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.2013.653
URN: urn:nbn:de:0030-drops-42241
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2013/4224/
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Wang, Qian ; Barras, Bruno

Semantics of Intensional Type Theory extended with Decidable Equational Theories

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Abstract

Incorporating extensional equality into a dependent intensional type system such as the Calculus of Constructions (CC) provides with stronger type-checking capabilities and makes the proof development closer to intuition. Since strong forms of extensionality generally leads to undecidable type-checking, it seems a reasonable trade-off to extend intensional equality with a decidable first-order theory, as experimented in earlier work on CoqMTU and its implementation CoqMT.
In this work, CoqMTU is extended with strong eliminations. The meta-theoretical study, particularly the part relying on semantic arguments, is more complex. A set-theoretical model of the equational theory is the key ingredient to derive the logical consistency of the formalism. Strong normalization, the main lemma from which type-decidability follows, is proved by attaching realizability information to the values of the model.
The approach we have followed is to first consider an abstract notion of first-order equational theory, and then instantiate it with a particular instance, Presburger Arithmetic. These results have been formalized using Coq.

BibTeX - Entry

@InProceedings{wang_et_al:LIPIcs:2013:4224,
  author =	{Qian Wang and Bruno Barras},
  title =	{{Semantics of Intensional Type Theory extended with Decidable Equational Theories}},
  booktitle =	{Computer Science Logic 2013 (CSL 2013)},
  pages =	{653--667},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-60-6},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{23},
  editor =	{Simona Ronchi Della Rocca},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2013/4224},
  URN =		{urn:nbn:de:0030-drops-42241},
  doi =		{10.4230/LIPIcs.CSL.2013.653},
  annote =	{Keywords: Calculus of Constructions, Extensional Type Theory, Intensional Type Theory, Model, Meta-theory, Consistency, Strong Normalization, Presburger Arithme}
}

Keywords: Calculus of Constructions, Extensional Type Theory, Intensional Type Theory, Model, Meta-theory, Consistency, Strong Normalization, Presburger Arithme
Collection: Computer Science Logic 2013 (CSL 2013)
Issue Date: 2013
Date of publication: 02.09.2013

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