ImProver: Agent-Based Automated Proof Optimization

Overview

ImProver is a LLM-powered AI agent for proof rewriting tasks, built around a general-purpose neural theorem proving framework. It allows for arbitrary Lean 4 code to be optimized for an arbitrary metric, enabling users to automatically optimize their formal proofs by providing a lambda and prompt. ImProver leverages advanced techniques in Lean metaprogramming, machine learning, and formal methods to deliver optimized proofs that meet user-defined criteria.

ImProver automates the process of optimizing Lean 4 proofs through the following steps:

1. Configuration and Setup: Define configuration files specifying Lean projects.
2. Building Process: Use Lean metaprogramming to generate proof data and cache it in JSON format.
3. Prompt Generation: Construct prompts using templates refined with theorem and metric data.
4. Retrieval-Augmented Generation (RAG): Enhance prompts with relevant examples and documentation.
5. Proof Generation and Refinement: Generate and iteratively refine proofs using language models.
6. Evaluation: Assess the correctness and quality of proofs using predefined and user-defined metrics.
7. Benchmarking: Compare different optimization strategies to determine the most effective approach.

Repository Structure

root/
├── .cache/ # Build Cache
├── .db/ # Vector DB Cache
├── .lake/ # Lean Environment
├── .trees/ # Proof tree export images
├── benchmark/ # Benchmarking tools
│ └── data/ # Benchmarking output
├── configs/ # Build configurations
├── evaluate/ # Correctness and metric evaluation
├── models/ # LLM interface
├── scripts/ # Build scripts
├── TrainingData/ # Metaprogramming
├── lake-manifest.json # Dependency info
├── lakefile.lean # Lean config
├── lean-toolchain # Lean version
└── README.md

Key Files

• scripts/build.py: Build environment and cache
• benchmark/tools.py: Run benchmarks with file-specified configuration.
• benchmark/extract.py: Run experimental measurements over benchmarking data.
• models/prompt.py: All prompting chains
• models/rag.py: All RAG code
• models/structures.py: All objects and datastructures
• evaluate/metrics: Metric definitions.
• evaluate/build_prooftree: Proof tree generation

Configuration

Configuration files specify the details of the Lean projects to be built. They are JSON arrays of objects, each representing a Lean project. Here is an example configuration file:

[
    {
        "path": "/Users/user/Desktop/lean-project",
        "lean": "leanprover/lean4:v4.9.0",
        "name": "LeanProject",
        "import_file": "LeanProject.lean",
        "imports": ["LeanProject"]
    },
    {
        "repo": "https://github.com/leanprover-community/mathlib4",
        "commit": "v4.9.0",
        "lean": "leanprover/lean4:v4.9.0",
        "name": "mathlib",
        "import_file": "Mathlib.lean",
        "imports": ["Mathlib"],
        "build": false
    }
]

Metrics

ImProver comes with several preinstalled metrics for evaluating proofs:

• Length Metric: Measures the length of the proof in terms of the number of steps.
• Readability Metric: Evaluates the degree of modularity by analyzing the proof tree for reusable, independent subproofs.
• Similarity Metric: Measures the structural and syntactic differences between two proofs. (Not used in ICLR)
• Completion Metric: Measures the completeness of a proof by evaluating the number of errors present. This is essentially used for full proof generation

Adding Custom Metrics

To add a new metric, create an instance of the Metric class by defining the following parameters:

Metric(name="Metric Name",
      prompts=prompts,
      examples=examples,
      minmax="minimize or maximize",
      score_fn=score_fn, #default = None; unary
      metric_fn=metric_fn, #default = None; binary
      cmp=comparison_fn, #default = None
      lock_refinement_state=False #default = False; Refinement compares to orginal or most recent instance
      )

Usage

• Setup the Environment: Clone the repository and set up the Lean environment by defining configuration files. Set the OPENAI_API_KEY environment variable.
• Build and Cache Proof Data: Run the build scripts to generate and cache proof data in JSON format.
  • python scripts/build.py --config CONFIG_PATH
• Setup Run Configuration: Use the provided tools and models to generate the parameter tuning for your desired tests.
• Evaluate and Benchmark: Assess the correctness and quality of the generated proofs using the evaluation and benchmarking tools.
  • python benchmark/tools.py

Acknowledgements

We would like to thank Kim Morrison for the Training Data repository and Sean Welleck for the Neural Theorem Proving (NTP) toolkit repository, which served as foundational resources for this project. Additionally, we would like to thank the Paperproof team for the Paperproof repository, which paved the way for our own prooftree generation and analysis system.

Special thanks goes to the L3 Lab, Hoskinson Center for Formal Mathematics, Convergent Research, the Lean FRO, and the OpenAI Researcher Access Program for their support.

Contributing

We welcome contributions from the community. If you have suggestions, bug reports, or want to contribute code, please open an issue or submit a pull request.

License

This project is licensed under the MIT License.

Contact

For any questions or inquiries, please contact:

• Riyaz Ahuja: [email protected]
• Jeremy Avigad: [email protected]
• Prasad Tetali: [email protected]
• Sean Welleck: [email protected]

BibTex

To cite ImProver, use the following BibTex:

@misc{ahuja2024improveragentbasedautomatedproof,
      title={ImProver: Agent-Based Automated Proof Optimization}, 
      author={Riyaz Ahuja and Jeremy Avigad and Prasad Tetali and Sean Welleck},
      year={2024},
      eprint={2410.04753},
      archivePrefix={arXiv},
      primaryClass={cs.AI},
      url={https://arxiv.org/abs/2410.04753}, 
}