Abstract
Instead of using program extraction mechanisms in various theorem provers, I suggest that users opt to create a database of formal proofs whose computational content is made explicit; this would be an alternative approach which, as libraries of formal mathematical proofs are constantly growing, would rely on future advances in automation and machine learning tools, so that as blocks of (sub)proofs get generated automatically, the preserved computational content would get recycled, recombined and would eventually manifest itself in different contexts. To this end, I do not suggest restricting to only constructive proofs, but I suggest that proof mined, possibly also non-constructive proofs with some explicit computational content should be preferable, if possible. To illustrate what kind of computational content in mathematical proofs may be of interest I give several very elementary examples (to be regarded as building blocks of proofs) and some samples of formalisations in Isabelle/HOL. Given the state of the art in automation and machine learning tools currently available for proof assistants, my suggestion is rather speculative, yet starting to build a database of formal proofs with explicit computational content would be a potentially useful first step.
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Notes
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L. S. van Benthem Jutting, as part of his PhD thesis in the late 1970s translated Edmund Landau’s Foundations of Analysis into AUTOMATH and checked its correctness [4].
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A recent work in this direction is the formalisation of several results from computable analysis in Coq by Steinberg, Théry and Thies [28].
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in this totally trivial example the set of zeros for both f and g is of course the one-element set \(\{ 0\}\).
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Ackowledgements
The author was supported by the ERC Advanced Grant ALEXANDRIA (Project 742178) led by Professor Lawrence C. Paulson FRS. I thank Wenda Li, Yiannos Stathopoulos and Lawrence Paulson for their very useful comments on a previous draft of this paper and Tobias Nipkow for informing me of reference [5].
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Koutsoukou-Argyraki, A. (2021). On Preserving the Computational Content of Mathematical Proofs: Toy Examples for a Formalising Strategy. In: De Mol, L., Weiermann, A., Manea, F., Fernández-Duque, D. (eds) Connecting with Computability. CiE 2021. Lecture Notes in Computer Science(), vol 12813. Springer, Cham. https://doi.org/10.1007/978-3-030-80049-9_26
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