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Confluence of Conditional Term Rewrite Systems via Transformations KarlGmeiner DepartmentofComputerScience,UASTechnikumWien,Austria [email protected] Conditionaltermrewritingisanintuitiveyetcomplexextensionoftermrewriting. Inordertoben- efit from the simpler framework of unconditional rewriting, transformations have been defined to eliminatetheconditionsofconditionaltermrewritesystems. Recent results provide confluence criteria for conditional term rewrite systems via transformations,yettheyarerestrictedtoCTRSswithcertainsyntacticpropertieslikeweakleft-linearity.These syntacticpropertiesimplythatthetransformationsaresoundforthegivenCTRS. Thispapershowshowtousetransformationstoproveconfluenceofoperationallyterminating, right-stabledeterministicconditionaltermrewritesystemswithoutthenecessityofsoundnessrestric- tions. For this purpose, it is shown that certain rewrite strategies, in particular almost U-eagerness andinnermostrewriting,alwaysimplysoundness. 1 Introduction 1.1 BackgroundandMotivation Conditionaltermrewritesystems(CTRSs)aretermrewritesystemsinwhichrewriterulesmaybebound to certain conditions. Such systems are a widely accepted extension of unconditional term rewrite systems(TRSs)thathasbeeninvestigatedfordecadesbuttheyaremorecomplexthanunconditionalTRSs. Severalpropertiesofunconditionalrewritingarenotsatisfiedanymoreorchangetheirintuitivemeaning andmanycriteriaforTRSscannotbeapplied. Thus,therehavebeeneffortstodeveloptransformations thatmapCTRSsintounconditionalTRSs,forinstancein[4,6,12,22,1]. TransformationsaresupposedtosimplifytheoriginalCTRSbyeliminatingtheconditions. Thisway, propertiesoftheCTRScanbeprovedbyusingthesimpler,unconditionalTRS.Yet,forthispurposeone mustensurethattherewriterelationofthetransformedTRSdoesnotgiverisetorewritesequencesthat arenotpossibleintheoriginalCTRS,apropertycalledsoundness. The aim of this paper is to prove that if the transformed TRS is confluent, then the CTRS is also confluent, without the necessity to also prove soundness. This main result is applicable to right-stable deterministicCTRSsthataretransformedintoterminatingTRSsanditsignificantlyimprovesother,similar confluence results like the ones in [11] and [17] because there are no syntactic restrictions required that imply soundness (in particular weak left-linearity). In fact, it also holds for CTRSs for which the used transformation is unsound. This result leads to a new method to prove confluence of CTRSs that canbeeasilyautomatedanditleavesspaceforfurtherimprovements. 1.2 OverviewandOutline In order to prove properties of CTRSs using transformations, one must prove that the transformation is suitable for the given purpose. [13] introduces the notions of soundness and completeness of a certain H.Cirstea,S.Escobar(Eds.):ThirdInternationalWorkshoponRewriting TechniquesforProgramTransformationsandEvaluation(WPTE’16). EPTCS235,2017,pp.32–45,doi:10.4204/EPTCS.235.3 K.Gmeiner 33 classoftransformations,so-calledunravelings. Informally,soundnessmeansthatifthetransformedTRS givesrisetoarewritesequenceinthetransformedTRSthenthisrewritesequenceisalsopossibleinthe originalCTRS.Completenessistheoppositeofsoundness,i.e.thatarewritesequenceintheCTRSalso existsinthetransformedTRS. Completeness is usually implied by the structure of transformations but soundness is more difficult toproveandnotsatisfiedingeneral. Yet,soundnessisneededtoprovepropertieslikenon-terminationor confluence. In many papers it is proved that certain syntactic properties like (weak) left-linearity imply soundnessforacertaintransformation(seee.g.