Theorem-Like Elements

Section 15.1 Theorem-Like Elements

The tags <theorem>, <algorithm>, <claim>, <corollary>, <fact>, <identity>, <lemma>, and <proposition> have the same structure in PreTeXt, so we will just illustrate <theorem> here.

<theorem>
    <title>Optional</title>
    <creator>I. Newton</creator>
  <statement>
    <p>
      Here's the statement of the theorem.
    </p>
  </statement>

  <proof>
    <p>
      You don't actually need a proof,
      but put it inside the <c>theorem</c>.
      You can actually put another <c>proof</c> right after this one if you want to.
    </p>
  </proof>
</theorem>

Listing 15.1.1. A theorem

The code in Listing 15.1.1 produces the following output:

Theorem 15.1.2 (I. Newton). Optional.

Here’s the statement of the theorem.

Proof.

You don’t actually need a proof, but put it inside the theorem. You can actually put another proof right after this one if you want to.

The <title> is optional and typically used for theorems with names. To give an attribution, one can use the optional <creator> tag. Cross references (see Section 22.1) can be made using the name or the number, depending on how the author codes them.

A theorem-like element can contain multipe <proof> elements. In such instances, it would be useful to use the <title> tag within your proof. By default, a <proof> is hidden in a knowl when using HTML output. Click the “Proof.” heading to expand the proof. Click it again to hide the proof. A <proof> can also be divided into <case>s, each of which can have a title. Although it has not always been so, you can author a <proof> all on its own within a division. The structure of such a detached <proof> is the same as for a <proof> contained within a theorem-like element.

You can use <definition> essentially like <theorem>, but a <definition> does not have a proof. You are encouraged to use the <term> tag to set off the word being defined. If you wish to include a list of notation as an appendix as your document, you might also add a <notation> tag such as shown in Listing 15.1.3. A <notation> tag has no effect unless you have an <appendix> with a <notation-list/> in it.

<definition>

  <notation>
    <usage><m>\binom{n}{k}</m></usage>
    <description>binomial coefficient</description>
  </notation>

  <statement>
    <p>
      The <term>binomial coefficient</term>
      <m>\binom{n}{k}</m> is the number of <m>k</m>-element subsets of an <m>n</m>-element set.
    </p>
  </statement>
</definition>

Listing 15.1.3. A definition with notation

The code in Listing 15.1.3 produces the following output:

Definition 15.1.4.

The binomial coefficient \(\binom{n}{k}\) is the number of \(k\)-element subsets of an \(n\)-element set.

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