Restating Theorems With The Same Numbering In LaTeX Using Thmtools

Hey guys! Ever found yourself in a situation where you need to restate a theorem in your LaTeX document and want it to have the same number as the original? It's a common challenge, especially when writing lengthy mathematical papers or theses. You introduce a theorem, build upon it, and then later, you need to bring it back into the spotlight without making your readers flip back and forth. That's where the power of the thmtools package comes in handy, along with the trusty amsthm package. Let's dive into how you can achieve this like a pro.

Setting the Stage: Packages and Preliminaries

Before we jump into the nitty-gritty, let's make sure we have our toolkit ready. We'll be using two key packages:

• amsthm: This package provides the foundational theorem-like environments. It's the bread and butter for defining theorems, lemmas, propositions, and more.
• thmtools: This is our secret weapon! It extends the capabilities of amsthm, allowing us to restate theorems and maintain their original numbering.

To get started, you'll need to include these packages in your LaTeX document's preamble. Just add the following lines after your \documentclass declaration:

\documentclass{article}
\usepackage{amsthm}
\usepackage{thmtools}

Next, we need to define our theorem environment. This is where amsthm shines. We use the \newtheorem command to create a new theorem environment. Let's create a simple theorem environment that's numbered within sections:

\newtheorem{theorem}{Theorem}[section]

Here's what this line of code does:

• \newtheorem{theorem}{Theorem}: This defines a new environment called theorem. Whenever you use \begin{theorem} and \end{theorem}, LaTeX will know you're referring to a theorem environment.
• [section]: This tells LaTeX to number the theorems within each section. So, the first theorem in Section 1 will be Theorem 1.1, the second will be Theorem 1.2, and so on.

Now that we have our packages loaded and our theorem environment defined, we're ready to explore the magic of restating theorems with the same numbering.

The Core Technique: restatable and Its Power

The star of the show here is the \begin{restatable} environment provided by thmtools. This environment allows you to define a theorem that can be restated later in your document with the same number. Here’s the basic structure:

\begin{restatable}{<theorem_environment>}{<label>}
  ... Theorem statement ...
\end{restatable}

Let's break down the components:

• <theorem_environment>: This is the name of the theorem environment you defined earlier (e.g., theorem, lemma, proposition).
• <label>: This is a unique label you'll use to refer to this specific theorem. It's like giving your theorem a secret name so you can call it back later. A good practice is to prefix your label with something descriptive, like thm: for theorems, to keep things organized.

Now, let's see this in action. Suppose we have a theorem about the fundamental group:

\begin{restatable}{theorem}{thm:fundamentalGroup}
  The fundamental group of a circle is isomorphic to the group of integers under addition.
\end{restatable}

This defines our theorem and gives it the label thm:fundamentalGroup. LaTeX will automatically assign it a number based on the current section.

Now, let's say we want to restate this theorem later in our document. We use the \cref command along with the label we assigned. Here's how:

\cref{thm:fundamentalGroup}

But simply referencing the theorem isn't enough to restate it. We need to use the starred version of the restatable environment: \begin{restatable*}. Here's how it looks:

\begin{restatable*}{theorem}{thm:fundamentalGroup}
  The fundamental group of a circle is isomorphic to the group of integers under addition.
\end{restatable*}

Notice the asterisk (*) after restatable. This tells thmtools that we're restating a theorem that was previously defined. The crucial part is that we use the same <theorem_environment> (theorem in this case) and the same <label> (thm:fundamentalGroup). This ensures that the restated theorem will have the same number as the original.

Putting It All Together: A Complete Example

To solidify your understanding, let's create a complete example that you can compile and run yourself. This example will demonstrate how to define a theorem, restate it, and reference it using thmtools and amsthm.

\documentclass{article}
\usepackage{amsthm}
\usepackage{thmtools}
\usepackage{hyperref} % For clickable references

\newtheorem{theorem}{Theorem}[section]

\begin{document}

\section{Introduction}

\begin{restatable}{theorem}{thm:pythagorean}
  The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
\end{restatable}

We can state \cref{thm:pythagorean} as follows:

\begin{restatable*}{theorem}{thm:pythagorean}
  In a right-angled triangle, the square of the length of the side opposite the right angle (the hypotenuse) is equal to the sum of the squares of the lengths of the other two sides.
\end{restatable*}

\section{Applications}

\begin{theorem}[A Simple Application] \label{thm:application}
  Using \cref{thm:pythagorean}, we can calculate the length of the hypotenuse if we know the lengths of the other two sides.
\end{theorem}

We can see that \cref{thm:pythagorean} is a fundamental result in geometry.

