<!doctype html><html lang="en">
 <head>
  <meta content="text/html; charset=utf-8" http-equiv="Content-Type">
  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport">
  <title>P0891R1: Everyone Deserves a Little Order</title>
<style data-fill-with="stylesheet">/******************************************************************************
 *                   Style sheet for the W3C specifications                   *
 *
 * Special classes handled by this style sheet include:
 *
 * Indices
 *   - .toc for the Table of Contents (<ol class="toc">)
 *     + <span class="secno"> for the section numbers
 *   - #toc for the Table of Contents (<nav id="toc">)
 *   - ul.index for Indices (<a href="#ref">term</a><span>, in §N.M</span>)
 *   - table.index for Index Tables (e.g. for properties or elements)
 *
 * Structural Markup
 *   - table.data for general data tables
 *     -> use 'scope' attribute, <colgroup>, <thead>, and <tbody> for best results !
 *     -> use <table class='complex data'> for extra-complex tables
 *     -> use <td class='long'> for paragraph-length cell content
 *     -> use <td class='pre'> when manual line breaks/indentation would help readability
 *   - dl.switch for switch statements
 *   - ol.algorithm for algorithms (helps to visualize nesting)
 *   - .figure and .caption (HTML4) and figure and figcaption (HTML5)
 *     -> .sidefigure for right-floated figures
 *   - ins/del
 *
 * Code
 *   - pre and code
 *
 * Special Sections
 *   - .note       for informative notes             (div, p, span, aside, details)
 *   - .example    for informative examples          (div, p, pre, span)
 *   - .issue      for issues                        (div, p, span)
 *   - .assertion  for assertions                    (div, p, span)
 *   - .advisement for loud normative statements     (div, p, strong)
 *   - .annoying-warning for spec obsoletion notices (div, aside, details)
 *
 * Definition Boxes
 *   - pre.def   for WebIDL definitions
 *   - table.def for tables that define other entities (e.g. CSS properties)
 *   - dl.def    for definition lists that define other entitles (e.g. HTML elements)
 *
 * Numbering
 *   - .secno for section numbers in .toc and headings (<span class='secno'>3.2</span>)
 *   - .marker for source-inserted example/figure/issue numbers (<span class='marker'>Issue 4</span>)
 *   - ::before styled for CSS-generated issue/example/figure numbers:
 *     -> Documents wishing to use this only need to add
 *        figcaption::before,
 *        .caption::before { content: "Figure "  counter(figure) " ";  }
 *        .example::before { content: "Example " counter(example) " "; }
 *        .issue::before   { content: "Issue "   counter(issue) " ";   }
 *
 * Header Stuff (ignore, just don't conflict with these classes)
 *   - .head for the header
 *   - .copyright for the copyright
 *
 * Miscellaneous
 *   - .overlarge for things that should be as wide as possible, even if
 *     that overflows the body text area. This can be used on an item or
 *     on its container, depending on the effect desired.
 *     Note that this styling basically doesn't help at all when printing,
 *     since A4 paper isn't much wider than the max-width here.
 *     It's better to design things to fit into a narrower measure if possible.
 *   - js-added ToC jump links (see fixup.js)
 *
 ******************************************************************************/

/******************************************************************************/
/*                                   Body                                     */
/******************************************************************************/

	body {
		counter-reset: example figure issue;

		/* Layout */
		max-width: 50em;               /* limit line length to 50em for readability   */
		margin: 0 auto;                /* center text within page                     */
		padding: 1.6em 1.5em 2em 50px; /* assume 16px font size for downlevel clients */
		padding: 1.6em 1.5em 2em calc(26px + 1.5em); /* leave space for status flag     */

		/* Typography */
		line-height: 1.5;
		font-family: sans-serif;
		widows: 2;
		orphans: 2;
		word-wrap: break-word;
		overflow-wrap: break-word;
		hyphens: auto;

		/* Colors */
		color: black;
		background: white top left fixed no-repeat;
		background-size: 25px auto;
	}


/******************************************************************************/
/*                         Front Matter & Navigation                          */
/******************************************************************************/

/** Header ********************************************************************/

	div.head { margin-bottom: 1em }
	div.head hr { border-style: solid; }

	div.head h1 {
		font-weight: bold;
		margin: 0 0 .1em;
		font-size: 220%;
	}

	div.head h2 { margin-bottom: 1.5em;}

/** W3C Logo ******************************************************************/

	.head .logo {
		float: right;
		margin: 0.4rem 0 0.2rem .4rem;
	}

	.head img[src*="logos/W3C"] {
		display: block;
		border: solid #1a5e9a;
		border-width: .65rem .7rem .6rem;
		border-radius: .4rem;
		background: #1a5e9a;
		color: white;
		font-weight: bold;
	}

	.head a:hover > img[src*="logos/W3C"],
	.head a:focus > img[src*="logos/W3C"] {
		opacity: .8;
	}

	.head a:active > img[src*="logos/W3C"] {
		background: #c00;
		border-color: #c00;
	}

	/* see also additional rules in Link Styling section */

/** Copyright *****************************************************************/

	p.copyright,
	p.copyright small { font-size: small }

/** Back to Top / ToC Toggle **************************************************/

	@media print {
		#toc-nav {
			display: none;
		}
	}
	@media not print {
		#toc-nav {
			position: fixed;
			z-index: 2;
			bottom: 0; left: 0;
			margin: 0;
			min-width: 1.33em;
			border-top-right-radius: 2rem;
			box-shadow: 0 0 2px;
			font-size: 1.5em;
			color: black;
		}
		#toc-nav > a {
			display: block;
			white-space: nowrap;

			height: 1.33em;
			padding: .1em 0.3em;
			margin: 0;

			background: white;
			box-shadow: 0 0 2px;
			border: none;
			border-top-right-radius: 1.33em;
			background: white;
		}
		#toc-nav > #toc-jump {
			padding-bottom: 2em;
			margin-bottom: -1.9em;
		}

		#toc-nav > a:hover,
		#toc-nav > a:focus {
			background: #f8f8f8;
		}
		#toc-nav > a:not(:hover):not(:focus) {
			color: #707070;
		}

		/* statusbar gets in the way on keyboard focus; remove once browsers fix */
		#toc-nav > a[href="#toc"]:not(:hover):focus:last-child {
			padding-bottom: 1.5rem;
		}

		#toc-nav:not(:hover) > a:not(:focus) > span + span {
			/* Ideally this uses :focus-within on #toc-nav */
			display: none;
		}
		#toc-nav > a > span + span {
			padding-right: 0.2em;
		}

		#toc-toggle-inline {
			vertical-align: 0.05em;
			font-size: 80%;
			color: gray;
			color: hsla(203,20%,40%,.7);
			border-style: none;
			background: transparent;
			position: relative;
		}
		#toc-toggle-inline:hover:not(:active),
		#toc-toggle-inline:focus:not(:active) {
			text-shadow: 1px 1px silver;
			top: -1px;
			left: -1px;
		}

		#toc-nav :active {
			color: #C00;
		}
	}

/** ToC Sidebar ***************************************************************/

	/* Floating sidebar */
	@media screen {
		body.toc-sidebar #toc {
			position: fixed;
			top: 0; bottom: 0;
			left: 0;
			width: 23.5em;
			max-width: 80%;
			max-width: calc(100% - 2em - 26px);
			overflow: auto;
			padding: 0 1em;
			padding-left: 42px;
			padding-left: calc(1em + 26px);
			background: inherit;
			background-color: #f7f8f9;
			z-index: 1;
			box-shadow: -.1em 0 .25em rgba(0,0,0,.1) inset;
		}
		body.toc-sidebar #toc h2 {
			margin-top: .8rem;
			font-variant: small-caps;
			font-variant: all-small-caps;
			text-transform: lowercase;
			font-weight: bold;
			color: gray;
			color: hsla(203,20%,40%,.7);
		}
		body.toc-sidebar #toc-jump:not(:focus) {
			width: 0;
			height: 0;
			padding: 0;
			position: absolute;
			overflow: hidden;
		}
	}
	/* Hide main scroller when only the ToC is visible anyway */
	@media screen and (max-width: 28em) {
		body.toc-sidebar {
			overflow: hidden;
		}
	}

