+(**************************************************************************)
+(* ___ *)
+(* ||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_lt.ma".
+
+(* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
+
+(* Constructions with ysucc *************************************************)
+
+(*** ylt_O_succ *)
+lemma ylt_zero_succ (y): 𝟎 < ↑y.
+#y @(ynat_split_nat_inf … y) -y
+/2 width=1 by ylt_inj/
+qed.
+
+(*** ylt_succ *)
+lemma ylt_succ_bi (x) (y): x < y → ↑x < ↑y.
+#x #y * -x -y
+/3 width=1 by ylt_inj, ylt_inf, nlt_succ_bi/
+qed.
+
+(*** ylt_succ_Y *)
+lemma ylt_succ_inf (x): x < ∞ → ↑x < ∞.
+#x @(ynat_split_nat_inf … x) -x //
+qed.
+
+(*** ylt_succ2_refl *)
+lemma ylt_succ_dx_refl (x) (y): x < y → x < ↑x.
+#x #y #H
+elim (ylt_des_gen_sn … H) -y #n #H destruct
+/2 width=1 by ylt_inj/
+qed.
+
+(* Inversions with ysucc ****************************************************)
+
+lemma ylt_inv_succ_inf (x): ↑x < ∞ → x < ∞.
+#x #H
+elim (ylt_des_gen_sn … H) -H #m0 #H
+elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct //
+qed-.
+
+(*** ylt_inv_succ *)
+lemma ylt_inv_succ_bi (x) (y): ↑x < ↑y → x < y.
+#x #y @(ynat_split_nat_inf … y) -y
+[ #n <ysucc_inj #H
+ elim (ylt_inv_inj_dx … H) -H #m0 #Hmn #H
+ elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct
+ /3 width=1 by ylt_inj, nlt_inv_succ_bi/
+| /2 width=1 by ylt_inv_succ_inf/
+]
+qed-.