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blog.youwen.dev

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<?xml version="1.0" encoding="utf-8"?>
<feed xmlns="http://www.w3.org/2005/Atom">
<title>gradient ascent</title>
<link href="https://blog.youwen.dev/atom.xml" rel="self" />
<link href="https://blog.youwen.dev" />
<id>https://blog.youwen.dev/atom.xml</id>
<author>
<name>Youwen Wu</name>
<email>youwenw@gmail.com</email>
</author>
<updated>2024-05-25T12:00:00Z</updated>
<entry>
<title>why I made my blog in haskell</title>
<link href="https://blog.youwen.dev/why-i-made-my-blog-in-haskell.html" />
<id>https://blog.youwen.dev/why-i-made-my-blog-in-haskell.html</id>
<published>2024-05-25T00:00:00Z</published>
<updated>2024-05-25T12:00:00Z</updated>
<summary type="html"><![CDATA[<article>
<header>
<h1 class="text-4xl">
<a href="./why-i-made-my-blog-in-haskell.html">why I made my blog in haskell</a>
</h1>
<p
class="mb-1 mt-2 italic font-light text-lg text-accent-light dark:text-accent-dark"
>
a purely functional...blog?
</p>
<div class="mt-2">2024-05-25</div>
<div class="mt-1 text-sm">
(last updated: 2024-05-25T12:00:00Z)
</div>
</header>
<main class="post mt-4"><p>Welcome! This is the first post on <em>gradient ascent</em> and also one that tests all
of the features.</p>
<p><img
alt="gradient ascent"
src="./images/gradient-ascent.jpg"
style="height: 200px; width: 100%; object-fit: cover"
/></p>
<p>I’ll be writing about computers, code, math, video games, and whatever else
here.</p>
<blockquote>
<p>A monad is just a monoid in the category of endofunctors, what’s the problem?</p>
</blockquote>
<h2 id="haskell">haskell?</h2>
<p>This entire blog is generated with <a href="https://jaspervdj.be/hakyll/">hakyll</a>. It’s
a library for generating static sites for Haskell, a purely functional
programming language. It’s a <em>library</em> because it doesn’t come with as many
batteries included as tools like Hugo or Astro. You set up most of the site
yourself by calling the library from Haskell.</p>
<p>Here’s a brief excerpt:</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="ot">main ::</span> <span class="dt">IO</span> ()</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>main <span class="ot">=</span> hakyllWith config <span class="op">$</span> <span class="kw">do</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a> forM_</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a> [ <span class="st">&quot;CNAME&quot;</span></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a> , <span class="st">&quot;favicon.ico&quot;</span></span>
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<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a> <span class="op">$</span> \f <span class="ot">-&gt;</span> match f <span class="op">$</span> <span class="kw">do</span></span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a> route idRoute</span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a> compile copyFileCompiler</span></code></pre></div>
<p>The code highlighting is also generated by hakyll.</p>
<hr />
<h2 id="why">why?</h2>
<p>Haskell is a purely functional language with no mutable state. Its syntax
actually makes it pretty elegant for declaring routes and “rendering” pipelines.</p>
<p>I originally wanted to build this entire blog myself. I had a working version
with the Svelte framework, complete with GFM rendering, table of contents, KaTeX
math, code highlighting, static generation, and other goodies. However, it
seemed like a little too much work to maintain. I switched to hakyll because</p>
<ol>
<li>Haskell is cool.</li>
<li>It comes with enough features that I don’t feel like I have to build
everything from scratch.</li>
<li>It comes with Pandoc, a Haskell library for converting between markdown
formats. It’s probably more powerful than anything you could do in <code>nodejs</code>.
It renders all of the markdown to HTML as well as the math.
