import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Basic
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Real.Basic
import Mathlib.Tactic

noncomputable section

namespace QFD.GA

open CliffordAlgebra
open scoped BigOperators

-- # Clifford Algebra Cl(3,3) - Eliminating EmergentAlgebra Axiom
-- This file formalizes the Clifford algebra Cl(3,3) with signature (+,+,+,-,-,-)
-- using Mathlib's `CliffordAlgebra` structure.

-- ## Purpose
-- Eliminates the axiom `generator_square` from EmergentAlgebra.lean by proving
-- that basis generators square to their metric signature:
--   eᵢ² = ηᵢᵢ
-- using Mathlib's `CliffordAlgebra.ι_sq_scalar` as the anchor lemma.

-- ## The Quadratic Form
-- For 6D phase space with signature (3,3):
-- - Q(e₁) = +1, Q(e₂) = +1, Q(e₃) = +1  (spacelike)
-- - Q(e₄) = -1, Q(e₅) = -1, Q(e₆) = -1  (timelike)
-- The Clifford algebra Cl(Q) is defined by the relation:
--   v · w + w · v = 2Q(v,w) · 1
-- For basis vectors:
--   eᵢ · eᵢ = Q(eᵢ) · 1

-- ## Strategy
-- 1. Define the quadratic form Q₃₃ : (Fin 6 → ℝ) → ℝ
-- 2. Use Mathlib's CliffordAlgebra Q₃₃
-- 3. Use the canonical generator ι : V → CliffordAlgebra Q₃₃
-- 4. Prove: ι(eᵢ) * ι(eᵢ) = algebraMap ℝ (CliffordAlgebra Q₃₃) (Q₃₃ eᵢ)

-- ## References
-- - Mathlib.LinearAlgebra.CliffordAlgebra.Basic
-- - Anchor lemma: `CliffordAlgebra.ι_sq_scalar`
-- - QFD Appendix Z.2: Clifford algebra structure
-- - QFD Appendix Z.4.A: Centralizer theorem

-- ## 1. The Signature (3,3) Quadratic Form

/--
The metric signature for 6D phase space Cl(3,3).
- Indices 0,1,2: +1 (spacelike)
- Indices 3,4,5: -1 (timelike)

This corresponds to:
- γ₁, γ₂, γ₃: spatial dimensions (x, p_x, etc.)
- γ₄: time-like (emergent spacetime)
- γ₅, γ₆: internal time-like (frozen by spectral gap)
-/
def signature33 : Fin 6 → ℝ
  | 0 => 1   -- γ₁: spacelike
  | 1 => 1   -- γ₂: spacelike
  | 2 => 1   -- γ₃: spacelike
  | 3 => -1  -- γ₄: timelike (emergent)
  | 4 => -1  -- γ₅: timelike (internal)
  | 5 => -1  -- γ₆: timelike (internal)

/--
The quadratic form Q₃₃ for the vector space V = Fin 6 → ℝ.

For a basis vector eᵢ (represented as Pi.single i 1):
  Q₃₃(eᵢ) = signature33(i)

For a general vector v = Σᵢ vᵢ eᵢ:
  Q₃₃(v) = Σᵢ signature33(i) · vᵢ²

Uses Mathlib's `QuadraticMap.weightedSumSquares` constructor to avoid
manual BilinForm API issues.
-/
def Q33 : QuadraticForm ℝ (Fin 6 → ℝ) :=
  QuadraticMap.weightedSumSquares ℝ signature33

-- ## 2. The Clifford Algebra Cl(3,3)

/--
The Clifford algebra over the quadratic form Q₃₃.
This is Mathlib's construction, satisfying all axioms by definition.
-/
abbrev Cl33 := CliffordAlgebra Q33

/--
The canonical linear map ι : V → Cl(3,3) that sends basis vectors
to Clifford generators.

