lean/OAI/Computability/MaxCut/Machines/AddressMachineLoop.lean

import Mathlib
import OAI.Computability.MaxCut.Encoding.TableKeysAddressGame
import OAI.Computability.MaxCut.Machines.MachineSingleOrbitProgram
import OAI.Computability.MaxCut.Machines.AddressMachineInitial
import OAI.Computability.MaxCut.Games.OccurrenceTupleOrder

namespace OAI

namespace MaxCutGames.Reduction.AddressMachineLoop

open Turing Foundations.Complexity Integration

noncomputable section

variable (k : Nat) {s d : Nat} (T : NoiseTables.Table s d)

def bits (F : SourceEncoding.Input)
    (digits : Fin k → Fin F.equations.length) : List Bool :=
  AddressOutcomeSpecs.tupleBits (AddressTupleBody.source F) k T
    (OccurrenceTupleOrder.readCoordinates digits)

def stateTapes (F : SourceEncoding.Input)
    (digits : Fin k → Fin F.equations.length) (output : List Bool) :
    AddressMachineProgram.Tape k T → List Bool :=
  AddressOdometerSchedule.setDigits F.equations.length (fun j => (digits j).val)
    (AddressOdometerSchedule.setAcc (AddressMachineInitial.initialized k T F) output)

theorem stateTapes_frame (F : SourceEncoding.Input)
    (digits : Fin k → Fin F.equations.length) (output : List Bool)
    (tape : AddressMachineProgram.Tape k T)
    (hc : ∀j, tape ≠ AddressMachineSpace.current k s d T.vectors.length j)
    (hr : ∀j, tape ≠ AddressMachineSpace.remaining k s d T.vectors.length j)
    (ha : tape ≠ AddressMachineSpace.headerTape k s d T.vectors.length .reversed) :
    stateTapes k T F digits output tape = AddressMachineInitial.initialized k T F tape := by
  rw [stateTapes, AddressOdometerSchedule.setDigits_other _ _ _ tape hc hr]
  exact AddressOdometerSchedule.setAcc_other _ _ tape ha

@[simp] theorem stateTapes_savedIndex (F : SourceEncoding.Input)
    (digits : Fin k → Fin F.equations.length) (output : List Bool) (j : Fin k) :
    stateTapes k T F digits output (.savedIndex j) = encodeWord (digits j.rev).val := by
  exact AddressOdometerSchedule.setDigits_savedIndex _ _ _ j

@[simp] theorem stateTapes_private (F : SourceEncoding.Input)
    (digits : Fin k → Fin F.equations.length) (output : List Bool)
    (tape : MachineTemplateAddress.Tape (4*k) (1+9*k)) :
    stateTapes k T F digits output (AddressMachineSpace.privateAddress k s d T.vectors.length tape) = [] := by
  rw [stateTapes_frame]
  · exact AddressMachineInitial.initialized_privateAddress k T F tape
  all_goals simp [AddressMachineSpace.privateAddress, AddressMachineSpace.current, AddressMachineSpace.remaining, AddressMachineSpace.headerTape]

@[simp] theorem stateTapes_header (F : SourceEncoding.Input)
    (digits : Fin k → Fin F.equations.length) (output : List Bool)
    (c : MachineAddressHeaders.Arithmetic.Control) (hc : c ≠ .reversed) :
    stateTapes k T F digits output (AddressMachineSpace.headerTape k s d T.vectors.length c) =
      AddressMachineInitial.initialized k T F (AddressMachineSpace.headerTape k s d T.vectors.length c) := by
  apply stateTapes_frame
  · exact AddressMachineSpace.headerTape_ne_current k s d T.vectors.length c
  · exact AddressMachineSpace.headerTape_ne_remaining k s d T.vectors.length c
  · intro h
    exact hc ((AddressMachineSpace.headerTape_eq_iff k s d T.vectors.length c .reversed).mp h)

