# Lean-STaR: Learning to Interleave Thinking and Proving

arXiv: https://arxiv.org/abs/2407.10040 · HTML: https://mlsanity.com/paper/2407.10040/

## Key facts

- Cited by 68 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 33 of its 61 references that are in the map.
- First posted on arXiv 14 Jul 2024.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 30 of 33)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Program of Thoughts Prompting: Disentangling Computation from Reasoning for Numerical Reasoning Tasks](https://mlsanity.com/paper/2211.12588/) (2022)
- [STaR: Bootstrapping Reasoning With Reasoning](https://mlsanity.com/paper/2203.14465/) (2022)
- [Show Your Work: Scratchpads for Intermediate Computation with Language Models](https://mlsanity.com/paper/2112.00114/) (2021)
- [LeanDojo: Theorem Proving with Retrieval-Augmented Language Models](https://mlsanity.com/paper/2306.15626/) (2023)
- [MiniF2F: a cross-system benchmark for formal Olympiad-level mathematics](https://mlsanity.com/paper/2109.00110/) (2021)
- [Generative Language Modeling for Automated Theorem Proving](https://mlsanity.com/paper/2009.03393/) (2020)
- [Llemma: An Open Language Model For Mathematics](https://mlsanity.com/paper/2310.10631/) (2023)
- [Beyond Human Data: Scaling Self-Training for Problem-Solving with Language Models](https://mlsanity.com/paper/2312.06585/) (2023)
- [Draft, Sketch, and Prove: Guiding Formal Theorem Provers with Informal Proofs](https://mlsanity.com/paper/2210.12283/) (2022)
- [Autoformalization with Large Language Models](https://mlsanity.com/paper/2205.12615/) (2022)
- [Thinking Fast and Slow with Deep Learning and Tree Search](https://mlsanity.com/paper/1705.08439/) (2017)
- [HyperTree Proof Search for Neural Theorem Proving](https://mlsanity.com/paper/2205.11491/) (2022)
- [ProofNet: Autoformalizing and Formally Proving Undergraduate-Level Mathematics](https://mlsanity.com/paper/2302.12433/) (2023)
- [Formal Mathematics Statement Curriculum Learning](https://mlsanity.com/paper/2202.01344/) (2022)
- [The Lean mathematical library](https://mlsanity.com/paper/1910.09336/) (2019)
- [Proof Artifact Co-training for Theorem Proving with Language Models](https://mlsanity.com/paper/2102.06203/) (2021)
- [Baldur: Whole-Proof Generation and Repair with Large Language Models](https://mlsanity.com/paper/2303.04910/) (2023)
- [LEGO-Prover: Neural Theorem Proving with Growing Libraries](https://mlsanity.com/paper/2310.00656/) (2023)
- [An In-Context Learning Agent for Formal Theorem-Proving](https://mlsanity.com/paper/2310.04353/) (2023)
- [NaturalProofs: Mathematical Theorem Proving in Natural Language](https://mlsanity.com/paper/2104.01112/) (2021)
- [NaturalProver: Grounded Mathematical Proof Generation with Language Models](https://mlsanity.com/paper/2205.12910/) (2022)
- [Don't Trust: Verify -- Grounding LLM Quantitative Reasoning with Autoformalization](https://mlsanity.com/paper/2403.18120/) (2024)
- [LIME: Learning Inductive Bias for Primitives of Mathematical Reasoning](https://mlsanity.com/paper/2101.06223/) (2021)
- [MUSTARD: Mastering Uniform Synthesis of Theorem and Proof Data](https://mlsanity.com/paper/2402.08957/) (2024)
- [Exploration of Neural Machine Translation in Autoformalization of Mathematics in Mizar](https://mlsanity.com/paper/1912.02636/) (2019)
- [Decomposing the Enigma: Subgoal-based Demonstration Learning for Formal Theorem Proving](https://mlsanity.com/paper/2305.16366/) (2023)

