How to Reason Coinductively Informally

Abstract

We start by giving an overview of the theory of indexed inductively and coinductively defined sets. We consider the theory of strictly positive indexed inductive definitions in a set theoretic setting. We show the equivalence between the definition as an indexed initial algebra, the definition via an induction principle, and the set theoretic definition of indexed inductive definitions. We review as well the equivalence of unique iteration, unique primitive recursion, and induction. Then we review the theory of indexed coinductively defined sets or final coalgebras. We construct indexed coinductively defined sets set theoretically, and show the equivalence between the category theoretic definition, the principle of unique coiteration, of unique corecursion, and of iteration together with bisimulation as equality. Bisimulation will be defined as an indexed coinductively defined set. Therefore proofs of bisimulation can be carried out corecursively. This fact can be considered together with bisimulation implying equality as the coinduction principle for the underlying coinductively defined set. Finally we introduce various schemata for reasoning about coinductively defined sets in an informal way: the schemata of corecursion, of indexed corecursion, of coinduction, and of corecursion for coinductively defined relations. This allows to reason about coinductively defined sets similarly as one does when reasoning about inductively defined sets using schemata of induction. We obtain the notion of a coinduction hypothesis, which is the dual of an induction hypothesis.

Dedicated to Gerhard Jäger on occasion of his 60th Birthday