This is Mathlib's `CliffordAlgebra.ι` - the universal property map.
-/
def ι33 : (Fin 6 → ℝ) →ₗ[ℝ] Cl33 := ι Q33

-- ## 3. Generator Squaring Theorem

/--
A basis vector eᵢ in V = (Fin 6 → ℝ), represented as Pi.single i 1.
-/
def basis_vector (i : Fin 6) : Fin 6 → ℝ := Pi.single i (1:ℝ)

/--
**Theorem EA-1**: The Basis Generator Squaring Property.

For each basis vector eᵢ, the Clifford generator ι(eᵢ) squares to
its metric signature:

  ι(eᵢ) · ι(eᵢ) = signature33(i) · 1

**Proof Strategy**:
1. Use Mathlib's anchor lemma `ι_sq_scalar`
2. Compute Q₃₃(eᵢ) = signature33(i) · 1² = signature33(i)
3. Apply the anchor lemma: ι(v) * ι(v) = algebraMap ℝ Cl33 (Q₃₃ v)

This eliminates the axiom `generator_square` from EmergentAlgebra.lean.
-/
theorem generator_squares_to_signature (i : Fin 6) :
    (ι33 (basis_vector i)) * (ι33 (basis_vector i)) =
    algebraMap ℝ Cl33 (signature33 i) := by
  -- Step 1: Apply the anchor lemma ι_sq_scalar
  -- This states: ι Q v * ι Q v = algebraMap R _ (Q v)
  unfold ι33
  rw [ι_sq_scalar]
  -- Step 2: Show Q₃₃(basis_vector i) = signature33(i)
  congr 1
  unfold Q33 basis_vector
  rw [QuadraticMap.weightedSumSquares_apply]
  -- The sum collapses because Pi.single i 1 j = 0 for j ≠ i
  classical
  simp only [Pi.single_apply]
  rw [Finset.sum_eq_single i]
  · simp
  · intro j _ hne; simp [hne]
  · intro h; exact absurd (Finset.mem_univ i) h

-- ## 4. Anticommutation Relations

/--
**Theorem EA-2**: Distinct basis generators anticommute.

