proposition: Lemma triple_mark1: forall root (g g1 g2: Graph), nothing root g g1 -> mark1 root g1 g2 -> let marked' := Union _ (marked g) (eq root) in let PV1 := reachable_by_through_set g nil (Complement _ marked') in (predicate_partialgraph g (Complement _ (Union _ PV1 (eq root)))) ~=~ (predicate_partialgraph g2 (Complement _ (Union _ PV1 (eq root)))) /\ Same_set (marked g2) (Union _ marked' PV1).
proof: Proof. intros. split. + destruct H, H0. rewrite <- H0. eapply si_stronger_partialgraph_simple; [| eauto]. apply Complement_Included_rev. apply right_Included_Union. + unfold PV1, marked'. rewrite Same_set_spec; intro n; rewrite !Union_spec. rewrite reachable_by_through_nil. destruct H0 as [_ [? ?]]. destruct_eq_dec root n. - subst n; tauto. - specialize (H1 n). destruct H. specialize (H3 n). tauto. Qed.
imports: Coq.Logic.ProofIrrelevance Coq.Classes.Morphisms VST.msl.Coqlib2 CertiGraph.lib.Coqlib CertiGraph.lib.EquivDec_ext CertiGraph.lib.Relation_ext CertiGraph.lib.relation_list CertiGraph.lib.EnumEnsembles CertiGraph.lib.List_ext CertiGraph.graph.graph_model CertiGraph.graph.path_lemmas CertiGraph.graph.reachable_computable CertiGraph.graph.reachable_ind CertiGraph.graph.graph_gen CertiGraph.graph.subgraph2 CertiGraph.graph.dual_graph