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760f90e | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 | import OAI.Computability.UniqueGames.Machines.MachineCompositionLemmas
import OAI.Computability.UniqueGames.Reduction.MachineTransfer
namespace OAI
section
namespace UniqueGamesTheorem.Foundations.Complexity.MachineAppendAt
open Turing
variable {K Λ σ β : Type} [DecidableEq K]
abbrev Alphabet (β : Type) (_ : K) := β
def appendTapes (source destination : K) (base : K → List β) : K → List β :=
Reduction.MachineTransfer.tapesAt source destination base []
(base destination ++ base source)
@[simp] theorem appendTapes_source (source destination : K)
(distinct : source ≠ destination) (base : K → List β) :
appendTapes source destination base source = [] := by
simp [appendTapes, distinct]
@[simp] theorem appendTapes_destination (source destination : K) (base : K → List β) :
appendTapes source destination base destination = base destination ++ base source := by
simp [appendTapes]
theorem appendTapes_other (source destination k : K)
(notSource : k ≠ source) (notDestination : k ≠ destination) (base : K → List β) :
appendTapes source destination base k = base k := by
simp [appendTapes, Reduction.MachineTransfer.tapesAt, notSource, notDestination]
theorem appendTapes_scratch (source destination scratch : K)
(sourceScratch : source ≠ scratch) (destinationScratch : destination ≠ scratch)
(base : K → List β) (scratchEmpty : base scratch = []) :
appendTapes source destination base scratch = [] := by
rw [appendTapes_other source destination scratch (Ne.symm sourceScratch)
(Ne.symm destinationScratch) base, scratchEmpty]
private def firstTapes (destination scratch : K) (base : K → List β) : K → List β :=
Reduction.MachineTransfer.tapesAt destination scratch base [] (base destination).reverse
private def secondTapes (source destination scratch : K) (base : K → List β) : K → List β :=
Reduction.MachineTransfer.tapesAt source scratch (firstTapes destination scratch base) []
((base source).reverse ++ (base destination).reverse)
private theorem finalTapes_eq (source destination scratch : K)
(sourceDestination : source ≠ destination) (sourceScratch : source ≠ scratch)
(destinationScratch : destination ≠ scratch) (base : K → List β)
(scratchEmpty : base scratch = []) :
Reduction.MachineTransfer.tapesAt scratch destination
(secondTapes source destination scratch base) [] (base destination ++ base source) =
appendTapes source destination base := by
funext k
by_cases hs : k = source
· subst k
simp [appendTapes, secondTapes, firstTapes, Reduction.MachineTransfer.tapesAt,
sourceDestination, sourceScratch]
· by_cases hd : k = destination
· subst k
simp [appendTapes, Reduction.MachineTransfer.tapesAt]
· by_cases ht : k = scratch
· subst k
simp [appendTapes, Reduction.MachineTransfer.tapesAt, Ne.symm sourceScratch,
Ne.symm destinationScratch, scratchEmpty]
· simp [appendTapes, secondTapes, firstTapes, Reduction.MachineTransfer.tapesAt,
hs, hd, ht]
theorem appendTrace (source destination scratch : K)
(sourceDestination : source ≠ destination) (sourceScratch : source ≠ scratch)
(destinationScratch : destination ≠ scratch)
(fallback : β) (firstLabel secondLabel thirdLabel : Λ) (exit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
(atFirst : program firstLabel = Reduction.MachineTransfer.loopAt destination scratch
id fallback firstLabel (some secondLabel))
(atSecond : program secondLabel = Reduction.MachineTransfer.loopAt source scratch
id fallback secondLabel (some thirdLabel))
(atThird : program thirdLabel = Reduction.MachineTransfer.loopAt scratch destination
id fallback thirdLabel exit)
(base : K → List β) (scratchEmpty : base scratch = [])
(ambient : σ) (register : Option β) :
(MachineComposition.advance (TM2.step program))^[
2 * ((base source).length + (base destination).length) + 3]
(some ⟨some firstLabel, (ambient, register), base⟩) =
some ⟨exit, (ambient, none), appendTapes source destination base⟩ := by
have first := Reduction.MachineTransfer.transferAt_fromTapes destination scratch
destinationScratch id fallback firstLabel (some secondLabel) program atFirst
base ambient register
change (MachineComposition.advance (TM2.step program))^[(base destination).length + 1]
(some ⟨some firstLabel, (ambient, register), base⟩) = _ at first
