We present a simple and easy to understand explanation of ML type inference and parametric polymorphism within the framework of type monomorphism, as in the first order typed lambda calculus. We prove the equivalence of this system with the standard interpretation using type polymorphism, and extend the equivalence to include polymorphic fixpoints. The monomorphic interpretation gives a purely combinatorial understanding of the type inference problem, and is a classic instance of quantifier elimination, as well as an example of Gentzenstyle cut elimination in the framework of the Curry-Howard propositions-as-types analogy. Supported by NSF Grant CCR-9017125, and grants from Texas Instruments and from the Tyson Foundation. 1 Introduction In his influential paper, "A theory of type polymorphism in programming," Robin Milner proposed an extension to the first order typed -calculus which has become known as the core of the ML programming language [Mil78, HMT90]. The extension augment...

for short, raw terms) is de ned inductively as follows: c 2 P, whenever c 2 , x 2 P, whenever x 2 X , (MN) 2 P, whenever M;N 2 P, hM; Ni 2 P, whenever M;N 2 P, 1 (M); 2 (M) 2 P, whenever M 2 P, (x: : M) 2 P, whenever x 2 X , 2 T , and M 2 P, (M) 2 P, whenever 2 T and M 2 P, (t: K: M) 2 P, whenever t 2 V, K 2 K, and M 2 P. The notions of substitution and -equivalence are extended in the obvious way. In order to deal with product types, it is necessary to add the following kind-checking rule: . : ? . : ? . : ? () The de nition of the relation ! ! does not have to be changed, since the congruence rule takes care of ), , and K . It is easy to see that corollary 6.18 and corollary 6.19 hold for the new class of types. Thus, every ( -equivalence class of) type that kind-checks has a unique -normal form. The following inference rules need to be added to the proof system used for typechecking terms. . M : . N :