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by Fairouz Kamareddine, Roel Bloo, Rob Nederpelt
http://www.macs.hw.ac.uk/~fairouz/papers/research-reports/Glasgow-TR-97-14.ps
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Abstract:
Typed-calculus uses two abstraction symbols ( and \Pi) which are usually treated in different ways: x::x has as type the abstraction \Pi x: : yet \Pi x: : has type 2 rather than an abstraction; moreover, ( x:A:B)C is allowed and fi-reduction evaluates it, but (\Pi x:A:B)C is rarely allowed. Furthermore, there is a general consensus that and \Pi are different abstraction operators. While we agree with this general consensus, we find it nonetheless important to allow \Pi- to act as an abstraction operator. Moreover, experience with AUTOMATH and the recent revivals of \Pi-reduction as in [KN 95b, PM 97], illustrate the elegance of giving \Pi-redexes some similar status to-redexes. Alas however, \Pireduction in the-cube faces serious problems as shown in [KN 95b, PM 97]: it is not safe as regards subject reduction, it does not satisfy type correctness, it loses the property that the type of an expression is well-formed and it fails to make any expression that contains a \Pi-redex well-formed. In this paper, we propose a solution to all those problems. The solution is to use a concept that is heavily present in most implementations of programming languages and theorem provers: abbreviations (viz. by means of a definition) or let expressions. We will show that the-cube extended with \Pi-conversion and abbreviations satisfies all the desirable properties of the cube and does not face any of the serious problems of \Pi-reduction. We believe that this extension of the-cube is very useful: it gives a full formal study of two concepts (\Pi-reduction and abbreviations) that are useful for theorem proving and programming languages. 1
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