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DOI: 10.4230/LIPIcs.TYPES.2016.13
On Subtyping in Type Theories with Canonical Objects

Authors: Georgiana Elena Lungu and Zhaohui Luo

Published in: LIPIcs, Volume 97, 22nd International Conference on Types for Proofs and Programs (TYPES 2016)

Abstract
How should one introduce subtyping into type theories with canonical objects such as Martin-Löf's type theory? It is known that the usual subsumptive subtyping is inadequate and it is understood, at least theoretically, that coercive subtyping should instead be employed. However, it has not been studied what the proper coercive subtyping mechanism is and how it should be used to capture intuitive notions of subtyping. In this paper, we introduce a type system with signatures where coercive subtyping relations can be specified, and argue that this provides a suitable subtyping mechanism for type theories with canonical objects. In particular, we show that the subtyping extension is well-behaved by relating it to the previous formulation of coercive subtyping. The paper then proceeds to study the connection with intuitive notions of subtyping. It first shows how a subsumptive subtyping system can be embedded faithfully. Then, it studies how Russell-style universe inclusions can be understood as coercions in our system. And finally, we study constructor subtyping as an example to illustrate that, sometimes, injectivity of coercions need be assumed in order to capture properly some notions of subtyping.
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Georgiana Elena Lungu and Zhaohui Luo. On Subtyping in Type Theories with Canonical Objects. In 22nd International Conference on Types for Proofs and Programs (TYPES 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 97, pp. 13:1-13:31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{lungu_et_al:LIPIcs.TYPES.2016.13,
  author =	{Lungu, Georgiana Elena and Luo, Zhaohui},
  title =	{{On Subtyping in Type Theories with Canonical Objects}},
  booktitle =	{22nd International Conference on Types for Proofs and Programs (TYPES 2016)},
  pages =	{13:1--13:31},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-065-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{97},
  editor =	{Ghilezan, Silvia and Geuvers, Herman and Ivetic, Jelena},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2016.13},
  URN =		{urn:nbn:de:0030-drops-98496},
  doi =		{10.4230/LIPIcs.TYPES.2016.13},
  annote =	{Keywords: subtyping, type theory, conservative extension, canonical objects}
}
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