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42 lines
1.1 KiB
42 lines
1.1 KiB
This sequent is provable.
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\begin{logicproof}{2}
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\lnot (a \lor b) & prem. \\
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a \lor \lnot a & LEM \\
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\begin{subproof}
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a & ass. \\
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a \lor b & $\lor$i1 3 \\
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\bot & $\lnot$e 4,1 \\
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\lnot a \land \lnot b & $\bot$e 5
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\end{subproof}
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\begin{subproof}
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\lnot a & ass. \\
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\begin{subproof}
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b & ass. \\
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a \lor b & $\lor$i2 8 \\
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\bot & $\lnot$e 9,1
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\end{subproof}
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\lnot b & $\lnot$i 8-10 \\
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\lnot a \land \lnot b & $\land$i 7, 11
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\end{subproof}
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\lnot a \land \lnot b & $\lor$e 2, 3-6, 7-12
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\end{logicproof}
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Alternate solution:
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\begin{logicproof}{2}
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\lnot(a \lor b) & prem \\
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\begin{subproof}
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a & ass \\
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a \lor b & $\lor\mathrm{i}_1$ 2 \\
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\bot & $\lnot$e 1,3
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\end{subproof}
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\lnot a & $\lnot$i 2--4 \\
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\begin{subproof}
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b & ass \\
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a \lor b & $\lor\mathrm{i}_2$ 6 \\
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\bot & $\lnot$e 1,7
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\end{subproof}
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\lnot b & $\lnot$i 6--8 \\
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\lnot a \land \lnot b & $\land$i 5,9
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\end{logicproof}
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