Coinductive characterizations of applicative structures

We discuss new ways of characterizing, as maximal xed points of monotone operators, observational congruences on-terms and, more in general, equivalences on applicative structures. These characterizations naturally induce new forms of coinduction principles, for reasoning on program equivalences, which are not based on Abramsky's applicative bisimulation. We discuss in particular, what we call, the cartesian coin-duction principle, which arises when we exploit the elementary observation that functional behaviours can be expressed as cartesian graphs. Using the paradigm of nal semantics, the soundness of this principle over an applicative structure can be expressed easily by saying that the applica-tive structure can be construed as a strongly extensional coalgebra for the functor (P()) (P()). In this paper, we present two general methods for showing the soundenss of this principle. The rst applies to approximable applicative structures. Many c.p.o.-models in the literature , and the corresponding quotient models of indexed terms turn out to be approximable applicative structures. The second method is based on Howe's congruence candidates, and it applies to many observational equivalences. Structures satisfying cartesian coinduction are precisely those applica-tive structures which can be modeled using the standard set-theoretic application in non-wellfounded theories of sets, where the Foundation Axiom is replaced by the Axiom X1 of Forti and Honsell.