Datasets:

hoskinson-center
/

proof-pile

text stringlengths 28 2.36M	meta stringlengths 20 188
\section{Invariance of the mutual information under spatial coupling}\label{sec:subadditivitystyle} In this section we prove that the mutual information remains unchanged under spatial coupling in a suitable asymptotic limit (Theorem \ref{LemmaGuerraSubadditivityStyle}). We will compare the mutual informations of th...	{"config": "arxiv", "file": "1812.02537/v5 arxiv/sections/5_interpolation_subadditivity_jeanRevision.tex"}
TITLE: Diverging improper integral QUESTION [6 upvotes]: When asked to evaluate $\int_{a}^{\infty}f(x)dx$, you split the interval based on the improper points. If there is another improper point other than $\infty$, at $b$, we will write: $\int_{a}^{\infty}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx+\int_{c}^{\infty}f...	{"set_name": "stack_exchange", "score": 6, "question_id": 13802}
TITLE: If $n$ vectors are linearly independent, is there only one way to write a vector as a linear combination of those vectors? QUESTION [2 upvotes]: I know the converse is true, because if you can write 0 in two ways you can keep adding 0 to get an infinite number of linear combinations that sum to the same thing. (...	{"set_name": "stack_exchange", "score": 2, "question_id": 87270}
TITLE: Permutations and Combinations. QUESTION [0 upvotes]: There are 4 red and 6 blue marbles in a bag, 3 are picked out at random, what is the probability that atleast one of them is red? My answer is 1/2 First I calculated the total number of ways that I can pick out three marbles at random: 10C3 = 120. Then I multi...	{"set_name": "stack_exchange", "score": 0, "question_id": 593868}
TITLE: Difficulty with a tricky matrix exponential step QUESTION [2 upvotes]: I am having great difficulty checking a step involving operations with 2x2 matrix exponentials. The expression I would like to simplify is $$\lim _{x\to \infty} e^{-iHt-Vt}$$ where, for some $\epsilon, x, y \in \mathbb{R}$, we define $$H=\beg...	{"set_name": "stack_exchange", "score": 2, "question_id": 3617895}
TITLE: Evaluating the limit $\lim_{n\to+\infty}(\sqrt[n]{n}-1)^n$ QUESTION [6 upvotes]: Evaluate the limit $$\lim_{n\to+\infty}(\sqrt[n]{n}-1)^n$$ I know the limit is 0 by looking at the graph of the function, but how can I algebraically show that that is the limit? REPLY [1 votes]: Since, $\frac{\log(x)}x\le\frac1e$,...	{"set_name": "stack_exchange", "score": 6, "question_id": 576009}
TITLE: $\sum_{k=0}^n {n \choose k} ^{2} = {2n \choose n}$ - Generating function $\sum_{k=0}^\infty \binom nk x^k = (1+x)^n$. QUESTION [2 upvotes]: As part of a preparatory course in the contest PUTNAM, I have to show $\sum_{k=0}^n {n \choose k} ^{2} = {2n \choose n}$. I know that I can use the identity $\sum_{k=0}^n {n...	{"set_name": "stack_exchange", "score": 2, "question_id": 1473827}
\begin{document} \articletype{} \title{Structure from Appearance: Topology with Shapes, without Points} \author{ \name{Alexandros Haridis\textsuperscript{a}\thanks{CONTACT A. Haridis. Email: charidis@mit.edu} } \affil{\textsuperscript{a}Department of Architecture, Massachusetts Institute of Technology, 77 Massachus...	{"config": "arxiv", "file": "1902.03974/interacttfssample.tex"}
TITLE: Quillen groupoid of a groupoid. QUESTION [1 upvotes]: For any category $\mathcal{C}$ we can define its Quillen's groupoid, denoted $\mathcal{Q}(\mathcal{C})$, as the category which have the same objects than $\mathcal{C}$ and the arrows between two objects are $\operatorname{Hom}_{\mathcal{Q}(\mathcal{C})}(c, c^...	{"set_name": "stack_exchange", "score": 1, "question_id": 1101005}
TITLE: How to prove that the intersection of $L^1(\mathbb{R})$ and $L^2(\mathbb{R})$ is dense in $L^2(\mathbb{R})$ QUESTION [5 upvotes]: How to prove that the intersection of $L^1(\mathbb{R})$ space and $L^2(\mathbb{R})$ space is dense in $L^2(\mathbb{R})$ space? REPLY [7 votes]: If $f \in L^2(\mathbb R)$, then defin...	{"set_name": "stack_exchange", "score": 5, "question_id": 541576}

