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We study a type checking algorithm that is able to type check a nontrivialsubclass of functional programs that use features such as higher-rank,impredicative and second-order types. The only place the algorithm requirestype annotation is before each function declaration. We prove the soundness ofthe type checking algorithm with respect to System $\mathbf{F}_{\omega}$, i.e.if the program is type checked, then the type checker will produce a well-typedannotated System $\mathbf{F}_{\omega}$ term. We extend the basic algorithm tohandle pattern matching and let-bindings. We implement a prototype type checkerand test it on a variety of functional programs.

A Type Checking Algorithm for Higher-rank, Impredicative and Second-order Types