`BookProof.ChapterAttentionFactorization.shannonEntropy_prodSoftmax` (beta : ℝ) (s₁ : Fin m₁ → ℝ) (s₂ : Fin m₂ → ℝ) (i₁ : Fin m₁) (i₂ : Fin m₂) : shannonEntropyProd...

Proved

BookProof.ChapterAttentionFactorization.shannonEntropy_prodSoftmax

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionFactorization.

BookProof.ChapterAttentionFactorization.shannonEntropy_prodSoftmax (beta : ℝ) (s₁ : Fin m₁ → ℝ) (s₂ : Fin m₂ → ℝ) (i₁ : Fin m₁) (i₂ : Fin m₂) : shannonEntropyProd (prodSoftmax beta s₁ s₂) = shannonEntropy (scoreSoftmax beta s₁) + shannonEntropy (scoreSoftmax beta s₂)

Formalization note: Lean 4 identifier BookProof.ChapterAttentionFactorization.shannonEntropy_prodSoftmax.

Preamble

-- Generated from ChapterAttentionFactorization.lean — theorem BookProof.ChapterAttentionFactorization.shannonEntropy_prodSoftmax
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionFactorization
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterAttentionEntropy
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionFactorization


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder
open BookProof.ChapterAttentionEntropy

variable {m₁ m₂ : ℕ}

Formal statement

theorem BookProof.ChapterAttentionFactorization.shannonEntropy_prodSoftmax (beta : ℝ) (s₁ : Fin m₁ → ℝ) (s₂ : Fin m₂ → ℝ)
    (i₁ : Fin m₁) (i₂ : Fin m₂) :
    shannonEntropyProd (prodSoftmax beta s₁ s₂)
      = shannonEntropy (scoreSoftmax beta s₁) + shannonEntropy (scoreSoftmax beta s₂) := by sorry

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