A mechanically assisted constructive proof in category theory

Abstract

Recently there has been a significant confluence of ideas from category theory, constructive type theory and functional programming. Our goal is to develop a toolkit that provides mechanical assistance in putting together constructive proofs of theorems in category theory and to extract executable programs from the result.

We describe a machine-assisted proof of the adjoint functor theorem from category theory. The theorem is a simple version of the adjoint functor theorem. It was chosen because of its interesting constructive content, the fact that the machine-checked proof is fairly large and because it is mathematically non-trivial. The category theory toolkit was built on top of the Nuprl system. The core of this toolkit is a collection of tactics for mechanizing “diagram chasing”. We selected Nuprl because the underlying type theory is constructive and one can automatically extract executable programs from the proofs and partly because of the large collection of tactics and techniques that were available to us. So far we have concentrated on proving the theorem rather than on extracting the computational content of the theorem.

Research supported in part by NSF Grant CCR-8818979.