Implementation of Bourbaki's Elements of Mathematics in Coq

213 Pages·2017·1.46 MB·English

by  José Grimm

Implementation of Bourbaki’s Elements of Mathematics
in Coq: Part One, Theory of Sets
José Grimm
To cite this version:
José Grimm. Implementation of Bourbaki’s Elements of Mathematics in Coq: Part One, Theory of
Sets. [Research Report] RR-6999, 2009, pp.209. ￿inria-00408143v3￿
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INSTITUTNATIONALDERECHERCHEENINFORMATIQUEETENAUTOMATIQUE
Implementation of Bourbaki’s Elements of
Mathematics in Coq:
Part One
Theory of Sets
JoséGrimm
N° 6999 — version 3
initialversionJuly2009—revisedversionMarch2010
Algorithmics,Programming,SoftwareandArchitecture
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ImplementationofBourbaki’sElementsofMathematicsinCoq:
PartOne
TheoryofSets
∗
JoséGrimm
Domain:Algorithmics,Programming,SoftwareandArchitecture
Équipe-ProjetApics
Rapportderecherche n°6999—version3—initialversionJuly2009—revisedversion
March2010—209pages
Abstract: WebelievethatitispossibletoputthewholeworkofBourbakiintoacomputer.
One of the objectives of the Gaia project concerns homological algebra (theory as well as
algorithms);inafirststepwewanttoimplementallninechaptersofthebookAlgebra. But
thisrequiresatheoryofsets(withaxiomofchoice,etc.)morepowerfulthanwhatisprovided
by Ensembles; we have chosen the work of Carlos Simpson as basis. This reports lists and
commentsalldefinitionsandtheoremsoftheChapter“TheoryofSets”.Thecode(including
almostallexercises)isavailableontheWeb,underhttp://www-sop.inria.fr/apics/gaia.
Version one was released in July 2009, version 2 in December 2009, version 3 in March
2010.Therearesmalldifferences,markedinfootnotes.
Key-words: Gaia,Coq,Bourbaki,FormalMathematics,Proofs,Sets
WorkdoneincollaborationwithAlbanQuadrat,basedonpreviousworkofCarlosSimpson(CNRS,Univer-
sityofNice-SophiaAntipolis)
∗
Email:[email protected]
CentrederechercheINRIASophiaAntipolis–Méditerranée
2004,routedesLucioles,BP93,06902SophiaAntipolisCedex
Téléphone:[phone]—Télécopie:[phone]
ImplémentationdesÉlémentsdemathématiquesdeBourbakien
Coq,
partie1
Théoriedesensembles
Résumé:Nouspensonsqu’ilestpossibledemettredansunordinateurl’ensembledel’œuvre
de Bourbaki. L’un des objectifs du projet Gaia concerne l’algèbre homologique (théorie et
algorithmes); dans une première étape nous voulons implémenter les neuf chapitres du
livre Algèbre. Au préalable, il faut implémenter la théorie des ensembles. Nous utilisons
l’AssistantdePreuveCoq;leschoixfondamentauxetaxiomessontceuxproposéesparCar-
losSimpson. CerapportlisteetcommentetouteslesdéfinitionsetthéorèmesduChapitre
théoriedesensembles. Presquetouslesexercisesontétérésolus. Lecodeestdisponiblesur
lesiteWebhttp://www-sop.inria.fr/apics/gaia.
Mots-clés: Gaia,Coq,Bourbaki,mathématiquesformelles,preuves,ensembles
Bourbaki:TheoryofsetsinCoqI(v3) 3
Chapter 1
Introduction
1.1 Objectives
Our objective (it will be called the Bourbaki Project in what follows) is to show that it
ispossibletoimplementtheworkofN.Bourbaki, “ÉlémentsdeMathématiques”[3], intoa
computer, and we have chosen the Coq Proof Assistant, see [4, 1]. All references are given
to the English version “Elements of Mathematics”[2], which is a translation of the French
version (the only major difference is that Bourbaki uses an axiom for the ordered pair in
theEnglishversionandatheoremintheFrenchone). Westartwiththefirstbook: theory
ofsets. Itisdividedintofourchapters, thefirstonedescribesformalmathematics(logical
connectors,quantifiers,axioms,theorems). ChaptersIIandIIIformthebasisofthetheory;
they define sets, unions, intersections, functions, products, equivalences, orders, integers,
cardinals,limits.Thelastchapterdescribesstructures.
