DROPS

Document

DOI: 10.4230/LIPIcs.CSL.2026.18

A Modular Framework for Proof-Search via Formalised Modal Completeness in HOL Light

Authors: Antonella Bilotta, Marco Maggesi, and Cosimo Perini Brogi

Published in: LIPIcs, Volume 363, 34th EACSL Annual Conference on Computer Science Logic (CSL 2026)

Abstract

We extend the existing HOL Light Library for Modal Systems (HOLMS) to support a modular implementation of modal reasoning within the HOL Light proof assistant. We deeply embed axiomatic calculi and relational semantics for seven normal modal logics (K, T, B, K4, S4, S5, GL) and formalise modal adequacy theorems for these systems. We then leverage those formalisations to implement a mechanism for automated reasoning via proof-search in the associated labelled sequent calculi, which we shallowly embed in HOL Light’s goal-stack mechanism. This way, we equip the general-purpose proof assistant with (semi)decision procedures for these logics that, in case of failure to construct a proof for the input formula, return a certified countermodel within the appropriate class for the logic under consideration. On the methodological side, we propose a precise measure of the modularity of our approach by systematically adopting Christopher Strachey’s distinction between ad hoc and parametric polymorphism throughout the library.

Cite as

Antonella Bilotta, Marco Maggesi, and Cosimo Perini Brogi. A Modular Framework for Proof-Search via Formalised Modal Completeness in HOL Light. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 18:1-18:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bilotta_et_al:LIPIcs.CSL.2026.18,
  author =	{Bilotta, Antonella and Maggesi, Marco and Perini Brogi, Cosimo},
  title =	{{A Modular Framework for Proof-Search via Formalised Modal Completeness in HOL Light}},
  booktitle =	{34th EACSL Annual Conference on Computer Science Logic (CSL 2026)},
  pages =	{18:1--18:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-411-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{363},
  editor =	{Guerrini, Stefano and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.18},
  URN =		{urn:nbn:de:0030-drops-254427},
  doi =		{10.4230/LIPIcs.CSL.2026.18},
  annote =	{Keywords: Modal logic, HOL Light, Labelled sequent calculi, Logical verification, Interactive theorem proving, Automated proof-search}
}

Document

DOI: 10.4230/LIPIcs.ITP.2025.7

A Verified Cost Model for Call-By-Push-Value

Authors: Zhuo Zoey Chen, Johannes Åman Pohjola, and Christine Rizkallah

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

Abstract

The call-by-push-value λ-calculus allows for syntactically specifying the order of evaluation as part of the term language. Hence, it serves as a unifying language for embedding various evaluation strategies including call-by-value and call-by-name. Given the impact of call-by-push-value, it is remarkable that its adequacy as a model for computational complexity theory has not yet been studied. In this paper, we show that the call-by-push-value λ-calculus is reasonable for both time and space complexity. A reasonable cost model can encode other reasonable cost models with polynomial overhead in time and constant factor overhead in space. We achieve this by encoding call-by-push-value λ-calculus into Turing machines, following a simulation strategy by Forster et al.; for the converse direction, we prove that Levy’s encoding of the call-by-value λ-calculus has reasonable complexity bounds. The main results have been formalised in the HOL4 theorem prover.

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Zhuo Zoey Chen, Johannes Åman Pohjola, and Christine Rizkallah. A Verified Cost Model for Call-By-Push-Value. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 7:1-7:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chen_et_al:LIPIcs.ITP.2025.7,
  author =	{Chen, Zhuo Zoey and \r{A}man Pohjola, Johannes and Rizkallah, Christine},
  title =	{{A Verified Cost Model for Call-By-Push-Value}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{7:1--7:19},
  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.2025.7},
  URN =		{urn:nbn:de:0030-drops-246067},
  doi =		{10.4230/LIPIcs.ITP.2025.7},
  annote =	{Keywords: lambda calculus, formalizations of computational models, computability theory, HOL, call-by-push-value reduction, time and space complexity, abstract machines}
}

