+(**************************************************************************)
+(* ___ *)
+(* ||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/nat_le_minus_plus.ma".
+include "ground/arith/nat_lt_minus.ma".
+
+(* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
+
+(* Constructions with nminus and nplus **************************************)
+
+(*** lt_plus_to_minus *)
+lemma nlt_minus_sn (o) (m) (n): m ≤ n → n < o + m → n - m < o.
+#o #m #n #Hmn #Ho
+lapply (nle_minus_sn … Ho) -Ho
+<nminus_succ_sn //
+qed.
+
+(*** lt_plus_to_minus_r *)
+lemma nlt_minus_dx (o) (m) (n): m + o < n → m < n - o.
+/2 width=1 by nle_minus_dx/ qed.
+
+(*** lt_inv_plus_l *)
+lemma nlt_minus_dx_full (o) (m) (n): m + o < n → ∧∧ o < n & m < n - o.
+/3 width=3 by nlt_minus_dx, le_nlt_trans, conj/ qed-.
+
+(* Inversions with nminus and nplus *****************************************)
+
+(*** lt_minus_to_plus *)
+lemma nlt_inv_minus_sn (o) (m) (n): m - o < n → m < n + o.
+#o #m #n #Ho
+@nle_inv_minus_sn
+@(nle_trans … Ho) -Ho //
+qed-.
+
+(*** lt_minus_to_plus_r *)
+lemma nlt_inv_minus_dx (o) (m) (n): m < n - o → m + o < n.
+#o #m #n #Ho
+lapply (nle_inv_minus_dx ???? Ho) //
+/3 width=2 by nlt_fwd_minus_dx, nlt_des_le/
+qed-.