Samplinglib
Lean gate not recorded for this source state main · 9bb7a7394ab0
Samplinglib mathematical lesson

Finite-integral function entropy

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

Read this first

Mathematical statement

For a measurable space X, a measure m, and a real function g, define its real function entropy by the difference of the integral of g log g and the mean of g times the logarithm of that mean. Applications below impose integrability explicitly; this literal real formula is not an extended-real entropy.

\[\operatorname{Ent}_\mu(g)=\int g\log g\,d\mu-m\log m,\qquad m=\int g\,d\mu.\]
Lean statement · entropy

For a measurable space X, a measure m, and a real function g, define its real function entropy by the difference of the integral of g log g and the mean of g times the logarithm of that mean. Applications below impose integrability explicitly; this literal real formula is not an extended-real entropy.

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)
noncomputable def entropy (μ : Measure X) (g : X → ℝ) : ℝ

Exact module and namespace context

Proof roadmap

  1. 1Use the literal real formula

    For a measurable space X, a measure m, and a real function g, define its real function entropy by the difference of the integral of g log g and the mean of g times the logarithm of that mean. Applications below impose integrability explicitly; this literal real formula is not an extended-real entropy.

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

For a measurable space X, a measure m, and a real function g, define its real function entropy by the difference of the integral of g log g and the mean of g times the logarithm of that mean. Applications below impose integrability explicitly; this literal real formula is not an extended-real entropy.

\[\operatorname{Ent}_\mu(g)=\int g\log g\,d\mu-m\log m,\qquad m=\int g\,d\mu.\]
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 · entropy

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)
noncomputable def entropy (μ : Measure X) (g : X → ℝ) : ℝ :=
  (∫ x, g x * Real.log (g x) ∂μ) - (∫ x, g x ∂μ) * Real.log (∫ x, g 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.
Sources, dependencies, and reuse

Source and reuse

Samplinglib parents called

    Mathlib API called (external library)

    No direct Mathlib call recorded; see the Samplinglib parents.

    Mathematical sources

    Samplinglib prose is not a quotation or a source-equivalence certificate. Definitions and aliases are explained as constructions, not counted as new mathematical proofs.