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DOI: 10.4230/LIPIcs.FSCD.2024.5

The Flower Calculus

Authors: Pablo Donato

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

We introduce the flower calculus, a deep inference proof system for intuitionistic first-order logic inspired by Peirce’s existential graphs. It works as a rewriting system over inductive objects called "flowers", that enjoy both a graphical interpretation as topological diagrams, and a textual presentation as nested sequents akin to coherent formulas. Importantly, the calculus dispenses completely with the traditional notion of symbolic connective, operating solely on nested flowers containing atomic predicates. We prove both the soundness of the full calculus and the completeness of an analytic fragment with respect to Kripke semantics. This provides to our knowledge the first analyticity result for a proof system based on existential graphs, adapting semantic cut-elimination techniques to a deep inference setting. Furthermore, the kernel of rules targetted by completeness is fully invertible, a desirable property for both automated and interactive proof search.

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Pablo Donato. The Flower Calculus. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 5:1-5:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{donato:LIPIcs.FSCD.2024.5,
  author =	{Donato, Pablo},
  title =	{{The Flower Calculus}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{5:1--5:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.5},
  URN =		{urn:nbn:de:0030-drops-203343},
  doi =		{10.4230/LIPIcs.FSCD.2024.5},
  annote =	{Keywords: deep inference, graphical calculi, existential graphs, intuitionistic logic, Kripke semantics, cut-elimination}
}

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DOI: 10.4230/LIPIcs.FSCD.2024.6

Delooping Generated Groups in Homotopy Type Theory

Authors: Camil Champin, Samuel Mimram, and Émile Oleon

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Homotopy type theory is a logical setting based on Martin-Löf type theory in which one can perform geometric constructions and proofs in a synthetic way. Namely, types can be interpreted as spaces (up to continuous deformation) and proofs as homotopy invariant constructions. In this context, loop spaces of pointed connected groupoids provide a natural representation of groups, and any group can be obtained as the loop space of such a type, which is then called a delooping of the group. There are two main methods to construct the delooping of an arbitrary group G. The first one consists in describing it as a pointed higher inductive type, whereas the second one consists in taking the connected component of the principal G-torsor in the type of sets equipped with an action of G. We show here that, when a presentation is known for the group, simpler variants of those constructions can be used to build deloopings. The resulting types are more amenable to computations and lead to simpler meta-theoretic reasoning. We also investigate, in this context, an abstract construction for the Cayley graph of a generated group and show that it encodes the relations of the group. Most of the developments performed in the article have been formalized using the cubical version of the Agda proof assistant.

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Camil Champin, Samuel Mimram, and Émile Oleon. Delooping Generated Groups in Homotopy Type Theory. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 6:1-6:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{champin_et_al:LIPIcs.FSCD.2024.6,
  author =	{Champin, Camil and Mimram, Samuel and Oleon, \'{E}mile},
  title =	{{Delooping Generated Groups in Homotopy Type Theory}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{6:1--6:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.6},
  URN =		{urn:nbn:de:0030-drops-203356},
  doi =		{10.4230/LIPIcs.FSCD.2024.6},
  annote =	{Keywords: homotopy type theory, delooping, group, generator, Cayley graph}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.10

Second-Order Generalised Algebraic Theories: Signatures and First-Order Semantics

Authors: Ambrus Kaposi and Szumi Xie

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Programming languages can be defined from the concrete to the abstract by abstract syntax trees, well-scoped syntax, well-typed (intrinsic) syntax, algebraic syntax (well-typed syntax quotiented by conversion). Another aspect is the representation of binding structure for which nominal approaches, De Bruijn indices/levels and higher order abstract syntax (HOAS) are available. In HOAS, binders are given by the function space of an internal language of presheaves. In this paper, we show how to combine the algebraic approach with the HOAS approach: following Uemura, we define languages as second-order generalised algebraic theories (SOGATs). Through a series of examples we show that non-substructural languages can be naturally defined as SOGATs. We give a formal definition of SOGAT signatures (using the syntax of a particular SOGAT) and define two translations from SOGAT signatures to GAT signatures (signatures for quotient inductive-inductive types), based on parallel and single substitutions, respectively.