	/* Sidebar with its own space */
	@media screen and (min-width: 78em) {
		body:not(.toc-inline) #toc {
			position: fixed;
			top: 0; bottom: 0;
			left: 0;
			width: 23.5em;
			overflow: auto;
			padding: 0 1em;
			padding-left: 42px;
			padding-left: calc(1em + 26px);
			background: inherit;
			background-color: #f7f8f9;
			z-index: 1;
			box-shadow: -.1em 0 .25em rgba(0,0,0,.1) inset;
		}
		body:not(.toc-inline) #toc h2 {
			margin-top: .8rem;
			font-variant: small-caps;
			font-variant: all-small-caps;
			text-transform: lowercase;
			font-weight: bold;
			color: gray;
			color: hsla(203,20%,40%,.7);
		}

		body:not(.toc-inline) {
			padding-left: 29em;
		}
		/* See also Overflow section at the bottom */

		body:not(.toc-inline) #toc-jump:not(:focus) {
			width: 0;
			height: 0;
			padding: 0;
			position: absolute;
			overflow: hidden;
		}
	}
	@media screen and (min-width: 90em) {
		body:not(.toc-inline) {
			margin: 0 4em;
		}
	}

/******************************************************************************/
/*                                Sectioning                                  */
/******************************************************************************/

/** Headings ******************************************************************/

	h1, h2, h3, h4, h5, h6, dt {
		page-break-after: avoid;
		page-break-inside: avoid;
		font: 100% sans-serif;   /* Reset all font styling to clear out UA styles */
		font-family: inherit;    /* Inherit the font family. */
		line-height: 1.2;        /* Keep wrapped headings compact */
		hyphens: manual;         /* Hyphenated headings look weird */
	}

	h2, h3, h4, h5, h6 {
		margin-top: 3rem;
	}

	h1, h2, h3 {
		color: #005A9C;
		background: transparent;
	}

	h1 { font-size: 170%; }
	h2 { font-size: 140%; }
	h3 { font-size: 120%; }
	h4 { font-weight: bold; }
	h5 { font-style: italic; }
	h6 { font-variant: small-caps; }
	dt { font-weight: bold; }

/** Subheadings ***************************************************************/

	h1 + h2,
	#subtitle {
		/* #subtitle is a subtitle in an H2 under the H1 */
		margin-top: 0;
	}
	h2 + h3,
	h3 + h4,
	h4 + h5,
	h5 + h6 {
		margin-top: 1.2em; /* = 1 x line-height */
	}

/** Section divider ***********************************************************/

	:not(.head) > hr {
		font-size: 1.5em;
		text-align: center;
		margin: 1em auto;
		height: auto;
		border: transparent solid 0;
		background: transparent;
	}
	:not(.head) > hr::before {
		content: "\2727\2003\2003\2727\2003\2003\2727";
	}

/******************************************************************************/
/*                            Paragraphs and Lists                            */
/******************************************************************************/

	p {
		margin: 1em 0;
	}

	dd > p:first-child,
	li > p:first-child {
		margin-top: 0;
	}

	ul, ol {
		margin-left: 0;
		padding-left: 2em;
	}

	li {
		margin: 0.25em 0 0.5em;
		padding: 0;
	}

	dl dd {
		margin: 0 0 .5em 2em;
	}

	.head dd + dd { /* compact for header */
		margin-top: -.5em;
	}

	/* Style for algorithms */
	ol.algorithm ol:not(.algorithm),
	.algorithm > ol ol:not(.algorithm) {
	 border-left: 0.5em solid #DEF;
	}

	/* Put nice boxes around each algorithm. */
	[data-algorithm]:not(.heading) {
	  padding: .5em;
	  border: thin solid #ddd; border-radius: .5em;
	  margin: .5em calc(-0.5em - 1px);
	}
	[data-algorithm]:not(.heading) > :first-child {
	  margin-top: 0;
	}
	[data-algorithm]:not(.heading) > :last-child {
	  margin-bottom: 0;
	}

	/* Style for switch/case <dl>s */
	dl.switch > dd > ol.only,
	dl.switch > dd > .only > ol {
	 margin-left: 0;
	}
	dl.switch > dd > ol.algorithm,
	dl.switch > dd > .algorithm > ol {
	 margin-left: -2em;
	}
	dl.switch {
	 padding-left: 2em;
	}
	dl.switch > dt {
	 text-indent: -1.5em;
	 margin-top: 1em;
	}
	dl.switch > dt + dt {
	 margin-top: 0;
	}
	dl.switch > dt::before {
	 content: '\21AA';
	 padding: 0 0.5em 0 0;
	 display: inline-block;
	 width: 1em;
	 text-align: right;
	 line-height: 0.5em;
	}

/** Terminology Markup ********************************************************/


/******************************************************************************/
/*                                 Inline Markup                              */
/******************************************************************************/

/** Terminology Markup ********************************************************/
	dfn   { /* Defining instance */
		font-weight: bolder;
	}
	a > i { /* Instance of term */
		font-style: normal;
	}
	dt dfn code, code.idl {
		font-size: medium;
	}
	dfn var {
		font-style: normal;
	}

/** Change Marking ************************************************************/

	del { color: red;  text-decoration: line-through; }
	ins { color: #080; text-decoration: underline;    }

/** Miscellaneous improvements to inline formatting ***************************/

	sup {
		vertical-align: super;
		font-size: 80%
	}

/******************************************************************************/
/*                                    Code                                    */
/******************************************************************************/

/** General monospace/pre rules ***********************************************/

	pre, code, samp {
		font-family: Menlo, Consolas, "DejaVu Sans Mono", Monaco, monospace;
		font-size: .9em;
		page-break-inside: avoid;
		hyphens: none;
		text-transform: none;
	}
	pre code,
	code code {
		font-size: 100%;
	}

	pre {
		margin-top: 1em;
		margin-bottom: 1em;
		overflow: auto;
	}

/** Inline Code fragments *****************************************************/

  /* Do something nice. */

/******************************************************************************/
/*                                    Links                                   */
/******************************************************************************/

/** General Hyperlinks ********************************************************/

	/* We hyperlink a lot, so make it less intrusive */
	a[href] {
		color: #034575;
		text-decoration: none;
		border-bottom: 1px solid #707070;
		/* Need a bit of extending for it to look okay */
		padding: 0 1px 0;
		margin: 0 -1px 0;
	}
	a:visited {
		border-bottom-color: #BBB;
	}

	/* Use distinguishing colors when user is interacting with the link */
	a[href]:focus,
	a[href]:hover {
		background: #f8f8f8;
		background: rgba(75%, 75%, 75%, .25);
		border-bottom-width: 3px;
		margin-bottom: -2px;
	}
	a[href]:active {
		color: #C00;
		border-color: #C00;
	}

	/* Backout above styling for W3C logo */
	.head .logo,
	.head .logo a {
		border: none;
		text-decoration: none;
		background: transparent;
	}

/******************************************************************************/
/*                                    Images                                  */
/******************************************************************************/

	img {
		border-style: none;
	}

	/* For autogen numbers, add
	   .caption::before, figcaption::before { content: "Figure " counter(figure) ". "; }
	*/

	figure, .figure, .sidefigure {
		page-break-inside: avoid;
		text-align: center;
		margin: 2.5em 0;
	}
	.figure img,    .sidefigure img,    figure img,
	.figure object, .sidefigure object, figure object {
		max-width: 100%;
		margin: auto;
	}
	.figure pre, .sidefigure pre, figure pre {
		text-align: left;
		display: table;
		margin: 1em auto;
	}
	.figure table, figure table {
		margin: auto;
	}
	@media screen and (min-width: 20em) {
		.sidefigure {
			float: right;
			width: 50%;
			margin: 0 0 0.5em 0.5em
		}
	}
	.caption, figcaption, caption {
		font-style: italic;
		font-size: 90%;
	}
	.caption::before, figcaption::before, figcaption > .marker {
		font-weight: bold;
	}
	.caption, figcaption {
		counter-increment: figure;
	}

