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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 *)
(************************************************************************)
(* Functional morphisms.
Author: Matthieu Sozeau
Institution: LRI, CNRS UMR 8623 - UniversitÃcopyright Paris Sud
91405 Orsay, France *)
(* $Id: Functions.v 11709 2008-12-20 11:42:15Z msozeau $ *)
Require Import Coq.Classes.RelationClasses.
Require Import Coq.Classes.Morphisms.
Set Implicit Arguments.
Unset Strict Implicit.
Class Injective `(m : Morphism (A -> B) (RA ++> RB) f) : Prop :=
injective : forall x y : A, RB (f x) (f y) -> RA x y.
Class Surjective `(m : Morphism (A -> B) (RA ++> RB) f) : Prop :=
surjective : forall y, exists x : A, RB y (f x).
Definition Bijective `(m : Morphism (A -> B) (RA ++> RB) (f : A -> B)) :=
Injective m /\ Surjective m.
Class MonoMorphism `(m : Morphism (A -> B) (eqA ++> eqB)) :=
monic :> Injective m.
Class EpiMorphism `(m : Morphism (A -> B) (eqA ++> eqB)) :=
epic :> Surjective m.
Class IsoMorphism `(m : Morphism (A -> B) (eqA ++> eqB)) :=
{ monomorphism :> MonoMorphism m ; epimorphism :> EpiMorphism m }.
Class AutoMorphism `(m : Morphism (A -> A) (eqA ++> eqA)) {I : IsoMorphism m}.
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