Definition thread_step_plus U c m c' m' :=
    exists n, thread_stepN (S n) U c m c' m'.Definition thread_step_star U c m c' m' :=
    exists n, thread_stepN n U c m c' m'.

Lemma thread_step_star_plus_trans : forall U c1 c2 c3 m1 m2 m3, thread_step_star U c1 m1 c2 m2 -> thread_step_plus U c2 m2 c3 m3 -> thread_step_plus U c1 m1 c3 m3.