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

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.

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.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} }

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**Published in:** Dagstuhl Reports, Volume 11, Issue 10 (2022)

At least from a practical and contemporary angle, the time-honoured question about the extent of intuitionistic mathematics rather is to which extent any given proof is effective, which proofs of which theorems can be rendered effective, and whether and how numerical information such as bounds and algorithms can be extracted from proofs. All this is ideally done by manipulating proofs mechanically or by adequate metatheorems, which includes proof translations, automated theorem proving, program extraction from proofs, proof analysis and proof mining. The question should thus be put as: What is the computational content of proofs?
Guided by this central question, the present Dagstuhl seminar puts a special focus on coherent and geometric theories and their generalisations. These are not only widespread in mathematics and non-classical logics such as temporal and modal logics, but also a priori amenable for constructivisation, e.g., by Barr’s Theorem, and last but not least particularly suited as a basis for automated theorem proving. Specific topics include categorical semantics for geometric theories, complexity issues of and algorithms for geometrisation of theories including speed-up questions, the use of geometric theories in constructive mathematics including finding algorithms, proof-theoretic presentation of sheaf models and higher toposes, and coherent logic for automatically readable proofs.

Thierry Coquand, Hajime Ishihara, Sara Negri, and Peter M. Schuster. Geometric Logic, Constructivisation, and Automated Theorem Proving (Dagstuhl Seminar 21472). In Dagstuhl Reports, Volume 11, Issue 10, pp. 151-172, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{coquand_et_al:DagRep.11.10.151, author = {Coquand, Thierry and Ishihara, Hajime and Negri, Sara and Schuster, Peter M.}, title = {{Geometric Logic, Constructivisation, and Automated Theorem Proving (Dagstuhl Seminar 21472)}}, pages = {151--172}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2022}, volume = {11}, number = {10}, editor = {Coquand, Thierry and Ishihara, Hajime and Negri, Sara and Schuster, Peter M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.11.10.151}, URN = {urn:nbn:de:0030-drops-159321}, doi = {10.4230/DagRep.11.10.151}, annote = {Keywords: automated theorem proving, categorical semantics, constructivisation, geometric logic, proof theory} }

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**Published in:** LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)

Cubical type theory provides a constructive justification of homotopy type theory and satisfies canonicity: every natural number is convertible to a numeral. A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices. In this paper we show by a sconing argument that if we remove these equations for the path lifting operation from the system, we still retain homotopy canonicity: every natural number is path equal to a numeral.

Thierry Coquand, Simon Huber, and Christian Sattler. Homotopy Canonicity for Cubical Type Theory. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{coquand_et_al:LIPIcs.FSCD.2019.11, author = {Coquand, Thierry and Huber, Simon and Sattler, Christian}, title = {{Homotopy Canonicity for Cubical Type Theory}}, booktitle = {4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)}, pages = {11:1--11:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-107-8}, ISSN = {1868-8969}, year = {2019}, volume = {131}, editor = {Geuvers, Herman}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.11}, URN = {urn:nbn:de:0030-drops-105188}, doi = {10.4230/LIPIcs.FSCD.2019.11}, annote = {Keywords: cubical type theory, univalence, canonicity, sconing, Artin glueing} }

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

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.

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.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} }

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

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.

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.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} }

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**Published in:** LIPIcs, Volume 69, 21st International Conference on Types for Proofs and Programs (TYPES 2015) (2018)

This paper presents a type theory in which it is possible to
directly manipulate $n$-dimensional cubes (points, lines, squares,
cubes, etc.) based on an interpretation of dependent type theory in
a cubical set model. This enables new ways to reason about identity
types, for instance, function extensionality is directly provable in
the system. Further, Voevodsky's univalence axiom is provable in
this system. We also explain an extension with some higher inductive
types like the circle and propositional truncation. Finally we
provide semantics for this cubical type theory in a constructive
meta-theory.