Anexampleofstructureisthenotionofrealvectorspace: itisdefinedonasetE, uses
theset(cid:82)ofrealnumbersasauxiliaryset,hassomecharacterization(therearetwolawson
E, a zero, and a action of (cid:82) over E), and has an axiom (the properties of the the laws, the
action,thezero,etc.). Acompleteexampleofastructureistheorder;givenasetA,wehave
as characterization s ∈P(A×A) and the axiom “s◦s =s and s∩s−1 =∆ ”. We shall see in
A
thesecondpartofthisreportthatanorderingsatisfiesthisaxiom,butitnotclearifthiskind
of construction is adapted to more complicated structure (for instance a left module on a
ring). Given two sets A and A(cid:48), with orderings s and s(cid:48), we can define σ(A,A(cid:48),s,s(cid:48)), the set
ofincreasingfunctionsfrom Ato A(cid:48). Anelementofthissetiscalledaσ-morphism. Inour
implementation, the“setoffunctions f suchthat...” doesnotexists; wemayconsiderthe
set of graphs of functions (this is well-defined), but we can also take another position: we
reallyneedσtobeasetifwetrytodonon-trivialsetoperationsonit,forinstanceifwewant
to define a bijection between σ and σ(cid:48); these are non-obvious problems, dealt with by the
theoryofcategories. Thereishoweveranotherpracticalproblem;Bourbakiveryoftensays:
letE beanorderedset; thisisashort-handforapair(A,s). Considernowamonoid(A,+).
Constructinganorderedmonoidistrivial: thecharacterizationistheproductofthecharac-
terizations, andtheaxiomistheconjunctionoftheaxioms. Theorderedmonoidcouldbe
(A,(s,+)). If f isamorphismfors,andu∈A,thenthemappingx(cid:55)→ f(x+u)isamorphism
fors,providedthat+iscompatiblewiths. IfwewanttoconvertthisintoatheoreminCoq,
theeasiestsolutionistodefineanobjectXequivalentto(A,(s,+)),awaytoextractX(cid:48)=(A,s)
andX(cid:48)(cid:48)=(A,+)fromX,anoperationsonAobtainedfromXorX(cid:48),andchangethedefinition
ofσ: itshoulddependonX(cid:48) ratherthanon Aand s. Thecompatibilityconditionisthena
propertyofX,σ(X,Y)andσ(X(cid:48),Y)areessentiallythesameobjects,if f ∈σ(X,Y)wecancon-
RR n°6999
4 JoséGrimm
sider f(cid:48)=x (cid:55)→ f(x+u),andshow f(cid:48)∈σ(X,Y). Fromthiswecandeducethemappingfrom
σ(X(cid:48),Y)intoσ(X(cid:48),Y)associatedto f (cid:55)→f(cid:48).
1.2 Background
WestartedwiththeworkofCarlosSimpson1,whohasimplementedtheGabriel-Zisman
localization of categories in a sequence of files: set.v, func.v, ord.v, comb.v, cat.v, and gz.v.
Onlythefirstthreefilesinthislistareusefulforourproject. Thefileord.zcontainsalotof
interestingmaterial,butifwewanttocloselyfollowBourbaki,itisbettertorestarteverything
fromscratch. Thefilefunc.v containsalotofinterestingconstructionsandtheorems, that
canbeusefulwhendealingwithcategories. Forinstance,itallowsustodefinemorphisms
on the category of left modules over a ring. The previous discussion about structures and
morphismexplainswhyonlyhalfofthisfileisused.
