Proof using.
intros.
induction H; simpl; auto.
concludes.
rewrite app_ass.
constructor 2 with x'; auto.
Qed.
Definition inductive (step : step_relation) (P : A -> Prop)  :=
forall (a a': A) (os : list trace),
P a ->
step a a' os ->
P a'.

Theorem step_star_inductive :
forall step P,
inductive step P ->
forall (a : A) a' os,
P a ->
(refl_trans_1n_trace step) a a' os ->
P a'.