We propose a refinement of the type theory underlying the LF logical framework by a form of subtypes and intersection types. This refinement preserves desirable features of LF, such as decidability of type-checking, and at the same time considerably simplifies the representations of many deductive systems. A subtheory can be applied directly to hereditary Harrop formulas which form the basis of Prolog and Isabelle. 1 Introduction Over the past two years we have carried out extensive experiments in the application of the LF Logical Framework [HHP93] to represent and implement deductive systems and their metatheory. Such systems arise naturally in the study of logic and the theory of programming languages. For example, we have formalized the operational semantics and type system of Mini-ML and implemented a proof of type preservation [MP91] and the correctness of a compiler to a variant of the Categorical Abstract Machine [HP92]. LF is based on a predicative type theory with dependent t...

We propose related algorithms for unification and constraint simplification in !& , a refinement of the simply-typed λ-calculus with subtypes and bounded intersection types. !& is intended as the basis of a logical framework in order to achieve more succinct and declarative axiomatizations of deductive systems than possible with the simply-typed λ-calculus. The unification and constraint simplification algorithms described here lay the groundwork for a mechanization of such frameworks as constraint logic programming languages and theorem provers.

In the following we describe the basic ideas underlying\Omega\Gamma mkrp, an interactive proof development environment [6]. The requirements for this system were derived from our experiences in proving an interrelated collection of theorems of a typical textbook on semi-groups and automata [3] with the first-order theorem prover mkrp [11]. An important finding was that although current automated theorem provers have evidently reached the power to solve non-trivial problems, they do not provide sufficient assistance for proving the theorems contained in such a textbook. On account of this, we believe that significantly more support for proof development can be provided by a system with the following two features: -- The system must provide a comfortable human-oriented problem-solving environment. In particular, a human user should be able to specify the problem to be solved in a natural way and communicate on proof