Proof using.
induction es; intros; simpl in *; intuition.
- break_match; simpl in *; auto.
break_if; simpl in *; auto.
do_bool.
find_insterU. conclude_using eauto. omega.
- repeat break_match; simpl in *; auto.
+ f_equal. eapply IHes; eauto.
intros. do_bool.
destruct (clientId_eq_dec (eClient a) (eClient e)).
* repeat find_rewrite.
find_rewrite_lem get_set_same. find_injection.
find_insterU. conclude_using eauto. intuition.
* find_erewrite_lem get_set_diff.
eauto.
+ f_equal. eapply IHes; eauto.
intros. do_bool.
destruct (clientId_eq_dec (eClient a) (eClient e)).
* repeat find_rewrite.
find_rewrite_lem get_set_same. find_injection.
match goal with
| H : forall _, ?x = _ \/ _ -> _ |- _ =>
specialize (H x)
end.
intuition.
* find_erewrite_lem get_set_diff.
eauto.
Qed.

Lemma deduplicate_log_snoc_split :
forall es e,
(forall e', In e' es -> eClient e' = eClient e -> eId e' < eId e) ->
deduplicate_log (es ++ [e]) = deduplicate_log es ++ [e].