Proof using.
induction l; simpl; intros.
- auto.
- break_if.
+ rewrite IHl.
* subst. now rewrite max_l by eauto.
* intros. subst. eapply le_trans; [| apply Max.le_max_l]. eauto.
+ auto.
Qed.

Lemma max_id_for_client_default_on_max :
forall c l x x',
max_id_for_client_default (max x x') c l =
max x (max_id_for_client_default x' c l).