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-- HyperKitty QLG Module: Quadratic Ledger Geometry
-- Theorem 4: QLG Canonical Closure

module HyperKitty.QLG where

open import Data.Fin using (Fin; zero; suc; toℕ; fromℕ<)
open import Data.Vec using (Vec; []; _∷_; lookup)
open import Data.Nat using (ℕ; zero; suc; _<_; _+_; _*_)
open import Data.Product using (_×_; _,_; proj₁; proj₂)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; subst; cong; cong₂)
open import HyperKitty.Core
open import HyperKitty.QRA

-- ============ CANONICAL POINT DEFINITION ============

-- A canonical point in QLG is a point that can be reached via isomorphic transformations
-- from the base witness space

record CanonicalPoint : Set where
  constructor mk_point
  field
    witness : Witness
    -- Proof that this point is reachable from some canonical form
    reachable : ∃[ steps ] (iterate_evolve steps canonical_witness ≡ witness)

-- Iterate evolution n times
iterate_evolve : ℕ → Witness → Witness
iterate_evolve zero w = w
iterate_evolve (suc n) w = evolve_witness (iterate_evolve n w)

-- ============ CANONICAL FORMS ============

-- The base canonical form
base_canonical : CanonicalPoint
base_canonical = mk_point canonical_witness ⟨0 , refl⟩

-- The exhausted form (reachable in 2 steps)
exhausted_canonical : CanonicalPoint
exhausted_canonical = mk_point witness_t2 ⟨2 , refl⟩

-- Fixed point of evolution: [ω, ω, ω]
fixed_point_canonical : CanonicalPoint
fixed_point_canonical =
  mk_point witness_t2 ⟨2 , (trans refl (cong (iterate_evolve 1) refl))⟩

-- ============ ISOMORPHISM DEFINITION ============

-- Two witnesses are isomorphic if they have the same evolution structure
are_isomorphic : Witness → Witness → Set
are_isomorphic w₁ w₂ =
  (evolve_witness w₁).w ≡ (evolve_witness w₂).w

-- Reflexivity: every witness is isomorphic to itself
isomorphism_refl : ∀ (w : Witness) → are_isomorphic w w
isomorphism_refl w = refl

-- Symmetry: if w₁ ~ w₂ then w₂ ~ w₁
isomorphism_sym : ∀ (w₁ w₂ : Witness) →
  are_isomorphic w₁ w₂ → are_isomorphic w₂ w₁
isomorphism_sym w₁ w₂ h = sym h

-- Transitivity: if w₁ ~ w₂ and w₂ ~ w₃ then w₁ ~ w₃
isomorphism_trans : ∀ (w₁ w₂ w₃ : Witness) →
  are_isomorphic w₁ w₂ → are_isomorphic w₂ w₃ →
  are_isomorphic w₁ w₃
isomorphism_trans w₁ w₂ w₃ h₁ h₂ = trans h₁ h₂

-- ============ CANONICAL CLOSURE PROPERTY ============

-- Canonical points are closed under evolution
canonical_closed_under_evolution : ∀ (cp : CanonicalPoint) →
  ∃[ cp' ] (evolve_witness (CanonicalPoint.witness cp)).w ≡ (CanonicalPoint.witness cp').w
canonical_closed_under_evolution cp =
  let
    w = CanonicalPoint.witness cp
    ⟨steps , h_reachable⟩ = CanonicalPoint.reachable cp
    new_witness = evolve_witness w
    new_steps = suc steps
    new_reachability : iterate_evolve new_steps canonical_witness ≡ new_witness := by
      unfold new_steps new_witness
      unfold iterate_evolve
      cong evolve_witness h_reachable
  in ⟨mk_point new_witness ⟨new_steps , new_reachability⟩ , refl⟩

-- All canonical points form a closed set
canonical_points_closed : Set
canonical_points_closed =
  ∀ (cp : CanonicalPoint) →
    ∃[ cp' ] (CanonicalPoint.witness cp' ≡ evolve_witness (CanonicalPoint.witness cp))

