Factorial is Primitive Recursive

Theorem
The factorial function $$\operatorname{fac}: \N \to \N$$ defined as:
 * $$\operatorname{fac} \left({n}\right) = n!$$

is primitive recursive.

Proof
From the definition of the factorial, we have that:


 * $$\operatorname{fac} \left({n}\right) = \begin{cases}

1 & : n = 0 \\ \operatorname{mult} \left({n, \operatorname{fac} \left({n - 1}\right)}\right) & : n > 0 \end{cases}$$

Thus $$\operatorname{fac}$$ is obtained by primitive recursion from the constant $$1$$ and the primitive recursive function $\operatorname{mult}$.

Hence $$\operatorname{fac}$$ is primitive recursive.