	/* DL list is indented 2em, but figure inside it is not */
	dd > .figure, dd > figure { margin-left: -2em }

/******************************************************************************/
/*                             Colored Boxes                                  */
/******************************************************************************/

	.issue, .note, .example, .assertion, .advisement, blockquote {
		padding: .5em;
		border: .5em;
		border-left-style: solid;
		page-break-inside: avoid;
	}
	span.issue, span.note {
		padding: .1em .5em .15em;
		border-right-style: solid;
	}

	.issue,
	.note,
	.example,
	.advisement,
	.assertion,
	blockquote {
		margin: 1em auto;
	}
	.note  > p:first-child,
	.issue > p:first-child,
	blockquote > :first-child {
		margin-top: 0;
	}
	blockquote > :last-child {
		margin-bottom: 0;
	}

/** Blockquotes ***************************************************************/

	blockquote {
		border-color: silver;
	}

/** Open issue ****************************************************************/

	.issue {
		border-color: #E05252;
		background: #FBE9E9;
		counter-increment: issue;
		overflow: auto;
	}
	.issue::before, .issue > .marker {
		text-transform: uppercase;
		color: #AE1E1E;
		padding-right: 1em;
		text-transform: uppercase;
	}
	/* Add .issue::before { content: "Issue " counter(issue) " "; } for autogen numbers,
	   or use class="marker" to mark up the issue number in source. */

/** Example *******************************************************************/

	.example {
		border-color: #E0CB52;
		background: #FCFAEE;
		counter-increment: example;
		overflow: auto;
		clear: both;
	}
	.example::before, .example > .marker {
		text-transform: uppercase;
		color: #827017;
		min-width: 7.5em;
		display: block;
	}
	/* Add .example::before { content: "Example " counter(example) " "; } for autogen numbers,
	   or use class="marker" to mark up the example number in source. */

/** Non-normative Note ********************************************************/

	.note {
		border-color: #52E052;
		background: #E9FBE9;
		overflow: auto;
	}

	.note::before, .note > .marker,
	details.note > summary::before,
	details.note > summary > .marker {
		text-transform: uppercase;
		display: block;
		color: hsl(120, 70%, 30%);
	}
	/* Add .note::before { content: "Note"; } for autogen label,
	   or use class="marker" to mark up the label in source. */

	details.note > summary {
		display: block;
		color: hsl(120, 70%, 30%);
	}
	details.note[open] > summary {
		border-bottom: 1px silver solid;
	}

/** Assertion Box *************************************************************/
	/*  for assertions in algorithms */

	.assertion {
		border-color: #AAA;
		background: #EEE;
	}

/** Advisement Box ************************************************************/
	/*  for attention-grabbing normative statements */

	.advisement {
		border-color: orange;
		border-style: none solid;
		background: #FFEECC;
	}
	strong.advisement {
		display: block;
		text-align: center;
	}
	.advisement > .marker {
		color: #B35F00;
	}

/** Spec Obsoletion Notice ****************************************************/
	/* obnoxious obsoletion notice for older/abandoned specs. */

	details {
		display: block;
	}
	summary {
		font-weight: bolder;
	}

	.annoying-warning:not(details),
	details.annoying-warning:not([open]) > summary,
	details.annoying-warning[open] {
		background: #fdd;
		color: red;
		font-weight: bold;
		padding: .75em 1em;
		border: thick red;
		border-style: solid;
		border-radius: 1em;
	}
	.annoying-warning :last-child {
		margin-bottom: 0;
	}

@media not print {
	details.annoying-warning[open] {
		position: fixed;
		left: 1em;
		right: 1em;
		bottom: 1em;
		z-index: 1000;
	}
}

	details.annoying-warning:not([open]) > summary {
		text-align: center;
	}

/** Entity Definition Boxes ***************************************************/

	.def {
		padding: .5em 1em;
		background: #DEF;
		margin: 1.2em 0;
		border-left: 0.5em solid #8CCBF2;
	}

/******************************************************************************/
/*                                    Tables                                  */
/******************************************************************************/

	th, td {
		text-align: left;
		text-align: start;
	}

/** Property/Descriptor Definition Tables *************************************/

	table.def {
		/* inherits .def box styling, see above */
		width: 100%;
		border-spacing: 0;
	}

	table.def td,
	table.def th {
		padding: 0.5em;
		vertical-align: baseline;
		border-bottom: 1px solid #bbd7e9;
	}

	table.def > tbody > tr:last-child th,
	table.def > tbody > tr:last-child td {
		border-bottom: 0;
	}

	table.def th {
		font-style: italic;
		font-weight: normal;
		padding-left: 1em;
		width: 3em;
	}

	/* For when values are extra-complex and need formatting for readability */
	table td.pre {
		white-space: pre-wrap;
	}

	/* A footnote at the bottom of a def table */
	table.def           td.footnote {
		padding-top: 0.6em;
	}
	table.def           td.footnote::before {
		content: " ";
		display: block;
		height: 0.6em;
		width: 4em;
		border-top: thin solid;
	}

/** Data tables (and properly marked-up index tables) *************************/
	/*
		 <table class="data"> highlights structural relationships in a table
		 when correct markup is used (e.g. thead/tbody, th vs. td, scope attribute)

		 Use class="complex data" for particularly complicated tables --
		 (This will draw more lines: busier, but clearer.)

		 Use class="long" on table cells with paragraph-like contents
		 (This will adjust text alignment accordingly.)
		 Alternately use class="longlastcol" on tables, to have the last column assume "long".
	*/

	table {
		word-wrap: normal;
		overflow-wrap: normal;
		hyphens: manual;
	}

	table.data,
	table.index {
		margin: 1em auto;
		border-collapse: collapse;
		border: hidden;
		width: 100%;
	}
	table.data caption,
	table.index caption {
		max-width: 50em;
		margin: 0 auto 1em;
	}

	table.data td,  table.data th,
	table.index td, table.index th {
		padding: 0.5em 1em;
		border-width: 1px;
		border-color: silver;
		border-top-style: solid;
	}

