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Canonical typing and Πconversion in the Barendregt Cube
, 1996
"... In this article, we extend the Barendregt Cube with \Piconversion (which is the analogue of betaconversion, on product type level) and study its properties. We use this extension to separate the problem of whether a term is typable from the problem of what is the type of a term. ..."
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In this article, we extend the Barendregt Cube with \Piconversion (which is the analogue of betaconversion, on product type level) and study its properties. We use this extension to separate the problem of whether a term is typable from the problem of what is the type of a term.
On \Piconversion in the lambdacube and the combination with abbreviations
, 1997
"... 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 fireduction evaluates it, but (\Pi x:A :B)C is rarely allowed. Fu ..."
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Cited by 4 (2 self)
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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 fireduction 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 \Pireduction as in [KN 95b, PM 97], illustrate the elegance of giving \Piredexes a status similar to redexes. 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 wellformed and it fails to make any expression that contains a \Piredex wellfor...
Definitions and \Piconversion in Type Theory
, 1997
"... In [KN 95b], the Barendregt Cube was extended with \Piconversion. The resulting system had only a Weak form of Subject Reduction. In this paper, the Cube is extended with explicit definitions. We show that the Cube extended with either explicit definitions alone or with both explicit definitions an ..."
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In [KN 95b], the Barendregt Cube was extended with \Piconversion. The resulting system had only a Weak form of Subject Reduction. In this paper, the Cube is extended with explicit definitions. We show that the Cube extended with either explicit definitions alone or with both explicit definitions and \Piconversion satisfies all its original properties including Subject Reduction. 1 Introduction Type theory has almost always been studied without \Piconversion (which is the analogue of ficonversion on product type level). That is, ! fi : ( x:A :b)C ! fi b[x := C] is always assumed but not ! \Pi : (\Pi x:A :B)C ! \Pi B[x := C]. The exception for this are some Automath languages in [NGV 95] and the current work of [KN 94] and [KN 95b]. We claim that ! \Pi is desirable for the following reasons: 1. \Pi is a kind of . In various higher order type theories, arrowtypes of the form A ! B are replaced by dependent products \Pi x:A :B, where B may contain x as a free variable, and thus may...