|
@ -1 +1,13 @@ |
|
|
\item TODO |
|
|
|
|
|
|
|
|
\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} |