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 the

notation have already b een imp orted see the end of Section

The remaining library directives are

clusters ArticleName ArticleName

a

imp orts from the listed articles the denitions of clusters that

state relationships among adjectives mo des and functors The

Mizar language is typ ed and dep endencies among clusters of

Bo olean attributes of ob jects are pro cessed automatically

definitions ArticleName ArticleName

requests denientia that can b e used in proving by denitional

expansion without mentioning the denitions name

theorems ArticleName ArticleName

enables referring to theorems and denitional theorems from the

listed articles

schemes ArticleName ArticleName

gives access to schemes which are theorems with second order free

variables Schemes can b e p erceived as inference rules that have to

b e proven b efore b eing used see Section

requirements ArticleName ArticleName

wfndtex no v p

On Equivalents of Wellfoundedness

gives access to the builtin features asso ciated with certain sp ecial

articles there is only one such article at the moment named ARYTM

Section

The Text Proper

Each section of Text Proper is a sequence TextItem s There are the

following kinds of TextItem s

a Reservation is used to reserve identiers for a typ e If a variable

has an identier reserved for a typ e and no explicit typ e is stated

for the variable then its typ e defaults to the typ e for which its

identier was reserved

a DenitionBlock contains a sequence of DenitionItem s each

dening or redening a constructor of Mizar phrases or a cluster

Each denition and redenition of a constructor requires a justi

cation of its correctness For instance when dening a functor

one has to justify conditions of its existence and uniqueness when

redening a functor one has to demonstrate that the redenition

is coherent with resp ect to the original denition of the functor

a StructureDenition intro duces a new structure which is an entity

that consists of a numb er of elds memb ers that are accessible

by selectors see Section

a Theorem announces a prop osition that is put into the data base

Section and can then b e referenced from other articles

a Scheme announces a scheme see Section which is a theo

rem with second order free variables and is accessible from other

articles

AuxiliaryItem s form those parts of a Mizar article that are lo cal

to the article and are not exp orted to the library les eg auxiliary

lemmas denitions of lo cal predicates functions and variables

Most TextItem s require a Justication which can b e either

a StraightforwardJustication a Proof or a SchemeJustication

Mizar p ermits a multitude of pro of structuresto o many to discuss

them herein the spirit of natural deduction see endnotes on page

The StraightforwardJustication takes the form

by Reference Reference

where a Reference is either a Private Reference a reference to a state

ment in the current article or a Library Reference a reference to a

theorem stored in the data base The latter has two forms

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

ArticleName TheoremNumber

or

ArticleName def DenitionNumber

where the latter refers to the so called denitional theorem a theorem

automatically created from a denition

The SchemeJustication is of the form

from SchemeName Reference Reference

SchemeName s are global and are not prexed with the name of the

article where they were intro duced The Reference s give the premises

of the scheme

Processing

The Accommodator creates the Environment les for an article based

on the Environment Directive s Then the Verier works with this lo cal

environment never contacting the data base The verier work is split

into three phases

The Parser is a relatively complicated program b ecause of the rich

Mizar syntax multiway overloading of names and the presence of

newly dened formats of phrase constructors and their precedences

The Analyzer identies constructors based on the available pat

terns and this involves pro cessing typ e information The Analyzer

also checks the correctness of pro of structures computes the for

mulae demonstrated in diuse statements and pro cesses clusters

of attributes see endnotes

The Checker checks the straightforward justications by treating

them as inferences of the form

premise premise premise conclusion

k

which are transformed into the conjunction

premise premise premise not conclusion

k

If the checker nds the conjunction contradictory then the original

inference is accepted The checker may not accept an inference

that is logically correct to get the inference accepted one has to

split it into a sequence of smaller ones or p ossibly use a pro of

structure The inference checker uses mo del elimination with stress

on pro cessing sp eed not p ower

wfndtex no v p

On Equivalents of Wellfoundedness

The Checker also checks the correctness of SchemeJustication s

by pattern matching the premises and the conclusion of the scheme

denition with the actual premises and the actual prop osition b eing

justied

Data Base

The Extractor program extracts all public information from an article

and stores it in library les This includes denitions theorems and

schemes justications and prop ositions not marked theorem are not

extracted The library les are used by the Accommodator to create an

Environment for an article to b e pro cessed by the Verier

The library les of the public data base are built from the articles

submitted to the MML and are distributed by the Mizar So ciety A

private data base may b e created by a user or a group of users using

the Extractor on articles stored in a private library Private data bases

have temp orary character once the author is sure of the quality of an

article which usually requires writing a couple of other articles using

it the article is submitted to the MML

Builtin notions

Several notions of set theory are built into the Mizar pro cessor and

their user interface is given in a sp ecial article named HIDDEN The

symb ols used in dening these notions are given in a sp ecial vo cabulary

also called HIDDEN The conceptual framework intro duced in HIDDEN is

automatically added to the lo cal environment of each article checked by

the Mizar pro cessor and so are the symb ols from vo cabulary HIDDEN

The rst notion dened in HIDDEN is a mo de Any with synonym

set

definition

mode Any

synonym set

end

Some years ago the mo des Any and set were distinct This mo de is the

ro ot of the Mizar typ e hierarchy the typ e of every ob ject ultimately

widens to set

Absolute equality is builtin as a reexive symmetric and transitive

predicate written in the common inx notation

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

definition let xy be Any

pred x y

reflexivity

symmetry

antonym x y

end

The ab ove denition also intro duces the inequality predicate as an

antonym of equality so that x y can b e written instead of

not x y

The elementho o d predicate the attribute empty and the nullary

functor for the empty set are builtin

definition let x be Any X be set

pred x X

asymmetry

end

definition let X be set

attr X is empty

end

definition

cluster empty set

cluster non empty set

end

definition

func empty set

end

Attributes are Bo olean valued and are constructors of adjectives With

the attribute empty we have two adjectives empty and non empty The

two clusters ab ove p ermit the use of empty set and non empty set

as typ e expressions when intro ducing sets The typ e of is empty set

The following mo de constructor

definition let X be set

mode Element of X

end

allows us to declare a set say x and declare its typ e as Element of X

even if X is empty However only when X is non empty will the checker

accept x X with no justication

The syntax and typ e of the p ower set functor is announced as

definition let X be set

func bool X non empty set

end

and its denition is given by an axiom in TARSKI see Section

wfndtex no v p

On Equivalents of Wellfoundedness

With the bool functor for p ower set available syntactically at this

moment the mo de Subset of X is intro duced

definition let X be set

mode Subset of X is Element of bool X

end

definition let X be non empty set

cluster non empty Subset of X

end

The last cluster denes a typ e expression non empty Subset of X

for non empty X which mirrors the fact that there are nonempty sub

sets of a nonempty set If we need one we can express its nonemptiness

by typing

The inclusion predicate is given its syntax here

definition let XY be set

pred X c Y

reflexivity

end

and its reexivity is announced the deniens is given in a redenition

of the predicate in TARSKI see Section

The last denition in HIDDEN

definition let D be non empty set X be non empty Subset of D

redefine mode Element of X Element of D

end

redenes the mo de Element of X for a nonempty subset X of a non

empty set D such that the mo de automatically widens to Element of D

Besides HIDDEN there is another sp ecial article named ARYTM that

contains a user interface to the builtin notions related to real and

natural numb ers however its contents are not automatically added to

the lo cal environment of any article

The sets REAL and NAT are intro duced in ARYTM

definition

func REAL non empty set

end

definition

func NAT non empty Subset of REAL

end

The syntax for addition multiplication and ordering for elements

of REAL are given

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

definition let xy be Element of REAL

func x y Element of REAL

commutativity

func x y Element of REAL

commutativity

pred x y

reflexivity

connectedness

synonym y  x antonym y x antonym x y

end

Finally the following two mo des intro duce a shorter notation for

elements of REAL and NAT

definition

mode Real is Element of REAL

mode Nat is Element of NAT

end

While the notions announced in HIDDEN are axiomatically dened

in TARSKI Section the notions intro duced in ARYTM used to

b e given their complete denitions in article AXIOMS see the end

of Section

Set theory of Mizar

The set theory axioms of Mizar are formalized in article TARSKI

the fundamental article of the MML which is not checked for correct

ness The set theory of Mizar is commonly referred to as the Tarski

Grothendieck set theory see It is basically ZF with the axiom of

choice replaced by Tarskis stronger axiom see b elow

REL and The article imp orts symb ols from two vo cabularies EQUI

OP The former intro duces  Mizar uses extended ASCI I as a FAM

predicate symb ol and the latter intro duces union as a functor symb ol

The remaining symb ols that are used in denitions are either built into

the parser or dened in the automatically imp orted vo cabulary HIDDEN

environ

REL FAM OP vocabulary EQUI

Recall that vo cabularies are indep endent of Mizar articles

The text prop er of TARSKI starts with the reservation

reserve xyzu for Any

NM XYZ for set

As a witness of the past b oth mo des Any and set are used although

they are now synonymous see Section

wfndtex no v p

On Equivalents of Wellfoundedness

The rst axiom named TARSKI for future reference states the

extensionality of set equality

theorem TARSKI

for x holds x X iff x Y implies X Y

The functors for forming singletons and unordered pairs are dened

by employing the builtin left functor bracket f and the right functor

bracket g This results in a familiar notation

definition let y

func f y g set means TARSKI def

x it iff x y

let z

func f y z g set means TARSKI def

x it iff x y or x z

commutativity

end

The two functors have dierent formats as they take dierent numb ers

of arguments The keyword it refers to the deniendum and can only

o ccur inside the deniens The free variable x in the denientiaits

typ e is reserved ab oveis b ound by a default universal quantier in

front of each formula

The following denition of two functorial clusters states that a sin

gleton and an unordered pair of typ e set by denition also have the

prop erty of b eing nonempty

definition let y

cluster f y g non empty

let z

cluster f y z g non empty

end

Once we request clusters from TARSKI the nonemptiness of singletons

and pairs is obvious to the verier

The set inclusion predicate intro duced as a reexive binary predi

cate in HIDDEN now obtains its deniens

definition let XY

redefine pred X c Y means TARSKI def

x X implies x Y

end

The union functor for a family of sets is dened as

2

As a result of the MML evolution the numb ering of axioms has gaps certain

axioms have b een canceled as spurious This also applies to theorems and denitions

