## Gradual computerisation/formalisation of mathematical texts into Mizar

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Venue: | From Insight to Proof: Festschrift in Honour of Andrzej Trybulec |

Citations: | 10 - 5 self |

### BibTeX

@INPROCEEDINGS{Kamareddine_gradualcomputerisation/formalisation,

author = {Fairouz Kamareddine and Manuel Maarek and Krzysztof Retel and J. B. Wells},

title = {Gradual computerisation/formalisation of mathematical texts into Mizar},

booktitle = {From Insight to Proof: Festschrift in Honour of Andrzej Trybulec},

year = {},

pages = {95--120},

publisher = {University}

}

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### Abstract

Abstract. We explain in this paper the gradual computerisation process of an ordinary mathematical text into more formal versions ending with a fully formalised Mizar text. The process is part of the MathLang–Mizar project and is divided into a number of steps (called aspects). The first three aspects (CGa, TSa and DRa) are the same for any MathLang–TP project where TP is any proof checker (e.g., Mizar, Coq, Isabelle, etc). These first three aspects are theoretically formalised and implemented and provide the mathematician and/or TP user with useful tools/automation. Using TSa, the mathematician edits his mathematical text just as he would use L ATEX, but at the same time he sees the mathematical text as it appears on his paper. TSa also gives the mathematician easy editing facilities to help assign to parts of the text, grammatical and mathematical roles and to relate different parts through a number of mathematical, rethorical and structural relations. MathLang would then automatically produce CGa and DRa versions of the text, checks

### Citations

1304 | Tarski Grothendieck set theory
- Trybulec
- 1990
(Show Context)
Citation Context ...[24]. MML consists of Mizar documents, which are called Articles within the Mizar community. This library is based on two axiomatic Articles: HIDDEN [4] which consists of built-in notions, and TARSKI =-=[22]-=- which presents axioms of the Tarski-Grothendieck set theory. All the other Articles of MML are consequences of those axioms and are verified by the Mizar system. The user while writing a new Mizar Ar... |

975 | An Introduction to the Theory of Numbers - Hardy, Wright - 1960 |

87 | An overview of the MIZAR project
- Rudnicki
- 1992
(Show Context)
Citation Context ...y to structure the skeleton of a document in a formal language Mizar (see Figure 6). 2.2 Mizar and Formal Proof Sketch The Mizar system (http://mizar.org) is a system for computer checked mathematics =-=[20,21,19,15]-=-. The ongoing development of the Mizar framework, 8 DRAFT r2462 -- DRAFT r2462 -- DRAFT r2462 -- DRAslead by Trybulec since 1973, has resulted in several things: the Mizar system, the Mizar language, ... |

60 |
The mathematical vernacular, a language for mathematics with typed sets. In: Workshop on Programming Logic (1987), reprinted in [10, F.3
- Bruijn
- 2006
(Show Context)
Citation Context ...checking tools and different proof checkers. Dividing the formalisation of mathematical texts into a number of stages was first proposed by N.G. de Bruijn to relate CML to his Mathematical Vernacular =-=[6]-=- (MV) and his proof checking system Automath. We call this principle de Bruijn’s path. The work may be subdivided. One can think of a first stage where a person with some mathematical training inserts... |

24 |
On the structure of Mizar types
- Bancerek
- 2003
(Show Context)
Citation Context ... to the Attribute in Mizar (see the table on the right). Furthermore, adjectives in CML and CGa are used to modify the characteristics of a noun. Similarly, in Mizar we use Adjectives to refine Types =-=[2]-=-. Other identifiers In CGa we have declared some identifiers to be terms, e.g. 0, 2, 4 (see line 10 of Figure 7), whereas in Mizar they are treated as Numerals, which have not been introduced inside M... |

23 | Flexible encoding of mathematics on the computer
- Kamareddine, Maarek, et al.
- 2004
(Show Context)
Citation Context ...ontrast to the formal versions of theorem provers as in [26]. Wiedijk’s comparison illustrates the need to assist the mathematician non-expert in theorem provers. We have already used this example in =-=[11]-=- to obtain what was then a CGa version. In this paper, we use this example to show all the versions of the proposed path (TSa, CGa, DRa, Mizar FPS skeleton, Mizar FPS and finally Mizar). 1.2 Notations... |

