Notation match_states':=
  (fun (idx s1 : state L1) (s2 : state L2) => idx = s1 /\ match_states s1 s2).

Lemma simulation_star: forall s1 t s1', Star L1 s1 t s1' -> forall i s2, match_states i s1 s2 -> exists i', exists s2', Star L2 s2 t s2' /\ match_states i' s1' s2'.