\end{document}

In this example:

1. We load the necessary packages: amsthm, thmtools, and hyperref (for clickable references). Hyperref is optional but highly recommended for making your document more navigable. It creates hyperlinks from your references to the actual theorem statements.
2. We define a theorem environment called theorem that's numbered within sections.
3. We define the Pythagorean theorem using \begin{restatable} and give it the label thm:pythagorean.
4. We restate the Pythagorean theorem later in the document using \begin{restatable*} and the same label. This ensures it has the same number.
5. We reference the theorem using \cref, which will generate a clickable link to the theorem statement.
6. We define another theorem (thm:application) that refers back to the Pythagorean theorem, demonstrating how you can build upon restated theorems.

Compile this LaTeX code, and you'll see that both instances of the Pythagorean theorem have the same number, and the references using \cref will correctly link to the theorem statements. This is super handy for keeping your document clear and consistent.

Advanced Techniques and Tips

Now that you've mastered the basics, let's explore some advanced techniques and tips for using thmtools effectively:

Customizing Theorem Styles

amsthm provides several predefined theorem styles (e.g., plain, definition, remark). You can customize the appearance of your theorems by using the \theoremstyle command. This command takes one of the predefined styles as an argument. For example:

\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]

This will create a definition environment with a different visual style than the default theorem style. Experiment with different styles to find what suits your document best. You can also define your own custom styles if you're feeling adventurous!

Using Theorem Options

When defining a theorem, you can include optional arguments within square brackets. These options can be used to add extra information to the theorem heading, such as a name or a source.

\begin{theorem}[Pythagorean Theorem] \label{thm:pythagorean}
  ...
\end{theorem}

This will display the theorem heading as "Theorem 1.1 (Pythagorean Theorem)". This can be particularly useful for well-known theorems or for citing the source of a theorem.

Working with Multiple Authors

If you're co-authoring a document, it's essential to have a consistent labeling scheme. Consider including your initials in the theorem labels to avoid conflicts. For example:

• thm:jr:pythagorean (for John's version of the Pythagorean theorem)
• thm:as:fundamentalGroup (for Alice's theorem on the fundamental group)

This simple convention can save you a lot of headaches down the road.

Dealing with Complex Theorem Statements

Sometimes, theorem statements can be quite long and complex. To improve readability, consider breaking them down into smaller chunks using displayed equations or itemized lists.

\begin{theorem}
  The following statements are equivalent:
  \begin{enumerate}
    \item Statement 1
    \item Statement 2
    \item Statement 3
  \end{enumerate}
\end{theorem}

This makes the theorem statement easier to digest and understand.

Common Pitfalls and How to Avoid Them

Even with the power of thmtools, there are a few common pitfalls to watch out for:

1. Forgetting the asterisk in \begin{restatable*}: This is the most common mistake. If you forget the asterisk, LaTeX will treat the restatement as a new theorem, and it will get a different number. Always double-check that you're using the starred version when restating.
2. Using the wrong label: Make sure you're using the same label for both the original theorem and its restatement. Typos in labels can lead to confusing errors. Copying and pasting the label is your friend here!
3. Not loading thmtools: If you forget to include \usepackage{thmtools} in your preamble, the restatable environment won't be defined, and you'll get an error. Always load the package before using its features.
4. Conflicting labels: If you use the same label for two different theorems, LaTeX will get confused. Make sure your labels are unique within your document. A good labeling convention (like including your initials) can help prevent this.

By being aware of these potential issues, you can avoid frustration and ensure that your theorems are numbered and referenced correctly.

Conclusion: Mastering Theorem Restatement

Using thmtools to restate theorems with the same numbering is a powerful technique for writing clear and organized mathematical documents. By understanding the restatable environment and following the best practices outlined in this article, you can streamline your writing process and create documents that are both elegant and easy to read. So go ahead, give it a try, and elevate your LaTeX skills to the next level! You've got this!