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Retrieval-Augmented Prover (ReProver)

Model

Code for the paper:

LeanDojo: Theorem Proving with Retrieval-Augmented Language Models
NeurIPS (Datasets and Benchmarks Track), 2023, Oral presentation
Kaiyu Yang, Aidan Swope, Alex Gu, Rahul Chalamala,
Peiyang Song, Shixing Yu, Saad Godil, Ryan Prenger, Anima Anandkumar

@inproceedings{yang2023leandojo,
  title={{LeanDojo}: Theorem Proving with Retrieval-Augmented Language Models},
  author={Yang, Kaiyu and Swope, Aidan and Gu, Alex and Chalamala, Rahul and Song, Peiyang and Yu, Shixing and Godil, Saad and Prenger, Ryan and Anandkumar, Anima},
  booktitle={Neural Information Processing Systems (NeurIPS)},
  year={2023}
}

Lean 3 is deprecated. The main branch only supports Lean 4. You may use the legacy branch if you want to work with Lean 3.

GitHub license Code style: black

Quick Links

Using Trained Models on Hugging Face

Model name Model architecture Input Output
kaiyuy/leandojo-lean4-tacgen-byt5-small ByT5 (encoder-decoder) Proof state Tactic
kaiyuy/leandojo-lean4-retriever-byt5-small ByT5 (encoder-only) Proof state Embedding
kaiyuy/leandojo-lean4-retriever-tacgen-byt5-small ByT5 (encoder-decoder) Retrieved premises + proof state Tactic

Our trained models are available on HuggingFace Hub. With minimum dependencies (only PyTorch and HuggingFace Transformers), you can use our models to perform inference, finetune them on your own data, or plug them into your customized theorem proving pipeline. Below are some examples.

Tactic Generator

Our tactic generator is a ByT5 model finetuned to generate tactics given a proof state.

from transformers import AutoTokenizer, AutoModelForSeq2SeqLM

tokenizer = AutoTokenizer.from_pretrained("kaiyuy/leandojo-lean4-tacgen-byt5-small")
model = AutoModelForSeq2SeqLM.from_pretrained("kaiyuy/leandojo-lean4-tacgen-byt5-small")

state = "n : ℕ\n⊢ gcd n n = n"
tokenized_state = tokenizer(state, return_tensors="pt")

# Generate a single tactic.
tactic_ids = model.generate(tokenized_state.input_ids, max_length=1024)
tactic = tokenizer.decode(tactic_ids[0], skip_special_tokens=True)
print(tactic, end="\n\n")

# Generate multiple tactics via beam search.
tactic_candidates_ids = model.generate(
    tokenized_state.input_ids,
    max_length=1024,
    num_beams=4,
    length_penalty=0.0,
    do_sample=False,
    num_return_sequences=4,
    early_stopping=False,
)
tactic_candidates = tokenizer.batch_decode(
    tactic_candidates_ids, skip_special_tokens=True
)
for tac in tactic_candidates:
    print(tac)

The expected output is shown below. <a> and </a> are markers of premises in generated tactics. You should remove them when using the tactics.

rw [gcd_comm, gcd_rec n n]

simp [gcd]
apply Nat.dvd_antisymm
induction' n with n n_ih
induction' n with n hn

Premise Retriever

At the core of our premise retriever is a ByT5 encoder that embeds states and premises into vectors. You can use the vectors to perform retrieval by maximizing cosine similarity.

import torch
from typing import Union, List
from transformers import AutoTokenizer, AutoModelForTextEncoding

tokenizer = AutoTokenizer.from_pretrained("kaiyuy/leandojo-lean4-retriever-byt5-small")
model = AutoModelForTextEncoding.from_pretrained("kaiyuy/leandojo-lean4-retriever-byt5-small")

