Proof using.
unfold correspond.
intros.
rewrite outputs_m_app, outputs_1_app in *.
repeat break_match.
rewrite app_ass.
simpl.
repeat find_rewrite.
rewrite processInputs_app in *.
repeat break_match.
repeat tuple_inversion.
simpl in *. break_match. tuple_inversion.
break_and. subst.
find_rewrite. tuple_inversion.
rewrite <- app_ass.
find_rewrite.
auto.
Qed.

Lemma inputs_m_backup_singleton :
forall i,
inputs_m [(Backup, inl i)] = [].