-lemma nexts_le: ∀h,k,l. k ≤ (next h)^l k.
-#h #k #l elim l -l // normalize #l #IHl
-lapply (next_lt h ((next h)^l k)) #H
-lapply (le_to_lt_to_lt … IHl H) -IHl -H /2 width=2 by lt_to_le/
+lemma nexts_le: ∀h,k,d. k ≤ (next h)^d k.
+#h #k #d elim d -d // normalize #d #IHd
+lapply (next_lt h ((next h)^d k)) #H
+lapply (le_to_lt_to_lt … IHd H) -IHd -H /2 width=2 by lt_to_le/