| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Machines.TreeFrame |
| import OAI.Combinatorics.MatchingCount.Tapes.Online |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineStack |
| open Turing MachineCost PhysicalStream StackComposition |
|
|
| variable {K Λ Λ' σ α : Type} {Γ : K → Type} [DecidableEq K] |
| namespace Executes |
| lemma relabel {M : Λ → TM2.Stmt Γ Λ σ} {coin : CoinTM1.Coins Λ} |
| {out : α → TM2.Cfg Γ Λ σ} {c : TM2.Cfg Γ Λ σ} {p : Experiment α} |
| (h : Executes M coin out c p) (N : Λ' → TM2.Stmt Γ Λ' σ) |
| (co : CoinTM1.Coins Λ') (f : Λ → Λ') (halt : Option Λ') |
| (hN : ∀ l,N (f l)=StackComposition.relabel f halt (M l)) |
| (hco : ∀ l,co (f l)=(coin l).map (fun g=>f ∘ g)) : |
| Executes N co (fun a=>mapCfg f halt (out a)) (mapCfg f halt c) p := by |
| induction h with |
| | done a => exact .done a |
| | read l v S g p hg rest ih => |
| apply Executes.read (f l) v S (f ∘ g) p |
| · rw [hco,hg]; rfl |
| · exact ih |
| | step l v S p hs rest ih => |
| apply Executes.step (f l) v S p |
| · rw [hco,hs]; rfl |
| · rw [hN,StackComposition.relabel_eval] |
| exact ih |
|
|
| lemma of_run (M : Λ → TM2.Stmt Γ Λ σ) (n : ℕ) (c : TM2.Cfg Γ Λ σ) |
| (v : σ) (S : ∀ k,List (Γ k)) (he : MachineCost.run (TM2.step M) n c=⟨none,v,S⟩) : |
| ∃ p : Experiment Unit,Executes M (fun _=>none) (fun _ : Unit=>⟨none,v,S⟩) c p ∧ |
| (∀ f : Unit → ℚ,p.mean f=f ()) ∧ p.cost≤n := by |
| induction n generalizing c with |
| | zero => cases he; exact ⟨.done (),.done (),fun _=>rfl,le_rfl⟩ |
| | succ n ih => |
| rcases c with ⟨l,v',S'⟩ |
| cases l with |
| | none => |
| rw [StackComposition.run_halted] at he |
| cases he |
| exact ⟨.done (),.done (),fun _=>rfl,Nat.zero_le _⟩ |
| | some l => |
| obtain ⟨p,hp,hm,hc⟩:=ih (TM2.stepAux (M l) v' S') he |
| exact ⟨.delay 1 p,.step l v' S' p rfl hp,hm,by simpa [Experiment.cost,Nat.add_comm] using Nat.succ_le_succ hc⟩ |
| end Executes |
| end MatchingFPRAS.OnlineStack |
|
|
| namespace MatchingFPRAS.OnlineProc |
| open Turing MachineCost PhysicalStream OnlineStack RegisterProc |
|
|
| variable {K : Type} [DecidableEq K] |
|
|
| structure Proc (K : Type) extends RegisterProc.Proc K where |
| coin : CoinTM1.Coins Label |
|
|
| def deterministic (P : RegisterProc.Proc K) : Proc K := ⟨P,fun _=>none⟩ |
| def seq (P Q : Proc K) : Proc K where |
| toProc := RegisterProc.seq P.toProc Q.toProc |
| coin := fun l=>match l with |
| | .inl l => (P.coin l).map (fun f b=>Sum.inl (f b)) |
| | .inr l => (Q.coin l).map (fun f b=>Sum.inr (f b)) |
|
|
| def Exec (P : Proc K) (S : Stack K) (p : Experiment (Stack K)) (bound : ℕ) : Prop := |
| ∃ q : Experiment (Stack K), |
| Executes P.code P.coin (fun T=>⟨none,none,T⟩) ⟨some default,none,S⟩ q ∧ |
| (∀ f : Stack K → ℚ,q.mean f=p.mean f) ∧ q.cost≤bound |
|
|
| lemma exec_deterministic {P : RegisterProc.Proc K} {S T : Stack K} {n : ℕ} |
| (h : RegisterProc.Exec P S T n) : Exec (deterministic P) S (.done T) n := by |
| obtain ⟨k,hk,he⟩:=h |
| obtain ⟨p,hp,hm,hc⟩:=Executes.of_run P.code k ⟨some default,none,S⟩ none T he |
| refine ⟨p.map (fun _=>T),?_,?_,?_⟩ |
| · apply hp.bind (fun _=>.done T) (fun T=>⟨none,none,T⟩) |
| intro _ |
| exact .done T |
| · intro f |
| rw [Experiment.mean_map,hm]; rfl |
| · rw [Experiment.cost_map] |
| exact hc.trans hk |
|
|
| lemma Exec.mono {P : Proc K} {S : Stack K} {p : Experiment (Stack K)} {n m : ℕ} |
| (h : Exec P S p n) (hn : n ≤ m) : Exec P S p m := by |
| obtain ⟨q,hq,hm,hc⟩:=h |
| exact ⟨q,hq,hm,hc.trans hn⟩ |
|
|
| lemma Exec.seq {P Q : Proc K} {S : Stack K} {p : Experiment (Stack K)} |
| {q : Stack K → Experiment (Stack K)} {n m : ℕ} |
| (hP : Exec P S p n) (hQ : ∀ T,Exec Q T (q T) m) : |
| Exec (OnlineProc.seq P Q) S (p.bind q) (n+m) := by |
| obtain ⟨p',hp',hpm,hpc⟩:=hP |
| choose q' hq' hqm hqc using hQ |
| have hp:=hp'.relabel (OnlineProc.seq P Q).code (OnlineProc.seq P Q).coin Sum.inl (some (Sum.inr default)) |
| (fun _=>rfl) (fun _=>rfl) |
| have hq T := (hq' T).relabel (OnlineProc.seq P Q).code (OnlineProc.seq P Q).coin Sum.inr none |
| (fun _=>rfl) (fun _=>rfl) |
| refine ⟨p'.bind q',hp.bind q' _ hq,?_,?_⟩ |
| · intro f |
| rw [Experiment.mean_bind,Experiment.mean_bind] |
| simp_rw [hqm] |
| exact hpm _ |
| · exact (Experiment.cost_bind_le p' q' m hqc).trans (Nat.add_le_add_right hpc m) |
| end MatchingFPRAS.OnlineProc |
|
|
| end OAI |
|
|