in other articles

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

definition let X

func union X set means TARSKI def

x it iff ex Y st x Y Y X

end

The p ower set functor bool is also intro duced syntactically in

HIDDEN and this axiom states its denition

theorem TARSKI

X bool Y iff for Z holds Z X iff Z c Y

Here is the regularity axiom

theorem TARSKI

x X implies ex Y st Y X not ex x st x X x Y

The replacement axiom is expressed as a scheme

scheme Fraenkel f A set PAny Any g

ex X st for x holds x X iff ex y st y A Pyx

provided for xyz st Pxy Pxz holds y z

where the premise of the scheme requires the predicate P to have the

functional prop erty This scheme is infrequently used as the Fraenkel

b

op erator is now built into Mizar

Tarski prop osed the following axiom

A For every set N there exists a system M of sets which satises

the following conditions

i N M

ii if X M and Y X then Y M

iii if X M and Z is the system of all subsets of X then

Z M

iv if X M and X and M do not have the same p otency

then X M

In order to state this axiom in Mizar we need the notion of equinu

merosity of two sets but b efore we dene it we intro duce a functor for

ordered pair using the builtin functor brackets and

definition let xy

func xy means TARSKI def

it f f xy g f x g g

end

The deniens is recast as a theorem

theorem TARSKI

xy f f xy g f x g g

wfndtex no v p

On Equivalents of Wellfoundedness

Note A reference to TARSKIdef refers to a dierent formula than

c

a reference to TARSKI

The equinumerosity of two sets is now dened note that we have

not constructed settheoretic functions yet

definition let XY

pred X  Y means TARSKI def

ex Z st

for x st x X ex y st y Y xy Z

for y st y Y ex x st x X xy Z

for xyzu st xy Z zu Z holds x z iff y u

end

Finally we can state the Tarski axiom in Mizar

theorem Tarski

ex M st N M

for XY holds X M Y c X implies Y M

for X holds X M implies bool X M

for X holds X c M implies X  M or X M

These are all the axioms of set theory that one can use when writing

a Mizar article

Before March the article AXIOMS used to b e the other fun

damental article with axioms dening the constructors intro duced in

ARYTM Section The axioms stated strong axiomatics of real num

b ers and dened NAT as an inductive set In March the con

struction of real numb ers from axioms of set theory was completed by

G Bancerek and A Trybulec and articles ARYTM and AXIOMS b ecame

normal ie fully checked Mizar articles Since the article AXIOMS is

not referenced directly in the sequel it is not discussed here

The Mizar Mathematical Library

At the b eginning of the Mizar group in Bialystok started col

lecting Mizar articles and organizing them into a library named the

Mizar Mathematical Library MML

As of this writing January the MML consists of Mizar

articles authored by p eople containing theorems and

denitions The nature of the articles varies most of them are Mizar

translations of basic mathematics The ma jority of MML theorems

state basic mathematical facts and the ma jority of denitions mirror

the essential mathematical to olkit Only some of the wellknown theo

rems from literature are currently in the MML Few of the theorems

are new results

The development of the Mizar library may b e p erceived as an exp er

iment in the so ciology of mathematics The acceptance criteria are quite

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

TARSKI ARYTM

BOOLE REAL

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ENUMSET NAT ANAL ZFMISC

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RELAT SQUARE SUBSET INT

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RELAT RELSET SETFAM STRUCT INT REAL FUNCT

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ORDINAL GRFUNC WELLORD PARTFUN

