| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.FiniteTrialLaw |
| import OAI.Combinatorics.MatchingCount.Algorithm.OutputLengthBound |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineStage |
| import OAI.Combinatorics.MatchingCount.Probability.OnlineLawBridge |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineLaw |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineStage |
| open PhysicalStream TreeTyped ExecutableSource NumericLadder TreeFraction ExecutableScaling |
| open UnitRefresh Transport Enlargement TreeTyped.Numbered ExecutableAdaptive Variation |
|
|
| 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 {n : ℕ} [Nontrivial (Fin n)] |
| variable (w : Fin n→Fin n→ℚ) (K D m : ℕ) |
| noncomputable local instance : Fintype (BagPM (RationalSource.bags w K)) := Fintype.ofFinite _ |
| noncomputable local instance : Fintype (RationalSource.Marked w K) := Fintype.ofFinite _ |
| noncomputable local instance : Fintype (SourceMarked (fun a b=>(w a b:ℝ)) K) := Fintype.ofFinite _ |
| noncomputable def realRawDraw (hw : ∀ a b,0<w a b) (hD : 0<D) |
| (hu : ∀ a b,a≠b→BottleneckAlgorithm.edge w a b≤(4:ℚ)^K) |
| (hq : 0<copies (n:=n) K D) (he : Even n) (η : ℚ) : |
| Raw w K D m η→SourceMarked (fun a b=>(w a b:ℝ)) K := |
| (RationalSource.sourceMarkedEquiv w K).symm ∘ rawDraw w K D m hw hD hu hq he η |
| lemma realRawDraw_law (hw : ∀ a b,0<w a b) (hD : 0<D) |
| (hu : ∀ a b,a≠b→BottleneckAlgorithm.edge w a b≤(4:ℚ)^K) |
| (hq : 0<copies (n:=n) K D) (he : Even n) (η : ℚ) : |
| imageLaw (realRawDraw w K D m hw hD hu hq he η)= |
| RationalSource.finiteSourceLaw w hw K D m hD Fintype.card_pos hu hq |
| (RationalSource.startMatching w K he) η := by |
| unfold realRawDraw |
| rw [imageLaw_output_equiv,rawDraw_law] |
| rfl |
| lemma realRawDraw_inverse (hw : ∀ a b,0<w a b) (hD : 0<D) |
| (hu : ∀ a b,a≠b→BottleneckAlgorithm.edge w a b≤(4:ℚ)^K) |
| (hq : 0<copies (n:=n) K D) (he : Even n) (η : ℚ) (b : Raw w K D m η) : |
| RationalSource.sourceMarkedEquiv w K (realRawDraw w K D m hw hD hu hq he η b)= |
| rawDraw w K D m hw hD hu hq he η b := |
| (RationalSource.sourceMarkedEquiv w K).apply_symm_apply _ |
|
|
| lemma finiteLaw_parameters (hw : ∀ a b,0<w a b) (D' : ℕ) (hDD : D=D') |
| (hD : 0<D) (hD' : 0<D') |
| (hu : ∀ a b,a≠b→BottleneckAlgorithm.edge w a b≤(4:ℚ)^K) |
| (hq : 0<copies (n:=n) K D) (hq' : 0<copies (n:=n) K D') |
| (M : BagPM (RationalSource.bags w K)) (η η' : ℚ) (hη : η=η') : |
| RationalSource.finiteSourceLaw w hw K D m hD Fintype.card_pos hu hq M η= |
| RationalSource.finiteSourceLaw w hw K D' m hD' Fintype.card_pos hu hq' M η' := by |
| subst D'; subst η'; rfl |
|
|
| variable {G : GraphInput} [Nontrivial (Fin G.n)] |
| noncomputable abbrev denominator (p : Trial.Parameters G) := 1000*p.K*p.S |
| noncomputable def fixedRequest {j : ℕ} (p : Trial.Parameters G) (s : FiniteTrial.ScaleState G j) : OnlineSample.Request := |
| (((choiceWidth s.activity p.K (FiniteTrial.sourceD (G:=G) p.K) (denominator p) (1/denominator p),p.N), |
| scansCount (n:=G.n) p.K (FiniteTrial.sourceD (G:=G) p.K) (denominator p)), |
| (markWidth s.activity p.K (1/denominator p), |
| List.ofFn (fun i=>index (orderedEnum (BagEdge (RationalSource.bags s.activity p.K)) i)))) |
| noncomputable def fixedData {j : ℕ} (p : Trial.Parameters G) (s : FiniteTrial.ScaleState G j) : SparseStageData := |
| data s p.K (FiniteTrial.sourceD (G:=G) p.K) |
| (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) |
| noncomputable def transition {j : ℕ} (p : Trial.Parameters G) (s : FiniteTrial.ScaleState G j) : |
| Experiment (Fraction×Vector) := execute (fixedData p s,(p.S,fixedRequest p s)) |
|
|
| lemma transition_mean {j : ℕ} (p : Trial.Parameters G) (hj : j≤p.K) |
| (s : FiniteTrial.ScaleState G j) (h : Fraction×Vector→ℚ) : |
| (transition p s).mean h= |
| qaverage (fun bits : Fin (FiniteTrial.stageBits p)→Bool=> |
| let x := RationalSource.sourceMarkedEquiv s.activity p.K ∘ s.drawStage p hj bits |
| h (repr (s.ratioEstimate x),vectorRep (s.update p.hn x).scale)) := by |
| 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 |
| unfold transition fixedData fixedRequest |
| simp only [Trial.Parameters.N] |
| rw [source_mean p.hn p.even s p.K hD (s.cap hj) _ _ (by simpa using hq)] |
| let raw := realRawDraw s.activity p.K (FiniteTrial.sourceD (G:=G) p.K) (denominator p) |
| s.activity_pos hD (s.cap hj) hq p.even (1/denominator p) |
| have hr : imageLaw raw=(s.runLaw p hj).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 p.S→Raw s.activity p.K (FiniteTrial.sourceD (G:=G) p.K) |
| (denominator p) (1/denominator p)=>fun i=>raw (bs i)) = imageLaw (s.drawStage p hj) := by |
| rw [imageLaw_componentwise,hr,FiniteTrial.ScaleState.drawStage_law] |
| rfl |
| have hmean := qaverage_eq_of_imageLaw_eq _ _ hp (fun y=> |
| let x:=RationalSource.sourceMarkedEquiv s.activity p.K ∘ y |
| h (repr (s.ratioEstimate x),vectorRep (s.update p.hn x).scale)) |
| apply Eq.trans _ hmean |
| apply qaverage_congr |
| intro bs |
| have hx : (fun i=>generatedSourceSample p.even s.activity s.activity_pos p.K |
| (FiniteTrial.sourceD (G:=G) p.K) hD (s.cap hj) |
| (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) (bs i).1 (fun e=>(bs i).2 ((orderedEnum _).symm e))) = |
| RationalSource.sourceMarkedEquiv s.activity p.K ∘ (fun i=>raw (bs i)) := by |
| funext i |
| rw [generated_eq_raw] |
| exact (realRawDraw_inverse ..).symm |
| rw [hx] |
| end MatchingFPRAS.OnlineStage |
|
|
| end OAI |
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