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 singleton_length : forall {A} n (a : A), length (list_singleton n a) = S n.