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(************************************************************************)
(* * The Rocq Prover / The Rocq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \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) *)
(************************************************************************)
From Stdlib Require Export Relation_Definitions.
From Stdlib Require Export Relation_Operators.
From Stdlib 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 A B f r H; 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.
intros A B f; split; red;
[ (* reflexivity *) reflexivity
| (* transitivity *) intros x y z; transitivity (f y); assumption
| (* symmetry *) intros; symmetry ; assumption ].
Qed.
|
