Proof using.
induction tr; intros; simpl in *.
- subst. auto.
- simpl in *. repeat break_match; subst; intuition.
Qed.

(* Specifications: *)
Lemma reachable_intro :
forall a tr,
step_async_star step_async_init a tr ->
reachable step_async step_async_init a.
Lemma reachable_intro :
forall a tr,
step_async_star step_async_init a tr ->
reachable step_async step_async_init a.