`BookProof.ChapterAttentionMixing.l1dist_pushIter_le` {P : Fin m → Fin m → ℝ} {p q : Fin m → ℝ} {eps : ℝ} (hP : IsStochastic P) (hmin : ∀ i j, eps ≤ P i j) (hp : IsProb p) (hq : Is

Proved

BookProof.ChapterAttentionMixing.l1dist_pushIter_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionMixing.

BookProof.ChapterAttentionMixing.l1dist_pushIter_le {P : Fin m → Fin m → ℝ} {p q : Fin m → ℝ} {eps : ℝ} (hP : IsStochastic P) (hmin : ∀ i j, eps ≤ P i j) (hp : IsProb p) (hq : IsProb q) (n : ℕ) : l1dist (pushIter P n p) (pushIter P n q) ≤ (1 - m * eps) ^ n * l1dist p q

Formalization note: Lean 4 identifier BookProof.ChapterAttentionMixing.l1dist_pushIter_le.

Preamble

-- Generated from ChapterAttentionMixing.lean — theorem BookProof.ChapterAttentionMixing.l1dist_pushIter_le
import Definitions.Def_ChapterSoftmaxSharpness
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionMixing
import Definitions.Def_ChapterDutchBook
import Definitions.Def_ChapterAttentionMarkov
open BookProof.ChapterAttentionMixing


open scoped BigOperators

open Filter Topology

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder
open BookProof.ChapterAttentionMarkov

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterAttentionMixing.l1dist_pushIter_le {P : Fin m → Fin m → ℝ} {p q : Fin m → ℝ} {eps : ℝ}
    (hP : IsStochastic P) (hmin : ∀ i j, eps ≤ P i j) (hp : IsProb p) (hq : IsProb q)
    (n : ℕ) :
    l1dist (pushIter P n p) (pushIter P n q) ≤ (1 - m * eps) ^ n * l1dist p q := by sorry

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