Proof using.
induction tr; intros; simpl in *; intuition.
repeat break_match; subst.
- find_apply_hyp_hyp. break_exists.
eauto 10.
- simpl in *. intuition.
+ find_inversion. simpl. eauto 10.
+ find_apply_lem_hyp in_remove.
find_apply_hyp_hyp. break_exists. eauto 10.
- do_in_app. intuition.
+ find_apply_lem_hyp in_dedup_was_in.
find_eapply_lem_hyp In_filterMap. break_exists. break_and.
break_match; discriminate.
+ find_eapply_lem_hyp in_remove_all_was_in.
find_apply_hyp_hyp. break_exists. eauto 10.
Qed.

Lemma in_applied_entries_in_IR :
forall log e client id env,
eClient e = client ->
eId e = id ->
In e log ->
(exists o, env (client, id) = Some o) ->
In (IRO (client, id)) (log_to_IR env log).