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WELLORD ORDINAL FUNCT MCART WELLSET FINSEQ

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ORDINAL DOMAIN EQREL MCART FUNCT BINOP FINSET

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FINSUB RECDEF TOPC RAT RLVECT FUNCOP CARD PRE

Figure Some of the MML articles from top levels

lib eral virtually every Mizar article submitted to the library commit

tee and accepted by the Mizar pro cessor is included in the library

Figure depicts the conceptual dep endencies b etween some of the

initial articles of the MML An article at a higher level towards the

b ottom uses concepts intro duced in articles at lower levels towards the

top An edge is drawn from an article to another article from which it

imp orts concepts only if their levels are adjacent Imp orts from articles

at still lower levels are not shown Only conceptual dep endencies are

shown that is if an article refers only to theorems from another article

then no dep endency app ears in Figure

The articles TARSKI and ARYTM are placed at the ro ot level num

b ered The sp ecial article HIDDEN which precedes all other articles is

not shown The other axiomatic le AXIOMS contains only theorems

and no new concepts and therefore is also not shown All articles at

wfndtex no v p

On Equivalents of Wellfoundedness

higher levels have b een checked for correctness At level there are

only two articles BOOLE ab out b o olean prop erties of sets denes

set op erations and REAL ab out basic prop erties of real numb ers

denes unary minus subtraction inverse etc For illustration let us

trace one path of how the article CARD at level dep ends on some

earlier articles

is the rst article ab out cardinal numb ers and uses the CARD

notion of finite set dened in FINSET in terms of a nite

A nite sequence with sequence which is dened in FINSEQ

elements from a set D is intro duced as a partial function from the nat

urals to D Partial functions are dened in PARTFUN in terms of

In this article Functionlike sets which are dened in FUNCT

Functionlike sets are used to intro duce the mo de Function For

dening this mo de the notion of Relationlike set is also used and

it is dened in RELAT where some prop erties of Relationlike

sets are expressed in terms of Cartesian pro ducts These in turn are

where prop erties of Cartesian pro ducts are dened in ZFMISC

stated using set op erations dened in BOOLE which is based only

on the set axioms from TARSKI and whatever is built into the

Mizar verier

Tracing concept dep endencies in the MML is a challenge and con

structing a lo cal environment for a new article requires a thorough

knowledge of the MML

Here are some b etter known theorems currently in the MML the

Zermelo theorem and the axiom of choice level the Kuratowski

Zorn lemma level the deduction theorem for predicate log

ic level Lagranges subgroup theorem level the Desar

gues theorem in pro jective space level the Hessenb erg the

orem level xp oint theorem for compact spaces level

the de lHospital theorem level Konigs lemma level

mean value theorems for real functions of one variable level

the Weierstrass theorem level HeineBorels covering theo

rem level the Brouwer xp oint theorem for intervals lev

el representation theorem for b o olean algebras level the

Steinitz theorem level the HahnBanach theorem level

the Tarski theorem on xp oints in complete lattices level

The level at which an article app ears in the MML should not b e

interpreted as an absolute measure of its conceptual complication Fre

quently the level reects the uncontrolled way in which the MML has

grown For instance the Tarski theorem ab ove could have b een proven

in an article of a much lower level but the article uses a simple notion

that has by chance b een intro duced in another article at a high level

that deals mainly with more advanced matters

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

Mizar abstracts

The source texts of Mizar articles tend to b e lengthy as they contain

complete pro ofs in a rather demanding formalism New articles strongly

dep end on already existing ones Therefore there is a need to provide

authors with a quick reference to the already collected articles The

solution is to automatically create an abstract for each Mizar article

Such an abstract includes a presentation of all the items that can b e

referenced from other articles but with all justications removed

Mizar abstracts are automatically typ eset using T X and then p eri

E

o dically published in a journal Formalized Mathematics a computer

assisted approach edited by R Matuszewski ISSN ISSN

for the initial volumes

How to learn Mizar

The Mizar language its pro cessor and the organization and contents

of the MML evolve Therefore there is not much in the way of written

do cumentation In the face of do cumentation shortages the b est way

to learn Mizar is to sp end approximately four weeks with a native

user of the system and coauthor a Mizar article However there are

numerous cases of Mizar users who advanced their knowledge of the

system by studying the MML and the existing do cuments

Wellfoundedness and its equivalents

T Franzen in presented his pro ofs that wellfoundedness is equiv

alent to the principle of mathematical induction and to the principle

of dening by recursion and used them as a background for discussing

the merits of the so called equational style of doing pro ofs We have

decided to formalize his pro ofs in a Mizar article Franzens pro ofs were

written by a mathematician having an argument with a computer sci

entist We were curious ab out the eort needed to formalize Franzens

pro ofs given the state of the MML at that time July The for

malization went quite smo othly once the mathematics was sorted out

Recall that a relational structure U is wel lfounded i every

nonempty subset of U has a minimal element ie

X UX m X x X x m x m

3

Available on the World Wide Web at httpmizarorg and several mirror

sites There is also a Mizar user service mailing list musmizaruwbialystokpl

wfndtex no v p

On Equivalents of Wellfoundedness

We do not assume that is transitive

We have proven and checked in Mizar that the following are equiv

alent

U is wellfounded

Wellfounded induction holds for U Section

For every set V there exist recursively dened functions from U

into V Section

Recursively dened functions on U are unique Section

There are no descending chains in U Section

Most of the ab ove are easy to prove and the pro ofs are wellknown

J Harrison presented his developments of similar theorems in HOL

Before we discuss the pro ofs we rst comment on setting up the con

ceptual environment for them and for illustration we unfold the de

nition of wellfounded relational structure down to the basic notions

The conceptual framework

Creating a conceptual framework for a new article requires a thor

ough knowledge of the MML Creating such an environment consists

of nding all the appropriate notions in the MML and preparing the

environment directives that imp ort the notions It is an iterative pro

cess parallel to writing the article itself Whenever the environment

directives change the Accommodator is run and creates a new environ

ment for the Verier b efore it checks the article The nal environment

directives for the article discussed b elow are presented in App endix A

Some of the articles forming the conceptual framework of our article

are listed in Figure see the description for Figure and some of the

notions are discussed in Section

The most advanced notions that we used alephs character of co

level not nality and regular cardinals come from article CARD

shown in Figure see b elow However most of the basic terminolo

gy comes from articles listed in Figure From some of those articles

we imp orted relatively simple purely technical notions that just hap

p ened to b e intro duced in articles dealing with dierent matters For

example consider the articles at level from which we imp ort the

following notions

attribute trivial for a set which is either empty or a singleton

article REALSET on algebraic elds

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

TARSKI ARYTM

BOOLE REAL

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RELAT SUBSET

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STRUCT RELAT FUNCT RELSET SETFAM

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. . . . .

PARTFUN WELLORD ORDINAL

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FINSEQ MCART ORDINAL FUNCT

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BINOP QC LANG FINSET DOMAIN

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PRE QC RLVECT TOPC FUNCOP LANG CARD

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SETWISEO QC ORDERS LANG FUNCT CARD VECTSP

