## Mathematical Knowledge Management in MIZAR (2001)

Venue: | Proc. of MKM 2001 |

Citations: | 9 - 0 self |

### BibTeX

@INPROCEEDINGS{Rudnicki01mathematicalknowledge,

author = {Piotr Rudnicki and Andrzej Trybulec},

title = {Mathematical Knowledge Management in MIZAR},

booktitle = {Proc. of MKM 2001},

year = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

We report on how mathematics is done in the Mizar system. Mizar oers a language for writing mathematics and provides software for proof-checking. Mizar is used to build Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowledge. The language and the checking software evolve and the evolution is driven by the growing MML.

### Citations

343 |
A Compendium of Continuous Lattices
- Gierz, Hoffmann, et al.
- 1980
(Show Context)
Citation Context ...ience may help in improving the system. There has been a focused eort to advance the contents of MML in some areas: the theory of continuous lattices through translation of the entire main text of [1=-=2-=-]. This theory involves diverse and recent mathematics, see [1] for the report on the Mizar developments. the proof of the Jordan curve theorem is being continued, articles in the series GOBOARD and ... |

311 | Finite sequences and tuples of elements of a non-empty sets
- ByliĆski
- 1990
(Show Context)
Citation Context ...re are two notions of a pair in MML: 1. [x,y] dened as ffx,yg,fxgg, that is the Kuratowski pair (see Section A) and 2.sdened as an Element of 2-tuples on a set, that is asnite sequence of length 2 ([6]). and there is a prevailing opinion that only one of them should be widely used: preferably the second, while thesrst should be avoided. Note that thesrst denition is used when dening relation, whi... |

119 |
Product of family of universal algebras
- Madras
- 1993
(Show Context)
Citation Context ...ulec Mathematical Knowledge Management in Mizar of structures, authors separately dene a binary Cartesian product of structures and only then a general product (see products of universal algebras in [=-=17-=-] or products of many sorted algebras in [18]). This is repetition of eort and if we decide to use the second variant for the binary Cartesian product then a lot of duplicated material could be elimin... |

63 |
The Mathematics of Metamathematics
- Rasiowa, Sikorski
- 1970
(Show Context)
Citation Context ...agement in Mizar mode that it constructs non empty types, for a cluster of adjectives we have to show that there are objects with all the corresponding properties. We borrowed the name \functor" =-=from [22]-=-, p. 148: ... some signs in the formalized language should correspond to the mappings and functions being examined. These signs are called functors, or|more precisely| m-argument functors provided the... |

37 |
Symbolic Logic. An Introduction
- Fitch
- 1952
(Show Context)
Citation Context ...ski [14] (see also [19]), which is considered an adequate reconstruction of proof styles used in mathematical publications. Similar systems were independently described by K. Ono [21] and F. B. Fitch =-=[10-=-]. the notion of an obvious inference instead of using somesxed set of inference rules. This notion has been addressed in the past by M. Davis [8], in the Kiev project [9] and recently by H. Friedman... |

17 |
An Outline of PC Mizar, Fondation Philippe le Hodey
- Muzalewski
- 1993
(Show Context)
Citation Context ...uage provides means for using free second order variables in forming schemes of theorems (innite families of theorems). An introductory information about Mizar with numerous examples can be found in [=-=23, 26,-=- 20]. Mizar oers a number of denitional facilities. Each denition denes a new constructor later used in syntactic constructions, and gives its syntax and meaning. In Mizar terminology, predicates are ... |

16 | Obvious logical inferences
- Davis
(Show Context)
Citation Context ...ependently described by K. Ono [21] and F. B. Fitch [10]. the notion of an obvious inference instead of using somesxed set of inference rules. This notion has been addressed in the past by M. Davis [=-=8]-=-, in the Kiev project [9] and recently by H. Friedman [11]. At the beginning of the project, there were plans to test the Mizar language and software in building libraries based on various axiomatics:... |

14 | Proof style
- Harrison
- 1997
(Show Context)
Citation Context ... intensity interaction: the entire text is processed each time. Such a low intensity interaction has its advantages as it stresses the division of responsibilities between the author and the checker (=-=[11, 13]-=-). The part of the Mizar language for structuring proofs is close to mathematical vernacular and can be used to record just an idea of a proof (with many unproven steps). Then one can check the overal... |

13 | On Equivalents of Well-foundedness. An experiment in Mizar
- Rudnicki, Trybulec
- 1999
(Show Context)
Citation Context ...uage provides means for using free second order variables in forming schemes of theorems (innite families of theorems). An introductory information about Mizar with numerous examples can be found in [=-=23, 26,-=- 20]. Mizar oers a number of denitional facilities. Each denition denes a new constructor later used in syntactic constructions, and gives its syntax and meaning. In Mizar terminology, predicates are ... |

11 |
On the rules of suppositions in formal logic
- Jaskowski
- 1967
(Show Context)
Citation Context ...enable computerized processing, in particular mechanical verication of correctness. For the above reasons, the logical basis of Mizar is formed by: the system of natural deduction by S. Jaskowski [14=-=-=-] (see also [19]), which is considered an adequate reconstruction of proof styles used in mathematical publications. Similar systems were independently described by K. Ono [21] and F. B. Fitch [10]. ... |

10 | Evidence Algorithm and Sequent Logical Inference Search
- Degtyarev, Lyaletski, et al.
- 1999
(Show Context)
Citation Context .... Ono [21] and F. B. Fitch [10]. the notion of an obvious inference instead of using somesxed set of inference rules. This notion has been addressed in the past by M. Davis [8], in the Kiev project [=-=9]-=- and recently by H. Friedman [11]. At the beginning of the project, there were plans to test the Mizar language and software in building libraries based on various axiomatics: ZFC for `normal' mathema... |

