Questionnaire-2024/decidability/0002.tex

\item
The soundness of natural deduction for propositional logic can be proven via a \emph{mathematical course-of-values induction on the length of the Natural Deduction proof}. Let $M(k)$ be the following assertion:

$M(k)\coloneqq$ „For all sequents $\phi_1,\phi_2,\dots,\phi_n\vdash \psi$ which have a proof of length $k$,
it is the case that $\phi_1,\phi_2,\dots,\phi_n\models \psi$  holds.”

Your tasks:
\begin{enumerate}
\setlength{\itemsep}{-0.1em}
\item Proof the induction base-case, i.e., $M(1)$ holds.
\item Explain the proof idea of the induction step:
$M(1)\wedge M(2) \land \dots \land M(k-1) \rightarrow M(k)$.
\end{enumerate}