feat(Analysis/SpecialFunctions): Erlang's loss formula

[matt-w-horn]

Adds Mathlib/Analysis/SpecialFunctions/ErlangB.lean. Real.erlangB a c is the blocking probability of an M/M/c/c loss system, defined by the standard recursion, with its range bounds, monotonicity in the load, antitonicity in the number of servers, and the closed form Real.erlangB_eq_div.

This upstreams a file that has been in use in Overload, a standalone Lean library on retry and overload dynamics, where it is sorry-free and axiom-audited: Overload/Queueing/Erlang.lean. The domain-specific parts of that file - the loop instances and the pinned numeric evaluations - stay downstream; what is here is the loss formula and its general properties.

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 37ffc7c7ef

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Analysis.SpecialFunctions.ErlangB (new file)	806

---

Declarations diff (regex)

+ erlangB
+ erlangB_antitone
+ erlangB_carried_le
+ erlangB_denom_pos
+ erlangB_eq_div
+ erlangB_le_one
+ erlangB_mem_Icc
+ erlangB_mono_load
+ erlangB_mul_sum
+ erlangB_nonneg
+ erlangB_succ
+ erlangB_succ_le
+ erlangB_zero

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)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 37ffc7c).

• +13 new declarations
• −0 removed declarations

+Real.erlangB
+Real.erlangB_antitone
+Real.erlangB_carried_le
+Real.erlangB_denom_pos
+Real.erlangB_eq_div
+Real.erlangB_le_one
+Real.erlangB_mem_Icc
+Real.erlangB_mono_load
+Real.erlangB_mul_sum
+Real.erlangB_nonneg
+Real.erlangB_succ
+Real.erlangB_succ_le
+Real.erlangB_zero

---

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 37ffc7c7ef
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