\subsection{Mathematical Induction}
To prove something of the form $\forall n \in \N. P$ (with $n$ free in $P$), we can use one of these two schemes

\subsubsection{Weak Mathematical Induction}
If $P$ is not defined yet (e.g. we just have a description of what we need to prove), give a full definition of $P$, which is our induction hypothesis.
Then state that we are proving $\forall n \in \N. P$ using \textit{weak} mathematical induction.

\shade{blue}{Base case} We show that $P[n \mapsto 0]$ holds.

\shade{green}{Step case} We proceed by stating the following:
``Let $m \geq 0$ be arbitrary. We assume $P[n \mapsto m]$ holds, and we prove $P[n \mapsto m + 1]$.''
Finally, do the actual proof. It is also possible to state $P[n \mapsto m]$ simply as $P(m)$

The same, but expressed as a Natural Deduction rule:
\[
    \begin{prooftree}
        \hypo{\Gamma \vdash P(0)}
        \hypo{\Gamma, P(n) \vdash P(n + 1)}
        \infer2[$n$ not free in $\Gamma$]{\Gamma \vdash \forall n. P(n)}
    \end{prooftree}
\]

\subsubsection{Strong Mathematical Induction}
If $P$ is not yet defined, then define it. It is our induction hypothesis.
Then state: ``We prove $\forall k. P(k)$ by strong (mathematical) induction.
Let $k$ be arbitary and let us assume $P(j)$ for all $j < k$''. Finally, do case distinction on the different values $k$ can take.

The same, but expressed as a Natural Deduction rule:
\[
    \begin{prooftree}
        \hypo{\Gamma, \forall m < n. P(m) \vdash P(n)}
        \infer1[$n$ not free in $\Gamma$, $m$ not free in $P(n)$]{\Gamma \vdash \forall n . P(n)}
    \end{prooftree}
\]