End of preview. Expand in Data Studio

Dataset Description

The proof-pile is a 13GB pre-training dataset of mathematical text that comprises 8.3 billion tokens (using the gpt-neox tokenizer). Models trained on this dataset are coming soon :) The dataset is composed of diverse sources of both informal and formal mathematics, namely

ArXiv.math (10GB)
Open-source math textbooks (50MB)
Formal mathematics libraries (500MB)
- Lean mathlib and other Lean repositories
- Isabelle AFP
- Coq mathematical components and other Coq repositories
- HOL Light
- set.mm
- Mizar Mathematical Library
Math Overflow and Math Stack Exchange (2.5GB)
Wiki-style sources (50MB)
- ProofWiki
- Wikipedia math articles
MATH dataset (6MB)

The construction of the dataset is reproducible using the code and instructions in the proof-pile Github repo.

Supported Tasks

This dataset is intended to be used for pre-training and fine-tuning language models. We envision models trained on the proof-pile will have many downstream applications, including informal quantitative reasoning, formal theorem proving, semantic search for formal mathematics, and autoformalization.

Languages

All informal mathematics in the proof-pile is written in English and LaTeX (arXiv articles in other languages are filtered out using languagedetect). Formal theorem proving languages represented in this dataset are Lean 3, Isabelle, Coq, HOL Light, Metamath, and Mizar.

Evaluation

The version of set.mm in this dataset has 10% of proofs replaced with the ? character in order to preserve a validation and test set for Metamath provers pre-trained on the proof-pile. The precise split can be found here: validation and test. The Lean mathlib commit used in this dataset is 6313863. Theorems created in subsequent commits can be used for evaluating Lean theorem provers.

This dataset contains only the training set of the MATH dataset. However, because this dataset contains ProofWiki, the Stacks Project, Trench's Analysis, and Stein's Number Theory, models trained on it cannot be evaluated on the NaturalProofs dataset.

Data Preprocessing

This section describes any significant filtering and transformations made to various subsets of the data.

arXiv.math. The arXiv.math dataset is large, heterogeneous, and contains a great deal of noise. We used the following heuristics when choosing which files from arXiv.math source folders to include in the dataset:

Keep only files with a .tex extension.
Only include files that use either a utf-8/16/32 or latin-1 text encoding.
Discard files that do not contain a part, chapter, section, sub...section, paragraph, or subparagraph heading.
Delete files that contain the keyword gnuplot. Gnuplot-latex is an old command line utility that generates blocks of entirely unintelligible source.
Include only articles in English, as determined by the langdetect library. \n", "\n",
Exclude files shorter than 280 characters (characters counted after substring removal described below).

In addition, we apply the following transformations to arXiv.math texts:

Delete everything outside of \begin{document} and \end{document}.
Delete everything including or after \Refs, \begin{thebibliography}, or \begin{bibdiv}
Delete comments.
Any more than three consecutive newlines are replaced by three consecutive newlines. In this notebook, we provide an analysis of the prevalence of noisy documents in the arXiv.math subset of the proof-pile.

Stack Exchange. We only include questions that have at least 5 upvotes and an answer. We format Stack Exchange posts as follows

QUESTION [{num_upvotes} upvotes]: {text of question}

REPLY [{num_upvotes} votes]: {text of reply}

REPLY [{num_upvotes} votes]: {text of reply}

.
.
.

set.mm. We converted set.mm into human-readable form by following the instructions in the mm-extract repo

Contributions

Authors: Zhangir Azerbayev, Edward Ayers, Bartosz Piotrowski.

We would like to thank Jeremy Avigad, Albert Jiang, and Wenda Li for their invaluable guidance, and the Hoskinson Center for Formal Mathematics for its support.

Downloads last month: 1,908

Size of the auto-converted Parquet files (First 5GB per split):

2.77 GB

Number of rows (First 5GB per split):

362,502

Models trained or fine-tuned on hoskinson-center/proof-pile