On Type-Cases, Union Elimination, and Occurrence Typing

We extend classic union and intersection type systems with a type-case construction and show that the combination of the union elimination rule of the former and the typing rules for type-cases of our extension encompasses occurrence typing. To apply this system in practice, we define a canonical form for the expressions of our extension, called MSC-form. We show that an expression of the extension is typable if and only if its MSC-form is, and reduce the problem of typing the latter to the one of reconstructing annotations for that term. We provide a sound algorithm that performs this reconstruction and a proof-of-concept implementation.