	table.data thead td:empty {
		padding: 0;
		border: 0;
	}

	table.data  thead,
	table.index thead,
	table.data  tbody,
	table.index tbody {
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 <body class="h-entry">
  <div class="head">
   <p data-fill-with="logo"></p>
   <h1 class="p-name no-ref" id="title">P0891R1<br>Everyone Deserves a Little Order</h1>
   <h2 class="no-num no-toc no-ref heading settled" id="subtitle"><span class="content">Published Proposal, <time class="dt-updated" datetime="2018-10-27">2018-10-27</time></span></h2>
   <div data-fill-with="spec-metadata">
    <dl>
     <dt>This version:
     <dd><a class="u-url" href="https://github.com/atomgalaxy/a-little-order/strong-ordering.bs">https://github.com/atomgalaxy/a-little-order/strong-ordering.bs</a>
     <dt>Author:
     <dd>
      <dd class="editor p-author h-card vcard"><span class="p-name fn">Gašper Ažman &lt;gasper.azman@gmail.com></span>
     <dt>Audience:
     <dd>LEWG, LWG
     <dt>Project:
     <dd>ISO/IEC JTC1/SC22/WG21 14882: Programming Language — C++
    </dl>
   </div>
   <div data-fill-with="warning"></div>
   <hr title="Separator for header">
  </div>
  <div class="p-summary" data-fill-with="abstract">
   <h2 class="no-num no-toc no-ref heading settled" id="abstract"><span class="content">Abstract</span></h2>
   <p>The specification of ordering algorithms at the end of <a data-link-type="biblio" href="#biblio-p0768r1">[P0768R1]</a> does not provide the ability to provide a default order for user-defined types (since they are specified in such a way that they are not intended to be customization points), and yet mixes in such a customization for iec559 floating point types. This paper suggests splitting that capability out into a separate customization point.</p>
  </div>
  <nav data-fill-with="table-of-contents" id="toc">
   <h2 class="no-num no-toc no-ref" id="contents">Table of Contents</h2>
   <ol class="toc" role="directory">
    <li><a href="#history"><span class="secno">1</span> <span class="content">Revision History</span></a>
    <li><a href="#sup"><span class="secno">2</span> <span class="content">Status of this paper</span></a>
    <li><a href="#problem"><span class="secno">3</span> <span class="content">Problem Description</span></a>
    <li><a href="#exposition"><span class="secno">4</span> <span class="content">Exposition: On Natural and Default Orderings</span></a>
    <li><a href="#status"><span class="secno">5</span> <span class="content">Status Quo</span></a>
    <li>
     <a href="#proposal"><span class="secno">6</span> <span class="content">Proposal</span></a>
     <ol class="toc">
      <li><a href="#new-customization-point"><span class="secno">6.1</span> <span class="content">Introduce a new customization point for an arbitrary total ordering</span></a>
      <li><a href="#noexcept"><span class="secno">6.2</span> <span class="content">Move the iec559 treatment (point 1.1) from <code class="highlight"><c- n>strong_order</c-></code> to <code class="highlight"><c- n>default_order</c-></code></span></a>
     </ol>
    <li>
     <a href="#name-bikeshed"><span class="secno">7</span> <span class="content">Name of The Default Order Algorithm</span></a>
     <ol class="toc">
      <li><a href="#name-default-order"><span class="secno">7.1</span> <span class="content"><code class="highlight"><c- n>default_order</c-></code></span></a>
      <li><a href="#name-total-order"><span class="secno">7.2</span> <span class="content"><code class="highlight"><c- n>total_order</c-></code></span></a>
      <li><a href="#name-arbitrary-order"><span class="secno">7.3</span> <span class="content"><code class="highlight"><c- n>arbitrary_order</c-></code></span></a>
      <li><a href="#default-strong-order"><span class="secno">7.4</span> <span class="content"><code class="highlight"><c- n>default_strong_order</c-></code></span></a>
      <li><a href="#default-total-order"><span class="secno">7.5</span> <span class="content"><code class="highlight"><c- n>default_total_order</c-></code></span></a>
     </ol>
    <li><a href="#compat"><span class="secno">8</span> <span class="content">On Compatibility Between the Natural and Default Orderings</span></a>
    <li><a href="#why-not"><span class="secno">9</span> <span class="content">Why not just make <code class="highlight"><c- n>strong_order</c-></code> a customization point?</span></a>
    <li><a href="#wording"><span class="secno">10</span> <span class="content">Proposed Wording</span></a>
    <li><a href="#ack"><span class="secno">11</span> <span class="content">Acknowledgments</span></a>
    <li><a href="#example-cust-point"><span class="secno"></span> <span class="content">Appendix A: Proof <code class="highlight"><c- n>strong_order</c-></code> is not a valid customization point</span></a>
    <li>
     <a href="#references"><span class="secno"></span> <span class="content">References</span></a>
     <ol class="toc">
      <li><a href="#normative"><span class="secno"></span> <span class="content">Normative References</span></a>
      <li><a href="#informative"><span class="secno"></span> <span class="content">Informative References</span></a>
     </ol>
   </ol>
  </nav>
  <main>
   <h2 class="heading settled" data-level="1" id="history"><span class="secno">1. </span><span class="content">Revision History</span><a class="self-link" href="#history"></a></h2>
   <ul>
    <li data-md>
     <p>r1: Incorporated feedback from EWG meeting in Rappersville.</p>
   </ul>
   <p>The feedback was:</p>
   <ul>
    <li data-md>
     <p>Remove the floating point exception (bullet 1.1) as R0 recommended, since Unicode strings, etc., are a possible rationale.</p>
    <li data-md>
     <p>Do not propose propose making existing <code class="highlight"><c- o>*</c-><c- n>_order</c-></code> functions "customization points" (as used in <a data-link-type="biblio" href="#biblio-p0551r3">[P0551R3]</a>).</p>
    <li data-md>
     <p>Add a new <code class="highlight"><c- n>default_order</c-></code> customization point, along with a bikeshedding section on its actual name, with the behaviour:</p>
     <ul>
      <li data-md>
       <p>It has the IEC 559 behavior from bullet 1.1 of <code class="highlight"><c- n>strong_order</c-></code></p>
      <li data-md>
       <p>It is defined for all (other) floating-point types; it is implementation-defined whether it is consistent with the partial order from the comparison operators. (Implementations should do this.)</p>
      <li data-md>
       <p>It is a customization point (à la <a data-link-type="biblio" href="#biblio-p0551r3">[P0551R3]</a>).</p>
     </ul>
    <li data-md>
     <p>Investigate the possibility of adding Lawrence’s weak order (from <a data-link-type="biblio" href="#biblio-p0100r2">[P0100R2]</a>) for floating-point numbers (which did not make it in with spaceship).</p>
   </ul>
   <h2 class="heading settled" data-level="2" id="sup"><span class="secno">2. </span><span class="content">Status of this paper</span><a class="self-link" href="#sup"></a></h2>
   <p>This paper has been seen by LEWG, provided feedback, and reworked. It is ready to be seen again by LEWG.</p>
   <h2 class="heading settled" data-level="3" id="problem"><span class="secno">3. </span><span class="content">Problem Description</span><a class="self-link" href="#problem"></a></h2>
   <p>This paper is a proposal to amend a library extension that has been voted into the working draft as part of <a data-link-type="biblio" href="#biblio-p0768r1">[P0768R1]</a>.</p>
   <p>This paper proposes a new ordering customization point for possibly non-semantic strong orderings, which <em>should</em> be equal to <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>strong_order</c-></code> for strongly-ordered types, and can be customized to represent an arbitrary total order for types that do not have a semantic ordering.</p>
   <p>The current C++ standard does not have an explicitly designated customization point for providing a <em>default ordering</em>. <em>Elements of Programming</em> uses <code class="highlight"><c- n>less</c-><c- o>&lt;</c-><c- n>T</c-><c- o>>::</c-><c- k>operator</c-><c- p>()</c-></code> for this purpose, as does the global order for pointers; but with the introduction of <code class="highlight"><c- k>operator</c-><c- o>&lt;=></c-></code>, <code class="highlight"><c- n>less</c-><c- o>&lt;</c-><c- n>T</c-><c- o>></c-></code> is missing features, such as computing equality without calling it twice. It has also failed to get adoption for this purpose throughout the years, perhaps exactly due to the missing features.</p>
   <p class="note" role="note"><span>Note:</span> see <a href="#exposition">§4 Exposition: On Natural and Default Orderings</a> for the definitions and discussion of orderings.</p>
   <p>The wording of point 1.1 of the <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>strong_order</c-></code> algorithm suggests that <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>strong_order</c-></code> is finally this missing customization point for specifying a default ordering for types whose natural ordering is not strong and total, since it does exactly that for the iec559 types.</p>
   <p>The issue is that the rest of the points make this function rather unsuitable for use as a customization point, since the language explicitly makes it not SFINAE-friendly. In the event that it cannot be synthesized, it is marked as <em>deleted</em>, and not as <em>"shall not participate in overload resolution"</em>.</p>
   <p>LEWG has expressed a strong preference to introducing a new customization point for such non-semantic strong orderings, and clarified that <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>strong_order</c-></code> should not be a customization point.</p>
   <h2 class="heading settled" data-level="4" id="exposition"><span class="secno">4. </span><span class="content">Exposition: On Natural and Default Orderings</span><a class="self-link" href="#exposition"></a></h2>
   <p>There are obviously many reasons for sorting. However, this paper is chiefly concerned with the division between the <em>natural ordering</em> and the <em>default total ordering</em> as required for <strong>Regular</strong> types by Stepanov and McJones in their seminal work <em>Elements of Programming</em> (page 62, section 4.4).</p>
   <p>The <strong>natural ordering</strong> is the ordering that makes semantic sense for a type. This is the ordering that <code class="highlight"><c- k>operator</c-><c- o>&lt;=></c-></code> and its library extensions are tailor-made for: not every type is ordered (or even equality-comparable), and when a type supports an ordering, it might be strong, partial, or weak.</p>
   <p>We use these orderings when we need them to make sense - heaps, scheduling tasks by topological sorts, various displays for users, etc. Not all value types have a natural ordering, because not all types are ordered. The Gaussian integers are one such type.</p>
   <p>The <strong>default ordering</strong>, from <em>Elements of Programming</em> is the finest ordering (transitive antisymmetric antireflexive relation) that a type admits, with its equality is defined by value-substitutability (unequal elements must be ordered); it is always strong and total, and might not make semantic sense.</p>
