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

2026-06-10

AI Summary

This work addresses the high computational cost in formal theorem proving caused by scarce verification data and excessively long reasoning traces. To this end, the authors propose a family of efficient Lean theorem provers featuring several key innovations: a novel diffusion-based Lean proof generation architecture, an Augmented Lean Formalization (ALF) data augmentation strategy, curriculum-based supervised fine-tuning with difficulty stratification, and a dynamic reasoning filtering mechanism. These components are integrated within an 8k-token context window that unifies autoregressive and diffusion-based generation paradigms. Experimental results demonstrate that their 4B-parameter model achieves 86.1% accuracy on MiniF2F-Test—surpassing DeepSeek-Prover-V2-671B (82.4%) despite using only 1/167 of its parameters—while their 32B model attains state-of-the-art performance among open-source systems at 93.0%, successfully solving 93 problems from the PutnamBench benchmark.

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.

Problem

• formal theorem proving
• Lean
• proof data scarcity
• reasoning trace length
• compute efficiency

Innovation

• diffusion-based prover
• curriculum SFT
• dynamic proof-reasoning filtering
• Augmented Lean Formalisation
• self-distillation