theorem stateTapes_ready (F : SourceEncoding.Input)
    (digits : Fin k → Fin F.equations.length) (output : List Bool) :
    AddressTupleBody.Ready F (OccurrenceTupleOrder.readCoordinates digits)
      (stateTapes k T F digits output) := by
  constructor
  · rw [stateTapes_frame]
    · exact AddressMachineInitial.initialized_source k T F
    all_goals simp [AddressMachineSpace.current, AddressMachineSpace.remaining, AddressMachineSpace.headerTape]
  · intro j
    exact stateTapes_savedIndex k T F digits output j
  · rw [stateTapes_frame]
    · exact AddressMachineInitial.initialized_index k T F
    all_goals simp [AddressMachineSpace.current, AddressMachineSpace.remaining, AddressMachineSpace.headerTape]
  · rw [stateTapes_frame]
    · exact AddressMachineInitial.initialized_work k T F
    all_goals simp [AddressMachineSpace.current, AddressMachineSpace.remaining, AddressMachineSpace.headerTape]
  · rw [stateTapes_frame]
    · exact AddressMachineInitial.initialized_scratch k T F
    all_goals simp [AddressMachineSpace.current, AddressMachineSpace.remaining, AddressMachineSpace.headerTape]
  · rw [stateTapes_frame]
    · exact AddressMachineInitial.initialized_copyScratch k T F
    all_goals simp [AddressMachineSpace.current, AddressMachineSpace.remaining, AddressMachineSpace.headerTape]
  · intro j slot
    rw [stateTapes_frame]
    · exact AddressMachineInitial.initialized_field k T F j slot
    all_goals simp [AddressMachineSpace.current, AddressMachineSpace.remaining, AddressMachineSpace.headerTape]

theorem stateTapes_clean (F : SourceEncoding.Input)
    (digits : Fin k → Fin F.equations.length) (output : List Bool) :
    MachineAddressEdge.Clean (AddressMachineSpace.addressEdgeSlots k s d T.vectors.length)
      (stateTapes k T F digits output) := by
  constructor
  · constructor
    · exact stateTapes_private k T F digits output .reversed
    · exact stateTapes_private k T F digits output .forward
    · exact stateTapes_private k T F digits output .copyScratch
    · exact stateTapes_private k T F digits output .accA
    · exact stateTapes_private k T F digits output .accB
    · exact stateTapes_private k T F digits output .counter
    · exact stateTapes_private k T F digits output .hornerScratch
    · intro j; exact stateTapes_private k T F digits output (.digit j)
  · exact stateTapes_private k T F digits output .output

theorem stateTapes_base (F : SourceEncoding.Input)
    (digits : Fin k → Fin F.equations.length) (output : List Bool) :
    stateTapes k T F digits output (AddressMachineSpace.headerTape k s d T.vectors.length .baseValue) =
      encodeWord (CanonicalAddress.base F.«variables» F.equations.length s d) := by
  rw [stateTapes_header _ _ _ _ _ _ (by decide), AddressMachineInitial.initialized_base]
  congr 1
  simp [MachineAddressHeaders.radix, MachineAddressHeaders.baseConstant,
    CanonicalAddress.base_eq, Nat.add_assoc]

theorem stateTapes_capacity (F : SourceEncoding.Input)
    (digits : Fin k → Fin F.equations.length) (output : List Bool) :
    stateTapes k T F digits output (AddressMachineSpace.headerTape k s d T.vectors.length .capacity) =
      encodeWord (AddressGame.bodyCapacity (AddressTupleBody.source F) k s d) := by
  rw [stateTapes_header _ _ _ _ _ _ (by decide), AddressMachineInitial.initialized_capacity]
  congr 1
  change MachineAddressHeaders.capacity k s d F.«variables» F.equations.length =
    CanonicalAddress.capacity F.«variables» F.equations.length k s d
  simp [MachineAddressHeaders.capacity, MachineAddressHeaders.radix,
    MachineAddressHeaders.baseConstant, CanonicalAddress.capacity,
    CanonicalAddress.base_eq, Nat.add_assoc]

/-- Appending actual tuple bytes is exactly the schedule's accumulator update. -/
theorem appended_stateTapes (F : SourceEncoding.Input)
    (digits : Fin k → Fin F.equations.length) (output payload : List Bool) :
    MachineAddressEdge.appended (AddressMachineSpace.addressEdgeSlots k s d T.vectors.length)
      (stateTapes k T F digits output) payload =
      stateTapes k T F digits (payload.reverse ++ output) := by
  change AddressOdometerSchedule.setAcc (stateTapes k T F digits output)
    (payload.reverse ++ stateTapes k T F digits output
      (AddressOdometerSchedule.accumulator k s d T.vectors.length)) = _
  simp only [stateTapes, AddressOdometerSchedule.setDigits_accumulator, AddressOdometerSchedule.setAcc_apply]
  rw [AddressOdometerSchedule.setDigits_setAcc, AddressOdometerSchedule.setAcc_setAcc,
    ← AddressOdometerSchedule.setDigits_setAcc]

