feat: Émile Borel's classic "Growth Lemma"

[kebekus]

Establish Émile Borel's classic Growth Lemma: if S : ℝ → ℝ is monotone on Set.Ici a and satisfies 1 ≤ S there, then

∀ᶠ r in volume.cofinite ⊓ atTop, S (r + (S r)⁻¹) ≤ 2 * S r.

In other words: The inequality S (r + (S r)⁻¹) ≤ 2 * S r holds for all sufficiently large r outside an exceptional set E of finite Lebesgue measure. In Value Distribution Theory, this statement is central to the proof of the "Lemma on the Logarithmic Derivatives".

This material is used in Project VD, formalizing Value Distribution Theory for meromorphic functions on the complex plane. Claude Code was used to create this PR.

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[github-actions]

PR summary 7462c959ea

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.MeasureTheory.Function.BorelGrowth (new file)	2299

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Declarations diff (regex)

+ MonotoneOn.eventually_le_two_mul

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 7462c95).

• +1 new declarations
• −0 removed declarations

+MonotoneOn.eventually_le_two_mul

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No changes to strong technical debt.

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

Current number	Change	Type (weak)
5004	1	exposed public sections

Current commit 7462c959ea
Reference commit 5a0753dc72

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).