matita/library/logic/connectives2.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 set "baseuri" "cic:/matita/logic/connectives2".
  16
  17 include "higher_order_defs/relations.ma".
  18
  19 theorem reflexive_iff: reflexive ? iff.
  20  unfold reflexive;
  21  intro;
  22  split;
  23  intro;
  24  assumption.
  25 qed.
  26
  27 theorem symmetric_iff: symmetric ? iff.
  28  unfold symmetric;
  29  intros;
  30  elim H;
  31  split;
  32  assumption.
  33 qed.
  34
  35 theorem transitive_iff: transitive ? iff.
  36  unfold transitive;
  37  intros;
  38  elim H;
  39  elim H1;
  40  split;
  41  intro;
  42  auto.
  43 qed.
  44