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On Girard’s “Candidats de Réductibilité
 Logic and Computer Science
, 1990
"... Abstract: We attempt to elucidate the conditions required on Girard’s candidates of reducibility (in French, “candidats de reductibilité”) in order to establish certain properties of various typed lambda calculi, such as strong normalization and ChurchRosser property. We present two generalizations ..."
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Abstract: We attempt to elucidate the conditions required on Girard’s candidates of reducibility (in French, “candidats de reductibilité”) in order to establish certain properties of various typed lambda calculi, such as strong normalization and ChurchRosser property. We present two generalizations of the candidates of reducibility, an untyped version in the line of Tait and Mitchell, and a typed version which is an adaptation of Girard’s original method. As an application of this general result, we give two proofs of strong normalization for the secondorder polymorphic lambda calculus under ⌘reduction (and thus underreduction). We present two sets of conditions for the typed version of the candidates. The first set consists of conditions similar to those used by Stenlund (basically the typed version of Tait’s conditions), and the second set consists of Girard’s original conditions. We also compare these conditions, and prove that Girard’s conditions are stronger than Tait’s conditions. We give a new proof of the ChurchRosser theorem for bothreduction and ⌘reduction, using the modified version of Girard’s method. We also compare various proofs that have appeared in the literature (see section 11). We conclude by sketching the extension of the above results to Girard’s higherorder polymorphic calculus F!, and in appendix 1, to F! with product types. i 1
Quantifier Elimination and Parametric Polymorphism in Programming Languages
 J. Functional Programming
, 1992
"... 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 ..."
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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 CurryHoward propositionsastypes analogy. Supported by NSF Grant CCR9017125, 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...
Appendix 1: Product Types in F !
"... 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) ..."
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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 kindchecking 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 kindchecks has a unique normal form. The following inference rules need to be added to the proof system used for typechecking terms. . M : . N :