Proof using.
intros.
unfold assoc_default in *.
break_match; intuition; eauto; omega.
Qed.

(* Specifications: *)
Lemma applyEntries_log_to_ks' :
forall l h st o st',
applyEntries h st l = (o, st') ->
a_equiv clientId_eq_dec (clientCache_to_ks (clientCache st'))
(log_to_ks' l (clientCache_to_ks (clientCache st))).