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An Algorithm for TypeChecking Dependent Types
 Science of Computer Programming
, 1996
"... We present a simple typechecker for a language with dependent types and let expressions, with a simple proof of correctness. Introduction Type Theory provides an interesting approach to the problem of (interactive) proofchecking. Instead of introducing, like in LCF [10], an abstract data type of t ..."
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Cited by 44 (4 self)
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We present a simple typechecker for a language with dependent types and let expressions, with a simple proof of correctness. Introduction Type Theory provides an interesting approach to the problem of (interactive) proofchecking. Instead of introducing, like in LCF [10], an abstract data type of theorems, it uses the proofsasprograms analogy and reduces the problem of proof checking to the problem of typechecking in a programming language with dependent types [5]. This approach presents several advantages, well described in [11,5], among those being the possibility of independent proof verification and of a uniform treatment for naming constants and theorems. It is crucial however for this approach to proofchecking to have a simple and reliable typechecking algorithm. Since the core part of such languages, like the ones described in [5,7], seems very simple, there may be some hope for such a short and simple typechecker for dependent types. Indeed, de Bruijn sketches such an al...
Weak Normalization Implies Strong Normalization in Generalized NonDependent Pure Type Systems
 Comput. Sci
, 1997
"... The BarendregtGeuversKlop conjecture states that every weakly normalizing pure type system is also strongly normalizing. We show that this is true for a uniform class of systems which includes, e.g., the left hand side of Barendregt's cube as well as the system U . This seems to be the first resu ..."
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Cited by 4 (3 self)
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The BarendregtGeuversKlop conjecture states that every weakly normalizing pure type system is also strongly normalizing. We show that this is true for a uniform class of systems which includes, e.g., the left hand side of Barendregt's cube as well as the system U . This seems to be the first result giving a positive answer to the conjecture not merely for some concrete systems for which strong normalization is known to hold, but for a uniform class of systems in which not all systems are strongly normalizing. 1.
Strong normalization from weak normalization in typed λcalculi
 Information and Computation
, 1997
"... For some typed λcalculi it is easier to prove weak normalization than strong normalization. Techniques to infer the latter from the former have been invented over the last twenty years by Nederpelt, Klop, Khasidashvili, Karr, de Groote, and Kfoury and Wells. However, these techniques infer strong n ..."
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Cited by 4 (1 self)
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For some typed λcalculi it is easier to prove weak normalization than strong normalization. Techniques to infer the latter from the former have been invented over the last twenty years by Nederpelt, Klop, Khasidashvili, Karr, de Groote, and Kfoury and Wells. However, these techniques infer strong normalization of one notion of reduction from weak normalization of a more complicated notion of reduction. This paper presents a new technique to infer strong normalization of a notion of reduction in a typed λcalculus from weak normalization of the same notion of reduction. The technique is demonstrated to work on some wellknown systems including secondorder λcalculus and the system of positive, recursive types. It gives hope for a positive answer to the BarendregtGeuvers conjecture stating that every pure type system which is weakly normalizing is also strongly normalizing. The paper also analyzes the relationship between the techniques mentioned above, and reviews, in less detail, other techniques for proving strong normalization.
Typechecking Injective Pure Type Systems
, 1993
"... Injective Pure Type Systems form a large class of Pure Type Systems for which one can compute by purely syntactic means two sorts elmt(\GammajM ) and sort(\GammajM ), where \Gamma is a pseudocontext and M is a pseudoterm, and such that for every sort s, \Gamma ` M : A \Gamma ` A : s ) elmt(\Gamm ..."
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Cited by 3 (1 self)
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Injective Pure Type Systems form a large class of Pure Type Systems for which one can compute by purely syntactic means two sorts elmt(\GammajM ) and sort(\GammajM ), where \Gamma is a pseudocontext and M is a pseudoterm, and such that for every sort s, \Gamma ` M : A \Gamma ` A : s ) elmt(\GammajM ) = s \Gamma ` M : s ) sort(\GammajM ) = s By eliminating the problematic clause in the (abstraction) rule in favor of constraints over elmt(:j:) and sort(:j:), we provide a sound and complete typechecking algorithm for injective Pure Type Systems. In addition, we prove Expansion Postponement for a variant of injective Pure Type Systems where the problematic clause in the (abstraction) rule is replaced in favor of constraints over elmt(:j:) and sort(:j:). 1
Pure type systems with corecursion on streams From finite to infinitary normalisation
 IN ICFP
, 2012
"... In this paper, we use types for ensuring that programs involving streams are wellbehaved. We extend pure type systems with a type constructor for streams, a modal operator next and a fixed point operator for expressing corecursion. This extension is called Pure Type Systems with Corecursion (CoPTS) ..."
