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\item \self Consider the following formula in $\mathcal{T}_{EUF}$. \begin{align} \phi_{EUF} \quad := \quad f(x)=f(y) \land f(y)=y \lor f(g(x))=f(f(y)) \land g(x)=x \\ \lor f(x) \neq f(y) \land y \neq g(f(y)) \land x \neq g(x) \end{align} Use Ackermann's reduction to compute an equisatisfiable formula in $\mathcal{T}_{E}$. Then perform the graph-based reduction on the outcome of Ackermann's reduction to construct an equisatisfiable propositional formula $\phi_{prop}$.
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