include "ground/arith/nat_pred_succ.ma".
include "ground/arith/nat_le.ma".
-(* NON-NEGATIVE INTEGERS ****************************************************)
+(* ORDER FOR NON-NEGATIVE INTEGERS ******************************************)
-(* Basic constructions with pred ********************************************)
+(* Constructions with npred *************************************************)
(*** le_pred_n *)
lemma nle_pred_sn_refl (m): ↓m ≤ m.
(*** monotonic_pred *)
lemma nle_pred_bi (m) (n): m ≤ n → ↓m ≤ ↓n.
-#m #n #H elim H -n
+#m #n #H elim H -n //
/2 width=3 by nle_trans/
qed.
+
+lemma nle_pred_sn (m) (n): m ≤ ↑n → ↓m ≤ n.
+#m #n elim m -m //
+/2 width=1 by nle_pred_bi/
+qed-.
+
+(* Destructions with npred **************************************************)
+
+lemma nle_inv_pred_sn (m) (n): ↓m ≤ n → m ≤ ↑n.
+#m #n elim m -m
+/2 width=1 by nle_succ_bi/
+qed-.