Section Equality.
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_simulation_atx:
simulation_atx Eqmatch_states;
Eqfsim_atx:
preserves_atx Eqmatch_states;
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),
(simulation_atx match_states) ->
(preserves_atx match_states) ->
(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.