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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,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*)
Require Export Relation_Definitions.
Require Export Relation_Operators.
Require Export Operators_Properties.
Lemma inverse_image_of_equivalence : (A,B:Set)(f:A->B)
(r:(relation B))(equivalence B r)->(equivalence A [x,y:A](r (f x) (f y))).
Intros; Split; Elim H; Red; Auto.
Intros _ equiv_trans _ x y z H0 H1; Apply equiv_trans with (f y); Assumption.
Qed.
Lemma inverse_image_of_eq : (A,B:Set)(f:A->B)
(equivalence A [x,y:A](f x)=(f y)).
Split; Red;
[ (* reflexivity *) Reflexivity
| (* transitivity *) Intros; Transitivity (f y); Assumption
| (* symmetry *) Intros; Symmetry; Assumption
].
Qed.
|
