Datasets:

roseking
/

openAI-math-dataset

import OAI.Computability.PerfectCompleteness.Machines.BinaryLiteralMachine

namespace OAI


namespace PerfectCompleteness.BinaryRenameLoop


open Turing
open UniqueGamesTheorem.Foundations.Complexity
open MachineComposition

abbrev Token := BinaryNameSearch.Token

def nextSlot (slot : Fin 3) : Fin 3 := if slot = 0 then 1 else if slot = 1 then 2 else 0

def clauseIncrement (slot : Fin 3) : Nat := if slot = 2 then 1 else 0

def finalSlot : Nat → Fin 3 → Fin 3
  | 0, slot => slot
  | n + 1, slot => finalSlot n (nextSlot slot)

def completedClauses : Nat → Fin 3 → Nat
  | 0, _ => 0
  | n + 1, slot => completedClauses n (nextSlot slot) + clauseIncrement slot

theorem finalSlot_three (n : Nat) : finalSlot (n + 3) 0 = finalSlot n 0 := by
  simp [finalSlot, nextSlot]

theorem completedClauses_three (n : Nat) :
    completedClauses (n + 3) 0 = completedClauses n 0 + 1 := by
  simp [completedClauses, nextSlot, clauseIncrement]

@[simp] theorem finalSlot_triples (n : Nat) : finalSlot (3 * n) 0 = 0 := by
  induction n with
  | zero => rfl
  | succ n ih => rw [Nat.mul_succ, finalSlot_three, ih]

@[simp] theorem completedClauses_triples (n : Nat) : completedClauses (3 * n) 0 = n := by
  induction n with
  | zero => rfl
  | succ n ih => rw [Nat.mul_succ, completedClauses_three, ih]

def outputBody (tokens input : List Token) : List Bool :=
  input.flatMap (fun token => BinaryLiteralMachine.outputWord tokens token.2 token.1)

@[simp] theorem outputBody_nil (tokens : List Token) : outputBody tokens [] = [] := rfl

@[simp] theorem outputBody_cons (tokens : List Token) (token : Token) (input : List Token) :
    outputBody tokens (token :: input) =
      BinaryLiteralMachine.outputWord tokens token.2 token.1 ++ outputBody tokens input := rfl

def steps (tokens : List Token) : List Token → Nat
  | [] => 1
  | token :: input => BinaryLiteralMachine.steps tokens token.2 + 2 + steps tokens input

theorem payload_length_le_stream (tokens : List Token) (bits : List Bool)
    (present : bits ∈ BinaryNameSearch.payloads tokens) :
    bits.length ≤ (BinaryNameSearch.stream tokens).length := by
  induction tokens with
  | nil => simp at present
  | cons token tokens ih =>
      rcases List.mem_cons.mp present with equal | remaining
      · rw [equal]
        simp only [BinaryNameSearch.stream_cons, List.length_append,
          BinaryNameSearch.tokenBits, List.length_cons, BinaryNameMachine.frame_length]
        omega
      · have tail := ih remaining
        simp only [BinaryNameSearch.stream_cons, List.length_append]
        omega

def perLiteralBound (streamLength : Nat) : Nat :=
  12 * streamLength ^ 2 + 26 * streamLength + 11

theorem literal_steps_le (tokens : List Token) (bits : List Bool)
    (present : bits ∈ BinaryNameSearch.payloads tokens) :
    BinaryLiteralMachine.steps tokens bits + 2 ≤
      perLiteralBound (BinaryNameSearch.stream tokens).length := by
  have body := BinaryLiteralMachine.steps_le tokens bits present
  have size := payload_length_le_stream tokens bits present
  have sumBound : (BinaryNameSearch.stream tokens).length + bits.length ≤
      2 * (BinaryNameSearch.stream tokens).length := by omega
  have sq := Nat.mul_self_le_mul_self sumBound
  unfold perLiteralBound
  nlinarith

