9 Search Results for "Dowek, Gilles"


Document
Expressing Ecumenical Systems in the λΠ-Calculus Modulo Theory

Authors: Emilie Grienenberger

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


Abstract
Systems in which classical and intuitionistic logics coexist are called ecumenical. Such a system allows for interoperability and hybridization between classical and constructive propositions and proofs. We study Ecumenical STT, a theory expressed in the logical framework of the λΠ-calculus modulo theory. We prove soudness and conservativity of four subtheories of Ecumenical STT with respect to constructive and classical predicate logic and simple type theory. We also prove the weak normalization of well-typed terms and thus the consistency of Ecumenical STT.

Cite as

Emilie Grienenberger. Expressing Ecumenical Systems in the λΠ-Calculus Modulo Theory. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 4:1-4:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{grienenberger:LIPIcs.TYPES.2022.4,
  author =	{Grienenberger, Emilie},
  title =	{{Expressing Ecumenical Systems in the \lambda\Pi-Calculus Modulo Theory}},
  booktitle =	{28th International Conference on Types for Proofs and Programs (TYPES 2022)},
  pages =	{4:1--4:23},
  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.2022.4},
  URN =		{urn:nbn:de:0030-drops-184479},
  doi =		{10.4230/LIPIcs.TYPES.2022.4},
  annote =	{Keywords: dependent types, predicate logic, higher order logic, constructivism, interoperability, ecumenical logics}
}
Document
Translating Proofs from an Impredicative Type System to a Predicative One

Authors: Thiago Felicissimo, Frédéric Blanqui, and Ashish Kumar Barnawal

Published in: LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)


Abstract
As the development of formal proofs is a time-consuming task, it is important to devise ways of sharing the already written proofs to prevent wasting time redoing them. One of the challenges in this domain is to translate proofs written in proof assistants based on impredicative logics, such as Coq, Matita and the HOL family, to proof assistants based on predicative logics like Agda, whenever impredicativity is not used in an essential way. In this paper we present an algorithm to do such a translation between a core impredicative type system and a core predicative one allowing prenex universe polymorphism like in Agda. It consists in trying to turn a potentially impredicative term into a universe polymorphic term as general as possible. The use of universe polymorphism is justified by the fact that mapping an impredicative universe to a fixed predicative one is not sufficient in most cases. During the algorithm, we need to solve unification problems modulo the max-successor algebra on universe levels. But, in this algebra, there are solvable problems having no most general solution. We however provide an incomplete algorithm whose solutions, when it succeeds, are most general ones. The proposed translation is of course partial, but in practice allows one to translate many proofs that do not use impredicativity in an essential way. Indeed, it was implemented in the tool Predicativize and then used to translate semi-automatically many non-trivial developments from Matita’s arithmetic library to Agda, including Bertrand’s Postulate and Fermat’s Little Theorem, which were not available in Agda yet.

Cite as

Thiago Felicissimo, Frédéric Blanqui, and Ashish Kumar Barnawal. Translating Proofs from an Impredicative Type System to a Predicative One. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{felicissimo_et_al:LIPIcs.CSL.2023.19,
  author =	{Felicissimo, Thiago and Blanqui, Fr\'{e}d\'{e}ric and Barnawal, Ashish Kumar},
  title =	{{Translating Proofs from an Impredicative Type System to a Predicative One}},
  booktitle =	{31st EACSL Annual Conference on Computer Science Logic (CSL 2023)},
  pages =	{19:1--19:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-264-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{252},
  editor =	{Klin, Bartek and Pimentel, Elaine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.19},
  URN =		{urn:nbn:de:0030-drops-174801},
  doi =		{10.4230/LIPIcs.CSL.2023.19},
  annote =	{Keywords: Type Theory, Impredicativity, Predicativity, Proof Translation, Universe Polymorphism, Unification Modulo Max, Agda, Dedukti}
}
Document
Linear Lambda-Calculus is Linear

Authors: Alejandro Díaz-Caro and Gilles Dowek

Published in: LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)


Abstract
We prove a linearity theorem for an extension of linear logic with addition and multiplication by a scalar: the proofs of some propositions in this logic are linear in the algebraic sense. This work is part of a wider research program that aims at defining a logic whose proof language is a quantum programming language.

