proposition: Lemma vertices_at1: forall (g: Graph) P x0 d, (forall x, P x <-> x0 = x) -> vgamma g x0 = d -> vertices_at P g = vertex_at x0 d.
proof: Proof. intros. replace (@vertex_at _ _ _ _ _ SGP x0 d) with (graph_vcell g x0). 2: { simpl. unfold graph_vcell; simpl. rewrite H0; auto. } erewrite vertices_at_Same_set. + apply pred_sepcon1. + rewrite Same_set_spec. intro x; specialize (H x). assert (x = x0 <-> x0 = x) by (split; congruence). tauto. Qed.
imports: Coq.Logic.Classical VST.msl.seplog VST.msl.log_normalize VST.msl.ramification_lemmas CertiGraph.lib.Coqlib CertiGraph.lib.EquivDec_ext CertiGraph.lib.EnumEnsembles CertiGraph.lib.List_ext CertiGraph.lib.Relation_ext CertiGraph.lib.Equivalence_ext CertiGraph.lib.Morphisms_ext CertiGraph.msl_ext.log_normalize CertiGraph.msl_ext.iter_sepcon CertiGraph.msl_ext.ramification_lemmas CertiGraph.msl_ext.abs_addr CertiGraph.msl_ext.seplog CertiGraph.graph.graph_model CertiGraph.graph.graph_gen CertiGraph.graph.path_lemmas CertiGraph.graph.reachable_computable CertiGraph.graph.reachable_ind CertiGraph.graph.subgraph2 CertiGraph.graph.dag CertiGraph.graph.FiniteGraph