Antifounded Coinduction in Type Theory

Capretta, Vene and myself have previously [2] studied antifounded algebras as algebras supporting a form of coinduction in a category-theoretic setting, arising as the dual concept of wellfounded coalgebras. Wellfounded coalgebras are a category theorist’s take on coalgebras supporting induction, introduced by Taylor [4]. Informally, wellfoundedness of a coalgebra means that a subobject of its carrier, called the wellfounded part, is isomorphic to the carrier. Under mild conditions, wellfounded coalgebras are the same as recursive coalgebras in the sense of Osius [3], i.e., coalgebras supporting recursion. Categorical wellfounded coalgebras are intimately related to the type-theoretic method of Bove and Capretta [1] for justifying function definitions by recursionand proofs by induction on a coalgebra. The idea of the method is this: One defines the wellfounded part of the coalgebra in terms of an inductively defined predicate of the carrier, to then rewrite the given “general” recursive definition or inductive proof on the carrier as a “structural ” recursive definition or inductive proof on the wellfounded part (these are systematic constructions). Separately, one establishes (in some way) that the carrier is contained in the wellfounded part