`BookProof.SymmetryEntropy.isDeterministicCol_iff_entropy_eq_zero` {U : Fin n → Fin n → ℂ} {a : Fin n} (hcol : ∑ k, ‖U k a‖ ^ 2 = 1) : IsDeterministicCol U a ↔ entropy (bornCol U a

Proved

BookProof.SymmetryEntropy.isDeterministicCol_iff_entropy_eq_zero

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSymmetryEntropy.

BookProof.SymmetryEntropy.isDeterministicCol_iff_entropy_eq_zero {U : Fin n → Fin n → ℂ} {a : Fin n} (hcol : ∑ k, ‖U k a‖ ^ 2 = 1) : IsDeterministicCol U a ↔ entropy (bornCol U a) = 0

Formalization note: Lean 4 identifier BookProof.SymmetryEntropy.isDeterministicCol_iff_entropy_eq_zero.

Preamble

-- Generated from ChapterSymmetryEntropy.lean — theorem BookProof.SymmetryEntropy.isDeterministicCol_iff_entropy_eq_zero
import Definitions.Def_ChapterMarkovEntropy
import Mathlib
import Definitions.Def_ChapterSymmetryEntropy
import Definitions.Def_ChapterIrreversible
import Definitions.Def_ChapterReconstruct
open BookProof.ChapterReconstruct
open BookProof.SymmetryEntropy



open Finset
open BookProof.ChapterMarkovEntropy (entropy)
open BookProof.ChapterReconstruct (IsDeterministicCol)

variable {n : ℕ}

Formal statement

theorem BookProof.SymmetryEntropy.isDeterministicCol_iff_entropy_eq_zero {U : Fin n → Fin n → ℂ} {a : Fin n}
    (hcol : ∑ k, ‖U k a‖ ^ 2 = 1) :
    IsDeterministicCol U a ↔ entropy (bornCol U a) = 0 := by sorry

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