Program Definition bupd (P: pred rmap): pred rmap :=
  fun m => forall c, joins (ghost_of m) (ghost_approx m c) ->
    exists b, joins b (ghost_approx m c) /\
    exists m', level m' = level m /\ resource_at m' = resource_at m /\ ghost_of m' = b /\ P m'.Definition ghost_fp_update_ND a B :=
  forall n c, joins (ghost_fmap (approx n) (approx n) a) c ->
    exists b, B b /\ joins (ghost_fmap (approx n) (approx n) b) c.Definition ghost_fp_update (a b : ghost) :=
  forall n c, joins (ghost_fmap (approx n) (approx n) a) c ->
               joins (ghost_fmap (approx n) (approx n) b) c.Instance ghost_fp_update_preorder: RelationClasses.PreOrder ghost_fp_update.

(* Specifications: *)
Lemma ghost_fp_update_approx: forall a b n, ghost_fp_update a b -> ghost_fp_update (ghost_fmap (approx n) (approx n) a) (ghost_fmap (approx n) (approx n) b).