Thisreportisdividedintwoparts.ThefirstpartdealswithimplementationofChapterII,
“Theoryofsets”,andthesecondpartwithchapterIII,“Orderedsets,cardinals;integers”of[2]
EachofthesixsectionsofBourbakigivesachapterinthisreport(weusethesametitlesasin
Bourbaki)butwestartwiththedescriptionofthetwofilesset.vandfunc.vbyCarlosSimpson
(itisasequenceofmodules). TheircontentcoversmostofSections1and2(“Collectivizing
relations”and“Orderedpairs”).
1.3 Notations
Choosingtractablenotationsisadifficulttask. Wewouldliketofollowthedefinitionsof
Bourbakiascloselyaspossible. Forinstancehedefinestheunionofafamily(Xι)ι∈I(Xι∈G).
ClassicFrenchtypographyusesitaliclower-caseletters,anduprightupper-caseletters,but
the current math tradition is to use italics for both upper- and lower-case letters for vari-
ables; constantslikepr andCarduseuprightfont. Thesetofintegersissometimesnoted
1
(cid:78); but Bourbaki uses only N. Some characters may have variants (for instance, the previ-
ousformulacontainsaFrakturvariantoftheletterG).IntheXMLversionofthisdocument
wedonotusetheUnicodecharacterU+1D50A(becausenotmostbrowsersdonothavethe
glyph),butacharacterwithvariant,sothatthereislittledifferencebetweenG,G,G,G,(cid:71),G.
InthisdocumentweuseonlyonevariantoftheGreekalphabet(Unicodeprovidesnormal,
italic,bold,bold-italic,sans-serifandsans-serifbolditalic;asaconsequence,theXMLver-
sionshowsgenerallyaslantedversionofGreekcharacters,wherethePdfdocumentusesan
uprightfont).
WecaneasilyreplacelowerGreeklettersbytheirLatinequivalents(thereislittlediffer-
ence between (Xι)ι∈I and (Xi)i∈I). We can replace these unreadable old German letters by
moresignificantones. WemustalsoreplaceIbysomethingelse,becausethisisareserved
keywordinCoq(andinisreservedtoo). Intheoriginalversion,C.Simpsonreservedthelet-
ters A,BandE. Thus,aphraselike: let AandBbetwosubsetsofasetE, andI=A×B,all
fouridentifiersarereservedlettersinSimpson’sframework. Notethat,traditionally,French
mathematiciansuseromanuprightuppercaselettersanditalicslowercaselettersforvari-
ables;
QuantitiesnamedR,B,X,Y,andZbySimpsonhavebeenrenamedtoRo,Bo,Xo,Yoand
Zo.QuantityAhasbeenremoved(itwasaprefixversionof&).QuantityEhasbeenrenamed
1http://math.unice.fr/~carlos/themes/verif.html
INRIA
Bourbaki:TheoryofsetsinCoqI(v3) 5
Bset then Set: this is the type of a Bourbaki set. It will still be denoted by E here. In our
framework,thereservedsingle-letteridentifiersareIJLOPQSVW.
CoqreservestheletterIasaproofofTrue, theletterOastheinteger0andtheletterS
forthefunctionn(cid:55)→n+1onintegers. Anorderedpairwithvaluesx and y isatermz that
hastwoprojectionspr z=x andpr z=y. Theconstructoriscalledbpair2 inCoq,andthe
1 2
destructorsarecalledpr1andpr2. WeshallreservethelettersJfortheconstructorandP,Q
forthedestructors,sothatJ(Pz)(Qz)=zforallpairsz(seesection2.6fordetails).
Bourbakihasasectiontitled“definitionofafunctionbymeansofaterm”. Anexample
wouldbe x (cid:55)→(x,x)(x ∈(cid:78)). ThiscorrespondstotheCoqexpressionfunx:nat=>(x,x). Ac-
cordingtotheCoqdocumentation,theexpression“definestheabstractionofthevariablex,
oftypenat,overtheterm(x,x).Itdenotesafunctionofthevariablexthatevaluatestotheex-
pression(x,x)”.Bourbakisays“amappingofAintoBisafunction f whosesourceisequalto
AandwhosetargetisequaltoB”. Thedistinctionbetweenthetermsfunctionandmapping
issubtle: thereisasectioncalled“setsofmappingsofonesetintoanother”; itcouldhave
been: “setsoffunctionswhosesourceisequaltosomegivensetandwhosetargetisequal
tosomeothergivenset”. Itisinterestingtonotethattheterm‘function’isusedonlyonce
intheexercisestoChapterIII,inacasewhere‘mapping’cannotbeusedbecauseBourbaki
doesnotspecifythesetB.
