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- ---
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- license: apache-2.0
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- ---
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ # Lemmas Dataset
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+ Lemmas dataset is a benchmark that consists of building blocks for 13 IMO problems, and also the complete formal proofs for 20 IMO problems. The topics cover a variety of concepts ranging from divisibility to finite sets and functions. All proof steps are written in Lean 4.
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+
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+ All files compile with no error in Lean v4.17.0.
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+
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+ The purpose of the dataset is to expose current theorem provers' ability in solving IMO problems and highlight their strengths and weaknesses.
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+
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+ The dataset is released under MIT license. Contributions are welcome through pull requests.
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+
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+ IMO problems formalized in this paper are as follows:
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+
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+ | \# | Year | Problem | Topic | in miniF2F | Lean proof publicly available | \# of lines of Lean4 code |
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+ |-------|------|---------|---------------|------------|-------------------------------|---------------------------|
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+ | 1 | 1959 | P1 | number theory | Yes | Yes | 9 |
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+ | 2 | 1960 | P2 | algebra | Yes | Yes | 40 |
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+ | 3 | 1962 | P2 | algebra | Yes | No | 60 |
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+ | 4 | 1963 | P5 | algebra | Yes | No | 50 |
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+ | 5 | 1964 | P2 | algebra | Yes | Yes | 50 |
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+ | 6 | 1965 | P2 | algebra | Yes | No | 210 |
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+ | 7 | 1968 | P5 | algebra | Yes | No | 30 |
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+ | 8 | 1969 | P2 | algebra | Yes | No | 150 |
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+ | 9 | 1974 | P3 | number theory | Yes | No | 510 |
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+ | 10 | 1981 | P6 | algebra | Yes | No | 40 |
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+ | 11 | 1982 | P1 | algebra | Yes | No | 75 |
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+ | 12 | 1983 | P6 | algebra | Yes | No | 180 |
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+ | 13 | 1984 | P6 | number theory | Yes | No | 380 |
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+ | 14 | 1985 | P6 | number theory | Yes | No | 1,310 |
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+ | 15 | 1992 | P1 | number theory | Yes | No | 480 |
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+ | 16 | 1997 | P5 | number theory | Yes | No | 390 |
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+ | 17 | 2007 | P6 | algebra | Yes | No | 570 |
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+ | 18 | 2022 | P2 | algebra | No | No | 260 |
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+ | 19 | 2022 | P5 | number theory | No | No | 640 |
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+ | 20 | 2023 | P4 | number theory | No | No | 450 |
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+ | total | | | | | | 5,884 |
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+
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+
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+ ## Performance of SOTA LLMs on the Lemmas Dataset
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+ | Problem | # of lemmas | DeepSeek Prover-v1.5-RL (@32) | Goedel-Prover (@32) | ReProver retrieval ✗ | ReProver retrieval ✓ | o3-mini (with 10 e.f.) |
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+ |:---------:|:--------------:|:-------------------------------------------:|:-------------------------------------------:|:-------------------------------:|:-------------------------------:|:-------------------------------:|
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+ | 1959-p1 | 4 | 3 (75.0\%) | 2 (50.0\%) | 2 (50.0\%) | 2 (50.0\%) | 2 (50.0\%) |
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+ | 1960-p2 | 9 | 7 (77.8\%) | 6 (66.7\%) | 3 (33.3\%) | 4 (44.4\%) | 1 (11.1\%) |
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+ | 1962-p2 | 14 | 13 (92.9\%) | 12 (85.7\%) | 7 (50.0\%) | 8 (57.1\%) | 6 (42.9\%) |
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+ | 1964-p2 | 9 | 5 (55.6\%) | 5 (55.6\%) | 5 (55.6\%) | 5 (55.6\%) | 3 (33.3\%) |
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+ | 1965-p2 | 73 | 48 (65.8\%) | 47 (64.4\%) | 47 (64.4\%) | 46 (63.0\%) | 12 (16.4\%) |
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+ | 1983-p6 | 53 | 25 (47.2\%) | 32 (60.4\%) | 28 (52.8\%) | 29 (54.7\%) | 18 (34.0\%) |
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+ | 1984-p6 | 64 | 31 (50.0\%) | 33 (51.6\%) | 25 (39.1\%) | 24 (37.5\%) | 13 (20.3\%) |
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+ | 1985-p6 | 427 | 116 (27.2\%) | 116 (27.2\%) | 89 (20.8\%) | 89 (20.8\%) | 84 (19.7\%) |
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+ | 1992-p1 | 91 | 48 (52.7\%) | 54 (59.3\%) | 35 (38.5\%) | 34 (37.4\%) | 25 (27.5\%) |
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+ | 1997-p5 | 122 | 51 (41.8\%) | 49 (40.2\%) | 48 (39.3\%) | 51 (41.8\%) | 30 (24.6\%) |
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+ | 2022-p2 | 61 | 34 (55.7\%) | 30 (49.2\%) | 24 (39.3\%) | 25 (41.0\%) | 25 (41.0\%) |
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+ | 2022-p5 | 265 | 89 (33.6\%) | 76 (28.7\%) | 80 (30.2\%) | 77 (29.1\%) | 60 (22.6\%) |
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+ | 2023-p4 | 137 | 52 (38.0\%) | 41 (29.9\%) | 43 (31.4\%) | 45 (32.8\%) | 37 (27.0\%) |
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+ | Total | 1,329 | 522 (39.3\%) | 504 (37.9\%) | 436 (32.8\%) | 439 (33.0\%) | 316 (23.8\%) |
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+
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+ ## Distribution of Proof Length of LLM-Generated Proofs
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+ ![LLM Comparison](assets/all_llms.png)
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+ The figure demonstrates that the SOTA LLMs tend to write shorter proofs, which is a potential challenge for lengthier proof procedures that require long-term planning.
