[Celioscopic differential diagnosis of gynecologic inflammations (proceedings)].

In this article, we introduce a ¿-notation that is useful for many concepts of the ¿-calculus. The new notation is a simple translation of the classical one. Yet, it provides many nice advantages.\nFirst, we show that definitions such as compatibility, the heart of a term and ß-redexes become simpler in item notation.\nSecond, we show that with this item notation, reduction can be generalised in a nice way. We find a relation ß which extends ¿ß, which is Church-Rosser and strongly normalising. This reduction relation may be the way to new reduction strategies. In classical notation, it is much harder to present this generalised reduction in a convincing manner.\nThird, we show that the item notation enables one to represent in a very simple way the canonical type t(G,A) of a term A in context G. This canonical type plays the role of a preference type and can be used to split G A : B into the two parts G A and t(G,A) = B. This means that the question is A typable with a type B is divided into two questions: is A typable and is B in the class of types of A. It turns out that calculating this preference type of A in item notation is a straightforward operation. One just goes through A from left to right performing very trivial steps on the items till the end variable (or heart) of A is reached.\nFourth, we can with this item notation, find the parts of a term t relevant for a variable occurrence x° in terms of binding, typing and substitution. Again, this part of t, t | x°, is very easy to find in item notation. Just take the part of t to the left of x° and remove all unmatched parentheses.\nFifth, we reflect on the status of variables and show that indeed it is easy to study this status in item notation.\nFinally, we show that for a substitution calculus à la de Bruijn with open terms, it is simpler to describe normal forms using item notation.\nThere are further advantages of item notation that are studied elsewhere. For example, in [9], we show that explicit substitution is easily built in item notation and that global and local strategies of substitution can be accommodated. In [10], we show that with item notation, one can give a unified approach to type theory.\nAn implementation of this item notation with most of the concepts discussed in this paper can be found in [15].