theorem steps_le (tokens input : List Token) (included : ∀ token ∈ input, token ∈ tokens) :
    steps tokens input ≤
      perLiteralBound (BinaryNameSearch.stream tokens).length * input.length + 1 := by
  induction input with
  | nil => simp [steps]
  | cons token input ih =>
      have present : token.2 ∈ BinaryNameSearch.payloads tokens :=
        List.mem_map.mpr ⟨token, included token (by simp), rfl⟩
      have head := literal_steps_le tokens token.2 present
      have tail := ih (fun t ht => included t (by simp [ht]))
      simp only [steps, List.length_cons]
      nlinarith

theorem steps_le_cubic (tokens input : List Token)
    (included : ∀ token ∈ input, token ∈ tokens)
    (countBound : input.length ≤ (BinaryNameSearch.stream tokens).length) :
    steps tokens input ≤
      12 * (BinaryNameSearch.stream tokens).length ^ 3 +
      26 * (BinaryNameSearch.stream tokens).length ^ 2 +
      11 * (BinaryNameSearch.stream tokens).length + 1 := by
  calc
    steps tokens input ≤
        perLiteralBound (BinaryNameSearch.stream tokens).length * input.length + 1 :=
      steps_le tokens input included
    _ ≤ perLiteralBound (BinaryNameSearch.stream tokens).length *
        (BinaryNameSearch.stream tokens).length + 1 :=
      Nat.add_le_add_right (Nat.mul_le_mul_left _ countBound) _
    _ = _ := by unfold perLiteralBound; ring

section Program

variable {K Λ A : Type} [DecidableEq K]

abbrev Alphabet (_ : K) := Bool
abbrev State (A : Type) := BinaryLiteralMachine.State (A × Fin 3)
abbrev clean (ambient : A) (slot : Fin 3) : State A :=
  BinaryLiteralMachine.clean (ambient, slot)

inductive Label
  | entry
  | literal (label : BinaryLiteralMachine.Label)
  | tally
  deriving DecidableEq, Fintype

def literalTapes (tape : Fin 11 → K) : Fin 9 → K := fun i => tape i.castSucc.castSucc

omit [DecidableEq K] in
theorem literalTapes_injective (tape : Fin 11 → K) (distinct : Function.Injective tape) :
    Function.Injective (literalTapes tape) := by
  intro i j h
  apply Fin.ext
  exact congrArg (fun i : Fin 11 => i.val) (distinct h)

def finish (exit : Option Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A) :=
  .load (fun state => clean state.1.1.1.1 state.1.1.1.2)
    (match exit with
      | none => .halt
      | some label => .goto fun _ => label)

def next (again : Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A) :=
  .load (fun state => clean state.1.1.1.1 (nextSlot state.1.1.1.2))
    (.goto fun _ => again)

def instruction (tape : Fin 11 → K) (labels : Label → Λ)
    (done malformed : Option Λ) : Label → TM2.Stmt (Alphabet (K := K)) Λ (State A)
  | .entry =>
      .pop (tape 0) (fun state head => (state.1, head))
        (.branch (fun state => state.2.isSome)
          (.push (tape 0) (fun state => state.2.getD false)
            (finish (some (labels (.literal .readSign)))))
          (finish done))
  | .literal l => BinaryLiteralMachine.instruction (literalTapes tape)
      (fun l => labels (.literal l)) (some (labels .tally)) malformed l
  | .tally =>
      .push (tape 9) (fun _ => true)
        (.branch (fun state => state.1.1.1.2 == (2 : Fin 3))
          (.push (tape 10) (fun _ => true) (next (labels .entry)))
          (next (labels .entry)))

def loopTapes (tape : Fin 11 → K) (base : K → List Bool)
    (cursor output variableCounter clauses : List Bool) : K → List Bool :=
  Function.update (Function.update (Function.update (Function.update base
    (tape 0) cursor) (tape 8) output) (tape 9) variableCounter) (tape 10) clauses

theorem loopTapes_cursor (tape : Fin 11 → K) (distinct : Function.Injective tape)
    (base : K → List Bool) (cursor output variableCounter clauses : List Bool) :
    loopTapes tape base cursor output variableCounter clauses (tape 0) = cursor := by
  have hd (i j : Fin 11) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
  simp [loopTapes, hd]

theorem loopTapes_output (tape : Fin 11 → K) (distinct : Function.Injective tape)
    (base : K → List Bool) (cursor output variableCounter clauses : List Bool) :
    loopTapes tape base cursor output variableCounter clauses (tape 8) = output := by
  have hd (i j : Fin 11) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
  simp [loopTapes, hd]

