Parametric Datatype−Genericity

Datatype-generic programs are programs that are parametrized by a datatype or type functor: whereas polymorphic programs abstract from the \"integers\" in \"lists of integers\", datatype-generic programs abstract from the \"lists of\". There are two main styles of datatype-generic programming: the Algebra of Programming approach, characterized by structured recursion operators arising from initial algebras and final coalgebras, and the Generic Haskell approach, characterized by case analysis over the structure of a datatype. We show that the former enjoys a kind of higher-order naturality, relating the behaviours of generic functions at different types; in contrast, the latter is more ad hoc, with no coherence required or provided between the various clauses of a definition. Moreover, the naturality properties arise \"for free\", simply from the parametrized types of the generic functions: we present a higher-order parametricity theorem for datatype-generic operators