Statements

Theorem-style blocks -- theorem, lemma, corollary, conjecture, definition, proof, example, remark -- and the problem-sheet family: problem, solution, hint

Statements are the LaTeX theorem-environment family as directives. Each renders with a typographic label in the tradition of a mathematics text: label in bold, body inline after it.

Kinds

Name	Renders as	Body style
`theorem`	Theorem.	italic
`lemma`	Lemma.	italic
`corollary`	Corollary.	italic
`conjecture`	Conjecture.	italic
`definition`	Definition.	upright
`proof`	Proof. ... $■$	upright, tombstone at the end
`example`	Example.	upright
`remark`	Remark.	upright
`problem`	Problem.	upright
`solution`	Solution. ... $■$	folded – opens on click
`hint`	Hint.	folded – opens on click

(note, warning, and caution are admonitions, not statements – they render as boxes, not as theorem text.)

Basic use

Result

Definition. A group is a set with an associative operation, an identity element, and inverses.

Markdown

:::definition
A group is a set with an associative operation, an identity element, and
inverses.
:::

Naming a statement

The [label] names the result – it renders in parentheses after the kind:

Result

Theorem (Lagrange). The order of a subgroup divides the order of the group.

Proof. Cosets partition the group and all have equal size.

Markdown

:::theorem[Lagrange]
The order of a subgroup divides the order of the group.
:::
 
:::proof
Cosets partition the group and all have equal size.
:::

The proof's tombstone ( $■$ ) is appended automatically – do not type your own QED mark.

Numbering a statement

A label made only of digits and dots is a number, not a name: [3] renders as Problem 3., [2.4] as Theorem 2.4. That is how a problem sheet keeps the numbering it was printed with.

A sheet is a publication of the Problem sheet type whose problems are :::problem blocks. Put the solution and any hint inside the problem they answer; both render folded, so a reader can try the problem before opening them:

Result

Problem 1. Prove that the fraction $\frac{21 n + 4}{14 n + 3}$ is irreducible for every natural $n$ .

Hint.

What does Euclid's algorithm say about $g cd (21 n + 4, 14 n + 3)$ ?

Solution.

$g cd (21 n + 4, 14 n + 3) = g cd (7 n + 1, 14 n + 3) = g cd (7 n + 1, 1) = 1$ .

Markdown

:::problem[1]{#p-4}
Prove that the fraction $\frac{21n+4}{14n+3}$ is irreducible for every
natural $n$.
 
:::hint
What does Euclid's algorithm say about $\gcd(21n+4, 14n+3)$?
:::
 
:::solution
$\gcd(21n+4, 14n+3) = \gcd(7n+1, 14n+3) = \gcd(7n+1, 1) = 1$.
:::
:::

The {#p-4} is an anchor: the block gets that id on the page, so …#p-4 links straight to it. Any statement may carry one (letters, digits, _ . : -, starting with a letter). Sheets migrated from LibreProblems carry p-<number> on every problem – those are the addresses the old problems.libretimes.io links redirect to, so leave them as they are.

A kind the list does not have

The vocabulary above is closed, so :::proposition is not a statement – it is an unrecognised directive and renders as the characters you typed. Use the generic kind instead and give it your own word:

Result

Proposition. Every ideal in a principal ideal domain is generated by one element.

Markdown

:::statement[Proposition]
Every ideal in a principal ideal domain is generated by one element.
:::

The word you write is used exactly as written and is not translated, unlike the kinds above. That is the trade: a fixed kind localizes, a custom one is yours.

A printed number and a printed word

Two attributes keep a source's own numbering and wording – a textbook's theorem numbers, a classic's proof heading:

Result

Theorem 3.17. Every bounded monotone sequence converges.

Dem. A bounded increasing sequence converges to its supremum.

Markdown

:::theorem{number=3.17}
Every bounded monotone sequence converges.
:::
 
:::proof{label="Dem."}
A bounded increasing sequence converges to its supremum.
:::

number sets the printed number: Theorem 3.17. Anything up to 32 characters: 3.17, IV.7, 1b, ✱2·17. When the number starts with a symbol (✱2·17, §4), the kind word is left out and it prints as the book prints it: ✱2·17. Quote it if it contains a space.
label replaces the kind word with your own, used exactly as written (not translated): Dem. instead of Proof. label="" prints no word. The block keeps everything else about its kind – the ∎ of a proof, the italic body of a theorem.
[3] (digits and dots in the label slot) still works and means number=3.

Labels are translated

The label text follows the publication's language, not the reader's interface language: in a Russian-language publication :::theorem renders as Теорема., in French as Théorème., and so on for German and Chinese. Write the directive name in English always; the rendering localizes.

Body content

A statement body is full Markdown: display math, lists, code, images, even a nested TikZ diagram (give the outer fence more colons – see nesting). Statements do not auto-number yet; give one an explicit number with a digits-only [label], or name it and refer to it by name.