65 Search Results for "Altenkirch, Thorsten"

Volume

LIPIcs, Volume 188

26th International Conference on Types for Proofs and Programs (TYPES 2020)

TYPES 2020, March 2-5, 2020, University of Turin, Italy

Editors: Ugo de'Liguoro, Stefano Berardi, and Thorsten Altenkirch

Volume

LIPIcs, Volume 38

13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)

TLCA 2015, July 1-3, 2015, Warsaw, Poland

Editors: Thorsten Altenkirch

Document
DOI: 10.4230/LIPIcs.ITP.2025.8
Abstract, Compositional Consistency: Isabelle/HOL Locales for Completeness à la Fitting

Authors: Asta Halkjær From and Anders Schlichtkrull

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
Smullyan and Fitting have used abstract consistency properties to great effect in unifying meta-theoretical results in logic. In this paper, we generalize these developments with the help of Isabelle/HOL. We use locales to decompose abstract consistency into general parts, and provide the textbook variants as special cases. Users can assemble their own consistency property for a given logic. The compositionality alleviates the absence of dependent types in Isabelle/HOL. We use our development to mechanize completeness of calculi for three logics: (1) first-order logic where we only instantiate universal quantifiers with already occurring terms, (2) second-order logic over general models, and (3) a recently developed strong hybrid logic with propositional quantification.
Cite as

Asta Halkjær From and Anders Schlichtkrull. Abstract, Compositional Consistency: Isabelle/HOL Locales for Completeness à la Fitting. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{from_et_al:LIPIcs.ITP.2025.8,
  author =	{From, Asta Halkj{\ae}r and Schlichtkrull, Anders},
  title =	{{Abstract, Compositional Consistency: Isabelle/HOL Locales for Completeness \`{a} la Fitting}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.8},
  URN =		{urn:nbn:de:0030-drops-246406},
  doi =		{10.4230/LIPIcs.ITP.2025.8},
  annote =	{Keywords: Logic, completeness, abstract consistency property, Isabelle/HOL, locales}
}
Document
DOI: 10.4230/LIPIcs.ITP.2025.36
Verifying Datalog Reasoning with Lean

Authors: Johannes Tantow, Lukas Gerlach, Stephan Mennicke, and Markus Krötzsch

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
Datalog is an essential logical rule language with many applications, and modern rule engines compute logical consequences for Datalog with high performance and scalability. While Datalog is rather simple and, in principle, explainable by design, such sophisticated implementations and optimizations are hard to verify. We therefore propose a certificate-based approach to validate results of Datalog reasoners in a formally verified checker for Datalog proofs. Using the proof assistant Lean, we implement such a checker and verify its correctness against direct formalizations of the Datalog semantics. We propose two JSON encodings for Datalog proofs: one using the widely supported Datalog proof trees, and one using directed acyclic graphs for succinctness. To evaluate the practical feasibility and performance of our approach, we validate proofs that we obtain by converting derivation traces of an existing Datalog reasoner into our tool-independent format.
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Johannes Tantow, Lukas Gerlach, Stephan Mennicke, and Markus Krötzsch. Verifying Datalog Reasoning with Lean. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 36:1-36:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{tantow_et_al:LIPIcs.ITP.2025.36,
  author =	{Tantow, Johannes and Gerlach, Lukas and Mennicke, Stephan and Kr\"{o}tzsch, Markus},
  title =	{{Verifying Datalog Reasoning with Lean}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{36:1--36:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.36},
  URN =		{urn:nbn:de:0030-drops-246342},
  doi =		{10.4230/LIPIcs.ITP.2025.36},
  annote =	{Keywords: Certifying Algorithms, Datalog, Formal Verification}
}
Document
DOI: 10.4230/LIPIcs.ITP.2025.33
Scott’s Representation Theorem and the Univalent Karoubi Envelope

Authors: Arnoud van der Leer, Kobe Wullaert, and Benedikt Ahrens

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
Lambek and Scott constructed a correspondence between simply-typed lambda calculi and Cartesian closed categories. Scott’s Representation Theorem is a cousin to this result for untyped lambda calculi. It states that every untyped lambda calculus arises from a reflexive object in some category. We present a formalization of Scott’s Representation Theorem in univalent foundations, in the (Rocq-)UniMath library. Specifically, we implement two proofs of that theorem, one by Scott and one by Hyland. We also explain the role of the Karoubi envelope - a categorical construction - in the proofs and the impact the chosen foundation has on this construction. Finally, we report on some automation we have implemented for the reduction of λ-terms.
Cite as

