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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Export Relation_Definitions.
Require Export Relation_Operators.
Require Export Operators_Properties.
Lemma inverse_image_of_equivalence :
forall (A B:Type) (f:A -> B) (r:relation B),
equivalence B r -> equivalence A (fun x y:A => r (f x) (f y)).
Proof.
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 :
forall (A B:Type) (f:A -> B), equivalence A (fun x y:A => f x = f y).
Proof.
split; red;
[ (* reflexivity *) reflexivity
| (* transitivity *) intros; transitivity (f y); assumption
| (* symmetry *) intros; symmetry ; assumption ].
Qed.
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