Section CARDINAL_EFFECT.
(* This section is really using the inductive type Message *)
Let Bag_of_message := Bag_of_Data Message.
Variable inc_fun : Site -> Z.

Lemma cardinal_collect_message :
forall (m : Message) (s1 s2 : Site) (b : Bag_of_message),
first Message (b s1 s2) = value Message m ->
cardinal (Collect_message Message b s1 s2 s1 s2) =
(cardinal (b s1 s2) - cardinal_count m)%Z.