`BookProof.ChapterSoftmaxMaxEntropy.shannonEntropy_le_crossEntropy` {p q : Fin m → ℝ} (hp0 : ∀ j, 0 ≤ p j) (hpsum : ∑ j, p j = 1) (hq0 : ∀ j, 0 < q j) (hqsum : ∑ j, q j ≤ 1) : shan

Proved

BookProof.ChapterSoftmaxMaxEntropy.shannonEntropy_le_crossEntropy

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxMaxEntropy.

BookProof.ChapterSoftmaxMaxEntropy.shannonEntropy_le_crossEntropy {p q : Fin m → ℝ} (hp0 : ∀ j, 0 ≤ p j) (hpsum : ∑ j, p j = 1) (hq0 : ∀ j, 0 < q j) (hqsum : ∑ j, q j ≤ 1) : shannonEntropy p ≤ crossEntropy p q

Formalization note: Lean 4 identifier BookProof.ChapterSoftmaxMaxEntropy.shannonEntropy_le_crossEntropy.

Preamble

-- Generated from ChapterSoftmaxMaxEntropy.lean — theorem BookProof.ChapterSoftmaxMaxEntropy.shannonEntropy_le_crossEntropy
import Definitions.Def_ChapterAttentionEntropy
import Definitions.Def_ChapterSoftmaxSharpness
import Mathlib
import Definitions.Def_ChapterSoftmaxMaxEntropy
open BookProof.ChapterSoftmaxMaxEntropy


open scoped BigOperators

noncomputable section


open BookProof.ChapterAttentionEntropy BookProof.ChapterSoftmaxSharpness

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterSoftmaxMaxEntropy.shannonEntropy_le_crossEntropy {p q : Fin m → ℝ} (hp0 : ∀ j, 0 ≤ p j)
    (hpsum : ∑ j, p j = 1) (hq0 : ∀ j, 0 < q j) (hqsum : ∑ j, q j ≤ 1) :
    shannonEntropy p ≤ crossEntropy p q := by sorry

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