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diff --git a/matita/matita/contribs/lambdadelta/basic_2/syntax/item_sh.ma b/matita/matita/contribs/lambdadelta/basic_2/syntax/item_sh.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "ground_2/lib/arith.ma".
+
+(* SORT HIERARCHY ***********************************************************)
+
+(* sort hierarchy specification *)
+record sh: Type[0] ≝ {
+   next   : nat → nat;     (* next sort in the hierarchy *)
+   next_lt: ∀s. s < next s (* strict monotonicity condition *)
+}.
+
+definition sh_N: sh ≝ mk_sh S ….
+// defined.
+
+(* Basic properties *********************************************************)
+
+lemma nexts_le: ∀h,s,n. s ≤ (next h)^n s.
+#h #s #n elim n -n // normalize #n #IH
+lapply (next_lt h ((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,s,n. s < (next h)^(⫯n) s.
+#h #s #n normalize
+lapply (nexts_le h s n) #H
+@(le_to_lt_to_lt … H) //
+qed.
+
+axiom nexts_dec: ∀h,s1,s2. Decidable (∃n. (next h)^n s1 = s2).
+
+axiom nexts_inj: ∀h,s,n1,n2. (next h)^n1 s = (next h)^n2 s → n1 = n2.