matita/matita/contribs/lambdadelta/basic_2/static/sh.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  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 (* Basic properties *********************************************************)
  26
  27 lemma nexts_le: ∀h,k,l. k ≤ (next h)^l k.
  28 #h #k #l elim l -l // normalize #l #IHl
  29 lapply (next_lt h ((next h)^l k)) #H
  30 lapply (le_to_lt_to_lt … IHl H) -IHl -H /2 width=2/
  31 qed.
  32
  33 axiom nexts_dec: ∀h,k1,k2. Decidable (∃l. (next h)^l k1 = k2).
  34
  35 axiom nexts_inj: ∀h,k,l1,l2. (next h)^l1 k = (next h)^l2 k → l1 = l2.
  36