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Proofs for free

Parametricity for dependent types

Part of: JFP Research Articles

Published online by Cambridge University Press:  30 March 2012

JEAN-PHILIPPE BERNARDY ,
PATRIK JANSSON  and
ROSS PATERSON
JEAN-PHILIPPE BERNARDY
Affiliation:
Chalmers University of Technology & University of Gothenburg, Sweden (e-mail: [email protected], [email protected])
PATRIK JANSSON
Affiliation:
Chalmers University of Technology & University of Gothenburg, Sweden (e-mail: [email protected], [email protected])
ROSS PATERSON
Affiliation:
City University, London, UK (e-mail: [email protected])
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Abstract

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Reynolds' abstraction theorem (Reynolds, J. C. (1983) Types, abstraction and parametric polymorphism, Inf. Process.83(1), 513–523) shows how a typing judgement in System F can be translated into a relational statement (in second-order predicate logic) about inhabitants of the type. We obtain a similar result for pure type systems (PTSs): for any PTS used as a programming language, there is a PTS that can be used as a logic for parametricity. Types in the source PTS are translated to relations (expressed as types) in the target. Similarly, values of a given type are translated to proofs that the values satisfy the relational interpretation. We extend the result to inductive families. We also show that the assumption that every term satisfies the parametricity condition generated by its type is consistent with the generated logic.

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Articles
Information
Journal of Functional Programming , Volume 22 , Issue 2 , March 2012 , pp. 107 - 152
Copyright
Copyright © Cambridge University Press 2012

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