Prove2Me

`BookProof.ChapterSoftmaxMaxEntropy.softmax_free_energy_le` (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) {p : Fin m → ℝ} (hp0 : ∀ j, 0 ≤ p j) (hpsum : ∑ j, p j = 1) : -logPartition beta

Proved
BookProof.ChapterSoftmaxMaxEntropy.softmax_free_energy_le

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_le (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) {p : Fin m → ℝ} (hp0 : ∀ j, 0 ≤ p j) (hpsum : ∑ j, p j = 1) : -logPartition beta s ≤ -(beta * ∑ j, p j * s j) - shannonEntropy p

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

Preamble
-- Generated from ChapterSoftmaxMaxEntropy.lean — theorem BookProof.ChapterSoftmaxMaxEntropy.softmax_free_energy_le
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_le (beta : ℝ) (s : Fin m → ℝ) (i : Fin m)
    {p : Fin m → ℝ} (hp0 : ∀ j, 0 ≤ p j) (hpsum : ∑ j, p j = 1) :
    -logPartition beta s ≤ -(beta * ∑ j, p j * s j) - shannonEntropy p := by sorry

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