| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineInitial |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineInitial |
| open TreeTyped TreeFraction ExecutableAdaptive ExecutableScaling ExecutableSource OnlineSchedule |
| open Transport Enlargement |
|
|
| variable {G : GraphInput} [Nontrivial (Fin G.n)] |
| variable (hn : 2≤G.n) (heven : Even G.n) (ε δ : ℚ) (he : 0<ε) (he1 : ε<1) |
| local notation "p" => FiniteTrial.parameters G hn heven ε he he1 |
| noncomputable def sourceSeed : Seed := (graphRep G,(TreeFraction.repr ε,TreeFraction.repr δ)) |
| lemma config_source : config (sourceSeed (G:=G) ε δ)= |
| ((sourceSeed (G:=G) ε δ,(p).K),((p).N,FiniteTrial.sourceD (G:=G) (p).K)) := by |
| have hl : (graphRep G).1.length=G.n := by simp [graphRep] |
| unfold config seedK seedN seedD sourceSeed |
| dsimp only |
| rw [hl,coolingSchedule_eq G.n hn ε he he1] |
| change ((_,(p).K),(vertexNumber (G.n,(p).K),distortion (G.n,vertexNumber (G.n,(p).K))))=_ |
| rw [distortion_source,vertexNumber_eq] |
| rfl |
| lemma data_source : data (config (sourceSeed (G:=G) ε δ))= |
| OnlineStage.fixedData p (FiniteTrial.initialScaleState G hn) := by |
| rw [config_source hn heven ε δ he he1] |
| unfold data sourceSeed OnlineStage.fixedData OnlineStage.data |
| dsimp only |
| rw [show (graphRep G).1=List.range G.n from rfl,scaleVector_source G hn,ladderSchedule_eq _ _ _ (p).N_pos] |
| rfl |
| lemma count_source : count (config (sourceSeed (G:=G) ε δ))=(p).S := by |
| rw [config_source hn heven ε δ he he1] |
| unfold count sourceSeed |
| dsimp only |
| have hl : (graphRep G).1.length=G.n := by simp [graphRep] |
| rw [hl] |
| exact sampleCount_source p |
| lemma trialInput_source : trialInput (config (sourceSeed (G:=G) ε δ))=OnlineAmplification.initial p := by |
| unfold trialInput |
| rw [data_source hn heven ε δ he he1,count_source hn heven ε δ he he1,config_source hn heven ε δ he he1] |
| unfold OnlineAmplification.initial OnlineTrial.represent OnlineTrial.product |
| simp only [FiniteTrial.firstLayer,Fin.prod_univ_zero,mul_one] |
| have hg : (sourceSeed (G:=G) ε δ).1.1.length=G.n := by simp [sourceSeed,graphRep] |
| rw [hg,GraphBridge.complete_count] |
| rw [TreeFraction.repr_nat] |
| lemma initializeInput_source (hd : 0≤δ) : initializeInput (sourceSeed (G:=G) ε δ)= |
| (2*Amplification.halfRepetitions δ+1,OnlineAmplification.initial p) := by |
| unfold initializeInput runInput |
| rw [trialInput_source hn heven ε δ he he1,config_source hn heven ε δ he he1] |
| dsimp only [sourceSeed] |
| rw [repetitions_repr δ hd] |
| end MatchingFPRAS.OnlineInitial |
|
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| end OAI |
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