flgrubm

The Category of Iterative Sets in Cubical Agda

Fabian Lukas Grubmüller — Wed, 22 Apr 2026 00:00:00 +0000

Iterative sets form a constructive Tarski-style universe V⁰ of h-sets that is itself an h-set. It arises naturally from the study of iterative multisets, where V⁰ is defined as a particular W-type for which the indexing function is restricted to embeddings. In previous work, Gratzer, Gylterud, Mörtberg, and Stenholm showed that V⁰ can be equipped with the structure of a Category with Families, that admits both Π- and Σ-structures. While their proofs were rather straightforward on paper, their Unimath Agda formalization faced significant obstacles, particularly for the Σ-structures.

In my master’s thesis, I explored whether a formalization of this using Cubical Agda would be easier, which ultimately turned out not to be the case. In this talk, I will present the results and challenges of this effort as well as the continuing work together with Anders Mörtberg and Peter Lumsdaine.

The Category of Iterative Sets in Cubical Agda

Fabian Lukas Grubmüller — Fri, 06 Feb 2026 00:00:00 +0000

Iterative sets form a constructive Tarski-style universe V⁰ of h-sets. This universe is closed under common type-theoretic constructions and is itself an h-set. It arises naturally from the study of iterative multisets, where V⁰ is defined as a specific type-indexed W-type for which the indexing function is restricted to embeddings, effectively collapsing higher structure.

In previous work, Gratzer, Gylterud, Mörtberg, and Stenholm showed that V⁰ is a model of dependent type theory, in particular a Category with Families (CwF), that admits both Π- and Σ-structures. While their proofs were rather straightforward on paper, often reducing to reflexivity, their formalization in standard Agda (using agda-unimath) faced significant obstacles. For the formalization of Π- and Σ-structures, they faced problems due to complex path algebra involving multiple layers of transport and function extensionality, leading them to abandon the formalization of the Σ-structure.

In this thesis, we explore whether a formalization in Cubical Agda, an extension of Agda for cubical type theory, is easier to accomplish. We implement the general properties of iterative sets, as well as CwF and Σ-structures. For the latter we use three distinct strategies: a naive translation of the prior work, a more cubical approach replacing equalities containing transport with heterogeneous path types, and a strategy that eliminates transport in favor of ad-hoc functions that can be later instantiated by the identity function. We find that while the cubical metatheory simplifies reasoning about extensionality, it also introduces new challenges. One of the main issues is the lack of a definitional J-rule in Cubical Agda, which means that certain terms do not compute definitionally, requiring manual handling of transport structures. Ultimately, we are also unable to finish the proof of the naturality condition for Σ-structures due to the inherent complexity of the goal types. However, our approach substantially simplifies the remaining proof goals, which makes us hopeful that the proof will be able to be completed in future work. We conclude that while Cubical Agda improves clarity in specific areas, we concede that the trade-off regarding the definitional behaviour of the Jrule makes the balance between benefits and downsides approximately equal.

(belongs to my master’s thesis</a>)

