Ruobing Zuo

Ruobing Zuo, Hanrui Zhao, Gaolei He et al.

Automated proving of polynomial inequalities is a fundamental challenge in automated mathematical reasoning, where rich algebraic structure and a rapidly growing certificate search space hinder scalability. Purely symbolic approaches provide strong guarantees but often scale poorly as the number of variables or the degree increases, due to expensive algebraic manipulations and rapidly growing intermediate expressions. In parallel, LLM-guided methods have made notable progress, particularly on competition-style inequalities with a small number of variables. To address the remaining scalability challenges, we propose NSPI, a neuro-symbolic framework that combines the complementary strengths of LLMs and symbolic computation for polynomial-inequality proving. Concretely, an LLM proposes a conjecture in the form of an approximate polynomial Sum-Of-Squares (SOS) decomposition; we refine it via symbolic computation to obtain an exact polynomial SOS representation, which directly proves the target inequality, and we further certify the proof in Lean, yielding an end-to-end pipeline from heuristic discovery to machine-checked proof. Experiments on challenging benchmarks involving polynomials with up to 10 variables demonstrate the effectiveness and scalability of the proposed method.

Shuming Shi, Ruobing Zuo, Gaolei He et al.

Automated theorem proving (ATP) is one of the most challenging mathematical reasoning tasks for Large Language Models (LLMs). Most existing LLM-based ATP methods rely on supervised fine-tuning, which results in a limited alignment between the theorem proving process and human preferences. Direct Preference Optimization (DPO), which aligns LLMs with human preferences, has shown positive effects for certain tasks. However, the lack of high-quality preference data for theorem proving presents a significant challenge. In this paper, we innovatively apply DPO to formal automated theorem proving and introduces a Curriculum Learning-based DPO Iterative Theorem Proving (CuDIP) method. Specifically, we propose a method for constructing preference data which utilizes LLMs and existing theorem proving data to enhance the diversity of the preference data while reducing the reliance on human preference annotations. We then integrate this preference data construction method with curriculum learning to iteratively fine-tune the theorem proving model through DPO. Experimental results on the MiniF2F and ProofNet datasets demonstrate the effectiveness of the proposed method.