this paper, the implementation has been used to verify an abstract version of sorting by insertion in (Tasistro 1997). In this latter work, dependent record types are used to express speciøcations of abstract data types. The theory here developed is a direct successor of the calculus of substitutions for type theory (Martin-L#f 1992; Tasistro 1997) in the sense that record types can be seen as type constructions corresponding to contexts of variables ¯record objects becoming then the counterpart to substitutions. Several theories of records have been developed in the context of systems without dependent types, mainly with the motivation of providing foundations for concepts that appear in object oriented programming. Then, for instance, there is by now a standard way of encoding objects in the sense of object oriented programming as recursively deøned records. The general motivation mentioned departs from ours, which, as far as the theory of programming is concerned, is limited to that of providing basic means that allow the use of dependent types for expressing speciøcations of abstract data types and modules in a general way. The problem of formulating a type system for object oriented programming raises a number of questions that are simply not relevant for our purposes. As to dependent record types, they have been implemented in PVS (Owre et al. 1993), which is a theorem proving system based on classical higher order logic. The subtyping that record types induce is, however, not a part of this implementation. In the original type theory, it is possible to encode each particular instance of inclusion between types ff and fi by using a coercion function that injects the objects of type ff into the type fi. In (Barthe 1996; Bailey 1996; Sa#bi 1997) different mechanisms...

. We present an example of formalization of systems of algebras using an extension of Martin-Lof's theory of types with record types and subtyping. This extension has been presented in [5]. In this paper we intend to illustrate all the features of the extended theory that we consider relevant for the task of formalizing algebraic constructions. We also provide code of the formalization as accepted by a type checker that has been implemented. 1. Introduction We shall use an extension of Martin-Lof's theory of logical types [14] with dependent record types and subtyping as the formal language in which constructions concerning systems of algebras are going to be represented. The original formulation of Martin-Lof's theory of types, from now on referred to as the logical framework, has been presented in [15, 7]. The system of types that this calculus embodies are the type Set (the type of inductively defined sets), dependent function types and for each set A, the type of the elements of A...