Remark rdec_trans_sigma_copy :
sigma_ctrl_copy s0 (bm (rec_dec_trans c0 s1 s2)) =
sigma_ctrl_copy s0 (bm c0).
Remark rdec_trans_sigma_dec :
sigma_ctrl_dec s0 (bm (rec_dec_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 rdec_trans_sigma_inc :
sigma_ctrl_inc s0 (bm (rec_dec_trans c0 s1 s2)) = sigma_ctrl_inc s0 (bm c0).

Lemma rdec_trans_sigma_ctrl :
sigma_ctrl s0 (bm (rec_dec_trans c0 s1 s2)) =
(if eq_site_dec s0 s2
then (sigma_ctrl s0 (bm c0) - 1)%Z
else sigma_ctrl s0 (bm c0)).