Proof using.
induction l; simpl; intuition; subst.
- break_if; try congruence.
rewrite Max.max_comm.
rewrite max_id_for_client_default_on_max.
apply Max.le_max_l.
- break_if; auto.
Qed.

Lemma max_id_for_client_default_ge_default :
forall l x c,
x <= max_id_for_client_default x c l.