Proof using.
intros.
apply le_antisym; auto using max_id_for_client_default_subset.
Qed.

Lemma log_to_ks'_assoc_default_ks :
forall l ks c i,
i <= assoc_default clientId_eq_dec
(log_to_ks' l (assoc_set clientId_eq_dec ks c i))
c 0.