Relational properties of recursively defined domains

Abstract

A mixed induction/coinduction property of relations on recursively defined domains is described, working within a general framework for relations on domains and for actions of type constructors on relations introduced by P.W. O'Hearn and R.D. Tennent (1993), and drawing upon P.J. Freyd's analysis (1991) of recursive types in terms of a simultaneous initiality/finality property. The utility of the mixed induction/coinducton property is demonstrated by deriving a number of families of proof principles from it. One instance of the relational framework yields a family of induction principles for admissible subsets of general recursively defined domains which extends the principle of structural induction for inductively defined sets. Another instance of the framework yields the coinduction principle studied elsewhere by the author, by which equalities between elements of recursively defined domains may be proved via bisimulations.<>