A fixedpoint approach to (co)inductive and (co)datatype definitions

This paper presents a fixedpoint approach to inductive definitions. Instead of using a syntactic test such as “strictly positive,” the approach lets definitions involve any operators that have been proved monotone. It is conceptually simple, which has allowed the easy implementation of mutual recursion and iterated definitions. It also handles coinductive definitions: simply replace the least fixedpoint by a greatest fixedpoint. The method has been implemented in two of Isabelle’s logics, zf set theory and higher-order logic. It should be applicable to any logic in which the Knaster-Tarski theorem can be proved. Examples include lists of n elements, the accessible part of a relation and the set of primitive recursive functions. One example of a coinductive definition is bisimulations for lazy lists. Recursive datatypes are examined in detail, as well as one example of a codatatype: lazy lists. The Isabelle package has been applied in several large case studies, including two proofs of the Church-Rosser theorem and a coinductive proof of semantic consistency. The package can be trusted because it proves theorems from definitions, instead of asserting desired properties as axioms. Copyright c © 2010 by Lawrence C. Paulson ∗J. Grundy and S. Thompson made detailed comments. Mads Tofte and the referees were also helpful. The research was funded by the SERC grants GR/G53279, GR/H40570 and by the ESPRIT Project 6453 “Types”.