-
Notifications
You must be signed in to change notification settings - Fork 22
/
p1189r0.html
executable file
·1222 lines (1107 loc) · 82 KB
/
p1189r0.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Adding <=> to library</title>
<style type="text/css">html {
position: relative;
max-width: 1024px;
height: 100%;
}
body {
font-family: Helvetica, arial, sans-serif;
font-size: 14px;
line-height: 1.6;
padding-top: 10px;
padding-bottom: 10px;
background-color: white;
padding: 30px;
}
body>*:first-child {
margin-top: 0 !important;
}
body>*:last-child {
margin-bottom: 0 !important;
}
a {
color: #4183C4;
}
a.absent {
color: #cc0000;
}
a.anchor {
display: block;
padding-left: 30px;
margin-left: -30px;
cursor: pointer;
position: absolute;
top: 0;
left: 0;
bottom: 0;
}
h1, h2, h3, h4, h5, h6 {
margin: 20px 0 10px;
padding: 0;
font-weight: bold;
-webkit-font-smoothing: antialiased;
cursor: text;
position: relative;
}
h1:hover a.anchor, h2:hover a.anchor, h3:hover a.anchor, h4:hover a.anchor, h5:hover a.anchor, h6:hover a.anchor {
background: url(data:image/png;base64,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) no-repeat 10px center;
text-decoration: none;
}
h1 tt, h1 code {
font-size: inherit;
}
h2 tt, h2 code {
font-size: inherit;
}
h3 tt, h3 code {
font-size: inherit;
}
h4 tt, h4 code {
font-size: inherit;
}
h5 tt, h5 code {
font-size: inherit;
}
h6 tt, h6 code {
font-size: inherit;
}
h1 {
font-size: 28px;
color: black;
}
h2 {
font-size: 24px;
border-bottom: 1px solid #cccccc;
color: black;
}
h3 {
font-size: 18px;
}
h4 {
font-size: 16px;
}
h5 {
font-size: 14px;
}
h6 {
color: #777777;
font-size: 14px;
}
p, blockquote, ol, dl, li, table, pre {
margin: 15px 0;
}
hr {
background: transparent url(data:image/png;base64,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) repeat-x 0 0;
border: 0 none;
color: #cccccc;
height: 4px;
padding: 0;
}
body>h2:first-child {
margin-top: 0;
padding-top: 0;
}
body>h1:first-child {
margin-top: 0;
padding-top: 0;
}
body>h1:first-child+h2 {
margin-top: 0;
padding-top: 0;
}
body>h3:first-child, body>h4:first-child, body>h5:first-child, body>h6:first-child {
margin-top: 0;
padding-top: 0;
}
a:first-child h1, a:first-child h2, a:first-child h3, a:first-child h4, a:first-child h5, a:first-child h6 {
margin-top: 0;
padding-top: 0;
}
h1 p, h2 p, h3 p, h4 p, h5 p, h6 p {
margin-top: 0;
}
li p.first {
display: inline-block;
}
li {
margin: 0;
}
ol {
padding-left: 30px;
margin: 5px;
counter-reset: item;
margin-left: -1px;
margin-bottom: -1px;
margin-top: -1px;
}
ol > li {
counter-increment: item;
margin-bottom: -1px;
margin-top: -1px;
}
ol ol > li {
display: block;
margin-bottom: -1px;
margin-top: -1px;
}
ol ol > li:before {
content: counters(item, ".") ". ";
margin-left: -30px;
margin-bottom: -1px;
margin-top: -1px;
}
ul :first-child, ol :first-child {
margin-top: 0;
}
ul ul {
margin-left: -15px;
}
dl {
padding: 0;
}
dl dt {
font-size: 14px;
font-weight: bold;
font-style: italic;
padding: 0;
margin: 15px 0 5px;
}
dl dt:first-child {
padding: 0;
}
dl dt> :first-child {
margin-top: 0;
}
dl dt> :last-child {
margin-bottom: 0;
}
dl dd {
margin: 0 0 15px;
padding: 0 15px;
}
dl dd> :first-child {
margin-top: 0;
}
dl dd> :last-child {
margin-bottom: 0;
}
blockquote {
border-left: 4px solid #dddddd;
padding: 0 15px;
color: #777777;
}
blockquote.ins {
border-left: 4px solid #00a000;
padding: 0 15px;
color: #00a000;
}
blockquote> :first-child {
margin-top: 0;
}
blockquote> :last-child {
margin-bottom: 0;
}
table {
padding: 0;
border-collapse: collapse;
}
table tr {
border-top: 1px solid #cccccc;
background-color: white;
margin: 0;
padding: 0;
}
table tr:nth-child(2n) {
background-color: #f8f8f8;
}
table tr th {
font-weight: bold;
border: 1px solid #cccccc;
margin: 0;
padding: 6px 13px;
}
table tr td {
