Proof using.
intros. induction tr1; simpl.
- auto.
- repeat break_match; subst; auto.
rewrite app_ass. auto using f_equal.
Qed.

Lemma correspond_preserved_primary_same_no_outputs :
forall sigma sigma' st tr_1 tr_m tr_m',
correspond st sigma tr_1 tr_m ->
sigma' Primary = sigma Primary ->
outputs_m tr_m' = [] ->
correspond st sigma' tr_1 (tr_m ++ tr_m').