arXiv subjects

Anders Mörtberg

Publications and source records attributed to Anders Mörtberg.

13 recordsLinked to original sources

arXiv2026

Automating Boundary Filling in Cubical Type Theories

When working in a proof assistant, automation is key to discharging routine proof goals such as equations between algebraic expressions. Homotopy type theory allows the user to reason about higher structures, such as topological spaces, using higher inductive types (HITs) and univalence. Cubical type theory provides computational support for HITs and univalence. A difficulty when working in cubical type theory is dealing with the complex combinatorics of higher structures, an infinite-dimensional generalisation of equational reasoning. To solve these higher-dimensional equations consists in constructing cubes with specified boundaries. We develop a simplified cubical language in which we isolate and study two automation problems: contortion solving, where we attempt to "contort" a cube to fit a given boundary, and the more general Kan solving, where we search for solutions that involve pasting multiple cubes together. Both problems are difficult in the general case-Kan solving is even undecidable-so we focus on heuristics that perform well on practical examples. Our language encompasses different variations of cubical type theory which differ in their "contortion theory", i.e., the class of contortions they support. We provide a solver for the contortion problem for the most complex contortion theories currently being researched, the Dedekind and De Morgan contortions, by utilizing a reformulation of contortions in terms of poset maps. We solve Kan problems using constraint satisfaction programming, which is applicable independently of the underlying contortion theory. We have implemented our algorithms in an experimental Haskell solver that can be used to automatically solve many goals a user of cubical type theory might face. We illustrate this with a case study establishing the Eckmann-Hilton theorem using our solver, as well as various benchmarks.

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arXiv2024

Formalising and Computing the Fourth Homotopy Group of the $3$-Sphere in Cubical Agda

Brunerie's 2016 PhD thesis contains the first synthetic proof in Homotopy Type Theory (HoTT) of the classical result that the fourth homotopy group of the 3-sphere is $\mathbb{Z}/2\mathbb{Z}$. The proof is one of the most impressive pieces of synthetic homotopy theory to date and uses a lot of advanced classical algebraic topology rephrased synthetically. Furthermore, the proof is fully constructive and the main result can be reduced to the question of whether a particular "Brunerie number" $β$ can be normalised to $\pm 2$. The question of whether Brunerie's proof could be formalised in a proof assistant, either by computing this number or by formalising the pen-and-paper proof, has since remained open. In this paper, we present a complete formalisation in Cubical Agda. We do this by modifying Brunerie's proof so that a key technical result, whose proof Brunerie only sketched in his thesis, can be avoided. We also present a formalisation of a new and much simpler proof that $β$ is $\pm 2$. This formalisation provides us with a sequence of simpler Brunerie numbers, one of which normalises very quickly to $-2$ in Cubical Agda, resulting in a fully formalised computer-assisted proof that $π_4(\mathbb{S}^3) \cong \mathbb{Z}/2\mathbb{Z}$.

math.AT↗

arXiv2024

Computational Synthetic Cohomology Theory in Homotopy Type Theory

This paper discusses the development of synthetic cohomology in Homotopy Type Theory (HoTT), as well as its computer formalisation. The objectives of this paper are (1) to generalise previous work on integral cohomology in HoTT by the current authors and Brunerie (2022) to cohomology with arbitrary coefficients and (2) to provide the mathematical details of, as well as extend, results underpinning the computer formalisation of cohomology rings by the current authors and Lamiaux (2023). With respect to objective (1), we provide new direct definitions of the cohomology group operations and of the cup product, which, just as in (Brunerie et al., 2022), enable significant simplifications of many earlier proofs in synthetic cohomology theory. In particular, the new definition of the cup product allows us to give the first complete formalisation of the axioms needed to turn the cohomology groups into a graded commutative ring. We also establish that this cohomology theory satisfies the HoTT formulation of the Eilenberg-Steenrod axioms for cohomology and study the classical Mayer-Vietoris and Gysin sequences. With respect to objective (2), we characterise the cohomology groups and rings of various spaces, including the spheres, torus, Klein bottle, real/complex projective planes, and infinite real projective space. All results have been formalised in Cubical Agda and we obtain multiple new numbers, similar to the famous `Brunerie number', which can be used as benchmarks for computational implementations of HoTT. Some of these numbers are infeasible to compute in Cubical Agda and hence provide new computational challenges and open problems which are much easier to define than the original Brunerie number.

