(** Constructing the backward simulation *)
Inductive f2b_match_states: f2b_index -> state L1 -> state L2 -> Prop :=
| f2b_match_at: forall i s1 s2,
match_states i s1 s2 ->
f2b_match_states (F2BI_after O) s1 s2
| f2b_match_before: forall s1 t s1' s2b s2 n s2a i,
Step L1 s1 t s1' ->  t <> E0 ->
Star L2 s2b E0 s2 ->
starN (step L2) (globalenv L2) n s2 t s2a ->
match_states i s1 s2b ->
f2b_match_states (F2BI_before n) s1 s2
| f2b_match_after: forall n s2 s2a s1 i,
starN (step L2) (globalenv L2) (S n) s2 E0 s2a ->
match_states i s1 s2a ->
f2b_match_states (F2BI_after (S n)) s1 s2.
Remark f2b_match_after':
forall n s2 s2a s1 i,
starN (step L2) (globalenv L2) n s2 E0 s2a ->
match_states i s1 s2a ->
f2b_match_states (F2BI_after n) s1 s2.
(** Backward simulation of L2 steps *)

Lemma f2b_simulation_step:
forall s2 t s2', Step L2 s2 t s2' ->
forall i s1, f2b_match_states i s1 s2 -> safe L1 s1 ->
exists i', exists s1',
(Plus L1 s1 t s1' \/ (Star L1 s1 t s1' /\ f2b_order i' i))
/\ f2b_match_states i' s1' s2'.