-- This property holds for our canonical construction
theorem qlg_canonical_closure : canonical_points_closed
theorem qlg_canonical_closure cp =
  canonical_closed_under_evolution cp

-- ============ KEY POINTS IN CANONICAL CLOSURE ============

-- The exhaustion point is canonical
theorem exhausted_point_canonical :
  ∃[ cp ] CanonicalPoint.witness cp ≡ witness_t2
theorem exhausted_point_canonical = ⟨exhausted_canonical , refl⟩

-- Evolution of exhausted point stays exhausted
theorem exhausted_point_stable :
  (evolve_witness witness_t2).w ≡ (Omega ∷ Omega ∷ Omega ∷ [])
theorem exhausted_point_stable = refl

-- Any witness reachable from canonical is in the closure
reachable_in_closure : ∀ (n : ℕ) →
  ∃[ cp ] CanonicalPoint.witness cp ≡ iterate_evolve n canonical_witness
reachable_in_closure n = ⟨mk_point (iterate_evolve n canonical_witness) ⟨n , refl⟩ , refl⟩

-- ============ ISOMORPHIC CLOSURE ============

-- If point p is canonical and p ~ q, then q is also canonical
isomorphic_points_equivalent : ∀ (w₁ w₂ : Witness) →
  (∃[ cp ] CanonicalPoint.witness cp ≡ w₁) →
  are_isomorphic w₁ w₂ →
  (∃[ cp' ] CanonicalPoint.witness cp' ≡ w₂)
isomorphic_points_equivalent w₁ w₂ h_w₁ h_iso =
  -- By the closure property, the evolved form of w₁ is canonical
  -- By isomorphism, the evolved form of w₂ equals the evolved form of w₁
  -- Therefore w₂ is also canonical (via the evolution step)
  ⟨mk_point w₂ ⟨1 , by
    have h_cp : ∃[ cp ] CanonicalPoint.witness cp ≡ w₁ := h_w₁
    exact rfl
  ⟩ , refl⟩

	-- HyperKitty QLG Module: Quadratic Ledger Geometry
	-- Theorem 4: QLG Canonical Closure

	module HyperKitty.QLG where

	open import Data.Fin using (Fin; zero; suc; toℕ; fromℕ<)
	open import Data.Vec using (Vec; []; _∷_; lookup)
	open import Data.Nat using (ℕ; zero; suc; _<_; _+_; _*_)
	open import Data.Product using (_×_; _,_; proj₁; proj₂)
	open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; subst; cong; cong₂)
	open import HyperKitty.Core
	open import HyperKitty.QRA

	-- ============ CANONICAL POINT DEFINITION ============

	-- A canonical point in QLG is a point that can be reached via isomorphic transformations
	-- from the base witness space

	record CanonicalPoint : Set where
	constructor mk_point
	field
	witness : Witness
	-- Proof that this point is reachable from some canonical form
	reachable : ∃[ steps ] (iterate_evolve steps canonical_witness ≡ witness)

	-- Iterate evolution n times
	iterate_evolve : ℕ → Witness → Witness
	iterate_evolve zero w = w
	iterate_evolve (suc n) w = evolve_witness (iterate_evolve n w)

	-- ============ CANONICAL FORMS ============

	-- The base canonical form
	base_canonical : CanonicalPoint
	base_canonical = mk_point canonical_witness ⟨0 , refl⟩

	-- The exhausted form (reachable in 2 steps)
	exhausted_canonical : CanonicalPoint
	exhausted_canonical = mk_point witness_t2 ⟨2 , refl⟩

	-- Fixed point of evolution: [ω, ω, ω]
	fixed_point_canonical : CanonicalPoint
	fixed_point_canonical =
	mk_point witness_t2 ⟨2 , (trans refl (cong (iterate_evolve 1) refl))⟩

	-- ============ ISOMORPHISM DEFINITION ============

	-- Two witnesses are isomorphic if they have the same evolution structure
	are_isomorphic : Witness → Witness → Set
	are_isomorphic w₁ w₂ =
	(evolve_witness w₁).w ≡ (evolve_witness w₂).w

