Minimum Function is Primitive Recursive

Theorem
The minimum function $$\min: \N^2 \to \N$$, defined as:

\min \left({n, m}\right) = \begin{cases} n: & n \le m \\ m: & m \le n \end{cases} $$ is primitive recursive‎.

Proof
From Sum Less Maximum is Minimum we have that:
 * $$\min \left({n, m}\right) = n + m - \max \left({n, m}\right)$$.

As $$n + m \ge \max \left({n, m}\right)$$, we have that:
 * $$\min \left({n, m}\right) = n + m \dot - \max \left({n, m}\right)$$

Hence we see that $$\min$$ is obtained by substitution from:
 * the primitive recursive function $n \dot - m$;
 * the primitive recursive function $\max \left({n, m}\right)$.

Hence the result.