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Project-declaredLean 4.32.0 ยท mathlib@81a5d257c8e4

Differential Entropy nonneg of prob le one

CanonicalEnsemble.differentialEntropy_nonneg_of_prob_le_one

physicsquantum field theoryrelativitybasic

Plain-language statement

General entropy non-negativity under a pointwise upper bound probability โ‰ค 1. This assumption holds automatically in the finite/counting case (since sums bound each term), but can fail in general (continuous) settings; hence we separate it as a hypothesis. Finite case: see CanonicalEnsemble.entropy_nonneg in Finite.

Exact Lean statement

lemma differentialEntropy_nonneg_of_prob_le_one
    (T : Temperature) [IsFiniteMeasure (๐“’.ฮผBolt T)] [NeZero ๐“’.ฮผ]
    (hInt : Integrable (fun i => Real.log (๐“’.probability T i)) (๐“’.ฮผProd T))
    (hP_le_one : โˆ€ i, ๐“’.probability T i โ‰ค 1) :
    0 โ‰ค ๐“’.differentialEntropy T

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
lemma differentialEntropy_nonneg_of_prob_le_one    (T : Temperature) [IsFiniteMeasure (๐“’.ฮผBolt T)] [NeZero ๐“’.ฮผ]    (hInt : Integrable (fun i => Real.log (๐“’.probability T i)) (๐“’.ฮผProd T))    (hP_le_one : โˆ€ i, ๐“’.probability T i โ‰ค 1) :    0 โ‰ค ๐“’.differentialEntropy T := by  rw [differentialEntropy]  refine mul_nonneg_of_nonpos_of_nonpos (neg_nonpos.mpr kB_nonneg) ?_  simpa using integral_mono_ae hInt (integrable_const 0) (Filter.Eventually.of_forall fun i =>    Real.log_nonpos (๐“’.probability_nonneg T i) (hP_le_one i))
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean:587-595

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Plain-language statement

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physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

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Plain-language statement

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Source project: Physlib

Person-level attribution pending.

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