Instance list_PCM (P : Ghost) : Ghost := { valid a := True; Join_G := list_join }.Definition ghost_list {P : Ghost} g l := own(RA := list_PCM P) g l NoneP.Definition list_incl {A} (l1 l2 : list (option A)) := (length l1 <= length l2)%nat /\
  forall n a, nth n l1 None = Some a -> nth n l2 None = Some a.

Lemma list_join_nth_error : forall {P : Ghost} l1 l2 l3 n, list_join l1 l2 l3 -> join (nth_error l1 n) (nth_error l2 n) (nth_error l3 n).