[13]). Soundness and confluence of the transformed system implies confluence of the original CTRS (see e.g. [11]), yet there is not yet a positive or a negative result whether soundness is essential (although confluenceofthetransformedCTRSdoesnotimplysoundnesswhichwasshownin[9]). Thispaperwill answerthisquestionbyfirstshowingthatinnermostderivationsarealwayssoundandthenshowthatthis infactimpliesconfluenceifthetransformedTRSisterminating. The following section recalls some basics and notions of (conditional) term rewriting. Section 3 introduces the most common unravelings of CTRSs. In Section 4 a rewrite strategy called almost U- eager derivations is introduced and it is proved that it implies soundness. Based on this, further results areshown,inparticularsoundnessofinnermostrewritesequences. TheseresultsareusedinSection5to proveconfluenceofCTRSs. Finally, theresultsaresummarizedandsimilarresultsintheliteratureand possibleperspectivesarediscussed. 2 Preliminaries This paper follows basic notions and notations as they are defined in [3] and [19]. Basic knowledge of (conditional)termrewritingisassumed. Somelesscommonnotionsarerecalledinthefollowing. ThesetofalltermsoverasignatureF andaninfinitebutcountablesetofvariablesV isdenotedas T (F,V). InthefollowingT isusedifF andV areclearfromcontext. Thesetofvariablesinaterm sisVar(s). ForasetofvariablesX,(cid:126)X denotesthesequenceofvariablesinX insomearbitrarybutfixed order. ThesetofpositionsofatermsisdenotedasPos(s),s| isthesubtermofsatposition pands[t] p p represents the term s after inserting the term t at position p. If p≤q (p<q), then q is below (strictly below) p. Otherwiseqisabove p(p≥q)orparallelto p(p(cid:107)q). A substitution σ is a mapping from variables to terms that is implicitly extended to terms. In the following, the common postfix notation sσ is used for the term s with the substitution σ applied. This notationisextendedtosubstitutions,i.e.στ correspondstoστ(x)=τ(σ(x)). Arewriteruleα isapairoftwoterms(l,r),denotedasl→r,whereVar(r)⊆Var(l). Atermrewrite system(TRS)isapairR =(F,R)ofasignatureandasetofrules. Inthefollowing,thesignaturewill oftenbeleftimplicitandslightlyabusingnotationR willbeusedinsteadofR. Arewritestepfromatermstoatermt ataposition pusingaruleα isdenotedass→p,α,R t. Some labels are skipped if they are clear from context or irrelevant. A single rewrite step is written as →, the transitive closure is →+, the reflexive and transitive closure is →∗. ← (←∗) is the inverse of → (→∗) and↔(↔∗)is←∪→((←∪→)∗). Arewritesequenceu→∗ vinsomeTRSR isnormalizingifvis R anormalforminR. Thesetofone-stepdescendantsq\Aofapositionqinatermsw.r.t.therewritestepA:s→ t is p,l→r 34 ConfluenceofCTRSsviaTransformations thesetofpositions  {q} ifq≤ por p(cid:107)q   q\A= (cid:8)p.q(cid:48).q(cid:48)(cid:48)|r| =l| (cid:9) ifl| isavariableandq= p.p(cid:48).q(cid:48)(cid:48) q(cid:48) p(cid:48) p(cid:48)  0/ otherwise Theone-stepdescendantrelationisdefinedas{(p,q)| p∈q\A}. Thedescendantrelationisthereflex- ive, transitive closure of the one-step descendant relation, extended to rewrite sequences. The ancestor relation is the inverse of the descendant relation. By slight abuse of terminology a term t| will be q(cid:48) referredtoasthe(one-step)descendantofaterms| ifq(cid:48) isa(one-step)descendantofq.1 q A conditional rule is a triple (l,r,c), usually denoted as l →r ⇐c where l,r are terms and c is a conjunctionofequationss =t ,...,s =t . Inthispaperweonlyconsiderorientedconditionalrulesin 1 1 k k which equality is defined as reducibility →∗. A conditional term rewrite system (CTRS) R over some signatureF consistsofconditionalrules. TheunderlyingTRSR containstheunconditionalpartofthe u conditionalrulesR ={l→r|l→r⇐c∈R}. u LetR bethefollowingTRSs: n R =0/ 0 R =(cid:8)lσ →rσ |l→r⇐c∈R andsσ →∗ tσ foralls→∗t ∈c(cid:9) n+1 R n ACTRSR givesrisetotherewritestepu→R vifthereisannsuchthatu→R v. Theminimalsuchn n isthedepthoftherewritestep. A conditional rule is of type 1 if there are no extra variables (Var(r)∪Var(c)⊆Var(l)). It is of type 3 if all extra variables occur in the conditions (Var(r)⊆Var(c)∪Var(l)). A normal conditional ruleisanoriented1-ruleinwhichforeveryconditions →∗t (i∈{1,...,k}),t isagroundnormalform i i i w.r.t.R . Adeterministicconditionalruleisanoriented3-rulel →r⇐s →∗t ,...,s →∗t suchthat u 1 1 k k Var(s)⊆Var(l,t ,...,t )foralli∈{1,...,k}. ACTRSisadeterministicCTRS(DCTRS)ifallrules i 1 i−1 aredeterministicconditionalrules. ACTRSisright-stableifforallconditionalrulesl →r⇐s →∗t ,...,s →∗t ,t iseitheralinear 1 1 k k i constructortermoragroundirreducibleterm(w.r.t.R ),andVar(t)∩Var(l,s ,t ,...,s ,t ,s)=0/ u i 1 1 i−1 i−1 i foralli∈{1,...,k}. Inthefollowingonlyright-stableDCTRSsareconsidered. 3 Unravelings Unravelings are a simple class of transformations from CTRSs to TRSs that was introduced in [13]. In the same paper Marchiori also introduces multiple specific unravelings, in particular the simultaneous unraveling U for normal 1-CTRSs. This unraveling splits a conditional rule α :l →r ⇐s →∗ sim 1 t ,...,s →∗t intotwounconditionalrules: 1 k k −−−−→ l→Uα(s ,...,s ,Var(l)) 1 k −−−−→ Uα(t ,...,t ,Var(l))→r 1 k The sequential unraveling that was introduced in [18] (a similar unraveling was already defined in [14])extendsthisapproachtoDCTRSs. 1 Fromthisdefinitionitfollowsthatt|p isaone-stepdescendantofs|p inarewritesteps→pt. Thiscaseissometimes excludedfromthedescendantrelation. K.Gmeiner 35 Definition1(sequentialunravelingU [18]). Givenadeterministicconditionalruleα :l→r⇐s →∗ seq 1 t ,...,s →∗t ,U translatestheruleintoasetofunconditionalrules: 1 k k seq  l→Uα(s ,X(cid:126) ) (introductionrule)  1 1 1       Uα(t ,X(cid:126) )→Uα(s ,X(cid:126) ) (switchrule)   1 .1 1 .2 2 2 .  Useq(α)= .. .. .. Ukα−1(tk−1,X(cid:126)k−1)→Ukα(sk,X(cid:126)k) (switchrule)       Uα(t ,X(cid:126) )→r (eliminationrule)  k k k where X = Var(l,t ,...,t ). For an unconditional rule α, U (α) = {α}. The unraveled CTRS i 1 i−1 seq Useq(R)thenisdefinedas(cid:83)α∈RUseq(α). In the following, U (F) denotes the signature of the unraveled TRS U (R). The new function seq seq symbols U (F)\F are U-symbols. Terms rooted by a U-symbol are U-terms. A terms s is a mixed seq term if it contains U-terms (s∈T (U (F),V), short U (T )), otherwise it is an original term (s∈ seq seq T ). In U-terms of some U (R), the first argument encodes the conditional argument while the other seq variableargumentscontainthevariablebindings. A rewrite step in the transformed TRS in which an introduction (switch/elimination) rule is applied isanintroductionstep(switchstep/eliminationstep). According to the original definition an unraveling U is complete, i.e. u→∗ v implies u→∗ v. It R U(R) issoundifu→∗ vimpliesu→∗ vforallu,v∈T . U(R) R TheunravelingU encodesall variablebindingsin itsU-termseven iftheyare notused anymore. seq In[5]thevariablebindingsareoptimized,leadingtotheoptimizedsequentialunravelingU ([15]). In opt this unraveling variables are not encoded if they are not required in a later condition or the right-hand sideoftheconditionalrule: (cid:110) (cid:111) U (α)= l→Uα(s ,X(cid:126) ),Uα(t ,X(cid:126) )→Uα(s ,X(cid:126) ),...,Uα(t ,X(cid:126) )→r opt 1 1 1 1 1 1 2 2 2 k k k whereX =Var(l,t ,...,t )∩Var(t ,s ...,s ,t ,r). i 1 i−1 i+1 i+2 k k Optimizingthevariablebindingsinunravelingshasadvantagesinsomecasesbecauselesstermshave to be considered in proofs. In [7] several soundness results for U are shown, in particular soundness opt for U-eager rewrite