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On Equivalents of Well-Foundedness On Equivalents of Wellfoundedness An experiment in Mizar Piotr Rudnicki Department of Computing Science University of Alberta Canada email piotrcsualbertaca Andrzej Trybulec Institute of Mathematics Warsaw University in Bialystok Poland email trybulecmathuwbedupl Received Accepted in nal form Abstract Four statements equivalent to wellfoundedness wellfounded induction existence of recursively dened functions uniqueness of recursively dened func tions and absence of descending chains have b een proved in Mizar and the pro ofs mechanicall y checked for correctness It seems not to b e widely known that the existence without the uniqueness assumption of recursively dened functions implies wellfoundedness In the pro of we used regular cardinals a fairly advanced notion of set theory The theory of cardinals in Mizar was develop ed earlier by G Bancerek With the current state of the Mizar system the pro ofs turned out to b e an exercise with only minor additions at the fundamental level We would like to stress the imp ortance of a systematic development of a mechanized data base for mathematics in the spirit of the QED Pro ject ENOD Experience Not Only Doctrine G Kreisel Key words Mizar QED Pro ject set theory wellfoundedness regular cardinals Abbreviations The Mizar Mathematical Library MML Octob er Piotr Rudnicki and Andrzej Trybulec Contents Intro duction Mizar The Mizar language Anatomy and pro cessing of a Mizar article Builtin notions Set theory of Mizar The Mizar Mathematical Library Mizar abstracts How to learn Mizar Wellfoundedness and its equivalents The conceptual framework A tour through the denitions Wellfounded induction Existence of recursively dened functions Uniqueness of recursively dened functions Nonexistence of descending chains Conclusions wfndtex no v p On Equivalents of Wellfoundedness Intro duction The pro ject Mizar started more than years ago under the leader ship of Andrzej Trybulec at the Plo ck Scientic So ciety Poland Its original goal was to design and implement a software environment that supp orts writing traditional mathematical pap ers where classical logic and set theory form the basis of all future developments The Mizar software veries correctness of mathematical texts written by humans Mechanical theorem proving has not b een one of the pro jects priorities The logical basis of Mizar is a system of natural deduction close to the comp osite system of logic develop ed by Stanislaw Jaskowski see also Katuzi Ono describ ed a similar sys tem John Harrison intro duced a Mizar mo de for HOL The set theory of Mizar is the TarskiGrothendieck system which is basically the ZermeloFraenkel set theory with the axiom of choice replaced by Tarskis stronger axiom of existence of arbitrar ily large strongly inaccessible cardinals Mizar is b oth the name of a formal languagedesigned by Andrzej Trybulecin which the mathematics is written and of an entire soft ware system that checks the texts for correctness and manages the data base of Mizar articles The systematic collection of Mizar articles started at the b eginning of and MMLthe Mizar Mathematical Library was b orn Development of the library is now the main eort in the pro ject The Mizar language and the asso ciated software evolve new and improved versions of the entire library are p erio dically pro duced It is worthwhile to note that in the pro cess of library evolution many denitions and theorems disapp ear either by b ecoming obvious for the improved language pro cessor or by b ecoming obsolete thanks to newer developments Mizar can b e seen as a trial run of the QED Pro ject which aims to build a computer system that eectively represents all imp ortant mathematical knowledge and techniques However with the mo dest means available Mizar captures only some asp ects of the entire QED idea Overview Section presents some asp ects of the Mizar system the structure of a Mizar article builtin notions the set theory of Mizar and the contents of the MML Although meant as an intro duction the presentation is quite technical at times In Section we discuss the main ideas b ehind the Mizar pro ofs of some equivalents of wellfoundedness it is probably the b est place to start reading this pap er First we present the conceptual framework of the pro ofs and unfold some denitions of the used notions Sections through discuss the actual pro ofs and are indep endent of each other We fo cus wfndtex no v p Piotr Rudnicki and Andrzej Trybulec on the state of the MML that p ermitted us to do the pro ofs with a routine eort by building up on previous contributions to the library The complete pro ofs are attached as App endices The Mizar text in the form discussed here was included into the MML as article WELLFND on February The Mizar system presented b elow is also as of February Because of the evolving nature of Mizar the current state of the article in the MML is dierent and is likely to change in the future Mizar At the moment there is no complete description of the Mizar system to which we could refer Such a description would require a text of substantial length In this section we briey present some asp ects of Mizar that should help in following the sequel The Mizar language Exp erience has shown that many p eople with some mathematical train ing develop a go o d idea ab out the nature of the Mizar language just by browsing through a sample article This is no big surprise since one of the original goals of the pro ject was to build an environment which supp orts the traditional ways that mathematicians work How ever some p eople nd the Mizar notation imp enetrable Therefore we make comments ab out the Mizar language when we use its more imp ortant or less intuitively clear features Because of the richness of the Mizar grammar even a sketchy pre sentation of the entire language is far b eyond the scop e of this pap er Anatomy and processing of a Mizar article A Mizar article is written as a text le and consists of two parts the Environment Declaration and the Text Proper see Figure The Envi ronment Declaration b egins with environ and consists of Directive s The Text Proper is a sequence of Section s each starting with begin and consisting of a sequence of TextItem s The division of the Text Proper into sections is only for editing purp oses and has no impact on the correctness of an article The two parts of an Article are pro cessed by two separate programs the Accommodator and the Verier The Accommodator pro cesses the 1 This division is used for sections when typ esetting Mizar abstracts in T X for E publication in Formalized Mathematics Section wfndtex no v p On Equivalents of Wellfoundedness environ Directive Directive begin TextItem TextItem begin TextItem TextItem Figure The overall structure of a Mizar article Environment Declaration and creates the Environment which consists of a numb er of working les in which the information requested in Directive s and imp orted from the data base is stored The Verier has no direct communication with the data base and checks the correctness of the Text Proper using only the information stored in the Environment les The ecient mechanism for imp orting the information from the MML into the lo cal Environment les is of utmost imp ortance and its design presents a substantial challenge the Accommodator evolves probably faster than other comp onents of Mizar The Environment directives There are two kinds of Directive s Vocabulary Directive s and Library Directive s A Vocabulary is a text le in extended ASCI I in which symb ols are dened The symb ols are qualied with their kind predicate functor mo de structure selector attribute or functor bracket and are used for lexical analysis A Vocabulary Directive has the form vocabulary VocabularyName VocabularyName and requests that all symb ols from the listed vo cabularies b e included in the lo cal environment Vo cabularies are indep endent of Mizar articles Library Directive s request information from the data base for inclu sion in the lo cal environment used later by the verier to check the article The conceptual framework of an article is imp orted by two directives of the following form wfndtex no v p Piotr Rudnicki and Andrzej Trybulec notation ArticleName ArticleName constructors ArticleName ArticleName In Mizar terminology predicates are constructors of atomic for mulae mo des are constructors of typ es functors are constructors of terms and attributes are constructors of adjectives A denition of a constructor gives its syntax and meaning The syntactic format of a constructor sp ecies the symb ol of the constructor and the place and numb er of arguments The format of a constructor together with the information ab out the typ es of arguments is called a pattern The for mats are used for parsing and the patterns for identifying constructors A constructor may b e represented by dierent patterns synonyms and antonyms are allowed and basically the same pattern can b e used for dierent constructors The constructors directive rst imp orts all constructors from the listed articles and then all other constructors needed to understand them As a result if a lo cal environment contains a constructor then it also contains the constructors o ccurring in its typ e and in typ es of its arguments The notation directive imp orts these formats and patterns from the listed articles that are used by the already imp orted constructors provided all the constructors needed to understand
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