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.. . METRIC

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REALSET

NORMSP FRAENKEL LANG CQC

Figure The environment articles for wellfoundedness from top levels

mo de sequence of X denoting a function from NAT to the carrier

of X for X b eing a sorted structure see Section article

on normed spaces NORMSP

a functor for creating functions on singleton domainsarticle

CQC LANG which formalizes the classical rst order language

attribute functional for a set whose elements are functions

article FRAENKEL ab out prop erties of the Fraenkel op erator

wfndtex no v p

On Equivalents of Wellfoundedness

We had to imp ort some notions from articles placed at even

higher levels not shown in Figure For instance from article

WAYBEL level ab out directed sets we imp ort the essential

attribute lower which is applicable to a subset of the carrier of a rela

tional structure we later redened this attribute by reformulating its

deniens see App endix B p

Finding appropriate library directives that create a working lo cal

environment dep ends on the organization of the MML and requires a

thorough knowledge of the library This pro cess would b e substantially

simplied with a reorganization of the MML and a design of such

reorganization is ongoing One of the p ossible solutions is to create

topical monographs derived from all articles submitted to the MML

A substantial eort in the pro ject mainly by Czeslaw Bylinski is

sp ent on developing new pro cedures for minimizing the size of the lo cal

environment for an article The environment is created by the Accom

modator using only the given environment directives and without exam

ining the text prop er of the article In such situations the created lo cal

environment should not contain notions that are unlikely to b e needed

for the Verier s work The essence of the problem can b e illustrated by

examining answers to the following simplied question When import

ing functors from an article should the Accommodator also import al l

constructors of functors that appear in the denientia Implementing

the naive answer yes leads to exp onential growth of the lo cal environ

ment The answer no forces the user to insert appropriate environment

directives when needed and this is sometimes troublesome The cur

rently implemented solution is closer to the latter Note that in our

at level but we did article we imp ort notions from article CARD

not need any notions at least not directly from articles at levels

through Actually the Accommodator imp orted constructors from

one of the articles at the intermediate level required to understand the

constructors from CARD although they were not explicitly requested

A similar remark applies to article WAYBEL

A tour through the definitions

Let us overview all the notions involved in the denition of a well

founded relational structure down to the basic notions This overview

illustrates the organization of the MML from the user viewp oint

At the time of writing our pro ofs July the MML included a

denition of a wellfounded relation but was missing a denition of a

wellfounded relational structure Thus we dened App endix B p

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

definition let R be RelStr

attr R is well founded means Lwell

the InternalRel of R is well founded in the carrier of R

end

founded applicable What is dened ab ove is an attribute named well

to ob jects of typ e RelStr This attribute can b e used to form atomic

sentences like R is well founded for R of typ e RelStr It can also b e

founded R used in typ e expressions to intro duce ob jects of typ e well

However b efore this can b e done we have to prove that there exists at

least one ob ject of this typ e This is achieved through dening

definition

cluster non empty well founded RelStr

existence proof

Here we show that there exists a RelStr that is nonempty

and wel lfounded

end

end

Note The double colon b egins a comment which is ended by a

newline The cluster assures the existence of nonempty wellfounded

relational structures structures with an empty carrier are not interest

ing

Two following subsections give the details of how relational struc

tures and wellfounded relations are dened in the MML

Relational structure

RelStr is dened in ORDERS

struct sorted RelStr 

carrier set

InternalRel Relation of the carrier



The structure RelStr is prexed by the structure sorted that is

and serves as the base for all dened in a sp ecial article STRUCT

structures with the eld carrier

definition

struct sorted  carrier set 

end

The eld carrier of RelStr is inherited from the base structure

sorted but it must b e rep eated in the denition of the derived struc

ture InternalRel the other eld of RelStr is dened to b e any ob ject

4

The article is sp ecial as it was prepared by the MML Committee Otherwise

it is a regular alb eit short Mizar article

wfndtex no v p

On Equivalents of Wellfoundedness

whose typ e widens to Relation of the carrier This typ e is dened

in RELSET X and Y are reserved for set

definition let X

mode Relation of X is Relation of XX

end

The denition of the mo de Relation with two arguments is given ear

lier in the same article

definition let XY

mode Relation of XY Subset of XY means RELSET def

not contradiction

end

X Y is the Cartesian pro duct of two sets dened in

X and X are reserved for set ZFMISC

definition let XX

func XX means ZFMISC def

z it iff ex xy st x X y X z xy

end

Wel lfounded relations

The notion of a wellfounded relation is intro duced in WELLORD We

use the denition of a relation which is wellfounded in a set

definition let R let X

well founded in X means WELLORD def pred R is

for Y st Y c X Y ex a st a Y RSega\Y

end

where R is reserved for Relation X Y and a are reserved for set The

inx functor Seg is describ ed b elow

The denitions of basic set op erations are intro duced in BOOLE

with the denition of set intersection as follows X and Y are reserved

for set

definition let XY

func X \ Y set means BOOLE def

x it iff x X x Y

commutativity

The source article BOOLEMIZ contains the pro of that the functor \ is

uniquely dened and that it is commutative

The mo de Relation is dened in RELAT

definition

mode Relation is Relationlike set

end

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

and the attribute Relationlike is dened as

definition let IT be set

attr IT is Relationlike means RELAT def

x IT implies ex yz st x yz

end

With this denition one can prove that any set of ordered pairs is

Relationlike

The set of all ob jects sets preceding a in R is written with the inx

functor Seg dened in WELLORD

definition let Ra

func RSega set means WELLORD def

x it iff x a xa R

end

where R is reserved for Relation a and x are reserved for set Note

that a do es not have to b elong to the eld of R

Wellfounded induction

