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A Bi-Directional Extensible Interface Between Lean and Mathematica

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Abstract

We implement a user-extensible ad hoc connection between the Lean proof assistant and the computer algebra system Mathematica. By reflecting the syntax of each system in the other and providing a flexible interface for extending translation, our connection allows for the exchange of arbitrary information between the two systems. We show how to make use of the Lean metaprogramming framework to verify certain Mathematica computations, so that the rigor of the proof assistant is not compromised. We also use Mathematica as an untrusted oracle to guide proof search in the proof assistant and interact with a Mathematica notebook from within a Lean session. In the other direction, we import and process Lean declarations from within Mathematica. The proof assistant library serves as a database of mathematical knowledge that the CAS can display and explore.

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Notes

  1. https://robertylewis.com/leanmm/.

  2. https://observablehq.com/@bryangingechen/hello-lean-prover.

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Acknowledgements

We acknowledge Jeremy Avigad, Jasmin Blanchette, Ian Ford, Johannes Hölzl, José Martín-García, Leonardo de Moura, James Mulnix, Michael Trott, and the Lean community for help, suggestions, and support.

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Authors and Affiliations

  1. Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

    Robert Y. Lewis

  2. Brown University, Providence, USA

    Robert Y. Lewis

  3. Australian National University, Canberra, Australia

    Minchao Wu

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  1. Robert Y. Lewis
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Correspondence to Robert Y. Lewis.

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Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This paper expands on a workshop paper by the first author [28], which describes an early version of one direction of the interface.

The first author receives support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 713999, Matryoshka) and from the Dutch Research Council (NWO) under the Vidi program (Project No. 016.Vidi.189.037, Lean Forward)

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Lewis, R.Y., Wu, M. A Bi-Directional Extensible Interface Between Lean and Mathematica. J Autom Reasoning 66, 215–238 (2022). https://doi.org/10.1007/s10817-021-09611-1

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