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-- Lean 4 Proof Sketch: Goldilocks Reduction Step Correctness
-- File: GoldilocksStep.lean

namespace Goldilocks

-- 1. State Definition
def Bit := Fin 2
def State := Bit × Bit × Bit -- (x, c_in, c_out)

-- 2. Gate Semantics (Classical Action on Basis States)
def X (s : State) : State :=
  (s.1 + 1, s.2.1, s.2.2)

def CNOT (control_target : Bit × Bit) (s : State) : State :=
  -- This is a helper; actual gates act on specific wires.
  -- We define specific wire actions below.
  s

-- Wire-specific Gates
-- Wire indices: 0 = x, 1 = c_in, 2 = c_out
def X_x (s : State) : State := ⟨(s.1 + 1 : Bit), s.2.1, s.2.2⟩
def CNOT_cin_x (s : State) : State := ⟨s.1 + s.2.1, s.2.1, s.2.2⟩
def CNOT_cin_cout (s : State) : State := ⟨s.1, s.2.1, s.2.2 + s.2.1⟩
def CCX_x_cin_cout (s : State) : State :=
  ⟨s.1, s.2.1, s.2.2 + (s.1 * s.2.1 : Bit)⟩

-- 3. Circuit Definitions
def lower_step (s : State) : State :=
  (X_x ∘ CNOT_cin_x ∘ X_x ∘ CCX_x_cin_cout ∘ X_x) s

def upper_step (s : State) : State :=
  (CNOT_cin_x ∘ CCX_x_cin_cout) s

-- 4. Specifications
def spec_lower (s : State) : State :=
  ⟨s.1 + 1 + s.2.1, s.2.1, s.1 * s.2.1⟩

def spec_upper (s : State) : State :=
  ⟨s.1 + s.2.1, s.2.1, s.1 * s.2.1⟩

-- 5. Correctness Theorems
theorem lower_step_correct : ∀ (s : State), s.2.2 = 0 → lower_step s = spec_lower s := by
  intro s h_zero
  -- Proof by exhaustive case analysis on 4 relevant states (c_out=0 fixed)
  -- x ∈ {0,1}, c_in ∈ {0,1}
  fin_cases s <;> simp_all [lower_step, spec_lower, X_x, CNOT_cin_x, CCX_x_cin_cout, Bit, Fin.ext_iff, Fin.val_zero, Fin.val_add, Fin.val_one]
  <;> norm_num at * <;> rfl

theorem upper_step_correct : ∀ (s : State), s.2.2 = 0 → upper_step s = spec_upper s := by
  intro s h_zero
  fin_cases s <;> simp_all [upper_step, spec_upper, CNOT_cin_x, CCX_x_cin_cout, Bit, Fin.ext_iff, Fin.val_zero, Fin.val_add, Fin.val_one]
  <;> norm_num at * <;> rfl

-- 6. Inverse Correctness (Uncomputation)
def lower_step_dag (s : State) : State :=
  (X_x ∘ CNOT_cin_x ∘ X_x ∘ CCX_x_cin_cout ∘ X_x) s -- Self-inverse check? No, reverse order.
  -- Actual Dagger:
  -- (X_x)⁻¹ ∘ (CCX)⁻¹ ∘ (X_x)⁻¹ ∘ (CNOT)⁻¹ ∘ (X_x)⁻¹
  -- X⁻¹ = X, CNOT⁻¹ = CNOT, CCX⁻¹ = CCX
  -- So: X_x ∘ CCX_x_cin_cout ∘ X_x ∘ CNOT_cin_x ∘ X_x

def lower_step_dag_correct (s : State) : State :=
  (X_x ∘ CCX_x_cin_cout ∘ X_x ∘ CNOT_cin_x ∘ X_x) s

theorem lower_step_dag_is_inverse : ∀ (s : State), lower_step_dag_correct (lower_step s) = s := by
  intro s
  fin_cases s <;> simp [lower_step, lower_step_dag_correct, X_x, CNOT_cin_x, CCX_x_cin_cout, Bit, Fin.ext_iff, Fin.val_zero, Fin.val_add, Fin.val_one]
  <;> norm_num <;> rfl

-- 7. Full 64-bit Composition Correctness (Sketch)
-- We model the Ping-Pong carry chain as a state machine.
-- State: (x_vec : Vector Bit 64, carry_in : Bit, carry_out : Bit, phase : Phase)
-- Phase: Lower (0..31), Upper (32..63), ReverseUpper, ReverseLower
-- Invariant: carry_out = 0 at step start.
-- Transition: Apply lower_step/upper_step, swap carry_in/carry_out.
-- Theorem: Final x_vec = x_init + (C * carry_init) mod 2^64, carry_in = carry_init, carry_out = 0.

