+(* lt and le trans *)
+theorem lt_to_le_to_lt: \forall n,m,p:nat. lt n m \to le m p \to lt n p.
+intros.elim H1.
+assumption.simplify.apply le_S.assumption.
+qed.
+
+theorem le_to_lt_to_lt: \forall n,m,p:nat. le n m \to lt m p \to lt n p.
+intros 4.elim H.
+assumption.apply H2.simplify.
+apply lt_to_le.assumption.
+qed.
+
+theorem ltn_to_ltO: \forall n,m:nat. lt n m \to lt O m.
+intros.apply le_to_lt_to_lt O n.
+apply le_O_n.assumption.
+qed.
+
+theorem lt_O_n_elim: \forall n:nat. lt O n \to
+\forall P:nat\to Prop. (\forall m:nat.P (S m)) \to P n.
+intro.elim n.apply False_ind.exact not_le_Sn_O O H.
+apply H2.
+qed.
+