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We introduce our implementation in HOL Light of the metatheory forG\"odel-L\"ob provability logic (GL), covering soundness and completenessw.r.t. possible world semantics and featuring a prototype of a theorem proverfor GL itself. The strategy we develop here to formalise the modal completenessproof overcomes the technical difficulty due to the non-compactness of GL andis an adaptation -- according to the formal language and tools at hand -- ofthe proof given in George Boolos' 1995 monograph. Our theorem prover for GLrelies then on this formalisation, is implemented as a tactic of HOL Light thatmimics the proof search in the labelled sequent calculus G3KGL, and works as adecision algorithm for the provability logic: if the algorithm positivelyterminates, the tactic succeeds in producing a HOL Light theorem stating thatthe input formula is a theorem of GL; if the algorithm negatively terminates,the tactic extracts a model falsifying the input formula. We discuss our codefor the formal proof of modal completeness and the design of our proof searchalgorithm. Furthermore, we propose some examples of the latter's interactiveand automated use.

Mechanising G\"odel-L\"ob provability logic in HOL Light