Datasets:

roseking
/

openAI-math-dataset

import OAI.Computability.UniqueGames.Machines.MachineDrainManyLemmas
import OAI.Computability.UniqueGames.Reduction.AddressMachineSpace

namespace OAI

section

namespace UniqueGamesTheorem.Integration.AddressTupleCleanup

open Turing UniqueGamesTheorem.Reduction UniqueGamesTheorem.Foundations.Complexity

variable (k s d noiseCount : Nat)

/-- The executable field order is slot-major within each tuple position. -/
def loadedFields : List (AddressMachineSpace.Tape k s d noiseCount) :=
  List.ofFn (fun i : Fin (4 * k) =>
    MachineSourceTuple.Tape.field ((CanonicalBodyMachine.fieldEquiv k).symm i).1
      ((CanonicalBodyMachine.fieldEquiv k).symm i).2)

@[simp] theorem loadedFields_length : (loadedFields k s d noiseCount).length = 4 * k := by
  simp only [loadedFields, List.length_ofFn]

theorem mem_loadedFields (tape : AddressMachineSpace.Tape k s d noiseCount) :
    tape ∈ loadedFields k s d noiseCount ↔
      ∃ position : Fin k, ∃ slot : Fin 4, tape = .field position slot := by
  unfold loadedFields
  constructor
  · intro h
    obtain ⟨i, hi⟩ := List.mem_ofFn.mp h
    exact ⟨((CanonicalBodyMachine.fieldEquiv k).symm i).1,
      ((CanonicalBodyMachine.fieldEquiv k).symm i).2, hi.symm⟩
  · rintro ⟨position, slot, rfl⟩
    apply List.mem_ofFn.mpr
    refine ⟨CanonicalBodyMachine.fieldEquiv k (position, slot), ?_⟩
    simp only [Equiv.symm_apply_apply]

abbrev Label := MachineDrainMany.Label (loadedFields k s d noiseCount)

noncomputable abbrev cleared (base : AddressMachineSpace.Tape k s d noiseCount → List Bool) :=
  MachineDrainMany.finalTapes (loadedFields k s d noiseCount) base

/-- Proof-side cost on the shared arena. Its arithmetic-header tape dimensions
use the fixed polynomial coefficient representation in `PolynomialMachine`.
The executable drain list, entry, and instructions do not evaluate this cost. -/
noncomputable abbrev steps (base : AddressMachineSpace.Tape k s d noiseCount → List Bool) :=
  MachineDrainMany.steps (loadedFields k s d noiseCount) base

noncomputable abbrev budget (base : AddressMachineSpace.Tape k s d noiseCount → List Bool) :=
  MachineDrainMany.lengthSum (loadedFields k s d noiseCount) base + 4 * k

@[simp] theorem cleared_field (base : AddressMachineSpace.Tape k s d noiseCount → List Bool)
    (position : Fin k) (slot : Fin 4) :
    cleared k s d noiseCount base (.field position slot) = [] :=
  MachineDrainMany.finalTapes_mem (loadedFields k s d noiseCount) base (.field position slot)
    ((mem_loadedFields k s d noiseCount _).2 ⟨position, slot, rfl⟩)

/-- Every tape outside the physical loaded fields is preserved verbatim. -/
theorem cleared_outside (base : AddressMachineSpace.Tape k s d noiseCount → List Bool)
    (tape : AddressMachineSpace.Tape k s d noiseCount)
    (outside : ∀ position : Fin k, ∀ slot : Fin 4, tape ≠ .field position slot) :
    cleared k s d noiseCount base tape = base tape := by
  apply MachineDrainMany.finalTapes_not_mem
  intro h
  obtain ⟨position, slot, same⟩ := (mem_loadedFields k s d noiseCount tape).1 h
  exact outside position slot same

@[simp] theorem cleared_source (base : AddressMachineSpace.Tape k s d noiseCount → List Bool) :
    cleared k s d noiseCount base .source = base .source := by
  apply cleared_outside
  intro position slot h
  cases h

@[simp] theorem cleared_savedIndex
    (base : AddressMachineSpace.Tape k s d noiseCount → List Bool) (position : Fin k) :
    cleared k s d noiseCount base (.savedIndex position) = base (.savedIndex position) := by
  apply cleared_outside
  intro other slot h
  cases h

