Pythagoras-Prover: Advancing Efficient Formal Proving via Augmented Lean Formalisation

2026-06-12 · Source: cs.AI updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Software Development & Engineering · Depth: Expert, extended

Summary

Pythagoras-Prover is a new open-source family of compute-efficient Lean theorem provers, featuring 4B and 32B parameter autoregressive models and a 4B diffusion-based model. It employs a compute-frugal data pipeline, including a stratified 800K Lean-verified corpus for curriculum supervised fine-tuning and dynamic proof-reasoning filtering. A key innovation, Augmented Lean Formalisation (ALF), expands the corpus by approximately 2.5 times through structured mutations and self-distillation, bypassing costly per-instance verification. Empirically, Pythagoras-Prover-4B outperforms DeepSeek-Prover-V2-671B on MiniF2F-Test at pass@32 (86.1% vs 82.4%), despite being 167 times smaller. The 32B model achieves 93.0% on MiniF2F-Test and solves 93 problems on PutnamBench, establishing a new open-source benchmark. The diffusion model, while 2.58 times faster, shows 63.25% accuracy.

Key takeaway

For Machine Learning Engineers developing theorem provers, you should prioritize data pipeline innovations over solely scaling model parameters. Implement curriculum learning with difficulty-stratified data and explore structured data augmentation like ALF to significantly boost performance on Lean proofs. This approach enables smaller models to surpass larger baselines, making advanced formal reasoning more accessible and computationally efficient for your projects.

Key insights

Compute-efficient Lean theorem proving is achievable through innovative data augmentation and training strategies, challenging reliance on frontier-scale models.

Principles

• Data quality and structure can substitute for raw model scale.
• Progressive skill acquisition improves proof generation.
• Perturbing formal statements reveals model robustness.

Method

Pythagoras-Prover uses curriculum supervised fine-tuning on a stratified Lean-verified corpus, dynamic proof-reasoning filtering, and Augmented Lean Formalisation (ALF) for corpus expansion via self-distillation.

In practice

• Stratify training data by difficulty for progressive learning.
• Use statement mutation to expand verified corpora efficiently.
• Evaluate provers on perturbed benchmarks to assess robustness.

Topics

• Lean Theorem Proving
• Automated Theorem Proving
• Augmented Lean Formalisation
• Diffusion Models
• Curriculum Learning
• MiniF2F-ALF Benchmark

Code references

• huggingface/open-r1
• project-numina/aimo-progress-prize

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

Related on AIssential

• Pythagoras-Prover: Advancing Efficient Formal Proving via Augmented Lean Formalisation — Compute-efficient Lean theorem provers can surpass larger models through strategic data augmentation and curriculum learning.
• I Solved an 'Impossible' Math Problem with AI — AI-assisted formal verification significantly lowers the barrier to entry for advanced mathematical theorem proving.
• From LLM-Generated Conjectures to Lean Formalizations: Automated Polynomial Inequality Proving via Sum-of-Squares Certificates — NSPI combines LLM conjecture generation with symbolic refinement and formal verification for scalable polynomial inequality proving.
• OProver: A Unified Framework for Agentic Formal Theorem Proving — Integrating agentic proving and compiler feedback directly into training significantly enhances formal theorem proving performance.
• Introduction to Lean for Programmers — Mathematical propositions can be treated as types, and their proofs as programs, via the Curry-Howard correspondence.
• TheoremBench: Evaluating LLMs on Theorem Proving in Formal Mathematics — TheoremBench evaluates LLMs on formal math proving, revealing performance gains with explicit premises and current prover inefficiencies.

Open in AIssential →

Editorial summary, takeaway, and curation by AIssential. Original article published by cs.AI updates on arXiv.org.