`BookProof.ChapterSoftmaxDivergence.klDiv_scoreSoftmax` (beta gamma : ℝ) (s : Fin m → ℝ) (i : Fin m) : klDiv (scoreSoftmax beta s) (scoreSoftmax gamma s) = logPartition gamma s - l

Proved

BookProof.ChapterSoftmaxDivergence.klDiv_scoreSoftmax

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxDivergence.

BookProof.ChapterSoftmaxDivergence.klDiv_scoreSoftmax (beta gamma : ℝ) (s : Fin m → ℝ) (i : Fin m) : klDiv (scoreSoftmax beta s) (scoreSoftmax gamma s) = logPartition gamma s - logPartition beta s - (gamma - beta) * meanScore beta s

Formalization note: Lean 4 identifier BookProof.ChapterSoftmaxDivergence.klDiv_scoreSoftmax.

Preamble

-- Generated from ChapterSoftmaxDivergence.lean — theorem BookProof.ChapterSoftmaxDivergence.klDiv_scoreSoftmax
import Definitions.Def_ChapterAttentionEntropy
import Mathlib
import Definitions.Def_ChapterSoftmaxDivergence
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxFluctuation
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxDivergence


open scoped BigOperators

noncomputable section


open BookProof.ChapterAttentionEntropy BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxFluctuation

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterSoftmaxDivergence.klDiv_scoreSoftmax (beta gamma : ℝ) (s : Fin m → ℝ) (i : Fin m) :
    klDiv (scoreSoftmax beta s) (scoreSoftmax gamma s)
      = logPartition gamma s - logPartition beta s
          - (gamma - beta) * meanScore beta s := by sorry

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