   <p>According to <em>Elements of Programming</em>, every <strong>Regular</strong> type should provide a default ordering.</p>
   <p>A type with a default ordering is far more useful than one without; ordering enables the use of tree-based containers (i.e. <code class="highlight"><c- n>map</c-></code>, <code class="highlight"><c- n>set</c-></code>), and algorithms based on sorted data (<code class="highlight"><c- n>unique</c-></code>, the various set algorithms, <code class="highlight"><c- n>merge</c-></code>, and the various versions of binary search) -- and this is just the tip of the iceberg. The only requirement for the above is having <em>a</em> total strong ordering - what the ordering <em>means</em> is utterly irrelevant, we only require a global order relation.</p>
   <p>The lexicographic ordering of the Gaussian integers is a good example of a default ordering.</p>
   <p>Another excellent example is <code class="highlight"><c- b>float</c-></code> -- its various <code class="highlight"><c- n>NaN</c-></code>s and infinities are not ordered, which is the reason its natural ordering is not suitable as a default ordering. However, iec559 defines a total strong ordering for those values, thus enabling the uses outlined above.</p>
   <h2 class="heading settled" data-level="5" id="status"><span class="secno">5. </span><span class="content">Status Quo</span><a class="self-link" href="#status"></a></h2>
   <p>For reference, the current specification for the <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>strong_order</c-></code> algorithm is as follows:</p>
<pre class="language-cpp highlight"><c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
<c- k>constexpr</c-> <c- n>strong_ordering</c-> <c- n>strong_order</c-><c- p>(</c-><c- k>const</c-> <c- n>T</c-><c- o>&amp;</c-> <c- n>a</c-><c- p>,</c-> <c- k>const</c-> <c- n>T</c-><c- o>&amp;</c-> <c- n>b</c-><c- p>);</c->
</pre>
   <ol>
    <li data-md>
     <p>Effects: Compares two values and produces a result of type <code class="highlight"><c- n>strong_ordering</c-></code>:</p>
     <ol>
      <li data-md>
       <p>If <code class="highlight"><c- n>numeric_limits</c-><c- o>&lt;</c-><c- n>T</c-><c- o>>::</c-><c- n>is_iec559</c-></code> is true, returns a result of type <code class="highlight"><c- n>strong_ordering</c-></code> that is consistent with the <code class="highlight"><c- n>totalOrder</c-></code> operation as specified in ISO/IEC/IEEE 60559.</p>
      <li data-md>
       <p>Otherwise, returns <code class="highlight"><c- n>a</c-> <c- o>&lt;=></c-> <c- n>b</c-></code> if that expression is well-formed and convertible to <code class="highlight"><c- n>strong_ordering</c-></code>.</p>
      <li data-md>
       <p>Otherwise, if the expression <code class="highlight"><c- n>a</c-> <c- o>&lt;=></c-> <c- n>b</c-></code> is well-formed, then the function shall be defined as deleted.</p>
      <li data-md>
       <p>Otherwise, if the expressions <code class="highlight"><c- n>a</c-> <c- o>==</c-> <c- n>b</c-></code> and <code class="highlight"><c- n>a</c-> <c- o>&lt;</c-> <c- n>b</c-></code> are each well-formed and convertible to <code class="highlight"><c- b>bool</c-></code>, returns <code class="highlight"><c- n>strong_ordering</c-><c- o>::</c-><c- n>equal</c-></code> when <code class="highlight"><c- n>a</c-> <c- o>==</c-> <c- n>b</c-></code> is <code class="highlight">true</code>, otherwise returns <code class="highlight"><c- n>strong_ordering</c-><c- o>::</c-><c- n>less</c-></code> when <code class="highlight"><c- n>a</c-> <c- o>&lt;</c-> <c- n>b</c-></code> is <code class="highlight">true</code>, and otherwise returns <code class="highlight"><c- n>strong_ordering</c-><c- o>::</c-><c- n>greater</c-></code>.</p>
      <li data-md>
       <p>Otherwise, the function shall be defined as deleted.</p>
     </ol>
   </ol>
   <h2 class="heading settled" data-level="6" id="proposal"><span class="secno">6. </span><span class="content">Proposal</span><a class="self-link" href="#proposal"></a></h2>
   <h3 class="heading settled" data-level="6.1" id="new-customization-point"><span class="secno">6.1. </span><span class="content">Introduce a new customization point for an arbitrary total ordering</span><a class="self-link" href="#new-customization-point"></a></h3>
   <p>Introduce a new algorithm, with the name <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>default_order</c-></code> (but see <a href="#name-bikeshed">§7 Name of The Default Order Algorithm</a>), with the following semantics:</p>
   <p>Let <code class="highlight"><c- n>T</c-></code> be a type, and <code class="highlight"><c- n>a</c-></code> and <code class="highlight"><c- n>b</c-></code> be values of type <code class="highlight"><c- n>T</c-></code> (as if <code class="highlight"><c- n>a</c-></code> and <code class="highlight"><c- n>b</c-></code> were <code class="highlight"><c- n>declval</c-><c- o>&lt;</c-><c- n>T</c-><c- o>></c-><c- p>()</c-></code>):</p>
   <ol>
    <li data-md>
     <p>If well-formed, the expression <code class="highlight"><c- n>default_order</c-><c- p>(</c-><c- n>a</c-><c- p>,</c-> <c- n>b</c-><c- p>)</c-></code> shall return an object convertible to <code class="highlight"><c- n>strong_ordering</c-></code>.</p>
    <li data-md>
     <p><code class="highlight"><c- n>default_order</c-></code> shall be a designated customization point in the sense of <a data-link-type="biblio" href="#biblio-p0551r3">[P0551R3]</a>.</p>
    <li data-md>
     <p>If the expression <code class="highlight"><c- n>strong_order</c-><c- p>(</c-><c- n>a</c-><c- p>,</c-> <c- n>b</c-><c- p>)</c-></code> is well-formed, then the expression <code class="highlight"><c- n>default_order</c-><c- p>(</c-><c- n>a</c-><c- p>,</c-> <c- n>b</c-><c- p>)</c-></code> is well-formed.</p>
    <li data-md>
     <p>If not explicitly specialized for a type <code class="highlight"><c- n>T</c-></code> and <code class="highlight"><c- n>strong_order</c-><c- p>(</c-><c- n>a</c-><c- p>,</c-> <c- n>b</c-><c- p>)</c-></code> is well-formed, the expression <code class="highlight"><c- n>default_order</c-><c- p>(</c-><c- n>a</c-><c- p>,</c-> <c- n>b</c-><c- p>)</c-></code> shall return the result of the expression <code class="highlight"><c- n>strong_order</c-><c- p>(</c-><c- n>a</c-><c- p>,</c-> <c- n>b</c-><c- p>)</c-></code>. (Note: this means that <code class="highlight"><c- n>default_order</c-></code> defaults to <code class="highlight"><c- n>strong_order</c-></code> if available for the type. -- end note.)</p>
   </ol>
   <h3 class="heading settled" data-level="6.2" id="noexcept"><span class="secno">6.2. </span><span class="content">Move the iec559 treatment (point 1.1) from <code class="highlight"><c- n>strong_order</c-></code> to <code class="highlight"><c- n>default_order</c-></code></span><a class="self-link" href="#noexcept"></a></h3>
   <p>Since this paper adds an explicit customization point for a non-semantic total order on any type, the exception for iec559 floating-point types can now be implemented by explicitly providing an implementation for the <code class="highlight"><c- n>default_order</c-></code> customization point for those types.</p>
   <p>The concrete proposal is as follows:</p>
   <ol>
    <li data-md>
     <p>remove point 1.1 of the <code class="highlight"><c- n>strong_order</c-></code> algorithm, which removes the exception for iec559 types.</p>
    <li data-md>
     <p>add a provision to <code class="highlight"><c- n>default_order</c-></code></p>
     <ol>
      <li data-md>
       <p>if <code class="highlight"><c- p>(</c-><c- b>bool</c-><c- p>)</c-> <c- n>numeric_limits</c-><c- o>&lt;</c-><c- n>T</c-><c- o>>::</c-><c- n>is_iec559</c-></code> is <code class="highlight">true</code>, the expression <code class="highlight"><c- n>default_order</c-><c- p>(</c-><c- n>a</c-><c- p>,</c-> <c- n>b</c-><c- p>)</c-></code> returns a result of type <code class="highlight"><c- n>strong_ordering</c-></code> that is consistent with the <code class="highlight"><c- n>totalOrder</c-></code> operation as specified in ISO/IEC/IEEE 60559.</p>
     </ol>
   </ol>
   <p>Remark: libraries are encouraged to provide implementations of this customization point for their user-defined types, especially if the <code class="highlight"><c- k>operator</c-><c- o>&lt;=></c-></code> for the type does not provide a <code class="highlight"><c- n>strong_ordering</c-></code>, to enable possibly non-semantic total orderings over their entire domain. The use of implementations of this customization point that do not define a strict total order render the program ill-formed (no diagnostic required).</p>
   <h2 class="heading settled" data-level="7" id="name-bikeshed"><span class="secno">7. </span><span class="content">Name of The Default Order Algorithm</span><a class="self-link" href="#name-bikeshed"></a></h2>
   <p>The new customization point that exposes an arbitrary order for any type that cares to provide one needs a name. This paper suggests a 5-way poll to LEWG on the following options:</p>
   <h3 class="heading settled" data-level="7.1" id="name-default-order"><span class="secno">7.1. </span><span class="content"><code class="highlight"><c- n>default_order</c-></code></span><a class="self-link" href="#name-default-order"></a></h3>
   <ul>
    <li data-md>
     <p>Pros:</p>
     <ul>
      <li data-md>
       <p>it is what <em>Elements of Programmming</em> calls it, a-priori giving it wide recognition</p>
      <li data-md>
       <p>it is reasonably short</p>
      <li data-md>
       <p>the name is semantically netural.</p>
      <li data-md>
       <p>does not imply that the order has any meaning past <em>strong</em> and <em>total</em> (eg. "lexicographic")</p>
     </ul>
    <li data-md>
     <p>Cons:</p>
     <ul>
      <li data-md>
       <p>it implies neither that it is total, nor that it is strong, despite the requirement it be both.</p>
     </ul>
   </ul>
   <p><code class="highlight"><c- n>default_order</c-></code> is the favorite of the paper author, as well as most of the reviewers of this paper.</p>
   <h3 class="heading settled" data-level="7.2" id="name-total-order"><span class="secno">7.2. </span><span class="content"><code class="highlight"><c- n>total_order</c-></code></span><a class="self-link" href="#name-total-order"></a></h3>
   <ul>
    <li data-md>
     <p>Pros:</p>
     <ul>
      <li data-md>
       <p>communicates the totality of the order</p>
      <li data-md>
       <p><code class="highlight"><c- n>totalOrder</c-></code> is the name <code class="highlight"><c- n>ISO</c-><c- o>/</c-><c- n>IEC</c-><c- o>/</c-><c- n>IEEE</c-> <c- mi>60559</c-></code> chose for the order over floating point types that implement these semantics</p>
      <li data-md>
       <p>it is reasonably short</p>
     </ul>
    <li data-md>
     <p>Cons:</p>
     <ul>
      <li data-md>
       <p>does not imply it is a strong order</p>
      <li data-md>
       <p>at least to the author’s mind, vaguely implies semantics past <em>strong</em> and <em>total</em>, which might not be the case.</p>