Cite as

Alejandro Díaz-Caro and Gilles Dowek. Linear Lambda-Calculus is Linear. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{diazcaro_et_al:LIPIcs.FSCD.2022.21,
  author =	{D{\'\i}az-Caro, Alejandro and Dowek, Gilles},
  title =	{{Linear Lambda-Calculus is Linear}},
  booktitle =	{7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)},
  pages =	{21:1--21:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-233-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{228},
  editor =	{Felty, Amy P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.21},
  URN =		{urn:nbn:de:0030-drops-163024},
  doi =		{10.4230/LIPIcs.FSCD.2022.21},
  annote =	{Keywords: Proof theory, Lambda calculus, Linear logic, Quantum computing}
}
Document
Adequate and Computational Encodings in the Logical Framework Dedukti

Authors: Thiago Felicissimo

Published in: LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)


Abstract
Dedukti is a very expressive logical framework which unlike most frameworks, such as the Edinburgh Logical Framework (LF), allows for the representation of computation alongside deduction. However, unlike LF encodings, Dedukti encodings proposed until now do not feature an adequacy theorem - i.e., a bijection between terms in the encoded system and in its encoding. Moreover, many of them also do not have a conservativity result, which compromises the ability of Dedukti to check proofs written in such encodings. We propose a different approach for Dedukti encodings which do not only allow for simpler conservativity proofs, but which also restore the adequacy of encodings. More precisely, we propose in this work adequate (and thus conservative) encodings for Functional Pure Type Systems. However, in contrast with LF encodings, ours is computational - that is, represents computation directly as computation. Therefore, our work is the first to present and prove correct an approach allowing for encodings that are both adequate and computational in Dedukti.

Cite as

Thiago Felicissimo. Adequate and Computational Encodings in the Logical Framework Dedukti. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 25:1-25:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{felicissimo:LIPIcs.FSCD.2022.25,
  author =	{Felicissimo, Thiago},
  title =	{{Adequate and Computational Encodings in the Logical Framework Dedukti}},
  booktitle =	{7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)},
  pages =	{25:1--25:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-233-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{228},
  editor =	{Felty, Amy P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.25},
  URN =		{urn:nbn:de:0030-drops-163064},
  doi =		{10.4230/LIPIcs.FSCD.2022.25},
  annote =	{Keywords: Type Theory, Logical Frameworks, Rewriting, Dedukti, Pure Type Systems}
}
Document
Some Axioms for Mathematics

Authors: Frédéric Blanqui, Gilles Dowek, Émilie Grienenberger, Gabriel Hondet, and François Thiré

Published in: LIPIcs, Volume 195, 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)


Abstract
The λΠ-calculus modulo theory is a logical framework in which many logical systems can be expressed as theories. We present such a theory, the theory {U}, where proofs of several logical systems can be expressed. Moreover, we identify a sub-theory of {U} corresponding to each of these systems, and prove that, when a proof in {U} uses only symbols of a sub-theory, then it is a proof in that sub-theory.

Cite as

Frédéric Blanqui, Gilles Dowek, Émilie Grienenberger, Gabriel Hondet, and François Thiré. Some Axioms for Mathematics. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{blanqui_et_al:LIPIcs.FSCD.2021.20,
  author =	{Blanqui, Fr\'{e}d\'{e}ric and Dowek, Gilles and Grienenberger, \'{E}milie and Hondet, Gabriel and Thir\'{e}, Fran\c{c}ois},
  title =	{{Some Axioms for Mathematics}},
  booktitle =	{6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)},
  pages =	{20:1--20:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-191-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{195},
  editor =	{Kobayashi, Naoki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2021.20},
  URN =		{urn:nbn:de:0030-drops-142581},
  doi =		{10.4230/LIPIcs.FSCD.2021.20},
  annote =	{Keywords: logical framework, axiomatic theory, dependent types, rewriting, interoperabilty}
}
Document
Encoding of Predicate Subtyping with Proof Irrelevance in the λΠ-Calculus Modulo Theory

Authors: Gabriel Hondet and Frédéric Blanqui

Published in: LIPIcs, Volume 188, 26th International Conference on Types for Proofs and Programs (TYPES 2020)


Abstract
The λΠ-calculus modulo theory is a logical framework in which various logics and type systems can be encoded, thus helping the cross-verification and interoperability of proof systems based on those logics and type systems. In this paper, we show how to encode predicate subtyping and proof irrelevance, two important features of the PVS proof assistant. We prove that this encoding is correct and that encoded proofs can be mechanically checked by Dedukti, a type checker for the λΠ-calculus modulo theory using rewriting.