theorem loopTapes_variables (tape : Fin 11 → K) (distinct : Function.Injective tape)
    (base : K → List Bool) (cursor output variableCounter clauses : List Bool) :
    loopTapes tape base cursor output variableCounter clauses (tape 9) = variableCounter := by
  have hd (i j : Fin 11) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
  simp [loopTapes, hd]

theorem loopTapes_clauses (tape : Fin 11 → K)
    (base : K → List Bool) (cursor output variableCounter clauses : List Bool) :
    loopTapes tape base cursor output variableCounter clauses (tape 10) = clauses := by
  simp [loopTapes]

private theorem update_cursor (tape : Fin 11 → K) (distinct : Function.Injective tape)
    (base : K → List Bool) (cursor output variableCounter clauses replacement : List Bool) :
    Function.update (loopTapes tape base cursor output variableCounter clauses) (tape 0) replacement =
      loopTapes tape base replacement output variableCounter clauses := by
  have hd (i j : Fin 11) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
  funext k
  by_cases h : k = tape 0
  · subst k; simp [loopTapes, hd]
  · simp [loopTapes, h, Function.update_apply]

private theorem update_output (tape : Fin 11 → K) (distinct : Function.Injective tape)
    (base : K → List Bool) (cursor output variableCounter clauses replacement : List Bool) :
    Function.update (loopTapes tape base cursor output variableCounter clauses) (tape 8) replacement =
      loopTapes tape base cursor replacement variableCounter clauses := by
  have hd (i j : Fin 11) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
  funext k
  by_cases h : k = tape 8
  · subst k; simp [loopTapes, hd]
  · simp [loopTapes, h, Function.update_apply]

private theorem update_variables (tape : Fin 11 → K) (distinct : Function.Injective tape)
    (base : K → List Bool) (cursor output variableCounter clauses replacement : List Bool) :
    Function.update (loopTapes tape base cursor output variableCounter clauses) (tape 9) replacement =
      loopTapes tape base cursor output replacement clauses := by
  have hd (i j : Fin 11) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
  funext k
  by_cases h : k = tape 9
  · subst k; simp [loopTapes, hd]
  · simp [loopTapes, h, Function.update_apply]

private theorem update_clauses (tape : Fin 11 → K)
    (base : K → List Bool) (cursor output variableCounter clauses replacement : List Bool) :
    Function.update (loopTapes tape base cursor output variableCounter clauses) (tape 10) replacement =
      loopTapes tape base cursor output variableCounter replacement := by
  simp [loopTapes]

theorem loopTapes_other (tape : Fin 11 → K) (base : K → List Bool)
    (cursor output variableCounter clauses : List Bool) (k : K)
    (h0 : k ≠ tape 0) (h8 : k ≠ tape 8) (h9 : k ≠ tape 9) (h10 : k ≠ tape 10) :
    loopTapes tape base cursor output variableCounter clauses k = base k := by
  simp [loopTapes, h0, h8, h9, h10]

@[simp] theorem stepAux_finish (exit : Option Λ) (state : State A) (base : K → List Bool) :
    TM2.stepAux (finish exit) state base =
      ⟨exit, clean state.1.1.1.1 state.1.1.1.2, base⟩ := by
  cases exit <;> rfl

private theorem joinTrace {X : Type*} {f : X → X} {a b c : X} {n m : Nat}
    (first : f^[n] a = b) (second : f^[m] b = c) : f^[n + m] a = c := by
  rw [Nat.add_comm, Function.iterate_add_apply, first, second]

variable (tape : Fin 11 → K) (distinct : Function.Injective tape)
variable (labels : Label → Λ) (done malformed : Option Λ)
variable (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A))
variable (atLabels : ∀ l, program (labels l) = instruction tape labels done malformed l)
variable (base : K → List Bool) (ambient : A)

