Type inference with expansion variables and intersection types in system E and an exact correspondence with β-reduction

System E is a recently designed type system for the ?-calculus with intersection types and expansion variables. During automatic type inference, expansion variables allow postponing decisions about which non-syntax-driven typing rules to use until the right information is available and allow implementing the choices via substitution. This paper uses expansion variables in a unification-based automatic type inference algorithm for System E that succeeds for every ß-normalizable ?-term. We have implemented and tested our algorithm and released our implementation publicly. Each step of our unification algorithm corresponds to exactly one ß-reduction step, and vice versa. This formally verifies and makes precise a step-for-step correspondence between type inference and ß-reduction. This also shows that type inference with intersection types and expansion variables can, in effect, carry out an arbitrary amount of partial evaluation of the program being analyzed.