Abstract. We introduce a new technique for proving termination of. term rewriting systems. The technique, a specialization of Zantema's

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1 Transforming Termination by Self-Labelling Aart Middeldorp, Hitoshi Ohsaki, Hans Zantema 2 Instite of Information Sciences and Electronics University of Tsukuba, Tsukuba 05, Japan 2 Department of Comper Science, Utrecht University P.O. Box 0.0, 50 TB Utrecht, The Netherlands Abstract. We introduce a new technique for proving termination of term rewriting systems. The technique, a specialization of Zantema's semantic labelling technique, is especially useful for establishing the correctness of transformation methods that attempt to prove termination by transforming term rewriting systems into systems whose termination is easier to prove. We apply the technique to modularity, distribion elimination, and currying, resulting in new results, shorter correctness proofs, and a positive solion to an open problem. Introduction Termination is an undecidable property of term rewriting systems. In the literature (Dershowitz [4] contains an early survey of termination techniques) several methods for proving termination are described that are quite successful in practice. We can distinguish roughly two kinds of termination methods:. basic methods like recursive path order and polynomial interpretations that apply directly to a given term rewriting system, and 2. methods that attempt to prove termination by transforming a given term rewriting system into a term rewriting system whose termination is easier to prove, e.g. by a method of the rst kind, and implies termination of the given system. Transformation orders (Bellegarde and Lescanne []), distribion elimination (Zantema []), and semantic labelling (Zantema []) are examples of methods of the second kind. The starting point of the present paper is the observation that semantic labelling is in a sense too powerful. We show that any terminating term rewriting system can be transformed by semantic labelling into a system whose termination can be shown by the recursive path order. The proof of this result gives rise to a new termination method which we name self-labelling. We show that self-labelling is especially useful for proving the correctness of termination methods of the second kind:. Using self-labelling we prove a new modularity result: the extension of any terminating term rewriting system with a terminating recursive program scheme that denes new functions is again terminating.

2 2. Using self-labelling we give a positive solion to an open problem in [] concerning distribion elimination: right-linearity is not necessary for the correctness of distribion elimination in the absence of distribion rules. The proof reveals how to improve distribion elimination in the absence of distribion rules.. Using self-labelling we give a short proof of the main result of Kennaway, Klop, Sleep, and de Vries [0], the correctness of currying, which for the purpose of this paper we view as a termination method of the second kind. The proofs of the above results are remarkably similar. The remainder of this paper is organized as follows. In the next section we recapitulate semantic labelling. In Sect. we show that every terminating term rewriting system can be transformed by semantic labelling into a term rewriting system whose termination is very easy to prove. This completeness result gives rise to the self-labelling technique. In Sect. 4 we obtain a new modularity result with self-labelling. In Sect. 5 we use self-labelling to solve the conjecture concerning distribion elimination. The self-labelling proof gives rise to a stronger result, which we explain in Sect. 6. Our nal illustration of the strength of self-labelling can be found in Sect. 7 where we present a short proof of the preservation of termination under currying. 2 Preliminaries We assume the reader is familiar with the basics of term rewriting (as expounded in [6, ]). This paper deals with the termination property. A term rewriting system (TRS for short) (F; R) is said to be terminating if it doesn't admit innite rewrite sequences. It is well-known that a TRS (F; R) is terminating if and only if there exists a reduction order a well-founded order that is closed under contexts and substitions on T (F; V) that orients the rewrite rules of R from left to right. Another well-known fact states that (! R [ B) + is a wellfounded order on T (F; V) for any terminating TRS (F; R). Here s B t if and only if t is a proper subterm of s. Observe that (! R [ B) + is in general not a reduction order as it lacks closure under contexts. In this paper we make use of the fact that termination (conuence) is preserved under signature extension, which follows from modularity considerations ([4, 6]). In this preliminary section we briey recall the ingredients of semantic labelling (Zantema []). Actually we present a special case which is sucient for our purposes. Let (F; R) be a TRS and A = (A; ff A g f2f ) an F-algebra with non-empty carrier A. Let be a well-founded order on A, write < for the union of and equality. We say that the pair (A; ) is a quasi-model for (F; R) if. the interpretation f A of every n-ary function symbol f 2 F is weakly monotone (with respect to ) in all its n coordinates, i.e., f A (a ; : : : ; a i ; : : : ; a n ) < f A (a ; : : : ; b; : : : ; a n ) for all a ; : : : ; a n ; b 2 A and i 2 f; : : : ; ng with a i b, and 2

