We are interested in developing a methodology for integrating mechanized reasoning systems such as Theorem Provers, Computer Algebra Systems, and Model Checkers. Our approach is to provide a framework for specifying mechanized reasoning systems and to use specifications as a starting point for integration. We build on top of the work presented in Giunchiglia et al. (1994) which introduces the notion of Open Mechanized Reasoning Systems (OMRS) as a specification framework for integrating reasoning systems. An OMRS specification consists of three components: the logic component, the control component, and the interaction component. In this paper we focus on the control level. We propose to specify the control component by first adding control knowledge to the data structures representing the logic by means of annotations and then by specifying proof strategies via tactics. To show the adequacy of the approach we present and discuss a structured specification of constraint contextual rewriting as a set of cooperating specialized reasoning modules.

Maple's evaluator, together with a feature that is usually known as the assume facility, is a combination of modules with specialised reasoning capabilities. These modules are identi ed, their interfaces are speci ed, and their interplay is reconstructed as Constraint Contextual Rewriting (CCR), a powerful form of conditional rewriting that incorporates the services provided by a decision procedure. Finally we show how Maple's evaluation process can be strengthened by borrowing ideas from CCR.

Abstract. We describe here a formal proof in the Coq system of the structure theorem for subresultants, which allows to prove formally the correctness of our implementation of the subresultants algorithm. Up to our knowledge it is the first mechanized proof of this result. 1

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We are interested in integrating mechanized reasoning systems such as, e.g., Theorem Provers and Computer Algebra Systems. Our approach to the problem is to provide a framework for specifying mechanized reasoning systems and to use specifications as a starting point for integration. We build on top of the work presented in [9] which introduces the notion of Open Mechanized Reasoning Systems (OMRS) as a specification framework for integrating reasoning systems. An OMRS specification consists of three components: the logic component, the control component, and the interaction component. In this paper we focus on the control level and propose to specify the control component by first adding control knowledge to the data structures representing the logic by means of annotations, and then by specifying proof strategies via tactics. To show the adequacy of the approach we present and discuss a structured specification of the top-level inference strategy of NQTHM as a set of cooperating specialized reasoning modules.

Now, although interactive provers may require manual guidance, it's desirable to provide quite high levels of automation so that the user avoids the tedious filling in of trivial details. Indeed, the most effective recent systems such as PVS do provide quite powerful automation for special theories felt to be particularly important in practice, e.g. linear arithmetic and propositional tautology checking. But what about the automation of pure, typically first order, logic? There have been attempts since at least SAM [4] to harness automation of pure logic in interactive systems. Yet a common view today is that automation of theories like linear arithmetic is far more significant in practice.

We describe an approach for formally verifying the linkage between a symbolic state enumeration system and a theorem proving system. This involves the following three stages of proof. Firstly we prove theorems about the correctness of the translation part of the symbolic state system. It interfaces between low level decision diagrams and high level description languages. We ensure that the semantics of a program is preserved in those of its translated form. Secondly we prove linkage theorems: theorems that justify introducing a result from a state enumeration system into a proof system. Finally we combine the translator correctness and linkage theorems. The resulting new linkage theorems convert results to a high level language from the low level decision diagrams that the result was actually proved about in the state enumeration system.They justify importing low-level external verification results into a theorem prover. We use a linkage between the HOL system and a simplified version of the MDG system to illustrate the ideas and consider a small example that integrates two applications from MDG and HOL to illustrate the linkage theorems.

This paper describes the integration of efficient external reasoners into proof planning. It shows how computer algebra systems and constraint solvers can be integrated, how the shortcuts produced by the external reasoners can simplify and guide a formal proof and how these shortcuts can be expanded to checkable proofs. The paper illustrates the integration and cooperation of the external reasoners with an example from proof planning limit theorems.

Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far. 1 Historical introduction The idea of reducing reasoning to mechanical calculation is an old dream [75]. Hobbes [55] made explicit the analogy in the slogan ‘Reason [...] is nothing but Reckoning’. This parallel was developed by Leibniz, who envisaged a ‘characteristica universalis’ (universal language) and a ‘calculus ratiocinator ’ (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. The characteristica universalis The dream of a truly universal language in Leibniz’s sense remains unrealized and probably unrealizable. But over the last few centuries a language that is at least adequate for

We describe a new formalisation in Isabelle/HOL of Intuitionistic Linear Logic and consider the support this provides for constructing plans by proving the achievability of given planning goals. The plans so found are provably correct, by construction. This representation of plans in linear logic provides a concise account of planning with sensing actions, allows the creation and deletion of objects, and solves the frame problem in an elegant way. Within this setting, we show how planning algorithms are implemented as search strategies within a theorem proving system. This allows us to provide a flexible methodology for developing search strategies that is independent of soundness issues. This feature is illustrated in two ways. Firstly, following ideas from logic programming, we show how a significant symmetry in search, caused by context splitting, can be pruned by using a derived inference rule. Secondly, we show how domain specific constraints on synthesis are supported and how they can be used to find contingent or conformant plans. We illustrate the approach with example planning scenarios. 1.