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.

#### MML Identifier: ASYMPT_0

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]

#### Contents (PDF format)

1. Preliminaries
2. A Notation for the order of"
3. Other Asymptotic Notation
4. Conditional Asymptotic Notation
5. Operations on Asymptotic Notation

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