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43 lines
1.4 KiB
43 lines
1.4 KiB
From a given formula $\varphi \lor \psi$, we want to proof some other formula $\chi$.
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We only know that $\varphi$ \emph{or} $\psi$ holds. It could be that both of them are true,
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but it could also be that only $\psi$ is true, or only $\varphi$ is true.
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Sine we don't know which sub-formula is true, we have to give two separate proofs:
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\begin{itemize}
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\item First box: We assume $\varphi$ is true and need to find a proof for $\chi$.
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\item Second box: We assume $\psi$ is true and need to find a proof for $\chi$.
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\end{itemize}
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Only if we can prove $\chi$ in the first and in the second box, then we can conclude that $\chi$ holds also outside of the box.
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The $\ore$ rules says that we can only derive $\chi$ from $ \varphi \lor \psi$ if we can derive $\chi$
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from the assumption $ \varphi$ as well as from the assumption $\psi$.
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Formally the rule is written as:
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\begin{center}
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\begin{prooftree}
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\AxiomC{\begin{tabular}{l}
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\vspace*{0.95ex}\\
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\vspace*{0.95ex}\\
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$ \varphi \lor \psi $\\
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\end{tabular}}
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\AxiomC{\begin{tabular}{|l|}
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\hline
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$ \varphi $ ass.\\
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\hspace*{0.2em}$ \vdots $\\
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$ \chi $\\
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\hline
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\end{tabular}}
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\AxiomC{\begin{tabular}{|l|}
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\hline
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$ \psi $ ass.\\
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\hspace*{0.2em}$ \vdots $\\
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$ \chi $\\
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\hline
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\end{tabular}}
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\RightLabel{$ \ore$}
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\TrinaryInfC{$ \chi $}
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\end{prooftree}
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\end{center}
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