Document

Short Paper

DOI: 10.4230/LIPIcs.ITP.2025.37

LeanLTL: A Unifying Framework for Linear Temporal Logics in Lean (Short Paper)

Authors: Eric Vin, Kyle A. Miller, and Daniel J. Fremont

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

Abstract

We propose LeanLTL, a unifying framework for linear temporal logics in Lean 4. LeanLTL supports reasoning about traces that represent either infinite or finite linear time. The library allows traditional LTL syntax to be combined with arbitrary Lean expressions, making it straightforward to define properties involving numerical or other types. We prove that standard flavors of LTL can be embedded in our framework. The library also provides automation for reasoning about LeanLTL formulas in a way that facilitates using Lean’s existing tactics. Finally, we provide examples illustrating the utility of the library in reasoning about systems that come from applications.

Cite as

Eric Vin, Kyle A. Miller, and Daniel J. Fremont. LeanLTL: A Unifying Framework for Linear Temporal Logics in Lean (Short Paper). In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 37:1-37:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vin_et_al:LIPIcs.ITP.2025.37,
  author =	{Vin, Eric and Miller, Kyle A. and Fremont, Daniel J.},
  title =	{{LeanLTL: A Unifying Framework for Linear Temporal Logics in Lean}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{37:1--37:9},
  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.2025.37},
  URN =		{urn:nbn:de:0030-drops-246356},
  doi =		{10.4230/LIPIcs.ITP.2025.37},
  annote =	{Keywords: Linear Temporal Logic, Interactive Theorem Proving, Lean 4}
}

Document

Invited Talk

DOI: 10.4230/LIPIcs.ITP.2025.40

Autosubst: On Mechanising Binders in a General-Purpose Proof Assistant (Invited Talk)

Authors: Kathrin Stark

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

Abstract

The question of handling binders effectively regularly comes up when mechanising results in a proof assistant. Since the beginning of proof assistants, many solutions have been suggested: these include de Bruijn syntax, locally nameless binders, nominal syntax or HOAS. But even today - 20 years after the POPLMark Challenge [Brian E. Aydemir et al., 2005] - new tools and approaches to binding are still being published. Binders are still mentioned as being a complication of mechanised proofs over pen-and-paper proofs and the source of uninteresting technicalities and heaps of boilerplate - particularly, when working in a proof assistant without native support for binders or quotients. Autosubst [Steven Schäfer et al., 2015; Steven Schäfer et al., 2015], initially developed a decade ago as a teaching tool, addresses this problem by automating boilerplate code generation for a de Bruijn representation in the Rocq prover. Autosubst translates a specification into the corresponding pure or scoped de Bruijn algebra: It hence generates a corresponding instantiation operation for parallel substitutions, and several equational substitution lemmas. Central to its usability is a rewriting tactic that automatically decides equality up to substitutions, requiring minimal input and knowledge from the user’s side. This greatly simplifies using the otherwise notoriously difficult de Bruijn representation. Since then, Autosubst has been successfully used in several projects. In this talk, I will give an overview of the background and history of Autosubst and give an overview on the work, tools, and formalisations around it [Kathrin Stark et al., 2019; Kathrin Stark, 2020; Andreas Abel et al., 2019]. Moreover, this talk will highlight some of the more recent extensions [Yannick Forster and Kathrin Stark, 2020] as well as outline future challenges and directions.