Arnoud van der Leer, Kobe Wullaert, and Benedikt Ahrens. Scott’s Representation Theorem and the Univalent Karoubi Envelope. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanderleer_et_al:LIPIcs.ITP.2025.33,
  author =	{van der Leer, Arnoud and Wullaert, Kobe and Ahrens, Benedikt},
  title =	{{Scott’s Representation Theorem and the Univalent Karoubi Envelope}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{33:1--33:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.33},
  URN =		{urn:nbn:de:0030-drops-246318},
  doi =		{10.4230/LIPIcs.ITP.2025.33},
  annote =	{Keywords: Lambda calculi, algebraic theories, categorical semantics, Karoubi envelope, formalization, Rocq-UniMath, univalent foundations}
}
Artifact
Software
DOI: 10.4230/artifacts.24710
stefaniatadama/formalising-inductive-coinductive-containers

Authors: Stefania Damato, Thorsten Altenkirch, and Axel Ljungström

Abstract
Cite as

Stefania Damato, Thorsten Altenkirch, Axel Ljungström. stefaniatadama/formalising-inductive-coinductive-containers (Software). Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@misc{dagstuhl-artifact-24710,
   title = {{stefaniatadama/formalising-inductive-coinductive-containers}}, 
   author = {Damato, Stefania and Altenkirch, Thorsten and Ljungstr\"{o}m, Axel},
   note = {Software, swhId: \href{https://archive.softwareheritage.org/swh:1:dir:a75b7ddaa46ca9a9f3ee6f65d22ba3b1c959d6d4;origin=https://github.com/stefaniatadama/formalising-inductive-coinductive-containers;visit=swh:1:snp:eebf2acbd002c4877bc6057e65e8ddef2b0fe79e;anchor=swh:1:rev:4e587677bdc0c7cb793ad8612c1e6e88c16877e7}{\texttt{swh:1:dir:a75b7ddaa46ca9a9f3ee6f65d22ba3b1c959d6d4}} (visited on 2025-09-22)},
   url = {https://github.com/stefaniatadama/formalising-inductive-coinductive-containers/blob/main/cubical/Cubical/Papers/Containers.agda},
   doi = {10.4230/artifacts.24710},
}
Document
DOI: 10.4230/LIPIcs.ITP.2025.17
Formalising Inductive and Coinductive Containers

Authors: Stefania Damato, Thorsten Altenkirch, and Axel Ljungström

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
Containers capture the concept of strictly positive data types in programming. The original development of containers is done in the internal language of locally cartesian closed categories (LCCCs) with disjoint coproducts and W-types, and uniqueness of identity proofs (UIP) is implicitly assumed throughout. Although it is claimed that these developments can also be interpreted in extensional Martin-Löf type theory, this interpretation is not made explicit. In this paper, we present a formalisation of the results that "containers preserve least and greatest fixed points" in Cubical Agda, thereby giving a formulation in intensional type theory. Our proofs do not make use of UIP and thereby generalise the original results from talking about container functors on Set to container functors on the wild category of types. Our main incentive for using Cubical Agda is that its path type restores the equivalence between bisimulation and coinductive equality. Thus, besides developing container theory in a more general setting, we also demonstrate the usefulness of Cubical Agda’s path type to coinductive proofs.
Cite as

Stefania Damato, Thorsten Altenkirch, and Axel Ljungström. Formalising Inductive and Coinductive Containers. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 17:1-17:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{damato_et_al:LIPIcs.ITP.2025.17,
  author =	{Damato, Stefania and Altenkirch, Thorsten and Ljungstr\"{o}m, Axel},
  title =	{{Formalising Inductive and Coinductive Containers}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{17:1--17:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.17},
  URN =		{urn:nbn:de:0030-drops-246151},
  doi =		{10.4230/LIPIcs.ITP.2025.17},
  annote =	{Keywords: type theory, container, initial algebra, terminal coalgebra, Cubical Agda}
}
Document
DOI: 10.4230/LIPIcs.CALCO.2025.4
Distributive Laws of Monadic Containers

Authors: Chris Purdy and Stefania Damato

Published in: LIPIcs, Volume 342, 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)

