Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.DirectedFeedback.Machines.TupleOrder
import OAI.Computability.DirectedFeedback.Machines.LoopInit

namespace OAI


namespace DFVSGames.Foundations.Hastad.SourceContextClear

open Turing Complexity MachineComposition

abbrev Tape := SourceContextLoad.Tape

variable {u : Nat} {Extra : Type}

def fieldAt (i : Fin (u * 6)) : Tape u Extra :=
  .field (finProdFinEquiv.symm i).1 (finProdFinEquiv.symm i).2

def fields : List (Tape u Extra) := List.ofFn fieldAt

theorem fieldAt_injective : Function.Injective (fieldAt (u := u) (Extra := Extra)) := by
  intro i j h
  have hpair := SourceContextLoad.Tape.field.inj h
  have hp : finProdFinEquiv.symm i = finProdFinEquiv.symm j :=
    Prod.ext hpair.1 hpair.2
  exact finProdFinEquiv.symm.injective hp

@[simp] theorem fields_length : (fields (u := u) (Extra := Extra)).length = 6 * u := by
  simp [fields, Nat.mul_comm]

theorem fields_nodup : (fields (u := u) (Extra := Extra)).Nodup := by
  exact List.nodup_ofFn.mpr fieldAt_injective

theorem field_mem (j : Fin u) (s : Fin 6) :
    (SourceContextLoad.Tape.field j s : Tape u Extra) ∈ fields := by
  apply List.mem_ofFn.mpr
  exact ⟨finProdFinEquiv (j, s), by simp [fieldAt]⟩

theorem mem_fields_iff (p : Tape u Extra) : p ∈ fields ↔ ∃ j s, p = .field j s := by
  constructor
  · intro h
    obtain ⟨i, hi⟩ := List.mem_ofFn.mp h
    exact ⟨(finProdFinEquiv.symm i).1, (finProdFinEquiv.symm i).2, hi.symm⟩
  · rintro ⟨j, s, rfl⟩
    exact field_mem j s

def outputTapes (base : Tape u Extra → List Bool) : Tape u Extra → List Bool
  | .field _ _ => []
  | p => base p

@[simp] theorem output_field (base : Tape u Extra → List Bool) (j : Fin u) (s : Fin 6) :
    outputTapes base (.field j s) = [] := rfl

theorem output_frame (base : Tape u Extra → List Bool) (p : Tape u Extra)
    (hp : ∀ j s, p ≠ .field j s) : outputTapes base p = base p := by
  cases p <;> simp_all [outputTapes]
abbrev Label (u : Nat) (Extra : Type) :=
  MachineDrainMany.Label (fields (u := u) (Extra := Extra)) ⊕ Unit

private def entryLabel_inline_SourceContextClear {K : Type} : (chosen : List K) → MachineDrainMany.Label chosen ⊕ Unit
  | [] => .inr ()
  | _ :: _ => .inl (.inl ())

def main : Label u Extra := entryLabel_inline_SourceContextClear fields

variable [DecidableEq Extra]

def program : Label u Extra → TM2.Stmt (fun _ : Tape u Extra => Bool)
    (Label u Extra) (Unit × Option Bool)
  | .inl l => MachineDrainMany.instruction fields Sum.inl (some (.inr ())) l
  | .inr _ => .load (fun _ => ((), none)) .halt

private theorem entryLabel_eq_inline_SourceContextClear {K : Type} (chosen : List K) :
    MachineDrainMany.entry chosen
      (Sum.inl : MachineDrainMany.Label chosen → MachineDrainMany.Label chosen ⊕ Unit)
      (some (.inr ())) = some (entryLabel_inline_SourceContextClear chosen) := by
  cases chosen <;> rfl
omit [DecidableEq Extra] in
private theorem entry_eq_inline_SourceContextClear :
    MachineDrainMany.entry (fields (u := u) (Extra := Extra)) Sum.inl (some (.inr ())) =
      some (main (u := u) (Extra := Extra)) := entryLabel_eq_inline_SourceContextClear fields