## Built on by (most cited first, 30 of 68)

- [Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models](https://mlsanity.com/paper/2503.09567/) (2025)
- [DeepSeek-Prover-V1.5: Harnessing Proof Assistant Feedback for Reinforcement Learning and Monte-Carlo Tree Search](https://mlsanity.com/paper/2408.08152/) (2024)
- [Kimina-Prover Preview: Towards Large Formal Reasoning Models with Reinforcement Learning](https://mlsanity.com/paper/2504.11354/) (2025)
- [Goedel-Prover: A Frontier Model for Open-Source Automated Theorem Proving](https://mlsanity.com/paper/2502.07640/) (2025)
- [BFS-Prover: Scalable Best-First Tree Search for LLM-based Automatic Theorem Proving](https://mlsanity.com/paper/2502.03438/) (2025)
- [Formal Mathematical Reasoning: A New Frontier in AI](https://mlsanity.com/paper/2412.16075/) (2024)
- [Rewarding the Unlikely: Lifting GRPO Beyond Distribution Sharpening](https://mlsanity.com/paper/2506.02355/) (2025)
- [STP: Self-play LLM Theorem Provers with Iterative Conjecturing and Proving](https://mlsanity.com/paper/2502.00212/) (2025)
- [OdysseyBench: Evaluating LLM Agents on Long-Horizon Complex Office Application Workflows](https://mlsanity.com/paper/2508.09124/) (2025)
- [FormalMATH: Benchmarking Formal Mathematical Reasoning of Large Language Models](https://mlsanity.com/paper/2505.02735/) (2025)
- [LeanAgent: Lifelong Learning for Formal Theorem Proving](https://mlsanity.com/paper/2410.06209/) (2024)
- [AlphaVerus: Bootstrapping Formally Verified Code Generation through Self-Improving Translation and Treefinement](https://mlsanity.com/paper/2412.06176/) (2024)
- [Solving Formal Math Problems by Decomposition and Iterative Reflection](https://mlsanity.com/paper/2507.15225/) (2025)
- [HunyuanProver: A Scalable Data Synthesis Framework and Guided Tree Search for Automated Theorem Proving](https://mlsanity.com/paper/2412.20735/) (2024)
- [Optimizing Temperature for Language Models with Multi-Sample Inference](https://mlsanity.com/paper/2502.05234/) (2025)
- [Prover Agent: An Agent-Based Framework for Formal Mathematical Proofs](https://mlsanity.com/paper/2506.19923/) (2025)
- [Chain-of-Reasoning: Towards Unified Mathematical Reasoning in Large Language Models via a Multi-Paradigm Perspective](https://mlsanity.com/paper/2501.11110/) (2025)
- [Leanabell-Prover: Posttraining Scaling in Formal Reasoning](https://mlsanity.com/paper/2504.06122/) (2025)
- [MA-LoT: Model-Collaboration Lean-based Long Chain-of-Thought Reasoning enhances Formal Theorem Proving](https://mlsanity.com/paper/2503.03205/) (2025)
- [Solving Inequality Proofs with Large Language Models](https://mlsanity.com/paper/2506.07927/) (2025)
- [Large Language Model Reasoning Failures](https://mlsanity.com/paper/2602.06176/) (2026)
- [Learning to Reason via Mixture-of-Thought for Logical Reasoning](https://mlsanity.com/paper/2505.15817/) (2025)
- [One Example Shown, Many Concepts Known! Counterexample-Driven Conceptual Reasoning in Mathematical LLMs](https://mlsanity.com/paper/2502.10454/) (2025)
- [LEAP: Supercharging LLMs for Formal Mathematics with Agentic Frameworks](https://mlsanity.com/paper/2606.03303/) (2026)
- [LeanMarathon: Toward Reliable AI Co-Mathematicians through Long-Horizon Lean Autoformalization](https://mlsanity.com/paper/2606.05400/) (2026)
- [SubgoalXL: Subgoal-based Expert Learning for Theorem Proving](https://mlsanity.com/paper/2408.11172/) (2024)
- [What makes math problems hard for reinforcement learning: a case study](https://mlsanity.com/paper/2408.15332/) (2024)
- [Goedel-Architect: Streamlining Formal Theorem Proving with Blueprint Generation and Refinement](https://mlsanity.com/paper/2606.06468/) (2026)
- [dafny-annotator: AI-Assisted Verification of Dafny Programs](https://mlsanity.com/paper/2411.15143/) (2024)
- [From Hypothesis to Publication: A Comprehensive Survey of AI-Driven Research Support Systems](https://mlsanity.com/paper/2503.01424/) (2025)

## Abstract (arXiv)

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 achieves state-of-the-art results on the miniF2F-test benchmark within the Lean theorem proving environment, significantly outperforming base models (boldsymbol43.4% rightarrow 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.