3 2. (A; ) and (F; R) are compatible, i.e., [](l) < [](r) for every rewrite rule l! r 2 R and assignment : V! A. Here [] denotes the unique homomorphism from T (F ; V) to A that extends, i.e., (t) if t 2 V, [](t) = f A ([](t ); : : : ; [](t n )) if t = f(t ; : : : ; t n ). The above takes care of the semantical content of semantic labelling. We now describe the labelling part. We label function symbols from F with elements of A. Formally, we consider the labelled signature F lab = ff a j f 2 F and a 2 Ag where each f a has the same arity as f. For every assignment we inductively dene a labelling function lab from T (F; V) to T (F lab ; V) as follows: t if t 2 V, lab (t) = f [](t) (lab (t ); : : : ; lab (t n )) if t = f(t ; : : : ; t n ). So function symbols in t are simply labelled by the value (under the assignment ) of the corresponding subterms. We dene the TRSs R lab and dec(f; ) over the signature F lab as follows: R lab = f lab (l)! lab (r) j l! r 2 R and : V! Ag; dec(f ; ) = ff a (x ; : : : ; x n )! f b (x ; : : : ; x n ) j f 2 F and a; b 2 A with a bg: The following theorem is a special case of the main result of Zantema []. Theorem. Let (F; R) be a TRS, A an F-algebra, and a well-founded order on the carrier of A. If (A; ) is a quasi-model then termination of (F; R) is equivalent to termination of (F lab ; R lab [ dec(f; )). Observe that in the above approach the labelling part of semantic labelling is completely determined by the semantics. This is not the case for semantic labelling as dened in [], b for our purpose it suces. If termination of (F lab ; R lab [ dec(f ; )) is proved by means of a recursive path order, as will be the case with self-labelling, then a corresponding termination ordering for (F; R) can be described as a semantic path order as dened in []. Self-Labelling In this section we show that every terminating TRS can be transformed by semantic labelling into a TRS whose termination is very easily established. The proof of this result forms the basis of a powerful technique for proving the correctness of transformation techniques for establishing termination. Denition 2. A TRS (F; R) is called precedence terminating if there exists a well-founded order A on F such that root(l) A f for every rewrite rule l! r 2 R and every function symbol f 2 Fun(r).

4 Lemma. Every precedence terminating TRS is terminating. Proof. Let (F; R) be a precedence terminating TRS. So there exists a wellfounded order A on F that satises the condition of Denition 2. An easy induction argument on the structure of r reveals that l A rpo r for every l! r 2 R. Here A rpo denotes the recursive path order (Dershowitz []) induced by the precedence A. Since A rpo is a reduction order, termination of (F; R) follows. The next result states that any terminating TRS can be transformed by semantic labelling into a precedence terminating TRS. Theorem 4. For every terminating TRS (F; R) there exists a quasi-model (A; ) such that (F lab ; R lab [ dec(f ; )) is precedence terminating. Proof. As F-algebra A we take the term algebra T (F; V). We equip T (F; V) with the well-founded order =! + R. (Well-foundedness is an immediate consequence of the termination of R.) Because rewriting is closed under contexts, all algebra operations are (strictly) monotone in all their coordinates. Because assignments in the term algebra T (F; V) are substitions and rewriting is closed under substitions, (A; ) is a quasi-model for (F; R). It remains to show that (F lab ; R lab [ dec(f; )) is precedence terminating. To this end we dene a wellfounded order A on F lab as follows: f s A g t if and only if s (! R [ B) + t. Let l! r be a rewrite rule of R lab [ dec(f; ).. If l! r 2 R lab then there exist an assignment : V! T (F ; V) and a rewrite rule l 0! r 0 2 R such that l = lab (l 0 ) and r = lab (r 0 ). The label of root(l) is [](l 0 ) = l 0. Let ` be the label of a function symbol in r. By construction ` = [](t) = t for some subterm t of r 0. Hence l 0! R r 0 D `. So root(l) A f for every f 2 Fun(r). 2. If l! r 2 dec(f; ) then l = f s (x ; : : : ; x n ) and r = f t (x ; : : : ; x n ) with s! + R t. Clearly root(l) = f s A f t. The particular use of semantic labelling in the above proof (i.e., choosing the term algebra as semantics and thus labelling function symbols with terms) is what we will call self-labelling. One may argue that Theorem 4 is completely useless, since the construction of the quasi-model in the proof relies on the fact that (F; R) is terminating. Nevertheless, in the following sections we will see how self-labelling gives rise to many new results and signicant simplications of existing results on the correctness of transformation techniques for establishing termination. Below we sketch the general framework. Let be a transformation on TRSs, designed to make the task of proving termination easier. In two of the three applications we give, the TRS (F; R) is a subsystem of (F; R). The crucial point is proving correctness of the transformation, i.e., proving that termination of (F; R) implies termination of the original TRS (F; R). Write (F; R) = (F 0 ; R 0 ). The basic idea is to label the TRS (F; R) with terms of (F 0 ; R 0 ). This is achieved by execing the following steps: 4

