Journal of Formalized Mathematics
Volume 13, 2001
University of Bialystok
Copyright (c) 2001 Association of Mizar Users

## The Set of Primitive Recursive Functions

Grzegorz Bancerek
University of Bialystok, Shinshu University, Nagano
Piotr Rudnicki
University of Alberta, Edmonton

### Summary.

We follow [31] in defining the set of primitive recursive functions. The important helper notion is the homogeneous function from finite sequences of natural numbers into natural numbers where homogeneous means that all the sequences in the domain are of the same length. The set of all such functions is then used to define the notion of a set closed under composition of functions and under primitive recursion. We call a set primitively recursively closed iff it contains the initial functions (nullary constant function returning 0, unary successor and projection functions for all arities) and is closed under composition and primitive recursion. The set of primitive recursive functions is then defined as the smallest set of functions which is primitive recursively closed. We show that this set can be obtained by primitive recursive approximation. We finish with showing that some simple and well known functions are primitive recursive.

This work has been supported by NSERC Grant OGP9207, NATO CRG 951368 and TYPES grant IST-1999-29001.

#### MML Identifier: COMPUT_1

The terminology and notation used in this paper have been introduced in the following articles [25] [11] [30] [27] [1] [32] [33] [8] [6] [12] [24] [16] [26] [9] [13] [3] [22] [4] [21] [29] [10] [19] [15] [18] [5] [7] [14] [28] [20] [17] [23] [2]

#### Contents (PDF format)

1. Preliminaries
2. Sets of Compatible Functions
3. Homogeneous Relations
4. Primitive Recursiveness
5. The Set of Primitive Recursive Functions
6. Examples

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