Cyril Cohen, Thierry Coquand, Simon Huber, and Anders Mörtberg. Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom. In 21st International Conference on Types for Proofs and Programs (TYPES 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 69, pp. 5:1-5:34, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{cohen_et_al:LIPIcs.TYPES.2015.5, author = {Cohen, Cyril and Coquand, Thierry and Huber, Simon and M\"{o}rtberg, Anders}, title = {{Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom}}, booktitle = {21st International Conference on Types for Proofs and Programs (TYPES 2015)}, pages = {5:1--5:34}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-030-9}, ISSN = {1868-8969}, year = {2018}, volume = {69}, editor = {Uustalu, Tarmo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2015.5}, URN = {urn:nbn:de:0030-drops-84754}, doi = {10.4230/LIPIcs.TYPES.2015.5}, annote = {Keywords: univalence axiom, dependent type theory, cubical sets} }

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**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

The Ackermann Award is the EACSL Outstanding Dissertation Award for Logic in Computer Science. It is presented during the annual conference of the EACSL (CSL'xx). This contribution reports on the 2016 edition of the award.

Thierry Coquand and Anuj Dawar. The Ackermann Award 2016. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 1:1-1:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{coquand_et_al:LIPIcs.CSL.2016.1, author = {Coquand, Thierry and Dawar, Anuj}, title = {{The Ackermann Award 2016}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {1:1--1:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.1}, URN = {urn:nbn:de:0030-drops-65419}, doi = {10.4230/LIPIcs.CSL.2016.1}, annote = {Keywords: Ackermann Award, Computer Science, Logic} }

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**Published in:** LIPIcs, Volume 52, 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)

In this paper, we show that Markov's principle is not derivable in
dependent type theory with natural numbers and one universe. One
tentative way to prove this would be to remark that Markov's principle
does not hold in a sheaf model of type theory over Cantor space, since
Markov's principle does not hold for the generic point of this model.
It is however not clear how to interpret the universe in a sheaf
model. Instead we design an extension of type theory, which
intuitively extends type theory by the addition of a generic point of
Cantor space. We then show the consistency of this extension by a
normalization argument. Markov's principle does not hold in this
extension, and it follows that it cannot be proved in type theory.

Thierry Coquand and Bassel Mannaa. The Independence of Markov’s Principle in Type Theory. In 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 52, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{coquand_et_al:LIPIcs.FSCD.2016.17, author = {Coquand, Thierry and Mannaa, Bassel}, title = {{The Independence of Markov’s Principle in Type Theory}}, booktitle = {1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)}, pages = {17:1--17:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-010-1}, ISSN = {1868-8969}, year = {2016}, volume = {52}, editor = {Kesner, Delia and Pientka, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2016.17}, URN = {urn:nbn:de:0030-drops-59939}, doi = {10.4230/LIPIcs.FSCD.2016.17}, annote = {Keywords: Forcing, Dependent type theory, Markov's Principle, Cantor Space} }

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

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.

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.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} }

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

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.

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.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} }

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**Published in:** LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

Report on the Ackermann Award 2012.

Thierry Coquand, Anuj Dawar, and Damian Niwinski. The Ackermann Award 2012. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{coquand_et_al:LIPIcs.CSL.2012.1, author = {Coquand, Thierry and Dawar, Anuj and Niwinski, Damian}, title = {{The Ackermann Award 2012}}, booktitle = {Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL}, pages = {1--5}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-42-2}, ISSN = {1868-8969}, year = {2012}, volume = {16}, editor = {C\'{e}gielski, Patrick and Durand, Arnaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.1}, URN = {urn:nbn:de:0030-drops-36575}, doi = {10.4230/LIPIcs.CSL.2012.1}, annote = {Keywords: Ackermann Award 2012} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 5021, Mathematics, Algorithms, Proofs (2006)

From 09.01.05 to 14.01.05, the Dagstuhl Seminar 05021 ``Mathematics, Algorithms, Proofs'' was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
LinkstFo extended abstracts or full papers are provided, if available.

