(***********************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
(*   \VV/  *************************************************************)
(*    //   *      This file is distributed under the terms of the      *)
(*         *       GNU Lesser General Public License Version 2.1       *)
(***********************************************************************)

(*i $Id$ i*)

Require Export Relation_Definitions.
Require Export Relation_Operators.
Require Export Operators_Properties.

Lemma inverse_image_of_equivalence :
 forall (A B:Set) (f:A -> B) (r:relation B),
   equivalence B r -> equivalence A (fun x y:A => r (f x) (f y)).
intros; split; elim H; red in |- *; auto.
intros _ equiv_trans _ x y z H0 H1; apply equiv_trans with (f y); assumption.
Qed.

Lemma inverse_image_of_eq :
 forall (A B:Set) (f:A -> B), equivalence A (fun x y:A => f x = f y).
split; red in |- *;
 [  (* reflexivity *) reflexivity
 |  (* transitivity *) intros; transitivity (f y); assumption
 |  (* symmetry *) intros; symmetry  in |- *; assumption ].
Qed.