CoFixpoint build_traceinf' (s1: state L) (t1: trace) (ST: Star L s0 t1 s1) : traceinf' :=
  match reacts' ST with
  | existT s2 (exist t2 (conj A B)) =>
      Econsinf' t2
                (build_traceinf' (star_trans ST A (eq_refl _)))
                B
  end.

Lemma forward_simulation_state_behaves: forall i s1 s2 beh1, match_states i s1 s2 -> state_behaves L1 s1 beh1 -> exists beh2, state_behaves L2 s2 beh2 /\ behavior_improves beh1 beh2.