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A reduction relation for which postponement of Kcontractions, Conservation and Preservation of Strong Normalisation hold
 Univ. of Glasgow, Glasgow
, 1996
"... Postponement of fi K contractions and the conservation theorem do not hold for ordinary fi but have been established by de Groote for a mixture of fi with another reduction relation. In this paper, de Groote's results are generalised for a single reduction relation fi e which generalises fi. ..."
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Postponement of fi K contractions and the conservation theorem do not hold for ordinary fi but have been established by de Groote for a mixture of fi with another reduction relation. In this paper, de Groote's results are generalised for a single reduction relation fi e which generalises fi. This then is used to solve an open problem of fi e : the Preservation of Strong Normalisation 1 . Keywords: Generalised fireduction, Postponement of Kcontractions, Generalised Conservation, Preservation of Strong Normalisation. 1 Introduction 1.1 Background and Motivation In the term (( x : y :N)P )Q, the function starting with x and the argument P result in the redex ( x : y :N)P . It is also the case that the function starting with y and the argument Q will result in another redex when the first redex is contracted. This idea has been exploited by many researchers and reduction has been extended so that the generalised redex based on the matching y and Q is given the same priority a...
On Strong Normalization in the Intersection Type Discipline (Extended Abstract)
"... We give a proof for the strong normalization result in the intersection type discipline, which we obtain by putting together some wellknown results and proof techniques. Our proof uses a variant of Klop's extended calculus, for which it is shown that strong normalization is equivalent to ..."
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We give a proof for the strong normalization result in the intersection type discipline, which we obtain by putting together some wellknown results and proof techniques. Our proof uses a variant of Klop's extended calculus, for which it is shown that strong normalization is equivalent to weak normalization. This is
Realizability and parametricity in pure type systems
 In the Proceedings of FoSSaCS 2011 (Saarbruecken
, 2011
"... Abstract. We describe a systematic method to build a logic from any programming language described as a Pure Type System (PTS). The formulas of this logic express properties about programs. We define a parametricity theory about programs and a realizability theory for the logic. The logic is express ..."
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Abstract. We describe a systematic method to build a logic from any programming language described as a Pure Type System (PTS). The formulas of this logic express properties about programs. We define a parametricity theory about programs and a realizability theory for the logic. The logic is expressive enough to internalize both theories. Thanks to the PTS setting, we abstract most idiosyncrasies specific to particular type theories. This confers generality to the results, and reveals parallels between parametricity and realizability. 1
Typed selfrepresentation
 IN PLDI
, 2009
"... Selfrepresentation – the ability to represent programs in their own language – has important applications in reflective languages and many other domains of programming language design. Although approaches to designing typed program representations for sublanguages of some base language have become ..."
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Selfrepresentation – the ability to represent programs in their own language – has important applications in reflective languages and many other domains of programming language design. Although approaches to designing typed program representations for sublanguages of some base language have become quite popular recently, the question whether a fully metacircular typed selfrepresentation is possible is still open. This paper makes a big step towards this aim by defining the F ∗ ω calculus, an extension of the higherorder polymorphic lambda calculus Fω that allows typed selfrepresentations. While the usability of these representations for metaprogramming is still limited, we believe that our approach makes a significant step towards a new generation of reflective languages that are both safe and efficient.
Currystyle types for nominal rewriting
, 2006
"... Abstract. We define a type inference system for Nominal Rewriting where the types associated to terms are polymorphic (built from a set of base data sorts, type variables, and userdefined type constructors). In contrast with standard term rewriting systems or the λcalculus, a typing environment fo ..."
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Abstract. We define a type inference system for Nominal Rewriting where the types associated to terms are polymorphic (built from a set of base data sorts, type variables, and userdefined type constructors). In contrast with standard term rewriting systems or the λcalculus, a typing environment for nominal rewriting includes type information for atoms, variables and function symbols. We give a syntaxdirected system of typing rules associating types to terms, and show that every typable term has a principal type in a given environment. Type inference is decidable; moreover, rewriting preserves types when rules are typable.
