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

Lemma fsim_properties_plus: fsim_properties L1 L2 _ (ltof _ ( fun _ => O)) (fun (idx s1 : state L1) (s2 : state L2) => idx = s1 /\ match_states s1 s2).