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Outline of a Proof Theory of Parametricity
 Proc. 5th International Symposium on Functional Programming Languages and Computer Architecture
, 1991
"... Reynolds' Parametricity Theorem (also known as the Abstraction Theorem), a result concerning the model theory of the second order polymorphic typed calculus (F 2 ), has recently been used by Wadler to prove some unusual and interesting properties of programs. We present a purely syntactic version o ..."
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Cited by 25 (2 self)
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Reynolds' Parametricity Theorem (also known as the Abstraction Theorem), a result concerning the model theory of the second order polymorphic typed calculus (F 2 ), has recently been used by Wadler to prove some unusual and interesting properties of programs. We present a purely syntactic version of the Parametricity Theorem, showing that it is simply an example of formal theorem proving in second order minimal logic over a first order equivalence theory on terms. We analyze the use of parametricity in proving program equivalences, and show that structural induction is still required: parametricity is not enough. As in Leivant's transparent presentation of Girard's Representation Theorem for F 2 , we show that algorithms can be extracted from the proofs, such that if a term can be proven parametric, we can synthesize from the proof an "equivalent" parametric term that is moreover F 2 typable. Given that Leivant showed how proofs of termination, based on inductive data types and s...
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...