Inwhatfollows,weshallusetheterm‘function’indifferentlyforS,orthemappingn(cid:55)→
n+1,ortheabstractionn=>Sn.GivenasetA,wecanconsiderthegraphg ofthismapping
whennisrestrictedtoA. ThisconstructionissoimportantthatwereservetheletterLforit.
GivenasetB,ifourmappingsendsAtoB,wecanconsiderthe(formal)functionf associated
tothemappingwithsourceAandtargetB. WeshalldenotethisbyBL.Thesetwoobjects f
and g havetheimportantpropertythat, ifn isin A, thereisanm denotedby f(n)or g(n)
suchthatm =n+1(wehavetheadditionalpropertythat f(n)isinB). Ashortnotationis
requiredforthemappingfrom(g,n)(cid:55)→g(n),or(f,n)(cid:55)→ f(n). WeshallusethelettersVand
Wrespectively. Inthisdocument,weshallusestandardnotations,forinstancepr andpr
1 2
forPandQ,whentheyexist,calligraphicletterslikeV orW forsomeobjectslikeVandW,
and a slanted font like is_function for the general case. Note that J x y is a Coq expression
meaningtheapplicationofJtobothargumentsxandy.
There a possibility to change the Coq parser and pretty printer so that (x,y) is read as
pair x y, and { x : A | P } is read as the set of all x in A satisfying the predicate P. We
shallnotusethisfeaturehere. Infact,thesearestandardnotationsinCoqfornotionsthat
arerelatedbutnotexactlyidenticaltoours.
1.4 Descriptionofformalmathematics
Termsandrelations. AmathematicaltheoryT isacollectionofwordsoverafinitealpha-
bet formed of letters, logical signs and specific signs. Logical signs are (cid:228), τ, ∨, ¬ (the first
twosignsarespecifictoBourbaki,theotherones,disjunctionandnegation,havetheirusual
meaning).Specificsignsare=,∈,lettersarex,y,A,A(cid:48),A(cid:48)(cid:48),A(cid:48)(cid:48)(cid:48),and“atanyplaceinthetextit
ispossibletointroducelettersotherthanthosewhichhaveappearedinpreviousarguments”
[2,p. 15](anynumberofprimesignsisallowed;thisisnotincontradictionwiththefinite-
ness of the alphabet). An assembly is a sequence of signs and links. Some assemblies are
well-formedaccordingtosomegrammarrules.InBackus-Naurformtheyare:
Term:=letter | τ (Relation) | SsignTerm ...Term
letter 1 n
2Thisiscalled‘pair’inSimpsonandinversion1ofthisreport
RR n°6999
6 JoséGrimm
Relation:=¬Relation | ∨RelationRelation | RsignTerm ...Term
1 n
Eachsignhastobefollowedbytheappropriatenumberofterms: (cid:228)takesnone,∈and=are
followed by two signs, and one can extend Bourbaki to non-standard analysis [7] by intro-
ducingaspecificsignst ofweight1qualifyingtherelationthatfollowstobestandard. Each
signissubstantificas(cid:228)(ityieldsaterm)orrelationalas=(ityieldsarelation).
We shall see below that τ (R) has to be interpreted as the expression where all occur-
x
rencesofxinRarereplacedby(cid:228)andlinkedtotheτ.Parenthesesareremoved.Thishasone
advantage: thereisnox inτ (R),hencesubstitutionrulesbecometrivial. Forinstance,the
x
function x (cid:55)→x+y isconstructedbyusingτ, itisidenticaltothefunction z (cid:55)→z+y. Ifwe
wanttoreplace y by z, weget x (cid:55)→x+z, butnot z (cid:55)→z+z. InCoq, thevariable y appears
freeinx(cid:55)→x+y,andthevariablexappearsboundinthesameexpression.Renamingbound
variablesiscalledα-conversion.Twoα-convertibletermsareconsideredequalinCoq.
