Definition bb_order : bb_index -> bb_index -> Prop := lex_ord (clos_trans _ order) order'.Inductive bb_match_states: bb_index -> state L1 -> state L3 -> Prop :=
  | bb_match_later: forall i1 i2 s1 s3 s2x s2y,
      match_states i1 s1 s2x -> Star L2 s2x E0 s2y -> match_states' i2 s2y s3 ->
      bb_match_states (i1, i2) s1 s3.Inductive bb_match_states: bb_index -> state L1 -> state L3 -> Prop :=
  | bb_match_later: forall i1 i2 s1 s3 s2x s2y,
      match_states i1 s1 s2x -> Star L2 s2x E0 s2y -> match_states' i2 s2y s3 ->
      bb_match_states (i1, i2) s1 s3.

Lemma bb_match_at: forall i1 i2 s1 s3 s2, match_states i1 s1 s2 -> match_states' i2 s2 s3 -> bb_match_states (i1, i2) s1 s3.