primrecapp list list listwhereappNilidjappConsxxsConsxappxsprimrecrev list listwhererevNilNiljrevConsxxsapprevxsConsxNilTheprimreccommandexpectstondanequationforeachofthedatatype ID: 145687
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2Background:DatatypesAfundamentalconceptinmostifnotalltypedfunctionalprogramminglanguagesisthatoffreelygenerateddatatypes.Theyarethemainwaytocreatenewtypesandcom-bineexistingtypesintomorecomplex(anduseful)types.Probablythemostpopularexampleisthedatatypeof(nite)listsoverelementtype ,whichcanbedenedbydatatype_new list(map:map)=null:NiljCons(hd: )(tl: list)Thedatatype_newcommandisprovidedbythenew(co)datatypepackage.Here,itintroducesthetype listgeneratedfreelybytwoconstructors:•Nil,whichrepresentsanemptylistandcanbetestedusingnull:: list)bool;•Cons,whichtakestwoargumentsoftypes and listandconstructsanewlistfromasingleitem(thenewlist'shead)andtherestofthelist(itstail).Thissimpleexampleshowsabasicconceptofdatatypes:Theycanrecursivelycon-tainanumberofothertypes,includingthemselves.Afunctionthattakesa(recursive)datatypeasoneofitsargumenttypescanthusre-applyitselftothenestedvalue.Thedatatype_newcommandsupportsthedenitionofmutuallyrecursivedata-types.Thesearecharacterizedbythesimultaneousspecicationoftwoormoredata-typesthatrecursivelycontaineachotherinsomeoftheirconstructorarguments.Atypicalexampleisthetypeofnitelybranchingtreesofnitedepthandassociated(nite)foreststhatdonotuseauniversallisttype:datatype_new tree0=Node0(tval0: )(children0: forest)and forest=FNiljFCons(fhd: tree0)(ftl: forest)Thecommandrequiresthateachsetofmutuallyrecursivedatatypeshaveidenticaltypearguments(here, ).Another,morecomplexexampleisarepresentationforasimplearithmeticexpressionlanguage:datatype_newexpr=XSumexpr_sumjXProdexpr_prodjXConstnatandexpr_sum=Sumexprexprandexpr_prod=ProdexprexprInadditiontomutualrecursion,itisoftendesirabletoincorporateexistingdatatypesintonewlydenedtypes.Thismakesitpossibletoreasonmoremodularlyaboutthedatatypeandtoreuseexistinglibraries.Forexample,themoremodularwaytospecifyatypesuchas tree0reusesthegenericlisttypeconstructorinsteadofintroducingadedicatedforesttype:datatype_new tree=Node(tval: )(children: treelist)Similarly,binarytrees,canbedenedviadatatype_new option=is_none:NonejSome(the: )datatype_new btree=BNode(bval: )(left: btreeoption)(right: btreeoption) primrecapp:: list) list) listwhereappNil=idjapp(Consxxs)=Consxappxsprimrecrev:: list) listwhererevNil=Niljrev(Consxxs)=app(revxs)(ConsxNil)Theprimreccommandexpectstondanequationforeachofthedatatype'scon-structors;otherwise,itprintsawarning.Thenonexhaustiveoptioncanbeusedtosup-pressthewarning:primrec(nonexhaustive)hd:: list) wherehd(Constx_)=x3.2MutualRecursionRecursionovermmutuallyrecursivedatatypes¯ 1,:::,¯ mgenerallyrequiresmmu-tuallyrecursivefunctions.Thisisachievedusingthesyntaxprimrecf1::)¯ 1))1and...fm::)¯ m))mwheref1:::(C11¯x11):::=:::...jf1:::(C1k1¯x1k1):::=:::...jfm:::(Cm1¯xm1):::=:::...jfm:::(Cmkm¯xmkm):::=:::Theright-handsidesmaycontainrecursivecallstoanyofthefunctionsfi.Forexam-ple,arithmeticexpressionsrepresentedbythedatatypesintroducedinSection2canbeevaluatedrecursivelyasfollows:primreceval::expr)natandeval_sum::expr_sum)natandeval_prod::expr_prod)natwhereeval(XSums)=eval_sumsjeval(XProdp)=eval_prodpjeval(XConstc)=cjeval_sum(Sumab)=evala+evalbjeval_prod(Prodab)=evalaevalb 