Automating Boundary Filling in Cubical Agda

Authors Maximilian Doré , Evan Cavallo , Anders Mörtberg

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Maximilian Doré
  • Department of Computer Science, University of Oxford, United Kingdom
Evan Cavallo
  • Department of Computer Science and Engineering, University of Gothenburg, Sweden
Anders Mörtberg
  • Department of Mathematics, Stockholm University, Sweden


We are grateful to Axel Ljungström for discussions about the solver and to him and Tom Jack for cubical versions of Eckmann-Hilton and syllepsis for us to test it with.

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Maximilian Doré, Evan Cavallo, and Anders Mörtberg. Automating Boundary Filling in Cubical Agda. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Theory of computation → Type theory
  • Computing methodologies → Theorem proving algorithms
  • Theory of computation → Constraint and logic programming
  • Theory of computation → Automated reasoning
  • Cubical Agda
  • Automated Reasoning
  • Constraint Satisfaction Programming


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