The Category of Iterative Sets in Cubical Agda

Fabian Lukas Grubmüller — Wed, 14 Jan 2026 00:00:00 +0000

Abstract</h2>
Iterative sets form a constructive Tarski-style universe V⁰ of h-sets. This universe is closed under common type-theoretic constructions and is itself an h-set. It arises naturally from the study of iterative multisets, where V⁰ is defined as a specific type-indexed W-type for which the indexing function is restricted to embeddings, effectively collapsing higher structure.
In previous work, Gratzer, Gylterud, Mörtberg, and Stenholm showed that V⁰ is a model of dependent type theory, in particular a Category with Families (CwF), that admits both Π- and Σ-structures. While their proofs were rather straightforward on paper, often reducing to reflexivity, their formalization in standard Agda (using agda-unimath) faced significant obstacles. For the formalization of Π- and Σ-structures, they faced problems due to complex path algebra involving multiple layers of transport and function extensionality, leading them to abandon the formalization of the Σ-structure.
In this thesis, we explore whether a formalization in Cubical Agda, an extension of Agda for cubical type theory, is easier to accomplish. We implement the general properties of iterative sets, as well as CwF and Σ-structures. For the latter we use three distinct strategies: a naive translation of the prior work, a more cubical approach replacing equalities containing transport with heterogeneous path types, and a strategy that eliminates transport in favor of ad-hoc functions that can be later instantiated by the identity function. We find that while the cubical metatheory simplifies reasoning about extensionality, it also introduces new challenges. One of the main issues is the lack of a definitional J-rule in Cubical Agda, which means that certain terms do not compute definitionally, requiring manual handling of transport structures. Ultimately, we are also unable to finish the proof of the naturality condition for Σ-structures due to the inherent complexity of the goal types. However, our approach substantially simplifies the remaining proof goals, which makes us hopeful that the proof will be able to be completed in future work. We conclude that while Cubical Agda improves clarity in specific areas, we concede that the trade-off regarding the definitional behaviour of the J-rule makes the balance between benefits and downsides approximately equal.
Cite</h2>
Hayagriva YAML</h3>
cubical-iterative-sets: type: Thesis title: The Category of Iterative Sets in Cubical Agda author: Grubmüller, Fabian Lukas date: 2026-01-14 organization: Stockholm University url: "https://grubmueller.dev/publications/2026-mt-iterative-sets/" note: Master thesis</code></pre>BibTeX</h3> @mastersthesis{cubical-iterative-sets, title = {The Category of Iterative Sets in Cubical Agda}, author = {Grubm\"{u}ller, Fabian Lukas}, year = {2026}, school = {Stockholm University}, type = {Master thesis}, note = {\url{https://grubmueller.dev/publications/2026-mt-iterative-sets/}}, }</code></pre>Notes</h2> Type: Master Thesis Supervisor: Anders Mörtberg Programme: Mathematics M.Sc. University: Stockholm University (joint with KTH Royal Institute of Technology)
A Predicative Approach to the Constructive Integration Theory of Locally Compact Metric Spaces Fabian Lukas Grubmüller — Thu, 13 Feb 2025 00:00:00 +0000 Abstract</h2> Based on the inherently impredicative approach of Bishop to constructive integration theory, we present a predicative version of the integration theory of locally compact metric spaces. For that, we first introduce locally compact metric spaces with a modulus of local compactness. This notion of local compactness is incompatible to Mandelkern’s but equivalent to both Bishop’s and Chan’s corresponding notions. Using our definition, we reconstruct the integration theory of continuous functions with compact support using set-indexed families of subsets, avoiding the impredicativity of the original constructive theory of Bishop and Cheng. We work within Bishop Set Theory, which provides an expressive framework for Bishop-style constructive mathematics and constitutes a minimal extension of Bishop’s original theory of sets. Cite</h2> Hayagriva YAML</h3> cp-integration-lcms: type: Article title: A Predicative Approach to the Constructive Integration Theory of Locally Compact Metric Spaces author: - Grubmüller, Fabian Lukas - Petrakis, Iosif date: 2025 serial-number: doi: 10.4115/jla.2025.17.FDS4 parent: type: Periodical title: Journal of Logic and Analysis volume: 17</code></pre>BibTeX</h3> @article{cp-integration-lcms, title = {A Predicative Approach to the Constructive Integration Theory of Locally Compact Metric Spaces}, author = {Grubmüller, Fabian Lukas and Petrakis, Iosif}, journal = {Journal of Logic and Analysis}, volume = {17}, year = {2025}, doi = {10.4115/jla.2025.17.FDS4}, url = {https://doi.org/10.4115/jla.2025.17.FDS4}, }</code></pre> Unification of Boolean Differential Rings Is Unitary Fabian Lukas Grubmüller — Tue, 10 Sep 2024 00:00:00 +0000 Abstract</h2> The theory of Boolean differential rings is a natural extension of the theory of Boolean rings, that additionaly provides an abstract notion of differential. Boolean rings are important and extensively studied concepts arising naturally in many parts of mathematics, especially logic, and computer science. One important result is that the theory of Boolean rings has the unitary unification type. We show that the unification of Boolean differential rings can be reduced to the unification of Boolean rings and that the theory of Boolean differential rings also has the unitary unification type, and we provide an algorithm that calculates a most general unifier. We also show that terms of Boolean differential rings have a flat normal form similar to the polynomial form of terms of Boolean rings and that terms of Boolean differential rings correspond to terms of Boolean rings in a way that respects both equivalences. Cite</h2> Hayagriva YAML</h3> bdr-unification: type: Thesis title: Unification of Boolean Differential Rings is Unitary author: Grubmüller, Fabian Lukas date: 2024-09-10 organization: Ludwig-Maximilians-Universität München serial-number: doi: 10.5282/ubm/epub.125897 note: Bachelor thesis</code></pre>BibTeX</h3> @mastersthesis{bdr-unification, title = {Unification of Boolean Differential Rings Is Unitary}, author = {Grubm\"{u}ller, Fabian Lukas}, year = {2024}, school = {Ludwig-Maximilians-Universit\"{a}t M\"{u}nchen}, type = {Bachelor thesis}, doi = {10.5282/ubm/epub.125897} }</code></pre>Notes</h2> Type: Bachelor Thesis Supervisor: Felix Weitkämper DPhil (Oxon) Programme: Computer Science B.Sc. University: Ludwig-Maximilians-Universität München Demystifying Proof Assistants: An Introduction to Interactive Theorem Proving Fabian Lukas Grubmüller — Tue, 23 Apr 2024 00:00:00 +0000 Interactive theorem proving (ITP) is a powerful technique for mathematicians to formally verify and develop proofs. I will give an elementary introduction to the topic of ITP, explore the benefits of ITP for rigour, clarity, as well as collaboration in mathematical research and give an overview of the different available proof assistants. In the second part, I will showcase how to develop some simple (undergraduate level) theory using Lean, a user-friendly and versatile proof assistant with an active community and a large library of already implemented theories. This talk is introductory, no prior knowledge about ITP is assumed. Towards a Constructive and Predicative Integration Theory of Locally Compact Metric Spaces Fabian Lukas Grubmüller — Mon, 06 Jun 2022 00:00:00 +0000 Abstract</h2> Bishop style constructive integration theory constitutes an important milestone in constructive mathematics as it demonstrates the actual feasibility of developing a rich theory of integration within the constructive framework. However, Bishop’s approach has the fundamental flaw that it allows impredicativity in the sense that it uses statements that contain quantification over the whole universe of sets. In this thesis, I work towards amending Bishop’s theory in order to remove this impredicativity. Furthermore, I try to increase clarity through the explicit use of moduli. First, I introduce the necessary fundamental notions of Bishop Set Theory as presented by Petrakis. Following Bishop’s book, I develop the theory of locally compact metric spaces. Lastly, I introduce a notion of integration on locally compact metric spaces and prove that the set of partial functions with compact support constitute an integration space in a sensible manner. Cite</h2> Hayagriva YAML</h3> cp-integration-lcms: type: Thesis title: Towards a Constructive and Predicative Integration Theory of Locally Compact Metric Spaces author: Grubmüller, Fabian Lukas date: 2022-06-06 organization: Ludwig-Maximilians-Universität München url: "https://grubmueller.dev/publications/2022-bt-cp-integration/" note: Bachelor thesis</code></pre>BibTeX</h3> @mastersthesis{cp-integration-lcms, type = {Bachelor thesis}, title = {Towards a Constructive and Predicative Integration Theory of Locally Compact Metric Spaces}, author = {Grubm\"{u}ller, Fabian Lukas}, year = {2022}, school = {LMU M\"{u}nchen}, note = {\url{https://grubmueller.dev/publications/2022-bt-cp-integration/}}, }</code></pre>Notes</h2> Type: Bachelor Thesis Supervisor: Dr. Iosif Petrakis Programme: Mathematics B.Sc. University: Ludwig-Maximilians-Universität München

