Gradual typing has emerged as the tonic for programmers wanting a mixture of static and dynamic typing, hoping to achieve the best of both. Sound gradual typing is the most potent brew, providing static type-checking and dynamic assertions. Contracts provide a lightweight form of gradual typing as they can be implemented as a library, rather than requiring a gradual type system.

Intersection and union types are well suited to dynamic languages: intersection encodes overloaded functions; union encodes uncertain data arising from branching code. We extend the untyped lambda calculus with contracts for monitoring higher-order intersection and union types, giving a uniform treatment to both. Each operator requires a single reduction rule that does not depend on the constituent types, or the context of the operator, unlike existing work.

We present a new method for defining contract satisfaction based on blame behaviour. A value positively satisfies a type if applying a contract of that type can never elicit positive blame. A continuation negatively satisfies a type if applying a contract of that type can never elicit negative blame. We supplement our definition of satisfaction with a series of monitoring properties that satisfying values and continuations should have. These properties ensure that the semantics of contracts are in alignment with the static types they represent.