| 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 |
|
|
