`BookProof.ProbabilityClockStochastic.preserves_prob_iff_isColumnStochastic` (M : Matrix (Fin 2) (Fin 2) ℝ) : (∀ v, IsProbabilityVector v → IsProbabilityVector (M.mulVec v)) ↔ IsCo

Proved

BookProof.ProbabilityClockStochastic.preserves_prob_iff_isColumnStochastic

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterProbabilityClockStochastic.

BookProof.ProbabilityClockStochastic.preserves_prob_iff_isColumnStochastic (M : Matrix (Fin 2) (Fin 2) ℝ) : (∀ v, IsProbabilityVector v → IsProbabilityVector (M.mulVec v)) ↔ IsColumnStochastic M

Formalization note: Lean 4 identifier BookProof.ProbabilityClockStochastic.preserves_prob_iff_isColumnStochastic.

Preamble

-- Generated from ChapterProbabilityClockStochastic.lean — theorem BookProof.ProbabilityClockStochastic.preserves_prob_iff_isColumnStochastic
import Mathlib
import Definitions.Def_ChapterProbabilityClockStochastic
open BookProof.ProbabilityClockStochastic



open Matrix
open scoped Norms.Operator

Formal statement

theorem BookProof.ProbabilityClockStochastic.preserves_prob_iff_isColumnStochastic (M : Matrix (Fin 2) (Fin 2) ℝ) :
    (∀ v, IsProbabilityVector v → IsProbabilityVector (M.mulVec v))
      ↔ IsColumnStochastic M := by sorry

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