Document

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

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.

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)

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

Metatheorems about type theories are often proven by interpreting the syntax into models constructed using categorical gluing. We propose to use only sconing (gluing along a global section functor) instead of general gluing. The sconing is performed internally to a presheaf category, and we recover the original glued model by externalization.
Our method relies on constructions involving two notions of models: first-order models (with explicit contexts) and higher-order models (without explicit contexts). Sconing turns a displayed higher-order model into a displayed first-order model.
Using these, we derive specialized induction principles for the syntax of type theory. The input of such an induction principle is a boilerplate-free description of its motives and methods, not mentioning contexts. The output is a section with computation rules specified in the same internal language. We illustrate our framework by proofs of canonicity and normalization for type theory.

Rafaël Bocquet, Ambrus Kaposi, and Christian Sattler. For the Metatheory of Type Theory, Internal Sconing Is Enough. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 18:1-18:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bocquet_et_al:LIPIcs.FSCD.2023.18, author = {Bocquet, Rafa\"{e}l and Kaposi, Ambrus and Sattler, Christian}, title = {{For the Metatheory of Type Theory, Internal Sconing Is Enough}}, booktitle = {8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)}, pages = {18:1--18:23}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.18}, URN = {urn:nbn:de:0030-drops-180029}, doi = {10.4230/LIPIcs.FSCD.2023.18}, annote = {Keywords: type theory, presheaves, canonicity, normalization, sconing, gluing} }

Document

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

It is well-known that extensional lambda calculus is equivalent to extensional combinatory logic. In this paper we describe a formalisation of this fact in Cubical Agda. The distinguishing features of our formalisation are the following: (i) Both languages are defined as generalised algebraic theories, the syntaxes are intrinsically typed and quotiented by conversion; we never mention preterms or break the quotients in our construction. (ii) Typing is a parameter, thus the un(i)typed and simply typed variants are special cases of the same proof. (iii) We define syntaxes as quotient inductive-inductive types (QIITs) in Cubical Agda; we prove the equivalence and (via univalence) the equality of these QIITs; we do not rely on any axioms, the conversion functions all compute and can be experimented with.

Thorsten Altenkirch, Ambrus Kaposi, Artjoms Šinkarovs, and Tamás Végh. Combinatory Logic and Lambda Calculus Are Equal, Algebraically. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{altenkirch_et_al:LIPIcs.FSCD.2023.24, author = {Altenkirch, Thorsten and Kaposi, Ambrus and \v{S}inkarovs, Artjoms and V\'{e}gh, Tam\'{a}s}, title = {{Combinatory Logic and Lambda Calculus Are Equal, Algebraically}}, booktitle = {8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)}, pages = {24:1--24:19}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.24}, URN = {urn:nbn:de:0030-drops-180086}, doi = {10.4230/LIPIcs.FSCD.2023.24}, annote = {Keywords: Combinatory logic, lambda calculus, quotient inductive types, Cubical Agda} }

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**Published in:** LIPIcs, Volume 239, 27th International Conference on Types for Proofs and Programs (TYPES 2021)

The setoid model of Martin-Löf’s type theory bootstraps extensional features of type theory from intensional type theory equipped with a universe of definitionally proof irrelevant (strict) propositions. Extensional features include a Prop-valued identity type with a strong transport rule and function extensionality. We show that a setoid model supporting these features can be defined in intensional type theory without any of these features. The key component is a point-free notion of propositions. Our construction suggests that strict algebraic structures can be defined along the same lines in intensional type theory.

István Donkó and Ambrus Kaposi. Internal Strict Propositions Using Point-Free Equations. In 27th International Conference on Types for Proofs and Programs (TYPES 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 239, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{donko_et_al:LIPIcs.TYPES.2021.6, author = {Donk\'{o}, Istv\'{a}n and Kaposi, Ambrus}, title = {{Internal Strict Propositions Using Point-Free Equations}}, booktitle = {27th International Conference on Types for Proofs and Programs (TYPES 2021)}, pages = {6:1--6:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-254-9}, ISSN = {1868-8969}, year = {2022}, volume = {239}, editor = {Basold, Henning and Cockx, Jesper and Ghilezan, Silvia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2021.6}, URN = {urn:nbn:de:0030-drops-167759}, doi = {10.4230/LIPIcs.TYPES.2021.6}, annote = {Keywords: Martin-L\"{o}f’s type theory, intensional type theory, function extensionality, setoid model, homotopy type theory} }

Document

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

Inductive-inductive types (IITs) are a generalisation of inductive types in type theory. They allow the mutual definition of types with multiple sorts where later sorts can be indexed by previous ones. An example is the Chapman-style syntax of type theory with conversion relations for each sort where e.g. the sort of types is indexed by contexts. In this paper we show that if a model of extensional type theory (ETT) supports indexed W-types, then it supports finitely branching IITs. We use a small internal type theory called the theory of signatures to specify IITs. We show that if a model of ETT supports the syntax for the theory of signatures, then it supports all IITs. We construct this syntax from indexed W-types using preterms and typing relations and prove its initiality following Streicher. The construction of the syntax and its initiality proof were formalised in Agda.

