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

Proved

BookProof.ChapterAttentionLocality.scoreSoftmax_alibi_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionLocality.

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

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

Preamble

-- Generated from ChapterAttentionLocality.lean — theorem BookProof.ChapterAttentionLocality.scoreSoftmax_alibi_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.scoreSoftmax_alibi_le {beta gamma Delta : ℝ} (hb : 0 ≤ beta) (s : Fin m → ℝ)
    (d : Fin m → ℝ) {j₀ : Fin m} (hd0 : d j₀ = 0) (hDelta : ∀ l, s l ≤ s j₀ + Delta)
    (l : Fin m) :
    scoreSoftmax beta (alibiScore s gamma d) l
      ≤ Real.exp (beta * Delta) * Real.exp (-(beta * (gamma * d l))) := by sorry

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