theorem finalTapes_eq (base : Tape u Extra → List Bool) :
    MachineDrainMany.finalTapes fields base = outputTapes base := by
  funext p
  rw [MachineDrainMany.finalTapes_apply]
  cases p <;> simp [mem_fields_iff, outputTapes]
private theorem lengthSum_update_inline_SourceContextClear {K : Type} [DecidableEq K] (chosen : List K)
    (base : K → List Bool) (source : K) (h : source ∉ chosen) :
    MachineDrainMany.lengthSum chosen (Function.update base source []) =
      MachineDrainMany.lengthSum chosen base := by
  unfold MachineDrainMany.lengthSum
  congr 1
  apply List.map_congr_left
  intro k hk
  rw [Function.update_of_ne (by intro he; subst k; exact h hk)]
private theorem steps_eq_of_nodup_inline_SourceContextClear {K : Type} [DecidableEq K] (chosen : List K)
    (h : chosen.Nodup) (base : K → List Bool) :
    MachineDrainMany.steps chosen base =
      MachineDrainMany.lengthSum chosen base + chosen.length := by
  induction chosen generalizing base with
  | nil => rfl
  | cons source chosen ih =>
      obtain ⟨hn, hd⟩ := List.nodup_cons.mp h
      rw [MachineDrainMany.steps, ih hd, lengthSum_update_inline_SourceContextClear chosen base source hn]
      simp only [MachineDrainMany.lengthSum, List.map_cons, List.sum_cons, List.length_cons]
      omega

private theorem trace_trans_inline_SourceContextClear {α : Type*} (f : α → α) {a b : Nat} {x y z : α}
    (first : f^[a] x = y) (second : f^[b] y = z) : f^[a + b] x = z := by
  rw [Nat.add_comm, Function.iterate_add_apply, first, second]

theorem clearTrace (base : Tape u Extra → List Bool) (register : Option Bool) :
    (advance (TM2.step program))^[MachineDrainMany.lengthSum fields base + 6 * u + 1]
      (some ⟨some main, ((), register), base⟩) =
      some ⟨none, ((), none), outputTapes base⟩ := by
  have drain := MachineDrainMany.trace fields Sum.inl (some (.inr ()))
    program (fun _ => rfl) base () register
  rw [entry_eq_inline_SourceContextClear, finalTapes_eq, steps_eq_of_nodup_inline_SourceContextClear fields fields_nodup, fields_length] at drain
  have last : (advance (TM2.step program))^[1]
      (some ⟨some (.inr ()), ((), MachineDrainMany.finalRegister (fields (u := u) (Extra := Extra)) register), outputTapes base⟩) =
      some ⟨none, ((), none), outputTapes base⟩ := rfl
  exact trace_trans_inline_SourceContextClear _ drain last

def clearInTime (base : Tape u Extra → List Bool) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some main, ((), register), base⟩ (some ⟨none, ((), none), outputTapes base⟩)
      (MachineDrainMany.lengthSum fields base + 6 * u + 1) where
  steps := MachineDrainMany.lengthSum fields base + 6 * u + 1
  evals_in_steps := clearTrace base register
  steps_le_m := Nat.le_refl _

private theorem lengthSum_le_selected_inline_SourceContextClear {K : Type} (chosen : List K)
    (base : K → List Bool) (B : Nat) (h : ∀ k ∈ chosen, (base k).length ≤ B) :
    MachineDrainMany.lengthSum chosen base ≤ chosen.length * B := by
  induction chosen with
  | nil => simp [MachineDrainMany.lengthSum]
  | cons source chosen ih =>
      have hs := h source (by simp)
      have ht := ih (by intro k hk; exact h k (by simp [hk]))
      simp only [MachineDrainMany.lengthSum, List.map_cons, List.sum_cons,
        List.length_cons, Nat.add_mul, Nat.one_mul] at ht ⊢
      omega
theorem clauseField_length_le {n : Nat} (c : Target.Clause n) (s : Fin 6) :
    (encodeWord ((clauseWords c)[s.val]'(by simp))).length ≤ n + 2 := by
  have h0 := (c)[0].variableIndex.isLt
  have h1 := (c)[1].variableIndex.isLt
  have h2 := (c)[2].variableIndex.isLt
  fin_cases s <;> simp [clauseWords, literalWords, encodeWord_length]
  all_goals first | omega | (split <;> omega)
theorem variables_add_two_le_input (F : Target.Formula) :
    F.«variables» + 2 ≤ (formulaBits F).length := by
  simp only [formulaBits, formulaWords, encodeWords_append, List.length_append,
    encodeWords, encodeWord_length, List.length_nil]
  omega

