/-
  QATAAUM Quantum Compiler - Semantic Relations
  
  This module defines semantic equivalence relations and denotational
  semantics for quantum circuits.
  
  Clean-room implementation based on:
  - Denotational semantics (Selinger 2004)
  - Quantum process algebra (Feng et al. 2007)
  - Categorical quantum mechanics (Abramsky & Coecke 2004)
-/

import QATAAUMVerification.Syntax
import Mathlib.Data.Complex.Basic
import Mathlib.LinearAlgebra.Matrix.Basic

namespace QATAAUMVerification

open Circuit Gate

/-! ## Semantic Equivalence -/

/-- Two circuits are semantically equivalent if they produce the same
    quantum state transformation (abstract definition) -/
axiom Circuit.semanticEquiv : Circuit → Circuit → Prop

notation:50 c1 " ≈ " c2 => Circuit.semanticEquiv c1 c2

/-- Semantic equivalence is reflexive -/
axiom semanticEquiv_refl (c : Circuit) : c ≈ c

/-- Semantic equivalence is symmetric -/
axiom semanticEquiv_symm {c1 c2 : Circuit} : c1 ≈ c2 → c2 ≈ c1

/-- Semantic equivalence is transitive -/
axiom semanticEquiv_trans {c1 c2 c3 : Circuit} : 
  c1 ≈ c2 → c2 ≈ c3 → c1 ≈ c3

/-- Empty circuits are semantically equivalent -/
theorem empty_semanticEquiv (n m : Nat) : 
    Circuit.empty n ≈ Circuit.empty m := by
  exact semanticEquiv_refl (Circuit.empty n)

/-- Composing semantically equivalent circuits preserves equivalence -/
axiom compose_preserves_semanticEquiv {c1 c2 c3 c4 : Circuit} :
  c1 ≈ c2 → c3 ≈ c4 → c1.compose c3 ≈ c2.compose c4

/-! ## Gate Cancellation -/

/-- X gate is self-inverse -/
axiom x_self_inverse (q : QubitId) :
  let g := Gate.mk GateType.X [q] none
  let c := (Circuit.empty 1).append g |>.append g
  c ≈ Circuit.empty 1

/-- Y gate is self-inverse -/
axiom y_self_inverse (q : QubitId) :
  let g := Gate.mk GateType.Y [q] none
  let c := (Circuit.empty 1).append g |>.append g
  c ≈ Circuit.empty 1

/-- Z gate is self-inverse -/
axiom z_self_inverse (q : QubitId) :
  let g := Gate.mk GateType.Z [q] none
  let c := (Circuit.empty 1).append g |>.append g
  c ≈ Circuit.empty 1

/-- H gate is self-inverse -/
axiom h_self_inverse (q : QubitId) :
  let g := Gate.mk GateType.H [q] none
  let c := (Circuit.empty 1).append g |>.append g
  c ≈ Circuit.empty 1

/-- CNOT gate is self-inverse -/
axiom cnot_self_inverse (q1 q2 : QubitId) :
  let g := Gate.mk GateType.CX [q1, q2] none
  let c := (Circuit.empty 2).append g |>.append g
  c ≈ Circuit.empty 2

/-! ## Rotation Folding -/

/-- Consecutive Z rotations can be folded -/
axiom rz_fold (q : QubitId) (θ1 θ2 : Angle) :
  let g1 := Gate.mk GateType.RZ [q] (some θ1)
  let g2 := Gate.mk GateType.RZ [q] (some θ2)
  let g3 := Gate.mk GateType.RZ [q] (some (θ1 + θ2))
  let c1 := (Circuit.empty 1).append g1 |>.append g2
  let c2 := (Circuit.empty 1).append g3
  c1 ≈ c2

/-- Consecutive X rotations can be folded -/
axiom rx_fold (q : QubitId) (θ1 θ2 : Angle) :
  let g1 := Gate.mk GateType.RX [q] (some θ1)
  let g2 := Gate.mk GateType.RX [q] (some θ2)
  let g3 := Gate.mk GateType.RX [q] (some (θ1 + θ2))
  let c1 := (Circuit.empty 1).append g1 |>.append g2
  let c2 := (Circuit.empty 1).append g3
  c1 ≈ c2

/-- Consecutive Y rotations can be folded -/
axiom ry_fold (q : QubitId) (θ1 θ2 : Angle) :
  let g1 := Gate.mk GateType.RY [q] (some θ1)
  let g2 := Gate.mk GateType.RY [q] (some θ2)
  let g3 := Gate.mk GateType.RY [q] (some (θ1 + θ2))
  let c1 := (Circuit.empty 1).append g1 |>.append g2
  let c2 := (Circuit.empty 1).append g3
  c1 ≈ c2

