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Untangling mechanized proofs

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Clément Pit-ClaudelAuthors Info & Claims

SLE 2020: Proceedings of the 13th ACM SIGPLAN International Conference on Software Language Engineering

Pages 155 - 174

https://doi.org/10.1145/3426425.3426940

Published: 15 November 2020 Publication History

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Abstract

Proof assistants like Coq, Lean, or HOL4 rely heavily on stateful meta-programs called scripts to assemble proofs. Unlike pen-and-paper proofs, proof scripts only describe the steps to take (induct on x, apply a theorem, …), not the states that these steps lead to; as a result, plain proof scripts are essentially incomprehensible without the assistance of an interactive user interface able to run the script and show the corresponding proof states.

Until now, the standard process to communicate a proof without forcing readers to execute its script was to manually copy-paste intermediate proof states into the script, as source code comments — a tedious and error-prone exercise. Additional prose (such as for a book or tutorial) was likewise embedded in comments, preserving executability at the cost of a mediocre text-editing experience.

This paper describes a new approach to the development and dissemination of literate proof scripts, with a focus on the Coq proof assistant. Specifically, we describe two contributions: a compiler that interleaves Coq’s output with the original proof script to produce interactive webpages that are complete, self-contained presentations of Coq proofs; and a new literate programming toolkit that allows authors to switch seamlessly between prose- and code-oriented views of the same sources, by translating back and forth between reStructuredText documents and literate Coq source files. In combination, these tools offer a new way to write, communicate, and preserve proofs, combining the flexibility of procedural proof scripts and the intelligibility of declarative proofs.

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Auxiliary Presentation Video (sle20main-p66-p-video.mp4)

A video introduction to Alectryon, from SLE 2020. Unlike pen-and-paper proofs, Coq proof scripts only describe the steps to take, not the states that these steps lead to; as a result, proof scripts are essentially incomprehensible without an interactive user interface to show proof goals. Until now, the standard process to share a self-contained Coq proof was to copy-paste goals into the proof script as comments. Additional prose was likewise embedded in comments. Alectryon is a way to develop and disseminate literate proof scripts. It combines a compiler that interleaves Coq’s output with the original proof script to produce interactive webpages, and a literate programming toolkit that allows authors to translate back and forth between reST and Coq. These tools offer a new way to write, communicate, and preserve proofs, combining the flexibility of procedural proof scripts and the intelligibility of declarative proofs.

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References

[1]

David Aspinall. 2000. Proof General: A Generic Tool for Proof Development. In Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2000, Susanne Graf and Michael Schwartzbach (Eds.). Lecture Notes in Computer Science, Vol. 1785. Springer Berlin Heidelberg, 38-43. https://doi.org/10.1007/3-540-46419-0_3

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