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)).