simp only [List.map_id, scratchEmpty, List.append_nil] at first
change (MachineComposition.advance (TM2.step program))^[(base destination).length + 1]
(some ⟨some firstLabel, (ambient, register), base⟩) =
some ⟨some secondLabel, (ambient, none), firstTapes destination scratch base⟩ at first
have firstSource : firstTapes destination scratch base source = base source := by
simp [firstTapes, Reduction.MachineTransfer.tapesAt, sourceDestination, sourceScratch]
have firstScratch : firstTapes destination scratch base scratch =
(base destination).reverse := by
simp [firstTapes]
have second := Reduction.MachineTransfer.transferAt_fromTapes source scratch
sourceScratch id fallback secondLabel (some thirdLabel) program atSecond
(firstTapes destination scratch base) ambient none
change (MachineComposition.advance (TM2.step program))^[
(firstTapes destination scratch base source).length + 1]
(some ⟨some secondLabel, (ambient, none), firstTapes destination scratch base⟩) = _
at second
rw [firstSource, firstScratch] at second
simp only [List.map_id] at second
change (MachineComposition.advance (TM2.step program))^[(base source).length + 1]
(some ⟨some secondLabel, (ambient, none), firstTapes destination scratch base⟩) =
some ⟨some thirdLabel, (ambient, none), secondTapes source destination scratch base⟩
at second
have secondScratch : secondTapes source destination scratch base scratch =
(base source).reverse ++ (base destination).reverse := by
simp [secondTapes]
have secondDestination : secondTapes source destination scratch base destination = [] := by
simp [secondTapes, firstTapes, Reduction.MachineTransfer.tapesAt,
Ne.symm sourceDestination, destinationScratch]
have third := Reduction.MachineTransfer.transferAt_fromTapes scratch destination
(Ne.symm destinationScratch) id fallback thirdLabel exit program atThird
(secondTapes source destination scratch base) ambient none
change (MachineComposition.advance (TM2.step program))^[
(secondTapes source destination scratch base scratch).length + 1]
(some ⟨some thirdLabel, (ambient, none), secondTapes source destination scratch base⟩) = _
at third
rw [secondScratch, secondDestination] at third
simp only [List.map_id, List.reverse_append, List.reverse_reverse, List.append_nil,
List.length_append, List.length_reverse] at third
rw [finalTapes_eq source destination scratch sourceDestination sourceScratch
destinationScratch base scratchEmpty] at third
rw [show 2 * ((base source).length + (base destination).length) + 3 =
((base source).length + (base destination).length + 1) +
(((base source).length + 1) + ((base destination).length + 1)) by omega]
rw [Function.iterate_add_apply]
rw [Function.iterate_add_apply (m := (base source).length + 1)
(n := (base destination).length + 1), first, second]
exact third
/-- The timed witness stores the exact three-loop transition count. -/
def appendInTime (source destination scratch : K)
(sourceDestination : source ≠ destination) (sourceScratch : source ≠ scratch)
(destinationScratch : destination ≠ scratch)
(fallback : β) (firstLabel secondLabel thirdLabel : Λ) (exit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K) β) Λ (σ × Option β))
(atFirst : program firstLabel = Reduction.MachineTransfer.loopAt destination scratch
id fallback firstLabel (some secondLabel))
(atSecond : program secondLabel = Reduction.MachineTransfer.loopAt source scratch
id fallback secondLabel (some thirdLabel))
(atThird : program thirdLabel = Reduction.MachineTransfer.loopAt scratch destination
id fallback thirdLabel exit)
(base : K → List β) (scratchEmpty : base scratch = [])
(ambient : σ) (register : Option β) :
StateTransition.EvalsToInTime (TM2.step program)
⟨some firstLabel, (ambient, register), base⟩
(some ⟨exit, (ambient, none), appendTapes source destination base⟩)
(2 * ((base source).length + (base destination).length) + 3) where
steps := 2 * ((base source).length + (base destination).length) + 3
evals_in_steps := appendTrace source destination scratch sourceDestination sourceScratch
destinationScratch fallback firstLabel secondLabel thirdLabel exit program atFirst
atSecond atThird base scratchEmpty ambient register
steps_le_m := Nat.le_refl _
end UniqueGamesTheorem.Foundations.Complexity.MachineAppendAt
end
end OAI
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