Remark add_reduce :
forall x y z w a : Z, x = a -> (x + y)%Z = (a + (y + z + w) - w - z)%Z.

Lemma sigma_weight_change_queue :
forall (s1 s2 s3 : Site) (b : Bag_of_message) (q : queue Message),
b s1 s2 = input Message dec (input Message (inc_dec s3) q) ->
sigma_weight (change_queue (queue Message) b s1 s2 q) =
(sigma_weight b - cardinal_count dec - cardinal_count (inc_dec s3))%Z.