José A. Alonso
A topological rewriting of Tarski’s mereogeometry. ~ Richard Dapoigny. https://hal.science/hal-05532641v1/file/RD_paper20381.pdf #CoqProver #ITP
Readings shared March 04, 2026. https://jaalonso.github.io/vestigium/posts/2026/03/05-readings_shared_02-04-26 #AI #AI4Math #CoqProver #FunctionalProgramming #Haskell #ITP #LeanProver #Math #RocqProver
A formally verified constructive proof of the consistency of Peano arithmetic using ordinal assignments. ~ Aaron Bryce, Rajeev Goré. https://arxiv.org/abs/2603.00487v1 #CoqProver #ITP #Math
DRAFT: A Formally Verified Constructive Proof of the Consistency of Peano Arithmetic Using Ordinal Assignments
Gentzen's 1936 proof of the consistency of Peano Arithmetic was a significant result in the foundations of mathematics. We provide here a modified version of the proof, based on Gödel's reformulation, and including additional details and minor corrections which are necessary to definitively prove the well-foundedness of the cut-elimination argument in a constructive environment. All results have been verified using the Coq theorem prover. NOTE TO READERS 26 February 2026: this is a draft which we had intended to submit to the Journal of Automated Reasoning with no particular time-line in our minds as the work was completed as part of Aaron's honours project at ANU in 2023. For that reason, we have used the Springer style files. We are putting it on arxiv as there appears to be some interest in this work as indicated by a post to https://proofassistants.stackexchange.com/questions/6462/how-far-is-gentzens-consistency-proof-of-peano-arithmetic-from-being-formalized in early February 2026. The Coq code is available here: https://github.com/aarondroidbryce/Gentzen/tree/master
Readings shared February 24, 2026. https://jaalonso.github.io/vestigium/posts/2026/02/25-readings_shared_02-24-26 #AI4Math #ATP #Agda #CoqProver #FunctionalProgramming #Haskell #ITP #IsabelleHOL #LLMs #LeanProver #Math #Reasoning #Vampire
Verifying the hashgraph consensus algorithm. ~ Karl Crary. https://arxiv.org/abs/2102.01167v1 #CoqProver #ITP
Verifying the Hashgraph Consensus Algorithm
The Hashgraph consensus algorithm is an algorithm for asynchronous Byzantine fault tolerance intended for distributed shared ledgers. Its main distinguishing characteristic is it achieves consensus without exchanging any extra messages; each participant's votes can be determined from public information, so votes need not be transmitted. In this paper, we discuss our experience formalizing the Hashgraph algorithm and its correctness proof using the Coq proof assistant. The paper is self-contained; it includes a complete discussion of the algorithm and its correctness argument in English.
Readings shared February 14, 2026. https://jaalonso.github.io/vestigium/posts/2026/02/15-readings_shared_02-14-26 #AI4Math #Agda #Clojure #CoqProver #FunctionalProgramming #Haskell #ITP #LeanProver #Lisp #Logic #Math #Minlog #Prolog #RustLang
Mechanized undecidability of higher-order beta-matching. ~ Andrej Dudenhefner. https://arxiv.org/abs/2602.02091 #ITP #CoqProver
Mechanized Undecidability of Higher-order beta-Matching (Extended Version)
Higher-order beta-matching is the following decision problem: given two simply typed lambda-terms, can the first term be instantiated to be beta-equivalent to the second term? This problem was formulated by Huet in the 1970s and shown undecidable by Loader in 2003 by reduction from lambda-definability. The present work provides a novel undecidability proof for higher-order beta-matching, in an effort to verify this result by means of a proof assistant. Rather than starting from lambda-definability, the presented proof encodes a restricted form of string rewriting as higher-order beta-matching. The particular approach is similar to Urzyczyn's undecidability result for intersection type inhabitation. The presented approach has several advantages. First, the proof is simpler to verify in full detail due to the simple form of rewriting systems, which serve as a starting point. Second, undecidability of the considered problem in string rewriting is already certified using the Coq proof assistant. As a consequence, we obtain a certified many-one reduction from the Halting Problem to higher-order beta-matching. Third, the presented approach identifies a uniform construction which shows undecidability of higher-order beta-matching, lambda-definability, and intersection type inhabitation. The presented undecidability proof is mechanized in the Coq proof assistant and contributed to the existing Coq Library of Undecidability Proofs.
Readings shared January 24, 2026. https://jaalonso.github.io/vestigium/posts/2026/01/25-readings_shared_01-24-26 #AI #AI4Math #ATP #CoqProver #HOL_Light #ITP #IsabelleHOL #LeanProver #Logic #Math #Prover9 #RocqProver
AI for mathematics: Progress, challenges, and prospects. ~ Haocheng Ju, Bin Dong. https://arxiv.org/abs/2601.13209v2 #AI #Math #AI4Math #ITP #IsabelleHOL #CoqProver #LeanProver
AI for Mathematics: Progress, Challenges, and Prospects
AI for Mathematics (AI4Math) has emerged as a distinct field that leverages machine learning to navigate mathematical landscapes historically intractable for early symbolic systems. While mid-20th-century symbolic approaches successfully automated formal logic, they faced severe scalability limitations due to the combinatorial explosion of the search space. The recent integration of data-driven approaches has revitalized this pursuit. In this review, we provide a systematic overview of AI4Math, highlighting its primary focus on developing AI models to support mathematical research. Crucially, we emphasize that this is not merely the application of AI to mathematical activities; it also encompasses the development of stronger AI systems where the rigorous nature of mathematics serves as a premier testbed for advancing general reasoning capabilities. We categorize existing research into two complementary directions: problem-specific modeling, involving the design of specialized architectures for distinct mathematical tasks, and general-purpose modeling, focusing on foundation models capable of broader reasoning, retrieval, and exploratory workflows. We conclude by discussing key challenges and prospects, advocating for AI systems that go beyond facilitating formal correctness to enabling the discovery of meaningful results and unified theories, recognizing that the true value of a proof lies in the insights and tools it offers to the broader mathematical landscape.