23 | Comparing mathematical provers
- Wiedijk
- 2003
(Show Context)
Citation Context ... processing. The Mizar system is accompanied by a library of mathematics – the Mizar Mathematical Library (MML), which is the biggest collection of digitalized mathematical texts verified by computer =-=[24]-=-. MML consists of Mizar documents, which are called Articles within the Mizar community. This library is based on two axiomatic Articles: HIDDEN [4] which consists of built-in notions, and TARSKI [22]... |

22 | Formal Proof Sketches
- Wiedijk
- 2004
(Show Context)
Citation Context ...y formalising or computer proof checking them. Such computerisations are not sufficiently detailed for correctness verification but are used as skeletons in the full formalisation (see Wiedijk’s work =-=[25]-=-). Although the computerised text remains at a low level to be fully automatically checked, it has a precise notion of correctness: it is syntactically correct according to the grammar language but ac... |

19 | P.: Mizar: the first 30 years
- Matuszewski, Rudnicki
- 2005
(Show Context)
Citation Context ...y to structure the skeleton of a document in a formal language Mizar (see Figure 6). 2.2 Mizar and Formal Proof Sketch The Mizar system (http://mizar.org) is a system for computer checked mathematics =-=[20,21,19,15]-=-. The ongoing development of the Mizar framework, 8 DRAFT r2462 -- DRAFT r2462 -- DRAFT r2462 -- DRAslead by Trybulec since 1973, has resulted in several things: the Mizar system, the Mizar language, ... |

18 | Toward an object-oriented structure for mathematical text
- Kamareddine, Maarek, et al.
- 2006
(Show Context)
Citation Context ...identifiers to a specific part of the text with a construction called context or local-scoping. We call step an expression which is either a phrase, a block or a local-scoping. MathLang’s type system =-=[13]-=- derives typing judgments to check whether the reasoning parts of a document are coherently built. CGa is intentionally elementary and results in a computable low level encoding. Corollary1. The Text ... |

16 | Mathlang: Experience-driven development of a new mathematical language
- Kamareddine, Maarek, et al.
- 2004
(Show Context)
Citation Context ... One would encode division elements (such as chapter, section) and mathematical labeling units (such as axiom, theorem, proof ) by this unique step construction (see the MathLang encoding examples in =-=[12,11,13]-=-). In our example in Figure 4, the proof paragraph is a step composed by several sub-steps. To enhance flexibility, CGa does not differentiate between these divisions, labels and any other kind of ste... |

15 |
The Mizar-QC/6000 Logic Information Language
- Trybulec
- 1978
(Show Context)
Citation Context ...y to structure the skeleton of a document in a formal language Mizar (see Figure 6). 2.2 Mizar and Formal Proof Sketch The Mizar system (http://mizar.org) is a system for computer checked mathematics =-=[20,21,19,15]-=-. The ongoing development of the Mizar framework, 8 DRAFT r2462 -- DRAFT r2462 -- DRAFT r2462 -- DRAslead by Trybulec since 1973, has resulted in several things: the Mizar system, the Mizar language, ... |

13 |
On the Rules of Supposition in Formal Logic. Studia Logica
- Jaśkowski
- 1934
(Show Context)
Citation Context ...table for the practical formalisation of mathematics. It is based on first–order logic with free second-order variables. Proofs are written in the style of natural deduction as proposed by Ja´skowski =-=[9]-=-. The language itself is also an attempt to approximate in a formal way the mathematical vernacular used in publications. On one hand, the Mizar language inherits the expressiveness, naturalness and f... |

13 | On Equivalents of Well-foundedness. An experiment in Mizar
- Rudnicki, Trybulec
- 1999
(Show Context)
Citation Context |

5 |
Checking mathematics with computer assistance
- Bruijn
- 1991
(Show Context)
Citation Context ...th just some elemenary mathematics training, or of a computer provided with some artificial intelligence. But we should not be too optimistic about that: programming such jobs is by no means trivial. =-=[5]-=- MV was proposed as a formal substitute for parts of CML. Nederpelt refined MV into another formal substitute for parts of CML, Weak Type Theory (WTT) whose underlying proof theory was developed by Ka... |

