Record fsim_properties_eq: Type :=
      {
        Eqindex: Type;
        Eqorder: Eqindex -> Eqindex -> Prop;
        Eqmatch_states: Eqindex -> state L1 -> state L2 -> Prop;  
        Eqfsim_order_wf: well_founded Eqorder;
        Eqfsim_match_meminj: forall i s1 s2, Eqmatch_states i s1 s2 ->  (get_mem1 s1) = (get_mem2 s2);
        (*    fsim_match_initial_states:
      forall s1 m1 f m2, initial_state L1 (s1,m1) -> Mem.inject f m1 m2 ->
      exists i, exists s2, initial_state L2 (s2,m2) /\ match_states i f (s1,m1) (s2,m2);*)
        Eqfsim_match_initial_states:
          forall s1, initial_state L1 s1 -> 
                exists i s2, initial_state L2 s2 /\ Eqmatch_states i s1 s2;
        Eqfsim_match_final_states:
          forall i s1 s2 r ,
            Eqmatch_states i s1 s2 -> final_state L1 s1 r -> (final_state L2 s2 r);
        Eqfsim_simulation:
          forall s1 t s1', Step L1 s1 t s1' ->
                      forall i s2, Eqmatch_states i s1 s2 ->
                              exists i', exists s2',
                                  (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ Eqorder i' i))
                                  /\ Eqmatch_states i' s1' s2';
        Eqfsim_public_preserved:
          forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
      }.Record fsim_properties_eq: Type :=
      {
        Eqindex: Type;
        Eqorder: Eqindex -> Eqindex -> Prop;
        Eqmatch_states: Eqindex -> state L1 -> state L2 -> Prop;  
        Eqfsim_order_wf: well_founded Eqorder;
        Eqfsim_match_meminj: forall i s1 s2, Eqmatch_states i s1 s2 ->  (get_mem1 s1) = (get_mem2 s2);
        (*    fsim_match_initial_states:
      forall s1 m1 f m2, initial_state L1 (s1,m1) -> Mem.inject f m1 m2 ->
      exists i, exists s2, initial_state L2 (s2,m2) /\ match_states i f (s1,m1) (s2,m2);*)
        Eqfsim_match_initial_states:
          forall s1, initial_state L1 s1 -> 
                exists i s2, initial_state L2 s2 /\ Eqmatch_states i s1 s2;
        Eqfsim_match_final_states:
          forall i s1 s2 r ,
            Eqmatch_states i s1 s2 -> final_state L1 s1 r -> (final_state L2 s2 r);
        Eqfsim_simulation:
          forall s1 t s1', Step L1 s1 t s1' ->
                      forall i s2, Eqmatch_states i s1 s2 ->
                              exists i', exists s2',
                                  (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ Eqorder i' i))
                                  /\ Eqmatch_states i' s1' s2';
        Eqfsim_public_preserved:
          forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
      }.Record fsim_properties_eq: Type :=
      {
        Eqindex: Type;
        Eqorder: Eqindex -> Eqindex -> Prop;
        Eqmatch_states: Eqindex -> state L1 -> state L2 -> Prop;  
        Eqfsim_order_wf: well_founded Eqorder;
        Eqfsim_match_meminj: forall i s1 s2, Eqmatch_states i s1 s2 ->  (get_mem1 s1) = (get_mem2 s2);
        (*    fsim_match_initial_states:
      forall s1 m1 f m2, initial_state L1 (s1,m1) -> Mem.inject f m1 m2 ->
      exists i, exists s2, initial_state L2 (s2,m2) /\ match_states i f (s1,m1) (s2,m2);*)
        Eqfsim_match_initial_states:
          forall s1, initial_state L1 s1 -> 
                exists i s2, initial_state L2 s2 /\ Eqmatch_states i s1 s2;
        Eqfsim_match_final_states:
          forall i s1 s2 r ,
            Eqmatch_states i s1 s2 -> final_state L1 s1 r -> (final_state L2 s2 r);
        Eqfsim_simulation:
          forall s1 t s1', Step L1 s1 t s1' ->
                      forall i s2, Eqmatch_states i s1 s2 ->
                              exists i', exists s2',
                                  (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ Eqorder i' i))
                                  /\ Eqmatch_states i' s1' s2';
        Eqfsim_public_preserved:
          forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
      }.Record fsim_properties_eq: Type :=
      {
        Eqindex: Type;
        Eqorder: Eqindex -> Eqindex -> Prop;
        Eqmatch_states: Eqindex -> state L1 -> state L2 -> Prop;  
        Eqfsim_order_wf: well_founded Eqorder;
        Eqfsim_match_meminj: forall i s1 s2, Eqmatch_states i s1 s2 ->  (get_mem1 s1) = (get_mem2 s2);
        (*    fsim_match_initial_states:
      forall s1 m1 f m2, initial_state L1 (s1,m1) -> Mem.inject f m1 m2 ->
      exists i, exists s2, initial_state L2 (s2,m2) /\ match_states i f (s1,m1) (s2,m2);*)
