Predecessor Function is Primitive Recursive

Theorem
The predecessor function $$\operatorname{pred}: \N \to \N$$ defined as:
 * $$\operatorname{pred} \left({n}\right) = \begin{cases}

0 & : n = 0 \\ n-1 & : n > 0 \end{cases}$$ is primitive recursive.

Proof
We can use Primitive Recursion on One Variable to find $$g: \N^2 \to \N$$ and $$h: \N \to \N$$ such that:


 * $$h \left({n}\right) = \begin{cases}

\operatorname{zero} \left({n}\right) & : n = 0 \\ g \left({n-1, h \left({n-1}\right)}\right) & : n > 0 \end{cases} $$

By setting:
 * $$g \left({n, m}\right) = \operatorname{pr}_1\left({n, m}\right)$$

we see that setting $$h = \operatorname{pred}$$ fits the pattern.

We have that the $$\operatorname{pr}$$ and $$\operatorname{zero}$$ functions are basic primitive recursive functions.

Hence the result.