Prove2Me

`BookProof.ChapterMaxEntropy.entropy_le_log_card` [Nonempty α] {p : α → ℝ} (hp : IsProb p) : entropy p ≤ Real.log (Fintype.card α)

Proved
BookProof.ChapterMaxEntropy.entropy_le_log_card

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterMaxEntropy.

BookProof.ChapterMaxEntropy.entropy_le_log_card [Nonempty α] {p : α → ℝ} (hp : IsProb p) : entropy p ≤ Real.log (Fintype.card α)

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

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


open Real BigOperators Finset


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

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