DROPS

Volume

LIPIcs, Volume 175

25th International Conference on Types for Proofs and Programs (TYPES 2019)

TYPES 2019, June 11-14, 2019, Oslo, Norway

Editors: Marc Bezem and Assia Mahboubi

Volume

LIPIcs, Volume 12

Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL

CSL 2011, September 12-15, 2011, Bergen, Norway

Editors: Marc Bezem

Document

Invited Talk

DOI: 10.4230/LIPIcs.FSCD.2025.1

Computation First: Rebuilding Constructivism with Effects (Invited Talk)

Authors: Liron Cohen

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

Abstract

Constructive logic and type theory have traditionally been grounded in pure, effect-free model of computation. This paper argues that such a restriction is not a foundational necessity but a historical artifact, and it advocates for a broader perspective of effectful constructivism, where computational effects, such as state, non-determinism, and exceptions, are directly and internally embedded in the logical and computational foundations. We begin by surveying examples where effects reshape logical principles, and then outline three approaches to effectful constructivism, focusing on realizability models: Monadic Combinatory Algebras, which extend classical partial combinatory algebras with effectful computation; Evidenced Frames, a flexible semantic structure capable of uniformly capturing a wide range of effects; and Effectful Higher-Order Logic (EffHOL), a syntactic approach that directly translates logical propositions into specifications for effectful programs. We further illustrate how concrete type theories can internalize effects, via the family of type theories TT^□_C. Together, these works demonstrate that effectful constructivism is not merely possible but a natural and robust extension of traditional frameworks.

Cite as

Liron Cohen. Computation First: Rebuilding Constructivism with Effects (Invited Talk). In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cohen:LIPIcs.FSCD.2025.1,
  author =	{Cohen, Liron},
  title =	{{Computation First: Rebuilding Constructivism with Effects}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{1:1--1:20},
  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.2025.1},
  URN =		{urn:nbn:de:0030-drops-236167},
  doi =		{10.4230/LIPIcs.FSCD.2025.1},
  annote =	{Keywords: Effectful constructivism, realizability, type theory, monadic combinatory algebras, evidenced frame}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2025.12

Ohana Trees and Taylor Expansion for the λI-Calculus: No variable gets left behind or forgotten!

Authors: Rémy Cerda, Giulio Manzonetto, and Alexis Saurin

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

Abstract

Although the λI-calculus is a natural fragment of the λ-calculus, obtained by forbidding the erasure, its equational theories did not receive much attention. The reason is that all proper denotational models studied in the literature equate all non-normalizable λI-terms, whence the associated theory is not very informative. The goal of this paper is to introduce a previously unknown theory of the λI-calculus, induced by a notion of evaluation trees that we call "Ohana trees". The Ohana tree of a λI-term is an annotated version of its Böhm tree, remembering all free variables that are hidden within its meaningless subtrees, or pushed into infinity along its infinite branches. We develop the associated theories of program approximation: the first approach - more classic - is based on finite trees and continuity, the second adapts Ehrhard and Regnier’s Taylor expansion. We then prove a Commutation Theorem stating that the normal form of the Taylor expansion of a λI-term coincides with the Taylor expansion of its Ohana tree. As a corollary, we obtain that the equality induced by Ohana trees is compatible with abstraction and application. We conclude by discussing the cases of Lévy-Longo and Berarducci trees, and generalizations to the full λ-calculus.

Cite as

Rémy Cerda, Giulio Manzonetto, and Alexis Saurin. Ohana Trees and Taylor Expansion for the λI-Calculus: No variable gets left behind or forgotten!. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cerda_et_al:LIPIcs.FSCD.2025.12,
  author =	{Cerda, R\'{e}my and Manzonetto, Giulio and Saurin, Alexis},
  title =	{{Ohana Trees and Taylor Expansion for the \lambdaI-Calculus: No variable gets left behind or forgotten!}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{12:1--12:20},
  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.2025.12},
  URN =		{urn:nbn:de:0030-drops-236277},
  doi =		{10.4230/LIPIcs.FSCD.2025.12},
  annote =	{Keywords: \lambda-calculus, program approximation, Taylor expansion, \lambdaI-calculus, persistent free variables, B\"{o}hm trees, Ohana trees}
}

Document

Invited Talk

DOI: 10.4230/LIPIcs.FSCD.2025.3

Unsolvable Terms in Filter Models (Invited Talk)

Authors: Mariangiola Dezani-Ciancaglini, Paola Giannini, and Furio Honsell

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

Abstract

Intersection type theories (itt’s) and filter models, i.e. λ-calculus models generated by itt’s, are reviewed in full generality. In this framework, which subsumes most λ-calculus models in the literature based on Scott-continuous functions, we discuss the interpretation of unsolvable terms. We give a necessary, but not sufficient, condition on an itt for the interpretation of some unsolvable term to be non-trivial in the filter model it generates. This result is obtained building on a type theoretic characterisation of the fine structure of unsolvables.

Cite as

Mariangiola Dezani-Ciancaglini, Paola Giannini, and Furio Honsell. Unsolvable Terms in Filter Models (Invited Talk). In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 3:1-3:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dezaniciancaglini_et_al:LIPIcs.FSCD.2025.3,
  author =	{Dezani-Ciancaglini, Mariangiola and Giannini, Paola and Honsell, Furio},
  title =	{{Unsolvable Terms in Filter Models}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{3:1--3:24},
  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.2025.3},
  URN =		{urn:nbn:de:0030-drops-236181},
  doi =		{10.4230/LIPIcs.FSCD.2025.3},
  annote =	{Keywords: \lambda-calculus, Intersection Types, Unsolvable Terms, Filter Models}
}

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

Track B: Automata, Logic, Semantics, and Theory of Programming

DOI: 10.4230/LIPIcs.ICALP.2025.145

Reducing Stochastic Games to Semidefinite Programming

Authors: Manuel Bodirsky, Georg Loho, and Mateusz Skomra

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract

We present a polynomial-time reduction from max-average constraints to the feasibility problem for semidefinite programs. This shows that Condon’s simple stochastic games, stochastic mean payoff games, and in particular mean payoff games and parity games can all be reduced to semidefinite programming.

Cite as

Manuel Bodirsky, Georg Loho, and Mateusz Skomra. Reducing Stochastic Games to Semidefinite Programming. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 145:1-145:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bodirsky_et_al:LIPIcs.ICALP.2025.145,
  author =	{Bodirsky, Manuel and Loho, Georg and Skomra, Mateusz},
  title =	{{Reducing Stochastic Games to Semidefinite Programming}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{145:1--145:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.145},
  URN =		{urn:nbn:de:0030-drops-235224},
  doi =		{10.4230/LIPIcs.ICALP.2025.145},
  annote =	{Keywords: Mean-payoff games, stochastic games, semidefinite programming, max-average constraints, max-atom problem}
}

Document

DOI: 10.4230/LIPIcs.ECOOP.2025.12

Automatic Goal Clone Detection in Rocq

Authors: Ali Ghanbari

Published in: LIPIcs, Volume 333, 39th European Conference on Object-Oriented Programming (ECOOP 2025)

Abstract

Proof engineering in Rocq is a labor-intensive process, and as proof developments grow in size, redundancy and maintainability become challenges. One such redundancy is goal cloning, i.e., proving α-equivalent goals multiple times, leading to wasted effort and bloated proof scripts. In this paper, we introduce clone-finder, a novel technique for detecting goal clones in Rocq proofs. By leveraging the formal notion of α-equivalence for Gallina terms, clone-finder systematically identifies duplicated proof goals across large Rocq codebases. We evaluate clone-finder on 40 real-world Rocq projects from the CoqGym dataset. Our results reveal that each project contains an average of 27.73 instances of goal clone. We observed that the clones can be categorized as either exact goal duplication, generalization, or α-equivalent goals with different proofs, each signifying varying levels duplicate effort. Our findings highlight significant untapped potential for proof reuse in Rocq-based formal verification projects, paving the way for future improvements in automated proof engineering.

Cite as

Ali Ghanbari. Automatic Goal Clone Detection in Rocq. In 39th European Conference on Object-Oriented Programming (ECOOP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 333, pp. 12:1-12:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ghanbari:LIPIcs.ECOOP.2025.12,
  author =	{Ghanbari, Ali},
  title =	{{Automatic Goal Clone Detection in Rocq}},
  booktitle =	{39th European Conference on Object-Oriented Programming (ECOOP 2025)},
  pages =	{12:1--12:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-373-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{333},
  editor =	{Aldrich, Jonathan 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.ECOOP.2025.12},
  URN =		{urn:nbn:de:0030-drops-233055},
  doi =		{10.4230/LIPIcs.ECOOP.2025.12},
  annote =	{Keywords: Clone Detection, Goal, Proof, Rocq, Gallina}
}

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.