include distinct atLabels

theorem peekStep (slot : Fin 3) (head : Bool) (tail output variableCounter clauses : List Bool) :
    TM2.step program
      ⟨some (labels .entry), clean ambient slot,
        loopTapes tape base (head :: tail) output variableCounter clauses⟩ =
      some ⟨some (labels (.literal .readSign)), clean ambient slot,
        loopTapes tape base (head :: tail) output variableCounter clauses⟩ := by
  have hd (i j : Fin 11) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
  change some (TM2.stepAux (program (labels .entry)) _ _) = _
  rw [atLabels .entry]
  simp [instruction, TM2.stepAux, clean, BinaryLiteralMachine.clean, BinaryNameCompare.clean,
    loopTapes_cursor tape distinct, update_cursor tape distinct]

theorem emptyStep (slot : Fin 3) (output variableCounter clauses : List Bool) :
    TM2.step program
      ⟨some (labels .entry), clean ambient slot,
        loopTapes tape base [] output variableCounter clauses⟩ =
      some ⟨done, clean ambient slot, loopTapes tape base [] output variableCounter clauses⟩ := by
  have hd (i j : Fin 11) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
  change some (TM2.stepAux (program (labels .entry)) _ _) = _
  rw [atLabels .entry]
  simp [instruction, TM2.stepAux, clean, BinaryLiteralMachine.clean, BinaryNameCompare.clean,
    loopTapes_cursor tape distinct, update_cursor tape distinct]

theorem tallyStep (slot : Fin 3) (cursor output variableCounter clauses : List Bool) :
    TM2.step program
      ⟨some (labels .tally), clean ambient slot,
        loopTapes tape base cursor output variableCounter clauses⟩ =
      some ⟨some (labels .entry), clean ambient (nextSlot slot),
        loopTapes tape base cursor output (true :: variableCounter)
          (List.replicate (clauseIncrement slot) true ++ clauses)⟩ := by
  have hd (i j : Fin 11) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
  change some (TM2.stepAux (program (labels .tally)) _ _) = _
  rw [atLabels .tally]
  fin_cases slot <;>
    simp [instruction, TM2.stepAux, clean, BinaryLiteralMachine.clean, BinaryNameCompare.clean,
      next, nextSlot, clauseIncrement, loopTapes_variables tape distinct, loopTapes_clauses,
      update_variables tape distinct, update_clauses]

