--- /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/ynat_succ.ma".
+include "ground/arith/ynat_le_minus1.ma".
+
+(* ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY ****************************)
+
+(* Constructions with yminus1 and ysucc *************************************)
+
+(*** yminus_succ1_inj *)
+lemma yminus1_succ_sn (x) (n):
+ yinj_nat n ≤ x → ↑(x - n) = ↑x - n.
+#x @(ynat_split_nat_inf … x) -x //
+#m #n #Hnm
+<yminus1_inj_sn <ysucc_inj <ysucc_inj <yminus1_inj_sn
+/4 width=1 by yle_inv_inj_bi, nminus_succ_sn, eq_f/
+qed-.