<ol>
<li>It supports KaTeX as well as MathML. I’m a little disappointed with the
KaTeX though. It doesn’t directly render it, but simply injects the KaTeX
files and renders it client-side.</li>
</ol></li>
</ol>
<h3 id="speaking-of-math">speaking of math</h3>
<p>We can have math inline, like so:
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mi>∞</mi><mi>∞</mi></msubsup><mspace width="0.167em"></mspace><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mspace width="0.167em"></mspace><mi>d</mi><mi>x</mi><mo>=</mo><msqrt><mi>π</mi></msqrt></mrow><annotation encoding="application/x-tex">\int_\infty^\infty \, e^{-x^2}\,dx = \sqrt{\pi}</annotation></semantics></math>. This site ships semantic
MathML math with its HTML, and the MathJax script to the client.</p>
<p>It’d be nice if MathML could just be used and supported across all browsers, but
unfortunately we still aren’t quite there yet. Firefox is the only one where
everything looks 80% of the way to LaTeX. On Safari and Chrome, even simple
equations like <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msqrt><mi>π</mi></msqrt><annotation encoding="application/x-tex">\sqrt{\pi}</annotation></semantics></math> render improperly.</p>
<p>Pros of MathML:</p>
<ul>
<li>A little more accessible</li>
<li>Can be rendered without additional stylesheets. I just installed the Latin
Modern font, but this isn’t even really necessary</li>
<li>Built-in to most browsers (#UseThePlatform)</li>
</ul>
<p>Cons:</p>
<ul>
<li>Isn’t fully standardized. Might look different on different browsers</li>
<li>Rendering quality isn’t as good as KaTeX</li>
</ul>
<p>This site has MathJax render all of the math so it looks nice and standardized
across browsers, but the math still displays regardless (like say if MathJax
couldn’t load due to slow network) because of MathML. Best of both worlds.</p>
<p>Let’s try it now. Here’s a simple theorem:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mi>n</mi></msup><mo>+</mo><msup><mi>b</mi><mi>n</mi></msup><mo>≠</mo><msup><mi>c</mi><mi>n</mi></msup><mspace width="0.167em"></mspace><mo>∀</mo><mspace width="0.167em"></mspace><mrow><mo stretchy="true" form="prefix">{</mo><mi>a</mi><mo>,</mo><mspace width="0.167em"></mspace><mi>b</mi><mo>,</mo><mspace width="0.167em"></mspace><mi>c</mi><mo stretchy="true" form="postfix">}</mo></mrow><mo>∈</mo><mstyle mathvariant="double-struck"><mi>ℤ</mi></mstyle><mo>∧</mo><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">
a^n + b^n \ne c^n \, \forall\,\left\{ a,\,b,\,c \right\} \in \mathbb{Z} \land n \ge 3
</annotation></semantics></math></p>
<p>The proof is trivial and will be left as an exercise to the reader.</p>
<h2 id="seems-a-little-overengineered">seems a little overengineered</h2>
<p>Probably is. Not as much as the old one, though.</p></main>
</article>
]]></summary>
</entry>
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<h1 class="text-4xl md:text-5xl font-serif font-medium">
<a
href="/"
class="dark:hover:text-muted-dark hover:text-muted-light transition-all duration-500 text-nowrap tracking-wide hover:tracking-wider"
>Gradient Ascent</a
>
</h1>
<div
class="w-full flex-grow flex-shrink rounded-lg h-1 bg-muted-light dark:bg-muted-dark mx-4"
></div>
</div>
<p class="mt-8 mb-3 px-1 italic font-light">
a web-log about computers, math, hacks, games, and life.
</p>
<a class="text-sm external-link" href="https://youwen.dev"
><em>by </em>Youwen Wu</a
>
<span class="ml-2 font-serif">|</span>
<button
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<h2 class="text-3xl">Latest</h2>
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<h3 class="text-2xl">why I made my blog in haskell</h3>
<p
class="italic text-accent-light dark:text-accent-dark my-1 group-hover:dark:text-muted-dark group-hover:text-muted-light transition-colors"
>
a purely functional...blog?
</p>
<p class="text-sm">2024-05-25</p>
</a>
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>
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<description><![CDATA[<article>
<header>
<h1 class="text-4xl">
<a href="./why-i-made-my-blog-in-haskell.html">why I made my blog in haskell</a>
</h1>
<p
class="mb-1 mt-2 italic font-light text-lg text-accent-light dark:text-accent-dark"
>
a purely functional...blog?
