Signatures + Propositions Used as Types ?

Languages that distinguish between types and structures use explicit components for the carrier type(s) in structures. Examples are the functional language Standard ML and most algebraic speciication systems. Hence, they have to use general sum types or signatures to give types to structures and to be able to build, for instance, the algebraic hierarchy. Furthermore, in most languages the modelling of properties that the elements of a signature { the structures { should fullll is not possible or computationally not relevant. This is a major drawback, especially in a computer algebra environment. This paper presents a calculus for building signatures with explicit type components which are needed for modelling many-sorted structures and structures of the same signature with identical carriers. Additionally, we provide the possibility of using propositions as components needed for modelling the properties of structures. Propositions, which are distinguished from booleans, are inhabited by their proofs and hence, they are types themselves. This idea stems from the \propositions as types" principle known from constructive logic and results in a coherent treatment of carrier types, operations and their properties. It allows us to express theories (speciications) while staying within the framework of signatures. The framework is used for the construction of function spaces between types with equality and the building of some parts of the algebraic hierarchy .