html"}, {"id": "Mathlib.Algebra.CharP.Subring", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 1, "macro_tier_score": 0.0045, "macro_tier_override": null, "x": 24.868, "z": 56.014, "size": 0.2446, "title": "Characteristic of subrings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/Subring.html"}, {"id": "Mathlib.FieldTheory.Perfect", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 3, "macro_tier_score": 0.1152, "macro_tier_override": null, "x": 86.197, "z": -47.523, "size": 0.3945, "title": "Perfect fields and rings", "summary": "In this file we define perfect fields, together with a generalisation to (commutative) rings in prime characteristic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Perfect.html"}, {"id": "Mathlib.RingTheory.Valuation.Integers", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 3, "macro_tier_score": 0.0634, "macro_tier_override": null, "x": 34.701, "z": 50.516, "size": 0.2611, "title": "Ring of integers under a given valuation", "summary": "The elements with valuation less than or equal to 1. TODO: Define characteristic predicate.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/Integers.html"}, {"id": "Mathlib.RingTheory.Derivation.Basic", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.0438, "macro_tier_override": null, "x": 38.555, "z": 50.007, "size": 0.3757, "title": "Derivations", "summary": "This file defines derivation. A derivation `D` from the `R`-algebra `A` to the `A`-module `M` is an `R`-linear map that satisfy the Leibniz rule `D (a * b) = a * D b + D a * b`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Derivation/Basic.html"}, {"id": "Mathlib.RingTheory.RingHom.Bijective", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 10.63, "z": 82.893, "size": 0.2, "title": "Meta properties of bijective ring homomorphisms", "summary": "We show some meta properties of bijective ring homomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/Bijective.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4674, "macro_tier_override": null, "x": 3.875, "z": 8.439, "size": 0.4289, "title": "Unbundled ordered monoid structures on the order dual.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Unbundled/OrderDual.html"}, {"id": "Mathlib.Algebra.Order.Group.Basic", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.3379, "macro_tier_override": null, "x": -3.313, "z": 14.483, "size": 0.391, "title": "Lemmas about the interaction of power operations with order", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Basic.html"}, {"id": "Mathlib.NumberTheory.ClassNumber.AdmissibleAbs", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -38.004, "z": -15.0, "size": 0.2484, "title": "Admissible absolute value on the integers", "summary": "This file defines an admissible absolute value `AbsoluteValue.absIsAdmissible` which we use to show the class number of the ring of integers of a number field is finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.html"}, {"id": "Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": 28.546, "z": 13.487, "size": 0.2703, "title": "Admissible absolute values", "summary": "This file defines a structure `AbsoluteValue.IsAdmissible` which we use to show the class number of the ring of integers of a global field is finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.html"}, {"id": "Mathlib.LinearAlgebra.Multilinear.DirectSum", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 35.278, "z": -43.123, "size": 0.2, "title": "Multilinear maps from direct sums", "summary": "This file describes multilinear maps on direct sums.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Multilinear/DirectSum.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Abelian", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 2, "macro_tier_score": 0.0283, "macro_tier_override": null, "x": 58.232, "z": 11.868, "size": 0.3743, "title": "The category of left R-modules is abelian.", "summary": "Additionally, two linear maps are exact in the categorical sense iff `range f = ker g`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Abelian.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Topology.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": 57.723, "z": 14.139, "size": 0.2654, "title": "The category `TopModuleCat R` of topological modules", "summary": "We define `TopModuleCat R`, the category of topological modules, and show that it has all limits and colimits. We also provide various adjunctions: - `TopModuleCat.withModuleTopologyAdj`: equipping the module topology is left adjoint to the forgetful functor into `ModuleCat R`. - `TopModuleCat.indiscreteAdj`: equipping the indiscrete topology is right adjoint to the forgetful functor into `ModuleCat R`. -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Topology/Basic.html"}, {"id": "Mathlib.Algebra.Regular.Pow", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 3, "macro_tier_score": 0.2733, "macro_tier_override": null, "x": 4.981, "z": 13.997, "size": 0.3056, "title": "Product of regular elements", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Regular/Pow.html"}, {"id": "Mathlib.Algebra.BigOperators.WithTop", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 20.205, "z": -13.216, "size": 0.2687, "title": "Sums in `WithTop`", "summary": "This file proves results about finite sums over monoids extended by a bottom or top element.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/WithTop.html"}, {"id": "Mathlib.Algebra.Group.Semiconj.Units", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4768, "macro_tier_override": null, "x": -12.963, "z": 0.982, "size": 0.4844, "title": "Semiconjugate elements of a semigroup", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Semiconj/Units.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.Synonym", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.3929, "macro_tier_override": null, "x": -6.705, "z": 3.197, "size": 0.4202, "title": "Group with zero structure on the order type synonyms", "summary": "Transfer algebraic instances from `α` to `αᵒᵈ` and `Lex α`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/Synonym.html"}, {"id": "Mathlib.Algebra.Lie.Solvable", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": 62.528, "z": 58.21, "size": 0.3442, "title": "Solvable Lie algebras", "summary": "Like groups, Lie algebras admit a natural concept of solvability. We define this here via the derived series and prove some related results. We also define the radical of a Lie algebra and prove that it is solvable when the Lie algebra is Noetherian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Solvable.html"}, {"id": "Mathlib.Algebra.Lie.Normalizer", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 78.802, "z": -12.942, "size": 0.2675, "title": "The normalizer of Lie submodules and subalgebras.", "summary": "Given a Lie module `M` over a Lie subalgebra `L`, the normalizer of a Lie submodule `N ⊆ M` is the Lie submodule with underlying set `{ m | ∀ (x : L), ⁅x, m⁆ ∈ N }`. The lattice of Lie submodules thus has two natural operations, the normalizer: `N ↦ N.normalizer` and the ideal operation: `N ↦ ⁅⊤, N⁆`; these are adjoint, i.e., they form a Galois connection. This adjointness is the reason that we may define nilpotency…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Normalizer.html"}, {"id": "Mathlib.LinearAlgebra.AnnihilatingPolynomial", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 2, "macro_tier_score": 0.0069, "macro_tier_override": null, "x": -23.256, "z": 89.899, "size": 0.2665, "title": "Annihilating Ideal", "summary": "Given a commutative ring `R` and an `R`-algebra `A`, every element `a : A` defines an ideal `Polynomial.annIdeal a ⊆ R[X]`. Simply put, this is the set of polynomials `p` where the polynomial evaluation `p(a)` is 0.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AnnihilatingPolynomial.html"}, {"id": "Mathlib.Algebra.Order.Archimedean.Real.Hom", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 1, "macro_tier_score": 0.0017, "macro_tier_override": null, "x": -1.132, "z": 27.834, "size": 0.3015, "title": "Uniqueness of ring homomorphisms to the real numbers", "summary": "This file contains results about ring homomorphisms to `ℝ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Archimedean/Real/Hom.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.Grading", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 55.768, "z": -25.415, "size": 0.2, "title": "Internal grading of an `AddMonoidAlgebra`", "summary": "In this file, we show that an `AddMonoidAlgebra` has an internal direct sum structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/Grading.html"}, {"id": "Mathlib.Algebra.Order.Group.Multiset", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.2997, "macro_tier_override": null, "x": -12.77, "z": 2.433, "size": 0.4932, "title": "Multisets form an ordered monoid", "summary": "This file contains the ordered monoid instance on multisets, and lemmas related to it. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Multiset.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.Homogeneous", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 2, "macro_tier_score": 0.0236, "macro_tier_override": null, "x": 1.055, "z": -74.279, "size": 0.3627, "title": "Homogeneous polynomials", "summary": "A multivariate polynomial `φ` is homogeneous of degree `n` if all monomials occurring in `φ` have degree `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/Homogeneous.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.Center", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -26.593, "z": 8.297, "size": 0.239, "title": "The center of a group with zero", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/Center.html"}, {"id": "Mathlib.Algebra.Group.TypeTags.Hom", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4666, "macro_tier_override": null, "x": -0.627, "z": -11.125, "size": 0.4395, "title": "Transport algebra morphisms between additive and multiplicative types.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/TypeTags/Hom.html"}, {"id": "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.3844, "macro_tier_override": null, "x": 11.911, "z": -21.0, "size": 0.317, "title": "Bundled non-unital subsemirings", "summary": "We define the `CompleteLattice` structure, and non-unital subsemiring `map`, `comap` and range (`srange`) of a `NonUnitalRingHom` etc.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/NonUnitalSubsemiring/Basic.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Center", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.3848, "macro_tier_override": null, "x": -8.64, "z": 18.512, "size": 0.2354, "title": "Center of a group with zero", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Center.html"}, {"id": "Mathlib.Algebra.Ring.Center", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.3848, "macro_tier_override": null, "x": 3.752, "z": -20.081, "size": 0.2354, "title": "Centers of rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Center.html"}, {"id": "Mathlib.Algebra.Ring.Centralizer", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.3848, "macro_tier_override": null, "x": 11.875, "z": 16.623, "size": 0.2354, "title": "Centralizers of rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Centralizer.html"}, {"id": "Mathlib.Algebra.Ring.Prod", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.3863, "macro_tier_override": null, "x": -11.206, "z": -14.81, "size": 0.37, "title": "Semiring, ring etc. structures on `R × S`", "summary": "In this file we define two-binop (`Semiring`, `Ring` etc) structures on `R × S`. We also prove trivial `simp` lemmas, and define the following operations on `RingHom`s and similarly for `NonUnitalRingHom`s: * `fst R S : R × S →+* R`, `snd R S : R × S →+* S`: projections `Prod.fst` and `Prod.snd` as `RingHom`s; * `f.prod g : R →+* S × T`: sends `x` to `(f x, g x)`; * `f.prod_map g : R × S → R' × S'`: `Prod.map f g`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Prod.html"}, {"id": "Mathlib.Algebra.Ring.Submonoid.Basic", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.3848, "macro_tier_override": null, "x": 10.291, "z": 7.943, "size": 0.2354, "title": "Lemmas about additive closures of `Subsemigroup`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Submonoid/Basic.html"}, {"id": "Mathlib.GroupTheory.Subsemigroup.Centralizer", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.4406, "macro_tier_override": null, "x": 21.365, "z": 6.339, "size": 0.2942, "title": "Centralizers in semigroups, as subsemigroups.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Subsemigroup/Centralizer.html"}, {"id": "Mathlib.RingTheory.NonUnitalSubsemiring.Defs", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.3877, "macro_tier_override": null, "x": 9.768, "z": -11.195, "size": 0.4328, "title": "Bundled non-unital subsemirings", "summary": "We define bundled non-unital subsemirings and some standard constructions: `subtype` and `inclusion` ring homomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/NonUnitalSubsemiring/Defs.html"}, {"id": "Mathlib.Algebra.Order.Nonneg.Floor", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 26.639, "z": -13.166, "size": 0.2938, "title": "Nonnegative elements are archimedean", "summary": "This file defines instances and prove some properties about the nonnegative elements `{x : α // 0 ≤ x}` of an arbitrary type `α`. This is used to derive algebraic structures on `ℝ≥0` and `ℚ≥0` automatically.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Nonneg/Floor.html"}, {"id": "Mathlib.GroupTheory.FixedPointFree", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 25.604, "z": -38.731, "size": 0.2, "title": "Fixed-point-free automorphisms", "summary": "This file defines fixed-point-free automorphisms and proves some basic properties. An automorphism `φ` of a group `G` is fixed-point-free if `1 : G` is the only fixed point of `φ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FixedPointFree.html"}, {"id": "Mathlib.GroupTheory.Perm.Cycle.Type", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 3, "macro_tier_score": 0.2301, "macro_tier_override": null, "x": 42.137, "z": -14.529, "size": 0.3763, "title": "Cycle Types", "summary": "In this file we define the cycle type of a permutation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Cycle/Type.html"}, {"id": "Mathlib.RingTheory.Bialgebra.MonoidAlgebra", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 26.776, "z": 65.295, "size": 0.2478, "title": "The bialgebra structure on monoid algebras", "summary": "Given a monoid `M`, a commutative semiring `R` and an `R`-bialgebra `A`, this file collects results about the `R`-bialgebra instance on `A[M]` inherited from the corresponding structure on its coefficients, building upon results in `Mathlib/RingTheory/Coalgebra/MonoidAlgebra.lean` about the coalgebra structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Bialgebra/MonoidAlgebra.html"}, {"id": "Mathlib.RingTheory.Bialgebra.Hom", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 1, "macro_tier_score": 0.004, "macro_tier_override": null, "x": -63.971, "z": -19.435, "size": 0.3102, "title": "Homomorphisms of `R`-bialgebras", "summary": "This file defines bundled homomorphisms of `R`-bialgebras. We simply mimic `Mathlib/Algebra/Algebra/Hom.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Bialgebra/Hom.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.MappingCocone", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -5.124, "z": 36.788, "size": 0.2822, "title": "The mapping cocone", "summary": "Given a morphism `φ : K ⟶ L` of cochain complexes, the mapping cone allows to obtain a triangle `K ⟶ L ⟶ mappingCone φ ⟶ ...`. In this file, we define the mapping cocone, which fits in a rotated triangle: `mappingCocone φ ⟶ K ⟶ L ⟶ ...`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/MappingCocone.html"}, {"id": "Mathlib.Algebra.Group.WithOne.Defs", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.433, "macro_tier_override": null, "x": 5.119, "z": -2.2, "size": 0.3024, "title": "Adjoining a zero/one to semigroups and related algebraic structures", "summary": "This file contains different results about adjoining an element to an algebraic structure which then behaves like a zero or a one. An example is adjoining a one to a semigroup to obtain a monoid. That this provides an example of an adjunction is proved in `Mathlib/Algebra/Category/MonCat/Adjunctions.lean`. Another result says that adjoining to a group an element `zero` gives a `GroupWithZero`. For more information…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/WithOne/Defs.html"}, {"id": "Mathlib.NumberTheory.ArithmeticFunction.LFunction", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -73.21, "z": 20.93, "size": 0.2276, "title": "Construction of L-functions", "summary": "This file constructs L-functions as formal Dirichlet series.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ArithmeticFunction/LFunction.html"}, {"id": "Mathlib.NumberTheory.ArithmeticFunction.Defs", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 2, "macro_tier_score": 0.0292, "macro_tier_override": null, "x": -36.487, "z": 6.953, "size": 0.3627, "title": "Arithmetic Functions and Dirichlet Convolution", "summary": "This file defines arithmetic functions, which are functions from `ℕ` to a specified type that map 0 to 0. In the literature, they are often instead defined as functions from `ℕ+`. These arithmetic functions are endowed with a multiplication, given by Dirichlet convolution, and pointwise addition, to form the Dirichlet ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ArithmeticFunction/Defs.html"}, {"id": "Mathlib.RingTheory.Flat.Domain", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -63.235, "z": -57.441, "size": 0.2604, "title": "Flat modules in domains", "summary": "We show that the tensor product of two injective linear maps is injective if the sources are flat and the ring is an integral domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/Domain.html"}, {"id": "Mathlib.Algebra.Ring.Int.Parity", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.4159, "macro_tier_override": null, "x": 19.128, "z": 7.173, "size": 0.351, "title": "Basic parity lemmas for the ring `ℤ`", "summary": "See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Int/Parity.html"}, {"id": "Mathlib.Algebra.Homology.HomologySequenceLemmas", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -18.272, "z": -3.32, "size": 0.2697, "title": "Consequences of the homology sequence", "summary": "Given a morphism `φ : S₁ ⟶ S₂` between two short exact sequences of homological complexes in an abelian category, we show the naturality of the homology sequence of `S₁` and `S₂` with respect to `φ` (see `HomologicalComplex.HomologySequence.δ_naturality`). Then, we shall show in this file that if two out of the three maps `φ.τ₁`, `φ.τ₂`, `φ.τ₃` are quasi-isomorphisms, then the third is. We also obtain more specific…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomologySequenceLemmas.html"}, {"id": "Mathlib.Algebra.Polynomial.Degree.Support", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 4, "macro_tier_score": 0.2872, "macro_tier_override": null, "x": -22.734, "z": 50.866, "size": 0.4473, "title": "Degree and support of univariate polynomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Degree/Support.html"}, {"id": "Mathlib.Algebra.Ring.Subsemiring.MulOpposite", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 2, "macro_tier_score": 0.0091, "macro_tier_override": null, "x": 27.2, "z": 6.018, "size": 0.2579, "title": "Subsemiring of opposite semirings", "summary": "For every semiring `R`, we construct an equivalence between subsemirings of `R` and that of `Rᵐᵒᵖ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Subsemiring/MulOpposite.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.Operations", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.2712, "macro_tier_override": null, "x": 10.094, "z": -58.566, "size": 0.3814, "title": "More operations on subalgebras", "summary": "In this file we relate the definitions in `Mathlib/RingTheory/Ideal/Operations.lean` to subalgebras. The contents of this file are somewhat random since both `Mathlib/Algebra/Algebra/Subalgebra/Basic.lean` and `Mathlib/RingTheory/Ideal/Operations.lean` are somewhat of a grab-bag of definitions, and this is whatever ends up in the intersection.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/Operations.html"}, {"id": "Mathlib.Algebra.Ring.Fin", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 3, "macro_tier_score": 0.271, "macro_tier_override": null, "x": 14.521, "z": -8.278, "size": 0.3721, "title": "Rings and `Fin`", "summary": "This file collects some basic results involving rings and the `Fin` type", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Fin.html"}, {"id": "Mathlib.RingTheory.Ideal.Quotient.Basic", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 4, "macro_tier_score": 0.2933, "macro_tier_override": null, "x": 28.09, "z": -43.761, "size": 0.4242, "title": "Ideal quotients", "summary": "This file defines ideal quotients as a special case of submodule quotients and proves some basic results about these quotients. See `RingCon.Quotient` for quotients of (possibly non-commutative) semirings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Quotient/Basic.html"}, {"id": "Mathlib.Algebra.Ring.Action.Submonoid", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 3, "macro_tier_score": 0.27, "macro_tier_override": null, "x": -4.446, "z": -21.838, "size": 0.3042, "title": "The subgroup of fixed points of an action", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Action/Submonoid.html"}, {"id": "Mathlib.Algebra.Order.Interval.Set.Instances", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": -4.072, "z": -20.019, "size": 0.3667, "title": "Algebraic instances for unit intervals", "summary": "For suitably structured underlying type `α`, we exhibit the structure of the unit intervals (`Set.Icc`, `Set.Ioc`, `Set.Ioc`, and `Set.Ioo`) from `0` to `1`. Note: Instances for the interval `Ici 0` are dealt with in `Mathlib/Algebra/Order/Nonneg/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Interval/Set/Instances.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.MulOppositeLemmas", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 4, "macro_tier_score": 0.4276, "macro_tier_override": null, "x": -21.032, "z": 15.286, "size": 0.3153, "title": "Mul-opposite subgroups", "summary": "This file contains a somewhat arbitrary assortment of results on the opposite subgroup `H.op` that rely on further theory to define. As such it is a somewhat arbitrary assortment of results, which might be organized and split up further.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/MulOppositeLemmas.html"}, {"id": "Mathlib.RingTheory.Polynomial.UniversalFactorizationRing", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 15.196, "z": 104.762, "size": 0.2428, "title": "Universal factorization ring", "summary": "Let `R` be a commutative ring and `p : R[X]` be monic of degree `n` and let `n = m + k`. We construct the universal ring of the following functors on `R-Alg`: - `S ↦ \"monic polynomials over S of degree n\"`: Represented by `R[X₁,...,Xₙ]`. See `MvPolynomial.mapEquivMonic`. - `S ↦ \"factorizations of p into (monic deg m) * (monic deg k) in S\"`: Represented by an `R`-algebra (`Polynomial.UniversalFactorizationRing`) that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.html"}, {"id": "Mathlib.RingTheory.Polynomial.Resultant.Basic", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 1, "macro_tier_score": 0.0041, "macro_tier_override": null, "x": 48.854, "z": -89.703, "size": 0.3194, "title": "Resultant of two polynomials", "summary": "This file contains basic facts about resultant of two polynomials over commutative rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Resultant/Basic.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Pointwise", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 2, "macro_tier_score": 0.0087, "macro_tier_override": null, "x": 19.819, "z": 13.787, "size": 0.33, "title": "Pointwise actions of equivariant maps", "summary": "- `image_smul_setₛₗ` : under a `σ`-equivariant map, one has `f '' (c • s) = (σ c) • f '' s`. - `preimage_smul_setₛₗ'` is a general version of the equality `f ⁻¹' (σ c • s) = c • f⁻¹' s`. It requires that `c` acts surjectively and `σ c` acts injectively and is provided with specific versions: - `preimage_smul_setₛₗ_of_isUnit_isUnit` when `c` and `σ c` are units - `IsUnit.preimage_smul_setₛₗ` when `σ` belongs to a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Pointwise.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Opposite", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4582, "macro_tier_override": null, "x": 11.136, "z": 0.391, "size": 0.4923, "title": "Opposites of groups with zero", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Opposite.html"}, {"id": "Mathlib.FieldTheory.RatFunc.Degree", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 80.841, "z": 68.343, "size": 0.2788, "title": "The degree of rational functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/RatFunc/Degree.html"}, {"id": "Mathlib.FieldTheory.RatFunc.AsPolynomial", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 2, "macro_tier_score": 0.0303, "macro_tier_override": null, "x": -86.997, "z": -56.988, "size": 0.3615, "title": "Generalities on the polynomial structure of rational functions", "summary": "* Main evaluation properties * Study of the X-adic valuation", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/RatFunc/AsPolynomial.html"}, {"id": "Mathlib.Algebra.Ring.Ext", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 3.915, "z": 8.42, "size": 0.2, "title": "Extensionality lemmas for rings and similar structures", "summary": "In this file we prove extensionality lemmas for the ring-like structures defined in `Mathlib/Algebra/Ring/Defs.lean`, ranging from `NonUnitalNonAssocSemiring` to `CommRing`. These extensionality lemmas take the form of asserting that two algebraic structures on a type are equal whenever the addition and multiplication defined by them are both the same.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Ext.html"}, {"id": "Mathlib.GroupTheory.GroupAction.MultiplePrimitivity", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 1, "macro_tier_score": 0.0041, "macro_tier_override": null, "x": 28.566, "z": -43.452, "size": 0.3158, "title": "Multiply preprimitive actions", "summary": "Let `G` be a group acting on a type `α`. * `MulAction.IsMultiplyPreprimitive` : The action is said to be `n`-primitive if, for every subset `s : Set α` with `n` elements, the actions f `stabilizer G s` on the complement of `s` is primitive. * `MulAction.is_zero_preprimitive` : any action is 0-primitive * `MulAction.is_one_preprimitive_iff` : an action is 1-primitive if and only if it is primitive *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/MultiplePrimitivity.html"}, {"id": "Mathlib.Algebra.Category.GrpWithZero", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -0.448, "z": -16.708, "size": 0.2, "title": "The category of groups with zero", "summary": "This file defines `GrpWithZero`, the category of groups with zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/GrpWithZero.html"}, {"id": "Mathlib.FieldTheory.Tower", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.0943, "macro_tier_override": null, "x": 52.311, "z": 35.364, "size": 0.2393, "title": "Finiteness of `IsScalarTower`", "summary": "We prove that given `IsScalarTower F K A`, if `A` is finite as a module over `F` then `A` is finite over `K`, and (as long as `A` is Noetherian over `F` and torsion-free over `K`) `K` is finite over `F`. In particular these conditions hold when `A`, `F`, and `K` are fields. The formulas for the dimensions are given elsewhere by `Module.finrank_mul_finrank`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Tower.html"}, {"id": "Mathlib.LinearAlgebra.Projectivization.Subspace", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 35.713, "z": 69.345, "size": 0.239, "title": "Subspaces of Projective Space", "summary": "In this file we define subspaces of a projective space, and show that the subspaces of a projective space form a complete lattice under inclusion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Projectivization/Subspace.html"}, {"id": "Mathlib.LinearAlgebra.SymplecticGroup", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -68.227, "z": 37.805, "size": 0.239, "title": "The Symplectic Group", "summary": "This file defines the symplectic group and proves elementary properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/SymplecticGroup.html"}, {"id": "Mathlib.RingTheory.Finiteness.Descent", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -91.05, "z": 46.296, "size": 0.2455, "title": "Descent of finiteness conditions under faithfully flat maps", "summary": "In this file we show that - `Algebra.FiniteType`: - `Algebra.FinitePresentation`: - `Module.Finite`: descend along faithfully flat base change.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Descent.html"}, {"id": "Mathlib.RingTheory.RingHom.FinitePresentation", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 2, "macro_tier_score": 0.0084, "macro_tier_override": null, "x": 17.5, "z": -98.748, "size": 0.3071, "title": "The meta properties of finitely-presented ring homomorphisms.", "summary": "The main result is `RingHom.finitePresentation_isLocal`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/FinitePresentation.html"}, {"id": "Mathlib.RingTheory.RingHom.FaithfullyFlat", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 2, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -67.646, "z": -74.037, "size": 0.317, "title": "Faithfully flat ring maps", "summary": "A ring map `f : R →+* S` is faithfully flat if `S` is faithfully flat as an `R`-algebra. This is the same as being flat and a surjection on prime spectra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/FaithfullyFlat.html"}, {"id": "Mathlib.NumberTheory.NumberField.CMField", "region_id": "algebra", "micro_elevation": 0.9737, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 134.405, "z": -28.675, "size": 0.2, "title": "CM-extension of number fields", "summary": "A CM-extension `K/F` of fields is an extension where `K` is totally complex, `F` is totally real and `K` is a quadratic extension of `F`. In this situation, the totally real subfield `F` is (isomorphic to) the maximal real subfield `K⁺` of `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/CMField.html"}, {"id": "Mathlib.FieldTheory.Galois.IsGaloisGroup", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 5, "macro_tier_score": 0.0103, "macro_tier_override": 5, "x": 109.993, "z": 58.176, "size": 0.2772, "title": "Galois Groups of Fields", "summary": "Given an action of a group `G` on an extension of fields `L/K`, the predicate `IsGaloisGroup G K L` states that `G` acts faithfully on `L` with fixed field `K`. In particular, we do not assume that `L` is an algebraic extension of `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Galois/IsGaloisGroup.html"}, {"id": "Mathlib.NumberTheory.NumberField.Cyclotomic.Embeddings", "region_id": "algebra", "micro_elevation": 0.9342, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -54.715, "z": 119.97, "size": 0.262, "title": "Cyclotomic extensions of `ℚ` are totally complex number fields.", "summary": "We prove that cyclotomic extensions of `ℚ` are totally complex, meaning that `NrRealPlaces K = 0` if `IsCyclotomicExtension {n} ℚ K` and `2 < n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Cyclotomic/Embeddings.html"}, {"id": "Mathlib.NumberTheory.NumberField.Units.Regulator", "region_id": "algebra", "micro_elevation": 0.9605, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 109.143, "z": -80.423, "size": 0.2622, "title": "Regulator of a number field", "summary": "We define and prove basic results about the regulator of a number field `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Units/Regulator.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.Pushforward", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 1.373, "z": -64.986, "size": 0.2846, "title": "Pushforward of presheaves of modules", "summary": "If `F : C ⥤ D`, the precomposition `F.op ⋙ _` induces a functor from presheaves over `D` to presheaves over `C`. When `R : Dᵒᵖ ⥤ RingCat`, we define the induced functor `pushforward₀ : PresheafOfModules.{v} R ⥤ PresheafOfModules.{v} (F.op ⋙ R)` on presheaves of modules. In case we have a morphism of presheaves of rings `S ⟶ F.op ⋙ R`, we also construct a functor `pushforward : PresheafOfModules.{v} R ⥤…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/Pushforward.html"}, {"id": "Mathlib.LinearAlgebra.GeneralLinearGroup.Basic", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 4, "macro_tier_score": 0.3113, "macro_tier_override": null, "x": -27.526, "z": 18.969, "size": 0.3893, "title": "The general linear group of linear maps", "summary": "The general linear group is defined to be the group of invertible linear maps from `M` to itself. See also `Matrix.GeneralLinearGroup`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/GeneralLinearGroup/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Eigenspace.Charpoly", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -50.064, "z": -95.374, "size": 0.241, "title": "Eigenvalues are the roots of the characteristic polynomial.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Eigenspace/Charpoly.html"}, {"id": "Mathlib.LinearAlgebra.Trace", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 3, "macro_tier_score": 0.0489, "macro_tier_override": null, "x": -78.696, "z": -28.129, "size": 0.3465, "title": "Trace of a linear map", "summary": "This file defines the trace of a linear map. See also `Mathlib/LinearAlgebra/Matrix/Trace.lean` for the trace of a matrix.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Trace.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Charpoly.Eigs", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -98.665, "z": -38.356, "size": 0.2387, "title": "Eigenvalues are characteristic polynomial roots.", "summary": "In fields we show that: * `Matrix.mem_spectrum_iff_isRoot_charpoly`: the roots of the characteristic polynomial are exactly the spectrum of the matrix. * `Matrix.det_eq_prod_roots_charpoly_of_splits`: the determinant (in the field of the matrix) is the product of the roots of the characteristic polynomial if the polynomial splits in the field of the matrix. * `Matrix.trace_eq_sum_roots_charpoly_of_splits`: the trace…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.html"}, {"id": "Mathlib.Algebra.Central.TensorProduct", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 39.932, "z": -71.294, "size": 0.239, "title": "Lemmas about tensor products of central algebras", "summary": "In this file we prove for algebras `B` and `C` over a field `K` that if `B ⊗[K] C` is a central algebra and `B, C` nontrivial, then both `B` and `C` are central algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Central/TensorProduct.html"}, {"id": "Mathlib.Algebra.Central.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 1, "macro_tier_score": 0.0042, "macro_tier_override": null, "x": 40.073, "z": 43.886, "size": 0.3235, "title": "Central Algebras", "summary": "In this file, we prove some basic results about central algebras over a commutative ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Central/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.RowCol", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 3, "macro_tier_score": 0.2608, "macro_tier_override": null, "x": 35.745, "z": -37.767, "size": 0.3581, "title": "Row and column matrices", "summary": "This file provides results about row and column matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/RowCol.html"}, {"id": "Mathlib.GroupTheory.Perm.List", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 3, "macro_tier_score": 0.2286, "macro_tier_override": null, "x": -5.866, "z": 19.569, "size": 0.2571, "title": "Permutations from a list", "summary": "A list `l : List α` can be interpreted as an `Equiv.Perm α` where each element in the list is permuted to the next one, defined as `formPerm`. When we have that `Nodup l`, we prove that `Equiv.Perm.support (formPerm l) = l.toFinset`, and that `formPerm l` is rotationally invariant, in `formPerm_rotate`. When there are duplicate elements in `l`, how and in what arrangement with respect to the other elements they…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/List.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Graded.External", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 22.72, "z": -58.914, "size": 0.2442, "title": "Graded tensor products over graded algebras", "summary": "The graded tensor product $A \\hat\\otimes_R B$ is imbued with a multiplication defined on homogeneous tensors by: $$(a \\otimes b) \\cdot (a' \\otimes b') = (-1)^{\\deg a' \\deg b} (a \\cdot a') \\otimes (b \\cdot b')$$ where $A$ and $B$ are algebras graded by `ℕ`, `ℤ`, or `ZMod 2` (or more generally, any index that satisfies `Module ι (Additive ℤˣ)`). The results for internally-graded algebras (via `GradedAlgebra`) are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Graded/External.html"}, {"id": "Mathlib.RingTheory.Finiteness.Subalgebra", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.1813, "macro_tier_override": null, "x": 8.34, "z": 56.965, "size": 0.3026, "title": "Subalgebras that are finitely generated as submodules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Subalgebra.html"}, {"id": "Mathlib.RingTheory.Finiteness.Bilinear", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 3, "macro_tier_score": 0.2036, "macro_tier_override": null, "x": -41.958, "z": -23.898, "size": 0.3129, "title": "Finitely generated submodules and bilinear maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Bilinear.html"}, {"id": "Mathlib.Algebra.Lie.CartanSubalgebra", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -39.303, "z": 80.012, "size": 0.2743, "title": "Cartan subalgebras", "summary": "Cartan subalgebras are one of the most important concepts in Lie theory. We define them here. The standard example is the set of diagonal matrices in the Lie algebra of matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/CartanSubalgebra.html"}, {"id": "Mathlib.Algebra.Homology.CommSq", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 0.421, "z": -1.809, "size": 0.2712, "title": "Relation between pullback/pushout squares and kernel/cokernel sequences", "summary": "Consider a commutative square in a preadditive category: ``` X₁ ⟶ X₂ | | v v X₃ ⟶ X₄ ``` In this file, we show that this is a pushout square iff the object `X₄` identifies to the cokernel of the difference map `X₁ ⟶ X₂ ⊞ X₃` via the obvious map `X₂ ⊞ X₃ ⟶ X₄`. Similarly, it is a pullback square iff the object `X₁` identifies to the kernel of the difference map `X₂ ⊞ X₃ ⟶ X₄` via the obvious map `X₁ ⟶ X₂ ⊞ X₃`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/CommSq.html"}, {"id": "Mathlib.Algebra.Order.Group.Abs", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.4126, "macro_tier_override": null, "x": 9.549, "z": -20.136, "size": 0.4097, "title": "Absolute values in ordered groups", "summary": "The absolute value of an element in a group which is also a lattice is its supremum with its negation. This generalizes the usual absolute value on real numbers (`|x| = max x (-x)`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Abs.html"}, {"id": "Mathlib.Algebra.Order.Ring.Int", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.4166, "macro_tier_override": null, "x": 21.08, "z": -11.769, "size": 0.4341, "title": "The integers form a linear ordered ring", "summary": "This file contains: * instances on `ℤ`. The stronger one is `Int.instLinearOrderedCommRing`. * basic lemmas about integers that involve order properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Int.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.RestrictionHomology", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 2, "macro_tier_score": 0.006, "macro_tier_override": null, "x": 1.45, "z": 14.786, "size": 0.2911, "title": "The homology of a restriction", "summary": "Under favourable circumstances, we may relate the homology of `K : HomologicalComplex C c'` in degree `j'` and that of `K.restriction e` in degree `j` when `e : Embedding c c'` is an embedding of complex shapes. See `restriction.sc'Iso` and `restriction.hasHomology`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/RestrictionHomology.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.Restriction", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 2, "macro_tier_score": 0.0065, "macro_tier_override": null, "x": 9.199, "z": -9.186, "size": 0.3295, "title": "The restriction functor of an embedding of complex shapes", "summary": "Given `c` and `c'` complex shapes on two types, and `e : c.Embedding c'` (satisfying `[e.IsRelIff]`), we define the restriction functor `e.restrictionFunctor C : HomologicalComplex C c' ⥤ HomologicalComplex C c`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/Restriction.html"}, {"id": "Mathlib.Algebra.Module.Presentation.Cokernel", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -17.086, "z": -53.03, "size": 0.2478, "title": "Presentation of a cokernel", "summary": "If `f : M₁ →ₗ[A] M₂` is a linear map between modules, `pres₂` is a presentation of `M₂` and `g₁ : ι → M₁` is a family of generators of `M₁` (which is expressed as `hg₁ : Submodule.span A (Set.range g₁) = ⊤`), then we provide a way to obtain a presentation of the cokernel of `f`. It requires an additional data `data : pres₂.CokernelData f g₁`, which consists of liftings of the images by `f` of the generators of `M₁`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Presentation/Cokernel.html"}, {"id": "Mathlib.Algebra.Ring.Action.Subobjects", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.3829, "macro_tier_override": null, "x": 22.249, "z": 1.279, "size": 0.2813, "title": "Instances of `MulSemiringAction` for subobjects", "summary": "These are defined in this file as `Semiring`s are not available yet where `Submonoid` and `Subgroup` are defined. Instances for `Subsemiring` and `Subring` are provided next to the other scalar actions instances for those subobjects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Action/Subobjects.html"}, {"id": "Mathlib.Algebra.Ring.Subsemiring.Defs", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.3848, "macro_tier_override": null, "x": 14.43, "z": 8.436, "size": 0.406, "title": "Bundled subsemirings", "summary": "We define bundled subsemirings and some standard constructions: `subtype` and `inclusion` ring homomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Subsemiring/Defs.html"}, {"id": "Mathlib.GroupTheory.Submonoid.Centralizer", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.4375, "macro_tier_override": null, "x": -19.03, "z": 14.858, "size": 0.312, "title": "Centralizers of magmas and monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Submonoid/Centralizer.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.MidpointZero", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -52.233, "z": 19.388, "size": 0.2, "title": "Midpoint of a segment for characteristic zero", "summary": "We collect lemmas that require that the underlying ring has characteristic zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/MidpointZero.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -39.612, "z": 36.49, "size": 0.3591, "title": "Midpoint of a segment", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Midpoint.html"}, {"id": "Mathlib.Algebra.Polynomial.Module.Basic", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.0638, "macro_tier_override": null, "x": -62.216, "z": 18.823, "size": 0.3011, "title": "Polynomial module", "summary": "In this file, we define the polynomial module for an `R`-module `M`, i.e. the `R[X]`-module `M[X]`. This is defined as a type alias `PolynomialModule R M := ℕ →₀ M`, since there might be different module structures on `ℕ →₀ M` of interest. See the docstring of `PolynomialModule` for details.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Module/Basic.html"}, {"id": "Mathlib.Algebra.Category.Grp.ZModuleEquivalence", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 3, "macro_tier_score": 0.0427, "macro_tier_override": null, "x": -35.177, "z": -30.302, "size": 0.3015, "title": null, "summary": "The forgetful functor from ℤ-modules to additive commutative groups is an equivalence of categories. TODO: either use this equivalence to transport the monoidal structure from `Module ℤ` to `Ab`, or, having constructed that monoidal structure directly, show this functor is monoidal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/ZModuleEquivalence.html"}, {"id": "Mathlib.Algebra.Polynomial.Degree.Lemmas", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 4, "macro_tier_score": 0.2799, "macro_tier_override": null, "x": -39.163, "z": 47.141, "size": 0.4204, "title": "Theory of degrees of polynomials", "summary": "Some of the main results include - `natDegree_comp_le` : The degree of the composition is at most the product of degrees", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Degree/Lemmas.html"}, {"id": "Mathlib.Algebra.Group.Pointwise.Set.ListOfFn", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.3643, "macro_tier_override": null, "x": 18.482, "z": -1.824, "size": 0.3067, "title": "Pointwise operations with lists of sets", "summary": "This file proves some lemmas about pointwise algebraic operations with lists of sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pointwise/Set/ListOfFn.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.List.Defs", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.4617, "macro_tier_override": null, "x": 4.8, "z": 2.828, "size": 0.3694, "title": "Sums and products from lists", "summary": "This file provides basic definitions for `List.prod`, `List.sum`, which calculate the product and sum of elements of a list and `List.alternatingProd`, `List.alternatingSum`, their alternating counterparts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/List/Defs.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.Finset.Lemmas", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4076, "macro_tier_override": null, "x": 15.246, "z": -6.851, "size": 0.3124, "title": "Miscellaneous lemmas on big operators", "summary": "The lemmas in this file have been moved out of `Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean` to reduce its imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Finset/Lemmas.html"}, {"id": "Mathlib.RingTheory.Congruence.Defs", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.3391, "macro_tier_override": null, "x": -14.397, "z": -3.668, "size": 0.391, "title": "Congruence relations on rings", "summary": "This file defines congruence relations on rings, which extend `Con` and `AddCon` on monoids and additive monoids. Most of the time you likely want to use the `Ideal.Quotient` API that is built on top of this.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Congruence/Defs.html"}, {"id": "Mathlib.Algebra.Ring.Subring.Basic", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 4, "macro_tier_score": 0.3847, "macro_tier_override": null, "x": 26.337, "z": -9.077, "size": 0.4447, "title": "Subrings", "summary": "We prove that subrings are a complete lattice, and that you can `map` (pushforward) and `comap` (pull back) them along ring homomorphisms. We define the `closure` construction from `Set R` to `Subring R`, sending a subset of `R` to the subring it generates, and prove that it is a Galois insertion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Subring/Basic.html"}, {"id": "Mathlib.RingTheory.Bialgebra.Equiv", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 1, "macro_tier_score": 0.0039, "macro_tier_override": null, "x": 45.534, "z": 51.462, "size": 0.3028, "title": "Isomorphisms of `R`-bialgebras", "summary": "This file defines bundled isomorphisms of `R`-bialgebras. We simply mimic the early parts of `Mathlib/Algebra/Algebra/Equiv.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Bialgebra/Equiv.html"}, {"id": "Mathlib.Algebra.Module.Presentation.Differentials", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.437, "z": 91.0, "size": 0.2, "title": "Presentation of the module of differentials", "summary": "Given a presentation of an `R`-algebra `S`, we obtain a presentation of the `S`-module `Ω[S⁄R]`. Assume `pres : Algebra.Presentation R S` is a presentation of `S` as an `R`-algebra. We then have a type `ι` for the generators, a type `σ` for the relations, and for each `r : σ` we have the relation `pres.relation r` in `pres.Ring` (which is a ring of polynomials over `R` with variables indexed by `ι`). Then,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Presentation/Differentials.html"}, {"id": "Mathlib.Algebra.Module.NatInt", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4524, "macro_tier_override": null, "x": -8.185, "z": -14.573, "size": 0.6016, "title": "Modules over `ℕ` and `ℤ`", "summary": "This file concerns modules where the scalars are the natural numbers or the integers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/NatInt.html"}, {"id": "Mathlib.Algebra.Ring.CompTypeclasses", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4277, "macro_tier_override": null, "x": -16.548, "z": -2.35, "size": 0.5393, "title": "Propositional typeclasses on several ring homs", "summary": "This file contains three typeclasses used in the definition of (semi)linear maps: * `RingHomId σ`, which expresses the fact that `σ₂₃ = id` * `RingHomCompTriple σ₁₂ σ₂₃ σ₁₃`, which expresses the fact that `σ₂₃.comp σ₁₂ = σ₁₃` * `RingHomInvPair σ₁₂ σ₂₁`, which states that `σ₁₂` and `σ₂₁` are inverses of each other * `RingHomSurjective σ`, which states that `σ` is surjective These typeclasses ensure that objects such…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/CompTypeclasses.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.CochainComplex", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 2, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -31.521, "z": 11.133, "size": 0.3373, "title": "Truncations on cochain complexes indexed by the integers.", "summary": "In this file, we introduce abbreviations for the canonical truncations `CochainComplex.truncLE`, `CochainComplex.truncGE` of cochain complexes indexed by `ℤ`, as well as the conditions `CochainComplex.IsStrictlyLE`, `CochainComplex.IsStrictlyGE`, `CochainComplex.IsLE`, and `CochainComplex.IsGE`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/CochainComplex.html"}, {"id": "Mathlib.Algebra.Homology.Single", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": 8.854, "z": 2.799, "size": 0.4058, "title": "Homological complexes supported in a single degree", "summary": "We define `single V j c : V ⥤ HomologicalComplex V c`, which constructs complexes in `V` of shape `c`, supported in degree `j`. In `ChainComplex.toSingle₀Equiv` we characterize chain maps to an `ℕ`-indexed complex concentrated in degree 0; they are equivalent to `{ f : C.X 0 ⟶ X // C.d 1 0 ≫ f = 0 }`. (This is useful translating between a projective resolution and an augmented exact complex of projectives.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Single.html"}, {"id": "Mathlib.FieldTheory.RatFunc.IntermediateField", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": 16.704, "z": -104.532, "size": 0.2589, "title": "Intermediate Fields of Rational Function Fields", "summary": "Results relating `IntermediateField` and `RatFunc`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/RatFunc/IntermediateField.html"}, {"id": "Mathlib.RingTheory.Adjoin.Polynomial.Bivariate", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": -100.276, "z": -1.479, "size": 0.2441, "title": "Bivariate polynomials and adjoining transcendental elements", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Adjoin/Polynomial/Bivariate.html"}, {"id": "Mathlib.FieldTheory.RatFunc.Valuation", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": 94.115, "z": -52.393, "size": 0.2441, "title": "Valuations on F(t)", "summary": "This file defines the valuation at infinity on the field of rational functions `F(t)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/RatFunc/Valuation.html"}, {"id": "Mathlib.Algebra.Homology.Augment", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -7.511, "z": 8.231, "size": 0.2, "title": "Augmentation and truncation of `ℕ`-indexed (co)chain complexes.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Augment.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.Finset", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 1, "macro_tier_score": 0.0027, "macro_tier_override": null, "x": -20.054, "z": -3.897, "size": 0.2942, "title": "`Finset.sup` in a group with zero", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/Finset.html"}, {"id": "Mathlib.RingTheory.RingHom.Flat", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 2, "macro_tier_score": 0.0113, "macro_tier_override": null, "x": 75.69, "z": 62.924, "size": 0.3543, "title": "Flat ring homomorphisms", "summary": "In this file we define flat ring homomorphisms and show their meta properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/Flat.html"}, {"id": "Mathlib.Algebra.Homology.Double", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -6.311, "z": -9.183, "size": 0.2442, "title": "A homological complex lying in two degrees", "summary": "Given `c : ComplexShape ι`, distinct indices `i₀` and `i₁` such that `hi₀₁ : c.Rel i₀ i₁`, we construct a homological complex `double f hi₀₁` for any morphism `f : X₀ ⟶ X₁`. It consists of the objects `X₀` and `X₁` in degrees `i₀` and `i₁`, respectively, with the differential `X₀ ⟶ X₁` given by `f`, and zero everywhere else.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Double.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 2, "macro_tier_score": 0.007, "macro_tier_override": null, "x": -55.09, "z": -77.046, "size": 0.3603, "title": "Conjugations", "summary": "This file defines the grade reversal and grade involution functions on multivectors, `reverse` and `involute`. Together, these operations compose to form the \"Clifford conjugate\", hence the name of this file. https://en.wikipedia.org/wiki/Clifford_algebra#Antiautomorphisms", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.html"}, {"id": "Mathlib.RingTheory.WittVector.Complete", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": -116.702, "z": 8.36, "size": 0.2796, "title": "The ring of Witt vectors is p-torsion free and p-adically complete", "summary": "In this file, we prove that the ring of Witt vectors `𝕎 k` is p-torsion free and p-adically complete when `k` is a perfect ring of characteristic `p`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/Complete.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.WeylGroup", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -29.621, "z": -111.269, "size": 0.2708, "title": "The Weyl group of a root pairing", "summary": "This file defines the Weyl group of a root pairing as the subgroup of automorphisms generated by reflection automorphisms. This deviates from the existing literature, which typically defines the Weyl group as the subgroup of linear transformations of the weight space generated by linear reflections. However, the automorphism group of a root pairing comes with a permutation representation on the set indexing roots…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/WeylGroup.html"}, {"id": "Mathlib.RepresentationTheory.Submodule", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -6.61, "z": 85.173, "size": 0.2708, "title": "Invariant submodules of a group representation", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Submodule.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Action", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.1379, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2892, "title": "Actions by matrices on vectors through `*ᵥ` and `ᵥ*`, cast as `Module`s", "summary": "This file provides the left- and right- module structures of square matrices on vectors, via `Matrix.mulVec` and `Matrix.vecMul`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Action.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Adjugate", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.2012, "macro_tier_override": null, "x": 72.346, "z": 3.47, "size": 0.3727, "title": "Cramer's rule and adjugate matrices", "summary": "The adjugate matrix is the transpose of the cofactor matrix. It is calculated with Cramer's rule, which we introduce first. The vectors returned by Cramer's rule are given by the linear map `cramer`, which sends a matrix `A` and vector `b` to the vector consisting of the determinant of replacing the `i`th column of `A` with `b` at index `i` (written as `(A.updateCol i b).det`). Using Cramer's rule, we can compute…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Adjugate.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Transvection", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.1957, "macro_tier_override": null, "x": -63.47, "z": -14.022, "size": 0.3062, "title": "Transvections", "summary": "Transvections are matrices of the form `1 + single i j c`, where `single i j c` is the basic matrix with a `c` at position `(i, j)`. Multiplying by such a transvection on the left (resp. on the right) amounts to adding `c` times the `j`-th row to the `i`-th row (resp `c` times the `i`-th column to the `j`-th column). Therefore, they are useful to present algorithms operating on rows and columns. Transvections are a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Transvection.html"}, {"id": "Mathlib.RingTheory.RootsOfUnity.Basic", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 3, "macro_tier_score": 0.1418, "macro_tier_override": null, "x": -75.249, "z": -11.638, "size": 0.3933, "title": "Roots of unity", "summary": "We define roots of unity in the context of an arbitrary commutative monoid, as a subgroup of the group of units.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RootsOfUnity/Basic.html"}, {"id": "Mathlib.Algebra.CharP.Algebra", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.116, "macro_tier_override": null, "x": 46.988, "z": -36.386, "size": 0.4294, "title": "Characteristics of algebras", "summary": "In this file we describe the characteristic of `R`-algebras. In particular we are interested in the characteristic of free algebras over `R` and the fraction field `FractionRing R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/Algebra.html"}, {"id": "Mathlib.FieldTheory.SplittingField.IsSplittingField", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 3, "macro_tier_score": 0.0957, "macro_tier_override": null, "x": 98.293, "z": 5.177, "size": 0.3647, "title": "Splitting fields", "summary": "This file introduces the notion of a splitting field of a polynomial and provides an embedding from a splitting field to any field that splits the polynomial. A polynomial `f : K[X]` splits over a field extension `L` of `K` if it is zero or all of its irreducible factors over `L` have degree `1`. A field extension of `K` of a polynomial `f : K[X]` is called a splitting field if it is the smallest field extension of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/SplittingField/IsSplittingField.html"}, {"id": "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.4006, "macro_tier_override": null, "x": -24.122, "z": 1.016, "size": 0.2872, "title": "Big operators on a multiset in ordered rings", "summary": "This file contains the results concerning the interaction of multiset big operators with ordered rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/BigOperators/Ring/Multiset.html"}, {"id": "Mathlib.Algebra.Order.BigOperators.GroupWithZero.Multiset", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4005, "macro_tier_override": null, "x": -18.181, "z": 3.789, "size": 0.2744, "title": "Big operators on a multiset in ordered groups with zeros", "summary": "This file contains the results concerning the interaction of multiset big operators with ordered groups with zeros.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/BigOperators/GroupWithZero/Multiset.html"}, {"id": "Mathlib.RingTheory.Ideal.Height", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 2, "macro_tier_score": 0.0064, "macro_tier_override": null, "x": -84.821, "z": -46.169, "size": 0.3222, "title": "The Height of an Ideal", "summary": "In this file, we define the height of a prime ideal and the height of an ideal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Height.html"}, {"id": "Mathlib.RepresentationTheory.FiniteIndex", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -85.04, "z": -53.157, "size": 0.2, "title": "(Co)induced representations of a finite index subgroup", "summary": "Given a commutative ring `k`, a finite index subgroup `S ≤ G`, and a `k`-linear `S`-representation `A`, this file defines an isomorphism $Ind_S^G(A) ≅ Coind_S^G(A)$. Given `g : G` and `a : A`, the forward map sends `⟦g ⊗ₜ[k] a⟧` to the function `G → A` supported at `sg` by `ρ(s)(a)` for `s : S` and which is 0 elsewhere. Meanwhile, the inverse sends `f : G → A` to `∑ᵢ ⟦gᵢ ⊗ₜ[k] f(gᵢ)⟧` for `1 ≤ i ≤ n`, where `g₁,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/FiniteIndex.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.Pullback", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 6.699, "z": -68.388, "size": 0.2463, "title": "Pullback of presheaves of modules", "summary": "Let `F : C ⥤ D` be a functor, `R : Dᵒᵖ ⥤ RingCat` and `S : Cᵒᵖ ⥤ RingCat` be presheaves of rings, and `φ : S ⟶ F.op ⋙ R` be a morphism of presheaves of rings, we introduce the pullback functor `pullback : PresheafOfModules S ⥤ PresheafOfModules R` as the left adjoint of `pushforward : PresheafOfModules R ⥤ PresheafOfModules S`. The existence of this left adjoint functor is obtained under suitable universe…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/Pullback.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": -6.144, "z": -66.575, "size": 0.2791, "title": "Pushforward of sheaves of modules", "summary": "Assume that categories `C` and `D` are equipped with Grothendieck topologies, and that `F : C ⥤ D` is a continuous functor. Then, if `φ : S ⟶ (F.sheafPushforwardContinuous RingCat.{u} J K).obj R` is a morphism of sheaves of rings, we construct the pushforward functor `pushforward φ : SheafOfModules.{v} R ⥤ SheafOfModules.{v} S`, and we show that they interact with the composition of morphisms similarly as…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Sheaf/PushforwardContinuous.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.Hom", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.3883, "macro_tier_override": null, "x": 4.476, "z": 14.167, "size": 0.2728, "title": "Zero-related `•` instances on group-like morphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/Hom.html"}, {"id": "Mathlib.Algebra.Group.Hom.Instances", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4792, "macro_tier_override": null, "x": -9.149, "z": 1.588, "size": 0.4094, "title": "Instances on spaces of monoid and group morphisms", "summary": "We endow the space of monoid morphisms `M →* N` with a `CommMonoid` structure when the target is commutative, through pointwise multiplication, and with a `CommGroup` structure when the target is a commutative group. We also prove the same instances for additive situations. Since these structures permit morphisms of morphisms, we also provide some composition-like operations. Finally, we provide the `Ring` structure…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Hom/Instances.html"}, {"id": "Mathlib.RingTheory.TwoSidedIdeal.Operations", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 3, "macro_tier_score": 0.1964, "macro_tier_override": null, "x": -29.831, "z": 40.304, "size": 0.3521, "title": "Operations on two-sided ideals", "summary": "This file defines operations on two-sided ideals of a ring `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TwoSidedIdeal/Operations.html"}, {"id": "Mathlib.FieldTheory.Fixed", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 3, "macro_tier_score": 0.0904, "macro_tier_override": null, "x": -34.094, "z": -90.354, "size": 0.3027, "title": "Fixed field under a group action.", "summary": "This is the basis of the Fundamental Theorem of Galois Theory. Given a (finite) group `G` that acts on a field `F`, we define `FixedPoints.subfield G F`, the subfield consisting of elements of `F` fixed by every element of `G`. This subfield is then normal and separable, and in addition if `G` acts faithfully on `F` then `finrank (FixedPoints.subfield G F) F = Fintype.card G`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Fixed.html"}, {"id": "Mathlib.Algebra.Ring.Action.Field", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 3, "macro_tier_score": 0.1046, "macro_tier_override": null, "x": 13.954, "z": -17.377, "size": 0.2882, "title": "Group action on fields", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Action/Field.html"}, {"id": "Mathlib.Algebra.Ring.Action.Invariant", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 3, "macro_tier_score": 0.091, "macro_tier_override": null, "x": -18.835, "z": 15.104, "size": 0.2411, "title": "Subrings invariant under an action", "summary": "If a monoid acts on a ring via a `MulSemiringAction`, then `IsInvariantSubring` is a predicate on subrings asserting that the subring is fixed elementwise by the action.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Action/Invariant.html"}, {"id": "Mathlib.FieldTheory.Finiteness", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 3, "macro_tier_score": 0.122, "macro_tier_override": null, "x": -10.58, "z": -73.529, "size": 0.3415, "title": "A module over a division ring is Noetherian if and only if it is finite.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Finiteness.html"}, {"id": "Mathlib.FieldTheory.Normal.Defs", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 3, "macro_tier_score": 0.091, "macro_tier_override": null, "x": -81.79, "z": 43.966, "size": 0.2411, "title": "Normal field extensions", "summary": "In this file we define normal field extensions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Normal/Defs.html"}, {"id": "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 3, "macro_tier_score": 0.1242, "macro_tier_override": null, "x": -2.934, "z": -79.804, "size": 0.3457, "title": "Finite and free modules using matrices", "summary": "We provide some instances for finite and free modules involving matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.html"}, {"id": "Mathlib.NumberTheory.SmoothNumbers", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -1.794, "z": -0.482, "size": 0.2731, "title": "Smooth numbers", "summary": "For `s : Finset ℕ` we define the set `Nat.factoredNumbers s` of \"`s`-factored numbers\" consisting of the positive natural numbers all of whose prime factors are in `s`, and we provide some API for this. We then define the set `Nat.smoothNumbers n` consisting of the positive natural numbers all of whose prime factors are strictly less than `n`. This is the special case `s = Finset.range n` of the set of `s`-factored…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/SmoothNumbers.html"}, {"id": "Mathlib.NumberTheory.ModularForms.Delta", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -111.405, "z": -35.753, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/Delta.html"}, {"id": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 113.196, "z": 21.091, "size": 0.2762, "title": "Slash action on E2", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/EisensteinSeries/E2/Transform.html"}, {"id": "Mathlib.RingTheory.Finiteness.Lattice", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 3, "macro_tier_score": 0.2299, "macro_tier_override": null, "x": 37.033, "z": 36.505, "size": 0.2864, "title": "Finite suprema of finite modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Lattice.html"}, {"id": "Mathlib.RingTheory.Finiteness.Defs", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 4, "macro_tier_score": 0.3189, "macro_tier_override": null, "x": -1.751, "z": -46.396, "size": 0.4291, "title": "Finiteness conditions in commutative algebra", "summary": "In this file we define a notion of finiteness that is common in commutative algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Defs.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.CategoryTheory", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 26.49, "z": -89.0, "size": 0.2, "title": "Category-theoretic interpretations of `CliffordAlgebra`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/CategoryTheory.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.QuadraticModuleCat", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 75.93, "z": 50.158, "size": 0.2607, "title": "The category of quadratic modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat.html"}, {"id": "Mathlib.GroupTheory.GroupAction.SubMulAction", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 4, "macro_tier_score": 0.4013, "macro_tier_override": null, "x": 26.261, "z": 13.904, "size": 0.5979, "title": "Sets invariant to a `MulAction`", "summary": "In this file we define `SubMulAction R M`; a subset of a `MulAction R M` which is closed with respect to scalar multiplication. For most uses, typically `Submodule R M` is more powerful.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/SubMulAction.html"}, {"id": "Mathlib.RingTheory.HahnSeries.PowerSeries", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": -5.448, "z": 68.499, "size": 0.2255, "title": "Comparison between Hahn series and power series", "summary": "If `Γ` is ordered and `R` has zero, then `R⟦Γ⟧` consists of formal series over `Γ` with coefficients in `R`, whose supports are partially well-ordered. With further structure on `R` and `Γ`, we can add further structure on `R⟦Γ⟧`. When `R` is a semiring and `Γ = ℕ`, then we get the more familiar semiring of formal power series with coefficients in `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HahnSeries/PowerSeries.html"}, {"id": "Mathlib.RingTheory.LocalRing.LocalSubring", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 42.019, "z": -56.699, "size": 0.2385, "title": "Local subrings of fields", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/LocalSubring.html"}, {"id": "Mathlib.Algebra.Ring.Periodic", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 1, "macro_tier_score": 0.0029, "macro_tier_override": null, "x": 1.144, "z": 1.463, "size": 0.3148, "title": "Periodicity", "summary": "In this file we define and then prove facts about periodic and antiperiodic functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Periodic.html"}, {"id": "Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleNumber", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 37.99, "z": -76.517, "size": 0.2, "title": "Liouville constants", "summary": "This file contains a construction of a family of Liouville numbers, indexed by a natural number $m$. The most important property is that they are examples of transcendental real numbers. This fact is recorded in `transcendental_liouvilleNumber`. More precisely, for a real number $m$, Liouville's constant is $$ \\sum_{i=0}^\\infty\\frac{1}{m^{i!}}. $$ The series converges only for $1 < m$. However, there is no…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleNumber.html"}, {"id": "Mathlib.Algebra.Homology.Localization", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": 15.466, "z": 10.282, "size": 0.2664, "title": "The category of homological complexes up to quasi-isomorphisms", "summary": "Given a category `C` with homology and any complex shape `c`, we define the category `HomologicalComplexUpToQuasiIso C c` which is the localized category of `HomologicalComplex C c` with respect to quasi-isomorphisms. When `C` is abelian, this will be the derived category of `C` in the particular case of the complex shape `ComplexShape.up ℤ`. Under suitable assumptions on `c` (e.g. chain complexes, or cochain…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Localization.html"}, {"id": "Mathlib.LinearAlgebra.FixedSubmodule", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": 53.079, "z": -9.123, "size": 0.2672, "title": "The fixed submodule of a linear map", "summary": "- `LinearMap.fixedSubmodule`: the submodule of a linear map consisting of its fixed points.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FixedSubmodule.html"}, {"id": "Mathlib.Algebra.Polynomial.Bivariate", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 2, "macro_tier_score": 0.0102, "macro_tier_override": null, "x": 20.508, "z": -96.269, "size": 0.3526, "title": "Bivariate polynomials", "summary": "This file introduces the notation `R[X][Y]` for the polynomial ring `R[X][X]` in two variables, and the notation `Y` for the second variable, in the `Polynomial.Bivariate` scope. It also defines `Polynomial.evalEval` for the evaluation of a bivariate polynomial at a point on the affine plane, which is a ring homomorphism (`Polynomial.evalEvalRingHom`), as well as the abbreviation `CC` to view a constant in the base…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Bivariate.html"}, {"id": "Mathlib.GroupTheory.ClassEquation", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 8.466, "z": 32.339, "size": 0.239, "title": "Class Equation", "summary": "This file establishes the class equation for finite groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/ClassEquation.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.Finite", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 4, "macro_tier_score": 0.3281, "macro_tier_override": null, "x": 18.023, "z": 18.74, "size": 0.3359, "title": "Subgroups", "summary": "This file provides some result on multiplicative and additive subgroups in the finite context.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/Finite.html"}, {"id": "Mathlib.RingTheory.Nilpotent.Exp", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -55.563, "z": -33.733, "size": 0.2632, "title": "Exponential map on algebras", "summary": "This file defines the exponential map `IsNilpotent.exp` on `ℚ`-algebras. The definition of `IsNilpotent.exp a` applies to any element `a` in an algebra over `ℚ`, though it yields meaningful (non-junk) values only when `a` is nilpotent. The main result is `IsNilpotent.exp_add_of_commute`, which establishes the expected connection between the additive and multiplicative structures of `A` for commuting nilpotent…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Nilpotent/Exp.html"}, {"id": "Mathlib.Algebra.Star.Pointwise", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 2, "macro_tier_score": 0.0317, "macro_tier_override": null, "x": 14.264, "z": 17.123, "size": 0.3091, "title": "Pointwise star operation on sets", "summary": "This file defines the star operation pointwise on sets and provides the basic API. Besides basic facts about how the star operation acts on sets (e.g., `(s ∩ t)⋆ = s⋆ ∩ t⋆`), if `s t : Set α`, then under suitable assumption on `α`, it is shown * `(s + t)⋆ = s⋆ + t⋆` * `(s * t)⋆ = t⋆ + s⋆` * `(s⁻¹)⋆ = (s⋆)⁻¹`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/Pointwise.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Card", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 20.011, "z": -84.962, "size": 0.2, "title": "Cardinal of the general linear group over finite rings", "summary": "This file computes the cardinal of the general linear group over finite rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Card.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Rank", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": -22.201, "z": 82.494, "size": 0.2814, "title": "Rank of matrices", "summary": "The rank of a matrix `A` is defined to be the rank of range of the linear map corresponding to `A`. This definition does not depend on the choice of basis, see `Matrix.rank_eq_finrank_range_toLin`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Rank.html"}, {"id": "Mathlib.RingTheory.PowerSeries.GaussNorm", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0027, "macro_tier_override": null, "x": -40.515, "z": 57.784, "size": 0.2899, "title": "Gauss norm for power series", "summary": "This file defines the Gauss norm for power series using the gaussNorm for multivariate power series. Given a power series `f` in `R⟦X⟧`, a function `v : R → ℝ` and a real number `c`, the Gauss norm is defined as the supremum of the set of all values of `v (f.coeff i) * c ^ i` for all `i : ℕ`. In case `f` is a polynomial, `v` is a non-negative function with `v 0 = 0` and `c ≥ 0`, `f.gaussNorm v c` reduces to the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/GaussNorm.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Order", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 2, "macro_tier_score": 0.0152, "macro_tier_override": null, "x": -37.843, "z": -57.355, "size": 0.3221, "title": "Formal power series (in one variable) - Order", "summary": "The `PowerSeries.order` of a formal power series `φ` is the multiplicity of the variable `X` in `φ`. If the coefficients form an integral domain, then `PowerSeries.order` is an additive valuation (`PowerSeries.order_mul`, `PowerSeries.min_order_le_order_add`). We prove that if the commutative ring `R` of coefficients is an integral domain, then the ring `R⟦X⟧` of formal power series in one variable over `R` is an…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Order.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.GaussNorm", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": -37.72, "z": 48.303, "size": 0.2754, "title": "Gauss norm for multivariate power series", "summary": "This file defines the Gauss norm for power series. Given a multivariate power series `f`, a function `v : R → ℝ` and a tuple `c` of real numbers, the Gauss norm is defined as the supremum of the set of all values of `v (coeff t f) * ∏ i : t.support, c i` for all `t : σ →₀ ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/GaussNorm.html"}, {"id": "Mathlib.RingTheory.Extension.Generators", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 2, "macro_tier_score": 0.0214, "macro_tier_override": null, "x": -3.459, "z": 85.359, "size": 0.2772, "title": "Generators of algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Extension/Generators.html"}, {"id": "Mathlib.RingTheory.Extension.Basic", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 2, "macro_tier_score": 0.0212, "macro_tier_override": null, "x": 32.586, "z": 76.958, "size": 0.249, "title": "Extension of algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Extension/Basic.html"}, {"id": "Mathlib.Algebra.Prime.Lemmas", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.3932, "macro_tier_override": null, "x": -13.847, "z": -12.376, "size": 0.3814, "title": "Associated, prime, and irreducible elements.", "summary": "In this file we define the predicate `Prime p` saying that an element of a commutative monoid with zero is prime. Namely, `Prime p` means that `p` isn't zero, it isn't a unit, and `p ∣ a * b → p ∣ a ∨ p ∣ b` for all `a`, `b`; In decomposition monoids (e.g., `ℕ`, `ℤ`), this predicate is equivalent to `Irreducible`, however this is not true in general. We also define an equivalence relation `Associated` saying that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Prime/Lemmas.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Cartan", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 56.799, "z": 27.587, "size": 0.278, "title": "Cartan matrices", "summary": "This file defines Cartan matrices for simple Lie algebras, both the exceptional types (E₆, E₇, E₈, F₄, G₂) and the classical infinite families (A, B, C, D).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Cartan.html"}, {"id": "Mathlib.Algebra.Category.Ring.Under.Limits", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 91.486, "z": -41.083, "size": 0.2736, "title": "Limits in `Under R` for a commutative ring `R`", "summary": "We show that `Under.pushout f` is left-exact, i.e. preserves finite limits, if `f : R ⟶ S` is flat.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/Under/Limits.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.RankNullity", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 3, "macro_tier_score": 0.2247, "macro_tier_override": null, "x": -73.546, "z": 10.466, "size": 0.3166, "title": "The rank nullity theorem", "summary": "In this file we provide the rank nullity theorem as a typeclass, and prove various corollaries of the theorem. The main definition is `HasRankNullity.{u} R`, which states that 1. Every `R`-module `M : Type u` has a linear independent subset of cardinality `Module.rank R M`. 2. `rank (M ⧸ N) + rank N = rank M` for every `R`-module `M : Type u` and every `N : Submodule R M`. The following instances are provided in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/RankNullity.html"}, {"id": "Mathlib.Algebra.Module.Submodule.Finsupp", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 4, "macro_tier_score": 0.3443, "macro_tier_override": null, "x": -34.509, "z": -25.174, "size": 0.3064, "title": "Results for pointwise instances on `Submodule`s using Finsupp", "summary": "This file provides the following results in the `Pointwise` locale: When we consider subsets of `R` acting on `M` - `Submodule.pointwiseSetDistribMulAction` : the action described above is distributive. - `Submodule.mem_set_smul` : `x ∈ s • N` iff `x` can be written as `r₀ n₀ + ... + rₖ nₖ` where `rᵢ ∈ s` and `nᵢ ∈ N`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/Finsupp.html"}, {"id": "Mathlib.Algebra.Module.Submodule.Pointwise", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 4, "macro_tier_score": 0.3842, "macro_tier_override": null, "x": 40.045, "z": -8.107, "size": 0.3752, "title": "Pointwise instances on `Submodule`s", "summary": "This file provides: * `Submodule.pointwiseNeg` and the actions * `Submodule.pointwiseDistribMulAction` * `Submodule.pointwiseMulActionWithZero` which matches the action of `Set.mulActionSet`. This file also provides: * `Submodule.pointwiseSetSMulSubmodule`: for `R`-module `M`, a `s : Set R` can act on `N : Submodule R M` by defining `s • N` to be the smallest submodule containing all `a • n` where `a ∈ s` and `n ∈…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/Pointwise.html"}, {"id": "Mathlib.NumberTheory.RatFunc.Ostrowski", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -23.026, "z": -109.025, "size": 0.2, "title": "Ostrowski's theorem for `K(X)`", "summary": "This file proves Ostrowski's theorem for the field of rational functions `K(X)`, where `K` is any field: if `v` is a discrete valuation on `K(X)` which is trivial on elements of `K`, then `v` is equivalent to either the `I`-adic valuation for some `I : HeightOneSpectrum K[X]`, or to the valuation at infinity `FunctionField.inftyValuation K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/RatFunc/Ostrowski.html"}, {"id": "Mathlib.Algebra.Star.TransferInstance", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -22.029, "z": -3.375, "size": 0.239, "title": "Transfer star (algebraic) structures across `Equiv`s", "summary": "This continues the pattern set in `Mathlib/Algebra/Group/TransferInstance.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/TransferInstance.html"}, {"id": "Mathlib.Algebra.BigOperators.Intervals", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.2777, "macro_tier_override": null, "x": 7.233, "z": 19.106, "size": 0.3697, "title": "Results about big operators over intervals", "summary": "We prove results about big operators over intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Intervals.html"}, {"id": "Mathlib.RingTheory.TensorProduct.Finite", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.1343, "macro_tier_override": null, "x": -57.683, "z": 29.963, "size": 0.4084, "title": "Finiteness of the tensor product of (sub)modules", "summary": "In this file we show that the supremum of two subalgebras that are finitely generated as modules, is again finitely generated.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/Finite.html"}, {"id": "Mathlib.Algebra.Order.Hom.TypeTags", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 3, "macro_tier_score": 0.1898, "macro_tier_override": null, "x": -15.896, "z": 5.165, "size": 0.2664, "title": "Order Monoid Isomorphisms on `Additive` and `Multiplicative`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Hom/TypeTags.html"}, {"id": "Mathlib.Algebra.Group.Equiv.TypeTags", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4534, "macro_tier_override": null, "x": -4.637, "z": -12.145, "size": 0.3396, "title": "Additive and multiplicative equivalences associated to `Multiplicative` and `Additive`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Equiv/TypeTags.html"}, {"id": "Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -50.355, "z": -19.106, "size": 0.239, "title": "The Klein Four subgroup of an alternating group on 4 letters", "summary": "Let `α` be a finite type such that `Nat.card α = 4`. * `alternatingGroup.kleinFour` : the subgroup of `alternatingGroup α` generated by permutations of cycle type (2, 2).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/SpecificGroups/Alternating/KleinFour.html"}, {"id": "Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 34.112, "z": -39.248, "size": 0.2442, "title": "Centralizer of an element in the alternating group", "summary": "Given a finite type `α`, our goal is to compute the cardinality of conjugacy classes in `alternatingGroup α`. * `AlternatingGroup.card_of_cycleType_mul_eq m` and `AlternatingGroup.card_of_cycleType m` compute the number of even permutations of given cycle type. * `Equiv.Perm.OnCycleFactors.odd_of_centralizer_le_alternatingGroup` : if `Subgroup.centralizer {g} ≤ alternatingGroup α`, then all members of the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/SpecificGroups/Alternating/Centralizer.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.Ext.TStructure", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 34.106, "z": -39.253, "size": 0.2478, "title": "Morphisms between bounded complexes are small", "summary": "Let `C` be an abelian category. Assuming `HasExt.{w} C`, we show that if two cochain complexes `K` and `L` are cohomologically in a single degree, then the type of morphisms from `K` to `L⟦n⟧` in the derived category is `w`-small for any `n : ℤ`, which we phrase here by saying that `HasSmallLocalizedShiftedHom.{w} (HomologicalComplex.quasiIso _ _) ℤ K L` hold.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/Ext/TStructure.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.KInjective", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 42.262, "z": -51.806, "size": 0.2472, "title": "Morphisms to K-injective complexes in the derived category", "summary": "In this file, we show that if `L : CochainComplex C ℤ` is K-injective, then for any `K : HomotopyCategory C (.up ℤ)`, the functor `DerivedCategory.Qh` induces a bijection from the type of morphisms `K ⟶ (HomotopyCategory.quotient _ _).obj L)` (i.e. homotopy classes of morphisms of cochain complexes) to the type of morphisms in the derived category. We obtain that a morphism between `K`-injective cochain complexes is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/KInjective.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 3.512, "z": 64.906, "size": 0.2478, "title": "Cochains from or to single complexes", "summary": "We introduce constructors `Cochain.fromSingleMk` and `Cocycle.fromSingleMk` for cochains and cocycles from a single complex. We also introduce similar definitions for cochains and cocycles to a single complex.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/HomComplexSingle.html"}, {"id": "Mathlib.Algebra.Group.Graph", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.3615, "macro_tier_override": null, "x": 22.721, "z": 8.163, "size": 0.4016, "title": "Vertical line test for group homs", "summary": "This file proves the vertical line test for monoid homomorphisms/isomorphisms. Let `f : G → H × I` be a homomorphism to a product of monoids. Assume that `f` is surjective on the first factor and that the image of `f` intersects every \"vertical line\" `{(h, i) | i : I}` at most once. Then the image of `f` is the graph of some monoid homomorphism `f' : H → I`. Furthermore, if `f` is also surjective on the second…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Graph.html"}, {"id": "Mathlib.RingTheory.WittVector.Defs", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": 16.578, "z": 104.552, "size": 0.2808, "title": "Witt vectors", "summary": "In this file we define the type of `p`-typical Witt vectors and ring operations on it. The ring axioms are verified in `Mathlib/RingTheory/WittVector/Basic.lean`. For a fixed commutative ring `R` and prime `p`, a Witt vector `x : 𝕎 R` is an infinite sequence `ℕ → R` of elements of `R`. However, the ring operations `+` and `*` are not defined in the obvious component-wise way. Instead, these operations are defined…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/Defs.html"}, {"id": "Mathlib.RingTheory.WittVector.StructurePolynomial", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": 99.264, "z": -31.029, "size": 0.3003, "title": "Witt structure polynomials", "summary": "In this file we prove the main theorem that makes the whole theory of Witt vectors work. Briefly, consider a polynomial `Φ : MvPolynomial idx ℤ` over the integers, with polynomials variables indexed by an arbitrary type `idx`. Then there exists a unique family of polynomials `φ : ℕ → MvPolynomial (idx × ℕ) Φ` such that for all `n : ℕ` we have (`wittStructureInt_existsUnique`) ``` bind₁ φ (wittPolynomial p ℤ n) =…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/StructurePolynomial.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Ordered", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 42.451, "z": -36.084, "size": 0.286, "title": "Ordered modules as affine spaces", "summary": "In this file we prove some theorems about `slope` and `lineMap` in the case when the module `E` acting on the codomain `PE` of a function is an ordered module over its domain `k`. We also prove inequalities that can be used to link convexity of a function on an interval to monotonicity of the slope, see section docstring below for details.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Ordered.html"}, {"id": "Mathlib.Algebra.BigOperators.Expect", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.004, "macro_tier_override": null, "x": -26.443, "z": -23.364, "size": 0.3774, "title": "Average over a finset", "summary": "This file defines `Finset.expect`, the average (aka expectation) of a function over a finset.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Expect.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.Finset.Indicator", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 2, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": 14.363, "z": -8.549, "size": 0.3247, "title": "Interaction of big operators with indicator functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Finset/Indicator.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Hermitian", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -60.03, "z": -52.666, "size": 0.2807, "title": "Hermitian matrices", "summary": "This file defines Hermitian matrices and some basic results about them. See also `IsSelfAdjoint`, which generalizes this definition to other star rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Hermitian.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Torsion", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 2, "macro_tier_score": 0.0101, "macro_tier_override": null, "x": 31.566, "z": 0.605, "size": 0.2553, "title": "Torsion-free monoids with zero", "summary": "We prove that if `M` is an `UniqueFactorizationMonoid` that can be equipped with a `NormalizationMonoid` structure and such that `Mˣ` is torsion-free, then `M` is torsion-free. Note. You need to import this file to get that the monoid of ideals of a Dedekind domain is torsion-free.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Torsion.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 3, "macro_tier_score": 0.2158, "macro_tier_override": null, "x": -29.665, "z": -1.724, "size": 0.4302, "title": "Unique factorization and normalization", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/NormalizedFactors.html"}, {"id": "Mathlib.Algebra.Homology.SingleHomology", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 2, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -3.96, "z": -14.32, "size": 0.3226, "title": "The homology of single complexes", "summary": "The main definition in this file is `HomologicalComplex.homologyFunctorSingleIso` which is a natural isomorphism `single C c j ⋙ homologyFunctor C c j ≅ 𝟭 C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/SingleHomology.html"}, {"id": "Mathlib.LinearAlgebra.Basis.Submodule", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 3, "macro_tier_score": 0.2574, "macro_tier_override": null, "x": -5.055, "z": 49.888, "size": 0.3506, "title": "Bases of submodules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Basis/Submodule.html"}, {"id": "Mathlib.RingTheory.Regular.Free", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 53.204, "z": 46.365, "size": 0.2, "title": "Freeness of `QuotSMulTop` by a regular element", "summary": "Let `M` be a finitely presented module over a commutative ring `R`. If `x` is in the Jacobson radical of `R` and `x` is `M`-regular, then `M/xM` is free over `R/(x)` if and only if `M` is free over `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Regular/Free.html"}, {"id": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.MDifferentiable", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 19.077, "z": 115.436, "size": 0.2478, "title": "MDifferentiability of the weight 2 Eisenstein series", "summary": "We show that the weight 2 Eisenstein series `E2` is MDifferentiable (i.e. holomorphic as a function `ℍ → ℂ`). The proof uses the relation between `E2` and the logarithmic derivative of the Dedekind eta function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/EisensteinSeries/E2/MDifferentiable.html"}, {"id": "Mathlib.RingTheory.Ideal.MinimalPrime.Colon", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 2, "macro_tier_score": 0.0095, "macro_tier_override": null, "x": 60.501, "z": 32.579, "size": 0.3047, "title": "Minimal primes over a colon ideal", "summary": "We prove that a minimal prime over an ideal of the form `N.colon {x}` in a Noetherian ring is itself an ideal of the form `N.colon {x'}`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.FreeCommRing", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 65.767, "z": 25.596, "size": 0.2403, "title": "Constructing Ring terms from MvPolynomial", "summary": "This file provides tools for constructing ring terms that can be evaluated to particular `MvPolynomial`s. The main motivation is in model theory. It can be used to construct first-order formulas whose realization is a property of an `MvPolynomial`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/FreeCommRing.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.LeftHomology", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 2, "macro_tier_score": 0.0252, "macro_tier_override": null, "x": 0.48, "z": 1.794, "size": 0.3197, "title": "Left Homology of short complexes", "summary": "Given a short complex `S : ShortComplex C`, which consists of two composable maps `f : X₁ ⟶ X₂` and `g : X₂ ⟶ X₃` such that `f ≫ g = 0`, we shall define here the \"left homology\" `S.leftHomology` of `S`. For this, we introduce the notion of \"left homology data\". Such an `h : S.LeftHomologyData` consists of the data of morphisms `i : K ⟶ X₂` and `π : K ⟶ H` such that `i` identifies `K` with the kernel of `g : X₂ ⟶…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.html"}, {"id": "Mathlib.RepresentationTheory.Maschke", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 82.332, "z": -34.176, "size": 0.2302, "title": "Maschke's theorem", "summary": "We prove **Maschke's theorem** for finite groups, in the formulation that every submodule of a `k[G]` module has a complement, when `k` is a field with `Fintype.card G` invertible in `k`. We do the core computation in greater generality. For any commutative ring `k` in which `Fintype.card G` is invertible, and a `k[G]`-linear map `i : V → W` which admits a `k`-linear retraction `π`, we produce a `k[G]`-linear…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Maschke.html"}, {"id": "Mathlib.Algebra.Group.TypeTags.Finite", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 3, "macro_tier_score": 0.2359, "macro_tier_override": null, "x": -7.492, "z": -8.249, "size": 0.3241, "title": "`Finite`, `Infinite` and `Fintype` are preserved by `Additive` and `Multiplicative`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/TypeTags/Finite.html"}, {"id": "Mathlib.RepresentationTheory.Semisimple", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 66.835, "z": -56.143, "size": 0.2516, "title": "Semisimple representations", "summary": "This file defines the typeclass `IsSemisimpleRepresentation` for semisimple monoid representations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Semisimple.html"}, {"id": "Mathlib.RepresentationTheory.Subrepresentation", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 2, "macro_tier_score": 0.021, "macro_tier_override": null, "x": 24.46, "z": 81.853, "size": 0.3361, "title": "Subrepresentations", "summary": "This file defines subrepresentations of a monoid representation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Subrepresentation.html"}, {"id": "Mathlib.RepresentationTheory.Basic", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 2, "macro_tier_score": 0.0262, "macro_tier_override": null, "x": -28.163, "z": 78.684, "size": 0.3826, "title": "Monoid representations", "summary": "This file introduces monoid representations and their characters and defines a few ways to construct representations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Basic.html"}, {"id": "Mathlib.GroupTheory.FiniteIndexNormalSubgroup", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 12.773, "z": -34.878, "size": 0.2765, "title": "Finite-index normal subgroups", "summary": "This file builds the lattice `FiniteIndexNormalSubgroup G` of finite-index normal subgroups of a group `G`, and its additive version `FiniteIndexNormalAddSubgroup`. This is used primarily in the definition of the profinite completion of a group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FiniteIndexNormalSubgroup.html"}, {"id": "Mathlib.LinearAlgebra.Projectivization.Constructions", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -31.242, "z": -71.471, "size": 0.239, "title": "Dot Product and Cross Product on Projective Spaces", "summary": "This file defines the dot product and cross product on projective spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Projectivization/Constructions.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Indicator", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4021, "macro_tier_override": null, "x": 5.623, "z": 9.62, "size": 0.3142, "title": "Indicator functions and support of a function in groups with zero", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Indicator.html"}, {"id": "Mathlib.GroupTheory.Congruence.Opposite", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.3291, "macro_tier_override": null, "x": -3.979, "z": -12.376, "size": 0.3286, "title": "Congruences on the opposite of a group", "summary": "This file defines the order isomorphism between the congruences on a group `G` and the congruences on the opposite group `Gᵒᵖ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Congruence/Opposite.html"}, {"id": "Mathlib.RingTheory.Flat.FaithfullyFlat.Descent", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -23.262, "z": -99.46, "size": 0.2469, "title": "Properties satisfying faithfully flat descent for rings", "summary": "We show the following properties of ring homomorphisms descend under faithfully flat ring maps: - injective - surjective - bijective", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/FaithfullyFlat/Descent.html"}, {"id": "Mathlib.RingTheory.RingHom.Injective", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 33.219, "z": -66.445, "size": 0.2613, "title": "Meta properties of injective ring homomorphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/Injective.html"}, {"id": "Mathlib.RingTheory.Finiteness.FinitePresentationLocal", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 2, "macro_tier_score": 0.0083, "macro_tier_override": null, "x": -56.179, "z": -80.823, "size": 0.2965, "title": "`Algebra.FinitePresentation` is local", "summary": "In this file we show that being a finitely presented algebra is local.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/FinitePresentationLocal.html"}, {"id": "Mathlib.RingTheory.Localization.Away.AdjoinRoot", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 2, "macro_tier_score": 0.0216, "macro_tier_override": null, "x": -45.368, "z": 85.253, "size": 0.2985, "title": null, "summary": "The `R`-`AlgEquiv` between the localization of `R` away from `r` and `R` with an inverse of `r` adjoined.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Away/AdjoinRoot.html"}, {"id": "Mathlib.Algebra.Order.Floor.Ring", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 1, "macro_tier_score": 0.0045, "macro_tier_override": null, "x": 19.735, "z": -24.644, "size": 0.3463, "title": "Lemmas on `Int.floor`, `Int.ceil` and `Int.fract`", "summary": "This file contains basic results on the integer-valued floor and ceiling functions, as well as the fractional part operator.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Floor/Ring.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughInjectives", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 1, "macro_tier_score": 0.0039, "macro_tier_override": null, "x": 16.185, "z": 53.312, "size": 0.3, "title": "Smallness of Ext-groups from the existence of enough injectives", "summary": "Let `C : Type u` be an abelian category (`Category.{v} C`) that has enough injectives. If `C` is locally `w`-small, i.e. the type of morphisms in `C` are `Small.{w}`, then we show that the condition `HasExt.{w}` holds, which means that for `X` and `Y` in `C`, and `n : ℕ`, we may define `Ext X Y n : Type w`. In particular, this holds for `w = v`. However, the main lemma `hasExt_of_enoughInjectives` is not made an…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/Ext/EnoughInjectives.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.ConjAct", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -7.777, "z": -30.599, "size": 0.239, "title": "Conjugation action of a group with zero on itself", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/ConjAct.html"}, {"id": "Mathlib.RingTheory.Localization.Integral", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 3, "macro_tier_score": 0.1294, "macro_tier_override": null, "x": 28.406, "z": -86.454, "size": 0.4313, "title": "Integral and algebraic elements of a fraction field", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Integral.html"}, {"id": "Mathlib.RingTheory.Localization.LocalizationLocalization", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0934, "macro_tier_override": null, "x": -70.57, "z": 0.535, "size": 0.4101, "title": "Localizations of localizations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/LocalizationLocalization.html"}, {"id": "Mathlib.RingTheory.Spectrum.Maximal.Localization", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 3, "macro_tier_score": 0.0743, "macro_tier_override": null, "x": 12.669, "z": 82.606, "size": 0.343, "title": "Maximal spectrum of a commutative (semi)ring", "summary": "Localization results.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Maximal/Localization.html"}, {"id": "Mathlib.RingTheory.Localization.AsSubring", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.0736, "macro_tier_override": null, "x": 70.234, "z": 17.699, "size": 0.2852, "title": "Localizations of domains as subalgebras of the fraction field.", "summary": "Given a domain `A` with fraction field `K`, and a submonoid `S` of `A` which does not contain zero, this file constructs the localization of `A` at `S` as a subalgebra of the field `K` over `A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/AsSubring.html"}, {"id": "Mathlib.RingTheory.Spectrum.Maximal.Basic", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.1042, "macro_tier_override": null, "x": 51.842, "z": 25.039, "size": 0.3385, "title": "Maximal spectrum of a commutative (semi)ring", "summary": "Basic properties the maximal spectrum of a ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Maximal/Basic.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Idempotent", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.3421, "macro_tier_override": null, "x": -11.417, "z": -6.218, "size": 0.304, "title": "Idempotent elements of a group with zero", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Idempotent.html"}, {"id": "Mathlib.Algebra.Group.ULift", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.3893, "macro_tier_override": null, "x": 3.644, "z": -8.541, "size": 0.4084, "title": "`ULift` instances for groups and monoids", "summary": "This file defines instances for group, monoid, semigroup and related structures on `ULift` types. (Recall `ULift α` is just a \"copy\" of a type `α` in a higher universe.) We also provide `MulEquiv.ulift : ULift R ≃* R` (and its additive analogue).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/ULift.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Subgroup", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 4, "macro_tier_score": 0.4195, "macro_tier_override": null, "x": 19.826, "z": 26.915, "size": 0.3674, "title": "Subgroups in a group with zero", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Subgroup.html"}, {"id": "Mathlib.Algebra.Group.Action.Units", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4682, "macro_tier_override": null, "x": -11.983, "z": 5.04, "size": 0.4945, "title": "Group actions on and by `Mˣ`", "summary": "This file provides the action of a unit on a type `α`, `SMul Mˣ α`, in the presence of `SMul M α`, with the obvious definition stated in `Units.smul_def`. This definition preserves `MulAction` and `DistribMulAction` structures too. Additionally, a `MulAction G M` for some group `G` satisfying some additional properties admits a `MulAction G Mˣ` structure, again with the obvious definition stated in `Units.coe_smul`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Units.html"}, {"id": "Mathlib.Algebra.Polynomial.Eval.SMul", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.1997, "macro_tier_override": null, "x": -40.035, "z": -41.374, "size": 0.348, "title": "Evaluating polynomials and scalar multiplication", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Eval/SMul.html"}, {"id": "Mathlib.Algebra.Ring.Semiconj", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.4191, "macro_tier_override": null, "x": -8.917, "z": -2.59, "size": 0.3408, "title": "Semirings and rings", "summary": "This file gives lemmas about semirings, rings and domains. This is analogous to `Mathlib/Algebra/Group/Basic.lean`, the difference being that the former is about `+` and `*` separately, while the present file is about their interaction. For the definitions of semirings and rings see `Mathlib/Algebra/Ring/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Semiconj.html"}, {"id": "Mathlib.Algebra.Group.Semiconj.Defs", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.5107, "macro_tier_override": null, "x": 3.286, "z": 4.499, "size": 0.7406, "title": "Semiconjugate elements of a semigroup", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Semiconj/Defs.html"}, {"id": "Mathlib.RingTheory.Valuation.Integral", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -94.524, "z": -6.024, "size": 0.2385, "title": "Integral elements over the ring of integers of a valuation", "summary": "The ring of integers is integrally closed inside the original ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/Integral.html"}, {"id": "Mathlib.RingTheory.Valuation.ValuationRing", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.0607, "macro_tier_override": null, "x": -48.431, "z": -43.353, "size": 0.324, "title": "Valuation Rings", "summary": "A valuation ring is a domain such that for every pair of elements `a b`, either `a` divides `b` or vice-versa. Any valuation ring induces a natural valuation on its fraction field, as we show in this file. Namely, given the following instances: `[CommRing A] [IsDomain A] [ValuationRing A] [Field K] [Algebra A K] [IsFractionRing A K]`, there is a natural valuation `Valuation A K` on `K` with values in `value_group A…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/ValuationRing.html"}, {"id": "Mathlib.NumberTheory.Cyclotomic.Basic", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 2, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 103.358, "z": 62.365, "size": 0.2749, "title": "Cyclotomic extensions", "summary": "Let `A` and `B` be commutative rings with `Algebra A B`. For `S : Set ℕ`, we define a class `IsCyclotomicExtension S A B` expressing the fact that `B` is obtained from `A` by adding `n`-th primitive roots of unity, for all nonzero `n ∈ S`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Cyclotomic/Basic.html"}, {"id": "Mathlib.RingTheory.HopfAlgebra.GroupLike", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -51.785, "z": -74.829, "size": 0.2, "title": "Group-like elements in a Hopf algebra", "summary": "This file proves that group-like elements in a Hopf algebra form a group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HopfAlgebra/GroupLike.html"}, {"id": "Mathlib.RingTheory.HopfAlgebra.Basic", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 1, "macro_tier_score": 0.003, "macro_tier_override": null, "x": 43.356, "z": -50.894, "size": 0.3209, "title": "Hopf algebras", "summary": "In this file we define `HopfAlgebra`, and provide instances for: * Commutative semirings: `CommSemiring.toHopfAlgebra`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HopfAlgebra/Basic.html"}, {"id": "Mathlib.RingTheory.Bialgebra.GroupLike", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -66.38, "z": -59.5, "size": 0.2478, "title": "Group-like elements in a bialgebra", "summary": "This file proves that group-like elements in a bialgebra form a monoid.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Bialgebra/GroupLike.html"}, {"id": "Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": 99.833, "z": 21.606, "size": 0.256, "title": "Cardinality of a transcendence basis", "summary": "This file concerns the cardinality of a transcendence basis.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AlgebraicIndependent/RankAndCardinality.html"}, {"id": "Mathlib.RingTheory.LinearDisjoint", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -0.856, "z": -89.14, "size": 0.256, "title": "Linearly disjoint subalgebras", "summary": "This file contains basics about linearly disjoint subalgebras. We adapt the definitions in . See the file `Mathlib/LinearAlgebra/LinearDisjoint.lean` for details.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LinearDisjoint.html"}, {"id": "Mathlib.GroupTheory.Congruence.BigOperators", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 3, "macro_tier_score": 0.1414, "macro_tier_override": null, "x": 23.903, "z": 10.23, "size": 0.3075, "title": "Interactions between `∑, ∏` and `(Add)Con`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Congruence/BigOperators.html"}, {"id": "Mathlib.GroupTheory.Congruence.Basic", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.3487, "macro_tier_override": null, "x": -18.566, "z": 0.445, "size": 0.3721, "title": "Congruence relations", "summary": "This file proves basic properties of the quotient of a type by a congruence relation. The second half of the file concerns congruence relations on monoids, in which case the quotient by the congruence relation is also a monoid. There are results about the universal property of quotients of monoids, and the isomorphism theorems for monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Congruence/Basic.html"}, {"id": "Mathlib.RingTheory.DiscreteValuationRing.Basic", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 3, "macro_tier_score": 0.0597, "macro_tier_override": null, "x": -57.01, "z": -47.628, "size": 0.3291, "title": "Discrete valuation rings", "summary": "This file defines discrete valuation rings (DVRs) and develops a basic interface for them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DiscreteValuationRing/Basic.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.Inverse", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 2, "macro_tier_score": 0.0078, "macro_tier_override": null, "x": 58.476, "z": 28.385, "size": 0.2395, "title": "Formal (multivariate) power series - Inverses", "summary": "This file defines multivariate formal power series and develops the basic properties of these objects, when it comes about multiplicative inverses. For `φ : MvPowerSeries σ R` and `u : Rˣ` is the constant coefficient of `φ`, `MvPowerSeries.invOfUnit φ u` is a formal power series such, and `MvPowerSeries.mul_invOfUnit` proves that `φ * invOfUnit φ u = 1`. The construction of the power series `invOfUnit` is done by…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/Inverse.html"}, {"id": "Mathlib.RingTheory.PowerSeries.NoZeroDivisors", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.008, "macro_tier_override": null, "x": -66.733, "z": 22.959, "size": 0.262, "title": "Power series over rings with no zero divisors", "summary": "This file proves, using the properties of orders of power series, that `R⟦X⟧` is an integral domain when `R` is. We then state various results about `R⟦X⟧` with `R` an integral domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/NoZeroDivisors.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 3, "macro_tier_score": 0.2017, "macro_tier_override": null, "x": 33.226, "z": 3.681, "size": 0.3368, "title": "Unique factorization and multiplicity", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/Multiplicity.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Comap", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 49.592, "z": -39.086, "size": 0.2, "title": "`comap` operation on `MvPolynomial`", "summary": "This file defines the `comap` function on `MvPolynomial`. `MvPolynomial.comap` is a low-tech example of a map of \"algebraic varieties,\" modulo the fact that `mathlib` does not yet define varieties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Comap.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Rename", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 3, "macro_tier_score": 0.2733, "macro_tier_override": null, "x": 59.222, "z": 15.773, "size": 0.3753, "title": "Renaming variables of polynomials", "summary": "This file establishes the `rename` operation on multivariate polynomials, which modifies the set of variables.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Rename.html"}, {"id": "Mathlib.RingTheory.Regular.Depth", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -88.813, "z": -7.672, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Regular/Depth.html"}, {"id": "Mathlib.RingTheory.Regular.LinearMap", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -61.106, "z": 62.33, "size": 0.274, "title": "Hom(N,M) is subsingleton iff there exists a smul regular element of M in ann(N)", "summary": "Let `M` and `N` be `R`-modules. In this section we prove that `Hom(N,M)` is subsingleton iff there exist `r : R`, such that `IsSMulRegular M r` and `r ∈ ann(N)`. This is the case if `Depth[I](M) = 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Regular/LinearMap.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Iwasawa", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 40.775, "z": 2.601, "size": 0.2552, "title": "Iwasawa criterion for simplicity", "summary": "- `IwasawaStructure` : the structure underlying the Iwasawa criterion. For a group `G`, this consists of an action of `G` on a type `α` and, for every `a : α`, of a subgroup `T a`, such that the following properties hold: - for all `a`, `T a` is commutative - for all `g : G` and `a : α`, `T (g • a) = MulAut.conj g • T a` - the subgroups `T a` generate `G` We then prove two versions of the Iwasawa criterion when…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Iwasawa.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Cardinal", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 2, "macro_tier_score": 0.0101, "macro_tier_override": null, "x": 6.237, "z": 66.566, "size": 0.2529, "title": "Cardinality of Multivariate Polynomial Ring", "summary": "The main result in this file is `MvPolynomial.cardinalMk_le_max`, which says that the cardinality of `MvPolynomial σ R` is bounded above by the maximum of `#R`, `#σ` and `ℵ₀`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Cardinal.html"}, {"id": "Mathlib.GroupTheory.Schreier", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -30.902, "z": -41.822, "size": 0.2, "title": "Schreier's Lemma", "summary": "In this file we prove Schreier's lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Schreier.html"}, {"id": "Mathlib.GroupTheory.Commutator.Finite", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 2, "macro_tier_score": 0.0069, "macro_tier_override": null, "x": 35.36, "z": -11.371, "size": 0.2647, "title": null, "summary": "The commutator of a finite direct product is contained in the direct product of the commutators.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Commutator/Finite.html"}, {"id": "Mathlib.Algebra.Order.Group.Action.End", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -12.808, "z": -2.227, "size": 0.239, "title": "Tautological action by relation automorphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Action/End.html"}, {"id": "Mathlib.Algebra.Order.Group.End", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2674, "title": "Relation isomorphisms form a group", "summary": "This file contains `Monoid` instances for `RelHom` and `OrderHom`, where multiplication is given by composition. Likewise there is a `Group` instance for `RelIso`. Because `OrderIso` is an abbreviation for `RelIso`, there is no need for an additional instance.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/End.html"}, {"id": "Mathlib.RingTheory.TensorProduct.IsBaseChangeFree", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.0082, "macro_tier_override": null, "x": -35.449, "z": 61.023, "size": 0.2887, "title": "Base change of a free module", "summary": "* `IsBaseChange.basis` : the natural basis of the base change of a module with a basis * `IsBaseChange.free` : a base change of a free module is free.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/IsBaseChangeFree.html"}, {"id": "Mathlib.RingTheory.NormTrace", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -83.86, "z": -30.233, "size": 0.2, "title": "Relation between norms and traces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/NormTrace.html"}, {"id": "Mathlib.Algebra.Algebra.Bilinear", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 4, "macro_tier_score": 0.3483, "macro_tier_override": null, "x": 17.179, "z": 49.081, "size": 0.4072, "title": "Facts about algebras involving bilinear maps and tensor products", "summary": "We move a few basic statements about algebras out of `Algebra.Algebra.Basic`, in order to avoid importing `LinearAlgebra.BilinearMap` and `LinearAlgebra.TensorProduct` unnecessarily.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Bilinear.html"}, {"id": "Mathlib.Algebra.Order.Group.Lattice", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4215, "macro_tier_override": null, "x": 18.477, "z": -1.875, "size": 0.3556, "title": "Lattice ordered groups", "summary": "Lattice ordered groups were introduced by [Birkhoff][birkhoff1942]. They form the algebraic underpinnings of vector lattices, Banach lattices, AL-space, AM-space etc. A lattice ordered group is a type `α` satisfying: * `Lattice α` * `CommGroup α` * `MulLeftMono α` * `MulRightMono α` This file establishes basic properties of lattice ordered groups. It is shown that when the group is commutative, the lattice is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Lattice.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 2, "macro_tier_score": 0.0202, "macro_tier_override": null, "x": 42.221, "z": 49.421, "size": 0.3575, "title": "Exactness of short complexes in concrete abelian categories", "summary": "If an additive concrete category `C` has an additive forgetful functor to `Ab` which preserves homology, then a short complex `S` in `C` is exact if and only if it is so after applying the functor `forget₂ C Ab`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.Ab", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 2, "macro_tier_score": 0.0196, "macro_tier_override": null, "x": -55.717, "z": -29.711, "size": 0.3169, "title": "Homology and exactness of short complexes of abelian groups", "summary": "In this file, the homology of a short complex `S` of abelian groups is identified with the quotient of `AddMonoidHom.ker S.g` by the image of the morphism `S.abToCycles : S.X₁ →+ AddMonoidHom.ker S.g` induced by `S.f`. The definitions are made in the `ShortComplex` namespace so as to enable dot notation. The names contain the prefix `ab` in order to allow similar constructions for other categories like `ModuleCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/Ab.html"}, {"id": "Mathlib.Algebra.Category.Ring.Limits", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 3, "macro_tier_score": 0.0707, "macro_tier_override": null, "x": 4.737, "z": 25.565, "size": 0.3231, "title": "The category of (commutative) rings has all limits", "summary": "Further, these limits are preserved by the forgetful functor --- that is, the underlying types are just the limits in the category of types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/Limits.html"}, {"id": "Mathlib.GroupTheory.DoubleCoset", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 31.79, "z": -10.339, "size": 0.2, "title": "Double cosets", "summary": "This file defines double cosets for two subgroups `H K` of a group `G` and the quotient of `G` by the double coset relation, i.e. `H \\ G / K`. We also prove that `G` can be written as a disjoint union of the double cosets and that if one of `H` or `K` is the trivial group (i.e. `⊥` ) then this is the usual left or right quotient of a group by a subgroup.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/DoubleCoset.html"}, {"id": "Mathlib.RingTheory.LocalProperties.IntegrallyClosed", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 3, "macro_tier_score": 0.0494, "macro_tier_override": null, "x": -57.129, "z": -75.547, "size": 0.3062, "title": "`IsIntegrallyClosed` is a local property", "summary": "In this file, we prove that `IsIntegrallyClosed` is a local property.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalProperties/IntegrallyClosed.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Unbundled.WithTop", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.3962, "macro_tier_override": null, "x": 9.886, "z": -8.443, "size": 0.4602, "title": "Adjoining top/bottom elements to ordered monoids.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Unbundled/WithTop.html"}, {"id": "Mathlib.Algebra.Order.Star.Prod", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -9.306, "z": -28.22, "size": 0.2691, "title": "Products of star-ordered rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Star/Prod.html"}, {"id": "Mathlib.LinearAlgebra.Transvection.Basic", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 1, "macro_tier_score": 0.0039, "macro_tier_override": null, "x": 38.246, "z": -90.695, "size": 0.3032, "title": "Transvections in a module", "summary": "* When `f : Module.Dual R V` and `v : V`, `LinearMap.transvection f v` is the linear map given by `x ↦ x + f x • v`, * `LinearMap.transvection.det` shows that the determinant of `LinearMap.transvection f v` is equal to `1 + f v`. * If, moreover, `f v = 0`, then `LinearEquiv.transvection` shows that it is a linear equivalence. * `LinearMap.transvections R V`: the set of transvections. * `LinearEquiv.dilatransvections…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Transvection/Basic.html"}, {"id": "Mathlib.Algebra.Polynomial.Smeval", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 2, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": 45.36, "z": -43.927, "size": 0.2303, "title": "Scalar-multiple polynomial evaluation", "summary": "This file defines polynomial evaluation via scalar multiplication. Our polynomials have coefficients in a semiring `R`, and we evaluate at a weak form of `R`-algebra, namely an additive commutative monoid with an action of `R` and a notion of natural number power. This is a generalization of `Algebra.Polynomial.Eval`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Smeval.html"}, {"id": "Mathlib.RingTheory.Derivation.Lie", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 1, "macro_tier_score": 0.0029, "macro_tier_override": null, "x": -75.616, "z": -19.142, "size": 0.3106, "title": "Lie Algebra Structure on Derivations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Derivation/Lie.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Free", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 8.097, "z": 70.106, "size": 0.2338, "title": "Exact sequences with free modules", "summary": "This file proves results about linear independence and span in exact sequences of modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Free.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.Support", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -7.825, "z": -32.5, "size": 0.2783, "title": "Supports of submonoids", "summary": "Let `G` be an (additive) group, and let `M` be a submonoid of `G`. The *support* of `M` is `M ∩ -M`, the largest subgroup of `G` contained in `M`. A submonoid `C` is *pointed*, or a *positive cone*, if it has zero support. A submonoid `C` is *spanning* if the subgroup it generates is `G` itself. The names for these concepts are taken from the theory of convex cones.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/Support.html"}, {"id": "Mathlib.RingTheory.WittVector.Teichmuller", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -109.479, "z": -4.53, "size": 0.2498, "title": "Teichmüller lifts", "summary": "This file defines `WittVector.teichmuller`, a monoid hom `R →* 𝕎 R`, which embeds `r : R` as the `0`-th component of a Witt vector whose other coefficients are `0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/Teichmuller.html"}, {"id": "Mathlib.RingTheory.WittVector.Basic", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 1, "macro_tier_score": 0.0032, "macro_tier_override": null, "x": 18.761, "z": -106.069, "size": 0.3337, "title": "Witt vectors", "summary": "This file verifies that the ring operations on `WittVector p R` satisfy the axioms of a commutative ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/Basic.html"}, {"id": "Mathlib.Algebra.BigOperators.Option", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0009, "macro_tier_override": null, "x": 13.128, "z": 10.346, "size": 0.3269, "title": "Lemmas about products and sums over finite sets in `Option α`", "summary": "In this file we prove formulas for products and sums over `Finset.insertNone s` and `Finset.eraseNone s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Option.html"}, {"id": "Mathlib.LinearAlgebra.Alternating.Basic", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.2073, "macro_tier_override": null, "x": 14.278, "z": 57.688, "size": 0.3417, "title": "Alternating Maps", "summary": "We construct the bundled function `AlternatingMap`, which extends `MultilinearMap` with all the arguments of the same type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Alternating/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Multilinear.Basis", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.2068, "macro_tier_override": null, "x": 47.571, "z": -32.428, "size": 0.3039, "title": "Multilinear maps in relation to bases.", "summary": "This file proves lemmas about the action of multilinear maps on basis vectors and constructs a basis for multilinear maps given bases on the domain and codomain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Multilinear/Basis.html"}, {"id": "Mathlib.NumberTheory.Zsqrtd.Basic", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 2, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": -45.862, "z": -43.403, "size": 0.2844, "title": "ℤ[√d]", "summary": "The ring of integers adjoined with a square root of `d : ℤ`. After defining the norm, we show that it is a linearly ordered commutative ring, as well as an integral domain. We provide the universal property, that ring homomorphisms `ℤ√d →+* R` correspond to choices of square roots of `d` in `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Zsqrtd/Basic.html"}, {"id": "Mathlib.LinearAlgebra.UnitaryGroup", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -70.175, "z": -34.052, "size": 0.351, "title": "The Unitary Group", "summary": "This file defines elements of the unitary group `Matrix.unitaryGroup n α`, where `α` is a `StarRing`. This consists of all `n` by `n` matrices with entries in `α` such that the star-transpose is its inverse. In addition, we define the group structure on `Matrix.unitaryGroup n α`, and the embedding into the general linear group `LinearMap.GeneralLinearGroup α (n → α)`. We also define the orthogonal group…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/UnitaryGroup.html"}, {"id": "Mathlib.LinearAlgebra.Eigenspace.Pi", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -99.745, "z": 45.356, "size": 0.2771, "title": "Simultaneous eigenvectors and eigenvalues for families of endomorphisms", "summary": "In finite dimensions, the theory of simultaneous eigenvalues for a family of linear endomorphisms `i ↦ f i` enjoys similar properties to that of a single endomorphism, provided the family obeys a compatibility condition. This condition is that the maximum generalised eigenspaces of each endomorphism are invariant under the action of all members of the family. It is trivially satisfied for commuting endomorphisms but…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Eigenspace/Pi.html"}, {"id": "Mathlib.LinearAlgebra.Eigenspace.Triangularizable", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 1, "macro_tier_score": 0.0041, "macro_tier_override": null, "x": 102.73, "z": -32.391, "size": 0.3151, "title": "Triangularizable linear endomorphisms", "summary": "This file contains basic results relevant to the triangularizability of linear endomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Eigenspace/Triangularizable.html"}, {"id": "Mathlib.LinearAlgebra.BilinearForm.Hom", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 3, "macro_tier_score": 0.0452, "macro_tier_override": null, "x": 53.18, "z": 8.517, "size": 0.3254, "title": "Bilinear form and linear maps", "summary": "This file describes the relation between bilinear forms and linear maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/BilinearForm/Hom.html"}, {"id": "Mathlib.LinearAlgebra.BilinearForm.Basic", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 3, "macro_tier_score": 0.0447, "macro_tier_override": null, "x": -0.447, "z": 46.427, "size": 0.2771, "title": "Bilinear form", "summary": "This file defines a bilinear form over a module. Basic ideas such as orthogonality are also introduced, as well as reflexive, symmetric, non-degenerate and alternating bilinear forms. Adjoints of linear maps with respect to a bilinear form are also introduced. A bilinear form on an `R`-(semi)module `M`, is a function from `M × M` to `R`, that is linear in both arguments. Comments will typically abbreviate…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/BilinearForm/Basic.html"}, {"id": "Mathlib.RingTheory.Congruence.BigOperators", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -5.583, "z": 27.292, "size": 0.2617, "title": "Interactions between `∑, ∏` and `RingCon`", "summary": "TODO: some of the typeclass assumptions in this file can be weakened if more instances are added for `RingCon.Quotient`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Congruence/BigOperators.html"}, {"id": "Mathlib.Algebra.ContinuedFractions.Computation.Basic", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -27.15, "z": -12.077, "size": 0.2433, "title": "Computable Continued Fractions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/ContinuedFractions/Computation/Basic.html"}, {"id": "Mathlib.Algebra.Group.WithOne.Map", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 3, "macro_tier_score": 0.2508, "macro_tier_override": null, "x": 6.09, "z": 4.254, "size": 0.2622, "title": "Adjoining a zero/one to semigroups and mapping", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/WithOne/Map.html"}, {"id": "Mathlib.Algebra.Order.AddGroupWithTop", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 3, "macro_tier_score": 0.2525, "macro_tier_override": null, "x": -8.021, "z": -14.664, "size": 0.3219, "title": "Linearly ordered commutative additive groups and monoids with a top element adjoined", "summary": "This file sets up a special class of linearly ordered commutative additive monoids that show up as the target of so-called “valuations” in algebraic number theory. Usually, in the informal literature, these objects are constructed by taking a linearly ordered commutative additive group Γ and formally adjoining a top element: `Γ ∪ {⊤}`. The disadvantage is that a type such as `ENNReal` is not of that form, whereas it…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/AddGroupWithTop.html"}, {"id": "Mathlib.Algebra.Order.Group.Int", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4153, "macro_tier_override": null, "x": -4.513, "z": 12.192, "size": 0.3113, "title": "The integers form a linear ordered group", "summary": "This file contains the instance necessary to show that the integers are a linear ordered additive group. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Int.html"}, {"id": "Mathlib.Algebra.Order.Group.Units", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 3, "macro_tier_score": 0.2508, "macro_tier_override": null, "x": 0.546, "z": -12.989, "size": 0.2622, "title": "The units of an ordered commutative monoid form an ordered commutative group", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Units.html"}, {"id": "Mathlib.Algebra.Order.Monoid.TypeTags", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 3, "macro_tier_score": 0.2508, "macro_tier_override": null, "x": 14.676, "z": 2.317, "size": 0.2622, "title": "Bundled ordered monoid structures on `Multiplicative α` and `Additive α`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/TypeTags.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 7.028, "z": 60.882, "size": 0.2594, "title": "Centroid of a simplex in affine space", "summary": "This file proves some basic properties of the centroid of a simplex in affine space. The definition of the centroid is based on `Finset.univ.centroid` applied to the set of vertices. For convenience, we use `Simplex.centroid` as an abbreviation. This file also defines `faceOppositeCentroid`, which is the centroid of the facet of the simplex obtained by removing one vertex. We prove several relations among the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Simplex/Centroid.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Centroid", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": -49.861, "z": 28.782, "size": 0.2936, "title": "Centroid of a Finite Set of Points in Affine Space", "summary": "This file defines the centroid of a finite set of points in an affine space over a division ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Centroid.html"}, {"id": "Mathlib.Algebra.Module.LocalizedModule.Exact", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 53.744, "z": 3.503, "size": 0.2475, "title": "Localization of modules is an exact functor", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LocalizedModule/Exact.html"}, {"id": "Mathlib.Algebra.Ring.Int.Units", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4139, "macro_tier_override": null, "x": -13.671, "z": -9.616, "size": 0.3617, "title": "Basic lemmas for `ℤˣ`.", "summary": "This file contains lemmas on the units of `ℤ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Int/Units.html"}, {"id": "Mathlib.RingTheory.LittleWedderburn", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 102.251, "z": -44.287, "size": 0.2, "title": "Wedderburn's Little Theorem", "summary": "This file proves Wedderburn's Little Theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LittleWedderburn.html"}, {"id": "Mathlib.Algebra.Star.RingQuot", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -45.305, "z": 16.704, "size": 0.2, "title": "The \\*-ring structure on suitable quotients of a \\*-ring.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/RingQuot.html"}, {"id": "Mathlib.Algebra.RingQuot", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -25.555, "z": -38.763, "size": 0.2478, "title": "Quotients of semirings", "summary": "In this file, we directly define the quotient of a semiring by any relation, by building a bigger relation that represents the ideal generated by that relation. We prove the universal properties of the quotient, and recommend avoiding relying on the actual definition, which is made irreducible for this purpose. Since everything runs in parallel for quotients of `R`-algebras, we do that case at the same time.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/RingQuot.html"}, {"id": "Mathlib.Algebra.BrauerGroup.Defs", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -53.94, "z": 48.337, "size": 0.2, "title": "Definition of Brauer group of a field K", "summary": "We introduce the definition of Brauer group of a field K, which is the quotient of the set of all finite-dimensional central simple algebras over K modulo the Brauer Equivalence where two central simple algebras `A` and `B` are Brauer Equivalent if there exist `n, m ∈ ℕ+` such that `Mₙ(A) ≃ₐ[K] Mₘ(B)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BrauerGroup/Defs.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.Finsupp", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 19.717, "z": 16.948, "size": 0.2, "title": "Connection between `Subgroup.closure` and `Finsupp.prod`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/Finsupp.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.Lattice", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4543, "macro_tier_override": null, "x": -9.555, "z": 15.925, "size": 0.4336, "title": "Lattice structure of subgroups", "summary": "We prove subgroups of a group form a complete lattice. There are also theorems about the subgroups generated by an element or a subset of a group, defined both inductively and as the infimum of the set of subgroups containing a given element/subset. Special thanks goes to Amelia Livingston and Yury Kudryashov for their help and inspiration.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/Lattice.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.Basic", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 1, "macro_tier_score": 0.003, "macro_tier_override": null, "x": 35.124, "z": 105.749, "size": 0.3208, "title": "Root data and root systems", "summary": "This file contains basic results for root systems and root data.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/Basic.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 1, "macro_tier_score": 0.0032, "macro_tier_override": null, "x": -64.1, "z": -88.867, "size": 0.3299, "title": "Nondegeneracy of the polarization on a finite root pairing", "summary": "We show that if the base ring of a finite root pairing is linearly ordered, then the canonical bilinear form is root-positive and positive-definite on the span of roots. From these facts, it is easy to show that Coxeter weights in a finite root pairing are bounded above by 4. Thus, the pairings of roots and coroots in a root pairing are restricted to the interval `[-4, 4]`. Furthermore, a linearly independent pair…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/Finite/Nondegenerate.html"}, {"id": "Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 75.714, "z": -71.3, "size": 0.2683, "title": "Valuations associated to discrete valuation rings", "summary": "Given a discrete valuation ring `A` with field of fractions `K`, the maximal ideal of `A` is a height-one prime, and the associated valuation `(maximalIdeal A).valuation K` is a rank-one discrete valuation on `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/Discrete/IsDiscreteValuationRing.html"}, {"id": "Mathlib.GroupTheory.FiniteAbelian.Basic", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -105.63, "z": 6.942, "size": 0.268, "title": "Structure of finite(ly generated) abelian groups", "summary": "* `AddCommGroup.equiv_free_prod_directSum_zmod` : Any finitely generated abelian group is the product of a power of `ℤ` and a direct sum of some `ZMod (p i ^ e i)` for some prime powers `p i ^ e i`. * `CommGroup.equiv_free_prod_prod_multiplicative_zmod` is a version for multiplicative groups. * `AddCommGroup.equiv_directSum_zmod_of_finite` : Any finite abelian group is a direct sum of some `ZMod (p i ^ e i)` for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FiniteAbelian/Basic.html"}, {"id": "Mathlib.Algebra.Module.PID", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 2, "macro_tier_score": 0.0069, "macro_tier_override": null, "x": -103.097, "z": 13.685, "size": 0.2717, "title": "Structure of finitely generated modules over a PID", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/PID.html"}, {"id": "Mathlib.RingTheory.Polynomial.GaussNorm", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -62.294, "z": 36.952, "size": 0.2646, "title": "Gauss norm for polynomials", "summary": "This file defines the Gauss norm for polynomials. Given a polynomial `p` in `R[X]`, a function `v : R → ℝ` and a real number `c`, the Gauss norm is defined as the supremum of the set of all values of `v (p.coeff i) * c ^ i` for all `i` in the support of `p`. This is mostly useful when `v` is an absolute value on `R` and `c` is set to be `1`, in which case the Gauss norm corresponds to the maximum of the absolute…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/GaussNorm.html"}, {"id": "Mathlib.RingTheory.Finiteness.Prod", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 3, "macro_tier_score": 0.0577, "macro_tier_override": null, "x": -14.438, "z": -46.077, "size": 0.3426, "title": "Finitely generated product (sub)modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Prod.html"}, {"id": "Mathlib.LinearAlgebra.Multilinear.Basic", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 3, "macro_tier_score": 0.2179, "macro_tier_override": null, "x": 43.45, "z": 25.028, "size": 0.4253, "title": "Multilinear maps", "summary": "We define multilinear maps as maps from `∀ (i : ι), M₁ i` to `M₂` which are linear in each coordinate. Here, `M₁ i` and `M₂` are modules over a ring `R`, and `ι` is an arbitrary type (although some statements will require it to be a fintype). This space, denoted by `MultilinearMap R M₁ M₂`, inherits a module structure by pointwise addition and multiplication.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Multilinear/Basic.html"}, {"id": "Mathlib.FieldTheory.IsSepClosed", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 3, "macro_tier_score": 0.0442, "macro_tier_override": null, "x": 92.324, "z": 68.808, "size": 0.3297, "title": "Separably Closed Field", "summary": "In this file we define the typeclass for separably closed fields and separable closures, and prove some of their properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/IsSepClosed.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.CartanMatrix", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -19.3, "z": -122.924, "size": 0.2613, "title": "Cartan matrices for root systems", "summary": "This file contains definitions and basic results about Cartan matrices of root pairings / systems.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/CartanMatrix.html"}, {"id": "Mathlib.Algebra.CharZero.Infinite", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": 1.086, "z": -1.507, "size": 0.2834, "title": "A characteristic-zero semiring is infinite", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharZero/Infinite.html"}, {"id": "Mathlib.Algebra.Module.Submodule.Union", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -81.606, "z": 4.214, "size": 0.2481, "title": "Unions of `Submodule`s", "summary": "This file is a home for results about unions of submodules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/Union.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.ZMatrix", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 58.782, "z": 67.017, "size": 0.2359, "title": "Z-matrices", "summary": "A matrix whose off-diagonal entries are all non-positive is known as a Z-matrix. Cartan matrices are examples of Z-matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/ZMatrix.html"}, {"id": "Mathlib.RepresentationTheory.Intertwining", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 2, "macro_tier_score": 0.0175, "macro_tier_override": null, "x": 74.58, "z": -45.352, "size": 0.3233, "title": "Intertwining maps", "summary": "This file gives defines intertwining maps of representations (aka equivariant linear maps).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Intertwining.html"}, {"id": "Mathlib.RingTheory.Trace.Basic", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 2, "macro_tier_score": 0.0323, "macro_tier_override": null, "x": -105.1, "z": -55.509, "size": 0.3533, "title": "Trace for (finite) ring extensions.", "summary": "Suppose we have an `R`-algebra `S` with a finite basis. For each `s : S`, the trace of the linear map given by multiplying by `s` gives information about the roots of the minimal polynomial of `s` over `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Trace/Basic.html"}, {"id": "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 2, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -92.93, "z": 54.467, "size": 0.2698, "title": "Results about `minpoly R x / (X - C x)`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Minpoly/MinpolyDiv.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 3, "macro_tier_score": 0.0374, "macro_tier_override": null, "x": 85.57, "z": -40.605, "size": 0.3202, "title": "The minimal polynomial divides the characteristic polynomial of a matrix.", "summary": "This also includes some miscellaneous results about `minpoly` on matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Charpoly/Minpoly.html"}, {"id": "Mathlib.LinearAlgebra.Vandermonde", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 3, "macro_tier_score": 0.0348, "macro_tier_override": null, "x": -66.139, "z": 44.754, "size": 0.2834, "title": "Vandermonde matrix", "summary": "This file defines the `vandermonde` matrix and gives its determinant. For each `CommRing R`, and function `v : Fin n → R` the matrix `vandermonde v` is defined to be `Fin n` by `Fin n` matrix `V` whose `i`th row is `[1, (v i), (v i)^2, ...]`. This matrix has determinant equal to the product of `v i - v j` over all unordered pairs `i,j`, and therefore is nonsingular if and only if `v` is injective. `vandermonde v` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Vandermonde.html"}, {"id": "Mathlib.GroupTheory.MonoidLocalization.DivPairs", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 22.252, "z": -1.229, "size": 0.2, "title": "Submonoid of pairs with quotient in a submonoid", "summary": "This file defines the submonoid of pairs whose quotient lies in a submonoid of the localization.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/MonoidLocalization/DivPairs.html"}, {"id": "Mathlib.GroupTheory.MonoidLocalization.Maps", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 3, "macro_tier_score": 0.2375, "macro_tier_override": null, "x": -12.471, "z": 16.181, "size": 0.3552, "title": "Mapping properties of monoid localizations", "summary": "Given an `S`-localization map `f : M →* N`, we can define `Submonoid.LocalizationMap.lift`, the homomorphism from `N` induced by a homomorphism from `M` which maps elements of `S` to invertible elements of the codomain. Similarly, given commutative monoids `P, Q`, a submonoid `T` of `P` and a localization map for `T` from `P` to `Q`, then a homomorphism `g : M →* P` such that `g(S) ⊆ T` induces a homomorphism of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/MonoidLocalization/Maps.html"}, {"id": "Mathlib.Algebra.Group.Pointwise.Finset.Interval", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -31.57, "z": 0.341, "size": 0.2, "title": "Pointwise operations on intervals", "summary": "This should be kept in sync with `Mathlib/Algebra/Order/Group/Pointwise/Interval.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pointwise/Finset/Interval.html"}, {"id": "Mathlib.Algebra.Polynomial.Degree.Operations", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 4, "macro_tier_score": 0.2911, "macro_tier_override": null, "x": 41.517, "z": -34.307, "size": 0.5021, "title": "Lemmas for calculating the degree of univariate polynomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Degree/Operations.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.ZPow", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": 21.345, "z": -75.023, "size": 0.2579, "title": "Integer powers of square matrices", "summary": "In this file, we define integer power of matrices, relying on the nonsingular inverse definition for negative powers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/ZPow.html"}, {"id": "Mathlib.FieldTheory.IntermediateField.ExtendRight", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -20.888, "z": 86.662, "size": 0.2, "title": "Extending intermediate fields to a larger extension", "summary": "Given a tower of field extensions `K ⊆ L ⊆ M` and an intermediate field `F` of `L/K`, this file defines `IntermediateField.extendRight F M`, the image of `F` under the inclusion `L ⊆ M`, as an intermediate field of `M/K`. It is canonically isomorphic to `F` as a `K`-algebra. The main motivation is to embed a subextension `F/K` of `L/K` into a larger extension `M/K`. This is useful for instance when one needs `M/K`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/IntermediateField/ExtendRight.html"}, {"id": "Mathlib.FieldTheory.IntermediateField.Basic", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.1351, "macro_tier_override": null, "x": -49.123, "z": 39.674, "size": 0.3947, "title": "Intermediate fields", "summary": "Let `L / K` be a field extension, given as an instance `Algebra K L`. This file defines the type of fields in between `K` and `L`, `IntermediateField K L`. An `IntermediateField K L` is a subfield of `L` which contains (the image of) `K`, i.e. it is a `Subfield L` and a `Subalgebra K L`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/IntermediateField/Basic.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Independent", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 1, "macro_tier_score": 0.0021, "macro_tier_override": null, "x": 48.482, "z": 31.048, "size": 0.3362, "title": "Affine independence", "summary": "This file defines affinely independent families of points.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Independent.html"}, {"id": "Mathlib.Algebra.Order.BigOperators.GroupWithZero.List", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4005, "macro_tier_override": null, "x": -14.075, "z": 9.015, "size": 0.2682, "title": "Big operators on a list in ordered groups with zeros", "summary": "This file contains the results concerning the interaction of list big operators with ordered groups with zeros.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/BigOperators/GroupWithZero/List.html"}, {"id": "Mathlib.Algebra.Homology.GrothendieckAbelian", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 18.159, "z": 3.895, "size": 0.2, "title": "Homological complexes in a Grothendieck abelian category", "summary": "Let `c : ComplexShape ι` be a complex shape with no loop, and such that `Small.{w} ι`. Then, if `C` is a Grothendieck abelian category (with `IsGrothendieckAbelian.{w} C`), the category `HomologicalComplex C c` is Grothendieck abelian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/GrothendieckAbelian.html"}, {"id": "Mathlib.RingTheory.Noetherian.Filter", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -16.517, "z": -51.262, "size": 0.2, "title": "Noetherian modules and finiteness of chains", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Noetherian/Filter.html"}, {"id": "Mathlib.LinearAlgebra.Transvection", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 39.952, "z": 91.985, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Transvection.html"}, {"id": "Mathlib.LinearAlgebra.Projectivization.PSL.PSL2", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.203, "z": -26.62, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Projectivization/PSL/PSL2.html"}, {"id": "Mathlib.GroupTheory.IsPerfect", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 15.62, "z": 49.599, "size": 0.239, "title": "Perfect groups", "summary": "A group `G` is perfect if it equals its commutator subgroup, that is `⁅G, G⁆ = G`. Among the basic results, we show that * a nontrivial perfect group is not solvable (`IsPerfect.not_isSolvable`); * an abelian perfect group is trivial (`IsPerfect.subsingleton_of_isMulCommutative`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/IsPerfect.html"}, {"id": "Mathlib.LinearAlgebra.Projectivization.PSL.Stabilizer", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 109.3, "z": -7.72, "size": 0.239, "title": "Stabilizer of a line in PSL(n, F)", "summary": "This file contains key constructions to prove that `PSL(n, F)` is simple via showing it has an Iwasawa structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Projectivization/PSL/Stabilizer.html"}, {"id": "Mathlib.RingTheory.Coalgebra.CoassocSimps", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.0071, "macro_tier_override": null, "x": -44.142, "z": 36.96, "size": 0.2856, "title": "Tactic to reassociate comultiplication in a coalgebra", "summary": "`coassoc_simps` is a simp set useful to prove tautologies on coalgebras. The general algorithm it follows is to push the associators `TensorProduct.assoc` and commutators `TensorProduct.comm` inwards (to the right) until they cancel against co-multiplications. The simp set makes the following choice of normal form * It regards `TensorProduct.map`, `TensorProduct.assoc`, `TensorProduct.comm` as the primitive…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Coalgebra/CoassocSimps.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Sheaf.LocallyFree", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 19.888, "z": 75.423, "size": 0.2, "title": "Locally Free Sheaves", "summary": "A sheaf of modules is locally free if it is locally isomorphic to a free module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Sheaf/LocallyFree.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.Ext.Map", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": 22.166, "z": -49.085, "size": 0.2255, "title": "Map between Ext groups induced by an exact functor", "summary": "In this file, we define the map `Ext^k (M, N) → Ext^k (F(M), F(N))`, where `F` is an exact functor between abelian categories. # Main Definition and results * `CategoryTheory.Abelian.Ext.mapExactFunctor` : The map between `Ext` induced by `CategoryTheory.LocalizerMorphism.smallShiftedHomMap`. * `CategoryTheory.Functor.mapExtAddHom` : Upgraded of `CategoryTheory.Abelian.Ext.mapExactFunctor` into an additive…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/Ext/Map.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.ExactFunctor", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 43.297, "z": -16.763, "size": 0.2413, "title": "An exact functor induces a functor on derived categories", "summary": "In this file, we show that if `F : C₁ ⥤ C₂` is an exact functor between abelian categories, then there is an induced triangulated functor `F.mapDerivedCategory : DerivedCategory C₁ ⥤ DerivedCategory C₂`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/ExactFunctor.html"}, {"id": "Mathlib.Algebra.Module.LinearMap.Prod", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -24.686, "z": -8.161, "size": 0.2809, "title": "Addition and subtraction are linear maps from the product space", "summary": "Note that these results use `IsLinearMap`, which is mostly discouraged.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LinearMap/Prod.html"}, {"id": "Mathlib.Algebra.Prime.Defs", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.3945, "macro_tier_override": null, "x": -15.707, "z": -5.715, "size": 0.326, "title": "Prime elements", "summary": "In this file we define the predicate `Prime p` saying that an element of a commutative monoid with zero is prime. Namely, `Prime p` means that `p` isn't zero, it isn't a unit, and `p ∣ a * b → p ∣ a ∨ p ∣ b` for all `a`, `b`; In decomposition monoids (e.g., `ℕ`, `ℤ`), this predicate is equivalent to `Irreducible` (see `irreducible_iff_prime`), however this is not true in general.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Prime/Defs.html"}, {"id": "Mathlib.RingTheory.Bezout", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.06, "macro_tier_override": null, "x": -24.856, "z": -58.046, "size": 0.25, "title": "Bézout rings", "summary": "A Bézout ring (Bezout ring) is a ring whose finitely generated ideals are principal. Notable examples include principal ideal rings, valuation rings, and the ring of algebraic integers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Bezout.html"}, {"id": "Mathlib.Algebra.Order.Ring.Synonym", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 4, "macro_tier_score": 0.389, "macro_tier_override": null, "x": -8.492, "z": -3.756, "size": 0.3341, "title": "Ring structure on the order type synonyms", "summary": "Transfer algebraic instances from `R` to `Rᵒᵈ` and `Lex R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Synonym.html"}, {"id": "Mathlib.RepresentationTheory.Homological.GroupHomology.FiniteCyclic", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 72.945, "z": -71.501, "size": 0.2, "title": "Group homology of a finite cyclic group", "summary": "Let `k` be a commutative ring, `G` a group and `A` a `k`-linear `G`-representation. Given endomorphisms `φ, ψ : A ⟶ A` such that `φ ∘ ψ = ψ ∘ φ = 0`, denote by `Chains(A, φ, ψ)` the periodic chain complex `... ⟶ A --φ--> A --ψ--> A --φ--> A --ψ--> A ⟶ 0`. When `G` is finite and generated by `g : G`, then `P := Chains(k[G], N, ρ(g) - Id)` (with `ρ` the left regular representation) is a projective resolution of `k` as…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/GroupHomology/FiniteCyclic.html"}, {"id": "Mathlib.RepresentationTheory.Homological.FiniteCyclic", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 40.02, "z": -89.926, "size": 0.2743, "title": "Projective resolution of `k` as a trivial `k`-linear representation of a finite cyclic group", "summary": "Let `k` be a commutative ring and `G` a finite commutative group. Given `g : G` and `A : Rep k G`, we can define a periodic chain complex in `Rep k G` given by `... ⟶ A --N--> A --(ρ(g) - 𝟙)--> A --N--> A --(ρ(g) - 𝟙)--> A ⟶ 0` where `N` is the norm map sending `a : A` to `∑ ρ(g)(a)` for all `g` in `G`. When `G` is generated by `g` and `A` is the left regular representation `k[G]`, this chain complex is a projective…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/FiniteCyclic.html"}, {"id": "Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": -85.892, "z": -51.769, "size": 0.2965, "title": "The low-degree homology of a `k`-linear `G`-representation", "summary": "Let `k` be a commutative ring and `G` a group. This file contains specialised API for the cycles and group homology of a `k`-linear `G`-representation `A` in degrees 0, 1 and 2. In `Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean`, we define the `n`th group homology of `A` to be the homology of a complex `inhomogeneousChains A`, whose objects are `(Fin n →₀ G) → A`; this is unnecessarily unwieldy…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.html"}, {"id": "Mathlib.FieldTheory.Minpoly.Basic", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 3, "macro_tier_score": 0.1464, "macro_tier_override": null, "x": -76.128, "z": -1.523, "size": 0.3451, "title": "Minimal polynomials", "summary": "This file defines the minimal polynomial of an element `x` of an `A`-algebra `B`, under the assumption that x is integral over `A`, and derives some basic properties such as irreducibility under the assumption `B` is a domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Minpoly/Basic.html"}, {"id": "Mathlib.RingTheory.Algebraic.Integral", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 3, "macro_tier_score": 0.1774, "macro_tier_override": null, "x": 79.478, "z": 40.371, "size": 0.4005, "title": "Algebraic elements and integral elements", "summary": "This file relates algebraic and integral elements of an algebra, by proving every integral element is algebraic and that every algebraic element over a field is integral.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Algebraic/Integral.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Lex", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 8.677, "z": -14.286, "size": 0.2338, "title": "Order homomorphisms for products of ordered monoids", "summary": "This file defines order homomorphisms for products of ordered monoids, for both the plain product and the lexicographic product. The product of ordered monoids `α × β` is an ordered monoid itself with both natural inclusions and projections, making it the coproduct as well.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Lex.html"}, {"id": "Mathlib.FieldTheory.Galois.Notation", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 5, "macro_tier_score": 0.1158, "macro_tier_override": 5, "x": -44.069, "z": 6.68, "size": 0.2964, "title": "Notation for Galois group", "summary": "The Galois group `Gal(L/K)` is implemented via `L ≃ₐ[K] L` in mathlib. We provide such a notation in this file. Although this notation works for all automorphism groups of algebras, we should only use this notation when `L/K` is an extension of fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Galois/Notation.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Opposite", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 9.706, "z": -64.272, "size": 0.239, "title": "`MulOpposite` distributes over `⊗`", "summary": "The main result in this file is: * `Algebra.TensorProduct.opAlgEquiv R S A B : Aᵐᵒᵖ ⊗[R] Bᵐᵒᵖ ≃ₐ[S] (A ⊗[R] B)ᵐᵒᵖ`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Opposite.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Canonical.Basic", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 7.72, "z": -12.694, "size": 0.2518, "title": "Extra lemmas about canonically ordered monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Canonical/Basic.html"}, {"id": "Mathlib.Algebra.Order.Group.Action", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4054, "macro_tier_override": null, "x": -8.273, "z": -7.464, "size": 0.3158, "title": "Results about `CovariantClass G α HSMul.hSMul LE.le`", "summary": "When working with group actions rather than modules, we drop the `0 < c` condition. Notably these are relevant for pointwise actions on set-like objects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Action.html"}, {"id": "Mathlib.Algebra.Order.Interval.Basic", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 15.716, "z": -20.713, "size": 0.2, "title": "Interval arithmetic", "summary": "This file defines arithmetic operations on intervals and prove their correctness. Note that this is full precision operations. The essentials of float operations can be found in `Data.FP.Basic`. We have not yet integrated these with the rest of the library.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Interval/Basic.html"}, {"id": "Mathlib.Algebra.Module.Submodule.Map", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 4, "macro_tier_score": 0.392, "macro_tier_override": null, "x": 25.154, "z": 24.746, "size": 0.4679, "title": "`map` and `comap` for `Submodule`s", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/Map.html"}, {"id": "Mathlib.Algebra.Module.Submodule.Basic", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 4, "macro_tier_score": 0.3904, "macro_tier_override": null, "x": 24.405, "z": -22.845, "size": 0.4053, "title": "Submodules of a module", "summary": "This file contains basic results on submodules that require further theory to be defined. As such it is a good target for organizing and splitting further.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/Basic.html"}, {"id": "Mathlib.Algebra.Module.Submodule.LinearMap", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 4, "macro_tier_score": 0.3907, "macro_tier_override": null, "x": 33.066, "z": -4.912, "size": 0.4196, "title": "Linear maps involving submodules of a module", "summary": "In this file we define a number of linear maps involving submodules of a module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/LinearMap.html"}, {"id": "Mathlib.Algebra.Order.Algebra", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -15.605, "z": -23.077, "size": 0.2482, "title": "Ordered algebras", "summary": "An ordered algebra is an ordered semiring, which is an algebra over an ordered commutative semiring, for which scalar multiplication is \"compatible\" with the two orders. The prototypical example is 2x2 matrices over the reals or complexes (or indeed any C^* algebra) where the ordering the one determined by the positive cone of positive operators, i.e. `A ≤ B` iff `B - A = star R * R` for some `R`. (We don't yet have…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Algebra.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.Submonoid", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 3, "macro_tier_score": 0.2307, "macro_tier_override": null, "x": -4.877, "z": -15.987, "size": 0.3445, "title": "The submonoid of positive elements", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/Submonoid.html"}, {"id": "Mathlib.GroupTheory.SemidirectProduct", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 1, "macro_tier_score": 0.0027, "macro_tier_override": null, "x": -37.282, "z": 11.449, "size": 0.2937, "title": "Semidirect product", "summary": "This file defines semidirect products of groups, and the canonical maps in and out of the semidirect product. The semidirect product of `N` and `G` given a hom `φ` from `G` to the automorphism group of `N` is the product of sets with the group `⟨n₁, g₁⟩ * ⟨n₂, g₂⟩ = ⟨n₁ * φ g₁ n₂, g₁ * g₂⟩`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/SemidirectProduct.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Unique", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": "One by one matrices", "summary": "This file proves that one by one matrices over a base are equivalent to the base itself under the canonical map that sends a one by one matrix `!![a]` to `a`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Unique.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.Matrix", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 48.166, "z": 28.004, "size": 0.2, "title": "Matrix subalgebras", "summary": "In this file we define the subalgebra of square matrices with entries in some subalgebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/Matrix.html"}, {"id": "Mathlib.LinearAlgebra.Complex.Determinant", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 79.327, "z": 31.707, "size": 0.2544, "title": "Determinants of maps in the complex numbers as a vector space over `ℝ`", "summary": "This file provides results about the determinants of maps in the complex numbers as a vector space over `ℝ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Complex/Determinant.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.TransferInstance", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.3657, "macro_tier_override": null, "x": -14.566, "z": -2.928, "size": 0.3907, "title": "Transfer algebraic structures across `Equiv`s", "summary": "This continues the pattern set in `Mathlib/Algebra/Group/TransferInstance.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/TransferInstance.html"}, {"id": "Mathlib.Algebra.Order.SuccPred.PartialSups", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.2911, "macro_tier_override": null, "x": 6.706, "z": -15.31, "size": 0.3074, "title": "`PartialSups` in a `SuccAddOrder`", "summary": "Basic results concerning `PartialSups` which follow with minimal assumptions beyond the fact that the `PartialSup` is defined over a `SuccAddOrder`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/SuccPred/PartialSups.html"}, {"id": "Mathlib.RingTheory.Algebraic.Cardinality", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": -20.378, "z": 71.437, "size": 0.2493, "title": "Cardinality of algebraic extensions", "summary": "This file contains results on cardinality of algebraic extensions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Algebraic/Cardinality.html"}, {"id": "Mathlib.Algebra.Polynomial.Cardinal", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 2, "macro_tier_score": 0.0102, "macro_tier_override": null, "x": 32.091, "z": 38.53, "size": 0.267, "title": "Cardinality of Polynomial Ring", "summary": "The result in this file is that the cardinality of `R[X]` is at most the maximum of `#R` and `ℵ₀`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Cardinal.html"}, {"id": "Mathlib.RingTheory.Algebraic.Defs", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.1847, "macro_tier_override": null, "x": -9.366, "z": 62.445, "size": 0.3064, "title": "Algebraic elements and algebraic extensions", "summary": "An element of an R-algebra is algebraic over R if it is the root of a nonzero polynomial. An R-algebra is algebraic over R if and only if all its elements are algebraic over R.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Algebraic/Defs.html"}, {"id": "Mathlib.Algebra.Group.Fin.Basic", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 2, "macro_tier_score": 0.0203, "macro_tier_override": null, "x": -7.215, "z": -1.767, "size": 0.3599, "title": "Fin is a group", "summary": "This file contains the additive and multiplicative monoid instances on `Fin n`. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Fin/Basic.html"}, {"id": "Mathlib.RingTheory.DiscreteValuationRing.TFAE", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 3, "macro_tier_score": 0.0494, "macro_tier_override": null, "x": -97.174, "z": 15.674, "size": 0.3062, "title": "Equivalent conditions for DVR", "summary": "In `IsDiscreteValuationRing.TFAE`, we show that the following are equivalent for a Noetherian local domain that is not a field `(R, m, k)`: - `R` is a discrete valuation ring - `R` is a valuation ring - `R` is a Dedekind domain - `R` is integrally closed with a unique prime ideal - `m` is principal - `dimₖ m/m² = 1` - Every nonzero ideal is a power of `m`. Also see…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DiscreteValuationRing/TFAE.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.Basic", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 3, "macro_tier_score": 0.0642, "macro_tier_override": null, "x": -94.136, "z": 21.555, "size": 0.3343, "title": "Dedekind rings and domains", "summary": "This file defines the notion of a Dedekind ring (domain), as a Noetherian integrally closed commutative ring (domain) of Krull dimension at most one.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/Basic.html"}, {"id": "Mathlib.RingTheory.Ideal.Cotangent", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 3, "macro_tier_score": 0.0584, "macro_tier_override": null, "x": 81.714, "z": -0.461, "size": 0.3156, "title": "The module `I ⧸ I ^ 2`", "summary": "In this file, we provide special API support for the module `I ⧸ I ^ 2`. The official definition is a quotient module of `I`, but the alternative definition as an ideal of `R ⧸ I ^ 2` is also given, and the two are `R`-equivalent as in `Ideal.cotangentEquivIdeal`. Additional support is also given to the cotangent space `m ⧸ m ^ 2` of a local ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Cotangent.html"}, {"id": "Mathlib.Algebra.Star.CentroidHom", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 10.464, "z": -27.811, "size": 0.2, "title": "Centroid homomorphisms on Star Rings", "summary": "When a (nonunital, non-associative) semiring is equipped with an involutive automorphism the center of the centroid becomes a star ring in a natural way and the natural mapping of the centre of the semiring into the centre of the centroid becomes a \\*-homomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/CentroidHom.html"}, {"id": "Mathlib.Algebra.Star.Subsemiring", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -15.076, "z": 23.425, "size": 0.239, "title": "Star subrings", "summary": "A \\*-subring is a subring of a \\*-ring which is closed under `*`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/Subsemiring.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.Defs", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4563, "macro_tier_override": null, "x": 9.817, "z": 5.273, "size": 0.5033, "title": "Subgroups", "summary": "This file defines multiplicative and additive subgroups as an extension of submonoids, in a bundled form. Special thanks goes to Amelia Livingston and Yury Kudryashov for their help and inspiration.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/Defs.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.SInteger", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -10.45, "z": -103.475, "size": 0.2, "title": "`S`-integers and `S`-units of fraction fields of Dedekind domains", "summary": "Let `K` be the field of fractions of a Dedekind domain `R`, and let `S` be a set of prime ideals in the height one spectrum of `R`. An `S`-integer of `K` is defined to have `v`-adic valuation at most one for all primes ideals `v` away from `S`, whereas an `S`-unit of `Kˣ` is defined to have `v`-adic valuation exactly one for all prime ideals `v` away from `S`. This file defines the subalgebra of `S`-integers of `K`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/SInteger.html"}, {"id": "Mathlib.Algebra.Group.Center", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.4466, "macro_tier_override": null, "x": 18.073, "z": -4.273, "size": 0.3919, "title": "Centers of magmas and semigroups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Center.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.Subsingleton", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 3, "macro_tier_score": 0.2567, "macro_tier_override": null, "x": 51.613, "z": -6.339, "size": 0.3, "title": "Dimension of trivial modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/Subsingleton.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.BilinearForm", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 3, "macro_tier_score": 0.0415, "macro_tier_override": null, "x": 48.004, "z": -70.667, "size": 0.2907, "title": "Bilinear form", "summary": "This file defines the conversion between bilinear forms and matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/BilinearForm.html"}, {"id": "Mathlib.Algebra.Divisibility.Finite", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -7.735, "z": 5.137, "size": 0.2, "title": "Divisibility in finite types", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Divisibility/Finite.html"}, {"id": "Mathlib.FieldTheory.IsAlgClosed.Spectrum", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0042, "macro_tier_override": null, "x": 83.525, "z": 65.035, "size": 0.3246, "title": "Spectrum mapping theorem", "summary": "This file develops and proves the spectral mapping theorem for polynomials over algebraically closed fields. In particular, if `a` is an element of a `𝕜`-algebra `A` where `𝕜` is a field, and `p : 𝕜[X]` is a polynomial, then the spectrum of `Polynomial.aeval a p` contains the image of the spectrum of `a` under `(fun k ↦ Polynomial.eval k p)`. When `𝕜` is algebraically closed, these are in fact equal (assuming either…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/IsAlgClosed/Spectrum.html"}, {"id": "Mathlib.LinearAlgebra.Basis.Bilinear", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 3, "macro_tier_score": 0.0469, "macro_tier_override": null, "x": 43.438, "z": 16.395, "size": 0.2769, "title": "Lemmas about bilinear maps with a basis over each argument", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Basis/Bilinear.html"}, {"id": "Mathlib.RingTheory.GradedAlgebra.Homogeneous.Submodule", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -24.602, "z": -56.132, "size": 0.2694, "title": "Homogeneous submodules of a graded module", "summary": "This file defines homogeneous submodule of a graded module `⨁ᵢ ℳᵢ` over graded ring `⨁ᵢ 𝒜ᵢ` and operations on them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/GradedAlgebra/Homogeneous/Submodule.html"}, {"id": "Mathlib.Algebra.GradedMulAction", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 1, "macro_tier_score": 0.0037, "macro_tier_override": null, "x": 15.537, "z": 10.174, "size": 0.2835, "title": "Additively-graded multiplicative action structures", "summary": "This module provides a set of heterogeneous typeclasses for defining a multiplicative structure over the sigma type `GradedMonoid A` such that `(•) : A i → M j → M (i +ᵥ j)`; that is to say, `A` has an additively-graded multiplicative action on `M`. The typeclasses are: * `GradedMonoid.GSMul A M` * `GradedMonoid.GMulAction A M` With the `SigmaGraded` scope open, these respectively imbue: * `SMul (GradedMonoid A)…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GradedMulAction.html"}, {"id": "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.016, "macro_tier_override": null, "x": 10.221, "z": -105.364, "size": 0.2907, "title": "Cyclotomic polynomials.", "summary": "For `n : ℕ` and an integral domain `R`, we define a modified version of the `n`-th cyclotomic polynomial with coefficients in `R`, denoted `cyclotomic' n R`, as `∏ (X - μ)`, where `μ` varies over the primitive `n`th roots of unity. If there is a primitive `n`th root of unity in `R` then this is the standard definition. We then define the standard cyclotomic polynomial `cyclotomic n R` with coefficients in any ring…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.html"}, {"id": "Mathlib.RingTheory.RootsOfUnity.Complex", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.0183, "macro_tier_override": null, "x": 79.744, "z": -4.264, "size": 0.3048, "title": "Complex roots of unity", "summary": "In this file we show that the `n`-th complex roots of unity are exactly the complex numbers `exp (2 * π * I * (i / n))` for `i ∈ Finset.range n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RootsOfUnity/Complex.html"}, {"id": "Mathlib.Algebra.CharP.CharAndCard", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 2, "macro_tier_score": 0.0091, "macro_tier_override": null, "x": 39.036, "z": -25.137, "size": 0.2696, "title": "Characteristic and cardinality", "summary": "We prove some results relating characteristic and cardinality of finite rings", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/CharAndCard.html"}, {"id": "Mathlib.Algebra.CharP.Basic", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 2, "macro_tier_score": 0.0192, "macro_tier_override": null, "x": -14.912, "z": 11.069, "size": 0.3631, "title": "Characteristic of semirings", "summary": "This file collects some fundamental results on the characteristic of rings that don't need the extra imports of `Mathlib/Algebra/CharP/Lemmas.lean`. As such, we can probably reorganize and find a better home for most of these lemmas.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Alternating.DomCoprod", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 23.014, "z": 56.801, "size": 0.2, "title": "Exterior product of alternating maps", "summary": "In this file we define `AlternatingMap.domCoprod` to be the exterior product of two alternating maps, taking values in the tensor product of the codomains of the original maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Alternating/DomCoprod.html"}, {"id": "Mathlib.GroupTheory.Perm.Basic", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 3, "macro_tier_score": 0.2286, "macro_tier_override": null, "x": -2.752, "z": -18.367, "size": 0.2634, "title": "Extra lemmas about permutations", "summary": "This file proves miscellaneous lemmas about `Equiv.Perm`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Multilinear.TensorProduct", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 2, "macro_tier_score": 0.0108, "macro_tier_override": null, "x": -49.858, "z": 14.774, "size": 0.3202, "title": "Constructions relating multilinear maps and tensor products.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Multilinear/TensorProduct.html"}, {"id": "Mathlib.Algebra.Algebra.Opposite", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 4, "macro_tier_score": 0.3551, "macro_tier_override": null, "x": 30.408, "z": -32.589, "size": 0.4937, "title": "Algebra structures on the multiplicative opposite", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Opposite.html"}, {"id": "Mathlib.Algebra.DualNumber", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 2, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 58.063, "z": -12.671, "size": 0.3059, "title": "Dual numbers", "summary": "The dual numbers over `R` are of the form `a + bε`, where `a` and `b` are typically elements of a commutative ring `R`, and `ε` is a symbol satisfying `ε^2 = 0` that commutes with every other element. They are a special case of `TrivSqZeroExt R M` with `M = R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/DualNumber.html"}, {"id": "Mathlib.RingTheory.RingHom.QuasiFinite", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -46.047, "z": 93.252, "size": 0.2303, "title": "The meta properties of quasi-finite ring homomorphisms.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/QuasiFinite.html"}, {"id": "Mathlib.RingTheory.ZariskisMainTheorem", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": 105.353, "z": -30.114, "size": 0.268, "title": "Algebraic Zariski's Main Theorem", "summary": "The statement of Zariski's main theorem is the following: Given a finite type `R`-algebra `S`, and `p` a prime of `S` such that `S` is quasi-finite at `R`, then there exists a `f ∉ p` such that `S[1/f]` is isomorphic to `R'[1/f]` where `R'` is the integral closure of `R` in `S`. We follow https://stacks.math.columbia.edu/tag/00PI and proceed in the following steps 1. `Algebra.ZariskisMainProperty.of_adjoin_eq_top`:…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/ZariskisMainTheorem.html"}, {"id": "Mathlib.Algebra.Module.Submodule.IterateMapComap", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 4, "macro_tier_score": 0.2754, "macro_tier_override": null, "x": 29.518, "z": -25.489, "size": 0.2944, "title": "Iterate maps and comaps of submodules", "summary": "Some preliminary work for establishing the strong rank condition for Noetherian rings. Given two linear maps `f i : N →ₗ[R] M` and a submodule `K : Submodule R N`, we can define `LinearMap.iterateMapComap f i n K : Submodule R N` to be `f⁻¹(i(⋯(f⁻¹(i(K)))))` (`n` times). If `f(K) ≤ i(K)`, then this sequence is non-decreasing (`LinearMap.iterateMapComap_le_succ`). On the other hand, if `f` is surjective, `i` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/IterateMapComap.html"}, {"id": "Mathlib.Algebra.Ring.ULift", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.3813, "macro_tier_override": null, "x": -9.707, "z": 13.607, "size": 0.3378, "title": "`ULift` instances for ring", "summary": "This file defines instances for ring, semiring and related structures on `ULift` types. (Recall `ULift R` is just a \"copy\" of a type `R` in a higher universe.) We also provide `ULift.ringEquiv : ULift R ≃+* R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/ULift.html"}, {"id": "Mathlib.Algebra.Ring.Subring.MulOpposite", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": 12.768, "z": -26.832, "size": 0.2504, "title": "Subring of opposite rings", "summary": "For every ring `R`, we construct an equivalence between subrings of `R` and that of `Rᵐᵒᵖ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Subring/MulOpposite.html"}, {"id": "Mathlib.FieldTheory.RatFunc.Luroth", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -97.413, "z": 45.971, "size": 0.2, "title": "Lüroth's theorem", "summary": "This file proves Lüroth's theorem, which says that for every field `K`, every intermediate field between `K` and the rational function field `K⟮X⟯` is either `K` or isomorphic to `K(X)` as an K-algebra, see `Luroth.algEquiv`. The proof depends on the following lemma on degrees of rational functions: Let `f` be a rational function, i.e. an element in the field `K⟮X⟯`. Let `p` be its numerator and `q` its denominator.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/RatFunc/Luroth.html"}, {"id": "Mathlib.Algebra.Polynomial.Basis", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": 23.246, "z": 44.43, "size": 0.2461, "title": "Basis of a polynomial ring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Basis.html"}, {"id": "Mathlib.FieldTheory.Relrank", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 102.132, "z": 1.579, "size": 0.239, "title": "Relative rank of subfields and intermediate fields", "summary": "This file contains basics about the relative rank of subfields and intermediate fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Relrank.html"}, {"id": "Mathlib.Algebra.Tropical.BigOperators", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -7.583, "z": -18.969, "size": 0.2, "title": "Tropicalization of finitary operations", "summary": "This file provides the \"big-op\" or notation-based finitary operations on tropicalized types. This allows easy conversion between sums to Infs and prods to sums. Results here are important for expressing that evaluation of tropical polynomials are the minimum over a finite piecewise collection of linear functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Tropical/BigOperators.html"}, {"id": "Mathlib.Algebra.Tropical.Basic", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -5.744, "z": -17.661, "size": 0.2585, "title": "Tropical algebraic structures", "summary": "This file defines algebraic structures of the (min-)tropical numbers, up to the tropical semiring. Some basic lemmas about conversion from the base type `R` to `Tropical R` are provided, as well as the expected implementations of tropical addition and tropical multiplication.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Tropical/Basic.html"}, {"id": "Mathlib.RingTheory.Ideal.Pointwise", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.1659, "macro_tier_override": null, "x": -35.714, "z": -47.501, "size": 0.3156, "title": "Pointwise instances on `Ideal`s", "summary": "This file provides the action `Ideal.pointwiseMulAction` which morally matches the action of `mulActionSet` (though here an extra `Ideal.span` is inserted). This action is available in the `Pointwise` locale.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Pointwise.html"}, {"id": "Mathlib.Algebra.Category.AlgCat.Monoidal", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -63.662, "z": -13.122, "size": 0.2612, "title": "The monoidal category structure on R-algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/AlgCat/Monoidal.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.Finite.G2", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -119.271, "z": 35.456, "size": 0.2782, "title": "Properties of the `𝔤₂` root system.", "summary": "The `𝔤₂` root pairing is special enough to deserve its own API. We provide one in this file. As an application we prove the key result that a crystallographic, reduced, irreducible root pairing containing two roots of Coxeter weight three is spanned by this pair of roots (and thus is two-dimensional). This result is usually proved only for pairs of roots belonging to a base (as a corollary of the fact that no node…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/Finite/G2.html"}, {"id": "Mathlib.RingTheory.Invariant.Galois", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 5, "macro_tier_score": 0.0047, "macro_tier_override": 5, "x": -118.153, "z": -44.591, "size": 0.2694, "title": "Invariant Extensions of Rings and Galois Theory", "summary": "Given an extension of rings `B/A` and an action of `G` on `B`, the predicate `Algebra.IsInvariant A B G` states that every fixed point of `B` lies in the image of `A`. This file relates this predicate `Algebra.IsInvariant` to Galois theory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Invariant/Galois.html"}, {"id": "Mathlib.RingTheory.Invariant.Basic", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 2, "macro_tier_score": 0.0149, "macro_tier_override": null, "x": 79.151, "z": -58.511, "size": 0.2969, "title": "Invariant Extensions of Rings", "summary": "Given an extension of rings `B/A` and an action of `G` on `B`, we introduce a predicate `Algebra.IsInvariant A B G` which states that every fixed point of `B` lies in the image of `A`. The main application is in algebraic number theory, where `G := Gal(L/K)` is the Galois group of some finite Galois extension of number fields, and `A := 𝓞K` and `B := 𝓞L` are their ring of integers. This main result in this file…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Invariant/Basic.html"}, {"id": "Mathlib.Algebra.Category.Grp.CartesianMonoidal", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -20.682, "z": -18.662, "size": 0.2357, "title": "Chosen finite products in `GrpCat` and friends", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/CartesianMonoidal.html"}, {"id": "Mathlib.Algebra.Category.Grp.EquivalenceGroupAddGroup", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 2, "macro_tier_score": 0.0191, "macro_tier_override": null, "x": -2.079, "z": -20.323, "size": 0.2695, "title": "Equivalence between `Group` and `AddGroup`", "summary": "This file contains two equivalences: * `groupAddGroupEquivalence` : the equivalence between `GrpCat` and `AddGrpCat` by sending `X : GrpCat` to `Additive X` and `Y : AddGrpCat` to `Multiplicative Y`. * `commGroupAddCommGroupEquivalence` : the equivalence between `CommGrpCat` and `AddCommGrpCat` by sending `X : CommGrpCat` to `Additive X` and `Y : AddCommGrpCat` to `Multiplicative Y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/EquivalenceGroupAddGroup.html"}, {"id": "Mathlib.Algebra.Regular.Defs", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 4, "macro_tier_score": 0.5061, "macro_tier_override": null, "x": -1.596, "z": -0.95, "size": 0.5593, "title": "Regular elements", "summary": "We introduce left-regular, right-regular and regular elements, along with their `to_additive` analogues add-left-regular, add-right-regular and add-regular elements. For monoids where _every_ element is regular, see `IsCancelMul` and nearby typeclasses.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Regular/Defs.html"}, {"id": "Mathlib.Algebra.Notation.Defs", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.5149, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.7184, "title": "Typeclasses for algebraic operations", "summary": "Notation typeclass for `Inv`, the multiplicative analogue of `Neg`. We also introduce notation classes `SMul` and `VAdd` for multiplicative and additive actions. We introduce the notation typeclass `Star` for algebraic structures with a star operation. Note: to accommodate diverse notational preferences, no default notation is provided for `Star.star`. `SMul` is typically, but not exclusively, used for scalar…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Notation/Defs.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Biproducts", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": -47.234, "z": 49.907, "size": 0.2545, "title": "The category of `R`-modules has finite biproducts", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Biproducts.html"}, {"id": "Mathlib.Algebra.Category.Grp.Biproducts", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -17.115, "z": -19.572, "size": 0.2701, "title": "The category of abelian groups has finite biproducts", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Biproducts.html"}, {"id": "Mathlib.Algebra.Group.Commute.Hom", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 1, "macro_tier_score": 0.0039, "macro_tier_override": null, "x": -9.251, "z": 0.806, "size": 0.3031, "title": "Multiplicative homomorphisms respect semiconjugation and commutation.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Commute/Hom.html"}, {"id": "Mathlib.Algebra.CharP.LinearMaps", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -64.453, "z": -8.423, "size": 0.2276, "title": "Characteristic of the ring of linear Maps", "summary": "This file contains properties of the characteristic of the ring of linear maps. The characteristic of the ring of linear maps is determined by its base ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/LinearMaps.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.Localization", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 3, "macro_tier_score": 0.176, "macro_tier_override": null, "x": -64.907, "z": 43.258, "size": 0.3199, "title": "Rank of localization", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/Localization.html"}, {"id": "Mathlib.Algebra.MvPolynomial.PDeriv", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.0301, "macro_tier_override": null, "x": 10.903, "z": -69.725, "size": 0.35, "title": "Partial derivatives of polynomials", "summary": "This file defines the notion of the formal *partial derivative* of a polynomial, the derivative with respect to a single variable. This derivative is not connected to the notion of derivative from analysis. It is based purely on the polynomial exponents and coefficients.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/PDeriv.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Derivation", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 2, "macro_tier_score": 0.0306, "macro_tier_override": null, "x": 67.552, "z": -12.587, "size": 0.3085, "title": "Derivations of multivariate polynomials", "summary": "In this file we prove that a derivation of `MvPolynomial σ R` is determined by its values on all monomials `MvPolynomial.X i`. We also provide a constructor `MvPolynomial.mkDerivation` that builds a derivation from its values on `X i`s and a linear equivalence `MvPolynomial.mkDerivationEquiv` between `σ → A` and `Derivation (MvPolynomial σ R) A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Derivation.html"}, {"id": "Mathlib.RingTheory.HopfAlgebra.Quotient", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -4.608, "z": -76.004, "size": 0.2, "title": "Hopf algebra structure on quotients by Hopf ideals", "summary": "A *Hopf ideal* of an `R`-Hopf algebra `A` is a biideal stable under the antipode. The quotient by a Hopf ideal inherits a Hopf algebra structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HopfAlgebra/Quotient.html"}, {"id": "Mathlib.RingTheory.Bialgebra.Quotient", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 58.446, "z": 36.136, "size": 0.2478, "title": "Bialgebra structure on quotients", "summary": "If `I` is a two-sided ideal of an `R`-bialgebra `A` whose underlying `R`-submodule is a coideal, then the quotient `A ⧸ I` inherits a bialgebra structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Bialgebra/Quotient.html"}, {"id": "Mathlib.NumberTheory.Padics.AddChar", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 80.051, "z": 24.004, "size": 0.2, "title": "Additive characters of `ℤ_[p]`", "summary": "We show that for any complete, ultrametric normed `ℤ_[p]`-algebra `R`, there is a bijection between continuous additive characters `ℤ_[p] → R` and topologically nilpotent elements of `R`, given by sending `κ` to the element `κ 1 - 1`. This is used to define the Mahler transform for `p`-adic measures. Note that if the norm on `R` is not strictly multiplicative, then the condition that `κ 1 - 1` be topologically…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/AddChar.html"}, {"id": "Mathlib.NumberTheory.Padics.MahlerBasis", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 77.431, "z": -26.111, "size": 0.239, "title": "The Mahler basis of continuous functions", "summary": "In this file we introduce the Mahler basis function `mahler k`, for `k : ℕ`, which is the unique continuous map `ℤ_[p] → ℤ_[p]` agreeing with `n ↦ n.choose k` for `n ∈ ℕ`. Using this, we prove Mahler's theorem, showing that for any continuous function `f` on `ℤ_[p]` (valued in a normed `ℤ_[p]`-module `E`), the Mahler series `x ↦ ∑' k, mahler k x • Δ^[n] f 0` converges (uniformly) to `f`, and this construction…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/MahlerBasis.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.3275, "macro_tier_override": null, "x": -10.981, "z": -21.501, "size": 0.4207, "title": "Variations on non-zero divisors in `AddMonoidAlgebra`s", "summary": "This file studies the interaction between typeclass assumptions on two Types `R` and `A` and whether `R[A]` has non-zero zero-divisors. For some background on related questions, see [Kaplansky's Conjectures](https://en.wikipedia.org/wiki/Kaplansky%27s_conjectures), especially the *zero divisor conjecture*. _Conjecture._ Let `K` be a field, and `G` a torsion-free group. The group ring `K[G]` does not contain…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.html"}, {"id": "Mathlib.Algebra.Group.UniqueProds.Basic", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.33, "macro_tier_override": null, "x": 22.28, "z": 0.526, "size": 0.3824, "title": "Unique products and related notions", "summary": "A group `G` has *unique products* if for any two non-empty finite subsets `A, B ⊆ G`, there is an element `g ∈ A * B` that can be written uniquely as a product of an element of `A` and an element of `B`. We call the formalization this property `UniqueProds`. Since the condition requires no property of the group operation, we define it for a Type simply satisfying `Mul`. We also introduce the analogous \"additive\"…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/UniqueProds/Basic.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.Opposite", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.3272, "macro_tier_override": null, "x": 19.608, "z": -5.734, "size": 0.352, "title": "Monoid algebras and the opposite ring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/Opposite.html"}, {"id": "Mathlib.RingTheory.LocalRing.MaximalIdeal.Defs", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 3, "macro_tier_score": 0.1196, "macro_tier_override": null, "x": -15.396, "z": -49.669, "size": 0.3297, "title": "Maximal ideal of local rings", "summary": "We define the maximal ideal of a local ring as the ideal of all nonunits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/MaximalIdeal/Defs.html"}, {"id": "Mathlib.RingTheory.AlgebraicIndependent.Adjoin", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 3, "macro_tier_score": 0.0612, "macro_tier_override": null, "x": 49.895, "z": 80.507, "size": 0.2675, "title": "Algebraic Independence", "summary": "This file concerns adjoining an algebraic independent family to a field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AlgebraicIndependent/Adjoin.html"}, {"id": "Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 3, "macro_tier_score": 0.0952, "macro_tier_override": null, "x": -66.618, "z": -64.689, "size": 0.3284, "title": "Adjoining Elements to Fields", "summary": "This file relates `IntermediateField.adjoin` to `Algebra.adjoin`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/IntermediateField/Adjoin/Algebra.html"}, {"id": "Mathlib.RingTheory.AlgebraicIndependent.Defs", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.0624, "macro_tier_override": null, "x": -61.969, "z": 29.691, "size": 0.2752, "title": "Algebraic Independence", "summary": "This file defines algebraic independence of a family of elements of an `R` algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AlgebraicIndependent/Defs.html"}, {"id": "Mathlib.Algebra.Polynomial.CancelLeads", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.1855, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2782, "title": "Cancel the leading terms of two polynomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/CancelLeads.html"}, {"id": "Mathlib.Algebra.Notation.Prod", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 4, "macro_tier_score": 0.4974, "macro_tier_override": null, "x": -1.854, "z": 0.106, "size": 0.5916, "title": "Arithmetic operators on (pairwise) product types", "summary": "This file provides only the notation for (componentwise) `0`, `1`, `+`, `*`, `•`, `^`, `⁻¹` on (pairwise) product types. See `Mathlib/Algebra/Group/Prod.lean` for the `Monoid` and `Group` instances. There is also an instance of the `Star` notation typeclass, but no default notation is included.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Notation/Prod.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.Torsion.Finite", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -28.867, "z": 68.448, "size": 0.2, "title": "Results relating rank and torsion.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/Torsion/Finite.html"}, {"id": "Mathlib.Algebra.AddTorsor.Defs", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 8.69, "z": 9.669, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/AddTorsor/Defs.html"}, {"id": "Mathlib.NumberTheory.ADEInequality", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": "The inequality `p⁻¹ + q⁻¹ + r⁻¹ > 1`", "summary": "In this file we classify solutions to the inequality `(p⁻¹ + q⁻¹ + r⁻¹ : ℚ) > 1`, for positive natural numbers `p`, `q`, and `r`. The solutions are exactly of the form. * `A' q r := {1,q,r}` * `D' r := {2,2,r}` * `E6 := {2,3,3}`, or `E7 := {2,3,4}`, or `E8 := {2,3,5}` This inequality shows up in Lie theory, in the classification of Dynkin diagrams, root systems, and semisimple Lie algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ADEInequality.html"}, {"id": "Mathlib.RingTheory.Valuation.LocalSubring", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 7.73, "z": 98.126, "size": 0.2403, "title": "Valuation subrings are exactly the maximal local subrings", "summary": "See `LocalSubring.isMax_iff`. Note that the order on local subrings is not merely inclusion but domination.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/LocalSubring.html"}, {"id": "Mathlib.NumberTheory.NumberField.Units.Basic", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 1.008, "z": 128.14, "size": 0.264, "title": "Units of a number field", "summary": "We prove some basic results on the group `(𝓞 K)ˣ` of units of the ring of integers `𝓞 K` of a number field `K` and its torsion subgroup.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Units/Basic.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Period", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -16.093, "z": 37.555, "size": 0.2, "title": "Period of a group action", "summary": "This module defines some helpful lemmas around [`MulAction.period`] and [`AddAction.period`]. The period of a point `a` by a group element `g` is the smallest `m` such that `g ^ m • a = a` (resp. `(m • g) +ᵥ a = a`) for a given `g : G` and `a : α`. If such an `m` does not exist, then by convention `MulAction.period` and `AddAction.period` return 0.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Period.html"}, {"id": "Mathlib.GroupTheory.Exponent", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 3, "macro_tier_score": 0.19, "macro_tier_override": null, "x": 34.765, "z": -17.677, "size": 0.29, "title": "Exponent of a group", "summary": "This file defines the exponent of a group, or more generally a monoid. For a group `G` it is defined to be the minimal `n≥1` such that `g ^ n = 1` for all `g ∈ G`. For a finite group `G`, it is equal to the lowest common multiple of the order of all elements of the group `G`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Exponent.html"}, {"id": "Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 55.475, "z": -40.55, "size": 0.2649, "title": "Monic polynomials of given degree", "summary": "This file defines the predicate `Polynomial.IsMonicOfDegree p n` that states that the polynomial `p` is monic and has degree `n` (i.e., `p.natDegree = n`.) We also provide some basic API.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Degree/IsMonicOfDegree.html"}, {"id": "Mathlib.Algebra.Lie.SkewAdjoint", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -58.424, "z": 62.329, "size": 0.239, "title": "Lie algebras of skew-adjoint endomorphisms of a bilinear form", "summary": "When a module carries a bilinear form, the Lie algebra of endomorphisms of the module contains a distinguished Lie subalgebra: the skew-adjoint endomorphisms. Such subalgebras are important because they provide a simple, explicit construction of the so-called classical Lie algebras. This file defines the Lie subalgebra of skew-adjoint endomorphisms cut out by a bilinear form on a module and proves some basic related…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/SkewAdjoint.html"}, {"id": "Mathlib.Algebra.Lie.Matrix", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 77.633, "z": -30.943, "size": 0.2622, "title": "Lie algebras of matrices", "summary": "An important class of Lie algebras are those arising from the associative algebra structure on square matrices over a commutative ring. This file provides some very basic definitions whose primary value stems from their utility when constructing the classical Lie algebras using matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Matrix.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 3, "macro_tier_score": 0.0478, "macro_tier_override": null, "x": 78.048, "z": -29.879, "size": 0.3504, "title": "Sesquilinear form", "summary": "This file defines the conversion between sesquilinear maps and matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/SesquilinearForm.html"}, {"id": "Mathlib.RingTheory.Radical.NatInt", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 5.688, "z": 62.887, "size": 0.2676, "title": "The radical in `ℕ` and `ℤ`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Radical/NatInt.html"}, {"id": "Mathlib.RingTheory.Radical.Basic", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": 15.004, "z": 33.978, "size": 0.2637, "title": "Radical of an element of a unique factorization normalization monoid", "summary": "This file defines the radical of an element `a` in a unique factorization normalization monoid as the product of normalized prime factors of `a` without duplication. This is different from the radical of an ideal. Lemmas relating to natural numbers and integers are in `Mathlib.RingTheory.Radical.NatInt`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Radical/Basic.html"}, {"id": "Mathlib.RingTheory.WittVector.DiscreteValuationRing", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -39.387, "z": -110.172, "size": 0.2649, "title": "Witt vectors over a perfect ring", "summary": "This file establishes that Witt vectors over a perfect field are a discrete valuation ring. When `k` is a perfect ring, a nonzero `a : 𝕎 k` can be written as `p^m * b` for some `m : ℕ` and `b : 𝕎 k` with nonzero 0th coefficient. When `k` is also a field, this `b` can be chosen to be a unit of `𝕎 k`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/DiscreteValuationRing.html"}, {"id": "Mathlib.RingTheory.WittVector.Domain", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 95.247, "z": 64.701, "size": 0.272, "title": "Witt vectors over a domain", "summary": "This file builds to the proof `WittVector.instIsDomain`, an instance that says if `R` is an integral domain, then so is `𝕎 R`. It depends on the API around iterated applications of `WittVector.verschiebung` and `WittVector.frobenius` found in `Identities.lean`. The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723) goes as follows: any…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/Domain.html"}, {"id": "Mathlib.GroupTheory.Subgroup.Saturated", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 19.368, "z": 14.414, "size": 0.2, "title": "Saturated subgroups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Subgroup/Saturated.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.DistribMulAction", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4262, "macro_tier_override": null, "x": -2.211, "z": 14.692, "size": 0.3707, "title": "Distributive actions by submonoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/DistribMulAction.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Determinant.Bird", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 0.311, "z": -9.281, "size": 0.2765, "title": "A division-free determinant algorithm", "summary": "This file defines `birdDet`, an implementation of an division-free algorithm for computing determinants. The algorithm runs in O(n^4) for an n-by-n matrix. This determinant algorithm comes from: Title: A simple division-free algorithm for computing determinants. Author: Richard S. Bird URL: https://doi.org/10.1016/j.ipl.2011.08.006", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Determinant/Bird.html"}, {"id": "Mathlib.Algebra.Category.Grp.LargeColimits", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -44.058, "z": 50.288, "size": 0.2502, "title": "Existence of \"big\" colimits in the category of additive commutative groups", "summary": "If `F : J ⥤ AddCommGrpCat.{w}` is a functor, we show that `F` admits a colimit if and only if `Colimits.Quot F` (the quotient of the direct sum of the commutative groups `F.obj j` by the relations given by the morphisms in the diagram) is `w`-small.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/LargeColimits.html"}, {"id": "Mathlib.Algebra.Category.Grp.Colimits", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 2, "macro_tier_score": 0.03, "macro_tier_override": null, "x": 23.789, "z": 10.493, "size": 0.3447, "title": "The category of additive commutative groups has all colimits.", "summary": "This file constructs colimits in the category of additive commutative groups, as quotients of finitely supported functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Colimits.html"}, {"id": "Mathlib.Algebra.Group.Subsemigroup.Defs", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.4705, "macro_tier_override": null, "x": -2.958, "z": 6.814, "size": 0.4947, "title": "Subsemigroups: definition", "summary": "This file defines bundled multiplicative and additive subsemigroups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subsemigroup/Defs.html"}, {"id": "Mathlib.RingTheory.Adjoin.Dimension", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.0891, "macro_tier_override": null, "x": -6.288, "z": -72.156, "size": 0.2836, "title": "Some results on dimensions of algebra adjoin", "summary": "This file contains some results on dimensions of `Algebra.adjoin`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Adjoin/Dimension.html"}, {"id": "Mathlib.RingTheory.Valuation.Archimedean", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 55.015, "z": -37.991, "size": 0.2483, "title": "Ring of integers under a given valuation in a multiplicatively archimedean codomain", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/Archimedean.html"}, {"id": "Mathlib.RingTheory.Grassmannian", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 100.876, "z": 16.043, "size": 0.2, "title": "Grassmannians", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Grassmannian.html"}, {"id": "Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 3, "macro_tier_score": 0.2344, "macro_tier_override": null, "x": 5.192, "z": -23.578, "size": 0.2944, "title": "Localizations of commutative monoids with zeroes", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/MonoidLocalization/MonoidWithZero.html"}, {"id": "Mathlib.RingTheory.LocalRing.Etale", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 39.852, "z": 115.913, "size": 0.2, "title": "Étale extensions of local rings", "summary": "We prove that a finite étale extension of local rings is monogenic (generated by a single element), and that the derivative of the minimal polynomial evaluated at the generator is a unit. These are parts 1 and 2 of Lemma 3.2 of [arXiv:2503.07846](https://arxiv.org/abs/2503.07846).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/Etale.html"}, {"id": "Mathlib.RingTheory.IsAdjoinRoot", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": 13.757, "z": -104.961, "size": 0.2565, "title": "A predicate on adjoining roots of polynomial", "summary": "This file defines a predicate `IsAdjoinRoot S f`, which states that the ring `S` can be constructed by adjoining a specified root of the polynomial `f : R[X]` to `R`. This predicate is useful when the same ring can be generated by adjoining the root of different polynomials, and you want to vary which polynomial you're considering. The results in this file are intended to mirror those in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IsAdjoinRoot.html"}, {"id": "Mathlib.RingTheory.Smooth.Flat", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 2, "macro_tier_score": 0.0073, "macro_tier_override": null, "x": -113.126, "z": 6.037, "size": 0.3085, "title": "Smooth algebras are flat", "summary": "Let `A` be a smooth `R`-algebra. In this file we show that then `A` is `R`-flat. The proof proceeds in two steps: 1. If `R` is Noetherian, let `R[X₁, ..., Xₙ] →ₐ[R] A` be surjective with kernel `I`. By formal smoothness we construct a section `A →ₐ[R] AdicCompletion I R[X₁, ..., Xₙ]` of the canonical map `AdicCompletion I R[X₁, ..., Xₙ] →ₐ[R] R[X₁, ..., Xₙ] ⧸ I ≃ₐ[R] A`. Since `R` is Noetherian, `AdicCompletion I R`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/Flat.html"}, {"id": "Mathlib.RingTheory.Unramified.LocalRing", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 2, "macro_tier_score": 0.0137, "macro_tier_override": null, "x": 48.325, "z": 110.621, "size": 0.2859, "title": "Unramified algebras over local rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Unramified/LocalRing.html"}, {"id": "Mathlib.Algebra.Module.LinearMap.DivisionRing", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 3, "macro_tier_score": 0.1778, "macro_tier_override": null, "x": -20.621, "z": 35.272, "size": 0.4184, "title": "Some lemmas about linear functionals on division rings", "summary": "This file proves some results on linear functionals on division semirings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LinearMap/DivisionRing.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.Acyclic", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 2, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -15.265, "z": -37.899, "size": 0.3015, "title": "The triangulated subcategory of acyclic complex in the homotopy category", "summary": "In this file, we define the triangulated subcategory `HomotopyCategory.subcategoryAcyclic C` of the homotopy category of cochain complexes in an abelian category `C`. In the lemma `HomotopyCategory.quasiIso_eq_subcategoryAcyclic_W` we obtain that the class of quasiisomorphisms `HomotopyCategory.quasiIso C (ComplexShape.up ℤ)` consists of morphisms whose cone belongs to the triangulated subcategory…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/Acyclic.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.HomComplexInduction", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -20.078, "z": 13.408, "size": 0.2493, "title": "Construction of cochains by induction", "summary": "Let `K` and `L` be cochain complexes in a preadditive category `C`. We provide an API to construct a cochain in `Cochain K L d` in the following situation. Assume that `X n : Set (Cochain K L d)` is a sequence of subsets of `Cochain K L d`, and `φ n : X n → X (n + 1)` is a sequence of maps such that for a certain `p₀ : ℕ` and any `x : X n`, `φ n x` and `x` coincide up to the degree `p₀ + n`, then we construct a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/HomComplexInduction.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 2, "macro_tier_score": 0.0225, "macro_tier_override": null, "x": -57.224, "z": 21.942, "size": 0.2734, "title": "Weighted homogeneous polynomials", "summary": "It is possible to assign weights (in a commutative additive monoid `M`) to the variables of a multivariate polynomial ring, so that monomials of the ring then have a weighted degree with respect to the weights of the variables. The weights are represented by a function `w : σ → M`, where `σ` are the indeterminates. A multivariate polynomial `φ` is weighted homogeneous of weighted degree `m : M` if all monomials…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.html"}, {"id": "Mathlib.RingTheory.HahnSeries.Addition", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": 5.936, "z": 25.314, "size": 0.2527, "title": "Additive properties of Hahn series", "summary": "If `Γ` is ordered and `R` has zero, then `R⟦Γ⟧` consists of formal series over `Γ` with coefficients in `R`, whose supports are partially well-ordered. With further structure on `R` and `Γ`, we can add further structure on `R⟦Γ⟧`. When `R` has an addition operation, `R⟦Γ⟧` also has addition by adding coefficients.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HahnSeries/Addition.html"}, {"id": "Mathlib.Algebra.Algebra.Prod", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 4, "macro_tier_score": 0.3702, "macro_tier_override": null, "x": 2.736, "z": 44.488, "size": 0.5449, "title": "The R-algebra structure on products of R-algebras", "summary": "The R-algebra structure on `(i : I) → A i` when each `A i` is an R-algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Prod.html"}, {"id": "Mathlib.RingTheory.ReesAlgebra", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 3, "macro_tier_score": 0.0589, "macro_tier_override": null, "x": 38.129, "z": -68.046, "size": 0.2522, "title": "Rees algebra", "summary": "The Rees algebra of an ideal `I` is the subalgebra `R[It]` of `R[t]` defined as `R[It] = ⨁ₙ Iⁿ tⁿ`. This is used to prove the Artin-Rees lemma, and will potentially enable us to calculate some blowup in the future.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/ReesAlgebra.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Transitive", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": -8.859, "z": 22.459, "size": 0.2483, "title": "Complements to pretransitive actions", "summary": "When `f : X →ₑ[φ] Y` is an equivariant map with respect to a map of monoids `φ: M → N`, - `MulAction.IsPretransitive.of_surjective_map` shows that the action of `N` on `Y` is pretransitive if that of `M` on `X` is pretransitive. - `MulAction.isPretransitive_congr` shows that when `φ` is surjective, the action of `N` on `Y` is pretransitive iff that of `M` on `X` is pretransitive. Given `MulAction G X` where `G` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Transitive.html"}, {"id": "Mathlib.LinearAlgebra.Reflection", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": 24.329, "z": 72.152, "size": 0.2815, "title": "Reflections in linear algebra", "summary": "Given an element `x` in a module `M` together with a linear form `f` on `M` such that `f x = 2`, the map `y ↦ y - (f y) • x` is an involutive endomorphism of `M`, such that: 1. the kernel of `f` is fixed, 2. the point `x` maps to `-x`. Such endomorphisms are often called reflections of the module `M`. When `M` carries an inner product for which `x` is perpendicular to the kernel of `f`, then (with mild assumptions)…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Reflection.html"}, {"id": "Mathlib.Algebra.EuclideanDomain.Basic", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.2947, "macro_tier_override": null, "x": -10.92, "z": -12.654, "size": 0.3265, "title": "Lemmas about Euclidean domains", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/EuclideanDomain/Basic.html"}, {"id": "Mathlib.LinearAlgebra.FiniteSpan", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": -38.736, "z": 4.531, "size": 0.2451, "title": "Additional results about finite spanning sets in linear algebra", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FiniteSpan.html"}, {"id": "Mathlib.Algebra.Order.AddTorsor", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -12.935, "z": -1.297, "size": 0.3103, "title": "Ordered scalar multiplication and vector addition", "summary": "This file defines ordered scalar multiplication and vector addition, and proves some properties. In the additive case, a motivating example is given by the additive action of `ℤ` on subsets of reals that are closed under integer translation. The order compatibility allows for a treatment of the `R((z))`-module structure on `(z ^ s) V((z))` for an `R`-module `V`, using the formalism of Hahn series. In the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/AddTorsor.html"}, {"id": "Mathlib.NumberTheory.MaricaSchoenheim", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": "The Marica-Schönheim special case of Graham's conjecture", "summary": "Graham's conjecture states that if $0 < a_1 < \\dots a_n$ are integers, then $\\max_{i, j} \\frac{a_i}{\\gcd(a_i, a_j)} \\ge n$. This file proves the conjecture when the $a_i$ are squarefree as a corollary of the Marica-Schönheim inequality.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/MaricaSchoenheim.html"}, {"id": "Mathlib.RingTheory.Adjoin.FGBaseChange", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 3, "macro_tier_score": 0.0513, "macro_tier_override": null, "x": 69.987, "z": 24.907, "size": 0.2726, "title": "Finitely generated subalgebras of a base change obtained from an element", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Adjoin/FGBaseChange.html"}, {"id": "Mathlib.Algebra.Category.Grp.ForgetCorepresentable", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 3, "macro_tier_score": 0.0889, "macro_tier_override": null, "x": -2.851, "z": -22.103, "size": 0.268, "title": "The forget functor is corepresentable", "summary": "It is shown that the forget functor `AddCommGrpCat.{u} ⥤ Type u` is corepresentable by `ULift ℤ`. Similar results are obtained for the variants `CommGrpCat`, `AddGrpCat` and `GrpCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/ForgetCorepresentable.html"}, {"id": "Mathlib.Algebra.Category.MonCat.ForgetCorepresentable", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 3, "macro_tier_score": 0.0889, "macro_tier_override": null, "x": -12.74, "z": -7.644, "size": 0.268, "title": "The forgetful functor is corepresentable", "summary": "The forgetful functor `AddCommMonCat.{u} ⥤ Type u` is corepresentable by `ULift ℕ`. Similar results are obtained for the variants `CommMonCat`, `AddMonCat` and `MonCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/MonCat/ForgetCorepresentable.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.Expand", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -11.905, "z": -78.966, "size": 0.257, "title": "Results on `MvPolynomial.expand`", "summary": "In this file we prove results about `MvPolynomial.expand` that require more than the basic API available in `Mathlib.Algebra.*`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/Expand.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.Basic", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.0995, "macro_tier_override": null, "x": -59.725, "z": -25.651, "size": 0.3236, "title": "Multivariate polynomials over commutative rings", "summary": "This file contains basic facts about multivariate polynomials over commutative rings, for example that the monomials form a basis.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/Basic.html"}, {"id": "Mathlib.Algebra.CharP.Frobenius", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 3, "macro_tier_score": 0.2077, "macro_tier_override": null, "x": 15.722, "z": 22.997, "size": 0.3636, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/Frobenius.html"}, {"id": "Mathlib.FieldTheory.SeparableDegree", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 3, "macro_tier_score": 0.0595, "macro_tier_override": null, "x": -88.767, "z": -61.017, "size": 0.3142, "title": "Separable degree", "summary": "This file contains basics about the separable degree of a field extension.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/SeparableDegree.html"}, {"id": "Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 3, "macro_tier_score": 0.0624, "macro_tier_override": null, "x": -18.867, "z": -92.817, "size": 0.2745, "title": "Transcendence basis", "summary": "This file defines the transcendence basis as a maximal algebraically independent subset.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.html"}, {"id": "Mathlib.RingTheory.Polynomial.SeparableDegree", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 3, "macro_tier_score": 0.06, "macro_tier_override": null, "x": -0.069, "z": 96.572, "size": 0.2558, "title": "Separable degree", "summary": "This file contains basics about the separable degree of a polynomial.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/SeparableDegree.html"}, {"id": "Mathlib.Algebra.Module.Lattice", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -15.435, "z": -78.352, "size": 0.2, "title": "Lattices", "summary": "Let `A` be an `R`-algebra and `V` an `A`-module. Then an `R`-submodule `M` of `V` is a lattice, if `M` is finitely generated and spans `V` as an `A`-module. The typical use case is `A = K` is the fraction field of an integral domain `R` and `V = ι → K` for some finite `ι`. The scalar multiple a lattice by a unit in `K` is again a lattice. This gives rise to a homothety relation. When `R` is a DVR and `ι = Fin 2`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Lattice.html"}, {"id": "Mathlib.NumberTheory.NumberField.Units.DirichletTheorem", "region_id": "algebra", "micro_elevation": 0.9474, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": 132.803, "z": -15.6, "size": 0.2814, "title": "Dirichlet theorem on the group of units of a number field", "summary": "This file is devoted to the proof of Dirichlet unit theorem that states that the group of units `(𝓞 K)ˣ` of units of the ring of integers `𝓞 K` of a number field `K` modulo its torsion subgroup is a free `ℤ`-module of rank `card (InfinitePlace K) - 1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.Colimits", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 1, "macro_tier_score": 0.0017, "macro_tier_override": null, "x": 58.851, "z": -22.882, "size": 0.3061, "title": "Colimits in categories of presheaves of modules", "summary": "In this file, it is shown that under suitable assumptions, colimits exist in the category `PresheafOfModules R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/Colimits.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.Homology", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 2, "macro_tier_score": 0.0259, "macro_tier_override": null, "x": -5.225, "z": -1.935, "size": 0.3671, "title": "Homology of short complexes", "summary": "In this file, we shall define the homology of short complexes `S`, i.e. diagrams `f : X₁ ⟶ X₂` and `g : X₂ ⟶ X₃` such that `f ≫ g = 0`. We shall say that `[S.HasHomology]` when there exists `h : S.HomologyData`. A homology data for `S` consists of compatible left/right homology data `left` and `right`. The left homology data `left` involves an object `left.H` that is a cokernel of the canonical map `S.X₁ ⟶ K` where…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/Homology.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.RightHomology", "region_id": "algebra", "micro_elevation": 0.0263, "macro_tier": 2, "macro_tier_score": 0.026, "macro_tier_override": null, "x": 2.927, "z": -2.286, "size": 0.368, "title": "Right Homology of short complexes", "summary": "In this file, we define the dual notions to those defined in `Algebra.Homology.ShortComplex.LeftHomology`. In particular, if `S : ShortComplex C` is a short complex consisting of two composable maps `f : X₁ ⟶ X₂` and `g : X₂ ⟶ X₃` such that `f ≫ g = 0`, we define `h : S.RightHomologyData` to be the datum of morphisms `p : X₂ ⟶ Q` and `ι : H ⟶ Q` such that `Q` identifies to the cokernel of `f` and `H` to the kernel…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/RightHomology.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.FunctorEquivalence", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.771, "z": 0.56, "size": 0.2, "title": "Short complexes in functor categories", "summary": "In this file, it is shown that if `J` and `C` are two categories (such that `C` has zero morphisms), then there is an equivalence of categories `ShortComplex.functorEquivalence J C : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/FunctorEquivalence.html"}, {"id": "Mathlib.GroupTheory.EckmannHilton", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -4.49, "z": -3.299, "size": 0.2585, "title": "Eckmann-Hilton argument", "summary": "The Eckmann-Hilton argument says that if a type carries two monoid structures that distribute over one another, then they are equal, and in addition commutative. The main application lies in proving that higher homotopy groups (`πₙ` for `n ≥ 2`) are commutative.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/EckmannHilton.html"}, {"id": "Mathlib.RingTheory.IntegralDomain", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 3, "macro_tier_score": 0.142, "macro_tier_override": null, "x": 45.643, "z": 58.611, "size": 0.3472, "title": "Integral domains", "summary": "Assorted theorems about integral domains.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IntegralDomain.html"}, {"id": "Mathlib.Algebra.Polynomial.Degree.SmallDegree", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 4, "macro_tier_score": 0.2845, "macro_tier_override": null, "x": -23.736, "z": 50.406, "size": 0.3121, "title": "Results on polynomials of specific small degrees", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Degree/SmallDegree.html"}, {"id": "Mathlib.RingTheory.Ideal.Quotient.Nilpotent", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.1065, "macro_tier_override": null, "x": -15.4, "z": -65.06, "size": 0.3432, "title": "Nilpotent elements in quotient rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Quotient/Nilpotent.html"}, {"id": "Mathlib.NumberTheory.AbelSummation", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2451, "title": "Abel's summation formula", "summary": "We prove several versions of Abel's summation formula.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/AbelSummation.html"}, {"id": "Mathlib.RingTheory.WittVector.FrobeniusFractionField", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -116.808, "z": 21.982, "size": 0.2676, "title": "Solving equations about the Frobenius map on the field of fractions of `𝕎 k`", "summary": "The goal of this file is to prove `WittVector.exists_frobenius_solution_fractionRing`, which says that for an algebraically closed field `k` of characteristic `p` and `a, b` in the field of fractions of Witt vectors over `k`, there is a solution `b` to the equation `φ b * a = p ^ m * b`, where `φ` is the Frobenius map. Most of this file builds up the equivalent theorem over `𝕎 k` directly, moving to the field of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/FrobeniusFractionField.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Combination", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": -6.224, "z": 55.366, "size": 0.3599, "title": "Affine combinations of points", "summary": "This file defines affine combinations of points.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Combination.html"}, {"id": "Mathlib.NumberTheory.ArithmeticFunction.Carmichael", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -27.251, "z": 100.367, "size": 0.2, "title": "The Carmichael function", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ArithmeticFunction/Carmichael.html"}, {"id": "Mathlib.RingTheory.ZMod.UnitsCyclic", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": -100.308, "z": 19.28, "size": 0.2557, "title": "Cyclicity of the units of `ZMod n`", "summary": "`ZMod.isCyclic_units_iff` : `(ZMod n)ˣ` is cyclic iff one of the following mutually exclusive cases happens: - `n = 0` (then `ZMod 0 ≃+* ℤ` and the group of units is cyclic of order 2); - `n = 1`, `2` or `4` - `n` is a power `p ^ e` of an odd prime number, or twice such a power (with `1 ≤ e`). The individual cases are proved by `inferInstance` and are also directly provided by : * `ZMod.isCyclic_units_zero` *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/ZMod/UnitsCyclic.html"}, {"id": "Mathlib.RingTheory.PolynomialLaw.Basic", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 46.088, "z": 62.929, "size": 0.2, "title": "Polynomial laws on modules", "summary": "Let `M` and `N` be a modules over a commutative ring `R`. A polynomial law `f : PolynomialLaw R M N`, with notation `f : M →ₚₗₗ[R] N`, is a “law” that assigns a natural map `PolynomialLaw.toFun' f S : S ⊗[R] M → S ⊗[R] N` for every `R`-algebra `S`. For type-theoretic reasons, if `R : Type u`, then the definition of the polynomial map `f` is restricted to `R`-algebras `S` such that `S : Type u`. Using the fact that a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PolynomialLaw/Basic.html"}, {"id": "Mathlib.RingTheory.Congruence.Hom", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": -55.849, "z": 13.978, "size": 0.3035, "title": "Congruence relations and ring homomorphisms", "summary": "This file contains elementary definitions involving congruence relations and morphisms for rings and semirings", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Congruence/Hom.html"}, {"id": "Mathlib.RingTheory.PowerSeries.WeierstrassPreparation", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -60.983, "z": -57.143, "size": 0.2, "title": "Weierstrass preparation theorem for power series over a complete local ring", "summary": "In this file we define Weierstrass division, Weierstrass factorization, and prove Weierstrass preparation theorem. We assume that a ring is adic complete with respect to some ideal. If such ideal is a maximal ideal, then by `isLocalRing_of_isAdicComplete_maximal`, such ring has only one maximal ideal, and hence it is a complete local ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/WeierstrassPreparation.html"}, {"id": "Mathlib.RingTheory.Polynomial.Eisenstein.Distinguished", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -50.476, "z": 64.262, "size": 0.2302, "title": "Distinguished polynomial", "summary": "In this file we define the predicate `Polynomial.IsDistinguishedAt` and develop the most basic lemmas about it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Eisenstein/Distinguished.html"}, {"id": "Mathlib.RingTheory.PowerSeries.CoeffMulMem", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 45.326, "z": 51.646, "size": 0.2302, "title": "Some results on the coefficients of multiplication of two power series", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/CoeffMulMem.html"}, {"id": "Mathlib.RingTheory.Invariant.Profinite", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 96.872, "z": 25.947, "size": 0.2, "title": "Invariant Extensions of Rings", "summary": "In this file we generalize the results in `Mathlib/RingTheory/Invariant/Basic.lean` to profinite groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Invariant/Profinite.html"}, {"id": "Mathlib.Algebra.Polynomial.Module.FiniteDimensional", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 91.722, "z": -14.48, "size": 0.2, "title": "Polynomial modules in finite dimensions", "summary": "This file is a place to collect results about the `R[X]`-module structure induced on an `R`-module by an `R`-linear endomorphism, which require the concept of finite-dimensionality.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Module/FiniteDimensional.html"}, {"id": "Mathlib.Algebra.Module.Torsion.Pi", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.3862, "macro_tier_override": null, "x": 20.064, "z": -9.701, "size": 0.28, "title": "Product of torsion-free modules", "summary": "This file shows that the product of torsion-free modules is torsion-free.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Torsion/Pi.html"}, {"id": "Mathlib.RingTheory.HahnSeries.Basic", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": 2.308, "z": 14.677, "size": 0.248, "title": "Hahn Series", "summary": "If `Γ` is ordered and `R` has zero, then the type `HahnSeries Γ R`, which we denote as `R⟦Γ⟧`, consists of formal series over `Γ` with coefficients in `R`, whose supports are partially well-ordered. With further structure on `R` and `Γ`, we can add further structure on `R⟦Γ⟧`, with the most studied case being when `Γ` is a linearly ordered abelian group and `R` is a field, in which case `R⟦Γ⟧` is a valued field,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HahnSeries/Basic.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Unbundled.MinMax", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4213, "macro_tier_override": null, "x": -9.924, "z": 5.067, "size": 0.3401, "title": "Lemmas about `min` and `max` in an ordered monoid.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.html"}, {"id": "Mathlib.Algebra.AddTorsor.Basic", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 7.623, "z": 20.941, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/AddTorsor/Basic.html"}, {"id": "Mathlib.RingTheory.Polynomial.Bernstein", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 4.357, "z": -72.298, "size": 0.2374, "title": "Bernstein polynomials", "summary": "The definition of the Bernstein polynomials ``` bernsteinPolynomial (R : Type*) [CommRing R] (n ν : ℕ) : R[X] := (choose n ν) * X^ν * (1 - X)^(n - ν) ``` and the fact that for `ν : Fin (n+1)` these are linearly independent over `ℚ`. We prove the basic identities * `(Finset.range (n + 1)).sum (fun ν ↦ bernsteinPolynomial R n ν) = 1` * `(Finset.range (n + 1)).sum (fun ν ↦ ν • bernsteinPolynomial R n ν) = n • X` *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Bernstein.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.IsSupported", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 2, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -8.098, "z": -18.755, "size": 0.3406, "title": "Support of homological complexes", "summary": "Given an embedding `e : c.Embedding c'` of complex shapes, we say that `K : HomologicalComplex C c'` is supported (resp. strictly supported) on `e` if `K` is exact in degree `i'` (resp. `K.X i'` is zero) whenever `i'` is not of the form `e.f i`. This defines two typeclasses `K.IsSupported e` and `K.IsStrictlySupported e`. We also define predicates `K.IsSupportedOutside e` and `K.IsStrictlySupportedOutside e` when…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/IsSupported.html"}, {"id": "Mathlib.Algebra.Category.MonCat.Shrink", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -0.577, "z": -16.705, "size": 0.2445, "title": "Shrinking a functor to `MonCat`", "summary": "For a functor `C ⥤ MonCat.{w'}` with `w`-small image, we shrink to a functor `C ⥤ MonCat.{w}`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/MonCat/Shrink.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 101.885, "z": -34.957, "size": 0.2685, "title": "The canonical bilinear form on a finite root pairing", "summary": "Given a finite root pairing, we define a canonical map from weight space to coweight space, and the corresponding bilinear form. This form is symmetric and Weyl-invariant, and if the base ring is linearly ordered, then the form is root-positive, positive-semidefinite on the weight space, and positive-definite on the span of roots. From these facts, it is easy to show that Coxeter weights in a finite root pairing are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/Finite/CanonicalBilinear.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.RootPositive", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -16.972, "z": 104.489, "size": 0.2653, "title": "Invariant and root-positive bilinear forms on root pairings", "summary": "This file contains basic results on Weyl-invariant inner products for root systems and root data. Given a root pairing we define a structure which contains a bilinear form together with axioms for reflection-invariance, symmetry, and strict positivity on all roots. We show that root-positive forms display the same sign behavior as the canonical pairing between roots and coroots. Root-positive forms show up naturally…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/RootPositive.html"}, {"id": "Mathlib.RingTheory.Valuation.Minpoly", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -80.524, "z": -46.243, "size": 0.2, "title": "Minimal polynomials.", "summary": "We prove some results about valuations of zero coefficients of minimal polynomials. Let `K` be a field with a valuation `v` and let `L` be a field extension of `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/Minpoly.html"}, {"id": "Mathlib.FieldTheory.Minpoly.Finite", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 3, "macro_tier_score": 0.0694, "macro_tier_override": null, "x": -53.858, "z": 66.314, "size": 0.31, "title": "Minimal polynomials on a finite algebra", "summary": "This file proves the bound on the degree of a minimal polynomial on an algebra that is finite as a module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Minpoly/Finite.html"}, {"id": "Mathlib.GroupTheory.OrderOfElement", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 3, "macro_tier_score": 0.2393, "macro_tier_override": null, "x": 18.418, "z": 32.255, "size": 0.3932, "title": "Order of an element", "summary": "This file defines the order of an element of a finite group. For a finite group `G` the order of `x ∈ G` is the minimal `n ≥ 1` such that `x ^ n = 1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/OrderOfElement.html"}, {"id": "Mathlib.RingTheory.ZMod.Torsion", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 74.102, "z": -70.302, "size": 0.2, "title": "Torsion group of `ZMod p` for prime `p`", "summary": "This file shows that the `ZMod p` has `p - 1` roots-of-unity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/ZMod/Torsion.html"}, {"id": "Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.0094, "macro_tier_override": null, "x": 53.252, "z": -59.511, "size": 0.3005, "title": "Commutative monoids with enough roots of unity", "summary": "We define a typeclass `HasEnoughRootsOfUnity M n` for a commutative monoid `M` and a natural number `n` that asserts that `M` contains a primitive `n`th root of unity and that the group of `n`th roots of unity in `M` is cyclic. Such monoids are suitable targets for homomorphisms from groups of exponent (dividing) `n`; for example, the homomorphisms can then be used to separate elements of the source group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RootsOfUnity/EnoughRootsOfUnity.html"}, {"id": "Mathlib.NumberTheory.RamificationInertia.Ramification", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 2, "macro_tier_score": 0.0103, "macro_tier_override": null, "x": -101.805, "z": -8.311, "size": 0.2739, "title": "Ramification index", "summary": "Given `P : Ideal S` lying over `p : Ideal R` for the ring extension `f : R →+* S` (assuming `P` and `p` are prime or maximal where needed), the **ramification index** `Ideal.ramificationIdx' p P` is the multiplicity of `P` in `map f p`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/RamificationInertia/Ramification.html"}, {"id": "Mathlib.Algebra.Polynomial.UnitTrinomial", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 39.348, "z": 56.333, "size": 0.257, "title": "Unit Trinomials", "summary": "This file defines irreducible trinomials and proves an irreducibility criterion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/UnitTrinomial.html"}, {"id": "Mathlib.Algebra.Order.Ring.Abs", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 4, "macro_tier_score": 0.4135, "macro_tier_override": null, "x": 16.923, "z": 19.739, "size": 0.4425, "title": "Absolute values in linear ordered rings.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Abs.html"}, {"id": "Mathlib.RingTheory.GradedAlgebra.AlgHom", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -63.139, "z": -0.738, "size": 0.239, "title": "`R`-linear homomorphisms of graded algebras", "summary": "This file defines bundled `R`-linear homomorphisms of graded `R`-algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/GradedAlgebra/AlgHom.html"}, {"id": "Mathlib.RingTheory.GradedAlgebra.RingHom", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 1, "macro_tier_score": 0.0017, "macro_tier_override": null, "x": -57.977, "z": -19.866, "size": 0.3014, "title": "Homomorphisms of graded (semi)rings", "summary": "This file defines bundled homomorphisms of graded (semi)rings. We use the same structure `GradedRingHom 𝒜 ℬ`, a.k.a. `𝒜 →+*ᵍ ℬ`, for both types of homomorphisms. We do **not** define a separate class of graded ring homomorphisms; instead, we use `[FunLike F A B] [GradedFunLike F 𝒜 ℬ] [RingHomClass F A B]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/GradedAlgebra/RingHom.html"}, {"id": "Mathlib.RingTheory.TotallySplit", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 79.006, "z": 93.713, "size": 0.239, "title": "Totally split algebras", "summary": "An `R`-algebra `S` is finite (totally) split if it is isomorphic to `Fin n → R` for some `n`. Geometrically, this corresponds to a trivial covering. Every totally split algebra is finite étale and conversely, every finite étale covering is étale locally totally split.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TotallySplit.html"}, {"id": "Mathlib.RingTheory.Flat.Rank", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -46.87, "z": -90.756, "size": 0.2471, "title": "Results for the rank of a finite flat algebra", "summary": "In this file we study a finite, flat `R`-algebra `S` and relate injectivity and bijectivity of `R → S` with the rank of `S` over `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/Rank.html"}, {"id": "Mathlib.RingTheory.TensorProduct.Pi", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 2, "macro_tier_score": 0.0191, "macro_tier_override": null, "x": 9.944, "z": 64.235, "size": 0.2709, "title": "Tensor product and products of algebras", "summary": "In this file we examine the behaviour of the tensor product with (finite) products. This is a direct application of `Mathlib/LinearAlgebra/TensorProduct/Pi.lean` to the algebra case.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/Pi.html"}, {"id": "Mathlib.GroupTheory.Perm.Closure", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 3, "macro_tier_score": 0.2287, "macro_tier_override": null, "x": -6.53, "z": -42.213, "size": 0.2785, "title": "Closure results for permutation groups", "summary": "* This file contains several closure results: * `closure_isCycle` : The symmetric group is generated by cycles * `closure_cycle_adjacent_swap` : The symmetric group is generated by a cycle and an adjacent transposition * `closure_cycle_coprime_swap` : The symmetric group is generated by a cycle and a coprime transposition * `closure_prime_cycle_swap` : The symmetric group is generated by a prime cycle and a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Closure.html"}, {"id": "Mathlib.GroupTheory.Perm.Cycle.Basic", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 3, "macro_tier_score": 0.2292, "macro_tier_override": null, "x": -20.752, "z": 35.195, "size": 0.3179, "title": "Cycles of a permutation", "summary": "This file starts the theory of cycles in permutations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Cycle/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -17.634, "z": 104.379, "size": 0.2806, "title": "Classification of elements of `GL (Fin 2) R`", "summary": "Here we classify `2 × 2` matrices over the reals (or more generally over `R` where `R` is a suitable ring, but `ℝ` is the motivating case), into the following classes: * scalars * parabolic elements (`Matrix.IsParabolic`) - one eigenvalue with non-semisimple generalized eigenspace * hyperbolic elements (`Matrix.IsHyperbolic`) - two distinct real eigenvalues * elliptic elements (`Matrix.IsElliptic`) - two distinct…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/FinTwo.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Charpoly.Disc", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 1, "macro_tier_score": 0.0017, "macro_tier_override": null, "x": 81.728, "z": -64.318, "size": 0.3002, "title": "The discriminant of a matrix", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Charpoly/Disc.html"}, {"id": "Mathlib.GroupTheory.GroupAction.DomAct.ActionHom", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 22.527, "z": -8.685, "size": 0.2, "title": "Action of `Mᵈᵐᵃ` on `α →[N] β` and `A →+[N] B`", "summary": "In this file we define action of `DomMulAct M = Mᵈᵐᵃ` on `α →[N] β` and on `A →+[N] B`. At the time of writing, these homomorphisms are not widely used in the library, so we put these instances into a separate file, not with the definition of `DomMulAct`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/DomAct/ActionHom.html"}, {"id": "Mathlib.FieldTheory.IntermediateField.Algebraic", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 3, "macro_tier_score": 0.0944, "macro_tier_override": null, "x": -48.582, "z": -76.948, "size": 0.2609, "title": "Results on finite dimensionality and algebraicity of intermediate fields.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/IntermediateField/Algebraic.html"}, {"id": "Mathlib.NumberTheory.EulerProduct.Basic", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -29.053, "z": 26.019, "size": 0.2447, "title": "Euler Products", "summary": "The main result in this file is `EulerProduct.eulerProduct_hasProd`, which says that if `f : ℕ → R` is norm-summable, where `R` is a complete normed commutative ring and `f` is multiplicative on coprime arguments with `f 0 = 0`, then `∏' p : Primes, ∑' e : ℕ, f (p^e)` converges to `∑' n, f n`. `ArithmeticFunction.IsMultiplicative.eulerProduct_hasProd` is a version for multiplicative arithmetic functions in the sense…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/EulerProduct/Basic.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.Isometry", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 2, "macro_tier_score": 0.0109, "macro_tier_override": null, "x": -67.65, "z": 55.159, "size": 0.3898, "title": "Isometric linear maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/Isometry.html"}, {"id": "Mathlib.Algebra.Lie.Weights.RootSystem", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -67.378, "z": -102.393, "size": 0.258, "title": "The root system associated with a Lie algebra", "summary": "We show that the roots of a finite-dimensional splitting semisimple Lie algebra over a field of characteristic 0 form a root system. We achieve this by studying root chains.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Weights/RootSystem.html"}, {"id": "Mathlib.Algebra.Lie.Weights.Killing", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -83.024, "z": 87.632, "size": 0.2497, "title": "Roots of Lie algebras with non-degenerate Killing forms", "summary": "The file contains definitions and results about roots of Lie algebras with non-degenerate Killing forms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Weights/Killing.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Submonoid", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 2, "macro_tier_score": 0.0134, "macro_tier_override": null, "x": -14.597, "z": 2.767, "size": 0.3475, "title": "Ordered instances on submonoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Submonoid.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Basic", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 3, "macro_tier_score": 0.0855, "macro_tier_override": null, "x": -0.109, "z": -13.0, "size": 0.4073, "title": "Ordered monoids", "summary": "This file develops some additional material on ordered monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Basic.html"}, {"id": "Mathlib.RingTheory.Valuation.PrimeMultiplicity", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 3, "macro_tier_score": 0.0589, "macro_tier_override": null, "x": -32.897, "z": 51.709, "size": 0.2557, "title": "`multiplicity` of a prime in an integral domain as an additive valuation", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/PrimeMultiplicity.html"}, {"id": "Mathlib.Algebra.Order.Group.Finset", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 1, "macro_tier_score": 0.004, "macro_tier_override": null, "x": 5.496, "z": 17.74, "size": 0.3104, "title": "`Finset.sup` in a group", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Finset.html"}, {"id": "Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -76.966, "z": 92.998, "size": 0.2, "title": "The graded ring of level-1 modular forms", "summary": "This file collects structural results about the graded ring `⨁ k, ModularForm 𝒮ℒ k` of level-1 modular forms, beyond those that fall out of the dimension formula directly.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/LevelOne/GradedRing.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 3, "macro_tier_score": 0.187, "macro_tier_override": null, "x": -31.531, "z": 1.596, "size": 0.3178, "title": "Building GCD out of unique factorization", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/GCDMonoid.html"}, {"id": "Mathlib.RingTheory.Norm.Basic", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 3, "macro_tier_score": 0.0331, "macro_tier_override": null, "x": 2.948, "z": 107.675, "size": 0.3317, "title": "Norm for (finite) ring extensions", "summary": "Suppose we have an `R`-algebra `S` with a finite basis. For each `s : S`, the determinant of the linear map given by multiplying by `s` gives information about the roots of the minimal polynomial of `s` over `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Norm/Basic.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.Linear", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 42.105, "z": 14.623, "size": 0.2942, "title": "The derived category of a linear abelian category is linear", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/Linear.html"}, {"id": "Mathlib.Algebra.Order.Group.Ideal", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 4.601, "z": 34.985, "size": 0.2221, "title": "Semigroup ideals in a canonically ordered and well-quasi-ordered monoid", "summary": "This file proves that in a canonically ordered and well-quasi-ordered monoid, any semigroup ideal is finitely generated, and the semigroup ideals satisfy the ascending chain condition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Ideal.html"}, {"id": "Mathlib.Algebra.Group.Ideal", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 8.843, "z": -32.238, "size": 0.2407, "title": "Semigroup ideals", "summary": "This file defines (left) semigroup ideals (also called monoid ideals sometimes), which are sets `s` in a semigroup such that `a * b ∈ s` whenever `b ∈ s`. Note that semigroup ideals are different from ring ideals (`Ideal` in Mathlib): a ring ideal is a semigroup ideal that is also an additive submonoid of the ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Ideal.html"}, {"id": "Mathlib.FieldTheory.Galois.GaloisClosure", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 5, "macro_tier_score": 0.0203, "macro_tier_override": 5, "x": -107.408, "z": -41.493, "size": 0.2795, "title": "Main definitions and results", "summary": "In a field extension `K/k` * `FiniteGaloisIntermediateField` : The type of intermediate fields of `K/k` that are finite and Galois over `k` * `adjoin` : The finite Galois intermediate field obtained from the normal closure of adjoining a finite `s : Set K` to `k`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Galois/GaloisClosure.html"}, {"id": "Mathlib.Algebra.Category.CoalgCat.ComonEquivalence", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 13.932, "z": 61.587, "size": 0.2, "title": "The category equivalence between `R`-coalgebras and comonoid objects in `R-Mod`", "summary": "Given a commutative ring `R`, this file defines the equivalence of categories between `R`-coalgebras and comonoid objects in the category of `R`-modules. We then use this to set up boilerplate for the `Coalgebra` instance on a tensor product of coalgebras defined in `Mathlib/RingTheory/Coalgebra/TensorProduct.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/CoalgCat/ComonEquivalence.html"}, {"id": "Mathlib.Algebra.Module.ZLattice.Summable", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 76.332, "z": -16.049, "size": 0.2239, "title": "Convergence of `p`-series on lattices", "summary": "Let `E` be a finite dimensional normed `ℝ`-space, and `L` a discrete subgroup of `E` of rank `d`. We show that `∑ z ∈ L, ‖z - x‖ʳ` is convergent for `r < -d`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/ZLattice/Summable.html"}, {"id": "Mathlib.Algebra.Module.ZLattice.Basic", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 1, "macro_tier_score": 0.0039, "macro_tier_override": null, "x": -67.591, "z": 35.062, "size": 0.3, "title": "ℤ-lattices", "summary": "Let `E` be a finite-dimensional vector space over a `NormedLinearOrderedField` `K` with a solid norm that is also a `FloorRing`, e.g. `ℝ`. A (full) `ℤ`-lattice `L` of `E` is a discrete subgroup of `E` such that `L` spans `E` over `K`. A `ℤ`-lattice `L` can be defined in two ways: * For `b` a basis of `E`, then `L = Submodule.span ℤ (Set.range b)` is a ℤ-lattice of `E` * As a `ℤ-submodule` of `E` with the additional…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/ZLattice/Basic.html"}, {"id": "Mathlib.LinearAlgebra.FreeModule.Int", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -23.333, "z": 66.603, "size": 0.2, "title": "Index of submodules of free ℤ-modules (considered as an `AddSubgroup`).", "summary": "This file provides lemmas about when a submodule of a free ℤ-module is a subgroup of finite index.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeModule/Int.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.Defs", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.3003, "macro_tier_override": null, "x": 6.92, "z": 21.184, "size": 0.3291, "title": "Unique factorization", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/Defs.html"}, {"id": "Mathlib.Algebra.Order.Group.Int.Sum", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 24.553, "z": 8.553, "size": 0.2, "title": "Sharp bounds for sums of bounded finsets of integers", "summary": "The sum of a finset of integers with cardinality `s` where all elements are at most `c` can be given a sharper upper bound than `#s * c`, because the elements are distinct. This file provides these sharp bounds, both in the upper-bounded and analogous lower-bounded cases.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Int/Sum.html"}, {"id": "Mathlib.FieldTheory.Finite.Trace", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 2, "macro_tier_score": 0.0069, "macro_tier_override": null, "x": -30.99, "z": 116.67, "size": 0.2599, "title": "The trace and norm maps for finite fields", "summary": "We state several lemmas about the trace and norm maps for finite fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Finite/Trace.html"}, {"id": "Mathlib.FieldTheory.Finite.GaloisField", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 5, "macro_tier_score": 0.0117, "macro_tier_override": 5, "x": 68.088, "z": 95.149, "size": 0.3052, "title": "Galois fields", "summary": "If `p` is a prime number, and `n` a natural number, then `GaloisField p n` is defined as the splitting field of `X^(p^n) - X` over `ZMod p`. It is a finite field with `p ^ n` elements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Finite/GaloisField.html"}, {"id": "Mathlib.RingTheory.FractionalIdeal.Basic", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 3, "macro_tier_score": 0.0512, "macro_tier_override": null, "x": -26.216, "z": 77.396, "size": 0.2615, "title": "Fractional ideals", "summary": "This file defines fractional ideals of an integral domain and proves basic facts about them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/FractionalIdeal/Basic.html"}, {"id": "Mathlib.LinearAlgebra.ExteriorPower.Pairing", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": 81.509, "z": -61.561, "size": 0.2484, "title": "The pairing between the exterior power of the dual and the exterior power", "summary": "We construct the pairing `exteriorPower.pairingDual : ⋀[R]^n (Module.Dual R M) →ₗ[R] (Module.Dual R (⋀[R]^n M))`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/ExteriorPower/Pairing.html"}, {"id": "Mathlib.LinearAlgebra.ExteriorPower.Basic", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 2, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -43.211, "z": 90.5, "size": 0.2745, "title": "Exterior powers", "summary": "We study the exterior powers of a module `M` over a commutative ring `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/ExteriorPower/Basic.html"}, {"id": "Mathlib.LinearAlgebra.TensorPower.Pairing", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -54.965, "z": -22.599, "size": 0.2572, "title": "The pairing between the tensor power of the dual and the tensor power", "summary": "We construct the pairing `TensorPower.pairingDual : ⨂[R]^n (Module.Dual R M) →ₗ[R] (Module.Dual R (⨂[R]^n M))`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorPower/Pairing.html"}, {"id": "Mathlib.RingTheory.PiTensorProduct", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -1.834, "z": -55.685, "size": 0.2593, "title": "Tensor product of `R`-algebras and rings", "summary": "If `(Aᵢ)` is a family of `R`-algebras then the `R`-tensor product `⨂ᵢ Aᵢ` is an `R`-algebra as well with structure map defined by `r ↦ r • 1`. In particular if we take `R` to be `ℤ`, then this collapses into the tensor product of rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PiTensorProduct.html"}, {"id": "Mathlib.Algebra.Lie.Abelian", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 1, "macro_tier_score": 0.0039, "macro_tier_override": null, "x": 3.552, "z": -77.92, "size": 0.375, "title": "Trivial Lie modules and Abelian Lie algebras", "summary": "The action of a Lie algebra `L` on a module `M` is trivial if `⁅x, m⁆ = 0` for all `x ∈ L` and `m ∈ M`. In the special case that `M = L` with the adjoint action, triviality corresponds to the concept of an Abelian Lie algebra. In this file we define these concepts and provide some related definitions and results.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Abelian.html"}, {"id": "Mathlib.Algebra.Lie.Quotient", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -33.917, "z": 68.173, "size": 0.2756, "title": "Quotients of Lie algebras and Lie modules", "summary": "Given a Lie submodule of a Lie module, the quotient carries a natural Lie module structure. In the special case that the Lie module is the Lie algebra itself via the adjoint action, the submodule is a Lie ideal and the quotient carries a natural Lie algebra structure. We define these quotient structures here. A notable omission at the time of writing (February 2021) is a statement and proof of the universal property…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Quotient.html"}, {"id": "Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 101.478, "z": 11.643, "size": 0.2541, "title": "Functoriality of group homology", "summary": "Given a commutative ring `k`, a group homomorphism `f : G →* H`, a `k`-linear `G`-representation `A`, a `k`-linear `H`-representation `B`, and a representation morphism `A ⟶ Res(f)(B)`, we get a chain map `inhomogeneousChains A ⟶ inhomogeneousChains B` and hence maps on homology `Hₙ(G, A) ⟶ Hₙ(H, B)`. We also provide extra API for these maps in degrees 0, 1, 2.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.html"}, {"id": "Mathlib.LinearAlgebra.SesquilinearForm.Star", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -15.653, "z": -87.759, "size": 0.2, "title": "Sesquilinear forms over a star ring", "summary": "This file provides some properties about sesquilinear forms `M →ₗ⋆[R] M →ₗ[R] R` when `R` is a `StarRing`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/SesquilinearForm/Star.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.PosDef", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 2, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 87.092, "z": 5.822, "size": 0.3122, "title": "Positive Definite Matrices", "summary": "This file defines positive (semi)definite matrices and connects the notion to positive definiteness of quadratic forms. In `Mathlib/Analysis/Matrix/Order.lean`, positive semi-definiteness is used to define the partial order on matrices on `ℝ` or `ℂ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/PosDef.html"}, {"id": "Mathlib.RingTheory.DividedPowerAlgebra.Init", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -24.98, "z": 66.003, "size": 0.2, "title": "The universal divided power algebra", "summary": "Let `R` be a (commutative) semiring and `M` be an `R`-module. In this file we define `Γ_R(M)`, the universal divided power algebra of `M`, as the ring quotient of the polynomial ring in the variables `ℕ × M` by the relation `DividedPowerAlgebra.Rel`. `DividedPowerAlgebra R M` satisfies a weak universal property for morphisms to rings with divided powers (`DividedPowerAlgebra.lift`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DividedPowerAlgebra/Init.html"}, {"id": "Mathlib.RingTheory.DividedPowers.Basic", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": -3.081, "z": 68.646, "size": 0.3061, "title": "Divided powers", "summary": "Let `A` be a commutative (semi)ring and `I` be an ideal of `A`. A *divided power* structure on `I` is the datum of operations `a n ↦ dpow a n` satisfying relations that model the intuitive formula `dpow n a = a ^ n / n !` and collected by the structure `DividedPowers`. The list of axioms is embedded in the structure: To avoid coercions, we rather consider `DividedPowers.dpow : ℕ → A → A`, extended by 0. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DividedPowers/Basic.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Free", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -57.79, "z": 13.862, "size": 0.2, "title": "Tensor product with free modules.", "summary": "This file contains lemmas about tensoring with free modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Free.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.MapDomain", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.3284, "macro_tier_override": null, "x": 6.354, "z": -17.451, "size": 0.3533, "title": "Maps of monoid algebras", "summary": "This file defines maps of monoid algebras along both the ring and monoid arguments.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/MapDomain.html"}, {"id": "Mathlib.Algebra.Homology.SpectralObject.Cycles", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 12.391, "z": 11.217, "size": 0.255, "title": "Kernel and cokernel of the differential of a spectral object", "summary": "Let `X` be a spectral object indexed by the category `ι` in the abelian category `C`. In this file, we introduce the kernel `X.cycles` and the cokernel `X.opcycles` of `X.δ`. These are defined when `f` and `g` are composable morphisms in `ι` and for any integer `n`. In the documentation, the kernel `X.cycles n f g` of `δ : H^n(g) ⟶ H^{n+1}(f)` shall be denoted `Z^n(f, g)`, and the cokernel `X.opcycles n f g` of `δ :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/SpectralObject/Cycles.html"}, {"id": "Mathlib.Algebra.Homology.SpectralObject.Basic", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -14.493, "z": 3.271, "size": 0.2641, "title": "Spectral objects in abelian categories", "summary": "In this file, we introduce the category `SpectralObject C ι` of spectral objects in an abelian category `C` indexed by the category `ι`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/SpectralObject/Basic.html"}, {"id": "Mathlib.Algebra.Homology.BifunctorFlip", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -26.635, "z": -8.163, "size": 0.2478, "title": "Action of the flip of a bifunctor on homological complexes", "summary": "Given `K₁ : HomologicalComplex C₁ c₁`, `K₂ : HomologicalComplex C₂ c₂`, a bifunctor `F : C₁ ⥤ C₂ ⥤ D`, and a complex shape `c` with `[TotalComplexShape c₁ c₂ c]` and `[TotalComplexShape c₂ c₁ c]`, we define an isomorphism `mapBifunctor K₂ K₁ F.flip c ≅ mapBifunctor K₁ K₂ F c` under the additional assumption `[TotalComplexShapeSymmetry c₁ c₂ c]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/BifunctorFlip.html"}, {"id": "Mathlib.Algebra.Homology.Bifunctor", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 12.17, "z": -22.976, "size": 0.2813, "title": "The action of a bifunctor on homological complexes", "summary": "Given a bifunctor `F : C₁ ⥤ C₂ ⥤ D` and complexes shapes `c₁ : ComplexShape I₁` and `c₂ : ComplexShape I₂`, we define a bifunctor `mapBifunctorHomologicalComplex F c₁ c₂` of type `HomologicalComplex C₁ c₁ ⥤ HomologicalComplex C₂ c₂ ⥤ HomologicalComplex₂ D c₁ c₂`. Then, when `K₁ : HomologicalComplex C₁ c₁`, `K₂ : HomologicalComplex C₂ c₂` and `c : ComplexShape J` are such that we have `TotalComplexShape c₁ c₂ c`, we…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Bifunctor.html"}, {"id": "Mathlib.Algebra.Homology.TotalComplexSymmetry", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -2.3, "z": -25.898, "size": 0.257, "title": "The symmetry of the total complex of a bicomplex", "summary": "Let `K : HomologicalComplex₂ C c₁ c₂` be a bicomplex. If we assume both `[TotalComplexShape c₁ c₂ c]` and `[TotalComplexShape c₂ c₁ c]`, we may form the total complex `K.total c` and `K.flip.total c`. In this file, we show that if we assume `[TotalComplexShapeSymmetry c₁ c₂ c]`, then there is an isomorphism `K.totalFlipIso c : K.flip.total c ≅ K.total c`. Moreover, if we also have `[TotalComplexShapeSymmetry c₂ c₁…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/TotalComplexSymmetry.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.FiniteDimensional", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 49.691, "z": -64.87, "size": 0.2594, "title": "The finite-dimensional space of matrices", "summary": "This file shows that `m` by `n` matrices form a finite-dimensional space. Note that this is proven more generally elsewhere over modules as `Module.Finite.matrix`; this file exists only to provide an entry in the instance list for `FiniteDimensional`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/FiniteDimensional.html"}, {"id": "Mathlib.RingTheory.Smooth.Field", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -58.263, "z": -107.84, "size": 0.2473, "title": "Smooth algebras over fields", "summary": "We show that separably generated extensions of fields are smooth. In particular finitely generated field extensions over perfect fields are smooth.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/Field.html"}, {"id": "Mathlib.NumberTheory.Harmonic.Int", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 60.803, "z": -17.034, "size": 0.2, "title": null, "summary": "The nth Harmonic number is not an integer. We formalize the proof using 2-adic valuations. This proof is due to Kürschák. Reference: https://kconrad.math.uconn.edu/blurbs/gradnumthy/padicharmonicsum.pdf", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Harmonic/Int.html"}, {"id": "Mathlib.RingTheory.Smooth.StandardSmoothOfFree", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": -109.094, "z": -10.231, "size": 0.2483, "title": "Standard smooth of free Kaehler differentials", "summary": "In this file we show a presentation independent characterization of being standard smooth: An `R`-algebra `S` of finite presentation is standard smooth if and only if `H¹(S/R) = 0` and `Ω[S⁄R]` is free on `{d sᵢ}ᵢ` for some `sᵢ : S`. From this we deduce relations of standard smooth with other local properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/StandardSmoothOfFree.html"}, {"id": "Mathlib.RingTheory.Extension.Cotangent.Basis", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": -86.899, "z": -63.649, "size": 0.2467, "title": "Basis of cotangent space can be realized as a presentation", "summary": "Let `S` be a finitely presented `R`-algebra and suppose `P : R[X] → S` generates `S` with kernel `I`. In this file we show `Algebra.Generators.exists_presentation_of_free`: If `I/I²` is free, there exists an `R`-presentation `P'` of `S` extending `P` with kernel `I'`, such that `I'/I'²` is free on the images of the relations of `P'`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Extension/Cotangent/Basis.html"}, {"id": "Mathlib.RingTheory.Extension.Cotangent.Free", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": -75.366, "z": -51.002, "size": 0.2467, "title": "Computation of Jacobian of presentations from basis of Cotangent", "summary": "Let `P` be a presentation of an `R`-algebra `S` with kernel `I = (fᵢ)`. In this file we provide lemmas to show that `P` is submersive when given suitable bases of `I/I²` and `Ω[S⁄R]`. We will later deduce from this a presentation-independent characterisation of standard smooth algebras (TODO @chrisflav).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Extension/Cotangent/Free.html"}, {"id": "Mathlib.RingTheory.Smooth.Locus", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.0072, "macro_tier_override": null, "x": 104.076, "z": -19.341, "size": 0.2957, "title": "Smooth locus of an algebra", "summary": "Most results in this file are proved for algebras of finite presentations. Some of them are true for arbitrary algebras but the proof is substantially harder.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/Locus.html"}, {"id": "Mathlib.Algebra.Group.Irreducible.Indecomposable", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -29.269, "z": -16.15, "size": 0.2319, "title": "Indecomposable elements of monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Irreducible/Indecomposable.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Schroder", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -60.763, "z": -32.088, "size": 0.2, "title": "Schröder Numbers Power Series", "summary": "This file defines lemmas and theorems about the power series for large and small Schröder numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Schroder.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.Pi", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.3881, "macro_tier_override": null, "x": 1.809, "z": -14.747, "size": 0.3484, "title": "Pi instances for multiplicative actions with zero", "summary": "This file defines instances for `MulActionWithZero` and related structures on `Pi` types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/Pi.html"}, {"id": "Mathlib.RingTheory.Teichmuller", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 12.646, "z": -101.358, "size": 0.2796, "title": "Teichmüller map", "summary": "Let `R` be an `I`-adically complete ring, and `p` be a prime number with `p ∈ I`. Then there is a canonical map `Perfection (R ⧸ I) p →*₀ R` that we shall call `Perfection.teichmuller`, such that it composed with the quotient map `R →+* R ⧸ I` is the \"0-th coefficient\" map `Perfection (R ⧸ I) p →+* R ⧸ I`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Teichmuller.html"}, {"id": "Mathlib.LinearAlgebra.SModEq.Pow", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -25.29, "z": 51.72, "size": 0.2575, "title": "Lemmas about SModEq related to powers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/SModEq/Pow.html"}, {"id": "Mathlib.RingTheory.Valuation.ValuativeRel.Basic", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 2, "macro_tier_score": 0.0072, "macro_tier_override": null, "x": -33.02, "z": 51.631, "size": 0.2996, "title": "Valuative Relations", "summary": "In this file we introduce a class called `ValuativeRel R` for a ring `R`. This bundles a relation `vle : R → R → Prop` on `R` which mimics a preorder on `R` arising from a valuation. We introduce the notation `x ≤ᵥ y` for this relation. Recall that the equivalence class of a valuation is *completely* characterized by such a preorder. Thus, we can think of `ValuativeRel R` as a way of saying that `R` is endowed with…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/ValuativeRel/Basic.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Adjunctions", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 1, "macro_tier_score": 0.0029, "macro_tier_override": null, "x": -50.172, "z": -31.853, "size": 0.31, "title": null, "summary": "The functor of forming finitely supported functions on a type with values in a `[Ring R]` is the left adjoint of the forgetful functor from `R`-modules to types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Adjunctions.html"}, {"id": "Mathlib.Algebra.Polynomial.Derivation", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.0371, "macro_tier_override": null, "x": 44.554, "z": -47.329, "size": 0.2998, "title": "Derivations of univariate polynomials", "summary": "In this file we prove that an `R`-derivation of `Polynomial R` is determined by its value on `X`. We also provide a constructor `Polynomial.mkDerivation` that builds a derivation from its value on `X`, and a linear equivalence `Polynomial.mkDerivationEquiv` between `A` and `Derivation (Polynomial R) A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Derivation.html"}, {"id": "Mathlib.Algebra.Order.BigOperators.Expect", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -24.19, "z": -28.186, "size": 0.2948, "title": "Order properties of the average over a finset", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/BigOperators/Expect.html"}, {"id": "Mathlib.Algebra.Order.Module.Rat", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -25.875, "z": 2.551, "size": 0.2552, "title": "Monotonicity of the action by rational numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Module/Rat.html"}, {"id": "Mathlib.Algebra.TrivSqZeroExt", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -42.818, "z": -41.213, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/TrivSqZeroExt.html"}, {"id": "Mathlib.NumberTheory.AlmostPrime", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 42.252, "z": 6.272, "size": 0.2, "title": "Almost prime numbers", "summary": "This file defines `Nat.IsAlmostPrime k n`, the predicate that `n` has exactly `k` prime factors counted with multiplicity. We also define `Nat.IsAtMostAlmostPrime`, the corresponding predicate with at most `k` prime factors, and `Nat.IsSemiprime`, the special case of `2`-almost-prime numbers. Both definitions use the arithmetic function `ArithmeticFunction.cardFactors`, written `Ω`. The terminology follows the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/AlmostPrime.html"}, {"id": "Mathlib.Algebra.Module.GradedModule", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -50.378, "z": 34.901, "size": 0.2, "title": "Graded Module", "summary": "Given an `R`-algebra `A` graded by `𝓐`, a graded `A`-module `M` is expressed as `DirectSum.Decomposition 𝓜` and `SetLike.GradedSMul 𝓐 𝓜`. Then `⨁ i, 𝓜 i` is an `A`-module and is isomorphic to `M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/GradedModule.html"}, {"id": "Mathlib.FieldTheory.Galois.NormalBasis", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 5, "macro_tier_score": 0.0, "macro_tier_override": 5, "x": -101.386, "z": -54.581, "size": 0.2, "title": "The normal basis theorem", "summary": "We prove the normal basis theorem `IsGalois.normalBasis`: every finite Galois extension has a basis that is an orbit under the Galois group action. The proof follows [ConradLinearChar] Keith Conrad, *Linear Independence of Characters*.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Galois/NormalBasis.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.Multiset.Basic", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 2, "macro_tier_score": 0.0065, "macro_tier_override": null, "x": -15.441, "z": -6.4, "size": 0.5418, "title": "Sums and products over multisets", "summary": "In this file we define products and sums indexed by multisets. This is later used to define products and sums indexed by finite sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Multiset/Basic.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.List.Lemmas", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -14.75, "z": -1.779, "size": 0.3577, "title": "Sums and products from lists", "summary": "This file provides further results about `List.prod`, `List.sum`, which calculate the product and sum of elements of a list and `List.alternatingProd`, `List.alternatingSum`, their alternating counterparts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/List/Lemmas.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.BaseChange", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -80.356, "z": -68.913, "size": 0.2685, "title": "Base change for root pairings", "summary": "When the coefficients are a field, root pairings behave well with respect to restriction and extension of scalars.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/BaseChange.html"}, {"id": "Mathlib.RingTheory.IsGaloisGroup.Basic", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 2, "macro_tier_score": 0.0113, "macro_tier_override": null, "x": -24.247, "z": 97.311, "size": 0.2565, "title": "Galois Groups of Rings", "summary": "Given an action of a group `G` on an extension of rings `B/A`, the predicate `IsGaloisGroup G A B` states that `G` acts faithfully on `B` with fixed ring `A`. This file develops some of the theory of this predicate without assuming Galois theory for fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IsGaloisGroup/Basic.html"}, {"id": "Mathlib.RingTheory.IsGaloisGroup.Defs", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 2, "macro_tier_score": 0.0123, "macro_tier_override": null, "x": -23.579, "z": 56.569, "size": 0.2492, "title": "Predicate for Galois Groups", "summary": "Given an action of a group `G` on an extension of fields `L/K`, we introduce a predicate `IsGaloisGroup G K L` saying that `G` acts faithfully on `L` with fixed field `K`. In particular, we do not assume that `L` is an algebraic extension of `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IsGaloisGroup/Defs.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.Prod", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.3894, "macro_tier_override": null, "x": 18.553, "z": -0.824, "size": 0.3582, "title": "Prod instances for multiplicative actions with zero", "summary": "This file defines instances for `MulActionWithZero` and related structures on `α × β`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/Prod.html"}, {"id": "Mathlib.RingTheory.RootsOfUnity.AlgebraicallyClosed", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -113.766, "z": 45.623, "size": 0.2403, "title": "Instances for HasEnoughRootsOfUnity", "summary": "We provide an instance for `HasEnoughRootsOfUnity F n` when `F` is a separably closed field and `n` is not divisible by the characteristic. In particular, when `F` has characteristic zero, this hold for all `n ≠ 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RootsOfUnity/AlgebraicallyClosed.html"}, {"id": "Mathlib.Algebra.Lie.Killing", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -73.895, "z": 90.713, "size": 0.2608, "title": "Lie algebras with non-degenerate Killing forms.", "summary": "In characteristic zero, the following three conditions are equivalent: 1. The solvable radical of a Lie algebra is trivial 2. A Lie algebra is a direct sum of its simple ideals 3. A Lie algebra has non-degenerate Killing form In positive characteristic, it is still true that 3 implies 2, and that 2 implies 1, but there are counterexamples to the remaining implications. Thus condition 3 is the strongest assumption.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Killing.html"}, {"id": "Mathlib.RingTheory.Artinian.Defs", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 3, "macro_tier_score": 0.0974, "macro_tier_override": null, "x": -23.887, "z": -23.386, "size": 0.331, "title": "Artinian rings and modules", "summary": "A module satisfying these equivalent conditions is said to be an *Artinian* R-module if every decreasing chain of submodules is eventually constant, or equivalently, if the relation `<` on submodules is well founded. A ring is said to be left (or right) Artinian if it is Artinian as a left (or right) module over itself, or simply Artinian if it is both left and right Artinian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Artinian/Defs.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.Trunc", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 2, "macro_tier_score": 0.0112, "macro_tier_override": null, "x": 23.464, "z": -60.618, "size": 0.2482, "title": "Formal (multivariate) power series - Truncation", "summary": "* `MvPowerSeries.truncFinset s p` restricts the support of a multivariate power series `p` to a finite set of monomials and obtains a multivariate polynomial. * `MvPowerSeries.trunc n φ` truncates a formal multivariate power series to the multivariate polynomial that has the same coefficients as `φ`, for all `m < n`, and `0` otherwise. Note that here, `m` and `n` have types `σ →₀ ℕ`, so that `m < n` means that `m ≠…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/Trunc.html"}, {"id": "Mathlib.RingTheory.LocalIso", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -89.535, "z": -36.191, "size": 0.2, "title": "Local isomorphisms", "summary": "A ring homomorphism is a local isomorphism if source locally (in the geometric sense) it is a standard open immersion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalIso.html"}, {"id": "Mathlib.Algebra.Order.Interval.Finset.Basic", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.2767, "macro_tier_override": null, "x": 10.627, "z": -10.383, "size": 0.3042, "title": "Algebraic properties of finset intervals", "summary": "This file provides results about the interaction of algebra with `Finset.Ixx`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Interval/Finset/Basic.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.Unitization", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 36.279, "z": 49.395, "size": 0.2, "title": "Relating unital and non-unital substructures", "summary": "This file relates various algebraic structures and provides maps (generally algebra homomorphisms), from the unitization of a non-unital subobject into the full structure. The range of this map is the unital closure of the non-unital subobject (e.g., `Algebra.adjoin`, `Subring.closure`, `Subsemiring.closure` or `StarAlgebra.adjoin`). When the underlying scalar ring is a field, for this map to be injective it…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/Unitization.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Basis", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 1, "macro_tier_score": 0.0021, "macro_tier_override": null, "x": -5.398, "z": -59.184, "size": 0.3349, "title": "Affine bases and barycentric coordinates", "summary": "Suppose `P` is an affine space modelled on the module `V` over the ring `k`, and `p : ι → P` is an affine-independent family of points spanning `P`. Given this data, each point `q : P` may be written uniquely as an affine combination: `q = w₀ p₀ + w₁ p₁ + ⋯` for some (finitely-supported) weights `wᵢ`. For each `i : ι`, we thus have an affine map `P →ᵃ[k] k`, namely `q ↦ wᵢ`. This family of maps is known as the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Basis.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Pointwise", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -39.914, "z": 38.871, "size": 0.2707, "title": "Pointwise instances on `AffineSubspace`s", "summary": "This file provides the additive action `AffineSubspace.pointwiseAddAction` in the `Pointwise` locale.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Pointwise.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.Basic", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 2, "macro_tier_score": 0.0069, "macro_tier_override": null, "x": 6.134, "z": -9.303, "size": 0.3563, "title": "Embeddings of complex shapes", "summary": "Given two complex shapes `c : ComplexShape ι` and `c' : ComplexShape ι'`, an embedding from `c` to `c'` (`e : c.Embedding c'`) consists of the data of an injective map `f : ι → ι'` such that for all `i₁ i₂ : ι`, `c.Rel i₁ i₂` implies `c'.Rel (e.f i₁) (e.f i₂)`. We define a type class `e.IsRelIff` to express that this implication is an equivalence. Other type classes `e.IsTruncLE` and `e.IsTruncGE` are introduced in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/Basic.html"}, {"id": "Mathlib.Algebra.Homology.Opposite", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 2, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": 16.356, "z": 8.798, "size": 0.3394, "title": "Opposite categories of complexes", "summary": "Given a preadditive category `V`, the opposite of its category of chain complexes is equivalent to the category of cochain complexes of objects in `Vᵒᵖ`. We define this equivalence, and another analogous equivalence (for a general category of homological complexes with a general complex shape). We then show that when `V` is abelian, if `C` is a homological complex, then the homology of `op(C)` is isomorphic to `op`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Opposite.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Ext.HasExt", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": 61.725, "z": -20.374, "size": 0.252, "title": "HasExt instance for Module Category", "summary": "If we assume `Small.{v} R`, the category `ModuleCat.{v} R` has enough projectives, which allows to introduce the instance `HasExt.{v} (ModuleCat.{v} R)`. As a result, `Ext`-groups in this category of modules are defined and belong to `Type v`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Ext/HasExt.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.Degree", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 4, "macro_tier_score": 0.2881, "macro_tier_override": null, "x": -27.009, "z": -42.248, "size": 0.3946, "title": "Lemmas about the `sup` and `inf` of the support of `AddMonoidAlgebra`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/Degree.html"}, {"id": "Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 2, "macro_tier_score": 0.0274, "macro_tier_override": null, "x": 0.308, "z": 94.715, "size": 0.3157, "title": "Properties of faithfully flat algebras", "summary": "An `A`-algebra `B` is faithfully flat if `B` is faithfully flat as an `A`-module. In this file we give equivalent characterizations of faithful flatness in the algebra case.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Flat/FaithfullyFlat/Algebra.html"}, {"id": "Mathlib.RingTheory.Coprime.Ideal", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.0813, "macro_tier_override": null, "x": 45.889, "z": -34.766, "size": 0.2833, "title": "An additional lemma about coprime ideals", "summary": "This lemma generalises `exists_sum_eq_one_iff_pairwise_coprime` to the case of non-principal ideals. It is on a separate file due to import requirements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Coprime/Ideal.html"}, {"id": "Mathlib.GroupTheory.Submonoid.Inverses", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 2, "macro_tier_score": 0.0214, "macro_tier_override": null, "x": -22.327, "z": 9.186, "size": 0.2743, "title": "Submonoid of inverses", "summary": "Given a submonoid `N` of a monoid `M`, we define the submonoid `N.leftInv` as the submonoid of left inverses of `N`. When `M` is commutative, we may define `fromCommLeftInv : N.leftInv →* N` since the inverses are unique. When `N ≤ IsUnit.Submonoid M`, this is precisely the pointwise inverse of `N`, and we may define `leftInvEquiv : S.leftInv ≃* S`. For the pointwise inverse of submonoids of groups, please refer to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Submonoid/Inverses.html"}, {"id": "Mathlib.RepresentationTheory.Action", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 2, "macro_tier_score": 0.0124, "macro_tier_override": null, "x": 56.679, "z": 68.805, "size": 0.2582, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Action.html"}, {"id": "Mathlib.RepresentationTheory.Equiv", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 2, "macro_tier_score": 0.0124, "macro_tier_override": null, "x": 55.896, "z": 69.442, "size": 0.2582, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Equiv.html"}, {"id": "Mathlib.Algebra.Symmetrized", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 50.96, "z": -59.053, "size": 0.2, "title": "Symmetrized algebra", "summary": "A commutative multiplication on a real or complex space can be constructed from any multiplication by \"symmetrization\" i.e. $$ a \\circ b = \\frac{1}{2}(ab + ba) $$ We provide the symmetrized version of a type `α` as `SymAlg α`, with notation `αˢʸᵐ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Symmetrized.html"}, {"id": "Mathlib.Algebra.Jordan.Basic", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -65.859, "z": 38.215, "size": 0.2676, "title": "Jordan rings", "summary": "Let `A` be a non-unital, non-associative ring. Then `A` is said to be a (commutative, linear) Jordan ring if the multiplication is commutative and satisfies a weak associativity law known as the Jordan Identity: for all `a` and `b` in `A`, ``` (a * b) * a^2 = a * (b * a^2) ``` i.e. the operators of multiplication by `a` and `a^2` commute. A more general concept of a (non-commutative) Jordan ring can also be defined,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Jordan/Basic.html"}, {"id": "Mathlib.Algebra.Group.ForwardDiff", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": 39.131, "z": -47.168, "size": 0.227, "title": "Forward difference operators and Newton series", "summary": "We define the forward difference operator, sending `f` to the function `x ↦ f (x + h) - f x` for a given `h` (for any additive semigroup, taking values in an abelian group). The notation `Δ_[h]` is defined for this operator, scoped in namespace `fwdDiff`. We prove two key formulae about this operator: * `shift_eq_sum_fwdDiff_iter`: the **Gregory-Newton formula**, expressing `f (y + n • h)` as a linear combination of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/ForwardDiff.html"}, {"id": "Mathlib.NumberTheory.Padics.ProperSpace", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": 9.246, "z": -79.321, "size": 0.227, "title": "Properness of the p-adic numbers", "summary": "In this file, we prove that `ℤ_[p]` is totally bounded and compact, and that `ℚ_[p]` is proper.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/ProperSpace.html"}, {"id": "Mathlib.RingTheory.QuasiFinite.Basic", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 2, "macro_tier_score": 0.0175, "macro_tier_override": null, "x": -79.887, "z": 63.651, "size": 0.3282, "title": "Quasi-finite algebras", "summary": "In this file, we define the notion of quasi-finite algebras and prove basic properties about them", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/QuasiFinite/Basic.html"}, {"id": "Mathlib.Algebra.Order.Archimedean.IndicatorCard", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 2.868, "z": -25.842, "size": 0.2403, "title": "Cardinality and limit of sum of indicators", "summary": "This file contains results relating the cardinality of subsets of ℕ and limits, limsups of sums of indicators.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Archimedean/IndicatorCard.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Vanishing", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 2, "macro_tier_score": 0.0245, "macro_tier_override": null, "x": 66.706, "z": -4.506, "size": 0.2455, "title": "Vanishing of elements in a tensor product of two modules", "summary": "Let $M$ and $N$ be modules over a commutative ring $R$. Recall that every element of $M \\otimes N$ can be written as a finite sum $\\sum_{i} m_i \\otimes n_i$ of pure tensors (`TensorProduct.exists_finset`). We would like to determine under what circumstances such an expression vanishes. Let us say that an expression $\\sum_{i \\in \\iota} m_i \\otimes n_i$ in $M \\otimes N$ *vanishes trivially*…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Vanishing.html"}, {"id": "Mathlib.GroupTheory.CommutingProbability", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 19.123, "z": -42.308, "size": 0.2, "title": "Commuting Probability", "summary": "This file introduces the commuting probability of finite groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/CommutingProbability.html"}, {"id": "Mathlib.GroupTheory.Abelianization.Finite", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -29.138, "z": 16.384, "size": 0.2427, "title": "The abelianization of a finite group is finite", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Abelianization/Finite.html"}, {"id": "Mathlib.GroupTheory.SpecificGroups.Dihedral", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -14.247, "z": 42.234, "size": 0.274, "title": "Dihedral Groups", "summary": "We define the dihedral groups `DihedralGroup n`, with elements `r i` and `sr i` for `i : ZMod n`. For `n ≠ 0`, `DihedralGroup n` represents the symmetry group of the regular `n`-gon. `r i` represents the rotations of the `n`-gon by `2πi/n`, and `sr i` represents the reflections of the `n`-gon. `DihedralGroup 0` corresponds to the infinite dihedral group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/SpecificGroups/Dihedral.html"}, {"id": "Mathlib.RingTheory.DividedPowers.Padic", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -74.101, "z": 24.354, "size": 0.2, "title": "Divided powers on `ℤ_[p]`", "summary": "Given a divided power algebra `(B, J, δ)` and an injective ring morphism `f : A →+* B`, if `I` is an `A`-ideal such that `I.map f = J` and such that for all `n : ℕ`, `x ∈ I`, the preimage of `hJ.dpow n (f x)` under `f` belongs to `I`, we get an induced divided power structure on `I`. We specialize this construction to the coercion map `ℤ_[p] →+* ℚ_[p]` to get a divided power structure on the ideal `(p) ⊆ ℤ_[p]`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DividedPowers/Padic.html"}, {"id": "Mathlib.RingTheory.DividedPowers.RatAlgebra", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -34.681, "z": 61.462, "size": 0.2478, "title": "Examples of divided power structures", "summary": "In this file we show that, for certain choices of a commutative (semi)ring `A` and an ideal `I` of `A`, the family of maps `ℕ → A → A` given by `fun n x ↦ x^n/n!` is a divided power structure on `I`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DividedPowers/RatAlgebra.html"}, {"id": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.Basic", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 46.323, "z": -45.599, "size": 0.252, "title": "Eisenstein series are Modular Forms", "summary": "We show that Eisenstein series of weight `k` and level `Γ(N)` with congruence condition `a : Fin 2 → ZMod N` are Modular Forms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/EisensteinSeries/Basic.html"}, {"id": "Mathlib.NumberTheory.ModularForms.Basic", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -34.421, "z": -52.937, "size": 0.27, "title": "Modular forms", "summary": "This file defines modular forms and proves some basic properties about them. Including constructing the graded ring of modular forms. We begin by defining modular forms and cusp forms as extension of `SlashInvariantForm`s then we define the space of modular forms, cusp forms and prove that the product of two modular forms is a modular form.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/Basic.html"}, {"id": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.MDifferentiable", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -6.177, "z": 36.626, "size": 0.2477, "title": "Holomorphicity of Eisenstein series", "summary": "We show that Eisenstein series of weight `k` and level `Γ(N)` with congruence condition `a : Fin 2 → ZMod N` are holomorphic on the upper half plane, which is stated as being MDifferentiable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/EisensteinSeries/MDifferentiable.html"}, {"id": "Mathlib.Algebra.Order.Antidiag.FinsuppEquiv", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 22.231, "z": 16.787, "size": 0.2, "title": "Equivalence between `Finset.finsuppAntidiag` and `Sym`", "summary": "This file collects further results about equivalence and cardinality related to `Finset.finsuppAntidiag`. This file is separated from `Mathlib.Algebra.Order.Antidiag.Finsupp` to reduce imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Antidiag/FinsuppEquiv.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.HomEquiv", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 2, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 11.375, "z": 21.295, "size": 0.3134, "title": "Relations between `extend` and `restriction`", "summary": "Given an embedding `e : Embedding c c'` of complex shapes satisfying `e.IsRelIff`, we obtain a bijection `e.homEquiv` between the type of morphisms `K ⟶ L.extend e` (with `K : HomologicalComplex C c'` and `L : HomologicalComplex C c`) and the subtype of morphisms `φ : K.restriction e ⟶ L` which satisfy a certain condition `e.HasLift φ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/HomEquiv.html"}, {"id": "Mathlib.GroupTheory.CoprodI", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 25.969, "z": 10.081, "size": 0.239, "title": "The coproduct (a.k.a. the free product) of groups or monoids", "summary": "Given an `ι`-indexed family `M` of monoids, we define their coproduct (a.k.a. free product) `Monoid.CoprodI M`. As usual, we use the suffix `I` for an indexed (co)product, leaving `Coprod` for the coproduct of two monoids. When `ι` and all `M i` have decidable equality, the free product bijects with the type `Monoid.CoprodI.Word M` of reduced words. This bijection is constructed by defining an action of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/CoprodI.html"}, {"id": "Mathlib.GroupTheory.FreeGroup.IsFreeGroup", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": 25.233, "z": -6.268, "size": 0.2663, "title": "Free groups structures on arbitrary types", "summary": "This file defines the notion of free basis of a group, which induces an isomorphism between the group and the free group generated by the basis. It also introduced a type class for groups which are free groups, i.e., for which some free basis exists. For the explicit construction of free groups, see `GroupTheory/FreeGroup`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FreeGroup/IsFreeGroup.html"}, {"id": "Mathlib.Algebra.Group.Commutator", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 4, "macro_tier_score": 0.4356, "macro_tier_override": null, "x": 5.357, "z": 1.533, "size": 0.3322, "title": "The bracket on a group given by commutator.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Commutator.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.MulOpposite", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.0067, "macro_tier_override": null, "x": 45.117, "z": 35.763, "size": 0.2245, "title": "Subalgebras of opposite rings", "summary": "For every ring `A` over a commutative ring `R`, we construct an equivalence between subalgebras of `A / R` and that of `Aᵐᵒᵖ / R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.Rank", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.0067, "macro_tier_override": null, "x": 12.122, "z": 78.933, "size": 0.2245, "title": "Some results on the ranks of subalgebras", "summary": "This file contains some results on the ranks of subalgebras, which are corollaries of `rank_mul_rank`. Since their proof essentially depends on the fact that a non-trivial commutative ring satisfies the strong rank condition, we put them into a separate file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/Rank.html"}, {"id": "Mathlib.LinearAlgebra.LinearDisjoint", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 2, "macro_tier_score": 0.0089, "macro_tier_override": null, "x": 58.332, "z": -57.225, "size": 0.2365, "title": "Linearly disjoint submodules", "summary": "This file contains basics about linearly disjoint submodules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/LinearDisjoint.html"}, {"id": "Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 3, "macro_tier_score": 0.1805, "macro_tier_override": null, "x": 45.162, "z": 58.982, "size": 0.3883, "title": "Properties of integral elements.", "summary": "We prove basic properties of integral elements in a ring extension.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.html"}, {"id": "Mathlib.RingTheory.TensorProduct.Nontrivial", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 2, "macro_tier_score": 0.0067, "macro_tier_override": null, "x": 84.066, "z": 15.201, "size": 0.2245, "title": "Nontriviality of tensor product of algebras", "summary": "This file contains some more results on nontriviality of tensor product of algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/Nontrivial.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.ClassGroup", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -100.592, "z": 17.74, "size": 0.227, "title": "The class group of a Unique Factorization Domain is trivial", "summary": "This file proves that the ideal class group of a Normalized GCD Domain is trivial. The main application is to Unique Factorization Domains, which are known to be Normalized GCD Domains.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/ClassGroup.html"}, {"id": "Mathlib.LinearAlgebra.Ray", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 1, "macro_tier_score": 0.0031, "macro_tier_override": null, "x": -19.743, "z": 42.022, "size": 0.329, "title": "Rays in modules", "summary": "This file defines rays in modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Ray.html"}, {"id": "Mathlib.RingTheory.SimpleModule.Rank", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -79.632, "z": 5.998, "size": 0.2, "title": "A module over a division ring is simple iff it has rank one", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SimpleModule/Rank.html"}, {"id": "Mathlib.Algebra.Order.Ring.StandardPart", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -75.222, "z": -63.482, "size": 0.239, "title": "Standard part function", "summary": "Given a finite element in a non-archimedean field, the standard part function rounds it to the unique closest real number. That is, it chops off any infinitesimals. Let `K` be a linearly ordered field. The subset of finite elements (i.e. those bounded by a natural number) is a `ValuationSubring`, which means we can construct its residue field `FiniteResidueField`, roughly corresponding to the finite elements…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/StandardPart.html"}, {"id": "Mathlib.LinearAlgebra.Lagrange", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 75.809, "z": 30.501, "size": 0.2446, "title": "Lagrange interpolation", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Lagrange.html"}, {"id": "Mathlib.Algebra.Star.UnitaryStarAlgAut", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -47.344, "z": -41.781, "size": 0.2863, "title": "The ⋆-algebra automorphism given by a unitary element", "summary": "This file defines the ⋆-algebra automorphism on `R` given by a unitary `u`, which is `Unitary.conjStarAlgAut S R u`, defined to be `x ↦ u * x * star u`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/UnitaryStarAlgAut.html"}, {"id": "Mathlib.Algebra.Star.NonUnitalSubsemiring", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 6.34, "z": -25.215, "size": 0.2541, "title": "Non-unital Star Subsemirings", "summary": "In this file we define `NonUnitalStarSubsemiring`s and the usual operations on them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/NonUnitalSubsemiring.html"}, {"id": "Mathlib.Algebra.Homology.SpectralObject.SpectralSequence", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 12.888, "z": -22.581, "size": 0.2676, "title": "The spectral sequence of a spectral object", "summary": "The main definition in this file is `Abelian.SpectralObject.spectralSequence`. Assume that `X` is a spectral object indexed by `ι` in an abelian category `C`, and that we have `data : SpectralSequenceDataCore ι c r₀` for a family of complex shapes `c : ℤ → ComplexShape κ` for a type `κ` and `r₀ : ℤ`. Then, under the assumption `X.HasSpectralSequence data` (see the file…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/SpectralObject/SpectralSequence.html"}, {"id": "Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -7.592, "z": -14.891, "size": 0.253, "title": "Shapes of spectral sequences obtained from a spectral object", "summary": "This file prepares for the construction of the spectral sequence of a spectral object in an abelian category which shall be conducted in the file `Mathlib/Algebra/Homology/SpectralObject/SpectralSequence.lean`. In this file, we introduce a structure `SpectralSequenceDataCore` which contains a recipe for the construction of the pages of the spectral sequence. For example, from a spectral object `X` indexed by `EInt`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/SpectralObject/HasSpectralSequence.html"}, {"id": "Mathlib.Algebra.Homology.SpectralSequence.Basic", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -14.837, "z": 0.769, "size": 0.253, "title": "Spectral sequences", "summary": "In this file, we define the category `SpectralSequence C c r₀` of spectral sequences in an abelian category `C` with `Eᵣ`-pages defined from `r₀ : ℤ` having differentials given by complex shapes `c : ℤ → ComplexShape κ`, where `κ` is the index type for the objects on each page (e.g. `κ := ℤ × ℤ` or `κ := ℕ × ℕ`). A spectral sequence is defined as the data of a sequence of homological complexes (the pages) and a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/SpectralSequence/Basic.html"}, {"id": "Mathlib.Algebra.Module.Congruence.Defs", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 4, "macro_tier_score": 0.2991, "macro_tier_override": null, "x": 6.621, "z": 27.059, "size": 0.3289, "title": "Congruence relations respecting scalar multiplication", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Congruence/Defs.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Pseudofunctor", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -41.952, "z": 44.677, "size": 0.2, "title": "The pseudofunctors which send a commutative ring to its category of modules", "summary": "In this file, we construct the pseudofunctors `CommRingCat.moduleCatRestrictScalarsPseudofunctor` and `RingCat.moduleCatRestrictScalarsPseudofunctor` which send a (commutative) ring to its category of modules: the contravariant functoriality is given by the restriction of scalars functors. We also define a pseudofunctor `CommRingCat.moduleCatExtendScalarsPseudofunctor`: the covariant functoriality is given by the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Pseudofunctor.html"}, {"id": "Mathlib.FieldTheory.RatFunc.Basic", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 2, "macro_tier_score": 0.0293, "macro_tier_override": null, "x": 30.367, "z": 85.785, "size": 0.2941, "title": "The field structure of rational functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/RatFunc/Basic.html"}, {"id": "Mathlib.GroupTheory.ResiduallyFinite", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 38.418, "z": 6.717, "size": 0.239, "title": "Residually Finite Groups", "summary": "In this file we define residually finite groups and prove some basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/ResiduallyFinite.html"}, {"id": "Mathlib.Algebra.AffineMonoid.UniqueSums", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 72.405, "z": -1.883, "size": 0.2, "title": "Affine monoids have unique sums", "summary": "In this file we show that finitely generated cancellative torsion-free commutative monoids have unique sums. This is a direct corollary of them embedding into `ℤⁿ` for some `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/AffineMonoid/UniqueSums.html"}, {"id": "Mathlib.Algebra.AffineMonoid.Embedding", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -69.993, "z": 9.021, "size": 0.2478, "title": "Affine monoids embed into `ℤⁿ`", "summary": "This file proves that finitely generated cancellative torsion-free commutative monoids embed into `ℤⁿ` for some `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/AffineMonoid/Embedding.html"}, {"id": "Mathlib.Algebra.FreeAbelianGroup.UniqueSums", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -32.407, "z": 18.15, "size": 0.2478, "title": "Free abelian groups have unique sums", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/FreeAbelianGroup/UniqueSums.html"}, {"id": "Mathlib.NumberTheory.Dioph", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -56.076, "z": 36.407, "size": 0.2, "title": "Diophantine functions and Matiyasevic's theorem", "summary": "Hilbert's tenth problem asked whether there exists an algorithm which for a given integer polynomial determines whether this polynomial has integer solutions. It was answered in the negative in 1970, the final step being completed by Matiyasevic who showed that the power function is Diophantine. Here a function is called Diophantine if its graph is Diophantine as a set. A subset `S ⊆ ℕ ^ α` in turn is called…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Dioph.html"}, {"id": "Mathlib.NumberTheory.PellMatiyasevic", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 62.943, "z": -16.224, "size": 0.239, "title": "Pell's equation and Matiyasevic's theorem", "summary": "This file solves Pell's equation, i.e. integer solutions to `x ^ 2 - d * y ^ 2 = 1` *in the special case that `d = a ^ 2 - 1`*. This is then applied to prove Matiyasevic's theorem that the power function is Diophantine, which is the last key ingredient in the solution to Hilbert's tenth problem. For the definition of Diophantine function, see `NumberTheory.Dioph`. For results on Pell's equation for arbitrary…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/PellMatiyasevic.html"}, {"id": "Mathlib.Algebra.Ring.Subring.Units", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 3, "macro_tier_score": 0.1651, "macro_tier_override": null, "x": -20.284, "z": 21.715, "size": 0.3343, "title": "Unit subgroups of a ring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Subring/Units.html"}, {"id": "Mathlib.LinearAlgebra.Eigenspace.Semisimple", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -73.881, "z": 80.918, "size": 0.2576, "title": "Eigenspaces of semisimple linear endomorphisms", "summary": "This file contains basic results relevant to the study of eigenspaces of semisimple linear endomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Eigenspace/Semisimple.html"}, {"id": "Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord", "region_id": "algebra", "micro_elevation": 0.9342, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 130.457, "z": 19.177, "size": 0.2484, "title": "Polar coordinate change of variables for the mixed space of a number field", "summary": "We define two polar coordinate changes of variables for the mixed space `ℝ^r₁ × ℂ^r₂` associated to a number field `K` of signature `(r₁, r₂)`. The first one is `mixedEmbedding.polarCoord` and has value in `realMixedSpace K` defined as `ℝ^r₁ × (ℝ ⨯ ℝ)^r₂`, the second is `mixedEmbedding.polarSpaceCoord` and has value in `polarSpace K` defined as `ℝ^(r₁+r₂) × ℝ^r₂`. The change of variables with the `polarSpace` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/PolarCoord.html"}, {"id": "Mathlib.NumberTheory.NumberField.AdeleRing", "region_id": "algebra", "micro_elevation": 0.9474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 45.506, "z": 125.734, "size": 0.2, "title": "The adele ring of a number field", "summary": "This file contains the formalisation of the adele ring of a number field as the direct product of the infinite adele ring and the finite adele ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/AdeleRing.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.FiniteAdeleRing", "region_id": "algebra", "micro_elevation": 0.9211, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -115.63, "z": -59.414, "size": 0.2478, "title": "The finite adèle ring of a Dedekind domain", "summary": "We define the ring of finite adèles of a Dedekind domain `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.html"}, {"id": "Mathlib.RingTheory.Localization.Algebra", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.1338, "macro_tier_override": null, "x": 61.018, "z": 31.6, "size": 0.3854, "title": "Localization of algebra maps", "summary": "In this file we provide constructors to localize algebra maps. Also we show that localization commutes with taking kernels for ring homomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Algebra.html"}, {"id": "Mathlib.RingTheory.PolynomialAlgebra", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.1904, "macro_tier_override": null, "x": 66.501, "z": -6.901, "size": 0.3203, "title": "Base change of polynomial algebras", "summary": "Given `[CommSemiring R] [Semiring A] [Algebra R A]` we show `A[X] ≃ₐ[R] (A ⊗[R] R[X])`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PolynomialAlgebra.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.Basis", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -32.662, "z": -80.946, "size": 0.2, "title": "Constructing a bilinear map from a quadratic map, given a basis", "summary": "This file provides an alternative to `QuadraticMap.associated`; unlike that definition, this one does not require `Invertible (2 : R)`. Unlike that definition, this only works in the presence of a basis.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/Basis.html"}, {"id": "Mathlib.Algebra.BigOperators.Sym", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2338, "title": "Lemmas on `Finset.sum` and `Finset.prod` involving `Finset.sym2` or `Finset.sym`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Sym.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.Basic", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 2, "macro_tier_score": 0.0137, "macro_tier_override": null, "x": -70.128, "z": -48.788, "size": 0.4186, "title": "Quadratic maps", "summary": "This file defines quadratic maps on an `R`-module `M`, taking values in an `R`-module `N`. An `N`-valued quadratic map on a module `M` over a commutative ring `R` is a map `Q : M → N` such that: * `QuadraticMap.map_smul`: `Q (a • x) = (a * a) • Q x` * `QuadraticMap.polar_add_left`, `QuadraticMap.polar_add_right`, `QuadraticMap.polar_smul_left`, `QuadraticMap.polar_smul_right`: the map `QuadraticMap.polar Q := fun x…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/Basic.html"}, {"id": "Mathlib.RingTheory.HahnSeries.Cardinal", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 60.145, "z": -11.773, "size": 0.2, "title": "Cardinality of Hahn series", "summary": "We define `HahnSeries.cardSupp` as the cardinality of the support of a Hahn series, and find bounds for the cardinalities of different operations. We also build the subgroups, subrings, etc. of Hahn series bounded by a given infinite cardinal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HahnSeries/Cardinal.html"}, {"id": "Mathlib.RingTheory.HahnSeries.Summable", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -56.193, "z": -19.342, "size": 0.271, "title": "Summable families of Hahn Series", "summary": "We introduce a notion of formal summability for families of Hahn series, and define a formal sum function. This theory is applied to characterize invertible Hahn series whose coefficients are in a commutative domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HahnSeries/Summable.html"}, {"id": "Mathlib.RingTheory.IdealFilter.Topology", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -61.486, "z": -14.372, "size": 0.2, "title": "Topologies associated to ideal filters", "summary": "This file constructs topological structures on a ring from an `IdealFilter` and characterizes uniform ideal filters in terms of ring filter bases.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IdealFilter/Topology.html"}, {"id": "Mathlib.RingTheory.IdealFilter.Basic", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 60.569, "z": -9.349, "size": 0.239, "title": "Ideal Filters", "summary": "An **ideal filter** is a filter in the lattice of ideals of a ring `A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IdealFilter/Basic.html"}, {"id": "Mathlib.Algebra.Order.Round", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 0, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -16.425, "z": 29.115, "size": 0.3125, "title": "Rounding", "summary": "This file defines the `round` function, which uses the `floor` or `ceil` function to round a number to the nearest integer.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Round.html"}, {"id": "Mathlib.RingTheory.Valuation.ValuativeRel.Trivial", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 18.911, "z": -60.245, "size": 0.2, "title": "Trivial Valuative Relations", "summary": "Trivial valuative relations relate all non-zero elements to each other. Equivalently, all elements are related to `1`: the relation is equal to the relation induced by the trivial valuation which sends all non-zero elements to `1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/ValuativeRel/Trivial.html"}, {"id": "Mathlib.LinearAlgebra.Multilinear.FiniteDimensional", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 62.369, "z": 52.797, "size": 0.2, "title": "Multilinear maps over finite-dimensional spaces", "summary": "The main results are that multilinear maps over finitely-generated, free modules are finitely-generated and free. * `Module.Finite.multilinearMap` * `Module.Free.multilinearMap` We do not put this in `LinearAlgebra.Multilinear.Basic` to avoid making the imports too large there.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Multilinear/FiniteDimensional.html"}, {"id": "Mathlib.LinearAlgebra.Multilinear.Curry", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 2, "macro_tier_score": 0.0151, "macro_tier_override": null, "x": 37.814, "z": 35.695, "size": 0.3803, "title": "Currying of multilinear maps", "summary": "We register isomorphisms corresponding to currying or uncurrying variables, transforming a multilinear function `f` on `n+1` variables into a linear function taking values in multilinear functions in `n` variables, and into a multilinear function in `n` variables taking values in linear functions. These operations are called `f.curryLeft` and `f.curryRight` respectively (with inverses `f.uncurryLeft` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Multilinear/Curry.html"}, {"id": "Mathlib.Algebra.Order.Ring.Idempotent", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 3, "macro_tier_score": 0.067, "macro_tier_override": null, "x": -12.604, "z": 10.978, "size": 0.2953, "title": "Boolean algebra structure on idempotents in a commutative (semi)ring", "summary": "We show that the idempotent in a commutative ring form a Boolean algebra, with complement given by `a ↦ 1 - a` and infimum given by multiplication. In a commutative semiring where subtraction is not available, it is still true that pairs of elements `(a, b)` satisfying `a * b = 0` and `a + b = 1` form a Boolean algebra (such elements are automatically idempotents, and such a pair is uniquely determined by either `a`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Idempotent.html"}, {"id": "Mathlib.RingTheory.Ideal.MinimalPrime.Localization", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0671, "macro_tier_override": null, "x": 41.414, "z": -57.143, "size": 0.3046, "title": "Minimal primes and localization", "summary": "We provide various results concerning the minimal primes above an ideal that require the theory of localizations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.html"}, {"id": "Mathlib.RingTheory.KrullDimension.Basic", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.0814, "macro_tier_override": null, "x": -65.902, "z": 19.46, "size": 0.37, "title": "Krull dimensions of (commutative) rings", "summary": "Given a commutative ring, its ring-theoretic Krull dimension is the order-theoretic Krull dimension of its prime spectrum. Unfolding this definition, it is the length of the longest sequence(s) of prime ideals ordered by strict inclusion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/KrullDimension/Basic.html"}, {"id": "Mathlib.Algebra.Homology.SpectralObject.FirstPage", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 25.384, "z": -11.475, "size": 0.2, "title": "The first page of the spectral sequence of a spectral object", "summary": "Let `ι` be a preordered type, `X` a spectral object in an abelian category indexed by `ι`. Let `data : SpectralSequenceDataCore ι c r₀`. Assume that `X.HasSpectralSequence data` holds. In this file, we introduce a property `data.HasFirstPageComputation` which allows to \"compute\" the objects of the `r₀`th page of the spectral sequence attached to `X` in terms of objects of the form `X.H`, and we compute the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/SpectralObject/FirstPage.html"}, {"id": "Mathlib.RingTheory.RootsOfUnity.Lemmas", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 98.301, "z": -63.453, "size": 0.2478, "title": "More results on primitive roots of unity", "summary": "(We put these in a separate file because of the `KummerExtension` import.) Assume that `μ` is a primitive `n`th root of unity in an integral domain `R`. Then $$ \\prod_{k=1}^{n-1} (1 - \\mu^k) = n \\,; $$ see `IsPrimitiveRoot.prod_one_sub_pow_eq_order` and its variant `IsPrimitiveRoot.prod_pow_sub_one_eq_order` in terms of `∏ (μ^k - 1)`. We use this to deduce that `n` is divisible by `(μ - 1)^k` in `ℤ[μ] ⊆ R` when `k <…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RootsOfUnity/Lemmas.html"}, {"id": "Mathlib.FieldTheory.KummerExtension", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": -114.874, "z": 7.886, "size": 0.2806, "title": "Kummer Extensions", "summary": "Let `K` be a field, `n` be an integer such that `K` contains a primitive `n`-th root of unity. Kummer theory is about the classification of finite extensions of `L` whose Galois group is cyclic of order `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/KummerExtension.html"}, {"id": "Mathlib.GroupTheory.PushoutI", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -38.698, "z": -4.845, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/PushoutI.html"}, {"id": "Mathlib.GroupTheory.Coprod.Basic", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -24.138, "z": -0.484, "size": 0.2669, "title": "Coproduct (free product) of two monoids or groups", "summary": "In this file we define `Monoid.Coprod M N` (notation: `M ∗ N`) to be the coproduct (a.k.a. free product) of two monoids. The same type is used for the coproduct of two monoids and for the coproduct of two groups. The coproduct `M ∗ N` has the following universal property: for any monoid `P` and homomorphisms `f : M →* P`, `g : N →* P`, there exists a unique homomorphism `fg : M ∗ N →* P` such that `fg ∘…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Coprod/Basic.html"}, {"id": "Mathlib.RingTheory.IsPrimary", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 3, "macro_tier_score": 0.1565, "macro_tier_override": null, "x": -61.176, "z": 3.683, "size": 0.3541, "title": "Primary submodules", "summary": "A proper submodule `S : Submodule R M` is primary iff `r • x ∈ S` implies `x ∈ S` or `∃ n : ℕ, r ^ n • (⊤ : Submodule R M) ≤ S`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IsPrimary.html"}, {"id": "Mathlib.NumberTheory.ModularForms.DimensionFormulas.LevelOne", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 119.46, "z": -17.367, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/DimensionFormulas/LevelOne.html"}, {"id": "Mathlib.Algebra.Order.Group.PosPart", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 10.934, "z": -19.419, "size": 0.2697, "title": "Positive & negative parts", "summary": "Mathematical structures possessing an absolute value often also possess a unique decomposition of elements into \"positive\" and \"negative\" parts which are in some sense \"disjoint\" (e.g. the Jordan decomposition of a measure). This file provides instances of `PosPart` and `NegPart`, the positive and negative parts of an element in a lattice ordered group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/PosPart.html"}, {"id": "Mathlib.Algebra.Lie.Submodule", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 2, "macro_tier_score": 0.0193, "macro_tier_override": null, "x": 2.7, "z": 72.379, "size": 0.4952, "title": "Lie submodules of a Lie algebra", "summary": "In this file we define Lie submodules, we construct the lattice structure on Lie submodules and we use it to define various important operations, notably the Lie span of a subset of a Lie module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Submodule.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Gershgorin", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 53.328, "z": -66.741, "size": 0.2582, "title": "Gershgorin's circle theorem", "summary": "This file gives the proof of Gershgorin's circle theorem `eigenvalue_mem_ball` on the eigenvalues of matrices and some applications.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Gershgorin.html"}, {"id": "Mathlib.RingTheory.Etale.Pi", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": 38.296, "z": 96.694, "size": 0.2705, "title": "Formal-étaleness of finite products of rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Etale/Pi.html"}, {"id": "Mathlib.Algebra.Category.Ring.Adjunctions", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.0701, "macro_tier_override": null, "x": -8.295, "z": 68.212, "size": 0.2694, "title": "Adjunctions in `CommRingCat`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/Adjunctions.html"}, {"id": "Mathlib.Algebra.Category.Ring.Instances", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.0701, "macro_tier_override": null, "x": -67.914, "z": 10.463, "size": 0.2694, "title": "Ring-theoretic results in terms of categorical language", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/Instances.html"}, {"id": "Mathlib.RingTheory.DividedPowers.DPMorphism", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 39.807, "z": -58.274, "size": 0.2478, "title": "Divided power morphisms", "summary": "Let `A` and `B` be commutative (semi)rings, let `I` be an ideal of `A` and let `J` be an ideal of `B`. Given divided power structures on `I` and `J`, a ring morphism `A →+* B` is a *divided power morphism* if it is compatible with these divided power structures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DividedPowers/DPMorphism.html"}, {"id": "Mathlib.Algebra.NoZeroSMulDivisors.Pi", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 15.389, "z": 16.12, "size": 0.2, "title": "Pi instances for NoZeroSMulDivisors", "summary": "This file defines instances for NoZeroSMulDivisors on Pi types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/NoZeroSMulDivisors/Pi.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.Finsupp", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 2, "macro_tier_score": 0.007, "macro_tier_override": null, "x": 18.768, "z": -25.388, "size": 0.281, "title": "Factors as finsupp", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/Finsupp.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.Moebius", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -34.591, "z": -28.109, "size": 0.2, "title": "The Moebius function on a unique factorization monoid", "summary": "We define the Moebius function on a unique factorization monoid.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/Moebius.html"}, {"id": "Mathlib.LinearAlgebra.DirectSum.Finite", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 2, "macro_tier_score": 0.0078, "macro_tier_override": null, "x": -39.754, "z": -39.035, "size": 0.2369, "title": "A finite direct sum of finite modules is finite", "summary": "This file defines a `Module.Finite` instance for a finite direct sum of finite modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/DirectSum/Finite.html"}, {"id": "Mathlib.Algebra.Order.Module.Basic", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -16.433, "z": 15.054, "size": 0.257, "title": "Further lemmas about monotonicity of scalar multiplication", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Module/Basic.html"}, {"id": "Mathlib.Algebra.Ring.Int.Field", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 3, "macro_tier_score": 0.1764, "macro_tier_override": null, "x": -12.558, "z": -3.36, "size": 0.2517, "title": "`ℤ` is not a field", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Int/Field.html"}, {"id": "Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 3, "macro_tier_score": 0.1764, "macro_tier_override": null, "x": -41.138, "z": -37.574, "size": 0.2517, "title": "Integral closure as a characteristic predicate", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Defs.html"}, {"id": "Mathlib.RingTheory.Polynomial.IntegralNormalization", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 3, "macro_tier_score": 0.1764, "macro_tier_override": null, "x": -77.67, "z": -18.566, "size": 0.2517, "title": "Theory of monic polynomials", "summary": "We define `integralNormalization`, which relate arbitrary polynomials to monic ones.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/IntegralNormalization.html"}, {"id": "Mathlib.Algebra.GCDMonoid.Basic", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.302, "macro_tier_override": null, "x": -23.775, "z": 4.203, "size": 0.3677, "title": "Monoids with normalization functions, `gcd`, and `lcm`", "summary": "This file defines extra structures on `CommMonoidWithZero`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GCDMonoid/Basic.html"}, {"id": "Mathlib.Algebra.Group.Nat.Range", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 2.111, "z": 7.122, "size": 0.2743, "title": "`Finset.range` and addition of natural numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Nat/Range.html"}, {"id": "Mathlib.LinearAlgebra.ExteriorAlgebra.Basis", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 73.887, "z": -75.807, "size": 0.2, "title": "Basis for `ExteriorAlgebra`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/ExteriorAlgebra/Basis.html"}, {"id": "Mathlib.LinearAlgebra.ExteriorAlgebra.Grading", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 58.401, "z": 74.567, "size": 0.2478, "title": "Results about the grading structure of the exterior algebra", "summary": "Many of these results are copied with minimal modification from the tensor algebra. The main result is `ExteriorAlgebra.gradedAlgebra`, which says that the exterior algebra is a ℕ-graded algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.html"}, {"id": "Mathlib.LinearAlgebra.ExteriorPower.Basis", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -37.957, "z": 96.827, "size": 0.2542, "title": "Constructs a basis for exterior powers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/ExteriorPower/Basis.html"}, {"id": "Mathlib.Algebra.Field.ULift", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": -3.827, "z": -18.173, "size": 0.2255, "title": "Field instances for `ULift`", "summary": "This file defines instances for fields, semifields, and related structures on `ULift` types. (Recall `ULift α` is just a \"copy\" of a type `α` in a higher universe.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/ULift.html"}, {"id": "Mathlib.Algebra.GroupWithZero.ULift", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.3809, "macro_tier_override": null, "x": -10.279, "z": 4.302, "size": 0.3073, "title": "`ULift` instances for groups and monoids with zero", "summary": "This file defines instances for group and monoid with zero and related structures on `ULift` types. (Recall `ULift α` is just a \"copy\" of a type `α` in a higher universe.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/ULift.html"}, {"id": "Mathlib.Algebra.Homology.Factorizations.CM5b", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -31.762, "z": 15.372, "size": 0.2298, "title": "Factorization lemma", "summary": "Let `C` be an abelian category with enough injectives. We show that any morphism `f : K ⟶ L` between bounded below cochain complexes in `C` can be factored as `i ≫ p` where `i : K ⟶ L'` is a monomorphism (with `L'` bounded below) and `p : L' ⟶ L` a quasi-isomorphism that is an epimorphism with a degreewise injective kernel. (This is part of the factorization axiom CM5 for a model category structure on bounded below…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Factorizations/CM5b.html"}, {"id": "Mathlib.Algebra.Polynomial.Eval.Algebra", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 4, "macro_tier_score": 0.2821, "macro_tier_override": null, "x": -51.975, "z": -1.627, "size": 0.3723, "title": "Evaluation of polynomials in an algebra", "summary": "This file concerns evaluating polynomials where the map is `algebraMap` TODO: merge with parts of `Mathlib/Algebra/Polynomial/AlgebraMap.lean`?", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Eval/Algebra.html"}, {"id": "Mathlib.NumberTheory.Pell", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 41.42, "z": 74.717, "size": 0.2, "title": "Pell's Equation", "summary": "*Pell's Equation* is the equation $x^2 - d y^2 = 1$, where $d$ is a positive integer that is not a square, and one is interested in solutions in integers $x$ and $y$. In this file, we aim at providing all of the essential theory of Pell's Equation for general $d$ (as opposed to the contents of `NumberTheory.PellMatiyasevic`, which is specific to the case $d = a^2 - 1$ for some $a > 1$). We begin by defining a type…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Pell.html"}, {"id": "Mathlib.NumberTheory.DiophantineApproximation.Basic", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 9.08, "z": 83.078, "size": 0.2585, "title": "Diophantine Approximation", "summary": "The first part of this file gives proofs of various versions of **Dirichlet's approximation theorem** and its important consequence that when $\\xi$ is an irrational real number, then there are infinitely many rationals $x/y$ (in lowest terms) such that $$\\left|\\xi - \\frac{x}{y}\\right| < \\frac{1}{y^2} \\,.$$ The proof is based on the pigeonhole principle. The second part of the file gives a proof of **Legendre's…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/DiophantineApproximation/Basic.html"}, {"id": "Mathlib.RingTheory.EuclideanDomain", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 3, "macro_tier_score": 0.1899, "macro_tier_override": null, "x": 42.916, "z": 46.318, "size": 0.2722, "title": "Lemmas about Euclidean domains", "summary": "Various about Euclidean domains are proved; all of them seem to be true more generally for principal ideal domains, so these lemmas should probably be reproved in more generality and this file perhaps removed?", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/EuclideanDomain.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Sheaf.Abelian", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 53.917, "z": -45.534, "size": 0.2618, "title": "The category of sheaves of modules is abelian", "summary": "In this file, it is shown that the category of sheaves of modules over a sheaf of rings `R` is an abelian category. More precisely, if `J` is a Grothendieck topology on a category `C` and `R : Sheaf J RingCat.{u}`, then `SheafOfModules.{v} R` is abelian if the conditions `HasSheafify J AddCommGrpCat.{v}` and `J.WEqualsLocallyBijective AddCommGrpCat.{v}` are satisfied. In particular, if `u = v` and `C : Type u` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Sheaf/Abelian.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Sheaf.Generators", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 0.217, "z": 74.286, "size": 0.2559, "title": "Generating sections of sheaves of modules", "summary": "In this file, given a sheaf of modules `M` over a sheaf of rings `R`, we introduce the structure `M.GeneratingSections` which consists of a family of (global) sections `s : I → M.sections` which generate `M`. We also introduce the structure `M.LocalGeneratorsData` which contains the data of a covering `X i` of the terminal object and the data of a `(M.over (X i)).GeneratingSections` for all `i`. This is used in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Sheaf/Generators.html"}, {"id": "Mathlib.Algebra.CharZero.Quotient", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -23.532, "z": -11.056, "size": 0.2673, "title": "Lemmas about quotients in characteristic zero", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharZero/Quotient.html"}, {"id": "Mathlib.Algebra.Order.Ring.Interval", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 22.399, "z": -16.562, "size": 0.2673, "title": "Intervals of integers in strict ordered rings", "summary": "These statements could perhaps be generalized, or there could be other variations provided (e.g., for `ℕ` instead of `ℤ`, or a version for locally finite `SuccOrder`s with strictly monotone functions), but for now these are the ones that have found utility in practice (e.g., for lemmas about `Real.Angle`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Interval.html"}, {"id": "Mathlib.Algebra.DirectSum.Finsupp", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 3, "macro_tier_score": 0.2741, "macro_tier_override": null, "x": 25.18, "z": 49.7, "size": 0.3579, "title": "Results on direct sums and finitely supported functions.", "summary": "1. The linear equivalence between finitely supported functions `ι →₀ M` and the direct sum of copies of `M` indexed by `ι`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/DirectSum/Finsupp.html"}, {"id": "Mathlib.Algebra.Central.Matrix", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 18.784, "z": -56.383, "size": 0.2, "title": "The matrix algebra is a central algebra", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Central/Matrix.html"}, {"id": "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 3, "macro_tier_score": 0.2744, "macro_tier_override": null, "x": -9.613, "z": 54.879, "size": 0.3715, "title": "Tensor products of direct sums", "summary": "This file shows that taking `TensorProduct`s commutes with taking `DirectSum`s in both arguments.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/DirectSum/TensorProduct.html"}, {"id": "Mathlib.RingTheory.Etale.StandardEtale", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -46.589, "z": -92.982, "size": 0.2433, "title": "Standard etale maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Etale/StandardEtale.html"}, {"id": "Mathlib.FieldTheory.IntermediateField.Adjoin.Defs", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.0946, "macro_tier_override": null, "x": 32.407, "z": 56.346, "size": 0.2774, "title": "Adjoining Elements to Fields", "summary": "In this file we introduce the notion of adjoining elements to fields. This isn't quite the same as adjoining elements to rings. For example, `K[x]` might not include `x⁻¹`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/IntermediateField/Adjoin/Defs.html"}, {"id": "Mathlib.Algebra.Order.Hom.Basic", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.402, "macro_tier_override": null, "x": 23.278, "z": -6.406, "size": 0.3099, "title": "Algebraic order homomorphism classes", "summary": "This file defines hom classes for common properties at the intersection of order theory and algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Hom/Basic.html"}, {"id": "Mathlib.Algebra.Homology.AlternatingConst", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 28.279, "z": -24.081, "size": 0.2629, "title": "The alternating constant complex", "summary": "Given an object `X : C` and endomorphisms `φ, ψ : X ⟶ X` such that `φ ∘ ψ = ψ ∘ φ = 0`, this file defines the periodic chain and cochain complexes `... ⟶ X --φ--> X --ψ--> X --φ--> X --ψ--> 0` and `0 ⟶ X --ψ--> X --φ--> X --ψ--> X --φ--> ...` (or more generally for any complex shape `c` on `ℕ` where `c.Rel i j` implies `i` and `j` have different parity). We calculate the homology of these periodic complexes. In…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/AlternatingConst.html"}, {"id": "Mathlib.RingTheory.AlgebraicIndependent.AlgebraicClosure", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 3, "macro_tier_score": 0.0594, "macro_tier_override": null, "x": -57.149, "z": 93.489, "size": 0.3065, "title": "Algebraic independence persists to the algebraic closure", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AlgebraicIndependent/AlgebraicClosure.html"}, {"id": "Mathlib.FieldTheory.AlgebraicClosure", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 3, "macro_tier_score": 0.0603, "macro_tier_override": null, "x": -96.856, "z": -47.133, "size": 0.2841, "title": "Relative Algebraic Closure", "summary": "In this file we construct the relative algebraic closure of a field extension.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/AlgebraicClosure.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Defs", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -7.333, "z": 21.045, "size": 0.2591, "title": "Matrices", "summary": "This file defines basic properties of matrices up to the module structure. Matrices with rows indexed by `m`, columns indexed by `n`, and entries of type `α` are represented with `Matrix m n α`. For the typical approach of counting rows and columns, `Matrix (Fin m) (Fin n) α` can be used.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Defs.html"}, {"id": "Mathlib.Algebra.Homology.LocalCohomology", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -33.6, "z": -55.643, "size": 0.2, "title": "Local cohomology.", "summary": "This file defines the `i`-th local cohomology module of an `R`-module `M` with support in an ideal `I` of `R`, where `R` is a commutative ring, as the direct limit of Ext modules: Given a collection of ideals cofinal with the powers of `I`, consider the directed system of quotients of `R` by these ideals, and take the direct limit of the system induced on the `i`-th Ext into `M`. One can, of course, take the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/LocalCohomology.html"}, {"id": "Mathlib.Algebra.Group.Hom.CompTypeclasses", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.4316, "macro_tier_override": null, "x": 7.284, "z": -1.458, "size": 0.359, "title": "Propositional typeclasses on several monoid homs", "summary": "This file contains typeclasses used in the definition of equivariant maps, in the spirit what was initially developed by Frédéric Dupuis and Heather Macbeth for linear maps. However, we do not expect that all maps should be guessed automatically, as it happens for linear maps. If `φ`, `ψ`… are monoid homs and `M`, `N`… are monoids, we add two instances: * `MonoidHom.CompTriple φ ψ χ`, which expresses that `ψ.comp φ…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Hom/CompTypeclasses.html"}, {"id": "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.2693, "macro_tier_override": null, "x": 45.621, "z": 35.118, "size": 0.3337, "title": "Trailing degree of univariate polynomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.html"}, {"id": "Mathlib.Algebra.Exact.Sequence", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 13.308, "z": -78.741, "size": 0.2676, "title": "Exactness of sequences", "summary": "In this file we provide some API for handling exact sequences.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Exact/Sequence.html"}, {"id": "Mathlib.RingTheory.AdicCompletion.AsTensorProduct", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": 26.974, "z": 79.1, "size": 0.2538, "title": "Adic completion as tensor product", "summary": "In this file we examine properties of the natural map `AdicCompletion I R ⊗[R] M →ₗ[AdicCompletion I R] AdicCompletion I M`. We show (in the `AdicCompletion` namespace): - `ofTensorProduct_bijective_of_pi_of_fintype`: it is an isomorphism if `M = R^n`. - `ofTensorProduct_surjective_of_finite`: it is surjective, if `M` is a finite `R`-module. - `ofTensorProduct_bijective_of_finite_of_isNoetherian`: it is an…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AdicCompletion/AsTensorProduct.html"}, {"id": "Mathlib.RingTheory.Ideal.Pure", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 68.991, "z": 47.165, "size": 0.2, "title": "Pure ideals", "summary": "An ideal `I` of a ring `R` is called pure if `R ⧸ I` is flat over `R` (see [Stacks 04PR](https://stacks.math.columbia.edu/tag/04PR)). In this file we show some properties of such ideals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Pure.html"}, {"id": "Mathlib.RingTheory.Ideal.IdempotentFG", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 2, "macro_tier_score": 0.0202, "macro_tier_override": null, "x": -51.148, "z": -30.261, "size": 0.2664, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/IdempotentFG.html"}, {"id": "Mathlib.RepresentationTheory.Invariants", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 1, "macro_tier_score": 0.004, "macro_tier_override": null, "x": -68.553, "z": 68.02, "size": 0.3103, "title": "Subspace of invariants a group representation", "summary": "This file introduces the subspace of invariants of a group representation and proves basic results about it. The main tool used is the average of all elements of the group, seen as an element of `k[G]`. The action of this special element gives a projection onto the subspace of invariants. In order for the definition of the average element to make sense, we need to assume for most of the results that the order of `G`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Invariants.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 2, "macro_tier_score": 0.0125, "macro_tier_override": null, "x": -30.085, "z": 9.575, "size": 0.2698, "title": "Multiplicative maps on unique factorization domains", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/Multiplicative.html"}, {"id": "Mathlib.Algebra.Lie.AdjointAction.JordanChevalley", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 99.178, "z": 31.303, "size": 0.2292, "title": "Jordan–Chevalley decomposition and the adjoint action", "summary": "This file contains results about the interaction between the adjoint action `LieAlgebra.ad` and the Jordan–Chevalley decomposition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/AdjointAction/JordanChevalley.html"}, {"id": "Mathlib.LinearAlgebra.Eigenspace.Matrix", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 1, "macro_tier_score": 0.0037, "macro_tier_override": null, "x": 11.122, "z": 80.955, "size": 0.2807, "title": "Eigenvalues, Eigenvectors and Spectrum for Matrices", "summary": "This file collects results about eigenvectors, eigenvalues and spectrum specific to matrices over a nontrivial commutative ring, nontrivial commutative ring without zero divisors, or field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Eigenspace/Matrix.html"}, {"id": "Mathlib.Algebra.Module.Submodule.RestrictScalars", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 4, "macro_tier_score": 0.3918, "macro_tier_override": null, "x": -1.058, "z": 35.27, "size": 0.4591, "title": "Restriction of scalars for submodules", "summary": "If semiring `S` acts on a semiring `R` and `M` is a module over both (compatibly with this action) then we can turn an `R`-submodule into an `S`-submodule by forgetting the action of `R`. We call this restriction of scalars for submodules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/RestrictScalars.html"}, {"id": "Mathlib.Algebra.Group.Action.Pretransitive", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.4526, "macro_tier_override": null, "x": -12.971, "z": -0.873, "size": 0.3551, "title": "Pretransitive group actions", "summary": "This file defines a typeclass for pretransitive group actions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Pretransitive.html"}, {"id": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 1, "macro_tier_score": 0.0021, "macro_tier_override": null, "x": 4.192, "z": 111.351, "size": 0.3305, "title": "Eisenstein series q-expansions", "summary": "We give the q-expansion of Eisenstein series of weight `k` and level 1. In particular, we prove `EisensteinSeries.q_expansion_bernoulli` which says that for even `k` with `3 ≤ k` Eisenstein series can be written as `1 - (2k / bernoulli k) ∑' n, σ_{k-1}(n) q^n` where `q = exp(2πiz)` and `σ_{k-1}(n)` is the sum of the `(k-1)`-th powers of the divisors of `n`. We need `k` to be even so that the Eisenstein series are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/EisensteinSeries/QExpansion.html"}, {"id": "Mathlib.NumberTheory.LSeries.HurwitzZetaValues", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -75.749, "z": -79.172, "size": 0.252, "title": "Special values of Hurwitz and Riemann zeta functions", "summary": "This file gives the formula for `ζ (2 * k)`, for `k` a non-zero integer, in terms of Bernoulli numbers. More generally, we give formulae for any Hurwitz zeta functions at any (strictly) negative integer in terms of Bernoulli polynomials. (Note that most of the actual work for these formulae is done elsewhere, in `Mathlib/NumberTheory/ZetaValues.lean`. This file has only those results which really need the definition…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/HurwitzZetaValues.html"}, {"id": "Mathlib.NumberTheory.TsumDivisorsAntidiagonal", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -17.3, "z": 39.055, "size": 0.252, "title": "Lemmas on infinite sums over the antidiagonal of the divisors function", "summary": "This file contains lemmas about the antidiagonal of the divisors function. It defines the map from `Nat.divisorsAntidiagonal n` to `ℕ+ × ℕ+` given by sending `n = a * b` to `(a, b)`. We then prove some identities about the infinite sums over this antidiagonal, such as `∑' n : ℕ+, n ^ k * r ^ n / (1 - r ^ n) = ∑' n : ℕ+, σ k n * r ^ n` which are used for Eisenstein series and their q-expansions. This is also a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/TsumDivisorsAntidiagonal.html"}, {"id": "Mathlib.LinearAlgebra.JordanChevalley", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 91.085, "z": -46.226, "size": 0.263, "title": "Jordan-Chevalley-Dunford decomposition", "summary": "Given a finite-dimensional linear endomorphism `f`, the Jordan-Chevalley-Dunford theorem provides a sufficient condition for there to exist a nilpotent endomorphism `n` and a semisimple endomorphism `s`, such that `f = n + s` and both `n` and `s` are polynomial expressions in `f`. The condition is that there exists a separable polynomial `P` such that the endomorphism `P(f)` is nilpotent. This condition is always…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/JordanChevalley.html"}, {"id": "Mathlib.NumberTheory.ModularForms.BoundedAtCusp", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -39.348, "z": -46.986, "size": 0.2381, "title": "Boundedness and vanishing at cusps", "summary": "We define the notions of \"bounded at c\" and \"vanishing at c\" for functions on `ℍ`, where `c` is an element of `OnePoint ℝ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/BoundedAtCusp.html"}, {"id": "Mathlib.NumberTheory.ModularForms.Cusps", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -43.17, "z": -40.843, "size": 0.2775, "title": "Cusps", "summary": "We define the cusps of a subgroup of `GL(2, ℝ)` as the fixed points of parabolic elements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/Cusps.html"}, {"id": "Mathlib.Algebra.Ring.Hom.InjSurj", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 4, "macro_tier_score": 0.3821, "macro_tier_override": null, "x": 5.762, "z": 11.653, "size": 0.3836, "title": "Pulling back rings along injective maps, and pushing them forward along surjective maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Hom/InjSurj.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -16.427, "z": -3.088, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/Unbundled/Basic.html"}, {"id": "Mathlib.FieldTheory.Cardinality", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -62.155, "z": -101.312, "size": 0.2, "title": "Cardinality of Fields", "summary": "In this file we show all the possible cardinalities of fields. All infinite cardinals can harbour a field structure, and so can all types with prime power cardinalities, and this is sharp.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Cardinality.html"}, {"id": "Mathlib.Algebra.Field.TransferInstance", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -13.893, "z": -19.745, "size": 0.2769, "title": "Transfer algebraic structures across `Equiv`s", "summary": "This continues the pattern set in `Mathlib/Algebra/Group/TransferInstance.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/TransferInstance.html"}, {"id": "Mathlib.RingTheory.Localization.Cardinality", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.009, "macro_tier_override": null, "x": -52.406, "z": 23.837, "size": 0.2425, "title": "Cardinality of localizations", "summary": "In this file, we establish the cardinality of localizations. In most cases, a localization has cardinality equal to the base ring. If there are zero-divisors, however, this is no longer true - for example, `ZMod 6` localized at `{2, 4}` is equal to `ZMod 3`, and if you have zero in your submonoid, then your localization is trivial (see `IsLocalization.uniqueOfZeroMem`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Cardinality.html"}, {"id": "Mathlib.RingTheory.Coalgebra.MulOpposite", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -53.276, "z": 21.822, "size": 0.2, "title": "MulOpposite of coalgebras", "summary": "Suppose `R` is a commutative semiring, and `A` is an `R`-coalgebra, then `Aᵐᵒᵖ` is an `R`-coalgebra, where we define the comultiplication and counit maps naturally.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Coalgebra/MulOpposite.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Range", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 3, "macro_tier_score": 0.0678, "macro_tier_override": null, "x": 33.416, "z": 0.947, "size": 0.262, "title": "The range of a MonoidWithZeroHom", "summary": "Given a `MonoidWithZeroHom` `f : A → B` whose codomain `B` is a `MonoidWithZero`, we define the type `MonoidWithZeroHom.valueMonoid` as the submonoid of `Bˣ` generated by the invertible elements in the range of `f`. For example, if `A = ℕ`, `f` is the natural cast to `B` where `B` is * `ℝ≥0`, then `MonoidWithZero.valueMonoid` are the positive natural numbers in `ℝ≥0`; * `WithZero ℤ`, then `MonoidWithZero.valueMonoid…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Range.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Submonoid.Instances", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 3, "macro_tier_score": 0.0689, "macro_tier_override": null, "x": -4.763, "z": -19.866, "size": 0.251, "title": "Instances for the range submonoid of a monoid with zero hom", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Submonoid/Instances.html"}, {"id": "Mathlib.Algebra.Module.LinearMap.Basic", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 4, "macro_tier_score": 0.3867, "macro_tier_override": null, "x": 15.467, "z": -20.899, "size": 0.3296, "title": "Further results on (semi)linear maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LinearMap/Basic.html"}, {"id": "Mathlib.RingTheory.LocalProperties.Submodule", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0724, "macro_tier_override": null, "x": 20.799, "z": 67.438, "size": 0.3591, "title": "Local properties of modules and submodules", "summary": "In this file, we show that several conditions on submodules can be checked on stalks.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalProperties/Submodule.html"}, {"id": "Mathlib.RingTheory.LocalRing.Length", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 2, "macro_tier_score": 0.0101, "macro_tier_override": null, "x": 12.577, "z": -95.75, "size": 0.2407, "title": "Lengths along extensions of local rings", "summary": "This file proves results relating lengths along extensions of local rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/Length.html"}, {"id": "Mathlib.Algebra.Notation", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2659, "title": "Notations for operations involving order and algebraic structure", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Notation.html"}, {"id": "Mathlib.FieldTheory.KrullTopology", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 2, "macro_tier_score": 0.0216, "macro_tier_override": null, "x": 65.227, "z": -94.887, "size": 0.2987, "title": "Krull topology", "summary": "We define the Krull topology on `Gal(L/K)` for an arbitrary field extension `L/K`. In order to do this, we first define a `GroupFilterBasis` on `Gal(L/K)`, whose sets are `E.fixingSubgroup` for all intermediate fields `E` with `E/K` finite dimensional.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/KrullTopology.html"}, {"id": "Mathlib.FieldTheory.Minpoly.ConjRootClass", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -43.333, "z": -96.583, "size": 0.2, "title": "Conjugate root classes", "summary": "In this file, we define the `ConjRootClass` of a field extension `L / K` as the quotient of `L` by the relation `IsConjRoot K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Minpoly/ConjRootClass.html"}, {"id": "Mathlib.FieldTheory.Minpoly.IsConjRoot", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": -63.588, "z": 82.297, "size": 0.2792, "title": "Conjugate roots", "summary": "Given two elements `x` and `y` of some `K`-algebra, these two elements are *conjugate roots* over `K` if they have the same minimal polynomial over `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Minpoly/IsConjRoot.html"}, {"id": "Mathlib.RepresentationTheory.AlgebraRepresentation.Basic", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": 92.481, "z": -51.51, "size": 0.2551, "title": "Basic facts about algebra representations", "summary": "This file collects basic general facts about algebra representations. The purpose of this file is to have general results so that when we prove a corresponding fact about group representations (or Lie algebra representations etc), we can deduce them as special cases of facts from this file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/AlgebraRepresentation/Basic.html"}, {"id": "Mathlib.Algebra.Homology.SpectralObject.Differentials", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -0.575, "z": -20.421, "size": 0.2722, "title": "Differentials of a spectral object", "summary": "Let `X` be a spectral object in an abelian category `C` indexed by a category `ι`. In this file, we construct the differentials `d : E^{n}(f₃, f₄, f₅) ⟶ E^{n+1}(f₁, f₂, f₃)` that are attached to families of five composable morphisms `f₁`, `f₂`, `f₃`, `f₄`, `f₅` in `ι`. We show that `d ≫ d = 0`. The homology of these differentials is computed in the file `Mathlib/Algebra/Homology/SpectralObject/Homology.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/SpectralObject/Differentials.html"}, {"id": "Mathlib.Algebra.Homology.SpectralObject.Page", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": -13.591, "z": -12.657, "size": 0.2946, "title": "Spectral objects in abelian categories", "summary": "Let `X` be a spectral object index by the category `ι` in the abelian category `C`. The purpose of this file is to introduce the homology `X.E` of the short complex `X.shortComplex` `(X.H n₀).obj (mk₁ f₃) ⟶ (X.H n₁).obj (mk₁ f₂) ⟶ (X.H n₂).obj (mk₁ f₁)` when `f₁`, `f₂` and `f₃` are composable morphisms in `ι` and the equalities `n₀ + 1 = n₁` and `n₁ + 1 = n₂` hold (both maps in the short complex are given by `X.δ`).…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/SpectralObject/Page.html"}, {"id": "Mathlib.RingTheory.LocalProperties.InjectiveDimension", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -83.175, "z": -19.495, "size": 0.2, "title": "Relation of Injective Dimension with Localizations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalProperties/InjectiveDimension.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.EnoughInjectives", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": -54.833, "z": -41.414, "size": 0.2239, "title": "Category of $R$-modules has enough injectives", "summary": "We lift enough injectives of abelian groups to arbitrary $R$-modules by adjoint functors `restrictScalars ⊣ coextendScalars`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/EnoughInjectives.html"}, {"id": "Mathlib.RingTheory.LocalProperties.Injective", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": 48.687, "z": -67.926, "size": 0.2239, "title": "Being injective is a local property", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalProperties/Injective.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Permanent", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 27.034, "z": 46.581, "size": 0.2, "title": "Permanent of a matrix", "summary": "This file defines the permanent of a matrix, `Matrix.permanent`, and some of its properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Permanent.html"}, {"id": "Mathlib.RingTheory.NonUnitalSubring.Defs", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.3838, "macro_tier_override": null, "x": -0.031, "z": -16.714, "size": 0.3514, "title": "`NonUnitalSubring`s", "summary": "Let `R` be a non-unital ring. This file defines the \"bundled\" non-unital subring type `NonUnitalSubring R`, a type whose terms correspond to non-unital subrings of `R`. This is the preferred way to talk about non-unital subrings in mathlib.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/NonUnitalSubring/Defs.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.Radical", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 90.926, "z": -3.691, "size": 0.2676, "title": "The radical of a quadratic form", "summary": "We define the radical of a quadratic form. This is a standard construction if 2 is invertible in the coefficient ring, but is more fiddly otherwise. We follow the account in Chapter II, §7 of [elman-karpenko-merkurjev-2008].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/Radical.html"}, {"id": "Mathlib.LinearAlgebra.Quotient.Bilinear", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -53.265, "z": 7.968, "size": 0.2649, "title": "Lifting bilinear forms to quotients", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Quotient/Bilinear.html"}, {"id": "Mathlib.NumberTheory.Harmonic.GammaDeriv", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": -20.415, "z": 0.751, "size": 0.2809, "title": "Derivative of Γ at positive integers", "summary": "We prove the formula for the derivative of `Real.Gamma` at a positive integer: `deriv Real.Gamma (n + 1) = Nat.factorial n * (-Real.eulerMascheroniConstant + harmonic n)`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Harmonic/GammaDeriv.html"}, {"id": "Mathlib.Algebra.Field.Action.ConjAct", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 22.498, "z": -24.725, "size": 0.2, "title": "Conjugation action of a field on itself", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/Action/ConjAct.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.LeftResolution", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.666, "z": -61.264, "size": 0.2, "title": "Functorial projective resolutions of modules", "summary": "The fact that an `R`-module `M` can be functorially written as a quotient of a projective `R`-module is expressed as the definition `ModuleCat.projectiveResolution`. Using the construction in the file `Mathlib/Algebra/Homology/LeftResolution/Basic.lean`, we may obtain a functor `(projectiveResolution R).chainComplexFunctor` which sends `M : ModuleCat R` to a projective resolution.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/LeftResolution.html"}, {"id": "Mathlib.Algebra.Homology.LeftResolution.Basic", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": -12.165, "z": 8.53, "size": 0.2858, "title": "Left resolutions", "summary": "Given a fully faithful functor `ι : C ⥤ A` to an abelian category, we introduce a structure `Abelian.LeftResolution ι` which gives a functor `F : A ⥤ C` and a natural epimorphism `π.app X : ι.obj (F.obj X) ⟶ X` for all `X : A`. This is used in order to construct a resolution functor `LeftResolution.chainComplexFunctor : A ⥤ ChainComplex C ℕ`. This shall be used in order to construct functorial flat resolutions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/LeftResolution/Basic.html"}, {"id": "Mathlib.RingTheory.SimpleRing.Defs", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 4, "macro_tier_score": 0.3292, "macro_tier_override": null, "x": -37.35, "z": 33.457, "size": 0.396, "title": "Simple rings", "summary": "A ring `R` is **simple** if it has only two two-sided ideals, namely `⊥` and `⊤`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SimpleRing/Defs.html"}, {"id": "Mathlib.RingTheory.TwoSidedIdeal.Lattice", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 4, "macro_tier_score": 0.332, "macro_tier_override": null, "x": 18.011, "z": 44.801, "size": 0.4613, "title": "The complete lattice structure on two-sided ideals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TwoSidedIdeal/Lattice.html"}, {"id": "Mathlib.RingTheory.TwoSidedIdeal.Instances", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -40.93, "z": -25.618, "size": 0.2, "title": "Additional instances for two-sided ideals.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TwoSidedIdeal/Instances.html"}, {"id": "Mathlib.RingTheory.TwoSidedIdeal.Basic", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 4, "macro_tier_score": 0.3321, "macro_tier_override": null, "x": 42.968, "z": 17.59, "size": 0.465, "title": "Two Sided Ideals", "summary": "In this file, for any `Ring R`, we reinterpret `I : RingCon R` as a two-sided-ideal of a ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TwoSidedIdeal/Basic.html"}, {"id": "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 61.077, "z": -45.469, "size": 0.2478, "title": "Jacobi's theta function", "summary": "This file defines the one-variable Jacobi theta function $$\\theta(\\tau) = \\sum_{n \\in \\mathbb{Z}} \\exp (i \\pi n ^ 2 \\tau),$$ and proves the modular transformation properties `θ (τ + 2) = θ τ` and `θ (-1 / τ) = (-I * τ) ^ (1 / 2) * θ τ`, using Poisson's summation formula for the latter. We also show that `θ` is differentiable on `ℍ`, and `θ(τ) - 1` has exponential decay as `im τ → ∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.html"}, {"id": "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 2, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 33.667, "z": -66.22, "size": 0.306, "title": "The two-variable Jacobi theta function", "summary": "This file defines the two-variable Jacobi theta function $$\\theta(z, \\tau) = \\sum_{n \\in \\mathbb{Z}} \\exp (2 i \\pi n z + i \\pi n ^ 2 \\tau),$$ and proves the functional equation relating the values at `(z, τ)` and `(z / τ, -1 / τ)`, using Poisson's summation formula. We also show holomorphy (jointly in both variables). Additionally, we show some analogous results about the derivative (in the `z`-variable)…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.Ideal.Basic", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 3, "macro_tier_score": 0.0526, "macro_tier_override": null, "x": 92.963, "z": -32.346, "size": 0.3686, "title": "Dedekind domains and invertible ideals", "summary": "In this file, we show a ring is a Dedekind domain iff all fractional ideals are invertible, and prove instances such as the unique factorization of ideals. Further results on the structure of ideals in a Dedekind domain are found in `Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/Ideal/Basic.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.Pointwise", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.0515, "macro_tier_override": null, "x": 57.348, "z": -5.075, "size": 0.2975, "title": "Pointwise actions on subalgebras.", "summary": "If `R'` acts on an `R`-algebra `A` (so that `R'` and `R` actions commute) then we get an `R'` action on the collection of `R`-subalgebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/Pointwise.html"}, {"id": "Mathlib.RingTheory.FractionalIdeal.Inverse", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 3, "macro_tier_score": 0.0515, "macro_tier_override": null, "x": -71.204, "z": -47.204, "size": 0.2975, "title": "Inverse operator for fractional ideals", "summary": "This file defines the notation `I⁻¹` where `I` is a not necessarily invertible fractional ideal. Note that this is somewhat misleading notation in case `I` is not invertible. The theorem that all nonzero fractional ideals are invertible in a Dedekind domain can be found in `Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/FractionalIdeal/Inverse.html"}, {"id": "Mathlib.RingTheory.Spectrum.Maximal.Defs", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 3, "macro_tier_score": 0.1377, "macro_tier_override": null, "x": -37.292, "z": -30.674, "size": 0.3577, "title": "Maximal spectrum of a commutative (semi)ring", "summary": "The maximal spectrum of a commutative (semi)ring is the type of all maximal ideals. It is naturally a subset of the prime spectrum endowed with the subspace topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Maximal/Defs.html"}, {"id": "Mathlib.RingTheory.IntegralClosure.GoingDown", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0045, "macro_tier_override": null, "x": -50.445, "z": -93.066, "size": 0.2376, "title": "Going down for integrally closed domains", "summary": "In this file, we provide the instance that any integral extension of `R ⊆ S` satisfies going down if `R` is integrally closed.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IntegralClosure/GoingDown.html"}, {"id": "Mathlib.RingTheory.Ideal.GoingDown", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 2, "macro_tier_score": 0.0177, "macro_tier_override": null, "x": 4.146, "z": 96.483, "size": 0.3379, "title": "Going down", "summary": "In this file we define a predicate `Algebra.HasGoingDown`: An `R`-algebra `S` satisfies `Algebra.HasGoingDown R S` if for every pair of prime ideals `p ≤ q` of `R` with `Q` a prime of `S` lying above `q`, there exists a prime `P ≤ Q` of `S` lying above `p`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/GoingDown.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 1, "macro_tier_score": 0.0044, "macro_tier_override": null, "x": 44.823, "z": 26.362, "size": 0.3983, "title": "Affine equivalences", "summary": "In this file we define `AffineEquiv k P₁ P₂` (notation: `P₁ ≃ᵃ[k] P₂`) to be the type of affine equivalences between `P₁` and `P₂`, i.e., equivalences such that both forward and inverse maps are affine maps. We define the following equivalences: * `AffineEquiv.refl k P`: the identity map as an `AffineEquiv`; * `e.symm`: the inverse map of an `AffineEquiv` as an `AffineEquiv`; * `e.trans e'`: composition of two…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.Opposite", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.4354, "macro_tier_override": null, "x": 14.857, "z": -0.054, "size": 0.3225, "title": "Scalar actions on and by `Mᵐᵒᵖ`", "summary": "This file defines the actions on the opposite type `SMul R Mᵐᵒᵖ`, and actions by the opposite type, `SMul Rᵐᵒᵖ M`. Note that `MulOpposite.smul` is provided in an earlier file as it is needed to provide the `NSMul.nsmul` and `ZSMul.zsmul` fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/Opposite.html"}, {"id": "Mathlib.Algebra.Homology.ComplexShapeSigns", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 3.792, "z": -6.388, "size": 0.2723, "title": "Signs in constructions on homological complexes", "summary": "In this file, we shall introduce various typeclasses which will allow the construction of the total complex of a bicomplex and of the monoidal category structure on categories of homological complexes (TODO). The most important definition is that of `TotalComplexShape c₁ c₂ c₁₂` given three complex shapes `c₁`, `c₂`, `c₁₂`: it allows the definition of a total complex functor `HomologicalComplex₂ C c₁ c₂ ⥤…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ComplexShapeSigns.html"}, {"id": "Mathlib.GroupTheory.GroupAction.SubMulAction.Closure", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 11.803, "z": -29.282, "size": 0.2546, "title": "Closure and finiteness of `SubMulAction` and `SubAddAction`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/SubMulAction/Closure.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Polynomial", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.2, "macro_tier_override": null, "x": 67.62, "z": 20.199, "size": 0.2863, "title": "Matrices of polynomials and polynomials of matrices", "summary": "In this file, we prove results about matrices over a polynomial ring. In particular, we give results about the polynomial given by `det (t * I + A)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Polynomial.html"}, {"id": "Mathlib.Algebra.Group.Hom.Basic", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.5037, "macro_tier_override": null, "x": -5.524, "z": -4.967, "size": 0.6544, "title": "Additional lemmas about monoid and group homomorphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Hom/Basic.html"}, {"id": "Mathlib.Algebra.Group.Irreducible.Lemmas", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.3928, "macro_tier_override": null, "x": -15.948, "z": -5.004, "size": 0.2809, "title": "More lemmas about irreducible elements", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Irreducible/Lemmas.html"}, {"id": "Mathlib.GroupTheory.Perm.Finite", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 3, "macro_tier_score": 0.2286, "macro_tier_override": null, "x": -22.127, "z": 32.116, "size": 0.2571, "title": "Permutations on `Fintype`s", "summary": "This file contains miscellaneous lemmas about `Equiv.Perm` and `Equiv.swap`, building on top of those in `Mathlib/Logic/Equiv/Basic.lean` and other files in `Mathlib/GroupTheory/Perm/*`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Finite.html"}, {"id": "Mathlib.Algebra.Group.Hom.End", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4128, "macro_tier_override": null, "x": -0.493, "z": 11.132, "size": 0.3614, "title": "Instances on spaces of monoid and group morphisms", "summary": "This file does two things involving `AddMonoid.End` and `Ring`. They are separate, and if someone would like to split this file in two that may be helpful. * We provide the `Ring` structure on `AddMonoid.End`. * Results about `AddMonoid.End R` when `R` is a ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Hom/End.html"}, {"id": "Mathlib.NumberTheory.Bertrand", "region_id": "algebra", "micro_elevation": 0.0263, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.614, "z": 3.663, "size": 0.2, "title": "Bertrand's Postulate", "summary": "This file contains a proof of Bertrand's postulate: That between any positive number and its double there is a prime. The proof follows the outline of the Erdős proof presented in \"Proofs from THE BOOK\": One considers the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various groups, then upper bounds the contribution of each group. This upper bounds the central binomial…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Bertrand.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.Ideal", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": -46.331, "z": -3.01, "size": 0.2305, "title": "Lemmas about ideals of `MonoidAlgebra` and `AddMonoidAlgebra`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/Ideal.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.StupidTrunc", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -6.66, "z": 23.206, "size": 0.2, "title": "The stupid truncation of homological complexes", "summary": "Given an embedding `e : c.Embedding c'` of complex shapes, we define a functor `stupidTruncFunctor : HomologicalComplex C c' ⥤ HomologicalComplex C c'` which sends `K` to `K.stupidTrunc e` which is defined as `(K.restriction e).extend e`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/StupidTrunc.html"}, {"id": "Mathlib.Algebra.Ring.Pointwise.Finset", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 18.196, "z": -12.868, "size": 0.2, "title": "Pointwise operations of sets in a ring", "summary": "This file proves properties of pointwise operations of sets in a ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Pointwise/Finset.html"}, {"id": "Mathlib.Algebra.Ring.Pointwise.Set", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 3.712, "z": 18.197, "size": 0.239, "title": "Pointwise operations of sets in a ring", "summary": "This file proves properties of pointwise operations of sets in a ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Pointwise/Set.html"}, {"id": "Mathlib.RingTheory.Localization.NumDen", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 3, "macro_tier_score": 0.066, "macro_tier_override": null, "x": -24.953, "z": -51.883, "size": 0.3024, "title": "Numerator and denominator in a localization", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/NumDen.html"}, {"id": "Mathlib.RingTheory.EssentialFiniteness", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 3, "macro_tier_score": 0.1355, "macro_tier_override": null, "x": 52.989, "z": -57.239, "size": 0.3552, "title": "Essentially of finite type algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/EssentialFiniteness.html"}, {"id": "Mathlib.NumberTheory.Modular", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2769, "title": "The action of the modular group SL(2, ℤ) on the upper half-plane", "summary": "We define the action of `SL(2,ℤ)` on `ℍ` (via restriction of the `SL(2,ℝ)` action in `Analysis.Complex.UpperHalfPlane`). We then define the standard fundamental domain (`ModularGroup.fd`, `𝒟`) for this action and show (`ModularGroup.exists_smul_mem_fd`) that any point in `ℍ` can be moved inside `𝒟`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Modular.html"}, {"id": "Mathlib.NumberTheory.ModularForms.QExpansion", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 1, "macro_tier_score": 0.0017, "macro_tier_override": null, "x": -33.522, "z": -59.984, "size": 0.3006, "title": "q-expansions of functions on the upper half plane", "summary": "We show that a function on the upper half plane with strict period `n` can be written as `τ ↦ F (𝕢 n τ)` where `F` is analytic on the open unit disc, and `𝕢 n` is the parameter `τ ↦ exp (2 * I * π * τ / n)`. As an application, we show that cusp forms decay exponentially to 0 as `im τ → ∞`. We also define the `q`-expansion of a function `f` on the upper half plane, either as a power series or as a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/QExpansion.html"}, {"id": "Mathlib.LinearAlgebra.Countable", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": -11.188, "z": 46.972, "size": 0.2466, "title": "Countable modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Countable.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Charpoly.Univ", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -73.744, "z": 39.32, "size": 0.2357, "title": "The universal characteristic polynomial", "summary": "In this file we define the universal characteristic polynomial `Matrix.charpoly.univ`, which is the characteristic polynomial of the matrix with entries `Xᵢⱼ`, and hence has coefficients that are multivariate polynomials. It is universal in the sense that one obtains the characteristic polynomial of a matrix `M` by evaluating the coefficients of `univ` at the entries of `M`. We use it to show that the coefficients…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Charpoly/Univ.html"}, {"id": "Mathlib.RingTheory.Polynomial.Morse", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -77.581, "z": -8.082, "size": 0.2, "title": "Galois Groups of Morse Polynomials", "summary": "This file proves that Morse polynomials have Galois group `S_n`. A Morse polynomial is a polynomial whose roots have at most one collision modulo each maximal ideal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Morse.html"}, {"id": "Mathlib.RingTheory.RingHom.Smooth", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 2, "macro_tier_score": 0.0069, "macro_tier_override": null, "x": 11.454, "z": 107.105, "size": 0.2636, "title": "Smooth ring homomorphisms", "summary": "In this file we define smooth ring homomorphisms and show their meta properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/Smooth.html"}, {"id": "Mathlib.FieldTheory.PurelyInseparable.AdjoinPthRoots", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -59.661, "z": -104.942, "size": 0.2, "title": "The extension adjoining all p-th roots to a field of characteristic p.", "summary": "In this file, we introduce the field extension adjoining all `p`-th roots to a field of (exponential) characteristic `p`. # Main definitions and results * `AdjoinPthRoots`: the field extension adjoining all `p`-th roots to a field of (exponential) characteristic `p`. * `AdjoinPthRoots.root`: for `k` a field of (exponential) characteristic `p`, the `p`-th root map `k → AdjoinPthRoots k`, mapping an element to its…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/PurelyInseparable/AdjoinPthRoots.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Dual", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.1328, "macro_tier_override": null, "x": 64.916, "z": -3.322, "size": 0.3245, "title": "Dual space, linear maps and matrices.", "summary": "This file contains some results about matrices and dual spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Dual.html"}, {"id": "Mathlib.Algebra.Module.Presentation.Finite", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -22.663, "z": 66.834, "size": 0.2, "title": "Characterization of finitely presented modules", "summary": "A module is finitely presented (in the sense of `Module.FinitePresentation`) iff it admits a presentation with finitely many generators and relations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Presentation/Finite.html"}, {"id": "Mathlib.RingTheory.RamificationInertia.Basic", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 2, "macro_tier_score": 0.01, "macro_tier_override": null, "x": -74.476, "z": -101.989, "size": 0.3446, "title": "Ramification index and inertia degree", "summary": "This file proves that the sum of ramification times inertia equals the degree of the extension. Typically this is only stated for extensions of Dedekind domains, but we prove it for any finite flat extension of an integral domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RamificationInertia/Basic.html"}, {"id": "Mathlib.RingTheory.Polynomial.ContentIdeal", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 44.61, "z": -63.985, "size": 0.239, "title": "The content ideal of a polynomial", "summary": "In this file we introduce the content ideal of a polynomial `p : R[X]` as the ideal generated by its coefficients, and we prove some basic properties about it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/ContentIdeal.html"}, {"id": "Mathlib.RingTheory.Polynomial.Content", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 3, "macro_tier_score": 0.1856, "macro_tier_override": null, "x": -72.077, "z": -24.552, "size": 0.2951, "title": "GCD structures on polynomials", "summary": "Definitions and basic results about polynomials over GCD domains, particularly their contents and primitive polynomials.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Content.html"}, {"id": "Mathlib.RingTheory.Congruence.Opposite", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 4, "macro_tier_score": 0.33, "macro_tier_override": null, "x": -44.191, "z": -5.817, "size": 0.3802, "title": "Congruences on the opposite ring", "summary": "This file defines the order isomorphism between the congruences on a ring `R` and the congruences on the opposite ring `Rᵐᵒᵖ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Congruence/Opposite.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Symmetric", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.2018, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.3426, "title": "Symmetric matrices", "summary": "This file contains the definition and basic results about symmetric matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Symmetric.html"}, {"id": "Mathlib.Algebra.Module.PointwisePi", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 7.804, "z": 18.879, "size": 0.2, "title": "Pointwise actions on sets in Pi types", "summary": "This file contains lemmas about pointwise actions on sets in Pi types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/PointwisePi.html"}, {"id": "Mathlib.Algebra.Category.FGModuleCat.Abelian", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -28.809, "z": -84.36, "size": 0.2302, "title": "`FGModuleCat K` is an abelian category.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/FGModuleCat/Abelian.html"}, {"id": "Mathlib.Algebra.Category.FGModuleCat.Colimits", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 2, "macro_tier_score": 0.008, "macro_tier_override": null, "x": -87.105, "z": -5.628, "size": 0.2653, "title": "`forget₂ (FGModuleCat K) (ModuleCat K)` creates all finite colimits.", "summary": "And hence `FGModuleCat K` has all finite colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/FGModuleCat/Colimits.html"}, {"id": "Mathlib.Algebra.Category.FGModuleCat.Limits", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 2, "macro_tier_score": 0.008, "macro_tier_override": null, "x": 81.527, "z": 31.182, "size": 0.2653, "title": "`forget₂ (FGModuleCat K) (ModuleCat K)` creates all finite limits.", "summary": "And hence `FGModuleCat K` has all finite limits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/FGModuleCat/Limits.html"}, {"id": "Mathlib.NumberTheory.Zsqrtd.ToReal", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 59.677, "z": 25.764, "size": 0.2, "title": "Image of `Zsqrtd` in `ℝ`", "summary": "This file defines `Zsqrtd.toReal` and related lemmas. It is in a separate file to avoid pulling in all of `Data.Real` into `Data.Zsqrtd`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Zsqrtd/ToReal.html"}, {"id": "Mathlib.RingTheory.Polynomial.Dickson", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -48.87, "z": 58.392, "size": 0.2, "title": "Dickson polynomials", "summary": "The (generalised) Dickson polynomials are a family of polynomials indexed by `ℕ × ℕ`, with coefficients in a commutative ring `R` depending on an element `a∈R`. More precisely, the they satisfy the recursion `dickson k a (n + 2) = X * (dickson k a n + 1) - a * (dickson k a n)` with starting values `dickson k a 0 = 3 - k` and `dickson k a 1 = X`. In the literature, `dickson k a n` is called the `n`-th Dickson…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Dickson.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Decomposition", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -53.842, "z": -20.387, "size": 0.239, "title": "Decomposition of tensor product", "summary": "In this file we show that if `ℳ` is a decomposition of an `R`-module `M` indexed by a type `ι`, then the `S`-module `S ⊗[R] M` has a decomposition `fun i ↦ (ℳ i).baseChange S` indexed by the same `ι`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Decomposition.html"}, {"id": "Mathlib.Algebra.Ring.NonZeroDivisors", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 4, "macro_tier_score": 0.3459, "macro_tier_override": null, "x": 10.358, "z": -21.808, "size": 0.3422, "title": "Non-zero divisors in a ring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/NonZeroDivisors.html"}, {"id": "Mathlib.Algebra.Order.Module.PositiveLinearMap", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": -10.454, "z": 23.806, "size": 0.2996, "title": "Positive linear maps", "summary": "This file defines positive linear maps as a linear map that is also an order homomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Module/PositiveLinearMap.html"}, {"id": "Mathlib.RingTheory.Localization.Basic", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 3, "macro_tier_score": 0.2348, "macro_tier_override": null, "x": -43.518, "z": 16.183, "size": 0.3904, "title": "Localizations of commutative rings", "summary": "This file contains various basic results on localizations. We characterize the localization of a commutative ring `R` at a submonoid `M` up to isomorphism; that is, a commutative ring `S` is the localization of `R` at `M` iff we can find a ring homomorphism `f : R →+* S` satisfying 3 properties: 1. For all `y ∈ M`, `f y` is a unit; 2. For all `z : S`, there exists `(x, y) : R × M` such that `z * f y = f x`; 3. For…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Basic.html"}, {"id": "Mathlib.RepresentationTheory.Continuous.Basic", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -52.076, "z": 72.351, "size": 0.2517, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Continuous/Basic.html"}, {"id": "Mathlib.GroupTheory.Perm.Fin", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 3, "macro_tier_score": 0.2092, "macro_tier_override": null, "x": 42.634, "z": 18.385, "size": 0.3211, "title": "Permutations of `Fin n`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Fin.html"}, {"id": "Mathlib.Algebra.Module.Submodule.EqLocus", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 4, "macro_tier_score": 0.3838, "macro_tier_override": null, "x": -19.276, "z": 33.904, "size": 0.3535, "title": "The submodule of elements `x : M` such that `f x = g x`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/EqLocus.html"}, {"id": "Mathlib.Algebra.Module.LinearMap.End", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 4, "macro_tier_score": 0.3912, "macro_tier_override": null, "x": -4.539, "z": 29.366, "size": 0.3927, "title": "Endomorphisms of a module", "summary": "In this file we define the type of linear endomorphisms of a module over a ring (`Module.End`). We set up the basic theory, including the action of `Module.End` on the module we are considering endomorphisms of.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LinearMap/End.html"}, {"id": "Mathlib.Algebra.Order.Group.Cyclic", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 3, "macro_tier_score": 0.0356, "macro_tier_override": null, "x": 37.431, "z": -20.578, "size": 0.2507, "title": "Cyclic linearly ordered groups", "summary": "This file contains basic results about cyclic linearly ordered groups and cyclic subgroups of linearly ordered groups. The definitions `LinearOrderedCommGroup.Subgroup.genLTOne` (*resp.* `LinearOrderedCommGroup.genLTOne`) yields a generator of a non-trivial subgroup of a linearly ordered commutative group with (*resp.* of a non-trivial linearly ordered commutative group) that is strictly less than `1`. The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Cyclic.html"}, {"id": "Mathlib.RingTheory.Valuation.ValuationSubring", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 3, "macro_tier_score": 0.0379, "macro_tier_override": null, "x": 63.382, "z": 72.862, "size": 0.3531, "title": "Valuation subrings of a field", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/ValuationSubring.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Pointwise.Set.Card", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -18.287, "z": 15.763, "size": 0.2, "title": "Cardinality of sets under pointwise group with zero operations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Pointwise/Set/Card.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Kernels", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 2, "macro_tier_score": 0.0285, "macro_tier_override": null, "x": -29.491, "z": 45.066, "size": 0.3184, "title": "The concrete (co)kernels in the category of modules are (co)kernels in the categorical sense.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Kernels.html"}, {"id": "Mathlib.Algebra.Order.Monoid.WithTop", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.3262, "macro_tier_override": null, "x": 3.61, "z": -14.412, "size": 0.3567, "title": "Adjoining top/bottom elements to ordered monoids.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/WithTop.html"}, {"id": "Mathlib.RingTheory.WittVector.InitTail", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": 60.478, "z": -93.59, "size": 0.304, "title": "`init` and `tail`", "summary": "Given a Witt vector `x`, we are sometimes interested in its components before and after an index `n`. This file defines those operations, proves that `init` is polynomial, and shows how that polynomial interacts with `MvPolynomial.bind₁`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/InitTail.html"}, {"id": "Mathlib.Algebra.Module.LocalizedModule.Int", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 3, "macro_tier_score": 0.0357, "macro_tier_override": null, "x": 4.231, "z": 48.101, "size": 0.2694, "title": "Integer elements of a localized module", "summary": "This is a mirror of the corresponding notion for localizations of rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LocalizedModule/Int.html"}, {"id": "Mathlib.NumberTheory.Niven", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -95.651, "z": 13.309, "size": 0.2302, "title": "Niven's Theorem", "summary": "This file proves Niven's theorem, stating that the only rational angles _in degrees_ which also have rational cosines, are 0, 30 degrees, and 90 degrees - up to reflection and shifts by π. Equivalently, the only rational numbers that occur as `cos(π * p / q)` are the five values `{-1, -1/2, 0, 1/2, 1}`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Niven.html"}, {"id": "Mathlib.RingTheory.Polynomial.RationalRoot", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 3, "macro_tier_score": 0.0648, "macro_tier_override": null, "x": 36.211, "z": -87.52, "size": 0.2974, "title": "Rational root theorem and integral root theorem", "summary": "This file contains the rational root theorem and integral root theorem. The rational root theorem (`num_dvd_of_is_root` and `den_dvd_of_is_root`) for a unique factorization domain `A` with localization `S`, states that the roots of `p : A[X]` in `A`'s field of fractions are of the form `x / y` with `x y : A`, `x ∣ p.coeff 0` and `y ∣ p.leadingCoeff`. The corollary is the integral root theorem…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/RationalRoot.html"}, {"id": "Mathlib.RingTheory.HahnSeries.Binomial", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 46.323, "z": -55.679, "size": 0.2, "title": "Binomial expansions of powers of Hahn Series", "summary": "We introduce binomial expansions using `embDomain`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HahnSeries/Binomial.html"}, {"id": "Mathlib.RingTheory.HahnSeries.HEval", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -54.479, "z": -41.878, "size": 0.2478, "title": "Evaluation of power series in Hahn Series", "summary": "We describe a class of ring homomorphisms from formal power series to Hahn series, given by substitution of the generating variable to an element of strictly positive order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HahnSeries/HEval.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Binomial", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -70.223, "z": 7.008, "size": 0.2478, "title": "Binomial Power Series", "summary": "We introduce formal power series of the form `(1 + X) ^ r`, where `r` is an element of a commutative binomial ring `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Binomial.html"}, {"id": "Mathlib.Algebra.Module.StablyFree.FreeOfInvertible", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -98.891, "z": 37.769, "size": 0.2, "title": null, "summary": "This file proves that a finite stably free module `M` is free if it is invertible.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/StablyFree/FreeOfInvertible.html"}, {"id": "Mathlib.Algebra.Module.StablyFree.Basic", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": 29.566, "z": -74.183, "size": 0.2255, "title": "Stably free modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/StablyFree/Basic.html"}, {"id": "Mathlib.LinearAlgebra.Alternating.Uncurry.Fin", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 62.42, "z": -9.534, "size": 0.262, "title": "Uncurrying alternating maps", "summary": "Given a function `f` which is linear in the first argument and is alternating form in the other `n` arguments, this file defines an alternating form `AlternatingMap.alternatizeUncurryFin f` in `n + 1` arguments. This function is given by ``` AlternatingMap.alternatizeUncurryFin f v = ∑ i : Fin (n + 1), (-1) ^ (i : ℕ) • f (v i) (removeNth i v) ``` Given an alternating map `f` of `n + 1` arguments, each term in the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Alternating/Uncurry/Fin.html"}, {"id": "Mathlib.RingTheory.PicardGroup", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": 103.45, "z": -10.696, "size": 0.2255, "title": "The Picard group of a commutative ring", "summary": "This file defines the Picard group `CommRing.Pic R` of a commutative ring `R` as the type of invertible `R`-modules (in the sense that `M` is invertible if there exists another `R`-module `N` such that `M ⊗[R] N ≃ₗ[R] R`) up to isomorphism, equipped with tensor product as multiplication.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PicardGroup.html"}, {"id": "Mathlib.RingTheory.SimpleModule.InjectiveProjective", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -42.468, "z": 46.729, "size": 0.2302, "title": null, "summary": "If `R` is a semisimple ring, then any `R`-module is both injective and projective.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SimpleModule/InjectiveProjective.html"}, {"id": "Mathlib.Algebra.Polynomial.HasseDeriv", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.1799, "macro_tier_override": null, "x": -10.138, "z": 69.84, "size": 0.2769, "title": "Hasse derivative of polynomials", "summary": "The `k`th Hasse derivative of a polynomial `∑ a_i X^i` is `∑ (i.choose k) a_i X^(i-k)`. It is a variant of the usual derivative, and satisfies `k! * (hasseDeriv k f) = derivative^[k] f`. The main benefit is that is gives an atomic way of talking about expressions such as `(derivative^[k] f).eval r / k!`, that occur in Taylor expansions, for example.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/HasseDeriv.html"}, {"id": "Mathlib.Algebra.Order.Module.Pointwise", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -16.923, "z": 19.739, "size": 0.2634, "title": "Bounds on scalar multiplication of set", "summary": "This file proves order properties of pointwise operations of sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Module/Pointwise.html"}, {"id": "Mathlib.GroupTheory.NoncommPiCoprod", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 3, "macro_tier_score": 0.2285, "macro_tier_override": null, "x": 37.233, "z": -11.607, "size": 0.2514, "title": "Canonical homomorphism from a finite family of monoids", "summary": "This file defines the construction of the canonical homomorphism from a family of monoids. Given a family of morphisms `ϕ i : N i →* M` for each `i : ι` where elements in the images of different morphisms commute, we obtain a canonical morphism `MonoidHom.noncommPiCoprod : (Π i, N i) →* M` that coincides with `ϕ`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/NoncommPiCoprod.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Basis", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 3, "macro_tier_score": 0.2723, "macro_tier_override": null, "x": 55.651, "z": 20.853, "size": 0.4288, "title": "Bases and dimensionality of tensor products of modules", "summary": "This file defines various bases on the tensor product of modules, and shows that the tensor product of free modules is again free.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Basis.html"}, {"id": "Mathlib.RingTheory.Ideal.Norm.RelNorm", "region_id": "algebra", "micro_elevation": 0.9211, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 66.596, "z": -111.649, "size": 0.2531, "title": "Ideal norms", "summary": "This file defines the relative ideal norm `Ideal.spanNorm R (I : Ideal S) : Ideal S` as the ideal spanned by the norms of elements in `I`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Norm/RelNorm.html"}, {"id": "Mathlib.NumberTheory.RamificationInertia.Galois", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 5, "macro_tier_score": 0.0048, "macro_tier_override": 5, "x": -40.763, "z": -121.488, "size": 0.2775, "title": "Ramification theory in Galois extensions of Dedekind domains", "summary": "In this file, we discuss the ramification theory in Galois extensions of Dedekind domains, which is also called Hilbert's Ramification Theory. Assume `B / A` is a finite extension of Dedekind domains, `K` is the fraction ring of `A`, `L` is the fraction ring of `K`, `L / K` is a Galois extension.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/RamificationInertia/Galois.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.Factorization", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 2, "macro_tier_score": 0.0072, "macro_tier_override": null, "x": -42.185, "z": -121.001, "size": 0.3006, "title": "Factorization of ideals and fractional ideals of Dedekind domains", "summary": "Every nonzero ideal `I` of a Dedekind domain `R` can be factored as a product `∏_v v^{n_v}` over the maximal ideals of `R`, where the exponents `n_v` are natural numbers. Similarly, every nonzero fractional ideal `I` of a Dedekind domain `R` can be factored as a product `∏_v v^{n_v}` over the maximal ideals of `R`, where the exponents `n_v` are integers. We define `FractionalIdeal.count K v I` (abbreviated as…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/Factorization.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.Instances", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": -14.356, "z": -103.006, "size": 0.2373, "title": "Instances for Dedekind domains", "summary": "This file contains various instances to work with localization of a ring extension. A very common situation in number theory is to have an extension of (say) Dedekind domains `R` and `S`, and to prove a property of this extension it is useful to consider the localization `Rₚ` of `R` at `P`, a prime ideal of `R`. One also works with the corresponding localization `Sₚ` of `S` and the fraction fields `K` and `L` of `R`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/Instances.html"}, {"id": "Mathlib.RingTheory.NormalClosure", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": 0.863, "z": 122.57, "size": 0.2373, "title": "Normal closure of an extension of domains", "summary": "We define the normal closure of an extension of domains `R ⊆ S` as a domain `T` such that `R ⊆ S ⊆ T` and the extension `Frac T / Frac R` is Galois, and prove several instances about it. Under the hood, `T` is defined as the `integralClosure` of `S` inside the `IntermediateField.normalClosure` of the extension `Frac S / Frac R` inside the `AlgebraicClosure` of `Frac S`. In particular, if `S` is a Dedekind domain,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/NormalClosure.html"}, {"id": "Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 70.255, "z": -48.605, "size": 0.2338, "title": "Liouville numbers with a given exponent", "summary": "We say that a real number `x` is a Liouville number with exponent `p : ℝ` if there exists a real number `C` such that for infinitely many denominators `n` there exists a numerator `m` such that `x ≠ m / n` and `|x - m / n| < C / n ^ p`. A number is a Liouville number in the sense of `Liouville` if it is `LiouvilleWith` any real exponent, see `forall_liouvilleWith_iff`. * If `p ≤ 1`, then this condition is trivial. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleWith.html"}, {"id": "Mathlib.RingTheory.DividedPowers.SubDPIdeal", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -29.816, "z": -66.008, "size": 0.2, "title": "Sub-divided power-ideals", "summary": "Let `A` be a commutative (semi)ring and let `I` be an ideal of `A` with a divided power structure `hI`. A subideal `J` of `I` is a *sub-dp-ideal* of `(I, hI)` if, for all `n ∈ ℕ > 0` and all `x ∈ J`, `hI.dpow n x ∈ J`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DividedPowers/SubDPIdeal.html"}, {"id": "Mathlib.Algebra.Order.Field.Subfield", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 10.927, "z": -25.625, "size": 0.2, "title": "Ordered instances on subfields", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Field/Subfield.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Submonoid.Pointwise", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 4, "macro_tier_score": 0.4201, "macro_tier_override": null, "x": 14.533, "z": -21.56, "size": 0.3364, "title": "Submonoids in a group with zero", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Submonoid/Pointwise.html"}, {"id": "Mathlib.RingTheory.Polynomial.Ideal", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.1488, "macro_tier_override": null, "x": -17.394, "z": 70.31, "size": 0.2686, "title": "Ideals in polynomial rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Ideal.html"}, {"id": "Mathlib.RingTheory.GradedAlgebra.Radical", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 7.421, "z": 64.576, "size": 0.2481, "title": null, "summary": "This file contains a proof that the radical of any homogeneous ideal is a homogeneous ideal", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/GradedAlgebra/Radical.html"}, {"id": "Mathlib.RingTheory.Coalgebra.GroupLike", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -81.869, "z": -30.272, "size": 0.257, "title": "Group-like elements in a coalgebra", "summary": "This file defines group-like elements in a coalgebra, i.e. elements `a` such that `ε a = 1` and `Δ a = a ⊗ₜ a`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Coalgebra/GroupLike.html"}, {"id": "Mathlib.RingTheory.Ideal.CotangentBaseChange", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 82.986, "z": 9.88, "size": 0.2397, "title": "Base change of cotangent spaces", "summary": "Given an `R`-algebra `S`, an ideal `I` of `S` and a flat `R`-algebra `T`, we show that the base change `T ⊗[R] I/I²` of the cotangent space of `I` is naturally isomorphic to the cotangent space of the extended ideal `I · (T ⊗[R] S)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/CotangentBaseChange.html"}, {"id": "Mathlib.RingTheory.TensorProduct.Quotient", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.1358, "macro_tier_override": null, "x": 48.511, "z": -46.007, "size": 0.3014, "title": "Interaction between quotients and tensor products for algebras", "summary": "This file proves algebra analogs of the isomorphisms in `Mathlib/LinearAlgebra/TensorProduct/Quotient.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/Quotient.html"}, {"id": "Mathlib.RingTheory.FractionalIdeal.Operations", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 3, "macro_tier_score": 0.0517, "macro_tier_override": null, "x": 45.579, "z": 70.049, "size": 0.3121, "title": "More operations on fractional ideals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/FractionalIdeal/Operations.html"}, {"id": "Mathlib.RingTheory.FractionalIdeal.Norm", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 57.19, "z": -110.509, "size": 0.2633, "title": "Fractional ideal norms", "summary": "This file defines the absolute ideal norm of a fractional ideal `I : FractionalIdeal R⁰ K` where `K` is a fraction field of `R`. The norm is defined by `FractionalIdeal.absNorm I = Ideal.absNorm I.num / |Algebra.norm ℤ I.den|` where `I.num` is an ideal of `R` and `I.den` an element of `R⁰` such that `I.den • I = I.num`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/FractionalIdeal/Norm.html"}, {"id": "Mathlib.Algebra.Order.Archimedean.Defs", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 3, "macro_tier_score": 0.1096, "macro_tier_override": null, "x": 0.013, "z": 20.429, "size": 0.331, "title": "Definitions of Archimedean monoids", "summary": "This file defines the archimedean property for ordered monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Archimedean/Defs.html"}, {"id": "Mathlib.RingTheory.Derivation.ToSquareZero", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 2, "macro_tier_score": 0.0224, "macro_tier_override": null, "x": 50.513, "z": 40.909, "size": 0.2571, "title": "Derivations into Square-Zero Ideals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Derivation/ToSquareZero.html"}, {"id": "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 2, "macro_tier_score": 0.0202, "macro_tier_override": null, "x": 21.242, "z": 80.828, "size": 0.2709, "title": "Dual submodule with respect to a bilinear form.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/BilinearForm/DualLattice.html"}, {"id": "Mathlib.Algebra.Homology.ImageToKernel", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.3402, "title": "Image-to-kernel comparison maps", "summary": "Whenever `f : A ⟶ B` and `g : B ⟶ C` satisfy `w : f ≫ g = 0`, we have `image_le_kernel f g w : imageSubobject f ≤ kernelSubobject g` (assuming the appropriate images and kernels exist). `imageToKernel f g w` is the corresponding morphism between objects in `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ImageToKernel.html"}, {"id": "Mathlib.RingTheory.Polynomial.Basic", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.2587, "macro_tier_override": null, "x": 61.282, "z": 35.0, "size": 0.3648, "title": "Ring-theoretic supplement of Algebra.Polynomial.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Basic.html"}, {"id": "Mathlib.Algebra.Module.LinearMap.Star", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 3, "macro_tier_score": 0.2743, "macro_tier_override": null, "x": 9.832, "z": -26.065, "size": 0.2925, "title": "Notation for star-linear maps", "summary": "This is in a separate file as it is not needed until much later, and avoids importing the theory of star operations unnecessarily early.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LinearMap/Star.html"}, {"id": "Mathlib.Algebra.Group.Subgroup.Even", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -14.527, "z": 3.113, "size": 0.2621, "title": "Squares and even elements", "summary": "This file defines the subgroup of squares / even elements in an abelian group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/Even.html"}, {"id": "Mathlib.Algebra.Group.Semiconj.Basic", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 3, "macro_tier_score": 0.2646, "macro_tier_override": null, "x": 6.987, "z": -2.523, "size": 0.312, "title": "Lemmas about semiconjugate elements of a group", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Semiconj/Basic.html"}, {"id": "Mathlib.Algebra.Ring.Subsemiring.Pointwise", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 3, "macro_tier_score": 0.0546, "macro_tier_override": null, "x": 26.353, "z": -9.03, "size": 0.2707, "title": "Pointwise instances on `Subsemiring`s", "summary": "This file provides the action `Subsemiring.PointwiseMulAction` which matches the action of `MulActionSet`. This actions is available in the `Pointwise` locale.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Subsemiring/Pointwise.html"}, {"id": "Mathlib.NumberTheory.Cyclotomic.Discriminant", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -93.754, "z": -84.608, "size": 0.2482, "title": "Discriminant of cyclotomic fields", "summary": "We compute the discriminant of a `p ^ n`-th cyclotomic extension.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Cyclotomic/Discriminant.html"}, {"id": "Mathlib.NumberTheory.ModularForms.CongruenceSubgroups", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 10.288, "z": 29.848, "size": 0.2841, "title": "Congruence subgroups", "summary": "This defines congruence subgroups of `SL(2, ℤ)` such as `Γ(N)`, `Γ₀(N)` and `Γ₁(N)` for `N` a natural number. It also contains basic results about congruence subgroups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.html"}, {"id": "Mathlib.NumberTheory.NumberField.House", "region_id": "algebra", "micro_elevation": 0.9474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 74.677, "z": -110.92, "size": 0.2, "title": "House of an algebraic number", "summary": "This file defines the house of an algebraic number `α`, which is the largest of the modulus of its conjugates.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/House.html"}, {"id": "Mathlib.NumberTheory.SiegelsLemma", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2478, "title": "Siegel's Lemma", "summary": "In this file we introduce and prove Siegel's Lemma in its most basic version. This is a fundamental tool in diophantine approximation and transcendence and says that there exists a \"small\" integral non-zero solution of a non-trivial underdetermined system of linear equations with integer coefficients.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/SiegelsLemma.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 2, "macro_tier_score": 0.021, "macro_tier_override": null, "x": 112.115, "z": 44.748, "size": 0.3359, "title": "Integral closure of Dedekind domains", "summary": "This file shows the integral closure of a Dedekind domain (in particular, the ring of integers of a number field) is a Dedekind domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/IntegralClosure.html"}, {"id": "Mathlib.Algebra.Order.Hom.Units", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -14.352, "z": 8.567, "size": 0.2397, "title": "Isomorphism of ordered monoids descends to units", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Hom/Units.html"}, {"id": "Mathlib.RingTheory.Finiteness.NilpotentKer", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 2, "macro_tier_score": 0.0167, "macro_tier_override": null, "x": -28.253, "z": 62.638, "size": 0.2452, "title": "Descend finiteness along quotients by nilpotent ideals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/NilpotentKer.html"}, {"id": "Mathlib.RingTheory.Noetherian.Nilpotent", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 2, "macro_tier_score": 0.0258, "macro_tier_override": null, "x": 61.02, "z": 27.324, "size": 0.2692, "title": "Nilpotent ideals in Noetherian rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Noetherian/Nilpotent.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.FullyFaithful", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 23.376, "z": -42.251, "size": 0.2914, "title": "The fully faithful embedding of the abelian category in its derived category", "summary": "In this file, we show that for any `n : ℤ`, the functor `singleFunctor C n : C ⥤ DerivedCategory C` is fully faithful.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/FullyFaithful.html"}, {"id": "Mathlib.Algebra.Category.BialgCat.Basic", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 70.489, "z": 3.434, "size": 0.2662, "title": "The category of bialgebras over a commutative ring", "summary": "We introduce the bundled category `BialgCat` of bialgebras over a fixed commutative ring `R` along with the forgetful functors to `CoalgCat` and `AlgCat`. This file mimics `Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/BialgCat/Basic.html"}, {"id": "Mathlib.NumberTheory.RamificationInertia.Basic", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 67.824, "z": -90.741, "size": 0.2, "title": "Ramification index and inertia degree", "summary": "Given `P : Ideal S` lying over `p : Ideal R` for the ring extension `f : R →+* S` (assuming `P` and `p` are prime or maximal where needed), the **ramification index** `Ideal.ramificationIdx f p P` is the multiplicity of `P` in `map f p`, and the **inertia degree** `Ideal.inertiaDeg f p P` is the degree of the field extension `(S / P) : (R / p)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/RamificationInertia/Basic.html"}, {"id": "Mathlib.NumberTheory.RamificationInertia.Inertia", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 2, "macro_tier_score": 0.0104, "macro_tier_override": null, "x": 91.801, "z": 63.16, "size": 0.286, "title": "Ramification index and inertia degree", "summary": "Given `P : Ideal S` lying over `p : Ideal R` for the ring extension `f : R →+* S` (assuming `P` and `p` are prime or maximal where needed), the **inertia degree** `Ideal.inertiaDeg' p P` is the degree of the field extension `(S / P) : (R / p)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/RamificationInertia/Inertia.html"}, {"id": "Mathlib.RingTheory.Coalgebra.Quotient", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 57.462, "z": -34.178, "size": 0.2465, "title": "Coalgebra structure on the quotient by a coideal", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Coalgebra/Quotient.html"}, {"id": "Mathlib.Algebra.GCDMonoid.PUnit", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -16.236, "z": 20.307, "size": 0.2, "title": "`PUnit` is a GCD monoid", "summary": "This file collects facts about algebraic structures on the one-element type, e.g. that it is has a GCD.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GCDMonoid/PUnit.html"}, {"id": "Mathlib.NumberTheory.Height.NumberField", "region_id": "algebra", "micro_elevation": 0.9474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -124.636, "z": -48.432, "size": 0.2, "title": "Heights over number fields", "summary": "We provide an instance of `Height.AdmissibleAbsValues` for algebraic number fields and set up some API.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Height/NumberField.html"}, {"id": "Mathlib.NumberTheory.NumberField.ProductFormula", "region_id": "algebra", "micro_elevation": 0.9342, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -99.49, "z": -86.535, "size": 0.2302, "title": "The Product Formula for number fields", "summary": "In this file we prove the Product Formula for number fields: for any non-zero element `x` of a number field `K`, we have `∏ |x|ᵥ=1` where the product runs over the equivalence classes of absolute values of `K`. The `|⬝|ᵥ` are normalized as follows: - for the infinite places, `|⬝|ᵥ` is the absolute value on `K` induced by the corresponding field embedding in `ℂ` and the usual absolute value on `ℂ`; - for the finite…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/ProductFormula.html"}, {"id": "Mathlib.Algebra.Order.Hom.Lattice", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -20.193, "z": 16.379, "size": 0.2302, "title": "Results on order homomorphism classes and lattice operations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Hom/Lattice.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Exp", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 2, "macro_tier_score": 0.007, "macro_tier_override": null, "x": -78.034, "z": -16.97, "size": 0.2784, "title": "Exponential Power Series", "summary": "This file defines the exponential power series `exp A = ∑ Xⁿ/n!` over ℚ-algebras and develops its key properties, including the fundamental differential equation `(exp A)' = exp A`, a uniqueness characterization, and the functional equation for multiplication.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Exp.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Derivative", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": 75.859, "z": -18.155, "size": 0.257, "title": "Definitions", "summary": "In this file we define an operation `derivative` (formal differentiation) on the ring of formal power series in one variable (over an arbitrary commutative semiring). Under suitable assumptions, we prove that two power series are equal if their derivatives are equal and their constant terms are equal. This will give us a simple tool for proving power series identities. For example, one can easily prove the power…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Derivative.html"}, {"id": "Mathlib.Algebra.Order.Star.Conjneg", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -6.607, "z": -28.971, "size": 0.2, "title": "Order properties of conjugation-negation", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Star/Conjneg.html"}, {"id": "Mathlib.Algebra.Polynomial.Monomial", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 4, "macro_tier_score": 0.2845, "macro_tier_override": null, "x": -42.226, "z": -27.044, "size": 0.3786, "title": "Univariate monomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Monomial.html"}, {"id": "Mathlib.NumberTheory.Ostrowski", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 31.303, "z": 4.109, "size": 0.2, "title": "Ostrowski’s Theorem", "summary": "Ostrowski's Theorem for the field `ℚ`: every absolute value on `ℚ` is equivalent to either a `p`-adic absolute value or to the standard Archimedean (Euclidean) absolute value.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Ostrowski.html"}, {"id": "Mathlib.GroupTheory.SchurZassenhaus", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 14.84, "z": 49.838, "size": 0.2302, "title": "The Schur-Zassenhaus Theorem", "summary": "In this file we prove the Schur-Zassenhaus theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/SchurZassenhaus.html"}, {"id": "Mathlib.Algebra.Algebra.IsSimpleRing", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -12.705, "z": 52.338, "size": 0.2948, "title": "Facts about algebras when the coefficient ring is a simple ring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/IsSimpleRing.html"}, {"id": "Mathlib.Algebra.BigOperators.Balance", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 0, "macro_tier_score": 0.001, "macro_tier_override": null, "x": 37.08, "z": -2.168, "size": 0.3334, "title": "Balancing a function", "summary": "This file defines the balancing of a function `f`, defined as `f` minus its average. This is the unique function `g` such that `f a - f b = g a - g b` for all `a` and `b`, and `∑ a, g a = 0`. This is particularly useful in Fourier analysis as `f` and `g` then have the same Fourier transform, except in the `0`-th frequency where the Fourier transform of `g` vanishes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Balance.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.DotProduct", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 2, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": 31.496, "z": 43.688, "size": 0.2799, "title": "Dot product of two vectors", "summary": "This file contains some results on the map `dotProduct`, which maps two vectors `v w : n → R` to the sum of the entrywise products `v i * w i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/DotProduct.html"}, {"id": "Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -21.934, "z": 76.787, "size": 0.2404, "title": "Homogeneous Localization", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.Finsupp", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -25.647, "z": -4.27, "size": 0.2409, "title": "Connection between `Submonoid.closure` and `Finsupp.prod`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/Finsupp.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.AlgClosed", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -14.676, "z": 104.836, "size": 0.2478, "title": "Quadratic forms over an algebraically closed field", "summary": "`equivalent_sum_squares`: A nondegenerate quadratic form over an algebraically closed field of characteristic not equal to 2 is equivalent to a sum of squares. TODO: generalize `QuadraticForm.isometryEquivSumSquares` to quadratically closed field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/AlgClosed.html"}, {"id": "Mathlib.Algebra.NonAssoc.LieAdmissible.Defs", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -30.914, "z": -32.109, "size": 0.2, "title": "Lie admissible rings and algebras", "summary": "We define a Lie-admissible ring as a nonunital nonassociative ring such that the associator satisfies the identity ``` associator x y z + associator z x y + associator y z x = associator y x z + associator z y x + associator x z y ```", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/NonAssoc/LieAdmissible/Defs.html"}, {"id": "Mathlib.Algebra.NonAssoc.PreLie.Basic", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 14.154, "z": -12.024, "size": 0.239, "title": "Pre-Lie rings and algebras", "summary": "In this file we introduce left and right pre-Lie rings, defined as a `NonUnitalNonAssocRing` where the associator `associator x y z := (x * y) * z - x * (y * z)` is left or right symmetric, respectively. We prove that every `Left(Right)PreLieRing L` is a `Right(Left)PreLieRing L` with the opposite `mul`. The equivalence is simple given by `op : L ≃* Lᵐᵒᵖ`. Everything holds for the algebra versions where `L` is also…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/NonAssoc/PreLie/Basic.html"}, {"id": "Mathlib.NumberTheory.Multiplicity", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 19.607, "z": -42.086, "size": 0.2, "title": "Multiplicity in Number Theory", "summary": "This file contains results in number theory relating to multiplicity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Multiplicity.html"}, {"id": "Mathlib.GroupTheory.GroupAction.IterateAct", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 8.485, "z": 7.222, "size": 0.2628, "title": "Monoid action by iterates of a map", "summary": "In this file we define `IterateMulAct f`, `f : α → α`, as a one field structure wrapper over `ℕ` that acts on `α` by iterates of `f`, `⟨n⟩ • x = f^[n] x`. It is useful to convert between definitions and theorems about maps and monoid actions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/IterateAct.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Shrink", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 4.626, "z": -16.062, "size": 0.2, "title": "Transfer group with zero structures from `α` to `Shrink α`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Shrink.html"}, {"id": "Mathlib.Algebra.GroupWithZero.TransferInstance", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 9.509, "z": -5.809, "size": 0.239, "title": "Transfer algebraic structures across `Equiv`s", "summary": "This continues the pattern set in `Mathlib/Algebra/Group/TransferInstance.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/TransferInstance.html"}, {"id": "Mathlib.RingTheory.Polynomial.ScaleRoots", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 3, "macro_tier_score": 0.1779, "macro_tier_override": null, "x": 63.351, "z": -45.506, "size": 0.3003, "title": "Scaling the roots of a polynomial", "summary": "This file defines `scaleRoots p s` for a polynomial `p` in one variable and a ring element `s` to be the polynomial with root `r * s` for each root `r` of `p` and proves some basic results about it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/ScaleRoots.html"}, {"id": "Mathlib.Algebra.Category.Ring.Under.Property", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -101.594, "z": -10.584, "size": 0.2, "title": "Properties of `P.Under ⊤ R` for `R : CommRingCat`", "summary": "In this file we translate ring theoretic properties of a property of ring homomorphisms `P` in properties of the category `P.Under ⊤ R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/Under/Property.html"}, {"id": "Mathlib.RingTheory.Ideal.Lattice", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 4, "macro_tier_score": 0.3376, "macro_tier_override": null, "x": 34.512, "z": 7.35, "size": 0.4223, "title": "The lattice of ideals in a ring", "summary": "Some basic results on lattice operations on ideals: `⊥`, `⊤`, `⊔`, `⊓`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Lattice.html"}, {"id": "Mathlib.Algebra.Category.Grp.Shrink", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -16.722, "z": 11.735, "size": 0.2445, "title": "Shrinking a functor to `GrpCat`", "summary": "For a functor `C ⥤ GrpCat.{w'}` with `w`-small image, we shrink to a functor `C ⥤ GrpCat.{w}`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Shrink.html"}, {"id": "Mathlib.GroupTheory.Perm.Subgroup", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 20.307, "z": -19.07, "size": 0.2, "title": "Lemmas about subgroups within the permutations (self-equivalences) of a type `α`", "summary": "This file provides extra lemmas about some `Subgroup`s that exist within `Equiv.Perm α`. `GroupTheory.Subgroup` depends on `GroupTheory.Perm.Basic`, so these need to be in a separate file. It also provides decidable instances on membership in these subgroups, since `MonoidHom.decidableMemRange` cannot be inferred without the help of a lambda. The presence of these instances induces a `Fintype` instance on the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Perm/Subgroup.html"}, {"id": "Mathlib.Algebra.Group.EvenFunction", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 2, "macro_tier_score": 0.0094, "macro_tier_override": null, "x": -12.424, "z": 11.181, "size": 0.3005, "title": "Even and odd functions", "summary": "We define even functions `α → β` assuming `α` has a negation, and odd functions assuming both `α` and `β` have negation. These definitions are `Function.Even` and `Function.Odd`; and they are `protected`, to avoid conflicting with the root-level definitions `Even` and `Odd` (which, for functions, mean that the function takes even resp. odd _values_, a wholly different concept).