A cosmology of datatypes : reusability and dependent types

This dissertation defends the idea of a closed dependent type theory whose inductive types are encoded in a universe. Each inductive definition arises by interpreting its description – itself a firstclass citizen in the type theory. Datatype-generic programming thus becomes ordinary programming. This approach is illustrated by several generic programs. We then introduce an elaboration of inductive definitions down to the universe of datatypes. By elaborating an inductive definition – a syntactic artefact – to its code – its type theoretic denotation – we obtain an internalised account of inductive types inside type theory. This is a small step toward bootstrapping, i.e. implementing the inductive fragment in the type theory itself. Building upon this universe of datatypes, ornaments let us treat datatypes as the combination of a structure and a logic: they relate datatypes through their common structure. We set out to rationalise this calculus of structures. We study a categorical model of ornaments, based on Cartesian morphisms of containers. We also illustrate the adequacy of our model by recasting the standard idioms into the categorical mould, and by translating the discovered categorical structures into type theoretic artefacts. Nonetheless, the extreme accuracy of these finely indexed datatypes is a curse for code reuse. Similar functions must be duplicated across similarly structured – but logically incompatible – indexed datatypes. We shall see how code reuse can be achieved by ornamenting functions. We thus capture the relationship between functions such as the addition of natural numbers and the concatenation of lists. We also demonstrate how the implementation of the former informs the implementation of the latter.