I'm asked to prove the following:
a) A function $f$ defined by the terms of $PA$ (Peano Arithmetic), that is, a function defined by $+$, $\cdot$ and $S(x)$ over any set (probably the natural numbers $\mathbb{N}$?) is a recursive function.
b) And if $f(x,y)$ is a recursive function then a function $g(f(x,y))$ defined by $g(f(x,y))= min \{ z \, | \, f(x,y) \hbox{ exists for each } y \leq z \land f(x,y)=0 \}$ and $g(f(x,y))$ undefined otherwise, then $g(f(x,y))$ is a recursive function.
Thanks in advance!