HybridProver: Augmenting Theorem Proving with LLM-Driven Proof Synthesis and Refinement

Formal methods is pivotal for verifying the reliability of critical systems through rigorous mathematical proofs. However, its adoption is hindered by labor-intensive manual proofs and the expertise required to use theorem provers. Recent advancements in large language models (LLMs) offer new opportunities for automated theorem proving. Two promising approaches are generating tactics step by step and generating a whole proof directly with an LLM. However, existing work makes no attempt to combine the two approaches. In this work, we introduce HybridProver, a dual-model proof synthesis framework that combines tactic-based generation and whole-proof synthesis to harness the benefits of both approaches. HybridProver generates whole proof candidates for evaluation directly, then extracts proof sketches from those candidates. It then uses a tactic-based generation model that integrates automated tools to complete the sketches via stepwise refinement. We implement HybridProver for the Isabelle theorem prover and fine-tune LLMs on our optimized Isabelle datasets. Evaluation on the miniF2F dataset illustrates HybridProver's effectiveness. We achieve a 59.4% success rate on miniF2F, where the previous SOTA is 56.1%. Our ablation studies show that this SOTA result is attributable to combining whole-proof and tactic-based generation. Additionally, we show how the dataset quality, training parameters, and sampling diversity affect the final result during automated theorem proving with LLMs. All of our code, datasets, and LLMs are open source.

MIRB: Mathematical Information Retrieval Benchmark

Mathematical Information Retrieval (MIR) is the task of retrieving information from mathematical documents and plays a key role in various applications, including theorem search in mathematical libraries, answer retrieval on math forums, and premise selection in automated theorem proving. However, a unified benchmark for evaluating these diverse retrieval tasks has been lacking. In this paper, we introduce MIRB (Mathematical Information Retrieval Benchmark) to assess the MIR capabilities of retrieval models. MIRB includes four tasks: semantic statement retrieval, question-answer retrieval, premise retrieval, and formula retrieval, spanning a total of 12 datasets. We evaluate 13 retrieval models on this benchmark and analyze the challenges inherent to MIR. We hope that MIRB provides a comprehensive framework for evaluating MIR systems and helps advance the development of more effective retrieval models tailored to the mathematical domain.

Ineq-Comp: Benchmarking Human-Intuitive Compositional Reasoning in Automated Theorem Proving on Inequalities

LLM-based formal proof assistants (e.g., in Lean) hold great promise for automating mathematical discovery. But beyond syntactic correctness, do these systems truly understand mathematical structure as humans do? We investigate this question through the lens of mathematical inequalities -- a fundamental tool across many domains. While modern provers can solve basic inequalities, we probe their ability to handle human-intuitive compositionality. We introduce Ineq-Comp, a benchmark built from elementary inequalities through systematic transformations, including variable duplication, algebraic rewriting, and multi-step composition. Although these problems remain easy for humans, we find that most provers -- including Goedel, STP, and Kimina-7B -- struggle significantly. DeepSeek-Prover-V2-7B shows relative robustness -- possibly because it is trained to decompose the problems into sub-problems -- but still suffers a 20\% performance drop (pass@32). Strikingly, performance remains poor for all models even when formal proofs of the constituent parts are provided in context, revealing that the source of weakness is indeed in compositional reasoning. Our results expose a persisting gap between the generalization behavior of current AI provers and human mathematical intuition.

LLM-based Automated Theorem Proving Hinges on Scalable Synthetic Data Generation

Recent advancements in large language models (LLMs) have sparked considerable interest in automated theorem proving and a prominent line of research integrates stepwise LLM-based provers into tree search. In this paper, we introduce a novel proof-state exploration approach for training data synthesis, designed to produce diverse tactics across a wide range of intermediate proof states, thereby facilitating effective one-shot fine-tuning of LLM as the policy model. We also propose an adaptive beam size strategy, which effectively takes advantage of our data synthesis method and achieves a trade-off between exploration and exploitation during tree search. Evaluations on the MiniF2F and ProofNet benchmarks demonstrate that our method outperforms strong baselines under the stringent Pass@1 metric, attaining an average pass rate of $60.74\%$ on MiniF2F and $21.18\%$ on ProofNet. These results underscore the impact of large-scale synthetic data in advancing automated theorem proving.

