thus obtained codewords are then communicated to Node 2 and Node 3 as follows. By leveraging the side information $Y^{n}$ available at Node 2, conveying the codewords $A^{n},$ $U^{n}$ and $\hat{X}_{1}^{n}$ to Node 2 requires rate $I(X,Y;U,A)+I(X,Y;\hat{X}_{1}|U,A)-I(U,A,\hat{X}_{1};Y)$ by the Wyner-Ziv theorem [@Elgammal p. 280]$,$ which equals the right-hand side of (\[eq:R1\]). Then, sequences $A^{n}$ and $U^{n}$ are sent by Node 2 to Node 3, which requires a rate equal to the right-hand side of (\[eq:R2\]). This follows from the rates of the used codebooks and from the Wyner-Ziv theorem, due to the side information $Z^{n}$ available at Node 3 upon application of the action sequence $A^{n}$. Finally, Node 3 produces $\hat{X}_{2}^{n}$ that leverages through a symbol-by-symbol function as $\hat{X}_{2i}=\textrm{f}(U_{i},Z_{i})$ for $i\in[1,n].$

Lossless Compression
--------------------

Suppose that the source sequence $X^{n}$ needs to be communicated [*losslessly*]{} at both Node 2 and Node 3, in the sense that $d_{j}(x,\hat{x}_{j})$ is the Hamming distortion measure for $j=1,2$ ($d_{j}(x,\hat{x}_{j})=0$ if $x=\hat{x}_{j}$ and $d_{j}(x,\hat{x}_{j})=1$ if $x\neq\hat{x}_{j}$) and $D_{1}=D_{2}=0$. We can establish the following immediate consequence of Proposition \[prop:RD\_action\_cascade\].

\[coro:cascade\_lossless\] The rate-distortion-cost region $\mathcal{R\mbox{\ensuremath{(0,0,\Gamma)}}}$ for the cascade source coding problem illustrated in Fig. \[fig:fig2\] with Hamming distortion metrics is given by the union of all rate 