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34
Metatheory and Reflection in Theorem Proving: A Survey and Critique
, 1995
"... One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an appro ..."
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Cited by 69 (2 self)
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One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fullyexpansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof ...
Reasoning Theories  Towards an Architecture for Open Mechanized Reasoning Systems
, 1994
"... : Our ultimate goal is to provide a framework and a methodology which will allow users, and not only system developers, to construct complex reasoning systems by composing existing modules, or to add new modules to existing systems, in a "plug and play" manner. These modules and systems ..."
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Cited by 47 (11 self)
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: Our ultimate goal is to provide a framework and a methodology which will allow users, and not only system developers, to construct complex reasoning systems by composing existing modules, or to add new modules to existing systems, in a "plug and play" manner. These modules and systems might be based on different logics; have different domain models; use different vocabularies and data structures; use different reasoning strategies; and have different interaction capabilities. This paper makes two main contributions towards our goal. First, it proposes a general architecture for a class of reasoning systems called Open Mechanized Reasoning Systems (OMRSs). An OMRS has three components: a reasoning theory component which is the counterpart of the logical notion of formal system, a control component which consists of a set of inference strategies, and an interaction component which provides an OMRS with the capability of interacting with other systems, including OMRSs and hum...
Integrating computer algebra into proof planning
 Journal of Automated Reasoning
, 1998
"... Abstract. Mechanised reasoning systems and computer algebra systems have different objectives. Their integration is highly desirable, since formal proofs often involve both of the two di erent tasks, proving and calculating. Even more importantly, proof and computation are often interwoven and not e ..."
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Cited by 41 (24 self)
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Abstract. Mechanised reasoning systems and computer algebra systems have different objectives. Their integration is highly desirable, since formal proofs often involve both of the two di erent tasks, proving and calculating. Even more importantly, proof and computation are often interwoven and not easily separable. In this contribution we advocate an integration of computer algebra into mechanised reasoning systems at the proof plan level. This approach allows to view the computer algebra algorithms as methods, that is, declarative representations of the problem solving knowledge speci c to a certain mathematical domain. Automation can be achieved in many cases bysearching for a hierarchic proof plan at the methodlevel using suitable domainspeci c control knowledge about the mathematical algorithms. In other words, the uniform framework of proof planning allows to solve a large class of problems that are not automatically solvable by separate systems. Our approach also gives an answer to the correctness problems inherent insuch an integration. We advocate an approach where the computer algebra system produces highlevel protocol information that can be processed by aninterface to derive proof plans. Such a proof plan in turn can be expanded to proofs at di erent levels of abstraction, so the approach iswellsuited for producing a highlevel verbalised explication as well as for a lowlevel machine checkable calculuslevel proof. We present an implementation of our ideas and exemplify them using an automatically solved example. Changes in the criterion of `rigour of the proof ' engender major revolutions in mathematics.
Integrating Automated and Interactive Theorem Proving
, 1998
"... Machine code ((Schellhorn and Ahrendt, 1997) and Chapter III.2.6). We use it as a reference or benchmark. Parts of it are repeated every now and then to evaluate the success of our integration concepts, see Section 7. In realistic applications in software verification, proof attempts are more likel ..."
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Cited by 31 (8 self)
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Machine code ((Schellhorn and Ahrendt, 1997) and Chapter III.2.6). We use it as a reference or benchmark. Parts of it are repeated every now and then to evaluate the success of our integration concepts, see Section 7. In realistic applications in software verification, proof attempts are more likely to fail than to go through. This is because specifications, programs, I_3_16mod_a.tex; 9/03/1998; 13:09; p.2 INTEGRATED THEOREM PROVING 549 or userdefined lemmas typically are erroneous. Correct versions usually are only obtained after a number of corrections and failed proof attempts. Therefore, the question is not only how to produce powerful theorem provers but also how to integrate proving and error correction. Current research on this and related topics is discussed in Section 8. There are different approaches of combining interactive methods with automated ones. Their relation to our approach is the subject of Section 9. Finally, in Section 10 we draw conclusions. 2. IDENTIFYING ...
The Refinement Calculator: Proof Support for Program Refinement
 Formal Methods Pacific ’97
, 1997
"... . We describe the Refinement Calculator, a tool which supports ..."
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Cited by 31 (4 self)
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. We describe the Refinement Calculator, a tool which supports
Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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Cited by 27 (0 self)
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
Elements of Mathematical Analysis in PVS
 Ninth international Conference on Theorem Proving in Higher Order Logics TPHOL
, 1996
"... . This paper presents the formalization of some elements of mathematical analysis using the PVS verification system. Our main motivation was to extend the existing PVS libraries and provide means of modelling and reasoning about hybrid systems. The paper focuses on several important aspects of PVS i ..."
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Cited by 20 (0 self)
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. This paper presents the formalization of some elements of mathematical analysis using the PVS verification system. Our main motivation was to extend the existing PVS libraries and provide means of modelling and reasoning about hybrid systems. The paper focuses on several important aspects of PVS including recent extensions of the type system and discusses their merits and effectiveness. We conclude by a brief comparison with similar developments using other theorem provers. 1 Introduction PVS is a specification and verification system whose ambition is to make formal proofs practical and applicable to large and complex problems. The system is based on a variant of higher order logic which includes complex typing mechanisms such as predicate subtypes or dependent types. It offers an expressive specification language coupled with a theorem prover designed for efficient interactive proof construction. In previous work we have applied PVS to the requirements analysis of a substantially ...
A TwoLevel Approach towards Lean ProofChecking
, 1996
"... We present a simple and effective methodology for equational reasoning in proof checkers. The method is based on a twolevel approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with inductive and oracle t ..."
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Cited by 20 (4 self)
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We present a simple and effective methodology for equational reasoning in proof checkers. The method is based on a twolevel approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with inductive and oracle types. The potential of our twolevel approach is illustrated by some examples developed in Lego.
Integrating Computer Algebra with Proof Planning
, 1996
"... . Mechanised reasoning systems and computer algebra systems have apparently different objectives. Their integration is, however, highly desirable, since in many formal proofs both of the two different tasks, proving and calculating, have to be performed. In the context of producing reliable proofs, ..."
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Cited by 15 (6 self)
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. Mechanised reasoning systems and computer algebra systems have apparently different objectives. Their integration is, however, highly desirable, since in many formal proofs both of the two different tasks, proving and calculating, have to be performed. In the context of producing reliable proofs, the question how to ensure correctness when integrating a computer algebra system into a mechanised reasoning system is crucial. In this contribution, we discuss the correctness problems that arise from such an integration and advocate an approach in which the calculations of the computer algebra system are checked at the calculus level of the mechanised reasoning system. We present an implementation which achieves this by adding a verbose mode to the computer algebra system which produces highlevel protocol information that can be processed by an interface to derive proof plans. Such a proof plan in turn can be expanded to proofs at different levels of abstraction, so the approach is well...