Remark del_trans_sigma_copy :
sigma_ctrl_copy s0 (bm (delete_trans c0 s1)) = sigma_ctrl_copy s0 (bm c0).
Remark del_trans_sigma_dec :
s0 <> owner ->
sigma_ctrl_dec s0 (bm (delete_trans c0 s1)) = sigma_ctrl_dec s0 (bm c0).
Remark del_trans_sigma_inc :
sigma_ctrl_inc s0 (bm (delete_trans c0 s1)) = sigma_ctrl_inc s0 (bm c0).

Lemma del_trans_sigma_ctrl :
s0 <> owner ->
sigma_ctrl s0 (bm (delete_trans c0 s1)) = sigma_ctrl s0 (bm c0).