Title: IneqSearch: Hybrid Reasoning for Olympiad Inequality Proofs
Venue: NeurIPS 2025
Accepted: yes
Keywords: Olympiad Inequality Proving

Neuro-Symbolic Method
Abstract: Mathematicians have long employed decomposition techniques to prove inequalities, yet automating this process remains a significant challenge in computational mathematics. We introduce IneqSearch, a hybrid reasoning system that integrates symbolic computation with large language models (LLMs) to address this challenge. IneqSearch reformulates inequality proving as a structured search problem: identifying appropriate combinations of theorems that decompose expressions into non-negative components. The system combines a symbolic solver for deductive reasoning with an LLM-based agent for constructive proof exploration, effectively implementing methodologies observed in formal mathematical practice. A key contribution of IneqSearch is its iterative learning mechanism that systematically incorporates newly proven results into its theorem database, enabling knowledge acquisition during practice that enhances its capabilities without requiring human intervention. In empirical evaluation on 437 Olympiad-level inequalities, IneqSearch successfully proves 342 problems, significantly outperforming existing methods and demonstrating the effectiveness of integrating symbolic and neural approaches for mathematical reasoning.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents IneqSearch, an inequality prover that utilizes decomposition techniques. Specifically, IneqSearch employs a symbolic solver to identify which theorems are involved in the proof and leverages Large Language Models (LLMs) to instantiate and establish a set of candidate theorems. Additionally, an iterative learning mechanism can be seamlessly integrated into the framework, allowing the theorem library to evolve automatically. Experimental results demonstrate that the proposed method achieves a significant improvement on two Olympiad-level benchmarks.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n- The decomposition-based framework, along with a neuro-symbolic methodology for inequality proving, shows great promise.\n- The framework incorporates an iterative learning strategy, offering an intriguing approach for the self-evolution of the neuro-symbolic system.\n- It demonstrates strong performance on competition-level benchmarks, indicating a promising direction for automatic inequality proving.\n\n### Weaknesses\n- Although the proposed method is extremely effective, it appears to be limited to the domain of inequality proving. It would be beneficial to discuss how this method could inspire advancements in other related fields, such as general mathematical reasoning or mathematical theorem proving.\n- The decomposition-based proving approach, particularly the SOS technique, generates proofs that lack human readability and explicitness. This limitation hinders the method's effectiveness as an educational tool for providing answers and affects its performance when used as training data to enhance models.\n- Evaluating the method on additional benchmarks could further demonstrate its effectiveness. For example, the 567Neq benchmark introduced in LIPS would be a valuable test case.\n- The ablation study indicates that the current iterative learning mechanism has limitations, as it only successfully solves eight additional problems when integrated into the method.\n\nQUESTIONS:\n- The authors mention that the natural language solutions generated by LLMs are manually checked by experts. What are the criteria used for evaluating these solutions? \n- The SOS and theorem searcher incorporate numerical optimization methods for semi-definite programming and linear programming. How is the discrepancy between numerical results and symbolic results addressed? For instance, if $λ_i$ is determined to be $1.414$, will it be converted to $\\sqrt{2}$?\n- Figure 1 lacks clarity; it would be beneficial to enhance the illustration of the workflow.\n- More experimental details are needed. For example, what is the average solving time? What prompts were used for generating natural language solutions? Additionally, will the code be made open-source in the end?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces IneqSearch, a hybrid reasoning system for automated proofs of multivariate polynomial inequalities common in math Olympiads. The symbolic core first normalizes the problem, then attempts to rewrite the inequality as a sum of elementary cyclic polynomials (ECPD) or a sum of squares (SOS). When this direct route fails, an LLM agent suggests transformations derived from standard lemmas (e.g., Cauchy-Schwarz) that create alternate but equivalent targets. A linear-programming search picks coefficients that express the transformed inequality as a non-negative combination of known theorems. Every successful proof is logged, checked, and appended to the theorem base, letting the system solve progressively harder problems in subsequent rounds.