/- SamplingOptimisationGeometryBench public-dev reference.
   Reuses the existing project proof; retain LICENSE and NOTICE. -/
import AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BernoulliLogSobolev
import AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianCompactEntropy
import AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianFlipEnergy
import Mathlib.Topology.Order.OrderClosed

open MeasureTheory ProbabilityTheory Filter
open scoped Topology BigOperators ENNReal
open AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities

example
    (f : ℝ → ℝ) (hf : ContDiff ℝ 2 f) (hs : HasCompactSupport f) :
    let γ : Measure ℝ := gaussianReal 0 1
    (∫ x, (f x)^2 * Real.log ((f x)^2) ∂γ) -
      (∫ x, (f x)^2 ∂γ) * Real.log (∫ x, (f x)^2 ∂γ) ≤
    2 * ∫ x, (deriv f x)^2 ∂γ := by
  -- Replace this comment with your proof.