`BookProof.ChapterAttentionLocality.farMass_le` {beta gamma Delta R : ℝ} (hb : 0 ≤ beta) (hg : 0 ≤ gamma) (s : Fin m → ℝ) (d : Fin m → ℝ) {j₀ : Fin m} (hd0 : d j₀ = 0) (hDelta : ∀

Proved

BookProof.ChapterAttentionLocality.farMass_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionLocality.

BookProof.ChapterAttentionLocality.farMass_le {beta gamma Delta R : ℝ} (hb : 0 ≤ beta) (hg : 0 ≤ gamma) (s : Fin m → ℝ) (d : Fin m → ℝ) {j₀ : Fin m} (hd0 : d j₀ = 0) (hDelta : ∀ l, s l ≤ s j₀ + Delta) : ∑ l ∈ (window d R)ᶜ, scoreSoftmax beta (alibiScore s gamma d) l ≤ (m : ℝ) * (Real.exp (beta * Delta) * Real.exp (-(beta * (gamma * R))))

Formalization note: Lean 4 identifier BookProof.ChapterAttentionLocality.farMass_le.

Preamble

-- Generated from ChapterAttentionLocality.lean — theorem BookProof.ChapterAttentionLocality.farMass_le
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionLocality
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionLocality


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

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

Formal statement

theorem BookProof.ChapterAttentionLocality.farMass_le {beta gamma Delta R : ℝ} (hb : 0 ≤ beta) (hg : 0 ≤ gamma)
    (s : Fin m → ℝ) (d : Fin m → ℝ) {j₀ : Fin m} (hd0 : d j₀ = 0)
    (hDelta : ∀ l, s l ≤ s j₀ + Delta) :
    ∑ l ∈ (window d R)ᶜ, scoreSoftmax beta (alibiScore s gamma d) l
      ≤ (m : ℝ) * (Real.exp (beta * Delta) * Real.exp (-(beta * (gamma * R)))) := by sorry

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