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11 (*        v         GNU General Public License Version 2                  *)
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14
15 include "ground_2/arith.ma".
16
17 (* SORT HIERARCHY ***********************************************************)
18
19 (* sort hierarchy specification *)
20 record sh: Type[0] ≝ {
21    next   : nat → nat;     (* next sort in the hierarchy *)
22    next_lt: ∀k. k < next k (* strict monotonicity condition *)
23 }.
24
25 definition sh_N: sh ≝ mk_sh S ….
26 // qed.
27
28 (* Basic properties *********************************************************)
29
30 lemma nexts_le: ∀h,k,l. k ≤ (next h)^l k.
31 #h #k #l elim l -l // normalize #l #IHl
32 lapply (next_lt h ((next h)^l k)) #H
33 lapply (le_to_lt_to_lt … IHl H) -IHl -H /2 width=2/
34 qed.
35
36 lemma nexts_lt: ∀h,k,l. k < (next h)^(l+1) k.
37 #h #k #l >iter_SO
38 lapply (nexts_le h k l) #H
39 @(le_to_lt_to_lt … H) //
40 qed.
41
42 axiom nexts_dec: ∀h,k1,k2. Decidable (∃l. (next h)^l k1 = k2).
43
44 axiom nexts_inj: ∀h,k,l1,l2. (next h)^l1 k = (next h)^l2 k → l1 = l2.
45