Constructive Representation of Nominal Sets in Agda

The theory of nominal sets provide a mathematical analysis of names that is based upon symmetry. It formalizes the informal reasoning we employ while working with languages involving name binding operators. The central ideas of the theory are support, freshness and name abstraction, which respectively encapsulate the ideas of name dependence, name independence and alpha equivalence. This theory has been developed within the framework of classical logic. Certain notions of this classical theory, like the smallest finite support for an element of a nominal set, are non-constructive. In this dissertation, we show that with appropriate modifications, a considerable portion of the theory of nominal sets can also be developed constructively. We show this by building from scratch, the theory of nominal sets, upto and including name abstraction sets, in the dependent type-theoretic environment of the programming language Agda. In our development, we replace the notion of unique smallest finite support with that of non-unique some finite support. In addition, all throughout our development, we work with setoids. This helps us in recovering extensionality of functions and in working with alpha-equivalence classes of terms. Though extensionality of functions could have been easily postulated in Agda, we refrained from using any postulate whatsoever at any level, since we cannot ensure constructivity of the whole development in case it rests even on just a single postulate. Our work can be extended and developed further to produce a nominal version of Agda which would make working with binding constructs in Agda much more easier.