Exploring the Structure of an Algebra Text with Locales

A Modular First Formalisation of Combinatorial Design Theory

“…A PBD is a clear specialisation of a tBD which can be defined formally using the for command in a locale definition to instantiate one parameter and simplify syntax. There are several variations on PBDs in the literature depending on block size properties and the value of λ, which are easy to specify by combining locales and the use of the sublocale declaration, following the functor proof pattern [3].…”

“…The "record + locale" approach first considered in (3.1) is based on Noschinski's graph theory library and the HOL-Algebra library. This approach uses a record to define structural elements and definitions, and locales for supporting concise syntax by parameter annotation [3]. It was originally designed when definitions could not be declared within a locale and is still widely used.…”

“…The small AFP development on matroids, another combinatorial structure, uses this more locale-centric approach [12], but more interesting is Ballarin's take to formalising algebra [3]. He uses locales to define structures as well as operations and relationships on multiple instances of a locale, similar to the design isomorphism definition.…”

“…This approach drew inspiration from both Noschinski and Ballarin [13,3], and yielded a number of benefits, preventing unnecessary duplication when new designs were introduced. More importantly, it increased the flexibility and extensibility of the library.…”

“…While locales have been available in the current form since the early 2000s, they typically have been used sparingly in mathematical contexts, or alongside other tools such as type classes and records. This project presented the opportunity to explore a locale-centric approach to formalising mathematical structures, building on Ballarin's prior work in algebra [3], and using ideas from Noschinski's graph theory library [13].…”