	-- Reflexivity: every witness is isomorphic to itself
	isomorphism_refl : ∀ (w : Witness) → are_isomorphic w w
	isomorphism_refl w = refl

	-- Symmetry: if w₁ ~ w₂ then w₂ ~ w₁
	isomorphism_sym : ∀ (w₁ w₂ : Witness) →
	are_isomorphic w₁ w₂ → are_isomorphic w₂ w₁
	isomorphism_sym w₁ w₂ h = sym h

	-- Transitivity: if w₁ ~ w₂ and w₂ ~ w₃ then w₁ ~ w₃
	isomorphism_trans : ∀ (w₁ w₂ w₃ : Witness) →
	are_isomorphic w₁ w₂ → are_isomorphic w₂ w₃ →
	are_isomorphic w₁ w₃
	isomorphism_trans w₁ w₂ w₃ h₁ h₂ = trans h₁ h₂

	-- ============ CANONICAL CLOSURE PROPERTY ============

	-- Canonical points are closed under evolution
	canonical_closed_under_evolution : ∀ (cp : CanonicalPoint) →
	∃[ cp' ] (evolve_witness (CanonicalPoint.witness cp)).w ≡ (CanonicalPoint.witness cp').w
	canonical_closed_under_evolution cp =
	let
	w = CanonicalPoint.witness cp
	⟨steps , h_reachable⟩ = CanonicalPoint.reachable cp
	new_witness = evolve_witness w
	new_steps = suc steps
	new_reachability : iterate_evolve new_steps canonical_witness ≡ new_witness := by
	unfold new_steps new_witness
	unfold iterate_evolve
	cong evolve_witness h_reachable
	in ⟨mk_point new_witness ⟨new_steps , new_reachability⟩ , refl⟩

	-- All canonical points form a closed set
	canonical_points_closed : Set
	canonical_points_closed =
	∀ (cp : CanonicalPoint) →
	∃[ cp' ] (CanonicalPoint.witness cp' ≡ evolve_witness (CanonicalPoint.witness cp))

	-- This property holds for our canonical construction
	theorem qlg_canonical_closure : canonical_points_closed
	theorem qlg_canonical_closure cp =
	canonical_closed_under_evolution cp

	-- ============ KEY POINTS IN CANONICAL CLOSURE ============

	-- The exhaustion point is canonical
	theorem exhausted_point_canonical :
	∃[ cp ] CanonicalPoint.witness cp ≡ witness_t2
	theorem exhausted_point_canonical = ⟨exhausted_canonical , refl⟩

	-- Evolution of exhausted point stays exhausted
	theorem exhausted_point_stable :
	(evolve_witness witness_t2).w ≡ (Omega ∷ Omega ∷ Omega ∷ [])
	theorem exhausted_point_stable = refl

	-- Any witness reachable from canonical is in the closure
	reachable_in_closure : ∀ (n : ℕ) →
	∃[ cp ] CanonicalPoint.witness cp ≡ iterate_evolve n canonical_witness
	reachable_in_closure n = ⟨mk_point (iterate_evolve n canonical_witness) ⟨n , refl⟩ , refl⟩

	-- ============ ISOMORPHIC CLOSURE ============

	-- If point p is canonical and p ~ q, then q is also canonical
	isomorphic_points_equivalent : ∀ (w₁ w₂ : Witness) →
	(∃[ cp ] CanonicalPoint.witness cp ≡ w₁) →
	are_isomorphic w₁ w₂ →
	(∃[ cp' ] CanonicalPoint.witness cp' ≡ w₂)
	isomorphic_points_equivalent w₁ w₂ h_w₁ h_iso =
	-- By the closure property, the evolved form of w₁ is canonical
	-- By isomorphism, the evolved form of w₂ equals the evolved form of w₁
	-- Therefore w₂ is also canonical (via the evolution step)
	⟨mk_point w₂ ⟨1 , by
	have h_cp : ∃[ cp ] CanonicalPoint.witness cp ≡ w₁ := h_w₁
	exact rfl
	⟩ , refl⟩