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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 ..."
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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.
Strong normalization for System F by HOAS on top of FOAS
"... Abstract—We present a point of view concerning HOAS (HigherOrder Abstract Syntax) and an extensive exercise in HOAS along this point of view. The point of view is that HOAS can be soundly and fruitfully regarded as a definitional extension on top of FOAS (FirstOrder Abstract Syntax). As such, HOAS ..."
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Abstract—We present a point of view concerning HOAS (HigherOrder Abstract Syntax) and an extensive exercise in HOAS along this point of view. The point of view is that HOAS can be soundly and fruitfully regarded as a definitional extension on top of FOAS (FirstOrder Abstract Syntax). As such, HOAS is not only an encoding technique, but also a higherorder view of a firstorder reality. A rich collection of concepts and proof principles is developed inside the standard mathematical universe to give technical life to this point of view. The exercise consists of a new proof of Strong Normalization for System F. HOAS makes our proof considerably more direct than previous proofs. The concepts and results presented here have been formalized in the theorem prover Isabelle/HOL.
Relational Limits in General Polymorphism
, 1993
"... Parametric models of polymorphic lambda calculus have the structure of enriched categories with cotensors and ends in some generalized sense, and thus have many categorical data types induced by them. The !order minimum model is a parametric model. 1 Introduction Higher order quantifier of polymor ..."
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Parametric models of polymorphic lambda calculus have the structure of enriched categories with cotensors and ends in some generalized sense, and thus have many categorical data types induced by them. The !order minimum model is a parametric model. 1 Introduction Higher order quantifier of polymorphic lambda calculus has several meanings. Two inventors of the calculus use different symbols. When Girard wrote V X:F (X) [10] (\PiX:F (X) in [12]), it corresponded to a higher order quantified formula 8X:F (X) via CurryHoward isomorphism. When Reynolds wrote \DeltaX:F (X) [33], it was the type of polymorphism, especially of parametric polymorphism [34]. The third interpretation leaded by categorical semantics is that the quantified type, we write 8X:F (X), is a kind of limits. The notation \PiX:F (X) suggests that it might be regarded as a product of all F (X) where X ranges over all types. That is to say, \PiX:F (X) is the collection of all sections (a section is a function sending a...
Free Theorems Involving . . .
, 2009
"... Free theorems are a charm, allowing the derivation of useful statements about programs from their (polymorphic) types alone. We show how to reap such theorems not only from polymorphism over ordinary types, but also from polymorphism over type constructors restricted by class constraints. Our prime ..."
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Free theorems are a charm, allowing the derivation of useful statements about programs from their (polymorphic) types alone. We show how to reap such theorems not only from polymorphism over ordinary types, but also from polymorphism over type constructors restricted by class constraints. Our prime application area is that of monads, which form the probably most popular type constructor class of Haskell. To demonstrate the broader scope, we also deal with a transparent way of introducing difference lists into a program, endowed with a neat and general correctness proof.