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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 ..."
Abstract

Cited by 89 (20 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.
Antisymmetry of higherorder subtyping and equality by subtyping
 Math. Struct. in Comput. Sci
, 2006
"... This paper gives the first proof that the subtyping relation of a higherorder lambda calculus, F ω ≤, is antisymmetric, establishing in the process that the subtyping relation is a partial order—reflexive, transitive, and antisymmetric up to βequality. While a subtyping relation is reflexive and ..."
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Cited by 1 (0 self)
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This paper gives the first proof that the subtyping relation of a higherorder lambda calculus, F ω ≤, is antisymmetric, establishing in the process that the subtyping relation is a partial order—reflexive, transitive, and antisymmetric up to βequality. While a subtyping relation is reflexive and transitive by definition, antisymmetry is a derived property. The result, which may seem obvious to the nonexpert, is technically challenging, and had been an open problem for almost a decade. In this context, typed operational semantics for subtyping offers a powerful new technology to solve the problem: of particular importance is our extended rule for the wellformedness of types with head variables. The paper also gives a presentation of F ω ≤ without a relation for βequality, apparently the first such, and shows its equivalence with the traditional presentation. 1
ii Acknowledgement
"... Hiermit versichere ich, dass ich die vorliegende Diplomarbeit selbständig verfasst und keine ..."
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Hiermit versichere ich, dass ich die vorliegende Diplomarbeit selbständig verfasst und keine
Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Language]: Mathematical LogicLambda calculus and related systems
"... 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 ..."
Abstract
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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 equivalance algorithm based on the shape of terms. Neither approach appears to scale well to richer languages with, for example, unit types or subtyping, and neither provides a notion of canonical form suitable for proving adequacy of encodings. In this paper we present a new, typedirected equivalence algorithm for the LF type theory that overcomes the weaknesses of previous approaches. The algorithm is practical, scales to richer languages, and yields a new notion of canonical form sufficient for adequate encodings of logical systems. The algorithm is proved complete by a Kripkestyle logical relations argument similar to that suggested by Coquand. Crucially, both the algorithm itself and the logical relations rely only on the shapes of types, ignoring dependencies on terms.