Ambrus Kaposi, András Kovács, and Ambroise Lafont. For Finitary Induction-Induction, Induction Is Enough. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 6:1-6:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kaposi_et_al:LIPIcs.TYPES.2019.6, author = {Kaposi, Ambrus and Kov\'{a}cs, Andr\'{a}s and Lafont, Ambroise}, title = {{For Finitary Induction-Induction, Induction Is Enough}}, booktitle = {25th International Conference on Types for Proofs and Programs (TYPES 2019)}, pages = {6:1--6:30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-158-0}, ISSN = {1868-8969}, year = {2020}, volume = {175}, editor = {Bezem, Marc and Mahboubi, Assia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.6}, URN = {urn:nbn:de:0030-drops-130707}, doi = {10.4230/LIPIcs.TYPES.2019.6}, annote = {Keywords: type theory, inductive types, inductive-inductive types} }

Document

**Published in:** LIPIcs, Volume 167, 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)

Inductive families of types are a feature of most languages based on dependent types. They are usually described either by syntactic schemes or by encodings of strictly positive functors such as combinator languages or containers. The former approaches are informal and give only external signatures, the latter approaches suffer from encoding overheads and do not directly represent mutual types.
In this paper we propose a direct method for describing signatures for mutual inductive families using a domain-specific type theory. A signature is a context (roughly speaking, a list of types) in this small type theory. Algebras, displayed algebras and sections are defined by models of this type theory: the standard model, the logical predicate and a logical relation interpretation, respectively. We reduce the existence of initial algebras for these signatures to the existence of the syntax of our domain-specific type theory. As this theory is very simple, its normal syntax can be encoded using indexed W-types. To the best of our knowledge, this is the first formalisation of the folklore fact that mutual inductive types can be reduced to indexed W-types.
The contents of this paper were formalised in the proof assistant Agda.

Ambrus Kaposi and Jakob von Raumer. A Syntax for Mutual Inductive Families. In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 23:1-23:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kaposi_et_al:LIPIcs.FSCD.2020.23, author = {Kaposi, Ambrus and von Raumer, Jakob}, title = {{A Syntax for Mutual Inductive Families}}, booktitle = {5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)}, pages = {23:1--23:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-155-9}, ISSN = {1868-8969}, year = {2020}, volume = {167}, editor = {Ariola, Zena M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2020.23}, URN = {urn:nbn:de:0030-drops-123451}, doi = {10.4230/LIPIcs.FSCD.2020.23}, annote = {Keywords: type theory, inductive types, mutual inductive types, W-types, Agda} }

Document

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

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.

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-dev.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

Complete Volume

**Published in:** LIPIcs, Volume 104, 23rd International Conference on Types for Proofs and Programs (TYPES 2017)

LIPIcs, Volume 104, TYPES'17, Complete Volume

23rd International Conference on Types for Proofs and Programs (TYPES 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 104, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{abel_et_al:LIPIcs.TYPES.2017, title = {{LIPIcs, Volume 104, TYPES'17, Complete Volume}}, booktitle = {23rd International Conference on Types for Proofs and Programs (TYPES 2017)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-071-2}, ISSN = {1868-8969}, year = {2019}, volume = {104}, editor = {Abel, Andreas and Nordvall Forsberg, Fredrik and Kaposi, Ambrus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2017}, URN = {urn:nbn:de:0030-drops-101671}, doi = {10.4230/LIPIcs.TYPES.2017}, annote = {Keywords: Theory of computation, Type theory, Proof theory, Program verification} }

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Front Matter

**Published in:** LIPIcs, Volume 104, 23rd International Conference on Types for Proofs and Programs (TYPES 2017)

Front Matter, Table of Contents, Preface, Conference Organization

23rd International Conference on Types for Proofs and Programs (TYPES 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 104, pp. 0:i-0:x, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{abel_et_al:LIPIcs.TYPES.2017.0, author = {Abel, Andreas and Nordvall Forsberg, Fredrik and Kaposi, Ambrus}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {23rd International Conference on Types for Proofs and Programs (TYPES 2017)}, pages = {0:i--0:x}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-071-2}, ISSN = {1868-8969}, year = {2019}, volume = {104}, editor = {Abel, Andreas and Nordvall Forsberg, Fredrik and Kaposi, Ambrus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2017.0}, URN = {urn:nbn:de:0030-drops-100488}, doi = {10.4230/LIPIcs.TYPES.2017.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)

