Section First.
Variable A : Type.
(* Take the first elements of a list *)
Fixpoint first_n (l : list A) (n : nat) {struct n} :
list A :=
match n with
| O => nil
| S n1 => match l with
| nil => nil
| a :: l1 => a :: first_n l1 n1
end
end.

(* Properties of first_n *)
Theorem first_n_app1 :
forall (n : nat) (l1 l2 : list A),
length l1 <= n -> first_n (l1 ++ l2) n = l1 ++ first_n l2 (n - length l1).