/-- The local-body premise of the odometer theorem is derived from the actual
placed tuple-body implementation and its actual uniform budget theorem. -/
theorem bodyTrace (F : SourceEncoding.Input) :
    AddressOdometerSchedule.BodyTrace (AddressMachineProgram.program k T) (AddressMachineProgram.bodyStart k T)
      (AddressMachineProgram.next k T 0) (AddressMachineSpace.initialState k).1 (AddressMachineInitial.initialized k T F)
      (bits k T F) ((AddressTupleBudget.timePolynomial k T).eval (SourceEncoding.inputBits F).length) := by
  intro digits output
  let base := stateTapes k T F digits output
  have ready := stateTapes_ready k T F digits output
  have clean := stateTapes_clean k T F digits output
  have actual := AddressTuplePolynomial.run_actual T (AddressMachineProgram.bodyLabels k T) (some (AddressMachineProgram.next k T 0))
    (AddressMachineProgram.program k T) (AddressMachineProgram.atBody k T) F
    (OccurrenceTupleOrder.readCoordinates digits) base ready clean (fun _ => false)
    (stateTapes_base k T F digits output) (stateTapes_capacity k T F digits output)
  refine ⟨actual.steps, actual.steps_le_m, ?_⟩
  have hemit : MachineAddressEdge.appended (AddressTupleBody.Slots k s d T.vectors.length)
      base (AddressOutcomeSpecs.tupleBits (AddressTupleBody.source F) k T
        (OccurrenceTupleOrder.readCoordinates digits)) =
      stateTapes k T F digits ((bits k T F digits).reverse ++ output) :=
    appended_stateTapes k T F digits output _
  exact actual.evals_in_steps.trans (congrArg
    (fun tapes => some (AddressOdometerSchedule.configuration
      (AddressMachineProgram.next k T 0) (AddressMachineSpace.initialState k).1 tapes)) hemit)

/-- The all-zero assignment is already physically present after initialization. -/
theorem setDigits_initialized_zero (F : SourceEncoding.Input) :
    AddressOdometerSchedule.setDigits F.equations.length (fun _ : Fin k => 0)
      (AddressMachineInitial.initialized k T F) = AddressMachineInitial.initialized k T F := by
  funext tape
  cases tape with
  | savedIndex j =>
    rw [AddressOdometerSchedule.setDigits_savedIndex, AddressMachineInitial.initialized_savedIndex]
  | extra e =>
    rcases e with e | e
    · rfl
    · rcases e with e | e
      · rfl
      · rcases e with j | e
        · change AddressOdometerSchedule.setDigits F.equations.length (fun _ : Fin k => 0)
            (AddressMachineInitial.initialized k T F) (AddressOdometerSchedule.remaining k s d T.vectors.length j) = _
          simp only [AddressOdometerSchedule.setDigits_remaining, Fin.isLt, dite_eq_left, Nat.sub_zero]
          exact (AddressMachineInitial.initialized_remaining k T F j).symm
        · rfl
  | _ => rfl

def finalTapes (F : SourceEncoding.Input) : AddressMachineProgram.Tape k T → List Bool :=
  (AddressOdometerSchedule.finalConfiguration (AddressMachineProgram.next k T k) (AddressMachineSpace.initialState k).1
    (AddressMachineInitial.initialized k T F) (bits k T F)).stk