5 . Turn the term algebra T (F 0 ; V) into an F-algebra A by choosing suitable interpretations for the function symbols in F nf 0 and taking term construction as interpretation of the function symbols in F \ F Equip the F-algebra A with the well-founded order =! + R 0.. Show that (A; ) is a quasi-model for (F; R). 4. Dene f s A g t if and only if s (! R 0 [ B) + t, for f; g 2 F \F 0 and extend this to a well-founded order A on F lab such that the TRS (F lab ; R lab [dec(f; )) is precedence terminating with respect to A. At this point termination of (F; R) and thus the correctness of the transformation is a consequence of Theorem. We would like to stress that the only creative step in this scheme is the choice of the interpretations for the function symbols that disappear during the transformation ; the choice of A will then be implied from the requirement of precedence termination. 4 Modularity Our rst application of self-labelling is a new modularity result. Modularity is concerned with the preservation of properties under combinations of TRSs. Recently the focus in modularity research (Ohlebusch [5] contains a recent overview) has shifted to so-called hierarchical combinations ([5, 2, ]). We will prove the following result: the combination of an arbitrary terminating TRS and a terminating recursive program scheme that denes new functions is terminating. A recursive program scheme (RPS for short) is a TRS (F; R) whose rewrite rules have the form f(x ; : : : ; x n )! t with x ; : : : ; x n pairwise distinct variables such that for every function symbol f 2 F there is at most one such rule. The subset of F consisting of all f such that there is a corresponding rule in R is denoted by F D. In the literature RPSs are assumed to be nite, b we don't need that restriction here. From a rewriting point of view RPSs are quite simple: every RPS is conuent and termination of RPSs is decidable. Moreover, the normals forms of an RPS (F; R) constite the set T (F n F D ; V) of terms that do not contain function symbols in F D. Below we make use of the following fact. Lemma 5. An RPS is terminating if and only if it is precedence terminating. Proof. The \if" direction is trivial. Let (F; R) be a terminating RPS. Dene a binary relation on F as follows: f g if and only if there exists a rewrite rule l! r in R such that root(l) = f and g 2 Fun(r). Termination of (F; R) implies that + is a well-founded order on F. Hence (F; R) is precedence terminating with respect to +. Theorem 6. Let (F; R) be a terminating TRS and (G; S) a terminating RPS satisfying F \ G D =?. Then (F [ G; R [ S) is terminating. 5

