Indexed Containers

We show that the syntactically rich notion of inductive families can be reduced to a core type theory with a fixed number of type constructors exploiting the novel notion of indexed containers. Indexed containers generalize simple containers, capturing strictly positive families instead of just strictly positive types, without having to extend the core type theory. Other applications of indexed containers include datatype-generic programming and reasoning about polymorphic functions. The construction presented here has been formalized using the Agda system.