5 | Development of the Theory of Continuous Lattices in Mizar
- Bancerek
(Show Context)
Citation Context ...d eort to advance the contents of MML in some areas: the theory of continuous lattices through translation of the entire main text of [12]. This theory involves diverse and recent mathematics, see [1] for the report on the Mizar developments. the proof of the Jordan curve theorem is being continued, articles in the series GOBOARD and JORDAN; reasoning about computations, articles in the series... |

4 |
The Reflection Theorem, Formalized Mathematics
- Bancerek
- 1990
(Show Context)
Citation Context ... As another example let us take the notion of Cartesian product. A binary Cartesian product can be expressed either as: 1. [:X,Y:] dened in [7], or 2. productsusing the generalized product dened in [2=-=-=-]. (If we want a Cartesian square, we can use the above as [:X,X:] or product ; the latter has a semantically equal counterpart available through a dierent notation: 2-tuples on X, see [6].) The quest... |

3 |
Wojciech Leonczuk, and Micha l Muzalewski. Abelian groups, and vector spaces
- Kusak
- 1990
(Show Context)
Citation Context ...a-structure suces. Once we want to develop a general theory and talk about relationship among structures, then the notion of a structure is indispensable. (In MML, theseld of reals is dened later in [=-=16-=-].) Another examples of widely use metastructures in MML: integer arithmetic, natural arithmetic,snite sets,snite sequences. Choosing notions The roles of classical and abstract mathematics are dieren... |

3 | Tarski Grothendieck set theory. Formalized Mathematics - Trybulec - 1990 |

2 |
Some basic properties of sets. Formalized Mathematics
- Bylinski
- 1990
(Show Context)
Citation Context ... but in order to dene them we need some notion of pairing. As another example let us take the notion of Cartesian product. A binary Cartesian product can be expressed either as: 1. [:X,Y:] dened in [7=-=-=-], or 2. productsusing the generalized product dened in [2]. (If we want a Cartesian square, we can use the above as [:X,X:] or product ; the latter has a semantically equal counterpart available thro... |

2 |
On a practical way of describing formal deductions
- Ono
- 1962
(Show Context)
Citation Context ...duction by S. Jaskowski [14] (see also [19]), which is considered an adequate reconstruction of proof styles used in mathematical publications. Similar systems were independently described by K. Ono [=-=21-=-] and F. B. Fitch [10]. the notion of an obvious inference instead of using somesxed set of inference rules. This notion has been addressed in the past by M. Davis [8], in the Kiev project [9] and re... |

1 | Zermelo theorem and axiom of choice. Formalized Mathematics - Bancerek - 1990 |

1 | On powers of cardinals. Formalized Mathematics - Bancerek - 1992 |

1 |
FOM: 107: Automated proof checking, May 2001. www.math.psu.edu/simpson/fom/postings
- Friedman
(Show Context)
Citation Context ... the notion of an obvious inference instead of using somesxed set of inference rules. This notion has been addressed in the past by M. Davis [8], in the Kiev project [9] and recently by H. Friedman [=-=11]-=-. At the beginning of the project, there were plans to test the Mizar language and software in building libraries based on various axiomatics: ZFC for `normal' mathematics, Peano axioms for theoretica... |

1 |
lgorzata Korolkiewicz. Homomorphisms of algebras. Quotient universal algebra. Formalized Mathematics
- Ma
- 1993
(Show Context)
Citation Context ...al notion of structure (for example a la Bourbaki) and then it would be possible to introduce a general notion of homomorphism, of substructure etc. in MML, this has been done for one-sorted algebras =-=[15-=-]. Since this is a simple case let us examine it a bit closer. The notion of a group may be dened in a number of ways using various constructs of Mizar or we can treat a group as a universal algebra. ... |

1 |
Products of many sorted algebras. Formalized Mathematics
- Madras
- 1996
(Show Context)
Citation Context ...zar of structures, authors separately dene a binary Cartesian product of structures and only then a general product (see products of universal algebras in [17] or products of many sorted algebras in [=-=18-=-]). This is repetition of eort and if we decide to use the second variant for the binary Cartesian product then a lot of duplicated material could be eliminated. This type of issues are subject to fre... |

1 |
Trybulec and Micha l J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics
- Wojciech
- 1991
(Show Context)
Citation Context ...p may be dened in a number of ways using various constructs of Mizar or we can treat a group as a universal algebra. In the former case, we have to introduce all the needed notions from scratch (see [=-=25]-=-), in the latter, one can employ the notions (e.g. isomorphism) previously developed for universal algebras. Certainly, using a more general apparatus (universal algebras) may cause various troubles. ... |

1 |
Mizar: an impression. www.cs.kun.nl/~freek/mizar/mizarintro.ps.gz. A MML Axiomatics The development of mathematics in Mizar is based on a version of the Zermelo-Fraenkel set theory. Several notions of set theory are built into the Mizar processor and thei
- Wiedijk
(Show Context)
Citation Context ...uage provides means for using free second order variables in forming schemes of theorems (innite families of theorems). An introductory information about Mizar with numerous examples can be found in [=-=23, 26,-=- 20]. Mizar oers a number of denitional facilities. Each denition denes a new constructor later used in syntactic constructions, and gives its syntax and meaning. In Mizar terminology, predicates are ... |