The pro of that wellfoundedness of a relational structure is equivalent

to the principle of induction for the structure is easy In order to facil

itate the work and keeping in mind the development of the data base

we rst proved the following scheme of minimum we could have done

all the pro ofs without it Given a structure R U the following

is valid

xP x

R is wellfounded

xP x y y x P y y x

In Mizar this scheme is written as

scheme WFMin fR non empty RelStr

x Element of R

Psetg

ex x being Element of R st Px

not ex y being Element of R

st x y Py yx the InternalRel of R

provided

Px and

R is well founded

The scheme is expressed in terms of three parameters Ra relational

structure xan element of Rs carrier and a one place predicate P

which is a second order free variable The scheme has two premises

which must b e supplied when it is used see b elow The predicate P and

the constants R and x are parameters of the scheme to b e automatically

wfndtex no v p

On Equivalents of Wellfoundedness

reconstructed in each application of the scheme They are reconstructed

by pattern matching the premises and the conclusion of the scheme

denition with the premises and the prop osition when the scheme is

used in a SchemeJustication

The validity of the scheme is proved in a straightforward way Let Z

b e the set of elements of Rs carrier which have prop erty P Z is a subset

of Rs carrier and b ecause of the rst premise Z is nonempty Then Z

has a minimal element a as the internal relation of R is wellfounded in

the carrier of R by the second premise a is a witness to the existential

thesis of the scheme Pa holds as a b elongs to Z and no other element

of R with prop erty P can precede a as a is minimal in Z The complete

pro of is in App endix C

Here is how T Franzen sketched the pro of of the following

Theorem U is wellfounded i the principle of mathe

matical induction holds for U

That the principle of mathematical induction holds for U

means that for every set S if x S whenever every y x b elongs

to S then U S

The theorem is easily proved In one direction if is well

founded supp ose M satises the condition that x b elongs to M

whenever every y x b elongs to M If M is not all of U U n M is

nonempty and has a minimal element x But x b eing minimal in

U n M every y x b elongs to M so x b elongs to M after all Thus

M must b e all of U Conversely if the principle of mathematical

induction holds supp ose a subset M of U has no minimal element

Then for any x if y b elongs to U n M for all y x x must also

b elong to U n M since otherwise x would b e a minimal element of

M So every x b elongs to U n M ie M is empty

In Mizar this theorem is formulated as

theorem WFInd WF iff WFInduction

for R being non empty RelStr holds

founded iff R is well

for S being set

st for x being Element of the carrier of R

st the InternalRel of RSeg x c S holds x S

holds the carrier of R c S

We prefer to have S as an arbitrary set rather than a Subset of the

carrier of R The complete pro of of the theorem is in App endix C

p and it closely follows the pro of of T Franzen alb eit in small

er steps Let c b e the shorthand for the carrier of R and r for the

d

InternalRel of R In one direction b etween hereby and the match

ing end after assumptions

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

hereby assume

R is well founded

let S be set such that

for x being Element of c st rSeg x c S holds x S

let us for the pro of by contradiction

assume not c c S

which gives an x with the following prop erties

x c not x S

e

as premises to the And now use the statements lab eled and

WFMin scheme to obtain an x

consider x being Element of R such that

x c not x S and

not ex y being Element of R

st x y y c not y S yx r

from WFMin

But now rSeg x is a subset of S as otherwise we would have an

element preceding x that is not in S which would contradict the min

imality of x As a result we have a contradiction b ecause of and

For the pro of of the converse see App endix C p assume the

principle of induction and assume that R is not wellfounded Then

there is a nonempty subset Y of c that has no minimal element For

any x b elonging to c if rSeg x is a subset of cnY then x do es not

b elong to Y since otherwise Y would have a minimal element Therefore

x b elongs to cnY and consequently c is a subset of cnY by the assumed

induction Y is an empty set which yields a contradiction

With the ab ove theorem we prove the more familiar version of the

induction scheme see App endix C p

scheme WFInduction fR non empty RelStr Psetg

for x being Element of R holds Px

provided

for x being Element of R

st for y being Element of R

st y x yx the InternalRel of R holds Py

holds Px

and

R is well founded

Existence of recursively defined functions

T Franzen discusses the following

wfndtex no v p

On Equivalents of Wellfoundedness

Theorem U is wellfounded i the principle of denition by

recursion holds for U

where the principle of denition by recursion is

U V

For any set V and function H U V there is a unique

function F U V satisfying F x H x F jfy y xg for

every x in U

where the notation F jM used ab ove denotes the restriction of the func

tion F to a subset M of the domain of F

In the pro cess of proving the ab ove theorem in Mizar we have

noticed that wellfoundedness is equivalent to the existence of recur

sively dened functions without assuming their uniqueness and sep

arately wellfoundedness is equivalent to the uniqueness of recursively

dened functions Therefore we formed two separate theorems similar

ly to J Harrison who noticed that wellfoundedness is equivalent

to just the uniqueness part of the recursion theorem However he did

not prove the equivalence of wellfoundedness and just the existence of

recursively dened functions

To facilitate formulation of the theorem in Mizar we dene a predi

cate saying that a function is recursively expressed by another function

definition

let R be non empty RelStr V be non empty set

H be Function of

the carrier of R PFuncsthe carrier of R V V

F be Function

recursively expressed by H means Lrecur pred F is

for x being Element of the carrier of R

holds Fx Hx Fthe InternalRel of RSeg x

end

We have departed slightly from the informal denition mentioned

ab ove The domain of H is dened to b e the Cartesian pro duct of the

carrier of R and the set of all partial functions from the carrier of

R to V instead of all binary relations from the carrier of R to V

since the second argument of H must b e a function anyway The set

of all partial functions from X to Y PFuncsX Y is originally dened

in The application of a settheoretic function F to an argument x

is written Fx dened in

5

It has b een noticed in part due to laziness There is no single quantier in

Mizar to say that something uniquely exists Expressing unique existence would

require us to typ e a long formula so at the rst try we stated just the existence