MPS-Prover: Advancing Stepwise Theorem Proving by Multi-Perspective Search and Data Curation

Automated Theorem Proving (ATP) in formal languages remains a formidable challenge in AI, demanding rigorous logical deduction and navigating vast search spaces. While large language models (LLMs) have shown promising performance, existing stepwise provers often suffer from biased search guidance, leading to inefficiencies and suboptimal proof strategies. This paper introduces the Multi-Perspective Search Prover (MPS-Prover), a novel stepwise ATP system designed to overcome these limitations. MPS-Prover incorporates two key innovations: a highly effective post-training data curation strategy that prunes approximately 40% of redundant training data without sacrificing performance, and a multi-perspective tree search mechanism. This search integrates a learned critic model with strategically designed heuristic rules to diversify tactic selection, prevent getting trapped in unproductive states, and enhance search robustness. Extensive evaluations demonstrate that MPS-Prover achieves state-of-the-art performance on multiple challenging benchmarks, including miniF2F and ProofNet, outperforming prior 7B parameter models. Furthermore, our analyses reveal that MPS-Prover generates significantly shorter and more diverse proofs compared to existing stepwise and whole-proof methods, highlighting its efficiency and efficacy. Our work advances the capabilities of LLM-based formal reasoning and offers a robust framework and a comprehensive analysis for developing more powerful theorem provers.

Accelerating Machine Learning Systems via Category Theory: Applications to Spherical Attention for Gene Regulatory Networks

How do we enable artificial intelligence models to improve themselves? This is central to exponentially improving generalized artificial intelligence models, which can improve their own architecture to handle new problem domains in an efficient manner that leverages the latest hardware. However, current automated compilation methods are poor, and efficient algorithms require years of human development. In this paper, we use neural circuit diagrams, based in category theory, to prove a general theorem related to deep learning algorithms, guide the development of a novel attention algorithm catered to the domain of gene regulatory networks, and produce a corresponding efficient kernel. The algorithm we propose, spherical attention, shows that neural circuit diagrams enable a principled and systematic method for reasoning about deep learning architectures and providing high-performance code. By replacing SoftMax with an $L^2$ norm as suggested by diagrams, it overcomes the special function unit bottleneck of standard attention while retaining the streaming property essential to high-performance. Our diagrammatically derived \textit{FlashSign} kernel achieves comparable performance to the state-of-the-art, fine-tuned FlashAttention algorithm on an A100, and $3.6\times$ the performance of PyTorch. Overall, this investigation shows neural circuit diagrams' suitability as a high-level framework for the automated development of efficient, novel artificial intelligence architectures.

APOLLO: Automated LLM and Lean Collaboration for Advanced Formal Reasoning

Formal reasoning and automated theorem proving constitute a challenging subfield of machine learning, in which machines are tasked with proving mathematical theorems using formal languages like Lean. A formal verification system can check whether a formal proof is correct or not almost instantaneously, but generating a completely correct formal proof with large language models (LLMs) remains a formidable task. The usual approach in the literature is to prompt the LLM many times (up to several thousands) until one of the generated proofs passes the verification system. In this work, we present APOLLO (Automated PrOof repair via LLM and Lean cOllaboration), a modular, model-agnostic pipeline that combines the strengths of the Lean compiler with an LLM's reasoning abilities to achieve better proof-generation results at a low sampling budget. Apollo directs a fully automated process in which the LLM generates proofs for theorems, a set of agents analyze the proofs, fix the syntax errors, identify the mistakes in the proofs using Lean, isolate failing sub-lemmas, utilize automated solvers, and invoke an LLM on each remaining goal with a low top-K budget. The repaired sub-proofs are recombined and reverified, iterating up to a user-controlled maximum number of attempts. On the miniF2F benchmark, we establish a new state-of-the-art accuracy of 75.0% among 7B-parameter models while keeping the sampling budget below one thousand. Moreover, Apollo raises the state-of-the-art accuracy for Goedel-Prover-SFT to 65.6% while cutting sample complexity from 25,600 to a few hundred. General-purpose models (o3-mini, o4-mini) jump from 3-7% to over 40% accuracy. Our results demonstrate that targeted, compiler-guided repair of LLM outputs yields dramatic gains in both efficiency and correctness, suggesting a general paradigm for scalable automated theorem proving.

