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_singleton : forall {P : Ghost} n a c l (Hn : (n < length l)%nat) (Hjoin: join (Some a) (nth n l None) (Some c)), list_join (list_singleton n a) l (replace_nth n l (Some c)).