`BookProof.ChapterA3.real_pauli` (hpf : PauliFundamental) (α β : Fin 4 → Matrix (Fin 4) (Fin 4) ℝ) (hα : IsCliffordR α) (hβ : IsCliffordR β) : ∃ S : Matrix (Fin 4) (Fin 4) ℝ, |S.de

Proved

BookProof.ChapterA3.real_pauli

by leonardopedro · Oct 6, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA3b.

BookProof.ChapterA3.real_pauli (hpf : PauliFundamental) (α β : Fin 4 → Matrix (Fin 4) (Fin 4) ℝ) (hα : IsCliffordR α) (hβ : IsCliffordR β) : ∃ S : Matrix (Fin 4) (Fin 4) ℝ, |S.det| = 1 ∧ (∀ μ, β μ = S * α μ * S⁻¹) ∧ (∀ S' : Matrix (Fin 4) (Fin 4) ℝ, |S'.det| = 1 → (∀ μ, β μ = S' * α μ * S'⁻¹) → S' = S ∨ S' = -S)

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

Preamble

-- Generated from ChapterA3b.lean — theorem BookProof.ChapterA3.real_pauli
import Mathlib
import Definitions.Def_ChapterA3b
import Definitions.Def_ChapterA3
open BookProof.ChapterA3


open Matrix
open scoped ComplexConjugate

Formal statement

theorem BookProof.ChapterA3.real_pauli (hpf : PauliFundamental)
    (α β : Fin 4 → Matrix (Fin 4) (Fin 4) ℝ)
    (hα : IsCliffordR α) (hβ : IsCliffordR β) :
    ∃ S : Matrix (Fin 4) (Fin 4) ℝ, |S.det| = 1 ∧ (∀ μ, β μ = S * α μ * S⁻¹) ∧
    (∀ S' : Matrix (Fin 4) (Fin 4) ℝ, |S'.det| = 1 → (∀ μ, β μ = S' * α μ * S'⁻¹) →
      S' = S ∨ S' = -S) := by sorry

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