+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground/arith/pnat_le_plus.ma".
+include "ground/arith/pnat_lt.ma".
+
+(* STRICT ORDER FOR POSITIVE INTEGERS ***************************************)
+
+(* Constructions with pplus *************************************************)
+
+lemma plt_plus_bi_dx (p) (q1) (q2): q1 < q2 → q1 + p < q2 + p.
+#p #q1 #q2 #H
+@plt_i >pplus_succ_sn /2 width=1 by ple_plus_bi_dx/
+qed.
+
+lemma plt_plus_bi_sn (p) (q1) (q2): q1 < q2 → p + q1 < p + q2.
+#p #q1 #q2 #H
+@plt_i >pplus_succ_dx /2 width=1 by ple_plus_bi_sn/
+qed.
+
+lemma plt_plus_dx_dx_refl (p) (q): p < p + q.
+/2 width=1 by ple_plus_bi_sn/ qed.
+
+lemma plt_plus_dx_sn_refl (p) (q): p < q + p.
+/2 width=1 by ple_plus_bi_dx/ qed.
+
+lemma plt_plus_dx_sn (r) (p) (q): q ≤ p → q < r + p.
+/2 width=3 by ple_plt_trans/ qed.
+
+lemma plt_plus_dx_dx (r) (p) (q): q ≤ p → q < p + r.
+/2 width=3 by ple_plt_trans/ qed.