You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
35 lines
1.4 KiB
35 lines
1.4 KiB
We start by computing $\skel$:
|
|
\begin{compactItemize}
|
|
\item $e_{0}\Leftrightarrow(x=y)$
|
|
\item $e_{1}\Leftrightarrow(y=f(y))$
|
|
\item $e_{2}\Leftrightarrow(y=f(x))$
|
|
\item $e_{3}\Leftrightarrow(z=f(z))$
|
|
\item $e_{4}\Leftrightarrow(f(z)=f(x))$
|
|
\end{compactItemize}
|
|
|
|
|
|
$$ \skel = e_{0} \land e_{1} \land \neg e_{2} \land e_{3} \land e_{4}$$
|
|
|
|
\hspace{-0.09cm}\scalebox{0.85}{
|
|
\begin{dplltabular}{6}
|
|
\dpllStep{1|2|3|4|5|6}
|
|
\dpllDecL{0|0|0|0|0|0}
|
|
\dpllAssi{ - |$e_{0}$|$e_{0}, e_{1}$|$e_{0}, e_{1}, \lnot e_{2}$|\makecell{$e_{0}, e_{1}, \lnot e_{2}, $ \\ $e_{3}$}|\makecell{$e_{0}, e_{1}, \lnot e_{2}, $ \\ $e_{3}, e_{4}$}}
|
|
\dpllClause{1}{$e_{0}$}{$e_{0}$|\done|\done|\done|\done|\done}
|
|
\dpllClause{2}{$e_{1}$}{$e_{1}$|$e_{1}$|\done|\done|\done|\done}
|
|
\dpllClause{3}{$\lnot e_{2}$}{$\lnot e_{2}$|$\lnot e_{2}$|$\lnot e_{2}$|\done|\done|\done}
|
|
\dpllClause{4}{$e_{3}$}{$e_{3}$|$e_{3}$|$e_{3}$|$e_{3}$|\done|\done}
|
|
\dpllClause{5}{$e_{4}$}{$e_{4}$|$e_{4}$|$e_{4}$|$e_{4}$|$e_{4}$|\done}
|
|
\dpllBCP{$e_{0}$|$e_{1}$|$\lnot e_{2}$|$e_{3}$|$e_{4}$| - }
|
|
\dpllPL{ - | - | - | - | - | - }
|
|
\dpllDeci{ - | - | - | - | - |SAT}
|
|
\end{dplltabular}
|
|
}
|
|
|
|
\vspace{0.5em}
|
|
The SAT solver has computed that $\skel$ is satisfiable, we are therefore going to check for consistency with the theory:
|
|
\begin{align*}
|
|
&\{x, y\}, \{y, f(y)\}, \{z, f(z)\}, \{f(z), f(x)\}\\
|
|
&\{f(y), x, y\}, \{f(x), f(z), z\}
|
|
\end{align*}
|
|
The $\EUF$-Solver returned SAT, therefore $\varphi$ is satisfiable.
|