| import Mathlib |
| import OAI.Computability.MinUncut.Graphs.UnboundedGraphBase |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace MinUncut.Costed.Arena |
| open _root_.Turing _root_.OAI.Turing Turing.ToPartrec _root_.Turing.PartrecToTM2 _root_.OAI.Turing.PartrecToTM2 Polynomial |
| open MinUncutGames.Foundations.Complexity MinUncutGames.BinaryEncoding |
|
|
| def initializedGraphMachine (c : Code) : FinTM2 := |
| MachineSequential.machine (codeMachine c) unboundedGraphMachine id .cons |
| lemma initializedGraphMachine_finite (c : Code) (k : (initializedGraphMachine c).K) : |
| Finite ((initializedGraphMachine c).Γ k) := |
| sequential_finiteAlphabet _ _ _ _ (codeMachine_finiteAlphabet _) unboundedGraphMachine_finiteAlphabet k |
|
|
| lemma magnitude_words (xs : List ℕ) : magnitude xs=(encodeWords xs).length := by |
| induction xs with |
| | nil => rfl |
| | cons a xs ih=> simp only [magnitude_cons,encodeWords,List.length_append,encodeWord_length,ih] |
|
|
| lemma exists_graph_initializer {f : ℕ → List ℕ → List ℕ} (P : ∀K,PolyProgram (f K)) |
| (he : Computable (fun K=>(P K).code)) : |
| ∃c : Code,∀K,∃p : Polynomial ℕ,∀xs n table k junk, |
| f K xs=n::((table : List Bool).map Bool.toNat++2::k::junk) → |
| Nonempty (TM2OutputsInTime (initializedGraphMachine c) |
| (trList (K::(encodeWords xs).map FrameDecode.symbol)) |
| (some (nameBits n++table++nameBits k)) (p.eval (encodeWords xs).length)) := by |
| obtain ⟨c,hc⟩:=Init.exists_initializer he |
| refine ⟨c,fun K=>?_⟩ |
| obtain ⟨p,hp⟩:=hc K |
| let q:Polynomial ℕ:=p.comp (C (K+1)+C 3*X) |
| let r:Polynomial ℕ:=q+1+C 2*(q+1)+unboundedGraphTime.comp ((P K).bound+1+q) |
| refine ⟨r,fun xs n table k junk ho=>?_⟩ |
| let v:=(encodeWords xs).map FrameDecode.symbol |
| let b:=Init.base (P K).code xs K v |
| let w:=(initialState (P K).code xs b).encode |
| have hs:magnitude (K::v)≤(C (K+1)+C 3*X : Polynomial ℕ).eval (encodeWords xs).length:=by |
| have hh:=FrameDecode.magnitude_symbols (encodeWords xs) |
| simp only [magnitude_cons,eval_add,eval_C,eval_mul,eval_X] |
| dsimp only [v] |
| omega |
| have hpbound:=evalNat_mono p hs |
| have hir: Runs c (K::v) w (q.eval (encodeWords xs).length):= |
| (hp xs).1.mono (by simpa only [q,eval_comp] using hpbound) |
| have his: magnitude w≤q.eval (encodeWords xs).length:= |
| (hp xs).2.trans (by simpa only [q,eval_comp] using hpbound) |
| have hg:Runs (P K).code xs (n::(table.map Bool.toNat++2::k::junk)) |
| ((P K).bound.eval (encodeWords xs).length):=by |
| have hh:=(P K).run xs |
| rw [ho,magnitude_words] at hh |
| exact hh |
| have hl:(n::(table.map Bool.toNat++2::k::junk)).length≤(P K).bound.eval (encodeWords xs).length:=by |
| have hh:=(magnitude_length (f K xs)).trans ((P K).size xs) |
| simpa only [ho,magnitude_words] using hh |
| let h2:=unboundedGraphMachine_outputs_base b hg hl |
| obtain ⟨t,ht,hr⟩:=hir |
| let h1:=hr.finiteMachine |
| let h2':TM2OutputsInTime unboundedGraphMachine ((trList w).map id) |
| (some (nameBits n++table++nameBits k)) |
| (unboundedGraphTime.eval (magnitude ((P K).bound.eval (encodeWords xs).length::w))) := by |
| have input_eq : @List.map (unboundedGraphMachine.Γ unboundedGraphMachine.k₀) |
| (unboundedGraphMachine.Γ unboundedGraphMachine.k₀) id (trList w) = trList w := |
| List.map_id _ |
| simpa only [input_eq] using h2 |
| have hh:=MachineSequential.execute (codeMachine c) unboundedGraphMachine id Γ'.cons |
| _ _ _ _ _ h1 h2' |
| have hwb: (trList w).length≤q.eval (encodeWords xs).length:= (bitsize_le_magnitude w).trans his |
| have hm:magnitude ((P K).bound.eval (encodeWords xs).length::w)≤ |
| (((P K).bound+1+q).eval (encodeWords xs).length):=by |
| simp only [magnitude_cons,eval_add,eval_one] |
| omega |
| have hb:=evalNat_mono unboundedGraphTime hm |
| simp only [eval_add,eval_one] at hb |
| exact ⟨⟨hh.toEvalsTo,hh.steps_le_m.trans (by |
| simp only [r,eval_add,eval_one,eval_mul,eval_C,eval_comp] |
| change t+1+2*((trList w).length+1)+_≤_ |
| omega)⟩⟩ |
| end MinUncut.Costed.Arena |
|
|
| end |
| end |
|
|
| end OAI |
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|
