Section FACTOR_BACKWARD_SIMULATION.
Variable L1: semantics.
Variable L2: semantics.
Context index order match_states (sim: bsim_properties L1 L2 index order match_states).
Hypothesis L1single: single_events L1.
Hypothesis L2wb: well_behaved_traces L2.
Inductive fbs_match: index -> state L1 -> (trace * state L2) -> Prop :=
| fbs_match_intro: forall i s1 t s2 s1',
Star L1 s1 t s1' -> match_states i s1' s2 ->
t = E0 \/ output_trace t ->
fbs_match i s1 (t, s2).
End FACTOR_BACKWARD_SIMULATION.

Lemma atomic_initial: forall L s,
initial_state L s ->  initial_state (atomic L) (E0, s).