(***************)
rewrite length_app. simpl in |- *. rewrite plus_n_SO. rewrite H. trivial with arith.
Qed.
(****************************************************************)
(* map *)
Fixpoint map (l : list A) : (A -> B) -> list B :=
(************************)
fun f : A -> B =>
match l return (list B) with
| nil => nil
| b :: l' => f b :: map l' f
end.

Lemma length_map :
forall (l : list A) (f : A -> B), length (map l f) = length l.