Write a theorem and its proof

Theorems, lemmas, definitions, proofs and the rest are statements: blocks that print a bold label at the start of their first line, the way a mathematics text does. This guide writes one from start to finish, numbers it, refers to it, and covers the cases a lecture course runs into: a kind the list does not have, a claim inside a proof, and a problem with its hint and solution.

Choose the kind

Pick the kind by what the block is in the argument: theorem for a main result, lemma for a step towards one, corollary for what follows from it, definition, example, remark, conjecture, and proof. The syntax reference lists every kind with how it renders. The label is printed in the document's language, so you always write the kind in English: a :::theorem in a Russian lecture prints Теорема.

Write a theorem with a name

1. Open the block with three colons and the kind, and put the result's name in square brackets if it has one: :::theorem[Lagrange].
2. Write the statement on the following lines. It is ordinary Markdown, with maths, lists and displays.
3. Close the block with a line of three colons.

:::theorem[Lagrange]
If $H$ is a subgroup of a finite group $G$, then $|H|$ divides $|G|$.
:::

The page prints Theorem (Lagrange). in bold and sets the statement in italics. The brackets hold the name, never the number: [Lagrange], not [Theorem 3].

Add its proof

1. Directly under the theorem, open a :::proof block, write the proof, and close it.

:::proof
The left cosets of $H$ partition $G$, and each has $|H|$ elements, so
$|G| = [G : H]\,|H|$.
:::

The page labels it Proof. and ends it with ∎, so do not type your own QED mark. To print a different word, such as Sketch of proof or the Dem. of an older text, give the proof a label: :::proof{label="Sketch of proof"}. It keeps its ∎.

Number your results

The published page does not number statements by itself yet: a theorem without a number is labelled Theorem. If the reader should be able to say "by Theorem 2.1", give each result its number.

1. Decide on the scheme, for example the section number and a running count: 2.1, 2.2, 2.3.
2. Write the number on the fence as number=: :::theorem{number=2.1}.
3. If the statement also has a name, keep the name in brackets: :::theorem[Lagrange]{number=2.1}.

:::definition{number=2.1}
The *index* $[G : H]$ of a subgroup $H$ is the number of its left cosets.
:::
 
:::theorem[Lagrange]{number=2.2}
If $H$ is a subgroup of a finite group $G$, then $|G| = [G : H]\,|H|$.
:::

The page prints Definition 2.1. and Theorem 2.2 (Lagrange). A number made only of digits and dots can also go in the brackets when there is no name: :::lemma[2.3] is Lemma 2.3.

A number does not have to be digits: number=IV.7, number=1b and number=A.3 print as written. A number that starts with a symbol, such as ✱2·17 or §4, is printed alone, without the word, the way a book prints it.

Refer to a result

1. Give the statement an anchor on its fence, together with its number: :::theorem[Lagrange]{#lagrange number=2.2}.
2. Link to it with the number in the link text: [Theorem 2.2](#lagrange).

:::theorem[Lagrange]{#lagrange number=2.2}
If $H$ is a subgroup of a finite group $G$, then $|H|$ divides $|G|$.
:::
 
By [Theorem 2.2](#lagrange), a group of prime order has no subgroups other
than itself and the trivial one.

A click on the link takes the reader to the theorem. Choose a name that says what the result is, such as #lagrange, rather than where it is, such as #theorem-2-2: the name stays right when the numbers change. Links from another work, and to figures and equations, are in Cite works and cross-reference.

Use a kind the list does not have

There is no :::proposition or :::claim. A misspelt or unknown kind is shown as typed, so you will see it in the preview. Two forms give you the word you want:

1. For a result that should look and behave like a theorem, keep the kind and change the word: :::theorem{label="Proposition"}.
2. For a block with a word of its own and no kind behind it, use :::statement[Claim]. Its body is upright, not italic.

Either way, the word is printed exactly as you wrote it and is not translated, so write it in the document's language.

Put a claim inside a proof

A statement can hold other blocks. Give the outer fence more colons than the one inside it:

1. Open the proof with four colons, ::::proof.
2. Write the inner block with three, :::statement[Claim] … :::.
3. Close the proof with four colons.

::::proof
We argue by induction on $n$.
 
:::statement[Claim]
For every $k \le n$, the map $\varphi_k$ is injective.
:::
 
The claim for $k = n$ is the theorem.
::::

The ∎ goes at the end of the whole proof, after the claim.

Set a problem with a hint and a solution

On a problem sheet, each problem is a :::problem, and its hint and solution go inside it. Both render folded, so a reader can try the problem before opening them.

1. Open the problem with four colons and its number: ::::problem[1].
2. Write the problem, then a :::hint and a :::solution with three colons each.
3. Close the problem with four colons.

::::problem[1]
Show that $\sqrt{2}$ is irrational.
 
:::hint
Suppose $\sqrt{2} = p/q$ in lowest terms, and look at the parity of $p$.
:::
 
:::solution
From $p^2 = 2q^2$, $p$ is even, say $p = 2r$. Then $q^2 = 2r^2$, so $q$ is
even too, contradicting lowest terms.
:::
::::

A problem migrated from an older sheet may carry an anchor such as {#p-412} on its fence. Leave it: old links to that problem arrive through it.

The result

In the preview, each statement starts with its bold label and the number and name you gave it; theorems, lemmas and corollaries have italic bodies, and definitions, examples and remarks upright ones; each proof ends with ∎; a link to a statement jumps to it; and a problem's hint and solution are closed until clicked. If a ::: line is shown as text instead, the kind is misspelt or the colons do not match; see Troubleshooting Markdown.