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62 lines
1.6 KiB
62 lines
1.6 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_{7}}
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\big)}_{x_3}
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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_{8}}
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\lor
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b
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)}_{x_6}
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\big)}_{x_4}
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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_{7}}
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\big)}_{x_3}
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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_{8}}
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\lor
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b
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)}_{x_6}
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\big)}_{x_4}
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\Big]}_{x_1}
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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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\varphi' = \ &x_\phi \land \\
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&\tseitinAnd{x_{\varphi}}{x_1}{x_2} \land \\
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&\tseitinNot{x_1}{x_2} \land \\
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&\tseitinOr{x_1}{x_3}{x_4} \land \\
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&\tseitinAnd{x_3}{a}{x_{7}} \land \\
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&\tseitinNot{x_4}{x_6} \land \\
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&\tseitinOr{x_6}{x_{8}}{b} \land \\
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&\tseitinNot{x_{7}}{b} \land \\
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&\tseitinNot{x_{8}}{a} \land
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\end{align*}
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