Proof using.
intros.
destruct l; simpl in *.
- discriminate.
- find_inversion. find_apply_lem_hyp inputs_1_nil_is_nil. subst.
destruct p. eauto.
Qed.

Lemma step_1_star_singleton_trace :
forall st st' i os,
step_1_star st st' [(i, os)] ->
step_1 st st' [(i, os)].