Inductive families

A general formulation of inductive and recursive definitions in Martin-Löf's type theory is presented. It extends Backhouse's ‘Do-It-Yourself Type Theory’ to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets are generated. The formulation is in natural deduction and is intended to be a natural generalisation to type theory of Martin-Löf's theory of iterated inductive definitions in predicate logic.Formal criteria are given for correct formation and introduction rules of a new set former capturing definition by strictly positive, iterated, generalised induction. Moreover, there is an inversion principle for deriving elimination and equality rules from the formation and introduction rules. Finally, there is an alternative schematic presentation of definition by recursion.The resulting theory is a flexible and powerful language for programming and constructive mathematics. We hint at the wealth of possible applications by showing several basic examples: predicate logic, generalised induction, and a formalisation of the untyped lambda calculus.