This paper presents the design, the implementation and experiments of the integration of syntactic, conditional possibly associative-commutative term rewriting into proof assistants based on constructive type theory. Our approach is called external since it consists in performing term rewriting in a speci c and ecient environment and to check the computations later in a proof assistant.

When rewriting and completion techniques are used for equational theorem proving, the axiom set is saturated with the aim to get a rewrite system that is terminating and confluent on ground terms. To reduce the computational effort it should (1) be powerful for rewriting and (2) create not too many critical pairs. These problems become especially important if some operators are associative and commutative (AC ). We show in this paper how these two goals can be reached to some extent by using ground joinable equations for simplification purposes and omitting them from the generation of new facts. For the special case of AC -operators we present a simple redundancy criterion which is easy to implement, efficient, and effective in practice, leading to significant speed-ups.

ion is in a precise sense a converse operation to application. Given 49 50 CHAPTER 5. PRIMITIVE BASIS OF HOL LIGHT a variable x and a term t, which may or may not contain x, one can construct the so-called lambda-abstraction x: t, which means `the function of x that yields t'. (In HOL's ASCII concrete syntax the backslash is used, e.g. \x. t.) For example, x: x + 1 is the function that adds one to its argument. Abstractions are not often seen in informal mathematics, but they have at least two merits. First, they allow one to write anonymous function-valued expressions without naming them (occasionally one sees x 7! t[x] used for this purpose), and since our logic is avowedly higher order, it's desirable to place functions on an equal footing with rstorder objects in this way. Secondly, they make variable dependencies and binding explicit; by contrast in informal mathematics one often writes f(x) in situations where one really means x: f(x). We should give some idea of how ordina...

In the last years, the development of automated theorem provers has been advancing in a so to speak Olympic spirit, following the motto "faster, higher, stronger"; and the Waldmeister system has been a part of that endeavour. We will survey the concepts underlying this prover, which implements Knuth-Bendix completion in its unfailing variant. The system architecture is based on a strict separation of active and passive facts, and is realized via speci cally tailored representations for each of the central data structures: indexing for the active facts, set-based compression for the passive facts, successor sets for the conjectures. In order to cope with large search spaces, specialized redundancy criteria are employed, and the empirically gained control knowledge is integrated to ease the use of the system. We conclude with a discussion of strengths and weaknesses, and a view of future prospects.

We propose a new method for deriving focused ordered resolution calculi, exemplified by chaining calculi for transitive relations. Previously, inference rules were postulated and a posteriori verified in semantic completeness proofs. We derive them from the theory axioms. Completeness of our calculi then follows from correctness of this synthesis. Our method clearly separates deductive and procedural aspects: relating ordered chaining to Knuth-Bendix completion for transitive relations provides the semantic background that drives the synthesis towards its goal. This yields a more restrictive and transparent chaining calculus. The method also supports the development of approximate focused calculi and a modular approach to theory hierarchies.

The architecture of the Waldmeister prover is based on a strict separation of active and passice facts...

this report we present an overview of dierent rewriting methods which allow one to automatically construct decision procedures for such kind of theories. The next section reviews main denitions and techniques in rewriting (rewriting, normal forms, termination, conuence, and completion); section 2 introduces non-computational equations; in section 3 we describe three main extensions of rewriting (as well as their combinations) for handling of non-computational equations; the report is concluded with a summary

this paper an optimised constructive decision procedure for AC equalities based on the syntacticness of AC theories. The original motivation for it comes from our work [5] to incorporate term rewriting into the Coq proof assistant [3] using ELAN [7]. The main idea is to perform term rewriting in ELAN and to only use Coq for checking purpose. When considering AC rewriting, proof checking requires an ecient method to prove AC equality in Coq using two axioms of associativity and commutativity or possibly a nite set of equalities derived from them