Theorem-like Environments in LaTeX

In this post, I will describe the setup I use for theorem-type environments in LaTeX (includes theorems, definitions, etc.).

A basic example of the code and output for a Theorem in a LaTeX document is shown here:

Code:

\begin{theorem}[Pythagorean]
  For a right triangle with sides $a$ and $b$, 
  and hypotenuse $c,$
  \[ a^2 + b^2 = c^2. \]
\end{theorem}

Output:

Given a right triangle with sides $a$ and $b$, and hypotenuse $c$, $$a^2 + b^2 = c^2.$$

The remainder of this document shows how to

• Setup the amsthm, hyperref, and cleveref packages for inserting and referencing Theorems, Definitions, and the like.
• Modify the style of Theorem environments to use slanted text instead of italics to improve readability in certain fonts (most notably, Times New Roman).
• Modify the QED mark or “Halmos Tombstone” (i.e., “$\square$”) that appears at the end of proofs and other environments.
• Define new theorem-like environments.
• Change the Theorem numbering scheme to include the chapter numbers.
• Create unnumbered theorem environments.
• Insert proofs, including modifying the introductory “Proof.” and moving the QED mark.

Packages

amsthm Package

The amsthm package defines macros and environments for creating theorem-like environments.

\usepackage{amsthm}

hyperref Package

The hyperref package provides hyperlinking. When hyperref is loaded, all references to theorem-like environments (inserted using \ref or \cref) automatically include links to the location where the environment is displayed. You will likely want to modify some of the hyperref options.

% Enable hyperlinks
\usepackage{hyperref}
\hypersetup{
    final,                   % Include links even if document is in 'draft' mode.
    hidelinks,               % Prevent boxes from being drawn around links in some viewers.
    breaklinks=true,         % Allow line breaks in the middle of links
    colorlinks=true,
    linkcolor=black,         % TOC, links to labeled equations and environments
    urlcolor=black!30!blue,  % URLS including links in references  
    anchorcolor=blue,        % I'm not sure what this does.
    citecolor=black!30!blue, % In-line citations  
}

cleveref Package

The cleveref package defines the \cref macro for inserting internal references to labeled object. In contrast to \ref, \cref inserts the type of environment referenced, so a reference \cref{my first theorem} will appear as “Theorem 1”, whereas \ref{my first theorem} only appears as “1”.

% Display "Theorem 1", "Lemma 2", etc. in cross references.  
% It's important to load this last after all the other packages!
\usepackage[
    noabbrev,   % E.g., "Figure" instead of "Fig."
    capitalise, % E.g., "Figure" instead of "figure"
    nameinlink  % Make "Figure 1" linked instead of just "1".
  ]{cleveref} 

Changing Theorem Body Text From Italic To Slanted

By default, the body amsthm theorem environments are italicized. I find large chunks of italic text hard to read, so I create and use a different theorem style that uses slanted text instead. This way, theorems are differentiated from surrounding text without compromising readability.

% Create a new amsthm theorem style that uses slanted text 
% instead of italics. I find this makes the text easier to read.
% See https://tex.stackexchange.com/a/417959/153678.
\usepackage{amssymb} % Provides \slshape
\newtheoremstyle{sltheorem}
                {}          % Space above
                {}          % Space below
                {\slshape}  % Body font
                {}          % Indent amount
                {\bfseries} % Head font
                {.}         % Punctuation after head
                { }         % Space after theorem head
                {}          % Theorem head spec
% Enable the new "sltheorem" theorem style.
\theoremstyle{sltheorem}

Discouraging Page Breaks in Theorems

By default, LaTeX will insert page breaks into the middle of a theorem statement just as readily as anywhere else in the document. This leads to unfortunate cases where one or two lines are on a separate page. LaTeX visual processor decides where to insert page breaks by a collection of penalties, with the goal of total minimizing the sum of the penalties. Thus, to discourage LaTeX from putting page breaks in a theorem, we can increase the penalties for page breaks between lines, and before and after display equations, which are stored in these macros:

\interlinepenalty   % Page break between lines.
\predisplaypenalty  % Page breaks before a display equation.
\postdisplaypenalty % Page break after equations.