/-! ## Commutation Relations -/

/-- Z gates on different qubits commute -/
axiom z_commute (q1 q2 : QubitId) (h : q1 ≠ q2) :
  let g1 := Gate.mk GateType.Z [q1] none
  let g2 := Gate.mk GateType.Z [q2] none
  let c1 := (Circuit.empty 2).append g1 |>.append g2
  let c2 := (Circuit.empty 2).append g2 |>.append g1
  c1 ≈ c2

/-- X gates on different qubits commute -/
axiom x_commute (q1 q2 : QubitId) (h : q1 ≠ q2) :
  let g1 := Gate.mk GateType.X [q1] none
  let g2 := Gate.mk GateType.X [q2] none
  let c1 := (Circuit.empty 2).append g1 |>.append g2
  let c2 := (Circuit.empty 2).append g2 |>.append g1
  c1 ≈ c2

/-! ## Optimization Correctness -/

/-- Self-inverse gate cancellation generalizes to arbitrary prefix circuits.
    If g is self-inverse, appending it twice to any circuit is a no-op. -/
axiom self_inverse_append (c : Circuit) (g : Gate)
    (h_inv : g.gateType = GateType.X ∨ g.gateType = GateType.Y ∨
             g.gateType = GateType.Z ∨ g.gateType = GateType.H) :
    c.append g |>.append g ≈ c

/-- Gate cancellation preserves semantics -/
theorem gate_cancellation_correct (c : Circuit) (g : Gate)
    (h_inv : g.gateType = GateType.X ∨ g.gateType = GateType.Y ∨
             g.gateType = GateType.Z ∨ g.gateType = GateType.H) :
    let c' := c.append g |>.append g
    c' ≈ c := by
  exact self_inverse_append c g h_inv

/-- Rotation folding generalizes to arbitrary prefix circuits. -/
axiom rotation_fold_append (c : Circuit) (q : QubitId)
    (gt : GateType) (θ1 θ2 : Angle)
    (h_rot : gt = GateType.RX ∨ gt = GateType.RY ∨ gt = GateType.RZ) :
    let g1 := Gate.mk gt [q] (some θ1)
    let g2 := Gate.mk gt [q] (some θ2)
    let g3 := Gate.mk gt [q] (some (θ1 + θ2))
    c.append g1 |>.append g2 ≈ c.append g3

/-- Rotation folding preserves semantics -/
theorem rotation_folding_correct (c : Circuit) (q : QubitId)
    (gt : GateType) (θ1 θ2 : Angle)
    (h_rot : gt = GateType.RX ∨ gt = GateType.RY ∨ gt = GateType.RZ) :
    let g1 := Gate.mk gt [q] (some θ1)
    let g2 := Gate.mk gt [q] (some θ2)
    let g3 := Gate.mk gt [q] (some (θ1 + θ2))
    let c1 := c.append g1 |>.append g2
    let c2 := c.append g3
    c1 ≈ c2 := by
  exact rotation_fold_append c q gt θ1 θ2 h_rot

/-! ## Pass Correctness -/

/-- A pass that preserves semantic equivalence is correct -/
def PassResult.semanticallyCorrect (pr : PassResult) : Prop :=
  pr.input ≈ pr.output

/-- Identity pass is semantically correct -/
theorem identity_semantically_correct (c : Circuit) :
    (PassResult.mk c c).semanticallyCorrect := by
  unfold PassResult.semanticallyCorrect
  exact semanticEquiv_refl c

/-- Composition of semantically correct passes is correct -/
theorem compose_passes_correct (pr1 pr2 : PassResult)
    (h1 : pr1.semanticallyCorrect)
    (h2 : pr2.semanticallyCorrect)
    (h_comp : pr1.output = pr2.input) :
    (PassResult.mk pr1.input pr2.output).semanticallyCorrect := by
  unfold PassResult.semanticallyCorrect at *
  rw [← h_comp] at h2
  exact semanticEquiv_trans h1 h2

/-! ## Witness Validation -/

/-- A witness certifies that a pass is correct -/
structure PassWitness where
  pass : PassResult
  preserves_qubits : pass.preservesQubits
  preserves_depth : pass.preservesDepth
  semantically_correct : pass.semanticallyCorrect

/-- Valid witness implies correct pass -/
theorem valid_witness_correct_pass (w : PassWitness) :
    w.pass.semanticallyCorrect := by
  exact w.semantically_correct

/-- Witness for identity pass -/
def identity_witness (c : Circuit) : PassWitness where
  pass := PassResult.mk c c
  preserves_qubits := by unfold PassResult.preservesQubits; rfl
  preserves_depth := by unfold PassResult.preservesDepth; rfl
  semantically_correct := identity_semantically_correct c

end QATAAUMVerification