An algorithm for checking incomplete proof objects in type theory with localization and unification

Abstract

A modular type/proof checking algorithm for incomplete proof objects is presented, where an incomplete proof object is represented as a term containing placeholders denoting the unfinished parts of the proof. The algorithm is designed for Martin-Löf 's type theory with explicit substitutions, but the general ideas can be adapted to similar theories. It is the kernel of the proof editor ALF.

The algorithm handles incomplete terms in such a way that the type checking problem is reduced to a unification problem, i.e., the problem of finding instantiations to the placeholders in the term. Placeholders are represented together with their expected type and local context. We show that checking the correctness of instantiations can be optimized by localization, which means that it is enough to check an instantiation of a placeholder relative to its expected type and local context.

We present a unification algorithm which partially solves the unification problem, and we apply this unification algorithm to the type checking algorithm. We show that the type checking algorithm with unification and with localization is sound, and hence when all placeholders in a proof object are filled in and checked locally, we do not have to type check the global proof object again to ensure type correctness. Finally, we define two basic operations on a type checking problem, insert and delete, and we show that the basic tactics “intro” and “refine” can be defined in terms of insert. The delete operation provides a local undo mechanism which is unique for ALF. The operations are shown to preserve the validity of a partially solved type checking problem, and hence the proof editing facilities are proved to construct valid proofs.