matita/matita/contribs/lambdadelta/ground/arith/ynat.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/arith/pnat.ma".
  16 include "ground/notation/functions/zero_0.ma".
  17 include "ground/notation/functions/infinity_0.ma".
  18
  19 (* NON-NEGATIVE INTEGERS WITH INFINITY **************************************)
  20
  21 (*** ynat *)
  22 inductive ynat: Type[0] ≝
  23 | yzero: ynat
  24 | yinj : pnat → ynat
  25 | yinf : ynat
  26 .
  27
  28 coercion yinj.
  29
  30 interpretation
  31   "zero (non-negative integers with infinity)"
  32   'Zero = yzero.
  33
  34 interpretation
  35   "infinity (non-negative integers with infinity)"
  36   'Infinity = yinf.
  37
  38 (* Inversions ***************************************************************)
  39
  40 (* Note: destruct *)
  41 (*** yinj_inj *)
  42 lemma eq_inv_yinj_bi (y1) (y2): yinj y1 = yinj y2 → y1 = y2.
  43 #x #y #H destruct //
  44 qed-.
  45
  46 (* Basic constructions ******************************************************)
  47
  48 (*** eq_ynat_dec *)
  49 lemma eq_ynat_dec (y1,y2:ynat): Decidable (y1 = y2).
  50 * [| #p1 |] *
  51 [1,9: /2 width=1 by or_introl/ |2,5,8: #p2 ]
  52 [2: elim (eq_pnat_dec p1 p2)
  53     /4 width=1 by eq_inv_yinj_bi, or_intror, or_introl/
  54 |*: @or_intror #H destruct
  55 ]
  56 qed-.