6 Proof. Let F 0 = F [ (G n G D ). Using the technique of self-labelling, we show how termination of (F [ G; R [ S) follows from termination of (F 0 ; R). We turn T (F 0 ; V) into an F [ G-algebra A by dening f A for every f 2 G D as follows: f A (t ; : : : ; t n ) = f(t ; : : : ; t n )# S for all terms t ; : : : ; t n 2 T (F 0 ; V). Here f(t ; : : : ; t n )# S denotes the (unique) normal form of f(t ; : : : ; t n ) with respect to the complete, i.e. conuent and terminating, RPS (F [ G; S). Note that f(t ; : : : ; t n )# S 2 T (F 0 ; V). As wellfounded order on T (F 0 ; V) we take! + R. We claim that (A; ) is a quasi-model for (F [ G; R [ S). First we show by induction on t 2 T (F [ G; V) that [](t) = t# S for all assignments : V! T (F 0 ; V). If t 2 V then [](t) = (t) = t = t# S because t 2 T (F 0 ; V) is a normal form with respect to S. Let t = f(t ; : : : ; t n ). We have [](t) = f A ([](t ); : : : ; [](t n )). From the induction hypothesis we obtain [](t i ) = t i # S for all i 2 f; : : : ; ng and thus [](t) = f A (t # S ; : : : ; t n # S ): If f =2 G D then [](t) = f(t # S ; : : : ; t n # S ) and if f 2 G D then [](t) = f(t # S ; : : : ; t n # S )# S : In both cases we have [](t) = f(t ; : : : ; t n )# S = t# S : The above property enables us to prove compatibility of (A; ) and (F [ G; R [ S). Let l! r 2 R [ S and : V! T (F 0 ; V). We have to show that [](l) < [](r). If l! r 2 R then [](l) = l! R r = [](r). If l! r 2 S then [](l) = l# S and [](r) = r# S. Because l! S r, conuence of S yields [](l) = [](r). We now show that every algebra operation is weakly monotone in all its coordinates. For f A with f 2 F 0 this is a consequence of closure under contexts of the rewrite relation! R. Let f be an n-ary function symbol in G D and s ; : : : ; s n ; t 2 T (F 0 ; V) such that s i t. Here i is an arbitrary element of f; : : : ; ng. We show that f A (s ; : : : ; s i ; : : : ; s n ) < f A (s ; : : : ; t; : : : ; s n ): To this end we make use of the fact that t# S = t# S for all terms t 2 T (F [ G; V) and assignments : V! T (F 0 ; V). This property is an easy consequence of the special structure of the left-hand sides of the rewrite rules of the RPS S. Let z be a fresh variable and dene s = f(s ; : : : ; z; : : : ; s n ). We have f A (s ; : : : ; s i ; : : : ; s n ) = s# S = s# S and f A (s ; : : : ; t; : : : ; s n ) = s# S = s# S : Here the substitions (assignments) and are dened by = fz 7! s i g and = fz 7! tg. Because (x) < (x) for every variable x, the desired s# S < s# S is a consequence of closure under contexts of the rewrite relation! R. It remains to show that R lab [S lab [dec(f [G; ) is precedence terminating. To this end we equip the labelled signature F lab [ G lab with a proper order A dened as follows: f s A g t if and only if. s (! R [ B) + t and either f; g 2 F 0 or f; g 2 G D, or 2. f 2 G D, g 2 G, and f I g. Here I is any well-founded order on G such that S is precedence terminating with respect to I. The existence of I is guaranteed by Lemma 5. From wellfoundedness of (! R [ B) + and I it follows that A is a well-founded order on F lab [ G lab. The rewrite rules in R lab [ dec(f [ G; ) are taken care of by the 6

7 rst clause of the denition of A, just as in the proof of Theorem 4. For the rules in S lab we use the second clause. We would like to remark that neither the results of Krishna Rao [2, ] nor the colorful theorems of Dershowitz [5] apply, because we don't p any restrictions on the base system R. One easily shows that S quasi-commes ([2]) over right-linear R, b this doesn't hold for arbitrary TRSs R. As a very special case of Theorem 6 we mention that the disjoint union of any terminating TRS R and the TRS S consisting of the single projection rule g(x; y)! x is terminating. This is to be contrasted with the celebrated counterexample of Toyama [7] against the preservation of termination under disjoint unions in which one of the TRSs consists of both projection rules g(x; y)! x and g(x; y)! y. 5 Distribion Elimination Our second application of self-labelling is the proof of a conjecture of Zantema [] concerning distribion elimination. Let (F; R) be a TRS and let e 2 F be a designated function symbol whose arity is at least one. A rewrite rule l! r 2 R is called a distribion rule for e if l = C[e(x ; : : : ; x n )] and r = e(c[x ]; : : : ; C[x n ]) for some non-empty context C in which e doesn't occur and pairwise dierent variables x ; : : : ; x n. Distribion elimination is a technique that transforms (F; R) by eliminating all distribion rules for e and removing the symbol e from the right-hand sides of the other rules. First we inductively dene a mapping E distr that assigns to every term in T (F; V) a non-empty subset of T (F n feg; V), as follows: ftg if t 2 V, >< n[ E distr (t) = E distr (t i ) if t = e(t ; : : : ; t n ), i= >: ff(s ; : : : ; s n ) j s i 2 E distr (t i )g if t = f(t ; : : : ; t n ) with f 6= e. The mapping E distr is illustrated in Fig., where we assume that the numbered contexts do not contain any occurrences of e. It is extended to rewrite systems as follows: E distr (R) = fl! r 0 j l! r 2 R is not a distribion rule for e and r 0 2 E distr (r)g. Observe that e does not occur in E distr (R) if and only if e does not occur in the left-hand sides of rewrite rules of R that are not distribion rules for e. One of the main results of Zantema [] is stated below. Theorem 7. Let (F; R) be a TRS and let e 2 F be a non-constant symbol which does not occur in the left-hand sides of rewrite rules of R that are not distribion rules for e.. If E distr (R) is terminating and right-linear then R is terminating. 7