math.AT↗

arXiv2024

The Category of Iterative Sets in Homotopy Type Theory and Univalent Foundations

When working in Homotopy Type Theory and Univalent Foundations, the traditional role of the category of sets, Set, is replaced by the category hSet of homotopy sets (h-sets); types with h-propositional identity types. Many of the properties of Set hold for hSet ((co)completeness, exactness, local cartesian closure, etc.). Notably, however, the univalence axiom implies that Ob(hSet) is not itself an h-set, but an h-groupoid. This is expected in univalent foundations, but it is sometimes useful to also have a stricter universe of sets, for example when constructing internal models of type theory. In this work, we equip the type of iterative sets V0, due to Gylterud (2018) as a refinement of the pioneering work of Aczel (1978) on universes of sets in type theory, with the structure of a Tarski universe and show that it satisfies many of the good properties of h-sets. In particular, we organize V0 into a (non-univalent strict) category and prove that it is locally cartesian closed. This enables us to organize it into a category with families with the structure necessary to model extensional type theory internally in HoTT/UF. We do this in a rather minimal univalent type theory with W-types, in particular we do not rely on any HITs, or other complex extensions of type theory. Furthermore, the construction of V0 and the model is fully constructive and predicative, while still being very convenient to work with as the decoding from V0 into h-sets commutes definitionally for all type constructors. Almost all of the paper has been formalized in Agda using the agda-unimath library of univalent mathematics.

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arXiv2023

A Univalent Formalization of Constructive Affine Schemes

We present a formalization of constructive affine schemes in the Cubical Agda proof assistant. This development is not only fully constructive and predicative, it also makes crucial use of univalence. By now schemes have been formalized in various proof assistants. However, most existing formalizations follow the inherently non-constructive approach of Hartshorne's classic "Algebraic Geometry" textbook, for which the construction of the so-called structure sheaf is rather straightforwardly formalizable and works the same with or without univalence. We follow an alternative approach that uses a point-free description of the constructive counterpart of the Zariski spectrum called the Zariski lattice and proceeds by defining the structure sheaf on formal basic opens and then lift it to the whole lattice. This general strategy is used in a plethora of textbooks, but formalizing it has proved tricky. The main result of this paper is that with the help of the univalence principle we can make this "lift from basis" strategy formal and obtain a fully formalized account of constructive affine schemes.

math.LO↗

arXiv2022

Computing Cohomology Rings in Cubical Agda

In Homotopy Type Theory, cohomology theories are studied synthetically using higher inductive types and univalence. This paper extends previous developments by providing the first fully mechanized definition of cohomology rings. These rings may be defined as direct sums of cohomology groups together with a multiplication induced by the cup product, and can in many cases be characterized as quotients of multivariate polynomial rings. To this end, we introduce appropriate definitions of direct sums and graded rings, which we then use to define both cohomology rings and multivariate polynomial rings. Using this, we compute the cohomology rings of some classical spaces, such as the spheres and the Klein bottle. The formalization is constructive so that it can be used to do concrete computations, and it relies on the Cubical Agda system which natively supports higher inductive types and computational univalence.

math.AT↗

arXiv2021

Implementing a Category-Theoretic Framework for Typed Abstract Syntax

In previous work ("From signatures to monads in UniMath"), we described a category-theoretic construction of abstract syntax from a signature, mechanized in the UniMath library based on the Coq proof assistant. In the present work, we describe what was necessary to generalize that work to account for simply-typed languages. First, some definitions had to be generalized to account for the natural appearance of non-endofunctors in the simply-typed case. As it turns out, in many cases our mechanized results carried over to the generalized definitions without any code change. Second, an existing mechanized library on $ω$-cocontinuous functors had to be extended by constructions and theorems necessary for constructing multi-sorted syntax. Third, the theoretical framework for the semantical signatures had to be generalized from a monoidal to a bicategorical setting, again to account for non-endofunctors arising in the typed case. This uses actions of endofunctors on functors with given source, and the corresponding notion of strong functors between actions, all formalized in UniMath using a recently developed library of bicategory theory. We explain what needed to be done to plug all of these ingredients together, modularly. The main result of our work is a general construction that, when fed with a signature for a simply-typed language, returns an implementation of that language together with suitable boilerplate code, in particular, a certified monadic substitution operation.

