`BookProof.ChapterAttentionMarkov.l1dist_push_le_of_min` {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 :

Proved

BookProof.ChapterAttentionMarkov.l1dist_push_le_of_min

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionMarkov.

BookProof.ChapterAttentionMarkov.l1dist_push_le_of_min {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) : l1dist (push P p) (push P q) ≤ (1 - m * eps) * l1dist p q

Formalization note: Lean 4 identifier BookProof.ChapterAttentionMarkov.l1dist_push_le_of_min.

Preamble

-- Generated from ChapterAttentionMarkov.lean — theorem BookProof.ChapterAttentionMarkov.l1dist_push_le_of_min
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_le_of_min {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) :
    l1dist (push P p) (push P q) ≤ (1 - m * eps) * l1dist p q := by sorry

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