Building a large library of mathematical knowledge is a complex and labour-intensive task. By examining current libraries of mathematics, we see that the human effort put in building them is not entirely directed towards tasks that need human creativity. Instead, a non-trivial amount of work is spent on providing definitions that could have been mechanically derived. In this work, we propose a generative approach to library building, so definitions that can be automatically derived are computed by meta-programs. We focus our attention on libraries of algebraic structures, like monoids, groups, and rings. These structures are highly inter-related and their commonalities have been well-studied in universal algebra. We use theory presentation combinators to build a library of algebraic structures. Definitions from universal algebra and programming languages meta-theory are used to derive library definitions of constructions, like homomorphisms and term languages, from algebraic theory presentations. The result is an interpreter that, given 227 theory expressions, builds a library of over 5000 definitions. This library is, then, exported to Agda and Lean. / Dissertation / Doctor of Philosophy (PhD)

This thesis aims at building a numerical linear algebra library with appropriate software engineering abstractions. Three areas of knowledge, namely, Numerical Linear Algebra (NLA), Software Engineering and Compiler Optimisation Techniques, are involved. Numerical simulation is widely used in a large number of distinct disciplines to help scientists understand and discover the world. The solutions to frequently occurring numerical problems have been implemented in subroutines, which were then grouped together to form libraries for ease of use. The design, implementation and maintenance of a NLA library require a great deal of work so that the other two topics, namely, software engineering and compiler optimisation techniques have emerged. Generally speaking, these both try to divide the system into smaller and controllable concerns, and allow the programmer to deal with fewer concerns at one time. Band matrix operation, as a new level of abstraction, is proposed for simplifying library implementation and enhancing extensibility for future functionality upgrades. Iteration Space Partitioning (ISP) is applied, in order to make the performance of this generalised implementation for band matrices comparable to that of the specialised implementations for dense and triangular matrices. The optimisation of ISP can be either programmed using the pointcut-advice model of Aspect-Oriented Programming, or integrated as part of a compiler. This naturally leads to a comparison of these two different techniques for resolving one fundamental problem. The thesis shows that software engineering properties of a library, such as modularity and extensibility, can be improved by the use of the appropriate level of abstraction, while performance is either not sacrificed at all, or at least the loss of performance is limited. In other words, the perceived trade-off between the use of high-level abstraction and fast execution is made less significant than previously assumed.