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Ambrus Kaposi and Szumi Xie. Second-Order Generalised Algebraic Theories: Signatures and First-Order Semantics. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kaposi_et_al:LIPIcs.FSCD.2024.10,
  author =	{Kaposi, Ambrus and Xie, Szumi},
  title =	{{Second-Order Generalised Algebraic Theories: Signatures and First-Order Semantics}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{10:1--10:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.10},
  URN =		{urn:nbn:de:0030-drops-203396},
  doi =		{10.4230/LIPIcs.FSCD.2024.10},
  annote =	{Keywords: Type theory, universal algebra, inductive types, quotient inductive types, higher-order abstract syntax, logical framework}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.22

Automating Boundary Filling in Cubical Agda

Authors: Maximilian Doré, Evan Cavallo, and Anders Mörtberg

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

When working in a proof assistant, automation is key to discharging routine proof goals such as equations between algebraic expressions. Homotopy Type Theory allows the user to reason about higher structures, such as topological spaces, using higher inductive types (HITs) and univalence. Cubical Agda is an extension of Agda with computational support for HITs and univalence. A difficulty when working in Cubical Agda is dealing with the complex combinatorics of higher structures, an infinite-dimensional generalisation of equational reasoning. To solve these higher-dimensional equations consists in constructing cubes with specified boundaries. We develop a simplified cubical language in which we isolate and study two automation problems: contortion solving, where we attempt to "contort" a cube to fit a given boundary, and the more general Kan solving, where we search for solutions that involve pasting multiple cubes together. Both problems are difficult in the general case - Kan solving is even undecidable - so we focus on heuristics that perform well on practical examples. We provide a solver for the contortion problem using a reformulation of contortions in terms of poset maps, while we solve Kan problems using constraint satisfaction programming. We have implemented our algorithms in an experimental Haskell solver that can be used to automatically solve goals presented by Cubical Agda. We illustrate this with a case study establishing the Eckmann-Hilton theorem using our solver, as well as various benchmarks - providing the ground for further study of proof automation in cubical type theories.

Cite as

Maximilian Doré, Evan Cavallo, and Anders Mörtberg. Automating Boundary Filling in Cubical Agda. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dore_et_al:LIPIcs.FSCD.2024.22,
  author =	{Dor\'{e}, Maximilian and Cavallo, Evan and M\"{o}rtberg, Anders},
  title =	{{Automating Boundary Filling in Cubical Agda}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{22:1--22:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.22},
  URN =		{urn:nbn:de:0030-drops-203514},
  doi =		{10.4230/LIPIcs.FSCD.2024.22},
  annote =	{Keywords: Cubical Agda, Automated Reasoning, Constraint Satisfaction Programming}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.26

On the Logical Structure of Some Maximality and Well-Foundedness Principles Equivalent to Choice Principles

Authors: Hugo Herbelin and Jad Koleilat

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

We study the logical structure of Teichmüller-Tukey lemma, a maximality principle equivalent to the axiom of choice and show that it corresponds to the generalisation to arbitrary cardinals of update induction, a well-foundedness principle from constructive mathematics classically equivalent to the axiom of dependent choice. From there, we state general forms of maximality and well-foundedness principles equivalent to the axiom of choice, including a variant of Zorn’s lemma. A comparison with the general class of choice and bar induction principles given by Brede and the first author is initiated.

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Hugo Herbelin and Jad Koleilat. On the Logical Structure of Some Maximality and Well-Foundedness Principles Equivalent to Choice Principles. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{herbelin_et_al:LIPIcs.FSCD.2024.26,
  author =	{Herbelin, Hugo and Koleilat, Jad},
  title =	{{On the Logical Structure of Some Maximality and Well-Foundedness Principles Equivalent to Choice Principles}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.26},
  URN =		{urn:nbn:de:0030-drops-203551},
  doi =		{10.4230/LIPIcs.FSCD.2024.26},
  annote =	{Keywords: axiom of choice, Teichm\"{u}ller-Tukey lemma, update induction, constructive reverse mathematics}
}

Document

DOI: 10.4230/LIPIcs.CSL.2024.16

A General Constructive Form of Higman’s Lemma

Authors: Stefano Berardi, Gabriele Buriola, and Peter Schuster

Published in: LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Abstract

In logic and computer science one often needs to constructivize a theorem ∀ f ∃ g.P(f,g), stating that for every infinite sequence f there is an infinite sequence g such that P(f,g). Here P is a computable predicate but g is not necessarily computable from f. In this paper we propose the following constructive version of ∀ f ∃ g.P(f,g): for every f there is a "long enough" finite prefix g₀ of g such that P(f,g₀), where "long enough" is expressed by membership to a bar which is a free parameter of the constructive version. Our approach with bars generalises the approaches to Higman’s lemma undertaken by Coquand-Fridlender, Murthy-Russell and Schwichtenberg-Seisenberger-Wiesnet. As a first test for our bar technique, we sketch a constructive theory of well-quasi orders. This includes yet another constructive version of Higman’s lemma: that every infinite sequence of words has an infinite ascending subsequence. As compared with the previous constructive versions of Higman’s lemma, our constructive proofs are closer to the original classical proofs.

