(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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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.