Definition zero_or_one_outputs_per_step_trace {A B C} tr :=
    forall x, In x tr -> @zero_or_one_outputs_per_step A B C x.

Lemma ZOOOPST_app : forall A B C xs ys, @zero_or_one_outputs_per_step_trace A B C xs -> zero_or_one_outputs_per_step_trace ys -> zero_or_one_outputs_per_step_trace (xs ++ ys).