Cite as

Kathrin Stark. Autosubst: On Mechanising Binders in a General-Purpose Proof Assistant (Invited Talk). In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 40:1-40:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{stark:LIPIcs.ITP.2025.40,
  author =	{Stark, Kathrin},
  title =	{{Autosubst: On Mechanising Binders in a General-Purpose Proof Assistant}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{40:1--40:2},
  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.2025.40},
  URN =		{urn:nbn:de:0030-drops-246385},
  doi =		{10.4230/LIPIcs.ITP.2025.40},
  annote =	{Keywords: Syntax, binders, Rocq}
}

Document

DOI: 10.4230/LIPIcs.ITP.2025.13

Barendregt’s Theory of the λ-Calculus, Refreshed and Formalized

Authors: Adrienne Lancelot, Beniamino Accattoli, and Maxime Vemclefs

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

Abstract

Barendregt’s book on the untyped λ-calculus refines the inconsistent view of β-divergence as representation of the undefined via the key concept of head reduction. In this paper, we put together recent revisitations of some key theorems laid out in Barendregt’s book, and we formalize them in the Abella proof assistant. Our work provides a compact and refreshed presentation of the core of the book. The formalization faithfully mimics pen-and-paper proofs. Two interesting aspects are the manipulation of contexts for the study of contextual equivalence and a formal alternative to the informal trick at work in Takahashi’s proof of the genericity lemma. As a by-product, we obtain an alternative definition of contextual equivalence that does not mention contexts.

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Adrienne Lancelot, Beniamino Accattoli, and Maxime Vemclefs. Barendregt’s Theory of the λ-Calculus, Refreshed and Formalized. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lancelot_et_al:LIPIcs.ITP.2025.13,
  author =	{Lancelot, Adrienne and Accattoli, Beniamino and Vemclefs, Maxime},
  title =	{{Barendregt’s Theory of the \lambda-Calculus, Refreshed and Formalized}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{13:1--13:22},
  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.2025.13},
  URN =		{urn:nbn:de:0030-drops-246114},
  doi =		{10.4230/LIPIcs.ITP.2025.13},
  annote =	{Keywords: lambda-calculus, head reduction, equational theory}
}

Document

DOI: 10.4230/LIPIcs.ITP.2025.11

Animating MRBNFs: Truly Modular Binding-Aware Datatypes in Isabelle/HOL

Authors: Jan van Brügge, Andrei Popescu, and Dmitriy Traytel

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

Abstract

Nominal Isabelle provides powerful tools for meta-theoretic reasoning about syntax of logics or programming languages, in which variables are bound. It has been instrumental to major verification successes, such as Gödel’s incompleteness theorems. However, the existing tooling is not compositional. In particular, it does not support nested recursion, linear binding patterns, or infinitely branching syntax. These limitations are fundamental in the way nominal datatypes and functions on them are constructed within Nominal Isabelle. Taking advantage of recent theoretical advancements that overcome these limitations through a modular approach using the concept of map-restricted bounded natural functor (MRBNF), we develop and implement a new definitional package for binding-aware datatypes in Isabelle/HOL, called MrBNF. We describe the journey from the user specification to the end-product types, constants and theorems the tool generates. We validate MrBNF in two formalization case studies that so far were out of reach of nominal approaches: (1) Mazza’s isomorphism between the finitary and the infinitary affine λ-calculus, and (2) the POPLmark 2B challenge, which involves non-free binders for linear pattern matching.

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Jan van Brügge, Andrei Popescu, and Dmitriy Traytel. Animating MRBNFs: Truly Modular Binding-Aware Datatypes in Isabelle/HOL. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanbrugge_et_al:LIPIcs.ITP.2025.11,
  author =	{van Br\"{u}gge, Jan and Popescu, Andrei and Traytel, Dmitriy},
  title =	{{Animating MRBNFs: Truly Modular Binding-Aware Datatypes in Isabelle/HOL}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{11:1--11:20},
  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.2025.11},
  URN =		{urn:nbn:de:0030-drops-246091},
  doi =		{10.4230/LIPIcs.ITP.2025.11},
  annote =	{Keywords: syntax with bindings, datatypes, inductive predicates, Isabelle/HOL}
}

