Journal of Formalized Mathematics
Volume 5, 1993
University of Bialystok
Copyright (c) 1993 Association of Mizar Users

## Euclid's Algorithm

Andrzej Trybulec
Warsaw University, Bialystok
Yatsuka Nakamura
Shinshu University, Nagano

### Summary.

The main goal of the paper is to prove the correctness of the Euclid's algorithm for {\bf SCM}. We define the Euclid's algorithm and describe the natural semantics of it. Eventually we prove that the Euclid's algorithm computes the Euclid's function. Let us observe that the Euclid's function is defined as a function mapping finite partial states to finite partial states of {\bf SCM} rather than pairs of integers to integers.

#### MML Identifier: AMI_4

The terminology and notation used in this paper have been introduced in the following articles [8] [7] [9] [2] [3] [11] [1] [4] [12] [5] [13] [6] [10]

#### Contents (PDF format)

1. Preliminaries
2. Euclid's Algorithm
3. The Natural Semantics of the Euclid's Algorithm

#### Bibliography

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[3] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. A classical first order language. Journal of Formalized Mathematics, 2, 1990.
[5] Rafal Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Journal of Formalized Mathematics, 2, 1990.
[6] Yatsuka Nakamura and Andrzej Trybulec. A mathematical model of CPU. Journal of Formalized Mathematics, 4, 1992.
[7] Andrzej Trybulec. Enumerated sets. Journal of Formalized Mathematics, 1, 1989.
[8] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[9] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[10] Andrzej Trybulec and Yatsuka Nakamura. Some remarks on the simple concrete model of computer. Journal of Formalized Mathematics, 5, 1993.
[11] Michal J. Trybulec. Integers. Journal of Formalized Mathematics, 2, 1990.
[12] Wojciech A. Trybulec. Groups. Journal of Formalized Mathematics, 2, 1990.
[13] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.