\begin{definition}\label{phase5_before_proof_failure}
    $\phasefivebeforeprooffailure = \emptyset$.
\end{definition}

\begin{proposition}\label{phase5_mismatched_assumption}
    For all $x$ we have if $x = x$, then $x = x$.
\end{proposition}
\begin{proof}
    Fix $x$.
    Assume $x = \emptyset$.
    Follows by assumption.
\end{proof}