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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | import OAI.Computability.BinPacking.Computation.MachineExpanderFamilyAffine
namespace OAI
namespace BinPackingGames.Foundations.Complexity.MachineExpanderTable
open Turing
open MachineComposition
open PCP.ExpanderTables PCP.ExpanderRowControl
variable {ρ : Type} {d : Nat}
@[simp] theorem clearRegister_position (state : State ρ d) :
(clearRegister state).2 = state.2 := rfl
@[simp] theorem clearRegister_register (state : State ρ d) :
(clearRegister state).1.2 = none := rfl
@[simp] theorem clearRegister_idempotent (state : State ρ d) :
clearRegister (clearRegister state) = clearRegister state := rfl
theorem boundaryTapes_reverse_update (vertex remaining : Nat)
(oldTable output countSuffix result nextOutput nextResult : List Bool) :
Function.update
(Function.update (boundaryTapes vertex remaining oldTable output countSuffix result)
(.inl .output) nextOutput)
(.inr .result) nextResult =
boundaryTapes vertex remaining oldTable nextOutput countSuffix nextResult := by
funext tape
rcases tape with row | extra
· cases row <;>
simp [boundaryTapes, rowTapes, MachineEmbedding.tapes, Function.update]
· cases extra <;>
simp [boundaryTapes, extraTapes, MachineEmbedding.tapes, Function.update]
variable [Fintype ρ]
theorem reverseOutputTrace (positive : 0 < d) (H : Table (cloudSize d) d)
(vertex remaining : Nat) (oldTable output countSuffix result : List Bool)
(state : State ρ d) :
(advance (TM2.step (program positive H)))^[output.length + 1]
(some ⟨some (.inr .reverseOutput), state,
boundaryTapes vertex remaining oldTable output countSuffix result⟩) =
some ⟨some (.inr .done), clearRegister state,
boundaryTapes vertex remaining oldTable [] countSuffix (output.reverse ++ result)⟩ := by
induction output generalizing result state with
| nil =>
simpa only [List.length_nil, Nat.zero_add, Function.iterate_one, advance_some,
List.reverse_nil, List.nil_append] using
reverseOutput_nilStep positive H
(boundaryTapes vertex remaining oldTable [] countSuffix result) state rfl
| cons symbol output ih =>
rw [List.length_cons, Function.iterate_succ_apply]
simp only [advance_some]
rw [reverseOutput_consStep positive H
(boundaryTapes vertex remaining oldTable (symbol :: output) countSuffix result)
state symbol output rfl]
change (advance (TM2.step (program positive H)))^[output.length + 1]
(some ⟨some (.inr .reverseOutput), ((state.1.1, some symbol), state.2),
Function.update
(Function.update
(boundaryTapes vertex remaining oldTable (symbol :: output) countSuffix result)
(.inl .output) output)
(.inr .result) (symbol :: result)⟩) = _
rw [boundaryTapes_reverse_update]
simpa only [clearRegister, List.reverse_cons, List.append_assoc,
List.singleton_append] using
ih (symbol :: result) ((state.1.1, some symbol), state.2)
theorem reverseOutputHaltTrace (positive : 0 < d) (H : Table (cloudSize d) d)
(vertex remaining : Nat) (oldTable output countSuffix result : List Bool)
(state : State ρ d) :
(advance (TM2.step (program positive H)))^[output.length + 2]
(some ⟨some (.inr .reverseOutput), state,
boundaryTapes vertex remaining oldTable output countSuffix result⟩) =
some ⟨none, clearRegister state,
boundaryTapes vertex remaining oldTable [] countSuffix (output.reverse ++ result)⟩ := by
rw [show output.length + 2 = (output.length + 1) + 1 by omega,
Function.iterate_succ_apply',
reverseOutputTrace positive H vertex remaining oldTable output countSuffix result state]
exact doneStep positive H
(boundaryTapes vertex remaining oldTable [] countSuffix (output.reverse ++ result))
(clearRegister state)
def reverseOutputInTime (positive : 0 < d) (H : Table (cloudSize d) d)
(vertex remaining : Nat) (oldTable output countSuffix result : List Bool)
(state : State ρ d) :
StateTransition.EvalsToInTime (TM2.step (program positive H))
⟨some (.inr .reverseOutput), state,
boundaryTapes vertex remaining oldTable output countSuffix result⟩
(some ⟨some (.inr .done), clearRegister state,
boundaryTapes vertex remaining oldTable [] countSuffix (output.reverse ++ result)⟩)
(output.length + 1) where
steps := output.length + 1
evals_in_steps := reverseOutputTrace positive H vertex remaining oldTable output countSuffix result state
steps_le_m := Nat.le_refl _
def reverseOutputHaltInTime (positive : 0 < d) (H : Table (cloudSize d) d)
(vertex remaining : Nat) (oldTable output countSuffix result : List Bool)
(state : State ρ d) :
StateTransition.EvalsToInTime (TM2.step (program positive H))
⟨some (.inr .reverseOutput), state,
boundaryTapes vertex remaining oldTable output countSuffix result⟩
(some ⟨none, clearRegister state,
boundaryTapes vertex remaining oldTable [] countSuffix (output.reverse ++ result)⟩)
(output.length + 2) where
steps := output.length + 2
evals_in_steps := reverseOutputHaltTrace positive H vertex remaining oldTable output countSuffix result state
steps_le_m := Nat.le_refl _
end BinPackingGames.Foundations.Complexity.MachineExpanderTable
end OAI
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