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\begin{datatype}\label{phase5_data}
Define $\phasefivedata$ inductively as follows.
\begin{enumerate}
\item $\phasefivezero \in \phasefivedata$.
\item $\phasefiveatom{n} \in \phasefivedata$ for $n \in \unions{\emptyset}$.
\item $\phasefivejoin{x}{y} \in \phasefivedata$ for $x \in \phasefivedata$ and $y \in \phasefivedata$.
\end{enumerate}
\end{datatype}
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