`BookProof.ChapterAttentionMarkov.push_nonneg` {P : Fin m → Fin m → ℝ} {p : Fin m → ℝ} (hP : IsStochastic P) (hp : ∀ j, 0 ≤ p j) (j : Fin m) : 0 ≤ push P p j

Proved

BookProof.ChapterAttentionMarkov.push_nonneg

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionMarkov.

BookProof.ChapterAttentionMarkov.push_nonneg {P : Fin m → Fin m → ℝ} {p : Fin m → ℝ} (hP : IsStochastic P) (hp : ∀ j, 0 ≤ p j) (j : Fin m) : 0 ≤ push P p j

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

Preamble

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


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m : ℕ}

Formal statement

theorem BookProof.ChapterAttentionMarkov.push_nonneg {P : Fin m → Fin m → ℝ} {p : Fin m → ℝ} (hP : IsStochastic P)
    (hp : ∀ j, 0 ≤ p j) (j : Fin m) : 0 ≤ push P p j := by sorry

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