/-- Actual complete tuple traversal, before the final output reversal. -/
def traversal (F : SourceEncoding.Input) :
    StateTransition.EvalsToInTime (TM2.step (AddressMachineProgram.program k T))
      ⟨some (AddressMachineProgram.bodyStart k T), AddressMachineSpace.initialState k, AddressMachineInitial.initialized k T F⟩
      (some ⟨some (AddressMachineProgram.finishStart k T), AddressMachineSpace.initialState k, finalTapes k T F⟩)
      (((AddressTupleBudget.timePolynomial k T).eval (SourceEncoding.inputBits F).length + 2*k) *
        F.equations.length^k) := by
  have actual := AddressOdometerSchedule.traversalInTime (AddressMachineProgram.program k T) (AddressMachineProgram.bodyStart k T)
    (AddressMachineProgram.next k T) (AddressMachineProgram.resetAt k T) (AddressMachineSpace.initialState k).1
    (List.length_pos_iff.mpr F.nonempty)
    ((AddressTupleBudget.timePolynomial k T).eval (SourceEncoding.inputBits F).length)
    (AddressMachineInitial.initialized k T F) (bits k T F)
    (by
      intro i hi
      simp only [AddressMachineProgram.next_lt k T i hi, AddressMachineProgram.resetAt_lt k T i hi,
        AddressOdometerSchedule.currentAt, AddressOdometerSchedule.remainingAt, hi, dite_eq_left]
      exact AddressMachineProgram.atCheck k T ⟨i,hi⟩)
    (by
      intro i hi
      simp only [AddressMachineProgram.resetAt_lt k T i hi,
        AddressOdometerSchedule.currentAt, AddressOdometerSchedule.remainingAt, hi, dite_eq_left]
      exact AddressMachineProgram.atReset k T ⟨i,hi⟩)
    (bodyTrace k T F)
  have run := Classical.choice actual
  have hstart : AddressOdometerSchedule.initialConfiguration (m := F.equations.length) k
      (AddressMachineProgram.bodyStart k T) (AddressMachineSpace.initialState k).1 (AddressMachineInitial.initialized k T F) =
      ⟨some (AddressMachineProgram.bodyStart k T), AddressMachineSpace.initialState k, AddressMachineInitial.initialized k T F⟩ := by
    simp only [AddressOdometerSchedule.initialConfiguration, AddressOdometerSchedule.configuration,
      setDigits_initialized_zero, AddressMachineSpace.initialState]
  have hfinish : AddressOdometerSchedule.finalConfiguration (AddressMachineProgram.next k T k) (AddressMachineSpace.initialState k).1
      (AddressMachineInitial.initialized k T F) (bits k T F) =
      ⟨some (AddressMachineProgram.finishStart k T), AddressMachineSpace.initialState k, finalTapes k T F⟩ := by
    simp only [AddressOdometerSchedule.finalConfiguration, AddressOdometerSchedule.configuration, finalTapes,
      AddressMachineProgram.next_ge k T k (Nat.le_refl _), AddressMachineSpace.initialState]
  rw [hstart, hfinish] at run
  exact run

/-- Exact output order, including repeated constraints and the original header. -/
theorem allBits_eq (F : SourceEncoding.Input) :
    AddressOdometerSchedule.allBits (bits k T F) =
      (FixedOutcomes.occurrenceTuples F.equations.length k).flatMap
        (AddressOutcomeSpecs.tupleBits (AddressTupleBody.source F) k T) := by
  exact (OccurrenceTupleOrder.functions_flatMap F.equations.length k _).symm

@[simp] theorem final_reversed (F : SourceEncoding.Input) :
    finalTapes k T F (AddressMachineSpace.headerTape k s d T.vectors.length .reversed) =
      ((FixedOutcomes.occurrenceTuples F.equations.length k).flatMap
        (AddressOutcomeSpecs.tupleBits (AddressTupleBody.source F) k T)).reverse ++
      (MachineAddressHeaders.Arithmetic.headerBits k s d T.vectors.length
        F.«variables» F.equations.length).reverse := by
  change (AddressOdometerSchedule.finalConfiguration _ _ _ _).stk (AddressOdometerSchedule.accumulator _ _ _ _) = _
  rw [AddressOdometerSchedule.final_accumulator]
  change (AddressOdometerSchedule.allBits (bits k T F)).reverse ++ _ = _
  rw [allBits_eq]
  congr 1
  exact AddressMachineInitial.initialized_reversed k T F

/-- The actual header machine emits exactly the target instance's three headers. -/
theorem headerBits_eq (F : SourceEncoding.Input) :
    MachineAddressHeaders.Arithmetic.headerBits k s d T.vectors.length
      F.«variables» F.equations.length =
      encodeWords [(AddressGame.tableOutput (AddressTupleBody.source F) k T).vertices, 2^s,
        (AddressGame.tableOutput (AddressTupleBody.source F) k T).constraints.length] := by
  have hcap : MachineAddressHeaders.capacity k s d F.«variables» F.equations.length =
      AddressGame.bodyCapacity (AddressTupleBody.source F) k s d := by
    change MachineAddressHeaders.capacity k s d F.«variables» F.equations.length =
      CanonicalAddress.capacity F.«variables» F.equations.length k s d
    simp [MachineAddressHeaders.capacity, MachineAddressHeaders.radix,
      MachineAddressHeaders.baseConstant, CanonicalAddress.capacity,
      CanonicalAddress.base_eq, Nat.add_assoc]
  have hedge : MachineAddressHeaders.edgeCount k s d T.vectors.length F.equations.length =
      (AddressGame.tableOutput (AddressTupleBody.source F) k T).constraints.length := by
    rw [AddressOutputSize.table_constraints_eq]
    change MachineAddressHeaders.edgeCount k s d T.vectors.length F.equations.length =
      Explicit.edgeCount F.equations.length k (s+d) T.vectors.length
    unfold MachineAddressHeaders.edgeCount MachineAddressHeaders.edgeFactor Explicit.edgeCount
    ring
  rw [MachineAddressHeaders.Arithmetic.headerBits, hcap, hedge]
  rfl

