openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Machines /BinaryNameSearch.lean
| import Mathlib.Tactic.Linarith | |
| import OAI.Computability.PerfectCompleteness.Machines.BinaryNameCompare | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.BinaryNameSearch | |
| open Turing | |
| open UniqueGamesTheorem.Foundations.Complexity | |
| open MachineComposition | |
| open UniqueGamesTheorem.Reduction.MachineTransfer | |
| abbrev Token := Bool × List Bool | |
| def tokenBits (token : Token) : List Bool := | |
| token.1 :: BinaryNameMachine.frame token.2 | |
| def stream : List Token → List Bool | |
| | [] => [] | |
| | token :: tokens => tokenBits token ++ stream tokens | |
| def payloads (tokens : List Token) : List (List Bool) := tokens.map Prod.snd | |
| def afterMatch : List Token → List Bool → List Token | |
| | [], _ => [] | |
| | token :: tokens, key => if token.2 = key then tokens else afterMatch tokens key | |
| def steps : List Token → List Bool → Nat | |
| | [], _ => 0 | |
| | token :: tokens, key => | |
| 1 + BinaryNameCompare.steps token.2.length key.length + | |
| if token.2 = key then 0 else 1 + steps tokens key | |
| def payloadSize (tokens : List Token) : Nat := (payloads tokens |>.map List.length).sum | |
| @[simp] theorem stream_nil : stream [] = [] := rfl | |
| @[simp] theorem stream_cons (token : Token) (tokens : List Token) : | |
| stream (token :: tokens) = tokenBits token ++ stream tokens := rfl | |
| @[simp] theorem payloads_nil : payloads [] = [] := rfl | |
| @[simp] theorem payloads_cons (token : Token) (tokens : List Token) : | |
| payloads (token :: tokens) = token.2 :: payloads tokens := rfl | |
| @[simp] theorem payloadSize_nil : payloadSize [] = 0 := rfl | |
| @[simp] theorem payloadSize_cons (token : Token) (tokens : List Token) : | |
| payloadSize (token :: tokens) = token.2.length + payloadSize tokens := rfl | |
| @[simp] theorem stream_length (tokens : List Token) : | |
| (stream tokens).length = 2 * payloadSize tokens + 2 * tokens.length := by | |
| induction tokens with | |
| | nil => rfl | |
| | cons token tokens ih => | |
| simp only [stream_cons, List.length_append, tokenBits, List.length_cons, | |
| BinaryNameMachine.frame_length, ih, payloadSize_cons] | |
| omega | |
| theorem steps_le (tokens : List Token) (key : List Bool) : | |
| steps tokens key ≤ 2 * payloadSize tokens + (3 * key.length + 6) * tokens.length := by | |
| induction tokens with | |
| | nil => simp [steps] | |
| | cons token tokens ih => | |
| have h := BinaryNameCompare.steps_le token.2.length key.length | |
| simp only [steps, payloadSize_cons, List.length_cons] | |
| split <;> nlinarith | |
| theorem steps_le_stream (tokens : List Token) (key : List Bool) : | |
| steps tokens key ≤ (3 * key.length + 7) * (stream tokens).length := by | |
| have h := steps_le tokens key | |
| rw [stream_length] | |
| nlinarith | |
| section Program | |
| variable {K Λ A : Type} [DecidableEq K] | |
| abbrev Alphabet (_ : K) := Bool | |
| abbrev State (A : Type) := BinaryNameCompare.State A | |
| abbrev clean (ambient : A) : State A := BinaryNameCompare.clean ambient | |
| inductive Label | |
| | sign | |
| | comparison (label : BinaryNameCompare.Label) | |
| | increment | |
| deriving DecidableEq, Fintype | |
| def compareTapes (tape : Fin 6 → K) : Fin 5 → K := fun i => tape i.castSucc | |
| omit [DecidableEq K] in | |
| theorem compareTapes_injective (tape : Fin 6 → K) (distinct : Function.Injective tape) : | |
| Function.Injective (compareTapes tape) := by | |
| intro i j h | |
| apply Fin.ext | |
| exact congrArg (fun i : Fin 6 => i.val) (distinct h) | |
| def exitAt (exit : Option Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A) := | |
