## 6 Conclusions
\] (A16)  
Substituting (A13) and (A14) into (A12), the approximate expression of \(Pr\Big{\{}R_{R_{n}U_{2}}^{x_{2},10}<R_{2}\Big{\}}\) can be obtained as  
\[ \begin{array}{l}Pr\Big{\{}R_{R_{n}U_{2}}^{x_{2},10}\geq R_{2}\Big{\}}=exp \Big{(}-\frac{\sigma^{2}\gamma_{0}}{\lambda_{R_{n}U_{2}}P_{2}}\Big{)}\bigg{(}1 -\frac{\lambda_{R_{n}R}\sigma^{2}\gamma_{0}}{\lambda_{R_{n}U_{2}}I_{0}+\lambda _{R_{n}R}\sigma^{2}\gamma_{0}}exp\Big{(}-\frac{I_{0}}{\lambda_{R_{n}R}P_{2}} \Big{)}\bigg{)}\\ \times exp\Big{(}-\frac{\sigma^{2}\gamma_{2}}{\lambda_{R_{n}U_{2}}P_{2}}\Big{)} \bigg{(}1-\frac{\lambda_{R_{n}R}\sigma^{2}\gamma_{2}}{\lambda_{R_{n}U_{2}}I_{0 }+\lambda_{R_{n}R}\sigma^{2}\gamma_{2}}exp\Big{(}-\frac{I_{0}}{\lambda_{R_{n}R }P_{2}}\Big{)}\bigg{)}\\ \times\frac{\sigma^{2}}{\lambda_{T_{1}}U_{1}P_{T}}exp\Big{(}\frac{\sigma^{2}}{ \lambda_{T_{1}}U_{1}P_{T}}\Big{)}\cdot(Q_{9}-Q_{10}).\\ \end{array} \] (A17)  
**(c)**\(\Theta=\) '01':  
Note that the optimal relay must ensure that U\({}_{1}\) successfully decodes \(x_{2}\), we have  
\[ \begin{array}{l}Pr\Big{\{}R_{R_{n}U_{2}}^{x_{2},01}\geq R_{2}\Big{\}}&=Pr \bigg{\{}\rho_{R}\big{|}h_{R_{n}U_{2}}\big{|}^{2}<\gamma_{2},\frac{\rho_{R}|h_{ R_{n}U_{1}}|^{2}}{\rho_{T}|h_{T_{1}U_{1}}|^{2}+1}\geq\gamma_{2}\Big{\}}\\ &=Pr\Big{\{}\rho_{R}|h_{R_{n}U_{2}}\big{|}^{2}<\gamma_{2}\Big{\}}\cdot Pr\Big{\{} \rho_{R}\big{|}h_{R_{n}U_{1}}\big{|}^{2}\geq\Big{(}\rho_{T}|h_{T_{1}U_{1}}|^{2}+ 1\Big{)}\gamma_{2}\Big{\}}\\ &=exp\Big{(}-\frac{\sigma^{2}\gamma_{2}}{\lambda_{R_{n}U_{2}}P_{2}}\Big{)} \bigg{(}1-\frac{\lambda_{R_{n}R}\sigma^{2}\gamma_{2}}{\lambda_{R_{n}U_{2}}I_{0 }+\lambda_{R_{n}R}\sigma^{2}\gamma_{2}}exp\Big{(}-\frac{I_{0}}{\lambda_{R_{n} R}P_{2}}\Big{)}\bigg{)}\\ &\times\frac{\sigma^{2}}{\lambda_{T_{1}}U_{1}P_{T}}exp\Big{(}\frac{\sigma^{2}}{ \lambda_{T_{1}}U_{1}P_{T}}\Big{)}\cdot(Q_{11}-Q_{12}).\\ \end{array} \] (A18)  
where \(Q_{11}=\frac{1}{48}exp(-a_{6}),Q_{12}=exp\Big{(}-\frac{I_{0}}{(\lambda_{R_{n} R}P_{2})}\Big{)}exp(a_{6}b_{6})\Big{[}\frac{1}{48}exp(-a_{6}-a_{6}b_{6})-b_{6}E_{1}(a_{6}+a_{6}b_{6 })\Big{]}.\) In \(Q_{11}\) and \(Q_{12}\), \(a_{6}=\frac{\sigma^{2}\left(\gamma_{2}\lambda_{T_{1}}U_{1}P_{T}+\lambda_{R_{n}U _{1}}P_{2}\right)}{\left(\lambda_{R_{n}U_{1}}P_{2}+\lambda_{T_{1}}U_{1}P_{T} \right)}\) and \(b_{6}=\frac{\lambda_{R_{n}U_{1}}I_{0}}{\lambda_{R_{n}R}\sigma^{2}\gamma_{2}}.\)  
**(d)**\(\Theta=\) '00':  
By using the same derivation method as **(b)**, we can obtain the approximate expression of \(Pr\Big{\{}R_{R_{n}U_{2}}^{x_{2},00}\geq R_{2}\Big{\}}\) as  
\[ \begin{array}{l}Pr\Big{\{}R_{R_{n}U_{2}}^{x_{2},00}\geq R_{2}\Big{\}}=exp \Big{(}-\frac{\sigma^{2}\gamma_{0}}{\lambda_{R_{n}U_{2}}P_{2}}\Big{)}\bigg{(}1- \frac{\lambda_{R_{n}R}\sigma^{2}\gamma_{0}}{\lambda_{R_{n}U_{2}}I_{0}+\lambda_{R_{n}R} \sigma^{2}\gamma_{0}}exp\Big{(}-\frac{I_{0}}{\lambda_{R_{n}R}R_{2}}\Big{)}\bigg{)} \\ \times exp\Big{(}-\frac{\sigma^{2}(1+\gamma_{1})\gamma_{2}}{\lambda_{R_{n}U_{2}} P_{2}}\Big{)}\bigg{(}1-\frac{\lambda_{R_{n}R}\sigma^{2}(1+\gamma_{1})\gamma_{2}}{ \lambda_{R_{n}U_{2}}I_{0}+\lambda_{R_{n}R}\sigma^{2}(1+\gamma_{1})\gamma_{2}} exp\Big{(}-\frac{I_{0}}{\lambda_{R_{n}R}R_{2}}\Big{)}\bigg{)}\\ \times\frac{\sigma^{2}}{\lambda_{T_{1}}U_{1}P_{T}}exp\Big{(}\frac{\sigma^{2}}{ \lambda_{T_{1}}U_{1}P_{T}}\Big{)}\cdot(Q_{9}-Q_{10}).\\ \end{array} \] (A19)  
Finally, using (A1)-(A19), the outage probability for U\({}_{2}\) in cooperative CR-NOMA networks can be obtained as Equation (28), which completes the proof of Theorem 1. \(\square\)