| 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 |
|
|
