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]

Preliminaries

Euclid's Algorithm

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.
Received October 8, 1993
[
Download a postscript version,
MML identifier index,
Mizar home page]