def CarryState := Bit × Bit -- (active_carry, passive_carry)

-- Ping-Pong Step for Lower Bits
def pp_lower_step (x_bit : Bit) (carries : CarryState) : (Bit × CarryState) :=
  let s : State := ⟨x_bit, carries.1, carries.2⟩
  let s' := lower_step s
  ⟨s'.1, ⟨s'.2.2, s'.2.1⟩⟩ -- Output x', Swap carries (new_active = old_c_out, new_passive = old_c_in)

-- Ping-Pong Step for Upper Bits
def pp_upper_step (x_bit : Bit) (carries : CarryState) : (Bit × CarryState) :=
  let s : State := ⟨x_bit, carries.1, carries.2⟩
  let s' := upper_step s
  ⟨s'.1, ⟨s'.2.2, s'.2.1⟩⟩

-- Reverse Steps
def pp_lower_step_dag (x_bit : Bit) (carries : CarryState) : (Bit × CarryState) :=
  let s : State := ⟨x_bit, carries.1, carries.2⟩
  let s' := lower_step_dag_correct s
  ⟨s'.1, ⟨s'.2.2, s'.2.1⟩⟩

def pp_upper_step_dag (x_bit : Bit) (carries : CarryState) : (Bit × CarryState) :=
  let s : State := ⟨x_bit, carries.1, carries.2⟩
  let s' := (CNOT_cin_x ∘ CCX_x_cin_cout)⁻¹ s -- Defined explicitly as upper_step_dg
  ⟨s'.1, ⟨s'.2.2, s'.2.1⟩⟩

-- Main Theorem Statement (Informal)
-- theorem goldilocks_reduce_correct (x_init : Vector Bit 64) (ctrl : Bit) :
-- let final_state := run_full_circuit x_init ctrl
-- final_state.x = (x_init + ctrl * (2^32 - 1)) % 2^64 ∧
-- final_state.carry_in = ctrl ∧
-- final_state.carry_out = 0 := by sorry

end Goldilocks

	-- Lean 4 Proof Sketch: Goldilocks Reduction Step Correctness
	-- File: GoldilocksStep.lean

	namespace Goldilocks

	-- 1. State Definition
	def Bit := Fin 2
	def State := Bit × Bit × Bit -- (x, c_in, c_out)

	-- 2. Gate Semantics (Classical Action on Basis States)
	def X (s : State) : State :=
	(s.1 + 1, s.2.1, s.2.2)

	def CNOT (control_target : Bit × Bit) (s : State) : State :=
	-- This is a helper; actual gates act on specific wires.
	-- We define specific wire actions below.
	s

	-- Wire-specific Gates
	-- Wire indices: 0 = x, 1 = c_in, 2 = c_out
	def X_x (s : State) : State := ⟨(s.1 + 1 : Bit), s.2.1, s.2.2⟩
	def CNOT_cin_x (s : State) : State := ⟨s.1 + s.2.1, s.2.1, s.2.2⟩
	def CNOT_cin_cout (s : State) : State := ⟨s.1, s.2.1, s.2.2 + s.2.1⟩
	def CCX_x_cin_cout (s : State) : State :=
	⟨s.1, s.2.1, s.2.2 + (s.1 * s.2.1 : Bit)⟩

	-- 3. Circuit Definitions
	def lower_step (s : State) : State :=
	(X_x ∘ CNOT_cin_x ∘ X_x ∘ CCX_x_cin_cout ∘ X_x) s

	def upper_step (s : State) : State :=
	(CNOT_cin_x ∘ CCX_x_cin_cout) s

	-- 4. Specifications
	def spec_lower (s : State) : State :=
	⟨s.1 + 1 + s.2.1, s.2.1, s.1 * s.2.1⟩

	def spec_upper (s : State) : State :=
	⟨s.1 + s.2.1, s.2.1, s.1 * s.2.1⟩

	-- 5. Correctness Theorems
	theorem lower_step_correct : ∀ (s : State), s.2.2 = 0 → lower_step s = spec_lower s := by
	intro s h_zero
	-- Proof by exhaustive case analysis on 4 relevant states (c_out=0 fixed)
	-- x ∈ {0,1}, c_in ∈ {0,1}
	fin_cases s <;> simp_all [lower_step, spec_lower, X_x, CNOT_cin_x, CCX_x_cin_cout, Bit, Fin.ext_iff, Fin.val_zero, Fin.val_add, Fin.val_one]
	<;> norm_num at * <;> rfl

	theorem upper_step_correct : ∀ (s : State), s.2.2 = 0 → upper_step s = spec_upper s := by
	intro s h_zero
	fin_cases s <;> simp_all [upper_step, spec_upper, CNOT_cin_x, CCX_x_cin_cout, Bit, Fin.ext_iff, Fin.val_zero, Fin.val_add, Fin.val_one]
	<;> norm_num at * <;> rfl