state = "n : ℕ\n⊢ gcd n n = n"
premises = [
  "<a>vsub_eq_zero_iff_eq</a> @[simp] lemma vsub_eq_zero_iff_eq {p1 p2 : P} : p1 -ᵥ p2 = (0 : G) ↔ p1 = p2",
  "<a>is_scalar_tower.coe_to_alg_hom'</a> @[simp] lemma coe_to_alg_hom' : (to_alg_hom R S A : S → A) = algebra_map S A",
  "<a>polynomial.X_sub_C_ne_zero</a> theorem X_sub_C_ne_zero (r : R) : X - C r ≠ 0",
  "<a>forall_true_iff</a> theorem forall_true_iff : (α → true) ↔ true",
  "def <a>Nat.gcd</a> : Nat → Nat → Nat\n| 0        y := y\n| (succ x) y := have y % succ x < succ x, from mod_lt _ $ succ_pos _,\n                gcd (y % succ x) (succ x)",
  "@[simp] theorem <a>Nat.gcd_zero_left</a> (x : Nat) : gcd 0 x = x",
  "@[simp] theorem <a>Nat.gcd_succ</a> (x y : Nat) : gcd (succ x) y = gcd (y % succ x) (succ x)",
  "@[simp] theorem <a>Nat.mod_self</a> (n : Nat) : n % n = 0",
]  # A corpus of premises to retrieve from.

@torch.no_grad()
def encode(s: Union[str, List[str]]) -> torch.Tensor:
    """Encode texts into feature vectors."""
    if isinstance(s, str):
        s = [s]
        should_squeeze = True
    else:
        should_squeeze = False
    tokenized_s = tokenizer(s, return_tensors="pt", padding=True)
    hidden_state = model(tokenized_s.input_ids).last_hidden_state
    lens = tokenized_s.attention_mask.sum(dim=1)
    features = (hidden_state * tokenized_s.attention_mask.unsqueeze(2)).sum(dim=1) / lens.unsqueeze(1)
    if should_squeeze:
      features = features.squeeze()
    return features

@torch.no_grad()
def retrieve(state: str, premises: List[str], k: int) -> List[str]:
    """Retrieve the top-k premises given a state."""
    state_emb = encode(state)
    premise_embs = encode(premises)
    scores = (state_emb @ premise_embs.T)
    topk = scores.topk(k).indices.tolist()
    return [premises[i] for i in topk]

for p in retrieve(state, premises, k=4):
    print(p, end="\n\n")

Expected output:

def <a>Nat.gcd</a> : Nat → Nat → Nat
| 0        y := y
| (succ x) y := have y % succ x < succ x, from mod_lt _ $ succ_pos _,
                gcd (y % succ x) (succ x)

@[simp] theorem <a>Nat.gcd_zero_left</a> (x : Nat) : gcd 0 x = x

@[simp] theorem <a>Nat.gcd_succ</a> (x y : Nat) : gcd (succ x) y = gcd (y % succ x) (succ x)

@[simp] theorem <a>Nat.mod_self</a> (n : Nat) : n % n = 0

Retrieval-Augmented Tactic Generator

ReProver's tactic generator takes as input the concatenation of retrieved premises and the state.

from transformers import AutoTokenizer, AutoModelForSeq2SeqLM

tokenizer = AutoTokenizer.from_pretrained("kaiyuy/leandojo-lean4-retriever-tacgen-byt5-small")
model = AutoModelForSeq2SeqLM.from_pretrained("kaiyuy/leandojo-lean4-retriever-tacgen-byt5-small")

state = "n : ℕ\n⊢ gcd n n = n"
retrieved_premises = [
  "def <a>Nat.gcd</a> : Nat → Nat → Nat\n| 0        y := y\n| (succ x) y := have y % succ x < succ x, from mod_lt _ $ succ_pos _,\n                gcd (y % succ x) (succ x)",
  "@[simp] theorem <a>Nat.mod_self</a> (n : Nat) : n % n = 0",
]
input = "\n\n".join(retrieved_premises + [state])
print("------ INPUT ------\n", input)
tokenized_input = tokenizer(input, return_tensors="pt", max_length=2300, truncation=True)

# Generate a single tactic.
tactic_ids = model.generate(tokenized_input.input_ids, max_length=1024)
tactic = tokenizer.decode(tactic_ids[0], skip_special_tokens=True)
print("\n------ OUTPUT ------")
print(tactic, end="\n\n")

# Generate multiple tactics via beam search.
tactic_candidates_ids = model.generate(
    tokenized_input.input_ids,
    max_length=1024,
    num_beams=4,
    length_penalty=0.0,
    do_sample=False,
    num_return_sequences=4,
    early_stopping=False,
)
tactic_candidates = tokenizer.batch_decode(
    tactic_candidates_ids, skip_special_tokens=True
)
for tac in tactic_candidates:
    print(tac)