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n•\tStrong empirical results (however, it needs to be evaluated on more datasets): IneqSearch achieves a 78.3% success rate on INEQ-437 and solves 100% of the MO-INT-20 dataset, outperforming both pure LLM-based systems and prior hybrid methods like AIPS and LIPS.\n\n•\tComprehensive evaluation, including ablation studies demonstrating the contribution of each system component (symbolic solver, inductive agent, iterative learning).\n\n•\tAblation study demonstrating the contribution of each system component (symbolic solver, inductive agent, iterative learning).\n\nWeaknesses:\n\n•\tInconsistent model coverage in Table 1. All three settings (natural-language, Lean 4, and formal verification) should be benchmarked on the same set of LLMs, yet the formal column reports results for only two models, one of which (“4o-mini”) never appears in the natural-language experiment, making direct cross-modal comparisons impossible.\n\n•\tLack of open-weight baselines. The study omits freely available models, hindering reproducibility and preventing the community from validating or extending the findings without paid API access. (also using deterministic sampling or setting a seed)\n\n•\tOrder-dependent accuracy. Reported success rates assume the exact sequence of previously posed questions; performance could drop under a different interaction history, but this sensitivity is neither analyzed nor controlled.\n\n\n•\tMissing comparison with retrieval-augmented generation (RAG). Given that the task involves answering from a fixed corpus, a RAG baseline is essential to demonstrate the advantage of the proposed method over standard retrieval + generation pipelines.\n\n•\tUnavailable code and data. The appendix does not include the implementation or evaluation scripts, so the reported results cannot be reproduced or independently verified.\n\n•\tWhile the iterative learning mechanism is promising, its scalability to larger or more diverse domains is untested; results are currently limited to algebraic inequalities within Olympiad settings.\n\n\n•\tNo analysis of failure modes by problem category (e.g. symmetric vs asymmetric inequalities, degree complexity), which would provide deeper insights for model improvement.\n\n•\tLack of statistical significance testing or variance reporting for experimental results, though the absolute performance gains are substantial.\n\n•\tCrucial implementation details, such as the LP search procedure, SDP solver configurations, and LLM prompting strategies, are missing, making it difficult for readers to reproduce or build upon the work. Additionally, the dataset used is limited.\n\nQuality:\n\nThe paper presents a well-motivated methodology grounded in mathematical practice, with good empirical validation and rigorous decomposition algorithms.\n\nClarity:\n\nThe writing is generally clear and well-organized. Diagrams (e.g. Figures 1–3) effectively illustrate the pipeline and proof construction. \n\nSignificance:\n\nThis work is significant for automated mathematical reasoning, particularly in algebraic inequality proofs, an area of interest for both AI for mathematics and AGI evaluation benchmarks. The hybrid design could inspire extensions to broader formal domains.\n\nOriginality:\n\nThe paper frames inequality proving as a decomposition-based search integrated with LLM-guided inductive exploration, and in its self-improving theorem acquisition paradigm to enrich the database of theorems.\n\nQUESTIONS:\n•\tThe solver currently targets multivariate polynomial inequalities in positive variables. What prevents it from handling rational functions, sign-changing variables, or transcendental terms?\n\n•\tHave you identified theoretical or practical obstacles to extending the approach beyond Olympiad-style problems?\n\n\n•\tHow sensitive is performance to the ordering in which new theorems are added?\n\n•\tWhat are the main failure modes of the system (e.g. failure to find decompositions, incorrect inductive transformations), and how might these be mitigated in future work?\n\n•\tHow large is the final theorem database after training, and what are its computational or memory implications for scaling to broader problem sets?\n\n•\tAre there plans to release the code and trained theorem database to enable reproducibility and benchmarking by the community?\n\nLIMITATIONS:\n•\tEvaluation is restricted to polynomial inequalities with positive variables under Olympiad assumptions; generality to broader mathematical domains is untested.