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Documents authored by Bezem, Marc

Document
DOI: 10.4230/LIPIcs.TYPES.2022.13
Type Theory with Explicit Universe Polymorphism

Authors: Marc Bezem, Thierry Coquand, Peter Dybjer, and Martín Escardó

Published in: LIPIcs, Volume 269, 28th International Conference on Types for Proofs and Programs (TYPES 2022)

Abstract
The aim of this paper is to refine and extend proposals by Sozeau and Tabareau and by Voevodsky for universe polymorphism in type theory. In those systems judgments can depend on explicit constraints between universe levels. We here present a system where we also have products indexed by universe levels and by constraints. Our theory has judgments for internal universe levels, built up from level variables by a successor operation and a binary supremum operation, and also judgments for equality of universe levels.
Cite as

Marc Bezem, Thierry Coquand, Peter Dybjer, and Martín Escardó. Type Theory with Explicit Universe Polymorphism. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bezem_et_al:LIPIcs.TYPES.2022.13,
  author =	{Bezem, Marc and Coquand, Thierry and Dybjer, Peter and Escard\'{o}, Mart{\'\i}n},
  title =	{{Type Theory with Explicit Universe Polymorphism}},
  booktitle =	{28th International Conference on Types for Proofs and Programs (TYPES 2022)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-285-3},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{269},
  editor =	{Kesner, Delia and P\'{e}drot, Pierre-Marie},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2022.13},
  URN =		{urn:nbn:de:0030-drops-184564},
  doi =		{10.4230/LIPIcs.TYPES.2022.13},
  annote =	{Keywords: type theory, universes in type theory, universe polymorphism, level-indexed products, constraint-indexed products}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.TYPES.2019
LIPIcs, Volume 175, TYPES 2019, Complete Volume

Authors: Marc Bezem and Assia Mahboubi

Published in: LIPIcs, Volume 175, 25th International Conference on Types for Proofs and Programs (TYPES 2019)

Abstract
LIPIcs, Volume 175, TYPES 2019, Complete Volume
Cite as

25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 1-256, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@Proceedings{bezem_et_al:LIPIcs.TYPES.2019,
  title =	{{LIPIcs, Volume 175, TYPES 2019, Complete Volume}},
  booktitle =	{25th International Conference on Types for Proofs and Programs (TYPES 2019)},
  pages =	{1--256},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-158-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{175},
  editor =	{Bezem, Marc and Mahboubi, Assia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019},
  URN =		{urn:nbn:de:0030-drops-130639},
  doi =		{10.4230/LIPIcs.TYPES.2019},
  annote =	{Keywords: LIPIcs, Volume 175, TYPES 2019, Complete Volume}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.TYPES.2019.0
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Marc Bezem and Assia Mahboubi

Published in: LIPIcs, Volume 175, 25th International Conference on Types for Proofs and Programs (TYPES 2019)

Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as

25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 0:i-0:x, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bezem_et_al:LIPIcs.TYPES.2019.0,
  author =	{Bezem, Marc and Mahboubi, Assia},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{25th International Conference on Types for Proofs and Programs (TYPES 2019)},
  pages =	{0:i--0:x},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-158-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{175},
  editor =	{Bezem, Marc and Mahboubi, Assia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.0},
  URN =		{urn:nbn:de:0030-drops-130640},
  doi =		{10.4230/LIPIcs.TYPES.2019.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2016.3
A Normalizing Computation Rule for Propositional Extensionality in Higher-Order Minimal Logic

Authors: Robin Adams, Marc Bezem, and Thierry Coquand

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

Abstract
The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A computation becomes "stuck" when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over Omega. There are proofs which inhabit propositions, which are the terms of type Omega. The canonical propositions are those constructed from false by implication. Thirdly, there are paths which inhabit equations M =_A N, where M and N are terms of type A. There are two ways to prove an equality: reflexivity, and propositional extensionality - logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda.
Cite as

Robin Adams, Marc Bezem, and Thierry Coquand. A Normalizing Computation Rule for Propositional Extensionality in Higher-Order Minimal Logic. In 22nd International Conference on Types for Proofs and Programs (TYPES 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 97, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{adams_et_al:LIPIcs.TYPES.2016.3,
  author =	{Adams, Robin and Bezem, Marc and Coquand, Thierry},
  title =	{{A Normalizing Computation Rule for Propositional Extensionality in Higher-Order Minimal Logic}},
  booktitle =	{22nd International Conference on Types for Proofs and Programs (TYPES 2016)},
  pages =	{3:1--3:20},
  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.2016.3},
  URN =		{urn:nbn:de:0030-drops-98581},
  doi =		{10.4230/LIPIcs.TYPES.2016.3},
  annote =	{Keywords: type theory, univalence, canonicity}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2016.6
Realizability at Work: Separating Two Constructive Notions of Finiteness

Authors: Marc Bezem, Thierry Coquand, Keiko Nakata, and Erik Parmann

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

Abstract
We elaborate in detail a realizability model for Martin-Löf dependent type theory with the purpose to analyze a subtle distinction between two constructive notions of finiteness of a set A. The two notions are: (1) A is Noetherian: the empty list can be constructed from lists over A containing duplicates by a certain inductive shortening process; (2) A is streamless: every enumeration of A contains a duplicate.
Cite as