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/EvenFunction.html"}, {"id": "Mathlib.Algebra.Order.Ring.Cone", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 24.122, "z": 1.02, "size": 0.2, "title": "Construct ordered rings from rings with a specified positive cone.", "summary": "In this file we provide the structure `RingCone` that encodes axioms of ordered rings in terms of the subset of non-negative elements. We also provide constructors that convert between cones in rings and the corresponding ordered rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Cone.html"}, {"id": "Mathlib.Algebra.Order.Group.Cone", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 8.2, "z": 14.565, "size": 0.2478, "title": "Construct ordered groups from groups with a specified positive cone.", "summary": "In this file we provide the structure `GroupCone` and the predicate `IsMaxCone` that encode the axioms of ordered groups in terms of the subset of non-negative elements. We also provide constructors that convert between cones in groups and the corresponding ordered groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Cone.html"}, {"id": "Mathlib.Algebra.Ring.Subsemiring.Order", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 21.202, "z": -6.866, "size": 0.2676, "title": "`Order`ed instances for `SubsemiringClass` and `Subsemiring`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Subsemiring/Order.html"}, {"id": "Mathlib.RingTheory.GradedAlgebra.Homogeneous.Maps", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -3.399, "z": 64.912, "size": 0.2478, "title": "Maps on homogeneous ideals", "summary": "In this file we define `HomogeneousIdeal.map` and `HomogeneousIdeal.comap`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/GradedAlgebra/Homogeneous/Maps.html"}, {"id": "Mathlib.Algebra.Order.Archimedean.Submonoid", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -0.718, "z": -24.132, "size": 0.2376, "title": "Submonoids of archimedean monoids", "summary": "This file defines the instances that show that the (mul)archimedean property is retained in a submonoid of the ambient group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Archimedean/Submonoid.html"}, {"id": "Mathlib.RingTheory.Valuation.Discrete.RankOne", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -75.512, "z": -65.995, "size": 0.2542, "title": "Discrete valuations have rank one", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/Discrete/RankOne.html"}, {"id": "Mathlib.RingTheory.Valuation.IsTrivialOn", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 2, "macro_tier_score": 0.0293, "macro_tier_override": null, "x": -50.309, "z": 64.392, "size": 0.2941, "title": "Basic lemmas on valuations that are trivial over a base ring", "summary": "This file contains additional results about `Valuation.IsTrivialOn` which is defined in `Mathlib.RingTheory.Valuation.Basic`. In what follows, we consider a `A`-algebra `B` and a valuation `v` over `B` which is trivial on `A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/IsTrivialOn.html"}, {"id": "Mathlib.Algebra.Order.Floor.Extended", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.239, "title": "Extended floor and ceil", "summary": "This file defines the extended floor and ceil functions `ENat.floor, ENat.ceil : ℝ≥0∞ → ℕ∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Floor/Extended.html"}, {"id": "Mathlib.NumberTheory.ModularForms.LevelOne", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -0.732, "z": -83.569, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/LevelOne.html"}, {"id": "Mathlib.RingTheory.Ideal.Prod", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 4, "macro_tier_score": 0.3038, "macro_tier_override": null, "x": -34.71, "z": 48.24, "size": 0.3447, "title": "Ideals in product rings", "summary": "For commutative rings `R` and `S` and ideals `I ≤ R`, `J ≤ S`, we define `Ideal.prod I J` as the product `I × J`, viewed as an ideal of `R × S`. In `ideal_prod_eq` we show that every ideal of `R × S` is of this form. Furthermore, we show that every prime ideal of `R × S` is of the form `p × S` or `R × p`, where `p` is a prime ideal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Prod.html"}, {"id": "Mathlib.RingTheory.Noetherian.UniqueFactorizationDomain", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 4, "macro_tier_score": 0.2933, "macro_tier_override": null, "x": -57.766, "z": 13.962, "size": 0.3063, "title": "Noetherian domains have unique factorization", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Noetherian/UniqueFactorizationDomain.html"}, {"id": "Mathlib.RingTheory.Regular.Category", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -41.345, "z": 54.885, "size": 0.2409, "title": "Categorical constructions for `IsSMulRegular`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Regular/Category.html"}, {"id": "Mathlib.FieldTheory.Extension", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 3, "macro_tier_score": 0.085, "macro_tier_override": null, "x": -93.082, "z": -42.06, "size": 0.3823, "title": "Extension of field embeddings", "summary": "`IntermediateField.exists_algHom_of_adjoin_splits'` is the main result: if E/L/F is a tower of field extensions, K is another extension of F, and `f` is an embedding of L/F into K/F, such that the minimal polynomials of a set of generators of E/L splits in K (via `f`), then `f` extends to an embedding of E/F into K/F.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Extension.html"}, {"id": "Mathlib.Algebra.Order.Group.Action.Flag", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -20.333, "z": -9.123, "size": 0.2, "title": "Action on flags", "summary": "Order isomorphisms act on flags.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Action/Flag.html"}, {"id": "Mathlib.RingTheory.QuasiFinite.Polynomial", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 1, "macro_tier_score": 0.0045, "macro_tier_override": null, "x": -106.545, "z": 15.836, "size": 0.2376, "title": "Quasi-finite primes in polynomial algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/QuasiFinite/Polynomial.html"}, {"id": "Mathlib.RingTheory.LocalRing.ResidueField.Polynomial", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -53.078, "z": -91.59, "size": 0.2536, "title": "Residue field of primes in polynomial algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/ResidueField/Polynomial.html"}, {"id": "Mathlib.RingTheory.QuasiFinite.Weakly", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": 87.645, "z": -55.988, "size": 0.2536, "title": "Weakly Quasi-finite primes", "summary": "The definition `Algebra.QuasiFiniteAt` is equivalent to the usual definition \"isolated in fibers\" mathematically for algebras of finite type, but this requires Zariski's main theorem to prove. Hence we introduce a weaker notion of being `Algebra.WeaklyQuasiFiniteAt` that we shall state Zariski's main theorem in terms of, and deduce from this that `Algebra.WeaklyQuasiFiniteAt` is equivalent to `Algebra.QuasiFiniteAt`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/QuasiFinite/Weakly.html"}, {"id": "Mathlib.Algebra.Group.Action.TypeTags", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.4526, "macro_tier_override": null, "x": -2.123, "z": -10.939, "size": 0.36, "title": "Additive and Multiplicative for group actions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/TypeTags.html"}, {"id": "Mathlib.Algebra.Category.Grp.EnoughInjectives", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 59.611, "z": 30.274, "size": 0.241, "title": "Category of abelian groups has enough injectives", "summary": "Given an abelian group `A`, then `i : A ⟶ ∏_{A⋆} ℚ ⧸ ℤ` defined by `i : a ↦ c ↦ c a` is an injective presentation for `A`, hence category of abelian groups has enough injectives.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/EnoughInjectives.html"}, {"id": "Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne", "region_id": "algebra", "micro_elevation": 0.9737, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -101.866, "z": 92.252, "size": 0.2541, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/NormLeOne.html"}, {"id": "Mathlib.NumberTheory.NumberField.ClassNumber", "region_id": "algebra", "micro_elevation": 0.9605, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -134.305, "z": 18.495, "size": 0.2754, "title": "Class numbers of number fields", "summary": "This file defines the class number of a number field as the (finite) cardinality of the class group of its ring of integers. It also proves some elementary results on the class number.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/ClassNumber.html"}, {"id": "Mathlib.RingTheory.Derivation.DifferentialRing", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 2, "macro_tier_score": 0.0124, "macro_tier_override": null, "x": -24.001, "z": 60.407, "size": 0.2585, "title": "Differential and Algebras", "summary": "This file defines derivations from a commutative ring to itself as a typeclass, which lets us use the x′ notation for the derivative of x.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Derivation/DifferentialRing.html"}, {"id": "Mathlib.Algebra.Order.Field.Pointwise", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -19.914, "z": 4.555, "size": 0.2523, "title": "Pointwise operations on ordered algebraic objects", "summary": "This file contains lemmas about the effect of pointwise operations on sets with an order structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Field/Pointwise.html"}, {"id": "Mathlib.Algebra.Module.LocalizedModule.AtPrime", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 3, "macro_tier_score": 0.0392, "macro_tier_override": null, "x": -46.521, "z": -12.936, "size": 0.2813, "title": "Localizations of modules at the complement of a prime ideal", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LocalizedModule/AtPrime.html"}, {"id": "Mathlib.NumberTheory.MahlerMeasure", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -99.182, "z": -78.175, "size": 0.2, "title": "Mahler measure of integer polynomials", "summary": "The main purpose of this file is to prove some facts about the Mahler measure of integer polynomials, in particular Northcott's Theorem for the Mahler measure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/MahlerMeasure.html"}, {"id": "Mathlib.Algebra.Polynomial.OfFn", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 34.37, "z": 52.97, "size": 0.2276, "title": "`Polynomial.ofFn` and `Polynomial.toFn`", "summary": "In this file we introduce `ofFn` and `toFn`, two functions that associate a polynomial to the vector of its coefficients and vice versa. We prove some basic APIs for these functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/OfFn.html"}, {"id": "Mathlib.RingTheory.SimpleRing.Principal", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 23.188, "z": -58.732, "size": 0.2276, "title": "A commutative simple ring is a principal ideal domain", "summary": "Indeed, it is a field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SimpleRing/Principal.html"}, {"id": "Mathlib.Algebra.HierarchyDesign", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": "Documentation of the algebraic hierarchy", "summary": "A library note giving advice on modifying the algebraic hierarchy. (It is not intended as a \"tour\".) This is ported directly from the Lean3 version, so may refer to files/types that currently only exist in mathlib3.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/HierarchyDesign.html"}, {"id": "Mathlib.NumberTheory.NumberField.Completion.LiesOverInstances", "region_id": "algebra", "micro_elevation": 0.9342, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 30.619, "z": -128.254, "size": 0.2478, "title": "`LiesOver` instances for completions of number fields", "summary": "If `L` and `K` are number fields such that `Algebra K L` then this algebra extends naturally to the completions of `K` and `L` at places, whenever the place of `L` lies over the place of `K`. This file contains the relevant instances and properties of this extension as `scoped` instances. These are scoped because they create non-defeq instance diamonds when `K = L`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Completion/LiesOverInstances.html"}, {"id": "Mathlib.NumberTheory.ZetaValues", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 75.918, "z": -76.414, "size": 0.2585, "title": "Critical values of the Riemann zeta function", "summary": "In this file we prove formulae for the critical values of `ζ(s)`, and more generally of Hurwitz zeta functions, in terms of Bernoulli polynomials.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ZetaValues.html"}, {"id": "Mathlib.RingTheory.Polynomial.Opposites", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 36.734, "z": 44.33, "size": 0.2, "title": "Interactions between `R[X]` and `Rᵐᵒᵖ[X]`", "summary": "This file contains the basic API for \"pushing through\" the isomorphism `opRingEquiv : R[X]ᵐᵒᵖ ≃+* Rᵐᵒᵖ[X]`. It allows going back and forth between a polynomial ring over a semiring and the polynomial ring over the opposite semiring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Opposites.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 3, "macro_tier_score": 0.0879, "macro_tier_override": null, "x": -35.857, "z": -71.355, "size": 0.2799, "title": null, "summary": "If `A` is a domain, and a finite-dimensional algebra over a field `F`, with prime dimension, then there are no non-trivial `F`-subalgebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/IsSimpleOrder.html"}, {"id": "Mathlib.Algebra.Algebra.Field", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 11.97, "z": -25.155, "size": 0.2338, "title": "Facts about `algebraMap` when the coefficient ring is a field.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Field.html"}, {"id": "Mathlib.RingTheory.Extension.Cotangent.BaseChange", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -87.547, "z": -24.834, "size": 0.2455, "title": "Base change for the naive cotangent complex", "summary": "This file shows that the cotangent space and first homology of the naive cotangent complex commute with base change.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Extension/Cotangent/BaseChange.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Prod", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 10.803, "z": -12.754, "size": 0.2434, "title": "Products of ordered monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Prod.html"}, {"id": "Mathlib.Algebra.Group.Translate", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -7.846, "z": 20.859, "size": 0.2, "title": "Translation operator", "summary": "This file defines the translation of a function from a group by an element of that group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Translate.html"}, {"id": "Mathlib.Algebra.Ring.Subring.Pointwise", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.0539, "macro_tier_override": null, "x": -32.277, "z": -14.257, "size": 0.3112, "title": "Pointwise instances on `Subring`s", "summary": "This file provides the action `Subring.pointwiseMulAction` which matches the action of `mulActionSet`. This actions is available in the `Pointwise` locale.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Subring/Pointwise.html"}, {"id": "Mathlib.RingTheory.Prime", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 8.549, "z": -20.581, "size": 0.2482, "title": "Prime elements in rings", "summary": "This file contains lemmas about prime elements of commutative rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Prime.html"}, {"id": "Mathlib.FieldTheory.SeparableClosure", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 3, "macro_tier_score": 0.0588, "macro_tier_override": null, "x": 105.393, "z": 36.179, "size": 0.3466, "title": "Separable closure", "summary": "This file contains basics about the (relative) separable closure of a field extension.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/SeparableClosure.html"}, {"id": "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.4119, "macro_tier_override": null, "x": 19.161, "z": -11.38, "size": 0.2994, "title": "Big operators on a multiset in ordered groups", "summary": "This file contains the results concerning the interaction of multiset big operators with ordered groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/BigOperators/Group/Multiset.html"}, {"id": "Mathlib.RingTheory.Adjoin.Tower", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 3, "macro_tier_score": 0.248, "macro_tier_override": null, "x": -58.944, "z": 45.212, "size": 0.3174, "title": "Adjoining elements and being finitely generated in an algebra tower", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Adjoin/Tower.html"}, {"id": "Mathlib.RingTheory.Frobenius", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 20.56, "z": 105.735, "size": 0.2, "title": "Frobenius elements", "summary": "In algebraic number theory, if `L/K` is a finite Galois extension of number fields, with rings of integers `𝓞L/𝓞K`, and if `q` is prime ideal of `𝓞L` lying over a prime ideal `p` of `𝓞K`, then there exists a **Frobenius element** `Frob p` in `Gal(L/K)` with the property that `Frob p x ≡ x ^ #(𝓞K/p) (mod q)` for all `x ∈ 𝓞L`. Following `Mathlib/RingTheory/Invariant/Basic.lean`, we develop the theory in the setting…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Frobenius.html"}, {"id": "Mathlib.Algebra.Category.Ring.LinearAlgebra", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 0, "macro_tier_score": 0.0009, "macro_tier_override": null, "x": -9.186, "z": -84.934, "size": 0.3256, "title": "Results on the category of rings requiring linear algebra", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/LinearAlgebra.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.Defs", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": -72.884, "z": 40.893, "size": 0.3017, "title": "Root data and root systems", "summary": "This file contains basic definitions for root systems and root data.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/Defs.html"}, {"id": "Mathlib.Algebra.MvPolynomial.SchwartzZippel", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -22.945, "z": -70.654, "size": 0.2, "title": "The Schwartz-Zippel lemma", "summary": "This file contains a proof of the [Schwartz-Zippel](https://en.wikipedia.org/wiki/Schwartz%E2%80%93Zippel_lemma) lemma. This lemma tells us that the probability that a nonzero multivariable polynomial over an integral domain evaluates to zero at a random point is bounded by the degree of the polynomial over the size of the field, or more generally, that a nonzero multivariable polynomial over any integral domain has…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/SchwartzZippel.html"}, {"id": "Mathlib.LinearAlgebra.Coevaluation", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": -72.313, "z": -41.895, "size": 0.2427, "title": "The coevaluation map on finite-dimensional vector spaces", "summary": "Given a finite-dimensional vector space `V` over a field `K` this describes the canonical linear map from `K` to `V ⊗ Dual K V` which corresponds to the identity function on `V`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Coevaluation.html"}, {"id": "Mathlib.Algebra.LieRinehartAlgebra.Defs", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 78.342, "z": -15.488, "size": 0.2676, "title": "Lie-Rinehart algebras", "summary": "This file defines Lie-Rinehart algebras and their morphisms. It also shows that the derivations of a commutative algebra over a commutative Ring form such a Lie-Rinehart algebra. Lie-Rinehart algebras appear in differential geometry as section spaces of Lie algebroids and singular foliations. The typical Cartan calculus of differential geometry can be restated fully in terms of the Chevalley-Eilenberg algebra of a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/LieRinehartAlgebra/Defs.html"}, {"id": "Mathlib.Algebra.Module.Presentation.Tautological", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 18.477, "z": -52.562, "size": 0.2, "title": "The tautological presentation of a module", "summary": "Given an `A`-module `M`, we provide its tautological presentation: * there is a generator `[m]` for each `m : M`; * the relations are `[m₁] + [m₂] - [m₁ + m₂] = 0` and `a • [m] - [a • m] = 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Presentation/Tautological.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.Finset.Preimage", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 1, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": 0.988, "z": 16.685, "size": 0.379, "title": "Sums and products over preimages of finite sets.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Finset/Preimage.html"}, {"id": "Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone", "region_id": "algebra", "micro_elevation": 0.9605, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -14.882, "z": -134.754, "size": 0.2484, "title": "Fundamental Cone", "summary": "Let `K` be a number field of signature `(r₁, r₂)`. We define an action of the units `(𝓞 K)ˣ` on the mixed space `ℝ^r₁ × ℂ^r₂` via the `mixedEmbedding`. The fundamental cone is a cone in the mixed space that is a fundamental domain for the action of `(𝓞 K)ˣ` modulo torsion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/FundamentalCone.html"}, {"id": "Mathlib.RingTheory.Finiteness.Small", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 3, "macro_tier_score": 0.0514, "macro_tier_override": null, "x": -25.935, "z": -73.563, "size": 0.2835, "title": "Smallness properties of modules and algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Finiteness/Small.html"}, {"id": "Mathlib.NumberTheory.MulChar.Basic", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 2, "macro_tier_score": 0.0163, "macro_tier_override": null, "x": 5.381, "z": -38.627, "size": 0.3203, "title": "Multiplicative characters of finite rings and fields", "summary": "Let `R` and `R'` be commutative rings. A *multiplicative character* of `R` with values in `R'` is a morphism of monoids from the multiplicative monoid of `R` into that of `R'` that sends non-units to zero. We use the namespace `MulChar` for the definitions and results.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/MulChar/Basic.html"}, {"id": "Mathlib.Algebra.Group.Submonoid.Units", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 3, "macro_tier_score": 0.168, "macro_tier_override": null, "x": 23.503, "z": 5.521, "size": 0.3746, "title": "Submonoid of units", "summary": "Given a submonoid `S` of a monoid `M`, we define the subgroup `S.units` as the units of `S` as a subgroup of `Mˣ`. That is to say, `S.units` contains all members of `S` which have a two-sided inverse within `S`, as terms of type `Mˣ`. We also define, for subgroups `S` of `Mˣ`, `S.ofUnits`, which is `S` considered as a submonoid of `M`. `Submonoid.units` and `Subgroup.ofUnits` form a Galois coinsertion. We also make…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Submonoid/Units.html"}, {"id": "Mathlib.LinearAlgebra.FreeProduct.Basic", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 62.277, "z": -10.424, "size": 0.2, "title": "The free product of $R$-algebras", "summary": "We define the free product of an indexed collection of (noncommutative) $R$-algebras `(i : ι) → A i`, with `Algebra R (A i)` for all `i` and `R` a commutative semiring, as the quotient of the tensor algebra on the direct sum `⨁ (i : ι), A i` by the relation generated by extending the relation * `aᵢ ⊗ₜ aᵢ' ~ aᵢ aᵢ'` for all `i : ι` and `aᵢ aᵢ' : A i` * `1ᵢ ~ 1ⱼ` for `1ᵢ := One.one (A i)` and for all `i, j : ι`. to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeProduct/Basic.html"}, {"id": "Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 15.062, "z": -70.846, "size": 0.2, "title": "The Hopf algebra structure on group algebras", "summary": "Given a group `G`, a commutative semiring `R` and an `R`-Hopf algebra `A`, this file collects results about the `R`-Hopf algebra instance on `A[G]`, building upon results in `Mathlib/RingTheory/Bialgebra/MonoidAlgebra.lean` about the bialgebra structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HopfAlgebra/MonoidAlgebra.html"}, {"id": "Mathlib.GroupTheory.Goursat", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 25.901, "z": -2.273, "size": 0.2478, "title": "Goursat's lemma for subgroups", "summary": "This file proves Goursat's lemma for subgroups. If `I` is a subgroup of `G × H` which projects fully on both factors, then there exist normal subgroups `G' ≤ G` and `H' ≤ H` such that `G' × H' ≤ I` and the image of `I` in `G ⧸ G' × H ⧸ H'` is the graph of an isomorphism `G ⧸ G' ≃ H ⧸ H'`. `G'` and `H'` can be explicitly constructed as `Subgroup.goursatFst I` and `Subgroup.goursatSnd I` respectively.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Goursat.html"}, {"id": "Mathlib.Algebra.Order.SuccPred.WithBot", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -16.255, "z": 3.893, "size": 0.2969, "title": "Algebraic properties of the successor function on `WithBot`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/SuccPred/WithBot.html"}, {"id": "Mathlib.Algebra.Lie.Derivation.BaseChange", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 36.049, "z": -75.397, "size": 0.2, "title": "LieDerivations of a Lie algebra created through BaseChange", "summary": "When, given an `R`-algebra `A` and an `R`-Lie algebra `L` the (Lie algebra) basechange `A ⊗[R] L`, both derivations of `A` and Lie derivations of `L` induce Lie derivations of `A ⊗[R] L`. Moreover, both these procedures are Lie algebra homomorphisms themselves.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Derivation/BaseChange.html"}, {"id": "Mathlib.Algebra.Category.Grp.Ulift", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -21.216, "z": -65.358, "size": 0.2239, "title": "Properties of the universe lift functor for groups", "summary": "This file shows that the functors `GrpCat.uliftFunctor` and `CommGrpCat.uliftFunctor` (as well as the additive versions) are fully faithful, preserve all limits and create small limits. Full faithfulness is pretty obvious. To prove that the functors preserve limits, we use the fact that the forgetful functor from `GrpCat` or `CommGrpCat` into `Type` creates all limits (because of the way limits are constructed in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Ulift.html"}, {"id": "Mathlib.Algebra.FreeMonoid.UniqueProds", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 3, "macro_tier_score": 0.2545, "macro_tier_override": null, "x": -24.088, "z": -1.636, "size": 0.3048, "title": "Free monoids have unique products", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/FreeMonoid/UniqueProds.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.Inversion", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -28.953, "z": 96.016, "size": 0.2302, "title": "Results about inverses in Clifford algebras", "summary": "This contains some basic results about the inversion of vectors, related to the fact that $ι(m)^{-1} = \\frac{ι(m)}{Q(m)}$.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.html"}, {"id": "Mathlib.Algebra.Lie.Derivation.Killing", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 116.245, "z": -24.786, "size": 0.2365, "title": "Derivations of finite-dimensional Killing Lie algebras", "summary": "This file establishes that all derivations of finite-dimensional Killing Lie algebras are inner.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Derivation/Killing.html"}, {"id": "Mathlib.RingTheory.Depth.Rees", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 48.353, "z": -83.595, "size": 0.2, "title": "The Rees theorem", "summary": "In this file we prove the Rees theorem for depth, which relates the vanishing of certain `Ext` groups and the length of a maximal regular sequence in a certain ideal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Depth/Rees.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Ext.Basic", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -35.645, "z": 56.563, "size": 0.2302, "title": "Some basic lemmas for manipulating `Ext` over `ModuleCat`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Ext/Basic.html"}, {"id": "Mathlib.RingTheory.Ideal.Basis", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 2, "macro_tier_score": 0.0112, "macro_tier_override": null, "x": -31.086, "z": -43.981, "size": 0.25, "title": "The basis of ideals", "summary": "Some results involving `Ideal` and `Basis`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Basis.html"}, {"id": "Mathlib.Algebra.Regular.ULift", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -11.104, "z": 12.493, "size": 0.2, "title": "Results about `IsRegular` and `ULift`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Regular/ULift.html"}, {"id": "Mathlib.RingTheory.Filtration", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 3, "macro_tier_score": 0.0581, "macro_tier_override": null, "x": 67.726, "z": 42.315, "size": 0.2865, "title": "`I`-filtrations of modules", "summary": "This file contains the definitions and basic results around (stable) `I`-filtrations of modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Filtration.html"}, {"id": "Mathlib.Algebra.Group.MinimalAxioms", "region_id": "algebra", "micro_elevation": 0.0395, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -2.858, "z": -4.782, "size": 0.2491, "title": "Minimal Axioms for a Group", "summary": "This file defines constructors to define a group structure on a Type, while proving only three equalities.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/MinimalAxioms.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Circulant", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -6.388, "z": -6.739, "size": 0.2, "title": "Circulant matrices", "summary": "This file contains the definition and basic results about circulant matrices. Given a vector `v : n → α` indexed by a type that is endowed with subtraction, `Matrix.circulant v` is the matrix whose `(i, j)`th entry is `v (i - j)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Circulant.html"}, {"id": "Mathlib.NumberTheory.ModularForms.JacobiTheta.Manifold", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -76.523, "z": 15.11, "size": 0.2, "title": "Manifold differentiability of the Jacobi theta function", "summary": "In this file we reformulate differentiability of the Jacobi theta function in terms of manifold differentiability.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/JacobiTheta/Manifold.html"}, {"id": "Mathlib.RingTheory.Etale.Finite", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -123.744, "z": -13.044, "size": 0.2, "title": "Category of finite étale `R`-algebras", "summary": "In this file we define the category of finite étale `R`-algebras over a ring `R`. For any geometric point `Ω` of `R`, we define a fiber functor sending a finite étale `R`-algebra `S` to the finite set of `R`-algebra homomorphisms `S →ₐ[R] Ω`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Etale/Finite.html"}, {"id": "Mathlib.LinearAlgebra.Span.TensorProduct", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 72.505, "z": 37.688, "size": 0.2761, "title": "The interaction of linear span and tensor product for mixed scalars.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Span/TensorProduct.html"}, {"id": "Mathlib.RingTheory.Etale.Weakly", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 57.5, "z": -99.759, "size": 0.2, "title": "Weakly étale algebras", "summary": "In this file we define weakly étale algebras. An `R`-algebra `S` is weakly étale if `S` is `R`-flat and the multiplication map `S ⊗[R] S → S` is flat.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Etale/Weakly.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Hadamard", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 2, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -18.695, "z": -54.452, "size": 0.2969, "title": "Hadamard product of matrices", "summary": "This file defines the Hadamard product `Matrix.hadamard` and contains basic properties about them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Hadamard.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.RightDerivedFunctorPlus", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 72.395, "z": 2.215, "size": 0.2, "title": "The right derived functor on the bounded below derived category", "summary": "If `F : C ⥤ D` is an additive functor between abelian categories, where `C` has enough injectives, we define the right derived functor `F.rightDerivedFunctorPlus : DerivedCategory.Plus C ⥤ DerivedCategory.Plus D` between the corresponding bounded below derived categories. TODO(@joelriou): show that this functor is triangulated and refactor the definiton of `Functor.rightDerived`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/RightDerivedFunctorPlus.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.DerivabilityStructureInjectives", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 9.35, "z": -69.95, "size": 0.2676, "title": "The injective derivability structure", "summary": "Let `C` be an abelian category with enough injectives. In this file, we define a localizer morphism `CochainComplex.Plus.localizerMorphism` (relative to quasi-isomorphisms) which is given by the (fully faithful) functor `CochainComplex.Plus (InjectiveObject C) ⥤ CochainComplex.Plus C`, and we show that it is a right derivability structure. (The proof proceeds by showing that up to equivalences of categories, this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/DerivabilityStructureInjectives.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.HomologySequence", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -41.155, "z": -17.115, "size": 0.2774, "title": "The homology sequence", "summary": "In this file, we construct `homologyFunctor C n : DerivedCategory C ⥤ C` for all `n : ℤ`, show that they are homological functors which form a shift sequence, and construct the long exact homology sequences associated to distinguished triangles in the derived category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/HomologySequence.html"}, {"id": "Mathlib.Algebra.Ring.Subgroup", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -35.275, "z": -0.867, "size": 0.2, "title": "Additive subgroups of rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Subgroup.html"}, {"id": "Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 86.337, "z": 61.253, "size": 0.239, "title": "Facts about the Gaussian integers relying on quadratic reciprocity.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.html"}, {"id": "Mathlib.NumberTheory.Zsqrtd.GaussianInt", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -32.98, "z": 56.013, "size": 0.2541, "title": "Gaussian integers", "summary": "The Gaussian integers are complex integer, complex numbers whose real and imaginary parts are both integers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Zsqrtd/GaussianInt.html"}, {"id": "Mathlib.Algebra.Module.SnakeLemma", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 2, "macro_tier_score": 0.0227, "macro_tier_override": null, "x": -34.862, "z": 41.052, "size": 0.2957, "title": "The snake lemma in terms of modules", "summary": "The snake lemma is proven in `Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean` for all abelian categories, but for definitional equality and universe issues we reprove them here for modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/SnakeLemma.html"}, {"id": "Mathlib.Algebra.Module.Submodule.Bilinear", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 4, "macro_tier_score": 0.3515, "macro_tier_override": null, "x": 42.559, "z": 13.242, "size": 0.3435, "title": "Images of pairs of submodules under bilinear maps", "summary": "This file provides `Submodule.map₂`, which is later used to implement `Submodule.mul`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Submodule/Bilinear.html"}, {"id": "Mathlib.GroupTheory.SpecificGroups.Quaternion", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -25.028, "z": -39.106, "size": 0.2, "title": "Quaternion Groups", "summary": "We define the (generalised) quaternion groups `QuaternionGroup n` of order `4n`, also known as dicyclic groups, with elements `a i` and `xa i` for `i : ZMod n`. The (generalised) quaternion groups can be defined by the presentation $\\langle a, x | a^{2n} = 1, x^2 = a^n, x^{-1}ax=a^{-1}\\rangle$. We write `a i` for $a^i$ and `xa i` for $x * a^i$. For `n=2` the quaternion group `QuaternionGroup 2` is isomorphic to the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/SpecificGroups/Quaternion.html"}, {"id": "Mathlib.RingTheory.Polynomial.Eisenstein.Criterion", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 2, "macro_tier_score": 0.0179, "macro_tier_override": null, "x": -63.412, "z": -45.421, "size": 0.2566, "title": "The Eisenstein criterion", "summary": "- `Polynomial.generalizedEisenstein` : Let `R` be an integral domain and let `K` an `R`-algebra which is a field Let `q : R[X]` be a monic polynomial which is prime in `K[X]`. Let `f : R[X]` be a polynomial of strictly positive degree satisfying the following properties: * the image of `f` in `K[X]` is a power of `q`. * the leading coefficient of `f` is not zero in `K` * the polynomial `f` is primitive. Assume…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Eisenstein/Criterion.html"}, {"id": "Mathlib.LinearAlgebra.Alternating.Curry", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 2, "macro_tier_score": 0.0073, "macro_tier_override": null, "x": 61.203, "z": -3.19, "size": 0.3075, "title": "Currying alternating forms", "summary": "In this file we define `AlternatingMap.curryLeft` which interprets an alternating map in `n + 1` variables as a linear map in the 0th variable taking values in the alternating maps in `n` variables.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Alternating/Curry.html"}, {"id": "Mathlib.FieldTheory.AbsoluteGaloisGroup", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -52.306, "z": -104.658, "size": 0.2, "title": "The topological abelianization of the absolute Galois group.", "summary": "We define the absolute Galois group of a field `K` and its topological abelianization.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/AbsoluteGaloisGroup.html"}, {"id": "Mathlib.Algebra.Order.CauSeq.BigOperators", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -31.569, "z": 0.407, "size": 0.2598, "title": "Cauchy sequences and big operators", "summary": "This file proves some more lemmas about basic Cauchy sequences that involve finite sums.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/CauSeq/BigOperators.html"}, {"id": "Mathlib.RingTheory.WittVector.Isocrystal", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 83.401, "z": 87.273, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/Isocrystal.html"}, {"id": "Mathlib.Algebra.Order.Ring.Prod", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -19.465, "z": -6.199, "size": 0.2357, "title": "Products of ordered rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Prod.html"}, {"id": "Mathlib.Algebra.Order.Disjointed", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -16.66, "z": -8.207, "size": 0.2357, "title": "`Disjointed` for functions on a `SuccAddOrder`", "summary": "This file contains material excised from `Mathlib/Order/Disjointed.lean` to avoid import dependencies from `Mathlib.Algebra.Order` into `Mathlib.Order`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Disjointed.html"}, {"id": "Mathlib.Algebra.Group.Fin.Tuple", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.3524, "macro_tier_override": null, "x": -7.395, "z": -0.708, "size": 0.3951, "title": "Algebraic properties of tuples", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Fin/Tuple.html"}, {"id": "Mathlib.Algebra.CharP.Reduced", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 3, "macro_tier_score": 0.1407, "macro_tier_override": null, "x": 26.026, "z": 14.34, "size": 0.3343, "title": "Results about characteristic p reduced rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/Reduced.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Images", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 57.147, "z": 22.14, "size": 0.2, "title": "The category of R-modules has images.", "summary": "Note that we don't need to register any of the constructions here as instances, because we get them from the fact that `ModuleCat R` is an abelian category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Images.html"}, {"id": "Mathlib.RingTheory.Derivation.MapCoeffs", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 2, "macro_tier_score": 0.0117, "macro_tier_override": null, "x": -95.214, "z": -16.14, "size": 0.3041, "title": "Coefficient-wise derivation on polynomials", "summary": "In this file we define applying a derivation on the coefficients of a polynomial, show this forms a derivation, and prove `apply_eval_eq`, which shows that for a derivation `D`, `D(p(x)) = (D.mapCoeffs p)(x) + D(x) * p'(x)`. `apply_aeval_eq` and `apply_aeval_eq'` are generalizations of that for algebras. We also have a special case for `DifferentialAlgebra`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Derivation/MapCoeffs.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Blocks", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": 27.239, "z": -25.252, "size": 0.2483, "title": "Blocks", "summary": "Given `SMul G X`, an action of a type `G` on a type `X`, we define - the predicate `MulAction.IsBlock G B` states that `B : Set X` is a block, which means that the sets `g • B`, for `g ∈ G`, are equal or disjoint. Under `Group G` and `MulAction G X`, this is equivalent to the classical definition `MulAction.IsBlock.def_one` - a bunch of lemmas that give examples of “trivial” blocks : ⊥, ⊤, singletons, and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Blocks.html"}, {"id": "Mathlib.GroupTheory.Solvable", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 3, "macro_tier_score": 0.0702, "macro_tier_override": null, "x": -30.622, "z": -13.409, "size": 0.2844, "title": "Solvable Groups", "summary": "In this file we introduce the notion of a solvable group. We define a solvable group as one whose derived series is eventually trivial. This requires defining the commutator of two subgroups and the derived series of a group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Solvable.html"}, {"id": "Mathlib.RingTheory.MvPowerSeries.Restricted", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -54.648, "z": -27.743, "size": 0.2, "title": "Multivariate restricted power series", "summary": "`IsRestricted` : We say a multivariate power series over a normed ring `R` is restricted for a tuple `c` if `‖coeff t f‖ * ∏ i ∈ t.support, c i ^ t i → 0` under the cofinite filter.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPowerSeries/Restricted.html"}, {"id": "Mathlib.Algebra.Order.Antidiag.Tendsto", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 20.227, "z": 2.867, "size": 0.2302, "title": "Antidiagonal tendsto", "summary": "`tendsto_sup'_antidiagonal_cofinite`: If a function `f : M × M → R` on a Finset `M`, that has the antidiagonal propertry, tends to to a filter `F` under the cofinite filter then so does the function assigning to `x : M` its supremum of its antidiagonal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Antidiag/Tendsto.html"}, {"id": "Mathlib.Algebra.Lie.SerreConstruction", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 59.124, "z": 53.681, "size": 0.2, "title": "Serre construction of Lie algebras from Cartan matrices", "summary": "This file provides the Serre construction of Lie algebras from Cartan matrices. Given a Cartan matrix `A`, we construct a Lie algebra as a quotient of the free Lie algebra on generators `{H_i, E_i, F_i}` by the Serre relations: $$ \\begin{align} [H_i, H_j] &= 0\\\\ [E_i, F_i] &= H_i\\\\ [E_i, F_j] &= 0 \\quad\\text{if $i \\ne j$}\\\\ [H_i, E_j] &= A_{ij}E_j\\\\ [H_i, F_j] &= -A_{ij}F_j\\\\ ad(E_i)^{1 - A_{ij}}(E_j) &= 0…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/SerreConstruction.html"}, {"id": "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 2, "macro_tier_score": 0.0067, "macro_tier_override": null, "x": 101.943, "z": -6.402, "size": 0.3427, "title": "Quadratic characters of finite fields", "summary": "This file defines the quadratic character on a finite field `F` and proves some basic statements about it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.html"}, {"id": "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 2, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 21.347, "z": 34.838, "size": 0.3195, "title": "Quadratic characters on ℤ/nℤ", "summary": "This file defines some quadratic characters on the rings ℤ/4ℤ and ℤ/8ℤ. We set them up to be of type `MulChar (ZMod n) ℤ`, where `n` is `4` or `8`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LegendreSymbol/ZModChar.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.Module", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 69.994, "z": 66.536, "size": 0.2526, "title": "Subsets of prime spectra related to modules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/Module.html"}, {"id": "Mathlib.LinearAlgebra.TensorAlgebra.Grading", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -49.158, "z": 36.599, "size": 0.2, "title": "Results about the grading structure of the tensor algebra", "summary": "The main result is `TensorAlgebra.gradedAlgebra`, which says that the tensor algebra is a ℕ-graded algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorAlgebra/Grading.html"}, {"id": "Mathlib.Algebra.Group.Equiv.Finite", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 8.166, "z": 4.421, "size": 0.2, "title": "Finite types with addition/multiplications", "summary": "This file contains basic results and instances for finite types that have an addition/multiplication operator.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Equiv/Finite.html"}, {"id": "Mathlib.Algebra.Category.Ring.FinitePresentation", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 64.013, "z": 47.746, "size": 0.2302, "title": "Finitely presentable objects in `Under R` with `R : CommRingCat`", "summary": "In this file, we show that finitely presented algebras are finitely presentable in `Under R`, i.e. `Hom_R(S, -)` preserves filtered colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/FinitePresentation.html"}, {"id": "Mathlib.Algebra.MvPolynomial.NoZeroDivisors", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": 29.985, "z": -74.015, "size": 0.2591, "title": "Multivariate polynomials over integral domains", "summary": "This file proves results about multivariate polynomials that hold when the coefficient (semi)ring has no zero divisors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/NoZeroDivisors.html"}, {"id": "Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": 25.528, "z": 4.935, "size": 0.2536, "title": "Locally Finite Linearly Ordered Abelian Groups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/LocallyFiniteOrder.html"}, {"id": "Mathlib.GroupTheory.QuotientGroup.Basic", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 4, "macro_tier_score": 0.3708, "macro_tier_override": null, "x": -32.778, "z": -6.565, "size": 0.4637, "title": "Quotients of groups by normal subgroups", "summary": "This file develops the basic theory of quotients of groups by normal subgroups. In particular, it proves Noether's first and second isomorphism theorems.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/QuotientGroup/Basic.html"}, {"id": "Mathlib.Algebra.CharP.Two", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 3, "macro_tier_score": 0.2385, "macro_tier_override": null, "x": -18.482, "z": -8.703, "size": 0.2548, "title": "Lemmas about rings of characteristic two", "summary": "This file contains results about `CharP R 2`, in the `CharTwo` namespace. The lemmas in this file with a `_sq` suffix are just special cases of the `_pow_char` lemmas elsewhere, with a shorter name for ease of discovery, and no need for a `[Fact (Prime 2)]` argument.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/Two.html"}, {"id": "Mathlib.Algebra.Expr", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": "Helpers to invoke functions involving algebra at tactic time", "summary": "This file provides instances on `x y : Q($α)` such that `x + y = q($x + $y)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Expr.html"}, {"id": "Mathlib.NumberTheory.Chebyshev", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 18.377, "z": 42.637, "size": 0.2, "title": "Chebyshev functions", "summary": "This file defines the Chebyshev functions `theta` and `psi`. These give logarithmically weighted sums of primes and prime powers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Chebyshev.html"}, {"id": "Mathlib.Algebra.Pointwise.Stabilizer", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": 14.239, "z": 21.755, "size": 0.2447, "title": "Stabilizer of a set under a pointwise action", "summary": "This file characterises the stabilizer of a set/finset under the pointwise action of a group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Pointwise/Stabilizer.html"}, {"id": "Mathlib.RingTheory.WittVector.WittPolynomial", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": -79.857, "z": -0.372, "size": 0.2659, "title": "Witt polynomials", "summary": "To endow `WittVector p R` with a ring structure, we need to study the so-called Witt polynomials. Fix a base value `p : ℕ`. The `p`-adic Witt polynomials are an infinite family of polynomials indexed by a natural number `n`, taking values in an arbitrary ring `R`. The variables of these polynomials are represented by natural numbers. The variable set of the `n`th Witt polynomial contains at most `n+1` elements `{0,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/WittVector/WittPolynomial.html"}, {"id": "Mathlib.RingTheory.Ideal.AssociatedPrime.Localization", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -79.091, "z": -32.292, "size": 0.2391, "title": "Associated primes of localized module", "summary": "This file mainly proves the relation between `Ass(S⁻¹M)` and `Ass(M)`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.MvPolynomial", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 67.675, "z": 45.798, "size": 0.2, "title": "Polynomial identities from evaluation at invertible matrices", "summary": "We prove `MvPolynomial.eq_of_eval_eq_on_gl`: two polynomials in `MvPolynomial (m × m) k` over an infinite field `k` are equal if their evaluations agree at every invertible matrix. The proof uses that the set of invertible matrices is Zariski-dense in `Matrix m m k`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/MvPolynomial.html"}, {"id": "Mathlib.Algebra.LinearRecurrence", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -58.386, "z": 42.861, "size": 0.2381, "title": "Linear recurrence", "summary": "Informally, a \"linear recurrence\" is an assertion of the form `∀ n : ℕ, u (n + d) = a 0 * u n + a 1 * u (n+1) + ... + a (d-1) * u (n+d-1)`, where `u` is a sequence, `d` is the *order* of the recurrence and the `a i` are its *coefficients*. In this file, we define the structure `LinearRecurrence` so that `LinearRecurrence.mk d a` represents the above relation, and we call a sequence `u` which verifies it a *solution*…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/LinearRecurrence.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.Ext.MapBijective", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -50.826, "z": 27.042, "size": 0.2, "title": "Bijections Between Ext", "summary": "In this file, we show that the maps between `Ext` induced by a fully faithful exact functor `F : C ⥤ D` are bijective when either 1. `F` preserves projective objects and `C` has enough projectives, or 2. `F` preserves injective objects and `C` has enough injectives.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/Ext/MapBijective.html"}, {"id": "Mathlib.Algebra.Order.Sub.Prod", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 11.543, "z": 5.98, "size": 0.2221, "title": "Products of `OrderedSub` types.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Sub/Prod.html"}, {"id": "Mathlib.Algebra.Homology.BifunctorShift", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -3.006, "z": -27.695, "size": 0.2, "title": "Behavior of the action of a bifunctor on cochain complexes with respect to shifts", "summary": "In this file, given cochain complexes `K₁ : CochainComplex C₁ ℤ`, `K₂ : CochainComplex C₂ ℤ` and a functor `F : C₁ ⥤ C₂ ⥤ D`, we define an isomorphism of cochain complexes in `D`: - `CochainComplex.mapBifunctorShift₁Iso K₁ K₂ F x` of type `mapBifunctor (K₁⟦x⟧) K₂ F ≅ (mapBifunctor K₁ K₂ F)⟦x⟧` for `x : ℤ`. - `CochainComplex.mapBifunctorShift₂Iso K₁ K₂ F y` of type `mapBifunctor K₁ (K₂⟦y⟧) F ≅ (mapBifunctor K₁ K₂…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/BifunctorShift.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.Lex", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 10.616, "z": 19.595, "size": 0.2, "title": "Order homomorphisms for products of linearly ordered groups with zero", "summary": "This file defines order homomorphisms for products of linearly ordered groups with zero, which is identified with the `WithZero` of the lexicographic product of the units of the groups. The product of linearly ordered groups with zero `WithZero (αˣ ×ₗ βˣ)` is a linearly ordered group with zero itself with natural inclusions but only one projection. One has to work with the lexicographic product of the units `αˣ ×ₗ…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/Lex.html"}, {"id": "Mathlib.LinearAlgebra.LeftExact", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 36.608, "z": 39.503, "size": 0.2, "title": "The Left Exactness of Hom", "summary": "If `M1 → M2 → M3 → 0` is an exact sequence of `R`-modules and `N` is an `R`-module, then `0 → (M3 →ₗ[R] N) → (M2 →ₗ[R] N) → (M1 →ₗ[R] N)` is exact. In this file, we show the exactness at `M2 →ₗ[R] N` (`exact_lcomp_of_exact_of_surjective`); the injectivity part is `LinearMap.lcomp_injective_of_surjective` in the file `Mathlib.LinearAlgebra.BilinearMap`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/LeftExact.html"}, {"id": "Mathlib.RingTheory.Adjoin.Field", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 3, "macro_tier_score": 0.0947, "macro_tier_override": null, "x": 78.616, "z": -56.087, "size": 0.2958, "title": "Adjoining elements to a field", "summary": "Some lemmas on the ring generated by adjoining an element to a field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Adjoin/Field.html"}, {"id": "Mathlib.GroupTheory.SpecificGroups.Alternating", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -46.177, "z": 14.115, "size": 0.2892, "title": "Alternating Groups", "summary": "The alternating group on a finite type `α` is the subgroup of the permutation group `Perm α` consisting of the even permutations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/SpecificGroups/Alternating.html"}, {"id": "Mathlib.RingTheory.ChainOfDivisors", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.0492, "macro_tier_override": null, "x": 9.674, "z": -33.934, "size": 0.2818, "title": "Chains of divisors", "summary": "The results in this file show that in the monoid `Associates M` of a `UniqueFactorizationMonoid` `M`, an element `a` is an n-th prime power iff its set of divisors is a strictly increasing chain of length `n + 1`, meaning that we can find a strictly increasing bijection between `Fin (n + 1)` and the set of factors of `a`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/ChainOfDivisors.html"}, {"id": "Mathlib.RingTheory.SimpleRing.Congr", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 48.526, "z": 23.364, "size": 0.2403, "title": "Simplicity is preserved by ring isomorphisms/surjective ring homomorphisms", "summary": "If `R` is a simple (non-assoc) ring and there exists surjective `f : R →+* S` where `S` is nontrivial, then `S` is also simple. If `R` is a simple (non-unital non-assoc) ring then any ring isomorphic to `R` is also simple.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SimpleRing/Congr.html"}, {"id": "Mathlib.Algebra.Lie.DirectSum", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 41.564, "z": -63.799, "size": 0.2, "title": "Direct sums of Lie algebras and Lie modules", "summary": "Direct sums of Lie algebras and Lie modules carry natural algebra and module structures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/DirectSum.html"}, {"id": "Mathlib.GroupTheory.IndexNormal", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2531, "title": "Subgroups of small index are normal", "summary": "* `Subgroup.normal_of_index_eq_smallest_prime_factor`: in a finite group `G`, a subgroup of index equal to the smallest prime factor of `Nat.card G` is normal. * `Subgroup.normal_of_index_two`: in a group `G`, a subgroup of index 2 is normal (This does not require `G` to be finite.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/IndexNormal.html"}, {"id": "Mathlib.RingTheory.Polynomial.Selmer", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 81.615, "z": 64.461, "size": 0.2, "title": "Irreducibility of Selmer Polynomials", "summary": "This file proves irreducibility of the Selmer polynomials `X ^ n - X - 1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Selmer.html"}, {"id": "Mathlib.RingTheory.Polynomial.GaussLemma", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 3, "macro_tier_score": 0.0464, "macro_tier_override": null, "x": -38.604, "z": -94.568, "size": 0.3315, "title": "Gauss's Lemma", "summary": "Gauss's Lemma is one of a few results pertaining to irreducibility of primitive polynomials.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/GaussLemma.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.AbsoluteValue", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": -54.861, "z": 31.264, "size": 0.2361, "title": "Absolute values and matrices", "summary": "This file proves some bounds on matrices involving absolute values.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/AbsoluteValue.html"}, {"id": "Mathlib.Algebra.Ring.MinimalAxioms", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 0.932, "z": 9.239, "size": 0.2562, "title": "Minimal Axioms for a Ring", "summary": "This file defines constructors to define a `Ring` or `CommRing` structure on a Type, while proving a minimum number of equalities.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/MinimalAxioms.html"}, {"id": "Mathlib.GroupTheory.GroupExtension.Basic", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -26.642, "z": -33.388, "size": 0.2, "title": "Basic lemmas about group extensions", "summary": "This file gives basic lemmas about group extensions. For the main definitions, see `Mathlib/GroupTheory/GroupExtension/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupExtension/Basic.html"}, {"id": "Mathlib.GroupTheory.GroupExtension.Defs", "region_id": "algebra", "micro_elevation": 0.2895, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 40.84, "z": -1.201, "size": 0.2478, "title": "Group Extensions", "summary": "This file defines extensions of multiplicative and additive groups and their associated structures such as splittings and equivalences.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupExtension/Defs.html"}, {"id": "Mathlib.Algebra.Category.BoolRing", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -22.235, "z": 1.511, "size": 0.2, "title": "The category of Boolean rings", "summary": "This file defines `BoolRing`, the category of Boolean rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/BoolRing.html"}, {"id": "Mathlib.Algebra.Ring.BooleanRing", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -8.094, "z": 14.624, "size": 0.239, "title": "Boolean rings", "summary": "A Boolean ring is a ring where multiplication is idempotent. They are equivalent to Boolean algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/BooleanRing.html"}, {"id": "Mathlib.RingTheory.RamificationInertia.Inertia", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 2, "macro_tier_score": 0.0104, "macro_tier_override": null, "x": 28.373, "z": -109.676, "size": 0.282, "title": "Inertia degree", "summary": "Given a prime ideal `q` of an `R`-algebra `S`, the inertia degree of `q` over `R` is defined to be the degree of the residue field of `q` over the residue field of its preimage `p` in `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RamificationInertia/Inertia.html"}, {"id": "Mathlib.NumberTheory.KummerDedekind", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 9.251, "z": -107.317, "size": 0.2434, "title": "Kummer-Dedekind theorem", "summary": "This file proves the Kummer-Dedekind theorem on the splitting of prime ideals in an extension of the ring of integers. This states the following: assume we are given - A prime ideal `I` of Dedekind domain `R` - An `R`-algebra `S` that is a Dedekind Domain - An `α : S` that is integral over `R` with minimal polynomial `f` If the conductor `𝓒` of `x` is such that `𝓒 ∩ R` is coprime to `I` then the prime factorisations…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/KummerDedekind.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.LinearDisjoint", "region_id": "algebra", "micro_elevation": 0.9342, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -127.481, "z": 33.696, "size": 0.2531, "title": "Disjoint extensions with coprime different ideals", "summary": "Let `A ⊆ B` be a finite extension of Dedekind domains and assume that `A ⊆ R₁, R₂ ⊆ B` are two subrings such that `Frac R₁ ⊔ Frac R₂ = Frac B`, `Frac R₁` and `Frac R₂` are linearly disjoint over `Frac A`, and that `𝓓(R₁/A)` and `𝓓(R₂/A)` are coprime where `𝓓` denotes the different ideal and `Frac R` denotes the fraction field of a domain `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/LinearDisjoint.html"}, {"id": "Mathlib.Algebra.Polynomial.Module.TensorProduct", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -47.855, "z": 46.689, "size": 0.2, "title": "PolynomialModule is isomorphic to a tensor product", "summary": "For a commutative ring `R` and an `R`-module `M`, we obtain an isomorphism between `R[X] ⊗[R] M` and `PolynomialModule R M` as `R[X]`-modules; this isomorphism is called `polynomialTensorProductLEquivPolynomialModule`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Module/TensorProduct.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.Generator", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -65.757, "z": 12.083, "size": 0.2399, "title": "Generators for the category of presheaves of modules", "summary": "In this file, given a presheaf of rings `R` on a category `C`, we study the set `freeYoneda R` of presheaves of modules of form `(free R).obj (yoneda.obj X)` for `X : C`, i.e. free presheaves of modules generated by the Yoneda presheaf represented by some `X : C` (the functor represented by such a presheaf of modules is the evaluation functor `M ↦ M.obj (op X)`, see `freeYonedaEquiv`). Lemmas…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/Generator.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.Order", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -21.915, "z": -51.224, "size": 0.2, "title": "Order instances on subalgebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/Order.html"}, {"id": "Mathlib.NumberTheory.NumberField.InfinitePlace.Completion", "region_id": "algebra", "micro_elevation": 0.9342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 124.012, "z": -44.808, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/InfinitePlace/Completion.html"}, {"id": "Mathlib.LinearAlgebra.Finsupp.Defs", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 4, "macro_tier_score": 0.3687, "macro_tier_override": null, "x": -12.133, "z": -29.147, "size": 0.3067, "title": "Properties of the module `α →₀ M`", "summary": "Given an `R`-module `M`, the `R`-module structure on `α →₀ M` is defined in `Mathlib/Data/Finsupp/SMul.lean`. In this file we define `LinearMap` versions of various maps: * `Finsupp.lsingle a : M →ₗ[R] ι →₀ M`: `Finsupp.single a` as a linear map; * `Finsupp.lapply a : (ι →₀ M) →ₗ[R] M`: the map `fun f ↦ f a` as a linear map; * `Finsupp.lsubtypeDomain (s : Set α) : (α →₀ M) →ₗ[R] (s →₀ M)`: restriction to a subtype…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Finsupp/Defs.html"}, {"id": "Mathlib.Algebra.Order.Chebyshev", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 41.644, "z": 9.505, "size": 0.2416, "title": "Chebyshev's sum inequality", "summary": "This file proves the Chebyshev sum inequality. Chebyshev's inequality states `(∑ i ∈ s, f i) * (∑ i ∈ s, g i) ≤ #s * ∑ i ∈ s, f i * g i` when `f g : ι → α` monovary, and the reverse inequality when `f` and `g` antivary.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Chebyshev.html"}, {"id": "Mathlib.Algebra.Order.Rearrangement", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 22.098, "z": 2.889, "size": 0.2388, "title": "Rearrangement inequality", "summary": "This file proves the rearrangement inequality and deduces the conditions for equality and strict inequality. The rearrangement inequality tells you that for two functions `f g : ι → α`, the sum `∑ i, f i * g (σ i)` is maximized over all `σ : Perm ι` when `g ∘ σ` monovaries with `f` and minimized when `g ∘ σ` antivaries with `f`. The inequality also tells you that `∑ i, f i * g (σ i) = ∑ i, f i * g i` if and only if…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Rearrangement.html"}, {"id": "Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -13.016, "z": 36.764, "size": 0.2478, "title": "Corollaries From Approximation Lemmas (`Algebra.ContinuedFractions.Computation.Approximations`)", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/ContinuedFractions/Computation/ApproximationCorollaries.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Embedding", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 2, "macro_tier_score": 0.0078, "macro_tier_override": null, "x": 12.727, "z": -15.98, "size": 0.2356, "title": "Group actions on embeddings", "summary": "This file provides a `MulAction G (α ↪ β)` instance that agrees with the `MulAction G (α → β)` instances defined by `Pi.mulAction`. Note that unlike the `Pi` instance, this requires `G` to be a group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Embedding.html"}, {"id": "Mathlib.NumberTheory.Padics.ValuativeRel", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.062, "z": 15.757, "size": 0.2, "title": "p-adic numbers with a valuative relation", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/ValuativeRel.html"}, {"id": "Mathlib.Algebra.Homology.Factorizations.CM5a", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -35.031, "z": -35.877, "size": 0.2301, "title": "Factorization lemma", "summary": "In this file, we show that if `f : K ⟶ L` is a morphism between bounded below cochain complexes in an abelian category with enough injectives, there exists a factorization `ι ≫ π = f` with `ι : K ⟶ K'` a monomorphism that is also a quasimorphism and `π : K' ⟶ L` a morphism which degreewise is an epimorphism with an injective kernel, while `K'` is also bounded below (with precise bounds depending on the available…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Factorizations/CM5a.html"}, {"id": "Mathlib.RingTheory.Polynomial.UniqueFactorization", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 3, "macro_tier_score": 0.1844, "macro_tier_override": null, "x": -68.461, "z": -37.38, "size": 0.2796, "title": "Unique factorization for univariate and multivariate polynomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/UniqueFactorization.html"}, {"id": "Mathlib.Algebra.Category.Grp.Injective", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 2, "macro_tier_score": 0.0302, "macro_tier_override": null, "x": 56.806, "z": -27.572, "size": 0.2702, "title": "Injective objects in the category of abelian groups", "summary": "In this file we prove that divisible groups are injective objects in category of (additive) abelian groups. The proof that the category of abelian groups has enough injective objects can be found in `Mathlib/Algebra/Category/Grp/EnoughInjectives.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Injective.html"}, {"id": "Mathlib.Algebra.Ring.Associated", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.302, "macro_tier_override": null, "x": 20.763, "z": 8.096, "size": 0.3683, "title": "Associated elements in rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Associated.html"}, {"id": "Mathlib.Algebra.Lie.CartanExists", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 34.612, "z": 103.962, "size": 0.2, "title": "Existence of Cartan subalgebras", "summary": "In this file we prove existence of Cartan subalgebras in finite-dimensional Lie algebras, following [barnes1967].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/CartanExists.html"}, {"id": "Mathlib.Algebra.Lie.Rank", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 107.71, "z": -1.042, "size": 0.2478, "title": "Rank of a Lie algebra and regular elements", "summary": "Let `L` be a Lie algebra over a nontrivial commutative ring `R`, and assume that `L` is finite free as `R`-module. Then the coefficients of the characteristic polynomial of `ad R L x` are polynomial in `x`. The *rank* of `L` is the smallest `n` for which the `n`-th coefficient is not the zero polynomial. Continuing to write `n` for the rank of `L`, an element `x` of `L` is *regular* if the `n`-th coefficient of the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Rank.html"}, {"id": "Mathlib.Algebra.Lie.Loop", "region_id": "algebra", "micro_elevation": 0.6447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -41.103, "z": 81.189, "size": 0.2, "title": "Loop Lie algebras and their central extensions", "summary": "Given a Lie algebra `L`, the loop algebra is the Lie algebra of maps from a circle into `L`. This can mean many different things, e.g., continuous maps, smooth maps, polynomial maps. In this file, we consider the simplest case of polynomial maps, meaning we take a base change with the ring of Laurent polynomials. Loop Lie algebras admit central extensions attached to invariant inner products on the base Lie algebra.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Loop.html"}, {"id": "Mathlib.Algebra.Lie.Cochain", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 56.319, "z": -56.617, "size": 0.2836, "title": "Lie algebra cohomology in low degree", "summary": "This file defines low degree cochains of Lie algebras with coefficients given by a module. They are useful in the construction of central extensions, so we treat these easier cases separately from the general theory of Lie algebra cohomology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Cochain.html"}, {"id": "Mathlib.FieldTheory.Finite.Polynomial", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 101.089, "z": 14.644, "size": 0.2659, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Finite/Polynomial.html"}, {"id": "Mathlib.LinearAlgebra.Goursat", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 39.188, "z": 34.181, "size": 0.2, "title": "Goursat's lemma for submodules", "summary": "Let `M, N` be modules over a ring `R`. If `L` is a submodule of `M × N` which projects fully onto both factors, then there exist submodules `M' ≤ M` and `N' ≤ N` such that `M' × N' ≤ L` and the image of `L` in `(M ⧸ M') × (N ⧸ N')` is the graph of an isomorphism `M ⧸ M' ≃ₗ[R] N ⧸ N'`. Equivalently, `L` is equal to the preimage in `M × N` of the graph of this isomorphism `M ⧸ M' ≃ₗ[R] N ⧸ N'`. `M'` and `N'` can be…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Goursat.html"}, {"id": "Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -57.534, "z": 2.083, "size": 0.2541, "title": "Maximal subgroups of the alternating group", "summary": "* `alternatingGroup.isCoatom_stabilizer`: if neither `s : Set α` nor its complement is empty, and if, moreover, `Nat.card α ≠ 2 * s.ncard`, then `stabilizer (alternatingGroup α) s` is a maximal subgroup of `alternatingGroup α`. This is the “intransitive case” of the O'Nan-Scott classification of maximal subgroups of the alternating groups. Compare with `Equiv.Perm.isCoatom_stabilizer` for the case of the permutation…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/SpecificGroups/Alternating/MaximalSubgroups.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 3, "macro_tier_score": 0.1867, "macro_tier_override": null, "x": 22.137, "z": -19.822, "size": 0.2903, "title": "Set of factors", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.html"}, {"id": "Mathlib.Algebra.Module.Presentation.Free", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -0.413, "z": 55.713, "size": 0.2, "title": "Presentation of free modules", "summary": "A module is free iff it admits a presentation with generators but no relation, see `Module.free_iff_exists_presentation`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Presentation/Free.html"}, {"id": "Mathlib.RingTheory.Smooth.NoetherianDescent", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": 10.768, "z": 110.908, "size": 0.2538, "title": "Smooth algebras have Noetherian models", "summary": "In this file, we show if `S` is a smooth `R`-algebra, there exists a `ℤ`-subalgebra of finite type `R₀` and a smooth `R₀`-algebra `S₀` such that `S ≃ₐ R ⊗[R₀] S₀`. The analogous result for etale algebras is also provided.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/NoetherianDescent.html"}, {"id": "Mathlib.Algebra.Tropical.Lattice", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -6.373, "z": -19.409, "size": 0.2, "title": "Order on tropical algebraic structure", "summary": "This file defines the orders induced on tropical algebraic structures by the underlying type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Tropical/Lattice.html"}, {"id": "Mathlib.Algebra.Ring.Aut", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 4, "macro_tier_score": 0.3815, "macro_tier_override": null, "x": -15.223, "z": 10.638, "size": 0.449, "title": "Ring automorphisms", "summary": "This file defines the automorphism group structure on `RingAut R := RingEquiv R R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Aut.html"}, {"id": "Mathlib.GroupTheory.QuotientGroup.Finite", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 4, "macro_tier_score": 0.2923, "macro_tier_override": null, "x": 33.26, "z": -11.784, "size": 0.3161, "title": "Deducing finiteness of a group.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/QuotientGroup/Finite.html"}, {"id": "Mathlib.RingTheory.Congruence.Basic", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 4, "macro_tier_score": 0.3343, "macro_tier_override": null, "x": -22.279, "z": -36.444, "size": 0.3729, "title": "Congruence relations on rings", "summary": "This file contains basic results concerning congruence relations on rings, which extend `Con` and `AddCon` on monoids and additive monoids. Most of the time you likely want to use the `Ideal.Quotient` API that is built on top of this.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Congruence/Basic.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.SingleFunctors", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 2, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 6.065, "z": -19.508, "size": 0.3297, "title": "Single functors from the homotopy category", "summary": "Let `C` be a preadditive category with a zero object. In this file, we put together all the single functors `C ⥤ CochainComplex C ℤ` along with their compatibilities with shifts into the definition `CochainComplex.singleFunctors C : SingleFunctors C (CochainComplex C ℤ) ℤ`. Similarly, we define `HomotopyCategory.singleFunctors C : SingleFunctors C (HomotopyCategory C (ComplexShape.up ℤ)) ℤ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/SingleFunctors.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 2, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -16.513, "z": -12.027, "size": 0.3189, "title": "Compatibilities of the homology functor with the shift", "summary": "This file studies how homology of cochain complexes behaves with respect to the shift: there is a natural isomorphism `(K⟦n⟧).homology a ≅ K.homology a` when `n + a = a'`. This is summarized by instances `(homologyFunctor C (ComplexShape.up ℤ) 0).ShiftSequence ℤ` in the `CochainComplex` and `HomotopyCategory` namespaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/ShiftSequence.html"}, {"id": "Mathlib.Algebra.SkewMonoidAlgebra.Lift", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -33.911, "z": -41.841, "size": 0.2, "title": "Lemmas about different kinds of \"lifts\" to `SkewMonoidAlgebra`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/SkewMonoidAlgebra/Lift.html"}, {"id": "Mathlib.Algebra.Lie.AdjointAction.Derivation", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 79.481, "z": 7.75, "size": 0.2533, "title": "Adjoint action of a Lie algebra on itself", "summary": "This file defines the *adjoint action* of a Lie algebra on itself, and establishes basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/AdjointAction/Derivation.html"}, {"id": "Mathlib.RingTheory.LocalProperties.Reduced", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 2, "macro_tier_score": 0.0062, "macro_tier_override": null, "x": 76.169, "z": -34.389, "size": 0.308, "title": "`IsReduced` is a local property", "summary": "In this file, we prove that `IsReduced` is a local property.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalProperties/Reduced.html"}, {"id": "Mathlib.Algebra.Central.End", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -20.957, "z": 57.592, "size": 0.239, "title": "`Module.End R M` is a central algebra", "summary": "This file shows that the algebra of endomorphisms on a free module is central.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Central/End.html"}, {"id": "Mathlib.Algebra.Category.FGModuleCat.EssentiallySmall", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -2.815, "z": 87.241, "size": 0.2, "title": "The category of finitely generated modules over a ring is essentially small", "summary": "This file proves that `FGModuleCat R`, the category of finitely generated modules over a ring `R`, is essentially small, by providing an explicit small model. However, for applications, it is recommended to use the standard `CategoryTheory.SmallModel (FGModuleCat R)` instead.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/FGModuleCat/EssentiallySmall.html"}, {"id": "Mathlib.Algebra.Category.FGModuleCat.Basic", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 2, "macro_tier_score": 0.0082, "macro_tier_override": null, "x": 85.365, "z": 3.323, "size": 0.2869, "title": "The category of finitely generated modules over a ring", "summary": "This introduces `FGModuleCat R`, the category of finitely generated modules over a ring `R`. It is implemented as a full subcategory on a subtype of `ModuleCat R`. When `K` is a field, `FGModuleCat K` is the category of finite-dimensional vector spaces over `K`. We first create the instance as a preadditive category. When `R` is commutative we then give the structure as an `R`-linear monoidal category. When `R` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/FGModuleCat/Basic.html"}, {"id": "Mathlib.RingTheory.Valuation.Quotient", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -7.425, "z": -62.705, "size": 0.2, "title": "The valuation on a quotient ring", "summary": "The support of a valuation `v : Valuation R Γ₀` is `supp v`. If `J` is an ideal of `R` with `h : J ⊆ supp v` then the induced valuation on `R / J` = `Ideal.Quotient J` is `onQuot v h`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/Quotient.html"}, {"id": "Mathlib.Algebra.Free", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 1, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -3.834, "z": 8.458, "size": 0.2617, "title": "Free constructions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Free.html"}, {"id": "Mathlib.Algebra.Ring.Subring.IntPolynomial", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -57.975, "z": -25.021, "size": 0.2, "title": "Polynomials over subrings.", "summary": "Given a field `K` with a subring `R`, in this file we construct a map from polynomials in `K[X]` with coefficients in `R` to `R[X]`. We provide several lemmas to deal with coefficients, degree, and evaluation of `Polynomial.int`. This is useful when dealing with integral elements in an extension of fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Subring/IntPolynomial.html"}, {"id": "Mathlib.RingTheory.SimpleRing.Matrix", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 3, "macro_tier_score": 0.1375, "macro_tier_override": null, "x": 27.859, "z": 60.777, "size": 0.3427, "title": null, "summary": "The matrix ring over a simple ring is simple", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SimpleRing/Matrix.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Ideal", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.1387, "macro_tier_override": null, "x": -6.605, "z": -64.664, "size": 0.3475, "title": "Ideals in a matrix ring", "summary": "This file defines left (resp. two-sided) ideals in a matrix semiring (resp. ring) over left (resp. two-sided) ideals in the base semiring (resp. ring). We also characterize Jacobson radicals of ideals in such rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Ideal.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.Complex", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 69.139, "z": 82.598, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/Complex.html"}, {"id": "Mathlib.FieldTheory.Laurent", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -19.007, "z": 104.138, "size": 0.2, "title": "Laurent expansions of rational functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Laurent.html"}, {"id": "Mathlib.LinearAlgebra.Projectivization.Collinear", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -64.906, "z": -46.524, "size": 0.2, "title": "Collinearity in Projective Space", "summary": "This file defines collinearity of points in projective space and proves the uniqueness of the line through two distinct points.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Projectivization/Collinear.html"}, {"id": "Mathlib.FieldTheory.SeparablyGenerated", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 103.301, "z": 46.506, "size": 0.2568, "title": "Separably generated extensions", "summary": "We aim to formalize the following result: Let `K/k` be a finitely generated field extension with characteristic `p > 0`, then TFAE 1. `K/k` is separably generated 2. If `{ sᵢ } ⊆ K` is an arbitrary `k`-linearly independent set, `{ sᵢᵖ } ⊆ K` is also `k`-linearly independent 3. `K ⊗ₖ k^{1/p}` is reduced 4. `K` is geometrically reduced over `k`. 5. `k` and `Kᵖ` are linearly disjoint over `kᵖ` in `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/SeparablyGenerated.html"}, {"id": "Mathlib.Algebra.CharP.IntermediateField", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 3, "macro_tier_score": 0.0452, "macro_tier_override": null, "x": -48.743, "z": -43.002, "size": 0.3255, "title": "Characteristic of intermediate fields", "summary": "This file contains some convenient instances for determining the characteristic of intermediate fields. Some char zero instances are not provided, since they are already covered by `SubsemiringClass.instCharZero`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharP/IntermediateField.html"}, {"id": "Mathlib.NumberTheory.NumberField.FractionalIdeal", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 86.639, "z": 91.881, "size": 0.264, "title": "Fractional ideals of number fields", "summary": "Prove some results on the fractional ideals of number fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/FractionalIdeal.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Monoidal.Adjunction", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 5.29, "z": -61.058, "size": 0.2, "title": "The monoidal adjunction between the extension and the restriction of scalars", "summary": "Let `f : R →+* S` be a morphism of commutative rings. We show that the functor `extendsScalars f : ModuleCat R ⥤ ModuleCat S` is monoidal, and deduce that `restrictScalars f : ModuleCat S ⥤ ModuleCat R` is lax monoidal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Monoidal/Adjunction.html"}, {"id": "Mathlib.RingTheory.RingHom.PurelyInseparable", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -24.159, "z": -116.377, "size": 0.274, "title": "Purely inseparable ring homomorphisms", "summary": "In this file we define purely inseparable ring homomorphisms and show their meta properties. Since purely inseparable is mainly used for fields, we cannot prove many general ring hom properties. E.g. we can't prove `StableUnderComposition IsPurelyInseparable`, since `IsPurelyInseparable.trans` requires the involved rings to be fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/PurelyInseparable.html"}, {"id": "Mathlib.Algebra.Regular.Opposite", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 4, "macro_tier_score": 0.4361, "macro_tier_override": null, "x": 6.852, "z": 2.869, "size": 0.3649, "title": "Results about `IsRegular` and `MulOpposite`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Regular/Opposite.html"}, {"id": "Mathlib.RingTheory.GradedAlgebra.Noetherian", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.795, "z": 63.139, "size": 0.2, "title": "The properties of a graded Noetherian ring.", "summary": "This file proves that the 0-th grade of a Noetherian ring is also a Noetherian ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/GradedAlgebra/Noetherian.html"}, {"id": "Mathlib.RingTheory.HahnSeries.HahnEmbedding", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 57.683, "z": 25.687, "size": 0.2, "title": "Hahn embedding theorem", "summary": "In this file, we prove the Hahn embedding theorem: every linearly ordered abelian group can be embedded as an ordered subgroup of `Lex ℝ⟦Ω⟧`, where `Ω` is the type of finite Archimedean classes of the group. The theorem is stated as `hahnEmbedding_isOrderedAddMonoid`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/HahnSeries/HahnEmbedding.html"}, {"id": "Mathlib.Algebra.Module.LinearMap.Rat", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 1, "macro_tier_score": 0.0045, "macro_tier_override": null, "x": 9.83, "z": 24.07, "size": 0.3438, "title": "Reinterpret an additive homomorphism as a `ℚ`-linear map.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LinearMap/Rat.html"}, {"id": "Mathlib.GroupTheory.DivisibleHull", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 33.265, "z": 46.989, "size": 0.2302, "title": "Divisible Hull of an abelian group", "summary": "This file constructs the divisible hull of an `AddCommMonoid` as a `ℕ`-module localized at `ℕ+` (implemented using `nonZeroDivisors ℕ`), which is a `ℚ≥0`-module. Furthermore, we show that * when `M` is a group, so is `DivisibleHull M`, which is also a `ℚ`-module * when `M` is linearly ordered and cancellative, so is `DivisibleHull M`, which is also an ordered `ℚ≥0`-module. * when `M` is a linearly ordered group,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/DivisibleHull.html"}, {"id": "Mathlib.FieldTheory.NormalizedTrace", "region_id": "algebra", "micro_elevation": 0.8553, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -91.48, "z": 78.763, "size": 0.2, "title": "Normalized trace", "summary": "This file defines the *normalized trace* map; that is, an `F`-linear map from the algebraic closure of `F` to `F` defined as the trace of an element from its adjoin extension divided by its degree. To avoid heavy imports, we define it here as a map from an arbitrary algebraic (equivalently integral) extension of `F`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/NormalizedTrace.html"}, {"id": "Mathlib.Algebra.Module.Bimodule", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 18.244, "z": -60.451, "size": 0.2, "title": "Bimodules", "summary": "One frequently encounters situations in which several sets of scalars act on a single space, subject to compatibility condition(s). A distinguished instance of this is the theory of bimodules: one has two rings `R`, `S` acting on an additive group `M`, with `R` acting covariantly (\"on the left\") and `S` acting contravariantly (\"on the right\"). The compatibility condition is just: `(r • m) • s = r • (m • s)` for all…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Bimodule.html"}, {"id": "Mathlib.Algebra.Homology.HomologicalBicomplex", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -8.841, "z": 2.841, "size": 0.2723, "title": "Bicomplexes", "summary": "Given a category `C` with zero morphisms and two complex shapes `c₁ : ComplexShape I₁` and `c₂ : ComplexShape I₂`, we define the type of bicomplexes `HomologicalComplex₂ C c₁ c₂` as an abbreviation for `HomologicalComplex (HomologicalComplex C c₂) c₁`. In particular, if `K : HomologicalComplex₂ C c₁ c₂`, then for each `i₁ : I₁`, `K.X i₁` is a column of `K`. In this file, we obtain the equivalence of categories…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomologicalBicomplex.html"}, {"id": "Mathlib.Algebra.Exact", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 9.459, "z": -53.021, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Exact.html"}, {"id": "Mathlib.NumberTheory.NumberField.Ideal.KummerDedekind", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 107.403, "z": 69.896, "size": 0.2361, "title": "Kummer-Dedekind criterion for the splitting of prime numbers", "summary": "In this file, we give a specialized version of the Kummer-Dedekind criterion for the case of the splitting of rational primes in number fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Ideal/KummerDedekind.html"}, {"id": "Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -88.272, "z": 79.595, "size": 0.2361, "title": "Factorization of cyclotomic polynomials over finite fields", "summary": "We compute the degree of the irreducible factors of the `n`-th cyclotomic polynomial over a finite field of characteristic `p`, where `p` and `n` are coprime.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Cyclotomic/Factorization.html"}, {"id": "Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 78.045, "z": -16.919, "size": 0.2361, "title": "Cyclotomic units.", "summary": "We gather miscellaneous results about units given by sums of powers of roots of unit, the so-called *cyclotomic units*.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RootsOfUnity/CyclotomicUnits.html"}, {"id": "Mathlib.RingTheory.Smooth.Quotient", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 62.897, "z": -89.722, "size": 0.2, "title": "Some lemmas about formally smooth under quotient", "summary": "In this file, we formalize the result [Stacks 031L] : For flat ring homomorphism `f : R →+* S`, `I` an ideal of `R` which is square zero, if `R ⧸ I →+* S ⧸ IS` is formally smooth, so is `f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/Quotient.html"}, {"id": "Mathlib.Algebra.Module.Presentation.RestrictScalars", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 6.423, "z": 57.213, "size": 0.2, "title": "Presentation of the restriction of scalars of a module", "summary": "Given a morphism of rings `A → B` and a `B`-module `M`, we obtain a presentation of `M` as a `A`-module from a presentation of `M` as `B`-module, a presentation of `B` as a `A`-module (and some additional data).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Presentation/RestrictScalars.html"}, {"id": "Mathlib.Algebra.Module.Presentation.DirectSum", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -53.858, "z": -14.263, "size": 0.2478, "title": "Presentation of a direct sum", "summary": "If `M : ι → Type _` is a family of `A`-modules, then the data of a presentation of each `M i`, we obtain a presentation of the module `⨁ i, M i`. In particular, from a presentation of an `A`-module `M`, we get a presentation of `ι →₀ M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Presentation/DirectSum.html"}, {"id": "Mathlib.Algebra.Order.Ring.Units", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -24.074, "z": -28.285, "size": 0.2, "title": "Lemmas about units of ordered rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Ring/Units.html"}, {"id": "Mathlib.GroupTheory.QuotientGroup.ModEq", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 21.111, "z": 15.178, "size": 0.2324, "title": "Congruence modulo multiples and congruence modulo `AddSubgroup.zmultiples _`", "summary": "In this file we show that in an additive commutative group, the congruence relation `a ≡ b [PMOD p]` is equivalent to the coercions of `a` and `b` to `G ⧸ AddSubgroup.zmultiples p` being equal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/QuotientGroup/ModEq.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 3.307, "z": 81.648, "size": 0.2, "title": "Irreducibility of linear and quadratic polynomials", "summary": "* `MvPolynomial.irreducible_of_totalDegree_eq_one`: a multivariate polynomial of `totalDegree` one is irreducible if its coefficients are relatively prime. * For `c : n →₀ R`, `MvPolynomial.sumSMulX c` is the linear polynomial $\\sum_i c_i X_i$ of $R[X_1\\dots,X_n]$. * `MvPolynomial.irreducible_sumSMulX` : if the support of `c` is nontrivial, if `R` is a domain, and if the only common divisors to all `c i` are units,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/IrreducibleQuadratic.html"}, {"id": "Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": 70.304, "z": 17.417, "size": 0.2476, "title": "Finitely generated module over Noetherian ring have finitely many associated primes.", "summary": "In this file we proved that any finitely generated module over a Noetherian ring have finitely many associated primes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/AssociatedPrime/Finiteness.html"}, {"id": "Mathlib.FieldTheory.Galois.Profinite", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 5, "macro_tier_score": 0.0, "macro_tier_override": 5, "x": 5.091, "z": -116.89, "size": 0.2, "title": "Galois Group as a profinite group", "summary": "In this file, we prove that given a field extension `K/k`, there is a continuous isomorphism between `Gal(K/k)` and the limit of `Gal(L/k)`, where `L` is a finite Galois intermediate field ordered by inverse inclusion, thus making `Gal(K/k)` profinite as a limit of finite groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Galois/Profinite.html"}, {"id": "Mathlib.NumberTheory.LSeries.PrimesInAP", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -74.77, "z": -85.108, "size": 0.2, "title": "Dirichlet's Theorem on primes in arithmetic progression", "summary": "The goal of this file is to prove **Dirichlet's Theorem**: If `q` is a positive natural number and `a : ZMod q` is invertible, then there are infinitely many prime numbers `p` such that `(p : ZMod q) = a`. The main steps of the proof are as follows. 1. Define `ArithmeticFunction.vonMangoldt.residueClass a` for `a : ZMod q`, which is a function `ℕ → ℝ` taking the value zero when `(n : ZMod q) ≠ a` and `Λ n` else…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/PrimesInAP.html"}, {"id": "Mathlib.RingTheory.DualNumber", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 38.933, "z": 54.353, "size": 0.2, "title": "Algebraic properties of dual numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DualNumber.html"}, {"id": "Mathlib.NumberTheory.Bernoulli", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 2, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 80.868, "z": 65.396, "size": 0.2763, "title": "Bernoulli numbers", "summary": "The Bernoulli numbers are a sequence of rational numbers that frequently show up in number theory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Bernoulli.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Orthogonal", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.245, "title": "Orthogonal", "summary": "This file contains definitions and properties concerning orthogonality of rows and columns.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Orthogonal.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.Pi", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 48.898, "z": 30.39, "size": 0.2, "title": "Products of subalgebras", "summary": "In this file we define the product of subalgebras as a subalgebra of the product algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/Pi.html"}, {"id": "Mathlib.Algebra.Homology.Functor", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 8.036, "z": -16.743, "size": 0.2, "title": "Complexes in functor categories", "summary": "We can view a complex valued in a functor category `T ⥤ V` as a functor from `T` to complexes valued in `V`. When `V` is abelian, a morphism of short complexes or homological complexes in the category `T ⥤ V` is a quasi-isomorphism iff it is so after evaluation at any `t : T`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Functor.html"}, {"id": "Mathlib.RepresentationTheory.Character", "region_id": "algebra", "micro_elevation": 0.7763, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 76.786, "z": 78.167, "size": 0.2302, "title": "Characters of representations", "summary": "This file introduces characters of representation and proves basic lemmas about how characters behave under various operations on representations. A key result is the orthogonality of characters for irreducible representations of finite group over an algebraically closed field whose characteristic doesn't divide the order of the group. It is the theorem `char_orthonormal`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Character.html"}, {"id": "Mathlib.RepresentationTheory.Irreducible", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 8.033, "z": -107.415, "size": 0.2422, "title": "Irreducible representations", "summary": "This file defines irreducible monoid representations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Irreducible.html"}, {"id": "Mathlib.Algebra.Lie.Semisimple.Defs", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 1, "macro_tier_score": 0.0027, "macro_tier_override": null, "x": 67.064, "z": 55.869, "size": 0.2918, "title": "Semisimple Lie algebras", "summary": "In this file we define simple and semisimple Lie algebras, together with related concepts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Semisimple/Defs.html"}, {"id": "Mathlib.Algebra.Category.AlgCat.Limits", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -24.237, "z": 56.29, "size": 0.2, "title": "The category of R-algebras has all limits", "summary": "Further, these limits are preserved by the forgetful functor --- that is, the underlying types are just the limits in the category of types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/AlgCat/Limits.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.BaseExists", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": 73.103, "z": 100.691, "size": 0.2429, "title": "Existence of bases for crystallographic root systems", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/BaseExists.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.Hom", "region_id": "algebra", "micro_elevation": 0.8026, "macro_tier": 1, "macro_tier_score": 0.0027, "macro_tier_override": null, "x": 38.632, "z": -106.496, "size": 0.2947, "title": "Morphisms of root pairings", "summary": "This file defines morphisms of root pairings, following the definition of morphisms of root data given in SGA III Exp. 21 Section 6.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/Hom.html"}, {"id": "Mathlib.NumberTheory.Transcendental.Liouville.Measure", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -66.03, "z": -57.088, "size": 0.2, "title": "Volume of the set of Liouville numbers", "summary": "In this file we prove that the set of Liouville numbers with exponent (irrationality measure) strictly greater than two is a set of Lebesgue measure zero, see `volume_iUnion_setOf_liouvilleWith`. Since this set is a residual set, we show that the filters `residual` and `ae volume` are disjoint. These filters correspond to two common notions of genericity on `ℝ`: residual sets and sets of full measure. The fact that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Transcendental/Liouville/Measure.html"}, {"id": "Mathlib.Algebra.Notation.Lemmas", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2974, "title": "Lemmas about inequalities with `1`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Notation/Lemmas.html"}, {"id": "Mathlib.RingTheory.KrullDimension.Polynomial", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 45.917, "z": -91.241, "size": 0.2, "title": "Krull dimension of polynomial ring", "summary": "This file proves properties of the Krull dimension of the polynomial ring over a commutative ring", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/KrullDimension/Polynomial.html"}, {"id": "Mathlib.Algebra.IsPrimePow", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 3, "macro_tier_score": 0.2456, "macro_tier_override": null, "x": -13.697, "z": 17.58, "size": 0.3038, "title": "Prime powers", "summary": "This file deals with prime powers: numbers which are positive integer powers of a single prime.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/IsPrimePow.html"}, {"id": "Mathlib.Algebra.Polynomial.CoeffMem", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -10.926, "z": 69.721, "size": 0.2414, "title": "Bounding the coefficients of the quotient and remainder of polynomials", "summary": "This file proves that, for polynomials `p q : R[X]`, the coefficients of `p /ₘ q` and `p %ₘ q` can be written as sums of products of coefficients of `p` and `q`. Precisely, we show that each summand needs at most one coefficient of `p` and `deg p` coefficients of `q`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/CoeffMem.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -54.115, "z": -45.299, "size": 0.2, "title": "Newton's Identities", "summary": "This file defines `MvPolynomial` power sums as a means of implementing Newton's identities. The combinatorial proof, due to Zeilberger, defines for `k : ℕ` a subset `pairs` of `(range k).powerset × range k` and a map `pairMap` such that `pairMap` is an involution on `pairs`, and a map `weight` which identifies elements of `pairs` with the terms of the summation in Newton's identities and which satisfies `weight ∘…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/Symmetric/NewtonIdentities.html"}, {"id": "Mathlib.Algebra.Colimit.Ring", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -69.562, "z": -11.896, "size": 0.2, "title": "Direct limit of rings, and fields", "summary": "See Atiyah-Macdonald PP.32-33, Matsumura PP.269-270 Generalizes the notion of \"union\", or \"gluing\", of incomparable rings or fields. It is constructed as a quotient of the free commutative ring instead of a quotient of the disjoint union so as to make the operations (addition etc.) \"computable\".", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Colimit/Ring.html"}, {"id": "Mathlib.RingTheory.Polynomial.Radical", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 42.548, "z": -63.147, "size": 0.2765, "title": "Radical of a polynomial", "summary": "This file proves some theorems on `radical` and `divRadical` of polynomials. See `Mathlib.RingTheory.Radical.Basic` for the definition of `radical` and `divRadical`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/Radical.html"}, {"id": "Mathlib.Algebra.Ring.Action.Rat", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.3237, "macro_tier_override": null, "x": -13.425, "z": -6.365, "size": 0.3991, "title": "Actions by nonnegative rational numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Action/Rat.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.SpectralObject", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 27.843, "z": 27.31, "size": 0.2, "title": "The spectral object with values in the homotopy category", "summary": "Let `C` be an additive category. In this file, we show that the mapping cone defines a spectral object with values in the homotopy category of `ℤ`-indexed cochain complexes `C` that is indexed by the category `CochainComplex C ℤ`. (It follows that to any functor `ι ⥤ CochainComplex C ℤ` (e.g. a filtered complex), there is an associated spectral object indexed by `ι`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/SpectralObject.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.Triangulated", "region_id": "algebra", "micro_elevation": 0.2632, "macro_tier": 2, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 10.644, "z": 35.585, "size": 0.2951, "title": "The triangulated structure on the homotopy category of complexes", "summary": "In this file, we show that for any additive category `C`, the pretriangulated category `HomotopyCategory C (ComplexShape.up ℤ)` is triangulated.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/Triangulated.html"}, {"id": "Mathlib.Algebra.Homology.HasNoLoop", "region_id": "algebra", "micro_elevation": 0.0526, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -7.329, "z": 1.214, "size": 0.2557, "title": "Complex shapes with no loop", "summary": "Let `c : ComplexShape ι`. We define a type class `c.HasNoLoop` which expresses that `¬ c.Rel i i` for all `i : ι`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HasNoLoop.html"}, {"id": "Mathlib.Algebra.Category.Ring.Small", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -80.792, "z": -12.246, "size": 0.2659, "title": "Smallness results on the category of `CommRing`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/Small.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 80.479, "z": 59.838, "size": 0.2523, "title": "Chevalley's theorem with complexity bound", "summary": "⚠ For general usage, see `Mathlib/RingTheory/Spectrum/Prime/Chevalley.lean`. Chevalley's theorem states that if `f : R → S` is a finitely presented ring hom between commutative rings, then the image of a constructible set in `Spec S` is a constructible set in `Spec R`. Constructible sets in the prime spectrum of `R[X]` are made of closed sets in the prime spectrum (using unions, intersections, and complements),…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -96.568, "z": -0.906, "size": 0.2414, "title": "Constructible sets in the prime spectrum", "summary": "This file provides tooling for manipulating constructible sets in the prime spectrum of a ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/ConstructibleSet.html"}, {"id": "Mathlib.NumberTheory.RamificationInertia.HilbertTheory", "region_id": "algebra", "micro_elevation": 0.9211, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -70.861, "z": -108.991, "size": 0.2, "title": "Decomposition and Inertia fields", "summary": "In this file, we develop Hilbert Theory on the splitting of prime ideals in a Galois extension. Let `L/K` be a Galois extension of fields. Let `A` and `B` be subrings of `K` `L` respectively with `K` fraction field of `A`, `L` fraction field of `B` and `B` the integral closure of `A` in `L`. For `P` a prime ideal of `B` lying over the prime ideal `p` of `A`, the decomposition field `D` of `P` in `L/K` is the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/RamificationInertia/HilbertTheory.html"}, {"id": "Mathlib.Algebra.Homology.Refinements", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 1, "macro_tier_score": 0.0046, "macro_tier_override": null, "x": -10.451, "z": -10.56, "size": 0.254, "title": "Refinements", "summary": "This file contains lemmas about \"refinements\" that are specific to the study of the homology of `HomologicalComplex`. General lemmas about refinements and the case of `ShortComplex` appear in the file `Mathlib/CategoryTheory/Abelian/Refinements.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Refinements.html"}, {"id": "Mathlib.LinearAlgebra.LinearIndependent.BaseChange", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 16.967, "z": 70.414, "size": 0.2438, "title": "Base change for linear independence", "summary": "This file is a place to collect base change results for linear independence.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/LinearIndependent/BaseChange.html"}, {"id": "Mathlib.Algebra.Order.Group.CompleteLattice", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 9.041, "z": -16.222, "size": 0.2, "title": "Distributivity of group operations over supremum/infimum", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/CompleteLattice.html"}, {"id": "Mathlib.NumberTheory.FactorisationProperties", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -29.66, "z": -1.804, "size": 0.2, "title": "Factorisation properties of natural numbers", "summary": "This file defines abundant, pseudoperfect, deficient, and weird numbers and formalizes their relations with prime and perfect numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/FactorisationProperties.html"}, {"id": "Mathlib.RingTheory.MatrixPolynomialAlgebra", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.1831, "macro_tier_override": null, "x": -49.917, "z": 47.223, "size": 0.2488, "title": "Algebra isomorphism between matrices of polynomials and polynomials of matrices", "summary": "We obtain the algebra isomorphism ``` def matPolyEquiv : Matrix n n R[X] ≃ₐ[R] (Matrix n n R)[X] ``` which is characterized by ``` coeff (matPolyEquiv m) k i j = coeff (m i j) k ``` We will use this algebra isomorphism to prove the Cayley-Hamilton theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MatrixPolynomialAlgebra.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 3, "macro_tier_score": 0.2, "macro_tier_override": null, "x": -8.324, "z": 68.209, "size": 0.2863, "title": "Matrices of multivariate polynomials", "summary": "In this file, we prove results about matrices over an `MvPolynomial` ring. In particular, we provide `Matrix.mvPolynomialX` which associates every entry of a matrix with a unique variable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/MvPolynomial.html"}, {"id": "Mathlib.Algebra.DualQuaternion", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -49.494, "z": -50.307, "size": 0.2, "title": "Dual quaternions", "summary": "Similar to the way that rotations in 3D space can be represented by quaternions of unit length, rigid motions in 3D space can be represented by dual quaternions of unit length.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/DualQuaternion.html"}, {"id": "Mathlib.LinearAlgebra.CrossProduct", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 53.039, "z": 34.264, "size": 0.2589, "title": "Cross products", "summary": "This module defines the cross product of vectors in $R^3$ for $R$ a commutative ring, as a bilinear map.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CrossProduct.html"}, {"id": "Mathlib.Algebra.FreeMonoid.Count", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 12.948, "z": 1.167, "size": 0.2, "title": "`List.count` as a bundled homomorphism", "summary": "In this file we define `FreeMonoid.countP`, `FreeMonoid.count`, `FreeAddMonoid.countP`, and `FreeAddMonoid.count`. These are `List.countP` and `List.count` bundled as multiplicative and additive homomorphisms from `FreeMonoid` and `FreeAddMonoid`. We do not use `to_additive` too much because it can't map `Multiplicative ℕ` to `ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/FreeMonoid/Count.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.Finset.Powerset", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 3, "macro_tier_score": 0.2155, "macro_tier_override": null, "x": 14.58, "z": 8.173, "size": 0.2856, "title": "Big operators", "summary": "In this file we prove theorems about products and sums over a `Finset.powerset`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Finset/Powerset.html"}, {"id": "Mathlib.Algebra.Homology.BifunctorHomotopy", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -29.596, "z": -2.656, "size": 0.2, "title": "The action of a bifunctor on homological complexes factors through homotopies", "summary": "Given a `TotalComplexShape c₁ c₂ c`, a functor `F : C₁ ⥤ C₂ ⥤ D`, we show in this file that up to homotopy the morphism `mapBifunctorMap f₁ f₂ F c` only depends on the homotopy classes of the morphism `f₁` in `HomologicalComplex C c₁` and of the morphism `f₂` in `HomologicalComplex C c₂`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/BifunctorHomotopy.html"}, {"id": "Mathlib.Algebra.Category.CommAlgCat.Monoidal", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 16.718, "z": 74.286, "size": 0.2429, "title": "The co-Cartesian monoidal category structure on commutative `R`-algebras", "summary": "This file provides the co-Cartesian-monoidal category structure on `CommAlgCat R` constructed explicitly using the tensor product.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/CommAlgCat/Monoidal.html"}, {"id": "Mathlib.Algebra.Order.Positive.Field", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -14.814, "z": -16.649, "size": 0.2, "title": "Algebraic structures on the set of positive numbers", "summary": "In this file we prove that the set of positive elements of a linear ordered field is a linear ordered commutative group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Positive/Field.html"}, {"id": "Mathlib.RingTheory.TwoSidedIdeal.Kernel", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 4, "macro_tier_score": 0.3292, "macro_tier_override": null, "x": -43.885, "z": 24.259, "size": 0.396, "title": "Kernel of a ring homomorphism as a two-sided ideal", "summary": "In this file we define the kernel of a ring homomorphism `f : R → S` as a two-sided ideal of `R`. We put this in a separate file so that we could import it in `Mathlib/RingTheory/SimpleRing/Basic.lean` without importing any finiteness result.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TwoSidedIdeal/Kernel.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Int", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Int.html"}, {"id": "Mathlib.Algebra.Module.LinearMap.Index", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 81.215, "z": -9.029, "size": 0.2, "title": "The index of a linear map", "summary": "In this file we define the index of a linear map and provide some basic API.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LinearMap/Index.html"}, {"id": "Mathlib.LinearAlgebra.Charpoly.Basic", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 2, "macro_tier_score": 0.0104, "macro_tier_override": null, "x": 60.84, "z": 70.15, "size": 0.2849, "title": "Characteristic polynomial", "summary": "We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and free `R`-module. The proof that `f.charpoly` is the characteristic polynomial of the matrix of `f` in any basis is in `LinearAlgebra/Charpoly/ToMatrix`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Charpoly/Basic.html"}, {"id": "Mathlib.RingTheory.Polynomial.HilbertPoly", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -54.261, "z": 50.737, "size": 0.2, "title": "Hilbert polynomials", "summary": "In this file, we formalise the following statement: if `F` is a field with characteristic `0`, then given any `p : F[X]` and `d : ℕ`, there exists some `h : F[X]` such that for any large enough `n : ℕ`, `h(n)` is equal to the coefficient of `Xⁿ` in the power series expansion of `p/(1 - X)ᵈ`. This `h` is unique and is denoted as `Polynomial.hilbertPoly p d`. For example, given `d : ℕ`, the power series expansion of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/HilbertPoly.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.SmallShiftedHom", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -18.202, "z": 62.4, "size": 0.2632, "title": "Cohomology of `HomComplex` and morphisms in the derived category", "summary": "Let `K` and `L` be two cochain complexes in an abelian category `C`. Given a class `x : HomComplex.CohomologyClass K L n`, we construct an element in the type `SmallShiftedHom (HomologicalComplex.quasiIso C (.up ℤ)) K L n`, and compute its image as a morphism `Q.obj K ⟶ (Q.obj L)⟦n⟧` in the derived category when `x` is given as the class of a cocycle.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/SmallShiftedHom.html"}, {"id": "Mathlib.RingTheory.NonUnitalSubring.Basic", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 4, "macro_tier_score": 0.382, "macro_tier_override": null, "x": -25.565, "z": 4.736, "size": 0.3059, "title": "`NonUnitalSubring`s", "summary": "Let `R` be a non-unital ring. We prove that non-unital subrings are a complete lattice, and that you can `map` (pushforward) and `comap` (pull back) them along ring homomorphisms. We define the `closure` construction from `Set R` to `NonUnitalSubring R`, sending a subset of `R` to the non-unital subring it generates, and prove that it is a Galois insertion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/NonUnitalSubring/Basic.html"}, {"id": "Mathlib.NumberTheory.LocalField.Basic", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 33.515, "z": 57.851, "size": 0.2, "title": "Definition of (Non-archimedean) local fields", "summary": "Given a topological field `K` equipped with an equivalence class of valuations (a `ValuativeRel`), we say that it is a non-archimedean local field if the topology comes from the given valuation, and it is locally compact and non-discrete.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LocalField/Basic.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Products", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": -18.272, "z": -52.633, "size": 0.2547, "title": "The concrete products in the category of modules are products in the categorical sense.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Products.html"}, {"id": "Mathlib.Algebra.Category.MonCat.Colimits", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -9.156, "z": 11.701, "size": 0.2, "title": "The category of monoids has all colimits.", "summary": "We do this construction knowing nothing about monoids. In particular, I want to claim that this file could be produced by a python script that just looks at what Lean 3's `#print monoid` printed a long time ago (it no longer looks like this due to the addition of `npow` fields): ``` structure monoid : Type u → Type u fields: monoid.mul : Π {M : Type u} [self : monoid M], M → M → M monoid.mul_assoc : ∀ {M : Type u}…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/MonCat/Colimits.html"}, {"id": "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 1, "macro_tier_score": 0.0037, "macro_tier_override": null, "x": -121.771, "z": -39.909, "size": 0.285, "title": "Quadratic characters of finite fields", "summary": "Further facts relying on Gauss sums.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.html"}, {"id": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.UniformConvergence", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 21.363, "z": -28.084, "size": 0.2465, "title": "Uniform convergence of Eisenstein series", "summary": "We show that the sum of `eisSummand` converges locally uniformly on `ℍ` to the Eisenstein series of weight `k` and level `Γ(N)` with congruence condition `a : Fin 2 → ZMod N`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.html"}, {"id": "Mathlib.Algebra.Ring.WithZero", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 1.271, "z": 16.666, "size": 0.2, "title": "Adjoining a zero to a semiring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/WithZero.html"}, {"id": "Mathlib.Algebra.Category.Ring.EqualizerPushout", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -101.948, "z": 6.32, "size": 0.2405, "title": "Equalizer of inclusions to pushouts in `CommRingCat`", "summary": "Given a map `f : R ⟶ S` in `CommRingCat`, we prove that the equalizer of the two maps `pushout.inl : S ⟶ pushout f f` and `pushout.inr : S ⟶ pushout f f` is canonically isomorphic to `R` when `R ⟶ S` is a faithfully flat ring map. Note that, under `CommRingCat.pushoutCoconeIsColimit`, the two maps `inl` and `inr` above can be described as `s ↦ s ⊗ₜ[R] 1` and `s ↦ 1 ⊗ₜ[R] s`, respectively.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Ring/EqualizerPushout.html"}, {"id": "Mathlib.RingTheory.TensorProduct.IncludeLeftSubRight", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -94.107, "z": -21.68, "size": 0.2545, "title": "Exactness properties of the difference map on tensor products", "summary": "For an `R`-algebra `S`, we collect some properties of the `R`-linear map `S →ₗ[R] S ⊗[R] S` given by `s ↦ s ⊗ₜ 1 - 1 ⊗ₜ s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/IncludeLeftSubRight.html"}, {"id": "Mathlib.Algebra.Homology.Monoidal", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -26.484, "z": 13.475, "size": 0.2, "title": "The monoidal category structure on homological complexes", "summary": "Let `c : ComplexShape I` with `I` an additive monoid. If `c` is equipped with the data and axioms `c.TensorSigns`, then the category `HomologicalComplex C c` can be equipped with a monoidal category structure if `C` is a monoidal category such that `C` has certain coproducts and both left/right tensoring commute with these. In particular, we obtain a monoidal category structure on `ChainComplex C ℕ` when `C` is an…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Monoidal.html"}, {"id": "Mathlib.Algebra.Homology.BifunctorAssociator", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 21.341, "z": -17.905, "size": 0.2338, "title": "The associator for actions of bifunctors on homological complexes", "summary": "In this file, we shall adapt the results of the file `CategoryTheory.GradedObject.Associator` to the case of homological complexes. Given functors `F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂`, `G : C₁₂ ⥤ C₃ ⥤ C₄`, `F : C₁ ⥤ C₂₃ ⥤ C₄`, `G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃` equipped with an isomorphism `associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃` (which informally means that we have natural isomorphisms `G(F₁₂(X₁, X₂), X₃) ≅ F(X₁, G₂₃(X₂,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/BifunctorAssociator.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.Jacobson", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 2, "macro_tier_score": 0.017, "macro_tier_override": null, "x": 88.515, "z": 43.051, "size": 0.28, "title": "The prime spectrum of a Jacobson ring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/Jacobson.html"}, {"id": "Mathlib.LinearAlgebra.GeneralLinearGroup.AlgEquiv", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 3, "macro_tier_score": 0.1374, "macro_tier_override": null, "x": 65.602, "z": 48.722, "size": 0.3369, "title": "Algebra isomorphisms between endomorphisms of projective modules are inner", "summary": "This file shows that given any algebra equivalence `f : End K V ≃ₐ[K] End K W`, there exists a linear equivalence `T : V ≃ₗ[K] W` such that `f x = T ∘ₗ x ∘ₗ T.symm`. In other words, for `V = W`, the map `MulSemiringAction.toAlgEquiv` from `GeneralLinearGroup K V` to `End K V ≃ₐ[K] End K V` is surjective. For the continuous versions, see `Mathlib/Analysis/Normed/Operator/ContinuousAlgEquiv.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/GeneralLinearGroup/AlgEquiv.html"}, {"id": "Mathlib.Algebra.Group.Nat.TypeTags", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 4.466, "z": 10.209, "size": 0.2, "title": "Lemmas about `Multiplicative ℕ`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Nat/TypeTags.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.Abelian", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": 60.232, "z": -24.438, "size": 0.2797, "title": "The category of presheaves of modules is abelian", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/Abelian.html"}, {"id": "Mathlib.NumberTheory.Rayleigh", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 45.668, "z": -69.991, "size": 0.2, "title": "Rayleigh's theorem on Beatty sequences", "summary": "This file proves Rayleigh's theorem on Beatty sequences. We start by proving `compl_beattySeq`, which is a generalization of Rayleigh's theorem, and eventually prove `Irrational.beattySeq_symmDiff_beattySeq_pos`, which is Rayleigh's theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Rayleigh.html"}, {"id": "Mathlib.Algebra.Group.Action.Sum", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 10.742, "z": -7.323, "size": 0.2, "title": "Sum instances for additive and multiplicative actions", "summary": "This file defines instances for additive and multiplicative actions on the binary `Sum` type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Sum.html"}, {"id": "Mathlib.Algebra.Algebra.StrictPositivity", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -56.984, "z": -27.201, "size": 0.2817, "title": "Strictly positive elements of an algebra", "summary": "This file introduces strictly positive elements of an algebra (also known as positive definite elements). This is mostly used for C⋆-algebras, but the basic definition makes sense in a more general context.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/StrictPositivity.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -30.217, "z": 97.572, "size": 0.2, "title": "The Pin group and the Spin group", "summary": "In this file we define `lipschitzGroup`, `pinGroup` and `spinGroup` and show they form a group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/SpinGroup.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.Star", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -1.146, "z": -96.566, "size": 0.2427, "title": "Star structure on `CliffordAlgebra`", "summary": "This file defines the \"clifford conjugation\", equal to `reverse (involute x)`, and assigns it the `star` notation. This choice is somewhat non-canonical; a star structure is also possible under `reverse` alone. However, defining it gives us access to constructions like `unitary`. Most results about `star` can be obtained by unfolding it via `CliffordAlgebra.star_def`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/Star.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Charpoly.FiniteField", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -16.954, "z": -100.727, "size": 0.2, "title": "Results on characteristic polynomials and traces over finite fields.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.CharP", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 1.74, "z": 0.65, "size": 0.2478, "title": "Matrices in prime characteristic", "summary": "In this file we prove that matrices over a ring of characteristic `p` with nonempty index type have the same characteristic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/CharP.html"}, {"id": "Mathlib.FieldTheory.JacobsonNoether", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -69.505, "z": -96.418, "size": 0.2, "title": "The Jacobson-Noether theorem", "summary": "This file contains a proof of the Jacobson-Noether theorem and some auxiliary lemmas. Here we discuss different cases of characteristics of the noncommutative division algebra `D` with center `k`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/JacobsonNoether.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Conj", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -2.464, "z": -20.28, "size": 0.2276, "title": "Conjugacy in a group with zero", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Conj.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -56.307, "z": 28.576, "size": 0.2, "title": "The colimit module of a presheaf of modules on a cofiltered category", "summary": "Given a colimit cocone `cR` for a presheaf of rings `R` on a cofiltered category `C`, `M` a presheaf of modules over `R`, and a colimit cocone `cM` for the underlying functor `Cᵒᵖ ⥤ AddCommGrpCat` of `M`, we define a structure of module over `cR.pt` on a type-synonym `PresheafOfModules.ModuleColimit` for `cM.pt`. This extends to a functor `PresheafOfModules.colimitFunctor : PresheafOfModules R ⥤ ModuleCat cR.pt`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/ColimitFunctor.html"}, {"id": "Mathlib.NumberTheory.Padics.WithVal", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -9.346, "z": -77.439, "size": 0.2338, "title": "Equivalence between `ℚ_[p]` and `(Rat.padicValuation p).Completion`", "summary": "The `p`-adic numbers are isomorphic as a field to the completion of the rationals at the `p`-adic valuation. This is implemented via `Valuation.Completion` using `Rat.padicValuation`, which is shorthand for `UniformSpace.Completion (WithVal (Rat.padicValuation p))`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/WithVal.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.PushforwardZeroMonoidal", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 60.675, "z": -28.079, "size": 0.2, "title": "The pushforward functor is monoidal", "summary": "If `F : C ⥤ D` is a functor and `R : Dᵒᵖ ⥤ CommRingCat` is a presheaf of commutative rings, then the pushforward functor from the category of presheaves of modules on `R` to the category of presheaves of modules on `F.op ⋙ R` is monoidal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/PushforwardZeroMonoidal.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.Monoidal", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -41.857, "z": 49.73, "size": 0.2478, "title": "The monoidal category structure on presheaves of modules", "summary": "Given a presheaf of commutative rings `R : Cᵒᵖ ⥤ CommRingCat`, we construct the monoidal category structure on the category of presheaves of modules `PresheafOfModules (R ⋙ forget₂ _ _)`. The tensor product `M₁ ⊗ M₂` is defined as the presheaf of modules which sends `X : Cᵒᵖ` to `M₁.obj X ⊗ M₂.obj X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/Monoidal.html"}, {"id": "Mathlib.RingTheory.Algebraic.StronglyTranscendental", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 1, "macro_tier_score": 0.0045, "macro_tier_override": null, "x": -49.66, "z": 84.984, "size": 0.2376, "title": "Strongly transcendental elements", "summary": "In this file, we provide basic properties for strongly transcendental elements in an algebra. This is a relatively niche notion, but is useful for proving Zariski's main theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Algebraic/StronglyTranscendental.html"}, {"id": "Mathlib.RingTheory.AdicCompletion.LocalRing", "region_id": "algebra", "micro_elevation": 0.6316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 46.161, "z": -76.261, "size": 0.2, "title": "Basic Properties of Complete Local Ring", "summary": "In this file we prove that for local ring `R` with finitely generated maximal ideal, `AdicCompletion (IsLocalRing.maximalIdeal R) R` is local ring with maximal ideal equal to `IsLocalRing.maximalIdeal R` mapped by algebra map. Furthermore, it is complete with respect to its maximal ideal. As a corollary, for Noetherian local ring `R`, `AdicCompletion (maximalIdeal R) R` is always a complete Noetherian local ring.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AdicCompletion/LocalRing.html"}, {"id": "Mathlib.Algebra.FreeAlgebra.Cardinality", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 14.519, "z": -57.628, "size": 0.2, "title": "Cardinality of free algebras", "summary": "This file contains some results about the cardinality of `FreeAlgebra`, parallel to that of `MvPolynomial`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/FreeAlgebra/Cardinality.html"}, {"id": "Mathlib.Algebra.Order.Star.Real", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 26.626, "z": 13.192, "size": 0.2994, "title": "`ℝ` and `ℝ≥0` are \\*-ordered rings.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Star/Real.html"}, {"id": "Mathlib.NumberTheory.Padics.HeightOneSpectrum", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 87.489, "z": 56.231, "size": 0.2, "title": "Isomorphisms between `adicCompletion ℚ` and `ℚ_[p]`", "summary": "Let `R` have field of fractions `ℚ`. If `v : HeightOneSpectrum R`, then `v.adicCompletion ℚ` is the uniform space completion of `ℚ` with respect to the `v`-adic valuation. On the other hand, `ℚ_[p]` is the `p`-adic numbers, defined as the completion of `ℚ` with respect to the `p`-adic norm using the completion of Cauchy sequences. This file constructs continuous `ℚ`-algebra isomorphisms between the two, as well as…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Padics/HeightOneSpectrum.html"}, {"id": "Mathlib.RingTheory.Valuation.Extension", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -86.634, "z": -46.722, "size": 0.2, "title": "Extension of Valuations", "summary": "In this file, we define the typeclass for valuation extensions and prove basic facts about the extension of valuations. Let `A` be an `R` algebra, equipped with valuations `vA` and `vR` respectively. Here, the extension of a valuation means that the pullback of valuation `vA` to `R` is equivalent to the valuation `vR` on `R`. We only require equivalence, not equality, of valuations here. Note that we do not require…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/Extension.html"}, {"id": "Mathlib.FieldTheory.AxGrothendieck", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": "Ax-Grothendieck", "summary": "This file proves that if `K` is an algebraically closed field, then any injective polynomial map `K^n → K^n` is also surjective.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/AxGrothendieck.html"}, {"id": "Mathlib.GroupTheory.MonoidLocalization.Lemmas", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -17.164, "z": -11.078, "size": 0.2, "title": "Lemmas about localizations of commutative monoids", "summary": "that requires additional imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/MonoidLocalization/Lemmas.html"}, {"id": "Mathlib.Algebra.MvPolynomial.Coeff", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 2, "macro_tier_score": 0.0113, "macro_tier_override": null, "x": -55.599, "z": -20.989, "size": 0.261, "title": "Formulas for coefficients of multivariate polynomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MvPolynomial/Coeff.html"}, {"id": "Mathlib.GroupTheory.MonoidLocalization.Finite", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 7.804, "z": 34.412, "size": 0.2403, "title": "Localization of a finitely generated submonoid", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/MonoidLocalization/Finite.html"}, {"id": "Mathlib.LinearAlgebra.ExteriorAlgebra.OfAlternating", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -17.28, "z": -96.901, "size": 0.268, "title": "Extending an alternating map to the exterior algebra", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.html"}, {"id": "Mathlib.NumberTheory.FLT.Four", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 62.956, "z": -22.507, "size": 0.2, "title": "Fermat's Last Theorem for the case n = 4", "summary": "There are no non-zero integers `a`, `b` and `c` such that `a ^ 4 + b ^ 4 = c ^ 4`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/FLT/Four.html"}, {"id": "Mathlib.NumberTheory.PythagoreanTriples", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 53.339, "z": 37.149, "size": 0.239, "title": "Pythagorean Triples", "summary": "The main result is the classification of Pythagorean triples. The final result is for general Pythagorean triples. It follows from the more interesting relatively prime case. We use the \"rational parametrization of the circle\" method for the proof. The parametrization maps the point `(x / z, y / z)` to the slope of the line through `(-1, 0)` and `(x / z, y / z)`. This quickly shows that `(x / z, y / z) = (2 * m * n…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/PythagoreanTriples.html"}, {"id": "Mathlib.Algebra.Homology.CochainComplexPlus", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 32.369, "z": -14.049, "size": 0.2448, "title": "Bounded below cochain complexes", "summary": "In this file, we consider the full subcategory `CochainComplex.Plus C` of `CochainComplex C ℤ` consisting of bounded below cochain complexes in a category `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/CochainComplexPlus.html"}, {"id": "Mathlib.NumberTheory.RamificationInertia.Valuation", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -66.444, "z": 80.009, "size": 0.2, "title": "Ramification theory for valuations", "summary": "- `A` is a Dedekind domain with field of fractions `K`. - `B` is a Dedekind domain with field of fractions `L`. - `L` is a field extension of `K`. - `v` is a height one prime ideal of `A`. - `w` is a height one prime ideal of `B` lying over `v`. This file establishes the relationship between the adic valuation on `K` associated to `v` and the adic valuation on `L` associated to `w`, in terms of the ramification…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/RamificationInertia/Valuation.html"}, {"id": "Mathlib.GroupTheory.Subsemigroup.Center", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 4, "macro_tier_score": 0.4457, "macro_tier_override": null, "x": 6.063, "z": -19.508, "size": 0.3444, "title": "Centers of semigroups, as subsemigroups.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Subsemigroup/Center.html"}, {"id": "Mathlib.NumberTheory.BernoulliPolynomials", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": 52.203, "z": -92.092, "size": 0.2386, "title": "Bernoulli polynomials", "summary": "The [Bernoulli polynomials](https://en.wikipedia.org/wiki/Bernoulli_polynomials) are an important tool obtained from Bernoulli numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/BernoulliPolynomials.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.Homeomorph", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -114.285, "z": -32.654, "size": 0.2, "title": "Purely inseparable extensions are universal homeomorphisms", "summary": "If `K` is a purely inseparable extension of `k`, the induced map `Spec K ⟶ Spec k` is a universal homeomorphism, i.e. it stays a homeomorphism after arbitrary base change.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/Homeomorph.html"}, {"id": "Mathlib.Algebra.Module.Torsion.PrimaryComponent", "region_id": "algebra", "micro_elevation": 0.9211, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -100.22, "z": 82.803, "size": 0.2, "title": "I-Primary Components of modules", "summary": "Let `A` be a commutative ring and `I`, an ideal of `A`. Given an `A`-Module `M` it's `I`-primary component is defined as $$M(I) := \\bigcup_{i : \\mathbb{N}} \\text{torsionBySet A M } I ^ i.$$ For `P : HeightOneSpectrum A`, the main result of this file is that $$M \\cong \\bigoplus_{P} M(P).$$", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Torsion/PrimaryComponent.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.BaseChange", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 65.402, "z": 71.055, "size": 0.2, "title": "The base change of a clifford algebra", "summary": "In this file we show the isomorphism * `CliffordAlgebra.equivBaseChange A Q` : `CliffordAlgebra (Q.baseChange A) ≃ₐ[A] (A ⊗[R] CliffordAlgebra Q)` with forward direction `CliffordAlgebra.toBaseChange A Q` and reverse direction `CliffordAlgebra.ofBaseChange A Q`. This covers a more general case of the complexification of clifford algebras (as described in §2.2 of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.Faithful", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 6.064, "z": 11.499, "size": 0.2, "title": "Faithful actions involving groups with zero", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/Faithful.html"}, {"id": "Mathlib.NumberTheory.SumTwoSquares", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 71.933, "z": 80.176, "size": 0.2, "title": "Sums of two squares", "summary": "Fermat's theorem on the sum of two squares. Every prime `p` congruent to 1 mod 4 is the sum of two squares; see `Nat.Prime.sq_add_sq` (which has the weaker assumption `p % 4 ≠ 3`). We also give the result that characterizes the (positive) natural numbers that are sums of two squares as those numbers `n` such that for every prime `q` congruent to 3 mod 4, the exponent of the largest power of `q` dividing `n` is even;…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/SumTwoSquares.html"}, {"id": "Mathlib.Algebra.Field.Power", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 7.433, "z": 21.01, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/Power.html"}, {"id": "Mathlib.Algebra.GradedMonoid", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 3, "macro_tier_score": 0.0385, "macro_tier_override": null, "x": -3.968, "z": -16.237, "size": 0.3215, "title": "Additively-graded multiplicative structures", "summary": "This module provides a set of heterogeneous typeclasses for defining a multiplicative structure over the sigma type `GradedMonoid A` such that `(*) : A i → A j → A (i + j)`; that is to say, `A` forms an additively-graded monoid. The typeclasses are: * `GradedMonoid.GOne A` * `GradedMonoid.GMul A` * `GradedMonoid.GMonoid A` * `GradedMonoid.GCommMonoid A` These respectively imbue: * `One (GradedMonoid A)` * `Mul…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GradedMonoid.html"}, {"id": "Mathlib.NumberTheory.NumberField.Cyclotomic.Three", "region_id": "algebra", "micro_elevation": 0.9868, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 137.879, "z": -19.758, "size": 0.2338, "title": "Third Cyclotomic Field", "summary": "We gather various results about the third cyclotomic field. The following notations are used in this file: `K` is a number field such that `IsCyclotomicExtension {3} ℚ K`, `ζ` is any primitive `3`-rd root of unity in `K`, `η` is the element in the units of the ring of integers corresponding to `ζ` and `λ = η - 1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Cyclotomic/Three.html"}, {"id": "Mathlib.FieldTheory.IntermediateField.Adjoin.Basic", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 3, "macro_tier_score": 0.0887, "macro_tier_override": null, "x": 37.925, "z": -92.839, "size": 0.3998, "title": "Adjoining Elements to Fields", "summary": "This file contains many results about adjoining elements to fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/IntermediateField/Adjoin/Basic.html"}, {"id": "Mathlib.RingTheory.Ideal.IsPrincipalPowQuotient", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -23.14, "z": 52.717, "size": 0.2304, "title": "Quotients of powers of principal ideals", "summary": "This file deals with taking quotients of powers of principal ideals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/IsPrincipalPowQuotient.html"}, {"id": "Mathlib.RingTheory.FormalGroup.Basic", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -72.954, "z": 21.806, "size": 0.2, "title": "Formal group laws over commutative ring", "summary": "Let `R` be a commutative ring, a one dimensional formal group law is a formal power series `F(X,Y) ∈ R⟦X,Y⟧` such that * `F(X,Y) = X + Y + higher order terms`. * `F(F(X,Y),Z) = F(X,F(Y,Z))`. Under this definition, we can prove that `F(X,0) = X` and `F(0,X) = X`. Moreover, there is a unique power series `i(X)` such that `F(X, i(X)) = 0`, which is considered to be the inverse of the formal group law `F(X,Y)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/FormalGroup/Basic.html"}, {"id": "Mathlib.Algebra.DirectSum.Ring", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 3, "macro_tier_score": 0.036, "macro_tier_override": null, "x": -17.787, "z": -5.341, "size": 0.2999, "title": "Additively-graded multiplicative structures on `⨁ i, A i`", "summary": "This module provides a set of heterogeneous typeclasses for defining a multiplicative structure over `⨁ i, A i` such that `(*) : A i → A j → A (i + j)`; that is to say, `A` forms an additively-graded ring. The five typeclasses are: * `DirectSum.GNonUnitalNonAssocSemiring A` * `DirectSum.GSemiring A` * `DirectSum.GRing A` * `DirectSum.GCommSemiring A` * `DirectSum.GCommRing A` Respectively, these five typeclasses…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/DirectSum/Ring.html"}, {"id": "Mathlib.RingTheory.RingHom.Etale", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 2, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -85.503, "z": -90.4, "size": 0.3046, "title": "Étale ring homomorphisms", "summary": "We show the meta properties of étale morphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/Etale.html"}, {"id": "Mathlib.LinearAlgebra.StdBasis", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 4, "macro_tier_score": 0.2985, "macro_tier_override": null, "x": -32.824, "z": -45.019, "size": 0.3556, "title": "The standard basis", "summary": "This file defines the standard basis `Pi.basis (s : ∀ j, Basis (ι j) R (M j))`, which is the `Σ j, ι j`-indexed basis of `Π j, M j`. The basis vectors are given by `Pi.basis s ⟨j, i⟩ j' = Pi.single j' (s j) i = if j = j' then s i else 0`. The standard basis on `R^η`, i.e. `η → R` is called `Pi.basisFun`. To give a concrete example, `Pi.single (i : Fin 3) (1 : R)` gives the `i`th unit basis vector in `R³`, and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/StdBasis.html"}, {"id": "Mathlib.RingTheory.Radical", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -26.552, "z": 59.33, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Radical.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Action.Pointwise.Finset", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 18.998, "z": -17.751, "size": 0.2, "title": "Pointwise operations of finsets in a group with zero", "summary": "This file proves properties of pointwise operations of finsets in a group with zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Action/Pointwise/Finset.html"}, {"id": "Mathlib.Algebra.Group.Pointwise.Finset.Density", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 15.172, "z": -18.78, "size": 0.2, "title": "Theorems about the density of pointwise operations on finsets.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pointwise/Finset/Density.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.Boundary", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -9.877, "z": -8.452, "size": 0.2716, "title": "Boundary of an embedding of complex shapes", "summary": "In the file `Mathlib/Algebra/Homology/Embedding/Basic.lean`, given `p : ℤ`, we have defined an embedding `embeddingUpIntGE p` of `ComplexShape.up ℕ` in `ComplexShape.up ℤ` which sends `n : ℕ` to `p + n`. The (canonical) truncation (`≥ p`) of `K : CochainComplex C ℤ` shall be defined as the extension to `ℤ` (see `Mathlib/Algebra/Homology/Embedding/Extend.lean`) of a certain cochain complex indexed by `ℕ`: `Q ⟶ K.X (p…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/Boundary.html"}, {"id": "Mathlib.FieldTheory.IsAlgClosed.Classification", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -60.781, "z": 86.669, "size": 0.2403, "title": "Classification of Algebraically closed fields", "summary": "This file contains results related to classifying algebraically closed fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/IsAlgClosed/Classification.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Log", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 58.833, "z": 56.71, "size": 0.2, "title": "Logarithmic Power Series", "summary": "This file defines the logarithmic power series `log A = ∑ (-1)^(n+1)/n · Xⁿ` over ℚ-algebras and establishes its key properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Log.html"}, {"id": "Mathlib.NumberTheory.NumberField.Completion.Ramification", "region_id": "algebra", "micro_elevation": 0.9474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 125.32, "z": 46.635, "size": 0.2, "title": "Ramification theory of completions of number fields", "summary": "This file studies the ramification of completions of number fields.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Completion/Ramification.html"}, {"id": "Mathlib.NumberTheory.LucasPrimality", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 72.661, "z": -22.763, "size": 0.257, "title": "The Lucas test for primes", "summary": "This file implements the Lucas test for primes (not to be confused with the Lucas-Lehmer test for Mersenne primes). A number `a` witnesses that `n` is prime if `a` has order `n-1` in the multiplicative group of integers mod `n`. This is checked by verifying that `a^(n-1) = 1 (mod n)` and `a^d ≠ 1 (mod n)` for any divisor `d | n - 1`. This test is the basis of the Pratt primality certificate.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LucasPrimality.html"}, {"id": "Mathlib.Algebra.Polynomial.Degree.Units", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 4, "macro_tier_score": 0.2852, "macro_tier_override": null, "x": 54.638, "z": 18.145, "size": 0.3555, "title": "Degree of polynomials that are units", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Degree/Units.html"}, {"id": "Mathlib.RingTheory.KrullDimension.LocalRing", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -8.378, "z": -98.072, "size": 0.2, "title": "The Krull dimension of a local ring", "summary": "In this file, we proved some results about the Krull dimension of a local ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/KrullDimension/LocalRing.html"}, {"id": "Mathlib.RingTheory.LaurentSeries", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -93.529, "z": 49.581, "size": 0.2, "title": "Laurent Series", "summary": "In this file we define `LaurentSeries R`, the formal Laurent series over `R`, here an *arbitrary* type with a zero. They are denoted `R⸨X⸩`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LaurentSeries.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Presheaf.Free", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -61.847, "z": 12.728, "size": 0.2344, "title": "The free presheaf of modules on a presheaf of sets", "summary": "In this file, given a presheaf of rings `R` on a category `C`, we construct the functor `PresheafOfModules.free : (Cᵒᵖ ⥤ Type u) ⥤ PresheafOfModules.{u} R` which sends a presheaf of types to the corresponding presheaf of free modules. `PresheafOfModules.freeAdjunction` shows that this functor is the left adjoint to the forget functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Presheaf/Free.html"}, {"id": "Mathlib.Algebra.MonoidAlgebra.ToDirectSum", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 57.572, "z": -0.219, "size": 0.2, "title": "Conversion between `AddMonoidAlgebra` and homogeneous `DirectSum`", "summary": "This module provides conversions between `AddMonoidAlgebra` and `DirectSum`. The latter is essentially a dependent version of the former. Note that since `DirectSum.instMul` combines indices additively, there is no equivalent to `MonoidAlgebra`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/MonoidAlgebra/ToDirectSum.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.QuadraticModuleCat.Monoidal", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -87.572, "z": -30.885, "size": 0.2478, "title": "The monoidal category structure on quadratic R-modules", "summary": "The monoidal structure is simply the structure on the underlying modules, where the tensor product of two modules is equipped with the form constructed via `QuadraticForm.tmul`. As with the monoidal structure on `ModuleCat`, the universe level of the modules must be at least the universe level of the ring, so that we have a monoidal unit. For now, we simplify by insisting both universe levels are the same.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat/Monoidal.html"}, {"id": "Mathlib.Algebra.Homology.Precylinder", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -14.067, "z": -17.285, "size": 0.2331, "title": "Precylinder and pre-path objects in the category of homological complexes", "summary": "In this file, we upgrade the definitions `HomologicalComplex.cylinder` and `HomologicalComplex.pathObject` to pre-cylinder objects and pre-path objects in the sense of homotopical algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Precylinder.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyFiber", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 9.871, "z": 17.886, "size": 0.2524, "title": "The homotopy fiber of a morphism of homological complexes", "summary": "In this file, we construct the homotopy fiber of a morphism `φ : F ⟶ G` between homological complexes. Moreover, we dualise the definition of the cylinder (which is a particular case of a homotopy cofiber) in order to define the path object of a homological complex.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyFiber.html"}, {"id": "Mathlib.NumberTheory.WellApproximable", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": "Well-approximable numbers and Gallagher's ergodic theorem", "summary": "Gallagher's ergodic theorem is a result in metric number theory. It thus belongs to that branch of mathematics concerning arithmetic properties of real numbers which hold almost everywhere with respect to the Lebesgue measure. Gallagher's theorem concerns the approximation of real numbers by rational numbers. The input is a sequence of distances `δ₁, δ₂, ...`, and the theorem concerns the set of real numbers `x` for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/WellApproximable.html"}, {"id": "Mathlib.Algebra.Group.Int.Even", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 4, "macro_tier_score": 0.4188, "macro_tier_override": null, "x": -14.127, "z": -8.933, "size": 0.4339, "title": "Parity of integers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Int/Even.html"}, {"id": "Mathlib.NumberTheory.GaussSum", "region_id": "algebra", "micro_elevation": 0.8947, "macro_tier": 2, "macro_tier_score": 0.0061, "macro_tier_override": null, "x": 53.91, "z": -114.202, "size": 0.3042, "title": "Gauss sums", "summary": "We define the Gauss sum associated to a multiplicative and an additive character of a finite field and prove some results about them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/GaussSum.html"}, {"id": "Mathlib.RingTheory.SurjectiveOnStalks", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0779, "macro_tier_override": null, "x": 50.89, "z": -48.894, "size": 0.2689, "title": "Ring Homomorphisms surjective on stalks", "summary": "In this file, we prove some results on ring homomorphisms surjective on stalks, to be used in the development of immersions in algebraic geometry. A ring homomorphism `R →+* S` is surjective on stalks if `R_p →+* S_q` is surjective for all pairs of primes `p = f⁻¹(q)`. We show that this property is stable under composition and base change, and that surjections and localizations satisfy this.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SurjectiveOnStalks.html"}, {"id": "Mathlib.GroupTheory.GroupAction.CardCommute", "region_id": "algebra", "micro_elevation": 0.2368, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -18.75, "z": -27.675, "size": 0.2553, "title": "Properties of group actions involving quotient groups", "summary": "This file proves cardinality properties of group actions which use the quotient group construction, notably * the class formula `MulAction.card_eq_sum_card_group_div_card_stabilizer'` * `card_comm_eq_card_conjClasses_mul_card` as well as their analogues for additive groups. See `Mathlib/GroupTheory/GroupAction/Quotient.lean` for the construction of isomorphisms used to prove these cardinality properties. These…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/CardCommute.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.KProjective", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 41.594, "z": -52.344, "size": 0.2338, "title": "Morphisms from K-projective complexes in the derived category", "summary": "In this file, we show that if `K : CochainComplex C ℤ` is K-projective, then for any `L : HomotopyCategory C (.up ℤ)`, the functor `DerivedCategory.Qh` induces a bijection from the type of morphisms `(HomotopyCategory.quotient _ _).obj K) ⟶ L` (i.e. homotopy classes of morphisms of cochain complexes) to the type of morphisms in the derived category. We obtain that a morphism between `K`-projective cochain complexes…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/KProjective.html"}, {"id": "Mathlib.FieldTheory.AbelRuffini", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -37.179, "z": 110.937, "size": 0.2, "title": "The Abel-Ruffini Theorem", "summary": "This file proves one direction of the Abel-Ruffini theorem, namely that if an element is solvable by radicals, then its minimal polynomial has solvable Galois group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/AbelRuffini.html"}, {"id": "Mathlib.Algebra.Homology.DifferentialObject", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -7.988, "z": 4.735, "size": 0.2, "title": "Homological complexes are differential graded objects.", "summary": "We verify that a `HomologicalComplex` indexed by an `AddCommGroup` is essentially the same thing as a differential graded object. This equivalence is probably not particularly useful in practice; it's here to check that definitions match up as expected.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DifferentialObject.html"}, {"id": "Mathlib.Algebra.Order.Sum", "region_id": "algebra", "micro_elevation": 0.0263, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.746, "z": -3.639, "size": 0.2, "title": "Interaction between `Sum.elim`, `≤`, and `0` or `1`", "summary": "This file provides basic API for part-wise comparison of `Sum.elim` vectors against `0` or `1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Sum.html"}, {"id": "Mathlib.LinearAlgebra.Projectivization.Cardinality", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -33.439, "z": -70.469, "size": 0.2, "title": "Cardinality of projective spaces", "summary": "We compute the cardinality of `ℙ k V` if `k` is a finite field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Projectivization/Cardinality.html"}, {"id": "Mathlib.RingTheory.Jacobson.Artinian", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 2, "macro_tier_score": 0.0167, "macro_tier_override": null, "x": 42.147, "z": 91.0, "size": 0.2452, "title": "Artinian rings over Jacobson rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Jacobson/Artinian.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Trace", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 3, "macro_tier_score": 0.204, "macro_tier_override": null, "x": 19.566, "z": 52.166, "size": 0.3442, "title": "Trace of a matrix", "summary": "This file defines the trace of a matrix, the map sending a matrix to the sum of its diagonal entries. See also `LinearAlgebra.Trace` for the trace of an endomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Trace.html"}, {"id": "Mathlib.Algebra.Order.Monoid.PNat", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -17.332, "z": -16.808, "size": 0.2327, "title": "Equivalence between `ℕ+` and `nonZeroDivisors ℕ`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/PNat.html"}, {"id": "Mathlib.Algebra.Order.Interval.Set.SuccPred", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 2.329, "z": -16.551, "size": 0.2, "title": "Set intervals in an additive successor-predecessor order", "summary": "This file proves relations between the various set intervals in an additive successor/predecessor order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Interval/Set/SuccPred.html"}, {"id": "Mathlib.NumberTheory.PrimesCongruentOne", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 109.044, "z": 22.937, "size": 0.2, "title": "Primes congruent to one", "summary": "We prove that, for any positive `k : ℕ`, there are infinitely many primes `p` such that `p ≡ 1 [MOD k]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/PrimesCongruentOne.html"}, {"id": "Mathlib.Algebra.Order.Antidiag.Nat", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -28.342, "z": 31.957, "size": 0.2, "title": "Sets of tuples with a fixed product", "summary": "This file defines the finite set of `d`-tuples of natural numbers with a fixed product `n` as `Nat.finMulAntidiag`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Antidiag/Nat.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 10.235, "z": -62.309, "size": 0.2, "title": "Totally unimodular matrices", "summary": "This file defines totally unimodular matrices and provides basic API for them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Determinant/TotallyUnimodular.html"}, {"id": "Mathlib.RingTheory.Extension.ExtendScalars", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 14.414, "z": -91.733, "size": 0.2, "title": "Extension of Scalars for Algebra Extensions", "summary": "This file provides APIs for extending the base ring of an algebra extension `P : Extension R S` to its own extension ring `P.Ring`. We introduce canonical maps and isomorphisms between the cotangent spaces and the first homology of naive cotangent complex associated with `P.extendScalars` and `P`. We provide commutativity results of these maps and ismorphisms (See…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Extension/ExtendScalars.html"}, {"id": "Mathlib.Algebra.NoZeroSMulDivisors.Prod", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/NoZeroSMulDivisors/Prod.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Defs", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -10.687, "z": -3.155, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/Unbundled/Defs.html"}, {"id": "Mathlib.Algebra.Lie.Subalgebra", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.0187, "macro_tier_override": null, "x": -18.723, "z": -68.043, "size": 0.4792, "title": "Lie subalgebras", "summary": "This file defines Lie subalgebras of a Lie algebra and provides basic related definitions and results.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Subalgebra.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Vec", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 1, "macro_tier_score": 0.0045, "macro_tier_override": null, "x": 62.481, "z": 17.922, "size": 0.2436, "title": "Vectorization of matrices", "summary": "This file defines `Matrix.vec A`, the vectorization of a matrix `A`, formed by stacking the columns of A into a single large column vector. Since mathlib indices matrices by arbitrary types rather than `Fin n`, the result of `Matrix.vec` on `A : Matrix m n R` is indexed by `n × m`. The `Fin (n * m)` interpretation can be restored by composing with `finProdFinEquiv.symm`: ```lean -- ![1, 2, 3, 4] #eval vec !![1, 3;…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Vec.html"}, {"id": "Mathlib.Algebra.Homology.ShortComplex.Linear", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 8.242, "z": -4.278, "size": 0.2, "title": "Homology of linear categories", "summary": "In this file, it is shown that if `C` is an `R`-linear category, then `ShortComplex C` is an `R`-linear category. Various homological notions are also shown to be linear.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ShortComplex/Linear.html"}, {"id": "Mathlib.Algebra.Order.Interval.Multiset", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -0.178, "z": 16.714, "size": 0.2, "title": "Algebraic properties of multiset intervals", "summary": "This file provides results about the interaction of algebra with `Multiset.Ixx`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Interval/Multiset.html"}, {"id": "Mathlib.Algebra.Group.Pointwise.Finset.BigOperators", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.665, "z": 22.224, "size": 0.2, "title": "Pointwise big operators on finsets", "summary": "This file contains basic results on applying big operators (product and sum) on finsets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Pointwise/Finset/BigOperators.html"}, {"id": "Mathlib.RingTheory.Polynomial.SmallDegreeVieta", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 79.73, "z": -4.514, "size": 0.2, "title": "Vieta's Formula for polynomial of small degrees.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/SmallDegreeVieta.html"}, {"id": "Mathlib.RingTheory.Polynomial.DegreeLT", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 1, "macro_tier_score": 0.0038, "macro_tier_override": null, "x": -82.688, "z": 21.468, "size": 0.2911, "title": "Polynomials with degree strictly less than `n`", "summary": "This file contains the properties of the submodule of polynomials of degree less than `n` in a (semi)ring `R`, denoted `R[X]_n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Polynomial/DegreeLT.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.Finset.Gaps", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -11.155, "z": -12.448, "size": 0.245, "title": "Sum of gaps", "summary": "This file proves that given a function `g` on `[a, b]`, `g b - g a` can be split according to a given finite collection of pairwise disjoint closed subintervals of `[a, b]`. It is the sum of two terms: - the sum of `g y - g x` for `[x, y]` in the collection, - the sum of `g y - g x` for `[x, y]` in the complement (modulo endpoints) of the union of the collection in `[a, b]`. We use `Finset.intervalGapsWithin` to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Finset/Gaps.html"}, {"id": "Mathlib.RingTheory.Localization.AtPrime.Extension", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -99.1, "z": -81.242, "size": 0.2, "title": "Primes in an extension of localization at prime", "summary": "Let `R ⊆ S` be an extension of Dedekind domains and `p` be a prime ideal of `R`. Let `Rₚ` be the localization of `R` at the complement of `p` and `Sₚ` the localization of `S` at the (image) of the complement of `p`. In this file, we study the relation between the (nonzero) prime ideals of `Sₚ` and the prime ideals of `S` above `p`. In particular, we prove that (under suitable conditions) they are in bijection and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/AtPrime/Extension.html"}, {"id": "Mathlib.Algebra.Polynomial.RuleOfSigns", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -26.101, "z": -69.55, "size": 0.2, "title": "Descartes' Rule of Signs", "summary": "We define the \"sign changes\" in the coefficients of a polynomial, and prove Descartes' Rule of Signs: a real polynomial has at most as many positive roots as there are sign changes. A sign change is when there is a positive coefficient followed by a negative coefficient, or vice versa, with any number of zero coefficients in between.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/RuleOfSigns.html"}, {"id": "Mathlib.Algebra.Module.LinearMap.Polynomial", "region_id": "algebra", "micro_elevation": 0.6842, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -89.533, "z": 36.196, "size": 0.2465, "title": "Characteristic polynomials of linear families of endomorphisms", "summary": "The coefficients of the characteristic polynomials of a linear family of endomorphisms are homogeneous polynomials in the parameters. This result is used in Lie theory to establish the existence of regular elements and Cartan subalgebras, and ultimately a well-defined notion of rank for Lie algebras. In this file we prove this result about characteristic polynomials. Let `L` and `M` be modules over a nontrivial…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LinearMap/Polynomial.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.ChangeOfRingsExact", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 23.218, "z": 64.674, "size": 0.2, "title": "Exactness of functors for change of rings.", "summary": "In this file we provide exactness of restrictScalars for general universe level using it preserves short exact sequences. Note : previously exactness of `ModuleCat.restrictScalars` is synthesized via being adjoint functor, however this needs the universe level to be some `max u v`, where `u` is the universe level of the ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/ChangeOfRingsExact.html"}, {"id": "Mathlib.Algebra.Ring.Identities", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": "Identities", "summary": "This file contains some \"named\" commutative ring identities.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Identities.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.Action.Synonym", "region_id": "algebra", "micro_elevation": 0.1053, "macro_tier": 4, "macro_tier_score": 0.3201, "macro_tier_override": null, "x": -13.924, "z": 5.184, "size": 0.3194, "title": "Actions by and on order synonyms", "summary": "This PR transfers group action with zero instances from a type `α` to `αᵒᵈ` and `Lex α`. Note that the `SMul` instances are already defined in `Mathlib/Algebra/Order/Group/Synonym.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/Action/Synonym.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.ExteriorPower", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -56.95, "z": 84.794, "size": 0.2, "title": "The exterior powers as functors on the category of modules", "summary": "In this file, given `M : ModuleCat R` and `n : ℕ`, we define `M.exteriorPower n : ModuleCat R`, and this extends to a functor `ModuleCat.exteriorPower.functor : ModuleCat R ⥤ ModuleCat R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/ExteriorPower.html"}, {"id": "Mathlib.LinearAlgebra.DirectSum.Basis", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 8.098, "z": 55.123, "size": 0.2, "title": "Bases for direct sum of modules", "summary": "This file defines a `Module.Free` instance for the direct sum of modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/DirectSum/Basis.html"}, {"id": "Mathlib.RingTheory.Valuation.FiniteField", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 41.24, "z": 95.475, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/FiniteField.html"}, {"id": "Mathlib.Algebra.Ring.Associator", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 3, "macro_tier_score": 0.0369, "macro_tier_override": null, "x": 16.712, "z": -0.26, "size": 0.2792, "title": "Associator in a ring", "summary": "If `R` is a non-associative ring, then `(x * y) * z - x * (y * z)` is called the `associator` of ring elements `x y z : R`. The associator vanishes exactly when `R` is associative. We prove variants of this statement also for the `AddMonoidHom` bundled version of the associator, as well as the bundled version of `mulLeft₃` and `mulRight₃`, the multiplications `(x * y) * z` and `x * (y * z)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Associator.html"}, {"id": "Mathlib.RingTheory.LocalRing.NonLocalRing", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -6.495, "z": 66.542, "size": 0.2, "title": "Non-local rings", "summary": "This file gathers some results about non-local rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/NonLocalRing.html"}, {"id": "Mathlib.NumberTheory.DiophantineApproximation.ContinuedFractions", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -68.679, "z": -50.807, "size": 0.2, "title": "Diophantine Approximation using continued fractions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/DiophantineApproximation/ContinuedFractions.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.EulerIdentity", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -26.581, "z": -71.354, "size": 0.2, "title": "Euler's homogeneous identity", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/EulerIdentity.html"}, {"id": "Mathlib.Algebra.Category.Grp.Subobject", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -36.934, "z": -44.164, "size": 0.2, "title": "The category of abelian groups is well-powered", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Subobject.html"}, {"id": "Mathlib.LinearAlgebra.BilinearForm.IsometryEquiv", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -50.13, "z": 24.314, "size": 0.2, "title": "Isometric equivalences with respect to bilinear forms", "summary": "In this file, we define isometry equivalences of bilinear spaces as linear equivalences that respect the associated bilinear forms. This file should be kept in sync with the corresponding file for quadratic maps, namely `Mathlib/LinearAlgebra/QuadraticForm/IsometryEquiv.lean`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/BilinearForm/IsometryEquiv.html"}, {"id": "Mathlib.LinearAlgebra.BilinearForm.Isometry", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -2.172, "z": 44.519, "size": 0.2478, "title": "Isometric linear maps", "summary": "In this file, we define isometries of bilinear spaces as linear maps that respect the associated bilinear forms. This file should be kept in sync with the corresponding file for quadratic maps, namely `Mathlib/LinearAlgebra/QuadraticForm/Isometry.lean`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/BilinearForm/Isometry.html"}, {"id": "Mathlib.Algebra.BigOperators.Group.Finset.Interval", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 0.622, "z": -18.561, "size": 0.2377, "title": "Sums/products over integer intervals", "summary": "This file contains some lemmas about sums and products over integer intervals `Ixx`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Group/Finset/Interval.html"}, {"id": "Mathlib.FieldTheory.CardinalEmb", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 26.275, "z": 115.918, "size": 0.2, "title": "Number of embeddings of an algebraic extension of infinite separable degree", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/CardinalEmb.html"}, {"id": "Mathlib.GroupTheory.GroupAction.OfQuotient", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -30.775, "z": 7.05, "size": 0.2, "title": "MulAction and MulDistribMulAction of quotient group on fixed points", "summary": "Given a `MulAction`/`MulDistribMulAction` of a group `G` on `A` and a normal subgroup `H` of `G`, there is a `MulAction`/`MulDistribMulAction` of the quotient group `G ⧸ H` on `fixedPoints H A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/OfQuotient.html"}, {"id": "Mathlib.RepresentationTheory.FinGroupCharZero", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 38.081, "z": 104.721, "size": 0.2, "title": "Applications of Maschke's theorem", "summary": "This proves some properties of representations that follow from Maschke's theorem. We prove that, if `G` is a finite group whose order is invertible in a field `k`, then every object of `Rep k G` (resp. `FDRep k G`) is injective and projective. We also give two simpleness criteria for an object `V` of `FDRep k G`, when `k` is an algebraically closed field in which the order of `G` is invertible: *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/FinGroupCharZero.html"}, {"id": "Mathlib.RingTheory.Spectrum.Prime.Chevalley", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -31.228, "z": 97.253, "size": 0.2658, "title": "Chevalley's theorem", "summary": "In this file we provide the usual (algebraic) version of Chevalley's theorem. For the proof see `Mathlib/RingTheory/Spectrum/Prime/ChevalleyComplexity.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Spectrum/Prime/Chevalley.html"}, {"id": "Mathlib.Algebra.Homology.DerivedCategory.Plus", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -68.511, "z": -5.285, "size": 0.2324, "title": "The bounded below derived category", "summary": "Let `C` be an abelian category. In this file, we show that the bounded below derived category `DerivedCategory.Plus C` (defined as a full subcategory of `DerivedCategory C`) is the localization of the bounded below homotopy category `HomotopyCategory.Plus C` with respect to quasi-isomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/DerivedCategory/Plus.html"}, {"id": "Mathlib.Algebra.Homology.HomotopyCategory.Plus", "region_id": "algebra", "micro_elevation": 0.3026, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -5.341, "z": -42.38, "size": 0.233, "title": "The triangulated subcategory of bounded below cochain complexes up to homotopy", "summary": "In this file, we introduce the triangulated full subcategory `HomotopyCategory.Plus C` of `HomotopyCategory C (.up ℤ)` consisting of bounded below cochain complexes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/HomotopyCategory/Plus.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Defs", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 4, "macro_tier_score": 0.3542, "macro_tier_override": null, "x": 45.647, "z": -8.486, "size": 0.4649, "title": "Tensor product of modules over commutative semirings", "summary": "This file constructs the tensor product of modules over commutative semirings. Given a semiring `R` and modules over it `M` and `N`, the standard construction of the tensor product is `TensorProduct R M N`. It is also a module over `R`. It comes with a canonical bilinear map `TensorProduct.mk R M N : M →ₗ[R] N →ₗ[R] TensorProduct R M N`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Defs.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Descent", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 98.142, "z": 28.31, "size": 0.2, "title": "Faithfully flat descent for modules", "summary": "In this file we show that extension of scalars by a faithfully flat ring homomorphism is comonadic. Then the general theory of descent implies that the pseudofunctor to `Cat` given by extension of scalars has effective descent relative to faithfully flat maps (TODO).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Descent.html"}, {"id": "Mathlib.Algebra.Order.Group.PiLex", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 2.392, "z": -12.778, "size": 0.2735, "title": "Lexicographic product of algebraic order structures", "summary": "This file proves that the lexicographic order on pi types is compatible with the pointwise algebraic operations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/PiLex.html"}, {"id": "Mathlib.Algebra.FreeMonoid.FreeSemigroup", "region_id": "algebra", "micro_elevation": 0.0921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -12.978, "z": 0.755, "size": 0.2, "title": "Relation between the free semigroup and the free monoid", "summary": "We provide some constructions relating the free semigroup and the free monoid on the same type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/FreeMonoid/FreeSemigroup.html"}, {"id": "Mathlib.Algebra.Group.WithOne.Basic", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 1, "macro_tier_score": 0.0023, "macro_tier_override": null, "x": -8.518, "z": 3.696, "size": 0.2385, "title": "More operations on `WithOne` and `WithZero`", "summary": "This file defines various bundled morphisms on `WithOne` and `WithZero` that were not available in `Algebra/Group/WithOne/Defs`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/WithOne/Basic.html"}, {"id": "Mathlib.Algebra.GroupWithZero.Submonoid.CancelMulZero", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -0.159, "z": -11.142, "size": 0.2, "title": "Submagmas with zero inherit cancellations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/GroupWithZero/Submonoid/CancelMulZero.html"}, {"id": "Mathlib.RingTheory.Perfectoid.BDeRham", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -67.141, "z": 102.549, "size": 0.2, "title": "The de Rham Period Ring $\\mathbb{B}_{dR}^+$ and $\\mathbb{B}_{dR}$", "summary": "In this file, we define the de Rham period ring $\\mathbb{B}_{dR}^+$ and the de Rham ring $\\mathbb{B}_{dR}$. We define a generalized version of these period rings following Scholze. When `R` is the ring of integers of `ℂₚ` (`PadicComplexInt`), they coincide with the classical de Rham period rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Perfectoid/BDeRham.html"}, {"id": "Mathlib.FieldTheory.Differential.Basic", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -107.321, "z": 9.208, "size": 0.2338, "title": "Differential Fields", "summary": "This file defines the logarithmic derivative `Differential.logDeriv` and proves properties of it. This is defined algebraically, compared to `logDeriv` which is analytical.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Differential/Basic.html"}, {"id": "Mathlib.GroupTheory.FreeGroup.GeneratorEquiv", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 68.541, "z": -4.892, "size": 0.2, "title": "Isomorphisms between free groups imply equivalences of their generators", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FreeGroup/GeneratorEquiv.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Catalan", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 68.63, "z": -3.408, "size": 0.2, "title": "Catalan Power Series", "summary": "We introduce the Catalan generating function as a formal power series over `ℕ`: `catalanSeries = ∑_{n ≥ 0} catalan n * X^n`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Catalan.html"}, {"id": "Mathlib.Algebra.Category.Grp.IsFinite", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -36.557, "z": 53.747, "size": 0.2, "title": "The Serre class of finite abelian groups", "summary": "In this file, we define `isFinite : ObjectProperty AddCommGrpCat` and show that it is a Serre class.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/IsFinite.html"}, {"id": "Mathlib.GroupTheory.RegularWreathProduct", "region_id": "algebra", "micro_elevation": 0.3553, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -50.053, "z": -3.006, "size": 0.2, "title": "Regular wreath product", "summary": "This file defines the regular wreath product of groups, and the canonical maps in and out of the product. The regular wreath product of `D` and `Q` is the product `(Q → D) × Q` with the group operation `⟨a₁, a₂⟩ * ⟨b₁, b₂⟩ = ⟨a₁ * (fun x ↦ b₁ (a₂⁻¹ * x)), a₂ * b₂⟩`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/RegularWreathProduct.html"}, {"id": "Mathlib.RingTheory.Nilpotent.GeometricallyReduced", "region_id": "algebra", "micro_elevation": 0.7632, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 64.013, "z": 86.631, "size": 0.2, "title": "Geometrically reduced algebras", "summary": "In this file we introduce geometrically reduced algebras. For a commutative ring `R` and an `R`-algebra `A`, we say that `A` is geometrically reduced (`IsGeometricallyReduced`) if for every prime ideal `p` of `R`, the base change of `A` to an algebraic closure of `κ(p)` is reduced. In the case of `R = k` a field, this is equivalent to `AlgebraicClosure k ⊗[k] A` being reduced.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Nilpotent/GeometricallyReduced.html"}, {"id": "Mathlib.Algebra.Polynomial.Degree.Monomial", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 3, "macro_tier_score": 0.2689, "macro_tier_override": null, "x": 38.973, "z": 37.171, "size": 0.3018, "title": "Degree of univariate monomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Degree/Monomial.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.OfBilinear", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -45.827, "z": 72.098, "size": 0.2, "title": "Root pairings made from bilinear forms", "summary": "A common construction of root systems is given by taking the set of all vectors in an integral lattice for which reflection yields an automorphism of the lattice. In this file, we generalize this construction, replacing the ring of integers with an arbitrary commutative ring and the integral lattice with an arbitrary reflexive module equipped with a bilinear form.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/OfBilinear.html"}, {"id": "Mathlib.RingTheory.Regular.ProjectiveDimension", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -85.955, "z": -15.188, "size": 0.2, "title": "ProjectiveDimension of quotient by regular element", "summary": "For `M` a finitely generated module over Noetherian local ring `R` and an `M`-regular element `x` contained in the unique maximal ideal of `R`, `projdim(M/xM) = projdim(M) + 1`. The analogous version for quotient regular sequence is also provided.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Regular/ProjectiveDimension.html"}, {"id": "Mathlib.LinearAlgebra.FreeModule.Finite.CardQuotient", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 2, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": -20.368, "z": 82.966, "size": 0.2715, "title": "Cardinal of quotient of free finite `ℤ`-modules by submodules of full rank", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/FreeModule/Finite/CardQuotient.html"}, {"id": "Mathlib.GroupTheory.MonoidLocalization.GrothendieckGroup", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -21.72, "z": -4.99, "size": 0.2545, "title": "Grothendieck group", "summary": "The Grothendieck group of a commutative monoid `M` is the \"smallest\" commutative group `G` containing `M`, in the sense that monoid homs `M → H` are in bijection with monoid homs `G → H` for any commutative group `H`. Note that \"Grothendieck group\" also refers to the analogous construction in an abelian category obtained by formally making the last term of each short exact sequence invertible.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/MonoidLocalization/GrothendieckGroup.html"}, {"id": "Mathlib.RingTheory.GradedAlgebra.TensorProduct", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -40.084, "z": -51.17, "size": 0.2, "title": "Tensor product of graded algebra", "summary": "In this file we show that if `𝒜` is a graded `R`-algebra, and `S` is any `R`-algebra, then `S ⊗[R] 𝒜` is a graded `S`-algebra with the grading `fun i ↦ (𝒜 i).baseChange S`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/GradedAlgebra/TensorProduct.html"}, {"id": "Mathlib.RingTheory.TensorProduct.IsBaseChangeRightExact", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 49.708, "z": 44.711, "size": 0.2, "title": "Lemmas about `IsBaseChange` under exact sequences", "summary": "In this file, we show that for an `R`-algebra `S` taking cokernels commutes with base change of modules from `R` to `S`. If `S` is a flat `R`-algebra, the same holds for kernels, see `Mathlib.RingTheory.Flat.IsBaseChange`. # Main Results For `S` an `R`-algebra, consider the following commutative diagram with exact rows, `M₁` `M₂` `M₃` `R`-modules, `N₁` `N₂` `N₃` `S`-modules, `R`-linear maps `f₁` `f₂` `i₁` `i₂` `i₃`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TensorProduct/IsBaseChangeRightExact.html"}, {"id": "Mathlib.Algebra.Category.Semigrp.Basic", "region_id": "algebra", "micro_elevation": 0.0658, "macro_tier": 0, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": -9.275, "z": 0.441, "size": 0.2239, "title": "Category instances for `Mul`, `Add`, `Semigroup` and `AddSemigroup`", "summary": "We introduce the bundled categories: * `MagmaCat` * `AddMagmaCat` * `Semigrp` * `AddSemigrp` along with the relevant forgetful functors between them. This closely follows `Mathlib/Algebra/Category/MonCat/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Semigrp/Basic.html"}, {"id": "Mathlib.RingTheory.LocalRing.Pullback", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -61.935, "z": 29.762, "size": 0.2, "title": "Local Ring Properties of Equalizers and Pullbacks", "summary": "In this file we provide basic lemmas for the equalizers the pullbacks and of ring homomorphisms and algebra homomorphisms. We show that they preserve the property of being a local ring under suitable conditions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/Pullback.html"}, {"id": "Mathlib.NumberTheory.FLT.Three", "region_id": "algebra", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -29.307, "z": 138.068, "size": 0.2, "title": "Fermat Last Theorem in the case `n = 3`", "summary": "The goal of this file is to prove Fermat's Last Theorem in the case `n = 3`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/FLT/Three.html"}, {"id": "Mathlib.NumberTheory.NumberField.Cyclotomic.PID", "region_id": "algebra", "micro_elevation": 0.9868, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 19.373, "z": -137.933, "size": 0.2338, "title": "Cyclotomic fields whose ring of integers is a PID.", "summary": "We prove that `ℤ [ζₚ]` is a PID for specific values of `p`. The result holds for `p ≤ 19`, but the proof is more and more involved.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Cyclotomic/PID.html"}, {"id": "Mathlib.Algebra.Ring.Divisibility.Lemmas", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 1, "macro_tier_score": 0.0037, "macro_tier_override": null, "x": -5.817, "z": 25.341, "size": 0.2771, "title": "Lemmas about divisibility in rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/Divisibility/Lemmas.html"}, {"id": "Mathlib.RingTheory.DedekindDomain.GaussLemma", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 101.873, "z": -20.929, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/DedekindDomain/GaussLemma.html"}, {"id": "Mathlib.Algebra.Category.MonCat.Adjunctions", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -17.938, "z": 4.81, "size": 0.2, "title": "Adjunctions regarding the category of monoids", "summary": "This file proves the adjunction between adjoining a unit to a semigroup and the forgetful functor from monoids to semigroups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/MonCat/Adjunctions.html"}, {"id": "Mathlib.Algebra.Star.TensorProduct", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 3, "macro_tier_score": 0.2539, "macro_tier_override": null, "x": 43.939, "z": -27.811, "size": 0.3448, "title": "The star structure on tensor products", "summary": "This file defines the `Star` structure on tensor products. This also defines the `StarAddMonoid` and `StarModule` instances for tensor products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/TensorProduct.html"}, {"id": "Mathlib.GroupTheory.Subsemigroup.Lemmas", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 17.019, "z": 14.388, "size": 0.2, "title": "Lemmas about subsemigroups", "summary": "This file collects various lemmas about subsemigroups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Subsemigroup/Lemmas.html"}, {"id": "Mathlib.Algebra.Lie.Weights.IsSimple", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -64.391, "z": -106.473, "size": 0.2, "title": "Lie ideals, invariant root submodules, and simple Lie algebras", "summary": "Given a semisimple Lie algebra, the lattice of ideals is order isomorphic to the lattice of Weyl-group-invariant submodules of the corresponding root system. In this file we provide `LieIdeal.toInvtRootSubmodule`, which constructs the invariant submodule associated to an ideal, and `LieAlgebra.IsKilling.invtSubmoduleToLieIdeal`, which constructs the ideal associated to an invariant submodule.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Weights/IsSimple.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Tannaka", "region_id": "algebra", "micro_elevation": 0.3289, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -10.961, "z": -45.117, "size": 0.2, "title": "Tannaka duality for rings", "summary": "A ring `R` is equivalent to the endomorphisms of the additive forgetful functor `Module R ⥤ AddCommGroup`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Tannaka.html"}, {"id": "Mathlib.RingTheory.FilteredAlgebra.Basic", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -15.64, "z": -13.143, "size": 0.2, "title": "The filtration on abelian groups and rings", "summary": "In this file, we define the concept of filtration for abelian groups, rings, and modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/FilteredAlgebra/Basic.html"}, {"id": "Mathlib.RingTheory.Smooth.Pi", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": 66.282, "z": -77.718, "size": 0.2468, "title": "Formal-smoothness of finite products of rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/Pi.html"}, {"id": "Mathlib.RingTheory.Artinian.Instances", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -59.539, "z": 47.466, "size": 0.2593, "title": "Instances related to Artinian rings", "summary": "We show that every reduced Artinian ring and the polynomial ring over it are decomposition monoids, and every reduced Artinian ring is semisimple.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Artinian/Instances.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.QuadraticModuleCat.Symmetric", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -63.687, "z": -70.107, "size": 0.2, "title": "The monoidal structure on `QuadraticModuleCat` is symmetric.", "summary": "In this file we show: * `QuadraticModuleCat.instSymmetricCategory : SymmetricCategory (QuadraticModuleCat.{u} R)`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat/Symmetric.html"}, {"id": "Mathlib.Algebra.LieRinehartAlgebra.Subalgebra", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -80.635, "z": 13.24, "size": 0.2, "title": "Lie-Rinehart subalgebras", "summary": "This file defines Lie-Rinehart subalgebras of a Lie-Rinehart algebra and provides basic related definitions and results.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/LieRinehartAlgebra/Subalgebra.html"}, {"id": "Mathlib.RingTheory.RingInvo", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -7.733, "z": -14.818, "size": 0.2, "title": "Ring involutions", "summary": "This file defines a ring involution as a structure extending `R ≃+* Rᵐᵒᵖ`, with the additional fact `f.involution : (f (f x).unop).unop = x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingInvo.html"}, {"id": "Mathlib.RingTheory.Congruence.Star", "region_id": "algebra", "micro_elevation": 0.3158, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 33.987, "z": -28.836, "size": 0.2, "title": "Helpers for working with star operators on quotients.", "summary": "TODO: consider defining `Star` versions of `RingCon`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Congruence/Star.html"}, {"id": "Mathlib.RingTheory.OrderOfVanishing.Noetherian", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 104.223, "z": -18.535, "size": 0.239, "title": "Order of vanishing in Noetherian rings.", "summary": "In this file we define various properties of the order of vanishing in Noetherian rings, including some API for computing the order of vanishing in discrete valuation rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/OrderOfVanishing/Noetherian.html"}, {"id": "Mathlib.FieldTheory.MvRatFunc.Rank", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 2, "macro_tier_score": 0.0067, "macro_tier_override": null, "x": -78.024, "z": -29.944, "size": 0.238, "title": "Rank of multivariate rational function field", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/MvRatFunc/Rank.html"}, {"id": "Mathlib.RingTheory.Algebraic.LinearIndependent", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 2, "macro_tier_score": 0.0078, "macro_tier_override": null, "x": -24.549, "z": 60.187, "size": 0.234, "title": "Linear independence of transcendental elements", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Algebraic/LinearIndependent.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Swap", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 1.362, "z": -1.263, "size": 0.2, "title": "Swap matrices", "summary": "A swap matrix indexed by `i` and `j` is the matrix that, when multiplying another matrix on the left (resp. on the right), swaps the `i`-th row with the `j`-th row (resp. the `i`-th column with the `j`-th column). Swap matrices are a special case of *elementary matrices*. For transvections see `Mathlib/LinearAlgebra/Matrix/Transvection.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Swap.html"}, {"id": "Mathlib.RingTheory.IntegralClosure.IsIntegral.AlmostIntegral", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": 5.274, "z": 87.127, "size": 0.2449, "title": "Almost integral elements", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/IntegralClosure/IsIntegral/AlmostIntegral.html"}, {"id": "Mathlib.NumberTheory.ModularForms.Bounds", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 31.858, "z": -65.047, "size": 0.2, "title": "Bounds for the norm of a modular form", "summary": "We prove bounds for the norm of a modular form `f τ` in terms of `im τ`, and deduce polynomial bounds for its q-expansion coefficients. The main results are * `ModularFormClass.exists_bound`: a modular form of weight `k` (for an arithmetic subgroup `Γ`) is bounded by a constant multiple of `max 1 (1 / (im τ) ^ k))`. * `CuspFormClass.exists_bound`: a cusp form of weight `k` (for an arithmetic subgroup `Γ`) is bounded…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/Bounds.html"}, {"id": "Mathlib.NumberTheory.ModularForms.Petersson", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 70.258, "z": -6.649, "size": 0.2478, "title": "The Petersson scalar product", "summary": "For `f, f'` functions `ℍ → ℂ`, we define `petersson k f f'` to be the function `τ ↦ conj (f τ) * f' τ * τ.im ^ k`. We show this function is (weight 0) invariant under `Γ` if `f, f'` are (weight `k`) invariant under `Γ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/Petersson.html"}, {"id": "Mathlib.Algebra.Category.Grp.Images", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 37.766, "z": 50.604, "size": 0.2, "title": "The category of commutative additive groups has images.", "summary": "Note that we don't need to register any of the constructions here as instances, because we get them from the fact that `AddCommGrpCat` is an abelian category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/Images.html"}, {"id": "Mathlib.NumberTheory.NumberField.Norm", "region_id": "algebra", "micro_elevation": 0.8816, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": 22.221, "z": -122.43, "size": 0.2734, "title": "Norm in number fields", "summary": "Given a finite extension of number fields, we define the norm morphism as a function between the rings of integers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/NumberField/Norm.html"}, {"id": "Mathlib.RingTheory.Noetherian.Orzech", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 4, "macro_tier_score": 0.2753, "macro_tier_override": null, "x": -10.434, "z": -62.276, "size": 0.3618, "title": "Noetherian rings have the Orzech property", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Noetherian/Orzech.html"}, {"id": "Mathlib.NumberTheory.FrobeniusNumber", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 51.826, "z": 50.598, "size": 0.2, "title": "Frobenius Number", "summary": "In this file we first define a predicate for Frobenius numbers, then solve the 2-variable variant of this problem. We also show the Frobenius number exists for any set of coprime natural numbers that doesn't contain 1. This is closely related to the fact that all ideals of ℕ are finitely generated, which we also prove in this file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/FrobeniusNumber.html"}, {"id": "Mathlib.RingTheory.Ideal.NatInt", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -23.53, "z": 66.534, "size": 0.2676, "title": "Prime ideals in ℕ and ℤ", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/NatInt.html"}, {"id": "Mathlib.Algebra.Homology.LeftResolution.Transport", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -16.699, "z": -0.71, "size": 0.239, "title": "Transport left resolutions along equivalences", "summary": "If `ι : C ⥤ A` is equipped with `Λ : LeftResolution ι` and `ι' : C' ⥤ A'` is a functor which corresponds to `ι` via equivalences of categories `A' ≌ A` and `C' ≌ C`, we define `Λ.transport .. : LeftResolution ι'`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/LeftResolution/Transport.html"}, {"id": "Mathlib.Algebra.Order.Field.Defs", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Field/Defs.html"}, {"id": "Mathlib.RingTheory.TwoSidedIdeal.BigOperators", "region_id": "algebra", "micro_elevation": 0.3421, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 15.759, "z": -45.642, "size": 0.2, "title": "Interactions between `∑, ∏` and two-sided ideals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/TwoSidedIdeal/BigOperators.html"}, {"id": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Defs", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": -50.761, "z": -66.39, "size": 0.2949, "title": "Eisenstein Series E2", "summary": "We define the Eisenstein series `E2` of weight `2` and level `1` as a limit of partial sums over non-symmetric intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/EisensteinSeries/E2/Defs.html"}, {"id": "Mathlib.Algebra.Group.UniqueProds.VectorSpace", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 30.998, "z": -46.295, "size": 0.2, "title": "A `ℚ`-vector space has `TwoUniqueSums`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/UniqueProds/VectorSpace.html"}, {"id": "Mathlib.GroupTheory.FreeGroup.Orbit", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -27.852, "z": 0.525, "size": 0.2, "title": null, "summary": "For any `w : α × Bool`, `FreeGroup.startsWith w` is the set of all elements of `FreeGroup α` that start with `w`. The main theorem `Orbit.duplicate` proves that applying `w⁻¹` to the orbit of `x` under the action of `FreeGroup.startsWith w` yields the orbit of `x` under the action of `FreeGroup.startsWith v` for every `v ≠ w⁻¹` (and the point `x`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FreeGroup/Orbit.html"}, {"id": "Mathlib.GroupTheory.GroupAction.SubMulAction.Combination", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 38.882, "z": -44.945, "size": 0.239, "title": "Combinations", "summary": "Combinations in a type are finite subsets of given cardinality. This file provides some API for handling them in the context of a group action. * `Set.powersetCard.subMulAction`: When a group `G` acts on `α`, the `SubMulAction` of `G` on `powersetCard α n`. This induces a `MulAction G (powersetCard α n)` instance. Then: * `Set.powerSetCard.mulActionHom_of_embedding`: the equivariant map from `Fin n ↪ α` to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/SubMulAction/Combination.html"}, {"id": "Mathlib.GroupTheory.Coxeter.Length", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 25.629, "z": 53.619, "size": 0.239, "title": "The length function, reduced words, and descents", "summary": "Throughout this file, `B` is a type and `M : CoxeterMatrix B` is a Coxeter matrix. `cs : CoxeterSystem M W` is a Coxeter system; that is, `W` is a group, and `cs` holds the data of a group isomorphism `W ≃* M.group`, where `M.group` refers to the quotient of the free group on `B` by the Coxeter relations given by the matrix `M`. See `Mathlib/GroupTheory/Coxeter/Basic.lean` for more details. Given any element $w \\in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Coxeter/Length.html"}, {"id": "Mathlib.Algebra.QuadraticAlgebra.NormDeterminant", "region_id": "algebra", "micro_elevation": 0.6053, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 78.039, "z": 34.758, "size": 0.2, "title": "Quadratic Algebra", "summary": "We prove that the expression for the norm of an element in a quadratic algebra comes from looking at the endomorphism defined by left multiplication by that element and taking its determinant.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/QuadraticAlgebra/NormDeterminant.html"}, {"id": "Mathlib.Algebra.CharZero.AddMonoidHom", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0137, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2864, "title": "Transporting `CharZero` across injective `AddMonoidHom`s", "summary": "This file exists in order to avoid adding extra imports to other files in this subdirectory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/CharZero/AddMonoidHom.html"}, {"id": "Mathlib.Algebra.Azumaya.Basic", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -59.729, "z": 58.454, "size": 0.2, "title": "Basic properties of Azumaya algebras", "summary": "In this file we prove basic facts about Azumaya algebras such as `R` is an Azumaya algebra over itself where `R` is a commutative ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Azumaya/Basic.html"}, {"id": "Mathlib.Algebra.Regular.Prod", "region_id": "algebra", "micro_elevation": 0.1184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 11.009, "z": -12.577, "size": 0.2, "title": "Results about `IsRegular` and `Prod`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Regular/Prod.html"}, {"id": "Mathlib.LinearAlgebra.TensorPower.Symmetric", "region_id": "algebra", "micro_elevation": 0.3947, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -53.011, "z": -17.146, "size": 0.2, "title": "Symmetric tensor power of a semimodule over a commutative semiring", "summary": "We define the `ι`-indexed symmetric tensor power of `M` as the `PiTensorProduct` quotiented by the relation that the `tprod` of `ι` elements is equal to the `tprod` of the same elements permuted by a permutation of `ι`. We denote this space by `Sym[R] ι M`, and the canonical multilinear map from `ι → M` to `Sym[R] ι M` by `⨂ₛ[R] i, f i`. We also reserve the notation `Sym[R]^n M` for the `n`th symmetric tensor power…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorPower/Symmetric.html"}, {"id": "Mathlib.Algebra.PresentedMonoid.Basic", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 15.811, "z": -9.743, "size": 0.2, "title": "Defining a monoid given by generators and relations", "summary": "Given relations `rels` on the free monoid on a type `α`, this file constructs the monoid given by generators `x : α` and relations `rels`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/PresentedMonoid/Basic.html"}, {"id": "Mathlib.GroupTheory.FreeGroup.NielsenSchreier", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 16.035, "z": 22.78, "size": 0.2, "title": "The Nielsen-Schreier theorem", "summary": "This file proves that a subgroup of a free group is itself free.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FreeGroup/NielsenSchreier.html"}, {"id": "Mathlib.Algebra.Module.LocalizedModule.IsLocalization", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 3, "macro_tier_score": 0.1899, "macro_tier_override": null, "x": -30.235, "z": 44.57, "size": 0.278, "title": "Equivalence between `IsLocalizedModule` and `IsLocalization`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LocalizedModule/IsLocalization.html"}, {"id": "Mathlib.LinearAlgebra.ConvexSpace", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/ConvexSpace.html"}, {"id": "Mathlib.NumberTheory.Height.Projectivization", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -58.041, "z": 52.11, "size": 0.2, "title": "Heights of points in projective space", "summary": "We define the multiplicative (`Projectivization.mulHeight`) and the logarithmic (`Projectivization.logHeight`) height of a point in a (finite-dimensional) projective space over a field that has a `Height.AdmissibleAbsValues` instance. The height is defined to be the height of any representative tuple; it does not depend on which representative is chosen.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Height/Projectivization.html"}, {"id": "Mathlib.Algebra.Lie.EngelSubalgebra", "region_id": "algebra", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 39.043, "z": 98.395, "size": 0.2465, "title": "Engel subalgebras", "summary": "This file defines Engel subalgebras of a Lie algebra and provides basic related properties. The Engel subalgebra `LieSubalgebra.Engel R x` consists of all `y : L` such that `(ad R L x)^n` kills `y` for some `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/EngelSubalgebra.html"}, {"id": "Mathlib.LinearAlgebra.QuadraticForm.Signature", "region_id": "algebra", "micro_elevation": 0.6579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -83.837, "z": 39.924, "size": 0.2, "title": "Signature of a quadratic form", "summary": "We define the signature of a quadratic form over a linearly ordered field, and show that it can be computed from any sum-of-squares representation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/QuadraticForm/Signature.html"}, {"id": "Mathlib.FieldTheory.Differential.Liouville", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -106.864, "z": -42.875, "size": 0.2, "title": "Liouville's theorem", "summary": "A proof of Liouville's theorem. Follows [Rosenlicht, M. Integration in finite terms][Rosenlicht_1972].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Differential/Liouville.html"}, {"id": "Mathlib.Algebra.Module.PUnit", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 4, "macro_tier_score": 0.3901, "macro_tier_override": null, "x": -5.12, "z": 21.69, "size": 0.3322, "title": "Instances on PUnit", "summary": "This file collects facts about module structures on the one-element type", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/PUnit.html"}, {"id": "Mathlib.Algebra.Homology.Embedding.ExtendHomotopy", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -14.82, "z": -19.059, "size": 0.2632, "title": "The extension functor on the homotopy categories", "summary": "Given an embedding of complex shapes `e : c.Embedding c'` and a preadditive category `C`, we define a fully faithful functor `e.extendHomotopyFunctor C : HomotopyCategory C c ⥤ HomotopyCategory C c'`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/Embedding/ExtendHomotopy.html"}, {"id": "Mathlib.LinearAlgebra.CliffordAlgebra.Equivs", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 80.766, "z": 56.26, "size": 0.2, "title": "Other constructions isomorphic to Clifford Algebras", "summary": "This file contains isomorphisms showing that other types are equivalent to some `CliffordAlgebra`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.html"}, {"id": "Mathlib.RingTheory.UniqueFactorizationDomain.Finite", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 3, "macro_tier_score": 0.1842, "macro_tier_override": null, "x": -14.324, "z": -28.135, "size": 0.2499, "title": "Finiteness of divisors", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/UniqueFactorizationDomain/Finite.html"}, {"id": "Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem", "region_id": "algebra", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 58.662, "z": 39.233, "size": 0.2, "title": "The Fundamental Theorem of Symmetric Polynomials", "summary": "In a polynomial ring in `n` variables over a commutative ring, the subalgebra of symmetric polynomials is freely generated by the first `n` elementary symmetric polynomials (excluding the 0th, which is simply 1). This is expressed as an isomorphism `MvPolynomial.esymmAlgEquiv` between `MvPolynomial (Fin n) R` and the symmetric subalgebra of any polynomial ring `MvPolynomial σ R` with `#σ = n`. The forward map is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/MvPolynomial/Symmetric/FundamentalTheorem.html"}, {"id": "Mathlib.LinearAlgebra.ConvexSpace.AffineSpace", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -8.867, "z": -56.885, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/ConvexSpace/AffineSpace.html"}, {"id": "Mathlib.Algebra.Order.WithTop.Untop0", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": 11.489, "z": 21.234, "size": 0.3084, "title": "Conversion from WithTop to Base Type", "summary": "For types α that are instances of `Zero`, we provide a convenient conversion, `WithTop.untop₀`, that maps elements `a : WithTop α` to `α`, by mapping `⊤` to zero. For settings where `α` has additional structure, we provide a large number of simplifier lemmas, akin to those that already exists for `ENat.toNat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/WithTop/Untop0.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Evaluation", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 2, "macro_tier_score": 0.0113, "macro_tier_override": null, "x": -24.235, "z": 68.254, "size": 0.261, "title": "Evaluation of power series", "summary": "Power series in one indeterminate are the particular case of multivariate power series, for the `Unit` type of indeterminates. This file provides a simpler syntax. Let `R`, `S` be types, with `CommRing R`, `CommRing S`. One assumes that `IsTopologicalRing R` and `IsUniformAddGroup R`, and that `S` is a complete and separated topological `R`-algebra, with `IsLinearTopology S S`, which means there is a basis of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Evaluation.html"}, {"id": "Mathlib.GroupTheory.Congruence.Star", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 22.16, "z": 2.363, "size": 0.2, "title": "Helpers for working with star operators on quotients.", "summary": "TODO: consider defining `Star` versions of `Con` and `AddCon`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Congruence/Star.html"}, {"id": "Mathlib.RingTheory.ClassGroup.ExtendedHom", "region_id": "algebra", "micro_elevation": 0.7368, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -75.922, "z": 71.078, "size": 0.2, "title": "Class group map induced by an extension of domains", "summary": "For an injective extension `A → B` of commutative domains (equivalently `Module.IsTorsionFree A B`), we construct the group homomorphism `ClassGroup.extendedHom : ClassGroup A →* ClassGroup B` given by pushing fractional ideals forward along the algebra map.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/ClassGroup/ExtendedHom.html"}, {"id": "Mathlib.LinearAlgebra.PiTensorProduct.DirectSum", "region_id": "algebra", "micro_elevation": 0.4079, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 35.276, "z": -45.499, "size": 0.2, "title": "Tensor products of direct sums", "summary": "This file shows that taking `PiTensorProduct`s commutes with taking `DirectSum`s in all arguments.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/PiTensorProduct/DirectSum.html"}, {"id": "Mathlib.RingTheory.SimpleRing.DivisionRing", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -67.86, "z": 45.523, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/SimpleRing/DivisionRing.html"}, {"id": "Mathlib.Algebra.Vertex.VertexOperator", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -26.312, "z": -55.351, "size": 0.2, "title": "Vertex operators", "summary": "In this file we introduce vertex operators as linear maps to Laurent series.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Vertex/VertexOperator.html"}, {"id": "Mathlib.Algebra.Vertex.HVertexOperator", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -20.999, "z": -55.596, "size": 0.2478, "title": "Vertex operators", "summary": "In this file we introduce heterogeneous vertex operators using Hahn series. When `R = ℂ`, `V = W`, and `Γ = ℤ`, then this is the usual notion of \"meromorphic left-moving 2D field\". The notion we use here allows us to consider composites and scalar-multiply by multivariable Laurent series.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Vertex/HVertexOperator.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.WithConv", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 11.844, "z": 82.729, "size": 0.2, "title": "The convolutive star ring on matrices", "summary": "In this file, we provide the star algebra instance on `WithConv (Matrix m n R)` given by the Hadamard product and intrinsic star (i.e., the star of each element in the matrix).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/WithConv.html"}, {"id": "Mathlib.Algebra.Order.PUnit", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.698, "z": 18.494, "size": 0.2, "title": "Instances on PUnit", "summary": "This file collects facts about ordered algebraic structures on the one-element type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/PUnit.html"}, {"id": "Mathlib.Algebra.Group.Action.Equidecomp", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 21.454, "z": -6.033, "size": 0.2, "title": "Equidecompositions", "summary": "This file develops the basic theory of equidecompositions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Action/Equidecomp.html"}, {"id": "Mathlib.RingTheory.Artinian.Algebra", "region_id": "algebra", "micro_elevation": 0.5395, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -37.44, "z": -66.303, "size": 0.2, "title": "Algebras over Artinian rings", "summary": "In this file we collect results about algebras over Artinian rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Artinian/Algebra.html"}, {"id": "Mathlib.GroupTheory.SpecificGroups.ZGroup", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -17.277, "z": -100.672, "size": 0.2, "title": "Z-Groups", "summary": "A Z-group is a group whose Sylow subgroups are all cyclic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/SpecificGroups/ZGroup.html"}, {"id": "Mathlib.GroupTheory.Coxeter.Inversion", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 37.659, "z": -48.351, "size": 0.2, "title": "Reflections, inversions, and inversion sequences", "summary": "Throughout this file, `B` is a type and `M : CoxeterMatrix B` is a Coxeter matrix. `cs : CoxeterSystem M W` is a Coxeter system; that is, `W` is a group, and `cs` holds the data of a group isomorphism `W ≃* M.group`, where `M.group` refers to the quotient of the free group on `B` by the Coxeter relations given by the matrix `M`. See `Mathlib/GroupTheory/Coxeter/Basic.lean` for more details. We define a *reflection*…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/Coxeter/Inversion.html"}, {"id": "Mathlib.NumberTheory.LSeries.ZetaZeros", "region_id": "algebra", "micro_elevation": 0.7895, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -43.261, "z": -102.689, "size": 0.2, "title": "Discreteness of the zeros of the Riemann zeta function", "summary": "We show that the zeros of the Riemann zeta function form a discrete subset of `ℂ`, so that in particular any compact subset of `ℂ` contains only finitely many zeros.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/LSeries/ZetaZeros.html"}, {"id": "Mathlib.Algebra.AddConstMap.Equiv", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 25.612, "z": 4.475, "size": 0.2, "title": "Equivalences conjugating `(· + a)` to `(· + b)`", "summary": "In this file we define `AddConstEquiv G H a b` (notation: `G ≃+c[a, b] H`) to be the type of equivalences such that `∀ x, f (x + a) = f x + b`. We also define the corresponding typeclass and prove some basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/AddConstMap/Equiv.html"}, {"id": "Mathlib.Algebra.Algebra.Subalgebra.Centralizer", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 59.753, "z": 25.586, "size": 0.2, "title": "Properties of centers and centralizers", "summary": "This file contains theorems about the center and centralizer of a subalgebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Subalgebra/Centralizer.html"}, {"id": "Mathlib.Algebra.AlgebraicCard", "region_id": "algebra", "micro_elevation": 0.5789, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 77.178, "z": 26.849, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/AlgebraicCard.html"}, {"id": "Mathlib.Algebra.Category.AlgCat.Symmetric", "region_id": "algebra", "micro_elevation": 0.4737, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -6.405, "z": -66.55, "size": 0.2, "title": "The monoidal structure on `AlgCat` is symmetric.", "summary": "In this file we show: * `AlgCat.instSymmetricCategory : SymmetricCategory (AlgCat.{u} R)`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/AlgCat/Symmetric.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Differentials.Presheaf", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 74.359, "z": 45.713, "size": 0.2, "title": "The presheaf of differentials of a presheaf of modules", "summary": "In this file, we define the type `M.Derivation φ` of derivations with values in a presheaf of `R`-modules `M` relative to a morphism of `φ : S ⟶ F.op ⋙ R` of presheaves of commutative rings over categories `C` and `D` that are related by a functor `F : C ⥤ D`. We formalize the notion of universal derivation. Geometrically, if `f : X ⟶ S` is a morphism of schemes (or more generally a morphism of commutative ringed…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Differentials/Presheaf.html"}, {"id": "Mathlib.Algebra.Category.ModuleCat.Ulift", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -21.974, "z": -65.107, "size": 0.2, "title": "Ulift functor for ModuleCat", "summary": "In this file, we define the obvious functor `ModuleCat.{v} R ⥤ ModuleCat.{max v v'} R` and prove it is exact, fully faithful and preserves projective and injective objects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/ModuleCat/Ulift.html"}, {"id": "Mathlib.Algebra.Field.Shrink", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -2.471, "z": -25.883, "size": 0.2, "title": "Transfer field structures from `α` to `Shrink α`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Field/Shrink.html"}, {"id": "Mathlib.Algebra.Homology.LeftResolution.Reduced", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -4.632, "z": -17.985, "size": 0.2, "title": "Left resolutions which preserve the zero object", "summary": "The structure `LeftResolution` allows to define a functorial resolution of an object (see `LeftResolution.chainComplexFunctor` in the file `Mathlib/Algebra/Homology/LeftResolution/Basic.lean`). In order to extend this resolution to complexes, we not only need the functoriality but also that zero morphisms are sent to zero. In this file, given `ι : C ⥤ A`, we extend `Λ : LeftResolution ι` to idempotent completions as…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/LeftResolution/Reduced.html"}, {"id": "Mathlib.Algebra.Lie.Character", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -86.211, "z": -13.661, "size": 0.2, "title": "Characters of Lie algebras", "summary": "A character of a Lie algebra `L` over a commutative ring `R` is a morphism of Lie algebras `L → R`, where `R` is regarded as a Lie algebra over itself via the ring commutator. For an Abelian Lie algebra (e.g., a Cartan subalgebra of a semisimple Lie algebra) a character is just a linear form.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Character.html"}, {"id": "Mathlib.Algebra.Lie.Classical", "region_id": "algebra", "micro_elevation": 0.6184, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -43.591, "z": 75.623, "size": 0.2, "title": "Classical Lie algebras", "summary": "This file is the place to find definitions and basic properties of the classical Lie algebras: * Aₗ = sl(l+1) * Bₗ ≃ so(l+1, l) ≃ so(2l+1) * Cₗ = sp(l) * Dₗ ≃ so(l, l) ≃ so(2l)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Classical.html"}, {"id": "Mathlib.Algebra.Lie.Graded", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -39.106, "z": 67.49, "size": 0.2, "title": "Graded Lie algebras", "summary": "This file defines typeclasses `SetLike.GradedBracket` and `GradedLieAlgebra`, for working with Lie algebras that are graded by a collection `ℒ` of submodules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Lie/Graded.html"}, {"id": "Mathlib.Algebra.Module.LinearMap.FiniteRange", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -64.865, "z": -4.194, "size": 0.2, "title": "`HasFiniteRange` predicate on linear maps, and the associated equivalence relation", "summary": "In this file, we define: * `LinearMap.HasFiniteRange`: a predicate expressing that a linear map has finitely generated range. * `LinearMap.HasNoetherianRange`: a predicate expressing that a linear map has noetherian range, i.e, all submodules of the range are finitely generated. This should be thought of as the \"better behaved\" version of `LinearMap.HasFiniteRange`: for example, `HasNoetherianRange` is always stable…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/LinearMap/FiniteRange.html"}, {"id": "Mathlib.Algebra.Order.Group.PartialSups", "region_id": "algebra", "micro_elevation": 0.1316, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -14.279, "z": -11.876, "size": 0.2, "title": "Results about `partialSups` of functions taking values in a `Group`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/PartialSups.html"}, {"id": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 2.231, "z": 20.307, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/GroupWithZero/Unbundled/OrderIso.html"}, {"id": "Mathlib.Algebra.Order.Monoid.Associated", "region_id": "algebra", "micro_elevation": 0.1579, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 3.787, "z": 21.962, "size": 0.2, "title": "Order on associates", "summary": "This file shows that divisibility makes associates into a canonically ordered monoid.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Monoid/Associated.html"}, {"id": "Mathlib.Algebra.Order.Star.Pi", "region_id": "algebra", "micro_elevation": 0.2105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -8.018, "z": 28.612, "size": 0.2, "title": "Pi-types of star-ordered rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Star/Pi.html"}, {"id": "Mathlib.Algebra.Polynomial.Degree.Definitions", "region_id": "algebra", "micro_elevation": 0.3816, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 18.615, "z": -50.539, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Polynomial/Degree/Definitions.html"}, {"id": "Mathlib.Algebra.Ring.IsFormallyReal", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -13.58, "z": 32.568, "size": 0.2, "title": "Formally real rings", "summary": "A ring `R` is *formally real* if, whenever `∑ i, x i ^ 2 = 0`, in fact `x i = 0` for all `i`. We define formally real rings in an index-free manner using the inductive predicate `IsSumNonzeroSq`, which asserts that an element is a finite sum of squares of nonzero elements. A ring is then formally real if `¬ IsSumNonzeroSq 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Ring/IsFormallyReal.html"}, {"id": "Mathlib.Algebra.Star.Free", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 42.516, "z": -41.524, "size": 0.2, "title": "A \\*-algebra structure on the free algebra.", "summary": "Reversing words gives a \\*-structure on the free monoid or on the free algebra on a type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/Free.html"}, {"id": "Mathlib.Algebra.TrivSqZeroExt.Ideal", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 4.309, "z": -59.273, "size": 0.2, "title": "The square zero ideal of the trivial square-zero extension", "summary": "- `TrivSqZeroExt.kerIdeal`: the ideal in the trivial square-zero extension - `TrivSqZeroExt.kerIdeal_sq `: this ideal has square zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/TrivSqZeroExt/Ideal.html"}, {"id": "Mathlib.GroupTheory.FinitelyPresentedGroup", "region_id": "algebra", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 22.445, "z": -27.227, "size": 0.2, "title": "Finitely Presented Groups", "summary": "This file defines finitely presented groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FinitelyPresentedGroup.html"}, {"id": "Mathlib.GroupTheory.GroupAction.Support", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -20.402, "z": 1.041, "size": 0.2, "title": "Support of an element under an action", "summary": "Given an action of a group `G` on a type `α`, we say that a set `s : Set α` supports an element `a : α` if, for all `g` that fix `s` pointwise, `g` fixes `a`. This is crucial in Fourier-Motzkin constructions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/Support.html"}, {"id": "Mathlib.GroupTheory.HNNExtension", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -34.403, "z": 18.37, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/HNNExtension.html"}, {"id": "Mathlib.GroupTheory.IndexNSmul", "region_id": "algebra", "micro_elevation": 0.5658, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -42.184, "z": 67.807, "size": 0.2, "title": "Lemmas about index and multiplication-by-n", "summary": "In this file we collect some results involving the multiplication-by-`n` map `nsmulAddMonoidHom n` (for a natural number `n`) on a commutative additive group and the (relative) index of subgroups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/IndexNSmul.html"}, {"id": "Mathlib.GroupTheory.SpecificGroups.Alternating.Simple", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 47.621, "z": 38.578, "size": 0.2, "title": "The alternating group is simple", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/SpecificGroups/Alternating/Simple.html"}, {"id": "Mathlib.LinearAlgebra.AffineSpace.Matrix", "region_id": "algebra", "micro_elevation": 0.5526, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -6.395, "z": 77.738, "size": 0.2, "title": "Matrix results for barycentric co-ordinates", "summary": "Results about the matrix of barycentric co-ordinates for a family of points in an affine space, with respect to some affine basis.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/AffineSpace/Matrix.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.HadamardMatrix", "region_id": "algebra", "micro_elevation": 0.5263, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 34.456, "z": -65.812, "size": 0.2, "title": "Hadamard matrices", "summary": "This file defines `Matrix.IsHadamard`, a unified notion that specializes to the classical real Hadamard matrices over `ℝ`/`ℤ` (where `star` is trivial and entries are `±1`) and to the complex Hadamard matrices over `ℂ` (where entries have unit norm). Basic results: conjugate-transpose closure, the order identity `n = s * star s` from constant row or column sums, the Sylvester (Kronecker) construction, and the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/HadamardMatrix.html"}, {"id": "Mathlib.LinearAlgebra.Matrix.Irreducible.Defs", "region_id": "algebra", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2, "title": "Irreducibility and primitivity of nonnegative matrices", "summary": "This file develops a graph-theoretic interface for studying the properties of nonnegative square matrices. We associate a directed graph (quiver) with a matrix `A`, where an edge `i ⟶ j` exists if and only if the entry `A i j` is strictly positive. This allows translating algebraic properties of the matrix (like powers) into graph-theoretic properties of its quiver (like the existence of paths).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Matrix/Irreducible/Defs.html"}, {"id": "Mathlib.LinearAlgebra.Multilinear.Pi", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.872, "z": -51.993, "size": 0.2, "title": "Interactions between (dependent) functions and multilinear maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Multilinear/Pi.html"}, {"id": "Mathlib.LinearAlgebra.RootSystem.RootPairingCat", "region_id": "algebra", "micro_elevation": 0.8158, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 6.312, "z": 114.971, "size": 0.2, "title": "The category of root pairings", "summary": "This file defines the category of root pairings, following the definition of category of root data given in SGA III Exp. 21 Section 6.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/RootSystem/RootPairingCat.html"}, {"id": "Mathlib.LinearAlgebra.TensorProduct.Matrix", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 62.875, "z": 16.488, "size": 0.2, "title": "Connections between `TensorProduct` and `Matrix`", "summary": "This file contains results about the matrices corresponding to maps between tensor product types, where the correspondence is induced by `Basis.tensorProduct` Notably, `TensorProduct.toMatrix_map` shows that taking the tensor product of linear maps is equivalent to taking the Kronecker product of their matrix representations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/TensorProduct/Matrix.html"}, {"id": "Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter", "region_id": "algebra", "micro_elevation": 0.8289, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 38.617, "z": -110.445, "size": 0.2, "title": "The cyclotomic character", "summary": "Let `L` be an integral domain and let `n : ℕ+` be a positive integer. If `μₙ` is the group of `n`th roots of unity in `L` then any field automorphism `g` of `L` induces an automorphism of `μₙ` which, being a cyclic group, must be of the form `ζ ↦ ζ^j` for some integer `j = j(g)`, well-defined in `ZMod d`, with `d` the cardinality of `μₙ`. The function `j` is a group homomorphism `(L ≃+* L) →* ZMod d`. Future work:…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.html"}, {"id": "Mathlib.NumberTheory.JacobiSum.Basic", "region_id": "algebra", "micro_elevation": 0.9079, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -61.324, "z": 112.518, "size": 0.2, "title": "Jacobi Sums", "summary": "This file defines the *Jacobi sum* of two multiplicative characters `χ` and `ψ` on a finite commutative ring `R` with values in another commutative ring `R'`: `jacobiSum χ ψ = ∑ x : R, χ x * ψ (1 - x)` (see `jacobiSum`) and provides some basic results and API lemmas on Jacobi sums.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/JacobiSum/Basic.html"}, {"id": "Mathlib.NumberTheory.ModularForms.Derivative", "region_id": "algebra", "micro_elevation": 0.8421, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 51.319, "z": -107.209, "size": 0.2, "title": "Derivatives of modular forms", "summary": "This file defines normalized derivative $D = \\frac{1}{2\\pi i} \\frac{d}{dz}$ and (Ramanujan-)Serre derivative $\\partial_k := D - \\frac{k}{12} E_2$ of modular forms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/Derivative.html"}, {"id": "Mathlib.NumberTheory.ModularForms.ProperlyDiscontinuous", "region_id": "algebra", "micro_elevation": 0.0132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.767, "z": -0.573, "size": 0.2, "title": "Arithmetic subgroups act properly discontinuously", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/ModularForms/ProperlyDiscontinuous.html"}, {"id": "Mathlib.NumberTheory.PowModTotient", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -101.926, "z": -6.664, "size": 0.2, "title": "Modular exponentiation with the totient function", "summary": "This file contains lemmas about modular exponentiation. In particular, it contains lemmas showing that an exponent can be reduced modulo the totient function when the base is coprime to the modulus.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/PowModTotient.html"}, {"id": "Mathlib.NumberTheory.SumFourSquares", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -33.893, "z": 96.357, "size": 0.2, "title": "Lagrange's four square theorem", "summary": "The main result in this file is `sum_four_squares`, a proof that every natural number is the sum of four square numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/SumFourSquares.html"}, {"id": "Mathlib.NumberTheory.Wilson", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 98.798, "z": 25.928, "size": 0.2, "title": "Wilson's theorem.", "summary": "This file contains a proof of Wilson's theorem. The heavy lifting is mostly done by the previous `wilsons_lemma`, but here we also prove the other logical direction. This could be generalized to similar results about finite abelian groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/Wilson.html"}, {"id": "Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality", "region_id": "algebra", "micro_elevation": 0.6711, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 89.732, "z": -30.318, "size": 0.2, "title": "Functoriality of continuous cohomology", "summary": "Given topological groups `G` and `H`, a continuous group homomorphism `φ : H →ₜ* G`, a topological representation `X` of `G`, a topological representation `Y` of `H`, and a morphism of topological `H`-representations `f : res φ X ⟶ Y`, we construct a cochain map `homogeneousCochains X ⟶ homogeneousCochains Y` and hence maps on continuous cohomology `Hⁿ(G, X) ⟶ Hⁿ(H, Y)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RepresentationTheory/Homological/ContCohomology/Functoriality.html"}, {"id": "Mathlib.RingTheory.AdicCompletion.Noetherian", "region_id": "algebra", "micro_elevation": 0.7105, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -82.329, "z": -57.266, "size": 0.2, "title": "Hausdorff-ness for Noetherian rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/AdicCompletion/Noetherian.html"}, {"id": "Mathlib.RingTheory.Algebraic.Pi", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -42.648, "z": 46.564, "size": 0.2, "title": "Algebraic functions", "summary": "This file defines algebraic functions as the image of the `algebraMap R[X] (R → S)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Algebraic/Pi.html"}, {"id": "Mathlib.RingTheory.Ideal.IsAugmentation", "region_id": "algebra", "micro_elevation": 0.4211, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.428, "z": -59.412, "size": 0.2, "title": "Augmentation ideals", "summary": "* `Ideal.IsAugmentation` : An ideal `I` of an `A`-algebra `S` is an augmentation ideal if its underlying submodule is a complement of `1 : Submodule A S`. * `Ideal.isAugmentation_subalgebra_iff` : If `S` is a subalgebra of an `R`-algebra `A`, then an ideal `I`of `A` is an augmentation ideal for the `R`-algebra structure if and only if it is an augmentation ideal for the `S`-algebra structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/IsAugmentation.html"}, {"id": "Mathlib.RingTheory.LocalProperties.FinitePresentation", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -44.956, "z": -56.789, "size": 0.2, "title": "`Module.FinitePresentation` is a local property", "summary": "In this file, we prove that `Module.FinitePresentation` is a local property.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalProperties/FinitePresentation.html"}, {"id": "Mathlib.RingTheory.LocalRing.MaximalIdeal.Square", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 52.017, "z": 50.4, "size": 0.2, "title": "Lemmas about square of maximal ideal of local ring", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/MaximalIdeal/Square.html"}, {"id": "Mathlib.RingTheory.LocalRing.Subring", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 23.114, "z": 15.55, "size": 0.2, "title": "Subrings of local rings", "summary": "We prove basic properties of subrings of local rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/LocalRing/Subring.html"}, {"id": "Mathlib.RingTheory.Localization.Rat", "region_id": "algebra", "micro_elevation": 0.4605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -64.92, "z": 3.24, "size": 0.2, "title": "Ring-theoretic fractions in `ℚ`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Localization/Rat.html"}, {"id": "Mathlib.RingTheory.PowerSeries.Restricted", "region_id": "algebra", "micro_elevation": 0.4868, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -68.373, "z": -6.848, "size": 0.2, "title": "Restricted power series", "summary": "`IsRestricted` : We say a power series over a normed ring `R` is restricted for a parameter `c` if `‖coeff R i f‖ * c ^ i → 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/PowerSeries/Restricted.html"}, {"id": "Mathlib.RingTheory.Valuation.AlgebraInstances", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -69.447, "z": -69.753, "size": 0.2, "title": "Algebra instances", "summary": "This file contains several `Algebra` and `IsScalarTower` instances related to extensions of a field with a valuation, as well as their unit balls.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/AlgebraInstances.html"}, {"id": "Mathlib.RingTheory.Valuation.RamificationGroup", "region_id": "algebra", "micro_elevation": 0.6974, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -11.763, "z": 97.724, "size": 0.2, "title": "Ramification groups", "summary": "The decomposition subgroup and inertia subgroups. TODO: Define higher ramification groups in lower numbering", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Valuation/RamificationGroup.html"}, {"id": "Mathlib.Algebra.Category.Grp.FiniteGrp", "region_id": "algebra", "micro_elevation": 0.1447, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 19.157, "z": 7.095, "size": 0.2443, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Category/Grp/FiniteGrp.html"}, {"id": "Mathlib.Algebra.Star.BigOperators", "region_id": "algebra", "micro_elevation": 0.1842, "macro_tier": 3, "macro_tier_score": 0.2602, "macro_tier_override": null, "x": -22.855, "z": -12.396, "size": 0.317, "title": "Big-operators lemmas about `star` algebraic operations", "summary": "These results are kept separate from `Algebra.Star.Basic` to avoid it needing to import `Finset`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Star/BigOperators.html"}, {"id": "Mathlib.FieldTheory.PrimeField", "region_id": "algebra", "micro_elevation": 0.4342, "macro_tier": 1, "macro_tier_score": 0.0036, "macro_tier_override": null, "x": -21.766, "z": -57.291, "size": 0.2696, "title": "Prime fields", "summary": "A prime field is a field that does not contain any nontrivial subfield. Prime fields are `ℚ` in characteristic `0` and `ZMod p` in characteristic `p` with `p` a prime number. Any field `K` contains a unique prime field: it is the smallest field contained in `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/PrimeField.html"}, {"id": "Mathlib.Algebra.Homology.ModelCategory.Injective", "region_id": "algebra", "micro_elevation": 0.3684, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 10.575, "z": -50.914, "size": 0.2324, "title": "The model category structure on bounded below complexes", "summary": "Let `C` be an abelian category with enough injectives. In this file, we define a model category structure on the category `CochainComplex.Plus C` of bounded below cochain complexes in `C`. The cofibrations are monomorphisms, the weak equivalences are quasi-isomorphisms and the fibrations are those morphisms that are degreewise epimorphisms with an injective kernel. The `ModelCategory` instance is scoped in the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ModelCategory/Injective.html"}, {"id": "Mathlib.Algebra.Homology.ModelCategory.Lifting", "region_id": "algebra", "micro_elevation": 0.2763, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -20.579, "z": 33.129, "size": 0.2301, "title": "Lifting properties in cochain complexes", "summary": "Let `C` be an abelian category. Consider a commutative diagram in the category `CochainComplex C ℤ`. ``` t A ⟶ X i| |p v v B ⟶ Y b ``` Assume that there exists a degreewise lifting `B.X n ⟶ X.X n` for any `n : ℤ`, that `Q` is a cokernel of `i`, and `K` is a kernel of `p`. In this situation, we construct a cocycle in `Cocycle Q K 1` and show that there exists a lifting `B ⟶ X` if this cocycle is a coboundary.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Homology/ModelCategory/Lifting.html"}, {"id": "Mathlib.Algebra.AddConstMap.Basic", "region_id": "algebra", "micro_elevation": 0.1711, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -15.705, "z": 18.336, "size": 0.2676, "title": "Maps (semi)conjugating a shift to a shift", "summary": "Denote by $S^1$ the unit circle `UnitAddCircle`. A common way to study a self-map $f\\colon S^1\\to S^1$ of degree `1` is to lift it to a map $\\tilde f\\colon \\mathbb R\\to \\mathbb R$ such that $\\tilde f(x + 1) = \\tilde f(x)+1$ for all `x`. In this file we define a structure and a typeclass for bundled maps satisfying `f (x + a) = f x + b`. We use parameters `a` and `b` instead of `1` to accommodate for two use cases: -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/AddConstMap/Basic.html"}, {"id": "Mathlib.RingTheory.Smooth.AdicCompletion", "region_id": "algebra", "micro_elevation": 0.7237, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": 10.657, "z": 101.586, "size": 0.2538, "title": "Formally smooth algebras and adic completion", "summary": "Let `A` be a formally smooth `R`-algebra. Then any algebra map `A →ₐ[R] S ⧸ I` lifts to an algebra map `A →ₐ[R] S` if `S` is `I`-adically complete. This is used in the proof that a smooth algebra over a Noetherian ring is flat (see `Mathlib.RingTheory.Smooth.Flat`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Smooth/AdicCompletion.html"}, {"id": "Mathlib.Algebra.Order.Group.Action.Synonym", "region_id": "algebra", "micro_elevation": 0.0789, "macro_tier": 4, "macro_tier_score": 0.3195, "macro_tier_override": null, "x": 4.039, "z": 10.385, "size": 0.2645, "title": "Actions by and on order synonyms", "summary": "This PR transfers group action instances from a type `α` to `αᵒᵈ` and `Lex α`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Action/Synonym.html"}, {"id": "Mathlib.RingTheory.RingHom.EssFiniteType", "region_id": "algebra", "micro_elevation": 0.5921, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 17.935, "z": -81.625, "size": 0.2659, "title": "Meta properties of essentially of finite type ring homomorphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RingHom/EssFiniteType.html"}, {"id": "Mathlib.RingTheory.RamificationInertia.Ramification", "region_id": "algebra", "micro_elevation": 0.8684, "macro_tier": 2, "macro_tier_score": 0.0102, "macro_tier_override": null, "x": -121.792, "z": -13.816, "size": 0.2666, "title": "Ramification index", "summary": "Let `S/R` be an extension of rings, and let `q` be a prime ideal of `S` lying over a prime ideal `p` of `R`. Let `Sq` be the localization of `S` and `q`, and let `pSq` be the image of `p` in `Sq`. Then the ramification index of `q` over `R` is defined to be the length of the quotient `Sq/pSq` as an `Sq`-module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/RamificationInertia/Ramification.html"}, {"id": "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "region_id": "algebra", "micro_elevation": 0.2237, "macro_tier": 4, "macro_tier_score": 0.3443, "macro_tier_override": null, "x": 15.589, "z": 27.455, "size": 0.3064, "title": "Pointwise monoid structures on SubMulAction", "summary": "This file provides `SubMulAction.Monoid` and weaker typeclasses, which show that `SubMulAction`s inherit the same pointwise multiplications as sets. To match `Submodule.idemSemiring`, we do not put these in the `Pointwise` locale.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/GroupAction/SubMulAction/Pointwise.html"}, {"id": "Mathlib.LinearAlgebra.Dimension.ErdosKaplansky", "region_id": "algebra", "micro_elevation": 0.5132, "macro_tier": 3, "macro_tier_score": 0.1756, "macro_tier_override": null, "x": 55.762, "z": 46.223, "size": 0.2922, "title": "Erdős-Kaplansky theorem", "summary": "* `rank_dual_eq_card_dual_of_aleph0_le_rank`: The **Erdős-Kaplansky Theorem** which says that the dimension of an infinite-dimensional dual space over a division ring has dimension equal to its cardinality.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/LinearAlgebra/Dimension/ErdosKaplansky.html"}, {"id": "Mathlib.Algebra.Module.Equiv.Opposite", "region_id": "algebra", "micro_elevation": 0.1974, "macro_tier": 4, "macro_tier_score": 0.3898, "macro_tier_override": null, "x": -14.638, "z": 23.701, "size": 0.3109, "title": "Module operations on `Mᵐᵒᵖ`", "summary": "This file contains definitions that build on top of the group action definitions in `Mathlib/Algebra/GroupWithZero/Action/Opposite.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Module/Equiv/Opposite.html"}, {"id": "Mathlib.Algebra.Algebra.Spectrum.Pi", "region_id": "algebra", "micro_elevation": 0.4474, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 63.135, "z": -1.021, "size": 0.2655, "title": "Spectrum and quasispectrum of products", "summary": "This file contains results regarding the spectra and quasispectra of (indexed) products of elements of a (non-unital) ring. The main result is that the (quasi)spectrum of a product is the union of the (quasi)spectra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Algebra/Spectrum/Pi.html"}, {"id": "Mathlib.Tactic.Determinant.Bird.Cert", "region_id": "tactic", "micro_elevation": 0.9167, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -82.143, "z": -131.884, "size": 0.268, "title": "Certificate-chain evaluator for `BirdDet.birdDet`", "summary": "This file contains an evaulator that computes the ring tactic normal form of `Mathlib.LinearAlgebra.Matrix.Determinant.Bird.birdDet` via iteratively unfolding its definition, using the ring tactic for ring operations, and caching intermediate certificates. The structure `Cert rα` carries the proof certificate and the evaluator builds larger certificates as `birdDet` is unfolded. The entrypoint of the evaluator…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Determinant/Bird/Cert.html"}, {"id": "Mathlib.Tactic.Determinant.Bird.Meta", "region_id": "tactic", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.019, "macro_tier_override": null, "x": -93.29, "z": -130.276, "size": 0.292, "title": "Reification support for the determinant tactic", "summary": "This file contains the meta-level parser, `refiyBirdDet`, used by `normalizeBirdDet` to turn `BirdDet.birdDet` calls into the context used by the certificate-chain evaluator.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Determinant/Bird/Meta.html"}, {"id": "Mathlib.Tactic.FastInstance", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.002, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3917, "title": "The `fast_instance%` and `inferInstanceAs%` term elaborators", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FastInstance.html"}, {"id": "Mathlib.Tactic.ComputeAsymptotics.Multiseries.Basic", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -107.221, "z": -155.94, "size": 0.208, "title": "Basic constructions for multiseries", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Basic.html"}, {"id": "Mathlib.Tactic.ComputeAsymptotics.Multiseries.Basis", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0189, "macro_tier_override": null, "x": -109.223, "z": -176.267, "size": 0.2832, "title": "Well-formed bases", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Basis.html"}, {"id": "Mathlib.Tactic.FieldSimp", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0213, "macro_tier_override": null, "x": -132.108, "z": -175.733, "size": 0.4249, "title": "`field_simp` tactic", "summary": "Tactic to clear denominators in algebraic expressions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FieldSimp.html"}, {"id": "Mathlib.Tactic.Linarith", "region_id": "tactic", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -82.129, "z": -199.087, "size": 0.45, "title": null, "summary": "We register `linarith` with the `hint` tactic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith.html"}, {"id": "Mathlib.Tactic.Ring", "region_id": "tactic", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0633, "macro_tier_override": null, "x": -72.178, "z": -171.672, "size": 0.5743, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Ring.html"}, {"id": "Mathlib.Tactic.ReduceModChar.Ext", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0186, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.234, "title": "`@[reduce_mod_char]` attribute", "summary": "This file registers `@[reduce_mod_char]` as a `simp` attribute.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ReduceModChar/Ext.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Elementwise", "region_id": "tactic", "micro_elevation": 0.3333, "macro_tier": 0, "macro_tier_score": 0.0136, "macro_tier_override": null, "x": -98.019, "z": -155.804, "size": 0.6949, "title": "Tools to reformulate category-theoretic lemmas in concrete categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Elementwise.html"}, {"id": "Mathlib.Tactic.ToAdditive", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.0506, "macro_tier_override": null, "x": -120.351, "z": -172.213, "size": 0.6937, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ToAdditive.html"}, {"id": "Mathlib.Tactic.Explode", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -105.767, "z": -175.696, "size": 0.208, "title": "Explode command", "summary": "This file contains the main code behind the `#explode` command. If you have a theorem with a name `hi`, `#explode hi` will display a Fitch table.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Explode.html"}, {"id": "Mathlib.Tactic.Explode.Pretty", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -104.773, "z": -167.379, "size": 0.268, "title": "Explode command: pretty", "summary": "This file contains UI code to render the Fitch table.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Explode/Pretty.html"}, {"id": "Mathlib.Tactic.NormNum", "region_id": "tactic", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0651, "macro_tier_override": null, "x": -73.186, "z": -158.853, "size": 0.6141, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Coherence.Basic", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0197, "macro_tier_override": null, "x": -116.572, "z": -170.631, "size": 0.344, "title": "The Core function for `monoidal` and `bicategory` tactics", "summary": "This file provides the function `BicategoryLike.main` for proving equalities in monoidal categories and bicategories. Using `main`, we will define the following tactics: - `monoidal` at `Mathlib/Tactic/CategoryTheory/Monoidal/Basic.lean` - `bicategory` at `Mathlib/Tactic/CategoryTheory/Bicategory/Basic.lean` The `main` first normalizes the both sides using `eval`, then compares the corresponding components. It…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Coherence/Basic.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Coherence.Normalize", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0383, "macro_tier_override": null, "x": -110.685, "z": -164.476, "size": 0.3482, "title": "Normalization of 2-morphisms in bicategories", "summary": "This file provides a function that normalizes 2-morphisms in bicategories. The function also used to normalize morphisms in monoidal categories. This is used in the string diagram widget given in `Mathlib/Tactic/Widget/StringDiagram.lean`, as well as `monoidal` and `bicategory` tactics. We say that the 2-morphism `η` in a bicategory is in normal form if 1. `η` is of the form `α₀ ≫ η₀ ≫ α₁ ≫ η₁ ≫ ... αₘ ≫ ηₘ ≫ αₘ₊₁`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Coherence/Normalize.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Coherence.PureCoherence", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0207, "macro_tier_override": null, "x": -112.434, "z": -169.967, "size": 0.3968, "title": "Coherence tactic", "summary": "This file provides a meta framework for the coherence tactic, which solves goals of the form `η = θ`, where `η` and `θ` are 2-morphism in a bicategory or morphisms in a monoidal category made up only of associators, unitors, and identities. The function defined here is a meta reimplementation of the formalized coherence theorems provided in the following files: - Mathlib.CategoryTheory.Monoidal.Free.Coherence -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Coherence/PureCoherence.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Monoidal.Basic", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.002, "macro_tier_override": null, "x": -97.242, "z": -163.99, "size": 0.391, "title": "`monoidal` tactic", "summary": "This file provides `monoidal` tactic, which solves equations in a monoidal category, where the two sides only differ by replacing strings of monoidal structural morphisms (that is, associators, unitors, and identities) with different strings of structural morphisms with the same source and target. In other words, `monoidal` solves equalities where both sides have the same string diagrams. The core function for the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Monoidal/Basic.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Monoidal.Normalize", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0378, "macro_tier_override": null, "x": -100.734, "z": -169.462, "size": 0.3153, "title": "Normalization of morphisms in monoidal categories", "summary": "This file provides the implementation of the normalization given in `Mathlib/Tactic/CategoryTheory/Coherence/Normalize.lean`. See this file for more details.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Monoidal/Normalize.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0216, "macro_tier_override": null, "x": -104.929, "z": -160.899, "size": 0.4338, "title": "Coherence tactic for monoidal categories", "summary": "We provide a `monoidal_coherence` tactic, which proves that any two morphisms (with the same source and target) in a monoidal category which are built out of associators and unitors are equal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Monoidal/PureCoherence.html"}, {"id": "Mathlib.Tactic.Linter.DocString", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -111.753, "z": -175.725, "size": 0.7902, "title": "The \"DocString\" style linter", "summary": "The \"DocString\" linter validates style conventions regarding doc-string formatting.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/DocString.html"}, {"id": "Mathlib.Tactic.Linter.Header", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.5493, "macro_tier_override": null, "x": -106.413, "z": -164.781, "size": 2.7602, "title": "The \"header\" linter", "summary": "The \"header\" style linter checks that a file starts with ``` /- Copyright ... Apache ... Authors ...", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/Header.html"}, {"id": "Mathlib.Tactic.RewriteSearch", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "The `rw_search` tactic has been removed from Mathlib.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/RewriteSearch.html"}, {"id": "Mathlib.Tactic.Generalize", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "Backwards compatibility shim for `generalize`.", "summary": "In https://github.com/leanprover/lean4/pull/3575 the transparency setting for `generalize` was changed to `instances`. This file provides a shim for the old setting, so that users who haven't updated their code yet can still use `generalize` with the old setting. This file can be removed once all uses of the compatibility shim have been removed from Mathlib. (Possibly we will instead add a `transparency` argument to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Generalize.html"}, {"id": "Mathlib.Tactic.Subsingleton", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0219, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4461, "title": "`subsingleton` tactic", "summary": "The `subsingleton` tactic closes `Eq` or `HEq` goals using an argument that the types involved are subsingletons. To first approximation, it does `apply Subsingleton.elim` but it also will try `proof_irrel_heq`, and it is careful not to accidentally specialize `Sort _` to `Prop`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Subsingleton.html"}, {"id": "Mathlib.Tactic.FinCases", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0038, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4609, "title": "The `fin_cases` tactic.", "summary": "Given a hypothesis of the form `h : x ∈ (A : List α)`, `x ∈ (A : Finset α)`, or `x ∈ (A : Multiset α)`, or a hypothesis of the form `h : A`, where `[Fintype A]` is available, `fin_cases h` will repeatedly call `cases` to split the goal into separate cases for each possible value.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FinCases.html"}, {"id": "Mathlib.Tactic.Linter.TextBased.UnicodeLinter", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0189, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2832, "title": "Tools for the unicode Linter", "summary": "The actual linter is defined in `TextBased.lean`. This file defines the allowlist and other tools used by the linter. **When changing, make sure to stay in sync with [style guide](https://github.com/leanprover-community/leanprover-community.github.io/blob/lean4/templates/contribute/style.md#unicode-usage)**", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/TextBased/UnicodeLinter.html"}, {"id": "Mathlib.Tactic.IntervalCases", "region_id": "tactic", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": -77.972, "z": -141.249, "size": 0.3666, "title": "Case bash on variables in finite intervals", "summary": "This file provides the tactic `interval_cases`. `interval_cases n` will: 1. inspect hypotheses looking for lower and upper bounds of the form `a ≤ n` or `a < n` and `n < b` or `n ≤ b`, including the bound `0 ≤ n` for `n : ℕ` automatically. 2. call `fin_cases` on the synthesised hypothesis `n ∈ Set.Ico a b`, assuming an appropriate `Fintype` instance can be found for the type of `n`. Currently, `n` must be of type…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/IntervalCases.html"}, {"id": "Mathlib.Tactic.BDSimp", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2942, "title": "`bdsimp` tactic", "summary": "`bdsimp` is a backward compatibility macro for `dsimp` that allows dependent rewrites inside depended-on positions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/BDSimp.html"}, {"id": "Mathlib.Tactic.Spread", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0432, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.8661, "title": "Macro for spread syntax (`__ := instSomething`) in structures.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Spread.html"}, {"id": "Mathlib.Tactic.Monotonicity.Attr", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.031, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.6736, "title": "The @[mono] attribute", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Monotonicity/Attr.html"}, {"id": "Mathlib.Tactic.Basic", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.4657, "macro_tier_override": null, "x": -112.153, "z": -170.436, "size": 0.4223, "title": "Basic tactics and utilities for tactic writing", "summary": "This file defines some basic utilities for tactic writing, and also - a dummy `variables` macro (which warns that the Lean 4 name is `variable`) - the `introv` tactic, which allows the user to automatically introduce the variables of a theorem and explicitly name the non-dependent hypotheses, - an `assumption` macro, calling the `assumption` tactic on all goals - the tactics `match_target` and `clear_aux_decl`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Basic.html"}, {"id": "Mathlib.Tactic.PPWithUniv", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.5159, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.735, "title": "Attribute to pretty-print universe level parameters by default", "summary": "This module contains the `pp_with_univ` attribute, which enables pretty-printing of universe parameters for the associated declaration. This is helpful for definitions like `Ordinal`, where the universe levels are both relevant and not deducible from the arguments.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/PPWithUniv.html"}, {"id": "Mathlib.Tactic.ExtendDoc", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.5128, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.6798, "title": "`extend_doc` command", "summary": "In a file where declaration `decl` is defined, writing ```lean extend_doc decl before \"I will be added as a prefix to the docs of `decl`\" after \"I will be added as a suffix to the docs of `decl`\" ``` does what is probably clear: it extends the doc-string of `decl` by adding the string of `before` at the beginning and the string of `after` at the end. At least one of `before` and `after` must appear, but either one…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ExtendDoc.html"}, {"id": "Mathlib.Tactic.Common", "region_id": "tactic", "micro_elevation": 0.9167, "macro_tier": 1, "macro_tier_score": 0.0191, "macro_tier_override": null, "x": -150.77, "z": -151.964, "size": 0.787, "title": "Common tactics, linters, and utilities", "summary": "This file imports all tactics which do not have significant theory imports, and hence can be imported very low in the theory import hierarchy, thereby making tactics widely available without needing specific imports. We include some commented out imports here, with an explanation of their theory requirements, to save some time for anyone wondering why they are not here. We also import theory-free linters, commands,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Common.html"}, {"id": "Mathlib.Tactic.Finiteness.Attr", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0037, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4567, "title": "Finiteness tactic attribute", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Finiteness/Attr.html"}, {"id": "Mathlib.Tactic.SetLike", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0021, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3966, "title": "SetLike Rule Set", "summary": "This module defines the `SetLike` and `SetLike!` Aesop rule sets. Aesop rule sets only become visible once the file in which they're declared is imported, so we must put this declaration into its own file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/SetLike.html"}, {"id": "Mathlib.Tactic.NormNum.NatLog", "region_id": "tactic", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -73.02, "z": -187.883, "size": 0.208, "title": "`norm_num` extensions for `Nat.log` and `Nat.clog`", "summary": "This module defines `norm_num` extensions for `Nat.log` and `Nat.clog`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/NatLog.html"}, {"id": "Mathlib.Tactic.ContinuousFunctionalCalculus", "region_id": "tactic", "micro_elevation": 0.5833, "macro_tier": 0, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": -83.063, "z": -155.585, "size": 0.3615, "title": "Tactics for the continuous functional calculus", "summary": "At the moment, these tactics are just wrappers, but potentially they could be more sophisticated.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ContinuousFunctionalCalculus.html"}, {"id": "Mathlib.Tactic.Positivity.Core", "region_id": "tactic", "micro_elevation": 0.3333, "macro_tier": 1, "macro_tier_score": 0.0982, "macro_tier_override": null, "x": -92.498, "z": -169.421, "size": 0.5182, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Positivity/Core.html"}, {"id": "Mathlib.Tactic.Nontriviality.Core", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0253, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.549, "title": "The `nontriviality` tactic.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Nontriviality/Core.html"}, {"id": "Mathlib.Tactic.Push.Attr", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1918, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.8721, "title": "The `@[push]` attribute for the `push` and `pull` tactics", "summary": "This file defines the `@[push]` attribute, so that it can be used without importing the tactic itself.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Push/Attr.html"}, {"id": "Mathlib.Tactic.Ring.PNat", "region_id": "tactic", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.0775, "macro_tier_override": null, "x": -76.759, "z": -174.674, "size": 0.4485, "title": "Additional instances for `ring` over `PNat`", "summary": "This adds some instances which enable `ring` to work on `PNat` even though it is not a commutative semiring, by lifting to `Nat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Ring/PNat.html"}, {"id": "Mathlib.Tactic.Ring.RingNF", "region_id": "tactic", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.1149, "macro_tier_override": null, "x": -126.539, "z": -140.629, "size": 0.4608, "title": "`ring_nf` tactic", "summary": "A tactic which uses `ring` to rewrite expressions. This can be used non-terminally to normalize ring expressions in the goal such as `⊢ P (x + x + x)` ~> `⊢ P (x * 3)`, as well as being able to prove some equations that `ring` cannot because they involve ring reasoning inside a subterm, such as `sin (x + y) + sin (y + x) = 2 * sin (x + y)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Ring/RingNF.html"}, {"id": "Mathlib.Tactic.LinearCombination", "region_id": "tactic", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0032, "macro_tier_override": null, "x": -128.563, "z": -199.134, "size": 0.44, "title": "`linear_combination` Tactic", "summary": "In this file, the `linear_combination` tactic is created. This tactic, which works over `CommRing`s, attempts to simplify the target by creating a linear combination of a list of equalities and subtracting it from the target. A `Syntax.Tactic` object can also be passed into the tactic, allowing the user to specify a normalization tactic. Over ordered algebraic objects (such as `LinearOrderedCommRing`), taking linear…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/LinearCombination.html"}, {"id": "Mathlib.Tactic.ToFun", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0203, "macro_tier_override": null, "x": -111.778, "z": -160.489, "size": 0.3771, "title": "The `to_fun` attribute", "summary": "Adding `@[to_fun]` to a lemma named `foo` creates a new lemma named `fun_foo`, which is obtained by running `pull fun _ ↦ _` on the type of `F`. This can be useful for generating the applied form of a continuity lemma from the unapplied form.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ToFun.html"}, {"id": "Mathlib.Tactic.Push", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.1677, "macro_tier_override": null, "x": -111.27, "z": -164.847, "size": 0.7931, "title": "The `push` and `pull` tactics", "summary": "The `push` tactic pushes a given constant inside expressions: it can be applied to goals as well as local hypotheses and also works as a `conv` tactic. The `pull` tactic does the reverse: it pulls the given constant towards the head of the expression.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Push.html"}, {"id": "Mathlib.Tactic.Translate.Attributes", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1569, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.5972, "title": "Combining attributes that generate declarations with translation attributes", "summary": "This file defines extensible support for writing e.g. `to_additive (attr := simps)` such that all of the declarations generated by `simps` will pairwise be added to the `to_additive` dictionary.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Translate/Attributes.html"}, {"id": "Mathlib.Tactic.NormNum.RealSqrt", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "`norm_num` extension for `Real.sqrt`", "summary": "This module defines a `norm_num` extension for `Real.sqrt` and `NNReal.sqrt`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/RealSqrt.html"}, {"id": "Mathlib.Tactic.ExistsI", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "The `existsi` tactic", "summary": "This file defines the `existsi` tactic: its purpose is to instantiate existential quantifiers. Internally, it applies the `refine` tactic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ExistsI.html"}, {"id": "Mathlib.Tactic.GRewrite", "region_id": "tactic", "micro_elevation": 0.3333, "macro_tier": 1, "macro_tier_score": 0.0456, "macro_tier_override": null, "x": -124.003, "z": -162.089, "size": 0.5925, "title": "The generalized rewriting tactic", "summary": "The `grw`/`grewrite` tactic is a generalization of the `rewrite` tactic that works with relations other than equality. The core implementation of `grewrite` is in the file `Mathlib/Tactic/GRewrite/Core.lean`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/GRewrite.html"}, {"id": "Mathlib.Tactic.Linarith.Verification", "region_id": "tactic", "micro_elevation": 0.5833, "macro_tier": 1, "macro_tier_score": 0.0377, "macro_tier_override": null, "x": -137.158, "z": -171.917, "size": 0.3099, "title": "Deriving a proof of false", "summary": "`linarith` uses an untrusted oracle to produce a certificate of unsatisfiability. It needs to do some proof reconstruction work to turn this into a proof term. This file implements the reconstruction.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith/Verification.html"}, {"id": "Mathlib.Tactic.Linarith.Parsing", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0379, "macro_tier_override": null, "x": -108.208, "z": -143.581, "size": 0.3214, "title": "Parsing input expressions into linear form", "summary": "`linarith` computes the linear form of its input expressions, assuming (without justification) that the type of these expressions is a commutative semiring. It identifies atoms up to ring-equivalence: that is, `(y*3)*x` will be identified `3*(x*y)`, where the monomial `x*y` is the linear atom. * Variables are represented by natural numbers. * Monomials are represented by `Monom := TreeMap ℕ ℕ`. The monomial `1` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith/Parsing.html"}, {"id": "Mathlib.Tactic.Peel", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0019, "macro_tier_override": null, "x": -102.4, "z": -173.194, "size": 0.386, "title": "The `peel` tactic", "summary": "`peel h with h' idents*` tries to apply `forall_imp` (or `Exists.imp`, or `Filter.Eventually.mp`, `Filter.Frequently.mp` and `Filter.Eventually.of_forall`) with the argument `h` and uses `idents*` to introduce variables with the supplied names, giving the \"peeled\" argument the name `h'`. One can provide a numeric argument as in `peel 4 h` which will peel 4 quantifiers off the expressions automatically name any…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Peel.html"}, {"id": "Mathlib.Tactic.Abel", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 1, "macro_tier_score": 0.029, "macro_tier_override": null, "x": -116.892, "z": -149.333, "size": 0.6348, "title": "The `abel` tactic", "summary": "Evaluate expressions in the language of additive, commutative monoids and groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Abel.html"}, {"id": "Mathlib.Tactic.Monotonicity", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -116.855, "z": -166.723, "size": 0.245, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Monotonicity.html"}, {"id": "Mathlib.Tactic.Monotonicity.Basic", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0187, "macro_tier_override": null, "x": -104.925, "z": -166.79, "size": 0.256, "title": "Monotonicity tactic", "summary": "The tactic `mono` applies monotonicity rules (collected through the library by being tagged `@[mono]`). The version of the tactic here is a cheap partial port of the `mono` tactic from Lean 3, which had many more options and features. It is implemented as a wrapper on top of `solve_by_elim`. Temporary syntax change: Lean 3 `mono` applied a single monotonicity rule, then applied local hypotheses and the `rfl` tactic…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Monotonicity/Basic.html"}, {"id": "Mathlib.Tactic.Monotonicity.Lemmas", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0187, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.256, "title": "Lemmas for the `mono` tactic", "summary": "The `mono` tactic works by throwing all lemmas tagged with the attribute `@[mono]` at the goal. In this file we tag a few foundational lemmas with the mono attribute. Lemmas in more advanced files are tagged in place.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Monotonicity/Lemmas.html"}, {"id": "Mathlib.Tactic.CategoryTheory.CategoryStar", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.5045, "title": "Support for `Category* C`.", "summary": "In the category theory library, it is common to introduce a (universe polymorphic) general category as follows: ```lean universe v u variable (C : Type u) [Category.{v} C] ``` We tend to put the universe level `v` of the morphisms ahead of the level `u` for objects because it makes it easier to specify explicit universes when needed. The elaborator `Category*` provides analogous behavior to `Type*` for introducing…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/CategoryStar.html"}, {"id": "Mathlib.Tactic.Linter.DocPrime", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -116.967, "z": -167.809, "size": 0.7902, "title": "The \"docPrime\" linter", "summary": "The \"docPrime\" linter emits a warning on declarations that have no doc-string and whose name ends with a `'`. Such declarations are expected to have a documented explanation for the presence of a `'` in their name. This may consist of discussion of the difference relative to an unprimed version of that declaration, or an explanation as to why no better naming scheme is possible.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/DocPrime.html"}, {"id": "Mathlib.Tactic.ClickSuggestions.SectionState", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.038, "macro_tier_override": null, "x": -85.603, "z": -176.083, "size": 0.3344, "title": "Infrastructure for searching and displaying sets of lemmas", "summary": "This is used for `apply`, `apply at`, `rw` and `grw` suggestions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ClickSuggestions/SectionState.html"}, {"id": "Mathlib.Tactic.ClickSuggestions.Util", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 1, "macro_tier_score": 0.0382, "macro_tier_override": null, "x": -90.689, "z": -177.586, "size": 0.3467, "title": "Various utilities used in `#click_suggestions`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ClickSuggestions/Util.html"}, {"id": "Mathlib.Tactic.Order", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": -101.044, "z": -158.599, "size": 0.3791, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Order.html"}, {"id": "Mathlib.Tactic.DeriveFintype", "region_id": "tactic", "micro_elevation": 0.3333, "macro_tier": 0, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": -124.923, "z": -170.81, "size": 0.3209, "title": "The `Fintype` derive handler", "summary": "This file defines a derive handler to automatically generate `Fintype` instances for structures and inductive types. The following is a prototypical example of what this can handle: ``` inductive MyOption (α : Type*) | none | some (x : α) deriving Fintype ``` This deriving handler does not attempt to process inductive types that are either recursive or that have indices. To get debugging information, do `set_option…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/DeriveFintype.html"}, {"id": "Mathlib.Tactic.ApplyFun", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0074, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.5648, "title": "The `apply_fun` tactic.", "summary": "Apply a function to an equality or inequality in either a local hypothesis or the goal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ApplyFun.html"}, {"id": "Mathlib.Tactic.Contrapose", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0218, "macro_tier_override": null, "x": -101.243, "z": -171.232, "size": 0.4435, "title": "Contrapose", "summary": "The `contrapose` tactic transforms the goal into its contrapositive when that goal is an implication or an iff. It also avoids creating a double negation if there already is a negation. * `contrapose` turns a goal `P → Q` into `¬ Q → ¬ P` and a goal `P ↔ Q` into `¬ P ↔ ¬ Q` * `contrapose!` runs `contrapose` and then pushes negations inside `P` and `Q` using `push Not` * `contrapose h` first reverts the local…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Contrapose.html"}, {"id": "Mathlib.Tactic.Convert", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.025, "macro_tier_override": null, "x": -104.727, "z": -167.715, "size": 0.541, "title": "The `convert` tactic.", "summary": "The `exact e` and `refine e` tactics require a term `e` whose type is definitionally equal to the goal. `convert e` is similar to `refine e`, but the type of `e` is not required to exactly match the goal. Instead, new goals are created for differences between the type of `e` and the goal using the same strategies as the `congr!` tactic. For example, in the proof state ```lean n : ℕ, e : Prime (2 * n + 1) ⊢ Prime (n…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Convert.html"}, {"id": "Mathlib.Tactic.Nontriviality", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0076, "macro_tier_override": null, "x": -104.986, "z": -169.582, "size": 0.5703, "title": "The `nontriviality` tactic.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Nontriviality.html"}, {"id": "Mathlib.Tactic.Linter.MinImports", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.0371, "macro_tier_override": null, "x": -102.363, "z": -178.543, "size": 0.2451, "title": "The `minImports` linter", "summary": "The `minImports` linter incrementally computes the minimal imports needed for each file to build. Whenever it detects that a new command requires an increase in the (transitive) imports that it computed so far, it emits a warning mentioning the bigger minimal imports. Unlike the related `#min_imports` command, the linter takes into account notation and tactic information. It also works incrementally, accumulating…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/MinImports.html"}, {"id": "Mathlib.Tactic.MinImports", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0782, "macro_tier_override": null, "x": -115.881, "z": -172.184, "size": 0.4711, "title": "`#min_imports in` a command to find minimal imports", "summary": "`#min_imports in stx` scans the syntax `stx` to find a collection of minimal imports that should be sufficient for `stx` to make sense. If `stx` is a command, then it also elaborates `stx` and, in case it is a declaration, then it also finds the imports implied by the declaration. Unlike the related `#find_home`, this command takes into account notation and tactic information.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/MinImports.html"}, {"id": "Mathlib.Tactic.Group", "region_id": "tactic", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -107.527, "z": -208.963, "size": 0.3479, "title": "`group` tactic", "summary": "Normalizes expressions in the language of groups. The basic idea is to use the simplifier to put everything into a product of group powers (`zpow` which takes a group element and an integer), then simplify the exponents using the `ring_nf` tactic. The process needs to be repeated since `ring_nf` can normalize an exponent to zero, leading to a factor that can be removed before collecting exponents again. The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Group.html"}, {"id": "Mathlib.Tactic.NormNum.Inv", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 1, "macro_tier_score": 0.2452, "macro_tier_override": null, "x": -102.715, "z": -187.617, "size": 0.483, "title": "`norm_num` plugins for `Rat.cast` and `⁻¹`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/Inv.html"}, {"id": "Mathlib.Tactic.NormNum.Pow", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 1, "macro_tier_score": 0.2461, "macro_tier_override": null, "x": -112.042, "z": -147.921, "size": 0.5092, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/Pow.html"}, {"id": "Mathlib.Tactic.GCongr.CoreAttrs", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0373, "macro_tier_override": null, "x": -116.956, "z": -167.583, "size": 0.2638, "title": "gcongr attributes for lemmas up in the import chain", "summary": "In this file we add `gcongr` attribute to lemmas in `Lean.Init`. We may add lemmas from other files imported by `Mathlib/Tactic/GCongr/Core` later.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/GCongr/CoreAttrs.html"}, {"id": "Mathlib.Tactic.GCongr.Core", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0854, "macro_tier_override": null, "x": -107.682, "z": -164.169, "size": 0.6523, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/GCongr/Core.html"}, {"id": "Mathlib.Tactic.Linter.DeprecatedModule", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/DeprecatedModule.html"}, {"id": "Mathlib.Tactic.NoncommRing", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0021, "macro_tier_override": null, "x": -133.32, "z": -168.577, "size": 0.394, "title": "The `noncomm_ring` tactic", "summary": "Solve goals in not necessarily commutative rings. This tactic is rudimentary, but useful for solving simple goals in noncommutative rings. One glaring flaw is that numeric powers are unfolded entirely with `pow_succ` and can easily exceed the maximum recursion depth. `noncomm_ring` is just a `simp only [some lemmas]` followed by `abel`. It automatically uses `abel1` to close the goal, and if that doesn't succeed,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NoncommRing.html"}, {"id": "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0186, "macro_tier_override": null, "x": -111.734, "z": -170.946, "size": 0.2442, "title": "A `coherence` tactic for bicategories", "summary": "We provide a `bicategory_coherence` tactic, which proves that any two 2-morphisms (with the same source and target) in a bicategory which are built out of associators and unitors are equal. This file mainly deals with the type class setup for the coherence tactic. The actual front end tactic is given in `Mathlib/Tactic/CategoryTheory/Coherence.lean` at the same time as the coherence tactic for monoidal categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/BicategoryCoherence.html"}, {"id": "Mathlib.Tactic.CategoryTheory.BicategoricalComp", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0376, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3038, "title": "Bicategorical composition `⊗≫` (composition up to associators)", "summary": "We provide `f ⊗≫ g`, the `bicategoricalComp` operation, which automatically inserts associators and unitors as needed to make the target of `f` match the source of `g`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/BicategoricalComp.html"}, {"id": "Mathlib.Tactic.Positivity.Basic", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -99.881, "z": -149.709, "size": 0.3527, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Positivity/Basic.html"}, {"id": "Mathlib.Tactic.Module", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.001, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3325, "title": "A tactic for normalization over modules", "summary": "This file provides the two tactics `match_scalars` and `module`. Given a goal which is an equality in a type `M` (with `M` an `AddCommMonoid`), the `match_scalars` tactic parses the LHS and RHS of the goal as linear combinations of `M`-atoms over some semiring `R`, and reduces the goal to the respective equalities of the `R`-coefficients of each atom. The `module` tactic does this and then runs the `ring` tactic on…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Module.html"}, {"id": "Mathlib.Tactic.NormNum.Ineq", "region_id": "tactic", "micro_elevation": 0.5833, "macro_tier": 1, "macro_tier_score": 0.1142, "macro_tier_override": null, "x": -91.233, "z": -190.695, "size": 0.4375, "title": "`norm_num` extensions for inequalities.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/Ineq.html"}, {"id": "Mathlib.Tactic.GRewrite.Elab", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.0632, "macro_tier_override": null, "x": -117.563, "z": -159.525, "size": 0.5701, "title": "The generalized rewriting tactic", "summary": "This file defines the tactics that use the backend defined in `Mathlib.Tactic.GRewrite.Core`: - `grewrite` - `grw` - `apply_rw` - `nth_grewrite` - `nth_grw`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/GRewrite/Elab.html"}, {"id": "Mathlib.Tactic.NormNum.Abs", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 1, "macro_tier_score": 0.0573, "macro_tier_override": null, "x": -111.994, "z": -188.291, "size": 0.3754, "title": "`norm_num` plugin for `abs`", "summary": "TODO: plugins for `mabs`, `norm`, `nnorm`, and `enorm`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/Abs.html"}, {"id": "Mathlib.Tactic.NormNum.DivMod", "region_id": "tactic", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.0575, "macro_tier_override": null, "x": -80.342, "z": -151.98, "size": 0.3875, "title": "`norm_num` extension for integer div/mod and divides", "summary": "This file adds support for the `%`, `/`, and `∣` (divisibility) operators on `ℤ` to the `norm_num` tactic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/DivMod.html"}, {"id": "Mathlib.Tactic.NormNum.OfScientific", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 1, "macro_tier_score": 0.0573, "macro_tier_override": null, "x": -88.76, "z": -164.057, "size": 0.3754, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/OfScientific.html"}, {"id": "Mathlib.Tactic.Inhabit", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0332, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.7144, "title": null, "summary": "Defines the `inhabit α` tactic, which tries to construct an `Inhabited α` instance, constructively or otherwise.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Inhabit.html"}, {"id": "Mathlib.Tactic.SimpRw", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0455, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.5894, "title": "The `simp_rw` tactic", "summary": "This file defines the `simp_rw` tactic: it functions as a mix of `simp` and `rw`. Like `rw`, it applies each rewrite rule in the given order, but like `simp` it repeatedly applies these rules and also under binders like `∀ x, ...`, `∃ x, ...` and `fun x ↦ ...`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/SimpRw.html"}, {"id": "Mathlib.Tactic.Simps.Basic", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0334, "macro_tier_override": null, "x": -106.477, "z": -171.469, "size": 0.7183, "title": "Simps attribute", "summary": "This file defines the `@[simps]` attribute, to automatically generate `simp` lemmas reducing a definition when projections are applied to it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Simps/Basic.html"}, {"id": "Mathlib.Tactic.SplitIfs", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0345, "macro_tier_override": null, "x": -106.701, "z": -164.592, "size": 0.7369, "title": null, "summary": "Tactic to split if-then-else expressions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/SplitIfs.html"}, {"id": "Mathlib.Tactic.TautoSet", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.355, "title": "The `tauto_set` tactic", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/TautoSet.html"}, {"id": "Mathlib.Tactic.CrossRefAttribute", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0312, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.6777, "title": "Cross-reference attributes", "summary": "This file provides attributes for tagging Mathlib results with cross-references to entries in external mathematical databases: * `@[stacks TAG]` — [Stacks Project](https://stacks.math.columbia.edu/tags) * `@[kerodon TAG]` — [Kerodon](https://kerodon.net/tag/) * `@[wikidata QID]` — [Wikidata](https://www.wikidata.org) Each attribute records the cross-reference in an environment extension and appends a link to the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CrossRefAttribute.html"}, {"id": "Mathlib.Tactic.NormNum.Basic", "region_id": "tactic", "micro_elevation": 0.3333, "macro_tier": 1, "macro_tier_score": 0.3443, "macro_tier_override": null, "x": -106.77, "z": -184.328, "size": 0.6439, "title": "`norm_num` basic plugins", "summary": "This file adds `norm_num` plugins for * constructors and constants * `Nat.cast`, `Int.cast`, and `mkRat` * `+`, `-`, `*`, and `/` * `Nat.succ`, `Nat.sub`, `Nat.mod`, and `Nat.div`. See other files in this directory for many more plugins.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/Basic.html"}, {"id": "Mathlib.Tactic.SuppressCompilation", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4491, "title": "Suppressing compilation to executable code in a file or in a section", "summary": "Currently, the compiler may spend a lot of time trying to produce executable code for complicated definitions. This is a waste of resources for definitions in area of mathematics that will never lead to executable code. The command `suppress_compilation` is a hack to disable code generation on all definitions (in a section or in a whole file). See the issue…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/SuppressCompilation.html"}, {"id": "Mathlib.Tactic.MkIffOfInductiveProp", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0324, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.6994, "title": "mk_iff_of_inductive_prop", "summary": "This file defines a command `mk_iff_of_inductive_prop` that generates `iff` rules for inductive `Prop`s. For example, when applied to `List.Chain`, it creates a declaration with the following type: ```lean ∀ {α : Type*} (R : α → α → Prop) (a : α) (l : List α), Chain R a l ↔ l = [] ∨ ∃ (b : α) (l' : List α), R a b ∧ Chain R b l ∧ l = b :: l' ``` This tactic can be called using either the `mk_iff_of_inductive_prop`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/MkIffOfInductiveProp.html"}, {"id": "Mathlib.Tactic.Algebraize", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.344, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Algebraize.html"}, {"id": "Mathlib.Tactic.ClickSuggestions.FindPremises", "region_id": "tactic", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.0374, "macro_tier_override": null, "x": -129.071, "z": -142.44, "size": 0.2829, "title": "Generating a shortlist of candidate lemmas for suggestions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ClickSuggestions/FindPremises.html"}, {"id": "Mathlib.Tactic.ClickSuggestions.Rewrite", "region_id": "tactic", "micro_elevation": 0.5833, "macro_tier": 1, "macro_tier_score": 0.0371, "macro_tier_override": null, "x": -88.599, "z": -188.375, "size": 0.2416, "title": "Support for `rw` suggestions in `#click_suggestions`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ClickSuggestions/Rewrite.html"}, {"id": "Mathlib.Tactic.ClickSuggestions.GRewrite", "region_id": "tactic", "micro_elevation": 0.5833, "macro_tier": 1, "macro_tier_score": 0.0371, "macro_tier_override": null, "x": -135.233, "z": -179.058, "size": 0.2416, "title": "Support for `grw` suggestions in `#click_suggestions`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ClickSuggestions/GRewrite.html"}, {"id": "Mathlib.Tactic.ClickSuggestions.Apply", "region_id": "tactic", "micro_elevation": 0.5833, "macro_tier": 1, "macro_tier_score": 0.0371, "macro_tier_override": null, "x": -96.526, "z": -193.954, "size": 0.2416, "title": "Support for `apply` suggestions in `#click_suggestions`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ClickSuggestions/Apply.html"}, {"id": "Mathlib.Tactic.ClickSuggestions.ApplyAt", "region_id": "tactic", "micro_elevation": 0.5833, "macro_tier": 1, "macro_tier_score": 0.0371, "macro_tier_override": null, "x": -92.863, "z": -144.332, "size": 0.2416, "title": "Support for `apply at` suggestions in `#click_suggestions`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ClickSuggestions/ApplyAt.html"}, {"id": "Mathlib.Tactic.HigherOrder", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "HigherOrder attribute", "summary": "This file defines the `@[higher_order]` attribute that applies to lemmas of the shape `∀ x, f (g x) = h x`. It derives an auxiliary lemma of the form `f ∘ g = h` for reasoning about higher-order functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/HigherOrder.html"}, {"id": "Mathlib.Tactic.FindSyntax", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "The `#find_syntax` command", "summary": "The `#find_syntax` command takes as input a string `str` and retrieves from the environment all the candidates for `syntax` terms that contain the string `str`. It also makes a very crude effort at regenerating what the syntax looks like, by inspecting the `ParserDescr`iptor of the corresponding parser.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FindSyntax.html"}, {"id": "Mathlib.Tactic.Linter.Lint", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -119.362, "z": -161.874, "size": 0.7902, "title": "Linters for Mathlib", "summary": "In this file we define additional linters for mathlib, which concern the *behaviour* of the linted code, and not issues of code style or formatting. Perhaps these should be moved to Batteries in the future.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/Lint.html"}, {"id": "Mathlib.Tactic.DeclarationNames", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0518, "macro_tier_override": null, "x": -116.95, "z": -167.496, "size": 0.9738, "title": null, "summary": "This file contains functions that are used by multiple linters.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/DeclarationNames.html"}, {"id": "Mathlib.Tactic.MoveAdd", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3117, "title": "`move_add` a tactic for moving summands in expressions", "summary": "The tactic `move_add` rearranges summands in expressions. The tactic takes as input a list of terms, each one optionally preceded by `←`. A term preceded by `←` gets moved to the left, while a term without `←` gets moved to the right. * Empty input: `move_add []` In this case, the effect of `move_add []` is equivalent to `simp only [← add_assoc]`: essentially the tactic removes all visible parentheses. * Singleton…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/MoveAdd.html"}, {"id": "Mathlib.Tactic.Lemma", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -110.507, "z": -176.097, "size": 0.7902, "title": "Support for `lemma` as a synonym for `theorem`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Lemma.html"}, {"id": "Mathlib.Tactic.Attr.Register", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0527, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.9849, "title": "Attributes used in `Mathlib`", "summary": "In this file we define all `simp`-like and `label`-like attributes used in `Mathlib`. We declare all of them in one file for two reasons: - in Lean 4, one cannot use an attribute in the same file where it was declared; - this way it is easy to see which simp sets contain a given lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Attr/Register.html"}, {"id": "Mathlib.Tactic.TacticAnalysis.Declarations", "region_id": "tactic", "micro_elevation": 0.3333, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -94.125, "z": -160.881, "size": 0.7902, "title": "Tactic linters", "summary": "This file defines passes to run from the tactic analysis framework.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/TacticAnalysis/Declarations.html"}, {"id": "Mathlib.Tactic.TacticAnalysis", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0227, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4736, "title": "Tactic analysis framework", "summary": "In this file we define a framework for analyzing sequences of tactics. This can be used for linting (for instance: report when two `rw` calls can be merged into one), but it can also be run in a more batch-like mode to report larger potential refactors (for instance: report when a sequence of three or more tactics can be replaced with `grind`, without taking more heartbeats than the original proof did).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/TacticAnalysis.html"}, {"id": "Mathlib.Tactic.ExtractGoal", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.0227, "macro_tier_override": null, "x": -117.029, "z": -159.011, "size": 0.4736, "title": "`extract_goal`: Format the current goal as a stand-alone example", "summary": "Useful for testing tactics or creating [minimal working examples](https://leanprover-community.github.io/mwe.html). ```lean example (i j k : Nat) (h₀ : i ≤ j) (h₁ : j ≤ k) : i ≤ k := by extract_goal /- theorem extracted_1 (i j k : Nat) (h₀ : i ≤ j) (h₁ : j ≤ k) : i ≤ k := sorry", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ExtractGoal.html"}, {"id": "Mathlib.Tactic.Linter.UnusedTacticExtension", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0352, "macro_tier_override": null, "x": -104.275, "z": -161.29, "size": 0.7472, "title": null, "summary": "This file defines the environment extension to keep track of which tactics are allowed to leave the tactic state unchanged and not trigger the unused tactic linter.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/UnusedTacticExtension.html"}, {"id": "Mathlib.Tactic.Field", "region_id": "tactic", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -86.139, "z": -144.518, "size": 0.4492, "title": "A tactic for proving algebraic goals in a field", "summary": "This file contains the `field` tactic, a finishing tactic which roughly consists of running `field_simp; ring1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Field.html"}, {"id": "Mathlib.Tactic.TFAE", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": -108.273, "z": -172.157, "size": 0.4256, "title": "The Following Are Equivalent (TFAE)", "summary": "This file provides the tactics `tfae_have` and `tfae_finish` for proving goals of the form `TFAE [P₁, P₂, ...]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/TFAE.html"}, {"id": "Mathlib.Tactic.Rify", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0403, "macro_tier_override": null, "x": -104.75, "z": -167.52, "size": 0.4441, "title": "`rify` tactic", "summary": "The `rify` tactic is used to shift propositions from `ℕ`, `ℤ`, `ℚ` or `ℝ≥0` to `ℝ`. Although less useful than its cousins `zify` and `qify`, it can be useful when your goal or context already involves real numbers. In the example below, assumption `hn` is about natural numbers, `hk` is about integers and involves casting a natural number to `ℤ`, and the conclusion is about real numbers. The proof uses `rify` to lift…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Rify.html"}, {"id": "Mathlib.Tactic.ToExpr", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0402, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4374, "title": "`ToExpr` instances for Mathlib", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ToExpr.html"}, {"id": "Mathlib.Tactic.Zify", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.076, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3859, "title": "`zify` tactic", "summary": "The `zify` tactic is used to shift propositions from `Nat` to `Int`. This is often useful since `Int` has well-behaved subtraction. ``` example (a b c x y z : Nat) (h : ¬ x*y*z < 0) : c < a + 3*b := by zify zify at h /- h : ¬↑x * ↑y * ↑z < 0 ⊢ ↑c < ↑a + 3 * ↑b", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Zify.html"}, {"id": "Mathlib.Tactic.Linter.Whitespace", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0381, "macro_tier_override": null, "x": -100.676, "z": -169.052, "size": 0.7941, "title": "The `whitespace` linter", "summary": "The `whitespace` linter emits a warning if * either a command does not start at the beginning of a line; * or the \"hypotheses segment\" of a declaration does not coincide with its pretty-printed version.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/Whitespace.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Bicategory.Basic", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": -96.654, "z": -166.377, "size": 0.363, "title": "`bicategory` tactic", "summary": "This file provides `bicategory` tactic, which solves equations in a bicategory, where the two sides only differ by replacing strings of bicategory structural morphisms (that is, associators, unitors, and identities) with different strings of structural morphisms with the same source and target. In other words, `bicategory` solves equalities where both sides have the same string diagrams. The core function for the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Bicategory/Basic.html"}, {"id": "Mathlib.Tactic.Choose", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0196, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3422, "title": "`choose` tactic", "summary": "Performs Skolemization, that is, given `h : ∀ a:α, ∃ b:β, p a b |- G` produces `f : α → β, hf: ∀ a, p a (f a) |- G`. TODO: switch to `rcases` syntax: `choose ⟨i, j, h₁ -⟩ := expr`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Choose.html"}, {"id": "Mathlib.Tactic.Positivity.Finset", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 1, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": -129.098, "z": -170.478, "size": 0.369, "title": "Positivity extensions for finsets", "summary": "This file provides a few `positivity` extensions that cannot be in either the finset files (because they don't know about ordered fields) or in `Tactic.Positivity.Basic` (because it doesn't want to know about finiteness).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Positivity/Finset.html"}, {"id": "Mathlib.Tactic.TypeStar", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0708, "macro_tier_override": null, "x": -116.403, "z": -171.1, "size": 0.9791, "title": "Support for `Sort*` and `Type*`.", "summary": "These elaborate as `Sort u` and `Type u` with a fresh implicit universe variable `u`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/TypeStar.html"}, {"id": "Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0191, "macro_tier_override": null, "x": -106.189, "z": -164.953, "size": 0.2991, "title": "Multiseries definitions", "summary": "In this file, we define the multiseries and its main properties: sortedness and approximation. A multiseries in a basis `[b₁, ..., bₙ]` represents a multivariate series: it is a formal series made from monomials `b₁ ^ e₁ * ... * bₙ ^ eₙ` where `e₁, ..., eₙ` are real numbers. We treat multivariate series in a basis `[b₁, ..., bₙ]` as a univariate series in the variable `b₁` (`basis_hd`) with coefficients being…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Defs.html"}, {"id": "Mathlib.Tactic.ComputeAsymptotics.Multiseries.Majorized", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0189, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2802, "title": "`Majorized` predicate", "summary": "This file defines the `Majorized` predicate, along with a few basic lemmas.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Majorized.html"}, {"id": "Mathlib.Tactic.GrindAttrs", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "Custom grind-sets", "summary": "In this file we declare custom grind attributes and tactics that call grind using only these grind attributes. These grind sets are helpful because they can contain a lot of specialized ways to prove a particular problem. Currently, this implements the `compactness` and `closedness` grind attribute and tactic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/GrindAttrs.html"}, {"id": "Mathlib.Tactic.Conv", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1753, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.5936, "title": null, "summary": "Additional `conv` tactics.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Conv.html"}, {"id": "Mathlib.Tactic.HaveI", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.3922, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4444, "title": "Variants of `haveI`/`letI` for use in do-notation.", "summary": "This file implements the `haveI'` and `letI'` macros which have the same semantics as `haveI` and `letI`, but are `doElem`s and can be used inside do-notation. They need an apostrophe after their name for disambiguation with the term variants. This is necessary because the do-notation has a hardcoded list of keywords which can appear both as term-mode and do-elem syntax (like for example `let` or `have`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/HaveI.html"}, {"id": "Mathlib.Tactic.NormNum.Core", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.3738, "macro_tier_override": null, "x": -96.675, "z": -166.235, "size": 0.4483, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/Core.html"}, {"id": "Mathlib.Tactic.Positivity", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.002, "macro_tier_override": null, "x": -84.288, "z": -169.091, "size": 0.3899, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Positivity.html"}, {"id": "Mathlib.Tactic.Widget.Calc", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.485, "z": -176.272, "size": 0.2647, "title": "Calc widget", "summary": "This file redefines the `calc` tactic so that it displays a widget panel allowing to create new calc steps with holes specified by selected sub-expressions in the goal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Widget/Calc.html"}, {"id": "Mathlib.Tactic.Widget.SelectPanelUtils", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0569, "macro_tier_override": null, "x": -110.61, "z": -171.766, "size": 0.3536, "title": "Selection panel utilities", "summary": "The main declaration is `mkSelectionPanelRPC` which helps creating rpc methods for widgets generating tactic calls based on selected sub-expressions in the main goal. There are also some minor helper functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Widget/SelectPanelUtils.html"}, {"id": "Mathlib.Tactic.Simproc.VecPerm", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "The vecPerm simproc", "summary": "The `vecPerm` simproc computes the new entries of a vector after applying a permutation to them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Simproc/VecPerm.html"}, {"id": "Mathlib.Tactic.Linter.DirectoryDependency", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.4892, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 2.5393, "title": "The `directoryDependency` linter", "summary": "The `directoryDependency` linter detects imports between directories that are supposed to be independent. By specifying that one directory does not import from another, we can improve the modularity of Mathlib.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/DirectoryDependency.html"}, {"id": "Mathlib.Tactic.Variable", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2756, "title": "The `variable?` command", "summary": "This defines a command like `variable` that automatically adds all missing typeclass arguments. For example, `variable? [Module R M]` is the same as `variable [Semiring R] [AddCommMonoid M] [Module R M]`, though if any of these three instance arguments can be inferred from previous variables then they will be omitted. An inherent limitation with this command is that variables are recorded in the scope as *syntax*.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Variable.html"}, {"id": "Mathlib.Tactic.Sat.FromLRAT", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -102.805, "z": -162.539, "size": 0.208, "title": "`lrat_proof` command", "summary": "Defines a macro for producing SAT proofs from CNF / LRAT files. These files are commonly used in the SAT community for writing proofs. Most SAT solvers support export to [DRAT](https://arxiv.org/abs/1610.06229) format, but this format can be expensive to reconstruct because it requires recomputing all unit propagation steps. The [LRAT](https://arxiv.org/abs/1612.02353) format solves this issue by attaching a proof…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Sat/FromLRAT.html"}, {"id": "Mathlib.Tactic.ENatToNat", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0193, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.315, "title": "`enat_to_nat`", "summary": "This file implements the `enat_to_nat` tactic that shifts `ENat`s in the context to `Nat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ENatToNat.html"}, {"id": "Mathlib.Tactic.CongrExclamation", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0399, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4278, "title": "The `congr!` tactic", "summary": "This is a more powerful version of the `congr` tactic that knows about more congruence lemmas and can apply to more situations. It is similar to the `congr'` tactic from Mathlib 3. The `congr!` tactic is used by the `convert` and `convert_to` tactics.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CongrExclamation.html"}, {"id": "Mathlib.Tactic.CategoryTheory.IsoReassoc", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.5257, "title": "Extension of `reassoc` to isomorphisms.", "summary": "We extend `reassoc` and `reassoc_of%` for equality of isomorphisms. Adding `@[reassoc]` to a lemma named `F` of shape `∀ .., f = g`, where `f g : X ≅ Y` in some category will create a new lemma named `F_assoc` of shape `∀ .. {Z : C} (h : Y ≅ Z), f ≪≫ h = g ≪≫ h` but with the conclusions simplified using basic proportions in isomorphisms in a category (`Iso.trans_refl`, `Iso.refl_trans`, `Iso.trans_assoc`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/IsoReassoc.html"}, {"id": "Mathlib.Tactic.Qify", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0565, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3295, "title": "`qify` tactic", "summary": "The `qify` tactic is used to shift propositions from `ℕ` or `ℤ` to `ℚ`. This is often useful since `ℚ` has well-behaved division. ``` example (a b c x y z : ℕ) (h : ¬ x*y*z < 0) : c < a + 3*b := by qify qify at h /- h : ¬↑x * ↑y * ↑z < 0 ⊢ ↑c < ↑a + 3 * ↑b", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Qify.html"}, {"id": "Mathlib.Tactic.Order.Preprocessing", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0189, "macro_tier_override": null, "x": -109.492, "z": -164.074, "size": 0.2797, "title": "Facts preprocessing for the `order` tactic", "summary": "In this file we implement the preprocessing procedure for the `order` tactic. See `Mathlib/Tactic/Order.lean` for details of preprocessing.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Order/Preprocessing.html"}, {"id": "Mathlib.Tactic.Order.CollectFacts", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3673, "title": "Facts collection for the `order` Tactic", "summary": "This file implements the collection of facts for the `order` tactic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Order/CollectFacts.html"}, {"id": "Mathlib.Tactic.Ring.Common", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.1902, "macro_tier_override": null, "x": -84.269, "z": -167.948, "size": 0.4991, "title": "`ring`-like tactics", "summary": "The core normalization procedure for ring-like tactics that solve equations in commutative (semi)rings where the exponents can also contain variables. Based on . More precisely, expressions of the following form are supported: - constants (non-negative integers) - variables - coefficients (living in `BaseType`; for `ring` this is a rational…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Ring/Common.html"}, {"id": "Mathlib.Tactic.ProxyType", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0189, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2835, "title": "Generating \"proxy types\"", "summary": "This module gives tools to create an equivalence between a given inductive type and a \"proxy type\" constructed from `Unit`, `PLift`, `Sigma`, `Empty`, and `Sum`. It works for any non-recursive inductive type without indices. The intended use case is for pulling typeclass instances across this equivalence. This reduces the problem of generating typeclass instances to that of writing typeclass instances for the above…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ProxyType.html"}, {"id": "Mathlib.Tactic.ScopedNS", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0213, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4229, "title": "`scoped[NS]` syntax", "summary": "This is a replacement for the `localized` command in mathlib. It is similar to `scoped`, but it scopes the syntax in the specified namespace instead of the current namespace.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ScopedNS.html"}, {"id": "Mathlib.Tactic.Linter.FindDeprecations", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "The `#clear_deprecations` command", "summary": "This file defines the `#clear_deprecations date₁ date₂ really` command. This function is intended for automated use by the `remove_deprecations` automation. It removes declarations that have been deprecated in the time range starting from `date₁` and ending with `date₂`. See the doc-string for the command for more information.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/FindDeprecations.html"}, {"id": "Mathlib.Tactic.CategoryTheory.CancelIso", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0009, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3262, "title": "Simproc for canceling morphisms with their inverses", "summary": "This module implements the `cancelIso` simproc, which simplifies the composition of a morphism and its inverse, given an expression of the form `f ≫ g`. Assuming `f` is not a composition (as `Category.assoc` is tagged `@[simp]`), if `g` is not a composition itself, it checks whether `f` is inverse to `g` by checking if `f` has an `IsIso` instance and then by running `push inv` on `inv f` and on `g`. If the check…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/CancelIso.html"}, {"id": "Mathlib.Tactic.Trace", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "Defines the `trace` tactic.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Trace.html"}, {"id": "Mathlib.Tactic.Continuity", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -105.158, "z": -169.966, "size": 0.2789, "title": "Continuity", "summary": "We define the `continuity` tactic using `aesop`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Continuity.html"}, {"id": "Mathlib.Tactic.FunProp", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0389, "macro_tier_override": null, "x": -96.386, "z": -189.259, "size": 0.3829, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FunProp.html"}, {"id": "Mathlib.Tactic.NormNum.BigOperators", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -129.128, "z": -165.993, "size": 0.2598, "title": "`norm_num` plugin for big operators", "summary": "This file adds `norm_num` plugins for `Finset.prod` and `Finset.sum`. The driving part of this plugin is `Mathlib.Meta.NormNum.evalFinsetBigop`. We repeatedly use `Finset.proveEmptyOrCons` to try to find a proof that the given set is empty, or that it consists of one element inserted into a strict subset, and evaluate the big operator on that subset until the set is completely exhausted.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/BigOperators.html"}, {"id": "Mathlib.Tactic.Find", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "The `#find` command and tactic.", "summary": "The `#find` command finds definitions & lemmas using pattern matching on the type. For instance: ```lean #find _ + _ = _ + _ #find ?n + _ = _ + ?n #find (_ : Nat) + _ = _ + _ #find Nat → Nat ``` Inside tactic proofs, there is a `#find` tactic with the same syntax, or the `find` tactic which looks for lemmas which are `apply`able against the current goal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Find.html"}, {"id": "Mathlib.Tactic.WithoutCDot", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "The `without_cdot()` elaborator", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/WithoutCDot.html"}, {"id": "Mathlib.Tactic.ComputeDegree", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.001, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3311, "title": "`compute_degree` and `monicity`: tactics for explicit polynomials", "summary": "This file defines two related tactics: `compute_degree` and `monicity`. Using `compute_degree` when the goal is of one of the seven forms * `natDegree f ≤ d` (or `<`), * `degree f ≤ d` (or `<`), * `natDegree f = d`, * `degree f = d`, * `coeff f d = r`, if `d` is the degree of `f`, tries to solve the goal. It may leave side-goals, in case it is not entirely successful. Using `monicity` when the goal is of the form…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ComputeDegree.html"}, {"id": "Mathlib.Tactic.Tauto", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0242, "macro_tier_override": null, "x": -112.856, "z": -168.583, "size": 0.521, "title": null, "summary": "The `tauto` tactic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Tauto.html"}, {"id": "Mathlib.Tactic.Measurability.Init", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.019, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2919, "title": "Measurability Rule Set", "summary": "This module defines the `Measurable` Aesop rule set which is used by the `measurability` tactic. Aesop rule sets only become visible once the file in which they're declared is imported, so we must put this declaration into its own file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Measurability/Init.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Slice", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.5152, "title": "The `slice` tactic", "summary": "Applies a tactic to an interval of terms from a term obtained by repeated application of `Category.comp`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Slice.html"}, {"id": "Mathlib.Tactic.Order.Graph.Basic", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0191, "macro_tier_override": null, "x": -109.81, "z": -164.142, "size": 0.2999, "title": "Graphs for the `order` tactic", "summary": "This module defines the `Graph` structure and basic operations on it. The `order` tactic uses `≤`-graphs, where the vertices represent atoms, and an edge `(x, y)` exists if `x ≤ y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Order/Graph/Basic.html"}, {"id": "Mathlib.Tactic.CategoryTheory.ToApp", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0019, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3869, "title": "The `to_app` attribute", "summary": "Adding `@[to_app]` to a lemma named `F` of shape `∀ .., η = θ`, where `η θ : f ⟶ g` are 2-morphisms in some bicategory, create a new lemma named `F_app`. This lemma is obtained by first specializing the bicategory in which the equality is taking place to `Cat`, then applying `toNatTrans_congr` and `NatTrans.congr_app` to obtain a proof of `∀ ... (X : Cat), η.toNatTrans.app X = θ.toNatTrans.app X`, and finally…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/ToApp.html"}, {"id": "Mathlib.Tactic.Ring.Basic", "region_id": "tactic", "micro_elevation": 0.5833, "macro_tier": 1, "macro_tier_score": 0.1907, "macro_tier_override": null, "x": -132.505, "z": -152.078, "size": 0.5159, "title": "`ring` tactic", "summary": "A tactic for solving equations in commutative (semi)rings, where the exponents can also contain variables. Based on . More precisely, expressions of the following form are supported: - constants (non-negative integers) - variables - coefficients (any rational number, embedded into the (semi)ring) - addition of expressions - multiplication of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Ring/Basic.html"}, {"id": "Mathlib.Tactic.TryThis", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1543, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.5331, "title": "'Try this' tactic macro", "summary": "This is a convenient shorthand intended for macro authors to be able to generate \"Try this\" recommendations. (It is not the main implementation of 'Try this', which is implemented in Lean core, see `Lean.Meta.Tactic.TryThis`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/TryThis.html"}, {"id": "Mathlib.Tactic.Lift", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.035, "macro_tier_override": null, "x": -120.988, "z": -169.439, "size": 0.7453, "title": "lift tactic", "summary": "This file defines the `lift` tactic, allowing the user to lift elements from one type to another under a specified condition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Lift.html"}, {"id": "Mathlib.Tactic.SuccessIfFailWithMsg", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "Success If Fail With Message", "summary": "This file implements a tactic that succeeds only if its argument fails with a specified message. It's mostly useful in tests, where we want to make sure that tactics fail in certain ways under circumstances.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/SuccessIfFailWithMsg.html"}, {"id": "Mathlib.Tactic.Translate.Core", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.1205, "macro_tier_override": null, "x": -112.525, "z": -169.779, "size": 0.9092, "title": "The translation attribute.", "summary": "Implementation of the translation attribute. This is used for `@[to_additive]` and `@[to_dual]`. See the docstring of `to_additive` for more information", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Translate/Core.html"}, {"id": "Mathlib.Tactic.Eqns", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1339, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4772, "title": "The `@[eqns]` attribute", "summary": "This file provides the `eqns` attribute as a way of overriding the default equation lemmas. For example ```lean4 def transpose {m n} (A : m → n → ℕ) : n → m → ℕ | i, j => A j i theorem transpose_apply {m n} (A : m → n → ℕ) (i j) : transpose A i j = A j i := rfl attribute [eqns transpose_apply] transpose theorem transpose_const {m n} (c : ℕ) : transpose (fun (i : m) (j : n) => c) = fun j i => c := by funext i j --…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Eqns.html"}, {"id": "Mathlib.Tactic.Translate.GuessName", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0964, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.463, "title": "Name generation APIs for `to_additive`-like attributes", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Translate/GuessName.html"}, {"id": "Mathlib.Tactic.Translate.Reorder", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0964, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.463, "title": "Reordering arguments in a translation", "summary": "This module defines reorders, which can be specified with `to_dual (reorder := ...)` or `to_additive (reorder := ...)`, to deal with definitions and theorems that need to have their arguments and/or universe parameters reordered. A reordering is specified using disjoint cycle notation. For example, `1 2 3, 4 5` will move the 1st argument to the 2nd, move the 2nd to the 3rd, and the 3rd to the 1st, and it will swap…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Translate/Reorder.html"}, {"id": "Mathlib.Tactic.Translate.UnfoldBoundary", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0964, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.463, "title": "Modify proof terms so that they don't rely on unfolding certain constants", "summary": "This file defines a procedure for inserting casts into (proof) terms in order to make them well typed in a setting where certain constants aren't allowed to be unfolded. We make use of `withCanUnfoldPred` in order to modify which constants can and cannot be unfolded. This way, `whnf` and `isDefEq` do not unfold these constants. So, the procedure is to check that an expression is well typed, analogous to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Translate/UnfoldBoundary.html"}, {"id": "Mathlib.Tactic.Says", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -116.789, "z": -166.378, "size": 0.2647, "title": "The `says` tactic combinator.", "summary": "If you write `X says`, where `X` is a tactic that produces a \"Try this: Y\" message, then you will get a message \"Try this: X says Y\". Once you've clicked to replace `X says` with `X says Y`, afterwards `X says Y` will only run `Y`. The typical usage case is: ``` simp? [X] says simp only [X, Y, Z] ``` If you use `set_option says.verify true` (set automatically during CI) then `X says Y` runs `X` and verifies that it…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Says.html"}, {"id": "Mathlib.Tactic.Coe", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.021, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.41, "title": "Additional coercion notation", "summary": "Defines notation for coercions. 1. `↑ t` is defined in core. 2. `(↑)` is equivalent to the eta-reduction of `(↑ ·)` 3. `⇑ t` is a coercion to a function type. 4. `(⇑)` is equivalent to the eta-reduction of `(⇑ ·)` 3. `↥ t` is a coercion to a type. 6. `(↥)` is equivalent to the eta-reduction of `(↥ ·)`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Coe.html"}, {"id": "Mathlib.Tactic.Linarith.Frontend", "region_id": "tactic", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.0407, "macro_tier_override": null, "x": -116.75, "z": -136.379, "size": 0.4562, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith/Frontend.html"}, {"id": "Mathlib.Tactic.FunProp.Theorems", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.0579, "macro_tier_override": null, "x": -99.587, "z": -160.003, "size": 0.4076, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FunProp/Theorems.html"}, {"id": "Mathlib.Tactic.FunProp.Decl", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0751, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.337, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FunProp/Decl.html"}, {"id": "Mathlib.Tactic.FunProp.Types", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0561, "macro_tier_override": null, "x": -105.618, "z": -175.635, "size": 0.3017, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FunProp/Types.html"}, {"id": "Mathlib.Tactic.NormNum.Eq", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.1122, "macro_tier_override": null, "x": -84.599, "z": -164.089, "size": 0.3382, "title": "`norm_num` extension for equalities", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/Eq.html"}, {"id": "Mathlib.Tactic.Linter.HaveLetLinter", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0371, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2451, "title": "The `have` vs `let` linter", "summary": "The `have` vs `let` linter flags uses of `have` to introduce a hypothesis whose Type is not `Prop`. The option for this linter is a natural number, but really there are only 3 settings: * `0` -- inactive; * `1` -- active only on noisy declarations; * `2` or more -- always active. TODO: * Also lint `let` vs `have`. * `haveI` may need to change to `let/letI`? * `replace`, `classical!`, `classical`, `tauto` internally…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/HaveLetLinter.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Bicategory.Normalize", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0376, "macro_tier_override": null, "x": -107.523, "z": -160.026, "size": 0.3026, "title": "Normalization of 2-morphisms in bicategories", "summary": "This file provides the implementation of the normalization given in `Mathlib/Tactic/CategoryTheory/Coherence/Normalize.lean`. See this file for more details.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Bicategory/Normalize.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0377, "macro_tier_override": null, "x": -104.776, "z": -167.36, "size": 0.313, "title": "Expressions for bicategories", "summary": "This file converts lean expressions representing 2-morphisms in bicategories into `Mor₂Iso` or `Mor` terms. The converted expressions are used in the coherence tactics and the string diagram widgets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Bicategory/Datatypes.html"}, {"id": "Mathlib.Tactic.Set", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0189, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2784, "title": "The `set` tactic", "summary": "This file defines the `set` tactic and its variant `set!`. `set a := t with h` is a variant of `let a := t`. It adds the hypothesis `h : a = t` to the local context and replaces `t` with `a` everywhere it can. `set a := t with ← h` will add `h : t = a` instead. `set! a := t with h` does not do any replacing.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Set.html"}, {"id": "Mathlib.Tactic.Finiteness", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 0, "macro_tier_score": 0.0019, "macro_tier_override": null, "x": -125.088, "z": -180.447, "size": 0.3848, "title": "Finiteness tactic", "summary": "This file implements a basic `finiteness` tactic, designed to solve goals of the form `*** < ∞` and (equivalently) `*** ≠ ∞` in the extended nonnegative reals (`ENNReal`, aka `ℝ≥0∞`). It works recursively according to the syntax of the expression. It is implemented as an `aesop` rule set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Finiteness.html"}, {"id": "Mathlib.Tactic.Linter.TextBased", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.544, "z": -172.182, "size": 0.208, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/TextBased.html"}, {"id": "Mathlib.Tactic.ArithMult.Init", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.019, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2913, "title": "`arith_mult` Rule Set", "summary": "This module defines the `IsMultiplicative` Aesop rule set which is used by the `arith_mult` tactic. Aesop rule sets only become visible once the file in which they're declared is imported, so we must put this declaration into its own file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ArithMult/Init.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Coherence.Datatypes", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0417, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4902, "title": "Datatypes for bicategory like structures", "summary": "This file defines the basic datatypes for bicategory like structures. We will use these datatypes to write tactics that can be applied to both monoidal categories and bicategories: - `Obj`: objects type - `Atom₁`: atomic 1-morphisms type - `Mor₁`: 1-morphisms type - `Atom`: atomic non-structural 2-morphisms type - `Mor₂`: 2-morphisms type - `AtomIso`: atomic non-structural 2-isomorphisms type - `Mor₂Iso`:…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Coherence/Datatypes.html"}, {"id": "Mathlib.Tactic.Linter.OverlappingInstances", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -108.038, "z": -164.085, "size": 0.7902, "title": "A linter for declarations with local instances that overlap", "summary": "If the same data can be obtained from two different instances, we risk having non-defeq versions of that data. This situation is known as an \"instance diamond\". This linter warns against instance diamonds in local contexts. This is a syntax linter. It is run on partially and fully elaborated declarations. To find diamonds, we compute the parent classes of each local instance. For classes that aren't structures, this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/OverlappingInstances.html"}, {"id": "Mathlib.Tactic.Linter.UnusedInstancesInType", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0268, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.5869, "title": "Linters for Unused Instances in Types", "summary": "This file declares linters which detect certain instance hypotheses in declarations that are unused in the remainder of the type. Currently, these linters only handle theorems. (This also includes `lemma`s and `instance`s of `Prop` classes.) - `unusedDecidableInType` linter (currently off by default): suggests replacing type-unused `Decidable*` instance hypotheses, and could therefore be replaced by `classical` in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/UnusedInstancesInType.html"}, {"id": "Mathlib.Tactic.ApplyAt", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0372, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2549, "title": "Apply at", "summary": "A tactic for applying functions at hypotheses.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ApplyAt.html"}, {"id": "Mathlib.Tactic.NormNum.ModEq", "region_id": "tactic", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -143.575, "z": -180.109, "size": 0.208, "title": "`norm_num` extensions for `Nat.ModEq` and `Int.ModEq`", "summary": "In this file we define `norm_num` extensions for `a ≡ b [MOD n]` and `a ≡ b [ZMOD n]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/ModEq.html"}, {"id": "Mathlib.Tactic.LinearCombination.Lemmas", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0195, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3332, "title": "Lemmas for the `linear_combination` tactic", "summary": "These should not be used directly in user code.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/LinearCombination/Lemmas.html"}, {"id": "Mathlib.Tactic.Ring.Compare", "region_id": "tactic", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.0195, "macro_tier_override": null, "x": -116.163, "z": -199.966, "size": 0.3332, "title": "Automation for proving inequalities in commutative (semi)rings", "summary": "This file provides automation for proving certain kinds of inequalities in commutative semirings: goals of the form `A ≤ B` and `A < B` for which the ring-normal forms of `A` and `B` differ by a nonnegative (resp. positive) constant. For example, `⊢ x + 3 + y < y + x + 4` is in scope because the normal forms of the LHS and RHS are, respectively, `3 + (x + y)` and `4 + (x + y)`, which differ by an additive constant.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Ring/Compare.html"}, {"id": "Mathlib.Tactic.Core", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1489, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.7894, "title": "Generally useful tactics.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Core.html"}, {"id": "Mathlib.Tactic.GCongr.ForwardAttr", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0774, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4444, "title": "Environment extension for the forward-reasoning part of the `gcongr` tactic", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/GCongr/ForwardAttr.html"}, {"id": "Mathlib.Tactic.GRewrite.Core", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0624, "macro_tier_override": null, "x": -116.328, "z": -171.284, "size": 0.5499, "title": "The generalized rewriting tactic", "summary": "This module defines the core of the `grw`/`grewrite` tactic. This file provides two implementations of the tactic: 1. The simple implementation uses `kabstract` to determine where to rewrite, and then calls `MVarId.gcongr` to prove that the rewrite is valid. This is used by `nth_grw` and `grw +useKAbstract`. 2. The more sophisticated implementation has its own congruence loop, applying `gcongr` lemmas to create the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/GRewrite/Core.html"}, {"id": "Mathlib.Tactic.Bound.Attribute", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": -112.881, "z": -168.274, "size": 0.3396, "title": "The `bound` attribute", "summary": "Any lemma tagged with `@[bound]` is registered as an apply rule for the `bound` tactic, by converting it to either `norm apply` or `safe apply `. The classification is based on the number and types of the lemma's hypotheses.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Bound/Attribute.html"}, {"id": "Mathlib.Tactic.SwapVar", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "Defines the `swap_var` tactic", "summary": "Swap the names of two hypotheses.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/SwapVar.html"}, {"id": "Mathlib.Tactic.NthRewrite", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.056, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2854, "title": "`nth_rewrite` tactic", "summary": "The tactic `nth_rewrite` and `nth_rw` are variants of `rewrite` and `rw` that only performs the `n`th possible rewrite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NthRewrite.html"}, {"id": "Mathlib.Tactic.DepRewrite", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0399, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4274, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/DepRewrite.html"}, {"id": "Mathlib.Tactic.ComputeAsymptotics.Lemmas", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "Conversion lemmas", "summary": "The main procedure of the `compute_asymptotics` tactic is able to compute limits of functions at `atTop` filter. This file contains lemmas we use to reduce other asymptotic goals to the case `Tendsto f atTop l`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ComputeAsymptotics/Lemmas.html"}, {"id": "Mathlib.Tactic.ToDual", "region_id": "tactic", "micro_elevation": 0.3333, "macro_tier": 1, "macro_tier_score": 0.0273, "macro_tier_override": null, "x": -92.76, "z": -164.905, "size": 0.9015, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ToDual.html"}, {"id": "Mathlib.Tactic.NormNum.Irrational", "region_id": "tactic", "micro_elevation": 0.9167, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -149.3, "z": -187.638, "size": 0.208, "title": "`norm_num` extension for `Irrational`", "summary": "This module defines a `norm_num` extension for `Irrational x ^ y` for rational `x` and `y`. It also supports `Irrational √x` expressions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/Irrational.html"}, {"id": "Mathlib.Tactic.NormNum.GCD", "region_id": "tactic", "micro_elevation": 0.8333, "macro_tier": 1, "macro_tier_score": 0.0195, "macro_tier_override": null, "x": -148.07, "z": -179.453, "size": 0.3351, "title": "`norm_num` extensions for GCD-adjacent functions", "summary": "This module defines some `norm_num` extensions for functions such as `Nat.gcd`, `Nat.lcm`, `Int.gcd`, and `Int.lcm`. Note that `Nat.coprime` is reducible and defined in terms of `Nat.gcd`, so the `Nat.gcd` extension also indirectly provides a `Nat.coprime` extension.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/GCD.html"}, {"id": "Mathlib.Tactic.Widget.Conv", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0372, "macro_tier_override": null, "x": -114.155, "z": -161.926, "size": 0.2521, "title": "Conv widget", "summary": "This is a slightly improved version of one of the examples that used to be in the ProofWidgets library. It defines a `conv?` tactic that displays a widget panel allowing to generate a `conv` call zooming to the subexpression selected in the goal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Widget/Conv.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Monoidal.Datatypes", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0389, "macro_tier_override": null, "x": -106.001, "z": -171.085, "size": 0.3823, "title": "Expressions for monoidal categories", "summary": "This file converts lean expressions representing morphisms in monoidal categories into `Mor₂Iso` or `Mor` terms. The converted expressions are used in the coherence tactics and the string diagram widgets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Monoidal/Datatypes.html"}, {"id": "Mathlib.Tactic.Explode.Datatypes", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.019, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.292, "title": "Explode command: datatypes", "summary": "This file contains datatypes used by the `#explode` command and their associated methods.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Explode/Datatypes.html"}, {"id": "Mathlib.Tactic.Bound.Init", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0198, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3533, "title": "Bound Rule Set", "summary": "This module defines the `Bound` Aesop rule set which is used by the `bound` tactic. Aesop rule sets only become visible once the file in which they're declared is imported, so we must put this declaration into its own file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Bound/Init.html"}, {"id": "Mathlib.Tactic.NormNum.PowMod", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0186, "macro_tier_override": null, "x": -94.22, "z": -148.375, "size": 0.234, "title": "`norm_num` handling for expressions of the form `a ^ b % m`.", "summary": "These expressions can often be evaluated efficiently in cases where first evaluating `a ^ b` and then reducing mod `m` is not feasible. We provide a function `evalNatPowMod` which is used by the `reduce_mod_char` tactic to efficiently evaluate powers in rings with positive characteristic. The approach taken here is identical to (and copied from) the development in `Mathlib/Tactic/NormNum/Pow.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/PowMod.html"}, {"id": "Mathlib.Tactic.ByContra", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0423, "macro_tier_override": null, "x": -104.276, "z": -174.915, "size": 0.5084, "title": "The `by_contra` tactic", "summary": "The `by_contra!` tactic is a variant of the `by_contra` tactic, for proofs of contradiction.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ByContra.html"}, {"id": "Mathlib.Tactic.Linarith.Lemmas", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0376, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3038, "title": "Lemmas for `linarith`.", "summary": "Those in the `Linarith` namespace should stay here. Those outside the `Linarith` namespace may be deleted as they are ported to mathlib4.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith/Lemmas.html"}, {"id": "Mathlib.Tactic.Linter.OldObtain", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -114.598, "z": -173.863, "size": 0.7902, "title": "The `oldObtain` linter, against stream-of-consciousness `obtain`", "summary": "The `oldObtain` linter flags any occurrences of \"stream-of-consciousness\" `obtain`, i.e. uses of the `obtain` tactic which do not immediately provide a proof.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/OldObtain.html"}, {"id": "Mathlib.Tactic.FieldSimp.Discharger", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 1, "macro_tier_score": 0.0375, "macro_tier_override": null, "x": -117.779, "z": -149.741, "size": 0.2888, "title": "Discharger for `field_simp` tactic", "summary": "The `field_simp` tactic (implemented downstream from this file) clears denominators in algebraic expressions. In order to do this, the denominators need to be certified as nonzero. This file contains the discharger which carries out these checks. Currently the discharger tries four strategies: 1. `assumption` 2. `positivity` 3. `norm_num` 4. `simp` with the same simp-context as the `field_simp` call which invoked it…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FieldSimp/Discharger.html"}, {"id": "Mathlib.Tactic.Linter.HashCommandLinter", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -101.421, "z": -164.574, "size": 0.7902, "title": "`#`-command linter", "summary": "The `#`-command linter produces a warning when a command starting with `#` is used *and* * either the command emits no message; * or `warningAsError` is set to `true`. The rationale behind this is that `#`-commands are intended to be transient: they provide useful information in development, but are not intended to be present in final code. Most of them are noisy and get picked up anyway by CI, but even the quiet…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/HashCommandLinter.html"}, {"id": "Mathlib.Tactic.Algebra.Basic", "region_id": "tactic", "micro_elevation": 0.8333, "macro_tier": 1, "macro_tier_score": 0.0374, "macro_tier_override": null, "x": -145.248, "z": -186.609, "size": 0.2766, "title": "The `algebra` tactic", "summary": "A suite of three tactics for solving equations in commutative algebras over commutative (semi)rings, where the exponents can also contain variables. Based largely on the implementation of `ring`. The `algebra` normal form mirrors that of `ring` except that the constants are expressions in the base ring that are kept in ring normal form.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Algebra/Basic.html"}, {"id": "Mathlib.Tactic.Algebra.Lemmas", "region_id": "tactic", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0376, "macro_tier_override": null, "x": -92.536, "z": -201.106, "size": 0.2975, "title": "Lemmas for the `algebra` tactic.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Algebra/Lemmas.html"}, {"id": "Mathlib.Tactic.Bound", "region_id": "tactic", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": -141.572, "z": -151.386, "size": 0.3814, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Bound.html"}, {"id": "Mathlib.Tactic.Linter.GlobalAttributeIn", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -112.212, "z": -160.674, "size": 0.7902, "title": "Linter for `attribute [...] in` declarations", "summary": "Linter for global attributes created via `attribute [...] in` declarations. The syntax `attribute [instance] instName in` can be used to accidentally create a global instance. This is **not** obvious from reading the code, and in fact happened twice during the port, hence, we lint against it. *Example*: before this was discovered, `Mathlib/Topology/Category/TopCat/Basic.lean` contained the following code: ```…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/GlobalAttributeIn.html"}, {"id": "Mathlib.Tactic.ClickSuggestions.TryPremises", "region_id": "tactic", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0372, "macro_tier_override": null, "x": -85.599, "z": -196.66, "size": 0.2521, "title": "generating lemma suggestions, given the the shortlist of candidate lemmas", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ClickSuggestions/TryPremises.html"}, {"id": "Mathlib.Tactic.PNatToNat", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "`pnat_to_nat`", "summary": "This file implements the `pnat_to_nat` tactic that shifts `PNat`s in the context to `Nat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/PNatToNat.html"}, {"id": "Mathlib.Tactic.Ext", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "Documentation for `ext` tactic", "summary": "This file contains a library note on the use of the `ext` tactic and `@[ext]` attribute.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Ext.html"}, {"id": "Mathlib.Tactic.FunProp.Mor", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0558, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2719, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FunProp/Mor.html"}, {"id": "Mathlib.Tactic.Simproc.Factors", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -84.359, "z": -170.212, "size": 0.208, "title": "`simproc` for `Nat.primeFactorsList`", "summary": "Note that since `norm_num` can only produce numerals, we can't register this as a `norm_num` extension.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Simproc/Factors.html"}, {"id": "Mathlib.Tactic.NormNum.Prime", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -114.363, "z": -187.77, "size": 0.2737, "title": "`norm_num` extensions on natural numbers", "summary": "This file provides a `norm_num` extension to prove that natural numbers are prime and compute its minimal factor. Todo: compute the list of all factors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/Prime.html"}, {"id": "Mathlib.Tactic.Measurability", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": -111.932, "z": -165.479, "size": 0.3213, "title": "Measurability", "summary": "We define the `measurability` tactic using `aesop`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Measurability.html"}, {"id": "Mathlib.Tactic.Widget.CongrM", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -102.428, "z": -162.975, "size": 0.2647, "title": "CongrM widget", "summary": "This file defines a `congrm?` tactic that displays a widget panel allowing to generate a `congrm` call with holes specified by selecting subexpressions in the goal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Widget/CongrM.html"}, {"id": "Mathlib.Tactic.StacksAttribute", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -104.911, "z": -166.832, "size": 0.208, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/StacksAttribute.html"}, {"id": "Mathlib.Tactic.Clear_", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "`clear_` tactic", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Clear_.html"}, {"id": "Mathlib.Tactic.ErwQuestion", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "The `erw?` tactic", "summary": "`erw? [r, ...]` calls `erw [r, ...]` (at hypothesis `h` if written `erw [r, ...] at h`), and then attempts to identify any subexpression which would block the use of `rw` instead. It does so by identifying subexpressions which are defeq, but not at reducible transparency.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ErwQuestion.html"}, {"id": "Mathlib.Tactic.FBinop", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0031, "macro_tier_override": null, "x": -108.261, "z": -164.049, "size": 0.4373, "title": "Elaborator for functorial binary operators", "summary": "`fbinop% f x y` elaborates `f x y` for `x : S α` and `y : S' β`, taking into account any coercions that the \"functors\" `S` and `S'` possess. While `binop%` tries to solve for a single minimal type, `fbinop%` tries to solve the parameterized problem of solving for a single minimal \"functor.\" The code is drawn from the Lean 4 core `binop%` elaborator. Two simplifications made were 1. It is assumed that every `f` has a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FBinop.html"}, {"id": "Mathlib.Tactic.Simps.NotationClass", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0526, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.7287, "title": "`@[notation_class]` attribute for `@[simps]`", "summary": "This declares the `@[notation_class]` attribute, which is used to give smarter default projections for `@[simps]`. We put this in a separate file so that we can already tag some declarations with this attribute in the file where we declare `@[simps]`. For further documentation, see `Tactic.Simps.Basic`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Simps/NotationClass.html"}, {"id": "Mathlib.Tactic.Linarith.Oracle.FourierMotzkin", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -103.522, "z": -192.057, "size": 0.208, "title": "The Fourier-Motzkin elimination procedure", "summary": "The Fourier-Motzkin procedure is a variable elimination method for linear inequalities. Given a set of linear inequalities `comps = {tᵢ Rᵢ 0}`, we aim to eliminate a single variable `a` from the set. We partition `comps` into `comps_pos`, `comps_neg`, and `comps_zero`, where `comps_pos` contains the comparisons `tᵢ Rᵢ 0` in which the coefficient of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith/Oracle/FourierMotzkin.html"}, {"id": "Mathlib.Tactic.Linarith.Datatypes", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 1, "macro_tier_score": 0.0389, "macro_tier_override": null, "x": -125.667, "z": -179.644, "size": 0.3808, "title": "Datatypes for `linarith`", "summary": "Some of the data structures here are used in multiple parts of the tactic. We split them into their own file. This file also contains a few convenient auxiliary functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith/Datatypes.html"}, {"id": "Mathlib.Tactic.Cases", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "Backward compatible implementation of lean 3 `cases` tactic", "summary": "This tactic is similar to the `cases` tactic in Lean 4 core, but the syntax for giving names is different: ``` example (h : p ∨ q) : q ∨ p := by cases h with | inl hp => exact Or.inr hp | inr hq => exact Or.inl hq example (h : p ∨ q) : q ∨ p := by cases' h with hp hq · exact Or.inr hp · exact Or.inl hq example (h : p ∨ q) : q ∨ p := by rcases h with hp | hq · exact Or.inr hp · exact Or.inl hq ``` Prefer `cases` or…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Cases.html"}, {"id": "Mathlib.Tactic.Linter.TacticDocumentation", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -115.13, "z": -162.931, "size": 0.7902, "title": "The `tacticDocs` linter", "summary": "The `tacticDocs` environment linter checks that all tactics defined in a module come with a (nonempty) docstring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/TacticDocumentation.html"}, {"id": "Mathlib.Tactic.OfNat", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0265, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.5783, "title": "The `ofNat()` macro", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/OfNat.html"}, {"id": "Mathlib.Tactic.Linarith.Oracle.SimplexAlgorithm.PositiveVector", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0374, "macro_tier_override": null, "x": -108.08, "z": -176.247, "size": 0.2858, "title": "`linarith` certificate search as an LP problem", "summary": "`linarith` certificate search can easily be reduced to the following problem: given the matrix `A` and the list `strictIndexes`, find the nonnegative vector `v` such that some of its coordinates from the `strictIndexes` are positive and `A v = 0`. The function `findPositiveVector` solves this problem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/PositiveVector.html"}, {"id": "Mathlib.Tactic.Linarith.Oracle.SimplexAlgorithm.Gauss", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0373, "macro_tier_override": null, "x": -107.872, "z": -164.12, "size": 0.2735, "title": "Gaussian Elimination algorithm", "summary": "The first step of `Linarith.SimplexAlgorithm.findPositiveVector` is finding initial feasible solution which is done by standard Gaussian Elimination algorithm implemented in this file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/Gauss.html"}, {"id": "Mathlib.Tactic.Linarith.Oracle.SimplexAlgorithm.SimplexAlgorithm", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0373, "macro_tier_override": null, "x": -106.923, "z": -164.469, "size": 0.2735, "title": "Simplex Algorithm", "summary": "To obtain required vector in `Linarith.SimplexAlgorithm.findPositiveVector` we run the Simplex Algorithm. We use Bland's rule for pivoting, which guarantees that the algorithm terminates.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/SimplexAlgorithm.html"}, {"id": "Mathlib.Tactic.ByCases", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0223, "macro_tier_override": null, "x": -100.71, "z": -166.896, "size": 0.4597, "title": "The `by_cases!` tactic", "summary": "The `by_cases!` tactic is a variant of the `by_cases` tactic that also calls `push Not` on the generated hypothesis that is a negation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ByCases.html"}, {"id": "Mathlib.Tactic.CasesM", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0398, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.4241, "title": "`casesm`, `cases_type`, `constructorm` tactics", "summary": "These tactics implement repeated `cases` / `constructor` on anything satisfying a predicate.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CasesM.html"}, {"id": "Mathlib.Tactic.ITauto", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "Intuitionistic tautology (`itauto`) decision procedure", "summary": "The `itauto` tactic will prove any intuitionistic tautology. It implements the well-known `G4ip` algorithm: [Dyckhoff, *Contraction-free sequent calculi for intuitionistic logic*][dyckhoff_1992]. All built in propositional connectives are supported: `True`, `False`, `And`, `Or`, `→`, `Not`, `Iff`, `Xor`, as well as `Eq` and `Ne` on propositions. Anything else, including definitions and predicate logical connectives…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ITauto.html"}, {"id": "Mathlib.Tactic.NormNum.Ordinal", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -121.771, "z": -183.897, "size": 0.208, "title": "`norm_num` extensions for Ordinals", "summary": "The default `norm_num` extensions for many operators requires a semiring, which without a right distributive law, ordinals do not have. We must therefore define new extensions for them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/Ordinal.html"}, {"id": "Mathlib.Tactic.GCongr", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -116.535, "z": -177.616, "size": 0.2647, "title": "Setup for the `gcongr` tactic", "summary": "The core implementation of the `gcongr` (\"generalized congruence\") tactic is in the file `Tactic.GCongr.Core`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/GCongr.html"}, {"id": "Mathlib.Tactic.Hint", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.4084, "macro_tier_override": null, "x": -106.411, "z": -160.282, "size": 0.336, "title": "The `hint` tactic.", "summary": "The `hint` tactic tries the kitchen sink: it runs every tactic registered via the `register_hint tac` command on the current goal, and reports which ones succeed.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Hint.html"}, {"id": "Mathlib.Tactic.Ring.NamePolyVars", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2976, "title": null, "summary": "The command `name_poly_vars` names variables in `MvPolynomial (Fin n) R` for the appropriate value of `n`. The notation introduced by this command is local. Usage: ```lean variable (R : Type) [CommRing R] name_poly_vars X, Y, Z over R #check Y -- Y : MvPolynomial (Fin 3) R ```", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Ring/NamePolyVars.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.019, "macro_tier_override": null, "x": -111.867, "z": -160.525, "size": 0.2948, "title": "Coherence tactic for bicategories", "summary": "We provide a `bicategory_coherence` tactic, which proves that any two morphisms (with the same source and target) in a bicategory which are built out of associators and unitors are equal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Bicategory/PureCoherence.html"}, {"id": "Mathlib.Tactic.Observe", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "The `observe` tactic.", "summary": "`observe hp : p` asserts the proposition `p`, and tries to prove it using `exact?`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Observe.html"}, {"id": "Mathlib.Tactic.Translate.ToAdditive", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0678, "macro_tier_override": null, "x": -110.861, "z": -160.191, "size": 0.6688, "title": "The `@[to_additive]` attribute.", "summary": "The `@[to_additive]` attribute is used to translate multiplicative declarations to their additive equivalent. See the docstrings of `to_additive` for more information.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Translate/ToAdditive.html"}, {"id": "Mathlib.Tactic.DeprecateTo", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "`deprecate to` -- a deprecation tool", "summary": "Writing ```lean deprecate to new_name new_name₂ ... new_nameₙ theorem old_name : True := .intro ``` where `new_name new_name₂ ... new_nameₙ` is a sequence of identifiers produces the `Try this` suggestion: ```lean theorem new_name : True := .intro @[deprecated (since := \"YYYY-MM-DD\")] alias old_name := new_name @[deprecated (since := \"YYYY-MM-DD\")] alias old_name₂ := new_name₂ ... @[deprecated (since :=…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/DeprecateTo.html"}, {"id": "Mathlib.Tactic.ModCases", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -106.924, "z": -164.468, "size": 0.208, "title": "`mod_cases` tactic", "summary": "The `mod_cases` tactic does case disjunction on `e % n`, where `e : ℤ` or `e : ℕ`, to yield `n` new subgoals corresponding to the possible values of `e` modulo `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ModCases.html"}, {"id": "Mathlib.Tactic.Linter.DeprecatedSyntaxLinter", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -102.144, "z": -172.856, "size": 0.7902, "title": "Linter against deprecated syntax", "summary": "`refine'`, `cases'` and `induction'` provide backward-compatible implementations of their unprimed equivalents in Lean 3 –`refine`, `cases` and `induction` respectively. They have been superseded by Lean 4 tactics: * `refine` and `apply` replace `refine'`. While they are similar, they handle metavariables slightly differently; this means that they are not completely interchangeable, nor can one completely replace…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/DeprecatedSyntaxLinter.html"}, {"id": "Mathlib.Tactic.ArithMult", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -109.723, "z": -164.12, "size": 0.2669, "title": "Multiplicativity", "summary": "We define the `arith_mult` tactic using aesop", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ArithMult.html"}, {"id": "Mathlib.Tactic.ClearExcept", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "The `clear*` tactic", "summary": "This file provides a variant of the `clear` tactic, which clears all hypotheses it can besides a provided list, class instances, and auxiliary declarations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ClearExcept.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Reassoc", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0205, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.8076, "title": "The `reassoc` attribute", "summary": "Adding `@[reassoc]` to a lemma named `F` of shape `∀ .., f = g`, where `f g : X ⟶ Y` in some category will create a new lemma named `F_assoc` of shape `∀ .. {Z : C} (h : Y ⟶ Z), f ≫ h = g ≫ h` but with the conclusions simplified using the axioms for a category (`Category.comp_id`, `Category.id_comp`, and `Category.assoc`). This is useful for generating lemmas which the simplifier can use even on expressions that are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Reassoc.html"}, {"id": "Mathlib.Tactic.CategoryTheory.Coherence", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -101.558, "z": -164.301, "size": 0.2393, "title": "A `coherence` tactic for monoidal categories", "summary": "We provide a `coherence` tactic, which proves equations where the two sides differ by replacing strings of monoidal structural morphisms with other such strings. (The replacements are always equalities by the monoidal coherence theorem.) A simpler version of this tactic is `pure_coherence`, which proves that any two morphisms (with the same source and target) in a monoidal category which are built out of associators…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/Coherence.html"}, {"id": "Mathlib.Tactic.Polynomial.Core", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0192, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3127, "title": "Setup for the `polynomial` tactic", "summary": "This file initializes the environment extensions and simp sets used by the `polynomial` tactic. These extensions let downstream users use their own polynomial-like types (such as `PowerSeries`) with the `polynomial` tactic suite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Polynomial/Core.html"}, {"id": "Mathlib.Tactic.Widget.CommDiag", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": null, "summary": "This module defines tactic/meta infrastructure for displaying commutative diagrams in the infoview.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Widget/CommDiag.html"}, {"id": "Mathlib.Tactic.Simproc.ExistsAndEq", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0251, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.5442, "title": "Simproc for `∃ a', ... ∧ a' = a ∧ ...`", "summary": "This module implements the `existsAndEq` simproc, which triggers on goals of the form `∃ a, P`. It checks whether `P` allows only one possible value for `a`, and if so, substitutes it, eliminating the leading quantifier. The procedure traverses the body, branching at each `∧` and entering existential quantifiers, searching for a subexpression of the form `a = a'` or `a' = a` for `a'` that is independent of `a`. If…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Simproc/ExistsAndEq.html"}, {"id": "Mathlib.Tactic.NormNum.Parity", "region_id": "tactic", "micro_elevation": 0.3333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -111.579, "z": -151.988, "size": 0.208, "title": "`norm_num` extensions for `Even` and `Odd`", "summary": "In this file we provide `norm_num` extensions for `Even n` and `Odd n`, where `n : ℕ` or `n : ℤ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/Parity.html"}, {"id": "Mathlib.Tactic.Linter.CommandStart", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -111.046, "z": -180.158, "size": 0.208, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/CommandStart.html"}, {"id": "Mathlib.Tactic.Continuity.Init", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0191, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2991, "title": "Continuity Rule Set", "summary": "This module defines the `Continuous` Aesop rule set which is used by the `continuity` tactic. Aesop rule sets only become visible once the file in which they're declared is imported, so we must put this declaration into its own file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Continuity/Init.html"}, {"id": "Mathlib.Tactic.Widget.LibraryRewrite", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -100.938, "z": -165.844, "size": 0.2647, "title": "Point & click library rewriting", "summary": "This file defines `rw??`, an interactive tactic that suggests rewrites for any expression selected by the user. `rw??` uses a (lazy) `RefinedDiscrTree` to lookup a list of candidate rewrite lemmas. It excludes lemmas that are automatically generated. Each lemma is then checked one by one to see whether it is applicable. For each lemma that works, the corresponding rewrite tactic is constructed and converted into a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Widget/LibraryRewrite.html"}, {"id": "Mathlib.Tactic.CategoryTheory.MonoidalComp", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0381, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3351, "title": "Monoidal composition `⊗≫` (composition up to associators)", "summary": "We provide `f ⊗≫ g`, the `monoidalComp` operation, which automatically inserts associators and unitors as needed to make the target of `f` match the source of `g`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/MonoidalComp.html"}, {"id": "Mathlib.Tactic.DSimpPercent", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3666, "title": null, "summary": "`dsimp% […] t` runs `dsimp […]` on term `t`. If `t` is a proof, then it runs `dsimp […]` on the type of `t` instead. For instance, instead of ``` have foo := ... dsimp at foo rw [foo] ``` one can do `rw [dsimp% foo]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/DSimpPercent.html"}, {"id": "Mathlib.Tactic.CancelDenoms.Core", "region_id": "tactic", "micro_elevation": 0.3333, "macro_tier": 1, "macro_tier_score": 0.0559, "macro_tier_override": null, "x": -106.079, "z": -151.977, "size": 0.2774, "title": "A tactic for canceling numeric denominators", "summary": "This file defines tactics that cancel numeric denominators from field Expressions. As an example, we want to transform a comparison `5*(a/3 + b/4) < c/3` into the equivalent `5*(4*a + 3*b) < 4*c`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CancelDenoms/Core.html"}, {"id": "Mathlib.Tactic.ComputeAsymptotics.Multiseries.Trimming", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -116.269, "z": -164.784, "size": 0.208, "title": "Trimming of multiseries", "summary": "A multiseries is *trimmed* when its leading coefficient (the head of its expansion) is itself trimmed and non-zero. For a trimmed multiseries, the leading monomial captures the main asymptotic behavior of the approximated function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Trimming.html"}, {"id": "Mathlib.Tactic.IrreducibleDef", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0192, "macro_tier_override": null, "x": -104.807, "z": -167.211, "size": 0.3131, "title": "Irreducible definitions", "summary": "This file defines an `irreducible_def` command, which works almost like the `def` command except that the introduced definition does not reduce to the value. Instead, the command adds a `_def` lemma which can be used for rewriting. ``` irreducible_def frobnicate (a b : Nat) := a + b example : frobnicate a 0 = a := by simp [frobnicate_def] ```", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/IrreducibleDef.html"}, {"id": "Mathlib.Tactic.FunProp.Core", "region_id": "tactic", "micro_elevation": 0.3333, "macro_tier": 1, "macro_tier_score": 0.0566, "macro_tier_override": null, "x": -121.057, "z": -157.282, "size": 0.3364, "title": "Tactic `fun_prop` for proving function properties like `Continuous f`, `Differentiable ℝ f`, ...", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FunProp/Core.html"}, {"id": "Mathlib.Tactic.ProdAssoc", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "Associativity of products", "summary": "This file constructs a term elaborator for \"obvious\" equivalences between iterated products. For example, ```lean (prod_assoc% : (α × β) × (γ × δ) ≃ α × (β × γ) × δ) ``` gives the \"obvious\" equivalence between `(α × β) × (γ × δ)` and `α × (β × γ) × δ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ProdAssoc.html"}, {"id": "Mathlib.Tactic.DeriveCountable", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "`Countable` deriving handler", "summary": "Adds a deriving handler for the `Countable` class.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/DeriveCountable.html"}, {"id": "Mathlib.Tactic.Polynomial.Basic", "region_id": "tactic", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -139.854, "z": -130.128, "size": 0.208, "title": "Polynomial", "summary": "An extensible tactic for proving equality of polynomial expressions implemented using `algebra`. To add support for a new polynomial-like type, one needs to do three things: * Implement a polynomial extension that lets `polynomial` infer the base ring from the algebraic type. For example: ``` @[polynomial_infer_base] def polynomialInferBase : PolynomialExt where infer e := do match_expr e with | Polynomial R _ =>…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Polynomial/Basic.html"}, {"id": "Mathlib.Tactic.Algebra.AlgebraNF", "region_id": "tactic", "micro_elevation": 0.9167, "macro_tier": 0, "macro_tier_score": 0.0186, "macro_tier_override": null, "x": -63.917, "z": -170.925, "size": 0.2392, "title": "The `algebra_nf` tactic", "summary": "This file contains helper functions for the (currently unimplemented) `algebra_nf` tactic. The defnitions in this file are currently only used by `polynomial_nf`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Algebra/AlgebraNF.html"}, {"id": "Mathlib.Tactic.ToLevel", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -105.336, "z": -165.925, "size": 0.2647, "title": "`ToLevel` class", "summary": "This module defines `Lean.ToLevel`, which is the `Lean.Level` analogue to `Lean.ToExpr`. **Warning:** Import `Mathlib/Tactic/ToExpr.lean` instead of this one if you are writing `ToExpr` instances. This ensures that you are using the universe polymorphic `ToExpr` instances that override the ones from Lean 4 core.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ToLevel.html"}, {"id": "Mathlib.Tactic.Widget.GCongr", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -100.912, "z": -170.267, "size": 0.208, "title": "GCongr widget", "summary": "This file defines a `gcongr?` tactic that displays a widget panel allowing to generate a `gcongr` call with holes specified by selecting subexpressions in the goal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Widget/GCongr.html"}, {"id": "Mathlib.Tactic.NormNum.NatFib", "region_id": "tactic", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -146.901, "z": -182.91, "size": 0.2389, "title": "`norm_num` extension for `Nat.fib`", "summary": "This `norm_num` extension uses a strategy parallel to that of `Nat.fastFib`, but it instead produces proofs of what `Nat.fib` evaluates to.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/NatFib.html"}, {"id": "Mathlib.Tactic.Linter.FlexibleLinter", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -116.849, "z": -169.519, "size": 0.7902, "title": "The \"flexible\" linter", "summary": "The \"flexible\" linter makes sure that a \"rigid\" tactic (such as `rw`) does not act on the output of a \"flexible\" tactic (such as `simp`). For example, this ensures that, if you want to use `simp [...]` in the middle of a proof, then you should replace `simp [...]` by one of * a `suffices \\\"expr after simp\\\" by simpa` line; * the output of `simp? [...]`, so that the final code contains `simp only [...]`; * something…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/FlexibleLinter.html"}, {"id": "Mathlib.Tactic.Widget.SelectInsertParamsClass", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0569, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3567, "title": "SelectInsertParamsClass", "summary": "Defines the basic class of parameters for a select and insert widget. This needs to be in a separate file in order to initialize the deriving handler.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Widget/SelectInsertParamsClass.html"}, {"id": "Mathlib.Tactic.Attr.Core", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0035, "macro_tier_override": null, "x": -110.737, "z": -171.7, "size": 0.4508, "title": "Simp tags for core lemmas", "summary": "In Lean 4, an attribute declared with `register_simp_attr` cannot be used in the same file. So, we declare all `simp` attributes used in `Mathlib` in `Mathlib/Tactic/Attr/Register` and tag lemmas from the core library and the `Batteries` library with these attributes in this file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Attr/Core.html"}, {"id": "Mathlib.Tactic.Use", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0196, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3385, "title": "The `use` tactic", "summary": "The `use` and `use!` tactics are for instantiating one-constructor inductive types just like the `exists` tactic, but they can be a little more flexible. `use` is the more restrained version for mathlib4, and `use!` is the exuberant version that more closely matches `use` from mathlib3. Note: The `use!` tactic is almost exactly the mathlib3 `use` except that it does not try applying `exists_prop`. See the failing…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Use.html"}, {"id": "Mathlib.Tactic.NormNum.NatFactorial", "region_id": "tactic", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -134.579, "z": -136.377, "size": 0.2598, "title": "`norm_num` extensions for factorials", "summary": "Extensions for `norm_num` that compute `Nat.factorial`, `Nat.ascFactorial` and `Nat.descFactorial`. This is done by reducing each of these to `ascFactorial`, which is computed using a divide and conquer strategy that improves performance and avoids exceeding the recursion depth.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/NatFactorial.html"}, {"id": "Mathlib.Tactic.Clean", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "`clean%` term elaborator", "summary": "Remove identity functions from a term.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Clean.html"}, {"id": "Mathlib.Tactic.Linter.UnusedTactic", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -115.318, "z": -157.716, "size": 0.7902, "title": "The unused tactic linter", "summary": "The unused linter makes sure that every tactic call actually changes *something*. The inner workings of the linter are as follows. The linter inspects the goals before and after each tactic execution. If they are not identical, the linter is happy. If they are identical, then the linter checks if the tactic is whitelisted. Possible reason for whitelisting are * tactics that emit messages, such as `have?`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/UnusedTactic.html"}, {"id": "Mathlib.Tactic.FunProp.Attr", "region_id": "tactic", "micro_elevation": 0.3333, "macro_tier": 1, "macro_tier_score": 0.0567, "macro_tier_override": null, "x": -93.543, "z": -173.996, "size": 0.3456, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FunProp/Attr.html"}, {"id": "Mathlib.Tactic.FunProp.Elab", "region_id": "tactic", "micro_elevation": 0.4167, "macro_tier": 1, "macro_tier_score": 0.0565, "macro_tier_override": null, "x": -91.807, "z": -179.467, "size": 0.3283, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FunProp/Elab.html"}, {"id": "Mathlib.Tactic.TypeCheck", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/TypeCheck.html"}, {"id": "Mathlib.Tactic.TermCongr", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0391, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3925, "title": "`congr(...)` congruence quotations", "summary": "This module defines a term elaborator for generating congruence lemmas from patterns written using quotation syntax. One can write `congr($hf $hx)` with `hf : f = f'` and `hx : x = x'` to get `f x = f' x'`. While in simple cases it might be possible to use `congr_arg` or `congr_fun`, congruence quotations are more general, since for example `f` could have implicit arguments, complicated dependent types, and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/TermCongr.html"}, {"id": "Mathlib.Tactic.Widget.StringDiagram", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -115.712, "z": -157.973, "size": 0.208, "title": "String Diagram Widget", "summary": "This file provides meta infrastructure for displaying string diagrams for morphisms in monoidal categories in the infoview. To enable the string diagram widget, you need to import this file and inserting `with_panel_widgets [Mathlib.Tactic.Widget.StringDiagram]` at the beginning of the proof. Alternatively, you can also write ```lean open Mathlib.Tactic.Widget show_panel_widgets [local StringDiagram] ``` to enable…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Widget/StringDiagram.html"}, {"id": "Mathlib.Tactic.Simproc.Divisors", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "Divisor Simprocs", "summary": "This file implements (d)simprocs to compute various objects related to divisors: - `Nat.divisors_ofNat`: computes `Nat.divisors n` for explicit values of `n` - `Nat.properDivisors_ofNat`: computes `Nat.properDivisors n` for explicit values of `n`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Simproc/Divisors.html"}, {"id": "Mathlib.Tactic.Linarith.NNRealPreprocessor", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -101.146, "z": -170.987, "size": 0.208, "title": "NNReal linarith preprocessing", "summary": "This file contains a `linarith` preprocessor for converting (in)equalities in `ℝ≥0` to `ℝ`. By overriding the behaviour of the placeholder preprocessor `nnrealToReal` (which does nothing unless this file is imported) `linarith` can still be used without importing `NNReal`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith/NNRealPreprocessor.html"}, {"id": "Mathlib.Tactic.Linter.EmptyLine", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -109.051, "z": -176.274, "size": 0.7902, "title": "The \"emptyLine\" linter", "summary": "The \"emptyLine\" linter emits a warning on empty lines inside a command, but outside of a doc-string/module-doc.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/EmptyLine.html"}, {"id": "Mathlib.Tactic.Substs", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.026, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.5679, "title": "The `substs` macro", "summary": "The `substs` macro is a deprecated version of the `subst` tactic that allowed for more than one hypothesis. Since `subst` now also supports multiple hypotheses, `substs` is deprecated.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Substs.html"}, {"id": "Mathlib.Tactic.NormNum.LegendreSymbol", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "A `norm_num` extension for Jacobi and Legendre symbols", "summary": "We extend the `norm_num` tactic so that it can be used to provably compute the value of the Jacobi symbol `J(a | b)` or the Legendre symbol `legendreSym p a` when the arguments are numerals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/LegendreSymbol.html"}, {"id": "Mathlib.Tactic.AdaptationNote", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -112.784, "z": -160.964, "size": 0.7902, "title": "Adaptation notes", "summary": "This file defines a `#adaptation_note` command. Adaptation notes are comments that are used to indicate that a piece of code has been changed to accommodate a change in Lean core. They typically require further action/maintenance to be taken in the future.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/AdaptationNote.html"}, {"id": "Mathlib.Tactic.Linter.AuxLemma", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -116.208, "z": -164.649, "size": 0.7902, "title": "The `auxLemma` linter", "summary": "The `auxLemma` linter flags explicit references to auto-generated \"auxiliary\" declarations such as `_proof_1`, `match_1`, or `_sizeOf_1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/AuxLemma.html"}, {"id": "Mathlib.Tactic.Linter.Multigoal", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -116.796, "z": -166.411, "size": 0.7902, "title": "The \"multiGoal\" linter", "summary": "The \"multiGoal\" linter emits a warning where there is more than a single goal in scope. There is an exception: a tactic that closes *all* remaining goals is allowed. There are a few tactics, such as `skip`, `swap` or the `try` combinator, that are intended to work specifically in such a situation. Otherwise, the mathlib style guide ask that goals be focused until there is only one active goal at a time. If this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/Multigoal.html"}, {"id": "Mathlib.Tactic.Linter.PrivateModule", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -101.178, "z": -171.07, "size": 0.7902, "title": "Private module linter", "summary": "This linter lints against nonempty modules that have only private declarations, and suggests adding `@[expose] public section` to the top or selectively marking declarations as `public`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/PrivateModule.html"}, {"id": "Mathlib.Tactic.Linter.Style", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -121.009, "z": -169.229, "size": 0.7902, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/Style.html"}, {"id": "Mathlib.Tactic.WLOG", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -106.811, "z": -160.171, "size": 0.2647, "title": "Without loss of generality tactic", "summary": "The tactic `wlog h : P` will add an assumption `h : P` to the main goal, and add a new goal that requires showing that the case `h : ¬ P` can be reduced to the case where `P` holds (typically by symmetry). `wlog! h : P` is a variant that will also call `push Not` at `h : ¬ P`. The new goal will be placed at the top of the goal stack.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/WLOG.html"}, {"id": "Mathlib.Tactic.RSuffices", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "`rsuffices` tactic", "summary": "The `rsuffices` tactic is an alternative version of `suffices`, that allows the usage of any syntax that would be valid in an `obtain` block. This tactic just calls `obtain` on the expression, and then `rotate_left`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/RSuffices.html"}, {"id": "Mathlib.Tactic.GuardGoalNums", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": null, "summary": "A tactic stub file for the `guard_goal_nums` tactic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/GuardGoalNums.html"}, {"id": "Mathlib.Tactic.RenameBVar", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "The `rename_bvar` tactic", "summary": "This file defines the `rename_bvar` tactic, for renaming bound variables.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/RenameBVar.html"}, {"id": "Mathlib.Tactic.Recall", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "`recall` command", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Recall.html"}, {"id": "Mathlib.Tactic.Recover", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "The `recover` tactic modifier", "summary": "This defines the `recover` tactic modifier, which can be used to debug cases where goals are not closed correctly. `recover tacs` for a tactic (or tactic sequence) `tacs` applies the tactics and then adds goals that are not closed, starting from the original goal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Recover.html"}, {"id": "Mathlib.Tactic.ComputeAsymptotics.Multiseries.Corecursion", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -111.116, "z": -171.468, "size": 0.208, "title": "Non-primitive corecursion for sequences", "summary": "Primitive corecursive definition of the form ``` def foo (x : X) := hd x :: foo (tlArg x) ``` (where hd and tlArg are arbitrary functions) can be encoded via the corecursor `Seq.corec`. It is not enough, however, to define multiplication and `powser` operation for multiseries. This file implements a more general form of corecursion in the spirit of [blanchette2015]. This is a bare minimum that needed for the tactic,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Corecursion.html"}, {"id": "Mathlib.Tactic.FunProp.ToBatteries", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0558, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2719, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FunProp/ToBatteries.html"}, {"id": "Mathlib.Tactic.Linarith.Preprocessing", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0377, "macro_tier_override": null, "x": -91.613, "z": -150.599, "size": 0.3099, "title": "Linarith preprocessing", "summary": "This file contains methods used to preprocess inputs to `linarith`. In particular, `linarith` works over comparisons of the form `t R 0`, where `R ∈ {<,≤,=}`. It assumes that expressions in `t` have integer coefficients and that the type of `t` has well-behaved subtraction.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith/Preprocessing.html"}, {"id": "Mathlib.Tactic.Translate.ToDual", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.042, "macro_tier_override": null, "x": -110.479, "z": -180.25, "size": 0.8502, "title": "The `@[to_dual]` attribute.", "summary": "The `@[to_dual]` attribute is used to translate declarations to their dual equivalent. See the docstrings of `to_dual` and `to_additive` for more information. Known limitations: - When combining `to_additive` and `to_dual`, we need to make sure that all translations are added. For example `attribute [to_dual (attr := to_additive) le_mul] mul_le` should generate `le_mul`, `le_add` and `add_le`, and in particular…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Translate/ToDual.html"}, {"id": "Mathlib.Tactic.Translate.TagUnfoldBoundary", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0572, "macro_tier_override": null, "x": -101.81, "z": -172.35, "size": 0.8026, "title": "Tagging of unfold boundaries for translation attributes", "summary": "The file `Mathlib.Tactic.Translate.UnfoldBoundary` defines how to add unfold boundaries in terms. In this file, we define the infrastructure for tagging declaration to be used for that. This is in a separate file, because we need to import `Mathlib.Tactic.Translate.Core` here.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Translate/TagUnfoldBoundary.html"}, {"id": "Mathlib.Tactic.Linter.CommandRanges", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "The \"commandRanges\" linter", "summary": "The \"commandRanges\" linter simply logs the `getRange?` and the `getRangeWithTrailing?` for each command. This is useful for the \"removeDeprecations\" automation, since it helps identifying the exact range of each declaration that should be removed. This linter is strictly tied to the `#clear_deprecations` command in `Mathlib/Tactic/Linter/FindDeprecations.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/CommandRanges.html"}, {"id": "Mathlib.Tactic.Order.ToInt", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0189, "macro_tier_override": null, "x": -105.09, "z": -166.377, "size": 0.2797, "title": "Translating linear orders to ℤ", "summary": "In this file we implement the translation of a problem in any linearly ordered type to a problem in `ℤ`. This allows us to use the `lia` tactic to solve it. While the core algorithm of the `order` tactic is complete for the theory of linear orders in the signature (`<`, `≤`), it becomes incomplete in the signature with lattice operations `⊓` and `⊔`. With these operations, the problem becomes NP-hard, and the idea…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Order/ToInt.html"}, {"id": "Mathlib.Tactic.Linter", "region_id": "tactic", "micro_elevation": 0.3333, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -122.334, "z": -177.275, "size": 0.2647, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter.html"}, {"id": "Mathlib.Tactic.Linter.PPRoundtrip", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0371, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2451, "title": "The \"ppRoundtrip\" linter", "summary": "The \"ppRoundtrip\" linter emits a warning when the syntax of a command differs substantially from the pretty-printed version of itself.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/PPRoundtrip.html"}, {"id": "Mathlib.Tactic.Linter.UpstreamableDecl", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0371, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2451, "title": "The `upstreamableDecl` linter", "summary": "The `upstreamableDecl` linter detects declarations that could be moved to a file higher up in the import hierarchy. This is intended to assist with splitting files.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/UpstreamableDecl.html"}, {"id": "Mathlib.Tactic.Linarith.Oracle.SimplexAlgorithm.Datatypes", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0381, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3353, "title": "Datatypes for the Simplex Algorithm implementation", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/Datatypes.html"}, {"id": "Mathlib.Tactic.CancelDenoms", "region_id": "tactic", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -87.173, "z": -143.566, "size": 0.208, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CancelDenoms.html"}, {"id": "Mathlib.Tactic.SudoSetOption", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "Defines the `sudo set_option` command.", "summary": "Allows setting undeclared options.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/SudoSetOption.html"}, {"id": "Mathlib.Tactic.FailIfNoProgress", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "Fail if no progress", "summary": "This implements the `fail_if_no_progress` tactic, which fails if no actual progress is made by the following tactic sequence. \"Actual progress\" means that either the number of goals has changed, that the number or presence of expressions in the context has changed, or that the type of some expression in the context or the type of the goal is no longer definitionally equal to what it used to be at reducible…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FailIfNoProgress.html"}, {"id": "Mathlib.Tactic.NormNum.NatSqrt", "region_id": "tactic", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -130.347, "z": -202.841, "size": 0.208, "title": "`norm_num` extension for `Nat.sqrt`", "summary": "This module defines a `norm_num` extension for `Nat.sqrt`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/NatSqrt.html"}, {"id": "Mathlib.Tactic.ClickSuggestions.Unfold", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0372, "macro_tier_override": null, "x": -106.562, "z": -143.676, "size": 0.2521, "title": "Interactive unfolding", "summary": "This file defines the interactive tactic `unfold?`. It allows you to shift-click on an expression in the goal, and then it suggests rewrites to replace the expression with an unfolded version. It can be used on its own, but it can also be used as part of the library rewrite tactic `rw??`, where these unfoldings are a subset of the suggestions. For example, if the goal contains `1+1`, then it will suggest rewriting…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ClickSuggestions/Unfold.html"}, {"id": "Mathlib.Tactic.NormNum.IsCoprime", "region_id": "tactic", "micro_elevation": 0.9167, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -141.062, "z": -136.779, "size": 0.208, "title": "`norm_num` extension for `IsCoprime`", "summary": "This module defines a `norm_num` extension for `IsCoprime` over `ℤ`. (While `IsCoprime` is defined over `ℕ`, since it uses Bezout's identity with `ℕ` coefficients it does not correspond to the usual notion of coprime.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/IsCoprime.html"}, {"id": "Mathlib.Tactic.FunProp.FunctionData", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.0559, "macro_tier_override": null, "x": -105.894, "z": -170.981, "size": 0.2818, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FunProp/FunctionData.html"}, {"id": "Mathlib.Tactic.ApplyCongr", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ApplyCongr.html"}, {"id": "Mathlib.Tactic.ApplyWith", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "The `applyWith` tactic", "summary": "The `applyWith` tactic is like `apply`, but allows passing a custom configuration to the underlying `apply` operation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ApplyWith.html"}, {"id": "Mathlib.Tactic.Check", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "`#check` tactic", "summary": "This module defines a tactic version of the `#check` command. While `#check t` is similar to `have := t`, it is a little more convenient since it elaborates `t` in a more tolerant way and so it can be possible to get a result. For example, `#check` allows metavariables. This module also defines the `#check'` tactic and command, which behaves like `#check` but only shows explicit arguments in the signature.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Check.html"}, {"id": "Mathlib.Tactic.ClearExclamation", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "`clear!` tactic", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ClearExclamation.html"}, {"id": "Mathlib.Tactic.ClickSuggestions", "region_id": "tactic", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -81.808, "z": -198.808, "size": 0.2647, "title": "Point & click suggestions", "summary": "This file defines `#click_suggestions`, a command that enables an interactive interface that gives lemma/tactic suggestions for any expression selected by the user. The library search uses a (lazy) `RefinedDiscrTree` to lookup a list of candidate rewrite lemmas. It excludes lemmas that are automatically generated. Each lemma is then checked one by one (in parallel) to see whether it is applicable. For each lemma…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ClickSuggestions.html"}, {"id": "Mathlib.Tactic.CongrM", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -104.752, "z": -167.509, "size": 0.2647, "title": "The `congrm` tactic", "summary": "The `congrm` tactic (\"`congr` with matching\") is a convenient frontend for `congr(...)` congruence quotations. Roughly, `congrm e` is `refine congr(e')`, where `e'` is `e` with every `?m` placeholder replaced by `$(?m)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CongrM.html"}, {"id": "Mathlib.Tactic.Constructor", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "The `fconstructor` and `econstructor` tactics", "summary": "The `fconstructor` and `econstructor` tactics are variants of the `constructor` tactic in Lean core, except that - `fconstructor` does not reorder goals - `econstructor` adds only non-dependent premises as new goals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Constructor.html"}, {"id": "Mathlib.Tactic.DefEqAbuse", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "The `#defeq_abuse` tactic and command combinators", "summary": "**WARNING:** `#defeq_abuse` is an experimental tool intended to assist with breaking changes to transparency handling (associated with `backward.isDefEq.respectTransparency`). Its syntax may change at any time, and it may not behave as expected. Please report unexpected behavior [on Zulip](https://leanprover.zulipchat.com/#narrow/channel/113488-general/topic/backward.2EisDefEq.2ErespectTransparency/with/575685551).…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/DefEqAbuse.html"}, {"id": "Mathlib.Tactic.DefEqTransformations", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "Tactics that transform types into definitionally equal types", "summary": "This module defines a standard wrapper that can be used to create tactics that change hypotheses and the goal to things that are definitionally equal. It then provides a number of tactics that transform local hypotheses and/or the target.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/DefEqTransformations.html"}, {"id": "Mathlib.Tactic.GuardHypNums", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": null, "summary": "A tactic stub file for the `guard_hyp_nums` tactic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/GuardHypNums.html"}, {"id": "Mathlib.Tactic.InferParam", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "Infer an optional parameter", "summary": "In this file we define a tactic `infer_param` that closes a goal with default value by using this default value.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/InferParam.html"}, {"id": "Mathlib.Tactic.Relation.Rfl", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "`Lean.MVarId.liftReflToEq`", "summary": "Convert a goal of the form `x ~ y` into the form `x = y`, where `~` is a reflexive relation, that is, a relation which has a reflexive lemma tagged with the attribute `@[refl]`. If this can't be done, returns the original `MVarId`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Relation/Rfl.html"}, {"id": "Mathlib.Tactic.Rename", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "The `rename'` tactic", "summary": "The `rename'` tactic renames one or several hypotheses.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Rename.html"}, {"id": "Mathlib.Tactic.SimpIntro", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "`simp_intro` tactic", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/SimpIntro.html"}, {"id": "Mathlib.Tactic.UnsetOption", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2647, "title": "The `unset_option` command", "summary": "This file defines an `unset_option` user command, which unsets user configurable options. For example, inputting `set_option blah 7` and then `unset_option blah` returns the user to the default state before any `set_option` command is called. This is helpful when the user does not know the default value of the option or it is cleaner not to write it explicitly, or for some options where the default behaviour is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/UnsetOption.html"}, {"id": "Mathlib.Tactic.Order.Graph.Tarjan", "region_id": "tactic", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0189, "macro_tier_override": null, "x": -102.733, "z": -162.618, "size": 0.2797, "title": "Tarjan's Algorithm", "summary": "This file implements Tarjan's algorithm for finding the strongly connected components (SCCs) of a graph.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Order/Graph/Tarjan.html"}, {"id": "Mathlib.Tactic.FieldSimp.Attr", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0375, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2888, "title": "Attribute grouping the `field_simp` simprocs", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FieldSimp/Attr.html"}, {"id": "Mathlib.Tactic.FieldSimp.Lemmas", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0375, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2888, "title": "Lemmas for the `field_simp` tactic", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/FieldSimp/Lemmas.html"}, {"id": "Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic", "region_id": "tactic", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -115.344, "z": -178.472, "size": 0.208, "title": "Computing limits of monomials", "summary": "In this file we define the `Monomial` structure, representing monomials in a basis, i.e. `coef * b₁ ^ e₁ * ... * bₙ ^ eₙ` where `[b₁, ..., bₙ]` is a well-formed basis. In the tactic implementation, we use `Monomial` to connect multiseries with real functions. In this file we show how to find a limit of `Monomial` and how to asymptotically compare two `Monomial`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Basic.html"}, {"id": "Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Predicates", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0186, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.248, "title": "Predicates on monomials", "summary": "In this file we define `UnitMonomial`: type to represent monomials without coefficient as a list of its exponents. `[e₁, e₂, ..., eₙ]` corresponds to `basis[0] ^ e₁ * ... * basis[n] ^ eₙ` where `basis` is the basis of functions. Then we define some predicates for these lists: 1. `FirstNonzeroIsPos li` means that the first non-zero element of the list `li` is positive. 2. `FirstNonzeroIsNeg li` means that the first…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Predicates.html"}, {"id": "Mathlib.Tactic.Polyrith", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "polyrith Tactic", "summary": "The `polyrith` tactic relied on an external Sage server which has been shut down. Hence this is no longer supported in Mathlib, but could be restored if someone wanted to provide an alternative backend (based on Sage or otherwise). In the meantime, the `grobner` tactic (which calls into the Grobner basis module of `grind`) can close goals requiring polynomial reasoning, but is not able to give a \"Try this:\"…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Polyrith.html"}, {"id": "Mathlib.Tactic.CategoryTheory.CheckCompositions", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": null, "summary": "The `check_compositions` tactic, which checks the typing of categorical compositions in the goal, reporting discrepancies at \"instances and reducible\" transparency. Reports from this tactic do not necessarily indicate a problem, although typically `simp` should reduce rather than increase the reported discrepancies. `check_compositions` may be useful in diagnosing uses of `erw` in the category theory library.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/CategoryTheory/CheckCompositions.html"}, {"id": "Mathlib.Tactic.Determinant.Bird", "region_id": "tactic", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -130.789, "z": -211.953, "size": 0.208, "title": "`norm_det` simproc and `eval_det` tactic", "summary": "A tactic for normalizing matrix determinants.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Determinant/Bird.html"}, {"id": "Mathlib.Tactic.Ring.NamePowerVars", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.2615, "title": null, "summary": "The command `name_power_vars` names variables in `MvPowerSeries (Fin n) R` for the appropriate value of `n`. The notation introduced by this command is local. Usage: ```lean variable (R : Type) [CommRing R] name_power_vars X, Y, Z over R #check Y -- Y : MvPowerSeries (Fin 3) R ```", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Ring/NamePowerVars.html"}, {"id": "Mathlib.Tactic.Have", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0188, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.268, "title": "Extending `have`, `let` and `suffices`", "summary": "This file extends the `have`, `let` and `suffices` tactics to allow the addition of hypotheses to the context without requiring their proofs to be provided immediately. As a style choice, this should not be used in mathlib; but is provided for downstream users who preferred the old style.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Have.html"}, {"id": "Mathlib.Tactic.Change", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "Tactic `change? term`", "summary": "This tactic is used to suggest a replacement of the goal by a definitionally equal term. `term` is intended to contain holes which get unified with the main goal and filled in explicitly in the suggestion. `term` can also be omitted, in which case `change?` simply suggests `change` with the main goal. This is helpful after tactics like `dsimp`, which can then be deleted.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Change.html"}, {"id": "Mathlib.Tactic.ReduceModChar", "region_id": "tactic", "micro_elevation": 0.5833, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -86.926, "z": -149.647, "size": 0.208, "title": "`reduce_mod_char` tactic", "summary": "Define the `reduce_mod_char` tactic, which traverses expressions looking for numerals `n`, such that the type of `n` is a ring of (positive) characteristic `p`, and reduces these numerals modulo `p`, to lie between `0` and `p`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/ReduceModChar.html"}, {"id": "Mathlib.Tactic.Linarith.Oracle.SimplexAlgorithm", "region_id": "tactic", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0377, "macro_tier_override": null, "x": -84.905, "z": -162.55, "size": 0.3099, "title": "The oracle based on Simplex Algorithm", "summary": "This file contains hooks to enable the use of the Simplex Algorithm in `linarith`. The algorithm's entry point is the function `Linarith.SimplexAlgorithm.findPositiveVector`. See the file `PositiveVector.lean` for details of how the procedure works.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm.html"}, {"id": "Mathlib.Tactic.NormNum.IsSquare", "region_id": "tactic", "micro_elevation": 0.9167, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -137.718, "z": -202.537, "size": 0.208, "title": "`norm_num` extension for `IsSquare`", "summary": "The extension in this file handles natural, integer, and rational numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/IsSquare.html"}, {"id": "Mathlib.Tactic.NormNum.Result", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.3712, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.3201, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/NormNum/Result.html"}, {"id": "Mathlib.Tactic.DeriveEncodable", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "`Encodable` deriving handler", "summary": "Adds a deriving handler for the `Encodable` class. The resulting `Encodable` instance should be considered to be opaque. The specific encoding used is an implementation detail.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/DeriveEncodable.html"}, {"id": "Mathlib.Tactic.Replace", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -106.699, "z": -171.611, "size": 0.208, "title": "Extending `replace`", "summary": "This file extends the `replace` tactic from the standard library to allow the addition of hypotheses to the context without requiring their proofs to be provided immediately. As a style choice, this should not be used in mathlib; but is provided for downstream users who preferred the old style.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Replace.html"}, {"id": "Mathlib.Tactic.Relation.Symm", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "`relSidesIfSymm?`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Relation/Symm.html"}, {"id": "Mathlib.Tactic.Linter.ValidatePRTitle", "region_id": "tactic", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -106.869, "z": -164.497, "size": 0.208, "title": "Checker for well-formed title and labels", "summary": "This script checks if a PR title matches [mathlib's commit conventions](https://leanprover-community.github.io/contribute/commit.html). Not all checks from the commit conventions are implemented: for instance, no effort is made to verify whether the title or body are written in present imperative tense.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Linter/ValidatePRTitle.html"}, {"id": "Mathlib.Tactic.DeriveTraversable", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "Deriving handler for `Traversable` instances", "summary": "This module gives deriving handlers for `Functor`, `LawfulFunctor`, `Traversable`, and `LawfulTraversable`. These deriving handlers automatically derive their dependencies, for example `deriving LawfulTraversable` all by itself gives all four.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/DeriveTraversable.html"}, {"id": "Mathlib.Tactic.LinearCombinationPrime", "region_id": "tactic", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -131.345, "z": -202.202, "size": 0.208, "title": "`linear_combination'` Tactic", "summary": "In this file, the `linear_combination'` tactic is created. This tactic, which works over `CommRing`s, attempts to simplify the target by creating a linear combination of a list of equalities and subtracting it from the target. A `Syntax.Tactic` object can also be passed into the tactic, allowing the user to specify a normalization tactic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/LinearCombinationPrime.html"}, {"id": "Mathlib.Tactic.Eval", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "The `eval%` term elaborator", "summary": "This file provides the `eval% x` term elaborator, which evaluates the constant `x` at compile-time in the interpreter, and interpolates it into the expression.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Eval.html"}, {"id": "Mathlib.Tactic.SetNotationForOrder", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "Set notation for order operations", "summary": "This file allows the use of `⊆` notation while the underlying constant is `≤`. Similarly for `⊂`/`<`, `⊇`/`≥` and `⊃`/`>`. A new copy of the `a ⊆ b` syntax is declared, which overwrites the original one. To elaborate this notation, `a` and `b` are elaborated, and if the type of `a` and `b` is tagged with `@[use_set_notation_for_order]`, `LE.le` is used instead of `Subset`. A new delaborator for `LE.le` is also added…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/SetNotationForOrder.html"}, {"id": "Mathlib.Tactic.Simproc.FinsetInterval", "region_id": "tactic", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.208, "title": "Simproc for intervals of natural numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/Simproc/FinsetInterval.html"}, {"id": "Mathlib.Topology.Instances.Sign", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -94.001, "z": 226.022, "size": 0.2749, "title": "Topology on `SignType`", "summary": "This file gives `SignType` the discrete topology, and proves continuity results for `SignType.sign` in an `OrderTopology`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/Sign.html"}, {"id": "Mathlib.Topology.Order.Basic", "region_id": "topology", "micro_elevation": 0.5116, "macro_tier": 4, "macro_tier_score": 0.1266, "macro_tier_override": null, "x": -92.541, "z": 223.851, "size": 0.5499, "title": "Theory of topology on ordered spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Basic.html"}, {"id": "Mathlib.Topology.IsClosedRestrict", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -124.26, "z": 152.336, "size": 0.2, "title": "Restriction of a closed compact set in a product space to a set of coordinates", "summary": "We show that the image of a compact closed set `s` in a product `Π i : ι, α i` by the restriction to a subset of coordinates `S : Set ι` is a closed set. The idea of the proof is to use `isClosedMap_snd_of_compactSpace`, which is the fact that if `X` is a compact topological space, then `Prod.snd : X × Y → Y` is a closed map. We remark that `s` is included in the set `Sᶜ.restrict ⁻¹' Sᶜ.restrict '' s`, and we build…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/IsClosedRestrict.html"}, {"id": "Mathlib.Topology.Maps.Proper.Basic", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 4, "macro_tier_score": 0.1548, "macro_tier_override": null, "x": -131.355, "z": 158.697, "size": 0.4599, "title": "Proper maps between topological spaces", "summary": "This file develops the basic theory of proper maps between topological spaces. A map `f : X → Y` between two topological spaces is said to be **proper** if it is continuous and satisfies the following equivalent conditions: 1. `f` is closed and has compact fibers. 2. `f` is **universally closed**, in the sense that for any topological space `Z`, the map `Prod.map f id : X × Z → Y × Z` is closed. 3. For any `ℱ :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Maps/Proper/Basic.html"}, {"id": "Mathlib.Topology.EMetricSpace.Defs", "region_id": "topology", "micro_elevation": 0.3721, "macro_tier": 4, "macro_tier_score": 0.0905, "macro_tier_override": null, "x": -97.543, "z": 215.642, "size": 0.3889, "title": "Extended metric spaces", "summary": "This file is devoted to the definition and study of `EMetricSpace`s, i.e., metric spaces in which the distance is allowed to take the value ∞. This extended distance is called `edist`, and takes values in `ℝ≥0∞`. Many definitions and theorems expected on emetric spaces are already introduced on uniform spaces and topological spaces. For example: open and closed sets, compactness, completeness, continuity and uniform…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/EMetricSpace/Defs.html"}, {"id": "Mathlib.Topology.UniformSpace.Basic", "region_id": "topology", "micro_elevation": 0.3488, "macro_tier": 4, "macro_tier_score": 0.1479, "macro_tier_override": null, "x": -83.731, "z": 181.478, "size": 0.5607, "title": "Basic results on uniform spaces", "summary": "Uniform spaces are a generalization of metric spaces and topological groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Basic.html"}, {"id": "Mathlib.Topology.UniformSpace.OfFun", "region_id": "topology", "micro_elevation": 0.2093, "macro_tier": 4, "macro_tier_score": 0.0893, "macro_tier_override": null, "x": -118.857, "z": 189.757, "size": 0.3168, "title": "Construct a `UniformSpace` from a `dist`-like function", "summary": "In this file we provide a constructor for `UniformSpace` given a `dist`-like function", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/OfFun.html"}, {"id": "Mathlib.Topology.MetricSpace.Contracting", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -160.98, "z": 206.267, "size": 0.2556, "title": "Contracting maps", "summary": "A Lipschitz continuous self-map with Lipschitz constant `K < 1` is called a *contracting map*. In this file we prove the Banach fixed point theorem, some explicit estimates on the rate of convergence, and some properties of the map sending a contracting map to its fixed point.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Contracting.html"}, {"id": "Mathlib.Topology.MetricSpace.Lipschitz", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 3, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -161.274, "z": 192.378, "size": 0.4615, "title": "Lipschitz continuous functions", "summary": "A map `f : α → β` between two (extended) metric spaces is called *Lipschitz continuous* with constant `K ≥ 0` if for all `x, y` we have `edist (f x) (f y) ≤ K * edist x y`. For a metric space, the latter inequality is equivalent to `dist (f x) (f y) ≤ K * dist x y`. There is also a version asserting this inequality only for `x` and `y` in some set `s`. Finally, `f : α → β` is called *locally Lipschitz continuous* if…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Lipschitz.html"}, {"id": "Mathlib.Topology.Algebra.Support", "region_id": "topology", "micro_elevation": 0.4884, "macro_tier": 4, "macro_tier_score": 0.1435, "macro_tier_override": null, "x": -113.467, "z": 159.929, "size": 0.4271, "title": "The topological support of a function", "summary": "In this file we define the topological support of a function `f`, `tsupport f`, as the closure of the support of `f`. Furthermore, we say that `f` has compact support if the topological support of `f` is compact.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Support.html"}, {"id": "Mathlib.Topology.MetricSpace.Basic", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 3, "macro_tier_score": 0.0883, "macro_tier_override": null, "x": -111.971, "z": 244.464, "size": 0.4989, "title": "Basic properties of metric spaces, and instances.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Basic.html"}, {"id": "Mathlib.Topology.Defs.Sequences", "region_id": "topology", "micro_elevation": 0.0465, "macro_tier": 4, "macro_tier_score": 0.1107, "macro_tier_override": null, "x": -107.483, "z": 189.922, "size": 0.3569, "title": "Sequences in topological spaces", "summary": "In this file we define sequential closure, continuity, compactness etc.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Defs/Sequences.html"}, {"id": "Mathlib.Topology.Defs.Filter", "region_id": "topology", "micro_elevation": 0.0233, "macro_tier": 4, "macro_tier_score": 0.502, "macro_tier_override": null, "x": -104.702, "z": 193.238, "size": 0.3875, "title": "Definitions about filters in topological spaces", "summary": "In this file we define filters in topological spaces, as well as other definitions that rely on `Filter`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Defs/Filter.html"}, {"id": "Mathlib.Topology.Sheaves.Stalks", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 2, "macro_tier_score": 0.0064, "macro_tier_override": null, "x": -151.14, "z": 211.004, "size": 0.3446, "title": "Stalks", "summary": "For a presheaf `F` on a topological space `X`, valued in some category `C`, the *stalk* of `F` at the point `x : X` is defined as the colimit of the composition of the inclusion of categories `(OpenNhds x)ᵒᵖ ⥤ (Opens X)ᵒᵖ` and the functor `F : (Opens X)ᵒᵖ ⥤ C`. For an open neighborhood `U` of `x`, we define the map `F.germ x : F.obj (op U) ⟶ F.stalk x` as the canonical morphism into this colimit. Taking stalks is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Stalks.html"}, {"id": "Mathlib.Topology.Algebra.Constructions.DomMulAct", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -76.424, "z": 222.955, "size": 0.239, "title": "Topological space structure on `Mᵈᵐᵃ` and `Mᵈᵃᵃ`", "summary": "In this file we define `TopologicalSpace` structure on `Mᵈᵐᵃ` and `Mᵈᵃᵃ` and prove basic theorems about these topologies. The topologies on `Mᵈᵐᵃ` and `Mᵈᵃᵃ` are the same as the topology on `M`. Formally, they are induced by `DomMulAct.mk.symm` and `DomAddAct.mk.symm`, since the types aren't definitionally equal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Constructions/DomMulAct.html"}, {"id": "Mathlib.Topology.Homeomorph.Lemmas", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 4, "macro_tier_score": 0.266, "macro_tier_override": null, "x": -132.494, "z": 161.741, "size": 0.64, "title": "Further properties of homeomorphisms", "summary": "This file proves further properties of homeomorphisms between topological spaces. Pretty much every topological property is preserved under homeomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homeomorph/Lemmas.html"}, {"id": "Mathlib.Topology.ContinuousMap.Lattice", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -52.95, "z": 217.38, "size": 0.2411, "title": "Continuous maps as a lattice ordered group", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Lattice.html"}, {"id": "Mathlib.Topology.ContinuousMap.Algebra", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 3, "macro_tier_score": 0.0757, "macro_tier_override": null, "x": -61.485, "z": 155.696, "size": 0.4233, "title": "Algebraic structures over continuous functions", "summary": "In this file we define instances of algebraic structures over the type `ContinuousMap α β` (denoted `C(α, β)`) of **bundled** continuous maps from `α` to `β`. For example, `C(α, β)` is a group when `β` is a group, a ring when `β` is a ring, etc. For each type of algebraic structure, we also define an appropriate subobject of `α → β` with carrier `{ f : α → β | Continuous f }`. For example, when `β` is a group, a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Algebra.html"}, {"id": "Mathlib.Topology.ContinuousMap.Ordered", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 3, "macro_tier_score": 0.0279, "macro_tier_override": null, "x": -67.411, "z": 194.54, "size": 0.3819, "title": "Bundled continuous maps into orders, with order-compatible topology", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Ordered.html"}, {"id": "Mathlib.Topology.Algebra.ContinuousMonoidHom", "region_id": "topology", "micro_elevation": 0.3256, "macro_tier": 4, "macro_tier_score": 0.1212, "macro_tier_override": null, "x": -110.476, "z": 170.478, "size": 0.3591, "title": "Continuous Monoid Homs", "summary": "This file defines the space of continuous homomorphisms between two topological groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/ContinuousMonoidHom.html"}, {"id": "Mathlib.Topology.Algebra.Group.Basic", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 4, "macro_tier_score": 0.1309, "macro_tier_override": null, "x": -62.413, "z": 168.168, "size": 0.5262, "title": "Topological groups", "summary": "This file defines the following typeclasses: * `IsTopologicalGroup`, `IsTopologicalAddGroup`: multiplicative and additive topological groups, i.e., groups with continuous `(*)` and `(⁻¹)` / `(+)` and `(-)`; * `ContinuousSub G` means that `G` has a continuous subtraction operation. There is an instance deducing `ContinuousSub` from `IsTopologicalGroup` but we use a separate typeclass because, e.g., `ℕ` and `ℝ≥0` have…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/Basic.html"}, {"id": "Mathlib.Topology.Algebra.Module.Spaces.CompactConvergenceCLM", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -115.279, "z": 253.467, "size": 0.2884, "title": "Topology of compact convergence on the space of continuous linear maps", "summary": "In this file, we define a type alias `CompactConvergenceCLM` for `E →L[𝕜] F`, endowed with the topology of uniform convergence on compact subsets. More concretely, `CompactConvergenceCLM` is an abbreviation for `UniformConvergenceCLM σ F {(S : Set E) | IsCompact S}`. We denote it by `E →SL_c[σ] F`. Here is a list of type aliases for `E →L[𝕜] F` endowed with various topologies : * `ContinuousLinearMap`: topology of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Spaces/CompactConvergenceCLM.html"}, {"id": "Mathlib.Topology.ContinuousMap.Compact", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 2, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": -140.297, "z": 139.995, "size": 0.4744, "title": "Continuous functions on a compact space", "summary": "Continuous functions `C(α, β)` from a compact space `α` to a metric space `β` are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`. This file transfers these structures, and restates some lemmas characterising these structures. If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact, you should restate it here. You can also use…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Compact.html"}, {"id": "Mathlib.Topology.ContinuousMap.Bounded.Star", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 2, "macro_tier_score": 0.0123, "macro_tier_override": null, "x": -139.177, "z": 242.239, "size": 0.3825, "title": "Star structures on bounded continuous functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Bounded/Star.html"}, {"id": "Mathlib.Topology.Sets.Compacts", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 3, "macro_tier_score": 0.0498, "macro_tier_override": null, "x": -90.545, "z": 234.78, "size": 0.4305, "title": "Compact sets", "summary": "We define a few types of compact sets in a topological space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sets/Compacts.html"}, {"id": "Mathlib.Topology.Instances.TrivSqZeroExt", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -82.411, "z": 138.27, "size": 0.2465, "title": "Topology on `TrivSqZeroExt R M`", "summary": "The type `TrivSqZeroExt R M` inherits the topology from `R × M`. Note that this is not the topology induced by the seminorm on the dual numbers suggested by [this Math.SE answer](https://math.stackexchange.com/a/1056378/1896), which instead induces the topology pulled back through the projection map `TrivSqZeroExt.fst : tsze R M → R`. Obviously, that topology is not Hausdorff and using it would result in `exp`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/TrivSqZeroExt.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.Basic", "region_id": "topology", "micro_elevation": 0.5116, "macro_tier": 4, "macro_tier_score": 0.0948, "macro_tier_override": null, "x": -137.795, "z": 181.446, "size": 0.3404, "title": "Lemmas on infinite sums and products in topological monoids", "summary": "This file contains many simple lemmas on `tsum`, `HasSum` etc, which are placed here in order to keep the basic file of definitions as short as possible. Results requiring a group (rather than monoid) structure on the target should go in `Group.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/Basic.html"}, {"id": "Mathlib.Topology.Algebra.IsUniformGroup.Constructions", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 3, "macro_tier_score": 0.0272, "macro_tier_override": null, "x": -148.961, "z": 218.693, "size": 0.3469, "title": "Constructions of new uniform groups from old ones", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/IsUniformGroup/Constructions.html"}, {"id": "Mathlib.Topology.Algebra.Module.ContinuousLinearMap.PiProd", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 2, "macro_tier_score": 0.0148, "macro_tier_override": null, "x": -67.893, "z": 149.135, "size": 0.4822, "title": "Continuous linear maps on products and Pi types", "summary": "In this file, we collect various constructions relating continuous linear maps with (binary or arbitrary) products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/ContinuousLinearMap/PiProd.html"}, {"id": "Mathlib.Topology.Algebra.Module.FiniteDimension", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 2, "macro_tier_score": 0.0081, "macro_tier_override": null, "x": -110.607, "z": 252.5, "size": 0.4269, "title": "Finite-dimensional topological vector spaces over complete fields", "summary": "Let `𝕜` be a complete nontrivially normed field, and `E` a topological vector space (TVS) over `𝕜` (i.e we have `[AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [IsTopologicalAddGroup E]` and `[ContinuousSMul 𝕜 E]`). If `E` is finite dimensional and Hausdorff, then all linear maps from `E` to any other TVS are continuous. When `E` is a normed space, this gets us the equivalence of norms in finite dimension.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/FiniteDimension.html"}, {"id": "Mathlib.Topology.Connected.Clopen", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 4, "macro_tier_score": 0.3288, "macro_tier_override": null, "x": -89.535, "z": 159.097, "size": 0.6478, "title": "Connected subsets and their relation to clopen sets", "summary": "In this file we show how connected subsets of a topological space are intimately connected to clopen sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Connected/Clopen.html"}, {"id": "Mathlib.Topology.Clopen", "region_id": "topology", "micro_elevation": 0.3488, "macro_tier": 4, "macro_tier_score": 0.3337, "macro_tier_override": null, "x": -86.785, "z": 176.714, "size": 0.5163, "title": "Clopen sets", "summary": "A clopen set is a set that is both closed and open.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Clopen.html"}, {"id": "Mathlib.Topology.Connected.Basic", "region_id": "topology", "micro_elevation": 0.5116, "macro_tier": 4, "macro_tier_score": 0.321, "macro_tier_override": null, "x": -119.472, "z": 160.462, "size": 0.4423, "title": "Connected subsets of topological spaces", "summary": "In this file we define connected subsets of a topological spaces and various other properties and classes related to connectivity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Connected/Basic.html"}, {"id": "Mathlib.Topology.UniformSpace.Completion", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 3, "macro_tier_score": 0.0272, "macro_tier_override": null, "x": -60.131, "z": 217.528, "size": 0.3457, "title": "Hausdorff completions of uniform spaces", "summary": "The goal is to construct a left-adjoint to the inclusion of complete Hausdorff uniform spaces into all uniform spaces. Any uniform space `α` gets a completion `Completion α` and a morphism (i.e. uniformly continuous map) `(↑) : α → Completion α` which solves the universal mapping problem of factorizing morphisms from `α` to any complete Hausdorff uniform space `β`. It means any uniformly continuous `f : α → β` gives…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Completion.html"}, {"id": "Mathlib.Topology.UniformSpace.AbstractCompletion", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 3, "macro_tier_score": 0.0273, "macro_tier_override": null, "x": -58.3, "z": 173.211, "size": 0.3502, "title": "Abstract theory of Hausdorff completions of uniform spaces", "summary": "This file characterizes Hausdorff completions of a uniform space α as complete Hausdorff spaces equipped with a map from α which has dense image and induces the original uniform structure on α. Assuming these properties we \"extend\" uniformly continuous maps from α to complete Hausdorff spaces to the completions of α. This is the universal property expected from a completion. It is then used to extend uniformly…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/AbstractCompletion.html"}, {"id": "Mathlib.Topology.Algebra.MetricSpace.Lipschitz", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 2, "macro_tier_score": 0.0118, "macro_tier_override": null, "x": -47.245, "z": 197.612, "size": 0.3584, "title": "Lipschitz continuous functions", "summary": "This file develops Lipschitz continuous functions further with some results that depend on algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/MetricSpace/Lipschitz.html"}, {"id": "Mathlib.Topology.Order.AtTopBotIxx", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 1, "macro_tier_score": 0.001, "macro_tier_override": null, "x": -101.278, "z": 229.1, "size": 0.3374, "title": "`Filter.atTop` and `Filter.atBot` for intervals in a linear order topology", "summary": "Let `X` be a linear order with order topology. Let `a` be a point that is either the bottom element of `X` or is not isolated on the left, see `Order.IsSuccPrelimit`. Then the `Filter.atTop` filter on `Set.Iio a` and `𝓝[<] a` are related by the coercion map via pushforward and pullback, see `map_coe_Iio_atTop` and `comap_coe_Iio_nhdsLT`. We prove several versions of this statement for `Set.Iio`, `Set.Ioi`, and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/AtTopBotIxx.html"}, {"id": "Mathlib.Topology.MetricSpace.Similarity", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -51.078, "z": 159.604, "size": 0.252, "title": "Similarities", "summary": "This file defines `Similar`, i.e., the equivalence between indexed sets of points in a metric space where all corresponding pairwise distances have the same ratio. The motivating example is triangles in the plane.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Similarity.html"}, {"id": "Mathlib.Topology.MetricSpace.Congruence", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -99.765, "z": 252.552, "size": 0.2755, "title": "Congruences", "summary": "This file defines `Congruent`, i.e., the equivalence between indexed families of points in a metric space where all corresponding pairwise distances are the same. The motivating example are triangles in the plane.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Congruence.html"}, {"id": "Mathlib.Topology.Instances.Discrete", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 3, "macro_tier_score": 0.0735, "macro_tier_override": null, "x": -75.045, "z": 211.855, "size": 0.3036, "title": "Instances related to the discrete topology", "summary": "We prove that the discrete topology is * first-countable, * second-countable for an encodable type, * equal to the order topology in linear orders which are also `PredOrder` and `SuccOrder`, * metrizable. When importing this file and `Data.Nat.SuccPred`, the instances `SecondCountableTopology ℕ` and `OrderTopology ℕ` become available.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/Discrete.html"}, {"id": "Mathlib.Topology.Algebra.IsUniformGroup.Basic", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 3, "macro_tier_score": 0.0232, "macro_tier_override": null, "x": -108.827, "z": 246.361, "size": 0.4074, "title": "Uniform structure on topological groups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/IsUniformGroup/Basic.html"}, {"id": "Mathlib.Topology.UniformSpace.CompleteSeparated", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 3, "macro_tier_score": 0.0479, "macro_tier_override": null, "x": -148.402, "z": 209.427, "size": 0.3328, "title": "Theory of complete separated uniform spaces.", "summary": "This file is for elementary lemmas that depend on both Cauchy filters and separation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/CompleteSeparated.html"}, {"id": "Mathlib.Topology.UniformSpace.HeineCantor", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 3, "macro_tier_score": 0.0267, "macro_tier_override": null, "x": -156.104, "z": 206.287, "size": 0.306, "title": "Compact separated uniform spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/HeineCantor.html"}, {"id": "Mathlib.Topology.Algebra.Group.Quotient", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 4, "macro_tier_score": 0.1025, "macro_tier_override": null, "x": -94.778, "z": 242.371, "size": 0.4525, "title": "Topology on the quotient group", "summary": "In this file we define topology on `G ⧸ N`, where `N` is a subgroup of `G`, and prove basic properties of this topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/Quotient.html"}, {"id": "Mathlib.Topology.Category.CompHaus.Projective", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -141.7, "z": 142.934, "size": 0.2649, "title": "CompHaus has enough projectives", "summary": "In this file we show that `CompHaus` has enough projectives.