Well-founded Functions and Extreme Predicates in Dafny: A Tutorial

A recursive function is well defined if its every recursive call corresponds a decrease in some well-founded order. Such well-founded functions are useful for example in computer programs when computing a value from some input. A boolean function can also be defined as an extreme solution to a recurrence relation, that is, as a least or greatest fixpoint of some functor. Such extreme predicates are useful for example in logic when encoding a set of inductive or coinductive inference rules. The verification-aware programming language Dafny supports both well-founded functions and extreme predicates. This tutorial describes the difference in general terms, and then describes novel syntactic support in Dafny for defining and proving lemmas with extreme predicates. Various examples and considerations are given. Although Dafny’s verifier has at its core a first-order SMT solver, Dafny’s logical encoding makes it possible to reason about fixpoints in an automated way.