1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "ground_2/lib/arith.ma".
17 (* SORT HIERARCHY ***********************************************************)
19 (* sort hierarchy specification *)
20 record sh: Type[0] ≝ {
21 next : nat → nat; (* next sort in the hierarchy *)
22 next_lt: ∀s. s < next s (* strict monotonicity condition *)
25 definition sh_N: sh ≝ mk_sh S ….
28 (* Basic properties *********************************************************)
30 lemma nexts_le: ∀h,s,n. s ≤ (next h)^n s.
31 #h #s #n elim n -n // normalize #n #IH
32 lapply (next_lt h ((next h)^n s)) #H
33 lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/
36 lemma nexts_lt: ∀h,s,n. s < (next h)^(⫯n) s.
38 lapply (nexts_le h s n) #H
39 @(le_to_lt_to_lt … H) //
42 axiom nexts_dec: ∀h,s1,s2. Decidable (∃n. (next h)^n s1 = s2).
44 axiom nexts_inj: ∀h,s,n1,n2. (next h)^n1 s = (next h)^n2 s → n1 = n2.