Document

DOI: 10.4230/LIPIcs.ITP.2025.10

An Isabelle/HOL Formalization of Semi-Thue and Conditional Semi-Thue Systems

Authors: Dohan Kim

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

Abstract

We present a formalized framework for semi-Thue and conditional semi-Thue systems for studying monoids and their word problem using the Isabelle/HOL proof assistant. We provide a formalized decision procedure for the word problem of monoids if they are finitely presented by complete semi-Thue systems. In particular, we present a new formalized method for checking confluence using (conditional) critical pairs for certain conditional semi-Thue systems. We propose and formalize an inference system for generating conditional equational theories and Thue congruences using conditional semi-Thue systems. Then we provide a new formalized decision procedure for the word problem of monoids which have finite complete (reductive) conditional presentations.

Cite as

Dohan Kim. An Isabelle/HOL Formalization of Semi-Thue and Conditional Semi-Thue Systems. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kim:LIPIcs.ITP.2025.10,
  author =	{Kim, Dohan},
  title =	{{An Isabelle/HOL Formalization of Semi-Thue and Conditional Semi-Thue Systems}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{10:1--10:20},
  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.2025.10},
  URN =		{urn:nbn:de:0030-drops-246081},
  doi =		{10.4230/LIPIcs.ITP.2025.10},
  annote =	{Keywords: semi-Thue systems, conditional semi-Thue systems, conditional string rewriting, monoids, word problem}
}

Document

DOI: 10.4230/OASIcs.Programming.2025.10

Shica - Improving the Programming Experience for Agent-Based, Distributed, Physical Computing Systems

Authors: Hiroto Shikada and Ian Piumarta

Published in: OASIcs, Volume 134, Companion Proceedings of the 9th International Conference on the Art, Science, and Engineering of Programming (Programming 2025)

Abstract

Agent-based and distributed computing systems play an important role in many fields. Programming these systems can be annoying because of the complexity of managing multiple asynchronous processes and state transitions, sometimes hidden inter-dependencies between program elements, and (often) unnecessarily terse, un-intuitive syntax. In this paper we describe the design and implementation of Shica, an experimental language designed for physical computing that is fun to program. Shica unifies state-based, event-based, and distributed programming along with some elements of context-oriented. We informally evaluate Shica’s characteristics including its suitability for deployment on resource-constrained, embedded devices and its contribution to improving the quality of the programming experience by maximizing scope, economy, and elegance of expression.

Cite as

Hiroto Shikada and Ian Piumarta. Shica - Improving the Programming Experience for Agent-Based, Distributed, Physical Computing Systems. In Companion Proceedings of the 9th International Conference on the Art, Science, and Engineering of Programming (Programming 2025). Open Access Series in Informatics (OASIcs), Volume 134, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{shikada_et_al:OASIcs.Programming.2025.10,
  author =	{Shikada, Hiroto and Piumarta, Ian},
  title =	{{Shica - Improving the Programming Experience for Agent-Based, Distributed, Physical Computing Systems}},
  booktitle =	{Companion Proceedings of the 9th International Conference on the Art, Science, and Engineering of Programming (Programming 2025)},
  pages =	{10:1--10:22},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-382-9},
  ISSN =	{2190-6807},
  year =	{2025},
  volume =	{134},
  editor =	{Edwards, Jonathan and Perera, Roly and Petricek, Tomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.Programming.2025.10},
  URN =		{urn:nbn:de:0030-drops-242948},
  doi =		{10.4230/OASIcs.Programming.2025.10},
  annote =	{Keywords: Programming Languages, run-time Systems, agent-based Programming}
}

Document

DOI: 10.4230/LIPIcs.CP.2025.20

Dependency-Curated Large Neighbourhood Search

Authors: Frej Knutar Lewander, Pierre Flener, and Justin Pearson

Published in: LIPIcs, Volume 340, 31st International Conference on Principles and Practice of Constraint Programming (CP 2025)

