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.

Theorem forward_simulation_behavior_improves: forall L1 L2, forward_simulation L1 L2 -> forall beh1, program_behaves L1 beh1 -> exists beh2, program_behaves L2 beh2 /\ behavior_improves beh1 beh2.