Definition close_tm_erase_all := fun x => close_tmco_erase_all (inl x).
Definition close_co_erase_all := fun x => close_tmco_erase_all (inr x).
Definition close_tm_erase_tm := fun x => proj1 (close_tm_erase_all x).
Definition close_co_erase_tm := fun x => proj1 (close_co_erase_all x).
*)
(* Pointless to close erased terms wrt co variables *)

Lemma close_co_erase_rec : ∀ x : covar,
(∀ (a : tm)         k, close_co_rec k x (erase a) = erase a) /\
(∀ (b : brs)        k, close_co_rec k x (erase b) = erase b) /\
(∀ _ : co, True) /\
(∀ (c : constraint) k, close_co_rec k x (erase c) = erase c).