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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.
We motivate and give semantics to theory presentation combinators as the foundational building blocks for a scalable library of theories. The key observation is that the category of contexts and fibered categories are the ideal theoretical tools for this purpose.
We define a notion of symmetric monoidal closed (SMC) theory, consisting of a SMC signature augmented with equations, and describe the classifying categories of such theories in terms of proof nets.
In this short note we study the semantics of two basic computational effects, exceptions and states, from a new point of view. In the handling of exceptions we dissociate the control from the elementary operation which recovers from the exception. In
This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program lo
When working with distant collaborators on the same documents, one often uses a version control system, which is a program tracking the history of files and helping importing modifications brought by others as patches. The implementation of such a sy