Abstract
Containers are used to carve out a class of strictly positive data types in terms of shapes and positions. They can be interpreted via a fully-faithful functor into endofunctors on Set. Monadic containers are those containers whose interpretation as a Set functor carries a monad structure. The category of containers is closed under container composition and is a monoidal category, whereas monadic containers do not in general compose. In this paper, we develop a characterisation of distributive laws of monadic containers. Distributive laws were introduced as a sufficient condition for the composition of the underlying functors of two monads to also carry a monad structure. Our development parallels Ahman and Uustalu’s characterisation of distributive laws of directed containers, i.e. containers whose Set functor interpretation carries a comonad structure. Furthermore, by combining our work with theirs, we construct characterisations of mixed distributive laws (i.e. of directed containers over monadic containers and vice versa), thereby completing the "zoo" of container characterisations of (co)monads and their distributive laws. We have found these characterisations amenable to development of existence and uniqueness proofs of distributive laws, particularly in the mechanised setting of Cubical Agda, in which most of the theory of this paper has been formalised.
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Chris Purdy and Stefania Damato. Distributive Laws of Monadic Containers. In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 4:1-4:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{purdy_et_al:LIPIcs.CALCO.2025.4,
  author =	{Purdy, Chris and Damato, Stefania},
  title =	{{Distributive Laws of Monadic Containers}},
  booktitle =	{11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)},
  pages =	{4:1--4:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-383-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{342},
  editor =	{C\^{i}rstea, Corina and Knapp, Alexander},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2025.4},
  URN =		{urn:nbn:de:0030-drops-235633},
  doi =		{10.4230/LIPIcs.CALCO.2025.4},
  annote =	{Keywords: distributive laws, monadic containers, monads, dependent types, cubical agda}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2025.27
What Does It Take to Certify a Conversion Checker?

Authors: Meven Lennon-Bertrand

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract
We report on a detailed exploration of the properties of conversion (definitional equality) in dependent type theory, with the goal of certifying decision procedures for it. While in that context the property of normalisation has attracted the most light, we instead emphasize the importance of injectivity properties, showing that they alone are both crucial and sufficient to certify most desirable properties of conversion checkers. We also explore the certification of a fully untyped conversion checker, with respect to a typed specification, and show that the story is mostly unchanged, although the exact injectivity properties needed are subtly different.
Cite as

Meven Lennon-Bertrand. What Does It Take to Certify a Conversion Checker?. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 27:1-27:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lennonbertrand:LIPIcs.FSCD.2025.27,
  author =	{Lennon-Bertrand, Meven},
  title =	{{What Does It Take to Certify a Conversion Checker?}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{27:1--27:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.27},
  URN =		{urn:nbn:de:0030-drops-236428},
  doi =		{10.4230/LIPIcs.FSCD.2025.27},
  annote =	{Keywords: Dependent types, Bidirectional typing, Certified software}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2025.29
Linear Logic Using Negative Connectives

Authors: Dale Miller

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract
In linear logic, the invertibility of a connective’s right-introduction rule is equivalent to the non-invertibility of its left-introduction rule. This duality motivates the concept of polarity: a connective is termed negative if its right-introduction rule is invertible, and positive otherwise. A two-sided sequent calculus for first-order linear logic featuring only negative connectives exhibits a compelling proof theory. Proof search in such a system unfolds through alternating phases of invertible (right-introduction) rules and non-invertible (left-introduction) rules, mirroring the processes of goal-reduction and backchaining, respectively. These phases are formalized here using the framework of multifocused proofs. We analyze linear logic by dissecting it into three sublogics: L₀ (first-order intuitionistic logic with conjunction, implication, and universal quantification); L₁ (an extension of L₀ incorporating linear implication which preserves its intuitionistic nature); and L₂ (which includes multiplicative falsity ⊥ and encompasses classical linear logic). It is worth noting that the single-conclusion restriction on sequents, a constraint imposed by Gentzen, is not a prerequisite for defining intuitionistic logic proofs within this framework, as it emerges naturally by restricting the formulas to those of L₀ and L₁. While multifocused proofs of L₂ sequents can accommodate parallel applications of left-introduction rules, proofs of L₀ and L₁ sequents cannot leverage such parallel rule applications. This notion of parallelism within proofs enables a novel approach to handling disjunctions and existential quantifiers in the natural deduction system for intuitionistic logic.
Cite as

