`BookProof.ChapterAttentionSparse.maskedSoftmax_sub_of_mem` (beta : ℝ) (s : Fin m → ℝ) {S : Finset (Fin m)} {j : Fin m} (hj : j ∈ S) : maskedSoftmax beta s S j - scoreSoftmax beta

Proved

BookProof.ChapterAttentionSparse.maskedSoftmax_sub_of_mem

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionSparse.

BookProof.ChapterAttentionSparse.maskedSoftmax_sub_of_mem (beta : ℝ) (s : Fin m → ℝ) {S : Finset (Fin m)} {j : Fin m} (hj : j ∈ S) : maskedSoftmax beta s S j - scoreSoftmax beta s j = scoreSoftmax beta s j * (1 - attendedMass beta s S) / attendedMass beta s S

Formalization note: Lean 4 identifier BookProof.ChapterAttentionSparse.maskedSoftmax_sub_of_mem.

Preamble

-- Generated from ChapterAttentionSparse.lean — theorem BookProof.ChapterAttentionSparse.maskedSoftmax_sub_of_mem
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionSparse
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterAttentionMasking
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionSparse


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder
open BookProof.ChapterAttentionMasking

variable {m : ℕ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]

Formal statement

theorem BookProof.ChapterAttentionSparse.maskedSoftmax_sub_of_mem (beta : ℝ) (s : Fin m → ℝ) {S : Finset (Fin m)}
    {j : Fin m} (hj : j ∈ S) :
    maskedSoftmax beta s S j - scoreSoftmax beta s j
      = scoreSoftmax beta s j * (1 - attendedMass beta s S) / attendedMass beta s S := by sorry

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