| match exit with | |
| | none => .halt | |
| | some label => .goto fun _ => label | |
| def finish (exit : Option Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A) := | |
| .load (fun state => clean state.1.1) (exitAt exit) | |
| def instruction (tape : Fin 6 → K) (labels : Label → Λ) | |
| (foundExit missingExit malformedExit : Option Λ) : | |
| Label → TM2.Stmt (Alphabet (K := K)) Λ (State A) | |
| | .sign => | |
| .pop (tape 0) (fun state head => (state.1, head)) | |
| (.branch (fun state => state.2.isSome) | |
| (finish (some (labels (.comparison .scan)))) (finish missingExit)) | |
| | .comparison l => BinaryNameCompare.instruction (compareTapes tape) | |
| (fun l => labels (.comparison l)) foundExit (some (labels .increment)) malformedExit l | |
| | .increment => .push (tape 5) (fun _ => true) (.goto fun _ => labels .sign) | |
| @[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, base⟩ := by | |
| cases exit <;> rfl | |
| private theorem update_source (source counter : K) (distinct : source ≠ counter) | |
| (base : K → List Bool) (input count replacement : List Bool) : | |
| Function.update (tapesAt source counter base input count) source replacement = | |
| tapesAt source counter base replacement count := by | |
| funext k | |
| by_cases hs : k = source | |
| · subst k; simp [tapesAt, distinct] | |
| · by_cases hc : k = counter | |
| · subst k; simp [tapesAt, Ne.symm distinct] | |
| · simp [tapesAt, hs, hc] | |
| private theorem update_counter (source counter : K) | |
| (base : K → List Bool) (input count replacement : List Bool) : | |
| Function.update (tapesAt source counter base input count) counter replacement = | |
| tapesAt source counter base input replacement := by | |
| simp [tapesAt] | |
| 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 6 → K) (distinct : Function.Injective tape) | |
| variable (labels : Label → Λ) (foundExit missingExit malformedExit : Option Λ) | |
| variable (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A)) | |
| variable (atLabels : ∀ l, program (labels l) = | |
| instruction tape labels foundExit missingExit malformedExit l) | |
| variable (base : K → List Bool) (ambient : A) | |
| include distinct atLabels | |
| theorem signStep (sign : Bool) (input counter : List Bool) : | |
| TM2.step program | |
| ⟨some (labels .sign), clean ambient, | |
| tapesAt (tape 0) (tape 5) base (sign :: input) counter⟩ = | |
| some ⟨some (labels (.comparison .scan)), clean ambient, | |
| tapesAt (tape 0) (tape 5) base input counter⟩ := by | |
| have hd : tape 0 ≠ tape 5 := fun h => (by decide : (0 : Fin 6) ≠ 5) (distinct h) | |
| change some (TM2.stepAux (program (labels .sign)) _ _) = _ | |
| rw [atLabels .sign] | |
| simp [instruction, TM2.stepAux, clean, BinaryNameCompare.clean, hd, update_source] | |
| theorem emptyStep (counter : List Bool) : | |
| TM2.step program | |
| ⟨some (labels .sign), clean ambient, | |
| tapesAt (tape 0) (tape 5) base [] counter⟩ = | |
| some ⟨missingExit, clean ambient, tapesAt (tape 0) (tape 5) base [] counter⟩ := by | |
| have hd : tape 0 ≠ tape 5 := fun h => (by decide : (0 : Fin 6) ≠ 5) (distinct h) | |
| change some (TM2.stepAux (program (labels .sign)) _ _) = _ | |
| rw [atLabels .sign] | |
| simp [instruction, TM2.stepAux, clean, BinaryNameCompare.clean, hd, update_source] | |
| omit distinct in | |
| theorem incrementStep (input counter : List Bool) : | |
| TM2.step program | |
| ⟨some (labels .increment), clean ambient, | |
| tapesAt (tape 0) (tape 5) base input counter⟩ = | |
| some ⟨some (labels .sign), clean ambient, | |
| tapesAt (tape 0) (tape 5) base input (true :: counter)⟩ := by | |
| change some (TM2.stepAux (program (labels .increment)) _ _) = _ | |