</p>
<div class="mt-2">2024-05-25</div>
<div class="mt-1 text-sm">
(last updated: 2024-05-25T12:00:00Z)
</div>
</header>
<main class="post mt-4"><p>Welcome! This is the first post on <em>gradient ascent</em> and also one that tests all
of the features.</p>
<p><img
alt="gradient ascent"
src="./images/gradient-ascent.jpg"
style="height: 200px; width: 100%; object-fit: cover"
/></p>
<p>I’ll be writing about computers, code, math, video games, and whatever else
here.</p>
<blockquote>
<p>A monad is just a monoid in the category of endofunctors, what’s the problem?</p>
</blockquote>
<h2 id="haskell">haskell?</h2>
<p>This entire blog is generated with <a href="https://jaspervdj.be/hakyll/">hakyll</a>. It’s
a library for generating static sites for Haskell, a purely functional
programming language. It’s a <em>library</em> because it doesn’t come with as many
batteries included as tools like Hugo or Astro. You set up most of the site
yourself by calling the library from Haskell.</p>
<p>Here’s a brief excerpt:</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="ot">main ::</span> <span class="dt">IO</span> ()</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>main <span class="ot">=</span> hakyllWith config <span class="op">$</span> <span class="kw">do</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a> forM_</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a> [ <span class="st">&quot;CNAME&quot;</span></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a> , <span class="st">&quot;favicon.ico&quot;</span></span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a> , <span class="st">&quot;robots.txt&quot;</span></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a> , <span class="st">&quot;_config.yml&quot;</span></span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a> , <span class="st">&quot;images/*&quot;</span></span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a> , <span class="st">&quot;out/*&quot;</span></span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a> , <span class="st">&quot;fonts/*&quot;</span></span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a> ]</span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a> <span class="op">$</span> \f <span class="ot">-&gt;</span> match f <span class="op">$</span> <span class="kw">do</span></span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a> route idRoute</span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a> compile copyFileCompiler</span></code></pre></div>
<p>The code highlighting is also generated by hakyll.</p>
<hr />
<h2 id="why">why?</h2>
<p>Haskell is a purely functional language with no mutable state. Its syntax
actually makes it pretty elegant for declaring routes and “rendering” pipelines.</p>
<p>I originally wanted to build this entire blog myself. I had a working version
with the Svelte framework, complete with GFM rendering, table of contents, KaTeX
math, code highlighting, static generation, and other goodies. However, it
seemed like a little too much work to maintain. I switched to hakyll because</p>
<ol>
<li>Haskell is cool.</li>
<li>It comes with enough features that I don’t feel like I have to build
everything from scratch.</li>
<li>It comes with Pandoc, a Haskell library for converting between markdown
formats. It’s probably more powerful than anything you could do in <code>nodejs</code>.
It renders all of the markdown to HTML as well as the math.
