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On the Conciliation of Traditional and Computer-Assisted Proofs

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Philosophy of Computing

Part of the book series: Philosophical Studies Series ((PSSP,volume 143))

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Abstract

A proof of a mathematical proposition or a program specification obtained by a formal verification process, using an interactive theorem prover, can be questioned as a true demonstration or as having the same purposes of a traditional pencil-and-paper proof. However, in our opinion the verification process of a software component exhibits the same construction phases as a purely mathematical one. A correspondence between both kinds of proofs enables us to give a proposal of what we call transitional proofs, a concept that outlines a conciliation between traditional paper-and-pencil and computer-assisted proofs, which can be useful in philosophical problems surrounding formalized mathematics and program verification with proof-assistants.

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Notes

  1. 1.

    An example is the symbol for “such that” outside a set definition, which may be a colon : , but also a kind of inverted epsilon \(\backepsilon \), and their usage is certainly non standard.

  2. 2.

    We consider here both the proof of a mathematical proposition and that of an algorithm or program fulfilling its specification.

  3. 3.

    Let us emphasize that we are giving a summary of Turner’s ideas, and as pointed by two anonymous referees this claim is most likely to be false, for is common for automated proofs to reveal faulty specifications that must be changed.

  4. 4.

    Desperately seeking software perfection, Xavier Leroy, Colloquium d’informatique, UPMC, Paris France, October 2015.

  5. 5.

    https://coq.inria.fr/.

  6. 6.

    “What You See Is What You Get”.

  7. 7.

    The direct way of implementing a forward reasoning step is by the backward reading of the cut rule, see page 93.

  8. 8.

    Either of a single self-contained proposition or of a more ambitious theorem requiring the development of several auxiliary results.

  9. 9.

    For instance by first proving r ⋅ r < r as an auxiliary lemma or at the beginning of the proof as in Theorem 5.

  10. 10.

    A typical example of this scenario arises in Group Theory where the simplicity of the alternating group A 5 can be proved either by a direct argument or as an application of the Sylow Theorems.

  11. 11.

    A finite collection for our purposes.

  12. 12.

    Please note that this is a general train of thought and does not refer to the specific deductive system in Sect. 3.5.

  13. 13.

    Of course there are good heuristics for specific deductive systems but this departs from the approach of this paper.

  14. 14.

    The full Coq development of our article González-Huesca et al. (2019) is available in https://bitbucket.org/luglzhuesca/mlogic-formalverif/src/master/S4.

  15. 15.

    Backus-Naur Form.

  16. 16.

    In González-Huesca et al. (2019) we proved the full equivalence between axiomatic (with the multiplicative style) and natural deduction systems for constructive S4. For a deep analysis of multiplicative and additive systems, the reader may consult (Plato 2014).

  17. 17.

    The guardeness condition of Coq.

  18. 18.

    Although sometimes it is easier to get a proof by modifying the proposition in a way that the original idea remains as a corollary of the new statement.

  19. 19.

    This means that the system that solved the problem, namely EQP, is a fully automated system. However, the full non-interactive proof was developed during 5 weeks but the development project that lead to the implementation of the EQP prover took 10 years. Thus, an interesting question would be to discuss, knowing this long effort from a research team, to what extent the final proof of the Robbins Conjecture can be considered as fully automated.

  20. 20.

    http://blogs.scientificamerican.com/guest-blog/2013/10/01/voevodskys-mathematical-revolution/.

  21. 21.

    See the summary of changes of Coq’s versions 8.X after the major revision of the development https://coq.inria.fr/distrib/current/refman/changes.html#recent-changes.

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Acknowledgements

This research has been funded by UNAM DGAPA PAPIIT grant IN119920. The authors would like to thank two anonymous referees as well as the volume editors for many helpful comments to improve the contents of this paper.

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  1. Departamento de Matemáticas, Facultad de Ciencias, UNAM, Mexico City, Mexico

    Favio E. Miranda-Perea & Lourdes del Carmen González Huesca

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  1. Favio E. Miranda-Perea
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Correspondence to Favio E. Miranda-Perea .

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Editors and Affiliations

  1. The Department of Philosophy and Religious Studies, Utrecht University, Utrecht, The Netherlands

    Björn Lundgren

  2. Institute of Philosophy, Czech Academy of Sciences, Prague, Czech Republic

    Nancy Abigail Nuñez Hernández

Appendix: CAP Counterparts for No Natural Between 0 and 1

Appendix: CAP Counterparts for No Natural Between 0 and 1

This appendix shows the Coq proofs for the different versions of the non-existence of a natural number between 0 and 1, discussed in Sect. 3.3.

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E. Miranda-Perea, F., Huesca, L.d.C.G. (2022). On the Conciliation of Traditional and Computer-Assisted Proofs. In: Lundgren, B., Nuñez Hernández, N.A. (eds) Philosophy of Computing. Philosophical Studies Series, vol 143. Springer, Cham. https://doi.org/10.1007/978-3-030-75267-5_3

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