/-- The final reversed tape is exactly the complete encoded actual instance. -/
theorem final_accumulator (F : SourceEncoding.Input) :
    finalTapes k T F (AddressMachineProgram.accumulator k T) =
      (gameBits (AddressGame.tableOutput (AddressTupleBody.source F) k T)).reverse := by
  rw [AddressOutcomeSpecs.gameBits_factor, ← headerBits_eq k T F, List.reverse_append]
  exact final_reversed k T F

theorem final_other (F : SourceEncoding.Input) (tape : AddressMachineProgram.Tape k T)
    (hc : ∀j, tape ≠ AddressMachineSpace.current k s d T.vectors.length j)
    (hr : ∀j, tape ≠ AddressMachineSpace.remaining k s d T.vectors.length j)
    (ha : tape ≠ AddressMachineSpace.headerTape k s d T.vectors.length .reversed) :
    finalTapes k T F tape = AddressMachineInitial.initialized k T F tape :=
  AddressOdometerSchedule.final_other _ _ _ _ tape hc hr ha

@[simp] theorem final_outputEmpty (F : SourceEncoding.Input) :
    finalTapes k T F (AddressMachineProgram.output k T) = [] := by
  rw [final_other]
  · exact AddressMachineInitial.initialized_finalOutput k T F
  · intro j
    exact Ne.symm (AddressMachineSpace.current_ne_finalOutput k s d T.vectors.length j)
  · intro j
    exact Ne.symm (AddressMachineSpace.remaining_ne_finalOutput k s d T.vectors.length j)
  · exact Ne.symm (AddressMachineSpace.headerTape_ne_finalOutput k s d T.vectors.length .reversed)

@[simp] theorem final_current (F : SourceEncoding.Input) (j : Fin k) :
    finalTapes k T F (AddressMachineSpace.current k s d T.vectors.length j) = encodeWord 0 :=
  AddressOdometerSchedule.final_current _ _ _ _ j

@[simp] theorem final_remaining (F : SourceEncoding.Input) (j : Fin k) :
    finalTapes k T F (AddressMachineSpace.remaining k s d T.vectors.length j) =
      encodeWord (F.equations.length-1) :=
  AddressOdometerSchedule.final_remaining _ _ _ _ j

/-- The exponent and coefficients depend only on fixed reduction parameters. -/
def timePolynomial : Polynomial Nat :=
  (AddressTupleBudget.timePolynomial k T + Polynomial.C (2*k)) * Polynomial.X^k

@[simp] theorem timePolynomial_eval (N : Nat) :
    (timePolynomial k T).eval N = ((AddressTupleBudget.timePolynomial k T).eval N + 2*k)*N^k := by
  simp only [timePolynomial, Polynomial.eval_mul, Polynomial.eval_add,
    Polynomial.eval_C, Polynomial.eval_pow, Polynomial.eval_X]

def inPolynomialTime (F : SourceEncoding.Input) :
    StateTransition.EvalsToInTime (TM2.step (AddressMachineProgram.program k T))
      ⟨some (AddressMachineProgram.bodyStart k T), AddressMachineSpace.initialState k, AddressMachineInitial.initialized k T F⟩
      (some ⟨some (AddressMachineProgram.finishStart k T), AddressMachineSpace.initialState k, finalTapes k T F⟩)
      ((timePolynomial k T).eval (SourceEncoding.inputBits F).length) := by
  have run := traversal k T F
  refine { steps := run.steps, evals_in_steps := run.evals_in_steps, steps_le_m := ?_ }
  apply run.steps_le_m.trans
  rw [timePolynomial_eval]
  exact Nat.mul_le_mul_left _ (Nat.pow_le_pow_left (SourceEncoding.inputBits_length_ge_equations F) k)

