Section max_list_map.
Variable (X : Type) (f : X -> nat).
Definition max_list_map (xs : list X) := max_list (map f xs).
(* This version may be useful sometimes *)
Definition max_list_map_rec (s : nat) (xs : list X) := max_list_rec s (map f xs).

Lemma max_list_map_ge (xs : list X) (x : X) :
In x xs ->
f x <= max_list_map xs.