Prove2Me

`BookProof.ChapterMaxEntropy.entropy_eq_log_card_iff` [Nonempty α] {p : α → ℝ} (hp : IsProb p) : entropy p = Real.log (Fintype.card α) ↔ p = uniform α

Proved
BookProof.ChapterMaxEntropy.entropy_eq_log_card_iff

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterMaxEntropy.

BookProof.ChapterMaxEntropy.entropy_eq_log_card_iff [Nonempty α] {p : α → ℝ} (hp : IsProb p) : entropy p = Real.log (Fintype.card α) ↔ p = uniform α

Formalization note: Lean 4 identifier BookProof.ChapterMaxEntropy.entropy_eq_log_card_iff.

Preamble
-- Generated from ChapterMaxEntropy.lean — theorem BookProof.ChapterMaxEntropy.entropy_eq_log_card_iff
import Mathlib
import Definitions.Def_ChapterMaxEntropy
import Definitions.Def_ChapterA3n
import Definitions.Def_ChapterDutchBook
import Definitions.Def_ChapterIrreversible
open BookProof.ChapterA3n
open BookProof.ChapterDutchBook
open BookProof.ChapterIrreversible
open BookProof.ChapterMaxEntropy


open Real BigOperators Finset


variable {α : Type*} [Fintype α]
Formal statement
theorem BookProof.ChapterMaxEntropy.entropy_eq_log_card_iff [Nonempty α] {p : α → ℝ} (hp : IsProb p) :
    entropy p = Real.log (Fintype.card α) ↔ p = uniform α := by sorry

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