proposition: Lemma reachable_valid_and_through_single: forall (g: Gph) {x y}, reachable g x y -> (vvalid g y /\ reachable_through_set g (x :: nil) y).
proof: Proof. intros. split. + apply reachable_foot_valid in H; auto. + exists x. split. - apply in_eq. - auto. Qed.
imports: Coq.Logic.ProofIrrelevance Coq.micromega.Lia CertiGraph.lib.Coqlib CertiGraph.lib.EnumEnsembles CertiGraph.lib.List_ext CertiGraph.lib.EquivDec_ext CertiGraph.graph.graph_model Coq.Lists.List Coq.Lists.ListDec