---
license: apache-2.0
language:
  - en
library_name: transformers
pipeline_tag: text-generation
tags:
  - lean4
  - theorem-proving
  - formal-mathematics
  - agentic
  - retrieval-augmented
datasets:
  - m-a-p/OProofs
---

# OProver

> A unified framework for **agentic formal theorem proving** in Lean 4.

- 📄 Paper: [OProver: A Unified Framework for Agentic Formal Theorem Proving](https://huggingface.co/papers/2605.17283) ([arXiv](https://arxiv.org/abs/2605.17283))
- 📚 Collection: [m-a-p/OProver](https://huggingface.co/collections/m-a-p/oprover)
- 📦 Dataset: [m-a-p/OProofs](https://huggingface.co/datasets/m-a-p/OProofs)

OProver treats theorem proving as a multi-round refinement loop. Given a target
theorem, the prover retrieves top-k compiler-verified proofs from a memory of
prior proofs, generates a proof attempt, runs the Lean 4 compiler, and—on
failure—revises the attempt using the compiler feedback in the next round.
The same retrieval and feedback signals are baked into training, so the
training-time interface matches the proving-time interaction.

## Highlights

- **State-of-the-art among open-weight whole-proof provers.** OProver-32B
  attains the best Pass@32 on MiniF2F (93.3), ProverBench (58.2), and
  PutnamBench (11.3), and the second-best on MathOlympiad (22.8) and
  ProofNet (33.2). Even OProver-8B beats Goedel-Prover-V2-32B on all five
  benchmarks despite having 4× fewer parameters.
- **Agentic proving is in the policy, not bolted on.** Retrieval, multi-turn
  compiler feedback, and iterative repair are all part of the trained policy.
  Ablations show feedback is the dominant driver: removing it costs OProver-32B
  4.9–7.4 Pass@32 points across benchmarks; removing retrieval on top adds
  0.5–1.7 more.
- **Co-evolving prover and corpus.** Each post-training iteration runs agentic
  rollouts; verified proofs are added to OProofs and re-indexed for retrieval,
  repair trajectories become SFT data, and unresolved hard cases provide RL
  signal. MiniF2F-Test Pass@32 grows monotonically: OProver-8B 79.5 → 86.2 →
  87.0 → 91.8 across rounds; OProver-32B 84.7 → 88.1 → 93.3.

## What's in this release

The OProver collection ([m-a-p/OProver](https://huggingface.co/collections/m-a-p/oprover)) bundles the paper,
the corpus, and seven model checkpoints covering both training stages and
both model sizes:

| Repo | Stage | Size | Note |
|---|---|---|---|
| [OProver-8B-Base](https://huggingface.co/m-a-p/OProver-8B-Base) | Continued pretraining only | 8B | Domain-adapted base; before SFT/RL |
| [OProver-32B-Base](https://huggingface.co/m-a-p/OProver-32B-Base) | Continued pretraining only | 32B | Domain-adapted base; before SFT/RL |
| [OProver-8B-Round1](https://huggingface.co/m-a-p/OProver-8B-Round1) | Post-training Round 1 | 8B | After first SFT+RL iteration |
| [OProver-8B-Round2](https://huggingface.co/m-a-p/OProver-8B-Round2) | Post-training Round 2 | 8B | After second SFT+RL iteration |
| [OProver-32B-Round1](https://huggingface.co/m-a-p/OProver-32B-Round1) | Post-training Round 1 | 32B | After first SFT+RL iteration |
| **[OProver-8B](https://huggingface.co/m-a-p/OProver-8B)** | Final | 8B | The 8B prover reported in the paper (Round 3) |
| **[OProver-32B](https://huggingface.co/m-a-p/OProver-32B)** | Final | 32B | The 32B prover reported in the paper (Round 2) |
| [OProofs](https://huggingface.co/datasets/m-a-p/OProofs) | Dataset | 6.86M proofs | Lean 4 corpus used for CPT, SFT, and the retrieval memory |

Use **OProver-8B** or **OProver-32B** for proving. The Base / Round-N
checkpoints are released for reproducibility and ablation studies.

## Dataset: OProofs

OProofs is a large-scale Lean 4 corpus that doubles as the retrieval memory at
proving time. It is built from three sources: public Lean resources
(NuminaMath-LEAN, Lean-Workbook, Leanabell-FormalStmt, Goedel-Pset, …),
large-scale autoformalization + agentic proof synthesis from informal math
mined on Common Crawl and GitHub, and traces from OProver's own agentic
proving runs.

![OProofs Lean 4 corpus statistics](./assets/oproofs.png)

## Training

Two stages, both anchored on OProofs.

**1. Continued pretraining (one-time).** A 65B-token mixture of formal
Lean (≈30%, from OProofs), code (≈20%, OpenCoder), mathematics (≈40%,
Nemotron-Math-4-Plus), and long-CoT (≈10%, ProLong-64K). AdamW, peak LR
5e-5, cosine with 3% warmup, batch 512, sequence length 8192. Output:
`OProver-{8B,32B}-Base`.

**2. Iterative post-training.** Each round runs:

1. Agentic proving with the current prover on a theorem pool, producing
   multi-round rollouts conditioned on retrieved proofs and Lean feedback.
2. **SFT** on round-level repair examples `(s, R, p_{t-1}, f_{t-1}) → p_t`,
   with cross-entropy loss only on the new attempt.
3. **GSPO RL** on hard cases (groups with non-trivial pass-rate). Per-round
   reward `r = 0.8 + 0.2·1[format ok]` if Lean-verified, else 0; advantages
   are pooled across the n×R rounds for the same theorem.
4. Newly verified proofs and repair trajectories are folded back into OProofs
   and re-indexed into the retrieval memory for the next round.

## Results

Pass@32 (n=64) across five Lean 4 benchmarks. **Bold** is best, _underlined_
is second-best.

![Main results: Pass@32 across five Lean 4 benchmarks](./assets/main_results.png)

OProver-32B reaches three best and two second-best across five benchmarks —
the most top placements of any model in the comparison, despite being a
32B dense model versus a 560B MoE or a 671B dense competitor.

## Loading

```python
from transformers import AutoModelForCausalLM, AutoTokenizer

# Pick any of: OProver-8B, OProver-32B, *-Base, *-Round1, *-Round2
name = "m-a-p/OProver-8B"
tok = AutoTokenizer.from_pretrained(name)
model = AutoModelForCausalLM.from_pretrained(name, torch_dtype="bfloat16", device_map="auto")
```

```python
from datasets import load_dataset
ds = load_dataset("m-a-p/OProofs", split="train")
```

OProver is trained against a multi-round agentic interface: at each round the
input includes the target Lean statement, top-k retrieved verified proofs, the
prior proof attempt, and the Lean compiler feedback. See the paper §2.1 for
the prompt template, and Appendix B for serialization details.

## Citation

```bibtex
@article{ma2026oprover,
  title   = {OProver: A Unified Framework for Agentic Formal Theorem Proving},
  author  = {David Ma and Kaijing Ma and Shawn Guo and Yunfeng Shi and Enduo Zhao and Jiajun Shi and Zhaoxiang Zhang and Gavin Cheung and Jiaheng Liu and Zili Wang},
  journal = {arXiv preprint arXiv:2605.17283},
  year    = {2026}
}
```