8 t = 2 e e 6 7 E distr(t) = >< 4 >= e 4 5 >: >; Fig.. The mapping E distr. 2. If E distr (R) is simply terminating and right-linear then R is simply terminating.. If E distr (R) is totally terminating then R is totally terminating. The following example from [] shows that right-linearity is essential in parts and 2. Example. Consider the TRS < f(a; b; x)! f(x; x; e(a; b)) = R = f(e(x; y); z; w)! e(f(x; z; w); f(y; z; w)) : f(x; e(y; z); w)! e(f(x; y; w); f(x; z; w)) The last two rules are distribion rules for e and e does not occur in the left-hand side of the rst rule. The TRS E distr (R) = ff(a; b; x)! f(x; x; a); f(a; b; x)! f(x; x; b)g can be shown to be simply terminating, while the term f(a; b; e(a; b)) admits an innite reduction in R. In [] it is conjectured that in the absence of distribion rules for e the rightlinearity assumption in part of Theorem 7 can be omitted. Before proving this conjecture with the technique of self-labelling, we show that a similar statement for simple termination doesn't hold, i.e., right-linearity is essential in part 2 of Theorem 7 even in the absence of distribion rules for e. Example 2. Let R 0 consist of the rst rule of the TRS R of Example. Simple termination of E distr (R 0 ) = E distr (R) was established in Example, b R 0 fails to be simply terminating as s = f(a; b; e(a; b))! R 0 f(e(a; b); e(a; b); e(a; b)) = t with s embedded in t. However, termination of R 0 follows from Theorem below. Theorem. Let (F; R) be a TRS and let e 2 F be a non-constant symbol which does not occur in the left-hand sides of rewrite rules of R. If E distr (R) is terminating then R is terminating. ; :

