Permutation of Variables of Primitive Recursive Function

Theorem
Let $f: \N^k \to \N$ be a primitive recursive function.

Let $\sigma$ be a permutation of $\left({1, 2, \ldots, k}\right)$.

Then the function $h: \N^k \to \N$ defined as:
 * $h \left({n_1, n_2, \ldots, n_k}\right) = f \left({n_{\sigma \left({1}\right)}, n_{\sigma \left({2}\right)}, \ldots, n_{\sigma \left({k}\right)}}\right)$

is also primitive recursive.

Proof
We have that:
 * $\forall j: 1 \le j \le k: n_{\sigma \left({j}\right)} = \operatorname{pr}^k_{\sigma \left({j}\right)}$.

Thus $h$ is obtained by substitution from $f$ and the projection functions $\operatorname{pr}^k_{\sigma \left({j}\right)}$.

The result follows.

It follows that if a function $h$ can be obtained from known primitive recursive functions by primitive recursion where a variable other than the last one is taken as the recursion variable, then $h$ is primitive recursive.