proposition: Lemma path_lastP : forall k x y l, path x y l -> S k = length l -> R (nth k l y) y.
proof: Proof. induction k. intros. destruct l. simpl in H0. lia. destruct l. simpl. simpl in H. intuition. simpl in H0. lia. intros. destruct l. simpl in H0. lia. simpl. simpl in H. simpl in H0. apply (IHk a y l); intuition. Qed.
imports: CoLoR(RelSub ListUtil ListNodup RelUtil LogicUtil ListShrink NatUtil ListOccur)