9 Proof. We turn the term algebra T (F n feg; V) into an F-algebra A by dening e A (t ; : : : ; t n ) = t for all terms t ; : : : ; t n 2 T (F nfeg; V). Here is an arbitrary b xed element of f; : : : ; ng. So e A is simply projection onto the -th coordinate. We equip A with the well-founded order =! + E distr. We show that (A; ) is a quasi-model for (R) (F; R). It is very easy to see that e A is weakly monotone in all its coordinates. All other operations are strictly monotone in all their coordinates (as! + E distr(r) is closed under contexts). Let " be the identity assignment from V to V. We denote ["](t) by hti. An easy induction proof shows that [](t) = hti for all terms t 2 T (F; V) and assignments : V! T (F n feg; V). Also the following two properties are easily shown by induction on the structure of t 2 T (F; V):. hti 2 E distr (t) and 2. if s E t then there exists a term t 0 2 E distr (t) such that hsi E t 0.. If t 2 V then hti = t and E distr (t) = ftg. For the induction step we distinguish two cases. If t = e(t ; : : : ; t n ) then hti = ht i and E distr (t) = S n i= E distr(t i ). We have ht i 2 E distr (t ) according to the induction hypothesis. Hence hti 2 E distr (t). If t = f(t ; : : : ; t n ) with f 6= e then hti = f(ht i; : : : ; ht n i) and E distr (t) = ff(s ; : : : ; s n ) j s i 2 E distr (t i )g. The induction hypothesis yields ht i i 2 E distr (t i ) for all i = ; : : : ; n. Hence also in this case we obtain the desired hti 2 E distr (t). 2. Observe that for s = t the statement follows from property because we can take t 0 = hti. This observation also takes care of the base of the induction. Suppose t = f(t ; : : : ; t n ) and let s be a proper subterm of t, so s is a subterm of t k for some k 2 f; : : : ; ng. From the induction hypothesis we obtain a term t 0 2 k E distr(t k ) such that hsi E t 0 k. Again we distinguish two cases. If f = e then E distr (t) = S n i= E distr(t i ) and thus we can take t 0 = t 0 k. If f 6= e then E distr (t) = ff(s ; : : : ; s n ) j s i 2 E distr (t i )g. Let t 0 = f(ht i; : : : ; t 0 k ; : : : ; ht ni): Using property we infer that t 0 2 E distr (t). Clearly hsi E t 0. Now let l! r be an arbitrary rewrite rule of R and : V! T (F n feg; V) an arbitrary assignment. We have [](l) = hli and [](r) = hri. Since e doesn't occur in l, hli = l and hence [](l) = l. Because hri 2 E distr (r), the rule l! hri belongs to E distr (R). Therefore l! Edistr (R) hri and thus also [](l) < [](r). Dene a well-founded order A on F lab as follows: f s A g t if and only if s (! Edistr(R) [ B) + t. We will show that (F lab ; R lab [ dec(f; )) is precedence terminating with respect to A. Let l! r be a rewrite rule in R lab [ dec(f; ). We distinguish two cases. If l! r 2 R lab then there exist an assignment : V! T (F n feg; V) and a rewrite rule l 0! r 0 2 R such that l = lab (l 0 ) and r = lab (r 0 ). The label of root(l) is [](l 0 ) = hl 0 i = l 0. Let ` be the label of a function symbol in r. By construction ` = [](t) = hti for some subterm t of r 0. According to property 2 above, hti is a subterm of some r 00 2 E distr (r). By denition l 0! r 00 2 E distr (R). Hence l 0! Edistr(R) r 00 D `. So root(l) A f for every f 2 Fun(r). If l! r 2 dec(f ; ) then l = f s (x ; : : : ; x n ) and r = f t (x ; : : : ; x n ) with f 2 F and s t. In this case we clearly have root(l) = f s A f t.

10 The only creative element in the above proof is the choice of e A. The rest is a roine verication of the two proof obligations of self-labelling. 6 Distribion Elimination Revisited In the proof of Theorem we saw that we can take any projection function as semantics for e. This freedom makes it possible to improve distribion elimination (in the absence of distribion rules) by reducing the size of E distr (R) while preserving correctness of the transformation. What are the essential properties of E distr that make the proof of Theorem work? A careful inspection reveals, apart from the obvious termination requirement for E distr (R), the following two properties:. hti 2 E distr (t), and 2. if s E t then there exists a term t 0 2 E distr (t) such that hsi E t 0, for every t 2 T (F; V). Below we dene a new transformation Edistr that satises these two properties. The transformation is parameterized by the argument positions of the function symbol e. The denition relies on the F-algebra dened in the proof of Theorem in that we use hti. Denition. Let (F ; R) be a TRS and let e 2 F be a function symbol whose arity is at least one. Fix 2 f; : : : ; arity(e)g. We inductively dene mappings and Edistr that assigns to every term in T (F; V) a subset of T (F n feg; V), as follows:? >< [ if t 2 V, (t ) [ Edistr(t i ) if t = e(t ; : : : ; t n ), (t) = i6= n[ >: (t i ) if t = f(t ; : : : ; t n ) with f 6= e, and i= E distr(t) = (t) [ fhtig: We extend the mapping E distr to R as follows: E distr (R) = fl! r0 j l! r 2 R is not a distribion rule for e and r 0 2 E distr (r)g. Figure 2 shows the eect of Edistr and E2 distr on the term t of Fig.. Observe that each numbered context occurs exactly once in each set. The following lemma states that Edistr has the two required properties. Lemma 0. Let (F; R) be a TRS and let e and be as above. For every t 2 T (F; V) we have. hti 2 Edistr (t), and 2. if s E t then there exists a term t 0 2 Edistr (t) such that hsi E t0. 0

