Structural Induction and Coinduction in a Fibrational Setting

Claudio Hermida, Bart Jacobs
1998 Information and Computation  
We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively de ned types. Our main results provide su cient 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
more » ... nary 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).
doi:10.1006/inco.1998.2725 fatcat:qwvx6yx3j5evdn7zrtdb6rox3i