omit [DecidableEq Extra] in
theorem loaded_field_length_le (F : Target.Formula)
    (tuple : Fin u → Fin F.clauses.length) (base : Tape u Extra → List Bool)
    (hfields : ∀ j s, base (.field j s) = []) (j : Fin u) (s : Fin 6) :
    (SourceContextLoad.stageTapes F tuple base u (.field j s)).length ≤
      (formulaBits F).length := by
  rw [SourceContextLoad.stageTapes_field, ite_eq_left j.isLt, hfields, List.append_nil]
  exact (clauseField_length_le _ s).trans (variables_add_two_le_input F)

omit [DecidableEq Extra] in
theorem loaded_cost_le (F : Target.Formula) (tuple : Fin u → Fin F.clauses.length)
    (base : Tape u Extra → List Bool) (hfields : ∀ j s, base (.field j s) = []) :
    MachineDrainMany.lengthSum fields (SourceContextLoad.stageTapes F tuple base u) + 6 * u + 1 ≤
      6 * u * ((formulaBits F).length + 1) + 1 := by
  have h := lengthSum_le_selected_inline_SourceContextClear fields (SourceContextLoad.stageTapes F tuple base u)
    (formulaBits F).length (by
      intro p hp
      obtain ⟨j, s, rfl⟩ := (mem_fields_iff p).mp hp
      exact loaded_field_length_le F tuple base hfields j s)
  rw [fields_length] at h
  rw [Nat.mul_add, Nat.mul_one]
  omega

def clearLoadedInTime (F : Target.Formula) (tuple : Fin u → Fin F.clauses.length)
    (base : Tape u Extra → List Bool) (hfields : ∀ j s, base (.field j s) = [])
    (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some main, ((), register), SourceContextLoad.stageTapes F tuple base u⟩
      (some ⟨none, ((), none), outputTapes (SourceContextLoad.stageTapes F tuple base u)⟩)
      (6 * u * ((formulaBits F).length + 1) + 1) where
  steps := MachineDrainMany.lengthSum fields (SourceContextLoad.stageTapes F tuple base u) + 6 * u + 1
  evals_in_steps := clearTrace _ register
  steps_le_m := loaded_cost_le F tuple base hfields

omit [DecidableEq Extra] in
theorem output_loaded_eq_base (F : Target.Formula) (tuple : Fin u → Fin F.clauses.length)
    (base : Tape u Extra → List Bool) (hfields : ∀ j s, base (.field j s) = [])
    (hindex : base .index = []) (hwork : base .work = []) :
    outputTapes (SourceContextLoad.stageTapes F tuple base u) = base := by
  funext p
  cases p with
  | field j s => exact (hfields j s).symm
  | index => exact (SourceContextLoad.stageTapes_clean F tuple base u hindex hwork).1.trans hindex.symm
  | work => exact (SourceContextLoad.stageTapes_clean F tuple base u hindex hwork).2.trans hwork.symm
  | _ => exact SourceContextLoad.stageTapes_frame F tuple base u _ (by simp) (by simp) (by simp)

def clearLoadedToBaseInTime (F : Target.Formula) (tuple : Fin u → Fin F.clauses.length)
    (base : Tape u Extra → List Bool) (hfields : ∀ j s, base (.field j s) = [])
    (hindex : base .index = []) (hwork : base .work = []) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some main, ((), register), SourceContextLoad.stageTapes F tuple base u⟩
      (some ⟨none, ((), none), base⟩)
      (6 * u * ((formulaBits F).length + 1) + 1) := by
  simpa only [output_loaded_eq_base F tuple base hfields hindex hwork] using
    clearLoadedInTime F tuple base hfields register

def machine (u : Nat) (Extra : Type) [DecidableEq Extra] [Fintype Extra] : FinTM2 where
  K := Tape u Extra
  k₀ := .formula
  k₁ := .formula
  Γ _ := Bool
  Λ := Label u Extra
  main := main
  σ := Unit × Option Bool
  initialState := ((), none)
  m := program

