Well-known techniques exist for proving the soundness of subtyping relations with respect to type safety. However, completeness has not been treated with widely applicable techniques, as far as we’re aware. This article develops techniques for stating and proving that a subtyping relation is complete with respect to type safety and applies the techniques to the study of iso-recursive subtyping. A new proof technique, induction on failing derivations, is provided that may be useful in other domains as well. The common subtyping rules for iso-recursive types—the “Amber rules”—are shown to be incomplete with respect to type safety. That is, there exist iso-recursive types τ 1 and τ 2 such that τ 1 can safely be considered a subtype of τ 2 , but τ 1 ⩽ τ 2 is not derivable with the Amber rules. New, algorithmic rules are defined for subtyping iso-recursive types, and the rules are proved sound and complete with respect to type safety. The fully implemented subtyping algorithm is optimized to run in O ( mn ) time, where m is the number of μ-terms in the types being considered and n is the size of the types being considered.

@article{LigattiBN17,
  title = {On Subtyping-Relation Completeness, with an Application to Iso-Recursive Types},
  author = {Jay Ligatti and Jeremy Blackburn and Michael Nachtigal},
  journal = {ACM Transactions on Programming Languages and Systems},
  volume = {39},
  number = {1},
  year = {2017},
  doi = {10.1145/2994596}
}