| 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 |
|
|
