Exploring the Structure of an Algebra Text with Locales

Abstract

Locales, the module system of the theorem prover Isabelle, were designed so that developments in abstract algebra could be represented faithfully and concisely. Whether these goals were met is assessed through a case study. Parts of an algebra textbook, Jacobson’s Basic Algebra, that are challenging structurally were formalised. Key parts of the formalisation are presented in greater detail. An analysis of the work from both qualitative and quantitative perspectives substantiates that the design goals were met. In particular, the size ratio of formal to “pen and paper” text does not increase when going further into the book. The analysis also yields guidance on locales including patterns of use, which are identified and described.

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Notes

  1. 1.

    Counted were the entries containing locale declarations—more precisely, where at least one theory file contained a line matching the regular expression .

  2. 2.

    Readers wishing to reproduce the examples in Isabelle should use bold, not regular, “1” (input token ).

  3. 3.

    In locale expressions outside locale declarations, the for clause retains its usual semantics as a binder.

  4. 4.

    Jacobson requires S to be non-vacuous, but this was not required in the formalisation.

  5. 5.

    Isabelle requires escaping the single quote character in syntax declarations; “ ” yields “ ”.

  6. 6.

    \(K_R(G)\) denotes the group of right translations \(k_R : G \rightarrow G\) for \(k \in K\). Jacobson leaves right translations as an exercise, which had to be formalised as well. For the corresponding work on left translations, see Sect. 3.5 above.

  7. 7.

    In reproducing Gunter’s definition I have changed \((G, {\textit{prod}} \ )\) to \(\mathcal {G}\). Her original definition \( inv \, (G, {\textit{prod}} \ ) \, x \equiv \varepsilon y. \, y \in G \wedge ({\textit{prod}} \ \, y \, x = id \, (G, {\textit{prod}} \ ))\) does not involve the projections \({ fst } \ \) and \({\textit{snd}} \ \), but many others of her definitions do. Further, since defined operations are well-defined on the carrier set definite choice is sufficient for making definitions. There is no need for the axiom of choice.

  8. 8.

    Proofs are preserved in Isabelle’s source code repository: https://isabelle.in.tum.de/repos/isabelle/file/7e6cdcd113a2/src/HOL/GroupTheory/.

  9. 9.

    Wiedijk [26] supposes that the mathematical text is available in computer-readable form and suggests the factor be computed by comparing sizes of compressed files of the formalisation and its mathematical source. I have chosen to simply compare numbers of lines as is common practice when measuring code size in computer science.

  10. 10.

    This paragraph reproduces my response to a post in the Isabelle Users mailing list: https://lists.cam.ac.uk/pipermail/cl-isabelle-users/2019-September/msg00074.html.

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Ballarin, C. Exploring the Structure of an Algebra Text with Locales. J Autom Reasoning 64, 1093–1121 (2020). https://doi.org/10.1007/s10817-019-09537-9

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Keywords

  • Abstract algebra
  • Case study
  • Isabelle
  • Locales
  • Module system
  • Theorem prover