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(* 02110-1301 USA                                                     *)


(* Contribution to the Coq Library   V6.3 (July 1999)                    *)

(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Functions.v                                *)
(****************************************************************************)

(* Basic Properties of functions over Set. G. Huet March 1996 *)

Section Mappings.
Variable X Y : Set.
Variable f : X -> Y.
Definition Injective := forall x x' : X, f x = f x' -> x = x'.
Definition Surjective := forall y : Y, exists x : X, y = f x.
End Mappings.

Notation One_one := (Injective _ _) (only parsing).
Notation Onto := (Surjective _ _) (only parsing).

Section Finiteness.
Variable X : Set.
Definition Finite := forall f : X -> X, Injective _ _ f -> Surjective _ _ f.
End Finiteness.

Section Composition.
Variable X Y Z : Set.
Variable f : X -> Y.
Variable g : Y -> Z.
Definition comp (x : X) := g (f x).
End Composition.

Notation "f 'o' g" := (comp _ _ _ f g) (at level 20, right associativity).

Hint Unfold Injective Surjective comp.

Section Preservation.
Variable X Y Z : Set.
Variable f : X -> Y.
Variable g : Y -> Z.

Lemma Injections_compose :
 Injective _ _ f -> Injective _ _ g -> Injective _ _ (f o g).
Proof.
(* Goal: forall (_ : Injective X Y f) (_ : Injective Y Z g), Injective X Z (comp X Y Z f g) *)
auto.
Qed.

Lemma Surjections_right : Surjective _ _ (f o g) -> Surjective _ _ g.
Proof.
(* Goal: forall _ : Surjective X Z (comp X Y Z f g), Surjective Y Z g *)
intro Sfg; red in |- *; intro z; elim (Sfg z).
(* Goal: forall (x : X) (_ : @eq Z z (comp X Y Z f g x)), @ex Y (fun x0 : Y => @eq Z z (g x0)) *)
intros x E; exists (f x); trivial.
Qed.

Lemma Surjections_compose :
 Surjective _ _ f -> Surjective _ _ g -> Surjective _ _ (f o g).
Proof.
(* Goal: forall (_ : Surjective X Y f) (_ : Surjective Y Z g), Surjective X Z (comp X Y Z f g) *)
unfold Surjective, comp in |- *; intros Sf Sg z.
(* Goal: @ex X (fun x : X => @eq Z z (g (f x))) *)
elim (Sg z); intros y E; elim (Sf y); intros x E'.
(* Goal: @ex X (fun x : X => @eq Z z (g (f x))) *)
exists x; rewrite E; rewrite E'; trivial.
Qed.

End Preservation.