`BookProof.ChapterSoftmaxMaxEntropy.crossEntropy_scoreSoftmax` (beta : ℝ) (s : Fin m → ℝ) {p : Fin m → ℝ} (hpsum : ∑ j, p j = 1) : crossEntropy p (scoreSoftmax beta s) = logPartiti

Proved

BookProof.ChapterSoftmaxMaxEntropy.crossEntropy_scoreSoftmax

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxMaxEntropy.

BookProof.ChapterSoftmaxMaxEntropy.crossEntropy_scoreSoftmax (beta : ℝ) (s : Fin m → ℝ) {p : Fin m → ℝ} (hpsum : ∑ j, p j = 1) : crossEntropy p (scoreSoftmax beta s) = logPartition beta s - beta * ∑ j, p j * s j

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

Preamble

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


open scoped BigOperators

noncomputable section


open BookProof.ChapterAttentionEntropy BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxFluctuation

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterSoftmaxMaxEntropy.crossEntropy_scoreSoftmax (beta : ℝ) (s : Fin m → ℝ) {p : Fin m → ℝ}
    (hpsum : ∑ j, p j = 1) :
    crossEntropy p (scoreSoftmax beta s) = logPartition beta s - beta * ∑ j, p j * s j := by sorry

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