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Parametricity, type equality, and higher-order polymorphism

Part of: JFP Research Articles

Published online by Cambridge University Press:  15 April 2010

DIMITRIOS VYTINIOTIS  and
STEPHANIE WEIRICH
DIMITRIOS VYTINIOTIS
Affiliation:
Microsoft Research Cambridge, 7 JJ Thomson Avenue, Cambridge CB3 0FBUK (e-mail: dimitris@microsoft.com)
STEPHANIE WEIRICH
Affiliation:
University of Pennsylvania, Department of Computer and Information Science, Levine Hall, 3330 Walnut St., Philadelphia PA 19104–6389, USA (e-mail: sweirich@cis.upenn.edu)
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Abstract

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Propositions that express type equality are a frequent ingredient of modern functional programming – they can encode generic functions, dynamic types, and GADTs. Via the Curry–Howard correspondence, these propositions are ordinary types inhabited by proof terms, computed using runtime type representations. In this paper we show that two examples of type equality propositions actually do reflect type equality; they are only inhabited when their arguments are equal and their proofs are unique (up to equivalence.) We show this result in the context of a strongly normalizing language with higher-order polymorphism and primitive recursion over runtime-type representations by proving Reynolds's abstraction theorem. We then use this theorem to derive “free” theorems about equality types.

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Type
Articles
Information
Journal of Functional Programming , Volume 20 , Issue 2 , March 2010 , pp. 175 - 210
Copyright
Copyright © Cambridge University Press 2010

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