}.
definition sh_N: sh ≝ mk_sh S ….
-// qed.
+// defined.
(* Basic properties *********************************************************)
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/
+lapply (le_to_lt_to_lt … IHl H) -IHl -H /2 width=2 by lt_to_le/
qed.
lemma nexts_lt: ∀h,k,l. k < (next h)^(l+1) k.