Volume 10, 1998

University of Bialystok

Copyright (c) 1998 Association of Mizar Users

**Jing-Chao Chen**- Shanghai Jiaotong University
**Yatsuka Nakamura**- Shinshu University, Nagano

- We present the bubble sorting algorithm using macro instructions such as the if-Macro (conditional branch macro instructions) and the Times-Macro (for-loop macro instructions) etc. The correctness proof of the program should include the proof of autonomic, halting and the correctness of the program result. In the three terms, we justify rigorously the correctness of the bubble sorting algorithm. In order to prove it is autonomic, we use the following theorem: if all variables used by the program are initialized, it is autonomic. This justification method probably reveals that autonomic concept is not important.

Contents (PDF format)

- [1]
Noriko Asamoto.
Conditional branch macro instructions of \SCMFSA. Part I.
*Journal of Formalized Mathematics*, 8, 1996. - [2]
Noriko Asamoto.
Conditional branch macro instructions of \SCMFSA. Part II.
*Journal of Formalized Mathematics*, 8, 1996. - [3]
Noriko Asamoto.
Constant assignment macro instructions of \SCMFSA. Part II.
*Journal of Formalized Mathematics*, 8, 1996. - [4]
Noriko Asamoto.
The \tt loop and \tt times macroinstruction for \SCMFSA.
*Journal of Formalized Mathematics*, 9, 1997. - [5]
Noriko Asamoto, Yatsuka Nakamura, Piotr Rudnicki, and Andrzej Trybulec.
On the composition of macro instructions. Part II.
*Journal of Formalized Mathematics*, 8, 1996. - [6]
Noriko Asamoto, Yatsuka Nakamura, Piotr Rudnicki, and Andrzej Trybulec.
On the composition of macro instructions. Part III.
*Journal of Formalized Mathematics*, 8, 1996. - [7]
Grzegorz Bancerek.
Cardinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Grzegorz Bancerek.
Sequences of ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Grzegorz Bancerek and Andrzej Trybulec.
Miscellaneous facts about functions.
*Journal of Formalized Mathematics*, 8, 1996. - [13]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Czeslaw Bylinski.
Partial functions.
*Journal of Formalized Mathematics*, 1, 1989. - [16]
Czeslaw Bylinski.
A classical first order language.
*Journal of Formalized Mathematics*, 2, 1990. - [17]
Czeslaw Bylinski.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [18]
Czeslaw Bylinski.
The modification of a function by a function and the iteration of the composition of a function.
*Journal of Formalized Mathematics*, 2, 1990. - [19]
Jing-Chao Chen and Yatsuka Nakamura.
Initialization halting concepts and their basic properties of \SCMFSA.
*Journal of Formalized Mathematics*, 10, 1998. - [20]
Jaroslaw Kotowicz.
Functions and finite sequences of real numbers.
*Journal of Formalized Mathematics*, 5, 1993. - [21]
Yatsuka Nakamura and Andrzej Trybulec.
A mathematical model of CPU.
*Journal of Formalized Mathematics*, 4, 1992. - [22]
Jan Popiolek.
Some properties of functions modul and signum.
*Journal of Formalized Mathematics*, 1, 1989. - [23]
Piotr Rudnicki and Andrzej Trybulec.
Memory handling for \SCMFSA.
*Journal of Formalized Mathematics*, 8, 1996. - [24]
Yasushi Tanaka.
On the decomposition of the states of SCM.
*Journal of Formalized Mathematics*, 5, 1993. - [25]
Andrzej Trybulec.
Enumerated sets.
*Journal of Formalized Mathematics*, 1, 1989. - [26]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [27]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [28]
Andrzej Trybulec and Agata Darmochwal.
Boolean domains.
*Journal of Formalized Mathematics*, 1, 1989. - [29]
Andrzej Trybulec and Yatsuka Nakamura.
Some remarks on the simple concrete model of computer.
*Journal of Formalized Mathematics*, 5, 1993. - [30]
Andrzej Trybulec and Yatsuka Nakamura.
Modifying addresses of instructions of \SCMFSA.
*Journal of Formalized Mathematics*, 8, 1996. - [31]
Andrzej Trybulec, Yatsuka Nakamura, and Noriko Asamoto.
On the compositions of macro instructions. Part I.
*Journal of Formalized Mathematics*, 8, 1996. - [32]
Andrzej Trybulec, Yatsuka Nakamura, and Piotr Rudnicki.
The \SCMFSA computer.
*Journal of Formalized Mathematics*, 8, 1996. - [33]
Michal J. Trybulec.
Integers.
*Journal of Formalized Mathematics*, 2, 1990. - [34]
Wojciech A. Trybulec.
Pigeon hole principle.
*Journal of Formalized Mathematics*, 2, 1990. - [35]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [36]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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