\subsection{Induction on Trees}
\mint{haskell}+data T t = Leaf t | Node1 (T t) | Node2 t (T t) (T t)+

\shade{blue}{Base Case} $T_0 = \{ \texttt{Leaf}\ a \divider a \in t \}$

\shade{green}{Step Case} $T_i = T_{i - 1} \cup \{ \texttt{Node1}\ s \divider s \in T_{i - 1} \} \cup \{ \texttt{Node2}\ a\ l\ r \divider a \in t \text{ and } l, r \in T_{i - 1} \}$

A natural deduction rule structural induction is:
\[
    \begin{prooftree}
        \hypo{\Gamma \vdash P[x \mapsto \texttt{Leaf}\ a]}
        \hypo{\Gamma, P[x \mapsto s] \vdash P[xs \mapsto \texttt{Node1}\ s]}
        \hypo{\Gamma, P[x \mapsto l], P[x \mapsto r] \vdash P[ \mapsto \texttt{Node2}\ a\ l\ r]}
        \infer3[$(*)$]{\Gamma \vdash \forall x \in \texttt{T}\ t. P}
    \end{prooftree}
\]
(*) $a$, $l$, $r$, $s$ not free in $\Gamma, P$