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-- P3Q Classical-Quantum Interface Formalization
-- Lean 4 - Core Invariants and Safety Properties

namespace P3Q

-- ──────────────────────────────────────────────
-- Classical Types (P4 Settlement Events)
-- ──────────────────────────────────────────────

structure SettlementEvent where
  eventId : UInt32
  eventType : UInt8
  params : Array UInt8
  deadline : UInt64

structure MeasurementResult where
  eventId : UInt32
  status : UInt8
  data : Array UInt8
  shotsCompleted: UInt32

-- ──────────────────────────────────────────────
-- Quantum Command/Response (Simulator Interface)
-- ──────────────────────────────────────────────

inductive QSimCmdType where
  | keygen256
  | nonce128
  | groverAES4
  | ampEstLeakage

structure QSimCommand where
  cmdType : QSimCmdType
  qubits : UInt16
  shots : UInt32
  params : Array UInt8
  eventId : UInt32

structure QSimResponse where
  eventId : UInt32
  status : UInt8
  data : Array UInt8
  shotsCompleted: UInt32

-- ──────────────────────────────────────────────
-- Handshake State Machine
-- ──────────────────────────────────────────────

inductive HandshakeState where
  | idle
  | translate
  | issueCmd
  | waitRsp
  | formatRsp
  | done

structure HandshakeContext where
  state : HandshakeState
  eventId : UInt32
  eventType : UInt8
  deadline : UInt64
  cycleCounter : UInt64

-- ──────────────────────────────────────────────
-- Validity Predicates
-- ──────────────────────────────────────────────

def ValidEventType (t : UInt8) : Bool :=
  t == 0x01 || t == 0x02 || t == 0x03 || t == 0x04

def ValidQubitCount (cmdType : QSimCmdType) (qubits : UInt16) : Bool :=
  match cmdType with
  | QSimCmdType.keygen256 => qubits == 256
  | QSimCmdType.nonce128 => qubits == 128
  | QSimCmdType.groverAES4 => qubits >= 512
  | QSimCmdType.ampEstLeakage => qubits >= 257

-- ──────────────────────────────────────────────
-- AES MixColumns Specification
-- ──────────────────────────────────────────────

def xtime (b : UInt8) : UInt8 :=
  let shifted := b <<< 1
  if b &&& 0x80 != 0 then
    shifted ^^^ 0x1B
  else
    shifted

def mixColumn (a0 a1 a2 a3 : UInt8) : UInt8 × UInt8 × UInt8 × UInt8 :=
  let t := a0 ^^^ a1 ^^^ a2 ^^^ a3
  let x0 := xtime (a0 ^^^ a1)
  let x1 := xtime (a1 ^^^ a2)
  let x2 := xtime (a2 ^^^ a3)
  let x3 := xtime (a3 ^^^ a0)
  (a0 ^^^ t ^^^ x0, a1 ^^^ t ^^^ x1, a2 ^^^ t ^^^ x2, a3 ^^^ t ^^^ x3)

-- ──────────────────────────────────────────────
-- Theorems
-- ──────────────────────────────────────────────

theorem handshake_preserves_event_id
  (ctx : HandshakeContext) (evt : SettlementEvent)
  (h_state : ctx.state = HandshakeState.idle)
  (h_id : ctx.eventId = evt.eventId) :
  ctx.eventId = evt.eventId := h_id

theorem xtime_bit7_clear (b : UInt8) :
  (xtime b) &&& 0x80 = 0 := by
  simp [xtime]
  split
  · next h => simp [h]
  · next h => simp [h]

theorem mixColumn_preserves_zero :
  mixColumn 0 0 0 0 = (0, 0, 0, 0) := by
  simp [mixColumn, xtime]

theorem mixColumn_preserves_uniform (a : UInt8) :
  mixColumn a a a a = (a, a, a, a) := by
  simp [mixColumn, xtime, UInt8.xor_self]

theorem canonical_vector_correct :
  let (y0, y1, y2, y3) := mixColumn 0xD4 0xBF 0x5D 0x30
  y0 == 0x04 && y1 == 0x66 && y2 == 0x81 && y3 == 0xE5 := by
  native_decide

-- ──────────────────────────────────────────────
-- Resource Bounds
-- ──────────────────────────────────────────────

structure ResourceEstimate where
  tCount : Nat
  tDepth : Nat
  logicalQubits : Nat
  cnotCount : Nat

def aes4_grover_iteration_resources : ResourceEstimate :=
  { tCount := 4400, tDepth := 350, logicalQubits := 1200, cnotCount := 7000 }

def grover_iterations_full_key : Nat := 2^64
def grover_iterations_64bit : Nat := 2^32

theorem full_key_grover_impractical :
  aes4_grover_iteration_resources.tCount * grover_iterations_full_key > 2^100 := by
  native_decide

theorem reduced_key_grover_feasible_sim_only :
  aes4_grover_iteration_resources.tCount * grover_iterations_64bit < 2^50 := by
  native_decide

end P3Q