`BookProof.ChapterAttentionSparse.l1dist_maskedSoftmax_eq_zero_iff_mass_one` (beta : ℝ) (s : Fin m → ℝ) {S : Finset (Fin m)} (hS : S.Nonempty) (i : Fin m) : l1dist (maskedSoftmax b

Proved

BookProof.ChapterAttentionSparse.l1dist_maskedSoftmax_eq_zero_iff_mass_one

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionSparse.

BookProof.ChapterAttentionSparse.l1dist_maskedSoftmax_eq_zero_iff_mass_one (beta : ℝ) (s : Fin m → ℝ) {S : Finset (Fin m)} (hS : S.Nonempty) (i : Fin m) : l1dist (maskedSoftmax beta s S) (scoreSoftmax beta s) = 0 ↔ attendedMass beta s S = 1

Formalization note: Lean 4 identifier BookProof.ChapterAttentionSparse.l1dist_maskedSoftmax_eq_zero_iff_mass_one.

Preamble

-- Generated from ChapterAttentionSparse.lean — theorem BookProof.ChapterAttentionSparse.l1dist_maskedSoftmax_eq_zero_iff_mass_one
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionSparse
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterAttentionMarkov
import Definitions.Def_ChapterAttentionMasking
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionSparse


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder
open BookProof.ChapterAttentionMarkov
open BookProof.ChapterAttentionMasking

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

Formal statement

theorem BookProof.ChapterAttentionSparse.l1dist_maskedSoftmax_eq_zero_iff_mass_one (beta : ℝ) (s : Fin m → ℝ)
    {S : Finset (Fin m)} (hS : S.Nonempty) (i : Fin m) :
    l1dist (maskedSoftmax beta s S) (scoreSoftmax beta s) = 0
      ↔ attendedMass beta s S = 1 := by sorry

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