OpenProver: Agentic and Interactive Theorem Proving with Lean 4

Abstract: In this system paper, we present OpenProver, an open-source system for LLM-driven automated theorem proving (ATP) with integrated Lean 4 formal verification. OpenProver integrates a Planner-Worker-Verifier architecture inspired by recent ATP agentic systems such as Aletheia. A Planner agent maintains a compact Whiteboard scratchpad and an unbounded Repository of intermediate findings, and decomposes mathematical work into parallel Workers. OpenProver is fully open-source, offers reproducible evaluation through automatic formal verification of generated proofs, and provides an interactive terminal interface for human-guided proof search. In interactive mode, OpenProver allows the human operator to monitor and steer the proof search process, motivated by the established human-AI synergy in interactive code generation. To showcase the potential for quantitative ablation experiments enabled by automatic formal verification, we evaluate OpenProver on ProofNet and compare it with a simple baseline. OpenProver is publicly available at https://github.com/kripner/OpenProver.

• The paper introduces OpenProver, a system that combines LLM-driven automated proof search with Lean 4's formal verification through an agentic PWV architecture.
• The system employs a Planner-Worker-Verifier design to decompose tasks, enable parallel exploration, and provide natural language feedback on Lean proofs.
• Evaluation on ProofNet theorems shows significant success rate improvements by leveraging agentic planning and interactive user guidance in formal verification.

OpenProver: Agentic and Interactive Theorem Proving with Lean 4

System Overview and Motivation

OpenProver introduces an open-source framework for LLM-driven automated theorem proving (ATP) that natively integrates Lean 4 for formal verification. Addressing both reproducibility and interactivity, the system integrates a Planner-Worker-Verifier (PWV) agentic architecture. This structure, inspired by recent agentic ATP systems such as Aletheia, facilitates decomposition of theorem proving tasks, parallel exploration, and real-time feedback within a formally verified context.

The core motivation for OpenProver is to provide a platform that not only automates proof search using LLMs but also allows fine-grained human supervision and reproducibility. This dual-mode operation is critical since autonomous systems do not yet consistently achieve expert-level performance. OpenProver’s inclusion of an interactive terminal interface extends its utility as a research tool, especially in collaborative human-AI mathematical investigations.

Architectural Details

OpenProver's architecture segregates responsibilities among three agent types:

• Planner: Responsible for maintaining a high-level objective, managing the Whiteboard (a succinct, persistent state artifact), and decomposing proof tasks for parallel execution.
• Workers: Execute isolated strategies including lemma proving, counterexample generation, and proof decomposition. Workers operate without access to each other’s reasoning traces to maximize approach diversity and independence.
• Verifiers: Assess and provide natural language feedback on Worker outputs without exposure to their reasoning processes, mitigating bias propagation and enhancing error detection.

The state management strategy employs a compact Markdown-based Whiteboard for persistent, high-value context and a hierarchical Repository for managing larger artifacts like Lean snippets, intermediate proofs, and literature search outputs. The system iteratively alternates between Planner coordination and distributed Worker exploration, checking outputs with Verifiers at each step, until a Lean-verified proof or token budget exhaustion.

Lean 4 Integration and Verification Workflow

A critical innovation in OpenProver is the coalescence of natural language theorem proving with formal Lean 4 verification at every proof stage. The system supports:

• Automated conversion of informal proofs into Lean code.
• Iterative repair cycles upon Lean verification failures, with Planner guidance on whether to amend the formal or informal proof.
• Verifier extension allowing Workers to check, search, and accumulate Lean snippets using primitives such as lean_verify and lean_search.

Only Lean code that passes all verification checks (i.e., absence of errors or warnings) is permanently retained in the Repository. This mechanism enforces trustworthy, fully auditable proof artifacts and enables robust automatic benchmarking across the full lifecycle of proof exploration, formalization, and checking.

Interactive User Interface

In interactive mode, OpenProver exposes full agent reasoning traces and history to users via a terminal UI. This design facilitates multiple modalities:

• Real-time user interruption of unproductive Workers or Planner trajectories.
• Injection of manual guidance to steer the proof search.
• Optional manual acceptance or rejection of Planner-generated action batches, with feedback propagation.

Such interactive controls enhance the system’s value as a collaborative ITP, recognizing the importance of human-AI synergy in domains where purely autonomous methods presently falter.

Evaluation Results

Autonomous OpenProver was evaluated on 185 formal theorems from ProofNet, comparing its performance to a simple linear Chain-of-Thought (CoT) baseline using two LLMs: Kimi-K2.5 and Leanstral. The system demonstrated substantial improvements in formal proof success rates, with Kimi-K2.5 achieving 57.3% versus the baseline's 36.8%, and Leanstral reaching 28.1% against the baseline's 21.1%, under equivalent 100,000 token-per-problem budgets.

Model	Baseline Success Rate	OpenProver Success Rate
Kimi-K2.5	36.8%	57.3%
Leanstral	21.1%	28.1%

These results empirically validate the advantage of agentic planning, parallel proof exploration, and formal verification feedback in enhancing LLM-driven ATP capabilities.

Implications and Future Directions

OpenProver’s design offers numerous practical and theoretical advantages:

• Reproducibility and Benchmarking: By embedding Lean 4 verification at every stage, OpenProver supports objective, reproducible empirical evaluation, an essential requirement for rigorous ATP research.
• Prompt-based Self-Improvement: By shifting agent behavior to prompt designs rather than static code, OpenProver aligns with open-ended prompt optimization strategies (e.g., Feedback Descent, AlphaEvolve). Automatic formal verification feedback could be used as a self-improvement signal for LLM-based agents.
• Seamless Human-AI Collaboration: The dual autonomous/interactive modality supports practical use-cases in mathematics and formal methods where human guidance is currently indispensable.
• Ecosystem Agnosticism: OpenProver’s independence from any fixed LLM backend broadens its applicability and longevity, allowing continual integration of improved LLMs and Lean library resources.

The primary challenge remains bridging the gap between informal proof discovery and formalization, a bottleneck likely to diminish as Lean's Mathlib and analogous resources mature.

Conclusion

OpenProver advances the state of agentic, LLM-driven automated theorem proving by tightly integrating Lean 4 formal verification within a reproducible, agentic architecture that supports both autonomous and interactive operation. Its empirical gains, system flexibility, and rigorous design provide a robust foundation for continued research in formalized mathematics, ATP evaluation, and co-evolution of agents and formal libraries. The system’s open-source nature further encourages community-driven development and benchmarking in the field.