Pythagoras-Prover: Advancing Efficient Formal Proving via Augmented Lean Formalisation
AuthorsJoshua Ong Jun Leang, Zheng Zhao, Mihaela Cătălina Stoian, Qiyuan Xu, Haonan Li, Wenda Li, Shay B. Cohen, Eleonora Giunchiglia
Resources
This paper makes Lean theorem proving cheaper and stronger by combining curriculum training, proof-trace filtering, and a novel data augmentation scheme, while also testing a diffusion-style prover.
Key results
Pass rate achieved by the 4B autoregressive prover.
Pythagoras-Prover-4B uses 167 times fewer parameters while achieving higher MiniF2F-Test pass@32.
Best reported MiniF2F-Test pass rate for the 32B model.
Problems solved out of 672 on PutnamBench.
Approximate multiplication of the training corpus through Augmented Lean Formalisation.
Pythagoras-Prover-Diffusion-4B generates proofs 2.58 times faster than the autoregressive 4B model.
What the paper found
Researchers from Imperial College London, the University of Edinburgh, Nanyang Technological University, and MBZUAI introduce Pythagoras-Prover, an open-source family of Lean 4 theorem provers designed to reduce dependence on frontier-scale models such as DeepSeek-Prover-V2. Its training pipeline combines Lean-verified data stratified into easy, medium, and hard tiers, LoRA-based curriculum fine-tuning, dynamic proof-reasoning filtering within an 8K-token context, reinforcement learning with Lean verification, and Augmented Lean Formalisation, or ALF. ALF generates structured variants through simplification, generalisation, lemma proposal, proof-step decomposition, and reformulation, then uses self-distillation to expand training data without verifying every mutation; the resulting corpus grows by 2.5 times. Pythagoras-Prover-4B reaches 86.1% pass@32 on MiniF2F-Test, surpassing DeepSeek-Prover-V2-671B despite using 167 times fewer parameters, while Pythagoras-Prover-32B achieves 93.03% at pass@2048 and solves 93 of 672 PutnamBench problems. The new MiniF2F-ALF benchmark exposes brittleness under controlled statement mutations, with the 32B model retaining the strongest performance. The paper also presents Pythagoras-Prover-Diffusion-4B, using block diffusion and tactic-level masking; although its accuracy is lower, it generates proofs 2.58 times faster than the autoregressive 4B model. OpenAI’s GPT-5.5 and Anthropic’s Claude Opus 4.6 assist in constructing and validating perturbation data, while the authors’ central result is that careful data engineering and formal verification can substitute substantially for raw model scale.
Original abstract
Modern Lean theorem provers achieve strong performance only with substantial training and inference compute, driven in part by scarce verified proof data and the long reasoning traces of formal proof search, making both supervised fine-tuning (SFT) and sampling expensive. We introduce Pythagoras-Prover, a compute-efficient open-source family of Lean theorem provers built for practical compute budgets. The family spans two generation paradigms: autoregressive models at 4B and 32B parameters, and a first proof-of-concept diffusion-based prover (4B) that iteratively refines Lean proofs at inference time. For training efficiency, we build a Lean-verified corpus stratified into easy, medium, and hard problems for curriculum SFT, so models acquire proof skills progressively from shorter, simpler proofs to longer, harder ones. During SFT, a dynamic proof-reasoning filtering scheme preserves informative proof traces while keeping each instance within an 8k-token context budget. We also introduce Augmented Lean Formalisation (ALF), which expands scarce verified corpora into variants of formal statements, populated via self-distillation for extra training signal without formally verifying every mutated instance. By perturbing known problems while preserving their formal character, ALF reduces reliance on any statement's surface form. Empirically, Pythagoras-Prover-4B surpasses DeepSeek-Prover-V2-671B at pass@32 on MiniF2F-Test (86.1% vs 82.4%) with ~167x fewer parameters, while Pythagoras-Prover-32B sets the open-source state of the art at 93.0% on MiniF2F-Test and solves 93 of 672 PutnamBench problems. We release MiniF2F-ALF, an ALF-mutated contamination-sensitive benchmark on which every evaluated model loses accuracy; here our 32B remains strongest and our 4B matches the prior state of the art, Goedel-Prover-V2-32B.
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