| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Tapes.GuardBits |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineGuard |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineTrialLaw |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineGuard |
| open PhysicalStream TreeTyped TreeFraction ExecutableSource ExecutableScaling OnlineSchedule OnlineStage |
| open Transport UnitRefresh Enlargement Variation TreeTyped.Numbered |
| open scoped Classical |
|
|
| noncomputable local instance {n : ℕ} [Nontrivial (Fin n)] (w : Fin n→Fin n→ℝ) (K : ℕ) : |
| Fintype (SourceMarked w K) := Fintype.ofFinite _ |
| 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)] |
|
|
| lemma guard_many_mean (p : Trial.Parameters G) (s : FiniteTrial.ScaleState G p.K) (h : Bool→ℚ) : |
| (test (fixedData p s,p.S)).mean h= |
| qaverage (fun b=>h (decide (FiniteTrial.guardPass p s (s.drawGuard p b)))) := by |
| classical |
| have hD : 0<FiniteTrial.sourceD (G:=G) p.K := by |
| rw [FiniteTrial.sourceD_eq]; exact Enlargement.sourceD_one p.K |
| have hq : 0<copies (n:=G.n) p.K (FiniteTrial.sourceD (G:=G) p.K) := by |
| simpa only [copies,vertexCount,FiniteTrial.sourceD_eq,Trial.Parameters.q,Trial.Parameters.N,Trial.Parameters.D] using p.q_pos |
| let raw := realRawDraw s.activity p.K (FiniteTrial.sourceD (G:=G) p.K) (denominator p) |
| s.activity_pos hD (s.cap le_rfl) hq p.even (1/denominator p) |
| have hr : imageLaw raw=(s.runLaw p le_rfl).mass := by |
| rw [realRawDraw_law] |
| dsimp only [FiniteTrial.ScaleState.runLaw] |
| apply finiteLaw_parameters |
| · exact FiniteTrial.sourceD_eq p.K |
| · simp only [denominator,Nat.cast_mul,Nat.cast_ofNat] |
| have hp : imageLaw (fun bs : Fin 16→Raw s.activity p.K (FiniteTrial.sourceD (G:=G) p.K) |
| (denominator p) (1/denominator p)=>fun i=>raw (bs i))=imageLaw (s.drawGuard p) := by |
| rw [imageLaw_componentwise,hr,FiniteTrial.ScaleState.drawGuard_law] |
| rfl |
| unfold test |
| rw [Experiment.mean_map,sourceRequest_source] |
| unfold fixedRequest |
| simp only [Trial.Parameters.N] |
| rw [OnlineSample.mean_many] |
| apply Eq.trans _ (qaverage_eq_of_imageLaw_eq _ _ hp (fun x=>h (decide (FiniteTrial.guardPass p s x)))) |
| apply qaverage_congr |
| intro bs |
| apply congrArg h |
| have hcert (b : Raw s.activity p.K (FiniteTrial.sourceD (G:=G) p.K) (denominator p) (1/denominator p)) : |
| sparseCertificate (fixedData p s,OnlineSample.sampleRep |
| (fun i=>index (orderedEnum (BagEdge (RationalSource.bags s.activity p.K)) i)) b)= |
| TreeFraction.repr (s.certificate p.K (raw b)) := by |
| have hc:=sparseCertificate_source p.hn p.even s p.K (FiniteTrial.sourceD (G:=G) p.K) |
| hD (s.cap le_rfl) |
| (List.replicate (tierCount (n:=G.n) p.K (FiniteTrial.sourceD (G:=G) p.K)) false) |
| (List.replicate (copies (n:=G.n) p.K (FiniteTrial.sourceD (G:=G) p.K)) false) |
| (by simpa using hq) b.1 (fun e=>b.2 ((orderedEnum _).symm e)) |
| dsimp only at hc |
| simp only [List.map_ofFn] at hc |
| change sparseCertificate (fixedData p s,_) = _ at hc |
| rw [generated_eq_raw] at hc |
| rw [FiniteTrial.ScaleState.certificate,realRawDraw_inverse] |
| simpa only [OnlineSample.sampleRep,OnlineSample.scanRep,OnlineSample.localRep, |
| List.map_ofFn,Function.comp_def,Equiv.symm_apply_apply,NumericLadder.localBitsRep] using hc |
| apply Bool.eq_iff_iff.mpr |
| rw [decide_eq_true_eq] |
| simp only [accept,List.any_eq_true,List.mem_ofFn,FiniteTrial.guardPass] |
| constructor |
| · rintro ⟨x,⟨i,rfl⟩,hi⟩ |
| rw [hcert,positive_repr (s.certificate_range p.K _).1,decide_eq_true_eq] at hi |
| exact ⟨i,hi⟩ |
| · rintro ⟨i,hi⟩ |
| refine ⟨_,⟨i,rfl⟩,?_⟩ |
| rw [hcert,positive_repr (s.certificate_range p.K _).1,decide_eq_true_eq] |
| exact hi |
|
|
| lemma checked_mean (p : Trial.Parameters G) (x : (FiniteTrial.layers p p.K le_rfl).Ω) (h : Fraction→ℚ) : |
| (checked (OnlineTrial.represent p (FiniteTrial.layers p p.K le_rfl) x)).mean h= |
| qaverage (fun b=>h (TreeFraction.repr (FiniteTrial.guardedOutput p ⟨x,((FiniteTrial.layers p p.K le_rfl).state x).drawGuard p b⟩))) := by |
| unfold checked OnlineTrial.represent |
| rw [Experiment.mean_map,guard_many_mean] |
| apply qaverage_congr |
| intro b |
| simp only [finish,decide_eq_true_eq,FiniteTrial.guardedOutput,OnlineTrial.product,FiniteTrial.output] |
| split_ifs <;> rfl |
| |
| lemma trial_mean (p : Trial.Parameters G) (h : Fraction→ℚ) : |
| (trial (List.replicate p.K false,OnlineTrial.represent p (FiniteTrial.firstLayer p) PUnit.unit)).mean h= |
| qaverage (fun b=>h (TreeFraction.repr (FiniteTrial.guardedBitOutput p b))) := by |
| unfold trial |
| rw [Experiment.mean_bind,OnlineTrial.sweep_mean p p.K le_rfl] |
| simp_rw [checked_mean] |
| change _=qaverage ((fun c : (Fin (p.K*FiniteTrial.stageBits p)→Bool)×(Fin (FiniteTrial.guardBits p)→Bool)=> |
| h (TreeFraction.repr (FiniteTrial.guardedOutput p ⟨FiniteTrial.drawLayer p p.K le_rfl c.1, |
| ((FiniteTrial.layers p p.K le_rfl).state (FiniteTrial.drawLayer p p.K le_rfl c.1)).drawGuard p c.2⟩))) ∘ |
| pairBitsEquiv (p.K*FiniteTrial.stageBits p) (FiniteTrial.guardBits p)) |
| rw [qaverage_equiv,qaverage_prod] |
| end MatchingFPRAS.OnlineGuard |
|
|
| end OAI |
|
|