\n\n•\tIterative learning risks compounding errors if incorrect proofs are integrated into the database; the safeguards to prevent this are not fully detailed.\n\n•\tCurrent symbolic solver methods may not generalize to inequalities involving transcendental or higher-order functions.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces IneqSearch, a novel hybrid reasoning system designed to automatically prove complex Olympiad-level inequalities. Similar to LIPS system in ICLR'25, this paper combines symbolic computation for rigorous deductive reasoning with large language models (LLMs) for innovative, constructive proof exploration. A key feature of IneqSearch is its iterative learning mechanism, allowing it to continuously expand its theorem database with newly proven results, thereby enhancing its capabilities over time without human intervention. Experimental results demonstrate that IneqSearch significantly outperforms existing methods by successfully solving a high percentage of challenging inequality problems, showcasing the power of integrating symbolic and neural approaches for mathematical reasoning.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n*   Quality & Significance: The paper introduces IneqSearch, a hybrid reasoning system that significantly outperforms existing methods on Olympiad-level inequalities. It successfully proves 342 out of 437 problems and achieves a 100% success rate on the challenging MO-INT-20 dataset. The system's strong performance on high-difficulty problems, which often require multi-step scaling and are non-symmetric or non-homogeneous, highlights its effectiveness where symbolic solvers and LLMs typically fail. An ablation study clearly demonstrates the complementary nature and individual contributions of its components, showing that each added feature boosts performance.\n*   Originality & Significance: IneqSearch's core strength and innovation lies in the ECPD and SOS methods. It reformulates inequality proving as a structured search problem focused on decomposing expressions into non-negative components, a strategy inspired by formal mathematical practice, which further enables effective iterative learning mechanism that autonomously incorporates newly proven results into its theorem database, allowing for continuous knowledge acquisition and enhanced capabilities without human intervention. \n\nWeaknesses:\n\n*   Clarity & Quality (Formal Proof System & Soundness): While the paper states that IneqSearch \"ensures soundness by constructing verifiable proofs\" and that \"every theorem in T has been rigorously validated, guaranteeing the system’s soundness\", the specific details of its formal proof system and verification mechanism are not fully elaborated in the provided excerpts. This leaves ambiguity regarding how its correctness is guaranteed compared to systems like Lean, which have automatically verifiable formal proofs.\n*   Clarity & Quality (Computational Budget & Comparison Fairness): The paper mentions that \"No GPU was used in the experiments\" and \"All computations can be executed on a standard personal computer\". However, key details regarding the computational complexity and exact runtime of the decomposition algorithms like ECPD and SOS are not clearly specified. Without these details, it is difficult to fully assess the fairness of comparisons with other systems, such as LIPS, especially concerning the computational budget required for the main results. The paper also states that the code is not yet ready for publication.\n*   Clarity & Quality (Trial-and-Error Process): The inductive agent proposes \"multiple possible choices at each step\", implying a trial-and-error approach. However, the paper does not explicitly specify the technical details of this trial-and-error process, including any backtracking mechanisms or how the cost associated with failed attempts is factored into the reported computational budget or performance metrics.\n*   Clarity (Theoretical Scope of Decomposition): While the system effectively uses ECPD and SOS for polynomial non-negativity and preprocessing converts fractional expressions to numerator-only forms, the paper does not clearly specify the theoretical completeness of these methods for *all* types of inequalities, particularly in relation to the broader mathematical context of Hilbert's Seventeenth Problem, which deals with *rational functions*. The system primarily operates on polynomials after preprocessing, which may limit its applicability to certain classes of inequalities compared to a universal theoretical claim.