Definition:Basic Primitive Recursive Function

Definition
The basic primitive recursive functions are:

Zero Function
The zero function $$\operatorname{zero}: \N \to \N$$, defined as:
 * $$\forall n \in \N: \operatorname{zero} \left({n}\right) = 0$$.

Successor Function
The successor function $$\operatorname{succ}: \N \to \N$$, defined as:
 * $$\forall n \in \N: \operatorname{succ} \left({n}\right) = n + 1$$.

Projection Function
The projection functions $$\operatorname{pr}_j: \N^k \to \N$$, defined as:
 * $$\forall \left({n_1, n_2, \ldots, n_k}\right) \in \N^k: \operatorname{pr}_j \left({\left({n_1, n_2, \ldots, n_k}\right)}\right) = n_j$$

where $$j \in \left[{1 \,. \, . \, k}\right]$$.

Identity Function
The identity function $$I_\N: \N \to \N$$, defined as:
 * $$\forall n \in \N: I_\N \left({n}\right) = n$$.

Note that this is an implementation of the projection function:
 * $$\operatorname{pr}_1: \N \to \N: \operatorname{pr}_1 \left({\left({n_1}\right)}\right) = n_1$$.

URM Computability
They are all URM computable by a single-instruction URM program.