Lambda-Calculus and Combinators, an Introduction

58 Computable functions
Definition 4.21 (Partial recursive functions) A function φ from
a subset of IN n into IN (n ≥ 0) is called partial recursive 4 iff there exist
primitive recursive ψ and χ such that, for all m1, ..., mn ∈ IN,
φ(m1,...,mn )=ψ µk[ χ(m1,...,mn,k)=0] ,
where µk[χ(m1,...,mn,k) = 0] is the least k such that χ(m1,...,mn ,k)
=0,ifsuchak exists, and is undefined if no such k exists.
Example 4.22 The subtraction function is partial recursive. Because
m1 − m2 = µk[((m2 + k) · m1) =0],
where · is the cut-off subtraction introduced in Example 4.10. Note
that when m1 0 for all k ≥ 0, so
µk[((m2 + k) · m1) = 0] does not exist. This agrees with m1 − m2 not
existing when m1