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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
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