3.8 Theorems

https://stackoverflow.com/questions/50379923/bookdown-remark-environment

Language internationalization: https://bookdown.org/yihui/bookdown/internationalization.html

Theorem environments in the bookdown package.

Table: Theorem environments in bookdown.

Environment Printed Name Label Prefix
theorem Theorem thm
lemma Lemma lem
corollary Corollary cor
proposition Proposition prp
conjecture Conjecture cnj
definition Definition def
example Example exm
exercise Exercise exr
hypothesis Hypothesis hyp

  • Definition 定义: an explanation of the mathematical meaning of a word.

  • Theorem 定理: A statement that has been proven to be true.

    是文章中重要的数学化的论述，一般有严格的数学证明。

  • Proposition 命题: A less important but nonetheless interesting true statement.

    经过证明且 intersting，但没有 Theorem 重要，比较常用。

  • Lemma 引理: A true statement used in proving other true statements (that is, a less important theorem that is helpful in the proof of other results).

    帮助证明 Theorem 的小结果。有时候可以将 Theorem 拆分成多个小的 Lemma 来逐步证明，以使得证明的思路更加清晰。很少情况下 Lemma 会以其自身的形式存在。

    • Lemmas are considered to be less important than propositions. But the distinction between categories is rather blurred.
    • There is no formal distinction among a lemma, a proposition, and a theorem.

  • Corollary 推论: A true statment that is a simple deduction from a Theorem or Proposition.

  • Proof: The explanation of why a statement is true.

  • Conjecture 猜想，猜测: A statement believed to be true, but for which we have no proof. (a statement that is being proposed to be a true statement).

  • Axiom 公理: A basic assumption about a mathematical situation. (a statement we assume to be true).

    不需要证明的论述，是其他所有 Theorem 的基础。

Usage

Theorems and proofs provide environments that are commonly used within articles and books in mathematics. To write a theorem, you can use the syntax below:

```{theorem, label, name="Theorem name"}
Here is my first theorem.
```

will be rendered as:

Theorem 3.1 (Theorem name) Here is my first theorem.

Refer to the theorem using \@ref(prefix:label). See the column Label Prefix in Table for the value of prefix for each environment. E.g., see theorem 3.1 (\@ref(thm:label)).

If you want to refer to a theorem, you should label it. The label can be provided as an ID to the block of the form #label.

Another example

```{theorem, thm-py, name="Pythagorean theorem"}
For a right triangle, if $c$ denotes the length of the hypotenuse and $a$ and $b$ denote the lengths of the other two sides, we have
  
  \begin{align*}
  c^2 = a^2+b^2
  \end{align*}
```

will be rendered as:

Theorem 3.2 (Pythagorean theorem) For a right triangle, if \(c\) denotes the length of the hypotenuse and \(a\) and \(b\) denote the lengths of the other two sides, we have

\[\begin{align*} c^2 = a^2+b^2 \end{align*}\]

Alternatively, you can use the syntax based on Pandoc’s fenced Div blocks. It can already be used in any R Markdown document to write custom blocks.

::: {.theorem #pyth name="Pythagorean theorem"}
For a right triangle, if $c$ denotes the length of the hypotenuse
and $a$ and $b$ denote the lengths of the other two sides, we have

$$a^2 + b^2 = c^2$$
:::

Theorem 3.3 (Pythagorean theorem) For a right triangle, if \(c\) denotes the length of the hypotenuse and \(a\) and \(b\) denote the lengths of the other two sides, we have

\[a^2 + b^2 = c^2\]

Apply Theorem 3.3, …

  • Variants of the theorem environments include: lemma, corollary, proposition, conjecture, definition, example, and exercise. The syntax for these environments is similar to the theorem environment, e.g., ```{lemma}.

  • The proof environment behaves similarly to theorem environments but is unnumbered. Variants of the proof environments include remark and solution.

    The proofenvironment behaves similarly to theorem environments but is unnumbered.

Customize math environment labels

You need to create a file _bookdown.yml in the same directory as your .Rmd.

In the configuration file _bookdown.yml

For example, if you want FIGURE x.x instead of Figure x.x, you can change fig to "FIGURE ":

language:
  label:
    fig: "FIGURE "

If you want to number proof,

  1. choose one of the predefined theorem like environments that you are not using otherwise, e.g. example or exercise.

  2. Redefine the printed name for that environment in _bookdown.yml (c.f. https://bookdown.org/yihui/bookdown/internationalization.html) via:

    language:
  label:
    exr: 'Proof '

    Here I changed the exercise environment leading word to “Proof”.

  3. In your Rmd files use {exercise, mylabel} environment.

    ```{exercise, mylabel}
my comment
```

In Remark \@ref(exr:mylabel) we discussed...

    Note that you have to use exercise and the corresponding label prefix exr.

Can specify environment style in style.css

.exercise {
    margin: 10px 5px 20px 5px; 
}
/* define a boxed environment */
.boxed {
    border: 1px solid #535353;
    padding-bottom: 20px;
}
<div class = "boxed">
```{exercise, proof2}
Show $\pi=\Phi \left(\frac{\mu}{\sigma}\right)$.
$$
\begin{aligned}[b]
P(r_t>0) &= P(\mu+e_t>0) \\
&= P(e_t>-\mu) \quad\quad\quad (\sigma>0, \text{dividing by a pos. number, inequality unchanged}) \\
&= P\left( \frac{e_t}{\sigma} > -\frac{\mu}{\sigma}\right) \quad\;\; e_t\sim N(0, \sigma^2), \text{ then } \frac{e_t}{\sigma}\sim N(0,1) \\
&= P \left( \frac{e_t}{\sigma} < \frac{\mu}{\sigma} \right) \\
&= \Phi \left(\frac{\mu}{\sigma} \right) 
\end{aligned}  \square
$$
```
</div>

box_theorem