Proof using.
intros.
pose proof decomposition_invariant.
find_apply_lem_hyp inductive_invariant_true_in_reachable.
unfold true_in_reachable in *. apply H0; auto.
Qed.

Lemma mutual_exclusion_invariant :
forall st,
reachable step_async step_async_init st ->
mutual_exclusion (nwState st).