end DFVSGames.Foundations.Hastad.SourceContextClear


namespace DFVSGames.Foundations.Hastad.SourceQueryOrder

open Turing Complexity Target SourceContexts SourceOccurrences
open SourceLocalSignature SourceAddressDescriptors SourceLoopOrder

def occurrenceBits (F : Formula) (u D : Nat) (p : SourceIndex F u D) : List Bool :=
  encodeWords (DFVSGames.Reduction.SourceEncoding.equationWords (sourceEquation F u D p))

def slotBits (F : Formula) (u D : Nat) (c : ClauseContext F u) (s : SlotContext u) :
    List Bool :=
  (testTapeEncoding u D).enumerate.flatMap (fun t => occurrenceBits F u D ((c, s), t))

def tupleBits (F : Formula) (u D : Nat) (c : ClauseContext F u) : List Bool :=
  (slotContextEncoding u).enumerate.flatMap (slotBits F u D c)

theorem equationBits_eq (F : Formula) (u D : Nat) (c : ClauseContext F u)
    (s : SlotContext u) (t : SourceQueryLoop.Query u D) :
    SourceTestAppend.equationBits u D (baseValue F u c (sampledVariables F c s))
      (ofContext F c s, t) = occurrenceBits F u D ((c, s), t) := by
  rw [occurrenceBits, sourceEquation_words]
  rfl

theorem prefixBits_eq_slotBits (F : Formula) (u D : Nat) (c : ClauseContext F u)
    (s : SlotContext u) (fallback : SourceQueryLoop.Query u D) :
    SourceQueryLoop.prefixBits u D (rankedQuery u D fallback)
      (baseValue F u c (sampledVariables F c s)) (ofContext F c s)
      (testTapeEncoding u D).size = slotBits F u D c s := by
  rw [query_prefixBits_eq_enumerate]
  unfold slotBits
  congr 1
  funext t
  exact equationBits_eq F u D c s t

variable {K Λ : Type} [DecidableEq K]

theorem sourceLoopInTime (F : Formula) (u D : Nat) (c : ClauseContext F u)
    (s : SlotContext u) (fallback initialQuery : SourceQueryLoop.Query u D)
    (layout : SourceTestAppend.Layout K)
    (labels : SourceQueryLoop.Label u D (testTapeEncoding u D).size → Λ)
    (exit : Option Λ)
    (p : Λ → TM2.Stmt (fun _ : K => Bool) Λ (SourceQueryLoop.State u D))
    (atLabels : ∀ l, p (labels l) = SourceQueryLoop.statement u D
      (testTapeEncoding u D).size (rankedQuery u D fallback) layout labels exit l)
    (base : K → List Bool)
    (sourceWords : ∀ side, base (layout.sources side) =
      encodeWord (baseValue F u c (sampledVariables F c s) side))
    (scratchEmpty : base layout.scratch = [])
    (temporaryEmpty : base layout.temporary = []) :
    ∃ lastQuery : SourceQueryLoop.Query u D,
      Nonempty (StateTransition.EvalsToInTime (TM2.step p)
        ⟨some (labels (.load ⟨0, Nat.zero_lt_succ _⟩)),
          ((ofContext F c s, initialQuery), none), base⟩
        (some ⟨exit, ((ofContext F c s, lastQuery), none),
          SourceTestAppend.resultTapes layout base (slotBits F u D c s)⟩)
        ((testTapeEncoding u D).size * (9 * nBits F u + 21) + 1)) := by
  have hbound : ∀ i, i < (testTapeEncoding u D).size →
      SourceTestAppend.steps u D (baseValue F u c (sampledVariables F c s))
        (ofContext F c s, rankedQuery u D fallback i) ≤ 9 * nBits F u + 20 := by
    intro i hi
    let run := SourceTestAppend.sourceAppendInTime F u D
      ((c, s), rankedQuery u D fallback i) layout
      (fun l => labels (.emit ⟨i, hi⟩ l))
      (some (labels (.load ⟨i + 1, by omega⟩))) p
      (fun l => atLabels (.emit ⟨i, hi⟩ l)) base sourceWords
      scratchEmpty temporaryEmpty none
    exact run.steps_le_m
  have h := SourceQueryLoop.loopInTime u D (testTapeEncoding u D).size
    (9 * nBits F u + 20) (rankedQuery u D fallback) layout labels exit p atLabels
    (ofContext F c s) initialQuery base (baseValue F u c (sampledVariables F c s))
    sourceWords scratchEmpty temporaryEmpty hbound
  rw [prefixBits_eq_slotBits] at h
  exact h

