| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineLadderSchedule |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineSampleCount |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineAmplification |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineInitial |
| open TreeTyped TreeFraction ExecutableAdaptive ExecutableScaling ExecutableSource OnlineSchedule |
|
|
| abbrev Seed := GraphData×(Fraction×Fraction) |
| abbrev Config := (Seed×ℕ)×(ℕ×ℕ) |
|
|
| def seedK (x : Seed) : Seed×ℕ := (x,coolingSchedule (x.1.1.length,x.2.1)) |
| def seedKRealizer : Realizer seedK := |
| (pair identity (composition (g:=coolingSchedule) (pair (composition first.fst listLength) second.fst) coolingScheduleRealizer)).congr (fun _=>rfl) |
| def seedN (x : Seed×ℕ) : (Seed×ℕ)×ℕ := (x,vertexNumber (x.1.1.1.length,x.2)) |
| def seedNRealizer : Realizer seedN := |
| (pair identity (composition (g:=vertexNumber) (pair (composition first.fst.fst listLength) second) vertexNumberRealizer)).congr (fun _=>rfl) |
| def seedD (x : (Seed×ℕ)×ℕ) : Config := |
| (x.1,(x.2,distortion (x.1.1.1.1.length,x.2))) |
| def seedDRealizer : Realizer seedD := |
| pair first (pair second (composition (pair (composition first.fst.fst.fst listLength) second) distortionRealizer)) |
| def config (x : Seed) : Config := seedD (seedN (seedK x)) |
| def configRealizer : Realizer config := composition (composition seedKRealizer seedNRealizer) seedDRealizer |
|
|
| def root (n : ℕ) : ℕ := findRoot (n-1,List.replicate n false) |
| def rootRealizer : Realizer root := |
| composition (pair (composition (pair identity (constant 1)) subtract) (composition identity fuelRealizer)) findRootRealizer |
| lemma root_eq (n : ℕ) : root n=(n-1).sqrt := by |
| apply findRoot_eq |
| simpa using (Nat.sqrt_le_self (n-1)).trans (Nat.sub_le n 1) |
| def initialScale (n : ℕ) : Fraction := TreeFraction.div ((1,1),(root n,1)) |
| def initialScaleRealizer : Realizer initialScale := |
| composition (pair (constant (1,1)) (pair rootRealizer (constant 1))) divRealizer |
| lemma initialScale_repr (n : ℕ) : initialScale n=TreeFraction.repr (Initializer.initialScale n) := by |
| unfold initialScale |
| rw [root_eq,←repr_nat ((n-1).sqrt),show ((1,1) : Fraction)=TreeFraction.repr (1:ℚ) from rfl,div_repr (by decide) (by positivity)] |
| rfl |
|
|
| def scaleVector (vs : List ℕ) : Vector := vs.map (fun i=>(i,initialScale vs.length)) |
| def scaleVectorRealizer : Realizer scaleVector := |
| (composition (pair (composition listLength initialScaleRealizer) identity) |
| (mapWith (pair second first))).congr (fun _=>rfl) |
| lemma scaleVector_source (G : GraphInput) (hn : 2≤G.n) : |
| scaleVector (List.range G.n)=vectorRep (FiniteTrial.initialScaleState G hn).scale := by |
| unfold scaleVector vectorRep |
| rw [List.length_range,initialScale_repr] |
| apply List.ext_getElem |
| · simp |
| · intro _ _ _ |
| simp [FiniteTrial.initialScaleState] |
|
|
| def nRealizer : Realizer (fun x : Config=>x.1.1.1.1.length) := composition first.fst.fst.fst listLength |
|
|
| def count (x : Config) : ℕ := sampleCount (x.1.1.1.1.length,((x.2.2,x.1.2),x.1.1.2.1)) |
| def countRealizer : Realizer count := |
| composition (pair nRealizer (pair (pair second.snd first.snd) first.fst.snd.fst)) sampleCountRealizer |
|
|
| def data (x : Config) : SparseStageData := |
| (((x.1.1.1,[]),(((List.replicate (2*x.1.2+2) false,List.replicate (2*x.1.2) false),x.2.2), |
| (List.replicate (ladderSchedule (x.2,x.1.2)) false,List.replicate (Transport.ladderQ x.2.1 x.2.2) false))), |
| scaleVector x.1.1.1.1) |
| def dataRealizer : Realizer data := |
| (pair (pair (pair first.fst.fst (constant [])) |
| (pair (pair (pair (composition (g:=fun n=>List.replicate n false) (((constant 2).mulNat first.snd).addNat (constant 2)) fuelRealizer) |
| (composition ((constant 2).mulNat first.snd) fuelRealizer)) second.snd) |
| (pair (composition (composition (pair second first.snd) ladderScheduleRealizer) fuelRealizer) |
| (composition (composition second qRealizer) fuelRealizer)))) |
| (composition first.fst.fst.fst scaleVectorRealizer)).congr (fun _=>rfl) |
|
|
| def trialInput (x : Config) : OnlineAmplification.TrialInput := |
| (List.replicate x.1.2 false,((data x,count x),(UnitRefresh.pairingCount x.1.1.1.1.length,1))) |
| def trialInputRealizer : Realizer trialInput := |
| pair (composition first.snd fuelRealizer) |
| (pair (pair dataRealizer countRealizer) (pair (composition nRealizer pairingNumber) (constant 1))) |
| def runInput (x : Config) : ℕ×OnlineAmplification.TrialInput := |
| (repetitions x.1.1.2.2,trialInput x) |
| def runInputRealizer : Realizer runInput := |
| (pair (composition (g:=repetitions) first.fst.snd.snd repetitionsRealizer) trialInputRealizer).congr (fun _=>rfl) |
|
|
| def initializeInput (x : Seed) : ℕ×OnlineAmplification.TrialInput := runInput (config x) |
| def initializeInputRealizer : Realizer initializeInput := composition configRealizer runInputRealizer |
| end MatchingFPRAS.OnlineInitial |
|
|
| end OAI |
|
|
