Mizar Light for HOL Light

There are two different approaches to formalizing proofs in a computer: the procedural approach (which is the one of the HOL system) and the declarative approach (which is the one of the Mizar system). Most provers are procedural. However declarative proofs are much closer in style to informal mathematical reasoning than procedural ones. There have been attempts to put declarative interfaces on top of procedural proof assistants, like John Harrison’s Mizar mode for HOL and Markus Wenzel’s Isar mode for Isabelle. However in those cases the declarative assistant is a layer on top of the procedural basis, having a separate syntax and a different ‘feel’ from the underlying system. This paper shows that the procedural and the declarative ways of proving are related and can be integrated seamlessly. It presents an implementation of the Mizar proof language on top of HOL that consists of only 41 lines of ML. This shows how close the procedural and declarative styles of proving really are.