border: 1px solid #cccccc;
margin: 0;
padding: 6px 13px;
}
table tr th :first-child, table tr td :first-child {
margin-top: 0;
}
table tr th :last-child, table tr td :last-child {
margin-bottom: 0;
}
td {
vertical-align: top;
}
img {
max-width: 100%;
}
span.frame {
display: block;
overflow: hidden;
}
span.frame>span {
border: 1px solid #dddddd;
display: block;
float: left;
overflow: hidden;
margin: 13px 0 0;
padding: 7px;
width: auto;
}
span.frame span img {
display: block;
float: left;
}
span.frame span span {
clear: both;
color: #333333;
display: block;
padding: 5px 0 0;
}
span.align-center {
display: block;
overflow: hidden;
clear: both;
}
span.align-center>span {
display: block;
overflow: hidden;
margin: 13px auto 0;
text-align: center;
}
span.align-center span img {
margin: 0 auto;
text-align: center;
}
span.align-right {
display: block;
overflow: hidden;
clear: both;
}
span.align-right>span {
display: block;
overflow: hidden;
margin: 13px 0 0;
text-align: right;
}
span.align-right span img {
margin: 0;
text-align: right;
}
span.float-left {
display: block;
margin-right: 13px;
overflow: hidden;
float: left;
}
span.float-left span {
margin: 13px 0 0;
}
span.float-right {
display: block;
margin-left: 13px;
overflow: hidden;
float: right;
}
span.float-right>span {
display: block;
overflow: hidden;
margin: 13px auto 0;
text-align: right;
}
code, tt {
margin: 0 2px;
padding: 0 5px;
white-space: nowrap;
border: 1px solid #eaeaea;
background-color: #f8f8f8;
border-radius: 3px;
}
pre code {
margin: 0;
padding: 0;
white-space: pre;
border: none;
background: transparent;
}
.highlight pre {
background-color: #f8f8f8;
border: 1px solid #cccccc;
font-size: 13px;
line-height: 19px;
overflow: auto;
padding: 6px 10px;
border-radius: 3px;
}
pre {
background-color: #f8f8f8;
border: 1px solid #cccccc;
font-size: 13px;
line-height: 19px;
overflow: auto;
overflow-x: hidden;
overflow-y: hidden;
padding: 6px 10px;
border-radius: 3px;
}
pre code, pre tt {
background-color: transparent;
border: none;
}
sup {
font-size: 0.83em;
vertical-align: super;
line-height: 0;
}
kbd {
display: inline-block;
padding: 3px 5px;
font-size: 11px;
line-height: 10px;
color: #555;
vertical-align: middle;
background-color: #fcfcfc;
border: solid 1px #ccc;
border-bottom-color: #bbb;
border-radius: 3px;
box-shadow: inset 0 -1px 0 #bbb
}
* {
-webkit-print-color-adjust: exact;
}
ins {
color: #00A000
}
del {
color: #A00000
}
a.self-link {
position: absolute;
top: 0;
left: calc(-1 * (3.5rem - 26px));
width: calc(3.5rem - 26px);
height: 2em;
text-align: center;
border: none;
transition: opacity .2s;
opacity: .5;
font-size: 83%;
}
a.self-link:hover {
opacity: 1;
}
a.self-link::before {
content: "§";
}
</style>
<style type="text/css">/* PrismJS 1.15.0
https://prismjs.com/download.html#themes=prism&languages=markup+css+clike+c+cpp&plugins=line-highlight */
/**
* prism.js default theme for JavaScript, CSS and HTML
* Based on dabblet (http://dabblet.com)
* @author Lea Verou
*/
code[class*="language-"],
pre[class*="language-"] {
color: black;
background: none;
text-shadow: 0 1px white;
font-family: Consolas, Monaco, 'Andale Mono', 'Ubuntu Mono', monospace;
text-align: left;
white-space: pre;
word-spacing: normal;
word-break: normal;
word-wrap: normal;
line-height: 1.5;
-moz-tab-size: 4;
-o-tab-size: 4;
tab-size: 4;
-webkit-hyphens: none;
-moz-hyphens: none;
-ms-hyphens: none;
hyphens: none;
}
pre[class*="language-"]::-moz-selection, pre[class*="language-"] ::-moz-selection,
code[class*="language-"]::-moz-selection, code[class*="language-"] ::-moz-selection {
text-shadow: none;
background: #b3d4fc;
}
pre[class*="language-"]::selection, pre[class*="language-"] ::selection,
code[class*="language-"]::selection, code[class*="language-"] ::selection {
text-shadow: none;
background: #b3d4fc;
}
