Proof using.
induction l; simpl; intros.
- auto.
- repeat break_match; do_bool; rewrite IHl; subst; auto.
+ rewrite get_set_same_default.
now rewrite max_r by auto.
+ now rewrite get_set_diff_default by auto.
+ now rewrite max_l by omega.
Qed.

(* Specifications: *)
Lemma max_id_for_client_default_le :
forall l x c,
(forall e, In e l -> eClient e = c -> eId e <= x) ->
max_id_for_client_default x c l = x.