matita/matita/contribs/lambdadelta/basic_2A/static/sh.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 include "ground_2A/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: ∀k. k < next k (* 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,k,d. k ≤ (next h)^d k.
  31 #h #k #d elim d -d // normalize #d #IHd
  32 lapply (next_lt h ((next h)^d k)) #H
  33 lapply (le_to_lt_to_lt … IHd H) -IHd -H /2 width=2 by lt_to_le/
  34 qed.
  35
  36 lemma nexts_lt: ∀h,k,d. k < (next h)^(d+1) k.
  37 #h #k #d >iter_SO
  38 lapply (nexts_le h k d) #H
  39 @(le_to_lt_to_lt … H) //
  40 qed.
  41
  42 axiom nexts_dec: ∀h,k1,k2. Decidable (∃d. (next h)^d k1 = k2).
  43
  44 axiom nexts_inj: ∀h,k,d1,d2. (next h)^d1 k = (next h)^d2 k → d1 = d2.