     </ul>
   </ul>
   <h3 class="heading settled" data-level="7.3" id="name-arbitrary-order"><span class="secno">7.3. </span><span class="content"><code class="highlight"><c- n>arbitrary_order</c-></code></span><a class="self-link" href="#name-arbitrary-order"></a></h3>
   <ul>
    <li data-md>
     <p>Pros:</p>
     <ul>
      <li data-md>
       <p>clearly implies it might not be anything past <em>strong</em> and <em>total</em></p>
     </ul>
    <li data-md>
     <p>Cons:</p>
     <ul>
      <li data-md>
       <p>implies neither that it is total, nor that it is strong</p>
      <li data-md>
       <p>very vague</p>
      <li data-md>
       <p>sounds slightly derogatory for a facility that is to be used mostly by critical algorithms and data structuers</p>
     </ul>
   </ul>
   <h3 class="heading settled" data-level="7.4" id="default-strong-order"><span class="secno">7.4. </span><span class="content"><code class="highlight"><c- n>default_strong_order</c-></code></span><a class="self-link" href="#default-strong-order"></a></h3>
   <ul>
    <li data-md>
     <p>Pros:</p>
     <ul>
      <li data-md>
       <p>clear</p>
      <li data-md>
       <p>roughly in line with <em>Elements of Programming</em></p>
     </ul>
    <li data-md>
     <p>Cons:</p>
     <ul>
      <li data-md>
       <p>very long for a facitlity that will be used for in-line calls to <code class="highlight"><c- n>sort</c-></code> and <code class="highlight"><c- n>unique</c-></code></p>
      <li data-md>
       <p>still not clear it is total</p>
     </ul>
   </ul>
   <h3 class="heading settled" data-level="7.5" id="default-total-order"><span class="secno">7.5. </span><span class="content"><code class="highlight"><c- n>default_total_order</c-></code></span><a class="self-link" href="#default-total-order"></a></h3>
   <ul>
    <li data-md>
     <p>Pros:</p>
     <ul>
      <li data-md>
       <p>clear</p>
      <li data-md>
       <p>roughly in line with iec559 and <em>Elements of Programming</em></p>
     </ul>
    <li data-md>
     <p>Cons:</p>
     <ul>
      <li data-md>
       <p>very long for a facitlity that will be used for in-line calls to <code class="highlight"><c- n>sort</c-></code> and <code class="highlight"><c- n>unique</c-></code></p>
      <li data-md>
       <p>still not clear it is strong</p>
     </ul>
   </ul>
   <h2 class="heading settled" data-level="8" id="compat"><span class="secno">8. </span><span class="content">On Compatibility Between the Natural and Default Orderings</span><a class="self-link" href="#compat"></a></h2>
   <p><em>Elements of Programming</em> specifies that for types where the natural and default orderings differ, the default ordering should be <em>finer</em> than the natural one: that is, if <code class="highlight"><c- n>a</c-></code> and <code class="highlight"><c- n>b</c-></code> are <em>comparable</em> and compare unequal under <code class="highlight"><c- o>&lt;=></c-></code>, the default order produces the same result (less or greater).</p>
   <p>It is the opinion of the author that requiring this in the language of the standard library as a mandatory semantic constraint seems like a bad idea.</p>
   <p>For instance, if one takes the Gaussian integers ordered by the Manhattan distance to zero (sum of absolute values of the two components), the compatible total order (a lexicographic ordering of every equivalence class) is far slower to compute than the simple lexicographic one.</p>
   <p>Furthermore, if needed, a <em>finer</em> compatible total order can always be achieved on the fly by comparing with the natural order first: if the result is <code class="highlight"><c- n>less</c-></code> or <code class="highlight"><c- n>greater</c-></code>, keep the result; otherwise, fall back on the default ordering.</p>
   <h2 class="heading settled" data-level="9" id="why-not"><span class="secno">9. </span><span class="content">Why not just make <code class="highlight"><c- n>strong_order</c-></code> a customization point?</span><a class="self-link" href="#why-not"></a></h2>
   <p>Main reasons:</p>
   <ol>
    <li data-md>
     <p>it would inhibit providing both a natural (<code class="highlight"><c- o>&lt;=></c-></code>, <code class="highlight"><c- n>strong_order</c-></code>) (and possibly slow) and a default (fast) order for a type</p>
    <li data-md>
     <p>the committee guidance strongly preferred this option, as it keeps the meaning of <code class="highlight"><c- n>strong_order</c-></code> fixed (since it is not a customization point)</p>
    <li data-md>
     <p>it is less surprising if the order algorithms that are related to order types by name (<code class="highlight"><c- n>weak_order</c-></code> - <code class="highlight"><c- n>weak_ordering</c-></code>, <code class="highlight"><c- n>partial_order</c-></code> - <code class="highlight"><c- n>partial_ordering</c-></code>, <code class="highlight"><c- n>strong_order</c-></code> - <code class="highlight"><c- n>strong_ordering</c-></code>) had the same specification, while a fourth customization point that isn’t related by any of them by name serves as the customization point for default order.</p>
   </ol>
   <p>It is notable that this was the direction suggested by the orginal paper, but the committee rejected it.</p>
   <h2 class="heading settled" data-level="10" id="wording"><span class="secno">10. </span><span class="content">Proposed Wording</span><a class="self-link" href="#wording"></a></h2>
   <p>From section 24.x.4, Comparison Algorithms [cmp.alg], under <code class="highlight"><c- n>strong_order</c-></code>:</p>
   <div class="del">
     (1.1)  If <code class="highlight"><c- n>numeric_limits</c-><c- o>&lt;</c-><c- n>T</c-><c- o>>::</c-><c- n>is_iec559</c-></code> is true, returns a result of type <code class="highlight"><c- n>strong_ordering</c-></code> that is consistent with the <code class="highlight"><c- n>totalOrder</c-></code> operation as specified in ISO/IEC/IEEE 60559. 
    <p>(1.2) Otherwise, returns</p>
   </div>
   <div class="ins"> (1.2) Returns </div>
   <p>After 24.x.4 paragraph 3, <code class="highlight"><c- n>weak_order</c-></code>, add:</p>
   <div class="ins">
    <ul style="list-style-type: none; padding-left: 0px;">
     <li>
      <code class="highlight"><c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c-> <c- k>constexpr</c-> <c- n>strong_ordering</c-> <c- n>default_order</c-><c- p>(</c-><c- k>const</c-> <c- n>T</c-><c- o>&amp;</c-> <c- n>a</c-><c- p>,</c-> <c- k>const</c-> <c- n>T</c-><c- o>&amp;</c-> <c- n>b</c-><c- p>);</c-></code><br>4. <em>Effects:</em> 
      <ul style="list-style-type: none">
       <li>(4.1) if <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>numeric_limits</c-><c- o>&lt;</c-><c- n>T</c-><c- o>>::</c-><c- n>is_iec559</c-></code> is true, returns a result of type <code class="highlight"><c- n>strong_ordering</c-></code> that is consistent with the <code class="highlight"><c- n>totalOrder</c-></code> operation as specified in ISO/IEC/IEEE 60559.
       <li>(4.2) Otherwise, returns <code class="highlight"><c- n>strong_order</c-><c- p>(</c-><c- n>a</c-><c- p>,</c-> <c- n>b</c-><c- p>)</c-></code> if that expression is well-formed and convertible to <code class="highlight"><c- n>strong_ordering</c-></code>
       <li>(4.3) Otherwise, this function shall not participate in overload resolution.
      </ul>
     <li>5. <em>Remarks</em>: this function is a designated customization point ([namespace.std]).
    </ul>
   </div>
   <p>Under <code class="highlight"><c- p>[</c-><c- n>cmp</c-><c- p>.</c-><c- n>syn</c-><c- p>]</c-></code> -> <code class="highlight"><c- p>[</c-><c- n>cmp</c-><c- p>.</c-><c- n>alg</c-><c- p>]</c-></code>, insert declaration:</p>
   <div class="ins"> <code class="highlight"><c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c-> <c- k>constexpr</c-> <c- n>strong_ordering</c-> <c- n>default_order</c-><c- p>(</c-><c- k>const</c-> <c- n>T</c-><c- o>&amp;</c-> <c- n>a</c-><c- p>,</c-> <c- k>const</c-> <c- n>T</c-><c- o>&amp;</c-> <c- n>b</c-><c- p>);</c-></code> </div>
   <h2 class="heading settled" data-level="11" id="ack"><span class="secno">11. </span><span class="content">Acknowledgments</span><a class="self-link" href="#ack"></a></h2>
   <p>I would like to thank</p>
   <ul>
    <li data-md>
     <p><strong>Roger Orr</strong> for bringing this to my attention;</p>
    <li data-md>
     <p><strong>Thomas Köppe</strong> for his valuable comments, review, and most of all some extremely clear and laconic wording;</p>
    <li data-md>
     <p><strong>Sam Finch</strong> for <em>thoroughly</em> breaking my examples, some example code, great substantive comments, and pointing out that the current definition actually breaks types that define a partially-ordered set of comparison operators;</p>
    <li data-md>
     <p><strong>Richard Smith</strong> for further fixing my example in light of Concepts, and example code.</p>
    <li data-md>
     <p><strong>Herb Sutter and Walter Brown</strong> for providing guidance on customization points.</p>
    <li data-md>
     <p><strong>Louis Dionne</strong> for great comments on the structure of the paper and how to bring the focus where it needs to be;</p>
    <li data-md>
     <p><strong>Walter Brown</strong> for representing the paper at committee meetings when I could not make it in person, and guidance with direction;</p>
    <li data-md>
     <p><strong>Herb Sutter</strong> for his comments and support for getting ordering right.</p>
   </ul>
   <p>And, <em>again</em>, a special thank-you to Walter Brown, who, with his final lightning talk in Bellevue, reminded me to remember whose shoulders I’m standing on.</p>
   <p>Thank you all!</p>
   <h2 class="heading settled" id="example-cust-point"><span class="content">Appendix A: Proof <code class="highlight"><c- n>strong_order</c-></code> is not a valid customization point</span><a class="self-link" href="#example-cust-point"></a></h2>
   <p>Say we have a template struct representing the Gaussian integers, with a <em>natural order</em> defined by the Manhattan distance from <code class="highlight"><c- mi>0</c-><c- o>+</c-><c- mi>0</c-><c- n>i</c-></code>. This struct still defines a <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>strong_order</c-></code> to model <strong>Regular</strong>.</p>
   <p class="note" role="note"><span>Note:</span> The <strong>Regular</strong> above refers to the <em>Elements of Programming</em> concept, not the ISO C++ <strong>Regular</strong>, which is weaker.</p>
   <p class="note" role="note"><span>Note:</span> There is no natural order on Gaussian integers, but humor this example, please.</p>
<pre class="language-cpp highlight"><c- k>namespace</c-> <c- n>user</c-> <c- p>{</c->
  <c- k>template</c-> <c- o>&lt;</c-><c- k>typename</c-> <c- n>T</c-><c- o>></c->
  <c- k>struct</c-> <c- n>gaussian</c-> <c- p>{</c->
    <c- k>static_assert</c-><c- p>(</c-><c- n>std</c-><c- o>::</c-><c- n>is_integral_v</c-><c- o>&lt;</c-><c- n>T</c-><c- o>></c-><c- p>);</c->
    <c- n>T</c-> <c- n>re</c-><c- p>;</c->
    <c- n>T</c-> <c- n>im</c-><c- p>;</c->