delaying stating the uniqueness condition until needed It wasnt needed after all

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

The theorem can now b e stated

theorem Well foundedness and existence

for R being non empty RelStr holds

R is well founded iff

for V being non empty set

H being Function of

the carrier of R PFuncsthe carrier of R V V

ex F being Function of the carrier of R V

st F is recursively expressed by H

In the pro of see App endix D p we rst intro duce some lo cal

notation c for the carrier of R r for the InternalRel of R and

we also intro duce a lo cal predicate PDR for Principle of Dening by

Recursion

defpred PDR means

for V being non empty set

H being Function of c PFuncsc V V

ex F being Function of c V

st F is recursively expressed by H

The rst implication is then stated as

founded implies PDR thus R is well

and its Mizar pro of is a routine alb eit lengthy construction of a func

tion Here is how T Franzen sketched the pro of

Essentially a pro of of the existence of the function F the unique

ness then b eing easily proved by induction must go as follows

Consider functions f dened on subsets of U with values in V For

such a function we say that Rf holds if

for any x in the domain of f fy y xg is included in the

domain of f

for any x in the domain of f f x H x f jfy y xg

We next establish that if Rf and Rg hold and x b elongs to

the intersection of the domains of f and g then f x g x This

is proved by induction using the wellfoundedness of We then

dene F as the union of all f such that Rf which is shown to b e

a maximal f with the prop erty Rf Then to establish that the

domain of F is in fact all of U we prove by induction that every x

b elongs to the domain of some f such that Rf Here if is not

assumed transitive we need to use the transitive closure of

In Mizar after making appropriate assumptions we intro duce fs

the subset of all partial functions from c to V whose domains are lower

sets of R see App endix B p for the denition of lower and which

wfndtex no v p

On Equivalents of Wellfoundedness

are recursively expressed by H on their domains This is done with the

premiseless scheme PFSeparation see App endix B p which was

proved using other schemes for subset separation

consider fs being Subset of PFuncsc V such that

for f being PartFunc of c V holds f fs iff

dom f is lower

for x being set st x dom f

holds fx Hx f rSeg x from PFSeparation

f

Let ufs b e the union of fs First in a unlab eled diuse statement

b etween now and the matching end we show that ufs satises the

g

conditions of the auxiliary theorem Funion see App endix B and then

we upgrade its typ e from set to Function

reconsider ufs as Function by Funion

From now on we can use ufs in contexts where a function is exp ect

ed Our goal is to show that ufs is the sought for function from c to V

therefore we characterize its domain and range

dom ufs c c

rng ufs c V

In the diuse statement lab eled we demonstrate that ufs on its

domain is recursively expressed by H

now let x be set

assume x dom ufs

thus ufsx

Hx ufs rSeg x by

end

In another diuse reasoning we show that the domain of ufs forms a

lower set of R

now let x y be set assume

x dom ufs y x r

hence y dom ufs by

end

Now in a longer pro of we show that

dom ufs c

which together with allows us to

reconsider ufs as Function of c V by

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

ufs is the sought for function from the carrier of R into V and since

at this p oint we are to prove an existential formula we

take ufs

and what remains to b e proven is that ufs is recursively expressed by H

We complete the pro of by proving its denitional expansion see lab el

Lrecur on page

let x be Element of c

thus thesis by

Note that with equality proves the exp ected equality namely

ufsx Hx ufs rSeg x

h

written here as thesis whichinside a pro ofdenotes the formula

that must b e concluded in order to complete the pro of

This completes the pro of that wellfoundedness implies the existence

of recursively dened functions

The pro of of the converse which is that the existence of recursive

ly dened functions on a relational structure without assuming their

uniqueness implies that the structure is wellfounded is more interest

ing The sketch of the pro of for U given by T Franzen is as

follows

Assuming the principle of denition by recursion to hold we get

pretty immediately that the relation is wellfounded For we can

then dene a rank function from U into an ordinal

rk x sup rk y

y x

and note that

y x implies rk y rk x

implying that is wellfounded

But of course this argument presupp oses quite a lot of set the

ory and in fact uses the replacement axiom

We have found his pro of quite sketchy and in fact we could not sort

out how to directly use the replacement axiom however we followed

his approach

The main steps of the pro of in Mizar are as follows the complete

pro of is in App endix D p

wfndtex no v p

On Equivalents of Wellfoundedness

assume

PDR

For the range typ e of the rank function we take the cardinal successor

of c in Mizar

reconsider ac alef Card c as infinite Cardinal

set V nextcard ac

The following rank function rk is guaranteed to exist by the assumption

therefore lab eled

consider rk being Function of c V such that

rk is recursively expressed by H by

but rst we have to construct function H This function is obtained with

the help of scheme KappaD

consider H being Function of c PFuncsc V V such that

for x being Element of c

p being Element of PFuncsc V

holds Hxp sup rng p from KappaD

The application of this scheme requires the following premise

for x being Element of c p being Element of PFuncsc V

holds sup rng p V

We were lucky to nd the development of the theory of cardinals in

the MML G Bancerek to b e suciently advanced to make the

pro of of an exercise The pro of hinges on the following prop erty of

regular cardinals the supremum of a subset X of a regular cardinal M

b elongs to M provided the cardinality of X is smaller than M This fact

has b een stated as a p otentially useful lemma in the preliminaries

App endix B p

theorem Reg

for M being regular Aleph X being set

st X c M Card X M holds sup X M

proof let M be regular Aleph

def cf M M by CARD

hence thesis by CARD

end

We found the key facts for proving the lemma in the MML

The complete pro of of is as follows

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

proof let x be Element of c p be Element of PFuncsc V

Card dom p c Card c Card rng p c Card dom p

by CARD then

Card rng p c Card c by BOOLE

Card c c ac by CARD then

Card rng p c ac by BOOLE

then ac V by CARD

Card rng p V by CARD

V is regular by CARD

hence sup rng p V by Reg

end

