Figure1:ThestartofadisproofattemptYourwindowmaynotlookexactlylikegure - PDF document

Figure1:ThestartofadisproofattemptYourwindowmaynotlookexactlylikegure
Figure1:ThestartofadisproofattemptYourwindowmaynotlookexactlylikegure

Figure1:ThestartofadisproofattemptYourwindowmaynotlookexactlylikegure - Description


Figure2Amultiworldsituation Figure3Asingleworldsituation Figure4Asimplecounterexample Figure5Anelaboratecounterexample13Whatsforced150colouringgreyingunderliningJapeevaluatesalltheform ID: 512759 Download Pdf

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Figure2:Amulti-worldsituation Figure3:Asingle-worldsituation Figure4:Asimplecounter-example Figure5:Anelaboratecounter-example1.3What'sforced?–colouring greying underliningJapeevaluatesalltheform

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Figure1:ThestartofadisproofattemptYourwindowmaynotlookexactlylikegure1,butitwillhavethesameparts.Otherillustrationsinthisdocumentshowtheimportantbitsoftheproofand/ordisproofsectionsofthewindow,withoutthewastebinandthescrollbarsandstuff.1.1AlternativesequentsIf,duringaproofattempt,youselectaconclusion(or,ifJapedoesn'twanttoletyoudothat,thereasonnexttotheconclusion)andthenhitEdit:Disprove,Japewillusethedisproofsequentconsistingoftheconclusionformulaandallthelinesaboveitashypothesisformulae.Ifyouselectsomeofthehypothesislines,youcanchoosethecollectionofhypothesisformulaethatappear.Youcanchooseanewdisproofsequentatanytime,eveninthemiddleofadisproofattempt.Ifyouwanttosetupanentirelynewchallenge,enteritintooneoftheconjecturespanels(Newbutton),hitProveandthenEdit:Disproveintheproofwindow.1.2SelectingasituationByselectingaworldyoudenea`situation'comprisingtheworldandalltheworldsyoucanreachaboveit.Therootworldofthesituationisringedinredtoshowthatitisselected.Thereisalwaysexactlyoneselectedworld,andthereforeexactlyonesituation.Figure2showsasituationwithmorethanoneworld;gure3showsasingle-worldsituationinsideamulti-worlddiagram.2 Figure2:Amulti-worldsituation Figure3:Asingle-worldsituation Figure4:Asimplecounter-example Figure5:Anelaboratecounter-example1.3What'sforced?–colouring,greying,underliningJapeevaluatesalltheformulaeandsub-formulaeinthesequentandcoloursthem.Thedefaultcolourisblack,anditmeansnotforced.Ifa(sub-)formula'svalueisnevercalledon–usuallythisistrueofquantiedpredicates,forexample–thenit'sgrey,meaningstrictlyirrrelevant.Anatomic(sub-)formulawhichisforcediscolouredviolet(thiscoversatomslikeE,atomicpredic-ateslikeS(j)andindividualslikeactualk).Whenanon-atomic(sub-)formulaisforced,itsconnectiveiscolouredviolet.Whenanentiresequenthypothesisorconclusionisforced,itisunderlinedinviolet.Sowhenallthepremisesareunderlinedandtheconclusionisnot,youhaveacounter-example;wheneverythingisunderlinedyouhaveanexample;otherwiseyouhavenothinginparticular.Figure4showsasamplecounter-example,andthecolouringgivesahintofwhythatis:Eisn'tforcedanywhere,sothehypothesisimplicationisforced;andthen,becauseE!FisforcedbutGisn't,theconclusionisn'tforced.Figure5showsanelaboratecounter-example,andthecolouringisn'tmuchhelpatall.2 2Ihavesomeideaswhichmighthelpinthiscase,butIhaven'timplementedthemyet.3 