Abstract

In large neighbourhood search (LNS), an incumbent initial solution is incrementally improved by selecting a subset of the variables, called the freeze set, and fixing them to their values in the incumbent solution, while a value for each remaining variable is found and assigned via solving (such as constraint programming-style propagation and search). Much research has been performed on finding generic and problem-specific LNS selection heuristics that select freeze sets that lead to high-quality solutions. In constraint-based local search (CBLS), the relations between the variables via the constraints are fundamental and well-studied, as they capture dependencies of the variables. In this paper, we apply these ideas from CBLS to the LNS context, presenting the novel dependency curation scheme, which exploits them to find a low-cardinality set of variables that the freeze set of any selection heuristic should be a subset of. The scheme often improves the overall performance of generic selection heuristics. Even when the scheme is used with a naïve generic selection heuristic that selects random freeze sets, the performance is competitive with more elaborate generic selection heuristics.

Cite as

Frej Knutar Lewander, Pierre Flener, and Justin Pearson. Dependency-Curated Large Neighbourhood Search. In 31st International Conference on Principles and Practice of Constraint Programming (CP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 340, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{knutarlewander_et_al:LIPIcs.CP.2025.20,
  author =	{Knutar Lewander, Frej and Flener, Pierre and Pearson, Justin},
  title =	{{Dependency-Curated Large Neighbourhood Search}},
  booktitle =	{31st International Conference on Principles and Practice of Constraint Programming (CP 2025)},
  pages =	{20:1--20:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-380-5},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{340},
  editor =	{de la Banda, Maria Garcia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2025.20},
  URN =		{urn:nbn:de:0030-drops-238810},
  doi =		{10.4230/LIPIcs.CP.2025.20},
  annote =	{Keywords: Combinatorial Optimisation, Large Neighbourhood Search (LNS), Constraint-Based Local Search (CBLS)}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2025.33

Solving Guarded Domain Equations in Presheaves over Ordinals and Mechanizing It

Authors: Sergei Stepanenko and Amin Timany

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

Abstract

Constructing solutions to recursive domain equations is a well-known, important problem in the study of programs and programming languages. Mathematically speaking, the problem is finding a fixed point (up to isomorphism) of a suitable functor over a suitable category. A particularly useful instance, inspired by the step-indexing technique, is where the functor is over (a subcategory of) the category of presheaves over the ordinal ω and the functors are locally-contractive, also known as guarded functors. This corresponds to step-indexing over natural numbers. However, for certain problems, e.g., when dealing with infinite non-determinism, one needs to employ trans-finite step-indexing, i.e., consider presheaf categories over higher ordinals. Prior work on trans-finite step-indexing either only considers a very narrow class of functors over a particularly restricted subcategory of presheaves over higher ordinals, or treats the problem very generally working with sheaves over an arbitrary complete Heyting algebra with a well-founded basis. In this paper we present a solution to the guarded domain equations problem over all guarded functors over the category of presheaves over ordinal numbers, as well as its mechanization in the Rocq Prover. As the categories of sheaves and presheaves over ordinals are equivalent, our main contribution is simplifying prior work from the setting of the category of sheaves to the setting of the category of presheaves and mechanizing it - presheaves are more amenable to mechanization in a proof assistant.

Cite as

Sergei Stepanenko and Amin Timany. Solving Guarded Domain Equations in Presheaves over Ordinals and Mechanizing It. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 33:1-33:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{stepanenko_et_al:LIPIcs.FSCD.2025.33,
  author =	{Stepanenko, Sergei and Timany, Amin},
  title =	{{Solving Guarded Domain Equations in Presheaves over Ordinals and Mechanizing It}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{33:1--33:24},
  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.2025.33},
  URN =		{urn:nbn:de:0030-drops-236486},
  doi =		{10.4230/LIPIcs.FSCD.2025.33},
  annote =	{Keywords: Domain Equations, Guarded Fixed Points, Fixed Points, Category Theory, Rocq, Presheaves, Ordinals}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2025.17

