proposition: Lemma vertices_at_ramif_1Q: forall {A: Type} (Pure: A -> Prop) (g: Graph) (P: V -> Prop) x (g': A -> Graph) (P': A -> V -> Prop) (x': A -> V), (exists F, Prop_join (eq x) F P /\ (forall a, Pure a -> Prop_join (eq (x' a)) F (P' a)) /\ (forall a, Pure a -> vertices_identical F g (g' a))) -> vertices_at P g |-- vertex_at x (vgamma g x) * (ALL a: A, !! Pure a --> (vertex_at (x' a) (vgamma (g' a) (x' a)) -* vertices_at (P' a) (g' a))).
proof: Proof. intros. unfold vertices_at. change (@vertex_at _ _ _ _ _ SGP x (vgamma g x)) with (graph_vcell g x). RAMIF_Q'.formalize. change ((fun a : A => vertex_at (x' a) (vgamma (g' a) (x' a)))) with ((fun a : A => graph_vcell (g' a) (x' a))). simpl. apply (pred_sepcon_ramif_1Q Pure P P' (graph_vcell g) (fun a => graph_vcell (g' a)) x x'). destruct H as [F [? [? ?]]]. exists F; split; [| split]; auto. intros. unfold graph_vcell; f_equal. specialize (H1 a H2). rewrite vertices_identical_spec in H1. apply H1; auto. 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