Definition rec_dec_trans (c : Config) (s1 s2 : Site) :=
  mkconfig (S (time c)) (dt c) (Dec_send_table (st c) s2) 
    (rt c) (Collect_message (bm c) s1 s2).

Lemma rdec_trans_ctrl_dec : forall s3 : Site, ctrl_dec s0 s3 (bm (rec_dec_trans c0 s1 s2)) = (if eq_site_dec s0 s2 then if eq_site_dec s3 s1 then (ctrl_dec s0 s3 (bm c0) - 1)%Z else ctrl_dec s0 s3 (bm c0) else ctrl_dec s0 s3 (bm c0)).