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 app_nth : forall {A} n l1 l2 (d : A), nth n (l1 ++ l2) d = if lt_dec n (length l1) then nth n l1 d else nth (n - length l1) l2 d.