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}.