Proof using.
induction tr; intros; simpl in *; intuition.
repeat break_match; subst; intuition.
- find_apply_hyp_hyp. break_exists_exists.
intuition.
- simpl in *. intuition; try congruence.
find_apply_lem_hyp in_remove.
find_apply_hyp_hyp. break_exists_exists.
intuition.
- do_in_app. intuition.
+ find_apply_lem_hyp in_dedup_was_in.
find_apply_lem_hyp In_filterMap.
break_exists. intuition.
break_match; try congruence.
find_inversion.
repeat eexists; intuition eauto.
+ find_apply_lem_hyp in_remove_all_was_in.
find_apply_hyp_hyp. break_exists_exists.
intuition.
Qed.

Lemma in_import_in_trace_I :
forall tr k,
In (I k) (import tr) ->
exists h i,
In (h, inl (ClientRequest (fst k) (snd k) i)) tr.