`BookProof.ChapterAttentionLocality.scoreSoftmax_alibi_antitone` {beta gamma : ℝ} (hb : 0 ≤ beta) (hg : 0 ≤ gamma) (c : ℝ) (d : Fin m → ℝ) {i j : Fin m} (hij : d i ≤ d j) : scoreSo

Proved

BookProof.ChapterAttentionLocality.scoreSoftmax_alibi_antitone

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionLocality.

BookProof.ChapterAttentionLocality.scoreSoftmax_alibi_antitone {beta gamma : ℝ} (hb : 0 ≤ beta) (hg : 0 ≤ gamma) (c : ℝ) (d : Fin m → ℝ) {i j : Fin m} (hij : d i ≤ d j) : scoreSoftmax beta (alibiScore (fun _ => c) gamma d) j ≤ scoreSoftmax beta (alibiScore (fun _ => c) gamma d) i

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

Preamble

-- Generated from ChapterAttentionLocality.lean — theorem BookProof.ChapterAttentionLocality.scoreSoftmax_alibi_antitone
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_antitone {beta gamma : ℝ} (hb : 0 ≤ beta) (hg : 0 ≤ gamma)
    (c : ℝ) (d : Fin m → ℝ) {i j : Fin m} (hij : d i ≤ d j) :
    scoreSoftmax beta (alibiScore (fun _ => c) gamma d) j
      ≤ scoreSoftmax beta (alibiScore (fun _ => c) gamma d) i := by sorry

View graph