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e320e82 | 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 | import Mathlib
import OAI.Combinatorics.IndependentSets.Machines.PoweringMachineGlobal
namespace OAI
namespace IndependentSetsGames.Foundations.Complexity.PoweringInitializeFrame
open Turing PCP
variable {vertices d : Nat}
def initialExtra (graph : PortTables.Table vertices d) (n : Nat) :
PoweringMachineInitialize.ExtraTape (PoweringMachineRowBody.capacity n) → List Bool :=
fun tape => if tape = .inl (PoweringMachineTapes.table (PoweringMachineRowBody.capacity n))
then PortTables.tableBits graph else []
@[simp] theorem initialExtra_table (graph : PortTables.Table vertices d) (n : Nat) :
initialExtra graph n (.inl (PoweringMachineTapes.table (PoweringMachineRowBody.capacity n))) =
PortTables.tableBits graph := by simp [initialExtra]
theorem initialExtra_other (graph : PortTables.Table vertices d) (n : Nat)
(tape : PoweringMachineInitialize.ExtraTape (PoweringMachineRowBody.capacity n))
(hne : tape ≠ .inl (PoweringMachineTapes.table (PoweringMachineRowBody.capacity n))) :
initialExtra graph n tape = [] := by simp [initialExtra, hne]
@[simp] theorem initialExtra_scratch (graph : PortTables.Table vertices d) (n : Nat) :
initialExtra graph n
(.inl (PoweringMachineTapes.scratch (PoweringMachineRowBody.capacity n))) = [] := by
simp [initialExtra, PoweringMachineTapes.table, PoweringMachineTapes.scratch]
def preparedTapes (graph : PortTables.Table vertices d) (n : Nat) :
PoweringMachineGlobal.Tape n → List Bool :=
PoweringMachineInitialize.mergeTapes
(PoweringMachineLoop.finalTapes (PoweringTableLayout.blockSize d n) vertices [])
(initialExtra graph n)
theorem prepared_commonRole (graph : PortTables.Table vertices d) (n : Nat)
(j : Fin 11) (hne : j ≠ 1) :
preparedTapes graph n (PoweringMachineGlobal.placement n (.inl j)) =
if j = 0 then PortTables.tableBits graph else [] := by
simp [preparedTapes, PoweringMachineGlobal.placement,
PoweringMachineInitialize.commonPlacement, PoweringMachineTapes.start,
PoweringMachineInitialize.mergeTapes, initialExtra, PoweringMachineTapes.table, hne]
theorem prepared_ready (graph : PortTables.Table vertices d) (n : Nat) :
PoweringMachineOuterLoop.Ready graph n (PoweringMachineGlobal.placement n)
(preparedTapes graph n) := by
constructor
· simpa using prepared_commonRole graph n 0 (by decide)
· simpa using prepared_commonRole graph n 5 (by decide)
· simpa using prepared_commonRole graph n 8 (by decide)
· simpa using prepared_commonRole graph n 9 (by decide)
· simpa using prepared_commonRole graph n 10 (by decide)
@[simp] theorem prepared_start (graph : PortTables.Table vertices d) (n : Nat) :
preparedTapes graph n (PoweringMachineGlobal.placement n (.inl 1)) =
encodeWord vertices := by
simpa only [preparedTapes, PoweringMachineGlobal.placement, PoweringMachineTapes.start] using
PoweringMachineInitialize.final_counter (PoweringMachineRowBody.capacity n)
(PoweringTableLayout.blockSize d n) vertices (initialExtra graph n)
theorem prepared_counter (graph : PortTables.Table vertices d) (n : Nat) :
PoweringMachineOuterLoop.counter (PoweringMachineGlobal.placement n)
(preparedTapes graph n) vertices = preparedTapes graph n := by
unfold PoweringMachineOuterLoop.counter MachineUnaryCounter.counterTapes
rw [List.append_nil, ← prepared_start graph n]
simp only [Function.update_eq_self]
@[simp] theorem prepared_output (graph : PortTables.Table vertices d) (n : Nat) :
preparedTapes graph n (PoweringMachineGlobal.outputTape n) = [] := by
