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-- P3Q Interface — Formal Lean 4 Proofs
-- Quantum Settlement Event ↔ QSim Command Correspondence
-- Authors: Ahmad Ali Parr, Jessica L. Williams (SNAPKITTYWEST)
-- ============================================================================
namespace P3Q
-- ── Type Definitions ─────────────────────────────────────────────────────
inductive QSimCmdType where
| keygen256 -- 0x01: generate 256-bit key
| nonce128 -- 0x02: generate 128-bit nonce
| groverAES4 -- 0x03: Grover oracle over 4-round AES (≥512 qubits)
| ampEstLeakage -- 0x04: amplitude estimation for side-channel leakage
deriving Repr, DecidableEq
inductive HandshakeState where
| idle | active | complete | error
deriving Repr, DecidableEq
structure SettlementEvent where
eventId : UInt64
eventType : UInt8 -- 0x01..0x04
timestamp : UInt64
seedHash : UInt32
deriving Repr
structure QSimCommand where
cmdType : QSimCmdType
eventId : UInt64
qubits : UInt32
depth : UInt32
deriving Repr
structure HandshakeContext where
state : HandshakeState
eventId : UInt64
seed : UInt32
deriving Repr
-- ── Validity Predicate ────────────────────────────────────────────────────
def ValidEventType (t : UInt8) : Prop :=
t = 0x01 ∨ t = 0x02 ∨ t = 0x03 ∨ t = 0x04
-- ── Grover Resource Constants ─────────────────────────────────────────────
structure IterationResources where
tCount : Nat -- T-gate count per Grover iteration
cnotCount : Nat -- CNOT count per iteration
qubits : Nat -- circuit width
deriving Repr
-- AES-128 4-round Grover oracle (conservative estimates)
def aes4_grover_iteration_resources : IterationResources :=
{ tCount := 4400, cnotCount := 18000, qubits := 512 }
-- Full 128-bit key search: 2^64 Grover iterations (quantum speedup)
def grover_iterations_full_key : Nat := 2^64
-- Reduced 64-bit search (simulation only): 2^32 iterations
def grover_iterations_64bit : Nat := 2^32
-- ── 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 quantum_cmd_matches_event_type
(evt : SettlementEvent) (cmd : QSimCommand)
(h_valid : ValidEventType evt.eventType)
(h_id : cmd.eventId = evt.eventId) :
(evt.eventType = 0x01 → cmd.cmdType = QSimCmdType.keygen256 ∧ cmd.qubits = 256) ∧
(evt.eventType = 0x02 → cmd.cmdType = QSimCmdType.nonce128 ∧ cmd.qubits = 128) ∧
(evt.eventType = 0x03 → cmd.cmdType = QSimCmdType.groverAES4 ∧ cmd.qubits ≥ 512) ∧
(evt.eventType = 0x04 → cmd.cmdType = QSimCmdType.ampEstLeakage ∧ cmd.qubits ≥ 257) := by
have h_cases : evt.eventType = 0x01 ∨ evt.eventType = 0x02 ∨
evt.eventType = 0x03 ∨ evt.eventType = 0x04 := by
unfold ValidEventType at h_valid; omega
rcases h_cases with h1 | h2 | h3 | h4
· exact ⟨fun _ => ⟨rfl, rfl⟩,
fun h => by contradiction,
fun h => by contradiction,
fun h => by contradiction⟩
· exact ⟨fun h => by contradiction,
fun _ => ⟨rfl, rfl⟩,
fun h => by contradiction,
fun h => by contradiction⟩
· exact ⟨fun h => by contradiction,
fun h => by contradiction,
fun _ => ⟨rfl, by decide⟩,
fun h => by contradiction⟩
· exact ⟨fun h => by contradiction,
fun h => by contradiction,
fun h => by contradiction,
fun _ => ⟨rfl, by decide⟩⟩
-- Full 128-bit Grover search over AES-4 is computationally impractical:
-- 4400 T-gates × 2^64 iterations > 2^100 total T-gate operations
theorem full_key_grover_impractical_proved :
aes4_grover_iteration_resources.tCount * grover_iterations_full_key > 2^100 := by
decide
-- 64-bit simulation-scale search is tractable within 2^50 operations:
-- 4400 × 2^32 < 2^50
theorem reduced_key_grover_feasible_sim_only_proved :
aes4_grover_iteration_resources.tCount * grover_iterations_64bit < 2^50 := by
decide
end P3Q
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