Library Coq.Classes.Functions
Require Import Coq.Classes.RelationClasses.
Require Import Coq.Classes.Morphisms.
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}.