end
end MaxCutGames.Reduction.AddressMachineLoop

/-!
The actual address machine's final bridge, cleanup, reversal, and halt. The
polynomial cleanup bound follows from space used by the supplied actual prefix
execution. The complete reduction supplies that prefix in `MachineAddressGame`.
-/

namespace MaxCutGames.Reduction.AddressMachineFinish

open Turing Foundations.Complexity Foundations.Hastad

noncomputable section

variable (k : Nat) {s d : Nat} (T : Integration.NoiseTables.Table s d)

def timePolynomial (prefixTime : Polynomial Nat) : Polynomial Nat :=
  SourceRuntimeSpace.completedTime (AddressMachineProgram.machine k T)
    (AddressMachineProgram.clearKeys k T).length (prefixTime + 1)

/-- Finish the concrete program, including its one-step finalizer entry bridge.
The runtime bound uses all actual intermediate tapes, with no output-size or
private-space assumption. -/
def completePrefix (input : List Bool) (prefixTime : Polynomial Nat)
    (base : AddressMachineProgram.Tape k T → List Bool)
    (execution : StateTransition.EvalsToInTime (AddressMachineProgram.machine k T).step
      (initList (AddressMachineProgram.machine k T) input)
      (some ⟨some (AddressMachineProgram.finishStart k T),
        AddressMachineSpace.initialState k, base⟩) (prefixTime.eval input.length))
    (outputEmpty : base (AddressMachineProgram.output k T) = []) :
    TM2OutputsInTime (AddressMachineProgram.machine k T) input
      (some (base (AddressMachineProgram.accumulator k T)).reverse)
      ((timePolynomial k T prefixTime).eval input.length) := by
  let bridge := AddressMachineProgram.finishStartInTime k T
    (AddressMachineSpace.initialState k) base
  let joined := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ execution bridge
  let prefixRun : StateTransition.EvalsToInTime (AddressMachineProgram.machine k T).step
      (initList (AddressMachineProgram.machine k T) input)
      (some ⟨SourceRuntimeFinish.entry (AddressMachineProgram.clearKeys k T)
          (AddressMachineProgram.finishLabels k T), AddressMachineSpace.initialState k, base⟩)
      ((prefixTime + 1).eval input.length) := {
    toEvalsTo := joined.toEvalsTo
    steps_le_m := by
      simpa only [Polynomial.eval_add, Polynomial.eval_one, Nat.add_comm] using
        joined.steps_le_m }
  let finishRun := SourceRuntimeFinish.finishInTime
    (AddressMachineProgram.clearKeys k T)
    (AddressMachineProgram.accumulator k T) (AddressMachineProgram.output k T)
    (AddressMachineProgram.accumulator_ne_output k T)
    (AddressMachineProgram.accumulator_not_mem_clearKeys k T)
    (AddressMachineProgram.output_not_mem_clearKeys k T)
    (AddressMachineProgram.clearKeys_covers k T)
    (AddressMachineSpace.initialState k).1 (AddressMachineProgram.finishLabels k T) none
    (AddressMachineProgram.program k T) (AddressMachineProgram.atFinish k T)
    base outputEmpty (AddressMachineSpace.initialState k).1 none
  let run := SourceRuntimeSpace.prefixAndFinishInTime (AddressMachineProgram.machine k T)
    input (prefixTime + 1) prefixRun
    (AddressMachineProgram.clearKeys k T)
    (AddressMachineProgram.accumulator k T) (AddressMachineProgram.output k T)
    (AddressMachineProgram.accumulator_not_mem_clearKeys k T)
    (AddressMachineProgram.output_not_mem_clearKeys k T)
    (AddressMachineProgram.clearKeys_covers k T) base outputEmpty (fun _ => rfl) finishRun
  change StateTransition.EvalsToInTime (AddressMachineProgram.machine k T).step
    (initList (AddressMachineProgram.machine k T) input)
    (some (haltList (AddressMachineProgram.machine k T)
      (base (AddressMachineProgram.accumulator k T)).reverse)) _
  rw [AddressMachineProgram.haltList_eq]
  exact run