Proceedings The 13th International Workshop on Theorem proving components for Educational software

The ThEdu series pursues the smooth transition from an intuitive way of doing mathematics at secondary school to a more formal approach to the subject in STEM education while favoring software support for this transition by exploiting the power of theorem-proving technologies. What follows is a brief description of how the present volume contributes to this enterprise. The 13th International Workshop on Theorem Proving Components for Educational Software (ThEdu'24), was a satellite event of the CADE29, part of IJCAR 2024, Nancy, France. ThEdu'24 was a vibrant workshop, with one invited talk by Jeremy Avigad (Carnegie Mellon University) and 14 submitted talks. An open call for papers was then issued and attracted 9 submissions. Eight of those submissions have been accepted by our reviewers. The resulting revised papers are collected in the present volume. The contributions in this volume are a faithful representation of the wide spectrum of ThEdu, ranging from those more focused on the automated deduction research, not losing track of the possible applications in an educational setting, to those focused on the applications, in educational settings, of automated deduction tools and methods. We, the volume editors, hope that this collection of papers will further promote the development of theorem-proving-based software and that it will allow to improve the mutual understanding between computer scientists, mathematicians, and stakeholders in education. While this volume goes to press, the next edition of the ThEdu workshop is being prepared: ThEdu'25 will be a satellite event of the 30th international Conference on Automated DEduction (CADE-30), July 28th - August 2nd, 2025, Stuttgart, Germany.

APE-Bench I: Towards File-level Automated Proof Engineering of Formal Math Libraries

Recent progress in large language models (LLMs) has shown promise in formal theorem proving, yet existing benchmarks remain limited to isolated, static proof tasks, failing to capture the iterative, engineering-intensive workflows of real-world formal mathematics libraries. Motivated by analogous advances in software engineering, we introduce the paradigm of Automated Proof Engineering (APE), which aims to automate proof engineering tasks such as feature addition, proof refactoring, and bug fixing using LLMs. To facilitate research in this direction, we present APE-Bench I, the first realistic benchmark built from real-world commit histories of Mathlib4, featuring diverse file-level tasks described in natural language and verified via a hybrid approach combining the Lean compiler and LLM-as-a-Judge. We further develop Eleanstic, a scalable parallel verification infrastructure optimized for proof checking across multiple versions of Mathlib. Empirical results on state-of-the-art LLMs demonstrate strong performance on localized edits but substantial degradation on handling complex proof engineering. This work lays the foundation for developing agentic workflows in proof engineering, with future benchmarks targeting multi-file coordination, project-scale verification, and autonomous agents capable of planning, editing, and repairing formal libraries.

Neural Theorem Proving: Generating and Structuring Proofs for Formal Verification

Formally verifying properties of software code has been a highly desirable task, especially with the emergence of LLM-generated code. In the same vein, they provide an interesting avenue for the exploration of formal verification and mechanistic interpretability. Since the introduction of code-specific models, despite their successes in generating code in Lean4 and Isabelle, the task of generalized theorem proving still remains far from being fully solved and will be a benchmark for reasoning capability in LLMs. In this work, we introduce a framework that generates whole proofs in a formal language to be used within systems that utilize the power of built-in tactics and off-the-shelf automated theorem provers. Our framework includes 3 components: generating natural language statements of the code to be verified, an LLM that generates formal proofs for the given statement, and a module employing heuristics for building the final proof. To train the LLM, we employ a 2-stage fine-tuning process, where we first use SFT-based training to enable the model to generate syntactically correct Isabelle code and then RL-based training that encourages the model to generate proofs verified by a theorem prover. We validate our framework using the miniF2F-test benchmark and the Isabelle proof assistant and design a use case to verify the correctness of the AWS S3 bucket access policy code. We also curate a dataset based on the FVEL\textsubscript{\textnormal{ER}} dataset for future training tasks.