We show how to write generic programs and proofs in MartinL of type theory. To this end we consider several extensions of MartinL of's logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introduction, elimination, and equality rules. These extensions are modeled on Dybjer and Setzer's finitely axiomatized theories of inductive-recursive definitions, which also have a universe of codes for sets, and generic formation, introduction, elimination, and equality rules.

With relational transformational programming in mind, an extension of a "lawless " relational theory of datatypes is proposed in order to study and manipulate quotient types within a Tarski-like calculus of relations. The extended notion of type, pertype (from partial equivalence relation), is shown to admit a complete lattice structure by constructing the order via a Galois connection. A pertyping of relations is developed and inductive pertypes generated by equations are discussed. Pertypes do occur in model theory for -calculus but we are unaware of manipulations with inductive "lawful" types based on a simple relational calculus. 1 Introduction Program construction and its theory are developing steadily towards an algebraic discipline combining methods from category theory, algebraic logic and lattice theory. The driving forces are the needs for uninterpreted manipulation, problem structuring and generalisation, and polymorphy, with a major role for calculational type theo...