Journal of Formalized Mathematics
Volume 11, 1999
University of Bialystok
Copyright (c) 1999
Association of Mizar Users
Asymptotic Notation. Part I: Theory
-
Richard Krueger
-
University of Alberta, Edmonton
-
Piotr Rudnicki
-
University of Alberta, Edmonton
-
Paul Shelley
-
University of Alberta, Edmonton
Summary.
-
The widely used textbook by Brassard and Bratley [3]
includes a chapter devoted to asymptotic notation (Chapter 3, pp. 79-97).
We have attempted to test how suitable the current version of Mizar is for
recording this type of material in its entirety. A more detailed report
on this experiment will be available separately.
This article presents the development of notions and a follow-up
article [11] includes examples and solutions to problems.
The preliminaries introduce a number of properties of real sequences,
some operations on real sequences, and a characterization of
convergence. The remaining sections in this article correspond to sections
of Chapter 3 of [3]. Section 2 defines the $O$ notation
and proves the threshold, maximum, and limit rules. Section 3 introduces
the $\Omega$ and $\Theta$ notations and their analogous rules.
Conditional asymptotic notation is defined in Section 4 where smooth
functions are also discussed. Section 5 defines some operations
on asymptotic notation (we have decided not to introduce the asymptotic
notation for functions of several variables as it is a straightforward
generalization of notions for unary functions).
This work has been supported by NSERC Grant OGP9207.
The terminology and notation used in this paper have been
introduced in the following articles
[14]
[18]
[2]
[16]
[7]
[4]
[5]
[15]
[1]
[9]
[8]
[12]
[13]
[6]
[17]
[10]
-
Preliminaries
-
A Notation for ``the order of"
-
Other Asymptotic Notation
-
Conditional Asymptotic Notation
-
Operations on Asymptotic Notation
Bibliography
- [1]
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Grzegorz Bancerek.
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Journal of Formalized Mathematics,
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- [3]
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\em Fundamentals of Algorithmics.
Prentice Hall, 1996.
- [4]
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Functions and their basic properties.
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1, 1989.
- [5]
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Journal of Formalized Mathematics,
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Journal of Formalized Mathematics,
1, 1989.
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Journal of Formalized Mathematics,
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Journal of Formalized Mathematics,
1, 1989.
- [10]
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The limit of a real function at infinity.
Journal of Formalized Mathematics,
2, 1990.
- [11]
Richard Krueger, Piotr Rudnicki, and Paul Shelley.
Asymptotic notation. Part II: Examples and problems.
Journal of Formalized Mathematics,
11, 1999.
- [12]
Rafal Kwiatek.
Factorial and Newton coefficients.
Journal of Formalized Mathematics,
2, 1990.
- [13]
Konrad Raczkowski and Andrzej Nedzusiak.
Real exponents and logarithms.
Journal of Formalized Mathematics,
2, 1990.
- [14]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
- [15]
Andrzej Trybulec.
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2, 1990.
- [16]
Andrzej Trybulec.
Subsets of real numbers.
Journal of Formalized Mathematics,
Addenda, 2003.
- [17]
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Journal of Formalized Mathematics,
1, 1989.
- [18]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
Received November 4, 1999
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