Introduction

In this paper, a method is presented for proof construction in Generalised Type Systems. An interactive system that implements the method has been developed. Generalised type systems (GTSs) provide a uniform way to describe and classify type theoretical systems, e.g. systems in the families of AUTOMATH, the Calculus of Constructions, LF. A method is presented to perform unification based top down proof construction for generalised type systems, thus offering a well-founded, elegant and powerful underlying formalism for a proof development system. It combines clause resolution with higher-order natural deduction style theorem proving. No theoretical contribution to generalised type systems is claimed.

A type theory presents a set of rules to derive types of objects in a given context with assumptions about the type of primitive objects. The objects and types are expressions in typed λ-calculus. The propositions as types paradigm provides a direct mapping between (higher-order) logic and type theory. In this interpretation, contexts correspond to theories, types correspond to propositions, and objects correspond to proofs of propositions. Type theory has successfully demonstrated its capabilities to formalise many parts of mathematics in a uniform and natural way. For many generalised type systems, like the systems in the so-called λ-cube, the typing relation is decidable. This permits automatic proof checking, and such proof checkers have been developed for specific type systems.

The problem addressed in this paper is to construct an object in a given context, given its type.