sequences. Formally, a derivation u → u → ··· → u in some U(R) is 0 p0 1 p1 pn−1 n U-eagerifU-termsareimmediatelyrewritten,i.e., p≤ p forallU-termsu| . i i p Yet, this optimization has some drawbacks. For instance, two terms that are not joinable in the original CTRS rewrite to the same mixed term because a variable binding is erased. Because of this phenomenon,themainresultofthispaperdoesnotholdforU . opt Example 2 (unsoundness for confluence of the optimized unraveling). Consider the following DCTRS anditstransformedterminatingTRSusingtheoptimizedunraveling:   a→s(b)    a→s(b)   (cid:38)       (cid:38)   s(c)  R = s(c) Uopt(R)=    s(x)→Uα(B) s(x)→A⇐B→∗C  1  Uα(C)→A  1 U (R)isconfluentsincetheonlycriticalpair(cid:104)s(b),s(c)(cid:105)givesrisetothecommonreductUα(B) opt 1 andthetransformedTRSisterminating. Yet, R isnotconfluentbecausearewritestos(b)ands(c)but theconditionoftheconditionalruleisneversatisfiedsothats(b)ands(c)areirreducible. 36 ConfluenceofCTRSsviaTransformations Since U preserves all variable bindings of the left-hand side of a conditional rule it is possible to seq extract these bindings and insert them into the corresponding left-hand side, thus obtaining the back- translationtb(definedin[8],similarmappingsareusedintheproofsin[13]and[19]). Definition 3 (back-translation tb). Let R =(F,R) be a CTRS, then tb:U (T )(cid:55)→T is defined as seq follows:  s ifsisavariable   f(tb(s ),...,tb(s )) ifs= f(s ,...,s )and f ∈F 1 k 1 k tb(s)= lσ ifs=Uiα(w,v1,...,vm)and   α =l→r⇐s →∗t ,...s →∗t 1 1 k k −−−−−−−−−−−−→ whereσ isdefinedasxσ =tb(v)whereVar(l,t ,...,t )=x ,...,x . i i 1 i−1 1 m Inthefollowingtbwillsometimesbeextendedtosubstitutions(xtb(σ)=tb(xσ)forx∈Dom(σ)). The back translation allows us to define soundness such that it also extends to mixed terms: A rewrite sequenceu→∗ (R)v(u∈T )issoundifu→∗ tb(v). U R seq 4 Soundness and Completeness of Transformations ThetransformationU isnotsoundforDCTRSsingeneral. ThiswasfirstshownbyMarchioriin[13] seq using a normal 1-CTRS that consists of multiple non-linear rules. For DCTRSs we presented another examplein[9]. Example4(unsoundness[9]). ConsiderthefollowingDCTRSanditsunraveling   a→c     a→c  (cid:37)(cid:38)       (cid:37)(cid:38)   b→d   b→d           s(c)→t(k)       s(c)→t(k)   (cid:38)  R = (cid:38) Useq(R)= t(l)  t(l)           g(x,x)→h(x,x)  g(x,x)→h(x,x)     f(x)→(cid:104)x,y(cid:105)⇐s(x)→∗t(y)  f(x)→U1α(s(x),x) Uα(t(y),x)→(cid:104)x,y(cid:105)  1 InU (R),thereisthefollowingreductionsequence: seq g(f(a),f(b))→∗g(Uα(s(c),d),Uα(s(c),d))→h(Uα(s(c),d),Uα(s(c),d))→∗h((cid:104)d,k(cid:105),(cid:104)d,l(cid:105)) 1 1 1 1 Yet, this derivation is not possible in R because there is no common reduct of f(a) and f(b) that rewritestoboth,(cid:104)d,k(cid:105)and(cid:104)d,l(cid:105). TheCTRSsofExample4andthecounterexampleof[13]aresyntacticallyverycomplex. Basedon thisobservationitwasshownthatmanysyntacticpropertiesimplysoundness: Left-linearity(normal1- CTRSs[13]/[19],DCTRSs[16]),weakleft-linearity,right-linearity(normal1-CTRSs[8],DCTRSs[9]), non-erasingness (normal 1-CTRSs [8], 2-DCTRSs [9], counterexample for 3-DCTRSs [9]) and weak right-linearity(DCTRSs[7]). The CTRS of Example 4 is not confluent and in [8] it is shown that the simultaneous unraveling is soundforconfluentnormal1-CTRSs. Yet,thisresultdoesnotholdforDCTRSs: K.Gmeiner 37 Example 5 (unsoundness for confluence [9]). Let R be the CTRS of Example 4 and R(cid:48) be the CTRS consistingoftheunconditionalrules R(cid:48)={c→e←d,k→e←l,s(e)→t(e)} Then,R∪R(cid:48) andU (R∪R(cid:48))areconfluent,yet,theargumentofExample4stillholdssothatthe seq reduction sequenceg(f(a),f(b))→∗ h((cid:104)d,k(cid:105),(cid:104)d,l(cid:105))in