@[simp] theorem cleared_extra (base : AddressMachineSpace.Tape k s d noiseCount → List Bool)
    (extra : AddressMachineSpace.Extra k s d noiseCount) :
    cleared k s d noiseCount base (.extra extra) = base (.extra extra) := by
  apply cleared_outside
  intro position slot h
  cases h

@[simp] theorem cleared_index (base : AddressMachineSpace.Tape k s d noiseCount → List Bool) :
    cleared k s d noiseCount base .index = base .index := by
  apply cleared_outside
  intro position slot h
  cases h

@[simp] theorem cleared_work (base : AddressMachineSpace.Tape k s d noiseCount → List Bool) :
    cleared k s d noiseCount base .work = base .work := by
  apply cleared_outside
  intro position slot h
  cases h

@[simp] theorem cleared_scratch (base : AddressMachineSpace.Tape k s d noiseCount → List Bool) :
    cleared k s d noiseCount base .scratch = base .scratch := by
  apply cleared_outside
  intro position slot h
  cases h

@[simp] theorem cleared_copyScratch (base : AddressMachineSpace.Tape k s d noiseCount → List Bool) :
    cleared k s d noiseCount base .copyScratch = base .copyScratch := by
  apply cleared_outside
  intro position slot h
  cases h

@[simp] theorem cleared_headerTape
    (base : AddressMachineSpace.Tape k s d noiseCount → List Bool)
    (header : MachineAddressHeaders.Arithmetic.Control) :
    cleared k s d noiseCount base (AddressMachineSpace.headerTape k s d noiseCount header) =
      base (AddressMachineSpace.headerTape k s d noiseCount header) := by
  simpa only [AddressMachineSpace.headerTape, AddressMachineSpace.headerArithmeticSlots] using
    cleared_extra k s d noiseCount base (.inl (.inl header))

@[simp] theorem cleared_current (base : AddressMachineSpace.Tape k s d noiseCount → List Bool)
    (position : Fin k) :
    cleared k s d noiseCount base (AddressMachineSpace.current k s d noiseCount position) =
      base (AddressMachineSpace.current k s d noiseCount position) := by
  simpa only [AddressMachineSpace.current] using
    cleared_savedIndex k s d noiseCount base position.rev

@[simp] theorem cleared_remaining (base : AddressMachineSpace.Tape k s d noiseCount → List Bool)
    (position : Fin k) :
    cleared k s d noiseCount base (AddressMachineSpace.remaining k s d noiseCount position) =
      base (AddressMachineSpace.remaining k s d noiseCount position) := by
  simpa only [AddressMachineSpace.remaining] using
    cleared_extra k s d noiseCount base (.inr (.inr (.inl position)))

@[simp] theorem cleared_finalOutput
    (base : AddressMachineSpace.Tape k s d noiseCount → List Bool) :
    cleared k s d noiseCount base (AddressMachineSpace.finalOutput k s d noiseCount) =
      base (AddressMachineSpace.finalOutput k s d noiseCount) := by
  simpa only [AddressMachineSpace.finalOutput] using
    cleared_extra k s d noiseCount base (.inr (.inr (.inr ())))

theorem steps_le (base : AddressMachineSpace.Tape k s d noiseCount → List Bool) :
    steps k s d noiseCount base ≤ budget k s d noiseCount base := by
  simpa only [loadedFields_length] using
    MachineDrainMany.steps_le (loadedFields k s d noiseCount) base

theorem steps_le_uniform (base : AddressMachineSpace.Tape k s d noiseCount → List Bool)
    (bound : Nat) (h : ∀ tape, (base tape).length ≤ bound) :
    steps k s d noiseCount base ≤ 4 * k * (bound + 1) := by
  simpa only [loadedFields_length] using
    MachineDrainMany.steps_le_uniform (loadedFields k s d noiseCount) base bound h

variable {Λ : Type}

abbrev entry (labels : Label k s d noiseCount → Λ) (exit : Option Λ) :=
  MachineDrainMany.entry (loadedFields k s d noiseCount) labels exit

