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

Proved

BookProof.ChapterSoftmaxMaxEntropy.shannonEntropy_le_of_meanScore_eq

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSoftmaxMaxEntropy.

BookProof.ChapterSoftmaxMaxEntropy.shannonEntropy_le_of_meanScore_eq (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) {p : Fin m → ℝ} (hp0 : ∀ j, 0 ≤ p j) (hpsum : ∑ j, p j = 1) (hmean : ∑ j, p j * s j = meanScore beta s) : shannonEntropy p ≤ shannonEntropy (scoreSoftmax beta s)

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

Preamble

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

View graph