`BookProof.ChapterAttentionFactorization.prodSoftmax_pos` (beta : ℝ) (s₁ : Fin m₁ → ℝ) (s₂ : Fin m₂ → ℝ) (j : Fin m₁ × Fin m₂) : 0 < prodSoftmax beta s₁ s₂ j

Proved

BookProof.ChapterAttentionFactorization.prodSoftmax_pos

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionFactorization.

BookProof.ChapterAttentionFactorization.prodSoftmax_pos (beta : ℝ) (s₁ : Fin m₁ → ℝ) (s₂ : Fin m₂ → ℝ) (j : Fin m₁ × Fin m₂) : 0 < prodSoftmax beta s₁ s₂ j

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

Preamble

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


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m₁ m₂ : ℕ}

Formal statement

theorem BookProof.ChapterAttentionFactorization.prodSoftmax_pos (beta : ℝ) (s₁ : Fin m₁ → ℝ) (s₂ : Fin m₂ → ℝ)
    (j : Fin m₁ × Fin m₂) : 0 < prodSoftmax beta s₁ s₂ j := by sorry

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