/-- Fixed concrete drain instructions on the actual shared address state. -/
def instruction (labels : Label k s d noiseCount → Λ) (exit : Option Λ) :
    Label k s d noiseCount → TM2.Stmt
      (fun _ : AddressMachineSpace.Tape k s d noiseCount => Bool) Λ (AddressMachineSpace.State k) :=
  MachineDrainMany.instruction (loadedFields k s d noiseCount) labels exit

theorem trace (labels : Label k s d noiseCount → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : AddressMachineSpace.Tape k s d noiseCount => Bool)
      Λ (AddressMachineSpace.State k))
    (atLabels : ∀ l, program (labels l) = instruction k s d noiseCount labels exit l)
    (base : AddressMachineSpace.Tape k s d noiseCount → List Bool)
    (control : AddressMachineSpace.Control k) (register : Option Bool) :
    (MachineComposition.advance (TM2.step program))^[steps k s d noiseCount base]
      (some ⟨entry k s d noiseCount labels exit, (control, register), base⟩) =
    some ⟨exit, (control, MachineDrainMany.finalRegister (loadedFields k s d noiseCount) register),
      cleared k s d noiseCount base⟩ :=
  MachineDrainMany.trace (loadedFields k s d noiseCount) labels exit program atLabels
    base control register

theorem trace_clean (labels : Label k s d noiseCount → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : AddressMachineSpace.Tape k s d noiseCount => Bool)
      Λ (AddressMachineSpace.State k))
    (atLabels : ∀ l, program (labels l) = instruction k s d noiseCount labels exit l)
    (base : AddressMachineSpace.Tape k s d noiseCount → List Bool)
    (control : AddressMachineSpace.Control k) :
    (MachineComposition.advance (TM2.step program))^[steps k s d noiseCount base]
      (some ⟨entry k s d noiseCount labels exit, (control, none), base⟩) =
    some ⟨exit, (control, none), cleared k s d noiseCount base⟩ := by
  simpa only [MachineDrainMany.finalRegister_none] using
    trace k s d noiseCount labels exit program atLabels base control none

noncomputable def execution (labels : Label k s d noiseCount → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : AddressMachineSpace.Tape k s d noiseCount => Bool)
      Λ (AddressMachineSpace.State k))
    (atLabels : ∀ l, program (labels l) = instruction k s d noiseCount labels exit l)
    (base : AddressMachineSpace.Tape k s d noiseCount → List Bool)
    (control : AddressMachineSpace.Control k) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨entry k s d noiseCount labels exit, (control, register), base⟩
      (some ⟨exit,
        (control, MachineDrainMany.finalRegister (loadedFields k s d noiseCount) register),
        cleared k s d noiseCount base⟩) (budget k s d noiseCount base) where
  steps := steps k s d noiseCount base
  evals_in_steps := trace k s d noiseCount labels exit program atLabels base control register
  steps_le_m := steps_le k s d noiseCount base

end UniqueGamesTheorem.Integration.AddressTupleCleanup

end

section

namespace UniqueGamesTheorem.Integration.AddressTupleLoaded

open UniqueGamesTheorem.Reduction UniqueGamesTheorem.Foundations.Complexity

abbrev Arena (k s d noiseCount : Nat) := AddressMachineSpace.Tape k s d noiseCount

variable {k s d noiseCount : Nat}

/-- The explicit field assignment performed by all `k` real lookup stages.
Unrelated tapes, including the loader's empty index/work tapes, are unchanged. -/
def fieldAssignment (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool) : Arena k s d noiseCount → List Bool
  | .field j slot => encodeWord
      ((SourceEncoding.equationWords F.equations[(tuple j).val])[slot.val]'(by simp)) ++
        baseArena (.field j slot)
  | tape => baseArena tape

theorem stageTapes_eq (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (hindex : baseArena .index = []) (hwork : baseArena .work = []) :
    MachineSourceTuple.stageTapes F tuple baseArena k = fieldAssignment F tuple baseArena := by
  funext tape
  cases tape with
  | index => exact (MachineSourceTuple.stageTapes_clean F tuple baseArena k hindex hwork).1.trans hindex.symm
  | work => exact (MachineSourceTuple.stageTapes_clean F tuple baseArena k hindex hwork).2.trans hwork.symm
  | field j slot =>
    simp only [MachineSourceTuple.stageTapes_field, ite_eq_left j.isLt, fieldAssignment]
  | source =>
    exact MachineSourceTuple.stageTapes_frame F tuple baseArena k _
      (by simp) (by simp) (by intros; simp)
  | scratch =>
    exact MachineSourceTuple.stageTapes_frame F tuple baseArena k _
      (by simp) (by simp) (by intros; simp)
  | copyScratch =>
    exact MachineSourceTuple.stageTapes_frame F tuple baseArena k _
      (by simp) (by simp) (by intros; simp)
  | savedIndex j =>
    exact MachineSourceTuple.stageTapes_frame F tuple baseArena k _
      (by simp) (by simp) (by intros; simp)
  | extra value =>
    exact MachineSourceTuple.stageTapes_frame F tuple baseArena k _
      (by simp) (by simp) (by intros; simp)

