be erroneous. However, evaluating these symbols using the decoding function $D_t$ of the network coding instance results in the correct values because $\delta_t=\delta_s^{(t)}$.

Sufficiency: Assume that there exists a $(\bm{\delta}_s,\mathcal{G})$-ICSIE together with its codeword $\bm{\sigma}$ by Lemma \[lemma:existence\]. Then, the encoding functions of the links and the decoding functions of the terminals in the corresponding network coding instance are defined using the decoding functions of the $(\bm{\delta}_s,\mathcal{G})$-ICSIE. For $e\in E$, $F_e$ is defined as a function whose output is $X_e=\hat{D}_e(\bm{\sigma},(\tilde{X}_{e^{\prime}} : e^{\prime}\in {\rm In}(e)))$. For $t\in T$, $D_t$ is defined as a function whose output is $\hat{D}_{t}(\bm{\sigma},(\tilde{X}_{e^{\prime}} : e^{\prime}\in {\rm In}(t)))$. Without loss of generality, we assume that $\bold{X}_S=\bold{\hat{X}}_S$ and then, there is a unique $\bold{\hat{X}}_E$ such that $\hat{F}(\bold{\hat{X}}_S,\bold{\hat{X}}_E)=\bm{\sigma}$ for any $\bold{\hat{X}}_S$ from Lemma \[lemma:existence\]. For $e \in E$, $\hat{D}_e(\bm{\sigma},(\tilde{X}_{e^{\prime}} : e^{\prime}\in {\rm In}(e)))=\hat{X}_e$ because $\delta_e=\delta_s^{(e)}$ and $\hat{X}_e$ is unique. For $t\in T$, $\hat{D}_{t}(\bm{\sigma},(\tilde{X}_{e^{\prime}} : e^{\prime}\in {\rm In}(t)))=\bold{\hat{X}}_{\mathcal{F}(t)}=\bold{X}_{\mathcal{F}(t)}$ because $\delta_t=\delta_s^{(t)}$.

\[theorem:equ\]

Thus, Theorem \[theorem:equ\] tells us that an NCLE problem is equivalent to the corresponding ICSIE problem when a network coding instance is given. The following example shows the equivalence between the two problems.

Suppose that a given network coding instance and the corresponding side information graph with $\bm{\delta}_s$ are 