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Types, Abstraction, and Parametric Polymorphism, Part 2
, 1991
"... The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and P ..."
Abstract

Cited by 53 (1 self)
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The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and PLcategory models of the polymorphic typed lambda calculus. Treatments of Kripke relations and of complete relations on domains are included.
The Impact of seq on Free TheoremsBased Program Transformations
 Fundamenta Informaticae
, 2006
"... Parametric polymorphism constrains the behavior of pure functional programs in a way that allows the derivation of interesting theorems about them solely from their types, i.e., virtually for free. Unfortunately, standard parametricity results — including socalled free theorems — fail for nonstrict ..."
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Cited by 13 (5 self)
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Parametric polymorphism constrains the behavior of pure functional programs in a way that allows the derivation of interesting theorems about them solely from their types, i.e., virtually for free. Unfortunately, standard parametricity results — including socalled free theorems — fail for nonstrict languages supporting a polymorphic strict evaluation primitive such as Haskell’s seq. A folk theorem maintains that such results hold for a subset of Haskell corresponding to a GirardReynolds calculus with fixpoints and algebraic datatypes even when seq is present provided the relations which appear in their derivations are required to be bottomreflecting and admissible. In this paper we show that this folklore is incorrect, but that parametricity results can be recovered in the presence of seq by restricting attention to leftclosed, total, and admissible relations instead. The key novelty of our approach is the asymmetry introduced by leftclosedness, which leads to “inequational” versions of standard parametricity results together with preconditions guaranteeing their validity even when seq is present. We use these results to derive criteria ensuring that both equational and inequational versions of short cut fusion and related program transformations based on free theorems hold in the presence of seq.