proposition: Lemma reachable_by_through_app: forall g P n l1 l2, reachable_by_through_set g (l1 ++ l2) P n <-> reachable_by_through_set g l1 P n \/ reachable_by_through_set g l2 P n.
proof: Proof. intros. unfold reachable_by_through_set. split; [intros [s [? ?]] | intros [[s [? ?]] | [s [? ?]]]]. + rewrite in_app_iff in H. destruct H; [left | right]; exists s; auto. + exists s. rewrite in_app_iff; auto. + exists s. rewrite in_app_iff; 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