flgrubm

The Category of Iterative Sets in Cubical Agda

The Category of Iterative Sets in Cubical Agda

The Category of Iterative Sets in Cubical Agda

Cite</h2>

Notes</h2>
Type:</strong> Master Thesis
Supervisor:</strong> Anders Mörtberg
Programme:</strong> Mathematics M.Sc.
University:</strong> Stockholm University (joint with KTH Royal Institute of Technology)</p>

A Predicative Approach to the Constructive Integration Theory of Locally Compact Metric Spaces

Cite</h2>

Unification of Boolean Differential Rings Is Unitary

Cite</h2>

Notes</h2>
Type:</strong> Bachelor Thesis
Supervisor:</strong> Felix Weitkämper DPhil (Oxon)
Programme:</strong> Computer Science B.Sc.
University:</strong> Ludwig-Maximilians-Universität München</p>

Demystifying Proof Assistants: An Introduction to Interactive Theorem Proving

Towards a Constructive and Predicative Integration Theory of Locally Compact Metric Spaces

Cite</h2>

Notes</h2>
Type:</strong> Bachelor Thesis
Supervisor:</strong> Dr. Iosif Petrakis
Programme:</strong> Mathematics B.Sc.
University:</strong> Ludwig-Maximilians-Universität München</p>

flgrubm

The Category of Iterative Sets in Cubical Agda

The Category of Iterative Sets in Cubical Agda

The Category of Iterative Sets in Cubical Agda

Cite</h2>

Notes</h2> Type:</strong> Master Thesis Supervisor:</strong> Anders Mörtberg Programme:</strong> Mathematics M.Sc. University:</strong> Stockholm University (joint with KTH Royal Institute of Technology)</p>

A Predicative Approach to the Constructive Integration Theory of Locally Compact Metric Spaces

Cite</h2>

Unification of Boolean Differential Rings Is Unitary

Cite</h2>

Notes</h2> Type:</strong> Bachelor Thesis Supervisor:</strong> Felix Weitkämper DPhil (Oxon) Programme:</strong> Computer Science B.Sc. University:</strong> Ludwig-Maximilians-Universität München</p>

Demystifying Proof Assistants: An Introduction to Interactive Theorem Proving

Towards a Constructive and Predicative Integration Theory of Locally Compact Metric Spaces

Cite</h2>

Notes</h2> Type:</strong> Bachelor Thesis Supervisor:</strong> Dr. Iosif Petrakis Programme:</strong> Mathematics B.Sc. University:</strong> Ludwig-Maximilians-Universität München</p>

Notes</h2>
Type:</strong> Master Thesis
Supervisor:</strong> Anders Mörtberg
Programme:</strong> Mathematics M.Sc.
University:</strong> Stockholm University (joint with KTH Royal Institute of Technology)</p>

Notes</h2>
Type:</strong> Bachelor Thesis
Supervisor:</strong> Felix Weitkämper DPhil (Oxon)
Programme:</strong> Computer Science B.Sc.
University:</strong> Ludwig-Maximilians-Universität München</p>

Notes</h2>
Type:</strong> Bachelor Thesis
Supervisor:</strong> Dr. Iosif Petrakis
Programme:</strong> Mathematics B.Sc.
University:</strong> Ludwig-Maximilians-Universität München</p>