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Modularity of Strong Normalization and Confluence in the algebraiclambdacube
, 1994
"... In this paper we present the algebraiccube, an extension of Barendregt's cube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraiccube, provided that the firstorder rewrite rules are nonduplicating and th ..."
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Cited by 25 (7 self)
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In this paper we present the algebraiccube, an extension of Barendregt's cube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraiccube, provided that the firstorder rewrite rules are nonduplicating and the higherorder rules satisfy the general schema of Jouannaud and Okada. This result is proven for the algebraic extension of the Calculus of Constructions, which contains all the systems of the algebraiccube. We also prove that local confluence is a modular property of all the systems in the algebraiccube, provided that the higherorder rules do not introduce critical pairs. This property and the strong normalization result imply the modularity of confluence. 1 Introduction Many different computational models have been developed and studied by theoretical computer scientists. One of the main motivations for the development This research was partially supported by ESPRIT Basic Research Act...
A module calculus for Pure Type Systems
, 1996
"... Several proofassistants rely on the very formal basis of Pure Type Systems. However, some practical issues raised by the development of large proofs lead to add other features to actual implementations for handling namespace management, for developing reusable proof libraries and for separate verif ..."
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Cited by 23 (3 self)
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Several proofassistants rely on the very formal basis of Pure Type Systems. However, some practical issues raised by the development of large proofs lead to add other features to actual implementations for handling namespace management, for developing reusable proof libraries and for separate verification of distincts parts of large proofs. Unfortunately, few theoretical basis are given for these features. In this paper we propose an extension of Pure Type Systems with a module calculus adapted from SMLlike module systems for programming languages. Our module calculus gives a theoretical framework addressing the need for these features. We show that our module extension is conservative, and that type inference in the module extension of a given PTS is decidable under some hypotheses over the considered PTS.
A Verified Typechecker
 PROCEEDINGS OF THE SECOND INTERNATIONAL CONFERENCE ON TYPED LAMBDA CALCULI AND APPLICATIONS, VOLUME 902 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1995
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Open Proofs and Open Terms: A Basis for Interactive Logic
 COMPUTER SCIENCE LOGIC: 16TH INTERNATIONAL WORKSHOP, CLS 2002, LECTURE NOTES IN COMPUTER SCIENCE 2471 (2002
, 2002
"... When proving a theorem, one makes intermediate claims, leaving parts temporarily unspecified. These `open' parts may be proofs but also terms. In interactive theorem proving systems, one prominently deals with these `unfinished proofs' and `open terms'. We study these `open phenomena& ..."
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Cited by 12 (1 self)
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When proving a theorem, one makes intermediate claims, leaving parts temporarily unspecified. These `open' parts may be proofs but also terms. In interactive theorem proving systems, one prominently deals with these `unfinished proofs' and `open terms'. We study these `open phenomena' from the point of view of logic. This amounts to finding a correctness criterion for `unfinished proofs' (where some parts may be left open, but the logical steps that have been made are still correct). Furthermore we want to capture the notion of `proof state'. Proof states are the objects that interactive theorem provers operate on and we want to understand them in terms of logic. In this paper we define `open higher order predicate logic', an extension of higher order logic with unfinished (open) proofs and open terms. Then we define a type theoretic variant of this open higher order logic together with a formulasastypes embedding from open higher order logic to this type theory. We show how this type theory nicely captures the notion of `proof state', which is now a typetheoretic context.
Proving Strong Normalization of CC by Modifying Realizability Semantics
 IN TYPES, VOLUME 806 OF LNCS
, 1994
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Pure type systems in rewriting logic
 In Collection to the honor of O.J. Dahl, volume 2635 of LNCS
, 2003
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The not so simple proofirrelevant model of CC
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2002
"... It is wellknown that the Calculus of Constructions (CC) bears a simple settheoretical model in which proofterms are mapped onto a single object—a property which is known as proofirrelevance. In this paper, we show that when going into the (generally omitted) technical details, this naive model r ..."
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Cited by 10 (1 self)
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It is wellknown that the Calculus of Constructions (CC) bears a simple settheoretical model in which proofterms are mapped onto a single object—a property which is known as proofirrelevance. In this paper, we show that when going into the (generally omitted) technical details, this naive model raises several unexpected difficulties related to the interpretation of the impredicative level, especially for the soundness property which is surprisingly difficult to be given a correct proof in this simple framework. We propose a way to tackle these difficulties, thus giving a (more) detailed elementary consistency proof of CC without going back to a translation to Fω. We also discuss some possible alternatives and possible extensions of our construction.
Modularity of Strong Normalization in the Algebraicλcube
, 1996
"... In this paper we present the algebraicλcube, an extension of Barendregt's λcube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraicλcube, provided that the firstorder rewrite rules are nonduplicating an ..."
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Cited by 8 (2 self)
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In this paper we present the algebraicλcube, an extension of Barendregt's λcube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraicλcube, provided that the firstorder rewrite rules are nonduplicating and the higherorder rules satisfy the general schema of Jouannaud and Okada. This result is proven for the algebraic extension of the Calculus of Constructions, which contains all the systems of the algebraicλcube. We also prove that local confluence is a modular property of all the systems in the algebraicλcube, provided that the higherorder rules do not introduce critical pairs. This property and the strong normalization result imply the modularity of confluence.
Refining the Barendregt Cube using Parameters
, 2001
"... The Barendregt Cube (introduced in [3]) is a framework in which eight important typed calculi are described in a uniform way. Moreover, many type systems (like Automath [18], LF [11], ML [17], and system F [10]) can be related to one of these eight systems. Furthermore, via the propositionsastype ..."
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Cited by 8 (4 self)
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The Barendregt Cube (introduced in [3]) is a framework in which eight important typed calculi are described in a uniform way. Moreover, many type systems (like Automath [18], LF [11], ML [17], and system F [10]) can be related to one of these eight systems. Furthermore, via the propositionsastypes principle, many logical systems can be described in the Barendregt Cube as well (see for instance [9]). However, there are important systems (including Automath, LF and ML) that cannot be adequately placed in the Barendregt Cube or in the larger framework of Pure Type Systems. In this paper we add a parameter mechanism to the systems of the Barendregt Cube. In doing so, we obtain a re nement of the Cube. In this re ned Barendregt Cube, systems like Automath, LF, and ML can be described more naturally and accurately than in the original Cube.
Monadic Type Systems: Pure Type Systems for Impure Settings (Preliminary Report)
 In Proceedings of the Second HOOTS Workshop
, 1997
"... Pure type systems and computational monads are two parameterized frameworks that have proved to be quite useful in both theoretical and practical applications. We join the foundational concepts of both of these to obtain monadic type systems. Essentially, monadic type systems inherit the parameteriz ..."
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Cited by 8 (2 self)
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Pure type systems and computational monads are two parameterized frameworks that have proved to be quite useful in both theoretical and practical applications. We join the foundational concepts of both of these to obtain monadic type systems. Essentially, monadic type systems inherit the parameterized higherorder type structure of pure type systems and the monadic term and type structure used to capture computational effects in the theory of computational monads. We demonstrate that monadic type systems nicely characterize previous work and suggest how they can support several new theoretical and practical applications. A technical foundation for monadic type systems is laid by recasting and scaling up the main results from pure type systems (confluence, subject reduction, strong normalisation for particular classes of systems, etc.) and from operational presentations of computational monads (notions of operational equivalence based on applicative similarity, coinduction proof techni...