Abstract
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.
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Notes
- 1.
Available as part of: https://github.com/leanprover/lean2.
- 2.
- 3.
- 4.
- 5.
It has 7 700 lines of code and 1 400 lines of comments. It is available at https://github.com/cmu-phil/Spectral.
- 6.
Although we use syntax inspired by the Lean syntax for inductive types, this is not valid syntax in Lean.
- 7.
For the n-truncation we treat the fact that the new type is n-truncated as a “path-constructor.” In [21, Sect. 7.3] it is explained that the fact that a type is n-truncated can be reduced to (recursive) path constructors.
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Acknowledgments
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.
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van Doorn, F., von Raumer, J., Buchholtz, U. (2017). Homotopy Type Theory in Lean. In: Ayala-Rincón, M., Muñoz, C.A. (eds) Interactive Theorem Proving. ITP 2017. Lecture Notes in Computer Science(), vol 10499. Springer, Cham. https://doi.org/10.1007/978-3-319-66107-0_30
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