Remark rinc_trans_sigma_copy :
 forall s0 : Site,
 sigma_ctrl_copy s0 (bm (rec_inc_trans c0 s1 s2)) =
 sigma_ctrl_copy s0 (bm c0).Remark rinc_trans_sigma_dec :
 forall s0 : Site,
 sigma_ctrl_dec s0 (bm (rec_inc_trans c0 s1 s2)) =
 (if eq_site_dec s0 s2
  then (sigma_ctrl_dec s0 (bm c0) + 1)%Z
  else sigma_ctrl_dec s0 (bm c0)).Remark rinc_trans_sigma_inc :
 forall s0 : Site,
 sigma_ctrl_inc s0 (bm (rec_inc_trans c0 s1 s2)) =
 (if eq_site_dec s0 s2
  then (sigma_ctrl_inc s0 (bm c0) - 1)%Z
  else sigma_ctrl_inc s0 (bm c0)).

Lemma rinc_trans_sigma_ctrl : forall s0 : Site, sigma_ctrl s0 (bm (rec_inc_trans c0 s1 s2)) = sigma_ctrl s0 (bm c0).