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DOI: 10.4230/LIPIcs.CSL.2012.381

Parametricity in an Impredicative Sort

Authors: Chantal Keller and Marc Lasson

Published in: LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

Abstract

Reynold's abstraction theorem is now a well-established result for a large class of type systems. We propose here a definition of relational parametricity and a proof of the abstraction theorem in the Calculus of Inductive Constructions (CIC), the underlying formal language of Coq, in which parametricity relations' codomain is the impredicative sort of propositions. To proceed, we need to refine this calculus by splitting the sort hierarchy to separate informative terms from non-informative terms. This refinement is very close to CIC, but with the property that typing judgments can distinguish informative terms. Among many applications, this natural encoding of parametricity inside CIC serves both theoretical purposes (proving the independence of propositions with respect to the logical system) as well as practical aspirations (proving properties of finite algebraic structures). We finally discuss how we can simply build, on top of our calculus, a new reflexive Coq tactic that constructs proof terms by parametricity.

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Chantal Keller and Marc Lasson. Parametricity in an Impredicative Sort. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 381-395, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{keller_et_al:LIPIcs.CSL.2012.381,
  author =	{Keller, Chantal and Lasson, Marc},
  title =	{{Parametricity in an Impredicative Sort}},
  booktitle =	{Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL},
  pages =	{381--395},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-42-2},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{16},
  editor =	{C\'{e}gielski, Patrick and Durand, Arnaud},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.381},
  URN =		{urn:nbn:de:0030-drops-36851},
  doi =		{10.4230/LIPIcs.CSL.2012.381},
  annote =	{Keywords: Calculus of Inductive Constructions, parametricity, impredicativity, Coq, universes}
}

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Parametricity in an Impredicative Sort

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