--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/sort.ma".
+
+(* SORT HIERARCHY ***********************************************************)
+
+record is_lt (h): Prop ≝
+{
+ next_lt: ∀s. s < ⫯[h]s (* strict monotonicity condition *)
+}.
+
+(* Basic properties *********************************************************)
+
+lemma nexts_le (h): is_lt h → ∀s,n. s ≤ (next h)^n s.
+#h #Hh #s #n elim n -n [ // ] normalize #n #IH
+lapply (next_lt … Hh ((next h)^n s)) #H
+lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/
+qed.
+
+lemma nexts_lt (h): is_lt h → ∀s,n. s < (next h)^(↑n) s.
+#h #Hh #s #n normalize
+lapply (nexts_le … Hh s n) #H
+@(le_to_lt_to_lt … H) /2 width=1 by next_lt/
+qed.