11 E distr(t) = >< >: >= >; E 2 distr(t) = >< >: >= >; Fig. 2. The mapping E distr. Proof. The rst statement holds by denition. The second statement we prove by induction on the structure of t 2 T (F; V). If s = t then the result follows from the rst statement. Hence we may assume that s C t. This is only possible if t is a non-variable term f(t ; : : : ; t n ). There exists a k 2 f; : : : ; ng such that s E t k. The induction hypothesis yields a term t 0 2 k E distr (t k) = (t k ) [ fht k ig such that hsi E t 0 k. We distinguish two cases. Suppose f = e. In this case we have [ Edistr(t) = (t) [ fhtig = (t ) [ fhtig [ Edistr(t i ): i6= If k = then t 0 2 k (t ) [ fht k ig = (t ) [ fhtig Edistr (t): If k 6= then t 0 2 k E distr (t k) Edistr (t). Hence in both cases we can take t0 = t 0. Suppose k f 6= e. We have n[ Edistr(t) = (t) [ fhtig = (t i ) [ ff(ht i; : : : ; ht n i)g: i= If t 0 2 k (t k) then clearly t 0 2 k E distr (t) and hence we can take t0 = t 0. If k t0 = k ht ki then we take t 0 = f(ht i; : : : ; ht n i) which satises hsi E t 0 E k t0. Hence we obtain the following result along the lines of the proof of Theorem. Theorem. Let (F; R) be a TRS and let e 2 F be a non-constant symbol which does not occur in the left-hand sides of rewrite rules of R. If Edistr (R) is terminating for some 2 f; : : : ; arity(e)g then R is terminating. Example. Consider the TRS R = ff(a)! f(e(a; b))g. Distribion elimination results in the non-terminating TRS E distr (R) = ff(a)! f(a); f(a)! f(b)g. The termination of the TRS E 2 distr(r) = f(a)! f(b) f(a)! a can be veried by recursive path order. Hence termination of R follows from Theorem. Observe that Edistr (R) fails to be terminating.

12 An obvious question is whether Edistr works in combination with distribion rules, i.e., does Theorem 7 hold for Edistr? The following example shows that the answer is negative. Example 4. Consider the non-terminating TRS < f(a; b)! f(e(a; b); e(a; b)) = R = f(e(x; y); z)! e(f(x; z); f(y; z)) : f(x; e(y; z))! e(f(x; y); f(x; z)) The TRS Edistr (R) is right-linear and (simply and totally) terminating for both choices of. For instance, E distr (R) = f(a; b)! f(a; a) : f(a; b)! b A natural question to ask is whether we need the assumption in Theorems and that e does not occur in the left-hand sides of the rewrite rules in R. In the proof of Theorem this assumption is only used to conclude that hli = l (where l is the left-hand side of a rewrite rule in R). We need this identity because the left-hand sides of rewrite rules in E distr (R) and Edistr (R) are of the form l rather than hli. This implies that we can completely remove the restriction that e does not occur in the left-hand sides of rules in R, provided we change Edistr (R) accordingly: E (R) = distr fhli! r0 j l! r 2 R and r 0 2 Edistr (r)g. This extension is useful since it enables us to conclude the termination of a non-simply terminating TRS like R = ff(e(a; b); a)! f(e(a; b); e(a; b))g by transforming it into the TRS f(b; a)! f(b; b) E 2 distr(r) = f(b; a)! a whose termination can be veried by recursive path order. The transformation Edistr is similar in spirit to the dummy elimination technique of Ferreira and Zantema []: a function symbol is eliminated from (the right-hand sides of) the rewrite rules witho duplicating the other parts. In dummy elimination this is achieved by introducing a fresh constant 5 to separate those parts, rather than gluing dierent parts together. The eect of dummy elimination E dummy on the term t of Fig. is shown in Fig.. Observe that E dummy (t) shares with Edistr (t) the characteristic that each numbered contexts occurs exactly once. An application of E dummy to the TRS R of Example results in the (terminating) TRS < f(a)! f(5) = E dummy (R) = f(a)! a : ; : f(a)! b A correctness proof of dummy elimination can be given along the lines of the proof of Theorem, because the two key properties identied earlier hold for ; : 2

