Lean is an advanced proof assistant designed to facilitate formal theorem proving by providing a variety of interactive feedback. In this paper, we explore methodologies to leverage proof assistant feedback to augment the capabilities of large language models in constructing formal proofs. First, we deploy online reinforcement learning using Lean verification outcomes as the reward signal to improve the proof completion policy. This straightforward approach shows great promise in enhancing the model's alignment with the formal verification system. In addition, we propose RMaxTS, a variant of Monte-Carlo tree search that employs an intrinsic-reward-driven exploration strategy to generate diverse proof paths. The tree structure is organized to represent the transitions of intermediate tactic states, extracted from the compilation messages given by Lean's tactic mode. The intrinsic reward is constructed to incentivize the discovery of novel tactic states, which helps to to mitigate the sparse-reward problem inherent in proof search. These techniques lead to a more efficient planning scheme for formal proof generation, achieving new state-of-the-art results on both miniF2F and ProofNet benchmarks.

The rapid progress in Deep Learning (DL) and Large Language Models (LLMs) has exponentially increased demands of computational power and bandwidth. This, combined with the high costs of faster computing chips and interconnects, has significantly inflated High Performance Computing (HPC) construction costs. To address these challenges, we introduce the Fire-Flyer AI-HPC architecture, a synergistic hardware-software co-design framework and its best practices. For DL training, we deployed the Fire-Flyer 2 with 10,000 PCIe A100 GPUs, achieved performance approximating the DGX-A100 while reducing costs by half and energy consumption by $40 \%$. We specifically engineered HFReduce to accelerate allreduce communication and implemented numerous measures to keep our Computation-Storage Integrated Network congestion-free. Through our software stack, including HaiScale, 3FS, and HAI-Platform, we achieved substantial scalability by overlapping computation and communication. Our system-oriented experience from DL training provides valuable insights to drive future advancements in AI-HPC.