-lemma nle_ge_dis (m) (n): ∨∨ m ≤ n | n ≤ m.
-#m @(nat_ind … m) -m [ /2 width=1 by or_introl/ ]
-#m #IH #n @(nat_ind … n) -n [ /2 width=1 by or_intror/ ]
-#n #_ elim (IH n) -IH /3 width=2 by nle_succ_bi, or_introl, or_intror/
+lemma nat_split_le_ge (m) (n): ∨∨ m ≤ n | n ≤ m.
+#m #n @(nat_ind_2_succ … m n) -m -n
+[ /2 width=1 by or_introl/
+| /2 width=1 by or_intror/
+| #m #n * /3 width=2 by nle_succ_bi, or_introl, or_intror/
+]