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 $\operatorname{pr}^k_j: \N^k \to \N$, defined as:
 * $\forall \left({n_1, n_2, \ldots, n_k}\right) \in \N^k: \operatorname{pr}^k_j \left({\left({n_1, n_2, \ldots, n_k}\right)}\right) = n_j$

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:
 * $\operatorname{pr}^1_1: \N \to \N: \operatorname{pr}^1_1 \left({\left({n_1}\right)}\right) = n_1$

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