Proof using.
intros.
induction log; simpl in *; auto.
- repeat break_match; simpl in *; constructor; auto.
Qed.

Lemma in_import_in_trace_O :
forall tr k,
In (O k) (import tr) ->
exists os h,
In (h, inr os) tr /\
exists o, In (ClientResponse (fst k) (snd k) o) os.