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+
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(*i $Id: Setoid_Prop.v 10739 2008-04-01 14:45:20Z herbelin $: i*)
+
+Require Import Setoid_tac.
+
+(** * A few examples on [iff] *)
+
+(** [iff] as a relation *)
+
+Add Relation Prop iff
+  reflexivity proved by iff_refl
+  symmetry proved by iff_sym
+  transitivity proved by iff_trans
+as iff_relation.
+
+(** [impl] as a relation *)
+
+Theorem impl_trans: transitive _ impl.
+Proof.
+  hnf; unfold impl; tauto.
+Qed.
+
+Add Relation Prop impl
+  reflexivity proved by impl_refl
+  transitivity proved by impl_trans
+as impl_relation.
+
+(** [impl] is a morphism *)
+
+Add Morphism impl with signature iff ==> iff ==> iff as Impl_Morphism.
+Proof.
+  unfold impl; tauto.
+Qed.
+
+(** [and] is a morphism *)
+
+Add Morphism and with signature iff ==> iff ==> iff as And_Morphism.
+ tauto.
+Qed.
+
+(** [or] is a morphism *)
+
+Add Morphism or with signature iff ==> iff ==> iff as Or_Morphism.
+Proof.
+  tauto.
+Qed.
+
+(** [not] is a morphism *)
+
+Add Morphism not with signature iff ==> iff as Not_Morphism.
+Proof.
+  tauto.
+Qed.
+
+(** The same examples on [impl] *)
+
+Add Morphism and with signature impl ++> impl ++> impl as And_Morphism2.
+Proof.
+  unfold impl; tauto.
+Qed.
+
+Add Morphism or with signature impl ++> impl ++> impl as Or_Morphism2.
+Proof.
+  unfold impl; tauto.
+Qed.
+
+Add Morphism not with signature impl --> impl as Not_Morphism2.
+Proof.
+  unfold impl; tauto.
+Qed.
+