5 |
The Seventeen Provers of the World, foreword by Dana S. Scott, volume 3600 of LNCS
- Wiedijk, editor
- 2006
(Show Context)
Citation Context ...s to deal with the whole mechanism (i.e. the Mizar system, MML, the Mizar Language, the MML search engines) and create a new Mizar document. 1.1 Example The example we use in this paper is taken from =-=[26]-=- where Wiedijk used Hardy and Wright’s version [8, Ch. IV] of Pythagoras’ theorem of irrationality of √ 2 to compare computer based theorem provers. Barendregt wrote a textual version [3] (reproduced ... |

4 |
A path to faithful formalizations of mathematics
- Jojgov, Nederpelt
- 2004
(Show Context)
Citation Context ...ial role in this research, the various levels (or aspects, or stages) of our proposed path are new. Another approach which follows the de Bruijn path principle is discussed by Jojgov and Nederpelt in =-=[10]-=-. However their description of a path from CML to type theory via WTT and type theory with open terms (TTOT) starts from a WTT-text which differs from (but represents) the original CML-text, and then ... |

3 |
Mizar built-in notions
- Committee
- 1989
(Show Context)
Citation Context ...igitalized mathematical texts verified by computer [24]. MML consists of Mizar documents, which are called Articles within the Mizar community. This library is based on two axiomatic Articles: HIDDEN =-=[4]-=- which consists of built-in notions, and TARSKI [22] which presents axioms of the Tarski-Grothendieck set theory. All the other Articles of MML are consequences of those axioms and are verified by the... |

2 |
10. Volume 3600 of Wiedijk [26
- Informal
- 2006
(Show Context)
Citation Context ... taken from [26] where Wiedijk used Hardy and Wright’s version [8, Ch. IV] of Pythagoras’ theorem of irrationality of √ 2 to compare computer based theorem provers. Barendregt wrote a textual version =-=[3]-=- (reproduced in Figure 8) of this proof which is said to be “informal” in contrast to the formal versions of theorem provers as in [26]. Wiedijk’s comparison illustrates the need to assist the mathema... |

2 | Comparing two user-friendly formal languages for mathematics: Weak type theory and Mizar
- Geleijnse
- 2004
(Show Context)
Citation Context ...959. Due to page constrains we do not attach the complete formalisation of our example in Mizar. However, it is available on-line: http://www.macs.hw.ac.uk/~retel/pythagoras/. Related work. Geleijnse =-=[7]-=- compared WTT and Mizar, presented CML examples in both WTT and Mizar and gave a correspondence between WTT and Mizar identifiers. His main approach was based on comparing these two languages. Our app... |

2 |
Formalising the natural language of mathematics: A mathematical vernacular
- Nederpelt, Kamareddine
- 2001
(Show Context)
Citation Context ...oposed as a formal substitute for parts of CML. Nederpelt refined MV into another formal substitute for parts of CML, Weak Type Theory (WTT) whose underlying proof theory was developed by Kamareddine =-=[17,14]-=-. MathLang started from de Bruijn’s path idea and Nederpelt’s WTT and was faced with the huge challenge of how to really create a path from original mathematical texts into fully formalised ones and h... |

2 | Corollary 1. uses C √ 2 /∈ Q justifies justifies Proof. Suppose √ 2 ∈ Q, i.e. √ 2 = p/q with p ∈ Z, q ∈ Z−{0 - unknown authors |

1 |
A proposed syntax for binders in mizar. http://www.cs.ru.nl/ ~freek/notes
- Wiedijk
(Show Context)
Citation Context ...rmula) | ex Qualified-Variables st Formula (holds Formula | Quantified-Formula) wants to introduce a new binder, has to do so in an indirect way, although the syntax for Mizar binders was proposed in =-=[23]-=-. Nonetheless, the Mizar language offers the two most essential binders: ∀ and ∃ which are given as QuantifiedFormula in the Mizar syntax (see the table on the right). Functions identifiers in 2 CML C... |

1 | Corollary 1. √ 2 /∈ Q Corollary 1. √ 2 C/∈ Q justifies Proof. Suppose √ 2 ∈ Q, i.e. √ 2 = p/q with p ∈ Z, q ∈ Z − {0}. Then √ 2 = m/n with m = |p|, n = |q| ̸= 0. It follows that m 2 - unknown authors |