+++ /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 "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.