end
end MaxCutGames.Reduction.AddressMachineFinish

/-!
# The complete addressed outer-reduction machine

The fixed finite machine reads an encoded nonempty E3LIN occurrence list and
writes the exact encoded addressed Unique Games instance. Its execution covers
all headers, all occurrence tuples, every fixed noise/coefficient outcome,
every permutation image, reversal, and complete tape cleanup.
-/

namespace MaxCutGames.Reduction.MachineAddressGame

open Turing Foundations.Complexity

noncomputable section

variable (k : Nat) {s d : Nat} (T : Integration.NoiseTables.Table s d)

def output (input : SourceEncoding.Input) : Foundations.Target.Instance (2 ^ s) :=
  AddressGame.tableOutput
    (ActualSource.Source.ofList input.equations input.nonempty) k T

def prefixPolynomial : Polynomial Nat :=
  AddressMachineInitial.timePolynomial k T + AddressMachineLoop.timePolynomial k T

def timePolynomial : Polynomial Nat :=
  AddressMachineFinish.timePolynomial k T (prefixPolynomial k T)

/-- The complete actual prefix, with both initialization and every tuple body
supplied by their constructed finite programs. -/
def prefixInTime (input : SourceEncoding.Input) :
    StateTransition.EvalsToInTime (AddressMachineProgram.machine k T).step
      (initList (AddressMachineProgram.machine k T) (SourceEncoding.inputBits input))
      (some ⟨some (AddressMachineProgram.finishStart k T), AddressMachineSpace.initialState k,
        AddressMachineLoop.finalTapes k T input⟩)
      ((prefixPolynomial k T).eval (SourceEncoding.inputBits input).length) := by
  let initial := AddressMachineInitial.inPolynomialTime k T input
  let loop := AddressMachineLoop.inPolynomialTime k T input
  let run := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ initial loop
  refine { toEvalsTo := run.toEvalsTo, steps_le_m := ?_ }
  simpa only [prefixPolynomial, Polynomial.eval_add, Nat.add_comm] using run.steps_le_m

/-- Literal `initList` to literal `haltList`, including all private tape and
finite-state cleanup. This theorem has no runtime premise. -/
def fullRunInTime (input : SourceEncoding.Input) :
    TM2OutputsInTime (AddressMachineProgram.machine k T) (SourceEncoding.inputBits input)
      (some (gameBits (output k T input)))
      ((timePolynomial k T).eval (SourceEncoding.inputBits input).length) := by
  let run := AddressMachineFinish.completePrefix k T (SourceEncoding.inputBits input)
    (prefixPolynomial k T) (AddressMachineLoop.finalTapes k T input)
    (prefixInTime k T input) (AddressMachineLoop.final_outputEmpty k T input)
  simpa only [AddressMachineLoop.final_accumulator, List.reverse_reverse, output,
    Integration.AddressTupleBody.source, timePolynomial] using run

def computableInPolyTime :
    TM2ComputableInPolyTime SourceEncoding.inputBits gameBits (output k T) where
  tm := AddressMachineProgram.machine k T
  inputAlphabet := Equiv.refl Bool
  outputAlphabet := Equiv.refl Bool
  time := timePolynomial k T
  outputsFun input := by
    change TM2OutputsInTime (AddressMachineProgram.machine k T)
      ((SourceEncoding.inputBits input).map id)
      (some ((gameBits (output k T input)).map id))
      ((timePolynomial k T).eval (SourceEncoding.inputBits input).length)
    have hi := @List.map_id ((AddressMachineProgram.machine k T).Γ
      (AddressMachineProgram.machine k T).k₀) (SourceEncoding.inputBits input)
    have ho := @List.map_id ((AddressMachineProgram.machine k T).Γ
      (AddressMachineProgram.machine k T).k₁) (gameBits (output k T input))
    erw [hi, ho]
    exact fullRunInTime k T input

/-- Every working tape of the actual certificate uses the finite Bool alphabet. -/
theorem workAlphabetFinite (tape : (computableInPolyTime k T).tm.K) :
    Finite ((computableInPolyTime k T).tm.Γ tape) := by
  change Finite Bool
  infer_instance

end
end MaxCutGames.Reduction.MachineAddressGame

/-!
The old addressed producer is reused only as a code generator. Its exact output
is the doubled codec of the new single-orbit game. The proved physical
postprocessor then yields the new game, with no premise about the value of the
legacy game and no legacy gadget or inverse theorem used.
-/

