Systems in this family follow the LCF approach as they are implemented as a library which defines an abstract data type of proven theorems such that new objects of this type can only be created using the functions in the library which correspond to inference rules in higher-order logic.
HOL systems use variants of classical higher-order logic, which has simple axiomatic foundations with few axioms and well-understood semantics.
A number of HOL systems (sharing essentially the same logic) remain active and in use: The CakeML project developed a formally proven compiler for ML.
[8] Previously, HOL was used to develop a formally proven Lisp implementation running on ARM, x86 and PowerPC.
[9] HOL was also used to formalize the semantics of x86 multiprocessors[10] as well as the machine code for Power ISA and ARM architectures.