Volume 10, 1998

University of Bialystok

Copyright (c) 1998 Association of Mizar Users

**Piotr Rudnicki**- University of Alberta, Edmonton
- This work was partially supported by NSERC Grant OGP9207 and NATO CRG 951368.

- We define a {\tt for} type (going up) macro instruction in terms of the {\tt while} macro. This gives an iterative macro with an explicit control variable. The {\tt for} macro is used to define a macro for the selection sort acting on a finite sequence location of {\SCMFSA}. On the way, a macro for finding a minimum in a section of an array is defined.

- General Preliminaries
- {\SCMFSA} Preliminaries
- The {\tt for-up} Macro Instruction
- Finding Minimum in a Section of an Array
- A Swap Macro Instruction
- Selection Sort

- [1]
Noriko Asamoto.
Conditional branch macro instructions of \SCMFSA. Part II.
*Journal of Formalized Mathematics*, 8, 1996. - [2]
Noriko Asamoto.
Constant assignment macro instructions of \SCMFSA. Part II.
*Journal of Formalized Mathematics*, 8, 1996. - [3]
Noriko Asamoto, Yatsuka Nakamura, Piotr Rudnicki, and Andrzej Trybulec.
On the composition of macro instructions. Part II.
*Journal of Formalized Mathematics*, 8, 1996. - [4]
Noriko Asamoto, Yatsuka Nakamura, Piotr Rudnicki, and Andrzej Trybulec.
On the composition of macro instructions. Part III.
*Journal of Formalized Mathematics*, 8, 1996. - [5]
Grzegorz Bancerek.
Cardinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Grzegorz Bancerek.
Sequences of ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Grzegorz Bancerek.
K\"onig's theorem.
*Journal of Formalized Mathematics*, 2, 1990. - [10]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Grzegorz Bancerek and Andrzej Trybulec.
Miscellaneous facts about functions.
*Journal of Formalized Mathematics*, 8, 1996. - [12]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Czeslaw Bylinski.
Partial functions.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Czeslaw Bylinski.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [16]
Jing-Chao Chen.
While macro instructions of \SCMFSA.
*Journal of Formalized Mathematics*, 9, 1997. - [17]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [18]
Jaroslaw Kotowicz.
Convergent real sequences. Upper and lower bound of sets of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [19]
Yatsuka Nakamura and Piotr Rudnicki.
Vertex sequences induced by chains.
*Journal of Formalized Mathematics*, 7, 1995. - [20]
Yatsuka Nakamura and Andrzej Trybulec.
A mathematical model of CPU.
*Journal of Formalized Mathematics*, 4, 1992. - [21]
Piotr Rudnicki.
On the composition of non-parahalting macro instructions.
*Journal of Formalized Mathematics*, 10, 1998. - [22]
Piotr Rudnicki and Andrzej Trybulec.
Memory handling for \SCMFSA.
*Journal of Formalized Mathematics*, 8, 1996. - [23]
Andrzej Trybulec.
Semilattice operations on finite subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [24]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [25]
Andrzej Trybulec.
On the sets inhabited by numbers.
*Journal of Formalized Mathematics*, 15, 2003. - [26]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [27]
Andrzej Trybulec and Agata Darmochwal.
Boolean domains.
*Journal of Formalized Mathematics*, 1, 1989. - [28]
Andrzej Trybulec and Yatsuka Nakamura.
Some remarks on the simple concrete model of computer.
*Journal of Formalized Mathematics*, 5, 1993. - [29]
Andrzej Trybulec and Yatsuka Nakamura.
Modifying addresses of instructions of \SCMFSA.
*Journal of Formalized Mathematics*, 8, 1996. - [30]
Andrzej Trybulec, Yatsuka Nakamura, and Noriko Asamoto.
On the compositions of macro instructions. Part I.
*Journal of Formalized Mathematics*, 8, 1996. - [31]
Andrzej Trybulec, Yatsuka Nakamura, and Piotr Rudnicki.
The \SCMFSA computer.
*Journal of Formalized Mathematics*, 8, 1996. - [32]
Michal J. Trybulec.
Integers.
*Journal of Formalized Mathematics*, 2, 1990. - [33]
Wojciech A. Trybulec.
Groups.
*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. - [37]
Edmund Woronowicz.
Relations defined on sets.
*Journal of Formalized Mathematics*, 1, 1989.

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