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DOI: 10.4230/LIPIcs.CSL.2016.23
URN: urn:nbn:de:0030-drops-65638
URL: http://drops.dagstuhl.de/opus/volltexte/2016/6563/
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Birkedal, Lars ; Bizjak, Aleš ; Clouston, Ranald ; Grathwohl, Hans Bugge ; Spitters, Bas ; Vezzosi, Andrea

Guarded Cubical Type Theory: Path Equality for Guarded Recursion

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LIPIcs-CSL-2016-23.pdf (0.6 MB)


Abstract

This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-Löf type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type checker. Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category.

BibTeX - Entry

@InProceedings{birkedal_et_al:LIPIcs:2016:6563,
  author =	{Lars Birkedal and Ale{\v{s}} Bizjak and Ranald Clouston and Hans Bugge Grathwohl and Bas Spitters and Andrea Vezzosi},
  title =	{{Guarded Cubical Type Theory: Path Equality for Guarded Recursion}},
  booktitle =	{25th EACSL Annual Conference on Computer Science Logic (CSL 2016)},
  pages =	{23:1--23:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-022-4},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{62},
  editor =	{Jean-Marc Talbot and Laurent Regnier},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6563},
  URN =		{urn:nbn:de:0030-drops-65638},
  doi =		{10.4230/LIPIcs.CSL.2016.23},
  annote =	{Keywords: Guarded Recursion, Dependent Type Theory, Cubical Type Theory, Denotational Semantics, Homotopy Type Theory}
}

Keywords: Guarded Recursion, Dependent Type Theory, Cubical Type Theory, Denotational Semantics, Homotopy Type Theory
Seminar: 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)
Issue Date: 2016
Date of publication: 25.08.2016


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