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+
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+ ## Citation
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+ The dataset, motivations and additional results are available [here](https://openreview.net/forum?id=CrKMqRAhBo)
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+ ```
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+ @article{
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+ yousefzadeh2025a,
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+ title={A Lean Dataset for International Math Olympiad: Small Steps towards Writing Math Proofs for Hard Problems},
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+ author={Roozbeh Yousefzadeh and Xuenan Cao and Azim Ospanov},
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+ journal={Transactions on Machine Learning Research},
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+ year={2025},
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+ url={https://openreview.net/forum?id=CrKMqRAhBo}
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+ }
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+ ```
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+ ## Versioning
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+ The Lemmas Dataset is explicitly written for the latest available Lean4 version at the moment - v4.17.0. Any subsequent release can impact the released lemmas and some proofs might need future revision.
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+
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+ ## Formalised IMO Problems Statistics
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+ | # | Year | Problem | Topic | in miniF2F | Lean proof publicly available | # of lemmas | # of lines of Lean4 proof |
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+ |:-------:|:------:|:---------:|:---------------:|:------------:|:-------------------------------:|:-------------:|:---------------------------:|
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+ | 1 | 1959 | p1 | number theory | Yes | Yes | 4 | 9 |
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+ | 2 | 1960 | p2 | algebra | Yes | Yes | 9 | 40 |
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+ | 3 | 1962 | p2 | algebra | Yes | No | 14 | 60 |
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+ | 4 | 1964 | p2 | algebra | Yes | Yes | 9 | 50 |
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+ | 5 | 1965 | p2 | algebra | Yes | No | 73 | 210 |
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+ | 6 | 1983 | p6 | algebra | Yes | No | 53 | 180 |
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+ | 7 | 1984 | p6 | number theory | Yes | No | 64 | 380 |
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+ | 8 | 1985 | p6 | number theory | Yes | No | 427 | 1,310 |
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+ | 9 | 1992 | p1 | number theory | Yes | No | 91 | 480 |
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+ | 10 | 1997 | p5 | number theory | Yes | No | 122 | 390 |
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+ | 11 | 2022 | p2 | algebra | No | No | 61 | 260 |
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+ | 12 | 2022 | p5 | number theory | No | No | 265 | 640 |
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+ | 13 | 2023 | p4 | number theory | No | No | 137 | 450 |
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+ | | total| | | | | 1,329 | 4,459 |
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+
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+ ## Additional Analysis of o3-mini Generated Proofs
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+ | \# | Problem | \# of lemmas | Correct proof in NL | Correct proof in Lean | Match between NL and Lean |
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+ |:-----:|:-------:|:------------:|:-------------------:|:---------------------:|:-------------------------:|
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+ | 1 | 1959-p1 | 4 | 100\% | 50.0\% | 100\% |
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+ | 2 | 1960-p2 | 9 | 55.6\% | 11.1\% | 100\% |
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+ | 3 | 1962-p2 | 14 | 92.9\% | 42.9\% | 100\% |
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+ | 4 | 1964-p2 | 9 | 77.8\% | 33.3\% | 100\% |
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+ | 5 | 1965-p2 | 73 | 97.3\% | 16.4\% | 100\% |
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+ | 6 | 1983-p6 | 53 | 64.2\% | 34.0\% | 100\% |
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+ | 7 | 1984-p6 | 64 | 73.4\% | 20.3\% | 100\% |
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+ | 8 | 1985-p6 | 427 | 75.2\% | 19.7\% | 95.6\% |
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+ | 9 | 1992-p1 | 91 | 87.6\% | 27.5\% | 100\% |
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+ | 10 | 1997-p5 | 122 | 69.7\% | 24.6\% | 100\% |
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+ | 11 | 2022-p2 | 61 | 77.0\% | 41.0\% | 100\% |
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+ | 12 | 2022-p5 | 265 | 63.4\% | 22.6\% | 92.8\% |
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+ | 13 | 2023-p4 | 137 | 88.3\% | 27.0\% | 92.7\% |
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+ | | total | 1,329 | 75.5\% | 23.8\% | 96.4\% |
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+