We formally characterize partial evaluation of functional programs as a normalization problem in an equational theory, and derive a type-based normalization-by-evaluation algorithm for computing normal forms in this setting. We then establish the correctness of this algorithm using a semantic argument based on Kripke logical relations. For simplicity, the results are stated for a nonstrict, purely functional language; but the methods are directly applicable to stating and proving correctness of type-directed partial evaluation in ML-like languages as well.

Abstract. We use a code generator—type-directed partial evaluation— to verify conversions between isomorphic types, or more precisely to verify that a composite function is the identity function at some complicated type. A typed functional language such as ML provides a natural support to express the functions and type-directed partial evaluation provides a convenient setting to obtain the normal form of their composition. However, off-the-shelf type-directed partial evaluation turns out to yield gigantic normal forms. We identify that this gigantism is due to redundancies, and that these redundancies originate in the handling of sums, which uses delimited continuations. We successfully eliminate these redundancies by extending type-directed partial evaluation with memoization capabilities. The result only works for pure functional programs, but it provides an unexpected use of code generation and it yields orders-of-magnitude improvements both in time and in space for type isomorphisms. 1