Gaining efficiency in completion-based theorem proving requires improvements on three levels: fast inference step execution, careful aggregation into an inference machine, and sophisticated control strategies, all that combined with space saving representation of derived facts.

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.

this paper we present two aspects of our recent work which aim at improving the system with respect to performance and ease of use. Section 2 describes a more powerful hypothesis handling. In Sect. 3 we investigate the control of the proof search and outline our current component of self-adaptation to the given proof problem

Waldmeister is a high performance theorem prover for unit equational first-- order logic. In the making of Waldmeister, we have applied an engineering approach, identifying the critical points with respect to efficiency in time and space. Our logical three--level system model consists of the basic operations on the lowest level, where we put great stress on efficient data structures and algorithms. For the mid--level, where the inference steps are aggregated into an inference machine, flexible adjustment has proven essential during experimental evaluation. The top--level holds control strategy and reduction ordering. Although at this level only employing standard strategies, really large proof tasks have been managed in reasonable time.

Program certification aims to provide explicit evidence that a program meets a specified level of safety. This evidence must be independently reproducible and verifiable. We have developed a system, based on theorem proving, that generates proofs that auto-generated aerospace code adheres to a number of safety policies. For certification purposes, these proofs need to be verified by a proof checker. Here, we describe and evaluate a semantic derivation verification approach to proof checking. The evaluation is based on 109 safety obligations that are attempted by EP and SPASS. Our system is able to verify 129 out of the 131 proofs found by the two provers. The majority of the proofs are checked completely in less than 15 seconds wall clock time. This shows that the proof checking task arising from a substantial prover application is practically tractable. 1

The interface between the Theorema system and external automated deduction systems is described. It provides a tool to access external provers within a Theorema session in the same way as \internal" Theorema provers. Currently 11 external systems are supported. The design of the interface allows combining external systems with each other as well as with \internal" Theorema provers.

Today the application area of... In this paper we present a system for equational deduction. Our prover Waldmeister avoids the main diseases today's provers suffer from: overindulgence of time and space (see also [HBF96]). In [KB70], Knuth and Bendix introduced the completion algorithm which tries to derive a set of convergent rewrite rules from a given set of equations. Its extension to unfailing completion in [BDP89] has turned out to be a valuable means of proving theorems in equational theories. Stated as an inference system, completion operates on a set of facts (rules and equations) which are used to simplify the hypothesis by rewriting. New facts can be derived via superposition of existing ones. Among these possible one-step derivations (critical pairs or cps) one is chosen to be added to the set of facts. If the hypothesis has not become trivial by now, the next iteration cycle starts. In [BH96] we elaborate...

The interface between Theorema and external automated deduction systems implements a link providing a user with a tool for using external provers within a Theorema session in the same way as "internal " Theorema provers. Currently it supports 11 external systems, and support of links to other systems can be easily done. Also, the TPTP (Thousands of Problems of Theorem Provers − problem library) problem converter to Theorema format is described. This research is supported by the Autsrian Science Foundation (FWF) project FO−1302 (SFB). 1.

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