4ImplementationofPrimitivelyRecursiveFunctionsIntheabsenceofprimrec,userscoulddenerecursivefunctionsusingsuitableargu-mentstorecursorsassociatedwiththedatatypesonwhichtheywanttorecurse.Usingthetheoremsassociatedwiththerecursors,itisthenpossibletoprovestatementsaboutthedenedfunctions.Theprimreccommandautomatesthisprocess:Givenhigh-levelspecicationsofafunction'sdesiredbehavior,itsynthesizesarecursor-baseddenitionandderivestheuserspecicationastheorems.4.1RecursorsRecursorsencodethemostgeneralformofprimitiverecursionoveradatatype.Forex-ample,thetypes listand treeintroducedinSection2areequippedwiththerecursorsrec_list::)( ) list)))) list)rec_tree::)( )( tree)list))) tree)characterizedbythefollowingequations:list.rec:rec_listn_Nil=nrec_listnc(Consyys)=cyys(rec_listncys)tree.rec:rec_treeenTEmpty=erec_treeen(Nodexts)=nx(map(t:(t;rec_treeent))ts)Givenarecursorthattakeskarguments,wewillrefertotherstk�1argumentsasitsbehavioralfunctions.Forlist,thebehavioralfunctionsnandcencodethebehaviorfortheNilandConscases,respectively.Noticethatcisgivennotonlythehead(y)andtail(ys)ofthelistbutalsotheresultoftherecursivecallofrec_listnconthetail.Fortree,theresultsoftherecursivecallsarepairedwiththeirargumentsinsidealist,asreectedinthetype( tree)list.Usingtherecursorforlists,thelengthfunctioncanbedenedasfollows:de nitionlength:: list)natwherelength=rec_list0(__r:1+r)Fromthisdenitionandthelist.recproperties,itiseasytoderiveahigh-levelcharac-terizationofthelengthfunction:lemmalength_simps[simp]:lengthNil=0length(Consxxs)=1+lengthxsunfoldinglength_deflist.recbysafeAsetofrecursorsassociatedwithmutuallyrecursivedatatypestakesonebehavioralfunctionperconstructorasarguments.Theyserveasadescriptionofhowtheconstruc-torargumentsandtheresultsofrecursivecallsarecombinedtogivethedesiredreturnvalues.Ingeneral: •Constructorargumentsonwhichrecursioncannotbeperformedarepassedasistothebehavioralfunctions.•Constructorargumentsxonwhichdirectormutualrecursioncanbeperformedarepassedtogetherwiththeresultyofpassingxtoarecursivecall.•Constructorargumentsonwhichnestedrecursionthroughamapfunctioncanbeperformedcontainpairs(x;y)thatcombinetheoriginalvaluexbeforetherecursion(theprevalue)andtherecursivecall'sresulty(thepostvalue).Inthemutualcase,eachdatatypepartakinginmutualrecursionhasitsownrecursor,buttherecursorsdifferonlyinthelastargumentandintheresulttype.Eachrecursorservesthusasitstype'sentrypointtomutualrecursion.4.2GeneralProcedureThegeneralprocedureimplementedbyprimrecperformsthefollowingsteps:1.Fromeachequation,extractthefollowinginformation:•thefunctionthattheequationtalksabout;•therecursivetype(deducedfromthefunction'sargumenttypesandthepositionoftheconstructorpattern);•thepattern-matchedconstructorandthenamesofitsarguments;•thenamesandtypesoftheotherarguments;•theequation'sright-handside.2.Querythedatatypedatabasetoobtaintherelevantinformation(constructors,recur-sors,theorems)abouttherecursivetypes.Ifnested-as-mutualrecursiontakesplace(ascanbedetectedfromtherecursivetypesandtheshapeoftherecursivecallsontheright-handsides),suitablerecursorsandtheoremsarederivedatthisstage[2].3.Traversetheright-handsideofeachequationinordertolocaterecursivecallsandreplacethembyanonrecursivetermthatwilleventuallyusetheadditionalargu-mentstherecursorprovidestodescribetherecursion.4.Introduce-abstractionsinfrontofeachofthesemodiedright-handsidesandusethemasbehavioralfunctions,llingupmissingspecicationswithunde