| rw [atLabels .increment] | |
| simp [instruction, TM2.stepAux, update_counter] | |
| theorem tokenTrace (token : Token) (key suffix counter : List Bool) | |
| (canonical : BinaryNameMachine.canonical token.2 = true) | |
| (keyWord : base (tape 1) = key.reverse) | |
| (candidateEmpty : base (tape 2) = []) (copyEmpty : base (tape 3) = []) | |
| (scratchEmpty : base (tape 4) = []) : | |
| (advance (TM2.step program))^[1 + BinaryNameCompare.steps token.2.length key.length] | |
| (some ⟨some (labels .sign), clean ambient, | |
| tapesAt (tape 0) (tape 5) base (tokenBits token ++ suffix) counter⟩) = | |
| some ⟨if token.2 = key then foundExit else some (labels .increment), clean ambient, | |
| tapesAt (tape 0) (tape 5) base suffix counter⟩ := by | |
| have hd (i j : Fin 6) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h) | |
| let initial := tapesAt (tape 0) (tape 5) base | |
| (BinaryNameMachine.frame token.2 ++ suffix) counter | |
| have first : (advance (TM2.step program))^[1] | |
| (some ⟨some (labels .sign), clean ambient, | |
| tapesAt (tape 0) (tape 5) base (tokenBits token ++ suffix) counter⟩) = | |
| some ⟨some (labels (.comparison .scan)), clean ambient, initial⟩ := by | |
| simpa only [Function.iterate_one, advance_some, tokenBits, List.cons_append] using | |
| signStep tape distinct labels foundExit missingExit malformedExit program atLabels | |
| base ambient token.1 (BinaryNameMachine.frame token.2 ++ suffix) counter | |
| have second := BinaryNameCompare.compareTrace (compareTapes tape) | |
| (compareTapes_injective tape distinct) (fun l => labels (.comparison l)) | |
| foundExit (some (labels .increment)) malformedExit program | |
| (fun l => atLabels (.comparison l)) initial token.2 suffix key.reverse canonical | |
| (by simp [initial, compareTapes, hd 0 5 (by decide)]) | |
| (by simpa [initial, compareTapes, tapesAt, | |
| hd 1 0 (by decide), hd 1 5 (by decide)] using keyWord) | |
| (by simpa [initial, compareTapes, tapesAt, hd 2 0 (by decide), hd 2 5 (by decide)] | |
| using candidateEmpty) | |
| (by simpa [initial, compareTapes, tapesAt, | |
| hd 3 0 (by decide), hd 3 5 (by decide)] using copyEmpty) | |
| (by simpa [initial, compareTapes, tapesAt, | |
| hd 4 0 (by decide), hd 4 5 (by decide)] using scratchEmpty) | |
| ambient | |
| have heq : token.2.reverse = key.reverse ↔ token.2 = key := by | |
| constructor | |
| · intro h | |
| have := congrArg List.reverse h | |
| simpa only [List.reverse_reverse] using this | |
| · intro h; rw [h] | |
| simp only [List.length_reverse, heq] at second | |
| rw [show Function.update initial (compareTapes tape 0) suffix = | |
| tapesAt (tape 0) (tape 5) base suffix counter by | |
| exact update_source _ _ (hd 0 5 (by decide)) base _ counter suffix] at second | |
| exact joinTrace first second | |
| theorem searchTrace (tokens : List Token) (key suffix counter : List Bool) | |
| (canonical : ∀ token ∈ tokens, BinaryNameMachine.canonical token.2 = true) | |
| (present : key ∈ payloads tokens) | |
| (keyWord : base (tape 1) = key.reverse) | |
| (candidateEmpty : base (tape 2) = []) (copyEmpty : base (tape 3) = []) | |
| (scratchEmpty : base (tape 4) = []) : | |
| (advance (TM2.step program))^[steps tokens key] | |
| (some ⟨some (labels .sign), clean ambient, | |
| tapesAt (tape 0) (tape 5) base (stream tokens ++ suffix) counter⟩) = | |
| some ⟨foundExit, clean ambient, | |
| tapesAt (tape 0) (tape 5) base (stream (afterMatch tokens key) ++ suffix) | |
| (List.replicate ((payloads tokens).idxOf key) true ++ counter)⟩ := by | |
| induction tokens generalizing counter with | |
| | nil => simp at present | |
| | cons token tokens ih => | |
| have headCanonical := canonical token (by simp) | |
