`BookProof.ChapterA3.nullConj_iff_form` (T : Matrix (Fin 2) (Fin 2) ℂ) : Tᴴ * (pauliσ 0 + pauliσ 3) * T = pauliσ 0 + pauliσ 3 ↔ T 0 1 = 0 ∧ Complex.normSq (T 0 0) = 1

Proved

BookProof.ChapterA3.nullConj_iff_form

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA4d.

BookProof.ChapterA3.nullConj_iff_form (T : Matrix (Fin 2) (Fin 2) ℂ) : Tᴴ * (pauliσ 0 + pauliσ 3) * T = pauliσ 0 + pauliσ 3 ↔ T 0 1 = 0 ∧ Complex.normSq (T 0 0) = 1

Formalization note: Lean 4 identifier BookProof.ChapterA3.nullConj_iff_form.

Preamble

-- Generated from ChapterA4d.lean — theorem BookProof.ChapterA3.nullConj_iff_form
import Mathlib
import Definitions.Def_ChapterA4d
import Definitions.Def_ChapterA3
import Definitions.Def_ChapterA3h
open BookProof.ChapterA3


open Matrix
open scoped ComplexConjugate

Formal statement

theorem BookProof.ChapterA3.nullConj_iff_form (T : Matrix (Fin 2) (Fin 2) ℂ) :
    Tᴴ * (pauliσ 0 + pauliσ 3) * T = pauliσ 0 + pauliσ 3 ↔
      T 0 1 = 0 ∧ Complex.normSq (T 0 0) = 1 := by sorry

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