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(* -*- coq-prog-args: ("-emacs-U" "-nois") -*- *)
(************************************************************************)
(* 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 11282 2008-07-28 11:51:53Z 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 ((m : Morphism (A -> B) (RA ++> RB) f)) => Surjective : 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 ((m : Morphism (A -> A) (eqA ++> eqA))) [ ! IsoMorphism m ] => AutoMorphism.
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