Making a Type Difference: Subtraction on Intersection Types as Generalized Record Operations.

In programming languages with records, objects, or traits, it is common to have operators that allow dropping, updating or renaming some components. These operators are useful for programmers to explicitly deal with conflicts and override or update some components. While such operators have been studied for record types, little work has been done to generalize and study their theory for other types. This paper shows that, given subtyping and disjointness relations, we can specify and derive algorithmic implementations for a general type difference operator that works for other types, including function types, record types and intersection types. When defined in this way, the type difference algebra has many desired properties that are expected from a subtraction operator. Together with a generic merge operator, using type difference we can generalize many operations on records formalized in the literature. To illustrate the usefulness of type difference we create an intermediate calculus with a rich set of operators on expressions of arbitrary type, and demonstrate applications of these operators in CP, a prototype language for Compositional Programming. The semantics of the calculus is given by elaborating into a calculus with disjoint intersection types and a merge operator. We have implemented type difference and all the operators in the CP language. Moreover, all the calculi and related proofs are mechanically formalized in the Coq theorem prover.