Solving Inequality Proofs with Large Language Models

Inequality proving, crucial across diverse scientific and mathematical fields, tests advanced reasoning skills such as discovering tight bounds and strategic theorem application. This makes it a distinct, demanding frontier for large language models (LLMs), offering insights beyond general mathematical problem-solving. Progress in this area is hampered by existing datasets that are often scarce, synthetic, or rigidly formal. We address this by proposing an informal yet verifiable task formulation, recasting inequality proving into two automatically checkable subtasks: bound estimation and relation prediction. Building on this, we release IneqMath, an expert-curated dataset of Olympiad-level inequalities, including a test set and training corpus enriched with step-wise solutions and theorem annotations. We also develop a novel LLM-as-judge evaluation framework, combining a final-answer judge with four step-wise judges designed to detect common reasoning flaws. A systematic evaluation of 29 leading LLMs on IneqMath reveals a surprising reality: even top models like o1 achieve less than 10% overall accuracy under step-wise scrutiny; this is a drop of up to 65.5% from their accuracy considering only final answer equivalence. This discrepancy exposes fragile deductive chains and a critical gap for current LLMs between merely finding an answer and constructing a rigorous proof. Scaling model size and increasing test-time computation yield limited gains in overall proof correctness. Instead, our findings highlight promising research directions such as theorem-guided reasoning and self-refinement. Code and data are available at https://ineqmath.github.io/.

Integrating Various Software Artifacts for Better LLM-based Bug Localization and Program Repair

LLMs have garnered considerable attention for their potential to streamline Automated Program Repair (APR). LLM-based approaches can either insert the correct code or directly generate patches when provided with buggy methods. However, most of LLM-based APR methods rely on a single type of software information, without fully leveraging different software artifacts. Despite this, many LLM-based approaches do not explore which specific types of information best assist in APR. Addressing this gap is crucial for advancing LLM-based APR techniques. We propose DEVLoRe to use issue content (description and message) and stack error traces to localize buggy methods, then rely on debug information in buggy methods and issue content and stack error to localize buggy lines and generate plausible patches which can pass all unit tests. The results show that while issue content is particularly effective in assisting LLMs with fault localization and program repair, different types of software artifacts complement each other. By incorporating different artifacts, DEVLoRe successfully locates 49.3% and 47.6% of single and non-single buggy methods and generates 56.0% and 14.5% plausible patches for the Defects4J v2.0 dataset, respectively. This outperforms current state-of-the-art APR methods. The source code and experimental results of this work for replication are available at https://github.com/XYZboom/DEVLoRe.