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

This article presents an alternative method of type-directed partial evaluation, which is simpler and more efficient than previous methods. Unlike previous methods, it is straightforwardly applicable to functional languages with various powerful type systems. As an extreme instance, this article mainly deals with a dynamically-typed functional language like Scheme. The key idea is to extend primitive value destructors such as function application and pair destruction (car and cdr), so that they generate residual code when their operands are dynamic. It unnecessitates an operation in type-directed partial evaluation called reflection, which was the major cause of complication and inefficiency in previous methods. We formalize our method as an extension of two-level -calculus, and prove it correct. Furthermore, we show that our type-directed partial evaluator can be derived from a simple online syntax-directed partial evaluator with higher-order abstract syntax, by the same transformatio...

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