Prove2Me

`BookProof.ChapterSoftmaxMaxEntropy.softmax_free_energy_eq` (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) : -(beta * ∑ j, scoreSoftmax beta s j * s j) - shannonEntropy (scoreSoftmax beta

Proved
BookProof.ChapterSoftmaxMaxEntropy.softmax_free_energy_eq

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxMaxEntropy.

BookProof.ChapterSoftmaxMaxEntropy.softmax_free_energy_eq (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) : -(beta * ∑ j, scoreSoftmax beta s j * s j) - shannonEntropy (scoreSoftmax beta s) = -logPartition beta s

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

Preamble
-- Generated from ChapterSoftmaxMaxEntropy.lean — theorem BookProof.ChapterSoftmaxMaxEntropy.softmax_free_energy_eq
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.softmax_free_energy_eq (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) :
    -(beta * ∑ j, scoreSoftmax beta s j * s j) - shannonEntropy (scoreSoftmax beta s)
      = -logPartition beta s := by sorry

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