Higher inductive-inductive types (HIITs) generalise inductive types of dependent type theories in two directions. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed over each other. On the other hand they support equality constructors, thus generalising higher inductive types of homotopy type theory. Examples that make use of both features are the Cauchy reals and the well-typed syntax of type theory where conversion rules are given as equality constructors. In this paper we propose a general definition of HIITs using a domain-specific type theory. A context in this small type theory encodes a HIIT by listing the type formation rules and constructors. The type of the elimination principle and its beta-rules are computed from the context using a variant of the syntactic logical relation translation. We show that for indexed W-types and various examples of HIITs the computed elimination principles are the expected ones. Showing that the thus specified HIITs exist is left as future work. The type theory specifying HIITs was formalised in Agda together with the syntactic translations. A Haskell implementation converts the types of sorts and constructors into valid Agda code which postulates the elimination principles and computation rules.

Ambrus Kaposi and András Kovács. A Syntax for Higher Inductive-Inductive Types. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kaposi_et_al:LIPIcs.FSCD.2018.20, author = {Kaposi, Ambrus and Kov\'{a}cs, Andr\'{a}s}, title = {{A Syntax for Higher Inductive-Inductive Types}}, booktitle = {3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)}, pages = {20:1--20:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-077-4}, ISSN = {1868-8969}, year = {2018}, volume = {108}, editor = {Kirchner, H\'{e}l\`{e}ne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.20}, URN = {urn:nbn:de:0030-drops-91906}, doi = {10.4230/LIPIcs.FSCD.2018.20}, annote = {Keywords: homotopy type theory, inductive-inductive types, higher inductive types, quotient inductive types, logical relations} }

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

Following the cubical set model of type theory which validates the
univalence axiom, cubical type theories have been developed that
interpret the identity type using an interval pretype. These theories start from a geometric view of equality. A proof of equality is encoded as a term in a context extended by the interval pretype. Our goal is to develop a cubical theory where the identity type is defined recursively over the type structure, and the geometry arises from these definitions. In this theory, cubes are present explicitly, e.g., a line is a telescope with 3 elements: two endpoints and the connecting equality. This is in line with Bernardy and Moulin's earlier work on internal parametricity. In this paper we present a naive syntax for internal parametricity and by replacing the parametric interpretation of the universe, we extend it to univalence. However, we do not know how to compute in this theory. As a second step, we present a version of the theory for parametricity with named dimensions which has an operational semantics. Extending this syntax to univalence is left as further work.

Thorsten Altenkirch and Ambrus Kaposi. Towards a Cubical Type Theory without an Interval. In 21st International Conference on Types for Proofs and Programs (TYPES 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 69, pp. 3:1-3:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{altenkirch_et_al:LIPIcs.TYPES.2015.3, author = {Altenkirch, Thorsten and Kaposi, Ambrus}, title = {{Towards a Cubical Type Theory without an Interval}}, booktitle = {21st International Conference on Types for Proofs and Programs (TYPES 2015)}, pages = {3:1--3:27}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2015.3}, URN = {urn:nbn:de:0030-drops-84739}, doi = {10.4230/LIPIcs.TYPES.2015.3}, annote = {Keywords: homotopy type theory, parametricity, univalence} }

Document

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

We develop normalisation by evaluation (NBE) for dependent types based
on presheaf categories. Our construction is formulated using internal
type theory using quotient inductive types. We use a typed
presentation hence there are no preterms or realizers in our
construction. NBE for simple types is using a logical relation between
the syntax and the presheaf interpretation. In our construction, we
merge the presheaf interpretation and the logical relation into a
proof-relevant logical predicate. We have formalized parts of the
construction in Agda.

Thorsten Altenkirch and Ambrus Kaposi. Normalisation by Evaluation for Dependent Types. In 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 52, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{altenkirch_et_al:LIPIcs.FSCD.2016.6, author = {Altenkirch, Thorsten and Kaposi, Ambrus}, title = {{Normalisation by Evaluation for Dependent Types}}, booktitle = {1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)}, pages = {6:1--6:16}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2016.6}, URN = {urn:nbn:de:0030-drops-59727}, doi = {10.4230/LIPIcs.FSCD.2016.6}, annote = {Keywords: normalisation by evaluation, dependent types, internal type theory, logical relations, Agda} }

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