For i ≠ j:
  ι(eᵢ) · ι(eⱼ) + ι(eⱼ) · ι(eᵢ) = 0

**Proof Strategy**:
Uses the fundamental Clifford relation:
  v · w + w · v = 2 * polar(v,w) · 1

For orthogonal basis vectors eᵢ, eⱼ (i≠j):
  polar(eᵢ, eⱼ) = 0 (diagonal quadratic form)
-/
theorem generators_anticommute (i j : Fin 6) (h_ne : i ≠ j) :
    (ι33 (basis_vector i)) * (ι33 (basis_vector j)) +
    (ι33 (basis_vector j)) * (ι33 (basis_vector i)) = 0 := by
  classical
  unfold ι33
  -- Fundamental Clifford relation: `ι v * ι w + ι w * ι v = polar Q v w`.
  rw [CliffordAlgebra.ι_mul_ι_add_swap]
  suffices hpolar : QuadraticMap.polar (⇑Q33) (basis_vector i) (basis_vector j) = 0 by
    simp [hpolar]
  -- Compute `polar` by expanding `Q(x+y) - Q x - Q y` for our diagonal `Q33`.
  have hQ_basis (k : Fin 6) : Q33 (basis_vector k) = signature33 k := by
    unfold Q33 basis_vector
    rw [QuadraticMap.weightedSumSquares_apply]
    have h0 : ∀ t : Fin 6, t ≠ k →
        signature33 t • (basis_vector k t * basis_vector k t) = 0 := by
      intro t ht
      simp [basis_vector, Pi.single_apply, ht]
    have hsum :
        (∑ t : Fin 6, signature33 t • (basis_vector k t * basis_vector k t)) =
          signature33 k • (basis_vector k k * basis_vector k k) := by
      simp only [Fintype.sum_eq_single (a := k)
        (f := fun t => signature33 t • (basis_vector k t * basis_vector k t)) h0]
    simp [Pi.single_apply, smul_eq_mul]
  have hQ_add :
      Q33 (basis_vector i + basis_vector j) = signature33 i + signature33 j := by
    unfold Q33 basis_vector
    rw [QuadraticMap.weightedSumSquares_apply]
    let f : Fin 6 → ℝ := fun t =>
      signature33 t • ((basis_vector i t + basis_vector j t) *
        (basis_vector i t + basis_vector j t))
    have h0 : ∀ t : Fin 6, t ≠ i ∧ t ≠ j → f t = 0 := by
      intro t ht
      have hi : basis_vector i t = 0 := by simp [basis_vector, Pi.single_apply, ht.1]
      have hj : basis_vector j t = 0 := by simp [basis_vector, Pi.single_apply, ht.2]
      simp [f, hi, hj]
    have hsum : (∑ t : Fin 6, f t) = f i + f j := by
      simpa using (Fintype.sum_eq_add (a := i) (b := j) (f := f) h_ne h0)
    have fi : f i = signature33 i := by
      simp [f, basis_vector, Pi.single_apply, h_ne, smul_eq_mul]
    have fj : f j = signature33 j := by
      have hji : j ≠ i := Ne.symm h_ne
      simp [f, basis_vector, Pi.single_apply, hji, smul_eq_mul]
    -- The sum in terms of f
    have hf_sum : (∑ x : Fin 6, f x) = signature33 i + signature33 j := by
      rw [hsum, fi, fj]
    -- Show goal by unfolding f and converting smul to mul
    simp only [f, basis_vector, smul_eq_mul] at hf_sum
    exact hf_sum
  unfold QuadraticMap.polar
  -- `Q33 (eᵢ + eⱼ) - Q33 eᵢ - Q33 eⱼ = 0`.
  -- We discharge the arithmetic with `ring` on the scalar identity.
  simp [hQ_add, hQ_basis]

/--
**Theorem**: Basis vectors are pairwise orthogonal with respect to Q33.

This uses Mathlib's `QuadraticMap.IsOrtho` API, which states that
v and w are orthogonal iff `polar Q v w = 0`.

For our diagonal quadratic form Q33, distinct basis vectors have zero polar product.
-/
theorem basis_isOrtho (i j : Fin 6) (h_ne : i ≠ j) :
    QuadraticMap.IsOrtho Q33 (basis_vector i) (basis_vector j) := by
  -- IsOrtho is defined as polar Q v w = 0
  -- We prove polar = 0 by computing Q(eᵢ + eⱼ) - Q(eᵢ) - Q(eⱼ)
  classical
  unfold QuadraticMap.IsOrtho
  -- Compute Q33 for individual basis vectors
  have hQ_basis (k : Fin 6) : Q33 (basis_vector k) = signature33 k := by
    unfold Q33 basis_vector
    rw [QuadraticMap.weightedSumSquares_apply]
    have h0 : ∀ t : Fin 6, t ≠ k →
        signature33 t • (basis_vector k t * basis_vector k t) = 0 := by
      intro t ht
      simp [basis_vector, Pi.single_apply, ht]
    have hsum :
        (∑ t : Fin 6, signature33 t • (basis_vector k t * basis_vector k t)) =
          signature33 k • (basis_vector k k * basis_vector k k) := by
      simp only [Fintype.sum_eq_single (a := k)
        (f := fun t => signature33 t • (basis_vector k t * basis_vector k t)) h0]
    simp [Pi.single_apply, smul_eq_mul]
  -- Compute Q33 for sum of basis vectors
  have hQ_add :
      Q33 (basis_vector i + basis_vector j) = signature33 i + signature33 j := by
    unfold Q33 basis_vector
    rw [QuadraticMap.weightedSumSquares_apply]
    let f : Fin 6 → ℝ := fun t =>
      signature33 t • ((basis_vector i t + basis_vector j t) *
        (basis_vector i t + basis_vector j t))
    have h0 : ∀ t : Fin 6, t ≠ i ∧ t ≠ j → f t = 0 := by
      intro t ht
      have hi : basis_vector i t = 0 := by simp [basis_vector, ht.1]
      have hj : basis_vector j t = 0 := by simp [basis_vector, ht.2]
      simp [f, hi, hj]
    have hsum : (∑ t : Fin 6, f t) = f i + f j := by
      simpa using (Fintype.sum_eq_add (a := i) (b := j) (f := f) h_ne h0)
    have fi : f i = signature33 i := by
      simp [f, basis_vector, Pi.single_apply, h_ne, smul_eq_mul]
    have fj : f j = signature33 j := by
      have hji : j ≠ i := Ne.symm h_ne
      simp [f, basis_vector, Pi.single_apply, hji, smul_eq_mul]
    have hf_sum : (∑ x : Fin 6, f x) = signature33 i + signature33 j := by
      rw [hsum, fi, fj]
    simp only [f, basis_vector, smul_eq_mul] at hf_sum
    exact hf_sum
  -- polar = Q(i+j) - Q(i) - Q(j) = 0
  simp [hQ_add, hQ_basis]

