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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                               *)
(*                                                                          *)
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(*                               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.

(* Specifications: *)
Lemma Injections_compose :
Injective _ _ f -> Injective _ _ g -> Injective _ _ (f o g).
Lemma Surjections_right : Surjective _ _ (f o g) -> Surjective _ _ g.
Lemma Surjections_compose :
Surjective _ _ f -> Surjective _ _ g -> Surjective _ _ (f o g).