Node 3 on the broadcast link of the rate $R_{b}$ by Node 1, while the message of rate $r_{2d}$ is sent by Node 2 to Node 3. This way, Node 1 and Node 2 cooperate to transmit $X$ to Node 3. As per the discussion above, the following inequalities have to be satisfied $$\begin{aligned}
r_{2b}+r_{2d}+r_{1b} & \geq & H(X|A,Z),\\
r_{1b}+r_{1d} & \geq & H(X|A,Y),\\
R_{1} & \geq & r_{1d},\\
R_{2} & \geq & r_{2d},\\
\textrm{and }R_{b} & \geq & r_{1b}+r_{2b}+I(X;A),\end{aligned}$$ Applying Fourier-Motzkin elimination [@Elgammal Appendix C] to the inequalities above, the inequalities in (\[eqn: RD\_BC\_lossless\]) are obtained.

Example: Switching-Dependent Side Information\[sub:lossless\_ex\]
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We now consider the special case of the model in Fig. \[fig:fig3\] in which the actions $A\in{\cal A}=\{0,1,2,3\}$ acts a switch that decides whether Node 2, Node 3 or either node gets to observe a side information $W$. The side information $W$ is jointly distributed with source $X$ according to the joint pmf $p(x,w)$. Moreover, defining as $\textrm{e}$ an erasure symbol, the conditional pmf $p(y,z|x,a)$ is as follows: $Y=Z=\textrm{e}$ for $A=0$ (neither Node 2 nor Node 3 observes the side information $W$); $Y=W$ and $Z=\textrm{e}$ for $A=1$ (only Node 2 observes the side information $W$); $Y=\textrm{e}$ and $Z=W$ 