TheAppendixtoChapterIdescribesanalgorithmthatdecideswhetheranassemblyisa
term,arelation,orisill-formed. Itworksintwostages. Inthefirststage,linksareignored. A
classicalresultincomputerscienceisthatthereexistsaprogram(calledaparser)thatrec-
ognizes all significant words (i.e., well-formed assemblies without links). We can associate
anumbertoeachsign(forinstance262to’a’,111to’=’)andthustoeachassembly(forin-
stance,262111262to’a=a’). ThiswillbecalledtheGödelnumberoftheassembly,see[5]for
anexample.TwodistinctassemblieshavedistinctGödelnumbers.ThesetofGödelnumbers
isarecursivelyenumerableset. GivenassembliesA ,A ,A ,etc,onecanformtheconcate-
1 2 3
nationA A A .... Ifeachassemblyisasignificantword,thereisauniquewaytorecoverA
1 2 3 i
fromtheconcatenation,hencefromtheGödelnumberoftheconcatenation.
Ademonstrativetext forBourbakiisasequenceofassemblies A A ...A , thatcontains
1 2 n
(cid:48) (cid:48) (cid:48) (cid:48)
aproof,whichisasubsequenceA A ...A ofrelations,whereeachA canbeshowntobe
1 2 m i
truebyapplicationofabasicderivationrulethatusesonlyA(cid:48) for j <i.EachA(cid:48) isatheorem.
j i
(cid:48) (cid:48) (cid:48)
Weshalluseavariant: aproof-pairisasequenceofrelationsA A ...A satisfyingthesame
1 2 m
(cid:48)
conditionsasabove,andatheoremisthelastrelationA inaproof-pair. Ifourbasicrules
m
are simple enough, the property of a number g to be the Gödel number of a proof-pair is
primitiverecursive. Fromthis,onecandeducetheexistenceofatruestatementthathasno
proof(thisisGödel’sTheorem).
Anassembly Acontaininglinksisanalyzedbyusingantecedents, whichareassemblies
of the form τ (R) (where x is some variable) that are identical to A if x is substituted in R
x
and links are added. The algorithm for deciding that an assembly with links is a term or a
relationisrathercomplicated. Bourbakigivesthreeexamplesofassemblieswithlinks; the
antecedentofthefirstoneisτ (x∈y)(thereisasinglelink);theantecedentofthesecondone
x
isτ (x∈A(cid:48) =⇒ x∈A(cid:48)(cid:48))(therearetwolinks);thethirdoneistheemptyset,seepicturebelow.
x
One can replace these links by the De Bruijn indices, so that the empty set would become
τ¬¬¬∈τ¬¬∈121. This has two drawbacks: the first one is that 121 could be understood
asoneintegerorasequenceofthreeintegers,thesecondisthatthisnotationassumesthat
integersarealreadydefined. Theremedytothefirstproblemwouldbetoinsertaseparator
(forinstanceasquare)andaremedytothesecondwouldbetouseabase-onerepresentation
ofintegers;theemptysetwouldbeτ¬¬¬∈τ¬¬∈(cid:228)−(cid:228)−−(cid:228)−. Thescopeofthesecondτ
isthescopeofitsoperator,thus¬¬∈(cid:228)(cid:228). Thismeansthatthetwosquaresareinthescope
of both τ, are are linked to the second and first τ respectively. The third square is in the
scopeofthefirstτonly,henceislinkedtothefirstτ. FormalmathematicsinBourbakiisso
complicatedthatthe(cid:228)symbolis,inreality,neverused.
