Proof using.
unfold zero_or_one_outputs_per_step_trace, zero_or_one_outputs_per_step.
simpl. intuition.
Qed.

(* Specifications: *)
Lemma ZOOOPST_head :
forall A B C x y,
@zero_or_one_outputs_per_step_trace A B C (x :: y) ->
zero_or_one_outputs_per_step x.