proposition: Lemma reachable_by_through_set_step: forall (g: Gph) l n2 n3 (P: Ensemble V), vvalid g n3 -> P n3 -> reachable_by_through_set g l P n2 -> step g n2 n3 -> reachable_by_through_set g l P n3.
proof: Proof. intros. eapply reachable_by_through_set_edge; eauto. split; [| split]; auto. apply reachable_by_through_set_foot_valid in H1; 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