theorem bodyTrace (tokens : List Token) (token : Token) (slot : Fin 3)
    (suffix output variableCounter clauses : List Bool)
    (canonical : BinaryNameMachine.canonical token.2 = true)
    (tokensCanonical : ∀ t ∈ tokens, BinaryNameMachine.canonical t.2 = true)
    (present : token.2 ∈ BinaryNameSearch.payloads tokens)
    (permanent : base (tape 1) = BinaryNameSearch.stream tokens)
    (empty : ∀ i : Fin 11, 2 ≤ i.val → i.val ≤ 7 → base (tape i) = []) :
    (advance (TM2.step program))^[BinaryLiteralMachine.steps tokens token.2 + 2]
      (some ⟨some (labels .entry), clean ambient slot,
        loopTapes tape base (BinaryNameSearch.tokenBits token ++ suffix) output variableCounter clauses⟩) =
      some ⟨some (labels .entry), clean ambient (nextSlot slot),
        loopTapes tape base suffix
          ((BinaryLiteralMachine.outputWord tokens token.2 token.1).reverse ++ output)
          (true :: variableCounter) (List.replicate (clauseIncrement slot) true ++ clauses)⟩ := by
  have hd (i j : Fin 11) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
  let initial := loopTapes tape base (BinaryNameSearch.tokenBits token ++ suffix)
    output variableCounter clauses
  have first : (advance (TM2.step program))^[1]
      (some ⟨some (labels .entry), clean ambient slot, initial⟩) =
      some ⟨some (labels (.literal .readSign)), clean ambient slot, initial⟩ := by
    simpa only [Function.iterate_one, advance_some, initial,
      BinaryNameSearch.tokenBits, List.cons_append] using
      peekStep tape distinct labels done malformed program atLabels base ambient slot token.1
        (BinaryNameMachine.frame token.2 ++ suffix) output variableCounter clauses
  have workEmpty (i : Fin 11) (lo : 2 ≤ i.val) (hi : i.val ≤ 7) :
      initial (tape i) = [] := by
    have h0 : i ≠ 0 := by intro h; subst i; omega
    have h8 : i ≠ 8 := by intro h; subst i; omega
    have h9 : i ≠ 9 := by intro h; subst i; omega
    have h10 : i ≠ 10 := by intro h; subst i; omega
    simpa [initial, loopTapes, hd i 0 h0, hd i 8 h8, hd i 9 h9, hd i 10 h10]
      using empty i lo hi
  have literal := BinaryLiteralMachine.literalTrace (literalTapes tape)
    (literalTapes_injective tape distinct) (fun l => labels (.literal l))
    (some (labels .tally)) malformed program (fun l => atLabels (.literal l))
    initial tokens token.1 token.2 suffix canonical tokensCanonical present
    (by simp [initial, literalTapes, loopTapes, BinaryNameSearch.tokenBits, hd])
    (by simpa [initial, literalTapes, loopTapes, hd] using permanent)
    (fun i lo hi => workEmpty i.castSucc.castSucc lo hi) (ambient, slot)
  have finalTapes :
      Function.update (Function.update initial (literalTapes tape 0) suffix)
        (literalTapes tape 8)
        ((BinaryLiteralMachine.outputWord tokens token.2 token.1).reverse ++
          initial (literalTapes tape 8)) =
      loopTapes tape base suffix
        ((BinaryLiteralMachine.outputWord tokens token.2 token.1).reverse ++ output)
        variableCounter clauses := by
    have out : initial (literalTapes tape 8) = output := by
      simp [initial, literalTapes, loopTapes, hd]
    rw [out]
    change Function.update (Function.update
      (loopTapes tape base (BinaryNameSearch.tokenBits token ++ suffix) output variableCounter clauses)
      (tape 0) suffix) (tape 8) _ = _
    rw [update_cursor tape distinct, update_output tape distinct]
  rw [finalTapes] at literal
  have tally : (advance (TM2.step program))^[1]
      (some ⟨some (labels .tally), clean ambient slot,
        loopTapes tape base suffix
          ((BinaryLiteralMachine.outputWord tokens token.2 token.1).reverse ++ output)
          variableCounter clauses⟩) =
      some ⟨some (labels .entry), clean ambient (nextSlot slot),
        loopTapes tape base suffix
          ((BinaryLiteralMachine.outputWord tokens token.2 token.1).reverse ++ output)
          (true :: variableCounter) (List.replicate (clauseIncrement slot) true ++ clauses)⟩ := by
    simpa only [Function.iterate_one, advance_some] using
      tallyStep tape distinct labels done malformed program atLabels base ambient slot suffix
        ((BinaryLiteralMachine.outputWord tokens token.2 token.1).reverse ++ output)
        variableCounter clauses
  have full := joinTrace (joinTrace first literal) tally
  simpa only [show 1 + BinaryLiteralMachine.steps tokens token.2 + 1 =
    BinaryLiteralMachine.steps tokens token.2 + 2 by omega] using full

