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Samplinglib mathematical lesson

An energy LSI on a fixed class of finite-integral tests

AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BoundedPerturbation.EnergyLogSobolev · def · Teaching coverage

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Mathematical statement

Fix a measurable space X, a measure m, a real energy density Γ(f)(x), a test-function predicate D(f), and C. The finite-integral energy LSI means: every D-test f for which f², f² log(f²), and Γ(f) are integrable, with Γ(f) nonnegative almost everywhere, satisfies Ent_m(f²) ≤ 2 C ∫Γ(f) dm. It asserts no integrability outside those tests.

\[\operatorname{Ent}_\mu(f^2)\le 2C\int\Gamma(f)\,d\mu,\qquad f\in D\text{ in the finite-integral domain}.\]
Lean statement · EnergyLogSobolev

Fix a measurable space X, a measure m, a real energy density Γ(f)(x), a test-function predicate D(f), and C. The finite-integral energy LSI means: every D-test f for which f², f² log(f²), and Γ(f) are integrable, with Γ(f) nonnegative almost everywhere, satisfies Ent_m(f²) ≤ 2 C ∫Γ(f) dm. It asserts no integrability outside those tests.

Braces mark parameters Lean can infer; square brackets request structures such as a measurable space or probability measure. Named hypotheses are mathematical premises, not facts established by this declaration. Section parameters are described in the mathematical hypotheses above; the module link retains their exact source context.

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def EnergyLogSobolev (μ : Measure X) (Γ : (X → ℝ) → X → ℝ)
    (admissible : (X → ℝ) → Prop) (C : ℝ) : Prop

Exact module and namespace context

Proof roadmap

  1. 1Use the literal real formula

    Fix a measurable space X, a measure m, a real energy density Γ(f)(x), a test-function predicate D(f), and C. The finite-integral energy LSI means: every D-test f for which f², f² log(f²), and Γ(f) are integrable, with Γ(f) nonnegative almost everywhere, satisfies Ent_m(f²) ≤ 2 C ∫Γ(f) dm. It asserts no integrability outside those tests.

Full assumptions and notation

All objects and hypotheses

  • Objects, quantifiers, constants and all finite-integral restrictions are stated in the adjacent statement and exact Lean.
  • The test predicate and energy density are fixed before changing the measure; no differential operator is transported.

Construction and meaning

1. Use the literal real formula

Fix a measurable space X, a measure m, a real energy density Γ(f)(x), a test-function predicate D(f), and C. The finite-integral energy LSI means: every D-test f for which f², f² log(f²), and Γ(f) are integrable, with Γ(f) nonnegative almost everywhere, satisfies Ent_m(f²) ≤ 2 C ∫Γ(f) dm. It asserts no integrability outside those tests.

\[\operatorname{Ent}_\mu(f^2)\le 2C\int\Gamma(f)\,d\mu,\qquad f\in D\text{ in the finite-integral domain}.\]
Corresponding Lean step

The complete definition body is displayed in the adjacent folded Lean disclosure.

Lean verification

The exact proof begins folded immediately after the mathematical argument; the statement is folded beside the claim above.

Lean construction · EnergyLogSobolev

See the adjacent exact source and the stepwise mathematical proof.

Braces mark parameters Lean can infer; square brackets request structures such as a measurable space or probability measure. Named hypotheses are mathematical premises, not facts established by this declaration. Section parameters are described in the mathematical hypotheses above; the module link retains their exact source context.

Download module (.lean)
def EnergyLogSobolev (μ : Measure X) (Γ : (X → ℝ) → X → ℝ)
    (admissible : (X → ℝ) → Prop) (C : ℝ) : Prop :=
  ∀ f : X → ℝ,
    admissible f →
    Integrable (fun x => (f x) ^ 2) μ →
    Integrable (fun x => (f x) ^ 2 * Real.log ((f x) ^ 2)) μ →
    Integrable (Γ f) μ →
    (0 ≤ᵐ[μ] Γ f) →
    entropy μ (fun x => (f x) ^ 2) ≤ 2 * C * ∫ x, Γ f x ∂μ

Exact module and namespace context

Scope and omitted-condition boundaries
  • Real-integral domain only; the extended-real entropy and arbitrary Sobolev closure are not supplied.
  • No geometric base LSI, gradient identification, PI, T2, diffusion, or model theorem is supplied.
  • No public manuscript identity or private application data is part of this contribution.
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