I am currently reading Cutland's Computability and would like to figure out how to solve Theorem 4.2 which states:
Let $x=(x_1 \dotsc x_n)$, and suppose that $f(x)$ and $g(x,y,z)$ are functions; then there is a unique function $h(x,y)$ satisfying the recursion equations $$h(x,0) = f(x),$$ $$h(x,y+1)=g(x,y,h(x,y))$$
I think I just have to show that $h(x,y)$ is well-defined for every $x,y$ and I know how to do this via induction for $n=1$ (induction on $x_1$, then in both induction base and induction step an induction on y), but not for an arbitrary n. I've heard of a concept called "simultaneous induction" but I don't know how to apply this here.
It is clear that for every $x$, the value of $h(x,0)$ is unique. This is because the value of $h(x,0)$ can never be defined in the second row, since $0$ is not a successor to any number.
Inductively, by using the fact that each nonzero natural number has a unique predecessor, you can also show that $h(x,y)$ is uniquely defined for $y>0$.