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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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | 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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