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/CompHaus/Projective.html"}, {"id": "Mathlib.Topology.Category.CompHaus.Basic", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 3, "macro_tier_score": 0.0223, "macro_tier_override": null, "x": -46.839, "z": 178.562, "size": 0.3634, "title": "The category of Compact Hausdorff Spaces", "summary": "We construct the category of compact Hausdorff spaces. The type of compact Hausdorff spaces is denoted `CompHaus`, and it is endowed with a category instance making it a full subcategory of `TopCat`. The fully faithful functor `CompHaus ⥤ TopCat` is denoted `compHausToTop`. **Note:** The file `Mathlib/Topology/Category/Compactum.lean` provides the equivalence between `Compactum`, which is defined as the category of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/CompHaus/Basic.html"}, {"id": "Mathlib.Topology.Algebra.Order.Field", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 3, "macro_tier_score": 0.0797, "macro_tier_override": null, "x": -97.355, "z": 137.385, "size": 0.3674, "title": "Topologies on linear ordered fields", "summary": "In this file we prove that a linear ordered field with order topology has continuous multiplication and division (apart from zero in the denominator). We also prove theorems like `Filter.Tendsto.mul_atTop`: if `f` tends to a positive number and `g` tends to positive infinity, then `f * g` tends to positive infinity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Order/Field.html"}, {"id": "Mathlib.Topology.Instances.RealVectorSpace", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -69.99, "z": 141.552, "size": 0.4098, "title": "Continuous additive maps are `ℝ`-linear", "summary": "In this file we prove that a continuous map `f : E →+ F` between two topological vector spaces over `ℝ` is `ℝ`-linear", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/RealVectorSpace.html"}, {"id": "Mathlib.Topology.Algebra.Module.Equiv", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 3, "macro_tier_score": 0.0174, "macro_tier_override": null, "x": -161.305, "z": 204.954, "size": 0.5556, "title": "Continuous linear equivalences", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Equiv.html"}, {"id": "Mathlib.Topology.Instances.Rat", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 2, "macro_tier_score": 0.0133, "macro_tier_override": null, "x": -148.891, "z": 151.601, "size": 0.4266, "title": "Topology on the rational numbers", "summary": "The structure of a metric space on `ℚ` is introduced in this file, induced from `ℝ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/Rat.html"}, {"id": "Mathlib.Topology.Category.CompactlyGenerated", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -160.209, "z": 221.099, "size": 0.2478, "title": "Compactly generated topological spaces", "summary": "This file defines the category of compactly generated topological spaces. These are spaces `X` such that a map `f : X → Y` is continuous whenever the composition `S → X → Y` is continuous for all compact Hausdorff spaces `S` mapping continuously to `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/CompactlyGenerated.html"}, {"id": "Mathlib.Topology.Compactness.CompactlyGeneratedSpace", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 2, "macro_tier_score": 0.011, "macro_tier_override": null, "x": -52.371, "z": 222.873, "size": 0.2985, "title": "Compactly generated topological spaces", "summary": "This file defines compactly generated topological spaces. A compactly generated space is a space `X` whose topology is coinduced by continuous maps from compact Hausdorff spaces to `X`. In such a space, a set `s` is closed (resp. open) if and only if for all compact Hausdorff space `K` and `f : K → X` continuous, `f ⁻¹' s` is closed (resp. open) in `K`. We provide two definitions. `UCompactlyGeneratedSpace.{u} X`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactness/CompactlyGeneratedSpace.html"}, {"id": "Mathlib.Topology.Separation.CompletelyRegular", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 2, "macro_tier_score": 0.016, "macro_tier_override": null, "x": -159.993, "z": 165.316, "size": 0.2872, "title": "Completely regular topological spaces.", "summary": "This file defines `CompletelyRegularSpace` and `T35Space`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/CompletelyRegular.html"}, {"id": "Mathlib.Topology.Hom.Open", "region_id": "topology", "micro_elevation": 0.5116, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -125.818, "z": 219.06, "size": 0.2, "title": "Continuous open maps", "summary": "This file defines bundled continuous open maps. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Hom/Open.html"}, {"id": "Mathlib.Topology.ContinuousMap.Basic", "region_id": "topology", "micro_elevation": 0.4884, "macro_tier": 4, "macro_tier_score": 0.2831, "macro_tier_override": null, "x": -125.613, "z": 166.139, "size": 0.5565, "title": "Continuous bundled maps", "summary": "In this file we define the type `ContinuousMap` of continuous bundled maps. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Basic.html"}, {"id": "Mathlib.Topology.Algebra.Star.Real", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 1, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -64.104, "z": 171.698, "size": 0.2915, "title": "Topological properties of conjugation on ℝ", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Star/Real.html"}, {"id": "Mathlib.Topology.Algebra.Star", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 3, "macro_tier_score": 0.0437, "macro_tier_override": null, "x": -78.302, "z": 226.558, "size": 0.3892, "title": "Continuity of `star`", "summary": "This file defines the `ContinuousStar` typeclass, along with instances on `Pi`, `Prod`, `MulOpposite`, and `Units`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Star.html"}, {"id": "Mathlib.Topology.MetricSpace.Pseudo.Constructions", "region_id": "topology", "micro_elevation": 0.4186, "macro_tier": 4, "macro_tier_score": 0.0897, "macro_tier_override": null, "x": -117.202, "z": 217.047, "size": 0.3425, "title": "Products of pseudometric spaces and other constructions", "summary": "This file constructs the supremum distance on binary products of pseudometric spaces and provides instances for type synonyms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Pseudo/Constructions.html"}, {"id": "Mathlib.Topology.Algebra.Star.LinearMap", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -163.942, "z": 207.321, "size": 0.2, "title": "Intrinsic star operation on continuous linear maps", "summary": "This file defines the star operation on continuous linear maps: `(star f) x = star (f (star x))`. This corresponds to a map being star-preserving, i.e., a map is self-adjoint iff it is star-preserving. This is the continuous version of the intrinsic star on linear maps (see `Mathlib/Algebra/Star/LinearMap.lean`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Star/LinearMap.html"}, {"id": "Mathlib.Topology.Algebra.Module.Star", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -58.417, "z": 228.865, "size": 0.2765, "title": "The star operation, bundled as a continuous star-linear equiv", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Star.html"}, {"id": "Mathlib.Topology.ContinuousOn", "region_id": "topology", "micro_elevation": 0.3256, "macro_tier": 4, "macro_tier_score": 0.4288, "macro_tier_override": null, "x": -88.097, "z": 205.781, "size": 0.6678, "title": "Neighborhoods and continuity relative to a subset", "summary": "This file develops API on the relative versions * `ContinuousOn` of `Continuous` * `ContinuousWithinAt` of `ContinuousAt` related to continuity, which are defined in previous definition files. Their basic properties studied in this file include the relationships between these restricted notions and the corresponding notions for the subtype equipped with the subspace topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousOn.html"}, {"id": "Mathlib.Topology.NhdsWithin", "region_id": "topology", "micro_elevation": 0.3023, "macro_tier": 4, "macro_tier_score": 0.4326, "macro_tier_override": null, "x": -110.447, "z": 172.094, "size": 0.6384, "title": "Neighborhoods relative to a subset", "summary": "This file develops API on the relative versions `nhdsWithin` and `nhdsSetWithin` of `nhds` and `nhdsSet`, which are defined in previous definition files. Their basic properties studied in this file include the relationship between neighborhood filters relative to a set and neighborhood filters in the corresponding subtype, and are in later files used to develop relative versions `ContinuousOn` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/NhdsWithin.html"}, {"id": "Mathlib.Topology.Sheaves.Presheaf", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": -102.241, "z": 232.32, "size": 0.3684, "title": "Presheaves on a topological space", "summary": "We define `TopCat.Presheaf C X` simply as `(TopologicalSpace.Opens X)ᵒᵖ ⥤ C`, and inherit the category structure with natural transformations as morphisms. We define * Given `{X Y : TopCat.{w}}` and `f : X ⟶ Y`, we define `TopCat.Presheaf.pushforward C f : X.Presheaf C ⥤ Y.Presheaf C`, with notation `f _* ℱ` for `ℱ : X.Presheaf C`. and for `ℱ : X.Presheaf C` provide the natural isomorphisms *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Presheaf.html"}, {"id": "Mathlib.Topology.UniformSpace.Compact", "region_id": "topology", "micro_elevation": 0.3721, "macro_tier": 4, "macro_tier_score": 0.132, "macro_tier_override": null, "x": -80.312, "z": 196.562, "size": 0.3797, "title": "Compact sets in uniform spaces", "summary": "* `compactSpace_uniformity`: On a compact uniform space, the topology determines the uniform structure, entourages are exactly the neighborhoods of the diagonal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Compact.html"}, {"id": "Mathlib.Topology.Compactness.Compact", "region_id": "topology", "micro_elevation": 0.3488, "macro_tier": 4, "macro_tier_score": 0.3748, "macro_tier_override": null, "x": -104.812, "z": 168.19, "size": 0.5005, "title": "Compact sets and compact spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactness/Compact.html"}, {"id": "Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 2, "macro_tier_score": 0.0089, "macro_tier_override": null, "x": -43.664, "z": 178.411, "size": 0.4589, "title": "Topology of bounded convergence on the space of continuous linear map", "summary": "In this file, we endow `E →L[𝕜] F` with the \"topology of bounded convergence\", or \"topology of uniform convergence on bounded sets\". This is declared as an instance. A key feature of the topology of bounded convergence is that, in the normed setting, it coincides with the operator norm topology. Note that, more generally, we defined the \"topology of `𝔖`-convergence\" for any `𝔖 : Set (Set E)` in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Spaces/ContinuousLinearMap.html"}, {"id": "Mathlib.Topology.Algebra.Module.LocallyConvex", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 2, "macro_tier_score": 0.0064, "macro_tier_override": null, "x": -81.63, "z": 130.15, "size": 0.3487, "title": "Locally convex topological modules", "summary": "A `LocallyConvexSpace` is a topological semimodule over an ordered semiring in which any point admits a neighborhood basis made of convex sets, or equivalently, in which convex neighborhoods of a point form a neighborhood basis at that point. In a module, this is equivalent to `0` satisfying such properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/LocallyConvex.html"}, {"id": "Mathlib.Topology.Category.LightProfinite.Injective", "region_id": "topology", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -47.716, "z": 227.276, "size": 0.239, "title": "Injectivity of light profinite spaces", "summary": "This file establishes that non-empty light profinite sets are injective in the category of profinite sets, and thus also in the category of light profinite sets. This is used in the proof that the null sequence module is internally projective in light condensed abelian groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/LightProfinite/Injective.html"}, {"id": "Mathlib.Topology.Category.LightProfinite.AsLimit", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -137.595, "z": 134.605, "size": 0.2682, "title": "Light profinite sets as limits of finite sets.", "summary": "We show that any light profinite set is isomorphic to a sequential limit of finite sets. The limit cone for `S : LightProfinite` is `S.asLimitCone`, the fact that it's a limit is `S.asLimit`. We also prove that the projection and transition maps in this limit are surjective.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/LightProfinite/AsLimit.html"}, {"id": "Mathlib.Topology.Category.CompHausLike.Limits", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 2, "macro_tier_score": 0.0073, "macro_tier_override": null, "x": -142.027, "z": 197.522, "size": 0.3936, "title": "Explicit limits and colimits", "summary": "This file collects some constructions of explicit limits and colimits in `CompHausLike P`, which may be useful due to their definitional properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/CompHausLike/Limits.html"}, {"id": "Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 2, "macro_tier_score": 0.0075, "macro_tier_override": null, "x": -106.051, "z": 252.757, "size": 0.4051, "title": "Topologies of uniform convergence on the space of continuous linear maps", "summary": "In this file, we define the \"topology of `𝔖`-convergence\" on `E →L[𝕜] F`, where `𝔖 : Set (Set E)`. It is the topology of uniform convergence on the elements of `𝔖`. Similarly to `UniformOnFun`, we define a type synonym `UniformConvergenceCLM` for `E →L[𝕜] F` endowed with this topology. The lemma `UniformOnFun.continuousSMul_of_image_bounded` tells us that this is a vector space topology if the continuous linear…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Spaces/UniformConvergenceCLM.html"}, {"id": "Mathlib.Topology.MetricSpace.Polish", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 2, "macro_tier_score": 0.0111, "macro_tier_override": null, "x": -113.784, "z": 128.085, "size": 0.3135, "title": "Polish spaces", "summary": "A topological space is Polish if its topology is second-countable and there exists a compatible complete metric. This is the class of spaces that is well-behaved with respect to measure theory. In this file, we establish the basic properties of Polish spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Polish.html"}, {"id": "Mathlib.Topology.Algebra.Valued.ValuativeRel", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -81.185, "z": 247.979, "size": 0.2338, "title": "Valuative Relations as Valued", "summary": "In this temporary file, we provide a helper instance for `Valued R Γ` derived from a `ValuativeRel R`, so that downstream files can refer to `ValuativeRel R`, to facilitate a refactor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Valued/ValuativeRel.html"}, {"id": "Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -134.014, "z": 139.776, "size": 0.2843, "title": "The topology on a ring induced by a valuation", "summary": "In this file, we define the non-Archimedean topology induced by a valuation on a ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/ValuativeRel/ValuativeTopology.html"}, {"id": "Mathlib.Topology.ContinuousMap.Sigma", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -113.476, "z": 231.492, "size": 0.2, "title": "Equivalence between `C(X, Σ i, Y i)` and `Σ i, C(X, Y i)`", "summary": "If `X` is a connected topological space, then for every continuous map `f` from `X` to the disjoint union of a collection of topological spaces `Y i` there exists a unique index `i` and a continuous map from `g` to `Y i` such that `f` is the composition of the natural embedding `Sigma.mk i : Y i → Σ i, Y i` with `g`. This defines an equivalence between `C(X, Σ i, Y i)` and `Σ i, C(X, Y i)`. In fact, this equivalence…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Sigma.html"}, {"id": "Mathlib.Topology.CompactOpen", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 4, "macro_tier_score": 0.1226, "macro_tier_override": null, "x": -66.484, "z": 199.3, "size": 0.4238, "title": "The compact-open topology", "summary": "In this file, we define the compact-open topology on the set of continuous maps between two topological spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/CompactOpen.html"}, {"id": "Mathlib.Topology.Algebra.Algebra.Equiv", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 2, "macro_tier_score": 0.0065, "macro_tier_override": null, "x": -97.299, "z": 250.715, "size": 0.3519, "title": "Isomorphisms of topological algebras", "summary": "This file contains an API for `ContinuousAlgEquiv R A B`, the type of continuous `R`-algebra isomorphisms with continuous inverses. Here `R` is a commutative (semi)ring, and `A` and `B` are `R`-algebras with topologies.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Algebra/Equiv.html"}, {"id": "Mathlib.Topology.Hom.ContinuousEvalConst", "region_id": "topology", "micro_elevation": 0.3023, "macro_tier": 4, "macro_tier_score": 0.1318, "macro_tier_override": null, "x": -117.106, "z": 175.392, "size": 0.3668, "title": "Bundled morphisms with continuous evaluation at a point", "summary": "In this file we define a typeclass saying that `F` is a type of bundled morphisms (in the sense of `DFunLike`) with a topology on `F` such that evaluation at a point is continuous in `f : F`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Hom/ContinuousEvalConst.html"}, {"id": "Mathlib.Topology.Algebra.SeparationQuotient.Section", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -154.328, "z": 221.925, "size": 0.3006, "title": "Algebraic operations on `SeparationQuotient`", "summary": "In this file we construct a section of the quotient map `E → SeparationQuotient E` as a continuous linear map `SeparationQuotient E →L[K] E`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/SeparationQuotient/Section.html"}, {"id": "Mathlib.Topology.Algebra.Module.UniformConvergence", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -121.251, "z": 137.723, "size": 0.3006, "title": "Algebraic facts about the topology of uniform convergence", "summary": "This file contains algebraic compatibility results about the uniform structure of uniform convergence / `𝔖`-convergence. They will mostly be useful for defining strong topologies on the space of continuous linear maps between two topological vector spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/UniformConvergence.html"}, {"id": "Mathlib.Topology.Order.LiminfLimsup", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 3, "macro_tier_score": 0.0326, "macro_tier_override": null, "x": -114.47, "z": 152.103, "size": 0.3585, "title": "Lemmas about liminf and limsup in an order topology.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/LiminfLimsup.html"}, {"id": "Mathlib.Topology.Order.Monotone", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 4, "macro_tier_score": 0.0948, "macro_tier_override": null, "x": -66.371, "z": 198.706, "size": 0.3358, "title": "Monotone functions on an order topology", "summary": "This file contains lemmas about limits and continuity for monotone / antitone functions on linearly-ordered sets (with the order topology). For example, we prove that a monotone function has left and right limits at any point (`Monotone.tendsto_nhdsLT`, `Monotone.tendsto_nhdsGT`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Monotone.html"}, {"id": "Mathlib.Topology.Order.MonotoneConvergence", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 3, "macro_tier_score": 0.0375, "macro_tier_override": null, "x": -72.198, "z": 176.558, "size": 0.3371, "title": "Bounded monotone sequences converge", "summary": "In this file we prove a few theorems of the form “if the range of a monotone function `f : ι → α` admits a least upper bound `a`, then `f x` tends to `a` as `x → ∞`”, as well as version of this statement for (conditionally) complete lattices that use `⨆ x, f x` instead of `IsLUB`. These theorems work for linear orders with order topologies as well as their products (both in terms of `Prod` and in terms of function…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/MonotoneConvergence.html"}, {"id": "Mathlib.Topology.Algebra.Module.PerfectPairing", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -161.093, "z": 196.256, "size": 0.3041, "title": "Continuous perfect pairings", "summary": "This file defines continuous perfect pairings. For a topological ring `R` and two topological modules `M` and `N`, a continuous perfect pairing is a continuous bilinear map `M × N → R` that is bijective in both arguments. We require continuity in the forward direction only so that we can put several different topologies on the continuous dual (e.g., strong, weak, weak-\\*). For example, if `M` is weakly reflexive…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/PerfectPairing.html"}, {"id": "Mathlib.Topology.Category.Compactum", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -143.398, "z": 241.105, "size": 0.2, "title": "Compacta and Compact Hausdorff Spaces", "summary": "Recall that, given a monad `M` on `Type*`, an *algebra* for `M` consists of the following data: - A type `X : Type*` - A \"structure\" map `M X → X`. This data must also satisfy a distributivity and unit axiom, and algebras for `M` form a category in an evident way. See the file `Mathlib/CategoryTheory/Monad/Algebra.lean` for a general version, as well as the following link. https://ncatlab.org/nlab/show/monad This…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Compactum.html"}, {"id": "Mathlib.Topology.Category.Profinite.Basic", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 2, "macro_tier_score": 0.012, "macro_tier_override": null, "x": -58.846, "z": 151.546, "size": 0.3688, "title": "The category of Profinite Types", "summary": "We construct the category of profinite topological spaces, often called profinite sets -- perhaps they could be called profinite types in Lean. The type of profinite topological spaces is called `Profinite`. It has a category instance and is a fully faithful subcategory of `TopCat`. The fully faithful functor is called `Profinite.toTop`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Profinite/Basic.html"}, {"id": "Mathlib.Topology.Algebra.SeparationQuotient.Hom", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -113.479, "z": 248.996, "size": 0.2478, "title": "Lift of `MonoidHom M N` to `MonoidHom (SeparationQuotient M) N`", "summary": "In this file we define the lift of a continuous monoid homomorphism `f` from `M` to `N` to `SeparationQuotient M`, assuming that `f` maps two inseparable elements to the same element.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/SeparationQuotient/Hom.html"}, {"id": "Mathlib.Topology.Algebra.GroupWithZero", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 3, "macro_tier_score": 0.0792, "macro_tier_override": null, "x": -102.538, "z": 143.186, "size": 0.3394, "title": "Topological group with zero", "summary": "In this file we define `ContinuousInv₀` to be a mixin typeclass a type with `Inv` and `Zero` (e.g., a `GroupWithZero`) such that `fun x ↦ x⁻¹` is continuous at all nonzero points. Any normed (semi)field has this property. Currently the only example of `ContinuousInv₀` in `mathlib` which is not a normed field is the type `NNReal` (a.k.a. `ℝ≥0`) of nonnegative real numbers. Then we prove lemmas about continuity of `x…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/GroupWithZero.html"}, {"id": "Mathlib.Topology.Algebra.Monoid", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 4, "macro_tier_score": 0.1339, "macro_tier_override": null, "x": -59.271, "z": 202.932, "size": 0.4562, "title": "Theory of topological monoids", "summary": "In this file we define mixin classes `ContinuousMul` and `ContinuousAdd`. While in many applications the underlying type is a monoid (multiplicative or additive), we do not require this in the definitions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Monoid.html"}, {"id": "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 2, "macro_tier_score": 0.0061, "macro_tier_override": null, "x": -164.212, "z": 216.282, "size": 0.3302, "title": "Topology on continuous multilinear maps", "summary": "In this file we define `TopologicalSpace` and `UniformSpace` structures on `ContinuousMultilinearMap 𝕜 E F`, where `E i` is a family of vector spaces over `𝕜` with topologies and `F` is a topological vector space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Multilinear/Topology.html"}, {"id": "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -54.878, "z": 223.946, "size": 0.2973, "title": "Images of (von Neumann) bounded sets under continuous multilinear maps", "summary": "In this file we prove that continuous multilinear maps send von Neumann bounded sets to von Neumann bounded sets. We prove 2 versions of the theorem: one assumes that the index type is nonempty, and the other assumes that the codomain is a topological vector space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Multilinear/Bounded.html"}, {"id": "Mathlib.Topology.Gluing", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -145.751, "z": 171.841, "size": 0.2641, "title": "Gluing Topological spaces", "summary": "Given a family of gluing data (see `Mathlib/CategoryTheory/GlueData.lean`), we can then glue them together. The construction should be \"sealed\" and considered as a black box, while only using the API provided.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Gluing.html"}, {"id": "Mathlib.Topology.Sheaves.AddCommGrpCat", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -117.317, "z": 138.291, "size": 0.2478, "title": "Sheaves of abelian groups.", "summary": "Results for sheaves of abelian groups on topological spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/AddCommGrpCat.html"}, {"id": "Mathlib.Topology.Sheaves.Abelian", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -63.411, "z": 158.289, "size": 0.2806, "title": "Sheaves over Abelian categories", "summary": "We provide instances for categories of sheaves over Abelian categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Abelian.html"}, {"id": "Mathlib.Topology.Algebra.Order.UpperLower", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -126.401, "z": 238.683, "size": 0.2329, "title": "Topological facts about upper/lower/order-connected sets", "summary": "The topological closure and interior of an upper/lower/order-connected set is an upper/lower/order-connected set (with the notable exception of the closure of an order-connected set).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Order/UpperLower.html"}, {"id": "Mathlib.Topology.Algebra.Group.Pointwise", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 4, "macro_tier_score": 0.1175, "macro_tier_override": null, "x": -126.434, "z": 236.939, "size": 0.4279, "title": "Pointwise operations on sets in topological groups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/Pointwise.html"}, {"id": "Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 4, "macro_tier_score": 0.0931, "macro_tier_override": null, "x": -61.734, "z": 225.485, "size": 0.7198, "title": "Continuous linear maps", "summary": "In this file we define the type of continuous (semi)linear maps between topological modules that are continuous, and endow it with its algebraic structure. Later files endow it with a topological structure, see the docstring of `Mathlib/Topology/Algebra/Module/Spaces/ContinuousLinearMap.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/ContinuousLinearMap/Basic.html"}, {"id": "Mathlib.Topology.Category.TopCat.Basic", "region_id": "topology", "micro_elevation": 0.5116, "macro_tier": 3, "macro_tier_score": 0.0544, "macro_tier_override": null, "x": -136.42, "z": 205.589, "size": 0.4044, "title": "Category instance for topological spaces", "summary": "We introduce the bundled category `TopCat` of topological spaces together with the functors `TopCat.discrete` and `TopCat.trivial` from the category of types to `TopCat` which equip a type with the corresponding discrete, resp. trivial, topology. For a proof that these functors are left, resp. right adjoint to the forgetful functor, see `Mathlib/Topology/Category/TopCat/Adjunctions.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/Basic.html"}, {"id": "Mathlib.Topology.Sets.Opens", "region_id": "topology", "micro_elevation": 0.5116, "macro_tier": 4, "macro_tier_score": 0.1989, "macro_tier_override": null, "x": -131.891, "z": 170.272, "size": 0.5332, "title": "Open sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sets/Opens.html"}, {"id": "Mathlib.Topology.Algebra.Order.ArchimedeanDiscrete", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -158.735, "z": 208.426, "size": 0.2439, "title": "Discreteness of subgroups in archimedean ordered groups", "summary": "This file contains some supplements to the results in `Mathlib/Topology/Algebra/Order/Archimedean.lean`, involving discreteness of subgroups, which require heavier imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Order/ArchimedeanDiscrete.html"}, {"id": "Mathlib.Topology.Algebra.Order.Archimedean", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 3, "macro_tier_score": 0.0164, "macro_tier_override": null, "x": -65.961, "z": 187.494, "size": 0.3192, "title": "Topology on archimedean groups and fields", "summary": "In this file we prove the following theorems: - `Rat.denseRange_cast`: the coercion from `ℚ` to a linear ordered archimedean field has dense range; - `AddSubgroup.dense_of_not_isolated_zero`, `AddSubgroup.dense_of_no_min`: two sufficient conditions for a subgroup of an archimedean linear ordered additive commutative group to be dense; - `AddSubgroup.dense_or_cyclic`: an additive subgroup of an archimedean linear…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Order/Archimedean.html"}, {"id": "Mathlib.Topology.Order.DenselyOrdered", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 4, "macro_tier_score": 0.0952, "macro_tier_override": null, "x": -138.189, "z": 171.077, "size": 0.3599, "title": "Order topology on a densely ordered set", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/DenselyOrdered.html"}, {"id": "Mathlib.Topology.MetricSpace.ThickenedIndicator", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 1, "macro_tier_score": 0.001, "macro_tier_override": null, "x": -42.171, "z": 205.46, "size": 0.3308, "title": "Thickened indicators", "summary": "This file is about thickened indicators of sets in (pseudo e)metric spaces. For a decreasing sequence of thickening radii tending to 0, the thickened indicators of a closed set form a decreasing pointwise converging approximation of the indicator function of the set, where the members of the approximating sequence are nonnegative bounded continuous functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/ThickenedIndicator.html"}, {"id": "Mathlib.Topology.MetricSpace.Ultra.Basic", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 2, "macro_tier_score": 0.0061, "macro_tier_override": null, "x": -86.67, "z": 229.845, "size": 0.3263, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Ultra/Basic.html"}, {"id": "Mathlib.Topology.Semicontinuity.Lindelof", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -81.775, "z": 253.271, "size": 0.2529, "title": "Envelopes of Semicontinuous functions on Hereditarily Lindelöf spaces", "summary": "In this file, we show that, if `X` is a hereditarily Lindelöf space and `𝓕` is any family of upper semicontinuous functions on `X`, then there is a countable subfamily `𝓕'` with the same infimum / lower envelope. Most importantly, this applies whenever `X` has a `SecondCountableTopology`. The codomain `E` of the functions is assumed to be linearly ordered, and to admit a countable order-dense subset. In particular…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Semicontinuity/Lindelof.html"}, {"id": "Mathlib.Topology.OpenPartialHomeomorph.Defs", "region_id": "topology", "micro_elevation": 0.3721, "macro_tier": 3, "macro_tier_score": 0.0429, "macro_tier_override": null, "x": -127.363, "z": 202.779, "size": 0.3507, "title": "Partial homeomorphisms: definitions", "summary": "This file defines homeomorphisms between open subsets of topological spaces. An element `e` of `OpenPartialHomeomorph X Y` is an extension of `PartialEquiv X Y`, i.e., it is a pair of functions `e.toFun` and `e.invFun`, inverse of each other on the sets `e.source` and `e.target`. Additionally, we require that these sets are open, and that the functions are continuous on them. Equivalently, they are homeomorphisms…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/OpenPartialHomeomorph/Defs.html"}, {"id": "Mathlib.Topology.PartialHomeomorph.Defs", "region_id": "topology", "micro_elevation": 0.3488, "macro_tier": 3, "macro_tier_score": 0.043, "macro_tier_override": null, "x": -82.627, "z": 184.185, "size": 0.3547, "title": "Partial homeomorphisms: definitions", "summary": "This file defines homeomorphisms between subsets of topological spaces. An element `e` of `PartialHomeomorph X Y` is an extension of `PartialEquiv X Y`, i.e., it is a pair of functions `e.toFun` and `e.invFun`, inverse of each other on the sets `e.source` and `e.target`. Additionally, we require that the functions are continuous on them. Equivalently, they are homeomorphisms there. As for `Equiv`s, we register a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/PartialHomeomorph/Defs.html"}, {"id": "Mathlib.Topology.Instances.Int", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 3, "macro_tier_score": 0.0749, "macro_tier_override": null, "x": -105.006, "z": 248.069, "size": 0.3891, "title": "Topology on the integers", "summary": "The structure of a metric space on `ℤ` is introduced in this file, induced from `ℝ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/Int.html"}, {"id": "Mathlib.Topology.MetricSpace.Bounded", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 4, "macro_tier_score": 0.0918, "macro_tier_override": null, "x": -60.277, "z": 159.83, "size": 0.4406, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Bounded.html"}, {"id": "Mathlib.Topology.Algebra.OpenSubgroup", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 2, "macro_tier_score": 0.0124, "macro_tier_override": null, "x": -111.112, "z": 138.795, "size": 0.3894, "title": "Open subgroups of a topological group", "summary": "This file builds the lattice `OpenSubgroup G` of open subgroups in a topological group `G`, and its additive version `OpenAddSubgroup`. This lattice has a top element, the subgroup of all elements, but no bottom element in general. The trivial subgroup which is the natural candidate bottom has no reason to be open (this happens only in discrete groups). Note that this notion is especially relevant in a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/OpenSubgroup.html"}, {"id": "Mathlib.Topology.Algebra.Ring.Basic", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 4, "macro_tier_score": 0.123, "macro_tier_override": null, "x": -155.457, "z": 180.968, "size": 0.4393, "title": "Topological (semi)rings", "summary": "A topological (semi)ring is a (semi)ring equipped with a topology such that all operations are continuous. Besides this definition, this file proves that the topological closure of a subring (resp. an ideal) is a subring (resp. an ideal) and defines products and quotients of topological (semi)rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Ring/Basic.html"}, {"id": "Mathlib.Topology.MetricSpace.GromovHausdorff", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -60.421, "z": 145.358, "size": 0.2, "title": "Gromov-Hausdorff distance", "summary": "This file defines the Gromov-Hausdorff distance on the space of nonempty compact metric spaces up to isometry. We introduce the space of all nonempty compact metric spaces, up to isometry, called `GHSpace`, and endow it with a metric space structure. The distance, known as the Gromov-Hausdorff distance, is defined as follows: given two nonempty compact spaces `X` and `Y`, their distance is the minimum Hausdorff…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/GromovHausdorff.html"}, {"id": "Mathlib.Topology.MetricSpace.Closeds", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -153.444, "z": 231.28, "size": 0.2302, "title": "Closed subsets", "summary": "This file defines the metric and emetric space structure on the types of closed subsets and nonempty compact subsets of a metric or emetric space. The Hausdorff distance induces an emetric space structure on the type of closed subsets of an emetric space, called `Closeds`. Its completeness, resp. compactness, resp. second-countability, follow from the corresponding properties of the original space. In a metric…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Closeds.html"}, {"id": "Mathlib.Topology.MetricSpace.Completion", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -159.666, "z": 214.971, "size": 0.304, "title": "The completion of a metric space", "summary": "Completion of uniform spaces are already defined in `Topology.UniformSpace.Completion`. We show here that the uniform space completion of a metric space inherits a metric space structure, by extending the distance to the completion and checking that it is indeed a distance, and that it defines the same uniformity as the already defined uniform structure on the completion", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Completion.html"}, {"id": "Mathlib.Topology.MetricSpace.GromovHausdorffRealized", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -108.222, "z": 130.69, "size": 0.2302, "title": "The Gromov-Hausdorff distance is realized", "summary": "In this file, we construct of a good coupling between nonempty compact metric spaces, minimizing their Hausdorff distance. This construction is instrumental to study the Gromov-Hausdorff distance between nonempty compact metric spaces. Given two nonempty compact metric spaces `X` and `Y`, we define `OptimalGHCoupling X Y` as a compact metric space, together with two isometric embeddings `optimalGHInjl` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/GromovHausdorffRealized.html"}, {"id": "Mathlib.Topology.MetricSpace.Kuratowski", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -61.687, "z": 173.213, "size": 0.2302, "title": "The Kuratowski embedding", "summary": "Any separable metric space can be embedded isometrically in `ℓ^∞(ℕ, ℝ)`. Any partially defined Lipschitz map into `ℓ^∞` can be extended to the whole space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Kuratowski.html"}, {"id": "Mathlib.Topology.Order.CountableSeparating", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -109.24, "z": 155.922, "size": 0.2541, "title": "Countably many infinite intervals separate points", "summary": "In this file we prove that in a linear order with second countable order topology, the points can be separated by countably many infinite intervals. We prove 4 versions of this statement (one for each of the infinite intervals), as well as provide convenience corollaries about `Filter.EventuallyEq`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/CountableSeparating.html"}, {"id": "Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Restrict", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 2, "macro_tier_score": 0.0136, "macro_tier_override": null, "x": -121.931, "z": 137.934, "size": 0.4393, "title": "Restrictions of continuous linear maps to submodules", "summary": "In this file, we collect the various operations of restrictions of `ContinuousLinearMap`s to subspaces of the domain/codomain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/ContinuousLinearMap/Restrict.html"}, {"id": "Mathlib.Topology.Order.LeftRightNhds", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 4, "macro_tier_score": 0.1184, "macro_tier_override": null, "x": -92.754, "z": 157.766, "size": 0.4628, "title": "Neighborhoods to the left and to the right on an `OrderTopology`", "summary": "We've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`. In an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable intervals to the right or to the left of `a`. We give now these characterizations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/LeftRightNhds.html"}, {"id": "Mathlib.Topology.Algebra.Nonarchimedean.Basic", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 2, "macro_tier_score": 0.0061, "macro_tier_override": null, "x": -65.923, "z": 153.128, "size": 0.3299, "title": "Nonarchimedean Topology", "summary": "In this file we set up the theory of nonarchimedean topological groups and rings. A nonarchimedean group is a topological group whose topology admits a basis of open neighborhoods of the identity element in the group consisting of open subgroups. A nonarchimedean ring is a topological ring whose underlying topological (additive) group is nonarchimedean.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Nonarchimedean/Basic.html"}, {"id": "Mathlib.Topology.VectorBundle.ContinuousAlternatingMap", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -116.423, "z": 245.276, "size": 0.2, "title": "The vector bundle of continuous alternating multilinear maps", "summary": "We define the topological vector bundle of continuous alternating maps between two vector bundles over the same base. Consider topological vector bundles with fibers `E₁ x`, `E₂ x`, `x : B`, with model fibers `F₁` and `F₂`, and a finite index type `ι`. If `F₁` and `F₂` are normed spaces over a nontrivially normed field `𝕜`, then we define a vector bundle with fiber `E₁ x [⋀^ι]→L[𝕜] E₂ x` with model fiber `F₁…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/VectorBundle/ContinuousAlternatingMap.html"}, {"id": "Mathlib.Topology.VectorBundle.Basic", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 2, "macro_tier_score": 0.0064, "macro_tier_override": null, "x": -62.203, "z": 223.518, "size": 0.3494, "title": "Vector bundles", "summary": "In this file we define (topological) vector bundles. Let `B` be the base space, let `F` be a normed space over a normed field `R`, and let `E : B → Type*` be a `FiberBundle` with fiber `F`, in which, for each `x`, the fiber `E x` is a topological vector space over `R`. To have a vector bundle structure on `Bundle.TotalSpace F E`, one should additionally have the following properties: * The bundle trivializations in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/VectorBundle/Basic.html"}, {"id": "Mathlib.Topology.Algebra.Module.ClosedSubmodule", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -77.245, "z": 244.39, "size": 0.2735, "title": "Closed submodules of a topological module", "summary": "This file builds the frame of closed `R`-submodules of a topological module `M`. One can turn `s : Submodule R E` + `hs : IsClosed s` into `s : ClosedSubmodule R E` in a tactic block by doing `lift s to ClosedSubmodule R E using hs`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/ClosedSubmodule.html"}, {"id": "Mathlib.Topology.Sets.Closeds", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 4, "macro_tier_score": 0.1176, "macro_tier_override": null, "x": -111.063, "z": 227.175, "size": 0.4336, "title": "Closed sets", "summary": "We define a few types of closed sets in a topological space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sets/Closeds.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.ENNReal", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 3, "macro_tier_score": 0.0261, "macro_tier_override": null, "x": -161.61, "z": 218.296, "size": 0.5089, "title": "Infinite sums in extended nonnegative reals", "summary": "This file proves results on infinite sums in `ℝ≥0∞`. In particular, we give lemmas relating sums of constants to the cardinality of the domain of these sums.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/ENNReal.html"}, {"id": "Mathlib.Topology.Category.Profinite.Projective", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -49.077, "z": 220.14, "size": 0.2, "title": "Profinite sets have enough projectives", "summary": "In this file we show that `Profinite` has enough projectives.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Profinite/Projective.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.Order", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 3, "macro_tier_score": 0.027, "macro_tier_override": null, "x": -81.644, "z": 140.314, "size": 0.3334, "title": "Infinite sum or product in an order", "summary": "This file provides lemmas about the interaction of infinite sums and products and order operations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/Order.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 3, "macro_tier_score": 0.0266, "macro_tier_override": null, "x": -64.506, "z": 226.394, "size": 0.2999, "title": "Infinite sums and products over `ℕ` and `ℤ`", "summary": "This file contains lemmas about `HasSum`, `Summable`, `tsum`, `HasProd`, `Multipliable`, and `tprod` applied to the important special cases where the domain is `ℕ` or `ℤ`. For instance, we prove the formula `∑ i ∈ range k, f i + ∑' i, f (i + k) = ∑' i, f i`, ∈ `sum_add_tsum_nat_add`, as well as several results relating sums and products on `ℕ` to sums and products on `ℤ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/NatInt.html"}, {"id": "Mathlib.Topology.FiberBundle.Constructions", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -69.315, "z": 152.058, "size": 0.2658, "title": "Standard constructions on fiber bundles", "summary": "This file contains several standard constructions on fiber bundles: * `Bundle.Trivial.fiberBundle 𝕜 B F`: the trivial fiber bundle with model fiber `F` over the base `B` * `FiberBundle.prod`: for fiber bundles `E₁` and `E₂` over a common base, a fiber bundle structure on their fiberwise product `E₁ ×ᵇ E₂` (the notation stands for `fun x ↦ E₁ x × E₂ x`). * `FiberBundle.pullback`: for a fiber bundle `E` over `B`, a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/FiberBundle/Constructions.html"}, {"id": "Mathlib.Topology.FiberBundle.Basic", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 3, "macro_tier_score": 0.0275, "macro_tier_override": null, "x": -109.04, "z": 243.204, "size": 0.36, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/FiberBundle/Basic.html"}, {"id": "Mathlib.Topology.LocallyConstant.Basic", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 3, "macro_tier_score": 0.0379, "macro_tier_override": null, "x": -95.844, "z": 229.78, "size": 0.3628, "title": "Locally constant functions", "summary": "This file sets up the theory of locally constant function from a topological space to a type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/LocallyConstant/Basic.html"}, {"id": "Mathlib.Topology.Connected.LocallyConnected", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 4, "macro_tier_score": 0.2784, "macro_tier_override": null, "x": -137.03, "z": 172.184, "size": 0.4073, "title": "Locally connected topological spaces", "summary": "A topological space is **locally connected** if each neighborhood filter admits a basis of connected *open* sets. Local connectivity is equivalent to each point having a basis of connected (not necessarily open) sets --- but in a non-trivial way, so we choose this definition and prove the equivalence later in `locallyConnectedSpace_iff_connected_basis`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Connected/LocallyConnected.html"}, {"id": "Mathlib.Topology.UniformSpace.Matrix", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -153.769, "z": 166.859, "size": 0.2918, "title": "Uniform space structure on matrices", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Matrix.html"}, {"id": "Mathlib.Topology.MetricSpace.Perfect", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 2, "macro_tier_score": 0.0063, "macro_tier_override": null, "x": -73.22, "z": 249.34, "size": 0.3374, "title": "Perfect Sets", "summary": "In this file we define properties of `Perfect` subsets of a metric space, including a version of the Cantor-Bendixson Theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Perfect.html"}, {"id": "Mathlib.Topology.Separation.CountableSeparatingOn", "region_id": "topology", "micro_elevation": 0.4186, "macro_tier": 1, "macro_tier_score": 0.001, "macro_tier_override": null, "x": -109.113, "z": 163.808, "size": 0.3322, "title": "Countable separating families of sets in topological spaces", "summary": "In this file we show that a T₀ topological space with second countable topology has a countable family of open (or closed) sets separating the points.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/CountableSeparatingOn.html"}, {"id": "Mathlib.Topology.OmegaCompletePartialOrder", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -139.633, "z": 212.931, "size": 0.2, "title": "Scott Topological Spaces", "summary": "A type of topological spaces whose notion of continuity is equivalent to continuity in ωCPOs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/OmegaCompletePartialOrder.html"}, {"id": "Mathlib.Topology.Order.ScottTopology", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 2, "macro_tier_score": 0.0108, "macro_tier_override": null, "x": -82.162, "z": 223.567, "size": 0.2827, "title": "Scott topology", "summary": "This file introduces the Scott topology on a preorder.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/ScottTopology.html"}, {"id": "Mathlib.Topology.Algebra.Equicontinuity", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -68.544, "z": 148.577, "size": 0.2978, "title": "Algebra-related equicontinuity criteria", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Equicontinuity.html"}, {"id": "Mathlib.Topology.Algebra.UniformConvergence", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 2, "macro_tier_score": 0.0061, "macro_tier_override": null, "x": -76.184, "z": 238.385, "size": 0.3252, "title": "Algebraic facts about the topology of uniform convergence", "summary": "This file contains algebraic compatibility results about the uniform structure of uniform convergence / `𝔖`-convergence. They will mostly be useful for defining strong topologies on the space of continuous linear maps between two topological vector spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/UniformConvergence.html"}, {"id": "Mathlib.Topology.UniformSpace.Equicontinuity", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 3, "macro_tier_score": 0.0324, "macro_tier_override": null, "x": -153.01, "z": 210.558, "size": 0.3452, "title": "Equicontinuity of a family of functions", "summary": "Let `X` be a topological space and `α` a `UniformSpace`. A family of functions `F : ι → X → α` is said to be *equicontinuous at a point `x₀ : X`* when, for any entourage `U` in `α`, there is a neighborhood `V` of `x₀` such that, for all `x ∈ V`, and *for all `i`*, `F i x` is `U`-close to `F i x₀`. In other words, one has `∀ U ∈ 𝓤 α, ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ U`. For maps between metric spaces, this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Equicontinuity.html"}, {"id": "Mathlib.Topology.Algebra.Polynomial", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 2, "macro_tier_score": 0.0067, "macro_tier_override": null, "x": -148.812, "z": 203.269, "size": 0.3651, "title": "Polynomials and limits", "summary": "In this file we prove the following lemmas. * `Polynomial.continuous_eval₂`: `Polynomial.eval₂` defines a continuous function. * `Polynomial.continuous_aeval`: `Polynomial.aeval` defines a continuous function; we also prove convenience lemmas `Polynomial.continuousAt_aeval`, `Polynomial.continuousWithinAt_aeval`, `Polynomial.continuousOn_aeval`. * `Polynomial.continuous`: `Polynomial.eval` defines a continuous…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Polynomial.html"}, {"id": "Mathlib.Topology.Algebra.Module.Spaces.WeakBilin", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 2, "macro_tier_score": 0.0118, "macro_tier_override": null, "x": -148.223, "z": 155.602, "size": 0.3569, "title": "Weak dual topology", "summary": "This file defines the weak topology given two vector spaces `E` and `F` over a commutative semiring `𝕜` and a bilinear form `B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜`. The weak topology on `E` is the coarsest topology such that for all `y : F` every map `fun x => B x y` is continuous.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Spaces/WeakBilin.html"}, {"id": "Mathlib.Topology.EMetricSpace.Diam", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 4, "macro_tier_score": 0.0898, "macro_tier_override": null, "x": -55.727, "z": 207.629, "size": 0.3494, "title": "Diameters of sets in extended metric spaces", "summary": "In this file we define the diameter of a set in the extended metric space as an extended nonnegative real number.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/EMetricSpace/Diam.html"}, {"id": "Mathlib.Topology.UniformSpace.UniformEmbedding", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 4, "macro_tier_score": 0.1481, "macro_tier_override": null, "x": -88.981, "z": 234.227, "size": 0.5669, "title": "Uniform embeddings of uniform spaces.", "summary": "Extension of uniform continuous functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/UniformEmbedding.html"}, {"id": "Mathlib.Topology.UniformSpace.Cauchy", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 4, "macro_tier_score": 0.1457, "macro_tier_override": null, "x": -64.292, "z": 175.104, "size": 0.5025, "title": "Theory of Cauchy filters in uniform spaces. Complete uniform spaces. Totally bounded subsets.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Cauchy.html"}, {"id": "Mathlib.Topology.UniformSpace.Separation", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 4, "macro_tier_score": 0.1439, "macro_tier_override": null, "x": -74.691, "z": 216.605, "size": 0.4444, "title": "Hausdorff properties of uniform spaces. Separation quotient.", "summary": "Two points of a topological space are called `Inseparable`, if their neighborhoods filter are equal. Equivalently, `Inseparable x y` means that any open set that contains `x` must contain `y` and vice versa. In a uniform space, points `x` and `y` are inseparable if and only if `(x, y)` belongs to all entourages, see `inseparable_iff_ker_uniformity`. A uniform space is a regular topological space, hence separation…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Separation.html"}, {"id": "Mathlib.Topology.Algebra.Valued.ValuedField", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -52.161, "z": 160.851, "size": 0.2584, "title": "Valued fields and their completions", "summary": "In this file we study the topology of a field `K` endowed with a valuation (in our application to adic spaces, `K` will be the valuation field associated to some valuation on a ring, defined in valuation.basic). We already know from valuation.topology that one can build a topology on `K` which makes it a topological ring. The first goal is to show `K` is a topological *field*, i.e. inversion is continuous at every…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Valued/ValuedField.html"}, {"id": "Mathlib.Topology.Algebra.Valued.ValuationTopology", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -57.574, "z": 225.14, "size": 0.3035, "title": "The topology on a valued ring", "summary": "In this file, we define the non-Archimedean topology induced by a valuation on a ring. The main definition is a `Valued` type class which equips a ring with a valuation taking values in a group with zero. Other instances are then deduced from this. *NOTE* (2025-07-02): The `Valued` class defined in this file will eventually get replaced with `ValuativeRel` from `Mathlib.RingTheory.Valuation.ValuativeRel.Basic`. New…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Valued/ValuationTopology.html"}, {"id": "Mathlib.Topology.Algebra.WithZeroTopology", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -74.545, "z": 151.795, "size": 0.2775, "title": "The topology on linearly ordered commutative groups with zero", "summary": "Let `Γ₀` be a linearly ordered commutative group to which we have adjoined a zero element. Then `Γ₀` may naturally be endowed with a topology that turns `Γ₀` into a topological monoid. Neighborhoods of zero are sets containing `{ γ | γ < γ₀ }` for some invertible element `γ₀` and every invertible element is open. In particular the topology is the following: \"a subset `U ⊆ Γ₀` is open if `0 ∉ U` or if there is an…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/WithZeroTopology.html"}, {"id": "Mathlib.Topology.Algebra.UniformField", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 2, "macro_tier_score": 0.006, "macro_tier_override": null, "x": -68.072, "z": 144.922, "size": 0.3227, "title": "Completion of topological fields", "summary": "The goal of this file is to prove the main part of Proposition 7 of Bourbaki GT III 6.8 : The completion `hat K` of a Hausdorff topological field is a field if the image under the mapping `x ↦ x⁻¹` of every Cauchy filter (with respect to the additive uniform structure) which does not have a cluster point at `0` is a Cauchy filter (with respect to the additive uniform structure). Bourbaki does not give any detail…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/UniformField.html"}, {"id": "Mathlib.Topology.ContinuousMap.Units", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -128.386, "z": 138.707, "size": 0.2739, "title": "Units of continuous functions", "summary": "This file concerns itself with `C(X, M)ˣ` and `C(X, Mˣ)` when `X` is a topological space and `M` has some monoid structure compatible with its topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Units.html"}, {"id": "Mathlib.Topology.Separation.Hausdorff", "region_id": "topology", "micro_elevation": 0.4651, "macro_tier": 4, "macro_tier_score": 0.3516, "macro_tier_override": null, "x": -120.074, "z": 219.084, "size": 0.5757, "title": "T₂ and T₂.₅ spaces.", "summary": "This file defines the T₂ (Hausdorff) condition, which is the most commonly-used among the various separation axioms, and the related T₂.₅ condition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/Hausdorff.html"}, {"id": "Mathlib.Topology.Order.Compact", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 4, "macro_tier_score": 0.0944, "macro_tier_override": null, "x": -88.75, "z": 230.771, "size": 0.3087, "title": "Compactness of a closed interval", "summary": "In this file we prove that a closed interval in a conditionally complete linear ordered type with order topology (or a product of such types) is compact. We prove the extreme value theorem (`IsCompact.exists_isMinOn`, `IsCompact.exists_isMaxOn`): a continuous function on a compact set takes its minimum and maximum values. We provide many variations of this theorem. We also prove that the image of a closed interval…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Compact.html"}, {"id": "Mathlib.Topology.Order.IntermediateValue", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 4, "macro_tier_score": 0.094, "macro_tier_override": null, "x": -132.782, "z": 221.357, "size": 0.2698, "title": "Intermediate Value Theorem", "summary": "In this file we prove the Intermediate Value Theorem: if `f : α → β` is a function defined on a connected set `s` that takes both values `≤ a` and values `≥ a` on `s`, then it is equal to `a` at some point of `s`. We also prove that intervals in a dense conditionally complete order are preconnected and any preconnected set is an interval. Then we specialize IVT to functions continuous on intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/IntermediateValue.html"}, {"id": "Mathlib.Topology.Order.LocalExtr", "region_id": "topology", "micro_elevation": 0.3488, "macro_tier": 4, "macro_tier_score": 0.0999, "macro_tier_override": null, "x": -92.127, "z": 171.96, "size": 0.3311, "title": "Local extrema of functions on topological spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/LocalExtr.html"}, {"id": "Mathlib.Topology.Algebra.ContinuousAffineMap", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 2, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": -105.722, "z": 133.737, "size": 0.3128, "title": "Continuous affine maps.", "summary": "This file defines a type of bundled continuous affine maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/ContinuousAffineMap.html"}, {"id": "Mathlib.Topology.Category.TopCat.GrothendieckTopology", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -69.812, "z": 162.831, "size": 0.2517, "title": "The Grothendieck topology on `TopCat`", "summary": "In this file we define the standard Grothendieck topology on `TopCat`. This is the topology generated by families of jointly surjective open embeddings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/GrothendieckTopology.html"}, {"id": "Mathlib.Topology.Baire.LocallyCompactRegular", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -90.432, "z": 146.976, "size": 0.258, "title": "Second Baire theorem", "summary": "In this file we prove that a locally compact regular topological space has Baire property.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Baire/LocallyCompactRegular.html"}, {"id": "Mathlib.Topology.Instances.Irrational", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -46.21, "z": 174.725, "size": 0.2695, "title": "Topology of irrational numbers", "summary": "In this file we prove the following theorems: * `IsGδ.setOf_irrational`, `dense_irrational`, `eventually_residual_irrational`: irrational numbers form a dense Gδ set; * `Irrational.eventually_forall_le_dist_cast_div`, `Irrational.eventually_forall_le_dist_cast_div_of_denom_le`; `Irrational.eventually_forall_le_dist_cast_rat_of_denom_le`: a sufficiently small neighborhood of an irrational number is disjoint with the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/Irrational.html"}, {"id": "Mathlib.Topology.IsLocalHomeomorph", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 3, "macro_tier_score": 0.0165, "macro_tier_override": null, "x": -133.242, "z": 231.102, "size": 0.3273, "title": "Local homeomorphisms", "summary": "This file defines local homeomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/IsLocalHomeomorph.html"}, {"id": "Mathlib.Topology.OpenPartialHomeomorph.Composition", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 3, "macro_tier_score": 0.0386, "macro_tier_override": null, "x": -125.891, "z": 233.718, "size": 0.3966, "title": "Partial homeomorphisms: composition", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/OpenPartialHomeomorph/Composition.html"}, {"id": "Mathlib.Topology.SeparatedMap", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 3, "macro_tier_score": 0.0214, "macro_tier_override": null, "x": -107.69, "z": 154.197, "size": 0.2968, "title": "Separated maps and locally injective maps out of a topological space.", "summary": "This module introduces a pair of dual notions `IsSeparatedMap` and `IsLocallyInjective`. A function from a topological space `X` to a type `Y` is a separated map if any two distinct points in `X` with the same image in `Y` can be separated by open neighborhoods. A constant function is a separated map if and only if `X` is a `T2Space`. A function from a topological space `X` is locally injective if every point of `X`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/SeparatedMap.html"}, {"id": "Mathlib.Topology.Bornology.Real", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 3, "macro_tier_score": 0.0733, "macro_tier_override": null, "x": -157.972, "z": 172.66, "size": 0.2845, "title": "The reals are equipped with their order bornology", "summary": "This file contains results related to the order bornology on (non-negative) real numbers. We prove that `ℝ` and `ℝ≥0` are equipped with the order topology and bornology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Bornology/Real.html"}, {"id": "Mathlib.Topology.Order.Rolle", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -145.795, "z": 175.861, "size": 0.2688, "title": "Rolle's Theorem (topological part)", "summary": "In this file we prove the purely topological part of Rolle's Theorem: a function that is continuous on an interval $[a, b]$, $a < b$, has a local extremum at a point $x ∈ (a, b)$ provided that $f(a)=f(b)$. We also prove several variations of this statement. In `Mathlib/Analysis/Calculus/LocalExtr/Rolle` we use these lemmas to prove several versions of Rolle's Theorem from calculus.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Rolle.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.Real", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 1, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": -105.652, "z": 127.471, "size": 0.3381, "title": "Infinite sum in the reals", "summary": "This file provides lemmas about Cauchy sequences in terms of infinite sums and infinite sums valued in the reals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/Real.html"}, {"id": "Mathlib.Topology.Compactification.StoneCech", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 3, "macro_tier_score": 0.0477, "macro_tier_override": null, "x": -71.273, "z": 168.703, "size": 0.3182, "title": "Stone-Čech compactification", "summary": "Construction of the Stone-Čech compactification using ultrafilters. For any topological space `α`, we build a compact Hausdorff space `StoneCech α` and a continuous map `stoneCechUnit : α → StoneCech α` which is minimal in the sense of the following universal property: for any compact Hausdorff space `β` and every map `f : α → β` such that `hf : Continuous f`, there is a unique map `stoneCechExtend hf : StoneCech α…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactification/StoneCech.html"}, {"id": "Mathlib.Topology.DenseEmbedding", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 4, "macro_tier_score": 0.2577, "macro_tier_override": null, "x": -129.005, "z": 222.535, "size": 0.4124, "title": "Dense embeddings", "summary": "This file defines three properties of functions: * `DenseRange f` means `f` has dense image; * `IsDenseInducing i` means `i` is also inducing, namely it induces the topology on its codomain; * `IsDenseEmbedding e` means `e` is further an embedding, namely it is injective and `Inducing`. The main theorem `continuous_extend` gives a criterion for a function `f : X → Z` to a T₃ space Z to extend along a dense embedding…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/DenseEmbedding.html"}, {"id": "Mathlib.Topology.Connected.TotallyDisconnected", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 4, "macro_tier_score": 0.2699, "macro_tier_override": null, "x": -124.171, "z": 159.413, "size": 0.4778, "title": "Totally disconnected and totally separated topological spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Connected/TotallyDisconnected.html"}, {"id": "Mathlib.Topology.Instances.Matrix", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 2, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": -127.316, "z": 143.381, "size": 0.3398, "title": "Topological properties of matrices", "summary": "This file is a place to collect topological results about matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/Matrix.html"}, {"id": "Mathlib.Topology.Category.Stonean.Basic", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -167.016, "z": 183.573, "size": 0.2979, "title": "Extremally disconnected sets", "summary": "This file develops some of the basic theory of extremally disconnected compact Hausdorff spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Stonean/Basic.html"}, {"id": "Mathlib.Topology.ExtremallyDisconnected", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -141.666, "z": 212.566, "size": 0.2649, "title": "Extremally disconnected spaces", "summary": "An extremally disconnected topological space is a space in which the closure of every open set is open. Such spaces are also called Stonean spaces. They are the projective objects in the category of compact Hausdorff spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ExtremallyDisconnected.html"}, {"id": "Mathlib.Topology.Algebra.Module.ContinuousLinearMap.RestrictScalars", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 2, "macro_tier_score": 0.008, "macro_tier_override": null, "x": -87.013, "z": 245.159, "size": 0.4248, "title": "Restriction of scalars for continuous linear maps", "summary": "In this file, we define and study `ContinuousLinearMap.restrictScalars`, which reinterprets a continuous `R`-linear map as a continuous `S`-linear map, for suitable `R` and `S`. This is the continuous version of `LinearMap.restrictScalars`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/ContinuousLinearMap/RestrictScalars.html"}, {"id": "Mathlib.Topology.Metrizable.CompletelyMetrizable", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 3, "macro_tier_score": 0.0162, "macro_tier_override": null, "x": -165.973, "z": 192.401, "size": 0.3021, "title": "Completely (pseudo)metrizable spaces", "summary": "A topological space is completely (pseudo)metrizable if one can endow it with a `(Pseudo)MetricSpace` structure which makes it complete and gives the same topology. This typeclass allows to state theorems which do not require a `(Pseudo)MetricSpace` structure to make sense without introducing such a structure. It is in particular useful in measure theory, where one often assumes that a space is a `PolishSpace`, i.e.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Metrizable/CompletelyMetrizable.html"}, {"id": "Mathlib.Topology.MetricSpace.Gluing", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 3, "macro_tier_score": 0.0213, "macro_tier_override": null, "x": -82.264, "z": 246.732, "size": 0.2922, "title": "Metric space gluing", "summary": "Gluing two metric spaces along a common subset. Formally, we are given ``` Φ Z ---> X | |Ψ v Y ``` where `hΦ : Isometry Φ` and `hΨ : Isometry Ψ`. We want to complete the square by a space `GlueSpace hΦ hΨ` and two isometries `toGlueL hΦ hΨ` and `toGlueR hΦ hΨ` that make the square commute. We start by defining a predistance on the disjoint union `X ⊕ Y`, for which points `Φ p` and `Ψ p` are at distance 0. The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Gluing.html"}, {"id": "Mathlib.Topology.Metrizable.Uniformity", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 3, "macro_tier_score": 0.0425, "macro_tier_override": null, "x": -121.59, "z": 139.469, "size": 0.323, "title": "Metrizable uniform spaces", "summary": "In this file we prove that a uniform space with countably generated uniformity filter is pseudometrizable: there exists a `PseudoMetricSpace` structure that generates the same uniformity. The proof follows [Sergey Melikhov, Metrizable uniform spaces][melikhov2011].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Metrizable/Uniformity.html"}, {"id": "Mathlib.Topology.EMetricSpace.BoundedVariation", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -126.238, "z": 132.781, "size": 0.2746, "title": "Functions of bounded variation", "summary": "We study functions of bounded variation. In particular, we show that a bounded variation function is a difference of monotone functions, and differentiable almost everywhere. This implies that Lipschitz functions from the real line into finite-dimensional vector space are also differentiable almost everywhere.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/EMetricSpace/BoundedVariation.html"}, {"id": "Mathlib.Topology.Category.Born", "region_id": "topology", "micro_elevation": 0.0465, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -106.459, "z": 188.971, "size": 0.2, "title": "The category of bornologies", "summary": "This defines `Born`, the category of bornologies.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Born.html"}, {"id": "Mathlib.Topology.Bornology.Hom", "region_id": "topology", "micro_elevation": 0.0233, "macro_tier": 3, "macro_tier_score": 0.0215, "macro_tier_override": null, "x": -105.79, "z": 192.967, "size": 0.3132, "title": "Locally bounded maps", "summary": "This file defines locally bounded maps between bornologies. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Bornology/Hom.html"}, {"id": "Mathlib.Topology.MetricSpace.ShrinkingLemma", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -115.969, "z": 134.8, "size": 0.2, "title": "Shrinking lemma in a proper metric space", "summary": "In this file we prove a few versions of the shrinking lemma for coverings by balls in a proper (pseudo) metric space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/ShrinkingLemma.html"}, {"id": "Mathlib.Topology.EMetricSpace.Paracompact", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -89.486, "z": 143.989, "size": 0.2812, "title": "(Extended) metric spaces are paracompact", "summary": "In this file we provide two instances: * `EMetric.instParacompactSpace`: a `PseudoEMetricSpace` is paracompact; formalization is based on [MR0236876]; * `EMetric.instNormalSpace`: an `EMetricSpace` is a normal topological space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/EMetricSpace/Paracompact.html"}, {"id": "Mathlib.Topology.MetricSpace.ProperSpace.Lemmas", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 1, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -54.004, "z": 215.967, "size": 0.2916, "title": "Proper spaces", "summary": "This file contains some more involved results about `ProperSpace`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.html"}, {"id": "Mathlib.Topology.ShrinkingLemma", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 2, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": -73.115, "z": 168.805, "size": 0.26, "title": "The shrinking lemma", "summary": "In this file we prove a few versions of the shrinking lemma. The lemma says that in a normal topological space a point finite open covering can be “shrunk”: for a point finite open covering `u : ι → Set X` there exists a refinement `v : ι → Set X` such that `closure (v i) ⊆ u i`. For finite or countable coverings this lemma can be proved without the axiom of choice, see…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ShrinkingLemma.html"}, {"id": "Mathlib.Topology.Category.Profinite.Nobeling.Successor", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -39.549, "z": 184.088, "size": 0.2338, "title": "The successor case in the induction for Nöbeling's theorem", "summary": "Here we assume that `o` is an ordinal such that `contained C (o+1)` and `o < I`. The element in `I` corresponding to `o` is called `term I ho`, but in this informal docstring we refer to it simply as `o`. This section follows the proof in [scholze2019condensed] quite closely. A translation of the notation there is as follows: ``` [scholze2019condensed] | This file `S₀` |`C0` `S₁` |`C1` `\\overline{S}` |`π C (ord I ·…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Profinite/Nobeling/Successor.html"}, {"id": "Mathlib.Topology.Category.Profinite.Nobeling.Basic", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -151.265, "z": 147.261, "size": 0.2831, "title": "Preliminaries for Nöbeling's theorem", "summary": "This file constructs basic objects and results concerning them that are needed in the proof of Nöbeling's theorem, which is in `Mathlib/Topology/Category/Profinite/Nobeling/Induction.lean`. See the section docstrings for more information.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Profinite/Nobeling/Basic.html"}, {"id": "Mathlib.Topology.Instances.ENat", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 2, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": -54.631, "z": 174.086, "size": 0.2679, "title": "Topology on extended natural numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/ENat.html"}, {"id": "Mathlib.Topology.ContinuousMap.ContinuousMapZero", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 2, "macro_tier_score": 0.0075, "macro_tier_override": null, "x": -81.523, "z": 251.497, "size": 0.4029, "title": "Continuous maps sending zero to zero", "summary": "This is the type of continuous maps from `X` to `R` such that `(0 : X) ↦ (0 : R)` for which we provide the scoped notation `C(X, R)₀`. We provide this as a dedicated type solely for the non-unital continuous functional calculus, as using various terms of type `Ideal C(X, R)` were overly burdensome on type class synthesis. Of course, one could generalize to maps between pointed topological spaces, but that goes…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/ContinuousMapZero.html"}, {"id": "Mathlib.Topology.UrysohnsLemma", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 3, "macro_tier_score": 0.0645, "macro_tier_override": null, "x": -47.587, "z": 183.053, "size": 0.3906, "title": "Urysohn's lemma", "summary": "In this file we prove Urysohn's lemma `exists_continuous_zero_one_of_isClosed`: for any two disjoint closed sets `s` and `t` in a normal topological space `X` there exists a continuous function `f : X → ℝ` such that * `f` equals zero on `s`; * `f` equals one on `t`; * `0 ≤ f x ≤ 1` for all `x`. We also give versions in a regular locally compact space where one assumes that `s` is compact and `t` is closed, in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UrysohnsLemma.html"}, {"id": "Mathlib.Topology.UniformSpace.UniformConvergence", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 4, "macro_tier_score": 0.1318, "macro_tier_override": null, "x": -148.337, "z": 178.427, "size": 0.3673, "title": "Uniform convergence", "summary": "A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a set `s` to a function `f` if, for all `ε > 0`, for all large enough `n`, one has for all `y ∈ s` the inequality `dist (f y, Fₙ y) < ε`. Under uniform convergence, many properties of the `Fₙ` pass to the limit, most notably continuity. We prove this in the file, defining the notion of uniform convergence in the more general setting…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/UniformConvergence.html"}, {"id": "Mathlib.Topology.Category.CompHaus.Limits", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -144.662, "z": 145.319, "size": 0.301, "title": "Explicit limits and colimits", "summary": "This file applies the general API for explicit limits and colimits in `CompHausLike P` (see the file `Mathlib/Topology/Category/CompHausLike/Limits.lean`) to the special case of `CompHaus`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/CompHaus/Limits.html"}, {"id": "Mathlib.Topology.ContinuousMap.Star", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 2, "macro_tier_score": 0.0111, "macro_tier_override": null, "x": -72.793, "z": 239.933, "size": 0.3111, "title": "Star structures on continuous maps.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Star.html"}, {"id": "Mathlib.Topology.Algebra.FilterBasis", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": -86.188, "z": 243.213, "size": 0.3677, "title": "Group and ring filter bases", "summary": "A `GroupFilterBasis` is a `FilterBasis` on a group with some properties relating the basis to the group structure. The main theorem is that a `GroupFilterBasis` on a group gives a topology on the group which makes it into a topological group with neighborhoods of the neutral element generated by the given basis.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/FilterBasis.html"}, {"id": "Mathlib.Topology.Algebra.Module.Basic", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 3, "macro_tier_score": 0.083, "macro_tier_override": null, "x": -114.478, "z": 244.067, "size": 0.4967, "title": "Theory of topological modules", "summary": "We use the class `ContinuousSMul` for topological (semi) modules and topological vector spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Basic.html"}, {"id": "Mathlib.Topology.Separation.PerfectlyNormal", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -54.18, "z": 222.838, "size": 0.2, "title": "Perfectly normal topological spaces.", "summary": "This file proves some properties of a perfectly normal space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/PerfectlyNormal.html"}, {"id": "Mathlib.Topology.Separation.GDelta", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 3, "macro_tier_score": 0.0169, "macro_tier_override": null, "x": -139.963, "z": 209.099, "size": 0.3535, "title": "Separation properties of topological spaces.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/GDelta.html"}, {"id": "Mathlib.Topology.Subpath", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2, "title": "Subpaths and concatenation of paths", "summary": "This file defines `Path.subpath` as a restriction of a path to a subinterval, reparameterized to have domain `[0, 1]` and possibly with a reverse of direction. It then defines `Path.concat` as a way to concatenate finite sequences of paths with compatible endpoints. The main result `Path.Homotopy.concatSubpath` shows that subpaths concatenate nicely. In particular: following the subpaths of `γ` from `t i` to `t (i +…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Subpath.html"}, {"id": "Mathlib.Topology.Algebra.Algebra", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 2, "macro_tier_score": 0.0125, "macro_tier_override": null, "x": -160.648, "z": 200.09, "size": 0.3937, "title": "Topological (sub)algebras", "summary": "A topological algebra over a topological semiring `R` is a topological semiring with a compatible continuous scalar multiplication by elements of `R`. We reuse typeclass `ContinuousSMul` for topological algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Algebra.html"}, {"id": "Mathlib.Topology.Algebra.UniformRing", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 3, "macro_tier_score": 0.0166, "macro_tier_override": null, "x": -113.062, "z": 134.31, "size": 0.3344, "title": "Completion of topological rings:", "summary": "This file endows the completion of a topological ring with a ring structure. More precisely, the instance `UniformSpace.Completion.ring` builds a ring structure on the completion of a ring endowed with a compatible uniform structure in the sense of `IsUniformAddGroup`. There is also a commutative version when the original ring is commutative. Moreover, if a topological ring is an algebra over a commutative semiring,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/UniformRing.html"}, {"id": "Mathlib.Topology.Sheaves.Over", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -59.495, "z": 193.149, "size": 0.2414, "title": "Opens and Over categories", "summary": "In this file, given a topological space `X`, and `U : Opens X`, we show that the category `Over U` (whose objects are the `V : Opens X` equipped with a morphism `V ⟶ U`) is equivalent to the category `Opens U`. This equivalence is bi-continuous, and thus induces an equivalence of sheaf categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Over.html"}, {"id": "Mathlib.Topology.Sheaves.SheafCondition.Sites", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 2, "macro_tier_score": 0.0067, "macro_tier_override": null, "x": -132.127, "z": 226.058, "size": 0.3639, "title": "Coverings and sieves; from sheaves on sites and sheaves on spaces", "summary": "In this file, we connect coverings in a topological space to sieves in the associated Grothendieck topology, in preparation of connecting the sheaf condition on sites to the various sheaf conditions on spaces. We also specialize results about sheaves on sites to sheaves on spaces; we show that the inclusion functor from a topological basis to `TopologicalSpace.Opens` is cover dense, that open maps induce…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/SheafCondition/Sites.html"}, {"id": "Mathlib.Topology.ContinuousMap.StoneWeierstrass", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 0, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": -146.137, "z": 140.436, "size": 0.3032, "title": "The Stone-Weierstrass theorem", "summary": "If a subalgebra `A` of `C(X, ℝ)`, where `X` is a compact topological space, separates points, then it is dense. We argue as follows. * In any subalgebra `A` of `C(X, ℝ)`, if `f ∈ A`, then `abs f ∈ A.topologicalClosure`. This follows from the Weierstrass approximation theorem on `[-‖f‖, ‖f‖]` by approximating `abs` uniformly thereon by polynomials. * This ensures that `A.topologicalClosure` is actually a sublattice:…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/StoneWeierstrass.html"}, {"id": "Mathlib.Topology.Constructions", "region_id": "topology", "micro_elevation": 0.2791, "macro_tier": 4, "macro_tier_score": 0.456, "macro_tier_override": null, "x": -115.971, "z": 176.5, "size": 0.6896, "title": "Constructions of new topological spaces from old ones", "summary": "This file constructs pi types, subtypes and quotients of topological spaces and sets up their basic theory, such as criteria for maps into or out of these constructions to be continuous; descriptions of the open sets, neighborhood filters, and generators of these constructions; and their behavior with respect to embeddings and other specific classes of maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Constructions.html"}, {"id": "Mathlib.Topology.Constructions.SumProd", "region_id": "topology", "micro_elevation": 0.2558, "macro_tier": 4, "macro_tier_score": 0.4512, "macro_tier_override": null, "x": -87.916, "z": 194.621, "size": 0.4414, "title": "Disjoint unions and products of topological spaces", "summary": "This file constructs sums (disjoint unions) and products of topological spaces and sets up their basic theory, such as criteria for maps into or out of these constructions to be continuous; descriptions of the open sets, neighborhood filters, and generators of these constructions; and their behavior with respect to embeddings and other specific classes of maps. We also provide basic homeomorphisms, to show that sums…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Constructions/SumProd.html"}, {"id": "Mathlib.Topology.WithTopology", "region_id": "topology", "micro_elevation": 0.093, "macro_tier": 4, "macro_tier_score": 0.4496, "macro_tier_override": null, "x": -104.372, "z": 197.927, "size": 0.3739, "title": "Basic lemmas and instances about the `WithTopology` type synonym", "summary": "`WithTopology X t` is a copy of `X` equipped with the topology `t`. This is useful for providing multiple topologies on the same type without causing instance conflicts. In this file we setup basic API about this type and transfer instances (basic, order) from `X` to `WithTopology X t`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/WithTopology.html"}, {"id": "Mathlib.Topology.Separation.Profinite", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 3, "macro_tier_score": 0.0268, "macro_tier_override": null, "x": -95.76, "z": 153.607, "size": 0.3196, "title": "Separation properties: profinite spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/Profinite.html"}, {"id": "Mathlib.Topology.Sober", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 3, "macro_tier_score": 0.0163, "macro_tier_override": null, "x": -129.588, "z": 220.024, "size": 0.3095, "title": "Sober spaces", "summary": "A quasi-sober space is a topological space where every irreducible closed subset has a generic point. A sober space is a quasi-sober space where every irreducible closed subset has a *unique* generic point. This is if and only if the space is T0, and thus sober spaces can be stated via `[QuasiSober α] [T0Space α]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sober.html"}, {"id": "Mathlib.Topology.Sets.OpenCover", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 3, "macro_tier_score": 0.064, "macro_tier_override": null, "x": -84.177, "z": 162.212, "size": 0.3647, "title": "Open covers", "summary": "We define `IsOpenCover` as a predicate on indexed families of open sets in a topological space `X`, asserting that their union is `X`. This is an example of a declaration whose name is actually longer than its content; but giving it a name serves as a way of standardizing API.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sets/OpenCover.html"}, {"id": "Mathlib.Topology.Order.LeftRightLim", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -111.563, "z": 151.51, "size": 0.2843, "title": "Left and right limits", "summary": "We define the (strict) left and right limits of a function. * `leftLim f x` is the strict left limit of `f` at `x` (using `f x` as a garbage value if `x` is isolated to its left). * `rightLim f x` is the strict right limit of `f` at `x` (using `f x` as a garbage value if `x` is isolated to its right). We develop a comprehensive API for monotone functions. Notably, * `Monotone.continuousAt_iff_leftLim_eq_rightLim`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/LeftRightLim.html"}, {"id": "Mathlib.Topology.Algebra.Module.LinearPMap", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -47.499, "z": 175.921, "size": 0.2338, "title": "Partially defined linear operators over topological vector spaces", "summary": "We define basic notions of partially defined linear operators, which we call unbounded operators for short. In this file we prove all elementary properties of unbounded operators that do not assume that the underlying spaces are normed.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/LinearPMap.html"}, {"id": "Mathlib.Topology.Algebra.Module.PerfectSpace", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -116.67, "z": 230.666, "size": 0.2962, "title": "Vector spaces over nontrivially normed fields are perfect spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/PerfectSpace.html"}, {"id": "Mathlib.Topology.MetricSpace.Ultra.ContinuousMaps", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -85.299, "z": 252.838, "size": 0.227, "title": "Ultrametric structure on continuous maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Ultra/ContinuousMaps.html"}, {"id": "Mathlib.Topology.Algebra.Module.LinearMapPiProd", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -141.037, "z": 146.384, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/LinearMapPiProd.html"}, {"id": "Mathlib.Topology.MetricSpace.CauSeqFilter", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -82.588, "z": 238.31, "size": 0.2728, "title": "Completeness in terms of `Cauchy` filters vs `isCauSeq` sequences", "summary": "In this file we apply `Metric.complete_of_cauchySeq_tendsto` to prove that a `NormedRing` is complete in terms of `Cauchy` filter if and only if it is complete in terms of `CauSeq` Cauchy sequences.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/CauSeqFilter.html"}, {"id": "Mathlib.Topology.Algebra.Group.AddTorsor", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -137.707, "z": 229.569, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/AddTorsor.html"}, {"id": "Mathlib.Topology.Algebra.Group.Torsor", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 3, "macro_tier_score": 0.0695, "macro_tier_override": null, "x": -61.582, "z": 213.631, "size": 0.3778, "title": "Topological torsors of groups", "summary": "This file defines topological torsors of additive and multiplicative groups, that is, torsors where `+ᵥ` and `-ᵥ` resp. `•` and `/ₛ` are continuous.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/Torsor.html"}, {"id": "Mathlib.Topology.Algebra.Module.Determinant", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -162.783, "z": 177.859, "size": 0.2733, "title": "The determinant of a continuous linear map.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Determinant.html"}, {"id": "Mathlib.Topology.Algebra.MulAction", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 4, "macro_tier_score": 0.1316, "macro_tier_override": null, "x": -136.831, "z": 223.981, "size": 0.3562, "title": "Continuous monoid action", "summary": "In this file we define class `ContinuousSMul`. We say `ContinuousSMul M X` if `M` acts on `X` and the map `(c, x) ↦ c • x` is continuous on `M × X`. We reuse this class for topological (semi)modules, vector spaces and algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/MulAction.html"}, {"id": "Mathlib.Topology.Order.SuccPred", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -114.221, "z": 226.479, "size": 0.2302, "title": "Order topologies of successor or predecessor orders", "summary": "This file proves miscellaneous results under the assumption of `OrderTopology` plus either of `SuccOrder` or `PredOrder`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/SuccPred.html"}, {"id": "Mathlib.Topology.Algebra.IsUniformGroup.DiscreteSubgroup", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -159.602, "z": 205.329, "size": 0.273, "title": "Discrete subgroups of topological groups", "summary": "Note that the instance `Subgroup.isClosed_of_discrete` does not live here, in order that it can be used in other files without requiring lots of group-theoretic imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/IsUniformGroup/DiscreteSubgroup.html"}, {"id": "Mathlib.Topology.Algebra.Group.ClosedSubgroup", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -99.793, "z": 138.675, "size": 0.3033, "title": "Closed subgroups of a topological group", "summary": "This file builds the frame of closed subgroups in a topological group `G`, and its additive version `ClosedAddSubgroup`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/ClosedSubgroup.html"}, {"id": "Mathlib.Topology.Algebra.NonUnitalAlgebra", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 2, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": -100.135, "z": 137.071, "size": 0.2628, "title": "Non-unital topological (sub)algebras", "summary": "A non-unital topological algebra over a topological semiring `R` is a topological (non-unital) semiring with a compatible continuous scalar multiplication by elements of `R`. We reuse typeclass `ContinuousSMul` to express the latter condition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/NonUnitalAlgebra.html"}, {"id": "Mathlib.Topology.Metrizable.Real", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 2, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": -54.16, "z": 212.454, "size": 0.3386, "title": "`ENNReal` is metrizable", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Metrizable/Real.html"}, {"id": "Mathlib.Topology.Order.MonotoneContinuity", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 3, "macro_tier_score": 0.079, "macro_tier_override": null, "x": -136.564, "z": 211.931, "size": 0.325, "title": "Continuity of monotone functions", "summary": "In this file we prove the following fact: if `f` is a monotone function on a neighborhood of `a` and the image of this neighborhood is a neighborhood of `f a`, then `f` is continuous at `a`, see `continuousWithinAt_of_monotoneOn_of_image_mem_nhds`, as well as several similar facts. We also prove that an `OrderIso` is continuous.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/MonotoneContinuity.html"}, {"id": "Mathlib.Topology.Order.Real", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 3, "macro_tier_score": 0.0804, "macro_tier_override": null, "x": -115.179, "z": 152.281, "size": 0.4005, "title": "The reals are equipped with their order topology", "summary": "This file contains results related to the order topology on (extended) (non-negative) real numbers. We - prove that `ℝ` and `ℝ≥0` are equipped with the order topology and bornology, - endow `EReal` with the order topology (and prove some very basic lemmas), - define the topology `ℝ≥0∞` (which is the order topology, *not* the `EMetricSpace` topology)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Real.html"}, {"id": "Mathlib.Topology.Algebra.ProperAction.Torsor", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -64.312, "z": 228.539, "size": 0.2676, "title": "The action underlying a topological torsor is proper.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/ProperAction/Torsor.html"}, {"id": "Mathlib.Topology.Algebra.ProperAction.Basic", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -53.068, "z": 203.935, "size": 0.2862, "title": "Proper group action", "summary": "In this file we define proper action of a group on a topological space, and we prove that in this case the quotient space is T2. We also give equivalent definitions of proper action using ultrafilters and show the transfer of proper action to a closed subgroup.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/ProperAction/Basic.html"}, {"id": "Mathlib.Topology.Algebra.Nonarchimedean.Completion", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -84.683, "z": 247.665, "size": 0.236, "title": "The completion of a nonarchimedean additive group", "summary": "The completion of a nonarchimedean additive group is a nonarchimedean additive group. The completion of a nonarchimedean ring is a nonarchimedean ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Nonarchimedean/Completion.html"}, {"id": "Mathlib.Topology.Algebra.Module.Spaces.PointwiseConvergenceCLM", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -101.74, "z": 255.823, "size": 0.2767, "title": "Topology of pointwise convergence on continuous linear maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Spaces/PointwiseConvergenceCLM.html"}, {"id": "Mathlib.Topology.Algebra.Module.Spaces.WeakDual", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 2, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": -130.977, "z": 243.433, "size": 0.3362, "title": "Weak dual topology", "summary": "We continue in the setting of `Mathlib/Topology/Algebra/Module/WeakBilin.lean`, which defines the weak topology given two vector spaces `E` and `F` over a commutative semiring `𝕜` and a bilinear form `B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜`. The weak topology on `E` is the coarsest topology such that for all `y : F` every map `fun x => B x y` is continuous. In this file, we consider two special cases. In the case that `F = E →L[𝕜]…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Spaces/WeakDual.html"}, {"id": "Mathlib.Topology.MetricSpace.Bilipschitz", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -161.463, "z": 204.264, "size": 0.2584, "title": "Bilipschitz equivalence", "summary": "A common pattern in Mathlib is to replace the topology, uniformity and bornology on a type synonym with those of the underlying type. The most common way to do this is to activate a local instance for something which puts a `PseudoMetricSpace` structure on the type synonym, prove that this metric is bilipschitz equivalent to the metric on the underlying type, and then use this to show that the uniformities and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Bilipschitz.html"}, {"id": "Mathlib.Topology.MetricSpace.Antilipschitz", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 3, "macro_tier_score": 0.0433, "macro_tier_override": null, "x": -48.807, "z": 197.489, "size": 0.3708, "title": "Antilipschitz functions", "summary": "We say that a map `f : α → β` between two (extended) metric spaces is `AntilipschitzWith K`, `K ≥ 0`, if for all `x, y` we have `edist x y ≤ K * edist (f x) (f y)`. For a metric space, the latter inequality is equivalent to `dist x y ≤ K * dist (f x) (f y)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Antilipschitz.html"}, {"id": "Mathlib.Topology.Algebra.Valued.WithVal", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -42.708, "z": 184.053, "size": 0.2763, "title": "Ring topologised by a valuation", "summary": "For a given valuation `v : Valuation R Γ₀` on a ring `R` taking values in `Γ₀`, this file defines the type synonym `WithVal v` of `R`. By assigning a `Valued (WithVal v) Γ₀` instance, `WithVal v` represents the ring `R` equipped with the topology coming from `v`. The type synonym `WithVal v` is in isomorphism to `R` as rings via `WithVal.equiv v`. This isomorphism should be used to explicitly map terms of `WithVal…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Valued/WithVal.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.Defs", "region_id": "topology", "micro_elevation": 0.4884, "macro_tier": 4, "macro_tier_score": 0.0953, "macro_tier_override": null, "x": -82.63, "z": 167.471, "size": 0.3697, "title": "Infinite sum and product in a topological monoid", "summary": "This file defines infinite products and sums for (possibly infinite) indexed families of elements in a commutative topological monoid (resp. add monoid). To handle convergence questions we use the formalism of *summation filters* (defined in the file `Mathlib/Topology/Algebra/InfiniteSum/SummationFilter.lean`). These are filters on the finite subsets of a given type, and we define a function to be *summable* for a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/Defs.html"}, {"id": "Mathlib.Topology.UniformSpace.Equiv", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 4, "macro_tier_score": 0.0904, "macro_tier_override": null, "x": -95.741, "z": 239.368, "size": 0.383, "title": "Uniform isomorphisms", "summary": "This file defines uniform isomorphisms between two uniform spaces. They are bijections with both directions uniformly continuous. We denote uniform isomorphisms with the notation `≃ᵤ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Equiv.html"}, {"id": "Mathlib.Topology.Sheaves.MayerVietoris", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2, "title": "Mayer-Vietoris squares", "summary": "Given two open subsets `U` and `V` of a topological space `T`, we construct the associated Mayer-Vietoris square: ``` U ⊓ V ---> U | | v v V ---> U ⊔ V ```", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/MayerVietoris.html"}, {"id": "Mathlib.Topology.KrullDimension", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -145.28, "z": 179.142, "size": 0.2953, "title": "The Krull dimension of a topological space", "summary": "The Krull dimension of a topological space is the order-theoretic Krull dimension applied to the collection of all its subsets that are closed and irreducible. Unfolding this definition, it is the length of longest series of closed irreducible subsets ordered by inclusion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/KrullDimension.html"}, {"id": "Mathlib.Topology.Defs.Induced", "region_id": "topology", "micro_elevation": 0.0698, "macro_tier": 4, "macro_tier_score": 0.456, "macro_tier_override": null, "x": -100.257, "z": 192.448, "size": 0.4256, "title": "Induced and coinduced topologies", "summary": "In this file we define the induced and coinduced topologies, as well as topology inducing maps, topological embeddings, and quotient maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Defs/Induced.html"}, {"id": "Mathlib.Topology.Maps.Basic", "region_id": "topology", "micro_elevation": 0.186, "macro_tier": 4, "macro_tier_score": 0.449, "macro_tier_override": null, "x": -97.622, "z": 201.888, "size": 0.3383, "title": "Specific classes of maps between topological spaces", "summary": "This file introduces the following properties of a map `f : X → Y` between topological spaces: * `IsOpenMap f` means the image of an open set under `f` is open. * `IsClosedMap f` means the image of a closed set under `f` is closed. (Open and closed maps need not be continuous.) * `IsInducing f` means the topology on `X` is the one induced via `f` from the topology on `Y`. These behave like embeddings except they…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Maps/Basic.html"}, {"id": "Mathlib.Topology.Order", "region_id": "topology", "micro_elevation": 0.1628, "macro_tier": 4, "macro_tier_score": 0.4502, "macro_tier_override": null, "x": -111.034, "z": 200.766, "size": 0.4038, "title": "Ordering on topologies and (co)induced topologies", "summary": "Topologies on a fixed type `α` are ordered, by reverse inclusion. That is, for topologies `t₁` and `t₂` on `α`, we write `t₁ ≤ t₂` if every set open in `t₂` is also open in `t₁`. (One also calls `t₁` *finer* than `t₂`, and `t₂` *coarser* than `t₁`.) Any function `f : α → β` induces * `TopologicalSpace.induced f : TopologicalSpace β → TopologicalSpace α`; * `TopologicalSpace.coinduced f : TopologicalSpace α →…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order.html"}, {"id": "Mathlib.Topology.NhdsSet", "region_id": "topology", "micro_elevation": 0.1163, "macro_tier": 4, "macro_tier_score": 0.4493, "macro_tier_override": null, "x": -109.463, "z": 198.043, "size": 0.3562, "title": "Neighborhoods of a set", "summary": "In this file we define the filter `𝓝ˢ s` or `nhdsSet s` consisting of all neighborhoods of a set `s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/NhdsSet.html"}, {"id": "Mathlib.Topology.Homotopy.Contractible", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 2, "macro_tier_score": 0.006, "macro_tier_override": null, "x": -145.881, "z": 140.231, "size": 0.3189, "title": "Contractible spaces", "summary": "In this file, we define `ContractibleSpace`, a space that is homotopy equivalent to `Unit`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/Contractible.html"}, {"id": "Mathlib.Topology.Convenient.HomSpace", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -141.551, "z": 209.419, "size": 0.2, "title": "The topological space of `X`-continuous maps", "summary": "Let `X i` be a family of topological spaces. Let `Z` and `T` be topological spaces. In this file, we endow the type `ContinuousMapGeneratedBy X Z T` of `X`-continuous maps `Z → T` with the coarsest topology which makes the precomposition maps `ContinuousMapGeneratedBy X Z T → C(X i, T)` continuous for any continuous map `X i → Z`, where `C(X i, T)` is endowed with the compact-open topology. If we assume that the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Convenient/HomSpace.html"}, {"id": "Mathlib.Topology.Convenient.ContinuousMapGeneratedBy", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 2, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": -96.721, "z": 226.769, "size": 0.2713, "title": "`X`-continuous maps", "summary": "Given a family `X i` of topological spaces, we introduce a predicate `ContinuousGeneratedBy X` on maps `g : Y ⟶ Z` saying that `g` is `X`-continuous, i.e. for any continuous map `f : X i → Y`, the composition `g ∘ f` is continuous.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Convenient/ContinuousMapGeneratedBy.html"}, {"id": "Mathlib.Topology.MetricSpace.Sequences", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -77.059, "z": 142.647, "size": 0.2474, "title": "Sequential compacts in metric spaces", "summary": "In this file we prove 2 versions of Bolzano-Weierstrass theorem for proper metric spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Sequences.html"}, {"id": "Mathlib.Topology.OpenPartialHomeomorph.Continuity", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 3, "macro_tier_score": 0.04, "macro_tier_override": null, "x": -87.573, "z": 231.974, "size": 0.4529, "title": "Partial homeomorphisms and continuity", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/OpenPartialHomeomorph/Continuity.html"}, {"id": "Mathlib.Topology.Category.TopCat.Monoidal", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 2, "macro_tier_score": 0.011, "macro_tier_override": null, "x": -164.152, "z": 206.506, "size": 0.3016, "title": "The cartesian monoidal structure on `TopCat`", "summary": "We define the cartesian monoidal category structure on `TopCat`. We also introduce the unit interval as an object `TopCat.I` of `TopCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/Monoidal.html"}, {"id": "Mathlib.Topology.Piecewise", "region_id": "topology", "micro_elevation": 0.3488, "macro_tier": 4, "macro_tier_score": 0.3655, "macro_tier_override": null, "x": -90.98, "z": 210.615, "size": 0.3281, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Piecewise.html"}, {"id": "Mathlib.Topology.MetricSpace.Pseudo.Basic", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 4, "macro_tier_score": 0.0899, "macro_tier_override": null, "x": -62.277, "z": 218.064, "size": 0.3557, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Pseudo/Basic.html"}, {"id": "Mathlib.Topology.EMetricSpace.Basic", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 4, "macro_tier_score": 0.0907, "macro_tier_override": null, "x": -150.085, "z": 173.931, "size": 0.3952, "title": "Extended metric spaces", "summary": "Further results about extended metric spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/EMetricSpace/Basic.html"}, {"id": "Mathlib.Topology.MetricSpace.Pseudo.Defs", "region_id": "topology", "micro_elevation": 0.3953, "macro_tier": 4, "macro_tier_score": 0.0922, "macro_tier_override": null, "x": -83.813, "z": 175.416, "size": 0.4568, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Pseudo/Defs.html"}, {"id": "Mathlib.Topology.Metrizable.Basic", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 4, "macro_tier_score": 0.1066, "macro_tier_override": null, "x": -109.26, "z": 143.326, "size": 0.4104, "title": "Metrizable Spaces", "summary": "In this file we define metrizable topological spaces, i.e., topological spaces for which there exists a metric space structure that generates the same topology. We define it without any reference to metric spaces in order to avoid importing the real numbers. For the proof that metrizable spaces admit a compatible metric, see `Mathlib/Topology/Metrizable/Uniformity.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Metrizable/Basic.html"}, {"id": "Mathlib.Topology.ContinuousMap.Interval", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -109.434, "z": 151.214, "size": 0.2302, "title": "Continuous bundled maps on intervals", "summary": "In this file we prove a few results about `ContinuousMap` when the domain is an interval.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Interval.html"}, {"id": "Mathlib.Topology.Order.ProjIcc", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 3, "macro_tier_score": 0.0549, "macro_tier_override": null, "x": -136.625, "z": 174.631, "size": 0.4264, "title": "Projection onto a closed interval", "summary": "In this file we prove that the projection `Set.projIcc f a b h` is a quotient map, and use it to show that `Set.IccExtend h f` is continuous if and only if `f` is continuous.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/ProjIcc.html"}, {"id": "Mathlib.Topology.Algebra.Indicator", "region_id": "topology", "micro_elevation": 0.3721, "macro_tier": 3, "macro_tier_score": 0.0372, "macro_tier_override": null, "x": -81.539, "z": 182.593, "size": 0.3132, "title": "Continuity of indicator functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Indicator.html"}, {"id": "Mathlib.Topology.ContinuousMap.ContinuousSqrt", "region_id": "topology", "micro_elevation": 1.0, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": -112.581, "z": 124.774, "size": 0.3163, "title": "Instances of `ContinuousSqrt`", "summary": "This provides the instances of `ContinuousSqrt` for `ℝ`, `ℝ≥0`, and `ℂ`, thereby yielding instances of `StarOrderedRing C(α, R)` and `StarOrderedRing C(α, R)₀` for any topological space `α` and `R` among `ℝ≥0`, `ℝ`, and `ℂ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/ContinuousSqrt.html"}, {"id": "Mathlib.Topology.Defs.Ultrafilter", "region_id": "topology", "micro_elevation": 0.0465, "macro_tier": 4, "macro_tier_score": 0.3707, "macro_tier_override": null, "x": -101.877, "z": 190.836, "size": 0.3283, "title": "Limit of an ultrafilter.", "summary": "* `Ultrafilter.lim f`: a limit of an ultrafilter `f`, defined as the limit of `(f : Filter X)` with a proof of `Nonempty X` deduced from existence of an ultrafilter on `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Defs/Ultrafilter.html"}, {"id": "Mathlib.Topology.Algebra.ContinuousAffineEquiv", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -128.181, "z": 246.456, "size": 0.2744, "title": "Continuous affine equivalences", "summary": "In this file, we define continuous affine equivalences, affine equivalences which are continuous with continuous inverse.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/ContinuousAffineEquiv.html"}, {"id": "Mathlib.Topology.Algebra.Valued.NormedValued", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -93.659, "z": 133.235, "size": 0.257, "title": "Correspondence between nontrivial nonarchimedean norms and rank one valuations", "summary": "Nontrivial nonarchimedean norms correspond to rank one valuations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Valued/NormedValued.html"}, {"id": "Mathlib.Topology.Order.IsLocallyClosed", "region_id": "topology", "micro_elevation": 0.5116, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -75.214, "z": 209.191, "size": 0.2396, "title": "Intervals are locally closed", "summary": "We prove that the intervals on a topological ordered space are locally closed.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/IsLocallyClosed.html"}, {"id": "Mathlib.Topology.Order.OrderClosed", "region_id": "topology", "micro_elevation": 0.4884, "macro_tier": 4, "macro_tier_score": 0.3339, "macro_tier_override": null, "x": -81.002, "z": 169.076, "size": 0.5211, "title": "Order-closed topologies", "summary": "In this file we introduce 3 typeclass mixins that relate topology and order structures: - `ClosedIicTopology` says that all the intervals $(-∞, a]$ (formally, `Set.Iic a`) are closed sets; - `ClosedIciTopology` says that all the intervals $[a, +∞)$ (formally, `Set.Ici a`) are closed sets; - `OrderClosedTopology` says that the set of points `(x, y)` such that `x ≤ y` is closed in the product topology. The last…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/OrderClosed.html"}, {"id": "Mathlib.Topology.LocallyClosed", "region_id": "topology", "micro_elevation": 0.3023, "macro_tier": 3, "macro_tier_score": 0.0424, "macro_tier_override": null, "x": -88.252, "z": 179.95, "size": 0.312, "title": "Locally closed sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/LocallyClosed.html"}, {"id": "Mathlib.Topology.Sheaves.LocallySurjective", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -55.08, "z": 205.475, "size": 0.2478, "title": "Locally surjective maps of presheaves.", "summary": "Let `X` be a topological space, `ℱ` and `𝒢` presheaves on `X`, `T : ℱ ⟶ 𝒢` a map. In this file we formulate two notions for what it means for `T` to be locally surjective: 1. For each open set `U`, each section `t : 𝒢(U)` is in the image of `T` after passing to some open cover of `U`. 2. For each `x : X`, the map of *stalks* `Tₓ : ℱₓ ⟶ 𝒢ₓ` is surjective. We prove that these are equivalent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/LocallySurjective.html"}, {"id": "Mathlib.Topology.Sheaves.Sheafify", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -131.042, "z": 145.293, "size": 0.2, "title": "Sheafification of `Type`-valued presheaves", "summary": "We construct the sheafification of a `Type`-valued presheaf, as the subsheaf of dependent functions into the stalks consisting of functions which are locally germs. We show that the stalks of the sheafification are isomorphic to the original stalks, via `stalkToFiber` which evaluates a germ of a dependent function at a point. We construct a morphism `toSheafify` from a presheaf to (the underlying presheaf of) its…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Sheafify.html"}, {"id": "Mathlib.Topology.Sheaves.LocalPredicate", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -77.846, "z": 147.638, "size": 0.2986, "title": "Functions satisfying a local predicate form a sheaf.", "summary": "At this stage, in `Mathlib/Topology/Sheaves/SheafOfFunctions.lean` we've proved that not-necessarily-continuous functions from a topological space into some type (or type family) form a sheaf. Why do the continuous functions form a sheaf? The point is just that continuity is a local condition, so one can use the lifting condition for functions to provide a candidate lift, then verify that the lift is actually…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/LocalPredicate.html"}, {"id": "Mathlib.Topology.Sheaves.Skyscraper", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -156.241, "z": 185.774, "size": 0.2655, "title": "Skyscraper (pre)sheaves", "summary": "A skyscraper (pre)sheaf `𝓕 : (Pre)Sheaf C X` is the (pre)sheaf with value `A` at point `p₀` that is supported only at open sets contain `p₀`, i.e. `𝓕(U) = A` if `p₀ ∈ U` and `𝓕(U) = *` if `p₀ ∉ U` where `*` is a terminal object of `C`. In terms of stalks, `𝓕` is supported at all specializations of `p₀`, i.e. if `p₀ ⤳ x` then `𝓕ₓ ≅ A` and if `¬ p₀ ⤳ x` then `𝓕ₓ ≅ *`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Skyscraper.html"}, {"id": "Mathlib.Topology.Algebra.AsymptoticCone", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2478, "title": "Asymptotic cone of a set", "summary": "This file defines the asymptotic cone of a set in a topological affine space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/AsymptoticCone.html"}, {"id": "Mathlib.Topology.Algebra.Order.Floor", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -135.054, "z": 216.654, "size": 0.2976, "title": "Topological facts about `Int.floor`, `Int.ceil` and `Int.fract`", "summary": "This file proves statements about limits and continuity of functions involving `floor`, `ceil` and `fract`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Order/Floor.html"}, {"id": "Mathlib.Topology.Sheaves.SheafOfFunctions", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -132.727, "z": 233.364, "size": 0.2799, "title": "Sheaf conditions for presheaves of (continuous) functions.", "summary": "We show that * `Top.Presheaf.toType_isSheaf`: not-necessarily-continuous functions into a type form a sheaf * `Top.Presheaf.toTypes_isSheaf`: in fact, these may be dependent functions into a type family For * `Top.sheafToTop`: continuous functions into a topological space form a sheaf please see `Mathlib/Topology/Sheaves/LocalPredicate.lean`, where we set up a general framework for constructing sub(pre)sheaves of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/SheafOfFunctions.html"}, {"id": "Mathlib.Topology.Homotopy.Path", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 2, "macro_tier_score": 0.0065, "macro_tier_override": null, "x": -50.006, "z": 158.348, "size": 0.3509, "title": "Homotopy between paths", "summary": "In this file, we define a `Homotopy` between two `Path`s. In addition, we define a relation `Homotopic` on `Path`s, and prove that it is an equivalence relation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/Path.html"}, {"id": "Mathlib.Topology.Algebra.Order.Group", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 3, "macro_tier_score": 0.0792, "macro_tier_override": null, "x": -82.605, "z": 221.954, "size": 0.3387, "title": "Topology on a linear ordered commutative group", "summary": "In this file we prove that a linear ordered commutative group with order topology is a topological group. We also prove continuity of `abs : G → G` and provide convenience lemmas like `ContinuousAt.abs`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Order/Group.html"}, {"id": "Mathlib.Topology.Instances.ZMultiples", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -48.79, "z": 171.812, "size": 0.2687, "title": null, "summary": "The subgroup \"multiples of `a`\" (`zmultiples a`) is a discrete subgroup of `ℝ`, i.e. its intersection with compact sets is finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/ZMultiples.html"}, {"id": "Mathlib.Topology.Algebra.Ring.Real", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 3, "macro_tier_score": 0.076, "macro_tier_override": null, "x": -133.444, "z": 141.253, "size": 0.4339, "title": "Topological algebra properties of ℝ", "summary": "This file defines topological field/(semi)ring structures on the (extended) (nonnegative) reals and shows the algebraic operations are (uniformly) continuous. It also includes a bit of more general topological theory of the reals, needed to define the structures and prove continuity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Ring/Real.html"}, {"id": "Mathlib.Topology.Separation.Regular", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 4, "macro_tier_score": 0.3065, "macro_tier_override": null, "x": -130.637, "z": 164.292, "size": 0.6173, "title": "Regular, normal, T₃, T₄ and T₅ spaces", "summary": "This file continues the study of separation properties of topological spaces, focusing on conditions strictly stronger than T₂.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/Regular.html"}, {"id": "Mathlib.Topology.Basic", "region_id": "topology", "micro_elevation": 0.0465, "macro_tier": 4, "macro_tier_score": 0.497, "macro_tier_override": null, "x": -101.775, "z": 191.375, "size": 0.401, "title": "Openness and closedness of a set", "summary": "This file provides lemmas relating to the predicates `IsOpen` and `IsClosed` of a set endowed with a topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Basic.html"}, {"id": "Mathlib.Topology.Continuous", "region_id": "topology", "micro_elevation": 0.1395, "macro_tier": 4, "macro_tier_score": 0.4557, "macro_tier_override": null, "x": -111.328, "z": 184.835, "size": 0.4163, "title": "Continuity in topological spaces", "summary": "For topological spaces `X` and `Y`, a function `f : X → Y` and a point `x : X`, `ContinuousAt f x` means `f` is continuous at `x`, and global continuity is `Continuous f`. There is also a version of continuity `PContinuous` for partially defined functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Continuous.html"}, {"id": "Mathlib.Topology.Compactification.OnePoint.Sphere", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -85.942, "z": 231.233, "size": 0.2, "title": "One-point compactification of Euclidean space is homeomorphic to the sphere.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactification/OnePoint/Sphere.html"}, {"id": "Mathlib.Topology.Compactification.OnePoint.Basic", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 3, "macro_tier_score": 0.0321, "macro_tier_override": null, "x": -76.372, "z": 222.908, "size": 0.3234, "title": "The one-point compactification", "summary": "We construct the one-point compactification of an arbitrary topological space `X` and prove some properties inherited from `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactification/OnePoint/Basic.html"}, {"id": "Mathlib.Topology.Instances.AddCircle.DenseSubgroup", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -158.498, "z": 213.703, "size": 0.239, "title": "Irrational rotation is minimal", "summary": "In this file we prove that the multiples of an irrational element of an `AddCircle` are dense. Moreover, an additive subgroup of the circle is dense iff it is not generated by a single element of finite additive order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/AddCircle/DenseSubgroup.html"}, {"id": "Mathlib.Topology.Instances.AddCircle.Defs", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 2, "macro_tier_score": 0.0111, "macro_tier_override": null, "x": -50.997, "z": 208.254, "size": 0.3086, "title": "The additive circle", "summary": "We define the additive circle `AddCircle p` as the quotient `𝕜 ⧸ ℤ ∙ p` for some period `p : 𝕜`. See also `Circle` and `Real.Angle`. For the normed group structure on `AddCircle`, see `AddCircle.NormedAddCommGroup` in a later file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/AddCircle/Defs.html"}, {"id": "Mathlib.Topology.Category.CompHausLike.Basic", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 3, "macro_tier_score": 0.0221, "macro_tier_override": null, "x": -124.748, "z": 161.625, "size": 0.3533, "title": "Categories of Compact Hausdorff Spaces", "summary": "We construct the category of compact Hausdorff spaces satisfying an additional property `P`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/CompHausLike/Basic.html"}, {"id": "Mathlib.Topology.Separation.Connected", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 2, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": -113.158, "z": 229.958, "size": 0.2716, "title": "Interaction of separation properties with connectedness properties", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/Connected.html"}, {"id": "Mathlib.Topology.MetricSpace.Algebra", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 2, "macro_tier_score": 0.0077, "macro_tier_override": null, "x": -107.054, "z": 249.595, "size": 0.4112, "title": "Compatibility of algebraic operations with metric space structures", "summary": "In this file we define mixin typeclasses `LipschitzMul`, `LipschitzAdd`, `IsBoundedSMul` expressing compatibility of multiplication, addition and scalar-multiplication operations with an underlying metric space structure. The intended use case is to abstract certain properties shared by normed groups and by `R≥0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Algebra.html"}, {"id": "Mathlib.Topology.Germ", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -145.608, "z": 192.447, "size": 0.2, "title": "Germs of functions between topological spaces", "summary": "In this file, we prove basic properties of germs of functions between topological spaces, with respect to the neighbourhood filter `𝓝 x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Germ.html"}, {"id": "Mathlib.Topology.Algebra.Field", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 3, "macro_tier_score": 0.0791, "macro_tier_override": null, "x": -58.949, "z": 164.757, "size": 0.3324, "title": "Topological fields", "summary": "A topological division ring is a topological ring whose inversion function is continuous at every non-zero element.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Field.html"}, {"id": "Mathlib.Topology.VectorBundle.FiniteDimensional", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -56.516, "z": 217.468, "size": 0.2416, "title": "Finite-rank vector bundles", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/VectorBundle/FiniteDimensional.html"}, {"id": "Mathlib.Topology.Category.TopCat.OpenNhds", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -143.653, "z": 179.193, "size": 0.286, "title": "The category of open neighborhoods of a point", "summary": "Given an object `X` of the category `TopCat` of topological spaces and a point `x : X`, this file builds the type `OpenNhds x` of open neighborhoods of `x` in `X` and endows it with the partial order given by inclusion and the corresponding category structure (as a full subcategory of the poset category `Set X`). This is used in `Topology.Sheaves.Stalks` to build the stalk of a sheaf at `x` as a limit over `OpenNhds…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/OpenNhds.html"}, {"id": "Mathlib.Topology.Category.TopCat.Opens", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 3, "macro_tier_score": 0.0167, "macro_tier_override": null, "x": -88.906, "z": 155.939, "size": 0.3397, "title": "The category of open sets in a topological space.", "summary": "We define `toTopCat : Opens X ⥤ TopCat` and `map (f : X ⟶ Y) : Opens Y ⥤ Opens X`, given by taking preimages of open sets. Unfortunately `Opens` isn't (usefully) a functor `TopCat ⥤ Cat`. (One can in fact define such a functor, but using it results in unresolvable `Eq.rec` terms in goals.) Really it's a 2-functor from (spaces, continuous functions, equalities) to (categories, functors, natural isomorphisms). We…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/Opens.html"}, {"id": "Mathlib.Topology.UniformSpace.Real", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 3, "macro_tier_score": 0.0734, "macro_tier_override": null, "x": -67.667, "z": 227.541, "size": 0.2923, "title": "The reals are complete", "summary": "This file provides the instances `CompleteSpace ℝ` and `CompleteSpace ℝ≥0`. Along the way, we add a shortcut instance for the natural topology on `ℝ≥0` (the one induced from `ℝ`), and add some basic API.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Real.html"}, {"id": "Mathlib.Topology.MetricSpace.Cauchy", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 4, "macro_tier_score": 0.0896, "macro_tier_override": null, "x": -78.322, "z": 149.185, "size": 0.3392, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Cauchy.html"}, {"id": "Mathlib.Topology.EMetricSpace.Lipschitz", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 3, "macro_tier_score": 0.0479, "macro_tier_override": null, "x": -52.666, "z": 202.086, "size": 0.3336, "title": "Lipschitz continuous functions", "summary": "A map `f : α → β` between two (extended) metric spaces is called *Lipschitz continuous* with constant `K ≥ 0` if for all `x, y` we have `edist (f x) (f y) ≤ K * edist x y`. For a metric space, the latter inequality is equivalent to `dist (f x) (f y) ≤ K * dist x y`. There is also a version asserting this inequality only for `x` and `y` in some set `s`. Finally, `f : α → β` is called *locally Lipschitz continuous* if…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/EMetricSpace/Lipschitz.html"}, {"id": "Mathlib.Topology.MetricSpace.HausdorffDistance", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 2, "macro_tier_score": 0.013, "macro_tier_override": null, "x": -119.631, "z": 132.404, "size": 0.4147, "title": "Hausdorff distance", "summary": "The Hausdorff distance on subsets of a metric (or emetric) space. Given two subsets `s` and `t` of a metric space, their Hausdorff distance is the smallest `d` such that any point of `s` is within `d` of a point in `t`, and conversely. This quantity is often infinite (think of `s` bounded and `t` unbounded), and therefore better expressed in the setting of emetric spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/HausdorffDistance.html"}, {"id": "Mathlib.Topology.MetricSpace.IsometricSMul", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 2, "macro_tier_score": 0.0137, "macro_tier_override": null, "x": -57.524, "z": 155.647, "size": 0.4434, "title": "Group actions by isometries", "summary": "In this file we define two typeclasses: - `IsIsometricSMul M X` says that `M` multiplicatively acts on a (pseudo extended) metric space `X` by isometries; - `IsIsometricVAdd` is an additive version of `IsIsometricSMul`. We also prove basic facts about isometric actions and define bundled isometries `IsometryEquiv.constSMul`, `IsometryEquiv.mulLeft`, `IsometryEquiv.mulRight`, `IsometryEquiv.divLeft`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/IsometricSMul.html"}, {"id": "Mathlib.Topology.Order.Lattice", "region_id": "topology", "micro_elevation": 0.5116, "macro_tier": 3, "macro_tier_score": 0.0651, "macro_tier_override": null, "x": -121.376, "z": 161.424, "size": 0.4175, "title": "Topological lattices", "summary": "In this file we define mixin classes `ContinuousInf` and `ContinuousSup`. We define the class `TopologicalLattice` as a topological space and lattice `L` extending `ContinuousInf` and `ContinuousSup`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Lattice.html"}, {"id": "Mathlib.Topology.Separation.Basic", "region_id": "topology", "micro_elevation": 0.3953, "macro_tier": 4, "macro_tier_score": 0.3672, "macro_tier_override": null, "x": -78.448, "z": 188.575, "size": 0.4169, "title": "Separation properties of topological spaces", "summary": "This file defines some of the weaker separation axioms (under the Kolmogorov classification), notably T₀, R₀, T₁ and R₁ spaces. For T₂ (Hausdorff) spaces and other stronger conditions, see the file `Mathlib/Topology/Separation/Hausdorff.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/Basic.html"}, {"id": "Mathlib.Topology.Category.CompHausLike.Cartesian", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -66.285, "z": 185.147, "size": 0.2546, "title": "Cartesian monoidal structure on `CompHausLike`", "summary": "If the predicate `P` is preserved under taking type-theoretic products and `PUnit` satisfies it, then `CompHausLike P` is a cartesian monoidal category. If the predicate `P` is preserved under taking type-theoretic sums, we provide an explicit coproduct cocone in `CompHausLike P`. When we have the dual of `CartesianMonoidalCategory`, this can be used to provide an instance of that on `CompHausLike P`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/CompHausLike/Cartesian.html"}, {"id": "Mathlib.Topology.Algebra.Ring.Ideal", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 3, "macro_tier_score": 0.0165, "macro_tier_override": null, "x": -129.121, "z": 144.261, "size": 0.3261, "title": "Ideals and quotients of topological rings", "summary": "In this file we define `Ideal.closure` to be the topological closure of an ideal in a topological ring. We also define a `TopologicalSpace` structure on the quotient of a topological ring by an ideal and prove that the quotient is a topological ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Ring/Ideal.html"}, {"id": "Mathlib.Topology.Category.TopCat.Limits.Pullbacks", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 2, "macro_tier_score": 0.011, "macro_tier_override": null, "x": -61.835, "z": 183.359, "size": 0.3061, "title": "Pullbacks and pushouts in the category of topological spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/Limits/Pullbacks.html"}, {"id": "Mathlib.Topology.MetricSpace.ProperSpace", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 3, "macro_tier_score": 0.089, "macro_tier_override": null, "x": -108.254, "z": 244.83, "size": 0.2946, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/ProperSpace.html"}, {"id": "Mathlib.Topology.MetricSpace.Pseudo.Lemmas", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 4, "macro_tier_score": 0.0915, "macro_tier_override": null, "x": -136.435, "z": 165.926, "size": 0.4292, "title": "Extra lemmas about pseudo-metric spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Pseudo/Lemmas.html"}, {"id": "Mathlib.Topology.MetricSpace.Pseudo.Pi", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 4, "macro_tier_score": 0.0904, "macro_tier_override": null, "x": -62.816, "z": 161.666, "size": 0.3827, "title": "Product of pseudometric spaces", "summary": "This file constructs the infinity distance on finite products of pseudometric spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Pseudo/Pi.html"}, {"id": "Mathlib.Topology.Maps.Proper.CompactlyGenerated", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 3, "macro_tier_score": 0.0165, "macro_tier_override": null, "x": -112.545, "z": 148.501, "size": 0.3262, "title": "A map is proper iff preimage of compact sets are compact", "summary": "This file proves that if `Y` is a Hausdorff and compactly generated space, a continuous map `f : X → Y` is proper if and only if preimage of compact sets are compact.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Maps/Proper/CompactlyGenerated.html"}, {"id": "Mathlib.Topology.Compactness.CompactlyCoherentSpace", "region_id": "topology", "micro_elevation": 0.4884, "macro_tier": 3, "macro_tier_score": 0.0788, "macro_tier_override": null, "x": -81.473, "z": 214.778, "size": 0.3123, "title": "Compactly coherent spaces and the compact coherentification", "summary": "In this file we will define compactly coherent spaces and the compact coherentification and prove basic properties about them. This is a weaker version of `CompactlyGeneratedSpace`. These notions agree on Hausdorff spaces. They are both referred to as compactly generated spaces in the literature while the compact coherentification is often called the k-ification.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactness/CompactlyCoherentSpace.html"}, {"id": "Mathlib.Topology.GDelta.MetrizableSpace", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 2, "macro_tier_score": 0.0074, "macro_tier_override": null, "x": -54.866, "z": 188.609, "size": 0.3993, "title": "`Gδ` sets and metrizable spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/GDelta/MetrizableSpace.html"}, {"id": "Mathlib.Topology.ContinuousMap.ZeroAtInfty", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -150.578, "z": 234.555, "size": 0.307, "title": "Continuous functions vanishing at infinity", "summary": "The type of continuous functions vanishing at infinity. When the domain is compact `C(α, β) ≃ C₀(α, β)` via the identity map. When the codomain is a metric space, every continuous map which vanishes at infinity is a bounded continuous function. When the domain is a locally compact space, this type has nice properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/ZeroAtInfty.html"}, {"id": "Mathlib.Topology.ContinuousMap.CocompactMap", "region_id": "topology", "micro_elevation": 0.5116, "macro_tier": 2, "macro_tier_score": 0.0061, "macro_tier_override": null, "x": -135.556, "z": 207.403, "size": 0.3237, "title": "Cocompact continuous maps", "summary": "The type of *cocompact continuous maps* are those which tend to the cocompact filter on the codomain along the cocompact filter on the domain. When the domain and codomain are Hausdorff, this is equivalent to many other conditions, including that preimages of compact sets are compact.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/CocompactMap.html"}, {"id": "Mathlib.Topology.MetricSpace.Isometry", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 3, "macro_tier_score": 0.0443, "macro_tier_override": null, "x": -119.629, "z": 247.73, "size": 0.4172, "title": "Isometries", "summary": "We define isometries, i.e., maps between emetric spaces that preserve the edistance (on metric spaces, these are exactly the maps that preserve distances), and prove their basic properties. We also introduce isometric bijections. Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the theory for `PseudoMetricSpace` and we specialize to `MetricSpace` when needed.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Isometry.html"}, {"id": "Mathlib.Topology.Category.Profinite.EffectiveEpi", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -157.632, "z": 228.323, "size": 0.2538, "title": "Effective epimorphisms in `Profinite`", "summary": "This file proves that `EffectiveEpi`, `Epi` and `Surjective` are all equivalent in `Profinite`. As a consequence we deduce from the material in `Mathlib/Topology/Category/CompHausLike/EffectiveEpi.lean` that `Profinite` is `Preregular` and `Precoherent`. We also prove that for a finite family of morphisms in `Profinite` with fixed target, the conditions jointly surjective, jointly epimorphic and effective epimorphic…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Profinite/EffectiveEpi.html"}, {"id": "Mathlib.Topology.Category.CompHaus.EffectiveEpi", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 2, "macro_tier_score": 0.0063, "macro_tier_override": null, "x": -90.051, "z": 130.816, "size": 0.3372, "title": "Effective epimorphisms in `CompHaus`", "summary": "This file proves that `EffectiveEpi`, `Epi` and `Surjective` are all equivalent in `CompHaus`. As a consequence we deduce from the material in `Mathlib/Topology/Category/CompHausLike/EffectiveEpi.lean` that `CompHaus` is `Preregular` and `Precoherent`. We also prove that for a finite family of morphisms in `CompHaus` with fixed target, the conditions jointly surjective, jointly epimorphic and effective epimorphic…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/CompHaus/EffectiveEpi.html"}, {"id": "Mathlib.Topology.Category.Profinite.Limits", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -86.357, "z": 251.529, "size": 0.2418, "title": "Explicit limits and colimits", "summary": "This file applies the general API for explicit limits and colimits in `CompHausLike P` (see the file `Mathlib/Topology/Category/CompHausLike/Limits.lean`) to the special case of `Profinite`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Profinite/Limits.html"}, {"id": "Mathlib.Topology.Instances.Real.Lemmas", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 2, "macro_tier_score": 0.006, "macro_tier_override": null, "x": -51.549, "z": 165.284, "size": 0.3162, "title": "Topological properties of ℝ", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/Real/Lemmas.html"}, {"id": "Mathlib.Topology.FiberBundle.Trivialization", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 3, "macro_tier_score": 0.0268, "macro_tier_override": null, "x": -79.807, "z": 235.075, "size": 0.3144, "title": "Trivializations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/FiberBundle/Trivialization.html"}, {"id": "Mathlib.Topology.MetricSpace.TransferInstance", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -151.203, "z": 162.349, "size": 0.2669, "title": "Transfer metric space structures across `Equiv`s", "summary": "In this file, we transfer a distance and (pseudo-)metric space structure across an equivalence.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/TransferInstance.html"}, {"id": "Mathlib.Topology.Order.LowerUpperTopology", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 3, "macro_tier_score": 0.0263, "macro_tier_override": null, "x": -130.56, "z": 216.961, "size": 0.2692, "title": "Lower and Upper topology", "summary": "This file introduces the lower topology on a preorder as the topology generated by the complements of the left-closed right-infinite intervals. For completeness we also introduce the dual upper topology, generated by the complements of the right-closed left-infinite intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/LowerUpperTopology.html"}, {"id": "Mathlib.Topology.EMetricSpace.Weak", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -104.335, "z": 251.199, "size": 0.2, "title": "Lemmas around weak (pseudo) extended metric spaces.", "summary": "In this file we show that whenever `some : α → Option α` is an open embedding and `α` is a `WeakPseudoEMetricSpace`, then `Option α` is as well in a natural manner. We then use this to prove `ℝ≥0` and `EReal` are weak extended metric spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/EMetricSpace/Weak.html"}, {"id": "Mathlib.Topology.Instances.Nat", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 3, "macro_tier_score": 0.0164, "macro_tier_override": null, "x": -53.154, "z": 217.79, "size": 0.3156, "title": "Topology on the natural numbers", "summary": "The structure of a metric space on `ℕ` is introduced in this file, induced from `ℝ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/Nat.html"}, {"id": "Mathlib.Topology.Order.WithTop", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -94.05, "z": 226.037, "size": 0.2873, "title": "Order topology on `WithTop ι`", "summary": "When `ι` is a topological space with the order topology, we also endow `WithTop ι` with the order topology. If `ι` is second countable, we prove that `WithTop ι` also is.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/WithTop.html"}, {"id": "Mathlib.Topology.UniformSpace.Defs", "region_id": "topology", "micro_elevation": 0.186, "macro_tier": 4, "macro_tier_score": 0.1441, "macro_tier_override": null, "x": -93.38, "z": 196.629, "size": 0.4486, "title": "Uniform spaces", "summary": "Uniform spaces are a generalization of metric spaces and topological groups. Many concepts directly generalize to uniform spaces, e.g. * uniform continuity (in this file) * completeness (in `Cauchy.lean`) * extension of uniform continuous functions to complete spaces (in `IsUniformEmbedding.lean`) * totally bounded sets (in `Cauchy.lean`) * totally bounded complete sets are compact (in `Cauchy.lean`) A uniform…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Defs.html"}, {"id": "Mathlib.Topology.Algebra.Module.LinearMap", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -53.569, "z": 164.77, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/LinearMap.html"}, {"id": "Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -75.29, "z": 239.673, "size": 0.2964, "title": "Idempotent continuous linear maps", "summary": "In this file, we study the idempotent elements (`IsIdempotentElem`) of the ring `M →L[R] M` of continuous endomorphisms of a topological `R`-module `M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/ContinuousLinearMap/Idempotent.html"}, {"id": "Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Quotient", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -143.076, "z": 233.173, "size": 0.2647, "title": "Continuous linear maps and quotient topological modules", "summary": "In this file, we collect various continuous linear maps associated to quotient spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/ContinuousLinearMap/Quotient.html"}, {"id": "Mathlib.Topology.ContinuousMap.LocallyConstant", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -75.135, "z": 241.412, "size": 0.2, "title": "The algebra morphism from locally constant functions to continuous functions.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/LocallyConstant.html"}, {"id": "Mathlib.Topology.LocallyConstant.Algebra", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 2, "macro_tier_score": 0.0108, "macro_tier_override": null, "x": -145.605, "z": 190.768, "size": 0.2806, "title": "Algebraic structure on locally constant functions", "summary": "This file puts algebraic structure (`Group`, `AddGroup`, etc) on the type of locally constant functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/LocallyConstant/Algebra.html"}, {"id": "Mathlib.Topology.NhdsKer", "region_id": "topology", "micro_elevation": 0.3721, "macro_tier": 3, "macro_tier_score": 0.0576, "macro_tier_override": null, "x": -113.208, "z": 168.043, "size": 0.2699, "title": "Neighborhoods kernel of a set", "summary": "In `Mathlib/Topology/Defs/Filter.lean`, `nhdsKer s` is defined to be the intersection of all neighborhoods of `s`. Note that this construction has no standard name in the literature. In this file we prove basic properties of this operation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/NhdsKer.html"}, {"id": "Mathlib.Topology.Inseparable", "region_id": "topology", "micro_elevation": 0.3488, "macro_tier": 4, "macro_tier_score": 0.3761, "macro_tier_override": null, "x": -96.271, "z": 169.828, "size": 0.3411, "title": "Inseparable points in a topological space", "summary": "In this file we prove basic properties of the following notions defined elsewhere. * `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`; * `Inseparable`: a relation saying that two points in a topological space have the same neighbourhoods; equivalently, they can't be separated by an open set; * `InseparableSetoid X`: same relation, as a `Setoid`; * `SeparationQuotient X`: the quotient of `X` by…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Inseparable.html"}, {"id": "Mathlib.Topology.Coherent", "region_id": "topology", "micro_elevation": 0.3488, "macro_tier": 4, "macro_tier_score": 0.0946, "macro_tier_override": null, "x": -115.974, "z": 212.4, "size": 0.3263, "title": "Topology generated by its restrictions to subsets", "summary": "We say that restrictions of the topology on `X` to sets from a family `S` generates the original topology, if either of the following equivalent conditions hold: - a set which is relatively open in each `s ∈ S` is open; - a set which is relatively closed in each `s ∈ S` is closed; - for any topological space `Y`, a function `f : X → Y` is continuous provided that it is continuous on each `s ∈ S`. We use the first…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Coherent.html"}, {"id": "Mathlib.Topology.Compactness.SigmaCompact", "region_id": "topology", "micro_elevation": 0.3953, "macro_tier": 4, "macro_tier_score": 0.3491, "macro_tier_override": null, "x": -96.811, "z": 166.313, "size": 0.5115, "title": "Sigma-compactness in topological spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactness/SigmaCompact.html"}, {"id": "Mathlib.Topology.Irreducible", "region_id": "topology", "micro_elevation": 0.4419, "macro_tier": 4, "macro_tier_score": 0.3472, "macro_tier_override": null, "x": -128.828, "z": 209.367, "size": 0.4493, "title": "Irreducibility in topological spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Irreducible.html"}, {"id": "Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -53.284, "z": 162.037, "size": 0.267, "title": "Adic topology", "summary": "Given a commutative ring `R` and an ideal `I` in `R`, this file constructs the unique topology on `R` which is compatible with the ring structure and such that a set is a neighborhood of zero if and only if it contains a power of `I`. This topology is non-archimedean: every neighborhood of zero contains an open subgroup, namely a power of `I`. It also studies the predicate `IsAdic` which states that a given…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.html"}, {"id": "Mathlib.Topology.Sheaves.Forget", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -69.616, "z": 217.738, "size": 0.2866, "title": "Checking the sheaf condition on the underlying presheaf of types.", "summary": "If `G : C ⥤ D` is a functor which reflects isomorphisms and preserves limits (we assume all limits exist in `C`), then checking the sheaf condition for a presheaf `F : Presheaf C X` is equivalent to checking the sheaf condition for `F ⋙ G`. The important special case is when `C` is a concrete category with a forgetful functor that preserves limits and reflects isomorphisms. Then to check the sheaf condition it…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Forget.html"}, {"id": "Mathlib.Topology.Sheaves.Sheaf", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 2, "macro_tier_score": 0.0069, "macro_tier_override": null, "x": -69.978, "z": 167.822, "size": 0.375, "title": "Sheaves", "summary": "We define sheaves on a topological space, with values in an arbitrary category. A presheaf on a topological space `X` is a sheaf precisely when it is a sheaf under the Grothendieck topology on `opens X`, which expands out to say: For each open cover `{ Uᵢ }` of `U`, and a family of compatible functions `A ⟶ F(Uᵢ)` for an `A : X`, there exists a unique gluing `A ⟶ F(U)` compatible with the restriction. See the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Sheaf.html"}, {"id": "Mathlib.Topology.Algebra.Group.Defs", "region_id": "topology", "micro_elevation": 0.3023, "macro_tier": 4, "macro_tier_score": 0.1314, "macro_tier_override": null, "x": -92.754, "z": 175.336, "size": 0.3483, "title": "Definitions about topological groups", "summary": "In this file we define mixin classes `ContinuousInv`, `IsTopologicalGroup`, and `ContinuousDiv`, as well as their additive versions. These classes say that the corresponding operations are continuous: - `ContinuousInv G` says that `(·⁻¹)` is continuous on `G`; - `IsTopologicalGroup G` says that `(· * ·)` is continuous on `G × G` and `(·⁻¹)` is continuous on `G`; - `ContinuousDiv G` says that `(· / ·)` is continuous…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/Defs.html"}, {"id": "Mathlib.Topology.Defs.Basic", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.5009, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.326, "title": "Basic definitions about topological spaces", "summary": "This file contains definitions about topology that do not require imports other than `Mathlib/Data/Set/Lattice.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Defs/Basic.html"}, {"id": "Mathlib.Topology.MetricSpace.Pseudo.Real", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 3, "macro_tier_score": 0.0266, "macro_tier_override": null, "x": -157.969, "z": 187.344, "size": 0.304, "title": "Lemmas about distances between points in intervals in `ℝ`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Pseudo/Real.html"}, {"id": "Mathlib.Topology.Category.TopCat.EpiMono", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 3, "macro_tier_score": 0.0267, "macro_tier_override": null, "x": -134.206, "z": 168.151, "size": 0.3051, "title": "Epi- and monomorphisms in `Top`", "summary": "This file shows that a continuous function is an epimorphism in the category of topological spaces if and only if it is surjective, and that a continuous function is a monomorphism in the category of topological spaces if and only if it is injective.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/EpiMono.html"}, {"id": "Mathlib.Topology.Category.TopCat.Adjunctions", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 3, "macro_tier_score": 0.0476, "macro_tier_override": null, "x": -125.053, "z": 161.829, "size": 0.3161, "title": "Adjunctions regarding the category of topological spaces", "summary": "This file shows that the forgetful functor from topological spaces to types has a left and right adjoint, given by `TopCat.discrete`, resp. `TopCat.trivial`, the functors which equip a type with the discrete, resp. trivial, topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/Adjunctions.html"}, {"id": "Mathlib.Topology.Baire.BaireMeasurable", "region_id": "topology", "micro_elevation": 0.3256, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -123.197, "z": 203.757, "size": 0.2, "title": "Baire category and Baire measurable sets", "summary": "This file defines some of the basic notions of Baire category and Baire measurable sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Baire/BaireMeasurable.html"}, {"id": "Mathlib.Topology.UniformSpace.Closeds", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -85.667, "z": 147.098, "size": 0.2516, "title": "Hausdorff uniformity", "summary": "This file defines the Hausdorff uniformity on the types of closed subsets, compact subsets and and nonempty compact subsets of a uniform space. This is the generalization of the uniformity induced by the Hausdorff metric to hyperspaces of uniform spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Closeds.html"}, {"id": "Mathlib.Topology.Sets.VietorisTopology", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 2, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": -148.94, "z": 175.321, "size": 0.2477, "title": "Vietoris topology", "summary": "This file defines the Vietoris topology on the types of compact subsets and nonempty compact subsets of a topological space. The Vietoris topology is generated by sets of the form $\\{K \\mid K \\subseteq U\\}$ and $\\{K \\mid K \\cap U \\ne \\emptyset\\}$, where $U$ is an open subset of the underlying space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sets/VietorisTopology.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.Module", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": -85.064, "z": 135.565, "size": 0.4213, "title": "Infinite sums in topological vector spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/Module.html"}, {"id": "Mathlib.Topology.ClusterPt", "region_id": "topology", "micro_elevation": 0.1163, "macro_tier": 4, "macro_tier_score": 0.4608, "macro_tier_override": null, "x": -97.214, "z": 190.146, "size": 0.4119, "title": "Lemmas on cluster and accumulation points", "summary": "In this file we prove various lemmas on [cluster points](https://en.wikipedia.org/wiki/Limit_point) (also known as limit points and accumulation points) of a filter and of a sequence. A filter `F` on `X` has `x` as a cluster point if `ClusterPt x F : 𝓝 x ⊓ F ≠ ⊥`. A map `f : α → X` clusters at `x` along `F : Filter α` if `MapClusterPt x F f : ClusterPt x (map f F)`. In particular the notion of cluster point of a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ClusterPt.html"}, {"id": "Mathlib.Topology.Neighborhoods", "region_id": "topology", "micro_elevation": 0.093, "macro_tier": 4, "macro_tier_score": 0.4779, "macro_tier_override": null, "x": -103.783, "z": 185.517, "size": 0.4661, "title": "Neighborhoods in topological spaces", "summary": "Each point `x` of `X` gets a neighborhood filter `𝓝 x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Neighborhoods.html"}, {"id": "Mathlib.Topology.Sets.Order", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -94.825, "z": 227.9, "size": 0.2, "title": "Clopen upper sets", "summary": "In this file we define the type of clopen upper sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sets/Order.html"}, {"id": "Mathlib.Topology.Separation.Lemmas", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -59.721, "z": 148.271, "size": 0.2, "title": "Further separation lemmas", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/Lemmas.html"}, {"id": "Mathlib.Topology.Algebra.Group.Extension", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -57.523, "z": 216.014, "size": 0.239, "title": "Short exact sequences of topological groups", "summary": "In this file, we define a short exact sequence of topological groups to be a closed embedding `φ` followed by an open quotient map `ψ` satisfying `φ.range = ψ.ker`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/Extension.html"}, {"id": "Mathlib.Topology.Covering.Quotient", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 2, "macro_tier_score": 0.0109, "macro_tier_override": null, "x": -142.439, "z": 151.74, "size": 0.2903, "title": "Covering maps to quotients by free and properly discontinuous group actions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Covering/Quotient.html"}, {"id": "Mathlib.Topology.Covering.Basic", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 2, "macro_tier_score": 0.0161, "macro_tier_override": null, "x": -51.664, "z": 193.292, "size": 0.2895, "title": "Covering Maps", "summary": "This file defines covering maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Covering/Basic.html"}, {"id": "Mathlib.Topology.Algebra.Affine", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 3, "macro_tier_score": 0.0691, "macro_tier_override": null, "x": -70.33, "z": 227.98, "size": 0.3578, "title": "Topological properties of affine spaces and maps", "summary": "This file contains a few facts regarding the continuity of affine maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Affine.html"}, {"id": "Mathlib.Topology.Instances.RatLemmas", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -46.987, "z": 167.765, "size": 0.2, "title": "Additional lemmas about the topology on rational numbers", "summary": "The structure of a metric space on `ℚ` (`Rat.MetricSpace`) is introduced elsewhere, induced from `ℝ`. In this file we prove some properties of this topological space and its one-point compactification.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/RatLemmas.html"}, {"id": "Mathlib.Topology.Semicontinuity.Basic", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 3, "macro_tier_score": 0.0168, "macro_tier_override": null, "x": -135.348, "z": 135.148, "size": 0.3458, "title": "Lower and Upper Semicontinuity", "summary": "This file develops key properties of upper and lower semicontinuous functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Semicontinuity/Basic.html"}, {"id": "Mathlib.Topology.EMetricSpace.PairReduction", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2, "title": "Pair Reduction", "summary": "The goal of this file is to prove the theorem `pair_reduction`. This is essentially Lemma 6.1 in [kratschmer_urusov2023] which is an extension of Lemma B.2.7. in [talagrand2014]. Given pseudometric spaces `T` and `E`, `c ≥ 0`, and a finite subset `J` of `T` such that `|J| ≤ aⁿ` for some `a ≥ 0` and `n : ℕ`, `pair_reduction` states that there exists a set `K ⊆ J²` such that for any function `f : T → E`: 1. `|K| ≤…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/EMetricSpace/PairReduction.html"}, {"id": "Mathlib.Topology.Algebra.UniformFilterBasis", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -113.978, "z": 136.034, "size": 0.2786, "title": "Uniform properties of neighborhood bases in topological algebra", "summary": "This file contains properties of filter bases on algebraic structures that also require the theory of uniform spaces. The only result so far is a characterization of Cauchy filters in topological groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/UniformFilterBasis.html"}, {"id": "Mathlib.Topology.Category.FinTopCat", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -138.776, "z": 179.449, "size": 0.239, "title": "Category of finite topological spaces", "summary": "Definition of the category of finite topological spaces with the canonical forgetful functors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/FinTopCat.html"}, {"id": "Mathlib.Topology.Algebra.Group.GroupTopology", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 4, "macro_tier_score": 0.1208, "macro_tier_override": null, "x": -76.157, "z": 150.618, "size": 0.3332, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/GroupTopology.html"}, {"id": "Mathlib.Topology.Category.TopCat.ULift", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -68.291, "z": 212.867, "size": 0.2586, "title": "Lifting topological spaces to a higher universe", "summary": "In this file, we construct the functor `uliftFunctor.{v, u} : TopCat.{u} ⥤ TopCat.{max u v}` which sends a topological space `X : Type u` to a homeomorphic space in `Type (max u v)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/ULift.html"}, {"id": "Mathlib.Topology.Category.TopCat.Limits.Basic", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 3, "macro_tier_score": 0.0435, "macro_tier_override": null, "x": -122.272, "z": 158.352, "size": 0.3824, "title": "The category of topological spaces has all limits and colimits", "summary": "Further, these limits and colimits are preserved by the forgetful functor --- that is, the underlying types are just the limits in the category of types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/Limits/Basic.html"}, {"id": "Mathlib.Topology.Instances.PNat", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -141.143, "z": 238.889, "size": 0.2, "title": "Topology on the positive natural numbers", "summary": "The structure of a metric space on `ℕ+` is introduced in this file, induced from `ℝ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/PNat.html"}, {"id": "Mathlib.Topology.Algebra.Localization", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -103.5, "z": 244.918, "size": 0.2, "title": "Localization of topological rings", "summary": "The topological localization of a topological commutative ring `R` at a submonoid `M` is the ring `Localization M` endowed with the final ring topology of the natural homomorphism sending `x : R` to the equivalence class of `(x, 1)` in the localization of `R` at an `M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Localization.html"}, {"id": "Mathlib.Topology.CWComplex.Classical.Basic", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.3017, "title": "CW complexes", "summary": "This file defines (relative) CW complexes and proves basic properties about them using the classical approach of Whitehead. A CW complex is a topological space that is made by gluing closed disks of different dimensions together.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/CWComplex/Classical/Basic.html"}, {"id": "Mathlib.Topology.Homeomorph.Quotient", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 2, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": -66.644, "z": 209.723, "size": 0.2675, "title": "Homeomorphisms between quotient spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homeomorph/Quotient.html"}, {"id": "Mathlib.Topology.VectorBundle.Constructions", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -150.887, "z": 161.856, "size": 0.3, "title": "Standard constructions on vector bundles", "summary": "This file contains several standard constructions on vector bundles: * `Bundle.Trivial.vectorBundle 𝕜 B F`: the trivial vector bundle with scalar field `𝕜` and model fiber `F` over the base `B` * `VectorBundle.prod`: for vector bundles `E₁` and `E₂` with scalar field `𝕜` over a common base, a vector bundle structure on their direct sum `E₁ ×ᵇ E₂` (the notation stands for `fun x ↦ E₁ x × E₂ x`). *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/VectorBundle/Constructions.html"}, {"id": "Mathlib.Topology.MetricSpace.ProperSpace.Real", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 3, "macro_tier_score": 0.0264, "macro_tier_override": null, "x": -161.592, "z": 209.784, "size": 0.2852, "title": "Second countability of the reals", "summary": "We prove that `EReal`, `ℝ≥0` and `ℝ≥0∞` are second countable. In the process, we also provide the instance `ProperSpace ℝ≥0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/ProperSpace/Real.html"}, {"id": "Mathlib.Topology.Algebra.ConstMulAction", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 4, "macro_tier_score": 0.1327, "macro_tier_override": null, "x": -126.37, "z": 153.447, "size": 0.4133, "title": "Monoid actions continuous in the second variable", "summary": "In this file we define class `ContinuousConstSMul`. We say `ContinuousConstSMul Γ T` if `Γ` acts on `T` and for each `γ`, the map `x ↦ γ • x` is continuous. (This differs from `ContinuousSMul`, which requires simultaneous continuity in both variables.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/ConstMulAction.html"}, {"id": "Mathlib.Topology.Algebra.Constructions", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 4, "macro_tier_score": 0.1539, "macro_tier_override": null, "x": -112.741, "z": 150.129, "size": 0.4264, "title": "Topological space structure on the opposite monoid and on the units group", "summary": "In this file we define `TopologicalSpace` structure on `Mᵐᵒᵖ`, `Mᵃᵒᵖ`, `Mˣ`, and `AddUnits M`. This file does not import definitions of a topological monoid and/or a continuous multiplicative action, so we postpone the proofs of `ContinuousMul Mᵐᵒᵖ` etc. till we have these definitions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Constructions.html"}, {"id": "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 4, "macro_tier_score": 0.0904, "macro_tier_override": null, "x": -150.684, "z": 171.307, "size": 0.3836, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/UniformConvergenceTopology.html"}, {"id": "Mathlib.Topology.UniformSpace.UniformApproximation", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 3, "macro_tier_score": 0.089, "macro_tier_override": null, "x": -63.497, "z": 166.308, "size": 0.2913, "title": "Uniform approximation", "summary": "In this file, we give lemmas ensuring that a function is continuous if it can be approximated uniformly by continuous functions. We give various versions, within a set or the whole space, at a single point or at all points, with locally uniform approximation or uniform approximation. All the statements are derived from a statement about locally uniform approximation within a set at a point, called…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/UniformApproximation.html"}, {"id": "Mathlib.Topology.Sets.CompactOpenCovered", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -102.734, "z": 143.177, "size": 0.2497, "title": "Compact open covered sets", "summary": "In this file we define the notion of a compact-open covered set with respect to a family of maps `fᵢ : X i → S`. A set `U` is compact-open covered by the family `fᵢ` if it is the finite union of images of compact open sets in the `X i`. This notion is not interesting, if the `fᵢ` are open maps (see `IsCompactOpenCovered.of_isOpenMap`). This is used to define the fpqc topology of schemes, there a cover is given by a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sets/CompactOpenCovered.html"}, {"id": "Mathlib.Topology.Spectral.Prespectral", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 3, "macro_tier_score": 0.0219, "macro_tier_override": null, "x": -126.277, "z": 149.843, "size": 0.3365, "title": "Prespectral spaces", "summary": "In this file, we define prespectral spaces as spaces whose lattice of compact opens forms a basis.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Spectral/Prespectral.html"}, {"id": "Mathlib.Topology.MetricSpace.Cover", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -168.7, "z": 184.442, "size": 0.2827, "title": "Covers in a metric space", "summary": "This file defines covers, aka nets, which are a quantitative notion of compactness in a metric space. A `ε`-cover of a set `s` is a set `N` such that every element of `s` is at distance at most `ε` to some element of `N`. In a proper metric space, sets admitting a finite cover are precisely the relatively compact sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Cover.html"}, {"id": "Mathlib.Topology.Order.UpperLowerSetTopology", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 3, "macro_tier_score": 0.0266, "macro_tier_override": null, "x": -141.545, "z": 200.028, "size": 0.3003, "title": "Upper and lower sets topologies", "summary": "This file introduces the upper set topology on a preorder as the topology where the open sets are the upper sets and the lower set topology on a preorder as the topology where the open sets are the lower sets. In general the upper set topology does not coincide with the upper topology and the lower set topology does not coincide with the lower topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/UpperLowerSetTopology.html"}, {"id": "Mathlib.Topology.Spectral.Hom", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 3, "macro_tier_score": 0.0211, "macro_tier_override": null, "x": -81.521, "z": 154.575, "size": 0.2713, "title": "Spectral maps", "summary": "This file defines spectral maps. A map is spectral when it's continuous and the preimage of a compact open set is compact open.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Spectral/Hom.html"}, {"id": "Mathlib.Topology.UniformSpace.LocallyUniformConvergence", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 4, "macro_tier_score": 0.1311, "macro_tier_override": null, "x": -147.569, "z": 211.346, "size": 0.3301, "title": "Locally uniform convergence", "summary": "We define a sequence of functions `Fₙ` to *converge locally uniformly* to a limiting function `f` with respect to a filter `p`, spelled `TendstoLocallyUniformly F f p`, if for any `x ∈ s` and any entourage of the diagonal `u`, there is a neighbourhood `v` of `x` such that `p`-eventually we have `(f y, Fₙ y) ∈ u` for all `y ∈ v`. It is important to note that this definition is somewhat non-standard; it is **not** in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/LocallyUniformConvergence.html"}, {"id": "Mathlib.Topology.Order.OrderClosedExtr", "region_id": "topology", "micro_elevation": 0.5116, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -89.087, "z": 222.303, "size": 0.239, "title": "Local maxima from monotonicity and antitonicity", "summary": "In this file we prove a lemma that is useful for the First Derivative Test in calculus, and its dual.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/OrderClosedExtr.html"}, {"id": "Mathlib.Topology.UniformSpace.DiscreteUniformity", "region_id": "topology", "micro_elevation": 0.3721, "macro_tier": 4, "macro_tier_score": 0.1422, "macro_tier_override": null, "x": -99.897, "z": 216.241, "size": 0.3662, "title": "Discrete uniformity", "summary": "The discrete uniformity is the smallest possible uniformity, the one for which the diagonal is an entourage of itself. It induces the discrete topology. It is complete.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/DiscreteUniformity.html"}, {"id": "Mathlib.Topology.Algebra.Star.Unitary", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -139.642, "z": 225.595, "size": 0.2, "title": "Topological properties of the unitary (sub)group", "summary": "* In a topological star monoid `R`, `unitary R` is a topological group * In a topological star monoid `R` which is T1, `unitary R` is closed as a subset of `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Star/Unitary.html"}, {"id": "Mathlib.Topology.Baire.CompleteMetrizable", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -43.775, "z": 177.907, "size": 0.2835, "title": "First Baire theorem", "summary": "In this file we prove that a completely pseudometrizable topological space is a Baire space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Baire/CompleteMetrizable.html"}, {"id": "Mathlib.Topology.Algebra.Algebra.Rat", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -149.156, "z": 181.492, "size": 0.2, "title": "Topological (sub)algebras over `Rat`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Algebra/Rat.html"}, {"id": "Mathlib.Topology.Algebra.Monoid.Defs", "region_id": "topology", "micro_elevation": 0.2791, "macro_tier": 4, "macro_tier_score": 0.1372, "macro_tier_override": null, "x": -87.073, "z": 185.708, "size": 0.377, "title": "Topological monoids - definitions", "summary": "In this file we define three mixin typeclasses: - `ContinuousMul M` says that the multiplication on `M` is continuous as a function on `M × M`; - `ContinuousAdd M` says that the addition on `M` is continuous as a function on `M × M`. - `SeparatelyContinuousMul M` says that the multiplication on `M` is continuous in each argument separately. This is strictly weaker than `ContinuousMul M`, but arises frequently in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Monoid/Defs.html"}, {"id": "Mathlib.Topology.ContinuousMap.Bounded.Normed", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 3, "macro_tier_score": 0.0171, "macro_tier_override": null, "x": -164.044, "z": 198.281, "size": 0.3609, "title": "Inheritance of normed algebraic structures by bounded continuous functions", "summary": "For various types of normed algebraic structures `β`, we show in this file that the space of bounded continuous functions from `α` to `β` inherits the same normed structure, by using pointwise operations and checking that they are compatible with the uniform distance.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Bounded/Normed.html"}, {"id": "Mathlib.Topology.ContinuousMap.Bounded.Basic", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 3, "macro_tier_score": 0.0277, "macro_tier_override": null, "x": -47.936, "z": 202.379, "size": 0.372, "title": "Bounded continuous functions", "summary": "The type of bounded continuous functions taking values in a metric space, with the uniform distance.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Bounded/Basic.html"}, {"id": "Mathlib.Topology.Perfect", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 3, "macro_tier_score": 0.0217, "macro_tier_override": null, "x": -85.811, "z": 225.876, "size": 0.3236, "title": "Perfect Sets", "summary": "In this file we define perfect subsets of a topological space, and prove some basic properties, including a version of the Cantor-Bendixson Theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Perfect.html"}, {"id": "Mathlib.Topology.MetricSpace.CantorScheme", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 2, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": -164.684, "z": 168.257, "size": 0.2726, "title": "(Topological) Schemes and their induced maps", "summary": "In topology, and especially descriptive set theory, one often constructs functions `(ℕ → β) → α`, where α is some topological space and β is a discrete space, as an appropriate limit of some map `List β → Set α`. We call the latter type of map a \"`β`-scheme on `α`\". This file develops the basic, abstract theory of these schemes and the functions they induce.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/CantorScheme.html"}, {"id": "Mathlib.Topology.Algebra.Module.ModuleTopology", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -99.997, "z": 132.367, "size": 0.2916, "title": "A \"module topology\" for modules over a topological ring", "summary": "If `R` is a topological ring acting on an additive abelian group `A`, we define the *module topology* to be the finest topology on `A` making both the maps `• : R × A → A` and `+ : A × A → A` continuous (with all the products having the product topology). Note that `- : A → A` is also automatically continuous as it is `a ↦ (-1) • a`. This topology was suggested by Will Sawin…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/ModuleTopology.html"}, {"id": "Mathlib.Topology.MetricSpace.PiNat", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 2, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": -167.032, "z": 183.695, "size": 0.339, "title": "Topological study of spaces `Π (n : ℕ), E n`", "summary": "When `E n` are topological spaces, the space `Π (n : ℕ), E n` is naturally a topological space (with the product topology). When `E n` are uniform spaces, it also inherits a uniform structure. However, it does not inherit a canonical metric space structure of the `E n`. Nevertheless, one can put a noncanonical metric space structure (or rather, several of them). This is done in this file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/PiNat.html"}, {"id": "Mathlib.Topology.UnitInterval", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 3, "macro_tier_score": 0.0555, "macro_tier_override": null, "x": -164.067, "z": 198.068, "size": 0.4465, "title": "The unit interval, as a topological space", "summary": "Use `open unitInterval` to turn on the notation `I := Set.Icc (0 : ℝ) (1 : ℝ)`. We provide basic instances, as well as a custom tactic for discharging `0 ≤ ↑x`, `0 ≤ 1 - ↑x`, `↑x ≤ 1`, and `1 - ↑x ≤ 1` when `x : I`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UnitInterval.html"}, {"id": "Mathlib.Topology.ContinuousMap.Ideals", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -167.479, "z": 177.302, "size": 0.2387, "title": "Ideals of continuous functions", "summary": "For a topological semiring `R` and a topological space `X` there is a Galois connection between `Ideal C(X, R)` and `Set X` given by sending each `I : Ideal C(X, R)` to `{x : X | ∀ f ∈ I, f x = 0}ᶜ` and mapping `s : Set X` to the ideal with carrier `{f : C(X, R) | ∀ x ∈ sᶜ, f x = 0}`, and we call these maps `ContinuousMap.setOfIdeal` and `ContinuousMap.idealOfSet`. As long as `R` is Hausdorff,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Ideals.html"}, {"id": "Mathlib.Topology.UniformSpace.Pi", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 4, "macro_tier_score": 0.1465, "macro_tier_override": null, "x": -148.683, "z": 174.645, "size": 0.525, "title": "Indexed product of uniform spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Pi.html"}, {"id": "Mathlib.Topology.Category.Profinite.Nobeling.Induction", "region_id": "topology", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -94.297, "z": 258.194, "size": 0.2, "title": "Nöbeling's theorem", "summary": "This file proves Nöbeling's theorem. For the overall proof outline see `Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Profinite/Nobeling/Induction.html"}, {"id": "Mathlib.Topology.Category.Profinite.Nobeling.Span", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -136.779, "z": 134.145, "size": 0.2338, "title": "The good products span", "summary": "Most of the argument is developing an API for `π C (· ∈ s)` when `s : Finset I`; then the image of `C` is finite with the discrete topology. In this case, there is a direct argument that the good products span. The general result is deduced from this. For the overall proof outline see `Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Profinite/Nobeling/Span.html"}, {"id": "Mathlib.Topology.Category.Profinite.Nobeling.ZeroLimit", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -39.305, "z": 196.755, "size": 0.2338, "title": "The zero and limit cases in the induction for Nöbeling's theorem", "summary": "This file proves the zero and limit cases of the ordinal induction used in the proof of Nöbeling's theorem. See the section docstrings for more information. For the overall proof outline see `Mathlib/Topology/Category/Profinite/Nobeling/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Profinite/Nobeling/ZeroLimit.html"}, {"id": "Mathlib.Topology.UniformSpace.Dini", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -156.123, "z": 206.222, "size": 0.2, "title": "Dini's Theorem", "summary": "This file proves Dini's theorem, which states that if `F n` is a monotone increasing sequence of continuous real-valued functions on a compact set `s` converging pointwise to a continuous function `f`, then `F n` converges uniformly to `f`. We generalize the codomain from `ℝ` to a normed lattice additive commutative group `G`. This theorem is true in a different generality as well: when `G` is a linearly ordered…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Dini.html"}, {"id": "Mathlib.Topology.UniformSpace.CompactConvergence", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 3, "macro_tier_score": 0.0738, "macro_tier_override": null, "x": -53.667, "z": 184.796, "size": 0.3261, "title": "Compact convergence (uniform convergence on compact sets)", "summary": "Given a topological space `α` and a uniform space `β` (e.g., a metric space or a topological group), the space of continuous maps `C(α, β)` carries a natural uniform space structure. We define this uniform space structure in this file and also prove its basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/CompactConvergence.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.Ring", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 3, "macro_tier_score": 0.0271, "macro_tier_override": null, "x": -58.04, "z": 220.143, "size": 0.3369, "title": "Infinite sum in a ring", "summary": "This file provides lemmas about the interaction between infinite sums and multiplication.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/Ring.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 3, "macro_tier_score": 0.0275, "macro_tier_override": null, "x": -69.926, "z": 151.518, "size": 0.3633, "title": "Topological sums and functorial constructions", "summary": "Lemmas on the interaction of `tprod`, `tsum`, `HasProd`, `HasSum` etc. with products, Sigma and Pi types, `MulOpposite`, etc.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/Constructions.html"}, {"id": "Mathlib.Topology.Convenient.OpenClosed", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -112.555, "z": 233.273, "size": 0.2, "title": "Open or closed subsets that are also `X`-generated spaces", "summary": "Let `X : ι → Type*` be a family of topological spaces. If all the opens (resp. closed) subsets of the `X i` are `X`-generated, then any open (resp. closed) subset of an `X`-generated space is `X`-generated.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Convenient/OpenClosed.html"}, {"id": "Mathlib.Topology.Convenient.GeneratedBy", "region_id": "topology", "micro_elevation": 0.5116, "macro_tier": 3, "macro_tier_score": 0.0164, "macro_tier_override": null, "x": -71.957, "z": 181.554, "size": 0.3167, "title": "The topology that is generated by a family of topological spaces", "summary": "Let `X : ι → Type u` be a family of topological spaces. Let `Y` be a topological space. We introduce a type synonym `WithGeneratedByTopology X Y` for `Y`. This type is endowed with the `X`-generated topology, which is coinduced by all continuous maps `X i → Y`. When the bijection `WithGeneratedByTopology X Y ≃ Y` is a homeomorphism, we say that `Y` is `X`-generated (typeclass `IsGeneratedBy X Y`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Convenient/GeneratedBy.html"}, {"id": "Mathlib.Topology.Algebra.SeparationQuotient.Basic", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 3, "macro_tier_score": 0.0173, "macro_tier_override": null, "x": -50.929, "z": 208.031, "size": 0.3737, "title": "Algebraic operations on `SeparationQuotient`", "summary": "In this file we define algebraic operations (multiplication, addition etc) on the separation quotient of a topological space with corresponding operation, provided that the original operation is continuous. We also prove continuity of these operations and show that they satisfy the same kind of laws (`Monoid` etc) as the original ones. Finally, we construct a section of the quotient map which is a continuous linear…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/SeparationQuotient/Basic.html"}, {"id": "Mathlib.Topology.Connected.LocPathConnected", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -44.793, "z": 164.934, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Connected/LocPathConnected.html"}, {"id": "Mathlib.Topology.Connected.LocallyPathConnected", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 3, "macro_tier_score": 0.0227, "macro_tier_override": null, "x": -40.755, "z": 188.498, "size": 0.3835, "title": "Locally path-connected spaces", "summary": "This file defines `LocallyPathConnectedSpace X`, a predicate class asserting that `X` is locally path-connected, in that each point has a basis of path-connected neighborhoods.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Connected/LocallyPathConnected.html"}, {"id": "Mathlib.Topology.MetricSpace.Holder", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2929, "title": "Hölder continuous functions", "summary": "In this file we define Hölder continuity on a set and on the whole space. We also prove some basic properties of Hölder continuous functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Holder.html"}, {"id": "Mathlib.Topology.MetricSpace.MetricSeparated", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -65.064, "z": 233.78, "size": 0.2834, "title": "Metric separation", "summary": "This file defines a few notions of separations of sets in a metric space. The first notion (`Metric.IsSeparated`) is quantitative and describes a single set: a set `s` is `ε`-separated if the distance between any two distinct elements is strictly greater than `ε` The second notion (`Metric.AreSeparated`) is qualitative and about two sets: Two sets `s` and `t` are separated if the distance between `x ∈ s` and `y ∈ t`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/MetricSeparated.html"}, {"id": "Mathlib.Topology.Category.TopCat.Limits.Products", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 3, "macro_tier_score": 0.0169, "macro_tier_override": null, "x": -97.643, "z": 233.347, "size": 0.3497, "title": "Products and coproducts in the category of topological spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/Limits/Products.html"}, {"id": "Mathlib.Topology.Algebra.Group.SubmonoidClosure", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -155.011, "z": 192.197, "size": 0.2, "title": "Topological closure of the submonoid closure", "summary": "In this file we prove several versions of the following statement: if `G` is a compact topological group and `s : Set G`, then the topological closures of `Submonoid.closure s` and `Subgroup.closure s` are equal. The proof is based on the following observation, see `mapClusterPt_self_zpow_atTop_pow`: each `x^m`, `m : ℤ` is a limit point (`MapClusterPt`) of the sequence `x^n`, `n : ℕ`, as `n → ∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/SubmonoidClosure.html"}, {"id": "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -157.448, "z": 216.104, "size": 0.2961, "title": "Continuous multilinear maps", "summary": "We define continuous multilinear maps as maps from `(i : ι) → M₁ i` to `M₂` which are multilinear and continuous, by extending the space of multilinear maps with a continuity assumption. Here, `M₁ i` and `M₂` are modules over a ring `R`, and `ι` is an arbitrary type, and all these spaces are also topological spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Multilinear/Basic.html"}, {"id": "Mathlib.Topology.Algebra.Order.LiminfLimsup", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -75.661, "z": 150.97, "size": 0.2636, "title": "Lemmas about liminf and limsup in an order topology.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Order/LiminfLimsup.html"}, {"id": "Mathlib.Topology.Semicontinuity.Defs", "region_id": "topology", "micro_elevation": 0.3488, "macro_tier": 3, "macro_tier_score": 0.0217, "macro_tier_override": null, "x": -127.056, "z": 199.477, "size": 0.3263, "title": "Semicontinuous maps", "summary": "A function `f` from a topological space `α` to an ordered space `β` is *lower semicontinuous* at a point `x` if, for any `y < f x`, for any `x'` close enough to `x`, one has `f x' > y`. In other words, `f` can jump up, but it cannot jump down. *Upper semicontinuous* functions are defined similarly. Upper and lower hemicontinuity (of functions `f : α → Set β`) are often defined in terms of sequential…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Semicontinuity/Defs.html"}, {"id": "Mathlib.Topology.Algebra.AffineSubspace", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -104.063, "z": 130.605, "size": 0.2639, "title": "Topology of affine subspaces.", "summary": "This file defines the embedding map from an affine subspace to the ambient space as a continuous affine map.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/AffineSubspace.html"}, {"id": "Mathlib.Topology.Order.LawsonTopology", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -68.8, "z": 172.824, "size": 0.2, "title": "Lawson topology", "summary": "This file introduces the Lawson topology on a preorder.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/LawsonTopology.html"}, {"id": "Mathlib.Topology.MetricSpace.Dilation", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 2, "macro_tier_score": 0.0072, "macro_tier_override": null, "x": -163.557, "z": 181.637, "size": 0.388, "title": "Dilations", "summary": "We define dilations, i.e., maps between emetric spaces that satisfy `edist (f x) (f y) = r * edist x y` for some `r ∉ {0, ∞}`. The value `r = 0` is not allowed because we want dilations of (e)metric spaces to be automatically injective. The value `r = ∞` is not allowed because this way we can define `Dilation.ratio f : ℝ≥0`, not `Dilation.ratio f : ℝ≥0∞`. Also, we do not often need maps sending distinct points to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Dilation.html"}, {"id": "Mathlib.Topology.Sheaves.Flasque", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -73.451, "z": 144.878, "size": 0.2, "title": "Flasque Sheaves", "summary": "We define and prove basic properties about flasque sheaves on topological spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Flasque.html"}, {"id": "Mathlib.Topology.Category.LightProfinite.Extend", "region_id": "topology", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -140.909, "z": 134.773, "size": 0.2465, "title": "Extending cones in `LightProfinite`", "summary": "Let `(Sₙ)_{n : ℕᵒᵖ}` be a sequential inverse system of finite sets and let `S` be its limit in `Profinite`. Let `G` be a functor from `LightProfinite` to a category `C` and suppose that `G` preserves the limit described above. Suppose further that the projection maps `S ⟶ Sₙ` are epimorphic for all `n`. Then `G.obj S` is isomorphic to a limit indexed by `StructuredArrow S toLightProfinite` (see…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/LightProfinite/Extend.html"}, {"id": "Mathlib.Topology.Instances.EReal.Lemmas", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 2, "macro_tier_score": 0.0063, "macro_tier_override": null, "x": -64.895, "z": 139.457, "size": 0.3429, "title": "Topological structure on `EReal`", "summary": "We prove basic properties of the topology on `EReal`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/EReal/Lemmas.html"}, {"id": "Mathlib.Topology.Category.TopCat.Limits.Konig", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 2, "macro_tier_score": 0.0062, "macro_tier_override": null, "x": -143.809, "z": 187.349, "size": 0.3339, "title": "Topological Kőnig's lemma", "summary": "A topological version of Kőnig's lemma is that the inverse limit of nonempty compact Hausdorff spaces is nonempty. (Note: this can be generalized further to inverse limits of nonempty compact T0 spaces, where all the maps are closed maps; see [Stone1979] --- however there is an erratum for Theorem 4 that the element in the inverse limit can have cofinally many components that are not closed points.) We give this in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/Limits/Konig.html"}, {"id": "Mathlib.Topology.Algebra.Nonarchimedean.Bases", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -112.4, "z": 135.8, "size": 0.2814, "title": "Neighborhood bases for non-archimedean rings and modules", "summary": "This file contains special families of filter bases on rings and modules that give rise to non-archimedean topologies. The main definition is `RingSubgroupsBasis` which is a predicate on a family of additive subgroups of a ring. The predicate ensures there is a topology `RingSubgroupsBasis.topology` which is compatible with a ring structure and admits the given family as a basis of neighborhoods of zero. In…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Nonarchimedean/Bases.html"}, {"id": "Mathlib.Topology.Algebra.TopologicallyNilpotent", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -101.347, "z": 247.958, "size": 0.2664, "title": "Topologically nilpotent elements", "summary": "Let `M` be a monoid with zero `M`, endowed with a topology. * `IsTopologicallyNilpotent a` says that `a : M` is *topologically nilpotent*, i.e., its powers converge to zero. * `IsTopologicallyNilpotent.map`: The image of a topologically nilpotent element under a continuous morphism of monoids with zero endowed with a topology is topologically nilpotent. * `IsTopologicallyNilpotent.zero`: `0` is topologically…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/TopologicallyNilpotent.html"}, {"id": "Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -48.102, "z": 173.871, "size": 0.2569, "title": "Character space of a topological algebra", "summary": "The character space of a topological algebra is the subset of elements of the weak dual that are also algebra homomorphisms. This space is used in the Gelfand transform, which gives an isomorphism between a commutative C⋆-algebra and continuous functions on the character space of the algebra. This, in turn, is used to construct the continuous functional calculus on C⋆-algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Spaces/CharacterSpace.html"}, {"id": "Mathlib.Topology.Algebra.UniformMulAction", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 3, "macro_tier_score": 0.0174, "macro_tier_override": null, "x": -142.63, "z": 154.109, "size": 0.3786, "title": "Multiplicative action on the completion of a uniform space", "summary": "In this file we define typeclasses `UniformContinuousConstVAdd` and `UniformContinuousConstSMul` and prove that a multiplicative action on `X` with uniformly continuous `(•) c` can be extended to a multiplicative action on `UniformSpace.Completion X`. In later files once the additive group structure is set up, we provide * `UniformSpace.Completion.DistribMulAction` * `UniformSpace.Completion.MulActionWithZero` *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/UniformMulAction.html"}, {"id": "Mathlib.Topology.Homotopy.Lifting", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -43.934, "z": 216.412, "size": 0.253, "title": "The homotopy lifting property for covering maps", "summary": "- `IsCoveringMap.exists_path_lifts`, `IsCoveringMap.liftPath`: any path in the base of a covering map lifts uniquely to the covering space (given a lift of the starting point). - `IsCoveringMap.liftHomotopy`: any homotopy `I × A → X` in the base of a covering map `E → X` can be lifted to a homotopy `I × A → E`, starting from a given lift of the restriction `{0} × A → X`. -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/Lifting.html"}, {"id": "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "region_id": "topology", "micro_elevation": 0.2791, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -86.998, "z": 185.935, "size": 0.2966, "title": "Maps equivariantly-homeomorphic to projection in a product", "summary": "This file contains the definition `IsHomeomorphicTrivialFiberBundle F p`, a Prop saying that a map `p : Z → B` between topological spaces is a \"trivial fiber bundle\" in the sense that there exists a homeomorphism `h : Z ≃ₜ B × F` such that `proj x = (h x).1`. This is an abstraction which is occasionally convenient in showing that a map is open, a quotient map, etc. This material was formerly linked to the main…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/FiberBundle/IsHomeomorphicTrivialBundle.html"}, {"id": "Mathlib.Topology.Algebra.Module.Complement", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -52.424, "z": 163.586, "size": 0.2733, "title": "Topological complements of submodules", "summary": "Let `M` be a topological `R`-module. Two submodules `p, q` of `M` are said to be *topological complements* (`Submodule.IsTopCompl`) if they are algebraic complements and the algebraic isomorphism `M ≃ p × q` is a homeomorphism. Not all submodules of `M` admit such a topological complements (even if they admit algebraic complements). In the literature, such a submodule is called *topologically complemented* or…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Complement.html"}, {"id": "Mathlib.Topology.Algebra.Monoid.AddChar", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -110.347, "z": 151.327, "size": 0.239, "title": "Additive characters of topological monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Monoid/AddChar.html"}, {"id": "Mathlib.Topology.Algebra.NonUnitalStarAlgebra", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -110.598, "z": 135.587, "size": 0.2929, "title": "Non-unital topological star (sub)algebras", "summary": "A non-unital topological star algebra over a topological semiring `R` is a topological (non-unital) semiring with a compatible continuous scalar multiplication by elements of `R` and a continuous `star` operation. We reuse typeclasses `ContinuousSMul` and `ContinuousStar` to express the latter two conditions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/NonUnitalStarAlgebra.html"}, {"id": "Mathlib.Topology.Algebra.StarSubalgebra", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -158.798, "z": 170.407, "size": 0.2929, "title": "Topological star (sub)algebras", "summary": "A topological star algebra over a topological semiring `R` is a topological semiring with a compatible continuous scalar multiplication by elements of `R` and a continuous star operation. We reuse typeclass `ContinuousSMul` for topological algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/StarSubalgebra.html"}, {"id": "Mathlib.Topology.Order.Bornology", "region_id": "topology", "micro_elevation": 0.0465, "macro_tier": 3, "macro_tier_score": 0.089, "macro_tier_override": null, "x": -102.607, "z": 193.825, "size": 0.2946, "title": "Bornology of order-bounded sets", "summary": "This file relates the notion of bornology-boundedness (sets that lie in a bornology) to the notion of order-boundedness (sets that are bounded above and below).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Bornology.html"}, {"id": "Mathlib.Topology.MetricSpace.Defs", "region_id": "topology", "micro_elevation": 0.4186, "macro_tier": 4, "macro_tier_score": 0.0907, "macro_tier_override": null, "x": -123.368, "z": 212.979, "size": 0.3957, "title": "Metric spaces", "summary": "This file defines metric spaces and shows some of their basic properties. Many definitions and theorems expected on metric spaces are already introduced on uniform spaces and topological spaces. This includes open and closed sets, compactness, completeness, continuity and uniform continuity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Defs.html"}, {"id": "Mathlib.Topology.ContinuousMap.Polynomial", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -151.134, "z": 231.597, "size": 0.2374, "title": "Constructions relating polynomial functions and continuous functions.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Polynomial.html"}, {"id": "Mathlib.Topology.Category.LightProfinite.Basic", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 2, "macro_tier_score": 0.0064, "macro_tier_override": null, "x": -132.313, "z": 249.752, "size": 0.3439, "title": "Light profinite spaces", "summary": "We construct the category `LightProfinite` of light profinite topological spaces. These are implemented as totally disconnected second countable compact Hausdorff spaces. This file also defines the category `LightDiagram`, which consists of those spaces that can be written as a sequential limit (in `Profinite`) of finite sets. We define an equivalence of categories `LightProfinite ≌ LightDiagram` and prove that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/LightProfinite/Basic.html"}, {"id": "Mathlib.Topology.Category.Profinite.AsLimit", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -42.642, "z": 198.748, "size": 0.289, "title": "Profinite sets as limits of finite sets.", "summary": "We show that any profinite set is isomorphic to the limit of its discrete (hence finite) quotients.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Profinite/AsLimit.html"}, {"id": "Mathlib.Topology.Category.Profinite.CofilteredLimit", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -52.43, "z": 157.438, "size": 0.2965, "title": "Cofiltered limits of profinite sets.", "summary": "This file contains some theorems about cofiltered limits of profinite sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Profinite/CofilteredLimit.html"}, {"id": "Mathlib.Topology.ClopenBox", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -132.395, "z": 221.716, "size": 0.2742, "title": "Clopen subsets in Cartesian products", "summary": "In general, a clopen subset in a Cartesian product of topological spaces cannot be written as a union of \"clopen boxes\", i.e. products of clopen subsets of the components (see [buzyakovaClopenBox] for counterexamples). However, when one of the factors is compact, a clopen subset can be written as such a union. Our argument in `TopologicalSpace.Clopens.exists_prod_subset` follows the one given in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ClopenBox.html"}, {"id": "Mathlib.Topology.Algebra.ProperConstSMul", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -61.168, "z": 203.983, "size": 0.2, "title": "Actions by proper maps", "summary": "In this file we define `ProperConstSMul M X` to be a mixin `Prop`-value class stating that `(c • ·)` is a proper map for all `c`. Note that this is **not** the same as a proper action (not yet in `Mathlib`) which requires `(c, x) ↦ (c • x, x)` to be a proper map. We also provide 4 instances: - for a continuous action on a compact Hausdorff space, - and for a continuous group action on a general space; - for the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/ProperConstSMul.html"}, {"id": "Mathlib.Topology.Algebra.IsUniformGroup.Defs", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 4, "macro_tier_score": 0.1248, "macro_tier_override": null, "x": -57.889, "z": 209.084, "size": 0.5005, "title": "Uniform structure on topological groups", "summary": "Given a topological group `G`, one can naturally build two uniform structures (the \"left\" and \"right\" ones) on `G` inducing its topology. This file defines typeclasses for groups equipped with either of these uniform structures, as well as a separate typeclass for the (very common) case where the given uniform structure coincides with **both** the left and right uniform structures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/IsUniformGroup/Defs.html"}, {"id": "Mathlib.Topology.Algebra.Module.Cardinality", "region_id": "topology", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -125.129, "z": 255.921, "size": 0.2536, "title": "Cardinality of open subsets of vector spaces", "summary": "Any nonempty open subset of a topological vector space over a nontrivially normed field has the same cardinality as the whole space. This is proved in `cardinal_eq_of_isOpen`. We deduce that a countable set in a nontrivial vector space over a complete nontrivially normed field has dense complement, in `Set.Countable.dense_compl`. This follows from the previous argument and the fact that a complete nontrivially…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Cardinality.html"}, {"id": "Mathlib.Topology.Instances.ENNReal.Lemmas", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 3, "macro_tier_score": 0.0286, "macro_tier_override": null, "x": -108.555, "z": 130.709, "size": 0.4142, "title": "Topology on extended non-negative reals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/ENNReal/Lemmas.html"}, {"id": "Mathlib.Topology.Algebra.Group.DiscontinuousSubgroup", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -66.374, "z": 167.611, "size": 0.239, "title": "Properly discontinuous actions of subgroups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/DiscontinuousSubgroup.html"}, {"id": "Mathlib.Topology.Algebra.Module.Alternating.Topology", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -107.242, "z": 125.942, "size": 0.2788, "title": "Topology on continuous alternating maps", "summary": "In this file we define `UniformSpace` and `TopologicalSpace` structures on the space of continuous alternating maps between topological vector spaces. The structures are induced by those on `ContinuousMultilinearMap`s, and most of the lemmas follow from the corresponding lemmas about `ContinuousMultilinearMap`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Alternating/Topology.html"}, {"id": "Mathlib.Topology.Algebra.Module.Alternating.Basic", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -123.34, "z": 135.096, "size": 0.2702, "title": "Continuous alternating multilinear maps", "summary": "In this file we define bundled continuous alternating maps and develop basic API about these maps, by reusing API about continuous multilinear maps and alternating maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Alternating/Basic.html"}, {"id": "Mathlib.Topology.Homotopy.HomotopyGroup", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2, "title": "`n`th homotopy group", "summary": "We define the `n`th homotopy group at `x : X`, `π_n X x`, as the equivalence classes of functions from the `n`-dimensional cube to the topological space `X` that send the boundary to the base point `x`, up to homotopic equivalence. Note that such functions are generalized loops `GenLoop (Fin n) x`; in particular `GenLoop (Fin 1) x ≃ Path x x`. We show that `π_0 X x` is equivalent to the path-connected components,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/HomotopyGroup.html"}, {"id": "Mathlib.Topology.Bornology.Constructions", "region_id": "topology", "micro_elevation": 0.0233, "macro_tier": 4, "macro_tier_score": 0.0956, "macro_tier_override": null, "x": -104.144, "z": 190.308, "size": 0.3812, "title": "Bornology structure on products and subtypes", "summary": "In this file we define `Bornology` and `BoundedSpace` instances on `α × β`, `Π i, X i`, and `{x // p x}`. We also prove basic lemmas about `Bornology.cobounded` and `Bornology.IsBounded` on these types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Bornology/Constructions.html"}, {"id": "Mathlib.Topology.Bornology.Basic", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.3745, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.4912, "title": "Basic theory of bornology", "summary": "We develop the basic theory of bornologies. Instead of axiomatizing bounded sets and defining bornologies in terms of those, we recognize that the cobounded sets form a filter and define a bornology as a filter of cobounded sets which contains the cofinite filter. This allows us to make use of the extensive library for filters, but we also provide the relevant connecting results for bounded sets. The specification…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Bornology/Basic.html"}, {"id": "Mathlib.Topology.GDelta.Basic", "region_id": "topology", "micro_elevation": 0.3023, "macro_tier": 4, "macro_tier_score": 0.2578, "macro_tier_override": null, "x": -93.471, "z": 174.827, "size": 0.4144, "title": "`Gδ` sets", "summary": "In this file we define `Gδ` sets and prove their basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/GDelta/Basic.html"}, {"id": "Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -100.986, "z": 127.585, "size": 0.2552, "title": "A profinite group is the projective limit of finite groups", "summary": "We define the topological group isomorphism between a profinite group and the projective limit of its quotients by open normal subgroups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Category/ProfiniteGrp/Limits.html"}, {"id": "Mathlib.Topology.Sheaves.Limits", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -67.261, "z": 169.163, "size": 0.2606, "title": "Presheaves in `C` have limits and colimits when `C` does.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Limits.html"}, {"id": "Mathlib.Topology.Order.Completion", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -127.959, "z": 219.355, "size": 0.2, "title": "Dense and continuous completion of a linear order", "summary": "Let `α` be a linear order. * `DedekindCut.continuous_principal`: the map `DedekindCut.principal : α → DedekindCut α` that embeds `α` in its Dedekind completion is continuous for the order topologies. * `Order.Fill α`: this is a type with a dense linear order endowed with a continuous order-embedding `Order.Fill.some` of `α`. It is defined as a subtype of `α × ℚ` and its order is induced by the lexicographic order. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Completion.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -104.438, "z": 252.766, "size": 0.2, "title": "Discrete Convolution", "summary": "Discrete convolution over monoids: `(f ⋆[L] g) x = ∑' (a, b) : mulFiber x, L (f a) (g b)` where `mulFiber x = {(a, b) | a * b = x}`. Additive monoids are also supported.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/DiscreteConvolution.html"}, {"id": "Mathlib.Topology.FiberPartition", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -65.401, "z": 181.744, "size": 0.244, "title": null, "summary": "This file provides some API surrounding `Function.Fiber` (see `Mathlib/Logic/Function/FiberPartition.lean`) in the presence of a topology on the domain of the function. Note: this API is designed to be useful when defining the counit of the adjunction between the functor which takes a set to the condensed set corresponding to locally constant maps to that set, and the forgetful functor from the category of condensed…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/FiberPartition.html"}, {"id": "Mathlib.Topology.Algebra.Group.Matrix", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -73.62, "z": 142.894, "size": 0.2824, "title": "Topology on matrix groups", "summary": "Lemmas about the topology of matrix groups, such as `GL(n, R)` and `SL(n, R)` for a topological ring `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/Matrix.html"}, {"id": "Mathlib.Topology.Metrizable.Urysohn", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": -161.529, "z": 173.387, "size": 0.3075, "title": "Urysohn's Metrization Theorem", "summary": "In this file we prove Urysohn's Metrization Theorem: a T₃ topological space with second countable topology `X` is metrizable. First we prove that `X` can be embedded into `l^∞`, then use this embedding to pull back the metric space structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Metrizable/Urysohn.html"}, {"id": "Mathlib.Topology.Separation.AlexandrovDiscrete", "region_id": "topology", "micro_elevation": 0.4419, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.908, "z": 221.17, "size": 0.2, "title": "T1 Alexandrov-discrete topology is discrete", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/AlexandrovDiscrete.html"}, {"id": "Mathlib.Topology.AlexandrovDiscrete", "region_id": "topology", "micro_elevation": 0.4186, "macro_tier": 3, "macro_tier_score": 0.0481, "macro_tier_override": null, "x": -77.629, "z": 198.86, "size": 0.3495, "title": "Alexandrov-discrete topological spaces", "summary": "This file defines Alexandrov-discrete spaces, aka finitely generated spaces. A space is Alexandrov-discrete if the (arbitrary) intersection of open sets is open. As such, the intersection of all neighborhoods of a set is a neighborhood itself. Hence every set has a minimal neighborhood, which we call the *neighborhoods kernel* of the set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/AlexandrovDiscrete.html"}, {"id": "Mathlib.Topology.UniformSpace.OfCompactT2", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -108.64, "z": 230.66, "size": 0.2478, "title": "Compact separated uniform spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/OfCompactT2.html"}, {"id": "Mathlib.Topology.Algebra.IsUniformGroup.Order", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": -135.84, "z": 150.285, "size": 0.3633, "title": "TendstoUniformlyOn on ordered spaces", "summary": "We gather some results about `TendstoUniformlyOn f g K` on ordered spaces, in particular bounding the values of `f` in terms of bounds on the limit `g`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/IsUniformGroup/Order.html"}, {"id": "Mathlib.Topology.Maps.OpenQuotient", "region_id": "topology", "micro_elevation": 0.2093, "macro_tier": 4, "macro_tier_score": 0.4489, "macro_tier_override": null, "x": -98.807, "z": 204.399, "size": 0.3303, "title": "Open quotient maps", "summary": "An open quotient map is an open map `f : X → Y` which is both an open map and a quotient map. Equivalently, it is a surjective continuous open map. We use the latter characterization as a definition. Many important quotient maps are open quotient maps, including - the quotient map from a topological space to its quotient by the action of a group; - the quotient map from a topological group to its quotient by a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Maps/OpenQuotient.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.Nonarchimedean", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -135.48, "z": 244.558, "size": 0.227, "title": "Infinite sums and products in nonarchimedean abelian groups", "summary": "Let `G` be a complete nonarchimedean abelian group and let `f : α → G` be a function. We prove that `f` is unconditionally summable if and only if `f a` tends to zero on the cofinite filter on `α` (`NonarchimedeanAddGroup.summable_iff_tendsto_cofinite_zero`). We also prove the analogous result in the multiplicative setting (`NonarchimedeanGroup.multipliable_iff_tendsto_cofinite_one`). We also prove that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.html"}, {"id": "Mathlib.Topology.Algebra.Monoid.FunOnFinite", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -74.717, "z": 229.722, "size": 0.27, "title": "Continuity of the functoriality of `X → M` when `X` is finite", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Monoid/FunOnFinite.html"}, {"id": "Mathlib.Topology.Baire.Lemmas", "region_id": "topology", "micro_elevation": 0.3256, "macro_tier": 4, "macro_tier_score": 0.2571, "macro_tier_override": null, "x": -83.51, "z": 186.823, "size": 0.3844, "title": "Baire spaces", "summary": "A topological space is called a *Baire space* if a countable intersection of dense open subsets is dense. Baire theorems say that all completely metrizable spaces and all locally compact regular spaces are Baire spaces. We prove the theorems in `Mathlib/Topology/Baire/CompleteMetrizable` and `Mathlib/Topology/Baire/LocallyCompactRegular`. In this file we prove some lemmas about Baire spaces. The good concept…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Baire/Lemmas.html"}, {"id": "Mathlib.Topology.Sheaves.Functors", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -90.81, "z": 234.867, "size": 0.2792, "title": "functors between categories of sheaves", "summary": "Show that the pushforward of a sheaf is a sheaf, and define the pushforward functor from the category of C-valued sheaves on X to that of sheaves on Y, given a continuous map between topological spaces X and Y.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Functors.html"}, {"id": "Mathlib.Topology.Algebra.Category.ProfiniteGrp.Completion", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -72.286, "z": 134.55, "size": 0.2, "title": "Profinite completion of groups", "summary": "We define the profinite completion of a group as the limit of its finite quotients, and prove its universal property.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Category/ProfiniteGrp/Completion.html"}, {"id": "Mathlib.Topology.Algebra.ProperAction.CompactlyGenerated", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -47.622, "z": 166.282, "size": 0.2776, "title": "When a proper action is properly discontinuous", "summary": "This file proves that if a discrete group acts on a T2 space `X` such that `X × X` is compactly generated, and if the action is continuous in the second variable, then the action is properly discontinuous if and only if it is proper. This is in particular true if `X` is first-countable or weakly locally compact.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/ProperAction/CompactlyGenerated.html"}, {"id": "Mathlib.Topology.Category.CompHausLike.EffectiveEpi", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 2, "macro_tier_score": 0.0063, "macro_tier_override": null, "x": -66.129, "z": 186.151, "size": 0.3389, "title": "Effective epimorphisms in `CompHausLike`", "summary": "In any category of compact Hausdorff spaces, continuous surjections are effective epimorphisms. We deduce that if the converse holds and explicit pullbacks exist, then `CompHausLike P` is preregular. If furthermore explicit finite coproducts exist, then `CompHausLike P` is precoherent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/CompHausLike/EffectiveEpi.html"}, {"id": "Mathlib.Topology.OpenPartialHomeomorph.Constructions", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 3, "macro_tier_score": 0.027, "macro_tier_override": null, "x": -137.898, "z": 227.294, "size": 0.3289, "title": "Constructions of new partial homeomorphisms from old", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/OpenPartialHomeomorph/Constructions.html"}, {"id": "Mathlib.Topology.Order.IsNormal", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -70.284, "z": 173.367, "size": 0.2302, "title": "A normal function is strictly monotone and continuous", "summary": "We defined the predicate `Order.IsNormal` in terms of `IsLUB`, which avoids having to import topology in order theory files. This file shows that the predicate is equivalent to the definition in the literature, being that of a strictly monotonic function, continuous in the order topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/IsNormal.html"}, {"id": "Mathlib.Topology.Order.IsLUB", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 4, "macro_tier_score": 0.0959, "macro_tier_override": null, "x": -136.566, "z": 171.439, "size": 0.3976, "title": "Properties of LUB and GLB in an order topology", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/IsLUB.html"}, {"id": "Mathlib.Topology.Algebra.LinearTopology", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -93.613, "z": 138.037, "size": 0.2752, "title": "Linear topologies on modules and rings", "summary": "Let `M` be a (left) module over a ring `R`. Following [Stacks: Definition 15.36.1](https://stacks.math.columbia.edu/tag/07E8), we say that a topology on `M` is *`R`-linear* if it is invariant by translations and admits a basis of neighborhoods of 0 consisting of (left) `R`-submodules. If `M` is an `(R, R')`-bimodule, we show that a topology is both `R`-linear and `R'`-linear if and only if there exists a basis of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/LinearTopology.html"}, {"id": "Mathlib.Topology.Constructible", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -147.843, "z": 169.039, "size": 0.2953, "title": "Constructible sets", "summary": "This file defines constructible sets, which are morally sets in a topological space which we can make out of finite unions and intersections of open and closed sets. Precisely, constructible sets are the Boolean subalgebra generated by open retrocompact sets, where a set is retrocompact if its intersection with every compact open set is compact. In a locally Noetherian space, all sets are retrocompact, in which case…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Constructible.html"}, {"id": "Mathlib.Topology.Spectral.Basic", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 2, "macro_tier_score": 0.0112, "macro_tier_override": null, "x": -112.315, "z": 143.7, "size": 0.3156, "title": "Spectral spaces", "summary": "A topological space is spectral if it is T0, compact, sober, quasi-separated, and its compact open subsets form an open basis. Prime spectra of commutative semirings are spectral spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Spectral/Basic.html"}, {"id": "Mathlib.Topology.Category.CompHaus.Frm", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 2, "macro_tier_score": 0.0109, "macro_tier_override": null, "x": -57.121, "z": 153.615, "size": 0.2918, "title": null, "summary": "The forgetful functor from `TopCatᵒᵖ` to `Frm`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/CompHaus/Frm.html"}, {"id": "Mathlib.Topology.LocallyFinsupp", "region_id": "topology", "micro_elevation": 0.4884, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -73.214, "z": 182.832, "size": 0.2985, "title": "Type of functions with locally finite support", "summary": "This file defines functions with locally finite support, provides supporting API. For suitable targets, it establishes functions with locally finite support as an instance of a lattice ordered commutative group. Throughout the present file, `X` denotes a topologically space and `U` a subset of `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/LocallyFinsupp.html"}, {"id": "Mathlib.Topology.Homotopy.Equiv", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -42.794, "z": 199.979, "size": 0.3068, "title": "Homotopy equivalences between topological spaces", "summary": "In this file, we define homotopy equivalences between topological spaces `X` and `Y` as a pair of functions `f : C(X, Y)` and `g : C(Y, X)` such that `f.comp g` and `g.comp f` are both homotopic to `ContinuousMap.id`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/Equiv.html"}, {"id": "Mathlib.Topology.LocallyFinsupp.Pushforward", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -154.484, "z": 184.417, "size": 0.2338, "title": "Pushforward of functions with locally finite support", "summary": "In this file we define the notion of the pushforward of a function with locally finite support between prespectral spaces along a spectral map. This is used for defining the (proper) pushforward of algebraic cycles in algebraic geometry.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/LocallyFinsupp/Pushforward.html"}, {"id": "Mathlib.Topology.UniformSpace.CompareReals", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -162.411, "z": 171.117, "size": 0.2, "title": "Comparison of Cauchy reals and Bourbaki reals", "summary": "In `Data.Real.Basic` real numbers are defined using the so called Cauchy construction (although it is due to Georg Cantor). More precisely, this construction applies to commutative rings equipped with an absolute value with values in a linear ordered field. On the other hand, in the `UniformSpace` folder, we construct completions of general uniform spaces, which allows to construct the Bourbaki real numbers. In this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/CompareReals.html"}, {"id": "Mathlib.Topology.UniformSpace.AbsoluteValue", "region_id": "topology", "micro_elevation": 0.2326, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -120.428, "z": 193.675, "size": 0.239, "title": "Uniform structure induced by an absolute value", "summary": "We build a uniform space structure on a commutative ring `R` equipped with an absolute value into a linear ordered field `𝕜`. Of course in the case `R` is `ℚ`, `ℝ` or `ℂ` and `𝕜 = ℝ`, we get the same thing as the metric space construction, and the general construction follows exactly the same path.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/AbsoluteValue.html"}, {"id": "Mathlib.Topology.Specialization", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -130.854, "z": 220.996, "size": 0.2676, "title": "Specialization order", "summary": "This file defines a type synonym for a topological space considered with its specialisation order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Specialization.html"}, {"id": "Mathlib.Topology.Homotopy.HSpaces", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -164.108, "z": 212.144, "size": 0.2, "title": "H-spaces", "summary": "This file defines H-spaces mainly following the approach proposed by Serre in his paper *Homologie singulière des espaces fibrés*. The idea beneath `H-spaces` is that they are topological spaces with a binary operation `⋀ : X → X → X` that is a homotopy-theoretic weakening of an operation that would make `X` into a topological monoid. In particular, there exists a \"neutral element\" `e : X` such that `fun x ↦ e ⋀ x`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/HSpaces.html"}, {"id": "Mathlib.Topology.Homotopy.Basic", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 2, "macro_tier_score": 0.012, "macro_tier_override": null, "x": -108.077, "z": 130.682, "size": 0.3699, "title": "Homotopy between functions", "summary": "In this file, we define a homotopy between two functions `f₀` and `f₁`. First we define `ContinuousMap.Homotopy` between the two functions, with no restrictions on the intermediate maps. Then, as in the formalisation in HOL-Analysis, we define `ContinuousMap.HomotopyWith f₀ f₁ P`, for homotopies between `f₀` and `f₁`, where the intermediate maps satisfy the predicate `P`. Finally, we define…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/Basic.html"}, {"id": "Mathlib.Topology.Path", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 3, "macro_tier_score": 0.0335, "macro_tier_override": null, "x": -134.53, "z": 138.27, "size": 0.4006, "title": "Paths in topological spaces", "summary": "This file introduces continuous paths and provides API for them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Path.html"}, {"id": "Mathlib.Topology.Instances.CantorSet", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2478, "title": "Ternary Cantor Set", "summary": "This file defines the Cantor ternary set and proves a few properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/CantorSet.html"}, {"id": "Mathlib.Topology.TietzeExtension", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -82.264, "z": 248.422, "size": 0.2539, "title": "Tietze extension theorem", "summary": "In this file we prove a few version of the Tietze extension theorem. The theorem says that a continuous function `s → ℝ` defined on a closed set in a normal topological space `Y` can be extended to a continuous function on the whole space. Moreover, if all values of the original function belong to some (finite or infinite, open or closed) interval, then the extension can be chosen so that it takes values in the same…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/TietzeExtension.html"}, {"id": "Mathlib.Topology.UrysohnsBounded", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -159.85, "z": 214.534, "size": 0.2483, "title": "Urysohn's lemma for bounded continuous functions", "summary": "In this file we reformulate Urysohn's lemma `exists_continuous_zero_one_of_isClosed` in terms of bounded continuous functions `X →ᵇ ℝ`. These lemmas live in a separate file because `Topology.ContinuousMap.Bounded` imports too many other files.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UrysohnsBounded.html"}, {"id": "Mathlib.Topology.OpenPartialHomeomorph.IsImage", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 3, "macro_tier_score": 0.0397, "macro_tier_override": null, "x": -142.511, "z": 217.139, "size": 0.442, "title": "Partial homeomorphisms: Images of sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/OpenPartialHomeomorph/IsImage.html"}, {"id": "Mathlib.Topology.EMetricSpace.Pi", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 4, "macro_tier_score": 0.0903, "macro_tier_override": null, "x": -144.233, "z": 160.628, "size": 0.3783, "title": "Indexed product of extended metric spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/EMetricSpace/Pi.html"}, {"id": "Mathlib.Topology.Instances.Shrink", "region_id": "topology", "micro_elevation": 0.2791, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -117.042, "z": 177.343, "size": 0.2363, "title": "Topological space structure on `Shrink X`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/Shrink.html"}, {"id": "Mathlib.Topology.Homeomorph.TransferInstance", "region_id": "topology", "micro_elevation": 0.2558, "macro_tier": 2, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": -91.811, "z": 202.895, "size": 0.2533, "title": "Transfer topological structure across `Equiv`s", "summary": "We show how to transport a topological space structure across an `Equiv` and prove that this make the equivalence a homeomorphism between the original space and the transported topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homeomorph/TransferInstance.html"}, {"id": "Mathlib.Topology.DiscreteQuotient", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -99.703, "z": 232.074, "size": 0.2934, "title": "Discrete quotients of a topological space.", "summary": "This file defines the type of discrete quotients of a topological space, denoted `DiscreteQuotient X`. To avoid quantifying over types, we model such quotients as setoids whose equivalence classes are clopen.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/DiscreteQuotient.html"}, {"id": "Mathlib.Topology.Algebra.Module.Simple", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -94.993, "z": 137.765, "size": 0.2733, "title": "The kernel of a linear function is closed or dense", "summary": "In this file we prove (`LinearMap.isClosed_or_dense_ker`) that the kernel of a linear function `f : M →ₗ[R] N` is either closed or dense in `M` provided that `N` is a simple module over `R`. This applies, e.g., to the case when `R = N` is a division ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Simple.html"}, {"id": "Mathlib.Topology.Algebra.SeparationQuotient.FiniteDimensional", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -116.093, "z": 248.543, "size": 0.2733, "title": "Separation quotient is a finite module", "summary": "In this file we show that the separation quotient of a finite module is a finite module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/SeparationQuotient/FiniteDimensional.html"}, {"id": "Mathlib.Topology.Sheaves.CommRingCat", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -160.499, "z": 208.004, "size": 0.2593, "title": "Sheaves of (commutative) rings.", "summary": "Results specific to sheaves of commutative rings including sheaves of continuous functions `TopCat.continuousFunctions` with natural operations of `pullback` and `map` and sub, quotient, and localization operations on sheaves of rings with - `SubmonoidPresheaf` : A subpresheaf with a submonoid structure on each of the components. - `LocalizationPresheaf` : The localization of a presheaf of commrings at a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/CommRingCat.html"}, {"id": "Mathlib.Topology.Category.Sequential", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -115.919, "z": 141.186, "size": 0.2446, "title": "The category of sequential topological spaces", "summary": "We define the category `Sequential` of sequential topological spaces. We follow the usual template for defining categories of topological spaces, by giving it the induced category structure from `TopCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Sequential.html"}, {"id": "Mathlib.Topology.Category.LightProfinite.Sequence", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -86.28, "z": 128.588, "size": 0.2446, "title": "The light profinite set classifying convergent sequences", "summary": "This file defines the light profinite set `ℕ∪{∞}`, defined as the one point compactification of `ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/LightProfinite/Sequence.html"}, {"id": "Mathlib.Topology.MetricSpace.UniformConvergence", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -163.512, "z": 165.462, "size": 0.2913, "title": "Metric structure on `α →ᵤ β` and `α →ᵤ[𝔖] β` for finite `𝔖`", "summary": "When `β` is a (pseudo, extended) metric space it is a uniform space, and therefore we may consider the type `α →ᵤ β` of functions equipped with the topology of uniform convergence. The natural (pseudo, extended) metric on this space is given by `fun f g ↦ ⨆ x, edist (f x) (g x)`, and this induces the existing uniformity. Unless `β` is a bounded space, this will not be a (pseudo) metric space (except in the trivial…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/UniformConvergence.html"}, {"id": "Mathlib.Topology.OpenPartialHomeomorph.Basic", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 3, "macro_tier_score": 0.0403, "macro_tier_override": null, "x": -119.833, "z": 231.246, "size": 0.4616, "title": "Partial homeomorphisms: basic theory", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/OpenPartialHomeomorph/Basic.html"}, {"id": "Mathlib.Topology.Connected.PathComponentOne", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -154.707, "z": 148.718, "size": 0.2, "title": "The path component of the identity in a locally path connected topological group", "summary": "This file defines the path component of the identity is an `OpenNormalSubgroup` when the ambient topological group is locally path connected. We place this in a separate file to avoid importing additional algebra into the topology hierarchy.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Connected/PathComponentOne.html"}, {"id": "Mathlib.Topology.Category.TopPair", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -52.474, "z": 154.587, "size": 0.239, "title": "Topological Pairs", "summary": "In this file we introduce `TopPair`, the category of topological pairs. It is defined as the category of arrows in `TopCat` which are topological embeddings. We provide the inclusion and diagonal functors `TopCat ⥤ TopPair` and show that they are left and right adjoint to the first projection functor, respectively. We also define for two morphisms of topological pairs `f, g : X ⟶ Y` the structure `Homotopy f g` of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopPair.html"}, {"id": "Mathlib.Topology.Homotopy.TopCat.Basic", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 2, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": -42.752, "z": 199.66, "size": 0.2724, "title": "Homotopies between morphisms in `TopCat`", "summary": "In this file, we define the type `TopCat.Homotopy` of homotopies between two morphisms in the category `TopCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/TopCat/Basic.html"}, {"id": "Mathlib.Topology.Category.Profinite.Extend", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -169.012, "z": 195.216, "size": 0.2565, "title": "Extending cones in `Profinite`", "summary": "Let `(Sᵢ)_{i : I}` be a family of finite sets indexed by a cofiltered category `I` and let `S` be its limit in `Profinite`. Let `G` be a functor from `Profinite` to a category `C` and suppose that `G` preserves the limit described above. Suppose further that the projection maps `S ⟶ Sᵢ` are epimorphic for all `i`. Then `G.obj S` is isomorphic to a limit indexed by `StructuredArrow S toProfinite` (see…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Profinite/Extend.html"}, {"id": "Mathlib.Topology.Order.ExtrClosure", "region_id": "topology", "micro_elevation": 0.5116, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -72.371, "z": 180.295, "size": 0.251, "title": "Maximum/minimum on the closure of a set", "summary": "In this file we prove several versions of the following statement: if `f : X → Y` has a (local or not) maximum (or minimum) on a set `s` at a point `a` and is continuous on the closure of `s`, then `f` has an extremum of the same type on `Closure s` at `a`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/ExtrClosure.html"}, {"id": "Mathlib.Topology.IndicatorConstPointwise", "region_id": "topology", "micro_elevation": 0.4186, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -101.969, "z": 219.724, "size": 0.2938, "title": "Pointwise convergence of indicator functions", "summary": "In this file, we prove the equivalence of three different ways to phrase that the indicator functions of sets converge pointwise.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/IndicatorConstPointwise.html"}, {"id": "Mathlib.Topology.Algebra.RestrictedProduct.Basic", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2629, "title": "Restricted products of sets, groups and rings", "summary": "We define the **restricted product** of `R : ι → Type*` of types, relative to a family of subsets `A : (i : ι) → Set (R i)` and a filter `𝓕 : Filter ι`. This is the set of all `x : Π i, R i` such that the set `{j | x j ∈ A j}` belongs to `𝓕`. We denote it by `Πʳ i, [R i, A i]_[𝓕]`. The main case of interest, which we shall refer to as the \"classical restricted product\", is that of `𝓕 = cofinite`. Recall that this is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/RestrictedProduct/Basic.html"}, {"id": "Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -104.634, "z": 143.13, "size": 0.2, "title": "The sheaf condition in terms of an equalizer of products", "summary": "Here we set up the machinery for the \"usual\" definition of the sheaf condition, e.g. as in https://stacks.math.columbia.edu/tag/0072 in terms of an equalizer diagram where the two objects are `∏ᶜ F.obj (U i)` and `∏ᶜ F.obj (U i) ⊓ (U j)`. We show that this sheaf condition is equivalent to the \"pairwise intersections\" sheaf condition when the presheaf is valued in a category with products, and thereby equivalent to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.html"}, {"id": "Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 2, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": -109.23, "z": 238.471, "size": 0.3098, "title": "Equivalent formulations of the sheaf condition", "summary": "We give an equivalent formulation of the sheaf condition. Given any indexed type `ι`, we define `overlap ι`, a category with objects corresponding to * individual open sets, `single i`, and * intersections of pairs of open sets, `pair i j`, with morphisms from `pair i j` to both `single i` and `single j`. Any open cover `U : ι → Opens X` provides a functor `diagram U : overlap ι ⥤ (Opens X)ᵒᵖ`. There is a canonical…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.html"}, {"id": "Mathlib.Topology.Category.DeltaGenerated", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -122.3, "z": 228.498, "size": 0.2, "title": "Delta-generated topological spaces", "summary": "This file defines the category `DeltaGenerated` of delta-generated spaces. This is a particular case of the construction in the file `Mathlib/Topology/Convenient/Category.Lean`: this is the category of `X`-generated spaces where `X` is the family of spaces `Fin n → ℝ` for all `n : ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/DeltaGenerated.html"}, {"id": "Mathlib.Topology.Compactness.DeltaGeneratedSpace", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -108.713, "z": 155.862, "size": 0.2478, "title": "Delta-generated topological spaces", "summary": "This file defines delta-generated spaces, as topological spaces whose topology is coinduced by all maps from Euclidean spaces into them. This is the strongest topological property that holds for all CW-complexes and is closed under quotients and disjoint unions; every delta-generated space is locally path-connected, sequential and in particular compactly generated. See…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactness/DeltaGeneratedSpace.html"}, {"id": "Mathlib.Topology.Convenient.Category", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 2, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": -143.913, "z": 188.416, "size": 0.2676, "title": "The category of `X`-generated spaces", "summary": "Let `X i` be a family of topological spaces. In this file, we define the category `GeneratedByTopCat X` of `X`-generated spaces: this is defined as a full subcategory of `TopCat`. We also introduce an equivalent category `ContinuousGeneratedByCat X` whose objects are all topological spaces, but morphisms from `Y` to `Z` identify to the type `ContinuousMapGeneratedBy X Y Z` of `X`-continuous maps from `Y` to `Z`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Convenient/Category.html"}, {"id": "Mathlib.Topology.ContinuousMap.CompactlySupported", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -126.193, "z": 252.265, "size": 0.2681, "title": "Compactly supported continuous functions", "summary": "In this file, we define the type `C_c(α, β)` of compactly supported continuous functions and the class `CompactlySupportedContinuousMapClass`, and prove basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/CompactlySupported.html"}, {"id": "Mathlib.Topology.PartitionOfUnity", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -124.835, "z": 247.761, "size": 0.2791, "title": "Continuous partition of unity", "summary": "In this file we define `PartitionOfUnity (ι X : Type*) [TopologicalSpace X] (s : Set X := univ)` to be a continuous partition of unity on `s` indexed by `ι`. More precisely, `f : PartitionOfUnity ι X s` is a collection of continuous functions `f i : C(X, ℝ)`, `i : ι`, such that * the supports of `f i` form a locally finite family of sets; * each `f i` is nonnegative; * `∑ᶠ i, f i x = 1` for all `x ∈ s`; * `∑ᶠ i, f i…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/PartitionOfUnity.html"}, {"id": "Mathlib.Topology.Algebra.Semigroup", "region_id": "topology", "micro_elevation": 0.4884, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -134.161, "z": 206.691, "size": 0.2338, "title": "Idempotents in topological semigroups", "summary": "This file provides a sufficient condition for a semigroup `M` to contain an idempotent (i.e. an element `m` such that `m * m = m `), namely that `M` is a nonempty compact Hausdorff space where right-multiplication by constants is continuous. We also state a corresponding lemma guaranteeing that a subset of `M` contains an idempotent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Semigroup.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.UniformOn", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 2, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": -50.871, "z": 207.839, "size": 0.3088, "title": "Infinite sum and products that converge uniformly", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/UniformOn.html"}, {"id": "Mathlib.Topology.Compactness.Lindelof", "region_id": "topology", "micro_elevation": 0.4186, "macro_tier": 4, "macro_tier_score": 0.3011, "macro_tier_override": null, "x": -133.069, "z": 192.639, "size": 0.4762, "title": "Lindelöf sets and Lindelöf spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactness/Lindelof.html"}, {"id": "Mathlib.Topology.Category.Stonean.EffectiveEpi", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -63.413, "z": 242.741, "size": 0.2538, "title": "Effective epimorphisms in `Stonean`", "summary": "This file proves that `EffectiveEpi`, `Epi` and `Surjective` are all equivalent in `Stonean`. As a consequence we deduce from the material in `Mathlib/Topology/Category/CompHausLike/EffectiveEpi.lean` that `Stonean` is `Preregular` and `Precoherent`. We also prove that for a finite family of morphisms in `Stonean` with fixed target, the conditions jointly surjective, jointly epimorphic and effective epimorphic are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Stonean/EffectiveEpi.html"}, {"id": "Mathlib.Topology.Category.Stonean.Limits", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -140.129, "z": 245.37, "size": 0.2483, "title": "Explicit limits and colimits", "summary": "This file applies the general API for explicit limits and colimits in `CompHausLike P` (see the file `Mathlib/Topology/Category/CompHausLike/Limits.lean`) to the special case of `Stonean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Stonean/Limits.html"}, {"id": "Mathlib.Topology.Order.PartialSups", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -94.887, "z": 157.077, "size": 0.2, "title": "Continuity of `partialSups`", "summary": "In this file we prove that `partialSups` of a sequence of continuous functions is continuous as well as versions for `Filter.Tendsto`, `ContinuousAt`, `ContinuousWithinAt`, and `ContinuousOn`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/PartialSups.html"}, {"id": "Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -163.957, "z": 170.792, "size": 0.2624, "title": "Category of Profinite Groups", "summary": "We say `G` is a profinite group if it is a topological group which is compact and totally disconnected.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.html"}, {"id": "Mathlib.Topology.Algebra.ClopenNhdofOne", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -159.277, "z": 184.793, "size": 0.2488, "title": "Existence of an open normal subgroup in any clopen neighborhood of the neutral element", "summary": "This file proves the lemma `IsTopologicalGroup.exist_openNormalSubgroup_sub_clopen_nhds_of_one`, which states that in a compact topological group, for any clopen neighborhood of 1, there exists an open normal subgroup contained within it. We then apply this lemma to show `ProfiniteGrp.closedSubgroup_eq_sInf_open`: any closed subgroup of a profinite group is the intersection of the open subgroups containing it. This…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/ClopenNhdofOne.html"}, {"id": "Mathlib.Topology.Sequences", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 3, "macro_tier_score": 0.0226, "macro_tier_override": null, "x": -101.724, "z": 141.663, "size": 0.3765, "title": "Sequences in topological spaces", "summary": "In this file we prove theorems about relations between closure/compactness/continuity etc. and their sequential counterparts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sequences.html"}, {"id": "Mathlib.Topology.Algebra.Valued.WithZeroMulInt", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -46.27, "z": 201.967, "size": 0.2, "title": "Topological results for integer-valued rings", "summary": "This file contains topological results for valuation rings taking values in the multiplicative integers with zero adjoined. These are useful for cases where there is a `Valued R ℤₘ₀` instance but no canonical base with which to embed this into `NNReal`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Valued/WithZeroMulInt.html"}, {"id": "Mathlib.Topology.VectorBundle.Riemannian", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -49.071, "z": 199.639, "size": 0.2624, "title": "Riemannian vector bundles", "summary": "Given a real vector bundle over a topological space whose fibers are all endowed with an inner product, we say that this bundle is Riemannian if the inner product depends continuously on the base point. We introduce a typeclass `[IsContinuousRiemannianBundle F E]` registering this property. Under this assumption, we show that the inner product of two continuous maps into the same fibers of the bundle is a continuous…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/VectorBundle/Riemannian.html"}, {"id": "Mathlib.Topology.VectorBundle.Hom", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -151.331, "z": 162.552, "size": 0.2983, "title": "The vector bundle of continuous (semi)linear maps", "summary": "We define the (topological) vector bundle of continuous (semi)linear maps between two vector bundles over the same base. Given bundles `E₁ E₂ : B → Type*`, normed spaces `F₁` and `F₂`, and a ring-homomorphism `σ` between their respective scalar fields, we define a vector bundle with fiber `E₁ x →SL[σ] E₂ x`. If the `E₁` and `E₂` are vector bundles with model fibers `F₁` and `F₂`, then this will be a vector bundle…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/VectorBundle/Hom.html"}, {"id": "Mathlib.Topology.Sheaves.PUnit", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -66.466, "z": 167.466, "size": 0.2522, "title": "Presheaves on `PUnit`", "summary": "Presheaves on `PUnit` satisfy sheaf condition iff its value at empty set is a terminal object.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/PUnit.html"}, {"id": "Mathlib.Topology.Instances.NNReal.Lemmas", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 3, "macro_tier_score": 0.0297, "macro_tier_override": null, "x": -117.03, "z": 133.416, "size": 0.4568, "title": "Topology on `ℝ≥0`", "summary": "The basic lemmas for the natural topology on `ℝ≥0` .", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/NNReal/Lemmas.html"}, {"id": "Mathlib.Topology.Category.TopCat.Yoneda", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -67.401, "z": 214.436, "size": 0.2599, "title": "Yoneda presheaves on topologically concrete categories", "summary": "This file develops some API for \"topologically concrete\" categories, defining universe polymorphic \"Yoneda presheaves\" on such categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/Yoneda.html"}, {"id": "Mathlib.Topology.Closure", "region_id": "topology", "micro_elevation": 0.0698, "macro_tier": 4, "macro_tier_score": 0.481, "macro_tier_override": null, "x": -106.669, "z": 196.034, "size": 0.3799, "title": "Interior, closure and frontier of a set", "summary": "This file provides lemmas relating to the functions `interior`, `closure` and `frontier` of a set endowed with a topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Closure.html"}, {"id": "Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -49.34, "z": 213.045, "size": 0.2422, "title": "The Arzelà–Ascoli theorem for bounded continuous functions", "summary": "Arzelà–Ascoli asserts that, on a compact space, a set of functions sharing a common modulus of continuity and taking values in a compact set forms a compact subset for the topology of uniform convergence. This file proves the theorem and several useful variations around it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Bounded/ArzelaAscoli.html"}, {"id": "Mathlib.Topology.MetricSpace.Thickening", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 2, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": -50.117, "z": 222.094, "size": 0.3693, "title": "Thickenings in pseudo-metric spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Thickening.html"}, {"id": "Mathlib.Topology.Compactness.CompactSystem", "region_id": "topology", "micro_elevation": 0.4884, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -136.807, "z": 199.641, "size": 0.2, "title": "Compact systems", "summary": "This file defines compact systems of sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactness/CompactSystem.html"}, {"id": "Mathlib.Topology.Instances.ZMod", "region_id": "topology", "micro_elevation": 0.186, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -112.314, "z": 201.779, "size": 0.2, "title": "Topology on `ZMod N`", "summary": "We equip `ZMod N` with the discrete topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/ZMod.html"}, {"id": "Mathlib.Topology.UniformSpace.Ultra.Constructions", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -76.797, "z": 231.284, "size": 0.2478, "title": "Products of ultrametric (nonarchimedean) uniform spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Ultra/Constructions.html"}, {"id": "Mathlib.Topology.UniformSpace.Ultra.Basic", "region_id": "topology", "micro_elevation": 0.3721, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -99.594, "z": 167.19, "size": 0.257, "title": "Ultrametric (nonarchimedean) uniform spaces", "summary": "Ultrametric (nonarchimedean) uniform spaces are ones that generalize ultrametric spaces by having a uniformity based on equivalence relations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Ultra/Basic.html"}, {"id": "Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -135.003, "z": 157.676, "size": 0.2943, "title": "Another version of the sheaf condition.", "summary": "Given a family of open sets `U : ι → Opens X` we can form the subcategory `{ V : Opens X // ∃ i, V ≤ U i }`, which has `iSup U` as a cocone. The sheaf condition on a presheaf `F` is equivalent to `F` sending the opposite of this cocone to a limit cone in `C`, for every `U`. This condition is particularly nice when checking the sheaf condition because we don't need to do any case bashing (depending on whether we're…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.html"}, {"id": "Mathlib.Topology.Spectral.ConstructibleTopology", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -153.327, "z": 204.573, "size": 0.2, "title": "Constructible topology", "summary": "In this file we define the constructible topology on a topological space. This is the topology generated by compact open subsets and their complements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Spectral/ConstructibleTopology.html"}, {"id": "Mathlib.Topology.JacobsonSpace", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -93.421, "z": 147.734, "size": 0.2696, "title": "Jacobson spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/JacobsonSpace.html"}, {"id": "Mathlib.Topology.Algebra.Order.Module", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -127.431, "z": 232.914, "size": 0.235, "title": "Continuous nonnegative scalar multiplication", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Order/Module.html"}, {"id": "Mathlib.Topology.Order.Category.FrameAdjunction", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -63.884, "z": 142.265, "size": 0.2, "title": "Adjunction between Locales and Topological Spaces", "summary": "This file defines the point functor from the category of locales to topological spaces and proves that it is right adjoint to the forgetful functor from topological spaces to locales.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Category/FrameAdjunction.html"}, {"id": "Mathlib.Topology.Category.Locale", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -99.005, "z": 129.31, "size": 0.2676, "title": "The category of locales", "summary": "This file defines `Locale`, the category of locales. This is the opposite of the category of frames.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Locale.html"}, {"id": "Mathlib.Topology.Algebra.Module.TransferInstance", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -45.378, "z": 191.015, "size": 0.2628, "title": "Transfer topological algebraic structures across `Equiv`s", "summary": "In this file, we construct a continuous linear equivalence `α ≃L[R] β` from an equivalence `α ≃ β`, where the continuous `R`-module structure on `α` is the one obtained by transporting an `R`-module structure on `β` back along `e`. We also specialize this construction to `Shrink α`. This continues the pattern set in `Mathlib/Algebra/Module/TransferInstance.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/TransferInstance.html"}, {"id": "Mathlib.Topology.Compactness.Bases", "region_id": "topology", "micro_elevation": 0.3721, "macro_tier": 4, "macro_tier_score": 0.1938, "macro_tier_override": null, "x": -103.459, "z": 166.664, "size": 0.3407, "title": "Topological bases in compact sets and compact spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactness/Bases.html"}, {"id": "Mathlib.Topology.Bases", "region_id": "topology", "micro_elevation": 0.3488, "macro_tier": 4, "macro_tier_score": 0.3789, "macro_tier_override": null, "x": -109.224, "z": 214.775, "size": 0.4639, "title": "Bases of topologies. Countability axioms.", "summary": "A topological basis on a topological space `t` is a collection of sets, such that all open sets can be generated as unions of these sets, without the need to take finite intersections of them. This file introduces a framework for dealing with these collections, and also what more we can say under certain countability conditions on bases, which are referred to as first- and second-countable. We also briefly cover the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Bases.html"}, {"id": "Mathlib.Topology.Algebra.GroupCompletion", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 2, "macro_tier_score": 0.0162, "macro_tier_override": null, "x": -92.295, "z": 138.331, "size": 0.2978, "title": "Completion of topological groups:", "summary": "This file endows the completion of a topological abelian group with a group structure. More precisely the instance `UniformSpace.Completion.addGroup` builds an abelian group structure on the completion of an abelian group endowed with a compatible uniform structure. Then the instance `UniformSpace.Completion.isUniformAddGroup` proves this group structure is compatible with the completed uniform structure. The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/GroupCompletion.html"}, {"id": "Mathlib.Topology.Maps.Strict.Basic", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -52.743, "z": 202.469, "size": 0.2733, "title": "Bourbaki Strict Maps", "summary": "This file defines Bourbaki strict maps (`Topology.IsStrictMap`) and proves some of their basic properties. A map `f : X → Y` between topological spaces is called *strict* in the sense of Bourbaki if the natural corestriction to its image (i.e., `Set.rangeFactorization f`) is a quotient map. Equivalently, these are precisely the maps for which the first isomorphism theorem yields a homeomorphism: the canonical…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Maps/Strict/Basic.html"}, {"id": "Mathlib.Topology.Order.Filter", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -81.027, "z": 218.668, "size": 0.2, "title": "Topology on filters of a space with order topology", "summary": "In this file we prove that `𝓝 (f x)` tends to `𝓝 Filter.atTop` provided that `f` tends to `Filter.atTop`, and similarly for `Filter.atBot`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Filter.html"}, {"id": "Mathlib.Topology.Filter", "region_id": "topology", "micro_elevation": 0.4186, "macro_tier": 2, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": -78.377, "z": 201.263, "size": 0.2617, "title": "Topology on the set of filters on a type", "summary": "This file introduces a topology on `Filter α`. It is generated by the sets `Set.Iic (𝓟 s) = {l : Filter α | s ∈ l}`, `s : Set α`. A set `s : Set (Filter α)` is open if and only if it is a union of a family of these basic open sets, see `Filter.isOpen_iff`. This topology has the following important properties. * If `X` is a topological space, then the map `𝓝 : X → Filter X` is a topology inducing map. * In…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Filter.html"}, {"id": "Mathlib.Topology.Algebra.LinearMapCompletion", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -76.414, "z": 240.349, "size": 0.2338, "title": "Completion of continuous (semi-)linear maps:", "summary": "This file has a declaration that enables a continuous (semi-)linear map between modules to be lifted to a continuous semilinear map between the completions of those modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/LinearMapCompletion.html"}, {"id": "Mathlib.Topology.Category.TopCat.EffectiveEpi", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -131.718, "z": 155.029, "size": 0.2405, "title": "Effective epimorphisms in `TopCat`", "summary": "This file proves the result `TopCat.effectiveEpi_iff_isQuotientMap`: The effective epimorphisms in `TopCat` are precisely the quotient maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/EffectiveEpi.html"}, {"id": "Mathlib.Topology.Compactness.LocallyCompact", "region_id": "topology", "micro_elevation": 0.3721, "macro_tier": 4, "macro_tier_score": 0.3668, "macro_tier_override": null, "x": -118.38, "z": 170.562, "size": 0.401, "title": "Locally compact spaces", "summary": "This file contains basic results about locally compact spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactness/LocallyCompact.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.Group", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 3, "macro_tier_score": 0.0269, "macro_tier_override": null, "x": -136.744, "z": 232.39, "size": 0.323, "title": "Infinite sums and products in topological groups", "summary": "Lemmas on topological sums in groups (as opposed to monoids).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/Group.html"}, {"id": "Mathlib.Topology.Instances.Complex", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -95.669, "z": 250.483, "size": 0.2734, "title": "Some results about the topology of ℂ", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/Complex.html"}, {"id": "Mathlib.Topology.Compactification.OnePoint.ProjectiveLine", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -137.359, "z": 221.166, "size": 0.2439, "title": "One-point compactification and projectivization", "summary": "We construct a set-theoretic equivalence between `OnePoint K` and the projectivization `ℙ K (Fin 2 → K)` for an arbitrary division ring `K`. TODO: Add the extension of this equivalence to a homeomorphism in the case `K = ℝ`, where `OnePoint ℝ` gets the topology of one-point compactification.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.html"}, {"id": "Mathlib.Topology.Covering.AddCircle", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -47.187, "z": 197.016, "size": 0.2614, "title": "Covering maps involving `AddCircle`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Covering/AddCircle.html"}, {"id": "Mathlib.Topology.MetricSpace.HausdorffDimension", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2, "title": "Hausdorff dimension", "summary": "The Hausdorff dimension of a set `X` in an (extended) metric space is the unique number `dimH s : ℝ≥0∞` such that for any `d : ℝ≥0` we have - `μH[d] s = 0` if `dimH s < d`, and - `μH[d] s = ∞` if `d < dimH s`. In this file we define `dimH s` to be the Hausdorff dimension of `s`, then prove some basic properties of Hausdorff dimension.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/HausdorffDimension.html"}, {"id": "Mathlib.Topology.Algebra.Group.CompactOpen", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -135.669, "z": 142.579, "size": 0.2478, "title": "The compact-open topology on continuous monoid morphisms.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/CompactOpen.html"}, {"id": "Mathlib.Topology.Algebra.Group.Compact", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -137.428, "z": 151.522, "size": 0.2756, "title": "Additional results on topological groups", "summary": "A result on topological groups that has been separated out as it requires more substantial imports developing positive compacts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/Compact.html"}, {"id": "Mathlib.Topology.UniformSpace.Ascoli", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -94.913, "z": 139.374, "size": 0.2403, "title": "Ascoli Theorem", "summary": "In this file, we prove the general **Arzela-Ascoli theorem**, and various related statements about the topology of equicontinuous subsets of `X →ᵤ[𝔖] α`, where `X` is a topological space, `𝔖` is a family of compact subsets of `X`, and `α` is a uniform space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Ascoli.html"}, {"id": "Mathlib.Topology.Category.LightProfinite.Cartesian", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -66.668, "z": 138.145, "size": 0.2408, "title": "Cartesian monoidal structure on `LightProfinite`", "summary": "This file defines the cartesian monoidal structure on `LightProfinite` given by the type-theoretic product.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/LightProfinite/Cartesian.html"}, {"id": "Mathlib.Topology.Algebra.Module.Compact", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -136.408, "z": 234.61, "size": 0.2478, "title": "Compact submodules", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/Compact.html"}, {"id": "Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -156.162, "z": 206.083, "size": 0.2403, "title": "Restricted products of topological spaces, topological groups and rings", "summary": "We endow a restricted product of topological spaces with a natural topology, which we describe below. We also show various compatibility results when we change filters, and extend the construction of restricted products of algebraic structures to the topological setting. In particular, with the theory of adeles in mind, we show that if each `R i` is a locally compact topological ring with open subring `A i`, and if…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.html"}, {"id": "Mathlib.Topology.Algebra.RestrictedProduct.Units", "region_id": "topology", "micro_elevation": 0.0233, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -105.964, "z": 192.826, "size": 0.2403, "title": "Units of restricted products", "summary": "This file contains results about the units of a restricted product. The restricted product `Πʳ i : ι, [R i, B i]_[𝓕]` of a family of types `R` with respect to a family of subsets `B` along a filter `𝓕` is defined in `Mathlib.Topology.Algebra.RestrictedProduct.Basic`. Here, we give conditions that characterize when an element of the restricted product is a unit, and provide an isomorphism between the units of the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/RestrictedProduct/Units.html"}, {"id": "Mathlib.Topology.MetricSpace.Equicontinuity", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 2, "macro_tier_score": 0.011, "macro_tier_override": null, "x": -88.643, "z": 242.397, "size": 0.3048, "title": "Equicontinuity in metric spaces", "summary": "This file contains various facts about (uniform) equicontinuity in metric spaces. Most importantly, we prove the usual characterization of equicontinuity of `F` at `x₀` in the case of (pseudo) metric spaces: `∀ ε > 0, ∃ δ > 0, ∀ x, dist x x₀ < δ → ∀ i, dist (F i x₀) (F i x) < ε`, and we prove that functions sharing a common (local or global) continuity modulus are (locally or uniformly) equicontinuous.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Equicontinuity.html"}, {"id": "Mathlib.Topology.DerivedSet", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -136.287, "z": 165.746, "size": 0.2, "title": "Derived set", "summary": "This file defines the derived set of a set, the set of all `AccPt`s of its principal filter, and proves some properties of it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/DerivedSet.html"}, {"id": "Mathlib.Topology.ContinuousMap.Weierstrass", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -100.566, "z": 255.755, "size": 0.2411, "title": "The Weierstrass approximation theorem for continuous functions on `[a,b]`", "summary": "We've already proved the Weierstrass approximation theorem in the sense that we've shown that the Bernstein approximations to a continuous function on `[0,1]` converge uniformly. Here we rephrase this more abstractly as `polynomialFunctions_closure_eq_top' : (polynomialFunctions I).topologicalClosure = ⊤` and then, by precomposing with suitable affine functions, `polynomialFunctions_closure_eq_top :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Weierstrass.html"}, {"id": "Mathlib.Topology.CWComplex.Classical.Subcomplex", "region_id": "topology", "micro_elevation": 0.0465, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -107.62, "z": 193.223, "size": 0.2, "title": "Subcomplexes", "summary": "In this file we discuss subcomplexes of CW complexes. The definition of subcomplexes is in the file `Mathlib/Topology/CWComplex/Classical/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/CWComplex/Classical/Subcomplex.html"}, {"id": "Mathlib.Topology.CWComplex.Classical.Finite", "region_id": "topology", "micro_elevation": 0.0233, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -104.525, "z": 190.161, "size": 0.2827, "title": "Finiteness notions on CW complexes", "summary": "In this file we define what it means for a CW complex to be finite dimensional, of finite type or finite. We define constructors with relaxed conditions for CW complexes of finite type and finite CW complexes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/CWComplex/Classical/Finite.html"}, {"id": "Mathlib.Topology.Bornology.Absorbs", "region_id": "topology", "micro_elevation": 0.0233, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -105.098, "z": 193.236, "size": 0.2959, "title": "Absorption of sets", "summary": "Let `M` act on `α`, let `A` and `B` be sets in `α`. We say that `A` *absorbs* `B` if for sufficiently large `a : M`, we have `B ⊆ a • A`. Formally, \"for sufficiently large `a : M`\" means \"for all but a bounded set of `a`\". Traditionally, this definition is formulated for the action of a (semi)normed ring on a module over that ring. We formulate it in a more general settings for two reasons: - this way we don't have…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Bornology/Absorbs.html"}, {"id": "Mathlib.Topology.Category.TopCommRingCat", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -86.253, "z": 141.799, "size": 0.239, "title": "Category of topological commutative rings", "summary": "We introduce the category `TopCommRingCat` of topological commutative rings together with the relevant forgetful functors to topological spaces and commutative rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCommRingCat.html"}, {"id": "Mathlib.Topology.Connected.PathConnected", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 3, "macro_tier_score": 0.028, "macro_tier_override": null, "x": -167.187, "z": 198.357, "size": 0.3856, "title": "Path connectedness", "summary": "Continuing from `Mathlib/Topology/Path.lean`, this file defines path components and path-connected spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Connected/PathConnected.html"}, {"id": "Mathlib.Topology.Sion", "region_id": "topology", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -169.695, "z": 173.335, "size": 0.2, "title": "Formalization of Sion's version of the von Neumann minimax theorem", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sion.html"}, {"id": "Mathlib.Topology.NoetherianSpace", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 3, "macro_tier_score": 0.0547, "macro_tier_override": null, "x": -128.824, "z": 226.55, "size": 0.4155, "title": "Noetherian space", "summary": "A Noetherian space is a topological space that satisfies any of the following equivalent conditions: - `WellFounded ((· > ·) : TopologicalSpace.Opens α → TopologicalSpace.Opens α → Prop)` - `WellFounded ((· < ·) : TopologicalSpace.Closeds α → TopologicalSpace.Closeds α → Prop)` - `∀ s : Set α, IsCompact s` - `∀ s : TopologicalSpace.Opens α, IsCompact s` The first is chosen as the definition, and the equivalence is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/NoetherianSpace.html"}, {"id": "Mathlib.Topology.Semicontinuity.Hemicontinuity", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -156.168, "z": 198.192, "size": 0.2455, "title": "Hemicontinuity", "summary": "This files provides basic facts about upper and lower hemicontinuity of correspondences `f : α → Set β`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Semicontinuity/Hemicontinuity.html"}, {"id": "Mathlib.Topology.Homotopy.Affine", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -169.704, "z": 202.951, "size": 0.2239, "title": "Affine homotopy between two continuous maps", "summary": "In this file we define `ContinuousMap.Homotopy.affine f g` to be the homotopy between `f` and `g` such that `affine f g (t, x) = AffineMap.lineMap (f x) (g x) t`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/Affine.html"}, {"id": "Mathlib.Topology.Algebra.Group.TopologicalAbelianization", "region_id": "topology", "micro_elevation": 0.7442, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -57.293, "z": 207.377, "size": 0.2478, "title": "The topological abelianization of a group.", "summary": "This file defines the topological abelianization of a topological group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/TopologicalAbelianization.html"}, {"id": "Mathlib.Topology.ContinuousMap.Periodic", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -49.537, "z": 174.533, "size": 0.2669, "title": "Sums of translates of a continuous function is a period continuous function.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Periodic.html"}, {"id": "Mathlib.Topology.Separation.LinearUpperLowerSetTopology", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -103.766, "z": 152.543, "size": 0.2, "title": "Linear upper or lower sets topologies are completely normal", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/LinearUpperLowerSetTopology.html"}, {"id": "Mathlib.Topology.UniformSpace.Path", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -43.037, "z": 201.632, "size": 0.2, "title": "Paths in uniform spaces", "summary": "In this file we define a `UniformSpace` structure on `Path`s between two points in a uniform space and prove that various functions associated with `Path`s are uniformly continuous. The uniform space structure is induced from the space of continuous maps `C(I, X)`, and corresponds to uniform convergence of paths on `I`, see `Path.hasBasis_uniformity`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Path.html"}, {"id": "Mathlib.Topology.Compactness.Paracompact", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 2, "macro_tier_score": 0.016, "macro_tier_override": null, "x": -80.102, "z": 161.373, "size": 0.2835, "title": "Paracompact topological spaces", "summary": "A topological space `X` is said to be paracompact if every open covering of `X` admits a locally finite refinement. The definition requires that each set of the new covering is a subset of one of the sets of the initial covering. However, one can ensure that each open covering `s : ι → Set X` admits a *precise* locally finite refinement, i.e., an open covering `t : ι → Set X` with the same index set such that `∀ i,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactness/Paracompact.html"}, {"id": "Mathlib.Topology.Category.Profinite.Product", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -61.516, "z": 236.89, "size": 0.251, "title": "Compact subsets of products as limits in `Profinite`", "summary": "This file exhibits a compact subset `C` of a product `(i : ι) → X i` of totally disconnected Hausdorff spaces as a cofiltered limit in `Profinite` indexed by `Finset ι`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Profinite/Product.html"}, {"id": "Mathlib.Topology.LocallyFinite", "region_id": "topology", "micro_elevation": 0.3488, "macro_tier": 4, "macro_tier_score": 0.3447, "macro_tier_override": null, "x": -118.197, "z": 211.048, "size": 0.3277, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/LocallyFinite.html"}, {"id": "Mathlib.Topology.QuasiSeparated", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 3, "macro_tier_score": 0.0496, "macro_tier_override": null, "x": -98.512, "z": 148.295, "size": 0.4224, "title": "Quasi-separated spaces", "summary": "A topological space is quasi-separated if the intersections of any pairs of compact open subsets are still compact. Notable examples include spectral spaces, Noetherian spaces, and Hausdorff spaces. A non-example is the interval `[0, 1]` with doubled origin: the two copies of `[0, 1]` are compact open subsets, but their intersection `(0, 1]` is not.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/QuasiSeparated.html"}, {"id": "Mathlib.Topology.ApproximateUnit", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -95.645, "z": 239.349, "size": 0.2466, "title": "Approximate units", "summary": "An *approximate unit* is a filter `l` such that multiplication on the left (or right) by `m : α` tends to `𝓝 m` along the filter, and additionally `l ≠ ⊥`. Examples of approximate units include: - The trivial approximate unit `pure 1` in a normed ring. - `𝓝 1` or `𝓝[≠] 1` in a normed ring (note that the latter is disjoint from `pure 1`). - In a C⋆-algebra, the filter generated by the sections `fun a ↦ {x | a ≤ x} ∩…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ApproximateUnit.html"}, {"id": "Mathlib.Topology.Instances.ENNReal.ENatENNReal", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -143.991, "z": 142.73, "size": 0.2, "title": "Topology lemma for `ENat.toENNReal`", "summary": "This file shows `ENat.toENNReal` is a closed embedding.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/ENNReal/ENatENNReal.html"}, {"id": "Mathlib.Topology.Sheaves.Alexandrov", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -109.155, "z": 144.889, "size": 0.2, "title": null, "summary": "Let `X` be a preorder `≤`, and consider the associated Alexandrov topology on `X`. Given a functor `F : X ⥤ C` to a complete category, we can extend `F` to a presheaf on (the topological space) `X` by taking the right Kan extension along the canonical functor `X ⥤ (Opens X)ᵒᵖ` sending `x : X` to the principal open `{y | x ≤ y}` in the Alexandrov topology. This file proves that this presheaf is a sheaf.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Alexandrov.html"}, {"id": "Mathlib.Topology.Connected.Separation", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -137.738, "z": 170.367, "size": 0.239, "title": "Separation and (dis)connectedness properties of topological spaces.", "summary": "This file provides an instance `T2Space X` given `TotallySeparatedSpace X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Connected/Separation.html"}, {"id": "Mathlib.Topology.Bornology.BoundedOperation", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.0266, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2993, "title": "Bounded operations", "summary": "This file introduces type classes for bornologically bounded operations. In particular, when combined with type classes which guarantee continuity of the same operations, we can equip bounded continuous functions with the corresponding operations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Bornology/BoundedOperation.html"}, {"id": "Mathlib.Topology.Algebra.ProperAction.AddTorsor", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -160.176, "z": 180.588, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/ProperAction/AddTorsor.html"}, {"id": "Mathlib.Topology.Connected.CardComponents", "region_id": "topology", "micro_elevation": 0.6744, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -86.124, "z": 233.046, "size": 0.2, "title": "Cardinality of connected components under open and closed maps", "summary": "Let `f : X → Y` be an open and closed map.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Connected/CardComponents.html"}, {"id": "Mathlib.Topology.LocalAtTarget", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 3, "macro_tier_score": 0.0272, "macro_tier_override": null, "x": -143.669, "z": 212.17, "size": 0.3437, "title": "Properties of maps that are local at the target or at the source.", "summary": "We show that the following properties of continuous maps are local at the target : - `Topology.IsInducing` - `IsOpenMap` - `IsClosedMap` - `Topology.IsEmbedding` - `Topology.IsOpenEmbedding` - `Topology.IsClosedEmbedding` - `GeneralizingMap` We show that the following properties of continuous maps are local at the source: - `IsOpenMap` - `GeneralizingMap`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/LocalAtTarget.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.TsumUniformlyOn", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -154.57, "z": 161.84, "size": 0.2292, "title": "Differentiability of sum of functions", "summary": "We prove some `HasSumUniformlyOn` versions of theorems from `Mathlib.Analysis.NormedSpace.FunctionSeries`. Alongside this we prove `derivWithin_tsum` which states that the derivative of a series of functions is the sum of the derivatives, under suitable conditions we also prove an `iteratedDerivWithin` version.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/TsumUniformlyOn.html"}, {"id": "Mathlib.Topology.Order.Category.AlexDisc", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -105.139, "z": 150.961, "size": 0.2, "title": "Category of Alexandrov-discrete topological spaces", "summary": "This defines `AlexDisc`, the category of Alexandrov-discrete topological spaces with continuous maps, and proves it's equivalent to the category of preorders.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Category/AlexDisc.html"}, {"id": "Mathlib.Topology.ContinuousMap.StarOrdered", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 2, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": -159.736, "z": 155.356, "size": 0.3095, "title": "Continuous functions as a star-ordered ring", "summary": "The type class `ContinuousSqrt` gives a sufficient condition on `R` to make `C(α, R)` and `C(α, R)₀` into a `StarOrderedRing` for any topological space `α`, thereby providing a means by which we can ensure `C(α, R)` has this property. This condition is satisfied by `ℝ≥0`, `ℝ`, and `ℂ`, and the instances can be found in the file `Mathlib/Topology/ContinuousMap/ContinuousSqrt.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/StarOrdered.html"}, {"id": "Mathlib.Topology.Partial", "region_id": "topology", "micro_elevation": 0.1163, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -106.365, "z": 199.375, "size": 0.2, "title": "Partial functions and topological spaces", "summary": "In this file we prove properties of `Filter.PTendsto` etc. in topological spaces. We also introduce `PContinuous`, a version of `Continuous` for partially defined functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Partial.html"}, {"id": "Mathlib.Topology.Homotopy.Product", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -60.954, "z": 240.641, "size": 0.2491, "title": "Product of homotopies", "summary": "In this file, we introduce definitions for the product of homotopies. We show that the products of relative homotopies are still relative homotopies. Finally, we specialize to the case of path homotopies, and provide the definition for the product of path classes. We show various lemmas associated with these products, such as the fact that path products commute with path composition, and that projection is the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/Product.html"}, {"id": "Mathlib.Topology.Sheaves.PresheafOfFunctions", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 1, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -104.268, "z": 233.968, "size": 0.2705, "title": "Presheaves of functions", "summary": "We construct some simple examples of presheaves of functions on a topological space. * `presheafToTypes X T`, where `T : X → Type`, is the presheaf of dependently-typed (not-necessarily continuous) functions * `presheafToType X T`, where `T : Type`, is the presheaf of (not-necessarily-continuous) functions to a fixed target type `T` * `presheafToTop X T`, where `T : TopCat`, is the presheaf of continuous functions…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/PresheafOfFunctions.html"}, {"id": "Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 2, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": -153.335, "z": 194.973, "size": 0.3112, "title": "The sheaf condition in terms of unique gluings", "summary": "We provide an alternative formulation of the sheaf condition in terms of unique gluings. We work with sheaves valued in a concrete category `C` admitting all limits, whose forgetful functor `C ⥤ Type` preserves limits and reflects isomorphisms. The usual categories of algebraic structures, such as `MonCat`, `AddCommGrpCat`, `RingCat`, `CommRingCat` etc. are all examples of this kind of category. A presheaf `F :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.html"}, {"id": "Mathlib.Topology.Category.CompHausLike.SigmaComparison", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -143.888, "z": 195.239, "size": 0.244, "title": "The sigma-comparison map", "summary": "This file defines the map `CompHausLike.sigmaComparison` associated to a presheaf `X` on `CompHausLike P`, and a finite family `S₁,...,Sₙ` of spaces in `CompHausLike P`, where `P` is stable under taking finite disjoint unions. The map `sigmaComparison` is the canonical map `X(S₁ ⊔ ... ⊔ Sₙ) ⟶ X(S₁) × ... × X(Sₙ)` induced by the inclusion maps `Sᵢ ⟶ S₁ ⊔ ... ⊔ Sₙ`, and it is an isomorphism when `X` preserves finite…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/CompHausLike/SigmaComparison.html"}, {"id": "Mathlib.Topology.MetricSpace.PartitionOfUnity", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -53.413, "z": 196.304, "size": 0.2626, "title": "Lemmas about (e)metric spaces that need partition of unity", "summary": "The main lemma in this file (see `Metric.exists_continuous_real_forall_closedBall_subset`) says the following. Let `X` be a metric space. Let `K : ι → Set X` be a locally finite family of closed sets, let `U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a positive continuous function `δ : C(X, ℝ)` such that for any `i` and `x ∈ K i`, we have `Metric.closedBall x (δ x) ⊆ U…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/PartitionOfUnity.html"}, {"id": "Mathlib.Topology.MetricSpace.CoveringNumbers", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -43.668, "z": 215.745, "size": 0.2, "title": "Covering numbers", "summary": "We define covering numbers of sets in a pseudo-metric space, which are minimal cardinalities of `ε`-covers of sets by closed balls. We also define the packing number, which is the maximal cardinality of an `ε`-separated set. We prove inequalities between these covering and packing numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/CoveringNumbers.html"}, {"id": "Mathlib.Topology.MetricSpace.Snowflaking", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -165.106, "z": 174.379, "size": 0.2, "title": "Snowflaking of a metric space", "summary": "Given a (pseudo) (extended) metric space `X` and a number `0 < α ≤ 1`, one can consider the metric given by `d x y = (dist x y) ^ α`. The metric space determined by this new metric is said to be the `α`-snowflaking (or `α`-snowflake) of `X`. In this file we define `Metric.Snowflaking X α hα₀ hα₁` to be a one-field structure wrapper around `X` with metric given by this formula. The use of the term *snowflaking*…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Snowflaking.html"}, {"id": "Mathlib.Topology.Category.LightProfinite.EffectiveEpi", "region_id": "topology", "micro_elevation": 1.0, "macro_tier": 1, "macro_tier_score": 0.0009, "macro_tier_override": null, "x": -168.531, "z": 169.636, "size": 0.3272, "title": "Effective epimorphisms in `LightProfinite`", "summary": "This file proves that `EffectiveEpi` and `Surjective` are equivalent in `LightProfinite`. As a consequence we deduce from the material in `Mathlib/Topology/Category/CompHausLike/EffectiveEpi.lean` that `LightProfinite` is `Preregular` and `Precoherent`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/LightProfinite/EffectiveEpi.html"}, {"id": "Mathlib.Topology.MetricSpace.DilationEquiv", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 1, "macro_tier_score": 0.0009, "macro_tier_override": null, "x": -149.997, "z": 232.877, "size": 0.3274, "title": "Dilation equivalence", "summary": "In this file we define `DilationEquiv X Y`, a type of bundled equivalences between `X` and `Y` such that `edist (f x) (f y) = r * edist x y` for some `r : ℝ≥0`, `r ≠ 0`. We also develop basic API about these equivalences.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/DilationEquiv.html"}, {"id": "Mathlib.Topology.Algebra.PontryaginDual", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -126.111, "z": 247.291, "size": 0.2, "title": "Pontryagin dual", "summary": "This file defines the Pontryagin dual of a topological group. The Pontryagin dual of a topological group `A` is the topological group of continuous homomorphisms `A →* Circle` with the compact-open topology. For example, `ℤ` and `Circle` are Pontryagin duals of each other. This is an example of Pontryagin duality, which states that a locally compact abelian topological group is canonically isomorphic to its double…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/PontryaginDual.html"}, {"id": "Mathlib.Topology.Category.TopCat.Sphere", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -77.441, "z": 219.606, "size": 0.239, "title": "Euclidean spheres", "summary": "This file defines the `n`-sphere `𝕊 n`, the `n`-disk `𝔻 n`, its boundary `∂𝔻 n` and its interior `𝔹 n` as objects in `TopCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/Sphere.html"}, {"id": "Mathlib.Topology.Homotopy.TopCat.ToSSet", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -93.697, "z": 128.45, "size": 0.253, "title": "The singular simplicial set functor preserves homotopies", "summary": "In this file, we define `TopCat.Homotopy.toSSet`, which shows that if two morphisms `f : X ⟶ Y` and `g : X ⟶ Y` in `TopCat` are homotopic (`h : Homotopy f g`), then so are their images by the functor `TopCat.toSSet : TopCat ⥤ SSet`. Indeed, if we apply the singular simplicial set functor to the morphism `h.h : X ⊗ I ⟶ Y` and use that this functor commutes with products, we obtain a morphism `toSSet.obj X ⊗…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/TopCat/ToSSet.html"}, {"id": "Mathlib.Topology.Homeomorph.Defs", "region_id": "topology", "micro_elevation": 0.2326, "macro_tier": 4, "macro_tier_score": 0.4498, "macro_tier_override": null, "x": -97.05, "z": 178.126, "size": 0.3851, "title": "Homeomorphisms", "summary": "This file defines homeomorphisms between two topological spaces. They are bijections with both directions continuous. We denote homeomorphisms with the notation `≃ₜ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homeomorph/Defs.html"}, {"id": "Mathlib.Topology.Separation.SeparatedNhds", "region_id": "topology", "micro_elevation": 0.1628, "macro_tier": 4, "macro_tier_score": 0.4494, "macro_tier_override": null, "x": -105.463, "z": 202.632, "size": 0.3652, "title": "Separated neighbourhoods", "summary": "This file defines the predicates `SeparatedNhds` and `HasSeparatingCover`, which are used in formulating separation axioms for topological spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/SeparatedNhds.html"}, {"id": "Mathlib.Topology.Order.Hom.Esakia", "region_id": "topology", "micro_elevation": 0.186, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -108.479, "z": 203.689, "size": 0.2, "title": "Esakia morphisms", "summary": "This file defines pseudo-epimorphisms and Esakia morphisms. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Hom/Esakia.html"}, {"id": "Mathlib.Topology.Order.Hom.Basic", "region_id": "topology", "micro_elevation": 0.1628, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -114.263, "z": 197.376, "size": 0.2478, "title": "Continuous order homomorphisms", "summary": "This file defines continuous order homomorphisms, that is maps which are both continuous and monotone. They are also called Priestley homomorphisms because they are the morphisms of the category of Priestley spaces. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Hom/Basic.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.SummationFilter", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.0941, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2847, "title": "Summation filters", "summary": "We define a `SummationFilter` on `β` to be a filter on the finite subsets of `β`. These are used in defining summability: if `L` is a summation filter, we define the `L`-sum of `f` to be the limit along `L` of the sums over finsets (if this limit exists). This file only develops the basic machinery of summation filters - the key definitions `HasSum`, `tsum` and `summable` (and their product variants) are in the file…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/SummationFilter.html"}, {"id": "Mathlib.Topology.Algebra.Ring.Compact", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -155.119, "z": 213.649, "size": 0.2, "title": "Compact Hausdorff Rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Ring/Compact.html"}, {"id": "Mathlib.Topology.Homotopy.TopCat.Path", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -148.462, "z": 236.704, "size": 0.2442, "title": "Paths between points of an object of `TopCat`", "summary": "This file introduces a structure `TopCat.Path` for paths between two points of an object `X : TopCat`. The data is defined using a morphism `I ⟶ X` in the category `TopCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/TopCat/Path.html"}, {"id": "Mathlib.Topology.Algebra.Group.OpenMapping", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -125.633, "z": 144.34, "size": 0.2, "title": "Open mapping theorem for morphisms of topological groups", "summary": "We prove that a continuous surjective group morphism from a sigma-compact group to a locally compact group is automatically open, in `MonoidHom.isOpenMap_of_sigmaCompact`. We deduce this from a similar statement for the orbits of continuous actions of sigma-compact groups on Baire spaces, given in `isOpenMap_smul_of_sigmaCompact`. Note that a sigma-compactness assumption is necessary. Indeed, let `G` be the real…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/OpenMapping.html"}, {"id": "Mathlib.Topology.Order.NhdsSet", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -137.444, "z": 207.115, "size": 0.2, "title": "Set neighborhoods of intervals", "summary": "In this file we prove basic theorems about `𝓝ˢ s`, where `s` is one of the intervals `Set.Ici`, `Set.Iic`, `Set.Ioi`, `Set.Iio`, `Set.Ico`, `Set.Ioc`, `Set.Ioo`, and `Set.Icc`. First, we prove lemmas in terms of filter equalities. Then we prove lemmas about `s ∈ 𝓝ˢ t`, where both `s` and `t` are intervals. Finally, we prove a few lemmas about filter bases of `𝓝ˢ (Iic a)` and `𝓝ˢ (Ici a)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/NhdsSet.html"}, {"id": "Mathlib.Topology.Compactness.LocallyFinite", "region_id": "topology", "micro_elevation": 0.3721, "macro_tier": 4, "macro_tier_score": 0.3454, "macro_tier_override": null, "x": -126.922, "z": 179.737, "size": 0.3706, "title": "Compact sets and compact spaces and locally finite functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactness/LocallyFinite.html"}, {"id": "Mathlib.Topology.Homotopy.TopCat.ZerothHomotopy", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -45.484, "z": 167.303, "size": 0.239, "title": "`ZerothHomotopy` and connected components of `TopCat.toSSet.obj X`", "summary": "In this file, given `X : TopCat`, we define a bijection `TopCat.zerothHomotopyEquiv` between `ZerothHomotopy X` and `(TopCat.toSSet.obj X).π₀`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/TopCat/ZerothHomotopy.html"}, {"id": "Mathlib.Topology.Algebra.Valued.LocallyCompact", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -42.523, "z": 185.757, "size": 0.2338, "title": "Necessary and sufficient conditions for a locally compact valued field", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Valued/LocallyCompact.html"}, {"id": "Mathlib.Topology.Sheaves.Module", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -141.453, "z": 162.167, "size": 0.2263, "title": "Specialized results for sheaves of modules over topological spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Module.html"}, {"id": "Mathlib.Topology.Algebra.Module.TopDualPairing", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -141.365, "z": 138.829, "size": 0.2338, "title": "Continuous Perfect Pairing for `topDualPairing`", "summary": "This file proves that `topDualPairing 𝕜 E` is a continuous perfect pairing when `E` is a finite-dimensional Hausdorff space over a complete nontrivially normed field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/TopDualPairing.html"}, {"id": "Mathlib.Topology.Algebra.Module.WeakDual", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -152.989, "z": 156.624, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/WeakDual.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.GroupCompletion", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -154.899, "z": 165.63, "size": 0.236, "title": "Infinite sums in the completion of a topological group", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/GroupCompletion.html"}, {"id": "Mathlib.Topology.UniformSpace.Ultra.Completion", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -57.772, "z": 216.493, "size": 0.2, "title": "Completions of ultrametric (nonarchimedean) uniform spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Ultra/Completion.html"}, {"id": "Mathlib.Topology.ExtendFrom", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -73.703, "z": 168.01, "size": 0.2591, "title": "Extending a function from a subset", "summary": "The main definition of this file is `extendFrom A f` where `f : X → Y` and `A : Set X`. This defines a new function `g : X → Y` which maps any `x₀ : X` to the limit of `f` as `x` tends to `x₀`, if such a limit exists. This is analogous to the way `IsDenseInducing.extend` \"extends\" a function `f : X → Z` to a function `g : Y → Z` along a dense inducing `i : X → Y`. The main theorem we prove about this definition is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ExtendFrom.html"}, {"id": "Mathlib.Topology.MetricSpace.Infsep", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -94.479, "z": 137.862, "size": 0.2, "title": "Infimum separation", "summary": "This file defines the extended infimum separation of a set. This is approximately dual to the diameter of a set, but where the extended diameter of a set is the supremum of the extended distance between elements of the set, the extended infimum separation is the infimum of the (extended) distance between *distinct* elements in the set. We also define the infimum separation as the cast of the extended infimum…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Infsep.html"}, {"id": "Mathlib.Topology.Category.LightProfinite.Limits", "region_id": "topology", "micro_elevation": 0.9767, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -167.03, "z": 213.28, "size": 0.2954, "title": "Explicit limits and colimits", "summary": "This file applies the general API for explicit limits and colimits in `CompHausLike P` (see the file `Mathlib/Topology/Category/CompHausLike/Limits.lean`) to the special case of `LightProfinite`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/LightProfinite/Limits.html"}, {"id": "Mathlib.Topology.Ultrafilter", "region_id": "topology", "micro_elevation": 0.1628, "macro_tier": 4, "macro_tier_score": 0.3707, "macro_tier_override": null, "x": -100.252, "z": 181.75, "size": 0.3283, "title": "Characterization of basic topological properties in terms of ultrafilters", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Ultrafilter.html"}, {"id": "Mathlib.Topology.Category.UniformSpace", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -118.779, "z": 140.273, "size": 0.2, "title": "The category of uniform spaces", "summary": "We construct the category of uniform spaces, show that the complete separated uniform spaces form a reflective subcategory, and hence possess all limits that uniform spaces do. TODO: show that uniform spaces actually have all limits!", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/UniformSpace.html"}, {"id": "Mathlib.Topology.Order.T5", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 3, "macro_tier_score": 0.0846, "macro_tier_override": null, "x": -109.524, "z": 152.802, "size": 0.3489, "title": "Linear order is a completely normal Hausdorff topological space", "summary": "In this file we prove that a linear order with order topology is a completely normal Hausdorff topological space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/T5.html"}, {"id": "Mathlib.Topology.Sheaves.Init", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2844, "title": "Rule sets related to topological (pre)sheaves", "summary": "This module defines the `Restrict` Aesop rule set. Aesop rule sets only become visible once the file in which they're declared is imported, so we must put this declaration into its own file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Init.html"}, {"id": "Mathlib.Topology.Algebra.MvPolynomial", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -146.228, "z": 225.258, "size": 0.2, "title": "Multivariate polynomials and continuity", "summary": "In this file we prove the following lemma: * `MvPolynomial.continuous_eval`: `MvPolynomial.eval` is continuous", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/MvPolynomial.html"}, {"id": "Mathlib.Topology.ContinuousMap.Defs", "region_id": "topology", "micro_elevation": 0.0233, "macro_tier": 4, "macro_tier_score": 0.449, "macro_tier_override": null, "x": -105.077, "z": 190.128, "size": 0.3422, "title": "Continuous bundled maps", "summary": "In this file we define the type `ContinuousMap` of continuous bundled maps. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/Defs.html"}, {"id": "Mathlib.Topology.Algebra.Module.IsWeak", "region_id": "topology", "micro_elevation": 0.8837, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -94.348, "z": 133.107, "size": 0.2, "title": "Weak topologies on modules", "summary": "Given a bilinear form `B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜`, the weak topology on `E` is the coarsest topology such that for all `y : F` every map `(B · y)` is continuous; equivalently, it is the topology on `E` induced by the map `(B · · : E → (F → 𝕜))`. This file defines a `Prop`-valued typeclass `LinearMap.IsWeak` expressing that an existing topology on `E` is the weak topology. Although this could be passed around explicitly…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/IsWeak.html"}, {"id": "Mathlib.Topology.Hom.ContinuousEval", "region_id": "topology", "micro_elevation": 0.3256, "macro_tier": 4, "macro_tier_score": 0.1259, "macro_tier_override": null, "x": -84.127, "z": 184.639, "size": 0.3249, "title": "Bundled maps with evaluation continuous in both variables", "summary": "In this file we define a class `ContinuousEval F X Y` saying that `F` is a bundled morphism class (in the sense of `FunLike`) with a topology such that `fun (f, x) : F × X ↦ f x` is a continuous function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Hom/ContinuousEval.html"}, {"id": "Mathlib.Topology.Maps.Proper.UniversallyClosed", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -143.444, "z": 170.776, "size": 0.2, "title": "A map is proper iff it is continuous and universally closed", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Maps/Proper/UniversallyClosed.html"}, {"id": "Mathlib.Topology.Order.ExtendFrom", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -126.714, "z": 157.301, "size": 0.2534, "title": "Lemmas about `extendFrom` in an order topology.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/ExtendFrom.html"}, {"id": "Mathlib.Topology.Instances.AddCircle.Real", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -42.868, "z": 208.319, "size": 0.2533, "title": "The additive circle over `ℝ`", "summary": "Results specific to the additive circle over `ℝ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Instances/AddCircle/Real.html"}, {"id": "Mathlib.Topology.Sheaves.Points", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -116.636, "z": 233.937, "size": 0.2, "title": "The standard conservative family of points for the site attached to a topological space", "summary": "If `X` is a topological space, any `x : X` defines a point of the site attached to `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/Points.html"}, {"id": "Mathlib.Topology.PreorderRestrict", "region_id": "topology", "micro_elevation": 0.3023, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -125.219, "z": 192.882, "size": 0.2, "title": "Continuity of the restriction function for functions indexed by a preorder", "summary": "We prove that the map which restricts a function `f : (i : α) → X i` to elements `≤ a` is continuous.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/PreorderRestrict.html"}, {"id": "Mathlib.Topology.Metrizable.ContinuousMap", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -150.686, "z": 218.866, "size": 0.2, "title": "Metrizability of `C(X, Y)`", "summary": "If `X` is a weakly locally compact σ-compact space and `Y` is a (pseudo)metrizable space, then `C(X, Y)` is a (pseudo)metrizable space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Metrizable/ContinuousMap.html"}, {"id": "Mathlib.Topology.Convenient.Localization", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -144.693, "z": 183.063, "size": 0.2, "title": "The category of `X`-generated spaces, as a localization", "summary": "Let `X i` be a family of topological spaces. In this file, we introduce a property of morphisms `morphismPropertyWithGeneratedByTopologyEquiv X` in the category `TopCat`: it consists of the morphisms corresponding to the canonical continuous maps `WithGeneratedByTopology X Z → Z` for all topological spaces `Z`. We show that the functor `TopCat.toContinuousGeneratedByCat X : TopCat ⥤ ContinuousGeneratedByCat X` makes…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Convenient/Localization.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.Field", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2687, "title": "Infinite sums and products in topological fields", "summary": "Lemmas on topological sums in rings with a strictly multiplicative norm, of which normed fields are the most familiar examples.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/Field.html"}, {"id": "Mathlib.Topology.SmallInductiveDimension", "region_id": "topology", "micro_elevation": 0.3721, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -81.867, "z": 201.575, "size": 0.2, "title": "Small inductive dimension", "summary": "The small inductive dimension of a space is inductively defined as follows. Empty spaces have small inductive dimension less than 0, and a topological space has dimension less than `n + 1` if it has a topological basis whose elements have frontiers of dimension strictly less `n`. In this file we formalize this notion, and characterize the cases `n = 0` and `n = 1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/SmallInductiveDimension.html"}, {"id": "Mathlib.Topology.Category.TopCat.Limits.Cofiltered", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -92.088, "z": 228.687, "size": 0.2557, "title": "Cofiltered limits in the category of topological spaces", "summary": "Given a *compatible* collection of topological bases for the factors in a cofiltered limit which contain `Set.univ` and are closed under intersections, the induced *naive* collection of sets in the limit is, in fact, a topological basis.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/TopCat/Limits/Cofiltered.html"}, {"id": "Mathlib.Topology.Algebra.Module.FiniteDimensionBilinear", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -78.546, "z": 133.12, "size": 0.2416, "title": "Building continuous bilinear maps in finite dimensions over complete fields", "summary": "Given a complete nontrivially normed field `𝕜` and finite dimensional T₂ topological vector spaces over `𝕜`, this file builds a continuous bilinear map from any bilinear function. This applies in particular to evaluation of linear maps between such spaces. Working with topological vector spaces instead of normed spaces is important for applications in the differential geometry part of Mathlib where we don’t want to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/FiniteDimensionBilinear.html"}, {"id": "Mathlib.Topology.Compactness.NhdsKer", "region_id": "topology", "micro_elevation": 0.3953, "macro_tier": 3, "macro_tier_score": 0.0527, "macro_tier_override": null, "x": -97.334, "z": 217.215, "size": 0.3089, "title": "Compactness of the neighborhoods kernel of a set", "summary": "In this file we prove that the neighborhoods kernel of a set (defined as the intersection of all of its neighborhoods) is a compact set if and only if the original set is a compact set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactness/NhdsKer.html"}, {"id": "Mathlib.Topology.Order.LeftRight", "region_id": "topology", "micro_elevation": 0.3488, "macro_tier": 4, "macro_tier_score": 0.3307, "macro_tier_override": null, "x": -104.375, "z": 168.195, "size": 0.4147, "title": "Left and right continuity", "summary": "In this file we prove a few lemmas about left and right continuous functions: * `continuousWithinAt_Ioi_iff_Ici`: two definitions of right continuity (with `(a, ∞)` and with `[a, ∞)`) are equivalent; * `continuousWithinAt_Iio_iff_Iic`: two definitions of left continuity (with `(-∞, a)` and with `(-∞, a]`) are equivalent; * `continuousAt_iff_continuous_left_right`, `continuousAt_iff_continuous_left'_right'` : a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/LeftRight.html"}, {"id": "Mathlib.Topology.NatEmbedding", "region_id": "topology", "micro_elevation": 0.6279, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -109.213, "z": 233.751, "size": 0.2, "title": "Infinite Hausdorff topological spaces", "summary": "In this file we prove several properties of infinite Hausdorff topological spaces. - `exists_seq_infinite_isOpen_pairwise_disjoint`: there exists a sequence of pairwise disjoint infinite open sets; - `exists_topology_isEmbedding_nat`: there exists a topological embedding of `ℕ` into the space; - `exists_infinite_discreteTopology`: there exists an infinite subset with discrete topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/NatEmbedding.html"}, {"id": "Mathlib.Topology.Algebra.IsOpenUnits", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -151.017, "z": 231.731, "size": 0.2, "title": "Topological monoids with open units", "summary": "We say that a topological monoid `M` has open units (`IsOpenUnits`) if `Mˣ` is open in `M` and has the subspace topology (i.e. inverse is continuous). Typical examples include monoids with discrete topology, topological groups (or fields), and rings `R` equipped with the `I`-adic topology where `I ≤ J(R)` (`IsOpenUnits.of_isAdic`). A non-example is `𝔸ₖ`, because the topology on ideles is not the induced topology…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/IsOpenUnits.html"}, {"id": "Mathlib.Topology.Compactness.CountablyCompact", "region_id": "topology", "micro_elevation": 0.7674, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -117.408, "z": 241.832, "size": 0.2, "title": "Countably compact sets", "summary": "A set `A` in a topological space is **countably compact** if every countably generated proper filter contained in `A` has a cluster point in `A`. Equivalently, every sequence in `A` has a cluster point in `A`, and every countable open cover of `A` admits a finite subcover.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Compactness/CountablyCompact.html"}, {"id": "Mathlib.Topology.Algebra.IntermediateField", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -93.822, "z": 245.373, "size": 0.2, "title": "Continuous actions related to intermediate fields", "summary": "In this file we define the instances related to continuous actions of intermediate fields. The topology on intermediate fields is already defined in earlier file `Mathlib/Topology/Algebra/Field.lean` as the subspace topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/IntermediateField.html"}, {"id": "Mathlib.Topology.Algebra.Group.Units", "region_id": "topology", "micro_elevation": 0.7209, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -96.972, "z": 239.587, "size": 0.2, "title": "Topological properties of units", "summary": "This file contains lemmas about the topology of units in topological monoids, including results about submonoid units and units of product spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Group/Units.html"}, {"id": "Mathlib.Topology.List", "region_id": "topology", "micro_elevation": 0.3023, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -88.969, "z": 178.994, "size": 0.2, "title": "Topology on lists and vectors", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/List.html"}, {"id": "Mathlib.Topology.MetricSpace.HolderNorm", "region_id": "topology", "micro_elevation": 0.0233, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -103.831, "z": 190.531, "size": 0.2, "title": "Hölder norm", "summary": "This file defines the Hölder (semi-)norm for Hölder functions alongside some basic properties. The `r`-Hölder norm of a function `f : X → Y` between two metric spaces is the least non-negative real number `C` for which `f` is `r`-Hölder continuous with constant `C`, i.e. it is the least `C` for which `WithHolder C r f` is true.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/HolderNorm.html"}, {"id": "Mathlib.Topology.Homotopy.LocallyContractible", "region_id": "topology", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -69.442, "z": 134.419, "size": 0.2, "title": "Strongly locally contractible spaces", "summary": "This file defines `LocallyContractibleSpace` and `StronglyLocallyContractibleSpace`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Homotopy/LocallyContractible.html"}, {"id": "Mathlib.Topology.Separation.DisjointCover", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -93.828, "z": 152.492, "size": 0.2338, "title": "Disjoint covers of profinite spaces", "summary": "We prove various results about covering profinite spaces by disjoint clopens, including * `TopologicalSpace.IsOpenCover.exists_finite_nonempty_disjoint_clopen_cover`: any open cover of a profinite space can be refined to a finite cover by pairwise disjoint nonempty clopens. * `ContinuousMap.exists_finite_approximation_of_mem_nhds_diagonal`: if `f : X → V` is continuous with `X` profinite, and `S` is a neighbourhood…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/DisjointCover.html"}, {"id": "Mathlib.Topology.MetricSpace.HausdorffAlexandroff", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -167.259, "z": 206.99, "size": 0.2, "title": "Hausdorff–Alexandroff Theorem", "summary": "In this file, we prove the Hausdorff–Alexandroff theorem, which states that every nonempty compact metric space is a continuous image of the Cantor set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/HausdorffAlexandroff.html"}, {"id": "Mathlib.Topology.ContinuousMap.SecondCountableSpace", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -145.586, "z": 193.234, "size": 0.2, "title": "Second countable topology on `C(X, Y)`", "summary": "In this file we prove that `C(X, Y)` with compact-open topology has second countable topology, if - both `X` and `Y` have second countable topology; - `X` is a locally compact space;", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/SecondCountableSpace.html"}, {"id": "Mathlib.Topology.Order.Priestley", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -82.812, "z": 224.02, "size": 0.2, "title": "Priestley spaces", "summary": "This file defines Priestley spaces. A Priestley space is an ordered compact topological space such that any two distinct points can be separated by a clopen upper set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Priestley.html"}, {"id": "Mathlib.Topology.UniformSpace.ProdApproximation", "region_id": "topology", "micro_elevation": 0.8605, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -66.964, "z": 147.867, "size": 0.2, "title": "Uniform approximation by products", "summary": "We show that if `X, Y` are compact Hausdorff spaces with `X` profinite, then any continuous function on `X × Y` valued in a ring (with a uniform structure) can be uniformly approximated by finite sums of functions of the form `f x * g y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/ProdApproximation.html"}, {"id": "Mathlib.Topology.CWComplex.Classical.Graph", "region_id": "topology", "micro_elevation": 0.0465, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -101.784, "z": 191.295, "size": 0.2, "title": "1-skeletons of CW complexes as graphs", "summary": "In this file we define the 1-skeleton of a CW complex as a graph.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/CWComplex/Classical/Graph.html"}, {"id": "Mathlib.Topology.MetricSpace.Ultra.TotallySeparated", "region_id": "topology", "micro_elevation": 0.6512, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -85.134, "z": 230.836, "size": 0.2, "title": "Ultrametric spaces are totally separated", "summary": "In a metric space with an ultrametric, the space is totally separated, hence totally disconnected.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Ultra/TotallySeparated.html"}, {"id": "Mathlib.Topology.UniformSpace.Uniformizable", "region_id": "topology", "micro_elevation": 0.9302, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -100.433, "z": 254.175, "size": 0.2, "title": "Uniformizable Spaces", "summary": "A topological space is uniformizable (there exists a uniformity that generates the same topology) iff it is completely regular.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/UniformSpace/Uniformizable.html"}, {"id": "Mathlib.Topology.Algebra.Module.StrongTopology", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -137.948, "z": 136.627, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Module/StrongTopology.html"}, {"id": "Mathlib.Topology.Algebra.Nonarchimedean.TotallyDisconnected", "region_id": "topology", "micro_elevation": 0.8372, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -150.879, "z": 159.055, "size": 0.2, "title": "Total separatedness of nonarchimedean groups", "summary": "In this file, we prove that a nonarchimedean group is a totally separated topological space. The fact that a nonarchimedean group is a totally disconnected topological space is implied by the fact that a nonarchimedean group is totally separated.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Nonarchimedean/TotallyDisconnected.html"}, {"id": "Mathlib.Topology.Algebra.ProperAction.ProperlyDiscontinuous", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -155.718, "z": 230.934, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/ProperAction/ProperlyDiscontinuous.html"}, {"id": "Mathlib.Topology.CWComplex.Abstract.Basic", "region_id": "topology", "micro_elevation": 0.6047, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -78.43, "z": 222.637, "size": 0.2, "title": "CW-complexes", "summary": "This file defines (relative) CW-complexes using a categorical approach.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/CWComplex/Abstract/Basic.html"}, {"id": "Mathlib.Topology.Category.Stonean.Adjunctions", "region_id": "topology", "micro_elevation": 0.9535, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -112.69, "z": 127.942, "size": 0.2, "title": "Adjunctions involving the category of Stonean spaces", "summary": "This file constructs the left adjoint `typeToStonean` to the forgetful functor from Stonean spaces to sets, using the Stone-Cech compactification. This allows to conclude that the monomorphisms in `Stonean` are precisely the injective maps (see `Stonean.mono_iff_injective`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Category/Stonean/Adjunctions.html"}, {"id": "Mathlib.Topology.ContinuousMap.BoundedCompactlySupported", "region_id": "topology", "micro_elevation": 0.907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -60.24, "z": 233.366, "size": 0.2, "title": "Compactly supported bounded continuous functions", "summary": "The two-sided ideal of compactly supported bounded continuous functions taking values in a metric space, with the uniform distance.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.html"}, {"id": "Mathlib.Topology.ContinuousMap.LocallyConvex", "region_id": "topology", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -37.745, "z": 186.479, "size": 0.2, "title": "The space of continuous maps is a locally convex space", "summary": "In this file we prove that the space of continuous maps from a topological space to a locally convex topological vector space is a locally convex topological vector space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/LocallyConvex.html"}, {"id": "Mathlib.Topology.ContinuousMap.T0Sierpinski", "region_id": "topology", "micro_elevation": 0.5349, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -68.945, "z": 189.331, "size": 0.2, "title": "Any T0 space embeds in a product of copies of the Sierpinski space.", "summary": "We consider `Prop` with the Sierpinski topology. If `X` is a topological space, there is a continuous map `productOfMemOpens` from `X` to `Opens X → Prop` which is the product of the maps `X → Prop` given by `x ↦ x ∈ u`. The map `productOfMemOpens` is always inducing. Whenever `X` is T0, `productOfMemOpens` is also injective and therefore an embedding.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/ContinuousMap/T0Sierpinski.html"}, {"id": "Mathlib.Topology.MetricSpace.BundledFun", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2, "title": "Pseudometrics as bundled functions", "summary": "This file defines a pseudometric as a bundled function. This allows one to define a semilattice on them, and to construct families of pseudometrics.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/BundledFun.html"}, {"id": "Mathlib.Topology.MetricSpace.Ultra.Pi", "region_id": "topology", "micro_elevation": 0.7907, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -128.633, "z": 144.015, "size": 0.2, "title": "Ultrametric distances on pi types", "summary": "This file contains results on the behavior of ultrametrics in products of ultrametric spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/MetricSpace/Ultra/Pi.html"}, {"id": "Mathlib.Topology.Order.HullKernel", "region_id": "topology", "micro_elevation": 0.5581, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -106.785, "z": 154.141, "size": 0.2, "title": "Hull-Kernel Topology", "summary": "Let `α` be a `CompleteLattice` and let `T` be a subset of `α`. The pair of maps `S → sInf (Subtype.val '' S)` and `a → T ↓∩ Ici a` are often referred to as the `kernel` and the `hull` respectively. They form an antitone Galois connection between the subsets of `T` and `α`. When `α` can be generated from `T` by taking infs, this becomes a Galois insertion and the relative topology (`Topology.lower`) on `T` takes on a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/HullKernel.html"}, {"id": "Mathlib.Topology.Separation.NotNormal", "region_id": "topology", "micro_elevation": 0.5814, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -75.981, "z": 165.275, "size": 0.2, "title": "Not normal topological spaces", "summary": "In this file we prove (see `IsClosed.not_normal_of_continuum_le_mk`) that a separable space with a discrete subspace of cardinality continuum is not a normal topological space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Separation/NotNormal.html"}, {"id": "Mathlib.Topology.Sets.BaseChangeNhds", "region_id": "topology", "micro_elevation": 0.6977, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -61.107, "z": 208.734, "size": 0.2, "title": "Base changes among different families of neighbourhoods", "summary": "This file builds base changes for `.compactsInside`, `openNhds`. It also contains the evidences that `openRcNhds_to_openNhds`and `openRcNhds_to_compactNhds` are initials functors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sets/BaseChangeNhds.html"}, {"id": "Mathlib.Topology.Sheaves.EtaleSpace", "region_id": "topology", "micro_elevation": 0.814, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -147.615, "z": 226.034, "size": 0.2, "title": "Etale space of a presheaf", "summary": "Given a presheaf `F` on a topological space `X`, its *etale space* is the space of pairs `(base, germ)`, where `base` is a point of `X`, and `germ` is an element of the stalk of `F` at `base`. This space is equipped with the following topology. For each open set `U` and a section `s` of `F` over `U`, the set of germs of `s` at points `x ∈ U` is an open set in the etale space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Sheaves/EtaleSpace.html"}, {"id": "Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt", "region_id": "topology", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -104.893, "z": 191.683, "size": 0.2551, "title": "Sums over symmetric integer intervals", "summary": "This file contains some lemmas about sums over symmetric integer intervals `Ixx -N N` used, for example in the definition of the Eisenstein series `E2`. In particular we define `symmetricIcc`, `symmetricIco`, `symmetricIoc` and `symmetricIoo` as `SummationFilter`s corresponding to the intervals `Icc -N N`, `Ico -N N`, `Ioc -N N` respectively. We also prove that these filters are all `NeBot` and `LeAtTop`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/InfiniteSum/ConditionalInt.html"}, {"id": "Mathlib.Topology.DiscreteSubset", "region_id": "topology", "micro_elevation": 0.4186, "macro_tier": 4, "macro_tier_score": 0.3521, "macro_tier_override": null, "x": -131.815, "z": 200.05, "size": 0.4396, "title": "Discrete subsets of topological spaces", "summary": "This file contains various additional properties of discrete subsets of topological spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/DiscreteSubset.html"}, {"id": "Mathlib.Topology.Algebra.Order.Support", "region_id": "topology", "micro_elevation": 0.5116, "macro_tier": 1, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -83.907, "z": 219.014, "size": 0.2531, "title": "The topological support of sup and inf of functions", "summary": "In a topological space `X` and a space `M` with `Sup` structure, for `f g : X → M` with compact support, we show that `f ⊔ g` has compact support. Similarly, in `β` with `Inf` structure, `f ⊓ g` has compact support if so do `f` and `g`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Algebra/Order/Support.html"}, {"id": "Mathlib.Data.Sign.Defs", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0217, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3282, "title": "Sign type", "summary": "This file defines the type of signs $\\{-1, 0, 1\\}$ and its basic arithmetic instances.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sign/Defs.html"}, {"id": "Mathlib.Data.Multiset.Fintype", "region_id": "foundations_data", "micro_elevation": 0.6765, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 169.825, "z": 50.109, "size": 0.2901, "title": "Multiset coercion to type", "summary": "This module defines a `CoeSort` instance for multisets and gives it a `Fintype` instance. It also defines `Multiset.toEnumFinset`, which is another way to enumerate the elements of a multiset. These coercions and definitions make it easier to sum over multisets using existing `Finset` theory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Fintype.html"}, {"id": "Mathlib.Data.Fintype.Card", "region_id": "foundations_data", "micro_elevation": 0.6471, "macro_tier": 2, "macro_tier_score": 0.2436, "macro_tier_override": null, "x": 250.896, "z": 61.051, "size": 0.7101, "title": "Cardinalities of finite types", "summary": "This file defines the cardinality `Fintype.card α` as the number of elements in `(univ : Finset α)`. We also include some elementary results on the values of `Fintype.card` on specific types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Card.html"}, {"id": "Mathlib.Data.Nat.Totient", "region_id": "foundations_data", "micro_elevation": 0.8824, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": 216.941, "z": -17.814, "size": 0.3118, "title": "Euler's totient function", "summary": "This file defines [Euler's totient function](https://en.wikipedia.org/wiki/Euler's_totient_function) `Nat.totient n` which counts the number of naturals less than `n` that are coprime with `n`. We prove the divisor sum formula, namely that `n` equals `φ` summed over the divisors of `n`. See `sum_totient`. We also prove two lemmas to help compute totients, namely `totient_mul` and `totient_prime_pow`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Totient.html"}, {"id": "Mathlib.Data.Nat.Cast.Field", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0111, "macro_tier_override": null, "x": 211.795, "z": 39.386, "size": 0.3114, "title": "Cast of naturals into fields", "summary": "This file concerns the canonical homomorphism `ℕ → F`, where `F` is a field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Cast/Field.html"}, {"id": "Mathlib.Data.Nat.Factorization.Basic", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 2, "macro_tier_score": 0.0225, "macro_tier_override": null, "x": 269.125, "z": 31.541, "size": 0.375, "title": "Basic lemmas on prime factorizations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Factorization/Basic.html"}, {"id": "Mathlib.Data.Nat.Factorization.Induction", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 1, "macro_tier_score": 0.0108, "macro_tier_override": null, "x": 265.239, "z": 62.657, "size": 0.2815, "title": "Induction principles involving factorizations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Factorization/Induction.html"}, {"id": "Mathlib.Data.Nat.Periodic", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 224.01, "z": -9.03, "size": 0.2465, "title": "Periodic Functions on ℕ", "summary": "This file identifies a few functions on `ℕ` which are periodic, and also proves a lemma about periodic predicates which helps determine their cardinality when filtering intervals over them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Periodic.html"}, {"id": "Mathlib.Data.Finset.Dedup", "region_id": "foundations_data", "micro_elevation": 0.4412, "macro_tier": 2, "macro_tier_score": 0.2827, "macro_tier_override": null, "x": 209.522, "z": 69.277, "size": 0.3596, "title": "Deduplicating Multisets to make Finsets", "summary": "This file concerns `Multiset.dedup` and `List.dedup` as a way to create `Finset`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Dedup.html"}, {"id": "Mathlib.Data.Finset.Defs", "region_id": "foundations_data", "micro_elevation": 0.2353, "macro_tier": 2, "macro_tier_score": 0.2874, "macro_tier_override": null, "x": 208.254, "z": 25.738, "size": 0.5328, "title": "Finite sets", "summary": "Terms of type `Finset α` are one way of talking about finite subsets of `α` in Mathlib. Below, `Finset α` is defined as a structure with 2 fields: 1. `val` is a `Multiset α` of elements; 2. `nodup` is a proof that `val` has no duplicates. Finsets in Lean are constructive in that they have an underlying `List` that enumerates their elements. In particular, any function that uses the data of the underlying list cannot…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Defs.html"}, {"id": "Mathlib.Data.Multiset.Dedup", "region_id": "foundations_data", "micro_elevation": 0.4118, "macro_tier": 2, "macro_tier_score": 0.2869, "macro_tier_override": null, "x": 236.959, "z": 26.673, "size": 0.5186, "title": "Erasing duplicates in a multiset.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Dedup.html"}, {"id": "Mathlib.Data.Multiset.Basic", "region_id": "foundations_data", "micro_elevation": 0.2353, "macro_tier": 2, "macro_tier_score": 0.2822, "macro_tier_override": null, "x": 207.364, "z": 26.085, "size": 0.334, "title": "Basic results on multisets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Basic.html"}, {"id": "Mathlib.Data.ENNReal.Inv", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 2, "macro_tier_score": 0.0328, "macro_tier_override": null, "x": 216.498, "z": 47.546, "size": 0.3661, "title": "Results about division in extended non-negative reals", "summary": "This file establishes basic properties related to the inversion and division operations on `ℝ≥0∞`. For instance, as a consequence of being a `DivInvOneMonoid`, `ℝ≥0∞` inherits a power operation with integer exponent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ENNReal/Inv.html"}, {"id": "Mathlib.Data.Real.Basic", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 2, "macro_tier_score": 0.0346, "macro_tier_override": null, "x": 223.016, "z": 47.073, "size": 0.4455, "title": "Real numbers from Cauchy sequences", "summary": "This file defines `ℝ` as the type of equivalence classes of Cauchy sequences of rational numbers. This choice is motivated by how easy it is to prove that `ℝ` is a commutative ring, by simply lifting everything to `ℚ`. The facts that the real numbers are an Archimedean floor ring, and a conditionally complete linear order, have been deferred to the file `Mathlib/Data/Real/Archimedean.lean`, in order to keep the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Real/Basic.html"}, {"id": "Mathlib.Data.Set.UnionLift", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": 210.085, "z": 42.321, "size": 0.4896, "title": "Union lift", "summary": "This file defines `Set.iUnionLift` to glue together functions defined on each of a collection of sets to make a function on the Union of those sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/UnionLift.html"}, {"id": "Mathlib.Data.PNat.Xgcd", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 216.976, "z": 53.533, "size": 0.2, "title": "Euclidean algorithm for ℕ", "summary": "This file sets up a version of the Euclidean algorithm that only works with natural numbers. Given `0 < a, b`, it computes the unique `(w, x, y, z, d)` such that the following identities hold: * `a = (w + x) d` * `b = (y + z) d` * `w * z = x * y + 1` `d` is then the gcd of `a` and `b`, and `a' := a / d = w + x` and `b' := b / d = y + z` are coprime. This story is closely related to the structure of SL₂(ℕ) (as a free…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PNat/Xgcd.html"}, {"id": "Mathlib.Data.PNat.Prime", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 205.603, "z": 31.771, "size": 0.2764, "title": "Primality and GCD on pnat", "summary": "This file extends the theory of `ℕ+` with `gcd`, `lcm` and `Prime` functions, analogous to those on `Nat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PNat/Prime.html"}, {"id": "Mathlib.Data.Nat.BitIndices", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.419, "z": 26.785, "size": 0.2342, "title": "Bit Indices", "summary": "Given `n : ℕ`, we define `Nat.bitIndices n`, which is the `List` of indices of `1`s in the binary expansion of `n`. If `s : Finset ℕ` and `n = ∑ i ∈ s, 2 ^ i`, then `Nat.bitIndices n` is the sorted list of elements of `s`. The lemma `sum_map_two_pow_bitIndices` proves that summing `2 ^ i` over this list gives `n`. This is used in `Combinatorics.colex` to construct a bijection `equivBitIndices : ℕ ≃ Finset ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/BitIndices.html"}, {"id": "Mathlib.Data.List.Sort", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 2, "macro_tier_score": 0.2555, "macro_tier_override": null, "x": 202.116, "z": 37.68, "size": 0.5154, "title": "Sorting algorithms on lists", "summary": "In this file we define the sorting algorithm `List.insertionSort r` and prove that we have `(l.insertionSort r l).Pairwise r` under suitable conditions on `r`. We then define `List.SortedLE`, `List.SortedGE`, `List.SortedLT` and `List.SortedGT`, predicates which are equivalent to `List.Pairwise` when the relation derives from a preorder (but which are defined in terms of the monotonicity predicates).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Sort.html"}, {"id": "Mathlib.Data.Finset.Update", "region_id": "foundations_data", "micro_elevation": 0.6471, "macro_tier": 1, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 172.421, "z": 28.379, "size": 0.346, "title": "Update a function on a set of values", "summary": "This file defines `Function.updateFinset`, the operation that updates a function on a (finite) set of values. This is a very specific function used for `MeasureTheory.marginal`, and possibly not that useful for other purposes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Update.html"}, {"id": "Mathlib.Data.Prod.TProd", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": 223.857, "z": 45.669, "size": 0.3155, "title": "Finite products of types", "summary": "This file defines the product of types over a list. For `l : List ι` and `α : ι → Type v` we define `List.TProd α l = l.foldr (fun i β ↦ α i × β) PUnit`. This type should not be used if `∀ i, α i` or `∀ i ∈ l, α i` can be used instead (in the last expression, we could also replace the list `l` by a set or a finset). This type is used as an intermediary between binary products and finitary products. The application…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Prod/TProd.html"}, {"id": "Mathlib.Data.Setoid.Basic", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1609, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.4907, "title": "Equivalence relations", "summary": "This file defines the complete lattice of equivalence relations on a type, results about the inductively defined equivalence closure of a binary relation, and the analogues of some isomorphism theorems for quotients of arbitrary types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Setoid/Basic.html"}, {"id": "Mathlib.Data.Finset.Grade", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 1, "macro_tier_score": 0.0108, "macro_tier_override": null, "x": 184.004, "z": 79.035, "size": 0.2828, "title": "Finsets and multisets form a graded order", "summary": "This file characterises atoms, coatoms and the covering relation in finsets and multisets. It also proves that they form a `ℕ`-graded order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Grade.html"}, {"id": "Mathlib.Data.Finset.Sups", "region_id": "foundations_data", "micro_elevation": 0.8235, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 263.837, "z": 60.974, "size": 0.2798, "title": "Set family operations", "summary": "This file defines a few binary operations on `Finset α` for use in set family combinatorics.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Sups.html"}, {"id": "Mathlib.Data.PNat.Notation", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.1254, "macro_tier_override": null, "x": 215.216, "z": 39.559, "size": 0.2804, "title": "Definition and notation for positive natural numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PNat/Notation.html"}, {"id": "Mathlib.Data.Nat.Notation", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.5408, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 1.0567, "title": "Notation `ℕ` for the natural numbers.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Notation.html"}, {"id": "Mathlib.Data.Real.ENatENNReal", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 220.263, "z": 30.919, "size": 0.352, "title": "Coercion from `ℕ∞` to `ℝ≥0∞`", "summary": "In this file we define a coercion from `ℕ∞` to `ℝ≥0∞` and prove some basic lemmas about this map.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Real/ENatENNReal.html"}, {"id": "Mathlib.Data.ENat.Basic", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 2, "macro_tier_score": 0.0376, "macro_tier_override": null, "x": 212.908, "z": 30.688, "size": 0.5391, "title": "Definition and basic properties of extended natural numbers", "summary": "In this file we define `ENat` (notation: `ℕ∞`) to be `WithTop ℕ` and prove some basic lemmas about this type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ENat/Basic.html"}, {"id": "Mathlib.Data.ENNReal.Basic", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.0344, "macro_tier_override": null, "x": 215.397, "z": 40.53, "size": 0.4385, "title": "Extended non-negative reals", "summary": "We define `ENNReal = ℝ≥0∞ := WithTop ℝ≥0` to be the type of extended nonnegative real numbers, i.e., the interval `[0, +∞]`. This type is used as the codomain of a `MeasureTheory.Measure`, and of the extended distance `edist` in an `EMetricSpace`. In this file we set up many of the instances on `ℝ≥0∞`, and provide relationships between `ℝ≥0∞` and `ℝ≥0`, and between `ℝ≥0∞` and `ℝ`. In particular, we provide a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ENNReal/Basic.html"}, {"id": "Mathlib.Data.Sigma.Basic", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0533, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3508, "title": "Sigma types", "summary": "This file proves basic results about sigma types. A sigma type is a dependent pair type. Like `α × β` but where the type of the second component depends on the first component. More precisely, given `β : ι → Type*`, `Sigma β` is made of stuff which is of type `β i` for some `i : ι`, so the sigma type is a disjoint union of types. For example, the sum type `X ⊕ Y` can be emulated using a sigma type, by taking `ι`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sigma/Basic.html"}, {"id": "Mathlib.Data.Set.Lattice", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.1432, "macro_tier_override": null, "x": 211.518, "z": 40.496, "size": 0.9181, "title": "The set lattice", "summary": "This file is a collection of results on the complete atomic Boolean algebra structure of `Set α`. Notation for the complete lattice operations can be found in `Mathlib/Order/SetNotation.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Lattice.html"}, {"id": "Mathlib.Data.Nat.Factorization.Defs", "region_id": "foundations_data", "micro_elevation": 0.8235, "macro_tier": 2, "macro_tier_score": 0.0235, "macro_tier_override": null, "x": 267.82, "z": 37.831, "size": 0.4179, "title": "Prime factorizations", "summary": "`n.factorization` is the finitely supported function `ℕ →₀ ℕ` mapping each prime factor of `n` to its multiplicity in `n`. For example, since 2000 = 2^4 * 5^3, * `factorization 2000 2` is 4 * `factorization 2000 5` is 3 * `factorization 2000 k` is 0 for all other `k : ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Factorization/Defs.html"}, {"id": "Mathlib.Data.Finset.Order", "region_id": "foundations_data", "micro_elevation": 0.2647, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 210.964, "z": 57.704, "size": 0.3375, "title": "Finsets of ordered types", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Order.html"}, {"id": "Mathlib.Data.NNReal.Basic", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 212.413, "z": 42.028, "size": 0.4589, "title": "Basic results on nonnegative real numbers", "summary": "This file contains all results on `NNReal` that do not directly follow from its basic structure. As a consequence, it is a bit of a random collection of results, and is a good target for cleanup.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/NNReal/Basic.html"}, {"id": "Mathlib.Data.Sigma.Order", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 214.837, "z": 41.76, "size": 0.2478, "title": "Orders on a sigma type", "summary": "This file defines two orders on a sigma type: * The disjoint sum of orders. `a` is less `b` iff `a` and `b` are in the same summand and `a` is less than `b` there. * The lexicographical order. `a` is less than `b` if its summand is strictly less than the summand of `b` or they are in the same summand and `a` is less than `b` there. We make the disjoint sum of orders the default set of instances. The lexicographic…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sigma/Order.html"}, {"id": "Mathlib.Data.Sigma.Lex", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0214, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2999, "title": "Lexicographic order on a sigma type", "summary": "This defines the lexicographical order of two arbitrary relations on a sigma type and proves some lemmas about `PSigma.Lex`, which is defined in core Lean. Given a relation in the index type and a relation on each summand, the lexicographical order on the sigma type relates `a` and `b` if their summands are related or they are in the same summand and related by the summand's relation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sigma/Lex.html"}, {"id": "Mathlib.Data.Nat.Choose.Sum", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0138, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.447, "title": "Sums of binomial coefficients", "summary": "This file includes variants of the binomial theorem and other results on sums of binomial coefficients. Theorems whose proofs depend on such sums may also go in this file for import reasons.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Choose/Sum.html"}, {"id": "Mathlib.Data.Set.Finite.Lattice", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 2, "macro_tier_score": 0.0356, "macro_tier_override": null, "x": 186.136, "z": 82.901, "size": 0.4803, "title": "Finiteness of unions and intersections", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Finite/Lattice.html"}, {"id": "Mathlib.Data.Set.Finite.Powerset", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 2, "macro_tier_score": 0.0556, "macro_tier_override": null, "x": 232.088, "z": -4.487, "size": 0.453, "title": "Finiteness of the powerset of a finite set", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Finite/Powerset.html"}, {"id": "Mathlib.Data.Set.Finite.Range", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 2, "macro_tier_score": 0.0594, "macro_tier_override": null, "x": 257.048, "z": 23.781, "size": 0.5633, "title": "Finiteness of `Set.range`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Finite/Range.html"}, {"id": "Mathlib.Data.Set.Lattice.Image", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.0571, "macro_tier_override": null, "x": 217.334, "z": 40.084, "size": 0.7271, "title": "The set lattice and (pre)images of functions", "summary": "This file contains lemmas on the interaction between the indexed union/intersection of sets and the image and preimage operations: `Set.image`, `Set.preimage`, `Set.image2`, `Set.kernImage`. It also covers `Set.MapsTo`, `Set.InjOn`, `Set.SurjOn`, `Set.BijOn`. In order to accommodate `Set.image2`, the file includes results on union/intersection in products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Lattice/Image.html"}, {"id": "Mathlib.Data.Fintype.Option", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 2, "macro_tier_score": 0.1321, "macro_tier_override": null, "x": 260.098, "z": 39.576, "size": 0.5578, "title": "fintype instances for option", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Option.html"}, {"id": "Mathlib.Data.Set.SymmDiff", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.3321, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.6069, "title": "Symmetric differences of sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/SymmDiff.html"}, {"id": "Mathlib.Data.Nat.PrimeFin", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 1, "macro_tier_score": 0.0216, "macro_tier_override": null, "x": 258.445, "z": 13.369, "size": 0.318, "title": "Prime numbers", "summary": "This file contains some results about prime numbers which depend on finiteness of sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/PrimeFin.html"}, {"id": "Mathlib.Data.Countable.Defs", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 2, "macro_tier_score": 0.0543, "macro_tier_override": null, "x": 210.534, "z": 35.346, "size": 0.3988, "title": "Countable and uncountable types", "summary": "In this file we define a typeclass `Countable` saying that a given `Sort*` is countable and a typeclass `Uncountable` saying that a given `Type*` is uncountable. See also `Encodable` for a version that singles out a specific encoding of elements of `α` by natural numbers. This file also provides a few instances of these typeclasses. More instances can be found in other files.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Countable/Defs.html"}, {"id": "Mathlib.Data.Nat.Factors", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 2, "macro_tier_score": 0.0322, "macro_tier_override": null, "x": 205.644, "z": 51.527, "size": 0.3317, "title": "Prime numbers", "summary": "This file deals with the factors of natural numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Factors.html"}, {"id": "Mathlib.Data.Nat.Prime.Infinite", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 1, "macro_tier_score": 0.0211, "macro_tier_override": null, "x": 203.753, "z": 39.91, "size": 0.2641, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Prime/Infinite.html"}, {"id": "Mathlib.Data.NNReal.Defs", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0548, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.4214, "title": "Nonnegative real numbers", "summary": "In this file we define `NNReal` (notation: `ℝ≥0`) to be the type of non-negative real numbers, a.k.a. the interval `[0, ∞)`. We also define the following operations and structures on `ℝ≥0`: * the order on `ℝ≥0` is the restriction of the order on `ℝ`; these relations define a conditionally complete linear order with a bottom element, `ConditionallyCompleteLinearOrderBot`; * `a + b` and `a * b` are the restrictions of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/NNReal/Defs.html"}, {"id": "Mathlib.Data.Finset.Insert", "region_id": "foundations_data", "micro_elevation": 0.4706, "macro_tier": 2, "macro_tier_score": 0.2884, "macro_tier_override": null, "x": 244.239, "z": 44.825, "size": 0.5591, "title": "Constructing finite sets by adding one element", "summary": "This file contains the definitions of `{a} : Finset α`, `insert a s : Finset α` and `Finset.cons`, all ways to construct a `Finset` by adding one element.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Insert.html"}, {"id": "Mathlib.Data.Finset.Attr", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2854, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.4722, "title": "Aesop rule set for finsets", "summary": "This file defines `finsetNonempty`, an aesop rule set to prove that a given finset is nonempty.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Attr.html"}, {"id": "Mathlib.Data.Finset.Empty", "region_id": "foundations_data", "micro_elevation": 0.2647, "macro_tier": 2, "macro_tier_score": 0.2838, "macro_tier_override": null, "x": 196.736, "z": 35.259, "size": 0.4144, "title": "Empty and nonempty finite sets", "summary": "This file defines the empty finite set ∅ and a predicate for nonempty `Finset`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Empty.html"}, {"id": "Mathlib.Data.Multiset.FinsetOps", "region_id": "foundations_data", "micro_elevation": 0.4412, "macro_tier": 2, "macro_tier_score": 0.2851, "macro_tier_override": null, "x": 203.996, "z": 12.814, "size": 0.463, "title": "Preparations for defining operations on `Finset`.", "summary": "The operations here ignore multiplicities, and prepare for defining the corresponding operations on `Finset`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/FinsetOps.html"}, {"id": "Mathlib.Data.List.Indexes", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 215.92, "z": 37.382, "size": 0.239, "title": "Lemmas about `List.*Idx` functions.", "summary": "Some specification lemmas for `List.mapIdx`, `List.mapIdxM`, `List.foldlIdx` and `List.foldrIdx`. As of 2025-01-29, these are not used anywhere in Mathlib. Moreover, with `List.enum` and `List.enumFrom` being replaced by `List.zipIdx` in Lean's `nightly-2025-01-29` release, they now use deprecated functions and theorems. Rather than updating this unused material, we are deprecating it. Anyone wanting to restore this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Indexes.html"}, {"id": "Mathlib.Data.Option.NAry", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.023, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3954, "title": "Binary map of options", "summary": "This file defines the binary map of `Option`. This is mostly useful to define pointwise operations on intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Option/NAry.html"}, {"id": "Mathlib.Data.Finite.Sigma", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 1, "macro_tier_score": 0.0037, "macro_tier_override": null, "x": 235.225, "z": 81.647, "size": 0.4563, "title": "Finiteness of sigma types", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finite/Sigma.html"}, {"id": "Mathlib.Data.Fintype.Pi", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 2, "macro_tier_score": 0.1586, "macro_tier_override": null, "x": 169.223, "z": 20.285, "size": 0.4065, "title": "Fintype instances for pi types", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Pi.html"}, {"id": "Mathlib.Data.Finset.Pi", "region_id": "foundations_data", "micro_elevation": 0.6176, "macro_tier": 2, "macro_tier_score": 0.1596, "macro_tier_override": null, "x": 178.112, "z": 60.799, "size": 0.4456, "title": "The Cartesian product of finsets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Pi.html"}, {"id": "Mathlib.Data.Set.Finite.Basic", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 2, "macro_tier_score": 0.1996, "macro_tier_override": null, "x": 252.282, "z": 66.249, "size": 0.6662, "title": "Finite sets", "summary": "This file provides `Fintype` instances for many set constructions. It also proves basic facts about finite sets and gives ways to manipulate `Set.Finite` expressions. Note that the instances in this file are selected somewhat arbitrarily on the basis of them not needing any imports beyond `Data.Fintype.Card` (which is required by `Finite.ofFinset`); they can certainly be organized better.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Finite/Basic.html"}, {"id": "Mathlib.Data.QPF.Multivariate.Constructions.Sigma", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 219.202, "z": 45.63, "size": 0.2, "title": "Dependent product and sum of QPFs are QPFs", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/QPF/Multivariate/Constructions/Sigma.html"}, {"id": "Mathlib.Data.QPF.Multivariate.Basic", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0119, "macro_tier_override": null, "x": 217.17, "z": 35.892, "size": 0.3655, "title": "Multivariate quotients of polynomial functors.", "summary": "Basic definition of multivariate QPF. QPFs form a compositional framework for defining inductive and coinductive types, their quotients and nesting. The idea is based on building ever larger functors. For instance, we can define a list using a shape functor: ```lean inductive ListShape (a b : Type) | nil : ListShape | cons : a -> b -> ListShape ``` This shape can itself be decomposed as a sum of product which are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/QPF/Multivariate/Basic.html"}, {"id": "Mathlib.Data.Multiset.AddSub", "region_id": "foundations_data", "micro_elevation": 0.2647, "macro_tier": 2, "macro_tier_score": 0.2932, "macro_tier_override": null, "x": 211.152, "z": 23.05, "size": 0.3666, "title": "Sum and difference of multisets", "summary": "This file defines the following operations on multisets: * `Add (Multiset α)` instance: `s + t` adds the multiplicities of the elements of `s` and `t` * `Sub (Multiset α)` instance: `s - t` subtracts the multiplicities of the elements of `s` and `t` * `Multiset.erase`: `s.erase x` reduces the multiplicity of `x` in `s` by one.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/AddSub.html"}, {"id": "Mathlib.Data.Multiset.Count", "region_id": "foundations_data", "micro_elevation": 0.2353, "macro_tier": 2, "macro_tier_score": 0.2925, "macro_tier_override": null, "x": 200.085, "z": 32.458, "size": 0.319, "title": "Counting multiplicity in a multiset", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Count.html"}, {"id": "Mathlib.Data.List.Count", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2931, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.359, "title": "Counting in lists", "summary": "This file proves basic properties of `List.countP` and `List.count`, which count the number of elements of a list satisfying a predicate and equal to a given element respectively.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Count.html"}, {"id": "Mathlib.Data.Int.ConditionallyCompleteOrder", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 222.059, "z": 44.914, "size": 0.2936, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/ConditionallyCompleteOrder.html"}, {"id": "Mathlib.Data.Int.Interval", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0121, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3742, "title": "Finite intervals of integers", "summary": "This file proves that `ℤ` is a `LocallyFiniteOrder` and calculates the cardinality of its intervals as finsets and fintypes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Interval.html"}, {"id": "Mathlib.Data.Int.SuccPred", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": 213.084, "z": 34.571, "size": 0.3177, "title": "Successors and predecessors of integers", "summary": "In this file, we show that `ℤ` is both an archimedean `SuccOrder` and an archimedean `PredOrder`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/SuccPred.html"}, {"id": "Mathlib.Data.Int.LeastGreatest", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 211.373, "z": 32.901, "size": 0.3401, "title": "Least upper bound and greatest lower bound properties for integers", "summary": "In this file we prove that a bounded above nonempty set of integers has the greatest element, and a counterpart of this statement for the least element.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/LeastGreatest.html"}, {"id": "Mathlib.Data.Nat.Find", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 2, "macro_tier_score": 0.3023, "macro_tier_override": null, "x": 214.251, "z": 46.167, "size": 0.6382, "title": "`Nat.find` and `Nat.findGreatest`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Find.html"}, {"id": "Mathlib.Data.Finsupp.Lex", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 1, "macro_tier_score": 0.0131, "macro_tier_override": null, "x": 177.789, "z": -3.253, "size": 0.4202, "title": "Lexicographic order on finitely supported functions", "summary": "This file defines the lexicographic order on `Finsupp`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Lex.html"}, {"id": "Mathlib.Data.ZMod.QuotientGroup", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0009, "macro_tier_override": null, "x": 211.57, "z": 40.849, "size": 0.3304, "title": "`ZMod n` and quotient groups / rings", "summary": "This file relates `ZMod n` to the quotient group `ℤ / AddSubgroup.zmultiples (n : ℤ)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ZMod/QuotientGroup.html"}, {"id": "Mathlib.Data.Fin.Basic", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.2217, "macro_tier_override": null, "x": 214.54, "z": 42.005, "size": 0.4306, "title": "The finite type with `n` elements", "summary": "`Fin n` is the type whose elements are natural numbers smaller than `n`. This file expands on the development in the core library.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Basic.html"}, {"id": "Mathlib.Data.PNat.Equiv", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 1, "macro_tier_score": 0.0116, "macro_tier_override": null, "x": 205.8, "z": 41.724, "size": 0.3429, "title": "The equivalence between `ℕ+` and `ℕ`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PNat/Equiv.html"}, {"id": "Mathlib.Data.Fintype.Order", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 1, "macro_tier_score": 0.002, "macro_tier_override": null, "x": 169.558, "z": 69.141, "size": 0.3881, "title": "Order structures on finite types", "summary": "This file provides order instances on fintypes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Order.html"}, {"id": "Mathlib.Data.Fin.VecNotation", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 1, "macro_tier_score": 0.0089, "macro_tier_override": null, "x": 221.329, "z": 34.689, "size": 0.5999, "title": "Matrix and vector notation", "summary": "This file defines notation for vectors and matrices. Given `a b c d : α`, the notation allows us to write `![a, b, c, d] : Fin 4 → α`. Nesting vectors gives coefficients of a matrix, so `![![a, b], ![c, d]] : Fin 2 → Fin 2 → α`. In later files we introduce `!![a, b; c, d]` as notation for `Matrix.of ![![a, b], ![c, d]]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/VecNotation.html"}, {"id": "Mathlib.Data.ZMod.Basic", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0275, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.5462, "title": "Integers mod `n`", "summary": "Definition of the integers mod n, and the field structure on the integers mod p.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ZMod/Basic.html"}, {"id": "Mathlib.Data.Prod.Basic", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.3432, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.8049, "title": "Extra facts about `Prod`", "summary": "This file proves various simple lemmas about `Prod`. It also defines better delaborators for product projections.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Prod/Basic.html"}, {"id": "Mathlib.Data.List.Nodup", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 2, "macro_tier_score": 0.3375, "macro_tier_override": null, "x": 209.93, "z": 47.317, "size": 0.712, "title": "Lists with no duplicates", "summary": "`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of this predicate.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Nodup.html"}, {"id": "Mathlib.Data.Set.Prod", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 2, "macro_tier_score": 0.3412, "macro_tier_override": null, "x": 210.084, "z": 31.277, "size": 0.7739, "title": "Sets in product and pi types", "summary": "This file proves basic properties of product of sets in `α × β` and in `Π i, α i`, and of the diagonal of a type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Prod.html"}, {"id": "Mathlib.Data.String.Defs", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0144, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.4684, "title": "Definitions for `String`", "summary": "This file defines a bunch of functions for the `String` datatype.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/String/Defs.html"}, {"id": "Mathlib.Data.PFunctor.Multivariate.W", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 209.334, "z": 36.269, "size": 0.2478, "title": "The W construction as a multivariate polynomial functor.", "summary": "W types are well-founded tree-like structures. They are defined as the least fixpoint of a polynomial functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PFunctor/Multivariate/W.html"}, {"id": "Mathlib.Data.PFunctor.Multivariate.Basic", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.0229, "macro_tier_override": null, "x": 211.407, "z": 37.088, "size": 0.3935, "title": "Multivariate polynomial functors.", "summary": "Multivariate polynomial functors are used for defining M-types and W-types. They map a type vector `α` to the type `Σ a : A, B a ⟹ α`, with `A : Type` and `B : A → TypeVec n`. They interact well with Lean's inductive definitions because they guarantee that occurrences of `α` are positive.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PFunctor/Multivariate/Basic.html"}, {"id": "Mathlib.Data.Sign.Basic", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 243.408, "z": 88.14, "size": 0.3333, "title": "Sign function", "summary": "This file defines the sign function for types with zero and a decidable less-than relation, and proves some basic theorems about it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sign/Basic.html"}, {"id": "Mathlib.Data.Fintype.BigOperators", "region_id": "foundations_data", "micro_elevation": 0.8235, "macro_tier": 2, "macro_tier_score": 0.1112, "macro_tier_override": null, "x": 262.207, "z": 16.197, "size": 0.5565, "title": null, "summary": "Results about \"big operations\" over a `Fintype`, and consequent results about cardinalities of certain types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/BigOperators.html"}, {"id": "Mathlib.Data.List.Destutter", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 201.908, "z": 38.785, "size": 0.2338, "title": "Destuttering of Lists", "summary": "This file proves theorems about `List.destutter` (in `Data.List.Defs`), which greedily removes all non-related items that are adjacent in a list, e.g. `[2, 2, 3, 3, 2].destutter (≠) = [2, 3, 2]`. Note that we make no guarantees of being the longest sublist with this property; e.g., `[123, 1, 2, 5, 543, 1000].destutter (<) = [123, 543, 1000]`, but a longer ascending chain could be `[1, 2, 5, 543, 1000]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Destutter.html"}, {"id": "Mathlib.Data.List.Chain", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 2, "macro_tier_score": 0.0227, "macro_tier_override": null, "x": 216.465, "z": 33.22, "size": 0.3842, "title": "Relation chain", "summary": "This file provides basic results about `List.IsChain` from Batteries. A list `[a₁, a₂, ..., aₙ]` satisfies `IsChain` with respect to the relation `r` if `r a₁ a₂` and `r a₂ a₃` and ... and `r aₙ₋₁ aₙ`. We write it `IsChain r [a₁, a₂, ..., aₙ]`. A graph-specialized version is in development and will hopefully be added under `combinatorics.` sometime soon.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Chain.html"}, {"id": "Mathlib.Data.List.Dedup", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 2, "macro_tier_score": 0.2841, "macro_tier_override": null, "x": 210.905, "z": 49.767, "size": 0.4284, "title": "Erasure of duplicates in a list", "summary": "This file proves basic results about `List.dedup` (definition in `Data.List.Defs`). `dedup l` returns `l` without its duplicates. It keeps the earliest (that is, rightmost) occurrence of each.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Dedup.html"}, {"id": "Mathlib.Data.Set.NAry", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 2, "macro_tier_score": 0.1625, "macro_tier_override": null, "x": 222.846, "z": 33.47, "size": 0.5355, "title": "N-ary images of sets", "summary": "This file defines `Set.image2`, the binary image of sets. This is mostly useful to define pointwise operations and `Set.seq`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/NAry.html"}, {"id": "Mathlib.Data.PEquiv", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 1, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 216.931, "z": 38.643, "size": 0.2758, "title": "Partial Equivalences", "summary": "In this file, we define partial equivalences `PEquiv`, which are a bijection between a subset of `α` and a subset of `β`. Notationally, a `PEquiv` is denoted by \"`≃.`\" (note that the full stop is part of the notation). The way we store these internally is with two functions `f : α → Option β` and the reverse function `g : β → Option α`, with the condition that if `f a` is `some b`, then `g b` is `some a`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PEquiv.html"}, {"id": "Mathlib.Data.Option.Basic", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.024, "macro_tier_override": null, "x": 214.875, "z": 39.059, "size": 0.438, "title": "Option of a type", "summary": "This file develops the basic theory of option types. If `α` is a type, then `Option α` can be understood as the type with one more element than `α`. `Option α` has terms `some a`, where `a : α`, and `none`, which is the added element. This is useful in multiple ways: * It is the prototype of addition of terms to a type. See for example `WithBot α` which uses `none` as an element smaller than all others. * It can be…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Option/Basic.html"}, {"id": "Mathlib.Data.Set.Basic", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.3841, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.7921, "title": "Basic properties of sets", "summary": "Sets in Lean are homogeneous; all their elements have the same type. Sets whose elements have type `X` are thus defined as `Set X := X → Prop`. Note that this function need not be decidable. The definition is in the module `Mathlib/Data/Set/Defs.lean`. This file provides some basic definitions related to sets and functions not present in the definitions file, as well as extra lemmas for functions defined in the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Basic.html"}, {"id": "Mathlib.Data.Finset.NAry", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 1, "macro_tier_score": 0.012, "macro_tier_override": null, "x": 165.695, "z": 62.128, "size": 0.3676, "title": "N-ary images of finsets", "summary": "This file defines `Finset.image₂`, the binary image of finsets. This is the finset version of `Set.image2`. This is mostly useful to define pointwise operations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/NAry.html"}, {"id": "Mathlib.Data.NNReal.Star", "region_id": "foundations_data", "micro_elevation": 0.2353, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 200.031, "z": 48.23, "size": 0.2905, "title": "The non-negative real numbers are a \\*-ring, with the trivial \\*-structure", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/NNReal/Star.html"}, {"id": "Mathlib.Data.Pi.Interval", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 171.326, "z": 77.831, "size": 0.267, "title": "Intervals in a pi type", "summary": "This file shows that (dependent) functions to locally finite orders equipped with the pointwise order are locally finite and calculates the cardinality of their intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Pi/Interval.html"}, {"id": "Mathlib.Data.Analysis.Filter", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.239, "title": "Computational realization of filters (experimental)", "summary": "This file provides infrastructure to compute with filters.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Analysis/Filter.html"}, {"id": "Mathlib.Data.Nat.Factorization.PrimePow", "region_id": "foundations_data", "micro_elevation": 0.8824, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": 270.786, "z": 29.74, "size": 0.3204, "title": "Prime powers and factorizations", "summary": "This file deals with factorizations of prime powers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Factorization/PrimePow.html"}, {"id": "Mathlib.Data.Fintype.Sets", "region_id": "foundations_data", "micro_elevation": 0.5882, "macro_tier": 2, "macro_tier_score": 0.2362, "macro_tier_override": null, "x": 178.349, "z": 57.074, "size": 0.5557, "title": "Subsets of finite types", "summary": "In a `Fintype`, all `Set`s are automatically `Finset`s, and there are only finitely many of them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Sets.html"}, {"id": "Mathlib.Data.Finset.BooleanAlgebra", "region_id": "foundations_data", "micro_elevation": 0.5588, "macro_tier": 2, "macro_tier_score": 0.2323, "macro_tier_override": null, "x": 231.519, "z": 72.601, "size": 0.4358, "title": "`Finset`s are a Boolean algebra", "summary": "This file provides the `BooleanAlgebra (Finset α)` instance, under the assumption that `α` is a `Fintype`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/BooleanAlgebra.html"}, {"id": "Mathlib.Data.Finset.SymmDiff", "region_id": "foundations_data", "micro_elevation": 0.5588, "macro_tier": 2, "macro_tier_score": 0.2313, "macro_tier_override": null, "x": 250.239, "z": 37.082, "size": 0.3949, "title": "Symmetric difference of finite sets", "summary": "This file concerns the symmetric difference operator `s Δ t` on finite sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/SymmDiff.html"}, {"id": "Mathlib.Data.Fintype.OfMap", "region_id": "foundations_data", "micro_elevation": 0.5588, "macro_tier": 2, "macro_tier_score": 0.2443, "macro_tier_override": null, "x": 225.615, "z": 75.261, "size": 0.4911, "title": "Constructors for `Fintype`", "summary": "This file contains basic constructors for `Fintype` instances, given maps from/to finite types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/OfMap.html"}, {"id": "Mathlib.Data.Nat.Log", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0218, "macro_tier_override": null, "x": 211.62, "z": 39.761, "size": 0.3339, "title": "Natural number logarithms", "summary": "This file defines two `ℕ`-valued analogs of the logarithm of `n` with base `b`: * `log b n`: Lower logarithm, or floor **log**. Greatest `k` such that `b^k ≤ n`. * `clog b n`: Upper logarithm, or **c**eil **log**. Least `k` such that `n ≤ b^k`. These are interesting because, for `1 < b`, `Nat.log b` and `Nat.clog b` are respectively right and left adjoints of `(b ^ ·)`. See `le_log_iff_pow_le` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Log.html"}, {"id": "Mathlib.Data.Finset.NoncommProd", "region_id": "foundations_data", "micro_elevation": 0.6471, "macro_tier": 1, "macro_tier_score": 0.0017, "macro_tier_override": null, "x": 234.534, "z": 77.595, "size": 0.3751, "title": "Products (respectively, sums) over a finset or a multiset.", "summary": "The regular `Finset.prod` and `Multiset.prod` require `[CommMonoid α]`. Often, there are collections `s : Finset α` where `[Monoid α]` and we know, in a dependent fashion, that for all the terms `∀ (x ∈ s) (y ∈ s), Commute x y`. This allows to still have a well-defined product over `s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/NoncommProd.html"}, {"id": "Mathlib.Data.PSigma.Order", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 215.258, "z": 39.654, "size": 0.2, "title": "Lexicographic order on a sigma type", "summary": "This file defines the lexicographic order on `Σₗ' i, α i`. `a` is less than `b` if its summand is strictly less than the summand of `b` or they are in the same summand and `a` is less than `b` there.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PSigma/Order.html"}, {"id": "Mathlib.Data.Nat.Choose.Factorization", "region_id": "foundations_data", "micro_elevation": 0.8824, "macro_tier": 1, "macro_tier_score": 0.0113, "macro_tier_override": null, "x": 155.157, "z": 41.271, "size": 0.3289, "title": "Factorization of Binomial Coefficients", "summary": "This file contains a few results on the multiplicity of prime factors within certain size bounds in binomial coefficients. These include: * `Nat.factorization_choose_le_log`: a logarithmic upper bound on the multiplicity of a prime in a binomial coefficient. * `Nat.factorization_choose_le_one`: Primes above `sqrt n` appear at most once in the factorization of `n` choose `k`. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Choose/Factorization.html"}, {"id": "Mathlib.Data.Nat.Choose.Central", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.0109, "macro_tier_override": null, "x": 223.931, "z": 45.52, "size": 0.292, "title": "Central binomial coefficients", "summary": "This file proves properties of the central binomial coefficients (that is, `Nat.choose (2 * n) n`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Choose/Central.html"}, {"id": "Mathlib.Data.Nat.Digits.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.0108, "macro_tier_override": null, "x": 206.861, "z": 30.774, "size": 0.2834, "title": "Digits of a natural number", "summary": "This provides lemma about the digits of natural numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Digits/Lemmas.html"}, {"id": "Mathlib.Data.PNat.Find", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 201.804, "z": 39.991, "size": 0.2, "title": "Explicit least witnesses to existentials on positive natural numbers", "summary": "Implemented via calling out to `Nat.find`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PNat/Find.html"}, {"id": "Mathlib.Data.PNat.Basic", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 1, "macro_tier_score": 0.0134, "macro_tier_override": null, "x": 212.823, "z": 50.087, "size": 0.4315, "title": "The positive natural numbers", "summary": "This file develops the type `ℕ+` or `PNat`, the subtype of natural numbers that are positive. It is defined in `Data.PNat.Defs`, but most of the development is deferred to here so that `Data.PNat.Defs` can have very few imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PNat/Basic.html"}, {"id": "Mathlib.Data.Fintype.Defs", "region_id": "foundations_data", "micro_elevation": 0.4118, "macro_tier": 2, "macro_tier_score": 0.2864, "macro_tier_override": null, "x": 228.61, "z": 62.992, "size": 0.5042, "title": "Finite types", "summary": "This file defines a typeclass to state that a type is finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Defs.html"}, {"id": "Mathlib.Data.List.Sublists", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 2, "macro_tier_score": 0.1048, "macro_tier_override": null, "x": 220.773, "z": 49.473, "size": 0.3102, "title": "sublists", "summary": "`List.Sublists` gives a list of all (not necessarily contiguous) sublists of a list. This file contains basic results on this function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Sublists.html"}, {"id": "Mathlib.Data.Nat.Factorial.DoubleFactorial", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 1, "macro_tier_score": 0.011, "macro_tier_override": null, "x": 205.822, "z": 38.935, "size": 0.3012, "title": "Double factorials", "summary": "This file defines the double factorial, `n‼ := n * (n - 2) * (n - 4) * ...`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Factorial/DoubleFactorial.html"}, {"id": "Mathlib.Data.Nat.Factorial.Basic", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 2, "macro_tier_score": 0.1273, "macro_tier_override": null, "x": 211.331, "z": 45.819, "size": 0.4022, "title": "Factorial and variants", "summary": "This file defines the factorial, along with the ascending and descending variants. For the proof that the factorial of `n` counts the permutations of an `n`-element set, see `Fintype.card_perm`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Factorial/Basic.html"}, {"id": "Mathlib.Data.Array.Defs", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": "Definitions on Arrays", "summary": "This file contains various definitions on `Array`. It does not contain proofs about these definitions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Array/Defs.html"}, {"id": "Mathlib.Data.PFunctor.Univariate.M", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 1, "macro_tier_score": 0.0213, "macro_tier_override": null, "x": 212.968, "z": 44.246, "size": 0.2884, "title": "M-types", "summary": "M types are potentially infinite tree-like structures. They are defined as the greatest fixpoint of a polynomial functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PFunctor/Univariate/M.html"}, {"id": "Mathlib.Data.PFunctor.Univariate.Basic", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0225, "macro_tier_override": null, "x": 212.965, "z": 42.27, "size": 0.3709, "title": "Polynomial Functors", "summary": "This file defines polynomial functors and the W-type construction as a polynomial functor. (For the M-type construction, see `Mathlib/Data/PFunctor/Univariate/M.lean`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PFunctor/Univariate/Basic.html"}, {"id": "Mathlib.Data.Sum.Basic", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.335, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.6657, "title": "Additional lemmas about sum types", "summary": "Most of the former contents of this file have been moved to Batteries.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sum/Basic.html"}, {"id": "Mathlib.Data.Option.Defs", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0332, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3862, "title": "Extra definitions on `Option`", "summary": "This file defines more operations involving `Option α`. Lemmas about them are located in other files under `Mathlib/Data/Option/`. Other basic operations on `Option` are defined in the core library.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Option/Defs.html"}, {"id": "Mathlib.Data.Nat.Cast.Defs", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.0997, "macro_tier_override": null, "x": 213.046, "z": 36.525, "size": 0.7434, "title": "Cast of natural numbers", "summary": "This file defines the *canonical* homomorphism from the natural numbers into an `AddMonoid` with a one. In additive monoids with one, there exists a unique such homomorphism and we store it in the `natCast : ℕ → R` field. Preferentially, the homomorphism is written as the coercion `Nat.cast`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Cast/Defs.html"}, {"id": "Mathlib.Data.Finset.Fold", "region_id": "foundations_data", "micro_elevation": 0.5588, "macro_tier": 2, "macro_tier_score": 0.2233, "macro_tier_override": null, "x": 185.593, "z": 64.623, "size": 0.4866, "title": "The fold operation for a commutative associative operation over a finset.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Fold.html"}, {"id": "Mathlib.Data.Nat.Cast.Basic", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1242, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.6167, "title": "Cast of natural numbers (additional theorems)", "summary": "This file proves additional properties about the *canonical* homomorphism from the natural numbers into an additive monoid with a one (`Nat.cast`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Cast/Basic.html"}, {"id": "Mathlib.Data.List.Infix", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 2, "macro_tier_score": 0.2832, "macro_tier_override": null, "x": 210.544, "z": 35.34, "size": 0.386, "title": "Prefixes, suffixes, infixes", "summary": "This file proves properties about * `List.isPrefix`: `l₁` is a prefix of `l₂` if `l₂` starts with `l₁`. * `List.isSuffix`: `l₁` is a suffix of `l₂` if `l₂` ends with `l₁`. * `List.isInfix`: `l₁` is an infix of `l₂` if `l₁` is a prefix of some suffix of `l₂`. * `List.inits`: The list of prefixes of a list. * `List.tails`: The list of prefixes of a list. * `insert` on lists All those (except `insert`) are defined in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Infix.html"}, {"id": "Mathlib.Data.Finset.Max", "region_id": "foundations_data", "micro_elevation": 0.6471, "macro_tier": 2, "macro_tier_score": 0.0585, "macro_tier_override": null, "x": 171.265, "z": 47.324, "size": 0.5402, "title": "Maximum and minimum of finite sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Max.html"}, {"id": "Mathlib.Data.Finsupp.Single", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.0235, "macro_tier_override": null, "x": 215.165, "z": 39.46, "size": 0.4205, "title": "Finitely supported functions on exactly one point", "summary": "This file contains definitions and basic results on defining/updating/removing `Finsupp`s using one point of the domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Single.html"}, {"id": "Mathlib.Data.NNRat.Encodable", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 214.818, "z": 39.001, "size": 0.2, "title": "The nonnegative rationals are `Encodable`.", "summary": "As a consequence we also get the instance `Countable ℚ≥0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/NNRat/Encodable.html"}, {"id": "Mathlib.Data.NNRat.Defs", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0329, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3747, "title": "Nonnegative rationals", "summary": "This file defines the nonnegative rationals as a subtype of `Rat` and provides its basic algebraic order structure. Note that `NNRat` is not declared as a `Semifield` here. See `Mathlib/Algebra/Field/Rat.lean` for that instance. We also define an instance `CanLift ℚ ℚ≥0`. This instance can be used by the `lift` tactic to replace `x : ℚ` and `hx : 0 ≤ x` in the proof context with `x : ℚ≥0` while replacing all…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/NNRat/Defs.html"}, {"id": "Mathlib.Data.DFinsupp.Module", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 2, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 261.371, "z": 24.324, "size": 0.4915, "title": "Group actions on `DFinsupp`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/Module.html"}, {"id": "Mathlib.Data.List.Enum", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 2, "macro_tier_score": 0.1672, "macro_tier_override": null, "x": 209.186, "z": 44.358, "size": 0.3004, "title": "Properties of `List.enum`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Enum.html"}, {"id": "Mathlib.Data.List.Basic", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.3549, "macro_tier_override": null, "x": 215.634, "z": 37.168, "size": 0.6473, "title": "Basic properties of lists", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Basic.html"}, {"id": "Mathlib.Data.Int.Fib.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.4706, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 198.891, "z": 67.865, "size": 0.2, "title": "Cassini and Catalan identities for the Fibonacci numbers", "summary": "Cassini's identity states that for `n : ℤ`, `fib (n + 1) * fib (n - 1) - fib n ^ 2` is equal to `(-1) ^ |n|`. And Catalan's identity states that for any integers `x` and `a`, we get `fib (x + a) ^ 2 - fib x * fib (x + 2 * a) = (-1) ^ |x| * fib a ^ 2`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Fib/Lemmas.html"}, {"id": "Mathlib.Data.Int.Fib.Basic", "region_id": "foundations_data", "micro_elevation": 0.4412, "macro_tier": 1, "macro_tier_score": 0.0108, "macro_tier_override": null, "x": 204.808, "z": 68.231, "size": 0.2776, "title": "Fibonacci numbers extended onto the integers", "summary": "This file defines the Fibonacci sequence on the integers. Definition of the sequence: `F₀ = 0`, `F₁ = 1`, and `Fₙ₊₂ = Fₙ₊₁ + Fₙ` (same as the natural number version `Nat.fib`, but here `n` is an integer).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Fib/Basic.html"}, {"id": "Mathlib.Data.Set.Inclusion", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.3319, "macro_tier_override": null, "x": 211.846, "z": 39.305, "size": 0.6015, "title": "Lemmas about `inclusion`, the injection of subtypes induced by `⊆`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Inclusion.html"}, {"id": "Mathlib.Data.Finset.Lattice.Pi", "region_id": "foundations_data", "micro_elevation": 0.6765, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 174.641, "z": 18.219, "size": 0.2, "title": "Lattice operations on finsets of functions", "summary": "This file is concerned with folding binary lattice operations over finsets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Lattice/Pi.html"}, {"id": "Mathlib.Data.Finset.Lattice.Prod", "region_id": "foundations_data", "micro_elevation": 0.6471, "macro_tier": 1, "macro_tier_score": 0.0121, "macro_tier_override": null, "x": 210.023, "z": 83.011, "size": 0.3744, "title": "Lattice operations on finsets of products", "summary": "This file is concerned with folding binary lattice operations over finsets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Lattice/Prod.html"}, {"id": "Mathlib.Data.Finsupp.Fintype", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 1, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": 173.71, "z": 0.427, "size": 0.3043, "title": "Finiteness and infiniteness of `Finsupp`", "summary": "Some lemmas on the combination of `Finsupp`, `Fintype` and `Infinite`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Fintype.html"}, {"id": "Mathlib.Data.Finset.Powerset", "region_id": "foundations_data", "micro_elevation": 0.6765, "macro_tier": 2, "macro_tier_score": 0.098, "macro_tier_override": null, "x": 250.262, "z": 15.016, "size": 0.4766, "title": "The powerset of a finset", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Powerset.html"}, {"id": "Mathlib.Data.PFun", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 1, "macro_tier_score": 0.0116, "macro_tier_override": null, "x": 200.287, "z": 43.797, "size": 0.3489, "title": "Partial functions", "summary": "This file defines partial functions. Partial functions are like functions, except they can also be \"undefined\" on some inputs. We define them as functions `α → Part β`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PFun.html"}, {"id": "Mathlib.Data.Part", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 2, "macro_tier_score": 0.0322, "macro_tier_override": null, "x": 205.726, "z": 39.588, "size": 0.3311, "title": "Partial values of a type", "summary": "This file defines `Part α`, the partial values of a type. `o : Part α` carries a proposition `o.Dom`, its domain, along with a function `get : o.Dom → α`, its value. The rule is then that every partial value has a value but, to access it, you need to provide a proof of the domain. `Part α` behaves the same as `Option α` except that `o : Option α` is decidably `none` or `some a` for some `a : α`, while the domain of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Part.html"}, {"id": "Mathlib.Data.Rel", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 2, "macro_tier_score": 0.0342, "macro_tier_override": null, "x": 210.815, "z": 29.032, "size": 0.429, "title": "Relations as sets of pairs", "summary": "This file provides API to regard relations between `α` and `β` as sets of pairs `Set (α × β)`. This is in particular useful in the study of uniform spaces, which are topological spaces equipped with a *uniformity*, namely a filter of pairs `α × α` whose elements can be viewed as \"proximity\" relations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rel.html"}, {"id": "Mathlib.Data.List.Duplicate", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 2, "macro_tier_score": 0.2444, "macro_tier_override": null, "x": 221.206, "z": 34.524, "size": 0.4942, "title": "List duplicates", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Duplicate.html"}, {"id": "Mathlib.Data.Set.Disjoint", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.3501, "macro_tier_override": null, "x": 212.032, "z": 39.07, "size": 0.538, "title": "Theorems about the `Disjoint` relation on `Set`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Disjoint.html"}, {"id": "Mathlib.Data.Real.Star", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 1, "macro_tier_score": 0.0113, "macro_tier_override": null, "x": 200.155, "z": 37.54, "size": 0.3286, "title": "The real numbers are a \\*-ring, with the trivial \\*-structure", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Real/Star.html"}, {"id": "Mathlib.Data.Seq.Parallel", "region_id": "foundations_data", "micro_elevation": 0.2647, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 207.949, "z": 56.992, "size": 0.2, "title": "Parallel computation", "summary": "Parallel computation of a computable sequence of computations by a diagonal enumeration. The important theorems of this operation are proven as terminates_parallel and exists_of_mem_parallel. (This operation is nondeterministic in the sense that it does not honor sequence equivalence (irrelevance of computation time).)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Seq/Parallel.html"}, {"id": "Mathlib.Data.WSeq.Relation", "region_id": "foundations_data", "micro_elevation": 0.2353, "macro_tier": 1, "macro_tier_score": 0.0213, "macro_tier_override": null, "x": 203.463, "z": 28.48, "size": 0.2955, "title": "Relations between and equivalence of weak sequences", "summary": "This file defines a relation between weak sequences as a relation between their `some` elements, ignoring computation time (`none` elements). Equivalence is then defined in the obvious way.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/WSeq/Relation.html"}, {"id": "Mathlib.Data.Set.CoeSort", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.2964, "macro_tier_override": null, "x": 213.949, "z": 38.509, "size": 0.493, "title": "Coercing sets to types.", "summary": "This file defines `Set.Elem s` as the type of all elements of the set `s`. More advanced theorems about these definitions are located in other files in `Mathlib/Data/Set`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/CoeSort.html"}, {"id": "Mathlib.Data.Set.BooleanAlgebra", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2616, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.9643, "title": "Sets are a complete atomic Boolean algebra.", "summary": "This file contains only the definition of the complete atomic Boolean algebra structure on `Set`. Indexed union/intersection are defined in `Mathlib.Order.SetNotation`; lemmas are available in `Mathlib/Data/Set/Lattice.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/BooleanAlgebra.html"}, {"id": "Mathlib.Data.Nat.Sqrt", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 2, "macro_tier_score": 0.0539, "macro_tier_override": null, "x": 218.157, "z": 36.937, "size": 0.3795, "title": "Properties of the natural number square root function.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Sqrt.html"}, {"id": "Mathlib.Data.Nat.Bits", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.0428, "macro_tier_override": null, "x": 215.087, "z": 43.92, "size": 0.3413, "title": "Additional properties of binary recursion on `Nat`", "summary": "This file documents additional properties of binary recursion, which allows us to more easily work with operations which do depend on the number of leading zeros in the binary representation of `n`. For example, we can more easily work with `Nat.bits` and `Nat.size`. See also: `Nat.bitwise`, `Nat.pow` (for various lemmas about `size` and `shiftLeft`/`shiftRight`), and `Nat.digits`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Bits.html"}, {"id": "Mathlib.Data.Nat.BinaryRec", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0602, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.5829, "title": "Binary recursion on `Nat`", "summary": "This file defines binary recursion on `Nat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/BinaryRec.html"}, {"id": "Mathlib.Data.List.Defs", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.3592, "macro_tier_override": null, "x": 211.955, "z": 39.159, "size": 0.502, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Defs.html"}, {"id": "Mathlib.Data.Fin.Tuple.Finset", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 244.483, "z": 0.5, "size": 0.2239, "title": "Fin-indexed tuples of finsets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Tuple/Finset.html"}, {"id": "Mathlib.Data.Finset.Prod", "region_id": "foundations_data", "micro_elevation": 0.6176, "macro_tier": 2, "macro_tier_score": 0.1745, "macro_tier_override": null, "x": 254.192, "z": 42.994, "size": 0.5766, "title": "Finsets in product types", "summary": "This file defines finset constructions on the product type `α × β`. Beware not to confuse with the `Finset.prod` operation which computes the multiplicative product.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Prod.html"}, {"id": "Mathlib.Data.Nat.ModEq", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 2, "macro_tier_score": 0.064, "macro_tier_override": null, "x": 210.436, "z": 47.553, "size": 0.3648, "title": "Congruences modulo a natural number", "summary": "This file defines the equivalence relation `a ≡ b [MOD n]` on the natural numbers, and proves basic properties about it such as the Chinese Remainder Theorem `modEq_and_modEq_iff_modEq_mul`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/ModEq.html"}, {"id": "Mathlib.Data.Int.GCD", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 2, "macro_tier_score": 0.0644, "macro_tier_override": null, "x": 212.799, "z": 46.183, "size": 0.3837, "title": "Extended GCD and divisibility over ℤ", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/GCD.html"}, {"id": "Mathlib.Data.Nat.GCD.Basic", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0739, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.33, "title": "Properties of `Nat.gcd`, `Nat.lcm`, and `Nat.Coprime`", "summary": "Definitions are provided in batteries. Generalizations of these are provided in a later file as `GCDMonoid.gcd` and `GCDMonoid.lcm`. Note that the global `IsCoprime` is not a straightforward generalization of `Nat.Coprime`, see `Nat.isCoprime_iff_coprime` for the connection between the two. Most of this file could be moved to batteries as well.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/GCD/Basic.html"}, {"id": "Mathlib.Data.Set.Function", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 2, "macro_tier_score": 0.3281, "macro_tier_override": null, "x": 205.636, "z": 49.039, "size": 0.8846, "title": "Functions over sets", "summary": "This file contains basic results on the following predicates of functions and sets: * `Set.EqOn f₁ f₂ s` : functions `f₁` and `f₂` are equal at every point of `s`; * `Set.MapsTo f s t` : `f` sends every point of `s` to a point of `t`; * `Set.InjOn f s` : restriction of `f` to `s` is injective; * `Set.SurjOn f s t` : every point in `s` has a preimage in `s`; * `Set.BijOn f s t` : `f` is a bijection between `s` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Function.html"}, {"id": "Mathlib.Data.ENat.Pow", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.0021, "macro_tier_override": null, "x": 216.589, "z": 29.156, "size": 0.3967, "title": "Powers of extended natural numbers", "summary": "We define the power of an extended natural `x : ℕ∞` by another extended natural `y : ℕ∞`. The definition is chosen such that `x ^ y` is the cardinality of `α → β`, when `β` has cardinality `x` and `α` has cardinality `y`: * When `y` is finite, it coincides with the exponentiation by natural numbers (e.g. `⊤ ^ 0 = 1`). * We set `0 ^ ⊤ = 0`, `1 ^ ⊤ = 1` and `x ^ ⊤ = ⊤` for `x > 1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ENat/Pow.html"}, {"id": "Mathlib.Data.Rat.Init", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.027, "macro_tier_override": null, "x": 214.806, "z": 41.791, "size": 0.5331, "title": "Basic definitions around the rational numbers", "summary": "This file declares `ℚ` notation for the rationals and defines the nonnegative rationals `ℚ≥0`. This file is eligible to upstreaming to Batteries.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Init.html"}, {"id": "Mathlib.Data.Nat.NthRoot.Defs", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.4161, "title": "Definition of `Nat.nthRoot`", "summary": "In this file we define `Nat.nthRoot n a` to be the floor of the `n`th root of `a`. The function is defined in terms of natural numbers with no dependencies outside of prelude.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/NthRoot/Defs.html"}, {"id": "Mathlib.Data.ENat.Defs", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.0641, "macro_tier_override": null, "x": 211.536, "z": 40.676, "size": 0.3675, "title": "Definition and notation for extended natural numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ENat/Defs.html"}, {"id": "Mathlib.Data.Fintype.Lattice", "region_id": "foundations_data", "micro_elevation": 0.6765, "macro_tier": 1, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": 168.812, "z": 38.151, "size": 0.3811, "title": "Lemmas relating fintypes and order/lattice structure.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Lattice.html"}, {"id": "Mathlib.Data.Fintype.Sum", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 2, "macro_tier_score": 0.129, "macro_tier_override": null, "x": 186.385, "z": 2.404, "size": 0.47, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Sum.html"}, {"id": "Mathlib.Data.Set.Subset", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 1, "macro_tier_score": 0.001, "macro_tier_override": null, "x": 203.873, "z": 38.795, "size": 0.3338, "title": "Sets in subtypes", "summary": "This file is about sets in `Set A` when `A` is a set. It defines notation `↓∩` for sets in a type pulled down to sets in a subtype, as an inverse operation to the coercion that lifts sets in a subtype up to sets in the ambient type. This module also provides lemmas for `↓∩` and this coercion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Subset.html"}, {"id": "Mathlib.Data.Set.Functor", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 1, "macro_tier_score": 0.0121, "macro_tier_override": null, "x": 213.856, "z": 32.626, "size": 0.3737, "title": "Functoriality of `Set`", "summary": "This file defines the functor structure of `Set`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Functor.html"}, {"id": "Mathlib.Data.Nat.Basic", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.4751, "macro_tier_override": null, "x": 217.281, "z": 39.682, "size": 0.75, "title": "Basic operations on the natural numbers", "summary": "This file builds on `Mathlib/Data/Nat/Init.lean` by adding basic lemmas on natural numbers depending on Mathlib definitions. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Basic.html"}, {"id": "Mathlib.Data.Finite.Prod", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 2, "macro_tier_score": 0.1613, "macro_tier_override": null, "x": 190.814, "z": -4.786, "size": 0.5018, "title": "Finiteness of products", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finite/Prod.html"}, {"id": "Mathlib.Data.Set.Finite.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 1, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": 164.881, "z": 41.521, "size": 0.3797, "title": "Lemmas on finiteness of sets", "summary": "This file should contain lemmas that prove some result under the *assumption* of `Set.Finite`. If your proof has as *result* `Set.Finite`, then it should go to a more specific file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Finite/Lemmas.html"}, {"id": "Mathlib.Data.List.Sym", "region_id": "foundations_data", "micro_elevation": 0.6765, "macro_tier": 2, "macro_tier_score": 0.0317, "macro_tier_override": null, "x": 196.012, "z": 81.548, "size": 0.2934, "title": "Unordered tuples of elements of a list", "summary": "Defines `List.sym` and the specialized `List.sym2` for computing lists of all unordered n-tuples from a given list. These are list versions of `Nat.multichoose`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Sym.html"}, {"id": "Mathlib.Data.Nat.Choose.Basic", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 2, "macro_tier_score": 0.116, "macro_tier_override": null, "x": 216.485, "z": 47.551, "size": 0.3602, "title": "Binomial coefficients", "summary": "This file defines binomial coefficients and proves simple lemmas (i.e. those not requiring more imports). For the lemma that `n.choose k` counts the `k`-element-subsets of an `n`-element set, see `Finset.card_powersetCard` in `Mathlib/Data/Finset/Powerset.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Choose/Basic.html"}, {"id": "Mathlib.Data.Sym.Sym2", "region_id": "foundations_data", "micro_elevation": 0.6471, "macro_tier": 2, "macro_tier_score": 0.0444, "macro_tier_override": null, "x": 245.851, "z": 68.302, "size": 0.4209, "title": "The symmetric square", "summary": "This file defines the symmetric square, which is `α × α` modulo swapping. This is also known as the type of unordered pairs. More generally, the symmetric square is the second symmetric power (see `Data.Sym.Basic`). The equivalence is `Sym2.equivSym`. From the point of view that an unordered pair is equivalent to a multiset of cardinality two (see `Sym2.equivMultiset`), there is a `Mem` instance `Sym2.Mem`, which is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sym/Sym2.html"}, {"id": "Mathlib.Data.Finite.Sum", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 1, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 166.19, "z": 51.647, "size": 0.3537, "title": "Finiteness of sum types", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finite/Sum.html"}, {"id": "Mathlib.Data.Matrix.Block", "region_id": "foundations_data", "micro_elevation": 0.9706, "macro_tier": 1, "macro_tier_score": 0.0133, "macro_tier_override": null, "x": 210.148, "z": -23.664, "size": 0.4279, "title": "Block Matrices", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/Block.html"}, {"id": "Mathlib.Data.Rat.NatSqrt.Defs", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2649, "title": null, "summary": "Rational approximation of the square root of a natural number. See also `Mathlib.Analysis.Rat.NatSqrt.Real` for comparisons with the real square root.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/NatSqrt/Defs.html"}, {"id": "Mathlib.Data.Opposite", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0197, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.6095, "title": "Opposites", "summary": "In this file we define a structure `Opposite α` containing a single field of type `α` and two bijections `op : α → αᵒᵖ` and `unop : αᵒᵖ → α`. If `α` is a category, then `αᵒᵖ` is the opposite category, with all arrows reversed.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Opposite.html"}, {"id": "Mathlib.Data.Nat.Choose.Dvd", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 225.401, "z": 46.908, "size": 0.2745, "title": "Divisibility properties of binomial coefficients", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Choose/Dvd.html"}, {"id": "Mathlib.Data.Nat.Squarefree", "region_id": "foundations_data", "micro_elevation": 0.8824, "macro_tier": 1, "macro_tier_score": 0.0009, "macro_tier_override": null, "x": 218.308, "z": -17.716, "size": 0.3282, "title": "Lemmas about squarefreeness of natural numbers", "summary": "A number is squarefree when it is not divisible by any squares except the squares of units.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Squarefree.html"}, {"id": "Mathlib.Data.Fin.Tuple.Reflection", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0113, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3262, "title": "Lemmas for tuples `Fin m → α`", "summary": "This file contains alternative definitions of common operators on vectors which expand definitionally to the expected expression when evaluated on `![]` notation. This allows \"proof by reflection\", where we prove `f = ![f 0, f 1]` by defining `FinVec.etaExpand f` to be equal to the RHS definitionally, and then prove that `f = etaExpand f`. The definitions in this file should normally not be used directly; the intent…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Tuple/Reflection.html"}, {"id": "Mathlib.Data.Nat.Cast.Order.Ring", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 1, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": 221.545, "z": 45.78, "size": 0.5091, "title": "Cast of natural numbers: lemmas about bundled ordered semirings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Cast/Order/Ring.html"}, {"id": "Mathlib.Data.Finsupp.Weight", "region_id": "foundations_data", "micro_elevation": 0.4412, "macro_tier": 1, "macro_tier_score": 0.0118, "macro_tier_override": null, "x": 195.802, "z": 17.191, "size": 0.3594, "title": "weights of Finsupp functions", "summary": "The theory of multivariate polynomials and power series is built on the type `σ →₀ ℕ` which gives the exponents of the monomials. Many aspects of the theory (degree, order, graded ring structure) require classifying these exponents according to their total sum `∑ i, f i`, or variants, and this file provides some API for that.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Weight.html"}, {"id": "Mathlib.Data.DFinsupp.Defs", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 2, "macro_tier_score": 0.0344, "macro_tier_override": null, "x": 261.915, "z": 36.785, "size": 0.438, "title": "Dependent functions with finite support", "summary": "For a non-dependent version see `Mathlib/Data/Finsupp/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/Defs.html"}, {"id": "Mathlib.Data.Rat.Cast.Defs", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 2, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 209.563, "z": 31.487, "size": 0.5376, "title": "Casts for Rational Numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Cast/Defs.html"}, {"id": "Mathlib.Data.Int.Cast.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1476, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.6674, "title": "Cast of integers (additional theorems)", "summary": "This file proves additional properties about the *canonical* homomorphism from the integers into an additive group with a one (`Int.cast`), particularly results involving algebraic homomorphisms or the order structure on `ℤ` which were not available in the import dependencies of `Data.Int.Cast.Basic`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Cast/Lemmas.html"}, {"id": "Mathlib.Data.Rat.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 2, "macro_tier_score": 0.1052, "macro_tier_override": null, "x": 219.716, "z": 45.004, "size": 0.3366, "title": "Further lemmas for the Rational Numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Lemmas.html"}, {"id": "Mathlib.Data.Matrix.Basic", "region_id": "foundations_data", "micro_elevation": 0.8824, "macro_tier": 1, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": 156.622, "z": 53.409, "size": 0.4735, "title": "Matrices", "summary": "This file contains basic results on matrices including bundled versions of matrix operators.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/Basic.html"}, {"id": "Mathlib.Data.Matrix.Mul", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 1, "macro_tier_score": 0.0132, "macro_tier_override": null, "x": 157.923, "z": 30.757, "size": 0.4257, "title": "Matrix multiplication", "summary": "This file defines vector and matrix multiplication", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/Mul.html"}, {"id": "Mathlib.Data.Nat.Prime.Factorial", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.011, "macro_tier_override": null, "x": 224.939, "z": 42.437, "size": 0.3019, "title": "Prime natural numbers and the factorial operator", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Prime/Factorial.html"}, {"id": "Mathlib.Data.Set.Order", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": 214.068, "z": 42.236, "size": 0.3191, "title": "Order structures and monotonicity lemmas for `Set`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Order.html"}, {"id": "Mathlib.Data.Finset.Lattice.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.2833, "macro_tier_override": null, "x": 197.925, "z": 69.552, "size": 0.3919, "title": "Lemmas about the lattice structure of finite sets", "summary": "This file contains many results on the lattice structure of `Finset α`, in particular the interaction between union, intersection, empty set and inserting elements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Lattice/Lemmas.html"}, {"id": "Mathlib.Data.Int.Range", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2817, "title": "Intervals in ℤ", "summary": "This file defines integer ranges. `range m n` is the set of integers greater than `m` and strictly less than `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Range.html"}, {"id": "Mathlib.Data.Nat.Factorial.NatCast", "region_id": "foundations_data", "micro_elevation": 0.4118, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 201.703, "z": 64.93, "size": 0.257, "title": "Invertibility of factorials", "summary": "This file contains lemmas providing sufficient conditions for the cast of `n!` to a (semi)ring `A` to be a unit.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Factorial/NatCast.html"}, {"id": "Mathlib.Data.Finset.NatAntidiagonal", "region_id": "foundations_data", "micro_elevation": 0.3824, "macro_tier": 2, "macro_tier_score": 0.0431, "macro_tier_override": null, "x": 195.15, "z": 57.803, "size": 0.362, "title": "Antidiagonals in ℕ × ℕ as finsets", "summary": "This file defines the antidiagonals of ℕ × ℕ as finsets: the `n`-th antidiagonal is the finset of pairs `(i, j)` such that `i + j = n`. This is useful for polynomial multiplication and more generally for sums going from `0` to `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/NatAntidiagonal.html"}, {"id": "Mathlib.Data.List.Shortlex", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 221.196, "z": 41.147, "size": 0.2, "title": "Shortlex ordering of lists.", "summary": "Given a relation `r` on `α`, the shortlex order on `List α` is defined by `L < M` iff * `L.length < M.length` * `L.length = M.length` and `L < M` under the lexicographic ordering over `r` on lists", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Shortlex.html"}, {"id": "Mathlib.Data.List.Lex", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 2, "macro_tier_score": 0.1257, "macro_tier_override": null, "x": 219.289, "z": 40.333, "size": 0.3122, "title": "Lexicographic ordering of lists.", "summary": "The lexicographic order on `List α` is defined by `L < M` iff * `[] < (a :: L)` for any `a` and `L`, * `(a :: L) < (b :: M)` where `a < b`, or * `(a :: L) < (a :: M)` where `L < M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Lex.html"}, {"id": "Mathlib.Data.Matrix.DualNumber", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/DualNumber.html"}, {"id": "Mathlib.Data.Finset.NatDivisors", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": "`Nat.divisors` as a multiplicative homomorphism", "summary": "The main definition of this file is `Nat.divisorsHom : ℕ →* Finset ℕ`, exhibiting `Nat.divisors` as a multiplicative homomorphism from `ℕ` to `Finset ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/NatDivisors.html"}, {"id": "Mathlib.Data.Finset.Lattice.Union", "region_id": "foundations_data", "micro_elevation": 0.6471, "macro_tier": 2, "macro_tier_score": 0.0953, "macro_tier_override": null, "x": 173.594, "z": 24.924, "size": 0.365, "title": "Relating `Finset.biUnion` with lattice operations", "summary": "This file shows `Finset.biUnion` could alternatively be defined in terms of `Finset.sup`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Lattice/Union.html"}, {"id": "Mathlib.Data.Multiset.Powerset", "region_id": "foundations_data", "micro_elevation": 0.4412, "macro_tier": 2, "macro_tier_score": 0.1064, "macro_tier_override": null, "x": 195.829, "z": 63.61, "size": 0.4023, "title": "The powerset of a multiset", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Powerset.html"}, {"id": "Mathlib.Data.Set.Pairwise.Lattice", "region_id": "foundations_data", "micro_elevation": 0.2353, "macro_tier": 2, "macro_tier_score": 0.0952, "macro_tier_override": null, "x": 222.375, "z": 27.651, "size": 0.3595, "title": "Relations holding pairwise", "summary": "In this file we prove many facts about `Pairwise` and the set lattice.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Pairwise/Lattice.html"}, {"id": "Mathlib.Data.Multiset.Sort", "region_id": "foundations_data", "micro_elevation": 0.4412, "macro_tier": 2, "macro_tier_score": 0.0537, "macro_tier_override": null, "x": 208.396, "z": 11.679, "size": 0.3709, "title": "Construct a sorted list from a multiset.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Sort.html"}, {"id": "Mathlib.Data.Multiset.Range", "region_id": "foundations_data", "micro_elevation": 0.4118, "macro_tier": 2, "macro_tier_score": 0.2937, "macro_tier_override": null, "x": 192.942, "z": 58.265, "size": 0.3936, "title": "`Multiset.range n` gives `{0, 1, ..., n-1}` as a multiset.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Range.html"}, {"id": "Mathlib.Data.Finite.Perm", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 203.631, "z": -7.196, "size": 0.2679, "title": "Properties of `Equiv.Perm` on `Finite` types", "summary": "Let `α` be a `Finite` type. * `Nat.card_perm`: cardinality of `Equiv.Perm α`. * `Equiv.Perm.isCyclic_of_card_le_two`: if `Nat.card α ≤ 2`, then `Equiv.Perm α` is cyclic. * `Equiv.Perm.isCyclic_iff_card_le_two`: `Equiv.Perm α` is cyclic iff `Nat.card α ≤ 2`. * `Equiv.Perm.isMulCommutative_iff_card_le_two`: `Equiv.Perm α` is commutative iff `Nat.card α ≤ 2`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finite/Perm.html"}, {"id": "Mathlib.Data.Fintype.Perm", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 1, "macro_tier_score": 0.0111, "macro_tier_override": null, "x": 171.611, "z": 60.998, "size": 0.3112, "title": "`Fintype` instances for `Equiv` and `Perm`", "summary": "Main declarations: * `permsOfFinset s`: The finset of permutations of the finset `s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Perm.html"}, {"id": "Mathlib.Data.Fin.SuccPred", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.1913, "macro_tier_override": null, "x": 209.595, "z": 40.82, "size": 0.462, "title": "Successors and predecessor operations of `Fin n`", "summary": "This file contains a number of definitions and lemmas related to `Fin.succ`, `Fin.pred`, and related operations on `Fin n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/SuccPred.html"}, {"id": "Mathlib.Data.Fin.Parity", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 185.869, "z": 0.392, "size": 0.2686, "title": "Parity in `Fin n`", "summary": "In this file we prove that an element `k : Fin n` is even in `Fin n` iff `n` is odd or `Fin.val k` is even. We also prove a lemma about parity of `Fin.succAbove i j + Fin.predAbove j i` which can be used to prove `d ∘ d = 0` for de Rham cohomologies.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Parity.html"}, {"id": "Mathlib.Data.ZMod.Defs", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 2, "macro_tier_score": 0.0345, "macro_tier_override": null, "x": 180.303, "z": 7.578, "size": 0.4411, "title": "Definition of `ZMod n` + basic results.", "summary": "This file provides the basic details of `ZMod n`, including its commutative ring structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ZMod/Defs.html"}, {"id": "Mathlib.Data.Finset.Interval", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 234.135, "z": 86.5, "size": 0.2302, "title": "Intervals of finsets as finsets", "summary": "This file provides the `LocallyFiniteOrder` instance for `Finset α` and calculates the cardinality of finite intervals of finsets. If `s t : Finset α`, then `Finset.Icc s t` is the finset of finsets which include `s` and are included in `t`. For example, `Finset.Icc {0, 1} {0, 1, 2, 3} = {{0, 1}, {0, 1, 2}, {0, 1, 3}, {0, 1, 2, 3}}` and `Finset.Icc {0, 1, 2} {0, 1, 3} = {}`. In addition, this file gives…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Interval.html"}, {"id": "Mathlib.Data.Fintype.EquivFin", "region_id": "foundations_data", "micro_elevation": 0.6765, "macro_tier": 2, "macro_tier_score": 0.2538, "macro_tier_override": null, "x": 203.74, "z": 84.024, "size": 0.8654, "title": "Equivalences between `Fintype`, `Fin` and `Finite`", "summary": "This file defines the bijection between a `Fintype α` and `Fin (Fintype.card α)`, and uses this to relate `Fintype` with `Finite`. From that we can derive properties of `Finite` and `Infinite`, and show some instances of `Infinite`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/EquivFin.html"}, {"id": "Mathlib.Data.Finset.Option", "region_id": "foundations_data", "micro_elevation": 0.6176, "macro_tier": 2, "macro_tier_score": 0.13, "macro_tier_override": null, "x": 253.267, "z": 49.402, "size": 0.5012, "title": "Finite sets in `Option α`", "summary": "In this file we define * `Option.toFinset`: construct an empty or singleton `Finset α` from an `Option α`; * `Finset.insertNone`: given `s : Finset α`, lift it to a finset on `Option α` using `Option.some` and then insert `Option.none`; * `Finset.eraseNone`: given `s : Finset (Option α)`, returns `t : Finset α` such that `x ∈ t ↔ some x ∈ s`. Then we prove some basic lemmas about these definitions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Option.html"}, {"id": "Mathlib.Data.Finset.Density", "region_id": "foundations_data", "micro_elevation": 0.6765, "macro_tier": 1, "macro_tier_score": 0.0019, "macro_tier_override": null, "x": 168.768, "z": 39.345, "size": 0.3863, "title": "Density of a finite set", "summary": "This defines the density of a `Finset` and provides induction principles for finsets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Density.html"}, {"id": "Mathlib.Data.NNRat.Order", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.0324, "macro_tier_override": null, "x": 213.945, "z": 38.508, "size": 0.3411, "title": "Bundled ordered algebra structures on `ℚ≥0`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/NNRat/Order.html"}, {"id": "Mathlib.Data.Rat.Cast.CharZero", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 2, "macro_tier_score": 0.0673, "macro_tier_override": null, "x": 201.918, "z": 42.064, "size": 0.495, "title": "Casts of rational numbers into characteristic zero fields (or division rings).", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Cast/CharZero.html"}, {"id": "Mathlib.Data.FunLike.Graded", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 216.296, "z": 43.047, "size": 0.2814, "title": "Class of grading-preserving functions and isomorphisms", "summary": "We define `GradedFunLike F 𝒜 ℬ` where `𝒜` and `ℬ` represent some sort of grading. This class assumes `FunLike A B` where `A` and `B` are the underlying types. We also define `GradedEquivLike E 𝒜 ℬ`, which is similar to `EquivLike`, where here `e : E` is required to satisfy `x ∈ 𝒜 i ↔ e x ∈ ℬ i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/FunLike/Graded.html"}, {"id": "Mathlib.Data.SetLike.Basic", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.2874, "macro_tier_override": null, "x": 214.53, "z": 42.012, "size": 0.5335, "title": "Typeclass for types with a set-like extensionality property", "summary": "The `Membership` typeclass is used to let terms of a type have elements. Many instances of `Membership` have a set-like extensionality property: things are equal iff they have the same elements. The `SetLike` typeclass provides a unified interface to define a `Membership` that is extensional in this way. The main use of `SetLike` is for algebraic subobjects (such as `Submonoid` and `Submodule`), whose non-proof data…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/SetLike/Basic.html"}, {"id": "Mathlib.Data.Multiset.ZeroCons", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 2, "macro_tier_score": 0.2948, "macro_tier_override": null, "x": 221.878, "z": 51.078, "size": 0.4394, "title": "Definition of `0` and `::ₘ`", "summary": "This file defines constructors for multisets: * `Zero (Multiset α)` instance: the empty multiset * `Multiset.cons`: add one element to a multiset * `Singleton α (Multiset α)` instance: multiset with one element It also defines the following predicates on multisets: * `Multiset.Rel`: `Rel r s t` lifts the relation `r` between two elements to a relation between `s` and `t`, s.t. there is a one-to-one mapping between…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/ZeroCons.html"}, {"id": "Mathlib.Data.Int.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 204.25, "z": 37.284, "size": 0.2, "title": "Miscellaneous lemmas about the integers", "summary": "This file contains lemmas about integers, which require further imports than `Data.Int.Basic` or `Data.Int.Order`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Lemmas.html"}, {"id": "Mathlib.Data.Int.Bitwise", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 206.0, "z": 38.199, "size": 0.2478, "title": "Bitwise operations on integers", "summary": "Possibly only of archaeological significance.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Bitwise.html"}, {"id": "Mathlib.Data.Int.Order.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2478, "title": "Further lemmas about the integers", "summary": "The distinction between this file and `Mathlib/Data/Int/Order/Basic.lean` is not particularly clear. They are separated by now to minimize the porting requirements for tactics during the transition to mathlib4. Please feel free to reorganize these two files.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Order/Lemmas.html"}, {"id": "Mathlib.Data.Finsupp.Fin", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 212.869, "z": 44.232, "size": 0.2727, "title": "`cons` and `tail` for maps `Fin n →₀ M`", "summary": "We interpret maps `Fin n →₀ M` as `n`-tuples of elements of `M`, We define the following operations: * `Finsupp.tail` : the tail of a map `Fin (n + 1) →₀ M`, i.e., its last `n` entries; * `Finsupp.cons` : adding an element at the beginning of an `n`-tuple, to get an `n + 1`-tuple; In this context, we prove some usual properties of `tail` and `cons`, analogous to those of `Data.Fin.Tuple.Basic`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Fin.html"}, {"id": "Mathlib.Data.Set.Notation", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0125, "macro_tier_override": null, "x": 208.739, "z": 43.814, "size": 0.3948, "title": "Set Notation", "summary": "This file defines two pieces of scoped notation related to sets and subtypes. The first is a coercion; for each `α : Type*` and `s : Set α`, `(↑) : Set s → Set α` is the function coercing `t : Set s` into a set in the ambient type; i.e. `↑t = Subtype.val '' t`. The second, for `s t : Set α`, is the notation `s ↓∩ t`, which denotes the intersection of `s` and `t` as a set in `Set s`. These notations are developed…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Notation.html"}, {"id": "Mathlib.Data.Set.Image", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 2, "macro_tier_score": 0.359, "macro_tier_override": null, "x": 209.803, "z": 47.252, "size": 1.0061, "title": "Images and preimages of sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Image.html"}, {"id": "Mathlib.Data.DFinsupp.Sigma", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 236.716, "z": 87.433, "size": 0.4082, "title": "`DFinsupp` on `Sigma` types", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/Sigma.html"}, {"id": "Mathlib.Data.DFinsupp.Submonoid", "region_id": "foundations_data", "micro_elevation": 0.8235, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": 214.577, "z": -14.02, "size": 0.4082, "title": "`DFinsupp` and submonoids", "summary": "This file mainly concerns the interaction between submonoids and products/sums of `DFinsupp`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/Submonoid.html"}, {"id": "Mathlib.Data.Rat.Encodable", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 1, "macro_tier_score": 0.011, "macro_tier_override": null, "x": 209.963, "z": 42.091, "size": 0.3031, "title": "The rationals are `Encodable`.", "summary": "As a consequence we also get the instance `Countable ℚ`. This is kept separate from `Data.Rat.Defs` in order to minimize imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Encodable.html"}, {"id": "Mathlib.Data.Rat.Cast.Order", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": 226.565, "z": 36.739, "size": 0.4167, "title": "Casts of rational numbers into linear ordered fields.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Cast/Order.html"}, {"id": "Mathlib.Data.Subtype", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.3777, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.6853, "title": "Subtypes", "summary": "This file provides basic API for subtypes, which are defined in core. A subtype is a type made from restricting another type, say `α`, to its elements that satisfy some predicate, say `p : α → Prop`. Specifically, it is the type of pairs `⟨val, property⟩` where `val : α` and `property : p val`. It is denoted `Subtype p` and notation `{val : α // p val}` is available. A subtype has a natural coercion to the parent…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Subtype.html"}, {"id": "Mathlib.Data.DFinsupp.Interval", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 165.532, "z": 70.056, "size": 0.239, "title": "Finite intervals of finitely supported functions", "summary": "This file provides the `LocallyFiniteOrder` instance for `Π₀ i, α i` when `α` itself is locally finite and calculates the cardinality of its finite intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/Interval.html"}, {"id": "Mathlib.Data.DFinsupp.BigOperators", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 1, "macro_tier_score": 0.0223, "macro_tier_override": null, "x": 168.36, "z": 13.557, "size": 0.3626, "title": "Dependent functions with finite support", "summary": "For a non-dependent version see `Mathlib/Data/Finsupp/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/BigOperators.html"}, {"id": "Mathlib.Data.DFinsupp.Order", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 2, "macro_tier_score": 0.0318, "macro_tier_override": null, "x": 191.475, "z": -7.261, "size": 0.2974, "title": "Pointwise order on finitely supported dependent functions", "summary": "This file lifts order structures on the `α i` to `Π₀ i, α i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/Order.html"}, {"id": "Mathlib.Data.Set.Card", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 2, "macro_tier_score": 0.037, "macro_tier_override": null, "x": 177.008, "z": 75.39, "size": 0.5219, "title": "Noncomputable Set Cardinality", "summary": "We define the cardinality of set `s` as a term `Set.encard s : ℕ∞` and a term `Set.ncard s : ℕ`. The latter takes the junk value of zero if `s` is infinite. Both functions are noncomputable, and are defined in terms of `ENat.card` (which takes a type as its argument); this file can be seen as an API for the same function in the special case where the type is a coercion of a `Set`, allowing for smoother interactions…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Card.html"}, {"id": "Mathlib.Data.Ordering.Basic", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0228, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3865, "title": "Helper definitions and instances for `Ordering`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Ordering/Basic.html"}, {"id": "Mathlib.Data.List.Cycle", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 261.826, "z": 45.043, "size": 0.2465, "title": "Cycles of a list", "summary": "Lists have an equivalence relation of whether they are rotational permutations of one another. This relation is defined as `IsRotated`. Based on this, we define the quotient of lists by the rotation relation, called `Cycle`. We also define a representation of concrete cycles, available when viewing them in a goal state or via `#eval`, when over representable types. For example, the cycle `(2 1 4 3)` will be shown as…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Cycle.html"}, {"id": "Mathlib.Data.Fintype.List", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 178.97, "z": 71.798, "size": 0.2462, "title": "Fintype instance for nodup lists", "summary": "The subtype of `{l : List α // l.Nodup}` over a `[Fintype α]` admits a `Fintype` instance.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/List.html"}, {"id": "Mathlib.Data.Fintype.Prod", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 2, "macro_tier_score": 0.1589, "macro_tier_override": null, "x": 209.771, "z": 86.891, "size": 0.4171, "title": "fintype instance for the product of two fintypes.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Prod.html"}, {"id": "Mathlib.Data.ULift", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1923, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.7238, "title": "Extra lemmas about `ULift` and `PLift`", "summary": "In this file we provide `Subsingleton`, `Unique`, `DecidableEq`, and `isEmpty` instances for `ULift α` and `PLift α`. We also prove `ULift.forall`, `ULift.exists`, `PLift.forall`, and `PLift.exists`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ULift.html"}, {"id": "Mathlib.Data.ENNReal.Real", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.0331, "macro_tier_override": null, "x": 214.728, "z": 44.064, "size": 0.3848, "title": "Maps between real and extended non-negative real numbers", "summary": "This file focuses on the functions `ENNReal.toReal : ℝ≥0∞ → ℝ` and `ENNReal.ofReal : ℝ → ℝ≥0∞` which were defined in `Data.ENNReal.Basic`. It collects all the basic results of the interactions between these functions and the algebraic and lattice operations, although a few may appear in earlier files. This file provides a `positivity` extension for `ENNReal.ofReal`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ENNReal/Real.html"}, {"id": "Mathlib.Data.Set.Operations", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.0829, "macro_tier_override": null, "x": 216.987, "z": 38.759, "size": 0.6242, "title": "Basic definitions about sets", "summary": "In this file we define various operations on sets. We also provide basic lemmas needed to unfold the definitions. More advanced theorems about these definitions are located in other files in `Mathlib/Data/Set`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Operations.html"}, {"id": "Mathlib.Data.List.Triplewise", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2478, "title": "Triplewise predicates on list.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Triplewise.html"}, {"id": "Mathlib.Data.ENNReal.Holder", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 1, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 206.14, "z": 46.784, "size": 0.2714, "title": "Hölder triples", "summary": "This file defines a new class: `ENNReal.HolderTriple` which takes arguments `p q r : ℝ≥0∞`, with `r` marked as a `semiOutParam`, and states that `p⁻¹ + q⁻¹ = r⁻¹`. This is exactly the condition for which **Hölder's inequality** is valid (see `MeasureTheory.MemLp.smul`). This allows us to declare a heterogeneous scalar multiplication (`HSMul`) instance on `MeasureTheory.Lp` spaces. In this file we provide many…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ENNReal/Holder.html"}, {"id": "Mathlib.Data.Nat.Prime.Defs", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 2, "macro_tier_score": 0.0544, "macro_tier_override": null, "x": 210.68, "z": 33.129, "size": 0.405, "title": "Prime numbers", "summary": "This file deals with prime numbers: natural numbers `p ≥ 2` whose only divisors are `p` and `1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Prime/Defs.html"}, {"id": "Mathlib.Data.Int.Cast.Basic", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 2, "macro_tier_score": 0.0411, "macro_tier_override": null, "x": 210.918, "z": 47.738, "size": 0.622, "title": "Cast of integers (additional theorems)", "summary": "This file proves additional properties about the *canonical* homomorphism from the integers into an additive group with a one (`Int.cast`). There is also `Mathlib.Data.Int.Cast.Lemmas`, which includes lemmas stated in terms of algebraic homomorphisms, and results involving the order structure of `ℤ`. By contrast, this file's only import beyond `Mathlib.Data.Int.Cast.Defs` is `Mathlib.Algebra.Group.Basic`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Cast/Basic.html"}, {"id": "Mathlib.Data.Finsupp.Defs", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0339, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.4188, "title": "Type of functions with finite support", "summary": "For any type `α` and any type `M` with zero, we define the type `Finsupp α M` (notation: `α →₀ M`) of finitely supported functions from `α` to `M`, i.e. the functions which are zero everywhere on `α` except on a finite set. Functions with finite support are used (at least) in the following parts of the library: * `MonoidAlgebra R M` and `AddMonoidAlgebra R M` are defined as `M →₀ R`; * polynomials and multivariate…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Defs.html"}, {"id": "Mathlib.Data.Tree.Basic", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 211.55, "z": 40.755, "size": 0.2705, "title": "Binary tree", "summary": "Provides binary tree storage for values of any type, with O(lg n) retrieval. See also `Lean.Data.RBTree` for red-black trees - this version allows more operations to be defined and is better suited for in-kernel computation. We also specialize for `BinaryTree Unit`, which is a binary tree without any additional data. We provide the notation `a △ b` for making a `BinaryTree Unit` with children `a` and `b`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Tree/Basic.html"}, {"id": "Mathlib.Data.List.Lookmap", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2501, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Lookmap.html"}, {"id": "Mathlib.Data.Finset.Piecewise", "region_id": "foundations_data", "micro_elevation": 0.5882, "macro_tier": 1, "macro_tier_score": 0.0031, "macro_tier_override": null, "x": 234.678, "z": 7.82, "size": 0.4363, "title": "Functions defined piecewise on a finset", "summary": "This file defines `Finset.piecewise`: Given two functions `f`, `g`, `s.piecewise f g` is a function which is equal to `f` on `s` and `g` on the complement.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Piecewise.html"}, {"id": "Mathlib.Data.ENat.Lattice", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 2, "macro_tier_score": 0.0239, "macro_tier_override": null, "x": 201.872, "z": 41.709, "size": 0.4342, "title": "Extended natural numbers form a complete linear order", "summary": "This instance is not in `Data.ENat.Basic` to avoid dependency on `Finset`s. We also restate some lemmas about `WithTop` for `ENat` to have versions that use `Nat.cast` instead of `WithTop.some`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ENat/Lattice.html"}, {"id": "Mathlib.Data.Countable.Small", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 205.864, "z": 42.051, "size": 0.3404, "title": "All countable types are small.", "summary": "That is, any countable type is equivalent to a type in any universe.