File size: 3,154 Bytes
2f887db | 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 | import Mathlib
import OAI.Computability.MinUncut.Graphs.GraphWordCodec
import OAI.Computability.MinUncut.Estimates.FiberPrefix
namespace OAI
section
namespace MinUncut.Costed.FrameDecode
open _root_.Turing _root_.OAI.Turing _root_.Turing.PartrecToTM2 _root_.OAI.Turing.PartrecToTM2 MinUncutGames.BinaryEncoding
inductive Mode
| flag | payload | suffix
deriving DecidableEq
protected abbrev Mode.enumList : List Mode := [.flag, .payload, .suffix]
protected theorem Mode.enumList_getElem?_ctorIdx_eq (x : Mode) :
Mode.enumList[x.ctorIdx]? = some x := by
cases x <;> rfl
protected theorem Mode.enumList_nodup : Mode.enumList.Nodup := by decide
instance : Fintype Mode where
elems := ⟨Mode.enumList, Mode.enumList_nodup⟩
complete x := by cases x <;> decide
def next : Mode → Bool → Mode
| .flag,true => .payload
| .flag,false => .suffix
| .payload,_ => .flag
| .suffix,_ => .suffix
def symbol (b : Bool) : ℕ := b.toNat+1
def emit : Mode → Bool → List Γ'
| .flag,true => []
| .flag,false => [.cons]
| .payload,b => [GraphCodec.bitSymbol b]
| .suffix,b => trNat (symbol b)++[.cons]
def machine : FinTM2 := WordTransducer.machine .flag next emit
lemma finiteAlphabet (k : machine.K) : Finite (machine.Γ k) :=
WordTransducer.finiteAlphabet .flag next emit k
lemma suffix_output (v : List Bool) :
WordTransducer.output next emit .suffix v=trList (v.map symbol) := by
induction v with
| nil => rfl
| cons b v ih =>
simp only [WordTransducer.output,next,emit,List.map_cons,trList,ih,List.append_assoc,List.singleton_append]
lemma frame_output (v w : List Bool) :
WordTransducer.output next emit .flag (frame v++w)=
v.map GraphCodec.bitSymbol++Γ'.cons::trList (w.map symbol) := by
induction v with
| nil => simpa only [frame,List.cons_append,List.nil_append,WordTransducer.output,next,emit,
List.singleton_append,List.map_nil] using congrArg (List.cons Γ'.cons) (suffix_output w)
| cons b v ih =>
simp only [frame,List.cons_append,WordTransducer.output,next,emit,List.nil_append,
List.map_cons,ih]
lemma output_spec (K : ℕ) (w : List Bool) :
WordTransducer.output next emit .flag (nameBits K++w)=trList (K::w.map symbol) := by
rw [nameBits,frame_output,←GraphCodec.trNat_bits]
rfl
def outputs (K : ℕ) (w : List Bool) :
TM2OutputsInTime machine (nameBits K++w) (some (trList (K::w.map symbol)))
(2*(nameBits K++w).length+(trList (K::w.map symbol)).length+2) := by
have h := WordTransducer.outputsInTime .flag next emit (nameBits K++w)
erw [output_spec] at h
exact h
lemma count_symbols (v : List Bool) : UnaryWords.count (v.map symbol)=v.length := by
induction v with
| nil => rfl
| cons b v ih =>
simp only [List.map_cons,UnaryWords.count,symbol,show b.toNat+1≠0 by omega,ite_false,ih,List.length_cons]
lemma magnitude_symbols (v : List Bool) : magnitude (v.map symbol)≤3*v.length := by
induction v with
| nil => rfl
| cons b v ih =>
have hb : b.toNat≤1 := by cases b <;> decide
simp only [List.map_cons,magnitude_cons,List.length_cons,symbol]
omega
end MinUncut.Costed.FrameDecode
end
end OAI
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