simp [preparedTapes, PoweringMachineGlobal.outputTape, PoweringMachineInitialize.finalOutput,
PoweringMachineInitialize.mergeTapes, initialExtra]
@[simp] theorem prepared_vertices (graph : PortTables.Table vertices d) (n : Nat) :
preparedTapes graph n (PoweringMachineGlobal.headerVertices n) = encodeWord vertices := rfl
@[simp] theorem prepared_darts (graph : PortTables.Table vertices d) (n : Nat) :
preparedTapes graph n (PoweringMachineGlobal.headerDarts n) =
encodeWord (PoweringTableLayout.blockSize d n * vertices) := rfl
@[simp] theorem prepared_source (graph : PortTables.Table vertices d) (n : Nat) :
preparedTapes graph n (.inl PoweringMachineLoop.HeaderTape.source) = [] := rfl
def preparedCfg (graph : PortTables.Table vertices d) (n : Nat) :
(PoweringMachineGlobal.machine d n).Cfg :=
⟨some (PoweringMachineGlobal.guardLabel d n),
PoweringMasterState.clean (PoweringMachineRowBody.bufferSize d n), preparedTapes graph n⟩
theorem initialGlobalTapes_eq (graph : PortTables.Table vertices d) (n : Nat) :
PoweringMachineInitialize.initialGlobalTapes (initialExtra graph n) =
fun tape => if tape = PoweringMachineGlobal.inputTape n
then PortTables.tableBits graph else [] := by
funext tape
cases tape with
| inl field =>
simp [PoweringMachineInitialize.initialGlobalTapes, PoweringMachineInitialize.mergeTapes,
PoweringMachineGlobal.inputTape, PoweringMachineGlobal.placement,
PoweringMachineInitialize.commonPlacement_table]
| inr field =>
simp [PoweringMachineInitialize.initialGlobalTapes, PoweringMachineInitialize.mergeTapes,
PoweringMachineGlobal.inputTape, PoweringMachineGlobal.placement,
PoweringMachineInitialize.commonPlacement_table, initialExtra]
private theorem machine_initialStacks (d n : Nat) (input : List Bool) :
(initList (PoweringMachineGlobal.machine d n) input).stk =
fun tape => if tape = PoweringMachineGlobal.inputTape n then input else [] := rfl
theorem initList_eq (graph : PortTables.Table vertices d) (n : Nat) :
initList (PoweringMachineGlobal.machine d n) (PortTables.tableBits graph) =
⟨some (PoweringMachineGlobal.main d n),
PoweringMasterState.clean (PoweringMachineRowBody.bufferSize d n),
PoweringMachineInitialize.initialGlobalTapes (initialExtra graph n)⟩ := by
have ht := (machine_initialStacks d n (PortTables.tableBits graph)).trans
(initialGlobalTapes_eq graph n).symm
exact congrArg (TM2.Cfg.mk _ _) ht
theorem tableBits_head (graph : PortTables.Table vertices d) :
PortTables.tableBits graph = encodeWord vertices ++
encodeWords (PortTables.tableWords graph).tail := rfl
def initializeInTime (graph : PortTables.Table vertices d) (n : Nat) :
StateTransition.EvalsToInTime (PoweringMachineGlobal.machine d n).step
(initList (PoweringMachineGlobal.machine d n) (PortTables.tableBits graph))
(some (preparedCfg graph n)) (3 * vertices + 5) := by
rw [initList_eq]
exact PoweringMachineInitialize.initializeAtInTime
(PoweringMachineRowBody.capacity n) (PoweringMachineRowBody.bufferSize d n)
(PoweringTableLayout.blockSize d n) vertices
(PoweringMachineGlobal.initLabels d n) (some (PoweringMachineGlobal.guardLabel d n))
(PoweringMachineGlobal.program d n) (PoweringMachineGlobal.atInit d n)
(initialExtra graph n) (encodeWords (PortTables.tableWords graph).tail)
(by rw [initialExtra_table, tableBits_head]) (initialExtra_scratch graph n)
(PoweringMasterState.clean (PoweringMachineRowBody.bufferSize d n))
end IndependentSetsGames.Foundations.Complexity.PoweringInitializeFrame
end OAI
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