theorem loopTrace (tokens input : List Token) (slot : Fin 3)
    (output variableCounter clauses : List Bool)
    (tokensCanonical : ∀ t ∈ tokens, BinaryNameMachine.canonical t.2 = true)
    (included : ∀ t ∈ input, t ∈ tokens)
    (permanent : base (tape 1) = BinaryNameSearch.stream tokens)
    (empty : ∀ i : Fin 11, 2 ≤ i.val → i.val ≤ 7 → base (tape i) = []) :
    (advance (TM2.step program))^[steps tokens input]
      (some ⟨some (labels .entry), clean ambient slot,
        loopTapes tape base (BinaryNameSearch.stream input) output variableCounter clauses⟩) =
      some ⟨done, clean ambient (finalSlot input.length slot),
        loopTapes tape base [] ((outputBody tokens input).reverse ++ output)
          (List.replicate input.length true ++ variableCounter)
          (List.replicate (completedClauses input.length slot) true ++ clauses)⟩ := by
  induction input generalizing slot output variableCounter clauses with
  | nil =>
      simpa only [steps, Function.iterate_one, advance_some, BinaryNameSearch.stream_nil,
        List.length_nil, outputBody_nil, List.reverse_nil, List.nil_append,
        finalSlot, completedClauses, List.replicate_zero] using
        emptyStep tape distinct labels done malformed program atLabels base ambient
          slot output variableCounter clauses
  | cons token input ih =>
      have member := included token (by simp)
      have present : token.2 ∈ BinaryNameSearch.payloads tokens :=
        List.mem_map.mpr ⟨token, member, rfl⟩
      have first := bodyTrace tape distinct labels done malformed program atLabels base ambient
        tokens token slot (BinaryNameSearch.stream input) output variableCounter clauses
        (tokensCanonical token member) tokensCanonical present permanent empty
      have rest := ih (nextSlot slot)
        ((BinaryLiteralMachine.outputWord tokens token.2 token.1).reverse ++ output)
        (true :: variableCounter) (List.replicate (clauseIncrement slot) true ++ clauses)
        (fun t ht => included t (by simp [ht]))
      have full := joinTrace first rest
      have variableEq : List.replicate input.length true ++ (true :: variableCounter) =
          List.replicate (input.length + 1) true ++ variableCounter := by
        rw [List.replicate_succ']
        simp only [List.append_assoc, List.singleton_append]
      have clauseEq : List.replicate (completedClauses input.length (nextSlot slot)) true ++
          (List.replicate (clauseIncrement slot) true ++ clauses) =
          List.replicate (completedClauses (input.length + 1) slot) true ++ clauses := by
        rw [← List.append_assoc, ← List.replicate_add]
        rfl
      simpa only [steps, BinaryNameSearch.stream_cons, List.length_cons, finalSlot,
        outputBody_cons, List.reverse_append, List.append_assoc, variableEq, clauseEq] using full

theorem triplesTrace (tokens input : List Token) (count : Nat)
    (length_eq : input.length = 3 * count) (output variableCounter clauses : List Bool)
    (tokensCanonical : ∀ t ∈ tokens, BinaryNameMachine.canonical t.2 = true)
    (included : ∀ t ∈ input, t ∈ tokens)
    (permanent : base (tape 1) = BinaryNameSearch.stream tokens)
    (empty : ∀ i : Fin 11, 2 ≤ i.val → i.val ≤ 7 → base (tape i) = []) :
    (advance (TM2.step program))^[steps tokens input]
      (some ⟨some (labels .entry), clean ambient 0,
        loopTapes tape base (BinaryNameSearch.stream input) output variableCounter clauses⟩) =
      some ⟨done, clean ambient 0,
        loopTapes tape base [] ((outputBody tokens input).reverse ++ output)
          (List.replicate (3 * count) true ++ variableCounter)
          (List.replicate count true ++ clauses)⟩ := by
  have h := loopTrace tape distinct labels done malformed program atLabels base ambient
    tokens input 0 output variableCounter clauses tokensCanonical included permanent empty
  simpa only [length_eq, finalSlot_triples, completedClauses_triples] using h

def triplesInTime (tokens input : List Token) (count : Nat)
    (length_eq : input.length = 3 * count) (output variableCounter clauses : List Bool)
    (tokensCanonical : ∀ t ∈ tokens, BinaryNameMachine.canonical t.2 = true)
    (included : ∀ t ∈ input, t ∈ tokens)
    (permanent : base (tape 1) = BinaryNameSearch.stream tokens)
    (empty : ∀ i : Fin 11, 2 ≤ i.val → i.val ≤ 7 → base (tape i) = []) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels .entry), clean ambient 0,
        loopTapes tape base (BinaryNameSearch.stream input) output variableCounter clauses⟩
      (some ⟨done, clean ambient 0,
        loopTapes tape base [] ((outputBody tokens input).reverse ++ output)
          (List.replicate (3 * count) true ++ variableCounter)
          (List.replicate count true ++ clauses)⟩)
      (perLiteralBound (BinaryNameSearch.stream tokens).length * input.length + 1) where
  steps := steps tokens input
  evals_in_steps := triplesTrace tape distinct labels done malformed program atLabels base ambient
    tokens input count length_eq output variableCounter clauses tokensCanonical included permanent empty
  steps_le_m := steps_le tokens input included

end Program

end PerfectCompleteness.BinaryRenameLoop

end OAI