feat(Analysis/FunctionalSpaces): the one-dimensional Poincaré inequality

[alejandro-soto-franco]

Adds the one-dimensional Poincaré inequality MeasureTheory.poincare_1d: for
f : ℝ → E continuous on [a, b], continuously differentiable on (a, b), with
f a = 0,

∫⁻ x in Icc a b, ‖f x‖ₑ ^ 2 ≤ ENNReal.ofReal ((b - a) ^ 2) * ∫⁻ x in Icc a b, ‖deriv f x‖ₑ ^ 2.

The statement uses lower Lebesgue integrals of extended norms, so no integrability
hypothesis on the derivative is needed (the bound is automatic when the right-hand
side is infinite), and it holds for an arbitrary normed space E. The general
(not necessarily complete) E is handled by wlog … : CompleteSpace E generalizing E
together with the completion-embedding idiom, as in
Mathlib/MeasureTheory/Integral/IntervalIntegral/DistLEIntegral.lean.

Also adds the supporting ENNReal.lintegral_sq_le_measure_mul_lintegral_sq
(Cauchy–Schwarz for the lower Lebesgue integral against the constant 1, the
p = q = 2 case of Hölder) to MeanInequalities.lean.

This is the base case for an n-dimensional convex-domain version (Fubini over
coordinate slices), planned as a follow-up. The statement form (extended-norm
Lebesgue integrals, arbitrary normed space) follows discussion with Sébastien
Gouëzel on Zulip.

---

AI assistance: this contribution was developed with the help of Claude Code Opus 4.8; the proof was
drafted and iteratively refined against the Lean compiler, then reviewed by me.

[github-actions]

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[github-actions]

PR summary 70eb62d408

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Analysis.FunctionalSpaces.PoincareInequality (new file)	2513

---

Declarations diff (regex)

+ MeasureTheory.poincare_1d
+ MeasureTheory.poincare_1d_uIcc
+ enorm_sub_le_lintegral_deriv_aux
+ enorm_sub_le_lintegral_deriv_of_contDiffOn_Ioo
+ rpow_lintegral_le_measure_rpow_mul_lintegral_rpow

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 70eb62d).

• +4 new declarations
• −0 removed declarations

+ENNReal.rpow_lintegral_le_measure_rpow_mul_lintegral_rpow
+MeasureTheory.poincare_1d
+MeasureTheory.poincare_1d_uIcc
+enorm_sub_le_lintegral_deriv_of_contDiffOn_Ioo

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 70eb62d408
Reference commit 07f4b8dcd0

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).

[eric-wieser]

Perhaps also worth having the UIcc version with enndist a b^2 as the scale factor?

[alejandro-soto-franco]

Good idea, adding it.

[alejandro-soto-franco]

Done, added.

[alejandro-soto-franco]

After conversations in the Zulip thread mentioned at the top, I refactored as follows:

• Generalised p = 2 to any 1 ≤ p. The power-mean step is a
  new helper ENNReal.rpow_lintegral_le_measure_rpow_mul_lintegral_rpow in MeanInequalities.
• Consolidated the FTC bound into IntervalIntegral/ContDiff.lean. I added the open-interval generalisation I used here and made the existing ..._Icc version by Gouëzel a corollary (signature unchanged).
• Switched to using the toComplₗᵢ completion idiom; thought it was more appropriate.
• Renamed away the _lp suffix as it is no longer relevant and did some have-folding.