maximum power levels do not impose any constrain and hence the IC conditions are satisfied. In the next proposition, we find the conditions under which IC cannot take place and we find the optimal power levels of $S$ and $R_t$.

\[prop:2\] For each pair of relays $R_r$ and $R_t$, interference cancellation is feasible if and only if $$\begin{aligned}
\label{eq:Pmax_condition}
P_{R_t}^{\max} \geq \gamma_0 \left[ \gamma_0 \left(\frac{\gamma_0 n_D}{g_{R_t D}}   + \frac{n_{R_r}}{g_{R_t R_r}}   \right) + \frac{n_{R_r}}{g_{R_t R_r}}   \right] \; .\end{aligned}$$ When interference cancellation is infeasible, the signal from $R_t$ can be decoded successfully at $D$ if and only if $$\begin{aligned}
\frac{\gamma_0 n_D}{g_{R_t D}}&\leq  P_{R_t}^{\max} \; .\label{subeq:ineq1} \end{aligned}$$ In addition, the signal from $S$ can be decoded successfully at $R_r$ if and only if $$\begin{aligned}
\gamma_0 \left(\frac{g_{R_t R_r}}{g_{S R_r}} P_{R_t}  +  \frac{n_{R_r}}{g_{S R_r}}\right) &\leq P_{S}^{\max} \; . \label{subeq:ineq2}\end{aligned}$$ When , and hold, the minimum power levels $P_{S}^\dagger $ and $P_{R_t}^\dagger $ are achieved when $\text{SNR}_{R_t D} = \text{SINR}_{S R_r} = \gamma_0$, and are given by

\[eq:P\_optimal2\] $$\begin{aligned}
P_{S}^\dagger &= \gamma_0 \left(\frac{g_{R_t R_r}}{g_{S R_r}} P_{R_t}  +  \frac{n_{R_r}}{g_{S R_r}}\right) \; , \label{eq:P_S_optimal2} \\
P_{R_t}^\dagger &=\frac{\gamma_0 n_D}{g_{R_t D}} \; \label{eq:P_Rt_optimal2}.\end{aligned}$$

Interference cancellation cannot take place when the maximum power of $R_t$ is not high enough, such that 