Proof using.
unfold no_output_at_backup_trace, no_output_at_backup.
intros. simpl in *.
find_insterU.
econcludes.
find_insterU.
simpl in *.
econcludes.
intuition.
repeat break_match; intuition eauto.
Qed.

Lemma outputs_m_revert_trace :
forall tr,
no_output_at_backup_trace tr ->
outputs_m tr = outputs_1 (revert_trace tr).