Bijection.v

(** * Bijective functions *)

(* Author: Maximilian Wuttke *)


Require Import PslBase.Base.


Section Bijection.
  Variable X Y : Type.

  (*
   *      f
   *   ------>
   * X         Y
   *   <------
   *      g
   *)

  Definition left_inverse  (f : X -> Y) (g : Y -> X) := forall x : X, g (f x) = x.
  Definition right_inverse (f : X -> Y) (g : Y -> X) := forall y : Y, f (g y) = y.
  Definition inverse (f : X -> Y) (g : Y -> X) := left_inverse f g /\ right_inverse f g.

  Definition injective (f : X -> Y) :=
    forall x y, f x = f y -> x = y.

  Lemma left_inv_inj (f : X -> Y) (g : Y -> X) : left_inverse f g -> injective f.
  Proof.
    intros HInv. hnf in *. intros x1 x2 Heq.
    enough (g (f x1) = g (f x2)) as L by now rewrite !HInv in L.
    f_equal. assumption.
  Qed.

  Definition surjective (f : X -> Y) :=
    forall y, exists x, f x = y.

  Lemma right_inv_surjective f g :
    right_inverse f g -> surjective f.
  Proof. intros HInv. hnf. eauto. Qed.

  Definition bijective (f : X -> Y) :=
    injective f /\ surjective f.
  
  Lemma inverse_bijective f g :
    inverse f g -> bijective f.
  Proof.
    intros (HInv1&HInv2). hnf. split.
    - eapply left_inv_inj; eauto.
    - eapply right_inv_surjective; eauto.
  Qed.

End Bijection.