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We start by translating $\varphi$ to $\hat{\varphi} = \skel$ and assign the following variables to the theory literals:\begin{itemize} \item $e_{0}\Leftrightarrow(f(a)=f(b))$ \item $e_{1}\Leftrightarrow(f(a)=c)$ \item $e_{2}\Leftrightarrow(a=b)$ \item $e_{3}\Leftrightarrow(f(c)=a)$\end{itemize}
$\hat{\varphi} = (\clause{\lnot e_{0}; e_{1}; e_{2}}) \land (\clause{\lnot e_{1}; e_{2}; \lnot e_{3}}) \land (\clause{e_{0}; \lnot e_{1}}) \land (\clause{\lnot e_{0}; e_{2}}) \land (\clause{\lnot e_{2}; \lnot e_{3}}) \land (\clause{e_{0}; e_{2}}) \land (\clause{\lnot e_{2}; \lnot e_{1}}) $
\hspace{-0.09cm}\scalebox{0.85}{ \begin{dplltabular}{5} \dpllStep{1|2|3|4|5} \dpllDecL{0|0|1|1|1} \dpllAssi{ - |$\lnot e_{3}$|$\lnot e_{3}, \lnot e_{0}$|$\lnot e_{3}, \lnot e_{0}, \lnot e_{1}$|\makecell{$\lnot e_{3}, \lnot e_{0}, \lnot e_{1}, $ \\ $e_{2}$}} \dpllClause{1}{$\lnot e_{0}, e_{1}, e_{2}$}{$\lnot e_{0}, e_{1}, e_{2}$|$\lnot e_{0}, e_{1}, e_{2}$|\done|\done|\done} \dpllClause{2}{$\lnot e_{1}, e_{2}, \lnot e_{3}$}{$\lnot e_{1}, e_{2}, \lnot e_{3}$|\done|\done|\done|\done} \dpllClause{3}{$e_{0}, \lnot e_{1}$}{$e_{0}, \lnot e_{1}$|$e_{0}, \lnot e_{1}$|$\lnot e_{1}$|\done|\done} \dpllClause{4}{$\lnot e_{0}, e_{2}$}{$\lnot e_{0}, e_{2}$|$\lnot e_{0}, e_{2}$|\done|\done|\done} \dpllClause{5}{$\lnot e_{2}, \lnot e_{3}$}{$\lnot e_{2}, \lnot e_{3}$|\done|\done|\done|\done} \dpllClause{6}{$e_{0}, e_{2}$}{$e_{0}, e_{2}$|$e_{0}, e_{2}$|$e_{2}$|$e_{2}$|\done} \dpllClause{7}{$\lnot e_{2}, \lnot e_{1}$}{$\lnot e_{2}, \lnot e_{1}$|$\lnot e_{2}, \lnot e_{1}$|$\lnot e_{2}, \lnot e_{1}$|\done|\done} \dpllBCP{ - | - |$\lnot e_{1}$|$e_{2}$| - } \dpllPL{$\lnot e_{3}$| - | - | - | - } \dpllDeci{ - |$\lnot e_{0}$| - | - |SAT} \end{dplltabular}}
$\Model_{\EUF} := (f(a) \neq f(b)) \land (f(a) \neq c) \land (a = b) \land (f(c) \neq a) $ \\Check if the assignment is consistent with the theory:
\begin{align*} &\{a, b\}, \{f(a)\}, \{f(b)\}, \{c\}, \{f(c)\}\end{align*}
$\Model_{\EUF}$ is consistent with the theory, \\$\Rightarrow \Model_{\EUF}$ is a satisfying assignment and $\varphi$ is SAT.
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