Definition ctrl_copy (s1 : Site) (bom : Bag_of_message) :=
  card_mess copy (bom s0 s1).Definition ctrl_dec (s1 : Site) (bom : Bag_of_message) :=
  card_mess dec (bom s1 s0).Definition ctrl_inc (s1 : Site) (bom : Bag_of_message) :=
  card_mess (inc_dec s0) (bom s1 owner).Definition sigma_ctrl_copy (bom : Bag_of_message) :=
  sigma Site LS (fun si : Site => ctrl_copy si bom).Definition sigma_ctrl_dec (bom : Bag_of_message) :=
  sigma Site LS (fun si : Site => ctrl_dec si bom).Definition sigma_ctrl_inc (bom : Bag_of_message) :=
  sigma Site LS (fun si : Site => ctrl_inc si bom).Definition sigma_ctrl (bom : Bag_of_message) :=
  (sigma_ctrl_copy bom + sigma_ctrl_dec bom + sigma_ctrl_inc bom)%Z.

Lemma ctrl_copy_pos : forall s0 s1 : Site, (ctrl_copy s0 s1 bom >= 0)%Z.