A Calculus of Coercions Proving the Strong Normalization of ML

We provide a strong normalization result for ML F , a type system generalizing ML with first-class polymorphism as in system F. The proof is achieved by translating ML F into a calculus of coercions, and showing that this calculus is just a decorated version of system F. Simulation results then entail strong normalization from the same property of system F. Introduction. ML F [3] is a type system for (extensions of) λ-calculus which enriches ML with the first class polymorphism of system F, providing a partial type annotation mechanism with an automatic type reconstructor. This extension allows to write system F programs, which is not possible in general in ML, while still being conservative: ML programs still typecheck without needing any annotation. An important feature are principal type schemata, lacking in system F, which are obtained by employing a downward bounded quantification ∀(α ≥ σ)τ, the so-called flexible quantifier. Such a type intuitively denotes that τ may be instantiated to any τ {σ /α}, provided that σ is an instantiation of σ. As already pointed out, system F is contained in ML F. It is not yet known, but it is conjectured [3], that the inclusion is strict. This makes the question of strong normalization (SN, i.e. whether λ-terms typed in ML F always terminate) a non-trivial one, to which we answer positively in this work. The result is proved via a suitable simulation in system F, with additional structure dealing with the complex type instantiations possible in ML F. Our starting point is xML F [5], the Church version of ML F : here type inference (and the rigid quantifier ∀(α = σ)τ we did not mention) is abandoned, with the aim of providing an internal language to which a compiler might map the surface language briefly presented above (denoted eML F from now on 1). Compared to Church-style system F, the type reduction → ι of xML F is more complex, and may a priori cause unexpected glitches: it could cause non-termination, or block the reduction of a β-redex. To prove that none of this happens, we use as target language of our translation a decoration of system F, the coercion calculus, which in our opinion has its own interest. Indeed, xML F has syntactic entities (the instantiations φ) which testify an instance relation between types, and it is not hard to regard them as coercions. The …