Summary: Under consideration for publication in J. Functional Programming 1
Pure Type Systems with Judgemental Equality
Department of Computer Science,
Royal Holloway, University of London
In a typing system, there are two approaches that may be taken to the notion of equality.
One can use some external relation of convertibility defined on the terms of the grammar,
such as #convertibility or ##convertibility; or one can introduce a judgement form for
equality into the rules of the typing system itself.
For quite some time, it has been an open problem whether the two systems produced
by these two choices are equivalent. This problem is essentially the problem of proving
that the Subject Reduction property holds in the system with judgemental equality.
In this paper, we shall prove that the equivalence holds for all functional Pure Type
Systems (PTSs). The proof essentially consists of proving the ChurchRosser Theorem for
a typed version of parallel onestep reduction. This method should generalise easily to
many typing systems which satisfy the Uniqueness of Types property.