Constructing Strictly Positive Families

In order to represent, compute and reason with advanced data types one must go beyond the traditional treatment of data types as being inductive types and, instead, consider them as inductive families. Strictly positive types (SPTs) form a grammar for defining inductive types and, consequently, a fundamental question in the the theory of inductive families is what constitutes a corresponding grammar for inductive families. This paper answers this question in the form of strictly positive families or SPFs. We show that these SPFs can be used to represent and compute with a variety of advanced data types, that generic programs can naturally be written over the universe of SPFs and that SPFs have a normal form in terms of indexed containers which are based upon the shapes and positions metaphor. Finally, we validate ou computational perspective by implementing SPFs in the programming language Epigram and, further, comment on how SPFs provide a meta-language for Epigram's data types.