/--
**Theorem**: Cleaner anticommutation using IsOrtho.

For orthogonal basis vectors: `eᵢ * eⱼ = -eⱼ * eᵢ`

This version directly uses Mathlib's `ι_mul_ι_comm_of_isOrtho` lemma,
providing a one-line proof compared to the manual polar computation.
-/
theorem generators_anticommute_alt (i j : Fin 6) (h_ne : i ≠ j) :
    ι33 (basis_vector i) * ι33 (basis_vector j) =
    -(ι33 (basis_vector j) * ι33 (basis_vector i)) := by
  unfold ι33
  exact CliffordAlgebra.ι_mul_ι_comm_of_isOrtho (basis_isOrtho i j h_ne)

-- ## 5. Connection to EmergentAlgebra.lean

/--
**Theorem EA-3**: Signature Values.
The signature function matches the metric definition from EmergentAlgebra.lean.
-/
theorem signature_values :
    signature33 0 = 1 ∧ signature33 1 = 1 ∧ signature33 2 = 1 ∧
    signature33 3 = -1 ∧ signature33 4 = -1 ∧ signature33 5 = -1 := by
  unfold signature33
  norm_num

-- ## Physical Summary
-- This file establishes the mathematical foundation for QFD's emergent spacetime:
-- 1. **Rigorous Definition**: Cl(3,3) defined via Mathlib's CliffordAlgebra
-- 2. **Generator Squaring**: eᵢ² = ηᵢᵢ (proven from ι_sq_scalar)
-- 3. **Anticommutation**: {eᵢ, eⱼ} = 0 for i ≠ j

-- ## Axiom Elimination Status
-- This file reduces the EmergentAlgebra axiom to:
-- - 1 placeholder: Computing Q₃₃(basis_vector i) (Pi.single calculation)
-- - Uses Mathlib anchor: `ι_sq_scalar`
-- The previous placeholder is now resolved; the former `generator_square`
-- axiom can be eliminated via this file.

-- ## Next Steps
-- 1. (Completed) Pi.single computation for `generator_squares_to_signature`
-- 2. (Completed) Anticommutation for distinct generators
-- 3. Import Cl33.lean into EmergentAlgebra.lean
-- 4. Replace axiom with theorem from this file

-- ## Overall Progress
-- - ✅ HardWall #1,#2: Proven in RickerAnalysis.lean
-- - ⚠️ HardWall #3: Physical constraint
-- - ✅ Quantization: Proven in GaussianMoments.lean
-- - ✅ EmergentAlgebra: Supported by this file (no placeholders)

-- Status: 4/5 axioms eliminated (80% complete)

-- ## References
-- - Mathlib anchor: `CliffordAlgebra.ι_sq_scalar`
-- - QFD Appendix Z.2: Clifford algebra Cl(3,3)
-- - QFD Appendix Z.4.A: Centralizer = Cl(3,1)

end QFD.GA

end
