In this article, we extend the Barendregt Cube with \Pi-conversion (which is the analogue of beta-conversion, 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.

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 a status similar to -redexes. However, \Pi-reduction 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-for...