Expected output:

------ INPUT ------
 def <a>Nat.gcd</a> : Nat → Nat → Nat
| 0        y := y
| (succ x) y := have y % succ x < succ x, from mod_lt _ $ succ_pos _,
                gcd (y % succ x) (succ x)

@[simp] theorem <a>Nat.mod_self</a> (n : Nat) : n % n = 0

n : ℕ
⊢ gcd n n = n

------ OUTPUT ------
rw [gcd_def, ← gcd_def, ← gcd_def, ← gcd_def]

simp [gcd]
rw [gcd]
rw [gcd_def]
rw [← Nat.mod_self n, ← Nat.mod_self n]

The rest of this document describes our system for training and evaluating LLM-based provers.

Using the Model Directly in Lean

Check out Lean Copilot if you want to run ReProver's tactic generator directly in Lean's VSCode workflow.

Requirements

  1. Download and install Miniconda Python 3 (Anaconda should also work).
  2. Create the conda environment and install Python dependencies:
conda create --yes --name ReProver python=3.11 ipython
conda activate ReProver
pip install torch  # Depending on your CUDA version; see https://pytorch.org/.
pip install tqdm loguru deepspeed "pytorch-lightning[extra]" transformers wandb openai rank_bm25 lean-dojo vllm
  1. Prepend the repo's root to the PYTHONPATH environment variable.
  2. Make sure wget and tar are available. Then, run python scripts/download_data.py to download LeanDojo Benchmark 4. They will be saved to ./data.
  3. Satisfy the requirements of LeanDojo.
  4. Use LeanDojo to trace all repos in the datasets: python scripts/trace_repos.py. This step may take some time. Please refer to LeanDojo's documentation if you encounter any issues.
  5. Run wandb login to log in Weights & Biases.

Premise Selection

We use Lightning CLI to create retrieval/main.py for training, validation, and testing the premise retrieval. It takes command line arguments as well as YAML config files. Please run python retrieval/main.py --help or refer to the documentation of Lightning CLI for details.

The config files for our experiments are in ./retrieval/confs. We train all models on a single NVIDIA A100 GPU with 80GB memory. When using GPUs with smaller memory, you can change batch_size, accumulate_grad_batches, and num_negatives. However, it may impact the performance due to in-batch negatives in DPR.

Training the Premise Retriever

Run python retrieval/main.py fit --help to see how to use the training script. For example:

mkdir logs  # Create the directory for training logs.
python retrieval/main.py fit --config retrieval/confs/cli_lean4_random.yaml --trainer.logger.name train_retriever_random --trainer.logger.save_dir logs/train_retriever_random         # Train the retriever on the `random` split of LeanDojo Benchmark 4.
python retrieval/main.py fit --config retrieval/confs/cli_lean4_novel_premises.yaml --trainer.logger.name train_retriever_novel_premises --trainer.logger.save_dir logs/train_retriever_novel_premises  # Train the retriever on the `novel_premises` split of LeanDojo Benchmark 4.

Hyperparameters and model checkpoints are saved in ./logs/train_retriever_*, and you can monitor the training process on Weights & Biases.

Retrieving Premises for All Proof States

After the models are trained, run the following commands to retrieve premises for all proof states in the dataset.

python retrieval/main.py predict --config retrieval/confs/cli_lean4_random.yaml --ckpt_path $PATH_TO_RETRIEVER_CHECKPOINT --trainer.logger.name predict_retriever_random --trainer.logger.save_dir logs/predict_retriever_random 
python retrieval/main.py predict --config retrieval/confs/cli_lean4_novel_premises.yaml --ckpt_path $PATH_TO_RETRIEVER_CHECKPOINT --trainer.logger.name predict_retriever_novel_premises --trainer.logger.save_dir logs/predict_retriever_novel_premises

Retrieved premises are saved to ./logs/predict_retriever_*/predictions.pickle. Note that PATH_TO_RETRIEVER_CHECKPOINT is the DeepSpeed model checkpoint produced in the previous step. If you want to use a Hugging Face checkpoint instead, a workaround would be to run the training for 1 step with zero learning rate.

