`BookProof.ChapterAttentionEntropy.shannonEntropy_scoreSoftmax_le_log_card` (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) : shannonEntropy (fun j => scoreSoftmax beta s j) ≤ Real.log m

Proved

BookProof.ChapterAttentionEntropy.shannonEntropy_scoreSoftmax_le_log_card

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionEntropy.

BookProof.ChapterAttentionEntropy.shannonEntropy_scoreSoftmax_le_log_card (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) : shannonEntropy (fun j => scoreSoftmax beta s j) ≤ Real.log m

Formalization note: Lean 4 identifier BookProof.ChapterAttentionEntropy.shannonEntropy_scoreSoftmax_le_log_card.

Preamble

-- Generated from ChapterAttentionEntropy.lean — theorem BookProof.ChapterAttentionEntropy.shannonEntropy_scoreSoftmax_le_log_card
import Definitions.Def_ChapterSoftmaxBorn
import Mathlib
import Definitions.Def_ChapterAttentionEntropy
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionEntropy


open scoped BigOperators

noncomputable section


open Filter Topology BookProof.ChapterSoftmaxBorn BookProof.ChapterSoftmaxSharpness

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterAttentionEntropy.shannonEntropy_scoreSoftmax_le_log_card (beta : ℝ) (s : Fin m → ℝ) (i : Fin m) :
    shannonEntropy (fun j => scoreSoftmax beta s j) ≤ Real.log m := by sorry

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