We extend the existing HOL Light Library for Modal Systems (HOLMS) to support a modular implementation of modal reasoning within the HOL Light proof assistant. We deeply embed axiomatic calculi and relational semantics for seven normal modal logics (K, T, B, K4, S4, S5, GL) and formalise modal adequacy theorems for these systems. We then leverage those formalisations to implement a mechanism for automated reasoning via proof-search in the associated labelled sequent calculi, which we shallowly embed in HOL Light’s goal-stack mechanism. This way, we equip the general-purpose proof assistant with (semi)decision procedures for these logics that, in case of failure to construct a proof for the input formula, return a certified countermodel within the appropriate class for the logic under consideration. On the methodological side, we propose a precise measure of the modularity of our approach by systematically adopting Christopher Strachey’s distinction between ad hoc and parametric polymorphism throughout the library.

Abstract This paper presents a certified theorem prover for Grzegorczyk logic (Grz) implemented in the general-purpose proof assistant HOL Light. Our prover builds on original HOL Light formalisations of modal adequacy for Grz with respect to finite partially ordered frames, and on the standard full and faithful translation of Grz into Gödel–Löb logic (GL). This formalised embedding allows us to extend the range of modal systems supported by the HOLMS library for automated modal reasoning, and constitutes a new methodology experimented in our framework, being the first logic added to the library through a modal translation. The deductive engine performs an automated proof search in the labelled sequent calculus for GL. When the proof search on the translated formula succeeds, the system returns a HOL Light theorem certifying provability of the original Grz formula. When proof search terminates negatively, the system constructs a verified GL countermodel and thus certifies that the original formula is not provable in Grz.

Abstract We present our library for universal algebra in the UniMath framework dealing with multi-sorted signatures, their algebras and the basics for equation systems. We show how to implement term algebras over a signature without resorting to general inductive constructions (currently not allowed in UniMath) still retaining the computational nature of the definition. We prove that our single sorted ground term algebras are instances of homotopy W-types. From this perspective, the library enriches UniMath with a computationally well-behaved implementation of a class of W-types. Moreover, we give neat constructions of the univalent categories of algebras and equational algebras by using the formalism of displayed categories and show that the term algebra over a signature is the initial object of the category of algebras. Finally, we showcase the computational relevance of our work by sketching some basic examples from algebra and propositional logic.

Abstract 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 $$\textsf{G3KGL}$$ 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.

This work presents a formalized proof of modal completeness for Gödel-Löb provability logic (GL) in the HOL Light theorem prover. We describe the code we developed, and discuss some details of our implementation, focusing on our choices in structuring proofs which make essential use of the tools of HOL Light and which differ in part from the standard strategies found in main textbooks covering the topic in an informal setting. Moreover, we propose a reflection on our own experience in using this specific theorem prover for this formalization task, with an analysis of pros and cons of reasoning within and about the formal system for GL we implemented in our code.