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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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Algorithm.FiniteStage
import OAI.Combinatorics.MatchingCount.Algorithm.RationalScale
import OAI.Combinatorics.MatchingCount.Algorithm.RationalStart
namespace OAI
namespace MatchingFPRAS.FiniteTrial
open Finset Switching Transport Enlargement Adaptive
open scoped Classical
variable {G : GraphInput} [Nontrivial (Fin G.n)]
attribute [local instance] NumberedHierarchy.labelOrder sym2Order bagEdgeOrder bagEdgeFintype
attribute [local instance] UnitRefresh.Computable.allowedDec prodOrder sumOrder
noncomputable local instance finiteTrialLawSourceLabelDecidableEq (w : Fin G.n → Fin G.n → ℚ) (K : ℕ) :
DecidableEq (SourceLabel (fun x y=>(w x y:ℝ)) K) := Classical.decEq _
noncomputable local instance finiteTrialLawSourceBagEdgeFintype (w : Fin G.n → Fin G.n → ℚ) (K : ℕ) :
Fintype (BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K)) := Fintype.ofFinite _
noncomputable local instance (w : Fin G.n → Fin G.n → ℚ) (K : ℕ) :
Fintype (SourceMarked (fun x y=>(w x y:ℝ)) K) := Fintype.ofFinite _
omit [Nontrivial (Fin G.n)] in
lemma ScaleState.cap {j K : ℕ} (s : ScaleState G j) (hj : j≤K) (a b : Fin G.n) (hab : a≠b) :
BottleneckAlgorithm.edge s.activity a b≤(4:ℚ)^K := by
have h:=s.toReal.cap hj a b hab
rw [←s.activity_cast] at h
have he : ((BottleneckAlgorithm.edge s.activity a b:ℚ):ℝ)=edgeActivity (fun a b=>(s.activity a b:ℝ)) a b := by
unfold BottleneckAlgorithm.edge edgeActivity
split_ifs <;> rfl
rw [←he] at h
exact_mod_cast h
noncomputable def ScaleState.runLaw (p : Trial.Parameters G) {j : ℕ} (hj : j≤p.K) (s : ScaleState G j) :
Law (SourceMarked (fun a b=>(s.activity a b:ℝ)) p.K) where
mass := RationalSource.finiteSourceLaw s.activity s.activity_pos p.K
(Enlargement.sourceD (V:=Fin G.n) p.K) (1000*p.K*p.S)
(sourceD_one p.K) Fintype.card_pos (s.cap hj) p.q_pos
(RationalSource.startMatching s.activity p.K p.even) (1/(1000*p.K*p.S))
nonneg := RationalSource.finiteSourceLaw_nonneg _ _ _ _ _ _ _ _ _ _ _
total := RationalSource.finiteSourceLaw_sum _ _ _ _ _ _ _ _ _ _ _
noncomputable def ScaleState.sampleLaw (p : Trial.Parameters G) {j : ℕ} (hj : j≤p.K) (s : ScaleState G j) :
Law (Fin p.S → SourceMarked (fun a b=>(s.activity a b:ℝ)) p.K) := (s.runLaw p hj).iid p.S
theorem ScaleState.sampleLaw_budget (p : Trial.Parameters G) {j : ℕ} (hj : j≤p.K) (s : ScaleState G j) :
(s.sampleLaw p hj).event
(fun x=>¬StageGood (fun a b=>(s.activity a b:ℝ)) (fun a b=>(s.nextActivity a b:ℝ)) p.K p.sigma x)
<13/(1000*p.K) := by
have h:=RationalSource.finite_source_stage_budget s.activity s.activity_pos
(fun a b=>(s.nextActivity a b:ℝ))
(fun a b=>by exact_mod_cast (s.nextActivity_pos a b).le)
(fun a b _=>by have h:=(s.toReal.cooling a b).2; rw [←s.activity_cast,←s.nextActivity_cast] at h; exact h)
(by rw [s.activity_cast]; exact s.toReal.partition_pos p.even)
p.K p.K p.S p.K_pos p.S_pos p.sigma p.sigma_pos.le (by simpa using p.S_bound)
(s.cap hj) p.q_pos (RationalSource.startMatching s.activity p.K p.even)
convert h using 1
unfold ScaleState.sampleLaw ScaleState.runLaw Law.event Law.iid Variation.event
rfl
end MatchingFPRAS.FiniteTrial
end OAI
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