Mechanized Undecidability of Higher-Order Beta-Matching

Authors: Andrej Dudenhefner

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

Abstract

Higher-order β-matching is the following decision problem: given two simply typed λ-terms, can the first term be instantiated to be β-equivalent to the second term? This problem was formulated by Huet in the 1970s and shown undecidable by Loader in 2003 by reduction from λ-definability. The present work provides a novel undecidability proof for higher-order β-matching, in an effort to verify this result by means of a proof assistant. Rather than starting from λ-definability, the presented proof encodes a restricted form of string rewriting as higher-order β-matching. The particular approach is similar to Urzyczyn’s undecidability result for intersection type inhabitation. The presented approach has several advantages. First, the proof is simpler to verify in full detail due to the simple form of rewriting systems, which serve as a starting point. Second, undecidability of the considered problem in string rewriting is already certified using the Coq proof assistant. As a consequence, we obtain a certified many-one reduction from the Halting Problem to higher-order β-matching. Third, the presented approach identifies a uniform construction which shows undecidability of higher-order β-matching, λ-definability, and intersection type inhabitation. The presented undecidability proof is mechanized in the Coq proof assistant and contributed to the existing Coq Library of Undecidability Proofs.

Cite as

Andrej Dudenhefner. Mechanized Undecidability of Higher-Order Beta-Matching. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dudenhefner:LIPIcs.FSCD.2025.17,
  author =	{Dudenhefner, Andrej},
  title =	{{Mechanized Undecidability of Higher-Order Beta-Matching}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{17:1--17:15},
  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.2025.17},
  URN =		{urn:nbn:de:0030-drops-236323},
  doi =		{10.4230/LIPIcs.FSCD.2025.17},
  annote =	{Keywords: lambda-calculus, simple types, undecidability, higher-order matching, mechanization, Coq}
}

Document

Invited Talk

DOI: 10.4230/LIPIcs.CSL.2025.3

Synthetic Mathematics for the Mechanisation of Computability Theory and Logic (Invited Talk)

Authors: Yannick Forster

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract

Mathematical practice in most areas of mathematics is based on the assumption that proofs could be made fully formal in a chosen foundation in principle. This assumption is backed by partial attempts at formalisation and by full mechanisation of various areas of mathematics in various proof assistants and various foundations. Areas that have been largely neglected for computer-assisted and machine-checked proofs are computability theory and logic: Fundamental results like Gödel’s second incompleteness theorem in its stronger forms due to Kleene and Rosser, Löb’s theorem, Post’s theorem connecting the arithmetical hierarchy and Turing jumps, or the Friedberg-Muĉnik theorem solving Post’s problem have not or only very recently been re-produced in proof assistants. This is due to the fact that making these arguments formal is several orders of magnitude more involved than formalising other areas of mathematics, due to the amount of invisible mathematics (a term coined by Andrej Bauer) involved. In computability theory, invisible arguments occur mainly behind proofs that a certain intuitively sketched procedure is computable in - citing Emil Post - "forbidding, diverse and alien formalisms in which this [...] work of Gödel, Church, Turing, Kleene, Rosser [...] is embodied.". For instance, there have been various approaches of formalising Turing machines, all to the ultimate dissatisfaction of the respective authors, and none going further than constructing a universal machine and proving the halting problem undecidable. Professional computability theorist and teachers of computability theory thus rely on the informal Church Turing thesis to carry out their work and only argue the computability of described algorithms informally. For computability theory, a way out was proposed in the 1980s by Fred Richman and developed during the last decade by Andrej Bauer: Synthetic computability theory, where one assumes axioms in a constructive foundation which essentially identify all (constructively definable) functions with computable functions. A drawback of the approach is that assuming such an axiom on top of the axiom of countable choice - which is routinely assumed in this branch of constructive mathematics and computable analysis - is that the law of excluded middle, i.e. classical logic, becomes invalid. Computability theory is however, as all mainstream branches of mathematics, making routine use of the axiom of excluded middle. In the case of logic, the invisible mathematics usually is either centered around encoding formulas and proofs as numbers using Gödel or similar encodings or about provability arguments that certain results can be proved in restricted proof systems such as Peano arithmetic. In several settings, synthetic computability arguments can be employed to mechanise these proofs. We observe that a slight foundational shift rectifies the situation: By basing synthetic computability theory in the Calculus of Inductive Constructions, the type theory underlying amongst others the Coq and Lean proof assistants, where countable choice is independent and thus not provable, axioms for synthetic computability are compatible with the law of excluded middle. This insight can be used to finally mechanise computability theory and logic, in an elegant, concise way where invisible arguments stay invisible: with Felix Jahn I have mechanised arguments related to many-one and truth-table reduction theory (published at CSL '23), Dominik Kirst and Benjamin Peters have presented Gödel’s first incompleteness theorem in this style (at CSL '23), and in collaboration with Dominik Kirst and Niklas Mück I have given a proof of Post’s hierarchy theorem (at CSL '24). In this invited talk, I will give a broader overview of this line of research investigating a mechanised synthetic approach to logic and computability theory. In particular, I will discuss a Coq library of undecidability proofs, results in the theory of reducibility degrees, constructive reverse analysis of theorems, as well as generalised incompleteness results such as Löb’s theorem.