Cite as

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

Complete Volume

DOI: 10.4230/LIPIcs.TYPES.2019

LIPIcs, Volume 175, TYPES 2019, Complete Volume

Authors: Marc Bezem and Assia Mahboubi

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

Abstract

LIPIcs, Volume 175, TYPES 2019, Complete Volume

Cite as

25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 1-256, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@Proceedings{bezem_et_al:LIPIcs.TYPES.2019,
  title =	{{LIPIcs, Volume 175, TYPES 2019, Complete Volume}},
  booktitle =	{25th International Conference on Types for Proofs and Programs (TYPES 2019)},
  pages =	{1--256},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019},
  URN =		{urn:nbn:de:0030-drops-130639},
  doi =		{10.4230/LIPIcs.TYPES.2019},
  annote =	{Keywords: LIPIcs, Volume 175, TYPES 2019, Complete Volume}
}

Document

Front Matter

DOI: 10.4230/LIPIcs.TYPES.2019.0

Front Matter, Table of Contents, Preface, Conference Organization

Authors: Marc Bezem and Assia Mahboubi

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

Abstract

Front Matter, Table of Contents, Preface, Conference Organization

Cite as

25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 0:i-0:x, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bezem_et_al:LIPIcs.TYPES.2019.0,
  author =	{Bezem, Marc and Mahboubi, Assia},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{25th International Conference on Types for Proofs and Programs (TYPES 2019)},
  pages =	{0:i--0:x},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.0},
  URN =		{urn:nbn:de:0030-drops-130640},
  doi =		{10.4230/LIPIcs.TYPES.2019.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2019.1

Making Isabelle Content Accessible in Knowledge Representation Formats

Authors: Michael Kohlhase, Florian Rabe, and Makarius Wenzel

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

Abstract

The libraries of proof assistants like Isabelle, Coq, HOL are notoriously difficult to interpret by external tools: de facto, only the prover itself can parse and process them adequately. In the case of Isabelle, an export of the library into a FAIR (Findable, Accessible, Interoperable, and Reusable) knowledge exchange format was already envisioned by the authors in 1999 but had previously proved too difficult. After substantial improvements of the Isabelle Prover IDE (PIDE) and the OMDoc/Mmt format since then, we are now able to deliver such an export. Concretely we present an integration of PIDE and Mmt that allows exporting all Isabelle libraries in OMDoc format. Our export covers the full Isabelle distribution and the Archive of Formal Proofs (AFP) - more than 12 thousand theories and locales resulting in over 65 GB of OMDoc/XML. Such a systematic export of Isabelle content to a well-defined interchange format like OMDoc enables many applications such as dependency management, independent proof checking, or library search.

Cite as

Michael Kohlhase, Florian Rabe, and Makarius Wenzel. Making Isabelle Content Accessible in Knowledge Representation Formats. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 1:1-1:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kohlhase_et_al:LIPIcs.TYPES.2019.1,
  author =	{Kohlhase, Michael and Rabe, Florian and Wenzel, Makarius},
  title =	{{Making Isabelle Content Accessible in Knowledge Representation Formats}},
  booktitle =	{25th International Conference on Types for Proofs and Programs (TYPES 2019)},
  pages =	{1:1--1:24},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.1},
  URN =		{urn:nbn:de:0030-drops-130651},
  doi =		{10.4230/LIPIcs.TYPES.2019.1},
  annote =	{Keywords: Isabelle, PIDE, OMDoc, MMT, library, export}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2019.2

Type Theory Unchained: Extending Agda with User-Defined Rewrite Rules

Authors: Jesper Cockx

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

Abstract

Dependently typed languages such as Coq and Agda can statically guarantee the correctness of our proofs and programs. To provide this guarantee, they restrict users to certain schemes - such as strictly positive datatypes, complete case analysis, and well-founded induction - that are known to be safe. However, these restrictions can be too strict, making programs and proofs harder to write than necessary. On a higher level, they also prevent us from imagining the different ways the language could be extended. In this paper I show how to extend a dependently typed language with user-defined higher-order non-linear rewrite rules. Rewrite rules are a form of equality reflection that is applied automatically by the typechecker. I have implemented rewrite rules as an extension to Agda, and I give six examples how to use them both to make proofs easier and to experiment with extensions of type theory. I also show how to make rewrite rules interact well with other features of Agda such as η-equality, implicit arguments, data and record types, irrelevance, and universe level polymorphism. Thus rewrite rules break the chains on computation and put its power back into the hands of its rightful owner: yours.

