Proof using.
induction tr; simpl; auto.
destruct a, s; simpl.
- rewrite IHtr; auto.
- destruct l; auto.
rewrite IHtr; auto.
Qed.

Lemma exported_filterMap_trace_non_empty_out :
forall tr tr' l tr1,
exported (get_input tr') (get_output tr') l tr1 ->
filterMap trace_non_empty_out tr = filterMap trace_non_empty_out tr' ->
exported (get_input tr) (get_output tr) l tr1.