Remark copy_trans_sigma_copy :
sigma_ctrl_copy s0 (bm (copy_trans c0 s1 s2)) =
(if eq_site_dec s0 s1
then  (sigma_ctrl_copy s0 (bm c0) + 1)%Z
else  sigma_ctrl_copy s0 (bm c0)).
Remark copy_trans_sigma_dec :
sigma_ctrl_dec s0 (bm (copy_trans c0 s1 s2)) = sigma_ctrl_dec s0 (bm c0).
Remark copy_trans_sigma_inc :
sigma_ctrl_inc s0 (bm (copy_trans c0 s1 s2)) = sigma_ctrl_inc s0 (bm c0).

Lemma copy_trans_sigma_ctrl :
sigma_ctrl s0 (bm (copy_trans c0 s1 s2)) =
(if eq_site_dec s0 s1
then  (sigma_ctrl s0 (bm c0) + 1)%Z
else  sigma_ctrl s0 (bm c0)).