#H destruct
qed.
+lemma yle_inv_succ_sn_lt (x:ynat) (y:ynat):
+ ↑x ≤ y → ∧∧ x ≤ ↓y & 0 < y.
+#x #y #H elim (yle_inv_succ1 … H) -H /3 width=2 by ylt_O1, conj/
+qed-.
+
(* Properties on predecessor ************************************************)
lemma ylt_pred: ∀m,n:ynat. m < n → 0 < m → ↓m < ↓n.
(* Properties on successor **************************************************)
-lemma ylt_O_succ: ∀n. 0 < ↑n.
+lemma ylt_O_succ: ∀x:ynat. 0 < ↑x.
* /2 width=1 by ylt_inj/
qed.