To check the default values of these penalties, you can print them in your document by prefixing each macro with \the, such as,

  interlinepenalty=\the\interlinepenalty  \\
 predisplaypenalty=\the\predisplaypenalty \\
postdisplaypenalty=\the\postdisplaypenalty

For the document I used to the above code, the results were interlinepenalty=0, predisplaypenalty=10000, and postdisplaypenalty=0.

To modify the penalties use the syntax \interlinepenalty=<whole number>. You don’t want to manually set the penalties within in each theorem, however, so use a theorem style to set the values, as follows:

\newtheoremstyle{breakResistantTheorem}
     {}          % Space above
     {}          % Space below
     {%          % Theorem body font (default is "\upshape")
         \interlinepenalty=100%    Discourage breaking between lines.
         \predisplaypenalty=10000% Never break right before a display equation.
         \postdisplaypenalty=100%  Discourage breaking after equations.
         \slshape% Use slanted font for Theorem body
     }           
     {}          % Indent amount
     {\bfseries} % Theorem head font % (default is \mdseries)
     {.}         % Punctuation after theorem head % default: no punctuation
     { }         % Space after theorem head
     {}          % Theorem head spec

My initial testing has shown that these penalties value seem to work well, but you can increase them if you find LaTeX is still inserting page breaks in theorems more often then you like or decrease them if your document ends up with a lot of extra vertical space. The range of valid penalties are 0 (no penalty) to 10000 (max penalty).

After creating the new theorem style, you must enable it before using \newtheorem, namely,

\theoremstyle{breakResistantTheorem}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
% etc...

Custom QED Marks at End of Environments

For long environments that are in upright font, I prefer to include a mark at the end of the environment to notify readers where the section ends. The following command, \setEnvironmentQed allows for defining the QED-like symbol to automatically insert at the end of all environments of a given type. For example, \setEnvironmentQed{definition}{\ensuremath{\blacksquare}} causes a black square to be inserted at the end of every definition environment.

% Define a macro for changing the QED symbol at the
% end of environments. This command allows for the
% use of \qedhere to insert the QED into, e.g., 
% equations or lists. 
\newcommand{\setEnvironmentQed}[2]{
  % #1: Environment name
  % #2: QED Symbol. Must be OK in text or math mode. 
  %     Use \ensuremath, if math is desired.
  \AtBeginEnvironment{#1}{%
    \pushQED{\qed}\renewcommand{\qedsymbol}{#2}%
  }
  \AtEndEnvironment{#1}{\popQED}
}

Here are several options for symbols that can be used to end environments:

Symbol	LaTeX Code	Notes
\(\square\)	\square	Traditionally used for the end of proofs
\(\blacksquare\)	\blacksquare
\(\triangle\)	\triangle
\(\blacktriangle\)	\blacktriangle
\(\triangledown\)	\triangledown
\(\blacktriangledown\)	\blacktriangledown
\(\circ\)	\circ
\(\bullet\)	\bullet
\(\diamond\)	\diamond	\blackdiamond is not a built-in macro, nor provided by the amssym package.
\(\lozenge\)	\lozenge
\(\blacklozenge\)	\blacklozenge

Environment Definitions

\providetheorem definition

In my LaTeX setup, I prefer to use the same preamble file for all of my documents, but I’ve found that some document classes or packages define the theorem environment and other related environments, which causes compilation errors. To avoid errors when trying to define to new theorem-like environments, \providetheorem first checks whether the environment is defined.

% Create a new macro analogous to "\providecommand", which 
% defines the given amsthm theorem-like environment only if 
% it does not already exist.
\newcommand{\providetheorem}[2]{
  % #1: Environment name.
  % #2: Display name
  \ifcsdef{#1}{
    % The #1 environment is already defined. 
    \ifcsdef{end#1}{}{
      \PackageError{providetheorem}{%
        % Error message:
        The command "#1" was already defined, but "#1end" was 
        undefined, indicating that "#1" is not an environment.
      }{}
    }
  }{
    % The #1 environment is not defined, yet, so we define it.
    \newtheorem{#1}{#2}
  }
}

Theorem Style

There are three default theorem styles: plain, definition, and remark, but I have replaced the plain style with sltheorem to use slanted text instead of italics.

The sltheorem (or plain) style displays a boldface label and slanted (or italic) body text.