INRIA
Bourbaki:TheoryofsetsinCoqI(v3) 7
Denoteby(B|x)Atheassemblyobtainedbyreplacingx,whereveritoccursinA,bytheas-
semblyB.Bourbakihassomecriteriaofsubstitutions,CS1,CS2,etc,thatarerulesaboutsub-
stitutions. ForinstanceCS3saysthatτx(A)andτx(cid:48)(A(cid:48))areidenticalifA(cid:48) is(x(cid:48)|x)Aprovided
thatx(cid:48) doesnotappearinA(informally: since x doesnotappearinτ (A),thenameofthe
x
variablex isirrelevant). FormativecriteriaCF1,CF2,etc.,giverulesaboutwell-formedness
ofassemblies. ForinstanceCF8saysthat(T|x)AisatermorarelationwheneverAisaterm
orarelation,T isaterm,xisaletter.
Abbreviationsareallowed,sothat∨¬canbereplacedby =⇒,and¬∈canbereplaced
by(cid:54)∈. Abbreviationsmaytakearguments,forinstance∧ABisthesameas¬∨¬A¬B. Aterm
mayappearmorethanonce,forinstance ⇐⇒ ABisthesameas∧ =⇒ AB =⇒ BA,andafter
expansion¬∨¬∨¬AB¬∨¬BA. Thelogicalconnectors¬,∨and∧arewritten~,\/,and/\
inCoq(weshalluse&insteadof∧,sinceitiseasiertotype). NotethatinCoq,A→Bisthe
typeofafunctionfromAtoBbutalsomeansA =⇒ B. Thereisnolimitonthenumberof
abbreviations(Bourbakiinvented(cid:59)asavariantofø).Unicodeprovidesalotofsymbols,but
fewofthemareavailableinLATEXorinWebbrowsers.
Starting with Section 2, Bourbaki switches to infix notation. For instance, whenever A
andBarerelationssois∨¬¬∨¬A¬BA,byvirtueofCF5andCF9. Usingabbreviations,this
relationcanbewrittenas =⇒ ∧ABA.Theinfixversionis(AandB) =⇒ A.Inordertoremove
ambiguities,parenthesesarerequired,butBourbakisays:“Sometimesweshallleaveoutthe
brackets”[2,p. 24],intheexampleabovethreepairsofbracketsareleftout. Insomecases
Bourbaki writes A∪B∪C. This can be interpreted as (A∪B)∪C or A∪(B∪C). These are
twodistinctobjectsthathappentobeequal:formally,therelation(A∪B)∪C=A∪(B∪C)is
true. SimilarlyA∨B∨Cisambiguous,butithappens,accordingtoC24,that(A∨B)∨Cand
A∨(B∨C)areequivalent(formally:relatedby ⇐⇒).InCoq,weuseunion2asprefixnotation
for∪, sowemustchosebetween∪(∪AB)C or∪A(∪BC). Functioncallsareleft-associative,
and brackets are required where indicated. We use \/ as infix notation for ∨, parentheses
maybeomitted,theoperatorisrightassociative.
Theoremsandproofs. Eachrelationcanbetrueorfalse. TosaythatP isfalseisthesame
as to say that ¬P is true. To say that P is either true or false is to say that P∨¬P is true.
A relation is true by assumption or deduction. A relation can be both true and false, case
where the current theory is called contradictory (and useless, since every property is then
true).TheremayberelationsPforwhichitisimpossibletodeducethatPistrueanditisalso
impossibletodeducethatPisfalse(Gödel’stheorem).Apropertycanbeindependentofthe
assumptions. ThismeansthatitisimpossibletodeduceP or¬P; inotherwords,addingP
or¬P doesnotmakethetheorycontradictory. Anexampleistheaxiomoffoundation(see
below),orthecontinuumhypothesis(everyuncountablesetcontainsasubsetwhichhasthe
powerofthecontinuum).
Some relations are true by assumption; these are called axioms. Anaxiom scheme is a
rulethatproducesaxioms.ThelistofaxiomsandschemesusedbyBourbakiaregivenatthe
endofthedocument. AtruerelationiscalledaTheorem(orProposition,Lemma,Remark,
etc). Aconjectureisarelationbelievedtobetrue,forwhichnoproofiscurrentlyfound. As
saidabove,inBourbaki,atheoremisarelationwithaproof,whichconsistsofasequenceof
truestatements,thetheoremisoneofthem,andeachstatementRinthesequenceiseither
RR n°6999