end DFVSGames.Foundations.Hastad.SourceQueryOrder


namespace DFVSGames.Foundations.Hastad.SourceTupleTape

open Turing Complexity SourceContexts SourceRuntimeModel

structure Snapshot where
  rank : List Bool := []
  rightBase : List Bool := []
  leftBase : List Bool := []
  zero : List Bool := []
  emitted : List Bool := []
  deriving DecidableEq, Inhabited

variable {u : Nat} {Extra : Type}

def tapes (base : Arena u Extra → List Bool) (s : Snapshot) : Arena u Extra → List Bool
  | .extra (.inr .rank) => s.rank
  | .extra (.inr .rightBase) => s.rightBase
  | .extra (.inr .leftBase) => s.leftBase
  | .extra (.inr .zero) => s.zero
  | .extra (.inr .accumulator) => s.emitted.reverse ++ base (workTape .accumulator)
  | k => base k

@[simp] theorem tapes_rank (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s (workTape .rank) = s.rank := rfl
@[simp] theorem tapes_rightBase (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s (workTape .rightBase) = s.rightBase := rfl
@[simp] theorem tapes_leftBase (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s (workTape .leftBase) = s.leftBase := rfl
@[simp] theorem tapes_zero (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s (workTape .zero) = s.zero := rfl
@[simp] theorem tapes_accumulator (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s (workTape .accumulator) = s.emitted.reverse ++ base (workTape .accumulator) := rfl
@[simp] theorem tapes_coefficient (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s (workTape .coefficient) = base (workTape .coefficient) := rfl
@[simp] theorem tapes_dummy (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s (workTape .dummy) = base (workTape .dummy) := rfl
@[simp] theorem tapes_leftBlock (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s (workTape .leftBlock) = base (workTape .leftBlock) := rfl
@[simp] theorem tapes_queryTemporary (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s (workTape .queryTemporary) = base (workTape .queryTemporary) := rfl
@[simp] theorem tapes_queryScratch (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s (workTape .queryScratch) = base (workTape .queryScratch) := rfl
@[simp] theorem tapes_field (base : Arena u Extra → List Bool) (s : Snapshot) (j : Fin u) (k : Fin 6) :
    tapes base s (.field j k) = base (.field j k) := rfl
@[simp] theorem tapes_current (base : Arena u Extra → List Bool) (s : Snapshot) (j : Fin u) :
    tapes base s (.current j) = base (.current j) := rfl
@[simp] theorem tapes_formula (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s .formula = base .formula := rfl
@[simp] theorem tapes_index (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s .index = base .index := rfl
@[simp] theorem tapes_work (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s .work = base .work := rfl
@[simp] theorem tapes_variableHeader (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s variableHeader = base variableHeader := rfl
@[simp] theorem tapes_clauseHeader (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes base s clauseHeader = base clauseHeader := rfl

variable [DecidableEq Extra]

