(** Exploiting forward simulation *)

Inductive f2b_transitions: state L1 -> state L2 -> Prop :=
  | f2b_trans_final: forall s1 s2 s1' r,
      Star L1 s1 E0 s1' ->
      final_state L1 s1' r ->
      final_state L2 s2 r ->
      f2b_transitions s1 s2
  | f2b_trans_step: forall s1 s2 s1' t s1'' s2' i' i'',
      Star L1 s1 E0 s1' ->
      Step L1 s1' t s1'' ->
      Plus L2 s2 t s2' ->
      match_states i' s1' s2 ->
      match_states i'' s1'' s2' ->
      f2b_transitions s1 s2.(** Exploiting forward simulation *)

Inductive f2b_transitions: state L1 -> state L2 -> Prop :=
  | f2b_trans_final: forall s1 s2 s1' r,
      Star L1 s1 E0 s1' ->
      final_state L1 s1' r ->
      final_state L2 s2 r ->
      f2b_transitions s1 s2
  | f2b_trans_step: forall s1 s2 s1' t s1'' s2' i' i'',
      Star L1 s1 E0 s1' ->
      Step L1 s1' t s1'' ->
      Plus L2 s2 t s2' ->
      match_states i' s1' s2 ->
      match_states i'' s1'' s2' ->
      f2b_transitions s1 s2.(** Exploiting forward simulation *)

Inductive f2b_transitions: state L1 -> state L2 -> Prop :=
  | f2b_trans_final: forall s1 s2 s1' r,
      Star L1 s1 E0 s1' ->
      final_state L1 s1' r ->
      final_state L2 s2 r ->
      f2b_transitions s1 s2
  | f2b_trans_step: forall s1 s2 s1' t s1'' s2' i' i'',
      Star L1 s1 E0 s1' ->
      Step L1 s1' t s1'' ->
      Plus L2 s2 t s2' ->
      match_states i' s1' s2 ->
      match_states i'' s1'' s2' ->
      f2b_transitions s1 s2.

Lemma fsim_simulation_not_E0: forall s1 t s1', Step L1 s1 t s1' -> t <> E0 -> forall i s2, match_states i s1 s2 -> exists i', exists s2', Plus L2 s2 t s2' /\ match_states i' s1' s2'.