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[helm.git] / matita / library / logic / coimplication.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/logic/coimplication".
+
+include "logic/connectives.ma".
+
+definition Iff : Prop \to Prop \to Prop \def
+   \lambda A,B. (A \to B) \land (B \to A).
+   
+ (*CSC: the URI must disappear: there is a bug now *)
+interpretation "logical iff" 'iff x y = (cic:/matita/logic/coimplication/Iff.con x y).
+
+notation > "hvbox(a break \liff b)" 
+  left associative with precedence 25
+for @{ 'iff $a $b }.
+
+notation < "hvbox(a break \leftrightarrow b)" 
+  left associative with precedence 25
+for @{ 'iff $a $b }.
+
+theorem iff_intro: \forall A,B. (A \to B) \to (B \to A) \to (A \liff B).
+ unfold Iff. intros. split; intros; autobatch.
+qed.
+
+theorem iff_refl: \forall A. A \liff A.
+ intros. apply iff_intro; intros; autobatch.
+qed.
+
+theorem iff_sym: \forall A,B. A \liff B \to B \liff A.
+ intros. elim H. apply iff_intro[assumption|assumption]
+qed.
+
+theorem iff_trans: \forall A,B,C. A \liff B \to B \liff C \to A \liff C.
+ intros. elim H. elim H1. apply iff_intro;intros;autobatch.
+qed.