Definition Bag_of_Data := Site -> Site -> queue Data.(* This is taken straight from Jean's file *)

Definition Post_message (m0 : Message) (b0 : Bag_of_message)
  (s0 s1 : Site) :=
  change_queue (queue Message) b0 s0 s1 (input Message m0 (b0 s0 s1)).Definition Collect_message (b0 : Bag_of_message) (s0 s1 : Site) :=
  change_queue (queue Message) b0 s0 s1 (first_out Message (b0 s0 s1)).

Lemma reduce_collect_message : forall (m : Message) (s1 s2 : Site) (b : Bag_of_message), first Message (b s1 s2) = value Message m -> reduce Message f (Collect_message Message b s1 s2 s1 s2) = (reduce Message f (b s1 s2) - f m)%Z.