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e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | 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
end OAI
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