ned.5.Foreachrecursor,permutetheresultingterm'sargumentsusing-abstractionstopulltheconstructortothefrontanddenethedesiredfunctionasequaltothisterm.6.Usingtherecursortheorems,provethatthedenitionfulllstheuserspecication(thefunction'scharacteristictheorems).4.3EliminatingRecursiveCallsThesecondstepoftheproceduredescribedabovedependsoninformationaboutthestructureofrecursionintheupcomingdenition.Thisposesachicken-and-eggprob-lem,sinceitisspecicallythisstepthatsuppliesalltheinformationabouttheinvolvedtypesandtheirproperties.Toovercomethisdifculty,theprocessofeliminatingrecur-sionfromthespecicationisdividedintotwosteps.First,everythingthatlookslikea shouldallowargumentsbeforetherecursiveargument.Thisismadepossiblebyper-mutingtheargument.Aspecicationf::1))k�1) )k+1))nisinternallyconvertedtof0:: )1))k�1)k+1))nOnceadeni-tionforf0isobtained,fcaneasilybedenedasf=(a1:::ak�1x:f0xa1:::ak�1)5Background:CodatatypesThesyntaxfordeningcodatatypesisalmostidenticaltothatfordatatypes.Butunlikedatatypes,codatatypescanbebuiltbyapplyingconstructorsinnitely,resultinginin-niteterms.Thisisreectedinthegeneratedcharacteristictheorems:Acoinduction(orbisimulation)principlereplacesinduction.Asaresult,thereneednotbeabasecasethatforces(primitively)recursivefunctionstoterminate.Thus,thetypeofinnitestreamsofdatacanbedenedusingthecommandcodatatype stream(map:smap)=SCons(shd: )(stl: stream)whereasthecorresponding“datatype”wouldbeempty(andhenceimpossibletodeneinHOL).Everyinstanceof streamhasaninnitenumberofconstructors.Thisimpliesthatanyfunctionthatconstructsastreamoneconstructoratatimemustcallitselfaninnitenumberoftimes.Otherexamplesarethecoinductivecounterpartstolistandnat:codatatype llist(map:lmap)=lnull:LNiljLCons(lhd: )(ltl: llist)codatatypeenat=EZerojESuc(epred:enat)Thetype llistrepresentslazy(orcoinductive)listswithaniteorinnitenumberofelements,whereasenatholdstheextendednaturalnumbers,consistingofnitetermsoftheformESuckEZeroandofthespecialvalueESuc(ESuc:::)representing¥.Analogouslytodatatypes,codatatypessupportmutuallycorecursiveandnestedco-recursivedenitions.Examplesincludeinnitelybranchingtreesofpotentiallyinnitedepth,whichcanbedenedbysimplysubstitutingcodatatypefordatatype_newandllistforlistinthemutualandnestedexamplesfromSection2:codatatype ltree0=LNode0(ltval0: )(lchildren0: lforest)and lforest=LFNiljLFCons(lfhd: ltree0)(lftl: lforest)codatatype ltree=LNode(ltval: )(lchildren: ltreellist)6SpecicationofPrimitivelyCorecursiveFunctionsCorecursivefunctionscanbespeciedusingprimcorecandprimcorecursive,whichsupportprimitivecorecursion,orusingthemoregeneralpartial_functioncommand.AlternativesbasedondomaintheoryandtopologyaredescribedbyLochbihlerandHölzl[6].Here,thefocusisonprimcorecandprimcorecursive.Whereasrecursivefunctionsconsumedatatypevaluesoneconstructoratatime,corecursivefunctionsproducepotentiallyinnitecodatatypevaluesoneconstructoratatime.Partlyreectingalackofagreementamongproponentsofcoinductivemethods,Isabellesupportsthreecompetingsyntaxesforspecifyingafunctionf: Syntacticrestrictionsontheselectorequationsensurethatprogressismadewitheachcorecursivecall—i.e.,thefunctionisproductive.Productivityguaranteesthatevenwhentheresultoflappisinnite,prexesofarbitrarynitelengthcanbecomputedbyexpandinganitenumberofcorecursivecalls.This,inturn,ensuresthatthefunctioniswelldened.Themainsyntacticrestrictionontheselectorequations,beyondthexedformatoftheleft-handside,isthatanycorecursivecallontheright-handsideeitheroccupiestheentireright-handsideorappearsasabranchina`if–then–else',`case–of