We introduce a generic definition of reduction orderings on term algebras containing associative-commutative (hereafter denoted AC) operators. These orderings are compatible with the AC theory hence makes them suitable for use in deduction systems where AC operators are built-in. Furthermore, they have the nice property of being total on AC classes of ground terms, a required property for example to avoid failure in ACcompletion, or to insure completeness of ordered strategies in first-order theorem proving with built-in AC operators. We show that the two definitions already known of such total and AC-compatible orderings [24, 25] are actually instances of our definition. Finally, we find new such orderings which have more properties, first an ordering based on an integer polynomial interpretation, answering positively to a question left open by Narendran and Rusinowitch, and second an ordering which allow to orient the distributivity axiom in the usual way, answering positively to a ...

We argue that most completion procedures for finitely presented algebras can be simulated by term completion procedures based on a generalized symmetrization process. Therefore we present three different constructive definitions of symmetrization procedures that can take the role of the orientation step in a symmetrization based completion procedure. We investigate confluence and compatibility properties of the symmetrized rules computed by the different symmetrization procedures. Based on semicompatibility properties we can present a generic version of the critical pair theorem that specializes to the critical pair theorems of Knuth-Bendix completion procedures and algebraic completion procedures like Buchberger's algorithm respectively. This critical pair theorem also applies to symmetrization based completion procedures using a normalized reduction relation if the result of the symmetrization is both semi-compatible and semi-stable. We conclude our paper showing how a generic Buchberger algorithm for polynomials over arbitrary finitely presented rings can be formulated as a symmetrization based completion procedure.

Abstract. In [17], Toyama proved that the union of two confluent term-rewriting systems that share absolutely no function symbols or constants is likewise confluent, a property called modularity. The proof of this beautiful modularity result, technically based on slicing terms into an homogeneous cap and a so called alien, possibly heterogeneous substitution, was later substantially simplified in [7, 9]. In this paper 3 we present a further simplification of the proof of Toyama’s result for confluence, which shows that the crux of the problem lies in two different properties: a cleaning lemma, whose goal is to anticipate the application of collapsing reductions; a modularity property of ordered completion, that allows to pairwise match the caps and alien substitutions of two equivalent terms obtained from the cleaning lemma. The approach allows for rules with extra variables on the right and scales up to rewriting modulo arbitrary sets of equations. 1

Abstract. In [19], Toyama proved that the union of two confluent term-rewriting systems that share absolutely no function symbols or constants is likewise confluent, a property called modularity. The proof of this beautiful modularity result, technically based on slicing terms into an homogeneous cap and a so called alien, possibly heterogeneous substitution, was later substantially simplified in [8, 12]. In this paper 3, we present a further simplification of the proof of Toyama’s result for confluence, which shows that the crux of the problem lies on two different properties: a cleaning lemma, whose goal is to anticipate the application of collapsing reductions and a modularity property of ordered completion that allows to pairwise match the caps and alien substitutions of two equivalent terms obtained from the cleaning lemma. The approach allows for arbitrary kinds of rules, and scales up to rewriting modulo arbitrary sets of equations. 1

: This paper extends the termination proof techniques based on rewrite orderings to a higher-order setting, by defining a recursive path ordering for simply typed higher-order terms in j-long fi-normal form. This ordering is powerful enough to show termination of several complex examples. Key words: higher-order rewriting ; typed lambda calculus; termination orderings. 1 Introduction Higher-order rewrite rules are used in various programming languages and logical systems with two different meanings. Some functional languages like ML or type theories like the calculus of inductive constructions use higher-order rewrite rules to define functions or recursors by first-order pattern matching. In this setting termination is known to be satisfied when all rules follow a generalized form of a primitive recursive schema of higher type [9, 1, 10]. In functional languages like Elf, or theorem provers like Isabelle, higher-order rewrite rules define functions by higher-order pattern matching. ...

Abstract. We introduce the notion of well-founded recursive order-sorted equational logic (OS) theories modulo axioms. Such theories de ne functions by well-founded recursion and are inherently terminating. Moreover, for well-founded recursive theories important properties such as con uence and su cient completeness are modular for so-called fair extensions. This enables us to incrementally check these properties for hierarchies of such theories that occur naturally in modular rule-based functional programs. Well-founded recursive OS theories modulo axioms contain only commutativity and associativity-commutativity axioms. In order to support arbitrary combinations of associativity, commutativity and identity axioms, we show how to eliminate identity and (under certain conditions) associativity (without commutativity) axioms by theory transformations in the last part of the paper. 1