The HOL system is a mechanized proof assistant for higher order logic that has been under continuous development since the mid-1980s, by an ever-changing group of developers and external contributors. We give a brief overview of various implementations of the HOL logic before focusing on the evolution of certain important features available in a recent implementation. We also illustrate how the module system of Standard ML provided security and modularity in the construction of the HOL kernel, as well as serving in a separate capacity as a useful representation medium for persistent, hierarchical logical theories.

We identify that a useful inductive data type ty with negative occurrences like ty!bool in the arguments of its constructors can have a set-theoretic interpretation when the negative occurrence models only fnite sets. Subsequently, we show how such data types can be manually added to higher order logic using equivalence sets.