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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 | 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
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