<ol>
<li>It supports KaTeX as well as MathML. I’m a little disappointed with the
KaTeX though. It doesn’t directly render it, but simply injects the KaTeX
files and renders it client-side.</li>
</ol></li>
</ol>
<h3 id="speaking-of-math">speaking of math</h3>
<p>We can have math inline, like so:
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mi>∞</mi><mi>∞</mi></msubsup><mspace width="0.167em"></mspace><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mspace width="0.167em"></mspace><mi>d</mi><mi>x</mi><mo>=</mo><msqrt><mi>π</mi></msqrt></mrow><annotation encoding="application/x-tex">\int_\infty^\infty \, e^{-x^2}\,dx = \sqrt{\pi}</annotation></semantics></math>. This site ships semantic
MathML math with its HTML, and the MathJax script to the client.</p>
<p>It’d be nice if MathML could just be used and supported across all browsers, but
unfortunately we still aren’t quite there yet. Firefox is the only one where
everything looks 80% of the way to LaTeX. On Safari and Chrome, even simple
equations like <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msqrt><mi>π</mi></msqrt><annotation encoding="application/x-tex">\sqrt{\pi}</annotation></semantics></math> render improperly.</p>
<p>Pros of MathML:</p>
<ul>
<li>A little more accessible</li>
<li>Can be rendered without additional stylesheets. I just installed the Latin
Modern font, but this isn’t even really necessary</li>
<li>Built-in to most browsers (#UseThePlatform)</li>
</ul>
<p>Cons:</p>
<ul>
<li>Isn’t fully standardized. Might look different on different browsers</li>
<li>Rendering quality isn’t as good as KaTeX</li>
</ul>
<p>This site has MathJax render all of the math so it looks nice and standardized
across browsers, but the math still displays regardless (like say if MathJax
couldn’t load due to slow network) because of MathML. Best of both worlds.</p>
<p>Let’s try it now. Here’s a simple theorem:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mi>n</mi></msup><mo>+</mo><msup><mi>b</mi><mi>n</mi></msup><mo>≠</mo><msup><mi>c</mi><mi>n</mi></msup><mspace width="0.167em"></mspace><mo>∀</mo><mspace width="0.167em"></mspace><mrow><mo stretchy="true" form="prefix">{</mo><mi>a</mi><mo>,</mo><mspace width="0.167em"></mspace><mi>b</mi><mo>,</mo><mspace width="0.167em"></mspace><mi>c</mi><mo stretchy="true" form="postfix">}</mo></mrow><mo>∈</mo><mstyle mathvariant="double-struck"><mi>ℤ</mi></mstyle><mo>∧</mo><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">
a^n + b^n \ne c^n \, \forall\,\left\{ a,\,b,\,c \right\} \in \mathbb{Z} \land n \ge 3
</annotation></semantics></math></p>
<p>The proof is trivial and will be left as an exercise to the reader.</p>
<h2 id="seems-a-little-overengineered">seems a little overengineered</h2>
<p>Probably is. Not as much as the old one, though.</p></main>
</article>
]]></description>
<pubDate>Sat, 25 May 2024 00:00:00 UT</pubDate>
<guid>https://blog.youwen.dev/why-i-made-my-blog-in-haskell.html</guid>
<dc:creator>Youwen Wu</dc:creator>
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a web-log about computers, math, hacks, games, and life.
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<a href="./why-i-made-my-blog-in-haskell.html">why I made my blog in haskell</a>
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a purely functional...blog?
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<div class="mt-2">2024-05-25</div>
<div class="mt-1 text-sm">
(last updated: 2024-05-25T12:00:00Z)
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<main class="post mt-4"><p>Welcome! This is the first post on <em>gradient ascent</em> and also one that tests all
of the features.</p>
<p><img
alt="gradient ascent"
src="./images/gradient-ascent.jpg"
style="height: 200px; width: 100%; object-fit: cover"
/></p>
<p>I’ll be writing about computers, code, math, video games, and whatever else
here.</p>
<blockquote>
<p>A monad is just a monoid in the category of endofunctors, what’s the problem?</p>
</blockquote>
<h2 id="haskell">haskell?</h2>
<p>This entire blog is generated with <a href="https://jaspervdj.be/hakyll/">hakyll</a>. It’s
a library for generating static sites for Haskell, a purely functional
programming language. It’s a <em>library</em> because it doesn’t come with as many
batteries included as tools like Hugo or Astro. You set up most of the site
yourself by calling the library from Haskell.</p>
<p>Here’s a brief excerpt:</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode haskell"><code class="sourceCode haskell"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="ot">main ::</span> <span class="dt">IO</span> ()</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>main <span class="ot">=</span> hakyllWith config <span class="op">$</span> <span class="kw">do</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a> forM_</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a> [ <span class="st">&quot;CNAME&quot;</span></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a> , <span class="st">&quot;favicon.ico&quot;</span></span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a> , <span class="st">&quot;robots.txt&quot;</span></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a> , <span class="st">&quot;_config.yml&quot;</span></span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a> , <span class="st">&quot;images/*&quot;</span></span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a> , <span class="st">&quot;out/*&quot;</span></span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a> , <span class="st">&quot;fonts/*&quot;</span></span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a> ]</span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a> <span class="op">$</span> \f <span class="ot">-&gt;</span> match f <span class="op">$</span> <span class="kw">do</span></span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a> route idRoute</span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a> compile copyFileCompiler</span></code></pre></div>
<p>The code highlighting is also generated by hakyll.</p>
<hr />
<h2 id="why">why?</h2>
<p>Haskell is a purely functional language with no mutable state. Its syntax
actually makes it pretty elegant for declaring routes and “rendering” pipelines.</p>
<p>I originally wanted to build this entire blog myself. I had a working version
with the Svelte framework, complete with GFM rendering, table of contents, KaTeX
math, code highlighting, static generation, and other goodies. However, it
seemed like a little too much work to maintain. I switched to hakyll because</p>
<ol>
<li>Haskell is cool.</li>
<li>It comes with enough features that I don’t feel like I have to build
everything from scratch.</li>
<li>It comes with Pandoc, a Haskell library for converting between markdown
formats. It’s probably more powerful than anything you could do in <code>nodejs</code>.