On \Piconversion in the lambdacube and the combination with abbreviations
, 1997
"... Typed calculus uses two abstraction symbols ( and \Pi) which are usually treated in different ways: x: :x has as type the abstraction \Pi x: :, yet \Pi x: : has type 2 rather than an abstraction; moreover, ( x:A :B)C is allowed and fireduction evaluates it, but (\Pi x:A :B)C is rarely allowed. Fu ..."
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Cited by 6 (3 self)
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Typed calculus uses two abstraction symbols ( and \Pi) which are usually treated in different ways: x: :x has as type the abstraction \Pi x: :, yet \Pi x: : has type 2 rather than an abstraction; moreover, ( x:A :B)C is allowed and fireduction evaluates it, but (\Pi x:A :B)C is rarely allowed. Furthermore, there is a general consensus that and \Pi are different abstraction operators. While we agree with this general consensus, we find it nonetheless important to allow \Pi to act as an abstraction operator. Moreover, experience with AUTOMATH and the recent revivals of \Pireduction as in [KN 95b, PM 97], illustrate the elegance of giving \Piredexes a status similar to redexes. However, \Pireduction in the cube faces serious problems as shown in [KN 95b, PM 97]: it is not safe as regards subject reduction, it does not satisfy type correctness, it loses the property that the type of an expression is wellformed and it fails to make any expression that contains a \Piredex wellfor...
Equivalences between Logics and their Representing Type Theories
, 1992
"... We propose a new framework for representing logics, called LF + and based on the Edinburgh Logical Framework. The new framework allows us to give, apparently for the first time, general definitions which capture how well a logic has been represented. These definitions are possible since we are abl ..."
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We propose a new framework for representing logics, called LF + and based on the Edinburgh Logical Framework. The new framework allows us to give, apparently for the first time, general definitions which capture how well a logic has been represented. These definitions are possible since we are able to distinguish in a generic way that part of the LF + entailment which corresponds to the underlying logic. This distinction does not seem to be possible with other frameworks. Using our definitions, we show that, for example, natural deduction firstorder logic can be wellrepresented in LF + , whereas linear and relevant logics cannot. We also show that our syntactic definitions of representation have a simple formulation as indexed isomorphisms, which both confirms that our approach is a natural one and provides a link between typetheoretic and categorical approaches to frameworks. 1 Introduction Much effort has been devoted to building systems for supporting the construction of f...
Certified higherorder recursive path ordering
 In RTA, LNCS
, 2006
"... Abstract. Recursive path ordering (RPO) is a wellknown reduction ordering introduced by Dershowitz [6], that is useful for proving termination of term rewriting systems (TRSs). Jouannaud and Rubio generalized this ordering to the higherorder case thus creating the higherorder recursive path order ..."
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Abstract. Recursive path ordering (RPO) is a wellknown reduction ordering introduced by Dershowitz [6], that is useful for proving termination of term rewriting systems (TRSs). Jouannaud and Rubio generalized this ordering to the higherorder case thus creating the higherorder recursive path ordering (HORPO) [8]. They proved that this ordering can be used for proving termination of higherorder TRSs which essentially comes down to proving wellfoundedness of the union of HORPO and βreduction relation of simply typed lambda calculus (λ →), [1]. This result entails wellfoundedness of RPO and termination of λ →. This paper describes author’s undertaking of providing a complete, axiomfree, fully constructive formalization of those results in the theorem prover Coq. Formalization is complete and hence it contains all the dependant results for λ → , multisets and multiset extension of the relation. Also decidability of HORPO has been proven and due to constructive nature of this proof a certified algorithm to verify whether two terms can be oriented with HORPO can be extracted from this proof. 1
Extensionality versus constructivity
 Mathematical logic Quarterly
, 2000
"... We will analyze some extensions of MartinLöf’s constructive type theory by means of extensional set constructors and we will show that often the most natural requirements over them lead to classical logic or even to inconsistency. 1 ..."
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We will analyze some extensions of MartinLöf’s constructive type theory by means of extensional set constructors and we will show that often the most natural requirements over them lead to classical logic or even to inconsistency. 1