proposition: Lemma uf_equiv_uf_set_in: forall (g1 g2: Graph) s, uf_equiv g1 g2 -> uf_set_in g1 s -> uf_set_in g2 s.
proof: Proof. intros. destruct H0; [left | right]; auto. destruct H0 as [rt [? ?]]. exists rt. split; auto. intros. rewrite H1. apply uf_equiv_the_same_root; auto. Qed.
imports: CertiGraph.lib.Coqlib CertiGraph.lib.EquivDec_ext CertiGraph.lib.List_ext CertiGraph.lib.Ensembles_ext CertiGraph.lib.EnumEnsembles CertiGraph.graph.graph_model CertiGraph.graph.path_lemmas CertiGraph.graph.graph_gen CertiGraph.graph.MathGraph CertiGraph.graph.LstGraph CertiGraph.graph.FiniteGraph CertiGraph.graph.reachable_computable