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Countable/Small.html"}, {"id": "Mathlib.Data.Fintype.Powerset", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 2, "macro_tier_score": 0.0355, "macro_tier_override": null, "x": 253.971, "z": 17.267, "size": 0.4773, "title": "fintype instance for `Set α`, when `α` is a fintype", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Powerset.html"}, {"id": "Mathlib.Data.Nat.Cast.Order.Basic", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 2, "macro_tier_score": 0.0572, "macro_tier_override": null, "x": 209.329, "z": 33.803, "size": 0.5033, "title": "Cast of natural numbers: lemmas about order", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Cast/Order/Basic.html"}, {"id": "Mathlib.Data.Set.Countable", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.0156, "macro_tier_override": null, "x": 225.055, "z": 41.625, "size": 0.5045, "title": "Countable sets", "summary": "In this file we define `Set.Countable s` as `Countable s` and prove basic properties of this definition. Note that this definition does not provide a computable encoding. For a noncomputable conversion to `Encodable s`, use `Set.Countable.nonempty_encodable`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Countable.html"}, {"id": "Mathlib.Data.List.Perm.Lattice", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 2, "macro_tier_score": 0.2931, "macro_tier_override": null, "x": 204.96, "z": 45.102, "size": 0.3597, "title": "List Permutations and list lattice operations.", "summary": "This file develops theory about the `List.Perm` relation and the lattice structure on lists.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Perm/Lattice.html"}, {"id": "Mathlib.Data.List.TakeDrop", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.2928, "macro_tier_override": null, "x": 209.787, "z": 39.115, "size": 0.3398, "title": "`Take` and `Drop` lemmas for lists", "summary": "This file provides lemmas about `List.take` and `List.drop` and related functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/TakeDrop.html"}, {"id": "Mathlib.Data.List.Lattice", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 2, "macro_tier_score": 0.2937, "macro_tier_override": null, "x": 212.32, "z": 34.671, "size": 0.3901, "title": "Lattice structure of lists", "summary": "This file proves basic properties about `List.disjoint`, `List.union`, `List.inter` and `List.bagInter`, which are defined in core Lean and `Data.List.Defs`. `l₁ ∪ l₂` is the list where all elements of `l₁` have been inserted in `l₂` in order. For example, `[0, 0, 1, 2, 2, 3] ∪ [4, 3, 3, 0] = [1, 2, 4, 3, 3, 0]`. `l₁ ∩ l₂` is the list of elements of `l₁` in order which are in `l₂`. For example, `[0, 0, 1, 2, 2, 3] ∩…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Lattice.html"}, {"id": "Mathlib.Data.Nat.GCD.Prime", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 220.46, "z": 47.13, "size": 0.2, "title": "Lemmas related to `Nat.Prime` and `lcm`", "summary": "This file contains lemmas related to `Nat.Prime`. These lemmas are kept separate from `Mathlib/Data/Nat/GCD/Basic.lean` in order to minimize imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/GCD/Prime.html"}, {"id": "Mathlib.Data.Set.Defs", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.3149, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.8468, "title": "Sets", "summary": "This file sets up the theory of sets whose elements have a given type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Defs.html"}, {"id": "Mathlib.Data.Nat.Init", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.4765, "macro_tier_override": null, "x": 215.353, "z": 40.827, "size": 0.5735, "title": "Basic operations on the natural numbers", "summary": "This file contains: * some basic lemmas about natural numbers * extra recursors: * `leRecOn`, `le_induction`: recursion and induction principles starting at non-zero numbers * `decreasing_induction`: recursion growing downwards * `le_rec_on'`, `decreasing_induction'`: versions with slightly weaker assumptions * `strong_rec'`: recursion based on strong inequalities * decidability instances on predicates about the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Init.html"}, {"id": "Mathlib.Data.Int.Notation", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.4959, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.7492, "title": "Notation `ℤ` for the integers.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Notation.html"}, {"id": "Mathlib.Data.Fintype.Parity", "region_id": "foundations_data", "micro_elevation": 0.6765, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 207.941, "z": -3.971, "size": 0.2582, "title": "The cardinality of `Fin 2` is even.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Parity.html"}, {"id": "Mathlib.Data.Int.Basic", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0145, "macro_tier_override": null, "x": 214.229, "z": 34.61, "size": 0.4723, "title": "Basic operations on the integers", "summary": "This file builds on `Data.Int.Init` by adding basic lemmas on integers. depending on Mathlib definitions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Basic.html"}, {"id": "Mathlib.Data.EReal.Operations", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 218.463, "z": 32.06, "size": 0.3389, "title": "Addition, negation, subtraction and multiplication on extended real numbers", "summary": "Addition and multiplication in `EReal` are problematic in the presence of `±∞`, but negation has a natural definition and satisfies the usual properties. In particular, it is an order-reversing isomorphism. The construction of `EReal` as `WithBot (WithTop ℝ)` endows a `LinearOrderedAddCommMonoid` structure on it. However, addition is badly behaved at `(⊥, ⊤)` and `(⊤, ⊥)`, so this cannot be upgraded to a group…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/EReal/Operations.html"}, {"id": "Mathlib.Data.Nat.Digits.Defs", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2418, "title": "Digits of a natural number", "summary": "This provides a basic API for extracting the digits of a natural number in a given base, and reconstructing numbers from their digits. We also prove some divisibility tests based on digits, in particular completing Theorem #85 from https://www.cs.ru.nl/~freek/100/. Also included is a bound on the length of `Nat.toDigits` from core.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Digits/Defs.html"}, {"id": "Mathlib.Data.Matrix.Bilinear", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/Bilinear.html"}, {"id": "Mathlib.Data.Finsupp.MonomialOrder", "region_id": "foundations_data", "micro_elevation": 0.9118, "macro_tier": 1, "macro_tier_score": 0.011, "macro_tier_override": null, "x": 206.71, "z": -19.483, "size": 0.2992, "title": "Monomial orders", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/MonomialOrder.html"}, {"id": "Mathlib.Data.Finsupp.WellFounded", "region_id": "foundations_data", "micro_elevation": 0.8824, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 175.537, "z": 84.682, "size": 0.3353, "title": "Well-foundedness of the lexicographic and product orders on `Finsupp`", "summary": "`Finsupp.Lex.wellFounded` and the two variants that follow it essentially say that if `(· > ·)` is a well order on `α`, `(· < ·)` is well-founded on `N`, and `0` is a bottom element in `N`, then the lexicographic `(· < ·)` is well-founded on `α →₀ N`. `Finsupp.Lex.wellFoundedLT_of_finite` says that if `α` is finite and equipped with a linear order and `(· < ·)` is well-founded on `N`, then the lexicographic `(· <…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/WellFounded.html"}, {"id": "Mathlib.Data.Nat.SuccPred", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.0558, "macro_tier_override": null, "x": 209.575, "z": 40.216, "size": 0.4589, "title": "Successors and predecessors of naturals", "summary": "In this file, we show that `ℕ` is both an archimedean `succOrder` and an archimedean `predOrder`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/SuccPred.html"}, {"id": "Mathlib.Data.Int.Init", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.019, "macro_tier_override": null, "x": 214.256, "z": 42.163, "size": 0.5937, "title": "Basic operations on the integers", "summary": "This file contains some basic lemmas about integers. See note [foundational algebra order theory]. This file should not depend on anything defined in Mathlib (except for notation), so that it can be upstreamed to Batteries easily.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Init.html"}, {"id": "Mathlib.Data.Set.MemPartition", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": 237.667, "z": -6.17, "size": 0.3216, "title": "Partitions based on membership of a sequence of sets", "summary": "Let `f : ℕ → Set α` be a sequence of sets. For `n : ℕ`, we can form the set of points that are in `f 0 ∪ f 1 ∪ ... ∪ f (n-1)`; then the set of points in `(f 0)ᶜ ∪ f 1 ∪ ... ∪ f (n-1)` and so on for all 2^n choices of a set or its complement. The at most 2^n sets we obtain form a partition of `univ : Set α`. We call that partition `memPartition f n` (the membership partition of `f`). For `n = 0` we set `memPartition…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/MemPartition.html"}, {"id": "Mathlib.Data.Int.Log", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 215.64, "z": 37.173, "size": 0.2645, "title": "Integer logarithms in a field with respect to a natural base", "summary": "This file defines two `ℤ`-valued analogs of the logarithm of `r : R` with base `b : ℕ`: * `Int.log b r`: Lower logarithm, or floor **log**. Greatest `k` such that `↑b^k ≤ r`. * `Int.clog b r`: Upper logarithm, or **c**eil **log**. Least `k` such that `r ≤ ↑b^k`. Note that `Int.log` gives the position of the left-most non-zero digit: ```lean #eval (Int.log 10 (0.09 : ℚ), Int.log 10 (0.10 : ℚ), Int.log 10 (0.11 : ℚ))…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Log.html"}, {"id": "Mathlib.Data.Nat.Factorial.Cast", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 1, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": 205.733, "z": 39.523, "size": 0.2553, "title": "Cast of factorials", "summary": "This file allows calculating factorials (including ascending and descending ones) as elements of a semiring. This is particularly crucial for `Nat.descFactorial` as subtraction on `ℕ` does **not** correspond to subtraction on a general semiring. For example, we can't rely on existing cast lemmas to prove `↑(a.descFactorial 2) = ↑a * (↑a - 1)`. We must use the fact that, whenever `↑(a - 1)` is not equal to `↑a - 1`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Factorial/Cast.html"}, {"id": "Mathlib.Data.List.ProdSigma", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 215.932, "z": 37.391, "size": 0.2571, "title": "Lists in product and sigma types", "summary": "This file proves basic properties of `List.product` and `List.sigma`, which are list constructions living in `Prod` and `Sigma` types respectively. Their definitions can be found in [`Data.List.Defs`](./defs). Beware, this is not about `List.prod`, the multiplicative product.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/ProdSigma.html"}, {"id": "Mathlib.Data.Seq.Defs", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 1, "macro_tier_score": 0.0219, "macro_tier_override": null, "x": 216.669, "z": 49.563, "size": 0.3418, "title": "Possibly infinite lists", "summary": "This file provides `Stream'.Seq α`, a type representing possibly infinite lists (referred here as sequences). It is encoded as an infinite stream of options such that if `f n = none`, then `f m = none` for all `m ≥ n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Seq/Defs.html"}, {"id": "Mathlib.Data.Finset.Preimage", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 1, "macro_tier_score": 0.0027, "macro_tier_override": null, "x": 250.869, "z": 71.398, "size": 0.4192, "title": "Preimage of a `Finset` under an injective map.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Preimage.html"}, {"id": "Mathlib.Data.Matrix.Invertible", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/Invertible.html"}, {"id": "Mathlib.Data.Finsupp.Notation", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 1, "macro_tier_score": 0.0108, "macro_tier_override": null, "x": 216.82, "z": 38.437, "size": 0.2851, "title": "Notation for `Finsupp`", "summary": "This file provides `fun₀ | 3 => a | 7 => b` notation for `Finsupp`, which desugars to `Finsupp.update` and `Finsupp.single`, in the same way that `{a, b}` desugars to `insert` and `singleton`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Notation.html"}, {"id": "Mathlib.Data.Int.WithZero", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 212.465, "z": 42.06, "size": 0.2824, "title": "WithZero", "summary": "In this file we provide some basic API lemmas for the `WithZero` construction and we define the morphism `WithZeroMultInt.toNNReal`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/WithZero.html"}, {"id": "Mathlib.Data.Nat.Prime.Int", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.0119, "macro_tier_override": null, "x": 224.74, "z": 43.344, "size": 0.3619, "title": "Prime numbers in the naturals and the integers", "summary": "TODO: This file can probably be merged with `Mathlib/Data/Int/NatPrime.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Prime/Int.html"}, {"id": "Mathlib.Data.Rat.Sqrt", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 212.35, "z": 30.735, "size": 0.2688, "title": "Square root on rational numbers", "summary": "This file defines the square root function on rational numbers `Rat.sqrt` and proves several theorems about it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Sqrt.html"}, {"id": "Mathlib.Data.Multiset.Replicate", "region_id": "foundations_data", "micro_elevation": 0.2941, "macro_tier": 2, "macro_tier_score": 0.2932, "macro_tier_override": null, "x": 229.091, "z": 51.94, "size": 0.3644, "title": "Repeating elements in multisets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Replicate.html"}, {"id": "Mathlib.Data.Multiset.Functor", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 213.528, "z": 42.332, "size": 0.239, "title": "Functoriality of `Multiset`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Functor.html"}, {"id": "Mathlib.Data.Multiset.Bind", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2155, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.5597, "title": "Bind operation for multisets", "summary": "This file defines a few basic operations on `Multiset`, notably the monadic bind.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Bind.html"}, {"id": "Mathlib.Data.Set.PowersetCard", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 243.599, "z": 83.349, "size": 0.2763, "title": "Combinations", "summary": "Combinations in a type are finite subsets of given cardinality. * `Set.powersetCard α n` is the set of all `Finset α` with cardinality `n`. The name is chosen in relation with `Finset.powersetCard` which corresponds to the analogous structure for subsets of given cardinality of a given `Finset`, as a `Finset`. * `Set.powersetCard.card` proves that the `Nat.card`-cardinality of this set is equal to `(Nat.card…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/PowersetCard.html"}, {"id": "Mathlib.Data.Nat.Multiplicity", "region_id": "foundations_data", "micro_elevation": 0.9118, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": 260.511, "z": 78.025, "size": 0.3149, "title": "Natural number multiplicity", "summary": "This file contains lemmas about the multiplicity function (the maximum prime power dividing a number) when applied to naturals, in particular calculating it for factorials and binomial coefficients.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Multiplicity.html"}, {"id": "Mathlib.Data.EReal.Basic", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 1, "macro_tier_score": 0.0127, "macro_tier_override": null, "x": 214.299, "z": 48.119, "size": 0.4008, "title": "The extended real numbers", "summary": "This file defines `EReal`, `ℝ` with a top element `⊤` and a bottom element `⊥`, implemented as `WithBot (WithTop ℝ)`. `EReal` is a `CompleteLinearOrder`, deduced by typeclass inference from the fact that `WithBot (WithTop L)` completes a conditionally complete linear order `L`. Coercions from `ℝ` (called `coe` in lemmas) and from `ℝ≥0∞` (`coe_ennreal`) are registered and their basic properties proved. The latter…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/EReal/Basic.html"}, {"id": "Mathlib.Data.ENNReal.Operations", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 2, "macro_tier_score": 0.0355, "macro_tier_override": null, "x": 218.466, "z": 37.403, "size": 0.4753, "title": "Properties of addition, multiplication and subtraction on extended non-negative real numbers", "summary": "In this file we prove elementary properties of algebraic operations on `ℝ≥0∞`, including addition, multiplication, natural powers and truncated subtraction, as well as how these interact with the order structure on `ℝ≥0∞`. Notably excluded from this list are inversion and division, the definitions and properties of which can be found in `Mathlib/Data/ENNReal/Inv.lean`. Note: the definitions of the operations…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ENNReal/Operations.html"}, {"id": "Mathlib.Data.Tree.Traversable", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.079, "z": 44.259, "size": 0.2, "title": "Traversable Binary Tree", "summary": "Provides a `Traversable` instance for the `Tree` type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Tree/Traversable.html"}, {"id": "Mathlib.Data.NNRat.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 220.978, "z": 46.547, "size": 0.2, "title": "Field and action structures on the nonnegative rationals", "summary": "This file provides additional results about `NNRat` that cannot live in earlier files due to import cycles.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/NNRat/Lemmas.html"}, {"id": "Mathlib.Data.Rat.Floor", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0222, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3565, "title": "Floor Function for Rational Numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Floor.html"}, {"id": "Mathlib.Data.Int.CharZero", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": 216.37, "z": 42.966, "size": 0.3706, "title": "Injectivity of `Int.Cast` into characteristic zero rings and fields.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/CharZero.html"}, {"id": "Mathlib.Data.Int.Cast.Field", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0108, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2865, "title": "Cast of integers into fields", "summary": "This file concerns the canonical homomorphism `ℤ → F`, where `F` is a field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Cast/Field.html"}, {"id": "Mathlib.Data.Int.Cast.Pi", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0219, "macro_tier_override": null, "x": 212.221, "z": 38.892, "size": 0.336, "title": "Cast of integers to function types", "summary": "This file provides a (pointwise) cast from `ℤ` to function types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Cast/Pi.html"}, {"id": "Mathlib.Data.Set.Equitable", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2492, "title": "Equitable functions", "summary": "This file defines equitable functions. A function `f` is equitable on a set `s` if `f a₁ ≤ f a₂ + 1` for all `a₁, a₂ ∈ s`. This is mostly useful when the codomain of `f` is `ℕ` or `ℤ` (or more generally a successor order).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Equitable.html"}, {"id": "Mathlib.Data.Ineq", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3457, "title": "`Ineq` datatype", "summary": "This file contains an enum `Ineq` (whose constructors are `eq`, `le`, `lt`), and operations involving it. The type `Ineq` is one of the fundamental objects manipulated by the `linarith` and `linear_combination` tactics.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Ineq.html"}, {"id": "Mathlib.Data.Fintype.Pigeonhole", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": 257.247, "z": 24.311, "size": 0.3231, "title": "Pigeonhole principles in finite types", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Pigeonhole.html"}, {"id": "Mathlib.Data.Finset.Union", "region_id": "foundations_data", "micro_elevation": 0.5882, "macro_tier": 2, "macro_tier_score": 0.2046, "macro_tier_override": null, "x": 224.369, "z": 77.7, "size": 0.5457, "title": "Unions of finite sets", "summary": "This file defines the union of a family `t : α → Finset β` of finsets bounded by a finset `s : Finset α`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Union.html"}, {"id": "Mathlib.Data.FunLike.Equiv", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 1, "macro_tier_score": 0.0074, "macro_tier_override": null, "x": 210.375, "z": 42.757, "size": 0.5653, "title": "Typeclass for a type `F` with an injective map to `A ≃ B`", "summary": "This typeclass is primarily for use by isomorphisms like `MonoidEquiv` and `LinearEquiv`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/FunLike/Equiv.html"}, {"id": "Mathlib.Data.Int.Order.Basic", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.1261, "macro_tier_override": null, "x": 211.547, "z": 40.039, "size": 0.3426, "title": "The order relation on the integers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Order/Basic.html"}, {"id": "Mathlib.Data.FunLike.Module", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": 210.437, "z": 45.377, "size": 0.4241, "title": "Module instances for `FunLike` types", "summary": "In this file we define various instances related to modules for `FunLike` types. Note that currently, these are not registered as instances, but only `abbrev`s to avoid long typeclass searches.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/FunLike/Module.html"}, {"id": "Mathlib.Data.Int.Cast.Defs", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 2, "macro_tier_score": 0.0468, "macro_tier_override": null, "x": 215.72, "z": 45.765, "size": 0.7298, "title": "Cast of integers", "summary": "This file defines the *canonical* homomorphism from the integers into an additive group with a one (typically a `Ring`). In additive groups with a one element, there exists a unique such homomorphism and we store it in the `intCast : ℤ → R` field. Preferentially, the homomorphism is written as a coercion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Cast/Defs.html"}, {"id": "Mathlib.Data.Nat.Choose.Bounds", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 1, "macro_tier_score": 0.0111, "macro_tier_override": null, "x": 203.83, "z": 41.705, "size": 0.3076, "title": "Inequalities for binomial coefficients", "summary": "This file proves exponential bounds on binomial coefficients. We might want to add here the bounds `n^r/r^r ≤ n.choose r ≤ e^r n^r/r^r` in the future.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Choose/Bounds.html"}, {"id": "Mathlib.Data.Set.Dissipate", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Dissipate.html"}, {"id": "Mathlib.Data.Multiset.UnionInter", "region_id": "foundations_data", "micro_elevation": 0.3824, "macro_tier": 2, "macro_tier_score": 0.2963, "macro_tier_override": null, "x": 238.204, "z": 35.281, "size": 0.488, "title": "Distributive lattice structure on multisets", "summary": "This file defines an instance `DistribLattice (Multiset α)` using the union and intersection operators: * `s ∪ t`: The multiset for which the number of occurrences of each `a` is the max of the occurrences of `a` in `s` and `t`. * `s ∩ t`: The multiset for which the number of occurrences of each `a` is the min of the occurrences of `a` in `s` and `t`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/UnionInter.html"}, {"id": "Mathlib.Data.Multiset.Filter", "region_id": "foundations_data", "micro_elevation": 0.3529, "macro_tier": 2, "macro_tier_score": 0.2963, "macro_tier_override": null, "x": 234.373, "z": 30.066, "size": 0.4884, "title": "Filtering multisets by a predicate", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Filter.html"}, {"id": "Mathlib.Data.Fin.FlagRange", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2338, "title": "Range of `f : Fin (n + 1) → α` as a `Flag`", "summary": "Let `f : Fin (n + 1) → α` be an `(n + 1)`-tuple `(f₀, …, fₙ)` such that - `f₀ = ⊥` and `fₙ = ⊤`; - `fₖ₊₁` weakly covers `fₖ` for all `0 ≤ k < n`; this means that `fₖ ≤ fₖ₊₁` and there is no `c` such that `fₖ` is well-founded when `∃ i, p i`, which allows us to implement searches on `ℕ`, starting at `0` and with an unknown upper-bound. It is similar to the well-founded relation constructed to define `Nat.find` with the difference that, in `Nat.Upto p`, `p`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Upto.html"}, {"id": "Mathlib.Data.Complex.BigOperators", "region_id": "foundations_data", "micro_elevation": 0.2353, "macro_tier": 1, "macro_tier_score": 0.0009, "macro_tier_override": null, "x": 225.119, "z": 50.676, "size": 0.324, "title": "Finite sums and products of complex numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Complex/BigOperators.html"}, {"id": "Mathlib.Data.List.Range", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 211.688, "z": 30.835, "size": 0.2571, "title": "Ranges of naturals as lists", "summary": "This file shows basic results about `List.iota`, `List.range`, `List.range'` and defines `List.finRange`. `finRange n` is the list of elements of `Fin n`. `iota n = [n, n - 1, ..., 1]` and `range n = [0, ..., n - 1]` are basic list constructions used for tactics. `range' a b = [a, ..., a + b - 1]` is there to help prove properties about them. Actual maths should use `List.Ico` instead.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Range.html"}, {"id": "Mathlib.Data.Rat.Defs", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 208.952, "z": 44.089, "size": 0.359, "title": "Basics for the Rational Numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Defs.html"}, {"id": "Mathlib.Data.Sym.Sym2.Finsupp", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 179.009, "z": 85.002, "size": 0.2338, "title": "Finitely supported functions from the symmetric square", "summary": "This file lifts functions `α →₀ M₀` to functions `Sym2 α →₀ M₀` by precomposing with multiplication.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sym/Sym2/Finsupp.html"}, {"id": "Mathlib.Data.DList.Instances", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": "Traversable instance for DLists", "summary": "This file provides the equivalence between `List α` and `DList α` and the traversable instance for `DList`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DList/Instances.html"}, {"id": "Mathlib.Data.List.OfFn", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 2, "macro_tier_score": 0.2551, "macro_tier_override": null, "x": 206.0, "z": 46.619, "size": 0.5033, "title": "Lists from functions", "summary": "Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list of length `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/OfFn.html"}, {"id": "Mathlib.Data.Set.Subsingleton", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 2, "macro_tier_score": 0.3433, "macro_tier_override": null, "x": 218.432, "z": 37.346, "size": 0.6238, "title": "Subsingleton", "summary": "Defines the predicate `Subsingleton s : Prop`, saying that `s` has at most one element. Also defines `Nontrivial s : Prop` : the predicate saying that `s` has at least two distinct elements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Subsingleton.html"}, {"id": "Mathlib.Data.Set.Piecewise", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 1, "macro_tier_score": 0.0174, "macro_tier_override": null, "x": 214.738, "z": 26.845, "size": 0.5545, "title": "Piecewise functions", "summary": "This file contains basic results on piecewise defined functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Piecewise.html"}, {"id": "Mathlib.Data.Finite.Card", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finite/Card.html"}, {"id": "Mathlib.Data.ZMod.Coprime", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 227.023, "z": 39.337, "size": 0.2338, "title": "Coprimality and vanishing", "summary": "We show that for prime `p`, the image of an integer `a` in `ZMod p` vanishes if and only if `a` and `p` are not coprime.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ZMod/Coprime.html"}, {"id": "Mathlib.Data.Vector.Defs", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.1882, "macro_tier_override": null, "x": 209.575, "z": 40.559, "size": 0.309, "title": null, "summary": "The type `List.Vector` represents lists with fixed length. TODO: The API of `List.Vector` is quite incomplete relative to `Vector`, and in particular does not use `x[i]` (that is `GetElem` notation) as the preferred accessor. Any combination of reducing the use of `List.Vector` in Mathlib, or modernising its API, would be welcome.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Vector/Defs.html"}, {"id": "Mathlib.Data.Finset.DenselyOrdered", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 168.391, "z": 22.225, "size": 0.2, "title": "Dense orders and finsets", "summary": "We prove that in a dense order, there's always an element lying in between any two finite sets of elements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/DenselyOrdered.html"}, {"id": "Mathlib.Data.Vector.Snoc", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 219.639, "z": 50.279, "size": 0.2676, "title": null, "summary": "This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single element to the back of a vector. It provides a collection of lemmas that show how different `Vector` operations reduce when their argument is `snoc xs x`. Also, an alternative, reverse, induction principle is added, that breaks down a vector into `snoc xs x` for its inductive case. Effectively doing induction from…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Vector/Snoc.html"}, {"id": "Mathlib.Data.Finsupp.Interval", "region_id": "foundations_data", "micro_elevation": 0.8824, "macro_tier": 1, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": 191.996, "z": 94.605, "size": 0.2561, "title": "Finite intervals of finitely supported functions", "summary": "This file provides the `LocallyFiniteOrder` instance for `ι →₀ α` when `α` itself is locally finite and calculates the cardinality of its finite intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Interval.html"}, {"id": "Mathlib.Data.Finset.Finsupp", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 1, "macro_tier_score": 0.021, "macro_tier_override": null, "x": 193.877, "z": -12.464, "size": 0.2491, "title": "Finitely supported product of finsets", "summary": "This file defines the finitely supported product of finsets as a `Finset (ι →₀ α)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Finsupp.html"}, {"id": "Mathlib.Data.Finsupp.Order", "region_id": "foundations_data", "micro_elevation": 0.2647, "macro_tier": 2, "macro_tier_score": 0.0336, "macro_tier_override": null, "x": 229.29, "z": 32.949, "size": 0.405, "title": "Pointwise order on finitely supported functions", "summary": "This file lifts order structures on `α` to `ι →₀ α`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Order.html"}, {"id": "Mathlib.Data.Set.Pairwise.List", "region_id": "foundations_data", "micro_elevation": 0.2353, "macro_tier": 1, "macro_tier_score": 0.0213, "macro_tier_override": null, "x": 225.838, "z": 49.799, "size": 0.289, "title": "Translating pairwise relations on sets to lists", "summary": "On a list with no duplicates, the condition of `Set.Pairwise` and `List.Pairwise` are equivalent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Pairwise/List.html"}, {"id": "Mathlib.Data.Set.Pairwise.Basic", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 2, "macro_tier_score": 0.2958, "macro_tier_override": null, "x": 226.947, "z": 38.608, "size": 0.472, "title": "Relations holding pairwise", "summary": "This file develops pairwise relations and defines pairwise disjoint indexed sets. We also prove many basic facts about `Pairwise`. It is possible that an intermediate file, with more imports than `Logic.Pairwise` but not importing `Data.Set.Function` would be appropriate to hold many of these basic facts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Pairwise/Basic.html"}, {"id": "Mathlib.Data.ZMod.Units", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": 213.432, "z": 42.334, "size": 0.3011, "title": "Lemmas about units in `ZMod`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ZMod/Units.html"}, {"id": "Mathlib.Data.Finsupp.PWO", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2358, "title": "Partial well ordering on finsupps", "summary": "This file contains the fact that finitely supported functions from a fintype are partially well-ordered when the codomain is a linear order that is well ordered. It is in a separate file for now so as to not add imports to the file `Order.WellFoundedSet`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/PWO.html"}, {"id": "Mathlib.Data.Nat.Count", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 1, "macro_tier_score": 0.0213, "macro_tier_override": null, "x": 192.552, "z": -3.473, "size": 0.2876, "title": "Counting on ℕ", "summary": "This file defines the `count` function, which gives, for any predicate on the natural numbers, \"how many numbers under `k` satisfy this predicate?\". We then prove several expected lemmas about `count`, relating it to the cardinality of other objects, and helping to evaluate it for specific `k`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Count.html"}, {"id": "Mathlib.Data.Finite.Set", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 247.752, "z": 72.011, "size": 0.2912, "title": "Lemmas about `Finite` and `Set`s", "summary": "In this file we prove two lemmas about `Finite` and `Set`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finite/Set.html"}, {"id": "Mathlib.Data.WSeq.Productive", "region_id": "foundations_data", "micro_elevation": 0.2647, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 228.497, "z": 49.326, "size": 0.2, "title": "Productive weak sequences", "summary": "This file defines the property of a weak sequence being productive as never stalling – the next output always comes after a finite time. Given a productive weak sequence, a regular sequence (`Seq`) can be derived from it using `toSeq`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/WSeq/Productive.html"}, {"id": "Mathlib.Data.Matrix.Cartan", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/Cartan.html"}, {"id": "Mathlib.Data.Int.ModEq", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 1, "macro_tier_score": 0.0216, "macro_tier_override": null, "x": 211.266, "z": 49.858, "size": 0.3152, "title": "Congruences modulo an integer", "summary": "This file defines the equivalence relation `a ≡ b [ZMOD n]` on the integers, similarly to how `Data.Nat.ModEq` defines them for the natural numbers. The notation is short for `n.ModEq a b`, which is defined to be `a % n = b % n` for integers `a b n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/ModEq.html"}, {"id": "Mathlib.Data.Set.FiniteExhaustion", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 202.699, "z": 91.753, "size": 0.2, "title": "Finite Exhaustions", "summary": "This file defines a structure called `FiniteExhaustion` which represents an exhaustion of a countable set by an increasing sequence of finite sets. Given a countable set `s`, `FiniteExhaustion.choice s` is a choice of a finite exhaustion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/FiniteExhaustion.html"}, {"id": "Mathlib.Data.QPF.Multivariate.Constructions.Comp", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 215.751, "z": 47.819, "size": 0.2, "title": "The composition of QPFs is itself a QPF", "summary": "We define composition between one `n`-ary functor and `n` `m`-ary functors and show that it preserves the QPF structure", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/QPF/Multivariate/Constructions/Comp.html"}, {"id": "Mathlib.Data.Nat.Factorial.SuperFactorial", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2417, "title": "Superfactorial", "summary": "This file defines the [superfactorial](https://en.wikipedia.org/wiki/Superfactorial) `sf n = 1! * 2! * 3! * ... * n!`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Factorial/SuperFactorial.html"}, {"id": "Mathlib.Data.Finset.Fin", "region_id": "foundations_data", "micro_elevation": 0.5882, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 180.427, "z": 60.883, "size": 0.2954, "title": "Finsets in `Fin n`", "summary": "A few constructions for Finsets in `Fin n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Fin.html"}, {"id": "Mathlib.Data.Finsupp.Multiset", "region_id": "foundations_data", "micro_elevation": 0.2647, "macro_tier": 2, "macro_tier_score": 0.0334, "macro_tier_override": null, "x": 200.614, "z": 28.516, "size": 0.3986, "title": "Equivalence between `Multiset` and `ℕ`-valued finitely supported functions", "summary": "This defines `Finsupp.toMultiset` the equivalence between `α →₀ ℕ` and `Multiset α`, along with `Multiset.toFinsupp` the reverse equivalence and `Finsupp.orderIsoMultiset` (the equivalence promoted to an order isomorphism).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Multiset.html"}, {"id": "Mathlib.Data.ENNReal.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.6471, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 189.828, "z": 4.754, "size": 0.2908, "title": "Some lemmas on extended non-negative reals", "summary": "These are some lemmas split off from `ENNReal.Basic` because they need a lot more imports. They are probably good targets for further cleanup or moves.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ENNReal/Lemmas.html"}, {"id": "Mathlib.Data.List.TakeWhile", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 218.892, "z": 42.507, "size": 0.239, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/TakeWhile.html"}, {"id": "Mathlib.Data.Finset.Card", "region_id": "foundations_data", "micro_elevation": 0.5588, "macro_tier": 2, "macro_tier_score": 0.2748, "macro_tier_override": null, "x": 246.742, "z": 24.391, "size": 0.7096, "title": "Cardinality of a finite set", "summary": "This defines the cardinality of a `Finset` and provides induction principles for finsets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Card.html"}, {"id": "Mathlib.Data.Multiset.Pi", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.1789, "macro_tier_override": null, "x": 211.977, "z": 41.648, "size": 0.3799, "title": "The Cartesian product of multisets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Pi.html"}, {"id": "Mathlib.Data.List.FinRange", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 2, "macro_tier_score": 0.2342, "macro_tier_override": null, "x": 208.128, "z": 30.018, "size": 0.5009, "title": "Lists of elements of `Fin n`", "summary": "This file develops some results on `finRange n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/FinRange.html"}, {"id": "Mathlib.Data.ZMod.QuotientRing", "region_id": "foundations_data", "micro_elevation": 0.8824, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 233.157, "z": -14.49, "size": 0.285, "title": "`ZMod n` and quotient groups / rings", "summary": "This file relates `ZMod n` to the quotient ring `ℤ ⧸ Ideal.span {(n : ℤ)}`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ZMod/QuotientRing.html"}, {"id": "Mathlib.Data.Nat.Set", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 1, "macro_tier_score": 0.0044, "macro_tier_override": null, "x": 203.794, "z": 39.371, "size": 0.4805, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Set.html"}, {"id": "Mathlib.Data.Int.DivMod", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2197, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3293, "title": "Basic lemmas about division and modulo for integers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/DivMod.html"}, {"id": "Mathlib.Data.Int.Star", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2359, "title": "Star ordered ring structure on `ℤ`", "summary": "This file shows that `ℤ` is a `StarOrderedRing`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Star.html"}, {"id": "Mathlib.Data.Finmap", "region_id": "foundations_data", "micro_elevation": 0.6765, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 186.485, "z": 4.742, "size": 0.2, "title": "Finite maps over `Multiset`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finmap.html"}, {"id": "Mathlib.Data.List.AList", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.0211, "macro_tier_override": null, "x": 220.332, "z": 30.969, "size": 0.2667, "title": "Association Lists", "summary": "This file defines association lists. An association list is a list where every element consists of a key and a value, and no two entries have the same key. The type of the value is allowed to be dependent on the type of the key. This type dependence is implemented using `Sigma`: The elements of the list are of type `Sigma β`, for some type index `β`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/AList.html"}, {"id": "Mathlib.Data.Nat.Nth", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 1, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": 163.638, "z": 31.932, "size": 0.2607, "title": "The `n`th Number Satisfying a Predicate", "summary": "This file defines a function for \"what is the `n`th number that satisfies a given predicate `p`\", and provides lemmas that deal with this function and its connection to `Nat.count`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Nth.html"}, {"id": "Mathlib.Data.List.OffDiag", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 2, "macro_tier_score": 0.169, "macro_tier_override": null, "x": 204.149, "z": 43.176, "size": 0.4043, "title": "Definition and basic properties of `List.offDiag`", "summary": "In this file we define `List.offDiag l` to be the product `l.product l` with the diagonal removed. The actual definition is more complicated to avoid assuming that equality on `α` is decidable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/OffDiag.html"}, {"id": "Mathlib.Data.SProd", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.3575, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.4459, "title": "Set Product Notation", "summary": "This file provides notation for a product of sets, and other similar types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/SProd.html"}, {"id": "Mathlib.Data.Finset.Range", "region_id": "foundations_data", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.2844, "macro_tier_override": null, "x": 245.977, "z": 46.245, "size": 0.4372, "title": "Finite sets made of a range of elements.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Range.html"}, {"id": "Mathlib.Data.Finset.Basic", "region_id": "foundations_data", "micro_elevation": 0.5294, "macro_tier": 2, "macro_tier_score": 0.27, "macro_tier_override": null, "x": 247.878, "z": 46.659, "size": 0.6162, "title": "Basic lemmas on finite sets", "summary": "This file contains lemmas on the interaction of various definitions on the `Finset` type. For an explanation of `Finset` design decisions, please see `Mathlib/Data/Finset/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Basic.html"}, {"id": "Mathlib.Data.Finset.Image", "region_id": "foundations_data", "micro_elevation": 0.5294, "macro_tier": 2, "macro_tier_score": 0.2987, "macro_tier_override": null, "x": 189.571, "z": 14.83, "size": 0.761, "title": "Image and map operations on finite sets", "summary": "This file provides the finite analog of `Set.image`, along with some other similar functions. Note there are two ways to take the image over a finset; via `Finset.image` which applies the function then removes duplicates (requiring `DecidableEq`), or via `Finset.map` which exploits injectivity of the function to avoid needing to deduplicate. Choosing between these is similar to choosing between `insert` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Image.html"}, {"id": "Mathlib.Data.Seq.Computation", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 1, "macro_tier_score": 0.0211, "macro_tier_override": null, "x": 206.693, "z": 44.219, "size": 0.2733, "title": "Coinductive formalization of unbounded computations.", "summary": "This file provides a `Computation` type where `Computation α` is the type of unbounded computations returning `α`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Seq/Computation.html"}, {"id": "Mathlib.Data.Stream.Init", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0213, "macro_tier_override": null, "x": 217.155, "z": 35.88, "size": 0.2945, "title": "Streams a.k.a. infinite lists a.k.a. infinite sequences", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Stream/Init.html"}, {"id": "Mathlib.Data.Multiset.Defs", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 2, "macro_tier_score": 0.2947, "macro_tier_override": null, "x": 209.716, "z": 29.345, "size": 0.4325, "title": "Multisets", "summary": "Multisets are finite sets with duplicates allowed. They are implemented here as the quotient of lists by permutation. This gives them computational content. This file contains the definition of `Multiset` and the basic predicates. Most operations have been split off into their own files. The goal is that we can define `Finset` with only importing `Multiset.Defs`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Defs.html"}, {"id": "Mathlib.Data.List.Rotate", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 204.283, "z": 43.591, "size": 0.2571, "title": "List rotation", "summary": "This file proves basic results about `List.rotate`, the list rotation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Rotate.html"}, {"id": "Mathlib.Data.Quot", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0178, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.5655, "title": "Quotient types", "summary": "This module extends the core library's treatment of quotient types (`Init.Core`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Quot.html"}, {"id": "Mathlib.Data.Finsupp.Basic", "region_id": "foundations_data", "micro_elevation": 0.2353, "macro_tier": 2, "macro_tier_score": 0.0455, "macro_tier_override": null, "x": 227.891, "z": 34.605, "size": 0.4611, "title": "Miscellaneous definitions, lemmas, and constructions using finsupp", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Basic.html"}, {"id": "Mathlib.Data.Finsupp.SMulWithZero", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0433, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3733, "title": "Scalar multiplication on `Finsupp`", "summary": "This file defines the pointwise scalar multiplication on `Finsupp`, assuming it preserves zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/SMulWithZero.html"}, {"id": "Mathlib.Data.List.Perm.Basic", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 2, "macro_tier_score": 0.2935, "macro_tier_override": null, "x": 208.398, "z": 46.291, "size": 0.3805, "title": "List Permutations", "summary": "This file develops theory about the `List.Perm` relation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Perm/Basic.html"}, {"id": "Mathlib.Data.List.Forall2", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 2, "macro_tier_score": 0.3482, "macro_tier_override": null, "x": 216.6, "z": 45.302, "size": 0.7171, "title": "Double universal quantification on a list", "summary": "This file provides an API for `List.Forall₂` (definition in `Data.List.Defs`). `Forall₂ R l₁ l₂` means that `l₁` and `l₂` have the same length, and whenever `a` is the nth element of `l₁`, and `b` is the nth element of `l₂`, then `R a b` is satisfied.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Forall2.html"}, {"id": "Mathlib.Data.List.InsertIdx", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2923, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3035, "title": "insertIdx", "summary": "Proves various lemmas about `List.insertIdx`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/InsertIdx.html"}, {"id": "Mathlib.Data.Finset.Slice", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 1, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 261.848, "z": 35.972, "size": 0.2669, "title": "`r`-sets and slice", "summary": "This file defines the `r`-th slice of a set family and provides a way to say that a set family is made of `r`-sets. An `r`-set is a finset of cardinality `r` (aka of *size* `r`). The `r`-th slice of a set family is the set family made of its `r`-sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Slice.html"}, {"id": "Mathlib.Data.List.Iterate", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 211.467, "z": 37.052, "size": 0.2514, "title": "iterate", "summary": "Proves various lemmas about `List.iterate`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Iterate.html"}, {"id": "Mathlib.Data.List.Zip", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 2, "macro_tier_score": 0.1054, "macro_tier_override": null, "x": 216.822, "z": 33.381, "size": 0.346, "title": "zip & unzip", "summary": "This file provides results about `List.zipWith`, `List.zip` and `List.unzip` (definitions are in core Lean). `zipWith f l₁ l₂` applies `f : α → β → γ` pointwise to a list `l₁ : List α` and `l₂ : List β`. It applies, until one of the lists is exhausted. For example, `zipWith f [0, 1, 2] [6.28, 31] = [f 0 6.28, f 1 31]`. `zip` is `zipWith` applied to `Prod.mk`. For example, `zip [a₁, a₂] [b₁, b₂, b₃] = [(a₁, b₁), (a₂,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Zip.html"}, {"id": "Mathlib.Data.Vector.Zip", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 220.939, "z": 31.443, "size": 0.2, "title": "The `zipWith` operation on vectors.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Vector/Zip.html"}, {"id": "Mathlib.Data.Complex.Basic", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 1, "macro_tier_score": 0.0123, "macro_tier_override": null, "x": 201.688, "z": 33.568, "size": 0.3832, "title": "The complex numbers", "summary": "The complex numbers are modelled as ℝ^2 in the obvious way and it is shown that they form a field of characteristic zero. For the result that the complex numbers are algebraically closed, see `Complex.isAlgClosed` in `Mathlib.Analysis.Complex.Polynomial.Basic`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Complex/Basic.html"}, {"id": "Mathlib.Data.Rat.Denumerable", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.974, "z": 34.582, "size": 0.2, "title": "Denumerability of ℚ", "summary": "This file proves that ℚ is denumerable. The fact that ℚ has cardinality ℵ₀ is proved in `Mathlib/Data/Rat/Cardinal.lean`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Denumerable.html"}, {"id": "Mathlib.Data.List.MinMax", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.001, "macro_tier_override": null, "x": 216.737, "z": 45.212, "size": 0.3363, "title": "Minimum and maximum of lists", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/MinMax.html"}, {"id": "Mathlib.Data.Multiset.Antidiagonal", "region_id": "foundations_data", "micro_elevation": 0.4706, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 196.324, "z": 66.342, "size": 0.2751, "title": "The antidiagonal on a multiset.", "summary": "The antidiagonal of a multiset `s` consists of all pairs `(t₁, t₂)` such that `t₁ + t₂ = s`. These pairs are counted with multiplicities.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Antidiagonal.html"}, {"id": "Mathlib.Data.Multiset.Sections", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 213.527, "z": 38.448, "size": 0.2751, "title": "Sections of a multiset", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Sections.html"}, {"id": "Mathlib.Data.List.ChainOfFn", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": "Lemmas about `IsChain` and `ofFn`", "summary": "This file provides lemmas involving both `List.IsChain` and `List.ofFn`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/ChainOfFn.html"}, {"id": "Mathlib.Data.Set.BoolIndicator", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0025, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.4123, "title": "Indicator function valued in bool", "summary": "See also `Set.indicator` and `Set.piecewise`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/BoolIndicator.html"}, {"id": "Mathlib.Data.List.Perm.Subperm", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 2, "macro_tier_score": 0.2925, "macro_tier_override": null, "x": 219.74, "z": 47.805, "size": 0.3246, "title": "List Sub-permutations", "summary": "This file develops theory about the `List.Subperm` relation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Perm/Subperm.html"}, {"id": "Mathlib.Data.PNat.Interval", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 210.052, "z": 33.402, "size": 0.2302, "title": "Finite intervals of positive naturals", "summary": "This file proves that `ℕ+` is a `LocallyFiniteOrder` and calculates the cardinality of its intervals as finsets and fintypes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PNat/Interval.html"}, {"id": "Mathlib.Data.Matrix.Auto", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 214.694, "z": 38.89, "size": 0.2, "title": "Automatically generated lemmas for working with concrete matrices", "summary": "In Mathlib3, this file contained \"magic\" lemmas which autogenerate to the correct size of matrix. For instance, `Matrix.of_mul_of_fin` could be used as: ```lean example {α} [AddCommMonoid α] [Mul α] (a₁₁ a₁₂ a₂₁ a₂₂ b₁₁ b₁₂ b₂₁ b₂₂ : α) : !![a₁₁, a₁₂; a₂₁, a₂₂] * !![b₁₁, b₁₂; b₂₁, b₂₂] = !![a₁₁ * b₁₁ + a₁₂ * b₂₁, a₁₁ * b₁₂ + a₁₂ * b₂₂; a₂₁ * b₁₁ + a₂₂ * b₂₁, a₂₁ * b₁₂ + a₂₂ * b₂₂] := by rw [of_mul_of_fin] ``` TODO:…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/Auto.html"}, {"id": "Mathlib.Data.Nat.Order.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": 206.321, "z": 43.472, "size": 0.3218, "title": "Further lemmas about the natural numbers", "summary": "The distinction between this file and `Mathlib/Algebra/Order/Ring/Nat.lean` is not particularly clear. They were separated for now to minimize the porting requirements for tactics during the transition to mathlib4. Please feel free to reorganize these two files.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Order/Lemmas.html"}, {"id": "Mathlib.Data.Matrix.Basis", "region_id": "foundations_data", "micro_elevation": 0.9118, "macro_tier": 1, "macro_tier_score": 0.0143, "macro_tier_override": null, "x": 181.439, "z": -10.65, "size": 0.4637, "title": "Matrices with a single non-zero element.", "summary": "This file provides `Matrix.single`. The matrix `Matrix.single i j c` has `c` at position `(i, j)`, and zeroes elsewhere.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/Basis.html"}, {"id": "Mathlib.Data.FunLike.Ring", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0022, "macro_tier_override": null, "x": 214.588, "z": 34.67, "size": 0.398, "title": "Ring instances for `FunLike` types", "summary": "In this file we define various instances related to ring for `FunLike` types. Note that currently, these are not registered as instances, but only `abbrev`s to avoid long typeclass searches.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/FunLike/Ring.html"}, {"id": "Mathlib.Data.Set.Card.Arithmetic", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 161.036, "z": 37.989, "size": 0.2543, "title": "Results using cardinal arithmetic", "summary": "This file contains results using cardinal arithmetic that are not in the main cardinal theory files. It has been separated out to not burden `Mathlib/Data/Set/Card.lean` with extra imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Card/Arithmetic.html"}, {"id": "Mathlib.Data.Finset.MulAntidiagonal", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 212.796, "z": 42.217, "size": 0.2527, "title": "Multiplication antidiagonal as a `Finset`.", "summary": "We construct the `Finset` of all pairs of an element in `s` and an element in `t` that multiply to `a`, given that `s` and `t` are well-ordered.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/MulAntidiagonal.html"}, {"id": "Mathlib.Data.Finset.SMulAntidiagonal", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 212.195, "z": 41.867, "size": 0.2836, "title": "Antidiagonal for scalar multiplication as a `Finset`.", "summary": "Given sets `G` and `P`, with an action of `G` on `P`, we construct, for any element `a` in `P`, the `Finset` of all pairs of an element in `s` and an element in `t` that scalar-multiply to `a`, assuming that set is finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/SMulAntidiagonal.html"}, {"id": "Mathlib.Data.Finset.Pairwise", "region_id": "foundations_data", "micro_elevation": 0.6471, "macro_tier": 1, "macro_tier_score": 0.0108, "macro_tier_override": null, "x": 253.212, "z": 56.14, "size": 0.2845, "title": "Relations holding pairwise on finite sets", "summary": "In this file we prove a few results about the interaction of `Set.PairwiseDisjoint` and `Finset`, as well as the interaction of `List.Pairwise Disjoint` and the condition of `Disjoint` on `List.toFinset`, in `Set` form.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Pairwise.html"}, {"id": "Mathlib.Data.Set.SMulAntidiagonal", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2723, "title": "Antidiagonal for scalar multiplication", "summary": "Given partially ordered sets `G` and `P`, with an action of `G` on `P`, we construct, for any element `a` in `P` and subsets `s` in `G` and `t` in `P`, the set of all pairs of an element in `s` and an element in `t` that scalar-multiply to `a`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/SMulAntidiagonal.html"}, {"id": "Mathlib.Data.Prod.Lex", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0118, "macro_tier_override": null, "x": 213.284, "z": 42.326, "size": 0.3598, "title": "Lexicographic order", "summary": "This file defines the lexicographic relation for pairs of orders, partial orders and linear orders.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Prod/Lex.html"}, {"id": "Mathlib.Data.Set.Pointwise.Support", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2361, "title": "Support of a function composed with a scalar action", "summary": "We show that the support of `x ↦ f (c⁻¹ • x)` is equal to `c • support f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Pointwise/Support.html"}, {"id": "Mathlib.Data.Fintype.Inv", "region_id": "foundations_data", "micro_elevation": 0.5588, "macro_tier": 1, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": 242.709, "z": 62.932, "size": 0.3065, "title": "Computable inverses for injective/surjective functions on finite types", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Inv.html"}, {"id": "Mathlib.Data.BitVec", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 1, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 181.287, "z": 3.976, "size": 0.2676, "title": "Basic Theorems About Bitvectors", "summary": "This file contains theorems about bitvectors which can only be stated in Mathlib or downstream because they refer to other notions defined in Mathlib. Please do not extend this file further: material about BitVec needed in downstream projects can either be PR'd to Lean, or kept downstream if it also relies on Mathlib.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/BitVec.html"}, {"id": "Mathlib.Data.Multiset.Lattice", "region_id": "foundations_data", "micro_elevation": 0.4706, "macro_tier": 2, "macro_tier_score": 0.2108, "macro_tier_override": null, "x": 191.105, "z": 18.771, "size": 0.4094, "title": "Lattice operations on multisets", "summary": "This file defines `Multiset.sup` and derives the dual `Multiset.inf` and their basic lemmas via `to_dual`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Lattice.html"}, {"id": "Mathlib.Data.Nat.Cast.Prod", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0109, "macro_tier_override": null, "x": 207.653, "z": 39.854, "size": 0.2968, "title": "The product of two `AddMonoidWithOne`s.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Cast/Prod.html"}, {"id": "Mathlib.Data.Rat.BigOperators", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 2, "macro_tier_score": 0.043, "macro_tier_override": null, "x": 222.451, "z": 50.6, "size": 0.3546, "title": "Casting lemmas for rational numbers involving sums and products", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/BigOperators.html"}, {"id": "Mathlib.Data.Fin.Rev", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0022, "macro_tier_override": null, "x": 216.445, "z": 35.382, "size": 0.3971, "title": "Reverse on `Fin n`", "summary": "This file contains lemmas about `Fin.rev : Fin n → Fin n` which maps `i` to `n - 1 - i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Rev.html"}, {"id": "Mathlib.Data.Bundle", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": 215.26, "z": 39.659, "size": 0.3007, "title": "Bundle", "summary": "Basic data structure to implement fiber bundles, vector bundles (maybe fibrations?), etc. This file should contain all possible results that do not involve any topology. We represent a bundle `E` over a base space `B` as a dependent type `E : B → Type*`. We define `Bundle.TotalSpace F E` to be the type of pairs `⟨b, x⟩`, where `b : B` and `x : E b`. This type is isomorphic to `Σ x, E x` and uses an extra argument…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Bundle.html"}, {"id": "Mathlib.Data.Nat.ChineseRemainder", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 215.165, "z": 49.957, "size": 0.2276, "title": "Chinese Remainder Theorem", "summary": "This file provides definitions and theorems for the Chinese Remainder Theorem. These are used in Gödel's Beta function, which is used in proving Gödel's incompleteness theorems.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/ChineseRemainder.html"}, {"id": "Mathlib.Data.Nat.Pairing", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": 221.134, "z": 39.155, "size": 0.3152, "title": "Naturals pairing function", "summary": "This file defines a pairing function for the naturals as follows: ```text 0 1 4 9 16 2 3 5 10 17 6 7 8 11 18 12 13 14 15 19 20 21 22 23 24 ``` It has the advantage of being monotone in both directions and sending `⟦0, n^2 - 1⟧` to `⟦0, n - 1⟧²`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Pairing.html"}, {"id": "Mathlib.Data.Set.MulAntidiagonal", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2618, "title": "Multiplication antidiagonal", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/MulAntidiagonal.html"}, {"id": "Mathlib.Data.Set.Monotone", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 1, "macro_tier_score": 0.0014, "macro_tier_override": null, "x": 210.538, "z": 27.102, "size": 0.3579, "title": "Monotone functions over sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Monotone.html"}, {"id": "Mathlib.Data.TwoPointing", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2478, "title": "Two-pointings", "summary": "This file defines `TwoPointing α`, the type of two pointings of `α`. A two-pointing is the data of two distinct terms. This is morally a Type-valued `Nontrivial`. Another type which is quite close in essence is `Sym2`. Categorically speaking, `prod` is a cospan in the category of types. This forms the category of bipointed types. Two-pointed types form a full subcategory of those.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/TwoPointing.html"}, {"id": "Mathlib.Data.Sym.Sym2.Order", "region_id": "foundations_data", "micro_elevation": 0.6765, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 198.544, "z": -1.751, "size": 0.2526, "title": "Sorting the elements of `Sym2`", "summary": "This file provides `Sym2.sortEquiv`, the forward direction of which is somewhat analogous to `Multiset.sort`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sym/Sym2/Order.html"}, {"id": "Mathlib.Data.Set.Restrict", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 2, "macro_tier_score": 0.3139, "macro_tier_override": null, "x": 209.009, "z": 49.03, "size": 0.6604, "title": "Restrict the domain of a function to a set", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Restrict.html"}, {"id": "Mathlib.Data.DFinsupp.Small", "region_id": "foundations_data", "micro_elevation": 0.8235, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 267.739, "z": 36.481, "size": 0.2898, "title": "Smallness of the `DFinsupp` type", "summary": "Let `π : ι → Type v`. If `ι` and all the `π i` are `w`-small, this provides a `Small.{w}` instance on `DFinsupp π`. As an application, `σ →₀ R` has a `Small.{v}` instance if `σ` and `R` have one.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/Small.html"}, {"id": "Mathlib.Data.Matrix.DMatrix", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2302, "title": "Dependent-typed matrices", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/DMatrix.html"}, {"id": "Mathlib.Data.List.Prime", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2567, "title": "Products of lists of prime elements.", "summary": "This file contains some theorems relating `Prime` and products of `List`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Prime.html"}, {"id": "Mathlib.Data.Matrix.Action", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/Action.html"}, {"id": "Mathlib.Data.Multiset.Fold", "region_id": "foundations_data", "micro_elevation": 0.4412, "macro_tier": 2, "macro_tier_score": 0.2223, "macro_tier_override": null, "x": 241.89, "z": 33.937, "size": 0.4542, "title": "The fold operation for a commutative associative operation over a multiset.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Fold.html"}, {"id": "Mathlib.Data.Nat.Choose.Vandermonde", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2399, "title": "Vandermonde's identity", "summary": "In this file we prove Vandermonde's identity (`Nat.add_choose_eq`): `(m + n).choose k = ∑ (i, j) ∈ antidiagonal k, m.choose i * n.choose j` We follow the algebraic proof from https://en.wikipedia.org/wiki/Vandermonde%27s_identity#Algebraic_proof .", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Choose/Vandermonde.html"}, {"id": "Mathlib.Data.List.SplitOn", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/SplitOn.html"}, {"id": "Mathlib.Data.Ordmap.Ordset", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 216.584, "z": 51.625, "size": 0.2, "title": "Verification of `Ordnode`", "summary": "This file uses the invariants defined in `Mathlib/Data/Ordmap/Invariants.lean` to construct `Ordset α`, a wrapper around `Ordnode α` which includes the correctness invariant of the type. It exposes parallel operations like `insert` as functions on `Ordset` that do the same thing but bundle the correctness proofs. The advantage is that it is possible to, for example, prove that the result of `find` on `insert` will…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Ordmap/Ordset.html"}, {"id": "Mathlib.Data.Ordmap.Invariants", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 1, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 205.697, "z": 46.238, "size": 0.2676, "title": "Invariants for the verification of `Ordnode`", "summary": "An `Ordnode`, defined in `Mathlib/Data/Ordmap/Ordnode.lean`, is an inductive type which describes a tree which stores the `size` at internal nodes. In this file we define the correctness invariant of an `Ordnode`, comprising: * `Ordnode.Sized t`: All internal `size` fields must match the actual measured size of the tree. (This is not hard to satisfy.) * `Ordnode.Balanced t`: Unless the tree has the form `()` or…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Ordmap/Invariants.html"}, {"id": "Mathlib.Data.Sym.Card", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 1, "macro_tier_score": 0.0111, "macro_tier_override": null, "x": 221.551, "z": -15.391, "size": 0.3131, "title": "Stars and bars", "summary": "In this file, we prove (in `Sym.card_sym_eq_multichoose`) that the function `multichoose n k` defined in `Data/Nat/Choose/Basic` counts the number of multisets of cardinality `k` over an alphabet of cardinality `n`. In conjunction with `Nat.multichoose_eq` proved in `Data/Nat/Choose/Basic`, which shows that `multichoose n k = choose (n + k - 1) k`, this is central to the \"stars and bars\" technique in combinatorics,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sym/Card.html"}, {"id": "Mathlib.Data.Multiset.OrderedMonoid", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0224, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3675, "title": "Multisets as ordered monoids", "summary": "The `IsOrderedCancelAddMonoid` and `CanonicallyOrderedAdd` instances on `Multiset α`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/OrderedMonoid.html"}, {"id": "Mathlib.Data.Finsupp.Sigma", "region_id": "foundations_data", "micro_elevation": 0.2647, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 224.903, "z": 53.62, "size": 0.2, "title": "Embedding a finitely supported function into a sigma type summand", "summary": "This file provides `Finsupp.embSigma`, which embeds a finitely supported function `ι k →₀ M` into the corresponding summand of `(Σ k, ι k) →₀ M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Sigma.html"}, {"id": "Mathlib.Data.List.Map2", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 209.703, "z": 41.392, "size": 0.2, "title": "Map₂ Lemmas", "summary": "This file contains additional lemmas about a number of list functions related to combining zipping Lists together. In particular, we include lemmas about: * `map₂Left'` * `map₂Right'` * `zipWith` * `zipLeft'` * `zipRight'`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Map2.html"}, {"id": "Mathlib.Data.Vector3", "region_id": "foundations_data", "micro_elevation": 0.5882, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 252.297, "z": 42.01, "size": 0.239, "title": "Alternate definition of `Vector` in terms of `Fin2`", "summary": "This file provides a scope `Vector3` which overrides the `[a, b, c]` notation to create a `Vector3` instead of a `List`. The `::` notation is also overloaded by this file to mean `Vector3.cons`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Vector3.html"}, {"id": "Mathlib.Data.Nat.Factorization.Divisors", "region_id": "foundations_data", "micro_elevation": 0.9118, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 153.524, "z": 34.216, "size": 0.2, "title": "Results about divisors and factorizations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Factorization/Divisors.html"}, {"id": "Mathlib.Data.Finsupp.Pointwise", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 212.168, "z": 41.843, "size": 0.2338, "title": "The pointwise product on `Finsupp`.", "summary": "For the convolution product on `Finsupp` when the domain has a binary operation, see the type synonyms `AddMonoidAlgebra` (which is in turn used to define `Polynomial` and `MvPolynomial`) and `MonoidAlgebra`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Pointwise.html"}, {"id": "Mathlib.Data.Stream.Defs", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.0316, "macro_tier_override": null, "x": 215.163, "z": 41.325, "size": 0.2741, "title": "Definition of `Stream'` and functions on streams", "summary": "A stream `Stream' α` is an infinite sequence of elements of `α`. One can also think about it as an infinite list. In this file we define `Stream'` and some functions that take and/or return streams. Note that we already have `Stream` to represent a similar object, hence the awkward naming.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Stream/Defs.html"}, {"id": "Mathlib.Data.Matrix.Composition", "region_id": "foundations_data", "micro_elevation": 0.9412, "macro_tier": 1, "macro_tier_score": 0.0118, "macro_tier_override": null, "x": 161.364, "z": 74.367, "size": 0.3558, "title": "Composition of matrices", "summary": "This file shows that `Mₙ(Mₘ(R)) ≃ Mₙₘ(R)`, `Mₙ(Rᵒᵖ) ≃ₐ[K] Mₙ(R)ᵒᵖ` and also different levels of equivalence when `R` is an `AddCommMonoid`, `Semiring`, and `Algebra` over a `CommSemiring K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/Composition.html"}, {"id": "Mathlib.Data.Nat.Prime.Nth", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 166.366, "z": 63.546, "size": 0.2578, "title": "The Nth primes", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Prime/Nth.html"}, {"id": "Mathlib.Data.Fintype.Sort", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": 208.591, "z": 88.736, "size": 0.3714, "title": "Sorting a finite type", "summary": "This file provides two equivalences for linearly ordered fintypes: * `monoEquivOfFin`: Order isomorphism between `α` and `Fin (card α)`. * `finSumEquivOfFinset`: Equivalence between `α` and `Fin m ⊕ Fin n` where `m` and `n` are respectively the cardinalities of some `Finset α` and its complement.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Sort.html"}, {"id": "Mathlib.Data.Int.Cast.Prod", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 213.546, "z": 30.672, "size": 0.2983, "title": "The product of two `AddGroupWithOne`s.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Cast/Prod.html"}, {"id": "Mathlib.Data.Set.Accumulate", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Accumulate.html"}, {"id": "Mathlib.Data.Real.Archimedean", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Real/Archimedean.html"}, {"id": "Mathlib.Data.DFinsupp.WellFounded", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 1, "macro_tier_score": 0.0111, "macro_tier_override": null, "x": 244.671, "z": -6.544, "size": 0.3083, "title": "Well-foundedness of the lexicographic and product orders on `DFinsupp` and `Pi`", "summary": "The primary results are `DFinsupp.Lex.wellFounded` and the two variants that follow it, which essentially say that if `(· > ·)` is a well order on `ι`, `(· < ·)` is well-founded on each `α i`, and `0` is a bottom element in `α i`, then the lexicographic `(· < ·)` is well-founded on `Π₀ i, α i`. The proof is modelled on the proof of `WellFounded.cutExpand`. The results are used to prove `Pi.Lex.wellFounded` and two…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/WellFounded.html"}, {"id": "Mathlib.Data.DFinsupp.Lex", "region_id": "foundations_data", "micro_elevation": 0.8235, "macro_tier": 1, "macro_tier_score": 0.0219, "macro_tier_override": null, "x": 164.156, "z": 63.431, "size": 0.3416, "title": "Lexicographic order on finitely supported dependent functions", "summary": "This file defines the lexicographic order on `DFinsupp`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/Lex.html"}, {"id": "Mathlib.Data.Matrix.Diagonal", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 1, "macro_tier_score": 0.0113, "macro_tier_override": null, "x": 217.739, "z": 49.115, "size": 0.3233, "title": "Diagonal matrices", "summary": "This file defines diagonal matrices and the `AddCommMonoidWithOne` structure on matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/Diagonal.html"}, {"id": "Mathlib.Data.List.Monad", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.3455, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3773, "title": "Monad instances for `List`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Monad.html"}, {"id": "Mathlib.Deprecated.Aliases", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": "Deprecated aliases can be dumped here if they are no longer used in Mathlib, to avoid needing their imports if they are otherwise unnecessary.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Deprecated/Aliases.html"}, {"id": "Mathlib.Data.SubtypeNeLift", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.236, "title": "Extending a function from the complement of a singleton", "summary": "In this file, we define `Function.subtypeNeLift` which allows to extend a (dependent) function defined on the complement of a singleton.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/SubtypeNeLift.html"}, {"id": "Mathlib.Data.ZMod.Aut", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 214.292, "z": 38.634, "size": 0.2594, "title": "Automorphism Group of `ZMod`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ZMod/Aut.html"}, {"id": "Mathlib.Data.QPF.Multivariate.Constructions.Prj", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 207.528, "z": 35.363, "size": 0.2, "title": null, "summary": "Projection functors are QPFs. The `n`-ary projection functors on `i` is an `n`-ary functor `F` such that `F (α₀..αᵢ₋₁, αᵢ, αᵢ₊₁..αₙ₋₁) = αᵢ`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/QPF/Multivariate/Constructions/Prj.html"}, {"id": "Mathlib.Data.FunLike.Group", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 1, "macro_tier_score": 0.0124, "macro_tier_override": null, "x": 214.033, "z": 44.235, "size": 0.3908, "title": "Group instances for `FunLike` types", "summary": "In this file we define various instances related to groups for `FunLike` types. For example given a `FunLike F α β` with `IsMulApply F α β` and `Semigroup β`, then `F` is naturally a semigroup. Note that currently, these are not registered as instances, but only `abbrev`s to avoid long typeclass searches. Moreover, we define the homomorphism `FunLike.coeMulHom : F →* α → β` that acts by coercion. This definition is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/FunLike/Group.html"}, {"id": "Mathlib.Data.Fin.Fin2", "region_id": "foundations_data", "micro_elevation": 0.5588, "macro_tier": 1, "macro_tier_score": 0.0223, "macro_tier_override": null, "x": 247.614, "z": 54.43, "size": 0.3644, "title": "Inductive type variant of `Fin`", "summary": "`Fin` is defined as a subtype of `ℕ`. This file defines an equivalent type, `Fin2`, which is defined inductively. This is useful for its induction principle and different definitional equalities.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Fin2.html"}, {"id": "Mathlib.Data.List.NodupEquivFin", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 2, "macro_tier_score": 0.2398, "macro_tier_override": null, "x": 207.602, "z": 28.11, "size": 0.6382, "title": "Equivalence between `Fin (length l)` and elements of a list", "summary": "Given a list `l`, * if `l` has no duplicates, then `List.Nodup.getEquiv` is the equivalence between `Fin (length l)` and `{x // x ∈ l}` sending `i` to `⟨get l i, _⟩` with the inverse sending `⟨x, hx⟩` to `⟨indexOf x l, _⟩`; * if `l` has no duplicates and contains every element of a type `α`, then `List.Nodup.getEquivOfForallMemList` defines an equivalence between `Fin (length l)` and `α`; if `α` does not have…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/NodupEquivFin.html"}, {"id": "Mathlib.Data.Set.Opposite", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 209.03, "z": 44.183, "size": 0.2, "title": "The opposite of a set", "summary": "The opposite of a set `s` is simply the set obtained by taking the opposite of each member of `s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Opposite.html"}, {"id": "Mathlib.Data.Multiset.DershowitzManna", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 215.097, "z": 39.344, "size": 0.2, "title": "Dershowitz-Manna ordering", "summary": "In this file we define the _Dershowitz-Manna ordering_ on multisets. Specifically, for two multisets `M` and `N` in a partial order `(S, <)`, `M` is smaller than `N` in the Dershowitz-Manna ordering if `M` can be obtained from `N` by replacing one or more elements in `N` by some finite number of elements from `S`, each of which is smaller (in the underling ordering over `S`) than one of the replaced elements from…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/DershowitzManna.html"}, {"id": "Mathlib.Data.ZMod.IntUnitsPower", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": 215.4, "z": 40.301, "size": 0.4244, "title": "The power operator on `ℤˣ` by `ZMod 2`, `ℕ`, and `ℤ`", "summary": "See also the related `negOnePow`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ZMod/IntUnitsPower.html"}, {"id": "Mathlib.Data.Nat.GCD.BigOperators", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0111, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3094, "title": "Lemmas about coprimality with big products.", "summary": "These lemmas are kept separate from `Data.Nat.GCD.Basic` in order to minimize imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/GCD/BigOperators.html"}, {"id": "Mathlib.Data.Sum.Order", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0109, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.291, "title": "Orders on a sum type", "summary": "This file defines the disjoint sum and the linear (aka lexicographic) sum of two orders and provides relation instances for `Sum.LiftRel` and `Sum.Lex`. We declare the disjoint sum of orders as the default set of instances. The linear order goes on a type synonym.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sum/Order.html"}, {"id": "Mathlib.Data.Fintype.Shrink", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 260.042, "z": 37.953, "size": 0.2615, "title": "Fintype instance for `Shrink`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Shrink.html"}, {"id": "Mathlib.Data.List.Flatten", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2745, "title": "Join of a list of lists", "summary": "This file proves basic properties of `List.flatten`, which concatenates a list of lists. It is defined in `Init.Prelude`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Flatten.html"}, {"id": "Mathlib.Data.Finsupp.Encodable", "region_id": "foundations_data", "micro_elevation": 0.8235, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 254.401, "z": 76.243, "size": 0.24, "title": "`Encodable` and `Countable` instances for `α →₀ β`", "summary": "In this file we provide instances for `Encodable (α →₀ β)` and `Countable (α →₀ β)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Encodable.html"}, {"id": "Mathlib.Data.List.Permutation", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.021, "macro_tier_override": null, "x": 202.921, "z": 35.395, "size": 0.2564, "title": "Permutations of a list", "summary": "In this file we prove properties about `List.Permutations`, a list of all permutations of a list. It is defined in `Data.List.Defs`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Permutation.html"}, {"id": "Mathlib.Data.Finsupp.SMul", "region_id": "foundations_data", "micro_elevation": 0.2647, "macro_tier": 2, "macro_tier_score": 0.0225, "macro_tier_override": null, "x": 200.978, "z": 52.647, "size": 0.3713, "title": "Declarations about scalar multiplication on `Finsupp`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/SMul.html"}, {"id": "Mathlib.Data.List.ReduceOption", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 218.18, "z": 36.969, "size": 0.2478, "title": "Properties of `List.reduceOption`", "summary": "In this file we prove basic lemmas about `List.reduceOption`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/ReduceOption.html"}, {"id": "Mathlib.Data.List.Intervals", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 212.046, "z": 32.745, "size": 0.2, "title": "Intervals in ℕ", "summary": "This file defines intervals of naturals. `List.Ico m n` is the list of integers greater than `m` and strictly less than `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Intervals.html"}, {"id": "Mathlib.Data.Nat.Cast.SetInterval", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 212.942, "z": 30.686, "size": 0.2, "title": "Images of intervals under `Nat.cast : ℕ → ℤ`", "summary": "In this file we prove that the image of each `Set.Ixx` interval under `Nat.cast : ℕ → ℤ` is the corresponding interval in `ℤ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Cast/SetInterval.html"}, {"id": "Mathlib.Data.ENNReal.BigOperators", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 204.673, "z": 44.543, "size": 0.2852, "title": "Properties of big operators extended non-negative real numbers", "summary": "In this file we prove elementary properties of sums and products on `ℝ≥0∞`, as well as how these interact with the order structure on `ℝ≥0∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ENNReal/BigOperators.html"}, {"id": "Mathlib.Data.Real.Pointwise", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0213, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2938, "title": "Pointwise operations on sets of reals", "summary": "This file relates `sInf (a • s)`/`sSup (a • s)` with `a • sInf s`/`a • sSup s` for `s : Set ℝ`. From these, it relates `⨅ i, a • f i` / `⨆ i, a • f i` with `a • (⨅ i, f i)` / `a • (⨆ i, f i)`, and provides lemmas about distributing `*` over `⨅` and `⨆`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Real/Pointwise.html"}, {"id": "Mathlib.Data.Int.Associated", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2471, "title": "Associated elements and the integers", "summary": "This file contains some results on equality up to units in the integers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Associated.html"}, {"id": "Mathlib.Data.Set.List", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 2, "macro_tier_score": 0.2932, "macro_tier_override": null, "x": 220.988, "z": 46.534, "size": 0.3644, "title": "Lemmas about `List`s and `Set.range`", "summary": "In this file we prove lemmas about range of some operations on lists.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/List.html"}, {"id": "Mathlib.Data.FunLike.Basic", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0222, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.6605, "title": "Typeclass for a type `F` with an injective map to `A → B`", "summary": "This typeclass is primarily for use by homomorphisms like `MonoidHom` and `LinearMap`. There is the \"D\"ependent version `DFunLike` and the non-dependent version `FunLike`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/FunLike/Basic.html"}, {"id": "Mathlib.Data.Nat.Dist", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2459, "title": "Distance function on ℕ", "summary": "This file defines a simple distance function on naturals from truncated subtraction.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Dist.html"}, {"id": "Mathlib.Data.Real.ConjExponents", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 204.045, "z": 47.274, "size": 0.2815, "title": "Real conjugate exponents", "summary": "This file defines Hölder triple and Hölder conjugate exponents in `ℝ` and `ℝ≥0`. Real numbers `p`, `q` and `r` form a *Hölder triple* if `0 < p` and `0 < q` and `p⁻¹ + q⁻¹ = r⁻¹` (which of course implies `0 < r`). We say `p` and `q` are *Hölder conjugate* if `p`, `q` and `1` are a Hölder triple. In this case, `1 < p` and `1 < q`. This property shows up often in analysis, especially when dealing with `L^p` spaces.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Real/ConjExponents.html"}, {"id": "Mathlib.Data.Finset.Attach", "region_id": "foundations_data", "micro_elevation": 0.3529, "macro_tier": 2, "macro_tier_score": 0.2842, "macro_tier_override": null, "x": 236.302, "z": 35.683, "size": 0.4323, "title": "Attaching a proof of membership to a finite set", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Attach.html"}, {"id": "Mathlib.Data.Finset.Disjoint", "region_id": "foundations_data", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.2842, "macro_tier_override": null, "x": 236.671, "z": 16.875, "size": 0.4323, "title": "Disjoint finite sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Disjoint.html"}, {"id": "Mathlib.Data.Finset.Erase", "region_id": "foundations_data", "micro_elevation": 0.3824, "macro_tier": 2, "macro_tier_score": 0.2842, "macro_tier_override": null, "x": 199.787, "z": 61.639, "size": 0.4323, "title": "Erasing an element from a finite set", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Erase.html"}, {"id": "Mathlib.Data.Finset.Filter", "region_id": "foundations_data", "micro_elevation": 0.3824, "macro_tier": 2, "macro_tier_score": 0.2839, "macro_tier_override": null, "x": 237.467, "z": 48.266, "size": 0.4176, "title": "Filtering a finite set", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Filter.html"}, {"id": "Mathlib.Data.Set.Sigma", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 2, "macro_tier_score": 0.128, "macro_tier_override": null, "x": 215.331, "z": 39.867, "size": 0.4331, "title": "Sets in sigma types", "summary": "This file defines `Set.sigma`, the indexed sum of sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Sigma.html"}, {"id": "Mathlib.Data.FunLike.IsApply", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 212.044, "z": 39.058, "size": 0.337, "title": "Typeclasses for `FunLike` and algebraic operations", "summary": "In this file we provide typeclasses for the compatibility of algebraic structures and `FunLike` instances. These instances encode the property that algebraic operations such as addition, subtraction, and negation are given by the pointwise operations, and moreover we provide classes for `1` acting as the identity and multiplication acting as composition. The algebraic `FunLike` typeclasses provide a `simp` lemma of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/FunLike/IsApply.html"}, {"id": "Mathlib.Data.String.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.987, "z": 38.52, "size": 0.2, "title": "Miscellaneous lemmas about strings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/String/Lemmas.html"}, {"id": "Mathlib.Data.Nat.Prime.Pow", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 204.762, "z": 32.621, "size": 0.2584, "title": "Prime numbers", "summary": "This file develops the theory of prime numbers: natural numbers `p ≥ 2` whose only divisors are `p` and `1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Prime/Pow.html"}, {"id": "Mathlib.Data.Int.Order.Units", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.012, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3676, "title": "Lemmas about units in `ℤ`, which interact with the order structure.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Order/Units.html"}, {"id": "Mathlib.Data.DFinsupp.Encodable", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 1, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": 163.131, "z": 44.95, "size": 0.2544, "title": "`Encodable` and `Countable` instances for `Π₀ i, α i`", "summary": "In this file we provide instances for `Encodable (Π₀ i, α i)` and `Countable (Π₀ i, α i)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/Encodable.html"}, {"id": "Mathlib.Data.List.DropRight", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 216.719, "z": 33.332, "size": 0.2, "title": "Dropping or taking from lists on the right", "summary": "Taking or removing element from the tail end of a list", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/DropRight.html"}, {"id": "Mathlib.Data.Nat.Factorization.LCM", "region_id": "foundations_data", "micro_elevation": 0.8824, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 171.638, "z": 81.022, "size": 0.2534, "title": "Lemmas about `factorizationLCMLeft`", "summary": "This file contains some lemmas about `factorizationLCMLeft`. These were split from `Mathlib.Data.Nat.Factorization.Basic` to reduce transitive imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Factorization/LCM.html"}, {"id": "Mathlib.Data.DFinsupp.Ext", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 1, "macro_tier_score": 0.0216, "macro_tier_override": null, "x": 258.478, "z": 63.346, "size": 0.3166, "title": "Extensionality principles for `DFinsupp`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/Ext.html"}, {"id": "Mathlib.Data.Nat.DvdSequence", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0211, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2706, "title": "Divisibility sequences", "summary": "A sequence `f : ℕ → ℕ` is a *divisibility sequence* if it satisfies `f a ∣ f b` whenever `a ∣ b`. A sequence `f : ℕ → ℕ` is a *strong divisibility sequence* if `gcd (f a) (f b) = f (gcd a b)`. This file defines divisibility sequences and strong divisibility sequences, and provides some basic API for these definitions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/DvdSequence.html"}, {"id": "Mathlib.Data.Nat.EvenOddRec", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.245, "title": "A recursion principle based on even and odd numbers.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/EvenOddRec.html"}, {"id": "Mathlib.Data.Int.CardIntervalMod", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 224.135, "z": 89.783, "size": 0.2, "title": "Counting elements in an interval with given residue", "summary": "The theorems in this file generalise `Nat.card_multiples` in `Mathlib/Data/Nat/Factorization/Basic.lean` to all integer intervals and any fixed residue (not just zero, which reduces to the multiples). Theorems are given for `Ico` and `Ioc` intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/CardIntervalMod.html"}, {"id": "Mathlib.Data.List.SplitBy", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 220.235, "z": 47.356, "size": 0.2, "title": "Split a list into contiguous runs of elements which pairwise satisfy a relation.", "summary": "This file provides the basic API for `List.splitBy` which is defined in Core. The main results are the following: - `List.flatten_splitBy`: the lists in `List.splitBy` join to the original list. - `List.nil_notMem_splitBy`: the empty list is not contained in `List.splitBy`. - `List.isChain_of_mem_splitBy`: any two adjacent elements in a list in `List.splitBy` are related by the specified relation. -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/SplitBy.html"}, {"id": "Mathlib.Data.Set.Insert", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.3583, "macro_tier_override": null, "x": 209.608, "z": 39.856, "size": 0.7123, "title": "Lemmas about insertion, singleton, and pairs", "summary": "This file provides extra lemmas about `insert`, `singleton`, and `pair`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Insert.html"}, {"id": "Mathlib.Data.Multiset.Interval", "region_id": "foundations_data", "micro_elevation": 0.8824, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 268.118, "z": 60.694, "size": 0.2, "title": "Finite intervals of multisets", "summary": "This file provides the `LocallyFiniteOrder` instance for `Multiset α` and calculates the cardinality of its finite intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Interval.html"}, {"id": "Mathlib.Data.DFinsupp.Multiset", "region_id": "foundations_data", "micro_elevation": 0.8235, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 262.568, "z": 63.841, "size": 0.239, "title": "Equivalence between `Multiset` and `ℕ`-valued finitely supported functions", "summary": "This defines `DFinsupp.toMultiset` the equivalence between `Π₀ a : α, ℕ` and `Multiset α`, along with `Multiset.toDFinsupp` the reverse equivalence.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/Multiset.html"}, {"id": "Mathlib.Data.Fintype.Fin", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.0116, "macro_tier_override": null, "x": 218.607, "z": 29.926, "size": 0.3429, "title": "The structure of `Fintype (Fin n)`", "summary": "This file contains some basic results about the `Fintype` instance for `Fin`, especially properties of `Finset.univ : Finset (Fin n)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Fin.html"}, {"id": "Mathlib.Data.Prod.PProd", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0021, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3962, "title": "Extra facts about `PProd`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Prod/PProd.html"}, {"id": "Mathlib.Data.Real.Hom", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Real/Hom.html"}, {"id": "Mathlib.Data.WSeq.Basic", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 1, "macro_tier_score": 0.0218, "macro_tier_override": null, "x": 222.121, "z": 50.882, "size": 0.3298, "title": "Partially defined possibly infinite lists", "summary": "This file provides a `WSeq α` type representing partially defined possibly infinite lists (referred here as weak sequences).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/WSeq/Basic.html"}, {"id": "Mathlib.Data.Fintype.Quotient", "region_id": "foundations_data", "micro_elevation": 0.4412, "macro_tier": 1, "macro_tier_score": 0.0118, "macro_tier_override": null, "x": 242.402, "z": 43.893, "size": 0.3573, "title": "Quotients of families indexed by a finite type", "summary": "This file proves some basic facts and defines lifting and recursion principle for quotients indexed by a finite type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Quotient.html"}, {"id": "Mathlib.Data.List.Pi", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.0322, "macro_tier_override": null, "x": 211.711, "z": 36.918, "size": 0.3273, "title": "The Cartesian product of lists", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Pi.html"}, {"id": "Mathlib.Data.Ordmap.Ordnode", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 205.715, "z": 41.085, "size": 0.2459, "title": "Ordered sets", "summary": "This file defines a data structure for ordered sets, supporting a variety of useful operations including insertion and deletion, logarithmic time lookup, set operations, folds, and conversion from lists. The `Ordnode α` operations all assume that `α` has the structure of a total preorder, meaning a `≤` operation that is * Transitive: `x ≤ y → y ≤ z → x ≤ z` * Reflexive: `x ≤ x` * Total: `x ≤ y ∨ y ≤ x` For example,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Ordmap/Ordnode.html"}, {"id": "Mathlib.Data.Multiset.Pairwise", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 220.995, "z": 29.063, "size": 0.2, "title": "Pairwise relations on a multiset", "summary": "This file provides basic results about `Multiset.Pairwise` (definitions are in `Mathlib/Data/Multiset/Defs.lean`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Pairwise.html"}, {"id": "Mathlib.Data.Ordering.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 211.534, "z": 40.124, "size": 0.2302, "title": "Some `Ordering` lemmas", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Ordering/Lemmas.html"}, {"id": "Mathlib.Data.Fin.Embedding", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0126, "macro_tier_override": null, "x": 208.361, "z": 37.56, "size": 0.396, "title": "Embeddings of `Fin n`", "summary": "`Fin n` is the type whose elements are natural numbers smaller than `n`. This file defines embeddings between `Fin n` and other types,", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Embedding.html"}, {"id": "Mathlib.Data.Real.Embedding", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2524, "title": "Embedding of archimedean groups into reals", "summary": "This file provides embedding of any archimedean groups into reals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Real/Embedding.html"}, {"id": "Mathlib.Data.Set.Enumerate", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 219.192, "z": 39.33, "size": 0.2, "title": "Set enumeration", "summary": "This file allows enumeration of sets given a choice function. The definition does not assume `sel` actually is a choice function, i.e. `sel s ∈ s` and `sel s = none ↔ s = ∅`. These assumptions are added to the lemmas needing them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Enumerate.html"}, {"id": "Mathlib.Data.Num.Prime", "region_id": "foundations_data", "micro_elevation": 0.2647, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 228.797, "z": 48.8, "size": 0.2, "title": "Primality for binary natural numbers", "summary": "This file defines versions of `Nat.minFac` and `Nat.Prime` for `Num` and `PosNum`. As with other `Num` definitions, they are not intended for general use (`Nat` should be used instead of `Num` in most cases) but they can be used in contexts where kernel computation is required, such as proofs by `rfl` and `decide`, as well as in `#reduce`. The default decidable instance for `Nat.Prime` is optimized for VM…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Num/Prime.html"}, {"id": "Mathlib.Data.Num.ZNum", "region_id": "foundations_data", "micro_elevation": 0.2353, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 198.118, "z": 42.927, "size": 0.2338, "title": "Properties of the `ZNum` representation of integers", "summary": "This file was split from `Mathlib/Data/Num/Lemmas.lean` to keep the former under 1500 lines.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Num/ZNum.html"}, {"id": "Mathlib.Data.Sum.Lattice", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 214.859, "z": 41.738, "size": 0.2, "title": "Lexicographic sum of lattices", "summary": "This file proves that we can combine two lattices `α` and `β` into a lattice `α ⊕ₗ β` where everything in `α` is declared smaller than everything in `β`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sum/Lattice.html"}, {"id": "Mathlib.Data.Setoid.Partition.Card", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 168.166, "z": 13.886, "size": 0.2483, "title": "Cardinality of parts of partitions", "summary": "* `Setoid.IsPartition.ncard_eq_finsum` on an ambient finite type, the cardinal of a set is the sum of the cardinalities of its trace on the parts of the partition", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Setoid/Partition/Card.html"}, {"id": "Mathlib.Data.Finite.Defs", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 2, "macro_tier_score": 0.2941, "macro_tier_override": null, "x": 211.454, "z": 43.721, "size": 0.4094, "title": "Definition of the `Finite` typeclass", "summary": "This file defines a typeclass `Finite` saying that `α : Sort*` is finite. A type is `Finite` if it is equivalent to `Fin n` for some `n`. We also define `Infinite α` as a typeclass equivalent to `¬Finite α`. The `Finite` predicate has no computational relevance and, being `Prop`-valued, gets to enjoy proof irrelevance -- it represents the mere fact that the type is finite. While the `Finite` class also represents…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finite/Defs.html"}, {"id": "Mathlib.Data.Sym.NatCard", "region_id": "foundations_data", "micro_elevation": 0.8824, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 156.562, "z": 27.635, "size": 0.2, "title": "`Nat.card` versions of `Fintype.card` lemmas on `Sym`", "summary": "Each of the lemmas assuming `[Fintype α]` and `Fintype.card` can be restated using `Nat.card` alone.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sym/NatCard.html"}, {"id": "Mathlib.Data.DFinsupp.FiniteInfinite", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.175, "z": 90.923, "size": 0.2, "title": "Finiteness and infiniteness of the `DFinsupp` type", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/FiniteInfinite.html"}, {"id": "Mathlib.Data.Bool.AllAny", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": "Boolean quantifiers", "summary": "This proves a few properties about `List.all` and `List.any`, which are the `Bool` universal and existential quantifiers. Their definitions are in core Lean.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Bool/AllAny.html"}, {"id": "Mathlib.Data.QPF.Univariate.Basic", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 216.468, "z": 35.396, "size": 0.2, "title": "Quotients of Polynomial Functors", "summary": "We assume the following: * `P`: a polynomial functor * `W`: its W-type * `M`: its M-type * `F`: a functor We define: * `q`: `QPF` data, representing `F` as a quotient of `P` The main goal is to construct: * `Fix`: the initial algebra with structure map `F Fix → Fix`. * `Cofix`: the final coalgebra with structure map `Cofix → F Cofix` We also show that the composition of qpfs is a qpf, and that the quotient of a qpf…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/QPF/Univariate/Basic.html"}, {"id": "Mathlib.Data.TypeVec", "region_id": "foundations_data", "micro_elevation": 0.5882, "macro_tier": 1, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": 243.588, "z": 64.952, "size": 0.3406, "title": "Tuples of types, and their categorical structure.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/TypeVec.html"}, {"id": "Mathlib.Data.NNRat.BigOperators", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": "Casting lemmas for non-negative rational numbers involving sums and products", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/NNRat/BigOperators.html"}, {"id": "Mathlib.Data.List.ToFinsupp", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": 218.748, "z": 42.845, "size": 0.2595, "title": "Lists as finsupp", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/ToFinsupp.html"}, {"id": "Mathlib.Data.Nat.Cast.Commute", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.4992, "title": "Cast of natural numbers: lemmas about `Commute`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Cast/Commute.html"}, {"id": "Mathlib.Data.Vector.Mem", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 202.397, "z": 44.081, "size": 0.2, "title": "Theorems about membership of elements in vectors", "summary": "This file contains theorems for membership in a `v.toList` for a vector `v`. Having the length available in the type allows some of the lemmas to be simpler and more general than the original version for lists. In particular we can avoid some assumptions about types being `Inhabited`, and make more general statements about `head` and `tail`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Vector/Mem.html"}, {"id": "Mathlib.Data.Fin.Tuple.NatAntidiagonal", "region_id": "foundations_data", "micro_elevation": 0.4118, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 238.894, "z": 30.723, "size": 0.2741, "title": "Collections of tuples of naturals with the same sum", "summary": "This file generalizes `List.Nat.Antidiagonal n`, `Multiset.Nat.Antidiagonal n`, and `Finset.Nat.Antidiagonal n` from the pair of elements `x : ℕ × ℕ` such that `n = x.1 + x.2`, to the sequence of elements `x : Fin k → ℕ` such that `n = ∑ i, x i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Tuple/NatAntidiagonal.html"}, {"id": "Mathlib.Data.Int.NatAbs", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2471, "title": "Lemmas about `Int.natAbs`", "summary": "This file contains some results on `Int.natAbs`, the absolute value of an integer as a natural number.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/NatAbs.html"}, {"id": "Mathlib.Data.Nat.Choose.Multinomial", "region_id": "foundations_data", "micro_elevation": 0.2941, "macro_tier": 0, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": 194.042, "z": 39.521, "size": 0.3026, "title": "Multinomial", "summary": "This file defines the multinomial coefficients and several small lemmas for manipulating them. - `Nat.multinomial`: the multinomial coefficient, Given a function `f : α → ℕ` and `s : Finset α`, this is the number of strings consisting of symbols from `s`, where `c ∈ s` appears with multiplicity `f c`. It is defined as `(∑ i ∈ s, f i)! / ∏ i ∈ s, (f i)!`. - `Multiset.countPerms`: multinomial coefficient associated…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Choose/Multinomial.html"}, {"id": "Mathlib.Data.List.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 207.305, "z": 32.869, "size": 0.2, "title": "Some lemmas about lists involving sets", "summary": "Split out from `Data.List.Basic` to reduce its dependencies.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Lemmas.html"}, {"id": "Mathlib.Data.Set.Semiring", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": 211.57, "z": 39.93, "size": 0.3064, "title": "Sets as a semiring under union", "summary": "This file defines `SetSemiring α`, an alias of `Set α`, which we endow with `∪` as addition and pointwise `*` as multiplication. If `α` is a (commutative) monoid, `SetSemiring α` is a (commutative) semiring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Semiring.html"}, {"id": "Mathlib.Data.Nat.Cast.Synonym", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 218.618, "z": 37.673, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Cast/Synonym.html"}, {"id": "Mathlib.Data.Rat.Cast.Lemmas", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 1, "macro_tier_score": 0.0116, "macro_tier_override": null, "x": 215.114, "z": 28.846, "size": 0.3433, "title": "Some exiled lemmas about casting", "summary": "These lemmas have been removed from `Mathlib/Data/Rat/Cast/Defs.lean` to avoiding needing to import `Mathlib/Algebra/Field/Basic.lean` there. In fact, these lemmas don't appear to be used anywhere in Mathlib, so perhaps this file can simply be deleted.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Cast/Lemmas.html"}, {"id": "Mathlib.Data.Set.Pairwise.Chain", "region_id": "foundations_data", "micro_elevation": 0.2647, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 230.943, "z": 40.931, "size": 0.2, "title": "Pairwise results for chains", "summary": "In this file `Pairwise` results are applied to chains of sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Pairwise/Chain.html"}, {"id": "Mathlib.Data.Nat.Factorial.BigOperators", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0221, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.349, "title": "Factorial with big operators", "summary": "This file contains some lemmas on factorials in combination with big operators. While in terms of semantics they could be in the `Basic.lean` file, importing `Algebra.BigOperators.Group.Finset` leads to a cyclic import.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Factorial/BigOperators.html"}, {"id": "Mathlib.Data.Int.NatPrime", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 211.13, "z": 51.817, "size": 0.2382, "title": "Lemmas about `Nat.Prime` using `Int`s", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/NatPrime.html"}, {"id": "Mathlib.Data.Matrix.PEquiv", "region_id": "foundations_data", "micro_elevation": 0.8824, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 225.107, "z": 97.523, "size": 0.2906, "title": "partial equivalences for matrices", "summary": "Using partial equivalences to represent matrices. This file introduces the function `PEquiv.toMatrix`, which returns a matrix containing ones and zeros. For any partial equivalence `f`, `f.toMatrix i j = 1 ↔ f i = some j`. The following important properties of this function are proved `toMatrix_trans : (f.trans g).toMatrix = f.toMatrix * g.toMatrix` `toMatrix_symm : f.symm.toMatrix = f.toMatrixᵀ` `toMatrix_refl :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/PEquiv.html"}, {"id": "Mathlib.Data.Multiset.NatAntidiagonal", "region_id": "foundations_data", "micro_elevation": 0.3529, "macro_tier": 2, "macro_tier_score": 0.0526, "macro_tier_override": null, "x": 200.059, "z": 59.48, "size": 0.2946, "title": "Antidiagonals in ℕ × ℕ as multisets", "summary": "This file defines the antidiagonals of ℕ × ℕ as multisets: the `n`-th antidiagonal is the multiset of pairs `(i, j)` such that `i + j = n`. This is useful for polynomial multiplication and more generally for sums going from `0` to `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/NatAntidiagonal.html"}, {"id": "Mathlib.Data.List.NatAntidiagonal", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 2, "macro_tier_score": 0.0628, "macro_tier_override": null, "x": 214.123, "z": 30.695, "size": 0.278, "title": "Antidiagonals in ℕ × ℕ as lists", "summary": "This file defines the antidiagonals of ℕ × ℕ as lists: the `n`-th antidiagonal is the list of pairs `(i, j)` such that `i + j = n`. This is useful for polynomial multiplication and more generally for sums going from `0` to `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/NatAntidiagonal.html"}, {"id": "Mathlib.Data.Set.Sups", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2575, "title": "Set family operations", "summary": "This file defines a few binary operations on `Set α` for use in set family combinatorics.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Sups.html"}, {"id": "Mathlib.Data.ZMod.ValMinAbs", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 214.506, "z": 38.753, "size": 0.2855, "title": "Absolute value in `ZMod n`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ZMod/ValMinAbs.html"}, {"id": "Mathlib.Data.Bool.Count", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 207.977, "z": 48.414, "size": 0.2, "title": "List of Booleans", "summary": "In this file we prove lemmas about the number of `false`s and `true`s in a list of Booleans. First we prove that the number of `false`s plus the number of `true` equals the length of the list. Then we prove that in a list with alternating `true`s and `false`s, the number of `true`s differs from the number of `false`s by at most one. We provide several versions of these statements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Bool/Count.html"}, {"id": "Mathlib.Data.Finset.PiInduction", "region_id": "foundations_data", "micro_elevation": 0.6765, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 246.853, "z": 70.109, "size": 0.2, "title": "Induction principles for `∀ i, Finset (α i)`", "summary": "In this file we prove a few induction principles for functions `Π i : ι, Finset (α i)` defined on a finite type. * `Finset.induction_on_pi` is a generic lemma that requires only `[Finite ι]`, `[DecidableEq ι]`, and `[∀ i, DecidableEq (α i)]`; this version can be seen as a direct generalization of `Finset.induction_on`. * `Finset.induction_on_pi_max` and `Finset.induction_on_pi_min`: generalizations of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/PiInduction.html"}, {"id": "Mathlib.Data.Finset.CastCard", "region_id": "foundations_data", "micro_elevation": 0.5882, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 238.666, "z": 69.982, "size": 0.2332, "title": "Cardinality of a finite set and subtraction", "summary": "This file contains results on the cardinality of a `Finset` and subtraction, by casting the cardinality as element of an `AddGroupWithOne`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/CastCard.html"}, {"id": "Mathlib.Data.List.PeriodicityLemma", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": "Periods of words (Lists)", "summary": "This file defines the notion of a period of a word (list) and proves the Periodicity Lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/PeriodicityLemma.html"}, {"id": "Mathlib.Data.Bracket", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3718, "title": "Bracket Notation", "summary": "This file provides notation which can be used for the Lie bracket, for the commutator of two subgroups, and for other similar operations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Bracket.html"}, {"id": "Mathlib.Data.Tree.RBMap", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Tree/RBMap.html"}, {"id": "Mathlib.Data.Matrix.ColumnRowPartitioned", "region_id": "foundations_data", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 275.524, "z": 63.082, "size": 0.2448, "title": "Block Matrices from Rows and Columns", "summary": "This file provides the basic definitions of matrices composed from columns and rows. The concatenation of two matrices with the same row indices can be expressed as `A = fromCols A₁ A₂` the concatenation of two matrices with the same column indices can be expressed as `B = fromRows B₁ B₂`. We then provide a few lemmas that deal with the products of these with each other and with block matrices", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Matrix/ColumnRowPartitioned.html"}, {"id": "Mathlib.Data.DFinsupp.NeLocus", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 1, "macro_tier_score": 0.0211, "macro_tier_override": null, "x": 184.244, "z": -0.843, "size": 0.2735, "title": "Locus of unequal values of finitely supported dependent functions", "summary": "Let `N : α → Type*` be a type family, assume that `N a` has a `0` for all `a : α` and let `f g : Π₀ a, N a` be finitely supported dependent functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/NeLocus.html"}, {"id": "Mathlib.Data.DFinsupp.Notation", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 163.573, "z": 32.322, "size": 0.2, "title": "Notation for `DFinsupp`", "summary": "This file extends the existing `fun₀ | 3 => a | 7 => b` notation to work for `DFinsupp`, which desugars to `DFinsupp.update` and `DFinsupp.single`, in the same way that `{a, b}` desugars to `insert` and `singleton`. Note that this syntax is for `Finsupp` by default, but works for `DFinsupp` if the expected type is correct.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/DFinsupp/Notation.html"}, {"id": "Mathlib.Data.Rel.Separated", "region_id": "foundations_data", "micro_elevation": 0.2353, "macro_tier": 1, "macro_tier_score": 0.0121, "macro_tier_override": null, "x": 222.824, "z": 52.802, "size": 0.3718, "title": "Uniform separation", "summary": "This file defines a notion of separation of a set relative to a relation. For a relation `R`, an `R`-separated set `s` is a set such that every pair of elements of `s` is `R`-unrelated. The concept of uniformly separated sets is used to define two further notions of separation: * Metric separation: `Metric.IsSeparated`, defined using the small distance relation. * Dynamical nets: `Dynamics.IsDynNetIn`, defined using…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rel/Separated.html"}, {"id": "Mathlib.Data.Finite.Vector", "region_id": "foundations_data", "micro_elevation": 0.8235, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 229.341, "z": 92.442, "size": 0.2478, "title": "Finiteness of vector types", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finite/Vector.html"}, {"id": "Mathlib.Data.Finsupp.Option", "region_id": "foundations_data", "micro_elevation": 0.2647, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": 196.027, "z": 41.847, "size": 0.4244, "title": "Operations on `Finsupp`s with an `Option` domain", "summary": "Similar to how `Finsupp.cons` and `Finsupp.tail` construct an object of type `Fin (n + 1) →₀ M` from a map `Fin n →₀ M` and vice versa, we define `Finsupp.optionElim` and `Finsupp.some` to construct `Option α →₀ M` from a map α →₀ M, and vice versa. As functions, these behave as `Option.elim'`, and as an application of `some` hence the names. We prove a variety of API lemmas, see `Mathlib/Data/Finsupp/Fin.lean` for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Option.html"}, {"id": "Mathlib.Data.FunLike.Embedding", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.017, "macro_tier_override": null, "x": 215.196, "z": 39.519, "size": 0.5432, "title": "Typeclass for a type `F` with an injective map to `A ↪ B`", "summary": "This typeclass is primarily for use by embeddings such as `RelEmbedding`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/FunLike/Embedding.html"}, {"id": "Mathlib.Data.Fin.Tuple.Embedding", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 222.176, "z": 36.095, "size": 0.2585, "title": "Constructions of embeddings of `Fin n` into a type", "summary": "* `Fin.Embedding.cons` : from an embedding `x : Fin n ↪ α` and `a : α` such that `a ∉ x.range`, construct an embedding `Fin (n + 1) ↪ α` by putting `a` at `0` * `Fin.Embedding.tail`: the tail of an embedding `x : Fin (n + 1) ↪ α` * `Fin.Embedding.snoc` : from an embedding `x : Fin n ↪ α` and `a : α` such that `a ∉ x.range`, construct an embedding `Fin (n + 1) ↪ α` by putting `a` at the end. * `Fin.Embedding.init`:…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Tuple/Embedding.html"}, {"id": "Mathlib.Data.Char", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 212.5, "z": 46.142, "size": 0.239, "title": "More `Char` instances", "summary": "This file provides a `LinearOrder` instance on `Char`. `Char` is the type of Unicode scalar values. Provides an additional definition to truncate a `Char` to `UInt8` and a theorem on conversion to `Nat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Char.html"}, {"id": "Mathlib.Data.Sigma.Interval", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 211.552, "z": 43.778, "size": 0.2, "title": "Finite intervals in a sigma type", "summary": "This file provides the `LocallyFiniteOrder` instance for the disjoint sum of orders `Σ i, α i` and calculates the cardinality of its finite intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sigma/Interval.html"}, {"id": "Mathlib.Data.Set.Constructions", "region_id": "foundations_data", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 236.843, "z": 17.047, "size": 0.3464, "title": "Constructions involving sets of sets.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Constructions.html"}, {"id": "Mathlib.Data.Int.Sqrt", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 1, "macro_tier_score": 0.001, "macro_tier_override": null, "x": 212.5, "z": 48.105, "size": 0.3314, "title": "Square root of integers", "summary": "This file defines the square root function on integers. `Int.sqrt z` is the greatest integer `r` such that `r * r ≤ z`. If `z ≤ 0`, then `Int.sqrt z = 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Int/Sqrt.html"}, {"id": "Mathlib.Data.UInt", "region_id": "foundations_data", "micro_elevation": 0.7647, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 212.886, "z": 90.921, "size": 0.2, "title": "Adds Mathlib specific instances to the `UIntX` data types.", "summary": "The `CommRing` instances (and the `NatCast` and `IntCast` instances from which they is built) are scoped in the `UIntX.CommRing` namespace, rather than available globally. As a result, the `ring` tactic will not work on `UIntX` types without `open scoped UIntX.Ring`. This is because the presence of these casting operations contradicts assumptions made by the expression tree elaborator, namely that coercions do not…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/UInt.html"}, {"id": "Mathlib.Data.FunLike.Fintype", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 265.154, "z": 31.361, "size": 0.2602, "title": "Finiteness of `DFunLike` types", "summary": "We show a type `F` with a `DFunLike F α β` is finite if both `α` and `β` are finite. This corresponds to the following two pairs of declarations: * `DFunLike.fintype` is a definition stating all `DFunLike`s are finite if their domain and codomain are. * `DFunLike.finite` is a lemma stating all `DFunLike`s are finite if their domain and codomain are. * `FunLike.fintype` is a non-dependent version of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/FunLike/Fintype.html"}, {"id": "Mathlib.Data.Nat.Digits.Div", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 212.243, "z": 53.941, "size": 0.2, "title": "Divisibility tests for natural numbers in terms of digits.", "summary": "We prove some divisibility tests based on digits, in particular completing Theorem #85 from https://www.cs.ru.nl/~freek/100/.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Digits/Div.html"}, {"id": "Mathlib.Data.List.Palindrome", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 214.356, "z": 38.666, "size": 0.2478, "title": "Palindromes", "summary": "This module defines *palindromes*, lists which are equal to their reverse. The main result is the `Palindrome` inductive type, and its associated `Palindrome.rec` induction principle. Also provided are conversions to and from other equivalent definitions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Palindrome.html"}, {"id": "Mathlib.Data.Fintype.CardEmbedding", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": 249.201, "z": -3.193, "size": 0.3096, "title": "Number of embeddings", "summary": "This file establishes the cardinality of `α ↪ β` in full generality.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/CardEmbedding.html"}, {"id": "Mathlib.Data.QPF.Multivariate.Constructions.Quot", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 217.609, "z": 46.964, "size": 0.2, "title": "The quotient of QPF is itself a QPF", "summary": "The quotients are here defined using a surjective function and its right inverse. They are very similar to the `abs` and `repr` functions found in the definition of `MvQPF`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/QPF/Multivariate/Constructions/Quot.html"}, {"id": "Mathlib.Data.EReal.Inv", "region_id": "foundations_data", "micro_elevation": 0.8824, "macro_tier": 1, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": 162.172, "z": 12.65, "size": 0.3045, "title": "Absolute value, sign, inversion and division on extended real numbers", "summary": "This file defines an absolute value and sign function on `EReal` and uses them to provide a `CommMonoidWithZero` instance, based on the absolute value and sign characterising all `EReal`s. Then it defines the inverse of an `EReal` as `⊤⁻¹ = ⊥⁻¹ = 0`, which leads to a `DivInvMonoid` instance and division.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/EReal/Inv.html"}, {"id": "Mathlib.Data.ZMod.Factorial", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 211.812, "z": 39.358, "size": 0.2, "title": "Facts about factorials in ZMod", "summary": "We collect facts about factorials in context of modular arithmetic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ZMod/Factorial.html"}, {"id": "Mathlib.Data.Nat.Choose.Lucas", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 214.162, "z": 38.578, "size": 0.2, "title": "Lucas's theorem", "summary": "This file contains a proof of [Lucas's theorem](https://en.wikipedia.org/wiki/Lucas's_theorem) about binomial coefficients, which says that for primes `p`, `n` choose `k` is congruent to product of `n_i` choose `k_i` modulo `p`, where `n_i` and `k_i` are the base-`p` digits of `n` and `k`, respectively.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Choose/Lucas.html"}, {"id": "Mathlib.Data.List.ModifyLast", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/ModifyLast.html"}, {"id": "Mathlib.Data.FinEnum", "region_id": "foundations_data", "micro_elevation": 0.7059, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 215.699, "z": 86.983, "size": 0.2478, "title": null, "summary": "Type class for finitely enumerable types. The property is stronger than `Fintype` in that it assigns each element a rank in a finite enumeration.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/FinEnum.html"}, {"id": "Mathlib.Data.List.SplitLengths", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": "Splitting a list to chunks of specified lengths", "summary": "This file defines splitting a list to chunks of given lengths, and some proofs about that.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/SplitLengths.html"}, {"id": "Mathlib.Data.Finsupp.AList", "region_id": "foundations_data", "micro_elevation": 0.2647, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 201.062, "z": 52.732, "size": 0.239, "title": "Connections between `Finsupp` and `AList`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/AList.html"}, {"id": "Mathlib.Data.Rat.Star", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 211.292, "z": 43.617, "size": 0.2, "title": "Star ordered ring structures on `ℚ` and `ℚ≥0`", "summary": "This file shows that `ℚ` and `ℚ≥0` are `StarOrderedRing`s. In particular, this means that every nonnegative rational number is a sum of squares.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Star.html"}, {"id": "Mathlib.Data.Nat.Fib.Basic", "region_id": "foundations_data", "micro_elevation": 0.4118, "macro_tier": 1, "macro_tier_score": 0.0219, "macro_tier_override": null, "x": 222.194, "z": 14.62, "size": 0.3382, "title": "Fibonacci numbers", "summary": "This file defines the Fibonacci sequence as `F₀ = 0, F₁ = 1, Fₙ₊₂ = Fₙ + Fₙ₊₁`. Furthermore, it proves results about the sequence and introduces methods to compute it quickly.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Fib/Basic.html"}, {"id": "Mathlib.Data.QPF.Multivariate.Constructions.Const", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 216.184, "z": 47.671, "size": 0.2, "title": "Constant functors are QPFs", "summary": "Constant functors map every type vectors to the same target type. This is a useful device for constructing data types from more basic types that are not actually functorial. For instance `Const n Nat` makes `Nat` into a functor that can be used in a functor-based data type specification.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/QPF/Multivariate/Constructions/Const.html"}, {"id": "Mathlib.Data.FP.Basic", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 206.967, "z": 44.668, "size": 0.2, "title": "Implementation of floating-point numbers (experimental).", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/FP/Basic.html"}, {"id": "Mathlib.Data.Semiquot", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 212.574, "z": 44.175, "size": 0.2276, "title": "Semiquotients", "summary": "A data type for semiquotients, which are classically equivalent to nonempty sets, but are useful for programming; the idea is that a semiquotient set `S` represents some (particular but unknown) element of `S`. This can be used to model nondeterministic functions, which return something in a range of values (represented by the predicate `S`) but are not completely determined.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Semiquot.html"}, {"id": "Mathlib.Data.Sum.Interval", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 211.987, "z": 41.66, "size": 0.2, "title": "Finite intervals in a disjoint union", "summary": "This file provides the `LocallyFiniteOrder` instance for the disjoint sum and linear sum of two orders and calculates the cardinality of their finite intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sum/Interval.html"}, {"id": "Mathlib.Data.Erased", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": "A type for VM-erased data", "summary": "This file defines a type `Erased α` which is classically isomorphic to `α`, but erased in the VM. That is, at runtime every value of `Erased α` is represented as `0`, just like types and proofs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Erased.html"}, {"id": "Mathlib.Data.PFunctor.Multivariate.M", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 207.642, "z": 39.979, "size": 0.2478, "title": "The M construction as a multivariate polynomial functor.", "summary": "M types are potentially infinite tree-like structures. They are defined as the greatest fixpoint of a polynomial functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PFunctor/Multivariate/M.html"}, {"id": "Mathlib.Data.PNat.Factors", "region_id": "foundations_data", "micro_elevation": 0.4706, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 244.067, "z": 34.891, "size": 0.2, "title": "Prime factors of nonzero naturals", "summary": "This file defines the factorization of a nonzero natural number `n` as a multiset of primes, the multiplicity of `p` in this factors multiset being the p-adic valuation of `n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PNat/Factors.html"}, {"id": "Mathlib.Data.FinEnum.Option", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 247.425, "z": 5.644, "size": 0.2, "title": "FinEnum instance for Option", "summary": "Provides a recursor for FinEnum types like `Fintype.truncRecEmptyOption`, but capable of producing non-truncated data.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/FinEnum/Option.html"}, {"id": "Mathlib.Data.Rat.NatSqrt.Real", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/NatSqrt/Real.html"}, {"id": "Mathlib.Data.Fin.Tuple.BubbleSortInduction", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 160.983, "z": 39.988, "size": 0.2, "title": "\"Bubble sort\" induction", "summary": "We implement the following induction principle `Tuple.bubble_sort_induction` on tuples with values in a linear order `α`. Let `f : Fin n → α` and let `P` be a predicate on `Fin n → α`. Then we can show that `f ∘ sort f` satisfies `P` if `f` satisfies `P`, and whenever some `g : Fin n → α` satisfies `P` and `g i > g j` for some `i < j`, then `g ∘ swap i j` also satisfies `P`. We deduce it from a stronger variant…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Tuple/BubbleSortInduction.html"}, {"id": "Mathlib.Data.Fin.Tuple.Sort", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 1, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": 171.95, "z": 15.13, "size": 0.2566, "title": "Sorting tuples by their values", "summary": "Given an `n`-tuple `f : Fin n → α` where `α` is ordered, we may want to turn it into a sorted `n`-tuple. This file provides an API for doing so, with the sorted `n`-tuple given by `f ∘ Tuple.sort f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Tuple/Sort.html"}, {"id": "Mathlib.Data.Analysis.Topology", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 211.539, "z": 40.694, "size": 0.2, "title": "Computational realization of topological spaces (experimental)", "summary": "This file provides infrastructure to compute with topological spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Analysis/Topology.html"}, {"id": "Mathlib.Data.Finsupp.NeLocus", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": "Locus of unequal values of finitely supported functions", "summary": "Let `α N` be two Types, assume that `N` has a `0` and let `f g : α →₀ N` be finitely supported functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/NeLocus.html"}, {"id": "Mathlib.Data.Fin.Tuple.Curry", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": "Currying and uncurrying of n-ary functions", "summary": "A function of `n` arguments can either be written as `f a₁ a₂ ⋯ aₙ` or `f' ![a₁, a₂, ⋯, aₙ]`. This file provides the currying and uncurrying operations that convert between the two, as n-ary generalizations of the binary `curry` and `uncurry`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Tuple/Curry.html"}, {"id": "Mathlib.Data.Num.Basic", "region_id": "foundations_data", "micro_elevation": 0.0294, "macro_tier": 1, "macro_tier_score": 0.0211, "macro_tier_override": null, "x": 211.814, "z": 41.426, "size": 0.2664, "title": "Binary representation of integers using inductive types", "summary": "Note: Unlike in Coq, where this representation is preferred because of the reliance on kernel reduction, in Lean this representation is discouraged in favor of the \"Peano\" natural numbers `Nat`, and the purpose of this collection of theorems is to show the equivalence of the different approaches.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Num/Basic.html"}, {"id": "Mathlib.Data.String.Basic", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 212.709, "z": 32.652, "size": 0.2, "title": "Strings", "summary": "Supplementary theorems about the `String` type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/String/Basic.html"}, {"id": "Mathlib.Data.Nat.Hyperoperation", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": "Hyperoperation sequence", "summary": "This file defines the Hyperoperation sequence. `hyperoperation 0 m k = k + 1` `hyperoperation 1 m k = m + k` `hyperoperation 2 m k = m * k` `hyperoperation 3 m k = m ^ k` `hyperoperation (n + 3) m 0 = 1` `hyperoperation (n + 1) m (k + 1) = hyperoperation n m (hyperoperation (n + 1) m k)`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Hyperoperation.html"}, {"id": "Mathlib.Data.Real.Sqrt", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Real/Sqrt.html"}, {"id": "Mathlib.Data.Sym.Sym2.Init", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0423, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.307, "title": "Sym2 Rule Set", "summary": "This module defines the `Sym2` Aesop rule set. Aesop rule sets only become visible once the file in which they're declared is imported, so we must put this declaration into its own file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Sym/Sym2/Init.html"}, {"id": "Mathlib.Data.Multiset.Sum", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2105, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.3983, "title": "Disjoint sum of multisets", "summary": "This file defines the disjoint sum of two multisets as `Multiset (α ⊕ β)`. Beware not to confuse with the `Multiset.sum` operation which computes the additive sum.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Multiset/Sum.html"}, {"id": "Mathlib.Data.LawfulXor", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": "The `LawfulXor` typeclass", "summary": "This file generalizes basic lemmas about the `^^^` operator across numeric types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/LawfulXor.html"}, {"id": "Mathlib.Data.Fin.Tuple.Take", "region_id": "foundations_data", "micro_elevation": 0.1471, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 221.498, "z": 34.93, "size": 0.2, "title": "Take operations on tuples", "summary": "We define the `take` operation on `n`-tuples, which restricts a tuple to its first `m` elements. * `Fin.take`: Given `h : m ≤ n`, `Fin.take m h v` for an `n`-tuple `v = (v 0, ..., v (n - 1))` is the `m`-tuple `(v 0, ..., v (m - 1))`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Tuple/Take.html"}, {"id": "Mathlib.Data.QPF.Multivariate.Constructions.Cofix", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 220.79, "z": 42.979, "size": 0.2, "title": "The final co-algebra of a multivariate qpf is again a qpf.", "summary": "For a `(n+1)`-ary QPF `F (α₀,..,αₙ)`, we take the least fixed point of `F` with regards to its last argument `αₙ`. The result is an `n`-ary functor: `Fix F (α₀,..,αₙ₋₁)`. Making `Fix F` into a functor allows us to take the fixed point, compose with other functors and take a fixed point again.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/QPF/Multivariate/Constructions/Cofix.html"}, {"id": "Mathlib.Data.Fin.Pigeonhole", "region_id": "foundations_data", "micro_elevation": 0.6765, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 255.328, "z": 24.726, "size": 0.2, "title": "Pigeonhole-like results for Fin", "summary": "This adapts Pigeonhole-like results from `Mathlib.Data.Fintype.Card` to the setting where the map has the type `f : Fin m → Fin n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fin/Pigeonhole.html"}, {"id": "Mathlib.Data.Rat.Cardinal", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Cardinal.html"}, {"id": "Mathlib.Data.Nat.Fib.Zeckendorf", "region_id": "foundations_data", "micro_elevation": 0.4412, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 235.027, "z": 60.006, "size": 0.2, "title": "Zeckendorf's Theorem", "summary": "This file proves Zeckendorf's theorem: Every natural number can be written uniquely as a sum of distinct non-consecutive Fibonacci numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Fib/Zeckendorf.html"}, {"id": "Mathlib.Data.ENat.BigOperators", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 226.098, "z": 45.426, "size": 0.2, "title": "Sum of suprema in `ENat`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/ENat/BigOperators.html"}, {"id": "Mathlib.Data.Finset.PImage", "region_id": "foundations_data", "micro_elevation": 0.6471, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 222.633, "z": -1.374, "size": 0.2, "title": "Image of a `Finset α` under a partially defined function", "summary": "In this file we define `Part.toFinset` and `Finset.pimage`. We also prove some trivial lemmas about these definitions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/PImage.html"}, {"id": "Mathlib.Data.Finsupp.BigOperators", "region_id": "foundations_data", "micro_elevation": 0.6765, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 258.05, "z": 37.228, "size": 0.2, "title": "Sums of collections of Finsupp, and their support", "summary": "This file provides results about the `Finsupp.support` of sums of collections of `Finsupp`, including sums of `List`, `Multiset`, and `Finset`. The support of the sum is a subset of the union of the supports: * `List.support_sum_subset` * `Multiset.support_sum_subset` * `Finset.support_sum_subset` The support of the sum of pairwise disjoint finsupps is equal to the union of the supports * `List.support_sum_eq` *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/BigOperators.html"}, {"id": "Mathlib.Data.Fintype.Units", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/Units.html"}, {"id": "Mathlib.Data.Holor", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 207.992, "z": 34.863, "size": 0.2, "title": "Basic properties of holors", "summary": "Holors are indexed collections of tensor coefficients. Confusingly, they are often called tensors in physics and in the neural network community. A holor is simply a multidimensional array of values. The size of a holor is specified by a `List ℕ`, whose length is called the dimension of the holor. The tensor product of `x₁ : Holor α ds₁` and `x₂ : Holor α ds₂` is the holor given by `(x₁ ⊗ x₂) (i₁ ++ i₂) = x₁ i₁ * x₂…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Holor.html"}, {"id": "Mathlib.Data.List.Sections", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 221.226, "z": 40.049, "size": 0.2, "title": "List sections", "summary": "This file proves some stuff about `List.sections` (definition in `Data.List.Defs`). A section of a list of lists `[l₁, ..., lₙ]` is a list whose `i`-th element comes from the `i`-th list.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/List/Sections.html"}, {"id": "Mathlib.Data.NNRat.Floor", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 214.551, "z": 44.121, "size": 0.2, "title": "Floor Function for Non-negative Rational Numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/NNRat/Floor.html"}, {"id": "Mathlib.Data.PNat.Order", "region_id": "foundations_data", "micro_elevation": 0.1765, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 217.482, "z": 51.336, "size": 0.2, "title": "Order related instances for `ℕ+`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/PNat/Order.html"}, {"id": "Mathlib.Data.QPF.Multivariate.Constructions.Fix", "region_id": "foundations_data", "micro_elevation": 0.1176, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 220.534, "z": 43.613, "size": 0.2, "title": "The initial algebra of a multivariate qpf is again a qpf.", "summary": "For an `(n+1)`-ary QPF `F (α₀,..,αₙ)`, we take the least fixed point of `F` with regards to its last argument `αₙ`. The result is an `n`-ary functor: `Fix F (α₀,..,αₙ₋₁)`. Making `Fix F` into a functor allows us to take the fixed point, compose with other functors and take a fixed point again.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/QPF/Multivariate/Constructions/Fix.html"}, {"id": "Mathlib.Data.Rat.Cast.OfScientific", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 227.064, "z": 40.498, "size": 0.2, "title": null, "summary": "The `OfScientific` instance for any characteristic zero field is well-behaved with respect to the field operations. It's probably possible, by adjusting the `OfScientific` instances, to make this more general, but it's not needed at present.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Rat/Cast/OfScientific.html"}, {"id": "Mathlib.Data.Real.StarOrdered", "region_id": "foundations_data", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.459, "z": 40.39, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Real/StarOrdered.html"}, {"id": "Mathlib.Data.Set.Finite.List", "region_id": "foundations_data", "micro_elevation": 0.8529, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 222.314, "z": 96.055, "size": 0.2, "title": "Finiteness of sets of lists", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Finite/List.html"}, {"id": "Mathlib.Data.Set.Finite.Monad", "region_id": "foundations_data", "micro_elevation": 0.7941, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 255.794, "z": 9.379, "size": 0.2, "title": "Finiteness of the Set monad operations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Finite/Monad.html"}, {"id": "Mathlib.Data.Tree.Get", "region_id": "foundations_data", "micro_elevation": 0.0588, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.424, "z": 44.277, "size": 0.2, "title": "Binary tree get operation", "summary": "In this file we define `Tree.indexOf`, `Tree.get`, and `Tree.getOrElse`. These definitions were moved from the main file to avoid a dependency on `Num`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Tree/Get.html"}, {"id": "Mathlib.Data.Vector.MapLemmas", "region_id": "foundations_data", "micro_elevation": 0.2059, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 221.578, "z": 29.473, "size": 0.2, "title": null, "summary": "This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Vector/MapLemmas.html"}, {"id": "Mathlib.Data.WSeq.Defs", "region_id": "foundations_data", "micro_elevation": 0.2353, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 225.307, "z": 50.459, "size": 0.2, "title": "Miscellaneous definitions concerning weak sequences", "summary": "These definitions, as well as those in `Mathlib/Data/WSeq/Productive.lean`, are not needed for the development of `Mathlib/Data/Seq/Parallel.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/WSeq/Defs.html"}, {"id": "Mathlib.Data.Fintype.WithTopBot", "region_id": "foundations_data", "micro_elevation": 0.7353, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 260.512, "z": 28.263, "size": 0.2742, "title": "Fintype instances for `WithTop α` and `WithBot α`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Fintype/WithTopBot.html"}, {"id": "Mathlib.Data.Bool.Set", "region_id": "foundations_data", "micro_elevation": 0.0882, "macro_tier": 1, "macro_tier_score": 0.0044, "macro_tier_override": null, "x": 218.999, "z": 42.207, "size": 0.4805, "title": "Booleans and set operations", "summary": "This file contains three trivial lemmas about `Bool`, `Set.univ`, and `Set.range`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Bool/Set.html"}, {"id": "Mathlib.Logic.Equiv.Functor", "region_id": "logic_set_theory", "micro_elevation": 0.0625, "macro_tier": 1, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": -32.54, "z": -209.872, "size": 0.2998, "title": "Functor and bifunctors can be applied to `Equiv`s.", "summary": "We define ```lean def Functor.mapEquiv (f : Type u → Type v) [Functor f] [LawfulFunctor f] : α ≃ β → f α ≃ f β ``` and ```lean def Bifunctor.mapEquiv (F : Type u → Type v → Type w) [Bifunctor F] [LawfulBifunctor F] : α ≃ β → α' ≃ β' → F α α' ≃ F β β' ```", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Functor.html"}, {"id": "Mathlib.Logic.Equiv.Defs", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.4227, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.9801, "title": "Equivalence between types", "summary": "In this file we define two types: * `Equiv α β` a.k.a. `α ≃ β`: a bijective map `α → β` bundled with its inverse map; we use this (and not equality!) to express that various `Type`s or `Sort`s are equivalent. * `Equiv.Perm α`: the group of permutations `α ≃ α`. More lemmas about `Equiv.Perm` can be found in `Mathlib/GroupTheory/Perm/`. Then we define * canonical isomorphisms between various types: e.g., -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Defs.html"}, {"id": "Mathlib.SetTheory.Cardinal.Cofinality.Ordinal", "region_id": "logic_set_theory", "micro_elevation": 0.8125, "macro_tier": 2, "macro_tier_score": 0.112, "macro_tier_override": null, "x": -11.002, "z": -200.741, "size": 0.3299, "title": "Cofinality of an ordinal", "summary": "This file contains the definition of the cofinality `Ordinal.cof o` of an ordinal. This is the cofinality of the ordinal `o` when viewed as a linear order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Cofinality/Ordinal.html"}, {"id": "Mathlib.SetTheory.Cardinal.Arithmetic", "region_id": "logic_set_theory", "micro_elevation": 0.75, "macro_tier": 3, "macro_tier_score": 0.1719, "macro_tier_override": null, "x": -11.455, "z": -204.136, "size": 0.5076, "title": "Cardinal arithmetic", "summary": "Arithmetic operations on cardinals are defined in `Mathlib/SetTheory/Cardinal/Order.lean`. However, proving the important theorem `c * c = c` for infinite cardinals and its corollaries requires the use of ordinal numbers. This is done within this file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Arithmetic.html"}, {"id": "Mathlib.Logic.Equiv.List", "region_id": "logic_set_theory", "micro_elevation": 0.1875, "macro_tier": 2, "macro_tier_score": 0.1136, "macro_tier_override": null, "x": -27.687, "z": -206.751, "size": 0.4117, "title": "Equivalences involving `List`-like types", "summary": "This file defines some additional constructive equivalences using `Encodable` and the pairing function on `ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/List.html"}, {"id": "Mathlib.Logic.Function.Basic", "region_id": "logic_set_theory", "micro_elevation": 0.0625, "macro_tier": 3, "macro_tier_score": 0.4323, "macro_tier_override": null, "x": -29.51, "z": -209.872, "size": 1.084, "title": "Miscellaneous function constructions and lemmas", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Function/Basic.html"}, {"id": "Mathlib.SetTheory.ZFC.Ordinal", "region_id": "logic_set_theory", "micro_elevation": 0.625, "macro_tier": 1, "macro_tier_score": 0.0559, "macro_tier_override": null, "x": -45.882, "z": -219.347, "size": 0.2806, "title": "Von Neumann ordinals", "summary": "This file works towards the development of von Neumann ordinals, i.e. transitive sets, well-ordered under `∈`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/ZFC/Ordinal.html"}, {"id": "Mathlib.SetTheory.ZFC.Cardinal", "region_id": "logic_set_theory", "micro_elevation": 0.3125, "macro_tier": 1, "macro_tier_score": 0.0557, "macro_tier_override": null, "x": -25.235, "z": -217.043, "size": 0.2579, "title": "Cardinalities of ZFC sets", "summary": "In this file, we define the cardinalities of ZFC sets as `ZFSet.{u} → Cardinal.{u}`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/ZFC/Cardinal.html"}, {"id": "Mathlib.SetTheory.ZFC.Rank", "region_id": "logic_set_theory", "micro_elevation": 0.5625, "macro_tier": 1, "macro_tier_score": 0.0557, "macro_tier_override": null, "x": -30.168, "z": -226.14, "size": 0.2579, "title": "Ordinal ranks of PSet and ZFSet", "summary": "In this file, we define the ordinal ranks of `PSet` and `ZFSet`. These ranks are the same as `IsWellFounded.rank` over `∈`, but are defined in a way that the universe levels of ranks are the same as the indexing types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/ZFC/Rank.html"}, {"id": "Mathlib.SetTheory.Cardinal.NatCard", "region_id": "logic_set_theory", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.0578, "macro_tier_override": null, "x": -31.742, "z": -224.425, "size": 0.402, "title": "Cardinality of finite types", "summary": "The cardinality of a finite type `α` is given by `Nat.card α`. This function has the \"junk value\" of `0` for infinite types, but to ensure the function has valid output, one just needs to know that it's possible to produce a `Finite` instance for the type. (Note: we could have defined a `Finite.card` that required you to supply a `Finite` instance, but (a) the function would be `noncomputable` anyway so there is no…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/NatCard.html"}, {"id": "Mathlib.Logic.Basic", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.4216, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.9678, "title": "Basic logic properties", "summary": "This file is one of the earliest imports in mathlib.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Basic.html"}, {"id": "Mathlib.Logic.Encodable.Basic", "region_id": "logic_set_theory", "micro_elevation": 0.0625, "macro_tier": 3, "macro_tier_score": 0.1698, "macro_tier_override": null, "x": -32.64, "z": -210.095, "size": 0.4364, "title": "Encodable types", "summary": "This file defines encodable (constructively countable) types as a typeclass. This is used to provide explicit encode/decode functions from and to `ℕ`, with the information that those functions are inverses of each other. The difference with `Denumerable` is that finite types are encodable. For infinite types, `Encodable` and `Denumerable` agree.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Encodable/Basic.html"}, {"id": "Mathlib.Logic.Equiv.Nat", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.1677, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.336, "title": "Equivalences involving `ℕ`", "summary": "This file defines some additional constructive equivalences using `Encodable` and the pairing function on `ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Nat.html"}, {"id": "Mathlib.Logic.Small.Basic", "region_id": "logic_set_theory", "micro_elevation": 0.125, "macro_tier": 3, "macro_tier_score": 0.2303, "macro_tier_override": null, "x": -32.028, "z": -213.976, "size": 0.5813, "title": "Instances and theorems for `Small`.", "summary": "In particular we prove `small_of_injective` and `small_of_surjective`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Small/Basic.html"}, {"id": "Mathlib.SetTheory.Cardinal.Basic", "region_id": "logic_set_theory", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.1703, "macro_tier_override": null, "x": -33.627, "z": -204.318, "size": 0.4554, "title": "Basic results on cardinal numbers", "summary": "We provide a collection of basic results on cardinal numbers, in particular focusing on finite/countable/small types and sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Basic.html"}, {"id": "Mathlib.SetTheory.Cardinal.Aleph", "region_id": "logic_set_theory", "micro_elevation": 0.6875, "macro_tier": 3, "macro_tier_score": 0.1715, "macro_tier_override": null, "x": -48.511, "z": -203.468, "size": 0.4947, "title": "Omega, aleph, and beth functions", "summary": "This file defines the `ω`, `ℵ`, and `ℶ` functions which enumerate certain kinds of ordinals and cardinals. Each is provided in two variants: the standard versions which only take infinite values, and \"preliminary\" versions which include finite values and are sometimes more convenient. * The function `Ordinal.preOmega` enumerates the initial ordinals, i.e. the smallest ordinals with any given cardinality. Thus…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Aleph.html"}, {"id": "Mathlib.Logic.Nontrivial.Basic", "region_id": "logic_set_theory", "micro_elevation": 0.1875, "macro_tier": 1, "macro_tier_score": 0.0078, "macro_tier_override": null, "x": -31.24, "z": -205.531, "size": 0.5751, "title": "Nontrivial types", "summary": "Results about `Nontrivial`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Nontrivial/Basic.html"}, {"id": "Mathlib.Logic.Unique", "region_id": "logic_set_theory", "micro_elevation": 0.125, "macro_tier": 2, "macro_tier_score": 0.0699, "macro_tier_override": null, "x": -30.393, "z": -207.305, "size": 0.7084, "title": "Types with a unique term", "summary": "In this file we define a typeclass `Unique`, which expresses that a type has a unique term. In other words, a type that is `Inhabited` and a `Subsingleton`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Unique.html"}, {"id": "Mathlib.Logic.IsEmpty.Basic", "region_id": "logic_set_theory", "micro_elevation": 0.125, "macro_tier": 1, "macro_tier_score": 0.0024, "macro_tier_override": null, "x": -27.897, "z": -212.115, "size": 0.4079, "title": null, "summary": "In this file we prove some basic properties about the typeclass `IsEmpty`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/IsEmpty/Basic.html"}, {"id": "Mathlib.Logic.Nontrivial.Defs", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.4027, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.6988, "title": "Nontrivial types", "summary": "A type is *nontrivial* if it contains at least two elements. This is useful in particular for rings (where it is equivalent to the fact that zero is different from one) and for vector spaces (where it is equivalent to the fact that the dimension is positive). We introduce a typeclass `Nontrivial` formalizing this property. Basic results about nontrivial types are in `Mathlib/Logic/Nontrivial/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Nontrivial/Defs.html"}, {"id": "Mathlib.Logic.Denumerable", "region_id": "logic_set_theory", "micro_elevation": 0.125, "macro_tier": 2, "macro_tier_score": 0.1147, "macro_tier_override": null, "x": -33.801, "z": -208.655, "size": 0.4537, "title": "Denumerable types", "summary": "This file defines denumerable (countably infinite) types as a typeclass extending `Encodable`. This is used to provide explicit encode/decode functions from and to `ℕ`, with the information that those functions are inverses of each other.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Denumerable.html"}, {"id": "Mathlib.SetTheory.Ordinal.Rank", "region_id": "logic_set_theory", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.056, "macro_tier_override": null, "x": -30.275, "z": -224.423, "size": 0.2909, "title": "Rank in a well-founded relation", "summary": "For `r` a well-founded relation, `IsWellFounded.rank r a` is recursively defined as the least ordinal greater than the ranks of all elements below `a`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/Rank.html"}, {"id": "Mathlib.SetTheory.Ordinal.Family", "region_id": "logic_set_theory", "micro_elevation": 0.4375, "macro_tier": 3, "macro_tier_score": 0.1689, "macro_tier_override": null, "x": -40.781, "z": -217.738, "size": 0.4024, "title": "Arithmetic on families of ordinals", "summary": "This file proves basic results about the suprema of families of ordinals. Various other basic arithmetic results are given in `Principal.lean` instead.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/Family.html"}, {"id": "Mathlib.Logic.Embedding.Basic", "region_id": "logic_set_theory", "micro_elevation": 0.25, "macro_tier": 2, "macro_tier_score": 0.1207, "macro_tier_override": null, "x": -26.756, "z": -205.291, "size": 0.6164, "title": "Injective functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Embedding/Basic.html"}, {"id": "Mathlib.Logic.Equiv.Set", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.2326, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.633, "title": "Equivalences and sets", "summary": "In this file we provide lemmas linking equivalences to sets. Some notable definitions are: * `Equiv.ofInjective`: an injective function is (noncomputably) equivalent to its range. * `Equiv.setCongr`: two equal sets are equivalent as types. * `Equiv.Set.union`: a disjoint union of sets is equivalent to their `Sum`. This file is separate from `Equiv/Basic` such that we do not require the full lattice structure on sets…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Set.html"}, {"id": "Mathlib.Logic.Encodable.Pi", "region_id": "logic_set_theory", "micro_elevation": 0.375, "macro_tier": 1, "macro_tier_score": 0.0032, "macro_tier_override": null, "x": -32.65, "z": -200.496, "size": 0.4385, "title": "Encodability of Pi types", "summary": "This file provides instances of `Encodable` for types of vectors and (dependent) functions: * `Encodable.List.Vector.encodable`: vectors of length `n` (represented by lists) are encodable * `Encodable.finArrow`: vectors of length `n` (represented by `Fin`-indexed functions) are encodable * `Encodable.fintypeArrow`, `Encodable.fintypePi`: (dependent) functions with finite domain and countable codomain are encodable", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Encodable/Pi.html"}, {"id": "Mathlib.SetTheory.Cardinal.Continuum", "region_id": "logic_set_theory", "micro_elevation": 0.8125, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -40.046, "z": -231.141, "size": 0.3143, "title": "Cardinality of continuum", "summary": "In this file we define `Cardinal.continuum` (notation: `𝔠`, localized in `Cardinal`) to be `2 ^ ℵ₀`. We also prove some `simp` lemmas about cardinal arithmetic involving `𝔠`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Continuum.html"}, {"id": "Mathlib.SetTheory.Cardinal.Rat", "region_id": "logic_set_theory", "micro_elevation": 0.3125, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -30.95, "z": -202.087, "size": 0.3115, "title": "Cardinality of ℚ", "summary": "This file proves that the Cardinality of ℚ is ℵ₀", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Rat.html"}, {"id": "Mathlib.SetTheory.Cardinal.HasCardinalLT", "region_id": "logic_set_theory", "micro_elevation": 1.0, "macro_tier": 1, "macro_tier_score": 0.001, "macro_tier_override": null, "x": -49.452, "z": -190.251, "size": 0.3374, "title": "The property of being of cardinality less than a cardinal", "summary": "Given `X : Type u` and `κ : Cardinal.{v}`, we introduce a predicate `HasCardinalLT X κ` expressing that `Cardinal.lift.{v} (Cardinal.mk X) < Cardinal.lift κ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/HasCardinalLT.html"}, {"id": "Mathlib.SetTheory.Cardinal.Regular", "region_id": "logic_set_theory", "micro_elevation": 0.9375, "macro_tier": 2, "macro_tier_score": 0.0572, "macro_tier_override": null, "x": -50.642, "z": -227.437, "size": 0.3708, "title": "Regular cardinals", "summary": "This file defines regular, singular, and inaccessible cardinals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Regular.html"}, {"id": "Mathlib.SetTheory.Ordinal.Principal", "region_id": "logic_set_theory", "micro_elevation": 0.625, "macro_tier": 3, "macro_tier_score": 0.1678, "macro_tier_override": null, "x": -38.319, "z": -226.26, "size": 0.343, "title": "Principal ordinals", "summary": "If `op` is a binary operation on ordinals, we say that an ordinal `o` is `op`-principal (or `op`-indecomposable) whenever `a < o` and `b < o` imply `op a b < o`. Most commonly, one talks of additive and multiplicative principal ordinals. Additive principal ordinals were originally called \"gamma numbers\" by Cantor, but this term now more commonly refers to the values given by `Ordinal.gamma`. Likewise, multiplicative…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/Principal.html"}, {"id": "Mathlib.SetTheory.Ordinal.FixedPoint", "region_id": "logic_set_theory", "micro_elevation": 0.5625, "macro_tier": 3, "macro_tier_score": 0.1681, "macro_tier_override": null, "x": -26.552, "z": -195.869, "size": 0.3626, "title": "Fixed points of normal functions", "summary": "We prove various statements about the fixed points of normal ordinal functions. We state them in two forms: as statements about indexed families of normal functions, and as statements about a single normal function. Moreover, we prove some lemmas about the fixed points of specific normal functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/FixedPoint.html"}, {"id": "Mathlib.Logic.Embedding.Set", "region_id": "logic_set_theory", "micro_elevation": 0.3125, "macro_tier": 2, "macro_tier_score": 0.0604, "macro_tier_override": null, "x": -38.118, "z": -205.825, "size": 0.4959, "title": "Interactions between embeddings and sets.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Embedding/Set.html"}, {"id": "Mathlib.SetTheory.Cardinal.Finsupp", "region_id": "logic_set_theory", "micro_elevation": 0.8125, "macro_tier": 2, "macro_tier_score": 0.0567, "macro_tier_override": null, "x": -28.666, "z": -232.918, "size": 0.3414, "title": "Results on the cardinality of finitely supported functions and multisets.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Finsupp.html"}, {"id": "Mathlib.Logic.Small.Set", "region_id": "logic_set_theory", "micro_elevation": 0.1875, "macro_tier": 3, "macro_tier_score": 0.1675, "macro_tier_override": null, "x": -29.052, "z": -205.919, "size": 0.3231, "title": "Results about `Small` on coerced sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Small/Set.html"}, {"id": "Mathlib.Logic.UnivLE", "region_id": "logic_set_theory", "micro_elevation": 0.125, "macro_tier": 3, "macro_tier_score": 0.1723, "macro_tier_override": null, "x": -33.711, "z": -208.538, "size": 0.5175, "title": "UnivLE", "summary": "A proposition expressing a universe inequality. `UnivLE.{u, v}` expresses that `u ≤ v`, in the form `∀ α : Type u, Small.{v} α`. This API indirectly provides an instance for `Small.{u, max u v}`, which could not be declared directly due to https://github.com/leanprover/lean4/issues/2297. See the doc-string for the comparison with an alternative stronger definition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/UnivLE.html"}, {"id": "Mathlib.SetTheory.Cardinal.Order", "region_id": "logic_set_theory", "micro_elevation": 0.0625, "macro_tier": 3, "macro_tier_score": 0.168, "macro_tier_override": null, "x": -31.707, "z": -209.107, "size": 0.3525, "title": "Order on cardinal numbers", "summary": "We define the order on cardinal numbers and show its basic properties, including the ordered semiring structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Order.html"}, {"id": "Mathlib.Logic.Equiv.Fin.Basic", "region_id": "logic_set_theory", "micro_elevation": 0.375, "macro_tier": 2, "macro_tier_score": 0.0625, "macro_tier_override": null, "x": -32.156, "z": -220.942, "size": 0.5538, "title": "Equivalences for `Fin n`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Fin/Basic.html"}, {"id": "Mathlib.SetTheory.Ordinal.Enum", "region_id": "logic_set_theory", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.1672, "macro_tier_override": null, "x": -41.465, "z": -201.725, "size": 0.2996, "title": "Enumerating sets of ordinals by ordinals", "summary": "The ordinals have the peculiar property that every subset bounded above is a small type, while themselves not being small. As a consequence of this, every unbounded subset of `Ordinal` is order isomorphic to `Ordinal`. We define this correspondence as `enumOrd`, and use it to then define an order isomorphism `enumOrdOrderIso`. This can be thought of as an ordinal analog of `Nat.nth`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/Enum.html"}, {"id": "Mathlib.Logic.Small.Defs", "region_id": "logic_set_theory", "micro_elevation": 0.0625, "macro_tier": 3, "macro_tier_score": 0.296, "macro_tier_override": null, "x": -29.321, "z": -210.453, "size": 0.7722, "title": "Small types", "summary": "A type is `w`-small if there exists an equivalence to some `S : Type w`. We provide a noncomputable model `Shrink α : Type w`, and `equivShrink α : α ≃ Shrink α`. A subsingleton type is `w`-small for any `w`. If `α ≃ β`, then `Small.{w} α ↔ Small.{w} β`. See `Mathlib/Logic/Small/Basic.lean` for further instances and theorems.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Small/Defs.html"}, {"id": "Mathlib.SetTheory.Cardinal.Pigeonhole", "region_id": "logic_set_theory", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -58.537, "z": -210.218, "size": 0.259, "title": "Infinite pigeonhole principle", "summary": "This file proves variants of the infinite pigeonhole principle.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Pigeonhole.html"}, {"id": "Mathlib.SetTheory.ZFC.VonNeumann", "region_id": "logic_set_theory", "micro_elevation": 0.8125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -30.112, "z": -188.348, "size": 0.2, "title": "Von Neumann hierarchy", "summary": "This file defines the von Neumann hierarchy of sets `V_ o` for ordinal `o`, which is recursively defined so that `V_ a = ⋃ b < a, powerset (V_ b)`. This stratifies the universal class, in the sense that `⋃ o, V_ o = univ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/ZFC/VonNeumann.html"}, {"id": "Mathlib.SetTheory.ZFC.Class", "region_id": "logic_set_theory", "micro_elevation": 0.6875, "macro_tier": 1, "macro_tier_score": 0.0557, "macro_tier_override": null, "x": -49.82, "z": -212.832, "size": 0.2478, "title": "ZFC classes", "summary": "Classes in set theory are usually defined as collections of elements satisfying some property. Here, however, we define `Class` as `Set ZFSet` to derive many instances automatically, most of them being the lifting of set operations to classes. The usual definition is then definitionally equal to ours.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/ZFC/Class.html"}, {"id": "Mathlib.Logic.Equiv.Pairwise", "region_id": "logic_set_theory", "micro_elevation": 0.1875, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -35.771, "z": -208.664, "size": 0.2, "title": "Interaction of equivalences with `Pairwise`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Pairwise.html"}, {"id": "Mathlib.Logic.Pairwise", "region_id": "logic_set_theory", "micro_elevation": 0.125, "macro_tier": 2, "macro_tier_score": 0.06, "macro_tier_override": null, "x": -28.393, "z": -212.901, "size": 0.483, "title": "Relations holding pairwise", "summary": "This file defines pairwise relations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Pairwise.html"}, {"id": "Mathlib.SetTheory.Ordinal.Arithmetic", "region_id": "logic_set_theory", "micro_elevation": 0.375, "macro_tier": 3, "macro_tier_score": 0.1694, "macro_tier_override": null, "x": -25.013, "z": -202.299, "size": 0.4205, "title": "Ordinal arithmetic", "summary": "Ordinals have an addition (corresponding to the disjoint union) that turns them into an additive monoid, and a multiplication (corresponding to the lexicographic order on the product) that turns them into a monoid. One can also define (truncated) subtraction and division operators. Ordinal powers and logarithms are defined in `Mathlib.SetTheory.Ordinal.Exponential`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/Arithmetic.html"}, {"id": "Mathlib.SetTheory.Ordinal.Basic", "region_id": "logic_set_theory", "micro_elevation": 0.3125, "macro_tier": 3, "macro_tier_score": 0.1685, "macro_tier_override": null, "x": -23.469, "z": -206.581, "size": 0.3837, "title": "Ordinals", "summary": "Ordinals are defined as equivalences of well-ordered sets under order isomorphism. They are endowed with a total order, where an ordinal is smaller than another one if it embeds into it as an initial segment (or, equivalently, in any way). This total order is well founded.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/Basic.html"}, {"id": "Mathlib.SetTheory.Cardinal.Finite", "region_id": "logic_set_theory", "micro_elevation": 0.4375, "macro_tier": 2, "macro_tier_score": 0.0629, "macro_tier_override": null, "x": -37.135, "z": -221.058, "size": 0.5631, "title": "Finite Cardinality Functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Finite.html"}, {"id": "Mathlib.Logic.Equiv.Option", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1139, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.4257, "title": "Equivalences for `Option α`", "summary": "We define * `Equiv.optionCongr`: the `Option α ≃ Option β` constructed from `e : α ≃ β` by sending `none` to `none`, and applying `e` elsewhere. * `Equiv.removeNone`: the `α ≃ β` constructed from `Option α ≃ Option β` by removing `none` from both sides.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Option.html"}, {"id": "Mathlib.Logic.OpClass", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.5238, "title": "Typeclasses for commuting heterogeneous operations", "summary": "The three classes in this file are for two-argument functions where one input is of type `α`, the output is of type `β` and the other input is of type `α` or `β`. They express the property that permuting arguments of type `α` does not change the result.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/OpClass.html"}, {"id": "Mathlib.Logic.Lemmas", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0022, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.4004, "title": "More basic logic properties", "summary": "A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is convenient to be able to use the `tauto` or `split_ifs` tactics.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Lemmas.html"}, {"id": "Mathlib.SetTheory.Ordinal.FundamentalSequence", "region_id": "logic_set_theory", "micro_elevation": 0.875, "macro_tier": 2, "macro_tier_score": 0.0564, "macro_tier_override": null, "x": -54.792, "z": -206.839, "size": 0.3208, "title": "Fundamental sequences", "summary": "A fundamental sequence for a countable limit ordinal `o` is a strictly monotone function `ℕ → Iio o` with cofinal range. We can generalize this notion to arbitrary ordinals by setting the domain as `Iio o.cof.card`. Note that for a countable limit ordinal, one has `o.cof.card = ω`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/FundamentalSequence.html"}, {"id": "Mathlib.Logic.Equiv.PartialEquiv", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.001, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.3355, "title": "Partial equivalences", "summary": "This file defines equivalences between subsets of given types. An element `e` of `PartialEquiv α β` is made of two maps `e.toFun` and `e.invFun` respectively from α to β and from β to α (just like equivs), which are inverse to each other on the subsets `e.source` and `e.target` of respectively α and β. They are designed in particular to define charts on manifolds. The main functionality is `e.trans f`, which…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/PartialEquiv.html"}, {"id": "Mathlib.Logic.Equiv.Basic", "region_id": "logic_set_theory", "micro_elevation": 0.1875, "macro_tier": 2, "macro_tier_score": 0.1254, "macro_tier_override": null, "x": -26.239, "z": -208.758, "size": 0.7065, "title": "Equivalence between types", "summary": "In this file we continue the work on equivalences begun in `Mathlib/Logic/Equiv/Defs.lean`, defining a lot of equivalences between various types and operations on these equivalences. More definitions of this kind can be found in other files. E.g., `Mathlib/Algebra/Group/TransferInstance.lean` does it for `Group`, `Mathlib/Algebra/Module/TransferInstance.lean` does it for `Module`, and similar files exist for other…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Basic.html"}, {"id": "Mathlib.Logic.Equiv.Sum", "region_id": "logic_set_theory", "micro_elevation": 0.125, "macro_tier": 2, "macro_tier_score": 0.1137, "macro_tier_override": null, "x": -32.48, "z": -213.802, "size": 0.4153, "title": "Equivalence between sum types", "summary": "In this file we continue the work on equivalences begun in `Mathlib/Logic/Equiv/Defs.lean`, defining * canonical isomorphisms between various types: e.g., - `Equiv.sumEquivSigmaBool` is the canonical equivalence between the sum of two types `α ⊕ β` and the sigma-type `Σ b, bif b then β else α`; - `Equiv.prodSumDistrib : α × (β ⊕ γ) ≃ (α × β) ⊕ (α × γ)` shows that type product and type sum satisfy the distributive…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Sum.html"}, {"id": "Mathlib.Logic.Function.Conjugate", "region_id": "logic_set_theory", "micro_elevation": 0.125, "macro_tier": 3, "macro_tier_score": 0.1806, "macro_tier_override": null, "x": -31.356, "z": -207.262, "size": 0.7005, "title": "Semiconjugate and commuting maps", "summary": "We define the following predicates: * `Function.Semiconj`: `f : α → β` semiconjugates `ga : α → α` to `gb : β → β` if `f ∘ ga = gb ∘ f`; * `Function.Semiconj₂`: `f : α → β` semiconjugates a binary operation `ga : α → α → α` to `gb : β → β → β` if `f (ga x y) = gb (f x) (f y)`; * `Function.Commute`: `f : α → α` commutes with `g : α → α` if `f ∘ g = g ∘ f`, or equivalently `Semiconj f g g`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Function/Conjugate.html"}, {"id": "Mathlib.Logic.Function.Iterate", "region_id": "logic_set_theory", "micro_elevation": 0.1875, "macro_tier": 1, "macro_tier_score": 0.0263, "macro_tier_override": null, "x": -32.741, "z": -205.82, "size": 0.8874, "title": "Iterations of a function", "summary": "In this file we prove simple properties of `Nat.iterate f n` a.k.a. `f^[n]`: * `iterate_zero`, `iterate_succ`, `iterate_succ'`, `iterate_add`, `iterate_mul`: formulas for `f^[0]`, `f^[n+1]` (two versions), `f^[n+m]`, and `f^[n*m]`; * `iterate_id` : `id^[n]=id`; * `Injective.iterate`, `Surjective.iterate`, `Bijective.iterate` : iterates of an injective/surjective/bijective function belong to the same class; *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Function/Iterate.html"}, {"id": "Mathlib.Logic.Nonempty", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.3952, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.5379, "title": "Nonempty types", "summary": "This file proves a few extra facts about `Nonempty`, which is defined in core Lean.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Nonempty.html"}, {"id": "Mathlib.Logic.Function.Defs", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.518, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.7693, "title": "General operations on functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Function/Defs.html"}, {"id": "Mathlib.Logic.ExistsUnique", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.3951, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.5347, "title": "`ExistsUnique`", "summary": "This file defines the `ExistsUnique` predicate, notated as `∃!`, and proves some of its basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/ExistsUnique.html"}, {"id": "Mathlib.Logic.IsEmpty.Defs", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.1748, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.5814, "title": "Types that are empty", "summary": "In this file we define a typeclass `IsEmpty`, which expresses that a type has no elements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/IsEmpty/Defs.html"}, {"id": "Mathlib.Logic.Function.DependsOn", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.3181, "title": "Functions depending only on some variables", "summary": "When dealing with a function `f : Π i, α i` depending on many variables, some operations may get rid of the dependency on some variables (see `Function.updateFinset` or `MeasureTheory.lmarginal` for example). However considering this new function as having a different domain with fewer points is not comfortable in Lean, as it requires the use of subtypes and can lead to tedious writing. On the other hand one wants…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Function/DependsOn.html"}, {"id": "Mathlib.Logic.Equiv.Prod", "region_id": "logic_set_theory", "micro_elevation": 0.0625, "macro_tier": 2, "macro_tier_score": 0.1132, "macro_tier_override": null, "x": -29.306, "z": -210.742, "size": 0.392, "title": "Equivalence between product types", "summary": "In this file we continue the work on equivalences begun in `Mathlib/Logic/Equiv/Defs.lean`, focusing on product types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Prod.html"}, {"id": "Mathlib.Logic.Encodable.Lattice", "region_id": "logic_set_theory", "micro_elevation": 0.125, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -34.398, "z": -211.359, "size": 0.3098, "title": "Lattice operations on encodable types", "summary": "Lemmas about lattice and set operations on encodable types", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Encodable/Lattice.html"}, {"id": "Mathlib.Logic.Function.Coequalizer", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.2522, "title": "Coequalizer of a pair of functions", "summary": "The coequalizer of two functions `f g : α → β` is the pair (`μ`, `p : β → μ`) that satisfies the following universal property: Every function `u : β → γ` with `u ∘ f = u ∘ g` factors uniquely via `p`. In this file we define the coequalizer and provide the basic API.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Function/Coequalizer.html"}, {"id": "Mathlib.Logic.Relator", "region_id": "logic_set_theory", "micro_elevation": 0.0625, "macro_tier": 2, "macro_tier_score": 0.1193, "macro_tier_override": null, "x": -31.481, "z": -209.028, "size": 0.585, "title": "Relator for functions, pairs, sums, and lists.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Relator.html"}, {"id": "Mathlib.Logic.Equiv.Fin.Rotate", "region_id": "logic_set_theory", "micro_elevation": 0.4375, "macro_tier": 1, "macro_tier_score": 0.001, "macro_tier_override": null, "x": -23.39, "z": -219.993, "size": 0.337, "title": "Cyclic permutations on `Fin n`", "summary": "This file defines * `finRotate`, which corresponds to the cycle `(1, ..., n)` on `Fin n` * `finCycle`, the permutation that adds a fixed number to each element of `Fin n` and proves various lemmas about them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Fin/Rotate.html"}, {"id": "Mathlib.Logic.Relation", "region_id": "logic_set_theory", "micro_elevation": 0.125, "macro_tier": 1, "macro_tier_score": 0.0043, "macro_tier_override": null, "x": -32.144, "z": -207.434, "size": 0.4785, "title": "Relation closures", "summary": "This file defines the reflexive, symmetric, transitive, reflexive transitive and equivalence closures of relations and proves some basic results on them. Note that this is about unbundled relations, that is terms of types of the form `α → β → Prop`. For the bundled version, see `Rel`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Relation.html"}, {"id": "Mathlib.SetTheory.Cardinal.Cofinality.Club", "region_id": "logic_set_theory", "micro_elevation": 0.5625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -16.823, "z": -216.839, "size": 0.2, "title": "Club sets and stationary sets", "summary": "A subset of a well-ordered type `α` is called a **club set** when it is closed in the order topology and cofinal. If `α` has no maximum, then an equivalent condition is that `α` is closed and unbounded; hence the name. A **stationary set** is a set which intersects all club sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Cofinality/Club.html"}, {"id": "Mathlib.SetTheory.Cardinal.Cofinality.Enum", "region_id": "logic_set_theory", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.1681, "macro_tier_override": null, "x": -30.482, "z": -224.433, "size": 0.3581, "title": "Enumerating a cofinal set", "summary": "We define a typeclass `IsRegularCardinalOrder` for well-ordered types, whose order type equals (the initial ordinal of) their cofinality. This notion does not appear in the literature, but intends to generalize the properties of intervals `Iio c.ord`, wherever `c` is a regular cardinal. Other instances of this typeclass include `ℕ`, `Ordinal`, and `Cardinal`. If `s` is a cofinal subset of a regular cardinal order…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Cofinality/Enum.html"}, {"id": "Mathlib.Logic.Small.List", "region_id": "logic_set_theory", "micro_elevation": 0.1875, "macro_tier": 2, "macro_tier_score": 0.1672, "macro_tier_override": null, "x": -26.641, "z": -213.406, "size": 0.2949, "title": "Instances for `Small (List α)` and `Small (Vector α)`.", "summary": "These must not be in `Logic.Small.Basic` as this is very low in the import hierarchy, and is used by category theory files which do not need everything imported by `Data.Vector.Basic`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Small/List.html"}, {"id": "Mathlib.SetTheory.Ordinal.Exponential", "region_id": "logic_set_theory", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.1674, "macro_tier_override": null, "x": -37.17, "z": -222.995, "size": 0.3133, "title": "Ordinal exponential", "summary": "In this file we define the power function and the logarithm function on ordinals. The two are related by the lemma `Ordinal.opow_le_iff_le_log : b ^ c ≤ x ↔ c ≤ log b x` for nontrivial inputs `b`, `c`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/Exponential.html"}, {"id": "Mathlib.SetTheory.Cardinal.SchroederBernstein", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1669, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.259, "title": "Schröder-Bernstein theorem, well-ordering of cardinals", "summary": "This file proves the Schröder-Bernstein theorem (see `schroeder_bernstein`), the well-ordering of cardinals (see `min_injective`) and the totality of their order (see `total`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/SchroederBernstein.html"}, {"id": "Mathlib.SetTheory.Cardinal.CountableCover", "region_id": "logic_set_theory", "micro_elevation": 0.8125, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -40.929, "z": -190.643, "size": 0.2374, "title": "Cardinality of a set with a countable cover", "summary": "Assume that a set `t` is eventually covered by a countable family of sets, all with cardinality `≤ a`. Then `t` itself has cardinality at most `a`. This is proved in `Cardinal.mk_subtype_le_of_countable_eventually_mem`. Versions are also given when `t = univ`, and with `= a` instead of `≤ a`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/CountableCover.html"}, {"id": "Mathlib.Logic.Equiv.Array", "region_id": "logic_set_theory", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -28.48, "z": -204.295, "size": 0.2, "title": "Equivalences involving `Array`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Array.html"}, {"id": "Mathlib.Logic.Godel.GodelBetaFunction", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.2, "title": "Gödel's Beta Function Lemma", "summary": "This file proves Gödel's Beta Function Lemma, used to prove the First Incompleteness Theorem. It permits quantification over finite sequences of natural numbers in formal theories of arithmetic. This Beta Function has no connection with the unrelated Beta Function defined in analysis. Note that `Nat.beta` and `Nat.unbeta` provide similar functionality to `Encodable.encodeList` and `Encodable.decodeList`. We define…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Godel/GodelBetaFunction.html"}, {"id": "Mathlib.Logic.Function.FiberPartition", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.2557, "title": null, "summary": "This file defines the type `f.Fiber` of fibers of a function `f : Y → Z`, and provides some API to work with and construct terms of this type. Note: this API is designed to be useful when defining the counit of the adjunction between the functor which takes a set to the condensed set corresponding to locally constant maps to that set, and the forgetful functor from the category of condensed sets to the category of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Function/FiberPartition.html"}, {"id": "Mathlib.SetTheory.Cardinal.UnivLE", "region_id": "logic_set_theory", "micro_elevation": 0.4375, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -19.78, "z": -214.983, "size": 0.2, "title": "UnivLE and cardinals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/UnivLE.html"}, {"id": "Mathlib.SetTheory.Ordinal.Univ", "region_id": "logic_set_theory", "micro_elevation": 0.375, "macro_tier": 2, "macro_tier_score": 0.1674, "macro_tier_override": null, "x": -40.632, "z": -214.449, "size": 0.3146, "title": "Universal ordinal and cardinal", "summary": "`Cardinal.univ` is the cardinality of the cardinals themselves. Likewise, `Ordinal.univ` is the order type of the ordinals. These are related via `Cardinal.univ.ord = Ordinal.univ` and `Ordinal.univ.card = Cardinal.univ`. The cardinal `Cardinal.univ` is strongly inaccessible. This reflects the fact that in ZFC, the cardinals form a proper class. See `IsInaccessible.univ` for a proof.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/Univ.html"}, {"id": "Mathlib.SetTheory.Cardinal.ToNat", "region_id": "logic_set_theory", "micro_elevation": 0.375, "macro_tier": 3, "macro_tier_score": 0.1699, "macro_tier_override": null, "x": -23.095, "z": -204.084, "size": 0.4413, "title": "Projection from cardinal numbers to natural numbers", "summary": "In this file we define `Cardinal.toNat` to be the natural projection `Cardinal → ℕ`, sending all infinite cardinals to zero. We also prove basic lemmas about this definition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/ToNat.html"}, {"id": "Mathlib.Logic.Equiv.Multiset", "region_id": "logic_set_theory", "micro_elevation": 0.25, "macro_tier": 2, "macro_tier_score": 0.1119, "macro_tier_override": null, "x": -37.904, "z": -210.681, "size": 0.3156, "title": "`Encodable` and `Denumerable` instances for `Multiset`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Multiset.html"}, {"id": "Mathlib.SetTheory.ZFC.Basic", "region_id": "logic_set_theory", "micro_elevation": 0.1875, "macro_tier": 1, "macro_tier_score": 0.056, "macro_tier_override": null, "x": -27.163, "z": -207.265, "size": 0.2859, "title": "A model of ZFC", "summary": "In this file, we model Zermelo-Fraenkel set theory (+ choice) using Lean's underlying type theory, building on the pre-sets defined in `Mathlib/SetTheory/ZFC/PSet.lean`. The theory of classes is developed in `Mathlib/SetTheory/ZFC/Class.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/ZFC/Basic.html"}, {"id": "Mathlib.Logic.Function.CompTypeclasses", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.315, "title": "Propositional typeclasses on several maps", "summary": "This file contains typeclasses that are used in the definition of equivariant maps in the spirit what was initially developed by Frédéric Dupuis and Heather Macbeth for linear maps. * `CompTriple φ ψ χ`, which expresses that `ψ.comp φ = χ` * `CompTriple.IsId φ`, which expresses that `φ = id` TODO : * align with RingHomCompTriple", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Function/CompTypeclasses.html"}, {"id": "Mathlib.SetTheory.Cardinal.Ordinal", "region_id": "logic_set_theory", "micro_elevation": 0.8125, "macro_tier": 2, "macro_tier_score": 0.0564, "macro_tier_override": null, "x": -8.753, "z": -212.633, "size": 0.3208, "title": "Ordinal arithmetic with cardinals", "summary": "This file collects results about the cardinality of different ordinal operations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Ordinal.html"}, {"id": "Mathlib.SetTheory.Cardinal.EventuallyConst", "region_id": "logic_set_theory", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -51.238, "z": -214.847, "size": 0.2, "title": "Eventually constant monotone functions", "summary": "This file proves variations of the following theorem: if `α` is a linear order and `β` is a partial order with `#β < cof α`, then any monotone function `f : α → β` must be eventually constant. In particular, this applies for functions from `Cardinal.{u}` or `Ordinal.{u}` into a `Small.{u}` type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/EventuallyConst.html"}, {"id": "Mathlib.SetTheory.Cardinal.Divisibility", "region_id": "logic_set_theory", "micro_elevation": 0.8125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -53.201, "z": -207.852, "size": 0.2255, "title": "Cardinal Divisibility", "summary": "We show basic results about divisibility in the cardinal numbers. This relation can be characterised in the following simple way: if `a` and `b` are both less than `ℵ₀`, then `a ∣ b` iff they are divisible as natural numbers. If `b` is greater than `ℵ₀`, then `a ∣ b` iff `a ≤ b`. This furthermore shows that all infinite cardinals are prime; recall that `a * b = max a b` if `ℵ₀ ≤ a * b`; therefore `a ∣ b * c = a ∣…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Divisibility.html"}, {"id": "Mathlib.Logic.Equiv.Fintype", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.3031, "title": "Equivalence between fintypes", "summary": "This file contains some basic results on equivalences where one or both sides of the equivalence are `Fintype`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Fintype.html"}, {"id": "Mathlib.SetTheory.Ordinal.FixedPointApproximants", "region_id": "logic_set_theory", "micro_elevation": 0.4375, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -38.116, "z": -220.413, "size": 0.2, "title": "Ordinal Approximants for the Fixed points on complete lattices", "summary": "This file sets up the ordinal-indexed approximation theory of fixed points of a monotone function in a complete lattice [Cousot1979]. The proof follows loosely the one from [Echenique2005]. However, the proof given here is not constructive as we use the non-constructive axiomatization of ordinals from mathlib. It still allows an approximation scheme indexed over the ordinals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/FixedPointApproximants.html"}, {"id": "Mathlib.Logic.Equiv.Embedding", "region_id": "logic_set_theory", "micro_elevation": 0.375, "macro_tier": 1, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -40.91, "z": -213.643, "size": 0.2607, "title": "Equivalences on embeddings", "summary": "This file shows some advanced equivalences on embeddings, useful for constructing larger embeddings from smaller ones.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Embedding.html"}, {"id": "Mathlib.Logic.Equiv.Bool", "region_id": "logic_set_theory", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -24.668, "z": -213.316, "size": 0.2, "title": "Equivalences involving `Bool`", "summary": "This file shows that `not : Bool → Bool` is an equivalence and derives some consequences", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Bool.html"}, {"id": "Mathlib.SetTheory.Cardinal.Defs", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1669, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.259, "title": "Cardinal Numbers", "summary": "We define cardinal numbers as a quotient of types under the equivalence relation of equinumerosity (i.e., existence of a bijection).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Defs.html"}, {"id": "Mathlib.SetTheory.Cardinal.Cofinality", "region_id": "logic_set_theory", "micro_elevation": 0.8125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -40.298, "z": -190.343, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Cofinality.html"}, {"id": "Mathlib.Logic.Function.ULift", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0029, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.43, "title": "`ULift` and `PLift`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Function/ULift.html"}, {"id": "Mathlib.SetTheory.Lists", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.2, "title": "A computable model of ZFA without infinity", "summary": "In this file we define finite hereditary lists. This is useful for calculations in naive set theory. We distinguish two kinds of ZFA lists: * Atoms. Directly correspond to an element of the original type. * Proper ZFA lists. Can be thought of (but are not implemented) as a list of ZFA lists (not necessarily proper). For example, `Lists ℕ` contains stuff like `23`, `[]`, `[37]`, `[1, [[2], 3], 4]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Lists.html"}, {"id": "Mathlib.SetTheory.Cardinal.Embedding", "region_id": "logic_set_theory", "micro_elevation": 0.5625, "macro_tier": 1, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -19.713, "z": -221.25, "size": 0.2697, "title": "Existence of embeddings from finite types", "summary": "Let `s : Set α` be a finite set. * `Fin.Embedding.exists_embedding_disjoint_range_of_add_le_ENat_card` If `s.ncard + n ≤ ENat.card α`, then there exists an embedding `Fin n ↪ α` whose range is disjoint from `s`. * `Fin.Embedding.exists_embedding_disjoint_range_of_add_le_Nat_card` If `α` is finite and `s.ncard + n ≤ Nat.card α`, then there exists an embedding `Fin n ↪ α` whose range is disjoint from `s`. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Embedding.html"}, {"id": "Mathlib.SetTheory.Cardinal.NatCount", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.2, "title": "Counting on ℕ", "summary": "This file provides lemmas about the relation of `Nat.count` with cardinality functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/NatCount.html"}, {"id": "Mathlib.SetTheory.Cardinal.Free", "region_id": "logic_set_theory", "micro_elevation": 0.875, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -23.022, "z": -187.978, "size": 0.239, "title": "Cardinalities of free constructions", "summary": "This file shows that all the free constructions over `α` have cardinality `max #α ℵ₀`, and are thus infinite, and specifically countable over countable generators. Combined with the ring `Fin n` for the finite cases, this lets us show that there is a `CommRing` of any cardinality.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Free.html"}, {"id": "Mathlib.SetTheory.Ordinal.Veblen", "region_id": "logic_set_theory", "micro_elevation": 0.625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -21.572, "z": -225.052, "size": 0.2, "title": "Veblen hierarchy", "summary": "We define the two-arguments Veblen function, which satisfies `veblen 0 a = ω ^ a` and that for `o ≠ 0`, `veblen o` enumerates the common fixed points of `veblen o'` for `o' < o`. We use this to define two important functions on ordinals: the epsilon function `ε_ o = veblen 1 o`, and the gamma function `Γ_ o` enumerating the fixed points of `veblen · 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/Veblen.html"}, {"id": "Mathlib.Logic.Equiv.Finset", "region_id": "logic_set_theory", "micro_elevation": 0.3125, "macro_tier": 2, "macro_tier_score": 0.057, "macro_tier_override": null, "x": -34.224, "z": -218.667, "size": 0.362, "title": "`Encodable` and `Denumerable` instances for `Finset`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Equiv/Finset.html"}, {"id": "Mathlib.SetTheory.Ordinal.Commute", "region_id": "logic_set_theory", "micro_elevation": 0.4375, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -27.836, "z": -222.294, "size": 0.2, "title": "Ordinal arithmetic commutativity", "summary": "Results on the commutativity of ordinal arithmetic operations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/Commute.html"}, {"id": "Mathlib.SetTheory.Ordinal.CantorNormalForm", "region_id": "logic_set_theory", "micro_elevation": 0.5625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -36.087, "z": -225.312, "size": 0.2, "title": "Cantor Normal Form", "summary": "The Cantor normal form of an ordinal is generally defined as its base `ω` expansion, with its non-zero exponents in decreasing order. Here, we more generally define a base `b` expansion `Ordinal.CNF` in this manner, which is well-behaved for any `b ≥ 2`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/CantorNormalForm.html"}, {"id": "Mathlib.Logic.IsEmpty", "region_id": "logic_set_theory", "micro_elevation": 0.125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -31.867, "z": -207.351, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/IsEmpty.html"}, {"id": "Mathlib.SetTheory.ZFC.PSet", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0557, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.2519, "title": "Pre-sets", "summary": "A pre-set is inductively defined by its indexing type and its members, which are themselves pre-sets. After defining pre-sets we define extensional equality over them, also inductively. We construct a `Setoid` instance from it, and in `Mathlib/SetTheory/ZFC/Basic.lean` we define ZFC sets as the quotient of pre-sets by extensional equality.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/ZFC/PSet.html"}, {"id": "Mathlib.Logic.Function.OfArity", "region_id": "logic_set_theory", "micro_elevation": 0.0625, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -29.673, "z": -209.623, "size": 0.2478, "title": "Function types of a given arity", "summary": "This provides `Function.OfArity`, such that `OfArity α β 2 = α → α → β`. Note that it is often preferable to use `(Fin n → α) → β` in place of `OfArity n α β`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Function/OfArity.html"}, {"id": "Mathlib.SetTheory.Cardinal.Cofinality.Basic", "region_id": "logic_set_theory", "micro_elevation": 0.3125, "macro_tier": 2, "macro_tier_score": 0.1671, "macro_tier_override": null, "x": -31.355, "z": -219.278, "size": 0.2926, "title": "Cofinality of an order", "summary": "This file contains the definition of the cofinality `Order.cof α` of an order. This is the smallest cardinality of a cofinal subset.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Cofinality/Basic.html"}, {"id": "Mathlib.SetTheory.Cardinal.Subfield", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.2543, "title": "Cardinality of the division ring generated by a set", "summary": "`Subfield.cardinalMk_closure_le_max`: the cardinality of the (sub-)division ring generated by a set is bounded by the cardinality of the set unless it is finite. The method used to prove this (via `WType`) can be easily generalized to other algebraic structures, but those cardinalities can usually be obtained by other means, using some explicit universal objects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/Subfield.html"}, {"id": "Mathlib.SetTheory.Descriptive.Tree", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.2, "title": "Trees in the sense of descriptive set theory", "summary": "This file defines trees of depth `ω` in the sense of descriptive set theory as sets of finite sequences that are stable under taking prefixes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Descriptive/Tree.html"}, {"id": "Mathlib.SetTheory.Ordinal.Topology", "region_id": "logic_set_theory", "micro_elevation": 0.5625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -35.083, "z": -195.75, "size": 0.2, "title": "Topology of ordinals", "summary": "We prove some miscellaneous results involving the order topology of ordinals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/Topology.html"}, {"id": "Mathlib.Logic.Hydra", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.2, "title": "Termination of a hydra game", "summary": "This file deals with the following version of the hydra game: each head of the hydra is labelled by an element in a type `α`, and when you cut off one head with label `a`, it grows back an arbitrary but finite number of heads, all labelled by elements smaller than `a` with respect to a well-founded relation `r` on `α`. We show that no matter how (in what order) you choose cut off the heads, the game always…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Hydra.html"}, {"id": "Mathlib.SetTheory.Ordinal.Notation", "region_id": "logic_set_theory", "micro_elevation": 0.6875, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -12.195, "z": -212.505, "size": 0.2, "title": "Ordinal notation", "summary": "Constructive ordinal arithmetic for ordinals below `ε₀`. We define a type `ONote`, with constructors `0 : ONote` and `ONote.oadd e n a` representing `ω ^ e * n + a`. We say that `o` is in Cantor normal form - `ONote.NF o` - if either `o = 0` or `o = ω ^ e * n + a` with `a < ω ^ e` and `a` in Cantor normal form. The type `NONote` is the type of ordinals below `ε₀` in Cantor normal form. Various operations (addition,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Ordinal/Notation.html"}, {"id": "Mathlib.SetTheory.Cardinal.ENat", "region_id": "logic_set_theory", "micro_elevation": 0.3125, "macro_tier": 3, "macro_tier_score": 0.1697, "macro_tier_override": null, "x": -22.561, "z": -209.169, "size": 0.4321, "title": "Conversion between `Cardinal` and `ℕ∞`", "summary": "In this file we define a coercion `Cardinal.ofENat : ℕ∞ → Cardinal` and a projection `Cardinal.toENat : Cardinal →+*o ℕ∞`. We also prove basic theorems about these definitions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/SetTheory/Cardinal/ENat.html"}, {"id": "Mathlib.Logic.Function.FromTypes", "region_id": "logic_set_theory", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0559, "macro_tier_override": null, "x": -31.025, "z": -210.686, "size": 0.2806, "title": "Function types of a given heterogeneous arity", "summary": "This provides `Function.FromTypes`, such that `FromTypes ![α, β] τ = α → β → τ`. Note that it is often preferable to use `((i : Fin n) → p i) → τ` in place of `FromTypes p τ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Logic/Function/FromTypes.html"}, {"id": "Mathlib.Control.Bifunctor", "region_id": "frontier", "micro_elevation": 0.1429, "macro_tier": 2, "macro_tier_score": 0.0103, "macro_tier_override": null, "x": -3.009, "z": 213.475, "size": 0.3073, "title": "Functors with two arguments", "summary": "This file defines bifunctors. A bifunctor is a function `F : Type* → Type* → Type*` along with a bimap which turns `F α β` into `F α' β'` given two functions `α → α'` and `β → β'`. It further * respects the identity: `bimap id id = id` * composes in the obvious way: `(bimap f' g') ∘ (bimap f g) = bimap (f' ∘ f) (g' ∘ g)`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Bifunctor.html"}, {"id": "Mathlib.Init", "region_id": "frontier", "micro_elevation": 0.1429, "macro_tier": 2, "macro_tier_score": 0.9904, "macro_tier_override": null, "x": -5.606, "z": 210.315, "size": 3.2, "title": null, "summary": "This is the root file in Mathlib: it is imported by virtually *all* Mathlib files. For this reason, the imports of this file are carefully curated. Any modification involving a change in the imports of this file should be discussed beforehand. Here are some general guidelines: * no bucket imports (e.g. `Batteries`/`Lean`/etc); * every import needs to have a comment explaining why the import is there; * strong…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init.html"}, {"id": "Mathlib.Util.Delaborators", "region_id": "frontier", "micro_elevation": 0.4286, "macro_tier": 1, "macro_tier_score": 0.007, "macro_tier_override": null, "x": -7.477, "z": 219.459, "size": 0.5552, "title": "Pi type notation", "summary": "Provides the `Π x : α, β x` notation as an alternative to Lean 4's built-in `(x : α) → β x` notation. To get all non-`∀` pi types to pretty print this way then do `open scoped PiNotation`. The notation also accepts extended binders, like `Π x ∈ s, β x` for `Π x, x ∈ s → β x`. This can be disabled with the `pp.mathlib.binderPredicates` option.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/Delaborators.html"}, {"id": "Mathlib.Control.Lawful", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.029, "macro_tier_override": null, "x": -5.252, "z": 203.433, "size": 0.249, "title": "Functor Laws, applicative laws, and monad Laws", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Lawful.html"}, {"id": "Mathlib.Util.DischargerAsTactic", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -9.211, "z": 210.981, "size": 0.275, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/DischargerAsTactic.html"}, {"id": "Mathlib.Util.FormatTable", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.2, "title": "Format Table", "summary": "This file provides a simple function for formatting a two-dimensional array of `String`s into a markdown-compliant table.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/FormatTable.html"}, {"id": "Mathlib.Util.CompileInductive", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": 3.97, "z": 205.69, "size": 0.5507, "title": "Define the `compile_inductive%` command.", "summary": "The command `compile_inductive% Foo` adds compiled code for the recursor `Foo.rec`, working around a bug in the core Lean compiler which does not support recursors. For technical reasons, the recursor code generated by `compile_inductive%` unfortunately evaluates the base cases eagerly. That is, `List.rec (unreachable!) (fun _ _ _ => 42) [42]` will panic. Similarly, `compile_def% Foo.foo` adds compiled code for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/CompileInductive.html"}, {"id": "Mathlib.Control.Monad.Basic", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0292, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.278, "title": "Monad", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Monad/Basic.html"}, {"id": "Mathlib.Control.Combinators", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.0503, "macro_tier_override": null, "x": -0.418, "z": 202.818, "size": 0.4011, "title": "Monad combinators, as in Haskell's Control.Monad.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Combinators.html"}, {"id": "Mathlib.Util.Simp", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -8.348, "z": 206.394, "size": 0.2, "title": "Additional simp utilities", "summary": "This file adds additional tools for metaprogramming with the `simp` tactic", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/Simp.html"}, {"id": "Mathlib.Util.AddRelatedDecl", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 5.344, "z": 209.645, "size": 0.4976, "title": "`addRelatedDecl`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/AddRelatedDecl.html"}, {"id": "Mathlib.Util.Qq", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0021, "macro_tier_override": null, "x": -3.837, "z": 202.897, "size": 0.3923, "title": "Extra `Qq` helpers", "summary": "This file contains some additional functions for using the quote4 library more conveniently.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/Qq.html"}, {"id": "Mathlib.Util.ElabWithoutMVars", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.001, "macro_tier_override": null, "x": -5.521, "z": 203.575, "size": 0.3357, "title": "`elabTermWithoutNewMVars`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/ElabWithoutMVars.html"}, {"id": "Mathlib.Control.EquivFunctor", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.3559, "title": "Functions functorial with respect to equivalences", "summary": "An `EquivFunctor` is a function from `Type → Type` equipped with the additional data of coherently mapping equivalences to equivalences. In categorical language, it is an endofunctor of the \"core\" of the category `Type`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/EquivFunctor.html"}, {"id": "Mathlib.Util.TermReduce", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 1.24, "z": 203.392, "size": 0.2877, "title": "Term elaborators for reduction", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/TermReduce.html"}, {"id": "Mathlib.Lean.Elab.Term", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.0121, "macro_tier_override": null, "x": -2.032, "z": 202.653, "size": 0.4133, "title": "Additions to `Lean.Elab.Term`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Elab/Term.html"}, {"id": "Mathlib.Util.Notation3", "region_id": "frontier", "micro_elevation": 0.4286, "macro_tier": 1, "macro_tier_score": 0.0085, "macro_tier_override": null, "x": 4.894, "z": 218.537, "size": 0.5912, "title": "The notation3 macro, simulating Lean 3's notation.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/Notation3.html"}, {"id": "Mathlib.Lean.FoldEnvironment", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 2.652, "z": 204.293, "size": 0.2416, "title": "Folding through the environment efficiently", "summary": "This file defines `foldImportedDecls`, a function for efficiently folding through the environment. It splits the environment into parts, each of which is folded over in a separate thread. We also provide `foldCurrFileDecls` which loops through the declarations of the current module, without any parallelism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/FoldEnvironment.html"}, {"id": "Mathlib.Lean.Meta.RefinedDiscrTree", "region_id": "frontier", "micro_elevation": 0.7143, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 8.219, "z": 194.776, "size": 0.2579, "title": null, "summary": "A discrimination tree for the purpose of unifying local expressions with library results. This data structure is based on `Lean.Meta.DiscrTree` and `Lean.Meta.LazyDiscrTree`, and includes many more features.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Meta/RefinedDiscrTree.html"}, {"id": "Mathlib.Lean.MessageData.Trace", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -8.779, "z": 212.632, "size": 0.2902, "title": "Utilities for analyzing `MessageData`", "summary": "Utility functions for working with trace messages. `withTraceNode` (in `Lean.Util.Trace`) stores a `TraceResult` in `TraceData.result?` and prepends emoji to the rendered header: - `✅️` (`checkEmoji`) for success - `❌️` (`crossEmoji`) for failure - `💥️` (`bombEmoji`) for exceptions The `traceResultOf` function provides backward-compatible parsing of rendered headers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/MessageData/Trace.html"}, {"id": "Mathlib.Util.PrintSorries", "region_id": "frontier", "micro_elevation": 0.1429, "macro_tier": 1, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -3.148, "z": 213.431, "size": 0.2647, "title": "Tracking uses of `sorry`", "summary": "This file provides a `#print sorries` command to help find out why a given declaration is not sorry-free. `#print sorries foo` returns a non-sorry-free declaration `bar` which `foo` depends on, if such a `bar` exists. The `#print sorries in CMD` combinator prints all sorries appearing in the declarations defined by the given command.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/PrintSorries.html"}, {"id": "Mathlib.Lean.Expr.Basic", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0248, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.5165, "title": "Additional operations on Expr and related types", "summary": "This file defines basic operations on the types expr, name, declaration, level, environment. This file is mostly for non-tactics.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Expr/Basic.html"}, {"id": "Mathlib.Lean.PrettyPrinter.Delaborator", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.0144, "macro_tier_override": null, "x": 2.62, "z": 215.671, "size": 0.495, "title": "Additions to the delaborator", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/PrettyPrinter/Delaborator.html"}, {"id": "Mathlib.Control.Functor", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0507, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.416, "title": "Functors", "summary": "This module provides additional lemmas, definitions, and instances for `Functor`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Functor.html"}, {"id": "Mathlib.Lean.Elab.InfoTree", "region_id": "frontier", "micro_elevation": 0.1429, "macro_tier": 1, "macro_tier_score": 0.0078, "macro_tier_override": null, "x": 0.9, "z": 207.695, "size": 0.5739, "title": "Additions to `Lean.Elab.InfoTree.Main`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Elab/InfoTree.html"}, {"id": "Mathlib.Util.ParseCommand", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0042, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.4736, "title": "`#parse` -- a command to parse text and log outputs", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/ParseCommand.html"}, {"id": "Mathlib.Control.Random", "region_id": "frontier", "micro_elevation": 0.7143, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -20.255, "z": 210.009, "size": 0.2, "title": "Rand Monad and Random Class", "summary": "This module provides tools for formulating computations guided by randomness and for defining objects that can be created randomly.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Random.html"}, {"id": "Mathlib.Control.ULiftable", "region_id": "frontier", "micro_elevation": 0.5714, "macro_tier": 2, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": -16.573, "z": 210.805, "size": 0.2757, "title": "Universe lifting for type families", "summary": "Some functors such as `Option` and `List` are universe polymorphic. Unlike type polymorphism where `Option α` is a function application and reasoning and generalizations that apply to functions can be used, `Option.{u}` and `Option.{v}` are not one function applied to two universe names but one polymorphic definition instantiated twice. This means that whatever works on `Option.{u}` is hard to transport over to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/ULiftable.html"}, {"id": "Mathlib.Control.Traversable.Lemmas", "region_id": "frontier", "micro_elevation": 0.7143, "macro_tier": 2, "macro_tier_score": 0.0296, "macro_tier_override": null, "x": 15.492, "z": 204.509, "size": 0.3134, "title": "Traversing collections", "summary": "This file proves basic properties of traversable and applicative functors and defines `PureTransformation F`, the natural applicative transformation from the identity functor to `F`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Traversable/Lemmas.html"}, {"id": "Mathlib.Control.Traversable.Instances", "region_id": "frontier", "micro_elevation": 0.7143, "macro_tier": 2, "macro_tier_score": 0.01, "macro_tier_override": null, "x": 14.733, "z": 217.434, "size": 0.2881, "title": "LawfulTraversable instances", "summary": "This file provides instances of `LawfulTraversable` for types from the core library: `Option`, `List` and `Sum`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Traversable/Instances.html"}, {"id": "Mathlib.Control.Applicative", "region_id": "frontier", "micro_elevation": 0.5714, "macro_tier": 2, "macro_tier_score": 0.0303, "macro_tier_override": null, "x": 10.064, "z": 218.301, "size": 0.3608, "title": "`applicative` instances", "summary": "This file provides `Applicative` instances for concrete functors: * `id` * `Functor.comp` * `Functor.const` * `Functor.add_const`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Applicative.html"}, {"id": "Mathlib.Control.Traversable.Basic", "region_id": "frontier", "micro_elevation": 0.5714, "macro_tier": 2, "macro_tier_score": 0.0306, "macro_tier_override": null, "x": 10.365, "z": 202.09, "size": 0.3757, "title": "Traversable type class", "summary": "Type classes for traversing collections. The concepts and laws are taken from Traversable collections are a generalization of functors. Whereas functors (such as `List`) allow us to apply a function to every element, it does not allow functions which external effects encoded in a monad. Consider for instance a functor `invite : email → IO…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Traversable/Basic.html"}, {"id": "Mathlib.Lean.Expr.ExtraRecognizers", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.3667, "title": "Additional Expr recognizers needing theory imports", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Expr/ExtraRecognizers.html"}, {"id": "Mathlib.Lean.Meta.KAbstractPositions", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -9.097, "z": 211.6, "size": 0.2578, "title": "Find the positions of a pattern in an expression", "summary": "This file defines some tools for dealing with subexpressions and occurrence numbers. This is used for creating a `rw` tactic call that rewrites a selected expression. `viewKAbstractSubExpr` takes an expression and a position in the expression, and returns the subexpression together with an optional occurrence number that would be required to find the subexpression using `kabstract` (which is what `rw` uses to find…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Meta/KAbstractPositions.html"}, {"id": "Mathlib.Lean.Expr", "region_id": "frontier", "micro_elevation": 0.1429, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -0.41, "z": 206.658, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Expr.html"}, {"id": "Mathlib.Util.LongNames", "region_id": "frontier", "micro_elevation": 0.4286, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 8.738, "z": 212.396, "size": 0.2, "title": "Commands `#long_names` and `#long_instances`", "summary": "For finding declarations with excessively long names.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/LongNames.html"}, {"id": "Mathlib.Lean.Name", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.0161, "macro_tier_override": null, "x": -7.795, "z": 214.39, "size": 0.5418, "title": "Additional functions on `Lean.Name`.", "summary": "We provide `allNames` and `allNamesByModule`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Name.html"}, {"id": "Mathlib.Lean.Meta.RefinedDiscrTree.Initialize", "region_id": "frontier", "micro_elevation": 0.4286, "macro_tier": 2, "macro_tier_score": 0.0104, "macro_tier_override": null, "x": 7.638, "z": 215.282, "size": 0.3185, "title": "Constructing a RefinedDiscrTree", "summary": "`RefinedDiscrTree` is lazy, so to add an entry, we need to compute the first `Key` and a `LazyEntry`. These are computed by `initializeLazyEntry`. We provide `RefinedDiscrTree.insert` for directly performing this insert. For initializing a `RefinedDiscrTree` using all imported constants, we provide `createImportedDiscrTree`, which loops through all imported constants, and does this with a parallel computation. There…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Meta/RefinedDiscrTree/Initialize.html"}, {"id": "Mathlib.Lean.Meta.RefinedDiscrTree.Basic", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.0119, "macro_tier_override": null, "x": -2.403, "z": 202.666, "size": 0.4049, "title": "Basic Definitions for `RefinedDiscrTree`", "summary": "We define * `Key`, the discrimination tree key * `LazyEntry`, the partial, lazy computation of a sequence of `Key`s * `Trie`, a node of the discrimination tree, which is indexed with `Key`s and stores an array of pending `LazyEntry`s * `RefinedDiscrTree`, the discrimination tree itself.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Meta/RefinedDiscrTree/Basic.html"}, {"id": "Mathlib.Control.Functor.Multivariate", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.001, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.3344, "title": "Functors between the category of tuples of types, and the category Type", "summary": "Features: * `MvFunctor n` : the type class of multivariate functors * `f <$$> x` : notation for map", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Functor/Multivariate.html"}, {"id": "Mathlib.Lean.Meta", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.0177, "macro_tier_override": null, "x": -9.213, "z": 210.966, "size": 0.582, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Meta.html"}, {"id": "Mathlib.Control.Fold", "region_id": "frontier", "micro_elevation": 0.8571, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 13.848, "z": 194.747, "size": 0.2, "title": "List folds generalized to `Traversable`", "summary": "Informally, we can think of `foldl` as a special case of `traverse` where we do not care about the reconstructed data structure and, in a state monad, we care about the final state. The obvious way to define `foldl` would be to use the state monad but it is nicer to reason about a more abstract interface with `foldMap` as a primitive and `foldMap_hom` as a defining property. ``` def foldMap {α ω} [One ω] [Mul ω] (f…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Fold.html"}, {"id": "Mathlib.Lean.Meta.CongrTheorems", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0034, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.4466, "title": "Additions to `Lean.Meta.CongrTheorems`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Meta/CongrTheorems.html"}, {"id": "Mathlib.Lean.Environment", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0163, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.5476, "title": "Additional utilities for `Lean.Environment`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Environment.html"}, {"id": "Mathlib.Lean.Elab.Tactic.Meta", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0199, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.6306, "title": "Additions to `Lean.Elab.Tactic.Meta`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Elab/Tactic/Meta.html"}, {"id": "Mathlib.Util.AtomM", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.0152, "macro_tier_override": null, "x": -4.255, "z": 216.918, "size": 0.5172, "title": "A monad for tracking and deduplicating atoms", "summary": "This monad is used by tactics like `ring` and `abel` to keep uninterpreted atoms in a consistent order, and also to allow unifying atoms up to a specified transparency mode. Note: this can become very expensive because it is using `isDefEq`. For performance reasons, consider whether `Lean.Meta.Canonicalizer.canon` can be used instead. After canonicalizing, a `HashMap Expr Nat` suffices to keep track of previously…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/AtomM.html"}, {"id": "Mathlib.Testing.Plausible.Functions", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Testing/Plausible/Functions.html"}, {"id": "Mathlib.Util.Export", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 4.263, "z": 213.81, "size": 0.2, "title": null, "summary": "A rudimentary export format, adapted from with support for Lean 4 kernel primitives.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/Export.html"}, {"id": "Mathlib.Util.DelabNonCanonical", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -5.125, "z": 203.371, "size": 0.2507, "title": null, "summary": "Delab checking canonicity. Provides a series of monadic functions in `DelabM` for delaborating expressions differently if their given instances differ (by definitional equality) with what is synthesized. Synthesized instances are considered 'canonical' for this purpose.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/DelabNonCanonical.html"}, {"id": "Mathlib.Util.AtLocation", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0042, "macro_tier_override": null, "x": 1.922, "z": 216.168, "size": 0.4761, "title": "Rewriting at specified locations", "summary": "Many metaprograms have the following general structure: the input is an expression `e` and the output is a new expression `e'`, together with a proof that `e = e'`. This file provides convenience functions to turn such a metaprogram into a variety of tactics: using the metaprogram to modify the goal, a specified hypothesis, or (via `Tactic.Location`) a combination of these.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/AtLocation.html"}, {"id": "Mathlib.Lean.GoalsLocation", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -0.698, "z": 217.175, "size": 0.2578, "title": null, "summary": "This file defines some functions for dealing with `SubExpr.GoalsLocation`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/GoalsLocation.html"}, {"id": "Mathlib.Util.PPOptions", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.0159, "macro_tier_override": null, "x": -2.658, "z": 202.686, "size": 0.5369, "title": null, "summary": "Mathlib-specific pretty printer options.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/PPOptions.html"}, {"id": "Mathlib.Control.Traversable.Equiv", "region_id": "frontier", "micro_elevation": 0.8571, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -2.388, "z": 231.914, "size": 0.2478, "title": "Transferring `Traversable` instances along isomorphisms", "summary": "This file allows to transfer `Traversable` instances along isomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Traversable/Equiv.html"}, {"id": "Mathlib.Lean.Thunk", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -9.043, "z": 211.82, "size": 0.2, "title": "Basic facts about `Thunk`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Thunk.html"}, {"id": "Mathlib.Util.AliasIn", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.2666, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/AliasIn.html"}, {"id": "Mathlib.Util.AtomM.Recurse", "region_id": "frontier", "micro_elevation": 0.4286, "macro_tier": 1, "macro_tier_score": 0.0047, "macro_tier_override": null, "x": -11.847, "z": 205.196, "size": 0.4901, "title": "Running `AtomM` metaprograms recursively", "summary": "Tactics such as `ring` and `abel` are implemented using the `AtomM` monad, which tracks \"atoms\" -- expressions which cannot be further parsed according to the arithmetic operations they handle -- to allow for consistent normalization relative to these atoms. This file provides methods to allow for such normalization to run recursively: the atoms themselves will have the normalization run on any of their subterms for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/AtomM/Recurse.html"}, {"id": "Mathlib.Lean.Meta.Simp", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.018, "macro_tier_override": null, "x": -8.482, "z": 206.646, "size": 0.7687, "title": "Helper functions for using the simplifier.", "summary": "[TODO] Needs documentation, cleanup, and possibly reunification of `mkSimpContext'` with core.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Meta/Simp.html"}, {"id": "Mathlib.Lean.Meta.RefinedDiscrTree.Lookup", "region_id": "frontier", "micro_elevation": 0.5714, "macro_tier": 2, "macro_tier_score": 0.0104, "macro_tier_override": null, "x": -16.577, "z": 210.74, "size": 0.3185, "title": "Matching with a RefinedDiscrTree", "summary": "This file defines the matching procedure for the `RefinedDiscrTree`. The main definitions are * The structure `MatchResult`, which contains the match results, ordered by matching score. * The (private) function `evalNode` which evaluates a node of the `RefinedDiscrTree` * The (private) function `getMatchLoop`, which is the main function that computes the matches. It implements the non-deterministic computation by…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Meta/RefinedDiscrTree/Lookup.html"}, {"id": "Mathlib.Util.Superscript", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": -8.155, "z": 206.07, "size": 0.3676, "title": "A parser for superscripts and subscripts", "summary": "This is intended for use in local notations. Basic usage is: ``` local syntax:arg term:max superscript(term) : term local macro_rules | `($a:term $b:superscript) => `($a ^ $b) ``` where `superscript(term)` indicates that it will parse a superscript, and the `$b:superscript` antiquotation binds the `term` argument of the superscript. Given a notation like this, the expression `2⁶⁴` parses and expands to `2 ^ 64`. The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/Superscript.html"}, {"id": "Mathlib.Control.Monad.Writer", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.029, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.249, "title": "Writer monads", "summary": "This file introduces monads for managing immutable, appendable state. Common applications are logging monads where the monad logs messages as the computation progresses.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Monad/Writer.html"}, {"id": "Mathlib.Lean.ContextInfo", "region_id": "frontier", "micro_elevation": 0.1429, "macro_tier": 2, "macro_tier_score": 0.012, "macro_tier_override": null, "x": -0.219, "z": 213.184, "size": 0.6657, "title": "Executing actions using the infotree", "summary": "This file contains helper functions for running `CoreM`, `MetaM` and tactic actions in the context of an infotree node.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/ContextInfo.html"}, {"id": "Mathlib.Util.TransImports", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -8.639, "z": 212.967, "size": 0.2647, "title": "The `#trans_imports` command", "summary": "`#trans_imports` reports how many transitive imports the current module has. The command takes an optional string input: `#trans_imports str` also shows the transitively imported modules whose name begins with `str`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/TransImports.html"}, {"id": "Mathlib.Lean.Expr.ReplaceRec", "region_id": "frontier", "micro_elevation": 0.4286, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.753, "z": 199.337, "size": 0.2, "title": "ReplaceRec", "summary": "We define a more flexible version of `Expr.replace` where we can use recursive calls even when replacing a subexpression. We completely mimic the implementation of `Expr.replace`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Expr/ReplaceRec.html"}, {"id": "Mathlib.Util.MemoFix", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": -9.166, "z": 208.676, "size": 0.2676, "title": "Fixpoint function with memoisation", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/MemoFix.html"}, {"id": "Mathlib.Util.WithWeakNamespace", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0026, "macro_tier_override": null, "x": -7.902, "z": 205.695, "size": 0.4171, "title": "Defines `with_weak_namespace` command.", "summary": "Changes the current namespace without causing scoped things to go out of scope.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/WithWeakNamespace.html"}, {"id": "Mathlib.Util.CountHeartbeats", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -7.415, "z": 205.089, "size": 0.2647, "title": null, "summary": "Defines a command wrapper that prints the number of heartbeats used in the enclosed command. For example ``` #count_heartbeats in theorem foo : 42 = 6 * 7 := rfl ``` will produce an info message containing a number around 51. If this number is above the current `maxHeartbeats`, we also print a `Try this:` suggestion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/CountHeartbeats.html"}, {"id": "Mathlib.Lean.Linter", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.5194, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.7902, "title": "Additional utilities and boilerplate for the `Linter` API", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Linter.html"}, {"id": "Mathlib.Lean.Meta.DiscrTree", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -4.423, "z": 216.86, "size": 0.2, "title": "Additions to `Lean.Meta.DiscrTree`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Meta/DiscrTree.html"}, {"id": "Mathlib.Lean.Elab.Tactic.Basic", "region_id": "frontier", "micro_elevation": 0.4286, "macro_tier": 1, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": -7.07, "z": 219.684, "size": 0.3059, "title": "Additions to `Lean.Elab.Tactic.Basic`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Elab/Tactic/Basic.html"}, {"id": "Mathlib.Lean.Meta.Basic", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -3.605, "z": 217.1, "size": 0.2843, "title": "Additions to `Lean.Meta.Basic`", "summary": "Likely these already exist somewhere. Pointers welcome.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Meta/Basic.html"}, {"id": "Mathlib.Control.Bitraversable.Instances", "region_id": "frontier", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -27.101, "z": 214.854, "size": 0.2, "title": "Bitraversable instances", "summary": "This file provides `Bitraversable` instances for concrete bifunctors: * `Prod` * `Sum` * `Functor.Const` * `flip` * `Function.bicompl` * `Function.bicompr`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Bitraversable/Instances.html"}, {"id": "Mathlib.Control.Bitraversable.Lemmas", "region_id": "frontier", "micro_elevation": 0.8571, "macro_tier": 1, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 14.764, "z": 195.761, "size": 0.2478, "title": "Bitraversable Lemmas", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Bitraversable/Lemmas.html"}, {"id": "Mathlib.Util.SynthesizeUsing", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": -8.163, "z": 213.857, "size": 0.3435, "title": "`SynthesizeUsing`", "summary": "This is a slight simplification of the `solve_aux` tactic in Lean3.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/SynthesizeUsing.html"}, {"id": "Mathlib.Control.Bitraversable.Basic", "region_id": "frontier", "micro_elevation": 0.7143, "macro_tier": 2, "macro_tier_score": 0.01, "macro_tier_override": null, "x": -1.733, "z": 228.259, "size": 0.2806, "title": "Bitraversable type class", "summary": "Type class for traversing bifunctors. Simple examples of `Bitraversable` are `Prod` and `Sum`. A more elaborate example is to define an a-list as: ``` def AList (key val : Type) := List (key × val) ``` Then we can use `f : key → IO key'` and `g : val → IO val'` to manipulate the `AList`'s key and value respectively with `bitraverse f g : AList key val → IO (AList key' val')`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Bitraversable/Basic.html"}, {"id": "Mathlib.Lean.Json", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 1.391, "z": 216.471, "size": 0.2, "title": "Json serialization typeclass for `PUnit` & `Fin n` & `Subtype p`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Json.html"}, {"id": "Mathlib.Lean.Expr.Rat", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": -6.719, "z": 204.409, "size": 0.3201, "title": "Additional operations on Expr and rational numbers", "summary": "This file defines some operations involving `Expr` and rational numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Expr/Rat.html"}, {"id": "Mathlib.Control.ULift", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0067, "macro_tier_override": null, "x": -9.277, "z": 209.725, "size": 0.5473, "title": "Monadic instances for `ULift` and `PLift`", "summary": "In this file we define `Monad` and `IsLawfulMonad` instances on `PLift` and `ULift`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/ULift.html"}, {"id": "Mathlib.Util.Tactic", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -5.593, "z": 216.323, "size": 0.2902, "title": "Miscellaneous helper functions for tactics.", "summary": "TODO: Ideally we would find good homes for everything in this file, eventually removing it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/Tactic.html"}, {"id": "Mathlib.Lean.MessageData.ForExprs", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 3.878, "z": 214.373, "size": 0.2, "title": "Tools for extracting `Expr`s from `MessageData` nodes", "summary": "This file provides `for (ppCtx, e) in msg.exprs do` notation, which iterates through the expressions `e` in a `msg : MessageData`. The surrounding monad must support `BaseIO` to handle `.ofLazy` `MessageData` nodes. `e` may be interpreted in a `MetaM` context using `ppCtx.runMetaM e`. Some helpers are provided implemented in terms of this.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/MessageData/ForExprs.html"}, {"id": "Mathlib.Util.WhatsNew", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 0.312, "z": 203.016, "size": 0.2647, "title": null, "summary": "Defines a command wrapper that prints the changes the command makes to the environment. ``` whatsnew in theorem foo : 42 = 6 * 7 := rfl ```", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/WhatsNew.html"}, {"id": "Mathlib.Control.EquivFunctor.Instances", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.2, "title": "`EquivFunctor` instances", "summary": "We derive some `EquivFunctor` instances, to enable `equiv_rw` to rewrite under these functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/EquivFunctor/Instances.html"}, {"id": "Mathlib.Lean.Meta.Tactic.Rewrite", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 4.596, "z": 206.731, "size": 0.2607, "title": "Additional declarations for `Lean.Meta.Tactic.Rewrite`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Meta/Tactic/Rewrite.html"}, {"id": "Mathlib.Control.Monad.Cont", "region_id": "frontier", "micro_elevation": 0.4286, "macro_tier": 2, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -9.712, "z": 217.743, "size": 0.256, "title": "Continuation Monad", "summary": "Monad encapsulating continuation passing programming style, similar to Haskell's `Cont`, `ContT` and `MonadCont`: ", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Monad/Cont.html"}, {"id": "Mathlib.Control.Basic", "region_id": "frontier", "micro_elevation": 0.4286, "macro_tier": 2, "macro_tier_score": 0.0435, "macro_tier_override": null, "x": -10.542, "z": 203.123, "size": 0.5003, "title": "Basic control operations", "summary": "Extends the theory on functors, applicatives and monads.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Basic.html"}, {"id": "Mathlib.Lean.EnvExtension", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -7.069, "z": 215.211, "size": 0.2, "title": "Helper function for environment extensions and attributes.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/EnvExtension.html"}, {"id": "Mathlib.Lean.LocalContext", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -9.28, "z": 209.839, "size": 0.2, "title": "Additional methods about `LocalContext`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/LocalContext.html"}, {"id": "Mathlib.Util.GetAllModules", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -4.985, "z": 216.633, "size": 0.2, "title": "Utility functions for finding all `.lean` files or modules in a project.", "summary": "TODO: `getLeanLibs` contains a hard-coded choice of which dependencies should be built and which ones should not. Could this be made more structural and robust, possibly with extra `Lake` support?", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/GetAllModules.html"}, {"id": "Mathlib.Util.SleepHeartbeats", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -9.231, "z": 209.116, "size": 0.2, "title": "Defines `sleep_heartbeats` tactic.", "summary": "This is useful for testing / debugging long running commands or elaboration in a somewhat precise manner.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/SleepHeartbeats.html"}, {"id": "Mathlib.Testing.Plausible.Sampleable", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.2, "title": null, "summary": "This module contains `Plausible.Shrinkable` and `Plausible.SampleableExt` instances for mathlib types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Testing/Plausible/Sampleable.html"}, {"id": "Mathlib.Lean.CoreM", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 2.705, "z": 215.602, "size": 0.2, "title": "Additional functions using `CoreM` state.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/CoreM.html"}, {"id": "Mathlib.Control.LawfulFix", "region_id": "frontier", "micro_elevation": 0.1429, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -5.503, "z": 209.042, "size": 0.2, "title": "Lawful fixed point operators", "summary": "This module defines the laws required of a `Fix` instance, using the theory of omega complete partial orders (ωCPO). Proofs of the lawfulness of all `Fix` instances in `Control.Fix` are provided.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/LawfulFix.html"}, {"id": "Mathlib.Control.Fix", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.239, "title": "Fixed point", "summary": "This module defines a generic `fix` operator for defining recursive computations that are not necessarily well-founded or productive. An instance is defined for `Part`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Control/Fix.html"}, {"id": "Mathlib.Lean.Exception", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 5.247, "z": 208.737, "size": 0.2, "title": "Additional methods for working with `Exception`s", "summary": "This file contains two additional methods for working with `Exception`s * `successIfFail`, a generalisation of `fail_if_success` to arbitrary `MonadError`s * `isFailedToSynthesize`: check if an exception is of the \"failed to synthesize\" form", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Exception.html"}, {"id": "Mathlib.Tactic", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic.html"}, {"id": "Mathlib.Testing.Plausible.Testable", "region_id": "frontier", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.965, "z": 209.969, "size": 0.2, "title": null, "summary": "This module contains `Plausible.Testable` and `Plausible.PrintableProb` instances for mathlib types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Testing/Plausible/Testable.html"}, {"id": "Mathlib.Util.AssertNoSorry", "region_id": "frontier", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.585, "z": 202.663, "size": 0.2, "title": "Defines the `assert_no_sorry` command.", "summary": "Throws an error if the given identifier uses sorryAx.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/AssertNoSorry.html"}, {"id": "Mathlib.Lean.Meta.RefinedDiscrTree.Encode", "region_id": "frontier", "micro_elevation": 0.4286, "macro_tier": 2, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": -9.917, "z": 202.406, "size": 0.3285, "title": "Encoding an `Expr` as a sequence of `Key`s", "summary": "We compute the encoding of an expression in a lazy way. This means computing only one `Key` at a time and storing the state of the remaining computation in a `LazyEntry`. Each step is computed by `evalLazyEntryWithEta : LazyEntry → MetaM (Option (List (Key × LazyEntry)))`. It returns `none` when the last `Key` has already been reached. The first step, which is used when initializing the tree, is computed by…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Lean/Meta/RefinedDiscrTree/Encode.html"}, {"id": "Mathlib.Analysis.AbsoluteValue.Equivalence", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 105.613, "z": -226.008, "size": 0.2808, "title": "Equivalence of real-valued absolute values", "summary": "Two absolute values `v₁, v₂ : AbsoluteValue R ℝ` are *equivalent* if there exists a positive real number `c` such that `v₁ x ^ c = v₂ x` for all `x : R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/AbsoluteValue/Equivalence.html"}, {"id": "Mathlib.Analysis.Normed.Field.WithAbs", "region_id": "analysis", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0142, "macro_tier_override": null, "x": 68.476, "z": -213.544, "size": 0.2709, "title": "WithAbs for fields", "summary": "This extends the `WithAbs` mechanism to fields, providing a type synonym for a field which depends on an absolute value. This is useful when dealing with several absolute values on the same field. In particular this allows us to define the completion of a field at a given absolute value.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Field/WithAbs.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Pow.Real", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 4, "macro_tier_score": 0.4084, "macro_tier_override": null, "x": 93.618, "z": -235.065, "size": 0.5182, "title": "Power function on `ℝ`", "summary": "We construct the power functions `x ^ y`, where `x` and `y` are real numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Pow/Real.html"}, {"id": "Mathlib.Analysis.Normed.Group.Indicator", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 83.978, "z": -211.754, "size": 0.2612, "title": "Indicator function and (e)norm", "summary": "This file contains a few simple lemmas about `Set.indicator`, `norm` and `enorm`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Indicator.html"}, {"id": "Mathlib.Analysis.Complex.LocallyUniformLimit", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 2, "macro_tier_score": 0.0424, "macro_tier_override": null, "x": 98.252, "z": -165.982, "size": 0.3118, "title": "Locally uniform limits of holomorphic functions", "summary": "This file gathers some results about locally uniform limits of holomorphic functions on an open subset of the complex plane.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/LocallyUniformLimit.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 3, "macro_tier_score": 0.2099, "macro_tier_override": null, "x": 65.058, "z": -262.305, "size": 0.3697, "title": "Derivatives of power function on `ℂ`, `ℝ`, `ℝ≥0`, and `ℝ≥0∞`", "summary": "We also prove differentiability and provide derivatives for the power functions `x ^ y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Pow/Deriv.html"}, {"id": "Mathlib.Analysis.Complex.HalfPlane", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 82.883, "z": -227.581, "size": 0.24, "title": "Half-planes in ℂ are open", "summary": "We state that open left, right, upper and lower half-planes in the complex numbers are open sets, where the bounding value of the real or imaginary part is given by an `EReal` `x`. So this includes the full plane and the empty set for `x = ⊤`/`x = ⊥`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/HalfPlane.html"}, {"id": "Mathlib.Analysis.SpecificLimits.Fibonacci", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 88.836, "z": -197.133, "size": 0.2, "title": "The ratio of consecutive Fibonacci numbers", "summary": "We prove that the ratio of consecutive Fibonacci numbers tends to the golden ratio.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecificLimits/Fibonacci.html"}, {"id": "Mathlib.Analysis.SpecificLimits.Normed", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 4, "macro_tier_score": 0.4486, "macro_tier_override": null, "x": 68.881, "z": -201.488, "size": 0.4735, "title": "A collection of specific limit computations", "summary": "This file contains important specific limit computations in (semi-)normed groups/rings/spaces, as well as such computations in `ℝ` when the natural proof passes through a fact about normed spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecificLimits/Normed.html"}, {"id": "Mathlib.Analysis.Convex.Caratheodory", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 84.313, "z": -203.264, "size": 0.2, "title": "Carathéodory's convexity theorem", "summary": "Convex hull can be regarded as a refinement of affine span. Both are closure operators but whereas convex hull takes values in the lattice of convex subsets, affine span takes values in the much coarser sublattice of affine subspaces. The cost of this refinement is that one no longer has bases. However Carathéodory's convexity theorem offers some compensation. Given a set `s` together with a point `x` in its convex…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Caratheodory.html"}, {"id": "Mathlib.Analysis.Convex.Combination", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 4, "macro_tier_score": 0.4188, "macro_tier_override": null, "x": 77.316, "z": -204.053, "size": 0.3962, "title": "Convex combinations", "summary": "This file defines convex combinations of points in a vector space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Combination.html"}, {"id": "Mathlib.Analysis.SpecificLimits.Basic", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.4643, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.5281, "title": "A collection of specific limit computations", "summary": "This file, by design, is independent of `NormedSpace` in the import hierarchy. It contains important specific limit computations in metric spaces, in ordered rings/fields, and in specific instances of these such as `ℝ`, `ℝ≥0` and `ℝ≥0∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecificLimits/Basic.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Projection", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 39.059, "z": -248.291, "size": 0.2, "title": "Projections in C⋆-algebras", "summary": "Here we collect results about projections specific to C⋆-algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Projection.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.0432, "macro_tier_override": null, "x": 25.424, "z": -203.972, "size": 0.3659, "title": "Facts about star-ordered rings that depend on the continuous functional calculus", "summary": "This file contains various basic facts about star-ordered rings (i.e. mainly C⋆-algebras) that depend on the continuous functional calculus. We also put an order instance on `A⁺¹ := Unitization ℂ A` when `A` is a C⋆-algebra via the spectral order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Projection", "region_id": "analysis", "micro_elevation": 0.0417, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 80.42, "z": -212.547, "size": 0.2478, "title": "Continuous functional calculus and projections", "summary": "This file collects some results related to projections, idempotents, and the continuous functional calculus.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Projection.html"}, {"id": "Mathlib.Analysis.Normed.Group.Basic", "region_id": "analysis", "micro_elevation": 0.0417, "macro_tier": 4, "macro_tier_score": 0.5081, "macro_tier_override": null, "x": 79.005, "z": -206.646, "size": 0.5823, "title": "(Semi)normed groups: basic theory", "summary": "We prove basic properties of (semi)normed groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Basic.html"}, {"id": "Mathlib.Analysis.Normed.Group.Defs", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 4, "macro_tier_score": 0.5018, "macro_tier_override": null, "x": 78.902, "z": -208.46, "size": 0.3797, "title": "(Semi)normed groups: definitions", "summary": "In this file we define 10 classes: * `Norm`, `NNNorm`: auxiliary classes endowing a type `α` with a function `norm : α → ℝ` (notation: `‖x‖`) and `nnnorm : α → ℝ≥0` (notation: `‖x‖₊`), respectively; * `Seminormed...Group`: A seminormed (additive) (commutative) group is an (additive) (commutative) group with a norm and a compatible pseudometric space structure: `∀ x y, dist x y = ‖x⁻¹ * y‖` or `∀ x y, dist x y = ‖-x…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Defs.html"}, {"id": "Mathlib.Analysis.SumIntegralExpDecay", "region_id": "analysis", "micro_elevation": 0.875, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 127.409, "z": -166.58, "size": 0.2, "title": "Bounds for sums and integrals of `x ^ k * exp (-c * x)`", "summary": "We bound the integral and sums of `x ^ k * exp (-c * x)` by `k ! / c ^ (k + 1)`, using the Gamma function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SumIntegralExpDecay.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "region_id": "analysis", "micro_elevation": 0.8542, "macro_tier": 3, "macro_tier_score": 0.0844, "macro_tier_override": null, "x": 88.172, "z": -147.61, "size": 0.3374, "title": "The Gamma function", "summary": "This file defines the `Γ` function (of a real or complex variable `s`). We define this by Euler's integral `Γ(s) = ∫ x in Ioi 0, exp (-x) * x ^ (s - 1)` in the range where this integral converges (i.e., for `0 < s` in the real case, and `0 < re s` in the complex case). We show that this integral satisfies `Γ(1) = 1` and `Γ(s + 1) = s * Γ(s)`; hence we can define `Γ(s)` for all `s` as the unique function satisfying…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Gamma/Basic.html"}, {"id": "Mathlib.Analysis.SumIntegralComparisons", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2617, "title": "Comparing sums and integrals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SumIntegralComparisons.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.CanonicalTensor", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 3, "macro_tier_score": 0.084, "macro_tier_override": null, "x": 105.166, "z": -245.832, "size": 0.312, "title": "Canonical tensors in real inner product spaces", "summary": "Given an `InnerProductSpace ℝ E`, this file defines two canonical tensors. * `InnerProductSpace.canonicalContravariantTensor E : E ⊗[ℝ] E →ₗ[ℝ] ℝ`. This is the element corresponding to the inner product. * If `E` is finite-dimensional, then `E ⊗[ℝ] E` is canonically isomorphic to its dual. Accordingly, there exists an element `InnerProductSpace.canonicalCovariantTensor E : E ⊗[ℝ] E` that corresponds to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/CanonicalTensor.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.PiL2", "region_id": "analysis", "micro_elevation": 0.5833, "macro_tier": 3, "macro_tier_score": 0.214, "macro_tier_override": null, "x": 120.221, "z": -195.372, "size": 0.5202, "title": "`L²` inner product space structure on finite products of inner product spaces", "summary": "The `L²` norm on a finite product of inner product spaces is compatible with an inner product $$ \\langle x, y\\rangle = \\sum \\langle x_i, y_i \\rangle. $$ This is recorded in this file as an inner product space instance on `PiLp 2`. This file develops the notion of a finite-dimensional Hilbert space over `𝕜 = ℂ, ℝ`, referred to as `E`. We define an `OrthonormalBasis 𝕜 ι E` as a linear isometric equivalence between `E`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/PiL2.html"}, {"id": "Mathlib.Analysis.Distribution.Distribution", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 55.425, "z": -253.398, "size": 0.2478, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/Distribution.html"}, {"id": "Mathlib.Analysis.Distribution.TestFunction", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 117.945, "z": -240.117, "size": 0.257, "title": "Continuously differentiable functions with compact support", "summary": "This file develops the basic theory of bundled `n`-times continuously differentiable functions with compact support contained in some open set `Ω`. More explicitly, given normed spaces `E` and `F`, an open set `Ω : Opens E` and `n : ℕ∞`, we are interested in the space `𝓓^{n}(Ω, F)` of maps `f : E → F` such that: - `f` is `n`-times continuously differentiable: `ContDiff ℝ n f`. - `f` has compact support:…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/TestFunction.html"}, {"id": "Mathlib.Analysis.Normed.Group.InfiniteSum", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 4, "macro_tier_score": 0.4624, "macro_tier_override": null, "x": 82.196, "z": -202.243, "size": 0.469, "title": "Infinite sums in (semi)normed groups", "summary": "In a complete (semi)normed group, - `summable_iff_vanishing_norm`: a series `∑' i, f i` is summable if and only if for any `ε > 0`, there exists a finite set `s` such that the sum `∑ i ∈ t, f i` over any finite set `t` disjoint with `s` has norm less than `ε`; - `Summable.of_norm_bounded`, `Summable.of_norm_bounded_eventually`: if `‖f i‖` is bounded above by a summable series `∑' i, g i`, then `∑' i, f i` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/InfiniteSum.html"}, {"id": "Mathlib.Analysis.Calculus.Implicit", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 1, "macro_tier_score": 0.0144, "macro_tier_override": null, "x": 54.358, "z": -233.427, "size": 0.2918, "title": "Implicit function theorem", "summary": "We prove three versions of the implicit function theorem. First we define a structure `ImplicitFunctionData` that holds arguments for the most general version of the implicit function theorem, see `ImplicitFunctionData.implicitFunction` and `ImplicitFunctionData.hasStrictFDerivAt_implicitFunction`. This version allows a user to choose a specific implicit function but provides only a little convenience over the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Implicit.html"}, {"id": "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 3, "macro_tier_score": 0.2369, "macro_tier_override": null, "x": 66.212, "z": -240.087, "size": 0.3166, "title": "Inverse function theorem", "summary": "In this file we prove the inverse function theorem. It says that if a map `f : E → F` has an invertible strict derivative `f'` at `a`, then it is locally invertible, and the inverse function has derivative `f' ⁻¹`. We define `HasStrictFDerivAt.toOpenPartialHomeomorph` that repacks a function `f` with a `hf : HasStrictFDerivAt f f' a`, `f' : E ≃L[𝕜] F`, into an `OpenPartialHomeomorph`. The `toFun` of this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Prod", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 4, "macro_tier_score": 0.2936, "macro_tier_override": null, "x": 49.523, "z": -210.217, "size": 0.3859, "title": "Derivative of the Cartesian product of functions", "summary": "For detailed documentation of the Fréchet derivative, see the module docstring of `Mathlib/Analysis/Calculus/FDeriv/Basic.lean`. This file contains the usual formulas (and existence assertions) for the derivative of Cartesian products of functions, and functions into Pi-types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Prod.html"}, {"id": "Mathlib.Analysis.Normed.Module.Complemented", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 99.846, "z": -228.438, "size": 0.2624, "title": "Complemented subspaces of Banach spaces", "summary": "A submodule `p` of a topological module `E` over `R` is called *complemented* (`Submodule.ClosedComplemented`) if there exists a continuous linear projection `f : E →ₗ[R] p`, `∀ x : p, f x = x`. All results in this file rely on the open mapping theorem, hence the Banach space assumption.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Complemented.html"}, {"id": "Mathlib.Analysis.LocallyConvex.HahnBanach", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 2, "macro_tier_score": 0.0702, "macro_tier_override": null, "x": 94.57, "z": -184.53, "size": 0.3199, "title": "Hahn-Banach theorem for polynormable spaces", "summary": "In this file, we prove the analytic Hahn-Banach theorem for polynormable spaces over a field satisfying `IsRCLikeNormedField`. For any continuous linear functional on a subspace, we can extend it to the entire space. Note that we cannot use `LocallyConvexSpace` because an `IsRCLikeNormedField` has no order structure. We prove * `Module.Dual.exists_continuous_extension_of_le_seminorm`: Hahn-Banach theorem for linear…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/HahnBanach.html"}, {"id": "Mathlib.Analysis.Convex.Cone.Extension", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.1254, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.285, "title": "Extension theorems", "summary": "We prove two extension theorems: * `riesz_extension`: [M. Riesz extension theorem](https://en.wikipedia.org/wiki/M._Riesz_extension_theorem) says that if `s` is a convex cone in a real vector space `E`, `p` is a submodule of `E` such that `p + s = E`, and `f` is a linear function `p → ℝ` which is nonnegative on `p ∩ s`, then there exists a globally defined linear function `g : E → ℝ` that agrees with `f` on `p`, and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Cone/Extension.html"}, {"id": "Mathlib.Analysis.LocallyConvex.AbsConvexOpen", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 3, "macro_tier_score": 0.0835, "macro_tier_override": null, "x": 72.133, "z": -191.35, "size": 0.2578, "title": "Absolutely convex open sets", "summary": "A set `s` in a commutative monoid `E` equipped with a topology is said to be an absolutely convex open set if it is absolutely convex and open. When `E` is a topological additive group, the topology coincides with the topology induced by the family of seminorms arising as gauges of absolutely convex open neighborhoods of zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/AbsConvexOpen.html"}, {"id": "Mathlib.Analysis.LocallyConvex.WeakDual", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 3, "macro_tier_score": 0.0835, "macro_tier_override": null, "x": 60.752, "z": -204.804, "size": 0.2578, "title": "Weak Dual in Topological Vector Spaces", "summary": "We prove that the weak topology induced by a bilinear form `B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜` is locally convex and we explicitly give a neighborhood basis in terms of the family of seminorms `fun x => ‖B x y‖` for `y : F`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/WeakDual.html"}, {"id": "Mathlib.Analysis.Normed.Module.RCLike.Extend", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 3, "macro_tier_score": 0.0835, "macro_tier_override": null, "x": 102.544, "z": -193.943, "size": 0.2578, "title": "Norm properties of the extension of continuous `ℝ`-linear functionals to `𝕜`-linear functionals", "summary": "This file shows that `StrongDual.extendRCLike` preserves the norm of the functional.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/RCLike/Extend.html"}, {"id": "Mathlib.Analysis.Analytic.WithLp", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 45.428, "z": -221.477, "size": 0.2, "title": "Analyticity on `WithLp`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/WithLp.html"}, {"id": "Mathlib.Analysis.Analytic.Linear", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 4, "macro_tier_score": 0.2649, "macro_tier_override": null, "x": 55.49, "z": -184.477, "size": 0.3375, "title": "Linear functions are analytic", "summary": "In this file we prove that a `ContinuousLinearMap` defines an analytic function with the formal power series `f x = f a + f (x - a)`. We also prove similar results for bilinear maps. We deduce this fact from the stronger result that continuous linear maps are continuously polynomial, i.e., they admit a finite power series.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/Linear.html"}, {"id": "Mathlib.Analysis.Normed.Lp.PiLp", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 3, "macro_tier_score": 0.224, "macro_tier_override": null, "x": 112.043, "z": -195.392, "size": 0.3809, "title": "`L^p` distance on finite products of metric spaces", "summary": "Given finitely many metric spaces, one can put the max distance on their product, but there is also a whole family of natural distances, indexed by a parameter `p : ℝ≥0∞`, that also induce the product topology. We define them in this file. For `0 < p < ∞`, the distance on `Π i, α i` is given by $$ d(x, y) = \\left(\\sum d(x_i, y_i)^p\\right)^{1/p}. $$, whereas for `p = 0` it is the cardinality of the set ${i | d (x_i,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Lp/PiLp.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno", "region_id": "analysis", "micro_elevation": 0.5625, "macro_tier": 3, "macro_tier_score": 0.2515, "macro_tier_override": null, "x": 120.26, "z": -201.18, "size": 0.3646, "title": "Faa di Bruno formula", "summary": "The Faa di Bruno formula gives the iterated derivative of `g ∘ f` in terms of those of `g` and `f`. It is expressed in terms of partitions `I` of `{0, ..., n-1}`. For such a partition, denote by `k` its number of parts, write the parts as `I₀, ..., Iₖ₋₁` ordered so that `max I₀ < ... < max Iₖ₋₁`, and let `iₘ` be the number of elements of `Iₘ`. Then `D^n (g ∘ f) (x) (v₀, ..., vₙ₋₁) = ∑_{I partition of {0, ..., n-1}}…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/FaaDiBruno.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Analytic", "region_id": "analysis", "micro_elevation": 0.5417, "macro_tier": 4, "macro_tier_score": 0.2713, "macro_tier_override": null, "x": 113.54, "z": -230.576, "size": 0.5641, "title": "Fréchet derivatives of analytic functions.", "summary": "A function expressible as a power series at a point has a Fréchet derivative there. Also the special case in terms of `deriv` when the domain is 1-dimensional. As an application, we show that continuous multilinear maps are smooth. We also compute their iterated derivatives, in `ContinuousMultilinearMap.iteratedFDeriv_eq`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Analytic.html"}, {"id": "Mathlib.Analysis.Asymptotics.Theta", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 4, "macro_tier_score": 0.448, "macro_tier_override": null, "x": 66.321, "z": -208.389, "size": 0.4537, "title": "Asymptotic equivalence up to a constant", "summary": "In this file we prove basic properties of the equivalence relation given by `f =Θ[l] g ↔ f =O[l] g ∧ g =O[l] f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Asymptotics/Theta.html"}, {"id": "Mathlib.Analysis.Asymptotics.Lemmas", "region_id": "analysis", "micro_elevation": 0.1458, "macro_tier": 4, "macro_tier_score": 0.4472, "macro_tier_override": null, "x": 69.323, "z": -209.28, "size": 0.421, "title": "Further basic lemmas about asymptotics", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Asymptotics/Lemmas.html"}, {"id": "Mathlib.Analysis.Normed.Module.Basic", "region_id": "analysis", "micro_elevation": 0.1667, "macro_tier": 4, "macro_tier_score": 0.4966, "macro_tier_override": null, "x": 75.331, "z": -198.267, "size": 0.6354, "title": "Normed spaces", "summary": "In this file we define (semi)normed spaces and algebras. We also prove some theorems about these definitions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Basic.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Exp", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 4, "macro_tier_score": 0.4054, "macro_tier_override": null, "x": 60.302, "z": -207.335, "size": 0.4156, "title": "Complex and real exponential", "summary": "In this file we prove continuity of `Complex.exp` and `Real.exp`. We also prove a few facts about limits of `Real.exp` at infinity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Exp.html"}, {"id": "Mathlib.Analysis.Complex.Asymptotics", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 4, "macro_tier_score": 0.4033, "macro_tier_override": null, "x": 96.198, "z": -201.088, "size": 0.2939, "title": "Lemmas about asymptotics and the natural embedding `ℝ → ℂ`", "summary": "In this file we prove several trivial lemmas about `Asymptotics.IsBigO` etc. and `(↑) : ℝ → ℂ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Asymptotics.html"}, {"id": "Mathlib.Analysis.Complex.Trigonometric", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 4, "macro_tier_score": 0.4036, "macro_tier_override": null, "x": 86.415, "z": -203.036, "size": 0.321, "title": "Trigonometric and hyperbolic trigonometric functions", "summary": "This file contains the definitions of the sine, cosine, tangent, hyperbolic sine, hyperbolic cosine, and hyperbolic tangent functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Trigonometric.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.ContinuousAlternatingMap", "region_id": "analysis", "micro_elevation": 0.5833, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 37.589, "z": -214.241, "size": 0.2465, "title": "Derivatives of operations on continuous alternating maps", "summary": "In this file we prove formulas for the derivatives of - `ContinuousAlternatingMap.compContinuousLinearMap`, the pullback of a continuous alternating map along a continuous linear map; - application of a `ContinuousAlternatingMap` as a function of both the map and the vectors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/ContinuousAlternatingMap.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.ContinuousMultilinearMap", "region_id": "analysis", "micro_elevation": 0.5625, "macro_tier": 1, "macro_tier_score": 0.0142, "macro_tier_override": null, "x": 68.415, "z": -170.06, "size": 0.2799, "title": "Derivatives of operations on continuous multilinear maps", "summary": "In this file, - `ι` is an index type (`Fin n` in many applications); - `E`, `F i`, `G i`, `H`, are normed spaces for each `i : ι`; - `f x` is a continuous multilinear map from `Π i, G i` to `H`, depending on a parameter `x : E`; - for each `i : ι`, `g i x` is a continuous linear map `F i → G i`, depending on a parameter `x : E`. Given this data, for each `x` we can define a continuous multilinear map from `Π i, F i`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/ContinuousMultilinearMap.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 3, "macro_tier_score": 0.1676, "macro_tier_override": null, "x": 113.096, "z": -251.324, "size": 0.3311, "title": "The functions `x ↦ x * log x` and `x ↦ - x * log x`", "summary": "The purpose of this file is to record basic analytic properties of - `x ↦ x * log x`, called `mul_log` in theorem statements - `x ↦ - x * log x`, named `negMulLog`, which is notably used in the theory of Shannon entropy.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.html"}, {"id": "Mathlib.Analysis.Convex.SimplicialComplex.AffineIndependentUnion", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 78.913, "z": -217.073, "size": 0.2, "title": "Simplicial complexes from affinely independent points", "summary": "This file provides constructions for simplicial complexes where the vertices are affinely independent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/SimplicialComplex/AffineIndependentUnion.html"}, {"id": "Mathlib.Analysis.Convex.SimplicialComplex.Basic", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 82.956, "z": -204.212, "size": 0.2338, "title": "Simplicial complexes", "summary": "In this file, we define simplicial complexes over `𝕜`-modules. A (pre-) abstract simplicial complex is a downwards-closed collection of nonempty finite sets, and a simplicial complex is such a collection identified with simplices closed by inclusion (of vertices) and intersection (of underlying sets) whose convex hulls \"glue nicely\", each finite set and its convex hull corresponding respectively to the vertices and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/SimplicialComplex/Basic.html"}, {"id": "Mathlib.Analysis.Normed.Module.RCLike.Basic", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 4, "macro_tier_score": 0.2652, "macro_tier_override": null, "x": 76.618, "z": -235.21, "size": 0.3552, "title": "Normed spaces over R or C", "summary": "This file is about results on normed spaces over the fields `ℝ` and `ℂ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/RCLike/Basic.html"}, {"id": "Mathlib.Analysis.RCLike.Basic", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 4, "macro_tier_score": 0.4495, "macro_tier_override": null, "x": 71.484, "z": -196.89, "size": 0.5003, "title": "`RCLike`: a typeclass for ℝ or ℂ", "summary": "This file defines the typeclass `RCLike` intended to have only two instances: ℝ and ℂ. It is meant for definitions and theorems which hold for both the real and the complex case, and in particular when the real case follows directly from the complex case by setting `re` to `id`, `im` to zero and so on. Its API follows closely that of ℂ. Applications include defining inner products and Hilbert spaces for both the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/RCLike/Basic.html"}, {"id": "Mathlib.Analysis.Normed.Operator.NormedSpace", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 4, "macro_tier_score": 0.3935, "macro_tier_override": null, "x": 95.82, "z": -228.061, "size": 0.4875, "title": "Operator norm for maps on normed spaces", "summary": "This file contains statements about operator norm for which it really matters that the underlying space has a norm (rather than just a seminorm).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/NormedSpace.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Integrals.LogTrigonometric", "region_id": "analysis", "micro_elevation": 0.8542, "macro_tier": 2, "macro_tier_score": 0.0419, "macro_tier_override": null, "x": 124.629, "z": -253.214, "size": 0.2601, "title": "Integral of `log ∘ sin`", "summary": "This file computes special values of the integral of `log ∘ sin`. Given that the indefinite integral involves the dilogarithm, this can be seen as computing special values of `Li₂`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Integrals/LogTrigonometric.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic", "region_id": "analysis", "micro_elevation": 0.8333, "macro_tier": 2, "macro_tier_score": 0.0424, "macro_tier_override": null, "x": 25.173, "z": -182.903, "size": 0.3156, "title": "Integrability for Logarithms of Meromorphic Functions", "summary": "We establish integrability for functions of the form `log ‖meromorphic‖`. In the real setting, these functions are interval integrable over every interval of the real line. This implies in particular that logarithms of trigonometric functions are interval integrable. In the complex setting, the functions are circle integrable over every circle in the complex plane.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Integrability/LogMeromorphic.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "region_id": "analysis", "micro_elevation": 0.8958, "macro_tier": 2, "macro_tier_score": 0.0282, "macro_tier_override": null, "x": 76.075, "z": -274.922, "size": 0.2835, "title": "Convexity properties of the Gamma function", "summary": "In this file, we prove that `Gamma` and `log ∘ Gamma` are convex functions on the positive real line. We then prove the Bohr-Mollerup theorem, which characterises `Gamma` as the *unique* positive-real-valued, log-convex function on the positive reals satisfying `f (x + 1) = x f x` and `f 1 = 1`. The proof of the Bohr-Mollerup theorem is bound up with the proof of (a weak form of) the Euler limit formula,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.html"}, {"id": "Mathlib.Analysis.Convolution", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.042, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2821, "title": "Convolution of functions", "summary": "This file defines the convolution on two functions, i.e. `x ↦ ∫ f(t)g(x - t) ∂t`. In the general case, these functions can be vector-valued, and have an arbitrary (additive) group as domain. We use a continuous bilinear operation `L` on these function values as \"multiplication\". The domain must be equipped with a Haar measure `μ` (though many individual results have weaker conditions on `μ`). For many applications…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convolution.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Comp", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 4, "macro_tier_score": 0.2942, "macro_tier_override": null, "x": 58.766, "z": -221.528, "size": 0.4147, "title": "The derivative of a composition (chain rule)", "summary": "For detailed documentation of the Fréchet derivative, see the module docstring of `Mathlib/Analysis/Calculus/FDeriv/Basic.lean`. This file contains the usual formulas (and existence assertions) for the derivative of composition of functions (the chain rule).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Comp.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Basic", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 4, "macro_tier_score": 0.2999, "macro_tier_override": null, "x": 79.736, "z": -186.682, "size": 0.5858, "title": "The Fréchet derivative: basic properties", "summary": "Let `E` and `F` be normed spaces, `f : E → F`, and `f' : E →L[𝕜] F` a continuous 𝕜-linear map, where `𝕜` is a non-discrete normed field. Then `HasFDerivWithinAt f f' s x` says that `f` has derivative `f'` at `x`, where the domain of interest is restricted to `s`. We also have `HasFDerivAt f f' x := HasFDerivWithinAt f f' x univ` Finally, `HasStrictFDerivAt f f' x` means that `f : E → F` has derivative `f' : E →L[𝕜]…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Basic.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Unitary.Connected", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 132.692, "z": -189.555, "size": 0.2, "title": "The unitary group in a unital C⋆-algebra is locally path connected", "summary": "When `A` is a unital C⋆-algebra and `u : unitary A` is a unitary element whose distance to `1` is less that `2`, the spectrum of `u` is contained in the slit plane, so the principal branch of the logarithm is continuous on the spectrum of `u` (or equivalently, `Complex.arg` is continuous on the spectrum). The continuous functional calculus can then be used to define a selfadjoint element `x` such that `u = exp (I •…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Unitary/Connected.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 2, "macro_tier_score": 0.0701, "macro_tier_override": null, "x": 103.816, "z": -195.967, "size": 0.3116, "title": "Maps on the unit circle", "summary": "In this file we prove some basic lemmas about `Circle.exp` and the restriction of `Complex.arg` to the unit circle. These two maps define a partial equivalence between `Circle` and `ℝ`, see `Circle.argPartialEquiv` and `Circle.argEquiv`, that sends the whole circle to `(-π, π]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Complex/Circle.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic", "region_id": "analysis", "micro_elevation": 0.5833, "macro_tier": 2, "macro_tier_score": 0.0279, "macro_tier_override": null, "x": 122.386, "z": -204.886, "size": 0.2516, "title": "The exponential and logarithm based on the continuous functional calculus", "summary": "This file defines the logarithm via the continuous functional calculus (CFC) and builds its API. This allows one to take logs of matrices, operators, elements of a C⋆-algebra, etc. It also shows that exponentials defined via the continuous functional calculus are equal to `NormedSpace.exp` (defined via power series) whenever the former are not junk values.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/ExpLog/Basic.html"}, {"id": "Mathlib.Analysis.Real.Pi.Bounds", "region_id": "analysis", "micro_elevation": 0.7917, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 108.847, "z": -259.713, "size": 0.2357, "title": "Pi", "summary": "This file contains lemmas which establish bounds on `Real.pi`. Notably, these include `pi_gt_sqrtTwoAddSeries` and `pi_lt_sqrtTwoAddSeries`, which bound `π` using series; numerical bounds on `π` such as `pi_gt_d2` and `pi_lt_d2` (more precise versions are given, too). See also `Mathlib/Analysis/Real/Pi/Leibniz.lean` and `Mathlib/Analysis/Real/Pi/Wallis.lean` for infinite formulas for `π`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Real/Pi/Bounds.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Log.Base", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 96.163, "z": -235.35, "size": 0.2968, "title": "Real logarithm base `b`", "summary": "In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We define this as the division of the natural logarithms of the argument and the base, so that we have a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x`, `logb 0 x = 0`, and `logb (-b) x = logb b x`. We prove some basic properties of this function and its relation to `rpow`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Log/Base.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.ConjSqrt", "region_id": "analysis", "micro_elevation": 0.5625, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 119.592, "z": -220.626, "size": 0.2571, "title": "Conjugating by the square root of a positive element in a C⋆-algebra", "summary": "This file defines `conjSqrt c a` as `sqrt c * a * sqrt c`, and develops API for this operation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow/ConjSqrt.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic", "region_id": "analysis", "micro_elevation": 0.5417, "macro_tier": 3, "macro_tier_score": 0.0704, "macro_tier_override": null, "x": 114.74, "z": -190.535, "size": 0.3313, "title": "Real powers defined via the continuous functional calculus", "summary": "This file defines real powers via the continuous functional calculus (CFC) and builds its API. This allows one to take real powers of matrices, operators, elements of a C⋆-algebra, etc. The square root is also defined via the non-unital CFC.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow/Basic.html"}, {"id": "Mathlib.Analysis.Fourier.ZMod", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 130.074, "z": -205.249, "size": 0.2521, "title": "Fourier theory on `ZMod N`", "summary": "Basic definitions and properties of the discrete Fourier transform for functions on `ZMod N` (taking values in an arbitrary `ℂ`-vector space).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/ZMod.html"}, {"id": "Mathlib.Analysis.Normed.Module.Connected", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 3, "macro_tier_score": 0.0838, "macro_tier_override": null, "x": 64.237, "z": -203.8, "size": 0.2905, "title": "Connectedness of subsets of vector spaces", "summary": "We show several results related to the (path)-connectedness of subsets of real vector spaces: * `Set.Countable.isPathConnected_compl_of_one_lt_rank` asserts that the complement of a countable set is path-connected in a space of dimension `> 1`. * `isPathConnected_compl_singleton_of_one_lt_rank` is the special case of the complement of a singleton. * `isPathConnected_sphere` shows that any sphere is path-connected in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Connected.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 3, "macro_tier_score": 0.1676, "macro_tier_override": null, "x": 105.001, "z": -192.14, "size": 0.3312, "title": "The `arctan` function.", "summary": "Inequalities, identities and `Real.tan` as an `OpenPartialHomeomorph` between `(-(π / 2), π / 2)` and the whole line. The result of `arctan x + arctan y` is given by `arctan_add`, `arctan_add_eq_add_pi` or `arctan_add_eq_sub_pi` depending on whether `x * y < 1` and `0 < x`. As an application of `arctan_add` we give four Machin-like formulas (linear combinations of arctangents equal to `π / 4 = arctan 1`), including…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 3, "macro_tier_score": 0.1681, "macro_tier_override": null, "x": 95.704, "z": -185.227, "size": 0.3601, "title": "Complex trigonometric functions", "summary": "Basic facts and derivatives for the complex trigonometric functions. Several facts about the real trigonometric functions have the proofs deferred here, rather than `Analysis.SpecialFunctions.Trigonometric.Basic`, as they are most easily proved by appealing to the corresponding fact for complex trigonometric functions, or require additional imports which are not available in that file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.html"}, {"id": "Mathlib.Analysis.Calculus.Monotone", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 1, "macro_tier_score": 0.0279, "macro_tier_override": null, "x": 50.4, "z": -228.307, "size": 0.242, "title": "Differentiability of monotone functions", "summary": "We show that a monotone function `f : ℝ → ℝ` is differentiable almost everywhere, in `Monotone.ae_differentiableAt`. (We also give a version for a function monotone on a set, in `MonotoneOn.ae_differentiableWithinAt`.) If the function `f` is continuous, this follows directly from general differentiation of measure theorems. Let `μ` be the Stieltjes measure associated to `f`. Then, almost everywhere, `μ [x, y] / Leb…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Monotone.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Slope", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 3, "macro_tier_score": 0.2515, "macro_tier_override": null, "x": 107.834, "z": -228.178, "size": 0.3658, "title": "Derivative as the limit of the slope", "summary": "In this file we relate the derivative of a function with its definition from a standard undergraduate course as the limit of the slope `(f y - f x) / (y - x)` as `y` tends to `𝓝[≠] x`. Since we are talking about functions taking values in a normed space instead of the base field, we use `slope f x y = (y - x)⁻¹ • (f y - f x)` instead of division. We also prove some estimates on the upper/lower limits of the slope in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Slope.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Basic", "region_id": "analysis", "micro_elevation": 0.2917, "macro_tier": 3, "macro_tier_score": 0.2111, "macro_tier_override": null, "x": 73.39, "z": -189.249, "size": 0.4219, "title": "Properties of inner product spaces", "summary": "This file proves many basic properties of inner product spaces (real or complex).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Basic.html"}, {"id": "Mathlib.Analysis.Complex.Basic", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 4, "macro_tier_score": 0.4087, "macro_tier_override": null, "x": 94.053, "z": -200.427, "size": 0.526, "title": "Normed space structure on `ℂ`.", "summary": "This file gathers basic facts of analytic nature on the complex numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Basic.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Defs", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 3, "macro_tier_score": 0.209, "macro_tier_override": null, "x": 60.272, "z": -211.438, "size": 0.3074, "title": "Inner product spaces", "summary": "This file defines inner product spaces. Hilbert spaces can be obtained using the set of assumptions `[RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E]`. For convenience, a variable alias `HilbertSpace` is provided so that one can write `variable? [HilbertSpace 𝕜 E]` and get this as a suggestion. An inner product space is a vector space endowed with an inner product. It generalizes the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Defs.html"}, {"id": "Mathlib.Analysis.Normed.Algebra.Basic", "region_id": "analysis", "micro_elevation": 0.5208, "macro_tier": 2, "macro_tier_score": 0.0417, "macro_tier_override": null, "x": 86.378, "z": -247.077, "size": 0.2387, "title": "Normed algebras", "summary": "This file contains basic facts about normed algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Algebra/Basic.html"}, {"id": "Mathlib.Analysis.Normed.Module.WeakDual", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 2, "macro_tier_score": 0.0558, "macro_tier_override": null, "x": 65.388, "z": -179.362, "size": 0.2721, "title": "Weak dual of normed space", "summary": "Let `E` be a normed space over a field `𝕜`. This file is concerned with properties of the weak-\\* topology on the dual of `E`. By the dual, we mean either of the type synonyms `StrongDual 𝕜 E` or `WeakDual 𝕜 E`, depending on whether it is viewed as equipped with its usual operator norm topology or the weak-\\* topology. It is shown that the canonical mapping `StrongDual 𝕜 E → WeakDual 𝕜 E` is continuous, and as a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/WeakDual.html"}, {"id": "Mathlib.Analysis.Normed.Algebra.Spectrum", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.0702, "macro_tier_override": null, "x": 53.169, "z": -184.677, "size": 0.3195, "title": "The spectrum of elements in a complete normed algebra", "summary": "This file contains the basic theory for the resolvent and spectrum of a Banach algebra. Theorems specific to *complex* Banach algebras, such as *Gelfand's formula* can be found in `Mathlib/Analysis/Normed/Algebra/GelfandFormula.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Algebra/Spectrum.html"}, {"id": "Mathlib.Analysis.Analytic.IteratedFDeriv", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 2, "macro_tier_score": 0.0559, "macro_tier_override": null, "x": 51.898, "z": -247.498, "size": 0.278, "title": "The iterated derivative of an analytic function", "summary": "If a function is analytic, written as `f (x + y) = ∑ pₙ (y, ..., y)` then its `n`-th iterated derivative at `x` is given by `(v₁, ..., vₙ) ↦ ∑ pₙ (v_{σ (1)}, ..., v_{σ (n)})` where the sum is over all permutations of `{1, ..., n}`. In particular, it is symmetric. This generalizes the result of `HasFPowerSeriesOnBall.factorial_smul` giving `D^n f (v, ..., v) = n! * pₙ (v, ..., v)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/IteratedFDeriv.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.Operations", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 4, "macro_tier_score": 0.2538, "macro_tier_override": null, "x": 109.325, "z": -244.572, "size": 0.46, "title": "Higher differentiability of usual operations", "summary": "We prove that the usual operations (addition, multiplication, difference, and so on) preserve `C^n` functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/Operations.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.CPolynomial", "region_id": "analysis", "micro_elevation": 0.5833, "macro_tier": 3, "macro_tier_score": 0.0976, "macro_tier_override": null, "x": 37.908, "z": -216.542, "size": 0.2809, "title": "Higher smoothness of continuously polynomial functions", "summary": "We prove that continuously polynomial functions are `C^∞`. In particular, this is the case of continuous multilinear maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/CPolynomial.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 2, "macro_tier_score": 0.0282, "macro_tier_override": null, "x": 36.163, "z": -203.915, "size": 0.2874, "title": "Gram-Schmidt Orthogonalization and Orthonormalization", "summary": "In this file we introduce Gram-Schmidt Orthogonalization and Orthonormalization. The Gram-Schmidt process takes a set of vectors as input and outputs a set of orthogonal vectors which have the same span.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 2, "macro_tier_score": 0.0428, "macro_tier_override": null, "x": 45.946, "z": -168.49, "size": 0.3402, "title": "Continuous functional calculus", "summary": "In this file we construct the `continuousFunctionalCalculus` for a normal element `a` of a (unital) C⋆-algebra over `ℂ`. This is a star algebra equivalence `C(spectrum ℂ a, ℂ) ≃⋆ₐ[ℂ] elemental ℂ a` which sends the (restriction of) the identity map `ContinuousMap.id ℂ` to the (unique) preimage of `a` under the coercion of `elemental ℂ a` to `A`. Being a star algebra equivalence between C⋆-algebras, this map is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Basic.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric", "region_id": "analysis", "micro_elevation": 0.5833, "macro_tier": 3, "macro_tier_score": 0.0705, "macro_tier_override": null, "x": 49.914, "z": -179.278, "size": 0.3346, "title": "Properties of `rpow` and `sqrt` over an algebra with an isometric CFC", "summary": "This file collects results about `CFC.rpow`, `CFC.nnrpow` and `CFC.sqrt` that use facts that rely on an isometric continuous functional calculus.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow/Isometric.html"}, {"id": "Mathlib.Analysis.LocallyConvex.StrongTopology", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.239, "title": "Local convexity of the strong topology", "summary": "In this file we prove that the strong topology on `E →L[ℝ] F` is locally convex provided that `F` is locally convex.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/StrongTopology.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 1, "macro_tier_score": 0.0143, "macro_tier_override": null, "x": 101.949, "z": -184.217, "size": 0.2885, "title": "Properties of the extended logarithm and exponential", "summary": "We prove that `log` and `exp` define order isomorphisms between `ℝ≥0∞` and `EReal`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Log/ENNRealLogExp.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Log.ERealExp", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 2, "macro_tier_score": 0.0279, "macro_tier_override": null, "x": 87.855, "z": -214.179, "size": 0.2528, "title": "Extended Nonnegative Real Exponential", "summary": "We define `exp` as an extension of the exponential of a real to the extended reals `EReal`. The function takes values in the extended nonnegative reals `ℝ≥0∞`, with `exp ⊥ = 0` and `exp ⊤ = ⊤`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Log/ERealExp.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 2, "macro_tier_score": 0.0279, "macro_tier_override": null, "x": 56.055, "z": -230.765, "size": 0.2528, "title": "Extended Nonnegative Real Logarithm", "summary": "We define `log` as an extension of the logarithm of a positive real to the extended nonnegative reals `ℝ≥0∞`. The function takes values in the extended reals `EReal`, with `log 0 = ⊥` and `log ⊤ = ⊤`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.html"}, {"id": "Mathlib.Analysis.Normed.Operator.Bilinear", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 4, "macro_tier_score": 0.3964, "macro_tier_override": null, "x": 102.518, "z": -213.329, "size": 0.5682, "title": "Operator norm: bilinear maps", "summary": "This file contains lemmas concerning operator norm as applied to bilinear maps `E × F → G`, interpreted as linear maps `E → F → G` as usual (and similarly for semilinear variants).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Bilinear.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Complex.CircleMap", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 85.578, "z": -182.687, "size": 0.2978, "title": "circleMap", "summary": "This file defines the circle map $θ ↦ c + R e^{θi}$, a parametrization of a circle.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Complex/CircleMap.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 3, "macro_tier_score": 0.1676, "macro_tier_override": null, "x": 110.761, "z": -253.072, "size": 0.3287, "title": "Non-integrable functions", "summary": "In this file we prove that the derivative of a function that tends to infinity is not interval integrable, see `not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_filter` and `not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_punctured`. Then we apply the latter lemma to prove that the function `fun x => x⁻¹` is integrable on `a..b` if and only if `a = b` or `0 ∉ [a, b]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/NonIntegrable.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "region_id": "analysis", "micro_elevation": 0.2917, "macro_tier": 4, "macro_tier_score": 0.4047, "macro_tier_override": null, "x": 93.485, "z": -226.042, "size": 0.3868, "title": "Trigonometric functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Dual", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 3, "macro_tier_score": 0.1122, "macro_tier_override": null, "x": 46.775, "z": -238.668, "size": 0.3414, "title": "The Fréchet-Riesz representation theorem", "summary": "We consider an inner product space `E` over `𝕜`, which is either `ℝ` or `ℂ`. We define `toDualMap`, a conjugate-linear isometric embedding of `E` into its dual, which maps an element `x` of the space to `fun y => ⟪x, y⟫`. Under the hypothesis of completeness (i.e., for Hilbert spaces), we upgrade this to `toDual`, a conjugate-linear isometric *equivalence* of `E` onto its dual; that is, we establish the surjectivity…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Dual.html"}, {"id": "Mathlib.Analysis.Normed.Group.NullSubmodule", "region_id": "analysis", "micro_elevation": 0.1458, "macro_tier": 3, "macro_tier_score": 0.1115, "macro_tier_override": null, "x": 90.236, "z": -206.594, "size": 0.2845, "title": "The null subgroup in a seminormed commutative group", "summary": "For any `SeminormedAddCommGroup M`, the quotient `SeparationQuotient M` by the null subgroup is defined as a `NormedAddCommGroup` instance in `Mathlib/Analysis/Normed/Group/Uniform.lean`. Here we define the null space as a subgroup.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/NullSubmodule.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "region_id": "analysis", "micro_elevation": 0.8333, "macro_tier": 2, "macro_tier_score": 0.0565, "macro_tier_override": null, "x": 21.735, "z": -227.43, "size": 0.3318, "title": "Estimates for the complex logarithm", "summary": "We show that `log (1+z)` differs from its Taylor polynomial up to degree `n` by at most `‖z‖^(n+1)/((n+1)*(1-‖z‖))` when `‖z‖ < 1`; see `Complex.norm_log_sub_logTaylor_le`. To this end, we derive the representation of `log (1+z)` as the integral of `1/(1+tz)` over the unit interval (`Complex.log_eq_integral`) and introduce notation `Complex.logTaylor n` for the Taylor polynomial up to degree `n-1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.html"}, {"id": "Mathlib.Analysis.Complex.Convex", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 3, "macro_tier_score": 0.1538, "macro_tier_override": null, "x": 99.321, "z": -205.253, "size": 0.3358, "title": "Theorems about convexity on the complex plane", "summary": "We show that the open and closed half-spaces in ℂ given by an inequality on either the real or imaginary part are all convex over ℝ. We also prove some results on star-convexity for the slit plane.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Convex.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Integrals.Basic", "region_id": "analysis", "micro_elevation": 0.8125, "macro_tier": 3, "macro_tier_score": 0.1689, "macro_tier_override": null, "x": 133.407, "z": -235.531, "size": 0.4001, "title": "Integration of specific interval integrals", "summary": "This file contains proofs of the integrals of various specific functions. This includes: * Integrals of simple functions, such as `id`, `pow`, `inv`, `exp`, `log` * Integrals of some trigonometric functions, such as `sin`, `cos`, `1 / (1 + x^2)` * The integral of `cos x ^ 2 - sin x ^ 2` * Reduction formulae for the integrals of `sin x ^ n` and `cos x ^ n` for `n ≥ 2` * The computation of `∫ x in 0..π, sin x ^ n` as…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Integrals/Basic.html"}, {"id": "Mathlib.Analysis.SpecificLimits.RCLike", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 2, "macro_tier_score": 0.0559, "macro_tier_override": null, "x": 82.333, "z": -226.124, "size": 0.2801, "title": "A collection of specific limit computations for `RCLike`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecificLimits/RCLike.html"}, {"id": "Mathlib.Analysis.Complex.Polynomial.Basic", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 3, "macro_tier_score": 0.0987, "macro_tier_override": null, "x": 37.245, "z": -229.623, "size": 0.3623, "title": "The fundamental theorem of algebra", "summary": "This file proves that every nonconstant complex polynomial has a root using Liouville's theorem. As a consequence, the complex numbers are algebraically closed. We also provide some specific results about the Galois groups of ℚ-polynomials with specific numbers of non-real roots. We also show that an irreducible real polynomial has degree at most two.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Polynomial/Basic.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "region_id": "analysis", "micro_elevation": 0.8333, "macro_tier": 3, "macro_tier_score": 0.0989, "macro_tier_override": null, "x": 74.418, "z": -148.85, "size": 0.3714, "title": "Evaluation of specific improper integrals", "summary": "This file contains some integrability results, and evaluations of integrals, over `ℝ` or over half-infinite intervals in `ℝ`. These lemmas are stated in terms of either `Iic` or `Ioi` (neglecting `Iio` and `Ici`) to match mathlib's conventions for integrals over finite intervals (see `intervalIntegral`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.html"}, {"id": "Mathlib.Analysis.Normed.Group.SeparationQuotient", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 77.156, "z": -200.84, "size": 0.2, "title": "Lifts of maps to separation quotients of seminormed groups", "summary": "For any `SeminormedAddCommGroup M`, a `NormedAddCommGroup` instance has been defined in `Mathlib/Analysis/Normed/Group/Uniform.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/SeparationQuotient.html"}, {"id": "Mathlib.Analysis.Normed.Group.Hom", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 4, "macro_tier_score": 0.4452, "macro_tier_override": null, "x": 81.635, "z": -216.968, "size": 0.3192, "title": "Normed groups homomorphisms", "summary": "This file gathers definitions and elementary constructions about bounded group homomorphisms between normed (abelian) groups (abbreviated to \"normed group homs\"). The main lemmas relate the boundedness condition to continuity and Lipschitzness. The main construction is to endow the type of normed group homs between two given normed groups with a group structure and a norm, giving rise to a normed group structure. We…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Hom.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Exponential", "region_id": "analysis", "micro_elevation": 0.5625, "macro_tier": 3, "macro_tier_score": 0.2509, "macro_tier_override": null, "x": 100.733, "z": -174.016, "size": 0.324, "title": "Calculus results on exponential in a Banach algebra", "summary": "In this file, we prove basic properties about the derivative of the exponential map `exp` in a Banach algebra `𝔸` over a field `𝕂`. We keep them separate from the main file `Analysis.Normed.Algebra.Exponential` in order to minimize dependencies.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Exponential.html"}, {"id": "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 2, "macro_tier_score": 0.0423, "macro_tier_override": null, "x": 90.243, "z": -164.995, "size": 0.3037, "title": "Swapping limits and derivatives via uniform convergence", "summary": "The purpose of this file is to prove that the derivative of the pointwise limit of a sequence of functions is the pointwise limit of the functions' derivatives when the derivatives converge _uniformly_. The formal statement appears as `hasFDerivAt_of_tendstoLocallyUniformlyOn`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/UniformLimitsDeriv.html"}, {"id": "Mathlib.Analysis.Calculus.MeanValue", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 3, "macro_tier_score": 0.2394, "macro_tier_override": null, "x": 47.112, "z": -239.048, "size": 0.4428, "title": "The mean value inequality and equalities", "summary": "In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/MeanValue.html"}, {"id": "Mathlib.Analysis.Real.Sqrt", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.4895, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.4478, "title": "Square root of a real number", "summary": "In this file we define * `NNReal.sqrt` to be the square root of a nonnegative real number. * `Real.sqrt` to be the square root of a real number, defined to be zero on negative numbers. Then we prove some basic properties of these functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Real/Sqrt.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Basic", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 4, "macro_tier_score": 0.446, "macro_tier_override": null, "x": 74.53, "z": -222.111, "size": 0.3683, "title": "Normed star rings and algebras", "summary": "A normed star group is a normed group with a compatible `star` which is isometric. A C⋆-ring is a normed star group that is also a ring and that verifies the stronger condition `‖x‖^2 ≤ ‖x⋆ * x‖` for all `x` (which actually implies equality). If a C⋆-ring is also a star algebra, then it is a C⋆-algebra. Note that the type classes corresponding to C⋆-algebras are defined in `Mathlib/Analysis/CStarAlgebra/Classes`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Basic.html"}, {"id": "Mathlib.Analysis.Normed.Operator.LinearIsometry", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 4, "macro_tier_score": 0.4751, "macro_tier_override": null, "x": 83.27, "z": -216.404, "size": 0.427, "title": "(Semi-)linear isometries", "summary": "In this file we define `LinearIsometry σ₁₂ E E₂` (notation: `E →ₛₗᵢ[σ₁₂] E₂`) to be a semilinear isometric embedding of `E` into `E₂` and `LinearIsometryEquiv` (notation: `E ≃ₛₗᵢ[σ₁₂] E₂`) to be a semilinear isometric equivalence between `E` and `E₂`. The notation for the associated purely linear concepts is `E →ₗᵢ[R] E₂`, `E ≃ₗᵢ[R] E₂`, and `E →ₗᵢ⋆[R] E₂`, `E ≃ₗᵢ⋆[R] E₂` for the star-linear versions. We also prove…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/LinearIsometry.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Mul", "region_id": "analysis", "micro_elevation": 0.5625, "macro_tier": 4, "macro_tier_score": 0.2682, "macro_tier_override": null, "x": 50.712, "z": -238.428, "size": 0.4776, "title": "Multiplicative operations on derivatives", "summary": "For detailed documentation of the Fréchet derivative, see the module docstring of `Mathlib/Analysis/Calculus/FDeriv/Basic.lean`. This file contains the usual formulas (and existence assertions) for the derivative of * multiplication of a function by a scalar function * product of finitely many scalar functions * taking the pointwise multiplicative inverse (i.e. `Inv.inv` or `Ring.inverse`) of a function", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Mul.html"}, {"id": "Mathlib.Analysis.Normed.Field.Lemmas", "region_id": "analysis", "micro_elevation": 0.1458, "macro_tier": 4, "macro_tier_score": 0.49, "macro_tier_override": null, "x": 87.104, "z": -201.592, "size": 0.4654, "title": "Normed fields", "summary": "In this file we continue building the theory of normed division rings and fields. Some useful results that relate the topology of the normed field to the discrete topology include: * `discreteTopology_or_nontriviallyNormedField` * `discreteTopology_of_bddAbove_range_norm`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Field/Lemmas.html"}, {"id": "Mathlib.Analysis.Normed.Group.Lemmas", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 3, "macro_tier_score": 0.2504, "macro_tier_override": null, "x": 87.585, "z": -209.026, "size": 0.2888, "title": "Further lemmas about normed groups", "summary": "This file contains further lemmas about normed groups, requiring heavier imports than `Mathlib/Analysis/Normed/Group/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Lemmas.html"}, {"id": "Mathlib.Analysis.Normed.Group.Uniform", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 4, "macro_tier_score": 0.4929, "macro_tier_override": null, "x": 83.626, "z": -204.646, "size": 0.5501, "title": "Normed groups are uniform groups", "summary": "This file proves lipschitzness of normed group operations and shows that normed groups are uniform groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Uniform.html"}, {"id": "Mathlib.Analysis.Convex.Approximation", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 89.257, "z": -230.417, "size": 0.2674, "title": "Approximation to convex functions", "summary": "In this file we show that a convex lower-semicontinuous function is the upper envelope of a family of continuous affine linear functions. We follow the proof in [N. Bourbaki, *Topological vector spaces*, Chapter II, §5][bourbaki1987].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Approximation.html"}, {"id": "Mathlib.Analysis.LocallyConvex.Separation", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 3, "macro_tier_score": 0.1121, "macro_tier_override": null, "x": 60.566, "z": -213.426, "size": 0.3328, "title": "Separation Hahn-Banach theorem", "summary": "In this file we prove the geometric Hahn-Banach theorem. For any two disjoint convex sets, there exists a continuous linear functional separating them, geometrically meaning that we can intercalate a plane between them. We provide many variations to stricten the result under more assumptions on the convex sets: * `geometric_hahn_banach_open`: One set is open. Weak separation. * `geometric_hahn_banach_open_point`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/Separation.html"}, {"id": "Mathlib.Analysis.Normed.Field.Approximation", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 85.386, "z": -177.999, "size": 0.2516, "title": "Approximate roots and polynomials in a normed field", "summary": "In this file, we prove several approximation lemmas on a normed field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Field/Approximation.html"}, {"id": "Mathlib.Analysis.Normed.Unbundled.RingSeminorm", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 2, "macro_tier_score": 0.029, "macro_tier_override": null, "x": 80.025, "z": -240.0, "size": 0.3454, "title": "Seminorms and norms on rings", "summary": "This file defines seminorms and norms on rings. These definitions are useful when one needs to consider multiple (semi)norms on a given ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Unbundled/RingSeminorm.html"}, {"id": "Mathlib.Analysis.Normed.Algebra.QuaternionExponential", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 36.212, "z": -222.674, "size": 0.2, "title": "Lemmas about `NormedSpace.exp` on `Quaternion`s", "summary": "This file contains results about `NormedSpace.exp` on `Quaternion ℝ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Algebra/QuaternionExponential.html"}, {"id": "Mathlib.Analysis.Quaternion", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 42.24, "z": -232.493, "size": 0.2478, "title": "Quaternions as a normed algebra", "summary": "In this file we define the following structures on the space `ℍ := ℍ[ℝ]` of quaternions: * inner product space; * normed ring; * normed space over `ℝ`. We show that the norm on `ℍ[ℝ]` agrees with the Euclidean norm of its components.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Quaternion.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Series", "region_id": "analysis", "micro_elevation": 0.5833, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 121.739, "z": -218.253, "size": 0.2478, "title": "Trigonometric functions as sums of infinite series", "summary": "In this file we express trigonometric functions in terms of their series expansion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.html"}, {"id": "Mathlib.Analysis.Analytic.RadiusLiminf", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 2, "macro_tier_score": 0.0557, "macro_tier_override": null, "x": 100.495, "z": -234.084, "size": 0.25, "title": "Representation of `FormalMultilinearSeries.radius` as a `liminf`", "summary": "In this file we prove that the radius of convergence of a `FormalMultilinearSeries` is equal to $\\liminf_{n\\to\\infty} \\frac{1}{\\sqrt[n]{‖p n‖}}$. This lemma can't go to `Analysis.Analytic.Basic` because this would create a circular dependency once we redefine `exp` using `FormalMultilinearSeries`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/RadiusLiminf.html"}, {"id": "Mathlib.Analysis.Analytic.ConvergenceRadius", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 4, "macro_tier_score": 0.266, "macro_tier_override": null, "x": 95.036, "z": -184.808, "size": 0.3957, "title": "Radius of convergence of a power series", "summary": "This file introduces the notion of the radius of convergence of a power series.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/ConvergenceRadius.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 4, "macro_tier_score": 0.381, "macro_tier_override": null, "x": 107.547, "z": -222.517, "size": 0.529, "title": "Power function on `ℝ≥0` and `ℝ≥0∞`", "summary": "We construct the power functions `x ^ y` where * `x` is a nonnegative real number and `y` is a real number; * `x` is a number from `[0, +∞]` (a.k.a. `ℝ≥0∞`) and `y` is a real number. We also prove basic properties of these functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Pow/NNReal.html"}, {"id": "Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic", "region_id": "analysis", "micro_elevation": 0.9375, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 145.8, "z": -190.349, "size": 0.2, "title": "Asymptotic Behavior of the Logarithmic Counting Function", "summary": "If `f` is meromorphic over a field `𝕜`, we show that the logarithmic counting function for the poles of `f` is asymptotically bounded if and only if `f` has only removable singularities. See Page 170f of [Lang, *Introduction to Complex Hyperbolic Spaces*][MR886677] for a detailed discussion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/ValueDistribution/LogCounting/Asymptotic.html"}, {"id": "Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic", "region_id": "analysis", "micro_elevation": 0.9167, "macro_tier": 1, "macro_tier_score": 0.0144, "macro_tier_override": null, "x": 139.689, "z": -240.002, "size": 0.2979, "title": "The Logarithmic Counting Function of Value Distribution Theory", "summary": "For nontrivially normed fields `𝕜`, this file defines the logarithmic counting function of a meromorphic function defined on `𝕜`. Also known as the `Nevanlinna counting function`, this is one of the three main functions used in Value Distribution Theory. The logarithmic counting function of a meromorphic function `f` is a logarithmically weighted measure of the number of times the function `f` takes a given value…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/ValueDistribution/LogCounting/Basic.html"}, {"id": "Mathlib.Analysis.Normed.Lp.Matrix", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.2096, "macro_tier_override": null, "x": 43.627, "z": -205.666, "size": 0.351, "title": "Matrices are isomorphic with linear maps between Lp spaces", "summary": "This file provides a `WithLp` version of `Matrix.toLin'`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Lp/Matrix.html"}, {"id": "Mathlib.Analysis.Calculus.ParametricIntegral", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 3, "macro_tier_score": 0.0983, "macro_tier_override": null, "x": 104.112, "z": -248.346, "size": 0.3418, "title": "Derivatives of integrals depending on parameters", "summary": "A parametric integral is a function with shape `f = fun x : H ↦ ∫ a : α, F x a ∂μ` for some `F : H → α → E`, where `H` and `E` are normed spaces and `α` is a measured space with measure `μ`. We already know from `continuous_of_dominated` in `Mathlib/MeasureTheory/Integral/Bochner/Basic.lean` how to guarantee that `f` is continuous using the dominated convergence theorem. In this file, we want to express the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ParametricIntegral.html"}, {"id": "Mathlib.Analysis.Complex.CauchyIntegral", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 3, "macro_tier_score": 0.1703, "macro_tier_override": null, "x": 119.093, "z": -230.082, "size": 0.457, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/CauchyIntegral.html"}, {"id": "Mathlib.Analysis.Normed.Module.Alternating.Basic", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 4, "macro_tier_score": 0.2644, "macro_tier_override": null, "x": 63.471, "z": -229.482, "size": 0.2962, "title": "Operator norm on the space of continuous alternating maps", "summary": "In this file we show that continuous alternating maps from a seminormed space to a (semi)normed space form a (semi)normed space. We also prove basic facts about this norm and define bundled versions of some operations on continuous alternating maps. Most proofs just invoke the corresponding fact about continuous multilinear maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Alternating/Basic.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Semisimple", "region_id": "analysis", "micro_elevation": 0.5625, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 117.642, "z": -226.077, "size": 0.2338, "title": "Semisimple operators on inner product spaces", "summary": "This file is a place to gather results related to semisimplicity of linear operators on inner product spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Semisimple.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Projection.Submodule", "region_id": "analysis", "micro_elevation": 0.5417, "macro_tier": 3, "macro_tier_score": 0.2095, "macro_tier_override": null, "x": 119.249, "z": -214.745, "size": 0.3438, "title": "Subspaces associated with orthogonal projections", "summary": "Here, the orthogonal projection is used to prove a series of more subtle lemmas about the orthogonal complement of subspaces of `E` (the orthogonal complement itself was defined in `Mathlib/Analysis/InnerProductSpace/Orthogonal.lean`) such that they admit orthogonal projections; the lemma `Submodule.sup_orthogonal_of_hasOrthogonalProjection`, stating that for a subspace `K` of `E` such that `K` admits an orthogonal…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Projection/Submodule.html"}, {"id": "Mathlib.Analysis.Real.Cardinality", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 3, "macro_tier_score": 0.1679, "macro_tier_override": null, "x": 81.064, "z": -208.457, "size": 0.3502, "title": "The cardinality of the reals", "summary": "This file shows that the real numbers have cardinality continuum, i.e. `#ℝ = 𝔠`. We show that `#ℝ ≤ 𝔠` by noting that every real number is determined by a Cauchy-sequence of the form `ℕ → ℚ`, which has cardinality `𝔠`. To show that `#ℝ ≥ 𝔠` we define an injection from `{0, 1} ^ ℕ` to `ℝ` with `f ↦ Σ n, f n * (1 / 3) ^ n`. We conclude that all intervals with distinct endpoints have cardinality continuum.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Real/Cardinality.html"}, {"id": "Mathlib.Analysis.Rat.NatSqrt.Real", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 80.561, "z": -210.942, "size": 0.2676, "title": null, "summary": "Comparisons between rational approximations to the square root of a natural number and the real square root.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Rat/NatSqrt/Real.html"}, {"id": "Mathlib.Analysis.Normed.Order.Hom.Ultra", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 81.402, "z": -215.458, "size": 0.2, "title": "Constructing nonarchimedean (ultrametric) normed groups from nonarchimedean normed homs", "summary": "This file defines constructions that upgrade `Add(Comm)Group` to `(Semi)NormedAdd(Comm)Group` using an `AddGroup(Semi)normClass` when the codomain is the reals and the hom is nonarchimedean.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Order/Hom/Ultra.html"}, {"id": "Mathlib.Analysis.Normed.Order.Hom.Basic", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 82.768, "z": -213.156, "size": 0.2478, "title": "Constructing (semi)normed groups from (semi)normed homs", "summary": "This file defines constructions that upgrade `(Comm)Group` to `(Semi)Normed(Comm)Group` using a `Group(Semi)normClass` when the codomain is the reals. See `Mathlib/Analysis/Normed/Order/Hom/Ultra.lean` for further upgrades to nonarchimedean normed groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Order/Hom/Basic.html"}, {"id": "Mathlib.Analysis.Analytic.CPolynomial", "region_id": "analysis", "micro_elevation": 0.5208, "macro_tier": 4, "macro_tier_score": 0.265, "macro_tier_override": null, "x": 112.301, "z": -229.684, "size": 0.3396, "title": "Properties of continuously polynomial functions", "summary": "We expand the API around continuously polynomial functions. Notably, we show that this class is stable under the usual operations (addition, subtraction, negation). We also prove that continuous multilinear maps are continuously polynomial, and so are continuous linear maps into continuous multilinear maps. In particular, such maps are analytic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/CPolynomial.html"}, {"id": "Mathlib.Analysis.Analytic.Inverse", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 4, "macro_tier_score": 0.265, "macro_tier_override": null, "x": 93.49, "z": -175.556, "size": 0.3396, "title": "Inverse of analytic functions", "summary": "We construct the left and right inverse of a formal multilinear series with invertible linear term, we prove that they coincide and study their properties (notably convergence). We deduce that the inverse of an analytic open partial homeomorphism is analytic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/Inverse.html"}, {"id": "Mathlib.Analysis.Analytic.Within", "region_id": "analysis", "micro_elevation": 0.5208, "macro_tier": 4, "macro_tier_score": 0.265, "macro_tier_override": null, "x": 117.223, "z": -201.544, "size": 0.3396, "title": "Properties of analyticity restricted to a set", "summary": "From `Mathlib/Analysis/Analytic/Basic.lean`, we have the definitions 1. `AnalyticWithinAt 𝕜 f s x` means a power series at `x` converges to `f` on `𝓝[insert x s] x`. 2. `AnalyticOn 𝕜 f s t` means `∀ x ∈ t, AnalyticWithinAt 𝕜 f s x`. This means there exists an extension of `f` which is analytic and agrees with `f` on `s ∪ {x}`, but `f` is allowed to be arbitrary elsewhere. Here we prove basic properties of these…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/Within.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Basic", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 4, "macro_tier_score": 0.2838, "macro_tier_override": null, "x": 67.202, "z": -233.792, "size": 0.5286, "title": "One-dimensional derivatives", "summary": "This file defines the derivative of a function `f : 𝕜 → F` where `𝕜` is a normed field and `F` is a normed space over this field. The derivative of such a function `f` at a point `x` is given by an element `f' : F`. The theory is developed analogously to the [Fréchet derivatives](./fderiv.html). We first introduce predicates defined in terms of the corresponding predicates for Fréchet derivatives: -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Basic.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 4, "macro_tier_score": 0.265, "macro_tier_override": null, "x": 52.28, "z": -230.984, "size": 0.3396, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.html"}, {"id": "Mathlib.Analysis.Normed.Module.Completion", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 4, "macro_tier_score": 0.2655, "macro_tier_override": null, "x": 73.332, "z": -184.502, "size": 0.3707, "title": "Normed space structure on the completion of a normed space", "summary": "If `E` is a normed space over `𝕜`, then so is `UniformSpace.Completion E`. In this file we provide necessary instances and define `UniformSpace.Completion.toComplₗᵢ` - coercion `E → UniformSpace.Completion E` as a bundled linear isometry. We also show that if `A` is a normed algebra over `𝕜`, then so is `UniformSpace.Completion A`. TODO: Generalise the results here from the concrete `completion` to any…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Completion.html"}, {"id": "Mathlib.Analysis.LConvolution", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2959, "title": "Convolution of functions using the Lebesgue integral", "summary": "In this file we define and prove properties about the convolution of two functions using the Lebesgue integral.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LConvolution.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.CStarMatrix", "region_id": "analysis", "micro_elevation": 0.8125, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 136.474, "z": -227.939, "size": 0.2478, "title": "Matrices with entries in a C⋆-algebra", "summary": "This file creates a type copy of `Matrix m n A` called `CStarMatrix m n A` meant for matrices with entries in a C⋆-algebra `A`. Its action on `C⋆ᵐᵒᵈ (n → A)` (via `Matrix.mulVec`) gives it the operator norm, and this norm makes `CStarMatrix n n A` a C⋆-algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/CStarMatrix.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Module.Constructions", "region_id": "analysis", "micro_elevation": 0.7917, "macro_tier": 2, "macro_tier_score": 0.0282, "macro_tier_override": null, "x": 32.593, "z": -242.778, "size": 0.282, "title": "Constructions of Hilbert C⋆-modules", "summary": "In this file we define the following constructions of `CStarModule`s where `A` denotes a C⋆-algebra. For some of the types listed below, the instance is declared on the type synonym `WithCStarModule E` (with the notation `C⋆ᵐᵒᵈ E`), instead of on `E` itself; we explain the reasoning behind each decision below. 1. `A` as a `CStarModule` over itself. 2. `C⋆ᵐᵒᵈ(A, E × F)` as a `CStarModule` over `A`, when `E` and `F`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Module/Constructions.html"}, {"id": "Mathlib.Analysis.Matrix.Normed", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 2, "macro_tier_score": 0.0702, "macro_tier_override": null, "x": 101.04, "z": -170.693, "size": 0.3146, "title": "Matrices as a normed space", "summary": "In this file we provide the following non-instances for norms on matrices: * The elementwise norm (with `open scoped Matrix.Norms.Elementwise`): * `Matrix.seminormedAddCommGroup` * `Matrix.normedAddCommGroup` * `Matrix.normedSpace` * `Matrix.isBoundedSMul` * `Matrix.normSMulClass` * The Frobenius norm (with `open scoped Matrix.Norms.Frobenius`): * `Matrix.frobeniusSeminormedAddCommGroup` *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Matrix/Normed.html"}, {"id": "Mathlib.Analysis.Polynomial.Order", "region_id": "analysis", "micro_elevation": 0.5208, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 89.375, "z": -172.622, "size": 0.2, "title": "Eventual sign of polynomials", "summary": "This file proves that a polynomial has a fixed sign beyond its largest or smallest root.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Polynomial/Order.html"}, {"id": "Mathlib.Analysis.Polynomial.Basic", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 53.298, "z": -234.525, "size": 0.2478, "title": "Limits related to polynomial and rational functions", "summary": "This file proves basic facts about limits of polynomial and rational functions. The main result is `Polynomial.isEquivalent_atTop_lead`, which states that for any polynomial `P` of degree `n` with leading coefficient `a`, the corresponding polynomial function is equivalent to `a * x^n` as `x` goes to +∞. We can then use this result to prove various limits for polynomial and rational functions, depending on the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Polynomial/Basic.html"}, {"id": "Mathlib.Analysis.LocallyConvex.Polar", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 2, "macro_tier_score": 0.0556, "macro_tier_override": null, "x": 91.723, "z": -216.616, "size": 0.239, "title": "Polar set", "summary": "In this file we define the polar set. There are different notions of the polar, we will define the *absolute polar*. The advantage over the real polar is that we can define the absolute polar for any bilinear form `B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜`, where `𝕜` is a normed commutative ring and `E` and `F` are modules over `𝕜`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/Polar.html"}, {"id": "Mathlib.Analysis.Complex.Exponential", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 4, "macro_tier_score": 0.4044, "macro_tier_override": null, "x": 82.3, "z": -216.788, "size": 0.3684, "title": "Exponential Function", "summary": "This file contains the definitions of the real and complex exponential function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Exponential.html"}, {"id": "Mathlib.Analysis.Oscillation", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2538, "title": "Oscillation", "summary": "In this file we define the oscillation of a function `f: E → F` at a point `x` of `E`. (`E` is required to be a TopologicalSpace and `F` a PseudoEMetricSpace.) The oscillation of `f` at `x` is defined to be the infimum of `diam f '' N` for all neighborhoods `N` of `x`. We also define `oscillationWithin f D x`, which is the oscillation at `x` of `f` restricted to `D`. We also prove some simple facts about…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Oscillation.html"}, {"id": "Mathlib.Analysis.RCLike.ContinuousMap", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 96.303, "z": -205.722, "size": 0.2, "title": "Mapping `C(X, ℝ)` to `C(X, 𝕜)` and back", "summary": "This file contains the definitions for `ContinuousMap.realToRCLike` and `ContinuousMap.rclikeToReal`, which map `C(X, ℝ)` to `C(X, 𝕜)` and back for any `RCLike 𝕜`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/RCLike/ContinuousMap.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Complex.Log", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 4, "macro_tier_score": 0.4078, "macro_tier_override": null, "x": 63.972, "z": -189.177, "size": 0.4998, "title": "The complex `log` function", "summary": "Basic properties, relationship with `exp`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Complex/Log.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 4, "macro_tier_score": 0.4052, "macro_tier_override": null, "x": 59.654, "z": -222.978, "size": 0.4092, "title": "The argument of a complex number.", "summary": "We define `arg : ℂ → ℝ`, returning a real number in the range $(-π, π]$, such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Complex/Arg.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Log.Basic", "region_id": "analysis", "micro_elevation": 0.2917, "macro_tier": 4, "macro_tier_score": 0.4071, "macro_tier_override": null, "x": 90.667, "z": -227.991, "size": 0.4778, "title": "Real logarithm", "summary": "In this file we define `Real.log` to be the logarithm of a real number. As usual, we extend it from its domain `(0, +∞)` to a globally defined function. We choose to do it so that `log 0 = 0` and `log (-x) = log x`. We prove some basic properties of this function and show that it is continuous.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Log/Basic.html"}, {"id": "Mathlib.Analysis.Convex.Radon", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 82.893, "z": -216.572, "size": 0.2, "title": "Radon's theorem on convex sets", "summary": "Radon's theorem states that any affine dependent set can be partitioned into two sets whose convex hulls intersect nontrivially. As a corollary, we prove Helly's theorem, which is a basic result in discrete geometry on the intersection of convex sets. Let `X₁, ⋯, Xₙ` be a finite family of convex sets in `ℝᵈ` with `n ≥ d + 1`. The theorem states that if any `d + 1` sets from this family intersect nontrivially, then…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Radon.html"}, {"id": "Mathlib.Analysis.Convex.NNReal", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 76.305, "z": -212.243, "size": 0.277, "title": "Specific lemmas about convexity over `ℝ≥0`", "summary": "This file collects some specific results about convexity over the ring `ℝ≥0`. Expand as needed.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/NNReal.html"}, {"id": "Mathlib.Analysis.Convex.Basic", "region_id": "analysis", "micro_elevation": 0.0417, "macro_tier": 4, "macro_tier_score": 0.45, "macro_tier_override": null, "x": 82.733, "z": -208.218, "size": 0.516, "title": "Convex sets", "summary": "In a 𝕜-vector space, we define the following property: * `Convex 𝕜 s`: A set `s` is convex if for any two points `x y ∈ s` it includes `segment 𝕜 x y`. We provide various equivalent versions, and prove that some specific sets are convex.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Basic.html"}, {"id": "Mathlib.Analysis.Convex.KreinMilman", "region_id": "analysis", "micro_elevation": 0.2917, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 94.5, "z": -193.906, "size": 0.2, "title": "The Krein-Milman theorem", "summary": "This file proves the Krein-Milman lemma and the Krein-Milman theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/KreinMilman.html"}, {"id": "Mathlib.Analysis.Convex.Exposed", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 73.824, "z": -205.051, "size": 0.239, "title": "Exposed sets", "summary": "This file defines exposed sets and exposed points for sets in a real vector space. An exposed subset of `A` is a subset of `A` that is the set of all maximal points of a functional (a continuous linear map `E → 𝕜`) over `A`. By convention, `∅` is an exposed subset of all sets. This allows for better functoriality of the definition (the intersection of two exposed subsets is exposed, faces of a polytope form a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Exposed.html"}, {"id": "Mathlib.Analysis.Complex.BorelCaratheodory", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 91.298, "z": -162.113, "size": 0.2, "title": "Borel-Carathéodory theorem", "summary": "This file proves the Borel-Carathéodory theorem: for any function `f` analytic on the open ball `|z| < R` such that `Re(f z) < M` for all `|z| < R`, we have `‖f z‖ ≤ 2 * M * ‖z‖ / (R - ‖z‖) + ‖f 0‖ * (R + ‖z‖) / (R - ‖z‖)`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/BorelCaratheodory.html"}, {"id": "Mathlib.Analysis.Complex.Schwarz", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 124.108, "z": -226.367, "size": 0.2676, "title": "Schwarz lemma", "summary": "In this file we prove several versions of the Schwarz lemma. * `Complex.norm_deriv_le_div_of_mapsTo_ball`. Let `f : ℂ → E` be a complex analytic function on an open disk with center `c` and a positive radius `R₁`. If `f` sends this ball to a closed ball with center `f c` and radius `R₂`, then the norm of the derivative of `f` at `c` is at most the ratio `R₂ / R₁`. * `Complex.dist_le_div_mul_dist_of_mapsTo_ball`. Let…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Schwarz.html"}, {"id": "Mathlib.Analysis.Normed.Lp.MeasurableSpace", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 64.934, "z": -242.853, "size": 0.287, "title": "Measurable space structure on `WithLp`", "summary": "If `X` is a measurable space, we set the measurable space structure on `WithLp p X` to be the same as the one on `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Lp/MeasurableSpace.html"}, {"id": "Mathlib.Analysis.Complex.TaylorSeries", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 1, "macro_tier_score": 0.0144, "macro_tier_override": null, "x": 62.111, "z": -167.469, "size": 0.295, "title": "Convergence of Taylor series of holomorphic functions", "summary": "We show that the Taylor series around some point `c : ℂ` of a function `f` that is complex differentiable on the open ball of radius `r` around `c` converges to `f` on that open ball; see `Complex.hasSum_taylorSeries_on_ball` and `Complex.taylorSeries_eq_on_ball` for versions (in terms of `HasSum` and `tsum`, respectively) for functions to a complete normed space over `ℂ`, and `Complex.taylorSeries_eq_on_ball'` for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/TaylorSeries.html"}, {"id": "Mathlib.Analysis.Analytic.Composition", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 4, "macro_tier_score": 0.2648, "macro_tier_override": null, "x": 58.498, "z": -237.21, "size": 0.3289, "title": "Composition of analytic functions", "summary": "In this file we prove that the composition of analytic functions is analytic. The argument is the following. Assume `g z = ∑' qₙ (z, ..., z)` and `f y = ∑' pₖ (y, ..., y)`. Then `g (f y) = ∑' qₙ (∑' pₖ (y, ..., y), ..., ∑' pₖ (y, ..., y)) = ∑' qₙ (p_{i₁} (y, ..., y), ..., p_{iₙ} (y, ..., y))`. For each `n` and `i₁, ..., iₙ`, define a `i₁ + ... + iₙ` multilinear function mapping `(y₀, ..., y_{i₁ + ... + iₙ - 1})` to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/Composition.html"}, {"id": "Mathlib.Analysis.Analytic.CPolynomialDef", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 4, "macro_tier_score": 0.2656, "macro_tier_override": null, "x": 103.477, "z": -233.434, "size": 0.3739, "title": null, "summary": "We specialize the theory of analytic functions to the case of functions that admit a development given by a *finite* formal multilinear series. We call them \"continuously polynomial\", which is abbreviated to `CPolynomial`. One reason to do that is that we no longer need a completeness assumption on the target space `F` to make the series converge, so some of the results are more general. The class of continuously…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/CPolynomialDef.html"}, {"id": "Mathlib.Analysis.Complex.UpperHalfPlane.Topology", "region_id": "analysis", "micro_elevation": 0.2917, "macro_tier": 1, "macro_tier_score": 0.0147, "macro_tier_override": null, "x": 101.283, "z": -210.655, "size": 0.3184, "title": "Topology on the upper half plane", "summary": "In this file we introduce a `TopologicalSpace` structure on the upper half plane and provide various instances.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/UpperHalfPlane/Topology.html"}, {"id": "Mathlib.Analysis.PSeries", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.0437, "macro_tier_override": null, "x": 99.665, "z": -178.719, "size": 0.3892, "title": "Convergence of `p`-series", "summary": "In this file we prove that the series `∑' k in ℕ, 1 / k ^ p` converges if and only if `p > 1`. The proof is based on the [Cauchy condensation test](https://en.wikipedia.org/wiki/Cauchy_condensation_test): `∑ k, f k` converges if and only if so does `∑ k, 2 ^ k f (2 ^ k)`. We prove this test in `NNReal.summable_condensed_iff` and `summable_condensed_iff_of_nonneg`, then use it to prove `summable_one_div_rpow`. After…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/PSeries.html"}, {"id": "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 3, "macro_tier_score": 0.2364, "macro_tier_override": null, "x": 59.599, "z": -181.033, "size": 0.2762, "title": "Rolle's Theorem", "summary": "In this file we prove Rolle's Theorem. The theorem says that for a function `f : ℝ → ℝ` such that * $f$ is differentiable on an open interval $(a, b)$, $a < b$; * $f$ is continuous on the corresponding closed interval $[a, b]$; * $f(a) = f(b)$, there exists a point $c∈(a, b)$ such that $f'(c)=0$. We prove four versions of this theorem. * `exists_hasDerivAt_eq_zero` is closest to the statement given above. It assumes…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/LocalExtr/Rolle.html"}, {"id": "Mathlib.Analysis.Calculus.LocalExtr.Basic", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 3, "macro_tier_score": 0.2368, "macro_tier_override": null, "x": 92.757, "z": -178.546, "size": 0.3145, "title": "Local extrema of differentiable functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/LocalExtr/Basic.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Multiplier", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 111.383, "z": -203.403, "size": 0.2, "title": "Multiplier Algebra of a C⋆-algebra", "summary": "Define the multiplier algebra of a C⋆-algebra as the algebra (over `𝕜`) of double centralizers, for which we provide the localized notation `𝓜(𝕜, A)`. A double centralizer is a pair of continuous linear maps `L R : A →L[𝕜] A` satisfying the intertwining condition `R x * y = x * L y`. There is a natural embedding `A → 𝓜(𝕜, A)` which sends `a : A` to the continuous linear maps `L R : A →L[𝕜] A` given by left and right…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Multiplier.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Unitization", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 2, "macro_tier_score": 0.0701, "macro_tier_override": null, "x": 65.967, "z": -184.207, "size": 0.312, "title": "The minimal unitization of a C⋆-algebra", "summary": "This file shows that when `E` is a C⋆-algebra (over a densely normed field `𝕜`), that the minimal `Unitization` is as well. In order to ensure that the norm structure is available, we must first show that every C⋆-algebra is a `RegularNormedAlgebra`. In addition, we show that in a `RegularNormedAlgebra` which is a `StarRing` for which the involution is isometric, that multiplication on the right is also an isometry…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Unitization.html"}, {"id": "Mathlib.Analysis.Analytic.ChangeOrigin", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 4, "macro_tier_score": 0.2658, "macro_tier_override": null, "x": 67.082, "z": -238.807, "size": 0.384, "title": "Changing origin in a power series", "summary": "If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that one. Indeed, one can write $$ f (x + y + z) = \\sum_{n} p_n (y + z)^n = \\sum_{n, k} \\binom{n}{k} p_n y^{n-k} z^k = \\sum_{k} \\Bigl(\\sum_{n} \\binom{n}{k} p_n y^{n-k}\\Bigr) z^k. $$ The corresponding power series has thus a `k`-th coefficient equal to $\\sum_{n} \\binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/ChangeOrigin.html"}, {"id": "Mathlib.Analysis.Normed.Ring.InfiniteSum", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 4, "macro_tier_score": 0.4448, "macro_tier_override": null, "x": 73.143, "z": -215.594, "size": 0.2754, "title": "Multiplying two infinite sums in a normed ring", "summary": "In this file, we prove various results about `(∑' x : ι, f x) * (∑' y : ι', g y)` in a normed ring. There are similar results proven in `Mathlib/Topology/Algebra/InfiniteSum/Ring.lean` (e.g. `tsum_mul_tsum`), but in a normed ring we get summability results which aren't true in general. We first establish results about arbitrary index types, `ι` and `ι'`, and then we specialize to `ι = ι' = ℕ` to prove the Cauchy…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Ring/InfiniteSum.html"}, {"id": "Mathlib.Analysis.Normed.Ring.Lemmas", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 4, "macro_tier_score": 0.4887, "macro_tier_override": null, "x": 83.653, "z": -216.207, "size": 0.4161, "title": "Normed rings", "summary": "In this file we continue building the theory of (semi)normed rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Ring/Lemmas.html"}, {"id": "Mathlib.Analysis.Complex.RiemannMapping", "region_id": "analysis", "micro_elevation": 0.7917, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 137.797, "z": -212.53, "size": 0.2, "title": "Riemann mapping theorem", "summary": "This file contains partial results towards Riemann Mapping Theorem. A complete proof is available at https://github.com/leanprover-community/mathlib4/pull/33505, though it may fail to compile with the latest Mathlib. It is being brought up to Mathlib code standards and merged in a series of smaller PRs. For now, all lemmas in this file are strictly weaker than the final theorem, so they're private.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/RiemannMapping.html"}, {"id": "Mathlib.Analysis.Complex.BranchLogRoot", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 101.87, "z": -261.476, "size": 0.2676, "title": "Branches of logarithm and `n`th root on simply connected domains", "summary": "In this file we prove that for a function `g : X → ℂ` defined on a locally path connected space that is continuous on an open simply connected set `U` and `0 ∉ g '' U`, there exist continuous branches of `log (g z)` and `ⁿ√(g z)` on `U`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/BranchLogRoot.html"}, {"id": "Mathlib.Analysis.Convex.DoublyStochasticMatrix", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.239, "title": "Doubly stochastic matrices", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/DoublyStochasticMatrix.html"}, {"id": "Mathlib.Analysis.Normed.Algebra.Ultra", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 72.46, "z": -220.994, "size": 0.2413, "title": "Normed algebra preserves ultrametricity", "summary": "This file contains the proof that a normed division ring over an ultrametric field is ultrametric.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Algebra/Ultra.html"}, {"id": "Mathlib.Analysis.Normed.Field.Ultra", "region_id": "analysis", "micro_elevation": 0.1458, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 80.621, "z": -220.177, "size": 0.2548, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Field/Ultra.html"}, {"id": "Mathlib.Analysis.BoundedVariation", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 97.59, "z": -241.576, "size": 0.2543, "title": "Almost everywhere differentiability of functions with locally bounded variation", "summary": "In this file we show that a bounded variation function is differentiable almost everywhere. This implies that Lipschitz functions from the real line into finite-dimensional vector spaces are also differentiable almost everywhere.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoundedVariation.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Equiv", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 4, "macro_tier_score": 0.2795, "macro_tier_override": null, "x": 50.304, "z": -197.592, "size": 0.3773, "title": "The derivative of a linear equivalence", "summary": "For detailed documentation of the Fréchet derivative, see the module docstring of `Mathlib/Analysis/Calculus/FDeriv/Basic.lean`. This file contains the usual formulas (and existence assertions) for the derivative of continuous linear equivalences. We also prove the usual formula for the derivative of the inverse function, assuming it exists. The inverse function theorem is in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Equiv.html"}, {"id": "Mathlib.Analysis.Normed.Field.Basic", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 4, "macro_tier_score": 0.4914, "macro_tier_override": null, "x": 83.932, "z": -216.046, "size": 0.5081, "title": "Normed division rings and fields", "summary": "In this file we define normed fields, and (more generally) normed division rings. We also prove some theorems about these definitions. Some useful results that relate the topology of the normed field to the discrete topology include: * `norm_eq_one_iff_ne_zero_of_discrete` Methods for constructing a normed field instance from a given real absolute value on a field are given in: * AbsoluteValue.toNormedField", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Field/Basic.html"}, {"id": "Mathlib.Analysis.Normed.Ring.Basic", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 4, "macro_tier_score": 0.489, "macro_tier_override": null, "x": 85.87, "z": -211.111, "size": 0.4265, "title": "Normed rings", "summary": "In this file we define (semi)normed rings. We also prove some theorems about these definitions. A normed ring instance can be constructed from a given real absolute value on a ring via `AbsoluteValue.toNormedRing`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Ring/Basic.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.InverseDeriv", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 117.033, "z": -247.878, "size": 0.241, "title": "derivatives of the inverse trigonometric functions", "summary": "Derivatives of `arcsin` and `arccos`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 4, "macro_tier_score": 0.4045, "macro_tier_override": null, "x": 79.719, "z": -232.381, "size": 0.3739, "title": "Inverse trigonometric functions.", "summary": "See also `Analysis.SpecialFunctions.Trigonometric.Arctan` for the inverse tan function. (This is delayed as it is easier to set up after developing complex trigonometric functions.) Basic inequalities on trigonometric functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 3, "macro_tier_score": 0.2234, "macro_tier_override": null, "x": 121.918, "z": -179.127, "size": 0.3456, "title": "Differentiability of trigonometric functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.html"}, {"id": "Mathlib.Analysis.Complex.JensenFormula", "region_id": "analysis", "micro_elevation": 0.8958, "macro_tier": 1, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": 48.513, "z": -152.08, "size": 0.3125, "title": "Jensen's Formula of Complex Analysis", "summary": "If a function `g : ℂ → ℂ` is analytic without zero on the closed ball with center `c` and radius `R`, then `log ‖g ·‖` is harmonic, and the mean value theorem of harmonic functions asserts that the circle average `circleAverage (log ‖g ·‖) c R` equals `log ‖g c‖`. Note that `g c` equals `meromorphicTrailingCoeffAt g c` and see `AnalyticOnNhd.circleAverage_log_norm_of_ne_zero` for the precise statement. Jensen's…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/JensenFormula.html"}, {"id": "Mathlib.Analysis.Complex.Harmonic.Poisson", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0143, "macro_tier_override": null, "x": 134.197, "z": -217.825, "size": 0.2873, "title": "Poisson Integral Formula", "summary": "This file establishes several versions of the **Poisson Integral Formula** for harmonic functions on arbitrary disks in the complex plane, formulated with the real part of the Herglotz–Riesz kernel of integration and with the Poisson kernel, respectively. TODO: Extend this formula to vector-valued harmonic functions", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Harmonic/Poisson.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage", "region_id": "analysis", "micro_elevation": 0.875, "macro_tier": 2, "macro_tier_score": 0.0284, "macro_tier_override": null, "x": 48.367, "z": -153.905, "size": 0.3083, "title": "Representation of `log⁺` as a Circle Average", "summary": "If `a` is any complex number, `circleAverage_log_norm_sub_const_eq_posLog` represents `log⁺ a` as the circle average of `log ‖· - a‖` over the unit circle.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Integrals/PosLogEqCircleAverage.html"}, {"id": "Mathlib.Analysis.Convex.Between", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 2, "macro_tier_score": 0.0295, "macro_tier_override": null, "x": 82.324, "z": -205.606, "size": 0.3786, "title": "Betweenness in affine spaces", "summary": "This file defines notions of a point in an affine space being between two given points.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Between.html"}, {"id": "Mathlib.Analysis.Normed.Affine.Isometry", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 3, "macro_tier_score": 0.251, "macro_tier_override": null, "x": 68.92, "z": -201.434, "size": 0.3367, "title": "Affine isometries", "summary": "In this file we define `AffineIsometry 𝕜 P P₂` to be an affine isometric embedding of normed add-torsors `P` into `P₂` over normed `𝕜`-spaces and `AffineIsometryEquiv` to be an affine isometric equivalence between `P` and `P₂`. We also prove basic lemmas and provide convenience constructors. The choice of these lemmas and constructors is closely modelled on those for the `LinearIsometry` and `AffineMap` theories.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Affine/Isometry.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Partial", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 36.12, "z": -222.363, "size": 0.2478, "title": "Partial derivatives", "summary": "Results in this file relate the partial derivatives of a bivariate function to its differentiability in the product space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Partial.html"}, {"id": "Mathlib.Analysis.Fourier.FourierTransformDeriv", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 2, "macro_tier_score": 0.042, "macro_tier_override": null, "x": 127.032, "z": -184.443, "size": 0.282, "title": "Derivatives of the Fourier transform", "summary": "In this file we compute the Fréchet derivative of the Fourier transform of `f`, where `f` is a function such that both `f` and `v ↦ ‖v‖ * ‖f v‖` are integrable. Here the Fourier transform is understood as an operator `(V → E) → (W → E)`, where `V` and `W` are normed `ℝ`-vector spaces and the Fourier transform is taken with respect to a continuous `ℝ`-bilinear pairing `L : V × W → ℝ` and a given reference measure…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/FourierTransformDeriv.html"}, {"id": "Mathlib.Analysis.Fourier.AddCircle", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 2, "macro_tier_score": 0.056, "macro_tier_override": null, "x": 34.1, "z": -185.5, "size": 0.2901, "title": "Fourier analysis on the additive circle", "summary": "This file contains basic results on Fourier series for functions on the additive circle `AddCircle T = ℝ / ℤ • T`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/AddCircle.html"}, {"id": "Mathlib.Analysis.Fourier.FourierTransform", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 2, "macro_tier_score": 0.0563, "macro_tier_override": null, "x": 58.155, "z": -198.687, "size": 0.3141, "title": "The Fourier transform", "summary": "We set up the Fourier transform for complex-valued functions on finite-dimensional spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/FourierTransform.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.Bounds", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 2, "macro_tier_score": 0.0418, "macro_tier_override": null, "x": 66.827, "z": -254.888, "size": 0.2565, "title": "Bounds on higher derivatives", "summary": "`norm_iteratedFDeriv_comp_le` gives the bound `n! * C * D ^ n` for the `n`-th derivative of `g ∘ f` assuming that the derivatives of `g` are bounded by `C` and the `i`-th derivative of `f` is bounded by `D ^ i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/Bounds.html"}, {"id": "Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 2, "macro_tier_score": 0.042, "macro_tier_override": null, "x": 37.72, "z": -192.139, "size": 0.2775, "title": "Integration by parts for line derivatives", "summary": "Let `f, g : E → ℝ` be two differentiable functions on a real vector space endowed with a Haar measure. Then `∫ f * g' = - ∫ f' * g`, where `f'` and `g'` denote the derivatives of `f` and `g` in a given direction `v`, provided that `f * g`, `f' * g` and `f * g'` are all integrable. In this file, we prove this theorem as well as more general versions where the multiplication is replaced by a general continuous…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/LineDeriv/IntegrationByParts.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity", "region_id": "analysis", "micro_elevation": 0.5625, "macro_tier": 2, "macro_tier_score": 0.0701, "macro_tier_override": null, "x": 73.851, "z": -168.859, "size": 0.3104, "title": "Continuity of the continuous functional calculus in each variable", "summary": "The continuous functional calculus is a map which takes a pair `a : A` (`A` is a C⋆-algebra) and a function `f : C(spectrum R a, R)` where `a` satisfies some predicate `p`, depending on `R` and returns another element of the algebra `A`. This is the map `cfcHom`. The class `ContinuousFunctionalCalculus` declares that `cfcHom` is a continuous map from `C(spectrum R a, R)` to `A`. However, users generally interact…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Continuity.html"}, {"id": "Mathlib.Analysis.Normed.Module.MStructure", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 84.545, "z": -209.24, "size": 0.2, "title": "M-structure", "summary": "A projection P on a normed space X is said to be an L-projection (`IsLprojection`) if, for all `x` in `X`, $\\|x\\| = \\|P x\\| + \\|(1 - P) x\\|$. A projection P on a normed space X is said to be an M-projection if, for all `x` in `X`, $\\|x\\| = max(\\|P x\\|,\\|(1 - P) x\\|)$. The L-projections on `X` form a Boolean algebra (`IsLprojection.Subtype.BooleanAlgebra`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/MStructure.html"}, {"id": "Mathlib.Analysis.Complex.UpperHalfPlane.Exp", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 60.549, "z": -164.824, "size": 0.259, "title": "Exp on the upper half plane", "summary": "This file contains lemmas about the exponential function on the upper half plane. Useful for q-expansions of modular forms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/UpperHalfPlane/Exp.html"}, {"id": "Mathlib.Analysis.Complex.Periodic", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 2, "macro_tier_score": 0.028, "macro_tier_override": null, "x": 38.928, "z": -232.871, "size": 0.2612, "title": "Periodic holomorphic functions", "summary": "We show that if `f : ℂ → ℂ` satisfies `f (z + h) = f z`, for some nonzero real `h`, then there is a function `F` such that `f z = F (exp (2 * π * I * z / h))` for all `z`; and if `f` is holomorphic at some `z`, then `F` is holomorphic at `exp (2 * π * I * z / h)`. We also show (using Riemann's removable singularity theorem) that if `f` is holomorphic and bounded for all sufficiently large `im z`, then `F` extends to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Periodic.html"}, {"id": "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 2, "macro_tier_score": 0.0289, "macro_tier_override": null, "x": 61.704, "z": -209.703, "size": 0.3404, "title": "The upper half plane", "summary": "This file defines `UpperHalfPlane` to be the upper half plane in `ℂ`. We define the notation `ℍ` for the upper half plane available in the locale `UpperHalfPlane` so as not to conflict with the quaternions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/UpperHalfPlane/Basic.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Polynomial", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 3, "macro_tier_score": 0.112, "macro_tier_override": null, "x": 38.895, "z": -189.521, "size": 0.3261, "title": "Derivatives of polynomials", "summary": "In this file we prove that derivatives of polynomials in the analysis sense agree with their derivatives in the algebraic sense. For a more detailed overview of one-dimensional derivatives in mathlib, see the module docstring of `Mathlib/Analysis/Calculus/Deriv/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Polynomial.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 3, "macro_tier_score": 0.2387, "macro_tier_override": null, "x": 91.314, "z": -160.552, "size": 0.4149, "title": "Complex and real exponential", "summary": "In this file we prove that `Complex.exp` and `Real.exp` are analytic functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ExpDeriv.html"}, {"id": "Mathlib.Analysis.Convex.Jensen", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 4, "macro_tier_score": 0.4176, "macro_tier_override": null, "x": 78.333, "z": -216.968, "size": 0.3304, "title": "Jensen's inequality and maximum principle for convex functions", "summary": "In this file, we prove the finite Jensen inequality and the finite maximum principle for convex functions. The integral versions are to be found in `Analysis.Convex.Integral`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Jensen.html"}, {"id": "Mathlib.Analysis.Convex.Function", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 4, "macro_tier_score": 0.4469, "macro_tier_override": null, "x": 82.602, "z": -213.278, "size": 0.41, "title": "Convex and concave functions", "summary": "This file defines convex and concave functions in vector spaces and proves the finite Jensen inequality. The integral version can be found in `Analysis.Convex.Integral`. A function `f : E → β` is `ConvexOn` a set `s` if `s` is itself a convex set, and for any two points `x y ∈ s`, the segment joining `(x, f x)` to `(y, f y)` is above the graph of `f`. Equivalently, `ConvexOn 𝕜 f s` means that the epigraph `{p : E ×…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Function.html"}, {"id": "Mathlib.Analysis.Complex.ExponentialBounds", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 38.229, "z": -176.373, "size": 0.2651, "title": "Bounds on specific values of the exponential", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/ExponentialBounds.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.0707, "macro_tier_override": null, "x": 66.89, "z": -196.775, "size": 0.3498, "title": "Uniqueness of the continuous functional calculus", "summary": "Let `s : Set 𝕜` be compact where `𝕜` is either `ℝ` or `ℂ`. By the Stone-Weierstrass theorem, the (star) subalgebra generated by polynomial functions on `s` is dense in `C(s, 𝕜)`. Moreover, this star subalgebra is generated by `X : 𝕜[X]` (i.e., `ContinuousMap.restrict s (.id 𝕜)`) alone. Consequently, any continuous star `𝕜`-algebra homomorphism with domain `C(s, 𝕜)`, is uniquely determined by its value on `X : 𝕜[X]`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unique.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 3, "macro_tier_score": 0.0707, "macro_tier_override": null, "x": 79.357, "z": -208.143, "size": 0.3493, "title": "The continuous functional calculus for non-unital algebras", "summary": "This file defines a generic API for the *continuous functional calculus* in *non-unital* algebras which is suitable in a wide range of settings. The design is intended to match as closely as possible that for unital algebras in `Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unital.lean`. Changes to either file should be mirrored in its counterpart whenever possible. The underlying reasons for the design…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.html"}, {"id": "Mathlib.Analysis.Matrix.LDL", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 23.875, "z": -214.917, "size": 0.2, "title": "LDL decomposition", "summary": "This file proves the LDL-decomposition of matrices: Any positive definite matrix `S` can be decomposed as `S = LDLᴴ` where `L` is a lower-triangular matrix and `D` is a diagonal matrix.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Matrix/LDL.html"}, {"id": "Mathlib.Analysis.Matrix.PosDef", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.042, "macro_tier_override": null, "x": 49.689, "z": -255.248, "size": 0.2831, "title": "Spectrum of positive (semi)definite matrices", "summary": "This file proves that eigenvalues of positive (semi)definite matrices are (nonnegative) positive.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Matrix/PosDef.html"}, {"id": "Mathlib.Analysis.Distribution.FourierSchwartz", "region_id": "analysis", "micro_elevation": 0.875, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 142.296, "z": -224.064, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/FourierSchwartz.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Bernstein", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 81.081, "z": -229.306, "size": 0.2672, "title": "Bernstein approximations and Weierstrass' theorem", "summary": "We prove that the Bernstein approximations ``` ∑ k : Fin (n+1), (n.choose k * x^k * (1-x)^(n-k)) • f (k/n : ℝ) ``` for a continuous function `f : C([0,1], E)` taking values in a locally convex vector space converge uniformly to `f` as `n` tends to infinity. This statement directly applies to the cases when the codomain is a (semi)normed space or, more generally, has a topology defined by a family of seminorms. Our…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Bernstein.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Affine", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 3, "macro_tier_score": 0.2508, "macro_tier_override": null, "x": 107.048, "z": -192.471, "size": 0.3175, "title": "The derivative of continuous affine maps", "summary": "For detailed documentation of the Fréchet derivative, see the module docstring of `Mathlib/Analysis/Calculus/FDeriv/Basic.lean`. This file contains the usual formulas (and existence assertions) for the derivative of continuous affine maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Affine.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Add", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 4, "macro_tier_score": 0.2813, "macro_tier_override": null, "x": 49.707, "z": -212.94, "size": 0.4516, "title": "Additive operations on derivatives", "summary": "For detailed documentation of the Fréchet derivative, see the module docstring of `Mathlib/Analysis/Calculus/FDeriv/Basic.lean`. This file contains the usual formulas (and existence assertions) for the derivative of * sum of finitely many functions * multiplication of a function by a scalar constant * negative of a function * subtraction of two functions", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Add.html"}, {"id": "Mathlib.Analysis.Normed.Group.AddTorsor", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 4, "macro_tier_score": 0.3502, "macro_tier_override": null, "x": 85.866, "z": -211.124, "size": 0.4315, "title": "Torsors of additive normed group actions.", "summary": "This file defines torsors of additive normed group actions, with a metric space structure. The motivating case is Euclidean affine spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/AddTorsor.html"}, {"id": "Mathlib.Analysis.RCLike.Lemmas", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 3, "macro_tier_score": 0.21, "macro_tier_override": null, "x": 63.965, "z": -183.614, "size": 0.3725, "title": "Further lemmas about `RCLike`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/RCLike/Lemmas.html"}, {"id": "Mathlib.Analysis.Normed.Module.FiniteDimension", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 4, "macro_tier_score": 0.2544, "macro_tier_override": null, "x": 104.362, "z": -193.926, "size": 0.48, "title": "Finite-dimensional normed spaces over complete fields", "summary": "Over a complete nontrivially normed field, in finite dimension, all norms are equivalent and all linear maps are continuous. Moreover, a finite-dimensional subspace is always complete and closed.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/FiniteDimension.html"}, {"id": "Mathlib.Analysis.Convex.LinearIsometry", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 91.947, "z": -223.356, "size": 0.2, "title": "(Strict) convexity and linear isometries", "summary": "In this file we prove some basic lemmas about (strict) convexity and linear isometries.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/LinearIsometry.html"}, {"id": "Mathlib.Analysis.Convex.ContinuousLinearEquiv", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 85.711, "z": -211.615, "size": 0.239, "title": "(Pre)images of strict convex sets under continuous linear equivalences", "summary": "In this file we prove that the (pre)image of a strict convex set under a continuous linear equivalence is a strict convex set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/ContinuousLinearEquiv.html"}, {"id": "Mathlib.Analysis.Convex.StrictConvexSpace", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 2, "macro_tier_score": 0.0567, "macro_tier_override": null, "x": 88.478, "z": -223.977, "size": 0.3465, "title": "Strictly convex spaces", "summary": "This file defines strictly convex spaces. A normed space is strictly convex if all closed balls are strictly convex. This does **not** mean that the norm is strictly convex (in fact, it never is).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/StrictConvexSpace.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Add", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 4, "macro_tier_score": 0.2664, "macro_tier_override": null, "x": 111.468, "z": -215.21, "size": 0.4112, "title": "One-dimensional derivatives of sums etc", "summary": "In this file we prove formulas about derivatives of `f + g`, `-f`, `f - g`, and `∑ i, f i x` for functions from the base field to a normed space over this field. For a more detailed overview of one-dimensional derivatives in mathlib, see the module docstring of `Analysis/Calculus/Deriv/Basic`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Add.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Basic", "region_id": "analysis", "micro_elevation": 0.1458, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 80.482, "z": -198.879, "size": 0.2611, "title": "Multiple angle formulas in terms of Chebyshev polynomials", "summary": "This file gives the trigonometric characterizations of Chebyshev polynomials, for the real (`Real.cos`) and complex (`Complex.cos`) cosine and the real (`Real.cosh`) and complex (`Complex.cosh`) hyperbolic cosine.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev/Basic.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Continuous", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 3, "macro_tier_score": 0.2098, "macro_tier_override": null, "x": 107.096, "z": -199.395, "size": 0.3644, "title": "Continuity of inner product", "summary": "We show that the inner product is continuous, `continuous_inner`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Continuous.html"}, {"id": "Mathlib.Analysis.Normed.Operator.BoundedLinearMaps", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 4, "macro_tier_score": 0.3363, "macro_tier_override": null, "x": 60.575, "z": -190.161, "size": 0.4314, "title": "Bounded linear maps", "summary": "This file defines a class stating that a map between normed vector spaces is (bi)linear and continuous. Instead of asking for continuity, the definition takes the equivalent condition (because the space is normed) that `‖f x‖` is bounded by a multiple of `‖x‖`. Hence the \"bounded\" in the name refers to `‖f x‖/‖x‖` rather than `‖f x‖` itself.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 69.819, "z": -154.089, "size": 0.248, "title": "Order properties of `Ring.inverse` in C⋆-algebras", "summary": "This file shows that `Ring.inverse` is convex on strictly positive operators.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow/RingInverseOrder.html"}, {"id": "Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 2, "macro_tier_score": 0.042, "macro_tier_override": null, "x": 69.332, "z": -236.445, "size": 0.279, "title": "Spectral theory of compact operators", "summary": "This file develops the spectral theory of compact operators on Banach spaces. The main result is the Fredholm alternative for compact operators.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Compact/FredholmAlternative.html"}, {"id": "Mathlib.Analysis.Normed.Module.RieszLemma", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 3, "macro_tier_score": 0.2505, "macro_tier_override": null, "x": 86.986, "z": -236.044, "size": 0.2976, "title": "Applications of the Hausdorff distance in normed spaces", "summary": "Riesz's lemma, stated for a normed space over a normed field: for any closed proper subspace `F` of `E`, there is a nonzero `x` such that `‖x - F‖` is at least `r * ‖x‖` for any `r < 1`. This is `riesz_lemma`. In a nontrivially normed field (with an element `c` of norm `> 1`) and any `R > ‖c‖`, one can guarantee `‖x‖ ≤ R` and `‖x - y‖ ≥ 1` for any `y` in `F`. This is `riesz_lemma_of_norm_lt`. For a normed space over…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/RieszLemma.html"}, {"id": "Mathlib.Analysis.Normed.Operator.Banach", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 4, "macro_tier_score": 0.3067, "macro_tier_override": null, "x": 82.357, "z": -183.742, "size": 0.3418, "title": "Banach open mapping theorem", "summary": "This file contains the Banach open mapping theorem, i.e., the fact that a bijective bounded linear map between Banach spaces has a bounded inverse.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Banach.html"}, {"id": "Mathlib.Analysis.Normed.Operator.Compact.Basic", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0422, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2959, "title": "Compact operators", "summary": "In this file we define compact linear operators between two topological vector spaces (TVS).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Compact/Basic.html"}, {"id": "Mathlib.Analysis.Convex.Piecewise", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 73.959, "z": -210.444, "size": 0.239, "title": "Convex and concave piecewise functions", "summary": "This file proves convex and concave theorems for piecewise functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Piecewise.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Mul", "region_id": "analysis", "micro_elevation": 0.5833, "macro_tier": 3, "macro_tier_score": 0.2524, "macro_tier_override": null, "x": 37.336, "z": -208.729, "size": 0.4058, "title": "Derivative of `f x * g x`", "summary": "In this file we prove formulas for `(f x * g x)'` and `(f x • g x)'`. For a more detailed overview of one-dimensional derivatives in mathlib, see the module docstring of `Mathlib/Analysis/Calculus/Deriv/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Mul.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Pow", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 4, "macro_tier_score": 0.2655, "macro_tier_override": null, "x": 123.562, "z": -202.269, "size": 0.3727, "title": "Derivative of `(f x) ^ n`, `n : ℕ`", "summary": "In this file we prove that the Fréchet derivative of `fun x => f x ^ n`, where `n` is a natural number, is `n * f x ^ (n - 1) * f'`. Additionally, we prove the case for non-commutative rings (with primed names like `deriv_pow'`), where the result is instead `∑ i ∈ Finset.range n, f x ^ (n.pred - i) * f' * f x ^ i`. For a more detailed overview of one-dimensional derivatives in mathlib, see the module docstring of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Pow.html"}, {"id": "Mathlib.Analysis.Normed.Group.HomCompletion", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 82.189, "z": -200.661, "size": 0.239, "title": "Completion of normed group homs", "summary": "Given two (semi) normed groups `G` and `H` and a normed group hom `f : NormedAddGroupHom G H`, we build and study a normed group hom `f.completion : NormedAddGroupHom (completion G) (completion H)` such that the diagram ``` f G -----------> H | | | | | | V V completion G -----------> completion H f.completion ``` commutes. The map itself comes from the general theory of completion of uniform spaces, but here we want…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/HomCompletion.html"}, {"id": "Mathlib.Analysis.Normed.Group.Completion", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 4, "macro_tier_score": 0.2786, "macro_tier_override": null, "x": 75.76, "z": -203.193, "size": 0.3225, "title": "Completion of a normed group", "summary": "In this file we prove that the completion of a (semi)normed group is a normed group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Completion.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Gamma.Deligne", "region_id": "analysis", "micro_elevation": 0.9375, "macro_tier": 0, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": 123.391, "z": -262.593, "size": 0.3141, "title": "Deligne's archimedean Gamma-factors", "summary": "In the theory of L-series one frequently encounters the following functions (of a complex variable `s`) introduced in Deligne's landmark paper *Valeurs de fonctions L et périodes d'intégrales*: $$ \\Gamma_{\\mathbb{R}}(s) = \\pi ^ {-s / 2} \\Gamma (s / 2) $$ and $$ \\Gamma_{\\mathbb{C}}(s) = 2 (2 \\pi) ^ {-s} \\Gamma (s). $$ These are the factors that need to be included in the Dedekind zeta function of a number field for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "region_id": "analysis", "micro_elevation": 0.9167, "macro_tier": 2, "macro_tier_score": 0.0291, "macro_tier_override": null, "x": 107.595, "z": -148.452, "size": 0.3528, "title": "The Beta function, and further properties of the Gamma function", "summary": "In this file we define the Beta integral, relate Beta and Gamma functions, and prove some refined properties of the Gamma function using these relations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Gamma/Beta.html"}, {"id": "Mathlib.Analysis.Normed.Unbundled.SeminormFromConst", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 0, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 101.328, "z": -185.701, "size": 0.2274, "title": "SeminormFromConst", "summary": "In this file, we prove [BGR, Proposition 1.3.2/2][bosch-guntzer-remmert] : starting from a power-multiplicative seminorm on a commutative ring `R` and a nonzero `c : R`, we create a new power-multiplicative seminorm for which `c` is multiplicative.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Unbundled/SeminormFromConst.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.AffineMap", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 3, "macro_tier_score": 0.2368, "macro_tier_override": null, "x": 99.921, "z": -236.472, "size": 0.31, "title": "Derivatives of affine maps", "summary": "In this file we prove formulas for one-dimensional derivatives of affine maps `f : 𝕜 →ᵃ[𝕜] E`. We also specialise some of these results to `AffineMap.lineMap` because it is useful to transfer MVT from dimension 1 to a domain in higher dimension.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/AffineMap.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Comp", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 3, "macro_tier_score": 0.2527, "macro_tier_override": null, "x": 92.007, "z": -235.862, "size": 0.4218, "title": "One-dimensional derivatives of compositions of functions", "summary": "In this file we prove the chain rule for the following cases: * `HasDerivAt.comp` etc: `f : 𝕜' → 𝕜'` composed with `g : 𝕜 → 𝕜'`; * `HasDerivAt.scomp` etc: `f : 𝕜' → E` composed with `g : 𝕜 → 𝕜'`; * `HasFDerivAt.comp_hasDerivAt` etc: `f : E → F` composed with `g : 𝕜 → E`; Here `𝕜` is the base normed field, `E` and `F` are normed spaces over `𝕜` and `𝕜'` is an algebra over `𝕜` (e.g., `𝕜'=𝕜` or `𝕜=ℝ`, `𝕜'=ℂ`). We also…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Comp.html"}, {"id": "Mathlib.Analysis.Polynomial.CauchyBound", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 70.848, "z": -209.799, "size": 0.2, "title": "Cauchy's bound on polynomial roots.", "summary": "The bound is given by `Polynomial.cauchyBound`, which for `a_n x^n + a_(n-1) x^(n - 1) + ⋯ + a_0` is `1 + max_(0 ≤ i < n) a_i / a_n`. The theorem that this gives a bound to polynomial roots is `Polynomial.IsRoot.norm_lt_cauchyBound`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Polynomial/CauchyBound.html"}, {"id": "Mathlib.Analysis.Normed.Module.Span", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 4, "macro_tier_score": 0.391, "macro_tier_override": null, "x": 90.766, "z": -218.002, "size": 0.3938, "title": "The span of a single vector", "summary": "The equivalence of `𝕜` and `𝕜 • x` for `x ≠ 0` are defined as continuous linear equivalence and isometry.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Span.html"}, {"id": "Mathlib.Analysis.Normed.Operator.ContinuousLinearMap", "region_id": "analysis", "micro_elevation": 0.1458, "macro_tier": 4, "macro_tier_score": 0.4747, "macro_tier_override": null, "x": 77.346, "z": -199.199, "size": 0.4097, "title": "Constructions of continuous linear maps between (semi-)normed spaces", "summary": "A fundamental fact about (semi-)linear maps between normed spaces over sensible fields is that continuity and boundedness are equivalent conditions. That is, for normed spaces `E`, `F`, a `LinearMap` `f : E →ₛₗ[σ] F` is the coercion of some `ContinuousLinearMap` `f' : E →SL[σ] F`, if and only if there exists a bound `C` such that for all `x`, `‖f x‖ ≤ C * ‖x‖`. We prove one direction in this file:…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/ContinuousLinearMap.html"}, {"id": "Mathlib.Analysis.Normed.Unbundled.SpectralNorm", "region_id": "analysis", "micro_elevation": 0.5417, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 66.861, "z": -246.903, "size": 0.2413, "title": "The spectral norm and the norm extension theorem", "summary": "This file shows that if `K` is a nonarchimedean normed field and `L/K` is an algebraic extension, then there is a natural extension of the norm on `K` to a `K`-algebra norm on `L`, the so-called *spectral norm*. The spectral norm of an element of `L` only depends on its minimal polynomial over `K`, so for `K ⊆ L ⊆ M` two extensions of `K`, the spectral norm on `M` restricts to the spectral norm on `L`. This work can…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Unbundled/SpectralNorm.html"}, {"id": "Mathlib.Analysis.Normed.Unbundled.InvariantExtension", "region_id": "analysis", "micro_elevation": 0.5208, "macro_tier": 0, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 51.126, "z": -184.678, "size": 0.2274, "title": "algNormOfAlgEquiv and invariantExtension", "summary": "Let `K` be a nonarchimedean normed field and `L/K` be a finite algebraic extension. In the comments, `‖ ⬝ ‖` denotes any power-multiplicative `K`-algebra norm on `L` extending the norm on `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Unbundled/InvariantExtension.html"}, {"id": "Mathlib.Analysis.Normed.Unbundled.IsPowMulFaithful", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 0, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 45.005, "z": -207.482, "size": 0.2274, "title": "Equivalent power-multiplicative norms", "summary": "In this file, we prove [BGR, Proposition 3.1.5/1][bosch-guntzer-remmert]: if `R` is a normed commutative ring and `f₁` and `f₂` are two power-multiplicative `R`-algebra norms on `S`, then if `f₁` and `f₂` are equivalent on every subring `R[y]` for `y : S`, it follows that `f₁ = f₂`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Unbundled/IsPowMulFaithful.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Projection.Reflection", "region_id": "analysis", "micro_elevation": 0.5417, "macro_tier": 3, "macro_tier_score": 0.2092, "macro_tier_override": null, "x": 118.473, "z": -200.18, "size": 0.3251, "title": "Reflection", "summary": "A linear isometry equivalence `K.reflection : E ≃ₗᵢ[𝕜] E` in constructed, by choosing for each `u : E`, `K.reflection u = 2 • K.starProjection u - u`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Projection/Reflection.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Projection.Basic", "region_id": "analysis", "micro_elevation": 0.5208, "macro_tier": 3, "macro_tier_score": 0.2105, "macro_tier_override": null, "x": 43.277, "z": -219.685, "size": 0.3994, "title": "The orthogonal projection", "summary": "Given a nonempty subspace `K` of an inner product space `E` such that `K` admits an orthogonal projection, this file constructs `K.orthogonalProjectionOnto : E →L[𝕜] K`, the orthogonal projection of `E` onto `K`. This map satisfies: for any point `u` in `E`, the point `v = K.orthogonalProjectionOnto u` in `K` minimizes the distance `‖u - v‖` to `u`. This file also defines `K.starProjection : E →L[𝕜] E` which is the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Projection/Basic.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Coalgebra", "region_id": "analysis", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 109.151, "z": -156.028, "size": 0.2, "title": "Finite-dimensional inner product space with a (co)algebra structure", "summary": "This file proves that a finite-dimensional inner product space has a coalgebra structure if it has an algebra structure, where the comultiplication and counit maps are given by taking adjoints of the multiplication and algebra linear maps, respectively. This is implemented by providing a linear equivalence between the inner product space and an algebra. And similarly, a finite-dimensional inner product space has an…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Coalgebra.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.TensorProduct", "region_id": "analysis", "micro_elevation": 0.8125, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 34.316, "z": -247.536, "size": 0.2478, "title": "Inner product space structure on tensor product spaces", "summary": "This file provides the inner product space structure on tensor product spaces. We define the inner product on `E ⊗ F` by `⟪a ⊗ₜ b, c ⊗ₜ d⟫ = ⟪a, c⟫ * ⟪b, d⟫`, when `E` and `F` are inner product spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/TensorProduct.html"}, {"id": "Mathlib.Analysis.Normed.Module.DoubleDual", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 94.377, "z": -241.478, "size": 0.2, "title": "The double dual of a normed space", "summary": "In this file we define the inclusion of a normed space into its double strong dual, prove that it is an isometry (for `𝕜 = ℝ` or `𝕜 = ℂ`), and use the corresponding weak-topology embedding together with Banach–Alaoglu to transfer compactness from the weak-star bidual back to the weak topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/DoubleDual.html"}, {"id": "Mathlib.Analysis.LocallyConvex.WeakSpace", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 96.982, "z": -224.805, "size": 0.2478, "title": "Closures of convex sets in locally convex spaces", "summary": "This file contains the standard result that if `E` is a vector space with two locally convex topologies, then the closure of a convex set is the same in either topology, provided they have the same collection of continuous linear functionals. In particular, the weak closure of a convex set in a locally convex space coincides with the closure in the original topology. Of course, we phrase this in terms of linear maps…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/WeakSpace.html"}, {"id": "Mathlib.Analysis.Complex.RemovableSingularity", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 2, "macro_tier_score": 0.0425, "macro_tier_override": null, "x": 115.234, "z": -180.442, "size": 0.3214, "title": "Removable singularity theorem", "summary": "In this file we prove Riemann's removable singularity theorem: if `f : ℂ → E` is complex differentiable in a punctured neighborhood of a point `c` and is bounded in a punctured neighborhood of `c` (or, more generally, $f(z) - f(c)=o((z-c)^{-1})$), then it has a limit at `c` and the function `update f c (limUnder (𝓝[≠] c) f)` is complex differentiable in a neighborhood of `c`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/RemovableSingularity.html"}, {"id": "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FiniteDimensional", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 107.719, "z": -196.919, "size": 0.2, "title": "A lemma about `ApproximatesLinearOn` that needs `FiniteDimensional`", "summary": "In this file we prove that in a real vector space, a function `f` that approximates a linear equivalence on a subset `s` can be extended to a homeomorphism of the whole space. This used to be the only lemma in `Mathlib/Analysis/Calculus/Inverse` depending on `FiniteDimensional`, so it was moved to a new file when the original file got split.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/InverseFunctionTheorem/FiniteDimensional.html"}, {"id": "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 3, "macro_tier_score": 0.2368, "macro_tier_override": null, "x": 104.778, "z": -221.244, "size": 0.3074, "title": "Non-linear maps close to affine maps", "summary": "In this file we study a map `f` such that `‖f x - f y - f' (x - y)‖ ≤ c * ‖x - y‖` on an open set `s`, where `f' : E →L[𝕜] F` is a continuous linear map and `c` is suitably small. Maps of this type behave like `f a + f' (x - a)` near each `a ∈ s`. When `f'` is onto, we show that `f` is locally onto. When `f'` is a continuous linear equiv, we show that `f` is a homeomorphism between `s` and `f '' s`. More precisely,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.html"}, {"id": "Mathlib.Analysis.Normed.Group.Tannery", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 69.977, "z": -222.973, "size": 0.2682, "title": "Tannery's theorem", "summary": "Tannery's theorem gives a sufficient criterion for the limit of an infinite sum (with respect to an auxiliary parameter) to equal the sum of the pointwise limits. See https://en.wikipedia.org/wiki/Tannery%27s_theorem. It is a special case of the dominated convergence theorem (with the measure chosen to be the counting measure); but we give here a direct proof, in order to avoid some unnecessary hypotheses that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Tannery.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Linear", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 3, "macro_tier_score": 0.2367, "macro_tier_override": null, "x": 51.833, "z": -221.186, "size": 0.3066, "title": "Derivatives of continuous linear maps from the base field", "summary": "In this file we prove that `f : 𝕜 →L[𝕜] E` (or `f : 𝕜 →ₗ[𝕜] E`) has derivative `f 1`. For a more detailed overview of one-dimensional derivatives in mathlib, see the module docstring of `Analysis/Calculus/Deriv/Basic`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Linear.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Linear", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 4, "macro_tier_score": 0.2938, "macro_tier_override": null, "x": 108.817, "z": -206.983, "size": 0.3944, "title": "The derivative of bounded linear maps", "summary": "For detailed documentation of the Fréchet derivative, see the module docstring of `Mathlib/Analysis/Calculus/FDeriv/Basic.lean`. This file contains the usual formulas (and existence assertions) for the derivative of bounded linear maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Linear.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Pow", "region_id": "analysis", "micro_elevation": 0.5833, "macro_tier": 4, "macro_tier_score": 0.2655, "macro_tier_override": null, "x": 122.157, "z": -215.935, "size": 0.3724, "title": "Fréchet Derivative of `f x ^ n`, `n : ℕ`", "summary": "In this file we prove that the Fréchet derivative of `fun x => f x ^ n`, where `n` is a natural number, is `n • f x ^ (n - 1)) • f'`. Additionally, we prove the case for non-commutative rings (with primed names like `fderiv_pow'`), where the result is instead `∑ i ∈ Finset.range n, f x ^ (n.pred - i) •> f' <• f x ^ i`. For detailed documentation of the Fréchet derivative, see the module docstring of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Pow.html"}, {"id": "Mathlib.Analysis.Real.OfDigits", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 81.29, "z": -224.71, "size": 0.257, "title": "Representation of reals in positional system", "summary": "This file defines `Real.ofDigits` and `Real.digits` functions which allows to work with the representations of reals as sequences of digits in positional system.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Real/OfDigits.html"}, {"id": "Mathlib.Analysis.Normed.Group.FunctionSeries", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 2, "macro_tier_score": 0.0429, "macro_tier_override": null, "x": 71.331, "z": -212.478, "size": 0.3498, "title": "Continuity of series of functions", "summary": "We show that series of functions are continuous when each individual function in the series is and additionally suitable uniform summable bounds are satisfied, in `continuous_tsum`. For smoothness of series of functions, see the file `Mathlib/Analysis/Calculus/SmoothSeries.lean`. TODO: update this to use `SummableUniformlyOn`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/FunctionSeries.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral", "region_id": "analysis", "micro_elevation": 0.875, "macro_tier": 2, "macro_tier_score": 0.0701, "macro_tier_override": null, "x": 96.592, "z": -147.741, "size": 0.307, "title": "Gaussian integral", "summary": "We prove various versions of the formula for the Gaussian integral: * `integral_gaussian`: for real `b` we have `∫ x:ℝ, exp (-b * x^2) = √(π / b)`. * `integral_gaussian_complex`: for complex `b` with `0 < re b` we have `∫ x:ℝ, exp (-b * x^2) = (π / b) ^ (1 / 2)`. * `integral_gaussian_Ioi` and `integral_gaussian_complex_Ioi`: variants for integrals over `Ioi 0`. * `Complex.Gamma_one_half_eq`: the formula `Γ (1 / 2) =…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Symmetric", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 2, "macro_tier_score": 0.0565, "macro_tier_override": null, "x": 94.974, "z": -255.919, "size": 0.3274, "title": "Symmetry of the second derivative", "summary": "We show that, over the reals, the second derivative is symmetric. The most precise result is `Convex.second_derivative_within_at_symmetric`. It asserts that, if a function is differentiable inside a convex set `s` with nonempty interior, and has a second derivative within `s` at a point `x`, then this second derivative at `x` is symmetric. Note that this result does not require continuity of the first derivative.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Symmetric.html"}, {"id": "Mathlib.Analysis.Convex.Continuous", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 69.126, "z": -241.239, "size": 0.2674, "title": "Convex functions are continuous", "summary": "This file proves that a convex function from a finite-dimensional real normed space to `ℝ` is continuous.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Continuous.html"}, {"id": "Mathlib.Analysis.Normed.Affine.Convex", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 1, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": 54.409, "z": -228.751, "size": 0.3102, "title": "Simplices in normed affine spaces", "summary": "We prove the following facts: * `exists_mem_interior_convexHull_affineBasis` : We can intercalate a simplex between a point and one of its neighborhoods. * `Convex.exists_subset_interior_convexHull_finset_of_isCompact`: We can intercalate a convex polytope between a compact convex set and one of its neighborhoods.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Affine/Convex.html"}, {"id": "Mathlib.Analysis.Calculus.VectorField", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 1, "macro_tier_score": 0.0143, "macro_tier_override": null, "x": 81.581, "z": -259.779, "size": 0.2886, "title": "Vector fields in vector spaces", "summary": "We study functions of the form `V : E → E` on a vector space, thinking of these as vector fields. We define several notions in this context, with the aim to generalize them to vector fields on manifolds. Notably, we define the pullback of a vector field under a map, as `VectorField.pullback 𝕜 f V x := (fderiv 𝕜 f x).inverse (V (f x))` (together with the same notion within a set). We also define the Lie bracket of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/VectorField.html"}, {"id": "Mathlib.Analysis.Normed.Unbundled.FiniteExtension", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 50.47, "z": -187.952, "size": 0.2416, "title": "Basis.norm", "summary": "In this file, we prove [BGR, Lemma 3.2.1./3][bosch-guntzer-remmert] : if `K` is a normed field with a nonarchimedean power-multiplicative norm and `L/K` is a finite extension, then there exists at least one power-multiplicative `K`-algebra norm on `L` extending the norm on `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Unbundled/FiniteExtension.html"}, {"id": "Mathlib.Analysis.Normed.Unbundled.AlgebraNorm", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 88.695, "z": -240.315, "size": 0.2641, "title": "Algebra norms", "summary": "We define algebra norms and multiplicative algebra norms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Unbundled/AlgebraNorm.html"}, {"id": "Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 62.873, "z": -182.501, "size": 0.2388, "title": "seminormFromBounded", "summary": "In this file, we prove [BGR, Proposition 1.2.1/2][bosch-guntzer-remmert] : given a nonzero additive group seminorm on a commutative ring `R` such that for some `c : ℝ` and every `x y : R`, the inequality `f (x * y) ≤ c * f x * f y)` is satisfied, we create a ring seminorm on `R`. In the file comments, we will use the expression `f is multiplicatively bounded` to indicate that this condition holds.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Unbundled/SeminormFromBounded.html"}, {"id": "Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 76.517, "z": -244.398, "size": 0.2388, "title": "smoothingSeminorm", "summary": "In this file, we prove [BGR, Proposition 1.3.2/1][bosch-guntzer-remmert]: if `μ` is a nonarchimedean seminorm on a commutative ring `R`, then `iInf (fun (n : PNat), (μ(x ^ (n : ℕ))) ^ (1 / (n : ℝ)))` is a power-multiplicative nonarchimedean seminorm on `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Unbundled/SmoothingSeminorm.html"}, {"id": "Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 83.055, "z": -233.712, "size": 0.2561, "title": "Definition of BoundedContinuousFunction.char", "summary": "Definition and basic properties of `BoundedContinuousFunction.char he hL w := fun v ↦ e (L v w)`, where `e` is a continuous additive character and `L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ` is a continuous bilinear map. In the special case `e = Circle.exp`, this is used to define the characteristic function of a measure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/BoundedContinuousFunctionChar.html"}, {"id": "Mathlib.Analysis.Complex.Harmonic.MeanValue", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 2, "macro_tier_score": 0.0422, "macro_tier_override": null, "x": 69.621, "z": -157.229, "size": 0.2954, "title": "The Mean Value Property of Harmonic Functions on the Complex Plane", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Harmonic/MeanValue.html"}, {"id": "Mathlib.Analysis.Complex.Harmonic.Analytic", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 2, "macro_tier_score": 0.042, "macro_tier_override": null, "x": 36.295, "z": -237.354, "size": 0.2748, "title": "Analyticity of Harmonic Functions", "summary": "If `f : ℂ → ℝ` is harmonic at `x`, we show that `∂f/∂1 - I • ∂f/∂I` is complex-analytic at `x`. If `f` is harmonic on an open ball, then it is the real part of a function `F : ℂ → ℂ` that is holomorphic on the ball. This implies in particular that harmonic functions are real-analytic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Harmonic/Analytic.html"}, {"id": "Mathlib.Analysis.Complex.MeanValue", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 2, "macro_tier_score": 0.0424, "macro_tier_override": null, "x": 102.419, "z": -169.715, "size": 0.3152, "title": "The Mean Value Property of Complex Differentiable Functions", "summary": "This file established the classic mean value properties of complex differentiable functions, computing the value of a function at the center of a circle as a circle average. It also provides generalized versions that computing the value of a function at arbitrary points of a disk as circle averages over suitable weighted functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/MeanValue.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Harmonic.HarmonicContOnCl", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 2, "macro_tier_score": 0.0419, "macro_tier_override": null, "x": 90.341, "z": -260.282, "size": 0.2638, "title": "Functions Harmonic on a Domain and Continuous on Its Closure", "summary": "Many theorems in harmonic analysis assume that a function is harmonic on a domain and is continuous on its closure. In this file we define a predicate `HarmonicContOnCl` that expresses this property and prove basic facts about this predicate.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Harmonic/HarmonicContOnCl.html"}, {"id": "Mathlib.Analysis.Convex.Contractible", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.0976, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2784, "title": "A convex set is contractible", "summary": "In this file we prove that a (star) convex set in a real topological vector space is a contractible topological space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Contractible.html"}, {"id": "Mathlib.Analysis.Complex.UnitDisc.Basic", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 56.364, "z": -203.514, "size": 0.253, "title": "Poincaré disc", "summary": "In this file we define `Complex.UnitDisc` to be the unit disc in the complex plane. We also introduce some basic operations on this disc.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/UnitDisc/Basic.html"}, {"id": "Mathlib.Analysis.Complex.Circle", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 3, "macro_tier_score": 0.0979, "macro_tier_override": null, "x": 90.684, "z": -189.34, "size": 0.3131, "title": "The circle", "summary": "This file defines `Circle` to be the metric sphere (`Metric.sphere`) in `ℂ` centred at `0` of radius `1`. We equip it with the following structure: * a submonoid of `ℂ` * a group * a topological group We furthermore define `Circle.exp` to be the natural map `fun t ↦ exp (t * I)` from `ℝ` to `Circle`, and show that this map is a group homomorphism. We define two additive characters onto the circle: *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Circle.html"}, {"id": "Mathlib.Analysis.Normed.Module.Ball.Action", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 77.247, "z": -196.097, "size": 0.2528, "title": "Multiplicative actions of/on balls and spheres", "summary": "Let `E` be a normed vector space over a normed field `𝕜`. In this file we define the following multiplicative actions. - The closed unit ball in `𝕜` acts on open balls and closed balls centered at `0` in `E`. - The unit sphere in `𝕜` acts on open balls, closed balls, and spheres centered at `0` in `E`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Ball/Action.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Integrability.Basic", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 3, "macro_tier_score": 0.1677, "macro_tier_override": null, "x": 29.666, "z": -184.129, "size": 0.3394, "title": "Integrability of Special Functions", "summary": "This file establishes basic facts about the interval integrability of special functions, including powers and the logarithm.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Integrability/Basic.html"}, {"id": "Mathlib.Analysis.Normed.Order.UpperLower", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 88.238, "z": -213.459, "size": 0.2478, "title": "Upper/lower/order-connected sets in normed groups", "summary": "The topological closure and interior of an upper/lower/order-connected set is an upper/lower/order-connected set (with the notable exception of the closure of an order-connected set). We also prove lemmas specific to `ℝⁿ`. Those are helpful to prove that order-connected sets in `ℝⁿ` are measurable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Order/UpperLower.html"}, {"id": "Mathlib.Analysis.Normed.Group.Pointwise", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 4, "macro_tier_score": 0.4596, "macro_tier_override": null, "x": 72.461, "z": -210.723, "size": 0.3514, "title": "Properties of pointwise addition of sets in normed groups", "summary": "We explore the relationships between pointwise addition of sets in normed groups, and the norm. Notably, we show that the sum of bounded sets remain bounded.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Pointwise.html"}, {"id": "Mathlib.Analysis.Asymptotics.Defs", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 4, "macro_tier_score": 0.4463, "macro_tier_override": null, "x": 78.414, "z": -200.527, "size": 0.3831, "title": "Asymptotics", "summary": "We introduce these relations: * `IsBigOWith c l f g` : \"f is big O of g along l with constant c\"; * `f =O[l] g` : \"f is big O of g along l\"; * `f =Θ[l] g` : \"f is big O of g along l and vice versa\"; * `f =o[l] g` : \"f is little o of g along l\"; * `f ~[l] g` : `f` and `g` are equivalent, i.e., `f - g =o[l] g`. Here `l` is any filter on the domain of `f` and `g`, which are assumed to be the same. The codomains of `f`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Asymptotics/Defs.html"}, {"id": "Mathlib.Analysis.Normed.Group.Bounded", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 4, "macro_tier_score": 0.4875, "macro_tier_override": null, "x": 85.907, "z": -210.964, "size": 0.3574, "title": "Boundedness in normed groups", "summary": "This file rephrases metric boundedness in terms of norms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Bounded.html"}, {"id": "Mathlib.Analysis.Normed.MulAction", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 4, "macro_tier_score": 0.4885, "macro_tier_override": null, "x": 86.952, "z": -215.448, "size": 0.4071, "title": "Lemmas for `IsBoundedSMul` over normed additive groups", "summary": "Lemmas which hold only in `NormedSpace α β` are provided in another file. Notably we prove that `NonUnitalSeminormedRing`s have bounded actions by left- and right- multiplication. This allows downstream files to write general results about `IsBoundedSMul`, and then deduce `const_mul` and `mul_const` results as an immediate corollary.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/MulAction.html"}, {"id": "Mathlib.Analysis.Normed.Operator.Compact", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 81.508, "z": -209.535, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Compact.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Calculus", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 3, "macro_tier_score": 0.1124, "macro_tier_override": null, "x": 46.6, "z": -174.007, "size": 0.3511, "title": "Calculus in inner product spaces", "summary": "In this file we prove that the inner product and square of the norm in an inner space are infinitely `ℝ`-smooth. In order to state these results, we need a `NormedSpace ℝ E` instance. Though we can deduce this structure from `InnerProductSpace 𝕜 E`, this instance may be not definitionally equal to some other “natural” instance. So, we assume `[NormedSpace ℝ E]`. We also prove that functions to a `EuclideanSpace` are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Calculus.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Inv", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 3, "macro_tier_score": 0.2369, "macro_tier_override": null, "x": 82.252, "z": -165.411, "size": 0.317, "title": "Derivatives of `x ↦ x⁻¹` and `f x / g x`", "summary": "In this file we prove `(x⁻¹)' = -1 / x ^ 2`, `((f x)⁻¹)' = -f' x / (f x) ^ 2`, and `(f x / g x)' = (f' x * g x - f x * g' x) / (g x) ^ 2` for different notions of derivative. For a more detailed overview of one-dimensional derivatives in mathlib, see the module docstring of `Analysis/Calculus/Deriv/Basic`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Inv.html"}, {"id": "Mathlib.Analysis.Calculus.Rademacher", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 122.292, "z": -230.517, "size": 0.2, "title": "Rademacher's theorem: a Lipschitz function is differentiable almost everywhere", "summary": "This file proves Rademacher's theorem: a Lipschitz function between finite-dimensional real vector spaces is differentiable almost everywhere with respect to the Lebesgue measure. This is the content of `LipschitzWith.ae_differentiableAt`. Versions for functions which are Lipschitz on sets are also given (see `LipschitzOnWith.ae_differentiableWithinAt`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Rademacher.html"}, {"id": "Mathlib.Analysis.Calculus.LineDeriv.Measurable", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 36.015, "z": -222.0, "size": 0.2338, "title": "Measurability of the line derivative", "summary": "We prove in `measurable_lineDeriv` that the line derivative of a function (with respect to a locally compact scalar field) is measurable, provided the function is continuous. In `measurable_lineDeriv_uncurry`, assuming additionally that the source space is second countable, we show that `(x, v) ↦ lineDeriv 𝕜 f x v` is also measurable. An assumption such as continuity is necessary, as otherwise one could alternate in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/LineDeriv/Measurable.html"}, {"id": "Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.042, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2742, "title": "Functions which vanish as distributions vanish as functions", "summary": "In a finite-dimensional normed real vector space endowed with a Borel measure, consider a locally integrable function whose integral against all compactly supported smooth functions vanishes. Then the function is almost everywhere zero. This is proved in `ae_eq_zero_of_integral_contDiff_smul_eq_zero`. A version for two functions having the same integral when multiplied by smooth compactly supported functions is also…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Range", "region_id": "analysis", "micro_elevation": 0.5417, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 98.291, "z": -174.407, "size": 0.2, "title": "Range of the continuous functional calculus", "summary": "This file contains results about the range of the continuous functional calculus, and consequences thereof.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Range.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances", "region_id": "analysis", "micro_elevation": 0.5208, "macro_tier": 3, "macro_tier_score": 0.0717, "macro_tier_override": null, "x": 76.011, "z": -171.654, "size": 0.3998, "title": "Instances of the continuous functional calculus", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Instances.html"}, {"id": "Mathlib.Analysis.Normed.Order.Lattice", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 4, "macro_tier_score": 0.4457, "macro_tier_override": null, "x": 87.01, "z": -215.379, "size": 0.3499, "title": "Normed lattice ordered groups", "summary": "Motivated by the theory of Banach Lattices, we then define `NormedLatticeAddCommGroup` as a lattice with a covariant normed group addition satisfying the solid axiom.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Order/Lattice.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Stirling", "region_id": "analysis", "micro_elevation": 0.8542, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 138.015, "z": -232.635, "size": 0.2, "title": "Stirling's formula", "summary": "This file proves Stirling's formula for the factorial. It states that $n!$ grows asymptotically like $\\sqrt{2\\pi n}(\\frac{n}{e})^n$. Also some _global_ bounds on the factorial function and the Stirling sequence are proved.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Stirling.html"}, {"id": "Mathlib.Analysis.Real.Pi.Wallis", "region_id": "analysis", "micro_elevation": 0.8333, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 140.042, "z": -199.217, "size": 0.2478, "title": "The Wallis formula for Pi", "summary": "This file establishes the Wallis product for `π` (`Real.tendsto_prod_pi_div_two`). Our proof is largely about analyzing the behaviour of the sequence `∫ x in 0..π, sin x ^ n` as `n → ∞`. See: https://en.wikipedia.org/wiki/Wallis_product The proof can be broken down into two pieces. The first step (carried out in `Mathlib/Analysis/SpecialFunctions/Integrals/Basic.lean`) is to use repeated integration by parts to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Real/Pi/Wallis.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Harmonic.Basic", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 2, "macro_tier_score": 0.0563, "macro_tier_override": null, "x": 37.247, "z": -236.006, "size": 0.3134, "title": "Harmonic Functions", "summary": "This file defines harmonic functions on real, finite-dimensional, inner product spaces `E`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Harmonic/Basic.html"}, {"id": "Mathlib.Analysis.Normed.Field.Dense", "region_id": "analysis", "micro_elevation": 0.5833, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 112.687, "z": -236.918, "size": 0.2302, "title": "Transfer algebraic properties from dense subfields", "summary": "In this file, we prove that algebraically closedness of a complete normed field of characteristic zero can be inherited from its dense subfields. Let `K` be a dense subfield of a complete normed field `L`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Field/Dense.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 4, "macro_tier_score": 0.2959, "macro_tier_override": null, "x": 111.599, "z": -204.636, "size": 0.4764, "title": "Limits and asymptotics of power functions at `+∞`", "summary": "This file contains results about the limiting behaviour of power functions at `+∞`. For convenience some results on asymptotics as `x → 0` (those which are not just continuity statements) are also located here.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Abs", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 54.894, "z": -165.967, "size": 0.2478, "title": "Derivative of the absolute value", "summary": "This file compiles basic derivability properties of the absolute value, and is largely inspired from `Mathlib/Analysis/InnerProductSpace/Calculus.lean`, which is the analogous file for norms derived from an inner product space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Abs.html"}, {"id": "Mathlib.Analysis.Analytic.Polynomial", "region_id": "analysis", "micro_elevation": 0.5208, "macro_tier": 2, "macro_tier_score": 0.0279, "macro_tier_override": null, "x": 82.001, "z": -171.5, "size": 0.2448, "title": "Polynomials are analytic", "summary": "This file combines the analysis and algebra libraries and shows that evaluation of a polynomial is an analytic function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/Polynomial.html"}, {"id": "Mathlib.Analysis.Analytic.Constructions", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 4, "macro_tier_score": 0.2667, "macro_tier_override": null, "x": 115.006, "z": -199.03, "size": 0.4249, "title": "Various ways to combine analytic functions", "summary": "We show that the following are analytic: 1. Cartesian products of analytic functions 2. Arithmetic on analytic functions: `mul`, `smul`, `inv`, `div` 3. Finite sums and products: `Finset.sum`, `Finset.prod`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/Constructions.html"}, {"id": "Mathlib.Analysis.Calculus.DiffContOnCl", "region_id": "analysis", "micro_elevation": 0.5833, "macro_tier": 3, "macro_tier_score": 0.1673, "macro_tier_override": null, "x": 116.308, "z": -187.168, "size": 0.3084, "title": "Functions differentiable on a domain and continuous on its closure", "summary": "Many theorems in complex analysis assume that a function is complex differentiable on a domain and is continuous on its closure. In this file we define a predicate `DiffContOnCl` that expresses this property and prove basic facts about this predicate.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/DiffContOnCl.html"}, {"id": "Mathlib.Analysis.Calculus.LineDeriv.Basic", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 2, "macro_tier_score": 0.0566, "macro_tier_override": null, "x": 118.53, "z": -231.12, "size": 0.3341, "title": "Line derivatives", "summary": "We define the line derivative of a function `f : E → F`, at a point `x : E` along a vector `v : E`, as the element `f' : F` such that `f (x + t • v) = f x + t • f' + o (t)` as `t` tends to `0` in the scalar field `𝕜`, if it exists. It is denoted by `lineDeriv 𝕜 f x v`. This notion is generally less well behaved than the full Fréchet derivative (for instance, the composition of functions which are line-differentiable…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/LineDeriv/Basic.html"}, {"id": "Mathlib.Analysis.BoxIntegral.Partition.Basic", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 1, "macro_tier_score": 0.0149, "macro_tier_override": null, "x": 79.67, "z": -211.022, "size": 0.3358, "title": "Partitions of rectangular boxes in `ℝⁿ`", "summary": "In this file we define (pre)partitions of rectangular boxes in `ℝⁿ`. A partition of a box `I` in `ℝⁿ` (see `BoxIntegral.Prepartition` and `BoxIntegral.Prepartition.IsPartition`) is a finite set of pairwise disjoint boxes such that their union is exactly `I`. We use `boxes : Finset (Box ι)` to store the set of boxes. Many lemmas about box integrals deal with pairwise disjoint collections of subboxes, so we define a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoxIntegral/Partition/Basic.html"}, {"id": "Mathlib.Analysis.BoxIntegral.Box.Basic", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0152, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.3549, "title": "Rectangular boxes in `ℝⁿ`", "summary": "In this file we define rectangular boxes in `ℝⁿ`. As usual, we represent `ℝⁿ` as the type of functions `ι → ℝ` (usually `ι = Fin n` for some `n`). When we need to interpret a box `[l, u]` as a set, we use the product `{x | ∀ i, l i < x i ∧ x i ≤ u i}` of half-open intervals `(l i, u i]`. We exclude `l i` because this way boxes of a partition are disjoint as sets in `ℝⁿ`. Currently, the only use cases for these…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoxIntegral/Box/Basic.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.PolarCoord", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 2, "macro_tier_score": 0.0699, "macro_tier_override": null, "x": 130.367, "z": -192.081, "size": 0.2925, "title": "Polar coordinates", "summary": "We define polar coordinates, as an open partial homeomorphism in `ℝ^2` between `ℝ^2 - (-∞, 0]` and `(0, +∞) × (-π, π)`. Its inverse is given by `(r, θ) ↦ (r cos θ, r sin θ)`. It satisfies the following change of variables formula (see `integral_comp_polarCoord_symm`): `∫ p in polarCoord.target, p.1 • f (polarCoord.symm p) = ∫ p, f p`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/PolarCoord.html"}, {"id": "Mathlib.Analysis.Normed.Affine.AsymptoticCone", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 58.924, "z": -187.514, "size": 0.2, "title": "Asymptotic cones in normed spaces", "summary": "In this file, we prove that the asymptotic cone of a set is non-trivial if and only if the set is unbounded.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Affine/AsymptoticCone.html"}, {"id": "Mathlib.Analysis.Real.Pi.Irrational", "region_id": "analysis", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 22.929, "z": -230.933, "size": 0.2, "title": "`Real.pi` is irrational", "summary": "The main result of this file is `irrational_pi`. The proof is adapted from https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational#Cartwright's_proof. The proof idea is as follows. * Define a sequence of integrals `I n θ = ∫ x in (-1)..1, (1 - x ^ 2) ^ n * cos (x * θ)`. * Give a recursion formula for `I (n + 2) θ * θ ^ 2` in terms of `I n θ` and `I (n + 1) θ`. Note we do not find it helpful to define `J` as in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Real/Pi/Irrational.html"}, {"id": "Mathlib.Analysis.Normed.Module.Shrink", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 80.892, "z": -194.325, "size": 0.247, "title": "Transfer normed algebraic structures from `α` to `Shrink α`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Shrink.html"}, {"id": "Mathlib.Analysis.Normed.Module.TransferInstance", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 1, "macro_tier_score": 0.0142, "macro_tier_override": null, "x": 83.971, "z": -196.413, "size": 0.2802, "title": "Transfer normed algebraic structures across `Equiv`s", "summary": "In this file, we transfer a (semi-)normed (additive) commutative group and normed space structures across an equivalence. This continues the pattern set in `Mathlib/Algebra/Module/TransferInstance.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/TransferInstance.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 2, "macro_tier_score": 0.0421, "macro_tier_override": null, "x": 108.172, "z": -251.159, "size": 0.2877, "title": "Euclidean distance on a finite-dimensional space", "summary": "When we define a smooth bump function on a normed space, it is useful to have a smooth distance on the space. Since the default distance is not guaranteed to be smooth, we define `toEuclidean` to be an equivalence between a finite-dimensional topological vector space and the standard Euclidean space of the same dimension. Then we define `Euclidean.dist x y = dist (toEuclidean x) (toEuclidean y)` and provide some…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/EuclideanDist.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Extend", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 3, "macro_tier_score": 0.2086, "macro_tier_override": null, "x": 41.521, "z": -234.216, "size": 0.2694, "title": "Extending differentiability to the boundary", "summary": "We investigate how differentiable functions inside a set extend to differentiable functions on the boundary. For this, it suffices that the function and its derivative admit limits there. A general version of this statement is given in `hasFDerivWithinAt_closure_of_tendsto_fderiv`. One-dimensional versions, in which one wants to obtain differentiability at the left endpoint or the right endpoint of an interval, are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Extend.html"}, {"id": "Mathlib.Analysis.RCLike.Extend", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 3, "macro_tier_score": 0.1254, "macro_tier_override": null, "x": 87.32, "z": -194.465, "size": 0.2871, "title": "Extending an `ℝ`-linear functional to a `𝕜`-linear functional", "summary": "In this file we provide a way to extend a (optionally, continuous) `ℝ`-linear map to a (continuous) `𝕜`-linear map in a way that bounds the norm by the norm of the original map, when `𝕜` is either `ℝ` (the extension is trivial) or `ℂ`. We formulate the extension uniformly, by assuming `RCLike 𝕜`. We motivate the form of the extension as follows. Note that `fc : F →ₗ[𝕜] 𝕜` is determined fully by `re fc`: for all `x :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/RCLike/Extend.html"}, {"id": "Mathlib.Analysis.Normed.Lp.WithLp", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.2229, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.3134, "title": "The `WithLp` type synonym", "summary": "`WithLp p V` is a copy of `V` with exactly the same vector space structure, but with the Lp norm instead of any existing norm on `V`; recall that by default `ι → R` and `R × R` are equipped with a norm defined as the supremum of the norms of their components. This file defines the vector space structure for all types `V`; the norm structure is built for different specializations of `V` in downstream files. Note that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Lp/WithLp.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Note", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2, "title": "Documentation concerning the continuous functional calculus", "summary": "A library note giving advice on developing and using the continuous functional calculus, as well as the organizational structure within Mathlib.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Note.html"}, {"id": "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 4, "macro_tier_score": 0.4037, "macro_tier_override": null, "x": 84.853, "z": -195.096, "size": 0.329, "title": "Balanced Core and Balanced Hull", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/BalancedCoreHull.html"}, {"id": "Mathlib.Analysis.LocallyConvex.Basic", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 4, "macro_tier_score": 0.4182, "macro_tier_override": null, "x": 70.199, "z": -219.136, "size": 0.3635, "title": "Local convexity", "summary": "This file defines absorbent and balanced sets. An absorbent set is one that \"surrounds\" the origin. The idea is made precise by requiring that any point belongs to all large enough scalings of the set. This is the vector world analog of a topological neighborhood of the origin. A balanced set is one that is everywhere around the origin. This means that `a • s ⊆ s` for all `a` of norm less than `1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/Basic.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Pi", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 0, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 113.234, "z": -213.746, "size": 0.2302, "title": "One-dimensional derivatives on pi-types.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Pi.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Pi", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 2, "macro_tier_score": 0.0279, "macro_tier_override": null, "x": 62.269, "z": -236.171, "size": 0.2516, "title": "Derivatives on pi-types.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Pi.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.ZPow", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 3, "macro_tier_score": 0.2371, "macro_tier_override": null, "x": 117.79, "z": -181.229, "size": 0.3321, "title": "Derivatives of `x ^ m`, `m : ℤ`", "summary": "In this file we prove theorems about (iterated) derivatives of `x ^ m`, `m : ℤ`. For a more detailed overview of one-dimensional derivatives in mathlib, see the module docstring of `Mathlib/Analysis/Calculus/Deriv/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/ZPow.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Shift", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 3, "macro_tier_score": 0.2369, "macro_tier_override": null, "x": 53.03, "z": -246.439, "size": 0.3184, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Shift.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 51.289, "z": -170.122, "size": 0.2478, "title": "Additive characters valued in the unit circle", "summary": "This file defines additive characters, valued in the unit circle, from either * the ring `ZMod N` for any non-zero natural `N`, * the additive circle `ℝ / T ⬝ ℤ`, for any real `T`. These results are separate from `Analysis.SpecialFunctions.Complex.Circle` in order to reduce the imports of that file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Complex/CircleAddChar.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 4, "macro_tier_score": 0.2949, "macro_tier_override": null, "x": 75.47, "z": -176.322, "size": 0.4429, "title": "Continuity of power functions", "summary": "This file contains lemmas about continuity of the power functions on `ℂ`, `ℝ`, `ℝ≥0`, and `ℝ≥0∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Pow/Continuity.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Prod", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 3, "macro_tier_score": 0.2086, "macro_tier_override": null, "x": 84.076, "z": -177.803, "size": 0.2734, "title": "Derivatives of functions taking values in product types", "summary": "In this file we prove lemmas about derivatives of functions `f : 𝕜 → E × F` and of functions `f : 𝕜 → (Π i, E i)`. For a more detailed overview of one-dimensional derivatives in mathlib, see the module docstring of `Mathlib/Analysis/Calculus/Deriv/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Prod.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 3, "macro_tier_score": 0.2242, "macro_tier_override": null, "x": 111.036, "z": -250.988, "size": 0.3879, "title": "Derivative and series expansion of real logarithm", "summary": "In this file we prove that `Real.log` is infinitely smooth at all nonzero `x : ℝ`. We also prove that the series `∑' n : ℕ, x ^ (n + 1) / (n + 1)` converges to `(-Real.log (1 - x))` for all `x : ℝ`, `|x| < 1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Log/Deriv.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 3, "macro_tier_score": 0.2091, "macro_tier_override": null, "x": 27.023, "z": -215.705, "size": 0.3184, "title": "Differentiability of the complex `log` function", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.html"}, {"id": "Mathlib.Analysis.Normed.Operator.Asymptotics", "region_id": "analysis", "micro_elevation": 0.2917, "macro_tier": 4, "macro_tier_score": 0.2935, "macro_tier_override": null, "x": 100.729, "z": -214.484, "size": 0.3814, "title": "Asymptotic statements about the operator norm", "summary": "This file contains lemmas about how operator norm on continuous linear maps interacts with `IsBigO`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Asymptotics.html"}, {"id": "Mathlib.Analysis.Normed.Operator.Basic", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 4, "macro_tier_score": 0.3974, "macro_tier_override": null, "x": 98.534, "z": -202.596, "size": 0.5909, "title": "Operator norm on the space of continuous linear maps", "summary": "Define the operator (semi)-norm on the space of continuous (semi)linear maps between (semi)-normed spaces, and prove its basic properties. In particular, show that this space is itself a semi-normed space. Since a lot of elementary properties don't require `‖x‖ = 0 → x = 0` we start setting up the theory for `SeminormedAddCommGroup`. Later we will specialize to `NormedAddCommGroup` in the file `NormedSpace.lean`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Basic.html"}, {"id": "Mathlib.Analysis.Convex.Side", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 77.72, "z": -226.136, "size": 0.2912, "title": "Sides of affine subspaces", "summary": "This file defines notions of two points being on the same or opposite sides of an affine subspace.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Side.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.lpSpace", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 105.408, "z": -235.807, "size": 0.2, "title": "`lp ∞ A` as a C⋆-algebra", "summary": "We place these here because, for reasons related to the import hierarchy, they should not be placed in earlier files.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/lpSpace.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Classes", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.0983, "macro_tier_override": null, "x": 63.097, "z": -216.533, "size": 0.3388, "title": "Classes of C⋆-algebras", "summary": "This file defines classes for complex C⋆-algebras. These are (unital or non-unital, commutative or noncommutative) Banach algebra over `ℂ` with an antimultiplicative conjugate-linear involution (`star`) satisfying the C⋆-identity `∥star x * x∥ = ∥x∥ ^ 2`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Classes.html"}, {"id": "Mathlib.Analysis.Normed.Lp.lpSpace", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 3, "macro_tier_score": 0.0978, "macro_tier_override": null, "x": 115.022, "z": -209.711, "size": 0.3043, "title": "ℓp space", "summary": "This file describes properties of elements `f` of a pi-type `∀ i, E i` with finite \"norm\", defined for `p : ℝ≥0∞` as the size of the support of `f` if `p=0`, `(∑' a, ‖f a‖^p) ^ (1/p)` for `0 < p < ∞` and `⨆ a, ‖f a‖` for `p=∞`. The Prop-valued `Memℓp f p` states that a function `f : ∀ i, E i` has finite norm according to the above definition; that is, `f` has finite support if `p = 0`, `Summable (fun a ↦ ‖f a‖^p)`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Lp/lpSpace.html"}, {"id": "Mathlib.Analysis.Convex.Integral", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 90.273, "z": -224.642, "size": 0.2523, "title": "Jensen's inequality for integrals", "summary": "In this file we prove several forms of Jensen's inequality for integrals. - for convex sets: `Convex.average_mem`, `Convex.set_average_mem`, `Convex.integral_mem`; - for convex functions: `ConvexOn.average_mem_epigraph`, `ConvexOn.map_average_le`, `ConvexOn.set_average_mem_epigraph`, `ConvexOn.map_set_average_le`, `ConvexOn.map_integral_le`; - for strictly convex sets:…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Integral.html"}, {"id": "Mathlib.Analysis.Complex.Arg", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 77.993, "z": -235.353, "size": 0.2546, "title": "Rays in the complex numbers", "summary": "This file links the definition `SameRay ℝ x y` with the equality of arguments of complex numbers, the usual way this is considered.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Arg.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Convex", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 2, "macro_tier_score": 0.0285, "macro_tier_override": null, "x": 90.625, "z": -189.309, "size": 0.3161, "title": "Convexity properties of inner product spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Convex.html"}, {"id": "Mathlib.Analysis.Complex.ValueDistribution.FirstMainTheorem", "region_id": "analysis", "micro_elevation": 0.9583, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 47.069, "z": -271.398, "size": 0.2478, "title": "The First Main Theorem of Value Distribution Theory", "summary": "The First Main Theorem of Value Distribution Theory is a two-part statement, establishing invariance of the characteristic function `characteristic f ⊤` under modifications of `f`. - If `f` is meromorphic on the complex plane, then the characteristic functions for the value `⊤` of the function `f` and `f⁻¹` agree up to a constant, see Proposition 2.1 on p. 168 of [Lang, *Introduction to Complex Hyperbolic…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/ValueDistribution/FirstMainTheorem.html"}, {"id": "Mathlib.Analysis.Complex.ValueDistribution.CharacteristicFunction", "region_id": "analysis", "micro_elevation": 0.9375, "macro_tier": 1, "macro_tier_score": 0.0142, "macro_tier_override": null, "x": 147.336, "z": -196.752, "size": 0.2806, "title": "The Characteristic Function of Value Distribution Theory", "summary": "This file defines the \"characteristic function\" attached to a meromorphic function defined on the complex plane. Also known as \"Nevanlinna Height\", this is one of the three main functions used in Value Distribution Theory. The characteristic function plays a role analogous to the height function in number theory: both measure the \"complexity\" of objects. For rational functions, the characteristic function grows like…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/ValueDistribution/CharacteristicFunction.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc", "region_id": "analysis", "micro_elevation": 0.7917, "macro_tier": 3, "macro_tier_score": 0.1676, "macro_tier_override": null, "x": 133.675, "z": -187.885, "size": 0.3276, "title": "Sinc function", "summary": "This file contains the definition of the sinc function and some of its properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Sinc.html"}, {"id": "Mathlib.Analysis.Analytic.OfScalars", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 4, "macro_tier_score": 0.2647, "macro_tier_override": null, "x": 97.785, "z": -234.259, "size": 0.3232, "title": "Scalar series", "summary": "This file contains API for analytic functions `∑ cᵢ • xⁱ` defined in terms of scalars `c₀, c₁, c₂, …`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/OfScalars.html"}, {"id": "Mathlib.Analysis.Normed.Field.UnitBall", "region_id": "analysis", "micro_elevation": 0.1667, "macro_tier": 3, "macro_tier_score": 0.1117, "macro_tier_override": null, "x": 84.277, "z": -198.125, "size": 0.3055, "title": "Algebraic structures on unit balls and spheres", "summary": "In this file we define algebraic structures (`Semigroup`, `CommSemigroup`, `Monoid`, `CommMonoid`, `Group`, `CommGroup`) on `Metric.ball (0 : 𝕜) 1`, `Metric.closedBall (0 : 𝕜) 1`, and `Metric.sphere (0 : 𝕜) 1`. In each case we use the weakest possible typeclass assumption on `𝕜`, from `NonUnitalSeminormedRing` to `NormedField`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Field/UnitBall.html"}, {"id": "Mathlib.Analysis.Normed.Ring.Units", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 4, "macro_tier_score": 0.3351, "macro_tier_override": null, "x": 65.167, "z": -205.993, "size": 0.3804, "title": "The group of units of a complete normed ring", "summary": "This file contains the basic theory for the group of units (invertible elements) of a complete normed ring (Banach algebras being a notable special case).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Ring/Units.html"}, {"id": "Mathlib.Analysis.Convex.StrictCombination", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 87.312, "z": -226.28, "size": 0.239, "title": "Convex combinations in strictly convex sets and spaces.", "summary": "This file proves lemmas about convex combinations of points in strictly convex sets and strictly convex spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/StrictCombination.