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Stefano Berardi, Gabriele Buriola, and Peter Schuster. A General Constructive Form of Higman’s Lemma. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{berardi_et_al:LIPIcs.CSL.2024.16,
  author =	{Berardi, Stefano and Buriola, Gabriele and Schuster, Peter},
  title =	{{A General Constructive Form of Higman’s Lemma}},
  booktitle =	{32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)},
  pages =	{16:1--16:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-310-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{288},
  editor =	{Murano, Aniello and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.16},
  URN =		{urn:nbn:de:0030-drops-196599},
  doi =		{10.4230/LIPIcs.CSL.2024.16},
  annote =	{Keywords: intuitionistic logic, constructive mathematics, formal proof, inductive predicate, bar induction, well quasi-order, Higman’s lemma}
}

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.

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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}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2022.14

A Univalent Formalization of Constructive Affine Schemes

Authors: Max Zeuner and Anders Mörtberg

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

Abstract

We present a formalization of constructive affine schemes in the Cubical Agda proof assistant. This development is not only fully constructive and predicative, it also makes crucial use of univalence. By now schemes have been formalized in various proof assistants. However, most existing formalizations follow the inherently non-constructive approach of Hartshorne’s classic "Algebraic Geometry" textbook, for which the construction of the so-called structure sheaf is rather straightforwardly formalizable and works the same with or without univalence. We follow an alternative approach that uses a point-free description of the constructive counterpart of the Zariski spectrum called the Zariski lattice and proceeds by defining the structure sheaf on formal basic opens and then lift it to the whole lattice. This general strategy is used in a plethora of textbooks, but formalizing it has proved tricky. The main result of this paper is that with the help of the univalence principle we can make this "lift from basis" strategy formal and obtain a fully formalized account of constructive affine schemes.

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Max Zeuner and Anders Mörtberg. A Univalent Formalization of Constructive Affine Schemes. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 14:1-14:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{zeuner_et_al:LIPIcs.TYPES.2022.14,
  author =	{Zeuner, Max and M\"{o}rtberg, Anders},
  title =	{{A Univalent Formalization of Constructive Affine Schemes}},
  booktitle =	{28th International Conference on Types for Proofs and Programs (TYPES 2022)},
  pages =	{14:1--14:24},
  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.2022.14},
  URN =		{urn:nbn:de:0030-drops-184574},
  doi =		{10.4230/LIPIcs.TYPES.2022.14},
  annote =	{Keywords: Affine Schemes, Homotopy Type Theory and Univalent Foundations, Cubical Agda, Constructive Mathematics}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2023.5

Homotopy Type Theory as Internal Languages of Diagrams of ∞-Logoses

Authors: Taichi Uemura

Published in: LIPIcs, Volume 260, 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)

Abstract

We show that certain diagrams of ∞-logoses are reconstructed in internal languages of their oplax limits via lex, accessible modalities, which enables us to use plain homotopy type theory to reason about not only a single ∞-logos but also a diagram of ∞-logoses. This also provides a higher dimensional version of Sterling’s synthetic Tait computability - a type theory for higher dimensional logical relations.

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Taichi Uemura. Homotopy Type Theory as Internal Languages of Diagrams of ∞-Logoses. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{uemura:LIPIcs.FSCD.2023.5,
  author =	{Uemura, Taichi},
  title =	{{Homotopy Type Theory as Internal Languages of Diagrams of ∞-Logoses}},
  booktitle =	{8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)},
  pages =	{5:1--5:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-277-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{260},
  editor =	{Gaboardi, Marco and van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.5},
  URN =		{urn:nbn:de:0030-drops-179897},
  doi =		{10.4230/LIPIcs.FSCD.2023.5},
  annote =	{Keywords: Homotopy type theory, ∞-logos, ∞-topos, oplax limit, Artin gluing, modality, synthetic Tait computability, logical relation}
}

Document

DOI: 10.4230/DagRep.11.10.151

Geometric Logic, Constructivisation, and Automated Theorem Proving (Dagstuhl Seminar 21472)

Authors: Thierry Coquand, Hajime Ishihara, Sara Negri, and Peter M. Schuster

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

Abstract

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.