Cite as

Jesper Cockx. Type Theory Unchained: Extending Agda with User-Defined Rewrite Rules. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 2:1-2:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cockx:LIPIcs.TYPES.2019.2,
  author =	{Cockx, Jesper},
  title =	{{Type Theory Unchained: Extending Agda with User-Defined Rewrite Rules}},
  booktitle =	{25th International Conference on Types for Proofs and Programs (TYPES 2019)},
  pages =	{2:1--2:27},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.2},
  URN =		{urn:nbn:de:0030-drops-130666},
  doi =		{10.4230/LIPIcs.TYPES.2019.2},
  annote =	{Keywords: Dependent types, Proof assistants, Rewrite rules, Higher-order rewriting, Agda}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2019.3

A Quantitative Understanding of Pattern Matching

Authors: Sandra Alves, Delia Kesner, and Daniel Ventura

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

Abstract

This paper shows that the recent approach to quantitative typing systems for programming languages can be extended to pattern matching features. Indeed, we define two resource-aware type systems, named U and E, for a λ-calculus equipped with pairs for both patterns and terms. Our typing systems borrow some basic ideas from [Antonio Bucciarelli et al., 2015], which characterises (head) normalisation in a qualitative way, in the sense that typability and normalisation coincide. But, in contrast to [Antonio Bucciarelli et al., 2015], our systems also provide quantitative information about the dynamics of the calculus. Indeed, system U provides upper bounds for the length of (head) normalisation sequences plus the size of their corresponding normal forms, while system E, which can be seen as a refinement of system U, produces exact bounds for each of them. This is achieved by means of a non-idempotent intersection type system equipped with different technical tools. First of all, we use product types to type pairs instead of the disjoint unions in [Antonio Bucciarelli et al., 2015], which turn out to be an essential quantitative tool because they remove the confusion between "being a pair" and "being duplicable". Secondly, typing sequents in system E are decorated with tuples of integers, which provide quantitative information about normalisation sequences, notably time (cf. length) and space (cf. size). Moreover, the time resource information is remarkably refined, because it discriminates between different kinds of reduction steps performed during evaluation, so that beta, substitution and matching steps are counted separately. Another key tool of system E is that the type system distinguishes between consuming (contributing to time) and persistent (contributing to space) constructors.

Cite as

Sandra Alves, Delia Kesner, and Daniel Ventura. A Quantitative Understanding of Pattern Matching. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 3:1-3:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{alves_et_al:LIPIcs.TYPES.2019.3,
  author =	{Alves, Sandra and Kesner, Delia and Ventura, Daniel},
  title =	{{A Quantitative Understanding of Pattern Matching}},
  booktitle =	{25th International Conference on Types for Proofs and Programs (TYPES 2019)},
  pages =	{3:1--3:36},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.3},
  URN =		{urn:nbn:de:0030-drops-130672},
  doi =		{10.4230/LIPIcs.TYPES.2019.3},
  annote =	{Keywords: Intersection Types, Pattern Matching, Exact Bounds}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2019.4

Big Step Normalisation for Type Theory

Authors: Thorsten Altenkirch and Colin Geniet

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

Abstract

Big step normalisation is a normalisation method for typed lambda-calculi which relies on a purely syntactic recursive evaluator. Termination of that evaluator is proven using a predicate called strong computability, similar to the techniques used to prove strong normalisation of β-reduction for typed lambda-calculi. We generalise big step normalisation to a minimalist dependent type theory. Compared to previous presentations of big step normalisation for e.g. the simply-typed lambda-calculus, we use a quotiented syntax of type theory, which crucially reduces the syntactic complexity introduced by dependent types. Most of the proof has been formalised using Agda.

Cite as

Thorsten Altenkirch and Colin Geniet. Big Step Normalisation for Type Theory. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{altenkirch_et_al:LIPIcs.TYPES.2019.4,
  author =	{Altenkirch, Thorsten and Geniet, Colin},
  title =	{{Big Step Normalisation for Type Theory}},
  booktitle =	{25th International Conference on Types for Proofs and Programs (TYPES 2019)},
  pages =	{4:1--4:20},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.4},
  URN =		{urn:nbn:de:0030-drops-130682},
  doi =		{10.4230/LIPIcs.TYPES.2019.4},
  annote =	{Keywords: Normalisation, big step normalisation, type theory, dependent types, Agda}
}

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