Dale Miller. Linear Logic Using Negative Connectives. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 29:1-29:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{miller:LIPIcs.FSCD.2025.29,
  author =	{Miller, Dale},
  title =	{{Linear Logic Using Negative Connectives}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{29:1--29:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.29},
  URN =		{urn:nbn:de:0030-drops-236442},
  doi =		{10.4230/LIPIcs.FSCD.2025.29},
  annote =	{Keywords: Linear logic, multifocused proofs, sequent calculus}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2025.22
Branch Sequentialization in Quantum Polytime

Authors: Emmanuel Hainry, Romain Péchoux, and Mário Silva

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract
Quantum algorithms leverage the use of quantumly-controlled data in order to achieve computational advantage. This implies that the programs use constructs depending on quantum data and not just classical data such as measurement outcomes. Current compilation strategies for quantum control flow involve compiling the branches of a quantum conditional, either in-depth or in-width, which in general leads to circuits of exponential size. This problem is coined as the branch sequentialization problem. We introduce and study a compilation technique for avoiding branch sequentialization on a language that is sound and complete for quantum polynomial time, thus, improving on existing polynomial-size-preserving compilation techniques.
Cite as

Emmanuel Hainry, Romain Péchoux, and Mário Silva. Branch Sequentialization in Quantum Polytime. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 22:1-22:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hainry_et_al:LIPIcs.FSCD.2025.22,
  author =	{Hainry, Emmanuel and P\'{e}choux, Romain and Silva, M\'{a}rio},
  title =	{{Branch Sequentialization in Quantum Polytime}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{22:1--22:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.22},
  URN =		{urn:nbn:de:0030-drops-236373},
  doi =		{10.4230/LIPIcs.FSCD.2025.22},
  annote =	{Keywords: Quantum Programs, Implicit Computational Complexity, Quantum Circuits}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2024.5
Implementing a Type Theory with Observational Equality, Using Normalisation by Evaluation

Authors: Matthew Sirman, Meven Lennon-Bertrand, and Neel Krishnaswami

Published in: LIPIcs, Volume 336, 30th International Conference on Types for Proofs and Programs (TYPES 2024)

Abstract
We report on an experimental implementation in Haskell of a dependent type theory featuring an observational equality type, based on Pujet et al.’s CCobs. We use normalisation by evaluation to produce an efficient normalisation function, which is used to implement a bidirectional type checker. To allow for greater expressivity, we extend the core CCobs calculus with quotient types and inductive types. To make the system usable, we explore various proof-assistant features, notably a rudimentary version of a "hole" system similar to Agda’s. While rather crude, this experience should inform other, more substantial implementation efforts of observational equality.
Cite as

Matthew Sirman, Meven Lennon-Bertrand, and Neel Krishnaswami. Implementing a Type Theory with Observational Equality, Using Normalisation by Evaluation. In 30th International Conference on Types for Proofs and Programs (TYPES 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 336, pp. 5:1-5:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{sirman_et_al:LIPIcs.TYPES.2024.5,
  author =	{Sirman, Matthew and Lennon-Bertrand, Meven and Krishnaswami, Neel},
  title =	{{Implementing a Type Theory with Observational Equality, Using Normalisation by Evaluation}},
  booktitle =	{30th International Conference on Types for Proofs and Programs (TYPES 2024)},
  pages =	{5:1--5:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-376-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{336},
  editor =	{M{\o}gelberg, Rasmus Ejlers and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2024.5},
  URN =		{urn:nbn:de:0030-drops-233673},
  doi =		{10.4230/LIPIcs.TYPES.2024.5},
  annote =	{Keywords: Dependent type theory, Bidirectional typing, Observational equality, Normalisation by evaluation}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2024.6
Data Types with Symmetries via Action Containers

Authors: Philipp Joram and Niccolò Veltri

Published in: LIPIcs, Volume 336, 30th International Conference on Types for Proofs and Programs (TYPES 2024)

Abstract
We study two kinds of containers for data types with symmetries in homotopy type theory, and clarify their relationship by introducing the intermediate notion of action containers. Quotient containers are set-valued containers with groups of permissible permutations of positions, interpreted as (possibly non-finitary) analytic functors on the category of sets. Symmetric containers encode symmetries in a groupoid of shapes, and are interpreted accordingly as polynomial functors on the 2-category of groupoids. Action containers are endowed with groups that act on their positions, with morphisms preserving the actions. We show that, as a category, action containers are equivalent to the free coproduct completion of a category of group actions. We derive that they model non-inductive single-variable strictly positive types in the sense of Abbott et al.: The category of action containers is closed under arbitrary (co)products and exponentiation with constants. We equip this category with the structure of a locally groupoidal 2-category, and prove that it locally embeds into the 2-category of symmetric containers. This follows from the embedding of a 2-category of groups into the 2-category of groupoids, extending the delooping construction.
Cite as

