Building on the Diamonds between Theories: Theory Presentation Combinators

To build a large library of mathematics, it seems more efficient to take advantage of the inherent structure of mathematical theories. Various theory presentation combinators have been proposed, and some have been implemented, in both legacy and current systems. Surprisingly, the ``standard library'' of most systems do not make pervasive use of these combinators. We present a set of combinators optimized for reuse, via the tiny theories approach. Our combinators draw their power from the inherent structure already present in the \emph{category of contexts} associated to a dependently typed language. The current work builds on ideas originating in CLEAR and Specware and their descendents (both direct and intellectual). Driven by some design criteria for user-centric library design, our library-building experience via the systematic use of combinators has fed back into the semantics of these combinators, and later into an updated syntax for them.