Inductive plus: state -> state -> Prop :=
  | plus_left: forall s1 s2 s3,
      CCstep s1 s2 -> star s2 s3 -> plus s1 s3.Inductive plus: state -> state -> Prop :=
  | plus_left: forall s1 s2 s3,
      CCstep s1 s2 -> star s2 s3 -> plus s1 s3.

Lemma plus_star_trans: forall {s1 s2 s3}, plus s1 s2 -> star s2 s3 -> plus s1 s3.