cs.PL↗

arXiv2020

Internalizing Representation Independence with Univalence

In their usual form, representation independence metatheorems provide an external guarantee that two implementations of an abstract interface are interchangeable when they are related by an operation-preserving correspondence. If our programming language is dependently-typed, however, we would like to appeal to such invariance results within the language itself, in order to obtain correctness theorems for complex implementations by transferring them from simpler, related implementations. Recent work in proof assistants has shown that Voevodsky's univalence principle allows transferring theorems between isomorphic types, but many instances of representation independence in programming involve non-isomorphic representations. In this paper, we develop techniques for establishing internal relational representation independence results in dependent type theory, by using higher inductive types to simultaneously quotient two related implementation types by a heterogeneous correspondence between them. The correspondence becomes an isomorphism between the quotiented types, thereby allowing us to obtain an equality of implementations by univalence. We illustrate our techniques by considering applications to matrices, queues, and finite multisets. Our results are all formalized in Cubical Agda, a recent extension of Agda which supports univalence and higher inductive types in a computationally well-behaved way.

cs.PL↗

arXiv2018

On Higher Inductive Types in Cubical Type Theory

Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly provable in the theory. This paper describes a constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some higher inductive types. It also extends cubical type theory by a syntax for the higher inductive types of spheres, torus, suspensions,truncations, and pushouts. All of these types are justified by the semantics and have judgmental computation rules for all constructors, including the higher dimensional ones, and the universes are closed under these type formers.

cs.LO↗

arXiv2016

From signatures to monads in UniMath

The term UniMath refers both to a formal system for mathematics, as well as a computer-checked library of mathematics formalized in that system. The UniMath system is a core dependent type theory, augmented by the univalence axiom. The system is kept as small as possible in order to ease verification of it - in particular, general inductive types are not part of the system. In this work, we partially remedy the lack of inductive types by constructing some datatypes and their associated induction principles from other type constructors. This involves a formalization of a category-theoretic result on the construction of initial algebras, as well as a mechanism to conveniently use the datatypes obtained. We also connect this construction to a previous formalization of substitution for languages with variable binding. Altogether, we construct a framework that allows us to concisely specify, via a simple notion of binding signature, a language with variable binding. From such a specification we obtain the datatype of terms of that language, equipped with a certified monadic substitution operation and a suitable recursion scheme. Using this we formalize the untyped lambda calculus and the raw syntax of Martin-Löf type theory.

cs.LO↗

arXiv2016

Cubical Type Theory: a constructive interpretation of the univalence axiom

This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.

cs.LO↗

arXiv2016

Formalized linear algebra over Elementary Divisor Rings in Coq

This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support essential operations of linear algebra, the classification theorem of finitely presented modules over such rings and the uniqueness of the Smith normal form up to multiplication by units. We present formally verified algorithms computing this normal form on a variety of coefficient structures including Euclidean domains and constructive principal ideal domains. We also study different ways to extend Bézout domains in order to be able to compute the Smith normal form of matrices. The extensions we consider are: adequacy (i.e. the existence of a gdco operation), Krull dimension $\leq 1$ and well-founded strict divisibility.

cs.LO↗

arXiv2012

Computing Persistent Homology within Coq/SSReflect

Persistent homology is one of the most active branches of Computational Algebraic Topology with applications in several contexts such as optical character recognition or analysis of point cloud data. In this paper, we report on the formal development of certified programs to compute persistent Betti numbers, an instrumental tool of persistent homology, using the Coq proof assistant together with the SSReflect extension. To this aim it has been necessary to formalize the underlying mathematical theory of these algorithms. This is another example showing that interactive theorem provers have reached a point where they are mature enough to tackle the formalization of nontrivial mathematical theories.

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