matita/matita/contribs/lambdadelta/basic_2/syntax/item_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/lib/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: ∀s. s < next s (* strict monotonicity condition *)
  23 }.
  24
  25 definition sh_N: sh ≝ mk_sh S ….
  26 // defined.
  27
  28 (* Basic properties *********************************************************)
  29
  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/
  34 qed.
  35
  36 lemma nexts_lt: ∀h,s,n. s < (next h)^(↑n) s.
  37 #h #s #n normalize
  38 lapply (nexts_le h s n) #H
  39 @(le_to_lt_to_lt … H) //
  40 qed.
  41
  42 axiom nexts_dec: ∀h,s1,s2. Decidable (∃n. (next h)^n s1 = s2).
  43
  44 axiom nexts_inj: ∀h,s,n1,n2. (next h)^n1 s = (next h)^n2 s → n1 = n2.