thetransformed TRS isstill unsound. Nonethe- less,thelasttermoftheunsoundderivationcanbefurtherreducedtotheirreducibletermh((cid:104)e,e(cid:105),(cid:104)e,e(cid:105)). Thederivationg(f(a),f(b))→∗h((cid:104)e,e(cid:105),(cid:104)e,e(cid:105))issound. Although the previous example shows that confluence of the transformed TRS is not sufficient for soundness, it also shows that (in contrast to Example 4) the last term of the unsound derivation can be furtherreduced. Infact,allnormalizingderivationsinconfluentDCTRSsaresound[9]. Theoriginaldefinitionofunravelingsin[13]statesthatanunravelingmustbecompleteandpreserve theoriginalsignature. Basedonthedefinitionoftheunravelingsitisnotsurprisingthatcompletenessis satisfied in all cases. In the following the proof of [13] is adapted to U . The proof will be useful to seq motivatearewritestrategythatimpliessoundness. Lemma 6 (completeness of Useq). Let R be an oriented CTRS and s,t be two terms such that s→R t, thens→+ t. Useq(R) Proof. By induction on the depth n of the rewrite step s→n,R t. If n=0, then the applied rule α is an unconditionalruleandα ∈U (R). seq Otherwise, let α :l →r ⇐s1 →∗ t1,...,sk →∗ tk be the rule applied in s→n,R t so that s=C[lσ] and t =C[rσ]. By the definition of the depth, sσ →∗ tσ for all i ∈ {1,...,k}. By the induction i R i hypothesis,therearederivationssσ →∗ tσ. Thus,nt−h1ereisthefollowingderivationinU (R): i Useq(R) i seq lσ →Uα(s σ,(cid:126)X σ)→∗Uα(t σ,(cid:126)X σ)→Uα(s σ,(cid:126)X σ)→∗Uα(t σ,(cid:126)X σ)→··· 1 1 1 1 1 1 2 2 2 2 2 2 →Uα(s σ,(cid:126)X σ)→∗Uα(t σ,(cid:126)X σ)→rσ 1 k k k k k The previous completeness result constructs a derivation in U (R) in which first the U-term is seq introduced,thentheconditionalargumentisrewrittenandfinallytheU-termiseliminated. Thedefinition oftheU-eagerrewritestrategyisbasedonsuchderivationsbutitalsoallowsrewritestepsinsidevariable bindings. InU-eagerderivations,afteraU-termisintroducedonlyrewritestepsinsidethisU-termareallowed until it is eliminated. Rewrite steps outside of U-terms are forbidden. The reason for this limitation is thatinaderivationinsomeU (R)oneobtainsmixedtermsthathavenomeaningintheoriginalCTRS. opt Forinstance,inExample2themixedtermUα(B)isacommonreductofs(b)ands(c). Yet,thereisno 1 suchtermintheoriginalCTRS. ForU ,mixedtermscanbeback-translatedtotheleft-handsideoftheconditionalrulebecauseall seq variable bindings are preserved. Therefore, U-eagerness for some U (R) can be generalized to also seq allow rewrite steps outside of U-terms even if they are not eliminated. In such almost U-eager rewrite sequences if a U-term is not rewritten it is considered to represent a failed conditional evaluation and thustheargumentsofsuchaU-termandtheU-termitselfmustnotberewrittenanymore. Rewritesteps abovesuchU-terms,includingerasingrewritesteps,areallowed. 38 ConfluenceofCTRSsviaTransformations Definition7(almostU-eagerderivations). LetR beaDCTRS.AderivationD:u → u → ···→ 0 p0 1 p1 pn−1 u inU (R)isalmostU-eager,ifforeveryrewritestepu → u ,ifthereisaU-termu| suchthat n seq i pi i+1 i q q≤ p,thenalsoq≤ p (i∈{1,...,n−1}). i i−1 This way, rewrite steps in U-terms are always grouped in such derivations which makes tracking termseasier. Furthermore,U-termsthatrepresentintermediateevaluationstepsofconditionsareisolated fromotherrewritesteps. RewritestepsaboveU-termsthatarerewritteninalaterrewritesteparevitally importantforunsoundness. ObservethattheunsoundderivationofExample4isnotalmostU-eagerand thatintheunsoundderivationanon-linearrewritestepisappliedaboveaU-term. The proof for soundness of almost U-eager derivations will use the same proof structure that was already used in [9]. First, we recall the following lemma that