Towards the Construction of a Model of Mizar Concepts

Grzegorz Bancerek

Formalized Mathematics, Tome 16 (2008), p. 207-230 / Harvested from The Polish Digital Mathematics Library

Résumé

The aim of this paper is to develop a formal theory of Mizar linguistic concepts following the ideas from [14] and [13]. The theory here presented is an abstract of the existing implementation of the Mizar system and is devoted to the formalization of Mizar expressions. The base idea behind the formalization is dependence on variables which is determined by variable-dependence (variables may depend on other variables). The dependence constitutes a Galois connection between opposite poset of dependence-closed set of variables and the sup-semilattice of widening of Mizar types (smooth type widening).In the paper the concepts strictly connected with Mizar expressions are formalized. Among them are quasi-loci, quasi-terms, quasi-adjectives, and quasi-types. The structural induction and operation of substitution are also introduced. The prefix quasi is used to indicate that some rules of construction of Mizar expressions may not be fulfilled. For example, variables, quasi-loci, and quasi-terms have no assigned types and, in result, there is no possibility to conduct type-checking of arguments. The other gaps concern inconsistent and out-of-context clusters of adjectives in types. Those rules are required in the Mizar identification process. However, the expression appearing in later processes of Mizar checker may not satisfy the rules. So, introduced apparatus is enough and adequate to describe data structures and algorithms from the Mizar checker (like equational classes).MML identifier: ABCMIZ 1, version: 7.9.01 4.101.1015

Détail

Publié le : 2008-01-01
EUDML-ID : urn:eudml:doc:266609

Bibliographie

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. | Zbl 06213858

[3] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421-427, 1990.

[4] Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.

[5] Grzegorz Bancerek. Tarski's classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.

[6] Grzegorz Bancerek. Complete lattices. Formalized Mathematics, 2(5):719-725, 1991.

[7] Grzegorz Bancerek. König's lemma. Formalized Mathematics, 2(3):397-402, 1991.

[8] Grzegorz Bancerek. Sets and functions of trees and joining operations of trees. Formalized Mathematics, 3(2):195-204, 1992.

[9] Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77-82, 1993.

[10] Grzegorz Bancerek. Terms over many sorted universal algebra. Formalized Mathematics, 5(2):191-198, 1996.

[11] Grzegorz Bancerek. Bounds in posets and relational substructures. Formalized Mathematics, 6(1):81-91, 1997.

[12] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Formalized Mathematics, 6(1):93-107, 1997.

[13] Grzegorz Bancerek. On semilattice structure of Mizar types. Formalized Mathematics, 11(4):355-369, 2003. | Zbl 1264.03040

[14] Grzegorz Bancerek. On the structure of Mizar types. In Herman Geuvers and Fairouz Kamareddine, editors, Electronic Notes in Theoretical Computer Science, volume 85. Elsevier, 2003. | Zbl 1264.03040

[15] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.

[16] Grzegorz Bancerek and Artur Korniłowicz. Yet another construction of free algebra. Formalized Mathematics, 9(4):779-785, 2001.

[17] Grzegorz Bancerek and Yatsuka Nakamura. Full adder circuit. Part I. Formalized Mathematics, 5(3):367-380, 1996.

[18] Grzegorz Bancerek and Piotr Rudnicki. On defining functions on trees. Formalized Mathematics, 4(1):91-101, 1993.

[19] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.

[20] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.

[21] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

[22] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.

[23] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.

[24] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

[25] Patricia L. Carlson and Grzegorz Bancerek. Context-free grammar - part 1. Formalized Mathematics, 2(5):683-687, 1991.

[26] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.

[27] Adam Grabowski and Robert Milewski. Boolean posets, posets under inclusion and products of relational structures. Formalized Mathematics, 6(1):117-121, 1997.

[28] Yatsuka Nakamura. Determinant of some matrices of field elements. Formalized Mathematics, 14(1):1-5, 2006. | Zbl 0264.94018

[29] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Formalized Mathematics, 5(2):167-172, 1996.

[30] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.

[31] Beata Perkowska. Free many sorted universal algebra. Formalized Mathematics, 5(1):67-74, 1996.

[32] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.

[33] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.

[34] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.

[35] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.

[36] Andrzej Trybulec. Many-sorted sets. Formalized Mathematics, 4(1):15-22, 1993.

[37] Andrzej Trybulec. Many sorted algebras. Formalized Mathematics, 5(1):37-42, 1996.

[38] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.

[39] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187-190, 1990.

[40] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics, 1(2):387-393, 1990.

[41] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

[42] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.

[43] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.