Chapter A1

import Definitions.Def_ChapterA
import Mathlib


/-!
# Chapter A, §A.1 scaffolding — anti-unitary operators and Def 8 structures

This file formalizes the self-contained infrastructure of Chapter A §A.1 of
`book.tex` (see `FORMALIZATION_ROADMAP.md`, work-package **N1**): the notion of an
**anti-unitary** operator (Note 4) and the two **Def 8** structures attached to a
system — a **C-conjugation** on a complex system and an **R-imaginary** operator
on a real system — together with the elementary structural lemmas about them.

An anti-unitary operator on a complex inner-product space is exactly a
conjugate-linear isometric equivalence, i.e. a `LinearIsometryEquiv` whose ring
homomorphism is complex conjugation `starRingEnd ℂ` (`V ≃ₗᵢ⋆[ℂ] V`).  We take
this as the definition of `AntiUnitary V`, so that the whole `LinearIsometryEquiv`
API is available.

The deep §A.1 propositions (Prop 11/12, the R-real/R-pseudoreal/R-complex
trichotomy) require a *complex inner-product* structure on the complexification
`W^c = ℂ ⊗_ℝ W` of a real Hilbert space, which is **not** available in Mathlib and
would have to be built from scratch; that remains an outstanding obstruction
recorded in `BookProof/STATUS.md`.  Everything in this file is `sorry`-free and
`axiom`-free.
-/

open scoped ComplexConjugate InnerProductSpace RealInnerProductSpace

namespace BookProof.ChapterA

/-! ## Note 4 / Def 8 — anti-unitary operators -/

/-- **Note 4 (anti-unitary operator).** An anti-unitary operator on a complex
inner-product space `V` is a conjugate-linear (semilinear over `starRingEnd ℂ`)
isometric equivalence of `V`. -/
abbrev AntiUnitary (V : Type*) [NormedAddCommGroup V] [InnerProductSpace ℂ V] :=
  V ≃ₗᵢ⋆[ℂ] V

namespace AntiUnitary

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





end AntiUnitary

/-! ## Def 8 — C-conjugation and R-imaginary structures on a system -/

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

/-- **Def 8.1 (C-conjugation).** On a complex system `(M, V)`, a *C-conjugation*
`θ` is an anti-unitary **involution** (`θ² = 1`) commuting with every `m ∈ M`. -/
def IsConjugation (M : System ℂ V) (θ : AntiUnitary V) : Prop :=
  (∀ x, θ (θ x) = x) ∧ (∀ m ∈ M.ops, ∀ x, θ (m x) = m (θ x))

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

/-- **Def 8.2 (R-imaginary).** On a real system `(M, W)`, an *R-imaginary*
operator `J` is a (linear) isometry with `J² = -1` commuting with every
`m ∈ M`. -/
def IsRImaginary (M : System ℝ W) (J : W ≃ₗᵢ[ℝ] W) : Prop :=
  (∀ x, J (J x) = -x) ∧ (∀ m ∈ M.ops, ∀ x, J (m x) = m (J x))

/-! ### Structural lemmas for a C-conjugation -/









/-! ### Structural lemmas for an R-imaginary operator -/





end BookProof.ChapterA