Section Maxl.
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).