| import OAI.Computability.BinPacking.Reductions.BinaryTokenMachine |
|
|
| namespace OAI |
|
|
| namespace BinPackingGap.FrameEmitMachine |
|
|
| open Turing BinPackingGames.Foundations.Complexity MachineComposition |
| open BinPackingGames.Reduction.MachineTransfer |
|
|
| variable {K Λ A : Type} [DecidableEq K] |
|
|
| abbrev Alphabet (_ : K) := Bool |
| abbrev State (A : Type) := A × Option Bool |
|
|
| def finish (exit : Option Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A) := |
| .load (fun state => (state.1, none)) |
| (match exit with | none => .halt | some label => .goto fun _ => label) |
|
|
| def instruction (source output : K) (again : Λ) (exit : Option Λ) : |
| TM2.Stmt (Alphabet (K := K)) Λ (State A) := |
| .pop source (fun state head => (state.1, head)) |
| (.branch (fun state => state.2.isSome) |
| (.push output (fun _ => true) |
| (.push output (fun state => state.2.getD false) (finish (some again)))) |
| (.push output (fun _ => false) (finish exit))) |
|
|
| variable (source output : K) (distinct : source ≠ output) |
| variable (again : Λ) (exit : Option Λ) |
| variable (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A)) |
| variable (atLabel : program again = instruction source output again exit) |
| variable (base : K → List Bool) (ambient : A) |
|
|
| include distinct in |
| private theorem update_source (input acc next : List Bool) : |
| Function.update (tapesAt source output base input acc) source next = |
| tapesAt source output base next acc := by |
| funext k |
| by_cases hs : k = source |
| · subst k; simp [tapesAt, distinct] |
| · by_cases ho : k = output |
| · subst k; simp [tapesAt, hs] |
| · simp [tapesAt, hs, ho] |
|
|
| private theorem update_output (input acc next : List Bool) : |
| Function.update (tapesAt source output base input acc) output next = |
| tapesAt source output base input next := by |
| simp [tapesAt] |
|
|
| include distinct atLabel |
|
|
| theorem step_nil (acc : List Bool) : |
| TM2.step program |
| ⟨some again, (ambient, none), tapesAt source output base [] acc⟩ = |
| some ⟨exit, (ambient, none), tapesAt source output base [] (false :: acc)⟩ := by |
| change some (TM2.stepAux (program again) _ _) = _ |
| rw [atLabel] |
| cases exit <;> |
| simp [instruction, finish, TM2.stepAux, tapesAt_src, tapesAt_dst, |
| distinct, update_source, update_output] |
|
|
| theorem step_cons (bit : Bool) (input acc : List Bool) : |
| TM2.step program |
| ⟨some again, (ambient, none), tapesAt source output base (bit :: input) acc⟩ = |
| some ⟨some again, (ambient, none), |
| tapesAt source output base input (bit :: true :: acc)⟩ := by |
| change some (TM2.stepAux (program again) _ _) = _ |
| rw [atLabel] |
| simp [instruction, finish, TM2.stepAux, tapesAt_src, tapesAt_dst, |
| distinct, update_source, update_output] |
|
|
| theorem trace (input acc : List Bool) : |
| (advance (TM2.step program))^[input.length + 1] |
| (some ⟨some again, (ambient, none), tapesAt source output base input acc⟩) = |
| some ⟨exit, (ambient, none), tapesAt source output base [] |
| ((BinPackingCompleteness.BinaryEncoding.frame input).reverse ++ acc)⟩ := by |
| induction input generalizing acc with |
| | nil => |
| simpa [advance, BinPackingCompleteness.BinaryEncoding.frame] using |
| step_nil source output distinct again exit program atLabel base ambient acc |
| | cons bit input ih => |
| rw [List.length_cons, Function.iterate_succ_apply] |
| simp only [advance_some] |
| rw [step_cons source output distinct again exit program atLabel base ambient] |
| simpa only [BinPackingCompleteness.BinaryEncoding.frame, List.reverse_cons, |
| List.append_assoc, List.cons_append, List.nil_append, List.singleton_append] using ih (bit :: true :: acc) |
|
|
| def inTime (input acc : List Bool) : |
| StateTransition.EvalsToInTime (TM2.step program) |
| ⟨some again, (ambient, none), tapesAt source output base input acc⟩ |
| (some ⟨exit, (ambient, none), tapesAt source output base [] |
| ((BinPackingCompleteness.BinaryEncoding.frame input).reverse ++ acc)⟩) |
| (input.length + 1) where |
| steps := input.length + 1 |
| evals_in_steps := trace source output distinct again exit program atLabel base ambient input acc |
| steps_le_m := Nat.le_refl _ |
|
|
| end BinPackingGap.FrameEmitMachine |
|
|
| end OAI |
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|
