File size: 4,205 Bytes
8df23f4 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 | 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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