theorem update_rank (base : Arena u Extra → List Bool) (s : Snapshot) (value : List Bool) :
    Function.update (tapes base s) (workTape .rank) value = tapes base {s with rank := value} := by
  funext k
  cases k <;> try rfl
  rename_i extra
  cases extra with
  | inl header => simp [tapes, workTape]
  | inr role => cases role <;> simp [tapes, workTape]
theorem update_rightBase (base : Arena u Extra → List Bool) (s : Snapshot) (value : List Bool) :
    Function.update (tapes base s) (workTape .rightBase) value = tapes base {s with rightBase := value} := by
  funext k
  cases k <;> try rfl
  rename_i extra
  cases extra with
  | inl header => simp [tapes, workTape]
  | inr role => cases role <;> simp [tapes, workTape]
theorem update_leftBase (base : Arena u Extra → List Bool) (s : Snapshot) (value : List Bool) :
    Function.update (tapes base s) (workTape .leftBase) value = tapes base {s with leftBase := value} := by
  funext k
  cases k <;> try rfl
  rename_i extra
  cases extra with
  | inl header => simp [tapes, workTape]
  | inr role => cases role <;> simp [tapes, workTape]
theorem update_zero (base : Arena u Extra → List Bool) (s : Snapshot) (value : List Bool) :
    Function.update (tapes base s) (workTape .zero) value = tapes base {s with zero := value} := by
  funext k
  cases k <;> try rfl
  rename_i extra
  cases extra with
  | inl header => simp [tapes, workTape]
  | inr role => cases role <;> simp [tapes, workTape]
theorem update_emitted (base : Arena u Extra → List Bool) (s : Snapshot) (value : List Bool) :
    Function.update (tapes base s) (workTape .accumulator)
      (value.reverse ++ base (workTape .accumulator)) = tapes base {s with emitted := value} := by
  funext k
  cases k <;> try rfl
  rename_i extra
  cases extra with
  | inl header => simp [tapes, workTape]
  | inr role => cases role <;> simp [tapes, workTape]
theorem query_result (base : Arena u Extra → List Bool) (s : Snapshot) (bits : List Bool) :
    SourceTestAppend.resultTapes queryLayout (tapes base s) bits =
      tapes base {s with emitted := s.emitted ++ bits} := by
  change Function.update (tapes base s) (workTape .accumulator)
    (bits.reverse ++ tapes base s (workTape .accumulator)) = _
  rw [tapes_accumulator]
  have h := update_emitted base s (s.emitted ++ bits)
  simpa only [List.reverse_append, List.append_assoc] using h

theorem horner_clause_result (base : Arena u Extra → List Bool) (s : Snapshot) (value : Nat) :
    MachineHorner.resultTapes clauseRankSlots (tapes base s) value =
      tapes base {s with rank := encodeWord value ++ s.rank} := by
  change Function.update (tapes base s) (workTape .rank)
    (encodeWord value ++ tapes base s (workTape .rank)) = _
  rw [tapes_rank, update_rank]

theorem horner_variable_result (base : Arena u Extra → List Bool) (s : Snapshot)
    (selected : SlotContext u) (value : Nat) :
    MachineHorner.resultTapes (variableRankSlots selected) (tapes base s) value =
      tapes base {s with rank := encodeWord value ++ s.rank} := by
  change Function.update (tapes base s) (workTape .rank)
    (encodeWord value ++ tapes base s (workTape .rank)) = _
  rw [tapes_rank, update_rank]

theorem horner_right_result (base : Arena u Extra → List Bool) (s : Snapshot) (value : Nat) :
    MachineHorner.resultTapes (baseSlots true) (tapes base s) value =
      tapes base {s with rightBase := encodeWord value ++ s.rightBase} := by
  change Function.update (tapes base s) (workTape .rightBase)
    (encodeWord value ++ tapes base s (workTape .rightBase)) = _
  rw [tapes_rightBase, update_rightBase]

theorem horner_left_result (base : Arena u Extra → List Bool) (s : Snapshot) (value : Nat) :
    MachineHorner.resultTapes (baseSlots false) (tapes base s) value =
      tapes base {s with leftBase := encodeWord value ++ s.leftBase} := by
  change Function.update (tapes base s) (workTape .leftBase)
    (encodeWord value ++ tapes base s (workTape .leftBase)) = _
  rw [tapes_leftBase, update_leftBase]

structure WorkClean (base : Arena u Extra → List Bool) : Prop where
  accA : base (workTape .accA) = []
  accB : base (workTape .accB) = []
  counter : base (workTape .counter) = []
  scratch : base (workTape .arithScratch) = []

omit [DecidableEq Extra] in
theorem clean_clause (base : Arena u Extra → List Bool) (s : Snapshot) (clean : WorkClean base) :
    MachineHorner.Clean clauseRankSlots (tapes base s) where
  accA := clean.accA
  accB := clean.accB
  counter := clean.counter
  scratch := clean.scratch

