Questionnaire-2024/equivalence_checking/0009_sol.tex

\begin{itemize}
  \item We start by construction $\varphi$: \\
    \begin{align*}
      \varphi &= \varphi_1 \oplus \varphi_2 \\
              &= [\varphi_1 \lor \varphi_2] \land \lnot [\varphi_1 \land \varphi_2] = \\
              &= [(a \land \lnot b) \lor (\lnot(\lnot a \lor b))] \land \lnot [(a \land \lnot b) \land (\lnot(\lnot a \lor b))]
    \end{align*}
\end{itemize}

$$
  \underbracket{
    \underbracket{\Big[
      \underbracket{\big(
        a
        \land
        \underbracket{\lnot b}_{x_{7}}
      \big)}_{x_3}
      \lor
      \underbracket{\big(
        \lnot
          \underbracket{(
            \underbracket{\lnot a}_{x_{8}}
            \lor
            b
          )}_{x_6}
      \big)}_{x_4}
    \Big]}_{x_1}
    \land
    \underbracket{
      \lnot
      \underbracket{\Big[
        \underbracket{\big(
          a
          \land
          \underbracket{\lnot b}_{x_{7}}
        \big)}_{x_3}
        \lor
        \underbracket{\big(
          \lnot
            \underbracket{(
              \underbracket{\lnot a}_{x_{8}}
              \lor
              b
            )}_{x_6}
        \big)}_{x_4}
      \Big]}_{x_1}
    }_{x_2}
  }_{x_\varphi}
    $$


\begin{align*}
  \varphi' = \ &x_\phi \land \\
                &\tseitinAnd{x_{\varphi}}{x_1}{x_2} \land \\
                &\tseitinNot{x_1}{x_2} \land \\
                &\tseitinOr{x_1}{x_3}{x_4} \land \\
                &\tseitinAnd{x_3}{a}{x_{7}} \land \\
                &\tseitinNot{x_4}{x_6} \land \\
                &\tseitinOr{x_6}{x_{8}}{b} \land \\
                &\tseitinNot{x_{7}}{b} \land \\
                &\tseitinNot{x_{8}}{a} \land
\end{align*}