',or`let–in'construct.Becauseofthisrestriction,thefollowingspecicationmustberejected:primcorecwrong::nat)natllistwhere:lnull(wrongn)jlhd(wrongn)=njltl(wrongn)=ltl(wrong(n+1))Somecodatatypesreusethesameselectorfunctionsforseveralconstructors.Ambi-guitiescanthenarisewhenconnectingtheselectorstothecorrespondingconstructors.Toresolvethisissue,primcorecacceptsselectorequationsoftheformget(f:::)=:::ofCwheregetisanambiguousselectorandCisaconstructor.Toavoidtheneedoftediousmanualspecicationofan`else'predicateforthediscriminatorformulas,asingleunderscore_isacceptedasacatch-allwildcard.Itisunderstoodastheimplicitnegationofallconditionsfortherelevantfunctioninpreviousequations.Thus,inPx=):::Qx=):::_=):::thelastequation'sconditionistakentobe:Px^:Qx.Arelatedfunctionalityisprovidedbythesequentialoption.Itcausesthediscrimi-natorformulaconditionsforafunctiontoapplyinsequence,ratherthanindependentlyofeachother.Thisrelievestheuser(orauto)fromhavingtoshowmutualexclusion,butthegeneratedtheoremsthenfeaturemorecomplicatedconditions.Finally,theexhaustiveoptionsignalsthatthediscriminatorformulapremisescoverallcases.Specifyingthisoptionaddsanotherproofobligation,whichagainiseithersolvedautomaticallybyprimcorecorlefttotheuserbyprimcorecursive.Inexchange,strongertheoremsaregeneratedaboutthediscriminators,with !inplaceof=).Anotherwaytoenablethisbehavioristospecify_asthelastcondition,inwhichcaseexhaustivenessissyntacticallytrivial.6.2TheConstructorViewTheconstructorviewcombinesthediscriminatorformulaandtheselectorequationsassociatedwithaconstructorinasingleequation.ThegeneralformisP1¯x=)=)Pn¯x=)f¯x=Ci(g1¯x):::(gk¯x) 6.5NestedCorecursionFornestedcorecursion,weconsideramoreelaborateexample:thedenitionofamonadicstructureforltree.Unfortunately,thestandardwaytodenethebindoper-ator =:: ltree)( )ltree))ltree,asisdoneforthemonadinstanceMonadTreeintheHaskelllibraryData.Tree,isnotprimitivelycorecursive:t =f=(casef(ltvalt)ofLNodebus)LNodeb(lappus(lmap(t:t =f)(lchildrent))))Theproblemisthatthecorecursivecallto =nestedthroughlmapoccursnotdirectlyasanargumenttotheproducedconstructorLNode,butasanargumenttolapp.Fortu-nately,wecanmakethespecicationprimitivelycorecursivebymovingthecorecursivecallpastlapp.Weusethesumtypetodistinguishbetweenthesubtreesusspawnedbyf(Inl)andthesubtreeslchildrenttowhichthecorecursivecallistobeapplied(Inr):primcorec =:: ltree)( )ltree))ltreewheret =f=(casef(ltvalt)ofLNodebus)LNodeb(lmap(ut:caseutofInlu)ujInrt)t =f)(lapp(lmapInlus)(lmapInr(lchildrent)))))Fromthelemmalmapf(lappxsys)=lapp(lmapfxs)(lmapfys),itiseasytoderivetheHaskell-stylenonprimitivelycorecursivespecication.GivenreturnxdenedasLNodexLNil,provingthemonadiclawsreturnx =f=fxt =return=tt =f =g=t =(x:fx =g)isaninterestingexerciseincoinduction.6.6Nested-as-MutualCorecursionTodemonstratetheconvenienceofnested-as-mutualcoinduction,letusdenethe =functionagain,butthistimemutuallywiththeternarymixxoperator_Ñ_ _thatreturnsalazylist,insteadofreusinglapp:primcorec =:: ltree)( )ltree))ltreeand_Ñ_ _::ltreellist) ltreellist)( )ltree))ltreellistwheret =f=(casef(ltvalt)ofLNodebus)LNodeb(usÑlchildrent f))jusÑts f=(caseusofLNil)(casetsofLNil)LNiljLConstts0)LCons(t =f)(usÑts0 f))jLConsuus0)LConsu(us0Ñts f)Thenewoperator'sintendedsemanticsis 7.1CorecursorsCorecursorsencodethemostgeneralformofprimitivecorecursionoveracodatatype.Thetypes llistand ltreeintroducedinSection5areequippedwiththecorecursorscorec_llist::( )bool))( )))( )bool))( )llist))( ) )) )llistcorec_ltree::( )))( )(ltree+ )llist)) )ltreecharacterizedbythefollowingequations:llist.corec:na=)corec_llistphseca=LNil:na=)corec_llistnhseca=LCons(ha)(ifsatheneaelsecorec_llistnhsec(ca))ltree.corec:corec_ltreelca=LNode(la)(lmap(x:casexofInlt)tjInrr)corec_treelcr)(ca))Givenacorecursorthattakeskarguments,wewillrefertotherstk�1argumentsasitsbehavioralfunctions.Forllist,thepredicatenindicateswhetheraLNilconstructorshouldbeproduced.Nextfollowsh,afunctionthatcomputestheheadofanonemptylistfromtheinputa.Thenextthreefunctions,s,e,andc,specifythelist'stail:s(“stop?”)isapredicatethatdetermineswhetherthecorecursionendswithanoncorecursivetermorcontinuesfurther;e(“end”)givesthenoncorecursivetailifsissatised;c(“continue”)computestheargumenttoacorecursivecallifsisnotsatised,specifyingacorecursivetail.Forltree,thebehavioralfunctionconstructsalistofsumvaluesthatspecifyhowtoproduceeachsubtreeindividually.AnInlvalueisinterpretedasaliteralnoncorecursiveresult,whereasanInrvalueispassedtoacorecursivecall.Ingeneral,ifthecodatatypehasnconstructors,thecorecursorexpectsn�1predicatespj:: )boolthataretestedinsequencetodeterminewhichconstructorshouldbeproduced.Additionally,foreachconstructorargumentoftype ,oneofthefollowingcasesapplies:•If doesnotcontain(oranyofitsmutuallycorecursivetypes),nocorecursionispossibleandthebehavioralfunctionissimplyafunctiong:: ) thatreturnstheconstructorargument'sunconditionalvalue.•If 'soutermosttypeconstructoris(oramutuallyrecursivetype),thisconstruc-torargumentallowsdirect(ormutual)corecursion.Thecorecursorexpectsthreebehavioralfunctions:s:: )bool(“stop?”),e:: ) (“end”),andc:: ) (“continue”).•If nestsunderoneormoreBNFs,thecorecursortakesoneargumentthatreturnsavalueof 'snestingtypewrappedaround+ .Thesumtyperepresentseitheranoncorecursiveconstantresultoracorecursivecall'sargument. ConstructorViewfromDestructorView.Essentially,thisjustusesthereductiondescribedinSection7.2inreverse.Eachconstructoranditsassociateddiscriminatorformulapremisesandselectorequationright-handsidesarecollectedandcombinedtoformaconstructorview.However,noequationisgeneratedifsomeselectorequationsaremissingforagivenconstructor(e.g.,iftheheadofalistisspeciedbutnotthetail).CodeViewfromConstructorView.Analogously,thisstepreversesthereductioninSection7.2.Ittakestheconstructor-viewright-handsidesalongwiththeirpreconditionsandbuildsan`if–then–elseif'treefromthem.Iftheexhaustiveoptionisnotspecied,thegeneratedtheoremwillhavean`else'branchcontainingCode.abort,whichthrowsanexceptionatruntimeifnoneofthecasesapply.Eitherway,thenewlyassembledtermsareprovedastheorems,exploitingtheconstructor-viewtheorems.7.4GeneralProcedureThegeneralprocedureimplementedbyprimcorecperformsthefollowingsteps:1.Usethefunctions'typestoquerytheinvolvedconstructors,discriminators,andselectorsfromthecodatatypedatabase.2.Fromeachsuppliedformula'sstructure,determinethekindofformula.Forcon-structororcodeequations,calltherespectivereductionfunctionstoextractthesameinternal,destructor-basedrepresentationfromallthreeinputsyntaxes.Duringthisstep,theinterpretationof`_'wildcardsandthesequentialoptionareperformedandanyimplicitdiscriminatorformulasaregenerated.Afterthisstep,foreach(speciedorgenerated)formula,wehave:•thefunction'sname,type,anditsarguments'namesandtypesasitoccuredinthisparticularterm;•theconstructorthatthisformulaisrelevantto;•theoriginaluserinput,andpossibly—ifthisformulawasobtainedbyreducingfromadifferentview—thereduction'spreimage.Thespeciceldsfordiscriminatorformulasandselectorequationsarealistofpremisesandtheright-handside.3.Theselectorequationright-handsidesarescannedforcorecursivecallsandtheirstructureisrecorded.4.Usingthisnewinformation,gettherestofthecodatatypes'information(core-cursors,theorems,typesofcorecursion,etc.)fromthecodatatypedatabase.Thenested-to-mutualreductionisperformedinthisstepifnecessary.5.Denitionsforthespeciedfunctionsareobtained.Thisinvolvestranslatingselec-torequationright-handsidestobehavioralfunctions.6.Anyexclusivenessandexhaustivenesspropertiesareassembled.7.Fromthedenitions,exclusivenessandexhaustivenesstheorems,andcodatatype-andcorecursor-relatedtheorems,provethefunctions'characteristictheoremsinallofthesyntaxstyles. Liketherestofthe(co)datatypepackage,thecommandsarefullydenitional:Theyanalyzethespecicationsenteredbytheuser,synthesizedenitionsintermsofinter-nal(co)recursors,andgeneratecharacteristictheorems,allofthiswithoutintroducingaxiomsorextendingthelogic.Nestedrecursionishandledtrulymodularly.Nonetheless,theolderapproachofreducingnestedrecursiontomutualrecursionisalsosupported.Thetwoapproachescanbearbitrarilycombinedforbothrecursiveandcorecursivefunctions.Internally,suitable(co)recursorsand(co)inductionprinciplesarederivedtomakethispossible.Forfuturework,weareinterestedinstrongerformsofcorecursionaswellasmixedrecursive–corecursivedenitions.Wehavesomeideasalready,butweneedasolidtheoreticalfoundationsothatIsabelle'sinferencekernelcanacceptthedenitions.Acknowledgment.TobiasNipkowmadethisworkpossiblebyallowingtheseniorauthorstoneglecttheirdutieswhiletheydesignedandimplementedthenew(co)datatypepackage.AndreiPopescumastermindedthemetatheorybehindthepackage,withoutwhichtherewouldbeno(co)datatypes,no(co)recursors,andhenceno(co)recursivefunctiondenitions.FlorianHaft-mann,ChristianUrban,andMakariusWenzelprovidedgeneraladviceonIsabelleandpackagewriting.JohannesHölzlandAndreasLochbihlergavepreciousfeedbackattheearlystagesofprimcorec'simplementation.BlanchetteissupportedbytheDeutscheForschungsgemeinschaft(DFG)projectHardeningtheHammer(grantNi491/14-1).TraytelissupportedbytheDFGprogramProgramandModelAnalysis(PUMA,doctorateprogram1480).Theseniorauthorsarelistedalphabeticallyregard-lessofindividualcontributionsorrelativeseniority.References1.Berghofer,S.,Wenzel,M.:InductivedatatypesinHOL—Lessonslearnedinformal-logicen-gineering.In:Bertot,Y.,Dowek,G.,Hirschowitz,A.,Paulin,C.,Théry,L.(eds.)TPHOLs'99.LNCS,vol.1690,pp.19–36.Springer(1999)2.Blanchette,J.C.,Hölzl,J.,Lochbihler,A.,Panny,L.,Popescu,A.,Traytel,D.:Trulymodular(co)datatypesforIsabelle/HOL.In:Klein,G.,Gamboa,R.(eds.)ITP2014.LNCS,Springer(2014)3.Blanchette,J.C.,Panny,L.,Popescu,A.,Traytel,D.:Dening(co)datatypesinIsabelle/HOL.http://isabelle.in.tum.de/dist/Isabelle/doc/datatypes.pdf(2013)4.Krauss,A.:Recursivedenitionsofmonadicfunctions.EPTCS43,1–13(2010)5.Krauss,A.:DeningrecursivefunctionsinIsabelle/HOL.http://isabelle.in.tum.de/doc/functions.pdf(2013)6.Lochbihler,A.,Hölzl,J.:Recursivefunctionsonlazylistsviadomainsandtopologies.In:Klein,G.,Gamboa,R.(eds.)ITP2014.LNCS,Springer(2014)7.Panny,L.:Primitively(Co)recursiveFunctionDenitionsforIsabelle/HOL.B.Sc.thesisdraft,TechnischeUniversitätMünchen(2014)8.Traytel,D.,Popescu,A.,Blanchette,J.C.:Foundational,compositional(co)datatypesforhigher-orderlogic:Categorytheoryappliedtotheoremproving.In:LICS2012,pp.596–605.IEEE(2012)