End REC_DEC_EFFECT.
Section REC_DEC_CTRL.
(* decompte de messages *)
Variable c0 : Config.
Variable s0 s1 s2 : Site.
Hypothesis first_mess : first Message (bm c0 s1 s2) = value Message dec.

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)).