Author name cluster

Anders Mörtberg

Possible papers associated with this exact author name in Arrow. This page groups case-insensitive exact name matches and is not a full identity disambiguation profile.

4 papers

2 author rows

FSCD Conference 2024 Conference Paper

Automating Boundary Filling in Cubical Agda

Maximilian Doré
Evan Cavallo
Anders Mörtberg

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 Agda is an extension of Agda with computational support for HITs and univalence. A difficulty when working in Cubical Agda 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. We provide a solver for the contortion problem using a reformulation of contortions in terms of poset maps, while we solve Kan problems using constraint satisfaction programming. We have implemented our algorithms in an experimental Haskell solver that can be used to automatically solve goals presented by Cubical Agda. We illustrate this with a case study establishing the Eckmann-Hilton theorem using our solver, as well as various benchmarks - providing the ground for further study of proof automation in cubical type theories.

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CSL Conference 2022 Conference Paper

Synthetic Integral Cohomology in Cubical Agda

Guillaume Brunerie
Axel Ljungström
Anders Mörtberg

This paper discusses the formalization of synthetic cohomology theory in a cubical extension of Agda which natively supports univalence and higher inductive types. This enables significant simplifications of many proofs from Homotopy Type Theory and Univalent Foundations as steps that used to require long calculations now hold simply by computation. To this end, we give a new definition of the group structure for cohomology with ℤ-coefficients, optimized for efficient computations. We also invent an optimized definition of the cup product which allows us to give the first complete formalization of the axioms needed to turn the integral cohomology groups into a graded commutative ring. Using this, we characterize the cohomology groups of the spheres, torus, Klein bottle and real/complex projective planes. As all proofs are constructive we can then use Cubical Agda to distinguish between spaces by computation.

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CSL Conference 2020 Conference Paper

Unifying Cubical Models of Univalent Type Theory

Evan Cavallo
Anders Mörtberg
Andrew W. Swan

We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure.

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FLAP Journal 2017 Journal Article

Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom.

Cyril Cohen
Thierry Coquand
Simon Huber
Anders Mörtberg

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.

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