\n*   Clarity (Limited Demonstrated Impact and Justification of Iterative Learning's Contribution, particularly ECPD's Role):\nThe paper identifies iterative learning as a distinguishing feature and a key contribution, enhancing IneqSearch's proving capability and scalability by autonomously incorporating solved inequalities into its knowledge base. However, the ablation study in Table 2 shows that the \"Iterative Learning\" mechanism is responsible for solving only an additional 8 problems (from 334 to 342). This comparatively modest gain raises questions regarding the practical significance of this component's overall impact. Furthermore, while Elementary Cyclic Polynomial Decomposition (ECPD) is mentioned as a core method for \"representing and manipulating expressions\" in inequality proofs, its specific and unique contribution to *enhancing iterative learning* (beyond its direct application in the solver) is not clearly demonstrated to be substantial or to significantly overcome limitations that other research (e.g., [1]) might suggest for symbolic library learning. \n\n[1] Berlot-Attwell, Ian, Frank Rudzicz, and Xujie Si. \"LLM Library Learning Fails: A LEGO-Prover Case Study.\" arXiv preprint arXiv:2504.03048 (2025).\n\nQUESTIONS:\n1.  Formal Proof System Details and Soundness: Please elaborate on the specific formal proof system and verification mechanism used by IneqSearch to \"ensure soundness by constructing verifiable proofs\" and guarantee that \"every theorem in T has been rigorously validated\". How does this internal verification process compare in rigor and automation to the \"automatically verified\" correctness of formal proofs in Lean4 mentioned for baseline solutions?\n    *   Criteria for Score Increase: A clear and detailed explanation of the underlying formal logic or automated verification framework (beyond just human readability of steps) that guarantees the soundness of IneqSearch's proofs would significantly increase the confidence in the system's fundamental \"Quality\" and \"Clarity\".\n\n2.  Computational Budget and Component Performance: What are the typical computational complexity and runtime (latency) characteristics of the Elementary Cyclic Polynomial Decomposition (ECPD) and Sum of Squares (SOS) decomposition algorithms, especially considering SOS is formulated as a semidefinite programming problem? Furthermore, what is the maximum or average time budget allowed per problem for IneqSearch to achieve the state-of-the-art results presented in Table 1? For instance, a comparison point like LIPS specifies a maximum of 1.5 hours per problem.\n    *   Criteria for Score Increase: Providing quantitative details on the computational cost of these core symbolic components and a clear time budget per problem for the reported results would greatly enhance the \"Clarity\" and \"Quality\" of the experimental evaluation, allowing for a more fair and comprehensive comparison of computational resources with other systems.\n\n3.  LLM Error Handling, Backtracking, and Strategy Management: Given that Large Language Models (LLMs) can exhibit \"critical consistency issues\" and \"hallucinations\", how does IneqSearch robustly manage errors introduced by the LLM's \"constructive proof exploration\"? Specifically, what are the mechanisms for backtracking when an LLM-proposed transformation or exploration direction (e.g., \"Inequality Adaption\" or \"Target Shift\" in Figure 2) proves incorrect or unprovable, and how does the system balance or prioritize these various strategies during the iterative trial-and-error process?\n    *   Criteria for Score Increase: A detailed explanation of the system's error recovery, backtracking, and adaptive strategy selection mechanisms would significantly improve the understanding of its \"Quality\" and robustness in handling the inherent unpredictability of LLM-driven exploration, which is crucial for its overall effectiveness as a hybrid system.\n\n4.  Unique Contribution of ECPD to Iterative Learning: Given that Table 2 shows a relatively modest gain (8 problems) from the 'Iterative Learning' component, and external research [1] suggests limitations in LLM-assisted symbolic library learning, could the authors elaborate on the *unique contribution* of ECPD within the iterative learning mechanism? Specifically, does ECPD's role extend beyond initial problem-solving to enhance the *quality* or *reusability* of theorems for subsequent learning, and are there further ablation studies that could demonstrate this?\n    *   Criteria for Score Increase: A clear explanation or additional experimental evidence that elucidates how ECPD, as a symbolic decomposition method, specifically contributes to the robustness, generalizability, or practical utility of the theorems acquired through iterative learning, thereby differentiating IneqSearch's approach from limitations observed in other LLM-assisted symbolic library learning studies, would enhance the understanding of its \"Originality & Significance\" and \"Quality\".