namespace MaxCutGames.Explicit.MachineSingleOrbitBridge

open Turing MaxCutGames.Reduction MaxCutGames.Foundations.Complexity
open ActualSource MachineSingleOrbitCodec

namespace SingleAddress

abbrev vertexCount := Decoder.TableKeysAddressGame.vertexCount
abbrev queryAddress := Decoder.TableKeysAddressGame.queryAddress
abbrev addressEdge := Decoder.TableKeysAddressGame.addressEdge
abbrev outputInstance := Decoder.TableKeysAddressGame.outputInstance
abbrev tableOutput := Decoder.TableKeysAddressGame.tableOutput

end SingleAddress

theorem old_addressEdge_eq_double (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (ω : ActualGame.Outcome S k g) :
    AddressGame.addressEdge S k g ω = doubleConstraint (SingleAddress.addressEdge S k g ω) := by
  unfold AddressGame.addressEdge doubleConstraint SingleAddress.addressEdge
    Decoder.TableKeysAddressGame.addressEdge
  congr 1
  apply Fin.ext
  exact Encoding.sideEquiv_true_val _

/-- Exact equality of the ordered, fully serialized constraint data. -/
theorem old_outputInstance_eq_double (S : Source) (k : Nat) {s d : Nat}
    (g : SplitGadget s d) (en : NoiseEnumeration g) :
    AddressGame.outputInstance S k g en = doubleInstance (SingleAddress.outputInstance S k g en) := by
  unfold AddressGame.outputInstance doubleInstance SingleAddress.outputInstance
    Decoder.TableKeysAddressGame.outputInstance
  congr 1
  simp only [List.map_map, Function.comp_def, ← old_addressEdge_eq_double]
  rfl

theorem old_tableOutput_eq_double (S : Source) (k : Nat) {s d : Nat}
    (T : MaxCutGames.Integration.NoiseTables.Table s d) :
    AddressGame.tableOutput S k T = doubleInstance (SingleAddress.tableOutput S k T) :=
  old_outputInstance_eq_double S k _ _

def output (k : Nat) {s d : Nat} (T : MaxCutGames.Integration.NoiseTables.Table s d)
    (input : SourceEncoding.Input) : MaxCutGames.Foundations.Target.Instance (2 ^ s) :=
  SingleAddress.tableOutput (Source.ofList input.equations input.nonempty) k T

theorem old_output_eq_double (k : Nat) {s d : Nat}
    (T : MaxCutGames.Integration.NoiseTables.Table s d) (input : SourceEncoding.Input) :
    MachineAddressGame.output k T input = doubleInstance (output k T input) :=
  old_tableOutput_eq_double _ k T

noncomputable section

/-- Reinterpret the existing producer using the proved exact codec equality. -/
def doubledComputableInPolyTime (k : Nat) {s d : Nat}
    (T : MaxCutGames.Integration.NoiseTables.Table s d) :
    TM2ComputableInPolyTime SourceEncoding.inputBits
      (fun I => gameBits (doubleInstance I)) (output k T) := by
  let old := MachineAddressGame.computableInPolyTime k T
  refine {
    tm := old.tm
    inputAlphabet := old.inputAlphabet
    outputAlphabet := old.outputAlphabet
    time := old.time
    outputsFun := ?_ }
  intro input
  rw [← old_output_eq_double k T input]
  exact old.outputsFun input

def computableInPolyTime (k : Nat) {s d : Nat}
    (T : MaxCutGames.Integration.NoiseTables.Table s d) :
    TM2ComputableInPolyTime SourceEncoding.inputBits gameBits (output k T) :=
  MachineSequential.composeBits (f := output k T) (g := id) (doubledComputableInPolyTime k T)
    (MachineSingleOrbitRuntime.computableInPolyTime (2 ^ s))

end

end MaxCutGames.Explicit.MachineSingleOrbitBridge

namespace MaxCutGames.Explicit.MachineSingleOrbitBridge

theorem workAlphabetFinite (k : Nat) {s d : Nat}
    (T : MaxCutGames.Integration.NoiseTables.Table s d)
    (tape : (computableInPolyTime k T).tm.K) :
    Finite ((computableInPolyTime k T).tm.Γ tape) := by
  rcases tape with first | second
  · change Finite Bool
    infer_instance
  · rcases second with localTape | bridgeTape
    · change Finite Bool
      infer_instance
    · change Finite Bool
      infer_instance

end MaxCutGames.Explicit.MachineSingleOrbitBridge

end OAI