Implementing theorem provers in a purely functional style

This paper discusses the principles of implementing an LCF-style proof assistant using a purely functional metalanguage. Two approaches are described; in both, signatures are treated as ordinary values, rather than as mutable components within an abstract datatype. The first approach treats the object logic as a partial algebra and represents it as a partial datatype, that is, a datatype in which the domains of the constructors are restricted by predicate functions. This results in a compact, executable specification of the logic. The second approach uses an abstract type to allow an efficient representation of the logic, whilst keeping the same interface. A case study describes how these principles were put into practice in implementing a fairly complex dependently-sorted logic.