states that rewrite steps in variable and conditionalargumentscanbeextractedfromderivations. Lemma 8 (extraction lemma of [9]). Let R be a DCTRS and D : u → u → ··· → u be a 0 p0 1 p1 pn−1 n derivationinU (R)(u ∈T ). Ifu | =Uα(w,(cid:126)Xσ )whereα istheconditionalrulel→r⇐s →∗ seq 0 n p i i i+1 1 t ,...,s →∗t ,thenthereisanindexmandapositionqsuchthatu | isanancestorofu | andthere 1 k k m q n p aresubstitutionsσ ,...,σ suchthatu | =lσ andthefollowingderivationscanbeextractedfromD: 1 i m q 1 • s σ →∗ t σ (j∈{1,...,i−1}), j j Useq(R) j j+1 • xσ →∗ xσ (j∈{1,...,i},x∈X ),and j Useq(R) j+1 j • sσ →∗ w. i i Useq(R) Furthermore, in the reductions above for every single rewrite step u → v there is an index m(cid:48) ∈ {m+1,...,n−1}andapositionq(cid:48) suchthatu | =uandu | =v. m(cid:48) q(cid:48) m(cid:48)+1 q(cid:48) Inthefollowingthisextractionlemmawillbeusedimplicitly. Next,amonotonyresultontbisshown. Lemma 9 (monotony of tb). Let R be a DCTRS. If u→p,Useq(R) v for u,v∈Useq(T ) and tb(u|p)→∗R tb(v| )thentb(u| )→∗ tb(v| )forallq∈Pos(u)anddescendantsq(cid:48) ofq. p q R q(cid:48) Proof. Bycasedistinctionon pandq: If p<qor p(cid:107)q,thenu| =v| ,hencealsotb(u| )=tb(v| ). q q(cid:48) q q(cid:48) Otherwise, if q≤ p, then there is only one descendant of u| which is v| . Let q.q(cid:48) = p. Then by q q inductionon|q(cid:48)|,ifq= pthentb(u| )→∗ tb(v| )isequivalenttotheassumptiontb(u| )→∗ tb(v| ). q R q(cid:48) p R p For the induction step, let q(cid:48) =i.q(cid:48)(cid:48). There are the following cases based on the term u| : If u| = q q f(u ,...,u ) where f ∈ F is an original symbol, then tb(u| ) = f(tb(u ),...,tb(u )) and tb(v| ) = 1 n q 1 n q f(tb(u ),tb(u ),tb(v| ),tb(u ),...,tb(u )). By the induction hypothesis tb(u)→∗ tb(v| ), thus 1 i−1 q.i i+1 n i q.i alsotb(u| )→∗tb(v| ). q q Theremainingcaseisthatu| isaU-termUα(w,x ,...x )σ. Ifi=1,thentherewritestepisinside q j 1 n theconditionalargumentsothatthevariablebindingsareunmodifiedandtb(u| )=tb(v| ). Otherwise, q q v| =Uα(w,x ,...x )σ(cid:48)wherex σ =x σ(cid:48)forall j∈{1,...,i−2,i,...,n}. Bytheinductionhypothesis, q j 1 n j j tb(x σ)→∗tb(x σ(cid:48)). Hence,tb(u| )=ltb(σ)wheretb(v| )=ltb(σ(cid:48))andthustb(u| )→∗tb(v| ). i−1 xi−1 q q q q Inthenextlemma,singlerewritestepsofaderivationaretranslatedusingtb. Lemma 10 (technical key lemma). Let R be a right-stable DCTRS and let u → u → ···→ 0 p0 1 p1 pn−1 u be an almost U-eager derivation in U (R) where u ∈ T . Then, tb(u| ) →∗ tb(u | ) (i ∈ n seq 0 i pi R i+1 pi {0,...,n−1}). K.Gmeiner 39 Proof. In the following, assume w.l.o.g. that for all substitutions, mapped terms do not share variables withthedomain,i.e.,Dom(σ)∩Var(xσ)=0/ forallx∈Dom(σ). Byinductiononthelengthofthederivationn: Ifn=0,theresultholdsvacuously. Otherwise tb(u| )→∗ tb(u | ) for all one-step descendants u | of u| by the induction hy- i q R i+1 q(cid:48) i+1 q(cid:48) i q pothesis and Lemma 9. Consequently also tb(u| )→∗ tb(u | ) for all descendants u | of u| (1≤ i q R j q(cid:48)(cid:48) j q(cid:48)(cid:48) i q i< j<n). Bycasedistinctionontheruleappliedinthelastrewritestepu → u : Iftheappliedruleis n−1 α,pn−1 n anunconditionaloriginalrulel→r∈R,thenu | =lσ,u | =rσ,tb(u | )=ltb(σ)and n−1 pn−1 n pn−1 n−1 pn−1 tb(u | )=rtb(σ). n pn−1 Iftheappliedruleisanintroductionruleoraswitchrule,tb(u | )=tb(u | ). n−1 pn−1 n pn−1 Finally, if the applied rule is an elimination rule, then by the definition of almost U-eagerness, all preceding rewrite steps are below p up to the introduction step of the U-term, i.e., if the conditional n−1 ruleisα :l →r⇐s →∗t ,...,s →∗t ,thenthereisanmsuchthatu | =lσ , p = p , p ≤ p 1 1 k k m pm 1 m n−1 m i foralli∈{m,...,n−1}andthederivationu | →∗ u | is m pm