(* This file is just for reference , 
    in order to show the real contents 
    of functions to be defined in other files.
    In this the argument X is explicitly written
    everywhere.
*)

(* Lists of nat  *)

Inductive natlist : Type :=
  | nil_nat : natlist
  | cons_nat : nat -> natlist ->  natlist.


(* Lists of something  *)

Inductive list (X: Type) : Type :=
  | nil : list X
  | cons :  X -> list X -> list X.

Fixpoint app (X:Type) (u v: list X) : list X :=
  match u with
  | nil => v
  | cons x u' => cons X x (app X u' v)
  end.

Fixpoint length (X:Type) (u: list X) : nat :=
  match u with
  | nil => O
  | cons x u' => S (length X u')
  end.

(* Example of a statement *)
Theorem length_morphism :
   forall X u v, 
   length X (app X u v) = length X u + length X v.