Absolute Difference Function is Primitive Recursive

Theorem
The absolute difference function $$\operatorname{adf}: \N^2 \to \N$$, defined as:
 * $$\operatorname{adf} \left({n, m}\right) = \left|{n - m}\right|$$

where $$\left|{a}\right|$$ is defined as the absolute value of $$a$$, is primitive recursive‎.

Proof
We note that $$\left|{n - m}\right| = \left({n \, \dot - \, m}\right) + \left({m \, \dot - \, n}\right) = \operatorname{add} \left({\left({n \, \dot - \, m}\right), \left({m \, \dot - \, n}\right)}\right)$$.

Next we note that:
 * $$m \, \dot - \, n = \operatorname{pr}^2_2 \left({n, m}\right) \, \dot - \, \operatorname{pr}^2_1 \left({n, m}\right)$$

where $$\operatorname{pr}^2_k$$ is the projection function.

Then:

$$ $$ $$ $$

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

Hence the result.