Cite as

Gabriel Hondet and Frédéric Blanqui. Encoding of Predicate Subtyping with Proof Irrelevance in the λΠ-Calculus Modulo Theory. In 26th International Conference on Types for Proofs and Programs (TYPES 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 188, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{hondet_et_al:LIPIcs.TYPES.2020.6,
  author =	{Hondet, Gabriel and Blanqui, Fr\'{e}d\'{e}ric},
  title =	{{Encoding of Predicate Subtyping with Proof Irrelevance in the \lambda\Pi-Calculus Modulo Theory}},
  booktitle =	{26th International Conference on Types for Proofs and Programs (TYPES 2020)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-182-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{188},
  editor =	{de'Liguoro, Ugo and Berardi, Stefano and 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.TYPES.2020.6},
  URN =		{urn:nbn:de:0030-drops-138853},
  doi =		{10.4230/LIPIcs.TYPES.2020.6},
  annote =	{Keywords: Predicate Subtyping, Logical Framework, PVS, Dedukti, Proof Irrelevance}
}
Document
Proof Normalisation in a Logic Identifying Isomorphic Propositions

Authors: Alejandro Díaz-Caro and Gilles Dowek

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


Abstract
We define a fragment of propositional logic where isomorphic propositions, such as A wedge B and B wedge A, or A ==> (B wedge C) and (A ==> B) wedge (A ==> C) are identified. We define System I, a proof language for this logic, and prove its normalisation and consistency.

Cite as

Alejandro Díaz-Caro and Gilles Dowek. Proof Normalisation in a Logic Identifying Isomorphic Propositions. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 14:1-14:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{diazcaro_et_al:LIPIcs.FSCD.2019.14,
  author =	{D{\'\i}az-Caro, Alejandro and Dowek, Gilles},
  title =	{{Proof Normalisation in a Logic Identifying Isomorphic Propositions}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{14:1--14:23},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.14},
  URN =		{urn:nbn:de:0030-drops-105210},
  doi =		{10.4230/LIPIcs.FSCD.2019.14},
  annote =	{Keywords: Simply typed lambda calculus, Isomorphisms, Logic, Cut-elimination, Proof-reduction}
}
Document
Models and Termination of Proof Reduction in the lambda Pi-Calculus Modulo Theory

Authors: Gilles Dowek

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)


Abstract
We define a notion of model for the lambda Pi-calculus modulo theory and prove a soundness theorem. We then define a notion of super-consistency and prove that proof reduction terminates in the lambda Pi-calculus modulo any super-consistent theory. We prove this way the termination of proof reduction in several theories including Simple type theory and the Calculus of constructions.

Cite as

Gilles Dowek. Models and Termination of Proof Reduction in the lambda Pi-Calculus Modulo Theory. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 109:1-109:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{dowek:LIPIcs.ICALP.2017.109,
  author =	{Dowek, Gilles},
  title =	{{Models and Termination of Proof Reduction in the lambda Pi-Calculus Modulo Theory}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{109:1--109:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.109},
  URN =		{urn:nbn:de:0030-drops-73919},
  doi =		{10.4230/LIPIcs.ICALP.2017.109},
  annote =	{Keywords: model, proof reduction, Simple type theory, Calculus of constructions}
}
Document
Universality of Proofs (Dagstuhl Seminar 16421)

Authors: Gilles Dowek, Catherine Dubois, Brigitte Pientka, and Florian Rabe

Published in: Dagstuhl Reports, Volume 6, Issue 10 (2017)


Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 16421 "Universality of Proofs" which took place October 16-21, 2016. The seminar was motivated by the fact that it is nowadays difficult to exchange proofs from one proof assistant to another one. Thus a formal proof cannot be considered as a universal proof, reusable in different contexts. The seminar aims at providing a comprehensive overview of the existing techniques for interoperability and going further into the development of a common objective and framework for proof developments that support the communication, reuse and interoperability of proofs. The seminar included participants coming from different fields of computer science such as logic, proof engineering, program verification, formal mathematics. It included overview talks, technical talks and breakout sessions. This report collects the abstracts of talks and summarizes the outcomes of the breakout sessions.

Cite as

Gilles Dowek, Catherine Dubois, Brigitte Pientka, and Florian Rabe. Universality of Proofs (Dagstuhl Seminar 16421). In Dagstuhl Reports, Volume 6, Issue 10, pp. 75-98, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@Article{dowek_et_al:DagRep.6.10.75,
  author =	{Dowek, Gilles and Dubois, Catherine and Pientka, Brigitte and Rabe, Florian},
  title =	{{Universality of Proofs (Dagstuhl Seminar 16421)}},
  pages =	{75--98},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2017},
  volume =	{6},
  number =	{10},
  editor =	{Dowek, Gilles and Dubois, Catherine and Pientka, Brigitte and Rabe, Florian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.6.10.75},
  URN =		{urn:nbn:de:0030-drops-69514},
  doi =		{10.4230/DagRep.6.10.75},
  annote =	{Keywords: Formal proofs, Interoperability, Logical frameworks, Logics, Proof formats, Provers, Reusability}
}
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