License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CSL.2018.6
URN: urn:nbn:de:0030-drops-96734
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9673/
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Angiuli, Carlo ; Hou (Favonia), Kuen-Bang ; Harper, Robert

Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities

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Abstract

We present a dependent type theory organized around a Cartesian notion of cubes (with faces, degeneracies, and diagonals), supporting both fibrant and non-fibrant types. The fibrant fragment validates Voevodsky's univalence axiom and includes a circle type, while the non-fibrant fragment includes exact (strict) equality types satisfying equality reflection. Our type theory is defined by a semantics in cubical partial equivalence relations, and is the first two-level type theory to satisfy the canonicity property: all closed terms of boolean type evaluate to either true or false.

BibTeX - Entry

@InProceedings{angiuli_et_al:LIPIcs:2018:9673,
  author =	{Carlo Angiuli and Kuen-Bang {Hou (Favonia)} and Robert Harper},
  title =	{{Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities}},
  booktitle =	{27th EACSL Annual Conference on Computer Science Logic  (CSL 2018)},
  pages =	{6:1--6:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-088-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{119},
  editor =	{Dan Ghica and Achim Jung},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9673},
  URN =		{urn:nbn:de:0030-drops-96734},
  doi =		{10.4230/LIPIcs.CSL.2018.6},
  annote =	{Keywords: Homotopy Type Theory, Two-Level Type Theory, Computational Type Theory, Cubical Sets}
}

Keywords: Homotopy Type Theory, Two-Level Type Theory, Computational Type Theory, Cubical Sets
Collection: 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)
Issue Date: 2018
Date of publication: 29.08.2018

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