this paper and the LF. In particular the idea of having an operator T : Prop ! Type appears already in De Bruijn's earlier work, as does the idea of having several judgements. The paper [24] describes the basic features of the LF. In this paper we are going to provide a broader illustration of its applicability and discuss to what extent it is successful. The analysis (of the formal presentation) of a system carried out through encoding often illuminates the system itself. This paper will also deal with this phenomenon.

Various formulations of constructive type theories have been proposed to serve as the basis for machine-assisted proof and as a theoretical basis for studying programming languages. Many of these calculi include a cumulative hierarchy of "universes," each a type of types closed under a collection of type-forming operations. Universes are of interest for a variety of reasons, some philosophical (predicative vs. impredicative type theories), some theoretical (limitations on the closure properties of type theories), and some practical (to achieve some of the advantages of a type of all types without sacrificing consistency.) The Generalized Calculus of Constructions (CC ! ) is a formal theory of types that includes such a hierarchy of universes. Although essential to the formalization of constructive mathematics, universes are tedious to use in practice, for one is required to make specific choices of universe levels and to ensure that all choices are consistent. In this pa...

. The underlying ideas, design principles and capabilities of a system called Disc Atinf (Defining and Implementing Strategies and Calculi), allowing to implement a large class of computational systems (proof procedures, constraints solving algorithms: : : ) are presented. Systems are described by sets of rules along with a strategy to guide the application of these rules. This formalism has been chosen because it fits well to the standard mathematical practice. Disc Atinf is user-oriented and very flexible. It allows to specify the source objects and the transformation process in a very easy way. Moreover it is very general. We describe the main features of Disc Atinf and gives examples of applications --- showing evidence of the usefulness of our system. Finally, we briefly compare our system with similar ones. Mathematicians, logicians and computer scientists usually spend a lot of time in implementing the computational systems they define. Such systems (typically proof procedure...