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

Lemma maxl_map_0 A l (f : A -> nat) : (* if n=0, we don't need l ≠ [] *) (forall x, In x l -> (f x) = 0) -> maxl (map f l) = 0.