zec
/
halo2
mirror of https://github.com/zcash/halo2.git
1
0
Fork 0
Code Issues Projects Releases Wiki Activity
halo2/user/tips-and-tricks.html

214 lines
77 KiB
HTML
Raw Normal View History

deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<!DOCTYPE HTML>
<html lang="en" class="sidebar-visible no-js light">
<head>
<!-- Book generated using mdBook -->
<meta charset="UTF-8">
<title>Tips and tricks - The halo2 Book</title>
<!-- Custom HTML head -->
<meta content="text/html; charset=utf-8" http-equiv="Content-Type">
<meta name="description" content="">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="theme-color" content="#ffffff" />
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
<link rel="icon" href="../favicon.svg">
<link rel="shortcut icon" href="../favicon.png">
<link rel="stylesheet" href="../css/variables.css">
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<link rel="stylesheet" href="../css/general.css">
<link rel="stylesheet" href="../css/chrome.css">
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
<link rel="stylesheet" href="../css/print.css" media="print">
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<!-- Fonts -->
<link rel="stylesheet" href="../FontAwesome/css/font-awesome.css">
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
<link rel="stylesheet" href="../fonts/fonts.css">
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<!-- Highlight.js Stylesheets -->
<link rel="stylesheet" href="../highlight.css">
<link rel="stylesheet" href="../tomorrow-night.css">
<link rel="stylesheet" href="../ayu-highlight.css">
<!-- Custom theme stylesheets -->
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
</head>
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<body>
<!-- Provide site root to javascript -->
<script type="text/javascript">
var path_to_root = "../";
var default_theme = window.matchMedia("(prefers-color-scheme: dark)").matches ? "navy" : "light";
</script>
<!-- Work around some values being stored in localStorage wrapped in quotes -->
<script type="text/javascript">
try {
var theme = localStorage.getItem('mdbook-theme');
var sidebar = localStorage.getItem('mdbook-sidebar');
if (theme.startsWith('"') && theme.endsWith('"')) {
localStorage.setItem('mdbook-theme', theme.slice(1, theme.length - 1));
}
if (sidebar.startsWith('"') && sidebar.endsWith('"')) {
localStorage.setItem('mdbook-sidebar', sidebar.slice(1, sidebar.length - 1));
}
} catch (e) { }
</script>
<!-- Set the theme before any content is loaded, prevents flash -->
<script type="text/javascript">
var theme;
try { theme = localStorage.getItem('mdbook-theme'); } catch(e) { }
if (theme === null || theme === undefined) { theme = default_theme; }
var html = document.querySelector('html');
html.classList.remove('no-js')
html.classList.remove('light')
html.classList.add(theme);
html.classList.add('js');
</script>
<!-- Hide / unhide sidebar before it is displayed -->
<script type="text/javascript">
var html = document.querySelector('html');
var sidebar = 'hidden';
if (document.body.clientWidth >= 1080) {
try { sidebar = localStorage.getItem('mdbook-sidebar'); } catch(e) { }
sidebar = sidebar || 'visible';
}
html.classList.remove('sidebar-visible');
html.classList.add("sidebar-" + sidebar);
</script>
<nav id="sidebar" class="sidebar" aria-label="Table of contents">
<div class="sidebar-scrollbox">
deploy: ec9dcefe9103fc23c13e8195120419d4d2f232a6
2022-10-18 09:58:24 -07:00
<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> PLONKish Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.3.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.4.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/dev-tools.html"><strong aria-hidden="true">2.1.</strong> Developer tools</a></li><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.2.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.3.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.4.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html" class="active"><strong aria-hidden="true">2.5.</strong> Tips and tricks</a></li><li class="chapter-item expanded "><a href="../user/wasm-port.html"><strong aria-hidden="true">2.6.</strong> WASM Guide</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/protocol.html"><strong aria-hidden="true">3.2.</strong> Protocol Description</a></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.3.</strong> Implementation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/implementation/proofs.html"><strong aria-hidden="true">3.3.1.</strong> Proofs</a></li><li class="chapter-item expanded "><a href="../design/implementation/fields.html"><strong aria-hidden="true">3.3.2.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../design/implementation/selector-combining.html"><strong aria-hidden="true">3.3.3.</strong> Selector combining</a></li></ol></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
</div>
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<div id="sidebar-resize-handle" class="sidebar-resize-handle"></div>
</nav>
<div id="page-wrapper" class="page-wrapper">
<div class="page">
<div id="menu-bar-hover-placeholder"></div>
<div id="menu-bar" class="menu-bar sticky bordered">
<div class="left-buttons">
