| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Machines.OnlineTree |
| import OAI.Combinatorics.MatchingCount.Machines.OnlineControl |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineTree |
| open TreeData TreeMachine RegisterProc TreeFunction TreeTyped |
| open Turing MachineCost PhysicalStream OnlineStack |
|
|
| variable {A B : Type} [Codec A] [Codec B] |
| namespace Case |
| open OnlineControl |
|
|
| inductive Point where | split | test | yesClear | noClear | yes | no | halt |
| deriving DecidableEq, Inhabited |
| instance : Fintype Point := ⟨{.split,.test,.yesClear,.noClear,.yes,.no,.halt},by intro p;cases p <;>simp⟩ |
| def code (P Q : OnlineProc.Proc (K Reg)) : Point→OnlineControl.Instr Reg Point |
| | .split => .call (OnlineProc.deterministic (TreeFunction.split 1 0 (by decide))) .test |
| | .test => .branch 1 .noClear .yesClear |
| | .yesClear => .call (OnlineProc.deterministic (TreeFunction.clear 1)) .yes |
| | .noClear => .call (OnlineProc.deterministic (TreeFunction.clear 1)) .no |
| | .yes => .call P .halt |
| | .no => .call Q .halt |
| | .halt => .halt |
| lemma runs {f g : A→Experiment B} (F : Realizer f) (G : Realizer g) (b : Bool) (a : A) (s : Data) : |
| ∃ q,At (code F.program G.program) (fun y=>env (encode y) .nil s) (Option.some Point.split) |
| (env (encode (b,a)) .nil s) q ∧ |
| (∀ h : B→ℚ,q.mean h=(if b then f a else g a).mean h) ∧ |
| q.cost≤(if b then F.cost a else G.cost a)+30 := by |
| let M:=code F.program G.program |
| let out:=fun y : B=>env (encode y) .nil s |
| have hend y:=OnlineControl.halt M out Point.halt rfl y |
| cases b with |
| | false => |
| obtain ⟨r,hr,hm,hc⟩:=G.correct a s |
| have hb:=call M out Point.no Point.halt G.program rfl (env (encode a) .nil s) |
| (fun y=>env (encode y) .nil s) r hr (fun y=>.delay 1 (.done y)) hend |
| obtain ⟨q,hq,hqm,hqc⟩:=deterministicCall M out Point.noClear Point.no |
| (TreeFunction.clear 1) rfl _ _ _ (clear1 (encode a) (encode false) s) _ hb |
| have ht:=nilBranch M out Point.test Point.noClear Point.yesClear 1 rfl _ rfl _ hq |
| obtain ⟨v,hv,hvm,hvc⟩:=deterministicCall M out Point.split Point.test |
| (TreeFunction.split 1 0 (by decide)) rfl _ _ _ (split10 (encode false) (encode a) .nil s) _ ht |
| refine ⟨v,hv,?_,?_⟩ |
| · intro h |
| rw [hvm,Experiment.mean,hqm,Experiment.mean,Experiment.mean_bind] |
| exact hm h |
| · have hh:=Experiment.cost_bind_le r (fun y=>.delay 1 (.done y)) 1 (fun _=>by rfl) |
| simp only [Experiment.cost] at hqc hvc |
| change v.cost≤G.cost a+30 |
| change q.cost≤1+2+1+(1+(r.bind (fun y=>.delay 1 (.done y))).cost) at hqc |
| change v.cost≤1+2*1+4+1+(1+q.cost) at hvc |
| omega |
| | true => |
| obtain ⟨r,hr,hm,hc⟩:=F.correct a s |
| have hb:=call M out Point.yes Point.halt F.program rfl (env (encode a) .nil s) |
| (fun y=>env (encode y) .nil s) r hr (fun y=>.delay 1 (.done y)) hend |
| obtain ⟨q,hq,hqm,hqc⟩:=deterministicCall M out Point.yesClear Point.yes |
| (TreeFunction.clear 1) rfl _ _ _ (clear1 (encode a) (encode true) s) _ hb |
| have ht:=consBranch M out Point.test Point.noClear Point.yesClear 1 rfl _ .nil .nil rfl _ hq |
| obtain ⟨v,hv,hvm,hvc⟩:=deterministicCall M out Point.split Point.test |
| (TreeFunction.split 1 0 (by decide)) rfl _ _ _ (split10 (encode true) (encode a) .nil s) _ ht |
| refine ⟨v,hv,?_,?_⟩ |
| · intro h |
| rw [hvm,Experiment.mean,hqm,Experiment.mean,Experiment.mean_bind] |
| exact hm h |
| · have hh:=Experiment.cost_bind_le r (fun y=>.delay 1 (.done y)) 1 (fun _=>by rfl) |
| simp only [Experiment.cost] at hqc hvc |
| change v.cost≤F.cost a+30 |
| change q.cost≤3+2+1+(1+(r.bind (fun y=>.delay 1 (.done y))).cost) at hqc |
| change v.cost≤1+2*3+4+1+(1+q.cost) at hvc |
| omega |
| end Case |
| noncomputable def caseBool {f g : A→Experiment B} (F : Realizer f) (G : Realizer g) : |
| Realizer (fun x : Bool×A=>if x.1 then f x.2 else g x.2) where |
| program := OnlineControl.compile (Case.code F.program G.program) |
| cost := fun x=>(if x.1 then F.cost x.2 else G.cost x.2)+30 |
| correct := fun x s=>Case.runs F G x.1 x.2 s |
| noncomputable def conditional {p : A→Bool} {f g : A→Experiment B} (P : TreeTyped.Realizer p) |
| (F : Realizer f) (G : Realizer g) : Realizer (fun a=>if p a then f a else g a) := |
| (composition (deterministic (TreeTyped.pair P TreeTyped.identity)) (caseBool F G)).congr (fun _ _=>rfl) |
| end MatchingFPRAS.OnlineTree |
|
|
| end OAI |
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|
