Four Decades of Mizar

Grabowski, Adam; Korniłowicz, Artur; Naumowicz, Adam

doi:10.1007/s10817-015-9345-1

Cited by 115 publications

(69 citation statements)

References 27 publications

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“…In particular we show that polynomial rings can be ordered in (at least) two different ways [8,5,4,9]. This is the continuation of the development of algebraic hierarchy in Mizar [2,3]. …”

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confidence: 73%

Ordered Rings and Fields

Schwarzweller

2017

Formalized Mathematics

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Summary.We introduce ordered rings and fields following Artin-Schreier's approach using positive cones. We show that such orderings coincide with total order relations and give examples of ordered (and non ordered) rings and fields. In particular we show that polynomial rings can be ordered in (at least) two different ways [8,5,4,9]. This is the continuation of the development of algebraic hierarchy in Mizar [2,3]. MSC: 12J15 03B35Keywords: commutative algebra; ordered fields; positive cones MML identifier: REALALG1, version: 8.1.05 5.40.1289 On Order RelationsLet X be a set and R be a binary relation on X. We say that R is strongly reflexive if and only if (Def. 1) R is reflexive in X.We say that R is totally connected if and only if (Def. 2) R is strongly connected in X.One can check that there exists a binary relation on X which is strongly reflexive and there exists a binary relation on X which is totally connected and every binary relation on X which is strongly reflexive is also reflexive and every binary relation on X which is totally connected is also strongly connected.Let X be a non empty set. One can check that every binary relation on X which is strongly reflexive is also non empty and every binary relation on X which is totally connected is also non empty. Now we state the propositions:

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“…In particular we show that polynomial rings can be ordered in (at least) two different ways [8,5,4,9]. This is the continuation of the development of algebraic hierarchy in Mizar [2,3]. …”

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confidence: 73%

Ordered Rings and Fields

Schwarzweller

2017

Formalized Mathematics

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“…We introduce one meta-level type for Mizar sets and one for Mizar types. We introduce the constants that correspond to sets being of particular types and to combine types (the Mizar soft type system allows intersection types [36]), the indefinite description operator, as well as the axioms that specify these constants. With the Isabelle syntax mechanisms, we allow defining Mizar like syntax for statements and definitions, which can later be used to specify the Tarski-Grothendieck foundations of set theory and translate the first few articles of the MML.…”

Section: Mmode For Coq or Declarative Proof Language (Dpl) For Coqmentioning

confidence: 99%

“…These 15 signatures are the direct ancestors of 57 other structure signatures in the MML. Furthermore, these 15 are directly used to define 293 Mizar types (this includes non-expandable types [36]), 291 attributes, 962 functors, and 91 predicates.…”

Section: Net Of Basic Algebraic Structuresmentioning

confidence: 99%

Progress in the Independent Certiﬁcation of Mizar Mathematical Library in Isabelle

Kaliszyk

Pąk

2017

Annals of Computer Science and Information Systems

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Abstract-The Mizar Mathematical Library is one of the largest collections of machine understandable formal proofs encompassing many areas of today mathematics including results from algebra, analysis, topology, and lattice theory. The Mizar system has so far been the only tool able to completely process, certify, and make use of these developments. In this paper, we present the progress in the development of an independent certification mechanism of Mizar proofs based on the Isabelle logical framework. The approach allows rechecking the Mizar formal proofs based on a more succinct and more precisely specified formal infrastructure. Additionally, it necessitates a full formal specification of the mechanisms that ensure the correctness of the defined objects, in particular, the proofs that such mechanisms are correct. The development already covers an important part of the Mizar library foundations. We improve the mechanism for defining Mizar structures and show that it permits simpler validation of proof developments involving such objects. To demonstrate this, we perform a complete translation of the Mizar net of basic algebraic structures including their attributes and certify all the corresponding proofs in Isabelle.

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“…Now we state the propositions: (14) Let us consider a sequence f of real numbers. Suppose there exists a natural number n such that for every natural number k such that k n holds f (k) = 0.…”

Section: Sequencesmentioning

confidence: 99%

“…We show also some preliminary constructions linking real sequences and finite sequences, where summing formulas are involved. In the Mizar [14] proof, we follow closely https: //en.wikipedia.org/wiki/Liouville_number. The aim is to show that all Liouville numbers are transcendental.…”

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confidence: 99%

Introduction to Liouville Numbers

Grabowski

Korniłowicz

2017

Formalized Mathematics

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Summary. The article defines Liouville numbers, originally introduced byJoseph Liouville in 1844 [17] as an example of an object which can be approximated "quite closely" by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 andIt is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is defined in Section 6 quite generally as the sumfor a finite sequence {a k } k∈N and b ∈ N. Based on this definition, we also introduced the so-called Liouville number as

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Four Decades of Mizar

Abstract: This special issue is dedicated to works related to Mizar, the theorem proving project started by Andrzej Trybulec in the 1970s, and other automated proof checking systems used for formalizing mathematics.

Cited by 115 publications

References 27 publications

Ordered Rings and Fields

Ordered Rings and Fields

Progress in the Independent Certiﬁcation of Mizar Mathematical Library in Isabelle

Introduction to Liouville Numbers

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