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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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Tapes.TreeSparseCertificate
import OAI.Combinatorics.MatchingCount.Algorithm.OnlineTrial
namespace OAI
namespace MatchingFPRAS.OnlineGuard
open PhysicalStream TreeTyped TreeFraction ExecutableSource NumericMarking OnlineSchedule
def positive (q : Fraction) : Bool := TreeFraction.less ((0,1),q)
def positiveRealizer : TreeTyped.Realizer positive :=
composition (pair (constant (0,1)) identity) lessRealizer
lemma polynomial_positive : PolynomialTime positiveRealizer :=
polynomial_composition (polynomial_pair (polynomial_constant _) polynomial_identity) polynomial_less
lemma positive_repr {q : ℚ} (hq : 0≤q) : positive (repr q)=decide (0<q) := by
rw [positive,less_val _ _ (by decide) q.pos,val_repr hq]
simp [val]
def accept (x : SparseStageData×List SampleBits) : Bool :=
x.2.any (fun b=>positive (sparseCertificate (x.1,b)))
def acceptRealizer : TreeTyped.Realizer accept :=
(composition (mapWith (composition sparseCertificateRealizer positiveRealizer)) any).congr (by
intro x; simp only [accept,Function.comp_def,List.any_map,Function.id_def])
lemma polynomial_accept : PolynomialTime acceptRealizer := by
unfold acceptRealizer
apply polynomial_congr
exact polynomial_composition (polynomial_mapWith (polynomial_composition polynomial_sparseCertificate polynomial_positive)) polynomial_any
def test (x : RequestInput) : Experiment Bool :=
(OnlineSample.many (16,sourceRequest x)).map (fun bs=>accept (x.1,bs))
noncomputable def testRealizer : OnlineTree.Realizer test :=
(OnlineTree.composition (OnlineTree.pair (OnlineTree.deterministic first)
(OnlineTree.composition (OnlineTree.deterministic (pair (constant 16) sourceRequestRealizer)) OnlineSample.manyRealizer))
(OnlineTree.deterministic acceptRealizer)).congr (by
intro x h; simp only [test,Experiment.mean_bind,Experiment.mean_map,Experiment.mean])
def finish (x : Fraction×Bool) : Fraction := if x.2 then x.1 else (0,1)
def finishRealizer : TreeTyped.Realizer finish :=
(choose second first (constant (0,1))).congr (fun _=>rfl)
lemma polynomial_finish : PolynomialTime finishRealizer := by
unfold finishRealizer; apply polynomial_congr; tree_cost
def checked (x : OnlineTrial.State) : Experiment Fraction :=
(test x.1).map (fun b=>finish (x.2,b))
noncomputable def checkedRealizer : OnlineTree.Realizer checked :=
(OnlineTree.composition (OnlineTree.pair (OnlineTree.deterministic second)
(OnlineTree.composition (OnlineTree.deterministic first) testRealizer))
(OnlineTree.deterministic finishRealizer)).congr (by
intro x h; simp only [checked,Experiment.mean_bind,Experiment.mean_map,Experiment.mean])
def trial (x : List Bool×OnlineTrial.State) : Experiment Fraction :=
(OnlineTrial.sweep x).bind checked
noncomputable def trialRealizer : OnlineTree.Realizer trial :=
OnlineTree.composition OnlineTrial.sweepRealizer checkedRealizer
end MatchingFPRAS.OnlineGuard
end OAI
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