\import{test/phase5/exact-producer.tex}

\begin{definition}\label{phase5_proof_seed}
    $\phasefiveproofseed = \emptyset$.
\end{definition}

\begin{proposition}\label{phase5_implicit_proof}
    $\phasefiveproofseed = \phasefiveproofseed$.
\end{proposition}

\begin{proposition}\label{phase5_structural_proof}
    For all $x$ we have if $x = x$, then $x = x$.
\end{proposition}
\begin{proof}
    Fix $x$.
    Assume $x = x$.
    We have $x = x$ by assumption.
    Follows by \cref{phase5_definition}.
\end{proof}

\begin{proposition}\label{phase5_subclaim_proof}
    For all $x$ we have $x = x$.
\end{proposition}
\begin{proof}
    Fix $x$.
    Show $x = x$.
    \begin{subproof}
        Follows by assumption.
    \end{subproof}
    Follows by assumption.
\end{proof}

\begin{proposition}\label{phase5_header_proof}
    Let $A$ be a set.
    Let $x \in A$.
    Then $x \in A$.
\end{proposition}
\begin{proof}
    Follows by assumption.
\end{proof}

\begin{proposition}\label{phase5_generalized_auto}
    Let $y$ be a set.
    Then $y = y$.
\end{proposition}