HOL Higher Order Logic denotes a family of interactive theorem proving systems using similar higher-order logics and implementation strategies. 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. As long as these functions are correctly implemented, all theorems proven in the system must be valid. As such, a large system can be built on top of a small trusted kernel. ML was originally demon slayer watch along with LCF as a meta-language for theorem proving systems; in fact, the name stands for "Meta-Language". HOL systems use variants demon slayer watch classical higher-order logicwhich has simple axiomatic foundations with few axioms and well-understood semantics.

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