Encoding minimalization

Now that we know how to convert recursive definitions into lambda terms, we can encode minimalization as follows. Assume that we want to define a function $f : \mathbb{N}^k \to \mathbb{N}$ by minimalization from a function $g : \mathbb{N}^{k{+1}} \to \mathbb{N}$ for which we already have an encoding $\langle{g}\rangle$. We begin with a recursive definition of a term $F$ as follows.

$$F = \lambda n x_1 x_2 \ldots x_k. \begin{array}[t]{ll} \ma... ...thrm{else~} F (\mathit{succ~} n)~x_1~x_2~\ldots~x_k \end{array}$$

Let $\tilde{F}$ be the lambda term corresponding to $F$ after unravelling the recursive definition. The encoding $\langle{f}\rangle$ of $f$ is then $\tilde{F}~\langle{0}\rangle$.

All that remains is to provide an encoding for the predicate iszero.

Encoding of iszero