Proof using.
unfold outputs_1.
induction tr; simpl; intros.
- auto.
- repeat break_match; subst.
+ eauto using NOABT_tail.
+ rewrite IHtr by eauto using NOABT_tail.
rewrite map_app. rewrite map_of_filterMap.
f_equal. apply filterMap_ext. intros.
repeat break_match; auto.
+ rewrite IHtr by eauto using NOABT_tail. auto.
+ find_copy_apply_lem_hyp NOABT_tail.
find_apply_lem_hyp NOABT_contra. intuition; break_exists;
subst; simpl; auto.
Qed.

Lemma NOABT_nil :
forall A,
@no_output_at_backup_trace A [].