Proof using.
induction l; intros; simpl.
- rewrite assoc_default_assoc_set. auto.
- break_if; simpl in *; eauto.
do_bool.
destruct (clientId_eq_dec (eClient a) c); simpl in *; auto.
+ subst.
find_rewrite_lem assoc_default_assoc_set.
rewrite assoc_set_assoc_set_same.
eauto using le_trans.
+ erewrite assoc_default_a_equiv;
[|eapply log_to_ks'_a_equiv;
eapply assoc_set_assoc_set_diff; auto].
eauto.
Qed.

(* Specifications: *)
Lemma log_to_ks'_assoc_default_assoc_default_le :
forall l ks c,
assoc_default clientId_eq_dec ks c 0 <=
assoc_default clientId_eq_dec (log_to_ks' l ks) c 0.