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Sigmoid", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 81.196, "z": -261.314, "size": 0.239, "title": "Sigmoid function", "summary": "In this file we define the sigmoid function `x : ℝ ↦ (1 + exp (-x))⁻¹` and prove some of its analytic properties. We then show that the sigmoid function can be seen as an order embedding from `ℝ` to `I = [0, 1]` and that this embedding is both a topological embedding and a measurable embedding. We also prove that the composition of this embedding with the measurable embedding from a standard Borel space `α` to `ℝ`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Sigmoid.html"}, {"id": "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 4, "macro_tier_score": 0.3924, "macro_tier_override": null, "x": 87.697, "z": -196.394, "size": 0.4508, "title": "Asymptotic equivalence", "summary": "In this file, we prove properties of the relation `IsEquivalent l u v`, which means that `u-v` is little o of `v` along the filter `l`. Unlike `Is(Little|Big)O` relations, this one requires `u` and `v` to have the same codomain `β`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Extreme", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 42.612, "z": -167.335, "size": 0.2, "title": "Extreme points of the closed unit ball in C⋆-algebras", "summary": "This file contains results on the extreme points of the closed unit ball in (unital) C⋆-algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Extreme.html"}, {"id": "Mathlib.Analysis.Convex.Extreme", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 2, "macro_tier_score": 0.0285, "macro_tier_override": null, "x": 85.632, "z": -211.821, "size": 0.3156, "title": "Extreme sets", "summary": "This file defines extreme sets and extreme points for sets in a module. An extreme set of `A` is a subset of `A` that is as far as it can get in any outward direction: If point `x` is in it and point `y ∈ A`, then the line passing through `x` and `y` leaves `A` at `x`. This is an analytic notion of \"being on the side of\". It is weaker than being exposed (see `IsExposed.isExtreme`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Extreme.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation", "region_id": "analysis", "micro_elevation": 0.8542, "macro_tier": 1, "macro_tier_score": 0.0142, "macro_tier_override": null, "x": 101.516, "z": -268.163, "size": 0.2743, "title": "Integral representations of `rpow`", "summary": "This file contains an integral representation of the `rpow` function between 0 and 1: we show that there exists a measure on ℝ such that `x ^ p = ∫ t, rpowIntegrand₀₁ p t x ∂μ` for the integrand `rpowIntegrand₀₁ p t x := t ^ p * (t⁻¹ - (t + x)⁻¹)`. This representation is useful for showing that `rpow` is operator monotone and operator concave in this range; that is, `cfc rpow` is monotone/concave. The integrand can…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow/IntegralRepresentation.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral", "region_id": "analysis", "micro_elevation": 0.5208, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 60.88, "z": -242.479, "size": 0.248, "title": "Integrals and the continuous functional calculus", "summary": "This file gives results about integrals of the form `∫ x, cfc (f x) a`. Most notably, we show that the integral commutes with the continuous functional calculus under appropriate conditions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Integral.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ApproximateUnit", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 88.641, "z": -153.834, "size": 0.248, "title": "Nonnegative contractions in a C⋆-algebra form an approximate unit", "summary": "This file shows that the collection of positive contractions (of norm strictly less than one) in a possibly non-unital C⋆-algebra form a directed set. The key step uses the continuous functional calculus applied with the functions `fun x : ℝ≥0, 1 - (1 + x)⁻¹` and `fun x : ℝ≥0, x * (1 - x)⁻¹`, which are inverses on the interval `{x : ℝ≥0 | x < 1}`. In addition, this file defines `IsIncreasingApproximateUnit` to be a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ApproximateUnit.html"}, {"id": "Mathlib.Analysis.Analytic.Binomial", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 0, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 93.631, "z": -264.221, "size": 0.2239, "title": "Binomial Series", "summary": "This file introduces the binomial series: $$ \\sum_{k=0}^{\\infty} \\; \\binom{a}{k} \\; x^k = 1 + a x + \\frac{a(a-1)}{2!} x^2 + \\frac{a(a-1)(a-2)}{3!} x^3 + \\cdots $$ where $a$ is an element of a normed field $\\mathbb{K}$, and $x$ is an element of a normed algebra over $\\mathbb{K}$.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/Binomial.html"}, {"id": "Mathlib.Analysis.Calculus.IteratedDeriv.ConvergenceOnBall", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.0278, "macro_tier_override": null, "x": 31.324, "z": -206.586, "size": 0.229, "title": "Taylor series converges to function on whole ball", "summary": "In this file we prove that if a function `f` is analytic on the ball of convergence of its Taylor series, then the series converges to `f` on this ball.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/IteratedDeriv/ConvergenceOnBall.html"}, {"id": "Mathlib.Analysis.Complex.OperatorNorm", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 1, "macro_tier_score": 0.0278, "macro_tier_override": null, "x": 94.082, "z": -187.807, "size": 0.229, "title": "The basic continuous linear maps associated to `ℂ`", "summary": "The continuous linear maps `Complex.reCLM` (real part), `Complex.imCLM` (imaginary part), `Complex.conjCLE` (conjugation), and `Complex.ofRealCLM` (inclusion of `ℝ`) were introduced in `Analysis.Complex.Basic`. This file contains a few calculations requiring more imports: the operator norm and (for `Complex.conjCLE`) the determinant.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/OperatorNorm.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Complex.Analytic", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.07, "macro_tier_override": null, "x": 77.381, "z": -154.75, "size": 0.2968, "title": "Various complex special functions are analytic", "summary": "`log`, and `cpow` are analytic, since they are differentiable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Complex/Analytic.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 1, "macro_tier_score": 0.0278, "macro_tier_override": null, "x": 85.856, "z": -178.083, "size": 0.229, "title": "Ordinary hypergeometric function in a Banach algebra", "summary": "In this file, we define `ordinaryHypergeometric`, the _ordinary_ or _Gaussian_ hypergeometric function in a topological algebra `𝔸` over a field `𝕂` given by: $$ _2\\mathrm{F}_1(a\\ b\\ c : \\mathbb{K}, x : \\mathbb{A}) = \\sum_{n=0}^{\\infty}\\frac{(a)_n(b)_n}{(c)_n} \\frac{x^n}{n!} \\,, $$ with $(a)_n$ is the ascending Pochhammer symbol (see `ascPochhammer`). This file contains the basic definitions over a general field `𝕂`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/OrdinaryHypergeometric.html"}, {"id": "Mathlib.Analysis.Normed.Operator.CompleteCodomain", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 0, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": 54.789, "z": -215.526, "size": 0.3781, "title": "Completeness of spaces of linear and multilinear maps", "summary": "If `E` is a nontrivial normed space over a nontrivially normed field `𝕜`, and `E` has a separating dual, then for any normed space `F`, the completeness of the space of continuous linear maps from `E` to `F` is equivalent to the completeness of `F`. A similar statement holds for spaces of continuous multilinear maps", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/CompleteCodomain.html"}, {"id": "Mathlib.Analysis.LocallyConvex.SeparatingDual", "region_id": "analysis", "micro_elevation": 0.2917, "macro_tier": 3, "macro_tier_score": 0.1128, "macro_tier_override": null, "x": 96.327, "z": -195.828, "size": 0.3756, "title": "Spaces with separating dual", "summary": "We introduce a typeclass `SeparatingDual R V`, registering that the points of the topological module `V` over `R` can be separated by continuous linear forms. This property is satisfied for normed spaces over `ℝ` or `ℂ` (by the analytic Hahn-Banach theorem) and for locally convex topological spaces over `ℝ` (by the geometric Hahn-Banach theorem). We show in `SeparatingDual.exists_ne_zero` that given any non-zero…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/SeparatingDual.html"}, {"id": "Mathlib.Analysis.Normed.Module.Multilinear.Basic", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 4, "macro_tier_score": 0.3362, "macro_tier_override": null, "x": 84.308, "z": -233.52, "size": 0.4263, "title": "Operator norm on the space of continuous multilinear maps", "summary": "When `f` is a continuous multilinear map in finitely many variables, we define its norm `‖f‖` as the smallest number such that `‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖` for all `m`. We show that it is indeed a norm, and prove its basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Multilinear/Basic.html"}, {"id": "Mathlib.Analysis.Complex.UpperHalfPlane.FunctionsBoundedAtInfty", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 57.134, "z": -209.324, "size": 0.2596, "title": "Bounded at infinity", "summary": "For complex-valued functions on the upper half plane, this file defines the filter `UpperHalfPlane.atImInfty` required for defining when functions are bounded at infinity and zero at infinity. Both of which are relevant for defining modular forms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/UpperHalfPlane/FunctionsBoundedAtInfty.html"}, {"id": "Mathlib.Analysis.Normed.Affine.Simplex", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 81.414, "z": -202.05, "size": 0.2478, "title": "Simplices in torsors over normed spaces.", "summary": "This file defines properties of simplices in a `NormedAddTorsor`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Affine/Simplex.html"}, {"id": "Mathlib.Analysis.Asymptotics.ExpGrowth", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 91.579, "z": -242.596, "size": 0.2881, "title": "Exponential growth", "summary": "This file defines the exponential growth of a sequence `u : ℕ → ℝ≥0∞`. This notion comes in two versions, using a `liminf` and a `limsup` respectively.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Asymptotics/ExpGrowth.html"}, {"id": "Mathlib.Analysis.Normed.Ring.InfiniteProd", "region_id": "analysis", "micro_elevation": 0.875, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 20.724, "z": -185.405, "size": 0.2448, "title": "Infinite products in normed rings", "summary": "This file proves a dominated convergence theorem for infinite products of terms of the form `(1 + f n k)` in a complete normed commutative ring, by reducing to the additive version (Tannery's theorem) via the formal expansion `∏ (1 + f i) = ∑ₛ ∏ᵢ∈ₛ f i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Ring/InfiniteProd.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Log.Summable", "region_id": "analysis", "micro_elevation": 0.8542, "macro_tier": 2, "macro_tier_score": 0.0281, "macro_tier_override": null, "x": 79.581, "z": -271.991, "size": 0.276, "title": "Summability of logarithms", "summary": "We give conditions under which the logarithms of a summable sequence are summable. We also use this to relate summability of `f` to multipliability of `1 + f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Log/Summable.html"}, {"id": "Mathlib.Analysis.Convex.Join", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 75.997, "z": -214.14, "size": 0.2478, "title": "Convex join", "summary": "This file defines the convex join of two sets. The convex join of `s` and `t` is the union of the segments with one end in `s` and the other in `t`. This is notably a useful gadget to deal with convex hulls of finite sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Join.html"}, {"id": "Mathlib.Analysis.Convex.Hull", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 4, "macro_tier_score": 0.4476, "macro_tier_override": null, "x": 84.316, "z": -210.99, "size": 0.4376, "title": "Convex hull", "summary": "This file defines the convex hull of a set `s` in a module. `convexHull 𝕜 s` is the smallest convex set containing `s`. In order theory speak, this is a closure operator.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Hull.html"}, {"id": "Mathlib.Analysis.Calculus.LogDerivUniformlyOn", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 31.934, "z": -217.762, "size": 0.2807, "title": "The Logarithmic derivative of an infinite product", "summary": "We show that if we have an infinite product of functions `f` that is locally uniformly convergent, then the logarithmic derivative of the product is the sum of the logarithmic derivatives of the individual functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/LogDerivUniformlyOn.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.GramMatrix", "region_id": "analysis", "micro_elevation": 0.7917, "macro_tier": 1, "macro_tier_score": 0.0142, "macro_tier_override": null, "x": 27.177, "z": -185.812, "size": 0.2716, "title": "Gram Matrices", "summary": "This file defines Gram matrices and proves their positive semidefiniteness. Results require `RCLike 𝕜`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/GramMatrix.html"}, {"id": "Mathlib.Analysis.Matrix.Order", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 2, "macro_tier_score": 0.0293, "macro_tier_override": null, "x": 24.63, "z": -220.168, "size": 0.3656, "title": "The partial order on matrices", "summary": "This file constructs the partial order and star ordered instances on matrices on `𝕜`. This allows us to use more general results from C⋆-algebras, like `CFC.sqrt`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Matrix/Order.html"}, {"id": "Mathlib.Analysis.Distribution.Sobolev", "region_id": "analysis", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 134.843, "z": -161.182, "size": 0.2, "title": "Sobolev spaces (Bessel potential spaces)", "summary": "In this file we define Sobolev spaces on normed vector spaces via the Fourier transform. These spaces are also known as Bessel potential spaces. The Bessel potential operator `besselPotential` is the Fourier multiplier with the symbol `x ↦ (1 + ‖x‖ ^ 2) ^ (s / 2)` and a tempered distribution `u` belongs to the Sobolev space `H ^ {s, p}` if `besselPotential E F s u` can be represented by a `Lp` function, informally…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/Sobolev.html"}, {"id": "Mathlib.Analysis.Distribution.FourierMultiplier", "region_id": "analysis", "micro_elevation": 0.9792, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 133.065, "z": -257.585, "size": 0.2478, "title": "Fourier multiplier on Schwartz functions and tempered distributions", "summary": "We define a Fourier multiplier as continuous linear maps on Schwartz functions and tempered distributions. The multiplier function is throughout assumed to have temperate growth.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/FourierMultiplier.html"}, {"id": "Mathlib.Analysis.Fourier.LpSpace", "region_id": "analysis", "micro_elevation": 0.9792, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 8.672, "z": -215.956, "size": 0.2478, "title": "The Fourier transform on $L^p$", "summary": "In this file we define the Fourier transform on $L^2$ as a linear isometry equivalence.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/LpSpace.html"}, {"id": "Mathlib.Analysis.Analytic.Uniqueness", "region_id": "analysis", "micro_elevation": 0.5208, "macro_tier": 3, "macro_tier_score": 0.2365, "macro_tier_override": null, "x": 54.395, "z": -237.74, "size": 0.2801, "title": "Uniqueness principle for analytic functions", "summary": "We show that two analytic functions which coincide around a point coincide on whole connected sets, in `AnalyticOnNhd.eqOn_of_preconnected_of_eventuallyEq`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/Uniqueness.html"}, {"id": "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 3, "macro_tier_score": 0.2378, "macro_tier_override": null, "x": 35.533, "z": -225.482, "size": 0.3735, "title": "One-dimensional iterated derivatives", "summary": "We define the `n`-th derivative of a function `f : 𝕜 → F` as a function `iteratedDeriv n f : 𝕜 → F`, as well as a version on domains `iteratedDerivWithin n f s : 𝕜 → F`, and prove their basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/IteratedDeriv/Defs.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Star", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 57.24, "z": -182.878, "size": 0.2, "title": "Star operations on derivatives", "summary": "This file contains the usual formulas (and existence assertions) for the derivative of the star operation. Most of the results in this file only apply when the field that the derivative is respect to has a trivial star operation; which as should be expected rules out `𝕜 = ℂ`. The exceptions are `HasDerivAt.conj_conj` and `DifferentiableAt.conj_conj`, showing that `conj ∘ f ∘ conj` is differentiable when `f` is (and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Star.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Star", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 46.813, "z": -204.738, "size": 0.2611, "title": "Star operations on derivatives", "summary": "This file contains the usual formulas (and existence assertions) for the Fréchet derivative of the star operation. For detailed documentation of the Fréchet derivative, see the module docstring of `Mathlib/Analysis/Calculus/FDeriv/Basic.lean`. Most of the results in this file only apply when the field that the derivative is respect to has a trivial star operation; which as should be expected rules out `𝕜 = ℂ`. The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Star.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Congr", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 4, "macro_tier_score": 0.2951, "macro_tier_override": null, "x": 66.282, "z": -189.373, "size": 0.4486, "title": "The Fréchet derivative: congruence properties", "summary": "Lemmas about congruence properties of the Fréchet derivative under change of function, set, etc.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Congr.html"}, {"id": "Mathlib.Analysis.Calculus.ParametricIntervalIntegral", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 127.193, "z": -210.82, "size": 0.2, "title": "Derivatives of interval integrals depending on parameters", "summary": "In this file we restate theorems about derivatives of integrals depending on parameters for interval integrals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ParametricIntervalIntegral.html"}, {"id": "Mathlib.Analysis.Normed.Operator.Conformal", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 2, "macro_tier_score": 0.0699, "macro_tier_override": null, "x": 66.467, "z": -211.827, "size": 0.2939, "title": "Conformal Linear Maps", "summary": "A continuous linear map between `R`-normed spaces `X` and `Y` `IsConformalMap` if it is a nonzero multiple of a linear isometry.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Conformal.html"}, {"id": "Mathlib.Analysis.Complex.Order", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 4, "macro_tier_score": 0.4034, "macro_tier_override": null, "x": 87.587, "z": -209.07, "size": 0.3086, "title": "The partial order on the complex numbers", "summary": "This order is defined by `z ≤ w ↔ z.re ≤ w.re ∧ z.im = w.im`. This is a natural order on `ℂ` because, as is well-known, there does not exist an order on `ℂ` making it into a linearly ordered field. However, the order described above is the canonical order stemming from the structure of `ℂ` as a ⋆-ring (i.e., it becomes a `StarOrderedRing`). Moreover, with this order `ℂ` satisfies `IsStrictOrderedRing` and the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Order.html"}, {"id": "Mathlib.Analysis.Complex.Conformal", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 2, "macro_tier_score": 0.0418, "macro_tier_override": null, "x": 49.065, "z": -196.599, "size": 0.2482, "title": "Conformal maps between complex vector spaces", "summary": "We prove the sufficient and necessary conditions for a real-linear map between complex vector spaces to be conformal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Conformal.html"}, {"id": "Mathlib.Analysis.Complex.HasPrimitives", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 2, "macro_tier_score": 0.0418, "macro_tier_override": null, "x": 34.375, "z": -206.615, "size": 0.2482, "title": "Primitives of Holomorphic Functions", "summary": "In this file, we give conditions under which holomorphic functions have primitives. The main goal is to prove that holomorphic functions on simply connected domains have primitives. As a first step, we prove that holomorphic functions on disks have primitives. The approach is based on Morera's theorem, that a continuous function (on a disk) whose integral round a rectangle vanishes on all rectangles contained in the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/HasPrimitives.html"}, {"id": "Mathlib.Analysis.Calculus.BumpFunction.Convolution", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 61.437, "z": -262.759, "size": 0.2676, "title": "Convolution with a bump function", "summary": "In this file we prove lemmas about convolutions `(φ.normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀`, where `φ : ContDiffBump 0` is a smooth bump function. We prove that this convolution is equal to `g x₀` if `g` is a constant on `Metric.ball x₀ φ.rOut`. We also provide estimates in the case if `g x` is close to `g x₀` on this ball.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/BumpFunction/Convolution.html"}, {"id": "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0142, "macro_tier_override": null, "x": 112.765, "z": -253.5, "size": 0.2737, "title": "Bump functions in finite-dimensional vector spaces", "summary": "Let `E` be a finite-dimensional real normed vector space. We show that any open set `s` in `E` is exactly the support of a smooth function taking values in `[0, 1]`, in `IsOpen.exists_contDiff_support_eq`. Then we use this construction to construct bump functions with nice behavior, by convolving the indicator function of `closedBall 0 1` with a function as above with `s = ball 0 D`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.html"}, {"id": "Mathlib.Analysis.Calculus.BumpFunction.Normed", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 31.931, "z": -217.742, "size": 0.253, "title": "Normed bump function", "summary": "In this file we define `ContDiffBump.normed f μ` to be the bump function `f` normalized so that `∫ x, f.normed μ x ∂μ = 1` and prove some properties of this function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/BumpFunction/Normed.html"}, {"id": "Mathlib.Analysis.Calculus.LogDeriv", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 3, "macro_tier_score": 0.2237, "macro_tier_override": null, "x": 31.962, "z": -217.92, "size": 0.3613, "title": "Logarithmic Derivatives", "summary": "We define the logarithmic derivative of a function `f` as `deriv f / f`. We then prove some basic facts about this, including how it changes under multiplication and composition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/LogDeriv.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Support", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 3, "macro_tier_score": 0.2511, "macro_tier_override": null, "x": 107.474, "z": -200.469, "size": 0.3421, "title": "Support of the derivative of a function", "summary": "In this file we prove that the (topological) support of a function includes the support of its derivative. As a corollary, we show that the derivative of a function with compact support has compact support.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Support.html"}, {"id": "Mathlib.Analysis.Convex.Visible", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 74.908, "z": -215.211, "size": 0.2, "title": "Points in sight", "summary": "This file defines the relation of visibility with respect to a set, and lower bounds how many elements of a set a point sees in terms of the dimension of that set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Visible.html"}, {"id": "Mathlib.Analysis.LocallyConvex.Barrelled", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 2, "macro_tier_score": 0.0557, "macro_tier_override": null, "x": 63.703, "z": -198.256, "size": 0.2516, "title": "Barrelled spaces and the Banach-Steinhaus theorem / Uniform Boundedness Principle", "summary": "This file defines barrelled spaces over a `NontriviallyNormedField`, and proves the Banach-Steinhaus theorem for maps from a barrelled space to a space equipped with a family of seminorms generating the topology (i.e. `WithSeminorms q` for some family of seminorms `q`). The more standard Banach-Steinhaus theorem for normed spaces is then deduced from that in `Mathlib/Analysis/Normed/Operator/BanachSteinhaus.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/Barrelled.html"}, {"id": "Mathlib.Analysis.LocallyConvex.WithSeminorms", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 4, "macro_tier_score": 0.3917, "macro_tier_override": null, "x": 62.022, "z": -206.131, "size": 0.4244, "title": "Topology induced by a family of seminorms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/WithSeminorms.html"}, {"id": "Mathlib.Analysis.Distribution.SchwartzSpace.Fourier", "region_id": "analysis", "micro_elevation": 0.9375, "macro_tier": 2, "macro_tier_score": 0.0424, "macro_tier_override": null, "x": 145.983, "z": -228.072, "size": 0.3158, "title": "Fourier transform on Schwartz functions", "summary": "This file constructs the Fourier transform as a continuous linear map acting on Schwartz functions, in `fourierTransformCLM`. It is also given as a continuous linear equiv, in `fourierTransformCLE`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/SchwartzSpace/Fourier.html"}, {"id": "Mathlib.Analysis.Distribution.SchwartzSpace.Deriv", "region_id": "analysis", "micro_elevation": 0.8542, "macro_tier": 2, "macro_tier_score": 0.0422, "macro_tier_override": null, "x": 135.632, "z": -181.166, "size": 0.2993, "title": "Derivatives of Schwartz functions", "summary": "In this file we define the various notions of derivatives of Schwartz functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/SchwartzSpace/Deriv.html"}, {"id": "Mathlib.Analysis.Fourier.Inversion", "region_id": "analysis", "micro_elevation": 0.9167, "macro_tier": 2, "macro_tier_score": 0.056, "macro_tier_override": null, "x": 54.667, "z": -147.466, "size": 0.2871, "title": "Fourier inversion formula", "summary": "In a finite-dimensional real inner product space, we show the Fourier inversion formula, i.e., `𝓕⁻ (𝓕 f) v = f v` if `f` and `𝓕 f` are integrable, and `f` is continuous at `v`. This is proved in `MeasureTheory.Integrable.fourier_inversion`. See also `Continuous.fourier_inversion` giving `𝓕⁻ (𝓕 f) = f` under an additional continuity assumption for `f`. We use the following proof. A naïve computation gives `𝓕⁻ (𝓕 f) v…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/Inversion.html"}, {"id": "Mathlib.Analysis.Distribution.SchwartzSpace.Basic", "region_id": "analysis", "micro_elevation": 0.8333, "macro_tier": 2, "macro_tier_score": 0.0419, "macro_tier_override": null, "x": 125.896, "z": -169.464, "size": 0.2655, "title": "Schwartz space", "summary": "This file defines the Schwartz space. Usually, the Schwartz space is defined as the set of smooth functions $f : ℝ^n → ℂ$ such that there exists $C_{αβ} > 0$ with $$|x^α ∂^β f(x)| < C_{αβ}$$ for all $x ∈ ℝ^n$ and for all multiindices $α, β$. In mathlib, we use a slightly different approach and define the Schwartz space as all smooth functions `f : E → F`, where `E` and `F` are real normed vector spaces such that for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/SchwartzSpace/Basic.html"}, {"id": "Mathlib.Analysis.Distribution.TemperateGrowth", "region_id": "analysis", "micro_elevation": 0.8125, "macro_tier": 2, "macro_tier_score": 0.0418, "macro_tier_override": null, "x": 20.847, "z": -215.259, "size": 0.2405, "title": "Functions and measures of temperate growth", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/TemperateGrowth.html"}, {"id": "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0418, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2405, "title": "ZeroAtInftyContinuousMapClass in normed additive groups", "summary": "In this file we give a characterization of the predicate `zero_at_infty` from `ZeroAtInftyContinuousMapClass`. A continuous map `f` is zero at infinity if and only if for every `ε > 0` there exists a `r : ℝ` such that for all `x : E` with `r < ‖x‖` it holds that `‖f x‖ < ε`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/ZeroAtInfty.html"}, {"id": "Mathlib.Analysis.Normed.Lp.SmoothApprox", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0418, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2405, "title": "Density of smooth compactly supported functions in `Lp`", "summary": "In this file, we prove that `Lp` functions can be approximated by smooth compactly supported functions for `p < ∞`. This result is recorded in `MeasureTheory.MemLp.exist_sub_eLpNorm_le`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Lp/SmoothApprox.html"}, {"id": "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 3, "macro_tier_score": 0.2225, "macro_tier_override": null, "x": 79.313, "z": -174.499, "size": 0.2766, "title": "Inverse function theorem, 1D case", "summary": "In this file we prove a version of the inverse function theorem for maps `f : 𝕜 → 𝕜`. We use `ContinuousLinearEquiv.unitsEquivAut` to translate `HasStrictDerivAt f f' a` and `f' ≠ 0` into `HasStrictFDerivAt f (_ : 𝕜 ≃L[𝕜] 𝕜) a`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/InverseFunctionTheorem/Deriv.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.Inverse", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 3, "macro_tier_score": 0.2514, "macro_tier_override": null, "x": 75.76, "z": -242.78, "size": 0.3593, "title": "Inverse function theorem - the easy half", "summary": "In this file we prove that `g' (f x) = (f' x)⁻¹` provided that `f` is strictly differentiable at `x`, `f' x ≠ 0`, and `g` is a local left inverse of `f` that is continuous at `f x`. This is the easy half of the inverse function theorem: the harder half states that `g` exists. For a more detailed overview of one-dimensional derivatives in mathlib, see the module docstring of `Analysis/Calculus/Deriv/Basic`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/Inverse.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 79.235, "z": -210.858, "size": 0.2, "title": "The arithmetic-geometric mean", "summary": "Starting with two nonnegative real numbers, repeatedly replace them with their arithmetic and geometric means. By the AM-GM inequality, the smaller number (geometric mean) will monotonically increase and the larger number (arithmetic mean) will monotonically decrease. The two monotone sequences converge to the same limit – the arithmetic-geometric mean (AGM). This file defines the AGM in the `NNReal` namespace and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ArithmeticGeometricMean.html"}, {"id": "Mathlib.Analysis.Distribution.DerivNotation", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 3, "macro_tier_score": 0.0839, "macro_tier_override": null, "x": 105.436, "z": -171.572, "size": 0.2974, "title": "Type classes for derivatives and the Laplacian", "summary": "In this file we define notation type classes for line derivatives, also known as partial derivatives, and for the Laplacian. Moreover, we provide type-classes that encode the linear structure. We also define the iterated line derivative and prove elementary properties. We define a Laplacian based on the sum of second derivatives formula and prove that the Laplacian thus defined is independent of the choice of basis.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/DerivNotation.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 2, "macro_tier_score": 0.0419, "macro_tier_override": null, "x": 34.4, "z": -176.377, "size": 0.2601, "title": "Construction of Harmonic Functions", "summary": "This file constructs examples of harmonic functions. If `f : ℂ → F` is complex-differentiable, then `f` is harmonic. If `F = ℂ`, then so is its real part, imaginary part, and complex conjugate. If `f` has no zero, then `log ‖f‖` is harmonic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Harmonic/Constructions.html"}, {"id": "Mathlib.Analysis.Normed.Operator.Mul", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 4, "macro_tier_score": 0.3494, "macro_tier_override": null, "x": 102.85, "z": -221.692, "size": 0.3979, "title": "Results about operator norms in normed algebras", "summary": "This file (split off from `OperatorNorm.lean`) contains results about the operator norm of multiplication and scalar-multiplication operations in normed algebras and normed modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Mul.html"}, {"id": "Mathlib.Analysis.Normed.Module.RCLike.Real", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 4, "macro_tier_score": 0.4618, "macro_tier_override": null, "x": 93.542, "z": -211.576, "size": 0.4501, "title": "Basic facts about real (semi)normed spaces", "summary": "In this file we prove some theorems about (semi)normed spaces over real numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/RCLike/Real.html"}, {"id": "Mathlib.Analysis.Convex.Topology", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 4, "macro_tier_score": 0.3911, "macro_tier_override": null, "x": 81.789, "z": -218.492, "size": 0.3984, "title": "Topological properties of convex sets", "summary": "We prove the following facts: * `Convex.interior` : interior of a convex set is convex; * `Convex.closure` : closure of a convex set is convex; * `closedConvexHull_closure_eq_closedConvexHull` : the closed convex hull of the closure of a set is equal to the closed convex hull of the set; * `Set.Finite.isCompact_convexHull` : convex hull of a finite set is compact; * `Set.Finite.isClosed_convexHull` : convex hull of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Topology.html"}, {"id": "Mathlib.Analysis.Convex.Strict", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 4, "macro_tier_score": 0.3896, "macro_tier_override": null, "x": 81.64, "z": -213.791, "size": 0.3117, "title": "Strictly convex sets", "summary": "This file defines strictly convex sets. A set is strictly convex if the open segment between any two distinct points lies in its interior.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Strict.html"}, {"id": "Mathlib.Analysis.Convex.StdSimplex", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 4, "macro_tier_score": 0.3895, "macro_tier_override": null, "x": 83.96, "z": -216.029, "size": 0.3089, "title": "The standard simplex", "summary": "In this file, given an ordered semiring `𝕜` and a finite type `ι`, we define `stdSimplex : Set (ι → 𝕜)` as the set of vectors with non-negative coordinates with total sum `1`. When `f : X → Y` is a map between finite types, we define the map `stdSimplex.map f : stdSimplex 𝕜 X → stdSimplex 𝕜 Y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/StdSimplex.html"}, {"id": "Mathlib.Analysis.Complex.SqrtDeriv", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 26.448, "z": -191.894, "size": 0.2594, "title": "Derivatives of `Complex.sqrt`", "summary": "This file proves that `Complex.sqrt` is differentiable on the slit plane `Complex.slitPlane` and computes its derivative.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/SqrtDeriv.html"}, {"id": "Mathlib.Analysis.Convex.Star", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 4, "macro_tier_score": 0.4454, "macro_tier_override": null, "x": 78.91, "z": -208.451, "size": 0.3335, "title": "Star-convex sets", "summary": "This file defines star-convex sets (aka star domains, star-shaped set, radially convex set). A set is star-convex at `x` if every segment from `x` to a point in the set is contained in the set. This is the prototypical example of a contractible set in homotopy theory (by scaling every point towards `x`), but has wider uses. Note that this has nothing to do with star rings, `Star` and co.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Star.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Measurable", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 1, "macro_tier_score": 0.0143, "macro_tier_override": null, "x": 111.172, "z": -225.502, "size": 0.287, "title": "Derivative is measurable", "summary": "In this file we prove that the derivative of any function with complete codomain is a measurable function. Namely, we prove: * `measurableSet_of_differentiableAt`: the set `{x | DifferentiableAt 𝕜 f x}` is measurable; * `measurable_fderiv`: the function `fderiv 𝕜 f` is measurable; * `measurable_fderiv_apply_const`: for a fixed vector `y`, the function `fun x ↦ fderiv 𝕜 f x y` is measurable; * `measurable_deriv`: the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Measurable.html"}, {"id": "Mathlib.Analysis.Calculus.TangentCone.Prod", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 0, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": 76.174, "z": -216.128, "size": 0.3488, "title": "Product of sets with unique differentiability property", "summary": "In this file we prove that the product of two sets with unique differentiability property has the same property, see `UniqueDiffOn.prod`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/TangentCone/Prod.html"}, {"id": "Mathlib.Analysis.AperiodicOrder.Delone.Basic", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2, "title": "Delone sets", "summary": "A **Delone set** `D ⊆ X` in a metric space is a set which is both: * **Uniformly Discrete**: there exists `packingRadius > 0` such that distinct points of `D` are separated by a distance strictly greater than `packingRadius`; * **Relatively Dense**: there exists `coveringRadius > 0` such that every point of `X` lies within distance `coveringRadius` of some point of `D`. The `DeloneSet` structure stores the set…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/AperiodicOrder/Delone/Basic.html"}, {"id": "Mathlib.Analysis.Convex.Cone.InnerDual", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 119.072, "z": -183.028, "size": 0.2, "title": "Inner dual cone of a set", "summary": "We define the inner dual cone of a set `s` in an inner product space to be the proper cone consisting of all points `y` such that `0 ≤ ⟪x, y⟫` for all `x ∈ s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Cone/InnerDual.html"}, {"id": "Mathlib.Analysis.Convex.Cone.Dual", "region_id": "analysis", "micro_elevation": 0.2917, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 62.064, "z": -197.967, "size": 0.2585, "title": "The topological dual of a cone and Farkas' lemma", "summary": "Given a continuous bilinear pairing `p` between two `R`-modules `M` and `N` and a set `s` in `M`, we define `ProperCone.dual p C` to be the proper cone in `N` consisting of all points `y` such that `0 ≤ p x y` for all `x ∈ s`. When the pairing is perfect, this gives us the algebraic dual of a cone. See `Mathlib/Geometry/Convex/Cone/Dual.lean` for that case. When the pairing is continuous and perfect (as a continuous…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Cone/Dual.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Adjoint", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 3, "macro_tier_score": 0.0845, "macro_tier_override": null, "x": 40.476, "z": -232.506, "size": 0.3452, "title": "Adjoint of operators on Hilbert spaces", "summary": "Given an operator `A : E →L[𝕜] F`, where `E` and `F` are Hilbert spaces, its adjoint `adjoint A : F →L[𝕜] E` is the unique operator such that `⟪x, A y⟫ = ⟪adjoint A x, y⟫` for all `x` and `y`. We then use this to put a C⋆-algebra structure on `E →L[𝕜] E` with the adjoint as the star operation. This construction is used to define an adjoint for linear maps (i.e. not continuous) between finite-dimensional spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Adjoint.html"}, {"id": "Mathlib.Analysis.Calculus.DerivativeTest", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 37.074, "z": -225.263, "size": 0.2, "title": "The First- and Second-Derivative Tests", "summary": "We prove the first-derivative test from calculus, in the strong form given on [Wikipedia](https://en.wikipedia.org/wiki/Derivative_test#First-derivative_test). The test is proved over the real numbers ℝ using `monotoneOn_of_deriv_nonneg` from `Mathlib/Analysis/Calculus/Deriv/MeanValue.lean`. We prove the second-derivative test using the first-derivative test. Source:…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/DerivativeTest.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.MeanValue", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 3, "macro_tier_score": 0.2378, "macro_tier_override": null, "x": 118.03, "z": -231.988, "size": 0.3736, "title": "Mean value theorem", "summary": "In this file we prove Cauchy's and Lagrange's mean value theorems, and deduce some corollaries. Cauchy's mean value theorem says that for two functions `f` and `g` that are continuous on `[a, b]` and are differentiable on `(a, b)`, there exists a point `c ∈ (a, b)` such that `f' c / g' c = (f b - f a) / (g b - g a)`. We formulate this theorem with both sides multiplied by the denominators, see…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/MeanValue.html"}, {"id": "Mathlib.Analysis.Matrix.Hermitian", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 2, "macro_tier_score": 0.0418, "macro_tier_override": null, "x": 44.805, "z": -241.039, "size": 0.2532, "title": "Hermitian matrices over ℝ and ℂ", "summary": "This file proves that Hermitian matrices over ℝ and ℂ are exactly the ones whose corresponding linear map is self-adjoint.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Matrix/Hermitian.html"}, {"id": "Mathlib.Analysis.RCLike.BoundedContinuous", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 64.695, "z": -226.515, "size": 0.249, "title": "Results on bounded continuous functions with `RCLike` values", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/RCLike/BoundedContinuous.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral", "region_id": "analysis", "micro_elevation": 0.8542, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 17.582, "z": -206.827, "size": 0.249, "title": "Properties of the integral of `mulExpNegMulSq`", "summary": "The mapping `mulExpNegMulSq` can be used to transform a function `g : E → ℝ` into a bounded function `mulExpNegMulSq ε ∘ g : E → ℝ = fun x => g x * Real.exp (-ε * g x * g x)`. This file contains results on the integral of `mulExpNegMulSq g ε` with respect to a finite measure `P`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/MulExpNegMulSqIntegral.html"}, {"id": "Mathlib.Analysis.Calculus.LocalExtr.Polynomial", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 55.187, "z": -249.724, "size": 0.2, "title": "Rolle's Theorem for polynomials", "summary": "In this file we use Rolle's Theorem to relate the number of real roots of a real polynomial and its derivative. Namely, we prove the following facts. * `Polynomial.card_roots_toFinset_le_card_roots_derivative_sdiff_roots_succ`: the number of roots of a real polynomial `p` is at most the number of roots of its derivative that are not roots of `p` plus one. * `Polynomial.card_roots_toFinset_le_derivative`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/LocalExtr/Polynomial.html"}, {"id": "Mathlib.Analysis.Normed.Operator.BanachSteinhaus", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 2, "macro_tier_score": 0.0558, "macro_tier_override": null, "x": 56.292, "z": -219.99, "size": 0.2638, "title": "The Banach-Steinhaus theorem: Uniform Boundedness Principle", "summary": "Herein we prove the Banach-Steinhaus theorem for normed spaces: any collection of bounded linear maps from a Banach space into a normed space which is pointwise bounded is uniformly bounded. Note that we prove the more general version about barrelled spaces in `Analysis.LocallyConvex.Barrelled`, and the usual version below is indeed deduced from the more general setup.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/BanachSteinhaus.html"}, {"id": "Mathlib.Analysis.Normed.Algebra.UnitizationL1", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 2, "macro_tier_score": 0.0696, "macro_tier_override": null, "x": 75.082, "z": -174.838, "size": 0.2455, "title": "Unitization equipped with the $L^1$ norm", "summary": "In another file, the `Unitization 𝕜 A` of a non-unital normed `𝕜`-algebra `A` is equipped with the norm inherited as the pullback via a map (closely related to) the left-regular representation of the algebra on itself (see `Unitization.instNormedRing`). However, this construction is only valid (and an isometry) when `A` is a `RegularNormedAlgebra`. Sometimes it is useful to consider the unitization of a non-unital…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Algebra/UnitizationL1.html"}, {"id": "Mathlib.Analysis.Normed.Lp.ProdLp", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 3, "macro_tier_score": 0.2234, "macro_tier_override": null, "x": 71.472, "z": -177.115, "size": 0.3463, "title": "`L^p` distance on products of two metric spaces", "summary": "Given two metric spaces, one can put the max distance on their product, but there is also a whole family of natural distances, indexed by a parameter `p : ℝ≥0∞`, that also induce the product topology. We define them in this file. For `0 < p < ∞`, the distance on `α × β` is given by $$ d(x, y) = \\left(d(x_1, y_1)^p + d(x_2, y_2)^p\\right)^{1/p}. $$ For `p = ∞` the distance is the supremum of the distances and `p = 0`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Lp/ProdLp.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Laplacian", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 3, "macro_tier_score": 0.0844, "macro_tier_override": null, "x": 123.943, "z": -230.605, "size": 0.3403, "title": "The Laplacian", "summary": "This file defines the Laplacian for functions `f : E → F` on real, finite-dimensional, inner product spaces `E`. In essence, we define the Laplacian of `f` as the second derivative, applied to the canonical covariant tensor of `E`, as defined and discussed in `Mathlib.Analysis.InnerProductSpace.CanonicalTensor`. We show that the Laplacian is `ℝ`-linear on continuously differentiable functions, and establish the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Laplacian.html"}, {"id": "Mathlib.Analysis.Normed.Group.Completeness", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 81.374, "z": -202.042, "size": 0.2, "title": "Completeness of normed groups", "summary": "This file includes a completeness criterion for normed additive groups in terms of convergent series.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Completeness.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 4, "macro_tier_score": 0.4068, "macro_tier_override": null, "x": 94.932, "z": -232.521, "size": 0.469, "title": "Power function on `ℂ`", "summary": "We construct the power functions `x ^ y`, where `x` and `y` are complex numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Pow/Complex.html"}, {"id": "Mathlib.Analysis.Normed.Module.Convex", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 4, "macro_tier_score": 0.3911, "macro_tier_override": null, "x": 64.793, "z": -210.678, "size": 0.3983, "title": "Metric properties of convex sets in normed spaces", "summary": "We prove the following facts: * `convexOn_norm`, `convexOn_dist` : norm and distance to a fixed point is convex on any convex set; * `convexOn_univ_norm`, `convexOn_univ_dist` : norm and distance to a fixed point is convex on the whole space; * `convexHull_ediam`, `convexHull_diam` : convex hull of a set has the same (e)metric diameter as the original set; * `isBounded_convexHull` : convex hull of a set is bounded…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Convex.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Module.Defs", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 2, "macro_tier_score": 0.028, "macro_tier_override": null, "x": 45.269, "z": -165.124, "size": 0.2584, "title": "Hilbert C⋆-modules", "summary": "A Hilbert C⋆-module is a complex module `E` together with a right `A`-module structure, where `A` is a C⋆-algebra, and with an `A`-valued inner product. This inner product satisfies the Cauchy-Schwarz inequality, and induces a norm that makes `E` a normed vector space over `ℂ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Module/Defs.html"}, {"id": "Mathlib.Analysis.Convex.Cone.Closure", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.235, "title": "Closure of cones", "summary": "We define the closures of convex and pointed cones. This construction is primarily needed for defining maps between proper cones. The current API is basic and should be extended as necessary.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Cone/Closure.html"}, {"id": "Mathlib.Analysis.Subadditive", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2, "title": "Convergence of subadditive sequences", "summary": "A subadditive sequence `u : ℕ → ℝ` is a sequence satisfying `u (m + n) ≤ u m + u n` for all `m, n`. We define this notion as `Subadditive u`, and prove in `Subadditive.tendsto_lim` that, if `u n / n` is bounded below, then it converges to a limit (that we denote by `Subadditive.lim` for convenience). This result is known as Fekete's lemma in the literature.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Subadditive.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.GelfandDuality", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 2, "macro_tier_score": 0.0421, "macro_tier_override": null, "x": 97.996, "z": -258.096, "size": 0.2839, "title": "Gelfand Duality", "summary": "The `gelfandTransform` is an algebra homomorphism from a topological `𝕜`-algebra `A` to `C(characterSpace 𝕜 A, 𝕜)`. In the case where `A` is a commutative complex Banach algebra, then the Gelfand transform is actually spectrum-preserving (`spectrum.gelfandTransform_eq`). Moreover, when `A` is a commutative C⋆-algebra over `ℂ`, then the Gelfand transform is a surjective isometry, and even an equivalence between…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/GelfandDuality.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Spectrum", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 2, "macro_tier_score": 0.0417, "macro_tier_override": null, "x": 129.182, "z": -219.872, "size": 0.2387, "title": "Spectral properties in C⋆-algebras", "summary": "In this file, we establish various properties related to the spectrum of elements in C⋆-algebras. In particular, we show that the spectrum of a unitary element is contained in the unit circle in `ℂ`, the spectrum of a selfadjoint element is real, the spectral radius of a selfadjoint element or normal element is its norm, among others. An essential feature of C⋆-algebras is **spectral permanence**. This is the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Spectrum.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousMap", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 2, "macro_tier_score": 0.0417, "macro_tier_override": null, "x": 66.476, "z": -224.014, "size": 0.2387, "title": "C⋆-algebras of continuous functions", "summary": "We place these here because, for reasons related to the import hierarchy, they cannot be placed in earlier files.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousMap.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Fuglede", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 2, "macro_tier_score": 0.0417, "macro_tier_override": null, "x": 123.508, "z": -227.86, "size": 0.2387, "title": "The Fuglede–Putnam–Rosenblum theorem", "summary": "Let `A` be a C⋆-algebra, and let `a b x : A`. The Fuglede–Putnam–Rosenblum theorem states that if `a` and `b` are normal and `x` intertwines `a` and `b` (i.e., `SemiconjBy x a b`, that is, `x * a = b * x`), then `x` also intertwines `star a` and `star b`. Fuglede's original result [fuglede1950] was for `a = b` (i.e., if `x` commutes with `a`, then `x` also commutes with `star a`), and Putnam [putnam1951] extended it…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Fuglede.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.LinearMap", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 3, "macro_tier_score": 0.209, "macro_tier_override": null, "x": 89.749, "z": -180.67, "size": 0.3056, "title": "Linear maps on inner product spaces", "summary": "This file studies linear maps on inner product spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/LinearMap.html"}, {"id": "Mathlib.Analysis.Calculus.LHopital", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 81.561, "z": -163.856, "size": 0.2, "title": "L'Hôpital's rule for 0/0 indeterminate forms", "summary": "In this file, we prove several forms of \"L'Hôpital's rule\" for computing 0/0 indeterminate forms. The proof of `HasDerivAt.lhopital_zero_right_on_Ioo` is based on the one given in the corresponding [Wikibooks](https://en.wikibooks.org/wiki/Calculus/L%27H%C3%B4pital%27s_Rule) chapter, and all other statements are derived from this one by composing by carefully chosen functions. Note that the filter `f'/g'` tends to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/LHopital.html"}, {"id": "Mathlib.Analysis.SumOverResidueClass", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 2, "macro_tier_score": 0.0562, "macro_tier_override": null, "x": 77.937, "z": -203.792, "size": 0.3077, "title": "Sums over residue classes", "summary": "We consider infinite sums over functions `f` on `ℕ`, restricted to a residue class mod `m`. The main result is `summable_indicator_mod_iff`, which states that when `f : ℕ → ℝ` is decreasing, then the sum over `f` restricted to any residue class mod `m ≠ 0` converges if and only if the sum over all of `ℕ` converges.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SumOverResidueClass.html"}, {"id": "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 3, "macro_tier_score": 0.0988, "macro_tier_override": null, "x": 83.398, "z": -244.404, "size": 0.367, "title": "A collection of specific asymptotic results", "summary": "This file contains specific lemmas about asymptotics which don't have their place in the general theory developed in `Mathlib/Analysis/Asymptotics/Defs.lean` and `Mathlib/Analysis/Asymptotics/Lemmas.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Asymptotics/SpecificAsymptotics.html"}, {"id": "Mathlib.Analysis.Fourier.Notation", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0557, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2557, "title": "Type classes for the Fourier transform", "summary": "In this file we define type classes for the Fourier transform and the inverse Fourier transform. We introduce the notation `𝓕` and `𝓕⁻` in these classes to denote the Fourier transform and the inverse Fourier transform, respectively. Moreover, we provide type-classes that encode the linear structure and the Fourier inversion theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/Notation.html"}, {"id": "Mathlib.Analysis.Asymptotics.LinearGrowth", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 1, "macro_tier_score": 0.0142, "macro_tier_override": null, "x": 81.319, "z": -208.798, "size": 0.2745, "title": "Linear growth", "summary": "This file defines the linear growth of a sequence `u : ℕ → R`. This notion comes in two versions, using a `liminf` and a `limsup` respectively. Most properties are developed for `R = EReal`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Asymptotics/LinearGrowth.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.Comp", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 4, "macro_tier_score": 0.2533, "macro_tier_override": null, "x": 40.979, "z": -230.277, "size": 0.442, "title": "Higher differentiability of composition", "summary": "We prove that the composition of `C^n` functions is `C^n`. We also expand the API around `C^n` functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/Comp.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.Basic", "region_id": "analysis", "micro_elevation": 0.5833, "macro_tier": 3, "macro_tier_score": 0.2515, "macro_tier_override": null, "x": 42.119, "z": -229.172, "size": 0.3646, "title": "Basic properties of continuously-differentiable functions", "summary": "This file continues the development of the API for `ContDiff`, `ContDiffAt`, etc, covering constants, products, composition with linear maps, etc.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/Basic.html"}, {"id": "Mathlib.Analysis.Polynomial.Norm", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 0, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 62.68, "z": -236.44, "size": 0.2295, "title": "Sup Norm of Polynomials", "summary": "In this file we define the sup norm on `Polynomial`s based on their coefficients as well as several basic results about this norm. We note that this is often called the _(naive) height_ of the polynomial in the literature. The sup norm is related to the Mahler measure of the polynomial. See `Mathlib/Analysis/Polynomial/MahlerMeasure.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Polynomial/Norm.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.Polynomial", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 60.837, "z": -166.361, "size": 0.2386, "title": "Higher smoothness of polynomials", "summary": "We prove that polynomials are `C^∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/Polynomial.html"}, {"id": "Mathlib.Analysis.Calculus.TangentCone.Real", "region_id": "analysis", "micro_elevation": 0.1458, "macro_tier": 4, "macro_tier_score": 0.2936, "macro_tier_override": null, "x": 90.442, "z": -211.619, "size": 0.3882, "title": "Unique differentiability property in real normed spaces", "summary": "In this file we prove that - `uniqueDiffOn_convex`: a convex set with nonempty interior in a real normed space has the unique differentiability property; - `uniqueDiffOn_Ioc` etc: intervals on the real line have the unique differentiability property.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/TangentCone/Real.html"}, {"id": "Mathlib.Analysis.Calculus.TangentCone.Basic", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 4, "macro_tier_score": 0.3077, "macro_tier_override": null, "x": 81.734, "z": -215.369, "size": 0.3969, "title": "Basic properties of tangent cones and sets with unique differentiability property", "summary": "In this file we prove basic lemmas about `tangentConeAt`, `UniqueDiffWithinAt`, and `UniqueDiffOn`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/TangentCone/Basic.html"}, {"id": "Mathlib.Analysis.Distribution.ContDiffMapSupportedIn", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 32.885, "z": -212.995, "size": 0.2493, "title": "Continuously differentiable functions supported in a given compact set", "summary": "This file develops the basic theory of bundled `n`-times continuously differentiable functions with support contained in a given compact set. Given `n : ℕ∞` and a compact subset `K` of a normed space `E`, we consider the type of bundled functions `f : E → F` (where `F` is a normed vector space) such that: - `f` is `n`-times continuously differentiable: `ContDiff ℝ n f`. - `f` vanishes outside of a compact set: `EqOn…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/ContDiffMapSupportedIn.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Defs", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 4, "macro_tier_score": 0.2935, "macro_tier_override": null, "x": 60.606, "z": -205.445, "size": 0.3814, "title": "The Fréchet derivative: definition", "summary": "Let `E` and `F` be normed spaces, `f : E → F`, and `f' : E →L[𝕜] F` a continuous 𝕜-linear map, where `𝕜` is a non-discrete normed field. Then `HasFDerivWithinAt f f' s x` says that `f` has derivative `f'` at `x`, where the domain of interest is restricted to `s`. We also have `HasFDerivAt f f' x := HasFDerivWithinAt f f' x univ` Finally, `HasStrictFDerivAt f f' x` means that `f : E → F` has derivative `f' : E →L[𝕜]…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Defs.html"}, {"id": "Mathlib.Analysis.Calculus.TangentCone.DimOne", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 4, "macro_tier_score": 0.2935, "macro_tier_override": null, "x": 84.937, "z": -217.214, "size": 0.3814, "title": "Unique differentiability property of a set in the base field", "summary": "In this file we prove that a set in the base field has the unique differentiability property at `x` iff `x` is an accumulation point of the set, see `uniqueDiffWithinAt_iff_accPt`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/TangentCone/DimOne.html"}, {"id": "Mathlib.Analysis.Analytic.Basic", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 4, "macro_tier_score": 0.2658, "macro_tier_override": null, "x": 104.759, "z": -227.267, "size": 0.3854, "title": "Analytic functions", "summary": "A function is analytic in one dimension around `0` if it can be written as a converging power series `Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a vector `(x, y) (x, y)`). A way to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/Basic.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Orthonormal", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 3, "macro_tier_score": 0.2091, "macro_tier_override": null, "x": 64.527, "z": -181.522, "size": 0.3158, "title": "Orthonormal sets", "summary": "This file defines orthonormal sets in inner product spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Orthonormal.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Subspace", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 3, "macro_tier_score": 0.2089, "macro_tier_override": null, "x": 67.935, "z": -240.806, "size": 0.3023, "title": "Subspaces of inner product spaces", "summary": "This file defines the inner-product structure on a subspace of an inner-product space, and proves some theorems about orthogonal families of subspaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Subspace.html"}, {"id": "Mathlib.Analysis.Normed.Module.Normalize", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 1, "macro_tier_score": 0.0145, "macro_tier_override": null, "x": 90.733, "z": -196.676, "size": 0.3026, "title": "Normalized vector", "summary": "Function that returns unit length vector that points in the same direction (if the given vector is nonzero vector) or returns zero vector (if the given vector is zero vector).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Normalize.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.RCLike", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 3, "macro_tier_score": 0.2369, "macro_tier_override": null, "x": 116.092, "z": -181.515, "size": 0.3191, "title": "Higher differentiability over `ℝ` or `ℂ`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/RCLike.html"}, {"id": "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 3, "macro_tier_score": 0.2371, "macro_tier_override": null, "x": 116.323, "z": -177.036, "size": 0.334, "title": "One-dimensional iterated derivatives", "summary": "This file contains a number of further results on `iteratedDerivWithin` that need more imports than are available in `Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.html"}, {"id": "Mathlib.Analysis.Complex.RealDeriv", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 3, "macro_tier_score": 0.2366, "macro_tier_override": null, "x": 84.174, "z": -162.492, "size": 0.2933, "title": "Real differentiability of complex-differentiable functions", "summary": "`HasDerivAt.real_of_complex` expresses that, if a function on `ℂ` is differentiable (over `ℂ`), then its restriction to `ℝ` is differentiable over `ℝ`, with derivative the real part of the complex derivative.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/RealDeriv.html"}, {"id": "Mathlib.Analysis.Normed.Module.Ray", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 2, "macro_tier_score": 0.0702, "macro_tier_override": null, "x": 66.467, "z": -216.559, "size": 0.3196, "title": "Rays in a real normed vector space", "summary": "In this file we prove some lemmas about the `SameRay` predicate in case of a real normed space. In this case, for two vectors `x y` in the same ray, the norm of their sum is equal to the sum of their norms and `‖y‖ • x = ‖x‖ • y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Ray.html"}, {"id": "Mathlib.Analysis.Normed.Module.Ball.Pointwise", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 4, "macro_tier_score": 0.418, "macro_tier_override": null, "x": 65.663, "z": -214.727, "size": 0.3531, "title": "Properties of pointwise scalar multiplication of sets in normed spaces.", "summary": "We explore the relationships between scalar multiplication of sets in vector spaces, and the norm. Notably, we express arbitrary balls as rescaling of other balls, and we show that the multiplication of bounded sets remain bounded.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Ball/Pointwise.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "region_id": "analysis", "micro_elevation": 0.7917, "macro_tier": 3, "macro_tier_score": 0.0977, "macro_tier_override": null, "x": 82.347, "z": -151.69, "size": 0.2939, "title": "Japanese Bracket", "summary": "In this file, we show that Japanese bracket $(1 + \\|x\\|^2)^{1/2}$ can be estimated from above and below by $1 + \\|x\\|$. The functions $(1 + \\|x\\|^2)^{-r/2}$ and $(1 + |x|)^{-r}$ are integrable provided that `r` is larger than the dimension.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/JapaneseBracket.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap", "region_id": "analysis", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 88.1, "z": -269.926, "size": 0.2, "title": "Completely positive maps", "summary": "A linear map `φ : A₁ →ₗ[ℂ] A₂` (where `A₁` and `A₂` are C⋆-algebras) is called *completely positive (CP)* if `CStarMatrix.map (Fin k) (Fin k) φ` (i.e. applying `φ` to all entries of a k × k matrix) is also positive for every `k : ℕ`. This file defines completely positive maps and develops their basic API.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/CompletelyPositiveMap.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.PositiveLinearMap", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 27.725, "z": -188.409, "size": 0.2585, "title": "Positive linear maps in C⋆-algebras", "summary": "This file develops the API for positive linear maps over C⋆-algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/PositiveLinearMap.html"}, {"id": "Mathlib.Analysis.Distribution.TemperedDistribution", "region_id": "analysis", "micro_elevation": 0.9583, "macro_tier": 2, "macro_tier_score": 0.0286, "macro_tier_override": null, "x": 145.85, "z": -185.605, "size": 0.3235, "title": "TemperedDistribution", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/TemperedDistribution.html"}, {"id": "Mathlib.Analysis.Normed.Ring.Ultra", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 71.273, "z": -212.302, "size": 0.2486, "title": "Ultrametric norms on rings where the norm of one is one", "summary": "This file contains results on the behavior of norms in ultrametric normed rings. The norm must send one to one.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Ring/Ultra.html"}, {"id": "Mathlib.Analysis.Convex.SpecificFunctions.Pow", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 65.847, "z": -180.833, "size": 0.2, "title": "Convexity properties of `rpow`", "summary": "We prove basic convexity properties of the `rpow` function. The proofs are elementary and do not require calculus, and as such this file has only moderate dependencies.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/SpecificFunctions/Pow.html"}, {"id": "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 3, "macro_tier_score": 0.2507, "macro_tier_override": null, "x": 97.22, "z": -184.407, "size": 0.3135, "title": "Collection of convex functions", "summary": "In this file we prove that the following functions are convex or strictly convex: * `strictConvexOn_exp` : The exponential function is strictly convex. * `strictConcaveOn_log_Ioi`, `strictConcaveOn_log_Iio`: `Real.log` is strictly concave on $(0, +∞)$ and $(-∞, 0)$ respectively. * `convexOn_rpow`, `strictConvexOn_rpow` : For `p : ℝ`, `fun x ↦ x ^ p` is convex on $[0, +∞)$ when `1 ≤ p` and strictly convex when `1 <…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/SpecificFunctions/Basic.html"}, {"id": "Mathlib.Analysis.RCLike.TangentCone", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 66.718, "z": -199.294, "size": 0.2789, "title": "Relationships between unique differentiability over `ℝ` and `ℂ`", "summary": "A set of unique differentiability for `ℝ` is also a set of unique differentiability for `ℂ` (or for a general field satisfying `IsRCLikeNormedField 𝕜`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/RCLike/TangentCone.html"}, {"id": "Mathlib.Analysis.Complex.AbelLimit", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 74.207, "z": -226.876, "size": 0.2478, "title": "Abel's limit theorem", "summary": "If a real or complex power series for a function has radius of convergence 1 and the series is only known to converge conditionally at 1, Abel's limit theorem gives the value at 1 as the limit of the function at 1 from the left. \"Left\" for complex numbers means within a fixed cone opening to the left with angle less than `π`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/AbelLimit.html"}, {"id": "Mathlib.Analysis.Complex.AbsMax", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 1, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": 46.506, "z": -240.644, "size": 0.3145, "title": "Maximum modulus principle", "summary": "In this file we prove several versions of the maximum modulus principle. There are several statements that can be called \"the maximum modulus principle\" for maps between normed complex spaces. They differ by assumptions on the domain (any space, a nontrivial space, a finite dimensional space), assumptions on the codomain (any space, a strictly convex space), and by conclusion (either equality of norms or of the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/AbsMax.html"}, {"id": "Mathlib.Analysis.Normed.Module.HahnBanach", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 2, "macro_tier_score": 0.0701, "macro_tier_override": null, "x": 110.342, "z": -206.937, "size": 0.3078, "title": "Hahn-Banach extension theorem", "summary": "In this file, we prove the analytic Hahn-Banach theorem for normed vector spaces. For any continuous linear functional on a subspace, we can extend it to the entire space without changing its norm. For Hahn-Banach theorems for locally convex spaces, see `Mathlib.Analysis.LocallyConvex.HahnBanach`. We prove * `exists_extension_norm_eq`: Hahn-Banach theorem for continuous linear functionals on normed spaces over `ℝ`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/HahnBanach.html"}, {"id": "Mathlib.Analysis.Calculus.ImplicitFunction.Bivariate", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 108.026, "z": -171.532, "size": 0.2, "title": "Implicit function theorem — curried bivariate", "summary": "This specialization of the implicit function theorem applies to a curried bivariate function `f : E₁ → E₂ → F` and assumes continuity of both its partial derivatives at `u : E₁ × E₂` as well as invertibility of `f₂ u.1 u.2 : E₂ →L[𝕜] F` its partial derivative with respect to the second argument. It proves the existence of `ψ : E₁ → E₂` such that for `v` in a neighbourhood of `u` we have `f v.1 v.2 = f u.1 u.2 ↔ ψ…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ImplicitFunction/Bivariate.html"}, {"id": "Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 112.621, "z": -226.012, "size": 0.2676, "title": "Implicit function theorem — domain a product space", "summary": "This specialization of the implicit function theorem applies to an uncurried bivariate function `f : E₁ × E₂ → F` and assumes strict differentiability of `f` at `u : E₁ × E₂` as well as invertibility of `f₂u : E₂ →L[𝕜] F` its partial derivative with respect to the second argument. It proves the existence of `ψ : E₁ → E₂` such that for `v` in a neighbourhood of `u` we have `f v = f u ↔ ψ v.1 = v.2`. This is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ImplicitFunction/ProdDomain.html"}, {"id": "Mathlib.Analysis.ConstantSpeed", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 96.121, "z": -205.012, "size": 0.2, "title": "Constant speed", "summary": "This file defines the notion of constant (and unit) speed for a function `f : ℝ → E` with pseudo-emetric structure on `E` with respect to a set `s : Set ℝ` and \"speed\" `l : ℝ≥0`, and shows that if `f` has locally bounded variation on `s`, it can be obtained (up to distance zero, on `s`), as a composite `φ ∘ (variationOnFromTo f s a)`, where `φ` has unit speed and `a ∈ s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/ConstantSpeed.html"}, {"id": "Mathlib.Analysis.Complex.ReImTopology", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.1815, "macro_tier_override": null, "x": 61.704, "z": -209.651, "size": 0.3294, "title": "Closure, interior, and frontier of preimages under `re` and `im`", "summary": "In this fact we use the fact that `ℂ` is naturally homeomorphic to `ℝ × ℝ` to deduce some topological properties of `Complex.re` and `Complex.im`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/ReImTopology.html"}, {"id": "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 48.095, "z": -206.967, "size": 0.2519, "title": "Divergence integral for Henstock-Kurzweil integral", "summary": "In this file we prove the Divergence Theorem for a Henstock-Kurzweil style integral. The theorem says the following. Let `f : ℝⁿ → Eⁿ` be a function differentiable on a closed rectangular box `I` with derivative `f' x : ℝⁿ →L[ℝ] Eⁿ` at `x ∈ I`. Then the divergence `fun x ↦ ∑ k, f' x eₖ k`, where `eₖ = Pi.single k 1` is the `k`-th basis vector, is integrable on `I`, and its integral is equal to the sum of integrals…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoxIntegral/DivergenceTheorem.html"}, {"id": "Mathlib.Analysis.BoxIntegral.Basic", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 1, "macro_tier_score": 0.0143, "macro_tier_override": null, "x": 108.393, "z": -220.544, "size": 0.2909, "title": "Integrals of Riemann, Henstock-Kurzweil, and McShane", "summary": "In this file we define the integral of a function over a box in `ℝⁿ`. The same definition works for Riemann, Henstock-Kurzweil, and McShane integrals. As usual, we represent `ℝⁿ` as the type of functions `ι → ℝ` for some finite type `ι`. A rectangular box `(l, u]` in `ℝⁿ` is defined to be the set `{x : ι → ℝ | ∀ i, l i < x i ∧ x i ≤ u i}`, see `BoxIntegral.Box`. Let `vol` be a box-additive function on boxes in `ℝⁿ`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoxIntegral/Basic.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.Defs", "region_id": "analysis", "micro_elevation": 0.5625, "macro_tier": 3, "macro_tier_score": 0.252, "macro_tier_override": null, "x": 116.281, "z": -228.881, "size": 0.3885, "title": "Higher differentiability", "summary": "A function is `C^1` on a domain if it is differentiable there, and its derivative is continuous. By induction, it is `C^n` if it is `C^{n-1}` and its (n-1)-th derivative is `C^1` there or, equivalently, if it is `C^1` and its derivative is `C^{n-1}`. It is `C^∞` if it is `C^n` for all n. Finally, it is `C^ω` if it is analytic (as well as all its derivative, which is automatic if the space is complete). We formalize…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/Defs.html"}, {"id": "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 3, "macro_tier_score": 0.1675, "macro_tier_override": null, "x": 28.977, "z": -193.998, "size": 0.3193, "title": "Collection of convex functions", "summary": "In this file we prove that certain specific functions are strictly convex, including the following: * `Even.strictConvexOn_pow` : For an even `n : ℕ` with `2 ≤ n`, `fun x => x ^ n` is strictly convex. * `strictConvexOn_pow` : For `n : ℕ`, with `2 ≤ n`, `fun x => x ^ n` is strictly convex on $[0,+∞)$. * `strictConvexOn_zpow` : For `m : ℤ` with `m ≠ 0, 1`, `fun x => x ^ m` is strictly convex on $[0, +∞)$. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/SpecificFunctions/Deriv.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Sqrt", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 3, "macro_tier_score": 0.2366, "macro_tier_override": null, "x": 76.72, "z": -256.645, "size": 0.2899, "title": "Smoothness of `Real.sqrt`", "summary": "In this file we prove that `Real.sqrt` is infinitely smooth at all points `x ≠ 0` and provide some dot-notation lemmas.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Sqrt.html"}, {"id": "Mathlib.Analysis.Convex.Deriv", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 3, "macro_tier_score": 0.1812, "macro_tier_override": null, "x": 57.626, "z": -169.672, "size": 0.3058, "title": "Convexity of functions and derivatives", "summary": "Here we relate convexity of functions `ℝ → ℝ` to properties of their derivatives.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Deriv.html"}, {"id": "Mathlib.Analysis.RCLike.Sqrt", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 2, "macro_tier_score": 0.056, "macro_tier_override": null, "x": 52.251, "z": -222.149, "size": 0.2916, "title": "Square root on `RCLike`", "summary": "This file contains the definitions `Complex.sqrt` and `RCLike.sqrt` and builds basic API.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/RCLike/Sqrt.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 2, "macro_tier_score": 0.0697, "macro_tier_override": null, "x": 80.322, "z": -189.73, "size": 0.2691, "title": "The continuous functional calculus on product types", "summary": "This file contains results about the continuous functional calculus on (indexed) product types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Pi.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 3, "macro_tier_score": 0.0703, "macro_tier_override": null, "x": 96.832, "z": -219.942, "size": 0.326, "title": "The positive (and negative) parts of a selfadjoint element in a C⋆-algebra", "summary": "This file defines the positive and negative parts of a selfadjoint element in a C⋆-algebra via the continuous functional calculus and develops the basic API, including the uniqueness of the positive and negative parts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/PosPart/Basic.html"}, {"id": "Mathlib.Analysis.Normed.Ring.WithAbs", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 1, "macro_tier_score": 0.0278, "macro_tier_override": null, "x": 78.765, "z": -200.473, "size": 0.2355, "title": "`WithAbs` type synonym", "summary": "`WithAbs v` is a copy of the semiring `R` with the same underlying ring structure, but assigned `v`-dependent instances (such as `NormedRing`) where `v` is an absolute value on `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Ring/WithAbs.html"}, {"id": "Mathlib.Analysis.Normed.Ring.TransferInstance", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 1, "macro_tier_score": 0.0279, "macro_tier_override": null, "x": 73.453, "z": -213.452, "size": 0.2375, "title": "Transfer normed algebraic structures across `Equiv`s", "summary": "In this file, we transfer a (semi-)normed ring structure across an equivalence. This continues the pattern set in `Mathlib/Algebra/Module/TransferInstance.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Ring/TransferInstance.html"}, {"id": "Mathlib.Analysis.Convex.Slope", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 4, "macro_tier_score": 0.3342, "macro_tier_override": null, "x": 78.263, "z": -215.377, "size": 0.3214, "title": "Slopes of convex functions", "summary": "This file relates convexity/concavity of functions in a linearly ordered field and the monotonicity of their slopes. The main use is to show convexity/concavity from monotonicity of the derivative.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Slope.html"}, {"id": "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 77.417, "z": -186.825, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.html"}, {"id": "Mathlib.Analysis.Normed.Operator.NNNorm", "region_id": "analysis", "micro_elevation": 0.2917, "macro_tier": 4, "macro_tier_score": 0.3963, "macro_tier_override": null, "x": 58.884, "z": -206.42, "size": 0.5663, "title": "Operator norm as an `NNNorm`", "summary": "Operator norm as an `NNNorm`, i.e. taking values in non-negative reals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/NNNorm.html"}, {"id": "Mathlib.Analysis.Complex.Positivity", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 105.837, "z": -170.01, "size": 0.2466, "title": "Nonnegativity of values of holomorphic functions", "summary": "We show that if `f` is holomorphic on an open disk `B(c,r)` and all iterated derivatives of `f` at `c` are nonnegative real, then `f z ≥ 0` for all `z ≥ c` in the disk; see `DifferentiableOn.nonneg_of_iteratedDeriv_nonneg`. We also provide a variant `Differentiable.nonneg_of_iteratedDeriv_nonneg` for entire functions and versions showing `f z ≥ f c` when all iterated derivatives except `f` itself are nonnegative.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Positivity.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.RestrictScalars", "region_id": "analysis", "micro_elevation": 0.5833, "macro_tier": 2, "macro_tier_score": 0.0557, "macro_tier_override": null, "x": 104.416, "z": -174.566, "size": 0.2504, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/RestrictScalars.html"}, {"id": "Mathlib.Analysis.ODE.PicardLindelof", "region_id": "analysis", "micro_elevation": 0.8333, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 105.777, "z": -264.741, "size": 0.2481, "title": "Picard-Lindelöf (Cauchy-Lipschitz) Theorem", "summary": "We prove the (local) existence of integral curves and flows to time-dependent vector fields. Let `f : ℝ → E → E` be a time-dependent (local) vector field on a Banach space, and let `t₀ : ℝ` and `x₀ : E`. If `f` is Lipschitz continuous in `x` within a closed ball around `x₀` of radius `a ≥ 0` at every `t` and continuous in `t` at every `x`, then there exists a (local) solution `α : ℝ → E` to the initial value problem…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/ODE/PicardLindelof.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Const", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 4, "macro_tier_score": 0.2952, "macro_tier_override": null, "x": 66.391, "z": -187.488, "size": 0.4505, "title": "Fréchet derivative of constant functions", "summary": "This file contains the usual formulas (and existence assertions) for the derivative of constant functions, including various special cases such as the functions `0`, `1`, `Nat.cast n`, `Int.cast z`, and other numerals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Const.html"}, {"id": "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 86.206, "z": -262.487, "size": 0.2361, "title": "Smooth bump functions in inner product spaces", "summary": "In this file we prove that a real inner product space has smooth bump functions, see `hasContDiffBump_of_innerProductSpace`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/BumpFunction/InnerProduct.html"}, {"id": "Mathlib.Analysis.Calculus.BumpFunction.Basic", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 47.693, "z": -175.07, "size": 0.2648, "title": "Infinitely smooth \"bump\" functions", "summary": "A smooth bump function is an infinitely smooth function `f : E → ℝ` supported on a ball that is equal to `1` on a ball of smaller radius. These functions have many uses in real analysis. E.g., - they can be used to construct a smooth partition of unity which is a very useful tool; - they can be used to approximate a continuous function by infinitely smooth functions. There are two classes of spaces where bump…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/BumpFunction/Basic.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 30.542, "z": -194.094, "size": 0.2553, "title": "Infinitely smooth transition function", "summary": "In this file we construct two infinitely smooth functions with properties that an analytic function cannot have: * `expNegInvGlue` is equal to zero for `x ≤ 0` and is strictly positive otherwise; it is given by `x ↦ exp (-1/x)` for `x > 0`; * `Real.smoothTransition` is equal to zero for `x ≤ 0` and is equal to one for `x ≥ 1`; it is given by `expNegInvGlue x / (expNegInvGlue x + expNegInvGlue (1 - x))`;", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/SmoothTransition.html"}, {"id": "Mathlib.Analysis.Normed.Group.Constructions", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 4, "macro_tier_score": 0.4881, "macro_tier_override": null, "x": 81.103, "z": -213.963, "size": 0.3886, "title": "Product of normed groups and other constructions", "summary": "This file constructs the infinity norm on finite products of normed groups and provides instances for type synonyms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Constructions.html"}, {"id": "Mathlib.Analysis.Normed.Group.Submodule", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 4, "macro_tier_score": 0.4877, "macro_tier_override": null, "x": 84.215, "z": -207.803, "size": 0.3669, "title": "Submodules of normed groups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Submodule.html"}, {"id": "Mathlib.Analysis.Convex.Gauge", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.1394, "macro_tier_override": null, "x": 61.744, "z": -210.752, "size": 0.3001, "title": "The Minkowski functional", "summary": "This file defines the Minkowski functional, aka gauge. The Minkowski functional of a set `s` is the function which associates each point to how much you need to scale `s` for `x` to be inside it. When `s` is symmetric, convex and absorbent, its gauge is a seminorm. Reciprocally, any seminorm arises as the gauge of some set, namely its unit ball. This induces the equivalence of seminorms and locally convex…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Gauge.html"}, {"id": "Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 33.958, "z": -220.109, "size": 0.2, "title": "Pontryagin duality for finite abelian groups", "summary": "This file proves the Pontryagin duality in case of finite abelian groups. This states that any finite abelian group is canonically isomorphic to its double dual (the space of complex-valued characters of its space of complex-valued characters). We first prove it for `ZMod n` and then extend to all finite abelian groups using the Structure Theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/FiniteAbelian/PontryaginDuality.html"}, {"id": "Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 44.141, "z": -181.176, "size": 0.2565, "title": "Orthogonality of characters of a finite abelian group", "summary": "This file proves that characters of a finite abelian group are orthogonal, and in particular that there are at most as many characters as there are elements of the group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/FiniteAbelian/Orthogonality.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Pochhammer", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 91.928, "z": -163.841, "size": 0.2, "title": "Pochhammer polynomials", "summary": "This file proves analysis theorems for Pochhammer polynomials.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Pochhammer.html"}, {"id": "Mathlib.Analysis.Normed.Ring.Finite", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 4, "macro_tier_score": 0.4449, "macro_tier_override": null, "x": 87.488, "z": -210.842, "size": 0.2948, "title": "Finite order elements in normed rings.", "summary": "A finite order element in a normed ring has norm 1. The values of additive characters on finite cancellative monoids have norm 1.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Ring/Finite.html"}, {"id": "Mathlib.Analysis.Complex.Polynomial.UnitTrinomial", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 115.938, "z": -176.61, "size": 0.2478, "title": "Irreducibility of unit trinomials", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Polynomial/UnitTrinomial.html"}, {"id": "Mathlib.Analysis.Convex.Measure", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 53.092, "z": -226.86, "size": 0.246, "title": "Convex sets are null-measurable", "summary": "Let `E` be a finite-dimensional real vector space, let `μ` be a Haar measure on `E`, let `s` be a convex set in `E`. Then the frontier of `s` has measure zero (see `Convex.addHaar_frontier`), hence `s` is a `NullMeasurableSet` (see `Convex.nullMeasurableSet`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Measure.html"}, {"id": "Mathlib.Analysis.Normed.Affine.AddTorsorBases", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 2, "macro_tier_score": 0.0426, "macro_tier_override": null, "x": 51.876, "z": -221.289, "size": 0.3275, "title": "Bases in normed affine spaces.", "summary": "This file contains results about bases in normed affine spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Affine/AddTorsorBases.html"}, {"id": "Mathlib.Analysis.Convex.Body", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 69.067, "z": -201.237, "size": 0.246, "title": "Convex bodies", "summary": "This file contains the definition of the type `ConvexBody V` consisting of convex, compact, nonempty subsets of a real topological vector space `V`. `ConvexBody V` is a module over the nonnegative reals (`NNReal`) and a pseudo-metric space. If `V` is a normed space, `ConvexBody V` is a metric space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Body.html"}, {"id": "Mathlib.Analysis.RCLike.Inner", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 1, "macro_tier_score": 0.0143, "macro_tier_override": null, "x": 62.225, "z": -249.984, "size": 0.2852, "title": "L2 inner product of finite sequences", "summary": "This file defines the weighted L2 inner product of functions `f g : ι → R` where `ι` is a fintype as `∑ i, conj (f i) * g i`. This convention (conjugation on the left) matches the inner product coming from `RCLike.innerProductSpace`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/RCLike/Inner.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric", "region_id": "analysis", "micro_elevation": 0.5417, "macro_tier": 3, "macro_tier_score": 0.0705, "macro_tier_override": null, "x": 43.116, "z": -195.054, "size": 0.3361, "title": "Isometric continuous functional calculus", "summary": "This file adds a class for an *isometric* continuous functional calculus. This is separate from the usual `ContinuousFunctionalCalculus` class because we prefer not to require a metric (or a norm) on the algebra for reasons discussed in the module documentation for that file. Of course, with a metric on the algebra and an isometric continuous functional calculus, the algebra must *be* a C⋆-algebra already. As such,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Isometric.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart", "region_id": "analysis", "micro_elevation": 0.5417, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 119.509, "z": -206.949, "size": 0.2, "title": "Interactions of the continuous functional calculus with the real and imaginary part", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/RealImaginaryPart.html"}, {"id": "Mathlib.Analysis.Normed.Group.SemiNormedGrp.Completion", "region_id": "analysis", "micro_elevation": 0.1458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 71.591, "z": -202.954, "size": 0.2, "title": "Completions of normed groups", "summary": "This file contains an API for completions of seminormed groups (basic facts about objects and morphisms).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/SemiNormedGrp/Completion.html"}, {"id": "Mathlib.Analysis.Normed.Group.SemiNormedGrp", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 89.124, "z": -209.639, "size": 0.2552, "title": "The category of seminormed groups", "summary": "We define `SemiNormedGrp`, the category of seminormed groups and normed group homs between them, as well as `SemiNormedGrp₁`, the subcategory of norm non-increasing morphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/SemiNormedGrp.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.ChebyshevGauss", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 129.724, "z": -236.049, "size": 0.2, "title": "Chebyshev polynomials over the reals: Chebyshev–Gauss", "summary": "The Chebyshev–Gauss property calculates an integral of a polynomial of degree `< 2 * n` with respect to the weight function `√(1 - x ^ 2)⁻¹` supported on `[-1, 1]` by a sum over appropriate evaluations of the polynomial.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev/ChebyshevGauss.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Orthogonality", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 34.283, "z": -179.214, "size": 0.2676, "title": "Chebyshev polynomials over the reals: orthogonality", "summary": "Chebyshev T polynomials are orthogonal with respect to `√(1 - x ^ 2)⁻¹`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev/Orthogonality.html"}, {"id": "Mathlib.Analysis.Complex.UpperHalfPlane.Measure", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 41.054, "z": -168.768, "size": 0.2, "title": "Invariant measure on the upper half-plane", "summary": "We equip the upper half-plane with a measure, defined as the restriction of the usual measure on `ℂ` weighted by the function `1 / (im z) ^ 2`. We show that this measure is invariant under the action of `GL(2, ℝ)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/UpperHalfPlane/Measure.html"}, {"id": "Mathlib.Analysis.Complex.UpperHalfPlane.Manifold", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.0143, "macro_tier_override": null, "x": 119.634, "z": -247.422, "size": 0.2861, "title": "Manifold structure on the upper half plane.", "summary": "In this file we define the complex manifold structure on the upper half-plane, and show it is invariant under Moebius transformations. We also calculate the derivative, and give an explicit formula for its Jacobian determinant over `ℝ` (used in proving that the action preserves a suitable measure).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/UpperHalfPlane/Manifold.html"}, {"id": "Mathlib.Analysis.Distribution.SchwartzSpace", "region_id": "analysis", "micro_elevation": 0.875, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 143.254, "z": -219.061, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/SchwartzSpace.html"}, {"id": "Mathlib.Analysis.Asymptotics.Completion", "region_id": "analysis", "micro_elevation": 0.1458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 90.567, "z": -208.222, "size": 0.2, "title": "Asymptotics in the completion of a normed space", "summary": "In this file we prove lemmas relating `f = O(g)` etc for composition of functions with coercion of a seminormed group to its completion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Asymptotics/Completion.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Positive", "region_id": "analysis", "micro_elevation": 0.7917, "macro_tier": 2, "macro_tier_score": 0.0286, "macro_tier_override": null, "x": 132.955, "z": -232.885, "size": 0.3214, "title": "Positive operators", "summary": "In this file we define when an operator in a Hilbert space is positive. We follow Bourbaki's choice of requiring self adjointness in the definition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Positive.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform", "region_id": "analysis", "micro_elevation": 0.8958, "macro_tier": 2, "macro_tier_score": 0.0566, "macro_tier_override": null, "x": 141.316, "z": -186.52, "size": 0.3347, "title": "Fourier transform of the Gaussian", "summary": "We prove that the Fourier transform of the Gaussian function is another Gaussian: * `integral_cexp_quadratic`: general formula for `∫ (x : ℝ), exp (b * x ^ 2 + c * x + d)` * `fourierIntegral_gaussian`: for all complex `b` and `t` with `0 < re b`, we have `∫ x:ℝ, exp (I * t * x) * exp (-b * x^2) = (π / b) ^ (1 / 2) * exp (-t ^ 2 / (4 * b))`. * `fourierIntegral_gaussian_pi`: a variant with `b` and `t` scaled to give a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.html"}, {"id": "Mathlib.Analysis.ODE.ExistUnique", "region_id": "analysis", "micro_elevation": 0.8542, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 125.696, "z": -166.967, "size": 0.253, "title": "Existence and uniqueness of solutions to ODEs", "summary": "This file collects the public-facing existence and uniqueness theorems for solutions to ODEs in normed spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/ODE/ExistUnique.html"}, {"id": "Mathlib.Analysis.ODE.Basic", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 63.167, "z": -185.973, "size": 0.2619, "title": "Integral curves of vector fields on a normed vector space", "summary": "Let `E` be a normed vector space and `v : ℝ → E → E` be a time-dependent vector field on `E`. An integral curve of `v` is a function `γ : ℝ → E` such that the derivative of `γ` at `t` equals `v t (γ t)`. The integral curve may only be defined for all `t` within some subset of `ℝ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/ODE/Basic.html"}, {"id": "Mathlib.Analysis.ODE.Gronwall", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 88.85, "z": -158.5, "size": 0.2481, "title": "Grönwall's inequality", "summary": "The main technical result of this file is the Grönwall-like inequality `norm_le_gronwallBound_of_norm_deriv_right_le`. It states that if `f : ℝ → E` satisfies `‖f a‖ ≤ δ` and `∀ x ∈ [a, b), ‖f' x‖ ≤ K * ‖f x‖ + ε`, then for all `x ∈ [a, b]` we have `‖f x‖ ≤ δ * exp (K * x) + (ε / K) * (exp (K * x) - 1)`. Then we use this inequality to prove some estimates on the possible rate of growth of the distance between two…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/ODE/Gronwall.html"}, {"id": "Mathlib.Analysis.Normed.Group.CocompactMap", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 82.495, "z": -213.35, "size": 0.2, "title": "Cocompact maps in normed groups", "summary": "This file gives a characterization of cocompact maps in terms of norm estimates.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/CocompactMap.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Log.RpowTendsto", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 76.424, "z": -239.791, "size": 0.2338, "title": "The logarithm as a limit of powers", "summary": "This file shows that the logarithm can be expressed as a limit of powers, namely that `p⁻¹ * (x ^ p - 1)` tends to `log x` as `p` tends to zero for positive `x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Log/RpowTendsto.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Orthogonal", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 3, "macro_tier_score": 0.2089, "macro_tier_override": null, "x": 107.306, "z": -187.595, "size": 0.2972, "title": "Orthogonal complements of submodules", "summary": "In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established. We make duplicates for `Submodule` and `ClosedSubmodule`. Some of the more subtle results about the orthogonal complement are delayed to `Mathlib/Analysis/InnerProductSpace/Projection/`. See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Orthogonal.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Affine", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 57.49, "z": -205.506, "size": 0.2836, "title": "Normed affine spaces over an inner product space", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Affine.html"}, {"id": "Mathlib.Analysis.Normed.Group.Ultra", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 1, "macro_tier_score": 0.0145, "macro_tier_override": null, "x": 77.962, "z": -216.876, "size": 0.3012, "title": "Ultrametric norms", "summary": "This file contains results on the behavior of norms in ultrametric groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Ultra.html"}, {"id": "Mathlib.Analysis.Calculus.IteratedDeriv.WithinZpow", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 126.236, "z": -194.129, "size": 0.2292, "title": "Derivatives of `x ^ m`, `m : ℤ` within an open set", "summary": "In this file we prove theorems about iterated derivatives of `x ^ m`, `m : ℤ` within an open set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/IteratedDeriv/WithinZpow.html"}, {"id": "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ContDiff", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 85.599, "z": -256.423, "size": 0.2478, "title": "Inverse function theorem, `C^r` case", "summary": "In this file we specialize the inverse function theorem to `C^r`-smooth functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/InverseFunctionTheorem/ContDiff.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 81.157, "z": -227.775, "size": 0.239, "title": "Conditions on unitary elements imposed by the continuous functional calculus", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unitary.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.0709, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.3621, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unital.html"}, {"id": "Mathlib.Analysis.Convex.Segment", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.4448, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2809, "title": "Segments in vector spaces", "summary": "In a 𝕜-vector space, we define the following objects and properties. * `segment 𝕜 x y`: Closed segment joining `x` and `y`. * `openSegment 𝕜 x y`: Open segment joining `x` and `y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Segment.html"}, {"id": "Mathlib.Analysis.Real.Hyperreal", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 81.439, "z": -209.985, "size": 0.2, "title": "Construction of the hyperreal numbers as an ultraproduct of real sequences", "summary": "We define the `Hyperreal` numbers as quotients of sequences `ℕ → ℝ` by an ultrafilter. These form a field, and we prove some of their basic properties. Note that most of the machinery that is usually defined for the specific purpose of non-standard analysis (infinitesimal and infinite elements, standard parts) has been generalized to other non-archimedean fields. In particular: - `ArchimedeanClass` can be used to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Real/Hyperreal.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 68.021, "z": -229.002, "size": 0.2, "title": "Chebyshev polynomials over the reals: some extremal properties", "summary": "* Chebyshev polynomials have largest leading coefficient, following proof in https://math.stackexchange.com/a/978145/1277 * Chebyshev polynomials maximize iterated derivatives at 1 and beyond", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev/Extremal.html"}, {"id": "Mathlib.Analysis.Complex.UpperHalfPlane.ProperAction", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 66.624, "z": -165.825, "size": 0.239, "title": "Transitivity and properness of actions", "summary": "We show that the actions of `SL(2, ℝ)` and `GL(2, ℝ)` on the upper half-plane are jointly continuous, and the action of `SL(2, ℝ)` is proper. (These results require more imports than in `UpperHalfPlane.Topology`, because they use the topology on the group as well) TODO: Show properness of the action of `PGL(2, ℝ)` once this is defined.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/UpperHalfPlane/ProperAction.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Complex.Arctan", "region_id": "analysis", "micro_elevation": 0.8542, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 141.824, "z": -218.317, "size": 0.2478, "title": "Complex arctangent", "summary": "This file defines the complex arctangent `Complex.arctan` as $$\\arctan z = -\\frac i2 \\log \\frac{1 + zi}{1 - zi}$$ and shows that it extends `Real.arctan` to the complex plane. Its Taylor series expansion $$\\arctan z = \\frac{(-1)^n}{2n + 1} z^{2n + 1},\\ |z|<1$$ is proved in `Complex.hasSum_arctan`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Complex/Arctan.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.JointEigenspace", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 102.438, "z": -161.17, "size": 0.2, "title": "Joint eigenspaces of commuting symmetric operators", "summary": "This file collects various decomposition results for joint eigenspaces of commuting symmetric operators on a finite-dimensional inner product space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/JointEigenspace.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Spectrum", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 2, "macro_tier_score": 0.0421, "macro_tier_override": null, "x": 89.933, "z": -260.364, "size": 0.2924, "title": "Spectral theory of self-adjoint operators", "summary": "This file covers the spectral theory of self-adjoint operators on an inner product space. The first part of the file covers general properties, true without any condition on boundedness or compactness of the operator or finite-dimensionality of the underlying space, notably: * `LinearMap.IsSymmetric.conj_eigenvalue_eq_self`: the eigenvalues are real * `LinearMap.IsSymmetric.orthogonalFamily_eigenspaces`: the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Spectrum.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.WithLp", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 3, "macro_tier_score": 0.1115, "macro_tier_override": null, "x": 100.081, "z": -252.268, "size": 0.2883, "title": "Derivatives on `WithLp`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/WithLp.html"}, {"id": "Mathlib.Analysis.Complex.ValueDistribution.Proximity.Basic", "region_id": "analysis", "micro_elevation": 0.8542, "macro_tier": 1, "macro_tier_score": 0.0143, "macro_tier_override": null, "x": 93.628, "z": -148.58, "size": 0.2909, "title": "The Proximity Function of Value Distribution Theory", "summary": "This file defines the \"proximity function\" attached to a meromorphic function defined on the complex plane. Also known as the `Nevanlinna Proximity Function`, this is one of the three main functions used in Value Distribution Theory. The proximity function is a logarithmically weighted measure quantifying how well a meromorphic function `f` approximates the constant function `a` on the circle of radius `R` in the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/ValueDistribution/Proximity/Basic.html"}, {"id": "Mathlib.Analysis.Normed.Algebra.GelfandFormula", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 2, "macro_tier_score": 0.0419, "macro_tier_override": null, "x": 37.155, "z": -186.246, "size": 0.2589, "title": "Gelfand's formula and other results on the spectrum in complex Banach algebras", "summary": "This file contains results on the spectrum of elements in a complex Banach algebra, including **Gelfand's formula** and the **Gelfand-Mazur theorem** and the fact that every element in a complex Banach algebra has nonempty spectrum.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Algebra/GelfandFormula.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Pow.Integral", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2, "title": "The integral of the real power of a nonnegative function", "summary": "In this file, we give a common application of the layer cake formula --- a representation of the integral of the p:th power of a nonnegative function `f`: `∫ f(ω)^p ∂μ(ω) = p * ∫ t^(p-1) * μ {ω | f(ω) ≥ t} dt`. A variant of the formula with measures of sets of the form `{ω | f(ω) > t}` instead of `{ω | f(ω) ≥ t}` is also included. Moreover, we prove that `‖x‖ ^ (-d + ε)` is locally integrable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Pow/Integral.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Unitary.Maps", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 93.675, "z": -216.212, "size": 0.2, "title": "Unitary maps in C⋆-algebras", "summary": "This file defines some basic maps by unitaries in C⋆-algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Unitary/Maps.html"}, {"id": "Mathlib.Analysis.Polynomial.Fourier", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 0, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 130.509, "z": -192.495, "size": 0.2295, "title": "Fourier Coefficients of Polynomials", "summary": "We define an algebra map from `ℂ[X]` to the `MeasureTheory.Lp` (with `p := 2`) space on the additive circle and show that it sends monomials to the Fourier basis. From this, we derive that polynomial coefficients match Fourier coefficients and prove Parseval's identity for polynomials.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Polynomial/Fourier.html"}, {"id": "Mathlib.Analysis.Meromorphic.Complex", "region_id": "analysis", "micro_elevation": 0.9375, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 61.774, "z": -275.623, "size": 0.2478, "title": "The Gamma function is meromorphic", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Meromorphic/Complex.html"}, {"id": "Mathlib.Analysis.Meromorphic.NormalForm", "region_id": "analysis", "micro_elevation": 0.7917, "macro_tier": 2, "macro_tier_score": 0.056, "macro_tier_override": null, "x": 136.192, "z": -223.385, "size": 0.2876, "title": "Normal form of meromorphic functions and continuous extension", "summary": "If a function `f` is meromorphic on `U` and if `g` differs from `f` only along a set that is codiscrete within `U`, then `g` is likewise meromorphic. The set of meromorphic functions is therefore huge, and `=ᶠ[codiscreteWithin U]` defines an equivalence relation. This file implements continuous extension to provide an API that allows picking the 'unique best' representative of any given equivalence class, where…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Meromorphic/NormalForm.html"}, {"id": "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 106.094, "z": -254.266, "size": 0.2, "title": "The Riemann-Lebesgue Lemma", "summary": "In this file we prove the Riemann-Lebesgue lemma, for functions on finite-dimensional real vector spaces `V`: if `f` is a function on `V` (valued in a complete normed space `E`), then the Fourier transform of `f`, viewed as a function on the dual space of `V`, tends to 0 along the cocompact filter. Here the Fourier transform is defined by `fun w : StrongDual ℝ V ↦ ∫ (v : V), exp (↑(2 * π * w v) * I) • f v`. This is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/RiemannLebesgueLemma.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "region_id": "analysis", "micro_elevation": 0.2917, "macro_tier": 1, "macro_tier_score": 0.0279, "macro_tier_override": null, "x": 100.0, "z": -216.897, "size": 0.2378, "title": "Limits of `P(x) / e ^ x` for a polynomial `P`", "summary": "In this file we prove that $\\lim_{x\\to\\infty}\\frac{P(x)}{e^x}=0$ for any polynomial `P`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/PolynomialExp.html"}, {"id": "Mathlib.Analysis.LocallyConvex.ContinuousOfBounded", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 65.27, "z": -220.38, "size": 0.2454, "title": "Continuity and Von Neumann boundedness", "summary": "This file proves that for two topological vector spaces `E` and `F` over nontrivially normed fields, if `E` is first countable, then every locally bounded linear map `E →ₛₗ[σ] F` is continuous (this is `LinearMap.continuous_of_locally_bounded`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.html"}, {"id": "Mathlib.Analysis.LocallyConvex.Bounded", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 4, "macro_tier_score": 0.3908, "macro_tier_override": null, "x": 91.402, "z": -197.266, "size": 0.3838, "title": "Von Neumann Boundedness", "summary": "This file defines natural or von Neumann bounded sets and proves elementary properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/Bounded.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 4, "macro_tier_score": 0.4039, "macro_tier_override": null, "x": 71.621, "z": -188.266, "size": 0.3448, "title": "The type of angles", "summary": "In this file we define `Real.Angle` to be the quotient group `ℝ/2πℤ` and prove a few simple lemmas about trigonometric functions and angles.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.html"}, {"id": "Mathlib.Analysis.Normed.Group.AddCircle", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 4, "macro_tier_score": 0.403, "macro_tier_override": null, "x": 82.063, "z": -226.16, "size": 0.2673, "title": "The additive circle as a normed group", "summary": "We define the normed group structure on `AddCircle p`, for `p : ℝ`. For example if `p = 1` then: `‖(x : AddCircle 1)‖ = |x - round x|` for any `x : ℝ` (see `UnitAddCircle.norm_eq`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/AddCircle.html"}, {"id": "Mathlib.Analysis.Normed.Field.ProperSpace", "region_id": "analysis", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 91.849, "z": -206.745, "size": 0.2593, "title": "Proper nontrivially normed fields", "summary": "Nontrivially normed fields are `ProperSpaces` when they are `WeaklyLocallyCompact`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Field/ProperSpace.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.OfCompLeft", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 3, "macro_tier_score": 0.2508, "macro_tier_override": null, "x": 69.608, "z": -187.476, "size": 0.3184, "title": "Inverse function theorem, the \"easy half\"", "summary": "In this file we prove several versions of the following theorem. Consider three functions `f : F → G`, `g : E → F`, and `h : E → G`, together with \"candidate derivatives\" `f' : F →L[𝕜] G`, `g' : E →L[𝕜] F`, and `h' : E →L[𝕜] G`. Suppose that - `f ∘ g = h` in a neighborhood of `a`; - `h` has derivative `h'` at `a`; - `f` has derivative `f'` at `g a`; - `g` is continuous at `a`; - either `f'` has a right inverse…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/OfCompLeft.html"}, {"id": "Mathlib.Analysis.Normed.Module.Multilinear.Curry", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 4, "macro_tier_score": 0.2654, "macro_tier_override": null, "x": 55.008, "z": -202.684, "size": 0.3664, "title": "Currying and uncurrying continuous multilinear maps", "summary": "We associate to a continuous multilinear map in `n+1` variables (i.e., based on `Fin n.succ`) two curried functions, named `f.curryLeft` (which is a continuous linear map on `E 0` taking values in continuous multilinear maps in `n` variables) and `f.curryRight` (which is a continuous multilinear map in `n` variables taking values in continuous linear maps on `E (last n)`). The inverse operations are called…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Multilinear/Curry.html"}, {"id": "Mathlib.Analysis.Meromorphic.RCLike", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 88.767, "z": -155.396, "size": 0.2478, "title": "Meromorphic Functions over the Real and Complex Numbers", "summary": "This file gathers results on meromorphic functions specifict to the real and complex numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Meromorphic/RCLike.html"}, {"id": "Mathlib.Analysis.Meromorphic.Order", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 2, "macro_tier_score": 0.0568, "macro_tier_override": null, "x": 93.432, "z": -261.128, "size": 0.3498, "title": "Orders of Meromorphic Functions", "summary": "This file defines the order of a meromorphic function `f` at a point `z₀`, as an element of `ℤ ∪ {∞}`. We characterize the order being `< 0`, or `= 0`, or `> 0`, as the convergence of the function to infinity, resp. a nonzero constant, resp. zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Meromorphic/Order.html"}, {"id": "Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 1, "macro_tier_score": 0.015, "macro_tier_override": null, "x": 63.09, "z": -219.867, "size": 0.3412, "title": "Group action on the upper half-plane", "summary": "We equip the upper half-plane with the structure of a `GL (Fin 2) ℝ` action by fractional linear transformations (composing with complex conjugation when needed to extend the action from the positive-determinant subgroup, so that `!![-1, 0; 0, 1]` acts as `z ↦ -conj z`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/UpperHalfPlane/MoebiusAction.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 2, "macro_tier_score": 0.0419, "macro_tier_override": null, "x": 123.733, "z": -215.682, "size": 0.2679, "title": "Absolute value defined via the continuous functional calculus", "summary": "This file defines the absolute value via the non-unital continuous functional calculus and provides basic API.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Abs.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute", "region_id": "analysis", "micro_elevation": 0.5417, "macro_tier": 2, "macro_tier_score": 0.0558, "macro_tier_override": null, "x": 88.529, "z": -248.208, "size": 0.265, "title": "Commuting with applications of the continuous functional calculus", "summary": "This file shows that if an element `b` commutes with both `a` and `star a`, then it commutes with `cfc f a` (or `cfcₙ f a`). In the case where `a` is selfadjoint, we may reduce the hypotheses.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Commute.html"}, {"id": "Mathlib.Analysis.SpecificLimits.FloorPow", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 49.929, "z": -214.534, "size": 0.239, "title": "Results on discretized exponentials", "summary": "We state several auxiliary results pertaining to sequences of the form `⌊c^n⌋₊`. * `tendsto_div_of_monotone_of_tendsto_div_floor_pow`: If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all `c > 1`, then `u n / n` tends to `l`. * `sum_div_nat_floor_pow_sq_le_div_sq`: The sum of `1/⌊c^i⌋₊^2` above a threshold `j` is comparable to `1/j^2`, up to a multiplicative constant.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecificLimits/FloorPow.html"}, {"id": "Mathlib.Analysis.Complex.SummableUniformlyOn", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 120.041, "z": -237.316, "size": 0.252, "title": "Differentiability of uniformly convergent series sums of functions", "summary": "We collect some results about the differentiability of infinite sums.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/SummableUniformlyOn.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent", "region_id": "analysis", "micro_elevation": 0.8958, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 37.291, "z": -159.848, "size": 0.252, "title": "Cotangent", "summary": "This file contains lemmas about the cotangent function, including useful series expansions. In particular, we prove that `π * cot (π * z) = π * I - 2 * π * I * ∑' n : ℕ, Complex.exp (2 * π * I * z) ^ n` as well as the infinite sum representation of cotangent (also known as the Mittag-Leffler expansion): `π * cot (π * z) = 1 / z + ∑' n : ℕ+, (1 / (z - n) + 1 / (z + n))`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Cotangent.html"}, {"id": "Mathlib.Analysis.Calculus.TangentCone.Seq", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 91.837, "z": -199.961, "size": 0.2, "title": "Tangent cone points as limits of sequences", "summary": "This file contains a few ways to describe `tangentConeAt` as the set of limits of certain sequences. In many cases, one can generalize results about the tangent cone by using `mem_tangentConeAt_of_seq` and `exists_fun_of_mem_tangentConeAt` instead of these lemmas.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/TangentCone/Seq.html"}, {"id": "Mathlib.Analysis.Complex.IsIntegral", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2298, "title": "Integral elements of ℂ", "summary": "This file proves that `Complex.I` is integral over ℤ and ℚ.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/IsIntegral.html"}, {"id": "Mathlib.Analysis.Normed.Affine.AddTorsor", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 2, "macro_tier_score": 0.043, "macro_tier_override": null, "x": 66.961, "z": -205.243, "size": 0.3547, "title": "Torsors of normed space actions.", "summary": "This file contains lemmas about normed additive torsors over normed spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Affine/AddTorsor.html"}, {"id": "Mathlib.Analysis.Normed.Module.Extr", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 93.387, "z": -199.473, "size": 0.251, "title": "(Local) maximums in a normed space", "summary": "In this file we prove the following lemma, see `IsMaxFilter.norm_add_sameRay`. If `f : α → E` is a function such that `norm ∘ f` has a maximum along a filter `l` at a point `c` and `y` is a vector on the same ray as `f c`, then the function `fun x => ‖f x + y‖` has a maximum along `l` at `c`. Then we specialize it to the case `y = f c` and to different special cases of `IsMaxFilter`: `IsMaxOn`, `IsLocalMaxOn`, and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Extr.html"}, {"id": "Mathlib.Analysis.Normed.Group.Int", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 4, "macro_tier_score": 0.4876, "macro_tier_override": null, "x": 75.534, "z": -213.694, "size": 0.3618, "title": "ℤ as a normed group", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Int.html"}, {"id": "Mathlib.Analysis.Normed.Group.Real", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 4, "macro_tier_score": 0.5034, "macro_tier_override": null, "x": 81.939, "z": -213.663, "size": 0.4473, "title": "Norms on `ℝ` and `ℝ≥0`", "summary": "We equip `ℝ`, `ℝ≥0`, and `ℝ≥0∞` with their natural norms / enorms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Real.html"}, {"id": "Mathlib.Analysis.Meromorphic.Divisor", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 2, "macro_tier_score": 0.0562, "macro_tier_override": null, "x": 130.377, "z": -184.278, "size": 0.3051, "title": "The Divisor of a meromorphic function", "summary": "This file defines the divisor of a meromorphic function and proves the most basic lemmas about those divisors. The lemma `MeromorphicOn.divisor_restrict` guarantees compatibility between restrictions of divisors and of meromorphic functions to subsets of their domain of definition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Meromorphic/Divisor.html"}, {"id": "Mathlib.Analysis.Meromorphic.IsolatedZeros", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.0559, "macro_tier_override": null, "x": 32.493, "z": -182.103, "size": 0.2833, "title": "Principles of Isolated Zeros and Identity Principles for Meromorphic Functions", "summary": "In line with results in `Mathlib.Analysis.Analytic.IsolatedZeros` and `Mathlib.Analysis.Analytic.Uniqueness`, this file establishes principles of isolated zeros and identity principles for meromorphic functions. Compared to the results for analytic functions, the principles established here are a little more complicated to state. This is because meromorphic functions can be modified at will along discrete subsets…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Meromorphic/IsolatedZeros.html"}, {"id": "Mathlib.Analysis.Convex.MetricSpace", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 78.012, "z": -223.1, "size": 0.2, "title": "Convex spaces with compatible metric structure", "summary": "A convex space has a compatible metric structure if `dist(∑ tᵢ xᵢ, ∑ tᵢ yᵢ) ≤ ∑ tᵢ dist(xᵢ, yᵢ)`. This is what one would expect from the triangle inequality. Note that there is a separate notion of [convex metric spaces](https://en.wikipedia.org/wiki/Convex_metric_space) in the literature that has little to do with this definition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/MetricSpace.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Projection.Minimal", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 3, "macro_tier_score": 0.209, "macro_tier_override": null, "x": 98.014, "z": -195.492, "size": 0.3127, "title": "Existence of minimizers (Hilbert projection theorem)", "summary": "This file shows the existence of minimizers (also known as the Hilbert projection theorem). This is the key tool that is used to define `Submodule.orthogonalProjection` in `Mathlib/Analysis/InnerProductSpace/Projection/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Projection/Minimal.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Symmetric", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.209, "macro_tier_override": null, "x": 112.629, "z": -193.066, "size": 0.3127, "title": "Symmetric linear maps in an inner product space", "summary": "This file defines and proves basic theorems about symmetric **not necessarily bounded** operators on an inner product space, i.e linear maps `T : E → E` such that `∀ x y, ⟪T x, y⟫ = ⟪x, T y⟫`. In comparison to `IsSelfAdjoint`, this definition works for non-continuous linear maps, and doesn't rely on the definition of the adjoint, which allows it to be stated in non-complete space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Symmetric.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional", "region_id": "analysis", "micro_elevation": 0.5625, "macro_tier": 3, "macro_tier_score": 0.2101, "macro_tier_override": null, "x": 93.006, "z": -170.515, "size": 0.3765, "title": "Orthogonal projections in finite-dimensional spaces", "summary": "This file contains results about orthogonal projections in finite-dimensional spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Projection/FiniteDimensional.html"}, {"id": "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 103.194, "z": -221.022, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/NormedSpace/OperatorNorm/Prod.html"}, {"id": "Mathlib.Analysis.Normed.Operator.Prod", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 1, "macro_tier_score": 0.0156, "macro_tier_override": null, "x": 71.164, "z": -232.255, "size": 0.373, "title": "Operator norm: Cartesian products", "summary": "Interaction of operator norm with Cartesian products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Prod.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.Deriv", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 3, "macro_tier_score": 0.238, "macro_tier_override": null, "x": 98.131, "z": -167.586, "size": 0.386, "title": "Higher differentiability in one dimension", "summary": "The general theory of higher derivatives in Mathlib is developed using the Fréchet derivative `fderiv`; but for maps defined on the field, the one-dimensional derivative `deriv` is often easier to use. In this file, we reformulate some higher smoothness results in terms of `deriv`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/Deriv.html"}, {"id": "Mathlib.Analysis.Normed.Module.Ball.RadialEquiv", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 76.928, "z": -196.166, "size": 0.2416, "title": "Homeomorphism between a normed space and sphere times `(0, +∞)`", "summary": "In this file we define a homeomorphism between nonzero elements of a normed space `E` and `Metric.sphere (0 : E) r × Set.Ioi (0 : ℝ)`, `r > 0`. One may think about it as generalization of polar coordinates to any normed space. We also specialize this definition to the case `r = 1` and prove", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Ball/RadialEquiv.html"}, {"id": "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 68.706, "z": -186.218, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.html"}, {"id": "Mathlib.Analysis.Normed.Operator.Completeness", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 2, "macro_tier_score": 0.0698, "macro_tier_override": null, "x": 85.655, "z": -185.826, "size": 0.2797, "title": "Operators on complete normed spaces", "summary": "This file contains statements about norms of operators on complete normed spaces, such as a version of the Banach-Alaoglu theorem (`ContinuousLinearMap.isCompact_image_coe_closedBall`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Completeness.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.WeakOperatorTopology", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 62.218, "z": -165.775, "size": 0.2, "title": "The weak operator topology in Hilbert spaces", "summary": "This file gives a few properties of the weak operator topology that are specific to operators on Hilbert spaces. This mostly involves using the Fréchet-Riesz representation to convert between applications of elements of the dual and inner products with vectors in the space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.html"}, {"id": "Mathlib.Analysis.LocallyConvex.WeakOperatorTopology", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 59.187, "z": -200.064, "size": 0.239, "title": "The weak operator topology", "summary": "This file defines a type copy of `E →L[𝕜] F` (where `E` and `F` are topological vector spaces) which is endowed with the weak operator topology (WOT) rather than the topology of bounded convergence (which is the usual one induced by the operator norm in the normed setting). The WOT is defined as the coarsest topology such that the functional `fun A => y (A x)` is continuous for any `x : E` and `y : StrongDual 𝕜 F`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/WeakOperatorTopology.html"}, {"id": "Mathlib.Analysis.Normed.Module.Ball.Homeomorph", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 3, "macro_tier_score": 0.1253, "macro_tier_override": null, "x": 63.834, "z": -205.06, "size": 0.2768, "title": "(Local) homeomorphism between a normed space and a ball", "summary": "In this file we show that a real (semi)normed vector space is homeomorphic to the unit ball. We formalize it in two ways: - as a `Homeomorph`, see `Homeomorph.unitBall`; - as an `OpenPartialHomeomorph` with `source = Set.univ` and `target = Metric.ball (0 : E) 1`. While the former approach is more natural, the latter approach provides us with a globally defined inverse function which makes it easier to say that this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Ball/Homeomorph.html"}, {"id": "Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 104.475, "z": -217.954, "size": 0.2478, "title": "Projective seminorm on the tensor of a finite family of normed spaces.", "summary": "Let `𝕜` be a normed field and `E` be a family of normed `𝕜`-vector spaces `Eᵢ`, indexed by a finite type `ι`. We define a seminorm on `⨂[𝕜] i, Eᵢ`, which we call the \"projective seminorm\". For `x` an element of `⨂[𝕜] i, Eᵢ`, its projective seminorm is the infimum over all expressions of `x` as `∑ j, ⨂ₜ[𝕜] mⱼ i` (with the `mⱼ` ∈ `Π i, Eᵢ`) of `∑ j, Π i, ‖mⱼ i‖`. In particular, every norm `‖.‖` on `⨂[𝕜] i, Eᵢ`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/PiTensorProduct/ProjectiveSeminorm.html"}, {"id": "Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn", "region_id": "analysis", "micro_elevation": 0.875, "macro_tier": 0, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 132.592, "z": -173.113, "size": 0.2292, "title": "Uniform convergence of products of functions", "summary": "We gather some results about the uniform convergence of infinite products, in particular those of the form `∏' i, (1 + f i x)` for a sequence `f` of complex-valued functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/MultipliableUniformlyOn.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 2, "macro_tier_score": 0.0282, "macro_tier_override": null, "x": 66.231, "z": -259.469, "size": 0.2843, "title": "Differentiability of hyperbolic trigonometric functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/DerivHyp.html"}, {"id": "Mathlib.Analysis.Normed.Lp.lpHolder", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 74.481, "z": -245.677, "size": 0.2, "title": "Hölder's inequality for `lp` spaces", "summary": "This file proves Hölder's inequality for `lp` spaces. We follow the established pattern for Hölder's inequality for `MeasureTheory.Lp` of generalizing multiplication to any continuous bilinear map. Since `lp` is a dependent Π-type, we actually need a uniformly bounded family of bilinear maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Lp/lpHolder.html"}, {"id": "Mathlib.Analysis.Real.Pi.Leibniz", "region_id": "analysis", "micro_elevation": 0.875, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 105.49, "z": -268.212, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Real/Pi/Leibniz.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.FiniteDimension", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 113.488, "z": -176.248, "size": 0.2, "title": "Higher differentiability in finite dimensions.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.html"}, {"id": "Mathlib.Analysis.Normed.Module.ContinuousInverse", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 53.135, "z": -203.958, "size": 0.2, "title": "Continuous linear maps with a continuous left/right inverse", "summary": "This file defines continuous linear maps which admit a continuous left/right inverse. We prove that both of these classes of maps are closed under products, composition and contain linear equivalences, and a sufficient criterion in finite dimension: a surjective linear map on a finite-dimensional space always admits a continuous right inverse; an injective linear map on a finite-dimensional space always admits a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/ContinuousInverse.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 3, "macro_tier_score": 0.2511, "macro_tier_override": null, "x": 58.247, "z": -220.561, "size": 0.3403, "title": "The derivative of the scalar restriction of a linear map", "summary": "For detailed documentation of the Fréchet derivative, see the module docstring of `Mathlib/Analysis/Calculus/FDeriv/Basic.lean`. This file contains the usual formulas (and existence assertions) for the derivative of the scalar restriction of a linear map.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.html"}, {"id": "Mathlib.Analysis.Fourier.Convolution", "region_id": "analysis", "micro_elevation": 0.9583, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 23.575, "z": -167.953, "size": 0.2, "title": "The Fourier transform of the convolution", "summary": "In this file we calculate the Fourier transform of a convolution.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/Convolution.html"}, {"id": "Mathlib.Analysis.MellinTransform", "region_id": "analysis", "micro_elevation": 0.8542, "macro_tier": 2, "macro_tier_score": 0.0425, "macro_tier_override": null, "x": 22.788, "z": -234.629, "size": 0.3215, "title": "The Mellin transform", "summary": "We define the Mellin transform of a locally integrable function on `Ioi 0`, and show it is differentiable in a suitable vertical strip.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/MellinTransform.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Choose", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 96.74, "z": -209.808, "size": 0.239, "title": "Binomial coefficients and factorial variants", "summary": "This file proves asymptotic theorems for binomial coefficients and factorial variants.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Choose.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict", "region_id": "analysis", "micro_elevation": 0.0417, "macro_tier": 2, "macro_tier_score": 0.0698, "macro_tier_override": null, "x": 77.027, "z": -210.265, "size": 0.2798, "title": "Restriction of the continuous functional calculus to a scalar subring", "summary": "The main declaration in this file is: + `SpectrumRestricts.cfc`: builds a continuous functional calculus over a subring of scalars. This is used for automatically deriving the continuous functional calculi on selfadjoint or positive elements from the one for normal elements. This will allow us to take an instance of the `ContinuousFunctionalCalculus ℂ A IsStarNormal` and produce both of the instances…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Restrict.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.CompareExp", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 44.754, "z": -219.309, "size": 0.2, "title": "Growth estimates on `x ^ y` for complex `x`, `y`", "summary": "Let `l` be a filter on `ℂ` such that `Complex.re` tends to infinity along `l` and `Complex.im z` grows at a subexponential rate compared to `Complex.re z`. Then - `Complex.isLittleO_log_abs_re`: `Real.log ∘ Complex.abs` is `o`-small of `Complex.re` along `l`; - `Complex.isLittleO_cpow_mul_exp`: $z^{a_1}e^{b_1 * z} = o\\left(z^{a_1}e^{b_1 * z}\\right)$ along `l` for any complex `a₁`, `a₂` and real `b₁ < b₂`. We use…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/CompareExp.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.SingularValues", "region_id": "analysis", "micro_elevation": 0.8125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 29.058, "z": -240.134, "size": 0.2, "title": "Singular values for finite-dimensional linear maps", "summary": "For a linear map `T` between finite-dimensional inner product spaces `E` and `F`, we define the singular values, which are the square roots of the eigenvalues of `T.adjoint ∘ₗ T`, arranged in descending order and repeated according to their multiplicity. With our definition, there are countably infinitely many singular values, but only the first rank(T) singular values are nonzero. The singular values are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/SingularValues.html"}, {"id": "Mathlib.Analysis.Asymptotics.TVS", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 4, "macro_tier_score": 0.2935, "macro_tier_override": null, "x": 95.394, "z": -199.696, "size": 0.3813, "title": "Asymptotics in a Topological Vector Space", "summary": "This file defines `Asymptotics.IsLittleOTVS`, `Asymptotics.IsBigOTVS`, and `Asymptotics.IsThetaTVS` as generalizations of `Asymptotics.IsLittleO`, `Asymptotics.IsBigO`, and `Asymptotics.IsTheta` from normed spaces to topological vector spaces. Given two functions `f` and `g` taking values in topological vector spaces over a normed field `K`, we say that $f = o(g)$ (resp., $f = O(g)$) if for any neighborhood of zero…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Asymptotics/TVS.html"}, {"id": "Mathlib.Analysis.Convex.EGauge", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 4, "macro_tier_score": 0.3059, "macro_tier_override": null, "x": 89.976, "z": -196.079, "size": 0.2743, "title": "The Minkowski functional, normed field version", "summary": "In this file we define `(egauge 𝕜 s ·)` to be the Minkowski functional (gauge) of the set `s` in a topological vector space `E` over a normed field `𝕜`, as a function `E → ℝ≥0∞`. It is defined as the infimum of the norms of `c : 𝕜` such that `x ∈ c • s`. In particular, for `𝕜 = ℝ≥0` this definition gives an `ℝ≥0∞`-valued version of `gauge` defined in `Mathlib/Analysis/Convex/Gauge.lean`. This definition can be used…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/EGauge.html"}, {"id": "Mathlib.Analysis.Normed.Affine.Ceva", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 67.758, "z": -200.443, "size": 0.2, "title": "Ceva's theorem.", "summary": "This file proves various versions of Ceva's theorem in a `NormedAddTorsor`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Affine/Ceva.html"}, {"id": "Mathlib.Analysis.Normed.Algebra.GelfandMazur", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 84.5, "z": -159.462, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Algebra/GelfandMazur.html"}, {"id": "Mathlib.Analysis.Polynomial.Factorization", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 98.74, "z": -164.534, "size": 0.2676, "title": "Factorization of monic polynomials of given degree", "summary": "This file contains two main results: * `Polynomial.IsMonicOfDegree.eq_mul_isMonicOfDegree_one_isMonicOfDegree` shows that a monic polynomial of positive degree over an algebraically closed field can be written as a monic polynomial of degree 1 times another monic factor. * `Polynomial.IsMonicOfDegree.eq_mul_isMonicOfDegree_two_isMonicOfDegree` shows that a monic polynomial of degree at least two over `ℝ` can be…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Polynomial/Factorization.html"}, {"id": "Mathlib.Analysis.Calculus.TaylorIntegral", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 90.83, "z": -257.059, "size": 0.2, "title": "Taylor's formula with an integral remainder in higher dimensions", "summary": "In this file we prove Taylor's formula with the remainder term in integral form. * `map_add_eq_sum_add_integral_iteratedFDeriv`: version for higher dimensions with `iteratedFDeriv` TODO: add a version that assumes `ContDiffOn f (closedBall x (‖y‖))`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/TaylorIntegral.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 4, "macro_tier_score": 0.2641, "macro_tier_override": null, "x": 78.473, "z": -177.576, "size": 0.2674, "title": "The derivative of bounded bilinear maps", "summary": "For detailed documentation of the Fréchet derivative, see the module docstring of `Mathlib/Analysis/Calculus/FDeriv/Basic.lean`. This file contains the usual formulas (and existence assertions) for the derivative of bounded bilinear maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Bilinear.html"}, {"id": "Mathlib.Analysis.Calculus.IteratedDeriv.Analytic", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 110.728, "z": -169.755, "size": 0.2, "title": "Iterated derivatives of analytic functions with power factors", "summary": "This file contains lemmas about the iterated derivative of a function that factors as a power of `(· - z₀)` times an analytic function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/IteratedDeriv/Analytic.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.MulOpposite", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 112.249, "z": -179.352, "size": 0.2, "title": "Inner product space on `Hᵐᵒᵖ`", "summary": "This file defines the inner product space structure on `Hᵐᵒᵖ` where we define the inner product naturally. We also define `OrthonormalBasis.mulOpposite`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/MulOpposite.html"}, {"id": "Mathlib.Analysis.BoxIntegral.Partition.Filter", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 86.019, "z": -208.685, "size": 0.2538, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoxIntegral/Partition/Filter.html"}, {"id": "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 76.446, "z": -212.425, "size": 0.2592, "title": "Induction on subboxes", "summary": "In this file we prove (see `BoxIntegral.Box.exists_taggedPartition_isHenstock_isSubordinate_homothetic`) that for every box `I` in `ℝⁿ` and a function `r : ℝⁿ → ℝ` positive on `I` there exists a tagged partition `π` of `I` such that * `π` is a Henstock partition; * `π` is subordinate to `r`; * each box in `π` is homothetic to `I` with coefficient of the form `1 / 2 ^ n`. Later we will use this lemma to prove that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoxIntegral/Partition/SubboxInduction.html"}, {"id": "Mathlib.Analysis.BoxIntegral.Partition.Split", "region_id": "analysis", "micro_elevation": 0.0417, "macro_tier": 1, "macro_tier_score": 0.0143, "macro_tier_override": null, "x": 79.715, "z": -206.497, "size": 0.2853, "title": "Split a box along one or more hyperplanes", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoxIntegral/Partition/Split.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 79.825, "z": -255.234, "size": 0.2, "title": "Continuously `k` times differentiable functions with pointwise Hölder continuous derivatives", "summary": "We say that a function is of class $C^{k+(α)}$ at a point `a`, where `k` is a natural number and `0 ≤ α ≤ 1`, if - it is of class $C^k$ at `a` in the sense of `ContDiffAt`; - its `k`th derivative satisfies $D^kf(x)-D^kf(a) = O(‖x - a‖ ^ α)$ as `x → a`. Note that the Hölder condition used in this definition fixes one of the points at `a`. In different sources, it is called *pointwise*, *local*, or *weak* Hölder…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiffHolder/Pointwise.html"}, {"id": "Mathlib.Analysis.Normed.Group.Continuity", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 4, "macro_tier_score": 0.4899, "macro_tier_override": null, "x": 77.311, "z": -213.239, "size": 0.462, "title": "Continuity of the norm on (semi)normed groups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Continuity.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "region_id": "analysis", "micro_elevation": 0.8333, "macro_tier": 2, "macro_tier_score": 0.0281, "macro_tier_override": null, "x": 140.159, "z": -199.921, "size": 0.2756, "title": "Euler's infinite product for the sine function", "summary": "This file proves the infinite product formula $$ \\sin \\pi z = \\pi z \\prod_{n = 1}^\\infty \\left(1 - \\frac{z ^ 2}{n ^ 2}\\right) $$ for any real or complex `z`. Our proof closely follows the article [Salwinski, *Euler's Sine Product Formula: An Elementary Proof*][salwinski2018]: the basic strategy is to prove a recurrence relation for the integrals `∫ x in 0..π/2, cos 2 z x * cos x ^ (2 * n)`, generalising the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.html"}, {"id": "Mathlib.Analysis.Normed.Field.Krasner", "region_id": "analysis", "micro_elevation": 0.5625, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 40.558, "z": -221.255, "size": 0.2516, "title": "Krasner's Lemma", "summary": "In this file, we prove Krasner's lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Field/Krasner.html"}, {"id": "Mathlib.Analysis.Convex.PartitionOfUnity", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 72.735, "z": -207.194, "size": 0.2627, "title": "Partition of unity and convex sets", "summary": "In this file we prove the following lemma, see `exists_continuous_forall_mem_convex_of_local`. Let `X` be a normal paracompact topological space (e.g., any extended metric space). Let `E` be a topological real vector space. Let `t : X → Set E` be a family of convex sets. Suppose that for each point `x : X`, there exists a neighborhood `U ∈ 𝓝 X` and a function `g : X → E` that is continuous on `U` and sends each `y ∈…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/PartitionOfUnity.html"}, {"id": "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 4, "macro_tier_score": 0.2652, "macro_tier_override": null, "x": 107.118, "z": -213.492, "size": 0.3555, "title": "Formal multilinear series", "summary": "In this file we define `FormalMultilinearSeries 𝕜 E F` to be a family of `n`-multilinear maps for all `n`, designed to model the sequence of derivatives of a function. In other files we use this notion to define `C^n` functions (called `contDiff` in `mathlib`) and analytic functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FormalMultilinearSeries.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 32.352, "z": -182.348, "size": 0.2, "title": "Chebyshev polynomials over the reals: roots and extrema", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev/RootsExtrema.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Arcosh", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 0, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 67.682, "z": -157.65, "size": 0.2302, "title": "Inverse of the cosh function", "summary": "In this file we define an inverse of cosh as a function from $[0, ∞)$ to $[1, ∞)$.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Arcosh.html"}, {"id": "Mathlib.Analysis.Convex.PathConnected", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 4, "macro_tier_score": 0.39, "macro_tier_override": null, "x": 75.461, "z": -210.187, "size": 0.3443, "title": "Segment between 2 points as a bundled path", "summary": "In this file we define `Path.segment a b : Path a b` to be the path going from `a` to `b` along the straight segment with constant velocity `b - a`. We also prove basic properties of this construction, then use it to show that a nonempty convex set is path connected. In particular, a topological vector space over `ℝ` is path connected.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/PathConnected.html"}, {"id": "Mathlib.Analysis.Polynomial.MahlerMeasure", "region_id": "analysis", "micro_elevation": 0.8958, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 51.6, "z": -150.494, "size": 0.2276, "title": "Mahler measure of complex polynomials", "summary": "In this file we define the Mahler measure of a polynomial over `ℂ[X]` and prove some basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Polynomial/MahlerMeasure.html"}, {"id": "Mathlib.Analysis.Convex.Intrinsic", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 62.817, "z": -236.527, "size": 0.239, "title": "Intrinsic frontier and interior", "summary": "This file defines the intrinsic frontier, interior and closure of a set in a normed additive torsor. These are also known as relative frontier, interior, closure. The intrinsic frontier/interior/closure of a set `s` is the frontier/interior/closure of `s` considered as a set in its affine span. The intrinsic interior is in general greater than the topological interior, the intrinsic frontier in general less than the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Intrinsic.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.BinaryEntropy", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 122.405, "z": -244.291, "size": 0.2, "title": "Properties of Shannon q-ary entropy and binary entropy functions", "summary": "The [binary entropy function](https://en.wikipedia.org/wiki/Binary_entropy_function) `binEntropy p := - p * log p - (1 - p) * log (1 - p)` is the Shannon entropy of a Bernoulli random variable with success probability `p`. More generally, the q-ary entropy function is the Shannon entropy of the random variable with possible outcomes `{1, ..., q}`, where outcome `1` has probability `1 - p` and all other outcomes are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/BinaryEntropy.html"}, {"id": "Mathlib.Analysis.MeanInequalities", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 3, "macro_tier_score": 0.2234, "macro_tier_override": null, "x": 73.815, "z": -178.14, "size": 0.3426, "title": "Mean value inequalities", "summary": "In this file we prove several inequalities for finite sums, including AM-GM inequality, HM-GM inequality, Young's inequality, Hölder inequality, and Minkowski inequality. Versions for integrals of some of these inequalities are available in `Mathlib/MeasureTheory/Integral/MeanInequalities.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/MeanInequalities.html"}, {"id": "Mathlib.Analysis.Normed.Group.Rat", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 4, "macro_tier_score": 0.4873, "macro_tier_override": null, "x": 79.11, "z": -201.965, "size": 0.3489, "title": "ℚ as a normed group", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Rat.html"}, {"id": "Mathlib.Analysis.ODE.DiscreteGronwall", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 71.157, "z": -260.57, "size": 0.2, "title": "Discrete Grönwall inequality", "summary": "Various forms of the discrete Grönwall inequality, bounding solutions to recurrence inequalities `u (n+1) ≤ c n * u n + b n` and `u (n+1) ≤ (1 + c n) * u n + b n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/ODE/DiscreteGronwall.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass", "region_id": "analysis", "micro_elevation": 0.7917, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 66.065, "z": -265.724, "size": 0.2, "title": "Weierstrass `℘` functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Elliptic/Weierstrass.html"}, {"id": "Mathlib.Analysis.Complex.Liouville", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 3, "macro_tier_score": 0.0977, "macro_tier_override": null, "x": 107.357, "z": -172.933, "size": 0.2969, "title": "Liouville's theorem", "summary": "In this file we prove Liouville's theorem: if `f : E → F` is complex differentiable on the whole space and its range is bounded, then the function is a constant. Various versions of this theorem are formalized in `Differentiable.apply_eq_apply_of_bounded`, `Differentiable.exists_const_forall_eq_of_bounded`, and `Differentiable.exists_eq_const_of_bounded`. The proof is based on the Cauchy integral formula for the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Liouville.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order", "region_id": "analysis", "micro_elevation": 0.8958, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 94.09, "z": -145.561, "size": 0.2, "title": "Order properties of the operator logarithm", "summary": "This file shows that the logarithm is operator monotone and concave, i.e. that `CFC.log` is monotone and concave on the strictly positive elements of a unital C⋆-algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/ExpLog/Order.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order", "region_id": "analysis", "micro_elevation": 0.875, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 143.927, "z": -207.25, "size": 0.2338, "title": "Order properties of `CFC.rpow`", "summary": "This file shows that `a ↦ a ^ p` is monotone for `p ∈ [0, 1]`, where `a` is an element of a C⋆-algebra. The proof makes use of the integral representation of `rpow` in `Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow/Order.html"}, {"id": "Mathlib.Analysis.Convex.FunctionTopology", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 73.891, "z": -209.599, "size": 0.2338, "title": "Topological properties of the set of convex/concave functions", "summary": "We prove the following facts: * `isClosed_setOf_convexOn` : the set of convex functions on a set is closed * `isClosed_setOf_concaveOn` : the set of concave functions on a set is closed", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/FunctionTopology.html"}, {"id": "Mathlib.Analysis.Calculus.LocalExtr.LineDeriv", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 49.764, "z": -243.817, "size": 0.2, "title": "Local extremum and line derivatives", "summary": "If `f` has a local extremum at a point, then the derivative at this point is zero. In this file we prove several versions of this fact for line derivatives.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/LocalExtr/LineDeriv.html"}, {"id": "Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 66.528, "z": -202.389, "size": 0.2, "title": "Kernels and cokernels in SemiNormedGrp₁ and SemiNormedGrp", "summary": "We show that `SemiNormedGrp₁` has cokernels (for which of course the `cokernel.π f` maps are norm non-increasing), as well as the easier result that `SemiNormedGrp` has cokernels. We also show that `SemiNormedGrp` has kernels. So far, I don't see a way to state nicely what we really want: `SemiNormedGrp` has cokernels, and `cokernel.π f` is norm non-increasing. The problem is that the limits API doesn't promise you…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.html"}, {"id": "Mathlib.Analysis.Normed.Group.Quotient", "region_id": "analysis", "micro_elevation": 0.1875, "macro_tier": 4, "macro_tier_score": 0.403, "macro_tier_override": null, "x": 73.742, "z": -221.739, "size": 0.2661, "title": "Quotients of seminormed groups", "summary": "For any `SeminormedAddCommGroup M` and any `S : AddSubgroup M`, we provide a `SeminormedAddCommGroup`, the group quotient `M ⧸ S`. If `S` is closed, we provide `NormedAddCommGroup (M ⧸ S)` (regardless of whether `M` itself is separated). The two main properties of these structures are the underlying topology is the quotient topology and the projection is a normed group homomorphism which is norm non-increasing…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Quotient.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.ContinuousLinearMap", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 47.997, "z": -174.788, "size": 0.239, "title": "`E →L[ℂ] E` as a C⋆-algebra", "summary": "We place this here because, for reasons related to the import hierarchy, it should not be placed in earlier files.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/ContinuousLinearMap.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Gamma.Digamma", "region_id": "analysis", "micro_elevation": 0.9583, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 30.473, "z": -259.125, "size": 0.2, "title": "The digamma function", "summary": "This file defines the digamma function as the logarithmic derivative of the Gamma function and proves some basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Gamma/Digamma.html"}, {"id": "Mathlib.Analysis.Matrix.Spectrum", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 2, "macro_tier_score": 0.0421, "macro_tier_override": null, "x": 102.208, "z": -161.064, "size": 0.2895, "title": "Spectral theory of Hermitian matrices", "summary": "This file proves the spectral theorem for matrices. The proof of the spectral theorem is based on the spectral theorem for linear maps (`LinearMap.IsSymmetric.eigenvectorBasis_apply_self_apply`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Matrix/Spectrum.html"}, {"id": "Mathlib.Analysis.Calculus.Conformal.InnerProduct", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 113.314, "z": -206.007, "size": 0.2, "title": "Conformal maps between inner product spaces", "summary": "A function between inner product spaces which has a derivative at `x` is conformal at `x` iff the derivative preserves inner products up to a scalar multiple.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Conformal/InnerProduct.html"}, {"id": "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 2, "macro_tier_score": 0.056, "macro_tier_override": null, "x": 84.021, "z": -177.796, "size": 0.2882, "title": "Conformal Maps", "summary": "A continuous linear map between real normed spaces `X` and `Y` is `ConformalAt` some point `x` if it is real differentiable at that point and its differential is a conformal linear map.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Conformal/NormedSpace.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.ConformalLinearMap", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 94.308, "z": -180.926, "size": 0.2478, "title": "Conformal maps between inner product spaces", "summary": "In an inner product space, a map is conformal iff it preserves inner products up to a scalar factor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/ConformalLinearMap.html"}, {"id": "Mathlib.Analysis.Convex.AmpleSet", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 97.325, "z": -215.321, "size": 0.2, "title": "Ample subsets of real vector spaces", "summary": "In this file we study ample sets in real vector spaces. A set is ample if all its connected component have full convex hull. Ample sets are an important ingredient for defining ample differential relations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/AmpleSet.html"}, {"id": "Mathlib.Analysis.Complex.IntegerCompl", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 0, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 66.149, "z": -223.702, "size": 0.2292, "title": "Integer Complement", "summary": "We define the complement of the integers in the complex plane and give some basic lemmas about it. We also show that the upper half plane embeds into the integer complement.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/IntegerCompl.html"}, {"id": "Mathlib.Analysis.Convex.Quasiconvex", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 78.991, "z": -203.519, "size": 0.239, "title": "Quasiconvex and quasiconcave functions", "summary": "This file defines quasiconvexity, quasiconcavity and quasilinearity of functions, which are generalizations of unimodality and monotonicity. Convexity implies quasiconvexity, concavity implies quasiconcavity, and monotonicity implies quasilinearity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Quasiconvex.html"}, {"id": "Mathlib.Analysis.Complex.CanonicalDecomposition", "region_id": "analysis", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 78.35, "z": -148.617, "size": 0.2, "title": "Canonical Decomposition", "summary": "If a function `f` is meromorphic on a compact set `U`, then it has only finitely many zeros and poles on the disk, and the theorem `MeromorphicOn.extract_zeros_poles` can be used to re-write `f` as `(∏ᶠ u, (· - u) ^ divisor f U u) • g`, where `g` is analytic without zeros on `U`. In case where `U` is a disk, one consider a similar decomposition, called *Finite Canonical Decomposition* or *Finite Blaschke Product*…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/CanonicalDecomposition.html"}, {"id": "Mathlib.Analysis.Meromorphic.FactorizedRational", "region_id": "analysis", "micro_elevation": 0.8125, "macro_tier": 2, "macro_tier_score": 0.0419, "macro_tier_override": null, "x": 136.178, "z": -190.24, "size": 0.2702, "title": "Factorized Rational Functions", "summary": "This file discusses functions `𝕜 → 𝕜` of the form `∏ᶠ u, (· - u) ^ d u`, where `d : 𝕜 → ℤ` is integer-valued. We show that these \"factorized rational functions\" are meromorphic in normal form, with divisor equal to `d`. Under suitable assumptions, we show that meromorphic functions are equivalent, modulo equality on codiscrete sets, to the product of a factorized rational function and an analytic function without…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Meromorphic/FactorizedRational.html"}, {"id": "Mathlib.Analysis.Calculus.SmoothSeries", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 49.358, "z": -245.481, "size": 0.2361, "title": "Smoothness of series", "summary": "We show that series of functions are differentiable, or smooth, when each individual function in the series is and additionally suitable uniform summable bounds are satisfied. More specifically, * `differentiable_tsum` ensures that a series of differentiable functions is differentiable. * `contDiff_tsum` ensures that a series of `C^n` functions is `C^n`. We also give versions of these statements which are localized…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/SmoothSeries.html"}, {"id": "Mathlib.Analysis.Calculus.ContDiff.Convolution", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 37.789, "z": -230.742, "size": 0.2361, "title": "Differentiability of a convolution of functions", "summary": "Criteria for a convolution of functions to be differentiable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ContDiff/Convolution.html"}, {"id": "Mathlib.Analysis.Complex.CoveringMap", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 125.109, "z": -178.362, "size": 0.253, "title": "Covering maps involving the complex plane", "summary": "In this file, we show that `Complex.exp` and `(· ^ n)` (for `n ≠ 0`) are a covering map on `{0}ᶜ`. We also show that any complex polynomial is a covering map on the set of regular values.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/CoveringMap.html"}, {"id": "Mathlib.Analysis.Normed.Module.Alternating.Curry", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 2, "macro_tier_score": 0.028, "macro_tier_override": null, "x": 103.44, "z": -195.327, "size": 0.2565, "title": "Currying continuous alternating forms", "summary": "In this file we define `ContinuousAlternatingMap.curryLeft` which interprets a continuous alternating map in `n + 1` variables as a continuous linear map in the 0th variable taking values in the continuous alternating maps in `n` variables.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Alternating/Curry.html"}, {"id": "Mathlib.Analysis.Calculus.AddTorsor.AffineMap", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 73.561, "z": -162.744, "size": 0.2634, "title": "Smooth affine maps", "summary": "This file contains results about smoothness of affine maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/AddTorsor/AffineMap.html"}, {"id": "Mathlib.Analysis.Fourier.PoissonSummation", "region_id": "analysis", "micro_elevation": 0.9583, "macro_tier": 1, "macro_tier_score": 0.0145, "macro_tier_override": null, "x": 12.748, "z": -189.781, "size": 0.3024, "title": "Poisson's summation formula", "summary": "We prove Poisson's summation formula `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the Fourier transform of `f`, under the following hypotheses: * `f` is a continuous function `ℝ → ℂ`. * The sum `∑ (n : ℤ), 𝓕 f n` is convergent. * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), ‖f(x + n)‖` is uniformly convergent on `K`. See `Real.tsum_eq_tsum_fourier` for this formulation. These hypotheses are potentially a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/PoissonSummation.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Gamma.Deriv", "region_id": "analysis", "micro_elevation": 0.875, "macro_tier": 2, "macro_tier_score": 0.0281, "macro_tier_override": null, "x": 102.007, "z": -269.606, "size": 0.2723, "title": "Derivative of the Gamma function", "summary": "This file shows that the (complex) `Γ` function is complex-differentiable at all `s : ℂ` with `s ∉ {-n : n ∈ ℕ}`, as well as the real counterpart.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Gamma/Deriv.html"}, {"id": "Mathlib.Analysis.Convex.Cone.Basic", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 80.688, "z": -210.883, "size": 0.2432, "title": "Proper cones", "summary": "We define a *proper cone* as a closed, pointed cone. Proper cones are used in defining conic programs which generalize linear programs. A linear program is a conic program for the positive cone. We then prove Farkas' lemma for conic programs following the proof in the reference below. Farkas' lemma is equivalent to strong duality. So, once we have the definitions of conic and linear programs, the results from this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Cone/Basic.html"}, {"id": "Mathlib.Analysis.LocallyConvex.PointwiseConvergence", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 64.122, "z": -221.39, "size": 0.2, "title": "The topology of pointwise convergence is locally convex", "summary": "We prove that the topology of pointwise convergence is induced by a family of seminorms and that it is locally convex in the topological sense * `PointwiseConvergenceCLM.seminorm`: the seminorms on `E →SLₚₜ[σ] F` given by `A ↦ ‖A x‖` for fixed `x : E`. * `PointwiseConvergenceCLM.withSeminorm`: the topology is induced by the seminorms. * `PointwiseConvergenceCLM.instLocallyConvexSpace`: `E →SLₚₜ[σ] F` is locally…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/PointwiseConvergence.html"}, {"id": "Mathlib.Analysis.Complex.Norm", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 4, "macro_tier_score": 0.4038, "macro_tier_override": null, "x": 80.471, "z": -215.606, "size": 0.3346, "title": "Norm on the complex numbers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Norm.html"}, {"id": "Mathlib.Analysis.NormedSpace.Real", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 72.785, "z": -196.106, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/NormedSpace/Real.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Log.InvLog", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 35.424, "z": -238.813, "size": 0.2523, "title": "Multiplicative inverse of real logarithm", "summary": "We prove properties of the function `x ↦ (log x)⁻¹`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Log/InvLog.html"}, {"id": "Mathlib.Analysis.Normed.Group.Seminorm", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.5003, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2733, "title": "Group seminorms", "summary": "This file defines norms and seminorms in a group. A group seminorm is a function to the reals which is positive-semidefinite and subadditive. A norm further only maps zero to zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Seminorm.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.028, "macro_tier_override": null, "x": 28.389, "z": -190.938, "size": 0.2682, "title": "C⋆-algebraic facts about `a⁺` and `a⁻`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/SpecialFunctions/PosPart.html"}, {"id": "Mathlib.Analysis.Analytic.IsolatedZeros", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 3, "macro_tier_score": 0.2368, "macro_tier_override": null, "x": 36.237, "z": -215.697, "size": 0.3126, "title": "Principle of isolated zeros", "summary": "This file proves the fact that the zeros of a non-constant analytic function of one variable are isolated. It also introduces a little bit of API in the `HasFPowerSeriesAt` namespace that is useful in this setup.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/IsolatedZeros.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Isometric", "region_id": "analysis", "micro_elevation": 0.5625, "macro_tier": 2, "macro_tier_score": 0.028, "macro_tier_override": null, "x": 85.287, "z": -250.321, "size": 0.2652, "title": "Facts about `CFC.posPart` and `CFC.negPart` involving norms", "summary": "This file collects various facts about the positive and negative parts of elements of a C⋆-algebra that involve the norm.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/PosPart/Isometric.html"}, {"id": "Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 106.142, "z": -232.844, "size": 0.2478, "title": "Results on `TrivSqZeroExt R M` related to the norm", "summary": "This file contains results about `NormedSpace.exp` for `TrivSqZeroExt`. It also contains a definition of the $ℓ^1$ norm, which defines $\\|r + m\\| \\coloneqq \\|r\\| + \\|m\\|$. This is not a particularly canonical choice of definition, but it is sufficient to provide a `NormedAlgebra` instance, and thus enables `NormedSpace.exp_add_of_commute` to be used on `TrivSqZeroExt`. If the non-canonicity becomes problematic in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Algebra/TrivSqZeroExt.html"}, {"id": "Mathlib.Analysis.Normed.Algebra.Exponential", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 4, "macro_tier_score": 0.2648, "macro_tier_override": null, "x": 108.392, "z": -227.316, "size": 0.3264, "title": "Exponential in a Banach algebra", "summary": "In this file, we define `NormedSpace.exp : 𝔸 → 𝔸`, the exponential map in a topological algebra `𝔸`. While for most interesting results we need `𝔸` to be normed algebra, we do not require this in the definition in order to make `NormedSpace.exp` independent of a particular choice of norm. The definition also does not require that `𝔸` be complete, but we need to assume it for most results. We then prove some basic…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Algebra/Exponential.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Module.Synonym", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.028, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2584, "title": "Type synonym for types with a `CStarModule` structure", "summary": "It is often the case that we want to construct a `CStarModule` instance on a type that is already endowed with a norm, but this norm is not the one associated to its `CStarModule` structure. For this reason, we create a type synonym `WithCStarModule` which is endowed with the requisite `CStarModule` instance. We also introduce the scoped notation `C⋆ᵐᵒᵈ` for this type synonym. The common use cases are, when `A` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Module/Synonym.html"}, {"id": "Mathlib.Analysis.SpecificLimits.ArithmeticGeometric", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 78.569, "z": -210.096, "size": 0.2589, "title": "Arithmetic-geometric sequences", "summary": "An arithmetic-geometric sequence is a sequence defined by the recurrence relation `u (n + 1) = a * u n + b`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecificLimits/ArithmeticGeometric.html"}, {"id": "Mathlib.Analysis.Normed.Field.TransferInstance", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 1, "macro_tier_score": 0.0278, "macro_tier_override": null, "x": 89.09, "z": -208.738, "size": 0.2355, "title": "Transfer normed algebraic structures across `Equiv`s", "summary": "In this file, we transfer a normed field structure across an equivalence. This continues the pattern set in `Mathlib/Algebra/Module/TransferInstance.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Field/TransferInstance.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Unitary.Span", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 93.597, "z": -154.833, "size": 0.2, "title": "Unitary elements span C⋆-algebras", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Unitary/Span.html"}, {"id": "Mathlib.Analysis.Complex.OpenMapping", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 31.242, "z": -210.404, "size": 0.2, "title": "The open mapping theorem for holomorphic functions", "summary": "This file proves the open mapping theorem for holomorphic functions, namely that an analytic function on a preconnected set of the complex plane is either constant or open. The main step is to show a local version of the theorem that states that if `f` is analytic at a point `z₀`, then either it is constant in a neighborhood of `z₀` or it maps any neighborhood of `z₀` to a neighborhood of its image `f z₀`. The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/OpenMapping.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.TwoDim", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 33.138, "z": -215.507, "size": 0.2653, "title": "Oriented two-dimensional real inner product spaces", "summary": "This file defines constructions specific to the geometry of an oriented two-dimensional real inner product space `E`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/TwoDim.html"}, {"id": "Mathlib.Analysis.Complex.Spectrum", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 2, "macro_tier_score": 0.0698, "macro_tier_override": null, "x": 97.451, "z": -214.926, "size": 0.2798, "title": "Some lemmas on the spectrum and quasispectrum of elements and positivity on `ℂ`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Spectrum.html"}, {"id": "Mathlib.Analysis.Calculus.TangentCone.Defs", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 4, "macro_tier_score": 0.3064, "macro_tier_override": null, "x": 83.733, "z": -212.146, "size": 0.3219, "title": "Tangent cone", "summary": "In this file, we define two predicates `UniqueDiffWithinAt 𝕜 s x` and `UniqueDiffOn 𝕜 s` ensuring that, if a function has two derivatives, then they have to coincide. As a direct definition of this fact (quantifying on all target types and all functions) would depend on universes, we use a more intrinsic definition: if all the possible tangent directions to the set `s` at the point `x` span a dense subset of the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/TangentCone/Defs.html"}, {"id": "Mathlib.Analysis.Normed.Order.Basic", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 77.633, "z": -205.613, "size": 0.2, "title": "Ordered normed spaces", "summary": "In this file, we define classes for fields and groups that are both normed and ordered. These are mostly useful to avoid diamonds during type class inference.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Order/Basic.html"}, {"id": "Mathlib.Analysis.Convex.Independent", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 80.52, "z": -217.13, "size": 0.2, "title": "Convex independence", "summary": "This file defines convex independent families of points. Convex independence is closely related to affine independence. In both cases, no point can be written as a combination of others. When the combination is affine (that is, any coefficients), this yields affine independence. When the combination is convex (that is, all coefficients are nonnegative), then this yields convex independence. In particular, affine…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Independent.html"}, {"id": "Mathlib.Analysis.Matrix.HermitianFunctionalCalculus", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.0419, "macro_tier_override": null, "x": 58.727, "z": -260.088, "size": 0.2679, "title": "Continuous Functional Calculus for Hermitian Matrices", "summary": "This file defines an instance of the continuous functional calculus for Hermitian matrices over an `RCLike` field `𝕜`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Matrix/HermitianFunctionalCalculus.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Matrix", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 2, "macro_tier_score": 0.0424, "macro_tier_override": null, "x": 123.797, "z": -191.904, "size": 0.3133, "title": "Analytic properties of the `star` operation on matrices", "summary": "This transports the operator norm on `EuclideanSpace 𝕜 n →L[𝕜] EuclideanSpace 𝕜 m` to `Matrix m n 𝕜`. See the file `Mathlib/Analysis/Matrix.lean` for many other matrix norms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Matrix.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.MulExpNegMulSq", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 28.234, "z": -211.708, "size": 0.2363, "title": "Definition of `mulExpNegMulSq` and properties", "summary": "`mulExpNegMulSq` is the mapping `fun (ε : ℝ) (x : ℝ) => x * Real.exp (- (ε * x * x))`. By composition, it can be used to transform a function `g : E → ℝ` into a bounded function `mulExpNegMulSq ε ∘ g : E → ℝ = fun x => g x * Real.exp (-ε * g x * g x)` with useful boundedness and convergence properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/MulExpNegMulSq.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Orientation", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 1, "macro_tier_score": 0.0143, "macro_tier_override": null, "x": 73.701, "z": -254.8, "size": 0.2853, "title": "Orientations of real inner product spaces.", "summary": "This file provides definitions and proves lemmas about orientations of real inner product spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Orientation.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Completion", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 2, "macro_tier_score": 0.028, "macro_tier_override": null, "x": 57.607, "z": -188.853, "size": 0.2585, "title": "Completion of an inner product space", "summary": "We show that the separation quotient and the completion of an inner product space are inner product spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Completion.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Artanh", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 58.507, "z": -217.336, "size": 0.2, "title": "Inverse of the tanh function", "summary": "In this file we define an inverse of tanh as a function from ℝ to (-1, 1).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Artanh.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.NormPow", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 0, "macro_tier_score": 0.0139, "macro_tier_override": null, "x": 120.484, "z": -248.736, "size": 0.2302, "title": "Properties about the powers of the norm", "summary": "In this file we prove that `x ↦ ‖x‖ ^ p` is continuously differentiable for an inner product space and for a real number `p > 1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/NormPow.html"}, {"id": "Mathlib.Analysis.Calculus.LagrangeMultipliers", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 2, "macro_tier_score": 0.0417, "macro_tier_override": null, "x": 50.042, "z": -227.73, "size": 0.2346, "title": "Lagrange multipliers", "summary": "In this file we formalize the [Lagrange multipliers](https://en.wikipedia.org/wiki/Lagrange_multiplier) method of solving conditional extremum problems: if a function `φ` has a local extremum at `x₀` on the set `f ⁻¹' {f x₀}`, `f x = (f₀ x, ..., fₙ₋₁ x)`, then the differentials of `fₖ` and `φ` are linearly dependent. First we formulate a geometric version of this theorem which does not rely on the target space being…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/LagrangeMultipliers.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Reproducing", "region_id": "analysis", "micro_elevation": 0.8125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 21.305, "z": -200.221, "size": 0.2, "title": "Reproducing Kernel Hilbert Spaces", "summary": "This file defines vector-valued reproducing Kernel Hilbert spaces, which are Hilbert spaces of functions, as well as characterizing these spaces in terms of infinite-dimensional positive semidefinite matrices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Reproducing.html"}, {"id": "Mathlib.Analysis.Normed.Group.ControlledClosure", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 67.525, "z": -218.298, "size": 0.2, "title": "Extending a backward bound on a normed group homomorphism from a dense set", "summary": "Possible TODO (from the PR's review, https://github.com/leanprover-community/mathlib/pull/8498): \"This feels a lot like the second step in the proof of the Banach open mapping theorem (`exists_preimage_norm_le`) ... wonder if it would be possible to refactor it using one of [the lemmas in this file].\"", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/ControlledClosure.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Rayleigh", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 2, "macro_tier_score": 0.0418, "macro_tier_override": null, "x": 120.798, "z": -180.179, "size": 0.241, "title": "The Rayleigh quotient", "summary": "The Rayleigh quotient of a self-adjoint operator `T` on an inner product space `E` is the function `fun x ↦ ⟪T x, x⟫ / ‖x‖ ^ 2`. The main results of this file are `IsSelfAdjoint.hasEigenvector_of_isMaxOn` and `IsSelfAdjoint.hasEigenvector_of_isMinOn`, which state that if `E` is complete, and if the Rayleigh quotient attains its global maximum/minimum over some sphere at the point `x₀`, then `x₀` is an eigenvector of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Rayleigh.html"}, {"id": "Mathlib.Analysis.MeanInequalitiesPow", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 3, "macro_tier_score": 0.0977, "macro_tier_override": null, "x": 61.502, "z": -183.418, "size": 0.2949, "title": "Mean value inequalities", "summary": "In this file we prove several mean inequalities for finite sums. Versions for integrals of some of these inequalities are available in `MeasureTheory.MeanInequalities`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/MeanInequalitiesPow.html"}, {"id": "Mathlib.Analysis.MellinInversion", "region_id": "analysis", "micro_elevation": 0.9375, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 38.671, "z": -154.825, "size": 0.2, "title": "Mellin inversion formula", "summary": "We derive the Mellin inversion formula as a consequence of the Fourier inversion formula.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/MellinInversion.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Log.PosLog", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 2, "macro_tier_score": 0.042, "macro_tier_override": null, "x": 84.195, "z": -187.072, "size": 0.2811, "title": "The Positive Part of the Logarithm", "summary": "This file defines the function `Real.posLog = r ↦ max 0 (log r)` and introduces the notation `log⁺`. For a finite length-`n` sequence `f i` of reals, it establishes the following standard estimates. - `theorem posLog_prod : log⁺ (∏ i, f i) ≤ ∑ i, log⁺ (f i)` - `theorem posLog_sum : log⁺ (∑ i, f i) ≤ log n + ∑ i, log⁺ (f i)` See `Mathlib/Analysis/SpecialFunctions/Integrals/PosLogEqCircleAverage.lean` for the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Log/PosLog.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 3, "macro_tier_score": 0.1681, "macro_tier_override": null, "x": 68.462, "z": -264.708, "size": 0.3619, "title": "Polynomial bounds for trigonometric functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 3, "macro_tier_score": 0.1674, "macro_tier_override": null, "x": 86.93, "z": -263.933, "size": 0.3154, "title": "Derivatives of the `tan` and `arctan` functions.", "summary": "Continuity and derivatives of the tangent and arctangent functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.html"}, {"id": "Mathlib.Analysis.Normed.Group.Subgroup", "region_id": "analysis", "micro_elevation": 0.0625, "macro_tier": 4, "macro_tier_score": 0.487, "macro_tier_override": null, "x": 76.265, "z": -212.187, "size": 0.3235, "title": "Subgroups of normed (semi)groups", "summary": "In this file, we prove that subgroups of a normed (semi)group are also normed (semi)groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/Subgroup.html"}, {"id": "Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints", "region_id": "analysis", "micro_elevation": 0.2917, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 87.424, "z": -189.543, "size": 0.2, "title": "Fixed points of isometries of the upper half-plane", "summary": "In this file we show that the scalar multiplication by an element `g : GL (Fin 2) ℝ` has the following set of fixed points, depending on `g`. - if `g` preserves orientation (i.e., has positive determinant) and is an elliptic matrix, then `z ↦ g • z` has a unique fixed point; - if `g` is a scalar matrix, then it acts by the identity map (proved upstream of this file); - if `g` preserves orientation, and is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/UpperHalfPlane/FixedPoints.html"}, {"id": "Mathlib.Analysis.Calculus.DSlope", "region_id": "analysis", "micro_elevation": 0.5833, "macro_tier": 3, "macro_tier_score": 0.2364, "macro_tier_override": null, "x": 107.96, "z": -177.331, "size": 0.2714, "title": "Slope of a differentiable function", "summary": "Given a function `f : 𝕜 → E` from a nontrivially normed field to a normed space over this field, `dslope f a b` is defined as `slope f a b = (b - a)⁻¹ • (f b - f a)` for `a ≠ b` and as `deriv f a` for `a = b`. In this file we define `dslope` and prove some basic lemmas about its continuity and differentiability.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/DSlope.html"}, {"id": "Mathlib.Analysis.Normed.Operator.Extend", "region_id": "analysis", "micro_elevation": 0.2917, "macro_tier": 1, "macro_tier_score": 0.015, "macro_tier_override": null, "x": 63.153, "z": -196.433, "size": 0.3385, "title": "Extension of continuous linear maps on Banach spaces", "summary": "In this file we provide two different ways to extend a continuous linear map defined on a dense subspace to the entire Banach space. * `ContinuousLinearMap.extend`: Extend `f : E →SL[σ₁₂] F` to a continuous linear map `Eₗ →SL[σ₁₂] F`, where `e : E →ₗ[𝕜] Eₗ` is a dense map that is `IsUniformInducing`. * `LinearMap.extendOfNorm`: Extend `f : E →ₛₗ[σ₁₂] F` to a continuous linear map `Eₗ →SL[σ₁₂] F`, where `e : E →ₗ[𝕜]…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Extend.html"}, {"id": "Mathlib.Analysis.Calculus.ImplicitContDiff", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 67.643, "z": -162.37, "size": 0.2, "title": "Implicit function theorem", "summary": "In this file, we apply the generalised implicit function theorem to the more familiar case and show that the implicit function preserves the smoothness class of the implicit equation. Let `E₁`, `E₂`, and `F` be real or complex Banach spaces. Let `f : E₁ × E₂ → F` be a function that is $C^n$ at a point `(u₁, u₂) : E₁ × E₂`, where `n ≥ 1`. Let `f'` be the derivative of `f` at `(u₁, u₂)`. If the map `y ↦ f' (0, y)` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/ImplicitContDiff.html"}, {"id": "Mathlib.Analysis.Seminorm", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 4, "macro_tier_score": 0.404, "macro_tier_override": null, "x": 94.598, "z": -213.836, "size": 0.3487, "title": "Seminorms", "summary": "This file defines seminorms. A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and subadditive. They are closely related to convex sets, and a topological vector space is locally convex if and only if its topology is induced by a family of seminorms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Seminorm.html"}, {"id": "Mathlib.Analysis.Meromorphic.TrailingCoefficient", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.0557, "macro_tier_override": null, "x": 131.754, "z": -227.632, "size": 0.2539, "title": "The Trailing Coefficient of a Meromorphic Function", "summary": "This file defines the trailing coefficient of a meromorphic function. If `f` is meromorphic at a point `x`, the trailing coefficient is defined as the (unique!) value `g x` for a presentation of `f` in the form `(z - x) ^ order • g z` with `g` analytic at `x`. The lemma `MeromorphicAt.tendsto_nhds_meromorphicTrailingCoeffAt` expresses the trailing coefficient as a limit.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Meromorphic/TrailingCoefficient.html"}, {"id": "Mathlib.Analysis.Complex.Harmonic.Liouville", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 32.566, "z": -233.913, "size": 0.2, "title": "Liouville's Theorem for Harmonic Functions on the Complex Plane", "summary": "A bounded harmonic function on the complex plane is constant.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Harmonic/Liouville.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.LaxMilgram", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 39.873, "z": -231.437, "size": 0.2, "title": "The Lax-Milgram Theorem", "summary": "We consider a Hilbert space `V` over `ℝ` equipped with a bounded bilinear form `B : V →L[ℝ] V →L[ℝ] ℝ`. Recall that a bilinear form `B : V →L[ℝ] V →L[ℝ] ℝ` is *coercive* iff `∃ C, (0 < C) ∧ ∀ u, C * ‖u‖ * ‖u‖ ≤ B u u`. Under the hypothesis that `B` is coercive we prove the Lax-Milgram theorem: that is, the map `InnerProductSpace.continuousLinearMapOfBilin` from `Analysis.InnerProductSpace.Dual` can be upgraded to a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/LaxMilgram.html"}, {"id": "Mathlib.Analysis.Calculus.DifferentialForm.Basic", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 63.564, "z": -257.047, "size": 0.2478, "title": "Exterior derivative of a differential form on a normed space", "summary": "In this file we define the exterior derivative of a differential form on a normed space. Under certain smoothness assumptions, we prove that this operation is linear in the form and the second exterior derivative of a form is zero. We represent a differential `n`-form on `E` taking values in `F` as `E → E [⋀^Fin n]→L[𝕜] F`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/DifferentialForm/Basic.html"}, {"id": "Mathlib.Analysis.Normed.Module.Alternating.Uncurry.Fin", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 104.006, "z": -193.383, "size": 0.2465, "title": "Uncurrying continuous alternating maps", "summary": "Given a continuous function `f` which is linear in the first argument and is alternating form in the other `n` arguments, this file defines a continuous alternating form `ContinuousAlternatingMap.alternatizeUncurryFin f` in `n + 1` arguments. This function is given by ``` ContinuousAlternatingMap.alternatizeUncurryFin f v = ∑ i : Fin (n + 1), (-1) ^ (i : ℕ) • f (v i) (removeNth i v) ``` Given a continuous…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Alternating/Uncurry/Fin.html"}, {"id": "Mathlib.Analysis.Calculus.Darboux", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 42.051, "z": -232.179, "size": 0.2, "title": "Darboux's theorem", "summary": "In this file we prove that the derivative of a differentiable function on an interval takes all intermediate values. The proof is based on the [Wikipedia](https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis)) page about this theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Darboux.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation", "region_id": "analysis", "micro_elevation": 0.9792, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 151.582, "z": -208.885, "size": 0.2838, "title": "Poisson summation applied to the Gaussian", "summary": "In `Real.tsum_exp_neg_mul_int_sq` and `Complex.tsum_exp_neg_mul_int_sq`, we use Poisson summation to prove the identity `∑' (n : ℤ), exp (-π * a * n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), exp (-π / a * n ^ 2)` for positive real `a`, or complex `a` with positive real part. (See also `NumberTheory.ModularForms.JacobiTheta`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Gaussian/PoissonSummation.html"}, {"id": "Mathlib.Analysis.Complex.Polynomial.GaussLucas", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 128.218, "z": -202.462, "size": 0.2, "title": "Gauss-Lucas Theorem", "summary": "In this file we prove Gauss-Lucas Theorem: the roots of the derivative of a nonconstant complex polynomial are included in the convex hull of the roots of the polynomial.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Polynomial/GaussLucas.html"}, {"id": "Mathlib.Analysis.Convex.Strict.Extreme", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 89.647, "z": -225.05, "size": 0.2, "title": "Extreme points of (strictly convex) sets", "summary": "This file collects some results of extreme points of (strictly convex) sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Strict/Extreme.html"}, {"id": "Mathlib.Analysis.Complex.Tietze", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 48.091, "z": -212.048, "size": 0.2, "title": "Finite-dimensional topological vector spaces over `ℝ` satisfy the Tietze extension property", "summary": "There are two main results here: - `RCLike.instTietzeExtensionTVS`: finite-dimensional topological vector spaces over `ℝ` (or `ℂ`) have the Tietze extension property. - `BoundedContinuousFunction.exists_norm_eq_restrict_eq`: when mapping into a finite-dimensional normed vector space over `ℝ` (or `ℂ`), the extension can be chosen to preserve the norm of the bounded continuous function it extends.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Tietze.html"}, {"id": "Mathlib.Analysis.PSeriesComplex", "region_id": "analysis", "micro_elevation": 0.5208, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 115.126, "z": -224.214, "size": 0.2844, "title": "Convergence of `p`-series (complex case)", "summary": "Here we show convergence of `∑ n : ℕ, 1 / n ^ p` for complex `p`. This is done in a separate file rather than in `Analysis.PSeries` in order to keep the prerequisites of the former relatively light.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/PSeriesComplex.html"}, {"id": "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 78.477, "z": -209.754, "size": 0.2614, "title": "Induction on subboxes", "summary": "In this file we prove the following induction principle for `BoxIntegral.Box`, see `BoxIntegral.Box.subbox_induction_on`. Let `p` be a predicate on `BoxIntegral.Box ι`, let `I` be a box. Suppose that the following two properties hold true. * Consider a smaller box `J ≤ I`. The hyperplanes passing through the center of `J` split it into `2 ^ n` boxes. If `p` holds true on each of these boxes, then it is true on `J`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.html"}, {"id": "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "region_id": "analysis", "micro_elevation": 0.0417, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 82.011, "z": -211.807, "size": 0.2614, "title": "Tagged partitions", "summary": "A tagged (pre)partition is a (pre)partition `π` enriched with a tagged point for each box of `π`. For simplicity we require that the function `BoxIntegral.TaggedPrepartition.tag` is defined on all boxes `J : Box ι` but use its values only on boxes of the partition. Given `π : BoxIntegral.TaggedPrepartition I`, we require that each `BoxIntegral.TaggedPrepartition π J` belongs to `BoxIntegral.Box.Icc I`. If for every…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoxIntegral/Partition/Tagged.html"}, {"id": "Mathlib.Analysis.BoxIntegral.UnitPartition", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 76.758, "z": -242.891, "size": 0.2511, "title": "Unit Partition", "summary": "Fix `n` a positive integer. `BoxIntegral.unitPartition.box` are boxes in `ι → ℝ` obtained by dividing the unit box uniformly into boxes of side length `1 / n` and translating the boxes by vectors `ν : ι → ℤ`. Let `B` be a `BoxIntegral`. A `unitPartition.box` is admissible for `B` (more precisely its index is admissible) if it is contained in `B`. There are finitely many admissible `unitPartition.box` for `B` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoxIntegral/UnitPartition.html"}, {"id": "Mathlib.Analysis.BoxIntegral.Integrability", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 1, "macro_tier_score": 0.0142, "macro_tier_override": null, "x": 109.894, "z": -198.179, "size": 0.278, "title": "McShane integrability vs Bochner integrability", "summary": "In this file we prove that any Bochner integrable function is McShane integrable (hence, it is Henstock and `GP` integrable) with the same integral. The proof is based on [Russel A. Gordon, *The integrals of Lebesgue, Denjoy, Perron, and Henstock*][Gordon55]. We deduce that the same is true for the Riemann integral for continuous functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoxIntegral/Integrability.html"}, {"id": "Mathlib.Analysis.Normed.Algebra.DualNumber", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 100.86, "z": -239.548, "size": 0.2, "title": "Results on `DualNumber R` related to the norm", "summary": "These are just restatements of similar statements about `TrivSqZeroExt R M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Algebra/DualNumber.html"}, {"id": "Mathlib.Analysis.BoxIntegral.Partition.Measure", "region_id": "analysis", "micro_elevation": 0.3958, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 61.489, "z": -187.266, "size": 0.2538, "title": "Box-additive functions defined by measures", "summary": "In this file we prove a few simple facts about rectangular boxes, partitions, and measures: - given a box `I : Box ι`, its coercion to `Set (ι → ℝ)` and `I.Icc` are measurable sets; - if `μ` is a locally finite measure, then `(I : Set (ι → ℝ))` and `I.Icc` have finite measure; - if `μ` is a locally finite measure, then `fun J ↦ μ.real J` is a box additive function. For the last statement, we both prove it as a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoxIntegral/Partition/Measure.html"}, {"id": "Mathlib.Analysis.Analytic.Order", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 3, "macro_tier_score": 0.2087, "macro_tier_override": null, "x": 126.556, "z": -190.598, "size": 0.2773, "title": "Vanishing Order of Analytic Functions", "summary": "This file defines the order of vanishing of an analytic function `f` at a point `z₀`, as an element of `ℕ∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Analytic/Order.html"}, {"id": "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Analytic", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 3, "macro_tier_score": 0.2085, "macro_tier_override": null, "x": 46.06, "z": -244.541, "size": 0.249, "title": "Analyticity of local inverses", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/InverseFunctionTheorem/Analytic.html"}, {"id": "Mathlib.Analysis.FunctionalSpaces.SobolevInequality", "region_id": "analysis", "micro_elevation": 0.7917, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 73.345, "z": -267.04, "size": 0.2, "title": "Gagliardo-Nirenberg-Sobolev inequality", "summary": "In this file we prove the Gagliardo-Nirenberg-Sobolev inequality. This states that for compactly supported `C¹`-functions between finite-dimensional vector spaces, we can bound the `L^p`-norm of `u` by the `L^q` norm of the derivative of `u`. The bound is up to a constant that is independent of the function `u`. Let `n` be the dimension of the domain. The main step in the proof, which we dubbed the \"grid-lines…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/FunctionalSpaces/SobolevInequality.html"}, {"id": "Mathlib.Analysis.Complex.ValueDistribution.Proximity.IntegralPresentation", "region_id": "analysis", "micro_elevation": 0.8958, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 18.817, "z": -186.084, "size": 0.2478, "title": "Integral Presentation of the Proximity Function", "summary": "If `f : ℂ → ℂ` is meromorphic, this file establishes a presentation of the proximity function `proximity f ⊤` as iterated circle averages. This statement can be used to compare the proximity- and logarithmic counting functions, and is one of the key ingredients in the proof of Cartan's classic formula for the characteristic function. See Section VI.2 of [Lang, *Introduction to Complex Hyperbolic Spaces*][MR886677]…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/ValueDistribution/Proximity/IntegralPresentation.html"}, {"id": "Mathlib.Analysis.Normed.Affine.MazurUlam", "region_id": "analysis", "micro_elevation": 0.2083, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 65.008, "z": -206.741, "size": 0.2, "title": "Mazur-Ulam Theorem", "summary": "Mazur-Ulam theorem states that an isometric bijection between two normed affine spaces over `ℝ` is affine. We formalize it in three definitions: * `IsometryEquiv.toRealLinearIsometryEquivOfMapZero` : given `E ≃ᵢ F` sending `0` to `0`, returns `E ≃ₗᵢ[ℝ] F` with the same `toFun` and `invFun`; * `IsometryEquiv.toRealLinearIsometryEquiv` : given `f : E ≃ᵢ F`, returns a linear isometry equivalence `g : E ≃ₗᵢ[ℝ] F` with…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Affine/MazurUlam.html"}, {"id": "Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv", "region_id": "analysis", "micro_elevation": 0.8125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 84.105, "z": -150.261, "size": 0.2, "title": "Continuous (star-)algebra equivalences between continuous endomorphisms are (isometrically) inner", "summary": "This file shows that continuous (star-)algebra equivalences between continuous endomorphisms are (isometrically) inner. See `Mathlib/LinearAlgebra/GeneralLinearGroup/AlgEquiv.lean` for the non-continuous version. The proof follows the same idea as the non-continuous version.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/ContinuousAlgEquiv.html"}, {"id": "Mathlib.Analysis.VonNeumannAlgebra.Basic", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 97.704, "z": -253.307, "size": 0.2, "title": "Von Neumann algebras", "summary": "We give the \"abstract\" and \"concrete\" definitions of a von Neumann algebra. We still have a major project ahead of us to show the equivalence between these definitions! An abstract von Neumann algebra `WStarAlgebra M` is a C⋆ algebra with a Banach space predual, per Sakai (1971). A concrete von Neumann algebra `VonNeumannAlgebra H` (where `H` is a Hilbert space) is a \\*-closed subalgebra of bounded operators on `H`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/VonNeumannAlgebra/Basic.html"}, {"id": "Mathlib.Analysis.Fourier.AddCircleMulti", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 48.577, "z": -166.443, "size": 0.2, "title": "Multivariate Fourier series", "summary": "In this file we define the Fourier series of an L² function on the `d`-dimensional unit circle, and show that it converges to the function in the L² norm. We also prove uniform convergence of the Fourier series if `f` is continuous and the sequence of its Fourier coefficients is summable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Fourier/AddCircleMulti.html"}, {"id": "Mathlib.Analysis.Normed.Affine.ContinuousAffineMap", "region_id": "analysis", "micro_elevation": 0.3542, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 89.068, "z": -185.279, "size": 0.2, "title": "Norm on the continuous affine maps between normed vector spaces.", "summary": "We define a norm on the space of continuous affine maps between normed vector spaces by defining the norm of `f : V →ᴬ[𝕜] W` to be `‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry: `(V →ᴬ[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`. The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with a vector space `W`, there is an exact sequence of `𝕜`-modules: `0…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Affine/ContinuousAffineMap.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.l2Space", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 2, "macro_tier_score": 0.0556, "macro_tier_override": null, "x": 38.641, "z": -225.106, "size": 0.2378, "title": "Hilbert sum of a family of inner product spaces", "summary": "Given a family `(G : ι → Type*) [Π i, InnerProductSpace 𝕜 (G i)]` of inner product spaces, this file equips `lp G 2` with an inner product space structure, where `lp G 2` consists of those dependent functions `f : Π i, G i` for which `∑' i, ‖f i‖ ^ 2`, the sum of the norms-squared, is summable. This construction is sometimes called the *Hilbert sum* of the family `G`. By choosing `G` to be `ι → 𝕜`, the Hilbert space…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/l2Space.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.MeanErgodic", "region_id": "analysis", "micro_elevation": 0.5625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 45.812, "z": -186.638, "size": 0.2, "title": "Von Neumann Mean Ergodic Theorem in a Hilbert Space", "summary": "In this file we prove the von Neumann Mean Ergodic Theorem for an operator in a Hilbert space. It says that for a contracting linear self-map `f : E →ₗ[𝕜] E` of a Hilbert space, the Birkhoff averages ``` birkhoffAverage 𝕜 f id N x = (N : 𝕜)⁻¹ • ∑ n ∈ Finset.range N, f^[n] x ``` converge to the orthogonal projection of `x` to the subspace of fixed points of `f`, see…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/MeanErgodic.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Exponential", "region_id": "analysis", "micro_elevation": 0.4792, "macro_tier": 2, "macro_tier_score": 0.0418, "macro_tier_override": null, "x": 48.253, "z": -194.671, "size": 0.2441, "title": "The exponential map from selfadjoint to unitary", "summary": "In this file, we establish various properties related to the map `fun a ↦ NormedSpace.exp ℂ A (I • a)` between the subtypes `selfAdjoint A` and `unitary A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Exponential.html"}, {"id": "Mathlib.Analysis.LocallyConvex.AbsConvex", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.0835, "macro_tier_override": null, "x": 66.804, "z": -222.2, "size": 0.2496, "title": "Absolutely convex sets", "summary": "A set `s` in a commutative monoid `E` is called absolutely convex or disked if it is convex and balanced. The importance of absolutely convex sets comes from the fact that every locally convex topological vector space has a basis consisting of absolutely convex sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/AbsConvex.html"}, {"id": "Mathlib.Analysis.Convex.TotallyBounded", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.0835, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2576, "title": "Totally Bounded sets and Convex Hulls", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/TotallyBounded.html"}, {"id": "Mathlib.Analysis.Complex.Cardinality", "region_id": "analysis", "micro_elevation": 0.0417, "macro_tier": 0, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": 81.664, "z": -206.99, "size": 0.3034, "title": "The cardinality of the complex numbers", "summary": "This file shows that the complex numbers have cardinality continuum, i.e. `#ℂ = 𝔠`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Cardinality.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.WithLp", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.1114, "macro_tier_override": null, "x": 46.321, "z": -195.263, "size": 0.269, "title": "Derivatives on `WithLp`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/WithLp.html"}, {"id": "Mathlib.Analysis.Convex.Cone.TensorProduct", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 57.157, "z": -210.58, "size": 0.2, "title": "Tensor Products of Pointed Cones", "summary": "This file proves that the minimal and maximal tensor products of pointed cones in finite-dimensional real vector spaces are equal when one cone is simplicial and generating and the other is proper (pointed and closed). Finite-dimensionality of the proper cone ambient space is by explicit declaration and is required for the `topDualPairing_isContPerfPair` instance (in `Topology.Algebra.Module.TopDualPairing`). The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Cone/TensorProduct.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.CompCLM", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 4, "macro_tier_score": 0.2642, "macro_tier_override": null, "x": 65.116, "z": -179.495, "size": 0.2729, "title": "Multiplicative operations on derivatives", "summary": "For detailed documentation of the Fréchet derivative, see the module docstring of `Mathlib/Analysis/Calculus/FDeriv/Basic.lean`. This file contains the usual formulas (and existence assertions) for the derivative of * composition of continuous linear maps * application of continuous (multi)linear maps to a constant", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/CompCLM.html"}, {"id": "Mathlib.Analysis.Normed.Module.Bases", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 78.639, "z": -182.143, "size": 0.2, "title": "Schauder Bases and Generalized Bases", "summary": "This file defines the theory of bases in Banach spaces, unifying the classical sequential notion with modern generalized bases.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Bases.html"}, {"id": "Mathlib.Analysis.Complex.ValueDistribution.Cartan", "region_id": "analysis", "micro_elevation": 0.9792, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 130.608, "z": -260.167, "size": 0.2, "title": "Cartan's Formula", "summary": "This file will, in the future, establish Cartan's classic formula, describing the characteristic function `characteristic f ⊤ r` as a sum of two circle averages, - `circleAverage (logCounting f · r) 0 1` and - `circleAverage (fun a ↦ log ‖meromorphicTrailingCoeffAt (f · - a) 0‖) 0 1`. As a corollary, Cartan's formula implies the (surprising, non-trival) fact that the characteristic function is monotone. At present,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/ValueDistribution/Cartan.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.StandardSubspace", "region_id": "analysis", "micro_elevation": 0.8125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 130.078, "z": -241.478, "size": 0.2, "title": "Standard subspaces of a Hilbert space", "summary": "This files defines standard subspaces of a complex Hilbert space: a standard subspace `S` of `H` is a closed real subspace `S` such that `S ⊓ i S = ⊥` and `S ⊔ i S = ⊤`. For a standard subspace, one can define a closable operator `x + i y ↦ x - i y` and develop an analogue of the Tomita-Takesaki modular theory for von Neumann algebras. By considering inclusions of standard subspaces, one can obtain unitary…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/StandardSubspace.html"}, {"id": "Mathlib.Analysis.Asymptotics.SuperpolynomialDecay", "region_id": "analysis", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 88.703, "z": -218.048, "size": 0.257, "title": "Super-Polynomial Function Decay", "summary": "This file defines a predicate `Asymptotics.SuperpolynomialDecay f` for a function satisfying one of the following equivalent definitions (the definition is in terms of the first condition): * `x ^ n * f` tends to `𝓝 0` for all (or sufficiently large) naturals `n` * `|x ^ n * f|` tends to `𝓝 0` for all naturals `n` (`superpolynomialDecay_iff_abs_tendsto_zero`) * `|x ^ n * f|` is bounded for all naturals `n`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal", "region_id": "analysis", "micro_elevation": 0.7917, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 93.418, "z": -153.222, "size": 0.2, "title": "The GNS (Gelfand-Naimark-Segal) construction", "summary": "This file contains the constructions and definitions that produce a ⋆-homomorphism from an arbitrary C⋆-algebra into the algebra of bounded linear operators on an appropriately constructed Hilbert space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/GelfandNaimarkSegal.html"}, {"id": "Mathlib.Analysis.CStarAlgebra.Hom", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 114.851, "z": -253.82, "size": 0.2, "title": "Properties of C⋆-algebra homomorphisms", "summary": "Here we collect properties of C⋆-algebra homomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Hom.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.OfNorm", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 91.438, "z": -231.048, "size": 0.2, "title": "Inner product space derived from a norm", "summary": "This file defines an `InnerProductSpace` instance from a norm that respects the parallelogram identity. The parallelogram identity is a way to express the inner product of `x` and `y` in terms of the norms of `x`, `y`, `x + y`, `x - y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/OfNorm.html"}, {"id": "Mathlib.Analysis.Complex.Isometry", "region_id": "analysis", "micro_elevation": 0.3333, "macro_tier": 2, "macro_tier_score": 0.0417, "macro_tier_override": null, "x": 82.192, "z": -185.257, "size": 0.2305, "title": "Isometries of the Complex Plane", "summary": "The lemma `linear_isometry_complex` states the classification of isometries in the complex plane. Specifically, isometries with rotations but without translation. The proof involves: 1. creating a linear isometry `g` with two fixed points, `g(0) = 0`, `g(1) = 1` 2. applying `linear_isometry_complex_aux` to `g` The proof of `linear_isometry_complex_aux` is separated in the following parts: 1. show that the real parts…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Isometry.html"}, {"id": "Mathlib.Analysis.Calculus.BumpFunction.SmoothApprox", "region_id": "analysis", "micro_elevation": 0.7917, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 124.296, "z": -172.279, "size": 0.2, "title": "Density of smooth functions in the space of continuous functions", "summary": "In this file we prove that smooth functions are dense in the set of continuous functions from a real finite-dimensional vector space to a Banach space, see `ContinuousMap.dense_setOf_contDiff`. We also prove several unbundled versions of this statement. The heavy part of the proof is done upstream in `ContDiffBump.dist_normed_convolution_le` and `HasCompactSupport.contDiff_convolution_left`. Here we wrap these…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/BumpFunction/SmoothApprox.html"}, {"id": "Mathlib.Analysis.Normed.Operator.FredholmAlternative", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 73.275, "z": -179.811, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/FredholmAlternative.html"}, {"id": "Mathlib.Analysis.NormedSpace.RCLike", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 90.79, "z": -234.735, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/NormedSpace/RCLike.html"}, {"id": "Mathlib.Analysis.Normed.Group.BallSphere", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 3, "macro_tier_score": 0.1115, "macro_tier_override": null, "x": 73.875, "z": -214.081, "size": 0.2835, "title": "Negation on spheres and balls", "summary": "In this file we define `InvolutiveNeg` and `ContinuousNeg` instances for spheres, open balls, and closed balls in a seminormed group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Group/BallSphere.html"}, {"id": "Mathlib.Analysis.Normed.Algebra.Unitization", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 2, "macro_tier_score": 0.0699, "macro_tier_override": null, "x": 101.246, "z": -226.848, "size": 0.287, "title": "Unitization norms", "summary": "Given a not-necessarily-unital normed `𝕜`-algebra `A`, it is frequently of interest to equip its `Unitization` with a norm which simultaneously makes it into a normed algebra and also satisfies two properties: - `‖1‖ = 1` (i.e., `NormOneClass`) - The embedding of `A` in `Unitization 𝕜 A` is an isometry. (i.e., `Isometry Unitization.inr`) One way to do this is to pull back the norm from `WithLp 1 (𝕜 × A)`, that is,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Algebra/Unitization.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 3, "macro_tier_score": 0.1671, "macro_tier_override": null, "x": 65.6, "z": -158.189, "size": 0.2889, "title": "Complex trigonometric functions", "summary": "Basic facts and derivatives for the complex trigonometric functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.html"}, {"id": "Mathlib.Analysis.Calculus.Gradient.Basic", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 97.394, "z": -167.275, "size": 0.2, "title": "Gradient", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Gradient/Basic.html"}, {"id": "Mathlib.Analysis.Distribution.Support", "region_id": "analysis", "micro_elevation": 0.9792, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 66.612, "z": -279.873, "size": 0.2, "title": "Support of distributions", "summary": "We define the support of a distribution, `dsupport u`, as the intersection of all closed sets for which `u` vanishes on the complement. For this we also define a predicate `IsVanishingOn` that asserts that a map `f : F → V` vanishes on `s : Set α` if for all `u : F` with `tsupport u ⊆ s` it follows that `f u = 0`. These definitions work independently of a specific class of distributions (classical, tempered, or…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Distribution/Support.html"}, {"id": "Mathlib.Analysis.Real.Spectrum", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0696, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2455, "title": "Some lemmas on the spectrum and quasispectrum of elements and positivity", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Real/Spectrum.html"}, {"id": "Mathlib.Analysis.Convex.Strong", "region_id": "analysis", "micro_elevation": 0.3125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 82.833, "z": -186.859, "size": 0.2, "title": "Uniformly and strongly convex functions", "summary": "In this file, we define uniformly convex functions and strongly convex functions. For a real normed space `E`, a uniformly convex function with modulus `φ : ℝ → ℝ` is a function `f : E → ℝ` such that `f (t • x + (1 - t) • y) ≤ t • f x + (1 - t) • f y - t * (1 - t) * φ ‖x - y‖` for all `t ∈ [0, 1]`. A `m`-strongly convex function is a uniformly convex function with modulus `fun r ↦ m / 2 * r ^ 2`. If `E` is an inner…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Strong.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.ProdL2", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 1, "macro_tier_score": 0.0143, "macro_tier_override": null, "x": 109.547, "z": -176.701, "size": 0.2824, "title": "`L²` inner product space structure on products of inner product spaces", "summary": "The `L²` norm on product of two inner product spaces is compatible with an inner product $$ \\langle x, y\\rangle = \\langle x_1, y_1 \\rangle + \\langle x_2, y_2 \\rangle. $$ This is recorded in this file as an inner product space instance on `WithLp 2 (E × F)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/ProdL2.html"}, {"id": "Mathlib.Analysis.Complex.Poisson", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 1, "macro_tier_score": 0.0142, "macro_tier_override": null, "x": 116.063, "z": -240.005, "size": 0.2741, "title": "Poisson Integral Formula", "summary": "We present two versions of the **Poisson Integral Formula** for ℂ-differentiable functions on arbitrary disks in the complex plane, formulated with the real part of the Herglotz–Riesz kernel of integration and with the Poisson kernel, respectively.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Poisson.html"}, {"id": "Mathlib.Analysis.NormedSpace.FunctionSeries", "region_id": "analysis", "micro_elevation": 0.1458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 77.483, "z": -199.165, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/NormedSpace/FunctionSeries.html"}, {"id": "Mathlib.Analysis.LocallyConvex.Montel", "region_id": "analysis", "micro_elevation": 0.4167, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 99.229, "z": -233.153, "size": 0.2, "title": "Montel spaces", "summary": "A Montel space is a topological vector space `E` that has the Heine-Borel property: every closed and (von Neumann) bounded set is compact. Note that we are not requiring that `E` is a barrelled space, so the usual definition of a Montel space would be `[MontelSpace 𝕜 E] [BarrelledSpace 𝕜 E]`. * `MontelSpace.finiteDimensional_of_normedSpace`: every normed Montel space is finite dimensional. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/Montel.html"}, {"id": "Mathlib.Analysis.Complex.Angle", "region_id": "analysis", "micro_elevation": 0.7917, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 22.362, "z": -203.965, "size": 0.2, "title": "Angle between complex numbers", "summary": "This file relates the Euclidean geometric notion of angle between complex numbers to the argument of their quotient. It also shows that the arc and chord distances between two unit complex numbers are equivalent up to a factor of `π / 2`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Angle.html"}, {"id": "Mathlib.Analysis.Normed.Field.Instances", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 76.751, "z": -200.982, "size": 0.2, "title": "A normed field is a completable topological field", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Field/Instances.html"}, {"id": "Mathlib.Analysis.Normed.Module.Dual", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 2, "macro_tier_score": 0.0557, "macro_tier_override": null, "x": 55.109, "z": -189.414, "size": 0.2423, "title": "Polar sets in the strong dual of a normed space", "summary": "In this file we study polar sets in the strong dual `StrongDual` of a normed space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/Dual.html"}, {"id": "Mathlib.Analysis.Calculus.TangentCone.ProperSpace", "region_id": "analysis", "micro_elevation": 0.2292, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 87.264, "z": -224.625, "size": 0.2, "title": "Tangent cone in a proper space", "summary": "In this file we prove that the tangent cone of a set in a proper normed space at an accumulation point of this set is nontrivial.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/TangentCone/ProperSpace.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas", "region_id": "analysis", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 79.984, "z": -209.532, "size": 0.2, "title": "Lemmas about `Nat.nthRoot`", "summary": "In this file we prove that `Nat.nthRoot n a` is indeed the floor of `ⁿ√a`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Pow/NthRootLemmas.html"}, {"id": "Mathlib.Analysis.Calculus.Deriv.CompMul", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 3, "macro_tier_score": 0.2364, "macro_tier_override": null, "x": 36.154, "z": -203.986, "size": 0.274, "title": "Derivative of `x ↦ f (cx)`", "summary": "In this file we prove that the derivative of `fun x ↦ f (c * x)` equals `c` times the derivative of `f` evaluated at `c * x`. Since Mathlib uses `0` as the fallback value for the derivatives whenever they are undefined, the theorems in this file require neither differentiability of `f`, nor assumptions like `UniqueDiffWithinAt 𝕜 s x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Deriv/CompMul.html"}, {"id": "Mathlib.Analysis.Convex.StrictConvexBetween", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 98.254, "z": -210.172, "size": 0.2613, "title": "Betweenness in affine spaces for strictly convex spaces", "summary": "This file proves results about betweenness for points in an affine space for a strictly convex space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/StrictConvexBetween.html"}, {"id": "Mathlib.Analysis.Calculus.LineDeriv.QuadraticMap", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 76.457, "z": -255.098, "size": 0.2, "title": "Quadratic forms are line (Gateaux) differentiable", "summary": "In this file we prove that a quadratic form is line differentiable, with the line derivative given by the polar bilinear form. Note that this statement does not need topology on the domain. In particular, it applies to discontinuous quadratic forms on infinite-dimensional spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/LineDeriv/QuadraticMap.html"}, {"id": "Mathlib.Analysis.Hofer", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 81.506, "z": -209.598, "size": 0.2, "title": "Hofer's lemma", "summary": "This is an elementary lemma about complete metric spaces. It is motivated by an application to the bubbling-off analysis for holomorphic curves in symplectic topology. We are *very* far away from having these applications, but the proof here is a nice example of a proof needing to construct a sequence by induction in the middle of the proof.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Hofer.html"}, {"id": "Mathlib.Analysis.Convex.Uniform", "region_id": "analysis", "micro_elevation": 0.25, "macro_tier": 2, "macro_tier_score": 0.0282, "macro_tier_override": null, "x": 73.728, "z": -192.354, "size": 0.2893, "title": "Uniformly convex spaces", "summary": "This file defines uniformly convex spaces, which are real normed vector spaces in which for all strictly positive `ε`, there exists some strictly positive `δ` such that `ε ≤ ‖x - y‖` implies `‖x + y‖ ≤ 2 - δ` for all `x` and `y` of norm at most than `1`. This means that the triangle inequality is strict with a uniform bound, as opposed to strictly convex spaces where the triangle inequality is strict but not…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Uniform.html"}, {"id": "Mathlib.Analysis.Complex.UpperHalfPlane.Metric", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 43.999, "z": -168.145, "size": 0.2, "title": "Metric on the upper half-plane", "summary": "In this file we define a `MetricSpace` structure on the `UpperHalfPlane`. We use hyperbolic (Poincaré) distance given by `dist z w = 2 * arsinh (dist (z : ℂ) w / (2 * √(z.im * w.im)))` instead of the induced Euclidean distance because the hyperbolic distance is invariant under holomorphic automorphisms of the upper half-plane. However, we ensure that the projection to `TopologicalSpace` is definitionally equal to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/UpperHalfPlane/Metric.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Arsinh", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 0, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 130.392, "z": -226.91, "size": 0.239, "title": "Inverse of the sinh function", "summary": "In this file we prove that sinh is bijective and hence has an inverse, arsinh.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Arsinh.html"}, {"id": "Mathlib.Analysis.Meromorphic.Basic", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 3, "macro_tier_score": 0.2092, "macro_tier_override": null, "x": 84.738, "z": -157.954, "size": 0.3259, "title": "Meromorphic functions", "summary": "Main statements: * `MeromorphicAt`: definition of meromorphy at a point * `MeromorphicAt.iff_eventuallyEq_zpow_smul_analyticAt`: `f` is meromorphic at `z₀` iff we have `f z = (z - z₀) ^ n • g z` on a punctured neighborhood of `z₀`, for some `n : ℤ` and `g` analytic at `z₀`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Meromorphic/Basic.html"}, {"id": "Mathlib.Analysis.ODE.Transform", "region_id": "analysis", "micro_elevation": 0.6042, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 103.475, "z": -172.115, "size": 0.2, "title": "Translation and scaling of integral curves", "summary": "New integral curves may be constructed by translating or scaling the domain of an existing integral curve.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/ODE/Transform.html"}, {"id": "Mathlib.Analysis.Convex.StoneSeparation", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 87.551, "z": -208.654, "size": 0.2, "title": "Stone's separation theorem", "summary": "This file proves Stone's separation theorem. This tells us that any two disjoint convex sets can be separated by a convex set whose complement is also convex. In locally convex real topological vector spaces, the Hahn-Banach separation theorems provide stronger statements: one may find a separating hyperplane, instead of merely a convex set whose complement is convex.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/StoneSeparation.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.Trace", "region_id": "analysis", "micro_elevation": 0.7292, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 29.699, "z": -227.263, "size": 0.2, "title": "Traces in inner product spaces", "summary": "This file contains various results about traces of linear operators in inner product spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Trace.html"}, {"id": "Mathlib.Analysis.Normed.Lp.LpEquiv", "region_id": "analysis", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 43.496, "z": -211.851, "size": 0.2, "title": "Equivalences among $L^p$ spaces", "summary": "In this file we collect a variety of equivalences among various $L^p$ spaces. In particular, when `α` is a `Fintype`, given `E : α → Type u` and `p : ℝ≥0∞`, if all `E i` for `i : α` are normed, additive commutative groups, there is a natural linear isometric equivalence `lpPiLpₗᵢ : lp E p ≃ₗᵢ PiLp p E`. In addition, when `α` is a discrete topological space, the bounded continuous functions `α →ᵇ β` correspond…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Lp/LpEquiv.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.LinearPMap", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 34.08, "z": -198.438, "size": 0.2, "title": "Partially defined linear operators on Hilbert spaces", "summary": "We will develop the basics of the theory of unbounded operators on Hilbert spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/LinearPMap.html"}, {"id": "Mathlib.Analysis.Calculus.AbsolutelyMonotone", "region_id": "analysis", "micro_elevation": 0.6875, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 101.296, "z": -164.0, "size": 0.2, "title": "Absolutely monotone functions", "summary": "A function `f : ℝ → ℝ` is *absolutely monotone* on a set `s` if its iterated derivatives are all nonnegative on `s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/AbsolutelyMonotone.html"}, {"id": "Mathlib.Analysis.Calculus.AddTorsor.Coord", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 66.237, "z": -162.761, "size": 0.2, "title": "Barycentric coordinates are smooth", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/AddTorsor/Coord.html"}, {"id": "Mathlib.Analysis.Calculus.DifferentialForm.VectorField", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 28.325, "z": -213.292, "size": 0.2, "title": "Evaluation of the derivative of differential forms on vector fields", "summary": "In this file we prove the following formula and its corollaries. If `ω` is a differentiable `k`-form and `V i` are `k + 1` differentiable vector fields, then $$ dω(V_0(x), \\dots, V_n(x)) = \\sum_{i=0}^k (-1)^i • D_x\\left(ω\\big(x; V_0(x), \\dots, \\widehat{V_i(x)}, \\dots, V_k(x)\\big)\\right)(V_i(x)) + \\sum_{0 \\le i < j\\le k} (-1)^{i + j} ω\\big(x; [V_i, V_j](x), V_0(x), …, \\widehat{V_i(x)}, …, \\widehat{V_j(x)}, …,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/DifferentialForm/VectorField.html"}, {"id": "Mathlib.Analysis.Calculus.FDeriv.Norm", "region_id": "analysis", "micro_elevation": 0.7083, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 30.584, "z": -225.105, "size": 0.2, "title": "Differentiability of the norm in a real normed vector space", "summary": "This file provides basic results about the differentiability of the norm in a real vector space. Most are of the following kind: if the norm has some differentiability property (`DifferentiableAt`, `ContDiffAt`, `HasStrictFDerivAt`, `HasFDerivAt`) at `x`, then so it has at `t • x` when `t ≠ 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/FDeriv/Norm.html"}, {"id": "Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 38.513, "z": -183.907, "size": 0.2, "title": "Iterated derivatives of compositions", "summary": "In this file we specialize Faà di Bruno's formula to one-dimensional domain to deduce formulae for `iteratedDerivWithin k (g ∘ f) s x` for `k = 2` and `k = 3`. We use - `vcomp` for lemmas about the composition of `g : E → F` with `f : 𝕜 → E`; - `scomp` for lemmas about the composition of `g : 𝕜 → E` with `f : 𝕜 → 𝕜`; - `comp` for lemmas about the composition of `g : 𝕜 → 𝕜` with `f : 𝕜 → 𝕜`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/IteratedDeriv/FaaDiBruno.html"}, {"id": "Mathlib.Analysis.Calculus.TangentCone.Pi", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 87.371, "z": -207.673, "size": 0.2, "title": "Indexed product of sets with unique differentiability property", "summary": "In this file we prove that the indexed product of a family sets with unique differentiability property has the same property, see `UniqueDiffOn.pi` and `UniqueDiffOn.univ_pi`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/TangentCone/Pi.html"}, {"id": "Mathlib.Analysis.Complex.Hadamard", "region_id": "analysis", "micro_elevation": 0.7708, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 74.124, "z": -153.47, "size": 0.2, "title": "Hadamard three-lines Theorem", "summary": "In this file we present a proof of Hadamard's three-lines Theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/Hadamard.html"}, {"id": "Mathlib.Analysis.Convex.BetweenList", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 74.88, "z": -206.203, "size": 0.2, "title": "Betweenness for lists of points.", "summary": "This file defines notions of lists of points in an affine space being in order on a line.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/BetweenList.html"}, {"id": "Mathlib.Analysis.Convex.Birkhoff", "region_id": "analysis", "micro_elevation": 0.1042, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 83.564, "z": -202.808, "size": 0.2, "title": "Birkhoff's theorem", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Birkhoff.html"}, {"id": "Mathlib.Analysis.Convex.Extrema", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 83.503, "z": -214.507, "size": 0.2, "title": "Minima and maxima of convex functions", "summary": "We show that if a function `f : E → β` is convex, then a local minimum is also a global minimum, and likewise for concave functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Extrema.html"}, {"id": "Mathlib.Analysis.Convex.GaugeRescale", "region_id": "analysis", "micro_elevation": 0.2708, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 85.381, "z": -190.477, "size": 0.2, "title": "\"Gauge rescale\" homeomorphism between convex sets", "summary": "Given two convex von Neumann bounded neighbourhoods of the origin in a real topological vector space, we construct a homeomorphism `gaugeRescaleHomeomorph` that sends the interior, the closure, and the frontier of one set to the interior, the closure, and the frontier of the other set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/GaugeRescale.html"}, {"id": "Mathlib.Analysis.InnerProductSpace.StarOrder", "region_id": "analysis", "micro_elevation": 0.8125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 138.959, "z": -202.329, "size": 0.2, "title": "Continuous linear maps on a Hilbert space are a `StarOrderedRing`", "summary": "In this file we show that the continuous linear maps on a complex Hilbert space form a `StarOrderedRing`. Note that they are already equipped with the Loewner partial order. We also prove that, with respect to this partial order, a map is positive if every element of the real spectrum is nonnegative. Consequently, when `H` is a Hilbert space, then `H →L[ℂ] H` is equipped with all the usual instances of the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/StarOrder.html"}, {"id": "Mathlib.Analysis.Normed.Algebra.MatrixExponential", "region_id": "analysis", "micro_elevation": 0.625, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 88.331, "z": -164.598, "size": 0.2, "title": "Lemmas about the matrix exponential", "summary": "In this file, we provide results about `NormedSpace.exp` on `Matrix`s over a topological or normed algebra. Note that generic results over all topological spaces such as `NormedSpace.exp_zero` can be used on matrices without issue, so are not repeated here. The topological results specific to matrices are: * `Matrix.exp_transpose` * `Matrix.exp_conjTranspose` * `Matrix.exp_diagonal` * `Matrix.exp_blockDiagonal` *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Algebra/MatrixExponential.html"}, {"id": "Mathlib.Analysis.Normed.Lp.Finsupp", "region_id": "analysis", "micro_elevation": 0.4583, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 46.597, "z": -206.604, "size": 0.2, "title": "Direct sum of metric spaces", "summary": "This files endows the direct sum `ι →₀ X` of `ι`-many copies of a metric space `X` with the L^p metric.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Lp/Finsupp.html"}, {"id": "Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 107.269, "z": -212.263, "size": 0.2, "title": "Injective seminorm on the tensor of a finite family of normed spaces.", "summary": "The purpose of this file is to develop the theory of the injective tensor norm. A first formalization turned out not to capture the common mathematical definition and is now deprecated. See https://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/injectiveSeminorm/with/568798633", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Module/PiTensorProduct/InjectiveSeminorm.html"}, {"id": "Mathlib.Analysis.Normed.Ring.Int", "region_id": "analysis", "micro_elevation": 0.125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 78.504, "z": -200.512, "size": 0.2, "title": "The integers as normed ring", "summary": "This file contains basic facts about the integers as normed ring. Recall that `‖n‖` denotes the norm of `n` as real number. This norm is always nonnegative, so we can bundle the norm together with this fact, to obtain a term of type `NNReal` (the nonnegative real numbers). The resulting nonnegative real number is denoted by `‖n‖₊`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Ring/Int.html"}, {"id": "Mathlib.Analysis.Real.Pi.Chudnovsky", "region_id": "analysis", "micro_elevation": 0.4375, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 74.813, "z": -241.103, "size": 0.2, "title": "Chudnovsky's formula for π", "summary": "This file defines the infinite sum in Chudnovsky's formula for computing `π⁻¹`. It does not (yet!) contain a proof; anyone is welcome to adopt this problem, but at present we are a long way off.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Real/Pi/Chudnovsky.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Log.Monotone", "region_id": "analysis", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 26.43, "z": -221.353, "size": 0.2, "title": "Logarithm Tonality", "summary": "In this file we describe the tonality of the logarithm function when multiplied by functions of the form `x ^ a`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Log/Monotone.html"}, {"id": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Meromorphic", "region_id": "analysis", "micro_elevation": 0.8125, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 137.396, "z": -194.241, "size": 0.2, "title": "Meromorphicity of `Complex.tan` and `Complex.tanh`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/SpecialFunctions/Trigonometric/Meromorphic.html"}, {"id": "Mathlib.Analysis.Convex.Mul", "region_id": "analysis", "micro_elevation": 0.0833, "macro_tier": 3, "macro_tier_score": 0.1113, "macro_tier_override": null, "x": 74.391, "z": -207.114, "size": 0.2552, "title": "Product of convex functions", "summary": "This file proves that the product of convex functions is convex, provided they monovary. As corollaries, we also prove that `x ↦ x ^ n` is convex * `Even.convexOn_pow`: for even `n : ℕ`. * `convexOn_pow`: over $[0, +∞)$ for `n : ℕ`. * `convexOn_zpow`: over $(0, +∞)$ For `n : ℤ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/Mul.html"}, {"id": "Mathlib.Analysis.Normed.Operator.Compact.FiniteDimension", "region_id": "analysis", "micro_elevation": 0.0208, "macro_tier": 2, "macro_tier_score": 0.0418, "macro_tier_override": null, "x": 80.631, "z": -210.911, "size": 0.241, "title": "Compact operators and finite dimensional spaces", "summary": "This file contains results linking `IsCompactOperator` with `FiniteDimensional`. The motivation for not including this in the same file as the definition of compact operators is that `Mathlib.Topology.Algebra.Module.FiniteDimension` is quite a heavy import to add there.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Compact/FiniteDimension.html"}, {"id": "Mathlib.Analysis.Complex.PhragmenLindelof", "region_id": "analysis", "micro_elevation": 0.6458, "macro_tier": 1, "macro_tier_score": 0.014, "macro_tier_override": null, "x": 116.273, "z": -239.755, "size": 0.2478, "title": "Phragmen-Lindelöf principle", "summary": "In this file we prove several versions of the Phragmen-Lindelöf principle, a version of the maximum modulus principle for an unbounded domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Complex/PhragmenLindelof.html"}, {"id": "Mathlib.Analysis.BoxIntegral.Partition.Additive", "region_id": "analysis", "micro_elevation": 0.375, "macro_tier": 1, "macro_tier_score": 0.0141, "macro_tier_override": null, "x": 100.882, "z": -227.286, "size": 0.2592, "title": "Box additive functions", "summary": "We say that a function `f : Box ι → M` from boxes in `ℝⁿ` to a commutative additive monoid `M` is *box additive* on subboxes of `I₀ : WithTop (Box ι)` if for any box `J`, `↑J ≤ I₀`, and a partition `π` of `J`, `f J = ∑ J' ∈ π.boxes, f J'`. We use `I₀ : WithTop (Box ι)` instead of `I₀ : Box ι` to use the same definition for functions box additive on subboxes of a box and for functions box additive on all boxes.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/BoxIntegral/Partition/Additive.html"}, {"id": "Mathlib.Analysis.Calculus.Taylor", "region_id": "analysis", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 94.549, "z": -256.055, "size": 0.2861, "title": "Taylor's theorem", "summary": "This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Calculus/Taylor.html"}, {"id": "Mathlib.CategoryTheory.Monad.Algebra", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 103, "macro_tier_score": 0.1537, "macro_tier_override": null, "x": -156.261, "z": -53.619, "size": 0.3556, "title": "Eilenberg-Moore (co)algebras for a (co)monad", "summary": "This file defines Eilenberg-Moore (co)algebras for a (co)monad, and provides the category instance for them. Further it defines the adjoint pair of free and forgetful functors, respectively from and to the original category, as well as the adjoint pair of forgetful and cofree functors, respectively from and to the original category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monad/Algebra.html"}, {"id": "Mathlib.CategoryTheory.Monad.Basic", "region_id": "category_theory", "micro_elevation": 0.3333, "macro_tier": 103, "macro_tier_score": 0.1534, "macro_tier_override": null, "x": 10.297, "z": 26.436, "size": 0.3329, "title": "Monads", "summary": "We construct the categories of monads and comonads, and their forgetful functors to endofunctors. (Note that these are the category theorist's monads, not the programmers monads. For the translation, see the file `Mathlib/CategoryTheory/Monad/Types.lean`.) For the fact that monads are \"just\" monoids in the category of endofunctors, see the file `CategoryTheory.Monad.EquivMon`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monad/Basic.html"}, {"id": "Mathlib.CategoryTheory.Functor.EpiMono", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 103, "macro_tier_score": 0.306, "macro_tier_override": null, "x": -137.659, "z": 215.683, "size": 0.8036, "title": "Preservation and reflection of monomorphisms and epimorphisms", "summary": "We provide typeclasses that state that a functor preserves or reflects monomorphisms or epimorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/EpiMono.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Extension", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -5.112, "z": 33.436, "size": 0.2839, "title": "Extensions and lifts in bicategories", "summary": "We introduce the concept of extensions and lifts within the bicategorical framework. These concepts are defined by commutative diagrams in the (1-)categorical context. Within the bicategorical framework, commutative diagrams are replaced by 2-morphisms. Depending on the orientation of the 2-morphisms, we define both left and right extensions (likewise for lifts). The use of left and right here is a common one in the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Extension.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Basic", "region_id": "category_theory", "micro_elevation": 0.0526, "macro_tier": 103, "macro_tier_score": 0.323, "macro_tier_override": null, "x": -108.502, "z": -165.102, "size": 0.4682, "title": "Bicategories", "summary": "In this file we define typeclass for bicategories. A bicategory `B` consists of * objects `a : B`, * 1-morphisms `f : a ⟶ b` between objects `a b : B`, and * 2-morphisms `η : f ⟶ g` between 1-morphisms `f g : a ⟶ b` between objects `a b : B`. We use `u`, `v`, and `w` as the universe variables for objects, 1-morphisms, and 2-morphisms, respectively. A typeclass for bicategories extends `CategoryTheory.CategoryStruct`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Basic.html"}, {"id": "Mathlib.CategoryTheory.Comma.StructuredArrow.Basic", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 103, "macro_tier_score": 0.2687, "macro_tier_override": null, "x": 22.883, "z": 32.436, "size": 0.5502, "title": "The category of \"structured arrows\"", "summary": "For `T : C ⥤ D`, a `T`-structured arrow with source `S : D` is just a morphism `S ⟶ T.obj Y`, for some `Y : C`. These form a category with morphisms `g : Y ⟶ Y'` making the obvious diagram commute. We prove that `𝟙 (T.obj Y)` is the initial object in `T`-structured objects with source `T.obj Y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/StructuredArrow/Basic.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Free.Coherence", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 2.709, "z": 33.436, "size": 0.2, "title": "The monoidal coherence theorem", "summary": "In this file, we prove the monoidal coherence theorem, stated in the following form: the free monoidal category over any type `C` is thin. We follow a proof described by Ilya Beylin and Peter Dybjer, which has been previously formalized in the proof assistant ALF. The idea is to declare a normal form (with regard to association and adding units) on objects of the free monoidal category and consider the discrete…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Free/Coherence.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Free.Basic", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -137.425, "z": -143.102, "size": 0.2807, "title": "The free monoidal category over a type", "summary": "Given a type `C`, the free monoidal category over `C` has as objects formal expressions built from (formal) tensor products of terms of `C` and a formal unit. Its morphisms are compositions and tensor products of identities, unitors and associators. In this file, we construct the free monoidal category and prove that it is a monoidal category. If `D` is a monoidal category, we construct the functor…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Free/Basic.html"}, {"id": "Mathlib.CategoryTheory.Category.Pointed", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 58.96, "z": 30.436, "size": 0.3088, "title": "The category of pointed types", "summary": "This defines `Pointed`, the category of pointed types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Pointed.html"}, {"id": "Mathlib.CategoryTheory.ConcreteCategory.Forget", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 103, "macro_tier_score": 0.1108, "macro_tier_override": null, "x": -56.748, "z": 22.567, "size": 0.5292, "title": "Forgetful functors", "summary": "A concrete category is a category `C` where the objects and morphisms correspond with types and (bundled) functions between these types, see the file `Mathlib.CategoryTheory.ConcreteCategory.Basic` Each concrete category `C` comes with a canonical faithful functor `forget C : C ⥤ Type*`. We impose no restrictions on the category `C`, so `Type` has the identity forgetful functor. We say that a concrete category `C`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ConcreteCategory/Forget.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Basic", "region_id": "category_theory", "micro_elevation": 0.386, "macro_tier": 103, "macro_tier_score": 0.3442, "macro_tier_override": null, "x": -112.421, "z": 96.123, "size": 0.8733, "title": "Adjunctions between functors", "summary": "`F ⊣ G` represents the data of an adjunction between two functors `F : C ⥤ D` and `G : D ⥤ C`. `F` is the left adjoint and `G` is the right adjoint. We provide various useful constructors: * `mkOfHomEquiv` * `mk'`: construct an adjunction from the data of a hom set equivalence, unit and counit natural transformations together with proofs of the equalities `homEquiv_unit` and `homEquiv_counit` relating them to each…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Basic.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Basic", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0157, "macro_tier_override": null, "x": 75.613, "z": 50.436, "size": 0.3616, "title": "Grothendieck Axioms", "summary": "This file defines some of the Grothendieck Axioms for abelian categories, and proves basic facts about them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Basic.html"}, {"id": "Mathlib.CategoryTheory.Abelian.FunctorCategory", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0247, "macro_tier_override": null, "x": 25.508, "z": 42.0, "size": 0.3285, "title": "If `D` is abelian, then the functor category `C ⥤ D` is also abelian.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/FunctorCategory.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.Filtered", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 102, "macro_tier_score": 0.029, "macro_tier_override": null, "x": 58.113, "z": 45.436, "size": 0.285, "title": "Constructing colimits from finite colimits and filtered colimits", "summary": "We construct colimits of size `w` from finite colimits and filtered colimits of size `w`. Since `w`-sized colimits are constructed from coequalizers and `w`-sized coproducts, it suffices to construct `w`-sized coproducts from finite coproducts and `w`-sized filtered colimits. The idea is simple: to construct coproducts of shape `α`, we take the colimit of the filtered diagram of all coproducts of finite subsets of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/Filtered.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.FunctorCategory", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 103, "macro_tier_score": 0.0817, "macro_tier_override": null, "x": 20.056, "z": 47.436, "size": 0.3198, "title": "Preservation of (co)limits in the functor category", "summary": "* Show that if `X ⨯ -` preserves colimits in `D` for any `X : D`, then the product functor `F ⨯ -` for `F : C ⥤ D` preserves colimits. The idea of the proof is simply that products and colimits in the functor category are computed pointwise, so pointwise preservation implies general preservation. * Show that `F ⋙ -` preserves limits if the target category has limits. * Show that `F : C ⥤ D` preserves limits of a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/FunctorCategory.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Countable", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 102, "macro_tier_score": 0.0197, "macro_tier_override": null, "x": 105.539, "z": 97.178, "size": 0.3087, "title": "Countable limits and colimits", "summary": "A typeclass for categories with all countable (co)limits. We also prove that all cofiltered limits over countable preorders are isomorphic to sequential limits, see `sequentialFunctor_initial`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Countable.html"}, {"id": "Mathlib.CategoryTheory.ConcreteCategory.Basic", "region_id": "category_theory", "micro_elevation": 0.1053, "macro_tier": 102, "macro_tier_score": 0.0061, "macro_tier_override": null, "x": -97.913, "z": 17.791, "size": 0.5307, "title": "Concrete categories", "summary": "A concrete category is a category `C` where the objects and morphisms correspond with types and (bundled) functions between these types. We define concrete categories using `class ConcreteCategory`. To convert an object to a type, write `ToType`. To convert a morphism to a (bundled) function, write `hom`. Each concrete category `C` comes with a canonical faithful functor `forget C : C ⥤ Type*`, see the file…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ConcreteCategory/Basic.html"}, {"id": "Mathlib.CategoryTheory.Sites.Monoidal", "region_id": "category_theory", "micro_elevation": 0.9649, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -266.194, "z": -95.756, "size": 0.2516, "title": "Monoidal category structure on categories of sheaves", "summary": "If `A` is a closed braided category with suitable limits, and `J` is a Grothendieck topology with `HasWeakSheafify J A`, then `Sheaf J A` can be equipped with a monoidal category structure. This is not made an instance as in some cases it may conflict with monoidal structure deduced from chosen finite products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Monoidal.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Basic", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 94.591, "z": 55.436, "size": 0.2476, "title": "Functor categories are monoidal closed", "summary": "Let `C` be a monoidal closed category. Let `J` be a category. In this file, we obtain that the category `J ⥤ C` is monoidal closed if `C` has suitable limits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/FunctorCategory/Basic.html"}, {"id": "Mathlib.CategoryTheory.Localization.Monoidal.Braided", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -17.106, "z": 37.436, "size": 0.2476, "title": "Localization of symmetric monoidal categories", "summary": "Let `C` be a monoidal category equipped with a class of morphisms `W` which is compatible with the monoidal category structure. The file `Mathlib.CategoryTheory.Localization.Monoidal.Basic` constructs a monoidal structure on the localized on `D` such that the localization functor is monoidal. In this file we promote this monoidal structure to a braided structure in the case where `C` is braided, in such a way that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Monoidal/Braided.html"}, {"id": "Mathlib.CategoryTheory.Sites.SheafHom", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 98.232, "z": 61.436, "size": 0.2476, "title": "Internal hom of sheaves", "summary": "In this file, given two sheaves `F` and `G` on a site `(C, J)` with values in a category `A`, we define a sheaf of types `sheafHom F G` which sends `X : C` to the type of morphisms between the restrictions of `F` and `G` to the categories `Over X`. We first define `presheafHom F G` when `F` and `G` are presheaves `Cᵒᵖ ⥤ A` and show that it is a sheaf when `G` is a sheaf. TODO: - turn both `presheafHom` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/SheafHom.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Injective.LiftingProperties", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 23.002, "z": 49.436, "size": 0.2714, "title": "Characterization of injective objects in terms of lifting properties", "summary": "An object `I` is injective iff the morphism `I ⟶ 0` has the right lifting property with respect to monomorphisms, `injective_iff_rlp_monomorphisms_zero`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Injective/LiftingProperties.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Injective.Basic", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 102, "macro_tier_score": 0.017, "macro_tier_override": null, "x": 18.099, "z": 103.894, "size": 0.4197, "title": "Injective objects and categories with enough injectives", "summary": "An object `J` is injective iff every morphism into `J` can be obtained by extending a monomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Injective/Basic.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.LiftingProperty", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 102, "macro_tier_score": 0.0203, "macro_tier_override": null, "x": 3.446, "z": 48.436, "size": 0.3482, "title": "Left and right lifting properties", "summary": "Given a morphism property `T`, we define the left and right lifting property with respect to `T`. We show that the left lifting property is stable under retracts, cobase change, coproducts, and composition, with dual statements for the right lifting property.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/LiftingProperty.html"}, {"id": "Mathlib.CategoryTheory.Functor.KanExtension.Dense", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 102, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": 54.865, "z": 55.436, "size": 0.2701, "title": "Dense functors", "summary": "A functor `F : C ⥤ D` is dense (`F.IsDense`) if `𝟭 D` is a pointwise left Kan extension of `F` along itself, i.e. any `Y : D` is the colimit of all `F.obj X` for all morphisms `F.obj X ⟶ Y` (which is the condition `F.DenseAt Y`). When `F` is full, we show that this is equivalent to saying that the restricted Yoneda functor `D ⥤ Cᵒᵖ ⥤ Type _` is fully faithful (see the lemma…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/KanExtension/Dense.html"}, {"id": "Mathlib.CategoryTheory.Functor.KanExtension.DenseAt", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 102, "macro_tier_score": 0.0192, "macro_tier_override": null, "x": 69.282, "z": 42.436, "size": 0.2539, "title": "Canonical colimits, or functors that are dense at an object", "summary": "Given a functor `F : C ⥤ D` and `Y : D`, we say that `F` is dense at `Y` (`F.DenseAt Y`), if `Y` identifies to the colimit of all `F.obj X` for `X : C` and `f : F.obj X ⟶ Y`, i.e. `Y` identifies to the colimit of the obvious functor `CostructuredArrow F Y ⥤ D`. In some references, it is also said that `Y` is a canonical colimit relatively to `F`. While `F.DenseAt Y` contains data, we also introduce the corresponding…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/KanExtension/DenseAt.html"}, {"id": "Mathlib.CategoryTheory.Limits.Presheaf", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 103, "macro_tier_score": 0.082, "macro_tier_override": null, "x": 129.702, "z": -68.661, "size": 0.3404, "title": "Colimit of representables", "summary": "In this file, We show that every presheaf of types on a category `C` (with `Category.{v₁} C`) is a colimit of representables. This result is also known as the density theorem, the co-Yoneda lemma and the Ninja Yoneda lemma. Three formulations are given: * `colimitOfRepresentable` uses the category of elements of a functor to types; * `isColimitTautologicalCocone` uses the category of costructured arrows for `yoneda…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Presheaf.html"}, {"id": "Mathlib.CategoryTheory.Generator.StrongGenerator", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 102, "macro_tier_score": 0.0196, "macro_tier_override": null, "x": -4.363, "z": 54.436, "size": 0.2961, "title": "Strong generators", "summary": "If `P : ObjectProperty C`, we say that `P` is a strong generator if it is a generator (in the sense that `IsSeparating P` holds) such that for any proper subobject `A ⊂ X`, there exists a morphism `G ⟶ X` which does not factor through `A` from an object satisfying `P`. The main result is the lemma `isStrongGenerator_iff_exists_extremalEpi` which says that if `P` is `w`-small, `C` is locally `w`-small and has…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Generator/StrongGenerator.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.Descent", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 149.261, "z": -26.328, "size": 0.2463, "title": "Descent of morphism properties", "summary": "Given morphism properties `P` and `Q` we say that `P` descends along `Q` (`P.DescendsAlong Q`), if whenever `Q` holds for `X ⟶ Z`, `P` holds for `X ×[Z] Y ⟶ X` implies `P` holds for `Y ⟶ Z`. Dually, we define `P.CodescendsAlong Q`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/Descent.html"}, {"id": "Mathlib.CategoryTheory.LiftingProperties.Over", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 187.122, "z": -80.661, "size": 0.2338, "title": "Lifting properties in Over categories", "summary": "In this file, we show that if `sq` is a commutative square in a category `Over S` for `S : C`, there is a lift for `sq` if there is a lift for the underlying commutative square in the category `C`. It follows that if `i` and `p` are morphisms in `Over S`, then `i` has the left lifting property with respect to `p` when `i.left` has the left lifting property with respect to `p.left`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/LiftingProperties/Over.html"}, {"id": "Mathlib.CategoryTheory.Comma.Over.Basic", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.26, "macro_tier_override": null, "x": 0.519, "z": 26.0, "size": 0.5697, "title": "Over and under categories", "summary": "Over (and under) categories are special cases of comma categories. * If `L` is the identity functor and `R` is a constant functor, then `Comma L R` is the \"slice\" or \"over\" category over the object `R` maps to. * Conversely, if `L` is a constant functor and `R` is the identity functor, then `Comma L R` is the \"coslice\" or \"under\" category under the object `L` maps to.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/Over/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Creates.Opposites", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 102, "macro_tier_score": 0.0196, "macro_tier_override": null, "x": 65.112, "z": 43.436, "size": 0.3, "title": "Limit creation properties of `Functor.op` and related constructions", "summary": "We formulate conditions about `F` which imply that `F.op`, `F.unop`, `F.leftOp` and `F.rightOp` create certain (co)limits and vice versa.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Creates/Opposites.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Opposites", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 103, "macro_tier_score": 0.0482, "macro_tier_override": null, "x": 9.296, "z": 42.436, "size": 0.3047, "title": "Limit preservation properties of `Functor.op` and related constructions", "summary": "We formulate conditions about `F` which imply that `F.op`, `F.unop`, `F.leftOp` and `F.rightOp` preserve certain (co)limits and vice versa.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Opposites.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Creates.Finite", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 103, "macro_tier_score": 0.12, "macro_tier_override": null, "x": 54.521, "z": 42.436, "size": 0.3326, "title": "Creation of finite limits", "summary": "This file defines the classes `CreatesFiniteLimits`, `CreatesFiniteColimits`, `CreatesFiniteProducts` and `CreatesFiniteCoproducts`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Creates/Finite.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Kernels", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 51.54, "z": 44.436, "size": 0.236, "title": "Horizontal maps in a pullback square have the same kernel", "summary": "Consider a commutative square: ``` t X₁ --> X₂ l| |r v v X₃ --> X₄ b ``` * If this is a pullback square, then the induced map `kernel t ⟶ kernel b` is an isomorphism. * If this is a pushout square, then the induced map `cokernel t ⟶ cokernel b` is an isomorphism. (Similar results for the (co)kernels of the vertical maps can be obtained by applying these results to the flipped square.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/IsPullback/Kernels.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Kernels", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 103, "macro_tier_score": 0.203, "macro_tier_override": null, "x": 70.58, "z": -68.766, "size": 0.5731, "title": "Kernels and cokernels", "summary": "In a category with zero morphisms, the kernel of a morphism `f : X ⟶ Y` is the equalizer of `f` and `0 : X ⟶ Y`. (Similarly the cokernel is the coequalizer.) The basic definitions are * `kernel : (X ⟶ Y) → C` * `kernel.ι : kernel f ⟶ X` * `kernel.condition : kernel.ι f ≫ f = 0` and * `kernel.lift (k : W ⟶ X) (h : k ≫ f = 0) : W ⟶ kernel f` (as well as the dual versions)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Kernels.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Defs", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 103, "macro_tier_score": 0.182, "macro_tier_override": null, "x": 25.247, "z": 35.0, "size": 0.5241, "title": "Pullback and pushout squares", "summary": "We provide another API for pullbacks and pushouts. `IsPullback fst snd f g` is the proposition that ``` P --fst--> X | | snd f | | v v Y ---g---> Z ``` is a pullback square. (And similarly for `IsPushout`.) We provide the glue to go back and forth to the usual `IsLimit` API for pullbacks, and prove `IsPullback (pullback.fst f g) (pullback.snd f g) f g` for the usual `pullback f g` provided by the `HasLimit` API.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/IsPullback/Defs.html"}, {"id": "Mathlib.CategoryTheory.Limits.Indization.ParallelPair", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 102, "macro_tier_score": 0.0192, "macro_tier_override": null, "x": -50.046, "z": 57.436, "size": 0.2522, "title": "Parallel pairs of natural transformations between ind-objects", "summary": "We show that if `A` and `B` are ind-objects and `f` and `g` are natural transformations between `A` and `B`, then there is a small filtered category `I` such that `A`, `B`, `f` and `g` are commonly presented by diagrams and natural transformations in `I ⥤ C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Indization/ParallelPair.html"}, {"id": "Mathlib.CategoryTheory.Comma.Final", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 102, "macro_tier_score": 0.0192, "macro_tier_override": null, "x": -38.045, "z": 55.436, "size": 0.2586, "title": "Finality of Projections in Comma Categories", "summary": "We show that `fst L R` is final if `R` is and that `snd L R` is initial if `L` is. As a corollary, we show that `Comma L R` with `L : A ⥤ T` and `R : B ⥤ T` is connected if `R` is final and `A` is connected. We then use this in a proof that derives finality of `map` between two comma categories on a quasi-commutative diagram of functors, some of which need to be final. Finally we prove filteredness of a `Comma L R`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/Final.html"}, {"id": "Mathlib.CategoryTheory.Limits.Indization.IndObject", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 102, "macro_tier_score": 0.0196, "macro_tier_override": null, "x": -13.911, "z": 56.436, "size": 0.296, "title": "Ind-objects", "summary": "For a presheaf `A : Cᵒᵖ ⥤ Type v` we define the type `IndObjectPresentation A` of presentations of `A` as a small filtered colimit of representable presheaves and define the predicate `IsIndObject A` asserting that there is at least one such presentation. A presheaf is an ind-object if and only if the category `CostructuredArrow yoneda A` is filtered and finally small. In this way, `CostructuredArrow yoneda A` can…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Indization/IndObject.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Limits", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.2159, "macro_tier_override": null, "x": 60.812, "z": 34.436, "size": 0.3683, "title": "Isomorphisms about functors which preserve (co)limits", "summary": "If `G` preserves limits, and `C` and `D` have limits, then for any diagram `F : J ⥤ C` we have a canonical isomorphism `preservesLimitsIso : G.obj (Limit F) ≅ Limit (F ⋙ G)`. We also show that we can commute `IsLimit.lift` of a preserved limit with `Functor.mapCone`: `(PreservesLimit.preserves t).lift (G.mapCone c₂) = G.map (t.lift c₂)`. The duals of these are also given. For functors which preserve (co)limits of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Limits.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Basic", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.2671, "macro_tier_override": null, "x": 8.593, "z": 33.436, "size": 0.7916, "title": "Preservation and reflection of (co)limits.", "summary": "There are various distinct notions of \"preserving limits\". The one we aim to capture here is: A functor F : C ⥤ D \"preserves limits\" if it sends every limit cone in C to a limit cone in D. Informally, F preserves all the limits which exist in C. Note that: * Of course, we do not want to require F to *strictly* take chosen limit cones of C to chosen limit cones of D. Indeed, the above definition makes no reference to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Basic.html"}, {"id": "Mathlib.CategoryTheory.Sites.Precoverage", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 103, "macro_tier_score": 0.0348, "macro_tier_override": null, "x": -26.756, "z": 45.436, "size": 0.3617, "title": "Precoverages", "summary": "A precoverage `K` on a category `C` is a set of presieves associated to every object `X : C`, called \"covering presieves\". There are no conditions on this set. Common extensions of a precoverage are: - `CategoryTheory.Coverage`: A coverage is a precoverage that satisfies a pullback compatibility condition, saying that whenever `S` is a covering presieve on `X` and `f : Y ⟶ X` is a morphism, then there exists some…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Precoverage.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Creates.Pullbacks", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 103, "macro_tier_score": 0.0337, "macro_tier_override": null, "x": -13.83, "z": 38.436, "size": 0.2817, "title": "Creation of limits and pullbacks", "summary": "We show some lemmas relating creation of (co)limits and pullbacks (resp. pushouts).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Creates/Pullbacks.html"}, {"id": "Mathlib.CategoryTheory.Sites.Sieves", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 103, "macro_tier_score": 0.0396, "macro_tier_override": null, "x": 13.425, "z": 43.436, "size": 0.3635, "title": "Theory of sieves", "summary": "- For an object `X` of a category `C`, a `Sieve X` is a predicate on morphisms to `X` which is closed under left-composition. - The complete lattice structure on sieves is given, as well as the Galois insertion given by downward-closing. - A `Sieve X` (functorially) induces a presheaf on `C` together with a monomorphism to the Yoneda embedding of `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Sieves.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 103, "macro_tier_score": 0.1523, "macro_tier_override": null, "x": 62.249, "z": -34.774, "size": 0.4913, "title": "Pullback and pushout squares", "summary": "We restate some results about pullbacks/pushouts in the language of `IsPullback` and `IsPushout`, among which the pasting lemmas", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/IsPullback/Basic.html"}, {"id": "Mathlib.CategoryTheory.Galois.Topology", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 3.865, "z": 51.436, "size": 0.2651, "title": "Topology of fundamental group", "summary": "In this file we define a natural topology on the automorphism group of a functor `F : C ⥤ FintypeCat`: It is defined as the subspace topology induced by the natural embedding of `Aut F` into `∀ X, Aut (F.obj X)` where `Aut (F.obj X)` carries the discrete topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Galois/Topology.html"}, {"id": "Mathlib.CategoryTheory.Galois.Prorepresentability", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": 50.178, "z": 50.436, "size": 0.2873, "title": "Pro-Representability of fiber functors", "summary": "We show that any fiber functor is pro-representable, i.e. there exists a pro-object `X : I ⥤ C` such that `F` is naturally isomorphic to the colimit of `X ⋙ coyoneda`. From this we deduce the canonical isomorphism of `Aut F` with the limit over the automorphism groups of all Galois objects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Galois/Prorepresentability.html"}, {"id": "Mathlib.CategoryTheory.Category.Basic", "region_id": "category_theory", "micro_elevation": 0.0175, "macro_tier": 102, "macro_tier_score": 0.0257, "macro_tier_override": null, "x": -108.27, "z": -167.102, "size": 0.6524, "title": "Categories", "summary": "Defines a category, as a type class parametrised by the type of objects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Basic.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Bimon_", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 6.052, "z": 37.436, "size": 0.2478, "title": "The category of bimonoids in a braided monoidal category.", "summary": "We define bimonoids in a braided monoidal category `C` as comonoid objects in the category of monoid objects in `C`. We verify that this is equivalent to the monoid objects in the category of comonoid objects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Bimon_.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Comon_", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 102, "macro_tier_score": 0.0217, "macro_tier_override": null, "x": 14.009, "z": 29.0, "size": 0.419, "title": "The category of comonoids in a monoidal category.", "summary": "We define comonoids in a monoidal category `C`, and show that they are equivalently monoid objects in the opposite category. We construct the monoidal structure on `Comon C`, when `C` is braided. An oplax monoidal functor takes comonoid objects to comonoid objects. That is, an oplax monoidal functor `F : C ⥤ D` induces a functor `Comon C ⥤ Comon D`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Comon_.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Bifunctor", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": -15.232, "z": 34.436, "size": 0.2513, "title": "Preservations of limits for bifunctors", "summary": "Let `G : C₁ ⥤ C₂ ⥤ C` a functor. We introduce a class `PreservesLimit₂ K₁ K₂ G` that encodes the hypothesis that the curried functor `F : C₁ × C₂ ⥤ C` preserves limits of the diagram `K₁ × K₂ : J₁ × J₂ ⥤ C₁ × C₂`. We give a basic API to extract isomorphisms $\\lim_{(j_1,j_2)} G(K_1(j_1), K_2(j_2)) \\simeq G(\\lim K_1, \\lim K_2)$ out of this typeclass.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Bifunctor.html"}, {"id": "Mathlib.CategoryTheory.Limits.Fubini", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": -13.937, "z": 33.436, "size": 0.2582, "title": "A Fubini theorem for categorical (co)limits", "summary": "We prove that $lim_{J × K} G = lim_J (lim_K G(j, -))$ for a functor `G : J × K ⥤ C`, when all the appropriate limits exist. We begin working with a functor `F : J ⥤ K ⥤ C`. We'll write `G : J × K ⥤ C` for the associated \"uncurried\" functor. In the first part, given a coherent family `D` of limit cones over the functors `F.obj j`, and a cone `c` over `G`, we construct a cone over the cone points of `D`. We then show…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Fubini.html"}, {"id": "Mathlib.CategoryTheory.Limits.FunctorCategory.EpiMono", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 103, "macro_tier_score": 0.0557, "macro_tier_override": null, "x": 118.516, "z": -16.83, "size": 0.4427, "title": "Monomorphisms and epimorphisms in functor categories", "summary": "A natural transformation `f : F ⟶ G` between functors `K ⥤ C` is a mono (resp. epi) iff for all `k : K`, `f.app k` is, at least when `C` has pullbacks (resp. pushouts), see `NatTrans.mono_iff_mono_app` and `NatTrans.epi_iff_epi_app`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FunctorCategory/EpiMono.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 103, "macro_tier_score": 0.1201, "macro_tier_override": null, "x": 34.73, "z": 36.0, "size": 0.523, "title": "Relating monomorphisms and epimorphisms to limits and colimits", "summary": "If `F` preserves (resp. reflects) pullbacks, then it preserves (resp. reflects) monomorphisms. We also provide the dual version for epimorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/EpiMono.html"}, {"id": "Mathlib.CategoryTheory.Monad.Comonadicity", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 45.595, "z": 41.0, "size": 0.2453, "title": "Comonadicity theorems", "summary": "We prove comonadicity theorems which can establish a given functor is comonadic. In particular, we show three versions of Beck's comonadicity theorem, and the coreflexive (crude) comonadicity theorem: `F` is a comonadic left adjoint if it has a right adjoint, and: * `C` has, `F` preserves and reflects `F`-split equalizers, see `CategoryTheory.Monad.comonadicOfHasPreservesReflectsFSplitEqualizers` * `F` creates…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monad/Comonadicity.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers", "region_id": "category_theory", "micro_elevation": 0.5789, "macro_tier": 103, "macro_tier_score": 0.0923, "macro_tier_override": null, "x": 131.118, "z": -74.661, "size": 0.3816, "title": "Preserving (co)equalizers", "summary": "Constructions to relate the notions of preserving (co)equalizers and reflecting (co)equalizers to concrete (co)forks. In particular, we show that `equalizerComparison f g G` is an isomorphism iff `G` preserves the limit of the parallel pair `f,g`, as well as the dual result.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.html"}, {"id": "Mathlib.CategoryTheory.Monad.Equalizer", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 102, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": 8.451, "z": 47.436, "size": 0.269, "title": "Special equalizers associated to a comonad", "summary": "Associated to a comonad `T : C ⥤ C` we have important equalizer constructions: Any coalgebra is an equalizer (in the category of coalgebras) of cofree coalgebras. Furthermore, this equalizer is coreflexive. In `C`, this fork diagram is a split equalizer (in particular, it is still an equalizer). This split equalizer is known as the Beck equalizer (as it features heavily in Beck's comonadicity theorem). This file is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monad/Equalizer.html"}, {"id": "Mathlib.CategoryTheory.Monad.Limits", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 102, "macro_tier_score": 0.0252, "macro_tier_override": null, "x": -29.731, "z": 155.931, "size": 0.3559, "title": "Limits and colimits in the category of (co)algebras", "summary": "This file shows that the forgetful functor `forget T : Algebra T ⥤ C` for a monad `T : C ⥤ C` creates limits and creates any colimits which `T` preserves. This is used to show that `Algebra T` has any limits which `C` has, and any colimits which `C` has and `T` preserves. This is generalised to the case of a monadic functor `D ⥤ C`. Dually, this file shows that the forgetful functor `forget T : Coalgebra T ⥤ C` for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monad/Limits.html"}, {"id": "Mathlib.CategoryTheory.Presentable.Adjunction", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 103.427, "z": 60.436, "size": 0.2276, "title": "Presentable objects and adjunctions", "summary": "If `adj : F ⊣ G` and `G` is `κ`-accessible for a regular cardinal `κ`, then `F` preserves `κ`-presentable objects. Moreover, if `G : D ⥤ C` is fully faithful, then `D` is locally `κ`-presentable (resp `κ`-accessible) if `C` is. In particular, if `e : C ≌ D` is an equivalence of categories and `C` is locally presentable (resp. accessible), then so is `D`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/Adjunction.html"}, {"id": "Mathlib.CategoryTheory.Presentable.LocallyPresentable", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 102, "macro_tier_score": 0.0148, "macro_tier_override": null, "x": 99.411, "z": 59.436, "size": 0.2973, "title": "Locally presentable and accessible categories", "summary": "In this file, we define the notion of locally presentable and accessible categories. We first define these notions for a category `C` relative to a fixed regular cardinal `κ` (typeclasses `IsCardinalLocallyPresentable C κ` and `IsCardinalAccessibleCategory C κ`). The existence of such a regular cardinal `κ` is asserted in the typeclasses `IsLocallyPresentable` and `IsAccessibleCategory`. We show that in a locally…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/LocallyPresentable.html"}, {"id": "Mathlib.CategoryTheory.Limits.Creates", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.2261, "macro_tier_override": null, "x": 12.507, "z": 27.0, "size": 0.5571, "title": "Creating (co)limits", "summary": "We say that `F` creates limits of `K` if, given any limit cone `c` for `K ⋙ F` (i.e. below), we can lift it to a cone \"above\", and further that `F` reflects limits for `K`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Creates.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 103, "macro_tier_score": 0.2637, "macro_tier_override": null, "x": 20.781, "z": 37.436, "size": 0.7374, "title": "HasPullback", "summary": "`HasPullback f g` and `pullback f g` provides API for `HasLimit` and `limit` in the case of pullbacks.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.html"}, {"id": "Mathlib.CategoryTheory.Core", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 16.85, "z": 27.436, "size": 0.2478, "title": "The core of a category", "summary": "The core of a category `C` is the (non-full) subcategory of `C` consisting of all objects, and all isomorphisms. We construct it as a `CategoryTheory.Groupoid`. `CategoryTheory.Core.inclusion : Core C ⥤ C` gives the faithful inclusion into the original category. Any functor `F` from a groupoid `G` into `C` factors through `CategoryTheory.Core C`, but this is not functorial with respect to `F`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Core.html"}, {"id": "Mathlib.CategoryTheory.Types.Basic", "region_id": "category_theory", "micro_elevation": 0.3333, "macro_tier": 103, "macro_tier_score": 0.3871, "macro_tier_override": null, "x": 10.263, "z": 228.969, "size": 0.9907, "title": "The category `Type`.", "summary": "In this section we define a `LargeCategory` structure on `Type u`, in such a way that it becomes a `ConcreteCategory`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Types/Basic.html"}, {"id": "Mathlib.CategoryTheory.ExtremalEpi", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 102, "macro_tier_score": 0.029, "macro_tier_override": null, "x": 29.569, "z": 53.436, "size": 0.2877, "title": "Extremal epimorphisms", "summary": "An extremal epimorphism `p : X ⟶ Y` is an epimorphism which does not factor through any proper subobject of `Y`. In case the category has equalizers, we show that a morphism `p : X ⟶ Y` which does not factor through any proper subobject of `Y` is automatically an epimorphism, and also an extremal epimorphism. We also show that a strong epimorphism is an extremal epimorphism, and that both notions coincide when the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ExtremalEpi.html"}, {"id": "Mathlib.CategoryTheory.Subobject.Lattice", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 103, "macro_tier_score": 0.0704, "macro_tier_override": null, "x": 74.393, "z": 52.436, "size": 0.4607, "title": "The lattice of subobjects", "summary": "We provide the `SemilatticeInf` with `OrderTop (Subobject X)` instance when `[HasPullback C]`, and the `SemilatticeSup (Subobject X)` instance when `[HasImages C] [HasBinaryCoproducts C]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subobject/Lattice.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.ShiftAdditive", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 66.504, "z": 49.436, "size": 0.2691, "title": "Properties of objects on preadditive categories equipped with shift", "summary": "In this file, we show that if `C` is a preadditive category equipped with a shift by an additive monoid `A`, and `P : ObjectProperty C` is stable under the shift, then the shift functors on the full subcategory associated to `P` are additive if the shift functors on `C` are. This instance is put in a separate file in order to reduce imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/ShiftAdditive.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.Shift", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 102, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 46.1, "z": 38.0, "size": 0.3311, "title": "Properties of objects on categories equipped with shift", "summary": "Given a predicate `P : ObjectProperty C` on objects of a category equipped with a shift by `A`, we define shifted properties of objects `P.shift a` for all `a : A`. We also introduce a typeclass `P.IsStableUnderShift A` to say that `P X` implies `P (X⟦a⟧)` for all `a : A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/Shift.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.AdditiveFunctor", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 103, "macro_tier_score": 0.0898, "macro_tier_override": null, "x": -40.96, "z": -11.383, "size": 0.5981, "title": "Additive Functors", "summary": "A functor between two preadditive categories is called *additive* provided that the induced map on hom types is a morphism of abelian groups. An additive functor between preadditive categories creates and preserves biproducts. Conversely, if `F : C ⥤ D` is a functor between preadditive categories, where `C` has binary biproducts, and if `F` preserves binary biproducts, then `F` is additive. We also define the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.html"}, {"id": "Mathlib.CategoryTheory.Category.Cat", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 103, "macro_tier_score": 0.3231, "macro_tier_override": null, "x": -74.853, "z": 33.023, "size": 0.5993, "title": "Category of categories", "summary": "This file contains the definition of the category `Cat` of all categories. In this category objects are categories and morphisms are functors between these categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Cat.html"}, {"id": "Mathlib.CategoryTheory.Category.Preorder", "region_id": "category_theory", "micro_elevation": 0.1754, "macro_tier": 103, "macro_tier_score": 0.3206, "macro_tier_override": null, "x": -20.858, "z": -28.488, "size": 0.5384, "title": "Preorders as categories", "summary": "We install a category instance on any preorder. This is not to be confused with the category _of_ preorders, defined in `Order.Category.Preorder`. We show that monotone functions between preorders correspond to functors of the associated categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Preorder.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Evaluation", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -12.646, "z": 34.436, "size": 0.2302, "title": "Adjunctions involving evaluation", "summary": "We show that evaluation of functors has adjoints, given the existence of (co)products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Evaluation.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Products", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.2616, "macro_tier_override": null, "x": -141.817, "z": 217.683, "size": 0.7022, "title": "Categorical (co)products", "summary": "This file defines (co)products as special cases of (co)limits. A product is the categorical generalization of the object `Π i, f i` where `f : ι → C`. It is a limit cone over the diagram formed by `f`, implemented by converting `f` into a functor `Discrete ι ⥤ C`. A coproduct is the dual concept.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Products.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.LocallyGroupoid", "region_id": "category_theory", "micro_elevation": 0.3684, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 52.255, "z": 28.436, "size": 0.2, "title": "(2,1)-categories", "summary": "A bicategory `B` is said to be locally groupoidal (or a (2,1)-category) if for every pair of objects `x, y`, the Hom-category `x ⟶ y` is a groupoid (which is expressed using the `CategoryTheory.IsGroupoid` typeclass). Given a bicategory `B`, we construct a bicategory `Pith B` which is obtained from `B` by discarding non-invertible 2-morphisms. This is realized in practice by applying `Core` to each hom-category of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/LocallyGroupoid.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 102, "macro_tier_score": 0.0104, "macro_tier_override": null, "x": 39.753, "z": 20.436, "size": 0.3229, "title": "Strong transformations of pseudofunctors", "summary": "There are three types of transformations between pseudofunctors, depending on the direction or invertibility of the 2-morphism witnessing the naturality condition. In this file we define strong transformations, which require the 2-morphism to be invertible.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Pseudo.html"}, {"id": "Mathlib.CategoryTheory.Profunctor.Basic", "region_id": "category_theory", "micro_elevation": 0.386, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -8.289, "z": 29.436, "size": 0.2, "title": "Profunctors", "summary": "A profunctor from a category `C` to a category `D` is a functor from `C` to a category of presheaves of sets on `D`. We define this as `Profunctor.{w} C D := C ⥤ Dᵒᵖ ⥤ Type w`. This file provides convenient constructors `ProfunctorCore` and `ProfunctorCore.Hom` for profunctors and natural transformations between them. We also define the identity profunctor `Profunctor.id` as the Yoneda bifunctor, the opposite of a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Profunctor/Basic.html"}, {"id": "Mathlib.CategoryTheory.Yoneda", "region_id": "category_theory", "micro_elevation": 0.3684, "macro_tier": 103, "macro_tier_score": 0.359, "macro_tier_override": null, "x": 3.023, "z": 21.0, "size": 0.9401, "title": "The Yoneda embedding", "summary": "Let `C : Type u₁` be a category (with `Category.{v₁} C`). We define the Yoneda embedding as a fully faithful functor `yoneda : C ⥤ Cᵒᵖ ⥤ Type v₁`, In addition to `yoneda`, we also define `uliftYoneda : C ⥤ Cᵒᵖ ⥤ Type (max w v₁)` with the additional universe parameter `w`. When `C` is locally `w`-small, one may also use `shrinkYoneda.{w} : C ⥤ Cᵒᵖ ⥤ Type w` from the file `Mathlib/CategoryTheory/ShrinkYoneda.lean`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Yoneda.html"}, {"id": "Mathlib.CategoryTheory.Limits.MorphismProperty", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 101, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": 220.698, "z": -0.496, "size": 0.3128, "title": "(Co)limits in subcategories of comma categories defined by morphism properties", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/MorphismProperty.html"}, {"id": "Mathlib.CategoryTheory.Products.Basic", "region_id": "category_theory", "micro_elevation": 0.1754, "macro_tier": 103, "macro_tier_score": 0.4561, "macro_tier_override": null, "x": 39.146, "z": 17.436, "size": 0.6884, "title": "Cartesian products of categories", "summary": "We define the category instance on `C × D` when `C` and `D` are categories. We define: * `sectL C Z` : the functor `C ⥤ C × D` given by `X ↦ ⟨X, Z⟩` * `sectR Z D` : the functor `D ⥤ C × D` given by `Y ↦ ⟨Z, Y⟩` * `fst` : the functor `⟨X, Y⟩ ↦ X` * `snd` : the functor `⟨X, Y⟩ ↦ Y` * `swap` : the functor `C × D ⥤ D × C` given by `⟨X, Y⟩ ↦ ⟨Y, X⟩` (and the fact that this is an equivalence) We further define `evaluation…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Products/Basic.html"}, {"id": "Mathlib.CategoryTheory.EqToHom", "region_id": "category_theory", "micro_elevation": 0.1579, "macro_tier": 103, "macro_tier_score": 0.4591, "macro_tier_override": null, "x": 23.716, "z": 16.436, "size": 0.654, "title": "Morphisms from equations between objects.", "summary": "When working categorically, sometimes one encounters an equation `h : X = Y` between objects. Your initial aversion to this is natural and appropriate: you're in for some trouble, and if there is another way to approach the problem that won't rely on this equality, it may be worth pursuing. You have two options: 1. Use the equality `h` as one normally would in Lean (e.g. using `rw` and `subst`). This may immediately…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/EqToHom.html"}, {"id": "Mathlib.CategoryTheory.Functor.Const", "region_id": "category_theory", "micro_elevation": 0.1579, "macro_tier": 103, "macro_tier_score": 0.4467, "macro_tier_override": null, "x": 18.767, "z": 16.436, "size": 0.4627, "title": "The constant functor", "summary": "`const J : C ⥤ (J ⥤ C)` is the functor that sends an object `X : C` to the functor `J ⥤ C` sending every object in `J` to `X`, and every morphism to `𝟙 X`. When `J` is nonempty, `const` is faithful. We have `(const J).obj X ⋙ F ≅ (const J).obj (F.obj X)` for any `F : C ⥤ D`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/Const.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Indization", "region_id": "category_theory", "micro_elevation": 0.9649, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -56.743, "z": 62.436, "size": 0.2478, "title": "AB axioms in the category of ind-objects", "summary": "We show that `Ind C` satisfies Grothendieck's axiom AB5 if `C` has finite limits and deduce that `Ind C` is Grothendieck abelian if `C` is small and abelian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Indization.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.FunctorCategory", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 34.128, "z": 51.436, "size": 0.257, "title": "AB axioms in functor categories", "summary": "This file proves that, when the relevant limits and colimits exist, exactness of limits and colimits carries over from `A` to the functor category `C ⥤ A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/FunctorCategory.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Types", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 102, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": -32.301, "z": 51.436, "size": 0.2868, "title": "The category of types satisfies Grothendieck's AB5 axiom", "summary": "This is of course just the well-known fact that filtered colimits commute with finite limits in the category of types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Types.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Indization", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 36.372, "z": 61.436, "size": 0.236, "title": "The category of ind-objects is abelian", "summary": "We show that if `C` is a small abelian category, then `Ind C` is an abelian category. In the file `Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Indization.lean`, we show that in this situation `Ind C` is in fact Grothendieck abelian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Indization.html"}, {"id": "Mathlib.CategoryTheory.Generator.Indization", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -48.435, "z": 61.436, "size": 0.236, "title": "Separating set in the category of ind-objects", "summary": "We construct a separating set in the category of ind-objects and conclude that if `C` is small and additive, then `Ind C` has a separator.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Generator/Indization.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Basic", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 71.867, "z": 48.0, "size": 0.3051, "title": "Grothendieck categories", "summary": "This file defines Grothendieck categories and proves basic facts about them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckCategory/Basic.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.Functor", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 88.677, "z": 50.436, "size": 0.2, "title": "Cartesian closed functors", "summary": "Define the exponential comparison morphisms for a functor which preserves binary products, and use them to define a Cartesian closed functor: one which (naturally) preserves exponentials. Define the Frobenius morphism, and show it is an isomorphism iff the exponential comparison is an isomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/Functor.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.Basic", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 103, "macro_tier_score": 0.1141, "macro_tier_override": null, "x": -93.22, "z": 169.789, "size": 0.4885, "title": "Categories with chosen finite products", "summary": "We introduce a class, `CartesianMonoidalCategory`, which bundles explicit choices for a terminal object and binary products in a category `C`. This is primarily useful for categories which have finite products with good definitional properties, such as the category of types. For better defeqs, we also extend `MonoidalCategory`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/Basic.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.Basic", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 103, "macro_tier_score": 0.1032, "macro_tier_override": null, "x": 14.969, "z": 29.0, "size": 0.4392, "title": "Closed monoidal categories", "summary": "Define (right) closed objects and (right) closed monoidal categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/Basic.html"}, {"id": "Mathlib.CategoryTheory.Comma.Over.Pullback", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 103, "macro_tier_score": 0.1398, "macro_tier_override": null, "x": 29.056, "z": 32.0, "size": 0.3777, "title": "Adjunctions related to the over category", "summary": "In a category with pullbacks, for any morphism `f : X ⟶ Y`, the functor `Over.map f : Over X ⥤ Over Y` has a right adjoint `Over.pullback f`. In a category with binary products, for any object `X` the functor `Over.forget X : Over X ⥤ C` has a right adjoint `Over.star X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/Over/Pullback.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Projective.Basic", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 91.167, "z": 52.436, "size": 0.2357, "title": "Projective objects in abelian categories", "summary": "In an abelian category, an object `P` is projective iff the functor `preadditiveCoyonedaObj P` preserves finite colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Projective/Basic.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Yoneda.Projective", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 10.133, "z": 51.436, "size": 0.2434, "title": null, "summary": "An object is projective iff the preadditive coyoneda functor on it preserves epimorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Yoneda/Projective.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Yoneda.Limits", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -33.248, "z": 51.436, "size": 0.2645, "title": "The Yoneda embedding for preadditive categories preserves limits", "summary": "The Yoneda embedding for preadditive categories preserves limits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Yoneda/Limits.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Functor", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 103, "macro_tier_score": 0.1547, "macro_tier_override": null, "x": 60.537, "z": 31.436, "size": 0.5563, "title": "(Lax) monoidal functors", "summary": "A lax monoidal functor `F` between monoidal categories `C` and `D` is a functor between the underlying categories equipped with morphisms * `ε : 𝟙_ D ⟶ F.obj (𝟙_ C)` (called the unit morphism) * `μ X Y : (F.obj X) ⊗ (F.obj Y) ⟶ F.obj (X ⊗ Y)` (called the tensorator, or strength). satisfying various axioms. This is implemented as a typeclass `F.LaxMonoidal`. Similarly, we define the typeclass `F.OplaxMonoidal`. For…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Functor.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.CoherenceLemmas", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 103, "macro_tier_score": 0.1256, "macro_tier_override": null, "x": 24.444, "z": 7.436, "size": 0.3774, "title": "Lemmas which are consequences of monoidal coherence", "summary": "These lemmas are all proved `by coherence`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Limits", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.2014, "macro_tier_override": null, "x": 43.418, "z": 42.004, "size": 0.5343, "title": "Adjunctions and limits", "summary": "A left adjoint preserves colimits (`CategoryTheory.Adjunction.leftAdjoint_preservesColimits`), and a right adjoint preserves limits (`CategoryTheory.Adjunction.rightAdjoint_preservesLimits`). Equivalences create and reflect (co)limits. (`CategoryTheory.Functor.createsLimitsOfIsEquivalence`, `CategoryTheory.Functor.createsColimitsOfIsEquivalence`, `CategoryTheory.Functor.reflectsLimits_of_isEquivalence`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Limits.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Mates", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 103, "macro_tier_score": 0.1843, "macro_tier_override": null, "x": 24.012, "z": 23.0, "size": 0.4463, "title": "Mate of natural transformations", "summary": "This file establishes the bijection between the 2-cells ``` L₁ R₁ C --→ D C ←-- D G ↓ ↗ ↓ H G ↓ ↘ ↓ H E --→ F E ←-- F L₂ R₂ ``` where `L₁ ⊣ R₁` and `L₂ ⊣ R₂`. The corresponding natural transformations are called mates. This bijection includes a number of interesting cases as specializations. For instance, in the special case where `G,H` are identity functors then the bijection preserves and reflects isomorphisms…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Mates.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Parametrized", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 103, "macro_tier_score": 0.1058, "macro_tier_override": null, "x": 12.1, "z": 30.436, "size": 0.3384, "title": "Adjunctions with a parameter", "summary": "Given bifunctors `F : C₁ ⥤ C₂ ⥤ C₃` and `G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂`, this file introduces the notation `F ⊣₂ G` for the adjunctions with a parameter (in `C₁`) between `F` and `G`. (See `MonoidalClosed.internalHomAdjunction₂` in the file `CategoryTheory.Closed.Monoidal` for an example of such an adjunction.) Note: this notion is weaker than the notion of \"adjunction of two variables\" which appears in the mathematical…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Parametrized.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Projective.Basic", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 102, "macro_tier_score": 0.0279, "macro_tier_override": null, "x": -52.19, "z": 166.789, "size": 0.4727, "title": "Projective objects and categories with enough projectives", "summary": "An object `P` is called *projective* if every morphism out of `P` factors through every epimorphism. A category `C` *has enough projectives* if every object admits an epimorphism from some projective object. `CategoryTheory.Projective.over X` picks an arbitrary such projective object, and `CategoryTheory.Projective.π X : CategoryTheory.Projective.over X ⟶ X` is the corresponding epimorphism. Given a morphism `f : X…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Projective/Basic.html"}, {"id": "Mathlib.CategoryTheory.ConcreteCategory.EpiMono", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 103, "macro_tier_score": 0.0351, "macro_tier_override": null, "x": -119.614, "z": 125.602, "size": 0.3782, "title": "Epi and mono in concrete categories", "summary": "In this file, we relate epimorphisms and monomorphisms in a concrete category `C` to surjective and injective morphisms, and we show that if `C` has strong epi mono factorizations and is such that `forget C` preserves both epi and mono, then any morphism in `C` can be factored in a functorial manner as a composition of a surjective morphism followed by an injective morphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ConcreteCategory/EpiMono.html"}, {"id": "Mathlib.CategoryTheory.Sites.CompatibleSheafification", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 102, "macro_tier_score": 0.0197, "macro_tier_override": null, "x": -5.753, "z": 51.436, "size": 0.3108, "title": null, "summary": "In this file, we prove that sheafification is compatible with functors which preserve the correct limits and colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/CompatibleSheafification.html"}, {"id": "Mathlib.CategoryTheory.Sites.CompatiblePlus", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0195, "macro_tier_override": null, "x": 5.959, "z": 50.436, "size": 0.2867, "title": null, "summary": "In this file, we prove that the plus functor is compatible with functors which preserve the correct limits and colimits. See `CategoryTheory/Sites/CompatibleSheafification` for the compatibility of sheafification, which follows easily from the content in this file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/CompatiblePlus.html"}, {"id": "Mathlib.CategoryTheory.Sites.ConcreteSheafification", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": 71.381, "z": 50.436, "size": 0.3388, "title": "Sheafification", "summary": "We construct the sheafification of a presheaf over a site `C` with values in `D` whenever `D` is a concrete category for which the forgetful functor preserves the appropriate (co)limits and reflects isomorphisms. We generally follow the approach of https://stacks.math.columbia.edu/tag/00W1", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/ConcreteSheafification.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Projective.Dimension", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2593, "title": "Projective dimension", "summary": "In an abelian category `C`, we shall say that `X : C` has projective dimension `< n` if all `Ext X Y i` vanish when `n ≤ i`. This defines a type class `HasProjectiveDimensionLT X n`. We also define a type class `HasProjectiveDimensionLE X n` as an abbreviation for `HasProjectiveDimensionLT X (n + 1)`. (Note that the fact that `X` is a zero object is equivalent to the condition `HasProjectiveDimensionLT X 0`, but…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Projective/Dimension.html"}, {"id": "Mathlib.CategoryTheory.Comma.Over.OverClass", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 101, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 176.018, "z": -40.328, "size": 0.2754, "title": "Typeclasses for `S`-objects and `S`-morphisms", "summary": "**Warning**: This is not usually how typeclasses should be used. This is only a sensible approach when the morphism is considered as a structure on `X`, typically in algebraic geometry. This is analogous to how we view ringhoms as structures via the `Algebra` typeclass. For other applications use unbundled arrows or `CategoryTheory.Over`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/Over/OverClass.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.Comma", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 102, "macro_tier_score": 0.0152, "macro_tier_override": null, "x": -23.346, "z": 55.008, "size": 0.33, "title": "Subcategories of comma categories defined by morphism properties", "summary": "Given functors `L : A ⥤ T` and `R : B ⥤ T` and morphism properties `P`, `Q` and `W` on `T`, `A` and `B` respectively, we define the subcategory `P.Comma L R Q W` of `Comma L R` where - objects are objects of `Comma L R` with the structural morphism satisfying `P`, and - morphisms are morphisms of `Comma L R` where the left morphism satisfies `Q` and the right morphism satisfies `W`. For an object `X : T`, this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/Comma.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 103, "macro_tier_score": 0.1595, "macro_tier_override": null, "x": 57.619, "z": 43.436, "size": 0.4078, "title": "Constructing binary product from pullbacks and terminal object.", "summary": "The product is the pullback over the terminal objects. In particular, if a category has pullbacks and a terminal object, then it has binary products. We also provide the dual.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/BinaryProducts.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 103, "macro_tier_score": 0.238, "macro_tier_override": null, "x": 36.1, "z": 39.436, "size": 0.4908, "title": "Pullbacks and monomorphisms", "summary": "This file provides some results about interactions between pullbacks and monomorphisms, as well as the dual statements between pushouts and epimorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Mono.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.Pasting", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 103, "macro_tier_score": 0.1639, "macro_tier_override": null, "x": -2.777, "z": 38.436, "size": 0.3912, "title": "Pasting lemma", "summary": "This file proves the pasting lemma for pullbacks. That is, given the following diagram: ``` X₁ - f₁ -> X₂ - f₂ -> X₃ | | | i₁ i₂ i₃ ∨ ∨ ∨ Y₁ - g₁ -> Y₂ - g₂ -> Y₃ ``` if the right square is a pullback, then the left square is a pullback iff the big square is a pullback.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Pasting.html"}, {"id": "Mathlib.CategoryTheory.Sites.Hypercover.One", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0199, "macro_tier_override": null, "x": -25.277, "z": 50.436, "size": 0.321, "title": "1-hypercovers", "summary": "Given a Grothendieck topology `J` on a category `C`, we define the type of `1`-hypercovers of an object `S : C`. They consist of a covering family of morphisms `X i ⟶ S` indexed by a type `I₀` and, for each tuple `(i₁, i₂)` of elements of `I₀`, a \"covering `Y j` of the fibre product of `X i₁` and `X i₂` over `S`\", a condition which is phrased here without assuming that the fibre product actually exists. The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Hypercover/One.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 103, "macro_tier_score": 0.0394, "macro_tier_override": null, "x": -137.59, "z": 225.683, "size": 0.3511, "title": "Products and coproducts in `C` and `Cᵒᵖ`", "summary": "We construct products and coproducts in the opposite categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Opposites/Products.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coverage", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0255, "macro_tier_override": null, "x": 60.235, "z": 49.436, "size": 0.3719, "title": "Coverages", "summary": "A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`, called \"covering presieves\". This collection must satisfy a certain \"pullback compatibility\" condition, saying that whenever `S` is a covering presieve on `X` and `f : Y ⟶ X` is a morphism, then there exists some covering sieve `T` on `Y` such that `T` factors through `S` along `f`. The main difference between a coverage and a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coverage.html"}, {"id": "Mathlib.CategoryTheory.Idempotents.Karoubi", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0072, "macro_tier_override": null, "x": 62.422, "z": 50.436, "size": 0.4074, "title": "The Karoubi envelope of a category", "summary": "In this file, we define the Karoubi envelope `Karoubi C` of a category `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Idempotents/Karoubi.html"}, {"id": "Mathlib.CategoryTheory.Idempotents.Basic", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": 86.291, "z": 49.436, "size": 0.343, "title": "Idempotent complete categories", "summary": "In this file, we define the notion of idempotent complete categories (also known as Karoubian categories, or pseudoabelian in the case of preadditive categories).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Idempotents/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.SigmaConst", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 101, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 133.322, "z": -34.774, "size": 0.2825, "title": "`sigmaConst.obj` preserves colimits", "summary": "Given an object `R` in a category `C` with coproducts of size `w`, the functor `sigmaConst.obj R : Type w ⥤ C` which sends a type `T` to the coproduct of copies of `R` indexed by `T` preserves all colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/SigmaConst.html"}, {"id": "Mathlib.CategoryTheory.Retract", "region_id": "category_theory", "micro_elevation": 0.3333, "macro_tier": 103, "macro_tier_score": 0.2906, "macro_tier_override": null, "x": -4.083, "z": 26.436, "size": 0.4953, "title": "Retracts", "summary": "Defines retracts of objects and morphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Retract.html"}, {"id": "Mathlib.CategoryTheory.EpiMono", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 103, "macro_tier_score": 0.3864, "macro_tier_override": null, "x": 45.325, "z": 25.436, "size": 0.8642, "title": "Facts about epimorphisms and monomorphisms.", "summary": "The definitions of `Epi` and `Mono` are in `CategoryTheory.Category`, since they are used by some lemmas for `Iso`, which is used everywhere.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/EpiMono.html"}, {"id": "Mathlib.CategoryTheory.Linear.Basic", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 103, "macro_tier_score": 0.0551, "macro_tier_override": null, "x": -7.397, "z": 2.113, "size": 0.4225, "title": "Linear categories", "summary": "An `R`-linear category is a category in which `X ⟶ Y` is an `R`-module in such a way that composition of morphisms is `R`-linear in both variables. Note that sometimes in the literature a \"linear category\" is further required to be abelian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Linear/Basic.html"}, {"id": "Mathlib.CategoryTheory.Category.Grpd", "region_id": "category_theory", "micro_elevation": 0.0175, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 24.272, "z": 8.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Grpd.html"}, {"id": "Mathlib.CategoryTheory.Category.Init", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 102, "macro_tier_score": 0.026, "macro_tier_override": null, "x": 24.444, "z": 7.436, "size": 0.398, "title": "Category Theory Rule Set", "summary": "This module defines the `CategoryTheory` Aesop rule set which is used by the `aesop_cat` tactic. Aesop rule sets only become visible once the file in which they're declared is imported, so we must put this declaration into its own file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Init.html"}, {"id": "Mathlib.CategoryTheory.Sites.CartesianMonoidal", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -252.455, "z": -105.756, "size": 0.2731, "title": "Chosen finite products on sheaves", "summary": "In this file, we put a `CartesianMonoidalCategory` instance on `A`-valued sheaves for a `GrothendieckTopology` whenever `A` has a `CartesianMonoidalCategory` instance.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/CartesianMonoidal.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.FunctorCategory", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 103, "macro_tier_score": 0.0722, "macro_tier_override": null, "x": 161.068, "z": -63.661, "size": 0.3189, "title": "Functor categories have chosen finite products", "summary": "If `C` is a category with chosen finite products, then so is `J ⥤ C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/FunctorCategory.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Subcategory", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 17.813, "z": 44.0, "size": 0.2696, "title": "Full monoidal subcategories", "summary": "Given a monoidal category `C` and a property of objects `P : ObjectProperty C` that is monoidal (i.e. it holds for the unit and is stable by `⊗`), we can put a monoidal structure on `P.FullSubcategory` (the category structure is defined in `Mathlib/CategoryTheory/ObjectProperty/FullSubcategory.lean`). When `C` is also braided/symmetric, the full monoidal subcategory also inherits the braided/symmetric structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Subcategory.html"}, {"id": "Mathlib.CategoryTheory.Sites.Limits", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0248, "macro_tier_override": null, "x": -106.332, "z": 234.683, "size": 0.3321, "title": "Limits and colimits of sheaves", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Limits.html"}, {"id": "Mathlib.CategoryTheory.Localization.Monoidal.Basic", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 102, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": 63.693, "z": 36.436, "size": 0.28, "title": "Localization of monoidal categories", "summary": "Let `C` be a monoidal category equipped with a class of morphisms `W` which is compatible with the monoidal category structure: this means `W` is multiplicative and stable by left and right whiskerings (this is the type class `W.IsMonoidal`). Let `L : C ⥤ D` be a localization functor for `W`. In the file, we construct a monoidal category structure on `D` such that the localization functor is monoidal. The structure…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Monoidal/Basic.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Braided.Multifunctor", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 63.886, "z": 35.436, "size": 0.2569, "title": "Constructing braided categories from natural transformations between multifunctors", "summary": "This file provides an alternative constructor for braided categories, given a braiding `β : -₁ ⊗ -₂ ≅ -₂ ⊗ -₁` as a natural isomorphism between bifunctors. The hexagon identities are phrased as equalities of natural transformations between trifunctors `(-₁ ⊗ -₂) ⊗ -₃ ⟶ -₂ ⊗ (-₃ ⊗ -₁)` and `-₁ ⊗ (-₂ ⊗ -₃) ⟶ (-₃ ⊗ -₁) ⊗ -₂`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Braided/Multifunctor.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.FullyFaithful", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 103, "macro_tier_score": 0.225, "macro_tier_override": null, "x": 131.325, "z": 154.664, "size": 0.5263, "title": "Adjoints of fully faithful functors", "summary": "A left adjoint is * faithful, if and only if the unit is a monomorphism * full, if and only if the unit is a split epimorphism * fully faithful, if and only if the unit is an isomorphism A right adjoint is * faithful, if and only if the counit is an epimorphism * full, if and only if the counit is a split monomorphism * fully faithful, if and only if the counit is an isomorphism This is Lemma 4.5.13 in Riehl's…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/FullyFaithful.html"}, {"id": "Mathlib.CategoryTheory.Sites.NonabelianCohomology.H1", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 24.444, "z": 7.436, "size": 0.2, "title": "The cohomology of a sheaf of groups in degree 1", "summary": "In this file, we shall define the cohomology in degree 1 of a sheaf of groups (TODO). Currently, given a presheaf of groups `G : Cᵒᵖ ⥤ GrpCat` and a family of objects `U : I → C`, we define 1-cochains/1-cocycles/H^1 with values in `G` over `U`. (This definition neither requires the assumption that `G` is a sheaf, nor that `U` covers the terminal object.) As we do not assume that `G` is a presheaf of abelian groups,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/NonabelianCohomology/H1.html"}, {"id": "Mathlib.CategoryTheory.Localization.DerivabilityStructure.OfLocalizedEquivalences", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -58.565, "z": 39.0, "size": 0.2324, "title": "Derivability structures deduced from localized equivalences", "summary": "Assume that we have a diagram of localizer morphisms, in the sense that we have an isomorphism `T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor`. ``` T W₁ ---> W₂ | | L| |R v v W₁' ---> W₂' B ``` In this file, we obtain the lemma `LocalizerMorphism.isRightDerivabilityStructure_of_isLocalizedEquivalence` which shows that if both `L` and `R` are localized equivalences (with `R.functor` essentially surjective), then `B`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/DerivabilityStructure/OfLocalizedEquivalences.html"}, {"id": "Mathlib.CategoryTheory.Localization.DerivabilityStructure.Basic", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 102, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": 72.861, "z": 45.436, "size": 0.3357, "title": "Derivability structures", "summary": "Let `Φ : LocalizerMorphism W₁ W₂` be a localizer morphism, i.e. `W₁ : MorphismProperty C₁`, `W₂ : MorphismProperty C₂`, and `Φ.functor : C₁ ⥤ C₂` is a functor which maps `W₁` to `W₂`. Following the definition introduced by Bruno Kahn and Georges Maltsiniotis in [Bruno Kahn and Georges Maltsiniotis, *Structures de dérivabilité*][KahnMaltsiniotis2008], we say that `Φ` is a right derivability structure if `Φ` has right…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/DerivabilityStructure/Basic.html"}, {"id": "Mathlib.CategoryTheory.GuitartExact.HorizontalComposition", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 28.147, "z": 45.436, "size": 0.2522, "title": "Horizontal composition of Guitart exact squares", "summary": "In this file, we show that the horizontal composition of Guitart exact squares is Guitart exact.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GuitartExact/HorizontalComposition.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.FreydCategory.Homotopy", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -40.038, "z": 50.436, "size": 0.2478, "title": "Homotopies in the arrow category", "summary": "We define left and right homotopies between morphisms of `Arrow V`, where `V` is a preadditive category. TODO: Define the preadditive categories `LeftFreyd V` (resp. `RightFreyd V`) obtained by taking the quotient of `Arrow V` by the left (resp. right) homotopy relation. If `V` has binary biproducts, this will have all kernels (resp. cokernels) and will be the category obtained by freely adjoining kernels (resp.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/FreydCategory/Homotopy.html"}, {"id": "Mathlib.CategoryTheory.Quotient", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 103, "macro_tier_score": 0.0888, "macro_tier_override": null, "x": 12.299, "z": 25.436, "size": 0.435, "title": "Quotient category", "summary": "Constructs the quotient of a category by an arbitrary family of relations on its hom-sets, by introducing a type synonym for the objects, and identifying homs as necessary. This is analogous to 'the quotient of a group by the normal closure of a subset', rather than 'the quotient of a group by a normal subgroup'. When taking the quotient by a congruence relation, `functor_map_eq_iff` says that no unnecessary…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Quotient.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Comma", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -15.171, "z": 49.436, "size": 0.257, "title": "The comma category is preadditive", "summary": "If we have additive functors `L : A ⥤ T` and `R : B ⥤ T` between preadditive categories, then there is a structure of preadditive category on `Comma L R` such that addition commutes with the left and right projections. We then apply this to `Arrow T` for `T` a preadditive category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Comma.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Colim", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -42.922, "z": 55.436, "size": 0.2655, "title": "Exactness of colimits", "summary": "In this file, we shall study exactness properties of colimits. First, we translate the assumption that `colim : (J ⥤ C) ⥤ C` preserves monomorphisms (resp. preserves epimorphisms, resp. is exact) into statements involving arbitrary cocones instead of the ones given by the colimit API. We also show that when an inductive system involves only monomorphisms, then the \"inclusion\" morphism into the colimit is also a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Colim.html"}, {"id": "Mathlib.CategoryTheory.Filtered.Final", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 103, "macro_tier_score": 0.0689, "macro_tier_override": null, "x": -203.606, "z": 185.017, "size": 0.3997, "title": "Final functors with filtered (co)domain", "summary": "If `C` is a filtered category, then the usual equivalent conditions for a functor `F : C ⥤ D` to be final can be restated. We show: * `final_iff_of_isFiltered`: a concrete description of finality which is sometimes a convenient way to show that a functor is final. * `final_iff_isFiltered_structuredArrow`: `F` is final if and only if `StructuredArrow d F` is filtered for all `d : D`, which strengthens the usual…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Filtered/Final.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.Limits", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 103, "macro_tier_score": 0.0729, "macro_tier_override": null, "x": 73.275, "z": -30.662, "size": 0.5352, "title": "Relation of morphism properties with limits", "summary": "The following predicates are introduces for morphism properties `P`: * `IsStableUnderBaseChange`: `P` is stable under base change if in all pullback squares, the left map satisfies `P` if the right map satisfies it. * `IsStableUnderCobaseChange`: `P` is stable under cobase change if in all pushout squares, the right map satisfies `P` if the left map satisfies it. We define `P.universally` for the class of morphisms…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/Limits.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Basic", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 103, "macro_tier_score": 0.0619, "macro_tier_override": null, "x": 10.317, "z": 5.113, "size": 0.4927, "title": "Abelian categories", "summary": "This file contains the definition and basic properties of abelian categories. There are many definitions of abelian category. Our definition is as follows: A category is called abelian if it is preadditive, has finite products, kernels, and cokernels, and if every monomorphism and epimorphism is normal. It should be noted that if we also assume finite coproducts, then preadditivity is actually a consequence of the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Basic.html"}, {"id": "Mathlib.CategoryTheory.Functor.ReflectsIso.Basic", "region_id": "category_theory", "micro_elevation": 0.0877, "macro_tier": 103, "macro_tier_score": 0.404, "macro_tier_override": null, "x": 23.145, "z": 12.436, "size": 0.5971, "title": "Functors which reflect isomorphisms", "summary": "A functor `F` reflects isomorphisms if whenever `F.map f` is an isomorphism, `f` was too. It is formalized as a `Prop`-valued typeclass `ReflectsIsomorphisms F`. Any fully faithful functor reflects isomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/ReflectsIso/Basic.html"}, {"id": "Mathlib.CategoryTheory.Whiskering", "region_id": "category_theory", "micro_elevation": 0.0702, "macro_tier": 103, "macro_tier_score": 0.485, "macro_tier_override": null, "x": 28.253, "z": 11.436, "size": 0.6945, "title": "Whiskering", "summary": "Given a functor `F : C ⥤ D` and functors `G H : D ⥤ E` and a natural transformation `α : G ⟶ H`, we can construct a new natural transformation `F ⋙ G ⟶ F ⋙ H`, called `whiskerLeft F α`. This is the same as the horizontal composition of `𝟙 F` with `α`. This operation is functorial in `F`, and we package this as `whiskeringLeft`. Here `(whiskeringLeft.obj F).obj G` is `F ⋙ G`, and `(whiskeringLeft.obj F).map α` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Whiskering.html"}, {"id": "Mathlib.CategoryTheory.Functor.OfSequence", "region_id": "category_theory", "micro_elevation": 0.193, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": -36.309, "z": 106.842, "size": 0.2716, "title": "Functors from the category of the ordered set `ℕ`", "summary": "In this file, we provide a constructor `Functor.ofSequence` for functors `ℕ ⥤ C` which takes as an input a sequence of morphisms `f : X n ⟶ X (n + 1)` for all `n : ℕ`. We also provide a constructor `NatTrans.ofSequence` for natural transformations between functors `ℕ ⥤ C` which allows to check the naturality condition only for morphisms `n ⟶ n + 1`. The duals of the above for functors `ℕᵒᵖ ⥤ C` are given by…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/OfSequence.html"}, {"id": "Mathlib.CategoryTheory.Groupoid.Discrete", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 39.251, "z": 25.436, "size": 0.2302, "title": "Discrete categories are groupoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Groupoid/Discrete.html"}, {"id": "Mathlib.CategoryTheory.Groupoid", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 103, "macro_tier_score": 0.417, "macro_tier_override": null, "x": 0.127, "z": 24.436, "size": 0.8906, "title": "Groupoids", "summary": "We define `Groupoid` as a typeclass extending `Category`, asserting that all morphisms have inverses. The instance `IsIso.ofGroupoid (f : X ⟶ Y) : IsIso f` means that you can then write `inv f` to access the inverse of any morphism `f`. `Groupoid.isoEquivHom : (X ≅ Y) ≃ (X ⟶ Y)` provides the equivalence between isomorphisms and morphisms in a groupoid. We provide a (non-instance) constructor `Groupoid.ofIsIso` from…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Groupoid.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Connected", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 102, "macro_tier_score": 0.01, "macro_tier_override": null, "x": 60.457, "z": 34.436, "size": 0.293, "title": "Connected shapes", "summary": "In this file we prove that various shapes are connected.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Connected.html"}, {"id": "Mathlib.CategoryTheory.IsConnected", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 103, "macro_tier_score": 0.1934, "macro_tier_override": null, "x": 0.038, "z": 25.436, "size": 0.4292, "title": "Connected category", "summary": "Define a connected category as a _nonempty_ category for which every functor to a discrete category is isomorphic to the constant functor. NB. Some authors include the empty category as connected, we do not. We instead are interested in categories with exactly one 'connected component'. We give some equivalent definitions: - A nonempty category for which every functor to a discrete category is constant on objects.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/IsConnected.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.2591, "macro_tier_override": null, "x": 45.501, "z": 33.436, "size": 0.5487, "title": "Wide pullbacks", "summary": "We define the category `WidePullbackShape`, (resp. `WidePushoutShape`) which is the category obtained from a discrete category of type `J` by adjoining a terminal (resp. initial) element. Limits of this shape are wide pullbacks (pushouts). The convenience method `wideCospan` (`wideSpan`) constructs a functor from this category, hitting the given morphisms. We use `WidePullbackShape` to define ordinary pullbacks…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.html"}, {"id": "Mathlib.CategoryTheory.LiftingProperties.ParametrizedAdjunction", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 188.302, "z": -68.661, "size": 0.2608, "title": "Lifting properties and parametrized adjunctions", "summary": "If we have a parametrized adjunction `adj₂ : F ⊣₂ G`, `sq₁₂ : F.PushoutObjObj f₁ f₂` and `sq₁₃ : G.PullbackObjObj f₁ f₃`, we show that `sq₁₂.ι` has the left lifting property with respect to `f₃` if and only if `f₂` has the left lifting property with respect to `sq₁₃.π`: this is the lemma `ParametrizedAdjunction.hasLiftingProperty_iff`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/LiftingProperties/ParametrizedAdjunction.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Braided.PushoutObjObj", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 162.182, "z": -68.661, "size": 0.2376, "title": "Pushout-tensor-products and the braiding", "summary": "In this file, we introduce a definition `Functor.PushoutObjObj.flipTensor` which switches the two morphisms involved in pushout-tensor-products in a braided category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Braided/PushoutObjObj.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.Braided", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 103.219, "z": -72.661, "size": 0.2542, "title": "Closed braided monoidal categories", "summary": "Interactions between monoidal closed and braided category structures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/Braided.html"}, {"id": "Mathlib.CategoryTheory.Monad.Types", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -125.192, "z": 215.683, "size": 0.2478, "title": "Convert from `Monad` (i.e. Lean's `Type`-based monads) to `CategoryTheory.Monad`", "summary": "This allows us to use these monads in category theory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monad/Types.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Limits.Colimits", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 67.858, "z": 47.0, "size": 0.2526, "title": "Tensor product of colimits", "summary": "In this file, we apply the `PreservesColimit₂` API to the bifunctor `curriedTensor C` on a monoidal category `C`. Given cocones `c₁` and `c₂` for functors `F₁ : J₁ ⥤ C` and `F₂ : J₂ ⥤ C`, we define a cocone `c₁.tensor₂ c₂` for the functor `J₁ × J₂ ⥤ C` obtained using the tensor product on `C`, and we obtain that it is a colimit cocone if both `c₁` and `c₂` are, and `PreservesColimit₂ F₁ F₂ (curriedTensor C)` holds.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Limits/Colimits.html"}, {"id": "Mathlib.CategoryTheory.Limits.Sifted", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 103, "macro_tier_score": 0.0723, "macro_tier_override": null, "x": -7.714, "z": 53.436, "size": 0.3282, "title": "Sifted categories", "summary": "A category `C` is sifted if `C` is nonempty and the diagonal functor `C ⥤ C × C` is final. Sifted categories can be characterized as those such that the colimit functor `(C ⥤ Type) ⥤ Type ` preserves finite products. We achieve this characterization in this file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Sifted.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Category", "region_id": "category_theory", "micro_elevation": 0.193, "macro_tier": 103, "macro_tier_score": 0.1534, "macro_tier_override": null, "x": 10.568, "z": -126.881, "size": 0.5238, "title": "Monoidal categories", "summary": "A monoidal category is a category equipped with a tensor product, unitors, and an associator. In the definition, we provide the tensor product as a pair of functions * `tensorObj : C → C → C` * `tensorHom : (X₁ ⟶ Y₁) → (X₂ ⟶ Y₂) → ((X₁ ⊗ X₂) ⟶ (Y₁ ⊗ Y₂))` and allow use of the overloaded notation `⊗` for both. The unitors and associator are provided componentwise. The tensor product can be expressed as a functor via…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Category.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.InducedBicategory", "region_id": "category_theory", "micro_elevation": 0.2632, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 4.006, "z": 22.436, "size": 0.2, "title": "Induced bicategories", "summary": "In this file we develop API for constructing a full sub-bicategory of a bicategory `C`, given a map `F : B → C`. The objects of the induced bicategory are the objects of `B`, while the 1-morphisms and 2-morphisms are taken as all corresponding morphisms in `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/InducedBicategory.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Functor.StrictPseudofunctor", "region_id": "category_theory", "micro_elevation": 0.2456, "macro_tier": 101, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": 19.968, "z": 21.436, "size": 0.2955, "title": "Strict pseudofunctors", "summary": "In this file we introduce the notion of strict pseudofunctors, which are pseudofunctors such that `mapId` and `mapComp` are given by `eqToIso _`. To a strict pseudofunctor between strict bicategories we can associate a functor between the underlying categories, see `StrictPseudofunctor.toFunctor`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Functor/StrictPseudofunctor.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.GabrielPopescu", "region_id": "category_theory", "micro_elevation": 1.0, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 41.619, "z": 64.436, "size": 0.2, "title": "The Gabriel-Popescu theorem", "summary": "We prove the following Gabriel-Popescu theorem: if `C` is a Grothendieck abelian category and `G` is a separator, then the functor `preadditiveCoyonedaObj G : C ⥤ ModuleCat (End G)ᵐᵒᵖ` sending `X` to `Hom(G, X)` is fully faithful and has an exact left adjoint. We closely follow the elementary proof given by Barry Mitchell.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckCategory/ModuleEmbedding/GabrielPopescu.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 90.543, "z": 51.436, "size": 0.2276, "title": "Pulling back connected colimits", "summary": "If `c` is a cocone over a functor `J ⥤ C` and `f : X ⟶ c.pt`, then for every `j : J` we can take the pullback of `c.ι.app j` and `f`. This gives a new cocone with cone point `X`. We show that if `c` is a colimit cocone, then this is again a colimit cocone as long as `J` is connected and `C` has exact colimits of shape `J`. From this we deduce a `hom_ext` principle for morphisms factoring through a colimit. Usually,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Connected.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Coseparator", "region_id": "category_theory", "micro_elevation": 0.9825, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 93.432, "z": 63.436, "size": 0.2276, "title": "Grothendieck categories have a coseparator", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckCategory/Coseparator.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Injective.Preserves", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -53.812, "z": 41.0, "size": 0.2524, "title": "Preservation of injective objects", "summary": "We define a typeclass `Functor.PreservesInjectiveObjects`. We restate the existing result that if `F ⊣ G` is an adjunction and `F` preserves monomorphisms, then `G` preserves injective objects. We show that the converse is true if the codomain of `F` has enough injectives.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Injective/Preserves.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.LiftToFinset", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -34.234, "z": 46.436, "size": 0.2276, "title": "Additional results about the `liftToFinset` construction", "summary": "If `f` is a family of objects of `C`, then there is a canonical cocone whose cocone point is the coproduct of `f` and whose legs are given by the inclusions of the finite subcoproducts. If `C` is preadditive, then we can describe the legs of this cocone as finite sums of projections followed by inclusions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/LiftToFinset.html"}, {"id": "Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four", "region_id": "category_theory", "micro_elevation": 0.0175, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -1.437, "z": 1.0, "size": 0.2645, "title": "The four and five lemmas", "summary": "Consider the following commutative diagram with exact rows in an abelian category `C`: ``` A ---f--> B ---g--> C ---h--> D ---i--> E | | | | | α β γ δ ε | | | | | v v v v v A' --f'-> B' --g'-> C' --h'-> D' --i'-> E' ``` We show: - the \"mono\" version of the four lemma: if `α` is an epimorphism and `β` and `δ` are monomorphisms, then `γ` is a monomorphism, - the \"epi\" version of the four lemma: if `β` and `δ` are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Refinements", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 102, "macro_tier_score": 0.006, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.3484, "title": "Refinements", "summary": "In order to prove injectivity/surjectivity/exactness properties for diagrams in the category of abelian groups, we often need to do diagram chases. Some of these can be carried out in more general abelian categories: for example, a morphism `X ⟶ Y` in an abelian category `C` is a monomorphism if and only if for all `A : C`, the induced map `(A ⟶ X) → (A ⟶ Y)` of abelian groups is a monomorphism, i.e. injective.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Refinements.html"}, {"id": "Mathlib.CategoryTheory.Sites.Localization", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0197, "macro_tier_override": null, "x": 25.685, "z": 50.436, "size": 0.3108, "title": "The sheaf category as a localized category", "summary": "In this file, it is shown that the category of sheaves `Sheaf J A` is a localization of the category `Presheaf J A` with respect to the class `J.W` of morphisms of presheaves which become isomorphisms after applying the sheafification functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Localization.html"}, {"id": "Mathlib.CategoryTheory.Localization.Bousfield", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0293, "macro_tier_override": null, "x": -36.69, "z": 49.436, "size": 0.3175, "title": "Bousfield localization", "summary": "Given a predicate `P : ObjectProperty C` on the objects of a category `C`, we define `W.isLocal : MorphismProperty C` as the class of morphisms `f : X ⟶ Y` such that for any `Z : C` such that `P Z`, the precomposition with `f` induces a bijection `(Y ⟶ Z) ≃ (X ⟶ Z)`. (This construction is part of the left Bousfield localization in the context of model categories.) When `G ⊣ F` is an adjunction with `F : C ⥤ D` fully…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Bousfield.html"}, {"id": "Mathlib.CategoryTheory.Sites.Sheafification", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0248, "macro_tier_override": null, "x": 85.407, "z": 49.436, "size": 0.3311, "title": "Sheafification", "summary": "Given a site `(C, J)` we define a typeclass `HasSheafify J A` saying that the inclusion functor from `A`-valued sheaves on `C` to presheaves admits a left exact left adjoint (sheafification). Note: to access the `HasSheafify` instance for suitable concrete categories, import the file `Mathlib/CategoryTheory/Sites/LeftExact.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Sheafification.html"}, {"id": "Mathlib.CategoryTheory.Functor.CurryingThree", "region_id": "category_theory", "micro_elevation": 0.2105, "macro_tier": 102, "macro_tier_score": 0.0196, "macro_tier_override": null, "x": 152.29, "z": -95.661, "size": 0.3025, "title": "Currying of functors in three variables", "summary": "We study the equivalence of categories `currying₃ : (C₁ ⥤ C₂ ⥤ C₃ ⥤ E) ≌ C₁ × C₂ × C₃ ⥤ E`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/CurryingThree.html"}, {"id": "Mathlib.CategoryTheory.Functor.Currying", "region_id": "category_theory", "micro_elevation": 0.193, "macro_tier": 103, "macro_tier_score": 0.2169, "macro_tier_override": null, "x": 17.457, "z": 18.436, "size": 0.4159, "title": "Curry and uncurry, as functors.", "summary": "We define `curry : ((C × D) ⥤ E) ⥤ (C ⥤ (D ⥤ E))` and `uncurry : (C ⥤ (D ⥤ E)) ⥤ ((C × D) ⥤ E)`, and verify that they provide an equivalence of categories `currying : (C ⥤ (D ⥤ E)) ≌ ((C × D) ⥤ E)`. This is used in `CategoryTheory.Category.Cat.CartesianClosed` to equip the category of small categories `Cat.{u, u}` with a Cartesian closed structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/Currying.html"}, {"id": "Mathlib.CategoryTheory.Functor.Trifunctor", "region_id": "category_theory", "micro_elevation": 0.0351, "macro_tier": 103, "macro_tier_score": 0.1505, "macro_tier_override": null, "x": 25.633, "z": 9.436, "size": 0.4265, "title": "Trifunctors obtained by composition of bifunctors", "summary": "Given two bifunctors `F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂` and `G : C₁₂ ⥤ C₃ ⥤ C₄`, we define the trifunctor `bifunctorComp₁₂ F₁₂ G : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄` which sends three objects `X₁ : C₁`, `X₂ : C₂` and `X₃ : C₃` to `G.obj ((F₁₂.obj X₁).obj X₂).obj X₃`. Similarly, given two bifunctors `F : C₁ ⥤ C₂₃ ⥤ C₄` and `G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃`, we define the trifunctor `bifunctorComp₂₃ F G₂₃ : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄` which sends three objects `X₁ :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/Trifunctor.html"}, {"id": "Mathlib.CategoryTheory.Products.Associator", "region_id": "category_theory", "micro_elevation": 0.193, "macro_tier": 103, "macro_tier_score": 0.0819, "macro_tier_override": null, "x": 29.099, "z": 18.436, "size": 0.3271, "title": null, "summary": "The associator functor `((C × D) × E) ⥤ (C × (D × E))` and its inverse form an equivalence.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Products/Associator.html"}, {"id": "Mathlib.CategoryTheory.Sites.SheafCohomology.Cech", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -18.833, "z": 47.436, "size": 0.2, "title": "Cech cohomology", "summary": "Given a family of objects `U : ι → C` in a category `C` that has finite products, we define a Cech complex functor `cechComplexFunctor : (Cᵒᵖ ⥤ A) ⥤ CochainComplex A ℕ` which sends a presheaf `P : Cᵒᵖ ⥤ A` in a preadditive category (where products exist) to the cochain complex which in degree `n` consists of the product, indexed by `i : Fin (n + 1) → ι`, of the value of `P` on the product of the objects `U (i a)`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/SheafCohomology/Cech.html"}, {"id": "Mathlib.CategoryTheory.Limits.FormalCoproducts.Cech", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 23.309, "z": 46.436, "size": 0.2617, "title": "The Cech object for formal coproducts", "summary": "Let `C` be a category that has finite products. In this file, we define a functor `cechFunctor : FormalCoproduct C ⥤ SimplicialObject (FormalCoproduct C)` which sends a formal coproduct of objects `U j` (for `j : ι`) to the simplicial object which sends `⦋n⦌` to the formal coproduct, indexed by `i : Fin (n + 1) → ι`, of the products of the objects `U (i a)` for all `a : Fin (n + 1)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FormalCoproducts/Cech.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.Extensions", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 24.444, "z": 7.436, "size": 0.2481, "title": "Properties of objects that are closed under extensions", "summary": "Given a category `C` and `P : ObjectProperty C`, we define a type class `P.IsClosedUnderExtensions` expressing that the property is closed under extensions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/Extensions.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Adjunction.Cat", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 222.807, "z": -43.328, "size": 0.247, "title": "Adjunctions in `Cat`", "summary": "We show that adjunctions in the bicategory `Cat` correspond to adjunctions between functors in the usual categorical sense.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Adjunction/Cat.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Adjunction.Adj", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 43.997, "z": 20.436, "size": 0.2567, "title": "The bicategory of adjunctions in a bicategory", "summary": "Given a bicategory `B`, we construct a bicategory `Adj B` that has essentially the same objects as `B` but whose `1`-morphisms are adjunctions (in the same direction as the left adjoints), and `2`-morphisms are tuples of mate maps between the left and right adjoints (where the map between right adjoints is in the opposite direction). Certain pseudofunctors to the bicategory `Adj Cat` are analogous to bifibered…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Adjunction/Adj.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.Products", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.2017, "macro_tier_override": null, "x": -70.964, "z": 219.683, "size": 0.3731, "title": "Products in `Type`", "summary": "We describe arbitrary products in the category of types, as well as binary products, and the terminal object.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/Products.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.2574, "macro_tier_override": null, "x": 63.418, "z": 34.436, "size": 0.6206, "title": "Binary (co)products", "summary": "We define a category `WalkingPair`, which is the index category for a binary (co)product diagram. A convenience method `pair X Y` constructs the functor from the walking pair, hitting the given objects. We define `prod X Y` and `coprod X Y` as limits and colimits of such functors. Typeclasses `HasBinaryProducts` and `HasBinaryCoproducts` assert the existence of (co)limits shaped as walking pairs. We include lemmas…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.Limits", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.2191, "macro_tier_override": null, "x": -15.244, "z": 121.842, "size": 0.4931, "title": "Limits in the category of types.", "summary": "We show that the category of types has all limits, by providing the usual concrete models.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/Limits.html"}, {"id": "Mathlib.CategoryTheory.Limits.Cones", "region_id": "category_theory", "micro_elevation": 0.386, "macro_tier": 103, "macro_tier_score": 0.2938, "macro_tier_override": null, "x": 42.486, "z": 29.436, "size": 0.5818, "title": "Cones and cocones", "summary": "We define `Cone F`, a cone over a functor `F`, and `F.cones : Cᵒᵖ ⥤ Type`, the functor associating to `X` the cones over `F` with cone point `X`. A cone `c` is defined by specifying its cone point `c.pt` and a natural transformation `c.π` from the constant `c.pt`-valued functor to `F`. We provide `c.w f : c.π.app j ≫ F.map f = c.π.app j'` for any `f : j ⟶ j'` as a wrapper for `c.π.naturality f` avoiding unneeded…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Cones.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.Coproducts", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.0583, "macro_tier_override": null, "x": 32.364, "z": -25.83, "size": 0.3417, "title": "Coproducts in `Type`", "summary": "If `F : J → Type max v u` (with `J : Type v`), we show that the coproduct of `F` exists in `Type max v u` and identifies to the sigma type `Σ j, F j`. Similarly, the binary coproduct of two types `X` and `Y` identifies to `X ⊕ Y`, and the initial object of `Type u` if `PEmpty`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/Coproducts.html"}, {"id": "Mathlib.CategoryTheory.Subobject.MonoOver", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 103, "macro_tier_score": 0.0965, "macro_tier_override": null, "x": 55.294, "z": 49.436, "size": 0.3502, "title": "Monomorphisms over a fixed object", "summary": "As preparation for defining `Subobject X`, we set up the theory for `MonoOver X := { f : Over X // Mono f.hom}`. Here `MonoOver X` is a thin category (a pair of objects has at most one morphism between them), so we can think of it as a preorder. However as it is not skeletal, it is not yet a partial order. `Subobject X` will be defined as the skeletalization of `MonoOver X`. We provide * `def pullback [HasPullbacks…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subobject/MonoOver.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Reflective", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 103, "macro_tier_score": 0.1122, "macro_tier_override": null, "x": 178.664, "z": -82.661, "size": 0.4188, "title": "Reflective functors", "summary": "Basic properties of reflective functors, especially those relating to their essential image. Note properties of reflective functors relating to limits and colimits are included in `Mathlib/CategoryTheory/Monad/Limits.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Reflective.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Restrict", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 103, "macro_tier_score": 0.0967, "macro_tier_override": null, "x": 58.526, "z": 30.436, "size": 0.3598, "title": "Restricting adjunctions", "summary": "`Adjunction.restrictFullyFaithful` shows that an adjunction can be restricted along fully faithful inclusions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Restrict.html"}, {"id": "Mathlib.CategoryTheory.Limits.FullSubcategory", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 103, "macro_tier_score": 0.1644, "macro_tier_override": null, "x": -17.346, "z": 48.436, "size": 0.413, "title": "Limits in full subcategories", "summary": "If a property of objects `P` is closed under taking limits, then limits in `FullSubcategory P` can be constructed from limits in `C`. More precisely, the inclusion creates such limits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FullSubcategory.html"}, {"id": "Mathlib.CategoryTheory.WithTerminal.Cone", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.1005, "macro_tier_override": null, "x": -33.087, "z": 28.0, "size": 0.2979, "title": "Relations between `Cone`, `WithTerminal` and `Over`", "summary": "Given categories `C` and `J`, an object `X : C` and a functor `K : J ⥤ Over X`, it has an obvious lift `liftFromOver K : WithTerminal J ⥤ C`, namely, send the terminal object to `X`. These two functors have equivalent categories of cones (`coneEquiv`). As a corollary, the limit of `K` is the limit of `liftFromOver K`, and vice-versa.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/WithTerminal/Cone.html"}, {"id": "Mathlib.CategoryTheory.Shift.CommShiftTwo", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 5.217, "z": 43.0, "size": 0.239, "title": "Commutation with shifts of functors in two variables", "summary": "We introduce a typeclass `Functor.CommShift₂Int` for a bifunctor `G : C₁ ⥤ C₂ ⥤ D` (with `D` a preadditive category) as the two variable analogue of `Functor.CommShift`. We require that `G` commutes with the shifts in both variables and that the two ways to identify `(G.obj (X₁⟦p⟧)).obj (X₂⟦q⟧)` to `((G.obj X₁).obj X₂)⟦p + q⟧` differ by the sign `(-1) ^ (p + q)`. This is implemented using a structure…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/CommShiftTwo.html"}, {"id": "Mathlib.CategoryTheory.Center.NegOnePow", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 49.381, "z": 47.436, "size": 0.2382, "title": "Powers of `-1` in the center of a preadditive category", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Center/NegOnePow.html"}, {"id": "Mathlib.CategoryTheory.Linear.LinearFunctor", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 103, "macro_tier_score": 0.041, "macro_tier_override": null, "x": 60.963, "z": 42.0, "size": 0.4305, "title": "Linear Functors", "summary": "An additive functor between two `R`-linear categories is called *linear* if the induced map on hom types is a morphism of `R`-modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Linear/LinearFunctor.html"}, {"id": "Mathlib.CategoryTheory.Shift.Twist", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 47.17, "z": 45.436, "size": 0.2382, "title": "Twisting a shift", "summary": "Given a category `C` equipped with a shift by a monoid `A`, we introduce a structure `t : TwistShiftData C A` which consists of a collection of invertible elements in the center of the category `C` (typically, `C` will be preadditive, and these will be signs), which allow to introduce a type synonym category `t.Category` with identical shift functors as `C` but where the isomorphisms `shiftFunctorAdd` have been…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/Twist.html"}, {"id": "Mathlib.CategoryTheory.Shift.Pullback", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": 16.955, "z": 49.436, "size": 0.2779, "title": "The pullback of a shift by a monoid morphism", "summary": "Given a shift by a monoid `B` on a category `C` and a monoid morphism `φ : A →+ B`, we define a shift by `A` on a category `PullbackShift C φ` which is a type synonym for `C`. If `F : C ⥤ D` is a functor between categories equipped with shifts by `B`, we define a type synonym `PullbackShift.functor F φ` for `F`. When `F` has a `CommShift` structure by `B`, we define a pulled back `CommShift` structure by `A` on…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/Pullback.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Action.LinearFunctor", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 5.394, "z": 20.436, "size": 0.2, "title": "Functors that are linear with respect to an action", "summary": "Given a monoidal category `C` acting on the left or on the right on categories `D` and `D'`, we introduce the following typeclasses on functors `F : D ⥤ D'` to express compatibility of `F` with the action of `C`: * `F.LaxLeftLinear C` bundles the \"lineator\" as a morphism `μₗ : c ⊙ₗ F.obj d ⟶ F.obj (c ⊙ₗ d)`. * `F.LaxRightLinear C` bundles the \"lineator\" as a morphism `μᵣ : F.obj d ⊙ᵣ c ⟶ F.obj (d ⊙ᵣ c)`. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Action/LinearFunctor.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Action.Basic", "region_id": "category_theory", "micro_elevation": 0.2105, "macro_tier": 101, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": 21.067, "z": 19.436, "size": 0.3173, "title": "Actions from a monoidal category on a category", "summary": "Given a monoidal category `C`, and a category `D`, we define a left action of `C` on `D` as the data of an object `c ⊙ₗ d` of `D` for every `c : C` and `d : D`, as well as the data required to turn `- ⊙ₗ -` into a bifunctor, along with structure natural isomorphisms `(- ⊗ -) ⊙ₗ - ≅ - ⊙ₗ - ⊙ₗ -` and `𝟙_ C ⊙ₗ - ≅ -`, subject to coherence conditions. We also define right actions, for these, the notation for the action…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Action/Basic.html"}, {"id": "Mathlib.CategoryTheory.Sites.Closed", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 102, "macro_tier_score": 0.0193, "macro_tier_override": null, "x": 39.424, "z": 48.436, "size": 0.2735, "title": "Closed sieves", "summary": "A natural closure operator on sieves is a closure operator on `Sieve X` for each `X` which commutes with pullback. We show that a Grothendieck topology `J` induces a natural closure operator, and define what the closed sieves are. The collection of `J`-closed sieves forms a presheaf which is a sheaf for `J`, and further this presheaf can be used to determine the Grothendieck topology from the sheaf predicate.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Closed.html"}, {"id": "Mathlib.CategoryTheory.Sites.SheafOfTypes", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 103, "macro_tier_score": 0.0302, "macro_tier_override": null, "x": 58.928, "z": 47.436, "size": 0.3734, "title": "Sheaves of types on a Grothendieck topology", "summary": "Defines the notion of a sheaf of types (usually called a sheaf of sets by mathematicians) on a category equipped with a Grothendieck topology, as well as a range of equivalent conditions useful in different situations. In `Mathlib/CategoryTheory/Sites/IsSheafFor.lean` it is defined what it means for a presheaf to be a sheaf *for* a particular sieve. Given a Grothendieck topology `J`, `P` is a sheaf if it is a sheaf…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/SheafOfTypes.html"}, {"id": "Mathlib.CategoryTheory.Sites.Hypercover.Homotopy", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 24.467, "z": 51.436, "size": 0.239, "title": "The category of `1`-hypercovers up to homotopy", "summary": "In this file we define the category of `1`-hypercovers up to homotopy. This is the category of `1`-hypercovers, but where morphisms are considered up to existence of a homotopy.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Hypercover/Homotopy.html"}, {"id": "Mathlib.CategoryTheory.Filtered.Basic", "region_id": "category_theory", "micro_elevation": 0.5789, "macro_tier": 103, "macro_tier_score": 0.2019, "macro_tier_override": null, "x": -66.363, "z": 224.683, "size": 0.5455, "title": "Filtered categories", "summary": "A category is filtered if every finite diagram admits a cocone. We give a simple characterisation of this condition as 1. for every pair of objects there exists another object \"to the right\", 2. for every pair of parallel morphisms there exists a morphism to the right so the compositions are equal, and 3. there exists some object. An important example of filtered category is given by nonempty directed types;…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Filtered/Basic.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.SingleObj", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 79.834, "z": 45.436, "size": 0.2302, "title": "`SingleObj α` is preadditive when `α` is a ring.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/SingleObj.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Basic", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 103, "macro_tier_score": 0.1391, "macro_tier_override": null, "x": 55.549, "z": 37.0, "size": 0.5217, "title": "Preadditive categories", "summary": "A preadditive category is a category in which `X ⟶ Y` is an abelian group in such a way that composition of morphisms is linear in both variables. This file contains a definition of preadditive category that directly encodes the definition given above. The definition could also be phrased as follows: A preadditive category is a category enriched over the category of Abelian groups. Once the general framework to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Basic.html"}, {"id": "Mathlib.CategoryTheory.SingleObj", "region_id": "category_theory", "micro_elevation": 0.3684, "macro_tier": 102, "macro_tier_score": 0.0245, "macro_tier_override": null, "x": 51.056, "z": 28.436, "size": 0.3137, "title": "Single-object category", "summary": "Single object category with a given monoid of endomorphisms. It is defined to facilitate transferring some definitions and lemmas (e.g., conjugacy etc.) from category theory to monoids and groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/SingleObj.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.Grp", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 102, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": 256.939, "z": -22.328, "size": 0.3253, "title": "Yoneda embedding of `Grp C`", "summary": "We show that group objects are exactly those whose yoneda presheaf is a presheaf of groups, by constructing the yoneda embedding `Grp C ⥤ Cᵒᵖ ⥤ GrpCat.{v}` and showing that it is fully faithful and its (essential) image is the representable functors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/Grp.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Grp", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 102, "macro_tier_score": 0.011, "macro_tier_override": null, "x": 59.425, "z": 44.0, "size": 0.3626, "title": "The category of groups in a Cartesian monoidal category", "summary": "We define group objects in Cartesian monoidal categories. We show that the associativity diagram of a group object is always Cartesian and deduce that morphisms of group objects commute with taking inverses. We show that a finite-product-preserving functor takes group objects to group objects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Grp.html"}, {"id": "Mathlib.CategoryTheory.Subobject.Classifier.Defs", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -13.541, "z": 52.436, "size": 0.2827, "title": "Subobject Classifier", "summary": "We define a structure containing the data of a subobject classifier in a category `C` as `CategoryTheory.Subobject.Classifier C`. c.f. the following Lean 3 code, where similar work was done: https://github.com/b-mehta/topos/blob/master/src/subobject_classifier.lean", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subobject/Classifier/Defs.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.RegularMono", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 103, "macro_tier_score": 0.1497, "macro_tier_override": null, "x": -203.932, "z": 147.184, "size": 0.3941, "title": "Definitions and basic properties of regular monomorphisms and epimorphisms.", "summary": "A regular monomorphism is a morphism that is the equalizer of some parallel pair. In this file, we give the following definitions. * `RegularMono f`, which is a structure carrying the data that exhibits `f` as a regular monomorphism. That is, it carries a fork and data specifying `f` as the equalizer of that fork. * `IsRegularMono f`, which is a `Prop`-valued class stating that `f` is a regular monomorphism. In…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/RegularMono.html"}, {"id": "Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 102, "macro_tier_score": 0.0102, "macro_tier_override": null, "x": -51.208, "z": 88.894, "size": 0.3138, "title": "Balanced categories and functors reflecting isomorphisms", "summary": "If a category is `C`, and a functor out of `C` reflects epimorphisms and monomorphisms, then the functor reflects isomorphisms. Furthermore, categories that admit a functor that `ReflectsIsomorphisms`, `PreservesEpimorphisms` and `PreservesMonomorphisms` are balanced.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/ReflectsIso/Balanced.html"}, {"id": "Mathlib.CategoryTheory.Subobject.Presheaf", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": -37.058, "z": 51.436, "size": 0.2587, "title": "Subobjects presheaf", "summary": "Following Section I.3 of [Sheaves in Geometry and Logic][MM92], we define the subobjects presheaf `Subobject.presheaf C` mapping any object `X` to its type of subobjects `Subobject X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subobject/Presheaf.html"}, {"id": "Mathlib.CategoryTheory.CodiscreteCategory", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 29.621, "z": 30.436, "size": 0.2, "title": "Codiscrete categories", "summary": "We define `Codiscrete A` as an alias for the type `A`, and use this type alias to provide a `Category` instance whose Hom types are `Unit`. `Codiscrete.functor` promotes a function `f : C → A` (for any category `C`) to a functor `f : C ⥤ Codiscrete A`. Similarly, `Codiscrete.natTrans` and `Codiscrete.natIso` promote `I`-indexed families of morphisms, or `I`-indexed families of isomorphisms to natural transformations…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/CodiscreteCategory.html"}, {"id": "Mathlib.CategoryTheory.Simple", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 102, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": 56.502, "z": 46.0, "size": 0.2752, "title": "Simple objects", "summary": "We define simple objects in any category with zero morphisms. A simple object is an object `Y` such that any monomorphism `f : X ⟶ Y` is either an isomorphism or zero (but not both). This is formalized as a `Prop`-valued typeclass `Simple X`. In some contexts, especially representation theory, simple objects are called \"irreducibles\". If a morphism `f` out of a simple object is nonzero and has a kernel, then that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Simple.html"}, {"id": "Mathlib.CategoryTheory.Monad.Coequalizer", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 55.379, "z": 47.436, "size": 0.2617, "title": "Special coequalizers associated to a monad", "summary": "Associated to a monad `T : C ⥤ C` we have important coequalizer constructions: Any algebra is a coequalizer (in the category of algebras) of free algebras. Furthermore, this coequalizer is reflexive. In `C`, this cofork diagram is a split coequalizer (in particular, it is still a coequalizer). This split coequalizer is known as the Beck coequalizer (as it features heavily in Beck's monadicity theorem). This file has…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monad/Coequalizer.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Reflexive", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 102, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": -4.026, "z": 46.436, "size": 0.2739, "title": "Reflexive coequalizers", "summary": "This file deals with reflexive pairs, which are pairs of morphisms with a common section. A reflexive coequalizer is a coequalizer of such a pair. These kind of coequalizers often enjoy nicer properties than general coequalizers, and feature heavily in some versions of the monadicity theorem. We also give some examples of reflexive pairs: for an adjunction `F ⊣ G` with counit `ε`, the pair `(FGε_B, ε_FGB)` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Reflexive.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 103, "macro_tier_score": 0.096, "macro_tier_override": null, "x": 34.707, "z": 39.436, "size": 0.316, "title": "Split coequalizers", "summary": "We define what it means for a triple of morphisms `f g : X ⟶ Y`, `π : Y ⟶ Z` to be a split coequalizer: there is a section `s` of `π` and a section `t` of `g`, which additionally satisfy `t ≫ f = π ≫ s`. In addition, we show that every split coequalizer is a coequalizer (`CategoryTheory.IsSplitCoequalizer.isCoequalizer`) and absolute (`CategoryTheory.IsSplitCoequalizer.map`) A pair `f g : X ⟶ Y` has a split…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/SplitCoequalizer.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.FunctorToTypes", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 5.479, "z": 36.436, "size": 0.2, "title": "Binary (co)products of type-valued functors", "summary": "Defines an explicit construction of binary products and coproducts of type-valued functors. Also defines isomorphisms to the categorical product and coproduct, respectively.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/FunctorToTypes.html"}, {"id": "Mathlib.CategoryTheory.Limits.FunctorCategory.Basic", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.2174, "macro_tier_override": null, "x": -9.975, "z": 35.436, "size": 0.4381, "title": "(Co)limits in functor categories.", "summary": "We show that if `D` has limits, then the functor category `C ⥤ D` also has limits (`CategoryTheory.Limits.functorCategoryHasLimits`), and the evaluation functors preserve limits (`CategoryTheory.Limits.evaluation_preservesLimits`) (and similarly for colimits). We also show that `F : D ⥤ K ⥤ C` preserves (co)limits if it does so for each `k : K` (`CategoryTheory.Limits.preservesLimits_of_evaluation` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FunctorCategory/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.Colimits", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.2171, "macro_tier_override": null, "x": -66.503, "z": 217.683, "size": 0.4262, "title": "Colimits in the category of types", "summary": "We show that the category of types has all colimits, by providing the usual concrete models.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/Colimits.html"}, {"id": "Mathlib.CategoryTheory.Functor.Basic", "region_id": "category_theory", "micro_elevation": 0.0351, "macro_tier": 102, "macro_tier_score": 0.0228, "macro_tier_override": null, "x": -106.48, "z": -166.102, "size": 0.7698, "title": "Functors", "summary": "Defines a functor between categories, extending a `Prefunctor` between quivers. Introduces, in the `CategoryTheory` scope, notations `C ⥤ D` for the type of all functors from `C` to `D`, `𝟭` for the identity functor and `⋙` for functor composition. TODO: Switch to using the `⇒` arrow.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/Basic.html"}, {"id": "Mathlib.CategoryTheory.Products.Unitor", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 103, "macro_tier_score": 0.4299, "macro_tier_override": null, "x": 8.823, "z": 20.436, "size": 0.6421, "title": "The left/right unitor equivalences `1 × C ≌ C` and `C × 1 ≌ C`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Products/Unitor.html"}, {"id": "Mathlib.CategoryTheory.Discrete.Basic", "region_id": "category_theory", "micro_elevation": 0.2105, "macro_tier": 103, "macro_tier_score": 0.4532, "macro_tier_override": null, "x": 39.071, "z": 19.436, "size": 0.7211, "title": "Discrete categories", "summary": "We define `Discrete α` as a structure containing a term `a : α` for any type `α`, and use this type alias to provide a `SmallCategory` instance whose only morphisms are the identities. There is an annoying technical difficulty that it has turned out to be inconvenient to allow categories with morphisms living in `Prop`, so instead of defining `X ⟶ Y` in `Discrete α` as `X = Y`, one might define it as `PLift (X =…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Discrete/Basic.html"}, {"id": "Mathlib.CategoryTheory.Action.Monoidal", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 41.959, "z": 45.0, "size": 0.2497, "title": "Induced monoidal structure on `Action V G`", "summary": "We show: * When `V` is monoidal, braided, or symmetric, so is `Action V G`. * When `V` is rigid and `G` is a group, `Action V G` is also rigid.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Action/Monoidal.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Linear", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0148, "macro_tier_override": null, "x": 25.935, "z": 43.0, "size": 0.2917, "title": "Linear monoidal categories", "summary": "A monoidal category is `MonoidalLinear R` if it is monoidal preadditive and tensor product of morphisms is `R`-linear in both factors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Linear.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 66.346, "z": 38.436, "size": 0.2288, "title": "Functors from a groupoid into a right/left rigid category form a right/left rigid category.", "summary": "(Using the pointwise monoidal structure on the functor category.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Rigid/FunctorCategory.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 69.42, "z": 38.436, "size": 0.2288, "title": "Transport rigid structures over a monoidal equivalence.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Rigid/OfEquivalence.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Transport", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.0303, "macro_tier_override": null, "x": 24.42, "z": 26.0, "size": 0.3744, "title": "Transport a monoidal structure along an equivalence.", "summary": "When `C` and `D` are equivalent as categories, we can transport a monoidal structure on `C` along the equivalence as `CategoryTheory.Monoidal.transport`, obtaining a monoidal structure on `D`. More generally, we can transport the lawfulness of a monoidal structure along a suitable faithful functor, as `CategoryTheory.Monoidal.induced`. The comparison is analogous to the difference between `Equiv.monoid` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Transport.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Types.Basic", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 103, "macro_tier_score": 0.0874, "macro_tier_override": null, "x": 7.465, "z": 43.0, "size": 0.3727, "title": "The category of types is a (symmetric) monoidal category", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Types/Basic.html"}, {"id": "Mathlib.CategoryTheory.Action.Concrete", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 68.684, "z": 37.436, "size": 0.2611, "title": "Constructors for `Action V G` for some concrete categories", "summary": "We construct `Action (Type*) G` from a `[MulAction G X]` instance and give some applications.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Action/Concrete.html"}, {"id": "Mathlib.CategoryTheory.Action.Limits", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": -2.073, "z": 51.436, "size": 0.2611, "title": "Categorical properties of `Action V G`", "summary": "We show: * When `V` has (co)limits so does `Action V G`. * When `V` is preadditive, linear, or abelian so is `Action V G`. * The forgetful functor `Action V G ⥤ V` preserves any (co)limit whose image in `V` exists, and reflects all (co)limits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Action/Limits.html"}, {"id": "Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 102, "macro_tier_score": 0.01, "macro_tier_override": null, "x": 25.49, "z": 45.436, "size": 0.2919, "title": "Formal Coproducts", "summary": "In this file we construct the category of formal coproducts given a category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FormalCoproducts/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.1838, "macro_tier_override": null, "x": -5.084, "z": 118.902, "size": 0.428, "title": "Preserving products", "summary": "Constructions to relate the notions of preserving products and reflecting products to concrete fans. In particular, we show that `piComparison G f` is an isomorphism iff `G` preserves the limit of `f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Products.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.2268, "macro_tier_override": null, "x": 21.578, "z": 27.0, "size": 0.4316, "title": "Zero objects", "summary": "A category \"has a zero object\" if it has an object which is both initial and terminal. Having a zero object provides zero morphisms, as the unique morphisms factoring through the zero object; see `CategoryTheory.Limits.Shapes.ZeroMorphisms`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.html"}, {"id": "Mathlib.CategoryTheory.Category.Cat.Terminal", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 175.899, "z": -80.661, "size": 0.2425, "title": "Terminal categories", "summary": "We prove that a category is terminal if its underlying type has a `Unique` structure and the category has an `IsDiscrete` instance. We then use this to provide various examples of terminal categories. TODO: Show the converse: that terminal categories have a unique object and are discrete. TODO: Provide an analogous characterization of terminal categories as codiscrete categories with a unique object.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Cat/Terminal.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.267, "macro_tier_override": null, "x": -88.267, "z": 121.842, "size": 0.7129, "title": "Initial and terminal objects in a category.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Terminal.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Grothendieck", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 57.798, "z": 36.436, "size": 0.2403, "title": "Colimits on Grothendieck constructions preserving limits", "summary": "We characterize the condition in which colimits on Grothendieck constructions preserve limits: By preserving limits on the Grothendieck construction's base category as well as on each of its fibers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Grothendieck.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Grothendieck", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.1876, "macro_tier_override": null, "x": -8.551, "z": 35.436, "size": 0.3837, "title": "(Co)limits on the (strict) Grothendieck Construction", "summary": "* Shows that if a functor `G : Grothendieck F ⥤ H`, with `F : C ⥤ Cat`, has a colimit, and every fiber of `G` has a colimit, then so does the fiberwise colimit functor `C ⥤ H`. * Vice versa, if each fiber of `G` has a colimit and the fiberwise colimit functor has a colimit, then `G` has a colimit. * Shows that colimits of functors on the Grothendieck construction are colimits of \"fibered colimits\", i.e. of applying…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Grothendieck.html"}, {"id": "Mathlib.CategoryTheory.Filtered.Small", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 102, "macro_tier_score": 0.0193, "macro_tier_override": null, "x": -13.577, "z": 41.436, "size": 0.2641, "title": "A functor from a small category to a filtered category factors through a small filtered category", "summary": "A consequence of this is that if `C` is filtered and finally small, then `C` is also \"finally filtered-small\", i.e., there is a final functor from a small filtered category to `C`. This is occasionally useful, for example in the proof of the recognition theorem for ind-objects (Proposition 6.1.5 in [Kashiwara2006]).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Filtered/Small.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Grp_", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 85.94, "z": 52.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Grp_.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts", "region_id": "category_theory", "micro_elevation": 0.5789, "macro_tier": 103, "macro_tier_score": 0.2068, "macro_tier_override": null, "x": 84.391, "z": -56.717, "size": 0.5495, "title": "Categories with finite (co)products", "summary": "Typeclasses representing categories with (co)products over finite indexing types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/FiniteProducts.html"}, {"id": "Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 103, "macro_tier_score": 0.0487, "macro_tier_override": null, "x": -13.02, "z": 98.894, "size": 0.337, "title": "Facts about (co)limits of functors into concrete categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/ConcreteCategory/Basic.html"}, {"id": "Mathlib.CategoryTheory.ConcreteCategory.Elementwise", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 101, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": 33.667, "z": 63.333, "size": 0.3199, "title": null, "summary": "In this file we provide various simp lemmas in its elementwise form via `Tactic.Elementwise`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ConcreteCategory/Elementwise.html"}, {"id": "Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 24.444, "z": 7.436, "size": 0.236, "title": "Long exact sequence for the kernel and cokernel of a composition", "summary": "If `f : X ⟶ Y` and `g : Y ⟶ Z` are composable morphisms in an abelian category, we construct a long exact sequence: `0 ⟶ ker f ⟶ ker (f ≫ g) ⟶ ker g ⟶ coker f ⟶ coker (f ≫ g) ⟶ coker g ⟶ 0`. This is obtained by applying the snake lemma to the following morphism of exact sequences, where the rows are the obvious split exact sequences ``` 0 ⟶ X ⟶ X ⊞ Y ⟶ Y ⟶ 0 |f |φ |g v v v 0 ⟶ Y ⟶ Y ⊞ Z ⟶ Z ⟶ 0 ``` and `φ` is given…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/DiagramLemmas/KernelCokernelComp.html"}, {"id": "Mathlib.CategoryTheory.Category.Quiv", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 103, "macro_tier_score": 0.0683, "macro_tier_override": null, "x": 43.858, "z": 30.436, "size": 0.3695, "title": "The category of quivers", "summary": "The category of (bundled) quivers, and the free/forgetful adjunction between `Cat` and `Quiv`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Quiv.html"}, {"id": "Mathlib.CategoryTheory.PathCategory.MorphismProperty", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 103, "macro_tier_score": 0.0672, "macro_tier_override": null, "x": -1.45, "z": 27.436, "size": 0.2982, "title": "Properties of morphisms in a path category.", "summary": "We provide a formulation of induction principles for morphisms in a path category in terms of `MorphismProperty`. This file is separate from `Mathlib/CategoryTheory/PathCategory/Basic.lean` in order to reduce transitive imports. We also define a morpism property `W.paths : MorphismProperty (Paths C)` for any `W : MorphismProperty C`, consisting of all paths in `C` that consist only of morphisms in `W`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/PathCategory/MorphismProperty.html"}, {"id": "Mathlib.CategoryTheory.Generator.Abelian", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -42.942, "z": 55.436, "size": 0.269, "title": "A complete abelian category with enough injectives and a separator has an injective coseparator", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Generator/Abelian.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Subobject", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": 85.182, "z": 54.436, "size": 0.275, "title": "Equivalence between subobjects and quotients in an abelian category", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Subobject.html"}, {"id": "Mathlib.CategoryTheory.Generator.Preadditive", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 42.236, "z": 54.436, "size": 0.2585, "title": "Separators in preadditive categories", "summary": "This file contains characterizations of separating sets and objects that are valid in all preadditive categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Generator/Preadditive.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Opposite", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0112, "macro_tier_override": null, "x": 38.172, "z": 43.0, "size": 0.3718, "title": "The opposite of an abelian category is abelian.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Opposite.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Finite", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 103, "macro_tier_score": 0.1869, "macro_tier_override": null, "x": 48.7, "z": 50.805, "size": 0.5277, "title": "Preservation of finite (co)limits.", "summary": "These functors are also known as left exact (flat) or right exact functors when the categories involved are abelian, or more generally, finitely (co)complete.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Finite.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 103, "macro_tier_score": 0.0684, "macro_tier_override": null, "x": 53.429, "z": 37.0, "size": 0.3758, "title": "Preserving (co)kernels", "summary": "Constructions to relate the notions of preserving (co)kernels and reflecting (co)kernels to concrete (co)forks. In particular, we show that `kernel_comparison f g G` is an isomorphism iff `G` preserves the limit of the parallel pair `f,0`, as well as the dual result.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 103, "macro_tier_score": 0.252, "macro_tier_override": null, "x": 62.679, "z": 39.436, "size": 0.7001, "title": "Categories with finite limits.", "summary": "A typeclass for categories with all finite (co)limits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/FiniteLimits.html"}, {"id": "Mathlib.CategoryTheory.Functor.Category", "region_id": "category_theory", "micro_elevation": 0.0175, "macro_tier": 103, "macro_tier_score": 0.5087, "macro_tier_override": null, "x": 25.943, "z": 8.436, "size": 0.6917, "title": "The category of functors and natural transformations between two fixed categories.", "summary": "We provide the category instance on `C ⥤ D`, with morphisms the natural transformations. At the end of the file, we provide the left and right unitors, and the associator, for functor composition. (In fact functor composition is definitionally associative, but very often relying on this causes extremely slow elaboration, so it is better to insert it explicitly.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/Category.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackObjObj", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 102, "macro_tier_score": 0.0103, "macro_tier_override": null, "x": 5.118, "z": 45.436, "size": 0.3182, "title": "Leibniz Constructions", "summary": "Let `F : C₁ ⥤ C₂ ⥤ C₃`. Given morphisms `f₁ : X₁ ⟶ Y₁` in `C₁` and `f₂ : X₂ ⟶ Y₂` in `C₂`, we introduce a structure `F.PushoutObjObj f₁ f₂` which contains the data of a pushout of `(F.obj Y₁).obj X₂` and `(F.obj X₁).obj Y₂` along `(F.obj X₁).obj X₂`. If `sq₁₂ : F.PushoutObjObj f₁ f₂`, we have a canonical \"inclusion\" `sq₁₂.ι : sq₁₂.pt ⟶ (F.obj Y₁).obj Y₂`. If `C₃` has pushouts, then we define the Leibniz pushout…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/PullbackObjObj.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Braided.Basic", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.1411, "macro_tier_override": null, "x": -40.622, "z": 27.0, "size": 0.5734, "title": "Braided and symmetric monoidal categories", "summary": "The basic definitions of braided monoidal categories, and symmetric monoidal categories, as well as braided functors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Braided/Basic.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Functor.Types", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -15.663, "z": 51.436, "size": 0.2, "title": "Convert from `Applicative` to `CategoryTheory.Functor.LaxMonoidal`", "summary": "This allows us to use Lean's `Type`-based applicative functors in category theory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Functor/Types.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.EilenbergMoore", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -30.029, "z": 49.436, "size": 0.2, "title": "Preadditive structure on algebras over a monad", "summary": "If `C` is a preadditive category and `T` is an additive monad on `C` then `Algebra T` is also preadditive. Dually, if `U` is an additive comonad on `C` then `Coalgebra U` is preadditive as well.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/EilenbergMoore.html"}, {"id": "Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0103, "macro_tier_override": null, "x": 86.312, "z": 50.436, "size": 0.3155, "title": "Chosen pullbacks along a morphism", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/LocallyCartesianClosed/ChosenPullbacksAlong.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Unique", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 103, "macro_tier_score": 0.0392, "macro_tier_override": null, "x": -137.436, "z": 135.184, "size": 0.3431, "title": "Uniqueness of adjoints", "summary": "This file shows that adjoints are unique up to natural isomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Unique.html"}, {"id": "Mathlib.CategoryTheory.Sites.CoproductSheafCondition", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 207.769, "z": -22.328, "size": 0.2517, "title": "The sheaf condition and universal coproducts", "summary": "In this file we show that if `{ fᵢ : Yᵢ ⟶ X }` is a family of morphisms and `∐ᵢ Yᵢ` is a universal coproduct, then any presheaf `F` that preserves products is a sheaf for the single object covering `{ ∐ᵢ Yᵢ ⟶ X }` if and only if it is a sheaf for `{ fᵢ : Yᵢ ⟶ X }ᵢ`. We provide both a version for a general coefficient category and one for type values presheafs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/CoproductSheafCondition.html"}, {"id": "Mathlib.CategoryTheory.Sites.Preserves", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 262.957, "z": -27.328, "size": 0.2755, "title": "Sheaves preserve products", "summary": "We prove that a presheaf which satisfies the sheaf condition with respect to certain presieves preserve \"the corresponding products\".", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Preserves.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Braided.Opposite", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 102, "macro_tier_score": 0.0199, "macro_tier_override": null, "x": 66.465, "z": 35.436, "size": 0.3225, "title": "If `C` is braided, so is `Cᵒᵖ`.", "summary": "Todo: we should also do `Cᵐᵒᵖ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Braided/Opposite.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.Retract", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 103, "macro_tier_score": 0.164, "macro_tier_override": null, "x": 28.749, "z": 46.436, "size": 0.3947, "title": "Properties of objects which are stable under retracts", "summary": "Given a category `C` and `P : ObjectProperty C` (i.e. `P : C → Prop`), this file introduces the type class `P.IsStableUnderRetracts`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/Retract.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 103, "macro_tier_score": 0.193, "macro_tier_override": null, "x": -22.278, "z": 45.436, "size": 0.4151, "title": "Binary biproducts", "summary": "We introduce the notion of binary biproducts. These are slightly unusual relative to the other shapes in the library, as they are simultaneously limits and colimits. (Zero objects are similar; they are \"biterminal\".) For results about biproducts in preadditive categories see `CategoryTheory.Preadditive.Biproducts`. In a category with zero morphisms, we model the (binary) biproduct of `P Q : C` using a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/BinaryBiproducts.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.ContainsZero", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 103, "macro_tier_score": 0.1631, "macro_tier_override": null, "x": 4.917, "z": 43.436, "size": 0.3449, "title": "Properties of objects which hold for a zero object", "summary": "Given a category `C` and `P : ObjectProperty C`, we define a type class `P.ContainsZero` expressing that there exists a zero object for which `P` holds. (We do not require that `P` holds for all zero objects, as in some applications (e.g. triangulated categories), `P` may not necessarily be closed under isomorphisms.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/ContainsZero.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.Small", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 103, "macro_tier_score": 0.1969, "macro_tier_override": null, "x": 33.959, "z": 32.436, "size": 0.3719, "title": "Smallness of a property of objects", "summary": "In this file, given `P : ObjectProperty C`, we define `ObjectProperty.Small.{w} P` as an abbreviation for `Small.{w} (Subtype P)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/Small.html"}, {"id": "Mathlib.CategoryTheory.Opposites", "region_id": "category_theory", "micro_elevation": 0.1404, "macro_tier": 103, "macro_tier_score": 0.4735, "macro_tier_override": null, "x": 18.03, "z": 15.436, "size": 0.743, "title": "Opposite categories", "summary": "We provide a category instance on `Cᵒᵖ`. The morphisms `X ⟶ Y` are defined to be the morphisms `unop Y ⟶ unop X` in `C`. Here `Cᵒᵖ` is an irreducible typeclass synonym for `C` (it is the same one used in the algebra library). We also provide various mechanisms for constructing opposite morphisms, functors, and natural transformations. Unfortunately, because we do not have a definitional equality `op (op X) = X`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Opposites.html"}, {"id": "Mathlib.CategoryTheory.Functor.Flat", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 103, "macro_tier_score": 0.0391, "macro_tier_override": null, "x": 0.559, "z": 49.436, "size": 0.3315, "title": "Representably flat functors", "summary": "We define representably flat functors as functors such that the category of structured arrows over `X` is cofiltered for each `X`. This concept is also known as flat functors as in [Elephant] Remark C2.3.7, and this name is suggested by Mike Shulman in https://golem.ph.utexas.edu/category/2011/06/flat_functors_and_morphisms_of.html to avoid confusion with other notions of flatness (e.g. see the notion of flat…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/Flat.html"}, {"id": "Mathlib.CategoryTheory.Filtered.Connected", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 103, "macro_tier_score": 0.097, "macro_tier_override": null, "x": 70.001, "z": 41.436, "size": 0.3806, "title": "Filtered categories are connected", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Filtered/Connected.html"}, {"id": "Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 103, "macro_tier_score": 0.044, "macro_tier_override": null, "x": 31.924, "z": 48.436, "size": 0.3414, "title": "Filtered colimits commute with finite limits.", "summary": "We show that for a functor `F : J × K ⥤ Type v`, when `J` is finite and `K` is filtered, the universal morphism `colimitLimitToLimitColimit F` comparing the colimit (over `K`) of the limits (over `J`) with the limit of the colimits is an isomorphism. (In fact, to prove that it is injective only requires that `J` has finitely many objects.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Filtered", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 103, "macro_tier_score": 0.0537, "macro_tier_override": null, "x": -92.598, "z": 49.342, "size": 0.3528, "title": "Preservation of filtered colimits and cofiltered limits.", "summary": "Typically forgetful functors from algebraic categories preserve filtered colimits (although not general colimits). See e.g. `Mathlib/Algebra/Category/MonCat/FilteredColimits.lean`. Note also that using the results in the file `Mathlib/CategoryTheory/Presentable/Directed.lean`, in order to show that a functor preserves filtered colimits, it would be sufficient to check that it preserves colimits indexed by nonempty…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Filtered.html"}, {"id": "Mathlib.CategoryTheory.Limits.Bicones", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 103, "macro_tier_score": 0.043, "macro_tier_override": null, "x": 33.977, "z": 30.436, "size": 0.2565, "title": "Bicones", "summary": "Given a category `J`, a walking `Bicone J` is a category whose objects are the objects of `J` and two extra vertices `Bicone.left` and `Bicone.right`. The morphisms are the morphisms of `J` and `left ⟶ j`, `right ⟶ j` for each `j : J` such that `(· ⟶ j)` and `(· ⟶ k)` commutes with each `f : j ⟶ k`. Given a diagram `F : J ⥤ C` and two `Cone F`s, we can join them into a diagram `Bicone J ⥤ C` via `biconeMk`. This is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Bicones.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Free", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -89.239, "z": -155.102, "size": 0.268, "title": "Free bicategories", "summary": "We define the free bicategory over a quiver. In this bicategory, the 1-morphisms are freely generated by the arrows in the quiver, and the 2-morphisms are freely generated by the formal identities, the formal unitors, and the formal associators modulo the relation derived from the axioms of a bicategory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Free.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preorder", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.0343, "macro_tier_override": null, "x": 5.24, "z": 70.011, "size": 0.3351, "title": "(Co)limits in a preorder category", "summary": "We provide basic results about (co)limits in the associated category of a preordered type. - We show that a functor `F` has a (co)limit iff it has a greatest lower bound (least upper bound). - We show maximal (minimal) elements correspond to terminal (initial) objects. - We show that (co)products correspond to infima (suprema).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preorder.html"}, {"id": "Mathlib.CategoryTheory.Sites.CoversTop", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 77.09, "z": 50.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/CoversTop.html"}, {"id": "Mathlib.CategoryTheory.Sites.CoversTop.Basic", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": 61.073, "z": 42.0, "size": 0.3389, "title": "Objects which cover the terminal object", "summary": "In this file, given a site `(C, J)`, we introduce the notion of a family of objects `Y : I → C` which \"cover the final object\": this means that for all `X : C`, the sieve `Sieve.ofObjects Y X` is covering for `J`. When there is a terminal object `X : C`, then `J.CoversTop Y` holds iff `Sieve.ofObjects Y X` is covering for `J`. We introduce a notion of compatible family of elements on objects `Y` and obtain…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/CoversTop/Basic.html"}, {"id": "Mathlib.CategoryTheory.Adhesive.Basic", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 102, "macro_tier_score": 0.0246, "macro_tier_override": null, "x": -6.217, "z": 48.436, "size": 0.3218, "title": "Adhesive categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adhesive/Basic.html"}, {"id": "Mathlib.CategoryTheory.Extensive", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 103, "macro_tier_score": 0.0356, "macro_tier_override": null, "x": -53.032, "z": 231.683, "size": 0.4018, "title": "Extensive categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Extensive.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.KernelPair", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 103, "macro_tier_score": 0.0962, "macro_tier_override": null, "x": 79.052, "z": 45.436, "size": 0.3287, "title": "Kernel pairs", "summary": "This file defines what it means for a parallel pair of morphisms `a b : R ⟶ X` to be the kernel pair for a morphism `f`. Some properties of kernel pairs are given, namely allowing one to transfer between the kernel pair of `f₁ ≫ f₂` to the kernel pair of `f₁`. It is also proved that if `f` is a coequalizer of some pair, and `a`,`b` is a kernel pair for `f` then it is a coequalizer of `a`,`b`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/KernelPair.html"}, {"id": "Mathlib.CategoryTheory.Limits.Indization.Category", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 102, "macro_tier_score": 0.0144, "macro_tier_override": null, "x": 95.36, "z": 59.436, "size": 0.2462, "title": "The category of Ind-objects", "summary": "We define the `v`-category of Ind-objects of a category `C`, called `Ind C`, as well as the functors `Ind.yoneda : C ⥤ Ind C` and `Ind.inclusion C : Ind C ⥤ Cᵒᵖ ⥤ Type v`. For a small filtered category `I`, we also define `Ind.lim I : (I ⥤ C) ⥤ Ind C` and show that it preserves finite limits and finite colimits. This file will mainly collect results about ind-objects (stated in terms of `IsIndObject`) and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Indization/Category.html"}, {"id": "Mathlib.CategoryTheory.Limits.ExactFunctor", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 103, "macro_tier_score": 0.0887, "macro_tier_override": null, "x": 48.881, "z": 42.436, "size": 0.4322, "title": "Bundled exact functors", "summary": "We say that a functor `F` is left exact if it preserves finite limits, it is right exact if it preserves finite colimits, and it is exact if it is both left exact and right exact. In this file, we define the categories of bundled left exact, right exact and exact functors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/ExactFunctor.html"}, {"id": "Mathlib.CategoryTheory.Limits.Indization.Equalizers", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 102, "macro_tier_score": 0.0191, "macro_tier_override": null, "x": -24.002, "z": 58.436, "size": 0.2326, "title": "Equalizers of ind-objects", "summary": "We show that if a category `C` has equalizers, then ind-objects are closed under equalizers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Indization/Equalizers.html"}, {"id": "Mathlib.CategoryTheory.Limits.Indization.LocallySmall", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 102, "macro_tier_score": 0.0191, "macro_tier_override": null, "x": 4.477, "z": 57.436, "size": 0.2326, "title": "There are only `v`-many natural transformations between Ind-objects", "summary": "We provide the instance `LocallySmall.{v} (FullSubcategory (IsIndObject (C := C)))`, which will serve as the basis for our definition of the category of Ind-objects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Indization/LocallySmall.html"}, {"id": "Mathlib.CategoryTheory.Limits.Indization.Products", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 102, "macro_tier_score": 0.0191, "macro_tier_override": null, "x": -37.029, "z": 58.436, "size": 0.2326, "title": "Ind-objects are closed under products", "summary": "We show that if `C` admits products indexed by `α`, then `IsIndObject` is closed under taking products in `Cᵒᵖ ⥤ Type v` indexed by `α`. This will imply that the functor `Ind C ⥤ Cᵒᵖ ⥤ Type v` creates products indexed by `α` and that the functor `C ⥤ Ind C` preserves them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Indization/Products.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Yoneda", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 102, "macro_tier_score": 0.0239, "macro_tier_override": null, "x": 52.096, "z": 37.436, "size": 0.2488, "title": "Yoneda preserves certain colimits", "summary": "Given a bifunctor `F : J ⥤ Cᵒᵖ ⥤ Type v`, we prove the isomorphism `Hom(YX, colim_j F(j, -)) ≅ colim_j Hom(YX, F(j, -))`, where `Y` is the Yoneda embedding. We state this in two ways. One is functorial in `X` and stated as a natural isomorphism of functors `yoneda.op ⋙ yoneda.obj (colimit F) ≅ yoneda.op ⋙ colimit (F ⋙ yoneda)`, and from this we deduce the more traditional preservation statement `PreservesColimit F…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Yoneda.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Ulift", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.2154, "macro_tier_override": null, "x": -65.63, "z": 219.683, "size": 0.3401, "title": "`ULift` creates small (co)limits", "summary": "This file shows that `uliftFunctor.{v, u}` preserves all limits and colimits, including those potentially too big to exist in `Type u`. As this functor is fully faithful, we also deduce that it creates `u`-small limits and colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Ulift.html"}, {"id": "Mathlib.CategoryTheory.Limits.FunctorToTypes", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 102, "macro_tier_score": 0.024, "macro_tier_override": null, "x": 45.049, "z": 36.436, "size": 0.2573, "title": "Concrete description of (co)limits in functor categories", "summary": "Some of the concrete descriptions of (co)limits in `Type v` extend to (co)limits in the functor category `K ⥤ Type v`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FunctorToTypes.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Indization", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": 81.7, "z": 60.436, "size": 0.2687, "title": "The category of ind-objects is preadditive", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Indization.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Transfer", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0289, "macro_tier_override": null, "x": 31.954, "z": 49.436, "size": 0.2778, "title": "Pulling back a preadditive structure along a fully faithful functor", "summary": "A preadditive structure on a category `D` transfers to a preadditive structure on `C` for a given fully faithful functor `F : C ⥤ D`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Transfer.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Opposite", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 103, "macro_tier_score": 0.0405, "macro_tier_override": null, "x": 14.634, "z": -11.83, "size": 0.4065, "title": "If `C` is preadditive, `Cᵒᵖ` has a natural preadditive structure.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Opposite.html"}, {"id": "Mathlib.CategoryTheory.Category.Cat.CartesianClosed", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 199.442, "z": -63.661, "size": 0.236, "title": "Cartesian closed structure on `Cat`", "summary": "The category of small categories is Cartesian closed, with the exponential at a category `C` defined by the functor category mapping out of `C`. Adjoint transposition is defined by currying and uncurrying. TODO: It would be useful to investigate and formalize further compatibilities along the lines of `Cat.ihom_obj` and `Cat.ihom_map`, relating currying of functors with currying in monoidal closed categories and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Cat/CartesianClosed.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.Cat", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": 94.357, "z": -64.661, "size": 0.2862, "title": "Chosen finite products in `Cat`", "summary": "This file proves that the Cartesian product of a pair of categories agrees with the product in `Cat`, and provides the associated `CartesianMonoidalCategory` instance.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/Cat.html"}, {"id": "Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 2.322, "z": 39.436, "size": 0.2553, "title": "Pointwise right derived functors", "summary": "We define pointwise right derived functors using the notion of pointwise left Kan extensions. We show that if `F : C ⥤ H` inverts `W : MorphismProperty C`, then it has a pointwise right derived functor. Note: the file `Mathlib/CategoryTheory/Functor/Derived/PointwiseLeftDerived.lean` was obtained by dualizing this file. These two files should be kept in sync.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/Derived/PointwiseRightDerived.html"}, {"id": "Mathlib.CategoryTheory.Functor.Derived.RightDerived", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": 47.519, "z": 38.436, "size": 0.2683, "title": "Right derived functors", "summary": "In this file, given a functor `F : C ⥤ H`, and `L : C ⥤ D` that is a localization functor for `W : MorphismProperty C`, we define `F.totalRightDerived L W : D ⥤ H` as the left Kan extension of `F` along `L`: it is defined if the type class `F.HasRightDerivedFunctor W` asserting the existence of a left Kan extension is satisfied. (The name `totalRightDerived` is to avoid name-collision with `Functor.rightDerived`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/Derived/RightDerived.html"}, {"id": "Mathlib.CategoryTheory.Functor.KanExtension.Pointwise", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 103, "macro_tier_score": 0.0873, "macro_tier_override": null, "x": 8.534, "z": 38.436, "size": 0.3664, "title": "Pointwise Kan extensions", "summary": "In this file, we define the notion of pointwise (left) Kan extension. Given two functors `L : C ⥤ D` and `F : C ⥤ H`, and `E : LeftExtension L F`, we introduce a cocone `E.coconeAt Y` for the functor `CostructuredArrow.proj L Y ⋙ F : CostructuredArrow L Y ⥤ H` the point of which is `E.right.obj Y`, and the type `E.IsPointwiseLeftKanExtensionAt Y` which expresses that `E.coconeAt Y` is a colimit. When this holds for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.html"}, {"id": "Mathlib.CategoryTheory.Localization.StructuredArrow", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 57.218, "z": 37.436, "size": 0.2625, "title": "Induction principles for structured and costructured arrows", "summary": "Assume that `L : C ⥤ D` is a localization functor for `W : MorphismProperty C`. Given `X : C` and a predicate `P` on `StructuredArrow (L.obj X) L`, we obtain the lemma `Localization.induction_structuredArrow` which shows that `P` holds for all structured arrows if `P` holds for the identity map `𝟙 (L.obj X)`, if `P` is stable by post-composition with `L.map f` for any `f` and if `P` is stable by post-composition…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/StructuredArrow.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.Retract", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 102, "macro_tier_score": 0.0203, "macro_tier_override": null, "x": -9.522, "z": 20.0, "size": 0.3493, "title": "Stability under retracts", "summary": "Given `P : MorphismProperty C`, we introduce a typeclass `P.IsStableUnderRetracts` which is the property that `P` is stable under retracts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/Retract.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.IsTerminal", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 103, "macro_tier_score": 0.2807, "macro_tier_override": null, "x": -3.84, "z": 31.436, "size": 0.6087, "title": "Initial and terminal objects in a category.", "summary": "In this file we define the predicates `IsTerminal` and `IsInitial` as well as the class `InitialMonoClass`. The classes `HasTerminal` and `HasInitial` and the associated notations for terminal and initial objects are defined in `Terminal.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/IsTerminal.html"}, {"id": "Mathlib.CategoryTheory.PEmpty", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 103, "macro_tier_score": 0.2742, "macro_tier_override": null, "x": 4.895, "z": 20.436, "size": 0.422, "title": "The empty category", "summary": "Defines a category structure on `PEmpty`, and the unique functor `PEmpty ⥤ C` for any category `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/PEmpty.html"}, {"id": "Mathlib.CategoryTheory.Limits.IsLimit", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 103, "macro_tier_score": 0.3031, "macro_tier_override": null, "x": 58.661, "z": 30.436, "size": 0.7599, "title": "Limits and colimits", "summary": "We set up the general theory of limits and colimits in a category. In this introduction we only describe the setup for limits; it is repeated, with slightly different names, for colimits. The main structures defined in this file is * `IsLimit c`, for `c : Cone F`, `F : J ⥤ C`, expressing that `c` is a limit cone, See also `CategoryTheory.Limits.HasLimits` which further builds: * `LimitCone F`, which consists of a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/IsLimit.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Mon_", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -6.854, "z": 36.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Mon_.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Mon", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.032, "macro_tier_override": null, "x": -34.848, "z": 28.0, "size": 0.4485, "title": "The category of monoids in a monoidal category.", "summary": "We define monoids in a monoidal category `C` and show that the category of monoids is equivalent to the category of lax monoidal functors from the unit monoidal category to `C`. We also show that if `C` is braided, then the category of monoids is naturally monoidal. We use the `to_additive` attribute in order to generate a parallel API for additive monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Mon.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 80.514, "z": 54.0, "size": 0.233, "title": "Localizing subcategories", "summary": "Let `C` be a pretriangulated category. If `A` and `B` are triangulated subcategories of `C`, we define predicates (typeclasses `IsVerdierRightLocalizing` and `IsVerdierLeftLocalizing`) saying that `A` is right `B`-localizing (or left `B`-localizing). When `B` is closed under isomorphisms, we show that this implies that the functor from the Verdier quotient `A/(A ⊓ B)` to `C/B` is fully faithful.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/LocalizingSubcategory.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Opposite.Subcategory", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -23.287, "z": 60.436, "size": 0.2524, "title": "The opposite of a triangulated subcategory", "summary": "In this file, we show that if `P : ObjectProperty C` is a triangulated subcategory of a pretriangulated category `C`, then `P.op` is a triangulated subcategory of `Cᵒᵖ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Opposite/Subcategory.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Opposite.Triangulated", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 85.79, "z": 60.436, "size": 0.2524, "title": "The opposite of a triangulated category is triangulated", "summary": "The pretriangulated structure on `Cᵒᵖ` was constructed in the file `CategoryTheory.Triangulated.Opposite.Pretriangulated`. Here, we show that `Cᵒᵖ` is triangulated if `C` is triangulated.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Opposite/Triangulated.html"}, {"id": "Mathlib.CategoryTheory.IsoCat", "region_id": "category_theory", "micro_elevation": 0.1754, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 37.338, "z": 17.436, "size": 0.2, "title": "Isomorphisms of categories", "summary": "An `IsoCat C D` is an isomorphism of categories: a pair of functors `C ⥤ D` and `D ⥤ C` whose composites are *equal* (not merely naturally isomorphic) to the identity functors. This is a strict notion, stronger than an equivalence of categories `C ≌ D`. We also define `Functor.IsIso` as a property saying that a functor is fully faithful and bijective on objects. We develop basic api for these two concepts. Unless…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/IsoCat.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Product", "region_id": "category_theory", "micro_elevation": 0.2632, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 40.667, "z": 22.436, "size": 0.2, "title": "Cartesian products of bicategories", "summary": "We define the bicategory instance on `B × C` when `B` and `C` are bicategories. We define: * `sectL B c` : the strictly unitary pseudofunctor `B ⥤ B × C` given by `X ↦ ⟨X, c⟩` * `sectR b C` : the strictly unitary pseudofunctor `C ⥤ B × C` given by `Y ↦ ⟨b, Y⟩` * `fst` : the strict pseudofunctor `⟨X, Y⟩ ↦ X` * `snd` : the strict pseudofunctor `⟨X, Y⟩ ↦ Y` * `swap` : the strict pseudofunctor `B × C ⥤ C × B` given by…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Product.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.Types", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 103, "macro_tier_score": 0.0717, "macro_tier_override": null, "x": -118.428, "z": -105.756, "size": 0.2701, "title": "Cartesian closure of Type", "summary": "Show that `Type u₁` is Cartesian closed, and `C ⥤ Type u₁` is Cartesian closed for `C` a small category in `Type u₁`. Note this implies that the category of presheaves on a small category `C` is Cartesian closed.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/Types.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.Cartesian", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 103, "macro_tier_score": 0.077, "macro_tier_override": null, "x": 197.138, "z": -64.661, "size": 0.3213, "title": "Cartesian closed categories", "summary": "A cartesian closed category is a category with `CartesianMonoidalCategory` and `MonoidalClosed` instances. There used to be a separate definition `CartesianClosed`, with its own API, but over time this ended up as a duplicate of the former. Now, `CartesianClosed` and the surrounding API has been deprecated, and the API for `MonoidalClosed` should be used instead. This file now contains a few basic constructions for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/Cartesian.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Functor.Pseudofunctor", "region_id": "category_theory", "micro_elevation": 0.2105, "macro_tier": 103, "macro_tier_score": 0.1224, "macro_tier_override": null, "x": 42.497, "z": 19.436, "size": 0.4438, "title": "Pseudofunctors", "summary": "A pseudofunctor is an oplax (or lax) functor whose `mapId` and `mapComp` are isomorphisms. We provide several constructors for pseudofunctors: * `Pseudofunctor.mk` : the default constructor, which requires `map₂_whiskerLeft` and `map₂_whiskerRight` instead of naturality of `mapComp`. * `Pseudofunctor.mkOfOplax` : construct a pseudofunctor from an oplax functor whose `mapId` and `mapComp` are isomorphisms. This…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Functor/Pseudofunctor.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Functor.Oplax", "region_id": "category_theory", "micro_elevation": 0.193, "macro_tier": 103, "macro_tier_score": 0.1212, "macro_tier_override": null, "x": 20.229, "z": 18.436, "size": 0.3974, "title": "Oplax functors", "summary": "An oplax functor `F` between bicategories `B` and `C` consists of * a function between objects `F.obj : B → C`, * a family of functions between 1-morphisms `F.map : (a ⟶ b) → (F.obj a ⟶ F.obj b)`, * a family of functions between 2-morphisms `F.map₂ : (f ⟶ g) → (F.map f ⟶ F.map g)`, * a family of 2-morphisms `F.mapId a : F.map (𝟙 a) ⟶ 𝟙 (F.obj a)`, * a family of 2-morphisms `F.mapComp f g : F.map (f ≫ g) ⟶ F.map f ≫…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Functor/Oplax.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Functor.Lax", "region_id": "category_theory", "micro_elevation": 0.193, "macro_tier": 103, "macro_tier_score": 0.1209, "macro_tier_override": null, "x": 40.568, "z": 18.436, "size": 0.3822, "title": "Lax functors", "summary": "A lax functor `F` between bicategories `B` and `C` consists of * a function between objects `F.obj : B → C`, * a family of functions between 1-morphisms `F.map : (a ⟶ b) → (F.obj a ⟶ F.obj b)`, * a family of functions between 2-morphisms `F.map₂ : (f ⟶ g) → (F.map f ⟶ F.map g)`, * a family of 2-morphisms `F.mapId a : 𝟙 (F.obj a) ⟶ F.map (𝟙 a)`, * a family of 2-morphisms `F.mapComp f g : F.map f ≫ F.map g ⟶ F.map (f…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Functor/Lax.html"}, {"id": "Mathlib.CategoryTheory.Limits.Final", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 103, "macro_tier_score": 0.1909, "macro_tier_override": null, "x": 170.924, "z": -73.661, "size": 0.5062, "title": "Final and initial functors", "summary": "A functor `F : C ⥤ D` is final if for every `d : D`, the comma category of morphisms `d ⟶ F.obj c` is connected. Dually, a functor `F : C ⥤ D` is initial if for every `d : D`, the comma category of morphisms `F.obj c ⟶ d` is connected. We show that right adjoints are examples of final functors, while left adjoints are examples of initial functors. For final functors, we prove that the following three statements are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Final.html"}, {"id": "Mathlib.CategoryTheory.Limits.VanKampen", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 103, "macro_tier_score": 0.0385, "macro_tier_override": null, "x": -5.898, "z": 46.436, "size": 0.2885, "title": "Universal colimits and van Kampen colimits", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/VanKampen.html"}, {"id": "Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 89.751, "z": 51.436, "size": 0.2729, "title": "`1`-hypercovers and (pre)sheaves of types", "summary": "In this file we provide some API for working with `1`-hypercovers for sheaves of types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Hypercover/SheafOfTypes.html"}, {"id": "Mathlib.CategoryTheory.Endomorphism", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 103, "macro_tier_score": 0.1556, "macro_tier_override": null, "x": -10.843, "z": 122.985, "size": 0.4389, "title": "Endomorphisms", "summary": "Definition and basic properties of endomorphisms and automorphisms of an object in a category. For each `X : C`, we provide `CategoryTheory.End X := X ⟶ X` with a monoid structure, and `CategoryTheory.Aut X := X ≅ X` with a group structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Endomorphism.html"}, {"id": "Mathlib.CategoryTheory.FinCategory.Basic", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 103, "macro_tier_score": 0.2619, "macro_tier_override": null, "x": 43.949, "z": 20.436, "size": 0.4917, "title": "Finite categories", "summary": "A category is finite in this sense if it has finitely many objects, and finitely many morphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/FinCategory/Basic.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.CatEnriched", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 94.187, "z": -60.661, "size": 0.2338, "title": "The strict bicategory associated to a Cat-enriched category", "summary": "If `C` is a type with an `EnrichedCategory Cat C` structure, then it has hom-categories, whose objects define 1-dimensional arrows on `C` and whose morphisms define 2-dimensional arrows between these. The enriched category axioms equip this data with the structure of a strict bicategory. We define a type alias `CatEnriched C` for a type `C` with an `EnrichedCategory Cat C` structure. We provide this with an instance…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/CatEnriched.html"}, {"id": "Mathlib.CategoryTheory.Enriched.Ordinary.Basic", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 102, "macro_tier_score": 0.0213, "macro_tier_override": null, "x": 123.48, "z": -61.661, "size": 0.4001, "title": "Enriched ordinary categories", "summary": "If `V` is a monoidal category, a `V`-enriched category `C` does not need to be a category. However, when we have both `Category C` and `EnrichedCategory V C`, we may require that the type of morphisms `X ⟶ Y` in `C` identify to `𝟙_ V ⟶ EnrichedCategory.Hom X Y`. This data shall be packaged in the typeclass `EnrichedOrdinaryCategory V C`. In particular, if `C` is a `V`-enriched category, it is shown that the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Enriched/Ordinary/Basic.html"}, {"id": "Mathlib.CategoryTheory.Sites.Hypercover.IsSheaf", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 102, "macro_tier_score": 0.0193, "macro_tier_override": null, "x": -17.902, "z": 51.436, "size": 0.2653, "title": "Characterization of sheaves using 1-hypercovers", "summary": "In this file, given a Grothendieck topology `J` on a category `C`, we define a type `J.OneHypercoverFamily` of families of 1-hypercovers. When `H : J.OneHypercoverFamily`, we define a predicate `H.IsGenerating` which means that any covering sieve contains the sieve generated by the underlying covering of one of the 1-hypercovers in the family. If this holds, we show in `OneHypercoverFamily.isSheaf_iff` that a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Hypercover/IsSheaf.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Functor", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 102, "macro_tier_score": 0.0104, "macro_tier_override": null, "x": 57.059, "z": 48.0, "size": 0.3264, "title": "Triangulated functors", "summary": "In this file, when `C` and `D` are categories equipped with a shift by `ℤ` and `F : C ⥤ D` is a functor which commutes with the shift, we define the induced functor `F.mapTriangle : Triangle C ⥤ Triangle D` on the categories of triangles. When `C` and `D` are pretriangulated, a triangulated functor is such a functor `F` which also sends distinguished triangles to distinguished triangles: this defines the typeclass…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Functor.html"}, {"id": "Mathlib.CategoryTheory.Sites.Equivalence", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 102, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": -115.96, "z": -98.756, "size": 0.3403, "title": "Equivalences of sheaf categories", "summary": "Given a site `(C, J)` and a category `D` which is equivalent to `C`, with `C` and `D` possibly large and possibly in different universes, we transport the Grothendieck topology `J` on `C` to `D` and prove that the sheaf categories are equivalent. We also prove that sheafification and the property `HasSheafCompose` transport nicely over this equivalence, and apply it to essentially small sites. We also provide…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Equivalence.html"}, {"id": "Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 102, "macro_tier_score": 0.0198, "macro_tier_override": null, "x": 175.054, "z": -16.328, "size": 0.316, "title": "Induced Topology", "summary": "We say that a functor `G : C ⥤ (D, K)` is locally dense if for each covering sieve `T` in `D` of some `X : C`, `T ∩ mor(C)` generates a covering sieve of `X` in `D`. A locally dense fully faithful functor then induces a topology on `C` via `{ T ∩ mor(C) | T ∈ K }`. Note that this is equal to the collection of sieves on `C` whose image generates a covering sieve. This construction would make `C` both cover-lifting…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/DenseSubsite/InducedTopology.html"}, {"id": "Mathlib.CategoryTheory.Sites.LocallyBijective", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 102, "macro_tier_score": 0.0207, "macro_tier_override": null, "x": 117.344, "z": 14.336, "size": 0.3734, "title": "Locally bijective morphisms of presheaves", "summary": "Let `C` be a category equipped with a Grothendieck topology `J`. Let `A` be a concrete category. In this file, we introduce a type class `J.WEqualsLocallyBijective A` which says that the class `J.W` (of morphisms of presheaves which become isomorphisms after sheafification) is the class of morphisms that are both locally injective and locally surjective (i.e. locally bijective). We prove that this holds iff for any…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/LocallyBijective.html"}, {"id": "Mathlib.CategoryTheory.Sites.PreservesLocallyBijective", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 102, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -2.054, "z": 57.436, "size": 0.284, "title": "Preserving and reflecting local injectivity and surjectivity", "summary": "This file proves that precomposition with a cocontinuous functor preserves local injectivity and surjectivity of morphisms of presheaves, and that precomposition with a cover-preserving and cover-dense functor reflects the same properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/PreservesLocallyBijective.html"}, {"id": "Mathlib.CategoryTheory.Limits.MonoCoprod", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 103, "macro_tier_score": 0.0482, "macro_tier_override": null, "x": 175.669, "z": -70.661, "size": 0.2985, "title": "Categories where inclusions into coproducts are monomorphisms", "summary": "If `C` is a category, the class `MonoCoprod C` expresses that left inclusions `A ⟶ A ⨿ B` are monomorphisms when `HasCoproduct A B` holds. If so, it is shown that right inclusions are also monomorphisms. More generally, we deduce that when suitable coproducts exist, then if `X : I → C` and `ι : J → I` is an injective map, then the canonical morphism `∐ (X ∘ ι) ⟶ ∐ X` is a monomorphism. It also follows that for any…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/MonoCoprod.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.1882, "macro_tier_override": null, "x": -16.807, "z": 35.436, "size": 0.4114, "title": "Preserving binary products", "summary": "Constructions to relate the notions of preserving binary products and reflecting binary products to concrete binary fans. In particular, we show that `ProdComparison G X Y` is an isomorphism iff `G` preserves the product of `X` and `Y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/BinaryProducts.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 103, "macro_tier_score": 0.2205, "macro_tier_override": null, "x": 26.508, "z": -70.766, "size": 0.5347, "title": "Zero morphisms and zero objects", "summary": "A category \"has zero morphisms\" if there is a designated \"zero morphism\" in each morphism space, and compositions of zero morphisms with anything give the zero morphism. (Notice this is extra structure, not merely a property.) A category \"has a zero object\" if it has an object which is both initial and terminal. Having a zero object provides zero morphisms, as the unique morphisms factoring through the zero object.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Braided.Transport", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 22.447, "z": 35.436, "size": 0.2, "title": "Transport a symmetric monoidal structure along an equivalence of categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Braided/Transport.html"}, {"id": "Mathlib.CategoryTheory.Idempotents.HomologicalComplex", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 104.715, "z": -63.661, "size": 0.2669, "title": "Idempotent completeness and homological complexes", "summary": "This file contains simplifications lemmas for categories `Karoubi (HomologicalComplex C c)` and the construction of an equivalence of categories `Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c`. When the category `C` is idempotent complete, it is shown that `HomologicalComplex (Karoubi C) c` is also idempotent complete.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Idempotents/HomologicalComplex.html"}, {"id": "Mathlib.CategoryTheory.Idempotents.KaroubiKaroubi", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 160.133, "z": -63.661, "size": 0.2473, "title": "Idempotence of the Karoubi envelope", "summary": "In this file, we construct the equivalence of categories `KaroubiKaroubi.equivalence C : Karoubi C ≌ Karoubi (Karoubi C)` for any category `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Idempotents/KaroubiKaroubi.html"}, {"id": "Mathlib.CategoryTheory.Presentable.Finite", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 102, "macro_tier_score": 0.0149, "macro_tier_override": null, "x": 80.275, "z": -3.83, "size": 0.3028, "title": "Finitely Presentable Objects", "summary": "We define finitely presentable objects as a synonym for `ℵ₀`-presentable objects, and link this definition with the preservation of filtered colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/Finite.html"}, {"id": "Mathlib.CategoryTheory.Presentable.Presheaf", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 128.804, "z": -53.661, "size": 0.2341, "title": "Categories of presheaves are locally presentable", "summary": "If `A` is a locally `κ`-presentable category and `C` is a small category, we show that `Cᵒᵖ ⥤ A` is also locally `κ`-presentable, under the additional assumption that `A` has pullbacks (a condition which should be automatically satisfied (TODO)).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/Presheaf.html"}, {"id": "Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 79.286, "z": 46.436, "size": 0.2738, "title": "Existence of pointwise right derived functors via derivability structures", "summary": "In this file, we show how a right derivability structure can be used in order to construct (pointwise) right derived functors. Let `Φ` be a right derivability structure from `W₁ : MorphismProperty C₁` to `W₂ : MorphismProperty C₂`. Let `F : C₂ ⥤ H` be a functor. Then, the lemma `hasPointwiseRightDerivedFunctor_iff_of_isRightDerivabilityStructure` says that `F` has a pointwise right derived functor with respect to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/DerivabilityStructure/PointwiseRightDerived.html"}, {"id": "Mathlib.CategoryTheory.GuitartExact.KanExtension", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 71.558, "z": 43.436, "size": 0.2553, "title": "Guitart exact squares and Kan extensions", "summary": "Given a Guitart exact square `w : T ⋙ R ⟶ L ⋙ B`, ``` T C₁ ⥤ C₂ L | | R v v C₃ ⥤ C₄ B ``` we show that an extension `F' : C₄ ⥤ D` of `F : C₂ ⥤ D` along `R` is a pointwise left Kan extension at `B.obj X₃` iff the composition `T ⋙ F'` is a pointwise left Kan extension at `X₃` of `B ⋙ F'`. When suitable (pointwise) left Kan extensions exist, we also show that the natural transformation of functors `(C₂ ⥤ D) ⥤ C₃ ⥤ D`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GuitartExact/KanExtension.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 33.76, "z": 45.436, "size": 0.2478, "title": "Chosen pullbacks", "summary": "Given two morphisms `f₁ : X₁ ⟶ S` and `f₂ : X₂ ⟶ S`, we introduce a structure `ChosenPullback f₁ f₂` which contains the data of pullback of `f₁` and `f₂`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/ChosenPullback.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.HomologicalFunctor", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 102, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -6.839, "z": 51.0, "size": 0.3187, "title": "Homological functors", "summary": "In this file, given a functor `F : C ⥤ A` from a pretriangulated category to an abelian category, we define the type class `F.IsHomological`, which is the property that `F` sends distinguished triangles in `C` to exact sequences in `A`. If `F` has been endowed with `[F.ShiftSequence ℤ]`, then we may think of the functor `F` as a `H^0`, and then the `H^n` functors are the functors `F.shift n : C ⥤ A`: we have…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Projective.Resolution", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.01, "macro_tier_override": null, "x": 84.63, "z": 49.436, "size": 0.2895, "title": "Abelian categories with enough projectives have projective resolutions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Projective/Resolution.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Projective.Resolution", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 102, "macro_tier_score": 0.01, "macro_tier_override": null, "x": -8.384, "z": 41.0, "size": 0.294, "title": "Projective resolutions", "summary": "A projective resolution `P : ProjectiveResolution Z` of an object `Z : C` consists of an `ℕ`-indexed chain complex `P.complex` of projective objects, along with a quasi-isomorphism `P.π` from `C` to the chain complex consisting just of `Z` in degree zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Projective/Resolution.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.Pushouts", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 6.979, "z": 45.436, "size": 0.2444, "title": "Pushouts in `Type`", "summary": "We describe the pushout of two maps `f : S ⟶ X₁` and `g : S ⟶ X₂` in the category of types as the quotient of `X₁ ⊕ X₂` by the equivalence relation generated by a relation. We also study the particular case when `f` is injective (in the file `CategoryTheory.Types.Monomorphisms`, it is deduced that monomorphisms are stable under cobase change in the category of types).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/Pushouts.