h_{13} X_3$$ Hence, the interference signal is an element of lattice $\Lambda_{i1} = h_{13}\Lambda_3$. In decoding the total interference, receiver $1$ sees a noise of $X_1 + Z_1$ whose power $\leq P_1 + \sigma_1^2$. Therefore, the interference can be successfully decoded if $$\begin{aligned}
R_2 \leq \frac{1}{2}\log\left(\frac{h_{12}^2P_2}{p^2 (P_1 + \sigma_1^2)}\right)\\
R_3 \leq \frac{1}{2}\log\left(\frac{h_{13}^2P_3}{P_1 + \sigma_1^2}\right).\end{aligned}$$ After decoding the interference, receiver $1$ decodes its message. The probability of decoding error decays with $n$ if $$R_1 \leq \frac{1}{2}\log\left(\frac{P_1}{\sigma_1^2}\right).$$ The analysis for receivers $2$ and $3$ are similar to that of receiver $1$ and the details are omitted here. If the channel gains satisfy (\[eqn : strong interference condition 1\]), then the rate constraints due to decoding interference are less binding than the rate constraints due to decoding the message at each receiver. Hence, the users can achieve rates given by $$R_i \leq \frac{1}{2}\log\left(\frac{P_i}{\sigma_i^2}\right), \quad i \in \{1,2,3\}.$$ This completes the outline of the proof when the channel gains satisfy (\[eqn : strong interference condition 1\]). Next, we analyze the degrees of freedom of non-symmetric ICs.

Degree of Freedom Analysis
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In this section, we briefly analyze the degrees of freedom of three user non-symmetric Gaussian ICs. We use the same layered lattice coding 