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(* 02110-1301 USA                                                     *)
(** list. Polymorphic lists with some operations.  @author Olga Caprotti and Martijn Oostdijk @version $Revision$ *)
(** * Polymorphic lists *)
Inductive list (A : Set) : Set :=
| Nil : list A
| Cons : A -> list A -> list A.
(** Length of a list *)
Fixpoint length (A : Set) (l : list A) {struct l} : nat :=
match l with
| Nil => 0
| Cons a r => S (length A r)
end.

(* Specifications: *)
Lemma length_0 : forall (A : Set) (l : list A), length A l = 0 -> l = Nil A.
Lemma length_S :
forall (A : Set) (l : list A) (n : nat),
length A l = S n ->
exists h : A, (exists t : list A, l = Cons A h t /\ length A t = n).