| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Graphs.TreeMatchingCheck |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.TreeTyped.SparseProgram |
| open TreeTyped |
|
|
| variable {E V C : Type} [Codec E] [Codec V] [Codec C] [LinearOrder E] [LinearOrder V] |
| attribute [local instance] Transport.Sparse.finsetOrder |
|
|
| def certify (p : C×(E×V)→Bool) (x : (C×Finset V)×Finset (Finset E)) : Finset (Finset E) := |
| x.2.filter (fun a=>matchingCheck p (x.1,a)=true) |
| def certifyRealizer {p : C×(E×V)→Bool} (P : Realizer p) : Realizer (certify p) := |
| (finsetFilterWith (matchingCheckRealizer P)).congr (fun _=>rfl) |
| lemma polynomial_certify {p : C×(E×V)→Bool} {P : Realizer p} (hP : PolynomialTime P) : |
| PolynomialTime (certifyRealizer P) := |
| polynomial_congr _ (polynomial_finsetFilterWith (polynomial_matchingCheck hP)) |
|
|
| variable [Fintype E] [Fintype V] (ends : E→Sym2 V) |
| attribute [local instance] Transport.Sparse.pmOrder |
|
|
| def pmEmbedding : @OrderEmbedding (Transport.PM (fun e v=>v∈ends e)) (Finset E) |
| (Transport.Sparse.pmOrder ends).toLE (Transport.Sparse.finsetOrder (E:=E)).toLE := |
| { toFun := Subtype.val |
| inj' := Subtype.val_injective |
| map_rel_iff' := Iff.rfl } |
|
|
| omit [Codec E] [Codec V] [Codec C] in |
| lemma certify_eq (p : C×(E×V)→Bool) (c : C) |
| (hp : ∀ e v,p (c,(e,v))=decide (v∈ends e)) (s : Finset (Finset E)) : |
| certify p ((c,Finset.univ),s)=(Transport.Sparse.certify ends s).map (pmEmbedding ends).toEmbedding := by |
| ext a |
| simp only [certify,Finset.mem_filter,matchingCheck_eq p c _ hp,decide_eq_true_eq, |
| Finset.mem_map,pmEmbedding] |
| constructor |
| · rintro ⟨ha,hm⟩ |
| exact ⟨⟨a,hm⟩,by simpa using ha,rfl⟩ |
| · rintro ⟨b,hb,rfl⟩ |
| exact ⟨by simpa using hb,b.property⟩ |
| end MatchingFPRAS.TreeTyped.SparseProgram |
|
|
| end OAI |
|
|
