proposition: Lemma reachable_partialgraph_reachable_equiv (g: PreGraph V E) (P: V -> Prop) (n: V): Included (reachable g n) P -> Same_set (reachable g n) (reachable (predicate_partialgraph g P) n).
proof: Proof. intros. rewrite Same_set_spec. intro. split. + intros. apply reachable_partialgraph_reachable in H0. unfold Ensembles.In in H0. rewrite <- reachable_by_eq_partialgraph_reachable in H0 |- *. eapply reachable_by_weaken; [| eauto]. auto. + apply predicate_partialgraph_reachable_included. Qed.
imports: Coq.Arith.Arith Coq.Logic.ProofIrrelevance Coq.Logic.FunctionalExtensionality Coq.Classes.Morphisms Coq.Classes.Equivalence CertiGraph.lib.Coqlib CertiGraph.lib.Ensembles_ext CertiGraph.lib.relation_list CertiGraph.lib.List_ext CertiGraph.lib.Equivalence_ext CertiGraph.lib.EquivDec_ext VST.msl.Coqlib2 CertiGraph.graph.graph_model CertiGraph.graph.path_lemmas CertiGraph.graph.reachable_ind CertiGraph.graph.graph_gen CertiGraph.graph.graph_relation