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Reviewing the classical and the de Bruijn notation for λcalculus and pure type systems
 Logic and Computation
, 2001
"... This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentat ..."
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This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λcalculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λcalculus is introduced and some of its advantages are outlined.
A CurryHowardDe Bruijn Isomorphism Modulo. Under submission
, 2006
"... The rewriting calculus combines in a unified setting the frameworks and capabilities of rewriting and calculus. Its most general typed version, called Pure Pattern Type Systems (P 2TS) and adapted from Barendregt’s cube, is especially interesting from a logical point of view. We show how to use a ..."
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The rewriting calculus combines in a unified setting the frameworks and capabilities of rewriting and calculus. Its most general typed version, called Pure Pattern Type Systems (P 2TS) and adapted from Barendregt’s cube, is especially interesting from a logical point of view. We show how to use a subset of P 2TS as a proofterm language for natural deduction modulo, extending the CurryHowardDe Bruijn isomorphism for this class of logical formalisms. The pattern matching featured in the calculus allows us to model any congruence given by a term rewriting system. We characterize how proofs can be denoted by P 2TS terms and we discuss the interest of our proofterm language for the issue of cut elimination. Finally, we explore some relations between our proofterm language and other formalisms: extraction of terms and/or rewrite rules from P 2TSterms, but also automated generation of proofterms by a rewritingbased language. 1
AUTOMATH and Pure Type Systems
, 1996
"... We study the position of Automath systems within the framework of the Pure Type Systems as discussed in [3]. ..."
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We study the position of Automath systems within the framework of the Pure Type Systems as discussed in [3].
Belief Revision In Type Theory
"... This paper explores belief revision for belief states in which an agent's beliefs as well as his justifications for these beliefs are explicitly represented in the context of type theory. This allows for a deductive perspective on belief revision which can be implemented using existing machiner ..."
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This paper explores belief revision for belief states in which an agent's beliefs as well as his justifications for these beliefs are explicitly represented in the context of type theory. This allows for a deductive perspective on belief revision which can be implemented using existing machinery for deductive reasoning.
Variants of the Basic Calculus of Constructions
, 2004
"... In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version i ..."
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In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version in early papers on the subject by Seldin. None of these results is very deep, but it seems useful to collect them in one place.
Pure Type Systems with de Bruijn indices
"... Nowadays, type theory has many applications and is used in many different disciplines. Within computer science, logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. A general framework called Pure Type Systems (PTSs for short) ..."
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Nowadays, type theory has many applications and is used in many different disciplines. Within computer science, logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. A general framework called Pure Type Systems (PTSs for short) has been introduced independently by Terlouw and Berardi in 1988 and 1989, in order to provide a unified formalism in which many type systems can be represented. In particular, PTSs allow the representation of the simple theory of types, the polymophic theory of types, the dependent theory of types and various other wellknown type systems such as the Edinburgh Logical Frameworks LF and the Automath system. Pure Type Systems are usually presented using variable names. In this article, we present a formulation of PTSs with de Bruijn indices. De Bruijn indices [6] avoid the problems caused by variable names during the implementation of type systems. We show that PTSs with variable names and PTSs with de Bruijn indices are isomorphic. This isomorphism enables us to answer questions about PTSs with de Bruijn indices including confluence, termination (strong normalisation) and safety (subject reduction).
Parameters in Pure Type Systems
, 2002
"... In this paper we study the addition of parameters to typed calculus with definitions. We show that the resulting systems have nice properties and illustrate that parameters allow for a better finetuning of the strength of type systems as well as staying closer to type systems used in practice i ..."
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In this paper we study the addition of parameters to typed calculus with definitions. We show that the resulting systems have nice properties and illustrate that parameters allow for a better finetuning of the strength of type systems as well as staying closer to type systems used in practice in theorem provers and programming languages.
Interpreting HOL in the Calculus of Constructions ⋆
"... The purpose of this paper is to consider a representation of the HOL theoremprover in the calculus of constructions with the property that consistency results from the calculus of constructions imply such results in HOL. This kind of representation is impossible using the propositionsastypes repr ..."
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The purpose of this paper is to consider a representation of the HOL theoremprover in the calculus of constructions with the property that consistency results from the calculus of constructions imply such results in HOL. This kind of representation is impossible using the propositionsastypes representation of logic and equality, but it is possible if a different representation is used.
Acknowledgement
, 2003
"... This work was completed with the support of Prof. Horst Reichel and Martin Morgenthal. ..."
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This work was completed with the support of Prof. Horst Reichel and Martin Morgenthal.
Automath and Pure Type Systems
, 2003
"... We study the position of the Automath systems within the framework of Pure Type Systems (PTSs). In [1,15], a rough relationship has been given between Automath and PTSs. That relationship ignores three of the most important features of Automath: de nitions, parameters and reduction, because at th ..."
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We study the position of the Automath systems within the framework of Pure Type Systems (PTSs). In [1,15], a rough relationship has been given between Automath and PTSs. That relationship ignores three of the most important features of Automath: de nitions, parameters and reduction, because at the time, PTSs did not have these features. Since, PTSs have been extended with these features and in view of this, we revisit the correspondence between Automath and PTSs. This paper gives the most accurate description of Automath as a PTS so far.