	-- 6. Inverse Correctness (Uncomputation)
	def lower_step_dag (s : State) : State :=
	(X_x ∘ CNOT_cin_x ∘ X_x ∘ CCX_x_cin_cout ∘ X_x) s -- Self-inverse check? No, reverse order.
	-- Actual Dagger:
	-- (X_x)⁻¹ ∘ (CCX)⁻¹ ∘ (X_x)⁻¹ ∘ (CNOT)⁻¹ ∘ (X_x)⁻¹
	-- X⁻¹ = X, CNOT⁻¹ = CNOT, CCX⁻¹ = CCX
	-- So: X_x ∘ CCX_x_cin_cout ∘ X_x ∘ CNOT_cin_x ∘ X_x

	def lower_step_dag_correct (s : State) : State :=
	(X_x ∘ CCX_x_cin_cout ∘ X_x ∘ CNOT_cin_x ∘ X_x) s

	theorem lower_step_dag_is_inverse : ∀ (s : State), lower_step_dag_correct (lower_step s) = s := by
	intro s
	fin_cases s <;> simp [lower_step, lower_step_dag_correct, X_x, CNOT_cin_x, CCX_x_cin_cout, Bit, Fin.ext_iff, Fin.val_zero, Fin.val_add, Fin.val_one]
	<;> norm_num <;> rfl

	-- 7. Full 64-bit Composition Correctness (Sketch)
	-- We model the Ping-Pong carry chain as a state machine.
	-- State: (x_vec : Vector Bit 64, carry_in : Bit, carry_out : Bit, phase : Phase)
	-- Phase: Lower (0..31), Upper (32..63), ReverseUpper, ReverseLower
	-- Invariant: carry_out = 0 at step start.
	-- Transition: Apply lower_step/upper_step, swap carry_in/carry_out.
	-- Theorem: Final x_vec = x_init + (C * carry_init) mod 2^64, carry_in = carry_init, carry_out = 0.

	def CarryState := Bit × Bit -- (active_carry, passive_carry)

	-- Ping-Pong Step for Lower Bits
	def pp_lower_step (x_bit : Bit) (carries : CarryState) : (Bit × CarryState) :=
	let s : State := ⟨x_bit, carries.1, carries.2⟩
	let s' := lower_step s
	⟨s'.1, ⟨s'.2.2, s'.2.1⟩⟩ -- Output x', Swap carries (new_active = old_c_out, new_passive = old_c_in)

	-- Ping-Pong Step for Upper Bits
	def pp_upper_step (x_bit : Bit) (carries : CarryState) : (Bit × CarryState) :=
	let s : State := ⟨x_bit, carries.1, carries.2⟩
	let s' := upper_step s
	⟨s'.1, ⟨s'.2.2, s'.2.1⟩⟩

	-- Reverse Steps
	def pp_lower_step_dag (x_bit : Bit) (carries : CarryState) : (Bit × CarryState) :=
	let s : State := ⟨x_bit, carries.1, carries.2⟩
	let s' := lower_step_dag_correct s
	⟨s'.1, ⟨s'.2.2, s'.2.1⟩⟩

	def pp_upper_step_dag (x_bit : Bit) (carries : CarryState) : (Bit × CarryState) :=
	let s : State := ⟨x_bit, carries.1, carries.2⟩
	let s' := (CNOT_cin_x ∘ CCX_x_cin_cout)⁻¹ s -- Defined explicitly as upper_step_dg
	⟨s'.1, ⟨s'.2.2, s'.2.1⟩⟩

	-- Main Theorem Statement (Informal)
	-- theorem goldilocks_reduce_correct (x_init : Vector Bit 64) (ctrl : Bit) :
	-- let final_state := run_full_circuit x_init ctrl
	-- final_state.x = (x_init + ctrl * (2^32 - 1)) % 2^64 ∧
	-- final_state.carry_in = ctrl ∧
	-- final_state.carry_out = 0 := by sorry

	end Goldilocks