Useq(R) n pn−1 lσ →Uα(s σ ,X(cid:126) σ )→∗Uα(t σ ,X(cid:126) σ )→Uα(s σ ,X(cid:126) σ )→∗··· 1 1 1 1 1 1 1 1 2 1 2 2 2 2 2 2 →∗Uα(t σ ,X(cid:126) σ )→rσ k k k+1 k k+1 k+1 Bytheinductionhypothesis,tb(xσ)→∗ tb(xσ )andtb(sσ)→∗ tb(tσ )forallx∈X. i R i+1 i i R i i+1 i Letσ bethecombinedsubstitutiontb(σ )tb(σ )···tb(σ ),thensσ →∗ s tb(σ)andt tb(σ )= 1 2 k+1 i R i i i i+1 tiσ by right-stability. Hence, the conditions are satisfied for σ and lσ →R rσ. Furthermore, lσ =lσ1. Thus,ltb(σ1)→R rσ →∗R rtb(σk+1). FinallyweprovesoundnessofalmostU-eagerrewritesequences. Lemma 11 (soundness of almost U-eager derivations). Let R be a right-stable DCTRS. If u → 0 p0 u ···→ u isanalmostU-eagerderivationinU (R)(u ∈T )thenu →∗ tb(u ). 1 pn−1 n seq 0 0 R n Proof. By induction on the length of the derivation, if n=0 the result holds vacuously. Otherwise, by Lemma 10, tb(u | )→∗ tb(u | ). By Lemma 9, tb(u )→∗ tb(u ). Since by the inductive n−1 pn−1 R n pn−1 n−1 R n hypothesisu →∗ tb(u )finallyu →∗ tb(u ). 0 R n−1 0 R n This result can be used to prove soundness for other rewrite strategies. Next, it is shown that inner- mostderivationscanbeconvertedintoalmostU-eagerderivations,thusprovingsoundnessofinnermost rewriting. Forthisreason,innermostderivationsaretranslatedintoalmostU-eagerderivations. Lemma12(innermosttoalmost-U-eager). LetR beaDCTRSandletu → u → ···→ u bean 0 p0 1 p1 pn−1 n innermostderivation(u ∈T ). Thenthereisaninnermost,almostU-eagerderivationu →∗ u . 0 0 Useq(R) n Proof. By induction on the length n of the derivation. If n=0 the result holds vacuously. Otherwise, bytheinductionhypothesis, thederivationu0 →∗Useq(R) un−2 →pn−2,Useq(R) un−1 isinnermostandalmost U-eager. By case distinction on the last rewrite step u → u : If p is not below a U-term, then n−1 pn−1 n n−1 u →∗ u isalreadyalmostU-eager. Otherwise,thereisaU-termu | andq≤ p . Ifthereare 0 Useq(R) n n−1 q n−1 multiple nested U-terms, let u | be the innermost such U-term. By case distinction on p and q: n−1 q n−2 Thecase p <qisnotpossiblebecauseoftheassumptionthatthederivationisinnermost. n−2 Ifq≤ p ,thenu →∗ u isalmostU-eager. n−2 0 Useq(R) n 40 ConfluenceofCTRSsviaTransformations If q(cid:107) p , then let m be the largest value such that p ,p ,...,p are parallel to q. Since n−2 n−m n−m+1 n−2 u ∈T , m<n. Then, u | =u | and q≤ p . Therefore, the following rewrite sequence in 0 n−m q n−1 q n−m−1 U (R)isinfactU-eager: seq u →∗u → u → u [u | ] → 0 n−m−1 pn−m−1 n−m pn−1 n−m n pn−1 pn−1 pn−m → u [u | ] → u [u | ] → ··· pn−m n−m+1 n pn−1 pn−1 pn−m+1 n−m+2 n pn−1 pn−1 pn−m+2 → u [u | ] → u pn−3 n−2 n pn−1 pn−1 pn−2 n SincealmostU-eagerrewritesequencesaresoundthisimpliessoundness. Lemma 13 (soundness of innermost derivations). Let R be a right-stable DCTRS. Let u→∗ v be Useq(R) aninnermostderivationsuchthatu∈T . Then,u→∗ tb(v). R Proof. By Lemma 12, there is an almost U-eager derivation u→∗ v. By Lemma 11, u→∗ tb(v). Useq(R) R Theorem 14 (soundness of innermost derivations). U is sound for innermost derivations for right- seq stableDCTRSs. Proof. ByLemma13,ifu→∗ visaninnermostderivation, thenthereisaderivationu→∗ tb(v). Useq(R) R Innermostderivationsarethereforesound. Nonetheless,innermostrewritingisnotsuitabletosimu- lateconditionalrewritingingeneralbecausetheyarenotcomplete. ThiscanbeeasilyseeninCTRSsin whichtheconditionsaresatisfiablebutnotinnermost-satisfiable. Example 15 (incompleteness of innermost rewriting). Consider the following CTRS and its unraveled TRS:  a→b    a→b          f(a)→b  R = f(a)→b U (R)= seq A→Uα(f(a))  A→B⇐ f(a)→∗b  1  Uα(b)→B  1 In R, the condition f(a)→∗ b is satisfied (although there is no innermost derivation f(a)→∗ b), R