Cite as

Yannick Forster. Synthetic Mathematics for the Mechanisation of Computability Theory and Logic (Invited Talk). In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 3:1-3:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{forster:LIPIcs.CSL.2025.3,
  author =	{Forster, Yannick},
  title =	{{Synthetic Mathematics for the Mechanisation of Computability Theory and Logic}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{3:1--3:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.3},
  URN =		{urn:nbn:de:0030-drops-227603},
  doi =		{10.4230/LIPIcs.CSL.2025.3},
  annote =	{Keywords: Synthetic mathematics, computability theory, logic}
}

Document

DOI: 10.4230/LIPIcs.CSL.2025.40

Completeness of First-Order Bi-Intuitionistic Logic

Authors: Dominik Kirst and Ian Shillito

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract

We provide a succinct and verified completeness proof for first-order bi-intuitionistic logic, relative to constant domain Kripke semantics. By doing so, we make up for the almost-50-year-old substantial mistakes in Rauszer’s foundational work, detected but unresolved by Shillito two years ago. Moreover, an even earlier but historically neglected proof by Klemke has been found to contain at least local errors by Olkhovikov and Badia, that remained unfixed due to the technical complexity of Klemke’s argument. To resolve this unclear situation once and for all, we give a succinct completeness proof, based on and dualising a standard proof for constant domain intuitionistic logic, and verify our constructions using the Coq proof assistant to guarantee correctness.

Cite as

Dominik Kirst and Ian Shillito. Completeness of First-Order Bi-Intuitionistic Logic. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 40:1-40:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kirst_et_al:LIPIcs.CSL.2025.40,
  author =	{Kirst, Dominik and Shillito, Ian},
  title =	{{Completeness of First-Order Bi-Intuitionistic Logic}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{40:1--40:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.40},
  URN =		{urn:nbn:de:0030-drops-227979},
  doi =		{10.4230/LIPIcs.CSL.2025.40},
  annote =	{Keywords: bi-intuitionistic logic, first-order logic, completeness, Coq proof assistant}
}

