Section ROOTED1.

Lemma rooted_first_out :
forall (s0 s1 s2 : Site) (q : queue Message) (m : Message),
first Message q = value Message m ->
rooted s0 s1 s2 (first_out Message q) =
(rooted s0 s1 s2 q - rooted_fun s0 s1 s2 m)%Z.