Bounded Summation is Primitive Recursive

Theorem

Let the function $f: \N^{k + 1} \to \N$ be primitive recursive.

Then so is the function $g: \N^{k + 1} \to \N$ defined as:

$\ds \map g {n_1, n_2, \ldots, n_k, z} = \begin{cases}

0 & : z = 0 \\ \sum_{y \mathop = 1}^z \map f {n_1, n_2, \ldots, n_k, y} & : z > 0 \end{cases}$

Proof

The function $g$ satisfies:

$\map g {n_1, n_2, \ldots, n_k, 0} = 0$

$\map g {n_1, n_2, \ldots, n_k, z + 1} = \map g {n_1, n_2, \ldots, n_k, z} + \map f {n_1, n_2, \ldots, n_k, z + 1}$.

Hence $g$ is defined by primitive recursion from:

the primitive recursive function $\operatorname{add}$

$f$, which is primitive recursive

constants, which are primitive recursive.

Hence the result.

$\blacksquare$

Note

The sum $\ds \sum_{y \mathop = 1}^z$ is referred to as a bounded sum to distinguish it from $\ds \sum_{y \mathop = 1}^\infty$ which is not.