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DOI: 10.4230/LIPIcs.TYPES.2016.9
URN: urn:nbn:de:0030-drops-98533
URL: http://drops.dagstuhl.de/opus/volltexte/2018/9853/
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Czajka, Lukasz

A Shallow Embedding of Pure Type Systems into First-Order Logic

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LIPIcs-TYPES-2016-9.pdf (0.6 MB)


Abstract

We define a shallow embedding of logical proof-irrelevant Pure Type Systems (piPTSs) into minimal first-order logic. In logical piPTSs a distinguished sort *^p of propositions is assumed. Given a context Gamma and a Gamma-proposition tau, i.e., a term tau such that Gamma |- tau : *^p, the embedding translates tau and Gamma into a first-order formula F_Gamma(tau) and a set of first-order axioms Delta_Gamma. The embedding is not complete in general, but it is strong enough to correctly translate most of piPTS propositions (by completeness we mean that if Gamma |- M : tau is derivable in the piPTS then F_Gamma(tau) is provable in minimal first-order logic from the axioms Delta_Gamma). We show the embedding to be sound, i.e., if F_Gamma(tau) is provable in minimal first-order logic from the axioms Delta_Gamma, then Gamma |- M : tau is derivable in the original system for some term M. The interest in the proposed embedding stems from the fact that it forms a basis of the translations used in the recently developed CoqHammer automation tool for dependent type theory.

BibTeX - Entry

@InProceedings{czajka:LIPIcs:2018:9853,
  author =	{Lukasz Czajka},
  title =	{{A Shallow Embedding of Pure Type Systems into First-Order Logic}},
  booktitle =	{22nd International Conference on Types for Proofs and  Programs (TYPES 2016)},
  pages =	{9:1--9:39},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-065-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{97},
  editor =	{Silvia Ghilezan and Herman Geuvers and Jelena Ivetić},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9853},
  URN =		{urn:nbn:de:0030-drops-98533},
  doi =		{10.4230/LIPIcs.TYPES.2016.9},
  annote =	{Keywords: pure type systems, first-order logic, hammers, proof automation, dependent type theory}
}

Keywords: pure type systems, first-order logic, hammers, proof automation, dependent type theory
Seminar: 22nd International Conference on Types for Proofs and Programs (TYPES 2016)
Issue Date: 2018
Date of publication: 24.10.2018


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