Large language models (LLMs) have significantly advanced formal theorem proving, yet the scarcity of high-quality training data constrains their capabilities in complex mathematical domains. Combinatorics, a cornerstone of mathematics, provides essential tools for analyzing discrete structures and solving optimization problems. However, its inherent complexity makes it particularly challenging for automated theorem proving (ATP) for combinatorial identities. To address this, we manually construct LeanComb, combinatorial identities benchmark in Lean, which is, to our knowledge, the first formalized theorem proving benchmark built for combinatorial identities. We develop an Automated Theorem Generator for Combinatorial Identities, ATG4CI, which combines candidate tactics suggested by a self-improving large language model with a Reinforcement Learning Tree Search approach for tactic prediction. By utilizing ATG4CI, we generate a LeanComb-Enhanced dataset comprising 260K combinatorial identities theorems, each with a complete formal proof in Lean, and experimental evaluations demonstrate that models trained on this dataset can generate more effective tactics, thereby improving success rates in automated theorem proving for combinatorial identities.

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.

Ultrasound-guided nerve block anesthesia (UGNB) is a high-tech visual nerve block anesthesia method that can observe the target nerve and its surrounding structures, the puncture needle's advancement, and local anesthetics spread in real-time. The key in UGNB is nerve identification. With the help of deep learning methods, the automatic identification or segmentation of nerves can be realized, assisting doctors in completing nerve block anesthesia accurately and efficiently. Here, we establish a public dataset containing 320 ultrasound images of brachial plexus (BP). Three experienced doctors jointly produce the BP segmentation ground truth and label brachial plexus trunks. We design a brachial plexus segmentation system (BPSegSys) based on deep learning. BPSegSys achieves experienced-doctor-level nerve identification performance in various experiments. We evaluate BPSegSys' performance in terms of intersection-over-union (IoU), a commonly used performance measure for segmentation experiments. Considering three dataset groups in our established public dataset, the IoU of BPSegSys are 0.5238, 0.4715, and 0.5029, respectively, which exceed the IoU 0.5205, 0.4704, and 0.4979 of experienced doctors. In addition, we show that BPSegSys can help doctors identify brachial plexus trunks more accurately, with IoU improvement up to 27%, which has significant clinical application value.