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Internal Parametricity for Cubical Type Theory

Authors Evan Cavallo , Robert Harper

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Author Details

Evan Cavallo
  • Carnegie Mellon University, Pittsburgh, PA, USA
Robert Harper
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

We gratefully acknowledge the support of the Air Force Office of Scientific Research through MURI grant FA9550-15-1-0053. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the AFOSR.

Acknowledgements

We thank Carlo Angiuli, Steve Awodey, Daniel Gratzer, Kuen-Bang Hou (Favonia), Dan Licata, Anders Mörtberg, Emily Riehl, Christian Sattler, and Jonathan Sterling for their comments and insights.

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Evan Cavallo and Robert Harper. Internal Parametricity for Cubical Type Theory. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.CSL.2020.13

Abstract

We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, and we give an account of the identity extension lemma for internal parametricity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
Keywords
  • parametricity
  • cubical type theory
  • higher inductive types

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