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

Lemma equivalent_filterMap_trace_non_empty_out :
forall tr tr' l,
equivalent key (import tr') l ->
filterMap trace_non_empty_out tr = filterMap trace_non_empty_out tr' ->
equivalent key (import tr) l.