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 replace_nth_app : forall {A} n l1 l2 (a : A), replace_nth n (l1 ++ l2) a = if lt_dec n (length l1) then replace_nth n l1 a ++ l2 else l1 ++ replace_nth (n - length l1) l2 a.