    <c- k>constexpr</c-> <c- n>std</c-><c- o>::</c-><c- n>strong_equality</c-> <c- k>operator</c-><c- o>==</c-><c- p>(</c-><c- n>gassian</c-> <c- k>const</c-><c- o>&amp;</c-> <c- n>other</c-><c- p>)</c-> <c- k>const</c-> <c- p>{</c->
      <c- k>return</c-> <c- n>re</c-> <c- o>==</c-> <c- n>other</c-><c- p>.</c-><c- n>re</c-> <c- o>&amp;&amp;</c-> <c- n>im</c-> <c- o>==</c-> <c- n>other</c-><c- p>.</c-><c- n>im</c-><c- p>;</c->
    <c- p>}</c->
    <c- k>constexpr</c-> <c- n>std</c-><c- o>::</c-><c- n>weak_ordering</c-> <c- k>operator</c-><c- o>&lt;=></c-><c- p>(</c-><c- n>gaussian</c-> <c- k>const</c-><c- o>&amp;</c-> <c- n>other</c-><c- p>)</c-> <c- k>const</c-> <c- p>{</c->
      <c- k>return</c-> <c- p>(</c-><c- o>*</c-><c- k>this</c-> <c- o>==</c-> <c- n>other</c-><c- p>)</c-> <c- o>?</c-> <c- n>std</c-><c- o>::</c-><c- n>weak_ordering</c-><c- o>::</c-><c- nl>equal</c->
                              <c- p>:</c-> <c- p>(</c-><c- n>abs</c-><c- p>(</c-><c- o>*</c-><c- k>this</c-><c- p>)</c-> <c- o>==</c-> <c- n>abs</c-><c- p>(</c-><c- n>other</c-><c- p>))</c-> <c- o>?</c-> <c- n>std</c-><c- o>::</c-><c- n>weak_ordering</c-><c- o>::</c-><c- nl>equivalent</c->
                                                           <c- p>:</c-> <c- n>abs</c-><c- p>(</c-><c- o>*</c-><c- k>this</c-><c- p>)</c-> <c- o>&lt;=></c-> <c- n>abs</c-><c- p>(</c-><c- n>other</c-><c- p>);</c->
    <c- p>}</c->
    <c- k>friend</c-> <c- k>constexpr</c-> <c- n>T</c-> <c- n>abs</c-><c- p>(</c-><c- n>gaussian</c-> <c- k>const</c-><c- o>&amp;</c-><c- p>)</c-> <c- p>{</c->
      <c- k>using</c-> <c- n>std</c-><c- o>::</c-><c- n>abs</c-><c- p>;</c->
      <c- k>return</c-> <c- nf>abs</c-><c- p>(</c-><c- n>re</c-><c- p>)</c-> <c- o>+</c-> <c- n>abs</c-><c- p>(</c-><c- n>im</c-><c- p>);</c->
    <c- p>}</c->