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.Pullbacks", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 103, "macro_tier_score": 0.0584, "macro_tier_override": null, "x": -29.159, "z": 43.436, "size": 0.3478, "title": "Pullbacks in the category of types", "summary": "In `Type*`, the pullback of `f : X ⟶ Z` and `g : Y ⟶ Z` is the subtype `{ p : X × Y // f p.1 = g p.2 }` of the product. We show some additional lemmas for pullbacks in the category of types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/Pullbacks.html"}, {"id": "Mathlib.CategoryTheory.Sites.Over", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 102, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 94.972, "z": 94.452, "size": 0.3887, "title": "Localization", "summary": "In this file, given a Grothendieck topology `J` on a category `C` and `X : C`, we construct a Grothendieck topology `J.over X` on the category `Over X`. In order to do this, we first construct a bijection `Sieve.overEquiv Y : Sieve Y ≃ Sieve Y.left` for all `Y : Over X`. Then, as it is stated in SGA 4 III 5.2.1, a sieve of `Y : Over X` is covering for `J.over X` if and only if the corresponding sieve of `Y.left` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Over.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.Equivalence", "region_id": "category_theory", "micro_elevation": 0.1404, "macro_tier": 103, "macro_tier_score": 0.1958, "macro_tier_override": null, "x": 26.211, "z": 15.436, "size": 0.2994, "title": "Equivalence of full subcategories", "summary": "The inclusion functor `P.FullSubcategory ⥤ Q.FullSubcategory` induced by an inequality `P ≤ Q` in `ObjectProperty C` is an equivalence iff `Q ≤ P.isoClosure`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/Equivalence.html"}, {"id": "Mathlib.CategoryTheory.Equivalence", "region_id": "category_theory", "micro_elevation": 0.1228, "macro_tier": 103, "macro_tier_score": 0.4854, "macro_tier_override": null, "x": 138.45, "z": -100.661, "size": 0.7811, "title": "Equivalence of categories", "summary": "An equivalence of categories `C` and `D` is a pair of functors `F : C ⥤ D` and `G : D ⥤ C` such that `η : 𝟭 C ≅ F ⋙ G` and `ε : G ⋙ F ≅ 𝟭 D`. In many situations, equivalences are a better notion of \"sameness\" of categories than the stricter isomorphism of categories. Recall that one way to express that two functors `F : C ⥤ D` and `G : D ⥤ C` are adjoint is using two natural transformations `η : 𝟭 C ⟶ F ⋙ G` and `ε…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Equivalence.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.Pullbacks", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 103, "macro_tier_score": 0.0582, "macro_tier_override": null, "x": 34.299, "z": 39.436, "size": 0.3386, "title": "Constructing pullbacks from binary products and equalizers", "summary": "If a category has binary products and equalizers, then it has pullbacks. Also, if a category has binary coproducts and coequalizers, then it has pushouts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/Pullbacks.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Equalizers", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 103, "macro_tier_score": 0.2581, "macro_tier_override": null, "x": 26.371, "z": 38.436, "size": 0.6348, "title": "Equalizers and coequalizers", "summary": "This file defines (co)equalizers as special cases of (co)limits. An equalizer is the categorical generalization of the subobject ${a ∈ A | f(a) = g(a)}$ known from abelian groups or modules. It is a limit cone over the diagram formed by `f` and `g`. A coequalizer is the dual concept.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Equalizers.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Monomorphisms", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 74.844, "z": 49.436, "size": 0.2, "title": "Monomorphisms are stable under cobase change", "summary": "In an abelian category `C`, the class of morphisms `monomorphisms C` is stable under cobase change and `epimorphisms C` is stable under base change.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Monomorphisms.html"}, {"id": "Mathlib.CategoryTheory.Enriched.Opposite", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 27.122, "z": 54.436, "size": 0.2, "title": "The opposite category of an enriched category", "summary": "When a monoidal category `V` is braided, we may define the opposite `V`-category of a `V`-category. The symmetry map is required to define the composition morphism. This file constructs the opposite `V`-category as an instance on the type `Cᵒᵖ` and constructs an equivalence between * `ForgetEnrichment V (Cᵒᵖ)`, the underlying category of the `V`-category `Cᵒᵖ`; and * `(ForgetEnrichment V C)ᵒᵖ`, the opposite category…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Enriched/Opposite.html"}, {"id": "Mathlib.CategoryTheory.FintypeCat", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 102, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": -122.908, "z": 127.513, "size": 0.3402, "title": "The category of finite types.", "summary": "We define the category of finite types, denoted `FintypeCat` as the full subcategory of types with a `Finite` instance. We also define `FintypeCat.Skeleton`, the standard skeleton of `FintypeCat` whose objects are `Fin n` for `n : ℕ`. We prove that the obvious inclusion functor `FintypeCat.Skeleton ⥤ FintypeCat` is an equivalence of categories in `FintypeCat.Skeleton.equivalence`. We prove that `FintypeCat.Skeleton`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/FintypeCat.html"}, {"id": "Mathlib.CategoryTheory.Center.Localization", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0096, "macro_tier_override": null, "x": -19.508, "z": 49.436, "size": 0.2464, "title": "Localization of the center of a category", "summary": "Given a localization functor `L : C ⥤ D` with respect to `W : MorphismProperty C`, we define a localization map `CatCenter C → CatCenter D` for the centers of these categories. In case `L` is an additive functor between preadditive categories, we promote this to a ring morphism `CatCenter C →+* CatCenter D`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Center/Localization.html"}, {"id": "Mathlib.CategoryTheory.Center.Preadditive", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 102, "macro_tier_score": 0.0147, "macro_tier_override": null, "x": -13.43, "z": 46.436, "size": 0.2906, "title": "The center of an additive category", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Center/Preadditive.html"}, {"id": "Mathlib.CategoryTheory.Localization.Predicate", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 103, "macro_tier_score": 0.0652, "macro_tier_override": null, "x": 57.447, "z": 32.436, "size": 0.4452, "title": "Predicate for localized categories", "summary": "In this file, a predicate `L.IsLocalization W` is introduced for a functor `L : C ⥤ D` and `W : MorphismProperty C`: it expresses that `L` identifies `D` with the localized category of `C` with respect to `W` (up to equivalence). We introduce a universal property `StrictUniversalPropertyFixedTarget L W E` which states that `L` inverts the morphisms in `W` and that all functors `C ⥤ E` inverting `W` uniquely factor…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Predicate.html"}, {"id": "Mathlib.CategoryTheory.Linear.Yoneda", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 31.612, "z": 44.0, "size": 0.2745, "title": "The Yoneda embedding for `R`-linear categories", "summary": "The Yoneda embedding for `R`-linear categories `C`, sends an object `X : C` to the `ModuleCat R`-valued presheaf on `C`, with value on `Y : Cᵒᵖ` given by `ModuleCat.of R (unop Y ⟶ X)`. TODO: `linearYoneda R C` is `R`-linear. TODO: In fact, `linearYoneda` itself is additive and `R`-linear.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Linear/Yoneda.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Yoneda.Basic", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 103, "macro_tier_score": 0.0352, "macro_tier_override": null, "x": -40.125, "z": 50.436, "size": 0.3816, "title": "The Yoneda embedding for preadditive categories", "summary": "The Yoneda embedding for preadditive categories sends an object `Y` to the presheaf sending an object `X` to the group of morphisms `X ⟶ Y`. At each point, we get an additional `End Y`-module structure. We also show that this presheaf is additive and that it is compatible with the normal Yoneda embedding in the expected way and deduce that the preadditive Yoneda embedding is fully faithful.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Yoneda/Basic.html"}, {"id": "Mathlib.CategoryTheory.ComposableArrows.One", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 16.228, "z": 17.0, "size": 0.274, "title": "Functors to `ComposableArrows C 1`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ComposableArrows/One.html"}, {"id": "Mathlib.CategoryTheory.ComposableArrows.Basic", "region_id": "category_theory", "micro_elevation": 0.2807, "macro_tier": 103, "macro_tier_score": 0.0417, "macro_tier_override": null, "x": 94.006, "z": -37.83, "size": 0.4535, "title": "Composable arrows", "summary": "If `C` is a category, the type of `n`-simplices in the nerve of `C` identifies to the type of functors `Fin (n + 1) ⥤ C`, which can be thought of as families of `n` composable arrows in `C`. In this file, we introduce and study this category `ComposableArrows C n` of `n` composable arrows in `C`. If `F : ComposableArrows C n`, we define `F.left` as the leftmost object, `F.right` as the rightmost object, and `F.hom :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ComposableArrows/Basic.html"}, {"id": "Mathlib.CategoryTheory.Category.KleisliCat", "region_id": "category_theory", "micro_elevation": 0.0351, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -3.662, "z": 211.969, "size": 0.2633, "title": "The Kleisli construction on the Type category", "summary": "Define the Kleisli category for (control) monads. `CategoryTheory/Monad/Kleisli` defines the general version for a monad on `C`, and demonstrates the equivalence between the two.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/KleisliCat.html"}, {"id": "Mathlib.CategoryTheory.Monad.Kleisli", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 59.049, "z": 30.436, "size": 0.257, "title": "Kleisli category on a (co)monad", "summary": "This file defines the Kleisli category on a monad `(T, η_ T, μ_ T)` as well as the co-Kleisli category on a comonad `(U, ε_ U, δ_ U)`. It also defines the Kleisli adjunction which gives rise to the monad `(T, η_ T, μ_ T)` as well as the co-Kleisli adjunction which gives rise to the comonad `(U, ε_ U, δ_ U)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monad/Kleisli.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.Opposite", "region_id": "category_theory", "micro_elevation": 0.1579, "macro_tier": 103, "macro_tier_score": 0.1968, "macro_tier_override": null, "x": 10.995, "z": 16.436, "size": 0.3675, "title": "The opposite of a property of objects", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/Opposite.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.CompleteLattice", "region_id": "category_theory", "micro_elevation": 0.1053, "macro_tier": 103, "macro_tier_score": 0.2265, "macro_tier_override": null, "x": 26.863, "z": 13.436, "size": 0.4188, "title": "ObjectProperty is a complete lattice", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/CompleteLattice.html"}, {"id": "Mathlib.CategoryTheory.Limits.HasLimits", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 103, "macro_tier_score": 0.3172, "macro_tier_override": null, "x": 23.942, "z": 25.0, "size": 1.1541, "title": "Existence of limits and colimits", "summary": "In `CategoryTheory.Limits.IsLimit` we defined `IsLimit c`, the data showing that a cone `c` is a limit cone. The two main structures defined in this file are: * `LimitCone F`, which consists of a choice of cone for `F` and the fact it is a limit cone, and * `HasLimit F`, asserting the mere existence of some limit cone for `F`. `HasLimit` is a propositional typeclass (it's important that it is a proposition merely…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/HasLimits.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 103, "macro_tier_score": 0.0639, "macro_tier_override": null, "x": -6.245, "z": 17.0, "size": 0.3908, "title": "Morphism properties that are inverted by a functor", "summary": "In this file, we introduce the predicate `P.IsInvertedBy F` which expresses that the morphisms satisfying `P : MorphismProperty C` are mapped to isomorphisms by a functor `F : C ⥤ D`. This is used in the localization of categories API (folder `CategoryTheory.Localization`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.Basic", "region_id": "category_theory", "micro_elevation": 0.2807, "macro_tier": 103, "macro_tier_score": 0.4097, "macro_tier_override": null, "x": 0.574, "z": 23.436, "size": 0.7107, "title": "Properties of morphisms", "summary": "We provide the basic framework for talking about properties of morphisms. The following meta-property is defined * `RespectsLeft P Q`: `P` respects the property `Q` on the left if `P f → P (i ≫ f)` where `i` satisfies `Q`. * `RespectsRight P Q`: `P` respects the property `Q` on the right if `P f → P (f ≫ i)` where `i` satisfies `Q`. * `Respects`: `P` respects `Q` if `P` respects `Q` both on the left and on the right.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/Basic.html"}, {"id": "Mathlib.CategoryTheory.EffectiveEpi.Basic", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.1498, "macro_tier_override": null, "x": -1.249, "z": 34.436, "size": 0.399, "title": "Effective epimorphisms", "summary": "We define the notion of effective epimorphism and effective epimorphic family of morphisms. A morphism is an *effective epi* if it is a joint coequalizer of all pairs of morphisms which it coequalizes. A family of morphisms with fixed target is *effective epimorphic* if it is initial among families of morphisms with its sources and a general fixed target, coequalizing every pair of morphisms it coequalizes (here,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/EffectiveEpi/Basic.html"}, {"id": "Mathlib.CategoryTheory.Sites.CartesianClosed", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -243.372, "z": -104.756, "size": 0.2576, "title": "Sheaf categories are Cartesian closed", "summary": "...if the underlying presheaf category is Cartesian closed, the target category has (chosen) finite products, and there exists a sheafification functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/CartesianClosed.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.Ideal", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 11.097, "z": 51.436, "size": 0.2607, "title": "Exponential ideals", "summary": "An exponential ideal of a Cartesian closed category `C` is a subcategory `D ⊆ C` such that for any `B : D` and `A : C`, the exponential `A ⟹ B` is in `D`: resembling ring-theoretic ideals. We define the notion here for inclusion functors `i : D ⥤ C` rather than explicit subcategories to preserve the principle of equivalence. We additionally show that if `C` is Cartesian closed and `i : D ⥤ C` is a reflective…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/Ideal.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Rigid.Basic", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 102, "macro_tier_score": 0.0104, "macro_tier_override": null, "x": 43.929, "z": 30.0, "size": 0.3246, "title": "Rigid (autonomous) monoidal categories", "summary": "This file defines rigid (autonomous) monoidal categories and the necessary theory about exact pairings and duals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Rigid/Basic.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.FunctorCategory", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.0826, "macro_tier_override": null, "x": -5.182, "z": 35.436, "size": 0.3729, "title": "Monoidal structure on `C ⥤ D` when `D` is monoidal.", "summary": "When `C` is any category, and `D` is a monoidal category, there is a natural \"pointwise\" monoidal structure on `C ⥤ D`. The initial intended application is tensor product of presheaves.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/FunctorCategory.html"}, {"id": "Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 102, "macro_tier_score": 0.0145, "macro_tier_override": null, "x": 27.716, "z": 47.436, "size": 0.2608, "title": "Extension of a functor from `Set.Iic j` to `Set.Iic (Order.succ j)`", "summary": "Given a linearly ordered type `J` with `SuccOrder J`, `j : J` that is not maximal, we define the extension of a functor `F : Set.Iic j ⥤ C` as a functor `Set.Iic (Order.succ j) ⥤ C` when an object `X : C` and a morphism `τ : F.obj ⟨j, _⟩ ⟶ X` is given.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/SmallObject/Iteration/ExtendToSucc.html"}, {"id": "Mathlib.CategoryTheory.SmallObject.Iteration.Basic", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 102, "macro_tier_score": 0.0151, "macro_tier_override": null, "x": -30.034, "z": 46.436, "size": 0.3239, "title": "Transfinite iterations of a successor structure", "summary": "In this file, we introduce the structure `SuccStruct` on a category `C`. It consists of the data of an object `X₀ : C`, a successor map `succ : C → C` and a morphism `toSucc : X ⟶ succ X` for any `X : C`. The map `toSucc` does not have to be natural in `X`. For any element `j : J` in a well-ordered type `J`, we would like to define the iteration of `Φ : SuccStruct C`, as a functor `F : J ⥤ C` such that `F.obj ⊥ =…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/SmallObject/Iteration/Basic.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Generators", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -50.183, "z": 58.436, "size": 0.2, "title": "Generators in triangulated categories", "summary": "We define the notions of strong and classical generators in (pre)triangulated categories. This is not to be confused with `ObjectProperty.IsStrongGenerator` defined in `CategoryTheory/Generator`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Generators.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.ClosureShift", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -7.899, "z": 47.436, "size": 0.2478, "title": "Closure operators and shifts", "summary": "In this file, we collect facts relating being stable under shifts with closure properties of object properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/ClosureShift.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Subcategory", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 102, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": 16.479, "z": 57.436, "size": 0.3314, "title": "Triangulated subcategories", "summary": "In this file, given a pretriangulated category `C` and `P : ObjectProperty C`, we introduce a typeclass `P.IsTriangulated` to express that `P` is a triangulated subcategory of `C`. When `P` is a triangulated subcategory, we introduce a class of morphisms `P.trW : MorphismProperty C` consisting of the morphisms whose \"cone\" belongs to `P` (up to isomorphisms), and we show that it has both calculus of left and right…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Subcategory.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Biproducts", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 103, "macro_tier_score": 0.0997, "macro_tier_override": null, "x": 48.215, "z": 40.0, "size": 0.4839, "title": "Basic facts about biproducts in preadditive categories.", "summary": "In (or between) preadditive categories, * Any biproduct satisfies the equality `total : ∑ j : J, biproduct.π f j ≫ biproduct.ι f j = 𝟙 (⨁ f)`, or, in the binary case, `total : fst ≫ inl + snd ≫ inr = 𝟙 X`. * Any (binary) `product` or (binary) `coproduct` is a (binary) `biproduct`. * In any category (with zero morphisms), if `biprod.map f g` is an isomorphism, then both `f` and `g` are isomorphisms. * If `f` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Biproducts.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 103, "macro_tier_score": 0.0691, "macro_tier_override": null, "x": 33.901, "z": 69.47, "size": 0.408, "title": "Constructing limits from products and equalizers.", "summary": "If a category has all products, and all equalizers, then it has all limits. Similarly, if it has all finite products, and all equalizers, then it has all finite limits. If a functor preserves all products and equalizers, then it preserves all limits. Similarly, if it preserves all finite products and equalizers, then it preserves all finite limits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/LimitsOfProductsAndEqualizers.html"}, {"id": "Mathlib.CategoryTheory.Abelian.NonPreadditive", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 103, "macro_tier_score": 0.0582, "macro_tier_override": null, "x": -25.477, "z": 46.436, "size": 0.3386, "title": "Every NonPreadditiveAbelian category is preadditive", "summary": "In mathlib, we define an abelian category as a preadditive category with finite products, kernels and cokernels, and in which every monomorphism and epimorphism is normal. While virtually every interesting abelian category has a natural preadditive structure (which is why it is included in the definition), preadditivity is not actually needed: Every category that has all of the other properties appearing in the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/NonPreadditive.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Preorder.Fin", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 102, "macro_tier_score": 0.0287, "macro_tier_override": null, "x": 57.654, "z": 32.436, "size": 0.2415, "title": "Limits and colimits indexed by `Fin`", "summary": "In this file, we show that `0 : Fin (n + 1)` is an initial object and `Fin.last n` is a terminal object. This allows to compute limits and colimits indexed by `Fin (n + 1)`, see `limitOfDiagramInitial` and `colimitOfDiagramTerminal` in the file `Limits.Shapes.IsTerminal`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Preorder/Fin.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 103, "macro_tier_score": 0.0621, "macro_tier_override": null, "x": 73.146, "z": 44.436, "size": 0.259, "title": "Definitions and basic properties of normal monomorphisms and epimorphisms.", "summary": "A normal monomorphism is a morphism that is the kernel of some other morphism. We give the construction `NormalMono → RegularMono` (`CategoryTheory.NormalMono.regularMono`) as well as the dual construction for normal epimorphisms. We show equivalences reflect normal monomorphisms (`CategoryTheory.equivalenceReflectsNormalMono`), and that the pullback of a normal monomorphism is normal…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Basic.html"}, {"id": "Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 101, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": 173.733, "z": -68.661, "size": 0.2978, "title": "Constructor for derivability structures", "summary": "In this file, we provide a constructor for right and left derivability structures. Assume that `W₁` and `W₂` are classes of morphisms in categories `C₁` and `C₂`, and that we have a localizer morphism `Φ : LocalizerMorphism W₁ W₂` that is a localized equivalence, i.e. `Φ.functor` induces an equivalence of categories between the localized categories. Assume moreover that `W₂` contains identities. Then, `Φ` is a right…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/DerivabilityStructure/Constructor.html"}, {"id": "Mathlib.CategoryTheory.Adhesive", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 5.485, "z": 49.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adhesive.html"}, {"id": "Mathlib.CategoryTheory.Limits.Opposites", "region_id": "category_theory", "micro_elevation": 0.5789, "macro_tier": 103, "macro_tier_score": 0.204, "macro_tier_override": null, "x": 55.152, "z": 40.436, "size": 0.5973, "title": "Limits in `C` give colimits in `Cᵒᵖ`.", "summary": "We construct limits and colimits in the opposite categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Opposites.html"}, {"id": "Mathlib.CategoryTheory.Localization.LocallySmall", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -16.028, "z": 34.436, "size": 0.2, "title": "Locally small localizations", "summary": "In this file, given `W : MorphismProperty C` and a universe `w`, we show that there exists a term in `HasLocalization.{w} W` if and only if there exists (or for all) localization functors `L : C ⥤ D` for `W`, the category `D` is locally `w`-small.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/LocallySmall.html"}, {"id": "Mathlib.CategoryTheory.Localization.HasLocalization", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 102, "macro_tier_score": 0.0198, "macro_tier_override": null, "x": -39.092, "z": 26.0, "size": 0.3173, "title": "Morphism properties equipped with a localized category", "summary": "If `C : Type u` is a category (with `[Category.{v} C]`), and `W : MorphismProperty C`, then the constructed localized category `W.Localization` is in `Type u` (the objects are essentially the same as those of `C`), but the morphisms are in `Type (max u v)`. In particular situations, it may happen that there is a localized category for `W` whose morphisms are in a lower universe like `v`: it shall be so for the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/HasLocalization.html"}, {"id": "Mathlib.CategoryTheory.EssentiallySmall", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 103, "macro_tier_score": 0.3033, "macro_tier_override": null, "x": 58.429, "z": 31.436, "size": 0.7623, "title": "Essentially small categories.", "summary": "A category given by `(C : Type u) [Category.{v} C]` is `w`-essentially small if there exists a `SmallModel C : Type w` equipped with `[SmallCategory (SmallModel C)]` and an equivalence `C ≌ SmallModel C`. A category is `w`-locally small if every hom type is `w`-small. The main theorem here is that a category is `w`-essentially small iff the type `Skeleton C` is `w`-small, and `C` is `w`-locally small.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/EssentiallySmall.html"}, {"id": "Mathlib.CategoryTheory.Localization.Triangulated", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -36.38, "z": 56.436, "size": 0.2691, "title": "Localization of triangulated categories", "summary": "If `L : C ⥤ D` is a localization functor for a class of morphisms `W` that is compatible with the triangulation on the category `C` and admits a left calculus of fractions, it is shown in this file that `D` can be equipped with a pretriangulated category structure, and that it is triangulated.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Triangulated.html"}, {"id": "Mathlib.CategoryTheory.Localization.CalculusOfFractions.ComposableArrows", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 102, "macro_tier_score": 0.0096, "macro_tier_override": null, "x": -17.18, "z": 36.436, "size": 0.2463, "title": "Essential surjectivity of the functor induced on composable arrows", "summary": "Assuming that `L : C ⥤ D` is a localization functor for a class of morphisms `W` that has a calculus of left *or* right fractions, we show in this file that the functor `L.mapComposableArrows n : ComposableArrows C n ⥤ ComposableArrows D n` is essentially surjective for any `n : ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/CalculusOfFractions/ComposableArrows.html"}, {"id": "Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0145, "macro_tier_override": null, "x": -23.531, "z": 49.436, "size": 0.2606, "title": "The preadditive category structure on the localized category", "summary": "In this file, it is shown that if `W : MorphismProperty C` has a left calculus of fractions, and `C` is preadditive, then the localized category is preadditive, and the localization functor is additive. Let `L : C ⥤ D` be a localization functor for `W`. We first construct an abelian group structure on `L.obj X ⟶ L.obj Y` for `X` and `Y` in `C`. The addition is defined using representatives of two morphisms in `L` as…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.html"}, {"id": "Mathlib.CategoryTheory.Shift.Localization", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 102, "macro_tier_score": 0.0101, "macro_tier_override": null, "x": 10.109, "z": 39.0, "size": 0.3003, "title": "The shift induced on a localized category", "summary": "Let `C` be a category equipped with a shift by a monoid `A`. A morphism property `W` on `C` satisfies `W.IsCompatibleWithShift A` when for all `a : A`, a morphism `f` is in `W` iff `f⟦a⟧'` is. When this compatibility is satisfied, then the corresponding localized category can be equipped with a shift by `A`, and the localization functor is compatible with the shift.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/Localization.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.EpiMono", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 102, "macro_tier_score": 0.0195, "macro_tier_override": null, "x": 24.444, "z": 7.436, "size": 0.2879, "title": "Properties of objects that are closed under subobjects and quotients", "summary": "Given a category `C` and `P : ObjectProperty C`, we define type classes `P.IsClosedUnderSubobjects` and `P.IsClosedUnderQuotients` expressing that `P` is closed under subobjects (resp. quotients).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/EpiMono.html"}, {"id": "Mathlib.CategoryTheory.Sites.Spaces", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 101, "macro_tier_score": 0.001, "macro_tier_override": null, "x": -44.402, "z": 234.683, "size": 0.3322, "title": "Grothendieck topology on a topological space", "summary": "Define the Grothendieck topology and the pretopology associated to a topological space, and show that the pretopology induces the topology. The covering (pre)sieves on `X` are those for which the union of domains contains `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Spaces.html"}, {"id": "Mathlib.CategoryTheory.Limits.Lattice", "region_id": "category_theory", "micro_elevation": 0.5789, "macro_tier": 102, "macro_tier_score": 0.01, "macro_tier_override": null, "x": -24.771, "z": 40.436, "size": 0.2958, "title": "Limits in lattice categories are given by infimums and supremums.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Lattice.html"}, {"id": "Mathlib.CategoryTheory.Limits.Set", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 198.29, "z": -72.661, "size": 0.247, "title": "The functor from `Set X` to types preserves filtered colimits", "summary": "Given `X : Type u`, the functor `Set.functorToTypes : Set X ⥤ Type u` which sends `A : Set X` to its underlying type preserves filtered colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Set.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.Filtered", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 103, "macro_tier_score": 0.0828, "macro_tier_override": null, "x": -46.257, "z": 49.342, "size": 0.383, "title": "Filtered colimits in the category of types.", "summary": "We give a characterisation of the equality in filtered colimits in `Type` as a lemma `CategoryTheory.Limits.Types.FilteredColimit.colimit_eq_iff`: `colimit.ι F i xi = colimit.ι F j xj ↔ ∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f xi = F.map g xj`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/Filtered.html"}, {"id": "Mathlib.CategoryTheory.Types.Set", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": 37.48, "z": 27.436, "size": 0.2704, "title": "The functor from `Set X` to types", "summary": "Given `X : Type u`, we define the functor `Set.functorToTypes : Set X ⥤ Type u` which sends `A : Set X` to its underlying type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Types/Set.html"}, {"id": "Mathlib.CategoryTheory.Sites.JointlySurjective", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 101, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 48.9, "z": 102.178, "size": 0.282, "title": "The jointly surjective precoverage", "summary": "In the category of types, the jointly surjective precoverage has the jointly surjective families as coverings. We show that this precoverage is stable under the standard constructions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/JointlySurjective.html"}, {"id": "Mathlib.CategoryTheory.Sites.MorphismProperty", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 68.626, "z": 105.178, "size": 0.3036, "title": "The site induced by a morphism property", "summary": "Let `C` be a category with pullbacks and `P` be a multiplicative morphism property which is stable under base change. Then `P` induces a pretopology, where coverings are given by presieves whose elements satisfy `P`. Standard examples of pretopologies in algebraic geometry, such as the étale site, are obtained from this construction by intersecting with the pretopology of surjective families.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/MorphismProperty.html"}, {"id": "Mathlib.CategoryTheory.Category.Cat.Op", "region_id": "category_theory", "micro_elevation": 0.3684, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -3.636, "z": 28.436, "size": 0.2, "title": "The dualizing functor on `Cat`", "summary": "We define a (strict) functor `opFunctor` and an equivalence assigning opposite categories to categories. We then show that this functor is strictly involutive and that it induces an equivalence on `Cat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Cat/Op.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 102, "macro_tier_score": 0.0068, "macro_tier_override": null, "x": 159.119, "z": -30.328, "size": 0.3911, "title": "Limits involving zero objects", "summary": "Binary products and coproducts with a zero object always exist, and pullbacks/pushouts over a zero object are products/coproducts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.html"}, {"id": "Mathlib.CategoryTheory.Subobject.HasCardinalLT", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -41.522, "z": 51.436, "size": 0.2391, "title": "Cardinality of Subobject", "summary": "If `X ⟶ Y` is a monomorphism, and the cardinality of `Subobject Y` is `< κ`, then the cardinality of `Subobject X` is also `< κ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subobject/HasCardinalLT.html"}, {"id": "Mathlib.CategoryTheory.Subobject.Basic", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 103, "macro_tier_score": 0.0843, "macro_tier_override": null, "x": -21.207, "z": 50.436, "size": 0.4438, "title": "Subobjects", "summary": "We define `Subobject X` as the quotient (by isomorphisms) of `MonoOver X := {f : Over X // Mono f.hom}`. Here `MonoOver X` is a thin category (a pair of objects has at most one morphism between them), so we can think of it as a preorder. However as it is not skeletal, it is not a partial order. There is a coercion from `Subobject X` back to the ambient category `C` (using choice to pick a representative), and for `P…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subobject/Basic.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Braided.Reflection", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -227.108, "z": -120.756, "size": 0.2322, "title": "Day's reflection theorem", "summary": "Let `D` be a symmetric monoidal closed category and let `C` be a reflective subcategory. Day's reflection theorem proves the equivalence of four conditions, which are all of the form that a map obtained by acting on the unit of the reflective adjunction, with the internal hom and tensor functors, is an isomorphism. Suppose that `C` is itself monoidal and that the reflector is a monoidal functor. Then we can apply…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Braided/Reflection.html"}, {"id": "Mathlib.CategoryTheory.Monad.Adjunction", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 102, "macro_tier_score": 0.0296, "macro_tier_override": null, "x": 51.06, "z": 33.436, "size": 0.3357, "title": "Adjunctions and (co)monads", "summary": "We develop the basic relationship between adjunctions and (co)monads. Given an adjunction `h : L ⊣ R`, we have `h.toMonad : Monad C` and `h.toComonad : Comonad D`. We then have `Monad.comparison (h : L ⊣ R) : D ⥤ h.toMonad.algebra` sending `Y : D` to the Eilenberg-Moore algebra for `L ⋙ R` with underlying object `R.obj X`, and dually `Comonad.comparison`. We say `R : D ⥤ C` is `MonadicRightAdjoint`, if it is a right…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monad/Adjunction.html"}, {"id": "Mathlib.CategoryTheory.Sites.PseudofunctorSheafOver", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -55.914, "z": 61.436, "size": 0.2, "title": "Sheaves on Over categories, as a pseudofunctor", "summary": "Given a Grothendieck topology `J` on a category `C` and a category `A`, we define the pseudofunctor `J.pseudofunctorOver A : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat` which sends `X : C` to the category of sheaves on `Over X` with values in `A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/PseudofunctorSheafOver.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Functor.LocallyDiscrete", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 92.1, "z": -15.664, "size": 0.267, "title": "Pseudofunctors from locally discrete bicategories", "summary": "This file provides various ways of constructing pseudofunctors from locally discrete bicategories. Firstly, we define the constructors `pseudofunctorOfIsLocallyDiscrete` and `oplaxFunctorOfIsLocallyDiscrete` for defining pseudofunctors and oplax functors from a locally discrete bicategory. In this situation, we do not need to care about the field `map₂`, because all the `2`-morphisms in `B` are identities. We also…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Functor/LocallyDiscrete.html"}, {"id": "Mathlib.CategoryTheory.Subfunctor.Equalizer", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 101, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": 90.856, "z": -68.661, "size": 0.3095, "title": "The equalizer of two morphisms of functors, as a subfunctor", "summary": "If `F₁` and `F₂` are type-valued functors, `A : Subfunctor F₁`, and `f` and `g` are two morphisms `A.toFunctor ⟶ F₂`, we introduce `Subcomplex.equalizer f g`, which is the subfunctor of `F₁` contained in `A` where `f` and `g` coincide.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subfunctor/Equalizer.html"}, {"id": "Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 9.367, "z": 43.0, "size": 0.255, "title": "Epimorphisms with an injective kernel", "summary": "In this file, we define the class of morphisms `epiWithInjectiveKernel` in an abelian category. We show that this property of morphisms is multiplicative. This shall be used in the file `Mathlib/Algebra/Homology/Factorizations/Basic.lean` in order to define morphisms of cochain complexes which satisfy this property degreewise.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/EpiWithInjectiveKernel.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.AdjointFunctorTheorems", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 90.237, "z": 54.436, "size": 0.2589, "title": "Adjoint functor theorem", "summary": "This file proves the (general) adjoint functor theorem, in the form: * If `G : D ⥤ C` preserves limits and `D` has limits, and satisfies the solution set condition, then it has a left adjoint: `isRightAdjoint_of_preservesLimits_of_solutionSetCondition`. We show that the converse holds, i.e. that if `G` has a left adjoint then it satisfies the solution set condition, see `solutionSetCondition_of_isRightAdjoint` (the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/AdjointFunctorTheorems.html"}, {"id": "Mathlib.CategoryTheory.Generator.Basic", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 103, "macro_tier_score": 0.0351, "macro_tier_override": null, "x": -35.585, "z": 46.0, "size": 0.3798, "title": "Separating and detecting sets", "summary": "There are several non-equivalent notions of a generator of a category. Here, we consider two of them: * We say that `P : ObjectProperty C` is a separating set if the functors `C(G, -)` for `G` such that `P G` are collectively faithful, i.e., if `h ≫ f = h ≫ g` for all `h` with domain satisfying `P` implies `f = g`. * We say that `P : ObjectProperty C` is a detecting set if the functors `C(G, -)` collectively reflect…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Generator/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.ConeCategory", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 103, "macro_tier_score": 0.111, "macro_tier_override": null, "x": 10.233, "z": 37.436, "size": 0.3628, "title": "Limits and the category of (co)cones", "summary": "This file contains results that stem from the limit API. For the definition and the category instance of `Cone`, please refer to `Mathlib/CategoryTheory/Limits/Cones.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/ConeCategory.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.WeaklyInitial", "region_id": "category_theory", "micro_elevation": 0.5789, "macro_tier": 102, "macro_tier_score": 0.0096, "macro_tier_override": null, "x": -3.827, "z": 40.436, "size": 0.2433, "title": "Constructions related to weakly initial objects", "summary": "This file gives constructions related to weakly initial objects, namely: * If a category has small products and a small weakly initial set of objects, then it has a weakly initial object. * If a category has wide equalizers and a weakly initial object, then it has an initial object. These are primarily useful to show the General Adjoint Functor Theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/WeaklyInitial.html"}, {"id": "Mathlib.CategoryTheory.Subobject.Comma", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 102, "macro_tier_score": 0.0096, "macro_tier_override": null, "x": -42.956, "z": 52.436, "size": 0.2433, "title": "Subobjects in the category of structured arrows", "summary": "We compute the subobjects of an object `A` in the category `StructuredArrow S T` for `T : C ⥤ D` and `S : D` as a subtype of the subobjects of `A.right`. We deduce that `StructuredArrow S T` is well-powered if `C` is.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subobject/Comma.html"}, {"id": "Mathlib.CategoryTheory.Functor.TypeValuedFlat", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 145.823, "z": -19.328, "size": 0.2478, "title": "Type-valued flat functors", "summary": "A functor `F : C ⥤ Type w` is a flat Type-valued functor if the category `F.Elements` is cofiltered. (This is not equivalent to saying that `F` is representably flat in the sense of the typeclass `RepresentablyFlat` defined in the file `Mathlib/CategoryTheory/Functor/Flat.lean`, see also https://golem.ph.utexas.edu/category/2011/06/flat_functors_and_morphisms_of.html for a clarification about the differences between…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/TypeValuedFlat.html"}, {"id": "Mathlib.CategoryTheory.Sites.Point.Conservative", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 71.696, "z": 114.178, "size": 0.3068, "title": "Conservative families of points", "summary": "Let `J` be a Grothendieck topology on a category `C`. Let `P : ObjectProperty J.Point` be a family of points. We say that `P` is a conservative family of points if the corresponding fiber functors `Sheaf J (Type w) ⥤ Type w` jointly reflect isomorphisms. Under suitable assumptions on the coefficient category `A`, this implies that the fiber functors `Sheaf J A ⥤ A` corresponding to the points in `P` jointly reflect…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Point/Conservative.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.LimitsClosure", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 102, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -30.964, "z": 44.436, "size": 0.2812, "title": "Closure of a property of objects under limits of certain shapes", "summary": "In this file, given a property `P` of objects in a category `C` and a family of categories `J : α → Type _`, we introduce the closure `P.limitsClosure J` of `P` under limits of shapes `J a` for all `a : α`, and under certain smallness assumptions, we show that it is essentially small.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/LimitsClosure.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.LimitsOfShape", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 103, "macro_tier_score": 0.1637, "macro_tier_override": null, "x": 40.495, "z": 43.436, "size": 0.3774, "title": "Objects that are limits of objects satisfying a certain property", "summary": "Given a property of objects `P : ObjectProperty C` and a category `J`, we introduce two properties of objects `P.strictLimitsOfShape J` and `P.limitsOfShape J`. The former contains exactly the objects of the form `limit F` for any functor `F : J ⥤ C` that has a limit and such that `F.obj j` satisfies `P` for any `j`, while the latter contains all the objects that are isomorphic to these \"chosen\" objects `limit F`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/LimitsOfShape.html"}, {"id": "Mathlib.CategoryTheory.Limits.Pi", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -10.902, "z": 33.436, "size": 0.2, "title": "Limits in the category of indexed families of objects.", "summary": "Given a functor `F : J ⥤ Π i, C i` into a category of indexed families, 1. we can assemble a collection of cones over `F ⋙ Pi.eval C i` into a cone over `F` 2. if all those cones are limit cones, the assembled cone is a limit cone, and 3. if we have limits for each of `F ⋙ Pi.eval C i`, we can produce a `HasLimit F` instance", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Pi.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Orthogonal", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -76.704, "z": 51.0, "size": 0.2493, "title": "Orthogonal of triangulated subcategories", "summary": "Let `P` be a triangulated subcategory of a pretriangulated category `C`. We show that `P.rightOrthogonal` (which consists of objects `Y` with no nonzero map `X ⟶ Y` with `X` satisfying `P`) is a triangulated subcategory. The dual result for `P.leftOrthogonal` is also obtained.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Orthogonal.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.Orthogonal", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 24.834, "z": 44.436, "size": 0.247, "title": "Orthogonal of a property of objects", "summary": "Let `P` be a property of objects in a category with zero morphisms. We define `P.rightOrthogonal` as the property of objects `Y` such that any map `f : X ⟶ Y` vanishes when `P X` holds. Similarly, we define `P.leftOrthogonal` as the property of objects `X` such that any map `f : X ⟶ Y` vanishes when `P Y` holds.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/Orthogonal.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.Local", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 103, "macro_tier_score": 0.0339, "macro_tier_override": null, "x": 32.473, "z": 48.436, "size": 0.3019, "title": "Objects that are local with respect to a property of morphisms", "summary": "Given `W : MorphismProperty C`, we define `W.isLocal : ObjectProperty C` which is the property of objects `Z` such that for any `f : X ⟶ Y` satisfying `W`, the precomposition with `f` gives a bijection `(Y ⟶ Z) ≃ (X ⟶ Z)`. (In the file `Mathlib/CategoryTheory/Localization/Bousfield.lean`, it is shown that this is part of a Galois connection, with \"dual\" construction `ObjectProperty.isLocal : ObjectProperty C →…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/Local.html"}, {"id": "Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime", "region_id": "category_theory", "micro_elevation": 0.9825, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 107.439, "z": 63.436, "size": 0.2, "title": "Descent data when we have pullbacks", "summary": "In this file, given a pseudofunctor `F` from `LocallyDiscrete Cᵒᵖ` to `Cat`, a family of maps `f i : X i ⟶ S` in the category `C`, chosen pullbacks `sq` and threefold wide pullbacks `sq₃` for these morphisms, we define a category `F.DescentData' sq sq₃` of objects over the `X i` equipped with a descent data relative to the morphisms `f i : X i ⟶ S`, where the data and compatibilities are expressed using the chosen…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Descent/DescentDataPrime.html"}, {"id": "Mathlib.CategoryTheory.Sites.Descent.DescentData", "region_id": "category_theory", "micro_elevation": 0.9649, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -26.702, "z": 62.436, "size": 0.3034, "title": "Descent data", "summary": "In this file, given a pseudofunctor `F` from `LocallyDiscrete Cᵒᵖ` to `Cat`, and a family of maps `f i : X i ⟶ S` in the category `C`, we define the category `F.DescentData f` of objects over the `X i` equipped with descent data relative to the morphisms `f i : X i ⟶ S`. We show that up to an equivalence, the category `F.DescentData f` is unchanged when we replace `S` by an isomorphic object, or the family `f i : X…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Descent/DescentData.html"}, {"id": "Mathlib.CategoryTheory.Limits.Presentation", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 103, "macro_tier_score": 0.1628, "macro_tier_override": null, "x": 56.009, "z": 42.436, "size": 0.3251, "title": "(Co)limit presentations", "summary": "Let `J` and `C` be categories and `X : C`. We define type `ColimitPresentation J X` that contains the data of objects `Dⱼ` and natural maps `sⱼ : Dⱼ ⟶ X` that make `X` the colimit of the `Dⱼ`. (See `Mathlib/CategoryTheory/Presentable/ColimitPresentation.lean` for the construction of a presentation of a colimit of objects that are equipped with presentations.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Presentation.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "region_id": "category_theory", "micro_elevation": 0.3684, "macro_tier": 103, "macro_tier_score": 0.2946, "macro_tier_override": null, "x": 28.923, "z": 28.436, "size": 0.599, "title": "Strong epimorphisms", "summary": "In this file, we define strong epimorphisms. A strong epimorphism is an epimorphism `f` which has the (unique) left lifting property with respect to monomorphisms. Similarly, a strong monomorphism is a monomorphism which has the (unique) right lifting property with respect to epimorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.html"}, {"id": "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 103, "macro_tier_score": 0.2946, "macro_tier_override": null, "x": 42.634, "z": 30.436, "size": 0.599, "title": "Lifting properties and adjunction", "summary": "In this file, we obtain `Adjunction.HasLiftingProperty_iff`, which states that when we have an adjunction `adj : G ⊣ F` between two functors `G : C ⥤ D` and `F : D ⥤ C`, then a morphism of the form `G.map i` has the left lifting property in `D` with respect to a morphism `p` if and only the morphism `i` has the left lifting property in `C` with respect to `F.map p`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/LiftingProperties/Adjunction.html"}, {"id": "Mathlib.CategoryTheory.Sites.MayerVietorisSquare", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -26.222, "z": 244.683, "size": 0.2617, "title": "Mayer-Vietoris squares", "summary": "The purpose of this file is to allow the formalization of long exact Mayer-Vietoris sequences in sheaf cohomology. If `X₄` is an open subset of a topological space that is covered by two open subsets `X₂` and `X₃`, it is known that there is a long exact sequence `... ⟶ H^q(X₄) ⟶ H^q(X₂) ⊞ H^q(X₃) ⟶ H^q(X₁) ⟶ H^{q+1}(X₄) ⟶ ...` where `X₁` is the intersection of `X₂` and `X₃`, and `H^q` are the cohomology groups with…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/MayerVietorisSquare.html"}, {"id": "Mathlib.CategoryTheory.Sites.Hypercover.Zero", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 102, "macro_tier_score": 0.0301, "macro_tier_override": null, "x": -0.392, "z": 46.436, "size": 0.3646, "title": "0-hypercovers", "summary": "Given a coverage `J` on a category `C`, we define the type of `0`-hypercovers of an object `S : C`. They consist of a covering family of morphisms `X i ⟶ S` indexed by a type `I₀` such that the induced presieve is in `J`. We define this with respect to a coverage and not to a Grothendieck topology, because this yields more control over the components of the cover.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Hypercover/Zero.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.Iso", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 103, "macro_tier_score": 0.2379, "macro_tier_override": null, "x": 64.212, "z": 38.436, "size": 0.4879, "title": "The pullback of an isomorphism", "summary": "This file provides some basic results about the pullback (and pushout) of an isomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Iso.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.PartialAdjoint", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 24.939, "z": 26.0, "size": 0.2485, "title": "Domain of definition of the partial left adjoint", "summary": "Given a functor `F : D ⥤ C`, we define a functor `F.partialLeftAdjoint : F.PartialLeftAdjointSource ⥤ D` which is defined on the full subcategory of `C` consisting of those objects `X : C` such that `F ⋙ coyoneda.obj (op X) : D ⥤ Type _` is corepresentable. For `X : F.PartialLeftAdjointSource` and `Y : D`, we have a natural bijection `(F.partialLeftAdjoint.obj X ⟶ Y) ≃ (X.obj ⟶ F.obj Y)` that is similar to what we…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/PartialAdjoint.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Pretriangulated", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 102, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 46.31, "z": 53.436, "size": 0.3465, "title": "Pretriangulated Categories", "summary": "This file contains the definition of pretriangulated categories and triangulated functors between them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Pretriangulated.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 103, "macro_tier_score": 0.1596, "macro_tier_override": null, "x": 37.469, "z": 42.436, "size": 0.4085, "title": "Constructing finite products from binary products and terminal.", "summary": "If a category has binary products and a terminal object then it has finite products. If a functor preserves binary products and the terminal object then it preserves finite products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/FiniteProductsOfBinaryProducts.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.TriangleShift", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 102, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": 44.156, "z": 52.436, "size": 0.287, "title": "The shift on the category of triangles", "summary": "In this file, it is shown that if `C` is a preadditive category with a shift by `ℤ`, then the category of triangles `Triangle C` is also endowed with a shift. We also show that rotating triangles three times identifies with the shift by `1`. The shift on the category of triangles was also obtained by Adam Topaz, Johan Commelin and Andrew Yang during the Liquid Tensor Experiment.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/TriangleShift.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.CommGrp_", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 77.006, "z": 53.436, "size": 0.2, "title": "Yoneda embedding of `CommGrp C`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/CommGrp_.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.CommMon_", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -24.052, "z": 51.436, "size": 0.239, "title": "Yoneda embedding of `CommMon C`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/CommMon_.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.CommGrp_", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 102, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": 67.456, "z": 52.436, "size": 0.2846, "title": "The category of commutative groups in a Cartesian monoidal category", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/CommGrp_.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax", "region_id": "category_theory", "micro_elevation": 0.2105, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": 35.362, "z": 19.436, "size": 0.2918, "title": "Transformations between lax functors", "summary": "Just as there are natural transformations between functors, there are transformations between lax functors. The equality in the naturality condition of a natural transformation gets replaced by a specified 2-morphism. Now, there are three possible types of transformations (between lax functors): * lax natural transformations; * oplax natural transformations; * strong natural transformations. These differ in the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Lax.html"}, {"id": "Mathlib.CategoryTheory.Localization.Adjunction", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 102, "macro_tier_score": 0.0291, "macro_tier_override": null, "x": 55.754, "z": 35.436, "size": 0.2946, "title": "Localization of adjunctions", "summary": "In this file, we show that if we have an adjunction `adj : G ⊣ F` such that both functors `G : C₁ ⥤ C₂` and `F : C₂ ⥤ C₁` induce functors `G' : D₁ ⥤ D₂` and `F' : D₂ ⥤ D₁` on localized categories, i.e. that we have localization functors `L₁ : C₁ ⥤ D₁` and `L₂ : C₂ ⥤ D₂` with respect to morphism properties `W₁` and `W₂` respectively, and 2-commutative diagrams `[CatCommSq G L₁ L₂ G']` and `[CatCommSq F L₂ L₁ F']`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Adjunction.html"}, {"id": "Mathlib.CategoryTheory.Sites.Point.Category", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 90.266, "z": 58.436, "size": 0.2596, "title": "The category of points of a site", "summary": "We define the category structure on the points of a site `(C, J)`: a morphism between `Φ₁ ⟶ Φ₂` between two points consists of a morphism `Φ₂.fiber ⟶ Φ₁.fiber` (SGA 4 IV 3.2).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Point/Category.html"}, {"id": "Mathlib.CategoryTheory.Sites.Point.Basic", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 102, "macro_tier_score": 0.0061, "macro_tier_override": null, "x": 146.579, "z": -17.328, "size": 0.3554, "title": "Points of a site", "summary": "Let `C` be a category equipped with a Grothendieck topology `J`. In this file, we define the notion of point of the site `(C, J)`, as a structure `GrothendieckTopology.Point`. Such a `Φ : J.Point` consists in a functor `Φ.fiber : C ⥤ Type w` such that the category `Φ.fiber.Elements` is cofiltered (and initially small) and such that if `x : Φ.fiber.obj X` and `R` is a covering sieve of `X`, then `x` belongs to the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Point/Basic.html"}, {"id": "Mathlib.CategoryTheory.Filtered.Flat", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 88.344, "z": 50.436, "size": 0.2, "title": "Pulling back filteredness along representably flat functors", "summary": "We show that if `F : C ⥤ D` is a representably coflat functor between two categories, filteredness of `D` implies filteredness of `C`. Dually, if `F` is representably flat, cofilteredness of `D` implies cofilteredness of `C`. Transferring (co)filteredness *along* representably (co)flat functors is given by `IsFiltered.of_final` and its dual, since every representably flat functor is final and every representably…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Filtered/Flat.html"}, {"id": "Mathlib.CategoryTheory.Filtered.CostructuredArrow", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 74.443, "z": 49.436, "size": 0.2478, "title": "Inferring Filteredness from Filteredness of Costructured Arrow Categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Filtered/CostructuredArrow.html"}, {"id": "Mathlib.CategoryTheory.PUnit", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 103, "macro_tier_score": 0.2746, "macro_tier_override": null, "x": 143.919, "z": -94.661, "size": 0.4376, "title": "The category `Discrete PUnit`", "summary": "We define `star : C ⥤ Discrete PUnit` sending everything to `PUnit.star`, show that any two functors to `Discrete PUnit` are naturally isomorphic, and construct the equivalence `(Discrete PUnit ⥤ C) ≌ C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/PUnit.html"}, {"id": "Mathlib.CategoryTheory.Distributive.Monoidal", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -0.088, "z": 50.436, "size": 0.239, "title": "Distributive monoidal categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Distributive/Monoidal.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.End", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 103, "macro_tier_score": 0.0442, "macro_tier_override": null, "x": -13.17, "z": 32.436, "size": 0.355, "title": "Endofunctors as a monoidal category.", "summary": "We give the monoidal category structure on `C ⥤ C`, and show that when `C` itself is monoidal, it embeds via a monoidal functor into `C ⥤ C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/End.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Preadditive", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0197, "macro_tier_override": null, "x": -38.416, "z": 49.436, "size": 0.3075, "title": "Preadditive monoidal categories", "summary": "A monoidal category is `MonoidalPreadditive` if it is preadditive and tensor product of morphisms is linear in both factors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Preadditive.html"}, {"id": "Mathlib.CategoryTheory.Limits.Elements", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 102, "macro_tier_score": 0.0242, "macro_tier_override": null, "x": 174.157, "z": -39.328, "size": 0.2799, "title": "Limits in the category of elements", "summary": "We show that if `C` has limits of shape `I` and `A : C ⥤ Type w` preserves limits of shape `I`, then the category of elements of `A` has limits of shape `I` and the forgetful functor `π : A.Elements ⥤ C` creates them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Elements.html"}, {"id": "Mathlib.CategoryTheory.DifferentialObject", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 24.037, "z": 37.0, "size": 0.2478, "title": "Differential objects in a category.", "summary": "A differential object in a category with zero morphisms and a shift is an object `X` equipped with a morphism `d : obj ⟶ obj⟦1⟧`, such that `d^2 = 0`. We build the category of differential objects, and some basic constructions such as the forgetful functor, zero morphisms and zero objects, and the shift functor on differential objects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/DifferentialObject.html"}, {"id": "Mathlib.CategoryTheory.Shift.Basic", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 103, "macro_tier_score": 0.0335, "macro_tier_override": null, "x": -1.025, "z": 43.436, "size": 0.4973, "title": "Shift", "summary": "A `Shift` on a category `C` indexed by a monoid `A` is nothing more than a monoidal functor from `A` to `C ⥤ C`. A typical example to keep in mind might be the category of complexes `⋯ → C_{n-1} → C_n → C_{n+1} → ⋯`. It has a shift indexed by `ℤ`, where we assign to each `n : ℤ` the functor `C ⥤ C` that re-indexes the terms, so the degree `i` term of `Shift n C` would be the degree `i+n`-th term of `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.2507, "macro_tier_override": null, "x": 12.85, "z": 34.436, "size": 0.5766, "title": "Preserving terminal object", "summary": "Constructions to relate the notions of preserving terminal objects and reflecting terminal objects to concrete objects. In particular, we show that `terminalComparison G` is an isomorphism iff `G` preserves terminal objects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Terminal.html"}, {"id": "Mathlib.CategoryTheory.Limits.Yoneda", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 103, "macro_tier_score": 0.2176, "macro_tier_override": null, "x": -12.846, "z": 36.436, "size": 0.4445, "title": "Limit properties relating to the (co)yoneda embedding.", "summary": "We calculate the colimit of `Y ↦ (X ⟶ Y)`, which is just `PUnit`. (This is used in characterising cofinal functors.) We also show the (co)yoneda embeddings preserve limits and jointly reflect them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Yoneda.html"}, {"id": "Mathlib.CategoryTheory.Abelian.RightDerived", "region_id": "category_theory", "micro_elevation": 0.0351, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 27.305, "z": 9.436, "size": 0.2, "title": "Right-derived functors", "summary": "We define the right-derived functors `F.rightDerived n : C ⥤ D` for any additive functor `F` out of a category with injective resolutions. We first define a functor `F.rightDerivedToHomotopyCategory : C ⥤ HomotopyCategory D (ComplexShape.up ℕ)` which is `injectiveResolutions C ⋙ F.mapHomotopyCategory _`. We show that if `X : C` and `I : InjectiveResolution X`, then `F.rightDerivedToHomotopyCategory.obj X` identifies…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/RightDerived.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Injective.Resolution", "region_id": "category_theory", "micro_elevation": 0.0175, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 25.225, "z": 8.436, "size": 0.2676, "title": "Abelian categories with enough injectives have injective resolutions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Injective/Resolution.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.Mon_", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 47.561, "z": 51.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/Mon_.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.Mon", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": -27.086, "z": 50.436, "size": 0.3412, "title": "Yoneda embedding of `Mon C`", "summary": "We show that monoid objects in Cartesian monoidal categories are exactly those whose yoneda presheaf is a presheaf of monoids, by constructing the yoneda embedding `Mon C ⥤ Cᵒᵖ ⥤ MonCat.{v}` and showing that it is fully faithful and its (essential) image is the representable functors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/Mon.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 103, "macro_tier_score": 0.2616, "macro_tier_override": null, "x": -5.721, "z": 36.436, "size": 0.702, "title": "PullbackCone", "summary": "This file provides API for interacting with cones (resp. cocones) in the case of pullbacks (resp. pushouts).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/PullbackCone.html"}, {"id": "Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -51.258, "z": 58.436, "size": 0.2676, "title": "Alternative constructor for points", "summary": "Let `J` be a Grothendieck topology on a category `C`. We provide a constructor `Point.ofIsCofiltered` for points for `J` which takes as inputs: - a functor `p : N ⥤ C` where `N` is cofiltered and initially small - the assumption that for any covering sieve `R` of `X`, any morphism `f : p.obj U ⟶ X`, there exists a morphism `g : Y ⟶ X` in `R`, a morphism `q : V ⟶ U` in `N` and a morphism `a : p.obj V ⟶ Y` such that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Point/OfIsCofiltered.html"}, {"id": "Mathlib.CategoryTheory.Sites.CoverLifting", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 102, "macro_tier_score": 0.0207, "macro_tier_override": null, "x": 6.011, "z": 55.436, "size": 0.3712, "title": "Cocontinuous functors between sites.", "summary": "We define cocontinuous functors between sites as functors that pull covering sieves back to covering sieves. This concept is also known as *cover-lifting* or *cover-reflecting functors*. We use the original terminology and definition of SGA 4 III 2.1. However, the notion of cocontinuous functor should not be confused with the general definition of cocontinuous functors between categories as functors preserving small…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/CoverLifting.html"}, {"id": "Mathlib.CategoryTheory.Subfunctor.OfSection", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 102, "macro_tier_score": 0.006, "macro_tier_override": null, "x": 172.349, "z": -68.661, "size": 0.3481, "title": "The subpresheaf generated by a section", "summary": "Given a presheaf of types `F : Cᵒᵖ ⥤ Type w` and a section `x : F.obj X`, we define `Subfunctor.ofSection x : Subfunctor F` as the subpresheaf of `F` generated by `x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subfunctor/OfSection.html"}, {"id": "Mathlib.CategoryTheory.Sites.EpiMono", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -98.845, "z": 69.464, "size": 0.2793, "title": "Morphisms of sheaves factor as a locally surjective followed by a locally injective morphism", "summary": "When morphisms in a concrete category `A` factor in a functorial manner as a surjective map followed by an injective map, we obtain that any morphism of sheaves in `Sheaf J A` factors in a functorial manner as a locally surjective morphism (which is epi) followed by a locally injective morphism (which is mono). Moreover, if we assume that the category of sheaves `Sheaf J A` is balanced (see `Sites.LeftExact`), then…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/EpiMono.html"}, {"id": "Mathlib.CategoryTheory.Balanced", "region_id": "category_theory", "micro_elevation": 0.3333, "macro_tier": 103, "macro_tier_score": 0.2897, "macro_tier_override": null, "x": 37.445, "z": 26.436, "size": 0.4675, "title": "Balanced categories", "summary": "A category is called balanced if any morphism that is both monic and epic is an isomorphism. Balanced categories arise frequently. For example, categories in which every monomorphism (or epimorphism) is strong are balanced. Examples of this are abelian categories and toposes, such as the category of types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Balanced.html"}, {"id": "Mathlib.CategoryTheory.FiberedCategory.Grothendieck", "region_id": "category_theory", "micro_elevation": 0.3684, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -1.813, "z": 28.436, "size": 0.2, "title": "The Grothendieck construction gives a fibered category", "summary": "In this file we show that the Grothendieck construction applied to a pseudofunctor `F` gives a fibered category over the base category. We also provide a `HasFibers` instance to `∫ᶜ F`, such that the fiber over `S` is the category `F(S)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/FiberedCategory/Grothendieck.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Grothendieck", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 33.37, "z": 25.436, "size": 0.2478, "title": "The Grothendieck and CoGrothendieck constructions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Grothendieck.html"}, {"id": "Mathlib.CategoryTheory.FiberedCategory.HasFibers", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 52.369, "z": 27.436, "size": 0.2478, "title": "Fibers of functors", "summary": "In this file we introduce a typeclass `HasFibers` for a functor `p : 𝒳 ⥤ 𝒮`, consisting of: - A collection of categories `Fib S` for every `S` in `𝒮` (the fiber categories) - Functors `ι : Fib S ⥤ 𝒳` such that `ι ⋙ p = const (Fib S) S` - The induced functor `Fib S ⥤ Fiber p S` is an equivalence. We also provide a canonical `HasFibers` instance, which uses the standard fibers `Fiber p S` (see…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/FiberedCategory/HasFibers.html"}, {"id": "Mathlib.CategoryTheory.GradedObject.Trifunctor", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 5.251, "z": 39.0, "size": 0.3112, "title": "The action of trifunctors on graded objects", "summary": "Given a trifunctor `F : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄` and types `I₁`, `I₂` and `I₃`, we define a functor `GradedObject I₁ C₁ ⥤ GradedObject I₂ C₂ ⥤ GradedObject I₃ C₃ ⥤ GradedObject (I₁ × I₂ × I₃) C₄` (see `mapTrifunctor`). When we have a map `p : I₁ × I₂ × I₃ → J` and suitable coproducts exist, we define a functor `GradedObject I₁ C₁ ⥤ GradedObject I₂ C₂ ⥤ GradedObject I₃ C₃ ⥤ GradedObject J C₄` (see `mapTrifunctorMap`) which…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GradedObject/Trifunctor.html"}, {"id": "Mathlib.CategoryTheory.GradedObject.Bifunctor", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 102, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -19.852, "z": 45.436, "size": 0.3224, "title": "The action of bifunctors on graded objects", "summary": "Given a bifunctor `F : C₁ ⥤ C₂ ⥤ C₃` and types `I` and `J`, we construct an obvious functor `mapBifunctor F I J : GradedObject I C₁ ⥤ GradedObject J C₂ ⥤ GradedObject (I × J) C₃`. When we have a map `p : I × J → K` and that suitable coproducts exist, we also get a functor `mapBifunctorMap F p : GradedObject I C₁ ⥤ GradedObject J C₂ ⥤ GradedObject K C₃`. In case `p : I × I → I` is the addition on a monoid and `F` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GradedObject/Bifunctor.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Skeleton", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 32.925, "z": 28.0, "size": 0.227, "title": "The monoid on the skeleton of a monoidal category", "summary": "The skeleton of a monoidal category is a monoid.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Skeleton.html"}, {"id": "Mathlib.CategoryTheory.Skeletal", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 103, "macro_tier_score": 0.2995, "macro_tier_override": null, "x": 158.07, "z": -84.661, "size": 0.4778, "title": "Skeleton of a category", "summary": "Define skeletal categories as categories in which any two isomorphic objects are equal. Construct the skeleton of an arbitrary category by taking isomorphism classes, and show it is a skeleton of the original category. In addition, construct the skeleton of a thin category as a partial ordering, and (noncomputably) show it is a skeleton of the original category. The advantage of this special case being handled…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Skeletal.html"}, {"id": "Mathlib.CategoryTheory.Sums.Products", "region_id": "category_theory", "micro_elevation": 0.2105, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 11.986, "z": 19.436, "size": 0.2, "title": "Functors out of sums of categories.", "summary": "This file records the universal property of sums of categories as an equivalence of categories `Sum.functorEquiv : A ⊕ A' ⥤ B ≌ (A ⥤ B) × (A' ⥤ B)`, and characterizes its precompositions with the left and right inclusion as corresponding to the projections on the product side.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sums/Products.html"}, {"id": "Mathlib.CategoryTheory.Sums.Associator", "region_id": "category_theory", "micro_elevation": 0.1579, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 37.181, "z": 16.436, "size": 0.2478, "title": "Associator for binary disjoint union of categories.", "summary": "The associator functor `((C ⊕ D) ⊕ E) ⥤ (C ⊕ (D ⊕ E))` and its inverse form an equivalence.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sums/Associator.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.FullSubcategory", "region_id": "category_theory", "micro_elevation": 0.0877, "macro_tier": 103, "macro_tier_score": 0.4792, "macro_tier_override": null, "x": 30.697, "z": 12.436, "size": 0.5743, "title": "The full subcategory associated to a property of objects", "summary": "Given a category `C` and `P : ObjectProperty C`, we define a category structure on the type `P.FullSubcategory` of objects in `C` satisfying `P`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/FullSubcategory.html"}, {"id": "Mathlib.CategoryTheory.InducedCategory", "region_id": "category_theory", "micro_elevation": 0.0702, "macro_tier": 103, "macro_tier_score": 0.4749, "macro_tier_override": null, "x": 21.744, "z": 11.436, "size": 0.4493, "title": "Induced categories and full subcategories", "summary": "Given a category `D` and a function `F : C → D` from a type `C` to the objects of `D`, there is an essentially unique way to give `C` a category structure such that `F` becomes a fully faithful functor, namely by taking $ Hom_C(X, Y) = Hom_D(FX, FY) $. We call this the category induced from `D` along `F`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/InducedCategory.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.Basic", "region_id": "category_theory", "micro_elevation": 0.0175, "macro_tier": 103, "macro_tier_score": 0.4773, "macro_tier_override": null, "x": 25.795, "z": 8.436, "size": 0.5261, "title": "Properties of objects in a category", "summary": "Given a category `C`, we introduce an abbreviation `ObjectProperty C` for predicates `C → Prop`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/Basic.html"}, {"id": "Mathlib.CategoryTheory.EffectiveEpi.Coproduct", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 70.72, "z": 38.436, "size": 0.2617, "title": "Effective epimorphic families and coproducts", "summary": "This file proves that an effective epimorphic family induces an effective epi from the coproduct if the coproduct exists, and the converse under some more conditions on the coproduct (that it interacts well with pullbacks).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/EffectiveEpi/Coproduct.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Transfer", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0198, "macro_tier_override": null, "x": 63.332, "z": 43.0, "size": 0.3148, "title": "Transferring \"abelian-ness\" across a functor", "summary": "If `C` is an additive category, `D` is an abelian category, we have `F : C ⥤ D` `G : D ⥤ C` (both preserving zero morphisms), `G` is left exact (that is, preserves finite limits), and further we have `adj : G ⊣ F` and `i : F ⋙ G ≅ 𝟭 C`, then `C` is also abelian. A particular example is the transfer of `Abelian` instances from a category `C` to `ShrinkHoms C`; see `ShrinkHoms.abelian`. In this case, we also transfer…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Transfer.html"}, {"id": "Mathlib.CategoryTheory.Groupoid.Grpd.Basic", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 173.93, "z": -80.661, "size": 0.2809, "title": "Category of groupoids", "summary": "This file contains the definition of the category `Grpd` of all groupoids. In this category objects are groupoids and morphisms are functors between these groupoids. We also provide two “forgetting” functors: `objects : Grpd ⥤ Type` and `forgetToCat : Grpd ⥤ Cat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Groupoid/Grpd/Basic.html"}, {"id": "Mathlib.CategoryTheory.Localization.Bifunctor", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 102, "macro_tier_score": 0.0193, "macro_tier_override": null, "x": -13.426, "z": 34.436, "size": 0.2708, "title": "Lifting of bifunctors", "summary": "In this file, in the context of the localization of categories, we extend the notion of lifting of functors to the case of bifunctors. As the localization of categories behaves well with respect to finite products of categories (when the classes of morphisms contain identities), all the definitions for bifunctors `C₁ ⥤ C₂ ⥤ E` are obtained by reducing to the case of functors `(C₁ × C₂) ⥤ E` by using currying and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Bifunctor.html"}, {"id": "Mathlib.CategoryTheory.Localization.Prod", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 102, "macro_tier_score": 0.0196, "macro_tier_override": null, "x": 45.729, "z": 33.436, "size": 0.3026, "title": "Localization of product categories", "summary": "In this file, it is shown that if functors `L₁ : C₁ ⥤ D₁` and `L₂ : C₂ ⥤ D₂` are localization functors for morphisms properties `W₁` and `W₂`, then the product functor `C₁ × C₂ ⥤ D₁ × D₂` is a localization functor for `W₁.prod W₂ : MorphismProperty (C₁ × C₂)`, at least if both `W₁` and `W₂` contain identities. This main result is the instance `Functor.IsLocalization.prod`. The proof proceeds by showing first…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Prod.html"}, {"id": "Mathlib.CategoryTheory.PathCategory.Basic", "region_id": "category_theory", "micro_elevation": 0.3333, "macro_tier": 103, "macro_tier_score": 0.0774, "macro_tier_override": null, "x": 141.644, "z": -88.661, "size": 0.3453, "title": "The category paths on a quiver.", "summary": "When `C` is a quiver, `paths C` is the category of paths.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/PathCategory/Basic.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.CoherentSheaves", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 15.514, "z": 51.436, "size": 0.2818, "title": "Sheaves for the coherent topology", "summary": "This file characterises sheaves for the coherent topology", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.html"}, {"id": "Mathlib.CategoryTheory.Sites.Canonical", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0205, "macro_tier_override": null, "x": -71.589, "z": 234.683, "size": 0.3615, "title": "The canonical topology on a category", "summary": "We define the finest (largest) Grothendieck topology for which a given presheaf `P` is a sheaf. This is well defined since if `P` is a sheaf for a topology `J`, then it is a sheaf for any coarser (smaller) topology. Nonetheless we define the topology explicitly by specifying its sieves: A sieve `S` on `X` is covering for `finestTopologySingle P` iff for any `f : Y ⟶ X`, `P` satisfies the sheaf axiom for `S.pullback…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Canonical.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.Basic", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.011, "macro_tier_override": null, "x": 81.451, "z": 50.436, "size": 0.3604, "title": "The Coherent, Regular and Extensive Grothendieck Topologies", "summary": "This file defines three related Grothendieck topologies on a category `C`. The first one is called the *coherent* topology. For that to exist, the category `C` must satisfy a condition called `Precoherent C`, which is essentially the minimal requirement for the coherent coverage to exist. It means that finite effective epimorphic families can be \"pulled back\". Given such a category, the coherent coverage is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/Basic.html"}, {"id": "Mathlib.CategoryTheory.Sites.EffectiveEpimorphic", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": 191.184, "z": -26.328, "size": 0.2884, "title": "Effective epimorphic sieves", "summary": "We define the notion of effective epimorphic (pre)sieves and provide some API for relating the notion with the notions of effective epimorphism and effective epimorphic family. More precisely, if `f` is a morphism, then `f` is an effective epi if and only if the sieve it generates is effective epimorphic; see `CategoryTheory.Sieve.effectiveEpimorphic_singleton`. The analogous statement for a family of morphisms is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.html"}, {"id": "Mathlib.CategoryTheory.Center.Basic", "region_id": "category_theory", "micro_elevation": 0.3333, "macro_tier": 102, "macro_tier_score": 0.0148, "macro_tier_override": null, "x": 51.357, "z": 26.436, "size": 0.293, "title": "The center of a category", "summary": "Given a category `C`, we introduce an abbreviation `CatCenter C` for the center of the category `C`, which is `End (𝟭 C)`, the type of endomorphisms of the identity functor of `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Center/Basic.html"}, {"id": "Mathlib.CategoryTheory.Shift.CommShift", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 102, "macro_tier_score": 0.0227, "macro_tier_override": null, "x": 27.181, "z": 44.436, "size": 0.458, "title": "Functors which commute with shifts", "summary": "Let `C` and `D` be two categories equipped with shifts by an additive monoid `A`. In this file, we define the notion of functor `F : C ⥤ D` which \"commutes\" with these shifts. The associated type class is `[F.CommShift A]`. The data consists of commutation isomorphisms `F.commShiftIso a : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a` for all `a : A` which satisfy a compatibility with the addition and the zero. After…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/CommShift.html"}, {"id": "Mathlib.CategoryTheory.Sites.EqualizerSheafCondition", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 102, "macro_tier_score": 0.0293, "macro_tier_override": null, "x": 4.272, "z": 46.436, "size": 0.3173, "title": "The equalizer diagram sheaf condition for a presieve", "summary": "In `Mathlib/CategoryTheory/Sites/IsSheafFor.lean` it is defined what it means for a presheaf to be a sheaf *for* a particular presieve. In this file we provide equivalent conditions in terms of equalizer diagrams. * In `Equalizer.Presieve.sheaf_condition`, the sheaf condition at a presieve is shown to be equivalent to that of https://stacks.math.columbia.edu/tag/00VM (and combined with `isSheaf_pretopology`, this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -37.123, "z": 49.436, "size": 0.2, "title": "Functors into a complete monoidal closed category form a monoidal closed category.", "summary": "TODO (in progress by Joël Riou): make a more explicit construction of the internal hom in functor categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/FunctorCategory/Complete.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Lifting.Right", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 57.572, "z": 48.436, "size": 0.2302, "title": "Adjoint lifting", "summary": "This file gives two constructions for building right adjoints: the adjoint triangle theorem and the adjoint lifting theorem. The adjoint triangle theorem concerns a functor `F : B ⥤ A` with a right adjoint `U` such that `η_X : X ⟶ UFX` is a regular mono. Then for any category `C` with equalizers of coreflexive pairs, a functor `L : C ⥤ B` has a right adjoint if (and only if) the composite `L ⋙ F` does. Note that the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Lifting/Right.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Groupoid", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -12.691, "z": 37.436, "size": 0.2302, "title": "Functors from a groupoid into a monoidal closed category form a monoidal closed category.", "summary": "(Using the pointwise monoidal structure on the functor category.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/FunctorCategory/Groupoid.html"}, {"id": "Mathlib.CategoryTheory.Category.GaloisConnection", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -71.923, "z": 214.683, "size": 0.2649, "title": "Galois connections between preorders are adjunctions.", "summary": "* `GaloisConnection.adjunction` is the adjunction associated to a Galois connection.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/GaloisConnection.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.EventuallyConstant", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 102, "macro_tier_score": 0.0248, "macro_tier_override": null, "x": 49.272, "z": 34.0, "size": 0.3324, "title": "Limits of eventually constant functors", "summary": "If `F : J ⥤ C` is a functor from a cofiltered category, and `j : J`, we introduce a property `F.IsEventuallyConstantTo j` which says that for any `f : i ⟶ j`, the induced morphism `F.map f` is an isomorphism. Under this assumption, it is shown that `F` admits `F.obj j` as a limit (`Functor.IsEventuallyConstantTo.isLimitCone`). A typeclass `Cofiltered.IsEventuallyConstant` is also introduced, and the dual results for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/EventuallyConstant.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Lifting.Left", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -29.461, "z": 48.436, "size": 0.2, "title": "Adjoint lifting", "summary": "This file gives two constructions for building left adjoints: the adjoint triangle theorem and the adjoint lifting theorem. The adjoint triangle theorem concerns a functor `U : B ⥤ C` with a left adjoint `F` such that `ε_X : FUX ⟶ X` is a regular epi. Then for any category `A` with coequalizers of reflexive pairs, a functor `R : A ⥤ B` has a left adjoint if (and only if) the composite `R ⋙ U` does. Note that the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Lifting/Left.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.TStructure.Basic", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 101, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -13.97, "z": 47.0, "size": 0.3122, "title": "t-structures on triangulated categories", "summary": "This file introduces the notion of t-structure on (pre)triangulated categories. The first example of t-structure shall be the canonical t-structure on the derived category of an abelian category (TODO). Given a t-structure `t : TStructure C`, we define typeclasses `t.IsLE X n` and `t.IsGE X n` in order to say that an object `X : C` is `≤ n` or `≥ n` for `t`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/TStructure/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.Multicoequalizer", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 101, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 167.421, "z": -70.661, "size": 0.2871, "title": "Multicoequalizers in the category of types", "summary": "Given `J : MultispanShape`, `d : MultispanIndex J (Type u)` and `c : d.multispan.CoconeTypes`, we obtain a lemma `isMulticoequalizer_iff` which gives a criteria for `c` to be a colimit (i.e. a multicoequalizer): it restates in a more explicit manner the injectivity and surjectivity conditions for the map `d.multispan.descColimitType c : d.multispan.ColimitType → c.pt`. We deduce a definition…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/Multicoequalizer.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.Quotient", "region_id": "category_theory", "micro_elevation": 0.3333, "macro_tier": 101, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 121.19, "z": -88.661, "size": 0.2751, "title": "Classes of morphisms induced on quotient categories", "summary": "Let `W : MorphismProperty C` and `homRel : HomRel C`. We assume that `homRel` is stable under pre- and postcomposition. We introduce a property `W.HasQuotient homRel` expressing that `W` induces a property of morphisms on the quotient category, i.e. `W f ↔ W g` when `homRel f g` holds. We denote `W.quotient homRel : MorphismProperty (Quotient homRel)` the induced property of morphisms: a morphism in `C` satisfies…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/Quotient.html"}, {"id": "Mathlib.CategoryTheory.Sites.LocallyInjective", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 102, "macro_tier_score": 0.0219, "macro_tier_override": null, "x": 13.863, "z": 53.436, "size": 0.4266, "title": "Locally injective morphisms of (pre)sheaves", "summary": "Let `C` be a category equipped with a Grothendieck topology `J`, and let `D` be a concrete category. In this file, we introduce the typeclass `Presheaf.IsLocallyInjective J φ` for a morphism `φ : F₁ ⟶ F₂` in the category `Cᵒᵖ ⥤ D`. This means that `φ` is locally injective. More precisely, if `x` and `y` are two elements of some `F₁.obj U` such the images of `x` and `y` in `F₂.obj U` coincide, then the equality `x =…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/LocallyInjective.html"}, {"id": "Mathlib.CategoryTheory.Sites.LeftExact", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 102, "macro_tier_score": 0.0202, "macro_tier_override": null, "x": -250.721, "z": -106.756, "size": 0.3469, "title": "Left exactness of sheafification", "summary": "In this file we show that sheafification commutes with finite limits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/LeftExact.html"}, {"id": "Mathlib.CategoryTheory.Sites.PreservesSheafification", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 102, "macro_tier_score": 0.0204, "macro_tier_override": null, "x": 66.419, "z": 52.436, "size": 0.3535, "title": "Functors which preserve sheafification", "summary": "In this file, given a Grothendieck topology `J` on `C` and `F : A ⥤ B`, we define a type class `J.PreservesSheafification F`. We say that `F` preserves the sheafification if whenever a morphism of presheaves `P₁ ⟶ P₂` induces an isomorphism on the associated sheaves, then the induced map `P₁ ⋙ F ⟶ P₂ ⋙ F` also induces an isomorphism on the associated sheaves. (Note: it suffices to check this property for the map…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/PreservesSheafification.html"}, {"id": "Mathlib.CategoryTheory.Sites.Subsheaf", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 102, "macro_tier_score": 0.0199, "macro_tier_override": null, "x": 69.317, "z": 51.436, "size": 0.3263, "title": "Subsheaf of types", "summary": "We define the subsheaf of a type-valued presheaf.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Subsheaf.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Limits.HasLimits", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 19.507, "z": 33.436, "size": 0.239, "title": "Compatibility lemmas for limits and colimits in a monoidal category", "summary": "For numerous simp lemmas of the form `f ≫ g = h`, we add accompanying simp lemmas of the form `Q ◁ f ≫ Q ◁ g = Q ◁ h` and `f ▷ Q ≫ g ▷ Q = h ▷ Q`. This file and `Mathlib.CategoryTheory.Monoidal.Limits.Shapes.Pullback` are needed to define a monoidal category structure in `Mathlib.CategoryTheory.Monoidal.Arrow`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Limits/HasLimits.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 103, "macro_tier_score": 0.2156, "macro_tier_override": null, "x": 41.196, "z": 35.0, "size": 0.6426, "title": "Preservation of zero objects and zero morphisms", "summary": "We define the class `PreservesZeroMorphisms` and show basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Zero.html"}, {"id": "Mathlib.CategoryTheory.Localization.CalculusOfFractions", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 102, "macro_tier_score": 0.015, "macro_tier_override": null, "x": 9.487, "z": 35.436, "size": 0.3125, "title": "Calculus of fractions", "summary": "Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967], given a morphism property `W : MorphismProperty C` on a category `C`, we introduce the class `W.HasLeftCalculusOfFractions`. The main result `Localization.exists_leftFraction` is that if `L : C ⥤ D` is a localization functor for `W`, then for any morphism `L.obj X ⟶ L.obj Y` in `D`, there exists an auxiliary object `Y' : C` and morphisms `g : X…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/CalculusOfFractions.html"}, {"id": "Mathlib.CategoryTheory.Localization.Opposite", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.0458, "macro_tier_override": null, "x": 44.317, "z": 34.436, "size": 0.4284, "title": "Localization of the opposite category", "summary": "If a functor `L : C ⥤ D` is a localization functor for `W : MorphismProperty C`, it is shown in this file that `L.op : Cᵒᵖ ⥤ Dᵒᵖ` is also a localization functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Opposite.html"}, {"id": "Mathlib.CategoryTheory.Limits.Filtered", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 103, "macro_tier_score": 0.044, "macro_tier_override": null, "x": -49.696, "z": 49.342, "size": 0.343, "title": "Filtered categories and limits", "summary": "In this file, we show that `C` is filtered if and only if for every functor `F : J ⥤ C` from a finite category there is some `X : C` such that `lim Hom(F·, X)` is nonempty. Furthermore, we define the type classes `HasCofilteredLimitsOfSize` and `HasFilteredColimitsOfSize`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Filtered.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.FunctorToTypes", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 101, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 81.935, "z": -60.661, "size": 0.2906, "title": "Functors to Type are closed.", "summary": "Show that `C ⥤ Type max w v u` is monoidal closed for `C` a category in `Type u` with morphisms in `Type v`, and `w` an arbitrary universe.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/FunctorToTypes.html"}, {"id": "Mathlib.CategoryTheory.Functor.FunctorHom", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 29.244, "z": 53.436, "size": 0.2757, "title": "Internal hom in functor categories", "summary": "Given functors `F G : C ⥤ D`, define a functor `functorHom F G` from `C` to `Type max v' v u`, which is a proxy for the \"internal hom\" functor Hom(F ⊗ coyoneda(-), G). This is used to show that the functor category `C ⥤ D` is enriched over `C ⥤ Type max v' v u`. This is also useful for showing that `C ⥤ Type max w v u` is monoidal closed. See `Mathlib/CategoryTheory/Closed/FunctorToTypes.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/FunctorHom.html"}, {"id": "Mathlib.CategoryTheory.Limits.Final.Type", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -16.564, "z": 35.0, "size": 0.239, "title": "Action of an initial functor on sections", "summary": "Given `F : C ⥤ D` and `P : D ⥤ Type w`, we define a map `sectionsPrecomp F : P.sections → (F ⋙ P).sections` and show that it is a bijection when `F` is initial. As `Functor.sections` identify to limits of functors to types (at least under suitable universe assumptions), this could be deduced from general results about limits and initial functors, but we provide a more down to earth proof. We also obtain the dual…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Final/Type.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Projective.Preserves", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 102, "macro_tier_score": 0.0149, "macro_tier_override": null, "x": 58.705, "z": 40.0, "size": 0.3027, "title": "Preservation of projective objects", "summary": "We define a typeclass `Functor.PreservesProjectiveObjects`. We restate the existing result that if `F ⊣ G` is an adjunction and `G` preserves monomorphisms, then `F` preserves projective objects. We show that the converse is true if the domain of `F` has enough projectives.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Projective/Preserves.html"}, {"id": "Mathlib.CategoryTheory.Idempotents.FunctorExtension", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0009, "macro_tier_override": null, "x": 140.64, "z": -9.83, "size": 0.324, "title": "Extension of functors to the idempotent completion", "summary": "In this file, we construct an extension `functorExtension₁` of functors `C ⥤ Karoubi D` to functors `Karoubi C ⥤ Karoubi D`. This results in an equivalence `karoubiUniversal₁ C D : (C ⥤ Karoubi D) ≌ (Karoubi C ⥤ Karoubi D)`. We also construct an extension `functorExtension₂` of functors `(C ⥤ D) ⥤ (Karoubi C ⥤ Karoubi D)`. Moreover, when `D` is idempotent complete, we get equivalences `karoubiUniversal₂ C D : C ⥤ D…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Idempotents/FunctorExtension.html"}, {"id": "Mathlib.CategoryTheory.Subfunctor.Finite", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 77.344, "z": 47.436, "size": 0.2, "title": "Subpresheaves that are generated by finitely many sections", "summary": "Given `F : Cᵒᵖ ⥤ Type w`, `G : Subfunctor F`, objects `X : ι → Cᵒᵖ` and sections `x : (i : ι) → F.obj (X i)`, we define a predicate `G.IsGeneratedBy x` saying that `G` is the subpresheaf generated by the sections `x i`. If this holds for a finite index type `ι`, we say that `G` is \"finite\", and this gives a type class `G.IsFinite`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subfunctor/Finite.html"}, {"id": "Mathlib.CategoryTheory.Products.Bifunctor", "region_id": "category_theory", "micro_elevation": 0.193, "macro_tier": 103, "macro_tier_score": 0.0765, "macro_tier_override": null, "x": 9.891, "z": 18.436, "size": 0.2778, "title": "Lemmas about functors out of product categories.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Products/Bifunctor.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Opposite.OpOp", "region_id": "category_theory", "micro_elevation": 0.9649, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -34.837, "z": 62.436, "size": 0.2, "title": "The triangulated equivalence `Cᵒᵖᵒᵖ ≌ C`.", "summary": "In this file, we show that if `C` is a pretriangulated category, then the functors `opOp C : C ⥤ Cᵒᵖᵒᵖ` and `unopUnop C : Cᵒᵖᵒᵖ ⥤ C` are triangulated. We also show that the unit and counit isomorphisms of the equivalence `opOpEquivalence C : Cᵒᵖᵒᵖ ≌ C` are compatible with shifts, which is summarized by the property `(opOpEquivalence C).IsTriangulated`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Opposite/OpOp.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Adjunction", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -21.889, "z": 61.436, "size": 0.2676, "title": "The adjoint functor is triangulated", "summary": "If a functor `F : C ⥤ D` between pretriangulated categories is triangulated, and if we have an adjunction `F ⊣ G`, then `G` is also a triangulated functor. We deduce the symmetric statement (if `G` is a triangulated functor, then so is `F`) using opposite categories. We then introduce a class `IsTriangulated` for adjunctions: an adjunction `F ⊣ G` is called triangulated if both `F` and `G` are triangulated, and if…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Adjunction.html"}, {"id": "Mathlib.CategoryTheory.Iso", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 103, "macro_tier_score": 0.518, "macro_tier_override": null, "x": -108.797, "z": -168.102, "size": 0.7685, "title": "Isomorphisms", "summary": "This file defines isomorphisms between objects of a category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Iso.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.DisjointCoproduct", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 103, "macro_tier_score": 0.0387, "macro_tier_override": null, "x": 24.956, "z": 45.436, "size": 0.3015, "title": "Disjoint coproducts", "summary": "Defines disjoint coproducts: coproducts where the intersection is initial and the coprojections are monic. Shows that a category with disjoint coproducts is `InitialMonoClass`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/DisjointCoproduct.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Subobject", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 45.752, "z": 56.436, "size": 0.2541, "title": "Subobjects in Grothendieck abelian categories", "summary": "We study the complete lattice of subobjects of `X : C` when `C` is a Grothendieck abelian category. In particular, for a functor `F : J ⥤ MonoOver X` from a filtered category, we relate the colimit of `F` (computed in `C`) and the supremum of the subobjects corresponding to the objects in the image of `F`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckCategory/Subobject.html"}, {"id": "Mathlib.CategoryTheory.Presentable.IsCardinalFiltered", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 102, "macro_tier_score": 0.0199, "macro_tier_override": null, "x": 55.811, "z": 55.436, "size": 0.3265, "title": "κ-filtered category", "summary": "If `κ` is a regular cardinal, we introduce the notion of `κ`-filtered category `J`: it means that any functor `A ⥤ J` from a small category such that `Arrow A` is of cardinality `< κ` admits a cocone. This generalizes the notion of filtered category. Indeed, we obtain the equivalence `IsCardinalFiltered J ℵ₀ ↔ IsFiltered J`. The API is mostly parallel to that of filtered categories. A preordered type `J` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/IsCardinalFiltered.html"}, {"id": "Mathlib.CategoryTheory.Localization.Preadditive", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -33.902, "z": 46.436, "size": 0.2, "title": "Localization of natural transformations to preadditive categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Preadditive.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.FunctorCategory", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 103, "macro_tier_score": 0.0994, "macro_tier_override": null, "x": 21.707, "z": 45.436, "size": 0.4733, "title": "Preadditive structure on functor categories", "summary": "If `C` and `D` are categories and `D` is preadditive, then `C ⥤ D` is also preadditive.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/FunctorCategory.html"}, {"id": "Mathlib.CategoryTheory.Action.Continuous", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 0.074, "z": 37.436, "size": 0.239, "title": "Topological subcategories of `Action V G`", "summary": "For a concrete category `V`, where the forgetful functor factors via `TopCat`, and a monoid `G`, equipped with a topological space instance, we define the full subcategory `ContAction V G` of all objects of `Action V G` where the induced action is continuous. We also define a category `DiscreteContAction V G` as the full subcategory of `ContAction V G`, where the underlying topological space is discrete. Finally we…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Action/Continuous.html"}, {"id": "Mathlib.CategoryTheory.Action.Basic", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 102, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": -41.792, "z": 29.0, "size": 0.2872, "title": "`Action V G`, the category of actions of a monoid `G` inside some category `V`.", "summary": "The prototypical example is `V = ModuleCat R`, where `Action (ModuleCat R) G` is the category of `R`-linear representations of `G`. We check `Action V G ≌ (CategoryTheory.SingleObj G ⥤ V)`, and construct the restriction functors `res {G H} [Monoid G] [Monoid H] (f : G →* H) : Action V H ⥤ Action V G`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Action/Basic.html"}, {"id": "Mathlib.CategoryTheory.Localization.Equivalence", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.0457, "macro_tier_override": null, "x": 29.98, "z": 33.436, "size": 0.4244, "title": "Localization functors are preserved through equivalences", "summary": "In `Mathlib/CategoryTheory/Localization/Predicate.lean`, the lemma `Localization.of_equivalence_target` already showed that the predicate of localized categories is unchanged when we replace the target category (i.e. the candidate localized category) by an equivalent category. In this file, we show the same for the source category (`Localization.of_equivalence_source`). More generally, `Localization.of_equivalences`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Equivalence.html"}, {"id": "Mathlib.CategoryTheory.CatCommSq", "region_id": "category_theory", "micro_elevation": 0.1404, "macro_tier": 103, "macro_tier_score": 0.0598, "macro_tier_override": null, "x": 36.479, "z": 15.436, "size": 0.4195, "title": "2-commutative squares of functors", "summary": "Similarly to `Mathlib/CategoryTheory/CommSq.lean`, which defines the notion of commutative squares, this file introduces the notion of 2-commutative squares of functors. If `T : C₁ ⥤ C₂`, `L : C₁ ⥤ C₃`, `R : C₂ ⥤ C₄`, `B : C₃ ⥤ C₄` are functors, then `[CatCommSq T L R B]` contains the datum of an isomorphism `T ⋙ R ≅ L ⋙ B`. Future work: using this notion in the development of the localization of categories (e.g.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/CatCommSq.html"}, {"id": "Mathlib.CategoryTheory.EffectiveEpi.Preserves", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 102, "macro_tier_score": 0.0101, "macro_tier_override": null, "x": 158.024, "z": -30.328, "size": 0.3001, "title": "Functors preserving effective epimorphisms", "summary": "This file concerns functors which preserve and/or reflect effective epimorphisms and effective epimorphic families.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/EffectiveEpi/Preserves.html"}, {"id": "Mathlib.CategoryTheory.EffectiveEpi.Comp", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 102, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": 41.015, "z": 35.436, "size": 0.2796, "title": "Composition of effective epimorphisms", "summary": "This file provides `EffectiveEpi` instances for certain compositions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/EffectiveEpi/Comp.html"}, {"id": "Mathlib.CategoryTheory.ConcreteCategory.ReflectsIso", "region_id": "category_theory", "micro_elevation": 0.3684, "macro_tier": 101, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -24.681, "z": 21.0, "size": 0.2961, "title": null, "summary": "A `forget₂ C D` forgetful functor between concrete categories `C` and `D` whose forgetful functors both reflect isomorphisms, itself reflects isomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ConcreteCategory/ReflectsIso.html"}, {"id": "Mathlib.CategoryTheory.HomCongr", "region_id": "category_theory", "micro_elevation": 0.0175, "macro_tier": 103, "macro_tier_score": 0.204, "macro_tier_override": null, "x": 22.945, "z": 8.436, "size": 0.4676, "title": "Conjugate morphisms by isomorphisms", "summary": "We define `CategoryTheory.Iso.homCongr : (X ≅ X₁) → (Y ≅ Y₁) → (X ⟶ Y) ≃ (X₁ ⟶ Y₁)`, cf. `Equiv.arrowCongr`, and `CategoryTheory.Iso.isoCongr : (f : X₁ ≅ X₂) → (g : Y₁ ≅ Y₂) → (X₁ ≅ Y₁) ≃ (X₂ ≅ Y₂)`. As corollaries, an isomorphism `α : X ≅ Y` defines - a monoid isomorphism `CategoryTheory.Iso.conj : End X ≃* End Y` by `α.conj f = α.inv ≫ f ≫ α.hom`; - a group isomorphism `CategoryTheory.Iso.conjAut : Aut X ≃* Aut Y`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/HomCongr.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Square", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 24.093, "z": 49.436, "size": 0.2567, "title": "Preservation of pullback/pushout squares", "summary": "If a functor `F : C ⥤ D` preserves suitable cospans (resp. spans), and `sq : Square C` is a pullback square (resp. a pushout square) then so is the square `sq.map F`. The lemma `Square.isPullback_iff_map_coyoneda_isPullback` also shows that a square is a pullback square iff it is so after the application of the functor `coyoneda.obj X` for all `X : Cᵒᵖ`. Similarly, a square is a pushout square iff the opposite…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Square.html"}, {"id": "Mathlib.CategoryTheory.Sites.Abelian", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 102, "macro_tier_score": 0.0101, "macro_tier_override": null, "x": -177.045, "z": -98.756, "size": 0.299, "title": "Category of sheaves is abelian", "summary": "Let `C, D` be categories and `J` be a Grothendieck topology on `C`, when `D` is abelian and sheafification is possible in `C`, `Sheaf J D` is abelian as well (`sheafIsAbelian`). Hence, `presheafToSheaf` is an additive functor (`presheafToSheaf_additive`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Abelian.html"}, {"id": "Mathlib.CategoryTheory.Sites.Adjunction", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -131.04, "z": -104.756, "size": 0.2812, "title": null, "summary": "In this file, we show that an adjunction `G ⊣ F` induces an adjunction between categories of sheaves. We also show that `G` preserves sheafification.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Adjunction.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax", "region_id": "category_theory", "micro_elevation": 0.2105, "macro_tier": 102, "macro_tier_score": 0.0108, "macro_tier_override": null, "x": 36.099, "z": 19.436, "size": 0.3487, "title": "Transformations between oplax functors", "summary": "Just as there are natural transformations between functors, there are transformations between oplax functors. The equality in the naturality condition of a natural transformation gets replaced by a specified 2-morphism. Now, there are three possible types of transformations (between oplax functors): * oplax natural transformations; * lax natural transformations; * strong natural transformations. These differ in the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Oplax.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -161.363, "z": 237.683, "size": 0.2874, "title": "Reflecting the property of being preregular", "summary": "We prove that given a fully faithful functor `F : C ⥤ D`, with `Preregular D`, such that for every object `X` of `D` there exists an object `W` of `C` with an effective epi `π : F.obj W ⟶ X`, the category `C` is `Preregular`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPreregular.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.Multiequalizer", "region_id": "category_theory", "micro_elevation": 0.5789, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 67.046, "z": 40.436, "size": 0.2673, "title": "Multiequalizers in Type", "summary": "Given `J : MulticospanShape` and `I : MulticospanIndex J (Type u)`, we define a type `I.sections`. When `c : Multifork I`, we show that `c` is a limit iff the canonical map `c.toSections : c.pt → I.sections` is a bijection.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/Multiequalizer.html"}, {"id": "Mathlib.CategoryTheory.SmallObject.TransfiniteIteration", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 102, "macro_tier_score": 0.0144, "macro_tier_override": null, "x": 48.697, "z": 59.436, "size": 0.2502, "title": "The transfinite iteration of a successor structure", "summary": "Given a successor structure `Φ : SuccStruct C` (see the file `Mathlib/CategoryTheory/SmallObject/Iteration/Basic.lean`) and a well-ordered type `J`, we define the iteration `Φ.iteration J : C`. It is defined as the colimit of a functor `Φ.iterationFunctor J : J ⥤ C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/SmallObject/TransfiniteIteration.html"}, {"id": "Mathlib.CategoryTheory.SmallObject.Iteration.Nonempty", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 102, "macro_tier_score": 0.0145, "macro_tier_override": null, "x": -22.547, "z": 48.436, "size": 0.2578, "title": "Existence of the iteration of a successor structure", "summary": "Given `Φ : SuccStruct C`, we show by transfinite induction that for any element `j` in a well-ordered set `J`, the type `Φ.Iteration j` is nonempty.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/SmallObject/Iteration/Nonempty.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.TransfiniteComposition", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 102, "macro_tier_score": 0.0248, "macro_tier_override": null, "x": 225.712, "z": -56.661, "size": 0.3315, "title": "Classes of morphisms that are stable under transfinite composition", "summary": "Given a well-ordered type `J`, `W : MorphismProperty C` and a morphism `f : X ⟶ Y`, we define a structure `W.TransfiniteCompositionOfShape J f` which expresses that `f` is a transfinite composition of shape `J` of morphisms in `W`. This structures extends `CategoryTheory.TransfiniteCompositionOfShape` which was defined in the file `CategoryTheory.Limits.Shape.Preorder.TransfiniteCompositionOfShape`. We use this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/TransfiniteComposition.html"}, {"id": "Mathlib.CategoryTheory.Idempotents.SimplicialObject", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 188.407, "z": -62.661, "size": 0.253, "title": "Idempotent completeness of categories of simplicial objects", "summary": "In this file, we provide an instance expressing that `SimplicialObject C` and `CosimplicialObject C` are idempotent complete categories when the category `C` is.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Idempotents/SimplicialObject.html"}, {"id": "Mathlib.CategoryTheory.Idempotents.FunctorCategories", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 204.362, "z": -63.661, "size": 0.2828, "title": "Idempotent completeness and functor categories", "summary": "In this file we define an instance `functor_category_isIdempotentComplete` expressing that a functor category `J ⥤ C` is idempotent complete when the target category `C` is. We also provide a fully faithful functor `karoubiFunctorCategoryEmbedding : Karoubi (J ⥤ C) ⥤ (J ⥤ Karoubi C)` for all categories `J` and `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Idempotents/FunctorCategories.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Internal.Types.Basic", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -16.415, "z": 51.436, "size": 0.26, "title": "`Mon Type u ≌ MonCat.{u}`", "summary": "The category of internal monoid objects in `Type` is equivalent to the category of \"native\" bundled monoids. Moreover, this equivalence is compatible with the forgetful functors to `Type`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Internal/Types/Basic.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.CommMon_", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 102, "macro_tier_score": 0.0103, "macro_tier_override": null, "x": -8.474, "z": 36.436, "size": 0.3173, "title": "The category of commutative monoids in a braided monoidal category.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/CommMon_.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.SheafComparison", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 102, "macro_tier_score": 0.0056, "macro_tier_override": null, "x": -125.24, "z": -98.756, "size": 0.3251, "title": "Categories of coherent sheaves", "summary": "Given a fully faithful functor `F : C ⥤ D` into a precoherent category, which preserves and reflects finite effective epi families, and satisfies the property `F.EffectivelyEnough` (meaning that to every object in `C` there is an effective epi from an object in the image of `F`), the categories of coherent sheaves on `C` and `D` are equivalent (see `CategoryTheory.coherentTopology.equivalence`). The main application…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.html"}, {"id": "Mathlib.CategoryTheory.Localization.HomEquiv", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 102, "macro_tier_score": 0.0102, "macro_tier_override": null, "x": -2.106, "z": 36.436, "size": 0.3113, "title": "Bijections between morphisms in two localized categories", "summary": "Given two localization functors `L₁ : C ⥤ D₁` and `L₂ : C ⥤ D₂` for the same class of morphisms `W : MorphismProperty C`, we define a bijection `Localization.homEquiv W L₁ L₂ : (L₁.obj X ⟶ L₁.obj Y) ≃ (L₂.obj X ⟶ L₂.obj Y)` between the types of morphisms in the two localized categories. More generally, given a localizer morphism `Φ : LocalizerMorphism W₁ W₂`, we define a map `Φ.homMap L₁ L₂ : (L₁.obj X ⟶ L₁.obj Y) ⟶…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/HomEquiv.html"}, {"id": "Mathlib.CategoryTheory.Localization.LocalizerMorphism", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 102, "macro_tier_score": 0.0209, "macro_tier_override": null, "x": -17.098, "z": 35.436, "size": 0.3832, "title": "Morphisms of localizers", "summary": "A morphism of localizers consists of a functor `F : C₁ ⥤ C₂` between two categories equipped with morphism properties `W₁` and `W₂` such that `F` sends morphisms in `W₁` to morphisms in `W₂`. If `Φ : LocalizerMorphism W₁ W₂`, and that `L₁ : C₁ ⥤ D₁` and `L₂ : C₂ ⥤ D₂` are localization functors for `W₁` and `W₂`, the induced functor `D₁ ⥤ D₂` is denoted `Φ.localizedFunctor L₁ L₂`; we introduce the condition…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/LocalizerMorphism.html"}, {"id": "Mathlib.CategoryTheory.Filtered.FinallySmall", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": 54.636, "z": 49.0, "size": 0.2859, "title": "Finally small filtered categories", "summary": "In this file, we show that if `C` is a filtered finally small category that is locally small, there exists a final functor `D ⥤ C` from a small filtered category. The dual result is also obtained.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Filtered/FinallySmall.html"}, {"id": "Mathlib.CategoryTheory.Limits.FinallySmall", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 102, "macro_tier_score": 0.0291, "macro_tier_override": null, "x": -34.647, "z": 55.436, "size": 0.2975, "title": "Finally small categories", "summary": "A category given by `(J : Type u) [Category.{v} J]` is `w`-finally small if there exists a `FinalModel J : Type w` equipped with `[SmallCategory (FinalModel J)]` and a final functor `FinalModel J ⥤ J`. This means that if a category `C` has colimits of size `w` and `J` is `w`-finally small, then `C` has colimits of shape `J`. In this way, the notion of \"finally small\" can be seen as a generalization of the notion of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FinallySmall.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Preorder.PrincipalSeg", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 102, "macro_tier_score": 0.0291, "macro_tier_override": null, "x": 56.053, "z": 30.436, "size": 0.2982, "title": "Cocones associated to principal segments", "summary": "If `f : α yoneda.obj X | | fst g | | v v F ------------ f --------------> G ``` In this file, we define a notion of relative…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/Representable.html"}, {"id": "Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Products", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 102, "macro_tier_score": 0.0287, "macro_tier_override": null, "x": 41.201, "z": 36.436, "size": 0.2457, "title": "(Co)products in functor categories", "summary": "Given `f : α → D ⥤ C`, we prove the isomorphisms `(∏ᶜ f).obj d ≅ ∏ᶜ (fun s => (f s).obj d)` and `(∐ f).obj d ≅ ∐ (fun s => (f s).obj d)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FunctorCategory/Shapes/Products.html"}, {"id": "Mathlib.CategoryTheory.Limits.Over", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 103, "macro_tier_score": 0.0816, "macro_tier_override": null, "x": -56.808, "z": 38.0, "size": 0.3114, "title": "Limits and colimits in the over and under categories", "summary": "Show that the forgetful functor `forget X : Over X ⥤ C` creates colimits, and hence `Over X` has any colimits that `C` has (as well as the dual that `forget X : Under X ⟶ C` creates limits). Note that the folder `CategoryTheory.Limits.Shapes.Constructions.Over` further shows that `forget X : Over X ⥤ C` creates connected limits (so `Over X` has connected limits), and that `Over X` has `J`-indexed products if `C` has…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Over.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 102, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": 52.996, "z": 43.436, "size": 0.2837, "title": "Equalizers as pullbacks of products", "summary": "Also see `CategoryTheory.Limits.Constructions.Equalizers` for very similar results.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Equalizer.html"}, {"id": "Mathlib.CategoryTheory.Limits.Skeleton", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 103, "macro_tier_score": 0.0817, "macro_tier_override": null, "x": 25.972, "z": 36.436, "size": 0.3172, "title": "(Co)limits of the skeleton of a category", "summary": "The skeleton of a category inherits all (co)limits the category has.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Skeleton.html"}, {"id": "Mathlib.CategoryTheory.LocallyDirected", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 170.369, "z": -40.328, "size": 0.2652, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/LocallyDirected.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.Local", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 102, "macro_tier_score": 0.01, "macro_tier_override": null, "x": 163.005, "z": -26.328, "size": 0.2897, "title": "Locality conditions on morphism properties", "summary": "In this file we define locality conditions on morphism properties in a category. Let `K` be a precoverage in a category `C` and `P` be a morphism property on `C` that respects isomorphisms. We say that - `P` is local at the target if for every `f : X ⟶ Y`, `P` holds for `f` if and only if it holds for the restrictions of `f` to `Uᵢ` for a `K`-cover `{Uᵢ}` of `Y`. - `P` is local at the source if for every `f : X ⟶…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/Local.html"}, {"id": "Mathlib.CategoryTheory.MarkovCategory.Positive", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -143.291, "z": 151.199, "size": 0.239, "title": "Positive Categories", "summary": "Markov categories where deletion is natural for all morphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MarkovCategory/Positive.html"}, {"id": "Mathlib.CategoryTheory.CopyDiscardCategory.Widesubcategory", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -235.477, "z": 151.199, "size": 0.239, "title": "Copy-discard structures on wide subcategories", "summary": "Given a monoidal category `C`, a morphism property `P : MorphismProperty C` satisfying `P.IsMonoidalStable` and a comonoid object `c : C`, we introduce a condition `P. IsStableUnderComonoid c` saying that `c` inherits a comonoid object structure in the category of `WideSubcategory P`. If `C` is a copy-discard category, if `P` is also stable under braiding and that this condition `P. IsStableUnderComonoid` holds for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/CopyDiscardCategory/Widesubcategory.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.LocallyDiscrete", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 102, "macro_tier_score": 0.0152, "macro_tier_override": null, "x": -1.032, "z": 24.436, "size": 0.325, "title": "Locally discrete bicategories", "summary": "A category `C` can be promoted to a strict bicategory `LocallyDiscrete C`. The objects and the 1-morphisms in `LocallyDiscrete C` are the same as the objects and the morphisms, respectively, in `C`, and the 2-morphisms in `LocallyDiscrete C` are the equalities between 1-morphisms. In other words, the category consisting of the 1-morphisms between each pair of objects `X` and `Y` in `LocallyDiscrete C` is defined as…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/LocallyDiscrete.html"}, {"id": "Mathlib.CategoryTheory.Category.Bipointed", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -84.754, "z": 39.342, "size": 0.2769, "title": "The category of bipointed types", "summary": "This defines `Bipointed`, the category of bipointed types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Bipointed.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Opposites.Kernels", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 101, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": 46.622, "z": 37.0, "size": 0.3197, "title": "Kernels and cokernels in `C` and `Cᵒᵖ`", "summary": "We construct kernels and cokernels in the opposite categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Opposites/Kernels.html"}, {"id": "Mathlib.CategoryTheory.Comma.StructuredArrow.Functor", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.0867, "macro_tier_override": null, "x": 65.477, "z": 35.436, "size": 0.3298, "title": "Structured Arrow Categories as strict functor to Cat", "summary": "Forming a structured arrow category `StructuredArrow d T` with `d : D` and `T : C ⥤ D` is strictly functorial in `S`, inducing a functor `Dᵒᵖ ⥤ Cat`. This file constructs said functor and proves that, in the dual case, we can precompose it with another functor `L : E ⥤ D` to obtain a category equivalent to `Comma L T`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/StructuredArrow/Functor.html"}, {"id": "Mathlib.CategoryTheory.Grothendieck", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.1878, "macro_tier_override": null, "x": -16.115, "z": 34.436, "size": 0.3945, "title": "The Grothendieck construction", "summary": "Given a functor `F : C ⥤ Cat`, the objects of `Grothendieck F` consist of dependent pairs `(b, f)`, where `b : C` and `f : F.obj b`, and a morphism `(b, f) ⟶ (b', f')` is a pair `β : b ⟶ b'` in `C`, and `φ : (F.map β).toFunctor.obj f ⟶ f'` `Grothendieck.functor` makes the Grothendieck construction into a functor from the functor category `C ⥤ Cat` to the over category `Over C` in the category of categories.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Grothendieck.html"}, {"id": "Mathlib.CategoryTheory.Limits.Connected", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 103, "macro_tier_score": 0.173, "macro_tier_override": null, "x": -16.848, "z": 39.436, "size": 0.3698, "title": "Connected limits", "summary": "A connected limit is a limit whose shape is a connected category. We show that constant functors from a connected category have a limit and a colimit. From this we deduce that a cocone `c` over a connected diagram is a colimit cocone if and only if `colimMap c.ι` is an isomorphism (where `c.ι : F ⟶ const c.pt` is the natural transformation that defines the cocone). We give examples of connected categories, and prove…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Connected.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.SequentialLimit", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -230.862, "z": -96.756, "size": 0.2499, "title": "Limits of epimorphisms in coherent topoi", "summary": "This file proves that a sequential limit of epimorphisms is epimorphic in the category of sheaves for the coherent topology on a preregular finitary extensive category where sequential limits of effective epimorphisms are effective epimorphisms. In other words, given epimorphisms of sheaves `⋯ ⟶ Xₙ₊₁ ⟶ Xₙ ⟶ ⋯ ⟶ X₀`, the projection map `lim Xₙ ⟶ X₀` is an epimorphism (see `coherentTopology.epi_π_app_zero_of_epi`).…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/SequentialLimit.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.LocallySurjective", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -178.258, "z": -97.756, "size": 0.2552, "title": "Locally surjective morphisms of coherent sheaves", "summary": "This file characterises locally surjective morphisms of presheaves for the coherent, regular and extensive topologies.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.html"}, {"id": "Mathlib.CategoryTheory.Sites.Subcanonical", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 281.552, "z": -20.328, "size": 0.3072, "title": "Subcanonical Grothendieck topologies", "summary": "This file provides some API for the Yoneda embedding into the category of sheaves for a subcanonical Grothendieck topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Subcanonical.html"}, {"id": "Mathlib.CategoryTheory.Abelian.CommSq", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -30.489, "z": 48.436, "size": 0.2532, "title": "The exact sequence attached to a pushout square", "summary": "Consider a pushout square in an abelian category: ``` t X₁ ⟶ X₂ l| |r v v X₃ ⟶ X₄ b ``` We study the associated exact sequence `X₁ ⟶ X₂ ⊞ X₃ ⟶ X₄ ⟶ 0`. We also show that the induced morphism `kernel t ⟶ kernel b` is an epimorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/CommSq.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.Square", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 102, "macro_tier_score": 0.0148, "macro_tier_override": null, "x": 57.955, "z": 41.0, "size": 0.3001, "title": "Commutative squares that are pushout or pullback squares", "summary": "In this file, we translate the `IsPushout` and `IsPullback` API for the objects of the category `Square C` of commutative squares in a category `C`. We also obtain lemmas which state in this language that a pullback of a monomorphism is a monomorphism (and similarly for pushouts of epimorphisms).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Square.html"}, {"id": "Mathlib.CategoryTheory.Square", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 102, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": 42.666, "z": 24.436, "size": 0.2807, "title": "The category of commutative squares", "summary": "In this file, we define a bundled version of `CommSq` which allows to consider commutative squares as objects in a category `Square C`. The four objects in a commutative square are numbered as follows: ``` X₁ --> X₂ | | v v X₃ --> X₄ ``` We define the flip functor, and two equivalences with the category `Arrow (Arrow C)`, depending on whether we consider a commutative square as a horizontal morphism between two…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Square.html"}, {"id": "Mathlib.CategoryTheory.Limits.WeakLimits.WeakPullbacks", "region_id": "category_theory", "micro_elevation": 0.5789, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 67.13, "z": 40.436, "size": 0.2, "title": "Weak pullbacks", "summary": "These are weak limits for diagrams of shape `WalkingCospan`. If a category has binary products and weak equalizers, then it has weak pullbacks (see `hasWeakPullbacks_of_hasBinaryProducts_of_hasWeakEqualizers`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/WeakLimits/WeakPullbacks.html"}, {"id": "Mathlib.CategoryTheory.Limits.WeakLimits.WeakEqualizers", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 102, "macro_tier_score": 0.01, "macro_tier_override": null, "x": 3.269, "z": 39.436, "size": 0.2884, "title": "Weak equalizers", "summary": "These are weak limits for diagrams of shape `WalkingParallelPair`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/WeakLimits/WeakEqualizers.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Functor.StrictlyUnitary", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 36.29, "z": 20.436, "size": 0.3108, "title": "Strictly unitary lax functors and pseudofunctors", "summary": "In this file, we define strictly unitary lax functors and strictly unitary pseudofunctors between bicategories. A lax functor `F` is said to be *strictly unitary* (sometimes, it is also called *normal*) if there is an equality `F.map (𝟙 X) = 𝟙 (F.obj X)` and the unit 2-morphism `𝟙 (F.obj X) ⟶ F.map (𝟙 X)` is the identity 2-morphism induced by this equality. A pseudofunctor is called *strictly unitary* (or a *normal…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Functor/StrictlyUnitary.html"}, {"id": "Mathlib.CategoryTheory.WithTerminal.FinCategory", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 19.785, "z": 35.436, "size": 0.239, "title": "`WithTerminal C` and `WithInitial C` are finite whenever `C` is", "summary": "If `C` has finitely many objects, then so do `WithTerminal C` and `WithInitial C`, and likewise if `C` has finitely many morphisms as well.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/WithTerminal/FinCategory.html"}, {"id": "Mathlib.CategoryTheory.WithTerminal.Basic", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.1055, "macro_tier_override": null, "x": 108.373, "z": -80.661, "size": 0.3176, "title": "`WithInitial` and `WithTerminal`", "summary": "Given a category `C`, this file constructs two objects: 1. `WithTerminal C`, the category built from `C` by formally adjoining a terminal object. 2. `WithInitial C`, the category built from `C` by formally adjoining an initial object. The terminal resp. initial object is `WithTerminal.star` resp. `WithInitial.star`, and the proofs that these are terminal resp. initial are in `WithTerminal.star_terminal` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/WithTerminal/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Pullbacks", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 102, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": 4.471, "z": 38.436, "size": 0.2739, "title": "Pullbacks in functor categories", "summary": "We prove the isomorphism `(pullback f g).obj d ≅ pullback (f.app d) (g.app d)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FunctorCategory/Shapes/Pullbacks.html"}, {"id": "Mathlib.CategoryTheory.Galois.Equivalence", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 9.243, "z": 56.436, "size": 0.2, "title": "Fiber functors induce an equivalence of categories", "summary": "Let `C` be a Galois category with fiber functor `F`. In this file we conclude that the induced functor from `C` to the category of finite, discrete `Aut F`-sets is an equivalence of categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Galois/Equivalence.html"}, {"id": "Mathlib.CategoryTheory.Galois.EssSurj", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -32.428, "z": 55.436, "size": 0.239, "title": "Essential surjectivity of fiber functors", "summary": "Let `F : C ⥤ FintypeCat` be a fiber functor of a Galois category `C` and denote by `H` the induced functor `C ⥤ Action FintypeCat (Aut F)`. In this file we show that the essential image of `H` consists of the finite `Aut F`-sets where the `Aut F` action is continuous.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Galois/EssSurj.html"}, {"id": "Mathlib.CategoryTheory.Shift.Quotient", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 101, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -8.222, "z": 39.0, "size": 0.2831, "title": "The shift on a quotient category", "summary": "Let `C` be a category equipped a shift by a monoid `A`. If we have a relation on morphisms `r : HomRel C` that is compatible with the shift (i.e. if two morphisms `f` and `g` are related, then `f⟦a⟧'` and `g⟦a⟧'` are also related for all `a : A`), then the quotient category `Quotient r` is equipped with a shift. The condition `r.IsCompatibleWithShift A` on the relation `r` is a class so that the shift can be…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/Quotient.html"}, {"id": "Mathlib.CategoryTheory.Shift.Induced", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 102, "macro_tier_score": 0.0148, "macro_tier_override": null, "x": 62.755, "z": 45.436, "size": 0.2977, "title": "Shift induced from a category to another", "summary": "In this file, we introduce a sufficient condition on a functor `F : C ⥤ D` so that a shift on `C` by a monoid `A` induces a shift on `D`. More precisely, when the functor `(D ⥤ D) ⥤ C ⥤ D` given by the precomposition with `F` is fully faithful, and that all the shift functors on `C` can be lifted to functors `D ⥤ D` (i.e. we have functors `s a : D ⥤ D` for all `a : A`, and isomorphisms `F ⋙ s a ≅ shiftFunctor C a ⋙…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/Induced.html"}, {"id": "Mathlib.CategoryTheory.Monad.EquivMon", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 65.861, "z": 36.436, "size": 0.2, "title": "The equivalence between `Monad C` and `Mon (C ⥤ C)`.", "summary": "A monad \"is just\" a monoid in the category of endofunctors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monad/EquivMon.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Images", "region_id": "category_theory", "micro_elevation": 0.5789, "macro_tier": 103, "macro_tier_score": 0.2168, "macro_tier_override": null, "x": 72.804, "z": 40.436, "size": 0.4138, "title": "Categorical images", "summary": "We define the categorical image of `f` as a factorisation `f = e ≫ m` through a monomorphism `m`, so that `m` factors through the `m'` in any other such factorisation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Images.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.Concrete", "region_id": "category_theory", "micro_elevation": 0.3684, "macro_tier": 103, "macro_tier_score": 0.0387, "macro_tier_override": null, "x": 8.767, "z": 28.436, "size": 0.3023, "title": "Morphism properties defined in concrete categories", "summary": "In this file, we define the class of morphisms `MorphismProperty.injective`, `MorphismProperty.surjective`, `MorphismProperty.bijective` in concrete categories, and show that it is stable under composition and respects isomorphisms. We introduce type-classes `HasSurjectiveInjectiveFactorization` and `HasFunctorialSurjectiveInjectiveFactorization` expressing that in a concrete category `C`, all morphisms can be…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/Concrete.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Injective.Extend", "region_id": "category_theory", "micro_elevation": 0.0175, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 22.968, "z": 8.436, "size": 0.2338, "title": "Injective resolutions as cochain complexes indexed by the integers", "summary": "Given an injective resolution `R` of an object `X` in an abelian category `C`, we define `R.cochainComplex : CochainComplex C ℤ`, which is the extension of `R.cocomplex : CochainComplex C ℕ`, and the quasi-isomorphism `R.ι' : (CochainComplex.singleFunctor C 0).obj X ⟶ R.cochainComplex`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Injective/Extend.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Injective.Resolution", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 24.444, "z": 7.436, "size": 0.2698, "title": "Injective resolutions", "summary": "An injective resolution `I : InjectiveResolution Z` of an object `Z : C` consists of an `ℕ`-indexed cochain complex `I.cocomplex` of injective objects, along with a quasi-isomorphism `I.ι` from the cochain complex consisting just of `Z` in degree zero to `I.cocomplex`. ``` Z ----> 0 ----> ... ----> 0 ----> ... | | | | | | v v v I⁰ ---> I¹ ---> ... ----> Iⁿ ---> ... ```", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Injective/Resolution.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Biproducts", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 103, "macro_tier_score": 0.0958, "macro_tier_override": null, "x": 65.883, "z": 46.436, "size": 0.3019, "title": "Preservation of biproducts", "summary": "We define the image of a (binary) bicone under a functor that preserves zero morphisms and define classes `PreservesBiproduct` and `PreservesBinaryBiproduct`. We then * show that a functor that preserves biproducts of a two-element type preserves binary biproducts, * construct the comparison morphisms between the image of a biproduct and the biproduct of the images and show that the biproduct is preserved if one of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Action.Opposites", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.46, "z": 33.436, "size": 0.2, "title": "Actions from the monoidal opposite of a category.", "summary": "In this file, given a monoidal category `C` and a category `D`, we construct a left `C`-action on `D` out of the data of a right `Cᴹᵒᵖ`-action on `D`. We also construct a right `C`-action on `D` from the data of a left `Cᴹᵒᵖ`-action on `D`. Conversely, given left/right `C`-actions on `D`, we construct a `Cᴹᵒᵖ` action with the conjugate variance. We construct similar actions for `Cᵒᵖ`, namely, left/right…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Action/Opposites.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Opposite", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 103, "macro_tier_score": 0.1354, "macro_tier_override": null, "x": -12.219, "z": 32.436, "size": 0.3913, "title": "Monoidal opposites", "summary": "We write `Cᵐᵒᵖ` for the monoidal opposite of a monoidal category `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Opposite.html"}, {"id": "Mathlib.CategoryTheory.CopyDiscardCategory.Basic", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 102, "macro_tier_score": 0.0154, "macro_tier_override": null, "x": 51.73, "z": 37.436, "size": 0.3442, "title": "Copy-Discard Categories", "summary": "Symmetric monoidal categories where every object has a commutative comonoid structure compatible with tensor products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/CopyDiscardCategory/Basic.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.Over", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 101, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": 278.608, "z": -22.328, "size": 0.3082, "title": "`CartesianMonoidalCategory` for `Over X`", "summary": "We provide a `CartesianMonoidalCategory (Over X)` instance via pullbacks, and provide simp lemmas for the induced `MonoidalCategory (Over X)` instance.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/Over.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.Over.Products", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 102, "macro_tier_score": 0.0148, "macro_tier_override": null, "x": -28.138, "z": 43.436, "size": 0.297, "title": "Products in the over category", "summary": "Shows that products in the over category can be derived from wide pullbacks in the base category. The main result is `over_product_of_widePullback`, which says that if `C` has `J`-indexed wide pullbacks, then `Over B` has `J`-indexed products. Note that the binary case is done separately to ensure defeqs with the pullback in the base category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/Over/Products.html"}, {"id": "Mathlib.CategoryTheory.Generator.HomologicalComplex", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 68.318, "z": 47.0, "size": 0.239, "title": "Generators of the category of homological complexes", "summary": "Let `c : ComplexShape ι` be a complex shape with no loop. If a category `C` has a separator, then `HomologicalComplex C c` has a separating family, and a separator when suitable coproducts exist.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Generator/HomologicalComplex.html"}, {"id": "Mathlib.CategoryTheory.EssentialImage", "region_id": "category_theory", "micro_elevation": 0.1053, "macro_tier": 103, "macro_tier_score": 0.4722, "macro_tier_override": null, "x": 15.418, "z": 13.436, "size": 0.5152, "title": "Essential image of a functor", "summary": "The essential image `essImage` of a functor consists of the objects in the target category which are isomorphic to an object in the image of the object function. This, for instance, allows us to talk about objects belonging to a subcategory expressed as a functor rather than a subtype, preserving the principle of equivalence. For example this lets us define exponential ideals. The essential image can also be seen as…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/EssentialImage.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.Mod_", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 41.532, "z": 52.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/Mod_.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.Mod", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 8.119, "z": 51.436, "size": 0.2676, "title": "Module objects in cartesian monoidal categories", "summary": "In this file we study module objects in a cartesian monoidal category `C` action on itself by `⊗`. In particular, for a monoid object `M : C` action on `X : C`, we equip `Z ⟶ X` with a `M ⟶ X` action for every `Z : C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/Mod.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Subcategory", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -26.255, "z": 50.436, "size": 0.2, "title": "Subcategories of abelian categories", "summary": "Let `C` be an abelian category. Given `P : ObjectProperty C` which contains zero, is closed under kernels, cokernels and finite products, we show that the full subcategory defined by `P` is abelian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Subcategory.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.FiniteProducts", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.01, "macro_tier_override": null, "x": -31.94, "z": 42.0, "size": 0.2906, "title": "Properties of objects that are stable under finite products", "summary": "We introduce typeclasses `IsClosedUnderBinaryProducts` and `IsClosedUnderFiniteProducts` expressing that `P : ObjectProperty C` is closed under binary products or finite products. We introduce a constructor for `P.IsClosedUnderFiniteProducts` assuming `P.IsClosedUnderBinaryProducts`, `P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)` and that `C` has finite products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/FiniteProducts.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.Kernels", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -6.064, "z": 49.436, "size": 0.2478, "title": "Objects that are (co)kernels of morphisms", "summary": "Given a morphism property `W` on a category, we introduce two object properties `kernels W` and `cokernels W`, consisting of all (co)kernels of morphisms satisfying `W`. Given an object property `P`, we also introduce two predicates `P.IsClosedUnderKernels` and `P.IsClosedUnderCokernels`, stating that all (co)kernels of morphisms between objects in `P` remain in `P`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/Kernels.html"}, {"id": "Mathlib.CategoryTheory.Join.Opposites", "region_id": "category_theory", "micro_elevation": 0.2105, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 6.407, "z": 19.436, "size": 0.2, "title": "Opposites of joins of categories", "summary": "This file constructs the canonical equivalence of categories `(C ⋆ D)ᵒᵖ ≌ Dᵒᵖ ⋆ Cᵒᵖ`. This equivalence is characterized in both directions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Join/Opposites.html"}, {"id": "Mathlib.CategoryTheory.Join.Basic", "region_id": "category_theory", "micro_elevation": 0.193, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 27.424, "z": 18.436, "size": 0.3068, "title": "Joins of categories", "summary": "Given categories `C, D`, this file constructs a category `C ⋆ D`. Its objects are either objects of `C` or objects of `D`, morphisms between objects of `C` are morphisms in `C`, morphisms between objects of `D` are morphisms in `D`, and finally, given `c : C` and `d : D`, there is a unique morphism `c ⟶ d` in `C ⋆ D`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Join/Basic.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Functor.Prelax", "region_id": "category_theory", "micro_elevation": 0.1754, "macro_tier": 103, "macro_tier_score": 0.121, "macro_tier_override": null, "x": 32.931, "z": 17.436, "size": 0.3869, "title": "Prelax functors", "summary": "This file defines lax prefunctors and prelax functors between bicategories. The point of these definitions is to provide some common API that will be helpful in the development of both lax and oplax functors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Functor/Prelax.html"}, {"id": "Mathlib.CategoryTheory.Galois.Full", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 71.983, "z": 54.436, "size": 0.2442, "title": "Fiber functors are (faithfully) full", "summary": "Any (fiber) functor `F : C ⥤ FintypeCat` factors via the forgetful functor from finite `Aut F`-sets to finite sets. The induced functor `H : C ⥤ Action FintypeCat (Aut F)` is faithfully full. The faithfulness follows easily from the faithfulness of `F`. In this file we show that `H` is also full.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Galois/Full.html"}, {"id": "Mathlib.CategoryTheory.Galois.Action", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -16.396, "z": 53.436, "size": 0.2788, "title": "Induced functor to finite `Aut F`-sets", "summary": "Any (fiber) functor `F : C ⥤ FintypeCat` factors via the forgetful functor from finite `Aut F`-sets to finite sets. In this file we collect basic properties of the induced functor `H : C ⥤ Action FintypeCat (Aut F)`. See `Mathlib/CategoryTheory/Galois/Full.lean` for the proof that `H` is (faithfully) full.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Galois/Action.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.WeakFactorizationSystem", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 101, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": 84.447, "z": -64.661, "size": 0.3512, "title": "Weak factorization systems", "summary": "In this file, we introduce the notion of weak factorization system, which is a property of two classes of morphisms `W₁` and `W₂` in a category `C`. The type class `IsWeakFactorizationSystem W₁ W₂` asserts that `W₁` is exactly `W₂.llp`, `W₂` is exactly `W₁.rlp`, and any morphism in `C` can be factored a `i ≫ p` with `W₁ i` and `W₂ p`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/WeakFactorizationSystem.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.RetractArgument", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0061, "macro_tier_override": null, "x": 36.712, "z": 49.436, "size": 0.3543, "title": "The retract argument", "summary": "Let `W₁` and `W₂` be classes of morphisms in a category `C` such that any morphism can be factored as a morphism in `W₁` followed by a morphism in `W₂` (this is `HasFactorization W₁ W₂`). If `W₁` has the left lifting property with respect to `W₂` (i.e. `W₁ ≤ W₂.llp`, or equivalently `W₂ ≤ W₁.rlp`), then `W₂.llp = W₁` if `W₁` is stable under retracts, and `W₁.rlp = W₂` if `W₂` is.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/RetractArgument.html"}, {"id": "Mathlib.CategoryTheory.CofilteredSystem", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 47.365, "z": 132.365, "size": 0.3103, "title": "Cofiltered systems", "summary": "This file deals with properties of cofiltered (and inverse) systems.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/CofilteredSystem.html"}, {"id": "Mathlib.CategoryTheory.Localization.Composition", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 43.299, "z": 29.0, "size": 0.2407, "title": "Composition of localization functors", "summary": "Given two composable functors `L₁ : C₁ ⥤ C₂` and `L₂ : C₂ ⥤ C₃`, it is shown in this file that under some suitable conditions on `W₁ : MorphismProperty C₁` `W₂ : MorphismProperty C₂` and `W₃ : MorphismProperty C₁`, then if `L₁ : C₁ ⥤ C₂` is a localization functor for `W₁`, then the composition `L₁ ⋙ L₂ : C₁ ⥤ C₃` is a localization functor for `W₃` if and only if `L₂ : C₂ ⥤ C₃` is a localization functor for `W₂`. The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Composition.html"}, {"id": "Mathlib.CategoryTheory.Enriched.Limits.HasConicalProducts", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 76.899, "z": 55.436, "size": 0.2676, "title": "Existence of conical products", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Enriched/Limits/HasConicalProducts.html"}, {"id": "Mathlib.CategoryTheory.Enriched.Limits.HasConicalLimits", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 101, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -41.713, "z": 54.436, "size": 0.3139, "title": "Existence of conical limits", "summary": "This file contains different statements about the (non-constructive) existence of conical limits. The main constructions are the following. - `HasConicalLimit`: there exists a conical limit for `F : J ⥤ C`. - `HasConicalLimitsOfShape J`: All functors `F : J ⥤ C` have conical limits. - `HasConicalLimitsOfSize.{v₁, u₁}`: For all small `J` all functors `F : J ⥤ C` have conical limits. - `HasConicalLimits`: `C` has all…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Enriched/Limits/HasConicalLimits.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.WeakKernels", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 34.252, "z": 54.436, "size": 0.2, "title": "Weak kernels in pretriangulated categories", "summary": "We prove that pretriangulated categories have weak kernels: if `f : X ⟶ Y` is a morphism in a pretriangulated category and if we complete it to a distinguished triangle `Z ⟶ X ⟶ Y ⟶ Z⟦1⟧`, then the first morphism `Z ⟶ Y` of that triangle is a weak kernel of `f`. TODO: Weak cokernels.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/WeakKernels.html"}, {"id": "Mathlib.CategoryTheory.Limits.WeakLimits.WeakKernels", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 24.694, "z": 45.436, "size": 0.2478, "title": "Weak kernels", "summary": "These are weak equalizers for functors of the form `parallelPair f 0`. If the category is preadditive, then weak equalizers exist if and only if weak kernels exist. (See `hasWeakEqualizer_of_hasWeakKernel` and `hasWeakKernel_of_hasWeakEqualizer`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/WeakLimits/WeakKernels.html"}, {"id": "Mathlib.CategoryTheory.FiberedCategory.BasedCategory", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 7.696, "z": 25.436, "size": 0.2, "title": "The bicategory of based categories", "summary": "In this file we define the type `BasedCategory 𝒮`, and give it the structure of a strict bicategory. Given a category `𝒮`, we define the type `BasedCategory 𝒮` as the type of categories `𝒳` equipped with a functor `𝒳.p : 𝒳 ⥤ 𝒮`. We also define a type of functors between based categories `𝒳` and `𝒴`, which we call `BasedFunctor 𝒳 𝒴` and denote as `𝒳 ⥤ᵇ 𝒴`. These are defined as functors between the underlying…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/FiberedCategory/BasedCategory.html"}, {"id": "Mathlib.CategoryTheory.FiberedCategory.HomLift", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 102, "macro_tier_score": 0.0061, "macro_tier_override": null, "x": 38.614, "z": 24.436, "size": 0.3553, "title": "HomLift", "summary": "Given a functor `p : 𝒳 ⥤ 𝒮`, this file provides API for expressing the fact that `p(φ) = f` for given morphisms `φ` and `f`. The reason this API is needed is because, in general, `p.map φ = f` does not make sense when the domain and/or codomain of `φ` and `f` are not definitionally equal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/FiberedCategory/HomLift.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Strict.Basic", "region_id": "category_theory", "micro_elevation": 0.1754, "macro_tier": 103, "macro_tier_score": 0.3222, "macro_tier_override": null, "x": 26.158, "z": 17.436, "size": 0.4396, "title": "Strict bicategories", "summary": "A bicategory is called `Strict` if the left unitors, the right unitors, and the associators are isomorphisms given by equalities.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Strict/Basic.html"}, {"id": "Mathlib.CategoryTheory.GradedObject.Monoidal", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -62.419, "z": 42.0, "size": 0.2585, "title": "The monoidal category structures on graded objects", "summary": "Assuming that `C` is a monoidal category and that `I` is an additive monoid, we introduce a partially defined tensor product on the category `GradedObject I C`: given `X₁` and `X₂` two objects in `GradedObject I C`, we define `GradedObject.Monoidal.tensorObj X₁ X₂` under the assumption `HasTensor X₁ X₂` that the coproduct of `X₁ i ⊗ X₂ j` for `i + j = n` exists for any `n : I`. Under suitable assumptions about the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GradedObject/Monoidal.html"}, {"id": "Mathlib.CategoryTheory.GradedObject.Unitor", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 24.806, "z": 48.436, "size": 0.2611, "title": "The left and right unitors", "summary": "Given a bifunctor `F : C ⥤ D ⥤ D`, an object `X : C` such that `F.obj X ≅ 𝟭 D` and a map `p : I × J → J` such that `hp : ∀ (j : J), p ⟨0, j⟩ = j`, we define an isomorphism of `J`-graded objects for any `Y : GradedObject J D`. `mapBifunctorLeftUnitor F X e p hp Y : mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`. Under similar assumptions, we also obtain a right unitor isomorphism `mapBifunctorMapObj F p X…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GradedObject/Unitor.html"}, {"id": "Mathlib.CategoryTheory.Galois.GaloisObjects", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 48.754, "z": 48.436, "size": 0.2479, "title": "Galois objects in Galois categories", "summary": "We define when a connected object of a Galois category `C` is Galois in a fiber-functor-independent way and show equivalent characterisations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Galois/GaloisObjects.html"}, {"id": "Mathlib.CategoryTheory.Galois.Basic", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -30.67, "z": 47.436, "size": 0.2712, "title": "Definition and basic properties of Galois categories", "summary": "We define the notion of a Galois category and a fiber functor as in SGA1, following the definitions in Lenstra's notes (see below for a reference).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Galois/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.SingleObj", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 2.738, "z": 34.436, "size": 0.2465, "title": "(Co)limits of functors out of `SingleObj M`", "summary": "We characterise (co)limits of shape `SingleObj M`. Currently only in the category of types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/SingleObj.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.DayConvolution.Braided", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 75.167, "z": 44.436, "size": 0.2, "title": "Braidings for Day convolution", "summary": "In this file, we show that if `C` is a braided monoidal category and `V` also a braided monoidal category, then the Day convolution monoidal structure on `C ⥤ V` is also braided monoidal. We prove it by constructing an explicit braiding isomorphism whenever sufficient Day convolutions exist, and we prove that it satisfies the forward and reverse hexagon identities. Furthermore, we show that when both `C` and `V` are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/DayConvolution/Braided.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.DayConvolution", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 78.546, "z": 43.436, "size": 0.3032, "title": "Day convolution monoidal structure", "summary": "Given functors `F G : C ⥤ V` between two monoidal categories, this file defines a typeclass `DayConvolution` on functors `F` `G` that contains a functor `F ⊛ G`, as well as the required data to exhibit `F ⊛ G` as a pointwise left Kan extension of `F ⊠ G` (see `Mathlib/CategoryTheory/Monoidal/ExternalProduct/Basic.lean` for the definition) along the tensor product of `C`. Such a functor is called a Day convolution of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/DayConvolution.html"}, {"id": "Mathlib.CategoryTheory.EffectiveEpi.Extensive", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -32.408, "z": 48.436, "size": 0.2903, "title": "Preserving and reflecting effective epis on extensive categories", "summary": "We prove that a functor between `FinitaryPreExtensive` categories preserves (resp. reflects) finite effective epi families if it preserves (resp. reflects) effective epis.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/EffectiveEpi/Extensive.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.Enrichment", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 2.805, "z": 54.436, "size": 0.2569, "title": "A closed monoidal category is enriched in itself", "summary": "From the data of a closed monoidal category `C`, we define a `C`-category structure for `C`. where the hom-object is given by the internal hom (coming from the closed structure). We use `scoped instance` to avoid potential issues where `C` may also have a `C`-category structure coming from another source (e.g. the type of simplicial sets `SSet.{v}` has an instance of `EnrichedCategory SSet.{v}` as a category of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/Enrichment.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.FunctorCategory", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 227.045, "z": -55.661, "size": 0.2629, "title": "Stability properties of morphism properties on functor categories", "summary": "Given `W : MorphismProperty C` and a category `J`, we study the stability properties of `W.functorCategory J : MorphismProperty (J ⥤ C)`. Under suitable assumptions, we also show that if monomorphisms in `C` are stable under transfinite compositions (or coproducts), then the same holds in the category `J ⥤ C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/FunctorCategory.html"}, {"id": "Mathlib.CategoryTheory.Types.Monomorphisms", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 140.133, "z": -66.661, "size": 0.2629, "title": "Stability properties of monomorphisms in `Type`", "summary": "In this file, we show that in the category `Type u`, monomorphisms are stable under cobase change, filtered colimits. After importing `Mathlib/CategoryTheory/MorphismProperty/TransfiniteComposition.lean`, the fact that monomorphisms are stable under transfinite composition will also be inferred automatically. (The stability by retracts holds in any category: it is shown in the file…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Types/Monomorphisms.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Injective.Ext", "region_id": "category_theory", "micro_elevation": 0.0351, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 21.969, "z": 9.436, "size": 0.2, "title": "Computing `Ext` using an injective resolution", "summary": "Given an injective resolution `R` of an object `Y` in an abelian category `C`, we provide an API in order to construct elements in `Ext X Y n` in terms of the complex `R.cocomplex` and to make computations in the `Ext`-group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Injective/Ext.html"}, {"id": "Mathlib.CategoryTheory.Functor.Derived.Adjunction", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 31.052, "z": 39.436, "size": 0.2, "title": "Derived adjunction", "summary": "Assume that functors `G : C₁ ⥤ C₂` and `F : C₂ ⥤ C₁` are part of an adjunction `adj : G ⊣ F`, that we have localization functors `L₁ : C₁ ⥤ D₁` and `L₂ : C₂ ⥤ D₂` with respect to classes of morphisms `W₁` and `W₂`, and that `G` admits a left derived functor `G' : D₁ ⥤ D₂` and `F` a right derived functor `F' : D₂ ⥤ D₁`. We show that there is an adjunction `G' ⊣ F'` under the additional assumption that `F'` and `G'`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/Derived/Adjunction.html"}, {"id": "Mathlib.CategoryTheory.Functor.Derived.LeftDerived", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 50.553, "z": 38.436, "size": 0.2617, "title": "Left derived functors", "summary": "In this file, given a functor `F : C ⥤ H`, and `L : C ⥤ D` that is a localization functor for `W : MorphismProperty C`, we define `F.totalLeftDerived L W : D ⥤ H` as the right Kan extension of `F` along `L`: it is defined if the type class `F.HasLeftDerivedFunctor W` asserting the existence of a right Kan extension is satisfied. (The name `totalLeftDerived` is to avoid name-collision with `Functor.leftDerived` which…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/Derived/LeftDerived.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.Equalizers", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 103, "macro_tier_score": 0.0385, "macro_tier_override": null, "x": 59.144, "z": 39.436, "size": 0.2818, "title": "Equalizers in Type", "summary": "The equalizer of a pair of maps `(g, h)` from `X` to `Y` is the subtype `{x : Y // g x = h x}`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/Equalizers.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Projective.Internal", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -191.866, "z": -110.756, "size": 0.2442, "title": "Internal projectivity", "summary": "This file defines internal projectivity of objects `P` in a category `C` as a class `InternallyProjective P`. This means that the functor taking internal homs out of `P` preserves epimorphisms. It also proves that a retract of an internally projective object is internally projective (see `InternallyProjective.ofRetract`). This property is important in the setting of light condensed abelian groups, when establishing…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Projective/Internal.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.Comon_", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -18.864, "z": 50.436, "size": 0.2478, "title": "Comonoid objects in a Cartesian monoidal category.", "summary": "The category of comonoid objects in a Cartesian monoidal category is equivalent to the category itself, via the forgetful functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/Comon_.html"}, {"id": "Mathlib.CategoryTheory.Equivalence.Symmetry", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -7.834, "z": 31.436, "size": 0.2, "title": "Functoriality of the symmetry of equivalences", "summary": "Using the calculus of mates in `Mathlib.CategoryTheory.Adjunction.Mates`, we prove that passing to the symmetric equivalence defines an equivalence between `C ≌ D` and `(D ≌ C)ᵒᵖ`, and provides the definition of the functor that takes an equivalence to its inverse.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Equivalence/Symmetry.html"}, {"id": "Mathlib.CategoryTheory.ComposableArrows.Two", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -20.852, "z": 17.0, "size": 0.274, "title": "API for compositions of two arrows", "summary": "Given morphisms `f : i ⟶ j`, `g : j ⟶ k`, and `fg : i ⟶ k` in a category `C` such that `f ≫ g = fg`, we define maps `twoδ₂Toδ₁ : mk₁ f ⟶ mk₁ fg` and `twoδ₁Toδ₀ : mk₁ fg ⟶ mk₁ g` in the category `ComposableArrows C 1`. The names are justified by the fact that `ComposableArrow.mk₂ f g` can be thought of as a `2`-simplex in the simplicial set `nerve C`, and its faces (numbered from `0` to `2`) are respectively `mk₁ g`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ComposableArrows/Two.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.ColimitType", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 103, "macro_tier_score": 0.2158, "macro_tier_override": null, "x": -5.598, "z": 27.436, "size": 0.3632, "title": "The colimit type of a functor to types", "summary": "Given a category `J` (with `J : Type u` and `[Category.{v} J]`) and a functor `F : J ⥤ Type w₀`, we introduce a type `F.ColimitType : Type (max u w₀)`, which satisfies a certain universal property of the colimit: it is defined as a suitable quotient of `Σ j, F.obj j`. This universal property is not expressed in a categorical way (as in general `Type (max u w₀)` is not the same as `Type u`). We also introduce a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/ColimitType.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Equivalence", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 103, "macro_tier_score": 0.1153, "macro_tier_override": null, "x": -1.317, "z": 36.436, "size": 0.336, "title": "Transporting existence of specific limits across equivalences", "summary": "For now, we only treat the case of initial and terminal objects, but other special shapes can be added in the future.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Equivalence.html"}, {"id": "Mathlib.CategoryTheory.Subfunctor.Basic", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 103, "macro_tier_score": 0.0496, "macro_tier_override": null, "x": 54.499, "z": 27.436, "size": 0.387, "title": "Subfunctor of types", "summary": "We define subfunctors of a type-valued functors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subfunctor/Basic.html"}, {"id": "Mathlib.CategoryTheory.ShrinkYoneda", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 103, "macro_tier_score": 0.2158, "macro_tier_override": null, "x": 9.755, "z": 25.0, "size": 0.3648, "title": "The Yoneda functor for locally small categories", "summary": "Let `C` be a locally `w`-small category. We define the Yoneda embedding `shrinkYoneda : C ⥤ Cᵒᵖ ⥤ Type w`. (See the file `CategoryTheory.Yoneda` for the other variants `yoneda` and `uliftYoneda`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ShrinkYoneda.html"}, {"id": "Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 42.759, "z": 52.436, "size": 0.2, "title": "Covers in subcanonical topologies", "summary": "In this file we provide API related to covers in subcanonical topologies.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Hypercover/Subcanonical.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Opposite.Triangle", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 92.005, "z": 52.436, "size": 0.3067, "title": "Triangles in the opposite category of a (pre)triangulated category", "summary": "Let `C` be a (pre)triangulated category. In `CategoryTheory.Triangulated.Opposite.Basic`, we have constructed a shift on `Cᵒᵖ` that will be part of a structure of (pre)triangulated category. In this file, we construct an equivalence of categories between `(Triangle C)ᵒᵖ` and `Triangle Cᵒᵖ`, called `CategoryTheory.Pretriangulated.triangleOpEquivalence`. It sends a triangle `X ⟶ Y ⟶ Z ⟶ X⟦1⟧` in `C` to the triangle…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Opposite/Triangle.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Basic", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0104, "macro_tier_override": null, "x": -23.378, "z": 50.436, "size": 0.3275, "title": "Triangles", "summary": "This file contains the definition of triangles in an additive category with an additive shift. It also defines morphisms between these triangles. TODO: generalise this to n-angles in n-angulated categories as in https://arxiv.org/abs/1006.4592", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Basic.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Opposite.Basic", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 102, "macro_tier_score": 0.0101, "macro_tier_override": null, "x": 90.26, "z": 51.436, "size": 0.3047, "title": "The shift on the opposite category of a pretriangulated category", "summary": "Let `C` be a (pre)triangulated category. We already have a shift on `Cᵒᵖ` given by `CategoryTheory.Shift.Opposite`, but this is not the shift that we want to make `Cᵒᵖ` into a (pre)triangulated category. The correct shift on `Cᵒᵖ` is obtained by combining the constructions in the files `CategoryTheory.Shift.Opposite` and `CategoryTheory.Shift.Pullback`. When the user opens `CategoryTheory.Pretriangulated.Opposite`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Opposite/Basic.html"}, {"id": "Mathlib.CategoryTheory.Sums.Basic", "region_id": "category_theory", "micro_elevation": 0.1404, "macro_tier": 102, "macro_tier_score": 0.0149, "macro_tier_override": null, "x": 31.468, "z": 15.436, "size": 0.3051, "title": "Binary disjoint unions of categories", "summary": "We define the category instance on `C ⊕ D` when `C` and `D` are categories. We define: * `inl_` : the functor `C ⥤ C ⊕ D` * `inr_` : the functor `D ⥤ C ⊕ D` * `swap` : the functor `C ⊕ D ⥤ D ⊕ C` (and the fact this is an equivalence) We provide an induction principle `Sum.homInduction` to reason and work with morphisms in this category. The sum of two functors `F : A ⥤ C` and `G : B ⥤ C` is a functor `A ⊕ B ⥤ C`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sums/Basic.html"}, {"id": "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 103, "macro_tier_score": 0.3172, "macro_tier_override": null, "x": 167.908, "z": -165.421, "size": 0.4284, "title": "Bundled types", "summary": "`Bundled c` provides a uniform structure for bundling a type equipped with a type class. We provide `Category` instances for these in `Mathlib/CategoryTheory/ConcreteCategory/UnbundledHom.lean` (for categories with unbundled homs, e.g. topological spaces) and in `Mathlib/CategoryTheory/ConcreteCategory/BundledHom.lean` (for categories with bundled homs, e.g. monoids). Note: this structure will be deprecated in the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ConcreteCategory/Bundled.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Rigid.Braided", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -45.099, "z": 31.0, "size": 0.2542, "title": "Deriving `RigidCategory` instance for braided and left/right rigid categories.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Rigid/Braided.html"}, {"id": "Mathlib.CategoryTheory.Sites.Descent.IsPrestack", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 101, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": 85.658, "z": 61.436, "size": 0.3166, "title": "Prestacks: descent of morphisms", "summary": "Let `C` be a category and `F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat`. Given `S : C`, and objects `M` and `N` in `F.obj (.mk (op S))`, we define a presheaf of types `F.presheafHom M N` on the category `Over S`: its sections on an object `T : Over S` corresponding to a morphism `p : X ⟶ S` are the type of morphisms `p^* M ⟶ p^* N`. We shall say that `F` satisfies the descent of morphisms for a Grothendieck topology `J` if these…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Descent/IsPrestack.html"}, {"id": "Mathlib.CategoryTheory.LocallyCartesianClosed.Sections", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 87.739, "z": 52.436, "size": 0.2, "title": "The section functor as a right adjoint to the toOver functor", "summary": "We show that in a cartesian monoidal category `C`, for any exponentiable object `I`, the functor `toOver I : C ⥤ Over I` mapping an object `X` to the projection `snd : X ⊗ I ⟶ I` in `Over I` has a right adjoint `sections I : Over I ⥤ C` whose object part is the object of sections of `X` over `I`. In particular, if `C` is cartesian closed, then for all objects `I` in `C`, `toOver I : C ⥤ Over I` has a right adjoint.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/LocallyCartesianClosed/Sections.html"}, {"id": "Mathlib.CategoryTheory.LocallyCartesianClosed.Over", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 26.41, "z": 51.436, "size": 0.2478, "title": "Cartesian monoidal structure on slices induced by chosen pullbacks", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/LocallyCartesianClosed/Over.html"}, {"id": "Mathlib.CategoryTheory.Category.Cat.AsSmall", "region_id": "category_theory", "micro_elevation": 0.3684, "macro_tier": 103, "macro_tier_score": 0.1912, "macro_tier_override": null, "x": -1.964, "z": 28.436, "size": 0.3107, "title": "Functorially embedding `Cat` into the category of small categories", "summary": "There is a canonical functor `asSmallFunctor` between the category of categories of any size and any larger category of small categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Cat/AsSmall.html"}, {"id": "Mathlib.CategoryTheory.Category.ULift", "region_id": "category_theory", "micro_elevation": 0.1754, "macro_tier": 103, "macro_tier_score": 0.3159, "macro_tier_override": null, "x": 30.674, "z": 17.436, "size": 0.5393, "title": "Basic API for ULift", "summary": "This file contains a very basic API for working with the categorical instance on `ULift C` where `C` is a type with a category instance. 1. `CategoryTheory.ULift.upFunctor` is the functorial version of the usual `ULift.up`. 2. `CategoryTheory.ULift.downFunctor` is the functorial version of the usual `ULift.down`. 3. `CategoryTheory.ULift.equivalence` is the categorical equivalence between `C` and `ULift C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/ULift.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 102, "macro_tier_score": 0.0248, "macro_tier_override": null, "x": -18.863, "z": 38.0, "size": 0.3337, "title": "Limits in concrete categories", "summary": "In this file, we combine the description of limits in `Types` and the API about the preservation of products and pullbacks in order to describe these limits in a concrete category `C`. If `F : J → C` is a family of objects in `C`, we define a bijection `Limits.Concrete.productEquiv F : ToType (∏ᶜ F) ≃ ∀ j, ToType (F j)`. Similarly, if `f₁ : X₁ ⟶ S` and `f₂ : X₂ ⟶ S` are two morphisms, the elements in `pullback f₁…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.html"}, {"id": "Mathlib.CategoryTheory.WithTerminal.Lemmas", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 75.127, "z": 41.436, "size": 0.2, "title": "Further lemmas on `WithTerminal`", "summary": "These lemmas and instances need more imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/WithTerminal/Lemmas.html"}, {"id": "Mathlib.CategoryTheory.Generator.Presheaf", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 102, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": 49.368, "z": 54.436, "size": 0.2704, "title": "Generators in the category of presheaves", "summary": "In this file, we show that if `A` is a category with zero morphisms that has a separator (and suitable coproducts), then the category of presheaves `Cᵒᵖ ⥤ A` also has a separator.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Generator/Presheaf.html"}, {"id": "Mathlib.CategoryTheory.Presentable.StrongGenerator", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 100.993, "z": 60.436, "size": 0.258, "title": "Locally presentable categories and strong generators", "summary": "In this file, we show that a category is locally `κ`-presentable iff it is cocomplete and has a strong generator consisting of `κ`-presentable objects. This is theorem 1.20 in the book by Adámek and Rosický. In particular, if a category is locally `κ`-presentable, it is also locally `κ'`-presentable for any regular cardinal `κ'` such that `κ ≤ κ'`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/StrongGenerator.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Yoneda.Injective", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -25.389, "z": 51.436, "size": 0.239, "title": null, "summary": "An object is injective iff the preadditive yoneda functor on it preserves epimorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Yoneda/Injective.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.PushoutProduct", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 102, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": 113.648, "z": -63.661, "size": 0.2817, "title": "Leibniz constructions associated to monoidal categories.", "summary": "In a monoidal category with pushouts, the pushout-product is the Leibniz functor associated to the tensor product. This is the bifunctor of arrow categories that sends `f : A ⟶ B` and `g : X ⟶ Y` to the canonical map from the pushout of `f ◁ X` and `A ▷ g` to `B ⊗ Y`, induced by the following diagram: ``` A ⊗ X --> B ⊗ X | | v v A ⊗ Y --> B ⊗ Y ``` In a monoidal closed category with pullbacks, the pullback-hom is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/PushoutProduct.html"}, {"id": "Mathlib.CategoryTheory.NatTrans", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 103, "macro_tier_score": 0.501, "macro_tier_override": null, "x": 24.444, "z": 7.436, "size": 0.5218, "title": "Natural transformations", "summary": "Defines natural transformations between functors. A natural transformation `α : NatTrans F G` consists of morphisms `α.app X : F.obj X ⟶ G.obj X`, and the naturality squares `α.naturality f : F.map f ≫ α.app Y = α.app X ≫ G.map f`, where `f : X ⟶ Y`. Note that we make `NatTrans.naturality` a simp lemma, with the preferred simp normal form pushing components of natural transformations to the left. See also…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/NatTrans.html"}, {"id": "Mathlib.CategoryTheory.Subobject.Limits", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 102, "macro_tier_score": 0.012, "macro_tier_override": null, "x": -27.837, "z": 46.0, "size": 0.4124, "title": "Specific subobjects", "summary": "We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which are the subobjects represented by the equalizer, kernel and image of (a pair of) morphism(s) and provide conditions for `P.factors f`, where `P` is one of these special subobjects. TODO: an iff characterisation of `(imageSubobject f).Factors h`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subobject/Limits.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Opposites.Filtered", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 102, "macro_tier_score": 0.0288, "macro_tier_override": null, "x": 76.766, "z": 42.436, "size": 0.2596, "title": "Filtered colimits and cofiltered limits in `C` and `Cᵒᵖ`", "summary": "We construct filtered colimits and cofiltered limits in the opposite categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Opposites/Filtered.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.Zero", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 86.671, "z": 51.436, "size": 0.2, "title": "A Cartesian closed category with zero object is trivial", "summary": "A Cartesian closed category with zero object is trivial: it is equivalent to the category with one object and one morphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/Zero.html"}, {"id": "Mathlib.CategoryTheory.Subfunctor.Image", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 103, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -8.351, "z": 45.436, "size": 0.4217, "title": "The image of a subfunctor", "summary": "Given a morphism of type-valued functors `p : F' ⟶ F`, we define its range `Subfunctor.range p`. More generally, if `G' : Subfunctor F'`, we define `G'.image p : Subfunctor F` as the image of `G'` by `f`, and if `G : Subfunctor F`, we define its preimage `G.preimage f : Subfunctor F'`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subfunctor/Image.html"}, {"id": "Mathlib.CategoryTheory.Subfunctor.Sieves", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 102, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": 19.786, "z": 46.436, "size": 0.2777, "title": "Sieves attached to subpresheaves", "summary": "Given a subpresheaf `G` of a presheaf of types `F : Cᵒᵖ ⥤ Type w` and a section `s : F.obj U`, we define a sieve `G.sieveOfSection s : Sieve (unop U)` and the associated compatible family of elements with values in `G.toFunctor`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subfunctor/Sieves.html"}, {"id": "Mathlib.CategoryTheory.Generator.Type", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 58.431, "z": 54.436, "size": 0.239, "title": "Generator of Type", "summary": "In this file, we show that `PUnit` is a separator of the category `Type u`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Generator/Type.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Additive", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": 27.085, "z": 51.436, "size": 0.274, "title": "Adjunctions between additive functors.", "summary": "This provides some results and constructions for adjunctions between functors on preadditive categories: * If one of the adjoint functors is additive, so is the other. * If one of the adjoint functors is additive, the equivalence `Adjunction.homEquiv` lifts to an additive equivalence `Adjunction.homAddEquiv`. * We also give a version of this additive equivalence as an isomorphism of `preadditiveYoneda` functors…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Additive.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Opposite.Functor", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 101.644, "z": 60.436, "size": 0.2649, "title": "Opposites of functors between pretriangulated categories,", "summary": "If `F : C ⥤ D` is a functor between pretriangulated categories, we prove that `F` is a triangulated functor if and only if `F.op` is a triangulated functor. In order to do this, we first show that a `CommShift` structure on `F` naturally gives one on `F.op` (for the shifts on `Cᵒᵖ` and `Dᵒᵖ` defined in `CategoryTheory.Triangulated.Opposite.Basic`), and we then prove that `F.mapTriangle.op` and `F.op.mapTriangle`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Opposite/Functor.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.CommGrp_", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -37.102, "z": 46.0, "size": 0.2357, "title": "Commutative group objects in additive categories.", "summary": "We construct an inverse of the forgetful functor `CommGrp C ⥤ C` if `C` is an additive category. This looks slightly strange because the additive structure of `C` maps to the multiplicative structure of the commutative group objects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/CommGrp_.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Limits.Cokernels", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 61.602, "z": 50.436, "size": 0.2, "title": "Tensor products of cokernels", "summary": "Let `c₁` and `c₂` be cokernel coforks for morphisms `f₁ : X₁ ⟶ Y₁` and `f₂ : X₂ ⟶ Y₂` in a monoidal preadditive category. We define a cokernel cofork for `(X₁ ⊗ Y₂) ⨿ (Y₁ ⊗ X₂) ⟶ Y₁ ⊗ Y₂` with point `c₁.pt ⊗ c₂.pt`, and show that it is colimit if `c₁` and `c₂` are colimit, and the cokernels of `f₁` and `f₂` are preserved by suitable tensor products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Limits/Cokernels.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 80.201, "z": 45.436, "size": 0.2478, "title": "Action of bifunctors on cokernels", "summary": "Let `c₁` (resp. `c₂`) be a cokernel cofork for a morphism `f₁ : X₁ ⟶ Y₁` in a category `C₁` (resp. `f₂ : X₂ ⟶ Y₂` in `C₂`). Given a bifunctor `F : C₁ ⥤ C₂ ⥤ C`, we construct a cokernel cofork with point `(F.obj c₁.pt).obj c₂.pt` for the obvious morphism `(F.obj X₁).obj Y₂ ⨿ (F.obj Y₁).obj X₂ ⟶ (F.obj Y₁).obj Y₂`, and show that it is a colimit when both coforks are colimit, the cokernel of `f₁` is preserved by `F.obj…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/BifunctorCokernel.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.CatCospanTransform", "region_id": "category_theory", "micro_elevation": 0.1579, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 15.889, "z": 16.436, "size": 0.2676, "title": "Morphisms of categorical cospans.", "summary": "Given `F : A ⥤ B`, `G : C ⥤ B`, `F' : A' ⥤ B'` and `G' : C' ⥤ B'`, this file defines `CatCospanTransform F G F' G'`, the category of \"categorical transformations\" from the (categorical) cospan `F G` to the (categorical) cospan `F' G'`. Such a transformation consists of a diagram ``` F G A ⥤ B ⥢ C H₁| |H₂ |H₃ v v v A'⥤ B'⥢ C' F' G' ``` with specified `CatCommSq`s expressing 2-commutativity of the squares. These…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Categorical/CatCospanTransform.html"}, {"id": "Mathlib.CategoryTheory.ConcreteCategory.Representable", "region_id": "category_theory", "micro_elevation": 0.386, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": -6.687, "z": 29.436, "size": 0.2731, "title": "Representable functors in concrete categories", "summary": "This file provides some API for the situation `(F ⋙ forget D).RepresentableBy Y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ConcreteCategory/Representable.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.StrictInitial", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.0484, "macro_tier_override": null, "x": 7.852, "z": 28.0, "size": 0.3185, "title": "Strict initial objects", "summary": "This file sets up the basic theory of strict initial objects: initial objects where every morphism to it is an isomorphism. This generalises a property of the empty set in the category of sets: namely that the only function to the empty set is from itself. We say `C` has strict initial objects if every initial object is strict, i.e. given any morphism `f : A ⟶ I` where `I` is initial, then `f` is an isomorphism.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.TStructure.TruncLEGT", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -13.052, "z": 59.436, "size": 0.2934, "title": "Truncations for a t-structure", "summary": "Let `t` be a t-structure on a (pre)triangulated category `C`. In this file, for any `n : ℤ`, we introduce the truncation functors `t.truncLE n : C ⥤ C` and `t.truncGT n : C ⥤ C`, as variants of the functors `t.truncLT n : C ⥤ C` and `t.truncGE n : C ⥤ C` introduced in the file `Mathlib/CategoryTheory/Triangulated/TStructure/TruncLTGE.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/TStructure/TruncLEGT.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": 3.775, "z": 58.436, "size": 0.3092, "title": "Truncations for a t-structure", "summary": "Let `t` be a t-structure on a (pre)triangulated category `C`. In this file, for any `n : ℤ`, we construct truncation functors `t.truncLT n : C ⥤ C`, `t.truncGE n : C ⥤ C` and natural transformations `t.truncLTι n : t.truncLT n ⟶ 𝟭 C`, `t.truncGEπ n : 𝟭 C ⟶ t.truncGE n` and `t.truncGEδLT n : t.truncGE n ⟶ t.truncLT n ⋙ shiftFunctor C (1 : ℤ)` which are part of a distinguished triangle `(t.truncLT n).obj X ⟶ X ⟶…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/TStructure/TruncLTGE.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Mod", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": 67.513, "z": 36.436, "size": 0.2858, "title": "The category of module objects over a monoid object.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Mod.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -46.522, "z": 55.436, "size": 0.2, "title": "Abelian subcategories of triangulated categories", "summary": "Let `ι : A ⥤ C` be a fully faithful additive functor where `A` is an additive category and `C` is a triangulated category. We show that `A` is an abelian category if the following conditions are satisfied: * For any object `X` and `Y` in `A`, there is no nonzero morphism `ι.obj X ⟶ (ι.obj Y)⟦n⟧` when `n < 0`. * Any morphism `f₁ : X₁ ⟶ X₂` in `A` is admissible, i.e. when we complete `ι.obj f₁` in a distinguished…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/TStructure/AbelianSubcategory.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Triangulated", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 102, "macro_tier_score": 0.01, "macro_tier_override": null, "x": -4.242, "z": 54.436, "size": 0.2912, "title": "Triangulated Categories", "summary": "This file contains the definition of triangulated categories, which are pretriangulated categories which satisfy the octahedron axiom.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Triangulated.html"}, {"id": "Mathlib.CategoryTheory.Abelian.SerreClass.MorphismProperty", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 56.725, "z": 49.436, "size": 0.2617, "title": "The class of isomorphisms modulo a Serre class", "summary": "Let `C` be an abelian category and `P : ObjectProperty C` a Serre class. We define `P.isoModSerre : MorphismProperty C`, which is the class of morphisms `f` such that `kernel f` and `cokernel f` satisfy `P`. We show that `P.isoModSerre` is multiplicative, satisfies the two out of three property and is stable under retracts. (Similarly, we define `P.monoModSerre` and `P.epiModSerre`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/SerreClass/MorphismProperty.html"}, {"id": "Mathlib.CategoryTheory.Abelian.SerreClass.Basic", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 52.099, "z": 37.0, "size": 0.2531, "title": "Serre classes", "summary": "For any abelian category `C`, we introduce a type class `IsSerreClass C` for Serre classes in `C` (also known as \"Serre subcategories\"). A Serre class is a property `P : ObjectProperty C` of objects in `C` which holds for a zero object, and is closed under subobjects, quotients and extensions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/SerreClass/Basic.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Functor.Cat", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 19.886, "z": 20.436, "size": 0.2633, "title": "Pseudofunctors to Cat", "summary": "In this file, we state naturality properties of `mapId'` and `mapComp'` for pseudofunctors to `Cat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Functor/Cat.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.SplitEqualizer", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 103, "macro_tier_score": 0.096, "macro_tier_override": null, "x": 64.605, "z": 39.436, "size": 0.3171, "title": "Split Equalizers", "summary": "We define what it means for a triple of morphisms `f g : X ⟶ Y`, `ι : W ⟶ X` to be a split equalizer: there is a retraction `r` of `ι` and a retraction `t` of `g`, which additionally satisfy `t ≫ f = r ≫ ι`. In addition, we show that every split equalizer is an equalizer (`CategoryTheory.IsSplitEqualizer.isEqualizer`) and absolute (`CategoryTheory.IsSplitEqualizer.map`) A pair `f g : X ⟶ Y` has a split equalizer if…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/SplitEqualizer.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Schur", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -12.985, "z": 47.0, "size": 0.2542, "title": "Schur's lemma", "summary": "We first prove the part of Schur's Lemma that holds in any preadditive category with kernels, that any nonzero morphism between simple objects is an isomorphism. Second, we prove Schur's lemma for `𝕜`-linear categories with finite-dimensional hom spaces, over an algebraically closed field `𝕜`: the hom space `X ⟶ Y` between simple objects `X` and `Y` is at most one dimensional, and is 1-dimensional iff `X` and `Y`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Schur.html"}, {"id": "Mathlib.CategoryTheory.Distributive.Cartesian", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 90.072, "z": 51.436, "size": 0.2, "title": "Distributive categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Distributive/Cartesian.html"}, {"id": "Mathlib.CategoryTheory.Comma.StructuredArrow.Small", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.0531, "macro_tier_override": null, "x": 48.302, "z": 33.436, "size": 0.3102, "title": "Small sets in the category of structured arrows", "summary": "Here we prove a technical result about small sets in the category of structured arrows that will be used in the proof of the Special Adjoint Functor Theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/StructuredArrow/Small.html"}, {"id": "Mathlib.CategoryTheory.Shift.Linear", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -50.039, "z": 44.0, "size": 0.2776, "title": "Localization of the linearity of the shift functors", "summary": "If `L : C ⥤ D` is a localization functor with respect to `W : MorphismProperty C` and both `C` and `D` have been equipped with `R`-linear category structures such that `L` is `R`-linear and the shift functors on `C` are `R`-linear, then the shift functors on `D` are `R`-linear.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/Linear.html"}, {"id": "Mathlib.CategoryTheory.Localization.Linear", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 10.748, "z": 50.436, "size": 0.2694, "title": "Localization of linear categories", "summary": "If `L : C ⥤ D` is an additive localization functor between preadditive categories, and `C` is `R`-linear, we show that `D` can also be equipped with an `R`-linear structure such that `L` is an `R`-linear functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Linear.html"}, {"id": "Mathlib.CategoryTheory.Limits.Indization.FilteredColimits", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 102, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -41.389, "z": 57.436, "size": 0.2739, "title": "Ind-objects are closed under filtered colimits", "summary": "We show that if `F : I ⥤ Cᵒᵖ ⥤ Type v` is a functor such that `I` is small and filtered and `F.obj i` is an ind-object for all `i`, then `colimit F` is also an ind-object. Our proof is a slight variant of the proof given in Kashiwara-Schapira.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Indization/FilteredColimits.html"}, {"id": "Mathlib.CategoryTheory.Galois.Examples", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 48.807, "z": 52.436, "size": 0.27, "title": "Examples of Galois categories and fiber functors", "summary": "We show that for a group `G` the category of finite `G`-sets is a `PreGaloisCategory` and that the forgetful functor to `FintypeCat` is a `FiberFunctor`. The connected finite `G`-sets are precisely the ones with transitive `G`-action.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Galois/Examples.html"}, {"id": "Mathlib.CategoryTheory.Presentable.CardinalFilteredPresentation", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 102, "macro_tier_score": 0.0152, "macro_tier_override": null, "x": 4.866, "z": 58.436, "size": 0.3308, "title": "Presentable generators", "summary": "Let `C` be a category, a regular cardinal `κ` and `P : ObjectProperty C`. We define a predicate `P.IsCardinalFilteredGenerator κ` saying that `P` consists of `κ`-presentable objects and that any object in `C` is a `κ`-filtered colimit of objects satisfying `P`. We show that if this condition is satisfied, then `P` is a strong generator (see `IsCardinalFilteredGenerator.isStrongGenerator`). Moreover, if `C` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/CardinalFilteredPresentation.html"}, {"id": "Mathlib.CategoryTheory.Presentable.Limits", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 102, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": -3.903, "z": 57.436, "size": 0.2796, "title": "Colimits of presentable objects", "summary": "In this file, we show that `κ`-accessible functors (to the category of types) are stable under limits indexed by a category `K` such that `HasCardinalLT (Arrow K) κ`. In particular, `κ`-presentable objects are stable by colimits indexed by a category `K` such that `HasCardinalLT (Arrow K) κ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/Limits.html"}, {"id": "Mathlib.CategoryTheory.Presentable.Retracts", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 102, "macro_tier_score": 0.0146, "macro_tier_override": null, "x": -11.618, "z": 57.436, "size": 0.2796, "title": "Presentable objects are stable under retracts", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/Retracts.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 103, "macro_tier_score": 0.0575, "macro_tier_override": null, "x": -15.63, "z": 45.436, "size": 0.2833, "title": "Normal mono categories with finite products and kernels have all equalizers.", "summary": "This, and the dual result, are used in the development of abelian categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Equalizers.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Images", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 103, "macro_tier_score": 0.0626, "macro_tier_override": null, "x": 78.0, "z": 44.436, "size": 0.3106, "title": "The abelian image and coimage.", "summary": "In an abelian category we usually want the image of a morphism `f` to be defined as `kernel (cokernel.π f)`, and the coimage to be defined as `cokernel (kernel.ι f)`. We make these definitions here, as `Abelian.image f` and `Abelian.coimage f` (without assuming the category is actually abelian), and later relate these to the usual categorical notions when in an abelian category. There is a canonical morphism…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Images.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Internal.Types.CommGrp_", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -61.873, "z": 46.0, "size": 0.2357, "title": "`CommGrp (Type u) ≌ CommGrpCat.{u}`", "summary": "The category of internal commutative group objects in `Type` is equivalent to the category of \"native\" bundled commutative groups. Moreover, this equivalence is compatible with the forgetful functors to `Type`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Internal/Types/CommGrp_.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 103, "macro_tier_score": 0.0595, "macro_tier_override": null, "x": -211.34, "z": 62.685, "size": 0.4045, "title": "Multi-(co)equalizers", "summary": "A *multiequalizer* is an equalizer of two morphisms between two products. Since both products and equalizers are limits, such an object is again a limit. This file provides the diagram whose limit is indeed such an object. In fact, it is well-known that any limit can be obtained as a multiequalizer. The dual construction (multicoequalizers) is also provided.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equifibered", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 103, "macro_tier_score": 0.043, "macro_tier_override": null, "x": 65.946, "z": 45.436, "size": 0.2561, "title": "Equifibered natural transformation", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Equifibered.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.Composition", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 103, "macro_tier_score": 0.1942, "macro_tier_override": null, "x": 155.074, "z": -90.661, "size": 0.4585, "title": "Compatibilities of properties of morphisms with respect to composition", "summary": "Given `P : MorphismProperty C`, we define the predicate `P.IsStableUnderComposition` which means that `P f → P g → P (f ≫ g)`. We also introduce the type classes `W.ContainsIdentities`, `W.IsMultiplicative`, and `W.HasTwoOutOfThreeProperty`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/Composition.html"}, {"id": "Mathlib.CategoryTheory.Sites.Point.Map", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -52.619, "z": 59.436, "size": 0.2, "title": "The image of a point by a cocontinuous functor", "summary": "Let `F : C ⥤ D` be a cocontinuous functor between sites `(C, J)` and `(D, K)`. Let `Φ` be a point of `(C, J)`. In this file, we define a point `Φ.map F K` of `(D, K)` and show that there are natural isomorphisms `(Φ.map F K).presheafFiber ≅ (Functor.whiskeringLeft _ _ A).obj F.op ⋙ Φ.presheafFiber` and `(Φ.map F K).sheafFiber ≅ F.sheafPushforwardContinuous A J K ⋙ Φ.sheafFiber` (the latter is defined only if `F` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Point/Map.html"}, {"id": "Mathlib.CategoryTheory.Category.Factorisation", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -2.543, "z": 27.0, "size": 0.2298, "title": "The Factorisation Category of a Category", "summary": "`Factorisation f` is the category containing as objects all factorisations of a morphism `f`. We show that `Factorisation f` always has an initial and a terminal object. TODO: Show that `Factorisation f` is isomorphic to a comma category in two ways. TODO: Make `MonoFactorisation f` a special case of a `Factorisation f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Factorisation.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.FunctorBicategory.Lax", "region_id": "category_theory", "micro_elevation": 0.2456, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 12.579, "z": 21.436, "size": 0.2, "title": "Bicategories of lax functors", "summary": "Given bicategories `B` and `C`, we give bicategory structures on `LaxFunctor B C` whose * objects are lax functors, * 1-morphisms are lax or oplax natural transformations, and * 2-morphisms are modifications.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/FunctorBicategory/Lax.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Modification.Lax", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 27.391, "z": 20.436, "size": 0.2676, "title": "Modifications between transformations of lax functors", "summary": "In this file we define modifications of lax and oplax transformations of lax functors. A modification `Γ` between lax transformations `η` and `θ` (of lax functors) consists of a family of 2-morphisms `Γ.app a : η.app a ⟶ θ.app a`, which for all 1-morphisms `f : a ⟶ b` satisfies the equation `app a ▷ G.map f ≫ θ.naturality f = η.naturality f ≫ F.map f ◁ app b`. Modifications between oplax transformations are defined…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Modification/Lax.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.Comparison", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -150.096, "z": 235.683, "size": 0.3182, "title": "Connections between the regular, extensive and coherent topologies", "summary": "This file compares the regular, extensive and coherent topologies.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/Comparison.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.Ring", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 79.597, "z": 54.436, "size": 0.2, "title": "Yoneda embedding of `RingCatObj C`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/Ring.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Ring", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 25.954, "z": 53.436, "size": 0.2478, "title": "Ring objects in cartesian monoidal categories", "summary": "If `C` is a cartesian monoidal category and `X : C`, we introduce a typeclass `RingObj X` which says that `X` is a ring object: it has a commutative additive group structure and a multiplicative monoid structure that is distributive over the additive structure. We also introduce a typeclass `CommRingObj X` which further requires that the multiplicative law is commutative. The categories of bundled ring objects and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Ring.html"}, {"id": "Mathlib.CategoryTheory.CommSq", "region_id": "category_theory", "micro_elevation": 0.2807, "macro_tier": 103, "macro_tier_score": 0.3031, "macro_tier_override": null, "x": 0.393, "z": 23.436, "size": 0.5769, "title": "Commutative squares", "summary": "This file provides an API for commutative squares in categories. If `top`, `left`, `right` and `bottom` are four morphisms which are the edges of a square, `CommSq top left right bottom` is the predicate that this square is commutative. The structure `CommSq` is extended in `Mathlib/CategoryTheory/Limits/Shapes/Pullback/IsPullback/Defs.lean` as `IsPullback` and `IsPushout` in order to define pullback and pushout…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/CommSq.html"}, {"id": "Mathlib.CategoryTheory.Sites.LocallySurjective", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 102, "macro_tier_score": 0.0221, "macro_tier_override": null, "x": 94.934, "z": 54.436, "size": 0.4358, "title": "Locally surjective morphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/LocallySurjective.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Multiequalizer", "region_id": "category_theory", "micro_elevation": 0.5789, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -5.683, "z": 40.436, "size": 0.2427, "title": "Preservation of multicoequalizers", "summary": "Let `J : MultispanShape` and `d : MultispanIndex J C`. If `F : C ⥤ D`, we define `d.map F : MultispanIndex J D` and an isomorphism of functors `(d.map F).multispan ≅ d.multispan ⋙ F` (see `MultispanIndex.multispanMapIso`). If `c : Multicofork d`, we define `c.map F : Multicofork (d.map F)` and obtain a bijection `IsColimit (F.mapCocone c) ≃ IsColimit (c.map F)` (see `Multicofork.isColimitMapEquiv`). As a result, if…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Multiequalizer.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.MultiequalizerPullback", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 32.446, "z": 43.436, "size": 0.2427, "title": "Multicoequalizers that are pushouts", "summary": "In this file, we show that a multicoequalizer for `I : MultispanIndex (.ofLinearOrder ι) C` is also a pushout when `ι` has exactly two elements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/MultiequalizerPullback.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.FiniteMultiequalizer", "region_id": "category_theory", "micro_elevation": 0.5789, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -39.657, "z": 33.0, "size": 0.2298, "title": "Finiteness instances on multi-spans", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/FiniteMultiequalizer.html"}, {"id": "Mathlib.CategoryTheory.Localization.FiniteProducts", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 77.778, "z": 48.436, "size": 0.2, "title": "The localized category has finite products", "summary": "In this file, it is shown that if `L : C ⥤ D` is a localization functor for `W : MorphismProperty C` and that `W` is stable under finite products, then `D` has finite products, and `L` preserves finite products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/FiniteProducts.html"}, {"id": "Mathlib.CategoryTheory.Localization.Pi", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -11.403, "z": 34.436, "size": 0.2276, "title": "Localization of product categories", "summary": "In this file, it is shown that if for all `j : J` (with `J` finite), functors `L j : C j ⥤ D j` are localization functors with respect to a class of morphisms `W j : MorphismProperty (C j)`, then the product functor `Functor.pi L : (∀ j, C j) ⥤ ∀ j, D j` is a localization functor for the product class of morphisms `MorphismProperty.pi W`. The proof proceeds by induction on the cardinal of `J` using the main result…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Pi.html"}, {"id": "Mathlib.CategoryTheory.Enriched.Limits.HasConicalTerminal", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 80.296, "z": 56.436, "size": 0.2, "title": "Existence of conical terminal objects", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Enriched/Limits/HasConicalTerminal.html"}, {"id": "Mathlib.CategoryTheory.Groupoid.Basic", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -2.094, "z": 25.436, "size": 0.2338, "title": null, "summary": "This file defines a few basic properties of groupoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Groupoid/Basic.html"}, {"id": "Mathlib.CategoryTheory.Conj", "region_id": "category_theory", "micro_elevation": 0.3333, "macro_tier": 102, "macro_tier_score": 0.01, "macro_tier_override": null, "x": 46.016, "z": -34.83, "size": 0.2878, "title": "Conjugate morphisms by isomorphisms", "summary": "An isomorphism `α : X ≅ Y` defines - a monoid isomorphism `CategoryTheory.Iso.conj : End X ≃* End Y` by `α.conj f = α.inv ≫ f ≫ α.hom`; - a group isomorphism `CategoryTheory.Iso.conjAut : Aut X ≃* Aut Y` by `α.conjAut f = α.symm ≪≫ f ≪≫ α` using `CategoryTheory.Iso.homCongr : (X ≅ X₁) → (Y ≅ Y₁) → (X ⟶ Y) ≃ (X₁ ⟶ Y₁)` and `CategoryTheory.Iso.isoCongr : (f : X₁ ≅ X₂) → (g : Y₁ ≅ Y₂) → (X₁ ≅ Y₁) ≃ (X₂ ≅ Y₂)` which are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Conj.html"}, {"id": "Mathlib.CategoryTheory.Sites.Sheaf", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 103, "macro_tier_score": 0.0327, "macro_tier_override": null, "x": 49.953, "z": 48.436, "size": 0.4734, "title": "Sheaves taking values in a category", "summary": "If C is a category with a Grothendieck topology, we define the notion of a sheaf taking values in an arbitrary category `A`. We follow the definition in https://stacks.math.columbia.edu/tag/00VR, noting that the presheaf of sets \"defined above\" can be seen in the comments between tags 00VQ and 00VR on the page . The advantage of this definition is that we need no…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Sheaf.html"}, {"id": "Mathlib.CategoryTheory.Shift.SingleFunctors", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 101, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -27.511, "z": 38.0, "size": 0.3101, "title": "Functors from a category to a category with a shift", "summary": "Given a category `C`, and a category `D` equipped with a shift by a monoid `A`, we define a structure `SingleFunctors C D A` which contains the data of functors `functor a : C ⥤ D` for all `a : A` and isomorphisms `shiftIso n a a' h : functor a' ⋙ shiftFunctor D n ≅ functor a` whenever `n + a = a'`. These isomorphisms should satisfy certain compatibilities with respect to the shift on `D`. This notion is similar to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/SingleFunctors.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Kan.Adjunction", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 27.74, "z": 36.436, "size": 0.2, "title": "Adjunctions as Kan extensions", "summary": "We show that adjunctions are realized as Kan extensions or Kan lifts. We also show that a left adjoint commutes with a left Kan extension. Under the assumption that `IsLeftAdjoint h`, the isomorphism `f⁺ (g ≫ h) ≅ f⁺ g ≫ h` can be accessed by `Lan.CommuteWith.lanCompIso f g h`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Kan/Adjunction.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Kan.HasKan", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 66.252, "z": 35.436, "size": 0.239, "title": "Existence of Kan extensions and Kan lifts in bicategories", "summary": "We provide the propositional typeclass `HasLeftKanExtension f g`, which asserts that there exists a left Kan extension of `g` along `f`. See `CategoryTheory.Bicategory.Kan.IsKan` for the definition of left Kan extensions. Under the assumption that `HasLeftKanExtension f g`, we define the left Kan extension `lan f g` by using the axiom of choice.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Kan/HasKan.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Adjunction.Basic", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": 24.444, "z": 7.436, "size": 0.2731, "title": "Adjunctions in bicategories", "summary": "For 1-morphisms `f : a ⟶ b` and `g : b ⟶ a` in a bicategory, an adjunction between `f` and `g` consists of a pair of 2-morphisms `η : 𝟙 a ⟶ f ≫ g` and `ε : g ≫ f ⟶ 𝟙 b` satisfying the triangle identities. The 2-morphism `η` is called the unit and `ε` is called the counit.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Adjunction/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.End", "region_id": "category_theory", "micro_elevation": 0.5789, "macro_tier": 102, "macro_tier_score": 0.0197, "macro_tier_override": null, "x": 8.387, "z": 40.436, "size": 0.3112, "title": "Ends and coends", "summary": "In this file, given a functor `F : Jᵒᵖ ⥤ J ⥤ C`, we define its end `end_ F`, which is a suitable multiequalizer of the objects `(F.obj (op j)).obj j` for all `j : J`. For this shape of limits, cones are named wedges: the corresponding type is `Wedge F`. We also introduce `coend F` as multicoequalizers of `(F.obj (op j)).obj j` for all `j : J`. In these cases, cocones are named cowedges.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/End.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 102, "macro_tier_score": 0.0291, "macro_tier_override": null, "x": 13.571, "z": 39.436, "size": 0.2953, "title": "Wide equalizers and wide coequalizers", "summary": "This file defines wide (co)equalizers as special cases of (co)limits. A wide equalizer for the family of morphisms `X ⟶ Y` indexed by `J` is the categorical generalization of the subobject `{a ∈ A | ∀ j₁ j₂, f(j₁, a) = f(j₂, a)}`. Note that if `J` has fewer than two morphisms this condition is trivial, so some lemmas and definitions assume `J` is nonempty.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Sheaf", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 101, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -90.247, "z": 4.103, "size": 0.2782, "title": "AB axioms in sheaf categories", "summary": "If `J` is a Grothendieck topology on a small category `C : Type v`, and `A : Type u₁` (with `Category.{v} A`) is a Grothendieck abelian category, then `Sheaf J A` is a Grothendieck abelian category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Sheaf.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.FunctorCategory.PreservesLimits", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 53.101, "z": 51.436, "size": 0.2, "title": "Preservation of limits, as a property of objects in the functor category", "summary": "We make the typeclass `PreservesLimitsOfShape K` (resp. `PreservesFiniteLimits`) a property of objects in the functor category `J ⥤ C`, and show that it is stable under colimits of shape `K'` when they commute to limits of shape `K` (resp. to finite limits).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/FunctorCategory/PreservesLimits.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Opposites.Pullbacks", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 103, "macro_tier_score": 0.2033, "macro_tier_override": null, "x": 64.977, "z": 41.436, "size": 0.5815, "title": "Pullbacks and pushouts in `C` and `Cᵒᵖ`", "summary": "We construct pullbacks and pushouts in the opposite categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Opposites/Pullbacks.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.Equivalence", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 68.646, "z": 60.436, "size": 0.2, "title": "Coherence and equivalence of categories", "summary": "This file proves that the coherent and regular topologies transfer nicely along equivalences of categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/Equivalence.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Widesubcategory", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -5.019, "z": 38.436, "size": 0.2765, "title": "Monoidal structures on wide subcategories", "summary": "Given a monoidal category `C` and a morphism property `P : MorphismProperty C`, this file introduces conditions on `P` ensuring that `WideSubcategory P` inherits additional structures. We define stability classes under associators, unitors, and braidings, and use them to construct monoidal, braided, and symmetric structures on `WideSubcategory P`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Widesubcategory.html"}, {"id": "Mathlib.CategoryTheory.Widesubcategory", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": -2.547, "z": 25.436, "size": 0.2563, "title": "Wide subcategories", "summary": "A wide subcategory of a category `C` is a subcategory containing all the objects of `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Widesubcategory.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Internal.Limits", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 39.299, "z": 37.436, "size": 0.2403, "title": "Limits of monoid objects.", "summary": "If `C` has limits (of a given shape), so does `Mon C`, and the forgetful functor preserves these limits. (This could potentially replace many individual constructions for concrete categories, in particular `MonCat`, `SemiRingCat`, `RingCat`, and `AlgCat R`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Internal/Limits.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Preorder.TransfiniteCompositionOfShape", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 102, "macro_tier_score": 0.0241, "macro_tier_override": null, "x": -21.637, "z": 57.436, "size": 0.2692, "title": "A structure to describe transfinite compositions", "summary": "Given a well-ordered type `J` and a morphism `f : X ⟶ Y` in a category, we introduce a structure `TransfiniteCompositionOfShape J f` expressing that `f` is a transfinite composition of shape `J`. This allows to extend this structure in order to require more properties or data for the morphisms `F.obj j ⟶ F.obj (Order.succ j)` which appear in the transfinite composition. See…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Preorder/TransfiniteCompositionOfShape.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Preorder", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 102, "macro_tier_score": 0.0287, "macro_tier_override": null, "x": -33.848, "z": 56.436, "size": 0.2415, "title": "Preservation of well order continuous functors", "summary": "Given a well-ordered type `J` and a functor `G : C ⥤ D`, we define a type class `PreservesWellOrderContinuousOfShape J G` saying that `G` preserves colimits of shape `Set.Iio j` for any limit element `j : J`. It follows that if `F : J ⥤ C` is well order continuous, then so is `F ⋙ G`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Preorder.html"}, {"id": "Mathlib.CategoryTheory.Sites.Finite", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 76.243, "z": 47.436, "size": 0.2, "title": "The Finite Pretopology", "summary": "In this file we define the finite pretopology on a category, which consists of presieves that contain only finitely many arrows.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Finite.html"}, {"id": "Mathlib.CategoryTheory.Sites.Pretopology", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 102, "macro_tier_score": 0.0295, "macro_tier_override": null, "x": 70.358, "z": 46.436, "size": 0.3317, "title": "Grothendieck pretopologies", "summary": "Definition and lemmas about Grothendieck pretopologies. A Grothendieck pretopology for a category `C` is a set of families of morphisms with fixed codomain, satisfying certain closure conditions. We show that a pretopology generates a genuine Grothendieck topology, and every topology has a maximal pretopology which generates it. The pretopology associated to a topological space is defined in `Spaces.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Pretopology.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Opposites", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 103, "macro_tier_score": 0.0388, "macro_tier_override": null, "x": 31.276, "z": 30.436, "size": 0.3157, "title": "Opposite adjunctions", "summary": "This file contains constructions to relate adjunctions of functors to adjunctions of their opposites.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Opposites.html"}, {"id": "Mathlib.CategoryTheory.Presentable.ColimitPresentation", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 102, "macro_tier_score": 0.0096, "macro_tier_override": null, "x": 81.378, "z": 58.436, "size": 0.2422, "title": "Presentation of a colimit of objects equipped with a presentation", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/ColimitPresentation.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Injective.Basic", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 84.52, "z": 52.436, "size": 0.2, "title": "Injective objects in abelian categories", "summary": "* Objects in an abelian category are injective if and only if the preadditive Yoneda functor on them preserves finite colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Injective/Basic.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Internal.Module", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 56.53, "z": 36.436, "size": 0.2, "title": "`Mon (ModuleCat R) ≌ AlgCat R`", "summary": "The category of internal monoid objects in `ModuleCat R` is equivalent to the category of \"native\" bundled `R`-algebras. Moreover, this equivalence is compatible with the forgetful functors to `ModuleCat R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Internal/Module.html"}, {"id": "Mathlib.CategoryTheory.Functor.KanExtension.Adjunction", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 103, "macro_tier_score": 0.0887, "macro_tier_override": null, "x": 67.033, "z": 74.011, "size": 0.4331, "title": "The Kan extension functor", "summary": "Given a functor `L : C ⥤ D`, we define the left Kan extension functor `L.lan : (C ⥤ H) ⥤ (D ⥤ H)` which sends a functor `F : C ⥤ H` to its left Kan extension along `L`. This is defined if all `F` have such a left Kan extension. It is shown that `L.lan` is the left adjoint to the functor `(D ⥤ H) ⥤ (C ⥤ H)` given by the precomposition with `L` (see `Functor.lanAdjunction`). Similarly, we define the right Kan…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/KanExtension/Adjunction.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Exact", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 102, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.328, "title": "Exact sequences in abelian categories", "summary": "In an abelian category, we get several interesting results related to exactness which are not true in more general settings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Exact.html"}, {"id": "Mathlib.CategoryTheory.Monad.Products", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.1432, "macro_tier_override": null, "x": -17.441, "z": 35.436, "size": 0.2791, "title": "Algebras for the coproduct monad", "summary": "The functor `Y ↦ X ⨿ Y` forms a monad, whose category of monads is equivalent to the under category of `X`. Similarly, `Y ↦ X ⨯ Y` forms a comonad, whose category of coalgebras is equivalent to the over category of `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monad/Products.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Mat", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 8.516, "z": 49.436, "size": 0.2, "title": "Matrices over a category.", "summary": "When `C` is a preadditive category, `Mat_ C` is the preadditive category whose objects are finite tuples of objects in `C`, and whose morphisms are matrices of morphisms from `C`. There is a functor `Mat_.embedding : C ⥤ Mat_ C` sending morphisms to one-by-one matrices. `Mat_ C` has finite biproducts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Mat.html"}, {"id": "Mathlib.CategoryTheory.Functor.KanExtension.Basic", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 103, "macro_tier_score": 0.0877, "macro_tier_override": null, "x": -2.773, "z": 37.436, "size": 0.3903, "title": "Kan extensions", "summary": "The basic definitions for Kan extensions of functors are introduced in this file. Part of API is parallel to the definitions for bicategories (see `CategoryTheory.Bicategory.Kan.IsKan`). (The bicategory API cannot be used directly here because it would not allow the universe polymorphism which is necessary for some applications.) Given a natural transformation `α : L ⋙ F' ⟶ F`, we define the property…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/KanExtension/Basic.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.Factorization", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 103, "macro_tier_score": 0.2164, "macro_tier_override": null, "x": 136.54, "z": -90.661, "size": 0.3971, "title": "The factorization axiom", "summary": "In this file, we introduce a type-class `HasFactorization W₁ W₂`, which, given two classes of morphisms `W₁` and `W₂` in a category `C`, asserts that any morphism in `C` can be factored as a morphism in `W₁` followed by a morphism in `W₂`. The data of such factorizations can be packaged in the type `FactorizationData W₁ W₂`. This shall be used in the formalization of model categories for which the CM5 axiom asserts…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/Factorization.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.Yoneda", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.2157, "macro_tier_override": null, "x": -13.824, "z": 34.436, "size": 0.3573, "title": "Cones and limits", "summary": "In this file, we give the natural isomorphism between cones on `F` with cone point `X` and the type `lim Hom(X, F·)`, and similarly the natural isomorphism between cocones on `F` with cocone point `X` and the type `lim Hom(F·, X)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/Yoneda.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.LocalClosure", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 80.82, "z": 51.436, "size": 0.2, "title": "Local closure of morphism properties", "summary": "We define the source local closure of a morphism property `P` w.r.t. a precoverage `K` as the weakest property containing `P` that is `K`-local on the source.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/LocalClosure.html"}, {"id": "Mathlib.CategoryTheory.Sites.Whiskering", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0208, "macro_tier_override": null, "x": 55.516, "z": 49.436, "size": 0.3791, "title": null, "summary": "In this file we construct the functor `Sheaf J A ⥤ Sheaf J B` between sheaf categories obtained by composition with a functor `F : A ⥤ B`. In order for the sheaf condition to be preserved, `F` must preserve the correct limits. The lemma `Presheaf.IsSheaf.comp` says that composition with such an `F` indeed preserves the sheaf condition. The functor between sheaf categories is called `sheafCompose J F`. Given a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Whiskering.html"}, {"id": "Mathlib.CategoryTheory.Filtered.Grothendieck", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 70.931, "z": 41.436, "size": 0.2, "title": "Filteredness of Grothendieck construction", "summary": "We show that if `F : C ⥤ Cat` is such that `C` is filtered and `F.obj c` is filtered for all `c : C`, then `Grothendieck F` is filtered.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Filtered/Grothendieck.html"}, {"id": "Mathlib.CategoryTheory.Linear.FunctorCategory", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 102, "macro_tier_score": 0.0096, "macro_tier_override": null, "x": 47.25, "z": 46.436, "size": 0.2393, "title": "Linear structure on functor categories", "summary": "If `C` and `D` are categories and `D` is `R`-linear, then `C ⥤ D` is also `R`-linear.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Linear/FunctorCategory.html"}, {"id": "Mathlib.CategoryTheory.Localization.Construction", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 103, "macro_tier_score": 0.065, "macro_tier_override": null, "x": 52.486, "z": 31.436, "size": 0.4353, "title": "Construction of the localized category", "summary": "This file constructs the localized category, obtained by formally inverting a class of maps `W : MorphismProperty C` in a category `C`. We first construct a quiver `LocQuiver W` whose objects are the same as those of `C` and whose maps are the maps in `C` and placeholders for the formal inverses of the maps in `W`. The localized category `W.Localization` is obtained by taking the quotient of the path category of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Construction.html"}, {"id": "Mathlib.CategoryTheory.Sites.Hypercover.ZeroFamily", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 155.635, "z": -27.328, "size": 0.258, "title": "Defining precoverages via pre-`0`-hypercovers", "summary": "A precoverage is a condition on all presieves. In some applications, it is practical to instead define a condition on all pre-`0`-hypercovers. Such a condition for every object is a pre-`0`-hypercover family if these conditions are invariant under deduplication.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Hypercover/ZeroFamily.html"}, {"id": "Mathlib.CategoryTheory.LiftingProperties.Basic", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 103, "macro_tier_score": 0.2937, "macro_tier_override": null, "x": 42.55, "z": 27.436, "size": 0.5783, "title": "Lifting properties", "summary": "This file defines the lifting property of two morphisms in a category and shows basic properties of this notion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/LiftingProperties/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Preorder.WellOrderContinuous", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 102, "macro_tier_score": 0.0287, "macro_tier_override": null, "x": 59.867, "z": 55.436, "size": 0.2548, "title": "Continuity of functors from well-ordered types", "summary": "Let `F : J ⥤ C` be a functor from a well-ordered type `J`. We introduce the typeclass `F.IsWellOrderContinuous` to say that if `m` is a limit element, then `F.obj m` is the colimit of the `F.obj j` for `j < m`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Preorder/WellOrderContinuous.html"}, {"id": "Mathlib.CategoryTheory.Filtered.OfColimitCommutesFiniteLimit", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 7.853, "z": 42.436, "size": 0.2403, "title": "If colimits of shape `K` commute with finite limits, then `K` is filtered.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Filtered/OfColimitCommutesFiniteLimit.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Kan.IsKan", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 44.304, "z": 34.436, "size": 0.2541, "title": "Kan extensions and Kan lifts in bicategories", "summary": "The left Kan extension of a 1-morphism `g : a ⟶ c` along a 1-morphism `f : a ⟶ b` is the initial object in the category of left extensions `LeftExtension f g`. The universal property can be accessed by the following definition and lemmas: * `LeftExtension.IsKan.desc`: the family of 2-morphisms out of the left Kan extension. * `LeftExtension.IsKan.fac`: the unit of any left extension factors through the left Kan…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Kan/IsKan.html"}, {"id": "Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 174.116, "z": -17.328, "size": 0.2524, "title": "Equivalence of categories of sheaves with a dense subsite that is 1-hypercover dense", "summary": "Let `F : C₀ ⥤ C` be a functor equipped with Grothendieck topologies `J₀` and `J`. Assume that `F` is a dense subsite. We introduce a typeclass `IsOneHypercoverDense.{w} F J₀ J` which roughly says that objects in `C` admits a `1`-hypercover consisting of objects in `C₀`. Under the assumption that the coefficient category `A` has limits of size `w`, we show that the restriction functor `sheafPushforwardContinuous F A…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/DenseSubsite/OneHypercoverDense.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Limits.Shapes.Pullback", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": -29.472, "z": 43.436, "size": 0.2583, "title": "Pullbacks and pushouts in a monoidal category", "summary": "For numerous simp lemmas of the form `f ≫ g = h`, we add accompanying simp lemmas of the form `Q ◁ f ≫ Q ◁ g = Q ◁ h` and `f ▷ Q ≫ g ▷ Q = h ▷ Q`. This file and `Mathlib.CategoryTheory.Monoidal.Limits.HasLimits` are needed to define a monoidal category structure in `Mathlib.CategoryTheory.Monoidal.Arrow`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Limits/Shapes/Pullback.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Discrete", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.1504, "macro_tier_override": null, "x": -4.132, "z": 33.436, "size": 0.4244, "title": "Monoids as discrete monoidal categories", "summary": "The discrete category on a monoid is a monoidal category. Multiplicative morphisms induce monoidal functors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Discrete.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Span.Basic", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -21.386, "z": 51.436, "size": 0.2, "title": "Bicategories of spans in a category", "summary": "In this file, given a category `C` and two morphism properties Wₗ and Wᵣ in C that are stable under compositions, contain identities and such that for any morphism `b : x₃ ⟶ x₄` in Wₗ and any morphism `r : x₂ → x₃` in Wᵣ, there exists a pullback square ``` t x₁ --> x₂ | | l | | r v v x₃ --> x₄ b ``` in `C` such that `t` satisfies `Wₗ` and `l` satisfies `Wᵣ`, we construct the bicategory of spans in C with left…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Span/Basic.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.ClosedUnderIsomorphisms", "region_id": "category_theory", "micro_elevation": 0.0351, "macro_tier": 103, "macro_tier_score": 0.4739, "macro_tier_override": null, "x": 21.853, "z": 9.436, "size": 0.4102, "title": "Properties of objects which are closed under isomorphisms", "summary": "Given a category `C` and `P : ObjectProperty C` (i.e. `P : C → Prop`), this file introduces the type class `P.IsClosedUnderIsomorphisms`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/ClosedUnderIsomorphisms.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.Over.Basic", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 102, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 200.971, "z": -38.494, "size": 0.3329, "title": "Limits in the over category", "summary": "Declare instances for limits in the over category: If `C` has finite wide pullbacks, `Over B` has finite limits, and if `C` has arbitrary wide pullbacks then `Over B` has limits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/Over/Basic.html"}, {"id": "Mathlib.CategoryTheory.Category.Cat.Adjunction", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 57.431, "z": 30.436, "size": 0.2, "title": "Adjunctions related to Cat, the category of categories", "summary": "The embedding `typeToCat: Type ⥤ Cat`, mapping a type to the corresponding discrete category, is left adjoint to the functor `Cat.objects`, which maps a category to its set of objects. Another functor `connectedComponents : Cat ⥤ Type` maps a category to the set of its connected components and functors to functions between those sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Cat/Adjunction.html"}, {"id": "Mathlib.CategoryTheory.ConnectedComponents", "region_id": "category_theory", "micro_elevation": 0.3333, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 30.621, "z": 26.436, "size": 0.239, "title": "Connected components of a category", "summary": "Defines a type `ConnectedComponents J` indexing the connected components of a category, and the full subcategories giving each connected component: `Component j : Type u₁`. We show that each `Component j` is in fact connected. We show every category can be expressed as a disjoint union of its connected components, in particular `Decomposed J` is the category (definitionally) given by the sigma-type of the connected…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ConnectedComponents.html"}, {"id": "Mathlib.CategoryTheory.FiberedCategory.Cocartesian", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 32.636, "z": 25.436, "size": 0.2, "title": "Co-Cartesian morphisms", "summary": "This file defines co-Cartesian resp. strongly co-Cartesian morphisms with respect to a functor `p : 𝒳 ⥤ 𝒮`. This file has been adapted from `Mathlib/CategoryTheory/FiberedCategory/Cartesian.lean`, please try to change them in sync.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/FiberedCategory/Cocartesian.html"}, {"id": "Mathlib.CategoryTheory.Sites.ConstantSheaf", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 102, "macro_tier_score": 0.0102, "macro_tier_override": null, "x": -250.697, "z": -99.756, "size": 0.3066, "title": "The constant sheaf", "summary": "We define the constant sheaf functor (the sheafification of the constant presheaf) `constantSheaf : D ⥤ Sheaf J D` and prove that it is left adjoint to evaluation at a terminal object (see `constantSheafAdj`). We also define a predicate on sheaves, `Sheaf.IsConstant`, saying that a sheaf is in the essential image of the constant sheaf functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/ConstantSheaf.html"}, {"id": "Mathlib.CategoryTheory.Endofunctor.Algebra", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 56.006, "z": 32.436, "size": 0.2478, "title": "Algebras of endofunctors", "summary": "This file defines (co)algebras of an endofunctor, and provides the category instance for them. It also defines the forgetful functor from the category of (co)algebras. It is shown that the structure map of the initial algebra of an endofunctor is an isomorphism. Furthermore, it is shown that for an adjunction `F ⊣ G` the category of algebras over `F` is equivalent to the category of coalgebras over `G`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Endofunctor/Algebra.html"}, {"id": "Mathlib.CategoryTheory.Limits.ConcreteCategory.WithAlgebraicStructures", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -29.438, "z": 43.436, "size": 0.2, "title": "Colimits in ModuleCat", "summary": "Let `C` be a concrete category and `F : J ⥤ C` a filtered diagram in `C`. We discuss some results about `colimit F` when objects and morphisms in `C` have some algebraic structures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/ConcreteCategory/WithAlgebraicStructures.html"}, {"id": "Mathlib.CategoryTheory.LiftingProperties.Limits", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 102, "macro_tier_score": 0.0195, "macro_tier_override": null, "x": -1.897, "z": 43.436, "size": 0.2877, "title": "Lifting properties and (co)limits", "summary": "In this file, we show some consequences of lifting properties in the presence of certain (co)limits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/LiftingProperties/Limits.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Injective.Dimension", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2239, "title": "Injective dimension", "summary": "In an abelian category `C`, we shall say that `X : C` has Injective dimension `< n` if all `Ext Y X i` vanish when `n ≤ i`. This defines a type class `HasInjectiveDimensionLT X n`. We also define a type class `HasInjectiveDimensionLE X n` as an abbreviation for `HasInjectiveDimensionLT X (n + 1)`. (Note that the fact that `X` is a zero object is equivalent to the condition `HasInjectiveDimensionLT X 0`, but this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Injective/Dimension.html"}, {"id": "Mathlib.CategoryTheory.Sites.CoversTop.Over", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -40.518, "z": 54.0, "size": 0.2559, "title": "CoversTop in over-categories", "summary": "This file contains a transitivity lemma for `GrothendieckTopology.CoversTop`: if a family `X : I → C` covers the top for `J`, and for each `i` a family `Y i` covers the top for the induced topology on `Over (X i)`, then the combined family covers the top for `J`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/CoversTop/Over.html"}, {"id": "Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Kernels", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -17.452, "z": 45.436, "size": 0.2, "title": "(Co)kernels in functor categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FunctorCategory/Shapes/Kernels.html"}, {"id": "Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 102, "macro_tier_score": 0.0144, "macro_tier_override": null, "x": -10.135, "z": 36.436, "size": 0.2505, "title": "Lemmas on fractions", "summary": "Let `W : MorphismProperty C`, and objects `X` and `Y` in `C`. In this file, we introduce structures like `W.LeftFraction₂ X Y` which consists of two left fractions with the \"same denominator\" which shall be important in the construction of the preadditive structure on the localized category when `C` is preadditive and `W` has a left calculus of fractions. When `W` has a left calculus of fractions, we generalize the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Lattice", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -28.216, "z": 42.436, "size": 0.2, "title": "Lattice Homs that Preserve Limits and Colimits", "summary": "This file provides instances for when OrderHom.toFunctor preserves limits/colimits. In particular, if `f` preserves finite infs/sups (i.e. is from a InfTopHomClass/SupBotHomClass) then `(toOrderHom f).toFunctor` preserves finite limits/colimits. If `f` preserves arbitrary infs/sups (i.e. is from a sInfHomClass/sSupHomClass) then `(toOrderHom f).toFunctor` preserves all limits/colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Lattice.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Monad.Basic", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -9.332, "z": 37.436, "size": 0.2, "title": "Comonads in a bicategory", "summary": "We define comonads in a bicategory `B` as comonoid objects in an endomorphism monoidal category. We show that this is equivalent to oplax functors from the trivial bicategory to `B`. From this, we show that comonads in `B` form a bicategory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Monad/Basic.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.FunctorBicategory.Oplax", "region_id": "category_theory", "micro_elevation": 0.2456, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 19.133, "z": 21.436, "size": 0.2338, "title": "Bicategories of oplax functors", "summary": "Given bicategories `B` and `C`, we give bicategory structures on `B ⥤ᵒᵖᴸ C` whose * objects are oplax functors, * 1-morphisms are lax or oplax natural transformations, and * 2-morphisms are modifications.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/FunctorBicategory/Oplax.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.End", "region_id": "category_theory", "micro_elevation": 0.2105, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 23.198, "z": 19.436, "size": 0.2585, "title": "Endomorphisms of an object in a bicategory, as a monoidal category.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/End.html"}, {"id": "Mathlib.CategoryTheory.Limits.IsConnected", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 103, "macro_tier_score": 0.0765, "macro_tier_override": null, "x": -28.038, "z": 42.436, "size": 0.2701, "title": "Colimits of connected index categories", "summary": "This file proves two characterizations of connected categories by means of colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/IsConnected.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Injective.InjectiveObject", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -62.293, "z": 42.0, "size": 0.2324, "title": "The full subcategory of injective objects", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Injective/InjectiveObject.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.EqToHom", "region_id": "category_theory", "micro_elevation": 0.1754, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 34.239, "z": 17.436, "size": 0.2, "title": "`eqToHom` in bicategories", "summary": "This file records some of the behavior of `eqToHom` 1-morphisms and 2-morphisms in bicategories. Given an equality of objects `h : x = y` in a bicategory, there is a 1-morphism `eqToHom h : x ⟶ y` just like in an ordinary category. The definitional property of this morphism is that if `h : x = x`, `eqToHom h = 𝟙 x`. This is implemented as the `eqToHom` morphism in the `CategoryStruct` underlying the bicategory.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/EqToHom.html"}, {"id": "Mathlib.CategoryTheory.Dialectica.Monoidal", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 82.923, "z": 53.436, "size": 0.2, "title": "The Dialectica category is symmetric monoidal", "summary": "We show that the category `Dial` has a symmetric monoidal category structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Dialectica/Monoidal.html"}, {"id": "Mathlib.CategoryTheory.Dialectica.Basic", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -38.975, "z": 51.436, "size": 0.239, "title": "Dialectica category", "summary": "We define the category `Dial` of the Dialectica interpretation, after [dialectica1989].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Dialectica/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.Over.Connected", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 102, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": -47.521, "z": 55.436, "size": 0.3354, "title": "Connected limits in the over category", "summary": "We show that the projection `CostructuredArrow K B ⥤ C` creates and preserves connected limits, without assuming that `C` has any limits. In particular, `CostructuredArrow K B` has any connected limit which `C` has. From this we deduce the corresponding results for the over category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/Over/Connected.html"}, {"id": "Mathlib.CategoryTheory.MarkovCategory.Basic", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -13.066, "z": 38.436, "size": 0.2541, "title": "Markov Categories", "summary": "Copy-discard categories where deletion is natural for all morphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MarkovCategory/Basic.html"}, {"id": "Mathlib.CategoryTheory.CopyDiscardCategory.Deterministic", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 102, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": -178.236, "z": 150.199, "size": 0.2846, "title": "Deterministic Morphisms in Copy-Discard Categories", "summary": "Morphisms that preserve the copy operation perfectly. A morphism `f : X → Y` is deterministic if copying then applying `f` to both copies equals applying `f` then copying: `f ≫ Δ[Y] = Δ[X] ≫ (f ⊗ f)`. In probabilistic settings, these are morphisms without randomness. In cartesian categories, all morphisms are deterministic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/CopyDiscardCategory/Deterministic.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Ext", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 63.401, "z": 45.0, "size": 0.2556, "title": "Ext", "summary": "We define `Ext R C n : Cᵒᵖ ⥤ C ⥤ ModuleCat R` for any `R`-linear abelian category `C` by (left) deriving in the first argument of the bifunctor `(X, Y) ↦ ModuleCat.of R (unop X ⟶ Y)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Ext.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Quadruple", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 81.468, "z": 45.436, "size": 0.2, "title": "Adjoint quadruples", "summary": "This file concerns adjoint quadruples `L ⊣ F ⊣ G ⊣ R` of functors `L G : C ⥤ D`, `F R : D ⥤ C`. We bundle the adjunctions in a structure `Quadruple L F G R` and make the two triples `Triple L F G` and `Triple F G R` accessible as `Quadruple.leftTriple` and `Quadruple.rightTriple`. Currently the only two results are the following: * When `F` and `R` are fully faithful, the components of the induced natural…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Quadruple.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Triple", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 59.966, "z": 34.436, "size": 0.2585, "title": "Adjoint triples", "summary": "This file concerns adjoint triples `F ⊣ G ⊣ H` of functors `F H : C ⥤ D`, `G : D ⥤ C`. We first prove that `F` is fully faithful iff `H` is, and then prove results about the two special cases where `G` is fully faithful or `F` and `H` are.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Triple.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.Assoc", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 103, "macro_tier_score": 0.0719, "macro_tier_override": null, "x": -23.658, "z": 39.436, "size": 0.2884, "title": "Associativity of pullbacks", "summary": "This file shows that pullbacks (and pushouts) are associative up to natural isomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Assoc.html"}, {"id": "Mathlib.CategoryTheory.SmallRepresentatives", "region_id": "category_theory", "micro_elevation": 0.2807, "macro_tier": 102, "macro_tier_score": 0.0096, "macro_tier_override": null, "x": 23.524, "z": 23.436, "size": 0.2471, "title": "Representatives of small categories", "summary": "Given a type `Ω : Type w`, we construct a structure `SmallCategoryOfSet Ω : Type w` which consists of the data and axioms that allows to define a category structure such that the type of objects and morphisms identify to subtypes of `Ω`. This allows to define a small family of small categories `SmallCategoryOfSet.categoryFamily : SmallCategoryOfSet Ω → Type w` which, up to equivalence, represents all categories such…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/SmallRepresentatives.html"}, {"id": "Mathlib.CategoryTheory.Comma.Arrow", "region_id": "category_theory", "micro_elevation": 0.2632, "macro_tier": 103, "macro_tier_score": 0.4333, "macro_tier_override": null, "x": 2.324, "z": 22.436, "size": 0.7066, "title": "The category of arrows", "summary": "The category of arrows, with morphisms commutative squares. We set this up as a specialization of the comma category `Comma L R`, where `L` and `R` are both the identity functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/Arrow.html"}, {"id": "Mathlib.CategoryTheory.Join.Sum", "region_id": "category_theory", "micro_elevation": 0.2105, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 37.641, "z": 19.436, "size": 0.2, "title": "Embedding of `C ⊕ D` into `C ⋆ D`", "summary": "This file constructs a canonical functor `Join.fromSum` from `C ⊕ D` to `C ⋆ D` and gives its characterization in terms of the canonical inclusions. We also provide `Faithful` and `EssSurj` instances on this functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Join/Sum.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.InheritedFromHom", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 29.928, "z": 25.436, "size": 0.2, "title": "Object properties transported along morphisms", "summary": "In this file we define the predicates `InheritedFromSource` and `InheritedFromTarget` for an object property `P` along a morphism property `Q`. `P` is inherited from the source (resp. target) along `Q` if for every morphism `f : X ⟶ Y` with `Q f`, `P X` implies `P Y` (resp. `P Y` implies `P X`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/InheritedFromHom.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.Projective.LiftingProperties", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.599, "z": 49.436, "size": 0.2, "title": "Characterization of projective objects in terms of lifting properties", "summary": "An object `P` is projective iff the morphism `0 ⟶ P` has the left lifting property with respect to epimorphisms, `projective_iff_llp_epimorphisms_zero`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/Projective/LiftingProperties.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.ExtensiveColimits", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -118.949, "z": -105.756, "size": 0.2302, "title": "Colimits in categories of extensive sheaves", "summary": "This file proves that `J`-shaped colimits of `A`-valued sheaves for the extensive topology are computed objectwise if `colim : J ⥤ A ⥤ A` preserves finite products. This holds for all shapes `J` if `A` is a preadditive category. This can also easily be applied to filtered `J` in the case when `A` is a category of sets, and eventually to sifted `J` once that API is developed.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveColimits.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 87.206, "z": 51.436, "size": 0.273, "title": "Sheaves for the extensive topology", "summary": "This file characterises sheaves for the extensive topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Opposites", "region_id": "category_theory", "micro_elevation": 0.1579, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 25.409, "z": 16.436, "size": 0.2478, "title": "Opposite bicategories", "summary": "We construct the 1-cell opposite of a bicategory `B`, called `Bᵒᵖ`. It is defined as follows * The objects of `Bᵒᵖ` correspond to objects of `B`. * The morphisms `X ⟶ Y` in `Bᵒᵖ` are the morphisms `Y ⟶ X` in `B`. * The 2-morphisms `f ⟶ g` in `Bᵒᵖ` are the 2-morphisms `f ⟶ g` in `B`. In other words, the directions of the 2-morphisms are preserved.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Opposites.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Pseudoelements", "region_id": "category_theory", "micro_elevation": 0.0175, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 25.939, "z": 8.436, "size": 0.2, "title": "Pseudoelements in abelian categories", "summary": "A *pseudoelement* of an object `X` in an abelian category `C` is an equivalence class of arrows ending in `X`, where two arrows are considered equivalent if we can find two epimorphisms with a common domain making a commutative square with the two arrows. While the construction shows that pseudoelements are actually subobjects of `X` rather than \"elements\", it is possible to chase these pseudoelements through…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Pseudoelements.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Coherence", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 7.832, "z": 27.436, "size": 0.2, "title": "The coherence theorem for bicategories", "summary": "In this file, we prove the coherence theorem for bicategories, stated in the following form: the free bicategory over any quiver is locally thin. The proof is almost the same as the proof of the coherence theorem for monoidal categories that has been previously formalized in mathlib, which is based on the proof described by Ilya Beylin and Peter Dybjer. The idea is to view a path on a quiver as a normal form of a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Coherence.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Whiskering", "region_id": "category_theory", "micro_elevation": 0.4035, "macro_tier": 102, "macro_tier_score": 0.0196, "macro_tier_override": null, "x": 173.136, "z": -84.661, "size": 0.2963, "title": null, "summary": "Given categories `C D E`, functors `F : D ⥤ E` and `G : E ⥤ D` with an adjunction `F ⊣ G`, we provide the induced adjunction between the functor categories `C ⥤ D` and `C ⥤ E`, and the functor categories `E ⥤ C` and `D ⥤ C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Whiskering.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Diagonal", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 103, "macro_tier_score": 0.0727, "macro_tier_override": null, "x": -30.519, "z": 46.436, "size": 0.3496, "title": "The diagonal object of a morphism.", "summary": "We provide various API and isomorphisms considering the diagonal object `Δ_{Y/X} := pullback f f` of a morphism `f : X ⟶ Y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Diagonal.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.EndoFunctor", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 64.43, "z": 49.436, "size": 0.2, "title": "Preadditive structure on algebras over a monad", "summary": "If `C` is a preadditive category and `F` is an additive endofunctor on `C` then `Algebra F` is also preadditive. Dually, the category `Coalgebra F` is also preadditive.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/EndoFunctor.html"}, {"id": "Mathlib.CategoryTheory.Limits.FintypeCat", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 24.639, "z": 42.436, "size": 0.2313, "title": "(Co)limits in the category of finite types", "summary": "We show that finite (co)limits exist in `FintypeCat` and that they are preserved by the natural inclusion `FintypeCat.incl`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FintypeCat.html"}, {"id": "Mathlib.CategoryTheory.GradedObject.Braiding", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -23.826, "z": 50.436, "size": 0.2, "title": "The braided and symmetric category structures on graded objects", "summary": "In this file, we construct the braiding `GradedObject.Monoidal.braiding : tensorObj X Y ≅ tensorObj Y X` for two objects `X` and `Y` in `GradedObject I C`, when `I` is a commutative additive monoid (and suitable coproducts exist in a braided category `C`). When `C` is a braided category and suitable assumptions are made, we obtain the braided category structure on `GradedObject I C` and show that it is symmetric if…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GradedObject/Braiding.html"}, {"id": "Mathlib.CategoryTheory.Quotient.Linear", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 101, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -6.851, "z": 43.0, "size": 0.2709, "title": "The quotient category is linear", "summary": "If `r : HomRel C` is a congruence on a preadditive category `C` which satisfies certain compatibilities, we have already defined a preadditive structure on `Quotient r` in the file `Mathlib/CategoryTheory/Quotient/Preadditive.lean` such that `functor r : C ⥤ Quotient r` is an additive functor. In this file, assuming moreover that `C` is an `R`-linear category and that the relation `r` is compatible with the scalar…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Quotient/Linear.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 102, "macro_tier_score": 0.006, "macro_tier_override": null, "x": 102.07, "z": 59.436, "size": 0.347, "title": "The pretriangulated structure on the opposite category", "summary": "In this file, we construct the pretriangulated structure on the opposite category `Cᵒᵖ` of a pretriangulated category `C`. The shift on `Cᵒᵖ` was constructed in `Mathlib.CategoryTheory.Triangulated.Opposite.Basic`, and is such that shifting by `n : ℤ` on `Cᵒᵖ` corresponds to the shift by `-n` on `C`. In `Mathlib.CategoryTheory.Triangulated.Opposite.Triangle`, we constructed an equivalence `(Triangle C)ᵒᵖ ≌ Triangle…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Opposite/Pretriangulated.html"}, {"id": "Mathlib.CategoryTheory.Comma.Over.StrictInitial", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 171.82, "z": -38.328, "size": 0.2268, "title": "`Over X` when `C` has strict initial objects", "summary": "In this file we define the canonical equivalence of `Over X` with `Discrete PUnit` when `C` has strict initial objects. We also provide the variants for `P.Over Q X` and the dual versions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/Over/StrictInitial.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Modification.Oplax", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 101, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": 4.943, "z": 20.436, "size": 0.3027, "title": "Modifications between transformations of oplax functors", "summary": "In this file we define modifications of lax, oplax, and strong transformations of oplax functors. A modification `Γ` between oplax transformations `η` and `θ` (of oplax functors) consists of a family of 2-morphisms `Γ.app a : η.app a ⟶ θ.app a`, which for all 1-morphisms `f : a ⟶ b` satisfies the equation `(F.map f ◁ app b) ≫ θ.naturality f = η.naturality f ≫ (app a ▷ G.map f)`. Modifications between lax and strong…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Modification/Oplax.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Yoneda", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 9.516, "z": 55.436, "size": 0.2465, "title": "Fullness of restrictions of `preadditiveCoyonedaObj`", "summary": "In this file we give a sufficient criterion for a restriction of the functor `preadditiveCoyonedaObj G` to be full: this is the case if `C` is an abelian category and `G : C` is a projective separator such that every object in the relevant subcategory is a quotient of `G`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Yoneda.html"}, {"id": "Mathlib.CategoryTheory.Pi.Basic", "region_id": "category_theory", "micro_elevation": 0.193, "macro_tier": 103, "macro_tier_score": 0.4447, "macro_tier_override": null, "x": 8.447, "z": 18.436, "size": 0.5437, "title": "Categories of indexed families of objects.", "summary": "We define the pointwise category structure on indexed families of objects in a category (and also the dependent generalization).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Pi/Basic.html"}, {"id": "Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 102, "macro_tier_score": 0.0203, "macro_tier_override": null, "x": -35.348, "z": 57.436, "size": 0.3486, "title": "The equivalence of categories of sheaves of a dense subsite", "summary": "If `G : C ⥤ D` exhibits `(C, J)` as a dense subsite of `(D, K)`, and `A` is a category with suitable limits, then the functor `G.sheafPushforwardContinuous A J K : Sheaf K A ⥤ Sheaf J A` is an equivalence of categories. The equivalence of categories can be obtained as `sheafEquiv J K G A` which is defined in the file `Mathlib/CategoryTheory/Sites/DenseSubsite/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/DenseSubsite/SheafEquiv.html"}, {"id": "Mathlib.CategoryTheory.Sites.DenseSubsite.Basic", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 102, "macro_tier_score": 0.0221, "macro_tier_override": null, "x": -126.477, "z": 240.683, "size": 0.4328, "title": "Dense subsites", "summary": "We define `IsCoverDense` functors into sites as functors such that there exists a covering sieve that factors through images of the functor for each object in `D`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/DenseSubsite/Basic.html"}, {"id": "Mathlib.CategoryTheory.UnivLE", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 4.984, "z": 27.436, "size": 0.2, "title": "Universe inequalities and essential surjectivity of `uliftFunctor`.", "summary": "We show `UnivLE.{max u v, v} ↔ EssSurj (uliftFunctor.{u, v} : Type v ⥤ Type max u v)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/UnivLE.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Preradical.Radical", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 90.981, "z": 52.436, "size": 0.2, "title": "Radicals", "summary": "In this file we define what it means for a preradical `Φ : Preradical C` on an abelian category `C` to be *radical*, and we define `Radical C` as the full subcategory of `Preradical C` consisting of radicals. Following Stenström, a preradical `Φ` is called radical if it coincides with its self colon. We encode this as the property that the natural transformation `toColon Φ Φ : Φ ⟶ Φ.colon Φ` is an isomorphism, and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Preradical/Radical.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Preradical.Colon", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -7.089, "z": 51.436, "size": 0.2676, "title": "The colon construction on preradicals", "summary": "Given preradicals `Φ` and `Ψ` on an abelian category `C`, this file defines their **colon** `Φ : Ψ` in the sense of Stenström. Following Stenström, one can realize the colon object `r : s` evaluated at `X : C` as the pullback of `X ⟶ X / r X` along `s (X / r X) ⟶ X / r X`. We encode this categorically by constructing `Φ : Ψ` as a pullback in the category of endofunctors of the canonical projection `Φ.π : 𝟭 C ⟶…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Preradical/Colon.html"}, {"id": "Mathlib.CategoryTheory.Functor.RegularEpi", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 198.883, "z": -69.661, "size": 0.2691, "title": "The category of type-valued sheaves is a regular epi category", "summary": "This file proves that when the target category `D` is a regular epi category (i.e. every epimorphism is regular) and has pushouts and kernel pairs of epimorphisms, the functor category `C ⥤ D` is a regular epi category. This is an instance that applies directly when `D` is `Type*`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/RegularEpi.html"}, {"id": "Mathlib.CategoryTheory.Comma.CardinalArrow", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 102, "macro_tier_score": 0.0195, "macro_tier_override": null, "x": -4.265, "z": 32.436, "size": 0.2907, "title": "Cardinal of Arrow", "summary": "We obtain various results about the cardinality of `Arrow C`. For example, if `C` is a (small) category, `Arrow C` is finite iff `FinCategory C` holds.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/CardinalArrow.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.AbelianImages", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 102, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -57.125, "z": 38.0, "size": 0.2845, "title": "Preservation of coimage-image comparisons", "summary": "If a functor preserves kernels and cokernels, then it preserves abelian images, abelian coimages and coimage-image comparisons.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/AbelianImages.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 247.117, "z": -33.328, "size": 0.2302, "title": "Filtered colimits of types", "summary": "In this file, given a functor `F : J ⥤ Type w₀` from a filtered category `J`, we compute the equivalence relation generated by `F.ColimitTypeRel` on `(j : J) × (F.obj j)`. Given `c : CoconeTypes F`, we deduce a lemma `Functor.CoconeTypes.injective_descColimitType_iff_of_isFiltered` which gives a concrete condition under which the map `F.descColimitType c : F.ColimitType → c.pt` is injective, which is an important…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/ColimitTypeFiltered.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.SpectralObject", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 60.387, "z": 49.0, "size": 0.2676, "title": "Spectral objects in triangulated categories", "summary": "In this file, we introduce the category `SpectralObject C ι` of spectral objects in a pretriangulated category `C` indexed by the category `ι`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/SpectralObject.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.SequentialProduct", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -243.085, "z": -111.756, "size": 0.2499, "title": "ℕ-indexed products as sequential limits", "summary": "Given sequences `M N : ℕ → C` of objects with morphisms `f n : M n ⟶ N n` for all `n`, this file exhibits `∏ M` as the limit of the tower ``` ⋯ → ∏_{n < m + 1} M n × ∏_{n ≥ m + 1} N n → ∏_{n < m} M n × ∏_{n ≥ m} N n → ⋯ → ∏ N ``` Further, we prove that the transition maps in this tower are epimorphisms, in the case when each `f n` is an epimorphism and `C` has finite biproducts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/SequentialProduct.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.PiProd", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 28.302, "z": 35.436, "size": 0.2333, "title": "A product as a binary product", "summary": "We write a product indexed by `I` as a binary product of the products indexed by a subset of `I` and its complement.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/PiProd.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.ParametrizedLimits", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -3.588, "z": 41.436, "size": 0.2484, "title": "Parametrized adjunctions and limits", "summary": "Given bifunctors `F : C₁ ⥤ C₂ ⥤ C₃`, `G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂` and a parametrized adjunction `adj₂ : F ⊣₂ G`, we show that for any `X₃ : C₃`, the functor `G.flip.obj X₃ : C₁ᵒᵖ ⥤ C₃` preserves limits of shape `J` if for any `X₂ : C₂`, the functor `F.flip.obj X₂ : C₁ ⥤ C₃` preserves colimits of shape `Jᵒᵖ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/ParametrizedLimits.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.InfSemilattice", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 89.124, "z": 50.436, "size": 0.2, "title": "The preorder category of a meet-semilattice with a greatest element is Cartesian monoidal", "summary": "The preorder category of a meet-semilattice `C` with a greatest element is Cartesian monoidal. A symmetric monoidal structure on the preorder category is automatically provided by the instance and `CartesianMonoidalCategory.toSymmetricCategory`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/InfSemilattice.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ColimCoyoneda", "region_id": "category_theory", "micro_elevation": 0.8772, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -50.784, "z": 57.436, "size": 0.2391, "title": "Morphisms to a colimit in a Grothendieck abelian category", "summary": "Let `C : Type u` be an abelian category `[Category.{v} C]` which satisfies `IsGrothendieckAbelian.{w} C`. We may expect that all the objects `X : C` are `κ`-presentable for some regular cardinal `κ`. However, we only prove a weaker result (which is enough in order to obtain the existence of enough injectives (TODO)): let `κ` be a big enough regular cardinal such that if `Y : J ⥤ C` is a functor from a `κ`-filtered…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckCategory/ColimCoyoneda.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Internal.FunctorCategory", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -11.848, "z": 37.436, "size": 0.2, "title": "`Mon (C ⥤ D) ≌ C ⥤ Mon D`", "summary": "When `D` is a monoidal category, monoid objects in `C ⥤ D` are the same thing as functors from `C` into the monoid objects of `D`. This is formalised as: * `monFunctorCategoryEquivalence : Mon (C ⥤ D) ≌ C ⥤ Mon D` The intended application is that as `Ring ≌ Mon Ab` (not yet constructed!), we have `presheaf Ring X ≌ presheaf (Mon Ab) X ≌ Mon (presheaf Ab X)`, and we can model a module over a presheaf of rings as a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.html"}, {"id": "Mathlib.CategoryTheory.Topos.Classifier", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -23.528, "z": 53.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Topos/Classifier.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Over", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -24.299, "z": 42.436, "size": 0.2, "title": "If a functor preserves limits, so does the induced functor in the `Over` or `Under` category", "summary": "Suppose we are given categories `C` and `D`, and object `X : C`, and a functor `F : C ⥤ D`. `F` induces a functor `Over.post F : Over X ⥤ Over (F.obj X)`. If `F` preserves limits of a certain shape `WithTerminal J`, then `Over.post F` preserves limits of shape `J`. As a corollary, if `F` preserves finite limits, or limits of a certain size, so does `Over.post F`. Dually, if `F` preserves certain colimits,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Over.html"}, {"id": "Mathlib.CategoryTheory.SmallObject.Construction", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 102, "macro_tier_score": 0.0096, "macro_tier_override": null, "x": 24.444, "z": 7.436, "size": 0.2419, "title": "Construction for the small object argument", "summary": "Given a family of morphisms `f i : A i ⟶ B i` in a category `C`, we define a functor `SmallObject.functor f : Arrow S ⥤ Arrow S` which sends an object given by arrow `πX : X ⟶ S` to the pushout `functorObj f πX`: ``` ∐ functorObjSrcFamily f πX ⟶ X | | | | v v ∐ functorObjTgtFamily f πX ⟶ functorObj f πX ``` where the morphism on the left is a coproduct (of copies of maps `f i`) indexed by a type `FunctorObjIndex f…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/SmallObject/Construction.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.RegularTopology", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 10.49, "z": 52.436, "size": 0.2742, "title": "Description of the covering sieves of the regular topology", "summary": "This file characterises the covering sieves of the regular topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -39.751, "z": 51.436, "size": 0.2959, "title": "Sheaves for the regular topology", "summary": "This file characterises sheaves for the regular topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.html"}, {"id": "Mathlib.CategoryTheory.Presentable.Basic", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 102, "macro_tier_score": 0.0156, "macro_tier_override": null, "x": 96.235, "z": 56.436, "size": 0.3559, "title": "Presentable objects", "summary": "A functor `F : C ⥤ D` is `κ`-accessible (`Functor.IsCardinalAccessible`) if it commutes with colimits of shape `J` where `J` is any `κ`-filtered category (that is essentially small relative to the universe `w` such that `κ : Cardinal.{w}`.). We also introduce another typeclass `Functor.IsAccessible` saying that there exists a regular cardinal `κ` such that `Functor.IsCardinalAccessible`. An object `X` of a category…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/Basic.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 103, "macro_tier_score": 0.1653, "macro_tier_override": null, "x": -0.229, "z": 47.436, "size": 0.4466, "title": "Objects that are colimits of objects satisfying a certain property", "summary": "Given a property of objects `P : ObjectProperty C` and a category `J`, we introduce two properties of objects `P.strictColimitsOfShape J` and `P.colimitsOfShape J`. The former contains exactly the objects of the form `colimit F` for any functor `F : J ⥤ C` that has a colimit and such that `F.obj j` satisfies `P` for any `j`, while the latter contains all the objects that are isomorphic to these \"chosen\" objects…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/ColimitsOfShape.html"}, {"id": "Mathlib.CategoryTheory.Localization.SmallHom", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 40.515, "z": 37.436, "size": 0.2679, "title": "Shrinking morphisms in localized categories", "summary": "Given a class of morphisms `W : MorphismProperty C`, and two objects `X` and `Y`, we introduce a type-class `HasSmallLocalizedHom.{w} W X Y` which expresses that in the localized category with respect to `W`, the type of morphisms from `X` to `Y` is `w`-small for a certain universe `w`. Under this assumption, we define `SmallHom.{w} W X Y : Type w` as the shrunk type. For any localization functor `L : C ⥤ D` for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/SmallHom.html"}, {"id": "Mathlib.CategoryTheory.Subterminal", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 68.973, "z": 50.436, "size": 0.2506, "title": "Subterminal objects", "summary": "Subterminal objects are the objects which can be thought of as subobjects of the terminal object. In fact, the definition can be constructed to not require a terminal object, by defining `A` to be subterminal iff for any `Z`, there is at most one morphism `Z ⟶ A`. An alternate definition is that the diagonal morphism `A ⟶ A ⨯ A` is an isomorphism. In this file we define subterminal objects and show the equivalence…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subterminal.html"}, {"id": "Mathlib.CategoryTheory.Sites.RegularEpi", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -168.028, "z": -105.756, "size": 0.2638, "title": "The category of type-valued sheaves is a regular epi category", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/RegularEpi.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Opposites.Equalizers", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 103, "macro_tier_score": 0.1486, "macro_tier_override": null, "x": 68.502, "z": 42.436, "size": 0.3306, "title": "Equalizers and coequalizers in `C` and `Cᵒᵖ`", "summary": "We construct equalizers and coequalizers in the opposite categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Opposites/Equalizers.html"}, {"id": "Mathlib.CategoryTheory.Localization.Monoidal.Functor", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -0.642, "z": 36.436, "size": 0.2403, "title": "Universal property of localized monoidal categories", "summary": "This file proves that, given a monoidal localization functor `L : C ⥤ D`, and a functor `F : D ⥤ E` to a monoidal category, such that `F` lifts along `L` to a monoidal functor `G`, then `F` is monoidal. See `CategoryTheory.Localization.Monoidal.functorMonoidalOfComp`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Monoidal/Functor.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Multifunctor", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 102, "macro_tier_score": 0.0096, "macro_tier_override": null, "x": 1.793, "z": 32.436, "size": 0.2385, "title": "Constructing monoidal functors from natural transformations between multifunctors", "summary": "This file provides alternative constructors for (op/lax) monoidal functors, given tensorators `μ : F - ⊗ F - ⟶ F (- ⊗ -)` / `δ : F (- ⊗ -) ⟶ F - ⊗ F -` as natural transformations between bifunctors. The associativity conditions are phrased as equalities of natural transformations between trifunctors `(F - ⊗ F -) ⊗ F - ⟶ F (- ⊗ (- ⊗ -))` / `F ((- ⊗ -) ⊗ -) ⟶ F - ⊗ (F - ⊗ F -)`, and the unitality conditions are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Multifunctor.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Internal.Types.Grp", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -28.109, "z": 52.436, "size": 0.2859, "title": "`Grp (Type u) ≌ GrpCat.{u}`", "summary": "The category of internal group objects in `Type` is equivalent to the category of \"native\" bundled groups. Moreover, this equivalence is compatible with the forgetful functors to `Type`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Internal/Types/Grp.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.Ind", "region_id": "category_theory", "micro_elevation": 0.9649, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 5.158, "z": 62.436, "size": 0.2, "title": "Ind and pro-properties", "summary": "Given a morphism property `P`, we define a morphism property `ind P` that is satisfied for `f : X ⟶ Y` if `Y` is a filtered colimit of `Yᵢ` and `fᵢ : X ⟶ Yᵢ` satisfy `P`. We show that `ind P` inherits stability properties from `P`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/Ind.html"}, {"id": "Mathlib.CategoryTheory.Comma.LocallySmall", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 102, "macro_tier_score": 0.024, "macro_tier_override": null, "x": -9.34, "z": 34.436, "size": 0.264, "title": "Comma categories are locally small", "summary": "We introduce instances showing that the various comma categories are locally small when the relevant categories that are involved are locally small.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/LocallySmall.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Over", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 8.484, "z": 35.0, "size": 0.2424, "title": "Forgetful functor from `Over X` preserves cofiltered limits", "summary": "Note that `Over.forget X : Over X ⥤ C` already preserves all colimits because it is a left adjoint. See `Mathlib/CategoryTheory/Comma/Over/Pullback.lean`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Over.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.Ind", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 68.515, "z": 61.436, "size": 0.2302, "title": "Ind and pro-properties", "summary": "Given an object property `P`, we define an object property `ind P` that is satisfied for `X` if `X` is a filtered colimit of `Xᵢ` and `Xᵢ` satisfies `P`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/Ind.html"}, {"id": "Mathlib.CategoryTheory.Subfunctor.Subobject", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 41.223, "z": 51.436, "size": 0.2, "title": "Comparison between `Subfunctor`, `MonoOver` and `Subobject`", "summary": "Given a type-valued functor `F : C ⥤ Type w`, we define an equivalence of categories `Subfunctor.equivalenceMonoOver F : Subfunctor F ≌ MonoOver F` and an order isomorphism `Subfunctor.orderIsoSubject F : Subfunctor F ≃o Subobject F`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subfunctor/Subobject.html"}, {"id": "Mathlib.CategoryTheory.Subobject.Types", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 16.194, "z": 52.436, "size": 0.2, "title": "`Type u` is well-powered", "summary": "By building a categorical equivalence `MonoOver α ≌ Set α` for any `α : Type u`, we deduce that `Subobject α ≃o Set α` and that `Type u` is well-powered. One would hope that for a particular concrete category `C` (`AddCommGroup`, etc) it's viable to prove `[WellPowered C]` without explicitly aligning `Subobject X` with the \"hand-rolled\" definition of subobjects. This may be possible using Lawvere theories, but it…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subobject/Types.html"}, {"id": "Mathlib.CategoryTheory.Subobject.WellPowered", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 103, "macro_tier_score": 0.0682, "macro_tier_override": null, "x": 58.277, "z": 51.436, "size": 0.3651, "title": "Well-powered categories", "summary": "A category `(C : Type u) [Category.{v} C]` is `[WellPowered.{w} C]` if `C` is locally small relative to `w` and for every `X : C`, we have `Small.{w} (Subobject X)`. The most common case is when `w = v`, in which case, it only involves the condition `Small.{v} (Subobject X)` (Note that in this situation `Subobject X : Type (max u v)`, so this is a nontrivial condition for large categories, but automatic for small…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subobject/WellPowered.html"}, {"id": "Mathlib.CategoryTheory.Sigma.Basic", "region_id": "category_theory", "micro_elevation": 0.0877, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 25.607, "z": 12.436, "size": 0.2541, "title": "Disjoint union of categories", "summary": "We define the category structure on a sigma-type (disjoint union) of categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sigma/Basic.html"}, {"id": "Mathlib.CategoryTheory.Enriched.Limits.HasConicalPullbacks", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 94.971, "z": 55.436, "size": 0.2, "title": "Existence of conical pullbacks", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Enriched/Limits/HasConicalPullbacks.html"}, {"id": "Mathlib.CategoryTheory.Sites.Plus", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0199, "macro_tier_override": null, "x": -21.725, "z": 49.436, "size": 0.3267, "title": "The plus construction for presheaves.", "summary": "This file contains the construction of `P⁺`, for a presheaf `P : Cᵒᵖ ⥤ D` where `C` is endowed with a Grothendieck topology `J`. See for details.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Plus.html"}, {"id": "Mathlib.CategoryTheory.Localization.SmallShiftedHom", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": 33.54, "z": 44.0, "size": 0.3049, "title": "Shrinking morphisms in localized categories equipped with shifts", "summary": "If `C` is a category equipped with a shift by an additive monoid `M`, and `W : MorphismProperty C` is compatible with the shift, we define a type-class `HasSmallLocalizedShiftedHom.{w} W X Y` which says that all the types of morphisms from `X⟦a⟧` to `Y⟦b⟧` in the localized category are `w`-small for a certain universe. Then, we define types `SmallShiftedHom.{w} W X Y m : Type w` for all `m : M`, and endow these with…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/SmallShiftedHom.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.OfHasFiniteProducts", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 25.305, "z": 50.436, "size": 0.2, "title": "The natural monoidal structure on any category with finite (co)products.", "summary": "A category with a monoidal structure provided in this way is sometimes called a (co-)Cartesian category, although this is also sometimes used to mean a finitely complete category. (See .) As this works with either products or coproducts, and sometimes we want to think of a different monoidal structure entirely, we don't set up either construct as an instance.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/OfHasFiniteProducts.html"}, {"id": "Mathlib.CategoryTheory.Join.Final", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -21.693, "z": 43.436, "size": 0.2, "title": "(Co)Finality of the inclusions in joins of categories", "summary": "This file records the fact that `inclLeft C D : C ⥤ C ⋆ D` is initial if `C` is connected. Dually, `inclRight : C ⥤ C ⋆ D` is final if `D` is connected.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Join/Final.html"}, {"id": "Mathlib.CategoryTheory.Functor.FullyFaithful", "region_id": "category_theory", "micro_elevation": 0.0526, "macro_tier": 103, "macro_tier_score": 0.4946, "macro_tier_override": null, "x": 21.564, "z": 10.436, "size": 0.5993, "title": "Full and faithful functors", "summary": "We define typeclasses `Full` and `Faithful`, decorating functors. These typeclasses carry no data. However, we also introduce a structure `Functor.FullyFaithful` which contains the data of the inverse map `(F.obj X ⟶ F.obj Y) ⟶ (X ⟶ Y)` of the map induced on morphisms by a functor `F`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/FullyFaithful.html"}, {"id": "Mathlib.CategoryTheory.NatIso", "region_id": "category_theory", "micro_elevation": 0.0351, "macro_tier": 103, "macro_tier_score": 0.4968, "macro_tier_override": null, "x": 26.524, "z": 9.436, "size": 0.6468, "title": "Natural isomorphisms", "summary": "For the most part, natural isomorphisms are just another sort of isomorphism. We provide some special support for extracting components: * if `α : F ≅ G`, then `α.app X : F.obj X ≅ G.obj X`, and building natural isomorphisms from components: * ``` NatIso.ofComponents (app : ∀ X : C, F.obj X ≅ G.obj X) (naturality : ∀ {X Y : C} (f : X ⟶ Y), F.map f ≫ (app Y).hom = (app X).hom ≫ G.map f) : F ≅ G ``` only needing to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/NatIso.html"}, {"id": "Mathlib.CategoryTheory.EffectiveEpi.Enough", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": 40.064, "z": 35.436, "size": 0.2896, "title": "Effectively enough objects in the image of a functor", "summary": "We define the class `F.EffectivelyEnough` on a functor `F : C ⥤ D` which says that for every object in `D`, there exists an effective epi to it from an object in the image of `F`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/EffectiveEpi/Enough.html"}, {"id": "Mathlib.CategoryTheory.EquivalenceRelation", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -26.647, "z": 48.436, "size": 0.2, "title": "Equivalence relations", "summary": "We define internal equivalence relations (sometimes called congruences) in any category `C`, as a structure on pairs of parallel morphisms `p₁, p₂ : R ⟶ X` . We also define effective and universally effective equivalence relations. We prove that equivalence relations on types provide internal equivalence relation structures in the category of types. In general, kernel pairs in any category are internal equivalence…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/EquivalenceRelation.html"}, {"id": "Mathlib.CategoryTheory.Localization.DerivabilityStructure.Derives", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 33.071, "z": 40.0, "size": 0.2324, "title": "Deriving functors using a derivability structure", "summary": "Let `Φ : LocalizerMorphism W₁ W₂` be a localizer morphism between classes of morphisms on categories `C₁` and `C₂`. Let `F : C₂ ⥤ H`. When `Φ` is a left or right derivability structure, it allows to derive the functor `F` (with respect to `W₂`) when `Φ.functor ⋙ F : C₁ ⥤ H` inverts `W₁` (this is the most favorable case when we can apply the lemma…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/DerivabilityStructure/Derives.html"}, {"id": "Mathlib.CategoryTheory.SmallObject.TransfiniteCompositionLifting", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": 116.254, "z": -55.661, "size": 0.2711, "title": "The left lifting property is stable under transfinite composition", "summary": "In this file, we show that if `W : MorphismProperty C`, then `W.llp.IsStableUnderTransfiniteCompositionOfShape J`, i.e. the class of morphisms which have the left lifting property with respect to `W` is stable under transfinite composition. The main technical lemma is `HasLiftingProperty.transfiniteComposition.hasLiftingProperty_ι_app_bot`. It corresponds to the particular case `W` contains only one morphism `p : X…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/SmallObject/TransfiniteCompositionLifting.html"}, {"id": "Mathlib.CategoryTheory.SmallObject.WellOrderInductionData", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 102, "macro_tier_score": 0.0193, "macro_tier_override": null, "x": 48.632, "z": 27.436, "size": 0.2684, "title": "Limits of inverse systems indexed by well-ordered types", "summary": "Given a functor `F : Jᵒᵖ ⥤ Type v` where `J` is a well-ordered type, we introduce a structure `F.WellOrderInductionData` which allows to show that the map `F.sections → F.obj (op ⊥)` is surjective. The data and properties in `F.WellOrderInductionData` consist of a section to the maps `F.obj (op (Order.succ j)) → F.obj (op j)` when `j` is not maximal, and, when `j` is limit, a section to the canonical map from `F.obj…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/SmallObject/WellOrderInductionData.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.Comma", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.1099, "macro_tier_override": null, "x": 63.303, "z": 34.436, "size": 0.2821, "title": "Properties of comma categories relating to adjunctions", "summary": "This file shows that for a functor `G : D ⥤ C` the data of an initial object in each `StructuredArrow` category on `G` is equivalent to a left adjoint to `G`, as well as the dual. Specifically, `adjunctionOfStructuredArrowInitials` gives the left adjoint assuming the appropriate initial objects exist, and `mkInitialOfLeftAdjoint` constructs the initial objects provided a left adjoint. The duals are also shown.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/Comma.html"}, {"id": "Mathlib.CategoryTheory.Localization.Quotient", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 101, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 124.723, "z": -78.661, "size": 0.2744, "title": "Localization of quotient categories", "summary": "Given a relation `homRel : HomRel C` on morphisms in a category `C` and `W : MorphismProperty C`, we introduce a property `homRel.FactorsThroughLocalization W` expressing that related morphisms are mapped to the same morphism in the localized category with respect to `W`. When `W` is compatible with `homRel` (i.e. there is a class of morphisms `W'` such that `hW : W = W'.inverseImage (Quotient.functor homRel)`), we…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Quotient.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.OverAdjunction", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 102, "macro_tier_score": 0.0101, "macro_tier_override": null, "x": 154.987, "z": -26.328, "size": 0.2997, "title": "Adjunction of pushforward and pullback in `P.Over Q X`", "summary": "Under suitable assumptions on `P`, `Q` and `f`, a morphism `f : X ⟶ Y` defines two functors: - `Over.map`: post-composition with `f` - `Over.pullback`: base-change along `f` such that `Over.map` is the left adjoint to `Over.pullback`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/OverAdjunction.html"}, {"id": "Mathlib.CategoryTheory.Enriched.Basic", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 102, "macro_tier_score": 0.0265, "macro_tier_override": null, "x": 76.434, "z": 52.436, "size": 0.4181, "title": "Enriched categories", "summary": "We set up the basic theory of `V`-enriched categories, for `V` an arbitrary monoidal category. We do not assume here that `V` is a concrete category, so there does not need to be an \"honest\" underlying category! Use `X ⟶[V] Y` to obtain the `V` object of morphisms from `X` to `Y`. This file contains the definitions of `V`-enriched categories and `V`-functors. We don't yet define the `V`-object of natural…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Enriched/Basic.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.HasCardinalLT", "region_id": "category_theory", "micro_elevation": 0.0351, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 22.981, "z": 9.436, "size": 0.2338, "title": "Properties of objects that are bounded by a cardinal", "summary": "Given `P : ObjectProperty C` and `κ : Cardinal`, we introduce a predicate `P.HasCardinalLT κ` saying that the cardinality of `Subtype P` is `< κ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/HasCardinalLT.html"}, {"id": "Mathlib.CategoryTheory.Localization.BousfieldTransfiniteComposition", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 63.113, "z": 59.436, "size": 0.2276, "title": "ObjectProperty.isLocal is stable under transfinite compositions", "summary": "If `P : ObjectProperty C`, then `P.isLocal : MorphismProperty C` is stable under transfinite compositions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/BousfieldTransfiniteComposition.html"}, {"id": "Mathlib.CategoryTheory.ConcreteCategory.BundledHom", "region_id": "category_theory", "micro_elevation": 0.0175, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 23.075, "z": 8.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ConcreteCategory/BundledHom.html"}, {"id": "Mathlib.CategoryTheory.Limits.FunctorCategory.Finite", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 102, "macro_tier_score": 0.0242, "macro_tier_override": null, "x": 19.654, "z": 42.436, "size": 0.2785, "title": "Functor categories have finite limits when the target category does", "summary": "These declarations cannot be in `Mathlib/CategoryTheory/Limits/FunctorCategory/Basic.lean` because that file shouldn't import `Mathlib/CategoryTheory/Limits/Shapes/FiniteProducts.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FunctorCategory/Finite.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Limits.Basic", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 21.518, "z": 36.436, "size": 0.2, "title": "`lim : (J ⥤ C) ⥤ C` is lax monoidal when `C` is a monoidal category.", "summary": "When `C` is a monoidal category, the limit functor `lim : (J ⥤ C) ⥤ C` is lax monoidal, i.e. there are morphisms * `(𝟙_ C) → limit (𝟙_ (J ⥤ C))` * `limit F ⊗ limit G ⟶ limit (F ⊗ G)` satisfying the laws of a lax monoidal functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Limits/Basic.html"}, {"id": "Mathlib.CategoryTheory.Presentable.EssentiallyLarge", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -48.049, "z": 59.436, "size": 0.2, "title": "Accessible categories are essentially large", "summary": "If a category `C` satisfies `HasCardinalFilteredGenerator C κ` for `κ : Cardinal.{w}` (e.g. it is locally `κ`-presentable or `κ`-accessible), then `C` is equivalent to a `w`-large category, i.e. a category whose type of objects is in `Type (w + 1)` and whose types of morphisms are in `Type w`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/EssentiallyLarge.html"}, {"id": "Mathlib.CategoryTheory.Quotient.Preadditive", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -51.271, "z": 42.0, "size": 0.2855, "title": "The quotient category is preadditive", "summary": "If an equivalence relation `r : HomRel C` on the morphisms of a preadditive category is compatible with the addition, then the quotient category `Quotient r` is also preadditive.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Quotient/Preadditive.html"}, {"id": "Mathlib.CategoryTheory.Limits.SmallComplete", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 23.142, "z": 34.436, "size": 0.2, "title": "Any small complete category is a preorder", "summary": "We show that any small category which has all (small) limits is a preorder: In particular, we show that if a small category `C` in universe `u` has products of size `u`, then for any `X Y : C` there is at most one morphism `X ⟶ Y`. Note that in Lean, a preorder category is strictly one where the morphisms are in `Prop`, so we instead show that the homsets are subsingleton.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/SmallComplete.html"}, {"id": "Mathlib.CategoryTheory.Presentable.SharplyLT.Basic", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 104.164, "z": 61.436, "size": 0.2, "title": "Sharply smaller regular cardinals", "summary": "In this file, we introduce the predicate `Cardinal.SharplyLT`. Given two regular cardinals `κ₁ < κ₂`, this condition can be described in different ways: (i) the category `CardinalDirectedPoset κ₁` (of `κ₁`-directed partially ordered types, with order embeddings as morphisms), is `κ₂`-accessible; (ii) any `κ₁`-accessible category is `κ₂`-accessible. (iii) for any type `X` of cardinality `< κ₂`, there exists a cofinal…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/SharplyLT/Basic.html"}, {"id": "Mathlib.CategoryTheory.Presentable.CardinalDirectedPoset", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -54.8, "z": 60.436, "size": 0.239, "title": "The κ-accessible category of κ-directed posets", "summary": "Given a regular cardinal `κ : Cardinal.{u}`, we define the category `CardinalDirectedPoset κ` of `κ`-directed partially ordered types (with order embeddings as morphisms), and we show that it is a `κ`-accessible category. The notion of `κ`-directed partially ordered type is implemented using the categorial notion `IsCardinalFiltered`: we may consider \"`κ`-directed\" and \"`κ`-filtered\" as synonyms. If `κ ≤ κ'` where…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/CardinalDirectedPoset.html"}, {"id": "Mathlib.CategoryTheory.Presentable.Dense", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 102, "macro_tier_score": 0.0145, "macro_tier_override": null, "x": 75.718, "z": 60.436, "size": 0.2574, "title": "`κ`-presentable objects form a dense subcategory", "summary": "In a `κ`-accessible category `C`, the inclusion of the full subcategory of `κ`-presentable objects is a dense functor. This expresses canonically any object `X : C` as a colimit of `κ`-presentable objects, and we show that this is a `κ`-filtered colimit.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/Dense.html"}, {"id": "Mathlib.CategoryTheory.Category.ReflQuiv", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 136.649, "z": -83.661, "size": 0.2425, "title": "The category of refl quivers", "summary": "The category `ReflQuiv` of (bundled) reflexive quivers, and the free/forgetful adjunction between `Cat` and `ReflQuiv`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/ReflQuiv.html"}, {"id": "Mathlib.CategoryTheory.Comma.Presheaf.Colimit", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 102, "macro_tier_score": 0.0192, "macro_tier_override": null, "x": 34.13, "z": 46.436, "size": 0.2553, "title": "Relative Yoneda preserves certain colimits", "summary": "In this file we turn the statement `yonedaYonedaColimit` from `CategoryTheory.Limits.Preserves.Yoneda` from a functor `F : J ⥤ Cᵒᵖ ⥤ Type v` into a statement about families of presheaves over `A`, i.e., functors `F : J ⥤ Over A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/Presheaf/Colimit.html"}, {"id": "Mathlib.CategoryTheory.Shift.ShiftedHom", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0149, "macro_tier_override": null, "x": 62.176, "z": 43.0, "size": 0.3086, "title": "Shifted morphisms", "summary": "Given a category `C` endowed with a shift by an additive monoid `M` and two objects `X` and `Y` in `C`, we consider the types `ShiftedHom X Y m` defined as `X ⟶ Y⟦m⟧` for all `m : M`, and the composition on these shifted hom.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/ShiftedHom.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.Equalizers", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 103, "macro_tier_score": 0.067, "macro_tier_override": null, "x": 29.115, "z": 43.436, "size": 0.2766, "title": "Constructing equalizers from pullbacks and binary products.", "summary": "If a category has pullbacks and binary products, then it has equalizers. TODO: generalize universe", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/Equalizers.html"}, {"id": "Mathlib.CategoryTheory.Sites.LocalSite", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 86.186, "z": 60.436, "size": 0.2, "title": "Local sites", "summary": "A site is called local if it has a terminal object whose only covering sieve is trivial - this makes it possible to define coconstant sheaves on it, giving its sheaf topos the structure of a local topos. This is one of the conditions of cohesive sites. Sheaves of types on any local site form a local topos (i.e. a topos whose global sections functor has a fully faithful right adjoint), and a subcanonical site is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/LocalSite.html"}, {"id": "Mathlib.CategoryTheory.Sites.GlobalSections", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -42.651, "z": 59.436, "size": 0.2338, "title": "Global sections of sheaves", "summary": "In this file we define a global sections functor `Sheaf.Γ : Sheaf J A ⥤ A` and show that it is isomorphic to several other constructions when they exist, most notably evaluation of sheaves on a terminal object and `Functor.sectionsFunctor`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/GlobalSections.html"}, {"id": "Mathlib.CategoryTheory.Sites.Point.Skyscraper", "region_id": "category_theory", "micro_elevation": 0.8947, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -36.093, "z": 58.436, "size": 0.2885, "title": "Skyscraper sheaves", "summary": "Let `Φ` be a point of a site `(C, J)`. In this file, we construct the skyscraper sheaf functor `skyscraperSheafFunctor : A ⥤ Sheaf J A` and show that it is a right adjoint to `Φ.sheafFiber : Sheaf J A ⥤ A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Point/Skyscraper.html"}, {"id": "Mathlib.CategoryTheory.Limits.EpiMono", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": 65.209, "z": 43.436, "size": 0.2651, "title": "Relation between mono/epi and pullback/pushout squares", "summary": "In this file, monomorphisms and epimorphisms are characterized in terms of pullback and pushout squares. For example, we obtain `mono_iff_isPullback` which asserts that a morphism `f : X ⟶ Y` is a monomorphism iff the obvious square ``` X ⟶ X | | v v X ⟶ Y ``` is a pullback square.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/EpiMono.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 79.186, "z": 44.436, "size": 0.2, "title": "Internal homs for day convolution", "summary": "Given a category `V` that is monoidal closed, a category `C` that is monoidal, a functor `C ⥤ V`, and given the data of suitable day convolutions and suitable ends of profunctors `c c₁ c₂ ↦ ihom (F c₁) (·.obj (c₂ ⊗ c))`, we prove that the data of the units of the left Kan extensions that define day convolutions and the data of the canonical morphisms to the aforementioned ends can be organised as data that exhibit…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/DayConvolution/Closed.html"}, {"id": "Mathlib.CategoryTheory.Sites.Pullback", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -136.852, "z": 238.683, "size": 0.2803, "title": "Pullback of sheaves", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Pullback.html"}, {"id": "Mathlib.CategoryTheory.Sites.Continuous", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 102, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": -169.146, "z": 237.683, "size": 0.3409, "title": "Continuous functors between sites.", "summary": "We define the notion of continuous functor between sites: these are functors `F` such that the precomposition with `F.op` preserves sheaves of types (and actually sheaves in any category).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Continuous.html"}, {"id": "Mathlib.CategoryTheory.Abelian.LeftDerived", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": -23.798, "z": 50.436, "size": 0.2638, "title": "Left-derived functors", "summary": "We define the left-derived functors `F.leftDerived n : C ⥤ D` for any additive functor `F` out of a category with projective resolutions. We first define a functor `F.leftDerivedToHomotopyCategory : C ⥤ HomotopyCategory D (ComplexShape.down ℕ)` which is `projectiveResolutions C ⋙ F.mapHomotopyCategory _`. We show that if `X : C` and `P : ProjectiveResolution X`, then `F.leftDerivedToHomotopyCategory.obj X`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/LeftDerived.html"}, {"id": "Mathlib.CategoryTheory.Sites.PreservesLimits", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 101, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -137.985, "z": -102.756, "size": 0.2734, "title": "Preservation of (co)limits by the sheaf Yoneda functor", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/PreservesLimits.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -4.584, "z": 46.436, "size": 0.2, "title": "Bi-Cartesian squares", "summary": "`BicartesianSq f g h i` is the proposition that ``` W ---f---> X | | g h | | v v Y ---i---> Z ``` is a pullback square *and* a pushout square. We show that the pullback and pushout squares for a biproduct are bi-Cartesian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/IsPullback/BicartesianSq.html"}, {"id": "Mathlib.CategoryTheory.Functor.Hom", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 103, "macro_tier_score": 0.3597, "macro_tier_override": null, "x": 1.124, "z": 27.436, "size": 0.8893, "title": null, "summary": "The hom functor, sending `(X, Y)` to the type `X ⟶ Y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/Hom.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Rigid.Functor", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 69.588, "z": 38.436, "size": 0.2, "title": "Dual Functors for Rigid Categories", "summary": "This file defines the left and right dual functors from a rigid monoidal category to `(Cᵒᵖ)ᴹᵒᵖ` (the monoidal opposite of the opposite category).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Rigid/Functor.html"}, {"id": "Mathlib.CategoryTheory.GradedObject.Associator", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -25.081, "z": 40.0, "size": 0.2755, "title": "The associator for actions of bifunctors on graded objects", "summary": "Given functors `F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂`, `G : C₁₂ ⥤ C₃ ⥤ C₄`, `F : C₁ ⥤ C₂₃ ⥤ C₄`, `G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃` equipped with an isomorphism `associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃` (which informally means that we have natural isomorphisms `G(F₁₂(X₁, X₂), X₃) ≅ F(X₁, G₂₃(X₂, X₃))`), a map `r : I₁ × I₂ × I₃ → J`, and data `ρ₁₂ : BifunctorComp₁₂IndexData r` and `ρ₂₃ : BifunctorComp₂₃IndexData r`, then if `X₁…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GradedObject/Associator.html"}, {"id": "Mathlib.CategoryTheory.GradedObject.Single", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 47.931, "z": 45.436, "size": 0.2621, "title": "The graded object in a single degree", "summary": "In this file, we define the functor `GradedObject.single j : C ⥤ GradedObject J C` which sends an object `X : C` to the graded object which is `X` in degree `j` and the initial object of `C` in other degrees.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GradedObject/Single.html"}, {"id": "Mathlib.CategoryTheory.Subobject.FactorThru", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 103, "macro_tier_score": 0.0678, "macro_tier_override": null, "x": -15.607, "z": 51.436, "size": 0.3437, "title": "Factoring through subobjects", "summary": "The predicate `h : P.Factors f`, for `P : Subobject Y` and `f : X ⟶ Y` asserts the existence of some `P.factorThru f : X ⟶ (P : C)` making the obvious diagram commute.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subobject/FactorThru.html"}, {"id": "Mathlib.CategoryTheory.Limits.Comma", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 103, "macro_tier_score": 0.0912, "macro_tier_override": null, "x": 10.292, "z": 44.436, "size": 0.3148, "title": "Limits and colimits in comma categories", "summary": "We build limits in the comma category `Comma L R` provided that the two source categories have limits and `R` preserves them. This is used to construct limits in the arrow category, structured arrow category and under category, and show that the appropriate forgetful functors create limits. The duals of all the above are also given.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Comma.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.SiteLocal", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 80.029, "z": 47.436, "size": 0.2, "title": "Locality conditions on object properties", "summary": "In this file we define locality conditions on object properties in a category. Let `K` be a precoverage in a category `C` and `P` be an object property that is closed under isomorphisms. We say that - `P` is local if for every `X : C`, `P` holds for `X` if and only if it holds for `Uᵢ` for a `K`-cover `{Uᵢ}` of `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/SiteLocal.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.ExternalProduct.Basic", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 103, "macro_tier_score": 0.0718, "macro_tier_override": null, "x": 67.112, "z": 36.436, "size": 0.2778, "title": "External product of diagrams in a monoidal category", "summary": "In a monoidal category `C`, given a pair of diagrams `K₁ : J₁ ⥤ C` and `K₂ : J₂ ⥤ C`, we introduce the external product `K₁ ⊠ K₂ : J₁ × J₂ ⥤ C` as the bifunctor `(j₁, j₂) ↦ K₁ j₁ ⊗ K₂ j₂`. The notation `- ⊠ -` is scoped to `MonoidalCategory.ExternalProduct`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/ExternalProduct/Basic.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Limits.Preserves", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 103, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 11.927, "z": 36.436, "size": 0.2513, "title": "Miscellany about preservation of (co)limits in monoidal categories", "summary": "This file records some `PreservesColimits` instances on tensor products in monoidal categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Limits/Preserves.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckCategory.HasExt", "region_id": "category_theory", "micro_elevation": 0.9825, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 261.036, "z": -11.328, "size": 0.2239, "title": "Ext in Grothendieck abelian categories", "summary": "Let `C : Type u` be an abelian category (with `Category.{v} C`). If `C` is a Grothendieck abelian category relatively to a universe `w`, the morphisms in `C` must be `w`-small, and as `C` has enough injectives, the `Ext`-groups are also `w`-small. If `C` is a nonzero category, it is possible to show that any `w`-small type `T` injects into a type of morphisms in `C` (consider the various inclusions `X ⟶ ∐ (fun (_ :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckCategory/HasExt.html"}, {"id": "Mathlib.CategoryTheory.Sites.SheafCohomology.Basic", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 133.538, "z": -14.328, "size": 0.2457, "title": "Sheaf cohomology", "summary": "Let `C` be a category equipped with a Grothendieck topology `J`. We define the cohomology types `Sheaf.H F n` of an abelian sheaf `F` on the site `(C, J)` for all `n : ℕ`. These abelian groups are defined as the `Ext`-groups from the constant abelian sheaf with values `ℤ` (actually `ULift ℤ`) to `F`. We also define `Sheaf.cohomologyPresheaf F n : Cᵒᵖ ⥤ AddCommGrpCat` which is the presheaf which sends `U` to the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/SheafCohomology/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.ConcreteCategory.Filtered", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 101, "macro_tier_score": 0.0055, "macro_tier_override": null, "x": -122.259, "z": 226.683, "size": 0.3168, "title": "Filtered colimits in concrete categories", "summary": "In this file, we provide analogues to some of the API in the `CategoryTheory.Limits.Types.FilteredColimit` namespace, for concrete categories for which the forgetful functor preserves filtered colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/ConcreteCategory/Filtered.html"}, {"id": "Mathlib.CategoryTheory.Abelian.SerreClass.Bousfield", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 49.512, "z": 50.436, "size": 0.2, "title": "Bousfield localizations with respect to Serre classes", "summary": "If `G : D ⥤ C` is an exact functor between abelian categories, with a fully faithful right adjoint `F`, then `G` identifies `C` to the localization of `D` with respect to the class of morphisms `G.kernel.isoModSerre`, i.e. `D` is the localization of `C` with respect to the Serre class `G.kernel` consisting of the objects in `D` that are sent to a zero object by `G`. (We also translate this in terms of a left…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/SerreClass/Bousfield.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 36.392, "z": 51.436, "size": 0.2, "title": "The right Freyd category", "summary": "Let `V` be a preadditive category. The right Freyd category of `V` is the quotient of `Arrow V` by the right homotopy relation. (This is simply called \"Freyd category\" in the reference.) This is a preadditive category with a fully faithful additive functor `RightFreyd.functor : V ⥤ RightFreyd V`. We also show that, if `V` has binary biproducts, then `RightFreyd V` has cokernels. In fact we construct, given a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/FreydCategory/RightFreyd.html"}, {"id": "Mathlib.CategoryTheory.Generator.Sheaf", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -31.593, "z": 55.436, "size": 0.2441, "title": "Generators in the category of sheaves", "summary": "In this file, we show that if `J : GrothendieckTopology C` and `A` is a preadditive category which has a separator (and suitable coproducts), then `Sheaf J A` has a separator.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Generator/Sheaf.html"}, {"id": "Mathlib.CategoryTheory.Sites.Descent.DescentDataAsCoalgebra", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 12.407, "z": 34.436, "size": 0.2, "title": "Descent data as coalgebras", "summary": "Let `F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Adj Cat` be a pseudofunctor to the bicategory of adjunctions in `Cat`. In particular, for any morphism `g : X ⟶ Y` in `C`, we have an adjunction `(g^*, g_*)` between a pullback functor and a pushforward functor. In this file, given a family of morphisms `f i : X i ⟶ S` indexed by a type `ι` in `C`, we introduce a category `F.DescentDataAsCoalgebra f` of descent data relative to the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Descent/DescentDataAsCoalgebra.html"}, {"id": "Mathlib.CategoryTheory.Limits.FunctorCategory.Filtered", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 102, "macro_tier_score": 0.0287, "macro_tier_override": null, "x": -23.048, "z": 42.436, "size": 0.2457, "title": "Functor categories have filtered colimits when the target category does", "summary": "These declarations cannot be in `Mathlib/CategoryTheory/Limits/FunctorCategory/Basic.lean` because that file shouldn't import `Mathlib/CategoryTheory/Limits/Filtered.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FunctorCategory/Filtered.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.CommComon_", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 23.912, "z": 37.436, "size": 0.2, "title": "The category of commutative comonoids in a braided monoidal category.", "summary": "We define the category of commutative comonoid objects in a braided monoidal category `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/CommComon_.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives", "region_id": "category_theory", "micro_elevation": 0.9649, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": 34.896, "z": 62.436, "size": 0.2854, "title": "Grothendieck abelian categories have enough injectives", "summary": "Let `C` be a Grothendieck abelian category. In this file, we formalize the theorem by Grothendieck that `C` has enough injectives. We recall that injective objects can be characterized in terms of lifting properties (see the file `Preadditive.Injective.LiftingProperties`): an object `I : C` is injective iff the morphism `I ⟶ 0` has the right lifting property with respect to all monomorphisms. The main technical…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckCategory/EnoughInjectives.html"}, {"id": "Mathlib.CategoryTheory.Limits.IndYoneda", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 102, "macro_tier_score": 0.0241, "macro_tier_override": null, "x": -25.691, "z": 41.436, "size": 0.274, "title": "Ind- and pro- (co)yoneda lemmas", "summary": "We define limit versions of the yoneda and coyoneda lemmas.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/IndYoneda.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Preorder.HasIterationOfShape", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 102, "macro_tier_score": 0.0292, "macro_tier_override": null, "x": -29.224, "z": 45.436, "size": 0.3085, "title": "An assumption for constructions by transfinite induction", "summary": "In this file, we introduce the typeclass `HasIterationOfShape J C` which is an assumption in order to do constructions by transfinite induction indexed by a well-ordered type `J` in a category `C` (see `CategoryTheory.SmallObject`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Preorder/HasIterationOfShape.html"}, {"id": "Mathlib.CategoryTheory.SmallObject.Iteration.FunctorOfCocone", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 102, "macro_tier_score": 0.0145, "macro_tier_override": null, "x": -35.718, "z": 47.436, "size": 0.2608, "title": "The functor from `Set.Iic j` deduced from a cocone", "summary": "Given a functor `F : Set.Iio j ⥤ C` and `c : Cocone F`, we define an extension of `F` as a functor `Set.Iic j ⥤ C` for which the top element is mapped to `c.pt`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/SmallObject/Iteration/FunctorOfCocone.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.CommaSites", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 180.454, "z": -13.328, "size": 0.2268, "title": "Sites on `P.Over ⊤ X`", "summary": "We provide some API for proving properties of `P.Over ⊤ X` in relation to precoverages. Consider a precoverage `K` on `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/CommaSites.html"}, {"id": "Mathlib.CategoryTheory.Sites.SubcanonicalOver", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 149.203, "z": -13.328, "size": 0.2268, "title": "Topology on `Over X` is subcanonical if the base is", "summary": "We show that if `J` is subcanonical, then also `J.over X` is subcanonical.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/SubcanonicalOver.html"}, {"id": "Mathlib.CategoryTheory.Elements", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.1915, "macro_tier_override": null, "x": -6.665, "z": 33.436, "size": 0.3338, "title": "The category of elements", "summary": "This file defines the category of elements, also known as (a special case of) the Grothendieck construction. Given a functor `F : C ⥤ Type`, an object of `F.Elements` is a pair `(X : C, x : F.obj X)`. A morphism `(X, x) ⟶ (Y, y)` is a morphism `f : X ⟶ Y` in `C`, so `F.map f` takes `x` to `y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Elements.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Hopf_", "region_id": "category_theory", "micro_elevation": 0.5439, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 20.386, "z": 38.436, "size": 0.2, "title": "The category of Hopf monoids in a braided monoidal category.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Hopf_.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Conv", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -19.809, "z": 37.436, "size": 0.2478, "title": "The convolution monoid.", "summary": "When `M : Comon C` and `N : Mon C`, the morphisms `M.X ⟶ N.X` form a monoid (in Type).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Conv.html"}, {"id": "Mathlib.CategoryTheory.FiberedCategory.Fiber", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 47.634, "z": 25.436, "size": 0.257, "title": "Fibers of functors", "summary": "In this file we define, for a functor `p : 𝒳 ⥤ 𝒴`, the fiber categories `Fiber p S` for every `S : 𝒮` as follows - An object in `Fiber p S` is a pair `(a, ha)` where `a : 𝒳` and `ha : p.obj a = S`. - A morphism in `Fiber p S` is a morphism `φ : a ⟶ b` in 𝒳 such that `p.map φ = 𝟙 S`. For any category `C` equipped with a functor `F : C ⥤ 𝒳` such that `F ⋙ p` is constant at `S`, we define a functor `inducedFunctor : C…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/FiberedCategory/Fiber.html"}, {"id": "Mathlib.CategoryTheory.Sites.IsSheafFor", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 103, "macro_tier_score": 0.0351, "macro_tier_override": null, "x": 63.749, "z": 45.436, "size": 0.3779, "title": "The sheaf condition for a presieve", "summary": "We define what it means for a presheaf `P : Cᵒᵖ ⥤ Type v` to be a sheaf *for* a particular presieve `R` on `X`: * A *family of elements* `x` for `P` at `R` is an element `x_f` of `P Y` for every `f : Y ⟶ X` in `R`. See `FamilyOfElements`. * The family `x` is *compatible* if, for any `f₁ : Y₁ ⟶ X` and `f₂ : Y₂ ⟶ X` both in `R`, and any `g₁ : Z ⟶ Y₁` and `g₂ : Z ⟶ Y₂` such that `g₁ ≫ f₁ = g₂ ≫ f₂`, the restriction of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/IsSheafFor.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.Connected", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 195.233, "z": -31.328, "size": 0.2302, "title": "Pullbacks commute with connected limits", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Connected.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Biproducts", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 103, "macro_tier_score": 0.1929, "macro_tier_override": null, "x": 44.587, "z": 44.436, "size": 0.4094, "title": "Biproducts and binary biproducts", "summary": "We introduce the notion of (finite) biproducts. Binary biproducts are defined in `CategoryTheory.Limits.Shapes.BinaryBiproducts`. These are slightly unusual relative to the other shapes in the library, as they are simultaneously limits and colimits. (Zero objects are similar; they are \"biterminal\".) For results about biproducts in preadditive categories see `CategoryTheory.Preadditive.Biproducts`. For biproducts…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Biproducts.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 7.654, "z": 35.436, "size": 0.2479, "title": "Constructors for combining (co)fans", "summary": "We provide constructors for combining (co)fans and show their (co)limit properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/CombinedProducts.html"}, {"id": "Mathlib.CategoryTheory.Functor.TwoSquare", "region_id": "category_theory", "micro_elevation": 0.1579, "macro_tier": 103, "macro_tier_score": 0.1873, "macro_tier_override": null, "x": 36.179, "z": 16.436, "size": 0.3676, "title": "2-squares of functors", "summary": "Given four functors `T`, `L`, `R` and `B`, a 2-square `TwoSquare T L R B` consists of a natural transformation `w : T ⋙ R ⟶ L ⋙ B`: ``` T C₁ ⥤ C₂ L | | R v v C₃ ⥤ C₄ B ``` We define operations to paste such squares horizontally and vertically and prove the interchange law of those two operations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/TwoSquare.html"}, {"id": "Mathlib.CategoryTheory.Abelian.ShortExact", "region_id": "category_theory", "micro_elevation": 0.0175, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 25.493, "z": 8.436, "size": 0.2, "title": "Short Exact Sequences in Abelian Categories", "summary": "This file contains lemmas about short exact sequences in abelian categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/ShortExact.html"}, {"id": "Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 75.565, "z": 61.436, "size": 0.2, "title": "The Mayer-Vietoris exact sequence in sheaf cohomology", "summary": "Let `C` be a category equipped with a Grothendieck topology `J`. Let `S : J.MayerVietorisSquare` be a Mayer-Vietoris square for `J`. Let `F` be an abelian sheaf on `(C, J)`. In this file, we obtain a long exact Mayer-Vietoris sequence: `... ⟶ H^n(S.X₄, F) ⟶ H^n(S.X₂, F) ⊞ H^n(S.X₃, F) ⟶ H^n(S.X₁, F) ⟶ H^{n + 1}(S.X₄, F) ⟶ ...`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/SheafCohomology/MayerVietoris.html"}, {"id": "Mathlib.CategoryTheory.RepresentedBy", "region_id": "category_theory", "micro_elevation": 0.386, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 34.872, "z": 29.436, "size": 0.2, "title": "`IsRepresentedBy` predicate", "summary": "In this file we define the predicate `IsRepresentedBy`: A presheaf `F` is represented by `X` with universal element `x : F.obj X` if the natural transformation `yoneda.obj X ⟶ F` induced by `x` is an isomorphism. For other declarations expressing a functor is representable, see also: - `CategoryTheory.Functor.RepresentableBy`: Structure bundling an explicit natural isomorphism `yoneda.obj X ⟶ F`. -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/RepresentedBy.html"}, {"id": "Mathlib.CategoryTheory.Limits.Unit", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 103, "macro_tier_score": 0.0908, "macro_tier_override": null, "x": 25.866, "z": 33.436, "size": 0.2723, "title": "`Discrete PUnit` has limits and colimits", "summary": "Mostly for the sake of constructing trivial examples, we show all (co)cones into `Discrete PUnit` are (co)limit (co)cones. We also show that such (co)cones exist, and that `Discrete PUnit` has all (co)limits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Unit.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Bimod", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 69.33, "z": 41.436, "size": 0.2, "title": "The category of bimodule objects over a pair of monoid objects.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Bimod.html"}, {"id": "Mathlib.CategoryTheory.Comma.Basic", "region_id": "category_theory", "micro_elevation": 0.2456, "macro_tier": 103, "macro_tier_score": 0.4316, "macro_tier_override": null, "x": 12.453, "z": 21.436, "size": 0.6753, "title": "Comma categories", "summary": "A comma category is a construction in category theory, which builds a category out of two functors with a common codomain. Specifically, for functors `L : A ⥤ T` and `R : B ⥤ T`, an object in `Comma L R` is a morphism `hom : L.obj left ⟶ R.obj right` for some objects `left : A` and `right : B`, and a morphism in `Comma L R` between `hom : L.obj left ⟶ R.obj right` and `hom' : L.obj left' ⟶ R.obj right'` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/Basic.html"}, {"id": "Mathlib.CategoryTheory.Limits.WeakLimits.Basic", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": 31.576, "z": 33.436, "size": 0.2747, "title": "Weak limits", "summary": "If `F : J ⥤ C` is a functor and `c : Cone F`, we say that `c` is a weak limit of `F` if every cone over `F` admits a (not necessarily unique) morphism to `c`. In other words, a weak limit satisfies the same \"versal property\" as a limit, without the uniqueness condition. In particular, weak limits are not unique, and they are not functorial. We set up some API for weak limits, mostly copied from that for limits,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/WeakLimits/Basic.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.HomOrthogonal", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -36.76, "z": 48.436, "size": 0.2, "title": "Hom orthogonal families.", "summary": "A family of objects in a category with zero morphisms is \"hom orthogonal\" if the only morphism between distinct objects is the zero morphism. We show that in any category with zero morphisms and finite biproducts, a morphism between biproducts drawn from a hom orthogonal family `s : ι → C` can be decomposed into a block diagonal matrix with entries in the endomorphism rings of the `s i`. When the category is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/HomOrthogonal.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq", "region_id": "category_theory", "micro_elevation": 0.0175, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 24.72, "z": 8.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.CompositionIso", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -5.286, "z": 24.0, "size": 0.2399, "title": "Compatibilities for left adjoints from compatibilities satisfied by right adjoints", "summary": "In this file, given isomorphisms between compositions of right adjoint functors, we obtain isomorphisms between the corresponding compositions of the left adjoint functors, and show that the left adjoint functors satisfy properties similar to the left/right unitality and the associativity of pseudofunctors if the right adjoint functors satisfy the corresponding properties. This is used in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/CompositionIso.html"}, {"id": "Mathlib.CategoryTheory.Adjunction.FullyFaithfulLimits", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -18.252, "z": 36.436, "size": 0.2, "title": "Preservation of colimits and reflective adjunctions", "summary": "Let `adj : F ⊣ G` be an adjunction with `G : D ⥤ C` full and faithful. We show that if colimits of shape `J` exist in `C`, then a functor `H : D ⥤ E` preserves colimits of shape `J` iff `F ⋙ H` does. In particular, a functor from a category of sheaves preserves colimits iff it does so after precomposition with the sheafification functor.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adjunction/FullyFaithfulLimits.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.ColimitsCardinalClosure", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -32.55, "z": 49.436, "size": 0.2497, "title": "Closure of a property of objects under colimits of bounded cardinality", "summary": "In this file, given `P : ObjectProperty C` and `κ : Cardinal.{w}`, we introduce the closure `P.colimitsCardinalClosure κ` of `P` under colimits of shapes given by categories `J` such that `Arrow J` is of cardinality `< κ`. If `C` is locally `w`-small and `P` is essentially `w`-small, we show that this closure `P.colimitsCardinalClosure κ` is also essentially `w`-small.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/ColimitsCardinalClosure.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 103, "macro_tier_score": 0.185, "macro_tier_override": null, "x": 53.14, "z": 42.436, "size": 0.4686, "title": "Preserving pullbacks", "summary": "Constructions to relate the notions of preserving pullbacks and reflecting pullbacks to concrete pullback cones. In particular, we show that `pullbackComparison G f g` is an isomorphism iff `G` preserves the pullback of `f` and `g`. The dual is also given.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -26.312, "z": 44.436, "size": 0.2, "title": "Day functors", "summary": "In this file, given a monoidal category `C` and a monoidal category `V`, we define a basic type synonym `DayFunctor C V` (denoted `C ⊛⥤ D`) for the category `C ⥤ V` and endow it with the monoidal structure coming from Day convolution. Such a setup is necessary as by default, the `MonoidalCategory` instance on `C ⥤ V` is the \"pointwise\" one, where the tensor product of `F` and `G` is the functor `x ↦ F.obj x ⊗ G.obj…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/DayConvolution/DayFunctor.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -38.861, "z": 51.436, "size": 0.2597, "title": "Description of the covering sieves of the extensive topology", "summary": "This file characterises the covering sieves of the extensive topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveTopology.html"}, {"id": "Mathlib.CategoryTheory.Discrete.SumsProducts", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 34.85, "z": 20.436, "size": 0.2, "title": "Sums and products of discrete categories.", "summary": "This file shows that binary products and binary sums of discrete categories are also discrete, both in the form of explicit equivalences and through the `IsDiscrete` typeclass.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Discrete/SumsProducts.html"}, {"id": "Mathlib.CategoryTheory.Functor.Functorial", "region_id": "category_theory", "micro_elevation": 0.0526, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 20.057, "z": 10.436, "size": 0.2, "title": "Unbundled functors, as a typeclass decorating the object-level function.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/Functorial.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Tor", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -29.618, "z": 51.436, "size": 0.2, "title": "Tor, the left-derived functor of tensor product", "summary": "We define `Tor C n : C ⥤ C ⥤ C`, by left-deriving in the second factor of `(X, Y) ↦ X ⊗ Y`. For now we have almost nothing to say about it! It would be good to show that this is naturally isomorphic to the functor obtained by left-deriving in the first factor, instead. For now we define `Tor'` by left-deriving in the first factor, but showing `Tor C n ≅ Tor' C n` will require a bit more theory! Possibly it's best to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Tor.html"}, {"id": "Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 102, "macro_tier_score": 0.0288, "macro_tier_override": null, "x": -8.06, "z": 43.436, "size": 0.2671, "title": "The IPC property", "summary": "Given a family of categories `I i` (`i : α`) and a family of functors `F i : I i ⥤ C`, we consider the diagram `pointwiseProduct F : Π i, I i ⥤ C` defined by `(Xᵢ)ᵢ ↦ ∏ᶜ Xᵢ`. Given a cocone `cᵢ` on `Fᵢ` for each `i`, there is a natural cocone on `pointwiseProduct F` with point `∏ᶜ cᵢ`. Similarly to the study of finite limits commuting with filtered colimits, we then study sufficient conditions for this cocone to be…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FilteredColimitCommutesProduct.html"}, {"id": "Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Terminal", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 102, "macro_tier_score": 0.0292, "macro_tier_override": null, "x": -19.105, "z": 36.436, "size": 0.3069, "title": "Initial and terminal objects in the category of functors", "summary": "We show that if a functor `F : C ⥤ D` is such that `F.obj X` is terminal for all `X`, then `F` is a terminal object.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FunctorCategory/Shapes/Terminal.html"}, {"id": "Mathlib.CategoryTheory.LiftingProperties.PushoutProduct", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 44.645, "z": 52.436, "size": 0.2, "title": "Lifting properties and pushout-products / pullback-homs", "summary": "Various equivalent lifting properties involving pushout-products and pullback-homs. For `f : A ⟶ B`, `g : K ⟶ L`, `h : X ⟶ Y` in a monoidal closed category with pushouts and pullbacks, `f □ g` lifts against `h` if and only if `g` lifts against `f ⋔ h`. Special cases are considered when any of `A = ∅`, `K = ∅`, or `Y = ⋆` are true.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/LiftingProperties/PushoutProduct.html"}, {"id": "Mathlib.CategoryTheory.Join.Pseudofunctor", "region_id": "category_theory", "micro_elevation": 0.2281, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 17.091, "z": 20.436, "size": 0.2, "title": "Pseudofunctoriality of categorical joins", "summary": "In this file, we promote the join construction to two pseudofunctors `Join.pseudofunctorLeft` and `Join.pseudofunctorRight`, expressing its pseudofunctoriality in each variable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Join/Pseudofunctor.html"}, {"id": "Mathlib.CategoryTheory.Sites.LocallyFullyFaithful", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 102, "macro_tier_score": 0.0204, "macro_tier_override": null, "x": 95.996, "z": 55.436, "size": 0.3584, "title": "Locally fully faithful functors into sites", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/LocallyFullyFaithful.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.TStructure.SpectralObject", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 86.263, "z": 61.436, "size": 0.2, "title": "Spectral objects attached to t-structures", "summary": "Let `C` be a triangulated category equipped with a t-structure `t`. We define a functor `t.ω₁ : ComposableArrows EInt 1 ⥤ C ⥤ C` which sends a map `a ⟶ b` in `EInt` (i.e. `a ≤ b`) to the functor `t.eTruncLT.obj b ⋙ t.eTruncGE.obj a`. (Roughly speaking, we \"keep\" the `t`-homology only in degree `n` such that `a ≤ n < b`.) When we have two composable morphisms `f : a ⟶ b` and `g : b ⟶ c` in `EInt`, we define a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/TStructure/SpectralObject.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.TStructure.ETrunc", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -54.555, "z": 60.436, "size": 0.2478, "title": "Truncations for a t-structure", "summary": "Let `t` be a t-structure on a triangulated category `C`. In this file, we extend the definition of the truncation functors `truncLT` and `truncGE` for indices in `ℤ` to `EInt`, as `t.eTruncLT : EInt ⥤ C ⥤ C` and `t.eTruncGE : EInt ⥤ C ⥤ C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/TStructure/ETrunc.html"}, {"id": "Mathlib.CategoryTheory.Localization.DerivabilityStructure.OfFunctorialResolutions", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -35.187, "z": 47.436, "size": 0.2, "title": "Functorial resolutions give derivability structures", "summary": "In this file, we provide a constructor for right derivability structures. We assume that `Φ : LocalizerMorphism W₁ W₂` is given by a fully faithful functor `Φ.functor : C₁ ⥤ C₂` and that we have a resolution functor `ρ : C₂ ⥤ C₁` with a natural transformation `i : 𝟭 C₂ ⟶ ρ ⋙ Φ.functor` such that `W₂ (i.app X₂)` for any `X₂ : C₂`. If we assume that `W₁` is induced by `W₂`, that `W₂` is multiplicative and has the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/DerivabilityStructure/OfFunctorialResolutions.html"}, {"id": "Mathlib.CategoryTheory.Thin", "region_id": "category_theory", "micro_elevation": 0.0351, "macro_tier": 103, "macro_tier_score": 0.301, "macro_tier_override": null, "x": 27.44, "z": 9.436, "size": 0.334, "title": "Thin categories", "summary": "A thin category (also known as a sparse category) is a category with at most one morphism between each pair of objects. Examples include posets, but also some indexing categories (diagrams) for special shapes of (co)limits. To construct a category instance one only needs to specify the `CategoryStruct` part, as the axioms hold for free. If `C` is thin, then the category of functors to `C` is also thin. Further, to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Thin.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic", "region_id": "category_theory", "micro_elevation": 0.1754, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 19.018, "z": 17.436, "size": 0.2, "title": "Categorical pullbacks", "summary": "This file defines the basic properties of categorical pullbacks. Given a pair of functors `(F : A ⥤ B, G : C ⥤ B)`, we define the category `CategoricalPullback F G` as the category of triples `(a : A, c : C, e : F.obj a ≅ G.obj b)`. The category `CategoricalPullback F G` sits in a canonical `CatCommSq`, and we formalize that this square is a \"limit\" in the following sense: functors `X ⥤ CategoricalPullback F G` are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Categorical/Basic.html"}, {"id": "Mathlib.CategoryTheory.Comma.StructuredArrow.CommaMap", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 102, "macro_tier_score": 0.024, "macro_tier_override": null, "x": 51.961, "z": 33.436, "size": 0.2611, "title": "Structured arrow categories on `Comma.map`", "summary": "We characterize structured arrow categories on arbitrary instances of `Comma.map` as a comma category itself.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/StructuredArrow/CommaMap.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.Transport", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 38.706, "z": 37.436, "size": 0.2, "title": "Transporting a closed monoidal structure along an equivalence of categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/Transport.html"}, {"id": "Mathlib.CategoryTheory.Adhesive.Subobject", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -34.268, "z": 51.436, "size": 0.2, "title": "Subobjects in adhesive categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adhesive/Subobject.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.OfBiproducts", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 75.0, "z": 46.436, "size": 0.2, "title": "Constructing a semiadditive structure from binary biproducts", "summary": "We show that any category with zero morphisms and binary biproducts is enriched over the category of commutative monoids.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/OfBiproducts.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Types.Coyoneda", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 102, "macro_tier_score": 0.0246, "macro_tier_override": null, "x": -32.042, "z": 51.436, "size": 0.322, "title": "`(𝟙_ C ⟶ -)` is a lax monoidal functor to `Type`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Types/Coyoneda.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Center", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 102, "macro_tier_score": 0.0246, "macro_tier_override": null, "x": -4.056, "z": 35.436, "size": 0.322, "title": "Half braidings and the Drinfeld center of a monoidal category", "summary": "We define `Center C` to be pairs `⟨X, b⟩`, where `X : C` and `b` is a half-braiding on `X`. We show that `Center C` is braided monoidal, and provide the monoidal functor `Center.forget` from `Center C` back to `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Center.html"}, {"id": "Mathlib.CategoryTheory.Sites.Precoverage.Generates", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -32.496, "z": 50.436, "size": 0.2, "title": "Generators of a Grothendieck topology", "summary": "Let `K` be a precoverage and `J` a Grothendieck topology on a category `C`. We say `K` generates `J` if for every presheaf `F` on `C`, it is a sheaf for `J` if and only if it is a sheaf for every covering in `K`. If `K` generates `J`, then `J` is the smallest Grothendieck topology containing `K`. The converse only holds if `K` is a coverage or a pretopology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Precoverage/Generates.html"}, {"id": "Mathlib.CategoryTheory.Sites.Precoverage.Subsheaf", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 80.875, "z": 46.436, "size": 0.2338, "title": "Sheafification of subpresheafs for precoverages", "summary": "Let `K` be a precoverage. In this file we define the `K`-sheafification of a subpresheaf. More generally, for a family of subsets `𝒮` of sections of a sheaf `F`, we construct the smallest subsheaf of `F` containing `𝒮`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Precoverage/Subsheaf.html"}, {"id": "Mathlib.CategoryTheory.Shift.InducedShiftSequence", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -4.741, "z": 43.0, "size": 0.2574, "title": "Induced shift sequences", "summary": "When `G : C ⥤ A` is a functor from a category equipped with a shift by a monoid `M`, we have defined in the file `Mathlib/CategoryTheory/Shift/ShiftSequence.lean` a type class `G.ShiftSequence M` which provides functors `G.shift a : C ⥤ A` for all `a : M`, isomorphisms `shiftFunctor C n ⋙ G.shift a ≅ G.shift a'` when `n + a = a'`, and isomorphisms `G.isoShift a : shiftFunctor C a ⋙ G ≅ G.shift a` for all `a`, all of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/InducedShiftSequence.html"}, {"id": "Mathlib.CategoryTheory.Shift.ShiftSequence", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0102, "macro_tier_override": null, "x": -38.727, "z": 49.436, "size": 0.3132, "title": "Sequences of functors from a category equipped with a shift", "summary": "Let `F : C ⥤ A` be a functor from a category `C` that is equipped with a shift by an additive monoid `M`. In this file, we define a typeclass `F.ShiftSequence M` which includes the data of a sequence of functors `F.shift a : C ⥤ A` for all `a : A`. For each `a : A`, we have an isomorphism `F.isoShift a : shiftFunctor C a ⋙ F ≅ F.shift a` which satisfies some coherence relations. This allows to state results (e.g.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/ShiftSequence.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Preorder.Basic", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 102, "macro_tier_score": 0.0288, "macro_tier_override": null, "x": -19.038, "z": 36.436, "size": 0.2602, "title": "Limits and colimits indexed by preorders", "summary": "In this file, we obtain the following very basic results about limits and colimits indexed by a preordered type `J`: * a least element in `J` implies the existence of all limits indexed by `J` * a greatest element in `J` implies the existence of all colimits indexed by `J`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Preorder/Basic.html"}, {"id": "Mathlib.CategoryTheory.Shift.ShiftedHomOpposite", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -34.725, "z": 52.436, "size": 0.2616, "title": "Shifted morphisms in the opposite category", "summary": "If `C` is a category equipped with a shift by `ℤ`, `X` and `Y` are objects of `C`, and `n : ℤ`, we define a bijection `ShiftedHom.opEquiv : ShiftedHom X Y n ≃ ShiftedHom (Opposite.op Y) (Opposite.op X) n`. We also introduce `ShiftedHom.opEquiv'` which produces a bijection `ShiftedHom X Y a' ≃ (Opposite.op (Y⟦a⟧) ⟶ (Opposite.op X)⟦n⟧)` when `n + a = a'`. The compatibilities that are obtained shall be used in order to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/ShiftedHomOpposite.html"}, {"id": "Mathlib.CategoryTheory.Subobject.ArtinianObject", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 47.144, "z": 54.436, "size": 0.2478, "title": "Artinian objects", "summary": "We shall say that an object `X` in a category `C` is Artinian (type class `IsArtinianObject X`) if the ordered type `Subobject X` satisfies the descending chain condition. The corresponding property of objects `isArtinianObject : ObjectProperty C` is always closed under subobjects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subobject/ArtinianObject.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Opposite.Mon_", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 68.386, "z": 37.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Opposite/Mon_.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Opposite.Mon", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 54.545, "z": 36.436, "size": 0.2676, "title": "Monoid objects internal to monoidal opposites", "summary": "In this file, we record the equivalence between `Mon C` and `Mon Cᴹᵒᵖ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Opposite/Mon.html"}, {"id": "Mathlib.CategoryTheory.CopyDiscardCategory.Cartesian", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 30.986, "z": 51.436, "size": 0.2, "title": "Cartesian Categories as Copy-Discard Categories", "summary": "Every cartesian monoidal category is a copy-discard category where: - Copy is the diagonal map - Discard is the unique map to terminal", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/CopyDiscardCategory/Cartesian.html"}, {"id": "Mathlib.CategoryTheory.Enriched.FunctorCategory", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": -33.644, "z": 54.436, "size": 0.2569, "title": "Functor categories are enriched", "summary": "If `C` is a `V`-enriched ordinary category, then `J ⥤ C` is also both a `V`-enriched ordinary category and a `J ⥤ V`-enriched ordinary category, provided `C` has suitable limits. We first define the `V`-enriched structure on `J ⥤ C` by saying that if `F₁` and `F₂` are in `J ⥤ C`, then `enrichedHom V F₁ F₂ : V` is a suitable limit involving `F₁.obj j ⟶[V] F₂.obj j` for all `j : C`. The `J ⥤ V` object of morphisms…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Enriched/FunctorCategory.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.LocalEpi", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -3.741, "z": 50.436, "size": 0.2, "title": "Local epimorphisms with respect to an object property", "summary": "Let `P` be an object property on a category `C`. We say that `f : X ⟶ Y` is a local epimorphism wrt. `P` if `f` cancels on the left for morphisms with codomain in `P`. If `C` is the category of presheafs on some category with Grothendieck topology `J` and `P` the property of being a sheaf for `J`, then being a local epimorphism wrt. `P` is being an epimorphism after sheafification.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/LocalEpi.html"}, {"id": "Mathlib.CategoryTheory.Center.Linear", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 102, "macro_tier_score": 0.0096, "macro_tier_override": null, "x": 82.141, "z": 47.436, "size": 0.2464, "title": "Center of a linear category", "summary": "If `C` is an `R`-linear category, we define a ring morphism `R →+* CatCenter C` and conversely, if `C` is a preadditive category, and `φ : R →+* CatCenter C` is a ring morphism, we define an `R`-linear structure on `C` attached to `φ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Center/Linear.html"}, {"id": "Mathlib.CategoryTheory.Limits.EssentiallySmall", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.0338, "macro_tier_override": null, "x": 28.549, "z": 34.436, "size": 0.2892, "title": "Limits over essentially small indexing categories", "summary": "If `C` has limits of size `w` and `J` is `w`-essentially small, then `C` has limits of shape `J`. See also the file `FinallySmall.lean` for more general results.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/EssentiallySmall.html"}, {"id": "Mathlib.CategoryTheory.Limits.ColimitLimit", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 103, "macro_tier_score": 0.0431, "macro_tier_override": null, "x": 66.581, "z": 36.436, "size": 0.2598, "title": "The morphism comparing a colimit of limits with the corresponding limit of colimits.", "summary": "For `F : J × K ⥤ C` there is always a morphism $\\colim_k \\lim_j F(j,k) → \\lim_j \\colim_k F(j, k)$. While it is not usually an isomorphism, with additional hypotheses on `J` and `K` it may be, in which case we say that \"colimits commute with limits\". The prototypical example, proved in `CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit`, is that when `C = Type`, filtered colimits commute with finite limits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/ColimitLimit.html"}, {"id": "Mathlib.CategoryTheory.Countable", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 102, "macro_tier_score": 0.0193, "macro_tier_override": null, "x": -7.596, "z": 32.436, "size": 0.2696, "title": "Countable categories", "summary": "A category is countable in this sense if it has countably many objects and countably many morphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Countable.html"}, {"id": "Mathlib.CategoryTheory.GlueData", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -141.43, "z": 228.683, "size": 0.24, "title": "Gluing data", "summary": "We define `GlueData` as a family of data needed to glue topological spaces, schemes, etc. We provide the API to realize it as a multispan diagram, and also state lemmas about its interaction with a functor that preserves certain pullbacks.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GlueData.html"}, {"id": "Mathlib.CategoryTheory.Category.TwoP", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 28.17, "z": 32.436, "size": 0.2, "title": "The category of two-pointed types", "summary": "This defines `TwoP`, the category of two-pointed types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/TwoP.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Presheaf", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 102, "macro_tier_score": 0.0193, "macro_tier_override": null, "x": -19.261, "z": 49.436, "size": 0.2641, "title": "Finite-limit-preserving presheaves", "summary": "In this file we prove that if `C` is a small finitely cocomplete category and `A : Cᵒᵖ ⥤ Type u` is a presheaf, then `CostructuredArrow yoneda A` is filtered (equivalently, the category of elements of `A` is cofiltered) if and only if `A` preserves finite limits. This is one of the keys steps of establishing the equivalence `Ind C ≌ (Cᵒᵖ ⥤ₗ Type u)` (here, `Cᵒᵖ ⥤ₗ Type u` is the category of left exact functors) for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Presheaf.html"}, {"id": "Mathlib.CategoryTheory.Abelian.FreydMitchell", "region_id": "category_theory", "micro_elevation": 1.0, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 108.932, "z": 64.436, "size": 0.2, "title": "The Freyd-Mitchell embedding theorem", "summary": "Let `C` be an abelian category. We construct a ring `FreydMitchell.EmbeddingRing C` and a functor `FreydMitchell.embedding : C ⥤ ModuleCat.{max u v} (EmbeddingRing C)` which is full, faithful and exact.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/FreydMitchell.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.Opposite", "region_id": "category_theory", "micro_elevation": 0.9825, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 91.752, "z": 63.436, "size": 0.2478, "title": "Embedding opposites of Grothendieck categories", "summary": "If `C` is Grothendieck abelian and `F : D ⥤ Cᵒᵖ` is a functor from a small category, we construct an object `G : Cᵒᵖ` such that `preadditiveCoyonedaObj G : Cᵒᵖ ⥤ ModuleCat (End G)ᵐᵒᵖ` is faithful and exact and its precomposition with `F` is full if `F` is.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckCategory/ModuleEmbedding/Opposite.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.CoherentTopology", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -4.029, "z": 52.436, "size": 0.2502, "title": "Description of the covering sieves of the coherent topology", "summary": "This file characterises the covering sieves of the coherent topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.html"}, {"id": "Mathlib.CategoryTheory.Comma.CatCommSq", "region_id": "category_theory", "micro_elevation": 0.2807, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 48.291, "z": 23.436, "size": 0.2807, "title": "2-commutative squares of categories of arrows", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/CatCommSq.html"}, {"id": "Mathlib.CategoryTheory.Limits.Final.ParallelPair", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 4.28, "z": 42.436, "size": 0.2453, "title": "Conditions for `parallelPair` to be initial", "summary": "In this file we give sufficient conditions on a category `C` and parallel morphisms `f g : X ⟶ Y` in `C` so that `parallelPair f g` becomes an initial functor. The conditions are that there is a morphism out of `X` to every object of `C` and that any two parallel morphisms out of `X` factor through the parallel pair `f`, `g` (`h₂ : ∀ ⦃Z : C⦄ (i j : X ⟶ Z), ∃ (a : Y ⟶ Z), i = f ≫ a ∧ j = g ≫ a`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Final/ParallelPair.html"}, {"id": "Mathlib.CategoryTheory.FinCategory.AsType", "region_id": "category_theory", "micro_elevation": 0.2456, "macro_tier": 103, "macro_tier_score": 0.2421, "macro_tier_override": null, "x": 6.828, "z": 21.436, "size": 0.4697, "title": "Finite categories are equivalent to categories in `Type 0`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/FinCategory/AsType.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Adjunction.Mate", "region_id": "category_theory", "micro_elevation": 0.0351, "macro_tier": 102, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": 22.176, "z": 9.436, "size": 0.2603, "title": "Mates in bicategories", "summary": "This file establishes the bijection between the 2-cells ``` l₁ r₁ c --→ d c ←-- d g ↓ ↗ ↓ h g ↓ ↘ ↓ h e --→ f e ←-- f l₂ r₂ ``` where `l₁ ⊣ r₁` and `l₂ ⊣ r₂`. The corresponding 2-morphisms are called mates. For the bicategory `Cat`, the definitions in this file are provided in `Mathlib/CategoryTheory/Adjunction/Mates.lean`, where you can find more detailed documentation about mates.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Adjunction/Mate.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.Cospan", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 103, "macro_tier_score": 0.26, "macro_tier_override": null, "x": -8.931, "z": 35.436, "size": 0.6717, "title": "Cospan & Span", "summary": "We define a category `WalkingCospan` (resp. `WalkingSpan`), which is the index category for the given data for a pullback (resp. pushout) diagram. Convenience methods `cospan f g` and `span f g` construct functors from the walking (co)span, hitting the given morphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Cospan.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Rotate", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": 7.233, "z": 51.436, "size": 0.274, "title": "Rotate", "summary": "This file adds the ability to rotate triangles and triangle morphisms. It also shows that rotation gives an equivalence on the category of triangles.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Rotate.html"}, {"id": "Mathlib.CategoryTheory.Limits.FormalCoproducts", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 77.499, "z": 46.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FormalCoproducts.html"}, {"id": "Mathlib.CategoryTheory.Idempotents.Biproducts", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -41.676, "z": 51.436, "size": 0.2, "title": "Biproducts in the idempotent completion of a preadditive category", "summary": "In this file, we define an instance expressing that if `C` is an additive category (i.e. is preadditive and has finite biproducts), then `Karoubi C` is also an additive category. We also obtain that for all `P : Karoubi C` where `C` is a preadditive category `C`, there is a canonical isomorphism `P ⊞ P.complement ≅ (toKaroubi C).obj P.X` in the category `Karoubi C` where `P.complement` is the formal direct factor of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Idempotents/Biproducts.html"}, {"id": "Mathlib.CategoryTheory.Preadditive.LeftExact", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 101, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 39.233, "z": 42.0, "size": 0.2857, "title": "Left exactness of functors between preadditive categories", "summary": "We show that a functor is left exact in the sense that it preserves finite limits, if it preserves kernels. The dual result holds for right exact functors and cokernels.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Preadditive/LeftExact.html"}, {"id": "Mathlib.CategoryTheory.Action", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -27.614, "z": 27.0, "size": 0.2338, "title": "Actions as functors and as categories", "summary": "From a multiplicative action M ↻ X, we can construct a functor from M to the category of types, mapping the single object of M to X and an element `m : M` to the map `X → X` given by multiplication by `m`. This functor induces a category structure on X -- a special case of the category of elements. A morphism `x ⟶ y` in this category is simply a scalar `m : M` such that `m • x = y`. In the case where M is a group,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Action.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.FunctorBicategory.Pseudo", "region_id": "category_theory", "micro_elevation": 0.2632, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 1.913, "z": 22.436, "size": 0.2478, "title": "The bicategory of pseudofunctors", "summary": "Given bicategories `B` and `C`, we define a bicategory structure on `Pseudofunctor B C` whose * objects are pseudofunctors, * 1-morphisms are strong natural transformations, and * 2-morphisms are modifications. We scope this instance to the `CategoryTheory.Pseudofunctor.StrongTrans` namespace to avoid potential future conflicts with other bicategory instances on `Pseudofunctor B C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/FunctorBicategory/Pseudo.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Modification.Pseudo", "region_id": "category_theory", "micro_elevation": 0.2456, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": 17.68, "z": 21.436, "size": 0.2806, "title": "Modifications between transformations of pseudofunctors", "summary": "In this file we define modifications of strong transformations of pseudofunctors. They are defined similarly to modifications of transformations of oplax functors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Modification/Pseudo.html"}, {"id": "Mathlib.CategoryTheory.Quotient.LocallySmall", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 186.105, "z": -82.661, "size": 0.2459, "title": "Quotient categories are locally small", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Quotient/LocallySmall.html"}, {"id": "Mathlib.CategoryTheory.Presentable.Type", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -28.226, "z": 61.436, "size": 0.2, "title": "Presentable objects in Type", "summary": "In this file, we show that if `κ : Cardinal.{u}` is a regular cardinal, then `X : Type u` is `κ`-presentable in the category of types iff `HasCardinalLT X κ` holds, i.e. the cardinal number of `X` is less than `κ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/Type.html"}, {"id": "Mathlib.CategoryTheory.Category.PartialFun", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 0.611, "z": 31.436, "size": 0.2, "title": "The category of types with partial functions", "summary": "This defines `PartialFun`, the category of types equipped with partial functions. This category is classically equivalent to the category of pointed types. The reason it doesn't hold constructively stems from the difference between `Part` and `Option`. Both can model partial functions, but the latter forces a decidable domain. Precisely, `PartialFunToPointed` turns a partial function `α →. β` into a function `Option…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/PartialFun.html"}, {"id": "Mathlib.CategoryTheory.Sites.Grothendieck", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 102, "macro_tier_score": 0.0291, "macro_tier_override": null, "x": 4.142, "z": 44.436, "size": 0.2983, "title": "Grothendieck topologies", "summary": "Definition and lemmas about Grothendieck topologies. A Grothendieck topology for a category `C` is a set of sieves on each object `X` satisfying certain closure conditions. Alternate versions of the axioms (in arrow form) are also described. Two explicit examples of Grothendieck topologies are given: * The dense topology * The atomic topology as well as the complete lattice structure on Grothendieck topologies…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Grothendieck.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.Comma", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 62.889, "z": 47.436, "size": 0.2, "title": "Properties of objects in comma categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/Comma.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.Images", "region_id": "category_theory", "micro_elevation": 0.5965, "macro_tier": 103, "macro_tier_score": 0.0479, "macro_tier_override": null, "x": -25.023, "z": 41.436, "size": 0.2715, "title": "Images in the category of types", "summary": "In this file, it is shown that the category of types has categorical images, and that these agree with the range of a function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/Images.html"}, {"id": "Mathlib.CategoryTheory.ComposableArrows.Four", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 101, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 25.034, "z": 17.0, "size": 0.2722, "title": "API for compositions of four arrows", "summary": "Given morphisms `f₁ : i₀ ⟶ i₁`, `f₂ : i₁ ⟶ i₂`, `f₃ : i₂ ⟶ i₃`, `f₄ : i₃ ⟶ i₄`, and their compositions `f₁₂ : i₀ ⟶ i₂`, `f₂₃ : i₁ ⟶ i₃` and `f₃₄ : i₂ ⟶ i₄`, we define maps `ComposableArrows.fourδ₄Toδ₃ : mk₃ f₁ f₂ f₃ ⟶ mk₂ f₁ f₂ f₃₄`, `fourδ₃Toδ₂ : mk₃ f₁ f₂ f₃₄ ⟶ mk₂ f₁ f₂₃ f₄`, `fourδ₂Toδ₁ : mk₃ f₁ f₂₃ f₄ ⟶ mk₂ f₁₂ f₃ f₄`, and `fourδ₁Toδ₀ : mk₃ f₁₂ f₃ f₄ ⟶ mk₂ f₂ f₃ f₄`. The names are justified by the fact that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ComposableArrows/Four.html"}, {"id": "Mathlib.CategoryTheory.Localization.OfQuotient", "region_id": "category_theory", "micro_elevation": 0.0, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 0.0, "z": 0.0, "size": 0.2331, "title": "Certain quotient categories are localizations", "summary": "Let `r : HomRel C` be a relation on morphisms in a category `C` and `W : MorphismProperty C`. We assume that `W` is inverted by the quotient functor `functor r : C ⥤ quotient r`. If any relation `r f₀ f₁` between morphisms `f₀ : X ⟶ Y` and `f₁ : X ⟶ Y` can be \"explained\" by the use of a homotopy involving a cylinder object (i.e. there exists an object `cylinder : C`, a morphism `π : cylinder ⟶ X` in `W`, a morphism…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/OfQuotient.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Arrow", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 89.917, "z": 52.436, "size": 0.2, "title": "Monoidal structure on the arrow category of a cartesian closed category.", "summary": "If `C` is a braided, cartesian closed category with pushouts and an initial object, then `Arrow C` has a symmetric monoidal category structure given by the pushout-product (the Leibniz construction given by the tensor product on `C`). If `C` also has pullbacks, then `Arrow C` has a monoidal closed structure given by the pullback-hom (the Leibniz construction given by the internal hom on `C`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Arrow.html"}, {"id": "Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 102, "macro_tier_score": 0.0099, "macro_tier_override": null, "x": -55.297, "z": 60.436, "size": 0.2842, "title": "Cardinals that are suitable for the small object argument", "summary": "In this file, given a class of morphisms `I : MorphismProperty C` and a regular cardinal `κ : Cardinal.{w}`, we define a typeclass `IsCardinalForSmallObjectArgument I κ` which requires certain smallness properties (`I` is `w`-small, `C` is locally `w`-small), the existence of certain colimits (pushouts, coproducts of size `w`, and the condition `HasIterationOfShape κ.ord.ToType C` about the existence of colimits…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/SmallObject/IsCardinalForSmallObjectArgument.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.IsSmall", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 102, "macro_tier_score": 0.0144, "macro_tier_override": null, "x": 1.733, "z": 24.436, "size": 0.2502, "title": "Small classes of morphisms", "summary": "A class of morphisms `W : MorphismProperty C` is `w`-small if the corresponding set in `Set (Arrow C)` is.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/IsSmall.html"}, {"id": "Mathlib.CategoryTheory.IsomorphismClasses", "region_id": "category_theory", "micro_elevation": 0.3684, "macro_tier": 103, "macro_tier_score": 0.301, "macro_tier_override": null, "x": 14.512, "z": 28.436, "size": 0.334, "title": "Objects of a category up to an isomorphism", "summary": "`IsIsomorphic X Y := Nonempty (X ≅ Y)` is an equivalence relation on the objects of a category. The quotient with respect to this relation defines a functor from our category to `Type`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/IsomorphismClasses.html"}, {"id": "Mathlib.CategoryTheory.Sites.InducedTopology", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 102, "macro_tier_score": 0.0195, "macro_tier_override": null, "x": 34.653, "z": 55.436, "size": 0.2892, "title": "Induced topologies", "summary": "In this file we study various topologies induced by a functor. Let `F : C ⥤ D` be a functor, `J` a Grothendieck topology on `C` and `K` a Grothendieck topology on `D`. - `CategoryTheory.Functor.inducedTopology F K`: The finest topology on `C` making `F` continuous. - `CategoryTheory.Functor.restrictedTopology F K`: The coarsest topology on `C` containing all sieves whose image generate a covering sieve of `K`. In…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/InducedTopology.html"}, {"id": "Mathlib.CategoryTheory.Sites.CoverPreserving", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 102, "macro_tier_score": 0.0203, "macro_tier_override": null, "x": 75.305, "z": 54.436, "size": 0.3476, "title": "Cover-preserving functors between sites.", "summary": "In order to show that a functor is continuous, we define cover-preserving functors between sites as functors that push covering sieves to covering sieves. Then, a cover-preserving and compatible-preserving functor is continuous.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/CoverPreserving.html"}, {"id": "Mathlib.CategoryTheory.LocallyCartesianClosed.ExponentiableMorphism", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 90.261, "z": 51.436, "size": 0.2, "title": "Exponentiable morphisms", "summary": "We define an exponentiable morphism `f : I ⟶ J` to be a morphism with a functorial choice of pullbacks, given by `ChosenPullbacksAlong f`, together with a right adjoint to the pullback functor `ChosenPullbacksAlong.pullback f : Over J ⥤ Over I`. We call this right adjoint the pushforward functor along `f`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/LocallyCartesianClosed/ExponentiableMorphism.html"}, {"id": "Mathlib.CategoryTheory.SmallObject.Basic", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 79.247, "z": -53.661, "size": 0.2547, "title": "The small object argument", "summary": "Let `C` be a category. A class of morphisms `I : MorphismProperty C` permits the small object argument (typeclass `HasSmallObjectArgument.{w} I` where `w` is an auxiliary universe) if there exists a regular cardinal `κ : Cardinal.{w}` such that `IsCardinalForSmallObjectArgument I κ` holds. This technical condition is defined in the file `Mathlib/CategoryTheory/SmallObject/IsCardinalForSmallObjectArgument.lean`. It…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/SmallObject/Basic.html"}, {"id": "Mathlib.CategoryTheory.Sites.Point.IsMonoidalW", "region_id": "category_theory", "micro_elevation": 1.0, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 97.205, "z": 64.436, "size": 0.2, "title": "Monoidal structure on sheaves using enough points", "summary": "Let `(C, J)` be a site with a conservative family of points. If `A` is a suitable monoidal category, we show that the class of morphisms `J.W : MorphismProperty (Cᵒᵖ ⥤ A)` is stable under tensor products, which allows to check the assumptions of `Sheaf.monoidalCategory` in the file `Mathlib/CategoryTheory/Sites/Monoidal.lean`, i.e. this can be used in order to construct the monoidal category structure on `Sheaf J A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Point/IsMonoidalW.html"}, {"id": "Mathlib.CategoryTheory.Sites.Point.Monoidal", "region_id": "category_theory", "micro_elevation": 0.9825, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 44.891, "z": 63.436, "size": 0.2478, "title": "Fiber functors are monoidal", "summary": "Let `Φ` be a point of a site `(C, J)`. Let `A` be a monoidal category where the tensor product commutes with filtered colimits in both variables. We show that the fiber functors `Φ.presheafFiber : (Cᵒᵖ ⥤ A) ⥤ A` and `Φ.sheafFiber : Sheaf J A ⥤ A` are monoidal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Point/Monoidal.html"}, {"id": "Mathlib.CategoryTheory.Sites.Point.Comap", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -9.501, "z": 59.436, "size": 0.2, "title": "Inverse image of a point by a continuous functor", "summary": "Let `F : C ⥤ D` be a representably flat continuous functor between sites `(C, J)` and `(D, K)`. Let `Φ` be a point of `(D, K)`. Assume `hF : CoverPreserving J K F`. In this file, we define a point `Φ.comap F hF` of the site `(C, J)` and construct an isomorphism `(Φ.comap F hF).sheafFiber ≅ F.sheafPullback A J K ⋙ Φ.sheafFiber`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Point/Comap.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Internal.Types.Grp_", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 93.654, "z": 53.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Internal/Types/Grp_.html"}, {"id": "Mathlib.CategoryTheory.GuitartExact.Opposite", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 102, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": -7.047, "z": 44.436, "size": 0.3401, "title": "The opposite of a Guitart exact square", "summary": "A `2`-square is Guitart exact iff the opposite (transposed) `2`-square is Guitart exact.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GuitartExact/Opposite.html"}, {"id": "Mathlib.CategoryTheory.Groupoid.VertexGroup", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 48.501, "z": 27.436, "size": 0.2338, "title": "Vertex group", "summary": "This file defines the vertex group (*aka* isotropy group) of a groupoid at a vertex.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Groupoid/VertexGroup.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Projective.Ext", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 1.702, "z": 50.436, "size": 0.2, "title": "Computing `Ext` using a projective resolution", "summary": "Given a projective resolution `R` of an object `X` in an abelian category `C`, we provide an API in order to construct elements in `Ext X Y n` in terms of the complex `R.complex` and to make computations in the `Ext`-group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Projective/Ext.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Projective.Extend", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 86.451, "z": 49.436, "size": 0.2338, "title": "Projective resolutions as cochain complexes indexed by the integers", "summary": "Given a projective resolution `R` of an object `X` in an abelian category `C`, we define `R.cochainComplex : CochainComplex C ℤ`, which is the extension of `R.complex : ChainComplex C ℕ`, and the quasi-isomorphism `R.π' : R.cochainComplex ⟶ (CochainComplex.singleFunctor C 0).obj X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Projective/Extend.html"}, {"id": "Mathlib.CategoryTheory.Category.Pairwise", "region_id": "category_theory", "micro_elevation": 0.4211, "macro_tier": 101, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -75.47, "z": 215.683, "size": 0.2859, "title": "The category of \"pairwise intersections\".", "summary": "Given `ι : Type v`, we build the diagram category `CategoryTheory.Pairwise ι` with objects `single i` and `pair i j`, for `i j : ι`, whose only non-identity morphisms are `left : pair i j ⟶ single i` and `right : pair i j ⟶ single j`. We use this later in describing (one formulation of) the sheaf condition. Given any function `U : ι → α`, where `α` is some complete lattice (e.g. `(Opens X)ᵒᵖ`), we produce a functor…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Pairwise.html"}, {"id": "Mathlib.CategoryTheory.DinatTrans", "region_id": "category_theory", "micro_elevation": 0.1579, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 20.826, "z": 16.436, "size": 0.2, "title": "Dinatural transformations", "summary": "Dinatural transformations are special kinds of transformations between functors `F G : Cᵒᵖ ⥤ C ⥤ D` which depend both covariantly and contravariantly on the same category (also known as difunctors). A dinatural transformation is a family of morphisms given only on *the diagonal* of the two functors, and is such that a certain naturality hexagon commutes. Note that dinatural transformations cannot be composed with…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/DinatTrans.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.Yoneda", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 101, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -62.119, "z": 53.0, "size": 0.3124, "title": "The Yoneda functors are homological", "summary": "Let `C` be a pretriangulated category. In this file, we show that the functors `preadditiveCoyoneda.obj A : C ⥤ AddCommGrpCat` for `A : Cᵒᵖ` and `preadditiveYoneda.obj B : Cᵒᵖ ⥤ AddCommGrpCat` for `B : C` are homological functors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/Yoneda.html"}, {"id": "Mathlib.CategoryTheory.Sites.Point.Over", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.532, "z": 61.436, "size": 0.2, "title": "Points of `Over` sites", "summary": "Given a point `Φ` of a site `(C, J)`, an object `X : C`, and `x : Φ.fiber.obj X`, we define a point `Φ.over x` of the site `(Over X, J.over X)`. We show that if `(C, J)` has enough points, then so does `(Over X, J.over X)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Point/Over.html"}, {"id": "Mathlib.CategoryTheory.ComposableArrows.Three", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 10.295, "z": 17.0, "size": 0.255, "title": "API for compositions of three arrows", "summary": "Given morphisms `f₁ : i ⟶ j`, `f₂ : j ⟶ k`, `f₃ : k ⟶ l`, and their compositions `f₁₂ : i ⟶ k` and `f₂₃ : j ⟶ l`, we define maps `ComposableArrows.threeδ₃Toδ₂ : mk₂ f₁ f₂ ⟶ mk₂ f₁ f₂₃`, `threeδ₂Toδ₁ : mk₂ f₁ f₂₃ ⟶ mk₂ f₁₂ f₃`, and `threeδ₁Toδ₀ : mk₂ f₁₂ f₃ ⟶ mk₂ f₂ f₃`. The names are justified by the fact that `ComposableArrow.mk₃ f₁ f₂ f₃` can be thought of as a `3`-simplex in the simplicial set `nerve C`, and its…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ComposableArrows/Three.html"}, {"id": "Mathlib.CategoryTheory.Abelian.Preradical.Basic", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -37.487, "z": 50.436, "size": 0.2459, "title": "Preradicals", "summary": "A **preradical** on an abelian category `C` is a monomorphism in the functor category `C ⥤ C` with codomain `𝟭 C`, i.e. an element of `MonoOver (𝟭 C)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/Preradical/Basic.html"}, {"id": "Mathlib.CategoryTheory.Topos.Sheaf", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 99.829, "z": 60.436, "size": 0.2, "title": "(Elementary) Sheaf Topos", "summary": "We define a subobject classifier for categories of sheaves of (large enough) types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Topos/Sheaf.html"}, {"id": "Mathlib.CategoryTheory.Limits.Weighted.HasWeightedLimit", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 52.958, "z": 34.436, "size": 0.2, "title": "Weighted limits", "summary": "In this file, we define weighted limits (in the non enriched case). Given a weight `W : J ⥤ Type w` and a functor `F : J ⥤ C`, the `W`-weighted limit of `J` is the limit of the functor `CategoryOfElements.π W ⋙ F : W.Elements ⥤ C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Weighted/HasWeightedLimit.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.TStructure.Heart", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 61.577, "z": 55.436, "size": 0.2, "title": "The heart of a t-structure", "summary": "Let `t` be a t-structure on a triangulated category `C`. We define the heart of `t` as a property `t.heart : ObjectProperty C`. As the the heart is usually identified to a particular category in the applications (e.g. the heart of the canonical t-structure on the derived category of an abelian category `A` identifies to `A`), instead of working with the full subcategory defined by `t.heart`, we introduce a typeclass…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/TStructure/Heart.html"}, {"id": "Mathlib.CategoryTheory.Functor.ReflectsIso.Limits", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 6.07, "z": 46.436, "size": 0.2, "title": "Exactness of families of functors which jointly reflect isomorphisms", "summary": "Let `Fᵢ : C ⥤ Dᵢ` be a conservative family of functors (i.e. they jointly reflect isomorphisms). Let `G : J ⥤ C` be a functor that has a limit that is preserved by the functors `Fᵢ`. In this file, we show that a cone for `G` is a limit if it is so after applying the functors `Fᵢ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/ReflectsIso/Limits.html"}, {"id": "Mathlib.CategoryTheory.Functor.ReflectsIso.Jointly", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 102, "macro_tier_score": 0.0101, "macro_tier_override": null, "x": -16.475, "z": 45.436, "size": 0.2982, "title": "Families of functors which jointly reflect isomorphisms", "summary": "Let `Fᵢ : C ⥤ Dᵢ` be a family of functors. The family is said to jointly reflect isomorphisms (resp. monomorphisms, resp. epimorphisms) if every `f : X ⟶ Y` in `C` for which `Fᵢ.map f` is an isomorphism (resp. monomorphism, resp. epimorphism) for all `i` is an isomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/ReflectsIso/Jointly.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.OfObjectProperty", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 102, "macro_tier_score": 0.0096, "macro_tier_override": null, "x": 54.361, "z": 48.436, "size": 0.2465, "title": "Morphism properties from object properties", "summary": "Given two object properties `P` and `Q`, we introduce a morphism property `ofObjectProperty P Q`, given by all morphisms whose source satisfies `P` and target satisfies `Q`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/OfObjectProperty.html"}, {"id": "Mathlib.CategoryTheory.Limits.Types.Coequalizers", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 47.758, "z": 193.674, "size": 0.2654, "title": "Coequalizers in Type", "summary": "The coequalizer of a pair of maps `(f, g)` from `X` to `Y` is the quotient of `Y` by `∀ x : Y, f x ~ g x`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Types/Coequalizers.html"}, {"id": "Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 102, "macro_tier_score": 0.0246, "macro_tier_override": null, "x": -16.568, "z": 48.436, "size": 0.3214, "title": "Grothendieck topology generated by a precoverage", "summary": "For any category `C`, we define the Grothendieck topology generated by a precoverage `J` on `C`. It is the smallest Grothendieck topology containing all the sieves generated by the covering presieves of `J`. The main definitions and theorems are: - `Precoverage.toGrothendieck`: The Grothendieck topology generated by the sieves generated by the covering presieves of `J`. - `Precoverage.toGrothendieck_eq_sInf`: The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/PrecoverageToGrothendieck.html"}, {"id": "Mathlib.CategoryTheory.Localization.CalculusOfFractions.OfAdjunction", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 100, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -0.033, "z": 71.011, "size": 0.2662, "title": "The calculus of fractions deduced from an adjunction", "summary": "If `G ⊣ F` is an adjunction, `F` is fully faithful, and `W` is a class of morphisms that is inverted by `G` and such that the morphism `adj.unit.app X` belongs to `W` for any object `X`, then `G` is a localization functor with respect to `W`. Moreover, if `W` is multiplicative, then `W` has a calculus of left fractions. This holds in particular if `W` is the inverse image of the class of isomorphisms by `G`. (The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/CalculusOfFractions/OfAdjunction.html"}, {"id": "Mathlib.CategoryTheory.GuitartExact.Over", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 76.182, "z": 43.436, "size": 0.2, "title": "Guitart exact squares involving `Over` categories", "summary": "Let `F : C ⥤ D` be a functor and `X : C`. One may consider the commutative square of categories where vertical functors are `Over.forget`: ``` Over.post F Over X ⥤ Over (F.obj X) | | v v C ⥤ D F ``` We show that this square is Guitart exact if for all `Y : C`, the binary product `X ⨯ Y` exists and `F` commutes with it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GuitartExact/Over.html"}, {"id": "Mathlib.CategoryTheory.GuitartExact.Basic", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 102, "macro_tier_score": 0.0105, "macro_tier_override": null, "x": 75.273, "z": 42.436, "size": 0.3305, "title": "Guitart exact squares", "summary": "Given four functors `T`, `L`, `R` and `B`, a 2-square `TwoSquare T L R B` consists of a natural transformation `w : T ⋙ R ⟶ L ⋙ B`: ``` T C₁ ⥤ C₂ L | | R v v C₃ ⥤ C₄ B ``` In this file, we define a typeclass `w.GuitartExact` which expresses that this square is exact in the sense of Guitart. This means that for any `X₃ : C₃`, the induced functor `CostructuredArrow L X₃ ⥤ CostructuredArrow R (B.obj X₃)` is final. It…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GuitartExact/Basic.html"}, {"id": "Mathlib.CategoryTheory.FiberedCategory.Fibered", "region_id": "category_theory", "micro_elevation": 0.3333, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 41.908, "z": 26.436, "size": 0.257, "title": "Fibered categories", "summary": "This file defines what it means for a functor `p : 𝒳 ⥤ 𝒮` to be (pre)fibered.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/FiberedCategory/Fibered.html"}, {"id": "Mathlib.CategoryTheory.FiberedCategory.Cartesian", "region_id": "category_theory", "micro_elevation": 0.3158, "macro_tier": 101, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": 16.75, "z": 25.436, "size": 0.2855, "title": "Cartesian morphisms", "summary": "This file defines Cartesian resp. strongly Cartesian morphisms with respect to a functor `p : 𝒳 ⥤ 𝒮`. This file has been adapted to `Mathlib/CategoryTheory/FiberedCategory/Cocartesian.lean`, please try to change them in sync.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/FiberedCategory/Cartesian.html"}, {"id": "Mathlib.CategoryTheory.Triangulated.TStructure.Induced", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 79.604, "z": 53.0, "size": 0.233, "title": "Induced t-structures", "summary": "Let `t` be a t-structure on a pretriangulated category `C`. If `P` is a triangulated subcategory of `C`, we introduce a typeclass `P.HasInducedTStructure t` which essentially says that up to isomorphisms `P` is stable by the application of the truncation functors. In particular, we show that the triangulated subcategory `t.plus` of `t`-bounded above objects can be endowed with a t-structure `t.onPlus`, and the same…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Triangulated/TStructure/Induced.html"}, {"id": "Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Images", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 82.803, "z": 46.436, "size": 0.2447, "title": "The category of type-valued functors has images", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FunctorCategory/Shapes/Images.html"}, {"id": "Mathlib.CategoryTheory.Pi.Monoidal", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 15.033, "z": 37.436, "size": 0.2, "title": "The pointwise monoidal structure on the product of families of monoidal categories", "summary": "Given a family of monoidal categories `C i`, we define a monoidal structure on `Π i, C i` where the tensor product is defined pointwise.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Pi/Monoidal.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Yoneda", "region_id": "category_theory", "micro_elevation": 0.2807, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 24.277, "z": 23.436, "size": 0.2, "title": "2-Yoneda embedding", "summary": "In this file we define the bicategorical Yoneda embedding.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Yoneda.html"}, {"id": "Mathlib.CategoryTheory.Sites.Descent.Precoverage", "region_id": "category_theory", "micro_elevation": 1.0, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 109.83, "z": 64.436, "size": 0.2, "title": "Characterization of (pre)stacks for a precoverage", "summary": "Let `F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat` be a pseudofunctor. Assuming `F` is a prestack for a Grothendieck topology `J`, we show that if `f : X i ⟶ S` and `f' : X' j ⟶ S` are two covering families of morphisms in `S` such that the sieve generated by `f'` is contained in the sieve generated by `f`, then the functor `F.DescentData f ⥤ F.DescentData f'` is fully faithful. It follows that if the descent is effective for the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Descent/Precoverage.html"}, {"id": "Mathlib.CategoryTheory.Sites.Descent.IsStack", "region_id": "category_theory", "micro_elevation": 0.9825, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 94.659, "z": 63.436, "size": 0.2676, "title": "Stacks: effectiveness of descent", "summary": "Let `C` be a category with a Grothendieck topology `J` and `F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat`. In this file, we define the typeclass `F.IsStack J` saying that `F` is a stack for `J`. (See the terminological note in the file `Mathlib/CategoryTheory/Sites/Descent/IsPrestack.lean`: we do not require that the categories `F.obj (.mk (op S))` are groupoids.) The typeclass `IsStack` extends `IsPrestack`. The effectiveness of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Descent/IsStack.html"}, {"id": "Mathlib.CategoryTheory.Shift.Opposite", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 102, "macro_tier_score": 0.0098, "macro_tier_override": null, "x": 85.578, "z": 50.436, "size": 0.2678, "title": "The (naive) shift on the opposite category", "summary": "If `C` is a category equipped with a shift by a monoid `A`, the opposite category can be equipped with a shift such that the shift functor by `n` is `(shiftFunctor C n).op`. This is the \"naive\" opposite shift, which we shall set on a category `OppositeShift C A`, which is a type synonym for `Cᵒᵖ`. However, for the application to (pre)triangulated categories, we would like to define the shift on `Cᵒᵖ` so that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/Opposite.html"}, {"id": "Mathlib.CategoryTheory.Limits.Shapes.Pullback.EquifiberedLimits", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -34.081, "z": 56.436, "size": 0.2, "title": "Functors equifibered over a fixed functor is closed under limits", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Shapes/Pullback/EquifiberedLimits.html"}, {"id": "Mathlib.CategoryTheory.Types.Epimorphisms", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 84.22, "z": 48.436, "size": 0.2596, "title": "Stability properties of epimorphisms in `Type`", "summary": "In this file, we show that in the category `Type u`, epimorphisms are stable under base change.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Types/Epimorphisms.html"}, {"id": "Mathlib.CategoryTheory.Noetherian", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 67.988, "z": 55.436, "size": 0.2, "title": "Artinian and Noetherian categories", "summary": "An Artinian category is a category in which objects do not have infinite decreasing sequences of subobjects. A Noetherian category is a category in which objects do not have infinite increasing sequences of subobjects. Note: In the file, `Mathlib/CategoryTheory/Subobject/ArtinianObject.lean`, it is shown that any nonzero Artinian object has a simple subobject.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Noetherian.html"}, {"id": "Mathlib.CategoryTheory.Subobject.NoetherianObject", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 20.617, "z": 53.436, "size": 0.2478, "title": "Noetherian objects", "summary": "We shall say that an object `X` in a category `C` is Noetherian (type class `IsNoetherianObject X`) if the ordered type `Subobject X` satisfies the ascending chain condition. The corresponding property of objects `isNoetherianObject : ObjectProperty C` is always closed under subobjects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subobject/NoetherianObject.html"}, {"id": "Mathlib.CategoryTheory.Sites.Coherent.ReflectsPrecoherent", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 18.026, "z": 53.436, "size": 0.2596, "title": "Reflecting the property of being precoherent", "summary": "We prove that given a fully faithful functor `F : C ⥤ D` which preserves and reflects finite effective epimorphic families, such that for every object `X` of `D` there exists an object `W` of `C` with an effective epi `π : F.obj W ⟶ X`, the category `C` is `Precoherent` whenever `D` is.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPrecoherent.html"}, {"id": "Mathlib.CategoryTheory.Category.Cat.Colimit", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -15.845, "z": 37.436, "size": 0.2, "title": "The category of small categories has all small colimits.", "summary": "In this file, the existence of colimits in `Cat` is deduced from the existence of colimits in the category of simplicial sets. Indeed, `Cat` identifies to a reflective subcategory of the category of simplicial sets (see `AlgebraicTopology.SimplicialSet.NerveAdjunction`), so that colimits in `Cat` can be computed by passing to nerves, taking the colimit in `SSet` and finally applying the homotopy category functor…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Cat/Colimit.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Strict.Pseudofunctor", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 101, "macro_tier_score": 0.005, "macro_tier_override": null, "x": 7.401, "z": 24.436, "size": 0.2633, "title": "Pseudofunctors from strict bicategory", "summary": "This file provides an API for pseudofunctors `F` from a strict bicategory `B`. In particular, this shall apply to pseudofunctors from locally discrete bicategories. Firstly, we study the compatibilities of the flexible variants `mapId'` and `mapComp'` of `mapId` and `mapComp` with respect to the composition with identities and the associativity. Secondly, given a commutative square `t ≫ r = l ≫ b` in `B`, we…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Strict/Pseudofunctor.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.Functor.Cat.ObjectProperty", "region_id": "category_theory", "micro_elevation": 0.2456, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 28.036, "z": 21.436, "size": 0.2, "title": "Properties of objects in target categories of a pseudofunctor to `Cat`", "summary": "Given `F : Pseudofunctor B Cat`, we introduce a type `F.ObjectProperty` which consists of properties `P` of objects for all categories `F.obj X` for `X : B`. The typeclass `P.IsClosedUnderMapObj` expresses that this property is preserved by the application of the functors `F.map`: this allows to define a sub-pseudofunctor `P.fullsubcategory : Pseudofunctor B Cat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/Functor/Cat/ObjectProperty.html"}, {"id": "Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory", "region_id": "category_theory", "micro_elevation": 0.4912, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -13.66, "z": 35.436, "size": 0.2, "title": "Free groupoid on a category", "summary": "This file defines the free groupoid on a category, the lifting of a functor to its unique extension as a functor from the free groupoid, and proves uniqueness of this extension.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Groupoid/FreeGroupoidOfCategory.html"}, {"id": "Mathlib.CategoryTheory.Sites.Hypercover.Saturate", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -25.827, "z": 52.436, "size": 0.2, "title": "Saturation of a `0`-hypercover", "summary": "Given a `0`-hypercover `E`, we define a `1`-hypercover `E.saturate`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Hypercover/Saturate.html"}, {"id": "Mathlib.CategoryTheory.Subfunctor.SubmonoidFunctor", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 8.326, "z": 45.436, "size": 0.2, "title": "Functors of submonoids", "summary": "Given a functor `M : C ⥤ MonCat`, we define a functor of submonoids `S` to be a family `Submonoid (M.obj U)` for all `U : C` that are compatible with the maps induced by `M`. We provide the complete lattice structure and the basic functoriality properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Subfunctor/SubmonoidFunctor.html"}, {"id": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Images", "region_id": "category_theory", "micro_elevation": 0.6491, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 80.111, "z": 44.436, "size": 0.2, "title": "Preserving images", "summary": "In this file, we show that if a functor preserves spans and cospans, then it preserves images.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Images.html"}, {"id": "Mathlib.CategoryTheory.Groupoid.Subgroupoid", "region_id": "category_theory", "micro_elevation": 0.3684, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 48.353, "z": 28.436, "size": 0.2, "title": "Subgroupoid", "summary": "This file defines subgroupoids as `structure`s containing the subsets of arrows and their stability under composition and inversion. Also defined are: * containment of subgroupoids is a complete lattice; * images and preimages of subgroupoids under a functor; * the notion of normality of subgroupoids and its stability under intersection and preimage; * compatibility of the above with…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Groupoid/Subgroupoid.html"}, {"id": "Mathlib.CategoryTheory.Functor.ReflectsIso.Exact", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -28.646, "z": 46.436, "size": 0.2, "title": "Exactness properties of functors which jointly reflect isomorphisms", "summary": "Let `Fᵢ : C ⥤ Dᵢ` be a family of exact functors between abelian categories. Assume that they jointly reflect isomorphisms. We show that a short complex in `C` is exact (resp. short exact) iff it is so after applying the functor `Fᵢ`. Similar results are obtained for the detection of quasi-isomorphisms between short complexes or homological complexes in `C`. (Corresponding results for a single functor are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/ReflectsIso/Exact.html"}, {"id": "Mathlib.CategoryTheory.ObjectProperty.ColimitsClosure", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 102, "macro_tier_score": 0.0193, "macro_tier_override": null, "x": 45.651, "z": 48.436, "size": 0.2713, "title": "Closure of a property of objects under colimits of certain shapes", "summary": "In this file, given a property `P` of objects in a category `C` and family of categories `J : α → Type _`, we introduce the closure `P.colimitsClosure J` of `P` under colimits of shapes `J a` for all `a : α`, and under certain smallness assumptions, we show that it is essentially small. (We deduce these results about the closure under colimits by dualising the results in the file…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ObjectProperty/ColimitsClosure.html"}, {"id": "Mathlib.CategoryTheory.Localization.Resolution", "region_id": "category_theory", "micro_elevation": 0.5088, "macro_tier": 101, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": 48.535, "z": 36.436, "size": 0.3003, "title": "Resolutions for a morphism of localizers", "summary": "Given a morphism of localizers `Φ : LocalizerMorphism W₁ W₂` (i.e. `W₁` and `W₂` are morphism properties on categories `C₁` and `C₂`, and we have a functor `Φ.functor : C₁ ⥤ C₂` which sends morphisms in `W₁` to morphisms in `W₂`), we introduce the notion of right resolutions of objects in `C₂`, for `X₂ : C₂`. A right resolution consists of an object `X₁ : C₁` and a morphism `w : X₂ ⟶ Φ.functor.obj X₁` that is in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Localization/Resolution.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.GrpLimits", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 93.86, "z": 54.436, "size": 0.2478, "title": "Limits in `Grp C`", "summary": "We show that `Grp C` has limits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/GrpLimits.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.ShrinkYoneda", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -14.304, "z": 53.436, "size": 0.2403, "title": "The Yoneda embedding for monoid objects for locally small categories", "summary": "Let `C` be a locally `w`-small category. We define the Yoneda embedding `shrinkYonedaMon : Mon C ⥤ Cᵒᵖ ⥤ MonCat.{w} w` and its `Grp` analogue.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/ShrinkYoneda.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.Grp_", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 13.468, "z": 53.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/Grp_.html"}, {"id": "Mathlib.CategoryTheory.MorphismProperty.HasCardinalLT", "region_id": "category_theory", "micro_elevation": 0.2982, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": 0.86, "z": 24.436, "size": 0.2338, "title": "Properties of morphisms that are bounded by a cardinal", "summary": "Given `P : MorphismProperty C` and `κ : Cardinal`, we introduce a predicate `P.HasCardinalLT κ` saying that the cardinality of `P.toSet` is `< κ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/MorphismProperty/HasCardinalLT.html"}, {"id": "Mathlib.CategoryTheory.Sites.LocalProperties", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 72.397, "z": 61.436, "size": 0.2, "title": "Local properties of sheaves", "summary": "In this file we study properties of sheaves that can be checked on a covering family of objects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/LocalProperties.html"}, {"id": "Mathlib.CategoryTheory.Adhesive.Over", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 45.944, "z": 56.436, "size": 0.2, "title": "Adhesive structure on slice categories", "summary": "The slice category `Over B` inherits the property of being adhesive from the base category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Adhesive/Over.html"}, {"id": "Mathlib.CategoryTheory.Limits.FormalCoproducts.ExtraDegeneracy", "region_id": "category_theory", "micro_elevation": 0.7018, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -27.071, "z": 47.436, "size": 0.2, "title": "Extradegeneracy for the Cech object", "summary": "Let `U : FormalCoproduct C`. Let `T` be a terminal object in `C`. In this file, we construct an isomorphism from the Cech object `U.cech` that is defined for objects in `FormalCoproduct` to the general Cech nerve construction applied to the morphism from `U` to the terminal object. This isomorphism is used in order to show that, as an augmented object (over `T`), the Cech object `U.cech` has an extra degeneracy when…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FormalCoproducts/ExtraDegeneracy.html"}, {"id": "Mathlib.CategoryTheory.Limits.Constructions.WidePullbackOfTerminal", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -5.824, "z": 34.436, "size": 0.239, "title": "Existence of wide pullbacks when the target object is terminal", "summary": "In this file, we show that the wide pullback of a family of arrows `objs j ⟶ B` exists when `B` is terminal and the product of the objects `objs j` exists.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/Constructions/WidePullbackOfTerminal.html"}, {"id": "Mathlib.CategoryTheory.Galois.IsFundamentalgroup", "region_id": "category_theory", "micro_elevation": 0.7895, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 77.435, "z": 52.436, "size": 0.2, "title": "Universal property of fundamental group", "summary": "Let `C` be a Galois category with fiber functor `F`. While in informal mathematics, we tend to identify known groups from other contexts (e.g. the absolute Galois group of a field) with the automorphism group `Aut F` of certain fiber functors `F`, this causes friction in formalization. Hence, in this file we develop conditions when a topological group `G` is canonically isomorphic to the automorphism group `Aut F`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Galois/IsFundamentalgroup.html"}, {"id": "Mathlib.CategoryTheory.RegularCategory.Basic", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -15.091, "z": 54.436, "size": 0.2, "title": "Regular categories", "summary": "A regular category is a category with finite limits such that each kernel pair has a coequalizer and such that regular epimorphisms are stable under pullback. These categories provide a good ground to develop the calculus of relations, as well as being the semantics for regular logic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/RegularCategory/Basic.html"}, {"id": "Mathlib.CategoryTheory.Abelian.SerreClass.Localization", "region_id": "category_theory", "micro_elevation": 0.7544, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -34.45, "z": 50.436, "size": 0.2, "title": "Localization with respect to a Serre class", "summary": "The main definition in this file is `ObjectProperty.SerreClassLocalization.abelian` which shows that if `L : C ⥤ D` is a localization functor with respect to the class of morphisms `P.isoModSerre` for a Serre class `P : ObjectProperty C` in the abelian category `C`, then `D` is an abelian category. We also show that a functor `G : D ⥤ E` to an abelian category is exact iff the composition `L ⋙ G` is.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/SerreClass/Localization.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Action.End", "region_id": "category_theory", "micro_elevation": 0.4561, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -14.66, "z": 33.436, "size": 0.2, "title": "Actions as monoidal functors to endofunctor categories", "summary": "In this file, we show that given a right action of a monoidal category `C` on a category `D`, the curried action functor `C ⥤ D ⥤ D` is monoidal. Conversely, given a monoidal functor `C ⥤ D ⥤ D`, we can define a right action of `C` on `D`. The corresponding results are also available for left actions: given a left action of `C` on `D`, composing `CategoryTheory.MonoidalCategory.MonoidalLeftAction.curriedAction C D`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Action/End.html"}, {"id": "Mathlib.CategoryTheory.Bicategory.SingleObj", "region_id": "category_theory", "micro_elevation": 0.4386, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 61.783, "z": 32.436, "size": 0.2, "title": "Promoting a monoidal category to a single object bicategory.", "summary": "A monoidal category can be thought of as a bicategory with a single object. The objects of the monoidal category become the 1-morphisms, with composition given by tensor product, and the morphisms of the monoidal category become the 2-morphisms. We verify that the endomorphisms of that single object recovers the original monoidal category. One could go much further: the bicategory of monoidal categories (equipped…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Bicategory/SingleObj.html"}, {"id": "Mathlib.CategoryTheory.Category.Cat.Limit", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 59.2, "z": 34.436, "size": 0.2, "title": "The category of small categories has all small limits.", "summary": "An object in the limit consists of a family of objects, which are carried to one another by the functors in the diagram. A morphism between two such objects is a family of morphisms between the corresponding objects, which are carried to one another by the action on morphisms of the functors in the diagram.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/Cat/Limit.html"}, {"id": "Mathlib.CategoryTheory.Category.RelCat", "region_id": "category_theory", "micro_elevation": 0.3509, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -4.635, "z": 27.436, "size": 0.2, "title": "Basics on the category of relations", "summary": "We define the category of types `CategoryTheory.RelCat` with binary relations as morphisms. Associating each function with the relation defined by its graph yields a faithful and essentially surjective functor `graphFunctor` that also characterizes all isomorphisms (see `rel_iso_iff`). By flipping the arguments to a relation, we construct an equivalence `opEquivalence` between `RelCat` and its opposite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Category/RelCat.html"}, {"id": "Mathlib.CategoryTheory.Comma.StructuredArrow.Final", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 75.19, "z": 42.436, "size": 0.2, "title": "Finality on Costructured Arrow categories", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/StructuredArrow/Final.html"}, {"id": "Mathlib.CategoryTheory.ConcreteCategory.UnbundledHom", "region_id": "category_theory", "micro_elevation": 0.0175, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 25.199, "z": 8.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/ConcreteCategory/UnbundledHom.html"}, {"id": "Mathlib.CategoryTheory.Enriched.EnrichedCat", "region_id": "category_theory", "micro_elevation": 0.807, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 17.991, "z": 53.436, "size": 0.2, "title": "The bicategory of `V`-enriched categories", "summary": "We define the bicategory `EnrichedCat V` of (bundled) `V`-enriched categories for a fixed monoidal category `V`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Enriched/EnrichedCat.html"}, {"id": "Mathlib.CategoryTheory.Enriched.HomCongr", "region_id": "category_theory", "micro_elevation": 0.8246, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -40.03, "z": 54.436, "size": 0.2, "title": "Congruence of enriched homs", "summary": "Recall that when `C` is both a category and a `V`-enriched category, we say it is an `EnrichedOrdinaryCategory` if it comes equipped with a sufficiently compatible equivalence between morphisms `X ⟶ Y` in `C` and morphisms `𝟙_ V ⟶ (X ⟶[V] Y)` in `V`. In such a `V`-enriched ordinary category `C`, isomorphisms in `C` induce isomorphisms between hom-objects in `V`. We define this isomorphism in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Enriched/HomCongr.html"}, {"id": "Mathlib.CategoryTheory.Functor.Derived.PointwiseLeftDerived", "region_id": "category_theory", "micro_elevation": 0.5614, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 48.982, "z": 39.436, "size": 0.2, "title": "Pointwise left derived functors", "summary": "We define pointwise left derived functors using the notion of pointwise right Kan extensions. We show that if `F : C ⥤ H` inverts `W : MorphismProperty C`, then it has a pointwise left derived functor. Note: this file was obtained by dualizing the definitions in the file `Mathlib/CategoryTheory/Functor/Derived/PointwiseRightDerived.lean`. These two files should be kept in sync.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/Derived/PointwiseLeftDerived.html"}, {"id": "Mathlib.CategoryTheory.Limits.FunctorCategory.BinaryBiproducts", "region_id": "category_theory", "micro_elevation": 0.6842, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -30.724, "z": 46.436, "size": 0.2, "title": "Biproducts in functor categories", "summary": "We show that if `C` has binary biproducts, then the functor category `D ⥤ C` also has binary biproducts (`CategoryTheory.Limits.BinaryBiproduct.functorCategoryHasBinaryBiproducts`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Limits/FunctorCategory/BinaryBiproducts.html"}, {"id": "Mathlib.CategoryTheory.Monad.Monadicity", "region_id": "category_theory", "micro_elevation": 0.7193, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 48.89, "z": 48.436, "size": 0.2, "title": "Monadicity theorems", "summary": "We prove monadicity theorems which can establish a given functor is monadic. In particular, we show three versions of Beck's monadicity theorem, and the reflexive (crude) monadicity theorem: `G` is a monadic right adjoint if it has a left adjoint, and: * `D` has, `G` preserves and reflects `G`-split coequalizers, see `CategoryTheory.Monad.monadicOfHasPreservesReflectsGSplitCoequalizers` * `G` creates `G`-split…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monad/Monadicity.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Cartesian.Normal", "region_id": "category_theory", "micro_elevation": 0.8421, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 69.944, "z": 55.436, "size": 0.2, "title": "Normal subgroup objects", "summary": "In this file we define normal subgroups of group objects in a cartesian monoidal category as a predicate on morphisms. A morphism `φ : H ⟶ G` of group objects is normal if it is mono, a monoid morphism and the conjugation map `(g, h) ↦ g * h * g⁻¹` factors through `φ`. This is applied in the study of group schemes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Cartesian/Normal.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Closed.InternalCurrying", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 64.636, "z": 37.436, "size": 0.2, "title": "The currying-uncurrying isomorphism between internal homs of a closed monoidal category", "summary": "For a closed monoidal category `C`, we construct the isomorphism of internal hom objects `C(x ⊗ y, z) ≅ C(y, C(x, z))` for any triple of objects `x y z : C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Closed/InternalCurrying.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.Mod_", "region_id": "category_theory", "micro_elevation": 0.5263, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -16.973, "z": 37.436, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/Mod_.html"}, {"id": "Mathlib.CategoryTheory.Presentable.Directed", "region_id": "category_theory", "micro_elevation": 0.8596, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 97.429, "z": 56.436, "size": 0.2, "title": "`κ`-filtered categories and `κ`-directed poset", "summary": "In this file, we formalize the proof by Deligne (SGA 4 I 8.1.6) that for any (small) filtered category `J`, there exists a final functor `F : α ⥤ J` where `α` is a directed partially ordered set (`IsFiltered.exists_directed`). The construction applies more generally to `κ`-filtered categories and `κ`-directed posets (`IsCardinalFiltered.exists_cardinal_directed`). Note: the argument by Deligne is reproduced (without…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/Directed.html"}, {"id": "Mathlib.CategoryTheory.Presentable.OrthogonalReflection", "region_id": "category_theory", "micro_elevation": 0.9474, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 101.324, "z": 61.436, "size": 0.2, "title": "The Orthogonal-reflection construction", "summary": "Given `W : MorphismProperty C` (which should be small) and assuming the existence of certain colimits in `C`, we construct a morphism `toSucc W Z : Z ⟶ succ W Z` for any `Z : C`. This morphism belongs to `W.isLocal.isLocal` and is an isomorphism iff `Z` belongs to `W.isLocal` (see the lemma `isIso_toSucc_iff`). The morphism `toSucc W Z : Z ⟶ succ W Z` is defined as a composition of two morphisms that are roughly…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Presentable/OrthogonalReflection.html"}, {"id": "Mathlib.CategoryTheory.Sites.Point.Presheaf", "region_id": "category_theory", "micro_elevation": 0.9298, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 100.604, "z": 60.436, "size": 0.2, "title": "Points of presheaf toposes", "summary": "Let `C` be a category. For the Grothendieck topology `⊥` on `C`, we know that the category of sheaves with values in `A` identifies to `Cᵒᵖ ⥤ A` (see `sheafBotEquivalence` in the file `Mathlib/CategoryTheory/Sites/Sheaf.lean`). In this file, we show that any `X : C` defines a point for this site, and that these points form a conservative family of points.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Point/Presheaf.html"}, {"id": "Mathlib.CategoryTheory.Sites.Types", "region_id": "category_theory", "micro_elevation": 0.7719, "macro_tier": 100, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 17.406, "z": 51.436, "size": 0.2, "title": "Grothendieck Topology and Sheaves on the Category of Types", "summary": "In this file we define a Grothendieck topology on the category of types, and construct the canonical functor that sends a type to a sheaf over the category of types, and make this an equivalence of categories. Then we prove that the topology defined is the canonical topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Types.html"}, {"id": "Mathlib.CategoryTheory.Shift.SingleFunctorsLift", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -62.65, "z": 42.0, "size": 0.2331, "title": "Lift of a \"single functor\" to a full subcategory", "summary": "Let `C`, `D` and `E` be categories. Let `A` be an additive monoid. Assume that `D` and `E` have shifts by `A` and that we have a fully faithful functor `G : D ⥤ A` which commutes with shifts. Given `F : SingleFunctors C E A`, and a family of functors `Φ a : C ⥤ D` with isomorphisms `Φ a ⋙ G ≅ F.functor a` for all `a : A`, we lift `F` in `SingleFunctor C D A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Shift/SingleFunctorsLift.html"}, {"id": "Mathlib.CategoryTheory.Comma.Presheaf.Basic", "region_id": "category_theory", "micro_elevation": 0.4737, "macro_tier": 103, "macro_tier_score": 0.0814, "macro_tier_override": null, "x": -16.04, "z": 34.436, "size": 0.2877, "title": "Computation of `Over A` for a presheaf `A`", "summary": "Let `A : Cᵒᵖ ⥤ Type v` be a presheaf. In this file, we construct an equivalence `e : Over A ≌ (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v` and show that there is a quasi-commutative diagram ``` CostructuredArrow yoneda A ⥤ Over A ⇘ ⥥ PSh(CostructuredArrow yoneda A) ``` where the top arrow is the forgetful functor forgetting the yoneda-costructure, the right arrow is the aforementioned equivalence and the diagonal arrow…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Comma/Presheaf/Basic.html"}, {"id": "Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Monomorphisms", "region_id": "category_theory", "micro_elevation": 0.9123, "macro_tier": 101, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -53.711, "z": 59.436, "size": 0.2391, "title": "Monomorphisms in Grothendieck abelian categories", "summary": "In this file, we show that in a Grothendieck abelian category, monomorphisms are stable under transfinite composition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Abelian/GrothendieckCategory/Monomorphisms.html"}, {"id": "Mathlib.CategoryTheory.Monoidal.ExternalProduct.KanExtension", "region_id": "category_theory", "micro_elevation": 0.614, "macro_tier": 101, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -20.204, "z": 42.436, "size": 0.2823, "title": "Preservation of pointwise left Kan extensions by external products", "summary": "We prove that if a functor `H' : D' ⥤ V` is a pointwise left Kan extension of `H : D ⥤ V` along `L : D ⥤ D'`, and if `K : E ⥤ V` is any functor such that for any `e : E`, the functor `tensorRight (K.obj e)` commutes with colimits of shape `CostructuredArrow L d`, then the functor `H' ⊠ K` is a pointwise left Kan extension of `H ⊠ K` along `L.prod (𝟭 E)`. We also prove a similar criterion to establish that `K ⊠ H'`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Monoidal/ExternalProduct/KanExtension.html"}, {"id": "Mathlib.CategoryTheory.Functor.KanExtension.Preserves", "region_id": "category_theory", "micro_elevation": 0.5789, "macro_tier": 102, "macro_tier_score": 0.0193, "macro_tier_override": null, "x": 27.086, "z": 40.436, "size": 0.2653, "title": "Preservation of Kan extensions", "summary": "Given functors `F : A ⥤ B`, `L : B ⥤ C`, and `G : B ⥤ D`, we introduce a typeclass `G.PreservesLeftKanExtension F L` which encodes the fact that the left Kan extension of `F` along `L` is preserved by the functor `G`. When the Kan extension is pointwise, it suffices that `G` preserves (co)limits of the relevant diagrams. We introduce the dual typeclass `G.PreservesRightKanExtension`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Functor/KanExtension/Preserves.html"}, {"id": "Mathlib.CategoryTheory.GuitartExact.Quotient", "region_id": "category_theory", "micro_elevation": 0.6667, "macro_tier": 100, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 20.438, "z": 38.0, "size": 0.2324, "title": "Guitart exact squares and quotient categories", "summary": "Consider a commutative square of categories given by a natural isomorphism `e : T ⋙ R ≅ L ⋙ B`: ``` T C₀ ----> H₀ | | L| |R v v C ----> H B ``` If both `T` and `B` are full and `T` is essentially surjective, we show that the `2`-square above is Guitart exact if, whenever two morphisms `f₀` and `f₁` in `L.obj X₀ ⟶ Y` (for `X₀ : C₀` and `Y : C`) become equal after applying `B`, there exists a precylinder object `P` of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GuitartExact/Quotient.html"}, {"id": "Mathlib.CategoryTheory.Galois.Decomposition", "region_id": "category_theory", "micro_elevation": 0.7368, "macro_tier": 101, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": 75.614, "z": 49.436, "size": 0.2524, "title": "Decomposition of objects into connected components and applications", "summary": "We show that in a Galois category every object is the (finite) coproduct of connected subobjects. This has many useful corollaries, in particular that the fiber of every object is represented by a Galois object.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Galois/Decomposition.html"}, {"id": "Mathlib.CategoryTheory.GuitartExact.VerticalComposition", "region_id": "category_theory", "micro_elevation": 0.6316, "macro_tier": 102, "macro_tier_score": 0.0107, "macro_tier_override": null, "x": 37.733, "z": 43.436, "size": 0.3453, "title": "Vertical composition of Guitart exact squares", "summary": "In this file, we show that the vertical composition of Guitart exact squares is Guitart exact.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/GuitartExact/VerticalComposition.html"}, {"id": "Mathlib.AlgebraicGeometry.ZariskisMainTheorem", "region_id": "algebraic_geometry", "micro_elevation": 0.9355, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 187.832, "z": -50.533, "size": 0.2338, "title": "Zariski's Main Theorem", "summary": "In this file we prove Grothendieck's reformulation of Zariski's main theorem, namely if `f : X ⟶ Y` is separated and of finite type, then the map from the quasi-finite locus `U ⊆ X` of `f` to the relative normalization `X'` of `Y` in `X` is an open immersion. We then have the following corollaries - `Scheme.Hom.isOpen_quasiFiniteAt` : If `f` is separated and of finite type, then the quasi-finite locus of `f` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/ZariskisMainTheorem.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.Etale", "region_id": "algebraic_geometry", "micro_elevation": 0.9032, "macro_tier": 3, "macro_tier_score": 0.046, "macro_tier_override": null, "x": 205.794, "z": -94.43, "size": 0.2972, "title": "Étale morphisms", "summary": "A morphism of schemes `f : X ⟶ Y` is étale if for each affine `U ⊆ Y` and `V ⊆ f ⁻¹' U`, the induced map `Γ(Y, U) ⟶ Γ(X, V)` is étale.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/Etale.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.FlatDescent", "region_id": "algebraic_geometry", "micro_elevation": 0.7742, "macro_tier": 2, "macro_tier_score": 0.0345, "macro_tier_override": null, "x": 195.452, "z": -85.189, "size": 0.2873, "title": "Properties of morphisms satisfying fpqc descent", "summary": "In this file we show some global properties satisfy fpqc descent. - universally closed (`AlgebraicGeometry.descendsAlong_universallyClosed_surjective_inf_flat_inf_quasicompact`) - universally open (`AlgebraicGeometry.descendsAlong_universallyOpen_surjective_inf_flat_inf_quasicompact`) - universally injective (`AlgebraicGeometry.descendsAlong_universallyInjective_surjective_inf_flat_inf_quasicompact`) - being an…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/FlatDescent.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite", "region_id": "algebraic_geometry", "micro_elevation": 0.871, "macro_tier": 2, "macro_tier_score": 0.0228, "macro_tier_override": null, "x": 231.376, "z": -84.367, "size": 0.2332, "title": "Quasi-finite morphisms", "summary": "We say that a morphism `f : X ⟶ Y` is locally quasi finite if `Γ(Y, U) ⟶ Γ(X, V)` is quasi-finite (in the mathlib sense) for every pair of affine opens that `f` maps one into the other. This is equivalent to all the fibers `f⁻¹(x)` having an open cover of `κ(x)`-finite schemes. Note that this does not require `f` to be quasi-compact nor locally of finite type. We prove that this is stable under composition and base…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/QuasiFinite.html"}, {"id": "Mathlib.AlgebraicGeometry.Normalization", "region_id": "algebraic_geometry", "micro_elevation": 0.871, "macro_tier": 2, "macro_tier_score": 0.0228, "macro_tier_override": null, "x": 184.552, "z": -70.637, "size": 0.2332, "title": "Relative Normalization", "summary": "Given a qcqs morphism `f : X ⟶ Y`, we define the relative normalization `f.normalization`, along with the maps that `f` factor into: - `f.toNormalization : X ⟶ f.normalization`: a dominant morphism - `f.fromNormalization : f.normalization ⟶ Y`: an integral morphism It satisfies the universal property: For any factorization `X ⟶ T ⟶ Y` with `T ⟶ Y` integral, the map `X ⟶ T` factors through `f.normalization` uniquely.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Normalization.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.Descent", "region_id": "algebraic_geometry", "micro_elevation": 0.7097, "macro_tier": 2, "macro_tier_score": 0.0342, "macro_tier_override": null, "x": 219.704, "z": -47.5, "size": 0.2469, "title": "Descent of morphism properties", "summary": "Let `P` and `P'` be morphism properties. In this file we show some results to deduce that `P` descends along `P'` from a codescent property of ring homomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/Descent.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.AffineAnd", "region_id": "algebraic_geometry", "micro_elevation": 0.6774, "macro_tier": 3, "macro_tier_score": 0.228, "macro_tier_override": null, "x": 205.904, "z": -87.394, "size": 0.3174, "title": "Affine morphisms with additional ring hom property", "summary": "In this file we define a constructor `affineAnd Q` for affine target morphism properties of schemes from a property of ring homomorphisms `Q`: A morphism `f : X ⟶ Y` with affine target satisfies `affineAnd Q` if it is an affine morphism (i.e. `X` is affine) and the induced ring map on global sections satisfies `Q`. `affineAnd Q` inherits most stability properties of `Q` and is local at the target if `Q` is local at…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/AffineAnd.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.LocalIso", "region_id": "algebraic_geometry", "micro_elevation": 0.4516, "macro_tier": 2, "macro_tier_score": 0.0342, "macro_tier_override": null, "x": 221.027, "z": -76.745, "size": 0.2463, "title": "Local isomorphisms", "summary": "A local isomorphism of schemes is a morphism that is source-locally an open immersion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/LocalIso.html"}, {"id": "Mathlib.AlgebraicGeometry.ResidueField", "region_id": "algebraic_geometry", "micro_elevation": 0.5806, "macro_tier": 4, "macro_tier_score": 0.3191, "macro_tier_override": null, "x": 193.217, "z": -66.935, "size": 0.3301, "title": "Residue fields of points", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/ResidueField.html"}, {"id": "Mathlib.AlgebraicGeometry.Stalk", "region_id": "algebraic_geometry", "micro_elevation": 0.5484, "macro_tier": 4, "macro_tier_score": 0.3301, "macro_tier_override": null, "x": 223.328, "z": -56.046, "size": 0.297, "title": "Stalks of a Scheme", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Stalk.html"}, {"id": "Mathlib.AlgebraicGeometry.RelativeGluing", "region_id": "algebraic_geometry", "micro_elevation": 0.6774, "macro_tier": 3, "macro_tier_score": 0.0571, "macro_tier_override": null, "x": 201.427, "z": -48.958, "size": 0.2717, "title": "Relative gluing", "summary": "In this file we show a relative gluing lemma (see https://stacks.math.columbia.edu/tag/01LH): If `{Uᵢ}` is a locally directed open cover of `S` and we have a compatible family of `Xᵢ` over `Uᵢ`, the `Xᵢ` glue to a morphism `f : X ⟶ S` such that `Xᵢ ≅ f⁻¹ Uᵢ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/RelativeGluing.html"}, {"id": "Mathlib.AlgebraicGeometry.Cover.Directed", "region_id": "algebraic_geometry", "micro_elevation": 0.6452, "macro_tier": 3, "macro_tier_score": 0.0684, "macro_tier_override": null, "x": 193.176, "z": -75.839, "size": 0.2667, "title": "Locally directed covers", "summary": "A locally directed `P`-cover of a scheme `X` is a cover `𝒰` with an ordering on the indices and compatible transition maps `𝒰ᵢ ⟶ 𝒰ⱼ` for `i ≤ j` such that every `x : 𝒰ᵢ ×[X] 𝒰ⱼ` comes from some `𝒰ₖ` for a `k ≤ i` and `k ≤ j`. Gluing along directed covers is easier, because the intersections `𝒰ᵢ ×[X] 𝒰ⱼ` can be covered by a subcover of `𝒰`. In particular, if `𝒰` is a Zariski cover, `X` naturally is the colimit of the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Cover/Directed.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective", "region_id": "algebraic_geometry", "micro_elevation": 0.6452, "macro_tier": 3, "macro_tier_score": 0.08, "macro_tier_override": null, "x": 226.86, "z": -78.951, "size": 0.287, "title": "Universally injective morphism", "summary": "A morphism of schemes `f : X ⟶ Y` is universally injective if `X ×[Y] Y' ⟶ Y'` is injective for all base changes `Y' ⟶ Y`. This is equivalent to the diagonal morphism being surjective (`AlgebraicGeometry.UniversallyInjective.iff_diagonal`). We show that being universally injective is local at the target, and is stable under compositions and base changes. We also prove that universally injective is equivalent to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/UniversallyInjective.html"}, {"id": "Mathlib.AlgebraicGeometry.LimitsOver", "region_id": "algebraic_geometry", "micro_elevation": 0.4194, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 208.607, "z": -79.921, "size": 0.2, "title": "(Co)limits in over categories", "summary": "We show that if `P` is a morphism property in `Scheme` that is local at the source, then colimits in `P.Over ⊤ X` for `X : Scheme` of locally directed diagrams of open immersions exist and agree with the colimit in `Scheme`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/LimitsOver.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.Basic", "region_id": "algebraic_geometry", "micro_elevation": 0.3871, "macro_tier": 4, "macro_tier_score": 0.3562, "macro_tier_override": null, "x": 221.591, "z": -72.477, "size": 0.4675, "title": "Properties of morphisms between Schemes", "summary": "We provide the basic framework for talking about properties of morphisms between Schemes. A `MorphismProperty Scheme` is a predicate on morphisms between schemes. For properties local at the target, its behaviour is entirely determined by its definition on morphisms into affine schemes, which we call an `AffineTargetMorphismProperty`. In this file, we provide API lemmas for properties local at the target, and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/Basic.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point", "region_id": "algebraic_geometry", "micro_elevation": 0.1613, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 206.515, "z": -69.478, "size": 0.2, "title": "Nonsingular points and the group law in Jacobian coordinates", "summary": "Let `W` be a Weierstrass curve over a field `F`. The nonsingular Jacobian points of `W` can be endowed with a group law, which is uniquely determined by the formulae in `Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian/Formula.lean` and follows from an equivalence with the nonsingular points in affine coordinates. This file defines the group law on nonsingular Jacobian points.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian/Point.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point", "region_id": "algebraic_geometry", "micro_elevation": 0.129, "macro_tier": 2, "macro_tier_score": 0.012, "macro_tier_override": null, "x": 214.867, "z": -66.912, "size": 0.3082, "title": "Nonsingular points and the group law in affine coordinates", "summary": "Let `W` be a Weierstrass curve over a field `F` given by a Weierstrass equation `W(X, Y) = 0` in affine coordinates. The type of nonsingular points in affine coordinates is an inductive, consisting of the unique point at infinity `𝓞` and nonsingular affine points `(x, y)`. It can be endowed with a group law, with `𝓞` as the identity nonsingular point, which is uniquely determined by the formulae in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/Affine/Point.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula", "region_id": "algebraic_geometry", "micro_elevation": 0.129, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 209.634, "z": -71.044, "size": 0.2478, "title": "Negation and addition formulae for nonsingular points in Jacobian coordinates", "summary": "Let `W` be a Weierstrass curve over a field `F`. The nonsingular Jacobian points on `W` can be given negation and addition operations defined by an analogue of the secant-and-tangent process in `Mathlib/AlgebraicGeometry/EllipticCurve/Affine/Formula.lean`, but the polynomials involved are `(2, 3, 1)`-homogeneous, so any instances of division become multiplication in the `Z`-coordinate. Most computational proofs are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian/Formula.