A General Formalised Framework for Reasoning About Display Calculi - Studia Logica

We encode Belnap’s basic theory of display calculi in the proof assistant Coq/Rocq version 8.18.0 and formalise the proof that Belnap’s conditions C2–C8 imply the cut-elimination theorem. Our framework allows us to formally prove meta-theoretic results such as Hilbert-completeness and derivability of explicit rules, such as cut, but also others if required. What makes our formalisation powerful is that it works entirely with an abstraction that can be instantiated to many possible logics and display calculi, although we make no attempt to precisely characterise the logics that can be properly displayed in our formalisation. For this reason, our work can be seen as a formalisation of a display calculus framework and a cut-elimination theorem for a wide range of display calculi. As examples, we apply our work to classical propositional logic (CPL), tense logic (Kt), and non-commutative non-associative Lambek calculus to obtain complete display calculi as well as formally proved cut-elimination theorems, but we believe our formalisation to be also applicable to many others. We also encoded a proof of the decidability of CPL that relies only on the structure of proofs within an additive display calculus for CPL. To our knowledge, our work is the first formalisation of a decidability result in display calculus. Moreover, all of our formal proofs are constructive as they never require the addition of the law of excluded middle in Coq’s environment (which is constructive by default), which means we were also able to extract computer programs from our proof of cut-elimination able to convert CPL/Kt/Lambek derivation trees into cut-free ones. As formal proofs are known to be significantly longer to write than usual pen-and-paper proofs, our work led to the development of a multitude of files of Coq code comprising more than 15,000 lines.

Greg Restall (@[email protected])

I’m looking forward to spending time today with @[email protected], @[email protected] and other folks at the LFCS at Edinburgh, and getting to talk about some weird substructural modal logic. https://consequently.org/presentation/2026/tlmh-edi/ #logic #prooftheory

Thoroughly Modal Hypersequents — consequently.org

LFCS Seminar: Tuesday 5th May: Greg Restall | LFCS | School of Informatics

Greg Restall University of St Andrews https://consequently.org

Intelligent Machinery : Free Download, Borrow, and Streaming : Internet Archive

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Cobblestone: A Divide-and-Conquer Approach for Automating Formal Verification

Formal verification using proof assistants, such as Coq, is an effective way of improving software quality, but requires significant effort and expertise. Machine learning can automatically synthesize proofs, but such tools are able to prove only a fraction of desired software properties. We introduce Cobblestone, a divide-and-conquer approach for proof synthesis. Cobblestone uses a large language model (LLM) to generate potential proofs, uses those proofs to break the problem into simpler parts, automatically identifies which of those parts were successfully proven, and iterates on the remaining parts to build a correct proof that is guaranteed to be sound, despite the reliance on unsound LLMs. We evaluate Cobblestone on four benchmarks of open-source Coq projects, controlling for training data leakage. Fully automatically, Cobblestone outperforms state-of-the-art non-LLM tools, and proves many theorems that other LLM-based tools cannot, and on many benchmarks, outperforms them. Each Cobblestone run costs only $1.25 and takes 14.7 minutes, on average. Cobblestone can also be used with external input, from a user or another tool, providing a proof structure or relevant lemmas. Evaluated with such an oracle, Cobblestone proves up to 58% of theorems. Overall, our research shows that tools can make use of partial progress and external input to more effectively automate formal verification.

Greg Restall (@[email protected])

Attached: 1 image Tomorrow, I get to give the last of my three talks on inferentialism. It’s time to buckle up your λs, and join in the search for some unicorns… https://consequently.org/presentation/2025/whl-a/ #prooftheory #semantics #linguistics