Questionnaire-2024/natural_deduction_propositional_logic/1010_sol.tex

This sequent is provable.

\begin{logicproof}{1}
    \lnot \lnot k \imp (l \land m) & \prem \\
    \lnot \lnot \lnot l \imp m & \prem \\
    m \lor \lnot m & $\LEM$ \\
    \begin{subproof}
        m & \assum \\
        \lnot \lnot m & $\lnot \negi 4$ \\
        l \lor \lnot \lnot m & $\ori{2} 5$ \\
        \lnot k \lor (l \lor \lnot \lnot m) & $\ori{2} 6$
    \end{subproof}
    \begin{subproof}
        \lnot m & \assum \\
        \lnot \lnot \lnot \lnot l & $\MT 2,8$ \\
        \lnot \lnot l & $\negnege 9$ \\
        l             & $\negnege 10$ \\
        l \lor \lnot \lnot m & $\ori{1} 11$ \\
        \lnot k \lor (l \lor \lnot \lnot m) & $\ori{2} 12$
    \end{subproof}
    \lnot k \lor (l \lor \lnot \lnot m) & $\ore 3, 4-7, 8-13$
\end{logicproof}