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.Integral", "region_id": "algebraic_geometry", "micro_elevation": 0.7742, "macro_tier": 3, "macro_tier_score": 0.114, "macro_tier_override": null, "x": 231.376, "z": -55.419, "size": 0.2842, "title": "Integral morphisms of schemes", "summary": "A morphism of schemes `f : X ⟶ Y` is integral if the preimage of an arbitrary affine open subset of `Y` is affine and the induced ring map is integral. It is equivalent to ask only that `Y` is covered by affine opens whose preimage is affine and the induced ring map is integral.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/Integral.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.Separated", "region_id": "algebraic_geometry", "micro_elevation": 0.7419, "macro_tier": 3, "macro_tier_score": 0.2281, "macro_tier_override": null, "x": 194.555, "z": -51.682, "size": 0.3251, "title": "Separated morphisms", "summary": "A morphism of schemes is separated if its diagonal morphism is a closed immersion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/Separated.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed", "region_id": "algebraic_geometry", "micro_elevation": 0.7419, "macro_tier": 3, "macro_tier_score": 0.114, "macro_tier_override": null, "x": 188.729, "z": -71.785, "size": 0.2756, "title": "Universally closed morphism", "summary": "A morphism of schemes `f : X ⟶ Y` is universally closed if `X ×[Y] Y' ⟶ Y'` is a closed map for all base change `Y' ⟶ Y`. This implies that `f` is topologically proper (`AlgebraicGeometry.Scheme.Hom.isProperMap`). We show that being universally closed is local at the target, and is stable under compositions and base changes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.Preimmersion", "region_id": "algebraic_geometry", "micro_elevation": 0.5161, "macro_tier": 4, "macro_tier_score": 0.3419, "macro_tier_override": null, "x": 196.554, "z": -60.899, "size": 0.3316, "title": "Preimmersions of schemes", "summary": "A morphism of schemes `f : X ⟶ Y` is a preimmersion if the underlying map of topological spaces is an embedding and the induced morphisms of stalks are all surjective. This is not a concept seen in the literature but it is useful for generalizing results on immersions to other maps including `Spec 𝒪_{X, x} ⟶ X` and inclusions of fibers `κ(x) ×ₓ Y ⟶ Y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/Preimmersion.html"}, {"id": "Mathlib.AlgebraicGeometry.Fiber", "region_id": "algebraic_geometry", "micro_elevation": 0.6774, "macro_tier": 3, "macro_tier_score": 0.0801, "macro_tier_override": null, "x": 195.348, "z": -80.92, "size": 0.3019, "title": "Scheme-theoretic fiber", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Fiber.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.Affine", "region_id": "algebraic_geometry", "micro_elevation": 0.6452, "macro_tier": 4, "macro_tier_score": 0.2629, "macro_tier_override": null, "x": 198.343, "z": -82.475, "size": 0.3653, "title": "Affine morphisms of schemes", "summary": "A morphism of schemes `f : X ⟶ Y` is affine if the preimage of an arbitrary affine open subset of `Y` is affine. It is equivalent to ask only that `Y` is covered by affine opens whose preimage is affine.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/Affine.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.FiniteType", "region_id": "algebraic_geometry", "micro_elevation": 0.4839, "macro_tier": 3, "macro_tier_score": 0.2398, "macro_tier_override": null, "x": 213.162, "z": -52.716, "size": 0.3429, "title": "Morphisms of finite type", "summary": "A morphism of schemes `f : X ⟶ Y` is locally of finite type if for each affine `U ⊆ Y` and `V ⊆ f ⁻¹' U`, The induced map `Γ(Y, U) ⟶ Γ(X, V)` is of finite type. A morphism of schemes is of finite type if it is both locally of finite type and quasi-compact. We show that these properties are local, and are stable under compositions and base change.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/FiniteType.html"}, {"id": "Mathlib.AlgebraicGeometry.GammaSpecAdjunction", "region_id": "algebraic_geometry", "micro_elevation": 0.2581, "macro_tier": 4, "macro_tier_score": 0.4208, "macro_tier_override": null, "x": 204.104, "z": -63.417, "size": 0.2763, "title": "Adjunction between `Γ` and `Spec`", "summary": "We define the adjunction `ΓSpec.adjunction : Γ ⊣ Spec` by defining the unit (`toΓSpec`, in multiple steps in this file) and counit (done in `Spec.lean`) and checking that they satisfy the left and right triangle identities. The constructions and proofs make use of maps and lemmas defined and proved in `Mathlib/AlgebraicGeometry/StructureSheaf.lean` extensively. Notice that since the adjunction is between…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/GammaSpecAdjunction.html"}, {"id": "Mathlib.AlgebraicGeometry.Restrict", "region_id": "algebraic_geometry", "micro_elevation": 0.2258, "macro_tier": 4, "macro_tier_score": 0.4439, "macro_tier_override": null, "x": 217.416, "z": -64.934, "size": 0.3174, "title": "Restriction of Schemes and Morphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Restrict.html"}, {"id": "Mathlib.AlgebraicGeometry.OpenImmersion", "region_id": "algebraic_geometry", "micro_elevation": 0.0968, "macro_tier": 4, "macro_tier_score": 0.4549, "macro_tier_override": null, "x": 210.466, "z": -64.412, "size": 0.2748, "title": "Open immersions of schemes", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/OpenImmersion.html"}, {"id": "Mathlib.AlgebraicGeometry.Scheme", "region_id": "algebraic_geometry", "micro_elevation": 0.0645, "macro_tier": 4, "macro_tier_score": 0.478, "macro_tier_override": null, "x": 211.378, "z": -65.405, "size": 0.318, "title": "The category of schemes", "summary": "A scheme is a locally ringed space such that every point is contained in some open set where there is an isomorphism of presheaves between the restriction to that open set, and the structure sheaf of `Spec R`, for some commutative ring `R`. A morphism of schemes is just a morphism of the underlying locally ringed spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Scheme.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap", "region_id": "algebraic_geometry", "micro_elevation": 0.4516, "macro_tier": 4, "macro_tier_score": 0.3439, "macro_tier_override": null, "x": 218.289, "z": -79.006, "size": 0.4322, "title": "Properties on the underlying functions of morphisms of schemes", "summary": "This file includes various results on properties of morphisms of schemes that come from properties of the underlying map of topological spaces, including - `Injective` - `Surjective` - `IsOpenMap` - `IsClosedMap` - `GeneralizingMap` - `IsEmbedding` - `IsOpenEmbedding` - `IsClosedEmbedding` - `DenseRange` (`IsDominant`)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/UnderlyingMap.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.SurjectiveOnStalks", "region_id": "algebraic_geometry", "micro_elevation": 0.4839, "macro_tier": 4, "macro_tier_score": 0.3414, "macro_tier_override": null, "x": 197.855, "z": -74.184, "size": 0.2979, "title": "Morphisms surjective on stalks", "summary": "We define the class of morphisms between schemes that are surjective on stalks. We show that this class is stable under composition and base change. We also show that (`AlgebraicGeometry.SurjectiveOnStalks.isEmbedding_pullback`) if `Y ⟶ S` is surjective on stalks, then for every `X ⟶ S`, `X ×ₛ Y` is a subset of `X × Y` (Cartesian product as topological spaces) with the induced topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/SurjectiveOnStalks.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass", "region_id": "algebraic_geometry", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0128, "macro_tier_override": null, "x": 210.948, "z": -67.328, "size": 0.3599, "title": "Weierstrass equations of elliptic curves", "summary": "This file defines the structure of an elliptic curve as a nonsingular Weierstrass curve given by a Weierstrass equation, which is mathematically accurate in many cases but also good for computation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.html"}, {"id": "Mathlib.AlgebraicGeometry.Geometrically.Connected", "region_id": "algebraic_geometry", "micro_elevation": 0.7419, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 188.935, "z": -72.708, "size": 0.2, "title": "Geometrically connected schemes", "summary": "In this file we define geometrically connected morphisms of schemes. A morphism `f : X ⟶ Y` is geometrically connected if for all `Spec K ⟶ Y` with `K` a field, `X ×[Y] Spec K` is connected. In the case where `Y = Spec K` for some field `K` this recovers the standard definition of a geometrically connected scheme over a field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Geometrically/Connected.html"}, {"id": "Mathlib.AlgebraicGeometry.Geometrically.Basic", "region_id": "algebraic_geometry", "micro_elevation": 0.7097, "macro_tier": 3, "macro_tier_score": 0.046, "macro_tier_override": null, "x": 189.272, "z": -67.305, "size": 0.2945, "title": "Geometrically-`P` schemes over a field", "summary": "In this file we define the basic interface for properties like geometrically reduced, geometrically irreducible, geometrically connected etc. In this file we treat an abstract property of schemes `P` and derive the general properties that are shared by all of these variants. A morphism of schemes `f : X ⟶ Y` is geometrically `P` if for any field `K` and morphism `Spec K ⟶ Y`, the base change `X ×[Y] Spec K`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Geometrically/Basic.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen", "region_id": "algebraic_geometry", "micro_elevation": 0.7097, "macro_tier": 3, "macro_tier_score": 0.0801, "macro_tier_override": null, "x": 189.334, "z": -68.961, "size": 0.3028, "title": "Universally open morphism", "summary": "A morphism of schemes `f : X ⟶ Y` is universally open if `X ×[Y] Y' ⟶ Y'` is an open map for all base change `Y' ⟶ Y`. We show that being universally open is local at the target, and is stable under compositions and base changes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/UniversallyOpen.html"}, {"id": "Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology", "region_id": "algebraic_geometry", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 210.948, "z": -67.328, "size": 0.2404, "title": "Projective spectrum of a graded ring", "summary": "The projective spectrum of a graded commutative ring is the subtype of all homogeneous ideals that are prime and do not contain the irrelevant ideal. It is naturally endowed with a topology: the Zariski topology.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.html"}, {"id": "Mathlib.AlgebraicGeometry.Modules.Tilde", "region_id": "algebraic_geometry", "micro_elevation": 0.3548, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 214.909, "z": -57.24, "size": 0.2, "title": "Construction of M^~", "summary": "Given any commutative ring `R` and `R`-module `M`, we construct the sheaf `M^~` of `𝒪_SpecR`-modules such that `M^~(U)` is the set of dependent functions that are locally fractions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Modules/Tilde.html"}, {"id": "Mathlib.AlgebraicGeometry.Modules.Sheaf", "region_id": "algebraic_geometry", "micro_elevation": 0.3226, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 216.649, "z": -75.363, "size": 0.2338, "title": "The category of sheaves of modules over a scheme", "summary": "In this file, we define the abelian category of sheaves of modules `X.Modules` over a scheme `X`, and study its basic functoriality.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Modules/Sheaf.html"}, {"id": "Mathlib.AlgebraicGeometry.Birational.Dominant", "region_id": "algebraic_geometry", "micro_elevation": 0.9032, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 231.042, "z": -86.23, "size": 0.2, "title": "Dominant rational maps", "summary": "This file defines `RationalMap.IsDominant` and establishes its connection to `IsDominant` on the underlying partial maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Birational/Dominant.html"}, {"id": "Mathlib.AlgebraicGeometry.Birational.RationalMap", "region_id": "algebraic_geometry", "micro_elevation": 0.871, "macro_tier": 2, "macro_tier_score": 0.0234, "macro_tier_override": null, "x": 210.416, "z": -40.731, "size": 0.3068, "title": "Rational maps between schemes", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Birational/RationalMap.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.BigZariski", "region_id": "algebraic_geometry", "micro_elevation": 0.6774, "macro_tier": 2, "macro_tier_score": 0.0233, "macro_tier_override": null, "x": 213.156, "z": -87.9, "size": 0.3021, "title": "The big Zariski site of schemes", "summary": "In this file, we define the Zariski topology, as a Grothendieck topology on the category `Scheme.{u}`: this is `Scheme.zariskiTopology.{u}`. If `X : Scheme.{u}`, the Zariski topology on `Over X` can be obtained as `Scheme.zariskiTopology.over X` (see `CategoryTheory.Sites.Over`.). TODO: * If `Y : Scheme.{u}`, define a continuous functor from the category of opens of `Y` to `Over Y`, and show that a presheaf on `Over…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/BigZariski.html"}, {"id": "Mathlib.AlgebraicGeometry.Cover.Sigma", "region_id": "algebraic_geometry", "micro_elevation": 0.4194, "macro_tier": 2, "macro_tier_score": 0.0229, "macro_tier_override": null, "x": 223.342, "z": -64.096, "size": 0.2517, "title": "Collapsing covers", "summary": "We define the endofunctor on `Scheme.Cover P` that collapses a cover to a single object cover.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Cover/Sigma.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.Pretopology", "region_id": "algebraic_geometry", "micro_elevation": 0.6452, "macro_tier": 2, "macro_tier_score": 0.023, "macro_tier_override": null, "x": 191.672, "z": -63.237, "size": 0.2729, "title": "Grothendieck topology defined by a morphism property", "summary": "Given a multiplicative morphism property `P` that is stable under base change, we define the associated (pre)topology on the category of schemes, where coverings are given by jointly surjective families of morphisms satisfying `P`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/Pretopology.html"}, {"id": "Mathlib.AlgebraicGeometry.SpreadingOut", "region_id": "algebraic_geometry", "micro_elevation": 0.8387, "macro_tier": 2, "macro_tier_score": 0.0231, "macro_tier_override": null, "x": 189.286, "z": -53.654, "size": 0.2843, "title": "Spreading out morphisms", "summary": "Under certain conditions, a morphism on stalks `Spec 𝒪_{X, x} ⟶ Spec 𝒪_{Y, y}` can be spread out into a neighborhood of `x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/SpreadingOut.html"}, {"id": "Mathlib.AlgebraicGeometry.FunctionField", "region_id": "algebraic_geometry", "micro_elevation": 0.4194, "macro_tier": 3, "macro_tier_score": 0.0915, "macro_tier_override": null, "x": 198.324, "z": -69.494, "size": 0.3042, "title": "Function field of integral schemes", "summary": "We define the function field of an irreducible scheme as the stalk of the generic point. This is a field when the scheme is integral.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/FunctionField.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.Small", "region_id": "algebraic_geometry", "micro_elevation": 0.6774, "macro_tier": 2, "macro_tier_score": 0.0116, "macro_tier_override": null, "x": 227.605, "z": -55.055, "size": 0.263, "title": "Small sites", "summary": "In this file we define the small sites associated to morphism properties and give generating pretopologies.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/Small.html"}, {"id": "Mathlib.AlgebraicGeometry.Cover.Over", "region_id": "algebraic_geometry", "micro_elevation": 0.4839, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 225.703, "z": -68.161, "size": 0.2514, "title": "Covers of schemes over a base", "summary": "In this file we define the typeclass `Cover.Over`. For a cover `𝒰` of an `S`-scheme `X`, the datum `𝒰.Over S` contains `S`-scheme structures on the components of `𝒰` and asserts that the component maps are morphisms of `S`-schemes. We provide instances of `𝒰.Over S` for standard constructions on covers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Cover/Over.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula", "region_id": "algebraic_geometry", "micro_elevation": 0.129, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 211.201, "z": -63.395, "size": 0.2478, "title": "Negation and addition formulae for nonsingular points in projective coordinates", "summary": "Let `W` be a Weierstrass curve over a field `F`. The nonsingular projective points on `W` can be given negation and addition operations defined by an analogue of the secant-and-tangent process in `Mathlib/AlgebraicGeometry/EllipticCurve/Affine/Formula.lean`, but the polynomials involved are homogeneous, so any instances of division become multiplication in the `Z`-coordinate. Most computational proofs are immediate…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/Projective/Formula.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula", "region_id": "algebraic_geometry", "micro_elevation": 0.0968, "macro_tier": 2, "macro_tier_score": 0.012, "macro_tier_override": null, "x": 211.194, "z": -70.273, "size": 0.3063, "title": "Negation and addition formulae for nonsingular points in affine coordinates", "summary": "Let `W` be a Weierstrass curve over a field `F` with coefficients `aᵢ`. The nonsingular affine points on `W` can be given negation and addition operations defined by a secant-and-tangent process. * Given a nonsingular affine point `P`, its *negation* `-P` is defined to be the unique third nonsingular point of intersection between `W` and the vertical line through `P`. Explicitly, if `P` is `(x, y)`, then `-P` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/Affine/Formula.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic", "region_id": "algebraic_geometry", "micro_elevation": 0.0968, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 208.822, "z": -65.274, "size": 0.257, "title": "Weierstrass equations and the nonsingular condition in projective coordinates", "summary": "A point on the unweighted projective plane over a commutative ring `R` is an equivalence class `[x : y : z]` of triples `(x, y, z) ≠ (0, 0, 0)` of elements in `R` such that `(x, y, z) ∼ (x', y', z')` if there is some unit `u` in `Rˣ` with `(x, y, z) = (ux', uy', uz')`. Let `W` be a Weierstrass curve over a commutative ring `R` with coefficients `aᵢ`. A *projective point* is a point on the unweighted projective plane…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/Projective/Basic.html"}, {"id": "Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper", "region_id": "algebraic_geometry", "micro_elevation": 0.9032, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 198.041, "z": -42.946, "size": 0.2, "title": "Properness of `Proj A`", "summary": "We show that `Proj 𝒜` is proper over `Spec 𝒜₀`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Proper.html"}, {"id": "Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic", "region_id": "algebraic_geometry", "micro_elevation": 0.3226, "macro_tier": 2, "macro_tier_score": 0.0116, "macro_tier_override": null, "x": 202.056, "z": -71.571, "size": 0.2676, "title": "Basic properties of the scheme `Proj A`", "summary": "The scheme `Proj 𝒜` for a graded ring `𝒜` is constructed in `Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean`. In this file we provide basic properties of the scheme.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.html"}, {"id": "Mathlib.AlgebraicGeometry.ValuativeCriterion", "region_id": "algebraic_geometry", "micro_elevation": 0.871, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 235.444, "z": -56.955, "size": 0.2478, "title": "Valuative criterion", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/ValuativeCriterion.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated", "region_id": "algebraic_geometry", "micro_elevation": 0.5161, "macro_tier": 4, "macro_tier_score": 0.2624, "macro_tier_override": null, "x": 214.625, "z": -51.999, "size": 0.3353, "title": "Quasi-separated morphisms", "summary": "A morphism of schemes `f : X ⟶ Y` is quasi-separated if the diagonal morphism `X ⟶ X ×[Y] X` is quasi-compact. A scheme is quasi-separated if the intersections of any two affine open sets is quasi-compact. (`AlgebraicGeometry.quasiSeparatedSpace_iff_affine`) We show that a morphism is quasi-separated if the preimage of every affine open is quasi-separated. We also show that this property is local at the target, and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact", "region_id": "algebraic_geometry", "micro_elevation": 0.4839, "macro_tier": 4, "macro_tier_score": 0.2741, "macro_tier_override": null, "x": 198.724, "z": -75.634, "size": 0.3594, "title": "Quasi-compact morphisms", "summary": "A morphism of schemes is quasi-compact if the preimages of quasi-compact open sets are quasi-compact. It suffices to check that preimages of affine open sets are compact (`quasiCompact_iff_forall_isAffineOpen`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.Immersion", "region_id": "algebraic_geometry", "micro_elevation": 0.7742, "macro_tier": 3, "macro_tier_score": 0.2172, "macro_tier_override": null, "x": 234.386, "z": -70.454, "size": 0.353, "title": "Immersions of schemes", "summary": "A morphism of schemes `f : X ⟶ Y` is an immersion if the underlying map of topological spaces is a locally closed embedding, and the induced morphisms of stalks are all surjective. This is true if and only if it can be factored into a closed immersion followed by an open immersion.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/Immersion.html"}, {"id": "Mathlib.AlgebraicGeometry.IdealSheaf.Functorial", "region_id": "algebraic_geometry", "micro_elevation": 0.7419, "macro_tier": 3, "macro_tier_score": 0.2168, "macro_tier_override": null, "x": 231.274, "z": -57.309, "size": 0.3251, "title": "Functorial constructions of ideal sheaves", "summary": "We define the pullback and pushforward of ideal sheaves in this file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/IdealSheaf/Functorial.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.MorphismProperty", "region_id": "algebraic_geometry", "micro_elevation": 0.129, "macro_tier": 4, "macro_tier_score": 0.4553, "macro_tier_override": null, "x": 207.031, "z": -67.768, "size": 0.3195, "title": "Site defined by a morphism property", "summary": "Given a multiplicative morphism property `P` that is stable under base change, we define the associated precoverage on the category of schemes, where coverings are given by jointly surjective families of morphisms satisfying `P`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/MorphismProperty.html"}, {"id": "Mathlib.AlgebraicGeometry.Limits", "region_id": "algebraic_geometry", "micro_elevation": 0.3548, "macro_tier": 4, "macro_tier_score": 0.3792, "macro_tier_override": null, "x": 210.645, "z": -78.162, "size": 0.4737, "title": "(Co)Limits of Schemes", "summary": "We construct various limits and colimits in the category of schemes. * The existence of fibred products was shown in `Mathlib/AlgebraicGeometry/Pullbacks.lean`. * `Spec ℤ` is the terminal object. * The preceding two results imply that `Scheme` has all finite limits. * The empty scheme is the (strict) initial object. * The disjoint union is the coproduct of a family of schemes, and the forgetful functor to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Limits.html"}, {"id": "Mathlib.AlgebraicGeometry.Pullbacks", "region_id": "algebraic_geometry", "micro_elevation": 0.3226, "macro_tier": 4, "macro_tier_score": 0.3883, "macro_tier_override": null, "x": 211.09, "z": -57.476, "size": 0.387, "title": "Fibred products of schemes", "summary": "In this file we construct the fibred product of schemes via gluing. We roughly follow [har77] Theorem 3.3. In particular, the main construction is to show that for an open cover `{ Uᵢ }` of `X`, if there exist fibred products `Uᵢ ×[Z] Y` for each `i`, then there exists a fibred product `X ×[Z] Y`. Then, for constructing the fibred product for arbitrary schemes `X, Y, Z`, we can use the construction to reduce to the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Pullbacks.html"}, {"id": "Mathlib.AlgebraicGeometry.RationalMap", "region_id": "algebraic_geometry", "micro_elevation": 0.9032, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 237.054, "z": -58.411, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/RationalMap.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.EtalePoint", "region_id": "algebraic_geometry", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 240.581, "z": -74.725, "size": 0.2, "title": "Points of the étale site", "summary": "In this file, we show that a morphism `Spec (.of Ω) ⟶ S` where `Ω` is a separably closed field defines a point on the small étale site of `S`. We show that these points form a conservative family.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/EtalePoint.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.AffineEtale", "region_id": "algebraic_geometry", "micro_elevation": 0.9677, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 226.652, "z": -42.287, "size": 0.2338, "title": "Affine étale site", "summary": "In this file we define the small affine étale site of a scheme `S`. The underlying category is the category of commutative rings `R` equipped with an étale structure morphism `Spec R ⟶ S`. We show that this category is essentially small, that it is a dense subsite of the small étale site, and that it is `1`-hypercover dense, which allows to show that if `S : Scheme.{u}`, then we can sheafify étale presheaves with…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/AffineEtale.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.Etale", "region_id": "algebraic_geometry", "micro_elevation": 0.9355, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 220.357, "z": -40.349, "size": 0.2427, "title": "The étale site", "summary": "In this file we define the big étale site, i.e. the étale topology as a Grothendieck topology on the category of schemes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/Etale.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic", "region_id": "algebraic_geometry", "micro_elevation": 0.1613, "macro_tier": 2, "macro_tier_score": 0.0116, "macro_tier_override": null, "x": 215.806, "z": -66.512, "size": 0.2676, "title": "Division polynomials of Weierstrass curves", "summary": "This file defines certain polynomials associated to division polynomials of Weierstrass curves. These are defined in terms of the auxiliary sequences for normalised elliptic divisibility sequences (EDS) as defined in `Mathlib/NumberTheory/EllipticDivisibilitySequence.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.html"}, {"id": "Mathlib.AlgebraicGeometry.Geometrically.Integral", "region_id": "algebraic_geometry", "micro_elevation": 0.9032, "macro_tier": 2, "macro_tier_score": 0.0342, "macro_tier_override": null, "x": 219.512, "z": -41.104, "size": 0.2526, "title": "Geometrically Integral Schemes", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Geometrically/Integral.html"}, {"id": "Mathlib.AlgebraicGeometry.Geometrically.Reduced", "region_id": "algebraic_geometry", "micro_elevation": 0.871, "macro_tier": 3, "macro_tier_score": 0.0457, "macro_tier_override": null, "x": 203.376, "z": -41.826, "size": 0.2705, "title": "Geometrically Reduced Schemes", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Geometrically/Reduced.html"}, {"id": "Mathlib.AlgebraicGeometry.Geometrically.Irreducible", "region_id": "algebraic_geometry", "micro_elevation": 0.7419, "macro_tier": 2, "macro_tier_score": 0.0456, "macro_tier_override": null, "x": 227.748, "z": -52.12, "size": 0.2587, "title": "Geometrically Irreducible Schemes", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Geometrically/Irreducible.html"}, {"id": "Mathlib.AlgebraicGeometry.Artinian", "region_id": "algebraic_geometry", "micro_elevation": 0.8387, "macro_tier": 3, "macro_tier_score": 0.0684, "macro_tier_override": null, "x": 235.84, "z": -73.377, "size": 0.2619, "title": "Artinian and Locally Artinian Schemes", "summary": "We define and prove basic properties about Artinian and locally Artinian Schemes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Artinian.html"}, {"id": "Mathlib.AlgebraicGeometry.Noetherian", "region_id": "algebraic_geometry", "micro_elevation": 0.8065, "macro_tier": 3, "macro_tier_score": 0.149, "macro_tier_override": null, "x": 203.033, "z": -90.653, "size": 0.3509, "title": "Noetherian and Locally Noetherian Schemes", "summary": "We introduce the concept of (locally) Noetherian schemes, giving definitions, equivalent conditions, and basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Noetherian.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.Affine.AddSubMap", "region_id": "algebraic_geometry", "micro_elevation": 0.1613, "macro_tier": 1, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 213.385, "z": -71.609, "size": 0.2478, "title": "The addition-and-subtraction map on x-coordinates", "summary": "We set up the endomorphism of `ℙ²` that on affine points with affine sum is equal to `(x(P) * x(Q) : x(P) + x(Q) : 1) ↦ (x(P+Q) * x(P-Q) : x(P+Q) + x(P-Q) : 1);` see `WeierstrassCurve.addSubMap` (this is on coordinate vectors). TODO: Show that the map really does what it is claimed to do. This will be used to eventually show the approximate parallelogram law for `K`-points on an elliptic curve `E`: `∃ C, ∀ P Q :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/Affine/AddSubMap.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.WeaklyEtale", "region_id": "algebraic_geometry", "micro_elevation": 0.9355, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 204.924, "z": -95.258, "size": 0.2268, "title": "Weakly étale morphisms", "summary": "A morphism of schemes is weakly étale if it is flat and its diagonal is flat. As the name suggests any étale morphism is weakly étale and every weakly étale morphism of finite presentation is étale.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/WeaklyEtale.html"}, {"id": "Mathlib.AlgebraicGeometry.Cover.Open", "region_id": "algebraic_geometry", "micro_elevation": 0.1935, "macro_tier": 4, "macro_tier_score": 0.455, "macro_tier_override": null, "x": 216.859, "z": -67.292, "size": 0.2901, "title": "Open covers of schemes", "summary": "This file provides the basic API for open covers of schemes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Cover/Open.html"}, {"id": "Mathlib.AlgebraicGeometry.Over", "region_id": "algebraic_geometry", "micro_elevation": 0.0968, "macro_tier": 4, "macro_tier_score": 0.455, "macro_tier_override": null, "x": 213.748, "z": -68.275, "size": 0.2901, "title": "Typeclasses for `S`-schemes and `S`-morphisms", "summary": "We define these as thin wrappers around `CategoryTheory/Comma/OverClass`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Over.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.ModelsWithJ", "region_id": "algebraic_geometry", "micro_elevation": 0.0323, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 211.168, "z": -66.368, "size": 0.2, "title": "Models of elliptic curves with prescribed j-invariant", "summary": "This file defines the Weierstrass equation over a field with prescribed j-invariant, proved that it is an elliptic curve, and that its j-invariant is equal to the given value. It is a modification of [silverman2009], Chapter III, Proposition 1.4 (c).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/ModelsWithJ.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.LocalFlatDescent", "region_id": "algebraic_geometry", "micro_elevation": 0.9355, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 220.426, "z": -94.283, "size": 0.239, "title": "Local properties satisfying fpqc descent", "summary": "In this file we provide instances that show that the following local properties satisfy fpqc descent: - locally of finite type - locally of finite presentation - smooth - formally unramified - étale", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/LocalFlatDescent.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.Representability", "region_id": "algebraic_geometry", "micro_elevation": 0.7419, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 232.966, "z": -72.687, "size": 0.2, "title": "Representability of schemes is a local property", "summary": "In this file we prove that a sheaf of types `F` on `Sch` is representable if it is locally representable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/Representability.html"}, {"id": "Mathlib.AlgebraicGeometry.GluingOneHypercover", "region_id": "algebraic_geometry", "micro_elevation": 0.7097, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 215.432, "z": -88.535, "size": 0.239, "title": "The 1-hypercover of a glue data", "summary": "In this file, given `D : Scheme.GlueData`, we construct a 1-hypercover `D.openHypercover` of the scheme `D.glued` in the big Zariski site. We use this 1-hypercover in order to define a constructor `D.sheafValGluedMk` for sections over `D.glued` of a sheaf of types over the big Zariski site.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/GluingOneHypercover.html"}, {"id": "Mathlib.AlgebraicGeometry.Gluing", "region_id": "algebraic_geometry", "micro_elevation": 0.2581, "macro_tier": 4, "macro_tier_score": 0.3985, "macro_tier_override": null, "x": 217.37, "z": -62.758, "size": 0.3205, "title": "Gluing Schemes", "summary": "Given a family of gluing data of schemes, we may glue them together. Also see the section about \"locally directed\" gluing, which is a special case where the conditions are easier to check.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Gluing.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree", "region_id": "algebraic_geometry", "micro_elevation": 0.1935, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.708, "z": -62.1, "size": 0.2, "title": "Division polynomials of Weierstrass curves", "summary": "This file computes the leading terms of certain polynomials associated to division polynomials of Weierstrass curves defined in `Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.SchemeTheoreticallyDominant", "region_id": "algebraic_geometry", "micro_elevation": 0.7742, "macro_tier": 2, "macro_tier_score": 0.0456, "macro_tier_override": null, "x": 193.55, "z": -51.313, "size": 0.254, "title": "Scheme-theoretically dominant morphisms", "summary": "In this file, we define scheme-theoretically dominant morphisms as morphisms with trivial kernel.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/SchemeTheoreticallyDominant.html"}, {"id": "Mathlib.AlgebraicGeometry.Group.Smooth", "region_id": "algebraic_geometry", "micro_elevation": 0.9677, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 181.404, "z": -66.414, "size": 0.2, "title": "Smoothness of group schemes", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Group/Smooth.html"}, {"id": "Mathlib.AlgebraicGeometry.AlgClosed.Basic", "region_id": "algebraic_geometry", "micro_elevation": 0.8387, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 213.085, "z": -92.855, "size": 0.2516, "title": "Schemes over algebraically closed fields", "summary": "We show that if `X` is locally of finite type over an algebraically closed field `k`, then the closed points of `X` are in bijection with the `k`-points of `X`. See `AlgebraicGeometry.pointEquivClosedPoint`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/AlgClosed/Basic.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.Fpqc", "region_id": "algebraic_geometry", "micro_elevation": 0.7742, "macro_tier": 2, "macro_tier_score": 0.0117, "macro_tier_override": null, "x": 187.461, "z": -70.074, "size": 0.2759, "title": "Fpqc topology", "summary": "In this file we define the fpqc topology and show it is subcanonical. It is the quasi-compact topology for flat morphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/Fpqc.html"}, {"id": "Mathlib.AlgebraicGeometry.EffectiveEpi", "region_id": "algebraic_geometry", "micro_elevation": 0.7097, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 189.286, "z": -68.095, "size": 0.2486, "title": "Effective epimorphisms in the category of schemes", "summary": "We collect results about effective epimorphisms in the category of schemes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EffectiveEpi.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.SheafQuasiCompact", "region_id": "algebraic_geometry", "micro_elevation": 0.7419, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 222.795, "z": -48.011, "size": 0.2486, "title": "Sheaves for the quasi-compact topology", "summary": "In this file we show that a presheaf is a sheaf in the `AlgebraicGeometry.Scheme.propQCTopology` if and only if it is a sheaf in the Zariski topology and a sheaf on single object `P`-coverings of affine schemes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/SheafQuasiCompact.html"}, {"id": "Mathlib.AlgebraicGeometry.IdealSheaf.IrreducibleComponent", "region_id": "algebraic_geometry", "micro_elevation": 0.8387, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 197.862, "z": -89.35, "size": 0.2, "title": "Subscheme structure on an irreducible component", "summary": "We define the subscheme structure on an irreducible component of a Noetherian scheme. Typically, one takes the reduced induced subscheme structure, but this will throw away information if the irreducible component is not already reduced. Instead, we take the closed subscheme defined by the kernel of the restriction to the complement of the union of the other irreducible components. For example, if `X` is irreducible…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/IdealSheaf/IrreducibleComponent.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.Flat", "region_id": "algebraic_geometry", "micro_elevation": 0.6774, "macro_tier": 3, "macro_tier_score": 0.1031, "macro_tier_override": null, "x": 230.388, "z": -74.411, "size": 0.322, "title": "Flat morphisms", "summary": "A morphism of schemes `f : X ⟶ Y` is flat if for each affine `U ⊆ Y` and `V ⊆ f ⁻¹' U`, the induced map `Γ(Y, U) ⟶ Γ(X, V)` is flat. This is equivalent to asking that all stalk maps are flat (see `AlgebraicGeometry.Flat.iff_flat_stalkMap`). We show that this property is local, and are stable under compositions and base change.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/Flat.html"}, {"id": "Mathlib.AlgebraicGeometry.Properties", "region_id": "algebraic_geometry", "micro_elevation": 0.3871, "macro_tier": 4, "macro_tier_score": 0.2747, "macro_tier_override": null, "x": 202.784, "z": -58.776, "size": 0.3905, "title": "Basic properties of schemes", "summary": "We provide some basic properties of schemes", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Properties.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point", "region_id": "algebraic_geometry", "micro_elevation": 0.1613, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 211.421, "z": -62.424, "size": 0.2, "title": "Nonsingular points and the group law in projective coordinates", "summary": "Let `W` be a Weierstrass curve over a field `F`. The nonsingular projective points of `W` can be endowed with a group law, which is uniquely determined by the formulae in `Mathlib/AlgebraicGeometry/EllipticCurve/Projective/Formula.lean` and follows from an equivalence with the nonsingular points in affine coordinates. This file defines the group law on nonsingular projective points.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/Projective/Point.html"}, {"id": "Mathlib.AlgebraicGeometry.PullbackCarrier", "region_id": "algebraic_geometry", "micro_elevation": 0.6129, "macro_tier": 4, "macro_tier_score": 0.2972, "macro_tier_override": null, "x": 204.652, "z": -49.699, "size": 0.379, "title": "Underlying topological space of fibre product of schemes", "summary": "Let `f : X ⟶ S` and `g : Y ⟶ S` be morphisms of schemes. In this file we describe the underlying topological space of `pullback f g`, i.e. the fiber product `X ×[S] Y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/PullbackCarrier.html"}, {"id": "Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme", "region_id": "algebraic_geometry", "micro_elevation": 0.5484, "macro_tier": 3, "macro_tier_score": 0.2279, "macro_tier_override": null, "x": 195.812, "z": -60.155, "size": 0.3079, "title": "Subscheme associated to an ideal sheaf", "summary": "We construct the subscheme associated to an ideal sheaf.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/IdealSheaf/Subscheme.html"}, {"id": "Mathlib.AlgebraicGeometry.IdealSheaf.Basic", "region_id": "algebraic_geometry", "micro_elevation": 0.5161, "macro_tier": 3, "macro_tier_score": 0.2275, "macro_tier_override": null, "x": 196.526, "z": -73.694, "size": 0.2691, "title": "Ideal sheaves on schemes", "summary": "We define ideal sheaves of schemes and provide various constructors for it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/IdealSheaf/Basic.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.Finite", "region_id": "algebraic_geometry", "micro_elevation": 0.8065, "macro_tier": 3, "macro_tier_score": 0.1031, "macro_tier_override": null, "x": 186.806, "z": -62.44, "size": 0.3221, "title": "Finite morphisms of schemes", "summary": "A morphism of schemes `f : X ⟶ Y` is finite if the preimage of an arbitrary affine open subset of `Y` is affine and the induced ring map is finite. It is equivalent to ask only that `Y` is covered by affine opens whose preimage is affine and the induced ring map is finite. Also see `AlgebraicGeometry.IsFinite.finite_preimage_singleton` in `Mathlib/AlgebraicGeometry/Fiber.lean` for the fact that finite morphisms have…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/Finite.html"}, {"id": "Mathlib.AlgebraicGeometry.StructureSheaf", "region_id": "algebraic_geometry", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.5005, "macro_tier_override": null, "x": 210.948, "z": -67.328, "size": 0.296, "title": "The structure sheaf on `PrimeSpectrum R`.", "summary": "We define the structure sheaf on `TopCat.of (PrimeSpectrum R)`, for an `R`-module `M` and prove basic properties about it. We define this as a subsheaf of the sheaf of dependent functions into the localizations, cut out by the condition that the function must be locally equal to a quotient of an element of `M` by an element of `R`. Because the condition \"is equal to a fraction\" passes to smaller open subsets, the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/StructureSheaf.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ", "region_id": "algebraic_geometry", "micro_elevation": 0.0968, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 212.166, "z": -64.635, "size": 0.2, "title": "Elliptic curves with same j-invariants are isomorphic", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms", "region_id": "algebraic_geometry", "micro_elevation": 0.0645, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 212.28, "z": -68.78, "size": 0.2478, "title": "Some normal forms of elliptic curves", "summary": "This file defines some normal forms of Weierstrass equations of elliptic curves.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation", "region_id": "algebraic_geometry", "micro_elevation": 0.5484, "macro_tier": 3, "macro_tier_score": 0.1716, "macro_tier_override": null, "x": 220.392, "z": -81.161, "size": 0.3425, "title": "Morphisms of finite presentation", "summary": "A morphism of schemes `f : X ⟶ Y` is locally of finite presentation if for each affine `U ⊆ Y` and `V ⊆ f ⁻¹' U`, The induced map `Γ(Y, U) ⟶ Γ(X, V)` is of finite presentation. A morphism of schemes is of finite presentation if it is both locally of finite presentation and quasi-compact. We do not provide a separate declaration for this, instead simply assume both conditions. We show that these properties are local,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/FinitePresentation.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.Proper", "region_id": "algebraic_geometry", "micro_elevation": 0.8387, "macro_tier": 3, "macro_tier_score": 0.0684, "macro_tier_override": null, "x": 236.322, "z": -70.846, "size": 0.2657, "title": "Proper morphisms", "summary": "A morphism of schemes is proper if it is separated, universally closed and (locally) of finite type. Note that we don't require quasi-compact, since this is implied by universally closed.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/Proper.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion", "region_id": "algebraic_geometry", "micro_elevation": 0.7097, "macro_tier": 3, "macro_tier_score": 0.2293, "macro_tier_override": null, "x": 195.003, "z": -52.645, "size": 0.3898, "title": "Closed immersions of schemes", "summary": "A morphism of schemes `f : X ⟶ Y` is a closed immersion if the underlying map of topological spaces is a closed immersion and the induced morphisms of stalks are all surjective.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.Constructors", "region_id": "algebraic_geometry", "micro_elevation": 0.4194, "macro_tier": 4, "macro_tier_score": 0.344, "macro_tier_override": null, "x": 198.781, "z": -63.325, "size": 0.4376, "title": "Constructors for properties of morphisms between schemes", "summary": "This file provides some constructors to obtain morphism properties of schemes from other morphism properties: - `AffineTargetMorphismProperty.diagonal` : Given an affine target morphism property `P`, `P.diagonal f` holds if `P (pullback.mapDesc f₁ f₂ f)` holds for two affine open immersions `f₁` and `f₂`. - `AffineTargetMorphismProperty.of`: Given a morphism property `P` of schemes, this is the restriction of `P` to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/Constructors.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.Affine", "region_id": "algebraic_geometry", "micro_elevation": 0.7097, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 193.771, "z": -80.549, "size": 0.2332, "title": "Small affine site induced by a morphism property", "summary": "Let `P` be a morphism property of schemes and `S` be a scheme. In this file we study the small affine `P`-site of `S`: its objects are rings `R` with a structure morphism `Spec R ⟶ S` that satisfies `P`. We don't make a separate definition for this site, but use `CategoryTheory.MorphismProperty.CostructuredArrow P ⊤ Scheme.Spec S`. Under suitable assumptions on `P`, the lemmas here can be used to show that the small…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/Affine.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic", "region_id": "algebraic_geometry", "micro_elevation": 0.0645, "macro_tier": 2, "macro_tier_score": 0.0126, "macro_tier_override": null, "x": 212.119, "z": -68.912, "size": 0.3463, "title": "Weierstrass equations and the nonsingular condition in affine coordinates", "summary": "Let `W` be a Weierstrass curve over a commutative ring `R` with coefficients `aᵢ`. An *affine point* on `W` is a tuple `(x, y)` of elements in `R` satisfying the *Weierstrass equation* `W(X, Y) = 0` in *affine coordinates*, where `W(X, Y) := Y² + a₁XY + a₃Y - (X³ + a₂X² + a₄X + a₆)`. It is *nonsingular* if its partial derivatives `W_X(x, y)` and `W_Y(x, y)` do not vanish simultaneously. This file defines polynomials…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/Affine/Basic.html"}, {"id": "Mathlib.AlgebraicGeometry.Spec", "region_id": "algebraic_geometry", "micro_elevation": 0.0323, "macro_tier": 4, "macro_tier_score": 0.4895, "macro_tier_override": null, "x": 211.913, "z": -67.524, "size": 0.3279, "title": "$Spec$ as a functor to locally ringed spaces.", "summary": "We define the functor $Spec$ from commutative rings to locally ringed spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Spec.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.LocalClosure", "region_id": "algebraic_geometry", "micro_elevation": 0.4194, "macro_tier": 2, "macro_tier_score": 0.0344, "macro_tier_override": null, "x": 222.242, "z": -73.369, "size": 0.2798, "title": "Local closure of morphism properties", "summary": "We define the source local closure of a property `P` w.r.t. a morphism property `W` and show it inherits stability properties from `P`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/LocalClosure.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.Smooth", "region_id": "algebraic_geometry", "micro_elevation": 0.8387, "macro_tier": 3, "macro_tier_score": 0.0573, "macro_tier_override": null, "x": 233.578, "z": -79.331, "size": 0.2885, "title": "Smooth morphisms", "summary": "In this file we define smooth morphisms. The main definitions are: - `AlgebraicGeometry.Smooth`: A morphism of schemes `f : X ⟶ Y` is smooth if for each affine `U ⊆ Y` and `V ⊆ f ⁻¹' U`, the induced map `Γ(Y, U) ⟶ Γ(X, V)` is smooth. - `AlgebraicGeometry.SmoothOfRelativeDimension`: A morphism of schemes `f : X ⟶ Y` is smooth of relative dimension `n` if for each `x : X` there exists an affine open neighborhood `V`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/Smooth.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.QuasiCompact", "region_id": "algebraic_geometry", "micro_elevation": 0.7097, "macro_tier": 2, "macro_tier_score": 0.0117, "macro_tier_override": null, "x": 231.188, "z": -75.085, "size": 0.2809, "title": "Quasi-compact precoverage", "summary": "In this file we define the quasi-compact precoverage. A cover is covering in the quasi-compact precoverage if it is a quasi-compact cover, i.e., if every affine open of the base can be covered by a finite union of images of quasi-compact opens of the components. The fpqc precoverage is the precoverage by flat covers that are quasi-compact in this sense.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/QuasiCompact.html"}, {"id": "Mathlib.AlgebraicGeometry.Cover.QuasiCompact", "region_id": "algebraic_geometry", "micro_elevation": 0.6774, "macro_tier": 2, "macro_tier_score": 0.0116, "macro_tier_override": null, "x": 223.307, "z": -50.734, "size": 0.258, "title": "Quasi-compact covers", "summary": "A cover of a scheme is quasi-compact if every affine open of the base can be covered by a finite union of images of quasi-compact opens of the components. This is used to define the fpqc (faithfully flat, quasi-compact) topology, where covers are given by flat covers that are quasi-compact.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Cover/QuasiCompact.html"}, {"id": "Mathlib.AlgebraicGeometry.PointsPi", "region_id": "algebraic_geometry", "micro_elevation": 0.8065, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 186.354, "z": -65.957, "size": 0.2, "title": "`Π Rᵢ`-Points of Schemes", "summary": "We show that the canonical map `X(Π Rᵢ) ⟶ Π X(Rᵢ)` (`AlgebraicGeometry.pointsPi`) is injective and surjective under various assumptions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/PointsPi.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange", "region_id": "algebraic_geometry", "micro_elevation": 0.0323, "macro_tier": 2, "macro_tier_score": 0.0124, "macro_tier_override": null, "x": 210.719, "z": -66.37, "size": 0.3397, "title": "Change of variables of Weierstrass curves", "summary": "This file defines admissible linear change of variables of Weierstrass curves.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/VariableChange.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic", "region_id": "algebraic_geometry", "micro_elevation": 0.0968, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 211.82, "z": -64.504, "size": 0.257, "title": "Weierstrass equations and the nonsingular condition in Jacobian coordinates", "summary": "A point on the projective plane over a commutative ring `R` with weights `(2, 3, 1)` is an equivalence class `[x : y : z]` of triples `(x, y, z) ≠ (0, 0, 0)` of elements in `R` such that `(x, y, z) ∼ (x', y', z')` if there is some unit `u` in `Rˣ` with `(x, y, z) = (u²x', u³y', uz')`. Let `W` be a Weierstrass curve over a commutative ring `R` with coefficients `aᵢ`. A *Jacobian point* is a point on the projective…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian/Basic.html"}, {"id": "Mathlib.AlgebraicGeometry.AffineScheme", "region_id": "algebraic_geometry", "micro_elevation": 0.2903, "macro_tier": 4, "macro_tier_score": 0.4099, "macro_tier_override": null, "x": 212.378, "z": -76.079, "size": 0.3233, "title": "Affine schemes", "summary": "We define the category of `AffineScheme`s as the essential image of `Spec`. We also define predicates about affine schemes and affine open sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/AffineScheme.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.IsIso", "region_id": "algebraic_geometry", "micro_elevation": 0.5161, "macro_tier": 3, "macro_tier_score": 0.2619, "macro_tier_override": null, "x": 226.364, "z": -70.622, "size": 0.2959, "title": "Being an isomorphism is local at the target", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/IsIso.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified", "region_id": "algebraic_geometry", "micro_elevation": 0.871, "macro_tier": 2, "macro_tier_score": 0.0457, "macro_tier_override": null, "x": 205.78, "z": -41.233, "size": 0.2647, "title": "Formally unramified morphisms", "summary": "A morphism of schemes `f : X ⟶ Y` is formally unramified if for each affine `U ⊆ Y` and `V ⊆ f ⁻¹' U`, the induced map `Γ(Y, U) ⟶ Γ(X, V)` is formally unramified. We show that these properties are local, and are stable under compositions and base change.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/FormallyUnramified.html"}, {"id": "Mathlib.AlgebraicGeometry.ColimitsOver", "region_id": "algebraic_geometry", "micro_elevation": 0.7097, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 219.003, "z": -47.205, "size": 0.2, "title": "Colimits in `P.Over ⊤ S`", "summary": "Let `P` be a morphism property in the category of schemes and `S` be a scheme. Let `D : J ⥤ P.Over ⊤ S` be a diagram and `𝒰` a locally directed open cover of `S` (e.g., the cover of all affine opens of `S`). Suppose the restrictions of `D` to `Dᵢ : J ⥤ P.Over ⊤ (𝒰.X i)` have a colimit for every `i`, then we show that also `D` has a colimit under the following assumptions: - `P` is local on the source. - For `i ⟶ j`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/ColimitsOver.html"}, {"id": "Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor", "region_id": "algebraic_geometry", "micro_elevation": 0.3548, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 213.146, "z": -77.941, "size": 0.2, "title": "Functoriality of Proj", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Functor.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.ElladicCohomology", "region_id": "algebraic_geometry", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 225.172, "z": -94.357, "size": 0.2, "title": "`ℓ`-adic cohomology of a scheme", "summary": "Let `X` be a scheme and `ℓ` be a prime number. In this file we define the sheaf associated to the topological group `ℤ_[ℓ]` on the pro-étale site of `X`. Its cohomology groups are the `ℓ`-adic cohomology groups of `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/ElladicCohomology.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.ConstantSheaf", "region_id": "algebraic_geometry", "micro_elevation": 0.8065, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 229.839, "z": -51.522, "size": 0.2239, "title": "Sheaf of continuous maps associated to topological space", "summary": "Given a topological space `T`, we consider the presheaf on `Scheme` given by `U ↦ C(U, T)` and show that it is a Zariski sheaf (TODO: show that it is a fpqc sheaf). When `T` is discrete, this is the constant sheaf associated to `T` (TODO).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/ConstantSheaf.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.Proetale", "region_id": "algebraic_geometry", "micro_elevation": 0.9677, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 211.041, "z": -37.77, "size": 0.2239, "title": "The pro-étale site", "summary": "In this file we define the big and small pro-étale site. The big pro-étale site is the category of schemes equipped with the pro-étale topology, which is the topology generated by fpqc covers of weakly étale morphisms. We prefer to work with weakly étale morphisms instead of pro-étale morphisms, since the property of being pro-étale is not well-behaved: it is not local on the target. We also define the small…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/Proetale.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties", "region_id": "algebraic_geometry", "micro_elevation": 0.4516, "macro_tier": 4, "macro_tier_score": 0.3414, "macro_tier_override": null, "x": 214.0, "z": -53.876, "size": 0.2926, "title": "Properties of morphisms from properties of ring homs.", "summary": "We provide the basic framework for talking about properties of morphisms that come from properties of ring homs. For `P` a property of ring homs, we have two ways of defining a property of scheme morphisms: Let `f : X ⟶ Y`, - `targetAffineLocally (affineAnd P)`: the preimage of an affine open `U = Spec A` is affine (`= Spec B`) and `A ⟶ B` satisfies `P`. (in `Mathlib/AlgebraicGeometry/Morphisms/AffineAnd.lean`) -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.html"}, {"id": "Mathlib.AlgebraicGeometry.AffineTransitionLimit", "region_id": "algebraic_geometry", "micro_elevation": 0.8387, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 198.277, "z": -89.592, "size": 0.2, "title": "Inverse limits of schemes with affine transition maps", "summary": "In this file, we develop API for inverse limits of schemes with affine transition maps, following EGA IV 8 and https://stacks.math.columbia.edu/tag/01YT.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/AffineTransitionLimit.html"}, {"id": "Mathlib.AlgebraicGeometry.QuasiAffine", "region_id": "algebraic_geometry", "micro_elevation": 0.8065, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 235.037, "z": -62.189, "size": 0.2302, "title": "Quasi-affine schemes", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/QuasiAffine.html"}, {"id": "Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski", "region_id": "algebraic_geometry", "micro_elevation": 0.7097, "macro_tier": 2, "macro_tier_score": 0.0342, "macro_tier_override": null, "x": 231.033, "z": -59.178, "size": 0.2371, "title": "The small affine Zariski site", "summary": "`X.AffineZariskiSite` is the small affine Zariski site of `X`, whose elements are affine open sets of `X`, and whose arrows are basic open sets `D(f) ⟶ U` for any `f : Γ(X, U)`. Every presieve on `U` is then given by a `Set Γ(X, U)` (`presieveOfSections_surjective`), and we endow `X.AffineZariskiSite` with `grothendieckTopology X`, such that `s : Set Γ(X, U)` is a cover if and only if `Ideal.span s = ⊤`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Sites/SmallAffineZariski.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.LFunction", "region_id": "algebraic_geometry", "micro_elevation": 0.1613, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 206.883, "z": -70.112, "size": 0.2, "title": "The L-function of a Weierstrass curve", "summary": "In this file, we define the L-function of a Weierstrass curve.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/LFunction.html"}, {"id": "Mathlib.AlgebraicGeometry.EllipticCurve.Reduction", "region_id": "algebraic_geometry", "micro_elevation": 0.0645, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 209.024, "z": -67.754, "size": 0.2276, "title": "Reduction of Weierstrass curves over local fields", "summary": "This file defines reduction of Weierstrass curves over local fields, or more generally, fraction fields of discrete valuation rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/EllipticCurve/Reduction.html"}, {"id": "Mathlib.AlgebraicGeometry.Cover.MorphismProperty", "region_id": "algebraic_geometry", "micro_elevation": 0.1613, "macro_tier": 4, "macro_tier_score": 0.4552, "macro_tier_override": null, "x": 213.612, "z": -71.471, "size": 0.3067, "title": "Covers of schemes", "summary": "This file provides the basic API for covers of schemes. A cover of a scheme `X` with respect to a morphism property `P` is a jointly surjective indexed family of scheme morphisms with target `X` all satisfying `P`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Cover/MorphismProperty.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion", "region_id": "algebraic_geometry", "micro_elevation": 0.4839, "macro_tier": 4, "macro_tier_score": 0.262, "macro_tier_override": null, "x": 199.243, "z": -58.304, "size": 0.3109, "title": "Open immersions", "summary": "A morphism is an open immersion if the underlying map of spaces is an open embedding `f : X ⟶ U ⊆ Y`, and the sheaf map `Y(V) ⟶ f _* X(V)` is an iso for each `V ⊆ U`. Most of the theories are developed in `AlgebraicGeometry/OpenImmersion`, and we provide the remaining theorems analogous to other lemmas in `AlgebraicGeometry/Morphisms/*`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.html"}, {"id": "Mathlib.AlgebraicGeometry.Birational.Birational", "region_id": "algebraic_geometry", "micro_elevation": 0.9677, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 234.444, "z": -85.261, "size": 0.2, "title": "Birationality and Rationality of schemes.", "summary": "This file defines partial isomorphisms between schemes and uses them to formalize birationality and rationality.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Birational/Birational.html"}, {"id": "Mathlib.AlgebraicGeometry.AffineSpace", "region_id": "algebraic_geometry", "micro_elevation": 0.9355, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 183.492, "z": -75.239, "size": 0.2478, "title": "Affine space", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/AffineSpace.html"}, {"id": "Mathlib.AlgebraicGeometry.Modules.Presheaf", "region_id": "algebraic_geometry", "micro_elevation": 0.0968, "macro_tier": 1, "macro_tier_score": 0.0114, "macro_tier_override": null, "x": 208.982, "z": -69.536, "size": 0.2263, "title": "The category of presheaves of modules over a scheme", "summary": "In this file, given a scheme `X`, we define the category of presheaves of modules over `X`. As categories of presheaves of modules are defined for presheaves of rings (and not presheaves of commutative rings), we also introduce a definition `X.ringCatSheaf` for the underlying sheaf of rings of `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Modules/Presheaf.html"}, {"id": "Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf", "region_id": "algebraic_geometry", "micro_elevation": 0.0323, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 210.001, "z": -67.055, "size": 0.2481, "title": "The structure sheaf on `ProjectiveSpectrum 𝒜`.", "summary": "In `Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean`, we have given a topology on `ProjectiveSpectrum 𝒜`; in this file we will construct a sheaf on `ProjectiveSpectrum 𝒜`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.html"}, {"id": "Mathlib.AlgebraicGeometry.Group.Abelian", "region_id": "algebraic_geometry", "micro_elevation": 0.9677, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 197.957, "z": -93.878, "size": 0.2, "title": "Abelian varieties", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Group/Abelian.html"}, {"id": "Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme", "region_id": "algebraic_geometry", "micro_elevation": 0.2903, "macro_tier": 1, "macro_tier_score": 0.0115, "macro_tier_override": null, "x": 202.558, "z": -70.2, "size": 0.253, "title": "Proj as a scheme", "summary": "This file is to prove that `Proj` is a scheme.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.html"}, {"id": "Mathlib.AlgebraicGeometry.OrderOfVanishing", "region_id": "algebraic_geometry", "micro_elevation": 0.8387, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 187.264, "z": -57.566, "size": 0.2, "title": "Order of vanishing in a scheme", "summary": "In this file we define the order of vanishing of an element of the function field of a locally Noetherian integral scheme at a point of codimension `1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/OrderOfVanishing.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.FlatMono", "region_id": "algebraic_geometry", "micro_elevation": 0.8065, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 186.36, "z": -68.797, "size": 0.2, "title": "Flat monomorphisms of finite presentation are open immersions", "summary": "We show the titular result `AlgebraicGeometry.IsOpenImmersion.of_flat_of_mono` by fpqc descent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/FlatMono.html"}, {"id": "Mathlib.AlgebraicGeometry.AlgebraicCycle.Basic", "region_id": "algebraic_geometry", "micro_elevation": 0.6129, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 192.245, "z": -68.131, "size": 0.2, "title": "Algebraic Cycles", "summary": "In this file we define algebraic cycles on a scheme `X` with coefficients in a type `R` and provide some basic API for working with them. We define an algebraic cycle on a scheme `X` with coefficients in a type `R` to be functions `c : X → R` whose support is locally finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/AlgebraicCycle/Basic.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.SmoothFiber", "region_id": "algebraic_geometry", "micro_elevation": 0.871, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 187.273, "z": -79.46, "size": 0.2, "title": "Smooth morphisms and their fibers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/SmoothFiber.html"}, {"id": "Mathlib.AlgebraicGeometry.Morphisms.FlatRank", "region_id": "algebraic_geometry", "micro_elevation": 0.8387, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 194.638, "z": -87.082, "size": 0.2, "title": "Rank of a finite flat morphism of schemes", "summary": "In this file we define the rank `AlgebraicGeometry.Scheme.Hom.finrank` of a finite flat morphism of schemes `f : X ⟶ Y`. It is locally constant and is characterized by the condition that the rank of `Spec S ⟶ Spec R` at some prime `p` of `R` is the rank of `S` as an `R`-algebra at `p`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicGeometry/Morphisms/FlatRank.html"}, {"id": "Mathlib.Probability.ProbabilityMassFunction.Binomial", "region_id": "probability", "micro_elevation": 0.68, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -177.311, "z": 101.094, "size": 0.2, "title": "The binomial distribution", "summary": "This file defines the probability mass function of the binomial distribution.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/ProbabilityMassFunction/Binomial.html"}, {"id": "Mathlib.Probability.Distributions.Binomial", "region_id": "probability", "micro_elevation": 0.64, "macro_tier": 1, "macro_tier_score": 0.0457, "macro_tier_override": null, "x": -205.263, "z": 126.804, "size": 0.2617, "title": "Binomial random variables", "summary": "This file defines the binomial distribution and binomial random variables, and computes their expectation and variance. For `n : ℕ` and `p : I`, the binomial distribution `Bin(n, p)` is defined as the cardinal of a random subset `U` of `Set.Iic n` such that each `k ∈ Set.Iic n` belongs to `U` independently with probability `p`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Binomial.html"}, {"id": "Mathlib.Probability.ProbabilityMassFunction.Constructions", "region_id": "probability", "micro_elevation": 0.08, "macro_tier": 1, "macro_tier_score": 0.0458, "macro_tier_override": null, "x": -185.054, "z": 120.057, "size": 0.2827, "title": "Specific Constructions of Probability Mass Functions", "summary": "This file gives a number of different `PMF` constructions for common probability distributions. `map` and `seq` allow pushing a `PMF α` along a function `f : α → β` (or distribution of functions `f : PMF (α → β)`) to get a `PMF β`. `ofFinset` and `ofFintype` simplify the construction of a `PMF α` from a function `f : α → ℝ≥0∞`, by allowing the \"sum equals 1\" constraint to be in terms of `Finset.sum` instead of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/ProbabilityMassFunction/Constructions.html"}, {"id": "Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily", "region_id": "probability", "micro_elevation": 0.96, "macro_tier": 0, "macro_tier_score": 0.0228, "macro_tier_override": null, "x": -181.68, "z": 147.862, "size": 0.239, "title": "Finite dimensional distributions of Brownian motion", "summary": "In this file we define `projectiveFamily : (I : Finset ℝ≥0) → Measure (I → ℝ)`. Each `projectiveFamily I` is the centered Gaussian measure over `I → ℝ` with covariance matrix given by `covMatrix I s t := min s t`. Note that we build a measure over `I → ℝ` rather than `EuclideanSpace I ℝ`. This is because we want to extend this family to a measure over `ℝ≥0 → ℝ` through the Kolmogorov's extension theorem, which is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/BrownianMotion/GaussianProjectiveFamily.html"}, {"id": "Mathlib.Probability.Distributions.Gaussian.Multivariate", "region_id": "probability", "micro_elevation": 0.92, "macro_tier": 1, "macro_tier_score": 0.0228, "macro_tier_override": null, "x": -170.096, "z": 141.263, "size": 0.2442, "title": "Multivariate Gaussian distributions", "summary": "In this file we define the standard Gaussian distribution over a Euclidean space and multivariate Gaussian distributions over `EuclideanSpace ℝ ι`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Gaussian/Multivariate.html"}, {"id": "Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic", "region_id": "probability", "micro_elevation": 0.88, "macro_tier": 2, "macro_tier_score": 0.0686, "macro_tier_override": null, "x": -182.26, "z": 145.495, "size": 0.2815, "title": "Gaussian random variables", "summary": "In this file we prove basic properties of Gaussian random variables.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Gaussian/HasGaussianLaw/Basic.html"}, {"id": "Mathlib.Probability.Kernel.Composition.RadonNikodym", "region_id": "probability", "micro_elevation": 0.24, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -189.07, "z": 112.105, "size": 0.2432, "title": "Radon-Nikodym derivative of a composition product", "summary": "We compute the Radon-Nikodym derivative of a composition product `μ ⊗ₘ κ` with respect to another composition product `ν ⊗ₘ η` in terms of the Radon-Nikodym derivatives `∂μ/∂ν` and `∂(μ ⊗ₘ κ)/∂(μ ⊗ₘ η)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/RadonNikodym.html"}, {"id": "Mathlib.Probability.Kernel.Composition.MeasureCompProd", "region_id": "probability", "micro_elevation": 0.2, "macro_tier": 3, "macro_tier_score": 0.4787, "macro_tier_override": null, "x": -193.173, "z": 120.912, "size": 0.363, "title": "Composition-Product of a measure and a kernel", "summary": "This operation, denoted by `⊗ₘ`, takes `μ : Measure α` and `κ : Kernel α β` and creates `μ ⊗ₘ κ : Measure (α × β)`. The integral of a function against `μ ⊗ₘ κ` is `∫⁻ x, f x ∂(μ ⊗ₘ κ) = ∫⁻ a, ∫⁻ b, f (a, b) ∂(κ a) ∂μ`. `μ ⊗ₘ κ` is defined as `((Kernel.const Unit μ) ⊗ₖ (Kernel.prodMkLeft Unit κ)) ()`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/MeasureCompProd.html"}, {"id": "Mathlib.Probability.Moments.Covariance", "region_id": "probability", "micro_elevation": 0.48, "macro_tier": 3, "macro_tier_score": 0.3419, "macro_tier_override": null, "x": -185.35, "z": 133.672, "size": 0.3336, "title": "Covariance", "summary": "We define the covariance of two real-valued random variables.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Moments/Covariance.html"}, {"id": "Mathlib.Probability.Independence.Integration", "region_id": "probability", "micro_elevation": 0.44, "macro_tier": 3, "macro_tier_score": 0.342, "macro_tier_override": null, "x": -174.64, "z": 123.464, "size": 0.3401, "title": "Integration in Probability Theory", "summary": "Integration results for independent random variables. Specifically, for two independent random variables X and Y over the extended non-negative reals, `E[X * Y] = E[X] * E[Y]`, and similar results.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/Integration.html"}, {"id": "Mathlib.Probability.Martingale.Convergence", "region_id": "probability", "micro_elevation": 0.2, "macro_tier": 3, "macro_tier_score": 0.273, "macro_tier_override": null, "x": -193.413, "z": 118.925, "size": 0.2724, "title": "Martingale convergence theorems", "summary": "The martingale convergence theorems are a collection of theorems characterizing the convergence of a martingale provided it satisfies some boundedness conditions. This file contains the almost everywhere martingale convergence theorem which provides an almost everywhere limit to an L¹ bounded submartingale. It also contains the L¹ martingale convergence theorem which provides an L¹ limit to a uniformly integrable…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Martingale/Convergence.html"}, {"id": "Mathlib.Probability.Martingale.Upcrossing", "region_id": "probability", "micro_elevation": 0.16, "macro_tier": 3, "macro_tier_score": 0.2732, "macro_tier_override": null, "x": -188.931, "z": 123.799, "size": 0.2948, "title": "Doob's upcrossing estimate", "summary": "Given a discrete real-valued submartingale $(f_n)_{n \\in \\mathbb{N}}$, denoting by $U_N(a, b)$ the number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing estimate (also known as Doob's inequality) states that $$(b - a) \\mathbb{E}[U_N(a, b)] \\le \\mathbb{E}[(f_N - a)^+].$$ Doob's upcrossing estimate is an important inequality and is central in proving the martingale convergence…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Martingale/Upcrossing.html"}, {"id": "Mathlib.Probability.Kernel.MeasurableLIntegral", "region_id": "probability", "micro_elevation": 0.08, "macro_tier": 3, "macro_tier_score": 0.5026, "macro_tier_override": null, "x": -189.372, "z": 117.869, "size": 0.4181, "title": "Measurability of the integral against a kernel", "summary": "The Lebesgue integral of a measurable function against a kernel is measurable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/MeasurableLIntegral.html"}, {"id": "Mathlib.Probability.Kernel.Basic", "region_id": "probability", "micro_elevation": 0.04, "macro_tier": 3, "macro_tier_score": 0.5049, "macro_tier_override": null, "x": -188.549, "z": 119.252, "size": 0.4974, "title": "Basic kernels", "summary": "This file contains basic results about kernels in general and definitions of some particular kernels.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Basic.html"}, {"id": "Mathlib.Probability.IdentDistrib", "region_id": "probability", "micro_elevation": 0.6, "macro_tier": 2, "macro_tier_score": 0.1828, "macro_tier_override": null, "x": -199.576, "z": 105.653, "size": 0.3324, "title": "Identically distributed random variables", "summary": "Two random variables defined on two (possibly different) probability spaces but taking value in the same space are *identically distributed* if their distributions (i.e., the image probability measures on the target space) coincide. We define this concept and establish its basic properties in this file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/IdentDistrib.html"}, {"id": "Mathlib.Probability.Martingale.Centering", "region_id": "probability", "micro_elevation": 0.16, "macro_tier": 1, "macro_tier_score": 0.0455, "macro_tier_override": null, "x": -184.193, "z": 122.922, "size": 0.2403, "title": "Centering lemma for stochastic processes", "summary": "Any `ℕ`-indexed stochastic process which is strongly adapted and integrable can be written as the sum of a martingale and a predictable process. This result is also known as **Doob's decomposition theorem**. From a process `f`, a filtration `ℱ` and a measure `μ`, we define two processes `martingalePart f ℱ μ` and `predictablePart f ℱ μ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Martingale/Centering.html"}, {"id": "Mathlib.Probability.Martingale.Basic", "region_id": "probability", "micro_elevation": 0.12, "macro_tier": 3, "macro_tier_score": 0.2736, "macro_tier_override": null, "x": -190.06, "z": 116.771, "size": 0.3285, "title": "Martingales", "summary": "A family of functions `f : ι → Ω → E` is a martingale with respect to a filtration `ℱ` if every `f i` is integrable, `f` is strongly adapted with respect to `ℱ` and for all `i ≤ j`, `μ[f j | ℱ i] =ᵐ[μ] f i`. On the other hand, `f : ι → Ω → E` is said to be a supermartingale with respect to the filtration `ℱ` if `f i` is integrable, `f` is strongly adapted with respect to `ℱ` and for all `i ≤ j`, `μ[f j | ℱ i] ≤ᵐ[μ]…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Martingale/Basic.html"}, {"id": "Mathlib.Probability.Independence.Kernel.IndepFun", "region_id": "probability", "micro_elevation": 0.32, "macro_tier": 3, "macro_tier_score": 0.41, "macro_tier_override": null, "x": -177.669, "z": 120.404, "size": 0.3287, "title": "Independence of random variables with respect to a kernel and a measure", "summary": "A family of random variables is independent if the corresponding `σ`-algebras are independent. Independence of families of sets and `σ`-algebras is covered in the `Indep` file. This file deals with independence of random variables specifically. Note that we define independence with respect to a kernel and a measure. This notion of independence is a generalization of both independence and conditional independence.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/Kernel/IndepFun.html"}, {"id": "Mathlib.Probability.Independence.Kernel.Indep", "region_id": "probability", "micro_elevation": 0.08, "macro_tier": 3, "macro_tier_score": 0.4093, "macro_tier_override": null, "x": -189.238, "z": 120.715, "size": 0.2681, "title": "Independence of families of sets with respect to a kernel and a measure", "summary": "A family of sets of sets `π : ι → Set (Set Ω)` is independent with respect to a kernel `κ : Kernel α Ω` and a measure `μ` on `α` if for any finite set of indices `s = {i_1, ..., i_n}`, for any sets `f i_1 ∈ π i_1, ..., f i_n ∈ π i_n`, then for `μ`-almost every `a : α`, `κ a (⋂ i in s, f i) = ∏ i ∈ s, κ a (f i)`. This notion of independence is a generalization of both independence and conditional independence. For…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/Kernel/Indep.html"}, {"id": "Mathlib.Probability.ConditionalProbability", "region_id": "probability", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.4325, "macro_tier_override": null, "x": -187.333, "z": 119.199, "size": 0.307, "title": "Conditional Probability", "summary": "This file defines conditional probability and includes basic results relating to it. Given some measure `μ` defined on a measure space on some type `Ω` and some `s : Set Ω`, we define the measure of `μ` conditioned on `s` as the restricted measure scaled by the inverse of the measure of `s`: `cond μ s = (μ s)⁻¹ • μ.restrict s`. The scaling ensures that this is a probability measure (when `μ` is a finite measure).…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/ConditionalProbability.html"}, {"id": "Mathlib.Probability.Kernel.Composition.MeasureComp", "region_id": "probability", "micro_elevation": 0.28, "macro_tier": 3, "macro_tier_score": 0.4791, "macro_tier_override": null, "x": -185.986, "z": 127.613, "size": 0.3834, "title": "Lemmas about the composition of a measure and a kernel", "summary": "Basic lemmas about the composition `κ ∘ₘ μ` of a kernel `κ` and a measure `μ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/MeasureComp.html"}, {"id": "Mathlib.Probability.Process.Stopping", "region_id": "probability", "micro_elevation": 0.08, "macro_tier": 3, "macro_tier_score": 0.2966, "macro_tier_override": null, "x": -189.261, "z": 117.714, "size": 0.3428, "title": "Stopping times, stopped processes and stopped values", "summary": "Definition and properties of stopping times.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Process/Stopping.html"}, {"id": "Mathlib.Probability.Process.Adapted", "region_id": "probability", "micro_elevation": 0.04, "macro_tier": 3, "macro_tier_score": 0.2966, "macro_tier_override": null, "x": -186.788, "z": 118.111, "size": 0.3437, "title": "Adapted and progressively measurable processes", "summary": "This file defines the related notions of a process `u` being (strongly) `Adapted` or `Progressive` (progressively measurable) with respect to a filtration `f`, and proves some basic facts about them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Process/Adapted.html"}, {"id": "Mathlib.Probability.Kernel.Composition.KernelLemmas", "region_id": "probability", "micro_elevation": 0.24, "macro_tier": 3, "macro_tier_score": 0.478, "macro_tier_override": null, "x": -194.482, "z": 117.704, "size": 0.3137, "title": "Lemmas relating different ways to compose kernels", "summary": "This file contains lemmas about the composition of kernels that involve several types of compositions/products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/KernelLemmas.html"}, {"id": "Mathlib.Probability.Kernel.Composition.CompProd", "region_id": "probability", "micro_elevation": 0.16, "macro_tier": 3, "macro_tier_score": 0.5014, "macro_tier_override": null, "x": -183.755, "z": 115.896, "size": 0.3583, "title": "Composition-product of kernels", "summary": "We define the composition-product `κ ⊗ₖ η` of two s-finite kernels `κ : Kernel α β` and `η : Kernel (α × β) γ`, a kernel from `α` to `β × γ`. A note on names: The composition-product `Kernel α β → Kernel (α × β) γ → Kernel α (β × γ)` is named composition in [kallenberg2021] and product on the wikipedia article on transition kernels. Most papers studying categories of kernels call composition the map we call…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/CompProd.html"}, {"id": "Mathlib.Probability.Kernel.Composition.Prod", "region_id": "probability", "micro_elevation": 0.2, "macro_tier": 3, "macro_tier_score": 0.478, "macro_tier_override": null, "x": -193.389, "z": 119.803, "size": 0.311, "title": "Product and composition of kernels", "summary": "We define the product `κ ×ₖ η` of s-finite kernels `κ : Kernel α β` and `η : Kernel α γ`, which is a kernel from `α` to `β × γ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/Prod.html"}, {"id": "Mathlib.Probability.UniformOn", "region_id": "probability", "micro_elevation": 0.04, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -188.492, "z": 119.57, "size": 0.2732, "title": "Classical probability", "summary": "The classical formulation of probability states that the probability of an event occurring in a finite probability space is the ratio of that event to all possible events. This notion can be expressed with measure theory using the counting measure. In particular, given the sets `s` and `t`, we define the probability of `t` occurring in `s` to be `|s|⁻¹ * |s ∩ t|`. With this definition, we recover the probability…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/UniformOn.html"}, {"id": "Mathlib.Probability.Kernel.Composition.ParallelComp", "region_id": "probability", "micro_elevation": 0.12, "macro_tier": 3, "macro_tier_score": 0.5011, "macro_tier_override": null, "x": -188.154, "z": 122.758, "size": 0.3426, "title": "Parallel composition of kernels", "summary": "Two kernels `κ : Kernel α β` and `η : Kernel γ δ` can be applied in parallel to give a kernel `κ ∥ₖ η` from `α × γ` to `β × δ`: `(κ ∥ₖ η) (a, c) = (κ a).prod (η c)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/ParallelComp.html"}, {"id": "Mathlib.Probability.Kernel.Composition.MapComap", "region_id": "probability", "micro_elevation": 0.08, "macro_tier": 3, "macro_tier_score": 0.5009, "macro_tier_override": null, "x": -184.96, "z": 119.743, "size": 0.3276, "title": "Map of a kernel by a measurable function", "summary": "We define the map and comap of a kernel along a measurable function, as well as some often useful particular cases.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/MapComap.html"}, {"id": "Mathlib.Probability.Martingale.OptionalStopping", "region_id": "probability", "micro_elevation": 0.16, "macro_tier": 1, "macro_tier_score": 0.0455, "macro_tier_override": null, "x": -192.027, "z": 120.492, "size": 0.2403, "title": "Optional stopping theorem (fair game theorem)", "summary": "The optional stopping theorem states that a strongly adapted integrable process `f` is a submartingale if and only if for all bounded stopping times `τ` and `π` such that `τ ≤ π`, the stopped value of `f` at `τ` has expectation smaller than its stopped value at `π`. This file also contains Doob's maximal inequality: given a non-negative submartingale `f`, for all `ε : ℝ≥0`, we have `ε • μ {ε ≤ f* n} ≤ ∫ ω in {ε ≤ f*…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Martingale/OptionalStopping.html"}, {"id": "Mathlib.Probability.Notation", "region_id": "probability", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.4787, "macro_tier_override": null, "x": -187.333, "z": 119.199, "size": 0.363, "title": "Notations for probability theory", "summary": "This file defines the following notations, for functions `X,Y`, measures `P, Q` defined on a measurable space `m0`, and another measurable space structure `m` with `hm : m ≤ m0`, - `P[X] = ∫ a, X a ∂P` - `𝔼[X] = ∫ a, X a` - `𝔼[X | m]`: conditional expectation of `X` with respect to the measure `volume` and the measurable space `m`. The similar `P[X|m]` for a measure `P` is defined in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Notation.html"}, {"id": "Mathlib.Probability.Process.HittingTime", "region_id": "probability", "micro_elevation": 0.12, "macro_tier": 3, "macro_tier_score": 0.2731, "macro_tier_override": null, "x": -184.0, "z": 120.694, "size": 0.2776, "title": "Hitting times", "summary": "Given a stochastic process, the hitting time provides the first time the process \"hits\" some subset of the state space. The hitting time is a stopping time in the case that the time index is discrete and the process is strongly adapted (this is true in a far more general setting however we have only proved it for the discrete case so far).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Process/HittingTime.html"}, {"id": "Mathlib.Probability.Moments.ComplexMGF", "region_id": "probability", "micro_elevation": 0.68, "macro_tier": 2, "macro_tier_score": 0.114, "macro_tier_override": null, "x": -192.621, "z": 99.193, "size": 0.2774, "title": "The complex-valued moment-generating function", "summary": "The moment-generating function (mgf) is `t : ℝ ↦ μ[fun ω ↦ rexp (t * X ω)]`. It can be extended to a complex function `z : ℂ ↦ μ[fun ω ↦ cexp (z * X ω)]`, which we call `complexMGF X μ`. That function is holomorphic on the vertical strip with base the interior of the interval of definition of the mgf. On the vertical line that goes through 0, `complexMGF X μ` is equal to the characteristic function. This allows us…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Moments/ComplexMGF.html"}, {"id": "Mathlib.Probability.Moments.Basic", "region_id": "probability", "micro_elevation": 0.64, "macro_tier": 2, "macro_tier_score": 0.1139, "macro_tier_override": null, "x": -170.391, "z": 128.807, "size": 0.262, "title": "Moments and moment-generating function", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Moments/Basic.html"}, {"id": "Mathlib.Probability.Moments.IntegrableExpMul", "region_id": "probability", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1137, "macro_tier_override": null, "x": -187.333, "z": 119.199, "size": 0.2438, "title": "Domain of the moment-generating function", "summary": "For `X` a real random variable and `μ` a finite measure, the set `{t | Integrable (fun ω ↦ exp (t * X ω)) μ}` is an interval containing zero. This is the set of points for which the moment-generating function `mgf X μ t` is well defined. We denote that set by `integrableExpSet X μ`. We prove the integrability of other functions for `t` in the interior of that interval.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Moments/IntegrableExpMul.html"}, {"id": "Mathlib.Probability.Process.Predictable", "region_id": "probability", "micro_elevation": 0.08, "macro_tier": 3, "macro_tier_score": 0.2731, "macro_tier_override": null, "x": -186.742, "z": 116.838, "size": 0.2785, "title": "Predictable σ-algebra", "summary": "This file defines the predictable σ-algebra associated to a filtration, as well as the notion of predictable processes. We prove that predictable processes are progressively measurable and adapted. We also give an equivalent characterization of predictability for discrete processes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Process/Predictable.html"}, {"id": "Mathlib.Probability.Kernel.MeasurableIntegral", "region_id": "probability", "micro_elevation": 0.12, "macro_tier": 2, "macro_tier_score": 0.2049, "macro_tier_override": null, "x": -188.217, "z": 122.743, "size": 0.2763, "title": "Measurability of the integral against a kernel", "summary": "The Bochner integral of a strongly measurable function against a kernel is strongly measurable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/MeasurableIntegral.html"}, {"id": "Mathlib.Probability.Kernel.Disintegration.Integral", "region_id": "probability", "micro_elevation": 0.36, "macro_tier": 2, "macro_tier_score": 0.2051, "macro_tier_override": null, "x": -177.86, "z": 113.695, "size": 0.3007, "title": "Lebesgue and Bochner integrals of conditional kernels", "summary": "Integrals of `ProbabilityTheory.Kernel.condKernel` and `MeasureTheory.Measure.condKernel`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Disintegration/Integral.html"}, {"id": "Mathlib.Probability.Kernel.Composition.IntegralCompProd", "region_id": "probability", "micro_elevation": 0.32, "macro_tier": 2, "macro_tier_score": 0.205, "macro_tier_override": null, "x": -184.418, "z": 128.491, "size": 0.2918, "title": "Bochner integral of a function against the composition and the composition-products of two kernels", "summary": "We prove properties of the composition and the composition-product of two kernels. If `κ` is a kernel from `α` to `β` and `η` is a kernel from `β` to `γ`, we can form their composition `η ∘ₖ κ : Kernel α γ`. We proved in `ProbabilityTheory.Kernel.lintegral_comp` that it verifies `∫⁻ c, f c ∂((η ∘ₖ κ) a) = ∫⁻ b, ∫⁻ c, f c ∂(η b) ∂(κ a)`. In this file, we prove the same equality for the Bochner integral. If `κ` is an…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/IntegralCompProd.html"}, {"id": "Mathlib.Probability.Kernel.Disintegration.StandardBorel", "region_id": "probability", "micro_elevation": 0.28, "macro_tier": 2, "macro_tier_score": 0.2277, "macro_tier_override": null, "x": -192.464, "z": 126.002, "size": 0.2904, "title": "Existence of disintegration of measures and kernels for standard Borel spaces", "summary": "Let `κ : Kernel α (β × Ω)` be a finite kernel, where `Ω` is a standard Borel space. Then if `α` is countable or `β` has a countably generated σ-algebra (for example if it is standard Borel), then there exists a `Kernel (α × β) Ω` called conditional kernel and denoted by `condKernel κ` such that `κ = fst κ ⊗ₖ condKernel κ`. We also define a conditional kernel for a measure `ρ : Measure (β × Ω)`, where `Ω` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Disintegration/StandardBorel.html"}, {"id": "Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Def", "region_id": "probability", "micro_elevation": 0.84, "macro_tier": 2, "macro_tier_score": 0.0689, "macro_tier_override": null, "x": -204.431, "z": 100.197, "size": 0.3142, "title": "Gaussian random variables", "summary": "In this file we define a predicate `HasGaussianLaw X P`, which states that under the probability measure `P`, the random variable `X` has a Gaussian distribution, i.e. `IsGaussian (P.map X)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Gaussian/HasGaussianLaw/Def.html"}, {"id": "Mathlib.Probability.Distributions.Gaussian.Basic", "region_id": "probability", "micro_elevation": 0.8, "macro_tier": 2, "macro_tier_score": 0.0694, "macro_tier_override": null, "x": -173.519, "z": 99.152, "size": 0.35, "title": "Gaussian distributions in Banach spaces", "summary": "We introduce a predicate `IsGaussian` for measures on a Banach space `E` such that the map by any continuous linear form is a Gaussian measure on `ℝ`. For Gaussian distributions in `ℝ`, see the file `Mathlib/Probability/Distributions/Gaussian/Real.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Gaussian/Basic.html"}, {"id": "Mathlib.Probability.Independence.Integrable", "region_id": "probability", "micro_elevation": 0.4, "macro_tier": 3, "macro_tier_score": 0.3413, "macro_tier_override": null, "x": -197.045, "z": 111.862, "size": 0.2838, "title": "Independence of functions implies that the measure is a probability measure", "summary": "If a nonzero function belongs to `ℒ^p` (in particular if it is integrable) and is independent of another function, then the space is a probability space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/Integrable.html"}, {"id": "Mathlib.Probability.Decision.Risk.Basic", "region_id": "probability", "micro_elevation": 0.32, "macro_tier": 1, "macro_tier_score": 0.0231, "macro_tier_override": null, "x": -191.244, "z": 128.118, "size": 0.2827, "title": "Basic properties of the risk of an estimator", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Decision/Risk/Basic.html"}, {"id": "Mathlib.Probability.Decision.Risk.Defs", "region_id": "probability", "micro_elevation": 0.16, "macro_tier": 1, "macro_tier_score": 0.023, "macro_tier_override": null, "x": -189.262, "z": 114.729, "size": 0.2719, "title": "Risk of an estimator", "summary": "An estimation problem is defined by a parameter space `Θ`, a data generating kernel `P : Kernel Θ 𝓧` and a loss function `ℓ : Θ → 𝓨 → ℝ≥0∞`. A (randomized) estimator is a kernel `κ : Kernel 𝓧 𝓨` that maps data to estimates of a quantity of interest that depends on the parameter. Often the quantity of interest is the parameter itself and `𝓨 = Θ`. The quality of an estimate `y` when data comes from the distribution…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Decision/Risk/Defs.html"}, {"id": "Mathlib.Probability.Kernel.Invariance", "region_id": "probability", "micro_elevation": 0.32, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -189.075, "z": 109.618, "size": 0.2, "title": "Invariance of measures along a kernel", "summary": "We say that a measure `μ` is invariant with respect to a kernel `κ` if its push-forward along the kernel `μ.bind κ` is the same measure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Invariance.html"}, {"id": "Mathlib.Probability.Density", "region_id": "probability", "micro_elevation": 0.4, "macro_tier": 3, "macro_tier_score": 0.3189, "macro_tier_override": null, "x": -175.289, "z": 120.966, "size": 0.316, "title": "Probability density function", "summary": "This file defines the probability density function of random variables, by which we mean measurable functions taking values in a Borel space. The probability density function is defined as the Radon–Nikodym derivative of the law of `X`. In particular, a measurable function `f` is said to the probability density function of a random variable `X` if for all measurable sets `S`, `ℙ(X ∈ S) = ∫ x in S, f x dx`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Density.html"}, {"id": "Mathlib.Probability.Independence.Basic", "region_id": "probability", "micro_elevation": 0.36, "macro_tier": 3, "macro_tier_score": 0.4107, "macro_tier_override": null, "x": -178.196, "z": 125.246, "size": 0.3695, "title": "Independence of sets of sets and measure spaces (σ-algebras)", "summary": "* A family of sets of sets `π : ι → Set (Set Ω)` is independent with respect to a measure `μ` if for any finite set of indices `s = {i_1, ..., i_n}`, for any sets `f i_1 ∈ π i_1, ..., f i_n ∈ π i_n`, `μ (⋂ i in s, f i) = ∏ i ∈ s, μ (f i)`. It will be used for families of π-systems. * A family of measurable space structures (i.e. of σ-algebras) is independent with respect to a measure `μ` (typically defined on a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/Basic.html"}, {"id": "Mathlib.Probability.Kernel.RadonNikodym", "region_id": "probability", "micro_elevation": 0.28, "macro_tier": 2, "macro_tier_score": 0.0685, "macro_tier_override": null, "x": -181.909, "z": 112.628, "size": 0.2745, "title": "Radon-Nikodym derivative and Lebesgue decomposition for kernels", "summary": "Let `α` and `γ` be two measurable spaces, where either `α` is countable or `γ` is countably generated. Let `κ, η : Kernel α γ` be finite kernels. Then there exists a function `Kernel.rnDeriv κ η : α → γ → ℝ≥0∞` jointly measurable on `α × γ` and a kernel `Kernel.singularPart κ η : Kernel α γ` such that * `κ = Kernel.withDensity η (Kernel.rnDeriv κ η) + Kernel.singularPart κ η`, * for all `a : α`, `Kernel.singularPart…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/RadonNikodym.html"}, {"id": "Mathlib.Probability.Kernel.Disintegration.Density", "region_id": "probability", "micro_elevation": 0.24, "macro_tier": 2, "macro_tier_score": 0.2277, "macro_tier_override": null, "x": -192.02, "z": 124.801, "size": 0.2865, "title": "Kernel density", "summary": "Let `κ : Kernel α (γ × β)` and `ν : Kernel α γ` be two finite kernels with `Kernel.fst κ ≤ ν`, where `γ` has a countably generated σ-algebra (true in particular for standard Borel spaces). We build a function `density κ ν : α → γ → Set β → ℝ` jointly measurable in the first two arguments such that for all `a : α` and all measurable sets `s : Set β` and `A : Set γ`, `∫ x in A, density κ ν a x s ∂(ν a) = (κ a).real (A…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Disintegration/Density.html"}, {"id": "Mathlib.Probability.Kernel.WithDensity", "region_id": "probability", "micro_elevation": 0.12, "macro_tier": 1, "macro_tier_score": 0.0684, "macro_tier_override": null, "x": -183.79, "z": 118.312, "size": 0.268, "title": "With Density", "summary": "For an s-finite kernel `κ : Kernel α β` and a function `f : α → β → ℝ≥0∞` which is finite everywhere, we define `withDensity κ f` as the kernel `a ↦ (κ a).withDensity (f a)`. This is an s-finite kernel.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/WithDensity.html"}, {"id": "Mathlib.Probability.Kernel.SetIntegral", "region_id": "probability", "micro_elevation": 0.12, "macro_tier": 2, "macro_tier_score": 0.1819, "macro_tier_override": null, "x": -188.641, "z": 122.609, "size": 0.2445, "title": "Integral against a kernel over a set", "summary": "This file contains lemmas about the integral against a kernel and over a set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/SetIntegral.html"}, {"id": "Mathlib.Probability.Kernel.Integral", "region_id": "probability", "micro_elevation": 0.08, "macro_tier": 2, "macro_tier_score": 0.182, "macro_tier_override": null, "x": -189.718, "z": 118.714, "size": 0.2558, "title": "Bochner integrals of kernels", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Integral.html"}, {"id": "Mathlib.Probability.ProductMeasure", "region_id": "probability", "micro_elevation": 0.52, "macro_tier": 2, "macro_tier_score": 0.1368, "macro_tier_override": null, "x": -198.097, "z": 107.6, "size": 0.2876, "title": "Infinite product of probability measures", "summary": "This file provides a definition for the product measure of an arbitrary family of probability measures. Given `μ : (i : ι) → Measure (X i)` such that each `μ i` is a probability measure, `Measure.infinitePi μ` is the only probability measure `ν` over `Π i, X i` such that `ν (Set.pi s t) = ∏ i ∈ s, μ i (t i)`, with `s : Finset ι` and such that `∀ i ∈ s, MeasurableSet (t i)` (see `eq_infinitePi` and `infinitePi_pi`).…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/ProductMeasure.html"}, {"id": "Mathlib.Probability.Kernel.IonescuTulcea.Traj", "region_id": "probability", "micro_elevation": 0.48, "macro_tier": 2, "macro_tier_score": 0.1597, "macro_tier_override": null, "x": -197.216, "z": 108.443, "size": 0.305, "title": "Ionescu-Tulcea theorem", "summary": "This file proves the *Ionescu-Tulcea theorem*. The idea of the statement is as follows: consider a family of kernels `κ : (n : ℕ) → Kernel (Π i : Iic n, X i) (X (n + 1))`. One can interpret `κ n` as a kernel which takes as an input the trajectory of a point started in `X 0` and moving `X 0 → X 1 → X 2 → ... → X n` and which outputs the distribution of the next position of the point in `X (n + 1)`. If `a b : ℕ` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/IonescuTulcea/Traj.html"}, {"id": "Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic", "region_id": "probability", "micro_elevation": 0.92, "macro_tier": 1, "macro_tier_score": 0.0456, "macro_tier_override": null, "x": -176.907, "z": 93.215, "size": 0.2442, "title": "Gaussian processes", "summary": "This file contains basic properties of Gaussian processes. In particular, in `IsGaussianProcess.of_isGaussianProcess`, we show that if a stochastic process `Y : S → Ω → F` is such that for each `s : S`, `Y s` can be written as a linear map applied to finitely many values of a certain Gaussian process, then `Y` is itself a Gaussian process.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Gaussian/IsGaussianProcess/Basic.html"}, {"id": "Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Def", "region_id": "probability", "micro_elevation": 0.88, "macro_tier": 1, "macro_tier_score": 0.0684, "macro_tier_override": null, "x": -176.039, "z": 143.482, "size": 0.2634, "title": "Gaussian processes", "summary": "In this file we define a **Gaussian process** as a stochastic process whose finite dimensional distributions are Gaussian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Gaussian/IsGaussianProcess/Def.html"}, {"id": "Mathlib.Probability.Process.FiniteDimensionalLaws", "region_id": "probability", "micro_elevation": 0.64, "macro_tier": 1, "macro_tier_score": 0.0456, "macro_tier_override": null, "x": -168.833, "z": 113.108, "size": 0.2455, "title": "Finite-dimensional distributions of a stochastic process", "summary": "For a stochastic process `X : T → Ω → 𝓧` and a finite measure `P` on `Ω`, the law of the process is `P.map (fun ω ↦ (X · ω))`, and its finite-dimensional distributions are `P.map (fun ω ↦ I.restrict (X · ω))` for `I : Finset T`. We show that two stochastic processes have the same laws if and only if they have the same finite-dimensional distributions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Process/FiniteDimensionalLaws.html"}, {"id": "Mathlib.Probability.Martingale.OptionalSampling", "region_id": "probability", "micro_elevation": 0.16, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -185.898, "z": 123.852, "size": 0.2, "title": "Optional sampling theorem", "summary": "If `τ` is a bounded stopping time and `σ` is another stopping time, then the value of a martingale `f` at the stopping time `min τ σ` is almost everywhere equal to `μ[stoppedValue f τ | hσ.measurableSpace]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Martingale/OptionalSampling.html"}, {"id": "Mathlib.Probability.ProbabilityMassFunction.Monad", "region_id": "probability", "micro_elevation": 0.04, "macro_tier": 1, "macro_tier_score": 0.0456, "macro_tier_override": null, "x": -186.454, "z": 120.042, "size": 0.2587, "title": "Monad Operations for Probability Mass Functions", "summary": "This file constructs two operations on `PMF` that give it a monad structure. `pure a` is the distribution where a single value `a` has probability `1`. `bind pa pb : PMF β` is the distribution given by sampling `a : α` from `pa : PMF α`, and then sampling from `pb a : PMF β` to get a final result `b : β`. `bindOnSupport` generalizes `bind` to allow binding to a partial function, so that the second argument only…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/ProbabilityMassFunction/Monad.html"}, {"id": "Mathlib.Probability.ProbabilityMassFunction.Basic", "region_id": "probability", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0461, "macro_tier_override": null, "x": -187.333, "z": 119.199, "size": 0.3059, "title": "Probability mass functions", "summary": "This file is about probability mass functions or discrete probability measures: a function `α → ℝ≥0∞` such that the values have (infinite) sum `1`. Construction of monadic `pure` and `bind` is found in `Mathlib/Probability/ProbabilityMassFunction/Monad.lean`, other constructions of `PMF`s are found in `Mathlib/Probability/ProbabilityMassFunction/Constructions.lean`. Given `p : PMF α`, `PMF.toOuterMeasure` constructs…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/ProbabilityMassFunction/Basic.html"}, {"id": "Mathlib.Probability.CondVar", "region_id": "probability", "micro_elevation": 0.56, "macro_tier": 1, "macro_tier_score": 0.0458, "macro_tier_override": null, "x": -200.928, "z": 108.924, "size": 0.2807, "title": "Conditional variance", "summary": "This file defines the variance of a real-valued random variable conditional to a sigma-algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/CondVar.html"}, {"id": "Mathlib.Probability.Distributions.Bernoulli", "region_id": "probability", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0458, "macro_tier_override": null, "x": -187.333, "z": 119.199, "size": 0.2734, "title": "Bernoulli distribution", "summary": "We define the **Bernoulli distribution** over an arbitrary measurable space `X`. Given `x y : X` and `p : I` (`I` is the `unitInterval`), `Ber(x, y, p) := toNNReal p • dirac x + toNNReal (σ p) • dirac y`. It is the measure which gives mass `p` to `{x}` and `1 - p` to `{y}`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Bernoulli.html"}, {"id": "Mathlib.Probability.Distributions.SetBernoulli", "region_id": "probability", "micro_elevation": 0.6, "macro_tier": 1, "macro_tier_score": 0.0684, "macro_tier_override": null, "x": -176.129, "z": 104.781, "size": 0.2589, "title": "Product of bernoulli distributions on a set", "summary": "This file defines the product of bernoulli distributions on a set as a measure on sets. For a set `u : Set ι` and `p` between `0` and `1`, this is the measure on `Set ι` such that each `i ∈ u` belongs to the random set with probability `p`, and each `i ∉ u` doesn't belong to it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/SetBernoulli.html"}, {"id": "Mathlib.Probability.Kernel.Representation", "region_id": "probability", "micro_elevation": 0.32, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -196.986, "z": 120.482, "size": 0.2, "title": "Representation of kernels", "summary": "This file contains results about isolation of kernels randomness. In particular, it shows that, when the target space is a standard Borel space, any Markov kernel can be represented as the image of the uniform measure on `[0,1]` by a deterministic map. It corresponds to Lemma 4.22 in [Foundations of Modern Probability][kallenberg2021].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Representation.html"}, {"id": "Mathlib.Probability.CDF", "region_id": "probability", "micro_elevation": 0.28, "macro_tier": 1, "macro_tier_score": 0.0459, "macro_tier_override": null, "x": -185.726, "z": 127.568, "size": 0.2895, "title": "Cumulative distribution function of a real probability measure", "summary": "The cumulative distribution function (cdf) of a probability measure over `ℝ` is a monotone, right continuous function with limit 0 at -∞ and 1 at +∞, such that `cdf μ x = μ (Iic x)` for all `x : ℝ`. Two probability measures are equal if and only if they have the same cdf.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/CDF.html"}, {"id": "Mathlib.Probability.Independence.CharacteristicFunction", "region_id": "probability", "micro_elevation": 0.4, "macro_tier": 2, "macro_tier_score": 0.0911, "macro_tier_override": null, "x": -198.281, "z": 113.878, "size": 0.2613, "title": "Links between independence and characteristic function", "summary": "Two random variables are independent if and only if their joint characteristic function is equal to the product of the characteristic functions. More specifically, prove this in Hilbert spaces for two variables and a finite family of variables. We prove the analogous statements in Banach spaces, with an arbitrary Lp norm, for the dual characteristic function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/CharacteristicFunction.html"}, {"id": "Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence", "region_id": "probability", "micro_elevation": 0.92, "macro_tier": 1, "macro_tier_score": 0.0684, "macro_tier_override": null, "x": -190.933, "z": 146.965, "size": 0.2624, "title": "Independence of Gaussian random variables", "summary": "In this file we prove some results linking Gaussian random variables and independence. It is a well known fact that if `(X, Y)` is Gaussian, then `X` and `Y` are independent if their covariance is zero. We prove many versions of this theorem in different settings: in Banach spaces, Hilbert spaces, and for families of real random variables. We also prove that independent Gaussian random variables are jointly Gaussian.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Gaussian/HasGaussianLaw/Independence.html"}, {"id": "Mathlib.Probability.Distributions.Gaussian.CharFun", "region_id": "probability", "micro_elevation": 0.88, "macro_tier": 2, "macro_tier_score": 0.0687, "macro_tier_override": null, "x": -192.426, "z": 92.908, "size": 0.294, "title": "Facts about Gaussian characteristic function", "summary": "In this file we prove that Gaussian measures over a Banach space `E` are exactly those measures `μ` such that there exist `m : E` and `f : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ` positive semidefinite (satisfying `f.toBilinForm.IsPosSemidef`) such that `charFunDual μ L = exp (L m * I - f L L / 2)`. We also prove that such `m` and `f` are unique and equal to `∫ x, x ∂μ` and `covarianceBilinDual μ`. We also…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Gaussian/CharFun.html"}, {"id": "Mathlib.Probability.Kernel.Category.Stoch", "region_id": "probability", "micro_elevation": 0.32, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -190.038, "z": 128.554, "size": 0.2, "title": "Stoch", "summary": "The category of measurable spaces with Markov kernels is a positive Markov category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Category/Stoch.html"}, {"id": "Mathlib.Probability.Kernel.Category.SFinKer", "region_id": "probability", "micro_elevation": 0.28, "macro_tier": 0, "macro_tier_score": 0.0228, "macro_tier_override": null, "x": -178.824, "z": 119.666, "size": 0.239, "title": "SFinKer", "summary": "The category of measurable spaces with s-finite kernels is a copy-discard category.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Category/SFinKer.html"}, {"id": "Mathlib.Probability.CentralLimitTheorem", "region_id": "probability", "micro_elevation": 0.8, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -177.991, "z": 141.682, "size": 0.2, "title": "Central limit theorem", "summary": "We prove the central limit theorem in dimension 1.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/CentralLimitTheorem.html"}, {"id": "Mathlib.Probability.Distributions.Gaussian.Real", "region_id": "probability", "micro_elevation": 0.76, "macro_tier": 2, "macro_tier_score": 0.0923, "macro_tier_override": null, "x": -164.619, "z": 114.838, "size": 0.3571, "title": "Gaussian distributions over ℝ", "summary": "We define a Gaussian measure over the reals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Gaussian/Real.html"}, {"id": "Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic", "region_id": "probability", "micro_elevation": 0.4, "macro_tier": 1, "macro_tier_score": 0.0458, "macro_tier_override": null, "x": -199.331, "z": 121.253, "size": 0.2807, "title": "Stochastic processes with independent increments", "summary": "A stochastic process `X : T → Ω → E` has independent increments if for any `n ≥ 1` and `t₁ ≤ ... ≤ tₙ`, the random variables `X t₂ - X t₁, ..., X tₙ - X tₙ₋₁` are independent. Equivalently, for any monotone sequence `(tₙ)`, the random variables `(X tₙ₊₁ - X tₙ)` are independent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/Process/HasIndepIncrements/Basic.html"}, {"id": "Mathlib.Probability.Moments.SubGaussian", "region_id": "probability", "micro_elevation": 0.8, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -168.461, "z": 134.58, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Moments/SubGaussian.html"}, {"id": "Mathlib.Probability.Kernel.Condexp", "region_id": "probability", "micro_elevation": 0.48, "macro_tier": 1, "macro_tier_score": 0.0457, "macro_tier_override": null, "x": -173.411, "z": 114.777, "size": 0.2667, "title": "Kernel associated with a conditional expectation", "summary": "We define `condExpKernel μ m`, a kernel from `Ω` to `Ω` such that for all integrable functions `f`, `μ[f | m] =ᵐ[μ] fun ω => ∫ y, f y ∂(condExpKernel μ m ω)`. This kernel is defined if `Ω` is a standard Borel space. In general, `μ⟦s | m⟧` maps a measurable set `s` to a function `Ω → ℝ≥0∞`, and for all `s` that map is unique up to a `μ`-null set. For all `a`, the map from sets to `ℝ≥0∞` that we obtain that way…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Condexp.html"}, {"id": "Mathlib.Probability.Moments.Tilted", "region_id": "probability", "micro_elevation": 0.76, "macro_tier": 1, "macro_tier_score": 0.0229, "macro_tier_override": null, "x": -201.294, "z": 137.639, "size": 0.2478, "title": "Results relating `Measure.tilted` to `mgf` and `cgf`", "summary": "For a random variable `X : Ω → ℝ` and a measure `μ`, the tilted measure `μ.tilted (t * X ·)` is linked to the moment-generating function (`mgf`) and the cumulant-generating function (`cgf`) of `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Moments/Tilted.html"}, {"id": "Mathlib.Probability.Kernel.Disintegration.Basic", "region_id": "probability", "micro_elevation": 0.24, "macro_tier": 2, "macro_tier_score": 0.2274, "macro_tier_override": null, "x": -184.575, "z": 112.436, "size": 0.2535, "title": "Disintegration of measures and kernels", "summary": "This file defines predicates for a kernel to \"disintegrate\" a measure or a kernel. This kernel is also called the \"conditional kernel\" of the measure or kernel. A measure `ρ : Measure (α × Ω)` is disintegrated by a kernel `ρCond : Kernel α Ω` if `ρ.fst ⊗ₘ ρCond = ρ`. A kernel `ρ : Kernel α (β × Ω)` is disintegrated by a kernel `κCond : Kernel (α × β) Ω` if `κ.fst ⊗ₖ κCond = κ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Disintegration/Basic.html"}, {"id": "Mathlib.Probability.Kernel.Disintegration.CondCDF", "region_id": "probability", "micro_elevation": 0.24, "macro_tier": 3, "macro_tier_score": 0.2508, "macro_tier_override": null, "x": -192.681, "z": 114.226, "size": 0.3191, "title": "Conditional cumulative distribution function", "summary": "Given `ρ : Measure (α × ℝ)`, we define the conditional cumulative distribution function (conditional cdf) of `ρ`. It is a function `condCDF ρ : α → ℝ → ℝ` such that if `ρ` is a finite measure, then for all `a : α` `condCDF ρ a` is monotone and right-continuous with limit 0 at -∞ and limit 1 at +∞, and such that for all `x : ℝ`, `a ↦ condCDF ρ a x` is measurable. For all `x : ℝ` and measurable set `s`, that function…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Disintegration/CondCDF.html"}, {"id": "Mathlib.Probability.Kernel.Composition.Lemmas", "region_id": "probability", "micro_elevation": 0.32, "macro_tier": 3, "macro_tier_score": 0.2506, "macro_tier_override": null, "x": -186.501, "z": 109.497, "size": 0.3026, "title": "Lemmas relating different ways to compose measures and kernels", "summary": "This file contains lemmas about the composition of measures and kernels that do not fit in any of the other files in this directory, because they involve several types of compositions/products.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/Lemmas.html"}, {"id": "Mathlib.Probability.Decision.Risk.Countable", "region_id": "probability", "micro_elevation": 0.36, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -181.008, "z": 128.145, "size": 0.2, "title": "Risk in countable spaces", "summary": "In countable spaces, we can write integrals as sums, hence we can write the average or Bayes risk with sums instead of integrals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Decision/Risk/Countable.html"}, {"id": "Mathlib.Probability.Kernel.Irreducible", "region_id": "probability", "micro_elevation": 0.16, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -188.593, "z": 114.496, "size": 0.2, "title": "Irreducibility of kernels", "summary": "A kernel `κ : Kernel α α` is `φ`-irreducible, for a given measure `φ` on `α`, if for every measurable set `A` with positive measure under `φ`, and for every `a : α`, there exists a positive integer `n` such that we have `(κ ^ n) a A > 0`. When the kernel `κ` is the transition kernel of a Markov chain, this precisely means that the Markov chain is `φ`-irreducible, that is, there is a positive probability of reaching…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Irreducible.html"}, {"id": "Mathlib.Probability.Kernel.Composition.Comp", "region_id": "probability", "micro_elevation": 0.12, "macro_tier": 3, "macro_tier_score": 0.5018, "macro_tier_override": null, "x": -187.775, "z": 115.574, "size": 0.3782, "title": "Composition of kernels", "summary": "We define the composition `η ∘ₖ κ` of kernels `κ : Kernel α β` and `η : Kernel β γ`, which is a kernel from `α` to `γ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/Comp.html"}, {"id": "Mathlib.Probability.Kernel.Disintegration.Unique", "region_id": "probability", "micro_elevation": 0.4, "macro_tier": 2, "macro_tier_score": 0.2049, "macro_tier_override": null, "x": -196.179, "z": 110.838, "size": 0.2814, "title": "Uniqueness of the conditional kernel", "summary": "We prove that the conditional kernels `ProbabilityTheory.Kernel.condKernel` and `MeasureTheory.Measure.condKernel` are almost everywhere unique.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Disintegration/Unique.html"}, {"id": "Mathlib.Probability.Kernel.Disintegration.CDFToKernel", "region_id": "probability", "micro_elevation": 0.2, "macro_tier": 2, "macro_tier_score": 0.2505, "macro_tier_override": null, "x": -183.847, "z": 124.189, "size": 0.2909, "title": "Building a Markov kernel from a conditional cumulative distribution function", "summary": "Let `κ : Kernel α (β × ℝ)` and `ν : Kernel α β` be two finite kernels. A function `f : α × β → StieltjesFunction ℝ` is called a conditional kernel CDF of `κ` with respect to `ν` if it is measurable, tends to 0 at -∞ and to 1 at +∞ for all `p : α × β`, `fun b ↦ f (a, b) x` is `(ν a)`-integrable for all `a : α` and `x : ℝ` and for all measurable sets `s : Set β`, `∫ b in s, f (a, b) x ∂(ν a) = (κ a).real (s ×ˢ Iic…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Disintegration/CDFToKernel.html"}, {"id": "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "region_id": "probability", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": -187.333, "z": 119.199, "size": 0.262, "title": "Measurable parametric Stieltjes functions", "summary": "We provide tools to build a measurable function `α → StieltjesFunction ℝ` with limits 0 at -∞ and 1 at +∞ for all `a : α` from a measurable function `f : α → ℚ → ℝ`. These measurable parametric Stieltjes functions are cumulative distribution functions (CDF) of transition kernels. The reason for going through `ℚ` instead of defining directly a Stieltjes function is that since `ℚ` is countable, building a measurable…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.html"}, {"id": "Mathlib.Probability.Kernel.Composition.AbsolutelyContinuous", "region_id": "probability", "micro_elevation": 0.32, "macro_tier": 2, "macro_tier_score": 0.0686, "macro_tier_override": null, "x": -194.091, "z": 112.188, "size": 0.2881, "title": "Absolute continuity of the composition of measures and kernels", "summary": "This file contains some results about the absolute continuity of the composition of measures and kernels which use an assumption `CountableOrCountablyGenerated α β` on the measurable spaces. Results that hold without that assumption are in files about the definitions of compositions and products, like `Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/AbsolutelyContinuous.html"}, {"id": "Mathlib.Probability.Process.Kolmogorov", "region_id": "probability", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -187.333, "z": 119.199, "size": 0.2, "title": "Stochastic processes satisfying the Kolmogorov condition", "summary": "A stochastic process `X : T → Ω → E` on an index space `T` and a measurable space `Ω` with measure `P` is said to satisfy the Kolmogorov condition with exponents `p, q` and constant `M` if for all `s, t : T`, the pair `(X s, X t)` is measurable for the Borel sigma-algebra on `E × E` and the following condition holds: `∫⁻ ω, edist (X s ω) (X t ω) ^ p ∂P ≤ M * edist s t ^ q`. This condition is the main assumption of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Process/Kolmogorov.html"}, {"id": "Mathlib.Probability.Kernel.IonescuTulcea.Maps", "region_id": "probability", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1819, "macro_tier_override": null, "x": -187.333, "z": 119.199, "size": 0.2456, "title": "Auxiliary maps for Ionescu-Tulcea theorem", "summary": "This file contains auxiliary maps which are used to prove the Ionescu-Tulcea theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/IonescuTulcea/Maps.html"}, {"id": "Mathlib.Probability.Kernel.CondDistrib", "region_id": "probability", "micro_elevation": 0.44, "macro_tier": 2, "macro_tier_score": 0.2051, "macro_tier_override": null, "x": -176.253, "z": 111.68, "size": 0.3015, "title": "Regular conditional probability distribution", "summary": "We define the regular conditional probability distribution of `Y : α → Ω` given `X : α → β`, where `Ω` is a standard Borel space. This is a `Kernel β Ω` such that for almost all `a`, `condDistrib` evaluated at `X a` and a measurable set `s` is equal to the conditional expectation `μ⟦Y ⁻¹' s | mβ.comap X⟧` evaluated at `a`. `μ⟦Y ⁻¹' s | mβ.comap X⟧` maps a measurable set `s` to a function `α → ℝ≥0∞`, and for all `s`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/CondDistrib.html"}, {"id": "Mathlib.Probability.Distributions.Poisson.Basic", "region_id": "probability", "micro_elevation": 0.6, "macro_tier": 0, "macro_tier_score": 0.0228, "macro_tier_override": null, "x": -193.953, "z": 136.216, "size": 0.239, "title": "Poisson distributions over ℕ", "summary": "Define the Poisson measure over the natural numbers. For `r : ℝ≥0`, `poissonMeasure r` is the measure which to `{n}` associates `exp (-r) * r ^ n / (n)!`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Poisson/Basic.html"}, {"id": "Mathlib.Probability.HasLaw", "region_id": "probability", "micro_elevation": 0.56, "macro_tier": 3, "macro_tier_score": 0.2966, "macro_tier_override": null, "x": -188.457, "z": 136.204, "size": 0.3452, "title": "Law of a random variable", "summary": "We introduce a predicate `HasLaw X μ P` stating that the random variable `X` has law `μ` under the measure `P`. This is expressed as `P.map X = μ`. We also require `X` to be `P`-almost-everywhere measurable. Indeed, if `X` is not almost-everywhere measurable then `P.map X` is defined to be `0`, so that `HasLaw X 0 P` would be true. The measurability hypothesis ensures nice interactions with operations on the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/HasLaw.html"}, {"id": "Mathlib.Probability.Distributions.TwoValued", "region_id": "probability", "micro_elevation": 0.6, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -201.181, "z": 107.299, "size": 0.2, "title": "Distributions on two values", "summary": "This file proves a few lemmas about random variables that take at most two values.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/TwoValued.html"}, {"id": "Mathlib.Probability.Distributions.Gaussian.Fernique", "region_id": "probability", "micro_elevation": 0.84, "macro_tier": 2, "macro_tier_score": 0.0688, "macro_tier_override": null, "x": -199.49, "z": 96.712, "size": 0.3053, "title": "Fernique's theorem for Gaussian measures", "summary": "We show that the product of two identical Gaussian measures is invariant under rotation. We then deduce Fernique's theorem, which states that for a Gaussian measure `μ`, there exists `C > 0` such that the function `x ↦ exp (C * ‖x‖ ^ 2)` is integrable with respect to `μ`. As a consequence, a Gaussian measure has finite moments of all orders.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Gaussian/Fernique.html"}, {"id": "Mathlib.Probability.Distributions.Fernique", "region_id": "probability", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0686, "macro_tier_override": null, "x": -187.333, "z": 119.199, "size": 0.2834, "title": "Fernique's theorem for rotation-invariant measures", "summary": "Let `μ` be a finite measure on a second-countable normed space `E` such that the product measure `μ.prod μ` on `E × E` is invariant by rotation of angle `-π/4`. Then there exists a constant `C > 0` such that the function `x ↦ exp (C * ‖x‖ ^ 2)` is integrable with respect to `μ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Fernique.html"}, {"id": "Mathlib.Probability.Moments.Variance", "region_id": "probability", "micro_elevation": 0.52, "macro_tier": 3, "macro_tier_score": 0.3418, "macro_tier_override": null, "x": -173.101, "z": 126.119, "size": 0.3255, "title": "Variance of random variables", "summary": "We define the variance of a real-valued random variable as `Var[X] = 𝔼[(X - 𝔼[X])^2]` (in the `ProbabilityTheory` locale).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Moments/Variance.html"}, {"id": "Mathlib.Probability.Martingale.BorelCantelli", "region_id": "probability", "micro_elevation": 0.24, "macro_tier": 1, "macro_tier_score": 0.0229, "macro_tier_override": null, "x": -184.635, "z": 112.412, "size": 0.2478, "title": "Generalized Borel-Cantelli lemma", "summary": "This file proves Lévy's generalized Borel-Cantelli lemma which is a generalization of the Borel-Cantelli lemmas. With this generalization, one can easily deduce the Borel-Cantelli lemmas by choosing appropriate filtrations. This file also contains the one-sided martingale bound which is required to prove the generalized Borel-Cantelli. **Note**: the usual Borel-Cantelli lemmas are not in this file. See…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Martingale/BorelCantelli.html"}, {"id": "Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj", "region_id": "probability", "micro_elevation": 0.24, "macro_tier": 2, "macro_tier_score": 0.1819, "macro_tier_override": null, "x": -185.067, "z": 126.143, "size": 0.2445, "title": "Consecutive composition of kernels", "summary": "This file is the first step towards Ionescu-Tulcea theorem, which allows for instance to construct the product of an infinite family of probability measures. The idea of the statement is as follows: consider a family of kernels `κ : (n : ℕ) → Kernel (Π i : Iic n, X i) (X (n + 1))`. One can interpret `κ n` as a kernel which takes as an input the trajectory of a point started in `X 0` and moving `X 0 → X 1 → X 2 → ...…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/IonescuTulcea/PartialTraj.html"}, {"id": "Mathlib.Probability.BrownianMotion.Basic", "region_id": "probability", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -166.722, "z": 141.589, "size": 0.2, "title": "Brownian motion", "summary": "In this file we define two predicates over stochastic processes `X : ℝ≥0 → Ω → ℝ` given a probability measure `P : Measure Ω`. `IsPreBrownianReal X P` means that `X` is a pre-Brownian motion. It means that it has the law of the Brownian motion, namely that its finite dimensional distributions are given by `projectiveFamily`. Then `IsBrownianReal X P` means that `X` is a Brownian motion, which means that it is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/BrownianMotion/Basic.html"}, {"id": "Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence", "region_id": "probability", "micro_elevation": 0.96, "macro_tier": 0, "macro_tier_score": 0.0228, "macro_tier_override": null, "x": -209.132, "z": 99.75, "size": 0.239, "title": "Independence of Gaussian processes", "summary": "This file contains properties about independence of Gaussian processes. More precisely, we prove different versions of the following statement: if some stochastic processes are jointly Gaussian, then they are independent if their marginals are uncorrelated.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Gaussian/IsGaussianProcess/Independence.html"}, {"id": "Mathlib.Probability.Independence.Process.HasIndepIncrements.IsGaussianProcess", "region_id": "probability", "micro_elevation": 0.96, "macro_tier": 0, "macro_tier_score": 0.0228, "macro_tier_override": null, "x": -166.202, "z": 99.025, "size": 0.239, "title": "A stochastic process with independent increments and Gaussian marginals is Gaussian", "summary": "We prove that a stochastic process with independent increments and Gaussian marginals is Gaussian. To do so, we first define `I.orderEmbOfFinWithBot : Fin (#I + 1) → T`, which is the map `(⊥, t₁, ..., tₙ)`, where `t₁ < ... < tₙ` are the elements of `I` and `⊥` is the smallest element of `T`. Assume then that `X` is a stochastic process with independent increments and Gaussian marginals, and such that `X ⊥ = 0`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/Process/HasIndepIncrements/IsGaussianProcess.html"}, {"id": "Mathlib.Probability.Process.Filtration", "region_id": "probability", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.2974, "macro_tier_override": null, "x": -187.333, "z": 119.199, "size": 0.3893, "title": "Filtrations", "summary": "This file defines filtrations of a measurable space and σ-finite filtrations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Process/Filtration.html"}, {"id": "Mathlib.Probability.IdentDistribIndep", "region_id": "probability", "micro_elevation": 0.64, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -178.614, "z": 101.783, "size": 0.2, "title": "Results about identically distributed random variables and independence", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/IdentDistribIndep.html"}, {"id": "Mathlib.Probability.Independence.InfinitePi", "region_id": "probability", "micro_elevation": 0.6, "macro_tier": 1, "macro_tier_score": 0.0458, "macro_tier_override": null, "x": -189.241, "z": 137.359, "size": 0.2827, "title": "Independence of an infinite family of random variables", "summary": "In this file we provide several results about independence of arbitrary families of random variables, relying on `Measure.infinitePi`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/InfinitePi.html"}, {"id": "Mathlib.Probability.Independence.Process.HasIndepIncrements", "region_id": "probability", "micro_elevation": 0.44, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -194.228, "z": 130.678, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/Process/HasIndepIncrements.html"}, {"id": "Mathlib.Probability.Distributions.Poisson.PoissonLimitThm", "region_id": "probability", "micro_elevation": 0.68, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -207.703, "z": 115.553, "size": 0.2, "title": "Poisson limit of binomial probabilities", "summary": "This file proves a Poisson limit theorem. Fix `k : ℕ`. Assuming `n * p n → r` as `n → ∞`, we show `PMF.binomial (p n) (h n) n (Fin.ofNat (n + 1) k) → poissonPMF r k`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Poisson/PoissonLimitThm.html"}, {"id": "Mathlib.Probability.StrongLaw", "region_id": "probability", "micro_elevation": 0.64, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -170.721, "z": 129.368, "size": 0.2, "title": "The strong law of large numbers", "summary": "We prove the strong law of large numbers, in `ProbabilityTheory.strong_law_ae`: If `X n` is a sequence of independent identically distributed integrable random variables, then `∑ i ∈ range n, X i / n` converges almost surely to `𝔼[X 0]`. We give here the strong version, due to Etemadi, that only requires pairwise independence. This file also contains the Lᵖ version of the strong law of large numbers provided by…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/StrongLaw.html"}, {"id": "Mathlib.Probability.Independence.Process.Basic", "region_id": "probability", "micro_elevation": 0.4, "macro_tier": 1, "macro_tier_score": 0.0684, "macro_tier_override": null, "x": -197.395, "z": 112.35, "size": 0.2589, "title": "Independence of stochastic processes", "summary": "We prove that a stochastic process $(X_s)_{s \\in S}$ is independent from a random variable $Y$ if for all $s_1, ..., s_p \\in S$ the family $(X_{s_1}, ..., X_{s_p})$ is independent from $Y$. We prove that two stochastic processes $(X_s)_{s \\in S}$ and $(Y_t)_{t \\in T}$ are independent if for all $s_1, ..., s_p \\in S$ and $t_1, ..., t_q \\in T$ the two families $(X_{s_1}, ..., X_{s_p})$ and $(Y_{t_1}, ..., Y_{t_q})$…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/Process/Basic.html"}, {"id": "Mathlib.Probability.Distributions.Beta", "region_id": "probability", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -187.333, "z": 119.199, "size": 0.2, "title": "Beta distributions over ℝ", "summary": "Define the beta distribution over the reals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Beta.html"}, {"id": "Mathlib.Probability.ProbabilityMassFunction.Integrals", "region_id": "probability", "micro_elevation": 0.12, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -183.848, "z": 120.291, "size": 0.2, "title": "Integrals with a measure derived from probability mass functions.", "summary": "This file connects `PMF` with `integral`. The main result is that the integral (i.e. the expected value) with regard to a measure derived from a `PMF` is a sum weighted by the `PMF`. It also provides the expected value for specific probability mass functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/ProbabilityMassFunction/Integrals.html"}, {"id": "Mathlib.Probability.BorelCantelli", "region_id": "probability", "micro_elevation": 0.44, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -189.66, "z": 132.386, "size": 0.2, "title": "The second Borel-Cantelli lemma", "summary": "This file contains the *second Borel-Cantelli lemma* which states that, given a sequence of independent sets `(sₙ)` in a probability space, if `∑ n, μ sₙ = ∞`, then the limsup of `sₙ` has measure 1. We employ a proof using Lévy's generalized Borel-Cantelli by choosing an appropriate filtration.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/BorelCantelli.html"}, {"id": "Mathlib.Probability.ConditionalExpectation", "region_id": "probability", "micro_elevation": 0.4, "macro_tier": 1, "macro_tier_score": 0.0229, "macro_tier_override": null, "x": -190.949, "z": 130.823, "size": 0.2478, "title": "Probabilistic properties of the conditional expectation", "summary": "This file contains some properties about the conditional expectation which does not belong in the main conditional expectation file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/ConditionalExpectation.html"}, {"id": "Mathlib.Probability.Moments.CovarianceBilin", "region_id": "probability", "micro_elevation": 0.68, "macro_tier": 2, "macro_tier_score": 0.0685, "macro_tier_override": null, "x": -180.922, "z": 99.523, "size": 0.2775, "title": "Covariance in Hilbert spaces", "summary": "Given a measure `μ` defined over a Banach space `E`, one can consider the associated covariance bilinear form which maps `L₁ L₂ : StrongDual ℝ E` to `cov[L₁, L₂; μ]`. This is called `covarianceBilinDual μ` and is defined in the `CovarianceBilinDual` file. In the special case where `E` is a Hilbert space, each `L : StrongDual ℝ E` can be represented as the scalar product against some element of `E`. This motivates…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Moments/CovarianceBilin.html"}, {"id": "Mathlib.Probability.Kernel.Composition.CompMap", "region_id": "probability", "micro_elevation": 0.16, "macro_tier": 3, "macro_tier_score": 0.4777, "macro_tier_override": null, "x": -191.659, "z": 116.966, "size": 0.2865, "title": "Lemmas about compositions and maps of kernels", "summary": "This file contains results that use both the composition of kernels and the map of a kernel by a function. Map and comap are particular cases of composition: they correspond to composition with a deterministic kernel. See `deterministic_comp_eq_map` and `comp_deterministic_eq_comap`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/CompMap.html"}, {"id": "Mathlib.Probability.Kernel.Deterministic", "region_id": "probability", "micro_elevation": 0.24, "macro_tier": 1, "macro_tier_score": 0.0457, "macro_tier_override": null, "x": -188.334, "z": 126.434, "size": 0.2598, "title": "Class `IsDeterministic` of deterministic kernels", "summary": "This file defines the class `IsDeterministic` of deterministic kernels, and proves some properties about them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Deterministic.html"}, {"id": "Mathlib.Probability.Process.PartitionFiltration", "region_id": "probability", "micro_elevation": 0.04, "macro_tier": 2, "macro_tier_score": 0.2275, "macro_tier_override": null, "x": -186.319, "z": 118.525, "size": 0.2602, "title": "Filtration built from the finite partitions of a countably generated measurable space", "summary": "In a countably generated measurable space `α`, we can build a sequence of finer and finer finite measurable partitions of the space such that the measurable space is generated by the union of all partitions. This sequence of partitions is defined in `MeasureTheory.MeasurableSpace.CountablyGenerated`. Here, we build the filtration of the measurable spaces generated by `countablePartition α n` for all `n : ℕ`, which…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Process/PartitionFiltration.html"}, {"id": "Mathlib.Probability.Independence.BoundedContinuousFunction", "region_id": "probability", "micro_elevation": 0.44, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -200.594, "z": 121.054, "size": 0.2, "title": "Characterizing independence via bounded continuous functions", "summary": "Given two random variables `X : Ω → E` and `Y : Ω → F` such that `E` and `F` are Borel spaces satisfying `HasOuterApproxClosed`, `X` and `Y` are independent if for any real bounded continuous functions `f` and `g`, `∫ ω, f (X ω) * g (Y ω) ∂P = (∫ ω, f (X ω) ∂P) * (∫ ω, g (Y ω) ∂P)`. Consider now `X : (s : S) → Ω → E s`, with `Fintype S` and each `E s` being a Borel space satisfying `HasOuterApproxClosed`. Then to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/BoundedContinuousFunction.html"}, {"id": "Mathlib.Probability.Moments.CovarianceBilinDual", "region_id": "probability", "micro_elevation": 0.56, "macro_tier": 2, "macro_tier_score": 0.091, "macro_tier_override": null, "x": -177.49, "z": 105.287, "size": 0.2439, "title": "Covariance in Banach spaces", "summary": "We define the covariance of a finite measure in a Banach space `E`, as a continuous bilinear form on `Dual ℝ E`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Moments/CovarianceBilinDual.html"}, {"id": "Mathlib.Probability.Distributions.Geometric", "region_id": "probability", "micro_elevation": 0.04, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -188.517, "z": 119.479, "size": 0.2, "title": "Geometric distributions", "summary": "We define the geometric distributions over natural numbers. For `0 < p ≤ 1`, `geometricMeasure p` is the measure which to `{n}` associates `(1 - p) ^ n * n`. As the parameter `p` needs to lie between `0` and `1`, we define `geometricMeasure p` with `p : unitInterval`. Imagine a certain experience which has success probability `p`. If you repeat this experience infintely many times and independently, the number of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Geometric.html"}, {"id": "Mathlib.Probability.Distributions.Cauchy", "region_id": "probability", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -187.333, "z": 119.199, "size": 0.2, "title": "Cauchy Distribution over ℝ", "summary": "Define the Cauchy distribution with location parameter `x₀` and scale parameter `γ`. Note that we use \"location\" and \"scale\" to refer to these parameters in theorem names.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Cauchy.html"}, {"id": "Mathlib.Probability.Kernel.Posterior", "region_id": "probability", "micro_elevation": 0.4, "macro_tier": 1, "macro_tier_score": 0.0229, "macro_tier_override": null, "x": -176.435, "z": 113.775, "size": 0.2478, "title": "Posterior kernel", "summary": "For `μ : Measure Ω` (called prior measure), seen as a measure on a parameter, and a kernel `κ : Kernel Ω 𝓧` that gives the conditional distribution of \"data\" in `𝓧` given the prior parameter, we can get the distribution of the data with `κ ∘ₘ μ`, and the joint distribution of parameter and data with `μ ⊗ₘ κ : Measure (Ω × 𝓧)`. The posterior distribution of the parameter given the data is a Markov kernel `κ†μ :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Posterior.html"}, {"id": "Mathlib.Probability.Kernel.CompProdEqIff", "region_id": "probability", "micro_elevation": 0.36, "macro_tier": 1, "macro_tier_score": 0.0684, "macro_tier_override": null, "x": -185.306, "z": 108.433, "size": 0.2615, "title": "Condition for two kernels to be equal almost everywhere", "summary": "We prove that two finite kernels `κ, η : Kernel α β` are `μ`-a.e. equal for a finite measure `μ` iff the composition-products `μ ⊗ₘ κ` and `μ ⊗ₘ η` are equal. The result requires `α` to be countable or `β` to be a countably generated measurable space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/CompProdEqIff.html"}, {"id": "Mathlib.Probability.Decision.BayesEstimator", "region_id": "probability", "micro_elevation": 0.44, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -175.782, "z": 125.974, "size": 0.2, "title": "Bayes estimator", "summary": "Let `Θ` be a parameter space, `𝓧` a data space, `𝓨` a prediction space, `P : Kernel Θ 𝓧` a data generating kernel, `π` a prior on the parameter space, and `ℓ : Θ → 𝓨 → ℝ≥0∞` a loss function. An estimator (a `Kernel 𝓧 𝓨`) is said to be a Bayes estimator if it attains the Bayes risk for the estimation problem. It can be written as a measurable function `x ↦ argmin_y P†π(x)[θ ↦ ℓ θ y]` for `(P ∘ₘ π)`-almost every `x`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Decision/BayesEstimator.html"}, {"id": "Mathlib.Probability.Kernel.Proper", "region_id": "probability", "micro_elevation": 0.12, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -186.711, "z": 122.798, "size": 0.2, "title": "Proper kernels", "summary": "This file defines properness of measure kernels. For two σ-algebras `𝓑 ≤ 𝓧`, a `𝓑, 𝓧`-kernel `π : X → Measure X` is proper if `∫ x, g x * f x ∂(π x₀) = g x₀ * ∫ x, f x ∂(π x₀)` for all `x₀ : X`, `𝓧`-measurable function `f` and `𝓑`-measurable function `g`. By the standard machine, this is equivalent to having that, for all `B ∈ 𝓑`, `π` restricted to `B` is the same as `π` times the indicator of `B`. This should be…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Proper.html"}, {"id": "Mathlib.Probability.Kernel.Composition.CompNotation", "region_id": "probability", "micro_elevation": 0.08, "macro_tier": 3, "macro_tier_score": 0.4781, "macro_tier_override": null, "x": -186.442, "z": 116.933, "size": 0.3246, "title": "Notation for the composition of a measure and a kernel", "summary": "This operation, for which we introduce the notation `∘ₘ`, takes `μ : Measure α` and `κ : Kernel α β` and creates `κ ∘ₘ μ : Measure β`. The integral of a function against `κ ∘ₘ μ` is `∫⁻ x, f x ∂(κ ∘ₘ μ) = ∫⁻ a, ∫⁻ b, f b ∂(κ a) ∂μ`. This file does not define composition but only introduces notation for `MeasureTheory.Measure.bind μ κ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Composition/CompNotation.html"}, {"id": "Mathlib.Probability.Independence.Conditional", "region_id": "probability", "micro_elevation": 0.52, "macro_tier": 1, "macro_tier_score": 0.0229, "macro_tier_override": null, "x": -175.812, "z": 130.049, "size": 0.2478, "title": "Conditional Independence", "summary": "We define conditional independence of sets/σ-algebras/functions with respect to a σ-algebra. Two σ-algebras `m₁` and `m₂` are conditionally independent given a third σ-algebra `m'` if for all `m₁`-measurable sets `t₁` and `m₂`-measurable sets `t₂`, `μ⟦t₁ ∩ t₂ | m'⟧ =ᵐ[μ] μ⟦t₁ | m'⟧ * μ⟦t₂ | m'⟧`. On standard Borel spaces, the conditional expectation with respect to `m'` defines a kernel…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/Conditional.html"}, {"id": "Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs", "region_id": "probability", "micro_elevation": 0.64, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -202.524, "z": 131.389, "size": 0.2, "title": "Binomial random graphs", "summary": "This file constructs the binomial distribution with parameter `p` on simple graphs with vertices `V`. This is the law `G(V, p)` of binomial random graphs with probability `p`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Combinatorics/BinomialRandomGraph/Defs.html"}, {"id": "Mathlib.Probability.Independence.ZeroOne", "region_id": "probability", "micro_elevation": 0.56, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -203.564, "z": 114.007, "size": 0.2, "title": "Kolmogorov's 0-1 law", "summary": "Let `s : ι → MeasurableSpace Ω` be an independent sequence of sub-σ-algebras. Then any set which is measurable with respect to the tail σ-algebra `limsup s atTop` has probability 0 or 1.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Independence/ZeroOne.html"}, {"id": "Mathlib.Probability.Kernel.Defs", "region_id": "probability", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.5044, "macro_tier_override": null, "x": -187.333, "z": 119.199, "size": 0.4819, "title": "Markov Kernels", "summary": "A kernel from a measurable space `α` to another measurable space `β` is a measurable map `α → MeasureTheory.Measure β`, where the measurable space instance on `measure β` is the one defined in `MeasureTheory.Measure.instMeasurableSpace`. That is, a kernel `κ` verifies that for all measurable sets `s` of `β`, `a ↦ κ a s` is measurable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Kernel/Defs.html"}, {"id": "Mathlib.Probability.Distributions.Pareto", "region_id": "probability", "micro_elevation": 0.32, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -196.625, "z": 122.113, "size": 0.2, "title": "Pareto distributions over ℝ", "summary": "Define the Pareto measure over the reals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Pareto.html"}, {"id": "Mathlib.Probability.Moments.MGFAnalytic", "region_id": "probability", "micro_elevation": 0.72, "macro_tier": 2, "macro_tier_score": 0.1145, "macro_tier_override": null, "x": -168.962, "z": 131.142, "size": 0.3261, "title": "The moment-generating function is analytic", "summary": "The moment-generating function `mgf X μ` of a random variable `X` with respect to a measure `μ` is analytic on the interior of `integrableExpSet X μ`, the interval on which it is defined.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Moments/MGFAnalytic.html"}, {"id": "Mathlib.Probability.Process.LocalProperty", "region_id": "probability", "micro_elevation": 0.12, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -188.947, "z": 122.475, "size": 0.2, "title": "Local properties of processes", "summary": "This file defines local and stable properties of stochastic processes with respect to a filtration. This is notably useful for local martingales.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Process/LocalProperty.html"}, {"id": "Mathlib.Probability.Distributions.Uniform", "region_id": "probability", "micro_elevation": 0.44, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -197.157, "z": 128.297, "size": 0.2, "title": "Uniform distributions and probability mass functions", "summary": "This file defines two related notions of uniform distributions, which will be unified in the future.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Uniform.html"}, {"id": "Mathlib.Probability.HasLawExists", "region_id": "probability", "micro_elevation": 0.64, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -181.978, "z": 100.473, "size": 0.2, "title": "Existence of Random Variables", "summary": "This file contains lemmas that state the existence of random variables with given distributions and a given dependency structure (currently only mutual independence is considered).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/HasLawExists.html"}, {"id": "Mathlib.Probability.Distributions.Exponential", "region_id": "probability", "micro_elevation": 0.36, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -188.159, "z": 130.124, "size": 0.2, "title": "Exponential distributions over ℝ", "summary": "Define the Exponential measure over the reals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Exponential.html"}, {"id": "Mathlib.Probability.HasCondDistrib", "region_id": "probability", "micro_elevation": 0.6, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -169.266, "z": 116.551, "size": 0.2, "title": "A predicate for having a specified conditional distribution", "summary": "We introduce a predicate `HasCondDistrib Y X κ P` stating that the conditional distribution of `Y` given `X` under the measure `P` is equal to the kernel `κ`. The statement uses `HasLaw` to express that the law of the pair `(X, Y)` under `P` is equal to `(P.map X) ⊗ₘ κ`, the product of the law of `X` under `P` and the kernel `κ`. The use of `HasLaw` also implies that `Y` and `X` are a.e. measurable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/HasCondDistrib.html"}, {"id": "Mathlib.Probability.Distributions.Gamma", "region_id": "probability", "micro_elevation": 0.32, "macro_tier": 1, "macro_tier_score": 0.023, "macro_tier_override": null, "x": -178.527, "z": 123.358, "size": 0.2676, "title": "Gamma distributions over ℝ", "summary": "Define the gamma measure over the reals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Distributions/Gamma.html"}, {"id": "Mathlib.Order.Northcott", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 1, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -221.915, "z": 2.847, "size": 0.2985, "title": "Northcott Functions", "summary": "In number theory, the height function `h` satisfies the *Northcott property* that the sets `{a | h a ≤ b}` are finite. This file extracts this notion as a typeclass and provides some API.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Northcott.html"}, {"id": "Mathlib.Order.Disjointed", "region_id": "order", "micro_elevation": 0.9231, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": -235.424, "z": 18.564, "size": 0.3688, "title": "Making a sequence disjoint", "summary": "This file defines the way to make a sequence of sets - or, more generally, a map from a partially ordered type `ι` into a (generalized) Boolean algebra `α` - into a *pairwise disjoint* sequence with the same partial sups. For a sequence `f : ℕ → α`, this new sequence will be `f 0`, `f 1 \\ f 0`, `f 2 \\ (f 0 ⊔ f 1) ⋯`. It is actually unique, as `disjointed_unique` shows.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Disjointed.html"}, {"id": "Mathlib.Order.ConditionallyCompleteLattice.Indexed", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 3, "macro_tier_score": 0.164, "macro_tier_override": null, "x": -182.287, "z": 1.456, "size": 0.504, "title": "Indexed sup / inf in conditionally complete lattices", "summary": "This file proves lemmas about `iSup` and `iInf` for functions valued in a conditionally complete, rather than complete, lattice. We add a prefix `c` to distinguish them from the versions for complete lattices, giving names `ciSup_xxx` or `ciInf_xxx`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ConditionallyCompleteLattice/Indexed.html"}, {"id": "Mathlib.Order.CompleteLatticeIntervals", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 2, "macro_tier_score": 0.0197, "macro_tier_override": null, "x": -222.066, "z": 11.116, "size": 0.3345, "title": "Subtypes of conditionally complete linear orders", "summary": "In this file we give conditions on a subset of a conditionally complete linear order, to ensure that the subtype is itself conditionally complete. We check that an `OrdConnected` set satisfies these conditions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompleteLatticeIntervals.html"}, {"id": "Mathlib.Order.Interval.Finset.Nat", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 3, "macro_tier_score": 0.0709, "macro_tier_override": null, "x": -193.843, "z": -8.163, "size": 0.5144, "title": "Finite intervals of naturals", "summary": "This file proves that `ℕ` is a `LocallyFiniteOrder` and calculates the cardinality of its intervals as finsets and fintypes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Finset/Nat.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Finset", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 1, "macro_tier_score": 0.001, "macro_tier_override": null, "x": -214.647, "z": 59.91, "size": 0.331, "title": "`Filter.atTop` and `Filter.atBot` filters and finite sets.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Finset.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Basic", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 2, "macro_tier_score": 0.0217, "macro_tier_override": null, "x": -169.335, "z": 8.826, "size": 0.573, "title": "Basic results on `Filter.atTop` and `Filter.atBot` filters", "summary": "In this file we prove many lemmas like “if `f → +∞`, then `f ± c → +∞`”.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Basic.html"}, {"id": "Mathlib.Order.Filter.Finite", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 2, "macro_tier_score": 0.0155, "macro_tier_override": null, "x": -221.589, "z": 39.69, "size": 0.3632, "title": "Results relating filters to finiteness", "summary": "This file proves that finitely many conditions eventually hold if each of them eventually holds.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Finite.html"}, {"id": "Mathlib.Order.Interval.Finset.Defs", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 3, "macro_tier_score": 0.1017, "macro_tier_override": null, "x": -165.571, "z": 37.56, "size": 0.4525, "title": "Locally finite orders", "summary": "This file defines locally finite orders. A locally finite order is an order for which all bounded intervals are finite. This allows to make sense of `Icc`/`Ico`/`Ioc`/`Ioo` as lists, multisets, or finsets. Further, if the order is bounded above (resp. below), then we can also make sense of the \"unbounded\" intervals `Ici`/`Ioi` (resp. `Iic`/`Iio`). Many theorems about these intervals can be found in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Finset/Defs.html"}, {"id": "Mathlib.Order.ConditionallyCompleteLattice.Basic", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 3, "macro_tier_score": 0.2273, "macro_tier_override": null, "x": -188.031, "z": 1.43, "size": 0.671, "title": "Theory of conditionally complete lattices", "summary": "A conditionally complete lattice is a lattice in which every non-empty bounded subset `s` has a least upper bound and a greatest lower bound, denoted below by `sSup s` and `sInf s`. Typical examples are `ℝ`, `ℕ`, and `ℤ` with their usual orders. The theory is very comparable to the theory of complete lattices, except that suitable boundedness and nonemptiness assumptions have to be added to most statements. We…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ConditionallyCompleteLattice/Basic.html"}, {"id": "Mathlib.Order.OmegaCompletePartialOrder", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 2, "macro_tier_score": 0.0247, "macro_tier_override": null, "x": -209.834, "z": 58.7, "size": 0.3572, "title": "Omega Complete Partial Orders", "summary": "An omega-complete partial order is a partial order with a supremum operation on increasing sequences indexed by natural numbers (which we call `ωSup`). In this sense, it is strictly weaker than join complete semi-lattices as only ω-sized totally ordered sets have a supremum. The concept of an omega-complete partial order (ωCPO) is useful for the formalization of the semantics of programming languages. Its notion of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/OmegaCompletePartialOrder.html"}, {"id": "Mathlib.Order.BoundedOrder.Basic", "region_id": "order", "micro_elevation": 0.1923, "macro_tier": 3, "macro_tier_score": 0.4449, "macro_tier_override": null, "x": -185.97, "z": 30.762, "size": 0.7196, "title": "⊤ and ⊥, bounded lattices and variants", "summary": "This file defines top and bottom elements (greatest and least elements) of a type, the bounded variants of different kinds of lattices, sets up the typeclass hierarchy between them and provides instances for `Prop` and `fun`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/BoundedOrder/Basic.html"}, {"id": "Mathlib.Order.Lattice", "region_id": "order", "micro_elevation": 0.2308, "macro_tier": 3, "macro_tier_score": 0.4461, "macro_tier_override": null, "x": -184.298, "z": 29.271, "size": 0.7394, "title": "(Semi-)lattices", "summary": "Semilattices are partially ordered sets with join (least upper bound, or `sup`) or meet (greatest lower bound, or `inf`) operations. Lattices are posets that are both join-semilattices and meet-semilattices. Distributive lattices are lattices which satisfy any of four equivalent distributivity properties, of `sup` over `inf`, on the left or on the right.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Lattice.html"}, {"id": "Mathlib.Order.Lex", "region_id": "order", "micro_elevation": 0.1154, "macro_tier": 2, "macro_tier_score": 0.0177, "macro_tier_override": null, "x": -189.539, "z": 30.059, "size": 0.458, "title": "Type synonyms", "summary": "This file provides two type synonyms for order theory: * `Lex α`: Type synonym of `α` to equip it with its lexicographic order. The precise meaning depends on the type we take the lex of. Examples include `Prod`, `Sigma`, `List`, `Finset`. * `Colex α`: Type synonym of `α` to equip it with its colexicographic order. The precise meaning depends on the type we take the colex of. Examples include `Finset`, `DFinsupp`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Lex.html"}, {"id": "Mathlib.Order.CompleteLattice.Chain", "region_id": "order", "micro_elevation": 0.4615, "macro_tier": 3, "macro_tier_score": 0.0568, "macro_tier_override": null, "x": -173.764, "z": 28.095, "size": 0.3128, "title": "Hausdorff's maximality principle", "summary": "This file proves Hausdorff's maximality principle.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompleteLattice/Chain.html"}, {"id": "Mathlib.Order.Preorder.Chain", "region_id": "order", "micro_elevation": 0.4231, "macro_tier": 3, "macro_tier_score": 0.2312, "macro_tier_override": null, "x": -213.814, "z": 26.682, "size": 0.4007, "title": "Chains and flags", "summary": "This file defines chains for an arbitrary relation and flags for an order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Preorder/Chain.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Defs", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 2, "macro_tier_score": 0.0158, "macro_tier_override": null, "x": -222.991, "z": 28.678, "size": 0.3782, "title": "Definition of `Filter.atTop` and `Filter.atBot` filters", "summary": "In this file we define the filters * `Filter.atTop`: corresponds to `n → +∞`; * `Filter.atBot`: corresponds to `n → -∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Defs.html"}, {"id": "Mathlib.Order.BooleanAlgebra.Set", "region_id": "order", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.3358, "macro_tier_override": null, "x": -200.971, "z": 52.803, "size": 0.595, "title": "Boolean algebra of sets", "summary": "This file proves that `Set α` is a Boolean algebra, and proves results about set difference and complement.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/BooleanAlgebra/Set.html"}, {"id": "Mathlib.Order.SymmDiff", "region_id": "order", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.2117, "macro_tier_override": null, "x": -178.542, "z": 46.9, "size": 0.5304, "title": "Symmetric difference and bi-implication", "summary": "This file defines the symmetric difference and bi-implication operators in (co-)Heyting algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SymmDiff.html"}, {"id": "Mathlib.Order.ConditionallyCompletePartialOrder.Basic", "region_id": "order", "micro_elevation": 0.0769, "macro_tier": 3, "macro_tier_score": 0.2272, "macro_tier_override": null, "x": -193.973, "z": 27.25, "size": 0.57, "title": "Basic results on conditionally complete partial orders", "summary": "This file contains some basic results on conditionally complete partial orders, and is intended to parallel the API for conditionally complete lattices where possible. For the reason, the theorems here are mostly protected within the `DirectedOn` namespace, unless such an assumption is unnecessary. Otherwise the names here share the same names as their counterparts in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ConditionallyCompletePartialOrder/Basic.html"}, {"id": "Mathlib.Order.ConditionallyCompletePartialOrder.Defs", "region_id": "order", "micro_elevation": 0.0385, "macro_tier": 3, "macro_tier_score": 0.2279, "macro_tier_override": null, "x": -193.373, "z": 31.724, "size": 0.4533, "title": "Conditionally complete partial orders", "summary": "This file defines *conditionally complete partial orders* with suprema, infima or both. These are partial orders where every nonempty, upwards (downwards) directed set which is bounded above (below) has a least upper bound (greatest lower bound). This class extends `SupSet` (`InfSet`) and the requirement is that `sSup` (`sInf`) must be the least upper bound. The three classes defined herein are: +…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ConditionallyCompletePartialOrder/Defs.html"}, {"id": "Mathlib.Order.GameAdd", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2984, "title": "Game addition relation", "summary": "This file defines, given relations `rα : α → α → Prop` and `rβ : β → β → Prop`, a relation `Prod.GameAdd` on pairs, such that `GameAdd rα rβ x y` iff `x` can be reached from `y` by decreasing either entry (with respect to `rα` and `rβ`). It is so called since it models the subsequency relation on the addition of combinatorial games. We also define `Sym2.GameAdd`, which is the unordered pair analog of `Prod.GameAdd`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/GameAdd.html"}, {"id": "Mathlib.Order.Interval.Set.Disjoint", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 2, "macro_tier_score": 0.0202, "macro_tier_override": null, "x": -171.415, "z": 46.553, "size": 0.3662, "title": "Extra lemmas about intervals", "summary": "This file contains lemmas about intervals that cannot be included into `Mathlib/Order/Interval/Set/Basic.lean` because this would create an `import` cycle. Namely, lemmas in this file can use definitions from `Data.Set.Lattice`, including `Disjoint`. We consider various intersections and unions of half infinite intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/Disjoint.html"}, {"id": "Mathlib.Order.Interval.Set.OrdConnectedLinear", "region_id": "order", "micro_elevation": 0.9231, "macro_tier": 1, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -193.16, "z": -11.676, "size": 0.2921, "title": "Order-connected subsets of linear orders", "summary": "In this file we provide some results about order-connected subsets of linear orders, together with some convenience lemmas for characterising closed intervals in certain concrete types such as `ℤ`, `ℕ`, and `Fin n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/OrdConnectedLinear.html"}, {"id": "Mathlib.Order.Lattice.Nat", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 3, "macro_tier_score": 0.0562, "macro_tier_override": null, "x": -227.956, "z": 54.163, "size": 0.4925, "title": "Conditionally complete linear order structure on `ℕ`", "summary": "In this file we * define a `ConditionallyCompleteLinearOrderBot` structure on `ℕ`; * prove a few lemmas about `iSup`/`iInf`/`Set.iUnion`/`Set.iInter` and natural numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Lattice/Nat.html"}, {"id": "Mathlib.Order.Bounds.Defs", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.3823, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.591, "title": "Definitions about upper/lower bounds", "summary": "In this file we define: * `upperBounds`, `lowerBounds` : the set of upper bounds (resp., lower bounds) of a set; * `BddAbove s`, `BddBelow s` : the set `s` is bounded above (resp., below), i.e., the set of upper (resp., lower) bounds of `s` is nonempty; * `IsLeast s a`, `IsGreatest s a` : `a` is a least (resp., greatest) element of `s`; for a partial order, it is unique if exists; * `IsLUB s a`, `IsGLB s a` : `a` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Bounds/Defs.html"}, {"id": "Mathlib.Order.SuccPred.LinearLocallyFinite", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -214.29, "z": 64.304, "size": 0.3402, "title": "Linear locally finite orders", "summary": "We prove that a `LinearOrder` which is a `LocallyFiniteOrder` also verifies * `SuccOrder` * `PredOrder` * `IsSuccArchimedean` * `IsPredArchimedean` * `Countable` Furthermore, we show that there is an `OrderIso` between such an order and a subset of `ℤ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SuccPred/LinearLocallyFinite.html"}, {"id": "Mathlib.Order.Interval.Set.Union", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -203.391, "z": 57.609, "size": 0.2532, "title": "Extra lemmas about unions of intervals", "summary": "This file contains lemmas about finite unions of intervals which can't be included with the lemmas concerning infinite unions in `Mathlib/Order/Interval/Set/Disjoint.lean` because we use `Finset.range`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/Union.html"}, {"id": "Mathlib.Order.Category.Preord", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0102, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.3263, "title": "Category of preorders", "summary": "This defines `Preord`, the category of preorders with monotone maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/Preord.html"}, {"id": "Mathlib.Order.Filter.Germ.Basic", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 2, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -173.397, "z": 7.174, "size": 0.3123, "title": "Germ of a function at a filter", "summary": "The germ of a function `f : α → β` at a filter `l : Filter α` is the equivalence class of `f` with respect to the equivalence relation `EventuallyEq l`: `f ≈ g` means `∀ᶠ x in l, f x = g x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Germ/Basic.html"}, {"id": "Mathlib.Order.Filter.Ultrafilter.Defs", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 2, "macro_tier_score": 0.0061, "macro_tier_override": null, "x": -194.674, "z": 71.31, "size": 0.3592, "title": "Ultrafilters", "summary": "An ultrafilter is a minimal (maximal in the set order) proper filter. In this file we define * `Ultrafilter.of`: an ultrafilter that is less than or equal to a given filter; * `Ultrafilter`: subtype of ultrafilters; * `pure x : Ultrafilter α`: `pure x` as an `Ultrafilter`; * `Ultrafilter.map`, `Ultrafilter.bind`, `Ultrafilter.comap` : operations on ultrafilters;", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Ultrafilter/Defs.html"}, {"id": "Mathlib.Order.Interval.Finset.Basic", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 3, "macro_tier_score": 0.079, "macro_tier_override": null, "x": -229.377, "z": 23.439, "size": 0.4771, "title": "Intervals as finsets", "summary": "This file provides basic results about all the `Finset.Ixx`, which are defined in `Order.Interval.Finset.Defs`. In addition, it shows that in a locally finite order `≤` and `<` are the transitive closures of, respectively, `⩿` and `⋖`, which then leads to a characterization of monotone and strictly functions whose domain is a locally finite order. In particular, this file proves: * `le_iff_transGen_wcovBy`: `≤` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Finset/Basic.html"}, {"id": "Mathlib.Order.Filter.Cofinite", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 2, "macro_tier_score": 0.0217, "macro_tier_override": null, "x": -194.242, "z": 66.007, "size": 0.572, "title": "The cofinite filter", "summary": "In this file we define `Filter.cofinite`: the filter of sets with finite complement and prove its basic properties. In particular, we prove that for `ℕ` it is equal to `Filter.atTop`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Cofinite.html"}, {"id": "Mathlib.Order.Types.Defs", "region_id": "order", "micro_elevation": 0.3846, "macro_tier": 1, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -177.903, "z": 35.828, "size": 0.239, "title": "Order types", "summary": "Order types are defined as the quotient of linear orders under order isomorphism. They are preordered by order embeddings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Types/Defs.html"}, {"id": "Mathlib.Order.Hom.Basic", "region_id": "order", "micro_elevation": 0.3462, "macro_tier": 3, "macro_tier_score": 0.4151, "macro_tier_override": null, "x": -209.444, "z": 36.874, "size": 0.9008, "title": "Order homomorphisms", "summary": "This file defines order homomorphisms, which are bundled monotone functions. A preorder homomorphism `f : α →o β` is a function `α → β` along with a proof that `∀ x y, x ≤ y → f x ≤ f y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Hom/Basic.html"}, {"id": "Mathlib.Order.Monotone.Extension", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -203.277, "z": 61.328, "size": 0.2449, "title": "Extension of a monotone function from a set to the whole space", "summary": "In this file we prove that if a function is monotone and is bounded on a set `s`, then it admits a monotone extension to the whole space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Monotone/Extension.html"}, {"id": "Mathlib.Order.CompleteLattice.Basic", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 3, "macro_tier_score": 0.0686, "macro_tier_override": null, "x": -223.124, "z": 40.661, "size": 0.7563, "title": "Theory of complete lattices", "summary": "This file contains basic results on complete lattices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompleteLattice/Basic.html"}, {"id": "Mathlib.Order.Iterate", "region_id": "order", "micro_elevation": 0.2308, "macro_tier": 3, "macro_tier_score": 0.129, "macro_tier_override": null, "x": -203.931, "z": 25.304, "size": 0.4244, "title": "Inequalities on iterates", "summary": "In this file we prove some inequalities comparing `f^[n] x` and `g^[n] x` where `f` and `g` are two self-maps that commute with each other. Current selection of inequalities is motivated by formalization of the rotation number of a circle homeomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Iterate.html"}, {"id": "Mathlib.Order.Part", "region_id": "order", "micro_elevation": 0.3846, "macro_tier": 2, "macro_tier_score": 0.0282, "macro_tier_override": null, "x": -209.469, "z": 40.526, "size": 0.2606, "title": "Monotonicity of monadic operations on `Part`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Part.html"}, {"id": "Mathlib.Order.ScottContinuity", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 2, "macro_tier_score": 0.0333, "macro_tier_override": null, "x": -174.434, "z": 52.75, "size": 0.3065, "title": "Scott continuity", "summary": "A function `f : α → β` between preorders is Scott continuous (referring to Dana Scott) if it distributes over `IsLUB`. Scott continuity corresponds to continuity in Scott topological spaces (defined in `Mathlib/Topology/Order/ScottTopology.lean`). It is distinct from the (more commonly used) continuity from topology (see `Mathlib/Topology/Basic.lean`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ScottContinuity.html"}, {"id": "Mathlib.Order.LiminfLimsup", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 1, "macro_tier_score": 0.0009, "macro_tier_override": null, "x": -162.259, "z": 51.924, "size": 0.3241, "title": "liminfs and limsups of functions and filters", "summary": "Defines the liminf/limsup of a function taking values in a conditionally complete lattice, with respect to an arbitrary filter. We define `limsSup f` (`limsInf f`) where `f` is a filter taking values in a conditionally complete lattice. `limsSup f` is the smallest element `a` such that, eventually, `u ≤ a` (and vice versa for `limsInf f`). To work with the Limsup along a function `u` use `limsSup (map u f)`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/LiminfLimsup.html"}, {"id": "Mathlib.Order.WellFoundedSet", "region_id": "order", "micro_elevation": 1.0, "macro_tier": 1, "macro_tier_score": 0.0022, "macro_tier_override": null, "x": -210.38, "z": 73.887, "size": 0.4009, "title": "Well-founded sets", "summary": "This file introduces versions of `WellFounded` and `WellQuasiOrdered` for sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/WellFoundedSet.html"}, {"id": "Mathlib.Order.KrullDimension", "region_id": "order", "micro_elevation": 1.0, "macro_tier": 1, "macro_tier_score": 0.0028, "macro_tier_override": null, "x": -158.79, "z": 59.186, "size": 0.423, "title": "Krull dimension of a preordered set and height of an element", "summary": "If `α` is a preordered set, then `krullDim α : WithBot ℕ∞` is defined to be `sup {n | a₀ < a₁ < ... < aₙ}`. In case that `α` is empty, then its Krull dimension is defined to be negative infinity; if the length of all series `a₀ < a₁ < ... < aₙ` is unbounded, then its Krull dimension is defined to be positive infinity. For `a : α`, its height (in `ℕ∞`) is defined to be `sup {n | a₀ < a₁ < ... < aₙ ≤ a}`, while its…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/KrullDimension.html"}, {"id": "Mathlib.Order.Interval.Set.OrderEmbedding", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 3, "macro_tier_score": 0.194, "macro_tier_override": null, "x": -200.676, "z": 1.237, "size": 0.4069, "title": "Preimages of intervals under order embeddings", "summary": "In this file we prove that the preimage of an interval in the codomain under an `OrderEmbedding` is an interval in the domain. Note that similar statements about images require the range to be order-connected.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/OrderEmbedding.html"}, {"id": "Mathlib.Order.Interval.Set.UnorderedInterval", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 3, "macro_tier_score": 0.1968, "macro_tier_override": null, "x": -175.327, "z": 51.166, "size": 0.5064, "title": "Intervals without endpoints ordering", "summary": "In any lattice `α`, we define `uIcc a b` to be `Icc (a ⊓ b) (a ⊔ b)`, which in a linear order is the set of elements lying between `a` and `b`. `Icc a b` requires the assumption `a ≤ b` to be meaningful, which is sometimes inconvenient. The interval as defined in this file is always the set of things lying between `a` and `b`, regardless of the relative order of `a` and `b`. For real numbers, `uIcc a b` is the same…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/UnorderedInterval.html"}, {"id": "Mathlib.Order.NonemptyFiniteChains", "region_id": "order", "micro_elevation": 0.0769, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -194.372, "z": 27.178, "size": 0.239, "title": "Nonempty finite chains in a partially ordered type", "summary": "Given a partially ordered type `X`, we introduce the type `NonemptyFiniteChains` of nonempty finite chains in `X`, i.e. nonempty finite subsets `A` of `X` such that all the elements in `A` are comparable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/NonemptyFiniteChains.html"}, {"id": "Mathlib.Order.Filter.ENNReal", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2824, "title": "Limsup and liminf of reals", "summary": "This file compiles filter-related results about `ℝ`, `ℝ≥0` and `ℝ≥0∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/ENNReal.html"}, {"id": "Mathlib.Order.Closure", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.0582, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.3973, "title": "Closure operators between preorders", "summary": "We define (bundled) closure operators on a preorder as monotone (increasing), extensive (inflationary) and idempotent functions. We define closed elements for the operator as elements which are fixed by it. Lower adjoints to a function between preorders `u : β → α` allow to generalise closure operators to situations where the closure operator we are dealing with naturally decomposes as `u ∘ l` where `l` is a worthy…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Closure.html"}, {"id": "Mathlib.Order.MinMax", "region_id": "order", "micro_elevation": 0.2692, "macro_tier": 3, "macro_tier_score": 0.2145, "macro_tier_override": null, "x": -201.267, "z": 20.146, "size": 0.5994, "title": "`max` and `min`", "summary": "This file proves basic properties about maxima and minima on a `LinearOrder`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/MinMax.html"}, {"id": "Mathlib.Order.Interval.Set.OrderIso", "region_id": "order", "micro_elevation": 0.5769, "macro_tier": 3, "macro_tier_score": 0.0482, "macro_tier_override": null, "x": -219.075, "z": 20.064, "size": 0.3642, "title": "Lemmas about images of intervals under order isomorphisms.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/OrderIso.html"}, {"id": "Mathlib.Order.SetIsMax", "region_id": "order", "micro_elevation": 0.1923, "macro_tier": 1, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -187.779, "z": 25.328, "size": 0.2515, "title": "Maximal elements of subsets", "summary": "Let `S : Set J` and `m : S`. If `m` is not a maximal element of `S`, then `↑m : J` is not maximal in `J`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SetIsMax.html"}, {"id": "Mathlib.Order.Max", "region_id": "order", "micro_elevation": 0.1538, "macro_tier": 3, "macro_tier_score": 0.4482, "macro_tier_override": null, "x": -187.867, "z": 32.041, "size": 0.4779, "title": "Minimal/maximal and bottom/top elements", "summary": "This file defines predicates for elements to be minimal/maximal or bottom/top and typeclasses saying that there are no such elements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Max.html"}, {"id": "Mathlib.Order.CompleteBooleanAlgebra", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 2, "macro_tier_score": 0.0348, "macro_tier_override": null, "x": -172.13, "z": 55.43, "size": 0.9917, "title": "Frames, completely distributive lattices and complete Boolean algebras", "summary": "In this file we define and provide API for (co)frames, completely distributive lattices and complete Boolean algebras. We distinguish two different distributivity properties: 1. `inf_iSup_eq : (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i` (finite `⊓` distributes over infinite `⨆`). This is required by `Frame`, `CompleteDistribLattice`, and `CompleteBooleanAlgebra` (`Coframe`, etc., require the dual property). 2. `iInf_iSup_eq : (⨅…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompleteBooleanAlgebra.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Archimedean", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 1, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -231.625, "z": 26.223, "size": 0.2881, "title": "`Filter.atTop` filter and archimedean (semi)rings/fields", "summary": "In this file we prove that for a linear ordered archimedean semiring `R` and a function `f : α → ℕ`, the function `Nat.cast ∘ f : α → R` tends to `Filter.atTop` along a filter `l` if and only if so does `f`. We also prove that `Nat.cast : ℕ → R` tends to `Filter.atTop` along `Filter.atTop`, as well as version of these two results for `ℤ` (and a ring `R`) and `ℚ` (and a field `R`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Archimedean.html"}, {"id": "Mathlib.Order.Monotone.Monovary", "region_id": "order", "micro_elevation": 0.2692, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -186.297, "z": 21.709, "size": 0.2699, "title": "Monovariance of functions", "summary": "Two functions *vary together* if a strict change in the first implies a change in the second. This is in some sense a way to say that two functions `f : ι → α`, `g : ι → β` are \"monotone together\", without actually having an order on `ι`. This condition comes up in the rearrangement inequality. See `Algebra.Order.Rearrangement`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Monotone/Monovary.html"}, {"id": "Mathlib.Order.Interval.Set.Basic", "region_id": "order", "micro_elevation": 0.5385, "macro_tier": 3, "macro_tier_score": 0.3345, "macro_tier_override": null, "x": -217.59, "z": 21.082, "size": 0.5651, "title": "Intervals", "summary": "In any preorder, we define intervals (which on each side can be either infinite, open or closed) using the following naming conventions: - `i`: infinite - `o`: open - `c`: closed Each interval has the name `I` + letter for left side + letter for right side. For instance, `Ioc a b` denotes the interval `(a, b]`. The definitions can be found in `Mathlib/Order/Interval/Set/Defs.lean`. This file contains basic facts on…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/Basic.html"}, {"id": "Mathlib.Order.SuccPred.Limit", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 2, "macro_tier_score": 0.0266, "macro_tier_override": null, "x": -173.473, "z": 63.168, "size": 0.4423, "title": "Successor and predecessor limits", "summary": "We define the predicate `Order.IsSuccPrelimit` for \"successor pre-limits\", values that don't cover any others. They are so named since they can't be the successors of anything smaller. We define `Order.IsPredPrelimit` analogously, and prove basic results. For some applications, it is desirable to exclude minimal elements from being successor limits, or maximal elements from being predecessor limits. As such, we also…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SuccPred/Limit.html"}, {"id": "Mathlib.Order.CompleteLattice.Lemmas", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 2, "macro_tier_score": 0.0198, "macro_tier_override": null, "x": -226.537, "z": 28.764, "size": 0.7212, "title": "Theory of complete lattices", "summary": "This file contains results on complete lattices that need more theory to develop.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompleteLattice/Lemmas.html"}, {"id": "Mathlib.Order.GaloisConnection.Basic", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 2, "macro_tier_score": 0.0354, "macro_tier_override": null, "x": -225.425, "z": 39.232, "size": 0.7481, "title": "Galois connections, insertions and coinsertions", "summary": "This file contains basic results on Galois connections, insertions and coinsertions in various order structures, and provides constructions that lift order structures from one type to another.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/GaloisConnection/Basic.html"}, {"id": "Mathlib.Order.Atoms", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 3, "macro_tier_score": 0.0754, "macro_tier_override": null, "x": -230.764, "z": 21.596, "size": 0.5077, "title": "Atoms, Coatoms, and Simple Lattices", "summary": "This module defines atoms, which are minimal non-`⊥` elements in bounded lattices, simple lattices, which are lattices with only two elements, and related ideas.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Atoms.html"}, {"id": "Mathlib.Order.TypeTags", "region_id": "order", "micro_elevation": 0.0385, "macro_tier": 3, "macro_tier_score": 0.2128, "macro_tier_override": null, "x": -193.368, "z": 31.717, "size": 0.413, "title": "Order-related type synonyms", "summary": "In this file we define `WithBot` and `WithTop`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/TypeTags.html"}, {"id": "Mathlib.Order.Filter.CountablyGenerated", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 2, "macro_tier_score": 0.0157, "macro_tier_override": null, "x": -176.301, "z": 58.686, "size": 0.3725, "title": "Countably generated filters", "summary": "In this file we define a typeclass `Filter.IsCountablyGenerated` saying that a filter is generated by a countable family of sets. We also define predicates `Filter.IsCountableBasis` and `Filter.HasCountableBasis` saying that a specific family of sets is a countable basis.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/CountablyGenerated.html"}, {"id": "Mathlib.Order.Filter.Ker", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 2, "macro_tier_score": 0.0147, "macro_tier_override": null, "x": -174.42, "z": 52.736, "size": 0.311, "title": "Kernel of a filter", "summary": "In this file we define the *kernel* `Filter.ker f` of a filter `f` to be the intersection of all its sets. We also prove that `Filter.principal` and `Filter.ker` form a Galois coinsertion and prove other basic theorems about `Filter.ker`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Ker.html"}, {"id": "Mathlib.Order.Filter.Pi", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 2, "macro_tier_score": 0.0147, "macro_tier_override": null, "x": -210.744, "z": 1.153, "size": 0.311, "title": "(Co)product of a family of filters", "summary": "In this file we prove some basic properties of two filters on `Π i, α i`. * `Filter.pi (f : Π i, Filter (α i))` to be the maximal filter on `Π i, α i` such that `∀ i, Filter.Tendsto (Function.eval i) (Filter.pi f) (f i)`. It is defined as `Π i, Filter.comap (Function.eval i) (f i)`. This is a generalization of binary products to indexed products. * `Filter.coprodᵢ (f : Π i, Filter (α i))`: a generalization of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Pi.html"}, {"id": "Mathlib.Order.Filter.Prod", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 2, "macro_tier_score": 0.0203, "macro_tier_override": null, "x": -168.486, "z": 48.528, "size": 0.537, "title": "Product and coproduct filters", "summary": "In this file we prove some basic properties of `f ×ˢ g` and `Filter.coprod f g`. The product of two filters is the largest filter `l` such that `Filter.Tendsto Prod.fst l f` and `Filter.Tendsto Prod.snd l g`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Prod.html"}, {"id": "Mathlib.Order.Heyting.Boundary", "region_id": "order", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.0147, "macro_tier_override": null, "x": -207.249, "z": 49.981, "size": 0.311, "title": "Co-Heyting boundary", "summary": "The boundary of an element of a co-Heyting algebra is the intersection of its Heyting negation with itself. The boundary in the co-Heyting algebra of closed sets coincides with the topological boundary.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Heyting/Boundary.html"}, {"id": "Mathlib.Order.Fin.Finset", "region_id": "order", "micro_elevation": 0.4231, "macro_tier": 1, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -191.731, "z": 11.499, "size": 0.2945, "title": "Order isomorphisms from Fin to finsets", "summary": "We define order isomorphisms like `Fin.orderIsoTriple` from `Fin 3` to the finset `{a, b, c}` when `a < b` and `b < c`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Fin/Finset.html"}, {"id": "Mathlib.Order.Fin.SuccAboveOrderIso", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2945, "title": "The order isomorphism `Fin (n + 1) ≃o {i}ᶜ`", "summary": "Given `i : Fin (n + 2)`, we show that `Fin.succAboveOrderEmb` induces an order isomorphism `Fin (n + 1) ≃o ({i}ᶜ : Finset (Fin (n + 2)))`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Fin/SuccAboveOrderIso.html"}, {"id": "Mathlib.Order.Category.BddDistLat", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 1, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -181.766, "z": 57.736, "size": 0.2757, "title": "The category of bounded distributive lattices", "summary": "This defines `BddDistLat`, the category of bounded distributive lattices. Note that this category is sometimes called [`DistLat`](https://ncatlab.org/nlab/show/DistLat) when being a lattice is understood to entail having a bottom and a top element.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/BddDistLat.html"}, {"id": "Mathlib.Order.Category.BddLat", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 1, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -223.018, "z": 32.277, "size": 0.2836, "title": "The category of bounded lattices", "summary": "This file defines `BddLat`, the category of bounded lattices. In literature, this is sometimes called `Lat`, the category of lattices, because being a lattice is understood to entail having a bottom and a top element.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/BddLat.html"}, {"id": "Mathlib.Order.Category.DistLat", "region_id": "order", "micro_elevation": 0.4615, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -173.626, "z": 29.744, "size": 0.2686, "title": "The category of distributive lattices", "summary": "This file defines `DistLat`, the category of distributive lattices. Note that [`DistLat`](https://ncatlab.org/nlab/show/DistLat) in the literature doesn't always correspond to `DistLat` as we don't require bottom or top elements. Instead, this `DistLat` corresponds to `BddDistLat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/DistLat.html"}, {"id": "Mathlib.Order.Nat", "region_id": "order", "micro_elevation": 0.2308, "macro_tier": 2, "macro_tier_score": 0.0202, "macro_tier_override": null, "x": -189.401, "z": 39.804, "size": 0.5339, "title": "The natural numbers form a linear order", "summary": "This file contains the linear order instance on the natural numbers. See note [foundational algebra order theory].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Nat.html"}, {"id": "Mathlib.Order.BourbakiWitt", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 2, "macro_tier_score": 0.0111, "macro_tier_override": null, "x": -220.402, "z": 8.984, "size": 0.3774, "title": "Bourbaki-Witt Theorem", "summary": "This file proves the Bourbaki-Witt Theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/BourbakiWitt.html"}, {"id": "Mathlib.Order.Disjoint", "region_id": "order", "micro_elevation": 0.3077, "macro_tier": 3, "macro_tier_score": 0.4214, "macro_tier_override": null, "x": -203.882, "z": 41.512, "size": 0.7173, "title": "Disjointness and complements", "summary": "This file defines `Disjoint`, `Codisjoint`, and the `IsCompl` predicate.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Disjoint.html"}, {"id": "Mathlib.Order.BoundedOrder.Lattice", "region_id": "order", "micro_elevation": 0.2692, "macro_tier": 3, "macro_tier_score": 0.429, "macro_tier_override": null, "x": -195.55, "z": 43.026, "size": 0.7662, "title": "Bounded lattices", "summary": "This file contains miscellaneous lemmas about lattices with top or bottom elements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/BoundedOrder/Lattice.html"}, {"id": "Mathlib.Order.Partition.Equipartition", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 1, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -168.612, "z": 61.743, "size": 0.2945, "title": "Finite equipartitions", "summary": "This file defines finite equipartitions, the partitions whose parts all are the same size up to a difference of `1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Partition/Equipartition.html"}, {"id": "Mathlib.Order.Bounds.Basic", "region_id": "order", "micro_elevation": 0.5769, "macro_tier": 3, "macro_tier_score": 0.3317, "macro_tier_override": null, "x": -168.326, "z": 31.709, "size": 0.6078, "title": "Upper / lower bounds", "summary": "In this file we prove various lemmas about upper/lower bounds of a set: monotonicity, behaviour under `∪`, `∩`, `insert`, and provide formulas for `∅`, `univ`, and intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Bounds/Basic.html"}, {"id": "Mathlib.Order.Interval.Set.Image", "region_id": "order", "micro_elevation": 0.5769, "macro_tier": 3, "macro_tier_score": 0.1924, "macro_tier_override": null, "x": -205.549, "z": 6.467, "size": 0.3191, "title": "Monotone functions on intervals", "summary": "This file shows many variants of the fact that a monotone function `f` sends an interval with endpoints `a` and `b` to the interval with endpoints `f a` and `f b`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/Image.html"}, {"id": "Mathlib.Order.Interval.Set.LinearOrder", "region_id": "order", "micro_elevation": 0.5769, "macro_tier": 3, "macro_tier_score": 0.2074, "macro_tier_override": null, "x": -177.032, "z": 11.031, "size": 0.3775, "title": "Interval properties in linear orders", "summary": "Since every pair of elements are comparable in a linear order, intervals over them are better behaved. This file collects their properties under this assumption.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/LinearOrder.html"}, {"id": "Mathlib.Order.Category.CompleteLat", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -167.857, "z": 17.43, "size": 0.2, "title": "The category of complete lattices", "summary": "This file defines `CompleteLat`, the category of complete lattices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/CompleteLat.html"}, {"id": "Mathlib.Order.Hom.CompleteLattice", "region_id": "order", "micro_elevation": 0.5769, "macro_tier": 2, "macro_tier_score": 0.0373, "macro_tier_override": null, "x": -187.634, "z": 56.192, "size": 0.4876, "title": "Complete lattice homomorphisms", "summary": "This file defines frame homomorphisms and complete lattice homomorphisms. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Hom/CompleteLattice.html"}, {"id": "Mathlib.Order.Defs.PartialOrder", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.5069, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.5512, "title": "Orders", "summary": "Defines classes for preorders and partial orders and proves some basic lemmas about them. We also define covering relations on a preorder. We say that `b` *covers* `a` if `a < b` and there is no element in between. We say that `b` *weakly covers* `a` if `a ≤ b` and there is no element between `a` and `b`. In a partial order this is equivalent to `a ⋖ b ∨ a = b`, in a preorder this is equivalent to `a ⋖ b ∨ (a ≤ b ∧…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Defs/PartialOrder.html"}, {"id": "Mathlib.Order.Interval.Finset.Box", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2606, "title": "Decomposing a locally finite ordered ring into boxes", "summary": "This file proves that any locally finite ordered ring can be decomposed into \"boxes\", namely differences of consecutive intervals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Finset/Box.html"}, {"id": "Mathlib.Order.Directed", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.3559, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.5133, "title": "Directed indexed families and sets", "summary": "This file defines directed indexed families and directed sets. An indexed family/set is directed iff each pair of elements has a shared upper bound.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Directed.html"}, {"id": "Mathlib.Order.SetNotation", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.3202, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.66, "title": "Notation classes for set supremum and infimum", "summary": "In this file we introduce notation for indexed suprema, infima, unions, and intersections.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SetNotation.html"}, {"id": "Mathlib.Order.Basic", "region_id": "order", "micro_elevation": 0.0769, "macro_tier": 3, "macro_tier_score": 0.5028, "macro_tier_override": null, "x": -192.872, "z": 33.644, "size": 0.7504, "title": "Basic definitions about `≤` and `<`", "summary": "This file proves basic results about orders, provides extensive dot notation, defines useful order classes and allows to transfer order instances.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Basic.html"}, {"id": "Mathlib.Order.Defs.LinearOrder", "region_id": "order", "micro_elevation": 0.0385, "macro_tier": 3, "macro_tier_score": 0.505, "macro_tier_override": null, "x": -193.91, "z": 32.209, "size": 0.6163, "title": "Orders", "summary": "Defines classes for linear orders and proves some basic lemmas about them. We intentionally avoid using `grind` in this fundamental file to keep the proofs understandable, rather than hiding the reasoning behind automation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Defs/LinearOrder.html"}, {"id": "Mathlib.Order.Defs.Prop", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.4928, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.3947, "title": "Order definitions for propositions", "summary": "This file defines orders on `Pi` and `Prop`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Defs/Prop.html"}, {"id": "Mathlib.Order.Notation", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.5069, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.5513, "title": "Notation classes for lattice operations", "summary": "In this file we introduce typeclasses and definitions for lattice operations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Notation.html"}, {"id": "Mathlib.Order.ModularLattice", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 3, "macro_tier_score": 0.0761, "macro_tier_override": null, "x": -159.503, "z": 29.246, "size": 0.3544, "title": "Modular Lattices", "summary": "This file defines (semi)modular lattices, a kind of lattice useful in algebra. For examples, look to the subobject lattices of abelian groups, submodules, and ideals, or consider any distributive lattice.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ModularLattice.html"}, {"id": "Mathlib.Order.SuccPred.Basic", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 3, "macro_tier_score": 0.107, "macro_tier_override": null, "x": -205.41, "z": -3.012, "size": 0.5995, "title": "Successor and predecessor", "summary": "This file defines successor and predecessor orders. `succ a`, the successor of an element `a : α` is the least element greater than `a`. `pred a` is the greatest element less than `a`. Typical examples include `ℕ`, `ℤ`, `ℕ+`, `Fin n`, but also `ENat`, the lexicographic order of a successor/predecessor order...", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SuccPred/Basic.html"}, {"id": "Mathlib.Order.BooleanGenerators", "region_id": "order", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -177.262, "z": -11.76, "size": 0.2721, "title": "Generators for Boolean algebras", "summary": "In this file, we provide an alternative constructor for Boolean algebras. A set of *Boolean generators* in a compactly generated complete lattice is a subset `S` such that * the elements of `S` are all atoms, and * the set `S` satisfies an atomicity condition: any compact element below the supremum of a subset `s` of generators is equal to the supremum of a subset of `s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/BooleanGenerators.html"}, {"id": "Mathlib.Order.CompactlyGenerated.Basic", "region_id": "order", "micro_elevation": 0.9615, "macro_tier": 2, "macro_tier_score": 0.0126, "macro_tier_override": null, "x": -188.595, "z": -13.036, "size": 0.4426, "title": "Compactness properties for complete lattices", "summary": "For complete lattices, there are numerous equivalent ways to express the fact that the relation `>` is well-founded. In this file we define three especially-useful characterisations and provide proofs that they are indeed equivalent to well-foundedness.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompactlyGenerated/Basic.html"}, {"id": "Mathlib.Order.Antisymmetrization", "region_id": "order", "micro_elevation": 0.3846, "macro_tier": 3, "macro_tier_score": 0.329, "macro_tier_override": null, "x": -184.681, "z": 45.162, "size": 0.3848, "title": "Turning a preorder into a partial order", "summary": "This file allows to make a preorder into a partial order by quotienting out the elements `a`, `b` such that `a ≤ b` and `b ≤ a`. `Antisymmetrization` is a functor from `Preorder` to `PartialOrder`. See `Preorder_to_PartialOrder`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Antisymmetrization.html"}, {"id": "Mathlib.Order.BoundedOrder.Monotone", "region_id": "order", "micro_elevation": 0.2308, "macro_tier": 3, "macro_tier_score": 0.3241, "macro_tier_override": null, "x": -205.369, "z": 31.488, "size": 0.372, "title": "Monotone functions on bounded orders", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/BoundedOrder/Monotone.html"}, {"id": "Mathlib.Order.Interval.Set.ProjIcc", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 1, "macro_tier_score": 0.003, "macro_tier_override": null, "x": -221.104, "z": 51.529, "size": 0.4342, "title": "Projection of a line onto a closed interval", "summary": "Given a linearly ordered type `α`, in this file we define * `Set.projIci (a : α)` to be the map `α → [a, ∞)` sending `(-∞, a]` to `a`, and each point `x ∈ [a, ∞)` to itself; * `Set.projIic (b : α)` to be the map `α → (-∞, b[` sending `[b, ∞)` to `b`, and each point `x ∈ (-∞, b]` to itself; * `Set.projIcc (a b : α) (h : a ≤ b)` to be the map `α → [a, b]` sending `(-∞, a]` to `a`, `[b, ∞)` to `b`, and each point `x ∈…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/ProjIcc.html"}, {"id": "Mathlib.Order.Interval.Set.OrdConnected", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 3, "macro_tier_score": 0.1836, "macro_tier_override": null, "x": -165.226, "z": 19.013, "size": 0.5302, "title": "Order-connected sets", "summary": "We say that a set `s : Set α` is `OrdConnected` if for all `x y ∈ s` it includes the interval `[[x, y]]`. If `α` is a `DenselyOrdered` `ConditionallyCompleteLinearOrder` with the `OrderTopology`, then this condition is equivalent to `IsPreconnected s`. If `α` is a linearly ordered field, then this condition is also equivalent to `Convex α s`. In this file we prove that intersection of a family of `OrdConnected` sets…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/OrdConnected.html"}, {"id": "Mathlib.Order.Irreducible", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2658, "title": "Irreducible and prime elements in an order", "summary": "This file defines irreducible and prime elements in an order and shows that in a well-founded lattice every element decomposes as a supremum of irreducible elements. An element is sup-irreducible (resp. inf-irreducible) if it isn't `⊥` and can't be written as the supremum of any strictly smaller elements. An element is sup-prime (resp. inf-prime) if it isn't `⊥` and is greater than the supremum of any two elements…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Irreducible.html"}, {"id": "Mathlib.Order.Filter.Curry", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 2, "macro_tier_score": 0.0091, "macro_tier_override": null, "x": -202.836, "z": 63.269, "size": 0.4836, "title": "Curried Filters", "summary": "This file provides an operation (`Filter.curry`) on filters which provides the equivalence `∀ᶠ a in l, ∀ᶠ b in l', p (a, b) ↔ ∀ᶠ c in (l.curry l'), p c` (see `Filter.eventually_curry_iff`). To understand when this operation might arise, it is helpful to think of `∀ᶠ` as a combination of the quantifiers `∃ ∀`. For instance, `∀ᶠ n in atTop, p n ↔ ∃ N, ∀ n ≥ N, p n`. A curried filter yields the quantifier order `∃ ∀ ∃…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Curry.html"}, {"id": "Mathlib.Order.Interval.Set.Monotone", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 1, "macro_tier_score": 0.001, "macro_tier_override": null, "x": -156.576, "z": 23.691, "size": 0.337, "title": "Monotonicity on intervals", "summary": "In this file we prove that `Set.Ici` etc. are monotone/antitone functions. We also prove some lemmas about functions monotone on intervals in `SuccOrder`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/Monotone.html"}, {"id": "Mathlib.Order.Hom.Set", "region_id": "order", "micro_elevation": 0.3846, "macro_tier": 3, "macro_tier_score": 0.2768, "macro_tier_override": null, "x": -199.449, "z": 47.726, "size": 0.523, "title": "Order homomorphisms and sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Hom/Set.html"}, {"id": "Mathlib.Order.Interval.Set.Defs", "region_id": "order", "micro_elevation": 0.0385, "macro_tier": 3, "macro_tier_score": 0.363, "macro_tier_override": null, "x": -194.091, "z": 32.302, "size": 0.5766, "title": "Intervals", "summary": "In any preorder `α`, we define intervals (which on each side can be either infinite, open, or closed) using the following naming conventions: - `i`: infinite - `o`: open - `c`: closed Each interval has the name `I` + letter for left side + letter for right side. For instance, `Ioc a b` denotes the interval `(a, b]`. We also define a typeclass `Set.OrdConnected` saying that a set includes `Set.Icc a b` whenever it…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/Defs.html"}, {"id": "Mathlib.Order.WellFounded", "region_id": "order", "micro_elevation": 0.0385, "macro_tier": 3, "macro_tier_score": 0.3507, "macro_tier_override": null, "x": -196.272, "z": 29.706, "size": 0.4973, "title": "Well-founded relations", "summary": "A relation is well-founded if it can be used for induction: for each `x`, `(∀ y, r y x → P y) → P x` implies `P x`. Well-founded relations can be used for induction and recursion, including construction of fixed points in the space of dependent functions `Π x : α, β x`. The predicate `WellFounded` is defined in the core library. In this file we prove some extra lemmas and provide a few new definitions:…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/WellFounded.html"}, {"id": "Mathlib.Order.JordanHolder", "region_id": "order", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -149.048, "z": 26.709, "size": 0.249, "title": "Jordan-Hölder Theorem", "summary": "This file proves the Jordan Hölder theorem for a `JordanHolderLattice`, a class also defined in this file. Examples of `JordanHolderLattice` include `Subgroup G` if `G` is a group, and `Submodule R M` if `M` is an `R`-module. Using this approach the theorem need not be proved separately for both groups and modules, the proof in this file can be applied to both.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/JordanHolder.html"}, {"id": "Mathlib.Order.RelSeries", "region_id": "order", "micro_elevation": 0.9615, "macro_tier": 2, "macro_tier_score": 0.01, "macro_tier_override": null, "x": -153.978, "z": 47.52, "size": 0.3076, "title": "Series of a relation", "summary": "If `r` is a relation on `α` then a relation series of length `n` is a series `a_0, a_1, ..., a_n` such that `r a_i a_{i+1}` for all `i < n`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/RelSeries.html"}, {"id": "Mathlib.Order.Hom.Order", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -216.322, "z": 56.436, "size": 0.2578, "title": "Lattice structure on order homomorphisms", "summary": "This file defines the lattice structure on order homomorphisms, which are bundled monotone functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Hom/Order.html"}, {"id": "Mathlib.Order.Fin.Basic", "region_id": "order", "micro_elevation": 0.3846, "macro_tier": 2, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -211.831, "z": 35.373, "size": 0.581, "title": "`Fin n` forms a bounded linear order", "summary": "This file contains the linear ordered instance on `Fin n`. `Fin n` is the type whose elements are natural numbers smaller than `n`. This file expands on the development in the core library.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Fin/Basic.html"}, {"id": "Mathlib.Order.Cover", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 3, "macro_tier_score": 0.1632, "macro_tier_override": null, "x": -227.533, "z": 38.097, "size": 0.4798, "title": "The covering relation", "summary": "This file proves properties of the covering relation in an order. We say that `b` *covers* `a` if `a < b` and there is no element in between. We say that `b` *weakly covers* `a` if `a ≤ b` and there is no element between `a` and `b`. In a partial order this is equivalent to `a ⋖ b ∨ a = b`, in a preorder this is equivalent to `a ⋖ b ∨ (a ≤ b ∧ b ≤ a)`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Cover.html"}, {"id": "Mathlib.Order.Preorder.Finite", "region_id": "order", "micro_elevation": 0.4231, "macro_tier": 3, "macro_tier_score": 0.0882, "macro_tier_override": null, "x": -214.177, "z": 32.133, "size": 0.4715, "title": "Finite preorders and finite sets in a preorder", "summary": "This file shows that non-empty finite sets in a preorder have minimal/maximal elements, and contrapositively that non-empty sets without minimal or maximal elements are infinite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Preorder/Finite.html"}, {"id": "Mathlib.Order.Monotone.Odd", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -224.093, "z": 37.292, "size": 0.2655, "title": "Monotonicity of odd functions", "summary": "An odd function on a linear ordered additive commutative group `G` is monotone on the whole group provided that it is monotone on `Set.Ici 0`, see `monotone_of_odd_of_monotoneOn_nonneg`. We also prove versions of this lemma for `Antitone`, `StrictMono`, and `StrictAnti`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Monotone/Odd.html"}, {"id": "Mathlib.Order.Monotone.Union", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 2, "macro_tier_score": 0.0095, "macro_tier_override": null, "x": -179.57, "z": 6.879, "size": 0.2522, "title": "Monotonicity on intervals", "summary": "In this file we prove that a function is (strictly) monotone (or antitone) on a linear order `α` provided that it is (strictly) monotone on `(-∞, a]` and on `[a, +∞)`. This is a special case of a more general statement where one deduces monotonicity on a union from monotonicity on each set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Monotone/Union.html"}, {"id": "Mathlib.Order.Filter.Cocardinal", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -231.538, "z": 43.352, "size": 0.2, "title": "The cocardinal filter", "summary": "In this file we define `Filter.cocardinal hc`: the filter of sets with cardinality less than a regular cardinal `c` that satisfies `Cardinal.aleph0 < c`. Such filters are `CardinalInterFilter` with cardinality `c`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Cocardinal.html"}, {"id": "Mathlib.Order.Filter.CardinalInter", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 1, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -218.942, "z": 2.522, "size": 0.2478, "title": "Filters with a cardinal intersection property", "summary": "In this file we define `CardinalInterFilter l c` to be the class of filters with the following property: for any collection of sets `s ∈ l` with cardinality strictly less than `c`, their intersection belongs to `l` as well.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/CardinalInter.html"}, {"id": "Mathlib.Order.Restriction", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -174.924, "z": -0.634, "size": 0.3161, "title": "Restriction of a function indexed by a preorder", "summary": "Given a preorder `α` and dependent function `f : (i : α) → π i` and `a : α`, one might want to consider the restriction of `f` to elements `≤ a`. This is defined in this file as `Preorder.restrictLe a f`. Similarly, if we have `a b : α`, `hab : a ≤ b` and `f : (i : ↑(Set.Iic b)) → π ↑i`, one might want to restrict it to elements `≤ a`. This is defined in this file as `Preorder.restrictLe₂ hab f`. We also provide…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Restriction.html"}, {"id": "Mathlib.Order.PartialSups", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 2, "macro_tier_score": 0.0158, "macro_tier_override": null, "x": -160.059, "z": 51.744, "size": 0.3795, "title": "The monotone sequence of partial supremums of a sequence", "summary": "For `ι` a preorder in which all bounded-above intervals are finite (such as `ℕ`), and `α` a `⊔`-semilattice, we define `partialSups : (ι → α) → ι →o α` by the formula `partialSups f i = (Finset.Iic i).sup' ⋯ f`, where the `⋯` denotes a proof that `Finset.Iic i` is nonempty. This is a way of spelling `⊔ k ≤ i, f k` which does not require a `α` to have a bottom element, and makes sense in conditionally-complete…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/PartialSups.html"}, {"id": "Mathlib.Order.Filter.CountableSeparatingOn", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 1, "macro_tier_score": 0.0016, "macro_tier_override": null, "x": -221.629, "z": 56.301, "size": 0.3681, "title": "Filters with countable intersections and countable separating families", "summary": "In this file we prove some facts about a filter with countable intersections property on a type with a countable family of sets that separates points of the space. The main use case is the `MeasureTheory.ae` filter and a space with countably generated σ-algebra but lemmas apply, e.g., to the `residual` filter and a T₀ topological space with second countable topology. To avoid repetition of lemmas for different…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/CountableSeparatingOn.html"}, {"id": "Mathlib.Order.Bounds.OrderIso", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 2, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -169.152, "z": 46.299, "size": 0.3143, "title": "Order isomorphisms and bounds.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Bounds/OrderIso.html"}, {"id": "Mathlib.Order.PropInstances", "region_id": "order", "micro_elevation": 0.3462, "macro_tier": 3, "macro_tier_score": 0.3561, "macro_tier_override": null, "x": -181.501, "z": 39.392, "size": 0.6307, "title": "The order on `Prop`", "summary": "Instances on `Prop` such as `DistribLattice`, `BoundedOrder`, `LinearOrder`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/PropInstances.html"}, {"id": "Mathlib.Order.Radical", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -225.558, "z": 54.435, "size": 0.2478, "title": "The radical of a lattice", "summary": "This file contains results on the order radical of a lattice: the infimum of the coatoms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Radical.html"}, {"id": "Mathlib.Order.Hom.WithTopBot", "region_id": "order", "micro_elevation": 0.5769, "macro_tier": 3, "macro_tier_score": 0.1752, "macro_tier_override": null, "x": -169.914, "z": 39.776, "size": 0.404, "title": "Adjoining `⊤` and `⊥` to order maps and lattice homomorphisms", "summary": "This file defines ways to adjoin `⊤` or `⊥` or both to order maps (homomorphisms, embeddings and isomorphisms) and lattice homomorphisms, and properties about the results. Some definitions cause a possibly unbounded lattice homomorphism to become bounded, so they change the type of the homomorphism.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Hom/WithTopBot.html"}, {"id": "Mathlib.Order.PrimeIdeal", "region_id": "order", "micro_elevation": 0.9615, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -179.854, "z": 72.237, "size": 0.2676, "title": "Prime ideals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/PrimeIdeal.html"}, {"id": "Mathlib.Order.PFilter", "region_id": "order", "micro_elevation": 0.9231, "macro_tier": 2, "macro_tier_score": 0.01, "macro_tier_override": null, "x": -157.697, "z": 51.189, "size": 0.3065, "title": "Order filters", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/PFilter.html"}, {"id": "Mathlib.Order.TransfiniteIteration", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 1, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -219.15, "z": -1.837, "size": 0.2802, "title": "Transfinite iteration of a function `I → I`", "summary": "Given `φ : I → I` where `[SupSet I]`, we define the `j`th transfinite iteration of `φ` for any `j : J`, with `J` a well-ordered type: this is `transfiniteIterate φ j : I → I`. If `i₀ : I`, then `transfiniteIterate φ ⊥ i₀ = i₀`; if `j` is a non-maximal element, then `transfiniteIterate φ (Order.succ j) i₀ = φ (transfiniteIterate φ j i₀)`; and if `j` is a limit element, `transfiniteIterate φ j i₀` is the supremum of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/TransfiniteIteration.html"}, {"id": "Mathlib.Order.WellQuasiOrder", "region_id": "order", "micro_elevation": 0.9615, "macro_tier": 1, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -234.549, "z": 11.447, "size": 0.2881, "title": "Well quasi-orders", "summary": "A well quasi-order (WQO) is a relation such that any infinite sequence contains an infinite subsequence of related elements. For a preorder, this is equivalent to having a well-founded order with no infinite antichains.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/WellQuasiOrder.html"}, {"id": "Mathlib.Order.OrderIsoNat", "region_id": "order", "micro_elevation": 0.9231, "macro_tier": 2, "macro_tier_score": 0.0343, "macro_tier_override": null, "x": -223.16, "z": 62.195, "size": 0.3689, "title": "Relation embeddings from the naturals", "summary": "This file allows translation from monotone functions `ℕ → α` to order embeddings `ℕ ↪ α` and defines the limit value of an eventually-constant sequence.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/OrderIsoNat.html"}, {"id": "Mathlib.Order.CompactlyGenerated.Intervals", "region_id": "order", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -176.631, "z": -11.493, "size": 0.249, "title": "Results about compactness properties for intervals in complete lattices", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompactlyGenerated/Intervals.html"}, {"id": "Mathlib.Order.Interval.Set.Infinite", "region_id": "order", "micro_elevation": 0.5769, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": -200.44, "z": 56.573, "size": 0.3198, "title": "Infinitude of intervals", "summary": "Bounded intervals in dense orders are infinite, as are unbounded intervals in orders that are unbounded on the appropriate side. We also prove that an unbounded preorder is an infinite type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/Infinite.html"}, {"id": "Mathlib.Order.DirectedInverseSystem", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -162.407, "z": 55.202, "size": 0.2708, "title": "Definition of direct systems, inverse systems, and cardinalities in specific inverse systems", "summary": "The first part of this file concerns directed systems: `DirectLimit` is defined as the quotient of the disjoint union (`Sigma` type) by an equivalence relation (`Setoid`): compare `CategoryTheory.Limits.Types.Quot`, which is a quotient by a plain relation. Recursion and induction principles for constructing functions from and to `DirectLimit` and proving things about elements in `DirectLimit`. In the second part we…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/DirectedInverseSystem.html"}, {"id": "Mathlib.Order.Monotone.Basic", "region_id": "order", "micro_elevation": 0.1923, "macro_tier": 3, "macro_tier_score": 0.4426, "macro_tier_override": null, "x": -186.01, "z": 31.536, "size": 0.5811, "title": "Monotonicity", "summary": "This file defines (strictly) monotone/antitone functions. Contrary to standard mathematical usage, \"monotone\"/\"mono\" here means \"increasing\", not \"increasing or decreasing\". We use \"antitone\"/\"anti\" to mean \"decreasing\".", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Monotone/Basic.html"}, {"id": "Mathlib.Order.Compare", "region_id": "order", "micro_elevation": 0.1538, "macro_tier": 3, "macro_tier_score": 0.4356, "macro_tier_override": null, "x": -188.153, "z": 33.078, "size": 0.3386, "title": "Comparison", "summary": "This file provides basic results about orderings and comparison in linear orders.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Compare.html"}, {"id": "Mathlib.Order.Monotone.Defs", "region_id": "order", "micro_elevation": 0.1154, "macro_tier": 3, "macro_tier_score": 0.4406, "macro_tier_override": null, "x": -197.122, "z": 25.921, "size": 0.5291, "title": "Monotonicity", "summary": "This file defines (strictly) monotone/antitone functions. Contrary to standard mathematical usage, \"monotone\"/\"mono\" here means \"increasing\", not \"increasing or decreasing\". We use \"antitone\"/\"anti\" to mean \"decreasing\".", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Monotone/Defs.html"}, {"id": "Mathlib.Order.RelClasses", "region_id": "order", "micro_elevation": 0.1538, "macro_tier": 3, "macro_tier_score": 0.4448, "macro_tier_override": null, "x": -201.811, "z": 29.797, "size": 0.5161, "title": "Unbundled relation classes", "summary": "In this file we prove some properties of `Is*` classes defined in `Mathlib/Order/Defs/Unbundled.lean`. The main difference between these classes and the usual order classes (`Preorder` etc) is that usual classes extend `LE` and/or `LT` while these classes take a relation as an explicit argument.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/RelClasses.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.ModEq", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -163.6, "z": 47.252, "size": 0.2754, "title": "Numbers are frequently ModEq to fixed numbers", "summary": "In this file we prove that `m ≡ d [MOD n]` frequently as `m → ∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/ModEq.html"}, {"id": "Mathlib.Order.CompleteLattice.Finset", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0163, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.4034, "title": "Lattice operations on finsets", "summary": "This file is concerned with how big lattice or set operations behave when indexed by a finset. See also `Mathlib/Data/Finset/Lattice/Fold.lean`, which is concerned with folding binary lattice operations over a finset.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompleteLattice/Finset.html"}, {"id": "Mathlib.Order.SetDissipate", "region_id": "order", "micro_elevation": 0.9615, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -238.957, "z": 31.171, "size": 0.3156, "title": "Dissipate", "summary": "The function `dissipate` takes `s : α → Set β` with `LE α` and returns `⋂ y ≤ x, s y`. It is related to `accumulate s := ⋃ y ≤ x, s y`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SetDissipate.html"}, {"id": "Mathlib.Order.Bounds.Image", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 3, "macro_tier_score": 0.07, "macro_tier_override": null, "x": -199.42, "z": 2.803, "size": 0.4872, "title": "Images of upper/lower bounds under monotone functions", "summary": "In this file we prove various results about the behaviour of bounds under monotone/antitone maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Bounds/Image.html"}, {"id": "Mathlib.Order.Filter.FilterProduct", "region_id": "order", "micro_elevation": 0.9231, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -157.761, "z": 10.065, "size": 0.239, "title": "Ultraproducts", "summary": "If `φ` is an ultrafilter, then the space of germs of functions `f : α → β` at `φ` is called the *ultraproduct*. In this file we prove properties of ultraproducts that rely on `φ` being an ultrafilter. Definitions and properties that work for any filter should go to `Order.Filter.Germ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/FilterProduct.html"}, {"id": "Mathlib.Order.Filter.Ring", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -217.629, "z": 3.723, "size": 0.2705, "title": "Lemmas about filters and ordered rings.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Ring.html"}, {"id": "Mathlib.Order.WithBot", "region_id": "order", "micro_elevation": 0.2692, "macro_tier": 3, "macro_tier_score": 0.2117, "macro_tier_override": null, "x": -187.636, "z": 20.608, "size": 0.6401, "title": "`WithBot`, `WithTop`", "summary": "Adding a `bot` or a `top` to an order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/WithBot.html"}, {"id": "Mathlib.Order.Filter.Pointwise", "region_id": "order", "micro_elevation": 0.9231, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -172.479, "z": 66.724, "size": 0.3094, "title": "Pointwise operations on filters", "summary": "This file defines pointwise operations on filters. This is useful because usual algebraic operations distribute over pointwise operations. For example, * `(f₁ * f₂).map m = f₁.map m * f₂.map m` * `𝓝 (x * y) = 𝓝 x * 𝓝 y`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Pointwise.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Map", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 1, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -169.148, "z": 11.902, "size": 0.2981, "title": "Map and comap of `Filter.atTop` and `Filter.atBot`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Map.html"}, {"id": "Mathlib.Order.Filter.NAry", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 1, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -196.226, "z": -2.846, "size": 0.2385, "title": "N-ary maps of filter", "summary": "This file defines the binary and ternary maps of filters. This is mostly useful to define pointwise operations on filters.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/NAry.html"}, {"id": "Mathlib.Order.ConditionallyCompleteLattice.Finset", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 3, "macro_tier_score": 0.0922, "macro_tier_override": null, "x": -212.384, "z": 2.099, "size": 0.449, "title": "Conditionally complete lattices and finite sets.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ConditionallyCompleteLattice/Finset.html"}, {"id": "Mathlib.Order.UpperLower.Basic", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 3, "macro_tier_score": 0.0678, "macro_tier_override": null, "x": -164.315, "z": 39.711, "size": 0.409, "title": "Properties of unbundled upper/lower sets", "summary": "This file proves results on `IsUpperSet` and `IsLowerSet`, including their interactions with set operations, images, preimages and order duals, and properties that reflect stronger assumptions on the underlying order (such as `PartialOrder` and `LinearOrder`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/UpperLower/Basic.html"}, {"id": "Mathlib.Order.Filter.Germ.OrderedMonoid", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -168.353, "z": 10.026, "size": 0.2662, "title": "Ordered monoid instances on the space of germs of a function at a filter", "summary": "For each of the following structures we prove that if `β` has this structure, then so does `Germ l β`: * `IsOrderedCancelMonoid` and `IsOrderedCancelAddMonoid`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Germ/OrderedMonoid.html"}, {"id": "Mathlib.Order.Partition.Finpartition", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 2, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": -167.163, "z": 3.368, "size": 0.3476, "title": "Finite partitions", "summary": "In this file, we define finite partitions. A finpartition of `a : α` is a finite set of pairwise disjoint parts `parts : Finset α` which does not contain `⊥` and whose supremum is `a`. Finpartitions of a finset are at the heart of Szemerédi's regularity lemma. They are also studied purely order theoretically in Sperner theory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Partition/Finpartition.html"}, {"id": "Mathlib.Order.SuccPred.Relation", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 1, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": -216.046, "z": 63.222, "size": 0.3784, "title": "Relations on types with a `SuccOrder`", "summary": "This file contains properties about relations on types with a `SuccOrder` and their closure operations (like the transitive closure).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SuccPred/Relation.html"}, {"id": "Mathlib.Order.CompletePartialOrder", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -188.482, "z": 63.645, "size": 0.2478, "title": "Complete Partial Orders", "summary": "This file considers complete partial orders (sometimes called directedly complete partial orders). These are partial orders for which every directed set has a least upper bound.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompletePartialOrder.html"}, {"id": "Mathlib.Order.IsNormal", "region_id": "order", "micro_elevation": 0.9231, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": -229.617, "z": 6.498, "size": 0.3174, "title": "Normal functions", "summary": "A normal function between well-orders is a strictly monotonic continuous function. Normal functions arise chiefly in the context of cardinal and ordinal-valued functions. We opt for an equivalent definition that's both simpler and often more convenient: a normal function is a strictly monotonic function `f` such that at successor limits `a`, `f a` is the least upper bound of `f b` with `b < a`. See…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/IsNormal.html"}, {"id": "Mathlib.Order.Grade", "region_id": "order", "micro_elevation": 0.4231, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -208.119, "z": 44.832, "size": 0.2705, "title": "Graded orders", "summary": "This file defines graded orders, also known as ranked orders. An `𝕆`-graded order is an order `α` equipped with a distinguished \"grade\" function `α → 𝕆` which should be understood as giving the \"height\" of the elements. Usual graded orders are `ℕ`-graded, cograded orders are `ℕᵒᵈ`-graded, but we can also grade by `ℤ`, and polytopes are naturally `Fin n`-graded. Visually, `grade ℕ a` is the height of `a` in the Hasse…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Grade.html"}, {"id": "Mathlib.Order.Filter.TendstoCofinite", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -220.489, "z": 3.926, "size": 0.3442, "title": "Functions tending to the cofinite filter", "summary": "This file introduces the typeclass `Filter.TendstoCofinite`, which represents functions `f : α → β` that tend to the cofinite filter along the cofinite filter. Functions of this class are precisely the valid index transformations for renaming variables in multivariate power series.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/TendstoCofinite.html"}, {"id": "Mathlib.Order.SuccPred.Archimedean", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 2, "macro_tier_score": 0.0339, "macro_tier_override": null, "x": -184.077, "z": 66.194, "size": 0.5253, "title": "Archimedean successor and predecessor", "summary": "* `IsSuccArchimedean`: `SuccOrder` where `succ` iterated to an element gives all the greater ones. * `IsPredArchimedean`: `PredOrder` where `pred` iterated to an element gives all the smaller ones.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SuccPred/Archimedean.html"}, {"id": "Mathlib.Order.Interval.Set.Nat", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -235.426, "z": 30.345, "size": 0.2441, "title": "Finite intervals of naturals", "summary": "This file calculates the cardinality of intervals of natural numbers as sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/Nat.html"}, {"id": "Mathlib.Order.OrderDual", "region_id": "order", "micro_elevation": 0.1154, "macro_tier": 3, "macro_tier_score": 0.4706, "macro_tier_override": null, "x": -198.788, "z": 34.175, "size": 0.5791, "title": "Order dual", "summary": "This file defines `OrderDual α`, a type synonym reversing the meaning of all inequalities, with notation `αᵒᵈ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/OrderDual.html"}, {"id": "Mathlib.Order.Atoms.Finite", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -158.409, "z": 44.313, "size": 0.2626, "title": "Atoms, Coatoms, Simple Lattices, and Finiteness", "summary": "This module contains some results on atoms and simple lattices in the finite context.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Atoms/Finite.html"}, {"id": "Mathlib.Order.Filter.Map", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 2, "macro_tier_score": 0.0251, "macro_tier_override": null, "x": -219.427, "z": 44.551, "size": 0.6458, "title": "Theorems about map and comap on filters.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Map.html"}, {"id": "Mathlib.Order.ZornAtoms", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 2, "macro_tier_score": 0.029, "macro_tier_override": null, "x": -232.555, "z": 39.887, "size": 0.3319, "title": "Zorn lemma for (co)atoms", "summary": "In this file we use Zorn's lemma to prove that a partial order is atomic if every nonempty chain `c`, `⊥ ∉ c`, has a lower bound not equal to `⊥`. We also prove the order dual version of this statement.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ZornAtoms.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Field", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": -179.71, "z": -5.125, "size": 0.3177, "title": "Convergence to ±infinity in linear ordered (semi)fields", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Field.html"}, {"id": "Mathlib.Order.InitialSeg", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 2, "macro_tier_score": 0.0152, "macro_tier_override": null, "x": -200.551, "z": 63.749, "size": 0.3431, "title": "Initial and principal segments", "summary": "This file defines initial and principal segment embeddings. Though these definitions make sense for arbitrary relations, they're intended for use with well orders. An initial segment is simply a lower set, i.e. if `x` belongs to the range, then any `y < x` also belongs to the range. A principal segment is a set of the form `Set.Iio x` for some `x`. An initial segment embedding `r ≼i s` is an order embedding `r ↪ s`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/InitialSeg.html"}, {"id": "Mathlib.Order.CompleteSublattice", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 2, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -172.731, "z": -1.3, "size": 0.302, "title": "Complete Sublattices", "summary": "This file defines complete sublattices. These are subsets of complete lattices which are closed under arbitrary suprema and infima. As a standard example one could take the complete sublattice of invariant submodules of some module with respect to a linear map.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompleteSublattice.html"}, {"id": "Mathlib.Order.RelIso.Set", "region_id": "order", "micro_elevation": 0.2308, "macro_tier": 2, "macro_tier_score": 0.0394, "macro_tier_override": null, "x": -195.438, "z": 20.106, "size": 0.3909, "title": "Interactions between relation homomorphisms and sets", "summary": "It is likely that there are better homes for many of these statement, in files further down the import graph.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/RelIso/Set.html"}, {"id": "Mathlib.Order.SupIndep", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 2, "macro_tier_score": 0.0149, "macro_tier_override": null, "x": -225.349, "z": 9.643, "size": 0.3249, "title": "Supremum independence", "summary": "In this file, we define supremum independence of indexed sets. An indexed family `f : ι → α` is sup-independent if, for all `a`, `f a` and the supremum of the rest are disjoint.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SupIndep.html"}, {"id": "Mathlib.Order.Filter.Partial", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -202.842, "z": -0.076, "size": 0.2478, "title": "`Tendsto` for relations and partial functions", "summary": "This file generalizes `Filter` definitions from functions to partial functions and relations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Partial.html"}, {"id": "Mathlib.Order.Filter.Tendsto", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 2, "macro_tier_score": 0.0225, "macro_tier_override": null, "x": -202.919, "z": 1.775, "size": 0.5897, "title": "Convergence in terms of filters", "summary": "The general notion of limit of a map with respect to filters on the source and target types is `Filter.Tendsto`. It is defined in terms of the order and the push-forward operation. For instance, anticipating on Topology.Basic, the statement: \"if a sequence `u` converges to some `x` and `u n` belongs to a set `M` for `n` large enough then `x` is in the closure of `M`\" is formalized as: `Tendsto u atTop (𝓝 x) → (∀ᶠ n…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Tendsto.html"}, {"id": "Mathlib.Order.Filter.Bases.Basic", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 2, "macro_tier_score": 0.0177, "macro_tier_override": null, "x": -186.553, "z": 1.812, "size": 0.4589, "title": "Basic results on filter bases", "summary": "A filter basis `B : FilterBasis α` on a type `α` is a nonempty collection of sets of `α` such that the intersection of two elements of this collection contains some element of the collection. Compared to filters, filter bases do not require that any set containing an element of `B` belongs to `B`. A filter basis `B` can be used to construct `B.filter : Filter α` such that a set belongs to `B.filter` if and only if…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Bases/Basic.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Tendsto", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 2, "macro_tier_score": 0.0166, "macro_tier_override": null, "x": -171.31, "z": 52.11, "size": 0.4173, "title": "Limits of `Filter.atTop` and `Filter.atBot`", "summary": "In this file we prove many lemmas on the combination of `Filter.atTop` and `Filter.atBot` and `Tendsto`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Tendsto.html"}, {"id": "Mathlib.Order.Interval.Finset.Fin", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 2, "macro_tier_score": 0.0058, "macro_tier_override": null, "x": -207.759, "z": -7.82, "size": 0.3396, "title": "Finite intervals in `Fin n`", "summary": "This file proves that `Fin n` is a `LocallyFiniteOrder` and calculates the cardinality of its intervals as Finsets and Fintypes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Finset/Fin.html"}, {"id": "Mathlib.Order.Interval.Finset.SuccPred", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 2, "macro_tier_score": 0.0057, "macro_tier_override": null, "x": -185.99, "z": 68.532, "size": 0.3379, "title": "Finset intervals in a successor-predecessor order", "summary": "This file proves relations between the various finset intervals in a successor/predecessor order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Finset/SuccPred.html"}, {"id": "Mathlib.Order.Category.FinPartOrd", "region_id": "order", "micro_elevation": 0.0769, "macro_tier": 1, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -193.747, "z": 27.313, "size": 0.2717, "title": "The category of finite partial orders", "summary": "This defines `FinPartOrd`, the category of finite partial orders. Note: `FinPartOrd` is *not* a subcategory of `BddOrd` because finite orders are not necessarily bounded.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/FinPartOrd.html"}, {"id": "Mathlib.Order.Category.PartOrd", "region_id": "order", "micro_elevation": 0.0385, "macro_tier": 2, "macro_tier_score": 0.0108, "macro_tier_override": null, "x": -193.05, "z": 30.916, "size": 0.3604, "title": "Category of partial orders", "summary": "This defines `PartOrd`, the category of partial orders with monotone maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/PartOrd.html"}, {"id": "Mathlib.Order.DirSupClosed", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 1, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -223.336, "z": 48.351, "size": 0.2867, "title": "Sets closed under directed suprema", "summary": "We say that a set `s` is closed under directed suprema whenever it contains all least upper bounds for nonempty, directed subsets. Conversely, a set `s` is inaccessible by directed suprema whenever its complement is closed under directed suprema. Equivalently, if the least upper bound of a nonempty directed set `t` is contained in `s`, then `t` and `s` must have nonempty intersection.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/DirSupClosed.html"}, {"id": "Mathlib.Order.Ideal", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 2, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": -158.088, "z": 48.08, "size": 0.3576, "title": "Order ideals, cofinal sets, and the Rasiowa–Sikorski lemma", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Ideal.html"}, {"id": "Mathlib.Order.RelIso.Basic", "region_id": "order", "micro_elevation": 0.1923, "macro_tier": 3, "macro_tier_score": 0.3998, "macro_tier_override": null, "x": -185.974, "z": 30.967, "size": 0.5627, "title": "Relation homomorphisms, embeddings, isomorphisms", "summary": "This file defines relation homomorphisms, embeddings, isomorphisms and order embeddings and isomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/RelIso/Basic.html"}, {"id": "Mathlib.Order.Category.LinOrd", "region_id": "order", "micro_elevation": 0.4615, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -174.165, "z": 35.526, "size": 0.2565, "title": "Category of linear orders", "summary": "This defines `LinOrd`, the category of linear orders with monotone maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/LinOrd.html"}, {"id": "Mathlib.Order.Category.Lat", "region_id": "order", "micro_elevation": 0.4231, "macro_tier": 2, "macro_tier_score": 0.0059, "macro_tier_override": null, "x": -179.78, "z": 43.008, "size": 0.3477, "title": "The category of lattices", "summary": "This defines `Lat`, the category of lattices. Note that `Lat` doesn't correspond to the literature definition of [`Lat`] (https://ncatlab.org/nlab/show/Lat) as we don't require bottom or top elements. Instead, `Lat` corresponds to `BddLat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/Lat.html"}, {"id": "Mathlib.Order.Fin.Tuple", "region_id": "order", "micro_elevation": 0.1923, "macro_tier": 1, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -203.55, "z": 31.895, "size": 0.2993, "title": "Order properties on tuples", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Fin/Tuple.html"}, {"id": "Mathlib.Order.Interval.Set.Fin", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 2, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": -199.439, "z": 60.352, "size": 0.2841, "title": "(Pre)images of set intervals under `Fin` operations", "summary": "In this file we prove basic lemmas about preimages and images of the intervals under the following operations: - `Fin.val`, - `Fin.castLE` (preimages only), - `Fin.castAdd`, - `Fin.cast`, - `Fin.castSucc`, - `Fin.natAdd`, - `Fin.addNat`, - `Fin.succ`, - `Fin.rev`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/Fin.html"}, {"id": "Mathlib.Order.Defs.Unbundled", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0179, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.7669, "title": "Orders", "summary": "Defines classes for preorders, partial orders, and linear orders and proves some basic lemmas about them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Defs/Unbundled.html"}, {"id": "Mathlib.Order.Filter.Basic", "region_id": "order", "micro_elevation": 0.5769, "macro_tier": 2, "macro_tier_score": 0.0196, "macro_tier_override": null, "x": -191.797, "z": 4.36, "size": 0.5158, "title": "Theory of filters on sets", "summary": "A *filter* on a type `α` is a collection of sets of `α` which contains the whole `α`, is upwards-closed, and is stable under intersection. They are mostly used to abstract two related kinds of ideas: * *limits*, including finite or infinite limits of sequences, finite or infinite limits of functions at a point or at infinity, etc... * *things happening eventually*, including things happening for large enough `n :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Basic.html"}, {"id": "Mathlib.Order.Category.BddOrd", "region_id": "order", "micro_elevation": 0.4231, "macro_tier": 1, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -177.753, "z": 21.364, "size": 0.2457, "title": "The category of bounded orders", "summary": "This defines `BddOrd`, the category of bounded orders.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/BddOrd.html"}, {"id": "Mathlib.Order.Hom.Bounded", "region_id": "order", "micro_elevation": 0.3846, "macro_tier": 3, "macro_tier_score": 0.2079, "macro_tier_override": null, "x": -211.96, "z": 34.877, "size": 0.4016, "title": "Bounded order homomorphisms", "summary": "This file defines (bounded) order homomorphisms. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Hom/Bounded.html"}, {"id": "Mathlib.Order.ConditionallyCompleteLattice.Group", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 1, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": -194.055, "z": -2.868, "size": 0.3388, "title": "Conditionally complete lattices and groups.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ConditionallyCompleteLattice/Group.html"}, {"id": "Mathlib.Order.Extension.Well", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2, "title": "Extend a well-founded order to a well-order", "summary": "This file constructs a well-order (linear well-founded order) which is an extension of a given well-founded order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Extension/Well.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Group", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 2, "macro_tier_score": 0.0065, "macro_tier_override": null, "x": -164.768, "z": 12.083, "size": 0.3812, "title": "Convergence to ±infinity in ordered commutative groups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Group.html"}, {"id": "Mathlib.Order.Filter.EventuallyConst", "region_id": "order", "micro_elevation": 0.9615, "macro_tier": 1, "macro_tier_score": 0.0018, "macro_tier_override": null, "x": -189.723, "z": -13.181, "size": 0.3782, "title": "Functions that are eventually constant along a filter", "summary": "In this file we define a predicate `Filter.EventuallyConst f l` saying that a function `f : α → β` is eventually equal to a constant along a filter `l`. We also prove some basic properties of these functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/EventuallyConst.html"}, {"id": "Mathlib.Order.Prod.Lex.Hom", "region_id": "order", "micro_elevation": 0.3846, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -207.802, "z": 18.728, "size": 0.2429, "title": "Order homomorphism for `Prod.Lex`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Prod/Lex/Hom.html"}, {"id": "Mathlib.Order.Interval.Basic", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2552, "title": "Order intervals", "summary": "This file defines (nonempty) closed intervals in an order (see `Set.Icc`). This is a prototype for interval arithmetic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Basic.html"}, {"id": "Mathlib.Order.Filter.SmallSets", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 2, "macro_tier_score": 0.0079, "macro_tier_override": null, "x": -203.761, "z": 64.857, "size": 0.4407, "title": "The filter of small sets", "summary": "This file defines the filter of small sets w.r.t. a filter `f`, which is the largest filter containing all powersets of members of `f`. `g` converges to `f.smallSets` if for all `s ∈ f`, eventually we have `g x ⊆ s`. An example usage is that if `f : ι → E → ℝ` is a family of nonnegative functions with integral 1, then saying that `fun i ↦ support (f i)` tendsto `(𝓝 0).smallSets` is a way of saying that `f` tends to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/SmallSets.html"}, {"id": "Mathlib.Order.Category.FinBddDistLat", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 1, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -190.009, "z": -0.746, "size": 0.239, "title": "The category of finite bounded distributive lattices", "summary": "This file defines `FinBddDistLat`, the category of finite distributive lattices with bounded lattice homomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/FinBddDistLat.html"}, {"id": "Mathlib.Order.Quotient", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -166.376, "z": 12.843, "size": 0.2342, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Quotient.html"}, {"id": "Mathlib.Order.ConditionallyCompletePartialOrder.Indexed", "region_id": "order", "micro_elevation": 0.1154, "macro_tier": 3, "macro_tier_score": 0.1612, "macro_tier_override": null, "x": -193.267, "z": 25.613, "size": 0.4038, "title": "Indexed sup / inf in conditionally complete lattices", "summary": "This file proves lemmas about `iSup` and `iInf` for functions valued in a conditionally complete partial order, as opposed to a conditionally complete lattice.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ConditionallyCompletePartialOrder/Indexed.html"}, {"id": "Mathlib.Order.FixedPoints", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -166.1, "z": 10.088, "size": 0.2501, "title": "Fixed point construction on complete lattices", "summary": "This file sets up the basic theory of fixed points of a monotone function in a complete lattice.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/FixedPoints.html"}, {"id": "Mathlib.Order.Zorn", "region_id": "order", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0528, "macro_tier_override": null, "x": -199.592, "z": 53.142, "size": 0.3567, "title": "Zorn's lemmas", "summary": "This file proves several formulations of Zorn's Lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Zorn.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Interval", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -232.68, "z": 22.01, "size": 0.2377, "title": "Limits of intervals along filters", "summary": "This file contains some lemmas about how filters `Ixx` behave as the endpoints tend to `±∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Interval.html"}, {"id": "Mathlib.Order.SupClosed", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 2, "macro_tier_score": 0.0398, "macro_tier_override": null, "x": -227.333, "z": 44.457, "size": 0.4068, "title": "Sets closed under join/meet", "summary": "This file defines predicates for sets closed under `⊔` and shows that each set in a join-semilattice generates a join-closed set and that a semilattice where every directed set has a least upper bound is automatically complete. All dually for `⊓`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SupClosed.html"}, {"id": "Mathlib.Order.Category.BoolAlg", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 1, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -162.175, "z": 22.819, "size": 0.2552, "title": "The category of Boolean algebras", "summary": "This defines `BoolAlg`, the category of Boolean algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/BoolAlg.html"}, {"id": "Mathlib.Order.Category.HeytAlg", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -219.544, "z": 10.718, "size": 0.2597, "title": "The category of Heyting algebras", "summary": "This file defines `HeytAlg`, the category of Heyting algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/HeytAlg.html"}, {"id": "Mathlib.Order.CompleteLattice.MulticoequalizerDiagram", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2427, "title": "Multicoequalizer diagrams in complete lattices", "summary": "We introduce the notion of bi-Cartesian square (`Lattice.BicartSq`) in a lattice `T`. This consists of elements `x₁`, `x₂`, `x₃` and `x₄` such that `x₂ ⊔ x₃ = x₄` and `x₂ ⊓ x₃ = x₁`. It shall be shown (TODO) that if `T := Set X`, then the image of the associated commutative square in the category `Type _` is a bi-Cartesian square in a categorical sense (both pushout and pullback). More generally, if `T` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompleteLattice/MulticoequalizerDiagram.html"}, {"id": "Mathlib.Order.Interval.Set.Final", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2415, "title": "Final functors between intervals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/Final.html"}, {"id": "Mathlib.Order.Interval.Finset.Gaps", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.256, "title": "Gaps of disjoint closed intervals", "summary": "This file defines `Finset.intervalGapsWithin` that computes the complement of the union of a collection of pairwise disjoint subintervals of `[a, b]`. If `LinearOrder α`, `F` is a finite subset of `α × α` such that for any `(x, y) ∈ F`, `a ≤ x ≤ y ≤ b` and all such `[x, y]`'s are pairwise disjoint, `h` is a proof of `F.card = k`, `i` is in `Fin (k + 1)`, we order `F` from left to right as `(x 0, y 0), ..., (x (k -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Finset/Gaps.html"}, {"id": "Mathlib.Order.Interval.Set.SuccPred", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 2, "macro_tier_score": 0.01, "macro_tier_override": null, "x": -170.803, "z": 2.4, "size": 0.3123, "title": "Set intervals in a successor-predecessor order", "summary": "This file proves relations between the various set intervals in a successor/predecessor order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/SuccPred.html"}, {"id": "Mathlib.Order.Hom.BoundedLattice", "region_id": "order", "micro_elevation": 0.5385, "macro_tier": 3, "macro_tier_score": 0.2124, "macro_tier_override": null, "x": -194.716, "z": 55.413, "size": 0.5484, "title": "Bounded lattice homomorphisms", "summary": "This file defines bounded lattice homomorphisms. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Hom/BoundedLattice.html"}, {"id": "Mathlib.Order.Preorder.Finsupp", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0011, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.3385, "title": "Pointwise order on finitely supported functions", "summary": "This file lifts order structures on `M` to `ι →₀ M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Preorder/Finsupp.html"}, {"id": "Mathlib.Order.UpperLower.Closure", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 2, "macro_tier_score": 0.0106, "macro_tier_override": null, "x": -167.722, "z": 56.035, "size": 0.3497, "title": "Upper and lower closures", "summary": "Upper (lower) closures generalise principal upper (lower) sets to arbitrary included sets. Indeed, they are equivalent to a union over principal upper (lower) sets, as shown in `coe_upperClosure` (`coe_lowerClosure`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/UpperLower/Closure.html"}, {"id": "Mathlib.Order.Completion", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -159.536, "z": 28.588, "size": 0.2302, "title": "Dedekind-MacNeille completion", "summary": "The Dedekind-MacNeille completion of a partial order is the smallest complete lattice into which it embeds. The theory of concept lattices allows for a simple construction. In fact, `DedekindCut α` is simply an abbreviation for `Concept α α (· ≤ ·)`. This means we don't need to reprove that this is a complete lattice; instead, the file simply proves that any order embedding into another complete lattice factors…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Completion.html"}, {"id": "Mathlib.Order.Heyting.Hom", "region_id": "order", "micro_elevation": 0.5769, "macro_tier": 2, "macro_tier_score": 0.0054, "macro_tier_override": null, "x": -216.14, "z": 46.39, "size": 0.3134, "title": "Heyting algebra morphisms", "summary": "A Heyting homomorphism between two Heyting algebras is a bounded lattice homomorphism that preserves Heyting implication. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Heyting/Hom.html"}, {"id": "Mathlib.Order.LatticeIntervals", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 3, "macro_tier_score": 0.0899, "macro_tier_override": null, "x": -172.889, "z": 12.836, "size": 0.3447, "title": "Intervals in Lattices", "summary": "In this file, we provide instances of lattice structures on intervals within lattices. Some of them depend on the order of the endpoints of the interval, and thus are not made global instances. These are probably not all of the lattice instances that could be placed on these intervals, but more can be added easily along the same lines when needed.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/LatticeIntervals.html"}, {"id": "Mathlib.Order.CompleteLattice.Group", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 1, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -164.129, "z": 39.056, "size": 0.2761, "title": "Complete lattices and groups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompleteLattice/Group.html"}, {"id": "Mathlib.Order.Filter.CountableInter", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 2, "macro_tier_score": 0.0109, "macro_tier_override": null, "x": -172.605, "z": 3.201, "size": 0.5337, "title": "Filters with countable intersection property", "summary": "In this file we define `CountableInterFilter` to be the class of filters with the following property: for any countable collection of sets `s ∈ l` their intersection belongs to `l` as well. Two main examples are the `residual` filter defined in `Mathlib/Topology/GDelta/Basic.lean` and the `MeasureTheory.ae` filter defined in `Mathlib/MeasureTheory/OuterMeasure/AE.lean`. We reformulate the definition in terms of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/CountableInter.html"}, {"id": "Mathlib.Order.SetAccumulate", "region_id": "order", "micro_elevation": 0.9231, "macro_tier": 2, "macro_tier_score": 0.0067, "macro_tier_override": null, "x": -200.613, "z": 72.677, "size": 0.3921, "title": "Accumulate", "summary": "The function `accumulate` takes `s : α → Set β` with `LE α` and returns `⋃ y ≤ x, s y`. It is related to `dissipate s := ⋂ y ≤ x, s y`. `accumulate` is closely related to the function `partialSups`, although these two functions have slightly different typeclass assumptions and API. `partialSups_eq_accumulate` shows that they coincide on `ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SetAccumulate.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Floor", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -188.964, "z": 67.316, "size": 0.2649, "title": "`a * c ^ n < (n - d)!` holds true for sufficiently large `n`.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Floor.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Ring", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 1, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -227.127, "z": 48.877, "size": 0.2907, "title": "Convergence to ±infinity in ordered rings", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Ring.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Finite", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 2, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -216.286, "z": 2.642, "size": 0.3061, "title": "Finiteness and `Filter.atTop` and `Filter.atBot` filters", "summary": "This file contains results on `Filter.atTop` and `Filter.atBot` that depend on the finiteness theory developed in Mathlib.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Finite.html"}, {"id": "Mathlib.Order.Interval.Set.InitialSeg", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -186.452, "z": 65.011, "size": 0.2671, "title": "Intervals as initial segments", "summary": "We show that `Iic` and `Iio` are respectively initial and principal segments, and that any principal segment `f` is order isomorphic to `Iio f.top`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/InitialSeg.html"}, {"id": "Mathlib.Order.Interval.Set.Limit", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -223.329, "z": 1.76, "size": 0.2298, "title": "Limit elements in Set.Ici", "summary": "If `J` is a linearly ordered type, `j : J`, and `m : Set.Ici j` is successor limit, then `↑m : J` is also successor limit.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/Limit.html"}, {"id": "Mathlib.Order.SuccPred.InitialSeg", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -154.931, "z": 22.885, "size": 0.2704, "title": "Initial segments and successors", "summary": "We establish some connections between initial segment embeddings and successors and predecessors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SuccPred/InitialSeg.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.BigOperators", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -218.049, "z": 57.284, "size": 0.2595, "title": "Two lemmas about limit of `Π b ∈ s, f b` along", "summary": "In this file we prove two auxiliary lemmas about `Filter.atTop : Filter (Finset _)` and `∏ b ∈ s, f b`. These lemmas are useful to build the theory of absolutely convergent series.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/BigOperators.html"}, {"id": "Mathlib.Order.Category.PartOrdEmb", "region_id": "order", "micro_elevation": 0.0769, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -192.33, "z": 33.21, "size": 0.2442, "title": "Category of partial orders, with order embeddings as morphisms", "summary": "This defines `PartOrdEmb`, the category of partial orders with order embeddings as morphisms. We also show that `PartOrdEmb` has filtered colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/PartOrdEmb.html"}, {"id": "Mathlib.Order.ScottContinuity.Prod", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -202.414, "z": -0.185, "size": 0.2676, "title": "Scott continuity on product spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ScottContinuity/Prod.html"}, {"id": "Mathlib.Order.Bounds.Lattice", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0097, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2757, "title": "Unions and intersections of bounds", "summary": "Some results about upper and lower bounds over collections of sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Bounds/Lattice.html"}, {"id": "Mathlib.Order.ULift", "region_id": "order", "micro_elevation": 0.1154, "macro_tier": 3, "macro_tier_score": 0.4413, "macro_tier_override": null, "x": -197.541, "z": 26.149, "size": 0.5466, "title": "Ordered structures on `ULift.{v} α`", "summary": "Once these basic instances are setup, the instances of more complex typeclasses should live next to the corresponding `Prod` instances.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ULift.html"}, {"id": "Mathlib.Order.Interval.Set.Pi", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 1, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -169.309, "z": 14.816, "size": 0.3472, "title": "Intervals in `pi`-space", "summary": "In this we prove various simple lemmas about intervals in `Π i, α i`. Closed intervals (`Ici x`, `Iic x`, `Icc x y`) are equal to products of their projections to `α i`, while (semi-)open intervals usually include the corresponding products as proper subsets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/Pi.html"}, {"id": "Mathlib.Order.Category.Frm", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -218.125, "z": 46.644, "size": 0.2624, "title": "The category of frames", "summary": "This file defines `Frm`, the category of frames.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/Frm.html"}, {"id": "Mathlib.Order.Interval.Set.IsoIoo", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -204.765, "z": 60.877, "size": 0.2483, "title": "Order isomorphism between a linear ordered field and `(-1, 1)`", "summary": "In this file we provide an order isomorphism `orderIsoIooNegOneOne` between the open interval `(-1, 1)` in a linear ordered field and the whole field.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/IsoIoo.html"}, {"id": "Mathlib.Order.Interval.Lex", "region_id": "order", "micro_elevation": 0.0385, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -195.073, "z": 32.43, "size": 0.2, "title": "The lexicographic order on intervals", "summary": "This order is compatible with the inclusion ordering, but is total. Under this ordering, `[(3, 3), (2, 2), (2, 3), (1, 1), (1, 2), (1, 3)]` is sorted.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Lex.html"}, {"id": "Mathlib.Order.GaloisConnection.Defs", "region_id": "order", "micro_elevation": 0.2308, "macro_tier": 3, "macro_tier_score": 0.3477, "macro_tier_override": null, "x": -200.833, "z": 39.399, "size": 0.3847, "title": "Galois connections, insertions and coinsertions", "summary": "Galois connections are order-theoretic adjoints, i.e. a pair of functions `u` and `l`, such that `∀ a b, l a ≤ b ↔ a ≤ u b`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/GaloisConnection/Defs.html"}, {"id": "Mathlib.Order.CompleteLattice.Defs", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 3, "macro_tier_score": 0.0551, "macro_tier_override": null, "x": -221.166, "z": 20.505, "size": 0.4591, "title": "Definition of complete lattices", "summary": "This file contains the definition of complete lattices with suprema/infima of arbitrary sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompleteLattice/Defs.html"}, {"id": "Mathlib.Order.CompleteLattice.SetLike", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -169.321, "z": 62.327, "size": 0.2676, "title": "`SetLike` instance for elements of `CompleteSublattice (Set X)`", "summary": "This file provides lemmas for the `SetLike` instance for elements of `CompleteSublattice (Set X)`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompleteLattice/SetLike.html"}, {"id": "Mathlib.Order.Category.Semilat", "region_id": "order", "micro_elevation": 0.5769, "macro_tier": 1, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -170.028, "z": 21.288, "size": 0.2457, "title": "The categories of semilattices", "summary": "This defines `SemilatSupCat` and `SemilatInfCat`, the categories of sup-semilattices with a bottom element and inf-semilattices with a top element.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/Semilat.html"}, {"id": "Mathlib.Order.CountableDenseLinearOrder", "region_id": "order", "micro_elevation": 0.9231, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -157.51, "z": 10.522, "size": 0.2276, "title": "The back and forth method and countable dense linear orders", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CountableDenseLinearOrder.html"}, {"id": "Mathlib.Order.Filter.Bases.Finite", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 2, "macro_tier_score": 0.0158, "macro_tier_override": null, "x": -192.249, "z": -1.007, "size": 0.38, "title": "Finiteness results on filter bases", "summary": "A filter basis `B : FilterBasis α` on a type `α` is a nonempty collection of sets of `α` such that the intersection of two elements of this collection contains some element of the collection.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Bases/Finite.html"}, {"id": "Mathlib.Order.Sublattice", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 2, "macro_tier_score": 0.015, "macro_tier_override": null, "x": -230.203, "z": 41.758, "size": 0.3334, "title": "Sublattices", "summary": "This file defines sublattices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Sublattice.html"}, {"id": "Mathlib.Order.UpperLower.Hom", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -193.644, "z": -6.391, "size": 0.2, "title": "`UpperSet.Ici` etc. as `Sup`/`sSup`/`Inf`/`sInf`-homomorphisms", "summary": "In this file we define `UpperSet.iciSupHom` etc. These functions are `UpperSet.Ici` and `LowerSet.Iic` bundled as `SupHom`s, `InfHom`s, `sSupHom`s, or `sInfHom`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/UpperLower/Hom.html"}, {"id": "Mathlib.Order.UpperLower.Principal", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 2, "macro_tier_score": 0.0336, "macro_tier_override": null, "x": -166.296, "z": 51.553, "size": 0.3247, "title": "Principal upper/lower sets", "summary": "The results in this file all assume that the underlying type is equipped with at least a preorder.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/UpperLower/Principal.html"}, {"id": "Mathlib.Order.Filter.Subsingleton", "region_id": "order", "micro_elevation": 0.9231, "macro_tier": 2, "macro_tier_score": 0.0053, "macro_tier_override": null, "x": -234.973, "z": 17.144, "size": 0.3027, "title": "Subsingleton filters", "summary": "We say that a filter `l` is a *subsingleton* if there exists a subsingleton set `s ∈ l`. Equivalently, `l` is either `⊥` or `pure a` for some `a`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Subsingleton.html"}, {"id": "Mathlib.Order.Filter.Lift", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 2, "macro_tier_score": 0.0071, "macro_tier_override": null, "x": -163.258, "z": 42.145, "size": 0.4083, "title": "Lift filters along filter and set functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Lift.html"}, {"id": "Mathlib.Order.Heyting.Regular", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -226.535, "z": 19.763, "size": 0.2, "title": "Heyting regular elements", "summary": "This file defines Heyting regular elements, elements of a Heyting algebra that are their own double complement, and proves that they form a Boolean algebra. From a logic standpoint, this means that we can perform classical logic within intuitionistic logic by simply double-negating all propositions. This is practical for synthetic computability theory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Heyting/Regular.html"}, {"id": "Mathlib.Order.Heyting.Basic", "region_id": "order", "micro_elevation": 0.3846, "macro_tier": 3, "macro_tier_score": 0.35, "macro_tier_override": null, "x": -178.708, "z": 37.965, "size": 0.474, "title": "Heyting algebras", "summary": "This file defines Heyting, co-Heyting and bi-Heyting algebras. A Heyting algebra is a bounded distributive lattice with an implication operation `⇨` such that `a ≤ b ⇨ c ↔ a ⊓ b ≤ c`. It also comes with a pseudo-complement `ᶜ`, such that `aᶜ = a ⇨ ⊥`. Co-Heyting algebras are dual to Heyting algebras. They have a difference `\\` and a negation `￢` such that `a \\ b ≤ c ↔ a ≤ b ⊔ c` and `￢a = ⊤ \\ a`. Bi-Heyting algebras…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Heyting/Basic.html"}, {"id": "Mathlib.Order.Bounded", "region_id": "order", "micro_elevation": 0.5769, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -219.08, "z": 41.293, "size": 0.2, "title": "Bounded and unbounded sets", "summary": "We prove miscellaneous lemmas about bounded and unbounded sets. Many of these are just variations on the same ideas, or similar results with a few minor differences. The file is divided into these different general ideas.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Bounded.html"}, {"id": "Mathlib.Order.Hom.Lattice", "region_id": "order", "micro_elevation": 0.3846, "macro_tier": 3, "macro_tier_score": 0.2133, "macro_tier_override": null, "x": -211.997, "z": 34.722, "size": 0.4344, "title": "Unbounded lattice homomorphisms", "summary": "This file defines unbounded lattice homomorphisms. _Bounded_ lattice homomorphisms are defined in `Mathlib/Order/Hom/BoundedLattice.lean`. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Hom/Lattice.html"}, {"id": "Mathlib.Order.Birkhoff", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -164.235, "z": 6.689, "size": 0.2269, "title": "Birkhoff representation", "summary": "This file proves two facts which together are commonly referred to as \"Birkhoff representation\": 1. Any nonempty finite partial order is isomorphic to the partial order of sup-irreducible elements in its lattice of lower sets. 2. Any nonempty finite distributive lattice is isomorphic to the lattice of lower sets of its partial order of sup-irreducible elements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Birkhoff.html"}, {"id": "Mathlib.Order.Booleanisation", "region_id": "order", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -217.484, "z": 34.259, "size": 0.2269, "title": "Adding complements to a generalized Boolean algebra", "summary": "This file embeds any generalized Boolean algebra into a Boolean algebra. This concretely proves that any equation holding true in the theory of Boolean algebras that does not reference `ᶜ` also holds true in the theory of generalized Boolean algebras. Put another way, one does not need the existence of complements to prove something which does not talk about complements.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Booleanisation.html"}, {"id": "Mathlib.Order.Hom.PowersetCard", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.27, "title": "Finite sets of an ordered type", "summary": "This file defines the isomorphism between ordered embeddings into a linearly ordered type and the finite sets of that type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Hom/PowersetCard.html"}, {"id": "Mathlib.Order.SuccPred.IntervalSucc", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -161.452, "z": 10.736, "size": 0.2425, "title": "Intervals `Ixx (f x) (f (Order.succ x))`", "summary": "In this file we prove * `Monotone.biUnion_Ico_Ioc_map_succ`: if `α` is a linear archimedean succ order and `β` is a linear order, then for any monotone function `f` and `m n : α`, the union of intervals `Set.Ioc (f i) (f (Order.succ i))`, `m ≤ i < n`, is equal to `Set.Ioc (f m) (f n)`; * `Monotone.pairwise_disjoint_on_Ioc_succ`: if `α` is a linear succ order, `β` is a preorder, and `f : α → β` is a monotone…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SuccPred/IntervalSucc.html"}, {"id": "Mathlib.Order.Copy", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 1, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -191.249, "z": -0.911, "size": 0.3551, "title": "Tooling to make copies of lattice structures", "summary": "Sometimes it is useful to make a copy of a lattice structure where one replaces the data parts with provably equal definitions that have better definitional properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Copy.html"}, {"id": "Mathlib.Order.PrimeSeparator", "region_id": "order", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -153.017, "z": 49.742, "size": 0.2, "title": "Separating prime filters and ideals", "summary": "In a distributive lattice, if $F$ is a filter, $I$ is an ideal, and $F$ and $I$ are disjoint, then there exists a prime ideal $J$ containing $I$ with $J$ still disjoint from $F$. This theorem is a crucial ingredient to [Stone's][Sto1938] duality for bounded distributive lattices. The construction of the separator relies on Zorn's lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/PrimeSeparator.html"}, {"id": "Mathlib.Order.Fin.Clamp", "region_id": "order", "micro_elevation": 0.4231, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -199.546, "z": 49.526, "size": 0.2416, "title": "Lemmas about `Fin.clamp`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Fin/Clamp.html"}, {"id": "Mathlib.Order.WithBotTop", "region_id": "order", "micro_elevation": 0.3077, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -181.779, "z": 25.198, "size": 0.2706, "title": "Adding both `⊥` and `⊤` to a type", "summary": "This files defines an abbreviation `WithBotTop ι` for `WithBot (WithTop ι)`. We also introduce an abbreviation `EInt` for `WithBotTop ℤ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/WithBotTop.html"}, {"id": "Mathlib.Order.Concept", "region_id": "order", "micro_elevation": 0.0385, "macro_tier": 1, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -196.549, "z": 30.428, "size": 0.2516, "title": "Formal concept analysis", "summary": "This file defines concept lattices. A concept of a relation `r : α → β → Prop` is a pair of sets `s : Set α` and `t : Set β` such that `s` is the set of all `a : α` that are related to all elements of `t`, and `t` is the set of all `b : β` that are related to all elements of `s`. Ordering the concepts of a relation `r` by inclusion on the first component gives rise to a *concept lattice*. Every concept lattice is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Concept.html"}, {"id": "Mathlib.Order.UpperLower.CompleteLattice", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 2, "macro_tier_score": 0.038, "macro_tier_override": null, "x": -182.86, "z": -0.68, "size": 0.3074, "title": "The complete lattice structure on `UpperSet`/`LowerSet`", "summary": "This file defines a completely distributive lattice structure on `UpperSet` and `LowerSet`, pulled back across the canonical injection (`UpperSet.carrier`, `LowerSet.carrier`) into `Set α`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/UpperLower/CompleteLattice.html"}, {"id": "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2797, "title": "Zero and Bounded at filter", "summary": "Given a filter `l` we define the notion of a function being `ZeroAtFilter` as well as being `BoundedAtFilter`. Alongside this we construct the `Submodule`, `AddSubmonoid` of functions that are `ZeroAtFilter`. Similarly, we construct the `Submodule` and `Subalgebra` of functions that are `BoundedAtFilter`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/ZeroAndBoundedAtFilter.html"}, {"id": "Mathlib.Order.Synonym", "region_id": "order", "micro_elevation": 0.1154, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -196.524, "z": 35.696, "size": 0.2, "title": "Type synonyms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Synonym.html"}, {"id": "Mathlib.Order.Filter.Interval", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 1, "macro_tier_score": 0.0012, "macro_tier_override": null, "x": -185.193, "z": 66.512, "size": 0.3472, "title": "Convergence of intervals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Interval.html"}, {"id": "Mathlib.Order.OrdContinuous", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 2, "macro_tier_score": 0.006, "macro_tier_override": null, "x": -176.742, "z": 56.852, "size": 0.3526, "title": "Order continuity", "summary": "We say that a function is *left order continuous* if it sends all least upper bounds to least upper bounds. The order dual notion is called *right order continuity*. For monotone functions `ℝ → ℝ` these notions correspond to the usual left and right continuity. We prove some basic lemmas (`map_sup`, `map_sSup` etc) and prove that a `RelIso` is both left and right order continuous.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/OrdContinuous.html"}, {"id": "Mathlib.Order.Filter.IndicatorFunction", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -222.068, "z": 14.333, "size": 0.3106, "title": "Indicator function and filters", "summary": "Properties of additive and multiplicative indicator functions involving `=ᶠ` and `≤ᶠ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/IndicatorFunction.html"}, {"id": "Mathlib.Order.Filter.Defs", "region_id": "order", "micro_elevation": 0.5385, "macro_tier": 2, "macro_tier_score": 0.0157, "macro_tier_override": null, "x": -174.136, "z": 44.267, "size": 0.3762, "title": "Definitions about filters", "summary": "A *filter* on a type `α` is a collection of sets of `α` which contains the whole `α`, is upwards-closed, and is stable under intersection. Filters are mostly used to abstract two related kinds of ideas: * *limits*, including finite or infinite limits of sequences, finite or infinite limits of functions at a point or at infinity, etc... * *things happening eventually*, including things happening for large enough `n :…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Defs.html"}, {"id": "Mathlib.Order.Filter.IsBounded", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 1, "macro_tier_score": 0.005, "macro_tier_override": null, "x": -225.306, "z": 51.789, "size": 0.2763, "title": "Lemmas about `Is(Co)Bounded(Under)`", "summary": "This file proves several lemmas about `IsBounded`, `IsBoundedUnder`, `IsCobounded` and `IsCoboundedUnder`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/IsBounded.html"}, {"id": "Mathlib.Order.SuccPred.CompleteLinearOrder", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 1, "macro_tier_score": 0.0051, "macro_tier_override": null, "x": -155.575, "z": 41.258, "size": 0.2869, "title": "Relation between `IsSuccPrelimit` and `iSup` in (conditionally) complete linear orders.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SuccPred/CompleteLinearOrder.html"}, {"id": "Mathlib.Order.Category.NonemptyFinLinOrd", "region_id": "order", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -201.066, "z": 52.776, "size": 0.3156, "title": "Nonempty finite linear orders", "summary": "This defines `NonemptyFinLinOrd`, the category of nonempty finite linear orders with monotone maps. This is the index category for simplicial objects. Note: `NonemptyFinLinOrd` is *not* a subcategory of `FinBddDistLat` because its morphisms do not preserve `⊥` and `⊤`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/NonemptyFinLinOrd.html"}, {"id": "Mathlib.Order.SaddlePoint", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -186.861, "z": 63.293, "size": 0.239, "title": "Saddle points of a map", "summary": "* `IsSaddlePointOn`. Let `f : E × F → β` be a map, where `β` is preordered. A pair `(a,b)` in `E × F` is a *saddle point* of `f` on `X × Y` if `f a y ≤ f x b` for all `x ∈ X` and all `y` in `Y`. * `isSaddlePointOn_iff`: if `β` is a complete linear order, then `(a, b) ∈ X × Y` is a saddle point on `X × Y` iff `⨆ y ∈ Y, f a y = ⨅ x ∈ X, f x b = f a b`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SaddlePoint.html"}, {"id": "Mathlib.Order.UpperLower.Fibration", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -196.14, "z": -2.849, "size": 0.2735, "title": "Upper/lower sets and fibrations", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/UpperLower/Fibration.html"}, {"id": "Mathlib.Order.Filter.ListTraverse", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -219.649, "z": 13.824, "size": 0.2338, "title": "Properties of `Traversable.traverse` on `List`s and `Filter`s", "summary": "In this file we prove basic properties (monotonicity, membership) for `Traversable.traverse f l`, where `f : β → Filter α` and `l : List β`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/ListTraverse.html"}, {"id": "Mathlib.Order.UpperLower.Relative", "region_id": "order", "micro_elevation": 0.0769, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -197.921, "z": 32.341, "size": 0.2587, "title": "Properties of relative upper/lower sets", "summary": "This file proves results on `IsRelUpperSet` and `IsRelLowerSet`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/UpperLower/Relative.html"}, {"id": "Mathlib.Order.BooleanSubalgebra", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -155.955, "z": 29.642, "size": 0.2638, "title": "Boolean subalgebras", "summary": "This file defines Boolean subalgebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/BooleanSubalgebra.html"}, {"id": "Mathlib.Order.Interval.Set.WithBotTop", "region_id": "order", "micro_elevation": 0.5769, "macro_tier": 3, "macro_tier_score": 0.1656, "macro_tier_override": null, "x": -207.078, "z": 54.164, "size": 0.3921, "title": "Intervals in `WithTop α` and `WithBot α`", "summary": "In this file we prove various lemmas about `Set.image`s and `Set.preimage`s of intervals under `some : α → WithTop α` and `some : α → WithBot α`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/WithBotTop.html"}, {"id": "Mathlib.Order.Interval.Set.SurjOn", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -167.621, "z": 22.942, "size": 0.2, "title": "Monotone surjective functions are surjective on intervals", "summary": "A monotone surjective function sends any interval in the domain onto the interval with corresponding endpoints in the range. This is expressed in this file using `Set.surjOn`, and provided for all permutations of interval endpoints.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/SurjOn.html"}, {"id": "Mathlib.Order.Hom.Lex", "region_id": "order", "micro_elevation": 0.4231, "macro_tier": 2, "macro_tier_score": 0.0194, "macro_tier_override": null, "x": -184.235, "z": 46.99, "size": 0.3108, "title": "Lexicographic order and order isomorphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Hom/Lex.html"}, {"id": "Mathlib.Order.PiLex", "region_id": "order", "micro_elevation": 0.1538, "macro_tier": 2, "macro_tier_score": 0.01, "macro_tier_override": null, "x": -200.185, "z": 26.11, "size": 0.3082, "title": "Lexicographic order on Pi types", "summary": "This file defines the lexicographic and colexicographic orders for Pi types. * In the lexicographic order, `a` is less than `b` if `a i = b i` for all `i` up to some point `k`, and `a k < b k`. * In the colexicographic order, `a` is less than `b` if `a i = b i` for all `i` above some point `k`, and `a k < b k`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/PiLex.html"}, {"id": "Mathlib.Order.Lattice.Congruence", "region_id": "order", "micro_elevation": 0.4231, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -180.223, "z": 17.84, "size": 0.2, "title": "Lattice Congruences", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Lattice/Congruence.html"}, {"id": "Mathlib.Order.Partition.Basic", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -173.354, "z": 63.09, "size": 0.2, "title": "Partitions", "summary": "A `Partition` of an element `a` in a complete lattice is an independent family of nontrivial elements whose supremum is `a`. An important special case is where `s : Set α`, where a `Partition s` corresponds to a partition of the elements of `s` into a family of nonempty sets. This is equivalent to a transitive and symmetric binary relation `r : α → α → Prop` where `s` is the set of all `x` for which `r x x`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Partition/Basic.html"}, {"id": "Mathlib.Order.Shrink", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -158.494, "z": 23.087, "size": 0.2692, "title": "Order instances on Shrink", "summary": "If `α : Type v` is `u`-small, we transport various order related instances on `α` to `Shrink.{u} α`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Shrink.html"}, {"id": "Mathlib.Order.SuccPred.WithBot", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 1, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": -222.055, "z": 55.847, "size": 0.306, "title": "Successor function on `WithBot`", "summary": "This file defines the successor of `a : WithBot α` as an element of `α`, and dually for `WithTop`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SuccPred/WithBot.html"}, {"id": "Mathlib.Order.ScottContinuity.Complete", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -163.358, "z": 42.418, "size": 0.2, "title": "Scott continuity on complete lattices", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ScottContinuity/Complete.html"}, {"id": "Mathlib.Order.Circular", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0049, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2577, "title": "Circular order hierarchy", "summary": "This file defines circular preorders, circular partial orders and circular orders.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Circular.html"}, {"id": "Mathlib.Order.UpperLower.LocallyFinite", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -169.229, "z": 1.425, "size": 0.2, "title": "Upper and lower sets in a locally finite order", "summary": "In this file we characterise the interaction of `UpperSet`/`LowerSet` and `LocallyFiniteOrder`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/UpperLower/LocallyFinite.html"}, {"id": "Mathlib.Order.Cofinal", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0239, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.3008, "title": "Cofinal sets", "summary": "A set `s` in an ordered type `α` is cofinal when for every `a : α` there exists an element of `s` greater or equal to it. This file provides a basic API for the `IsCofinal` predicate. For the cofinality of a set as a cardinal, see `Mathlib/SetTheory/Cardinal/Cofinality/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Cofinal.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.CompleteLattice", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 1, "macro_tier_score": 0.0013, "macro_tier_override": null, "x": -165.574, "z": 47.18, "size": 0.3527, "title": "`Filter.atTop` and `Filter.atBot` in (conditionally) complete lattices", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/CompleteLattice.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.CountablyGenerated", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 1, "macro_tier_score": 0.0015, "macro_tier_override": null, "x": -231.854, "z": 32.408, "size": 0.3648, "title": "Convergence to infinity and countably generated filters", "summary": "In this file we prove that - `Filter.atTop` and `Filter.atBot` filters on a countable type are countably generated; - `Filter.exists_seq_tendsto`: if `f` is a nontrivial countably generated filter, then there exists a sequence that converges. to `f`; - `Filter.tendsto_iff_seq_tendsto`: convergence along a countably generated filter is equivalent to convergence along all sequences that converge to this filter.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/CountablyGenerated.html"}, {"id": "Mathlib.Order.Nucleus", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 1, "macro_tier_score": 0.0048, "macro_tier_override": null, "x": -211.061, "z": 53.8, "size": 0.2478, "title": "Nucleus", "summary": "Locales are the dual concept to frames. Locale theory is a branch of point-free topology, where intuitively locales are like topological spaces which may or may not have enough points. Sublocales of a locale generalize the concept of subspaces in topology to the point-free setting. A nucleus is an endomorphism of a frame which corresponds to a sublocale.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Nucleus.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Prod", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 1, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -169.699, "z": 5.828, "size": 0.296, "title": "`Filter.atTop` and `Filter.atBot` filters on products", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Prod.html"}, {"id": "Mathlib.Order.Minimal", "region_id": "order", "micro_elevation": 0.3846, "macro_tier": 3, "macro_tier_score": 0.1432, "macro_tier_override": null, "x": -206.423, "z": 17.383, "size": 0.4329, "title": "Minimality and Maximality", "summary": "This file proves basic facts about minimality and maximality of an element with respect to a predicate `P` on an ordered type `α`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Minimal.html"}, {"id": "Mathlib.Order.ConditionallyCompleteLattice.Defs", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 3, "macro_tier_score": 0.2207, "macro_tier_override": null, "x": -167.701, "z": 38.703, "size": 0.5209, "title": "Definitions of conditionally complete lattices", "summary": "A conditionally complete lattice is a lattice in which every non-empty bounded subset `s` has a least upper bound and a greatest lower bound, denoted below by `sSup s` and `sInf s`. Typical examples are `ℝ`, `ℕ`, and `ℤ` with their usual orders. The theory is very comparable to the theory of complete lattices, except that suitable boundedness and nonemptiness assumptions have to be added to most statements. We…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/ConditionallyCompleteLattice/Defs.html"}, {"id": "Mathlib.Order.BooleanAlgebra.Defs", "region_id": "order", "micro_elevation": 0.4231, "macro_tier": 3, "macro_tier_score": 0.3503, "macro_tier_override": null, "x": -200.804, "z": 49.164, "size": 0.4837, "title": "(Generalized) Boolean algebras", "summary": "This file sets up the theory of (generalized) Boolean algebras. A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras generalize the (classical) logic of propositions and the lattice of subsets of a set. Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which do not necessarily have a top element (`⊤`) (and hence not all elements…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/BooleanAlgebra/Defs.html"}, {"id": "Mathlib.Order.BooleanAlgebra.Basic", "region_id": "order", "micro_elevation": 0.4615, "macro_tier": 3, "macro_tier_score": 0.3516, "macro_tier_override": null, "x": -215.073, "z": 24.492, "size": 0.5239, "title": "Basic properties of Boolean algebras", "summary": "This file provides some basic definitions, functions as well as lemmas for functions and type classes related to Boolean algebras as defined in `Mathlib/Order/BooleanAlgebra/Defs.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/BooleanAlgebra/Basic.html"}, {"id": "Mathlib.Order.Filter.Extr", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 1, "macro_tier_score": 0.002, "macro_tier_override": null, "x": -191.969, "z": 64.126, "size": 0.3894, "title": "Minimum and maximum w.r.t. a filter and on a set", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Extr.html"}, {"id": "Mathlib.Order.SemiconjSup", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -161.412, "z": 27.294, "size": 0.2478, "title": "Semiconjugate by `sSup`", "summary": "In this file we prove two facts about semiconjugate (families of) functions. First, if an order isomorphism `fa : α → α` is semiconjugate to an order embedding `fb : β → β` by `g : α → β`, then `fb` is semiconjugate to `fa` by `y ↦ sSup {x | g x ≤ y}`, see `Semiconj.symm_adjoint`. Second, consider two actions `f₁ f₂ : G → α → α` of a group on a complete lattice by order isomorphisms. Then the map `x ↦ ⨆ g : G, (f₁…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SemiconjSup.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Monoid", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 1, "macro_tier_score": 0.0052, "macro_tier_override": null, "x": -186.872, "z": -1.926, "size": 0.2974, "title": "Convergence to ±infinity in ordered commutative monoids", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Monoid.html"}, {"id": "Mathlib.Order.Interval.Set.OrdConnectedComponent", "region_id": "order", "micro_elevation": 0.7308, "macro_tier": 1, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -161.99, "z": 23.631, "size": 0.2879, "title": "Order connected components of a set", "summary": "In this file we define `Set.ordConnectedComponent s x` to be the set of `y` such that `Set.uIcc x y ⊆ s` and prove some basic facts about this definition. At the moment of writing, this construction is used only to prove that any linear order with order topology is a T₅ space, so we only add API needed for this lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/OrdConnectedComponent.html"}, {"id": "Mathlib.Order.Antichain", "region_id": "order", "micro_elevation": 0.6154, "macro_tier": 3, "macro_tier_score": 0.1802, "macro_tier_override": null, "x": -217.022, "z": 48.148, "size": 0.4168, "title": "Antichains", "summary": "This file defines antichains. An antichain is a set where any two distinct elements are not related. If the relation is `(≤)`, this corresponds to incomparability and usual order antichains. If the relation is `G.Adj` for `G : SimpleGraph α`, this corresponds to independent sets of `G`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Antichain.html"}, {"id": "Mathlib.Order.Interval.Set.SuccOrder", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -160.97, "z": 45.895, "size": 0.2692, "title": "Successors in intervals", "summary": "If `j` is an element of a partially ordered set equipped with a successor function, then for any element `i : Set.Iic j` which is not the maximum, we have `↑(Order.succ i) = Order.succ ↑i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Set/SuccOrder.html"}, {"id": "Mathlib.Order.UpperLower.Prod", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -194.746, "z": -8.175, "size": 0.2, "title": "Upper and lower set product", "summary": "The Cartesian product of sets carries over to upper and lower sets in a natural way. This file defines said product over the types `UpperSet` and `LowerSet` and proves some of its properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/UpperLower/Prod.html"}, {"id": "Mathlib.Order.Interval.Multiset", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 3, "macro_tier_score": 0.068, "macro_tier_override": null, "x": -179.981, "z": 64.689, "size": 0.4157, "title": "Intervals as multisets", "summary": "This file defines intervals as multisets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Multiset.html"}, {"id": "Mathlib.Order.Extension.Linear", "region_id": "order", "micro_elevation": 0.5385, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -219.023, "z": 25.702, "size": 0.2, "title": "Extend a partial order to a linear order", "summary": "This file constructs a linear order which is an extension of the given partial order, using Zorn's lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Extension/Linear.html"}, {"id": "Mathlib.Order.Types.Arithmetic", "region_id": "order", "micro_elevation": 0.4231, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -175.669, "z": 34.075, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Types/Arithmetic.html"}, {"id": "Mathlib.Order.SuccPred.Tree", "region_id": "order", "micro_elevation": 0.8462, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -169.531, "z": 1.163, "size": 0.2, "title": "Rooted trees", "summary": "This file proves basic results about rooted trees, represented using the ancestorship order. This is a `PartialOrder`, with `PredOrder` with the immediate parent as a predecessor, and an `OrderBot` which is the root. We also have an `IsPredArchimedean` assumption to prevent infinite dangling chains.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/SuccPred/Tree.html"}, {"id": "Mathlib.Order.Monotone.MonovaryOrder", "region_id": "order", "micro_elevation": 0.3077, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -182.858, "z": 23.132, "size": 0.2, "title": "Interpreting monovarying functions as monotone functions", "summary": "This file proves that monovarying functions to linear orders can be made simultaneously monotone by setting the correct order on their shared indexing type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Monotone/MonovaryOrder.html"}, {"id": "Mathlib.Order.TeichmullerTukey", "region_id": "order", "micro_elevation": 0.5385, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -181.053, "z": 51.239, "size": 0.2, "title": "Teichmuller-Tukey", "summary": "This file defines the notion of being of finite character for a family of sets and proves the Teichmuller-Tukey lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/TeichmullerTukey.html"}, {"id": "Mathlib.Order.Filter.Ultrafilter.Basic", "region_id": "order", "micro_elevation": 0.9231, "macro_tier": 1, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -214.241, "z": 68.357, "size": 0.2968, "title": "Ultrafilters", "summary": "An ultrafilter is a minimal (maximal in the set order) proper filter. In this file we define * `hyperfilter`: the ultrafilter extending the cofinite filter.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/Ultrafilter/Basic.html"}, {"id": "Mathlib.Order.Filter.AtTopBot.Disjoint", "region_id": "order", "micro_elevation": 0.6538, "macro_tier": 2, "macro_tier_score": 0.0153, "macro_tier_override": null, "x": -187.231, "z": 1.627, "size": 0.3497, "title": "Disjointness of `Filter.atTop` and `Filter.atBot`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Filter/AtTopBot/Disjoint.html"}, {"id": "Mathlib.Order.Category.FinBoolAlg", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -213.776, "z": 60.483, "size": 0.2, "title": "The category of finite Boolean algebras", "summary": "This file defines `FinBoolAlg`, the category of finite Boolean algebras.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/FinBoolAlg.html"}, {"id": "Mathlib.Order.Height", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2, "title": "Maximal length of chains", "summary": "This file contains lemmas to work with the maximal lengths of chains of arbitrary relations. See `Order.height` for a definition specialized to finding the height of an element in a preorder.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Height.html"}, {"id": "Mathlib.Order.Comparable", "region_id": "order", "micro_elevation": 0.4231, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -212.938, "z": 23.714, "size": 0.2, "title": "Comparability and incomparability relations", "summary": "Two values in a preorder are said to be comparable (`SymmRel`) whenever `a ≤ b` or `b ≤ a`. We define both the comparability and incomparability relations. In a linear order, `SymmGen (· ≤ ·) a b` is always true, and `IncompRel (· ≤ ·) a b` is always false.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Comparable.html"}, {"id": "Mathlib.Order.Interval.Finset.DenselyOrdered", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -170.282, "z": 2.85, "size": 0.2536, "title": "Linear locally finite orders are densely ordered iff they are trivial", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Interval/Finset/DenselyOrdered.html"}, {"id": "Mathlib.Order.Sublocale", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -208.094, "z": 65.314, "size": 0.2, "title": "Sublocale", "summary": "Locales are the dual concept to frames. Locale theory is a branch of point-free topology, where intuitively locales are like topological spaces which may or may not have enough points. Sublocales of a locale generalize the concept of subspaces in topology to the point-free setting.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Sublocale.html"}, {"id": "Mathlib.Order.Std", "region_id": "order", "micro_elevation": 0.0769, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -197.702, "z": 28.669, "size": 0.2, "title": "Converting Std order typeclasses into Mathlib ones", "summary": "This file provides factories for creating Mathlib order typeclasses (`PartialOrder`, `LinearOrder`) from Std ones. When all instances are present, the factories may be used without arguments: ```lean instance : LinearOrder X := .ofStd X ``` Otherwise, it may be necessary to provide some instances manually: ```lean instance : LinearOrder X := .ofStd X { lawfulOrderOrd := sorry } ``` When existing instances of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Std.html"}, {"id": "Mathlib.Order.Circular.ZMod", "region_id": "order", "micro_elevation": 0.0385, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -193.115, "z": 30.157, "size": 0.2, "title": "The circular order on `ZMod n`", "summary": "This file defines the circular order on `ZMod n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Circular/ZMod.html"}, {"id": "Mathlib.Order.Category.OmegaCompletePartialOrder", "region_id": "order", "micro_elevation": 0.7692, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -161.903, "z": 43.557, "size": 0.2, "title": "Category of types with an omega complete partial order", "summary": "In this file, we bundle the class `OmegaCompletePartialOrder` into a concrete category and prove that continuous functions also form an `OmegaCompletePartialOrder`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Category/OmegaCompletePartialOrder.html"}, {"id": "Mathlib.Order.CompleteLattice.PiLex", "region_id": "order", "micro_elevation": 0.6923, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -176.519, "z": 56.697, "size": 0.2, "title": "Complete linear order instance on lexicographically ordered pi types", "summary": "We show that for `α` a family of complete linear orders, the lexicographically ordered type of dependent functions `Πₗ i, α i` is itself a complete linear order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CompleteLattice/PiLex.html"}, {"id": "Mathlib.Order.CountableSupClosed", "region_id": "order", "micro_elevation": 0.8077, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -159.457, "z": 19.429, "size": 0.2, "title": "Sets closed under countable join/meet", "summary": "This file defines predicates for sets closed under countable supremum and dually for countable infimum.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/CountableSupClosed.html"}, {"id": "Mathlib.Order.KonigLemma", "region_id": "order", "micro_elevation": 0.8846, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -233.942, "z": 19.8, "size": 0.2, "title": "Kőnig's infinity lemma", "summary": "Kőnig's infinity lemma is most often stated as a graph theory result: every infinite, locally finite connected graph contains an infinite path. It has links to computability and proof theory, and it has a number of formulations. In practice, most applications are not to an abstract graph, but to a concrete collection of objects that are organized in a graph-like way, often where the graph is a rooted tree…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/KonigLemma.html"}, {"id": "Mathlib.Order.Rel.GaloisConnection", "region_id": "order", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -194.801, "z": 30.685, "size": 0.2, "title": "The Galois Connection Induced by a Relation", "summary": "In this file, we show that an arbitrary relation `R` between a pair of types `α` and `β` defines a pair `toDual ∘ R.leftDual` and `R.rightDual ∘ ofDual` of adjoint order-preserving maps between the corresponding posets `Set α` and `(Set β)ᵒᵈ`. We define `R.leftFixedPoints` (resp. `R.rightFixedPoints`) as the set of fixed points `J` (resp. `I`) of `Set α` (resp. `Set β`) such that `rightDual (leftDual J) = J` (resp.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Rel/GaloisConnection.html"}, {"id": "Mathlib.Order.Set", "region_id": "order", "micro_elevation": 0.0769, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -196.669, "z": 27.686, "size": 0.2, "title": "`Set.range` on `WithBot` and `WithTop`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Set.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSeminorm.Indicator", "region_id": "measure_theory", "micro_elevation": 0.537, "macro_tier": 3, "macro_tier_score": 0.2817, "macro_tier_override": null, "x": -204.786, "z": -84.054, "size": 0.2886, "title": "ℒp seminorms and indicator functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSeminorm/Indicator.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "region_id": "measure_theory", "micro_elevation": 0.5185, "macro_tier": 4, "macro_tier_score": 0.2848, "macro_tier_override": null, "x": -201.996, "z": -89.333, "size": 0.4528, "title": "Basic theorems about ℒp space", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSeminorm/Basic.html"}, {"id": "Mathlib.MeasureTheory.Integral.Lebesgue.Sub", "region_id": "measure_theory", "micro_elevation": 0.463, "macro_tier": 3, "macro_tier_score": 0.282, "macro_tier_override": null, "x": -179.709, "z": -57.382, "size": 0.3139, "title": "Subtraction of Lebesgue integrals", "summary": "In this file we first show that Lebesgue integrals can be subtracted with the expected results – `∫⁻ f - ∫⁻ g ≤ ∫⁻ (f - g)`, with equality if `g ≤ f` almost everywhere. Then we prove variants of the monotone convergence theorem that use this subtraction in their proofs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Lebesgue/Sub.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.Constructions", "region_id": "measure_theory", "micro_elevation": 0.037, "macro_tier": 4, "macro_tier_score": 0.412, "macro_tier_override": null, "x": -181.365, "z": -77.042, "size": 0.5212, "title": "Constructions for measurable spaces and functions", "summary": "This file provides several ways to construct new measurable spaces and functions from old ones: `Quotient`, `Subtype`, `Prod`, `Pi`, etc.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/Constructions.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.Basic", "region_id": "measure_theory", "micro_elevation": 0.0185, "macro_tier": 4, "macro_tier_score": 0.407, "macro_tier_override": null, "x": -180.879, "z": -79.456, "size": 0.3155, "title": "Measurable spaces and measurable functions", "summary": "This file provides properties of measurable spaces and the functions and isomorphisms between them. The definition of a measurable space is in `Mathlib/MeasureTheory/MeasurableSpace/Defs.lean`. A measurable space is a set equipped with a σ-algebra, a collection of subsets closed under complementation and countable union. A function between measurable spaces is measurable if the preimage of each measurable subset is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/Basic.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.Instances", "region_id": "measure_theory", "micro_elevation": 0.0185, "macro_tier": 4, "macro_tier_score": 0.407, "macro_tier_override": null, "x": -180.267, "z": -77.905, "size": 0.3155, "title": "Measurable-space typeclass instances", "summary": "This file provides measurable-space instances for a selection of standard countable types, in each case defining the Σ-algebra to be `⊤` (the discrete measurable-space structure).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/Instances.html"}, {"id": "Mathlib.MeasureTheory.Constructions.Projective", "region_id": "measure_theory", "micro_elevation": 0.3148, "macro_tier": 1, "macro_tier_score": 0.0316, "macro_tier_override": null, "x": -166.302, "z": -77.675, "size": 0.2847, "title": "Projective measure families and projective limits", "summary": "A family of measures indexed by finite sets of `ι` is projective if, for finite sets `J ⊆ I`, the projection from `∀ i : I, α i` to `∀ i : J, α i` maps `P I` to `P J`. A measure `μ` is the projective limit of such a family of measures if for all `I : Finset ι`, the projection from `∀ i, α i` to `∀ i : I, α i` maps `μ` to `P I`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/Projective.html"}, {"id": "Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.NNReal", "region_id": "measure_theory", "micro_elevation": 0.8519, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -219.733, "z": -75.592, "size": 0.2, "title": "Riesz–Markov–Kakutani representation theorem for `ℝ≥0`", "summary": "This file proves the Riesz-Markov-Kakutani representation theorem on a locally compact T2 space `X` for `ℝ≥0`-linear functionals `Λ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/NNReal.html"}, {"id": "Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real", "region_id": "measure_theory", "micro_elevation": 0.8333, "macro_tier": 2, "macro_tier_score": 0.0629, "macro_tier_override": null, "x": -195.016, "z": -43.115, "size": 0.2894, "title": "Riesz–Markov–Kakutani representation theorem for real-linear functionals", "summary": "The Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures. There are many closely related variations of the theorem. This file contains the proof of the version where the space is a locally compact T2 space, the linear functionals are real and the continuous functions have compact support.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Real.html"}, {"id": "Mathlib.MeasureTheory.Measure.Haar.Disintegration", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -160.888, "z": -116.23, "size": 0.2338, "title": "Pushing a Haar measure by a linear map", "summary": "We show that the push-forward of an additive Haar measure in a vector space under a surjective linear map is proportional to the Haar measure on the target space, in `LinearMap.exists_map_addHaar_eq_smul_addHaar`. We deduce disintegration properties of the Haar measure: to check that a property is true ae, it suffices to check that it is true ae along all translates of a given vector subspace. See…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Haar/Disintegration.html"}, {"id": "Mathlib.MeasureTheory.Measure.Haar.Unique", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 2, "macro_tier_score": 0.1281, "macro_tier_override": null, "x": -216.683, "z": -57.549, "size": 0.4372, "title": "Uniqueness of Haar measure in locally compact groups", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Haar/Unique.html"}, {"id": "Mathlib.MeasureTheory.Integral.CircleAverage", "region_id": "measure_theory", "micro_elevation": 0.9444, "macro_tier": 1, "macro_tier_score": 0.0009, "macro_tier_override": null, "x": -140.738, "z": -95.413, "size": 0.3276, "title": "Circle Averages", "summary": "For a function `f` on the complex plane, this file introduces the definition `Real.circleAverage f c R` as a shorthand for the average of `f` on the circle with center `c` and radius `R`, equipped with the rotation-invariant measure of total volume one. Like `IntervalAverage`, this notion exists as a convenience. It avoids notationally inconvenient compositions of `f` with `circleMap` and avoids the need to manually…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/CircleAverage.html"}, {"id": "Mathlib.MeasureTheory.Integral.CircleIntegral", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 2, "macro_tier_score": 0.0328, "macro_tier_override": null, "x": -145.44, "z": -102.345, "size": 0.3693, "title": "Integral over a circle in `ℂ`", "summary": "In this file we define `∮ z in C(c, R), f z` to be the integral $\\oint_{|z-c|=|R|} f(z)\\,dz$ and prove some properties of this integral. We give definition and prove most lemmas for a function `f : ℂ → E`, where `E` is a complex Banach space. For this reason, some lemmas use, e.g., `(z - c)⁻¹ • f z` instead of `f z / (z - c)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/CircleIntegral.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntervalAverage", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 2, "macro_tier_score": 0.0318, "macro_tier_override": null, "x": -168.185, "z": -116.572, "size": 0.2956, "title": "Integral average over an interval", "summary": "In this file we introduce notation `⨍ x in a..b, f x` for the average `⨍ x in Ι a b, f x` of `f` over the interval `Ι a b = Set.Ioc (min a b) (max a b)` w.r.t. the Lebesgue measure, then prove formulas for this average: * `interval_average_eq`: `⨍ x in a..b, f x = (b - a)⁻¹ • ∫ x in a..b, f x`; * `interval_average_eq_div`: `⨍ x in a..b, f x = (∫ x in a..b, f x) / (b - a)`; * `exists_eq_interval_average_of_measure`:…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntervalAverage.html"}, {"id": "Mathlib.MeasureTheory.Group.Arithmetic", "region_id": "measure_theory", "micro_elevation": 0.3519, "macro_tier": 4, "macro_tier_score": 0.3159, "macro_tier_override": null, "x": -173.141, "z": -64.353, "size": 0.4467, "title": "Typeclasses for measurability of operations", "summary": "In this file we define classes `MeasurableMul` etc. and prove dot-style lemmas (`Measurable.mul`, `AEMeasurable.mul` etc). For binary operations we define two typeclasses: - `MeasurableMul` says that both left and right multiplication are measurable; - `MeasurableMul₂` says that `fun p : α × α => p.1 * p.2` is measurable, and similarly for other binary operations. The reason for introducing these classes is that in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/Arithmetic.html"}, {"id": "Mathlib.MeasureTheory.Measure.AEMeasurable", "region_id": "measure_theory", "micro_elevation": 0.3333, "macro_tier": 4, "macro_tier_score": 0.3174, "macro_tier_override": null, "x": -190.838, "z": -67.123, "size": 0.4968, "title": "Almost everywhere measurable functions", "summary": "A function is almost everywhere measurable if it coincides almost everywhere with a measurable function. This property, called `AEMeasurable f μ`, is defined in the file `MeasureSpaceDef`. We discuss several of its properties that are analogous to properties of measurable functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/AEMeasurable.html"}, {"id": "Mathlib.MeasureTheory.Integral.Bochner.Set", "region_id": "measure_theory", "micro_elevation": 0.8148, "macro_tier": 4, "macro_tier_score": 0.2854, "macro_tier_override": null, "x": -208.946, "z": -103.197, "size": 0.4739, "title": "Set integral", "summary": "In this file we prove some properties of `∫ x in s, f x ∂μ`. Recall that this notation is defined as `∫ x, f x ∂(μ.restrict s)`. In `integral_indicator` we prove that for a measurable function `f` and a measurable set `s` this definition coincides with another natural definition: `∫ x, indicator s f x ∂μ = ∫ x in s, f x ∂μ`, where `indicator s f x` is equal to `f x` for `x ∈ s` and is zero otherwise. Since `∫ x in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Bochner/Set.html"}, {"id": "Mathlib.MeasureTheory.Integral.CompactlySupported", "region_id": "measure_theory", "micro_elevation": 0.7963, "macro_tier": 2, "macro_tier_score": 0.0627, "macro_tier_override": null, "x": -215.278, "z": -66.631, "size": 0.2532, "title": "Integrating compactly supported continuous functions", "summary": "This file contains definitions and lemmas related to integrals of compactly supported continuous functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/CompactlySupported.html"}, {"id": "Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic", "region_id": "measure_theory", "micro_elevation": 0.4259, "macro_tier": 2, "macro_tier_score": 0.0627, "macro_tier_override": null, "x": -181.307, "z": -59.067, "size": 0.2532, "title": "Riesz–Markov–Kakutani representation theorem", "summary": "This file prepares technical definitions and results for the Riesz-Markov-Kakutani representation theorem on a locally compact T2 space `X`. As a special case, the statements about linear functionals on bounded continuous functions follows. Actual theorems, depending on the linearity (`ℝ`, `ℝ≥0` or `ℂ`), are proven in separate files (`Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Real.lean`,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Basic.html"}, {"id": "Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym", "region_id": "measure_theory", "micro_elevation": 0.8519, "macro_tier": 2, "macro_tier_score": 0.0961, "macro_tier_override": null, "x": -155.242, "z": -108.299, "size": 0.4053, "title": "Radon-Nikodym theorem", "summary": "This file proves the Radon-Nikodym theorem. The Radon-Nikodym theorem states that, given measures `μ, ν`, if `HaveLebesgueDecomposition μ ν`, then `μ` is absolutely continuous with respect to `ν` if and only if there exists a measurable function `f : α → ℝ≥0∞` such that `μ = fν`. In particular, we have `f = rnDeriv μ ν`. The Radon-Nikodym theorem will allow us to define many important concepts in probability theory,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Decomposition/RadonNikodym.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSpace.DomAct.Basic", "region_id": "measure_theory", "micro_elevation": 0.6296, "macro_tier": 1, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -207.573, "z": -89.367, "size": 0.239, "title": "Action of `Mᵈᵐᵃ` on `Lᵖ` spaces", "summary": "In this file we define action of `Mᵈᵐᵃ` on `MeasureTheory.Lp E p μ` If `f : α → E` is a function representing an equivalence class in `Lᵖ(α, E)`, `M` acts on `α`, and `c : M`, then `(.mk c : Mᵈᵐᵃ) • [f]` is represented by the function `a ↦ f (c • a)`. We also prove basic properties of this action.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.html"}, {"id": "Mathlib.MeasureTheory.Function.AEEqFun.DomAct", "region_id": "measure_theory", "micro_elevation": 0.5926, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -193.134, "z": -54.396, "size": 0.2541, "title": "Action of `DomMulAct` and `DomAddAct` on `α →ₘ[μ] β`", "summary": "If `M` acts on `α` by measure-preserving transformations, then `Mᵈᵐᵃ` acts on `α →ₘ[μ] β` by sending each function `f` to `f ∘ (DomMulAct.mk.symm c • ·)`. We define this action and basic instances about it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/AEEqFun/DomAct.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSpace.Indicator", "region_id": "measure_theory", "micro_elevation": 0.6111, "macro_tier": 4, "macro_tier_score": 0.2841, "macro_tier_override": null, "x": -208.723, "z": -80.626, "size": 0.4282, "title": "Indicator of a set as an element of `Lp`", "summary": "For a set `s` with `(hs : MeasurableSet s)` and `(hμs : μ s < ∞)`, we build `indicatorConstLp p hs hμs c`, the element of `Lp` corresponding to `s.indicator (fun _ => c)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSpace/Indicator.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSeminorm.Count", "region_id": "measure_theory", "micro_elevation": 0.5556, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -205.27, "z": -85.598, "size": 0.2, "title": "`L^p`-seminorms on `count` and `dirac`", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSeminorm/Count.html"}, {"id": "Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv", "region_id": "measure_theory", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0316, "macro_tier_override": null, "x": -180.729, "z": -78.619, "size": 0.275, "title": "Integrals of functions of Radon-Nikodym derivatives", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Decomposition/IntegralRNDeriv.html"}, {"id": "Mathlib.MeasureTheory.Measure.LogLikelihoodRatio", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 1, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -216.813, "z": -59.531, "size": 0.2579, "title": "Log-likelihood Ratio", "summary": "The likelihood ratio between two measures `μ` and `ν` is their Radon-Nikodym derivative `μ.rnDeriv ν`. The logarithm of that function is often used instead: this is the log-likelihood ratio. This file contains a definition of the log-likelihood ratio (llr) and its properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.html"}, {"id": "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "region_id": "measure_theory", "micro_elevation": 0.6296, "macro_tier": 3, "macro_tier_score": 0.2827, "macro_tier_override": null, "x": -188.722, "z": -106.408, "size": 0.363, "title": "Convergence in measure", "summary": "We define convergence in measure which is one of the many notions of convergence in probability. A sequence of functions `f` is said to converge in measure to some function `g` if for all `ε > 0`, the measure of the set `{x | ε ≤ edist (f i x) (g x)}` tends to 0 as `i` converges along some given filter `l`. Convergence in measure is most notably used in the formulation of the weak law of large numbers and is also…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConvergenceInMeasure.html"}, {"id": "Mathlib.MeasureTheory.Function.Egorov", "region_id": "measure_theory", "micro_elevation": 0.4815, "macro_tier": 3, "macro_tier_score": 0.282, "macro_tier_override": null, "x": -202.021, "z": -84.587, "size": 0.3178, "title": "Egorov theorem", "summary": "This file contains the Egorov theorem which states that an almost everywhere convergent sequence on a finite measure space converges uniformly except on an arbitrarily small set. This theorem is useful for the Vitali convergence theorem as well as theorems regarding convergence in measure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/Egorov.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSpace.Complete", "region_id": "measure_theory", "micro_elevation": 0.6111, "macro_tier": 3, "macro_tier_score": 0.282, "macro_tier_override": null, "x": -196.98, "z": -101.501, "size": 0.3178, "title": "`Lp` is a complete space", "summary": "In this file we show that `Lp` is a complete space for `1 ≤ p`, in `MeasureTheory.Lp.instCompleteSpace`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSpace/Complete.html"}, {"id": "Mathlib.MeasureTheory.Group.GeometryOfNumbers", "region_id": "measure_theory", "micro_elevation": 0.8519, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -142.082, "z": -72.547, "size": 0.2461, "title": "Geometry of numbers", "summary": "In this file we prove some of the fundamental theorems in the geometry of numbers, as studied by Hermann Minkowski.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/GeometryOfNumbers.html"}, {"id": "Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -155.681, "z": -44.256, "size": 0.2461, "title": "Volume of balls", "summary": "Let `E` be a finite-dimensional normed `ℝ`-vector space equipped with a Haar measure `μ`. We prove that `μ (Metric.ball 0 1) = (∫ (x : E), Real.exp (- ‖x‖ ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)` for any real number `p` with `0 < p`, see `MeasureTheory.measure_unitBall_eq_integral_div_gamma`. We also prove the corresponding result to compute `μ {x : E | g x < 1}` where `g : E → ℝ` is a function defining a norm…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.html"}, {"id": "Mathlib.MeasureTheory.Integral.Prod", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 2, "macro_tier_score": 0.1281, "macro_tier_override": null, "x": -189.826, "z": -38.823, "size": 0.4363, "title": "Integration with respect to the product measure", "summary": "In this file we prove Fubini's theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Prod.html"}, {"id": "Mathlib.MeasureTheory.Group.Integral", "region_id": "measure_theory", "micro_elevation": 0.7963, "macro_tier": 2, "macro_tier_score": 0.1262, "macro_tier_override": null, "x": -210.31, "z": -100.121, "size": 0.347, "title": "Bochner Integration on Groups", "summary": "We develop properties of integrals with a group as domain. This file contains properties about integrability and Bochner integration.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/Integral.html"}, {"id": "Mathlib.MeasureTheory.Measure.DiracProba", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -171.886, "z": -118.472, "size": 0.2, "title": "Dirac deltas as probability measures and embedding of a space into probability measures on it", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/DiracProba.html"}, {"id": "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 2, "macro_tier_score": 0.0629, "macro_tier_override": null, "x": -210.071, "z": -51.475, "size": 0.2808, "title": "Probability measures", "summary": "This file defines the type of probability measures on a given measurable space. When the underlying space has a topology and the measurable space structure (sigma algebra) is finer than the Borel sigma algebra, then the type of probability measures is equipped with the topology of convergence in distribution (weak convergence of measures). The topology of convergence in distribution is the coarsest topology w.r.t.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/ProbabilityMeasure.html"}, {"id": "Mathlib.MeasureTheory.Function.ConvergenceInDistribution", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -221.819, "z": -85.567, "size": 0.2338, "title": "Convergence in distribution", "summary": "We introduce a definition of convergence in distribution of random variables: this is the weak convergence of the laws of the random variables. In Mathlib terms this is a `Tendsto` in the `ProbabilityMeasure` type. We also state results relating convergence in probability (`TendstoInMeasure`) and convergence in distribution.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConvergenceInDistribution.html"}, {"id": "Mathlib.MeasureTheory.Measure.Portmanteau", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 2, "macro_tier_score": 0.063, "macro_tier_override": null, "x": -177.856, "z": -119.341, "size": 0.2985, "title": "Characterizations of weak convergence of finite measures and probability measures", "summary": "This file will provide portmanteau characterizations of the weak convergence of finite measures and of probability measures, i.e., the standard characterizations of convergence in distribution.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Portmanteau.html"}, {"id": "Mathlib.MeasureTheory.Covering.OneDim", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 1, "macro_tier_score": 0.0315, "macro_tier_override": null, "x": -209.133, "z": -109.113, "size": 0.2641, "title": "Covering theorems for Lebesgue measure in one dimension", "summary": "We have a general theory of covering theorems for doubling measures, developed notably in `DensityTheorem.lean`. In this file, we expand the API for this theory in one dimension, by showing that intervals belong to the relevant Vitali family.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Covering/OneDim.html"}, {"id": "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "region_id": "measure_theory", "micro_elevation": 0.3704, "macro_tier": 4, "macro_tier_score": 0.317, "macro_tier_override": null, "x": -179.088, "z": -61.689, "size": 0.4853, "title": "Borel (measurable) space", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSpace.Basic", "region_id": "measure_theory", "micro_elevation": 0.5926, "macro_tier": 3, "macro_tier_score": 0.2826, "macro_tier_override": null, "x": -155.084, "z": -69.509, "size": 0.354, "title": "Lp space", "summary": "This file provides the space `Lp E p μ` as the subtype of elements of `α →ₘ[μ] E` (see `MeasureTheory.AEEqFun`) such that `eLpNorm f p μ` is finite. For `1 ≤ p`, `eLpNorm` defines a norm and `Lp` is a complete metric space (the latter is proved at `Mathlib/MeasureTheory/Function/LpSpace/Complete.lean`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSpace/Basic.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "region_id": "measure_theory", "micro_elevation": 0.537, "macro_tier": 3, "macro_tier_score": 0.2816, "macro_tier_override": null, "x": -187.204, "z": -102.418, "size": 0.2791, "title": "Chebyshev-Markov inequality in terms of Lp seminorms", "summary": "In this file we formulate several versions of the Chebyshev-Markov inequality in terms of the `MeasureTheory.eLpNorm` seminorm.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "region_id": "measure_theory", "micro_elevation": 0.5741, "macro_tier": 3, "macro_tier_score": 0.2816, "macro_tier_override": null, "x": -162.962, "z": -59.141, "size": 0.2791, "title": "Compare Lp seminorms for different values of `p`", "summary": "In this file we compare `MeasureTheory.eLpNorm'` and `MeasureTheory.eLpNorm` for different exponents.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "region_id": "measure_theory", "micro_elevation": 0.537, "macro_tier": 3, "macro_tier_score": 0.2816, "macro_tier_override": null, "x": -156.169, "z": -80.88, "size": 0.2791, "title": "Triangle inequality for `Lp`-seminorm", "summary": "In this file we prove several versions of the triangle inequality for the `Lp` seminorm, as well as simple corollaries.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 2, "macro_tier_score": 0.0634, "macro_tier_override": null, "x": -164.74, "z": -117.103, "size": 0.3239, "title": "Integrals of periodic functions", "summary": "In this file we prove that the half-open interval `Ioc t (t + T)` in `ℝ` is a fundamental domain of the action of the subgroup `ℤ ∙ T` on `ℝ`. A consequence is `AddCircle.measurePreserving_mk`: the covering map from `ℝ` to the \"additive circle\" `ℝ ⧸ (ℤ ∙ T)` is measure-preserving, with respect to the restriction of Lebesgue measure to `Ioc t (t + T)` (upstairs) and with respect to Haar measure (downstairs). Another…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntervalIntegral/Periodic.html"}, {"id": "Mathlib.MeasureTheory.Integral.ExpDecay", "region_id": "measure_theory", "micro_elevation": 1.0, "macro_tier": 1, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": -135.522, "z": -86.71, "size": 0.3012, "title": "Integrals with exponential decay at ∞", "summary": "As easy special cases of general theorems in the library, we prove the following test for integrability: * `integrable_of_isBigO_exp_neg`: If `f` is continuous on `[a,∞)`, for some `a ∈ ℝ`, and there exists `b > 0` such that `f(x) = O(exp(-b x))` as `x → ∞`, then `f` is integrable on `(a, ∞)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/ExpDecay.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.Pi", "region_id": "measure_theory", "micro_elevation": 0.0556, "macro_tier": 3, "macro_tier_score": 0.2195, "macro_tier_override": null, "x": -183.268, "z": -78.873, "size": 0.3126, "title": "Bases of the indexed product σ-algebra", "summary": "In this file we prove several versions of the following lemma: given a finite indexed collection of measurable spaces `α i`, if the σ-algebra on each `α i` is generated by `C i`, then the sets `{x | ∀ i, x i ∈ s i}`, where `s i ∈ C i`, generate the σ-algebra on the indexed product of `α i`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/Pi.html"}, {"id": "Mathlib.MeasureTheory.Constructions.BorelSpace.WithTop", "region_id": "measure_theory", "micro_elevation": 0.3889, "macro_tier": 1, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": -165.055, "z": -70.058, "size": 0.3043, "title": "Borel measurable space on `WithTop`", "summary": "For `ι` a linear order with the order topology, we define the Borel measurable space on `WithTop ι`. We then prove that the natural inclusion `ι → WithTop ι` is measurable, and that the function `WithTop.untopA : WithTop ι → ι` (which sends `⊤` to an arbitrary element of `ι`) is measurable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/BorelSpace/WithTop.html"}, {"id": "Mathlib.MeasureTheory.OuterMeasure.AE", "region_id": "measure_theory", "micro_elevation": 0.037, "macro_tier": 4, "macro_tier_score": 0.3793, "macro_tier_override": null, "x": -179.448, "z": -77.501, "size": 0.4791, "title": "The “almost everywhere” filter of co-null sets.", "summary": "If `μ` is an outer measure or a measure on `α`, then `MeasureTheory.ae μ` is the filter of co-null sets: `s ∈ ae μ ↔ μ sᶜ = 0`. In this file we define the filter and prove some basic theorems about it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/OuterMeasure/AE.html"}, {"id": "Mathlib.MeasureTheory.OuterMeasure.Basic", "region_id": "measure_theory", "micro_elevation": 0.0185, "macro_tier": 4, "macro_tier_score": 0.3793, "macro_tier_override": null, "x": -181.299, "z": -79.251, "size": 0.4795, "title": "Outer Measures", "summary": "An outer measure is a function `μ : Set α → ℝ≥0∞`, from the powerset of a type to the extended nonnegative real numbers that satisfies the following conditions: 1. `μ ∅ = 0`; 2. `μ` is monotone; 3. `μ` is countably subadditive. This means that the outer measure of a countable union is at most the sum of the outer measure on the individual sets. Note that we do not need `α` to be measurable to define an outer measure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/OuterMeasure/Basic.html"}, {"id": "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "region_id": "measure_theory", "micro_elevation": 0.8333, "macro_tier": 2, "macro_tier_score": 0.0318, "macro_tier_override": null, "x": -214.977, "z": -61.538, "size": 0.2985, "title": "Basic properties of Haar measures on real vector spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Haar/NormedSpace.html"}, {"id": "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "region_id": "measure_theory", "micro_elevation": 0.6481, "macro_tier": 2, "macro_tier_score": 0.0637, "macro_tier_override": null, "x": -206.759, "z": -64.182, "size": 0.3461, "title": "Volume forms and measures on inner product spaces", "summary": "A volume form induces a Lebesgue measure on general finite-dimensional real vector spaces. In this file, we discuss the specific situation of inner product spaces, where an orientation gives rise to a canonical volume form. We show that the measure coming from this volume form gives measure `1` to the parallelepiped spanned by any orthonormal basis, and that it coincides with the canonical `volume` from the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.html"}, {"id": "Mathlib.MeasureTheory.Function.EssSup", "region_id": "measure_theory", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.2816, "macro_tier_override": null, "x": -180.729, "z": -78.619, "size": 0.2843, "title": "Essential supremum and infimum", "summary": "We define the essential supremum and infimum of a function `f : α → β` with respect to a measure `μ` on `α`. The essential supremum is the infimum of the constants `c : β` such that `f x ≤ c` almost everywhere. TODO: The essential supremum of functions `α → ℝ≥0∞` is used in particular to define the norm in the `L∞` space (see `Mathlib/MeasureTheory/Function/LpSeminorm/Defs.lean`). There is a different quantity which…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/EssSup.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSeminorm.SMul", "region_id": "measure_theory", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.2814, "macro_tier_override": null, "x": -158.586, "z": -91.293, "size": 0.2574, "title": "Scalar multiplication on ℒp space", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSeminorm/SMul.html"}, {"id": "Mathlib.MeasureTheory.Integral.MeanInequalities", "region_id": "measure_theory", "micro_elevation": 0.463, "macro_tier": 3, "macro_tier_score": 0.2817, "macro_tier_override": null, "x": -164.568, "z": -64.804, "size": 0.2907, "title": "Mean value inequalities for integrals", "summary": "In this file we prove several inequalities on integrals, notably the Hölder inequality and the Minkowski inequality. The versions for finite sums are in `Analysis.MeanInequalities`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/MeanInequalities.html"}, {"id": "Mathlib.MeasureTheory.Covering.LiminfLimsup", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -212.505, "z": -105.581, "size": 0.2478, "title": "Liminf, limsup, and uniformly locally doubling measures.", "summary": "This file is a place to collect lemmas about liminf and limsup for subsets of a metric space carrying a uniformly locally doubling measure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Covering/LiminfLimsup.html"}, {"id": "Mathlib.MeasureTheory.Covering.DensityTheorem", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 2, "macro_tier_score": 0.0632, "macro_tier_override": null, "x": -184.776, "z": -37.998, "size": 0.3087, "title": "Uniformly locally doubling measures and Lebesgue's density theorem", "summary": "Lebesgue's density theorem states that given a set `S` in a sigma compact metric space with locally-finite uniformly locally doubling measure `μ` then for almost all points `x` in `S`, for any sequence of closed balls `B₀, B₁, B₂, ...` containing `x`, the limit `μ (S ∩ Bⱼ) / μ (Bⱼ) → 1` as `j → ∞`. In this file we combine general results about existence of Vitali families for uniformly locally doubling measures with…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Covering/DensityTheorem.html"}, {"id": "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "region_id": "measure_theory", "micro_elevation": 0.8148, "macro_tier": 2, "macro_tier_score": 0.0629, "macro_tier_override": null, "x": -143.317, "z": -79.426, "size": 0.2842, "title": "Spaces where indicators of closed sets have decreasing approximations by continuous functions", "summary": "In this file we define a typeclass `HasOuterApproxClosed` for topological spaces in which indicator functions of closed sets have sequences of bounded continuous functions approximating them from above. All pseudo-emetrizable spaces have this property, see `instHasOuterApproxClosed`. In spaces with the `HasOuterApproxClosed` property, finite Borel measures are uniquely characterized by the integrals of bounded…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.html"}, {"id": "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "region_id": "measure_theory", "micro_elevation": 0.7963, "macro_tier": 2, "macro_tier_score": 0.0628, "macro_tier_override": null, "x": -161.668, "z": -109.829, "size": 0.2695, "title": "Integration of bounded continuous functions", "summary": "In this file, some results are collected about integrals of bounded continuous functions. They are mostly specializations of results in general integration theory, but they are used directly in this specialized form in some other files, in particular in those related to the topology of weak convergence of probability measures and finite measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntegrableOn", "region_id": "measure_theory", "micro_elevation": 0.6852, "macro_tier": 3, "macro_tier_score": 0.2838, "macro_tier_override": null, "x": -151.682, "z": -90.722, "size": 0.4151, "title": "Functions integrable on a set and at a filter", "summary": "We define `IntegrableOn f s μ := Integrable f (μ.restrict s)` and prove theorems like `integrableOn_union : IntegrableOn f (s ∪ t) μ ↔ IntegrableOn f s μ ∧ IntegrableOn f t μ`. Next we define a predicate `IntegrableAtFilter (f : α → E) (l : Filter α) (μ : Measure α)` saying that `f` is integrable at some set `s ∈ l` and prove that a measurable function is integrable at `l` with respect to `μ` provided that `f` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntegrableOn.html"}, {"id": "Mathlib.MeasureTheory.Function.SpecialFunctions.Arctan", "region_id": "measure_theory", "micro_elevation": 0.3889, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -169.341, "z": -92.378, "size": 0.2, "title": "Measurability of arctan", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/SpecialFunctions/Arctan.html"}, {"id": "Mathlib.MeasureTheory.OuterMeasure.OfAddContent", "region_id": "measure_theory", "micro_elevation": 0.3333, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -195.082, "z": -83.942, "size": 0.2445, "title": "Carathéodory's extension theorem", "summary": "Let `C` be a semiring of sets and `m` an additive content on `C`, which is sigma-subadditive. Then all sets in the sigma-algebra generated by `C` are Carathéodory measurable with respect to the outer measure induced by `m`. The induced outer measure is equal to `m` on `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/OuterMeasure/OfAddContent.html"}, {"id": "Mathlib.MeasureTheory.Measure.AddContent", "region_id": "measure_theory", "micro_elevation": 0.1111, "macro_tier": 2, "macro_tier_score": 0.0941, "macro_tier_override": null, "x": -180.509, "z": -83.717, "size": 0.2816, "title": "Additive Contents", "summary": "An additive content `m` on a set of sets `C` is a set function with value 0 at the empty set which is finitely additive on `C`. That means that for any finset `I` of pairwise disjoint sets in `C` such that `⋃₀ I ∈ C`, `m (⋃₀ I) = ∑ s ∈ I, m s`. Mathlib also has a definition of contents over compact sets: see `MeasureTheory.Content`. A `Content` is in particular an `AddContent` on the set of compact sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/AddContent.html"}, {"id": "Mathlib.MeasureTheory.Measure.Trim", "region_id": "measure_theory", "micro_elevation": 0.3148, "macro_tier": 4, "macro_tier_score": 0.3461, "macro_tier_override": null, "x": -182.052, "z": -64.222, "size": 0.4069, "title": "Restriction of a measure to a sub-σ-algebra", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Trim.html"}, {"id": "Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue", "region_id": "measure_theory", "micro_elevation": 0.5926, "macro_tier": 3, "macro_tier_score": 0.1905, "macro_tier_override": null, "x": -191.58, "z": -103.577, "size": 0.4317, "title": "Lebesgue decomposition", "summary": "This file proves the Lebesgue decomposition theorem. The Lebesgue decomposition theorem states that, given two σ-finite measures `μ` and `ν`, there exists a σ-finite measure `ξ` and a measurable function `f` such that `μ = ξ + fν` and `ξ` is mutually singular with respect to `ν`. The Lebesgue decomposition provides the Radon-Nikodym theorem readily.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Decomposition/Lebesgue.html"}, {"id": "Mathlib.MeasureTheory.Measure.Prod", "region_id": "measure_theory", "micro_elevation": 0.5185, "macro_tier": 3, "macro_tier_score": 0.2826, "macro_tier_override": null, "x": -160.108, "z": -66.71, "size": 0.3574, "title": "The product measure", "summary": "In this file we define and prove properties about the binary product measure. If `α` and `β` have s-finite measures `μ` resp. `ν` then `α × β` can be equipped with an s-finite measure `μ.prod ν` that satisfies `(μ.prod ν) s = ∫⁻ x, ν {y | (x, y) ∈ s} ∂μ`. We also have `(μ.prod ν) (s ×ˢ t) = μ s * ν t`, i.e. the measure of a rectangle is the product of the measures of the sides. We also prove Tonelli's theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Prod.html"}, {"id": "Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 2, "macro_tier_score": 0.0319, "macro_tier_override": null, "x": -207.493, "z": -45.575, "size": 0.3075, "title": "Characteristic Function of a Finite Measure", "summary": "This file defines the characteristic function of a finite measure on a topological vector space `V`. The characteristic function of a finite measure `P` on `V` is the mapping `W → ℂ, w => ∫ v, e (L v w) ∂P`, where `e` is a continuous additive character and `L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ` is a bilinear map. A typical example is `V = W = ℝ` and `L v w = v * w`. The integral is expressed as `∫ v, char he hL w v ∂P`, where…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/CharacteristicFunction/Basic.html"}, {"id": "Mathlib.MeasureTheory.Integral.TorusIntegral", "region_id": "measure_theory", "micro_elevation": 0.9444, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -137.39, "z": -80.346, "size": 0.2, "title": "Integral over a torus in `ℂⁿ`", "summary": "In this file we define the integral of a function `f : ℂⁿ → E` over a torus `{z : ℂⁿ | ∀ i, z i ∈ Metric.sphere (c i) (R i)}`. In order to do this, we define `torusMap (c : ℂⁿ) (R θ : ℝⁿ)` to be the point in `ℂⁿ` given by $z_k=c_k+R_ke^{θ_ki}$, where $i$ is the imaginary unit, then define `torusIntegral f c R` as the integral over the cube $[0, (fun _ ↦ 2π)] = \\{θ\\|∀ k, 0 ≤ θ_k ≤ 2π\\}$ of the Jacobian of the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/TorusIntegral.html"}, {"id": "Mathlib.MeasureTheory.OuterMeasure.Defs", "region_id": "measure_theory", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.377, "macro_tier_override": null, "x": -180.729, "z": -78.619, "size": 0.3914, "title": "Definitions of an outer measure and the corresponding `FunLike` class", "summary": "In this file we define `MeasureTheory.OuterMeasure α` to be the type of outer measures on `α`. An outer measure is a function `μ : Set α → ℝ≥0∞`, from the powerset of a type to the extended nonnegative real numbers that satisfies the following conditions: 1. `μ ∅ = 0`; 2. `μ` is monotone; 3. `μ` is countably subadditive. This means that the outer measure of a countable union is at most the sum of the outer measure…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/OuterMeasure/Defs.html"}, {"id": "Mathlib.MeasureTheory.Covering.Besicovitch", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 2, "macro_tier_score": 0.1253, "macro_tier_override": null, "x": -203.038, "z": -44.432, "size": 0.278, "title": "Besicovitch covering theorems", "summary": "The topological Besicovitch covering theorem ensures that, in a nice metric space, there exists a number `N` such that, from any family of balls with bounded radii, one can extract `N` families, each made of disjoint balls, covering together all the centers of the initial family. By \"nice metric space\", we mean a technical property stated as follows: there exists no satellite configuration of `N + 1` points (with a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Covering/Besicovitch.html"}, {"id": "Mathlib.MeasureTheory.Covering.Differentiation", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 2, "macro_tier_score": 0.1259, "macro_tier_override": null, "x": -160.628, "z": -44.069, "size": 0.3303, "title": "Differentiation of measures", "summary": "On a second countable metric space with a measure `μ`, consider a Vitali family (i.e., for each `x` one has a family of sets shrinking to `x`, with a good behavior with respect to covering theorems). Consider also another measure `ρ`. Then, for almost every `x`, the ratio `ρ a / μ a` converges when `a` shrinks to `x` along the Vitali family, towards the Radon-Nikodym derivative of `ρ` with respect to `μ`. This is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Covering/Differentiation.html"}, {"id": "Mathlib.MeasureTheory.Measure.WithDensity", "region_id": "measure_theory", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.2827, "macro_tier_override": null, "x": -155.218, "z": -78.229, "size": 0.364, "title": "Measure with a given density with respect to another measure", "summary": "For a measure `μ` on `α` and a function `f : α → ℝ≥0∞`, we define a new measure `μ.withDensity f`. On a measurable set `s`, that measure has value `∫⁻ a in s, f a ∂μ`. An important result about `withDensity` is the Radon-Nikodym theorem. It states that, given measures `μ, ν`, if `HaveLebesgueDecomposition μ ν` then `μ` is absolutely continuous with respect to `ν` if and only if there exists a measurable function `f…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/WithDensity.html"}, {"id": "Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion", "region_id": "measure_theory", "micro_elevation": 0.3148, "macro_tier": 3, "macro_tier_score": 0.2819, "macro_tier_override": null, "x": -169.894, "z": -88.192, "size": 0.3072, "title": "Method of exhaustion", "summary": "If `μ, ν` are two measures with `ν` s-finite, then there exists a set `s` such that `μ` is sigma-finite on `s`, and for all sets `t ⊆ sᶜ`, either `ν t = 0` or `μ t = ∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Decomposition/Exhaustion.html"}, {"id": "Mathlib.MeasureTheory.Group.Convolution", "region_id": "measure_theory", "micro_elevation": 0.537, "macro_tier": 3, "macro_tier_score": 0.2818, "macro_tier_override": null, "x": -159.304, "z": -66.402, "size": 0.2959, "title": "The multiplicative and additive convolution of measures", "summary": "In this file we define and prove properties about the convolutions of two measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/Convolution.html"}, {"id": "Mathlib.MeasureTheory.Constructions.Polish.Basic", "region_id": "measure_theory", "micro_elevation": 0.4444, "macro_tier": 4, "macro_tier_score": 0.3156, "macro_tier_override": null, "x": -170.744, "z": -60.817, "size": 0.435, "title": "The Borel sigma-algebra on Polish spaces", "summary": "We discuss several results pertaining to the relationship between the topology and the Borel structure on Polish spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/Polish/Basic.html"}, {"id": "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "region_id": "measure_theory", "micro_elevation": 0.4259, "macro_tier": 4, "macro_tier_score": 0.3145, "macro_tier_override": null, "x": -185.072, "z": -97.692, "size": 0.3882, "title": "Measurable functions in (pseudo-)metrizable Borel spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.html"}, {"id": "Mathlib.MeasureTheory.Integral.DominatedConvergence", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 2, "macro_tier_score": 0.1282, "macro_tier_override": null, "x": -172.487, "z": -117.732, "size": 0.4404, "title": "The dominated convergence theorem", "summary": "This file collects various results related to the Lebesgue dominated convergence theorem for the Bochner integral.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/DominatedConvergence.html"}, {"id": "Mathlib.MeasureTheory.Measure.Tilted", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 2, "macro_tier_score": 0.0318, "macro_tier_override": null, "x": -151.889, "z": -50.943, "size": 0.3031, "title": "Exponentially tilted measures", "summary": "The exponential tilting of a measure `μ` on `α` by a function `f : α → ℝ` is the measure with density `x ↦ exp (f x) / ∫ y, exp (f y) ∂μ` with respect to `μ`. This is sometimes also called the Esscher transform. The definition is mostly used for `f` linear, in which case the exponentially tilted measure belongs to the natural exponential family of the base measure. Exponentially tilted measures for general `f` can…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Tilted.html"}, {"id": "Mathlib.MeasureTheory.Measure.Complex", "region_id": "measure_theory", "micro_elevation": 0.3519, "macro_tier": 2, "macro_tier_score": 0.0626, "macro_tier_override": null, "x": -164.819, "z": -81.444, "size": 0.2446, "title": "Complex measure", "summary": "This file defines a complex measure to be a vector measure with codomain `ℂ`. Then we prove some elementary results about complex measures. In particular, we prove that a complex measure is always in the form `s + it` where `s` and `t` are signed measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Complex.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.Basic", "region_id": "measure_theory", "micro_elevation": 0.3333, "macro_tier": 2, "macro_tier_score": 0.0948, "macro_tier_override": null, "x": -194.74, "z": -72.451, "size": 0.3384, "title": "Vector-valued measures", "summary": "This file defines vector-valued measures, which are σ-additive functions from a set to an additive monoid `M` such that it maps the empty set and non-measurable sets to zero. In the case that `M = ℝ`, we called the vector measure a signed measure and write `SignedMeasure α`. Similarly, when `M = ℂ`, we call the measure a complex measure and write `ComplexMeasure α` (defined in `MeasureTheory/Measure/Complex`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/Basic.html"}, {"id": "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "region_id": "measure_theory", "micro_elevation": 0.3889, "macro_tier": 2, "macro_tier_score": 0.0635, "macro_tier_override": null, "x": -187.742, "z": -62.194, "size": 0.3312, "title": "Measurable functions in normed spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/BorelSpace/ContinuousLinearMap.html"}, {"id": "Mathlib.MeasureTheory.Measure.GiryMonad", "region_id": "measure_theory", "micro_elevation": 0.5, "macro_tier": 4, "macro_tier_score": 0.3172, "macro_tier_override": null, "x": -185.205, "z": -56.097, "size": 0.4895, "title": "The Giry monad", "summary": "Let X be a measurable space. The collection of all measures on X again forms a measurable space. This construction forms a monad on measurable spaces and measurable functions, called the Giry monad. Note that most sources use the term \"Giry monad\" for the restriction to *probability* measures. Here we include all measures on X. See also `Mathlib/MeasureTheory/Category/MeasCat.lean`, containing an upgrade of the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/GiryMonad.html"}, {"id": "Mathlib.MeasureTheory.Integral.Lebesgue.Countable", "region_id": "measure_theory", "micro_elevation": 0.4815, "macro_tier": 4, "macro_tier_score": 0.3146, "macro_tier_override": null, "x": -201.981, "z": -84.727, "size": 0.395, "title": "Lebesgue integral over finite and countable types, sets and measures", "summary": "The lemmas in this file require at least one of the following of the Lebesgue integral: * The type of the set of integration is finite or countable * The set of integration is finite or countable * The measure is finite, s-finite or sigma-finite", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Lebesgue/Countable.html"}, {"id": "Mathlib.MeasureTheory.Integral.Average", "region_id": "measure_theory", "micro_elevation": 0.8519, "macro_tier": 3, "macro_tier_score": 0.1581, "macro_tier_override": null, "x": -194.06, "z": -41.839, "size": 0.3836, "title": "Integral average of a function", "summary": "In this file we define `MeasureTheory.average μ f` (notation: `⨍ x, f x ∂μ`) to be the average value of `f` with respect to measure `μ`. It is defined as `∫ x, f x ∂((μ univ)⁻¹ • μ)`, so it is equal to zero if `f` is not integrable or if `μ` is an infinite measure. If `μ` is a probability measure, then the average of any function is equal to its integral. For the average on a set, we use `⨍ x in s, f x ∂μ` (notation…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Average.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic", "region_id": "measure_theory", "micro_elevation": 0.8519, "macro_tier": 3, "macro_tier_score": 0.1598, "macro_tier_override": null, "x": -189.689, "z": -40.538, "size": 0.4534, "title": "Integral over an interval", "summary": "In this file we define `∫ x in a..b, f x ∂μ` to be `∫ x in Ioc a b, f x ∂μ` if `a ≤ b` and `-∫ x in Ioc b a, f x ∂μ` if `b ≤ a`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntervalIntegral/Basic.html"}, {"id": "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "region_id": "measure_theory", "micro_elevation": 0.1296, "macro_tier": 4, "macro_tier_score": 0.3131, "macro_tier_override": null, "x": -174.821, "z": -77.885, "size": 0.3058, "title": "Sequence of measurable functions associated to a sequence of a.e.-measurable functions", "summary": "We define here tools to prove statements about limits (infi, supr...) of sequences of `AEMeasurable` functions. Given a sequence of a.e.-measurable functions `f : ι → α → β` with hypothesis `hf : ∀ i, AEMeasurable (f i) μ`, and a pointwise property `p : α → (ι → β) → Prop` such that we have `hp : ∀ᵐ x ∂μ, p x (fun n ↦ f n x)`, we define a sequence of measurable functions `aeSeq hf p` and a measurable set `aeSeqSet…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/AEMeasurableSequence.html"}, {"id": "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "region_id": "measure_theory", "micro_elevation": 0.1111, "macro_tier": 4, "macro_tier_score": 0.377, "macro_tier_override": null, "x": -177.101, "z": -82.208, "size": 0.3902, "title": "Measure spaces", "summary": "This file defines measure spaces, the almost-everywhere filter and `AEMeasurable` functions. See `MeasureTheory.MeasureSpace` for their properties and for extended documentation. Given a measurable space `α`, a measure on `α` is a function that sends measurable sets to the extended nonnegative reals that satisfies the following conditions: 1. `μ ∅ = 0`; 2. `μ` is countably additive. This means that the measure of a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/MeasureSpaceDef.html"}, {"id": "Mathlib.MeasureTheory.Integral.Pi", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 2, "macro_tier_score": 0.0323, "macro_tier_override": null, "x": -191.735, "z": -38.426, "size": 0.3367, "title": "Integration with respect to a finite product of measures", "summary": "On a finite product of measure spaces, we show that a product of integrable functions each depending on a single coordinate is integrable, in `MeasureTheory.integrable_fintype_prod`, and that its integral is the product of the individual integrals, in `MeasureTheory.integral_fintype_prod_eq_prod`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Pi.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSpace.CompleteOfCompleteLp", "region_id": "measure_theory", "micro_elevation": 0.7407, "macro_tier": 1, "macro_tier_score": 0.0317, "macro_tier_override": null, "x": -210.024, "z": -61.326, "size": 0.288, "title": "If an `Lp` space is complete, so is the target space", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSpace/CompleteOfCompleteLp.html"}, {"id": "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "region_id": "measure_theory", "micro_elevation": 0.7222, "macro_tier": 2, "macro_tier_score": 0.0644, "macro_tier_override": null, "x": -157.291, "z": -102.089, "size": 0.3854, "title": "Finitely strongly measurable functions in `Lp`", "summary": "Functions in `Lp` for `0 < p < ∞` are finitely strongly measurable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/StronglyMeasurable/Lp.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "region_id": "measure_theory", "micro_elevation": 0.537, "macro_tier": 1, "macro_tier_score": 0.0315, "macro_tier_override": null, "x": -173.214, "z": -102.11, "size": 0.2653, "title": "Lp seminorm with respect to trimmed measure", "summary": "In this file we prove basic properties of the Lp-seminorm of a function with respect to the restriction of a measure to a sub-σ-algebra.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSeminorm/Trim.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.WithDensity", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 2, "macro_tier_score": 0.0627, "macro_tier_override": null, "x": -218.697, "z": -66.122, "size": 0.2653, "title": "Vector measure defined by an integral", "summary": "Given a measure `μ` and an integrable function `f : α → E`, we can define a vector measure `v` such that for all measurable sets `s`, `v s = ∫ x in s, f x ∂μ`. This definition is useful for the Radon-Nikodym theorem for signed measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/WithDensity.html"}, {"id": "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "region_id": "measure_theory", "micro_elevation": 0.8519, "macro_tier": 2, "macro_tier_score": 0.0636, "macro_tier_override": null, "x": -187.717, "z": -117.112, "size": 0.3395, "title": "From equality of integrals to equality of functions", "summary": "This file provides various statements of the general form \"if two functions have the same integral on all sets, then they are equal almost everywhere\". The different lemmas use various hypotheses on the class of functions, on the target space or on the possible finiteness of the measure. This file is about Bochner integrals. See the file `AEEqOfLIntegral` for Lebesgue integrals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/AEEqOfIntegral.html"}, {"id": "Mathlib.MeasureTheory.Function.ContinuousMapDense", "region_id": "measure_theory", "micro_elevation": 0.7963, "macro_tier": 1, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": -189.106, "z": -114.217, "size": 0.2729, "title": "Approximation in Lᵖ by continuous functions", "summary": "This file proves that bounded continuous functions are dense in `Lp E p μ`, for `p < ∞`, if the domain `α` of the functions is a normal topological space and the measure `μ` is weakly regular. It also proves the same results for approximation by continuous functions with compact support when the space is locally compact and `μ` is regular. The result is presented in several versions. First concrete versions giving…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ContinuousMapDense.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions", "region_id": "measure_theory", "micro_elevation": 0.6111, "macro_tier": 1, "macro_tier_score": 0.0316, "macro_tier_override": null, "x": -172.108, "z": -105.328, "size": 0.2773, "title": "Continuous functions in Lp space", "summary": "When `α` is a topological space equipped with a finite Borel measure, there is a bounded linear map from the normed space of bounded continuous functions (`α →ᵇ E`) to `Lp E p μ`. We construct this as `BoundedContinuousFunction.toLp`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSpace/ContinuousFunctions.html"}, {"id": "Mathlib.MeasureTheory.Integral.Bochner.Basic", "region_id": "measure_theory", "micro_elevation": 0.7778, "macro_tier": 4, "macro_tier_score": 0.2852, "macro_tier_override": null, "x": -180.638, "z": -42.9, "size": 0.4656, "title": "Bochner integral", "summary": "The Bochner integral extends the definition of the Lebesgue integral to functions that map from a measure space into a Banach space (complete normed vector space). It is constructed here using the L1 Bochner integral constructed in the file `Mathlib/MeasureTheory/Integral/Bochner/L1.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Bochner/Basic.html"}, {"id": "Mathlib.MeasureTheory.Measure.Hausdorff", "region_id": "measure_theory", "micro_elevation": 0.6296, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -209.454, "z": -75.304, "size": 0.2552, "title": "Hausdorff measure and metric (outer) measures", "summary": "In this file we define the `d`-dimensional Hausdorff measure on an (extended) metric space `X` and the Hausdorff dimension of a set in an (extended) metric space. Let `μ d δ` be the maximal outer measure such that `μ d δ s ≤ (ediam s) ^ d` for every set of diameter less than `δ`. Then the Hausdorff measure `μH[d] s` of `s` is defined as `⨆ δ > 0, μ d δ s`. By Carathéodory's theorem…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Hausdorff.html"}, {"id": "Mathlib.MeasureTheory.Constructions.SubmoduleQuotient", "region_id": "measure_theory", "micro_elevation": 0.0556, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -181.053, "z": -76.089, "size": 0.2, "title": "Measurability on the quotient of a module by a submodule", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/SubmoduleQuotient.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn", "region_id": "measure_theory", "micro_elevation": 0.3519, "macro_tier": 2, "macro_tier_score": 0.0626, "macro_tier_override": null, "x": -191.524, "z": -66.596, "size": 0.2501, "title": "Hahn decomposition", "summary": "This file proves the Hahn decomposition theorem (signed version). The Hahn decomposition theorem states that, given a signed measure `s`, there exist complementary, measurable sets `i` and `j`, such that `i` is positive and `j` is negative with respect to `s`; that is, `s` restricted on `i` is non-negative and `s` restricted on `j` is non-positive. The Hahn decomposition theorem leads to many other results in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/Decomposition/Hahn.html"}, {"id": "Mathlib.MeasureTheory.Measure.Haar.DistribChar", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -142.753, "z": -97.752, "size": 0.2, "title": "The distributive character of Haar measures", "summary": "Given a group `G` acting by additive morphisms on a locally compact additive commutative group `A`, and an element `g : G`, one can pull back the Haar measure `μ` of `A` along the map `(g • ·) : A → A` to get another Haar measure `μ'` on `A`. By unicity of Haar measures, there exists some nonnegative real number `r` such that `μ' = r • μ`. We can thus define a map `distribHaarChar : G → ℝ≥0` sending `g` to its…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Haar/DistribChar.html"}, {"id": "Mathlib.MeasureTheory.Measure.MeasureSpace", "region_id": "measure_theory", "micro_elevation": 0.1667, "macro_tier": 4, "macro_tier_score": 0.3772, "macro_tier_override": null, "x": -188.096, "z": -76.543, "size": 0.401, "title": "Measure spaces", "summary": "The definition of a measure and a measure space are in `MeasureTheory.MeasureSpaceDef`, with only a few basic properties. This file provides many more properties of these objects. This separation allows the measurability tactic to import only the file `MeasureSpaceDef`, and to be available in `MeasureSpace` (through `MeasurableSpace`). Given a measurable space `α`, a measure on `α` is a function that sends…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/MeasureSpace.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated", "region_id": "measure_theory", "micro_elevation": 0.0556, "macro_tier": 4, "macro_tier_score": 0.3756, "macro_tier_override": null, "x": -178.466, "z": -79.797, "size": 0.3002, "title": "Measurably generated filters", "summary": "We say that a filter `f` is measurably generated if every set `s ∈ f` includes a measurable set `t ∈ f`. This property is useful, e.g., to extract a measurable witness of `Filter.Eventually`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/MeasurablyGenerated.html"}, {"id": "Mathlib.MeasureTheory.Measure.NullMeasurable", "region_id": "measure_theory", "micro_elevation": 0.1481, "macro_tier": 4, "macro_tier_score": 0.3756, "macro_tier_override": null, "x": -182.637, "z": -85.15, "size": 0.3002, "title": "Null measurable sets and complete measures", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/NullMeasurable.html"}, {"id": "Mathlib.MeasureTheory.Measure.Stieltjes", "region_id": "measure_theory", "micro_elevation": 0.4074, "macro_tier": 3, "macro_tier_score": 0.2508, "macro_tier_override": null, "x": -189.253, "z": -95.275, "size": 0.3196, "title": "Stieltjes measures on the real line", "summary": "Consider a function `f : ℝ → ℝ` which is monotone and right-continuous. Then one can define a corresponding measure, giving mass `f b - f a` to the interval `(a, b]`. We implement more generally this notion for `f : R → ℝ` where `R` is a conditionally complete dense linear order.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Stieltjes.html"}, {"id": "Mathlib.MeasureTheory.Constructions.BorelSpace.Order", "region_id": "measure_theory", "micro_elevation": 0.3889, "macro_tier": 4, "macro_tier_score": 0.3151, "macro_tier_override": null, "x": -179.109, "z": -96.406, "size": 0.4159, "title": "Borel sigma algebras on spaces with orders", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/BorelSpace/Order.html"}, {"id": "Mathlib.MeasureTheory.Measure.Typeclasses.Probability", "region_id": "measure_theory", "micro_elevation": 0.2963, "macro_tier": 4, "macro_tier_score": 0.3156, "macro_tier_override": null, "x": -193.946, "z": -81.855, "size": 0.4377, "title": "Classes for probability measures", "summary": "We introduce the following typeclasses for measures: * `IsZeroOrProbabilityMeasure μ`: `μ univ = 0 ∨ μ univ = 1`; * `IsProbabilityMeasure μ`: `μ univ = 1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Typeclasses/Probability.html"}, {"id": "Mathlib.MeasureTheory.Group.Prod", "region_id": "measure_theory", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.2195, "macro_tier_override": null, "x": -155.218, "z": -78.211, "size": 0.3158, "title": "Measure theory in the product of groups", "summary": "In this file we show properties about measure theory in products of measurable groups and properties of iterated integrals in measurable groups. These lemmas show the uniqueness of left invariant measures on measurable groups, up to scaling. In this file we follow the proof and refer to the book *Measure Theory* by Paul Halmos. The idea of the proof is to use the translation invariance of measures to prove `μ(t) = c…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/Prod.html"}, {"id": "Mathlib.MeasureTheory.SpecificCodomains.ContinuousMap", "region_id": "measure_theory", "micro_elevation": 0.7037, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -197.371, "z": -106.323, "size": 0.2532, "title": "Specific results about `ContinuousMap`-valued integration", "summary": "In this file, we collect a few results regarding integrability, on a measure space `(X, μ)`, of a `C(Y, E)`-valued function, where `Y` is a compact topological space and `E` is a normed group. These are all elementary from a mathematical point of view, but they require a bit of care in order to be conveniently usable. In particular, to accommodate the need of families `f : X → Y → E` such that `f x` is only…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/SpecificCodomains/ContinuousMap.html"}, {"id": "Mathlib.MeasureTheory.Function.L1Space.AEEqFun", "region_id": "measure_theory", "micro_elevation": 0.6852, "macro_tier": 3, "macro_tier_score": 0.2828, "macro_tier_override": null, "x": -194.863, "z": -50.505, "size": 0.3653, "title": "`L¹` space", "summary": "In this file we establish an API between `Integrable` and the space `L¹` of equivalence classes of integrable functions, already defined as a special case of `L^p` spaces for `p = 1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/L1Space/AEEqFun.html"}, {"id": "Mathlib.MeasureTheory.Function.L1Space.Integrable", "region_id": "measure_theory", "micro_elevation": 0.6667, "macro_tier": 4, "macro_tier_score": 0.286, "macro_tier_override": null, "x": -164.521, "z": -104.594, "size": 0.4919, "title": "Integrable functions", "summary": "In this file, the predicate `Integrable` is defined and basic properties of integrable functions are proved. Such a predicate is already available under the name `MemLp 1`. We give a direct definition which is easier to use, and show that it is equivalent to `MemLp 1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/L1Space/Integrable.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "region_id": "measure_theory", "micro_elevation": 0.0741, "macro_tier": 4, "macro_tier_score": 0.3151, "macro_tier_override": null, "x": -183.472, "z": -80.632, "size": 0.4164, "title": "Countably generated measurable spaces", "summary": "We say a measurable space is countably generated if it can be generated by a countable set of sets. In such a space, we can also build a sequence of finer and finer finite measurable partitions of the space such that the measurable space is generated by the union of all partitions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.Embedding", "region_id": "measure_theory", "micro_elevation": 0.0556, "macro_tier": 4, "macro_tier_score": 0.41, "macro_tier_override": null, "x": -178.668, "z": -77.116, "size": 0.4608, "title": "Measurable embeddings and equivalences", "summary": "A measurable equivalence between measurable spaces is an equivalence which respects the σ-algebras, that is, for which both directions of the equivalence are measurable functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/Embedding.html"}, {"id": "Mathlib.MeasureTheory.Constructions.ClosedCompactCylinders", "region_id": "measure_theory", "micro_elevation": 0.3889, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -191.049, "z": -64.043, "size": 0.2, "title": "Cylinders with closed compact bases", "summary": "We define the set of all cylinders with closed compact bases. Those sets play a role in the proof of Kolmogorov's extension theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/ClosedCompactCylinders.html"}, {"id": "Mathlib.MeasureTheory.Constructions.Cylinders", "region_id": "measure_theory", "micro_elevation": 0.0556, "macro_tier": 2, "macro_tier_score": 0.0325, "macro_tier_override": null, "x": -182.115, "z": -76.478, "size": 0.3514, "title": "π-systems of cylinders and square cylinders", "summary": "The instance `MeasurableSpace.pi` on `∀ i, α i`, where each `α i` has a `MeasurableSpace` `m i`, is defined as `⨆ i, (m i).comap (fun a => a i)`. That is, a function `g : β → ∀ i, α i` is measurable iff for all `i`, the function `b ↦ g b i` is measurable. We define two π-systems generating `MeasurableSpace.pi`, cylinders and square cylinders.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/Cylinders.html"}, {"id": "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "region_id": "measure_theory", "micro_elevation": 0.5926, "macro_tier": 3, "macro_tier_score": 0.2196, "macro_tier_override": null, "x": -165.203, "z": -56.268, "size": 0.3271, "title": "Additive Haar measure constructed from a basis", "summary": "Given a basis of a finite-dimensional real vector space, we define the corresponding Lebesgue measure, which gives measure `1` to the parallelepiped spanned by the basis.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Haar/OfBasis.html"}, {"id": "Mathlib.MeasureTheory.Group.LIntegral", "region_id": "measure_theory", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.2197, "macro_tier_override": null, "x": -170.239, "z": -55.362, "size": 0.3306, "title": "Lebesgue Integration on Groups", "summary": "We develop properties of integrals with a group as domain. This file contains properties about Lebesgue integration.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/LIntegral.html"}, {"id": "Mathlib.MeasureTheory.Group.Measure", "region_id": "measure_theory", "micro_elevation": 0.537, "macro_tier": 3, "macro_tier_score": 0.2215, "macro_tier_override": null, "x": -178.838, "z": -54.028, "size": 0.4233, "title": "Measures on Groups", "summary": "We develop some properties of measures on (topological) groups * We define properties on measures: measures that are left or right invariant w.r.t. multiplication. * We define the measure `μ.inv : A ↦ μ(A⁻¹)` and show that it is right invariant iff `μ` is left invariant. * We define a class `IsHaarMeasure μ`, requiring that the measure `μ` is left-invariant, finite on compact sets, and positive on open sets. We also…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/Measure.html"}, {"id": "Mathlib.MeasureTheory.Constructions.HaarToSphere", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -220.262, "z": -91.803, "size": 0.2528, "title": "Generalized polar coordinate change", "summary": "Consider an `n`-dimensional normed space `E` and an additive Haar measure `μ` on `E`. Then `μ.toSphere` is the measure on the unit sphere such that `μ.toSphere s` equals `n • μ (Set.Ioo 0 1 • s)`. If `n ≠ 0`, then `μ` can be represented (up to `homeomorphUnitSphereProd`) as the product of `μ.toSphere` and the Lebesgue measure on `(0, +∞)` taken with density `fun r ↦ r ^ n`. One can think about this fact as a version…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/HaarToSphere.html"}, {"id": "Mathlib.MeasureTheory.Integral.Gamma", "region_id": "measure_theory", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -180.729, "z": -78.619, "size": 0.2356, "title": "Integrals involving the Gamma function", "summary": "In this file, we collect several integrals over `ℝ` or `ℂ` that evaluate in terms of the `Real.Gamma` function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Gamma.html"}, {"id": "Mathlib.MeasureTheory.Measure.ContinuousPreimage", "region_id": "measure_theory", "micro_elevation": 0.4074, "macro_tier": 1, "macro_tier_score": 0.0315, "macro_tier_override": null, "x": -187.458, "z": -61.161, "size": 0.2698, "title": "Continuity of the preimage of a set under a measure-preserving continuous function", "summary": "In this file we prove that the preimage of a null measurable set `s : Set Y` under a measure-preserving continuous function `f : C(X, Y)` is continuous in `f` in the sense that `μ ((f a ⁻¹' s) ∆ (g ⁻¹' s))` tends to zero as `f a` tends to `g`. As a corollary, we show that for a continuous family of continuous maps `f z : C(X, Y)`, a null measurable set `s`, and a null measurable set `t` of finite measure, the set of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/ContinuousPreimage.html"}, {"id": "Mathlib.MeasureTheory.Measure.Regular", "region_id": "measure_theory", "micro_elevation": 0.3889, "macro_tier": 3, "macro_tier_score": 0.2198, "macro_tier_override": null, "x": -192.737, "z": -91.84, "size": 0.3354, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Regular.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 2, "macro_tier_score": 0.0328, "macro_tier_override": null, "x": -198.673, "z": -115.287, "size": 0.3694, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntervalIntegral/FundThmCalculus.html"}, {"id": "Mathlib.MeasureTheory.Measure.FiniteMeasureProd", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -169.509, "z": -119.636, "size": 0.2, "title": "Products of finite measures and probability measures", "summary": "This file introduces binary products of finite measures and probability measures. The constructions are obtained from special cases of products of general measures. Taking products nevertheless has specific properties in the cases of finite measures and probability measures, notably the fact that the product measures depend continuously on their factors in the topology of weak convergence when the underlying space…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/FiniteMeasureProd.html"}, {"id": "Mathlib.MeasureTheory.Measure.LevyProkhorovMetric", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 2, "macro_tier_score": 0.0319, "macro_tier_override": null, "x": -220.96, "z": -67.753, "size": 0.3051, "title": "The Lévy-Prokhorov distance on spaces of finite measures and probability measures", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.html"}, {"id": "Mathlib.MeasureTheory.Measure.CharacteristicFunction", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -146.979, "z": -52.752, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/CharacteristicFunction.html"}, {"id": "Mathlib.MeasureTheory.Group.IntegralConvolution", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -151.665, "z": -108.484, "size": 0.2561, "title": "Bochner integrals of convolutions", "summary": "This file contains results about the Bochner integrals of convolutions of measures. These results are not placed in the main convolution file because we don't want to import Bochner integrals over there.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/IntegralConvolution.html"}, {"id": "Mathlib.MeasureTheory.Measure.FiniteMeasureExt", "region_id": "measure_theory", "micro_elevation": 0.8333, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -142.836, "z": -73.254, "size": 0.2561, "title": "Extensionality of finite measures", "summary": "The main result is `ext_of_forall_mem_subalgebra_integral_eq_of_pseudoEMetric_complete_countable`: Let `A` be a StarSubalgebra of `C(E, 𝕜)` that separates points and whose elements are bounded. If the integrals of all elements of `A` with respect to two finite measures `P, P'` coincide, then the measures coincide. In other words: If a subalgebra separates points, it separates finite measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/FiniteMeasureExt.html"}, {"id": "Mathlib.MeasureTheory.Covering.VitaliFamily", "region_id": "measure_theory", "micro_elevation": 0.2222, "macro_tier": 2, "macro_tier_score": 0.1256, "macro_tier_override": null, "x": -172.164, "z": -73.07, "size": 0.3003, "title": "Vitali families", "summary": "On a metric space `X` with a measure `μ`, consider for each `x : X` a family of measurable sets with nonempty interiors, called `setsAt x`. This family is a Vitali family if it satisfies the following property: consider a (possibly non-measurable) set `s`, and for any `x` in `s` a subfamily `f x` of `setsAt x` containing sets of arbitrarily small diameter. Then one can extract a disjoint subfamily covering almost…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Covering/VitaliFamily.html"}, {"id": "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "region_id": "measure_theory", "micro_elevation": 0.4074, "macro_tier": 2, "macro_tier_score": 0.1253, "macro_tier_override": null, "x": -180.108, "z": -97.319, "size": 0.2687, "title": "Measurability criterion for ennreal-valued functions", "summary": "Consider a function `f : α → ℝ≥0∞`. If the level sets `{f < p}` and `{q < f}` have measurable supersets which are disjoint up to measure zero when `p` and `q` are finite numbers satisfying `p < q`, then `f` is almost-everywhere measurable. This is proved in `ENNReal.aemeasurable_of_exist_almost_disjoint_supersets`, and deduced from an analogous statement for any target space which is a complete linear dense order,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/AEMeasurableOrder.html"}, {"id": "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "region_id": "measure_theory", "micro_elevation": 0.9444, "macro_tier": 2, "macro_tier_score": 0.0317, "macro_tier_override": null, "x": -212.128, "z": -48.696, "size": 0.2928, "title": "Conditional expectation of indicator functions", "summary": "This file proves some results about the conditional expectation of an indicator function and as a corollary, also proves several results about the behaviour of the conditional expectation on a restricted measure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.html"}, {"id": "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 2, "macro_tier_score": 0.0327, "macro_tier_override": null, "x": -142.701, "z": -59.591, "size": 0.3591, "title": "Conditional expectation", "summary": "We build the conditional expectation of an integrable function `f` with value in a Banach space with respect to a measure `μ` (defined on a measurable space structure `m₀`) and a measurable space structure `m` with `hm : m ≤ m₀` (a sub-sigma-algebra). This is an `m`-strongly measurable function `μ[f | m]` which is integrable and verifies `∫ x in s, μ[f | m] x ∂μ = ∫ x in s, f x ∂μ` for all `m`-measurable sets `s`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.html"}, {"id": "Mathlib.MeasureTheory.OuterMeasure.Operations", "region_id": "measure_theory", "micro_elevation": 0.037, "macro_tier": 4, "macro_tier_score": 0.3753, "macro_tier_override": null, "x": -180.645, "z": -76.921, "size": 0.2681, "title": "Operations on outer measures", "summary": "In this file we define algebraic operations (addition, scalar multiplication) on the type of outer measures on a type. We also show that outer measures on a type `α` form a complete lattice.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/OuterMeasure/Operations.html"}, {"id": "Mathlib.MeasureTheory.Category.MeasCat", "region_id": "measure_theory", "micro_elevation": 0.5185, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -203.973, "z": -73.444, "size": 0.2, "title": "The category of measurable spaces", "summary": "Measurable spaces and measurable functions form a (concrete) category `MeasCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Category/MeasCat.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -144.711, "z": -57.659, "size": 0.2, "title": "Displacement is at most the integral of the speed", "summary": "In this file we prove several version of the following fact: the displacement (`dist (f a) (f b)`) is at most the integral of `‖deriv f‖` over `[a, b]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntervalIntegral/DistLEIntegral.html"}, {"id": "Mathlib.MeasureTheory.Integral.Lebesgue.Map", "region_id": "measure_theory", "micro_elevation": 0.463, "macro_tier": 4, "macro_tier_score": 0.3132, "macro_tier_override": null, "x": -201.372, "z": -83.712, "size": 0.3122, "title": "Behavior of the Lebesgue integral under maps", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Lebesgue/Map.html"}, {"id": "Mathlib.MeasureTheory.Integral.Lebesgue.Markov", "region_id": "measure_theory", "micro_elevation": 0.463, "macro_tier": 4, "macro_tier_score": 0.3135, "macro_tier_override": null, "x": -190.867, "z": -59.931, "size": 0.3347, "title": "Markov's inequality", "summary": "The classical form of Markov's inequality states that for a nonnegative random variable `X` and real number `ε > 0`, `P(X ≥ ε) ≤ E(X) / ε`. Multiplying both sides by the measure of the space gives the measure-theoretic form: ``` μ { x | ε ≤ f x } ≤ (∫⁻ a, f a ∂μ) / ε ``` This file proves a few variants of the inequality and other lemmas that depend on it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Lebesgue/Markov.html"}, {"id": "Mathlib.MeasureTheory.Measure.Count", "region_id": "measure_theory", "micro_elevation": 0.3333, "macro_tier": 4, "macro_tier_score": 0.3132, "macro_tier_override": null, "x": -168.258, "z": -69.741, "size": 0.3122, "title": "Counting measure", "summary": "In this file we define the counting measure `MeasureTheory.Measure.count` as `MeasureTheory.Measure.sum MeasureTheory.Measure.dirac` and prove basic properties of this measure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Count.html"}, {"id": "Mathlib.MeasureTheory.Measure.MutuallySingular", "region_id": "measure_theory", "micro_elevation": 0.2778, "macro_tier": 4, "macro_tier_score": 0.313, "macro_tier_override": null, "x": -193.449, "z": -77.644, "size": 0.2982, "title": "Mutually singular measures", "summary": "Two measures `μ`, `ν` are said to be mutually singular (`MeasureTheory.Measure.MutuallySingular`, localized notation `μ ⟂ₘ ν`) if there exists a measurable set `s` such that `μ s = 0` and `ν sᶜ = 0`. The measurability of `s` is an unnecessary assumption (see `MeasureTheory.Measure.MutuallySingular.mk`) but we keep it because this way `rcases (h : μ ⟂ₘ ν)` gives us a measurable set and usually it is easy to prove…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/MutuallySingular.html"}, {"id": "Mathlib.MeasureTheory.Measure.Restrict", "region_id": "measure_theory", "micro_elevation": 0.2593, "macro_tier": 4, "macro_tier_score": 0.3849, "macro_tier_override": null, "x": -187.56, "z": -68.867, "size": 0.6222, "title": "Restricting a measure to a subset or a subtype", "summary": "Given a measure `μ` on a type `α` and a subset `s` of `α`, we define a measure `μ.restrict s` as the restriction of `μ` to `s` (still as a measure on `α`). We investigate how this notion interacts with usual operations on measures (sum, pushforward, pullback), and on sets (inclusion, union, Union). We also study the relationship between the restriction of a measure to a subtype (given by the pullback under…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Restrict.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.SetIntegral", "region_id": "measure_theory", "micro_elevation": 0.8148, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -156.465, "z": -107.107, "size": 0.2478, "title": "Set integral", "summary": "In this file we prove properties of `∫ᵛ x in s, f x ∂[B; μ]`. Recall that this notation is defined as `∫ᵛ x, f x ∂[B; μ.restrict s]`. The API in this file is modelled on the API for the Bochner integral.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/SetIntegral.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.Integral", "region_id": "measure_theory", "micro_elevation": 0.7963, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -199.109, "z": -110.235, "size": 0.257, "title": "Integral of vector-valued function against vector measure", "summary": "We extend the definition of the Bochner integral (of vector-valued function against `ℝ≥0∞`-valued measure) to vector measures through a bilinear pairing. Let `E`, `F` be normed vector spaces, and `G` be a Banach space (complete normed vector space). We fix a continuous linear pairing `B : E →L[ℝ] F →L[ℝ] G` and an `F`-valued vector measure `μ` on a measurable space `X`. For an integrable function `f : X → E` with…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/Integral.html"}, {"id": "Mathlib.MeasureTheory.Function.StronglyMeasurable.ENNReal", "region_id": "measure_theory", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.1881, "macro_tier_override": null, "x": -158.044, "z": -75.062, "size": 0.3037, "title": "Finitely strongly measurable functions with value in ENNReal", "summary": "A measurable function with finite Lebesgue integral can be approximated by simple functions whose support has finite measure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/StronglyMeasurable/ENNReal.html"}, {"id": "Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable", "region_id": "measure_theory", "micro_elevation": 0.4815, "macro_tier": 4, "macro_tier_score": 0.2841, "macro_tier_override": null, "x": -160.217, "z": -86.879, "size": 0.4281, "title": "Strongly measurable and finitely strongly measurable functions", "summary": "A function `f` is said to be almost everywhere strongly measurable if `f` is almost everywhere equal to a strongly measurable function, i.e. the sequential limit of simple functions. It is said to be almost everywhere finitely strongly measurable with respect to a measure `μ` if the supports of those simple functions have finite measure. Almost everywhere strongly measurable functions form the largest class of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/StronglyMeasurable/AEStronglyMeasurable.html"}, {"id": "Mathlib.MeasureTheory.Integral.Lebesgue.Add", "region_id": "measure_theory", "micro_elevation": 0.4444, "macro_tier": 4, "macro_tier_score": 0.3158, "macro_tier_override": null, "x": -198.297, "z": -68.228, "size": 0.4429, "title": "Monotone convergence theorem and addition of Lebesgue integrals", "summary": "The monotone convergence theorem (aka Beppo Levi lemma) states that the Lebesgue integral and supremum can be swapped for a pointwise monotone sequence of functions. This file first proves several variants of this theorem, then uses it to show that the Lebesgue integral is additive (assuming one of the functions is at least `AEMeasurable`) and respects multiplication by a constant.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Lebesgue/Add.html"}, {"id": "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas", "region_id": "measure_theory", "micro_elevation": 0.5741, "macro_tier": 3, "macro_tier_score": 0.2829, "macro_tier_override": null, "x": -193.441, "z": -101.717, "size": 0.3706, "title": "Strongly measurable and finitely strongly measurable functions", "summary": "This file contains some further development of strongly measurable and finitely strongly measurable functions, started in `Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.Variation.Defs", "region_id": "measure_theory", "micro_elevation": 0.3704, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -193.492, "z": -67.375, "size": 0.2565, "title": "Total variation for vector-valued measures", "summary": "This file contains the definition of variation for any `VectorMeasure` in an `ENormedAddCommMonoid`, in particular, any `NormedAddCommGroup`. Given a vector-valued measure `μ` we consider the problem of finding a countably additive function `f` such that, for any set `E`, `‖μ(E)‖ ≤ f(E)`. This suggests defining `f(E)` as the supremum over partitions `{Eᵢ}` of `E`, of the quantity `∑ᵢ, ‖μ(Eᵢ)‖`. Indeed any solution…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/Variation/Defs.html"}, {"id": "Mathlib.MeasureTheory.Measure.PreVariation", "region_id": "measure_theory", "micro_elevation": 0.3519, "macro_tier": 1, "macro_tier_score": 0.0315, "macro_tier_override": null, "x": -195.6, "z": -72.297, "size": 0.2602, "title": "Pre-variation of a subadditive set function", "summary": "Given a σ-subadditive `ℝ≥0∞`-valued set function `f`, we define the pre-variation as the supremum over finite measurable partitions of the sum of `f` on the parts. This construction yields a measure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/PreVariation.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts", "region_id": "measure_theory", "micro_elevation": 0.963, "macro_tier": 2, "macro_tier_score": 0.0328, "macro_tier_override": null, "x": -224.702, "z": -83.33, "size": 0.3662, "title": "Integration by parts and by substitution", "summary": "We derive additional integration techniques from FTC-2: * `intervalIntegral.integral_mul_deriv_eq_deriv_mul` - integration by parts * `intervalIntegral.integral_comp_mul_deriv''` - integration by substitution Versions of the change of variables formula for monotone and antitone functions, but with much weaker assumptions on the integrands and not restricted to intervals, can be found in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntervalIntegral/IntegrationByParts.html"}, {"id": "Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent", "region_id": "measure_theory", "micro_elevation": 0.3333, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -170.192, "z": -89.724, "size": 0.2445, "title": "Additive content built from a projective family of measures", "summary": "Let `P` be a projective family of measures on a family of measurable spaces indexed by `ι`. That is, for each finite set `I` of indices, `P I` is a measure on `Π j : I, α j`, and for `J ⊆ I`, the projection from `Π i : I, α i` to `Π i : J, α i` maps `P I` to `P J`. We build an additive content `projectiveFamilyContent` on the measurable cylinders, by setting `projectiveFamilyContent s = P I S` for `s = cylinder I…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/ProjectiveFamilyContent.html"}, {"id": "Mathlib.MeasureTheory.SetAlgebra", "region_id": "measure_theory", "micro_elevation": 0.0556, "macro_tier": 2, "macro_tier_score": 0.0627, "macro_tier_override": null, "x": -178.205, "z": -78.25, "size": 0.266, "title": "Algebra of sets", "summary": "In this file we define the notion of algebra of sets and give its basic properties. An algebra of sets is a family of sets containing the empty set and closed by complement and binary union. It is therefore similar to a `σ`-algebra, except that it is not necessarily closed by countable unions. We also define the algebra of sets generated by a family of sets and give its basic properties, and we prove that it is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/SetAlgebra.html"}, {"id": "Mathlib.MeasureTheory.Constructions.UnitInterval", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -196.373, "z": -118.16, "size": 0.2519, "title": "The canonical measure on the unit interval", "summary": "This file provides a `MeasureTheory.MeasureSpace` instance on `unitInterval`, and shows it is a probability measure with value zero on singletons. It also contains some basic results on the volume of various interval sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/UnitInterval.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.Variation.Basic", "region_id": "measure_theory", "micro_elevation": 0.3889, "macro_tier": 1, "macro_tier_score": 0.0316, "macro_tier_override": null, "x": -182.081, "z": -60.811, "size": 0.277, "title": "Properties of variation", "summary": "We prove basic properties of `variation` for `μ : VectorMeasure X V` in `ENormedAddCommMonoid V` on `MeasurableSpace X`. It is defined as the supremum over partitions `{Eᵢ}` of `E`, of the quantity `∑ᵢ, ‖μ(Eᵢ)‖`. This definition allows one to define the integral against such vector-valued measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/Variation/Basic.html"}, {"id": "Mathlib.MeasureTheory.Measure.Dirac", "region_id": "measure_theory", "micro_elevation": 0.3148, "macro_tier": 4, "macro_tier_score": 0.3145, "macro_tier_override": null, "x": -174.972, "z": -65.357, "size": 0.3905, "title": "Dirac measure", "summary": "In this file we define the Dirac measure `MeasureTheory.Measure.dirac a` and prove some basic facts about it.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Dirac.html"}, {"id": "Mathlib.MeasureTheory.Integral.Lebesgue.Basic", "region_id": "measure_theory", "micro_elevation": 0.4259, "macro_tier": 4, "macro_tier_score": 0.3144, "macro_tier_override": null, "x": -161.189, "z": -79.522, "size": 0.3841, "title": "Lower Lebesgue integral for `ℝ≥0∞`-valued functions", "summary": "We define the lower Lebesgue integral of an `ℝ≥0∞`-valued function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Lebesgue/Basic.html"}, {"id": "Mathlib.MeasureTheory.Function.SimpleFunc", "region_id": "measure_theory", "micro_elevation": 0.4074, "macro_tier": 4, "macro_tier_score": 0.3143, "macro_tier_override": null, "x": -193.793, "z": -92.013, "size": 0.3804, "title": "Simple functions", "summary": "A function `f` from a measurable space to any type is called *simple*, if every preimage `f ⁻¹' {x}` is measurable, and the range is finite. In this file, we define simple functions and establish their basic properties; and we construct a sequence of simple functions approximating an arbitrary Borel measurable function `f : α → ℝ≥0∞`. The theorem `Measurable.ennreal_induction` shows that in order to prove something…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/SimpleFunc.html"}, {"id": "Mathlib.MeasureTheory.OuterMeasure.OfFunction", "region_id": "measure_theory", "micro_elevation": 0.0556, "macro_tier": 4, "macro_tier_score": 0.3753, "macro_tier_override": null, "x": -178.241, "z": -79.185, "size": 0.2747, "title": "Outer measures from functions", "summary": "Given an arbitrary function `m : Set α → ℝ≥0∞` that sends `∅` to `0` we can define an outer measure on `α` that on `s` is defined to be the infimum of `∑ᵢ, m (sᵢ)` for all collections of sets `sᵢ` that cover `s`. This is the unique maximal outer measure that is at most the given function. Given an outer measure `m`, the Carathéodory-measurable sets are the sets `s` such that for all sets `t` we have `m t = m (t ∩ s)…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/OuterMeasure/OfFunction.html"}, {"id": "Mathlib.MeasureTheory.Function.Piecewise", "region_id": "measure_theory", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -173.83, "z": -56.718, "size": 0.2, "title": "Measurability of piecewise functions", "summary": "In this file, we prove some results about measurability of functions defined by using `IndexedPartition.piecewise`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/Piecewise.html"}, {"id": "Mathlib.MeasureTheory.Measure.AbsolutelyContinuous", "region_id": "measure_theory", "micro_elevation": 0.2037, "macro_tier": 4, "macro_tier_score": 0.3793, "macro_tier_override": null, "x": -171.452, "z": -77.41, "size": 0.4768, "title": "Absolute Continuity of Measures", "summary": "We say that `μ` is absolutely continuous with respect to `ν`, or that `μ` is dominated by `ν`, if `ν(A) = 0` implies that `μ(A) = 0`. We denote that by `μ ≪ ν`. It is equivalent to an inequality of the almost everywhere filters of the measures: `μ ≪ ν ↔ ae μ ≤ ae ν`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/AbsolutelyContinuous.html"}, {"id": "Mathlib.MeasureTheory.Measure.Map", "region_id": "measure_theory", "micro_elevation": 0.1852, "macro_tier": 4, "macro_tier_score": 0.3798, "macro_tier_override": null, "x": -184.863, "z": -86.052, "size": 0.4933, "title": "Pushforward of a measure", "summary": "In this file we define the pushforward `MeasureTheory.Measure.map f μ` of a measure `μ` along an almost everywhere measurable map `f`. If `f` is not a.e. measurable, then we define `map f μ` to be zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Map.html"}, {"id": "Mathlib.MeasureTheory.Measure.LevyConvergence", "region_id": "measure_theory", "micro_elevation": 0.963, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -182.246, "z": -122.818, "size": 0.2338, "title": "Lévy's convergence theorem", "summary": "This file contains developments related to Lévy's convergence theorem, which links convergence of characteristic functions and convergence in distribution in finite dimensional inner product spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/LevyConvergence.html"}, {"id": "Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion", "region_id": "measure_theory", "micro_elevation": 0.9444, "macro_tier": 1, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -140.153, "z": -93.945, "size": 0.2372, "title": "Taylor expansion of the characteristic function", "summary": "This file provides the Taylor expansion of the characteristic function of a measure at `0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/CharacteristicFunction/TaylorExpansion.html"}, {"id": "Mathlib.MeasureTheory.Measure.IntegralCharFun", "region_id": "measure_theory", "micro_elevation": 0.9444, "macro_tier": 1, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -139.756, "z": -64.389, "size": 0.2372, "title": "Integrals of characteristic functions", "summary": "This file contains results about integrals of characteristic functions, and lemmas relating the measure of some sets to integrals of characteristic functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/IntegralCharFun.html"}, {"id": "Mathlib.MeasureTheory.Measure.Prokhorov", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 1, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -138.214, "z": -77.762, "size": 0.2372, "title": "Prokhorov theorem", "summary": "We prove statements about the compactness of sets of finite measures or probability measures, notably several versions of Prokhorov theorem on tight sets of probability measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Prokhorov.html"}, {"id": "Mathlib.MeasureTheory.Measure.TightNormed", "region_id": "measure_theory", "micro_elevation": 0.5556, "macro_tier": 1, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -162.313, "z": -96.278, "size": 0.2372, "title": "Tight sets of measures in normed spaces", "summary": "Criteria for tightness of sets of measures in normed and inner product spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/TightNormed.html"}, {"id": "Mathlib.MeasureTheory.Function.SpecialFunctions.Inner", "region_id": "measure_theory", "micro_elevation": 0.4074, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -170.735, "z": -94.437, "size": 0.2, "title": "Measurability of scalar products", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.html"}, {"id": "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "region_id": "measure_theory", "micro_elevation": 0.3889, "macro_tier": 3, "macro_tier_score": 0.2824, "macro_tier_override": null, "x": -191.908, "z": -92.548, "size": 0.3415, "title": "Equip `ℂ` with the Borel sigma-algebra", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/BorelSpace/Complex.html"}, {"id": "Mathlib.MeasureTheory.Measure.Tight", "region_id": "measure_theory", "micro_elevation": 0.537, "macro_tier": 2, "macro_tier_score": 0.0628, "macro_tier_override": null, "x": -193.353, "z": -57.432, "size": 0.2754, "title": "Tight sets of measures", "summary": "A set of measures is tight if for all `0 < ε`, there exists a compact set `K` such that for all measures in the set, the complement of `K` has measure at most `ε`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Tight.html"}, {"id": "Mathlib.MeasureTheory.Integral.Layercake", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 2, "macro_tier_score": 0.063, "macro_tier_override": null, "x": -160.531, "z": -44.126, "size": 0.2989, "title": "The layer cake formula / Cavalieri's principle / tail probability formula", "summary": "In this file we prove the following layer cake formula. Consider a non-negative measurable function `f` on a measure space. Apply pointwise to it an increasing absolutely continuous function `G : ℝ≥0 → ℝ≥0` vanishing at the origin, with derivative `G' = g` on the positive real line (in other words, `G` a primitive of a non-negative locally integrable function `g` on the positive real line). Then the integral of the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Layercake.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.BoundedVariation", "region_id": "measure_theory", "micro_elevation": 0.4259, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -198.736, "z": -86.26, "size": 0.2, "title": "Vector valued Stieltjes measure associated to a bounded variation function", "summary": "Let `α` be a dense linear order with compact segments (e.g. `ℝ` or `ℝ≥0`), and `f : α → E` a bounded variation function taking values in a complete additive normed group. We associate to `f` a vector measure, called `BoundedVariationOn.vectorMeasure`. It gives mass `f.rightLim b - f.leftLim a` to the interval `[a, b]` (with similar formulas for other types of intervals). For the construction, we define first an…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/BoundedVariation.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.AddContent", "region_id": "measure_theory", "micro_elevation": 0.3519, "macro_tier": 1, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -196.449, "z": -82.362, "size": 0.239, "title": "Constructing a vector measure from an additive content", "summary": "Consider a content defined on a semiring of sets. We investigate in this file whether it is possible to extend it to a (countably additive) vector measure on the whole sigma-algebra. We show that this is possible when the content is dominated by a finite measure, see `exists_extension_of_isSetSemiring_of_le_measure_of_generateFrom`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/AddContent.html"}, {"id": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "region_id": "measure_theory", "micro_elevation": 0.6111, "macro_tier": 3, "macro_tier_score": 0.222, "macro_tier_override": null, "x": -204.469, "z": -63.651, "size": 0.4422, "title": "Lebesgue measure on the real line and on `ℝⁿ`", "summary": "We show that the Lebesgue measure on the real line (constructed as a particular case of additive Haar measure on inner product spaces) coincides with the Stieltjes measure associated to the function `x ↦ x`. We deduce properties of this measure on `ℝ`, and then of the product Lebesgue measure on `ℝⁿ`. In particular, we prove that they are translation invariant. We show that, on `ℝⁿ`, a linear map acts on Lebesgue…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Lebesgue/Basic.html"}, {"id": "Mathlib.MeasureTheory.Integral.Marginal", "region_id": "measure_theory", "micro_elevation": 0.5741, "macro_tier": 3, "macro_tier_score": 0.2194, "macro_tier_override": null, "x": -168.146, "z": -55.452, "size": 0.3041, "title": "Marginals of multivariate functions", "summary": "In this file, we define a convenient way to compute integrals of multivariate functions, especially if you want to write expressions where you integrate only over some of the variables that the function depends on. This is common in induction arguments involving integrals of multivariate functions. This constructions allows working with iterated integrals and applying Tonelli's theorem and Fubini's theorem, without…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Marginal.html"}, {"id": "Mathlib.MeasureTheory.SetSemiring", "region_id": "measure_theory", "micro_elevation": 0.037, "macro_tier": 3, "macro_tier_score": 0.1571, "macro_tier_override": null, "x": -179.5, "z": -77.444, "size": 0.3236, "title": "Semirings and rings of sets", "summary": "A semi-ring of sets `C` (in the sense of measure theory) is a family of sets containing `∅`, stable by intersection and such that for all `s, t ∈ C`, `t \\ s` is equal to a disjoint union of finitely many sets in `C`. Note that a semi-ring of sets may not contain unions. An important example of a semi-ring of sets is intervals in `ℝ`. The intersection of two intervals is an interval (possibly empty). The union of two…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/SetSemiring.html"}, {"id": "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 2, "macro_tier_score": 0.0942, "macro_tier_override": null, "x": -194.436, "z": -39.265, "size": 0.2949, "title": "Satellite configurations for Besicovitch covering lemma in vector spaces", "summary": "The Besicovitch covering theorem ensures that, in a nice metric space, there exists a number `N` such that, from any family of balls with bounded radii, one can extract `N` families, each made of disjoint balls, covering together all the centers of the initial family. A key tool in the proof of this theorem is the notion of a satellite configuration, i.e., a family of `N + 1` balls, where the first `N` balls all…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.html"}, {"id": "Mathlib.MeasureTheory.Function.SpecialFunctions.RCLike", "region_id": "measure_theory", "micro_elevation": 0.4074, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -162.12, "z": -76.675, "size": 0.2, "title": "Measurability of the basic `RCLike` functions", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.html"}, {"id": "Mathlib.MeasureTheory.Function.L2Space", "region_id": "measure_theory", "micro_elevation": 0.8519, "macro_tier": 2, "macro_tier_score": 0.0317, "macro_tier_override": null, "x": -143.828, "z": -65.626, "size": 0.2928, "title": "`L^2` space", "summary": "If `E` is an inner product space over `𝕜` (`ℝ` or `ℂ`), then `Lp E 2 μ` (defined in `Mathlib/MeasureTheory/Function/LpSpace/Basic.lean`) is also an inner product space, with inner product defined as `inner f g := ∫ a, ⟪f a, g a⟫ ∂μ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/L2Space.html"}, {"id": "Mathlib.MeasureTheory.Integral.Bochner.L1", "region_id": "measure_theory", "micro_elevation": 0.7593, "macro_tier": 4, "macro_tier_score": 0.2848, "macro_tier_override": null, "x": -194.992, "z": -110.438, "size": 0.4532, "title": "Bochner integral", "summary": "The Bochner integral extends the definition of the Lebesgue integral to functions that map from a measure space into a Banach space (complete normed vector space). It is constructed here for L1 functions by extending the integral on simple functions. See the file `Mathlib/MeasureTheory/Integral/Bochner/Basic.lean` for the integral of functions and corresponding API.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Bochner/L1.html"}, {"id": "Mathlib.MeasureTheory.Function.SpecialFunctions.Sinc", "region_id": "measure_theory", "micro_elevation": 0.6852, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -164.85, "z": -105.786, "size": 0.2535, "title": "Measurability and integrability of the sinc function", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/SpecialFunctions/Sinc.html"}, {"id": "Mathlib.MeasureTheory.Function.AEEqOfLIntegral", "region_id": "measure_theory", "micro_elevation": 0.5741, "macro_tier": 3, "macro_tier_score": 0.1886, "macro_tier_override": null, "x": -195.223, "z": -100.642, "size": 0.3418, "title": "From equality of integrals to equality of functions", "summary": "This file provides various statements of the general form \"if two functions have the same integral on all sets, then they are equal almost everywhere\". The different lemmas use various hypotheses on the class of functions, on the target space or on the possible finiteness of the measure. This file is about Lebesgue integrals. See the file `AEEqOfIntegral` for Bochner integrals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/AEEqOfLIntegral.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntervalIntegral.MeanValue", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -212.601, "z": -53.112, "size": 0.2, "title": "First mean value theorem for interval integrals", "summary": "We prove versions of the first mean value theorem for interval integrals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntervalIntegral/MeanValue.html"}, {"id": "Mathlib.MeasureTheory.Integral.MeanValue", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -151.539, "z": -51.312, "size": 0.2478, "title": "First mean value theorem for set integrals", "summary": "We prove versions of the first mean value theorem for set integrals.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/MeanValue.html"}, {"id": "Mathlib.MeasureTheory.Measure.Typeclasses.Finite", "region_id": "measure_theory", "micro_elevation": 0.2778, "macro_tier": 4, "macro_tier_score": 0.3515, "macro_tier_override": null, "x": -168.208, "z": -76.177, "size": 0.5729, "title": "Classes for finite measures", "summary": "We introduce the following typeclasses for measures: * `IsFiniteMeasure μ`: `μ univ < ∞`; * `IsLocallyFiniteMeasure μ` : `∀ x, ∃ s ∈ 𝓝 x, μ s < ∞`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Typeclasses/Finite.html"}, {"id": "Mathlib.MeasureTheory.Function.Jacobian", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 2, "macro_tier_score": 0.0632, "macro_tier_override": null, "x": -149.167, "z": -50.123, "size": 0.3145, "title": "Change of variables in higher-dimensional integrals", "summary": "Let `μ` be a Lebesgue measure on a finite-dimensional real vector space `E`. Let `f : E → E` be a function which is injective and differentiable on a measurable set `s`, with derivative `f'`. Then we prove that `f '' s` is measurable, and its measure is given by the formula `μ (f '' s) = ∫⁻ x in s, |(f' x).det| ∂μ` (where `(f' x).det` is almost everywhere measurable, but not Borel-measurable in general). This…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/Jacobian.html"}, {"id": "Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -219.032, "z": -97.088, "size": 0.2, "title": "Scaling Haar measure by a continuous isomorphism", "summary": "If `G` is a locally compact topological group and `μ` is a regular Haar measure on `G`, then an isomorphism `φ : G ≃ₜ* G` scales this measure by some positive real constant which we call `mulEquivHaarChar φ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Haar/MulEquivHaarChar.html"}, {"id": "Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus", "region_id": "measure_theory", "micro_elevation": 0.8333, "macro_tier": 1, "macro_tier_score": 0.0315, "macro_tier_override": null, "x": -204.439, "z": -108.661, "size": 0.2693, "title": "Fundamental theorem of calculus for set integrals", "summary": "This file proves a version of the [Fundamental theorem of calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus) for set integrals. See `Filter.Tendsto.integral_sub_linear_isLittleO_ae` and its corollaries. Namely, consider a measurably generated filter `l`, a measure `μ` finite at this filter, and a function `f` that has a finite limit `c` at `l ⊓ ae μ`. Then `∫ x in s, f x ∂μ = μ s • c + o(μ s)`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Bochner/FundThmCalculus.html"}, {"id": "Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory", "region_id": "measure_theory", "micro_elevation": 0.7963, "macro_tier": 1, "macro_tier_score": 0.0315, "macro_tier_override": null, "x": -187.751, "z": -42.73, "size": 0.2693, "title": "Vitali-Carathéodory theorem", "summary": "Vitali-Carathéodory theorem asserts the following. Consider an integrable function `f : α → ℝ` on a space with a regular measure. Then there exists a function `g : α → EReal` such that `f x < g x` everywhere, `g` is lower semicontinuous, and the integral of `g` is arbitrarily close to that of `f`. This theorem is proved in this file, as `exists_lt_lower_semicontinuous_integral_lt`. Symmetrically, there exists `g <…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Bochner/VitaliCaratheodory.html"}, {"id": "Mathlib.MeasureTheory.Group.Pointwise", "region_id": "measure_theory", "micro_elevation": 0.3704, "macro_tier": 3, "macro_tier_score": 0.2191, "macro_tier_override": null, "x": -180.779, "z": -95.629, "size": 0.2816, "title": "Pointwise set operations on `MeasurableSet`s", "summary": "In this file we prove several versions of the following fact: if `s` is a measurable set, then so is `a • s`. Note that the pointwise product of two measurable sets need not be measurable, so there is no `MeasurableSet.mul` etc.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/Pointwise.html"}, {"id": "Mathlib.MeasureTheory.Group.AEStabilizer", "region_id": "measure_theory", "micro_elevation": 0.4259, "macro_tier": 1, "macro_tier_score": 0.0316, "macro_tier_override": null, "x": -174.537, "z": -60.065, "size": 0.2801, "title": "A.e. stabilizer of a set", "summary": "In this file we define the a.e. stabilizer of a set under a measure-preserving group action. The a.e. stabilizer `MulAction.aestabilizer G μ s` of a set `s` is the set of the elements `g : G` such that `s` is a.e.-invariant under `(g • ·)`. For a measure-preserving group action, this set is a subgroup of `G`. If the set is null or conull, then this subgroup is the whole group. The converse is true for an ergodic…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/AEStabilizer.html"}, {"id": "Mathlib.MeasureTheory.Group.Action", "region_id": "measure_theory", "micro_elevation": 0.4074, "macro_tier": 3, "macro_tier_score": 0.2199, "macro_tier_override": null, "x": -191.608, "z": -63.397, "size": 0.3456, "title": "Measures invariant under group actions", "summary": "A measure `μ : Measure α` is said to be *invariant* under an action of a group `G` if scalar multiplication by `c : G` is a measure-preserving map for all `c`. In this file we define a typeclass for measures invariant under action of an (additive or multiplicative) group and prove some basic properties of such measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/Action.html"}, {"id": "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "region_id": "measure_theory", "micro_elevation": 0.6296, "macro_tier": 3, "macro_tier_score": 0.1893, "macro_tier_override": null, "x": -191.653, "z": -105.392, "size": 0.3823, "title": "Relationship between the Haar and Lebesgue measures", "summary": "We prove that the Haar measure and Lebesgue measure are equal on `ℝ` and on `ℝ^ι`, in `MeasureTheory.addHaarMeasure_eq_volume` and `MeasureTheory.addHaarMeasure_eq_volume_pi`. We deduce basic properties of any Haar measure on a finite-dimensional real vector space: * `map_linearMap_addHaar_eq_smul_addHaar`: a linear map rescales the Haar measure by the absolute value of its determinant. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.html"}, {"id": "Mathlib.MeasureTheory.Constructions.Polish.EmbeddingReal", "region_id": "measure_theory", "micro_elevation": 0.463, "macro_tier": 1, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -161.065, "z": -86.705, "size": 0.262, "title": "A Polish Borel space is measurably equivalent to a set of reals", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/Polish/EmbeddingReal.html"}, {"id": "Mathlib.MeasureTheory.Function.UniformIntegrable", "region_id": "measure_theory", "micro_elevation": 0.6852, "macro_tier": 2, "macro_tier_score": 0.0323, "macro_tier_override": null, "x": -181.32, "z": -47.158, "size": 0.3355, "title": "Uniform integrability", "summary": "This file contains the definitions for uniform integrability (both in the measure theory sense as well as the probability theory sense). This file also contains the Vitali convergence theorem which establishes a relation between uniform integrability, convergence in measure and Lp convergence. Uniform integrability plays a vital role in the theory of martingales and most notably is used to formulate the martingale…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/UniformIntegrable.html"}, {"id": "Mathlib.MeasureTheory.Function.LocallyIntegrable", "region_id": "measure_theory", "micro_elevation": 0.7037, "macro_tier": 3, "macro_tier_score": 0.2821, "macro_tier_override": null, "x": -211.94, "z": -87.006, "size": 0.3261, "title": "Locally integrable functions", "summary": "A function is called *locally integrable* (`MeasureTheory.LocallyIntegrable`) if it is integrable on a neighborhood of every point. More generally, it is *locally integrable on `s`* if it is locally integrable on a neighbourhood within `s` of any point of `s`. This file contains properties of locally integrable functions, and integrability results on compact sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LocallyIntegrable.html"}, {"id": "Mathlib.MeasureTheory.Integral.Bochner.SumMeasure", "region_id": "measure_theory", "micro_elevation": 0.7963, "macro_tier": 3, "macro_tier_score": 0.2823, "macro_tier_override": null, "x": -160.1, "z": -48.423, "size": 0.3348, "title": "Integral with respect to a sum of measures", "summary": "In this file we prove that a function `f` is integrable with respect to a countable sum of measures `Measure.sum μ` if and only if `f` is integrable with respect to each `μ i` and the sequence `fun i ↦ ∫ x, ‖f x‖ ∂μ i` is summable. We then show that under this integrability condition, `∫ x, f x ∂Measure.sum μ = ∑' i, ∫ f x ∂μ i`. We specialize these results to the case where each measure is a Dirac mass, i.e. `μ i =…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Bochner/SumMeasure.html"}, {"id": "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "region_id": "measure_theory", "micro_elevation": 0.7407, "macro_tier": 1, "macro_tier_score": 0.0315, "macro_tier_override": null, "x": -174.839, "z": -112.124, "size": 0.2681, "title": "Functions a.e. measurable with respect to a sub-σ-algebra", "summary": "A function `f` verifies `AEStronglyMeasurable[m] f μ` if it is `μ`-a.e. equal to an `m`-strongly measurable function. This is similar to `AEStronglyMeasurable`, but the `MeasurableSpace` structures used for the measurability statement and for the measure are different. We define `lpMeas F 𝕜 m p μ`, the subspace of `Lp F p μ` containing functions `f` verifying `AEStronglyMeasurable[m] f μ`, i.e. functions which are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.html"}, {"id": "Mathlib.MeasureTheory.Measure.OpenPos", "region_id": "measure_theory", "micro_elevation": 0.3889, "macro_tier": 3, "macro_tier_score": 0.282, "macro_tier_override": null, "x": -163.771, "z": -84.223, "size": 0.32, "title": "Measures positive on nonempty opens", "summary": "In this file we define a typeclass for measures that are positive on nonempty opens, see `MeasureTheory.Measure.IsOpenPosMeasure`. Examples include (additive) Haar measures, as well as measures that have positive density with respect to a Haar measure. We also prove some basic facts about these measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/OpenPos.html"}, {"id": "Mathlib.MeasureTheory.Measure.Doubling", "region_id": "measure_theory", "micro_elevation": 0.1296, "macro_tier": 3, "macro_tier_score": 0.2817, "macro_tier_override": null, "x": -184.358, "z": -83.339, "size": 0.2924, "title": "Uniformly locally doubling measures", "summary": "A uniformly locally doubling measure `μ` on a metric space is a measure for which there exists a constant `C` such that for all sufficiently small radii `ε`, and for any centre, the measure of a ball of radius `2 * ε` is bounded by `C` times the measure of the concentric ball of radius `ε`. This file records basic facts about uniformly locally doubling measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Doubling.html"}, {"id": "Mathlib.MeasureTheory.Constructions.BorelSpace.Real", "region_id": "measure_theory", "micro_elevation": 0.4074, "macro_tier": 4, "macro_tier_score": 0.3149, "macro_tier_override": null, "x": -162.035, "z": -79.391, "size": 0.407, "title": "Borel (measurable) spaces ℝ, ℝ≥0, ℝ≥0∞", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/BorelSpace/Real.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.Prod", "region_id": "measure_theory", "micro_elevation": 0.0741, "macro_tier": 4, "macro_tier_score": 0.3131, "macro_tier_override": null, "x": -179.456, "z": -75.465, "size": 0.3021, "title": "The product sigma algebra", "summary": "This file talks about the measurability of operations on binary functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/Prod.html"}, {"id": "Mathlib.MeasureTheory.Measure.Typeclasses.NullSingletonClass", "region_id": "measure_theory", "micro_elevation": 0.2778, "macro_tier": 4, "macro_tier_score": 0.3462, "macro_tier_override": null, "x": -168.104, "z": -80.447, "size": 0.4105, "title": "Measures having value zero on singletons", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Typeclasses/NullSingletonClass.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.Invariants", "region_id": "measure_theory", "micro_elevation": 0.0185, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -181.573, "z": -78.724, "size": 0.2, "title": "σ-algebra of sets invariant under a self-map", "summary": "In this file we define `MeasurableSpace.invariants (f : α → α)` to be the σ-algebra of sets `s : Set α` such that - `s` is measurable w.r.t. the canonical σ-algebra on `α`; - and `f ⁻¹' s = s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/Invariants.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.Defs", "region_id": "measure_theory", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.5016, "macro_tier_override": null, "x": -180.729, "z": -78.619, "size": 0.3713, "title": "Measurable spaces and measurable functions", "summary": "This file defines measurable spaces and measurable functions. A measurable space is a set equipped with a σ-algebra, a collection of subsets closed under complementation and countable union. A function between measurable spaces is measurable if the preimage of each measurable subset is measurable. σ-algebras on a fixed set `α` form a complete lattice. Here we order σ-algebras by writing `m₁ ≤ m₂` if every set which…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/Defs.html"}, {"id": "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "region_id": "measure_theory", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.0007, "macro_tier_override": null, "x": -211.155, "z": -75.203, "size": 0.3139, "title": "Lebesgue measure on `ℂ`", "summary": "In this file, we consider the Lebesgue measure on `ℂ` defined as the push-forward of the volume on `ℝ²` under the natural isomorphism and prove that it is equal to the measure `volume` of `ℂ` coming from its `InnerProductSpace` structure over `ℝ`. For that, we consider the two frequently used ways to represent `ℝ²` in `mathlib`: `ℝ × ℝ` and `Fin 2 → ℝ`, define measurable equivalences (`MeasurableEquiv`) to both…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Lebesgue/Complex.html"}, {"id": "Mathlib.MeasureTheory.Function.LpOrder", "region_id": "measure_theory", "micro_elevation": 0.6481, "macro_tier": 3, "macro_tier_score": 0.2827, "macro_tier_override": null, "x": -161.633, "z": -101.453, "size": 0.36, "title": "Order related properties of Lp spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpOrder.html"}, {"id": "Mathlib.MeasureTheory.PiSystem", "region_id": "measure_theory", "micro_elevation": 0.0185, "macro_tier": 4, "macro_tier_score": 0.4708, "macro_tier_override": null, "x": -180.56, "z": -77.786, "size": 0.3898, "title": "Induction principles for measurable sets, related to π-systems and λ-systems.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/PiSystem.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict", "region_id": "measure_theory", "micro_elevation": 0.0556, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": -178.597, "z": -80.021, "size": 0.3169, "title": "Measurability of the restriction function for functions indexed by a preorder", "summary": "We prove that the map which restricts a function `f : (i : α) → X i` to elements `≤ a` is measurable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/PreorderRestrict.html"}, {"id": "Mathlib.MeasureTheory.Measure.Decomposition.Hahn", "region_id": "measure_theory", "micro_elevation": 0.2963, "macro_tier": 3, "macro_tier_score": 0.1886, "macro_tier_override": null, "x": -188.237, "z": -67.271, "size": 0.3388, "title": "Unsigned Hahn decomposition theorem", "summary": "This file proves the unsigned version of the Hahn decomposition theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Decomposition/Hahn.html"}, {"id": "Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap", "region_id": "measure_theory", "micro_elevation": 0.8333, "macro_tier": 4, "macro_tier_score": 0.2872, "macro_tier_override": null, "x": -191.9, "z": -42.015, "size": 0.5276, "title": "Continuous linear maps composed with integration", "summary": "The goal of this file is to prove that integration commutes with continuous linear maps. This holds for simple functions. The general result follows from the continuity of all involved operations on the space `L¹`. Note that composition by a continuous linear map on `L¹` is not just the composition, as we are dealing with classes of functions, but it has already been defined as `ContinuousLinearMap.compLp`. We take…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Bochner/ContinuousLinearMap.html"}, {"id": "Mathlib.MeasureTheory.Constructions.Pi", "region_id": "measure_theory", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.2206, "macro_tier_override": null, "x": -189.637, "z": -54.711, "size": 0.3809, "title": "Indexed product measures", "summary": "In this file we define and prove properties about finite products of measures (and at some point, countable products of measures).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/Pi.html"}, {"id": "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "region_id": "measure_theory", "micro_elevation": 0.4259, "macro_tier": 3, "macro_tier_score": 0.2826, "macro_tier_override": null, "x": -164.693, "z": -67.418, "size": 0.3552, "title": "Measurability of real and complex functions", "summary": "We show that most standard real and complex functions are measurable, notably `exp`, `cos`, `sin`, `cosh`, `sinh`, `log`, `pow`, `arcsin`, `arccos`. See also `MeasureTheory.Function.SpecialFunctions.Arctan` and `MeasureTheory.Function.SpecialFunctions.Inner`, which have been split off to minimize imports.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "region_id": "measure_theory", "micro_elevation": 0.9815, "macro_tier": 2, "macro_tier_score": 0.0332, "macro_tier_override": null, "x": -209.614, "z": -44.016, "size": 0.3854, "title": "Links between an integral and its \"improper\" version", "summary": "In its current state, mathlib only knows how to talk about definite (\"proper\") integrals, in the sense that it treats integrals over `[x, +∞)` the same as it treats integrals over `[y, z]`. For example, the integral over `[1, +∞)` is **not** defined to be the limit of the integral over `[1, x]` as `x` tends to `+∞`, which is known as an **improper integral**. Indeed, the \"proper\" definition is stronger than the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntegralEqImproper.html"}, {"id": "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 1, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": -203.875, "z": -114.291, "size": 0.2856, "title": "Properties of integration with respect to the Lebesgue measure", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Lebesgue/Integral.html"}, {"id": "Mathlib.MeasureTheory.Order.Group.Lattice", "region_id": "measure_theory", "micro_elevation": 0.3704, "macro_tier": 1, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -165.535, "z": -86.264, "size": 0.2556, "title": "Measurability results on groups with a lattice structure.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Order/Group/Lattice.html"}, {"id": "Mathlib.MeasureTheory.Group.ModularCharacter", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -178.978, "z": -121.107, "size": 0.2, "title": "Modular character of a locally compact group", "summary": "On a locally compact group, there is a natural homomorphism `G → ℝ≥0*`, which for `g : G` gives the value `μ (· * g⁻¹) / μ`, where `μ` is an (inner regular) Haar measure. This file defines this homomorphism, called the modular character, and shows that it is independent of the chosen Haar measure. TODO: Show that the character is continuous.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/ModularCharacter.html"}, {"id": "Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving", "region_id": "measure_theory", "micro_elevation": 0.2222, "macro_tier": 4, "macro_tier_score": 0.383, "macro_tier_override": null, "x": -189.798, "z": -73.939, "size": 0.5783, "title": "Quasi-Measure-Preserving Functions", "summary": "A map `f : α → β` is said to be *quasi-measure-preserving* (a.k.a. non-singular) w.r.t. measures `μa` and `μb` if it is measurable and `μb s = 0` implies `μa (f ⁻¹' s) = 0`. That last condition can also be written `μa.map f ≪ μb` (the map of `μa` by `f` is absolutely continuous with respect to `μb`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/QuasiMeasurePreserving.html"}, {"id": "Mathlib.MeasureTheory.OuterMeasure.BorelCantelli", "region_id": "measure_theory", "micro_elevation": 0.0556, "macro_tier": 4, "macro_tier_score": 0.3785, "macro_tier_override": null, "x": -182.139, "z": -80.746, "size": 0.4521, "title": "Borel-Cantelli lemma, part 1", "summary": "In this file we show one implication of the **Borel-Cantelli lemma**: if `s i` is a countable family of sets such that `∑' i, μ (s i)` is finite, then a.e. all points belong to finitely many sets of the family. We prove several versions of this lemma: - `MeasureTheory.ae_finite_setOf_mem`: as stated above; - `MeasureTheory.measure_limsup_cofinite_eq_zero`: in terms of `Filter.limsup` along `Filter.cofinite`; -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/OuterMeasure/BorelCantelli.html"}, {"id": "Mathlib.MeasureTheory.Measure.Sub", "region_id": "measure_theory", "micro_elevation": 0.2963, "macro_tier": 3, "macro_tier_score": 0.1886, "macro_tier_override": null, "x": -172.555, "z": -89.498, "size": 0.3388, "title": "Subtraction of measures", "summary": "In this file we define `μ - ν` to be the least measure `τ` such that `μ ≤ τ + ν`. It is equivalent to `(μ - ν) ⊔ 0` if `μ` and `ν` were signed measures. Compare with `ENNReal.instSub`. Specifically, note that if you have `α = {1,2}`, and `μ {1} = 2`, `μ {2} = 0`, and `ν {2} = 2`, `ν {1} = 0`, then `(μ - ν) {1, 2} = 2`. However, if `μ ≤ ν`, and `ν univ ≠ ∞`, then `(μ - ν) + ν = μ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Sub.html"}, {"id": "Mathlib.MeasureTheory.Measure.Typeclasses.SFinite", "region_id": "measure_theory", "micro_elevation": 0.2963, "macro_tier": 4, "macro_tier_score": 0.3479, "macro_tier_override": null, "x": -171.732, "z": -88.828, "size": 0.4746, "title": "Classes for s-finite measures", "summary": "We introduce the following typeclasses for measures: * `SFinite μ`: the measure `μ` can be written as a countable sum of finite measures; * `SigmaFinite μ`: there exists a countable collection of sets that cover `univ` where `μ` is finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Typeclasses/SFinite.html"}, {"id": "Mathlib.MeasureTheory.Function.AEEqFun", "region_id": "measure_theory", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.2827, "macro_tier_override": null, "x": -171.497, "z": -57.594, "size": 0.3615, "title": "Almost everywhere equal functions", "summary": "We build a space of equivalence classes of functions, where two functions are treated as identical if they are almost everywhere equal. We form the set of equivalence classes under the relation of being almost everywhere equal, which is sometimes known as the `L⁰` space. To use this space as a basis for the `L^p` spaces and for the Bochner integral, we consider equivalence classes of strongly measurable functions…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/AEEqFun.html"}, {"id": "Mathlib.MeasureTheory.Measure.MeasuredSets", "region_id": "measure_theory", "micro_elevation": 0.2963, "macro_tier": 1, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -193.711, "z": -74.541, "size": 0.2382, "title": "Measured sets", "summary": "Consider a measure `μ` on a measurable space. One can define an extended distance on the space of measurable sets, by `edist s t := μ (s ∆ t)`. In this file, we introduce this definition on the type synonym `MeasuredSets μ`, and we prove that `μ` is a continuous function on this space. We also give a density criterion for this distance, in `exists_measure_symmDiff_lt_of_generateFrom_isSetRing`: given a ring of sets…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/MeasuredSets.html"}, {"id": "Mathlib.MeasureTheory.Measure.Comap", "region_id": "measure_theory", "micro_elevation": 0.2407, "macro_tier": 4, "macro_tier_score": 0.3838, "macro_tier_override": null, "x": -174.469, "z": -87.733, "size": 0.5971, "title": "Pullback of a measure", "summary": "In this file we define the pullback `MeasureTheory.Measure.comap f μ` of a measure `μ` along an injective map `f` such that the image of any measurable set under `f` is a null-measurable set. If `f` does not have these properties, then we define `comap f μ` to be zero. In the future, we may decide to redefine `comap f μ` so that it gives meaningful results, e.g., for covering maps like `(↑) : ℝ → AddCircle (1 : ℝ)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Comap.html"}, {"id": "Mathlib.MeasureTheory.Measure.Content", "region_id": "measure_theory", "micro_elevation": 0.4074, "macro_tier": 3, "macro_tier_score": 0.2191, "macro_tier_override": null, "x": -189.214, "z": -95.295, "size": 0.2801, "title": "Contents", "summary": "In this file we work with *contents*. A content `λ` is a function from a certain class of subsets (such as the compact subsets) to `ℝ≥0` that is * additive: If `K₁` and `K₂` are disjoint sets in the domain of `λ`, then `λ(K₁ ∪ K₂) = λ(K₁) + λ(K₂)`; * subadditive: If `K₁` and `K₂` are in the domain of `λ`, then `λ(K₁ ∪ K₂) ≤ λ(K₁) + λ(K₂)`; * monotone: If `K₁ ⊆ K₂` are in the domain of `λ`, then `λ(K₁) ≤ λ(K₂)`. We…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Content.html"}, {"id": "Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut", "region_id": "measure_theory", "micro_elevation": 0.963, "macro_tier": 1, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": -139.203, "z": -63.409, "size": 0.2932, "title": "Pull-out property of the conditional expectation", "summary": "Let `Ω` be endowed with a measurable space structure `mΩ`, and let `m : MeasurableSpace Ω` such that `m ≤ mΩ`. Let `μ` be a measure over `Ω`. Let `B : F →L[ℝ] E →L[ℝ] G` a continuous bilinear map, `f : Ω → F` and `g : Ω → E` such that `fun ω ↦ B (f ω) (g ω)` is integrable, `g` is integrable and `f` is `AEStronglyMeasurable` with respect to `m`. The **pull-out** property of the conditional expectation states that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConditionalExpectation/PullOut.html"}, {"id": "Mathlib.MeasureTheory.Function.Holder", "region_id": "measure_theory", "micro_elevation": 0.7963, "macro_tier": 2, "macro_tier_score": 0.0319, "macro_tier_override": null, "x": -216.73, "z": -72.195, "size": 0.3118, "title": "Continuous bilinear maps on `MeasureTheory.Lp` spaces", "summary": "Given a continuous bilinear map `B : E →L[𝕜] F →L[𝕜] G`, we define an associated map `ContinuousLinearMap.holder : Lp E p μ → Lp F q μ → Lp G r μ` where `p q r` are a Hölder triple. We bundle this into a bilinear map `ContinuousLinearMap.holderₗ` and a continuous bilinear map `ContinuousLinearMap.holderL` under some additional assumptions. We also declare a heterogeneous scalar multiplication (`HSMul`) instance on…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/Holder.