Note the usage of the key fact that a cardinal successor of an innite

cardinal is regular theorem CARD

Using the rank function rk we prove that R is wellfounded by de

nitional expansion Section

let Y be set assume

Y c c Y

and let m b e the inmum of the rk image of Y formally

o

set m inf rk Y

o

rk Y is nonempty so from the prop erties of ordinals stated in

o

we have that m rk Y Therefore consider an a that rk maps to m

a dom rk a Y rka m

It remains to b e proven that a is minimal in Y therefore

take a

thus a Y by

assume rSega \ Y

and we have to show a contradiction From the last assumption there

is an e such that

e rSeg a e Y

o

rke is an ordinal b elonging to rk Y and thus is a sup erset of the

inmum

m c rke

Since rk is recursively expressed by H see we would like to show

rka sup rng rk rSeg a as stated in However in order

to use we have to help the Mizar typ e checker by intro ducing a

new ob ject with the typ e appropriate to substitute for p in

reconsider rkra rk rSeg a as Element of PFuncscV

wfndtex no v p

On Equivalents of Wellfoundedness

Now we infer

rka Ha rkra by Lrecur

sup rng rkra by

Because of rke rng rkra and thus b elongs to the supremum

of rng rkra which by and is equal to m

We have a contradiction by as no set can b e a subset of one of

its elements theorem

Uniqueness of recursively defined functions

The pro of that uniqueness of recursively dened functions is equiva

lent to wellfoundedness is split into two implications as the conditions

under which the implications hold are slightly dierent The complete

pro ofs are in App endix E

Uniqueness implies wel lfoundedness

The formal statement of the theorem in Mizar is a bit long

for R being non empty RelStr

V being non trivial set

st for H being Function of

the carrier of R PFuncsthe carrier of R V V

F F being Function of the carrier of R V

recursively expressed by H st F is

recursively expressed by H F is

holds F F

holds R is well founded

The ab ove theorem do es not hold for trivial range typ es empty or

singleton of recursively dened functions

The pro of of the theorem is straightforward Since V is nontrivial it

has at least two elements call them a and a a a We construct

two functions F and F from cthe carrier of Rinto V F maps

each element of c to a while F maps the wellfounded part of c to a

and the remaining elements to a The wellfounded part of R is dened

as follows see App endix B p

let R be RelStr

foundedPart R Subset of R means LwfPart func well

it union fS where S is Subset of R S is well founded lowerg

existence proof

show that it is a Subset of R

end

uniqueness obvious

end

Note how the deniens uses the Fraenkel op erator see endnote on

page

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

Let wfp b e a shorthand for the wellfounded part of R F is formally

intro duced

set F c a

while in order to dene F we rst intro duce two auxiliary functions

and make F their union function overriding

F c a

F wfp a

F F F

First we prove that F and F are indeed functions from the

carrier of R into V and then that they are recursively expressed by

the same H Therefore by assumption they are equal If R were not well

founded then there would b e an x in the nonwellfounded part of R

But then Fx a and Fx a which leads to a contradiction

Wel lfoundedness implies uniqueness

The pro of of the following theorem requires only a routine application

of induction the scheme WFMin in this case

for R being non empty well founded RelStr

V being non empty set

H being Function of

the carrier of R PFuncsthe carrier of R V V

F F being Function of the carrier of R V

recursively expressed by H st F is

F is recursively expressed by H

holds F F

After intro ducing the arbitrary but xed constants for R V H F

and F App endix E p we

assume that

F is recursively expressed by H and

F is recursively expressed by H

and switch to a pro of by contradiction

assume F F then

consider x being Element of c such that

Fx Fx

Now we are ready to apply the scheme WFmin but rst we have to state

the premises to the scheme such that they exactly match the premis

es exp ected by the scheme denition when checking the correctness

of a scheme application Mizar attempts only a rudimentary pattern

matching Here are the premises as required by WFmin

wfndtex no v p

On Equivalents of Wellfoundedness

reconsider x as Element of R by

Fx Fx by

R is well founded

and from WFmin we have a minimal x where F and F dier

Fx Fx and

not ex y being Element of R

st x y Fy Fy yx r from WFMin

With it is simple to show that F rSeg x F rSeg x

and then

Lrecur Fx Hx F rSeg x by

Fx by Lrecur

which contradicts

Nonexistence of descending chains

The Mizar denition of a descending chain takes the form of dening

an attribute

definition

let R be RelStr f be sequence of R

attr f is descending means Lchain

for n being Nat

holds fn fn fn fn the InternalRel of R

end

We embark on proving the following

theorem omega chains

for R being non empty RelStr holds

founded iff not ex f being sequence of R st f is descending R is well

Again we use the shorthand c for the carrier of R and r for the

InternalRel of R The pro of by contradiction of the rst implication

is stated in a diuse conclusion b etween hereby and the matching end

see endnotes and starts

hereby assume R is well founded then

well founded in c by Lwell r is

given f being sequence of R such that

f is descending

rng f is nonempty and is a subset of a wellfounded set c so let a

b e a minimal element of rng f For a natural n we have fn a but

then fn precedes a in rng f which contradicts the minimality

of a in rng f The complete pro of is in App endix F

The pro of of the converse is also by contradiction and starts as

follows

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

assume

not ex f being sequence of R st f is descending

assume not R is well founded then

well founded in c by Lwell then not r is

consider Y being set such that

Y c c Y and

for a being set holds

not a Y or rSega \ Y by

and now we construct fa descending sequence of Y We do so with

the help of a scheme LambdaRecEx that assures existence of recur

sively dened functions on natural numb ers We will set fn to

an arbitrary element of rSegfn but for this we need an indenite

description op erator We had to dene one as there is no such op erator

in the Mizar language and there was no appropriate functor in the

MML

definition let S be set assume

contradiction

func choose S Element of S means

not contradiction whatever no need for a label

correctness by

end

This denition may lo ok a bit unusual as it is based on a condition that

can never b e satised But this only means that we will not b e able to

use the deniensand we are not planning to The deniens in that

