MathCoPilot: An Interactive System for Human-AI Symbiotic Paradigm of Mathematical Research

2026-07-16

AI Summary

This work proposes a novel human–AI symbiotic paradigm for theorem proving, addressing the limitation of current large language model–driven provers that primarily operate autonomously and struggle to collaborate with mathematicians in high-level research. In this framework, mathematicians guide the overall research direction, while AI agents iteratively handle formalization and proof details under continuous supervision, facilitated by an interactive proof blueprint. The system integrates three core capabilities: interactive proof blueprints, adaptive knowledge-base retrieval coupled with Lean 4 verification, and topic-aware paper retrieval with automatic formalization. Experiments demonstrate strong performance on a subset of FormalMATH and two partial differential equation theorems requiring substantial domain expertise, effectively solving undergraduate-level problems, though challenges remain for theorems demanding deeper mathematical insight.

Abstract

Existing LLM-based theorem provers have achieved impressive results on formal mathematics benchmarks, yet they remain confined to acting as autonomous agents that prove a stated proposition. In this paper, we propose MathCoPilot, a human-in-the-loop system that embodies a new human--AI symbiotic paradigm for mathematical research, in which the mathematician steers the high-level mathematical direction while AI agents carry out the detailed formalization and proof work under continuous human guidance. MathCoPilot unifies three core capabilities: (1) an interactive workbench where the mathematician and AI agents collaborate through a living proof blueprint that decomposes a proof into navigable steps the human can directly inspect, direct, and refine; (2) automated proving skill orchestration with adaptive knowledge base search and Lean-integrated iterative verification; and (3) topic-driven paper retrieval and automated formalization into a verified Lean knowledge base. Using MathCoPilot, we systematically compare four state-of-the-art LLMs, including Gemini~3.1~Pro, GPT-5.4, and Claude~Opus~4.7, on a FormalMATH subset and on two real PDE theorems requiring deep domain expertise, evaluating their ability to produce verified Lean~4 proofs and to identify errors in deliberately incorrect proofs. Our results show that while current models can handle undergraduate-level problems with high success rates under favorable autoformalization conditions, substantial challenges remain for domain-specific theorems requiring genuine mathematical understanding.

Problem

• human-AI symbiosis
• mathematical research
• interactive theorem proving
• formalization
• large language models

Innovation

• human-AI symbiosis
• interactive theorem proving
• proof blueprint
• autoformalization
• Lean verification