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum", "region_id": "measure_theory", "micro_elevation": 0.7407, "macro_tier": 2, "macro_tier_score": 0.032, "macro_tier_override": null, "x": -147.373, "z": -85.299, "size": 0.3138, "title": "Pointwise convergence of infinite sums in `Lᵖ`", "summary": "If a series in `Lᵖ` is converging in norm, then the series also converges pointwise almost everywhere.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSpace/InfiniteSum.html"}, {"id": "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 1, "macro_tier_score": 0.0316, "macro_tier_override": null, "x": -220.671, "z": -77.082, "size": 0.2745, "title": "Uniqueness of the conditional expectation", "summary": "Two Lp functions `f, g` which are almost everywhere strongly measurable with respect to a σ-algebra `m` and verify `∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ` for all `m`-measurable sets `s` are equal almost everywhere. This proves the uniqueness of the conditional expectation, which is not yet defined in this file but is introduced in `Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.html"}, {"id": "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "region_id": "measure_theory", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -163.956, "z": -62.937, "size": 0.2544, "title": "Inner products of strongly measurable functions are strongly measurable.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/StronglyMeasurable/Inner.html"}, {"id": "Mathlib.MeasureTheory.Integral.PeakFunction", "region_id": "measure_theory", "micro_elevation": 1.0, "macro_tier": 1, "macro_tier_score": 0.0006, "macro_tier_override": null, "x": -226.07, "z": -85.923, "size": 0.3008, "title": "Integrals against peak functions", "summary": "A sequence of peak functions is a sequence of functions with average one concentrating around a point `x₀`. Given such a sequence `φₙ`, then `∫ φₙ g` tends to `g x₀` in many situations, with a whole zoo of possible assumptions on `φₙ` and `g`. This file is devoted to such results. Such functions are also called approximations of unity, or approximations of identity.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/PeakFunction.html"}, {"id": "Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -157.844, "z": -44.815, "size": 0.2478, "title": "Results about both conditional expectations", "summary": "For non-negative real functions, we have two versions of the conditional expectation: `condExp` and `condLExp`, built from the Bochner and Lebesgue integrals respectively. In this file, we gather results that involve both versions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConditionalExpectation/LebesgueBochner.html"}, {"id": "Mathlib.MeasureTheory.Function.ConditionalLExpectation", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 1, "macro_tier_score": 0.0316, "macro_tier_override": null, "x": -172.192, "z": -117.669, "size": 0.2806, "title": "Conditional Lebesgue expectation", "summary": "We define the conditional expectation of a `ℝ≥0∞`-valued function using the Lebesgue integral. Given a measure `P : Measure[mΩ₀] Ω` and a sub-σ-algebra `mΩ` of `mΩ₀` (meaning `hm : mΩ ≤ mΩ₀`) and a function `X : Ω → ℝ≥0∞`, if `P.trim hm` is σ-finite, then the conditional (Lebesgue) expectation `P⁻[X|mΩ]` of `X` is the `mΩ`-measurable function such that for all `mΩ`-measurable sets `s`, `∫⁻ ω in s, P⁻[X|mΩ] ω ∂P = ∫⁻…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConditionalLExpectation.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable", "region_id": "measure_theory", "micro_elevation": 0.0185, "macro_tier": 4, "macro_tier_score": 0.3754, "macro_tier_override": null, "x": -181.113, "z": -77.86, "size": 0.2819, "title": "Measurability modulo a filter", "summary": "In this file we consider the general notion of measurability modulo a σ-filter. Two important instances of this construction are null-measurability with respect to a measure, where the filter is the collection of co-null sets, and Baire-measurability with respect to a topology, where the filter is the collection of comeager (residual) sets. (not to be confused with measurability with respect to the sigma algebra of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/EventuallyMeasurable.html"}, {"id": "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "region_id": "measure_theory", "micro_elevation": 0.9815, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": -184.239, "z": -123.557, "size": 0.3188, "title": "Conditional expectation of real-valued functions", "summary": "This file proves some results regarding the conditional expectation of real-valued functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.html"}, {"id": "Mathlib.MeasureTheory.Measure.EverywherePos", "region_id": "measure_theory", "micro_elevation": 0.5556, "macro_tier": 2, "macro_tier_score": 0.1256, "macro_tier_override": null, "x": -157.366, "z": -88.872, "size": 0.3021, "title": "Everywhere positive sets in measure spaces", "summary": "A set `s` in a topological space with a measure `μ` is *everywhere positive* (also called *self-supporting*) if any neighborhood `n` of any point of `s` satisfies `μ (s ∩ n) > 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/EverywherePos.html"}, {"id": "Mathlib.MeasureTheory.Topology", "region_id": "measure_theory", "micro_elevation": 0.2963, "macro_tier": 3, "macro_tier_score": 0.1573, "macro_tier_override": null, "x": -173.149, "z": -89.92, "size": 0.34, "title": "Theorems combining measure theory and topology", "summary": "This file gathers theorems that combine measure theory and topology, and cannot easily be added to the existing files without introducing massive dependencies between the subjects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Topology.html"}, {"id": "Mathlib.MeasureTheory.Measure.Haar.Extension", "region_id": "measure_theory", "micro_elevation": 0.8519, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -215.029, "z": -97.435, "size": 0.2, "title": "Haar measures on group extensions", "summary": "In this file, if `1 → A → B → C → 1` is a short exact sequence of topological groups, we construct a Haar measure on `B` from Haar measures on `A` and `C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Haar/Extension.html"}, {"id": "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "region_id": "measure_theory", "micro_elevation": 0.463, "macro_tier": 4, "macro_tier_score": 0.285, "macro_tier_override": null, "x": -199.255, "z": -68.186, "size": 0.4602, "title": "Strongly measurable and finitely strongly measurable functions", "summary": "A function `f` is said to be strongly measurable if `f` is the sequential limit of simple functions. It is said to be finitely strongly measurable with respect to a measure `μ` if the supports of those simple functions have finite measure. If the target space has a second countable topology, strongly measurable and measurable are equivalent. If the measure is sigma-finite, strongly measurable and finitely strongly…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.html"}, {"id": "Mathlib.MeasureTheory.Function.SimpleFuncDense", "region_id": "measure_theory", "micro_elevation": 0.4444, "macro_tier": 3, "macro_tier_score": 0.283, "macro_tier_override": null, "x": -171.197, "z": -96.668, "size": 0.3773, "title": "Density of simple functions", "summary": "Show that each Borel measurable function can be approximated pointwise by a sequence of simple functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/SimpleFuncDense.html"}, {"id": "Mathlib.MeasureTheory.Constructions.BorelSpace.Metric", "region_id": "measure_theory", "micro_elevation": 0.3889, "macro_tier": 4, "macro_tier_score": 0.3133, "macro_tier_override": null, "x": -175.498, "z": -95.696, "size": 0.3228, "title": "Borel sigma algebras on (pseudo-)metric spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.html"}, {"id": "Mathlib.MeasureTheory.Measure.SubFinite", "region_id": "measure_theory", "micro_elevation": 0.5741, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -203.091, "z": -92.584, "size": 0.2, "title": "Results about subtraction of finite measures", "summary": "The content of this file is not placed in `MeasureTheory.Measure.Sub` because it uses tools that are not imported in the other file: the Hahn decomposition of finite measures and measures built with `withDensity`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/SubFinite.html"}, {"id": "Mathlib.MeasureTheory.Integral.CircleTransform", "region_id": "measure_theory", "micro_elevation": 0.9444, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -181.622, "z": -35.255, "size": 0.2, "title": "Circle integral transform", "summary": "In this file we define the circle integral transform of a function `f` with complex domain. This is defined as $(2πi)^{-1}\\frac{f(x)}{x-w}$ where `x` moves along a circle. We then prove some basic facts about these functions. These results are useful for proving that the uniform limit of a sequence of holomorphic functions is holomorphic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/CircleTransform.html"}, {"id": "Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne", "region_id": "measure_theory", "micro_elevation": 0.463, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -184.332, "z": -99.573, "size": 0.2503, "title": null, "summary": "We introduce the typeclass `IsZeroOneMeasure` for measures that only take the values `0` and `1`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Typeclasses/ZeroOne.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSpace.DomAct.Continuous", "region_id": "measure_theory", "micro_elevation": 0.7407, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -174.608, "z": -112.083, "size": 0.2, "title": "Continuity of the action of `Mᵈᵐᵃ` on `MeasureSpace.Lp E p μ`", "summary": "In this file we prove that under certain conditions, the action of `Mᵈᵐᵃ` on `MeasureTheory.Lp E p μ` is continuous in both variables. Recall that `Mᵈᵐᵃ` acts on `MeasureTheory.Lp E p μ` by `mk c • f = MeasureTheory.Lp.compMeasurePreserving (c • ·) _ f`. This action is defined, if `M` acts on `X` by measure-preserving maps. If `M` acts on `X` by continuous maps preserving a locally finite measure which is inner…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSpace/DomAct/Continuous.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSpace.ContinuousCompMeasurePreserving", "region_id": "measure_theory", "micro_elevation": 0.7222, "macro_tier": 1, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -190.944, "z": -110.175, "size": 0.239, "title": "Continuity of `MeasureTheory.Lp.compMeasurePreserving`", "summary": "In this file we prove that the composition of an `L^p` function `g` with a continuous measure-preserving map `f` is continuous in both arguments. First, we prove it for indicator functions, in terms of convergence of `μ ((f a ⁻¹' s) ∆ (g ⁻¹' s))` to zero. Then we prove the continuity of the function of two arguments defined on `MeasureTheory.Lp E p ν × {f : C(X, Y) // MeasurePreserving f μ ν}`. Finally, we provide…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSpace/ContinuousCompMeasurePreserving.html"}, {"id": "Mathlib.MeasureTheory.Measure.FiniteMeasurePi", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -166.682, "z": -38.483, "size": 0.2, "title": "Products of finite measures and probability measures", "summary": "This file introduces finite products of finite measures and probability measures. The constructions are obtained from special cases of products of general measures. Taking products nevertheless has specific properties in the cases of finite measures and probability measures, notably the fact that the product measures depend continuously on their factors in the topology of weak convergence when the underlying space…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/FiniteMeasurePi.html"}, {"id": "Mathlib.MeasureTheory.Group.Defs", "region_id": "measure_theory", "micro_elevation": 0.2037, "macro_tier": 3, "macro_tier_score": 0.2819, "macro_tier_override": null, "x": -189.79, "z": -80.945, "size": 0.3088, "title": "Definitions about invariant measures", "summary": "In this file we define typeclasses for measures invariant under (scalar) multiplication. - `MeasureTheory.SMulInvariantMeasure M α μ` says that the measure `μ` is invariant under scalar multiplication by `c : M`; - `MeasureTheory.VAddInvariantMeasure M α μ` is the additive version of this typeclass; - `MeasureTheory.Measure.IsMulLeftInvariant μ`, `MeasureTheory.Measure.IsMulRightInvariant μ` say that the measure `μ`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/Defs.html"}, {"id": "Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence", "region_id": "measure_theory", "micro_elevation": 0.4815, "macro_tier": 3, "macro_tier_score": 0.2817, "macro_tier_override": null, "x": -159.713, "z": -71.745, "size": 0.2872, "title": "Dominated convergence theorem", "summary": "Lebesgue's dominated convergence theorem states that the limit and Lebesgue integral of a sequence of (almost everywhere) measurable functions can be swapped if the functions are pointwise dominated by a fixed function. This file provides a few variants of the result.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Lebesgue/DominatedConvergence.html"}, {"id": "Mathlib.MeasureTheory.SpecificCodomains.Pi", "region_id": "measure_theory", "micro_elevation": 0.8519, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -141.615, "z": -77.851, "size": 0.2556, "title": "Integrability in a product space", "summary": "We prove that `f : X → Π i, E i` is in `Lᵖ` if and only if for all `i`, `f · i` is in `Lᵖ`. We do the same for `f : X → (E × F)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/SpecificCodomains/Pi.html"}, {"id": "Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms", "region_id": "measure_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -180.729, "z": -78.619, "size": 0.2, "title": "NoAtoms", "summary": "This file is deprecated. Please use `Mathlib.MeasureTheory.Measure.Typeclasses.NullSingletonClass` instead.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Typeclasses/NoAtoms.html"}, {"id": "Mathlib.MeasureTheory.Order.Lattice", "region_id": "measure_theory", "micro_elevation": 0.3519, "macro_tier": 4, "macro_tier_score": 0.3133, "macro_tier_override": null, "x": -175.749, "z": -93.992, "size": 0.3215, "title": "Typeclasses for measurability of lattice operations", "summary": "In this file we define classes `MeasurableSup` and `MeasurableInf` and prove dot-style lemmas (`Measurable.sup`, `AEMeasurable.sup` etc). For binary operations we define two typeclasses: - `MeasurableSup` says that both left and right sup are measurable; - `MeasurableSup₂` says that `fun p : α × α => p.1 ⊔ p.2` is measurable, and similarly for other binary operations. The reason for introducing these classes is that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Order/Lattice.html"}, {"id": "Mathlib.MeasureTheory.Integral.LebesgueNormedSpace", "region_id": "measure_theory", "micro_elevation": 0.5741, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -164.417, "z": -99.332, "size": 0.2, "title": "A lemma about measurability with density under scalar multiplication in normed spaces", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/LebesgueNormedSpace.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.WithDensityVec", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -217.494, "z": -60.877, "size": 0.2, "title": "Vector measure with density with respect to a vector measure", "summary": "Given a vector measure `μ`, a function `f` and a pairing `B`, we define the vector measure with density `f` and pairing `B`, denoted `μ.withDensity f B`. It associates to a measurable set the mass `∫ᵛ x in s, f x ∂[B; μ]`. This file implements the basic property of this notion. Notably, we show in `variation_withDensity` that the variation of the vector measure `μ.withDensity f B` is the positive measure with…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/WithDensityVec.html"}, {"id": "Mathlib.MeasureTheory.Integral.CurveIntegral.Basic", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 1, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -209.066, "z": -106.811, "size": 0.2239, "title": "Integral of a 1-form along a path", "summary": "In this file we define the integral of a 1-form along a path indexed by `[0, 1]` and prove basic properties of this operation. The integral `∫ᶜ x in γ, ω x` is defined as $\\int_0^1 \\omega(\\gamma(t))(\\gamma'(t))$. More precisely, we use - `Path.extend γ t` instead of `γ t`, because both derivatives and `intervalIntegral` expect globally defined functions; - `derivWithin γ.extend (Set.Icc 0 1) t`, not `deriv γ.extend…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/CurveIntegral/Basic.html"}, {"id": "Mathlib.MeasureTheory.Group.FoelnerFilter", "region_id": "measure_theory", "micro_elevation": 0.4259, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -189.94, "z": -95.876, "size": 0.2, "title": "Følner sequences and filters - definitions and properties", "summary": "This file defines Følner sequences and filters for measurable spaces acted on by a group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/FoelnerFilter.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntervalIntegral.ContDiff", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 1, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": -179.7, "z": -36.959, "size": 0.2566, "title": "Fundamental theorem of calculus for `C^1` functions", "summary": "We give versions of the second fundamental theorem of calculus under the strong assumption that the function is `C^1` on the interval. This is restrictive, but satisfied in many situations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntervalIntegral/ContDiff.html"}, {"id": "Mathlib.MeasureTheory.Measure.Haar.Basic", "region_id": "measure_theory", "micro_elevation": 0.5741, "macro_tier": 3, "macro_tier_score": 0.2193, "macro_tier_override": null, "x": -186.704, "z": -52.941, "size": 0.2955, "title": "Haar measure", "summary": "In this file we prove the existence of Haar measure for a locally compact Hausdorff topological group. We follow the write-up by Jonathan Gleason, *Existence and Uniqueness of Haar Measure*. This is essentially the same argument as in https://en.wikipedia.org/wiki/Haar_measure#A_construction_using_compact_subsets. We construct the Haar measure first on compact sets. For this we define `(K : U)` as the (smallest)…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Haar/Basic.html"}, {"id": "Mathlib.MeasureTheory.Function.Floor", "region_id": "measure_theory", "micro_elevation": 0.4074, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -199.427, "z": -77.946, "size": 0.2396, "title": "Measurability of `⌊x⌋` etc", "summary": "In this file we prove that `Int.floor`, `Int.ceil`, `Int.fract`, `Nat.floor`, and `Nat.ceil` are measurable under some assumptions on the (semi)ring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/Floor.html"}, {"id": "Mathlib.MeasureTheory.Integral.Asymptotics", "region_id": "measure_theory", "micro_elevation": 0.8519, "macro_tier": 2, "macro_tier_score": 0.0317, "macro_tier_override": null, "x": -216.397, "z": -94.692, "size": 0.2904, "title": "Bounding of integrals by asymptotics", "summary": "We establish integrability of `f` from `f = O(g)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Asymptotics.html"}, {"id": "Mathlib.MeasureTheory.Constructions.AddChar", "region_id": "measure_theory", "micro_elevation": 0.0185, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -181.261, "z": -77.956, "size": 0.2, "title": "Measurable space instance for additive characters", "summary": "This file endows `AddChar A M` with the discrete measurable space structure whenever `A` is a finite discrete measurable space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/AddChar.html"}, {"id": "Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -148.623, "z": -50.736, "size": 0.2, "title": "Poincaré lemma for 1-forms", "summary": "In this file we prove Poincaré lemma for 1-forms for convex sets. Namely, we show that a closed 1-form on a convex subset of a normed space is exact. We also prove that the integrals of a closed 1-form along 2 curves that are joined by a `C²`-smooth homotopy are equal. In the future, this will allow us to prove Poincaré lemma for simply connected open sets and, more generally, for simply connected locally convex…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/CurveIntegral/Poincare.html"}, {"id": "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 2, "macro_tier_score": 0.0318, "macro_tier_override": null, "x": -145.531, "z": -56.31, "size": 0.303, "title": "Divergence theorem for Bochner integral", "summary": "In this file we prove the Divergence theorem for Bochner integral on a box in `ℝⁿ⁺¹ = Fin (n + 1) → ℝ`. More precisely, we prove the following theorem. Let `E` be a complete normed space. If `f : ℝⁿ⁺¹ → Eⁿ⁺¹` is continuous on a rectangular box `[a, b] : Set ℝⁿ⁺¹`, `a ≤ b`, differentiable on its interior with derivative `f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹`, and the divergence `fun x ↦ ∑ i, f' x eᵢ i` is integrable on `[a,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/DivergenceTheorem.html"}, {"id": "Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -152.958, "z": -109.69, "size": 0.2, "title": "Radon-Nikodym derivatives and conditional expectations", "summary": "We express the Radon-Nikodym derivative of the pushforward of measures in terms of the conditional expectation of the Radon-Nikodym derivative of the original measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConditionalExpectation/RadonNikodym.html"}, {"id": "Mathlib.MeasureTheory.Measure.WithDensityFinite", "region_id": "measure_theory", "micro_elevation": 0.3333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -165.733, "z": -81.696, "size": 0.2, "title": "s-finite measures can be written as `withDensity` of a finite measure", "summary": "If `μ` is an s-finite measure, then there exists a finite measure `μ.toFinite` such that a set is `μ`-null iff it is `μ.toFinite`-null. In particular, `MeasureTheory.ae μ.toFinite = MeasureTheory.ae μ` and `μ.toFinite = 0` iff `μ = 0`. As a corollary, `μ` can be represented as `μ.toFinite.withDensity (μ.rnDeriv μ.toFinite)`. Our definition of `MeasureTheory.Measure.toFinite` ensures some extra properties: - if `μ`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/WithDensityFinite.html"}, {"id": "Mathlib.MeasureTheory.Integral.Regular", "region_id": "measure_theory", "micro_elevation": 0.8333, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -183.066, "z": -116.819, "size": 0.2437, "title": "Integrals of continuous functions with respect to regular measures", "summary": "When a measure is regular, one may express the measure of compact sets and of open sets in terms of the integral of continuous functions equal to 1 on the compact set, or to 0 outside of the open set respectively.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Regular.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSeminorm.Prod", "region_id": "measure_theory", "micro_elevation": 0.537, "macro_tier": 2, "macro_tier_score": 0.1264, "macro_tier_override": null, "x": -205.292, "z": -80.841, "size": 0.3607, "title": "ℒp spaces and products", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSeminorm/Prod.html"}, {"id": "Mathlib.MeasureTheory.Measure.ResolventTransform", "region_id": "measure_theory", "micro_elevation": 0.4259, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -186.624, "z": -59.968, "size": 0.2, "title": "Resolvent Transform of a Measure", "summary": "Given a normed algebra `A` over a normed field `𝕜`, and `μ : Measure 𝕜`, we define the resolvent transform of `μ` by the formula `resolventTransform μ a = ∫ x, resolvent a x ∂μ = ∫ x, (↑ₐ x - a)⁻¹ʳ ∂μ` This is not a standard notion in the literature, but specializes to a few standard notions, namely the case `𝕜 = ℝ` and `A = ℂ` is the Stieltjes transform, and the case `𝕜 = A = ℂ` is the Cauchy transform, given by…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/ResolventTransform.html"}, {"id": "Mathlib.MeasureTheory.Measure.Support", "region_id": "measure_theory", "micro_elevation": 0.4074, "macro_tier": 1, "macro_tier_score": 0.0313, "macro_tier_override": null, "x": -197.769, "z": -70.892, "size": 0.239, "title": "Support of a Measure", "summary": "This file develops the theory of the **support** of a measure `μ` on a topological measurable space. The support is defined as the set of points whose every open neighborhood has positive measure. We give equivalent characterizations, prove basic measure-theoretic properties, and study interactions with sums, restrictions, and absolute continuity. Under various Lindelöf conditions, the support is conull, and various…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Support.html"}, {"id": "Mathlib.MeasureTheory.Measure.AEDisjoint", "region_id": "measure_theory", "micro_elevation": 0.1296, "macro_tier": 4, "macro_tier_score": 0.3754, "macro_tier_override": null, "x": -174.806, "z": -79.22, "size": 0.2819, "title": "Almost everywhere disjoint sets", "summary": "We say that sets `s` and `t` are `μ`-a.e. disjoint (see `MeasureTheory.AEDisjoint`) if their intersection has measure zero. This assumption can be used instead of `Disjoint` in most theorems in measure theory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/AEDisjoint.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation", "region_id": "measure_theory", "micro_elevation": 0.4074, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -167.415, "z": -65.474, "size": 0.2, "title": "The semivariation of a vector measure", "summary": "The semivariation of a vector measure is the supremum of the variations of its push-forwards to `ℝ` through all linear forms of norm at most `1`. The interest of this notion is that, in the reals, any set has nonnegative or nonpositive measure, so that the variation is realized by a subset (up to a factor of at most `2`). This property is inherited by the semivariation in general: one has the inequalities ``` ‖μ s‖ₑ…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/Variation/Semivariation.html"}, {"id": "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 1, "macro_tier_score": 0.0317, "macro_tier_override": null, "x": -149.292, "z": -104.661, "size": 0.288, "title": "Conditional expectation in L2", "summary": "This file contains one step of the construction of the conditional expectation, which is completed in `Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean`. See that file for a description of the full process. We build the conditional expectation of an `L²` function, as an element of `L²`. This is the orthogonal projection on the subspace of almost everywhere `m`-measurable functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.html"}, {"id": "Mathlib.MeasureTheory.Covering.Vitali", "region_id": "measure_theory", "micro_elevation": 0.3889, "macro_tier": 2, "macro_tier_score": 0.0629, "macro_tier_override": null, "x": -198.529, "z": -80.086, "size": 0.2852, "title": "Vitali covering theorems", "summary": "The topological Vitali covering theorem, in its most classical version, states the following. Consider a family of balls `(B (x_i, r_i))_{i ∈ I}` in a metric space, with uniformly bounded radii. Then one can extract a disjoint subfamily indexed by `J ⊆ I`, such that any `B (x_i, r_i)` is included in a ball `B (x_j, 5 r_j)`. We prove this theorem in `Vitali.exists_disjoint_subfamily_covering_enlargement_closedBall`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Covering/Vitali.html"}, {"id": "Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral", "region_id": "measure_theory", "micro_elevation": 0.5741, "macro_tier": 3, "macro_tier_score": 0.2827, "macro_tier_override": null, "x": -173.791, "z": -53.184, "size": 0.36, "title": "Function with finite integral", "summary": "In this file we define the predicate `HasFiniteIntegral`, which is then used to define the predicate `Integrable` in the corresponding file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/L1Space/HasFiniteIntegral.html"}, {"id": "Mathlib.MeasureTheory.Integral.Lebesgue.Norm", "region_id": "measure_theory", "micro_elevation": 0.4444, "macro_tier": 3, "macro_tier_score": 0.2815, "macro_tier_override": null, "x": -195.784, "z": -64.837, "size": 0.2729, "title": "Interactions between the Lebesgue integral and norms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Lebesgue/Norm.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -201.318, "z": -112.881, "size": 0.256, "title": "Some properties of the interval integral of `fun x ↦ slope f x (x + c)`, given a constant `c : ℝ`", "summary": "This file proves that: * `IntervalIntegrable.intervalIntegrable_slope`: If `f` is interval integrable on `a..(b + c)` where `a ≤ b` and `0 ≤ c`, then `fun x ↦ slope f x (x + c)` is interval integrable on `a..b`. * `MonotoneOn.intervalIntegrable_slope`: If `f` is monotone on `a..(b + c)` where `a ≤ b` and `0 ≤ c`, then `fun x ↦ slope f x (x + c)` is interval integrable on `a..b`. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntervalIntegral/Slope.html"}, {"id": "Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -222.309, "z": -75.832, "size": 0.239, "title": "Characterization of a finite measure by the integrals of products of bounded functions", "summary": "Given two finite families of Borel spaces `(i : ι) → X i` and `(j : κ) → Y j` satisfying `HasOuterApproxClosed`, a finite measure `μ` over `(Π i, X i) × (Π j, Y j)` is determined by the values `∫ p, (Π i, f i (p.1 i)) * (Π j, g j (p.2 j)) ∂μ`, for `f : (i : ι) → X i → ℝ` and `g : (j : κ) → Y j → ℝ` any families of bounded continuous functions. In particular, if `μ` and `ν` are two finite measures over `Π i, X i` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/HasOuterApproxClosedProd.html"}, {"id": "Mathlib.MeasureTheory.Measure.Haar.Quotient", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 2, "macro_tier_score": 0.0628, "macro_tier_override": null, "x": -177.879, "z": -37.897, "size": 0.2764, "title": "Haar quotient measure", "summary": "In this file, we consider properties of fundamental domains and measures for the action of a subgroup `Γ` of a topological group `G` on `G` itself. Let `μ` be a measure on `G ⧸ Γ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Haar/Quotient.html"}, {"id": "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 2, "macro_tier_score": 0.032, "macro_tier_override": null, "x": -140.959, "z": -66.17, "size": 0.3138, "title": "Conditional expectation in L1", "summary": "This file contains two more steps of the construction of the conditional expectation, which is completed in `Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean`. See that file for a description of the full process. The conditional expectation of an `L²` function is defined in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`. In this file, we perform two steps. * Show that the conditional…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity", "region_id": "measure_theory", "micro_elevation": 0.537, "macro_tier": 3, "macro_tier_score": 0.2817, "macro_tier_override": null, "x": -171.689, "z": -55.673, "size": 0.286, "title": "Monotonicity and ℒp seminorms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSeminorm/Monotonicity.html"}, {"id": "Mathlib.MeasureTheory.Integral.SetToL1", "region_id": "measure_theory", "micro_elevation": 0.7407, "macro_tier": 4, "macro_tier_score": 0.2845, "macro_tier_override": null, "x": -213.651, "z": -70.052, "size": 0.4421, "title": "Extension of a linear function from indicators to L1", "summary": "Given `T : Set α → E →L[ℝ] F` with `DominatedFinMeasAdditive μ T C`, we construct an extension of `T` to integrable simple functions, which are finite sums of indicators of measurable sets with finite measure, then to integrable functions, which are limits of integrable simple functions. The main result is a continuous linear map `(α →₁[μ] E) →L[ℝ] F`. This extension process is used to define the Bochner integral in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/SetToL1.html"}, {"id": "Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable", "region_id": "measure_theory", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.2823, "macro_tier_override": null, "x": -202.295, "z": -86.506, "size": 0.3343, "title": "Results about strongly measurable functions", "summary": "In measure theory it is often assumed that some space is a `PolishSpace`, i.e. a separable and completely metrizable topological space, because it ensures a nice interaction between the topology and the measurable space structure. Moreover a strongly measurable function whose codomain is a metric space is measurable and has a separable range (see `stronglyMeasurable_iff_measurable_separable`). Therefore if the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/Polish/StronglyMeasurable.html"}, {"id": "Mathlib.MeasureTheory.Integral.FinMeasAdditive", "region_id": "measure_theory", "micro_elevation": 0.7222, "macro_tier": 3, "macro_tier_score": 0.2823, "macro_tier_override": null, "x": -147.709, "z": -75.491, "size": 0.3343, "title": "Additivity on measurable sets with finite measure", "summary": "Let `T : Set α → E →L[ℝ] F` be additive for measurable sets with finite measure, in the sense that for `s, t` two such sets, `Disjoint s t → T (s ∪ t) = T s + T t`. `T` is akin to a bilinear map on `Set α × E`, or a linear map on indicator functions. This property is named `FinMeasAdditive` in this file. We also define `DominatedFinMeasAdditive`, which requires in addition that the norm on every set is less than the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/FinMeasAdditive.html"}, {"id": "Mathlib.MeasureTheory.Measure.FiniteMeasure", "region_id": "measure_theory", "micro_elevation": 0.8519, "macro_tier": 2, "macro_tier_score": 0.0628, "macro_tier_override": null, "x": -158.654, "z": -110.917, "size": 0.271, "title": "Finite measures", "summary": "This file defines the type of finite measures on a given measurable space. When the underlying space has a topology and the measurable space structure (sigma algebra) is finer than the Borel sigma algebra, then the type of finite measures is equipped with the topology of weak convergence of measures. The topology of weak convergence is the coarsest topology w.r.t. which for every bounded continuous `ℝ≥0`-valued…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/FiniteMeasure.html"}, {"id": "Mathlib.MeasureTheory.Group.AddCircle", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -196.286, "z": -118.195, "size": 0.236, "title": "Measure-theoretic results about the additive circle", "summary": "The file is a place to collect measure-theoretic results about the additive circle.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/AddCircle.html"}, {"id": "Mathlib.MeasureTheory.SpecificCodomains.WithLp", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -199.374, "z": -113.977, "size": 0.2439, "title": "Integrability in `WithLp`", "summary": "We prove that `f : X → PiLp q E` is in `Lᵖ` if and only if for all `i`, `f · i` is in `Lᵖ`. We do the same for `f : X → WithLp q (E × F)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/SpecificCodomains/WithLp.html"}, {"id": "Mathlib.MeasureTheory.Constructions.SimpleGraph", "region_id": "measure_theory", "micro_elevation": 0.0741, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -183.896, "z": -79.863, "size": 0.239, "title": "Sigma-algebra on simple graphs", "summary": "In this file, we pull back the sigma-algebra on `V → V → Prop` to a sigma-algebra on `SimpleGraph V` and prove that common operations are measurable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Constructions/SimpleGraph.html"}, {"id": "Mathlib.MeasureTheory.Group.FundamentalDomain", "region_id": "measure_theory", "micro_elevation": 0.8333, "macro_tier": 2, "macro_tier_score": 0.0943, "macro_tier_override": null, "x": -203.199, "z": -47.639, "size": 0.2951, "title": "Fundamental domain of a group action", "summary": "A set `s` is said to be a *fundamental domain* of an action of a group `G` on a measurable space `α` with respect to a measure `μ` if * `s` is a measurable set; * the sets `g • s` over all `g : G` cover almost all points of the whole space; * the sets `g • s`, are pairwise a.e. disjoint, i.e., `μ (g₁ • s ∩ g₂ • s) = 0` whenever `g₁ ≠ g₂`; we require this for `g₂ = 1` in the definition, then deduce it for any two `g₁…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/FundamentalDomain.html"}, {"id": "Mathlib.MeasureTheory.Measure.Real", "region_id": "measure_theory", "micro_elevation": 0.3148, "macro_tier": 3, "macro_tier_score": 0.2827, "macro_tier_override": null, "x": -192.886, "z": -70.794, "size": 0.3597, "title": "Measures as real-valued functions", "summary": "Given a measure `μ`, we have defined `μ.real` as the function sending a set `s` to `(μ s).toReal`. In this file, we develop a basic API around this notion. We essentially copy relevant lemmas from the files `MeasureSpaceDef.lean`, `NullMeasurable.lean` and `MeasureSpace.lean`, and adapt them by replacing in their name `measure` with `measureReal`. Many lemmas require an assumption that some set has finite measure.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/Real.html"}, {"id": "Mathlib.MeasureTheory.SpecificCodomains.ContinuousMapZero", "region_id": "measure_theory", "micro_elevation": 0.7222, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -196.991, "z": -49.711, "size": 0.2361, "title": "Specific results about `ContinuousMapZero`-valued integration", "summary": "In this file, we collect a few results regarding integrability, on a measure space `(X, μ)`, of a `C(Y, E)₀`-valued function, where `Y` is a compact topological space with a distinguished `0`, and `E` is a normed group. The structure of this file is largely similar to that of `Mathlib.MeasureTheory.SpecificCodomains.ContinuousMap`, which contains a more detailed module docstring.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/SpecificCodomains/ContinuousMapZero.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.Decomposition.RadonNikodym", "region_id": "measure_theory", "micro_elevation": 0.9074, "macro_tier": 1, "macro_tier_score": 0.0315, "macro_tier_override": null, "x": -159.586, "z": -42.709, "size": 0.2641, "title": "Radon-Nikodym derivatives of vector measures", "summary": "This file contains results about Radon-Nikodym derivatives of signed measures that depend both on the Lebesgue decomposition of signed measures and the theory of Radon-Nikodym derivatives of usual measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/Decomposition/RadonNikodym.html"}, {"id": "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen", "region_id": "measure_theory", "micro_elevation": 0.963, "macro_tier": 1, "macro_tier_score": 0.0315, "macro_tier_override": null, "x": -219.632, "z": -57.587, "size": 0.2641, "title": "Conditional Jensen's Inequality", "summary": "This file contains the conditional Jensen's inequality. We follow the proof in [Hytonen_VanNeerven_Veraar_Wies_2016].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondJensen.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm", "region_id": "measure_theory", "micro_elevation": 0.7963, "macro_tier": 1, "macro_tier_score": 0.0315, "macro_tier_override": null, "x": -216.425, "z": -86.567, "size": 0.2641, "title": "Real-valued Lᵖ norm", "summary": "This file proves theorems about `MeasureTheory.lpNorm`, a real-valued version of `MeasureTheory.eLpNorm`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSeminorm/LpNorm.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -191.567, "z": -117.977, "size": 0.245, "title": "`f'` is interval integrable for certain classes of functions `f`", "summary": "This file proves that: * `MonotoneOn.intervalIntegrable_deriv`: If `f` is monotone on `a..b`, then `f'` is interval integrable on `a..b`. * `MonotoneOn.intervalIntegral_deriv_mem_uIcc`: If `f` is monotone on `a..b`, then the integral of `f'` on `a..b` is in `uIcc 0 (f b - f a)`. * `BoundedVariationOn.intervalIntegrable_deriv`: If `f` has bounded variation on `a..b`, then `f'` is interval integrable on `a..b`. *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntervalIntegral/DerivIntegrable.html"}, {"id": "Mathlib.MeasureTheory.Function.AbsolutelyContinuous", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -216.745, "z": -61.282, "size": 0.256, "title": "Absolutely Continuous Functions", "summary": "This file defines absolutely continuous functions on a closed interval `uIcc a b` and proves some basic properties about absolutely continuous functions. A function `f` is *absolutely continuous* on `uIcc a b` if for any `ε > 0`, there is `δ > 0` such that for any finite disjoint collection of intervals `uIoc (a i) (b i)` for `i < n` where `a i`, `b i` are all in `uIcc a b` for `i < n`, if `∑ i ∈ range n, dist (a i)…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/AbsolutelyContinuous.html"}, {"id": "Mathlib.MeasureTheory.Measure.RegularityCompacts", "region_id": "measure_theory", "micro_elevation": 0.4074, "macro_tier": 2, "macro_tier_score": 0.0628, "macro_tier_override": null, "x": -198.184, "z": -85.358, "size": 0.2774, "title": "Inner regularity of finite measures", "summary": "The main result of this file is `InnerRegularCompactLTTop_of_pseudoEMetricSpace_completeSpace_secondCountable`: A finite measure `μ` on a `PseudoEMetricSpace E` and `CompleteSpace E` with `SecondCountableTopology E` is inner regular with respect to compact sets. In other words, a finite measure on such a space is a tight measure. Finite measures on Polish spaces are an important special case, which makes the result…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/RegularityCompacts.html"}, {"id": "Mathlib.MeasureTheory.Measure.SeparableMeasure", "region_id": "measure_theory", "micro_elevation": 0.7222, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -168.895, "z": -47.634, "size": 0.2, "title": "Separable measure", "summary": "The goal of this file is to give a sufficient condition on the measure space `(X, μ)` and the `NormedAddCommGroup E` for the space `MeasureTheory.Lp E p μ` to have `SecondCountableTopology` when `1 ≤ p < ∞`. To do so we define the notion of a `MeasureTheory.MeasureDense` family and a separable measure (`MeasureTheory.IsSeparable`). We prove that if `X` is `MeasurableSpace.CountablyGenerated` and `μ` is s-finite,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Measure/SeparableMeasure.html"}, {"id": "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "region_id": "measure_theory", "micro_elevation": 0.7037, "macro_tier": 4, "macro_tier_score": 0.2845, "macro_tier_override": null, "x": -166.499, "z": -107.635, "size": 0.4424, "title": "Density of simple functions", "summary": "Show that each `Lᵖ` Borel measurable function can be approximated in `Lᵖ` norm by a sequence of simple functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule", "region_id": "measure_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -180.729, "z": -78.619, "size": 0.2, "title": "The trapezoidal rule", "summary": "This file contains a definition of integration on `[[a, b]]` via the trapezoidal rule, along with an error bound in terms of a bound on the second derivative of the integrand.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntervalIntegral/TrapezoidalRule.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue", "region_id": "measure_theory", "micro_elevation": 0.8889, "macro_tier": 2, "macro_tier_score": 0.0627, "macro_tier_override": null, "x": -140.967, "z": -69.374, "size": 0.2634, "title": "Lebesgue decomposition", "summary": "This file proves the Lebesgue decomposition theorem for signed measures. The Lebesgue decomposition theorem states that, given two σ-finite measures `μ` and `ν`, there exists a σ-finite measure `ξ` and a measurable function `f` such that `μ = ξ + fν` and `ξ` is mutually singular with respect to `ν`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/Decomposition/Lebesgue.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan", "region_id": "measure_theory", "micro_elevation": 0.3704, "macro_tier": 2, "macro_tier_score": 0.0627, "macro_tier_override": null, "x": -197.023, "z": -73.738, "size": 0.2592, "title": "Jordan decomposition", "summary": "This file proves the existence and uniqueness of the Jordan decomposition for signed measures. The Jordan decomposition theorem states that, given a signed measure `s`, there exists a unique pair of mutually singular measures `μ` and `ν`, such that `s = μ - ν`. The Jordan decomposition theorem for measures is a corollary of the Hahn decomposition theorem and is useful for the Lebesgue decomposition theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/Decomposition/Jordan.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.NCard", "region_id": "measure_theory", "micro_elevation": 0.0556, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -182.386, "z": -80.56, "size": 0.25, "title": "Measurability of `Set.encard` and `Set.ncard`", "summary": "In this file we prove that `Set.encard` and `Set.ncard` are measurable functions, provided that the ambient space is countable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/NCard.html"}, {"id": "Mathlib.MeasureTheory.Function.LpSeminorm.Defs", "region_id": "measure_theory", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.2823, "macro_tier_override": null, "x": -187.486, "z": -56.674, "size": 0.3402, "title": "ℒp space", "summary": "This file describes properties of almost everywhere strongly measurable functions with finite `p`-seminorm, denoted by `eLpNorm f p μ` and defined for `p:ℝ≥0∞` as `0` if `p=0`, `(∫ ‖f a‖^p ∂μ) ^ (1/p)` for `0 < p < ∞` and `essSup ‖f‖ μ` for `p=∞`. The Prop-valued `MemLp f p μ` states that a function `f : α → E` has finite `p`-seminorm and is almost everywhere strongly measurable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/LpSeminorm/Defs.html"}, {"id": "Mathlib.MeasureTheory.Function.JacobianOneDim", "region_id": "measure_theory", "micro_elevation": 0.9444, "macro_tier": 2, "macro_tier_score": 0.0321, "macro_tier_override": null, "x": -172.694, "z": -35.996, "size": 0.3228, "title": "Change of variable formulas for integrals in dimension 1", "summary": "We record in this file versions of the general change of variables formula in integrals for functions from `ℝ` to `ℝ`. This makes it possible to replace the determinant of the Fréchet derivative with the one-dimensional derivative. We also give more specific versions of these theorems for monotone and antitone functions: this makes it possible to drop the injectivity assumption of the general theorems, as the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/JacobianOneDim.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntervalIntegral.LebesgueDifferentiationThm", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 1, "macro_tier_score": 0.0314, "macro_tier_override": null, "x": -145.131, "z": -101.879, "size": 0.245, "title": "Lebesgue Differentiation Theorem (Interval Version)", "summary": "This file proves the interval version of the Lebesgue Differentiation Theorem. There are two versions in this file. * `LocallyIntegrable.ae_hasDerivAt_integral` is the global version. It states that if `f : ℝ → E` is locally integrable (`E` a Banach space), then for almost every `x`, for any `c : ℝ`, the derivative of `∫ (t : ℝ) in c..x, f t` at `x` is equal to `f x`. * `IntervalIntegrable.ae_hasDerivAt_integral` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntervalIntegral/LebesgueDifferentiationThm.html"}, {"id": "Mathlib.MeasureTheory.OuterMeasure.Induced", "region_id": "measure_theory", "micro_elevation": 0.0926, "macro_tier": 4, "macro_tier_score": 0.3763, "macro_tier_override": null, "x": -176.675, "z": -79.903, "size": 0.3506, "title": "Induced Outer Measure", "summary": "We can extend a function defined on a subset of `Set α` to an outer measure. The underlying function is called `extend`, and the measure it induces is called `inducedOuterMeasure`. Some lemmas below are proven twice, once in the general case, and once where the function `m` is only defined on measurable sets (i.e. when `P = MeasurableSet`). In the latter cases, we can remove some hypotheses in the statement. The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/OuterMeasure/Induced.html"}, {"id": "Mathlib.MeasureTheory.OuterMeasure.Caratheodory", "region_id": "measure_theory", "micro_elevation": 0.0741, "macro_tier": 4, "macro_tier_score": 0.3754, "macro_tier_override": null, "x": -181.906, "z": -75.428, "size": 0.2886, "title": "The Carathéodory σ-algebra of an outer measure", "summary": "Given an outer measure `m`, the Carathéodory-measurable sets are the sets `s` such that for all sets `t` we have `m t = m (t ∩ s) + m (t \\ s)`. This forms a measurable space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/OuterMeasure/Caratheodory.html"}, {"id": "Mathlib.MeasureTheory.VectorMeasure.Decomposition.JordanSub", "region_id": "measure_theory", "micro_elevation": 0.3889, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -162.95, "z": -80.32, "size": 0.2, "title": "Jordan decomposition from signed measure subtraction", "summary": "This file develops the Jordan decomposition of the signed measure `μ - ν` for finite measures `μ` and `ν`, expressing it as the pair `(μ - ν, ν - μ)` of mutually singular finite measures. The key tool is the Hahn decomposition theorem, which yields a measurable partition of the space where `μ ≤ ν` and `ν ≤ μ`, and the measure difference behaves like a signed measure difference.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/VectorMeasure/Decomposition/JordanSub.html"}, {"id": "Mathlib.MeasureTheory.Function.FactorsThrough", "region_id": "measure_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -180.729, "z": -78.619, "size": 0.2445, "title": "Factorization of a map from measurability", "summary": "Consider `f : X → Y` and `g : X → Z` and assume that `g` is measurable with respect to the pullback along `f`. Then `g` factors through `f`, which means that (if `Z` is nonempty) there exists `h : Y → Z` such that `g = h ∘ f`. If `Z` is completely metrizable, the factorization map `h` can be taken to be measurable. This is the content of the [Doob-Dynkin lemma](https://en.wikipedia.org/wiki/Doob–Dynkin_lemma): see…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/FactorsThrough.html"}, {"id": "Mathlib.MeasureTheory.Group.MeasurableEquiv", "region_id": "measure_theory", "micro_elevation": 0.3704, "macro_tier": 3, "macro_tier_score": 0.2191, "macro_tier_override": null, "x": -191.127, "z": -92.081, "size": 0.2818, "title": "(Scalar) multiplication and (vector) addition as measurable equivalences", "summary": "In this file we define the following measurable equivalences: * `MeasurableEquiv.smul`: if a group `G` acts on `α` by measurable maps, then each element `c : G` defines a measurable automorphism of `α`; * `MeasurableEquiv.vadd`: additive version of `MeasurableEquiv.smul`; * `MeasurableEquiv.smul₀`: if a group with zero `G` acts on `α` by measurable maps, then each nonzero element `c : G` defines a measurable…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Group/MeasurableEquiv.html"}, {"id": "Mathlib.MeasureTheory.Function.Intersectivity", "region_id": "measure_theory", "micro_elevation": 0.8704, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -166.197, "z": -115.856, "size": 0.2, "title": "Bergelson's intersectivity lemma", "summary": "This file proves the Bergelson intersectivity lemma: In a finite measure space, a sequence of events that have measure at least `r` has an infinite subset whose finite intersections all have positive volume. This is in some sense a finitary version of the second Borel-Cantelli lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/Intersectivity.html"}, {"id": "Mathlib.MeasureTheory.Integral.Indicator", "region_id": "measure_theory", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -176.413, "z": -101.173, "size": 0.2, "title": "Results about indicator functions, their integrals, and measures", "summary": "This file has a few measure-theoretic or integration-related results on indicator functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/Indicator.html"}, {"id": "Mathlib.MeasureTheory.Order.UpperLower", "region_id": "measure_theory", "micro_elevation": 0.9259, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -177.812, "z": -121.043, "size": 0.2, "title": "Order-connected sets are null-measurable", "summary": "This file proves that order-connected sets in `ℝⁿ` under the pointwise order are null-measurable. Recall that `x ≤ y` iff `∀ i, x i ≤ y i`, and `s` is order-connected iff `∀ x y ∈ s, ∀ z, x ≤ z → z ≤ y → z ∈ s`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Order/UpperLower.html"}, {"id": "Mathlib.MeasureTheory.MeasurableSpace.Card", "region_id": "measure_theory", "micro_elevation": 0.0185, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -179.998, "z": -79.054, "size": 0.2, "title": "Cardinal of sigma-algebras", "summary": "If a sigma-algebra is generated by a set of sets `s`, then the cardinality of the sigma-algebra is bounded by `(max #s 2) ^ ℵ₀`. This is stated in `MeasurableSpace.cardinal_generate_measurable_le` and `MeasurableSpace.cardinalMeasurableSet_le`. In particular, if `#s ≤ 𝔠`, then the generated sigma-algebra has cardinality at most `𝔠`, see `MeasurableSpace.cardinal_measurableSet_le_continuum`. For the proof, we rely on…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/MeasurableSpace/Card.html"}, {"id": "Mathlib.MeasureTheory.Function.UnifTight", "region_id": "measure_theory", "micro_elevation": 0.7037, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -154.269, "z": -60.064, "size": 0.2, "title": "Uniform tightness", "summary": "This file contains the definitions for uniform tightness for a family of Lp functions. It is used as a hypothesis to the version of Vitali's convergence theorem for Lp spaces that works also for spaces of infinite measure. This version of Vitali's theorem is also proved later in the file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Function/UnifTight.html"}, {"id": "Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun", "region_id": "measure_theory", "micro_elevation": 0.9444, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": -223.79, "z": -83.819, "size": 0.2425, "title": "Fundamental theorem of calculus and integration by parts for absolutely continuous functions", "summary": "This file proves that: * `AbsolutelyContinuousOnInterval.integral_deriv_eq_sub`: If `f` is absolutely continuous on `uIcc a b`, then *Fundamental Theorem of Calculus* holds for `f'` on `a..b`, i.e. `∫ (x : ℝ) in a..b, deriv f x = f b - f a`. * `AbsolutelyContinuousOnInterval.integral_mul_deriv_eq_deriv_mul`: *Integration by Parts* holds for absolutely continuous functions, i.e. if `f` and `g` are absolutely…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/MeasureTheory/Integral/IntervalIntegral/AbsolutelyContinuousFun.html"}, {"id": "Mathlib.Combinatorics.SetFamily.KruskalKatona", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 62.22, "z": 172.573, "size": 0.2, "title": "Kruskal-Katona theorem", "summary": "This file proves the Kruskal-Katona theorem. This is a sharp statement about how many sets of size `k - 1` are covered by a family of sets of size `k`, given only its size.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SetFamily/KruskalKatona.html"}, {"id": "Mathlib.Combinatorics.Colex", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.239, "title": "Colexicographic order", "summary": "We define the colex order for finite sets, and give a couple of important lemmas and properties relating to it. The colex ordering likes to avoid large values: If the biggest element of `t` is bigger than all elements of `s`, then `s < t`. In the special case of `ℕ`, it can be thought of as the \"binary\" ordering. That is, order `s` based on $∑_{i ∈ s} 2^i$. It's defined here on `Finset α` for any linear order `α`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Colex.html"}, {"id": "Mathlib.Combinatorics.SetFamily.Compression.UV", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 62.309, "z": 174.661, "size": 0.239, "title": "UV-compressions", "summary": "This file defines UV-compression. It is an operation on a set family that reduces its shadow. UV-compressing `a : α` along `u v : α` means replacing `a` by `(a ⊔ u) \\ v` if `a` and `u` are disjoint and `v ≤ a`. In some sense, it's moving `a` from `v` to `u`. UV-compressions are immensely useful to prove the Kruskal-Katona theorem. The idea is that compressing a set family might decrease the size of its shadow, so…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SetFamily/Compression/UV.html"}, {"id": "Mathlib.Combinatorics.SetFamily.Intersecting", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1002, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2617, "title": "Intersecting families", "summary": "This file defines intersecting families and proves their basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SetFamily/Intersecting.html"}, {"id": "Mathlib.Combinatorics.SetFamily.Shadow", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1004, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2813, "title": "Shadows", "summary": "This file defines shadows of a set family. The shadow of a set family is the set family of sets we get by removing any element from any set of the original family. If one pictures `Finset α` as a big hypercube (each dimension being membership of a given element), then taking the shadow corresponds to projecting each finset down once in all available directions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SetFamily/Shadow.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walk.Traversal", "region_id": "combinatorics", "micro_elevation": 0.2222, "macro_tier": 3, "macro_tier_score": 0.3009, "macro_tier_override": null, "x": 55.151, "z": 177.331, "size": 0.3266, "title": "Traversing walks", "summary": "Functions that help access different parts of a walk.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walk/Traversal.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walk.Basic", "region_id": "combinatorics", "micro_elevation": 0.1667, "macro_tier": 3, "macro_tier_score": 0.3017, "macro_tier_override": null, "x": 67.18, "z": 180.983, "size": 0.3728, "title": "Walks", "summary": "In a simple graph, a *walk* is a finite sequence of adjacent vertices, and can be thought of equally well as a sequence of directed edges. **Warning:** graph theorists mean something different by \"path\" than do homotopy theorists. A \"walk\" in graph theory is a \"path\" in homotopy theory. Another warning: some graph theorists use \"path\" and \"simple path\" for \"walk\" and \"path.\" Some definitions and theorems have…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walk/Basic.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite", "region_id": "combinatorics", "micro_elevation": 0.8333, "macro_tier": 2, "macro_tier_score": 0.1003, "macro_tier_override": null, "x": 93.122, "z": 172.934, "size": 0.2743, "title": "Complete Multipartite Graphs", "summary": "A graph is complete multipartite iff non-adjacency is transitive.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/CompleteMultipartite.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Extremal.Turan", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 2, "macro_tier_score": 0.1003, "macro_tier_override": null, "x": 41.642, "z": 181.683, "size": 0.2679, "title": "Turán's theorem", "summary": "In this file we prove Turán's theorem, the first important result of extremal graph theory, which states that the `r + 1`-cliquefree graph on `n` vertices with the most edges is the complete `r`-partite graph with part sizes as equal as possible (`turanGraph n r`). The forward direction of the proof performs \"Zykov symmetrisation\", which first shows constructively that non-adjacency is an equivalence relation in a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Extremal/Turan.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Hasse", "region_id": "combinatorics", "micro_elevation": 0.7778, "macro_tier": 2, "macro_tier_score": 0.1009, "macro_tier_override": null, "x": 82.367, "z": 155.891, "size": 0.3237, "title": "The Hasse diagram as a graph", "summary": "This file defines the Hasse diagram of an order (graph of `CovBy`, the covering relation) and the path graph on `n` vertices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Hasse.html"}, {"id": "Mathlib.Combinatorics.Matroid.Dual", "region_id": "combinatorics", "micro_elevation": 0.1667, "macro_tier": 2, "macro_tier_score": 0.1011, "macro_tier_override": null, "x": 64.773, "z": 170.672, "size": 0.3376, "title": "Matroid Duality", "summary": "For a matroid `M` on ground set `E`, the collection of complements of the bases of `M` is the collection of bases of another matroid on `E` called the 'dual' of `M`. The map from `M` to its dual is an involution, interacts nicely with minors, and preserves many important matroid properties such as representability and connectivity. This file defines the dual matroid `M✶` of `M`, and gives associated API. The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Dual.html"}, {"id": "Mathlib.Combinatorics.Matroid.IndepAxioms", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 2, "macro_tier_score": 0.1011, "macro_tier_override": null, "x": 64.412, "z": 172.665, "size": 0.3432, "title": "Matroid Independence and Basis axioms", "summary": "Matroids in mathlib are defined axiomatically in terms of bases, but can be described just as naturally via their collections of independent sets, and in fact such a description, being more 'verbose', can often be useful. As well as this, the definition of a `Matroid` uses an unwieldy 'maximality' axiom that can be dropped in cases where there is some finiteness assumption. This file provides several ways to do…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/IndepAxioms.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walks.Maps", "region_id": "combinatorics", "micro_elevation": 0.3889, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 49.139, "z": 175.101, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walks/Maps.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walk.Maps", "region_id": "combinatorics", "micro_elevation": 0.3333, "macro_tier": 3, "macro_tier_score": 0.302, "macro_tier_override": null, "x": 73.448, "z": 182.791, "size": 0.388, "title": "Mapping walks between graphs", "summary": "Functions that map walks between different graphs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walk/Maps.html"}, {"id": "Mathlib.Combinatorics.Additive.SmallTripling", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 66.39, "z": 174.096, "size": 0.2338, "title": "Small tripling implies small powers", "summary": "This file shows that a set with small tripling has small powers, even in non-abelian groups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/SmallTripling.html"}, {"id": "Mathlib.Combinatorics.Additive.PluenneckeRuzsa", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 2, "macro_tier_score": 0.1003, "macro_tier_override": null, "x": 65.148, "z": 176.753, "size": 0.2753, "title": "The Plünnecke-Ruzsa inequality", "summary": "This file proves Ruzsa's triangle inequality, the Plünnecke-Petridis lemma, and the Plünnecke-Ruzsa inequality.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/PluenneckeRuzsa.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Basic", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 3, "macro_tier_score": 0.4016, "macro_tier_override": null, "x": 64.81, "z": 177.629, "size": 0.3705, "title": "Simple graphs", "summary": "This module defines simple graphs on a vertex type `V` as an irreflexive symmetric relation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Basic.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.DegreeSum", "region_id": "combinatorics", "micro_elevation": 0.2778, "macro_tier": 3, "macro_tier_score": 0.2006, "macro_tier_override": null, "x": 58.255, "z": 167.701, "size": 0.3004, "title": "Degree-sum formula and handshaking lemma", "summary": "The degree-sum formula is that the sum of the degrees of the vertices in a finite graph is equal to twice the number of edges. The handshaking lemma, a corollary, is that the number of odd-degree vertices is even.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/DegreeSum.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Finite", "region_id": "combinatorics", "micro_elevation": 0.2222, "macro_tier": 3, "macro_tier_score": 0.3012, "macro_tier_override": null, "x": 63.471, "z": 168.446, "size": 0.3455, "title": "Definitions for finite and locally finite graphs", "summary": "This file defines finite versions of `edgeSet`, `neighborSet` and `incidenceSet` and proves some of their basic properties. It also defines the notion of a locally finite graph, which is one whose vertices have finite degree. The design for finiteness is that each definition takes the smallest finiteness assumption necessary. For example, `SimpleGraph.neighborFinset v` only requires that `v` have finitely many…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Finite.html"}, {"id": "Mathlib.Combinatorics.Derangements.Exponential", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 62.607, "z": 180.473, "size": 0.2, "title": "Derangement exponential series", "summary": "This file proves that the probability of a permutation on n elements being a derangement is 1/e. The specific lemma is `numDerangements_tendsto_inv_e`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Derangements/Exponential.html"}, {"id": "Mathlib.Combinatorics.Derangements.Finite", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 63.374, "z": 174.487, "size": 0.2338, "title": "Derangements on fintypes", "summary": "This file contains lemmas that describe the cardinality of `derangements α` when `α` is a fintype.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Derangements/Finite.html"}, {"id": "Mathlib.Combinatorics.Quiver.Prefunctor", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 3, "macro_tier_score": 0.1105, "macro_tier_override": null, "x": 65.009, "z": 175.709, "size": 0.6352, "title": "Morphisms of quivers", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/Prefunctor.html"}, {"id": "Mathlib.Combinatorics.Derangements.Basic", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2372, "title": "Derangements on types", "summary": "In this file we define `derangements α`, the set of derangements on a type `α`. We also define some equivalences involving various subtypes of `Perm α` and `derangements α`: * `derangementsOptionEquivSigmaAtMostOneFixedPoint`: An equivalence between `derangements (Option α)` and the sigma-type `Σ a : α, {f : Perm α // fixedPoints f ⊆ a}`. * `derangementsRecursionEquiv`: An equivalence between `derangements (Option…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Derangements/Basic.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 39.014, "z": 176.866, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Coloring/VertexColoring.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 2, "macro_tier_score": 0.1014, "macro_tier_override": null, "x": 66.064, "z": 154.548, "size": 0.3604, "title": "Graph Coloring", "summary": "This module defines colorings of simple graphs (also known as proper colorings in the literature). A graph coloring is the attribution of \"colors\" to all of its vertices such that adjacent vertices have different colors. A coloring can be represented as a homomorphism into a complete graph, whose vertices represent the colors.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Coloring/Vertex.html"}, {"id": "Mathlib.Combinatorics.Quiver.Basic", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.111, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.644, "title": "Quivers", "summary": "This module defines quivers. A quiver on a type `V` of vertices assigns to every pair `a b : V` of vertices a type `a ⟶ b` of arrows from `a` to `b`. This is a generalization of `Digraph V`, which can be thought of as \"a proposition `a ⟶ b` of arrows\".", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/Basic.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Girth", "region_id": "combinatorics", "micro_elevation": 0.9444, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 39.625, "z": 201.309, "size": 0.2, "title": "Girth of a simple graph", "summary": "This file defines the girth and the extended girth of a simple graph as the length of its smallest cycle, they give `0` or `∞` respectively if the graph is acyclic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Girth.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Acyclic", "region_id": "combinatorics", "micro_elevation": 0.8889, "macro_tier": 2, "macro_tier_score": 0.1005, "macro_tier_override": null, "x": 94.704, "z": 170.099, "size": 0.2955, "title": "Acyclic graphs and trees", "summary": "This module introduces *acyclic graphs* (a.k.a. *forests*) and *trees*.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Acyclic.html"}, {"id": "Mathlib.Combinatorics.Enumerative.Partition.GenFun", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 67.172, "z": 176.269, "size": 0.2478, "title": "Generating functions for partitions", "summary": "This file defines generating functions related to partitions. Given a character function $f(i, c)$ of a part $i$ and the number of occurrences of the part $c$, the related generating function is $$ G_f(X) = \\sum_{n = 0}^{\\infty} \\left(\\sum_{p \\in P_{n}} \\prod_{i \\in p} f(i, \\#i)\\right) X^n = \\prod_{i = 1}^{\\infty}\\left(1 + \\sum_{j = 1}^{\\infty} f(i, j) X^{ij}\\right) $$ where $P_n$ is all partitions of $n$, $\\#i$ is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/Partition/GenFun.html"}, {"id": "Mathlib.Combinatorics.Enumerative.Partition.Basic", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 2, "macro_tier_score": 0.1012, "macro_tier_override": null, "x": 64.122, "z": 174.723, "size": 0.3452, "title": "Partitions", "summary": "A partition of a natural number `n` is a way of writing `n` as a sum of positive integers, where the order does not matter: two sums that differ only in the order of their summands are considered the same partition. This notion is closely related to that of a composition of `n`, but in a composition of `n` the order does matter. A summand of the partition is called a part.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/Partition/Basic.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.2002, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2531, "title": "Equitabilising a partition", "summary": "This file allows to blow partitions up into parts of controlled size. Given a partition `P` and `a b m : ℕ`, we want to find a partition `Q` with `a` parts of size `m` and `b` parts of size `m + 1` such that all parts of `P` are \"as close as possible\" to unions of parts of `Q`. By \"as close as possible\", we mean that each part of `P` can be written as the union of some parts of `Q` along with at most `m` other…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.html"}, {"id": "Mathlib.Combinatorics.Enumerative.Stirling", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "Stirling Numbers", "summary": "This file defines Stirling numbers of the first and second kinds, proves their fundamental recurrence relations, and establishes some of their key properties and identities.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/Stirling.html"}, {"id": "Mathlib.Combinatorics.Enumerative.Composition", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1011, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.3436, "title": "Compositions", "summary": "A composition of a natural number `n` is a decomposition `n = i₀ + ... + i_{k-1}` of `n` into a sum of positive integers. Combinatorially, it corresponds to a decomposition of `{0, ..., n-1}` into non-empty blocks of consecutive integers, where the `iⱼ` are the lengths of the blocks. This notion is closely related to that of a partition of `n`, but in a composition of `n` the order of the `iⱼ`s matters. We implement…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/Composition.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Diam", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 48.716, "z": 195.837, "size": 0.239, "title": "Diameter of a simple graph", "summary": "This module defines the eccentricity of vertices, the diameter, and the radius of a simple graph.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Diam.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Metric", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 2, "macro_tier_score": 0.1007, "macro_tier_override": null, "x": 48.941, "z": 193.451, "size": 0.3107, "title": "Graph metric", "summary": "This module defines the `SimpleGraph.edist` function, which takes pairs of vertices to the length of the shortest walk between them, or `⊤` if they are disconnected. It also defines `SimpleGraph.dist` which is the `ℕ`-valued version of `SimpleGraph.edist`, and `SimpleGraph.ball` which is the open ball in the graph extended metric.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Metric.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walks.Operations", "region_id": "combinatorics", "micro_elevation": 0.3333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 56.164, "z": 186.328, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walks/Operations.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walk.Operations", "region_id": "combinatorics", "micro_elevation": 0.2778, "macro_tier": 3, "macro_tier_score": 0.3012, "macro_tier_override": null, "x": 70.779, "z": 183.047, "size": 0.3475, "title": "Operations on walks", "summary": "Operations on walks that produce a new walk in the same graph.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walk/Operations.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Coloring.Constructions", "region_id": "combinatorics", "micro_elevation": 0.9444, "macro_tier": 1, "macro_tier_score": 0.1003, "macro_tier_override": null, "x": 33.545, "z": 159.367, "size": 0.2676, "title": "Concrete colorings of common graphs", "summary": "This file defines colorings for some common graphs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Coloring/Constructions.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Bipartite", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 2, "macro_tier_score": 0.1005, "macro_tier_override": null, "x": 50.436, "z": 197.008, "size": 0.2941, "title": "Bipartite graphs", "summary": "This file proves results about bipartite simple graphs, including several double-counting arguments.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Bipartite.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Circulant", "region_id": "combinatorics", "micro_elevation": 0.8889, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 33.376, "z": 188.712, "size": 0.2459, "title": "Definition of circulant graphs", "summary": "This file defines and proves several fact about circulant graphs. A circulant graph over type `G` with jumps `s : Set G` is a graph in which two vertices `u` and `v` are adjacent if and only if `u - v ∈ s` or `v - u ∈ s`. The elements of `s` are called jumps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Circulant.html"}, {"id": "Mathlib.Combinatorics.Quiver.SingleObj", "region_id": "combinatorics", "micro_elevation": 0.2222, "macro_tier": 1, "macro_tier_score": 0.1002, "macro_tier_override": null, "x": 57.477, "z": 182.191, "size": 0.261, "title": "Single-object quiver", "summary": "Single object quiver with a given arrows type.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/SingleObj.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Tutte", "region_id": "combinatorics", "micro_elevation": 0.7778, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 46.43, "z": 153.823, "size": 0.2, "title": "Tutte's theorem", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Tutte.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.UniversalVerts", "region_id": "combinatorics", "micro_elevation": 0.7222, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 53.78, "z": 152.07, "size": 0.2478, "title": "Universal Vertices", "summary": "This file defines the set of universal vertices: those vertices that are connected to all others. In addition, it describes results when considering connected components of the graph where universal vertices are deleted. This particular graph plays a role in the proof of Tutte's Theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/UniversalVerts.html"}, {"id": "Mathlib.Combinatorics.Quiver.Schreier", "region_id": "combinatorics", "micro_elevation": 0.2778, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 53.257, "z": 178.288, "size": 0.2, "title": "Schreier Graphs", "summary": "This module defines Schreier graphs as quivers with labelled edges. Given a monoid `M` acting on a type `V` and a map `ι : S → M`, the Schreier graph has vertices `V` and a directed edge `x → ι(s) • x` for each `x : V` and `s : S`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/Schreier.html"}, {"id": "Mathlib.Combinatorics.Quiver.Covering", "region_id": "combinatorics", "micro_elevation": 0.2222, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 62.158, "z": 168.501, "size": 0.2338, "title": "Covering", "summary": "This file defines coverings of quivers as prefunctors that are bijective on the so-called stars and costars at each vertex of the domain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/Covering.html"}, {"id": "Mathlib.Combinatorics.Tiling.Tile", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "Tiles for tilings", "summary": "This file defines some basic concepts related to individual tiles for tilings in a discrete context (with definitions in a continuous context to be developed separately but analogously). Work in the field of tilings does not generally try to define or state things in any kind of maximal generality, so it is necessary to adapt definitions and statements from the literature to produce something that seems…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Tiling/Tile.html"}, {"id": "Mathlib.Combinatorics.Hypergraph.Basic", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "Undirected hypergraphs", "summary": "An *undirected hypergraph* (here abbreviated as *hypergraph*) `H` is a generalization of a graph (see `Mathlib.Combinatorics.Graph` or `Mathlib.Combinatorics.SimpleGraph`) and consists of a set of *vertices*, usually denoted `V` or `V(H)`, and a set of *hyperedges*, here called *edges* and denoted `E` or `E(H)`. In contrast with a graph, where edges are unordered pairs of vertices, in hypergraphs, edges are…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Hypergraph/Basic.html"}, {"id": "Mathlib.Combinatorics.Matroid.Rank.Cardinal", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 1, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 45.028, "z": 163.79, "size": 0.2555, "title": "Cardinal-valued rank", "summary": "In a finitary matroid, all bases have the same cardinality. In fact, something stronger holds: if each of `I` and `J` is a basis for a set `X`, then `#(I \\ J) = #(J \\ I)` and (consequently) `#I = #J`. This file introduces a typeclass `InvariantCardinalRank` that applies to any matroid such that this property holds for all `I`, `J` and `X`. A matroid satisfying this condition has a well-defined cardinality-valued…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Rank/Cardinal.html"}, {"id": "Mathlib.Combinatorics.Matroid.Rank.ENat", "region_id": "combinatorics", "micro_elevation": 0.5556, "macro_tier": 1, "macro_tier_score": 0.1002, "macro_tier_override": null, "x": 81.185, "z": 185.411, "size": 0.2599, "title": "`ℕ∞`-valued rank", "summary": "If the 'cardinality' of `s : Set α` is taken to mean the `ℕ∞`-valued term `Set.encard s`, then all bases of any `M : Matroid α` have the same cardinality, and for each `X : Set α` with `X ⊆ M.E`, all `M`-bases for `X` have the same cardinality. The 'rank' of a matroid is the cardinality of all its bases, and the 'rank' of a set `X` in a matroid `M` is the cardinality of each `M`-basis of `X`. This file defines these…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Rank/ENat.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Operations", "region_id": "combinatorics", "micro_elevation": 0.3889, "macro_tier": 3, "macro_tier_score": 0.3009, "macro_tier_override": null, "x": 74.964, "z": 168.816, "size": 0.328, "title": "Local graph operations", "summary": "This file defines some single-graph operations that modify a finite number of vertices and proves basic theorems about them. When the graph itself has a finite number of vertices we also prove theorems about the number of edges in the modified graphs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Operations.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Subgraph", "region_id": "combinatorics", "micro_elevation": 0.3333, "macro_tier": 3, "macro_tier_score": 0.3016, "macro_tier_override": null, "x": 52.348, "z": 181.867, "size": 0.3682, "title": "Subgraphs of a simple graph", "summary": "A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Subgraph.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Copy", "region_id": "combinatorics", "micro_elevation": 0.3889, "macro_tier": 3, "macro_tier_score": 0.3007, "macro_tier_override": null, "x": 49.1, "z": 175.581, "size": 0.3129, "title": "Containment of graphs", "summary": "This file introduces the concept of one simple graph containing a copy of another. For two simple graphs `G` and `H`, a *copy* of `G` in `H` is a (not necessarily induced) subgraph of `H` isomorphic to `G`. If there exists a copy of `G` in `H`, we say that `H` *contains* `G`. This is equivalent to saying that there is an injective graph homomorphism `G → H` between them (this is **not** the same as a graph…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Copy.html"}, {"id": "Mathlib.Combinatorics.Matroid.Minor.Contract", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 1, "macro_tier_score": 0.1003, "macro_tier_override": null, "x": 71.423, "z": 197.014, "size": 0.2676, "title": "Matroid Contraction", "summary": "Instead of deleting the elements of `X : Set α` from `M : Matroid α`, we can contract them. The *contraction* of `X` from `M`, denoted `M ／ X`, is the matroid on ground set `M.E \\ X` in which a set `I` is independent if and only if `I ∪ J` is independent in `M`, where `J` is an arbitrarily chosen basis for `X`. Contraction corresponds to contracting edges in graphic matroids (hence the name) and corresponds to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Minor/Contract.html"}, {"id": "Mathlib.Combinatorics.Matroid.Minor.Delete", "region_id": "combinatorics", "micro_elevation": 0.5556, "macro_tier": 1, "macro_tier_score": 0.1002, "macro_tier_override": null, "x": 56.871, "z": 157.374, "size": 0.2649, "title": "Matroid Deletion", "summary": "For `M : Matroid α` and `X : Set α`, the *deletion* of `X` from `M` is the matroid `M ＼ X` with ground set `M.E \\ X`, in which a subset of `M.E \\ X` is independent if and only if it is independent in `M`. The deletion `M ＼ X` is equal to the restriction `M ↾ (M.E \\ X)`, but is of special importance in the theory because it is the dual notion of *contraction*, and thus plays a more central and natural role than…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Minor/Delete.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walk.Decomp", "region_id": "combinatorics", "micro_elevation": 0.4444, "macro_tier": 3, "macro_tier_score": 0.3015, "macro_tier_override": null, "x": 79.184, "z": 177.932, "size": 0.3616, "title": "Decomposing walks", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walk/Decomp.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks", "region_id": "combinatorics", "micro_elevation": 0.3889, "macro_tier": 3, "macro_tier_score": 0.3017, "macro_tier_override": null, "x": 49.116, "z": 175.352, "size": 0.3765, "title": "Subwalks", "summary": "We define a relation on walks stating that one walk is the subwalk of another.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walk/Subwalks.html"}, {"id": "Mathlib.Combinatorics.Quiver.Cast", "region_id": "combinatorics", "micro_elevation": 0.1667, "macro_tier": 2, "macro_tier_score": 0.1003, "macro_tier_override": null, "x": 59.924, "z": 171.388, "size": 0.2704, "title": "Rewriting arrows and paths along vertex equalities", "summary": "This file defines `Hom.cast` and `Path.cast` (and associated lemmas) in order to allow rewriting arrows and paths along equalities of their endpoints.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/Cast.html"}, {"id": "Mathlib.Combinatorics.Quiver.Path", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 3, "macro_tier_score": 0.1045, "macro_tier_override": null, "x": 67.054, "z": 175.496, "size": 0.4847, "title": "Paths in quivers", "summary": "Given a quiver `V`, we define the type of paths from `a : V` to `b : V` as an inductive family. We define composition of paths and the action of prefunctors on paths.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/Path.html"}, {"id": "Mathlib.Combinatorics.Quiver.Path.Weight", "region_id": "combinatorics", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 65.372, "z": 170.874, "size": 0.2, "title": "Path weights in a Quiver", "summary": "This file defines the weight of a path in a quiver. The weight of a path is the product of the weights of its edges, where weights are taken from a monoid.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/Path/Weight.html"}, {"id": "Mathlib.Combinatorics.Quiver.ReflQuiver", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.278, "title": "Reflexive Quivers", "summary": "This module defines reflexive quivers. A reflexive quiver, or \"refl quiver\" for short, extends a quiver with a specified endoarrow on each term in its type of objects. We also introduce morphisms between reflexive quivers, called reflexive prefunctors or \"refl prefunctors\" for short. Note: Currently Category does not extend ReflQuiver, although it could. (TODO: do this)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/ReflQuiver.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Hamiltonian", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 81.871, "z": 161.238, "size": 0.2, "title": "Hamiltonian Graphs", "summary": "In this file we introduce Hamiltonian paths, cycles and graphs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Hamiltonian.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 2, "macro_tier_score": 0.1005, "macro_tier_override": null, "x": 42.214, "z": 183.647, "size": 0.293, "title": "Edge Connectivity", "summary": "This file defines k-edge-connectivity for simple graphs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Connectivity/EdgeConnectivity.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Partition", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 40.619, "z": 185.15, "size": 0.2, "title": "Graph partitions", "summary": "This module provides an interface for dealing with partitions on simple graphs. A partition of a graph `G`, with vertices `V`, is a set `P` of disjoint nonempty subsets of `V` such that: * The union of the subsets in `P` is `V`. * Each element of `P` is an independent set. (Each subset contains no pair of adjacent vertices.) Graph partitions are graph colorings that do not name their colors. They are adjoint in the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Partition.html"}, {"id": "Mathlib.Combinatorics.Additive.RuzsaCovering", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2338, "title": "Ruzsa's covering lemma", "summary": "This file proves the Ruzsa covering lemma. This says that, for `A`, `B` finsets, we can cover `A` with at most `#(A + B) / #B` copies of `B - B`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/RuzsaCovering.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits", "region_id": "combinatorics", "micro_elevation": 0.8889, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 87.952, "z": 155.963, "size": 0.2, "title": "The Erdős-Stone-Simonovits theorem", "summary": "This file proves the **Erdős-Stone-Simonovits theorem** for simple graphs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Extremal/ErdosStoneSimonovits.html"}, {"id": "Mathlib.Combinatorics.Pigeonhole", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 3, "macro_tier_score": 0.3004, "macro_tier_override": null, "x": 61.224, "z": 175.918, "size": 0.2829, "title": "Pigeonhole principles", "summary": "Given pigeons (possibly infinitely many) in pigeonholes, the pigeonhole principle states that, if there are more pigeons than pigeonholes, then there is a pigeonhole with two or more pigeons. There are a few variations on this statement, and the conclusion can be made stronger depending on how many pigeons you know you might have. The basic statements of the pigeonhole principle appear in the following locations: *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Pigeonhole.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Density", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 3, "macro_tier_score": 0.2002, "macro_tier_override": null, "x": 62.645, "z": 180.478, "size": 0.2666, "title": "Edge density", "summary": "This file defines the number and density of edges of a relation/graph.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Density.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 2, "macro_tier_score": 0.1003, "macro_tier_override": null, "x": 41.895, "z": 170.338, "size": 0.2761, "title": "Connectivity of subgraphs and induced graphs", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Prod", "region_id": "combinatorics", "micro_elevation": 0.7222, "macro_tier": 1, "macro_tier_score": 0.1002, "macro_tier_override": null, "x": 86.649, "z": 164.997, "size": 0.2661, "title": "Graph products", "summary": "This file defines the box product of graphs and other product constructions. The box product of `G` and `H` is the graph on the product of the vertices such that `x` and `y` are related iff they agree on one component and the other one is related via either `G` or `H`. For example, the box product of two edges is a square.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Prod.html"}, {"id": "Mathlib.Combinatorics.Quiver.Push", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 2, "macro_tier_score": 0.1036, "macro_tier_override": null, "x": 67.012, "z": 175.345, "size": 0.4538, "title": "Pushing a quiver structure along a map", "summary": "Given a map `σ : V → W` and a `Quiver` instance on `V`, this file defines a `Quiver` instance on `W` by associating to each arrow `v ⟶ v'` in `V` an arrow `σ v ⟶ σ v'` in `W`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/Push.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Dart", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 3, "macro_tier_score": 0.3019, "macro_tier_override": null, "x": 65.31, "z": 179.884, "size": 0.3829, "title": "Darts in graphs", "summary": "A `Dart` or half-edge or bond in a graph is an ordered pair of adjacent vertices, regarded as an oriented edge. This file defines darts and proves some of their basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Dart.html"}, {"id": "Mathlib.Combinatorics.Quiver.Path.Decomposition", "region_id": "combinatorics", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 68.04, "z": 180.03, "size": 0.2, "title": "Path Decomposition and Boundary Crossing", "summary": "This section provides lemmas for decomposing non-empty paths and for reasoning about paths that cross the boundary of a given set of vertices `S`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/Path/Decomposition.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected", "region_id": "combinatorics", "micro_elevation": 0.5556, "macro_tier": 2, "macro_tier_score": 0.1016, "macro_tier_override": null, "x": 56.387, "z": 195.434, "size": 0.367, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Connectivity/Connected.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Extremal.TuranDensity", "region_id": "combinatorics", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 53.607, "z": 161.101, "size": 0.2, "title": "Turán density", "summary": "This file defines the **Turán density** of a simple graph.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Extremal/TuranDensity.html"}, {"id": "Mathlib.Combinatorics.Enumerative.DoubleCounting", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.5008, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.3209, "title": "Double counting", "summary": "This file gathers a few double counting arguments.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/DoubleCounting.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Extremal.Basic", "region_id": "combinatorics", "micro_elevation": 0.4444, "macro_tier": 1, "macro_tier_score": 0.1002, "macro_tier_override": null, "x": 59.137, "z": 192.073, "size": 0.255, "title": "Extremal graph theory", "summary": "This file introduces basic definitions for extremal graph theory, including extremal numbers.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Extremal/Basic.html"}, {"id": "Mathlib.Combinatorics.Matroid.Constructions", "region_id": "combinatorics", "micro_elevation": 0.2778, "macro_tier": 2, "macro_tier_score": 0.1008, "macro_tier_override": null, "x": 73.211, "z": 176.71, "size": 0.3218, "title": "Some constructions of matroids", "summary": "This file defines some very elementary examples of matroids, namely those with at most one base.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Constructions.html"}, {"id": "Mathlib.Combinatorics.Matroid.Minor.Restrict", "region_id": "combinatorics", "micro_elevation": 0.2222, "macro_tier": 2, "macro_tier_score": 0.1009, "macro_tier_override": null, "x": 58.032, "z": 182.694, "size": 0.3306, "title": "Matroid Restriction", "summary": "Given `M : Matroid α` and `R : Set α`, the independent sets of `M` that are contained in `R` are the independent sets of another matroid `M ↾ R` with ground set `R`, called the 'restriction' of `M` to `R`. For `I ⊆ R ⊆ M.E`, `I` is a basis of `R` in `M` if and only if `I` is a base of the restriction `M ↾ R`, so this construction relates `Matroid.IsBasis` to `Matroid.IsBase`. If `N M : Matroid α` satisfy `N = M ↾ R`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Minor/Restrict.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Connectivity.WalkCounting", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 52.814, "z": 156.92, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Connectivity/WalkCounting.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walk.Counting", "region_id": "combinatorics", "micro_elevation": 0.5556, "macro_tier": 2, "macro_tier_score": 0.1008, "macro_tier_override": null, "x": 80.876, "z": 166.964, "size": 0.3177, "title": "Counting walks of a given length", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walk/Counting.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.CycleGraph", "region_id": "combinatorics", "micro_elevation": 0.8333, "macro_tier": 2, "macro_tier_score": 0.1004, "macro_tier_override": null, "x": 60.322, "z": 206.532, "size": 0.2852, "title": "Definition of cycle graphs", "summary": "This file defines and proves several fact about cycle graphs on `n` vertices and the cycle around the cycle graph when `n ≥ 3`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/CycleGraph.html"}, {"id": "Mathlib.Combinatorics.Young.YoungDiagram", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1003, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2676, "title": "Young diagrams", "summary": "A Young diagram is a finite set of up-left justified boxes: ```text □□□□□ □□□ □□□ □ ``` This Young diagram corresponds to the [5, 3, 3, 1] partition of 12. We represent it as a lower set in `ℕ × ℕ` in the product partial order. We write `(i, j) ∈ μ` to say that `(i, j)` (in matrix coordinates) is in the Young diagram `μ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Young/YoungDiagram.html"}, {"id": "Mathlib.Combinatorics.Additive.Energy", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "Additive energy", "summary": "This file defines the additive energy of two finsets of a group. This is a central quantity in additive combinatorics.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/Energy.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Init", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.4003, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2768, "title": "SimpleGraph Rule Set", "summary": "This module defines the `SimpleGraph` Aesop rule set which is used by the `aesop_graph` tactic. Aesop rule sets only become visible once the file in which they're declared is imported, so we must put this declaration into its own file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Init.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Ends.Defs", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 1, "macro_tier_score": 0.1003, "macro_tier_override": null, "x": 64.419, "z": 198.581, "size": 0.2676, "title": "Ends", "summary": "This file contains a definition of the ends of a simple graph, as sections of the inverse system assigning, to each finite set of vertices, the connected components of its complement.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Ends/Defs.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Hall", "region_id": "combinatorics", "micro_elevation": 0.7222, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 38.846, "z": 186.14, "size": 0.2, "title": "Hall's Marriage Theorem", "summary": "This file derives Hall's Marriage Theorem for bipartite graphs from the combinatorial formulation in `Mathlib/Combinatorics/Hall/Basic.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Hall.html"}, {"id": "Mathlib.Combinatorics.Hall.Basic", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 3, "macro_tier_score": 0.2003, "macro_tier_override": null, "x": 64.36, "z": 174.876, "size": 0.2676, "title": "Hall's Marriage Theorem", "summary": "Given a list of finite subsets $X_1, X_2, \\dots, X_n$ of some given set $S$, P. Hall in [Hall1935] gave a necessary and sufficient condition for there to be a list of distinct elements $x_1, x_2, \\dots, x_n$ with $x_i\\in X_i$ for each $i$: it is when for each $k$, the union of every $k$ of these subsets has at least $k$ elements. Rather than a list of finite subsets, one may consider indexed families `t : ι → Finset…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Hall/Basic.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Matching", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 2, "macro_tier_score": 0.1003, "macro_tier_override": null, "x": 48.454, "z": 157.335, "size": 0.269, "title": "Matchings", "summary": "A *matching* for a simple graph is a set of disjoint pairs of adjacent vertices, and the set of all the vertices in a matching is called its *support* (and sometimes the vertices in the support are said to be *saturated* by the matching). A *perfect matching* is a matching whose support contains every vertex of the graph. In this module, we represent a matching as a subgraph whose vertices are each incident to at…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Matching.html"}, {"id": "Mathlib.Combinatorics.Configuration", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 60.473, "z": 173.486, "size": 0.2, "title": "Configurations of Points and lines", "summary": "This file introduces abstract configurations of points and lines, and proves some basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Configuration.html"}, {"id": "Mathlib.Combinatorics.Enumerative.InclusionExclusion", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0008, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.3166, "title": "Inclusion-exclusion principle", "summary": "This file proves several variants of the inclusion-exclusion principle. The inclusion-exclusion principle says that the sum/integral of a function over a finite union of sets can be calculated as the alternating sum over `n > 0` of the sum/integral of the function over the intersection of `n` of the sets. By taking complements, it also says that the sum/integral of a function over a finite intersection of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/InclusionExclusion.html"}, {"id": "Mathlib.Combinatorics.SetFamily.LYM", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 64.982, "z": 177.327, "size": 0.2, "title": "Lubell-Yamamoto-Meshalkin inequality and Sperner's theorem", "summary": "This file proves the local LYM and LYM inequalities as well as Sperner's theorem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SetFamily/LYM.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walks.Basic", "region_id": "combinatorics", "micro_elevation": 0.2222, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 65.903, "z": 184.05, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walks/Basic.html"}, {"id": "Mathlib.Combinatorics.Young.SemistandardTableau", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 62.479, "z": 178.382, "size": 0.2, "title": "Semistandard Young tableaux", "summary": "A semistandard Young tableau is a filling of a Young diagram by natural numbers, such that the entries are weakly increasing left-to-right along rows (i.e. for fixed `i`), and strictly-increasing top-to-bottom along columns (i.e. for fixed `j`). An example of an SSYT of shape `μ = [4, 2, 1]` is: ```text 0 0 0 2 1 1 2 ``` We represent a semistandard Young tableau as a function `ℕ → ℕ → ℕ`, which is required to be…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Young/SemistandardTableau.html"}, {"id": "Mathlib.Combinatorics.Additive.Dissociation", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2478, "title": "Dissociation and span", "summary": "This file defines dissociation and span of sets in groups. These are analogs to the usual linear independence and linear span of sets in a vector space but where the scalars are only allowed to be `0` or `±1`. In characteristic 2 or 3, the two pairs of concepts are actually equivalent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/Dissociation.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Sum", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.1002, "macro_tier_override": null, "x": 40.658, "z": 185.251, "size": 0.2642, "title": "Disjoint sum of graphs", "summary": "This file defines the disjoint sum of graphs. The disjoint sum of `G : SimpleGraph V` and `H : SimpleGraph W` is a graph on `V ⊕ W` where `u` and `v` are adjacent if and only if they are both in `G` and adjacent in `G`, or they are both in `H` and adjacent in `H`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Sum.html"}, {"id": "Mathlib.Combinatorics.Quiver.ConnectedComponent", "region_id": "combinatorics", "micro_elevation": 0.2222, "macro_tier": 1, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 55.193, "z": 175.309, "size": 0.2516, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/ConnectedComponent.html"}, {"id": "Mathlib.Combinatorics.Quiver.Subquiver", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 2, "macro_tier_score": 0.1003, "macro_tier_override": null, "x": 65.061, "z": 177.128, "size": 0.2724, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/Subquiver.html"}, {"id": "Mathlib.Combinatorics.Quiver.Symmetric", "region_id": "combinatorics", "micro_elevation": 0.1667, "macro_tier": 3, "macro_tier_score": 0.1082, "macro_tier_override": null, "x": 60.572, "z": 181.942, "size": 0.5831, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/Symmetric.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walks.Traversal", "region_id": "combinatorics", "micro_elevation": 0.2778, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 53.534, "z": 173.546, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walks/Traversal.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Clique", "region_id": "combinatorics", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.3008, "macro_tier_override": null, "x": 72.664, "z": 158.758, "size": 0.3185, "title": "Graph cliques", "summary": "This file defines cliques in simple graphs. A clique is a set of vertices that are pairwise adjacent.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Clique.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Connectivity.Finite", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 2, "macro_tier_score": 0.1006, "macro_tier_override": null, "x": 70.581, "z": 155.64, "size": 0.3013, "title": "Counting walks of a given length", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Connectivity/Finite.html"}, {"id": "Mathlib.Combinatorics.SetFamily.FourFunctions", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2478, "title": "The four functions theorem and corollaries", "summary": "This file proves the four functions theorem. The statement is that if `f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)` for all `a`, `b` in a finite distributive lattice, then `(∑ x ∈ s, f₁ x) * (∑ x ∈ t, f₂ x) ≤ (∑ x ∈ s ⊼ t, f₃ x) * (∑ x ∈ s ⊻ t, f₄ x)` where `s ⊼ t = {a ⊓ b | a ∈ s, b ∈ t}`, `s ⊻ t = {a ⊔ b | a ∈ s, b ∈ t}`. The proof uses Birkhoff's representation theorem to restrict to the case where the finite…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SetFamily/FourFunctions.html"}, {"id": "Mathlib.Combinatorics.Additive.AP.Three.Behrend", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 67.134, "z": 177.075, "size": 0.2478, "title": "Behrend's bound on Roth numbers", "summary": "This file proves Behrend's lower bound on Roth numbers. This says that we can find a subset of `{1, ..., n}` of size `n / exp (O (sqrt (log n)))` which does not contain arithmetic progressions of length `3`. The idea is that the sphere (in the `n`-dimensional Euclidean space) doesn't contain arithmetic progressions (literally) because the corresponding ball is strictly convex. Thus we can take integer points on that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/AP/Three/Behrend.html"}, {"id": "Mathlib.Combinatorics.Additive.AP.Three.Defs", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 3, "macro_tier_score": 0.2001, "macro_tier_override": null, "x": 62.753, "z": 178.458, "size": 0.2494, "title": "Sets without arithmetic progressions of length three and Roth numbers", "summary": "This file defines sets without arithmetic progressions of length three, aka 3AP-free sets, and the Roth number of a set. The corresponding notion, sets without geometric progressions of length three, are called 3GP-free sets. The Roth number of a finset is the size of its biggest 3AP-free subset. This is a more general definition than the one often found in mathematical literature, where the `n`-th Roth number is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/AP/Three/Defs.html"}, {"id": "Mathlib.Combinatorics.Enumerative.Catalan.Basic", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1007, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.3124, "title": "Catalan numbers", "summary": "The Catalan numbers (http://oeis.org/A000108) are probably the most ubiquitous sequence of integers in mathematics. They enumerate several important objects like binary trees, Dyck paths, and triangulations of convex polygons.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/Catalan/Basic.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.FiveWheelLike", "region_id": "combinatorics", "micro_elevation": 0.8889, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 65.597, "z": 208.584, "size": 0.2, "title": "Five-wheel like graphs", "summary": "This file defines an `IsFiveWheelLike` structure in a graph, and describes properties of these structures as well as graphs which avoid this structure. These have two key uses: * We use them to prove that a maximally `Kᵣ₊₁`-free graph is `r`-colorable iff it is complete-multipartite: `colorable_iff_isCompleteMultipartite_of_maximal_cliqueFree`. * They play a key role in Brandt's proof of the Andrásfai-Erdős-Sós…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/FiveWheelLike.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Triangle.Basic", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 3, "macro_tier_score": 0.3005, "macro_tier_override": null, "x": 84.017, "z": 183.87, "size": 0.2925, "title": "Triangles in graphs", "summary": "A *triangle* in a simple graph is a `3`-clique, namely a set of three vertices that are pairwise adjacent. This module defines and proves properties about triangles in simple graphs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.html"}, {"id": "Mathlib.Combinatorics.Schnirelmann", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "Schnirelmann density", "summary": "We define the Schnirelmann density of a set `A` of natural numbers as $inf_{n > 0} |A ∩ {1,...,n}| / n$. As this density is very sensitive to changes in small values, we must exclude `0` from the infimum, and from the intersection.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Schnirelmann.html"}, {"id": "Mathlib.Combinatorics.Enumerative.Schroder", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2478, "title": "Schröder numbers", "summary": "The Schröder numbers (https://oeis.org/A006318) are a sequence of integers that appear in various combinatorial contexts.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/Schroder.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Paths", "region_id": "combinatorics", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.3019, "macro_tier_override": null, "x": 53.037, "z": 161.47, "size": 0.3869, "title": "Trail, Path, and Cycle", "summary": "In a simple graph, * A *trail* is a walk whose edges each appear no more than once. * A *circuit* is a nonempty trail whose first and last vertices are the same. * A *path* is a trail whose vertices appear no more than once. * A *cycle* is a nonempty trail whose first and last vertices are the same and whose vertices except for the first appear no more than once. **Warning:** graph theorists mean something different…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Paths.html"}, {"id": "Mathlib.Combinatorics.Digraph.Orientation", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 64.288, "z": 180.347, "size": 0.2, "title": "Graph Orientation", "summary": "This module introduces conversion operations between `Digraph`s and `SimpleGraph`s, by forgetting the edge orientations of `Digraph`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Digraph/Orientation.html"}, {"id": "Mathlib.Combinatorics.Digraph.Basic", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2478, "title": "Digraphs", "summary": "This module defines directed graphs on a vertex type `V`, which is the same notion as a relation `V → V → Prop`. While this might be too simple of a notion to deserve the grandeur of a new definition, the intention here is to develop relations using the language of graph theory. Note that in this treatment, a digraph may have self loops. The type `Digraph V` is structurally equivalent to `Quiver.{0} V`, but a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Digraph/Basic.html"}, {"id": "Mathlib.Combinatorics.KatonaCircle", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "The Katona circle method", "summary": "This file provides tooling to use the Katona circle method, which is double-counting ways to order `n` elements on a circle under some condition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/KatonaCircle.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walks.Decomp", "region_id": "combinatorics", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 80.351, "z": 170.821, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walks/Decomp.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.UnitDistance.Basic", "region_id": "combinatorics", "micro_elevation": 0.4444, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 48.2, "z": 170.534, "size": 0.2, "title": "Unit-distance graph embeddings", "summary": "An embedding of a graph into some metric space is _unit-distance_ if the distance between any two adjacent vertices is 1. The space in question is usually the Euclidean plane, but can also be higher-dimensional Euclidean space or the sphere (cf. [Frankl_2020]). We do not require nonadjacent vertices to not be distance 1 apart as [hong2014] does.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/UnitDistance/Basic.html"}, {"id": "Mathlib.Combinatorics.Graph.Delete", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 60.014, "z": 173.97, "size": 0.2, "title": "Deletion of edges and vertices", "summary": "This file defines the deletion of edges and vertices from a graph.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Graph/Delete.html"}, {"id": "Mathlib.Combinatorics.Graph.Subgraph", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 2, "macro_tier_score": 0.1008, "macro_tier_override": null, "x": 62.881, "z": 174.494, "size": 0.317, "title": "Subgraphs of multigraphs", "summary": "This file develops the basic theory of subgraphs for multigraphs `Graph α β`: the subgraph relation, standard classes of subgraphs (spanning, induced, closed), and the bottom element `⊥`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Graph/Subgraph.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.StronglyRegular", "region_id": "combinatorics", "micro_elevation": 0.7222, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 43.406, "z": 159.339, "size": 0.2, "title": "Strongly regular graphs", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/StronglyRegular.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.AdjMatrix", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 1, "macro_tier_score": 0.1002, "macro_tier_override": null, "x": 83.238, "z": 167.192, "size": 0.2552, "title": "Adjacency Matrices", "summary": "This module defines the adjacency matrix of a graph, and provides theorems connecting graph properties to computational properties of the matrix.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/AdjMatrix.html"}, {"id": "Mathlib.Combinatorics.Graph.Lattice", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 60.463, "z": 173.495, "size": 0.2, "title": "Intersection and union of graphs", "summary": "This file defines the lattice-like structures on graphs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Graph/Lattice.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Star", "region_id": "combinatorics", "micro_elevation": 0.9444, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 30.198, "z": 167.339, "size": 0.2, "title": "Star Graphs", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Star.html"}, {"id": "Mathlib.Combinatorics.Matroid.Sum", "region_id": "combinatorics", "micro_elevation": 0.3889, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 68.885, "z": 163.622, "size": 0.2, "title": "Sums of matroids", "summary": "The *sum* `M` of a collection `M₁, M₂, ..` of matroids is a matroid on the disjoint union of the ground sets of the summands, in which the independent sets are precisely the unions of independent sets of the summands. We can ask for such a sum both for pairs and for arbitrary indexed collections of matroids, and we can also ask for the 'disjoint union' to be either set-theoretic or type-theoretic. To this end, we…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Sum.html"}, {"id": "Mathlib.Combinatorics.Matroid.Map", "region_id": "combinatorics", "micro_elevation": 0.3333, "macro_tier": 2, "macro_tier_score": 0.1007, "macro_tier_override": null, "x": 58.692, "z": 187.703, "size": 0.3104, "title": "Maps between matroids", "summary": "This file defines maps and comaps, which move a matroid on one type to a matroid on another using a function between the types. The constructions are (up to isomorphism) just combinations of restrictions and parallel extensions, so are not mathematically difficult. Because a matroid `M : Matroid α` is defined with am embedded ground set `M.E : Set α` which contains all the structure of `M`, there are several types…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Map.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Coloring", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 44.326, "z": 161.375, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Coloring.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Triangle.Counting", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 3, "macro_tier_score": 0.2001, "macro_tier_override": null, "x": 41.023, "z": 176.521, "size": 0.2332, "title": "Triangle counting lemma", "summary": "In this file, we prove the triangle counting lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform", "region_id": "combinatorics", "micro_elevation": 0.1667, "macro_tier": 3, "macro_tier_score": 0.2002, "macro_tier_override": null, "x": 58.33, "z": 172.859, "size": 0.2647, "title": "Graph uniformity and uniform partitions", "summary": "In this file we define uniformity of a pair of vertices in a graph and uniformity of a partition of vertices of a graph. Both are also known as ε-regularity. Finsets of vertices `s` and `t` are `ε`-uniform in a graph `G` if their edge density is at most `ε`-far from the density of any big enough `s'` and `t'` where `s' ⊆ s`, `t' ⊆ t`. The definition is pretty technical, but it amounts to the edges between `s` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.html"}, {"id": "Mathlib.Combinatorics.HalesJewett", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "The Hales-Jewett theorem", "summary": "We prove the Hales-Jewett theorem. We deduce Van der Waerden's theorem and the multidimensional Hales-Jewett theorem as corollaries. The Hales-Jewett theorem is a result in Ramsey theory dealing with *combinatorial lines*. Given an 'alphabet' `α : Type*` and `a b : α`, an example of a combinatorial line in `α^5` is `{ (a, x, x, b, x) | x : α }`. See `Combinatorics.Line` for a precise general definition. The…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/HalesJewett.html"}, {"id": "Mathlib.Combinatorics.Enumerative.Pentagonal", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "Pentagonal numbers", "summary": "This file introduces (generalized) pentagonal numbers $k(3k-1)/2$ for integer $k$. Some sources, such as A001318 in the OEIS, order generalized pentagonal numbers by indices $k = 0, 1, -1, 2, -2, \\cdots$ to form a strictly monotone sequence. This file doesn't follow this convention, but implicitly shows the monotonicity in `pentagonal_lt_pentagonal_neg` and `pentagonal_neg_lt_pentagonal_add_one`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/Pentagonal.html"}, {"id": "Mathlib.Combinatorics.Matroid.Circuit", "region_id": "combinatorics", "micro_elevation": 0.4444, "macro_tier": 2, "macro_tier_score": 0.1007, "macro_tier_override": null, "x": 47.516, "z": 172.677, "size": 0.3141, "title": "Matroid IsCircuits", "summary": "A 'Circuit' of a matroid `M` is a minimal set `C` that is dependent in `M`. A matroid is determined by its set of circuits, and often the circuits offer a more compact description of a matroid than the collection of independent sets or bases. In matroids arising from graphs, circuits correspond to graphical cycles.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Circuit.html"}, {"id": "Mathlib.Combinatorics.Matroid.Closure", "region_id": "combinatorics", "micro_elevation": 0.3889, "macro_tier": 2, "macro_tier_score": 0.1012, "macro_tier_override": null, "x": 71.294, "z": 164.995, "size": 0.3473, "title": "Matroid Closure", "summary": "A flat (`IsFlat`) of a matroid `M` is a combinatorial analogue of a subspace of a vector space, and is defined to be a subset `F` of the ground set of `M` such that for each basis `I` for `F`, every set having `I` as a basis is contained in `F`. The *closure* of a set `X` in a matroid `M` is the intersection of all flats of `M` containing `X`. This is a combinatorial analogue of the linear span of a set of vectors.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Closure.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.LapMatrix", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 85.919, "z": 184.525, "size": 0.2, "title": "Laplacian Matrix", "summary": "This module defines the Laplacian matrix of a graph, and proves some of its elementary properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/LapMatrix.html"}, {"id": "Mathlib.Combinatorics.Enumerative.Partition.Glaisher", "region_id": "combinatorics", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 64.388, "z": 170.579, "size": 0.2, "title": "Glaisher's theorem", "summary": "This file proves Glaisher's theorem: the number of partitions of an integer $n$ into parts not divisible by $d$ is equal to the number of partitions in which no part is repeated $d$ or more times.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/Partition/Glaisher.html"}, {"id": "Mathlib.Combinatorics.SetFamily.HarrisKleitman", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 61.163, "z": 176.196, "size": 0.2478, "title": "Harris-Kleitman inequality", "summary": "This file proves the Harris-Kleitman inequality. This relates `#𝒜 * #ℬ` and `2 ^ card α * #(𝒜 ∩ ℬ)` where `𝒜` and `ℬ` are upward- or downcard-closed finite families of finsets. This can be interpreted as saying that any two lower sets (resp. any two upper sets) correlate in the uniform measure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SetFamily/HarrisKleitman.html"}, {"id": "Mathlib.Combinatorics.SetFamily.Compression.Down", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2494, "title": "Down-compressions", "summary": "This file defines down-compression. Down-compressing `𝒜 : Finset (Finset α)` along `a : α` means removing `a` from the elements of `𝒜`, when the resulting set is not already in `𝒜`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SetFamily/Compression/Down.html"}, {"id": "Mathlib.Combinatorics.Hindman", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "Hindman's theorem on finite sums", "summary": "We prove Hindman's theorem on finite sums, using idempotent ultrafilters. Given an infinite sequence `a₀, a₁, a₂, …` of positive integers, the set `FS(a₀, …)` is the set of positive integers that can be expressed as a finite sum of `aᵢ`'s, without repetition. Hindman's theorem asserts that whenever the positive integers are finitely colored, there exists a sequence `a₀, a₁, a₂, …` such that `FS(a₀, …)` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Hindman.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walks.Counting", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 73.586, "z": 156.969, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walks/Counting.html"}, {"id": "Mathlib.Combinatorics.Quiver.Path.Vertices", "region_id": "combinatorics", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 59.232, "z": 181.075, "size": 0.239, "title": "Path Vertices", "summary": "This file provides lemmas for reasoning about the vertices of a path.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/Path/Vertices.html"}, {"id": "Mathlib.Combinatorics.Additive.FreimanHom", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.2004, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2828, "title": "Freiman homomorphisms", "summary": "In this file, we define Freiman homomorphisms and isomorphisms. An `n`-Freiman homomorphism from `A` to `B` is a function `f : α → β` such that `f '' A ⊆ B` and `f x₁ * ... * f xₙ = f y₁ * ... * f yₙ` for all `x₁, ..., xₙ, y₁, ..., yₙ ∈ A` such that `x₁ * ... * xₙ = y₁ * ... * yₙ`. In particular, any `MulHom` is a Freiman homomorphism. Note a `0`- or `1`-Freiman homomorphism is simply a map, thus a `2`-Freiman…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/FreimanHom.html"}, {"id": "Mathlib.Combinatorics.Matroid.Init", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1004, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2808, "title": "Matroid Rule Set", "summary": "This module defines the `Matroid` Aesop rule set which is used by the `aesop_mat` tactic. Aesop rule sets only become visible once the file in which they're declared is imported, so we must put this declaration into its own file.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Init.html"}, {"id": "Mathlib.Combinatorics.Hall.Finite", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.3002, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.253, "title": "Hall's Marriage Theorem for finite index types", "summary": "This module proves the basic form of Hall's theorem. In contrast to the theorem described in `Combinatorics.Hall.Basic`, this version requires that the indexed family `t : ι → Finset α` have `ι` be finite. The `Combinatorics.Hall.Basic` module applies a compactness argument to this version to remove the `Finite` constraint on `ι`. The modules are split like this since the generalized statement depends on the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Hall/Finite.html"}, {"id": "Mathlib.Combinatorics.Matroid.Basic", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 2, "macro_tier_score": 0.1004, "macro_tier_override": null, "x": 62.455, "z": 178.373, "size": 0.2853, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Basic.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Maps", "region_id": "combinatorics", "micro_elevation": 0.1667, "macro_tier": 3, "macro_tier_score": 0.3004, "macro_tier_override": null, "x": 67.508, "z": 172.307, "size": 0.2865, "title": "Maps between graphs", "summary": "This file defines two functions and three structures relating graphs. The structures directly correspond to the classification of functions as injective, surjective and bijective, and have corresponding notation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Maps.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Ends.Properties", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 40.591, "z": 167.896, "size": 0.2, "title": "Properties of the ends of graphs", "summary": "This file is meant to contain results about the ends of (locally finite connected) graphs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Ends/Properties.html"}, {"id": "Mathlib.Combinatorics.SetFamily.Kleitman", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 66.82, "z": 178.144, "size": 0.2, "title": "Kleitman's bound on the size of intersecting families", "summary": "An intersecting family on `n` elements has size at most `2ⁿ⁻¹`, so we could naïvely think that two intersecting families could cover all `2ⁿ` sets. But actually that's not case because for example none of them can contain the empty set. Intersecting families are in some sense correlated. Kleitman's bound stipulates that `k` intersecting families cover at most `2ⁿ - 2ⁿ⁻ᵏ` sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SetFamily/Kleitman.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.IncMatrix", "region_id": "combinatorics", "micro_elevation": 0.2778, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 64.387, "z": 166.503, "size": 0.2, "title": "Incidence matrix of a simple graph", "summary": "This file defines the unoriented incidence matrix of a simple graph.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/IncMatrix.html"}, {"id": "Mathlib.Combinatorics.Graph.Basic", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1006, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.3017, "title": "Multigraphs", "summary": "A multigraph is a set of vertices and a set of edges, together with incidence data that associates each edge `e` with an unordered pair `s(x,y)` of vertices called the *ends* of `e`. The pair of `e` and `s(x,y)` is called a *link*. The vertices `x` and `y` may be equal, in which case `e` is a *loop*. There may be more than one edge with the same ends. If a multigraph has no loops and has at most one edge for every…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Graph/Basic.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.1002, "macro_tier_override": null, "x": 47.702, "z": 157.937, "size": 0.257, "title": "Representation of components by a set of vertices", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Connectivity/Represents.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.DeleteEdges", "region_id": "combinatorics", "micro_elevation": 0.2778, "macro_tier": 3, "macro_tier_score": 0.3017, "macro_tier_override": null, "x": 67.314, "z": 167.328, "size": 0.3771, "title": "Edge deletion", "summary": "This file defines operations deleting the edges of a simple graph and proves theorems in the finite case.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/DeleteEdges.html"}, {"id": "Mathlib.Combinatorics.Optimization.ValuedCSP", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "General-Valued Constraint Satisfaction Problems", "summary": "General-Valued CSP is a very broad class of problems in discrete optimization. General-Valued CSP subsumes Min-Cost-Hom (including 3-SAT for example) and Finite-Valued CSP.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Optimization/ValuedCSP.html"}, {"id": "Mathlib.Combinatorics.Enumerative.Bell", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 1, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 59.661, "z": 174.488, "size": 0.2685, "title": "Bell numbers for multisets", "summary": "For `n : ℕ`, the `n`th Bell number is the number of partitions of a set of cardinality `n`. Here, we define a refinement of these numbers, that count, for any `m : Multiset ℕ`, the number of partitions of a set of cardinality `m.sum` whose parts have cardinalities given by `m`. The definition presents it as a natural number. * `Multiset.bell`: number of partitions of a set whose parts have cardinalities a given…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/Bell.html"}, {"id": "Mathlib.Combinatorics.Quiver.Arborescence", "region_id": "combinatorics", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 64.882, "z": 170.704, "size": 0.2338, "title": "Arborescences", "summary": "A quiver `V` is an arborescence (or directed rooted tree) when we have a root vertex `root : V` such that for every `b : V` there is a unique path from `root` to `b`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Quiver/Arborescence.html"}, {"id": "Mathlib.Combinatorics.Matroid.Rank.Finite", "region_id": "combinatorics", "micro_elevation": 0.4444, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 59.593, "z": 160.791, "size": 0.2436, "title": "Finite-rank sets", "summary": "`Matroid.IsRkFinite M X` means that every basis of the set `X` in the matroid `M` is finite, or equivalently that the restriction of `M` to `X` is `Matroid.RankFinite`. Sets in a matroid with `IsRkFinite` are the largest class of sets for which one can do nontrivial integer arithmetic involving the rank function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Rank/Finite.html"}, {"id": "Mathlib.Combinatorics.Matroid.Loop", "region_id": "combinatorics", "micro_elevation": 0.5, "macro_tier": 2, "macro_tier_score": 0.1006, "macro_tier_override": null, "x": 59.968, "z": 158.662, "size": 0.3001, "title": "Matroid loops and coloops", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Loop.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk", "region_id": "combinatorics", "micro_elevation": 0.2222, "macro_tier": 3, "macro_tier_score": 0.2003, "macro_tier_override": null, "x": 55.838, "z": 179.839, "size": 0.2678, "title": "Chunk of the increment partition for Szemerédi Regularity Lemma", "summary": "In the proof of Szemerédi Regularity Lemma, we need to partition each part of a starting partition to increase the energy. This file defines those partitions of parts and shows that they locally increase the energy. This entire file is internal to the proof of Szemerédi Regularity Lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Regularity.Bound", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.2002, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2531, "title": "Numerical bounds for Szemerédi Regularity Lemma", "summary": "This file gathers the numerical facts required by the proof of Szemerédi's regularity lemma. This entire file is internal to the proof of Szemerédi Regularity Lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.html"}, {"id": "Mathlib.Combinatorics.Additive.ErdosGinzburgZiv", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "The Erdős–Ginzburg–Ziv theorem", "summary": "This file proves the Erdős–Ginzburg–Ziv theorem as a corollary of Chevalley-Warning. This theorem states that among any (not necessarily distinct) `2 * n - 1` elements of `ZMod n`, we can find `n` elements of sum zero.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/ErdosGinzburgZiv.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walk.Chord", "region_id": "combinatorics", "micro_elevation": 0.2222, "macro_tier": 1, "macro_tier_score": 0.1002, "macro_tier_override": null, "x": 55.431, "z": 174.223, "size": 0.2561, "title": "Chords of walks", "summary": "This file defines chords and chordless walks in a simple graph.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walk/Chord.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Trails", "region_id": "combinatorics", "micro_elevation": 0.5556, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 81.328, "z": 167.858, "size": 0.2, "title": "Trails and Eulerian trails", "summary": "This module contains additional theory about trails, including Eulerian trails (also known as Eulerian circuits).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Trails.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Triangle.Removal", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 87.15, "z": 179.138, "size": 0.2338, "title": "Triangle removal lemma", "summary": "In this file, we prove the triangle removal lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Triangle/Removal.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma", "region_id": "combinatorics", "micro_elevation": 0.3333, "macro_tier": 3, "macro_tier_score": 0.2001, "macro_tier_override": null, "x": 51.091, "z": 176.948, "size": 0.2332, "title": "Szemerédi's Regularity Lemma", "summary": "In this file, we prove Szemerédi's Regularity Lemma (aka SRL). This is a landmark result in combinatorics roughly stating that any sufficiently big graph behaves like a random graph. This is useful because random graphs are well-behaved in many aspects. More precisely, SRL states that for any `ε > 0` and integer `l` there exists a bound `M` such that any graph on at least `l` vertices can be partitioned into at…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Regularity/Lemma.html"}, {"id": "Mathlib.Combinatorics.SetFamily.AhlswedeZhang", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "The Ahlswede-Zhang identity", "summary": "This file proves the Ahlswede-Zhang identity, which is a nontrivial relation between the size of the \"truncated unions\" of a set family. It sharpens the Lubell-Yamamoto-Meshalkin inequality `Finset.lubell_yamamoto_meshalkin_inequality_sum_card_div_choose`, by making explicit the correction term. For a set family `𝒜` over a ground set of size `n`, the Ahlswede-Zhang identity states that the sum of `|⋂ B ∈ 𝒜, B ⊆ A,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SetFamily/AhlswedeZhang.html"}, {"id": "Mathlib.Combinatorics.Additive.Corner.Roth", "region_id": "combinatorics", "micro_elevation": 0.7222, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 37.464, "z": 171.578, "size": 0.2, "title": "The corners theorem and Roth's theorem", "summary": "This file proves the corners theorem and Roth's theorem on arithmetic progressions of length three.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/Corner/Roth.html"}, {"id": "Mathlib.Combinatorics.Additive.Corner.Defs", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 62.92, "z": 178.485, "size": 0.2338, "title": "Corners", "summary": "This file defines corners, namely triples of the form `(x, y), (x, y + d), (x + d, y)`, and the property of being corner-free.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/Corner/Defs.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 3, "macro_tier_score": 0.2002, "macro_tier_override": null, "x": 39.987, "z": 169.693, "size": 0.2585, "title": "Construct a tripartite graph from its triangles", "summary": "This file contains the construction of a simple graph on `α ⊕ β ⊕ γ` from a list of triangles `(a, b, c)` (with `a` in the first component, `b` in the second, `c` in the third). We call * `t : Finset (α × β × γ)` the set of *triangle indices* (its elements are not triangles within the graph but instead index them). * *explicit* a triangle of the constructed graph coming from a triangle index. * *accidental* a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.html"}, {"id": "Mathlib.Combinatorics.Nullstellensatz", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "Alon's Combinatorial Nullstellensatz", "summary": "This is a formalization of Noga Alon's Combinatorial Nullstellensatz. It follows [Alon_1999]. We consider a family `S : σ → Finset R` of finite subsets of a domain `R` and a multivariate polynomial `f` in `MvPolynomial σ R`. The combinatorial Nullstellensatz gives combinatorial constraints for the vanishing of `f` at any `x : σ → R` such that `x s ∈ S s` for all `s`. -…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Nullstellensatz.html"}, {"id": "Mathlib.Combinatorics.Additive.VerySmallDoubling", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 62.242, "z": 178.28, "size": 0.2, "title": "Sets with very small doubling", "summary": "For a finset `A` in a group, its *doubling* is `#(A * A) / #A`. This file characterises sets with * no doubling as the sets which are either empty or translates of a subgroup. For the converse, use the existing facts from the pointwise API: `∅ ^ 2 = ∅` (`Finset.empty_pow`), `(a • H) ^ 2 = a ^ 2 • H ^ 2 = a ^ 2 • H` (`smul_pow`, `coe_set_pow`). * doubling strictly less than `3 / 2` as the sets that are contained in a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/VerySmallDoubling.html"}, {"id": "Mathlib.Combinatorics.Additive.Convolution", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2302, "title": "Convolution", "summary": "This file defines convolution of finite subsets `A` and `B` of group `G` as the map `A ⋆ B : G → ℕ` that maps `x ∈ G` to the number of distinct representations of `x` in the form `x = ab` for `a ∈ A`, `b ∈ B`. It is shown how convolution behaves under the change of order of `A` and `B`, as well as under the left and right actions on `A`, `B`, and the function argument.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/Convolution.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.VertexCover", "region_id": "combinatorics", "micro_elevation": 0.6111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 78.941, "z": 160.978, "size": 0.2, "title": "Vertex cover", "summary": "A *vertex cover* of a simple graph is a set of vertices such that every edge is incident to at least one of the vertices in the set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/VertexCover.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Regularity.Increment", "region_id": "combinatorics", "micro_elevation": 0.2778, "macro_tier": 3, "macro_tier_score": 0.2003, "macro_tier_override": null, "x": 72.97, "z": 178.687, "size": 0.2741, "title": "Increment partition for Szemerédi Regularity Lemma", "summary": "In the proof of Szemerédi Regularity Lemma, we need to partition each part of a starting partition to increase the energy. This file defines the partition obtained by gluing the parts partitions together (the *increment partition*) and shows that the energy globally increases. This entire file is internal to the proof of Szemerédi Regularity Lemma.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.html"}, {"id": "Mathlib.Combinatorics.Additive.ETransform", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2478, "title": "e-transforms", "summary": "e-transforms are a family of transformations of pairs of finite sets that aim to reduce the size of the sumset while keeping some invariant the same. This file defines a few of them, to be used as internals of other proofs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/ETransform.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Regularity.Energy", "region_id": "combinatorics", "micro_elevation": 0.1667, "macro_tier": 3, "macro_tier_score": 0.2003, "macro_tier_override": null, "x": 63.2, "z": 170.451, "size": 0.2678, "title": "Energy of a partition", "summary": "This file defines the energy of a partition. The energy is the auxiliary quantity that drives the induction process in the proof of Szemerédi's Regularity Lemma. As long as we do not have a suitable equipartition, we will find a new one that has an energy greater than the previous one plus some fixed constant.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.html"}, {"id": "Mathlib.Combinatorics.Extremal.RuzsaSzemeredi", "region_id": "combinatorics", "micro_elevation": 0.7222, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 48.612, "z": 198.226, "size": 0.2, "title": "The Ruzsa-Szemerédi problem", "summary": "This file proves the lower bound of the Ruzsa-Szemerédi problem. The problem is to find the maximum number of edges that a graph on `n` vertices can have if all edges belong to at most one triangle. The lower bound comes from turning the big 3AP-free set from Behrend's construction into a graph that has the property that every triangle gives a (possibly trivial) arithmetic progression on the original set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Connectivity.WalkDecomp", "region_id": "combinatorics", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 45.09, "z": 175.241, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Connectivity/WalkDecomp.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Coloring.EdgeLabeling", "region_id": "combinatorics", "micro_elevation": 0.2778, "macro_tier": 1, "macro_tier_score": 0.1003, "macro_tier_override": null, "x": 71.55, "z": 182.027, "size": 0.2676, "title": "Edge labelings", "summary": "This module defines labelings of the edges of a graph.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Coloring/EdgeLabeling.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Cayley", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 60.643, "z": 173.342, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Cayley.html"}, {"id": "Mathlib.Combinatorics.SetFamily.Shatter", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 65.14, "z": 176.165, "size": 0.2, "title": "Shattering families", "summary": "This file defines the shattering property and VC-dimension of set families.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SetFamily/Shatter.html"}, {"id": "Mathlib.Combinatorics.Additive.Randomisation", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 65.152, "z": 176.252, "size": 0.2, "title": "Randomising by a function of dissociated support", "summary": "This file proves that a function from a finite abelian group can be randomised by a function of dissociated support. Precisely, for `G` a finite abelian group and two functions `c : AddChar G ℂ → ℝ` and `d : AddChar G ℂ → ℝ` such that `{ψ | d ψ ≠ 0}` is dissociated, the product of the `c ψ` over `ψ` is the same as the average over `a` of the product of the `c ψ + Re (d ψ * ψ a)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/Randomisation.html"}, {"id": "Mathlib.Combinatorics.Enumerative.Catalan", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 64.438, "z": 180.3, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/Catalan.html"}, {"id": "Mathlib.Combinatorics.Enumerative.Catalan.Tree", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 2, "macro_tier_score": 0.1004, "macro_tier_override": null, "x": 63.82, "z": 174.589, "size": 0.2827, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/Catalan/Tree.html"}, {"id": "Mathlib.Combinatorics.Enumerative.DyckWord", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 66.197, "z": 173.855, "size": 0.2, "title": "Dyck words", "summary": "A Dyck word is a sequence consisting of an equal number `n` of symbols of two types such that for all prefixes one symbol occurs at least as many times as the other. If the symbols are `(` and `)` the latter restriction is equivalent to balanced brackets; if they are `U = (1, 1)` and `D = (1, -1)` the sequence is a lattice path from `(0, 0)` to `(0, 2n)` and the restriction requires the path to never go below the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/DyckWord.html"}, {"id": "Mathlib.Combinatorics.Graph.Maps", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 61.12, "z": 173.015, "size": 0.2, "title": "Maps between graphs", "summary": "This file defines vertex map between graphs `Graph α β`. Morphisms between graphs will also be defined in this file in the future.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Graph/Maps.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.ConcreteColorings", "region_id": "combinatorics", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 95.222, "z": 159.662, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/ConcreteColorings.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.LineGraph", "region_id": "combinatorics", "micro_elevation": 0.4444, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 47.26, "z": 173.951, "size": 0.2, "title": "LineGraph", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/LineGraph.html"}, {"id": "Mathlib.Combinatorics.Additive.CovBySMul", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1001, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2338, "title": "Relation of covering by cosets", "summary": "This file defines a predicate for a set to be covered by at most `K` cosets of another set. This is a fundamental relation to study in additive combinatorics.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/CovBySMul.html"}, {"id": "Mathlib.Combinatorics.Additive.DoublingConst", "region_id": "combinatorics", "micro_elevation": 0.1111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 59.223, "z": 175.628, "size": 0.2, "title": "Doubling and difference constants", "summary": "This file defines the doubling and difference constants of two finsets in a group.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/DoublingConst.html"}, {"id": "Mathlib.Combinatorics.Additive.CauchyDavenport", "region_id": "combinatorics", "micro_elevation": 0.0556, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 64.858, "z": 175.417, "size": 0.2, "title": "The Cauchy-Davenport theorem", "summary": "This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups. Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`, where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that `|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup (in which case taking `s` and `t` to be that…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/CauchyDavenport.html"}, {"id": "Mathlib.Combinatorics.Additive.ApproximateSubgroup", "region_id": "combinatorics", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 65.904, "z": 171.114, "size": 0.2, "title": "Approximate subgroups", "summary": "This file defines approximate subgroups of a group, namely symmetric sets `A` such that `A * A` can be covered by a small number of translates of `A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/ApproximateSubgroup.html"}, {"id": "Mathlib.Combinatorics.Additive.SubsetSum", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "Subset sums", "summary": "This file defines the subset sum of a finite subset of a commutative monoid.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Additive/SubsetSum.html"}, {"id": "Mathlib.Combinatorics.Compactness", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "Combinatorial compactness and the Rado selection lemma", "summary": "This file contains compactness arguments for constructing infinite objects from finite approximations. The main result is a formalization of Rado's selection principle, as an application of compactness to combinatorics. We give four versions, depending on whether the \"partial\" functions are defined locally or globally, and whether we use `Finset` or `Set.Finite`. The precise formulation of the lemma is therefore…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Compactness.html"}, {"id": "Mathlib.Combinatorics.Enumerative.IncidenceAlgebra", "region_id": "combinatorics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 63.154, "z": 176.487, "size": 0.2, "title": "Incidence algebras", "summary": "Given a locally finite order `α` the incidence algebra over `α` is the type of functions from non-empty intervals of `α` to some algebraic codomain. This algebra has a natural multiplication operation whereby the product of two such functions is defined on an interval by summing over all divisions into two subintervals the product of the values of the original pair of functions. This structure allows us to interpret…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Enumerative/IncidenceAlgebra.html"}, {"id": "Mathlib.Combinatorics.Matroid.Minor.Order", "region_id": "combinatorics", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 40.317, "z": 184.32, "size": 0.2, "title": "Matroid Minors", "summary": "A matroid `N = M ／ C ＼ D` obtained from a matroid `M` by a contraction then a delete, (or equivalently, by any number of contractions/deletions in any order) is a *minor* of `M`. This gives a partial order on `Matroid α` that is ubiquitous in matroid theory, and interacts nicely with duality and linear representations. Although we provide a `PartialOrder` instance on `Matroid α` corresponding to the minor order, we…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/Matroid/Minor/Order.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.EdgeLabeling", "region_id": "combinatorics", "micro_elevation": 0.3333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 53.834, "z": 184.158, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/EdgeLabeling.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Finsubgraph", "region_id": "combinatorics", "micro_elevation": 0.3889, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 61.156, "z": 190.428, "size": 0.2, "title": "Homomorphisms from finite subgraphs", "summary": "This file defines the type of finite subgraphs of a `SimpleGraph` and proves a compactness result for homomorphisms to a finite codomain.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Finsubgraph.html"}, {"id": "Mathlib.Combinatorics.SimpleGraph.Walks.Subwalks", "region_id": "combinatorics", "micro_elevation": 0.4444, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 66.877, "z": 192.145, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Combinatorics/SimpleGraph/Walks/Subwalks.html"}, {"id": "Mathlib.Dynamics.FixedPoints.Topology", "region_id": "dynamics", "micro_elevation": 0.4, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 181.241, "z": 130.38, "size": 0.2489, "title": "Topological properties of fixed points", "summary": "Currently this file contains two lemmas: - `isFixedPt_of_tendsto_iterate`: if `f^n(x) → y` and `f` is continuous at `y`, then `f y = y`; - `isClosed_fixedPoints`: the set of fixed points of a continuous map is a closed set.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/FixedPoints/Topology.html"}, {"id": "Mathlib.Dynamics.Transitive", "region_id": "dynamics", "micro_elevation": 0.2, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 185.154, "z": 129.797, "size": 0.2, "title": "Topologically transitive monoid actions", "summary": "In this file we define an action of a monoid `M` on a topological space `α` to be *topologically transitive* if for any pair of nonempty open sets `U` and `V` in `α` there exists an `m : M` such that `(m • U) ∩ V` is nonempty. We also provide an additive version of this definition and prove basic facts about topologically transitive actions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Transitive.html"}, {"id": "Mathlib.Dynamics.Minimal", "region_id": "dynamics", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1256, "macro_tier_override": null, "x": 187.033, "z": 132.176, "size": 0.3009, "title": "Minimal action of a group", "summary": "In this file we define an action of a monoid `M` on a topological space `α` to be *minimal* if the `M`-orbit of every point `x : α` is dense. We also provide an additive version of this definition and prove some basic facts about minimal actions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Minimal.html"}, {"id": "Mathlib.Dynamics.PeriodicPts.Defs", "region_id": "dynamics", "micro_elevation": 0.4, "macro_tier": 2, "macro_tier_score": 0.1256, "macro_tier_override": null, "x": 185.423, "z": 138.023, "size": 0.2996, "title": "Periodic points", "summary": "A point `x : α` is a periodic point of `f : α → α` of period `n` if `f^[n] x = x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/PeriodicPts/Defs.html"}, {"id": "Mathlib.Dynamics.FixedPoints.Basic", "region_id": "dynamics", "micro_elevation": 0.2, "macro_tier": 2, "macro_tier_score": 0.5011, "macro_tier_override": null, "x": 185.536, "z": 129.539, "size": 0.3428, "title": "Fixed points of a self-map", "summary": "We prove some simple lemmas about `IsFixedPt` and `∘`, `iterate`, and `Semiconj`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/FixedPoints/Basic.html"}, {"id": "Mathlib.Dynamics.Ergodic.RadonNikodym", "region_id": "dynamics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1251, "macro_tier_override": null, "x": 187.033, "z": 132.176, "size": 0.239, "title": "Radon-Nikodym derivative of invariant measures", "summary": "Given two finite invariant measures of a self-map, we prove that their singular parts, their absolutely continuous parts, and their Radon-Nikodym derivatives are invariant too. For the first two theorems, we only assume that one of the measures is finite and the other is σ-finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Ergodic/RadonNikodym.html"}, {"id": "Mathlib.Dynamics.Ergodic.Action.Regular", "region_id": "dynamics", "micro_elevation": 0.6, "macro_tier": 1, "macro_tier_score": 0.1251, "macro_tier_override": null, "x": 180.138, "z": 126.244, "size": 0.2442, "title": "Regular action of a group on itself is ergodic", "summary": "In this file we prove that the left and right actions of a group on itself are ergodic.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Ergodic/Action/Regular.html"}, {"id": "Mathlib.Dynamics.Ergodic.Action.Basic", "region_id": "dynamics", "micro_elevation": 0.4, "macro_tier": 1, "macro_tier_score": 0.1252, "macro_tier_override": null, "x": 183.938, "z": 137.391, "size": 0.2557, "title": "Ergodic group actions", "summary": "A group action of `G` on a space `α` with measure `μ` is called *ergodic*, if for any (null) measurable set `s`, if it is a.e.-invariant under each scalar multiplication `(g • ·)`, `g : G`, then it is either null or conull.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Ergodic/Action/Basic.html"}, {"id": "Mathlib.Dynamics.FixedPoints.Prufer", "region_id": "dynamics", "micro_elevation": 0.4, "macro_tier": 0, "macro_tier_score": 0.1251, "macro_tier_override": null, "x": 192.273, "z": 129.124, "size": 0.236, "title": "Results about pointwise operations on sets with iteration.", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/FixedPoints/Prufer.html"}, {"id": "Mathlib.Dynamics.FixedPoints.Defs", "region_id": "dynamics", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.5006, "macro_tier_override": null, "x": 187.033, "z": 132.176, "size": 0.3065, "title": "Fixed points of a self-map", "summary": "In this file we define the set `Function.fixedPoints` of fixed points of a function `f : α → α`. The related predicate `IsFixedPt` is defined in `Mathlib.Logic.Function.Defs`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/FixedPoints/Defs.html"}, {"id": "Mathlib.Dynamics.Ergodic.MeasurePreserving", "region_id": "dynamics", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1266, "macro_tier_override": null, "x": 187.033, "z": 132.176, "size": 0.3698, "title": "Measure-preserving maps", "summary": "We say that `f : α → β` is a measure-preserving map w.r.t. measures `μ : Measure α` and `ν : Measure β` if `f` is measurable and `map f μ = ν`. In this file we define the predicate `MeasureTheory.MeasurePreserving` and prove its basic properties. We use the term \"measure preserving\" because in many applications `α = β` and `μ = ν`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Ergodic/MeasurePreserving.html"}, {"id": "Mathlib.Dynamics.BirkhoffSum.Average", "region_id": "dynamics", "micro_elevation": 0.6, "macro_tier": 2, "macro_tier_score": 0.1253, "macro_tier_override": null, "x": 182.215, "z": 124.461, "size": 0.2743, "title": "Birkhoff average", "summary": "In this file we define `birkhoffAverage f g n x` to be $$ \\frac{1}{n}\\sum_{k=0}^{n-1}g(f^{[k]}(x)), $$ where `f : α → α` is a self-map on some type `α`, `g : α → M` is a function from `α` to a module over a division semiring `R`, and `R` is used to formalize division by `n` as `(n : R)⁻¹ • _`. While we need an auxiliary division semiring `R` to define `birkhoffAverage`, the definition does not depend on the choice…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/BirkhoffSum/Average.html"}, {"id": "Mathlib.Dynamics.BirkhoffSum.Basic", "region_id": "dynamics", "micro_elevation": 0.4, "macro_tier": 1, "macro_tier_score": 0.1253, "macro_tier_override": null, "x": 191.619, "z": 128.208, "size": 0.2679, "title": "Birkhoff sums", "summary": "In this file we define `birkhoffSum f g n x` to be the sum `∑ k ∈ Finset.range n, g (f^[k] x)`. This sum (more precisely, the corresponding average `n⁻¹ • birkhoffSum f g n x`) appears in various ergodic theorems saying that these averages converge to the \"space average\" `⨍ x, g x ∂μ` in some sense. See also `birkhoffAverage` defined in `Dynamics/BirkhoffSum/Average`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/BirkhoffSum/Basic.html"}, {"id": "Mathlib.Dynamics.BirkhoffSum.NormedSpace", "region_id": "dynamics", "micro_elevation": 0.8, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 177.658, "z": 124.482, "size": 0.2478, "title": "Birkhoff average in a normed space", "summary": "In this file we prove some lemmas about the Birkhoff average (`birkhoffAverage`) of a function which takes values in a normed space over `ℝ` or `ℂ`. At the time of writing, all lemmas in this file are motivated by the proof of the von Neumann Mean Ergodic Theorem, see `LinearIsometry.tendsto_birkhoffAverage_orthogonalProjection`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/BirkhoffSum/NormedSpace.html"}, {"id": "Mathlib.Dynamics.TopologicalEntropy.CoverEntropy", "region_id": "dynamics", "micro_elevation": 0.2, "macro_tier": 2, "macro_tier_score": 0.1257, "macro_tier_override": null, "x": 184.062, "z": 132.781, "size": 0.3139, "title": "Topological entropy via covers", "summary": "We implement Bowen-Dinaburg's definitions of the topological entropy, via covers. All is stated in the vocabulary of uniform spaces. For compact spaces, the uniform structure is canonical, so the topological entropy depends only on the topological structure. This will give a clean proof that the topological entropy is a topological invariant of the dynamics. A notable choice is that we define the topological entropy…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/TopologicalEntropy/CoverEntropy.html"}, {"id": "Mathlib.Dynamics.TopologicalEntropy.DynamicalEntourage", "region_id": "dynamics", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1254, "macro_tier_override": null, "x": 187.033, "z": 132.176, "size": 0.2881, "title": "Dynamical entourages", "summary": "Bowen-Dinaburg's definition of topological entropy of a transformation `T` in a metric space `(X, d)` relies on the so-called dynamical balls. These balls are sets `B (x, ε, n) = { y | ∀ k < n, d(T^[k] x, T^[k] y) < ε }`. We implement Bowen-Dinaburg's definitions in the more general context of uniform spaces. Dynamical balls are replaced by what we call dynamical entourages. This file collects all general lemmas…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/TopologicalEntropy/DynamicalEntourage.html"}, {"id": "Mathlib.Dynamics.PeriodicPts.Lemmas", "region_id": "dynamics", "micro_elevation": 0.6, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 193.49, "z": 125.77, "size": 0.2548, "title": "Extra lemmas about periodic points", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/PeriodicPts/Lemmas.html"}, {"id": "Mathlib.Dynamics.Ergodic.Action.OfMinimal", "region_id": "dynamics", "micro_elevation": 0.8, "macro_tier": 1, "macro_tier_score": 0.1251, "macro_tier_override": null, "x": 197.311, "z": 138.616, "size": 0.239, "title": "Ergodicity from minimality", "summary": "In this file we prove that the left shift `(a * ·)` on a compact topological group `G` is ergodic with respect to the Haar measure if and only if it is minimal, i.e., the powers `a ^ n` are dense in `G`. The proof of the more difficult \"if minimal, then ergodic\" implication is based on the ergodicity of the left action of a group on itself and the following fact that we prove in `ergodic_smul_of_denseRange_pow`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Ergodic/Action/OfMinimal.html"}, {"id": "Mathlib.Dynamics.Ergodic.Extreme", "region_id": "dynamics", "micro_elevation": 0.6, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 192.174, "z": 139.681, "size": 0.2, "title": "Ergodic measures as extreme points", "summary": "In this file we prove that a finite measure `μ` is an ergodic measure for a self-map `f` iff it is an extreme point of the set of invariant measures of `f` with the same total volume. We also specialize this result to probability measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Ergodic/Extreme.html"}, {"id": "Mathlib.Dynamics.Ergodic.Function", "region_id": "dynamics", "micro_elevation": 0.4, "macro_tier": 1, "macro_tier_score": 0.1251, "macro_tier_override": null, "x": 190.655, "z": 127.312, "size": 0.239, "title": "Functions invariant under a (quasi)ergodic map", "summary": "In this file we prove that an a.e. strongly measurable function `g : α → X` that is a.e. invariant under a (quasi)ergodic map is a.e. equal to a constant. We prove several versions of this statement with slightly different measurability assumptions. We also formulate a version for `MeasureTheory.AEEqFun` functions with all a.e. equalities replaced with equalities in the quotient space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Ergodic/Function.html"}, {"id": "Mathlib.Dynamics.Ergodic.Ergodic", "region_id": "dynamics", "micro_elevation": 0.2, "macro_tier": 2, "macro_tier_score": 0.1254, "macro_tier_override": null, "x": 190.064, "z": 132.095, "size": 0.2884, "title": "Ergodic maps and measures", "summary": "Let `f : α → α` be measure preserving with respect to a measure `μ`. We say `f` is ergodic with respect to `μ` (or `μ` is ergodic with respect to `f`) if the only measurable sets `s` such that `f⁻¹' s = s` are either almost empty or full. In this file we define ergodic maps / measures together with quasi-ergodic maps / measures and provide some basic API. Quasi-ergodicity is a weaker condition than ergodicity for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Ergodic/Ergodic.html"}, {"id": "Mathlib.Dynamics.Newton", "region_id": "dynamics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 187.033, "z": 132.176, "size": 0.2514, "title": "Newton-Raphson method", "summary": "Given a single-variable polynomial `P` with derivative `P'`, Newton's method concerns iteration of the rational map: `x ↦ x - P(x) / P'(x)`. Over a field, it can serve as a root-finding algorithm. It is also useful in proving results such as Hensel's lemma and the Jordan-Chevalley decomposition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Newton.html"}, {"id": "Mathlib.Dynamics.TopologicalEntropy.Semiconj", "region_id": "dynamics", "micro_elevation": 0.4, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 191.158, "z": 127.731, "size": 0.2, "title": "Topological entropy of the image of a set under a semiconjugacy", "summary": "Consider two dynamical systems `(X, S)` and `(Y, T)` together with a semiconjugacy `φ`: ``` X ---S--> X | | φ φ | | v v Y ---T--> Y ``` We relate the topological entropy of a subset `F ⊆ X` with the topological entropy of its image `φ '' F ⊆ Y`. The best-known theorem is that, if all maps are uniformly continuous, then `coverEntropy T (φ '' F) ≤ coverEntropy S F`. This is theorem…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/TopologicalEntropy/Semiconj.html"}, {"id": "Mathlib.Dynamics.Ergodic.AddCircleAdd", "region_id": "dynamics", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 198.309, "z": 122.043, "size": 0.2, "title": "Ergodicity of an irrational rotation", "summary": "In this file we prove that rotation of `AddCircle p` by `a` is ergodic if and only if `a` has infinite order (in other words, if `a / p` is irrational).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Ergodic/AddCircleAdd.html"}, {"id": "Mathlib.Dynamics.FixedPoints.Support", "region_id": "dynamics", "micro_elevation": 0.4, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 183.444, "z": 127.288, "size": 0.2, "title": "Support of a self-map", "summary": "Given a self-map `f : X → X` of a topological space `X`, the closure of the non-fixed points of `f` is often called the support of `f`. Since the word \"support\" is used to label various concepts throughout mathematics, we will use the term \"fixed support\" for the concept described above. This file contains basic definitions and API for the fixed support.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/FixedPoints/Support.html"}, {"id": "Mathlib.Dynamics.BirkhoffSum.QuasiMeasurePreserving", "region_id": "dynamics", "micro_elevation": 0.8, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 176.551, "z": 138.277, "size": 0.2, "title": "Birkhoff sum and average for quasi-measure-preserving maps", "summary": "Given a map `f` and measure `μ`, under the assumption of `QuasiMeasurePreserving f μ μ` we prove: - `birkhoffSum_ae_eq_of_ae_eq`: if observables `φ` and `ψ` are `μ`-a.e. equal then the corresponding `birkhoffSum f` are `μ`-a.e. equal. - `birkhoffAverage_ae_eq_of_ae_eq`: if observables `φ` and `ψ` are `μ`-a.e. equal then the corresponding `birkhoffAverage R f` are `μ`-a.e. equal.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/BirkhoffSum/QuasiMeasurePreserving.html"}, {"id": "Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber", "region_id": "dynamics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 187.033, "z": 132.176, "size": 0.2, "title": "Translation number of a monotone real map that commutes with `x ↦ x + 1`", "summary": "Let `f : ℝ → ℝ` be a monotone map such that `f (x + 1) = f x + 1` for all `x`. Then the limit $$ \\tau(f)=\\lim_{n\\to\\infty}{f^n(x)-x}{n} $$ exists and does not depend on `x`. This number is called the *translation number* of `f`. Different authors use different notation for this number: `τ`, `ρ`, `rot`, etc In this file we define a structure `CircleDeg1Lift` for bundled maps with these properties, define translation…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.html"}, {"id": "Mathlib.Dynamics.Ergodic.AddCircle", "region_id": "dynamics", "micro_elevation": 0.6, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 181.302, "z": 125.113, "size": 0.2478, "title": "Ergodic maps of the additive circle", "summary": "This file contains proofs of ergodicity for maps of the additive circle.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Ergodic/AddCircle.html"}, {"id": "Mathlib.Dynamics.Ergodic.Conservative", "region_id": "dynamics", "micro_elevation": 0.2, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 189.729, "z": 130.789, "size": 0.2, "title": "Conservative systems", "summary": "In this file we define `f : α → α` to be a *conservative* system w.r.t. a measure `μ` if `f` is non-singular (`MeasureTheory.QuasiMeasurePreserving`) and for every measurable set `s` of positive measure at least one point `x ∈ s` returns back to `s` after some number of iterations of `f`. There are several properties that look like they are stronger than this one but actually follow from it: *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Ergodic/Conservative.html"}, {"id": "Mathlib.Dynamics.Flow", "region_id": "dynamics", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.1253, "macro_tier_override": null, "x": 187.033, "z": 132.176, "size": 0.2676, "title": "Flows and invariant sets", "summary": "This file defines a flow on a topological space `α` by a topological monoid `τ` as a continuous monoid-action of `τ` on `α`. Anticipating the cases where `τ` is one of `ℕ`, `ℤ`, `ℝ⁺`, or `ℝ`, we use additive notation for the monoids, though the definition does not require commutativity. A subset `s` of `α` is invariant under a family of maps `ϕₜ : α → α` if `ϕₜ s ⊆ s` for all `t`. In many cases `ϕ` will be a flow on…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/Flow.html"}, {"id": "Mathlib.Dynamics.TopologicalEntropy.Subset", "region_id": "dynamics", "micro_elevation": 0.6, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 181.284, "z": 139.226, "size": 0.2, "title": "Topological entropy of subsets: monotonicity, closure, union", "summary": "This file contains general results about the topological entropy of various subsets of the same dynamical system `(X, T)`. We prove that: - the topological entropy `CoverEntropy T F` of `F` is monotone in `F`: the larger the subset, the larger its entropy. - the topological entropy of a subset equals the entropy of its closure. - the entropy of the union of two sets is the maximum of their entropies. We generalize…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/TopologicalEntropy/Subset.html"}, {"id": "Mathlib.Dynamics.TopologicalEntropy.NetEntropy", "region_id": "dynamics", "micro_elevation": 0.4, "macro_tier": 1, "macro_tier_score": 0.1253, "macro_tier_override": null, "x": 191.535, "z": 136.24, "size": 0.2676, "title": "Topological entropy via nets", "summary": "We implement Bowen-Dinaburg's definitions of the topological entropy, via nets. The major design decisions are the same as in `Mathlib/Dynamics/TopologicalEntropy/CoverEntropy.lean`, and are explained in detail there: use of uniform spaces, definition of the topological entropy of a subset, and values taken in `EReal`. Given a map `T : X → X` and a subset `F ⊆ X`, the topological entropy is loosely defined using…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/TopologicalEntropy/NetEntropy.html"}, {"id": "Mathlib.Dynamics.OmegaLimit", "region_id": "dynamics", "micro_elevation": 0.2, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 186.262, "z": 135.109, "size": 0.2, "title": "ω-limits", "summary": "For a function `ϕ : τ → α → β` where `β` is a topological space, we define the ω-limit under `ϕ` of a set `s` in `α` with respect to filter `f` on `τ`: an element `y : β` is in the ω-limit of `s` if the forward images of `s` intersect arbitrarily small neighbourhoods of `y` frequently \"in the direction of `f`\". In practice `ϕ` is often a continuous monoid-act, but the definition requires only that `ϕ` has a coercion…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/OmegaLimit.html"}, {"id": "Mathlib.Dynamics.SymbolicDynamics.Basic", "region_id": "dynamics", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 187.033, "z": 132.176, "size": 0.2, "title": "Symbolic dynamics on cancellative monoids", "summary": "This file develops a minimal API for symbolic dynamics over a **left-cancellative monoid** `G`—formally, a structure carrying `[Monoid G]` and `[IsLeftCancelMul G]` (which becomes `[AddMonoid G]` and `[IsLeftCancelAdd G]` in the additive form). Throughout the documentation we use the **additive** notations, which are the most common in symbolic dynamics, although all the notions introduced are defined in the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Dynamics/SymbolicDynamics/Basic.html"}, {"id": "Mathlib.Condensed.Solid", "region_id": "condensed", "micro_elevation": 0.5714, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -176.456, "z": -151.381, "size": 0.2, "title": "Solid modules", "summary": "This file contains the definition of a solid `R`-module: `CondensedMod.isSolid R`. Solid modules groups were introduced in [scholze2019condensed], Definition 5.1.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Solid.html"}, {"id": "Mathlib.Condensed.Functors", "region_id": "condensed", "micro_elevation": 0.4286, "macro_tier": 2, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": -190.835, "z": -153.802, "size": 0.2585, "title": "Functors from categories of topological spaces to condensed sets", "summary": "This file defines the embedding of the test objects (compact Hausdorff spaces) into condensed sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Functors.html"}, {"id": "Mathlib.Condensed.Limits", "region_id": "condensed", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.1074, "macro_tier_override": null, "x": -185.561, "z": -155.063, "size": 0.2658, "title": "Limits in categories of condensed objects", "summary": "This file adds some instances for limits in condensed sets and condensed modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Limits.html"}, {"id": "Mathlib.Condensed.TopCatAdjunction", "region_id": "condensed", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -189.414, "z": -150.407, "size": 0.2, "title": "The adjunction between condensed sets and topological spaces", "summary": "This file defines the functor `condensedSetToTopCat : CondensedSet.{u} ⥤ TopCat.{u + 1}` which is left adjoint to `topCatToCondensedSet : TopCat.{u + 1} ⥤ CondensedSet.{u}`. We prove that the counit is bijective (but not in general an isomorphism) and conclude that the right adjoint is faithful. The counit is an isomorphism for compactly generated spaces, and we conclude that the functor `topCatToCondensedSet` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/TopCatAdjunction.html"}, {"id": "Mathlib.Condensed.TopComparison", "region_id": "condensed", "micro_elevation": 0.1429, "macro_tier": 2, "macro_tier_score": 0.1076, "macro_tier_override": null, "x": -184.525, "z": -152.845, "size": 0.2857, "title": "The functor from topological spaces to condensed sets", "summary": "This file builds on the API from the file `TopCat.Yoneda`. If the forgetful functor to `TopCat` has nice properties, like preserving pullbacks and finite coproducts, then this Yoneda presheaf satisfies the sheaf condition for the regular and extensive topologies respectively. We apply this API to `CompHaus` and define the functor `topCatToCondensedSet : TopCat.{u + 1} ⥤ CondensedSet.{u}`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/TopComparison.html"}, {"id": "Mathlib.Condensed.Light.Sequence", "region_id": "condensed", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -199.431, "z": -145.821, "size": 0.2, "title": "The free light condensed `R`-module `R[ℕ∪∞]` is internally projective", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/Sequence.html"}, {"id": "Mathlib.Condensed.Light.InternallyProjective", "region_id": "condensed", "micro_elevation": 0.8571, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": -176.826, "z": -140.733, "size": 0.239, "title": "Characterization of internal projectivity in light condensed modules", "summary": "This file gives an explicit condition on light condensed modules over a ring `R` to be internally projective, namely the following: `internallyProjective_iff_tensor_condition`: `P : LightCondMod R` is internally projective if and only if, for all `A B : LightCondMod R`, for all epimorphisms `e : A ⟶ B`, for all `S : LightProfinite` and all morphisms `g : P ⊗ R[S] ⟶ B`, there exists a `S' : LightProfinite` with a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/InternallyProjective.html"}, {"id": "Mathlib.Condensed.Light.EffectiveEpi", "region_id": "condensed", "micro_elevation": 0.8571, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": -197.44, "z": -154.818, "size": 0.239, "title": "The functor from light profinite sets to light condensed sets preserves effective epimorphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/EffectiveEpi.html"}, {"id": "Mathlib.Condensed.Light.Explicit", "region_id": "condensed", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -182.057, "z": -153.843, "size": 0.2, "title": "The explicit sheaf condition for light condensed sets", "summary": "We give an explicit description of light condensed sets: * `LightCondensed.ofSheafLightProfinite`: A finite-product-preserving presheaf on `LightProfinite`, satisfying `EqualizerCondition`. The property `EqualizerCondition` is defined in `Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean` and it says that for any effective epi `X ⟶ B` (in this case that is equivalent to being a continuous surjection), the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/Explicit.html"}, {"id": "Mathlib.Condensed.Light.Module", "region_id": "condensed", "micro_elevation": 0.1429, "macro_tier": 2, "macro_tier_score": 0.0723, "macro_tier_override": null, "x": -185.021, "z": -148.592, "size": 0.3272, "title": "Light condensed `R`-modules", "summary": "This file defines light condensed modules over a ring `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/Module.html"}, {"id": "Mathlib.Condensed.Light.Monoidal", "region_id": "condensed", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": -188.327, "z": -153.641, "size": 0.2442, "title": "Closed symmetric monoidal structure on light condensed modules", "summary": "We define a symmetric monoidal structure on light condensed modules by localizing the symmetric monoidal structure on the presheaf category. By Day's reflection theorem, we obtain a closed structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/Monoidal.html"}, {"id": "Mathlib.Condensed.Discrete.Characterization", "region_id": "condensed", "micro_elevation": 0.5714, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -183.213, "z": -159.212, "size": 0.2, "title": "Characterizing discrete condensed sets and `R`-modules.", "summary": "This file proves a characterization of discrete condensed sets, discrete condensed `R`-modules over a ring `R`, discrete light condensed sets, and discrete light condensed `R`-modules over a ring `R`. see `CondensedSet.isDiscrete_tfae`, `CondensedMod.isDiscrete_tfae`, `LightCondSet.isDiscrete_tfae`, and `LightCondMod.isDiscrete_tfae`. Informally, we can say: The following conditions characterize a condensed set `X`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Discrete/Characterization.html"}, {"id": "Mathlib.Condensed.Discrete.Colimit", "region_id": "condensed", "micro_elevation": 0.4286, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": -181.91, "z": -145.094, "size": 0.2478, "title": "The condensed set given by left Kan extension from `FintypeCat` to `Profinite`.", "summary": "This file provides the necessary API to prove that a condensed set `X` is discrete if and only if for every profinite set `S = limᵢSᵢ`, `X(S) ≅ colimᵢX(Sᵢ)`, and the analogous result for light condensed sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Discrete/Colimit.html"}, {"id": "Mathlib.Condensed.Discrete.Module", "region_id": "condensed", "micro_elevation": 0.4286, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": -188.625, "z": -156.212, "size": 0.2478, "title": "Discrete condensed `R`-modules", "summary": "This file provides the necessary API to prove that a condensed `R`-module is discrete if and only if the underlying condensed set is (both for light condensed and condensed). That is, it defines the functor `CondensedMod.LocallyConstant.functor` which takes an `R`-module to the condensed `R`-modules given by locally constant maps to it, and proves that this functor is naturally isomorphic to the constant sheaf…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Discrete/Module.html"}, {"id": "Mathlib.Condensed.Light.Epi", "region_id": "condensed", "micro_elevation": 0.7143, "macro_tier": 2, "macro_tier_score": 0.0361, "macro_tier_override": null, "x": -175.083, "z": -146.632, "size": 0.28, "title": "Epimorphisms of light condensed objects", "summary": "This file characterises epimorphisms in light condensed sets and modules as the locally surjective morphisms. Here, the condition of locally surjective is phrased in terms of continuous surjections of light profinite sets. Further, we prove that the functor `lim : Discrete ℕ ⥤ LightCondMod R` preserves epimorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/Epi.html"}, {"id": "Mathlib.Condensed.Discrete.LocallyConstant", "region_id": "condensed", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0359, "macro_tier_override": null, "x": -180.779, "z": -150.397, "size": 0.2615, "title": "The sheaf of locally constant maps on `CompHausLike P`", "summary": "This file proves that under suitable conditions, the functor from the category of sets to the category of sheaves for the coherent topology on `CompHausLike P`, given by mapping a set to the sheaf of locally constant maps to it, is left adjoint to the \"underlying set\" functor (evaluation at the point). We apply this to prove that the constant sheaf functor into (light) condensed sets is isomorphic to the functor of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Discrete/LocallyConstant.html"}, {"id": "Mathlib.Condensed.Equivalence", "region_id": "condensed", "micro_elevation": 0.1429, "macro_tier": 2, "macro_tier_score": 0.2505, "macro_tier_override": null, "x": -187.256, "z": -150.599, "size": 0.2912, "title": "Sheaves on CompHaus are equivalent to sheaves on Stonean", "summary": "The forgetful functor from extremally disconnected spaces `Stonean` to compact Hausdorff spaces `CompHaus` has the marvellous property that it induces an equivalence of categories between sheaves on these two sites. With the terminology of nLab, `Stonean` is a *dense subsite* of `CompHaus`: see https://ncatlab.org/nlab/show/dense+sub-site Since Stonean spaces are the projective objects in `CompHaus`, which has…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Equivalence.html"}, {"id": "Mathlib.Condensed.Light.CartesianClosed", "region_id": "condensed", "micro_elevation": 0.1429, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -186.915, "z": -151.932, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/CartesianClosed.html"}, {"id": "Mathlib.Condensed.Light.Basic", "region_id": "condensed", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0719, "macro_tier_override": null, "x": -185.096, "z": -150.756, "size": 0.2917, "title": "Light condensed objects", "summary": "This file defines the category of light condensed objects in a category `C`, following the work of Clausen-Scholze (see https://www.youtube.com/playlist?list=PLx5f8IelFRgGmu6gmL-Kf_Rl_6Mm7juZO).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/Basic.html"}, {"id": "Mathlib.Condensed.Light.Instances", "region_id": "condensed", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0717, "macro_tier_override": null, "x": -185.096, "z": -150.756, "size": 0.2727, "title": "`HasSheafify` instances", "summary": "In this file, we obtain a `HasSheafify (coherentTopology LightProfinite.{u}) (Type u)` instance (and similarly for other concrete categories). These instances are not obtained automatically because `LightProfinite.{u}` is a large category, but as it is essentially small, the instances can be obtained using the results in the file `Mathlib/CategoryTheory/Sites/Equivalence.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/Instances.html"}, {"id": "Mathlib.Condensed.Light.TopCatAdjunction", "region_id": "condensed", "micro_elevation": 0.4286, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": -190.491, "z": -147.136, "size": 0.2408, "title": "The adjunction between light condensed sets and topological spaces", "summary": "This file defines the functor `lightCondSetToTopCat : LightCondSet.{u} ⥤ TopCat.{u}` which is left adjoint to `topCatToLightCondSet : TopCat.{u} ⥤ LightCondSet.{u}`. We prove that the counit is bijective (but not in general an isomorphism) and conclude that the right adjoint is faithful. The counit is an isomorphism for sequential spaces, and we conclude that the functor `topCatToLightCondSet` is fully faithful when…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/TopCatAdjunction.html"}, {"id": "Mathlib.Condensed.Light.TopComparison", "region_id": "condensed", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": -183.13, "z": -146.897, "size": 0.2446, "title": "The functor from topological spaces to light condensed sets", "summary": "We define the functor `topCatToLightCondSet : TopCat.{u} ⥤ LightCondSet.{u}`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/TopComparison.html"}, {"id": "Mathlib.Condensed.Basic", "region_id": "condensed", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.5007, "macro_tier_override": null, "x": -185.096, "z": -150.756, "size": 0.3098, "title": "Condensed Objects", "summary": "This file defines the category of condensed objects in a category `C`, following the work of Clausen-Scholze and Barwick-Haine.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Basic.html"}, {"id": "Mathlib.Condensed.Light.Functors", "region_id": "condensed", "micro_elevation": 0.5714, "macro_tier": 1, "macro_tier_score": 0.0359, "macro_tier_override": null, "x": -178.02, "z": -145.759, "size": 0.2499, "title": "Functors from categories of topological spaces to light condensed sets", "summary": "This file defines the embedding of the test objects (light profinite sets) into light condensed sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/Functors.html"}, {"id": "Mathlib.Condensed.Explicit", "region_id": "condensed", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.1073, "macro_tier_override": null, "x": -181.933, "z": -153.716, "size": 0.2611, "title": "The explicit sheaf condition for condensed sets", "summary": "We give the following three explicit descriptions of condensed objects: * `Condensed.ofSheafStonean`: A finite-product-preserving presheaf on `Stonean`. * `Condensed.ofSheafProfinite`: A finite-product-preserving presheaf on `Profinite`, satisfying `EqualizerCondition`. * `Condensed.ofSheafCompHaus`: A finite-product-preserving presheaf on `CompHaus`, satisfying `EqualizerCondition`. The property…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Explicit.html"}, {"id": "Mathlib.Condensed.Light.Limits", "region_id": "condensed", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.0359, "macro_tier_override": null, "x": -184.897, "z": -155.083, "size": 0.2499, "title": "Limits in categories of light condensed objects", "summary": "This file adds some instances for limits in light condensed sets and modules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/Limits.html"}, {"id": "Mathlib.Condensed.AB", "region_id": "condensed", "micro_elevation": 0.4286, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -184.126, "z": -144.332, "size": 0.2, "title": "AB axioms in condensed modules", "summary": "This file proves that the category of condensed modules over a ring satisfies Grothendieck's axioms AB5, AB4, and AB4`*`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/AB.html"}, {"id": "Mathlib.Condensed.CartesianClosed", "region_id": "condensed", "micro_elevation": 0.1429, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -185.957, "z": -148.769, "size": 0.2, "title": "Condensed sets form a Cartesian closed category", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/CartesianClosed.html"}, {"id": "Mathlib.Condensed.Light.AB", "region_id": "condensed", "micro_elevation": 0.8571, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -187.075, "z": -137.913, "size": 0.2, "title": "Grothendieck's AB axioms for light condensed modules", "summary": "The category of light condensed `R`-modules over a ring satisfies the countable version of Grothendieck's AB4\\* axiom", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/AB.html"}, {"id": "Mathlib.Condensed.Module", "region_id": "condensed", "micro_elevation": 0.1429, "macro_tier": 2, "macro_tier_score": 0.2866, "macro_tier_override": null, "x": -187.118, "z": -151.533, "size": 0.3228, "title": "Condensed `R`-modules", "summary": "This file defines condensed modules over a ring `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Module.html"}, {"id": "Mathlib.Condensed.Epi", "region_id": "condensed", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": -186.644, "z": -146.711, "size": 0.2338, "title": "Epimorphisms of condensed objects", "summary": "This file characterises epimorphisms of condensed sets and condensed `R`-modules for any ring `R`, as those morphisms which are objectwise surjective on `Stonean` (see `CondensedSet.epi_iff_surjective_on_stonean` and `CondensedMod.epi_iff_surjective_on_stonean`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Epi.html"}, {"id": "Mathlib.Condensed.Discrete.Basic", "region_id": "condensed", "micro_elevation": 0.1429, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": -187.223, "z": -150.346, "size": 0.244, "title": "Discrete-underlying adjunction", "summary": "Given a category `C` with sheafification with respect to the coherent topology on compact Hausdorff spaces, we define a functor `C ⥤ Condensed C` which associates to an object of `C` the corresponding \"discrete\" condensed object (see `Condensed.discrete`). In `Condensed.discreteUnderlyingAdj` we prove that this functor is left adjoint to the forgetful functor from `Condensed C` to `C`. We also give the variant…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Discrete/Basic.html"}, {"id": "Mathlib.Condensed.EffectiveEpi", "region_id": "condensed", "micro_elevation": 0.5714, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -183.842, "z": -159.328, "size": 0.2, "title": "The functor from compact Hausdorff spaces to condensed sets preserves effective epimorphisms", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/EffectiveEpi.html"}, {"id": "Mathlib.Condensed.Light.Small", "region_id": "condensed", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": -182.614, "z": -147.206, "size": 0.2, "title": "Equivalence of light condensed objects with sheaves on a small site", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Condensed/Light/Small.html"}, {"id": "Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField", "region_id": "geometry", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0005, "macro_tier_override": null, "x": 126.769, "z": 123.936, "size": 0.297, "title": "Residue fields of points", "summary": "Any point `x` of a locally ringed space `X` comes with a natural residue field, namely the residue field of the stalk at `x`. Moreover, for every open subset of `X` containing `x`, we have a canonical evaluation map from `Γ(X, U)` to the residue field of `X` at `x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.html"}, {"id": "Mathlib.Geometry.RingedSpace.Stalks", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 3, "macro_tier_score": 0.5001, "macro_tier_override": null, "x": 124.707, "z": 131.473, "size": 0.2496, "title": "Stalks for presheafed spaces", "summary": "This file lifts constructions of stalks and pushforwards of stalks to work with the category of presheafed spaces. Additionally, we prove that restriction of presheafed spaces does not change the stalks.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/RingedSpace/Stalks.html"}, {"id": "Mathlib.Geometry.RingedSpace.PresheafedSpace", "region_id": "geometry", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.5003, "macro_tier_override": null, "x": 126.245, "z": 131.664, "size": 0.2706, "title": "Presheafed spaces", "summary": "Introduces the category of topological spaces equipped with a presheaf (taking values in an arbitrary target category `C`). We further describe how to apply functors and natural transformations to the values of the presheaves.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/RingedSpace/PresheafedSpace.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic", "region_id": "geometry", "micro_elevation": 0.8, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 140.375, "z": 152.027, "size": 0.2676, "title": "Covariant derivatives", "summary": "This file defines covariant derivatives (aka Koszul connections) on vector bundles over manifolds. There are versions of the story: a local unbundled one and a global bundled one. The local version is used by the global version but also (in other files) when seeing a global object in a local trivialization. In the whole file `M` is a manifold over any nontrivially normed field `𝕜` and `V` is a vector bundle over `M`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/CovariantDerivative/Basic.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.Hom", "region_id": "geometry", "micro_elevation": 0.7, "macro_tier": 2, "macro_tier_score": 0.2506, "macro_tier_override": null, "x": 122.89, "z": 153.09, "size": 0.3064, "title": "Homs of `C^n` vector bundles over the same base space", "summary": "Here we show that the bundle of continuous linear maps is a `C^n` vector bundle. We also show that applying a smooth family of linear maps to a smooth family of vectors gives a smooth result, in several versions. Note that we only do this for bundles of linear maps, not for bundles of arbitrary semilinear maps. Indeed, semilinear maps are typically not smooth. For instance, complex conjugation is not…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/Hom.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.Tangent", "region_id": "geometry", "micro_elevation": 0.55, "macro_tier": 2, "macro_tier_score": 0.2507, "macro_tier_override": null, "x": 122.736, "z": 114.988, "size": 0.309, "title": "Tangent bundles", "summary": "This file defines the tangent bundle as a `C^n` vector bundle. Let `M` be a manifold with model `I` on `(E, H)`. The tangent space `TangentSpace I (x : M)` has already been defined as a type synonym for `E`, and the tangent bundle `TangentBundle I M` as an abbrev of `Bundle.TotalSpace E (TangentSpace I : M → Type _)`. In this file, when `M` is `C^1`, we construct a vector bundle structure on `TangentBundle I M`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/Tangent.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.Tensoriality", "region_id": "geometry", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": 149.309, "z": 128.841, "size": 0.253, "title": "The tensoriality criterion", "summary": "Given vector bundles `V` and `W` over a manifold `M`, one can construct a section of the hom-bundle `Π x, V x →L[𝕜] W x` from a *tensorial* operation sending sections of `V` to sections of `W`. This file provides this construction. In fact, we define tensoriality, and provide the above criterion, in slightly greater generality: for operations sending sections of `V` to a vector space `A` (which in the above…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/Tensoriality.html"}, {"id": "Mathlib.Geometry.Convex.Set", "region_id": "geometry", "micro_elevation": 0.1, "macro_tier": 2, "macro_tier_score": 0.2505, "macro_tier_override": null, "x": 128.016, "z": 134.205, "size": 0.2954, "title": "Convex sets", "summary": "This file defines convex sets in a convex space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/Set.html"}, {"id": "Mathlib.Geometry.Convex.ConvexSpace.Prod", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 125.545, "z": 130.282, "size": 0.2782, "title": "Product of convex spaces", "summary": "This file defines the cartesian product of convex spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/ConvexSpace/Prod.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection", "region_id": "geometry", "micro_elevation": 0.6, "macro_tier": 2, "macro_tier_score": 0.2506, "macro_tier_override": null, "x": 118.806, "z": 148.7, "size": 0.3072, "title": "`C^n` sections", "summary": "In this file we define the type `ContMDiffSection` of `n` times continuously differentiable sections of a vector bundle over a manifold `M` and prove that it's a module over the base field. In passing, we prove that binary and finite sums, differences and scalar products of `C^n` sections are `C^n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/ContMDiffSection.html"}, {"id": "Mathlib.Geometry.Manifold.Algebra.SMul", "region_id": "geometry", "micro_elevation": 0.55, "macro_tier": 3, "macro_tier_score": 0.251, "macro_tier_override": null, "x": 127.282, "z": 114.655, "size": 0.3371, "title": "Cⁿ monoid actions", "summary": "In this file we define Cⁿ actions (e.g. by Lie groups or monoids) on manifolds: we say `ContMDiffSMul I I' n G M` if `G` acts multiplicatively on `M` and the action map `fun p : G × M ↦ p.1 • p.2` is Cⁿ. We also provide API for additive actions using `@[to_additive]`. We also provide `ContMDiffSMul` instances for scalar multiplication in normed spaces and for the action of the monoid `E →L[𝕜] E` of continuous linear…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Algebra/SMul.html"}, {"id": "Mathlib.Geometry.Manifold.Algebra.LieGroup", "region_id": "geometry", "micro_elevation": 0.55, "macro_tier": 3, "macro_tier_score": 0.2512, "macro_tier_override": null, "x": 132.922, "z": 147.341, "size": 0.3482, "title": "Lie groups", "summary": "A Lie group is a group that is also a `C^n` manifold, in which the group operations of multiplication and inversion are `C^n` maps. Regularity of the group multiplication means that multiplication is a `C^n` mapping of the product manifold `G` × `G` into `G`. Note that, since a manifold here is not second-countable and Hausdorff, a Lie group here is not guaranteed to be second-countable (even though it can be proved…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Algebra/LieGroup.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.Basic", "region_id": "geometry", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.2512, "macro_tier_override": null, "x": 140.623, "z": 125.898, "size": 0.3444, "title": "`C^n` vector bundles", "summary": "This file defines `C^n` vector bundles over a manifold. Let `E` be a topological vector bundle, with model fiber `F` and base space `B`. We consider `E` as carrying a charted space structure given by its trivializations -- these are charts to `B × F`. Then, by \"composition\", if `B` is itself a charted space over `H` (e.g. a smooth manifold), then `E` is also a charted space over `H × F`. Now, we define…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/Basic.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.Pullback", "region_id": "geometry", "micro_elevation": 0.55, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 113.665, "z": 143.158, "size": 0.2, "title": "Pullbacks of `C^n` vector bundles", "summary": "This file defines pullbacks of `C^n` vector bundles over a manifold.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/Pullback.html"}, {"id": "Mathlib.Geometry.Manifold.ContMDiffMap", "region_id": "geometry", "micro_elevation": 0.45, "macro_tier": 3, "macro_tier_score": 0.252, "macro_tier_override": null, "x": 112.897, "z": 135.691, "size": 0.3915, "title": "`C^n` bundled maps", "summary": "In this file we define the type `ContMDiffMap` of `n` times continuously differentiable bundled maps.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/ContMDiffMap.html"}, {"id": "Mathlib.Geometry.RingedSpace.OpenImmersion", "region_id": "geometry", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.5004, "macro_tier_override": null, "x": 121.596, "z": 137.859, "size": 0.2897, "title": "Open immersions of structured spaces", "summary": "We say that a morphism of presheafed spaces `f : X ⟶ Y` is an open immersion if the underlying map of spaces is an open embedding `f : X ⟶ U ⊆ Y`, and the sheaf map `Y(V) ⟶ f _* X(V)` is an iso for each `V ⊆ U`. Abbreviations are also provided for `SheafedSpace`, `LocallyRingedSpace` and `Scheme`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/RingedSpace/OpenImmersion.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle", "region_id": "geometry", "micro_elevation": 0.2, "macro_tier": 2, "macro_tier_score": 0.2504, "macro_tier_override": null, "x": 130.838, "z": 127.504, "size": 0.2799, "title": "Oriented angles in right-angled triangles.", "summary": "This file proves basic geometric results about distances and oriented angles in (possibly degenerate) right-angled triangles in real inner product spaces and Euclidean affine spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Oriented.Affine", "region_id": "geometry", "micro_elevation": 0.15, "macro_tier": 2, "macro_tier_score": 0.2508, "macro_tier_override": null, "x": 126.329, "z": 136.31, "size": 0.3222, "title": "Oriented angles.", "summary": "This file defines oriented angles in Euclidean affine spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle", "region_id": "geometry", "micro_elevation": 0.1, "macro_tier": 2, "macro_tier_score": 0.2504, "macro_tier_override": null, "x": 128.704, "z": 129.78, "size": 0.2876, "title": "Right-angled triangles", "summary": "This file proves basic geometric results about distances and angles in (possibly degenerate) right-angled triangles in real inner product spaces and Euclidean affine spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.html"}, {"id": "Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing", "region_id": "geometry", "micro_elevation": 0.35, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 135.343, "z": 125.764, "size": 0.2652, "title": "Gluing structured spaces", "summary": "Given a family of gluing data of structured spaces (presheafed spaces, sheafed spaces, or locally ringed spaces), we may glue them together. The construction should be \"sealed\" and considered as a black box, while only using the API provided.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.html"}, {"id": "Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits", "region_id": "geometry", "micro_elevation": 0.3, "macro_tier": 1, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": 127.866, "z": 140.816, "size": 0.2641, "title": "Colimits of LocallyRingedSpace", "summary": "We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`. It then follows that `LocallyRingedSpace` has all colimits, and `forgetToSheafedSpace` preserves them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.html"}, {"id": "Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions", "region_id": "geometry", "micro_elevation": 0.45, "macro_tier": 3, "macro_tier_score": 0.251, "macro_tier_override": null, "x": 120.299, "z": 119.053, "size": 0.3336, "title": "Differentiability of specific functions", "summary": "In this file, we establish differentiability results for - continuous linear maps and continuous linear equivalences - the identity - constant functions - products - arithmetic operations (such as addition and scalar multiplication).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.html"}, {"id": "Mathlib.Geometry.Manifold.MFDeriv.FDeriv", "region_id": "geometry", "micro_elevation": 0.4, "macro_tier": 2, "macro_tier_score": 0.2506, "macro_tier_override": null, "x": 118.944, "z": 121.649, "size": 0.3049, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/MFDeriv/FDeriv.html"}, {"id": "Mathlib.Geometry.Manifold.MFDeriv.NormedSpace", "region_id": "geometry", "micro_elevation": 0.6, "macro_tier": 2, "macro_tier_score": 0.2504, "macro_tier_override": null, "x": 123.645, "z": 113.257, "size": 0.2808, "title": "Equivalence of manifold differentiability with the basic definition for functions between", "summary": "vector spaces The API in this file is mostly copied from `Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean`, providing the same statements for higher smoothness. In this file, we do the same for differentiability.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/MFDeriv/NormedSpace.html"}, {"id": "Mathlib.Geometry.Manifold.Bordism", "region_id": "geometry", "micro_elevation": 0.85, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 152.577, "z": 132.057, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Bordism.html"}, {"id": "Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary", "region_id": "geometry", "micro_elevation": 0.8, "macro_tier": 2, "macro_tier_score": 0.2505, "macro_tier_override": null, "x": 134.793, "z": 108.398, "size": 0.2901, "title": "Interior and boundary of a manifold", "summary": "Define the interior and boundary of a manifold.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/IsManifold/InteriorBoundary.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 3, "macro_tier_score": 0.251, "macro_tier_override": null, "x": 126.879, "z": 133.077, "size": 0.3341, "title": "Angles between points", "summary": "This file defines unoriented angles in Euclidean affine spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic", "region_id": "geometry", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.2509, "macro_tier_override": null, "x": 126.245, "z": 131.664, "size": 0.3301, "title": "Angles between vectors", "summary": "This file defines unoriented angles in real inner product spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.html"}, {"id": "Mathlib.Geometry.Euclidean.Volume.Measure", "region_id": "geometry", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 126.245, "z": 131.664, "size": 0.2, "title": "Volume measure for Euclidean geometry", "summary": "In this file we introduce a `d`-dimensional measure for `n`-dimensional Euclidean affine space, namely `MeasureTheory.Measure.euclideanHausdorffMeasure d` with notation `μHE[d]`. This is the suitable measure to describe area and volume in an environment of arbitrary dimension. It is characterized by the following properties: * Coincides with Lebesgue measure when `d = n`. * Preserved through isometry, and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Volume/Measure.html"}, {"id": "Mathlib.Geometry.Manifold.Instances.UnitsOfNormedAlgebra", "region_id": "geometry", "micro_elevation": 0.6, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 120.043, "z": 114.139, "size": 0.2, "title": "Units of a normed algebra", "summary": "We construct the Lie group structure on the group of units of a complete normed `𝕜`-algebra `R`. The group of units `Rˣ` has a natural `C^n` manifold structure modelled on `R` given by its embedding into `R`. Together with the smoothness of the multiplication and inverse of its elements, `Rˣ` forms a Lie group. An important special case of this construction is the general linear group. For a normed space `V` over a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Instances/UnitsOfNormedAlgebra.html"}, {"id": "Mathlib.Geometry.Euclidean.Congruence", "region_id": "geometry", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 120.793, "z": 137.166, "size": 0.2, "title": "Triangle congruence", "summary": "This file proves the classical triangle congruence criteria for (possibly degenerate) triangles in real inner product spaces and Euclidean affine spaces. We prove SSS, SAS, ASA, and AAS congruence.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Congruence.html"}, {"id": "Mathlib.Geometry.Euclidean.Triangle", "region_id": "geometry", "micro_elevation": 0.2, "macro_tier": 2, "macro_tier_score": 0.2505, "macro_tier_override": null, "x": 132.376, "z": 130.77, "size": 0.2977, "title": "Triangles", "summary": "This file proves basic geometrical results about distances and angles in (possibly degenerate) triangles in real inner product spaces and Euclidean affine spaces. More specialized results, and results developed for simplices in general rather than just for triangles, are in separate files. Definitions and results that make sense in more general affine spaces rather than just in the Euclidean case go under…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Triangle.html"}, {"id": "Mathlib.Geometry.Manifold.StructureGroupoid", "region_id": "geometry", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2507, "macro_tier_override": null, "x": 126.245, "z": 131.664, "size": 0.3132, "title": "Structure groupoids", "summary": "This file contains the definitions and properties of structure groupoids, i.e., sets of open partial homeomorphisms stable under composition and inverse. These are used to define charted spaces (and hence manifolds). See the file `Mathlib.Geometry.Manifold.ChartedSpace` for more details.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/StructureGroupoid.html"}, {"id": "Mathlib.Geometry.Euclidean.NinePointCircle", "region_id": "geometry", "micro_elevation": 0.3, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 134.591, "z": 135.754, "size": 0.2, "title": "Nine-point circle", "summary": "This file defines the nine-point circle of a triangle, and its higher dimension analogue, the 3(n+1)-point sphere of a simplex. Specifically for triangles, we show that it passes through nine specific points as desired.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/NinePointCircle.html"}, {"id": "Mathlib.Geometry.Euclidean.MongePoint", "region_id": "geometry", "micro_elevation": 0.2, "macro_tier": 0, "macro_tier_score": 0.2501, "macro_tier_override": null, "x": 127.445, "z": 125.585, "size": 0.239, "title": "Monge point and orthocenter", "summary": "This file defines the orthocenter of a triangle, via its n-dimensional generalization, the Monge point of a simplex.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/MongePoint.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Sphere", "region_id": "geometry", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 132.204, "z": 136.612, "size": 0.2757, "title": "Angles in circles and spheres", "summary": "This file proves results about angles in circles and spheres.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Sphere.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Unoriented.Projection", "region_id": "geometry", "micro_elevation": 0.1, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 126.143, "z": 134.76, "size": 0.2719, "title": "Angles and orthogonal projection.", "summary": "This file proves lemmas relating to angles involving orthogonal projections.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Unoriented/Projection.html"}, {"id": "Mathlib.Geometry.Euclidean.Sphere.Ptolemy", "region_id": "geometry", "micro_elevation": 0.35, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 115.796, "z": 128.763, "size": 0.2, "title": "Ptolemy's theorem", "summary": "This file proves Ptolemy's theorem on the lengths of the diagonals and sides of a cyclic quadrilateral.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Sphere/Ptolemy.html"}, {"id": "Mathlib.Geometry.Euclidean.Sphere.Power", "region_id": "geometry", "micro_elevation": 0.3, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 121.428, "z": 123.714, "size": 0.2676, "title": "Power of a point (intersecting chords and secants)", "summary": "This file proves basic geometrical results about power of a point (intersecting chords and secants) in spheres in real inner product spaces and Euclidean affine spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Sphere/Power.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Bisector", "region_id": "geometry", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.2501, "macro_tier_override": null, "x": 130.803, "z": 125.402, "size": 0.2478, "title": "Angle bisectors.", "summary": "This file proves lemmas relating to bisecting angles.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Bisector.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Oriented.Projection", "region_id": "geometry", "micro_elevation": 0.2, "macro_tier": 1, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": 121.664, "z": 135.837, "size": 0.257, "title": "Oriented angles and orthogonal projection.", "summary": "This file proves lemmas relating to oriented angles involving orthogonal projections.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Oriented/Projection.html"}, {"id": "Mathlib.Geometry.Euclidean.Circumcenter", "region_id": "geometry", "micro_elevation": 0.15, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 130.496, "z": 129.788, "size": 0.2778, "title": "Circumcenter and circumradius", "summary": "This file proves some lemmas on points equidistant from a set of points, and defines the circumradius and circumcenter of a simplex. There are also some definitions for use in calculations where it is convenient to work with affine combinations of vertices together with the circumcenter.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Circumcenter.html"}, {"id": "Mathlib.Geometry.Euclidean.Sphere.Basic", "region_id": "geometry", "micro_elevation": 0.1, "macro_tier": 2, "macro_tier_score": 0.2509, "macro_tier_override": null, "x": 126.623, "z": 128.589, "size": 0.33, "title": "Spheres", "summary": "This file defines and proves basic results about spheres and cospherical sets of points in Euclidean affine spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Sphere/Basic.html"}, {"id": "Mathlib.Geometry.Euclidean.Inversion.Calculus", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 126.35, "z": 130.118, "size": 0.2, "title": "Derivative of the inversion", "summary": "In this file we prove a formula for the derivative of `EuclideanGeometry.inversion c R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Inversion/Calculus.html"}, {"id": "Mathlib.Geometry.Euclidean.Inversion.Basic", "region_id": "geometry", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 126.245, "z": 131.664, "size": 0.273, "title": "Inversion in an affine space", "summary": "In this file we define inversion in a sphere in an affine space. This map sends each point `x` to the point `y` such that `y -ᵥ c = (R / dist x c) ^ 2 • (x -ᵥ c)`, where `c` and `R` are the center and the radius of the sphere. In many applications, it is convenient to assume that the inversion swaps the center and the point at infinity. In order to stay in the original affine space, we define the map so that it…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Inversion/Basic.html"}, {"id": "Mathlib.Geometry.Manifold.PartitionOfUnity", "region_id": "geometry", "micro_elevation": 0.65, "macro_tier": 2, "macro_tier_score": 0.2507, "macro_tier_override": null, "x": 106.151, "z": 130.312, "size": 0.3149, "title": "Smooth partition of unity", "summary": "In this file we define two structures, `SmoothBumpCovering` and `SmoothPartitionOfUnity`. Both structures describe coverings of a set by a locally finite family of supports of smooth functions with some additional properties. The former structure is mostly useful as an intermediate step in the construction of a smooth partition of unity but some proofs that traditionally deal with a partition of unity can use a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/PartitionOfUnity.html"}, {"id": "Mathlib.Geometry.Diffeology.Basic", "region_id": "geometry", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 126.245, "z": 131.664, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Diffeology/Basic.html"}, {"id": "Mathlib.Geometry.Convex.Cone.DualFinite", "region_id": "geometry", "micro_elevation": 0.15, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 129.778, "z": 134.682, "size": 0.2, "title": "Duals of finitely generated cones", "summary": "This file defines the notion of dually finitely generated cones. A cone is dually finitely generated (or `DualFG` for short) if it is the dual of a finite set, or equivalently, of a finitely generated cone. In geometric terms, a cone is dually finitely generated if it can be written as the intersection of finitely many halfspaces. This is also known as an H-cone. This is the counterpart to `FG` (finitely generated)…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/Cone/DualFinite.html"}, {"id": "Mathlib.Geometry.Convex.Cone.Dual", "region_id": "geometry", "micro_elevation": 0.1, "macro_tier": 2, "macro_tier_score": 0.2504, "macro_tier_override": null, "x": 127.702, "z": 134.397, "size": 0.2831, "title": "The algebraic dual of a cone", "summary": "Given a bilinear pairing `p` between two `R`-modules `M` and `N` and a set `s` in `M`, we define `PointedCone.dual p s` to be the pointed cone in `N` consisting of all points `y` such that `0 ≤ p x y` for all `x ∈ s`. When the pairing is perfect, this gives us the algebraic dual of a cone. This is developed here. When the pairing is continuous and perfect (as a continuous pairing), this gives us the topological dual…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/Cone/Dual.html"}, {"id": "Mathlib.Geometry.Manifold.Instances.Sphere", "region_id": "geometry", "micro_elevation": 0.9, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 130.892, "z": 159.158, "size": 0.2676, "title": "Manifold structure on the sphere", "summary": "This file defines stereographic projection from the sphere in an inner product space `E`, and uses it to put an analytic manifold structure on the sphere.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Instances/Sphere.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation", "region_id": "geometry", "micro_elevation": 0.1, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 128.622, "z": 129.677, "size": 0.2756, "title": "Rotations by oriented angles.", "summary": "This file defines rotations by oriented angles in real inner product spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.html"}, {"id": "Mathlib.Geometry.Manifold.Algebra.Monoid", "region_id": "geometry", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.2524, "macro_tier_override": null, "x": 114.301, "z": 121.798, "size": 0.4085, "title": "`C^n` monoid", "summary": "A `C^n` monoid is a monoid that is also a `C^n` manifold, in which multiplication is a `C^n` map of the product manifold `G` × `G` into `G`. In this file we define the basic structures to talk about `C^n` monoids: `ContMDiffMul` and its additive counterpart `ContMDiffAdd`. These structures are general enough to also talk about `C^n` semigroups.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Algebra/Monoid.html"}, {"id": "Mathlib.Geometry.Manifold.MFDeriv.Basic", "region_id": "geometry", "micro_elevation": 0.35, "macro_tier": 3, "macro_tier_score": 0.2519, "macro_tier_override": null, "x": 121.147, "z": 141.235, "size": 0.3855, "title": "Basic properties of the manifold Fréchet derivative", "summary": "In this file, we show various properties of the manifold Fréchet derivative, mimicking the API for Fréchet derivatives. - basic properties of unique differentiability sets - various general lemmas about the manifold Fréchet derivative - deducing differentiability from smoothness, - deriving continuity from differentiability on manifolds, - congruence lemmas for derivatives on manifolds - composition lemmas and the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/MFDeriv/Basic.html"}, {"id": "Mathlib.Geometry.Manifold.ContMDiff.Constructions", "region_id": "geometry", "micro_elevation": 0.35, "macro_tier": 3, "macro_tier_score": 0.2512, "macro_tier_override": null, "x": 126.946, "z": 142.485, "size": 0.3451, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/ContMDiff/Constructions.html"}, {"id": "Mathlib.Geometry.Manifold.ContMDiff.Basic", "region_id": "geometry", "micro_elevation": 0.3, "macro_tier": 3, "macro_tier_score": 0.2512, "macro_tier_override": null, "x": 119.384, "z": 125.393, "size": 0.3494, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/ContMDiff/Basic.html"}, {"id": "Mathlib.Geometry.Euclidean.Projection", "region_id": "geometry", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2508, "macro_tier_override": null, "x": 126.245, "z": 131.664, "size": 0.3167, "title": "Orthogonal projection in affine spaces", "summary": "This file defines orthogonal projection onto an affine subspace, and reflection of a point in an affine subspace.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Projection.html"}, {"id": "Mathlib.Geometry.Euclidean.Sphere.SecondInter", "region_id": "geometry", "micro_elevation": 0.15, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 121.605, "z": 131.939, "size": 0.2, "title": "Second intersection of a sphere and a line", "summary": "This file defines and proves basic results about the second intersection of a sphere with a line through a point on that sphere.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Sphere/SecondInter.html"}, {"id": "Mathlib.Geometry.Euclidean.Simplex", "region_id": "geometry", "micro_elevation": 0.1, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 123.165, "z": 131.326, "size": 0.2, "title": "Simplices in Euclidean spaces.", "summary": "This file defines properties of simplices in a Euclidean space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Simplex.html"}, {"id": "Mathlib.Geometry.Convex.Cone.Face.Basic", "region_id": "geometry", "micro_elevation": 0.1, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 128.981, "z": 130.21, "size": 0.2, "title": "Faces of pointed cones", "summary": "This file defines what it means for a pointed cone to be a face of another pointed cone and establishes basic properties of this relation. A subcone `F` of a cone `C` is a face if any two points in `C` that have a positive combination in `F` are also in `F`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/Cone/Face/Basic.html"}, {"id": "Mathlib.Geometry.Convex.Cone.Pointed", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 3, "macro_tier_score": 0.2513, "macro_tier_override": null, "x": 126.116, "z": 133.207, "size": 0.3537, "title": "Pointed cones", "summary": "A *pointed cone* is defined to be a submodule of a module where the scalars are restricted to be nonnegative. This is equivalent to saying that, as a set, a pointed cone is a convex cone which contains `0`. This is a bundled version of `ConvexCone.Pointed`. We choose the submodule definition as it allows us to use the `Module` API to work with convex cones.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/Cone/Pointed.html"}, {"id": "Mathlib.Geometry.Manifold.LocalDiffeomorph", "region_id": "geometry", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": 142.444, "z": 148.323, "size": 0.2617, "title": "Local diffeomorphisms between manifolds", "summary": "In this file, we define `C^n` local diffeomorphisms between manifolds. A `C^n` map `f : M → N` is a **local diffeomorphism at `x`** iff there are neighbourhoods `s` and `t` of `x` and `f x`, respectively, such that `f` restricts to a diffeomorphism between `s` and `t`. `f` is called a **local diffeomorphism on `s`** iff it is a local diffeomorphism at every `x ∈ s`, and a **local diffeomorphism** iff it is a local…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/LocalDiffeomorph.html"}, {"id": "Mathlib.Geometry.Manifold.Diffeomorph", "region_id": "geometry", "micro_elevation": 0.7, "macro_tier": 3, "macro_tier_score": 0.2515, "macro_tier_override": null, "x": 106.447, "z": 122.808, "size": 0.3652, "title": "Diffeomorphisms", "summary": "This file implements diffeomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Diffeomorph.html"}, {"id": "Mathlib.Geometry.Manifold.DerivationBundle", "region_id": "geometry", "micro_elevation": 0.7, "macro_tier": 0, "macro_tier_score": 0.2501, "macro_tier_override": null, "x": 132.068, "z": 110.772, "size": 0.2478, "title": "Derivation bundle", "summary": "In this file we define the derivations at a point of a manifold on the algebra of smooth functions. Moreover, we define the differential of a function in terms of derivations. The content of this file is not meant to be regarded as an alternative definition to the current tangent bundle but rather as a purely algebraic theory that provides a purely algebraic definition of the Lie algebra for a Lie group. This theory…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/DerivationBundle.html"}, {"id": "Mathlib.Geometry.Manifold.Algebra.SmoothFunctions", "region_id": "geometry", "micro_elevation": 0.65, "macro_tier": 1, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": 110.784, "z": 118.758, "size": 0.2658, "title": "Algebraic structures over `C^n` functions", "summary": "In this file, we define instances of algebraic structures over `C^n` functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Algebra/SmoothFunctions.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Torsion", "region_id": "geometry", "micro_elevation": 0.95, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 155.67, "z": 132.353, "size": 0.2, "title": "Torsion of an affine connection", "summary": "We define the torsion tensor of an affine connection, i.e. a covariant derivative on the tangent bundle `TM` of some manifold `M`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/CovariantDerivative/Torsion.html"}, {"id": "Mathlib.Geometry.Manifold.VectorField.LieBracket", "region_id": "geometry", "micro_elevation": 0.9, "macro_tier": 1, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": 153.82, "z": 127.522, "size": 0.2617, "title": "Lie brackets of vector fields on manifolds", "summary": "We define the Lie bracket of two vector fields, denoted with `VectorField.mlieBracket I V W x`, as the pullback in the manifold of the corresponding notion in the model space (through `extChartAt I x`). The main results are the following: * `VectorField.mpullback_mlieBracket` states that the pullback of the Lie bracket is the Lie bracket of the pullbacks. * `VectorField.leibniz_identity_mlieBracket` is the Leibniz…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorField/LieBracket.html"}, {"id": "Mathlib.Geometry.Euclidean.Similarity", "region_id": "geometry", "micro_elevation": 0.25, "macro_tier": 1, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": 120.851, "z": 137.222, "size": 0.2649, "title": "Triangle Similarity", "summary": "This file contains theorems about similarity of triangles, including conditions for similarity based on sides and angles.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Similarity.html"}, {"id": "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "region_id": "geometry", "micro_elevation": 0.45, "macro_tier": 2, "macro_tier_score": 0.2507, "macro_tier_override": null, "x": 129.654, "z": 145.183, "size": 0.3158, "title": "Smoothness of charts and local structomorphisms", "summary": "We show that the model with corners, charts, extended charts and their inverses are `C^n`, and that local structomorphisms are `C^n` with `C^n` inverses.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/ContMDiff/Atlas.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear", "region_id": "geometry", "micro_elevation": 0.45, "macro_tier": 2, "macro_tier_score": 0.2504, "macro_tier_override": null, "x": 127.787, "z": 117.807, "size": 0.2859, "title": "The groupoid of `C^n`, fiberwise-linear maps", "summary": "This file contains preliminaries for the definition of a `C^n` vector bundle: an associated `StructureGroupoid`, the groupoid of `contMDiffFiberwiseLinear` functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.html"}, {"id": "Mathlib.Geometry.Convex.Hull", "region_id": "geometry", "micro_elevation": 0.15, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 129.831, "z": 128.708, "size": 0.2, "title": "Convex hull", "summary": "This file defines the convex hull of a set in a convex space. `convexHull R s` is the smallest convex set containing `s`. In order theory speak, this is a closure operator.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/Hull.html"}, {"id": "Mathlib.Geometry.Manifold.ContMDiff.Defs", "region_id": "geometry", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.2525, "macro_tier_override": null, "x": 118.518, "z": 132.212, "size": 0.4107, "title": "`C^n` functions between manifolds", "summary": "We define `Cⁿ` functions between manifolds, as functions which are `Cⁿ` in charts, and prove basic properties of these notions. Here, `n` can be finite, or `∞`, or `ω`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/ContMDiff/Defs.html"}, {"id": "Mathlib.Geometry.Manifold.IsManifold.ExtChartAt", "region_id": "geometry", "micro_elevation": 0.2, "macro_tier": 3, "macro_tier_score": 0.2518, "macro_tier_override": null, "x": 128.817, "z": 126.026, "size": 0.3794, "title": "Extended charts in smooth manifolds", "summary": "In a `C^n` manifold with corners with the model `I` on `(E, H)`, the charts take values in the model space `H`. However, we also need to use extended charts taking values in the model vector space `E`. These extended charts are not `OpenPartialHomeomorph` as the target is not open in `E` in general, but we can still register them as `PartialEquiv`s.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/IsManifold/ExtChartAt.html"}, {"id": "Mathlib.Geometry.Manifold.LocalInvariantProperties", "region_id": "geometry", "micro_elevation": 0.15, "macro_tier": 3, "macro_tier_score": 0.2519, "macro_tier_override": null, "x": 129.805, "z": 134.65, "size": 0.3869, "title": "Local properties invariant under a groupoid", "summary": "We study properties of a triple `(g, s, x)` where `g` is a function between two spaces `H` and `H'`, `s` is a subset of `H` and `x` is a point of `H`. Our goal is to register how such a property should behave to make sense in charted spaces modelled on `H` and `H'`. The main examples we have in mind are the properties \"`g` is differentiable at `x` within `s`\", or \"`g` is smooth at `x` within `s`\". We want to develop…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/LocalInvariantProperties.html"}, {"id": "Mathlib.Geometry.Manifold.Sheaf.Basic", "region_id": "geometry", "micro_elevation": 0.2, "macro_tier": 0, "macro_tier_score": 0.2501, "macro_tier_override": null, "x": 120.897, "z": 128.534, "size": 0.2329, "title": "Generic construction of a sheaf from a `LocalInvariantProp` on a manifold", "summary": "This file constructs the sheaf-of-types of functions `f : M → M'` (for charted spaces `M`, `M'`) which satisfy the lifted property `LiftProp P` associated to some locally invariant (in the sense of `StructureGroupoid.LocalInvariantProp`) property `P` on the model spaces of `M` and `M'`. For example, differentiability and smoothness are locally invariant properties in this sense, so this construction can be used to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Sheaf/Basic.html"}, {"id": "Mathlib.Geometry.Convex.Cone.Basic", "region_id": "geometry", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 126.245, "z": 131.664, "size": 0.2783, "title": "Convex cones", "summary": "In an `R`-module `M`, we define a convex cone as a set `s` such that `a • x + b • y ∈ s` whenever `x, y ∈ s` and `a, b > 0`. We prove that convex cones form a `CompleteLattice`, and define their images (`ConvexCone.map`) and preimages (`ConvexCone.comap`) under linear maps. We define pointed, blunt, flat and salient cones, and prove the correspondence between convex cones and ordered modules. We define…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/Cone/Basic.html"}, {"id": "Mathlib.Geometry.Convex.ConvexSpace.AffineSpace", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 125.388, "z": 132.954, "size": 0.2302, "title": "Affine spaces are convex spaces", "summary": "This file shows that every affine space is a convex space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/ConvexSpace/AffineSpace.html"}, {"id": "Mathlib.Geometry.Convex.ConvexSpace.Defs", "region_id": "geometry", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2507, "macro_tier_override": null, "x": 126.245, "z": 131.664, "size": 0.3113, "title": "Convex spaces", "summary": "This file defines convex spaces as an algebraic structure supporting finite convex combinations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/ConvexSpace/Defs.html"}, {"id": "Mathlib.Geometry.Euclidean.PerpBisector", "region_id": "geometry", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2506, "macro_tier_override": null, "x": 126.245, "z": 131.664, "size": 0.3001, "title": "Perpendicular bisector of a segment", "summary": "We define `AffineSubspace.perpBisector p₁ p₂` to be the perpendicular bisector of the segment `[p₁, p₂]`, as a bundled affine subspace. We also prove that a point belongs to the perpendicular bisector if and only if it is equidistant from `p₁` and `p₂`, as well as a few linear equations that define this subspace.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/PerpBisector.html"}, {"id": "Mathlib.Geometry.Manifold.Algebra.Structures", "region_id": "geometry", "micro_elevation": 0.6, "macro_tier": 2, "macro_tier_score": 0.2505, "macro_tier_override": null, "x": 144.183, "z": 126.786, "size": 0.2961, "title": "`C^n` structures", "summary": "In this file we define `C^n` structures that build on Lie groups. We prefer using the term `ContMDiffRing` instead of Lie mainly because Lie ring has currently another use in mathematics.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Algebra/Structures.html"}, {"id": "Mathlib.Geometry.Convex.ConvexSpace.Module", "region_id": "geometry", "micro_elevation": 0.2, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 120.169, "z": 130.445, "size": 0.2302, "title": "Modules are convex spaces", "summary": "This file shows that every module over ordered coefficients is a convex space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/ConvexSpace/Module.html"}, {"id": "Mathlib.Geometry.Convex.Star", "region_id": "geometry", "micro_elevation": 0.15, "macro_tier": 1, "macro_tier_score": 0.2501, "macro_tier_override": null, "x": 129.661, "z": 134.814, "size": 0.2516, "title": "Star-convex sets", "summary": "This file defines star-convex sets in a convex space. A set is star-convex at `x` if every segment from `x` to a point in the set is contained in the set. This is the prototypical example of a contractible set in homotopy theory (by scaling every point towards `x`), but has wider uses. Note that this has nothing to do with star rings, `Star` and co.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/Star.html"}, {"id": "Mathlib.Geometry.Manifold.Sheaf.Smooth", "region_id": "geometry", "micro_elevation": 0.7, "macro_tier": 0, "macro_tier_score": 0.2501, "macro_tier_override": null, "x": 128.4, "z": 110.083, "size": 0.2478, "title": "The sheaf of smooth functions on a manifold", "summary": "The sheaf of `𝕜`-smooth functions from a manifold `M` to a manifold `N` can be defined as a sheaf of types using the construction `StructureGroupoid.LocalInvariantProp.sheaf` from the file `Mathlib/Geometry/Manifold/Sheaf/Basic.lean`. In this file we write that down (a one-liner), then do the work of upgrading this to a sheaf of [groups]/[abelian groups]/[rings]/[commutative rings] when `N` carries more algebraic…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Sheaf/Smooth.html"}, {"id": "Mathlib.Geometry.Manifold.Instances.Real", "region_id": "geometry", "micro_elevation": 0.85, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 104.217, "z": 117.23, "size": 0.2731, "title": "Constructing examples of manifolds over ℝ", "summary": "We introduce the necessary bits to be able to define manifolds modelled over `ℝ^n`, boundaryless or with boundary or with corners. As a concrete example, we construct explicitly the manifold with boundary structure on the real interval `[x, y]`, and prove that its boundary is indeed `{x, y}` whenever `x < y`. As a corollary, a product `M × [x, y]` with a manifold `M` without boundary has boundary `M × {x, y}`. More…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Instances/Real.html"}, {"id": "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "region_id": "geometry", "micro_elevation": 0.4, "macro_tier": 3, "macro_tier_score": 0.2525, "macro_tier_override": null, "x": 132.495, "z": 120.962, "size": 0.4106, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.html"}, {"id": "Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace", "region_id": "geometry", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 143.403, "z": 147.334, "size": 0.2, "title": "Smooth manifolds as locally ringed spaces", "summary": "This file equips a smooth manifold with the structure of a locally ringed space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Sheaf/LocallyRingedSpace.html"}, {"id": "Mathlib.Geometry.Manifold.VectorField.Pullback", "region_id": "geometry", "micro_elevation": 0.85, "macro_tier": 2, "macro_tier_score": 0.2504, "macro_tier_override": null, "x": 112.228, "z": 109.368, "size": 0.2882, "title": "Vector fields in manifolds", "summary": "We study functions of the form `V : Π (x : M), TangentSpace I x` on a manifold, i.e., vector fields. We define the pullback of a vector field under a map, as `VectorField.mpullback I I' f V x := (mfderiv I I' f x).inverse (V (f x))` (together with the same notion within a set). Note that the pullback uses the junk-value pattern: if the derivative of the map is not invertible, then pullback is given the junk value…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorField/Pullback.html"}, {"id": "Mathlib.Geometry.Manifold.ContMDiffMFDeriv", "region_id": "geometry", "micro_elevation": 0.8, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 135.583, "z": 154.623, "size": 0.2746, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/ContMDiffMFDeriv.html"}, {"id": "Mathlib.Geometry.Manifold.SmoothApprox", "region_id": "geometry", "micro_elevation": 0.7, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 104.579, "z": 132.64, "size": 0.2545, "title": "Approximation of continuous functions by smooth functions", "summary": "In this file, we deduce from the existence of smooth partitions of unity that any continuous map from a real σ-compact finite dimensional manifold `M` to a real normed space `F` can be approximated uniformly by smooth functions. More precisely, we strengthen this result in three ways : * instead of a single number `ε > 0`, one may prescribe the precision of the approximation using an arbitrary continuous positive…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/SmoothApprox.html"}, {"id": "Mathlib.Geometry.Euclidean.Inversion.ImageHyperplane", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 126.148, "z": 130.117, "size": 0.2, "title": "Image of a hyperplane under inversion", "summary": "In this file we prove that the inversion with center `c` and radius `R ≠ 0` maps a sphere passing through the center to a hyperplane, and vice versa. More precisely, it maps a sphere with center `y ≠ c` and radius `dist y c` to the hyperplane `AffineSubspace.perpBisector c (EuclideanGeometry.inversion c R y)`. The exact statements are a little more complicated because `EuclideanGeometry.inversion c R` sends the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.html"}, {"id": "Mathlib.Geometry.Euclidean.Incenter", "region_id": "geometry", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.2501, "macro_tier_override": null, "x": 124.534, "z": 124.109, "size": 0.2478, "title": "Incenters and excenters of simplices.", "summary": "This file defines the insphere and exspheres of a simplex (tangent to the faces of the simplex), and the center and radius of such spheres. The terms \"exsphere\", \"excenter\" and \"exradius\" are used in this file in a general sense where a `Finset` `signs` of indices is given that determine, up to negating all the signs, which vertices of the simplex lie on the same side of the opposite face as the excenter and which…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Incenter.html"}, {"id": "Mathlib.Geometry.Euclidean.Altitude", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 1, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": 126.592, "z": 133.173, "size": 0.2633, "title": "Altitudes of a simplex", "summary": "This file defines the altitudes of a simplex and their feet.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Altitude.html"}, {"id": "Mathlib.Geometry.Euclidean.SignedDist", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 0, "macro_tier_score": 0.2501, "macro_tier_override": null, "x": 126.341, "z": 133.21, "size": 0.2329, "title": "Signed distance to an affine subspace in a Euclidean space.", "summary": "This file defines the signed distance between two points, in the direction of a given vector, and the signed distance between an affine subspace and a point, in the direction of a given reference point.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/SignedDist.html"}, {"id": "Mathlib.Geometry.Euclidean.Sphere.Tangent", "region_id": "geometry", "micro_elevation": 0.2, "macro_tier": 1, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": 130.291, "z": 136.357, "size": 0.2649, "title": "Tangency for spheres.", "summary": "This file defines notions of spheres being tangent to affine subspaces and other spheres.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Sphere/Tangent.html"}, {"id": "Mathlib.Geometry.Manifold.HasGroupoid", "region_id": "geometry", "micro_elevation": 0.1, "macro_tier": 3, "macro_tier_score": 0.2523, "macro_tier_override": null, "x": 129.032, "z": 133.017, "size": 0.4022, "title": "Charted spaces with a given structure groupoid", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/HasGroupoid.html"}, {"id": "Mathlib.Geometry.Manifold.ChartedSpace", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 3, "macro_tier_score": 0.2523, "macro_tier_override": null, "x": 126.645, "z": 130.167, "size": 0.4054, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/ChartedSpace.html"}, {"id": "Mathlib.Geometry.Group.Growth.LinearLowerBound", "region_id": "geometry", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 126.245, "z": 131.664, "size": 0.2, "title": "Linear lower bound on the growth of a generating set", "summary": "This file proves that the growth of a set generating an infinite group is at least linear.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Group/Growth/LinearLowerBound.html"}, {"id": "Mathlib.Geometry.Manifold.IsManifold.Basic", "region_id": "geometry", "micro_elevation": 0.15, "macro_tier": 3, "macro_tier_score": 0.2513, "macro_tier_override": null, "x": 121.92, "z": 129.963, "size": 0.3541, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/IsManifold/Basic.html"}, {"id": "Mathlib.Geometry.Manifold.Notation", "region_id": "geometry", "micro_elevation": 0.3, "macro_tier": 3, "macro_tier_score": 0.2524, "macro_tier_override": null, "x": 131.486, "z": 139.34, "size": 0.4097, "title": "Elaborators for differential geometry", "summary": "This file defines custom elaborators for differential geometry to allow for more compact notation. We introduce a class of elaborators for handling differentiability on manifolds, and the elaborator `T%` for converting dependent sections of fibre bundles into non-dependent functions into the total space. All of these elaborators are scoped to the `Manifold` namespace.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Notation.html"}, {"id": "Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation", "region_id": "geometry", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 139.821, "z": 150.522, "size": 0.2, "title": "Left invariant derivations", "summary": "In this file we define the concept of left invariant derivations for a Lie group. The concept is analogous to the more classical concept of left invariant vector fields, and it holds that the derivation associated to a vector field is left invariant iff the field is. Moreover we prove that `LeftInvariantDerivation I G` has the structure of a Lie algebra, hence implementing one of the possible definitions of the Lie…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Algebra/LeftInvariantDerivation.html"}, {"id": "Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential", "region_id": "geometry", "micro_elevation": 0.65, "macro_tier": 2, "macro_tier_score": 0.2508, "macro_tier_override": null, "x": 144.933, "z": 139.167, "size": 0.3174, "title": "Unique derivative sets in manifolds", "summary": "In this file, we prove various properties of unique derivative sets in manifolds. * `image_denseRange`: suppose `f` is differentiable on `s` and its derivative at every point of `s` has dense range. If `s` has the unique differential property, then so does `f '' s`. * `uniqueMDiffOn_preimage`: the unique differential property is preserved by local diffeomorphisms * `uniqueDiffOn_target_inter`: the unique…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/MFDeriv/UniqueDifferential.html"}, {"id": "Mathlib.Geometry.Manifold.MFDeriv.Defs", "region_id": "geometry", "micro_elevation": 0.25, "macro_tier": 3, "macro_tier_score": 0.2512, "macro_tier_override": null, "x": 132.231, "z": 136.579, "size": 0.3444, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/MFDeriv/Defs.html"}, {"id": "Mathlib.Geometry.Manifold.Submersion", "region_id": "geometry", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 133.697, "z": 145.244, "size": 0.2, "title": "Smooth submersions", "summary": "In this file, we define `C^n` submersions between `C^n` manifolds. As in the case of immersions, the correct definition in the infinite-dimensional setting differs from the classical finite-dimensional one (which is usually phrased in terms of surjectivity of the `mfderiv`). Future work will prove that our definition implies the latter, and that both are equivalent for finite-dimensional manifolds. Our definition is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Submersion.html"}, {"id": "Mathlib.Geometry.Manifold.LocalSourceTargetProperty", "region_id": "geometry", "micro_elevation": 0.2, "macro_tier": 0, "macro_tier_score": 0.2501, "macro_tier_override": null, "x": 122.159, "z": 127.005, "size": 0.247, "title": "Local properties of smooth functions which depend on both the source and target", "summary": "In this file, we consider local properties of functions between manifolds, which depend on both the source and the target: more precisely, properties `P` of functions `f : M → N` such that `f` has property `P` if and only if there is a suitable pair of charts on `M` and `N`, respectively, such that `f` read in these charts has a particular form. The motivating examples of this general description are immersions and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/LocalSourceTargetProperty.html"}, {"id": "Mathlib.Geometry.Manifold.MFDeriv.Tangent", "region_id": "geometry", "micro_elevation": 0.75, "macro_tier": 2, "macro_tier_score": 0.2508, "macro_tier_override": null, "x": 103.689, "z": 126.078, "size": 0.3226, "title": "Derivatives of maps in the tangent bundle", "summary": "This file contains properties of derivatives which need the manifold structure of the tangent bundle. Notably, it includes formulas for the tangent maps to charts, and unique differentiability statements for subsets of the tangent bundle.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/MFDeriv/Tangent.html"}, {"id": "Mathlib.Geometry.RingedSpace.Basic", "region_id": "geometry", "micro_elevation": 0.15, "macro_tier": 3, "macro_tier_score": 0.5008, "macro_tier_override": null, "x": 122.663, "z": 128.702, "size": 0.3195, "title": "Ringed spaces", "summary": "We introduce the category of ringed spaces, as an alias for `SheafedSpace CommRingCat`. The facts collected in this file are typically stated for locally ringed spaces, but never actually make use of the locality of stalks. See for instance .", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/RingedSpace/Basic.html"}, {"id": "Mathlib.Geometry.RingedSpace.SheafedSpace", "region_id": "geometry", "micro_elevation": 0.1, "macro_tier": 3, "macro_tier_score": 0.5002, "macro_tier_override": null, "x": 123.598, "z": 133.274, "size": 0.2577, "title": "Sheafed spaces", "summary": "Introduces the category of topological spaces equipped with a sheaf (taking values in an arbitrary target category `C`). We further describe how to apply functors and natural transformations to the values of the presheaves.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/RingedSpace/SheafedSpace.html"}, {"id": "Mathlib.Geometry.Manifold.Immersion", "region_id": "geometry", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.2501, "macro_tier_override": null, "x": 133.105, "z": 117.774, "size": 0.2478, "title": "Smooth immersions", "summary": "In this file, we define `C^n` immersions between `C^n` manifolds. The correct definition in the infinite-dimensional setting differs from the standard finite-dimensional definition (concerning the `mfderiv` being injective): future pull requests will prove that our definition implies the latter, and that both are equivalent for finite-dimensional manifolds. This definition can be conveniently formulated in terms of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Immersion.html"}, {"id": "Mathlib.Geometry.Manifold.IntegralCurve.UniformTime", "region_id": "geometry", "micro_elevation": 0.95, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 153.736, "z": 142.177, "size": 0.2, "title": "Uniform time lemma for the global existence of integral curves", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/IntegralCurve/UniformTime.html"}, {"id": "Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique", "region_id": "geometry", "micro_elevation": 0.9, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 145.926, "z": 111.911, "size": 0.2676, "title": "Existence and uniqueness of integral curves", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/IntegralCurve/ExistUnique.html"}, {"id": "Mathlib.Geometry.Manifold.IntegralCurve.Transform", "region_id": "geometry", "micro_elevation": 0.85, "macro_tier": 1, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": 108.141, "z": 150.789, "size": 0.253, "title": "Translation and scaling of integral curves", "summary": "New integral curves may be constructed by translating or scaling the domain of an existing integral curve. This file mirrors `Mathlib/Analysis/ODE/Transform`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/IntegralCurve/Transform.html"}, {"id": "Mathlib.Geometry.RingedSpace.LocallyRingedSpace", "region_id": "geometry", "micro_elevation": 0.2, "macro_tier": 3, "macro_tier_score": 0.5016, "macro_tier_override": null, "x": 121.858, "z": 127.287, "size": 0.3687, "title": "The category of locally ringed spaces", "summary": "We define (bundled) locally ringed spaces (as `SheafedSpace CommRing` along with the fact that the stalks are local rings), and morphisms between these (morphisms in `SheafedSpace` with `IsLocalHom` on the stalk maps).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/RingedSpace/LocallyRingedSpace.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality", "region_id": "geometry", "micro_elevation": 0.25, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 133.81, "z": 133.326, "size": 0.2, "title": "The Triangle Inequality for Angles", "summary": "This file contains the proof that angles obey the triangle inequality.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Unoriented/TriangleInequality.html"}, {"id": "Mathlib.Geometry.Euclidean.Basic", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 1, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": 126.064, "z": 130.125, "size": 0.2613, "title": "Euclidean spaces", "summary": "This file makes some definitions and proves very basic geometrical results about real inner product spaces and Euclidean affine spaces. Results about real inner product spaces that involve the norm and inner product but not angles generally go in `Analysis.NormedSpace.InnerProduct`. Results with longer proofs or more geometrical content generally go in separate files.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Basic.html"}, {"id": "Mathlib.Geometry.Manifold.IntegralCurve.Basic", "region_id": "geometry", "micro_elevation": 0.8, "macro_tier": 2, "macro_tier_score": 0.2504, "macro_tier_override": null, "x": 151.026, "z": 131.179, "size": 0.2833, "title": "Integral curves of vector fields on a manifold", "summary": "Let `M` be a manifold and `v : (x : M) → TangentSpace I x` be a vector field on `M`. An integral curve of `v` is a function `γ : ℝ → M` such that the derivative of `γ` at `t` equals `v (γ t)`. The integral curve may only be defined for all `t` within some subset of `ℝ`. This is the first of a series of files, organised as follows: * `Mathlib/Geometry/Manifold/IntegralCurve/Basic.lean` (this file): Basic definitions…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/IntegralCurve/Basic.html"}, {"id": "Mathlib.Geometry.Manifold.Metrizable", "region_id": "geometry", "micro_elevation": 0.2, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 122.12, "z": 127.039, "size": 0.2, "title": "Metrizability of a σ-compact manifold", "summary": "In this file we show that a σ-compact Hausdorff topological manifold over a finite-dimensional real vector space is metrizable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Metrizable.html"}, {"id": "Mathlib.Geometry.Manifold.MFDeriv.Atlas", "region_id": "geometry", "micro_elevation": 0.6, "macro_tier": 2, "macro_tier_score": 0.2509, "macro_tier_override": null, "x": 137.837, "z": 146.196, "size": 0.3271, "title": "Differentiability of models with corners and (extended) charts", "summary": "In this file, we analyse the differentiability of charts, models with corners and extended charts. We show that * models with corners are differentiable * charts are differentiable on their source * `mdifferentiableOn_extChartAt`: `extChartAt` is differentiable on its source Suppose an open partial homeomorphism `e` is differentiable. This file shows * `OpenPartialHomeomorph.MDifferentiable.mfderiv`: its derivative…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/MFDeriv/Atlas.html"}, {"id": "Mathlib.Geometry.Convex.Cone.Simplicial", "region_id": "geometry", "micro_elevation": 0.1, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 127.981, "z": 134.23, "size": 0.2338, "title": "Simplicial cones", "summary": "A **simplicial cone** is a pointed convex cone that equals the conic hull of a finite linearly independent set of vectors. We do not require that the generators span the ambient module. However, when the cone is also generating, its generators linearly span the module.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/Cone/Simplicial.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable", "region_id": "geometry", "micro_elevation": 0.65, "macro_tier": 2, "macro_tier_score": 0.2506, "macro_tier_override": null, "x": 109.762, "z": 143.235, "size": 0.3003, "title": "Differentiability of functions in vector bundles", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/MDifferentiable.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Oriented.Basic", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 1, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 124.756, "z": 131.236, "size": 0.2685, "title": "Oriented angles.", "summary": "This file defines oriented angles in real inner product spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.Riemannian", "region_id": "geometry", "micro_elevation": 0.75, "macro_tier": 1, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": 149.375, "z": 129.438, "size": 0.2617, "title": "Riemannian vector bundles", "summary": "Given a vector bundle over a manifold whose fibers are all endowed with a scalar product, we say that this bundle is Riemannian if the scalar product depends smoothly on the base point. We introduce a typeclass `[IsContMDiffRiemannianBundle IB n F E]` registering this property. Under this assumption, we show that the scalar product of two smooth maps into the same fibers of the bundle is a smooth function. If the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/Riemannian.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.LocalFrame", "region_id": "geometry", "micro_elevation": 0.7, "macro_tier": 0, "macro_tier_score": 0.2501, "macro_tier_override": null, "x": 117.855, "z": 151.663, "size": 0.2416, "title": "Local frames in a vector bundle", "summary": "Let `V → M` be a finite rank smooth vector bundle with standard fiber `F`. A family of sections `s i` of `V → M` is called a **C^k local frame** on a set `U ⊆ M` iff each section `s i` is `C^k` on `U`, and the section values `s i x` form a basis for each `x ∈ U`. We define a predicate `IsLocalFrame` for a collection of sections to be a local frame on a set, and define basic notions (such as the coefficients of a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/LocalFrame.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Unoriented.CrossProduct", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 125.008, "z": 130.73, "size": 0.2, "title": "Norm of cross-products", "summary": "This file proves `InnerProductGeometry.norm_withLpEquiv_crossProduct`, relating the norm of the cross-product of two real vectors with their individual norms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Unoriented/CrossProduct.html"}, {"id": "Mathlib.Geometry.Manifold.ConformalGroupoid", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 127.792, "z": 131.739, "size": 0.2, "title": "Conformal Groupoid", "summary": "In this file we define the groupoid of conformal maps on normed spaces.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/ConformalGroupoid.html"}, {"id": "Mathlib.Geometry.Manifold.Riemannian.Basic", "region_id": "geometry", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 130.049, "z": 100.915, "size": 0.2, "title": "Riemannian manifolds", "summary": "A Riemannian manifold `M` is a real manifold such that its tangent spaces are endowed with an inner product, depending smoothly on the point, and such that `M` has an emetric space structure for which the distance is the infimum of lengths of paths. We register a Prop-valued typeclass `IsRiemannianManifold I M` recording this fact, building on top of `[EMetricSpace M] [RiemannianBundle (fun (x : M) ↦ TangentSpace I…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Riemannian/Basic.html"}, {"id": "Mathlib.Geometry.Manifold.Riemannian.PathELength", "region_id": "geometry", "micro_elevation": 0.95, "macro_tier": 0, "macro_tier_score": 0.2501, "macro_tier_override": null, "x": 97.429, "z": 125.665, "size": 0.239, "title": "Lengths of paths in manifolds", "summary": "Consider a manifold in which the tangent spaces have an enormed structure. Then one defines `pathELength γ a b` as the length of the path `γ : ℝ → M` between `a` and `b`, i.e., the integral of the norm of its derivative on `Icc a b`. We give several ways to write this quantity (as an integral over `Icc`, or `Ioo`, or the subtype `Icc`, using either `mfderiv` or `mfderivWithin`). We show that this notion is invariant…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Riemannian/PathELength.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Incenter", "region_id": "geometry", "micro_elevation": 0.3, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 121.935, "z": 123.428, "size": 0.2, "title": "Angles and incenters and excenters.", "summary": "This file proves lemmas relating incenters and excenters of a simplex to angle bisection.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Incenter.html"}, {"id": "Mathlib.Geometry.Convex.Cone.TensorProduct", "region_id": "geometry", "micro_elevation": 0.15, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 130.375, "z": 129.534, "size": 0.2338, "title": "Tensor products of cones", "summary": "Given ordered modules `M` and `N`, there are in general several distinct possible orderings of the tensor product module `M ⊗ N`. Since the ordering of an ordered module can be represented by its cone of nonnegative elements, there are likewise multiple ways to construct a cone in `M ⊗ N` from cones in `M` and `N`. Such constructions are referred to as tensor products of cones. \"Sufficiently nice\" candidates for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Convex/Cone/TensorProduct.html"}, {"id": "Mathlib.Geometry.Manifold.Complex", "region_id": "geometry", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 113.247, "z": 123.234, "size": 0.2, "title": "Holomorphic functions on complex manifolds", "summary": "Thanks to the rigidity of complex-differentiability compared to real-differentiability, there are many results about complex manifolds with no analogue for manifolds over a general normed field. For now, this file contains just two (closely related) such results:", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Complex.html"}, {"id": "Mathlib.Geometry.Manifold.WhitneyEmbedding", "region_id": "geometry", "micro_elevation": 0.9, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 99.562, "z": 139.761, "size": 0.2, "title": "Whitney embedding theorem", "summary": "In this file we prove a version of the Whitney embedding theorem: for any compact real manifold `M`, for sufficiently large `n` there exists a smooth embedding `M → ℝ^n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/WhitneyEmbedding.html"}, {"id": "Mathlib.Geometry.Manifold.SmoothEmbedding", "region_id": "geometry", "micro_elevation": 0.75, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 141.393, "z": 114.043, "size": 0.2, "title": "Smooth embeddings", "summary": "In this file, we define `C^n` embeddings between `C^n` manifolds. This will be useful to define embedded submanifolds.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/SmoothEmbedding.html"}, {"id": "Mathlib.Geometry.Group.Growth.QuotientInter", "region_id": "geometry", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 126.245, "z": 131.664, "size": 0.2, "title": "Growth in the quotient and intersection with a subgroup", "summary": "For a group `G` and a subgroup `H ≤ G`, this file gives upper and lower bounds on the growth of a finset by its growth in `H` and `G ⧸ H`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Group/Growth/QuotientInter.html"}, {"id": "Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 3, "macro_tier_score": 0.5001, "macro_tier_override": null, "x": 126.879, "z": 130.251, "size": 0.2496, "title": "`PresheafedSpace C` has colimits.", "summary": "If `C` has limits, then the category `PresheafedSpace C` has colimits, and the forgetful functor to `TopCat` preserves these colimits. When restricted to a diagram where the underlying continuous maps are open embeddings, this says that we can glue presheafed spaces. Given a diagram `F : J ⥤ PresheafedSpace C`, we first build the colimit of the underlying topological spaces, as `colimit (F ⋙ PresheafedSpace.forget…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.html"}, {"id": "Mathlib.Geometry.Manifold.PoincareConjecture", "region_id": "geometry", "micro_elevation": 0.95, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 96.952, "z": 128.791, "size": 0.2, "title": "Statement of the generalized Poincaré conjecture", "summary": "https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conjecture The mathlib notation `≃ₕ` stands for a homotopy equivalence, `≃ₜ` stands for a homeomorphism, and `≃ₘ⟮𝓡 n, 𝓡 n⟯` stands for a diffeomorphism, where `𝓡 n` is the `n`-dimensional Euclidean space viewed as a model space.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/PoincareConjecture.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Metric", "region_id": "geometry", "micro_elevation": 0.85, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 132.009, "z": 105.967, "size": 0.2, "title": "Metric connections", "summary": "This file defines connections on a Riemannian vector bundle which are compatible with the ambient metric. A bundled connection `∇` on a Riemannian vector bundle `(V, g)` is compatible with the metric `g` if and only if the differentiated metric tensor `∇ g` (defined by `(X, σ, τ) ↦ 𝓛_X g(σ, τ) - g(∇_X σ, τ) - g(σ, ∇_X τ)`) vanishes on all differentiable vector fields `X` and differentiable sections `σ`, `τ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/CovariantDerivative/Metric.html"}, {"id": "Mathlib.Geometry.Manifold.Instances.Icc", "region_id": "geometry", "micro_elevation": 0.9, "macro_tier": 0, "macro_tier_score": 0.2501, "macro_tier_override": null, "x": 127.988, "z": 103.834, "size": 0.2342, "title": "Manifold structure on real intervals", "summary": "The manifold structure on real intervals is defined in `Mathlib.Geometry.Manifold.Instances.Real`. We relate it to the manifold structure on the real line, by showing that the inclusion (`contMDiff_subtype_coe_Icc`) and projection (`contMDiffOn_projIcc`) are smooth, and showing that a function defined on the interval is smooth iff its composition with the projection is smooth on the interval in `ℝ` (see…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Instances/Icc.html"}, {"id": "Mathlib.Geometry.Polygon.Basic", "region_id": "geometry", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 126.245, "z": 131.664, "size": 0.2, "title": "Polygons", "summary": "This file defines polygons in affine spaces. For the special case `n = 3`, an interconversion is provided with `Affine.Triangle`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Polygon/Basic.html"}, {"id": "Mathlib.Geometry.Euclidean.Angle.Unoriented.Conformal", "region_id": "geometry", "micro_elevation": 0.05, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 127.061, "z": 130.347, "size": 0.2, "title": "Angles and conformal maps", "summary": "This file proves that conformal maps preserve angles.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Angle/Unoriented/Conformal.html"}, {"id": "Mathlib.Geometry.Manifold.GroupLieAlgebra", "region_id": "geometry", "micro_elevation": 0.95, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 132.288, "z": 160.47, "size": 0.2, "title": "The Lie algebra of a Lie group", "summary": "Given a Lie group, we define `GroupLieAlgebra I G` as its tangent space at the identity, and we endow it with a Lie bracket, as follows. Given two vectors `v, w : GroupLieAlgebra I G`, consider the associated left-invariant vector fields `mulInvariantVectorField v` (given at a point `g` by the image of `v` under the derivative of left-multiplication by `g`) and `mulInvariantVectorField w`. Then take their Lie…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/GroupLieAlgebra.html"}, {"id": "Mathlib.Geometry.Manifold.Instances.Quotient", "region_id": "geometry", "micro_elevation": 0.1, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 127.426, "z": 128.799, "size": 0.2, "title": "Quotients of manifolds", "summary": "This file contains results about quotients of manifolds by group actions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/Instances/Quotient.html"}, {"id": "Mathlib.Geometry.Manifold.VectorBundle.SmoothSection", "region_id": "geometry", "micro_elevation": 0.65, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 146.151, "z": 134.712, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/VectorBundle/SmoothSection.html"}, {"id": "Mathlib.Geometry.Manifold.BumpFunction", "region_id": "geometry", "micro_elevation": 0.6, "macro_tier": 1, "macro_tier_score": 0.2502, "macro_tier_override": null, "x": 139.746, "z": 118.885, "size": 0.2626, "title": "Smooth bump functions on a smooth manifold", "summary": "In this file we define `SmoothBumpFunction I c` to be a bundled smooth \"bump\" function centered at `c`. It is a structure that consists of two real numbers `0 < rIn < rOut` with small enough `rOut`. We define a coercion to function for this type, and for `f : SmoothBumpFunction I c`, the function `⇑f` written in the extended chart at `c` has the following properties: * `f x = 1` in the closed ball of radius `f.rIn`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Manifold/BumpFunction.html"}, {"id": "Mathlib.Geometry.Euclidean.Sphere.OrthRadius", "region_id": "geometry", "micro_elevation": 0.15, "macro_tier": 2, "macro_tier_score": 0.2505, "macro_tier_override": null, "x": 129.826, "z": 128.702, "size": 0.2901, "title": "Spaces orthogonal to the radius vector in spheres.", "summary": "This file defines the affine subspace orthogonal to the radius vector at a point.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Geometry/Euclidean/Sphere/OrthRadius.html"}, {"id": "Mathlib.Computability.Ackermann", "region_id": "computability", "micro_elevation": 0.5714, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 126.108, "z": 182.482, "size": 0.2, "title": "Ackermann function", "summary": "In this file, we define the two-argument Ackermann function `ack`. Despite having a recursive definition, we show that this isn't a primitive recursive function.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/Ackermann.html"}, {"id": "Mathlib.Computability.PartrecCode", "region_id": "computability", "micro_elevation": 0.4286, "macro_tier": 2, "macro_tier_score": 0.309, "macro_tier_override": null, "x": 128.473, "z": 186.566, "size": 0.352, "title": "Gödel Numbering for Partial Recursive Functions.", "summary": "This file defines `Nat.Partrec.Code`, an inductive datatype describing code for partial recursive functions on ℕ. It defines an encoding for these codes, and proves that the constructors are primitive recursive with respect to the encoding. It also defines the evaluation of these codes as partial functions using `PFun`, and proves that a function is partially recursive (as defined by `Nat.Partrec`) if and only if it…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/PartrecCode.html"}, {"id": "Mathlib.Computability.TuringDegree", "region_id": "computability", "micro_elevation": 0.5714, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 132.036, "z": 193.448, "size": 0.2, "title": "Turing degrees", "summary": "This file defines Turing reducibility and equivalence, proves that Turing equivalence is an equivalence relation, and defines Turing degrees as the quotient under this relation.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/TuringDegree.html"}, {"id": "Mathlib.Computability.RecursiveIn", "region_id": "computability", "micro_elevation": 0.4286, "macro_tier": 1, "macro_tier_score": 0.0387, "macro_tier_override": null, "x": 129.948, "z": 189.36, "size": 0.2676, "title": "Oracle computability", "summary": "This file defines oracle computability using partial recursive functions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/RecursiveIn.html"}, {"id": "Mathlib.Computability.Language", "region_id": "computability", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2314, "macro_tier_override": null, "x": 123.466, "z": 190.995, "size": 0.3048, "title": "Languages", "summary": "This file contains the definition and operations on formal languages over an alphabet. Note that \"strings\" are implemented as lists over the alphabet. Union and concatenation define a [Kleene algebra](https://en.wikipedia.org/wiki/Kleene_algebra) over the languages. In addition to that, we define a reversal of a language and prove that it behaves well with respect to other language operations.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/Language.html"}, {"id": "Mathlib.Computability.Halting", "region_id": "computability", "micro_elevation": 0.7143, "macro_tier": 1, "macro_tier_score": 0.1157, "macro_tier_override": null, "x": 124.819, "z": 202.056, "size": 0.279, "title": "Computability theory and the halting problem", "summary": "A universal partial recursive function, Rice's theorem, and the halting problem.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/Halting.html"}, {"id": "Mathlib.Computability.RE", "region_id": "computability", "micro_elevation": 0.5714, "macro_tier": 2, "macro_tier_score": 0.1544, "macro_tier_override": null, "x": 128.284, "z": 183.496, "size": 0.2992, "title": "Computable and Recursively Enumerable Predicates", "summary": "This file defines computable (`ComputablePred`) and recursively enumerable (`REPred`) predicates. It also provides basic closure properties and Post's theorem on the equivalence of recursive, r.e., and co-r.e. sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/RE.html"}, {"id": "Mathlib.Computability.TMConfig", "region_id": "computability", "micro_elevation": 0.8571, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 124.963, "z": 204.283, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/TMConfig.html"}, {"id": "Mathlib.Computability.TuringMachine.Config", "region_id": "computability", "micro_elevation": 0.7143, "macro_tier": 1, "macro_tier_score": 0.1157, "macro_tier_override": null, "x": 129.786, "z": 200.172, "size": 0.279, "title": "Modelling partial recursive functions using Turing machines", "summary": "The files `Config` and `ToPartrec` define a simplified basis for partial recursive functions, and a `Turing.TM2` model Turing machine for evaluating these functions. This amounts to a constructive proof that every `Partrec` function can be evaluated by a Turing machine.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/TuringMachine/Config.html"}, {"id": "Mathlib.Computability.ContextFreeGrammar", "region_id": "computability", "micro_elevation": 0.1429, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 125.694, "z": 191.002, "size": 0.2, "title": "Context-Free Grammars", "summary": "This file contains the definition of a context-free grammar, which is a grammar that has a single nonterminal symbol on the left-hand side of each rule. We restrict nonterminals of a context-free grammar to `Type` because universe polymorphism would be cumbersome and unnecessary; we can always restrict a context-free grammar to the finitely many nonterminal symbols that are referred to by its finitely many rules.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/ContextFreeGrammar.html"}, {"id": "Mathlib.Computability.Primrec", "region_id": "computability", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 127.834, "z": 191.88, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/Primrec.html"}, {"id": "Mathlib.Computability.Primrec.List", "region_id": "computability", "micro_elevation": 0.1429, "macro_tier": 2, "macro_tier_score": 0.5008, "macro_tier_override": null, "x": 121.274, "z": 191.397, "size": 0.3233, "title": "Primitive recursive functions on Lists", "summary": "The primitive recursive functions are defined in `Mathlib.Computability.Primrec.Basic`. This file contains definitions and theorems about primitive recursive functions that relate to operation on lists.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/Primrec/List.html"}, {"id": "Mathlib.Computability.Encoding", "region_id": "computability", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0772, "macro_tier_override": null, "x": 123.466, "z": 190.995, "size": 0.2742, "title": "Encodings", "summary": "This file contains the definition of an encoding, a map from a type to strings in an alphabet, used in defining computability by Turing machines. It also contains several examples:", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/Encoding.html"}, {"id": "Mathlib.Computability.Primrec.Basic", "region_id": "computability", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.5005, "macro_tier_override": null, "x": 123.466, "z": 190.995, "size": 0.2932, "title": "The primitive recursive functions", "summary": "The primitive recursive functions are the least collection of functions `ℕ → ℕ` which are closed under projections (using the `pair` pairing function), composition, zero, successor, and primitive recursion (i.e. `Nat.rec` where the motive is `C n := ℕ`). We can extend this definition to a large class of basic types by using canonical encodings of types as natural numbers (Gödel numbering), which we implement through…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/Primrec/Basic.html"}, {"id": "Mathlib.Computability.DFA", "region_id": "computability", "micro_elevation": 0.1429, "macro_tier": 1, "macro_tier_score": 0.1157, "macro_tier_override": null, "x": 122.29, "z": 192.888, "size": 0.2743, "title": "Deterministic Finite Automata", "summary": "A Deterministic Finite Automaton (DFA) is a state machine which decides membership in a particular `Language`, by following a path uniquely determined by an input string. We define regular languages to be ones for which a DFA exists, other formulations are later proved equivalent. Note that this definition allows for automata with infinite states, a `Fintype` instance must be supplied for true DFAs.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/DFA.html"}, {"id": "Mathlib.Computability.TMComputable", "region_id": "computability", "micro_elevation": 0.5714, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 125.817, "z": 182.397, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/TMComputable.html"}, {"id": "Mathlib.Computability.TuringMachine.Computable", "region_id": "computability", "micro_elevation": 0.4286, "macro_tier": 1, "macro_tier_score": 0.0387, "macro_tier_override": null, "x": 129.835, "z": 188.963, "size": 0.2676, "title": "Computable functions", "summary": "This file contains the definition of a Turing machine with some finiteness conditions (bundling the definition of TM2 in `StackTuringMachine.lean`), a definition of when a TM gives a certain output (in a certain time), and the definition of computability (in polynomial time or any time function) of a function between two types that have an encoding (as in `Encoding.lean`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/TuringMachine/Computable.html"}, {"id": "Mathlib.Computability.Reduce", "region_id": "computability", "micro_elevation": 0.8571, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 132.937, "z": 200.435, "size": 0.2, "title": "Strong reducibility and degrees.", "summary": "This file defines the notions of computable many-one reduction and one-one reduction between sets, and shows that the corresponding degrees form a semilattice.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/Reduce.html"}, {"id": "Mathlib.Computability.EpsilonNFA", "region_id": "computability", "micro_elevation": 0.4286, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 116.865, "z": 189.933, "size": 0.2, "title": "Epsilon Nondeterministic Finite Automata", "summary": "This file contains the definition of an epsilon Nondeterministic Finite Automaton (`εNFA`), a state machine which determines whether a string (implemented as a list over an arbitrary alphabet) is in a regular set by evaluating the string over every possible path, also having access to ε-transitions, which can be followed without reading a character. Since this definition allows for automata with infinite states, a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/EpsilonNFA.html"}, {"id": "Mathlib.Computability.NFA", "region_id": "computability", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0386, "macro_tier_override": null, "x": 127.653, "z": 192.524, "size": 0.2478, "title": "Nondeterministic Finite Automata", "summary": "A Nondeterministic Finite Automaton (NFA) is a state machine which decides membership in a particular `Language`, by following every possible path that describes an input string. We show that DFAs and NFAs can decide the same languages, by constructing an equivalent DFA for every NFA, and vice versa. As constructing a DFA from an NFA uses an exponential number of states, we re-prove the pumping lemma instead of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/NFA.html"}, {"id": "Mathlib.Computability.AkraBazzi.AkraBazzi", "region_id": "computability", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 122.391, "z": 186.67, "size": 0.2, "title": "Divide-and-conquer recurrences and the Akra-Bazzi theorem", "summary": "A divide-and-conquer recurrence is a function `T : ℕ → ℝ` that satisfies a recurrence relation of the form `T(n) = ∑_{i=0}^{k-1} a_i T(r_i(n)) + g(n)` for sufficiently large `n`, where `r_i(n)` is a function such that `‖r_i(n) - b_i n‖ ∈ o(n / (log n)^2)` for every `i`, the coefficients `a_i` are positive, and the coefficients `b_i` are real numbers in `(0, 1)`. (This assumption can be relaxed to `O(n / (log…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/AkraBazzi/AkraBazzi.html"}, {"id": "Mathlib.Computability.AkraBazzi.SumTransform", "region_id": "computability", "micro_elevation": 0.1429, "macro_tier": 0, "macro_tier_score": 0.0386, "macro_tier_override": null, "x": 125.595, "z": 190.337, "size": 0.2478, "title": "Akra-Bazzi theorem: the sum transform", "summary": "We develop further preliminaries required for the theorem, up to the sum transform.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/AkraBazzi/SumTransform.html"}, {"id": "Mathlib.Computability.RegularExpressions", "region_id": "computability", "micro_elevation": 0.1429, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 121.313, "z": 191.571, "size": 0.2, "title": "Regular Expressions", "summary": "This file contains the formal definition for regular expressions and basic lemmas. Note these are regular expressions in terms of formal language theory. Note this is different to regexes used in computer science such as the POSIX standard.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/RegularExpressions.html"}, {"id": "Mathlib.Computability.TuringMachine", "region_id": "computability", "micro_elevation": 0.4286, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 124.464, "z": 197.606, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/TuringMachine.html"}, {"id": "Mathlib.Computability.TuringMachine.StackTuringMachine", "region_id": "computability", "micro_elevation": 0.2857, "macro_tier": 1, "macro_tier_score": 0.1159, "macro_tier_override": null, "x": 120.134, "z": 193.955, "size": 0.2952, "title": "Turing machines", "summary": "The files `PostTuringMachine.lean` and `StackTuringMachine.lean` define a sequence of simple machine languages, starting with Turing machines and working up to more complex languages based on Wang B-machines. `PostTuringMachine.lean` covers the TM0 model and TM1 model; `StackTuringMachine.lean` adds the TM2 model.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/TuringMachine/StackTuringMachine.html"}, {"id": "Mathlib.Computability.PostTuringMachine", "region_id": "computability", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 127.349, "z": 188.808, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/PostTuringMachine.html"}, {"id": "Mathlib.Computability.TuringMachine.PostTuringMachine", "region_id": "computability", "micro_elevation": 0.1429, "macro_tier": 2, "macro_tier_score": 0.193, "macro_tier_override": null, "x": 124.102, "z": 193.131, "size": 0.312, "title": "Turing machines", "summary": "The files `PostTuringMachine.lean` and `StackTuringMachine.lean` define a sequence of simple machine languages, starting with Turing machines and working up to more complex languages based on Wang B-machines. `PostTuringMachine.lean` covers the TM0 model and TM1 model; `StackTuringMachine.lean` adds the TM2 model.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/TuringMachine/PostTuringMachine.html"}, {"id": "Mathlib.Computability.PartrecBasis", "region_id": "computability", "micro_elevation": 0.5714, "macro_tier": 1, "macro_tier_score": 0.1157, "macro_tier_override": null, "x": 117.079, "z": 184.777, "size": 0.2701, "title": "A simplified basis for partial recursive functions", "summary": "This file defines `Nat.Partrec'`, an inductive predicate that provides an alternative, structural basis for partial recursive functions using vectors. It establishes the equivalence between this vector-based definition and the standard `Partrec` definition.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/PartrecBasis.html"}, {"id": "Mathlib.Computability.TMToPartrec", "region_id": "computability", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 112.753, "z": 179.656, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/TMToPartrec.html"}, {"id": "Mathlib.Computability.TuringMachine.ToPartrec", "region_id": "computability", "micro_elevation": 0.8571, "macro_tier": 1, "macro_tier_score": 0.0387, "macro_tier_override": null, "x": 130.616, "z": 179.696, "size": 0.2676, "title": "Modelling partial recursive functions using Turing machines", "summary": "The files `Config` and `ToPartrec` define a simplified basis for partial recursive functions, and a `Turing.TM2` model Turing machine for evaluating these functions. This amounts to a constructive proof that every `Partrec` function can be evaluated by a Turing machine.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/TuringMachine/ToPartrec.html"}, {"id": "Mathlib.Computability.AkraBazzi.GrowsPolynomially", "region_id": "computability", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0386, "macro_tier_override": null, "x": 123.466, "z": 190.995, "size": 0.2465, "title": "Akra-Bazzi theorem: the polynomial growth condition", "summary": "This file defines and develops an API for the polynomial growth condition that appears in the statement of the Akra-Bazzi theorem: for the theorem to hold, the function `g` must satisfy the condition that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b*n` and `n` for any constant `b ∈ (0,1)`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/AkraBazzi/GrowsPolynomially.html"}, {"id": "Mathlib.Computability.Tape", "region_id": "computability", "micro_elevation": 0.1429, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 123.534, "z": 188.768, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/Tape.html"}, {"id": "Mathlib.Computability.TuringMachine.Tape", "region_id": "computability", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.2313, "macro_tier_override": null, "x": 123.466, "z": 190.995, "size": 0.298, "title": "Turing machine tapes", "summary": "This file defines the notion of a Turing machine tape, and the operations on it. A tape is a bidirectional infinite sequence of cells, each of which stores an element of a given alphabet `Γ`. All but finitely many of the cells are required to hold the blank symbol `default : Γ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/TuringMachine/Tape.html"}, {"id": "Mathlib.Computability.Partrec", "region_id": "computability", "micro_elevation": 0.2857, "macro_tier": 2, "macro_tier_score": 0.4249, "macro_tier_override": null, "x": 119.051, "z": 191.605, "size": 0.3801, "title": "The partial recursive functions", "summary": "The partial recursive functions are defined similarly to the primitive recursive functions, but now all functions are partial, implemented using the `Part` monad, and there is an additional operation, called μ-recursion, which performs unbounded minimization: `μ f` returns the least natural number `n` for which `f n = 0`, or diverges if such `n` doesn't exist.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/Partrec.html"}, {"id": "Mathlib.Computability.StateTransition", "region_id": "computability", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.1926, "macro_tier_override": null, "x": 123.466, "z": 190.995, "size": 0.271, "title": "State Transition Systems", "summary": "This file contains simple definitions and lemmas for reasoning about state transition systems defined by a function `σ → Option σ`, where `σ` is the type of states.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/StateTransition.html"}, {"id": "Mathlib.Computability.MyhillNerode", "region_id": "computability", "micro_elevation": 0.2857, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 127.711, "z": 192.352, "size": 0.2, "title": "Myhill–Nerode theorem", "summary": "This file proves the Myhill–Nerode theorem using left quotients. Given a language `L` and a word `x`, the *left quotient* of `L` by `x` is the set of suffixes `y` such that `x ++ y` is in `L`. The *Myhill–Nerode theorem* shows that each left quotient, in fact, corresponds to the state of an automaton that matches `L`, and that `L` is regular if and only if there are finitely many such states.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/Computability/MyhillNerode.html"}, {"id": "Mathlib.ModelTheory.Skolem", "region_id": "model_theory", "micro_elevation": 0.6364, "macro_tier": 2, "macro_tier_score": 0.1603, "macro_tier_override": null, "x": 158.409, "z": -167.107, "size": 0.2713, "title": "Skolem Functions and Downward Löwenheim–Skolem", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Skolem.html"}, {"id": "Mathlib.ModelTheory.ElementarySubstructures", "region_id": "model_theory", "micro_elevation": 0.5455, "macro_tier": 2, "macro_tier_score": 0.2208, "macro_tier_override": null, "x": 164.113, "z": -158.073, "size": 0.3184, "title": "Elementary Substructures", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/ElementarySubstructures.html"}, {"id": "Mathlib.ModelTheory.Ultraproducts", "region_id": "model_theory", "micro_elevation": 0.4545, "macro_tier": 2, "macro_tier_score": 0.1603, "macro_tier_override": null, "x": 171.073, "z": -159.3, "size": 0.2713, "title": "Ultraproducts and Łoś's Theorem", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Ultraproducts.html"}, {"id": "Mathlib.ModelTheory.Quotients", "region_id": "model_theory", "micro_elevation": 0.3636, "macro_tier": 3, "macro_tier_score": 0.2402, "macro_tier_override": null, "x": 163.042, "z": -168.012, "size": 0.2632, "title": "Quotients of First-Order Structures", "summary": "This file defines prestructures and quotients of first-order structures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Quotients.html"}, {"id": "Mathlib.ModelTheory.LanguageMap", "region_id": "model_theory", "micro_elevation": 0.0909, "macro_tier": 3, "macro_tier_score": 0.4805, "macro_tier_override": null, "x": 169.076, "z": -166.152, "size": 0.2947, "title": "Language Maps", "summary": "Maps between first-order languages in the style of the [Flypitch project](https://flypitch.github.io/), as well as several important maps between structures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/LanguageMap.html"}, {"id": "Mathlib.ModelTheory.Basic", "region_id": "model_theory", "micro_elevation": 0.0, "macro_tier": 3, "macro_tier_score": 0.5007, "macro_tier_override": null, "x": 167.908, "z": -165.421, "size": 0.3101, "title": "Basics on First-Order Structures", "summary": "This file defines first-order languages and structures in the style of the [Flypitch project](https://flypitch.github.io/), as well as several important maps between structures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Basic.html"}, {"id": "Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs", "region_id": "model_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.02, "macro_tier_override": null, "x": 167.908, "z": -165.421, "size": 0.2221, "title": "Linear and semilinear sets", "summary": "This file defines linear and semilinear sets. In an `AddCommMonoid`, a linear set is a coset of a finitely generated additive submonoid, and a semilinear set is a finite union of linear sets. We prove that semilinear sets are closed under union, projection, set addition and additive closure. We also prove that any semilinear set can be decomposed into a finite union of proper linear sets, which are linear sets with…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Arithmetic/Presburger/Semilinear/Defs.html"}, {"id": "Mathlib.ModelTheory.DirectLimit", "region_id": "model_theory", "micro_elevation": 0.5455, "macro_tier": 2, "macro_tier_score": 0.0602, "macro_tier_override": null, "x": 163.676, "z": -172.525, "size": 0.2549, "title": "Direct Limits of First-Order Structures", "summary": "This file constructs the direct limit of a directed system of first-order embeddings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/DirectLimit.html"}, {"id": "Mathlib.ModelTheory.FinitelyGenerated", "region_id": "model_theory", "micro_elevation": 0.4545, "macro_tier": 2, "macro_tier_score": 0.0801, "macro_tier_override": null, "x": 164.959, "z": -171.649, "size": 0.2422, "title": "Finitely Generated First-Order Structures", "summary": "This file defines what it means for a first-order (sub)structure to be finitely or countably generated, similarly to other finitely-generated objects in the algebra library.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/FinitelyGenerated.html"}, {"id": "Mathlib.ModelTheory.Types", "region_id": "model_theory", "micro_elevation": 0.8182, "macro_tier": 1, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": 168.88, "z": -153.055, "size": 0.239, "title": "Type Spaces", "summary": "This file defines the space of complete types over a first-order theory. (Note that types in model theory are different from types in type theory.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Types.html"}, {"id": "Mathlib.ModelTheory.Satisfiability", "region_id": "model_theory", "micro_elevation": 0.7273, "macro_tier": 2, "macro_tier_score": 0.141, "macro_tier_override": null, "x": 158.987, "z": -171.901, "size": 0.3368, "title": "First-Order Satisfiability", "summary": "This file deals with the satisfiability of first-order theories, as well as equivalence over them.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Satisfiability.html"}, {"id": "Mathlib.ModelTheory.Algebra.Field.Basic", "region_id": "model_theory", "micro_elevation": 0.4545, "macro_tier": 1, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": 161.351, "z": -167.542, "size": 0.2445, "title": "The First-Order Theory of Fields", "summary": "This file defines the first-order theory of fields as a theory over the language of rings.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Algebra/Field/Basic.html"}, {"id": "Mathlib.ModelTheory.Algebra.Ring.Basic", "region_id": "model_theory", "micro_elevation": 0.3636, "macro_tier": 1, "macro_tier_score": 0.0203, "macro_tier_override": null, "x": 165.656, "z": -160.389, "size": 0.2767, "title": "First-Order Language of Rings", "summary": "This file defines the first-order language of rings, as well as defining instance of `Add`, `Mul`, etc. on terms in the language.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Algebra/Ring/Basic.html"}, {"id": "Mathlib.ModelTheory.Substructures", "region_id": "model_theory", "micro_elevation": 0.3636, "macro_tier": 3, "macro_tier_score": 0.3208, "macro_tier_override": null, "x": 170.937, "z": -160.815, "size": 0.3204, "title": "First-Order Substructures", "summary": "This file defines substructures of first-order structures in a similar manner to the various substructures appearing in the algebra library.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Substructures.html"}, {"id": "Mathlib.ModelTheory.Semantics", "region_id": "model_theory", "micro_elevation": 0.2727, "macro_tier": 3, "macro_tier_score": 0.4213, "macro_tier_override": null, "x": 166.04, "z": -161.732, "size": 0.3513, "title": "Basics on First-Order Semantics", "summary": "This file defines the interpretations of first-order terms, formulas, sentences, and theories in a style inspired by the [Flypitch project](https://flypitch.github.io/).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Semantics.html"}, {"id": "Mathlib.ModelTheory.Encoding", "region_id": "model_theory", "micro_elevation": 0.2727, "macro_tier": 3, "macro_tier_score": 0.3402, "macro_tier_override": null, "x": 172.015, "z": -165.899, "size": 0.2648, "title": "Encodings and Cardinality of First-Order Syntax", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Encoding.html"}, {"id": "Mathlib.ModelTheory.Complexity", "region_id": "model_theory", "micro_elevation": 0.9091, "macro_tier": 0, "macro_tier_score": 0.02, "macro_tier_override": null, "x": 173.419, "z": -152.789, "size": 0.2276, "title": "Quantifier Complexity", "summary": "This file defines quantifier complexity of first-order formulas, and constructs prenex normal forms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Complexity.html"}, {"id": "Mathlib.ModelTheory.Equivalence", "region_id": "model_theory", "micro_elevation": 0.8182, "macro_tier": 2, "macro_tier_score": 0.0403, "macro_tier_override": null, "x": 163.326, "z": -153.894, "size": 0.2722, "title": "Equivalence of Formulas", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Equivalence.html"}, {"id": "Mathlib.ModelTheory.Algebra.Ring.FreeCommRing", "region_id": "model_theory", "micro_elevation": 0.4545, "macro_tier": 1, "macro_tier_score": 0.0202, "macro_tier_override": null, "x": 163.778, "z": -159.904, "size": 0.26, "title": "Making a term in the language of rings from an element of the FreeCommRing", "summary": "This file defines the function `FirstOrder.Ring.termOfFreeCommRing` which constructs a `Language.ring.Term α` from an element of `FreeCommRing α`. The theorem `FirstOrder.Ring.realize_termOfFreeCommRing` shows that the term constructed when realized in a ring `R` is equal to the lift of the element of `FreeCommRing α` to `R`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Algebra/Ring/FreeCommRing.html"}, {"id": "Mathlib.ModelTheory.Algebra.Ring.Definability", "region_id": "model_theory", "micro_elevation": 0.5455, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 170.179, "z": -157.47, "size": 0.2478, "title": "Definable Subsets in the language of rings", "summary": "This file proves that the set of zeros of a multivariable polynomial is a definable subset.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Algebra/Ring/Definability.html"}, {"id": "Mathlib.ModelTheory.Definability", "region_id": "model_theory", "micro_elevation": 0.3636, "macro_tier": 3, "macro_tier_score": 0.2606, "macro_tier_override": null, "x": 162.691, "z": -167.204, "size": 0.3064, "title": "Definable Sets", "summary": "This file defines what it means for a set over a first-order structure to be definable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Definability.html"}, {"id": "Mathlib.ModelTheory.ElementaryMaps", "region_id": "model_theory", "micro_elevation": 0.4545, "macro_tier": 3, "macro_tier_score": 0.2405, "macro_tier_override": null, "x": 164.793, "z": -171.568, "size": 0.2905, "title": "Elementary Maps Between First-Order Structures", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/ElementaryMaps.html"}, {"id": "Mathlib.ModelTheory.Order", "region_id": "model_theory", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 153.928, "z": -159.554, "size": 0.2, "title": "Ordered First-Ordered Structures", "summary": "This file defines ordered first-order languages and structures, as well as their theories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Order.html"}, {"id": "Mathlib.ModelTheory.Fraisse", "region_id": "model_theory", "micro_elevation": 0.7273, "macro_tier": 0, "macro_tier_score": 0.02, "macro_tier_override": null, "x": 168.118, "z": -176.444, "size": 0.2276, "title": "Fraïssé Classes and Fraïssé Limits", "summary": "This file pertains to the ages of countable first-order structures. The age of a structure is the class of all finitely-generated structures that embed into it. Of particular interest are Fraïssé classes, which are exactly the ages of countable ultrahomogeneous structures. To each is associated a unique (up to nonunique isomorphism) Fraïssé limit - the countable ultrahomogeneous structure with that age.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Fraisse.html"}, {"id": "Mathlib.ModelTheory.Syntax", "region_id": "model_theory", "micro_elevation": 0.1818, "macro_tier": 3, "macro_tier_score": 0.4615, "macro_tier_override": null, "x": 165.348, "z": -166.444, "size": 0.363, "title": "Basics on First-Order Syntax", "summary": "This file defines first-order terms, formulas, sentences, and theories in a style inspired by the [Flypitch project](https://flypitch.github.io/).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Syntax.html"}, {"id": "Mathlib.ModelTheory.PartialEquiv", "region_id": "model_theory", "micro_elevation": 0.6364, "macro_tier": 1, "macro_tier_score": 0.0401, "macro_tier_override": null, "x": 167.469, "z": -175.058, "size": 0.2417, "title": "Partial Isomorphisms", "summary": "This file defines partial isomorphisms between first-order structures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/PartialEquiv.html"}, {"id": "Mathlib.ModelTheory.Bundled", "region_id": "model_theory", "micro_elevation": 0.6364, "macro_tier": 2, "macro_tier_score": 0.1804, "macro_tier_override": null, "x": 167.056, "z": -175.031, "size": 0.2826, "title": "Bundled First-Order Structures", "summary": "This file bundles types together with their first-order structure.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Bundled.html"}, {"id": "Mathlib.ModelTheory.Graph", "region_id": "model_theory", "micro_elevation": 0.8182, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 164.301, "z": -153.553, "size": 0.2, "title": "First-Order Structures in Graph Theory", "summary": "This file defines first-order languages, structures, and theories in graph theory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Graph.html"}, {"id": "Mathlib.ModelTheory.Algebra.Field.IsAlgClosed", "region_id": "model_theory", "micro_elevation": 0.8182, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 178.231, "z": -158.544, "size": 0.2478, "title": "The First-Order Theory of Algebraically Closed Fields", "summary": "This file defines the theory of algebraically closed fields of characteristic `p`, as well as proving completeness of the theory and the Lefschetz Principle.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Algebra/Field/IsAlgClosed.html"}, {"id": "Mathlib.ModelTheory.Algebra.Field.CharP", "region_id": "model_theory", "micro_elevation": 0.5455, "macro_tier": 1, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": 162.091, "z": -171.298, "size": 0.2403, "title": "First-order theory of fields", "summary": "This file defines the first-order theory of fields of characteristic `p` as a theory over the language of rings", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Algebra/Field/CharP.html"}, {"id": "Mathlib.ModelTheory.Arithmetic.Presburger.Basic", "region_id": "model_theory", "micro_elevation": 0.3636, "macro_tier": 1, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": 172.281, "z": -168.777, "size": 0.239, "title": "Presburger arithmetic", "summary": "This file defines the first-order language of Presburger arithmetic as (0,1,+).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Arithmetic/Presburger/Basic.html"}, {"id": "Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic", "region_id": "model_theory", "micro_elevation": 0.0909, "macro_tier": 1, "macro_tier_score": 0.0201, "macro_tier_override": null, "x": 168.087, "z": -166.787, "size": 0.239, "title": "Semilinear sets are closed under intersection, set difference and complement", "summary": "This file proves that the semilinear sets in any commutative monoid are closed under intersection and set difference. They are also closed under complement if the monoid is finitely generated. We prove these results on `ℕ ^ k` first (which are private APIs in this file) and then generalize to any commutative monoid.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Arithmetic/Presburger/Semilinear/Basic.html"}, {"id": "Mathlib.ModelTheory.Arithmetic.Presburger.Definability", "region_id": "model_theory", "micro_elevation": 0.4545, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 161.088, "z": -166.412, "size": 0.2, "title": "Presburger definability and semilinear sets", "summary": "This file formalizes the classical result that Presburger definable sets are the same as semilinear sets. As an application of this result, we show that the graph of multiplication is not Presburger definable.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Arithmetic/Presburger/Definability.html"}, {"id": "Mathlib.ModelTheory.Topology.Types", "region_id": "model_theory", "micro_elevation": 0.9091, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 155.591, "z": -171.606, "size": 0.2, "title": "Topology on the space of complete types", "summary": "This file defines a topological structure on the type `CompleteType T α` (note that these are types from model theory and not types from type theory). The topology is generated by sets of the form `{p : CompleteType T α | ∃ φ, φ ∈ p}`. Note that the contents of this file are separate from `Mathlib/ModelTheory/Types.lean` to avoid importing files from the Topology folder there.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/ModelTheory/Topology/Types.html"}, {"id": "Mathlib.InformationTheory.KullbackLeibler.KLFun", "region_id": "information_theory", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.5004, "macro_tier_override": null, "x": 137.897, "z": 76.141, "size": 0.2806, "title": "The real function `fun x ↦ x * log x + 1 - x`", "summary": "We define `klFun x = x * log x + 1 - x`. That function is notable because the Kullback-Leibler divergence is an f-divergence for `klFun`. That is, the Kullback-Leibler divergence is an integral of `klFun` composed with a Radon-Nikodym derivative. For probability measures, any function `f` that differs from `klFun` by an affine function of the form `x ↦ a * (x - 1)` would give the same value for the integral `∫ x, f…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/InformationTheory/KullbackLeibler/KLFun.html"}, {"id": "Mathlib.InformationTheory.KullbackLeibler.ChainRule", "region_id": "information_theory", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 136.013, "z": 70.057, "size": 0.2, "title": "Chain rule for the Kullback-Leibler divergence", "summary": "Suppose that we have two finite joint measures on a product `𝓧 × 𝓨`, which can be decomposed as `μ ⊗ₘ κ` and `ν ⊗ₘ η`, where `μ` and `ν` are measures on `𝓧` and `κ` and `η` are Markov kernels from `𝓧` to `𝓨`. Then we can express the Kullback-Leibler divergence between these two joint measures as a sum of `klDiv μ ν` and the conditional Kullback-Leibler divergence between the kernels `κ` and `η`, averaged over `μ`.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/InformationTheory/KullbackLeibler/ChainRule.html"}, {"id": "Mathlib.InformationTheory.KullbackLeibler.Basic", "region_id": "information_theory", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.5001, "macro_tier_override": null, "x": 138.919, "z": 73.125, "size": 0.2478, "title": "Kullback-Leibler divergence", "summary": "The Kullback-Leibler divergence is a measure of the difference between two measures.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/InformationTheory/KullbackLeibler/Basic.html"}, {"id": "Mathlib.InformationTheory.Coding.KraftMcMillan", "region_id": "information_theory", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 140.825, "z": 77.392, "size": 0.2, "title": "Kraft-McMillan Inequality", "summary": "This file proves the Kraft-McMillan inequality for uniquely decodable codes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/InformationTheory/Coding/KraftMcMillan.html"}, {"id": "Mathlib.InformationTheory.Coding.UniquelyDecodable", "region_id": "information_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.5001, "macro_tier_override": null, "x": 137.897, "z": 76.141, "size": 0.2478, "title": "Uniquely Decodable Codes", "summary": "This file defines uniquely decodable codes and proves basic properties.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/InformationTheory/Coding/UniquelyDecodable.html"}, {"id": "Mathlib.InformationTheory.Hamming", "region_id": "information_theory", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 137.897, "z": 76.141, "size": 0.2, "title": "Hamming spaces", "summary": "The Hamming metric counts the number of places two members of a (finite) Pi type differ. The Hamming norm is the same as the Hamming metric over additive groups, and counts the number of places a member of a (finite) Pi type differs from zero. This is a useful notion in various applications, but in particular it is relevant in coding theory, in which it is fundamental for defining the minimum distance of a code.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/InformationTheory/Hamming.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Homology.Nondegenerate", "region_id": "algebraic_topology", "micro_elevation": 0.7778, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 125.208, "z": -98.892, "size": 0.2, "title": "Computing homology using nondegenerate simplices", "summary": "In this file, we introduce the normalized chain complex `X.normalizedChainComplex R` of a simplicial set `X` with coefficients in `R` (where `R` is an object of a preadditive category `C` with coproducts). The `n`-chains of this complex identify to the coproduct of copies of `R` indexed by the nondegenerate `n`-simplices of `X`. In particular, we deduce that the homology is zero in degree `≥ d` when `X` has…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Homology/Nondegenerate.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Homology.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0718, "macro_tier_override": null, "x": 142.688, "z": -92.636, "size": 0.2826, "title": "Simplicial homology", "summary": "In this file, we define the homology of simplicial sets. For any preadditive category `C` with coproducts of size `w` and any object `R : C`, the simplicial chain complex of a simplicial set `X` is denoted `X.chainComplex R`, and its homology in degree `n : ℕ` is `X.homology R n`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Homology/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Splitting", "region_id": "algebraic_topology", "micro_elevation": 0.3333, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 139.304, "z": -112.589, "size": 0.2302, "title": "The splitting of a simplicial set", "summary": "Let `X` be a simplicial set. The fact that any simplex `x : X _⦋n⦌` can be written in a unique way as `X.map f.op y` for an epimorphism `f : ⦋n⦌ ⟶ ⦋m⦌` and a nondegenerate simplex `y : X _⦋m⦌` is translated in this file as the data of a splitting of `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Splitting.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject", "region_id": "algebraic_topology", "micro_elevation": 0.7222, "macro_tier": 3, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 172.481, "z": -106.519, "size": 0.2561, "title": "Split simplicial objects in preadditive categories", "summary": "In this file we define a functor `nondegComplex : SimplicialObject.Split C ⥤ ChainComplex C ℕ` when `C` is a preadditive category with finite coproducts, and get an isomorphism `toKaroubiNondegComplexFunctorIsoN₁ : nondegComplex ⋙ toKaroubi _ ≅ forget C ⋙ DoldKan.N₁`. (See `Equivalence.lean` for the general strategy of proof of the Dold-Kan equivalence.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.SubcomplexOp", "region_id": "algebraic_topology", "micro_elevation": 0.2778, "macro_tier": 4, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 157.565, "z": -104.798, "size": 0.2793, "title": "The opposite of a subcomplex", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/SubcomplexOp.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Op", "region_id": "algebraic_topology", "micro_elevation": 0.2222, "macro_tier": 4, "macro_tier_score": 0.2514, "macro_tier_override": null, "x": 147.066, "z": -100.679, "size": 0.3613, "title": "The covariant involution of the category of simplicial sets", "summary": "In this file, we define the covariant involution `opFunctor : SSet ⥤ SSet` of the category of simplicial sets that is induced by the covariant involution `SimplexCategory.op : SimplexCategory ⥤ SimplexCategory`. We use an abbreviation `X.op` for `opFunctor.obj X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Op.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Subcomplex", "region_id": "algebraic_topology", "micro_elevation": 0.2222, "macro_tier": 4, "macro_tier_score": 0.252, "macro_tier_override": null, "x": 150.29, "z": -100.541, "size": 0.3892, "title": "Subcomplexes of a simplicial set", "summary": "Given a simplicial set `X`, this file defines the type `X.Subcomplex` of subcomplexes of `X` as an abbreviation for `Subfunctor X`. It also introduces a coercion from `X.Subcomplex` to `SSet`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Subcomplex.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PushoutProduct", "region_id": "algebraic_topology", "micro_elevation": 0.8889, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 161.102, "z": -133.96, "size": 0.2, "title": "Anodyne extensions and pushout-products, fibrations and pullbacks", "summary": "The main result in this file is that if `i : X₁ ⟶ Y₁` is a monomorphism in `SSet` and `j : X₂ ⟶ Y₂` is an anodyne extension, then the map from the pushout-product of `i` and `j` into `Y₁ ⊗ Y₂` is an anodyne extension (`SSet.anodyneExtensions_pushoutObjObjι`). This is closely related to the lemma `SSet.fibration_pullbackObjObjπ` which says that if `i : X₁ ⟶ Y₁` is a monomorphism and `p : E ⟶ B` is a fibration, then…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/PushoutProduct.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.8333, "macro_tier": 1, "macro_tier_score": 0.0359, "macro_tier_override": null, "x": 122.152, "z": -103.504, "size": 0.2571, "title": "Anodyne extensions", "summary": "Anodyne extensions form a property of morphisms in the category of simplicial sets. It contains horn inclusions and it is closed under coproducts, pushouts, transfinite compositions and retracts. Equivalently, using the small object argument, anodyne extensions can be defined (and are defined here) as the class of morphisms that satisfy the left lifting property with respect to the class of fibrations (for the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd", "region_id": "algebraic_topology", "micro_elevation": 0.7778, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 144.576, "z": -82.708, "size": 0.2376, "title": "A pairing for the pushout-product of a horn inclusion and a boundary inclusion", "summary": "Let `l : Fin (m + 2)` and `n : ℕ`. In this file, we construct a regular pairing for the subcomplex `unionProd Λ[m + 1, l] ∂Δ[n]` of `Δ[m + 1] ⊗ Δ[n]`. It follows immediately that the inclusion of the union of `Λ[m + 1, l] ⊗ Δ[n]` and `Δ[m + 1] ⊗ ∂Δ[n]` in `Δ[m + 1] ⊗ Δ[n]` is a (strong) anodyne extension (which is inner when `l ≠ 0` and `l ≠ Fin.last _`). The main construction works only when `l ≠ Fin.last _`, i.e.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/UnionProd.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.KanComplex", "region_id": "algebraic_topology", "micro_elevation": 0.6667, "macro_tier": 3, "macro_tier_score": 0.0723, "macro_tier_override": null, "x": 127.505, "z": -110.9, "size": 0.3242, "title": "Kan complexes", "summary": "In this file, the abbreviation `KanComplex` is introduced for fibrant objects in the category `SSet` which is equipped with Kan fibrations. In `Mathlib/AlgebraicTopology/Quasicategory/Basic.lean` we show that every Kan complex is a quasicategory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/KanComplex.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.PushoutProduct", "region_id": "algebraic_topology", "micro_elevation": 0.5556, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 144.589, "z": -125.219, "size": 0.2376, "title": "Pushout-products of simplicial sets", "summary": "Results about pushout-products and pullback-homs in the category of simplicial sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/PushoutProduct.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.CompStruct", "region_id": "algebraic_topology", "micro_elevation": 0.2778, "macro_tier": 4, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 151.315, "z": -98.918, "size": 0.2695, "title": "Edges and \"triangles\" in simplicial sets", "summary": "Given a simplicial set `X`, we introduce two types: * Given `0`-simplices `x₀` and `x₁`, we define `Edge x₀ x₁` which is the type of `1`-simplices with faces `x₁` and `x₀` respectively; * Given `0`-simplices `x₀`, `x₁`, `x₂`, edges `e₀₁ : Edge x₀ x₁`, `e₁₂ : Edge x₁ x₂`, `e₀₂ : Edge x₀ x₂`, a structure `CompStruct e₀₁ e₁₂ e₀₂` which records the data of a `2`-simplex with faces `e₁₂`, `e₀₂` and `e₀₁` respectively.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/CompStruct.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.CompStructTruncated", "region_id": "algebraic_topology", "micro_elevation": 0.2222, "macro_tier": 4, "macro_tier_score": 0.2507, "macro_tier_override": null, "x": 152.632, "z": -101.41, "size": 0.3149, "title": "Edges and \"triangles\" in truncated simplicial sets", "summary": "Given a `2`-truncated simplicial set `X`, we introduce two types: * Given `0`-simplices `x₀` and `x₁`, we define `Edge x₀ x₁` which is the type of `1`-simplices with faces `x₁` and `x₀` respectively; * Given `0`-simplices `x₀`, `x₁`, `x₂`, edges `e₀₁ : Edge x₀ x₁`, `e₁₂ : Edge x₁ x₂`, `e₀₂ : Edge x₀ x₂`, a structure `CompStruct e₀₁ e₁₂ e₀₂` which records the data of a `2`-simplex with faces `e₁₂`, `e₀₂` and `e₀₁`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/CompStructTruncated.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Subdivision", "region_id": "algebraic_topology", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 152.475, "z": -91.751, "size": 0.2, "title": "The subdivision functors", "summary": "In this file, we define the subdivision functor `sd : SSet ⥤ SSet` and its right adjoint `ex`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Subdivision.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex", "region_id": "algebraic_topology", "micro_elevation": 0.4444, "macro_tier": 4, "macro_tier_score": 0.2526, "macro_tier_override": null, "x": 146.147, "z": -93.462, "size": 0.418, "title": "The standard simplex", "summary": "We define the standard simplices `Δ[n]` as simplicial sets. See files `SimplicialSet.Boundary` and `SimplicialSet.Horn` for their boundaries `∂Δ[n]` and horns `Λ[n, i]`. (The notations are available via `open Simplicial`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/StdSimplex.html"}, {"id": "Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 2, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 149.478, "z": -102.254, "size": 0.2676, "title": "Epi-mono factorization in the simplex category presented by generators and relations", "summary": "This file aims to establish that there is a nice epi-mono factorization in `SimplexCategoryGenRel`. More precisely, we introduce two morphism properties `P_δ` and `P_σ` that single out morphisms that are compositions of `δ i` (resp. `σ i`). The main result of this file is `exists_P_σ_P_δ_factorization`, which asserts that every morphism as a decomposition of a `P_σ` followed by a `P_δ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/EpiMono.html"}, {"id": "Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.1111, "macro_tier": 2, "macro_tier_score": 0.0362, "macro_tier_override": null, "x": 145.448, "z": -108.452, "size": 0.2918, "title": "Presentation of the simplex category by generators and relations.", "summary": "We introduce `SimplexCategoryGenRel` as the category presented by generating morphisms `δ i : [n] ⟶ [n + 1]` and `σ i : [n + 1] ⟶ [n]` and subject to the simplicial identities, and we provide induction principles for reasoning about objects and morphisms in this category. This category admits a canonical functor `toSimplexCategory` to the usual simplex category. The fact that this functor is an equivalence will be…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.AlternatingFaceMapComplex", "region_id": "algebraic_topology", "micro_elevation": 0.2222, "macro_tier": 4, "macro_tier_score": 0.1796, "macro_tier_override": null, "x": 150.429, "z": -100.567, "size": 0.3365, "title": "The alternating face map complex of a simplicial object in a preadditive category", "summary": "We construct the alternating face map complex, as a functor `alternatingFaceMapComplex : SimplicialObject C ⥤ ChainComplex C ℕ` for any preadditive category `C`. For any simplicial object `X` in `C`, this is the homological complex `... → X_2 → X_1 → X_0` where the differentials are alternating sums of faces. The dual version `alternatingCofaceMapComplex : CosimplicialObject C ⥤ CochainComplex C ℕ` is also…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Inner.PushoutProduct", "region_id": "algebraic_topology", "micro_elevation": 0.9444, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 128.526, "z": -130.645, "size": 0.2, "title": "Inner anodyne extensions and pushout-products, inner fibrations and pullbacks", "summary": "This file is mirrored from `SSet/AnodyneExtensions/PushoutProduct`. The main result in this file is that if `i : X₁ ⟶ Y₁` is a monomorphism in `SSet` and `j : X₂ ⟶ Y₂` is an inner anodyne extension, then the pushout-product of `i` and `j` is an inner anodyne extension (`SSet.innerAnodyneExtensions_pushoutObjObjι`). This is closely related to the lemma `SSet.innerFibration_pullbackObjObjπ` which says that if `i : X₁…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/Inner/PushoutProduct.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Inner.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.8889, "macro_tier": 0, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 120.89, "z": -100.624, "size": 0.2276, "title": "Inner anodyne extensions", "summary": "Much of this file is mirrored from `Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Basic`. *Inner* anodyne extensions form a property of morphisms in the category of simplicial sets. It contains *inner* horn inclusions and it is closed under coproducts, pushouts, transfinite compositions and retracts. Equivalently, using the small object argument, inner anodyne extensions can be defined (and are defined…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/Inner/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.CechNerve", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 149.836, "z": -102.299, "size": 0.2431, "title": "The Čech Nerve", "summary": "This file provides a definition of the Čech nerve associated to an arrow, provided the base category has the correct wide pullbacks. Several variants are provided, given `f : Arrow C`: 1. `f.cechNerve` is the Čech nerve, considered as a simplicial object in `C`. 2. `f.augmentedCechNerve` is the augmented Čech nerve, considered as an augmented simplicial object in `C`. 3. `SimplicialObject.cechNerve` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/CechNerve.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialObject.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.1111, "macro_tier": 4, "macro_tier_score": 0.5052, "macro_tier_override": null, "x": 145.591, "z": -106.391, "size": 0.5061, "title": "Simplicial objects in a category.", "summary": "A simplicial object in a category `C` is a `C`-valued presheaf on `SimplexCategory`. (Similarly, a cosimplicial object is a functor `SimplexCategory ⥤ C`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialObject/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Finite", "region_id": "algebraic_topology", "micro_elevation": 0.3889, "macro_tier": 4, "macro_tier_score": 0.2506, "macro_tier_override": null, "x": 153.512, "z": -119.492, "size": 0.3081, "title": "Finite simplicial sets", "summary": "A simplicial set is finite (`SSet.Finite`) if it has finitely many nondegenerate simplices.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Finite.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.NerveNondegenerate", "region_id": "algebraic_topology", "micro_elevation": 0.3889, "macro_tier": 4, "macro_tier_score": 0.2505, "macro_tier_override": null, "x": 139.83, "z": -98.899, "size": 0.2945, "title": "The nondegenerate simplices in the nerve of a partially ordered type", "summary": "In this file, we show that if `X` is a partially ordered type, then an `n`-simplex `s` of the nerve is nondegenerate iff the monotone map `s.obj : Fin (n + 1) → X` is strictly monotone.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/NerveNondegenerate.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.FunctorN", "region_id": "algebraic_topology", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.0721, "macro_tier_override": null, "x": 131.13, "z": -104.67, "size": 0.3112, "title": "Construction of functors N for the Dold-Kan correspondence", "summary": "In this file, we construct functors `N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ)` and `N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ)` for any preadditive category `C`. (The indices of these functors are the number of occurrences of `Karoubi` at the source or the target.) In the case `C` is additive, the functor `N₂` shall be the functor of the equivalence…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/FunctorN.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.PInfty", "region_id": "algebraic_topology", "micro_elevation": 0.5, "macro_tier": 4, "macro_tier_score": 0.073, "macro_tier_override": null, "x": 133.008, "z": -104.465, "size": 0.3657, "title": "Construction of the projection `PInfty` for the Dold-Kan correspondence", "summary": "In this file, we construct the projection `PInfty : K[X] ⟶ K[X]` by passing to the limit the projections `P q` defined in `Projections.lean`. This projection is a critical tool in this formalisation of the Dold-Kan correspondence, because in the case of abelian categories, `PInfty` corresponds to the projection on the normalized Moore subcomplex, with kernel the degenerate subcomplex. (See `Equivalence.lean` for the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/PInfty.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.Homotopy", "region_id": "algebraic_topology", "micro_elevation": 0.3333, "macro_tier": 2, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 152.57, "z": -97.412, "size": 0.2751, "title": "Homotopies in model categories", "summary": "In this file, we relate left and right homotopies between morphisms `X ⟶ Y` in model categories. In particular, if `X` is cofibrant and `Y` is fibrant, these notions coincide (for arbitrary choices of good cylinders or good path objects). Using the factorization lemma by K. S. Brown, we deduce versions of the Whitehead theorem (`LeftHomotopyClass.whitehead` and `RightHomotopyClass.whitehead`) which assert that when…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/Homotopy.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.BrownLemma", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 2, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 144.553, "z": -110.803, "size": 0.2683, "title": "The factorization lemma by K. S. Brown", "summary": "In a model category, any morphism `f : X ⟶ Y` between cofibrant objects can be factored as `i ≫ p` with `i` a cofibration and `p` a trivial fibration which has a section `s` that is a cofibration. In order to state this, we introduce a structure `CofibrantBrownFactorization f` with the data of such morphisms `i`, `p` and `s` with the expected properties, and show it is nonempty. Moreover, if `f` is a weak…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/BrownLemma.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.RightHomotopy", "region_id": "algebraic_topology", "micro_elevation": 0.2778, "macro_tier": 2, "macro_tier_score": 0.0364, "macro_tier_override": null, "x": 157.15, "z": -103.769, "size": 0.3135, "title": "Right homotopies in model categories", "summary": "We introduce the types `PrepathObject.RightHomotopy` and `PathObject.RightHomotopy` of homotopies between morphisms `X ⟶ Y` relative to a (pre)path object of `Y`. Given two morphisms `f` and `g`, we introduce the relation `RightHomotopyRel f g` asserting the existence of a path object `P` and a right homotopy `P.RightHomotopy f g`, and we define the quotient type `RightHomotopyClass X Y`. We show that if `Y` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/RightHomotopy.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.IsUniquelyCodimOneFace", "region_id": "algebraic_topology", "micro_elevation": 0.3333, "macro_tier": 2, "macro_tier_score": 0.0361, "macro_tier_override": null, "x": 138.761, "z": -103.988, "size": 0.2819, "title": "Simplices that are uniquely codimensional one faces", "summary": "Let `X` be a simplicial set. If `x : X _⦋d⦌` and `y : X _⦋d + 1⦌`, we say that `x` is uniquely a `1`-codimensional face of `y` if there exists a unique `i : Fin (d + 2)` such that `X.δ i y = x`. In this file, we extend this to a predicate `IsUniquelyCodimOneFace` involving two terms in the type `X.S` of simplices of `X`. This is used in the file…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/IsUniquelyCodimOneFace.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Simplices", "region_id": "algebraic_topology", "micro_elevation": 0.2778, "macro_tier": 4, "macro_tier_score": 0.2509, "macro_tier_override": null, "x": 156.298, "z": -112.984, "size": 0.3284, "title": "The preordered type of simplices of a simplicial set", "summary": "In this file, we define the type `X.S` of simplices of a simplicial set `X`, where a simplex consists of the data of `dim : ℕ` and `simplex : X _⦋dim⦌`. We endow this type with a preorder defined by `x ≤ y ↔ Subcomplex.ofSimplex x.simplex ≤ Subcomplex.ofSimplex y.simplex`. In particular, as a preordered type, `X.S` is a category, but this is not what is called \"the category of simplices of `X`\" in the literature…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Simplices.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Pairing", "region_id": "algebraic_topology", "micro_elevation": 0.4444, "macro_tier": 2, "macro_tier_score": 0.0363, "macro_tier_override": null, "x": 159.351, "z": -97.557, "size": 0.3024, "title": "Pairings", "summary": "In this file, we introduce the definition of a pairing for a subcomplex `A` of a simplicial set `X`, following the ideas by Sean Moss, *Another approach to the Kan-Quillen model structure*, who gave a complete combinatorial characterization of strong (inner) anodyne extensions. Strong (inner) anodyne extensions are transfinite compositions of pushouts of coproducts of (inner) horn inclusions, i.e. this is similar to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/Pairing.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplicesSubcomplex", "region_id": "algebraic_topology", "micro_elevation": 0.3889, "macro_tier": 2, "macro_tier_score": 0.0361, "macro_tier_override": null, "x": 139.799, "z": -116.391, "size": 0.2819, "title": "The type of nondegenerate simplices not in a subcomplex", "summary": "In this file, given a subcomplex `A` of a simplicial set `X`, we introduce the type `A.N` of nondegenerate simplices of `X` that are not in `A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/NonDegenerateSimplicesSubcomplex.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 4, "macro_tier_score": 0.2524, "macro_tier_override": null, "x": 149.827, "z": -113.024, "size": 0.4068, "title": "Simplicial sets", "summary": "A simplicial set is just a simplicial object in `Type`, i.e. a `Type`-valued presheaf on the simplex category. (One might be tempted to call these \"simplicial types\" when working in type-theoretic foundations, but this would be unnecessarily confusing given the existing notion of a simplicial type in homotopy type theory.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialObject.Coskeletal", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 3, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 143.928, "z": -109.65, "size": 0.2552, "title": "Coskeletal simplicial objects", "summary": "The identity natural transformation exhibits a simplicial object `X` as a right extension of its restriction along `(Truncated.inclusion n).op` recorded by `rightExtensionInclusion X n`. The simplicial object `X` is *n-coskeletal* if `rightExtensionInclusion X n` is a right Kan extension. When the ambient category admits right Kan extensions along `(Truncated.inclusion n).op`, then when `X` is `n`-coskeletal, the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialObject/Coskeletal.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.NReflectsIso", "region_id": "algebraic_topology", "micro_elevation": 0.6111, "macro_tier": 2, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 163.384, "z": -93.916, "size": 0.2722, "title": "N₁ and N₂ reflect isomorphisms", "summary": "In this file, it is shown that the functors `N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ)` and `N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ)` reflect isomorphisms for any preadditive category `C`. (See `Equivalence.lean` for the general strategy of proof of the Dold-Kan equivalence.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/NReflectsIso.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.Decomposition", "region_id": "algebraic_topology", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.0717, "macro_tier_override": null, "x": 163.986, "z": -117.781, "size": 0.2744, "title": "Decomposition of the Q endomorphisms", "summary": "In this file, we obtain a lemma `decomposition_Q` which expresses explicitly the projection `(Q q).f (n+1) : X _⦋n+1⦌ ⟶ X _⦋n+1⦌` (`X : SimplicialObject C` with `C` a preadditive category) as a sum of terms which are postcompositions with degeneracies. (TODO @joelriou: when `C` is abelian, define the degenerate subcomplex of the alternating face map complex of `X` and show that it is a complement to the normalized…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/Decomposition.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Presentable", "region_id": "algebraic_topology", "micro_elevation": 0.6667, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 148.35, "z": -129.37, "size": 0.2382, "title": "Finite simplicial sets are presentable", "summary": "In this file, we show that finite simplicial sets are finitely presentable, which will allow the use of the small object argument in `SSet`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Presentable.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.FiniteColimits", "region_id": "algebraic_topology", "micro_elevation": 0.4444, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 135.311, "z": -112.435, "size": 0.2341, "title": "Finite colimits of finite simplicial sets are finite", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/FiniteColimits.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.FiniteProd", "region_id": "algebraic_topology", "micro_elevation": 0.6111, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 157.582, "z": -89.706, "size": 0.2341, "title": "A binary product of finite simplicial sets is finite", "summary": "If `X₁` and `X₂` are respectively of dimensions `≤ d₁` and `≤ d₂`, then `X₁ ⊗ X₂` has dimension `≤ d₁ + d₂`. We also show that if `X₁` and `X₂` are finite, then `X₁ ⊗ X₂` is also finite.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/FiniteProd.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.RegularEpi", "region_id": "algebraic_topology", "micro_elevation": 0.2222, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 143.304, "z": -112.154, "size": 0.2341, "title": "The category of simplicial sets is a regular epi category", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/RegularEpi.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Nonsingular", "region_id": "algebraic_topology", "micro_elevation": 0.6111, "macro_tier": 2, "macro_tier_score": 0.0361, "macro_tier_override": null, "x": 168.089, "z": -102.074, "size": 0.2789, "title": "Nonsingular simplicial sets", "summary": "In this file, we introduce a typeclass `SSet.Nonsingular` for a simplicial set `X : SSet`: it says that for any non-degenerate simplex `x : X _⦋n⦌`, the corresponding morphism `Δ[n] ⟶ X` is a monomorphism. This notion is useful in the context of the study of the subdivision functor (TODO @joelriou). The condition `SSet.Nonsingular` is a weaker condition compared to the notion of \"polyhedral complex\" which appears in…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Nonsingular.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.ProdStdSimplex", "region_id": "algebraic_topology", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.0725, "macro_tier_override": null, "x": 160.381, "z": -121.718, "size": 0.3372, "title": "Binary product of standard simplices", "summary": "In this file, we show that `Δ[p] ⊗ Δ[q]` identifies to the nerve of `ULift (Fin (p + 1) × Fin (q + 1))`. We relate the `n`-simplices of `Δ[p] ⊗ Δ[q]` to order preserving maps `Fin (n + 1) →o Fin (p + 1) × Fin (q + 1)`, Via this bijection, a simplex in `Δ[p] ⊗ Δ[q]` is nondegenerate iff the corresponding monotone map `Fin (n + 1) →o Fin (p + 1) × Fin (q + 1)` is injective (or a strict mono). We also show that the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/ProdStdSimplex.html"}, {"id": "Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.0, "macro_tier": 2, "macro_tier_score": 0.0365, "macro_tier_override": null, "x": 148.98, "z": -107.661, "size": 0.3175, "title": "Fundamental groupoid of a space", "summary": "Given a topological space `X`, we can define the fundamental groupoid of `X` to be the category with objects being points of `X`, and morphisms `x ⟶ y` being paths from `x` to `y`, quotiented by homotopy equivalence. With this, the fundamental group of `X` based at `x` is just the automorphism group of `x`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialObject.Op", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 4, "macro_tier_score": 0.2507, "macro_tier_override": null, "x": 145.172, "z": -103.791, "size": 0.3151, "title": "The covariant involution of the category of simplicial objects", "summary": "In this file, we define the covariant involution `SimplicialObject.opFunctor` of the category of simplicial objects that is induced by the covariant involution `SimplexCategory.rev : SimplexCategory ⥤ SimplexCategory`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialObject/Op.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.DerivabilityStructureFibrant", "region_id": "algebraic_topology", "micro_elevation": 0.4444, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 162.32, "z": -113.291, "size": 0.2324, "title": "The right derivability structure attached to a model category", "summary": "We show that the inclusion of the full subcategory of fibrant objects in a model category is a right derivability structure. This is Corollaire 10.10 in [the paper by Kahn and Maltsiniotis][KahnMaltsiniotis2008].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/DerivabilityStructureFibrant.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy", "region_id": "algebraic_topology", "micro_elevation": 0.3889, "macro_tier": 2, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 151.758, "z": -120.022, "size": 0.2695, "title": "The homotopy category of fibrant objects", "summary": "Let `C` be a model category. By using the left homotopy relation, we introduce the homotopy category `FibrantObject.HoCat C` of fibrant objects in `C`, and we define a fibrant resolution functor `FibrantObject.HoCat.resolution : C ⥤ FibrantObject.HoCat C`. This file was obtained by dualizing the definitions in `Mathlib/AlgebraicTopology/ModelCategory/CofibrantObjectHomotopy.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/FibrantObjectHomotopy.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.EquivalenceAdditive", "region_id": "algebraic_topology", "micro_elevation": 0.8889, "macro_tier": 1, "macro_tier_score": 0.0359, "macro_tier_override": null, "x": 149.708, "z": -78.712, "size": 0.253, "title": "The Dold-Kan equivalence for additive categories.", "summary": "This file defines `Preadditive.DoldKan.equivalence` which is the equivalence of categories `Karoubi (SimplicialObject C) ≌ Karoubi (ChainComplex C ℕ)`. (See `Equivalence.lean` for the general strategy of proof of the Dold-Kan equivalence.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/EquivalenceAdditive.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.NCompGamma", "region_id": "algebraic_topology", "micro_elevation": 0.8333, "macro_tier": 2, "macro_tier_score": 0.0361, "macro_tier_override": null, "x": 172.864, "z": -120.568, "size": 0.2833, "title": "The unit isomorphism of the Dold-Kan equivalence", "summary": "In order to construct the unit isomorphism of the Dold-Kan equivalence, we first construct natural transformations `Γ₂N₁.natTrans : N₁ ⋙ Γ₂ ⟶ toKaroubi (SimplicialObject C)` and `Γ₂N₂.natTrans : N₂ ⋙ Γ₂ ⟶ 𝟭 (SimplicialObject C)`. It is then shown that `Γ₂N₂.natTrans` is an isomorphism by using that it becomes an isomorphism after the application of the functor `N₂ : Karoubi (SimplicialObject C) ⥤ Karoubi…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/NCompGamma.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplicesColimit", "region_id": "algebraic_topology", "micro_elevation": 0.3889, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 141.647, "z": -97.33, "size": 0.239, "title": "Any simplicial set is the colimit of its monogenous subcomplexes", "summary": "Let `X` be a simplicial set. The definition `SSet.isColimitCoconeN` shows that `X` is the colimit of the monogenous subcomplexes of `X` (the index category of this colimit is the partially ordered type `X.N` of nondegenerate simplices of `X`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/NonDegenerateSimplicesColimit.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices", "region_id": "algebraic_topology", "micro_elevation": 0.3333, "macro_tier": 4, "macro_tier_score": 0.2509, "macro_tier_override": null, "x": 140.887, "z": -114.901, "size": 0.3302, "title": "The partially ordered type of non degenerate simplices of a simplicial set", "summary": "In this file, we introduce the partially ordered type `X.N` of non degenerate simplices of a simplicial set `X`. We obtain an embedding `X.orderEmbeddingN : X.N ↪o X.Subcomplex` which sends a non degenerate simplex to the subcomplex of `X` it generates. Given an arbitrary simplex `x : X.S`, we show that there is a unique non degenerate `x.toN : X.N` such that `x.toN.subcomplex = x.subcomplex`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/NonDegenerateSimplices.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.SubcomplexColimits", "region_id": "algebraic_topology", "micro_elevation": 0.2778, "macro_tier": 4, "macro_tier_score": 0.1793, "macro_tier_override": null, "x": 140.013, "z": -108.878, "size": 0.3123, "title": "Colimits involving subcomplexes of a simplicial set", "summary": "If `X` is a simplicial set, and we have subcomplexes `A`, `U i` (for `i : ι`) and `V i j` which satisfy `Subcomplex.MulticoequalizerDiagram A U V` (an abbreviation for `CompleteLattice.MulticoequalizerDiagram`), we show that the simplicial sset corresponding to `A` is the multicoequalizer of the `U i` along the `V i j`. Similarly, bicartesian squares in the lattice `Subcomplex X` give pushout squares in the category…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/SubcomplexColimits.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.TopAdj", "region_id": "algebraic_topology", "micro_elevation": 0.5556, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 135.449, "z": -119.681, "size": 0.2653, "title": "Properties of the geometric realization", "summary": "In this file, we introduce some API in order to study the geometric realization functor (and its right adjoint the singular simplicial set functor): * `SimplexCategory.toTopHomeo`: the homeomorphism between the geometric realization of `Δ[n]` and `stdSimplex ℝ (Fin (n + 1))`; * `TopCat.toSSetObj₀Equiv : toSSet.obj X _⦋0⦌ ≃ X` for `X : TopCat`; * `SSet.stdSimplex.toTopObjIsoI : |Δ[1]| ≅ TopCat.I`; *…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/TopAdj.html"}, {"id": "Mathlib.AlgebraicTopology.SingularSet", "region_id": "algebraic_topology", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0717, "macro_tier_override": null, "x": 164.291, "z": -113.22, "size": 0.2661, "title": "The singular simplicial set of a topological space and geometric realization of a simplicial set", "summary": "The *singular simplicial set* `TopCat.toSSet.obj X` of a topological space `X` has `n`-simplices which identify to continuous maps `stdSimplex ℝ (Fin (n + 1)) → X`, where `stdSimplex ℝ (Fin (n + 1))` is the standard topological `n`-simplex, defined as the subtype of `Fin (n + 1) → ℝ` consisting of functions `f` such that `0 ≤ f i` for all `i` and `∑ i, f i = 1`. The *geometric realization* functor `SSet.toTop` is…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SingularSet.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Monoidal", "region_id": "algebraic_topology", "micro_elevation": 0.5, "macro_tier": 4, "macro_tier_score": 0.181, "macro_tier_override": null, "x": 132.706, "z": -106.985, "size": 0.408, "title": "The monoidal category structure on simplicial sets", "summary": "This file defines an instance of chosen finite products for the category `SSet`. It follows from the fact the `SSet` if a category of functors to the category of types and that the category of types have chosen finite products. As a result, we obtain a monoidal category structure on `SSet`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Monoidal.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.HornColimits", "region_id": "algebraic_topology", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 135.287, "z": -119.495, "size": 0.2574, "title": "Horns as colimits", "summary": "In this file, we express horns as colimits: * horns in `Δ[2]` are pushouts of two copies of `Δ[1]`; * horns in `Δ[n]` are multicoequalizers of copies of the standard simplex of dimension `n-1` (a dedicated API is provided for inner horns in `Δ[3]`).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/HornColimits.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Horn", "region_id": "algebraic_topology", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0725, "macro_tier_override": null, "x": 156.894, "z": -121.898, "size": 0.3357, "title": "Horns", "summary": "This file introduces horns `Λ[n, i]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Horn.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.IsCofibrant", "region_id": "algebraic_topology", "micro_elevation": 0.1111, "macro_tier": 4, "macro_tier_score": 0.1087, "macro_tier_override": null, "x": 152.039, "z": -105.725, "size": 0.3693, "title": "Fibrant and cofibrant objects in a model category", "summary": "Once a category `C` has been endowed with a `CategoryWithCofibrations C` instance, it is possible to define the property `IsCofibrant X` for any `X : C` as an abbreviation for `Cofibration (initial.to X : ⊥_ C ⟶ X)`. (Fibrant objects are defined similarly.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/IsCofibrant.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.CategoryWithFibrations", "region_id": "algebraic_topology", "micro_elevation": 0.6111, "macro_tier": 3, "macro_tier_score": 0.072, "macro_tier_override": null, "x": 164.09, "z": -120.625, "size": 0.3012, "title": "Cofibrations and fibrations in the category of simplicial sets", "summary": "We endow `SSet` with `CategoryWithCofibrations` and `CategoryWithFibrations` instances. Cofibrations are monomorphisms, and fibrations are morphisms having the right lifting property with respect to horn inclusions. We have an instance `mono_of_cofibration` (but only a lemma `cofibration_of_mono`). Then, when stating lemmas about cofibrations of simplicial sets, it is advisable to use the assumption `[Mono f]`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/CategoryWithFibrations.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.Bifibrant", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 2, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 148.107, "z": -102.302, "size": 0.2751, "title": "Bifibrant objects", "summary": "In this file, we introduce the full subcategories `CofibrantObject C`, `FibrantObject C` and `BifibrantObject C` of a model category `C` which respectively consist of cofibrant objects, fibrant objects, and bifibrant objects, where \"bifibrant\" means both cofibrant and fibrant.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/Bifibrant.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.Projections", "region_id": "algebraic_topology", "micro_elevation": 0.4444, "macro_tier": 3, "macro_tier_score": 0.0722, "macro_tier_override": null, "x": 147.567, "z": -122.071, "size": 0.3177, "title": "Construction of projections for the Dold-Kan correspondence", "summary": "In this file, we construct endomorphisms `P q : K[X] ⟶ K[X]` for all `q : ℕ`. We study how they behave with respect to face maps with the lemmas `HigherFacesVanish.of_P`, `HigherFacesVanish.comp_P_eq_self` and `comp_P_eq_self_iff`. Then, we show that they are projections (see `P_f_idem` and `P_idem`). They are natural transformations (see `natTransP` and `P_f_naturality`) and are compatible with the application of…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/Projections.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.Opposite", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 152.525, "z": -111.774, "size": 0.2, "title": "The opposite of a model category structure", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/Opposite.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.1111, "macro_tier": 2, "macro_tier_score": 0.0367, "macro_tier_override": null, "x": 146.953, "z": -110.659, "size": 0.3362, "title": "Model categories", "summary": "We introduce a typeclass `ModelCategory C` expressing that `C` is equipped with classes of morphisms named \"fibrations\", \"cofibrations\" and \"weak equivalences\" which satisfy the axioms of (closed) model categories as they appear for example in *Simplicial Homotopy Theory* by Goerss and Jardine. We also provide an alternate constructor `ModelCategory.mk'` which uses a formulation of the axioms using weak…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Degenerate", "region_id": "algebraic_topology", "micro_elevation": 0.2778, "macro_tier": 4, "macro_tier_score": 0.2513, "macro_tier_override": null, "x": 145.265, "z": -115.912, "size": 0.3507, "title": "Degenerate simplices", "summary": "Given a simplicial set `X` and `n : ℕ`, we define the sets `X.degenerate n` and `X.nonDegenerate n` of degenerate or non-degenerate simplices of dimension `n`. Any simplex `x : X _⦋n⦌` can be written in a unique way as `X.map f.op y` for an epimorphism `f : ⦋n⦌ ⟶ ⦋m⦌` and a non-degenerate `m`-simplex `y` (see lemmas `exists_nonDegenerate`, `unique_nonDegenerate_dim`, `unique_nonDegenerate_simplex` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Degenerate.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.FundamentalLemma", "region_id": "algebraic_topology", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 158.763, "z": -120.685, "size": 0.2, "title": "The fundamental lemma of homotopical algebra", "summary": "Let `C` be a model category. Let `L : C ⥤ H` be a localization functor with respect to weak equivalences in `C`. We obtain the fundamental lemma of homotopical algebra: if `X` is cofibrant and `Y` fibrant, the map `(X ⟶ Y) → (L.obj X ⟶ L.obj Y)` identifies `L.obj X ⟶ L.obj Y` to the quotient of `X ⟶ Y` by the homotopy relation (in this case, the left and right homotopy relations coincide).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/FundamentalLemma.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy", "region_id": "algebraic_topology", "micro_elevation": 0.4444, "macro_tier": 2, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 147.108, "z": -93.303, "size": 0.2676, "title": "The homotopy category of bifibrant objects", "summary": "We construct the homotopy category `BifibrantObject.HoCat C` of bifibrant objects in a model category `C` and show that the functor `BifibrantObject.toHoCat : BifibrantObject C ⥤ BifibrantObject.HoCat C` is a localization functor with respect to weak equivalences. We also show that certain localizer morphisms are localized weak equivalences, which can be understood by saying that we obtain the same localized…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/BifibrantObjectHomotopy.html"}, {"id": "Mathlib.AlgebraicTopology.FundamentalGroupoid.PUnit", "region_id": "algebraic_topology", "micro_elevation": 0.0556, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 147.183, "z": -107.453, "size": 0.2459, "title": "Fundamental groupoid of punit", "summary": "The fundamental groupoid of punit is naturally isomorphic to `CategoryTheory.Discrete PUnit`", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/FundamentalGroupoid/PUnit.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialObject.II", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 151.237, "z": -102.722, "size": 0.2, "title": "A construction by Gabriel and Zisman", "summary": "In this file, we construct a cosimplicial object `SimplexCategory.II` in `SimplexCategoryᵒᵖ`, i.e. a functor `SimplexCategory ⥤ SimplexCategoryᵒᵖ`. If we identify `SimplexCategory` with the category of finite nonempty linearly ordered types, this functor could be interpreted as the contravariant functor which sends a finite nonempty linearly ordered type `T` to `T →o Fin 2` (with `f ≤ g ↔ ∀ i, g i ≤ f i`, which…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialObject/II.html"}, {"id": "Mathlib.AlgebraicTopology.MooreComplex", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 4, "macro_tier_score": 0.2145, "macro_tier_override": null, "x": 143.551, "z": -107.7, "size": 0.2582, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/MooreComplex.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialObject.Split", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 3, "macro_tier_score": 0.0718, "macro_tier_override": null, "x": 154.255, "z": -108.948, "size": 0.2861, "title": "Split simplicial objects", "summary": "In this file, we introduce the notion of split simplicial object. If `C` is a category that has finite coproducts, a splitting `s : Splitting X` of a simplicial object `X` in `C` consists of the datum of a sequence of objects `s.N : ℕ → C` (which we shall refer to as \"nondegenerate simplices\") and a sequence of morphisms `s.ι n : s.N n → X _⦋n⦌` that have the property that a certain canonical map identifies `X _⦋n⦌`…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialObject/Split.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.GammaCompN", "region_id": "algebraic_topology", "micro_elevation": 0.7778, "macro_tier": 2, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 127.672, "z": -93.95, "size": 0.2722, "title": "The counit isomorphism of the Dold-Kan equivalence", "summary": "The purpose of this file is to construct natural isomorphisms `N₁Γ₀ : Γ₀ ⋙ N₁ ≅ toKaroubi (ChainComplex C ℕ)` and `N₂Γ₂ : Γ₂ ⋙ N₂ ≅ 𝟭 (Karoubi (ChainComplex C ℕ))`. (See `Equivalence.lean` for the general strategy of proof of the Dold-Kan equivalence.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/GammaCompN.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex", "region_id": "algebraic_topology", "micro_elevation": 0.7778, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 173.814, "z": -102.626, "size": 0.2382, "title": "The relative cell complex attached to a rank function for a pairing", "summary": "Let `A` be a subcomplex of a simplicial set `X`. Let `P : A.Pairing` be a proper pairing (in the sense of Moss) and `f : P.RankFunction ι` be a rank function. We show that the inclusion `A.ι` is a relative cell complex with basic cells given by horn inclusions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/RelativeCellComplex.html"}, {"id": "Mathlib.AlgebraicTopology.RelativeCellComplex.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.0556, "macro_tier": 3, "macro_tier_score": 0.0717, "macro_tier_override": null, "x": 148.741, "z": -105.867, "size": 0.2756, "title": "Relative cell complexes", "summary": "In this file, we define a structure `RelativeCellComplex` which expresses that a morphism `f : X ⟶ Y` is a transfinite composition of morphisms, all of which consist in attaching cells. Here, we allow a different family of authorized cells at each step. For example, (relative) CW-complexes are defined in the file `Mathlib/Topology/CWComplex/Abstract/Basic.lean` by requiring that at the `n`th step, we attach…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/RelativeCellComplex/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Rank", "region_id": "algebraic_topology", "micro_elevation": 0.7222, "macro_tier": 2, "macro_tier_score": 0.0362, "macro_tier_override": null, "x": 127.356, "z": -98.39, "size": 0.2906, "title": "Rank functions for pairings", "summary": "We introduce types of (weak) rank functions for a pairing `P` of a subcomplex `A` of a simplicial set `X`. These are functions `f : P.II → α` such that `P.AncestralRel x y` implies `f x < f y` (in the weak case, we require this only under the additional condition that `x` and `y` are of the same dimension). Such rank functions are used in order to show that the ancestrality relation on `P.II` is well founded, i.e.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/Rank.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.SubcomplexEvaluation", "region_id": "algebraic_topology", "micro_elevation": 0.2778, "macro_tier": 2, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 157.99, "z": -106.814, "size": 0.2493, "title": "The evaluation functor on subcomplexes", "summary": "We define an evaluation functor `SSet.Subcomplex.evaluation : X.Subcomplex ⥤ Set (X.obj j)` when `X : SSet` and `j : SimplexCategoryᵒᵖ`. We use it to show that the functor `Subcomplex.toSSetFunctor : X.Subcomplex ⥤ SSet` preserves filtered colimits.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/SubcomplexEvaluation.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Op", "region_id": "algebraic_topology", "micro_elevation": 0.5, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 149.209, "z": -123.948, "size": 0.2379, "title": "The opposite of a pairing", "summary": "Let `A` be a subcomplex of a simplicial set `X`. If `P` is a pairing of `A`, we construct a pairing `P.op` for the subcomplex `A.op` of `X.op`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/Op.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Boundary", "region_id": "algebraic_topology", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0717, "macro_tier_override": null, "x": 156.515, "z": -93.219, "size": 0.272, "title": "The boundary of the standard simplex", "summary": "We introduce the boundary `∂Δ[n]` of the standard simplex `Δ[n]`. (These notations become available by doing `open Simplicial`.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Boundary.html"}, {"id": "Mathlib.AlgebraicTopology.SimplexCategory.Defs", "region_id": "algebraic_topology", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.5007, "macro_tier_override": null, "x": 148.98, "z": -107.661, "size": 0.3156, "title": "The simplex category", "summary": "We construct a skeletal model of the simplex category, with an object `⦋n⦌` for each `n : ℕ`, and morphisms `⦋n⦌ ⟶ ⦋m⦌` identify to monotone maps from `Fin (n + 1)` to `Fin (m + 1)`. In `Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean`, we show that this category is equivalent to `NonemptyFinLinOrd`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplexCategory/Defs.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Path", "region_id": "algebraic_topology", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.072, "macro_tier_override": null, "x": 163.758, "z": -97.211, "size": 0.3034, "title": "Paths in simplicial sets", "summary": "A path in a simplicial set `X` of length `n` is a directed path comprised of `n + 1` 0-simplices and `n` 1-simplices, together with identifications between 0-simplices and the sources and targets of the 1-simplices. We define this construction first for truncated simplicial sets in `SSet.Truncated.Path`. A path in a simplicial set `X` is then defined as a 1-truncated path in the 1-truncation of `X`. An `n`-simplex…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Path.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Dimension", "region_id": "algebraic_topology", "micro_elevation": 0.3333, "macro_tier": 4, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 138.122, "z": -107.545, "size": 0.2693, "title": "Dimension of a simplicial set", "summary": "For a simplicial set `X` and `d : ℕ`, we introduce a typeclass `X.HasDimensionLT d` saying that the dimension of `X` is `< d`, i.e. all nondegenerate simplices of `X` are of dimension `< d`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Dimension.html"}, {"id": "Mathlib.AlgebraicTopology.ExtraDegeneracy", "region_id": "algebraic_topology", "micro_elevation": 0.5, "macro_tier": 0, "macro_tier_score": 0.0004, "macro_tier_override": null, "x": 154.937, "z": -122.822, "size": 0.2883, "title": "Augmented simplicial objects with an extra degeneracy", "summary": "In simplicial homotopy theory, in order to prove that the connected components of a simplicial set `X` are contractible, it suffices to construct an extra degeneracy as it is defined in *Simplicial Homotopy Theory* by Goerss-Jardine p. 190. It consists of a series of maps `π₀ X → X _⦋0⦌` and `X _⦋n⦌ → X _⦋n+1⦌` which behave formally like an extra degeneracy `σ (-1)`. It can be thought as a datum associated to the…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ExtraDegeneracy.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialObject.DeltaZeroIter", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 143.563, "z": -107.296, "size": 0.2431, "title": "Iterations of `δ 0` and `σ 0`", "summary": "This file introduces morphisms `δ₀Iter i` and `σ₀Iter i` for simplicial objects: they are obtained as the `i`th iteration of `δ 0` or `σ 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialObject/DeltaZeroIter.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialObject.Homotopy", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 4, "macro_tier_score": 0.1432, "macro_tier_override": null, "x": 153.567, "z": -104.755, "size": 0.2828, "title": "Simplicial homotopies of simplicial objects", "summary": "This file defines the notion of a combinatorial simplicial homotopy between two morphisms of simplicial objects.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialObject/Homotopy.html"}, {"id": "Mathlib.AlgebraicTopology.TopologicalSimplex", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 4, "macro_tier_score": 0.1073, "macro_tier_override": null, "x": 145.561, "z": -103.443, "size": 0.2524, "title": "Topological simplices", "summary": "We define the natural functor from `SimplexCategory` to `TopCat` sending `⦋n⦌` to the topological `n`-simplex. This is used to define `TopCat.toSSet` in `AlgebraicTopology.SingularSet`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/TopologicalSimplex.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.StrictSegal", "region_id": "algebraic_topology", "micro_elevation": 0.6111, "macro_tier": 3, "macro_tier_score": 0.0718, "macro_tier_override": null, "x": 161.513, "z": -123.13, "size": 0.2852, "title": "Strict Segal simplicial sets", "summary": "A simplicial set `X` satisfies the `StrictSegal` condition if for all `n`, the map `X.spine n : X _⦋n⦌ → X.Path n` is an equivalence, with equivalence inverse `spineToSimplex {n : ℕ} : Path X n → X _⦋n⦌`. Examples of `StrictSegal` simplicial sets are given by nerves of categories. TODO: Show that these are the only examples: that a `StrictSegal` simplicial set is isomorphic to the nerve of its homotopy category.…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/StrictSegal.html"}, {"id": "Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0003, "macro_tier_override": null, "x": 147.742, "z": -112.947, "size": 0.2678, "title": "Simply connected spaces", "summary": "This file defines simply connected spaces. A topological space is simply connected if its fundamental groupoid is equivalent to `Unit`. We also define the corresponding predicate for sets.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.html"}, {"id": "Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup", "region_id": "algebraic_topology", "micro_elevation": 0.0556, "macro_tier": 1, "macro_tier_score": 0.0359, "macro_tier_override": null, "x": 149.238, "z": -105.869, "size": 0.257, "title": "Fundamental group of a space", "summary": "Given a topological space `X` and a basepoint `x`, the fundamental group is the automorphism group of `x` i.e. the group with elements being loops based at `x` (quotiented by homotopy equivalence).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/FundamentalGroupoid/FundamentalGroup.html"}, {"id": "Mathlib.AlgebraicTopology.SimplexCategory.Rev", "region_id": "algebraic_topology", "micro_elevation": 0.1111, "macro_tier": 4, "macro_tier_score": 0.2862, "macro_tier_override": null, "x": 145.435, "z": -108.391, "size": 0.2887, "title": "The covariant involution of the simplex category", "summary": "In this file, we introduce the functor `rev : SimplexCategory ⥤ SimplexCategory` which, via the equivalence between the simplex category and the category of nonempty finite linearly ordered types, corresponds to the *covariant* functor which sends a type `α` to `αᵒᵈ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplexCategory/Rev.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.Homotopies", "region_id": "algebraic_topology", "micro_elevation": 0.3333, "macro_tier": 3, "macro_tier_score": 0.0719, "macro_tier_override": null, "x": 139.568, "z": -113.076, "size": 0.2955, "title": "Construction of homotopies for the Dold-Kan correspondence", "summary": "(The general strategy of proof of the Dold-Kan correspondence is explained in `Equivalence.lean`.) The purpose of the files `Homotopies.lean`, `Faces.lean`, `Projections.lean` and `PInfty.lean` is to construct an idempotent endomorphism `PInfty : K[X] ⟶ K[X]` of the alternating face map complex for each `X : SimplicialObject C` when `C` is a preadditive category. In the case `C` is abelian, this `PInfty` shall be…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/Homotopies.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.Notations", "region_id": "algebraic_topology", "micro_elevation": 0.2778, "macro_tier": 3, "macro_tier_score": 0.0718, "macro_tier_override": null, "x": 149.162, "z": -116.708, "size": 0.2783, "title": "Notations for the Dold-Kan equivalence", "summary": "This file defines the notation `K[X] : ChainComplex C ℕ` for the alternating face map complex of `(X : SimplicialObject C)` where `C` is a preadditive category, as well as `N[X]` for the normalized subcomplex in the case `C` is an abelian category. (See `Equivalence.lean` for the general strategy of proof of the Dold-Kan equivalence.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/Notations.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Nonempty", "region_id": "algebraic_topology", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 163.005, "z": -115.946, "size": 0.2627, "title": "Nonempty simplicial sets", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Nonempty.html"}, {"id": "Mathlib.AlgebraicTopology.SimplexCategory.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.0556, "macro_tier": 4, "macro_tier_score": 0.5041, "macro_tier_override": null, "x": 150.474, "z": -106.639, "size": 0.4714, "title": "Basic properties of the simplex category", "summary": "In `Mathlib/AlgebraicTopology/SimplexCategory/Defs.lean`, we define the simplex category with objects `ℕ` and morphisms `n ⟶ m` the monotone maps from `Fin (n + 1)` to `Fin (m + 1)`. In this file, we define the generating maps for the simplex category, show that this category is equivalent to `NonemptyFinLinOrd`, and establish basic properties of its epimorphisms and monomorphisms.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplexCategory/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms", "region_id": "algebraic_topology", "micro_elevation": 0.2222, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 156.116, "z": -106.436, "size": 0.2, "title": "Normal forms for morphisms in `SimplexCategoryGenRel`.", "summary": "In this file, we establish that `P_δ` and `P_σ` morphisms in `SimplexCategoryGenRel` each admits a normal form. In both cases, the normal forms are encoded as an integer `m`, and a strictly increasing list of integers `[i₀,…,iₙ]` such that `iₖ ≤ m + k` for all `k`. We define a predicate `isAdmissible m : List ℕ → Prop` encoding this property. And provide some lemmas to help work with such lists. Normal forms for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/NormalForms.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.Faces", "region_id": "algebraic_topology", "micro_elevation": 0.3889, "macro_tier": 3, "macro_tier_score": 0.0723, "macro_tier_override": null, "x": 141.549, "z": -97.4, "size": 0.3273, "title": "Study of face maps for the Dold-Kan correspondence", "summary": "In this file, we obtain the technical lemmas that are used in the file `Projections.lean` in order to get basic properties of the endomorphisms `P q : K[X] ⟶ K[X]` with respect to face maps (see `Homotopies.lean` for the role of these endomorphisms in the overall strategy of proof). The main lemma in this file is `HigherFacesVanish.induction`. It is based on two technical lemmas `HigherFacesVanish.comp_Hσ_eq` and…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/Faces.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.Compatibility", "region_id": "algebraic_topology", "micro_elevation": 0.0, "macro_tier": 1, "macro_tier_score": 0.0359, "macro_tier_override": null, "x": 148.98, "z": -107.661, "size": 0.253, "title": "Tools for compatibilities between Dold-Kan equivalences", "summary": "The purpose of this file is to introduce tools which will enable the construction of the Dold-Kan equivalence `SimplicialObject C ≌ ChainComplex C ℕ` for a pseudoabelian category `C` from the equivalence `Karoubi (SimplicialObject C) ≌ Karoubi (ChainComplex C ℕ)` and the two equivalences `SimplicialObject C ≌ Karoubi (SimplicialObject C)` and `ChainComplex C ℕ ≌ Karoubi (ChainComplex C ℕ)`. It is certainly possible…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/Compatibility.html"}, {"id": "Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 147.243, "z": -102.517, "size": 0.2478, "title": "The Augmented simplex category", "summary": "This file defines the `AugmentedSimplexCategory` as the category obtained by adding an initial object to `SimplexCategory` (using `CategoryTheory.WithInitial`). This definition provides a canonical full and faithful inclusion functor `inclusion : SimplexCategory ⥤ AugmentedSimplexCategory`. We prove that functors out of `AugmentedSimplexCategory` are equivalent to augmented cosimplicial objects and that functors out…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplexCategory/Augmented/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.Instances", "region_id": "algebraic_topology", "micro_elevation": 0.0556, "macro_tier": 4, "macro_tier_score": 0.1099, "macro_tier_override": null, "x": 147.56, "z": -108.782, "size": 0.4212, "title": "Consequences of model category axioms", "summary": "In this file, we deduce basic properties of fibrations, cofibrations, and weak equivalences from the axioms of model categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/Instances.html"}, {"id": "Mathlib.AlgebraicTopology.Reedy.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 148.98, "z": -107.661, "size": 0.2, "title": "Reedy categories", "summary": "In this file, we introduce the definition of a Reedy structure on a category `C` equipped with two classes of morphisms `W₁` and `W₂` (these are sometimes denoted `C₋` and `C₊` in the literature).", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/Reedy/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.SimplexCategory.Truncated", "region_id": "algebraic_topology", "micro_elevation": 0.1111, "macro_tier": 4, "macro_tier_score": 0.2865, "macro_tier_override": null, "x": 150.196, "z": -111.07, "size": 0.3162, "title": "Properties of the truncated simplex category", "summary": "We prove that for `n > 0`, the inclusion functor from the `n`-truncated simplex category to the untruncated simplex category, and the inclusion functor from the `n`-truncated to the `m`-truncated simplex category, for `n ≤ m` are initial.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplexCategory/Truncated.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.Over", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 143.847, "z": -105.891, "size": 0.2, "title": "The model category structure on Over categories", "summary": "Let `C` be a model category. For any `S : C`, we define a model category structure on the category `Over S`: a morphism `X ⟶ Y` in `Over S` is a cofibration (resp. a fibration, a weak equivalence) if the underlying morphism `f.left : X.left ⟶ Y.left` is. (Apart from the existence of (finite) limits from `Mathlib.CategoryTheory.Limits.Constructions.Over.Basic`, the verification of the axioms is straightforward.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/Over.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.Normalized", "region_id": "algebraic_topology", "micro_elevation": 0.6111, "macro_tier": 3, "macro_tier_score": 0.0718, "macro_tier_override": null, "x": 132.085, "z": -97.13, "size": 0.2812, "title": "Comparison with the normalized Moore complex functor", "summary": "In this file, we show that when the category `A` is abelian, there is an isomorphism `N₁_iso_normalizedMooreComplex_comp_toKaroubi` between the functor `N₁ : SimplicialObject A ⥤ Karoubi (ChainComplex A ℕ)` defined in `FunctorN.lean` and the composition of `normalizedMooreComplex A` with the inclusion `ChainComplex A ℕ ⥤ Karoubi (ChainComplex A ℕ)`. This isomorphism shall be used in `Equivalence.lean` in order to…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/Normalized.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PairingCore", "region_id": "algebraic_topology", "micro_elevation": 0.6667, "macro_tier": 2, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 144.847, "z": -86.339, "size": 0.2758, "title": "Helper structure in order to construct pairings", "summary": "In this file, we introduce a helper structure `Subcomplex.PairingCore` in order to construct a pairing for a subcomplex of a simplicial set. The main difference with `Subcomplex.Pairing` are that we provide an index type `ι` and a function `dim : ι → ℕ` which allow to parametrize type (II) and (I) simplices in such a way that, *definitionally*, their dimensions are respectively `dim s` or `dim s + 1` for `s : ι`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/PairingCore.html"}, {"id": "Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter", "region_id": "algebraic_topology", "micro_elevation": 0.1111, "macro_tier": 3, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 151.1, "z": -110.595, "size": 0.2554, "title": "Iterations of `δ 0` and `σ 0`", "summary": "This file introduces morphisms `δ₀Iter i` and `σ₀Iter i` in the simplex category: they are obtained as the `i`th iteration of `δ 0` or `σ 0`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplexCategory/DeltaZeroIter.html"}, {"id": "Mathlib.AlgebraicTopology.SingularHomology.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.5556, "macro_tier": 1, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 143.593, "z": -90.382, "size": 0.2658, "title": "Singular homology", "summary": "In this file, we define the singular chain complex and singular homology of a topological space. We also calculate the homology of a totally disconnected space as an example.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SingularHomology/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Nerve", "region_id": "algebraic_topology", "micro_elevation": 0.3333, "macro_tier": 4, "macro_tier_score": 0.2503, "macro_tier_override": null, "x": 159.42, "z": -110.65, "size": 0.2777, "title": "The nerve of a category", "summary": "This file provides the definition of the nerve of a category `C`, which is a simplicial set `nerve C` (see [goerss-jardine-2009], Example I.1.4). By definition, the type of `n`-simplices of `nerve C` is `ComposableArrows C n`, which is the category `Fin (n + 1) ⥤ C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Nerve.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Homotopy", "region_id": "algebraic_topology", "micro_elevation": 0.6667, "macro_tier": 3, "macro_tier_score": 0.0717, "macro_tier_override": null, "x": 145.51, "z": -86.221, "size": 0.268, "title": "Simplicial homotopies", "summary": "In this file, we define the notion of homotopy (`SSet.Homotopy`) between morphisms `f : X ⟶ Y` and `g : X ⟶ Y` of simplicial sets: it involves a morphism `X ⊗ Δ[1] ⟶ Y` inducing both `f` and `g`. (This definition is a particular case of `SSet.RelativeMorphism.Homotopy` that is defined in the file `Mathlib/AlgebraicTopology/SimplicialSet/RelativeMorphism.lean`). We show that from `H : SSet.Homotopy f g`, we can…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Homotopy.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.ProdStdSimplexOne", "region_id": "algebraic_topology", "micro_elevation": 0.6111, "macro_tier": 2, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 162.656, "z": -122.129, "size": 0.2531, "title": "Binary products `Δ[n] ⊗ Δ[1]`", "summary": "In this file, we define a bijection `SSet.prodStdSimplex.nonDegenerateEquiv₁` between `Fin (p + 1)` and the type of nondegenerate `(p + 1)`-simplices of `Δ[p] ⊗ Δ[1]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/ProdStdSimplexOne.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.RelativeMorphism", "region_id": "algebraic_topology", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.0717, "macro_tier_override": null, "x": 164.314, "z": -117.276, "size": 0.2714, "title": "Relative morphisms of simplicial sets", "summary": "Given two simplicial sets `X` and `Y`, and subcomplexes `A` of `X`, and `B` of `Y`, we introduce a type `RelativeMorphism A B φ` of morphisms `X ⟶ Y` which induce a given morphism of simplicial sets `A ⟶ B`. We define homotopies between these relative morphisms and introduce the quotient type of homotopy classes. This is used in the file `Mathlib/AlgebraicTopology/SimplicialSet/Homotopy.lean` in order to define…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/RelativeMorphism.html"}, {"id": "Mathlib.AlgebraicTopology.RelativeCellComplex.AttachCells", "region_id": "algebraic_topology", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.1077, "macro_tier_override": null, "x": 148.98, "z": -107.661, "size": 0.3036, "title": "Attaching cells", "summary": "Given a family of morphisms `g a : A a ⟶ B a` and a morphism `f : X₁ ⟶ X₂`, we introduce a structure `AttachCells g f` which expresses that `X₂` is obtained from `X₁` by attaching cells of the form `g a`. It means that there is a pushout diagram of the form ``` ⨿ i, A (π i) -----> X₁ | |f v v ⨿ i, B (π i) -----> X₂ ``` In other words, the morphism `f` is a pushout of coproducts of morphisms of the form `g a : A a ⟶…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/RelativeCellComplex/AttachCells.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.JoyalTrick", "region_id": "algebraic_topology", "micro_elevation": 0.0556, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 147.655, "z": -108.894, "size": 0.2, "title": "A trick by Joyal", "summary": "In order to construct a model category, we may sometimes have basically proven all the axioms with the exception of the left lifting property of cofibrations with respect to trivial fibrations. A trick by Joyal allows to obtain this lifting property under suitable assumptions, namely that cofibrations are stable under composition and cobase change. (The dual result is also formalized.)", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/JoyalTrick.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.CategoryWithCofibrations", "region_id": "algebraic_topology", "micro_elevation": 0.0, "macro_tier": 4, "macro_tier_score": 0.1087, "macro_tier_override": null, "x": 148.98, "z": -107.661, "size": 0.3664, "title": "Categories with classes of fibrations, cofibrations, weak equivalences", "summary": "We introduce typeclasses `CategoryWithFibrations`, `CategoryWithCofibrations` and `CategoryWithWeakEquivalences` to express that a category `C` is equipped with classes of morphisms named \"fibrations\", \"cofibrations\" or \"weak equivalences\".", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/CategoryWithCofibrations.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.Degeneracies", "region_id": "algebraic_topology", "micro_elevation": 0.6111, "macro_tier": 2, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 147.546, "z": -87.804, "size": 0.2491, "title": "Behaviour of `P_infty` with respect to degeneracies", "summary": "For any `X : SimplicialObject C` where `C` is an abelian category, the projector `PInfty : K[X] ⟶ K[X]` is supposed to be the projection on the normalized subcomplex, parallel to the degenerate subcomplex, i.e. the subcomplex generated by the images of all `X.σ i`. In this file, we obtain `degeneracy_comp_P_infty` which states that if `X : SimplicialObject C` with `C` a preadditive category, `θ : ⦋n⦌ ⟶ Δ'` is a…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/Degeneracies.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal", "region_id": "algebraic_topology", "micro_elevation": 0.6667, "macro_tier": 2, "macro_tier_score": 0.0715, "macro_tier_override": null, "x": 129.419, "z": -117.098, "size": 0.2425, "title": "Coskeletal simplicial sets", "summary": "In this file, we prove that if `X` is `StrictSegal` then `X` is 2-coskeletal, i.e. `X ≅ (cosk 2).obj X`. In particular, nerves of categories are 2-coskeletal. This isomorphism follows from the fact that `rightExtensionInclusion X 2` is a right Kan extension. In fact, we show that when `X` is `StrictSegal` then `(rightExtensionInclusion X n).IsPointwiseRightKanExtension` holds. As an example,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Coskeletal.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Homology.HomologyZero", "region_id": "algebraic_topology", "micro_elevation": 0.6111, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 143.768, "z": -126.875, "size": 0.239, "title": "Homology of simplicial sets in degree 0", "summary": "The main definition in this file is `SSet.homology₀Iso` which is an isomorphism `X.homology R 0 ≅ ∐ (fun (_ : π₀ X) ↦ R)` for any simplicial set `X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Homology/HomologyZero.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.PiZero", "region_id": "algebraic_topology", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 145.358, "z": -125.393, "size": 0.2624, "title": "Connected components of simplicial sets", "summary": "In this file, we define the type `π₀ X` of connected components of a simplicial sets. We also introduce typeclasses `IsPreconnected X` and `IsConnected X`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/PiZero.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.Transport", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 148.482, "z": -113.067, "size": 0.2, "title": "Transport a model category via an equivalence", "summary": "Given an equivalence of categories `e : C ≌ D`, we transport a model category structure on `D` in order to obtain a model category structure on `C`. More precisely, we assume that `C` has been equipped with notions of cofibrations, fibrations and weak equivalences and that these properties of morphisms are the inverse images of the corresponding properties of morphisms by the functor `e.functor : C ⥤ D`. Under these…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/Transport.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialComplex.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0002, "macro_tier_override": null, "x": 148.98, "z": -107.661, "size": 0.2526, "title": "Abstract Simplicial complexes", "summary": "In this file, we define abstract simplicial complexes. An abstract simplicial complex is a downwards-closed collection of nonempty finite sets containing every singleton. These are defined first defining `PreAbstractSimplicialComplex`, which does not require the presence of singletons, and then defining `AbstractSimplicialComplex` as an extension. This is related to the geometrical notion of simplicial complexes,…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialComplex/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Skeleton", "region_id": "algebraic_topology", "micro_elevation": 0.5556, "macro_tier": 3, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 150.901, "z": -89.664, "size": 0.2574, "title": "The skeleton of a simplicial set", "summary": "In this file, we define the skeleton `X.skeleton : ℕ →o X.Subcomplex` of a simplicial set `X`. For any `n : ℕ`, `X.skeleton n` is the subcomplex of `X` that is generated by (non-degenerate) simplices of dimension `< n`. If `i : X ⟶ Y` is a monomorphism, we define `skeletonOfMono i : ℕ →o Y.Subcomplex` so that `skeletonOfMono i n = Subcomplex.range i ⊔ Y.skeleton n`. We show that this filtration is part of a relative…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Skeleton.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.DerivabilityStructureCofibrant", "region_id": "algebraic_topology", "micro_elevation": 0.4444, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 145.082, "z": -93.716, "size": 0.2, "title": "The left derivability structure attached to a model category", "summary": "We show that the inclusion of the full subcategory of cofibrant objects in a model category is a left derivability structure. This is the dual to Corollaire 10.10 in [the paper by Kahn and Maltsiniotis][KahnMaltsiniotis2008].", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/DerivabilityStructureCofibrant.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy", "region_id": "algebraic_topology", "micro_elevation": 0.3889, "macro_tier": 1, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 159.829, "z": -114.205, "size": 0.2662, "title": "The homotopy category of cofibrant objects", "summary": "Let `C` be a model category. By using the right homotopy relation, we introduce the homotopy category `CofibrantObject.HoCat C` of cofibrant objects in `C`, and we define a cofibrant resolution functor `CofibrantObject.HoCat.resolution : C ⥤ CofibrantObject.HoCat C`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/CofibrantObjectHomotopy.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialCategory.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.5556, "macro_tier": 4, "macro_tier_score": 0.1077, "macro_tier_override": null, "x": 162.913, "z": -96.108, "size": 0.3005, "title": "Simplicial categories", "summary": "A simplicial category is a category `C` that is enriched over the category of simplicial sets in such a way that morphisms in `C` identify to the `0`-simplices of the enriched hom.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialCategory/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.FundamentalGroupoid.Product", "region_id": "algebraic_topology", "micro_elevation": 0.0556, "macro_tier": 1, "macro_tier_score": 0.0359, "macro_tier_override": null, "x": 150.698, "z": -108.232, "size": 0.2563, "title": "Fundamental groupoid preserves products", "summary": "In this file, we give the following definitions/theorems: - `FundamentalGroupoidFunctor.piIso` An isomorphism between Π i, (π Xᵢ) and π (Πi, Xᵢ), whose inverse is precisely the product of the maps π (Π i, Xᵢ) → π (Xᵢ), each induced by the projection in `Top` Π i, Xᵢ → Xᵢ. - `FundamentalGroupoidFunctor.prodIso` An isomorphism between πX × πY and π (X × Y), whose inverse is precisely the product of the maps π (X × Y)…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/FundamentalGroupoid/Product.html"}, {"id": "Mathlib.AlgebraicTopology.Quasicategory.StrictSegal", "region_id": "algebraic_topology", "micro_elevation": 0.7778, "macro_tier": 2, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 127.321, "z": -94.511, "size": 0.2676, "title": "Strict Segal simplicial sets are quasicategories", "summary": "In `AlgebraicTopology.SimplicialSet.StrictSegal`, we define the strict Segal condition on a simplicial set `X`. We say that `X` is strict Segal if its simplices are uniquely determined by their spine. In this file, we prove that any simplicial set satisfying the strict Segal condition is a quasicategory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/Quasicategory/StrictSegal.html"}, {"id": "Mathlib.AlgebraicTopology.Quasicategory.Basic", "region_id": "algebraic_topology", "micro_elevation": 0.7222, "macro_tier": 3, "macro_tier_score": 0.0721, "macro_tier_override": null, "x": 126.581, "z": -100.46, "size": 0.3101, "title": "Quasicategories", "summary": "In this file we define quasicategories, a common model of infinity categories. We show that every Kan complex is a quasicategory. In `Mathlib/AlgebraicTopology/Quasicategory/Nerve.lean`, we show that the nerve of a category is a quasicategory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/Quasicategory/Basic.html"}, {"id": "Mathlib.AlgebraicTopology.SingularHomology.HomotopyInvariance", "region_id": "algebraic_topology", "micro_elevation": 0.7778, "macro_tier": 2, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 131.367, "z": -89.445, "size": 0.2676, "title": "Homotopy invariance of singular homology", "summary": "In this file, we show that for any homotopy `H : TopCat.Homotopy f g` between two morphisms `f : X ⟶ Y` and `g : X ⟶ Y` in `TopCat`, the corresponding morphisms on the singular chain complexes are homotopic, and in particular the induced morphisms on singular homology are equal. The proof proceeds by observing that this result is a particular case of the homotopy invariance of the homology of simplicial sets (see…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SingularHomology/HomotopyInvariance.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Homology.HomotopyInvariance", "region_id": "algebraic_topology", "micro_elevation": 0.7222, "macro_tier": 1, "macro_tier_score": 0.0359, "macro_tier_override": null, "x": 156.813, "z": -129.847, "size": 0.253, "title": "Homotopy invariance of simplicial homology", "summary": "This file proves that homotopic morphisms of simplicial sets induce the same maps on singular homology (with coefficients in an object `R` of a preadditive category `C` with coproducts). First, in the case where the homotopy between two morphisms of simplicial sets `f : X ⟶ Y` and `g : X ⟶ Y` is given as combinatorial simplicial homotopy (`SimplicialObject.Homotopy`), i.e. as family of morphisms `X _⦋n⦌ ⟶ Y _⦋n +…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Homology/HomotopyInvariance.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.PathObject", "region_id": "algebraic_topology", "micro_elevation": 0.2222, "macro_tier": 2, "macro_tier_score": 0.0363, "macro_tier_override": null, "x": 147.831, "z": -114.808, "size": 0.3023, "title": "Path objects", "summary": "We introduce a notion of path object for an object `A : C` in a model category. It consists of an object `P`, a weak equivalence `ι : A ⟶ P` equipped with two retractions `p₀` and `p₁`. This notion shall be important in the definition of \"right homotopies\" in model categories. This file dualizes the definitions in the file `Mathlib/AlgebraicTopology/ModelCategory/Cylinder.lean`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/PathObject.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.Cylinder", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 2, "macro_tier_score": 0.0368, "macro_tier_override": null, "x": 147.696, "z": -112.936, "size": 0.3375, "title": "Cylinders", "summary": "We introduce a notion of cylinder for an object `A : C` in a model category. It consists of an object `I`, a weak equivalence `π : I ⟶ A` equipped with two sections `i₀` and `i₁`. This notion shall be important in the definition of \"left homotopies\" in model categories.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/Cylinder.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.Monomorphisms", "region_id": "algebraic_topology", "micro_elevation": 0.2222, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 145.205, "z": -101.483, "size": 0.2, "title": "Monomorphisms of simplicial sets", "summary": "In this file, we show that the class of monomorphisms in `SSet` is stable under coproducts, pushouts, filtered colimits and transfinite compositions.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/Monomorphisms.html"}, {"id": "Mathlib.AlgebraicTopology.SingularHomology.HomotopyInvarianceTopCat", "region_id": "algebraic_topology", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 150.139, "z": -80.537, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SingularHomology/HomotopyInvarianceTopCat.html"}, {"id": "Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal", "region_id": "algebraic_topology", "micro_elevation": 0.2222, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 141.939, "z": -105.98, "size": 0.2, "title": "Monoidal structure on the augmented simplex category", "summary": "This file defines a monoidal structure on `AugmentedSimplexCategory`. The tensor product of objects is characterized by the fact that the initial object `star` is also the unit, and the fact that `⦋m⦌ ⊗ ⦋n⦌ = ⦋m + n + 1⦌` for `n m : ℕ`. Through the (not in mathlib) equivalence between `AugmentedSimplexCategory` and the category of finite ordinals, the tensor products corresponds to ordinal sum. As the unit of this…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplexCategory/Augmented/Monoidal.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.StdSimplexOne", "region_id": "algebraic_topology", "micro_elevation": 0.5, "macro_tier": 3, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 165.258, "z": -107.058, "size": 0.2589, "title": "Simplices in `Δ[1]`", "summary": "We define a bijection `SSet.stdSimplex.objMk₁` between `Fin (n + 2)` and `Δ[1] _⦋n⦌` for any `n : ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/StdSimplexOne.html"}, {"id": "Mathlib.AlgebraicTopology.Quasicategory.Nerve", "region_id": "algebraic_topology", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 155.629, "z": -81.339, "size": 0.2, "title": "The nerve of a category is a quasicategory", "summary": "In `AlgebraicTopology.Quasicategory.StrictSegal`, we show that any strict Segal simplicial set is a quasicategory. In `AlgebraicTopology.SimplicialSet.StrictSegal`, we show that the nerve of a category satisfies the strict Segal condition. In this file, we prove as a direct consequence that the nerve of a category is a quasicategory.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/Quasicategory/Nerve.html"}, {"id": "Mathlib.AlgebraicTopology.Quasicategory.InnerFibration", "region_id": "algebraic_topology", "micro_elevation": 0.7778, "macro_tier": 1, "macro_tier_score": 0.0359, "macro_tier_override": null, "x": 129.754, "z": -124.165, "size": 0.251, "title": "Inner fibrations", "summary": "Inner fibrations of simplicial sets are the morphisms in `SSet` which have the right lifting property with respect to all inner horn inclusions. Basic consequences of inner fibrations with respect to the definition of quasi-categories are formalized.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/Quasicategory/InnerFibration.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.EquivalencePseudoabelian", "region_id": "algebraic_topology", "micro_elevation": 0.9444, "macro_tier": 2, "macro_tier_score": 0.036, "macro_tier_override": null, "x": 176.841, "z": -120.717, "size": 0.2676, "title": "The Dold-Kan correspondence for pseudoabelian categories", "summary": "In this file, for any idempotent complete additive category `C`, the Dold-Kan equivalence `Idempotents.DoldKan.Equivalence C : SimplicialObject C ≌ ChainComplex C ℕ` is obtained. It is deduced from the equivalence `Preadditive.DoldKan.Equivalence` between the respective idempotent completions of these categories using the fact that when `C` is idempotent complete, then both `SimplicialObject C` and `ChainComplex C…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.HomotopyEquivalence", "region_id": "algebraic_topology", "micro_elevation": 0.6667, "macro_tier": 2, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 144.023, "z": -128.806, "size": 0.2491, "title": "The normalized Moore complex and the alternating face map complex are homotopy equivalent", "summary": "In this file, when the category `A` is abelian, we obtain the homotopy equivalence `homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex` between the normalized Moore complex and the alternating face map complex of a simplicial object in `A`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/HomotopyEquivalence.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialObject.ChainHomotopy", "region_id": "algebraic_topology", "micro_elevation": 0.2778, "macro_tier": 2, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 148.908, "z": -98.612, "size": 0.2481, "title": "Simplicial homotopies induce chain homotopies", "summary": "Given a simplicial homotopy between morphisms of simplicial objects in a preadditive category, we construct a chain homotopy between the induced morphisms on the alternating face map complexes. Concretely, if `H : Homotopy f g` gives maps `H.h i : X _⦋n⦌ ⟶ Y _⦋n+1⦌` indexed by `i : Fin (n + 1)`, we define the degree-`n` component of the chain homotopy as the opposite of alternating sum `∑ i, (-1)^i • H.h i`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialObject/ChainHomotopy.html"}, {"id": "Mathlib.AlgebraicTopology.ModelCategory.LeftHomotopy", "region_id": "algebraic_topology", "micro_elevation": 0.2222, "macro_tier": 2, "macro_tier_score": 0.0361, "macro_tier_override": null, "x": 142.048, "z": -105.574, "size": 0.2878, "title": "Left homotopies in model categories", "summary": "We introduce the types `Precylinder.LeftHomotopy` and `Cylinder.LeftHomotopy` of homotopies between morphisms `X ⟶ Y` relative to a (pre)cylinder of `X`. Given two morphisms `f` and `g`, we introduce the relation `LeftHomotopyRel f g` asserting the existence of a cylinder object `P` and a left homotopy `P.LeftHomotopy f g`, and we define the quotient type `LeftHomotopyClass X Y`. We show that if `X` is a cofibrant…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/ModelCategory/LeftHomotopy.html"}, {"id": "Mathlib.AlgebraicTopology.Quasicategory.StrictBicategory", "region_id": "algebraic_topology", "micro_elevation": 0.8333, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 130.37, "z": -127.426, "size": 0.2, "title": "The strict bicategory of quasicategories", "summary": "In this file we define a strict bicategory `QCat.strictBicategory` whose objects are quasicategories. This strict category is defined from `QCat.catEnrichedOrdinaryCategory` which is the `Cat`-enriched ordinary category of quasicategories whose hom-categories are the homotopy categories of the simplicial internal homs, defined by applying `hoFunctor : SSet ⥤ Cat`. As an enriched ordinary category, there is an…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/Quasicategory/StrictBicategory.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialCategory.SimplicialObject", "region_id": "algebraic_topology", "micro_elevation": 0.6111, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 166.153, "z": -97.588, "size": 0.2338, "title": "The category of simplicial objects is simplicial", "summary": "In `CategoryTheory.Functor.FunctorHom`, it was shown that a category of functors `C ⥤ D` is enriched over a suitable category `C ⥤ Type _` of functors to types. In this file, we deduce that `SimplicialObject D` is enriched over `SSet.{v} D` (when `D : Type u` and `[Category.{v} D]`) and that `SimplicialObject D` is actually a simplicial category. In particular, the category of simplicial sets is a simplicial…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialCategory/SimplicialObject.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal", "region_id": "algebraic_topology", "micro_elevation": 0.7778, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 140.229, "z": -131.44, "size": 0.2338, "title": "The homotopy category functor is monoidal", "summary": "Given `2`-truncated simplicial sets `X` and `Y`, we introduce ad operation `Truncated.Edge.tensor : Edge x x' → Edge y y' → Edge (x, y) (x', y')`. We use this in order to construct an equivalence of categories `(X ⊗ Y).HomotopyCategory ≌ X.HomotopyCategory × Y.HomotopyCategory`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/HoFunctorMonoidal.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat", "region_id": "algebraic_topology", "micro_elevation": 0.7222, "macro_tier": 3, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 126.887, "z": -99.569, "size": 0.256, "title": "The homotopy category of a simplicial set", "summary": "The homotopy category of a simplicial set is defined as a quotient of the free category on its underlying reflexive quiver (equivalently its one truncation). The quotient imposes an additional hom relation on this free category, asserting that `f ≫ g = h` whenever `f`, `g`, and `h` are respectively the 2nd, 0th, and 1st faces of a 2-simplex. In fact, the associated functor `SSet.hoFunctor : SSet.{u} ⥤ Cat.{u, u} :=…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/HomotopyCat.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction", "region_id": "algebraic_topology", "micro_elevation": 0.7778, "macro_tier": 0, "macro_tier_score": 0.0001, "macro_tier_override": null, "x": 172.357, "z": -117.437, "size": 0.2478, "title": "The adjunction between the nerve and the homotopy category functor", "summary": "We define an adjunction `nerveAdjunction : hoFunctor ⊣ nerveFunctor` between the functor that takes a simplicial set to its homotopy category and the functor that takes a category to its nerve. Up to natural isomorphism, this is constructed as the composite of two other adjunctions, namely `nerve₂Adj : hoFunctor₂ ⊣ nerveFunctor₂` between analogously-defined functors involving the category of 2-truncated simplicial…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RankNat", "region_id": "algebraic_topology", "micro_elevation": 0.7778, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 149.987, "z": -82.342, "size": 0.2382, "title": "Existence of a rank function to natural numbers", "summary": "In this file, we show that if `P : A.Pairing` is a regular pairing of subcomplex `A` of a simplicial set `X`, then there exists a rank function for `P` with values in `ℕ`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/RankNat.html"}, {"id": "Mathlib.AlgebraicTopology.SimplexCategory.SemiSimplexCategory", "region_id": "algebraic_topology", "micro_elevation": 0.1111, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 151.824, "z": -109.901, "size": 0.239, "title": "The semi-simplex category", "summary": "We define a category `SemiSimplexCategory` so that semi-simplicial objects can be defined (TODO) as functors from `SemiSimplexCategoryᵒᵖ` similarly as simplicial objects are functors from `SimplexCategory`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplexCategory/SemiSimplexCategory.html"}, {"id": "Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps", "region_id": "algebraic_topology", "micro_elevation": 0.1111, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 152.193, "z": -105.993, "size": 0.2459, "title": "Homotopic maps induce naturally isomorphic functors", "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.html"}, {"id": "Mathlib.AlgebraicTopology.EilenbergSteenrod", "region_id": "algebraic_topology", "micro_elevation": 0.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 148.98, "z": -107.661, "size": 0.2, "title": "Eilenberg-Steenrod homology theories", "summary": "In this file we introduce the Eilenberg-Steenrod axioms for homology theories. The data for a homology theory is bundled in a structure `HomologyPretheory` consisting of functors `Hₚ i : TopPair ⥤ C` and `H i : TopCat ⥤ C` which represent the `i`th relative and regular homology, respectively, (indexed by a `ComplexShape`) and a proof that they agree on `TopCat`. They also require boundary morphisms `δ i j : Hₚ i ⟶…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/EilenbergSteenrod.html"}, {"id": "Mathlib.AlgebraicTopology.SimplexCategory.MorphismProperty", "region_id": "algebraic_topology", "micro_elevation": 0.1667, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 154.292, "z": -106.534, "size": 0.236, "title": "Properties of morphisms in the simplex category", "summary": "In this file, we show that morphisms in the simplex category are generated by faces and degeneracies. This is stated by saying that if `W : MorphismProperty SimplexCategory` is multiplicative, and contains faces and degeneracies, then `W = ⊤`. This statement is deduced from a similar statement for the category `SimplexCategory.Truncated d`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplexCategory/MorphismProperty.html"}, {"id": "Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated", "region_id": "algebraic_topology", "micro_elevation": 0.2778, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 152.886, "z": -115.824, "size": 0.2, "title": "2-truncated quasicategories and homotopy relations", "summary": "We define 2-truncated quasicategories `Quasicategory₂` by three horn-filling properties, and the left and right homotopy relations `HomotopicL` and `HomotopicR` on the edges in a 2-truncated simplicial set. We prove that for 2-truncated quasicategories, both homotopy relations are equivalence relations, and that the left and right homotopy relations coincide. For a 2-truncated quasicategory `A`, we define a category…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/Quasicategory/TwoTruncated.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.Equivalence", "region_id": "algebraic_topology", "micro_elevation": 1.0, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 165.355, "z": -135.824, "size": 0.2, "title": null, "summary": null, "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/Equivalence.html"}, {"id": "Mathlib.AlgebraicTopology.SimplexCategory.ToMkOne", "region_id": "algebraic_topology", "micro_elevation": 0.1111, "macro_tier": 3, "macro_tier_score": 0.0716, "macro_tier_override": null, "x": 152.584, "z": -107.316, "size": 0.2612, "title": "Morphisms to `⦋1⦌`", "summary": "We define a bijective map `SimplexCategory.toMk₁ : Fin (n + 2) → `⦋n⦌ ⟶ ⦋1⦌`. This is used in the file `Mathlib.AlgebraicTopology.SimplicialSet.StdSimplexOne` in the study of simplices in the simplicial set `Δ[1]`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplexCategory/ToMkOne.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialNerve", "region_id": "algebraic_topology", "micro_elevation": 0.6111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 164.095, "z": -94.703, "size": 0.2, "title": "The simplicial nerve of a simplicial category", "summary": "This file defines the simplicial nerve (sometimes called homotopy coherent nerve) of a simplicial category. We define the *simplicial thickening* of a linear order `J` as the simplicial category whose hom objects `i ⟶ j` are given by the nerve of the poset of \"paths\" from `i` to `j` in `J`. This is the poset of subsets of the interval `[i, j]` in `J`, containing the endpoints. The simplicial nerve of a simplicial…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialNerve.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct", "region_id": "algebraic_topology", "micro_elevation": 0.6111, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 143.223, "z": -126.719, "size": 0.2, "title": "Pointed simplices", "summary": "Given a simplicial set `X`, `n : ℕ` and `x : X _⦋0⦌`, we introduce the type `X.PtSimplex n x` of morphisms `Δ[n] ⟶ X` which send `∂Δ[n]` to `x`. We introduce structures `PtSimplex.RelStruct` and `PtSimplex.MulStruct` which will be used in the definition of homotopy groups of Kan complexes.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/KanComplex/MulStruct.html"}, {"id": "Mathlib.AlgebraicTopology.SimplicialSet.NonsingularColimit", "region_id": "algebraic_topology", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 163.821, "z": -123.518, "size": 0.2, "title": "Nonsingular simplicial sets, as colimits of standard simplices", "summary": "In the file `Mathlib/AlgebraicTopology/SimplicialSet/NonDegenerateSimplicesColimit.lean`, it was shown that any simplicial set `X` is the colimit (indexed by the type `X.N` of nondegenerate simplices) of its monogenous subcomplexes. In this file, we assume that `X` is nonsingular, in which case its monogenous subcomplexes identify to standard simplices. This allows to show that `X` is the colimit of `Δ[x.dim]` for…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SimplicialSet/NonsingularColimit.html"}, {"id": "Mathlib.AlgebraicTopology.SingularHomology.HomologyZero", "region_id": "algebraic_topology", "micro_elevation": 0.6667, "macro_tier": 0, "macro_tier_score": 0.0, "macro_tier_override": null, "x": 137.263, "z": -89.374, "size": 0.2, "title": "Singular homology in degree 0", "summary": "The main definition in this file is `TopCat.singularHomology₀Iso` which is an isomorphism `((singularHomologyFunctor C 0).obj R).obj X ≅ ∐ (fun (_ : ZerothHomotopy X) ↦ R)` for any `X : TopCat`.", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/SingularHomology/HomologyZero.html"}, {"id": "Mathlib.AlgebraicTopology.DoldKan.FunctorGamma", "region_id": "algebraic_topology", "micro_elevation": 0.5556, "macro_tier": 1, "macro_tier_score": 0.0358, "macro_tier_override": null, "x": 163.9, "z": -97.415, "size": 0.2473, "title": "Construction of the inverse functor of the Dold-Kan equivalence", "summary": "In this file, we construct the functor `Γ₀ : ChainComplex C ℕ ⥤ SimplicialObject C` which shall be the inverse functor of the Dold-Kan equivalence in the case of abelian categories, and more generally pseudoabelian categories. By definition, when `K` is a `ChainComplex`, `Γ₀.obj K` is a simplicial object which sends `Δ : SimplexCategoryᵒᵖ` to a certain coproduct indexed by the set `Splitting.IndexSet Δ` whose…", "doc_url": "https://leanprover-community.github.io/mathlib4_docs/Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.html"}], "edges": [{"from": "Mathlib.Algebra.Order.Module.HahnEmbedding", "to": "Mathlib.Algebra.Order.Module.Equiv", "is_bridge": false}, {"from": "Mathlib.Algebra.Order.Module.Equiv", "to": "Mathlib.Algebra.Module.Equiv.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift", "to": "Mathlib.Algebra.Module.Equiv.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.Lie.Basic", "to": "Mathlib.Algebra.Module.Equiv.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.GeneralLinearGroup.Basic", "to": "Mathlib.Algebra.Module.Equiv.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.FiniteSpan", "to": "Mathlib.Algebra.Module.Equiv.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.Finsupp.LSum", "to": "Mathlib.Algebra.Module.Equiv.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.Module.Submodule.Invariant", "to": "Mathlib.Algebra.Module.Equiv.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.TensorProduct.Defs", "to": "Mathlib.Algebra.Module.Equiv.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.Algebra.Basic", "to": "Mathlib.Algebra.Module.Equiv.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.Category.ModuleCat.Semi", "to": "Mathlib.Algebra.Module.Equiv.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.Order.Module.Equiv", "to": "Mathlib.Algebra.Order.Group.Equiv", "is_bridge": false}, {"from": "Mathlib.Algebra.Order.GroupWithZero.Lex", "to": "Mathlib.Algebra.Order.Group.Equiv", "is_bridge": false}, {"from": "Mathlib.Algebra.Order.Module.Equiv", "to": "Mathlib.Algebra.Order.Module.Synonym", "is_bridge": false}, {"from": "Mathlib.Algebra.Order.Monovary", "to": "Mathlib.Algebra.Order.Module.Synonym", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.AffineSpace.Ordered", "to": "Mathlib.Algebra.Order.Module.Synonym", "is_bridge": false}, {"from": "Mathlib.Algebra.Order.Rearrangement", "to": "Mathlib.Algebra.Order.Module.Synonym", "is_bridge": false}, {"from": "Mathlib.Analysis.Convex.Star", "to": "Mathlib.Algebra.Order.Module.Synonym", "is_bridge": true}, {"from": "Mathlib.Tactic.Determinant.Bird", "to": "Mathlib.Tactic.Determinant.Bird.Cert", "is_bridge": false}, {"from": "Mathlib.Tactic.Determinant.Bird.Cert", "to": "Mathlib.Tactic.Determinant.Bird.Meta", "is_bridge": false}, {"from": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "to": "Mathlib.Topology.Instances.Sign", "is_bridge": true}, {"from": "Mathlib.Geometry.Euclidean.Incenter", "to": "Mathlib.Topology.Instances.Sign", "is_bridge": true}, {"from": "Mathlib.Topology.Instances.Sign", "to": "Mathlib.Data.Sign.Defs", "is_bridge": true}, {"from": "Mathlib.Data.Sign.Basic", "to": "Mathlib.Data.Sign.Defs", "is_bridge": false}, {"from": "Mathlib.GroupTheory.DivisibleHull", "to": "Mathlib.Data.Sign.Defs", "is_bridge": true}, {"from": "Mathlib.Analysis.Normed.Module.Normalize", "to": "Mathlib.Data.Sign.Defs", "is_bridge": true}, {"from": "Mathlib.Topology.Instances.Sign", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.Topology.Instances.Discrete", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.Topology.Order.SuccPred", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.Topology.Order.ProjIcc", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.Topology.Order.Completion", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.Topology.FiberBundle.Trivialization", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.Analysis.Convex.Strict", "to": "Mathlib.Topology.Order.Basic", "is_bridge": true}, {"from": "Mathlib.Topology.Order.Filter", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.Topology.Algebra.IsUniformGroup.Order", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.Topology.Order.NhdsSet", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.Topology.Order.T5", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.Topology.Order.CountableSeparating", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.Topology.Order.LeftRightNhds", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.Topology.Order.WithTop", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.Topology.Order.MonotoneConvergence", "to": "Mathlib.Topology.Order.Basic", "is_bridge": false}, {"from": "Mathlib.NumberTheory.NumberField.Units.Regulator", "to": "Mathlib.LinearAlgebra.Matrix.Determinant.Misc", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.Matrix.Determinant.Misc", "to": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.Matrix.Reindex", "to": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.Matrix.Polynomial", "to": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.Matrix.Cartan", "to": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.Matrix.AbsoluteValue", "to": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "to": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular", "to": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.Matrix.Kronecker", "to": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.CrossProduct", "to": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.Matrix.Determinant.Misc", "to": "Mathlib.Algebra.Ring.NegOnePow", "is_bridge": false}, {"from": "Mathlib.Algebra.Homology.EulerCharacteristic", "to": "Mathlib.Algebra.Ring.NegOnePow", "is_bridge": false}, {"from": "Mathlib.RingTheory.Polynomial.Chebyshev", "to": "Mathlib.Algebra.Ring.NegOnePow", "is_bridge": false}, {"from": "Mathlib.CategoryTheory.Center.NegOnePow", "to": "Mathlib.Algebra.Ring.NegOnePow", "is_bridge": true}, {"from": "Mathlib.RingTheory.Binomial", "to": "Mathlib.Algebra.Ring.NegOnePow", "is_bridge": false}, {"from": "Mathlib.Algebra.Homology.HomotopyCategory.HomComplex", "to": "Mathlib.Algebra.Ring.NegOnePow", "is_bridge": false}, {"from": "Mathlib.Algebra.Homology.ComplexShapeSigns", "to": "Mathlib.Algebra.Ring.NegOnePow", "is_bridge": false}, {"from": "Mathlib.Algebra.Ring.Periodic", "to": "Mathlib.Algebra.Ring.NegOnePow", "is_bridge": false}, {"from": "Mathlib.Algebra.Homology.HomotopyCategory.Shift", "to": "Mathlib.Algebra.Ring.NegOnePow", "is_bridge": false}, {"from": "Mathlib.CategoryTheory.Triangulated.TriangleShift", "to": "Mathlib.Algebra.Ring.NegOnePow", "is_bridge": true}, {"from": "Mathlib.Algebra.Field.NegOnePow", "to": "Mathlib.Algebra.Ring.NegOnePow", "is_bridge": false}, {"from": "Mathlib.Analysis.Convex.DoublyStochasticMatrix", "to": "Mathlib.LinearAlgebra.Matrix.Stochastic", "is_bridge": true}, {"from": "Mathlib.LinearAlgebra.Matrix.Stochastic", "to": "Mathlib.LinearAlgebra.Matrix.Permutation", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.Matrix.Swap", "to": "Mathlib.LinearAlgebra.Matrix.Permutation", "is_bridge": false}, {"from": "Mathlib.RingTheory.RingHom.FinitePresentation", "to": "Mathlib.RingTheory.RingHom.FiniteType", "is_bridge": false}, {"from": "Mathlib.AlgebraicGeometry.Morphisms.FiniteType", "to": "Mathlib.RingTheory.RingHom.FiniteType", "is_bridge": true}, {"from": "Mathlib.RingTheory.RingHom.FiniteType", "to": "Mathlib.RingTheory.FiniteStability", "is_bridge": false}, {"from": "Mathlib.RingTheory.QuasiFinite.Basic", "to": "Mathlib.RingTheory.FiniteStability", "is_bridge": false}, {"from": "Mathlib.RingTheory.Unramified.Basic", "to": "Mathlib.RingTheory.FiniteStability", "is_bridge": false}, {"from": "Mathlib.RingTheory.Spectrum.Prime.Chevalley", "to": "Mathlib.RingTheory.FiniteStability", "is_bridge": false}, {"from": "Mathlib.RingTheory.RingHom.FiniteType", "to": "Mathlib.RingTheory.Finiteness.FiniteTypeLocal", "is_bridge": false}, {"from": "Mathlib.RingTheory.Finiteness.FinitePresentationLocal", "to": "Mathlib.RingTheory.Finiteness.FiniteTypeLocal", "is_bridge": false}, {"from": "Mathlib.RingTheory.RingHom.FiniteType", "to": "Mathlib.RingTheory.Localization.InvSubmonoid", "is_bridge": false}, {"from": "Mathlib.RingTheory.Unramified.Locus", "to": "Mathlib.RingTheory.Localization.InvSubmonoid", "is_bridge": false}, {"from": "Mathlib.AlgebraicGeometry.AffineScheme", "to": "Mathlib.RingTheory.Localization.InvSubmonoid", "is_bridge": true}, {"from": "Mathlib.RingTheory.QuasiFinite.Basic", "to": "Mathlib.RingTheory.Localization.InvSubmonoid", "is_bridge": false}, {"from": "Mathlib.RingTheory.RingHom.FiniteType", "to": "Mathlib.RingTheory.RingHom.Finite", "is_bridge": false}, {"from": "Mathlib.RingTheory.DedekindDomain.Instances", "to": "Mathlib.RingTheory.RingHom.Finite", "is_bridge": false}, {"from": "Mathlib.RingTheory.IntegralClosure.IntegralRestrict", "to": "Mathlib.RingTheory.RingHom.Finite", "is_bridge": false}, {"from": "Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set", "to": "Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic", "to": "Mathlib.Algebra.GroupWithZero.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.GroupWithZero.Pointwise.Finset", "to": "Mathlib.Algebra.GroupWithZero.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.GroupWithZero.Units.Basic", "to": "Mathlib.Algebra.GroupWithZero.Basic", "is_bridge": false}, {"from": "Mathlib.Data.Sym.Sym2.Finsupp", "to": "Mathlib.Algebra.GroupWithZero.Basic", "is_bridge": true}, {"from": "Mathlib.Algebra.GroupWithZero.Indicator", "to": "Mathlib.Algebra.GroupWithZero.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.GroupWithZero.Action.ConjAct", "to": "Mathlib.Algebra.GroupWithZero.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.GroupWithZero.Hom", "to": "Mathlib.Algebra.GroupWithZero.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic", "to": "Mathlib.Algebra.Group.Pointwise.Set.Basic", "is_bridge": false}, {"from": "Mathlib.GroupTheory.Coset.Defs", "to": "Mathlib.Algebra.Group.Pointwise.Set.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.Group.Pointwise.Set.BigOperators", "to": "Mathlib.Algebra.Group.Pointwise.Set.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.Order.Group.Pointwise.Interval", "to": "Mathlib.Algebra.Group.Pointwise.Set.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.Group.Pointwise.Set.Lattice", "to": "Mathlib.Algebra.Group.Pointwise.Set.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.Star.Conjneg", "to": "Mathlib.Algebra.Group.Pointwise.Set.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.Order.Group.Pointwise.Bounds", "to": "Mathlib.Algebra.Group.Pointwise.Set.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.Group.Pointwise.Set.Small", "to": "Mathlib.Algebra.Group.Pointwise.Set.Basic", "is_bridge": false}, {"from": "Mathlib.LinearAlgebra.Span.Defs", "to": "Mathlib.Algebra.Group.Pointwise.Set.Basic", "is_bridge": false}, {"from": "Mathlib.Algebra.Group.Pointwise.Set.ListOfFn", "to": "Mathlib.Algebra.Group.Pointwise.Set.Basic", "is_bridge": false}, 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1}, {"from_region": "algebra", "to_region": "geometry", "edge_count": 1}, {"from_region": "probability", "to_region": "tactic", "edge_count": 1}, {"from_region": "information_theory", "to_region": "probability", "edge_count": 1}, {"from_region": "probability", "to_region": "frontier", "edge_count": 1}, {"from_region": "algebraic_topology", "to_region": "combinatorics", "edge_count": 1}, {"from_region": "dynamics", "to_region": "combinatorics", "edge_count": 1}, {"from_region": "analysis", "to_region": "dynamics", "edge_count": 1}, {"from_region": "algebra", "to_region": "algebraic_geometry", "edge_count": 1}, {"from_region": "topology", "to_region": "geometry", "edge_count": 1}, {"from_region": "algebraic_topology", "to_region": "analysis", "edge_count": 1}, {"from_region": "model_theory", "to_region": "category_theory", "edge_count": 1}, {"from_region": "algebraic_topology", "to_region": "frontier", "edge_count": 1}, {"from_region": "geometry", "to_region": "algebraic_topology", "edge_count": 1}, {"from_region": "model_theory", "to_region": "computability", "edge_count": 1}, {"from_region": "probability", "to_region": "foundations_data", "edge_count": 1}, {"from_region": "analysis", "to_region": "algebraic_topology", "edge_count": 1}]} \ No newline at end of file diff --git a/kingdom/schema.md b/kingdom/schema.md index 6793732..e90066c 100644 --- a/kingdom/schema.md +++ b/kingdom/schema.md @@ -51,6 +51,9 @@ | `macro_tier_override` | int \| `null` | 非 `null` 时表示这个层级来自 `kingdom/tier_overrides.yaml` 的人工指定,而不是启发式计算 | | `x`, `z` | float | 节点在地图平面上的坐标。地面山脉节点:`map_center` + 一个半径不超过 `footprint_radius` 的局部极坐标偏移(半径 = 归一化的 `micro_elevation`,角度 = 节点名的稳定哈希),保证同一座山的所有节点都落在它自己的 `footprint_radius` 圆内,不会侵入相邻山脉。天空层节点:其**依赖方各区域 `map_center` 按引用次数加权的质心**——被代数大量引用的范畴论节点,坐标会漂移到代数山上空 | | `size` | float | 节点大小,直接复用 `gen_graph.py` 里已经在算的全局反向 PageRank 半径公式 | +| `title` | string \| `null` | 从 mathlib4 源码里对应 `.lean` 文件的模块级 `/-! ... -/` 文档注释中提取的标题(真实内容,不是生成的)。文件不存在或没有模块文档时为 `null` | +| `summary` | string \| `null` | 同一个文档注释里标题之后的正文,截到下一个 `##` 子标题之前,并裁剪到 `SUMMARY_MAX_LEN`(420 字符)。同样是真实的 mathlib 源码内容 | +| `doc_url` | string | 对应的 [mathlib4_docs](https://leanprover-community.github.io/mathlib4_docs/) 页面链接,按 `id` 里的点号转斜杠拼出来,不依赖 `title`/`summary` 是否提取成功,总是有值 | ### `edges[]` —— import 依赖 diff --git a/script/gen_kingdom.py b/script/gen_kingdom.py index 8458083..a6267e9 100644 --- a/script/gen_kingdom.py +++ b/script/gen_kingdom.py @@ -41,6 +41,38 @@ LOCAL_SPREAD_FACTOR = 2.6 # per-region footprint radius = this * sqrt(node_count) MOUNTAIN_MARGIN = 10.0 # minimum gap kept between any two mountains' footprints GOLDEN_ANGLE = math.pi * (3 - math.sqrt(5)) +MATHLIB_DOCS_BASE = 'https://leanprover-community.github.io/mathlib4_docs/' +SUMMARY_MAX_LEN = 420 +MODULE_DOC_RE = re.compile(r'/-!(.*?)-/', re.S) + + +def module_doc(node_id): + """Pull the module's own `/-! ... -/` doc-string straight out of the + mathlib4 source (real content, not a fabricated description), plus the + corresponding mathlib4_docs URL. Returns (title, summary, doc_url).""" + rel_path = node_id.replace('.', '/') + '.lean' + doc_url = MATHLIB_DOCS_BASE + rel_path[:-len('.lean')] + '.html' + abs_path = os.path.join(mathlib_src_path, rel_path) + if not os.path.exists(abs_path): + return None, None, doc_url + with open(abs_path, encoding='utf-8', errors='ignore') as f: + text = f.read(6000) + m = MODULE_DOC_RE.search(text) + if not m: + return None, None, doc_url + kept_lines = [] + for line in m.group(1).split('\n'): + if line.strip().startswith('##'): + break + kept_lines.append(line) + kept = '\n'.join(kept_lines) + title_m = re.match(r'\s*#\s*(.+)', kept) + title = title_m.group(1).strip() if title_m else None + body = re.sub(r'^\s*#\s*.+', '', kept, count=1) + body = re.sub(r'\s+', ' ', body).strip() + if len(body) > SUMMARY_MAX_LEN: + body = body[:SUMMARY_MAX_LEN].rsplit(' ', 1)[0] + '…' + return title, (body or None), doc_url # --------------------------------------------------------------------------- @@ -318,6 +350,7 @@ def cluster_key(node, depth=3): for n, d in node_data.items(): rank = page_rank[n] size = 0.2 + 3 * ((rank - pr_min) / (pr_max - pr_min)) ** 0.5 if pr_max > pr_min else 0.2 + title, summary, doc_url = module_doc(n) nodes_out.append({ 'id': n, 'region_id': d['region_id'], @@ -328,6 +361,9 @@ def cluster_key(node, depth=3): 'x': round(d['x'], 3), 'z': round(d['z'], 3), 'size': round(size, 4), + 'title': title, + 'summary': summary, + 'doc_url': doc_url, }) edges_out = [] From 4d285407cc5f66be90a8192dced3ba7458a7fe43 Mon Sep 17 00:00:00 2001 From: Henry Date: Thu, 9 Jul 2026 10:27:49 -0700 Subject: [PATCH 4/4] Add self-contained 3D kingdom viewer and unified 2d/3d launcher kingdom/viewer/index.html is a single self-contained page (Three.js and the full kingdom dataset inlined, no network access needed) rendering the mountains as a progressively-disclosed 3D scene: collapsed tier platforms by default, click-to-expand detail, search, and a click-to- highlight explorer for a concept's direct dependencies. Replaces the separate per-folder run.sh/run.bat (which was a source of confusion -- easy to launch the wrong one) with a single pair of scripts at the repo root that take a 2d/3d argument and dispatch to either the bgfx explorer or the 3D viewer, opening the latter in an app-mode browser window when Edge/Chrome is available. Co-Authored-By: Claude Sonnet 5 --- README.md | 17 +- kingdom/viewer/index.html | 991 ++++++++++++++++++++++++++++++++++++++ run.bat | 53 ++ run.sh | 90 ++++ 4 files changed, 1148 insertions(+), 3 deletions(-) create mode 100644 kingdom/viewer/index.html create mode 100644 run.bat create mode 100644 run.sh diff --git a/README.md b/README.md index 837317f..c527a50 100644 --- a/README.md +++ b/README.md @@ -37,15 +37,26 @@ Clone this repo: git clone https://github.com/Crispher/MathlibExplorer ``` -Go to the binary folder of your platform: +Run it with one command from the repo root, picking `2d` for the original bgfx import-graph explorer or `3d` for the experimental [3D "mathematical kingdom" map](./kingdom/): ``` -cd MathlibExplorer/release/bin_{YOUR_PLATFORM} +cd MathlibExplorer + +# macOS / Linux +./run.sh 2d +./run.sh 3d + +# Windows +run.bat 2d +run.bat 3d ``` -Run the executable: +(There is currently no prebuilt Linux binary for the 2D explorer, only macOS and Windows; `run.sh 2d` will tell you so instead of failing silently. The 3D map only needs a browser and works everywhere.) + +Or run the 2D explorer's executable directly: ``` +cd MathlibExplorer/release/bin_{YOUR_PLATFORM} ./MathlibExplorer ``` diff --git a/kingdom/viewer/index.html b/kingdom/viewer/index.html new file mode 100644 index 0000000..7367385 --- /dev/null +++ b/kingdom/viewer/index.html @@ -0,0 +1,991 @@ +数学王国地图 + + +
正在测绘数学王国…
+
+ +
+

数学王国地图

+

mathlib4 import 依赖 · 启发式抽象海拔 · 实验性数据原型

+
+ +
+

搜寻概念

+ +
+
+ +
+

山脉索引

+
+
+ +
+
+ + +
+ + +
+ 点击发光平台 = 展开/收起该层细节 · 点击星体/高亮线 = 查看详情 · 拖拽 = 平移 · 右键拖拽 = 环视 · WASD = 飞行 · Q/E = 升降 · 滚轮 = 缩放 +
+
大海拔为图论启发式估算,不代表精确的数学教学阶梯。
+
+ +
+

探测器

+
点击一座发光平台展开它的细节,或点击展开后的星体查看单个概念。
+
+ +
+ + + + diff --git a/run.bat b/run.bat new file mode 100644 index 0000000..58ce4cc --- /dev/null +++ b/run.bat @@ -0,0 +1,53 @@ +@echo off +REM One-command launcher for Windows. +REM +REM Usage: +REM run.bat 2d - launch the original bgfx import-graph explorer +REM run.bat 3d - launch the 3D "mathematical kingdom" map + +setlocal +set DIR=%~dp0 +set MODE=%1 + +if /i "%MODE%"=="2d" goto :run2d +if /i "%MODE%"=="3d" goto :run3d + +echo Usage: run.bat [2d^|3d] +echo 2d - launch the original bgfx import-graph explorer +echo 3d - launch the 3D "mathematical kingdom" map +exit /b 1 + +:run2d +set EXE=%DIR%release\bin_win64\MathlibExplorer.exe +if not exist "%EXE%" ( + echo No prebuilt MathlibExplorer.exe found at "%EXE%" + exit /b 1 +) +cd /d "%DIR%release\bin_win64" +start "" "MathlibExplorer.exe" +goto :eof + +:run3d +set HTML=%DIR%kingdom\viewer\index.html +if not exist "%HTML%" ( + echo No 3D viewer found at "%HTML%" + exit /b 1 +) +set FILEURL=file:///%HTML:\=/% + +REM Prefer an "app mode" window (no tabs/address bar) for a closer-to-native +REM feel matching the bgfx MathlibExplorer window; fall back to whatever the +REM system's default browser association is. +set EDGE=%ProgramFiles(x86)%\Microsoft\Edge\Application\msedge.exe +set CHROME=%ProgramFiles%\Google\Chrome\Application\chrome.exe + +if exist "%EDGE%" ( + start "" "%EDGE%" --app="%FILEURL%" + goto :eof +) +if exist "%CHROME%" ( + start "" "%CHROME%" --app="%FILEURL%" + goto :eof +) + +start "" "%HTML%" diff --git a/run.sh b/run.sh new file mode 100644 index 0000000..4bcaebe --- /dev/null +++ b/run.sh @@ -0,0 +1,90 @@ +#!/usr/bin/env bash +# One-command launcher for macOS / Linux. +# +# Usage: +# ./run.sh 2d - launch the original bgfx import-graph explorer +# ./run.sh 3d - launch the 3D "mathematical kingdom" map + +set -euo pipefail + +DIR="$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)" +MODE="${1:-}" + +usage() { + echo "Usage: ./run.sh [2d|3d]" >&2 + echo " 2d - launch the original bgfx import-graph explorer" >&2 + echo " 3d - launch the 3D \"mathematical kingdom\" map" >&2 + exit 1 +} + +run_2d() { + local OS BIN_DIR + OS="$(uname -s)" + case "$OS" in + Darwin) BIN_DIR="$DIR/release/bin_osx" ;; + Linux) BIN_DIR="$DIR/release/bin_linux" ;; + *) + echo "Unsupported OS: $OS" >&2 + exit 1 + ;; + esac + if [ ! -d "$BIN_DIR" ]; then + available="$(cd "$DIR/release" && ls -d bin_* 2>/dev/null | tr '\n' ' ')" + echo "No prebuilt MathlibExplorer binary for $OS yet." >&2 + echo "Available platform builds: ${available:-none}" >&2 + if [ "$OS" = "Linux" ]; then + echo "There is currently no Linux build in this repo (only macOS and Windows are published)." >&2 + fi + exit 1 + fi + local EXE="$BIN_DIR/MathlibExplorer" + [ -x "$EXE" ] || chmod +x "$EXE" + cd "$BIN_DIR" + exec "./MathlibExplorer" +} + +run_3d() { + local HTML="$DIR/kingdom/viewer/index.html" + if [ ! -f "$HTML" ]; then + echo "No 3D viewer found at $HTML" >&2 + exit 1 + fi + local URL="file://$HTML" + + # Prefer an "app mode" window (no tabs/address bar) for a closer-to-native + # feel matching the bgfx MathlibExplorer window; fall back to whatever the + # system's default browser association is. + if [ "$(uname -s)" = "Darwin" ]; then + if [ -d "/Applications/Google Chrome.app" ]; then + open -na "Google Chrome" --args --app="$URL" + return + fi + if [ -d "/Applications/Microsoft Edge.app" ]; then + open -na "Microsoft Edge" --args --app="$URL" + return + fi + open "$HTML" + return + fi + + for b in google-chrome google-chrome-stable chromium chromium-browser microsoft-edge; do + if command -v "$b" >/dev/null 2>&1; then + "$b" --app="$URL" >/dev/null 2>&1 & + return + fi + done + + if command -v xdg-open >/dev/null 2>&1; then + xdg-open "$HTML" + else + echo "Could not find a way to open a browser automatically." >&2 + echo "Please open this file manually: $HTML" >&2 + exit 1 + fi +} + +case "$MODE" in + 2d) run_2d ;; + 3d) run_3d ;; + *) usage ;; +esac