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

Lemma forallb_map : forall A B (l : list A) (p : B -> bool) (f : A -> B), (forall x, In x l -> p (f x) = true) -> forallb p (map f l) = true.