We introduce our implementation in HOL Light of the metatheory for Gödel–Löb provability logic (GL), covering soundness and completeness w.r.t. possible world semantics and featuring a prototype of a theorem prover for GL itself. The strategy we develop here to formalise the modal completeness proof overcomes the technical difficulty due to the non-compactness of GL and is an adaptation—according to the formal language and tools at hand—of the proof given in George Boolos’ 1995 monograph. Our theorem prover for GL relies then on this formalisation, is implemented as a tactic of HOL Light that mimics the proof search in the labelled sequent calculus \(\textsf{G3KGL}\), and works as a decision algorithm for the provability 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, we propose some examples of the latter’s interactive and automated use.

我们在 HOL Light 中介绍了哥德尔-勒布可证明性逻辑 (GL) 元理论的实现，涵盖了可能的世界语义的健全性和完整性，并以 GL 本身的定理证明者原型为特色。我们在这里开发的形式化模态完整性证明的策略克服了由于 GL 的非紧性而导致的技术困难，并且是根据现有的形式语言和工具对 George Boolos 1995 年专着中给出的证明进行的改编。我们的 GL 定理证明器依赖于这种形式化，被实现为 HOL Light 的策略，模仿标记序列微积分中的证明搜索\(\textsf{G3KGL}\)，并用作可证明性逻辑的决策算法：如果算法肯定终止，则该策略成功产生 HOL Light 定理，表明输入公式是 GL 的定理；如果算法否定终止，则该策略会提取伪造输入公式的模型。我们讨论模态完整性的形式证明代码以及证明搜索算法的设计。此外，我们提出了后者的交互式和自动化使用的一些示例。