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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.
-