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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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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...
HPGP: HighPerformance Generic Programming for Computational Mathematics by CompileTime Instantiation of Higher Order Functors
, 1997
"... A functor is a parameterized program module i.e. a function that takes modules as arguments and returns a module as its result. A higherorder functor deals in the same way with modules whose components are functors themselves. We propose to develop a generic compilation system for the construction ..."
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A functor is a parameterized program module i.e. a function that takes modules as arguments and returns a module as its result. A higherorder functor deals in the same way with modules whose components are functors themselves. We propose to develop a generic compilation system for the construction of highperformance mathematical software libraries for scientific and technical application domains. This system has the following features: 1. It is based on a powerful higherorder functor language. 2. It is an open library that can be retargeted to any core language. 3. It is able to resolve functor instantiation at compiletime. The functor language is expressive enough to build all types and type constructors without referring to the core language (thus maximizing flexibility) and to express all interactions between modules by parameterization (thus maximizing reusability). By compiletime instantiation, genericity does not cause any execution overhead; by automatically sharing instant...
Irrelevance, Polymorphism, and Erasure in Type Theory
, 2008
"... Dependent type theory is a proven technology for verified functional programming in which programs and their correctness proofs may be developed using the same rules in a single formal system. In practice, large portions of programs developed in this way have no computational relevance to the ultima ..."
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Dependent type theory is a proven technology for verified functional programming in which programs and their correctness proofs may be developed using the same rules in a single formal system. In practice, large portions of programs developed in this way have no computational relevance to the ultimate result of the program and should therefore be removed prior to program execution. In previous work on identifying and removing irrelevant portions of programs, computational irrelevance is usually treated as an intrinsic property of program expressions. We find that such an approach forces programmers to maintain two copies of commonly used datatypes: a computationally relevant one and a computationally irrelevant one. We instead develop an extrinsic notion of computational irrelevance and find that it yields several benefits including (1) avoidance of the above mentioned code duplication problem; (2) an identification of computational irrelevance with a highly general form of parametric polymorphism; and (3) an elective (i.e., user2 directed) notion of proof irrelevance. We also develop a program analysis for identifying irrelevant expressions and show how previously studied types embodying computational irrelevance (including subset types and squash types) are expressible in the extension of type theory developed herein.