File size: 3,999 Bytes
e5a97c6 | 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 | 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 C D : Type} [Codec A] [Codec B] [Codec C] [Codec D]
namespace Second
open OnlineControl
inductive Point where | split | save | body | pop | cons | halt
deriving DecidableEq, Inhabited
instance : Fintype Point := ⟨{.split,.save,.body,.pop,.cons,.halt},by intro p; cases p <;> simp⟩
def code (P : OnlineProc.Proc (K Reg)) : Point → OnlineControl.Instr Reg Point
| .split => .call (OnlineProc.deterministic (TreeFunction.split 1 0 (by decide))) .save
| .save => .call (OnlineProc.deterministic (TreeFunction.cons 1 2 (by decide))) .body
| .body => .call P .pop
| .pop => .call (OnlineProc.deterministic (TreeFunction.split 1 2 (by decide))) .cons
| .cons => .call (OnlineProc.deterministic (TreeFunction.cons 1 0 (by decide))) .halt
| .halt => .halt
lemma runs {f : B → Experiment C} (F : Realizer f) (a : A) (b : B) (s : Data) :
∃ q,At (code F.program) (fun ac : A×C=>env (encode ac) .nil s) (Option.some Point.split)
(env (encode (a,b)) .nil s) q ∧
(∀ g : A×C → ℚ,q.mean g=(f b).mean (fun c=>g (a,c))) ∧
q.cost ≤ F.cost b+8*TreeFunction.size (encode a)+22 := by
let M:=code F.program
let out:=fun ac : A×C=>env (encode ac) .nil s
have hend c:=OnlineControl.halt M out Point.halt rfl (a,c)
have hcons c:=deterministicCall M out Point.cons Point.halt
(TreeFunction.cons 1 0 (by decide)) rfl _ _ _ (cons10 (encode c) (encode a) s) _ (hend c)
choose pc hc hmc hcc using hcons
have hpop c:=deterministicCall M out Point.pop Point.cons
(TreeFunction.split 1 2 (by decide)) rfl _ _ _ (split12 (encode c) .nil (encode a) s) _ (hc c)
choose pp hp hmp hcp using hpop
obtain ⟨r,hr,hmr,hcr⟩:=F.correct b (.cons (encode a) s)
have hbody:=call M out Point.body Point.pop F.program rfl
(env (encode b) .nil (.cons (encode a) s)) (fun c=>env (encode c) .nil (.cons (encode a) s))
r hr pp hp
obtain ⟨p2,hp2,hm2,hc2⟩:=deterministicCall M out Point.save Point.body
(TreeFunction.cons 1 2 (by decide)) rfl _ _ _ (cons12 (encode b) (encode a) s) _ hbody
obtain ⟨p1,hp1,hm1,hc1⟩:=deterministicCall M out Point.split Point.save
(TreeFunction.split 1 0 (by decide)) rfl _ _ _ (split10 (encode a) (encode b) .nil s) _ hp2
refine ⟨p1,hp1,?_,?_⟩
· intro g
rw [hm1,hm2,Experiment.mean,Experiment.mean_bind]
simp_rw [hmp,hmc,Experiment.mean]
exact hmr _
· have hb : ∀ c,(pp c).cost ≤ 4*TreeFunction.size (encode a)+11 := by
intro c
have h1:=hcc c
have h2:=hcp c
simp only [Experiment.cost,size_nil] at h1 h2
omega
have hbind:=Experiment.cost_bind_le r pp _ hb
simp only [Experiment.cost,size_nil] at hc1 hc2
omega
end Second
noncomputable def mapSecond {f : B → Experiment C} (F : Realizer f) :
Realizer (fun ab : A×B=>(f ab.2).map (Prod.mk ab.1)) where
program := OnlineControl.compile (Second.code F.program)
cost := fun ab=>F.cost ab.2+8*TreeFunction.size (encode ab.1)+22
correct := by
intro ab s
obtain ⟨q,hq,hm,hc⟩:=Second.runs F ab.1 ab.2 s
exact ⟨q,hq,fun g=>by rw [Experiment.mean_map]; exact hm g,hc⟩
noncomputable def pair {f : A → Experiment B} {g : A → Experiment C}
(F : Realizer f) (G : Realizer g) :
Realizer (fun a=>(f a).bind (fun b=>(g a).map (Prod.mk b))) :=
(composition (deterministic (TreeTyped.pair (TreeTyped.identity (A:=A)) TreeTyped.identity))
(composition (mapSecond F)
(composition (deterministic (TreeTyped.pair (TreeTyped.second (A:=A) (B:=B)) TreeTyped.first))
(mapSecond G)))).congr (by
intro a h
simp only [Experiment.mean_bind,Experiment.mean,Experiment.mean_map,id_eq])
end MatchingFPRAS.OnlineTree
end OAI
|