Code:

\begin{theorem}[Pythagorean Theorem]
  \label{result:pythagorean}
  The sum of the squares of the legs of a right 
  triangle equals the square of the hypotenuse.
\end{theorem}

Output:

I use the theorem style for any environments that make truth claims, namely theorem, proposition, lemma, corollary, and conjecture.

%%--------------------------------------------------%%
%| Define environments that use the sltheorem style |%
%%--------------------------------------------------%%
\theoremstyle{sltheorem}%

%%% Define theorem environment %%%
\providetheorem{theorem}{Theorem}
\crefname{theorem}{Theorem}{Theorems}%

%%% Define proposition environment %%%
\providetheorem{proposition}{Proposition}
\crefname{proposition}{Proposition}{Propositions}%

%%% Define lemma environment %%%
\providetheorem{lemma}{Lemma}
\crefname{lemma}{Lemma}{Lemmas}%

%%% Define corollary environment %%%
\providetheorem{corollary}{Corollary}
\crefname{corollary}{Corollary}{Corollaries}%

%%% Define conjecture environment %%%
\providetheorem{conjecture}{Conjecture}
\crefname{conjecture}{Conjecture}{Conjectures}%

Definition Style

The definition style displays a boldface label and upright body text.

Code:

\begin{definition}
  \label{def:ring center}
  Let $R$ be a ring.
  The \emph{center} of $R$ is the subring of $R$ 
  that contains all elements $c \in R$ such that 
  $c x = x c$ for every $x$ in $R$.
\end{definition}

Output:

I use the definition style define the following environments: definition, problem, example, and assumption.

%%---------------------------------------------------%%
%| Define environments that use the definition style |%
%%---------------------------------------------------%%
\theoremstyle{definition}%

%%% Define definition environment %%%
\providetheorem{definition}{Definition}
\crefname{definition}{Definition}{Definitions}%

%%% Define problem environment %%%
\providetheorem{problem}{Problem}
\crefname{problem}{Problem}{Problems}%

%%% Define example environment %%%
\providetheorem{example}{Example}

% Add a mark at the end of each example.
\setEnvironmentQed{example}{\ensuremath{\blacksquare}}

%%% Define assumption environment %%%
% Note: We use singular "Assumption" even when there are 
% multiple assumptions within a particular block.
\providetheorem{assumption}{Assumption}
\crefname{assumption}{Assumption}{Assumptions}

% Add a mark at the end of each assumption.
\setEnvironmentQed{assumption}{\ensuremath{\blacksquare}}

In definitions, you should emphasize the defined term to make it stand out. The preferred way to do this is to surround the term with \emph{}, which will typically typeset the text in italics. Unlike \textit{}, however, the \emph macro will switch to upright text if the surrounding text is already italicized, ensuring that the term highlighted even if you use italic text in your definition environments.

Remark Style

The remark style displays an italic label and upright body text.

Code:

\begin{remark}
  Here is a remark.
\end{remark}

Output:

I use the remark style only for the remark environment.

%%-----------------------------------------------%%
%| Define environments that use the remark style |%
%%-----------------------------------------------%%
\theoremstyle{remark}%

%%% Define remark environment %%%
\providetheorem{remark}{Remark}

Theorem-like Environment Numbering

This section describes how to modify the automatic numbering of Theorem-like environments, such as using unified numbering for multiple types of environments, chapter-based numbering, and using no numbering at all.

Unified Numbering of Different Environment Types

Some authors prefer to use one counter for all the results in their document, so instead of

• Theorem 1
• Theorem 2
• Lemma 1
• Theorem 3
• Proposition 1

the numbering would be

• Theorem 1
• Theorem 2
• Lemma 3
• Theorem 4
• Proposition 5

The benefit of this method is that it is easier for the reader navigate to a particular result via bisection, but it may confuse readers who try to find “Lemma 1” and “Lemma 2”.

To make LaTeX use a unified numbering, you create the first environment, such as theorem, and then pass the name theorem as an optional argument when creating the other environments:

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
%                          ^--- optional argument.

Chapter-based Numbering

For document classes that use chapters, I prefer to number the theorem-like environments on a chapter-by-chapter basis (e.g., “Theorem 1.1”, “Theorem 1.2” in Chapter 1 and “Theorem 2.1”, “Theorem 2.2” in Chapter 2.) The following code automatically enables chapter-based numbering when the chapter counter is defined.