\n\n[1] Berlot-Attwell, Ian, Frank Rudzicz, and Xujie Si. \"LLM Library Learning Fails: A LEGO-Prover Case Study.\" arXiv preprint arXiv:2504.03048 (2025).\n\nLIMITATIONS:\nI generally like this paper, especially on its significant results, the introduction of ECPD, and the surprising effect of iterative learning mechanism. All my concerns are mainly around the clarity of this paper, and I hope this can be addressed in the future version.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduce IneqSearch, a neuro-symbolic system, automates the proof of Olympiad inequalities using symbolic computation and LLM-based heuristic construction. The system decomposes the target inequality into a sum of non-negative expressions, which are either established theorems in a database or constructed by LLMs. Proved inequalities are then added to the database, leading to a self-improving system. The system’s performance is evaluated on two datasets, demonstrating its significant superiority over existing baselines.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n+ The task of proving Olympiad inequalities addressed by the paper is challenging.\n+ The proposed approach is mathematically well-founded.\n+ The paper collects a large dataset of Olympiad inequalities, which is valuable for further development.\n+ The performance of the experiment is impressive, significantly surpassing existing baselines.\n\nWeaknesses:\n- The design of the LLM-based inductive agent is not adequately presented. Given that the agent is a crucial component of this neuro-symbolic system, the paper should provide more detailed information about the agent, including the prompt and some example inputs.\n- In addition to the absence of agent information, the paper requires further justification for the design of the agent. As evidenced in Table 2, the performance of the symbolic component alone is already impressive. The combination of ECPD and SOS effectively resolves 257 problems, closely approximating the performance of LIPS, which addresses 273 problems. Without additional justification, it appears that the majority of the improvement is attributed to the symbolic component, which is limited to inequality proving problems. As a result, its influence on broader neuro-symbolic systems may be reduced.\n\nQUESTIONS:\n- It is unclear to me how the generated proofs are verified. Does IneqSearch generate proofs in Lean4?\n- In the evaluation, other methods for comparison are evaluated for three shots. How does it count for one shot? For example, in LIPS, each attempt is allocated 90 minutes. Does one shot of LIPS take 90 minutes in the evaluation?\n\nLIMITATIONS:\nWhile the details of the agent design are absent, the potential application of paper is limited to automated inequality proving.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces IneqSearch, a hybrid system for proving Olympiad-level algebraic inequalities. A symbolic solver first tries direct decompositions and searches a growing theorem database; when that fails, an LLM-based “inductive agent” suggests transformations. Successfully proven sub-results are fed back into the database so the system “learns” over multiple passes. On the authors’ new INEQ-437 benchmark and the MO-INT-20 set, IneqSearch solves 342/437 (78 %) and 20/20 (100 %) inequalities, substantially outperforming state-of-the-art LLMs and prior hybrids such as LIPS and AIPS. The system outputs human-readable proof logs and formal Lean 4 certificates.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* Strong empirical results \n* The system has a notion of memory (theorem database) which allows for self-improvement\n\nWeaknesses:\n*  Evaluation is almost entirely on an in-house benchmark created by the authors (except MO-INT-20 which is a small set); external sets such as IneqMath are missing. \n* Heavy reliance on hand-crafted ECPD/SOS + linear programming - may not generalise.\n* Domain scope is narrow (inequality problems)\n* The approach doesn't make proper use of stronger LLMs (the improvement from 4o-mini to o3-mini is marginal)\n\nQUESTIONS:\n1. How does IneqSearch perform on publicly used inequality suites such as IneqMath? [Sheng et al., 2025] \n2. Could you quantify IneqSearch’s compute footprint (e.g., wall-clock time per problem, total GPU/CPU hours, number of LLM calls) and compare it directly to the baselines you evaluate against (LIPS, AIPS, pure-LLM solvers)?\n3. Can you provide more details on INEQ-437's construction (e.g., difficulty distribution)?\n\nLIMITATIONS:\nPartially addressed - the paper contains a dedicated “Limitations” appendix, but it skips compute-resource discussion and omits the lack of external benchmarks; these should be added. Otherwise the section is adequate.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}