Structural Induction and Coinduction in a Fibrational Setting

We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coin-ductively deened types. Our main results provide suucient criteria for the validity of such principles: in the presence of comprehension, the induction principle for initial algebras is admissible, and dually, in the presence of quotient types, the coinduction principle for terminal coalgebras is admissible. After giving an alternative formulation of induction in terms of binary relations, we combine both principles and obtain a mixed induction/coinduction principle which allows us to reason about minimal solutions X = (X) where X may occur both positively and negatively in the type constructor. We further strengthen these logical principles to deal with contexts and prove that such strengthening is valid when the (abstract) logic we consider is contextually/functionally complete. All the main results follow from a basic result about adjunctions between`categories of algebras' (inserters).