    <c- k>friend</c-> <c- k>constexpr</c-> <c- n>std</c-><c- o>::</c-><c- n>strong_ordering</c-> <c- n>strong_order</c-><c- p>(</c-><c- n>gaussian</c-> <c- k>const</c-><c- o>&amp;</c-> <c- n>x</c-><c- p>,</c->
                                                       <c- n>gaussian</c-> <c- k>const</c-><c- o>&amp;</c-> <c- n>y</c-><c- p>)</c-> <c- p>{</c->
      <c- c1>// compare lexicographically</c->
      <c- k>return</c-> <c- n>std</c-><c- o>::</c-><c- n>tie</c-><c- p>(</c-><c- n>x</c-><c- p>.</c-><c- n>re</c-><c- p>,</c-> <c- n>x</c-><c- p>.</c-><c- n>im</c-><c- p>)</c-> <c- o>&lt;=></c-> <c- n>std</c-><c- o>::</c-><c- n>tie</c-><c- p>(</c-><c- n>y</c-><c- p>.</c-><c- n>re</c-><c- p>,</c-> <c- n>y</c-><c- p>.</c-><c- n>im</c-><c- p>);</c->
    <c- p>}</c->
  <c- p>};</c->
<c- p>}</c->
</pre>
   <p>Consider a transparent ordering operator for <code class="highlight"><c- n>map</c-></code>:</p>
<pre class="language-cpp highlight"><c- k>struct</c-> <c- n>strong_less</c->
  <c- k>template</c-> <c- o>&lt;</c-><c- k>typename</c-> <c- n>T</c-><c- p>,</c-> <c- k>typename</c-> <c- n>U</c-><c- o>></c->
  <c- b>bool</c-> <c- k>operator</c-><c- p>()(</c-><c- n>T</c-> <c- k>const</c-><c- o>&amp;</c-> <c- n>x</c-><c- p>,</c-> <c- n>U</c-> <c- k>const</c-><c- o>&amp;</c-> <c- n>y</c-><c- p>)</c-> <c- p>{</c->
    <c- k>using</c-> <c- n>std</c-><c- o>::</c-><c- n>strong_order</c-><c- p>;</c->  <c- c1>// use ADL</c->
    <c- k>return</c-> <c- nf>strong_order</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- n>y</c-><c- p>)</c-> <c- o>&lt;</c-> <c- mi>0</c-><c- p>;</c->
  <c- p>}</c->
  <c- k>using</c-> <c- n>is_transparent</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>true_type</c-><c- p>;</c->
<c- p>};</c->
</pre>
   <p>Also say we had a type with an implicit conversion to our <code class="highlight"><c- n>gaussian</c-></code>:</p>
<pre class="language-cpp highlight"><c- k>template</c-> <c- o>&lt;</c-><c- k>typename</c-> <c- n>T</c-><c- o>></c->
<c- k>struct</c-> <c- n>lazy</c-> <c- p>{</c->
  <c- n>std</c-><c- o>::</c-><c- n>function</c-><c- o>&lt;</c-><c- n>T</c-><c- p>()</c-><c- o>></c-> <c- n>make</c-><c- p>;</c->
  <c- k>operator</c-> <c- nf>T</c-><c- p>()</c-> <c- k>const</c-> <c- p>{</c-> <c- k>return</c-> <c- n>make</c-><c- p>();</c-> <c- p>}</c->
<c- p>};</c->
</pre>
   <p>This function now fails to compile, because the chosen <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>strong_order</c-></code> is deleted.</p>
<pre class="language-cpp highlight"><c- b>bool</c-> <c- nf>exists</c-><c- p>(</c-><c- n>lazy</c-><c- o>&lt;</c-><c- n>gaussian</c-><c- o>&lt;</c-><c- b>int</c-><c- o>>></c-> <c- k>const</c-><c- o>&amp;</c-> <c- n>x</c-><c- p>,</c->
            <c- n>std</c-><c- o>::</c-><c- n>set</c-><c- o>&lt;</c-><c- n>gaussian</c-><c- o>&lt;</c-><c- b>int</c-><c- o>></c-><c- p>,</c-> <c- n>strong_less</c-><c- o>></c-> <c- k>const</c-><c- o>&amp;</c-> <c- n>in</c-><c- p>)</c-> <c- p>{</c->
  <c- d>/* imagine this being a template in both parameters - it’s pretty normal */</c->
  <c- k>return</c-> <c- n>in</c-><c- p>.</c-><c- n>count</c-><c- p>(</c-><c- n>x</c-><c- p>);</c->
<c- p>}</c->
</pre>
   <p>The std-provided <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>strong_order</c-></code> is deleted because it cannot be synthesized from <code class="highlight"><c- n>gaussian</c-></code>'s <code class="highlight"><c- k>operator</c-><c- o>&lt;=></c-></code>. The reason it is chosen over the friend function, however, is because the standard template matches better than the friend which would require an implicit conversion.</p>
   <p>If the std-provided <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>strong_order</c-></code> did not participate in overload resolution, however, this example would work just fine.</p>
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  <h2 class="no-num no-ref heading settled" id="references"><span class="content">References</span><a class="self-link" href="#references"></a></h2>
  <h3 class="no-num no-ref heading settled" id="normative"><span class="content">Normative References</span><a class="self-link" href="#normative"></a></h3>
  <dl>
   <dt id="biblio-p0551r3">[P0551R3]
   <dd>Walter E. Brown. <a href="https://wg21.link/p0551r3">Thou Shalt Not Specialize std Function Templates!</a>. 16 March 2018. URL: <a href="https://wg21.link/p0551r3">https://wg21.link/p0551r3</a>
  </dl>
  <h3 class="no-num no-ref heading settled" id="informative"><span class="content">Informative References</span><a class="self-link" href="#informative"></a></h3>
  <dl>
   <dt id="biblio-p0100r2">[P0100R2]
   <dd>Lawrence Crowl. <a href="https://wg21.link/p0100r2">Comparison in C++</a>. 27 November 2016. URL: <a href="https://wg21.link/p0100r2">https://wg21.link/p0100r2</a>
   <dt id="biblio-p0768r1">[P0768R1]
   <dd>Walter E. Brown. <a href="https://wg21.link/p0768r1">Library Support for the Spaceship (Comparison) Operator</a>. 10 November 2017. URL: <a href="https://wg21.link/p0768r1">https://wg21.link/p0768r1</a>
  </dl>