End SIMULATION_PLUS.
(** Lock-step simulation: each transition in the first semantics corresponds to exactly one transition in the second semantics. *)
Section SIMULATION_STEP.
Hypothesis simulation:
forall s1 t s1', Step L1 s1 t s1' ->
forall s2, match_states s1 s2 ->
exists s2', Step L2 s2 t s2' /\ match_states s1' s2'.

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