theorem fieldAssignment_frame (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool) (tape : Arena k s d noiseCount)
    (hfield : ∀ j slot, tape ≠ .field j slot) :
    fieldAssignment F tuple baseArena tape = baseArena tape := by
  cases tape <;> simp_all [fieldAssignment]

theorem loaded_frame (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (hindex : baseArena .index = []) (hwork : baseArena .work = [])
    (tape : Arena k s d noiseCount) (hfield : ∀ j slot, tape ≠ .field j slot) :
    MachineSourceTuple.stageTapes F tuple baseArena k tape = baseArena tape := by
  rw [stageTapes_eq F tuple baseArena hindex hwork]
  exact fieldAssignment_frame F tuple baseArena tape hfield

theorem loaded_field (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (hempty : ∀ j slot, baseArena (.field j slot) = []) (j : Fin k) (slot : Fin 4) :
    MachineSourceTuple.stageTapes F tuple baseArena k (.field j slot) =
      encodeWord ((SourceEncoding.equationWords F.equations[(tuple j).val])[slot.val]'(by simp)) := by
  rw [MachineSourceTuple.stageTapes_field, ite_eq_left j.isLt, hempty, List.append_nil]
theorem loaded_source (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (hsource : baseArena .source = SourceEncoding.inputBits F) :
    MachineSourceTuple.stageTapes F tuple baseArena k .source = SourceEncoding.inputBits F :=
  (MachineSourceTuple.output_source F tuple baseArena).trans hsource
theorem loaded_extra (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (extra : AddressMachineSpace.Extra k s d noiseCount) :
    MachineSourceTuple.stageTapes F tuple baseArena k (.extra extra) = baseArena (.extra extra) :=
  MachineSourceTuple.stageTapes_frame F tuple baseArena k _
    (by simp) (by simp) (by intros; simp)
theorem loaded_header (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (header : MachineAddressHeaders.Arithmetic.Control) :
    MachineSourceTuple.stageTapes F tuple baseArena k
      (AddressMachineSpace.headerTape k s d noiseCount header) =
        baseArena (AddressMachineSpace.headerTape k s d noiseCount header) :=
  AddressMachineSpace.tupleOutput_headerTape k s d noiseCount F tuple baseArena header
theorem loaded_radix (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool) (B : Nat)
    (hB : baseArena (AddressMachineSpace.headerTape k s d noiseCount .baseValue) = encodeWord B) :
    MachineSourceTuple.stageTapes F tuple baseArena k
      (MachineAddressEdge.addressSlots (AddressMachineSpace.addressEdgeSlots k s d noiseCount)
        .radix) = encodeWord B := by
  rw [AddressMachineSpace.addressSlots_radix, loaded_header, hB]
theorem loaded_capacity (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool) (C : Nat)
    (hC : baseArena (AddressMachineSpace.headerTape k s d noiseCount .capacity) = encodeWord C) :
    MachineSourceTuple.stageTapes F tuple baseArena k
      (MachineAddressEdge.capacityTape (AddressMachineSpace.addressEdgeSlots k s d noiseCount)) =
        encodeWord C := by
  rw [AddressMachineSpace.addressSlots_capacity, loaded_header, hC]
theorem loaded_current (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool) (j : Fin k) :
    MachineSourceTuple.stageTapes F tuple baseArena k
      (AddressMachineSpace.current k s d noiseCount j) =
        baseArena (AddressMachineSpace.current k s d noiseCount j) :=
  MachineSourceTuple.output_savedIndex F tuple baseArena j.rev
theorem loaded_remaining (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool) (j : Fin k) :
    MachineSourceTuple.stageTapes F tuple baseArena k
      (AddressMachineSpace.remaining k s d noiseCount j) =
        baseArena (AddressMachineSpace.remaining k s d noiseCount j) :=
  loaded_extra F tuple baseArena _
theorem loaded_finalOutput (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool) :
    MachineSourceTuple.stageTapes F tuple baseArena k