denition can b e an arbitrary syntactically correct formula The functor

choose applied to a set S returns a set whose typ e is Element of S

The builtin pro cessing of this typ e guarantees that a set x whose typ e

is Element of S satises x S only for nonempty S

We intro duce f as follows

consider f being Function such that

dom f NAT and

f choose Y and

for n being Element of NAT x being set

st x fn holds fn choose rSegx \ Y

from LambdaRecEx

The completion of the pro of is now routine We show that f forms a

descending sequence with values in c First we prove that rng f c c

using the denitional expansion of c and this allows us to state

then f is Function of NAT c by FUNCT

reconsider f as sequence of R by NORMSP def

and the remaining conditions required for f to b e a descending sequence

are easily proven This contradicts assumption

wfndtex no v p

On Equivalents of Wellfoundedness

Conclusions

We have proven in Mizar some statements equivalent to the well

foundedness of a relational structure One of the statements name

ly that the existence of recursively dened functions without requir

ing uniqueness on a relational structure implies that the structure is

wellfounded seems not to b e widely known Its pro of required some

advanced notions of set theory and we were fortunate to nd the need

ed notions already formalized in the MML The development of such

a data base for mathematics is a challenge The current organization

of the MML makes searching for needed facts an arduous task Pro ofs

in Mizar tend to b e lengthy either due to the lack of expressive p ower

of the language or the level of detail required by the verier

The Mizar exp erience indicates that computerized supp ort for

mathematics aiming at the QED goals cannot b e designed once and

then simply implemented A system of mechanized supp ort for math

ematics is likely to succeed if it has an evolutionary nature The main

comp onents of such a systemthe authoring language the checking

software and the organization of the data basemust evolve as more

exp erience is collected At this moment it seems dicult to extrap o

late the exp erience with Mizar to the fully edged goals of the QED

Pro ject However the p eople involved in Mizar are optimistic

Acknowledgments

The research was funded by the NSERC OGP Grant No OGP

Trybulecs visit was made p ossible by the NATO Collab orative

Research Grant No CRG

Kevin Charter help ed in improving the presentation We thank Bob

Boyer and Paul Jackson for their remarks

Notes

a

Clusters are used to express the following relationshi ps

i Conditional clusters are used to say that any ob ject having a prop erty expressed

by an attribute also has another prop erty expressed by another attribute For

example

cluster non trivial non empty set

says that every non trivial set is non empty

ii Functorial clusters say that a term constructed with a sp ecic functor has a

certain prop erty expressed by an attribute For example

cluster f y z g non empty

says that an ordered pair is a nonempty set Section

wfndtex no v p

Piotr Rudnicki and Andrzej Trybulec

iii Existential clusters are used to dene new typ e expressions that involve

attributes For example

cluster non empty well founded RelStr

see Section

Each dened cluster needs to satisfy certain correctness conditions The relationships

expressed by clusters are automatically pro cessed by the Analyzer

b

In Mizar the Fraenkel op erator denes a set as a functional image of elements

of a nonempty set that satisfy some stated condition Its typical use is

f x where x is Element of X x g

where x is a term X is known to b e a nonempty set and x is a formula

c

The denitiona l theorem named TARSKIdef is generated automatically and

contains an additional quantier With the default quantiers stated explicitly

TARSKI denotes the following formula

for x y holds xy f f xy g f x g g

while TARSKIdef denotes

for x y IT holds IT xy iff IT f f xy g f x g g

Although logically equivalent the formulae are dierent and in some contexts it

matters for the Checker

d

Mizar p ermits many pro of structures when proving an equivalence iff

among them the following are p ermitted

proof proof proof

thus implies thus implies hereby

thus implies assume assume

end thus thus

end end

assume

thus

end

The last column illustrates a typical use of the hereby end construct

e

The lab eling scheme is derived from L Lamp orts metho d of writing pro ofs

see also A lab el of the form nk lab els item k app earing at the nesting level

k is used for assumptions n The form n

f

A diuse statement is a counterpart of a pro of When writing a pro of a prop osi

tion is rst stated and then proven In a diuse statement the formula b eing proven

is not stated explicitly instead it is automatically recovered from the reasoning In

the following example

P

proof N now

a proof structure a proof structure

appropriate for appropriate for

end end

references to P or to N refer to the same formula

g

The binary inx predicate symb ol  o ccurring in theorem Funion in App endix B

is overloaded In the environment of this article when the typ es of b oth its arguments

are functions f and g f  g is the tolerance predicate dened in meaning that

f and g are equal on the intersection of their domains and not the equinumerosity

relation of two sets as dened in see page

wfndtex no v p

On Equivalents of Wellfoundedness

h

The keyword thesis can b e used inside a pro of and denotes the yet unproven

part of the original prop osition The pro of structuring constructs must agree with

the current thesis and they change the meaning of thesis The pro of structuring

constructs are

Generalization let appropriate for thesis that starts with a universal

quantier

Assumption assume appropriate when thesis is an implicati on

Existentialassumption given appropriate when thesis is an implication

with an existentially quantied antecedent

Exemplication take appropriate for thesis starting with an existential

quantier

Conclusion thus or hence appropriate when thesis can b e seen as a

conjunction even with only one conjunct

DiuseConclusion hereby end which is a conclusion written as a diuse

statement hereby in some sense plays the role of thus and now

The pro of structuring constructs change the meaning of thesis in a natural

way By assuming not thesis one switches to a pro of by contradiction and what

remains to b e proven the new thesis is contradiction

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App endix

App endices are available on the WEB

httpwwwcsualbertacapiotrMizarWfndappendixdvi

Address for correspondence

Piotr Rudnicki

Department of Computing Science

University of Alb erta

Edmonton Alb erta Canada TJ H

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