| 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 |
| |
