| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.TreeGeneratedUpdate |
| import OAI.Combinatorics.MatchingCount.Probability.OnlineSampleLaw |
|
|
| 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) |
| abbrev vertexCount := Fintype.card (SourceVertex (Fin n) K) |
| abbrev labelCount := Fintype.card (RationalSource.Label w K) |
| abbrev edgeCount := Fintype.card (BagEdge (RationalSource.bags w K)) |
| abbrev tierCount := Transport.endpoint (vertexCount (n:=n) K) D K Fintype.card_pos |
| abbrev copies := ladderQ (vertexCount (n:=n) K) D |
| abbrev scansCount := discreteLadderSteps (vertexCount (n:=n) K) D K (tierCount (n:=n) K D) m |
| abbrev choiceWidth (η : ℚ) := bitPrecision (scansCount (n:=n) K D m * |
| scanErrorNumerator (SourceVertex (Fin n) K) (RationalSource.Label w K) (RationalSource.bags w K) |
| (tierCount (n:=n) K D) (copies (n:=n) K D)) η |
| abbrev markWidth (η : ℚ) := bitPrecision (6*edgeCount w K) η |
| abbrev Raw (η : ℚ) := OnlineSample.Random (vertexCount (n:=n) K) (choiceWidth w K D m η) |
| (scansCount (n:=n) K D m) (edgeCount w K) (markWidth w K η) |
|
|
| def rawEquiv (η : ℚ) : |
| ((Fin (scansCount (n:=n) K D m*scanBitCount (vertexCount (n:=n) K) (choiceWidth w K D m η))→Bool) × |
| (BagEdge (RationalSource.bags w K)→Fin (markWidth w K η)→Bool)) ≃ Raw w K D m η := |
| Equiv.prodCongr (runBitsEquiv (SourceVertex (Fin n) K) (choiceWidth w K D m η) (scansCount (n:=n) K D m)) |
| (Equiv.arrowCongr (orderedEnum (BagEdge (RationalSource.bags w K))).symm (Equiv.refl _)) |
| lemma raw_scan (η : ℚ) (bs : Raw w K D m η) : |
| runBitsEquiv (SourceVertex (Fin n) K) (choiceWidth w K D m η) (scansCount (n:=n) K D m) |
| ((rawEquiv w K D m η).symm bs).1=bs.1 := |
| congrArg Prod.fst ((rawEquiv w K D m η).apply_symm_apply bs) |
| lemma raw_marks (η : ℚ) (bs : Raw w K D m η) (e : BagEdge (RationalSource.bags w K)) : |
| ((rawEquiv w K D m η).symm bs).2 e=bs.2 ((orderedEnum _).symm e) := rfl |
| noncomputable def rawDraw (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→RationalSource.Marked w K := |
| RationalSource.markedSampler w hw K D m hD Fintype.card_pos hu hq |
| (RationalSource.startMatching w K he) η ∘ (rawEquiv w K D m η).symm |
|
|
| noncomputable def core (hw : ∀ a b,0<w a b) (hD : 0<D) |
| (hu : ∀ a b,a≠b→BottleneckAlgorithm.edge w a b≤(4:ℚ)^K) (he : Even n) |
| (T q : ℕ) (hq : 0<q) {k r km : ℕ} |
| (bits : Fin r→ScanBits (SourceVertex (Fin n) K) k) |
| (mb : BagEdge (RationalSource.bags w K)→Fin km→Bool) : RationalSource.Marked w K := |
| TableMarking.reveal (BagEdge.Incident (bags:=RationalSource.bags w K)) |
| (ladderRun (RationalSource.bags_adjacent w K hu) (RationalSource.tree_acyclic w K) |
| (fun l : RationalSource.Label w K=>l.1.val) (RationalSource.parent w K) |
| (RationalSource.bags_ranked w K hu) (RationalSource.bags_nonempty w K) |
| (RationalSource.parent_rank w K) hq (RationalSource.bagWeight w K D) |
| (RationalSource.startMatching w K he) k r (fun _=>RationalSource.startMatching w K he) bits |
| (topSlot T q hq)) |
| (fun e=>(RationalSource.markingTable w hw K D hD e).draw km (mb e)) |
| lemma generated_eq_core (hw : ∀ a b,0<w a b) (hD : 0<D) |
| (hu : ∀ a b,a≠b→BottleneckAlgorithm.edge w a b≤(4:ℚ)^K) (he : Even n) |
| (ft fq : List Bool) (hq : 0<fq.length) {k r km : ℕ} |
| (bits : Fin r→ScanBits (SourceVertex (Fin n) K) k) |
| (mb : BagEdge (RationalSource.bags w K)→Fin km→Bool) : |
| generatedSourceSample he w hw K D hD hu ft fq hq bits mb= |
| core w K D hw hD hu he ft.length fq.length hq bits mb := rfl |
|
|
| lemma generated_eq_raw (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) (bs : Raw w K D m η) : |
| generatedSourceSample he w hw K D hD hu |
| (List.replicate (tierCount (n:=n) K D) false) (List.replicate (copies (n:=n) K D) false) |
| (by simpa only [List.length_replicate] using hq) bs.1 |
| (fun e=>bs.2 ((orderedEnum (BagEdge (RationalSource.bags w K))).symm e)) = |
| rawDraw w K D m hw hD hu hq he η bs := by |
| rw [generated_eq_core] |
| simp only [List.length_replicate] |
| unfold core rawDraw RationalSource.markedSampler TableMarking.draw RationalSource.topSampler |
| RationalSource.sampler finiteLadder ladderBitRun |
| simp only [Function.comp_apply,raw_scan,raw_marks] |
|
|
| noncomputable local instance : Fintype (BagPM (RationalSource.bags w K)) := Fintype.ofFinite _ |
| noncomputable local instance : Fintype (RationalSource.Marked w K) := Fintype.ofFinite _ |
| lemma rawDraw_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 (rawDraw w K D m hw hD hu hq he η)= |
| RationalSource.markedSamplerLaw w hw K D m hD Fintype.card_pos hu hq |
| (RationalSource.startMatching w K he) η := by |
| unfold rawDraw RationalSource.markedSamplerLaw |
| apply Eq.trans (imageLaw_comp_equiv (rawEquiv w K D m η).symm _) |
| exact imageLaw_instances _ _ _ |
| end MatchingFPRAS.OnlineStage |
|
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| end OAI |
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