Fixpoint maxl (l : list nat) : nat :=
  match l with
  | nil => 0
  | n :: l' => max n (maxl l')
  end.

Lemma maxl_map_le_iff (A:Type): forall (l:list A) f n, maxl (map f l) <= n <-> (forall m, In m l -> (f m) <= n).