| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineHistorySize |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.PhysicalStream.Experiment |
| lemma all_of_qaverage {A B : Type} [Fintype B] [Nonempty B] (p : Experiment A) (f : B→A) |
| (hm : ∀ h,p.mean h=qaverage (h ∘ f)) (P : A→Prop) (hf : ∀ b,P (f b)) : p.All P := by |
| classical |
| rw [all_indicator,hm] |
| calc |
| _=qaverage (fun _ : B=>(0:ℚ)) := by |
| apply qaverage_congr |
| intro b |
| simp only [Function.comp_apply,ite_eq_left (hf b)] |
| _=0 := qaverage_const 0 |
| end MatchingFPRAS.PhysicalStream.Experiment |
|
|
| namespace MatchingFPRAS.OnlineTrial |
| open TreeTyped PhysicalStream Polynomial OnlineGraph OnlineSample |
|
|
| variable {G : GraphInput} [Nontrivial (Fin G.n)] |
| noncomputable def initial (f : Edges→Edges) (p : Trial.Parameters G) : List Bool×State := |
| (List.replicate p.K false,state f (represent p (FiniteTrial.firstLayer p) PUnit.unit)) |
| def History (f : Edges→Edges) (p : Trial.Parameters G) (k : ℕ) (s : State) : Prop := |
| ∃ (j : ℕ) (hj : j≤p.K) (x : (FiniteTrial.layers p j hj).Ω),j+k=p.K ∧ |
| s=state f (represent p (FiniteTrial.layers p j hj) x) |
| lemma history_initial (f : Edges→Edges) (p : Trial.Parameters G) : History f p p.K (initial f p).2 := by |
| exact ⟨0,by omega,PUnit.unit,by omega,rfl⟩ |
| lemma history_size (f : Edges→Edges) (p : Trial.Parameters G) (k : ℕ) (s : State) |
| (hs : History f p k s) : stateSize s≤ statePolynomial.eval (sizeNorm p (graph f (ExecutableAdaptive.graphRep G))) := by |
| obtain ⟨j,hj,x,_,rfl⟩:=hs |
| exact state_weight_bound f p j hj x |
| lemma history_step (f : Edges→Edges) (hf : ∀ es e,e∈f es↔e∈es) (p : Trial.Parameters G) |
| (k : ℕ) (s : State) (hs : History f p (k+1) s) : (step s).All (History f p k) := by |
| classical |
| obtain ⟨j,hj,x,hjk,rfl⟩:=hs |
| apply Experiment.all_of_qaverage _ |
| (fun b=>state f (represent p ((FiniteTrial.layers p j hj).append p hj) |
| ⟨x,((FiniteTrial.layers p j hj).state x).drawStage p hj b⟩)) |
| · intro h |
| rw [OnlineGraph.step_mean f hf,step_mean p hj] |
| rfl |
| · intro b |
| exact ⟨j+1,by omega,⟨x,((FiniteTrial.layers p j hj).state x).drawStage p hj b⟩,by omega,rfl⟩ |
| lemma sweep_history (f : Edges→Edges) (hf : ∀ es e,e∈f es↔e∈es) (p : Trial.Parameters G) : |
| (sweep (initial f p)).All (History f p 0) := by |
| have hall : ∀ as : List Bool,∀ s,History f p as.length s→ |
| (OnlineTree.foldM body as s).All (History f p 0) := by |
| intro as |
| induction as with |
| | nil => intro s hs;exact hs |
| | cons a as ih => |
| intro s hs |
| rw [OnlineTree.foldM,Experiment.all_bind] |
| exact (history_step f hf p as.length s hs).mono (fun v hv=>ih v hv) |
| exact hall _ _ (by simpa only [initial,List.length_replicate] using history_initial f p) |
|
|
| open TreeTyped PhysicalStream Polynomial OnlineGraph OnlineSample |
|
|
| lemma sweep_bound : ∃ q : Polynomial ℕ,∀ (G : GraphInput) [Nontrivial (Fin G.n)] |
| (f : Edges→Edges) (_hf : ∀ es e,e∈f es↔e∈es) (p : Trial.Parameters G), |
| sweepRealizer.cost (initial f p)+(sweep (initial f p)).worst weight≤ |
| q.eval (sizeNorm p (graph f (ExecutableAdaptive.graphRep G))) := by |
| obtain ⟨q,hq⟩:=work_body |
| let C := q.comp (2*statePolynomial+2) |
| let M : Polynomial ℕ := 2*X+1 |
| refine ⟨X*(C+14*M+34)+10+statePolynomial,?_⟩ |
| intro G inst f hf p |
| let n:=sizeNorm p (graph f (ExecutableAdaptive.graphRep G)) |
| have hk : p.K ≤ n := (sizeNorm_fields p _).2.1 |
| have hw : weight (initial f p).1≤M.eval n := by |
| simp only [initial,weight_replicate,show weight false=1 from rfl,M,Polynomial.eval_add, |
| Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X,Polynomial.eval_one] |
| omega |
| have hc : ∀ a∈(initial f p).1,∀ k<(initial f p).1.length,∀ s,History f p (k+1) s→ |
| bodyRealizer.cost (a,s)≤C.eval n := by |
| intro a ha k hk s hs |
| have hsz : stateSize s ≤ statePolynomial.eval n := history_size f p (k+1) s hs |
| have hw:=weight_le_stateSize s |
| have hb:=hq (a,s) |
| have ha' : weight a=1 := by |
| have he : a=false := (List.mem_replicate.mp ha).2 |
| rw [he];rfl |
| simp only [weight_prod,ha'] at hb |
| have hm : stateSize s+(1+1+weight s)≤(2*statePolynomial+2).eval n := by |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat] |
| omega |
| exact (Nat.le_add_right _ _).trans (hb.trans (by simpa only [C,Polynomial.eval_comp] using polynomial_eval_monotone q hm)) |
| have hh:=OnlineTree.fold_cost_bound body bodyRealizer.cost (initial f p).1 (initial f p).2 |
| (History f p) (C.eval n) (M.eval n) |
| (by simpa only [initial,List.length_replicate] using history_initial f p) hw |
| (fun a ha k hk s hs=>history_step f hf p k s hs) hc |
| have hout : (sweep (initial f p)).worst weight≤ statePolynomial.eval n := by |
| apply Experiment.worst_le _ _ _ |
| exact hh.2.mono (fun s hs=>(weight_le_stateSize s).trans (history_size f p 0 s hs)) |
| have hmul:=Nat.mul_le_mul_right (C.eval n+14*M.eval n+34) hk |
| change OnlineTree.Fold.cost body bodyRealizer.cost (initial f p).1 (initial f p).2+_≤_ |
| have hlen : (initial f p).1.length=p.K := List.length_replicate |
| rw [hlen] at hh |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X] |
| change _ ≤ n*(C.eval n+14*M.eval n+34)+10+statePolynomial.eval n |
| omega |
| end MatchingFPRAS.OnlineTrial |
|
|
| end OAI |
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