Homotopy Type Theory in Lean

  • Floris van Doorn
  • Jakob von Raumer
  • Ulrik Buchholtz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10499)


We discuss the homotopy type theory library in the Lean proof assistant. The library is especially geared toward synthetic homotopy theory. Of particular interest is the use of just a few primitive notions of higher inductive types, namely quotients and truncations, and the use of cubical methods.


Homotopy type theory Formalized mathematics Lean Proof assistants 



We wish to thank the members of the HoTT group at Carnegie Mellon University for many fruitful discussions and Lean hacking sessions, and in particular Steve Awodey and Jeremy Avigad who have been very supportive of our work. Additionally, we deeply appreciate all the times Leonardo de Moura fixed an issue in the Lean kernel to accommodate our library. Lastly, we want to thank all contributors to the HoTT library and the Spectral repository, most notably Egbert Rijke and Mike Shulman.

The first and second authors gratefully acknowledge the support of the Air Force Office of Scientific Research through MURI grant FA9550-15-1-0053. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the AFOSR.


  1. 1.
    Ahrens, B., Kapulkin, K., Shulman, M.: Univalent categories and the Rezk completion. Mathe. Struct. Comput. Sci. 25(5), 1010–1039 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Angiuli, C., Harper, R., Wilson, T.: Computational higher-dimensional type theory. In: Proceedings of the 44th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2017. ACM (2017)Google Scholar
  3. 3.
    Awodey, S., Warren, M.A.: Homotopy theoretic models of identity types. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 146, no. 1, pp. 45–55 (2009)Google Scholar
  4. 4.
    Bauer, A., Gross, J., LeFanu Lumsdaine, P., Shulman, M., Sozeau, M., Spitters, B.: The HoTT library: a formalization of homotopy type theory in Coq. ArXiv e-prints, October 2016Google Scholar
  5. 5.
    Brunerie, G., Hou (Favonia), K.B., Cavallo, E., Finster, E., Cockx, J., Sattler, C., Jeris, C., Shulman, M., et al.: Homotopy Type Theory in Agda (2017). Code library.
  6. 6.
    Buchholtz, U., Rijke, E.: The cayley-dickson construction in homotopy type theory. ArXiv e-prints, October 2016Google Scholar
  7. 7.
    Cohen, C., Coquand, T., Huber, S., Mörtberg, A.: Cubical type theory. Code library.
  8. 8.
    Cohen, C., Coquand, T., Huber, S., Mörtberg, A.: Cubical type theory: a constructive interpretation of the univalence axiom. In: 21st International Conference on Types for Proofs and Programs (TYPES 2015). LIPIcs. Leibniz International Proceedings in Informatics, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2016, to appear)Google Scholar
  9. 9.
    van Doorn, F.: Constructing the propositional truncation using non-recursive hits. In: Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs, pp. 122–129. ACM (2016)Google Scholar
  10. 10.
    Dybjer, P.: Inductive families. Formal Aspects Comput. 6(4), 440–465 (1994)CrossRefzbMATHGoogle Scholar
  11. 11.
    Goguen, H., McBride, C., McKinna, J.: Eliminating dependent pattern matching. In: Futatsugi, K., Jouannaud, J.-P., Meseguer, J. (eds.) Algebra, Meaning, and Computation. LNCS, vol. 4060, pp. 521–540. Springer, Heidelberg (2006). doi: 10.1007/11780274_27 CrossRefGoogle Scholar
  12. 12.
    Hedberg, M.: A coherence theorem for Martin-Löf’s type theory. J. Funct. Program. 8(4), 413–436 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hofmann, M., Streicher, T.: The groupoid interpretation of type theory. In: Twentyfive Years of Constructive Type Theory (Venice, 1995). Oxford Logic Guides, vol. 36, pp. 83–111. Oxford University Press, New York (1998)Google Scholar
  14. 14.
    Kapulkin, C., Lumsdaine, P.L.: The simplicial model of univalent foundations (after voevodsky) (2012, preprint)Google Scholar
  15. 15.
    Licata, D.: Running circles around (in) your proof assistant; or, quotients that compute. blog post, April 2011.
  16. 16.
    Licata, D., Brunerie, G.: A cubical approach to synthetic homotopy theory. In: Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), LICS 2015, pp. 92–103. IEEE Computer Society, Washington, DC (2015)Google Scholar
  17. 17.
    de Moura, L., Ebner, G., Roesch, J., Ullrich, S.: The Lean theorem prover. Slides, January 2017.
  18. 18.
    de Moura, L., Kong, S., Avigad, J., van Doorn, F., van Raumer, J.: The lean theorem prover (system description). In: Felty, A.P., Middeldorp, A. (eds.) CADE 2015. LNCS, vol. 9195, pp. 378–388. Springer, Cham (2015). doi: 10.1007/978-3-319-21401-6_26 CrossRefGoogle Scholar
  19. 19.
    Rijke, E.: The join construction. ArXiv e-prints, January 2017Google Scholar
  20. 20.
    Spitters, B., van der Weegen, E.: Developing the algebraic hierarchy with type classes in Coq. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 490–493. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14052-5_35 CrossRefGoogle Scholar
  21. 21.
    The Univalent Foundations Program: Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study (2013).
  22. 22.
    Voevodsky, V.: A very short note on the homotopy \(\lambda \)-calculus (2006).
  23. 23.
    Voevodsky, V., Mörtberg, A., Ahrens, B., Lelay, C., Pannila, T., Matthes, R.: UniMath: Univalent Mathematics (2017). Code library.

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Floris van Doorn
    • 1
  • Jakob von Raumer
    • 2
  • Ulrik Buchholtz
    • 3
  1. 1.Carnegie Mellon UniversityPittsburghUSA
  2. 2.University of NottinghamNottinghamUK
  3. 3.TU DarmstadtDarmstadtGermany

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