Proof using.
induction l; simpl; intuition.
break_if; intuition.
rewrite max_id_for_client_default_on_max.
apply Max.le_max_l.
Qed.

Lemma max_id_for_client_default_subset :
forall l l' x c,
(forall e, In e l -> In e l') ->
max_id_for_client_default x c l <= max_id_for_client_default x c l'.