Encoding types in ML-like languages

A Hindley-Milner type system such as ML's seems to prohibit type-indexed values, i.e., functions that map a family of types to a family of values. Such functions generally perform case analysis on the input types and return values of possibly different types. The goal of our work is to demonstrate how to program with type-indexed values within a Hindley-Milner type system.Our first approach is to interpret an input type as its corresponding value, recursively. This solution is type-safe, in the sense that the ML type system statically prevents any mismatch between the input type and function arguments that depend on this type.Such specific type interpretations, however, prevent us from combining different type-indexed values that share the same type. To meet this objection, we focus on finding a value-independent type encoding that can be shared by different functions. We propose and compare two solutions. One requires first-class and higher-order polymorphism, and, thus, is not implementable in the core language of ML, but it can be programmed using higher-order functors in Standard ML of New Jersey. Its usage, however, is clumsy. The other approach uses embedding/projection functions. It appears to be more practical.We demonstrate the usefulness of type-indexed values through examples including type-directed partial evaluation, C printf-like formatting, and subtype coercions. Finally, we discuss the tradeoffs between our approach and some other solutions based on more expressive typing disciplines.