      (AddressMachineSpace.finalOutput k s d noiseCount) =
        baseArena (AddressMachineSpace.finalOutput k s d noiseCount) :=
  loaded_extra F tuple baseArena _
theorem loaded_loaderWork (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (hindex : baseArena .index = []) (hwork : baseArena .work = [])
    (hscratch : baseArena .scratch = []) (hcopy : baseArena .copyScratch = []) :
    MachineSourceTuple.stageTapes F tuple baseArena k .index = [] ∧
      MachineSourceTuple.stageTapes F tuple baseArena k .work = [] ∧
      MachineSourceTuple.stageTapes F tuple baseArena k .scratch = [] ∧
      MachineSourceTuple.stageTapes F tuple baseArena k .copyScratch = [] := by
  have clean := MachineSourceTuple.stageTapes_clean F tuple baseArena k hindex hwork
  refine ⟨clean.1, clean.2, ?_, ?_⟩
  · exact (MachineSourceTuple.stageTapes_frame F tuple baseArena k .scratch
      (by simp) (by simp) (by intros; simp)).trans hscratch
  · exact (MachineSourceTuple.stageTapes_frame F tuple baseArena k .copyScratch
      (by simp) (by simp) (by intros; simp)).trans hcopy
theorem loaded_privateAddress (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (tape : MachineTemplateAddress.Tape (4 * k) (1 + 9 * k)) :
    MachineSourceTuple.stageTapes F tuple baseArena k
      (AddressMachineSpace.privateAddress k s d noiseCount tape) =
        baseArena (AddressMachineSpace.privateAddress k s d noiseCount tape) :=
  loaded_extra F tuple baseArena _
/-- Cleanliness of the actual edge machine is preserved by the actual loader.
All of these local work tapes occupy the private disjoint arena block. -/
theorem loaded_clean (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (clean : MachineAddressEdge.Clean (AddressMachineSpace.addressEdgeSlots k s d noiseCount)
      baseArena) :
    MachineAddressEdge.Clean (AddressMachineSpace.addressEdgeSlots k s d noiseCount)
      (MachineSourceTuple.stageTapes F tuple baseArena k) := by
  constructor
  · constructor
    · exact (loaded_privateAddress F tuple baseArena .reversed).trans clean.address.reversed
    · exact (loaded_privateAddress F tuple baseArena .forward).trans clean.address.forward
    · exact (loaded_privateAddress F tuple baseArena .copyScratch).trans clean.address.copyScratch
    · exact (loaded_privateAddress F tuple baseArena .accA).trans clean.address.accA
    · exact (loaded_privateAddress F tuple baseArena .accB).trans clean.address.accB
    · exact (loaded_privateAddress F tuple baseArena .counter).trans clean.address.counter
    · exact (loaded_privateAddress F tuple baseArena .hornerScratch).trans clean.address.hornerScratch
    · intro j
      exact (loaded_privateAddress F tuple baseArena (.digit j)).trans (clean.address.digit j)
  · exact (loaded_privateAddress F tuple baseArena .output).trans clean.work
theorem loaded_rhs (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (hempty : ∀ j slot, baseArena (.field j slot) = []) (j : Fin k) :
    MachineSourceTuple.stageTapes F tuple baseArena k
      (AddressMachineSpace.rhsField k s d noiseCount j) =
        encodeWord (if F.equations[(tuple j).val].rhs then 1 else 0) := by
  rw [AddressMachineSpace.tupleOutput_rhs]
  change _ ++ baseArena (.field j 3) = _
  rw [hempty, List.append_nil]
theorem loaded_name (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (hempty : ∀ j slot, baseArena (.field j slot) = []) (j : Fin k) (slot : Fin 3) :
    MachineSourceTuple.stageTapes F tuple baseArena k (MachineSourceTuple.nameField j slot) =
      encodeWord (MachineSourceTuple.equationName F.equations[(tuple j).val] slot) := by
  rw [MachineSourceTuple.output_name]
  change _ ++ baseArena (.field j (MachineSourceTuple.nameSlot slot)) = _
  rw [hempty, List.append_nil]
theorem loaded_occurrence (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (hsaved : ∀ j, baseArena (.savedIndex j) = encodeWord (tuple j).val) (j : Fin k) :
