README.md

metadata

license: cc-by-4.0
language:
  - en
tags:
  - lean4
  - mathlib
  - formal-theorem-proving
  - category-theory
  - benchmark
  - proof-generation
pretty_name: LeanCat
size_categories:
  - n<1K
task_categories:
  - text-generation
configs:
  - config_name: default
    data_files:
      - split: test
        path: data/leancat_records.jsonl

LeanCat: A Lean Dataset for Evaluating Library-Grounded Category-Theoretic Reasoning

This is an anonymized review artifact.

LeanCat is a dataset of 100 statement-level problems in Lean 4 (mathlib), designed to stress-test abstraction-heavy, library-grounded reasoning in formal mathematics. This repository contains Part I: 1-Category Theory.

Overview

LeanCat addresses a critical gap in automated theorem proving evaluation datasets by focusing on category theory - the unifying language of modern mathematics that requires sophisticated abstraction and library navigation skills. While existing benchmarks target olympiad-style problems or undergraduate mathematics, LeanCat challenges AI systems with research-level categorical reasoning.

Repository Structure

LeanCat/
├── CAT_statement/          # Formal Lean 4 statements of dataset problems
├── problems/               # Natural language problem descriptions (Markdown)
├── data/                   # JSONL records for Hugging Face dataset viewing
│   └── leancat_records.jsonl # One record per dataset problem
├── configs/                # Evaluation protocol configuration
├── prompts/                # Prompt templates used by baseline protocols
├── scripts/                # Dataset validation utilities
├── .github/               
├── CAT_statement.lean     # Main Lean 4 file containing all statement imports
├── DATASET_CARD.md        # Dataset documentation and responsible-use notes
├── EVALUATION.md          # Proof validation and reporting protocol
├── lakefile.lean          
├── lean-toolchain         # Use Lean version 4.19.0
├── metadata.json          # Problem metadata (difficulty, tags, refs)
├── lake-manifest.json    
├── .gitignore            
└── LICENSE               

Quick Start

1. Install Lean via elan: https://leanprover-community.github.io/get_started.html
2. Build the project:

# Build with lake
lake build

The dataset artifact builds with Lean 4.19.0 and mathlib 4.19.0.

To validate the artifact structure, run:

python scripts/validate_dataset.py

For Hugging Face hosting and programmatic loading, the derived JSONL index is provided at data/leancat_records.jsonl. It contains one record per problem, including the problem id, metadata fields, source file paths, natural-language statement, and Lean formal statement. Regenerate it after metadata or statement edits with:

python scripts/generate_hf_records.py

The NeurIPS Croissant metadata file with Responsible AI fields is provided as croissant.json. This file is the validated submission version for OpenReview; the Hugging Face Croissant API may still expose the platform's automatically generated core-metadata version.

To run the provided evaluation drivers with an OpenAI-compatible chat API:

$env:OPENAI_API_KEY="your_key"
$env:OPENAI_BASE_URL="https://api.openai.com/v1"
python scripts/passk.py --start 1 --end 1 --model gpt-5.2 -k 4
python scripts/leanbridge.py --start 1 --end 1 --model gpt-5.2 --max-iterations 4

On Linux/macOS:

export OPENAI_API_KEY="your_key"
export OPENAI_BASE_URL="https://api.openai.com/v1"
python scripts/passk.py --start 1 --end 1 --model gpt-5.2 -k 4
python scripts/leanbridge.py --start 1 --end 1 --model gpt-5.2 --max-iterations 4

leanbridge.py uses the local LeanExplore backend by default. Install and prepare it with pip install lean-explore[local] and lean-explore data fetch. The local LeanExplore service is initialized once per process and reused across queries. Use --search-backend none for a no-search refinement loop. Outputs are written under results/, which is ignored by git.

Evaluation Protocol

Each Lean file contains a dataset statement with one or more sorry placeholders. A problem is solved when the placeholders are replaced by a proof and the file is accepted by Lean under the pinned toolchain and dependencies in this artifact. The aggregate project can be checked with lake build.

See EVALUATION.md for the full proof validity and reporting protocol. See DATASET_CARD.md for dataset documentation and responsible-use notes.

Benchmark Content

Problem Categories (100 problems total for 1-Category Theory)

1. Basic Category Properties (Problems 1-18): Fundamental results about categories, morphisms, monomorphisms, epimorphisms, initial/terminal objects

2. Adjunctions (Problems 19–29): Adjoint functors, universal properties, comma categories

3. Reflective and Coreflective Subcategories (Problems 30-33): Subcategory properties and classifications

4. Concrete Categories (Problems 34-41): Categories with faithful forgetful functors to Set

5. Limits and Colimits (Problems 42-73): The largest cluster covering limits, colimits, and related constructions

6. Cocompletions (Problems 74-78): Recent work on cocompletions requiring new definitions

7. Abelian Categories (Problems 79-90): Homological algebra concepts, kernels, cokernels, exact sequences

8. Monads (Problems 91-100): Monads, Kleisli and Eilenberg-Moore categories

Difficulty Distribution

• Easy: 20 problems (≤5.54/10 difficulty score)
• Medium: 40 problems (5.54-7.8/10 difficulty score)
• High: 40 problems (≥7.8/10 difficulty score)

License

Dataset contents, including LeanCat problem statements, natural-language descriptions, and metadata, are released under CC BY 4.0. Evaluation scripts and software code are released under the repository MIT license.