`BookProof.ChapterAttentionMarkov.l1dist_push_attentionMatrix_le` {beta D : ℝ} (hb : 0 ≤ beta) {S : Fin m → Fin m → ℝ} {p q : Fin m → ℝ} (hp : IsProb p) (hq : IsProb q) (hD : ∀ i j

Proved

BookProof.ChapterAttentionMarkov.l1dist_push_attentionMatrix_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionMarkov.

BookProof.ChapterAttentionMarkov.l1dist_push_attentionMatrix_le {beta D : ℝ} (hb : 0 ≤ beta) {S : Fin m → Fin m → ℝ} {p q : Fin m → ℝ} (hp : IsProb p) (hq : IsProb q) (hD : ∀ i j l, S i l ≤ S i j + D) : l1dist (push (attentionMatrix beta S) p) (push (attentionMatrix beta S) q) ≤ (1 - Real.exp (-(beta * D))) * l1dist p q

Formalization note: Lean 4 identifier BookProof.ChapterAttentionMarkov.l1dist_push_attentionMatrix_le.

Preamble

-- Generated from ChapterAttentionMarkov.lean — theorem BookProof.ChapterAttentionMarkov.l1dist_push_attentionMatrix_le
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionMarkov
import Definitions.Def_ChapterDutchBook
open BookProof.ChapterDutchBook
open BookProof.ChapterAttentionMarkov


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterAttentionMarkov.l1dist_push_attentionMatrix_le {beta D : ℝ} (hb : 0 ≤ beta)
    {S : Fin m → Fin m → ℝ} {p q : Fin m → ℝ} (hp : IsProb p) (hq : IsProb q)
    (hD : ∀ i j l, S i l ≤ S i j + D) :
    l1dist (push (attentionMatrix beta S) p) (push (attentionMatrix beta S) q)
      ≤ (1 - Real.exp (-(beta * D))) * l1dist p q := by sorry

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