| have tailCanonical : ∀ t ∈ tokens, BinaryNameMachine.canonical t.2 = true := | |
| fun t ht => canonical t (by simp [ht]) | |
| have first := tokenTrace tape distinct labels foundExit missingExit malformedExit | |
| program atLabels base ambient token key (stream tokens ++ suffix) counter | |
| headCanonical keyWord candidateEmpty copyEmpty scratchEmpty | |
| by_cases same : token.2 = key | |
| · simpa only [steps, same, ite_true, Nat.add_zero, stream_cons, | |
| List.append_assoc, afterMatch, payloads_cons, List.idxOf_cons_self, | |
| List.replicate_zero, List.nil_append] using first | |
| · have remaining : key ∈ payloads tokens := by | |
| rcases List.mem_cons.mp present with equal | remaining | |
| · exact False.elim (same equal.symm) | |
| · exact remaining | |
| have increment : (advance (TM2.step program))^[1] | |
| (some ⟨some (labels .increment), clean ambient, | |
| tapesAt (tape 0) (tape 5) base (stream tokens ++ suffix) counter⟩) = | |
| some ⟨some (labels .sign), clean ambient, | |
| tapesAt (tape 0) (tape 5) base (stream tokens ++ suffix) (true :: counter)⟩ := by | |
| simpa only [Function.iterate_one, advance_some] using | |
| incrementStep tape labels foundExit missingExit malformedExit program | |
| atLabels base ambient (stream tokens ++ suffix) counter | |
| have rest := ih (true :: counter) tailCanonical remaining | |
| simp only [same, ite_false] at first | |
| have full := joinTrace first (joinTrace increment rest) | |
| have counterEq : List.replicate ((payloads tokens).idxOf key) true ++ | |
| (true :: counter) = | |
| List.replicate ((payloads tokens).idxOf key + 1) true ++ counter := by | |
| rw [List.replicate_succ'] | |
| simp only [List.append_assoc, List.singleton_append] | |
| have indexEq : (token.2 :: payloads tokens).idxOf key = | |
| (payloads tokens).idxOf key + 1 := by | |
| simp only [List.idxOf_cons, beq_false_of_ne same, Bool.false_eq_true, ite_false] | |
| simpa only [steps, same, ite_false, stream_cons, List.append_assoc, afterMatch, | |
| payloads_cons, indexEq, counterEq] using full | |
| def searchInTime (tokens : List Token) (key suffix counter : List Bool) | |
| (canonical : ∀ token ∈ tokens, BinaryNameMachine.canonical token.2 = true) | |
| (present : key ∈ payloads tokens) | |
| (keyWord : base (tape 1) = key.reverse) | |
| (candidateEmpty : base (tape 2) = []) (copyEmpty : base (tape 3) = []) | |
| (scratchEmpty : base (tape 4) = []) : | |
| StateTransition.EvalsToInTime (TM2.step program) | |
| ⟨some (labels .sign), clean ambient, | |
| tapesAt (tape 0) (tape 5) base (stream tokens ++ suffix) counter⟩ | |
| (some ⟨foundExit, clean ambient, | |
| tapesAt (tape 0) (tape 5) base (stream (afterMatch tokens key) ++ suffix) | |
| (List.replicate ((payloads tokens).idxOf key) true ++ counter)⟩) | |
| ((3 * key.length + 7) * (stream tokens).length) where | |
| steps := steps tokens key | |
| evals_in_steps := searchTrace tape distinct labels foundExit missingExit malformedExit | |
| program atLabels base ambient tokens key suffix counter canonical present keyWord | |
| candidateEmpty copyEmpty scratchEmpty | |
| steps_le_m := steps_le_stream tokens key | |
| end Program | |
| def machine : FinTM2 where | |
| K := Fin 6 | |
| k₀ := 0 | |
| k₁ := 5 | |
| Γ _ := Bool | |
| Λ := Label ⊕ Fin 3 | |
| main := .inl .sign | |
| σ := State Unit | |
| initialState := clean () | |
| m label := match label with | |
| | .inl l => instruction id Sum.inl (some (.inr 0)) (some (.inr 1)) (some (.inr 2)) l | |
| | .inr _ => .halt | |
| theorem machine_finiteAlphabet (k : machine.K) : Finite (machine.Γ k) := by | |
| change Finite Bool | |
| infer_instance | |
| end PerfectCompleteness.BinaryNameSearch | |
| end | |
| end OAI | |