    MachineSourceTuple.stageTapes F tuple baseArena k (.savedIndex j) = encodeWord (tuple j).val :=
  MachineSourceTuple.output_savedIndex_unary F tuple baseArena hsaved j
def canonicalValues (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (j : Fin k) : Fin 4 → Nat :=
  CanonicalBodyTemplate.recordFields
    (MachineSourceTuple.equationName F.equations[(tuple j).val]) (tuple j).val
/-- The three names and the distinct saved occurrence field are exact unary
values; the RHS remains on its separate field tape. -/
theorem loaded_canonicalField (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (hsaved : ∀ j, baseArena (.savedIndex j) = encodeWord (tuple j).val)
    (hempty : ∀ j slot, baseArena (.field j slot) = []) (j : Fin k) (slot : Fin 4) :
    MachineSourceTuple.stageTapes F tuple baseArena k
      (AddressMachineSpace.canonicalField k s d noiseCount j slot) =
        encodeWord (canonicalValues F tuple j slot) := by
  fin_cases slot
  · simpa [AddressMachineSpace.canonicalField, canonicalValues, CanonicalBodyTemplate.recordFields,
      MachineSourceTuple.nameField, MachineSourceTuple.nameSlot, MachineSourceTuple.equationName] using
      loaded_name F tuple baseArena hempty j 0
  · simpa [AddressMachineSpace.canonicalField, canonicalValues, CanonicalBodyTemplate.recordFields,
      MachineSourceTuple.nameField, MachineSourceTuple.nameSlot, MachineSourceTuple.equationName] using
      loaded_name F tuple baseArena hempty j 1
  · simpa [AddressMachineSpace.canonicalField, canonicalValues, CanonicalBodyTemplate.recordFields,
      MachineSourceTuple.nameField, MachineSourceTuple.nameSlot, MachineSourceTuple.equationName] using
      loaded_name F tuple baseArena hempty j 2
  · simpa [AddressMachineSpace.canonicalField, canonicalValues, CanonicalBodyTemplate.recordFields] using
      loaded_occurrence F tuple baseArena hsaved j
/-- Literal input function used by the generic field-template machine. -/
theorem loaded_savedFields (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (baseArena : Arena k s d noiseCount → List Bool)
    (hsaved : ∀ j, baseArena (.savedIndex j) = encodeWord (tuple j).val)
    (hempty : ∀ j slot, baseArena (.field j slot) = []) :
    (fun i => MachineSourceTuple.stageTapes F tuple baseArena k
      (MachineAddressEdge.addressSlots (AddressMachineSpace.addressEdgeSlots k s d noiseCount)
        (.field i))) = CanonicalBodyMachine.savedFields (canonicalValues F tuple) := by
  funext i
  change MachineSourceTuple.stageTapes F tuple baseArena k
    (AddressMachineSpace.canonicalField k s d noiseCount
      ((CanonicalBodyMachine.fieldEquiv k).symm i).1
      ((CanonicalBodyMachine.fieldEquiv k).symm i).2) = _
  exact loaded_canonicalField F tuple baseArena hsaved hempty _ _
end UniqueGamesTheorem.Integration.AddressTupleLoaded
end
section
/-!
# The exact frame after a tuple body and its field cleanup
Loading replaces the initially empty equation-field tapes. Appending writes
only the shared reversed-output header, which is outside those fields. The
actual field drains therefore restore the original arena with precisely the
same appended bytes. Saved occurrence indices and the odometer survive.
-/
namespace UniqueGamesTheorem.Integration.AddressTupleFrame
open UniqueGamesTheorem.Reduction UniqueGamesTheorem.Foundations.Complexity
variable {k s d noiseCount : Nat}
abbrev Arena (k s d noiseCount : Nat) := AddressMachineSpace.Tape k s d noiseCount
/-- Clearing the physical fields commutes with an append to any other tape. -/
theorem cleared_outputTapes (base : Arena k s d noiseCount → List Bool)
    (output : Arena k s d noiseCount) (bits : List Bool)
    (outside : ∀ j : Fin k, ∀ slot : Fin 4, output ≠ .field j slot) :
    AddressTupleCleanup.cleared k s d noiseCount
        (MachineFieldTemplate.outputTapes base output bits) =