Document

DOI: 10.4230/LIPIcs.CSL.2024.29

The Kleene-Post and Post’s Theorem in the Calculus of Inductive Constructions

Authors: Yannick Forster, Dominik Kirst, and Niklas Mück

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

Abstract

The Kleene-Post theorem and Post’s theorem are two central and historically important results in the development of oracle computability theory, clarifying the structure of Turing reducibility degrees. They state, respectively, that there are incomparable Turing degrees and that the arithmetical hierarchy is connected to the relativised form of the halting problem defined via Turing jumps. We study these two results in the calculus of inductive constructions (CIC), the constructive type theory underlying the Coq proof assistant. CIC constitutes an ideal foundation for the formalisation of computability theory for two reasons: First, like in other constructive foundations, computable functions can be treated via axioms as a purely synthetic notion rather than being defined in terms of a concrete analytic model of computation such as Turing machines. Furthermore and uniquely, CIC allows consistently assuming classical logic via the law of excluded middle or weaker variants on top of axioms for synthetic computability, enabling both fully classical developments and taking the perspective of constructive reverse mathematics on computability theory. In the present paper, we give a fully constructive construction of two Turing-incomparable degrees à la Kleene-Post and observe that the classical content of Post’s theorem seems to be related to the arithmetical hierarchy of the law of excluded middle due to Akama et. al. Technically, we base our investigation on a previously studied notion of synthetic oracle computability and contribute the first consistency proof of a suitable enumeration axiom. All results discussed in the paper are mechanised and contributed to the Coq library of synthetic computability.

Cite as

Yannick Forster, Dominik Kirst, and Niklas Mück. The Kleene-Post and Post’s Theorem in the Calculus of Inductive Constructions. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{forster_et_al:LIPIcs.CSL.2024.29,
  author =	{Forster, Yannick and Kirst, Dominik and M\"{u}ck, Niklas},
  title =	{{The Kleene-Post and Post’s Theorem in the Calculus of Inductive Constructions}},
  booktitle =	{32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)},
  pages =	{29:1--29:20},
  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.2024.29},
  URN =		{urn:nbn:de:0030-drops-196728},
  doi =		{10.4230/LIPIcs.CSL.2024.29},
  annote =	{Keywords: Constructive mathematics, Computability theory, Logical foundations, Constructive type theory, Interactive theorem proving, Coq proof assistant}
}

Document

DOI: 10.4230/LIPIcs.CSL.2023.21

Constructive and Synthetic Reducibility Degrees: Post’s Problem for Many-One and Truth-Table Reducibility in Coq

Authors: Yannick Forster and Felix Jahn

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

Abstract

We present a constructive analysis and machine-checked theory of one-one, many-one, and truth-table reductions based on synthetic computability theory in the Calculus of Inductive Constructions, the type theory underlying the proof assistant Coq. We give elegant, synthetic, and machine-checked proofs of Post’s landmark results that a simple predicate exists, is enumerable, undecidable, but many-one incomplete (Post’s problem for many-one reducibility), and a hypersimple predicate exists, is enumerable, undecidable, but truth-table incomplete (Post’s problem for truth-table reducibility). In synthetic computability, one assumes axioms allowing to carry out computability theory with all definitions and proofs purely in terms of functions of the type theory with no mention of a model of computation. Proofs can focus on the essence of the argument, without having to sacrifice formality. Synthetic computability also clears the lense for constructivisation. Our constructively careful definition of simple and hypersimple predicates allows us to not assume classical axioms, not even Markov’s principle, still yielding the expected strong results.

Cite as

Yannick Forster and Felix Jahn. Constructive and Synthetic Reducibility Degrees: Post’s Problem for Many-One and Truth-Table Reducibility in Coq. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{forster_et_al:LIPIcs.CSL.2023.21,
  author =	{Forster, Yannick and Jahn, Felix},
  title =	{{Constructive and Synthetic Reducibility Degrees: Post’s Problem for Many-One and Truth-Table Reducibility in Coq}},
  booktitle =	{31st EACSL Annual Conference on Computer Science Logic (CSL 2023)},
  pages =	{21:1--21:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-264-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{252},
  editor =	{Klin, Bartek and Pimentel, Elaine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.21},
  URN =		{urn:nbn:de:0030-drops-174820},
  doi =		{10.4230/LIPIcs.CSL.2023.21},
  annote =	{Keywords: type theory, computability theory, constructive mathematics, Coq}
}

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