Figure1isneitheracounter-examplenoranexample.Thehypothesesaren'tforcedbecause,asatomicformulae,theyaren'tpresentattherootworld;theconclusionisforcedbecauseeveryindividualinthesituation(therearen'tany..)forcesaversionofR(x)!S(x).Similarremarksapplytogures2and3.2MakingdiagramsTobeginyourdisproofJapepresentsyouwiththesimplestdiagram:theisolatedemptyworld.Youaddpartsofyourdiagram–worlds,formulaeandlines–usingdrag-and-dropmousegestures.Youcandeletethesamecomponentsbydraggingthemtothewastebin.Youcanaddtileswithdouble-clicks.YoucanUndoyouractionstoanydegreethatyoulike,andRedolikewise.32.1MakingnewworldsBydraggingonaworldwiththemiddlemousebuttonorbyholdingdownthealtshiftkeywhiledraggingwithasingle-buttonmouse,youdragacopyoftheworldwithalineattachingittotheworlditcamefrom.Thenewworldstayswhereyouputit.Ifit'sabovetheworldyoudraggeditfrom,theywillbeconnectedbyaline.Ifit'slevelorbelow,thennoline.Japepreservesmonotonicity,sothenewworldwillcontainalltheformulaethattheoriginalworlddid.2.2DraggingworldsYoucandragworldsabouttheplacetomakeyourdiagramlooknicer.Youusetheleftmousebuttononamultibuttonmouse.Japewilldeletealineifyoudragaworldbelowitsparent.Youcandragaworldontoaline(whichlightsuptoshowyouareoverit)oranotherworld(whichlightsupditto)andJapedoestheobviousthing,addingwhateverformulaearenecessarytowhateverworldsinordertomaintainmonotonicity.Ifyoudragaworldtothewastebinit'sdeleted,alongwithalltheformulaeattachedtoitandallthelinesconnectedtoit.Thebinlightsupwhenitwillaccepttheworld;itwon'tacceptthecurrently-selectedworld.2.3MakinglinesIfyouwanttomakealinebetweenworldAandworldB,whereAisbelowB,make-world-dragfromAtoBanddropthenewworldontoB.Japemakesthemonotonicityadjustments.DraggingfromBtoAhassomeeffect,butperhapsnotaveryusefuleffect(butthere'salwaysUndo!). 3UndoandRedoapplytothelastpaneyouclicked,eitherproofordisproof.4 Figure6:Acounter-example Figure7:::Eforcedhere Figure8:::E!Eforcedhere Figure9::Eforcedhere Figure10:Multipleforcing Figure11:Toomuchforcing4ExploringreasonsThehardestthingtoexplaintoanoviceiswhyaparticularformulaisforcedinaparticularsituation.Figure6showsaparticularlyludicrousformulaanditscounter-example,whichreallyneedsexplanation.Japeprovidessomemechanismswhichcanbesomehelp(inversionsv6.2.a1.2andlater).First,thecolouringofatomic(sub-)formulaeandconnectivescanhelp.Butwiththenegativeconnectives–!and:–andwiththequantiers,weoftenneedmorehelp.Second,Japeallowsyoutotext-select–alt+press-and-drag,ormiddlebuttonpress-and-drag–asubfor-mula(orselectaformula)inthesequent.Itwillthencolourvioletalltheworldswhichforcetheselected(sub)formula.Forexample,gure7showswhere::Eisforcedinourdifcultexample.Figures8and9showinformationaboutsomeothersubformulae.4.1MultipleselectionsIfyouchoosemorethanoneformula,asingure10,forexample,Japeshowsyouwheretheyareallforced.Ifyouchoosetoomanyformulae,asingure11,youmayndthatthereisnosubsituationwhichforces6 Figure12:Supportforaquantierthemall.Butthat'sexplanationtoo!4.2QuantierreasonsTheworld-colouringtrickworkswell,butitisn'tenoughwhenyouaredealingwithquantiers.Youwouldliketoknowwhichindividualformulaeinthetreesupportthequantier,andwhichdonot.Japecandothat(inversionsv6.2.a1.3andlater)bycolouringindividualsinthetree.Ingure12,forexample,worldswhichforcetheselectedquantierareviolet;individualswhichsupportthequantier(thosewhichgenerateaforcedformulaifyouinstantiatethequantiedformulawiththem)areviolettoo.YoucanworkoutthatR(k)!R(j)@R(j)ateachoftheworldsofgure12.Thisisn'tenoughingeneral.Youwouldliketobeabletoseethoseinstantiationsandplaywiththem–seewhytheyareforced(ornot,asthecasemaybe).IthinkIcanseehowtobuildaJapewhichcoulddothat,butitdoesn'tdoitatpresent.HowtocomplainIfJapedoesn'tworkforyou,itdoesn'twork.It'sbesttosendyourtalesofwoetobugs@jape.org.uk.Onedaythere'llbeamechanismtoletyouseeotherpeople'smessages,butatthemomentit'saprivateconversation.7

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