matita/matita/contribs/lambdadelta/ground/arith/nat.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/notation/functions/zero_0.ma".
  16 include "ground/arith/pnat.ma".
  17
  18 (* NON-NEGATIVE INTEGERS ****************************************************)
  19
  20 (*** nat *)
  21 inductive nat: Type[0] ≝
  22 | nzero: nat
  23 | ninj : pnat → nat
  24 .
  25
  26 coercion ninj.
  27
  28 interpretation
  29   "zero (non-negative integers)"
  30   'Zero = nzero.
  31
  32 (* Basic inversions *********************************************************)
  33
  34 (* Note: destruct *)
  35 lemma eq_inv_ninj_bi: injective … ninj.
  36 #p #q #H destruct //
  37 qed-.
  38
  39 (* Basic constructions ******************************************************)
  40
  41 (*** eq_nat_dec *)
  42 lemma eq_nat_dec (n1,n2:nat): Decidable (n1 = n2).
  43 * [| #p1 ] * [2,4: #p2 ]
  44 [1,4: @or_intror #H destruct
  45 | elim (eq_pnat_dec p1 p2)
  46   /4 width=1 by eq_inv_ninj_bi, or_intror, or_introl/
  47 | /2 width=1 by or_introl/
  48 ]
  49 qed-.