Optimizing the Cost-Quality Tradeoff of Agentic Theorem Provers in Lean

2026-06-03

AI Summary

This work addresses the high computational cost of generating formal proofs in Lean using large language models, where substantial resources are often wasted on unsuccessful attempts. The authors propose an action-routing agent comprising a data plane and a control plane: the data plane handles natural language decomposition of lemmas, their formalization, and proof sampling, while the control plane leverages historical failure trajectories to estimate success probability and cost, dynamically deciding whether to continue or restart a proof attempt. By treating failed proof trajectories as scheduling signals, this approach enables cost-aware resource allocation—a strategy introduced here for the first time. Evaluated on a subset of PutnamBench, the method reduces average computational cost by 25.8% compared to a fixed-step baseline while preserving the original proof success rate.

Abstract

Large language models (LLMs) are increasingly used in workflows for generating formal proofs in Lean. These workflows often decompose problems into smaller lemmas, sample many proof attempts, and use compiler feedback to guide search. However, they can be prohibitively expensive, often spending substantial compute on attempts that ultimately fail. In this work, we address this problem with an action routing agent that consists of a data plane and a control plane. The data plane generates natural-language lemma decompositions, formalizes them in Lean, and samples proof attempts for the resulting theorem and lemma targets. The control plane observes previous failed Lean attempts, estimates both the likelihood of success and cost of another attempt, and decides whether to continue proving the current target or restart from a new breakdown. On a subset of PutnamBench, our agent decreases the cost by $25.8\%$ over a fixed-step baseline on average, preserving performance while using substantially less compute. These results suggest that failed Lean trajectories provide actionable signals for cost-aware resource allocation in agentic theorem proving.

Problem

• agentic theorem proving
• cost-quality tradeoff
• formal proofs
• Lean
• large language models

Innovation

• action routing agent
• cost-quality tradeoff
• agentic theorem proving
• Lean
• failed trajectory utilization