`BookProof.ChapterA.conjugation_decomp` (θ : AntiUnitary V) (x : V) : x = (2⁻¹ : ℂ) • (x + θ x) + (2⁻¹ : ℂ) • (x - θ x)

Proved

BookProof.ChapterA.conjugation_decomp

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA1.

BookProof.ChapterA.conjugation_decomp (θ : AntiUnitary V) (x : V) : x = (2⁻¹ : ℂ) • (x + θ x) + (2⁻¹ : ℂ) • (x - θ x)

Formalization note: Lean 4 identifier BookProof.ChapterA.conjugation_decomp.

Preamble

-- Generated from ChapterA1.lean — theorem BookProof.ChapterA.conjugation_decomp
import Mathlib
import Definitions.Def_ChapterA1
import Definitions.Def_ChapterA
open BookProof.ChapterA

variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℝ W] [CompleteSpace W]


open scoped ComplexConjugate InnerProductSpace RealInnerProductSpace

Formal statement

theorem BookProof.ChapterA.conjugation_decomp (θ : AntiUnitary V) (x : V) :
    x = (2⁻¹ : ℂ) • (x + θ x) + (2⁻¹ : ℂ) • (x - θ x) := by sorry

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