A Compendium of Continuous Lattices in MIZAR

This paper reports on the MIZAR formalization of the theory of continuous lattices as presented in Gierz et al.: A Compendium of Continuous Lattices, 1980. By a MIZAR formalization we mean a formulation of theorems, definitions, and proofs written in the MIZAR language whose correctness is verified by the MIZAR processor. This effort was originally motivated by the question of whether or not the MIZAR system was sufficiently developed for the task of expressing advanced mathematics. The current state of the formalization – 57 MIZAR articles written by 16 authors – indicates that in principle the MIZAR system has successfully met the challenge. To our knowledge it is the most sizable effort aimed at mechanically checking some substantial and relatively recent field of advanced mathematics. However, it does not mean that doing mathematics in MIZAR is as simple as doing mathematics traditionally (if doing mathematics is simple at all). The work of formalizing the material of the Gierz et al. compendium has (i) prompted many improvements of the MIZAR proof checking system, (ii) caused numerous revisions of the the MIZAR data base, and (iii) contributed to the “to do” list of further changes to the MIZAR system.

[1]  Yasunari Shidama,et al.  Lim-inf Convergence and its Compactness , 2002 .

[2]  Grzegorz Bancerek,et al.  Lawson Topology in Continuous Lattices 1 , 2004 .

[3]  B. Balkay,et al.  Introduction to lattice theory , 1965 .

[4]  Adam Grabowski,et al.  Boolean Posets, Posets under Inclusion and Products of Relational Structures 1 , 1997 .

[5]  A. Zalewska,et al.  Properties of Binary Relations , 1990 .

[6]  Andrzej Trybulec.  Categories without Uniqueness of cod and dom , 1996 .

[7]  Czeslaw Bylinski.  Some Basic Properties of Sets , 2004 .

[8]  Grzegorz Bancerek.  Arithmetic of Non-Negative Rational Numbers 1 , 1998 .

[9]  Grzegorz Bancerek,et al.  Duality in Relation Structures 1 , 1994 .

[10]  Wojciech A. Trybulec.  Partially Ordered Sets , 1990 .

[11]  Piotr Rudnicki,et al.  Commutative Algebra in the Mizar System , 2001, J. Symb. Comput..

[12]  Grzegorz Bancerek,et al.  Development of the theory of continuous lattices in Mizar , 2001 .

[13]  Freek Wiedijk.  Mizar : An Impression , 1999 .

[14]  Grzegorz Bancerek,et al.  The "Way-Below" Relation 1 , 1997 .

[15]  Andrzej Grzegorczyk.  Zarys arytmetyki teoretycznej , 1971 .

[16]  G. Bancerek.  Konig's Theorem , 1990 .

[17]  Martin D. Davis,et al.  Obvious Logical Inferences , 1981, IJCAI.

[18]  Andrzej Trybulec,et al.  Functors for Alternative Categories , 1996 .

[19]  Grzegorz Bancerek,et al.  Directed Sets, Nets, Ideals, Filters, and Maps 1 , 1997 .

[20]  Robert Milewski,et al.  Algebraic Lattices , 1994 .

[21]  K. Hofmann,et al.  A Compendium of Continuous Lattices , 1980 .

[22]  Edmund Woronowicz.  Relations Defined on Sets , 1990 .

[23]  Horst Herrlich,et al.  Galois Connections , 1985, Mathematical Foundations of Programming Semantics.

[24]  Alexander Yu,et al.  The Cantor Set , 2007 .

[25]  Grzegorz Bancerek,et al.  The \way-below" Relation , 1996 .

[26]  Klaus Keimel,et al.  Continuous Lattices and Domains: The Scott Topology , 2003 .

[27]  A. Trybulec.  Tarski Grothendieck Set Theory , 1990 .

[28]  Zinaida Trybulec,et al.  Boolean Properties of Sets , 1990 .

[29]  Jozef Bia,et al.  Group and Field Definitions , 1990 .

[30]  Czesław Bylí,et al.  Introduction to Categories and Functors , 1989 .

[31]  Artur Korniłowicz,et al.  Cartesian Products of Relations and Relational Structures 1 , 1996 .

[32]  Alexander V. Lyaletski,et al.  Evidence Algorithm and Sequent Logical Inference Search , 1999, LPAR.

[33]  Agata Darmochwa,et al.  Topological Spaces and Continuous Functions , 1990 .

[34]  Krzysztof Grabczewski,et al.  Mechanizing Set Theory: Cardinal Arithmetic and the Axiom of Choice , 2001, ArXiv.

[35]  Rp Rob Nederpelt,et al.  Selected papers on Automath , 1994 .

[36]  Freek Wiedijk,et al.  The De Bruijn Factor , 2000 .

[37]  Grzegorz Bancerek,et al.  The Lawson Topology 1 , 1998 .

[38]  Grzegorz Bancerek,et al.  Bounds in Posets and Relational Substructures 1 , 1997 .

[39]  Piotr Rudnicki,et al.  On Equivalents of Well-Foundedness , 1999, Journal of Automated Reasoning.

[40]  S. Zukowski.  Introduction to Lattice Theory , 1990 .

[41]  G. Bancerek.  Prime Ideals and Filters 1 , 2004 .

[42]  Tobias Nipkow,et al.  Winskel is (almost) Right: Towards a Mechanized Semantics Textbook , 1996, Formal Aspects of Computing.

[43]  Lawrence C. Paulson,et al.  Mechanizing set theory , 1996, Journal of Automated Reasoning.

[44]  Grzegorz Bancerek,et al.  Tarski's Classes and Ranks , 1990 .

[45]  Grzegorz Bancerek,et al.  Continuous Lattices of Maps between T 0 Spaces 1 , 1999 .

[46]  Piotr Rudnicki.  Obvious inferences , 2004, Journal of Automated Reasoning.

[47]  R. Sikorski,et al.  The mathematics of metamathematics , 1963 .

[48]  D. Barton,et al.  Grundlagen der Analysis , 1934 .

[49]  Sam Alfred Pearsall.  The Cantor set , 1999 .

[50]  Beata Madras,et al.  Product of Family of Universal Algebras , 2004 .

[51]  P. Rudnicki.  Kernel Projections and Quotient Lattices , 1998 .

[52]  Grzegorz Bancerek,et al.  Cartesian Product of Functions , 1991 .

[53]  Artur Korniłowicz,et al.  Journal of Formalized Mathematics on the Topological Properties of Meet-continuous Lattices , 2022 .

[54]  Jacques D. Fleuriot,et al.  A Combination of Nonstandard Analysis and Geometry Theorem Proving, with Application to Newton's Principia , 1998, CADE.

[55]  van Ls Bert Benthem Jutting,et al.  Checking Landau's “Grundlagen” in the Automath System: Appendices 3 and 4 (The PN-lines; Excerpt for “Satz 27”) , 1994 .