<button id="sidebar-toggle" class="icon-button" type="button" title="Toggle Table of Contents" aria-label="Toggle Table of Contents" aria-controls="sidebar">
<i class="fa fa-bars"></i>
</button>
<button id="theme-toggle" class="icon-button" type="button" title="Change theme" aria-label="Change theme" aria-haspopup="true" aria-expanded="false" aria-controls="theme-list">
<i class="fa fa-paint-brush"></i>
</button>
<ul id="theme-list" class="theme-popup" aria-label="Themes" role="menu">
<li role="none"><button role="menuitem" class="theme" id="light">Light (default)</button></li>
<li role="none"><button role="menuitem" class="theme" id="rust">Rust</button></li>
<li role="none"><button role="menuitem" class="theme" id="coal">Coal</button></li>
<li role="none"><button role="menuitem" class="theme" id="navy">Navy</button></li>
<li role="none"><button role="menuitem" class="theme" id="ayu">Ayu</button></li>
</ul>
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
<button id="search-toggle" class="icon-button" type="button" title="Search. (Shortkey: s)" aria-label="Toggle Searchbar" aria-expanded="false" aria-keyshortcuts="S" aria-controls="searchbar">
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<i class="fa fa-search"></i>
</button>
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
</div>
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<h1 class="menu-title">The halo2 Book</h1>
<div class="right-buttons">
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
<a href="../print.html" title="Print this book" aria-label="Print this book">
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<i id="print-button" class="fa fa-print"></i>
</a>
</div>
</div>
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
<div id="search-wrapper" class="hidden">
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<form id="searchbar-outer" class="searchbar-outer">
deploy: 53ac47fceceba6ebc2c3d54d3a1f62d4b874b8db
2021-06-05 03:42:23 -07:00
<input type="search" id="searchbar" name="searchbar" placeholder="Search this book ..." aria-controls="searchresults-outer" aria-describedby="searchresults-header">
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
</form>
<div id="searchresults-outer" class="searchresults-outer hidden">
<div id="searchresults-header" class="searchresults-header"></div>
<ul id="searchresults">
</ul>
</div>
</div>
<!-- Apply ARIA attributes after the sidebar and the sidebar toggle button are added to the DOM -->
<script type="text/javascript">
document.getElementById('sidebar-toggle').setAttribute('aria-expanded', sidebar === 'visible');
document.getElementById('sidebar').setAttribute('aria-hidden', sidebar !== 'visible');
Array.from(document.querySelectorAll('#sidebar a')).forEach(function(link) {
link.setAttribute('tabIndex', sidebar === 'visible' ? 0 : -1);
});
</script>
<div id="content" class="content">
<main>
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
deploy: 53ac47fceceba6ebc2c3d54d3a1f62d4b874b8db
2021-06-05 03:42:23 -07:00
<h1 id="tips-and-tricks"><a class="header" href="#tips-and-tricks">Tips and tricks</a></h1>
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<p>This section contains various ideas and snippets that you might find useful while writing
halo2 circuits.</p>
deploy: 53ac47fceceba6ebc2c3d54d3a1f62d4b874b8db
2021-06-05 03:42:23 -07:00
<h2 id="small-range-constraints"><a class="header" href="#small-range-constraints">Small range constraints</a></h2>
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<p>A common constraint used in R1CS circuits is the boolean constraint: <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>.
This constraint can only be satisfied by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span> or <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>.</p>
<p>In halo2 circuits, you can similarly constrain a cell to have one of a small set of
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
values. For example, to constrain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">a</span></span></span></span> to the range <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0..5</span><span class="mclose">]</span></span></span></span>, you would create a gate of
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
the form:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44445em;vertical-align:0em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></p>
deploy: 6e762bdde4b3d44dfe260c48dab592b8fd266463
2022-05-05 19:49:43 -07:00
<p>while to constrain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span></span></span> to be either 7 or 13, you would use:</p>
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">7</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">13</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></p>
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<blockquote>
<p>The underlying principle here is that we create a polynomial constraint with roots at
each value in the set of possible values we want to allow. In R1CS circuits, the maximum
supported polynomial degree is 2 (due to all constraints being of the form <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.46528em;vertical-align:0em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span></span></span>).
