Mastering Theorem Environments — theorem, definition, proof, and Friends
A complete guide to Folio's 18+ built-in theorem environments. Covers theorem, definition, proof, lemma, corollary, example, remark, and more — usage, numbering, named arguments, and italic/roman styles explained.
1. Theorem Environment Basics
Folio comes with amsthm-equivalent theorem environments built in. No \newtheorem declarations needed.
Basic syntax:
\begin{theorem}[Fermat's Little Theorem]
Let $p$ be a prime and $\gcd(a, p) = 1$.
Then $a^{p-1} \equiv 1 \pmod{p}$.
\end{theorem}Result:
2. Complete List of Theorem Environments
Folio provides the following environments. All are automatically numbered (except proof).
2.1. Theorem-Type (Italic Body)
Environments for stating theorem-like assertions. The body text is displayed in italic:
theorem environment. For any prime and integer with , we have . lemma environment. If is composite, then has a prime factor at most . proposition environment. The composition of continuous functions is continuous. corollary environment. The order of a finite group is divisible by the order of each element. claim environment. This construction is unique. fact environment. is a transcendental number. hypothesis environment. . 2.2. Definition-Type (Roman Body)
Environments for definitions and examples. The body text is displayed in roman (upright) style:
definition environment. A subset of a group is a subgroup if itself forms a group under the operation of . example environment. In , the order of is . remark environment. The converse of this theorem does not hold in general. note environment. Unless otherwise stated, all rings are assumed to be commutative. exercise environment. Find all Sylow -subgroups of . problem environment. Prove that . assumption environment. Assume is continuous on . observation environment. The determinant changes sign when two rows are swapped. question environment. Can every even integer be expressed as the sum of two primes? solution environment. Substituting gives . answer environment. The number of ways is . □2.3. Proof Environment
The proof environment is unnumbered and automatically appends a QED symbol ():
Named proofs are also supported:
\begin{proof}[Proof of Theorem 1.1] you can specify a custom name. □
3. Environment Reference Table
| Environment | Display Label | Body Style | Numbered |
theorem |
Theorem | Italic | Yes |
lemma |
Lemma | Italic | Yes |
proposition |
Proposition | Italic | Yes |
corollary |
Corollary | Italic | Yes |
claim |
Claim | Italic | Yes |
fact |
Fact | Italic | Yes |
hypothesis |
Hypothesis | Italic | Yes |
definition |
Definition | Roman | Yes |
example |
Example | Roman | Yes |
remark |
Remark | Roman | Yes |
note |
Note | Roman | Yes |
exercise |
Exercise | Roman | Yes |
problem |
Problem | Roman | Yes |
assumption |
Assumption | Roman | Yes |
observation |
Observation | Roman | Yes |
question |
Question | Roman | Yes |
solution |
Solution | Roman | Yes |
answer |
Answer | Roman | Yes |
proof |
Proof | Roman | No |
4. Combining with Cross-References
You can label theorem environments with \label and reference them with \ref:
\begin{theorem}[Lagrange's Theorem]
\label{thm:lagrange}
For a subgroup $H$ of a finite group $G$, $|H|$ divides $|G|$.
\end{theorem}
By Theorem \ref{thm:lagrange}...By Theorem 21, the possible orders of subgroups of a group of order are .
5. Summary
Folio's theorem environments provide amsthm-equivalent functionality with zero configuration. Using 19 different environments, you can clearly express the structure of mathematical arguments. Next, we'll look at code blocks and algorithms.
Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.