\ifcsname thechapter\endcsname
    % If the document class uses chapters, 
    % then update numbering to use chapters.
    \numberwithin{assumption}{chapter}
    \numberwithin{definition}{chapter}
    \numberwithin{remark}{chapter}
    \numberwithin{example}{chapter}
    \numberwithin{proposition}{chapter}
    \numberwithin{theorem}{chapter}
    \numberwithin{lemma}{chapter}
    \numberwithin{corollary}{chapter}
    \numberwithin{conjecture}{chapter}
    \numberwithin{application}{chapter}
\fi

Unnumbered Environments

In some contexts, such as presentations, you may wish to use unnumbered environments (e.g., “Theorem” instead “Theorem 1”). To omit theorem numbering, define theorem environments using \newtheorem* macro instead of \newtheorem. If you already have theorem defined, you must use a different name for the unnumbered theorem environment, such as theorem*.

\newtheorem*{theorem*}{Theorem}

It is a bad idea, however, to have mixture of numbered and unnumbered theorems in the same document, so you may wish to simply replace \newtheorem{theorem}{Theorem} with \newtheorem*{theorem}{Theorem} so that \begin{theorem}...\end{theorem} inserts unnumbered theorems.

Proof Environment

The amsthm package also defines a proof environment. I’ve found the default definition to be exactly how I want it, with a white square inserted at the end of the proof.

The usage is as follows:

Code:

\begin{proof}
  It is true because I say so.
\end{proof}

Output:

You can change the “proof” label by using an optional environment argument.

Code:

\begin{proof}[Proof Sketch]
  This example is too small to contain the proof.
\end{proof}

Output:

Using the optional argument is particularly useful if a theorem and its proof are separated by other text:

Code:

\begin{proof}[Proof of \cref{result:an earlier theorem}]
  This proof does not occur immediately 
  after \cref{result:an earlier theorem}.
\end{proof}

Output:

Note that the QED symbol in above example appears at the end of the next line because the first line fills the text width.

If the proof ends with an equation or a list, then the QED symbol will be placed below the equation or list, even if there is space to the side, which wastes space and looks bad:

Code:

\begin{proof}
  The conclusion follow directly from this equation:
  \[
    1 + 1 = 2.
  \] 
\end{proof}

Output:

To fix this problem, place \qedhere inside the equation or list (at the end) to display the QED symbol at the correct location:

Code:

\begin{proof}
  The conclusion follow directly from this equation:
  \[
    1 + 1 = 2. \qedhere
  \] 
\end{proof}

Output:

Displaying Environment Labels in the Margin

When working on a draft, it is convenient to have the labels assigned to an theorem-like environment (via \label) in the document itself.
To show these labels, along with labels for equations, figures, tables, and whatnot, use the showlabels package. By default, showlabels displays each label in the right margin with somewhat unfortunate formatting. In particular, labels use inconsistent text weight (some labels are bold while others are not)and long labels extend off the page, without line breaks, even between words.

The following configuration fixes all of these problems by showing each label using a small teletype font in a box that is the width of the right margin, with line breaks after words, as needed. Some extra care is taken to avoid warnings from underfull and overfull boxes.

\usepackage[right]{showlabels}
% Using "\upshape" ensures that the label is displayed without italics (e.g., if inside a theorem), and "\ttfamily" displays label with a    font, since it is essentially code.
\renewcommand{\showlabelfont}{\scriptsize\upshape\ttfamily}
\renewcommand{\showlabelsetlabel}[1]{%
  % We use 95% of the marginparwidth because it looks slightly nicer to not have the box extend to the edge of the page.
  % "\raggedright" prevents underfull hbox warnings when the label doesn't fit nicely in a line.
  \fbox{\parbox{0.95\marginparwidth}{%
    \raggedright% Prevent underfull box warnings.
    \hfuzz=1000em% Prevent overfull box warnings (unless more than 1000em too wide!) 
    \showlabelfont #1}}%
}

The result looks like this:

Of course, the labels should typically be hidden from the final version of the document (e.g., before publication). Fortunately, showlabels automatically hides the labels when the document is compiled with the final option passed to the document class, such as

\documentclass[final]{article}