`BookProof.ChapterE.stochastic_uniform_to_vertex_singular` (M : Matrix (Fin 2) (Fin 2) ℝ) (hnonneg : ∀ i j, 0 ≤ M i j) (hcol : ∀ j, ∑ i, M i j = 1) (huniform : M *ᵥ ![1 / 2, 1 / 2]

Proved

BookProof.ChapterE.stochastic_uniform_to_vertex_singular

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterE.

BookProof.ChapterE.stochastic_uniform_to_vertex_singular (M : Matrix (Fin 2) (Fin 2) ℝ) (hnonneg : ∀ i j, 0 ≤ M i j) (hcol : ∀ j, ∑ i, M i j = 1) (huniform : M *ᵥ ![1 / 2, 1 / 2] = ![1, 0]) : M.det = 0

Formalization note: Lean 4 identifier BookProof.ChapterE.stochastic_uniform_to_vertex_singular.

Preamble

-- Generated from ChapterE.lean — theorem BookProof.ChapterE.stochastic_uniform_to_vertex_singular
import Mathlib
import Definitions.Def_ChapterE
open BookProof.ChapterE


open scoped Matrix BigOperators
open Filter
open scoped Topology

Formal statement

theorem BookProof.ChapterE.stochastic_uniform_to_vertex_singular
    (M : Matrix (Fin 2) (Fin 2) ℝ)
    (hnonneg : ∀ i j, 0 ≤ M i j)
    (hcol : ∀ j, ∑ i, M i j = 1)
    (huniform : M *ᵥ ![1 / 2, 1 / 2] = ![1, 0]) :
    M.det = 0 := by sorry

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