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On Equivalence and Canonical Forms in the LF Type Theory
 ACM Transactions on Computational Logic
, 2001
"... Decidability of definitional equality and conversion of terms into canonical form play a central role in the metatheory of a typetheoretic logical framework. Most studies of definitional equality are based on a confluent, stronglynormalizing notion of reduction. Coquand has considered a different ..."
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Cited by 83 (16 self)
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Decidability of definitional equality and conversion of terms into canonical form play a central role in the metatheory of a typetheoretic logical framework. Most studies of definitional equality are based on a confluent, stronglynormalizing notion of reduction. Coquand has considered a different approach, directly proving the correctness of a practical equivalence algorithm based on the shape of terms. Neither approach appears to scale well to richer languages with unit types or subtyping, and neither directly addresses the problem of conversion to canonical form.
A concurrent logical framework I: Judgments and properties
, 2003
"... The Concurrent Logical Framework, or CLF, is a new logical framework in which concurrent computations can be represented as monadic objects, for which there is an intrinsic notion of concurrency. It is designed as a conservative extension of the linear logical framework LLF with the synchronous con ..."
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Cited by 73 (25 self)
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The Concurrent Logical Framework, or CLF, is a new logical framework in which concurrent computations can be represented as monadic objects, for which there is an intrinsic notion of concurrency. It is designed as a conservative extension of the linear logical framework LLF with the synchronous connectives# of intuitionistic linear logic, encapsulated in a monad. LLF is itself a conservative extension of LF with the asynchronous connectives #, & and #.
Deciding Type Equivalence in a Language with Singleton Kinds
 In TwentySeventh ACM Symposium on Principles of Programming Languages
, 2000
"... Work on the TILT compiler for Standard ML led us to study a language with singleton kinds: S(A) is the kind of all types provably equivalent to the type A. Singletons are interesting because they provide a very general form of definitions for type variables, allow finegrained control of type comput ..."
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Cited by 39 (6 self)
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Work on the TILT compiler for Standard ML led us to study a language with singleton kinds: S(A) is the kind of all types provably equivalent to the type A. Singletons are interesting because they provide a very general form of definitions for type variables, allow finegrained control of type computations, and allow many equational constraints to be expressed within the type system.
Extensional equivalence and singleton types
 ACM Transactions on Computational Logic
"... We study the λΠΣS ≤ calculus, which contains singleton types S(M) classifying terms of base type provably equivalent to the term M. The system includes dependent types for pairs and functions (Σ and Π) and a subtyping relation induced by regarding singletons as subtypes of the base type. The decidab ..."
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Cited by 35 (7 self)
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We study the λΠΣS ≤ calculus, which contains singleton types S(M) classifying terms of base type provably equivalent to the term M. The system includes dependent types for pairs and functions (Σ and Π) and a subtyping relation induced by regarding singletons as subtypes of the base type. The decidability of type checking for this language is nonobvious, since to type check we must be able to determine equivalence of wellformed terms. But in the presence of singleton types, the provability of an equivalence judgment Γ ⊢ M1 ≡ M2: A can depend both on the typing context Γ and on the particular type A at which M1 and M2 are compared. We show how to prove decidability of term equivalence, hence of type checking, in λΠΣS ≤ by exhibiting a typedirected algorithm for directly computing normal forms. The correctness of normalization is shown using an unusual variant of Kripke logical relations organized around sets; rather than defining a logical equivalence relation, we work directly with (subsets of) the corresponding equivalence classes. We then provide a more efficient algorithm for checking type equivalence without constructing normal forms. We also show that type checking, subtyping, and all other judgments of the system are decidable.
Coercive Subtyping in Type Theory
 Proc. of CSL'96, the 1996 Annual Conference of the European Association for Computer Science Logic, Utrecht. LNCS 1258
, 1996
"... We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as MartinLof's type theory [NPS90] and the type theory UTT [Luo94]. In this approach, subtyping with specified implicit coercions is treated as a feature at the level of the logical framework; ..."
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Cited by 25 (14 self)
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We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as MartinLof's type theory [NPS90] and the type theory UTT [Luo94]. In this approach, subtyping with specified implicit coercions is treated as a feature at the level of the logical framework; in particular, subsumption and coercion are combined in such a way that the meaning of an object being in a supertype is given by coercive definition rules for the definitional equality. It is shown that this provides a conceptually simple and uniform framework to understand subtyping and coercion relations in type theories with sophisticated type structures such as inductive types and universes. The use of coercive subtyping in formal development and in reasoning about subsets of objects is discussed in the context of computerassisted formal reasoning. 1 Introduction A type in type theory is often intuitively thought of as a set. For example, types in MartinLof's type theory [ML84, NPS90...
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.
Normalization by evaluation for MartinLöf type theory with one universe
 IN 23RD CONFERENCE ON THE MATHEMATICAL FOUNDATIONS OF PROGRAMMING SEMANTICS, MFPS XXIII, ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE
, 2007
"... ..."
An Implementation of LF with Coercive Subtyping & Universes
 Journal of Automated Reasoning
"... . We present `Plastic', an implementation of LF with Coercive Subtyping, and focus on its implementation of Universes. LF is a variant of MartinLof's logical framework, with explicitly typed abstractions. We outline the system of LF with its extensions of inductive types and coercions. Plastic is ..."
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Cited by 15 (9 self)
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. We present `Plastic', an implementation of LF with Coercive Subtyping, and focus on its implementation of Universes. LF is a variant of MartinLof's logical framework, with explicitly typed abstractions. We outline the system of LF with its extensions of inductive types and coercions. Plastic is the first implementation of this extended system; we discuss motivations and basic architecture, and give examples of its use. LF is used to specify type theories. The theory UTT includes a hierarchy of universes which is specified in Tarski style. We outline the theory of these universes and explain how they are implemented in Plastic. Of particular interest is the relationship between universes and inductive types, and the relationship between universes and coercive subtyping. We claim that the combination of Tarskistyle universes together with coercive subtyping provides an ideal formulation of universes which is both semantically clear and practical to use. Keywords: type theory, un...
Short Proofs of Normalization for the simplytyped λcalculus, permutative conversions and Gödel's T
 TO APPEAR: ARCHIVE FOR MATHEMATICAL LOGIC
, 1998
"... Inductive characterizations of the sets of terms, the subset of strongly normalizing terms and normal forms are studied in order to reprove weak and strong normalization for the simplytyped λcalculus and for an extension by sum types with permutative conversions. The analogous treatment of a new sy ..."
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Cited by 15 (1 self)
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Inductive characterizations of the sets of terms, the subset of strongly normalizing terms and normal forms are studied in order to reprove weak and strong normalization for the simplytyped λcalculus and for an extension by sum types with permutative conversions. The analogous treatment of a new system with generalized applications inspired by von Plato's generalized elimination rules in natural deduction shows the flexibility of the approach which does not use the strong computability/candidate style a la Tait and Girard. It is also shown that the extension of the system with permutative conversions by rules is still strongly normalizing, and likewise for an extension of the system of generalized applications by a rule of "immediate simplification". By introducing an innitely branching inductive rule the method even extends to Gödel's T.