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Gradual typing has emerged as the tonic for programmers with a thirst for a blend of static and dynamic typing. 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 static and dynamic languages: intersection encodes overloaded functions; union encodes uncertain data arising from branching code. We extend the untyped lambda calculus with contracts for monitoringdoi:10.1145/3276504 fatcat:d7ppbeyxgzh5hf34d2wph2t3k4