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68 lines
1.9 KiB
68 lines
1.9 KiB
\begin{itemize}
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\item We start by construction $\varphi$: \\
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\begin{align*}
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\varphi &= \varphi_1 \oplus \varphi_2 \\
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&= [\varphi_1 \lor \varphi_2] \land \lnot [\varphi_1 \land \varphi_2] = \\
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&= [(a \land \lnot b) \lor (\lnot(\lnot a \lor b))] \land \lnot [(a \land \lnot b) \land (\lnot(\lnot a \lor b))]
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\end{align*}
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\end{itemize}
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$$
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\underbracket{
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\underbracket{\Big[
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\underbracket{\big(
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a
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\land
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\underbracket{\lnot b}_{x_{10}}
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\big)}_{x_4}
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\lor
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\underbracket{\big(
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\lnot
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\underbracket{(
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\underbracket{\lnot a}_{x_{11}}
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\lor
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b
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)}_{x_8}
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\big)}_{x_5}
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\Big]}_{x_1}
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\land
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\underbracket{
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\lnot
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\underbracket{\Big[
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\underbracket{\big(
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a
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\land
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\underbracket{\lnot b}_{x_{12}}
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\big)}_{x_6}
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\lor
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\underbracket{\big(
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\lnot
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\underbracket{(
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\underbracket{\lnot a}_{x_{13}}
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\lor
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b
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)}_{x_9}
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\big)}_{x_7}
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\Big]}_{x_3}
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}_{x_2}
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}_{x_\varphi}
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$$
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\begin{align*}
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CNF(\phi) = \ &x_\phi \land \\
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&\tseitinAnd{x_{\varphi}}{x_1}{x_2} \land \\
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&\tseitinNot{x_2}{x_3} \land \\
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&\tseitinOr{x_1}{x_4}{x_5} \land \\
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&\tseitinOr{x_3}{x_6}{x_7} \land \\
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&\tseitinAnd{x_4}{a}{x_{10}} \land \\
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&\tseitinNot{x_5}{x_8} \land \\
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&\tseitinAnd{x_6}{a}{x_{12}} \land \\
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&\tseitinNot{x_7}{x_9} \land \\
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&\tseitinOr{x_8}{x_{11}}{b} \land \\
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&\tseitinOr{x_9}{x_{13}}{b} \land \\
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&\tseitinNot{x_{10}}{b} \land \\
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&\tseitinNot{x_{11}}{a} \land \\
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&\tseitinNot{x_{12}}{b} \land \\
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&\tseitinNot{x_{13}}{a}
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\end{align*}
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