Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof theoretic account of the combination of computations and deductions. The congruence on propositions is handled via rewrite rules and equational axioms. Rewrite rules apply to terms and also directly to atomic propositions. The second contribution is to give a complete proof search method, called Extended Narrowing and Resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higher-order logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higher-order logic subsumes full higher-order resolution.

We describe the application of proof orderings---a technique for reasoning about inference systems---to various rewrite-based theorem-proving methods, including re#nements of the standard Knuth-Bendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a congruence; ordered completion #a refutationally complete extension of standard completion#; and a proof by consistency procedure for proving inductive theorems. # This is a substantially revised version of the paper, #Orderings for equational proofs," co-authored with J. Hsiang and presented at the Symp. on Logic in Computer Science #Boston, Massachusetts, June 1986#. It includes material from the paper #Proof by consistency in equational theories," by the #rst author, presented at the ThirdAnnual Symp. on Logic in Computer Science #Edinburgh, Scotland, July 1988#. This researchwas supported in part by the National Science Foundation under grants CCR-89-01322, CCR-90-07195, and CCR-90-24271. 1 ...

Simultaneous rigid E-unification was introduced in 1987 by Gallier, Raatz and Snyder. It is used in the area of automated reasoning with equality in extension procedures, like the tableau method or the connection method. Many articles in this area assumed the existence of an algorithm for the simultaneous rigid E-unification problem. There were several faulty proofs of the decidability of this problem. In this paper we prove that simultaneous rigid E-unification is undecidable. As a consequence, we obtain the undecidability of the 9 -fragment of intuitionistic logic with equality. 1 Introduction Simultaneous rigid E-unification plays a crucial role in automatic proof methods for first order logic with equality based on sequent calculi, such as semantic tableaux [13], the connection method [7] (also known as the mating method [1]), model elimination [25] and a dozen other procedures. All these methods are based on the Herbrand theorem and express the idea that the proof-search can ...

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The notion of simultaneous rigid E-unification was introduced in 1987 in the area of automated theorem proving with equality in sequent-based methods, for example the connection method or the tableau method. Recently, simultaneous rigid E-unification was shown undecidable. Despite the importance of this notion, for example in theorem proving in intuitionistic logic, very little is known of its decidable fragments. We prove decidability results for fragments of monadic simultaneous rigid E-unification and show the connections between this notion and some algorithmic problems of logic and computer science. 1 Introduction Simultaneous rigid E-unification plays a crucial role in automatic proof methods for first-order logic with equality based on sequent calculi, such as semantic tableaux [13], the connection method [6] (also known as the mating method [1]), model elimination [26] and a dozen other procedures. All these methods are based on the Herbrand theorem and express the idea that ...