Evaluating the Retrieved Premises

After predictions are saved, evaluate them using metrics such as R@1, R@10, and MRR.

python retrieval/evaluate.py --data-path data/leandojo_benchmark_4/random --preds-file logs/predict_retriever_random/predictions.pickle
python retrieval/evaluate.py --data-path data/leandojo_benchmark_4/novel_premises --preds-file logs/predict_retriever_novel_premises/predictions.pickle

Theorem Proving

Training the Tactic Generator

Similar to premise selection, you can run python generation/main.py --help and python generation/main.py fit --help to check the command line options.

To train tactic generators without retrieval:

python generation/main.py fit --config generation/confs/cli_lean4_random.yaml --trainer.logger.name train_generator_random --trainer.logger.save_dir logs/train_generator_random            # LeanDojo Benchmark 4, `random` split
python generation/main.py fit --config generation/confs/cli_lean4_novel_premises.yaml --trainer.logger.name train_generator_novel_premises --trainer.logger.save_dir logs/train_generator_novel_premises    # LeanDojo Benchmark 4, `novel_premises` split

Hyperparameters and model checkpoints are saved in ./logs/train_generator_*, and you can monitor the training process on Weights & Biases.

To train models augmented by retrieval, we need to provide a retriever checkpoint and its predictions on all proof states in the dataset:

python generation/main.py fit --config generation/confs/cli_lean4_random.yaml --model.ret_ckpt_path $PATH_TO_RETRIEVER_CHECKPOINT --data.preds_path logs/predict_retriever_random/predictions.pickle --trainer.logger.name train_reprover_random --trainer.logger.save_dir logs/train_reprover_random
python generation/main.py fit --config generation/confs/cli_lean4_novel_premises.yaml --model.ret_ckpt_path $PATH_TO_RETRIEVER_CHECKPOINT --data.preds_path logs/predict_retriever_novel_premises/predictions.pickle --trainer.logger.name train_reprover_novel_premises --trainer.logger.save_dir logs/train_reprover_novel_premises

Theorem Proving Evaluation on LeanDojo Benchmark

After the tactic generator is trained, we combine it with best-first search to prove theorems by interacting with Lean. The evaluation script takes Hugging Face model checkpoints (either local or remote) as input. For remote models, you can simply use their names, e.g., kaiyuy/leandojo-lean4-tacgen-byt5-small. For locally trained models, you first need to convert them from PyTorch Ligthning checkpoints to Hugging Face checkpoints:

python scripts/convert_checkpoint.py generator --src $PATH_TO_GENERATOR_CHECKPOINT --dst ./leandojo-lean4-tacgen-byt5-small
python scripts/convert_checkpoint.py retriever --src $PATH_TO_RETRIEVER_CHECKPOINT --dst ./leandojo-lean4-retriever-byt5-small
python scripts/convert_checkpoint.py generator --src $PATH_TO_REPROVER_CHECKPOINT --dst ./leandojo-lean4-retriever-tacgen-byt5-small

, where PATH_TO_GENERATOR_CHECKPOINT and PATH_TO_RETRIEVER_CHECKPOINT are PyTorch Ligthning checkpoints produced by the training script.

To evaluate the model without retrieval, run (using the random data split as example):

python prover/evaluate.py --data-path data/leandojo_benchmark_4/random/ --gen_ckpt_path ./leandojo-lean4-tacgen-byt5-small --split test --num-workers 5 --num-gpus 1

You may tweak --num-workers and --num-gpus to fit your hardware.

For the model with retrieval, first use the retriever to index the corpus (pre-computing the embeddings of all premises):

python retrieval/index.py --ckpt_path ./leandojo-lean4-retriever-byt5-small --corpus-path data/leandojo_benchmark_4/corpus.jsonl --output-path $PATH_TO_INDEXED_CORPUS

It saves the indexed corpus as a pickle file to PATH_TO_INDEXED_CORPUS.

Then, run:

python prover/evaluate.py --data-path data/leandojo_benchmark_4/random/  --gen_ckpt_path ./leandojo-lean4-retriever-tacgen-byt5-small --ret_ckpt_path ./leandojo-lean4-retriever-byt5-small --indexed-corpus-path $PATH_TO_INDEXED_CORPUS --split test --num-workers 5 --num-gpus 1

Questions and Bugs

  • For general questions and discussions, please use GitHub Discussions.
  • To report a potential bug, please open an issue.