Philipp Joram and Niccolò Veltri. Data Types with Symmetries via Action Containers. In 30th International Conference on Types for Proofs and Programs (TYPES 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 336, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{joram_et_al:LIPIcs.TYPES.2024.6,
  author =	{Joram, Philipp and Veltri, Niccol\`{o}},
  title =	{{Data Types with Symmetries via Action Containers}},
  booktitle =	{30th International Conference on Types for Proofs and Programs (TYPES 2024)},
  pages =	{6:1--6:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-376-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{336},
  editor =	{M{\o}gelberg, Rasmus Ejlers and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2024.6},
  URN =		{urn:nbn:de:0030-drops-233681},
  doi =		{10.4230/LIPIcs.TYPES.2024.6},
  annote =	{Keywords: Containers, Homotopy Type Theory, Agda, 2-categories}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2024.7
Synthetic 1-Categories in Directed Type Theory

Authors: Jacob Neumann and Thorsten Altenkirch

Published in: LIPIcs, Volume 336, 30th International Conference on Types for Proofs and Programs (TYPES 2024)

Abstract
The field of directed type theory seeks to design type theories capable of reasoning synthetically about (higher) categories, by generalizing the symmetric identity types of Martin-Löf Type Theory to asymmetric hom-types. We articulate the directed type theory of the category model, with appropriate modalities for keeping track of variances and a powerful directed-J rule capable of proving results about arbitrary terms of hom-types; we put this rule to use in making several constructions in synthetic 1-category theory. Because this theory is expressed entirely in terms of generalized algebraic theories, we know automatically that this directed type theory admits a syntax model.
Cite as

Jacob Neumann and Thorsten Altenkirch. Synthetic 1-Categories in Directed Type Theory. In 30th International Conference on Types for Proofs and Programs (TYPES 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 336, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{neumann_et_al:LIPIcs.TYPES.2024.7,
  author =	{Neumann, Jacob and Altenkirch, Thorsten},
  title =	{{Synthetic 1-Categories in Directed Type Theory}},
  booktitle =	{30th International Conference on Types for Proofs and Programs (TYPES 2024)},
  pages =	{7:1--7:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-376-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{336},
  editor =	{M{\o}gelberg, Rasmus Ejlers and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2024.7},
  URN =		{urn:nbn:de:0030-drops-233694},
  doi =		{10.4230/LIPIcs.TYPES.2024.7},
  annote =	{Keywords: Semantics, directed type theory, homotopy type theory, category theory, generalized algebraic theories}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2022.10
The Münchhausen Method in Type Theory

Authors: Thorsten Altenkirch, Ambrus Kaposi, Artjoms Šinkarovs, and Tamás Végh

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

Abstract
In one of his long tales, after falling into a swamp, Baron Münchhausen salvaged himself and the horse by lifting them both up by his hair. Inspired by this, the paper presents a technique to justify very dependent types. Such types reference the term that they classify, e.g. x : F x. While in most type theories this is not allowed, we propose a technique on salvaging the meaning of both the term and the type. The proposed technique does not refer to preterms or typing relations and works in a completely algebraic setting, e.g categories with families. With a series of examples we demonstrate our technique. We use Agda to demonstrate that our examples are implementable within a proof assistant.
Cite as

Thorsten Altenkirch, Ambrus Kaposi, Artjoms Šinkarovs, and Tamás Végh. The Münchhausen Method in Type Theory. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{altenkirch_et_al:LIPIcs.TYPES.2022.10,
  author =	{Altenkirch, Thorsten and Kaposi, Ambrus and \v{S}inkarovs, Artjoms and V\'{e}gh, Tam\'{a}s},
  title =	{{The M\"{u}nchhausen Method in Type Theory}},
  booktitle =	{28th International Conference on Types for Proofs and Programs (TYPES 2022)},
  pages =	{10:1--10:20},
  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.2022.10},
  URN =		{urn:nbn:de:0030-drops-184534},
  doi =		{10.4230/LIPIcs.TYPES.2022.10},
  annote =	{Keywords: type theory, proof assistants, very dependent types}
}
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