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).