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Cited by 3 (2 self)
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In this paper, we use types for ensuring that programs involving streams are wellbehaved. We extend pure type systems with a type constructor for streams, a modal operator next and a fixed point operator for expressing corecursion. This extension is called Pure Type Systems with Corecursion (CoPTS). The typed lambda calculus for reactive programs defined by Krishnaswami and Benton can be obtained as a CoPTS. CoPTS’s allow us to study a wide range of typed lambda calculi extended with corecursion using only one framework. In particular, we study this extension for the calculus of constructions which is the underlying formal language of Coq. We use the machinery of infinitary rewriting and formalize the idea of wellbehaved programs using the concept of infinitary normalization. We study the properties of infinitary weak and strong normalization for CoPTS’s. The set of finite and infinite terms is defined as a metric completion. We shed new light on the meaning of the modal operator by connecting the modality with the depth used to define the metric. This connection is the key to the proofs of infinitary weak and strong normalization.
Parametricity and variants of Girard's J operator
, 1999
"... The GirardReynolds polymorphic calculus is generally regarded as a calculus of parametric polymorphism in which all wellformed terms are strongly normalizing with respect to fireductions. Girard demonstrated that the additional of a simple "nonparametric" operation, J , to the calculus allows t ..."
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The GirardReynolds polymorphic calculus is generally regarded as a calculus of parametric polymorphism in which all wellformed terms are strongly normalizing with respect to fireductions. Girard demonstrated that the additional of a simple "nonparametric" operation, J , to the calculus allows the definition of a nonnormalizing term. Since the type of J is not inhabited by any closed term, one might suspect that this may play a role in defining a nonnormalizing term using it. We demonstrate that this is not the case by giving a simple variant, J 0 , of J whose type is otherwise inhabited and which causes normalization to fail. It appears that impredicativity is essential to the argument; predicative variants of the polymorphic calculus admit nonparametric operations without sacrificing normalization. Key words: Formal semantics, functional programming, programming calculi, programming languages, theory of computation. 1 This research was sponsored by the Advanced Research P...
On the Equational Theory of NonNormalising Pure Type Systems
, 2001
"... this paper we are chieAEy concerned with the specications U \Gamma , U and . The denitions are taken from e.g. [1] ..."
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this paper we are chieAEy concerned with the specications U \Gamma , U and . The denitions are taken from e.g. [1]
On Fixed point and Looping Combinators in Type Theory
"... Abstract. The type theories λU and λU − are known to be logically inconsistent. For λU, this is known as Girard’s paradox [Gir72]; for λU − the inconsistency was proved by Coquand [Coq94]. It is also known that the inconsistency gives rise to a so called ”looping combinator”: a family of terms Ln su ..."
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Abstract. The type theories λU and λU − are known to be logically inconsistent. For λU, this is known as Girard’s paradox [Gir72]; for λU − the inconsistency was proved by Coquand [Coq94]. It is also known that the inconsistency gives rise to a so called ”looping combinator”: a family of terms Ln such that Lnf is convertible with f(Ln+1f). It was unclear whether a fixed point combinator exists in these systems. Later, Hurkens [Hur95] has given a simpler version of the paradox in λU − , giving rise to an actual proof term that can be analyzed. In the present paper we analyze the proof of Hurkens and we study the looping combinator that arises from it: it is a real looping combinator (not a fixed point combinator) but in the Curry version of λU − it is a fixedpoint combinator. We also analyze the possibility of typing a fixed point combinator in λU − and we prove that the Church and Turing fixed point combinators cannot be typed in λU −. 1