      MachineFieldTemplate.outputTapes
        (AddressTupleCleanup.cleared k s d noiseCount base) output bits := by
  funext tape
  by_cases h : tape = output
  · subst tape
    rw [AddressTupleCleanup.cleared_outside k s d noiseCount _ output outside,
      MachineFieldTemplate.outputTapes_output, MachineFieldTemplate.outputTapes_output,
      AddressTupleCleanup.cleared_outside k s d noiseCount base output outside]
  · rw [MachineFieldTemplate.outputTapes_other _ output bits tape h]
    change MachineDrainMany.finalTapes (AddressTupleCleanup.loadedFields k s d noiseCount)
        (MachineFieldTemplate.outputTapes base output bits) tape =
      MachineDrainMany.finalTapes (AddressTupleCleanup.loadedFields k s d noiseCount) base tape
    rw [MachineDrainMany.finalTapes_apply, MachineDrainMany.finalTapes_apply,
      MachineFieldTemplate.outputTapes_other base output bits tape h]
/-- The actual edge-output slot is a header tape, not a loaded equation field. -/
theorem cleared_appended (base : Arena k s d noiseCount → List Bool) (bits : List Bool) :
    AddressTupleCleanup.cleared k s d noiseCount
        (MachineAddressEdge.appended (AddressMachineSpace.addressEdgeSlots k s d noiseCount)
          base bits) =
      MachineAddressEdge.appended (AddressMachineSpace.addressEdgeSlots k s d noiseCount)
        (AddressTupleCleanup.cleared k s d noiseCount base) bits := by
  apply cleared_outputTapes
  intro j slot
  simp [AddressMachineSpace.addressSlots_output, AddressMachineSpace.headerTape]
/-- Cleanup erases exactly the fields assigned by the tuple loader. -/
theorem cleared_fieldAssignment (F : SourceEncoding.Input)
    (tuple : Fin k → Fin F.equations.length) (base : Arena k s d noiseCount → List Bool)
    (hfields : ∀ j slot, base (.field j slot) = []) :
    AddressTupleCleanup.cleared k s d noiseCount
      (AddressTupleLoaded.fieldAssignment F tuple base) = base := by
  funext tape
  cases tape <;> simp [AddressTupleLoaded.fieldAssignment, hfields]
theorem cleared_loaded (F : SourceEncoding.Input)
    (tuple : Fin k → Fin F.equations.length) (base : Arena k s d noiseCount → List Bool)
    (hindex : base .index = []) (hwork : base .work = [])
    (hfields : ∀ j slot, base (.field j slot) = []) :
    AddressTupleCleanup.cleared k s d noiseCount
      (MachineSourceTuple.stageTapes F tuple base k) = base := by
  rw [AddressTupleLoaded.stageTapes_eq F tuple base hindex hwork]
  exact cleared_fieldAssignment F tuple base hfields
/-- Exact endpoint of load, edge-byte append, and actual loaded-field cleanup. -/
theorem cleared_appended_loaded (F : SourceEncoding.Input)
    (tuple : Fin k → Fin F.equations.length) (base : Arena k s d noiseCount → List Bool)
    (hindex : base .index = []) (hwork : base .work = [])
    (hfields : ∀ j slot, base (.field j slot) = []) (bits : List Bool) :
    AddressTupleCleanup.cleared k s d noiseCount
        (MachineAddressEdge.appended (AddressMachineSpace.addressEdgeSlots k s d noiseCount)
          (MachineSourceTuple.stageTapes F tuple base k) bits) =
      MachineAddressEdge.appended (AddressMachineSpace.addressEdgeSlots k s d noiseCount)
        base bits := by
  rw [cleared_appended, cleared_loaded F tuple base hindex hwork hfields]
/-- Literal final arena; this abbreviation contains no execution assumption. -/
noncomputable abbrev finalTapes (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
    (base : Arena k s d noiseCount → List Bool) (bits : List Bool) :
    Arena k s d noiseCount → List Bool :=
  AddressTupleCleanup.cleared k s d noiseCount
    (MachineAddressEdge.appended (AddressMachineSpace.addressEdgeSlots k s d noiseCount)
      (MachineSourceTuple.stageTapes F tuple base k) bits)
section Frame
variable (F : SourceEncoding.Input) (tuple : Fin k → Fin F.equations.length)
  (base : Arena k s d noiseCount → List Bool)
