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The structural induction principle says that a property $P(f)$ holds for every primitive recursive function if it holds for every initial function of recursion theory and is preserved by function composition in recursion theory and primitive recursion. This is justified because primitive recursive functions are, by definition, the smallest class closed under those constructors; equivalently, every such function has a finite construction tree, and ordinary induction on its height proves $P$.

Take $P(f)$ to mean that $f$ is total. The zero, successor, and projection functions are total. A composition of total functions is total. Finally, suppose $g$ and $h$ are total and $f$ is defined from them by primitive recursion. For fixed $\mathbf{x}$, induction on $n$ proves that $f(\mathbf{x},n)$ exists: the value at zero is $g(\mathbf{x})$, and from the existing value at $n$, totality of $h$ gives the value at $n+1$. Therefore every primitive recursive function is total.

A partial recursive function is obtained from the three kinds of initial function of recursion theory—
the zero function, successor function, and projection function—by finitely many applications of function composition in recursion theory, primitive recursion, and unbounded minimization. Explicitly, primitive recursion has the formwhile minimization takes the least $y$ for which $g(\mathbf{x},y)=0$ and is undefined if no such $y$ is found.

For the function in part (i), defineThe initial value $1=S(Z(0))$ and the identically zero recursion step are primitive recursive, so this is a primitive-recursive definition of the required zero test. It uses no minimization.