Proof using.
induction l; simpl; intros.
- auto.
- break_if; repeat rewrite IHl; auto with *.
zify. omega.
Qed.

Lemma max_id_for_client_default_or_entry :
forall c l x,
max_id_for_client_default x c l = x \/
exists e, In e l /\ eClient e = c /\ max_id_for_client_default x c l = eId e.