omit [DecidableEq Extra] in
theorem clean_variable (base : Arena u Extra → List Bool) (s : Snapshot) (selected : SlotContext u)
    (clean : WorkClean base) : MachineHorner.Clean (variableRankSlots selected) (tapes base s) where
  accA := clean.accA
  accB := clean.accB
  counter := clean.counter
  scratch := clean.scratch

omit [DecidableEq Extra] in
theorem clean_base (base : Arena u Extra → List Bool) (s : Snapshot) (right : Bool)
    (clean : WorkClean base) : MachineHorner.Clean (baseSlots right) (tapes base s) where
  accA := clean.accA
  accB := clean.accB
  counter := clean.counter
  scratch := clean.scratch

omit [DecidableEq Extra] in
theorem tapes_coordinateOutput {arity : Nat} (j : Fin u) (clause : Target.Clause arity)
    (base : Arena u Extra → List Bool) (s : Snapshot) :
    tapes (SourceContextLoad.coordinateOutput j clause base) s =
      SourceContextLoad.coordinateOutput j clause (tapes base s) := by
  funext k
  cases k <;> simp [tapes, SourceContextLoad.coordinateOutput, workTape]
  rename_i extra
  cases extra with
  | inl header => rfl
  | inr role => cases role <;> rfl
omit [DecidableEq Extra] in
theorem tapes_stageTapes (F : Target.Formula) (c : ClauseContext F u)
    (base : Arena u Extra → List Bool) (s : Snapshot) (r : Nat) :
    tapes (SourceContextLoad.stageTapes F c base r) s =
      SourceContextLoad.stageTapes F c (tapes base s) r := by
  induction r with
  | zero => rfl
  | succ r ih =>
    simp only [SourceContextLoad.stageTapes]
    split
    · rw [tapes_coordinateOutput, ih]
    · exact ih

theorem final_tapes (base : Arena u Extra → List Bool) (emitted : List Bool)
    (hrank : base (workTape .rank) = []) (hright : base (workTape .rightBase) = [])
    (hleft : base (workTape .leftBase) = []) (hzero : base (workTape .zero) = []) :
    tapes base {emitted := emitted} =
      Function.update base (workTape .accumulator) (emitted.reverse ++ base (workTape .accumulator)) := by
  funext k
  cases k <;> try rfl
  rename_i extra
  cases extra with
  | inl header => simp [tapes, workTape]
  | inr role =>
    cases role <;> simp_all [tapes, workTape]
theorem empty_tapes (base : Arena u Extra → List Bool)
    (hrank : base (workTape .rank) = []) (hright : base (workTape .rightBase) = [])
    (hleft : base (workTape .leftBase) = []) (hzero : base (workTape .zero) = []) :
    tapes base {} = base := by
  simpa using final_tapes base [] hrank hright hleft hzero

omit [DecidableEq Extra] in
theorem clear_loaded (F : Target.Formula) (c : ClauseContext F u)
    (base : Arena u Extra → List Bool) (s : Snapshot)
    (hfields : ∀ j k, base (.field j k) = []) (hindex : base .index = []) (hwork : base .work = []) :
    SourceContextClear.outputTapes (tapes (SourceContextLoad.stageTapes F c base u) s) = tapes base s := by
  rw [tapes_stageTapes]
  exact SourceContextClear.output_loaded_eq_base F c (tapes base s) hfields hindex hwork

def clearLoadedSnapshotInTime (F : Target.Formula) (c : ClauseContext F u)
    (base : Arena u Extra → List Bool) (s : Snapshot)
    (hfields : ∀ j k, base (.field j k) = []) (hindex : base .index = []) (hwork : base .work = [])
    (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step SourceContextClear.program)
      ⟨some SourceContextClear.main, ((), register), tapes (SourceContextLoad.stageTapes F c base u) s⟩
      (some ⟨none, ((), none), tapes base s⟩)
      (6 * u * ((formulaBits F).length + 1) + 1) := by
  rw [tapes_stageTapes]
  exact SourceContextClear.clearLoadedToBaseInTime F c (tapes base s) hfields hindex hwork register

end DFVSGames.Foundations.Hastad.SourceTupleTape

end OAI