Lemma Par_lc1 : forall G D a a' , Par G D a a' -> lc_tm a.
  intros.  induction H; auto. all: lc_solve.
Qed.

(* FIXME: find a good place for this tactic. *)
(* cannot move this to ett_ind.v because need Toplevel_lc ??? *)
Ltac lc_toplevel_inversion :=
  ma

Lemma Par_open_tm_wrt_co_preservation: forall G D B1 B2 c, Par G D (open_tm_wrt_co B1 (g_Var_f c)) B2 -> exists B', B2 = open_tm_wrt_co B' (g_Var_f c) /\ Par G D (open_tm_wrt_co B1 (g_Var_f c)) (open_tm_wrt_co B' (g_Var_f c)).