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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: Relations.v 8642 2006-03-17 10:09:02Z notin $ 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.
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