Hint Extern 1 (lc_co (g_AbsCong _ _ _)) =>
  let x1 := fresh in
  pick_fresh x1;
  apply (lc_g_AbsCong_exists x1).Hint Extern 1 (lc_co (g_CAbsCong _ _ _)) =>
  let c1 := fresh in
  pick_fresh c1;
  apply (lc_g_CAbsCong_exists c1).

Lemma co_subst_co_brs_open_brs_wrt_tm_var : forall brs1 g1 c1 x1, lc_co g1 -> open_brs_wrt_tm (co_subst_co_brs g1 c1 brs1) (a_Var_f x1) = co_subst_co_brs g1 c1 (open_brs_wrt_tm brs1 (a_Var_f x1)).