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DOI: 10.4230/LIPIcs.FSCD.2024.21

Impredicativity, Cumulativity and Product Covariance in the Logical Framework Dedukti

Authors: Thiago Felicissimo and Théo Winterhalter

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Proof assistants such as Coq implement a type theory featuring three important features: impredicativity, cumulativity and product covariance. This combination has proven difficult to be expressed in the logical framework Dedukti, and previous attempts have failed in providing an encoding that is proven confluent, sound and conservative. In this work we solve this longstanding open problem by providing an encoding of these three features that we prove to be confluent, sound and to satisfy a restricted (but, we argue, strong enough) form of conservativity. Our proof of confluence is a contribution by itself, and combines various criteria and proof techniques from rewriting theory. Our proof of soundness also contributes a new strategy in which the result is shown in terms of an inverse translation function, fixing a common flaw made in some previous encoding attempts.

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Thiago Felicissimo and Théo Winterhalter. Impredicativity, Cumulativity and Product Covariance in the Logical Framework Dedukti. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 21:1-21:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{felicissimo_et_al:LIPIcs.FSCD.2024.21,
  author =	{Felicissimo, Thiago and Winterhalter, Th\'{e}o},
  title =	{{Impredicativity, Cumulativity and Product Covariance in the Logical Framework Dedukti}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{21:1--21:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.21},
  URN =		{urn:nbn:de:0030-drops-203503},
  doi =		{10.4230/LIPIcs.FSCD.2024.21},
  annote =	{Keywords: Dedukti, Rewriting, Confluence, Dependent types, Cumulativity, Universes}
}

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Impredicativity, Cumulativity and Product Covariance in the Logical Framework Dedukti

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