Cite as

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}
}

Document

DOI: 10.4230/LIPIcs.CSL.2020.14

Unifying Cubical Models of Univalent Type Theory

Authors: Evan Cavallo, Anders Mörtberg, and Andrew W Swan

Published in: LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)

Abstract

We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure.

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Evan Cavallo, Anders Mörtberg, and Andrew W Swan. Unifying Cubical Models of Univalent Type Theory. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cavallo_et_al:LIPIcs.CSL.2020.14,
  author =	{Cavallo, Evan and M\"{o}rtberg, Anders and Swan, Andrew W},
  title =	{{Unifying Cubical Models of Univalent Type Theory}},
  booktitle =	{28th EACSL Annual Conference on Computer Science Logic (CSL 2020)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-132-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{152},
  editor =	{Fern\'{a}ndez, Maribel and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.14},
  URN =		{urn:nbn:de:0030-drops-116578},
  doi =		{10.4230/LIPIcs.CSL.2020.14},
  annote =	{Keywords: Cubical Set Models, Cubical Type Theory, Homotopy Type Theory, Univalent Foundations}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2019.11

Homotopy Canonicity for Cubical Type Theory

Authors: Thierry Coquand, Simon Huber, and Christian Sattler

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

Abstract

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.

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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}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2019.25

Gluing for Type Theory

Authors: Ambrus Kaposi, Simon Huber, and Christian Sattler

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

Abstract

The relationship between categorical gluing and proofs using the logical relation technique is folklore. In this paper we work out this relationship for Martin-Löf type theory and show that parametricity and canonicity arise as special cases of gluing. The input of gluing is two models of type theory and a pseudomorphism between them and the output is a displayed model over the first model. A pseudomorphism preserves the categorical structure strictly, the empty context and context extension up to isomorphism, and there are no conditions on preservation of type formers. We look at three examples of pseudomorphisms: the identity on the syntax, the interpretation into the set model and the global section functor. Gluing along these result in syntactic parametricity, semantic parametricity and canonicity, respectively.

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Ambrus Kaposi, Simon Huber, and Christian Sattler. Gluing for Type Theory. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 25:1-25:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kaposi_et_al:LIPIcs.FSCD.2019.25,
  author =	{Kaposi, Ambrus and Huber, Simon and Sattler, Christian},
  title =	{{Gluing for Type Theory}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{25:1--25:19},
  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.2019.25},
  URN =		{urn:nbn:de:0030-drops-105323},
  doi =		{10.4230/LIPIcs.FSCD.2019.25},
  annote =	{Keywords: Martin-L\"{o}f type theory, logical relations, parametricity, canonicity, quotient inductive types}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2019.31

Cubical Syntax for Reflection-Free Extensional Equality

Authors: Jonathan Sterling, Carlo Angiuli, and Daniel Gratzer

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

Abstract

We contribute XTT, a cubical reconstruction of Observational Type Theory [Altenkirch et al., 2007] which extends Martin-Löf’s intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental version of the unicity of identity proofs principle (UIP): any two elements of the same equality type are judgmentally equal. Moreover, we conjecture that the typing relation can be decided in a practical way. In this paper, we establish an algebraic canonicity theorem using a novel extension of the logical families or categorical gluing argument inspired by Coquand and Shulman [Coquand, 2018; Shulman, 2015]: every closed element of boolean type is derivably equal to either true or false.

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Jonathan Sterling, Carlo Angiuli, and Daniel Gratzer. Cubical Syntax for Reflection-Free Extensional Equality. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 31:1-31:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{sterling_et_al:LIPIcs.FSCD.2019.31,
  author =	{Sterling, Jonathan and Angiuli, Carlo and Gratzer, Daniel},
  title =	{{Cubical Syntax for Reflection-Free Extensional Equality}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{31:1--31:25},
  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.2019.31},
  URN =		{urn:nbn:de:0030-drops-105387},
  doi =		{10.4230/LIPIcs.FSCD.2019.31},
  annote =	{Keywords: Dependent type theory, extensional equality, cubical type theory, categorical gluing, canonicity}
}

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.

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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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