        Eqfsim_match_initial_states:
          forall s1, initial_state L1 s1 -> 
                exists i s2, initial_state L2 s2 /\ Eqmatch_states i s1 s2;
        Eqfsim_match_final_states:
          forall i s1 s2 r ,
            Eqmatch_states i s1 s2 -> final_state L1 s1 r -> (final_state L2 s2 r);
        Eqfsim_simulation:
          forall s1 t s1', Step L1 s1 t s1' ->
                      forall i s2, Eqmatch_states i s1 s2 ->
                              exists i', exists s2',
                                  (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ Eqorder i' i))
                                  /\ Eqmatch_states i' s1' s2';
        Eqfsim_public_preserved:
          forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
      }.Record fsim_properties_eq: Type :=
      {
        Eqindex: Type;
        Eqorder: Eqindex -> Eqindex -> Prop;
        Eqmatch_states: Eqindex -> state L1 -> state L2 -> Prop;  
        Eqfsim_order_wf: well_founded Eqorder;
        Eqfsim_match_meminj: forall i s1 s2, Eqmatch_states i s1 s2 ->  (get_mem1 s1) = (get_mem2 s2);
        (*    fsim_match_initial_states:
      forall s1 m1 f m2, initial_state L1 (s1,m1) -> Mem.inject f m1 m2 ->
      exists i, exists s2, initial_state L2 (s2,m2) /\ match_states i f (s1,m1) (s2,m2);*)
        Eqfsim_match_initial_states:
          forall s1, initial_state L1 s1 -> 
                exists i s2, initial_state L2 s2 /\ Eqmatch_states i s1 s2;
        Eqfsim_match_final_states:
          forall i s1 s2 r ,
            Eqmatch_states i s1 s2 -> final_state L1 s1 r -> (final_state L2 s2 r);
        Eqfsim_simulation:
          forall s1 t s1', Step L1 s1 t s1' ->
                      forall i s2, Eqmatch_states i s1 s2 ->
                              exists i', exists s2',
                                  (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ Eqorder i' i))
                                  /\ Eqmatch_states i' s1' s2';
        Eqfsim_public_preserved:
          forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
      }.Record fsim_properties_eq: Type :=
      {
        Eqindex: Type;
        Eqorder: Eqindex -> Eqindex -> Prop;
        Eqmatch_states: Eqindex -> state L1 -> state L2 -> Prop;  
        Eqfsim_order_wf: well_founded Eqorder;
        Eqfsim_match_meminj: forall i s1 s2, Eqmatch_states i s1 s2 ->  (get_mem1 s1) = (get_mem2 s2);
        (*    fsim_match_initial_states:
      forall s1 m1 f m2, initial_state L1 (s1,m1) -> Mem.inject f m1 m2 ->
      exists i, exists s2, initial_state L2 (s2,m2) /\ match_states i f (s1,m1) (s2,m2);*)
        Eqfsim_match_initial_states:
          forall s1, initial_state L1 s1 -> 
                exists i s2, initial_state L2 s2 /\ Eqmatch_states i s1 s2;
        Eqfsim_match_final_states:
          forall i s1 s2 r ,
            Eqmatch_states i s1 s2 -> final_state L1 s1 r -> (final_state L2 s2 r);
        Eqfsim_simulation:
          forall s1 t s1', Step L1 s1 t s1' ->
                      forall i s2, Eqmatch_states i s1 s2 ->
                              exists i', exists s2',
                                  (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ Eqorder i' i))
                                  /\ Eqmatch_states i' s1' s2';
        Eqfsim_public_preserved:
          forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
      }.Record fsim_properties_eq: Type :=
      {
        Eqindex: Type;
        Eqorder: Eqindex -> Eqindex -> Prop;
        Eqmatch_states: Eqindex -> state L1 -> state L2 -> Prop;  
        Eqfsim_order_wf: well_founded Eqorder;
        Eqfsim_match_meminj: forall i s1 s2, Eqmatch_states i s1 s2 ->  (get_mem1 s1) = (get_mem2 s2);
        (*    fsim_match_initial_states:
      forall s1 m1 f m2, initial_state L1 (s1,m1) -> Mem.inject f m1 m2 ->
      exists i, exists s2, initial_state L2 (s2,m2) /\ match_states i f (s1,m1) (s2,m2);*)
        Eqfsim_match_initial_states:
          forall s1, initial_state L1 s1 -> 
                exists i s2, initial_state L2 s2 /\ Eqmatch_states i s1 s2;
        Eqfsim_match_final_states:
          forall i s1 s2 r ,
            Eqmatch_states i s1 s2 -> final_state L1 s1 r -> (final_state L2 s2 r);
        Eqfsim_simulation:
          forall s1 t s1', Step L1 s1 t s1' ->
                      forall i s2, Eqmatch_states i s1 s2 ->
                              exists i', exists s2',
