Volume 7, 1995

University of Bialystok

Copyright (c) 1995 Association of Mizar Users

**Yatsuka Nakamura**- Shinshu University, Nagano
**Grzegorz Bancerek**- Institute of Mathematics, Polish Academy of Sciences

- We continue the formalisation of circuits started in [11],[12],[10], [13]. Our goal was to work out the notation of combining circuits which could be employed to prove the properties of real circuits.

This work was written while the second author visited Shinshu University, July--August 1994.

- Combining of Many Sorted Signatures
- Combining of Circuits
- Signatures with One Operation
- Unsplit Condition
- One Gate Circuits
- Boolean Circuits

- [1]
Grzegorz Bancerek.
K\"onig's theorem.
*Journal of Formalized Mathematics*, 2, 1990. - [2]
Grzegorz Bancerek.
Tarski's classes and ranks.
*Journal of Formalized Mathematics*, 2, 1990. - [3]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski.
Partial functions.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Czeslaw Bylinski.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [8]
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. - [9]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto.
Introduction to circuits, I.
*Journal of Formalized Mathematics*, 6, 1994. - [11]
Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto.
Preliminaries to circuits, I.
*Journal of Formalized Mathematics*, 6, 1994. - [12]
Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto.
Preliminaries to circuits, II.
*Journal of Formalized Mathematics*, 6, 1994. - [13]
Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto.
Introduction to circuits, II.
*Journal of Formalized Mathematics*, 7, 1995. - [14]
Andrzej Trybulec.
Binary operations applied to functions.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [16]
Andrzej Trybulec.
Tuples, projections and Cartesian products.
*Journal of Formalized Mathematics*, 1, 1989. - [17]
Andrzej Trybulec.
Many-sorted sets.
*Journal of Formalized Mathematics*, 5, 1993. - [18]
Andrzej Trybulec.
Many sorted algebras.
*Journal of Formalized Mathematics*, 6, 1994. - [19]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [20]
Wojciech A. Trybulec.
Groups.
*Journal of Formalized Mathematics*, 2, 1990. - [21]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [22]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [23]
Edmund Woronowicz.
Many-argument relations.
*Journal of Formalized Mathematics*, 2, 1990.

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