It renders all of the markdown to HTML as well as the math.
<ol>
<li>It supports KaTeX as well as MathML. I’m a little disappointed with the
KaTeX though. It doesn’t directly render it, but simply injects the KaTeX
files and renders it client-side.</li>
</ol></li>
</ol>
<h3 id="speaking-of-math">speaking of math</h3>
<p>We can have math inline, like so:
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mi>∞</mi><mi>∞</mi></msubsup><mspace width="0.167em"></mspace><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mspace width="0.167em"></mspace><mi>d</mi><mi>x</mi><mo>=</mo><msqrt><mi>π</mi></msqrt></mrow><annotation encoding="application/x-tex">\int_\infty^\infty \, e^{-x^2}\,dx = \sqrt{\pi}</annotation></semantics></math>. This site ships semantic
MathML math with its HTML, and the MathJax script to the client.</p>
<p>It’d be nice if MathML could just be used and supported across all browsers, but
unfortunately we still aren’t quite there yet. Firefox is the only one where
everything looks 80% of the way to LaTeX. On Safari and Chrome, even simple
equations like <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msqrt><mi>π</mi></msqrt><annotation encoding="application/x-tex">\sqrt{\pi}</annotation></semantics></math> render improperly.</p>
<p>Pros of MathML:</p>
<ul>
<li>A little more accessible</li>
<li>Can be rendered without additional stylesheets. I just installed the Latin
Modern font, but this isn’t even really necessary</li>
<li>Built-in to most browsers (#UseThePlatform)</li>
</ul>
<p>Cons:</p>
<ul>
<li>Isn’t fully standardized. Might look different on different browsers</li>
<li>Rendering quality isn’t as good as KaTeX</li>
</ul>
<p>This site has MathJax render all of the math so it looks nice and standardized
across browsers, but the math still displays regardless (like say if MathJax
couldn’t load due to slow network) because of MathML. Best of both worlds.</p>
<p>Let’s try it now. Here’s a simple theorem:</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mi>n</mi></msup><mo>+</mo><msup><mi>b</mi><mi>n</mi></msup><mo>≠</mo><msup><mi>c</mi><mi>n</mi></msup><mspace width="0.167em"></mspace><mo>∀</mo><mspace width="0.167em"></mspace><mrow><mo stretchy="true" form="prefix">{</mo><mi>a</mi><mo>,</mo><mspace width="0.167em"></mspace><mi>b</mi><mo>,</mo><mspace width="0.167em"></mspace><mi>c</mi><mo stretchy="true" form="postfix">}</mo></mrow><mo>∈</mo><mstyle mathvariant="double-struck"><mi>ℤ</mi></mstyle><mo>∧</mo><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">
a^n + b^n \ne c^n \, \forall\,\left\{ a,\,b,\,c \right\} \in \mathbb{Z} \land n \ge 3
</annotation></semantics></math></p>
<p>The proof is trivial and will be left as an exercise to the reader.</p>
<h2 id="seems-a-little-overengineered">seems a little overengineered</h2>
<p>Probably is. Not as much as the old one, though.</p></main>
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