proposition: Lemma spanning_tree_vvalid_s: forall (g1 : Graph) (root : V) (P: V -> Prop) (g2 : Graph) x, spanning_tree g1 root P g2 -> (vvalid g1 x <-> vvalid g2 x).
proof: Proof. intros. destruct (classic (g1 |= root ~o~> x satisfying P)) as [r | n]. + destruct H as [? [? [? ?]]]. split; intros. - specialize (H1 _ r). apply reachable_foot_valid in H1. auto. - apply reachable_by_foot_valid in r. auto. + eapply not_reachable_ST_vvalid; eauto. Qed.
imports: Coq.Classes.Morphisms CertiGraph.lib.Coqlib CertiGraph.lib.EquivDec_ext CertiGraph.lib.EnumEnsembles CertiGraph.lib.List_ext CertiGraph.lib.relation_list CertiGraph.graph.graph_model CertiGraph.graph.path_lemmas CertiGraph.graph.graph_gen CertiGraph.graph.graph_relation CertiGraph.graph.subgraph2 CertiGraph.graph.reachable_ind Coq.Logic.Classical