Traditional language model-based theorem proving assumes that by training on a sufficient amount of formal proof data, a model will learn to prove theorems. Our key observation is that a wealth of informal information that is not present in formal proofs can be useful for learning to prove theorems. For instance, humans think through steps of a proof, but this thought process is not visible in the resulting code. We present Lean-STaR, a framework for training language models to produce informal thoughts prior to each step of a proof, thereby boosting the model's theorem-proving capabilities. Lean-STaR uses retrospective ground-truth tactics to generate synthetic thoughts for training the language model. At inference time, the trained model directly generates the thoughts prior to the prediction of the tactics in each proof step. Building on the self-taught reasoner framework, we then apply expert iteration to further fine-tune the model on the correct proofs it samples and verifies using the Lean solver. Lean-STaR significantly outperform base models (43.4% → 46.3%, Pass@64). We also analyze the impact of the augmented thoughts on various aspects of the theorem proving process, providing insights into their effectiveness.

Mobile Edge Computing (MEC) holds excellent potential in Congestion Management (CM) of 6G vehicular networks. A reasonable schedule of MEC ensures a more reliable and efficient CM system. Unfortunately, existing parallel and sequential models cannot cope with scarce computing resources and constrained channels, especially during traffic rush hour. In this paper, we propose a channel-constrained multi-core sequential model (CCMSM) for task offloading and resource allocation. The CCMSM incorporates a utility index that couples system energy consumption and delay, applying Genetic Algorithm combining Sparrow Search Algorithm (GA-SSA) in the branching optimization. Furthermore, we prove that the system delay is the shortest with the FCFS computing strategy in the MEC server. Simulation demonstrates that the proposed CCMSM achieves a higher optimization level and exhibits better robustness and resilient scalability for traffic spikes.