Recent advances in automated theorem proving leverages language models to explore expanded search spaces by step-by-step proof generation. However, such approaches are usually based on short-sighted heuristics (e.g., log probability or value function scores) that potentially lead to suboptimal or even distracting sub-goals, preventing us from finding longer proofs. To address this challenge, we propose POETRY (PrOvE Theorems RecursivelY), which proves theorems in a recursive, level-by-level manner in the Isabelle theorem prover. Unlike previous step-by-step methods, POETRY searches for a verifiable sketch of the proof at each level and focuses on solving the current level's theorem or conjecture. Detailed proofs of intermediate conjectures within the sketch are temporarily replaced by a placeholder tactic called sorry, deferring their proofs to subsequent levels. This approach allows the theorem to be tackled incrementally by outlining the overall theorem at the first level and then solving the intermediate conjectures at deeper levels. Experiments are conducted on the miniF2F and PISA datasets and significant performance gains are observed in our POETRY approach over state-of-the-art methods. POETRY on miniF2F achieves an average proving success rate improvement of 5. 1% . Moreover, we observe a substantial increase in the maximum proof length found by POETRY, from 10 to 26 .

In theorem proving, the task of selecting useful premises from alarge library to unlock the proof of a given conjecture is crucially important. This presents a challenge foralltheorem provers,especially theonesbasedonlanguage models, due to their relative inability to reason over huge volumes of premises in text form.

Premise selection is a key bottleneck for scaling theorem proving in large formal libraries. Yet existing language-based methods often treat premises in isolation, ignoring the web of dependencies that connects them. We present a graph-augmented approach that combines dense text embeddings of Lean formalizations with graph neural networks over a heterogeneous dependency graph capturing both state-premise and premise-premise relations. On the LeanDojo Benchmark, our method outperforms the ReProver language-based baseline by over 25\% across standard retrieval metrics. These results suggest that relational information is beneficial for premise selection.

Nowadays, formal theorem provers have made monumental progress on high-school and competition-level mathematics, but few of them generalize to more advanced mathematics. In this paper, we present REAL-Prover, a new open-source stepwise theorem prover for Lean 4 to push this boundary. This prover, based on our fine-tuned large language model (REAL-Prover-v1) and integrated with a retrieval system (Leansearch-PS), notably boosts performance on solving college-level mathematics problems. To train REAL-Prover-v1, we developed HERALD-AF, a data extraction pipeline that converts natural language math problems into formal statements, and a new open-source Lean 4 interactive environment (Jixia-interactive) to facilitate synthesis data collection. In our experiments, our prover using only supervised fine-tune achieves competitive results with a 23.7% success rate (Pass@64) on the ProofNet dataset-comparable to state-of-the-art (SOTA) models. To further evaluate our approach, we introduce FATE-M, a new benchmark focused on algebraic problems, where our prover achieves a SOTA success rate of 56.7% (Pass@64).

Current state-of-the-art methods for automated Knowledge Base (KB) completion use neural link prediction models to learn distributed vector representations of symbols ( i.e. subsymbolic representations)

We present Lean Finder, a semantic search engine for Lean and mathlib that understands and aligns with the intents of mathematicians. We further align Lean Finder with mathematicians' preferences using In addition, Lean Finder is compatible with LLM-based theorem provers, bridging retrieval with formal reasoning. Advances in Lean and mathlib (De Moura et al., 2015; Moura & Ullrich, 2021) are turning mathematical discovery into a collaborative and verifiable research workflow. Despite these advances, state-of-the-art LLMs still cannot solve math research problems. Lean's syn tax, gram mar, and tac tics in cur a steep learn ing curve. All experiments and data processing were conducted outside Meta. Figure 1: In the evaluation with user queries, real users preferred Lean Finder in 81.6% of cases, compared with Consider the two queries below. Lean search engines handle (Gao et al., 2024a;b; Ju & Dong, 2025; Asher, 2025): Denote L/K a field extension, x, y in L are algebraic elements over K with the same minimal polynomial. I'm working with algebraic elements over a field extension and I have two elements, say x and y in L. I know x is algebraic over K, and I've shown that y is a root of the minimal polynomial of x. Does this imply that the minimal polynomials of x and y are actually equal? T arget Statement 2: 1 theorem eq_of_root {x y: L} (hx: IsAlgebraic K x) (h_ev: Polynomial.aeval y (minpoly K x) = 0): minpoly K y = minpoly K x):= -- proof omitted for brevity This user latent (motivation, perspective, abstraction) cannot be inferred or encoded by a purely syntactic informalization. Addressing this challenge calls for Lean search engines that can understand a mathematician's intent, not merely We defer a more rigorous analysis in Section 2.2, and ask our core question: Our approach analyzes and clusters public discussions, then synthesizes queries that simulate user intents (Section 3.1).

Recent advances in automated theorem proving leverages language models to explore expanded search spaces by step-by-step proof generation. However, such approaches are usually based on short-sighted heuristics (e.g., log probability or value function scores) that potentially lead to suboptimal or even distracting sub-goals, preventing us from finding longer proofs. To address this challenge, we propose POETRY (PrOvE Theorems RecursivelY), which proves theorems in a recursive, level-by-level manner in the Isabelle theorem prover. Unlike previous step-by-step methods, POETRY searches for a verifiable sketch of the proof at each level and focuses on solving the current level's theorem or conjecture. Detailed proofs of intermediate conjectures within the sketch are temporarily replaced by a placeholder tactic called sorry, deferring their proofs to subsequent levels. This approach allows the theorem to be tackled incrementally by outlining the overall theorem at the first level and then solving the intermediate conjectures at deeper levels. Experiments are conducted on the miniF2F and PISA datasets and significant performance gains are observed in our POETRY approach over state-of-the-art methods. POETRY on miniF2F achieves an average proving success rate improvement of 5. 1% . Moreover, we observe a substantial increase in the maximum proof length found by POETRY, from 10 to 26 .