`BookProof.ChapterAttentionSparse.one_sub_attendedMass_le` (beta : ℝ) (s : Fin m → ℝ) (S : Finset (Fin m)) (i : Fin m) {eps : ℝ} (h : ∀ l ∉ S, scoreSoftmax beta s l ≤ eps) : 1 - at

Proved

BookProof.ChapterAttentionSparse.one_sub_attendedMass_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionSparse.

BookProof.ChapterAttentionSparse.one_sub_attendedMass_le (beta : ℝ) (s : Fin m → ℝ) (S : Finset (Fin m)) (i : Fin m) {eps : ℝ} (h : ∀ l ∉ S, scoreSoftmax beta s l ≤ eps) : 1 - attendedMass beta s S ≤ (m - S.card : ℝ) * eps

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

Preamble

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


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

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

Formal statement

theorem BookProof.ChapterAttentionSparse.one_sub_attendedMass_le (beta : ℝ) (s : Fin m → ℝ) (S : Finset (Fin m)) (i : Fin m)
    {eps : ℝ} (h : ∀ l ∉ S, scoreSoftmax beta s l ≤ eps) :
    1 - attendedMass beta s S ≤ (m - S.card : ℝ) * eps := by sorry

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