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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.Algorithm.OnlineTrial
import OAI.Combinatorics.MatchingCount.Algorithm.OnlineRequestLaw
import OAI.Combinatorics.MatchingCount.Probability.OnlineLayerLaw
namespace OAI
namespace MatchingFPRAS.OnlineTree
open PhysicalStream
lemma foldM_append_mean {A B : Type} (f : A×B→Experiment B) (as bs : List A) (b : B) (h : B→ℚ) :
(foldM f (as++bs) b).mean h=(foldM f as b).mean (fun c=>(foldM f bs c).mean h) := by
induction as generalizing b with
| nil => rfl
| cons a as ih => simp only [List.cons_append,foldM,Experiment.mean_bind,ih]
end MatchingFPRAS.OnlineTree
namespace MatchingFPRAS.OnlineTrial
open PhysicalStream TreeTyped TreeFraction ExecutableSource ExecutableScaling OnlineSchedule OnlineStage
open Transport UnitRefresh Enlargement
attribute [local instance] NumberedHierarchy.labelOrder
attribute [local instance] Transport.sym2Order Transport.bagEdgeOrder Transport.bagEdgeFintype
attribute [local instance] UnitRefresh.Computable.allowedDec Transport.prodOrder Transport.sumOrder
variable {G : GraphInput} [Nontrivial (Fin G.n)]
noncomputable def represent {j : ℕ} (p : Trial.Parameters G) (L : FiniteTrial.Layer p j) (x : L.Ω) : State :=
((fixedData p (L.state x),p.S),repr (product p L x))
lemma advance_source {j : ℕ} (p : Trial.Parameters G) (s : FiniteTrial.ScaleState G j)
(xs : Fin p.S→RationalSource.Marked s.activity p.K) :
advance (fixedData p s,vectorRep (s.update p.hn xs).scale)=fixedData p (s.update p.hn xs) := by
unfold advance fixedData data
simp only [List.replicate_succ]
lemma carry_represent {j : ℕ} (p : Trial.Parameters G) (hj : j≤p.K) (L : FiniteTrial.Layer p j)
(x : (L.append p hj).Ω) :
carry (represent p L x.1,
(repr ((L.state x.1).ratioEstimate (RationalSource.sourceMarkedEquiv (L.state x.1).activity p.K ∘ x.2)),
vectorRep ((L.state x.1).update p.hn (RationalSource.sourceMarkedEquiv (L.state x.1).activity p.K ∘ x.2)).scale))=
represent p (L.append p hj) x := by
unfold carry represent
dsimp only
rw [advance_source,mul_repr (product_nonneg p L _) ((L.state x.1).ratioEstimate_nonneg x.2),←product_append]
rfl
lemma step_mean {j : ℕ} (p : Trial.Parameters G) (hj : j≤p.K) (L : FiniteTrial.Layer p j)
(x : L.Ω) (h : State→ℚ) :
(step (represent p L x)).mean h=
qaverage (fun b=>h (represent p (L.append p hj) ⟨x,(L.state x).drawStage p hj b⟩)) := by
unfold step
rw [Experiment.mean_map]
have hp : prepare (represent p L x).1 = (fixedData p (L.state x),(p.S,fixedRequest p (L.state x))) := by
change (fixedData p (L.state x),(p.S,sourceRequest (fixedData p (L.state x),p.S)))=_
rw [sourceRequest_source]
rw [hp]
change (transition p (L.state x)).mean (fun y=>h (carry (represent p L x,y)))=_
rw [transition_mean p hj]
apply qaverage_congr
intro b
have hc := carry_represent p hj L ⟨x,(L.state x).drawStage p hj b⟩
dsimp only at hc ⊢
exact congrArg h hc
lemma sweep_mean (p : Trial.Parameters G) (j : ℕ) (hj : j≤p.K) (h : State→ℚ) :
(sweep (List.replicate j false,represent p (FiniteTrial.firstLayer p) PUnit.unit)).mean h=
qaverage (fun b=>h (represent p (FiniteTrial.layers p j hj) (FiniteTrial.drawLayer p j hj b))) := by
induction j generalizing h with
| zero =>
change h (represent p (FiniteTrial.firstLayer p) PUnit.unit)=qaverage (fun _=>h (represent p (FiniteTrial.firstLayer p) PUnit.unit))
rw [qaverage_const]
| succ j ih =>
change (OnlineTree.foldM body (List.replicate (j+1) false) _).mean h=_
rw [List.replicate_add,OnlineTree.foldM_append_mean]
simp only [List.replicate_one,OnlineTree.foldM,body,Experiment.mean_bind,Experiment.mean]
change (sweep (List.replicate j false,represent p (FiniteTrial.firstLayer p) PUnit.unit)).mean (fun x=>(step x).mean h)=_
rw [ih (by omega)]
rw [mean_layer_succ p hj (fun x=>h (represent p (FiniteTrial.layers p (j+1) hj) x))]
apply qaverage_congr
intro b
exact step_mean p (by omega) (FiniteTrial.layers p j (by omega)) (FiniteTrial.drawLayer p j (by omega) b) h
end MatchingFPRAS.OnlineTrial
end OAI
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