--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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.ma".
+include "ground/arith/nat_lt_pred.ma".
+
+(* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
+
+(* Rewrites with nminus *****************************************************)
+
+(*** minus_pred_pred *)
+lemma nminus_pred_bi (m) (n): 𝟎 < m → 𝟎 < n → n - m = ↓n - ↓m.
+#m #n #Hm #Hn
+>(nlt_inv_zero_sn … Hm) in ⊢ (??%?); -Hm
+>(nlt_inv_zero_sn … Hn) in ⊢ (??%?); -Hn
+//
+qed-.
+
+(* Constructions with nminus ************************************************)
+
+(*** monotonic_lt_minus_l *)
+lemma nlt_minus_sn_bi (o) (m) (n): o ≤ m → m < n → m - o < n - o.
+#o #m #n #Hom #Hmn
+lapply (nle_minus_sn_bi … o Hmn) -Hmn
+<(nminus_succ_sn … Hom) //
+qed.