Type Theory based on Dependent Inductive and Coinductive Types

In this talk, we will develop a type theory that is based solely on dependent inductive and coinductive types. By this we mean that the only way to form new types is by specifying the type of their corresponding constructors or destructors, respectively. From such a specification, we get the corresponding recursion and corecursion principles. One might be tempted to think that such a theory is relatively weak as, for example, there is no function space type. However, as it turns out, the function space is definable as a coinductive type. In fact, we can encode the connectives of intuitionistic predicate logic: falsity, conjunction, disjunction, dependent function space, existential quantification, and equality. Further, well-known types like natural numbers, vectors etc. arise as well. The presented type theory is based on ideas from categorical logic that have been investigated before by the first author, and it extends Hagino’s categorical data types to a dependently typed setting. By basing the type theory on concepts from category theory we maintain the duality between inductive and coinductive types.