Proof using.
unfold correspond.
intros.
subst.
rewrite outputs_m_app.
repeat find_rewrite.
simpl in *.
repeat break_match. repeat find_inversion.
find_rewrite. find_inversion.
rewrite app_ass. auto.
Qed.

Lemma inputs_m_inr_singleton :
forall h l,
inputs_m [(h, inr l)] = [].