therefore,ArewritestoB. Thisderivationisinnermost(yet,noticethattheconditionalevaluationisnot). Nonetheless, in U (R) the only innermost derivation starting from A is A→Uα(f(a))→Uα(f(b)) seq 1 1 wherethelasttermisirreducible. Inparticular,thereisnoinnermostderivationforA→∗ B. Useq(R) Nonetheless,weobtaincompletenessifthetransformedTRSisconfluentandterminating: Proposition 16 (completeness for innermost rewriting). Let R be a right-stable DCTRS such that U (R) is confluent and terminating. Then, if u→∗ v (u,v∈T ) such that v is irreducible (w.r.t. R), seq R thenthereisaninnermostderivationu→∗ v(cid:48) suchthattb(v(cid:48))=v. Useq(R) Proof. Because of completeness of U , there is a derivation u→∗ v. By confluence and termi- seq Useq(R) nation there is a unique normal form w ∈ U (T ) of u and v in U (R) and there is an innermost seq seq derivationu→∗ w. Useq(R) Finally, the assumption that v is a normal form in R and Lemma 13 imply that tb(v)=w(cid:48) for all w(cid:48)∈U (T )suchthatv→∗ w(cid:48). seq Useq(R) K.Gmeiner 41 Next, we prove soundness for DCTRSs that are transformed into confluent and terminating TRSs. For this purpose, observe that if a TRS is confluent and terminating, then for every derivation u→∗ v suchthatvisanormalformthereisaninnermostderivationu→∗v. Thisobservationcanbecombined withTheorem14thatstatesthatinnermost,normalizingrewritesequencesinsomeright-stableDCTRS aresound: Lemma 17 (soundness for confluent and terminating TRSs). Let R be a right-stable DCTRS such that U (R) is terminating and confluent and let u→∗ v be a normalizing rewrite sequence (u∈T ). seq Useq(R) Then,u→∗ tb(v). R Proof. U (R) is terminating and confluent, and v is a normal form in the derivation u →∗ v. seq Useq(R) Therefore,thereisaninnermostderivationu→∗ v. ByLemma13thisimpliesu→∗ tb(v). Useq(R) R Theorem 18 (soundness for normalizing rewrite sequences). Let R be a right-stable DCTRS such that U (R)isconfluentandterminating. ThenU issoundforreductionstonormalforms. seq seq Proof. StraightforwardfromLemma17. The previous theorem is interesting because it shows that [9, Theorem 9] (soundness for reductions tonormalformsofconfluentDCTRSs),alsoholdsifonlythetransformedTRSisknowntobeconfluent. 5 Confluence of Conditional Term Rewrite Systems OurgoalistoprovethatifU (R)isconfluent,thenalsoR isconfluent. Forthispurposeweintroduce seq anothersoundnessproperty,soundnessforjoinability. Definition 19 (soundness for joinability). An unraveling U is sound for joinability for a CTRS R if for alltermsu,v∈T suchthatu↓U(R)valsou↓R v. Soundness for joinability is important in connection with confluence because it allows us to prove confluenceofaDCTRSviaconfluenceofthetransformedTRS. Thereisanimportantconnectionbetweensoundnessforjoinabilityandconfluence. Lemma20(soundnessforjoinabilityandconfluence). LetR beaCTRSsuchthatU (R)isconfluent seq andU issoundforjoinability,thenR isconfluent. seq Proof. Considertwotermsu,v∈T suchthatu↔∗ v. SinceU iscompletebyLemma6,u↔∗ v. R seq Useq(R) Useq(R)isconfluentsothatu↓Useq(R)v. Bysoundnessforjoinabilitythisimpliesu↓R v. Itremainstoprovesoundnessforjoinabilityofright-stableDCTRSsforwhichthetransformedTRS isconfluent. Theorem18showsthatconfluenceandterminationofthetransformedTRSimplessound- nessfornormalizingderivations. Sinceeverytermisterminatingthisimpliessoundnessforjoinability: Lemma 21 (soundness for joinability). Let R be a right-stable DCTRS such that U (R) is confluent seq andterminating,andletu↓Useq(R)v(u,v∈T ),thenu↓R v. Proof. SinceU (R)isconfluentandterminating,u→∗ vimpliesthatthereisanirreducibleterm seq Useq(R) w ∈ U (T ) such that u →∗ w ←∗ v. Since R is right-stable, Lemma 17 implies u →∗ seq Useq(R) Useq(R) R tb(w)←∗ v. R Thusweobtainourmainresult: