feat(Analysis/SpecificLimits): asymptotics of counting functions

[matt-w-horn]

Adds Mathlib/Analysis/SpecificLimits/Counting.lean. If the n-th point of an increasing sequence satisfies a n / n → τ with τ > 0, then the associated counting function satisfies c t / t → 1 / τ.

This generalizes results in use in Overload, a standalone Lean library on retry and overload dynamics, where they are sorry-free and axiom-audited: Overload/Queueing/Little.lean. There they are stated over a sample-path structure and used to prove Little's law; the counting-rate core needs none of that, so it is stated here for a bare sequence.

Generated by Claude Fable based on a spec I provided, then tested and reviewed extremely thoroughly myself. I performed adversarial testing on the output, including a full set of suites used in the above library. Dependencies of this have been thoroughly tested as well.

Assisted-by: Claude Fable 5

[github-actions]

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[github-actions]

PR summary f7bc0bfbfb

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Analysis.SpecificLimits.Counting (new file)	1491

---

Declarations diff (regex)

+ tendsto_comp_count_div_atTop
+ tendsto_count_atTop
+ tendsto_count_div_atTop

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean -- pending)

Computed after the build finishes.

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (weak)
5048	1	exposed public sections

Current commit f7bc0bfbfb
Reference commit 375d54da29

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[matt-w-horn]

LLM-generated