                                  (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ Eqorder i' i))
                                  /\ Eqmatch_states i' s1' s2';
        Eqfsim_public_preserved:
          forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
      }.Record fsim_properties_eq: Type :=
      {
        Eqindex: Type;
        Eqorder: Eqindex -> Eqindex -> Prop;
        Eqmatch_states: Eqindex -> state L1 -> state L2 -> Prop;  
        Eqfsim_order_wf: well_founded Eqorder;
        Eqfsim_match_meminj: forall i s1 s2, Eqmatch_states i s1 s2 ->  (get_mem1 s1) = (get_mem2 s2);
        (*    fsim_match_initial_states:
      forall s1 m1 f m2, initial_state L1 (s1,m1) -> Mem.inject f m1 m2 ->
      exists i, exists s2, initial_state L2 (s2,m2) /\ match_states i f (s1,m1) (s2,m2);*)
        Eqfsim_match_initial_states:
          forall s1, initial_state L1 s1 -> 
                exists i s2, initial_state L2 s2 /\ Eqmatch_states i s1 s2;
        Eqfsim_match_final_states:
          forall i s1 s2 r ,
            Eqmatch_states i s1 s2 -> final_state L1 s1 r -> (final_state L2 s2 r);
        Eqfsim_simulation:
          forall s1 t s1', Step L1 s1 t s1' ->
                      forall i s2, Eqmatch_states i s1 s2 ->
                              exists i', exists s2',
                                  (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ Eqorder i' i))
                                  /\ Eqmatch_states i' s1' s2';
        Eqfsim_public_preserved:
          forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
      }.Record fsim_properties_eq: Type :=
      {
        Eqindex: Type;
        Eqorder: Eqindex -> Eqindex -> Prop;
        Eqmatch_states: Eqindex -> state L1 -> state L2 -> Prop;  
        Eqfsim_order_wf: well_founded Eqorder;
        Eqfsim_match_meminj: forall i s1 s2, Eqmatch_states i s1 s2 ->  (get_mem1 s1) = (get_mem2 s2);
        (*    fsim_match_initial_states:
      forall s1 m1 f m2, initial_state L1 (s1,m1) -> Mem.inject f m1 m2 ->
      exists i, exists s2, initial_state L2 (s2,m2) /\ match_states i f (s1,m1) (s2,m2);*)
        Eqfsim_match_initial_states:
          forall s1, initial_state L1 s1 -> 
                exists i s2, initial_state L2 s2 /\ Eqmatch_states i s1 s2;
        Eqfsim_match_final_states:
          forall i s1 s2 r ,
            Eqmatch_states i s1 s2 -> final_state L1 s1 r -> (final_state L2 s2 r);
        Eqfsim_simulation:
          forall s1 t s1', Step L1 s1 t s1' ->
                      forall i s2, Eqmatch_states i s1 s2 ->
                              exists i', exists s2',
                                  (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ Eqorder i' i))
                                  /\ Eqmatch_states i' s1' s2';
        Eqfsim_public_preserved:
          forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
      }.Record fsim_properties_eq: Type :=
      {
        Eqindex: Type;
        Eqorder: Eqindex -> Eqindex -> Prop;
        Eqmatch_states: Eqindex -> state L1 -> state L2 -> Prop;  
        Eqfsim_order_wf: well_founded Eqorder;
        Eqfsim_match_meminj: forall i s1 s2, Eqmatch_states i s1 s2 ->  (get_mem1 s1) = (get_mem2 s2);
        (*    fsim_match_initial_states:
      forall s1 m1 f m2, initial_state L1 (s1,m1) -> Mem.inject f m1 m2 ->
      exists i, exists s2, initial_state L2 (s2,m2) /\ match_states i f (s1,m1) (s2,m2);*)
        Eqfsim_match_initial_states:
          forall s1, initial_state L1 s1 -> 
                exists i s2, initial_state L2 s2 /\ Eqmatch_states i s1 s2;
        Eqfsim_match_final_states:
          forall i s1 s2 r ,
            Eqmatch_states i s1 s2 -> final_state L1 s1 r -> (final_state L2 s2 r);
        Eqfsim_simulation:
          forall s1 t s1', Step L1 s1 t s1' ->
                      forall i s2, Eqmatch_states i s1 s2 ->
                              exists i', exists s2',
                                  (Plus L2 s2 t s2' \/ (Star L2 s2 t s2' /\ Eqorder i' i))
                                  /\ Eqmatch_states i' s1' s2';
        Eqfsim_public_preserved:
          forall id, Senv.public_symbol (symbolenv L2) id = Senv.public_symbol (symbolenv L1) id
      }.

Lemma sim_eqSim': forall index order (match_states:index -> state L1 -> state L2 -> Prop), (forall i s1 s2, match_states i s1 s2 -> (get_mem1 s1) = (get_mem2 s2)) -> fsim_properties L1 L2 index order match_states -> fsim_properties_eq.