In halo2 circuits, you can use arbitrary-degree polynomials - with the proviso that
higher-degree constraints are more expensive to use.</p>
</blockquote>
<p>Note that the roots don't have to be constants; for example <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span> will constrain <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">a</span></span></span></span> to be equal to one of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">}</span></span></span></span> where the latter can be arbitrary polynomials, as long as the whole expression stays within the maximum degree bound.</p>
deploy: 53ac47fceceba6ebc2c3d54d3a1f62d4b874b8db
2021-06-05 03:42:23 -07:00
<h2 id="small-set-interpolation"><a class="header" href="#small-set-interpolation">Small set interpolation</a></h2>
deploy: 3f856e3066a1197af975f63a060108084ca49f82
2020-12-22 12:47:31 -08:00
<p>We can use Lagrange interpolation to create a polynomial constraint that maps
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> for small sets of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>. </p>
<p>For instance, say we want to map a 2-bit value to a &quot;spread&quot; version interleaved
with zeros. We first precompute the evaluations at each point:</p>
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.800000000000001em;vertical-align:-2.1500000000000004em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6500000000000004em;"><span style="top:-4.8100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">00</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0000</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">01</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0001</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">10</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0100</span></span></span><span style="top:-1.2099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">11</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0101</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6500000000000004em;"><span style="top:-4.8100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span></span><span style="top:-1.2099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6500000000000004em;"><span style="top:-4.8100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.27777
deploy: 3f856e3066a1197af975f63a060108084ca49f82
2020-12-22 12:47:31 -08:00
<p>Then, we construct the Lagrange basis polynomial for each point using the
identity:
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.698551em;vertical-align:-1.438221em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">m</span><span class="mrel mtight">&lt;</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mpunct mtight">,</span><span class="mspace mtight" style="margin-right:0.3252777777777778em;"></span><span class="mord mathnormal mtight">m</span><span class="mrel mtight"><span class="mrel mtight"><span class="mord vbox mtight"><span class="thinbox mtight"><span class="rlap mtight"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mord mtight"><span class="mrel mtight"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel mtight">=</span></span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∏</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.438221em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2603300000000002em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="to
deploy: c9b606212ed89947e854cc637f850c96a485fd9f
2021-02-19 07:25:17 -08:00
where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> is the number of data points. (<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span> in our example above.)</p>
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<p>Recall that the Lagrange basis polynomial <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span> evaluates to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> at
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span> at all other <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathnormal">i</span><span class="mord">.</span></span></span></span></p>
deploy: 3f856e3066a1197af975f63a060108084ca49f82
2020-12-22 12:47:31 -08:00
<p>Continuing our example, we get four Lagrange basis polynomials:</p>
deploy: c9b606212ed89947e854cc637f850c96a485fd9f
2021-02-19 07:25:17 -08:00
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.933000000000001em;vertical-align:-3.2165000000000004em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.7165000000000004em;"><span style="top:-5.7165em;"><span class="pstrut" style="height:3.01em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-3.9154999999999998em;"><span class="pstrut" style="height:3.01em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-2.1144999999999996em;"><span class="pstrut" style="height:3.01em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-0.31349999999999945em;"><span class="pstrut" style="height:3.01em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.2165000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></spa
deploy: 3f856e3066a1197af975f63a060108084ca49f82
2020-12-22 12:47:31 -08:00
<p>Our polynomial constraint is then</p>
deploy: 71fae3cf0a49ed51089260b47ec68c9fc3dc1821
2021-05-03 15:55:00 -07:00
<p><span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4000000000000004em;vertical-align:-0.9500000000000004em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">+</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><sp
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
</main>
<nav class="nav-wrapper" aria-label="Page navigation">
<!-- Mobile navigation buttons -->
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
<a rel="prev" href="../user/gadgets.html" class="mobile-nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left">
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<i class="fa fa-angle-left"></i>
</a>
deploy: ec9dcefe9103fc23c13e8195120419d4d2f232a6
2022-10-18 09:58:24 -07:00
<a rel="next" href="../user/wasm-port.html" class="mobile-nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<i class="fa fa-angle-right"></i>
</a>
<div style="clear: both"></div>
</nav>
</div>
</div>
<nav class="nav-wide-wrapper" aria-label="Page navigation">
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
<a rel="prev" href="../user/gadgets.html" class="nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left">
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<i class="fa fa-angle-left"></i>
</a>
deploy: ec9dcefe9103fc23c13e8195120419d4d2f232a6
2022-10-18 09:58:24 -07:00
<a rel="next" href="../user/wasm-port.html" class="nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<i class="fa fa-angle-right"></i>
</a>
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
</nav>
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
</div>
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
<script type="text/javascript">
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
window.playground_copyable = true;
</script>
deploy: 6eca11b37a2dded48d50e5554a7c9f0d7b78a9ef
2021-10-14 03:44:50 -07:00
<script src="../elasticlunr.min.js" type="text/javascript" charset="utf-8"></script>
deploy: 69d987644cf8939b9400523e480b6570336e1be5
2020-12-13 09:44:09 -08:00
<script src="../mark.min.js" type="text/javascript" charset="utf-8"></script>
<script src="../searcher.js" type="text/javascript" charset="utf-8"></script>
<script src="../clipboard.min.js" type="text/javascript" charset="utf-8"></script>
<script src="../highlight.js" type="text/javascript" charset="utf-8"></script>
<script src="../book.js" type="text/javascript" charset="utf-8"></script>
<!-- Custom JS scripts -->
</body>
</html>