A theorem prover and countermodel constructor for provability logic in HOL Light
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We introduce our implementation in HOL Light of a prototype of a general theorem prover for normal modal logics. In the present work, we start by considering Gödel-Löb provability logic (GL). The methodology consists of a shallow embedding of a labelled sequent calculus for GL whose validity relies on our formalised proof in HOL Light of modal completeness for GL w.r.t. possible world semantics, that we present in the first part of the present work. Our theorem prover for GL is thus implemented as a tactic of HOL Light that formalises the proof search in the corresponding labelled sequent calculus, and works as a decision algorithm for that logic: if the algorithm positively terminates, the tactic succeeds in producing a HOL Light theorem stating that the input formula is a theorem of GL; if the algorithm negatively terminates, the tactic extracts a model falsifying the input formula. We discuss our code for the formal proof of modal completeness, and the design of our proof search algorithm. Furthermore some examples of both interactive and automated use of the latter are shown.
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