  (hindex : base .index = []) (hwork : base .work = [])
  (hfields : ∀ j slot, base (.field j slot) = []) (bits : List Bool)
include hindex hwork hfields
theorem finalTapes_other (tape : Arena k s d noiseCount)
    (hne : tape ≠ MachineAddressEdge.outputTape
      (AddressMachineSpace.addressEdgeSlots k s d noiseCount)) :
    finalTapes F tuple base bits tape = base tape := by
  rw [finalTapes, cleared_appended_loaded F tuple base hindex hwork hfields bits]
  exact MachineAddressEdge.appended_other _ base bits tape hne
theorem finalTapes_source : finalTapes F tuple base bits .source = base .source := by
  apply finalTapes_other F tuple base hindex hwork hfields bits
  simp [AddressMachineSpace.addressSlots_output, AddressMachineSpace.headerTape]
theorem finalTapes_current (j : Fin k) :
    finalTapes F tuple base bits (AddressMachineSpace.current k s d noiseCount j) =
      base (AddressMachineSpace.current k s d noiseCount j) := by
  apply finalTapes_other F tuple base hindex hwork hfields bits
  simpa only [AddressMachineSpace.addressSlots_output] using
    (AddressMachineSpace.headerTape_ne_current k s d noiseCount .reversed j).symm
theorem finalTapes_remaining (j : Fin k) :
    finalTapes F tuple base bits (AddressMachineSpace.remaining k s d noiseCount j) =
      base (AddressMachineSpace.remaining k s d noiseCount j) := by
  apply finalTapes_other F tuple base hindex hwork hfields bits
  simpa only [AddressMachineSpace.addressSlots_output] using
    (AddressMachineSpace.headerTape_ne_remaining k s d noiseCount .reversed j).symm
/-- All arithmetic headers except the reversed-output tape remain literal. -/
theorem finalTapes_header (header : MachineAddressHeaders.Arithmetic.Control)
    (hne : header ≠ .reversed) :
    finalTapes F tuple base bits (AddressMachineSpace.headerTape k s d noiseCount header) =
      base (AddressMachineSpace.headerTape k s d noiseCount header) := by
  apply finalTapes_other F tuple base hindex hwork hfields bits
  intro same
  apply hne
  exact (AddressMachineSpace.headerTape_eq_iff k s d noiseCount header .reversed).mp same
theorem finalTapes_radix :
    finalTapes F tuple base bits (MachineAddressEdge.addressSlots
        (AddressMachineSpace.addressEdgeSlots k s d noiseCount) .radix) =
      base (MachineAddressEdge.addressSlots
        (AddressMachineSpace.addressEdgeSlots k s d noiseCount) .radix) := by
  rw [finalTapes, cleared_appended_loaded F tuple base hindex hwork hfields bits]
  exact MachineAddressEdge.appended_address _ base bits .radix
theorem finalTapes_capacity :
    finalTapes F tuple base bits (MachineAddressEdge.capacityTape
        (AddressMachineSpace.addressEdgeSlots k s d noiseCount)) =
      base (MachineAddressEdge.capacityTape
        (AddressMachineSpace.addressEdgeSlots k s d noiseCount)) := by
  rw [finalTapes, cleared_appended_loaded F tuple base hindex hwork hfields bits]
  exact MachineAddressEdge.appended_capacity _ base bits
theorem finalTapes_output :
    finalTapes F tuple base bits (MachineAddressEdge.outputTape
        (AddressMachineSpace.addressEdgeSlots k s d noiseCount)) =
      bits.reverse ++ base (MachineAddressEdge.outputTape
        (AddressMachineSpace.addressEdgeSlots k s d noiseCount)) := by
  rw [finalTapes, cleared_appended_loaded F tuple base hindex hwork hfields bits]
  exact MachineAddressEdge.appended_output _ base bits
theorem finalTapes_clean
    (clean : MachineAddressEdge.Clean
      (AddressMachineSpace.addressEdgeSlots k s d noiseCount) base) :
    MachineAddressEdge.Clean (AddressMachineSpace.addressEdgeSlots k s d noiseCount)
      (finalTapes F tuple base bits) := by
  rw [finalTapes, cleared_appended_loaded F tuple base hindex hwork hfields bits]
  exact MachineAddressEdge.Clean.appended _ base clean bits
end Frame
end UniqueGamesTheorem.Integration.AddressTupleFrame
end
end OAI