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24 lines
1.2 KiB
24 lines
1.2 KiB
\item \self Consider the following natural deduction proof for the sequent $$\quad \exists x \; P(x) \lor \exists x \; Q(x) \quad \ent \quad \exists x \; (P(x)\lor Q(x)).$$
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Is the proof correct? If not, explain the error in the proof and either show how to correctly prove the sequent, or give a counterexample that proves the sequent invalid.
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\setlength\subproofhorizspace{1.7em}
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\begin{logicproof}{2}
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\exists x \; P(x) \lor \exists x \; Q(x) & prem.\\
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\begin{subproof}
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\exists x \; P(x) & ass.\\
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\begin{subproof}
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\llap{$x_0\enspace \;$} P(x_0) & ass.\\
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P(x_0) \lor Q(x_0) & $\lor \mathrm{i}_1$ 3
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\end{subproof}
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\exists x \; (P(x) \lor Q(x)) & $\exists \mathrm{e}$ 2,3-4
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\end{subproof}
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\begin{subproof}
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\exists x \; Q(x) & ass.\\
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\begin{subproof}
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\llap{$x_0\enspace \;$} Q(x_0) & ass.\\
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P(x_0) \lor Q(x_0) & $\lor \mathrm{i}_2$ 7
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\end{subproof}
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\exists x \; (P(x)\lor Q(x)) & $\exists \mathrm{e}$ 6,7-8
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\end{subproof}
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\exists x \; (P(x)\lor Q(x)) & $\lor \mathrm{e}$ 1,2-5,6-9
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\end{logicproof}
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