Marc Bezem, Thierry Coquand, Keiko Nakata, and Erik Parmann. Realizability at Work: Separating Two Constructive Notions of Finiteness. In 22nd International Conference on Types for Proofs and Programs (TYPES 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 97, pp. 6:1-6:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bezem_et_al:LIPIcs.TYPES.2016.6,
  author =	{Bezem, Marc and Coquand, Thierry and Nakata, Keiko and Parmann, Erik},
  title =	{{Realizability at Work: Separating Two Constructive Notions of Finiteness}},
  booktitle =	{22nd International Conference on Types for Proofs and Programs (TYPES 2016)},
  pages =	{6:1--6:23},
  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.2016.6},
  URN =		{urn:nbn:de:0030-drops-98541},
  doi =		{10.4230/LIPIcs.TYPES.2016.6},
  annote =	{Keywords: Type theory, realizability, constructive notions of finiteness}
}
Document
DOI: 10.4230/LIPIcs.TLCA.2015.92
Non-Constructivity in Kan Simplicial Sets

Authors: Marc Bezem, Thierry Coquand, and Erik Parmann

Published in: LIPIcs, Volume 38, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)

Abstract
We give an analysis of the non-constructivity of the following basic result: if X and Y are simplicial sets and Y has the Kan extension property, then Y^X also has the Kan extension property. By means of Kripke countermodels we show that even simple consequences of this basic result, such as edge reversal and edge composition, are not constructively provable. We also show that our unprovability argument will have to be refined if one strengthens the usual formulation of the Kan extension property to one with explicit horn-filler operations.
Cite as

Marc Bezem, Thierry Coquand, and Erik Parmann. Non-Constructivity in Kan Simplicial Sets. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 92-106, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bezem_et_al:LIPIcs.TLCA.2015.92,
  author =	{Bezem, Marc and Coquand, Thierry and Parmann, Erik},
  title =	{{Non-Constructivity in Kan Simplicial Sets}},
  booktitle =	{13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)},
  pages =	{92--106},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-87-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{38},
  editor =	{Altenkirch, Thorsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TLCA.2015.92},
  URN =		{urn:nbn:de:0030-drops-51579},
  doi =		{10.4230/LIPIcs.TLCA.2015.92},
  annote =	{Keywords: Constructive logic, simplicial sets, semantics of simple types}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2013.107
A Model of Type Theory in Cubical Sets

Authors: Marc Bezem, Thierry Coquand, and Simon Huber

Published in: LIPIcs, Volume 26, 19th International Conference on Types for Proofs and Programs (TYPES 2013)

Abstract
We present a model of type theory with dependent product, sum, and identity, in cubical sets. We describe a universe and explain how to transform an equivalence between two types into an equality. We also explain how to model propositional truncation and the circle. While not expressed internally in type theory, the model is expressed in a constructive metalogic. Thus it is a step towards a computational interpretation of Voevodsky's Univalence Axiom.
Cite as

Marc Bezem, Thierry Coquand, and Simon Huber. A Model of Type Theory in Cubical Sets. In 19th International Conference on Types for Proofs and Programs (TYPES 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 26, pp. 107-128, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{bezem_et_al:LIPIcs.TYPES.2013.107,
  author =	{Bezem, Marc and Coquand, Thierry and Huber, Simon},
  title =	{{A Model of Type Theory in Cubical Sets}},
  booktitle =	{19th International Conference on Types for Proofs and Programs (TYPES 2013)},
  pages =	{107--128},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-72-9},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{26},
  editor =	{Matthes, Ralph and Schubert, Aleksy},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2013.107},
  URN =		{urn:nbn:de:0030-drops-46284},
  doi =		{10.4230/LIPIcs.TYPES.2013.107},
  annote =	{Keywords: Models of dependent type theory, cubical sets, Univalent Foundations}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.CSL.2011
LIPIcs, Volume 12, CSL'11, Complete Volume

Authors: Marc Bezem

Published in: LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

Abstract
LIPIcs, Volume 12, CSL'11, Complete Volume
Cite as

Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@Proceedings{bezem:LIPIcs.CSL.2011,
  title =	{{LIPIcs, Volume 12, CSL'11, Complete Volume}},
  booktitle =	{Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-32-3},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{12},
  editor =	{Bezem, Marc},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011},
  URN =		{urn:nbn:de:0030-drops-41066},
  doi =		{10.4230/LIPIcs.CSL.2011},
  annote =	{Keywords: Conference Proceedings; Software; Theory of Computation; Graph Theory; Probability and Statistics; Artificial Intelligence}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.CSL.2011.i
Frontmatter, Table of Contents, Preface, Conference Organization

Authors: Marc Bezem

Published in: LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

Abstract
Frontmatter, Table of Contents, Preface, Conference Organization
Cite as

Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. i-xix, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{bezem:LIPIcs.CSL.2011.i,
  author =	{Bezem, Marc},
  title =	{{Frontmatter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL},
  pages =	{i--xix},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-32-3},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{12},
  editor =	{Bezem, Marc},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.i},
  URN =		{urn:nbn:de:0030-drops-32144},
  doi =		{10.4230/LIPIcs.CSL.2011.i},
  annote =	{Keywords: Frontmatter, Table of Contents, Preface, Conference Organization}
}
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