Plain-English statement
Independently compiled at proof commit 9ba98ec42d9fa6abd78c045315cdd8d795e37c3c; current five packet-bound audits ASTIS-RT-20261020-BoundedEntropy-1 through -5 accept only this finite-integral scope. The zero-mean case is proved separately; no extended-real entropy claim.
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Proof architecture
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Lean statement
theorem entropy_le_mul_of_measure_le {μ ν : Measure X}
[IsProbabilityMeasure μ] [IsProbabilityMeasure ν] {g : X → ℝ} {b : ℝ}
(hb : 0 ≤ b) (hν : ν ≤ ENNReal.ofReal b • μ)
(hg0 : 0 ≤ᵐ[μ] g) (hg : Integrable g μ)
(hlog : Integrable (fun x => g x * Real.log (g x)) μ) :
entropy ν g ≤ b * entropy μ g := by
have hνμ : ν ≪ μ := Measure.absolutelyContinuous_of_le_smul hν
have hgn : Integrable g ν := (hg.smul_measure ENNReal.ofReal_ne_top).mono_measure hν
have hln : Integrable (fun x => g x * Real.log (g x)) ν :=
(hlog.smul_measure ENNReal.ofReal_ne_top).mono_measure hν
have hgn0 : 0 ≤ᵐ[ν] g := hνμ.ae_le hg0
have hm0 : 0 ≤ ∫ x, g x ∂μ := integral_nonneg_of_ae hg0
rcases hm0.eq_or_lt with hm | hm
· have hgzero : g =ᵐ[μ] 0 := (integral_eq_zero_iff_of_nonneg_ae hg0 hg).1 hm.symm
have hgnzero : g =ᵐ[ν] 0 := hνμ.ae_le hgzero
have hlogzero : (fun x => g x * Real.log (g x)) =ᵐ[μ] 0 :=
hgzero.mono fun x hx => by simp [hx]
have hlognzero : (fun x => g x * Real.log (g x)) =ᵐ[ν] 0 :=
hgnzero.mono fun x hx => by simp [hx]
simp [entropy, integral_congr_ae hgzero, integral_congr_ae hgnzero,
integral_congr_ae hlogzero, integral_congr_ae hlognzero]
· calc
entropy ν g ≤ ∫ x, cost (g x) (∫ y, g y ∂μ) ∂ν :=
entropy_le_integral_cost hgn0 hgn hln hm
_ ≤ ∫ x, cost (g x) (∫ y, g y ∂μ) ∂(ENNReal.ofReal b • μ) :=
integral_mono_measure hν
(by filter_upwards [Measure.ae_smul_measure hg0 (ENNReal.ofReal b)] with x hx
exact cost_nonneg hx hm)
((integrable_cost hg hlog hm).smul_measure ENNReal.ofReal_ne_top)
_ = b * entropy μ g := by
rw [integral_smul_measure, ENNReal.toReal_ofReal hb, smul_eq_mul,
integral_cost hg hlog hm]
simp [cost]
/- The energy functional is held fixed while the probability law changes.
For the usual gradient form choose Γ f = ‖∇f‖². -/Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/BoundedPerturbation.lean:84published source at ed709a7f320d
Proof architecture
For probability measures and nonnegative g with finite entropy integrals, nu <= b mu gives Ent_nu(g) <= b Ent_mu(g).
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `have` creates a named intermediate mathematical fact.
- `calc` records an equality or inequality chain matching a paper calculation.
- `rw` rewrites by an established identity.
- `simp` normalizes through registered definitional and theorem rewrites.
- `exact` closes the current goal with an already typed term.
- `filter_upwards` moves an almost-everywhere or eventual statement into a pointwise local context.
Why the statement has this shape
The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.
Hidden assumptions and non-claims
- Measurability is represented explicitly or must be supplied by a dependency.
- Integrability is an input or proved output; a displayed integral alone does not supply it.
- Almost-everywhere hypotheses depend on the stated measure and representative.