Thierry Coquand, Henri Lombardi, and Marie-Françoise Roy. 05021 Abstracts Collection – Mathematics, Algorithms, Proofs. In Mathematics, Algorithms, Proofs. Dagstuhl Seminar Proceedings, Volume 5021, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{coquand_et_al:DagSemProc.05021.1, author = {Coquand, Thierry and Lombardi, Henri and Roy, Marie-Fran\c{c}oise}, title = {{05021 Abstracts Collection – Mathematics, Algorithms, Proofs}}, booktitle = {Mathematics, Algorithms, Proofs}, pages = {1--15}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {5021}, editor = {Thierry Coquand and Henri Lombardi and Marie-Fran\c{c}oise Roy}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05021.1}, URN = {urn:nbn:de:0030-drops-3038}, doi = {10.4230/DagSemProc.05021.1}, annote = {Keywords: Constructive mathematics, computer algebra, proof systems} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 5021, Mathematics, Algorithms, Proofs (2006)

This workshop was the third MAP meeting,
a continuation of the seminar "Verification and constructive algebra" held in Dagstuhl from 6 to 10 January 2003.
The goal of these meetings is to bring together people from the communities of formal proofs, constructive
mathematics and computer algebra (in a wide meaning).
The special
emphasis of the present meeting was on the constructive mathematics and efficient proofs in computer algebra.

Thierry Coquand. 05021 Executive Summary – Mathematics, Algorithms, Proofs. In Mathematics, Algorithms, Proofs. Dagstuhl Seminar Proceedings, Volume 5021, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{coquand:DagSemProc.05021.2, author = {Coquand, Thierry}, title = {{05021 Executive Summary – Mathematics, Algorithms, Proofs}}, booktitle = {Mathematics, Algorithms, Proofs}, pages = {1--3}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {5021}, editor = {Thierry Coquand and Henri Lombardi and Marie-Fran\c{c}oise Roy}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05021.2}, URN = {urn:nbn:de:0030-drops-4385}, doi = {10.4230/DagSemProc.05021.2}, annote = {Keywords: Constructive mathematics, computer algebra, proof systems} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 5021, Mathematics, Algorithms, Proofs (2006)

An element or an ideal of a commutative ring is nilregular if and only if
it is regular modulo the nilradical. We prove that if the ring is
Noetherian, then every nilregular ideal contains a nilregular element. In
constructive mathematics, this proof can then be seen as an algorithm to
produce nilregular elements of nilregular ideals whenever the ring is coherent,
Noetherian, and discrete. As an application, we give a constructive proof of
the Eisenbud--Evans--Storch theorem that every algebraic set in
$n$--dimensional affine space is the intersection of $n$ hypersurfaces.
The input of the algorithm is an arbitrary finite list of polynomials,
which need not arrive in a special form such as a Gr"obner basis.
We dispense with prime ideals when defining concepts or carrying out proofs.

Thierry Coquand, Henri Lombardi, and Peter Schuster. A Nilregular Element Property. In Mathematics, Algorithms, Proofs. Dagstuhl Seminar Proceedings, Volume 5021, pp. 1-6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{coquand_et_al:DagSemProc.05021.4, author = {Coquand, Thierry and Lombardi, Henri and Schuster, Peter}, title = {{A Nilregular Element Property}}, booktitle = {Mathematics, Algorithms, Proofs}, pages = {1--6}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {5021}, editor = {Thierry Coquand and Henri Lombardi and Marie-Fran\c{c}oise Roy}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05021.4}, URN = {urn:nbn:de:0030-drops-2784}, doi = {10.4230/DagSemProc.05021.4}, annote = {Keywords: Lists of generators, polynomial ideals, Krull dimension, Zariski topology, commutative Noetherian rings, constructive algebra} }

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**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Thierry Coquand, Henri Lombardi, and Marie-Françoise Roy. Verification and Constructive Algebra (Dagstuhl Seminar 03021). Dagstuhl Seminar Report 362, pp. 1-17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2004)

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@TechReport{coquand_et_al:DagSemRep.362, author = {Coquand, Thierry and Lombardi, Henri and Roy, Marie-Fran\c{c}oise}, title = {{Verification and Constructive Algebra (Dagstuhl Seminar 03021)}}, pages = {1--17}, ISSN = {1619-0203}, year = {2004}, type = {Dagstuhl Seminar Report}, number = {362}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.362}, URN = {urn:nbn:de:0030-drops-152425}, doi = {10.4230/DagSemRep.362}, }