13 E dummy(t) = < : = ; Fig.. The mapping E dummy. E dummy as well, provided we change the interpretation of e to e A (t ; : : : ; t n ) = 5 for all t ; : : : ; t n 2 T ((F n feg) [ f5g; V). A thorough investigation of the relative strength of (variants of) distribion elimination and dummy elimination can be found in Ferreira [7]. 7 Currying In this nal section we show that the main result of Kennaway, Klop, Sleep, and de Vries [0] the preservation of termination under currying is easily proved by self-labelling. Currying is the transformation on TRSs dened below. Denition 2. With every TRS (F; R) we associate a TRS ; ) as follows: the signature contains all function symbols of F together with. function symbols f i of arity i for every f 2 F of arity n with 0 6 i < n, 2. a binary function called application, and is the extension of R with all rewrite i (x ; : : : ; x i ); y)! f i+ (x ; : : : ; x i ; y) with f 2 F of arity n > and 0 6 i < n. Here x ; : : : ; x i ; y are pairwise dierent variables and f i+ denotes f if i + = n. Clearly termination of implies termination of R. Theorem Kennaway et al. [0]. If R is a terminating TRS then is terminating. The proof in [0] is rather involved. We present a self-labelling proof. Proof. Let F 0 = n f@g. The question is how termination of ; ) follows from termination of (F 0 ; R). We turn T (F 0 ; V) into an -algebra A by A (s; t) by induction on the structure of s, as follows: < t if s 2 A (s; t) = f : i+ (s ; : : : ; s i ; t) if s = f i (s ; : : : ; s i ) with i < arity(f), f(@ A (s ; t); : : : A (s n ; t)) if s = f(s ; : : : ; s n ). As well-founded order on T (F 0 ; V) we take! + R. We show that (A; ) is a quasi-model for ; ). We claim that every algebra operation is strictly monotone in all its coordinates. Here we consider

14 only the rst coordinate A, which is the most interesting case. Before proceeding we mention the following fact, which is easily proved by induction on the structure of s: if s 2 T (F; V), t 2 T (F 0 ; V), and 2 6(F 0 ; V) A (s; t) = A (; t). Here the A (; t) is dened as the mapping that assigns to every variable x the A (x; t). We show A (s; t)! A (u; t) whenever s; t; u 2 T (F 0 ; V) with s! R u by induction on the structure of s. Strict monotonicity A in its rst coordinate follows from this by an obvious induction argument. Since s cannot be a variable, we have either s = f i (s ; : : : ; s i ) with i < arity(f) or s = f(s ; : : : ; s n ). In the former case we A (s; t) = f i+ (s ; : : : ; s i ; t). Moreover, as s is root-stable, u must be of the form f i (s ; : : : ; u j ; : : : ; s i ) with s j! R u j. A (s; t)! R f i+ (s ; : : : ; u j ; : : : ; s i ; t) A (u; t): Suppose s = f(s ; : : : ; s n ). If the rewrite step from s to u takes place in one of the arguments of s then v = f(s ; : : : ; u j ; : : : ; s n ) with s j! R u j. From the induction hypothesis we A (s j ; t)! A (u j ; t) and A (s; t) = f(@ A (s ; t); : : : A (s j ; t); : : : A (s n ; t))! R f(@ A (s ; t); : : : A (u j ; t); : : : A (s n ; t)) A (u; t): If the rewrite step from s to u takes place at the root of s then s = l and u = r for some rewrite rule l! r 2 R and substition 2 6(F 0 ; V). Because l and r do not contain function symbols from F 0 n F, we A (s; t) = A (; t) A (u; t) = A (; t) from the above fact. Therefore also in this case we A (s; t)! A (u; t). In order to conclude that (A; ) is a quasimodel for ; ), it remains to show that [](l) < [](r) for every rewrite rule l! r 2 and assignment from V to T (F 0 ; V). If l! r 2 R then [](l) = l! R r = [](r). Otherwise l i (x ; : : : ; x i ); y) and r = f i+ (x ; : : : ; x i ; y) for some f 2 F and i < arity(f), in which case we have [](l) = f i+ (x ; : : : ; x i ; y) = [](r) by denition. To conclude our proof we show that ) lab [ ; ) is precedence terminating with respect to the well-founded order A dened as follows: f s A g t if and only if. s (! R [ B) + t and either f; g or f; g or 2. f and g It is easy to see that A is indeed a well-founded order. Clearly ) lab = R lab [ n R) lab. The rewrite rules in R lab ; ) are taken care of by the rst clause of the denition of A, just as in the proof of Theorem 4. For the rules in n R) lab we use the second clause. The reader is invited to compare our proof with the one of Kennaway et al. [0]. 4

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