# Definition:Primitive Recursion/One Variable

## Definition

Let $a \in \N$ be a natural number.

Let $g: \N^2 \to \N$ be a function.

Then the function $h: \N \to \N$ is obtained from the constant $a$ and $g$ by primitive recursion if and only if:

$\forall n \in \N: \map h n = \begin {cases} a & : n = 0 \\ \map g {n - 1, \map h {n - 1} } & : n > 0 \end{cases}$

## Also see

It can be seen that this is a special case of primitive recursion on several variables, with $k = 0$ and $f$ replaced by the constant function $f_a$.