MiniF2F

MiniF2F is a formal mathematics benchmark (translated across multiple formal systems) consisting of exercise statements from olympiads (AMC, AIME, IMO) as well as high-school and undergraduate maths classes.

The goal of the project is to provide a shared benchmark to evaluate and directly compare automated theorem proving systems based on the formal systems targeted, initially Lean, Isabelle, and Metamath (targeting also Hol Light).

The benchmark (released under permissive licenses (MIT for Metamath, Apache for Lean)) is a work in progress and contributions are welcome and encouraged through pull requests.

Citation

The initial version of the benchmark is described in detail in the following pre-print:

@article{zheng2021minif2f,
  title={MiniF2F: a cross-system benchmark for formal Olympiad-level mathematics},
  author={Zheng, Kunhao and Han, Jesse Michael and Polu, Stanislas},
  journal={arXiv preprint arXiv:2109.00110},
  year={2021}
}

The original repo is miniF2F.

It has then seen significant fixes and improvements, notably the addition of an informal statement and an informal proof for each problem. The curation of the informal component is described in the following paper. To cite it:

@inproceedings{
  anonymous2023draft,
  title={Draft, Sketch, and Prove: Guiding Formal Theorem Provers with Informal Proofs},
  author={Anonymous},
  booktitle={Submitted to The Eleventh International Conference on Learning Representations},
  year={2023},
  url={https://openreview.net/forum?id=SMa9EAovKMC},
  note={under review}
}

We decided to start a separate repository, instead of submitting PRs, for better maintainence of the dataset.

Statistics

	Test	Valid
Lean	244	244
Metamath	244	244
Isabelle	244	244
Hol Light	165	165
Informal	244	244

Example problem statement (mathd_algebra_17)

Informal

Solve for $a$: $\sqrt{4+\sqrt{16+16a}}+ \sqrt{1+\sqrt{1+a}} = 6.$ Show that it is 8.

Lean

theorem mathd_algebra_17
  (a : ℝ)
  (h₀ : real.sqrt (4 + real.sqrt (16 + 16 * a)) + real.sqrt (1 + real.sqrt (1 + a)) = 6) :
  a = 8 :=
begin
  sorry
end

Isabelle

theorem mathd_algebra_17:
  fixes a :: real
  assumes "1 + a>0"
  assumes "sqrt (4 + sqrt (16 + 16 * a)) 
    + sqrt (1 + sqrt (1 + a)) = 6" 
  shows "a = 8"
  sorry

 Hol Light

let mathd-algebra-17 = `!a. sqrt (&4 + sqrt (&16 + &16 * a)) + sqrt (&1 + sqrt (&1 + a)) = &6 /\ &0 <= (&1 + a) ==> a = &8`;;

Structure

Each problem is represented by a unique name and a file for each of the formal systems we target. Each file consists at minima in the problem statement and optionally one or more example proofs associated with it. The benchmark is divided in two splits:

• valid: validation set that can be used while designing automated theorem proving systems (early-stopping, reinforcement learning, data-augmentation, curriculum design, ...).
• test: held-out test set reserved for final evaluation.

Naming conventions are still a work in progress. Olympiads problems are generally named after their competition year and problem number (eg. imo-1990-p3 or aime-1983-p2). Problems coming from a particular dataset (eg the MATH dataset) are named to ease their retrieval (eg. mathd-algebra-125). Other problems are prefixed by a category hint and a unique name in the style of Metamath naming conventions (eg. induction-11div10tonmn1ton).

Each exercise file complies to the following system-specific conventions.

Lean

To install the project make sure you have elan installed, then in the directory where you want the project installed run:

git clone https://github.com/openai/miniF2F
cd miniF2F
leanpkg configure
leanproject get-mathlib-cache
leanproject build

Since having one file per statement causes slowness in Lean parsing stage, all Lean statements are exceptionally aggregated in two files (valid.lean and test.lean). These files contain a list of the problem statements defined as theorems. Optionally, proofs for these statements are provided as well as potential lemmas to support the ground-truth proof.

No theorem should appear that do not correspond to a problem statement; use lemma instead.

Please use lean/scripts/lint_style.py to check all the statements pass the linter. You can also make use of lean/scripts/simple_formatter.sh to enforce a few basic formatting rules.

The lean folder is released under the Apache License (so that it is aligned with Lean's mathlib license).

Metamath

Each file contains the problem statement with the same name as the problem unique name. The statement is commented (using Metamath convention) if provided without proof.

The metamath folder is released under the MIT License.

HOL Light

Each file contains the problem statement defined as a HOL Light term whose name must match the file name.

The hollight folder is released under the FreeBSD License.

Isabelle

Each file contains the problem statement defined as a theorem whose name must match the file name, optionally with a proof for it as well as the necessary imports.

The isabelle folder is released under the Apache License.

Informal

Each file contains the problem statement and the proof written in natural mathematical language. The data come from the following sources: (1) The MATH dataset (2) The Art of Problem Solving website (3) Albert Qiaochu Jiang, Timothée Lacroix, Guillaume Lample, Sean Welleck, Jiacheng Liu, and Marie-Anne Lachaux.

Code of Conduct and Contributions

MiniF2F is meant to serve as a shared and useful resource for the machine learning community working on formal mathematics.

There is no obligation tied with the use and reporting of a result based on miniF2F. But if you're using it and discovering new proofs (manually or automatically) please contribute them back to the benchmark.

All contributions, such as new statements for later versions, addition of missing statements for existing versions, bug fixes, additional proofs are all welcome.

Versioning

A version of miniF2F is defined by a frozen set of statements. The goal for each version is to get full coverage on all formal systems for that version even if that might not be the case when the version is frozen.

When reporting a result based on miniF2F please always specify the version you used. The current version is v2, frozen as of October 2022, including 488 statements (fully translated to Lean, Isabelle, and Metamath but still WIP in other formal systems).

Each version will live in its own branch to allow later additions of translated statements or fixes to existing statements as needed. The main branch remains reserved for active development and should not be used when reporting results.

Active version

• Version v2
• Freeze date: October 2022
• Branch: v2

Previous versions

• Version: v1
• Freeze date: August 2021
• Branch: v1