matita/library/logic/coimplication.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/coimplication".
  16
  17 include "logic/connectives.ma".
  18
  19 definition Iff : Prop \to Prop \to Prop \def
  20    \lambda A,B. (A \to B) \land (B \to A).
  21
  22  (*CSC: the URI must disappear: there is a bug now *)
  23 interpretation "logical iff" 'iff x y = (cic:/matita/logic/coimplication/Iff.con x y).
  24
  25 notation > "hvbox(a break \liff b)"
  26   left associative with precedence 25
  27 for @{ 'iff $a $b }.
  28
  29 notation < "hvbox(a break \leftrightarrow b)"
  30   left associative with precedence 25
  31 for @{ 'iff $a $b }.
  32
  33 theorem iff_intro: \forall A,B. (A \to B) \to (B \to A) \to (A \liff B).
  34  unfold Iff. intros. split; intros; autobatch.
  35 qed.
  36
  37 theorem iff_refl: \forall A. A \liff A.
  38  intros. apply iff_intro; intros; autobatch.
  39 qed.
  40
  41 theorem iff_sym: \forall A,B. A \liff B \to B \liff A.
  42  intros. elim H. apply iff_intro[assumption|assumption]
  43 qed.
  44
  45 theorem iff_trans: \forall A,B,C. A \liff B \to B \liff C \to A \liff C.
  46  intros. elim H. elim H1. apply iff_intro;intros;autobatch.
  47 qed.