+(*** le_inv_S1 *)
+lemma nle_inv_succ_sn (m) (n):
+ ↑m ≤ n → ∧∧ m ≤ ↓n & n = ↑↓n.
+#m #n * -n
+[ /2 width=3 by nle_refl, conj/
+| #n #Hn /3 width=1 by nle_des_succ_sn, conj/
+]
+qed-.
+
+lemma nle_inv_succ_dx (m) (n):
+ m ≤ ↑n → ∨∨ 𝟎 = m | ∧∧ ↓m ≤ n & m = ↑↓m.
+#m #n @(nat_ind_succ … m) -m
+[ /2 width=1 by or_introl/
+| #m #_ #H0
+ /4 width=1 by nle_inv_succ_bi, or_intror, conj/
+]
+qed-.
+