@media print {
code[class*="language-"],
pre[class*="language-"] {
text-shadow: none;
}
}
/* Code blocks */
pre[class*="language-"] {
padding: 1em;
margin: .5em 0;
overflow: auto;
}
:not(pre) > code[class*="language-"],
pre[class*="language-"] {
background: #f8f8f8;
}
/* Inline code */
:not(pre) > code[class*="language-"] {
padding: .1em;
border-radius: .3em;
white-space: normal;
}
.token.comment,
.token.prolog,
.token.doctype,
.token.cdata {
color: slategray;
}
.token.punctuation {
color: #999;
}
.namespace {
opacity: .7;
}
.token.property,
.token.tag,
.token.boolean,
.token.number,
.token.constant,
.token.symbol,
.token.deleted {
color: #905;
}
.token.selector,
.token.attr-name,
.token.string,
.token.char,
.token.builtin,
.token.inserted {
color: #690;
}
.token.operator,
.token.entity,
.token.url,
.language-css .token.string,
.style .token.string {
color: #9a6e3a;
}
.token.atrule,
.token.attr-value,
.token.keyword {
color: #07a;
}
.token.function,
.token.class-name {
color: #DD4A68;
}
.token.regex,
.token.important,
.token.variable {
color: #e90;
}
.token.important,
.token.bold {
font-weight: bold;
}
.token.italic {
font-style: italic;
}
.token.entity {
cursor: help;
}
pre[data-line] {
position: relative;
padding: 1em 0 1em 3em;
}
.line-highlight {
position: absolute;
left: 0;
right: 0;
padding: inherit 0;
margin-top: 1em; /* Same as .prism’s padding-top */
background: hsla(24, 20%, 50%,.08);
background: linear-gradient(to right, hsla(24, 20%, 50%,.1) 70%, hsla(24, 20%, 50%,0));
pointer-events: none;
line-height: inherit;
white-space: pre;
}
.line-highlight:before,
.line-highlight[data-end]:after {
content: attr(data-start);
position: absolute;
top: .4em;
left: .6em;
min-width: 1em;
padding: 0 .5em;
background-color: hsla(24, 20%, 50%,.4);
color: hsl(24, 20%, 95%);
font: bold 65%/1.5 sans-serif;
text-align: center;
vertical-align: .3em;
border-radius: 999px;
text-shadow: none;
box-shadow: 0 1px white;
}
.line-highlight[data-end]:after {
content: attr(data-end);
top: auto;
bottom: .4em;
}
.line-numbers .line-highlight:before,
.line-numbers .line-highlight:after {
content: none;
}
</style>
<script type="text/javascript">/* PrismJS 1.15.0
https://prismjs.com/download.html#themes=prism&languages=markup+css+clike+c+cpp+nasm+rust&plugins=line-highlight */
var _self="undefined"!=typeof window?window:"undefined"!=typeof WorkerGlobalScope&&self instanceof WorkerGlobalScope?self:{},Prism=function(){var e=/\blang(?:uage)?-([\w-]+)\b/i,t=0,n=_self.Prism={manual:_self.Prism&&_self.Prism.manual,disableWorkerMessageHandler:_self.Prism&&_self.Prism.disableWorkerMessageHandler,util:{encode:function(e){return e instanceof r?new r(e.type,n.util.encode(e.content),e.alias):"Array"===n.util.type(e)?e.map(n.util.encode):e.replace(/&/g,"&").replace(/</g,"<").replace(/\u00a0/g," ")},type:function(e){return Object.prototype.toString.call(e).match(/\[object (\w+)\]/)[1]},objId:function(e){return e.__id||Object.defineProperty(e,"__id",{value:++t}),e.__id},clone:function(e,t){var r=n.util.type(e);switch(t=t||{},r){case"Object":if(t[n.util.objId(e)])return t[n.util.objId(e)];var a={};t[n.util.objId(e)]=a;for(var l in e)e.hasOwnProperty(l)&&(a[l]=n.util.clone(e[l],t));return a;case"Array":if(t[n.util.objId(e)])return t[n.util.objId(e)];var a=[];return t[n.util.objId(e)]=a,e.forEach(function(e,r){a[r]=n.util.clone(e,t)}),a}return e}},languages:{extend:function(e,t){var r=n.util.clone(n.languages[e]);for(var a in t)r[a]=t[a];return r},insertBefore:function(e,t,r,a){a=a||n.languages;var l=a[e];if(2==arguments.length){r=arguments[1];for(var i in r)r.hasOwnProperty(i)&&(l[i]=r[i]);return l}var o={};for(var s in l)if(l.hasOwnProperty(s)){if(s==t)for(var i in r)r.hasOwnProperty(i)&&(o[i]=r[i]);o[s]=l[s]}var u=a[e];return a[e]=o,n.languages.DFS(n.languages,function(t,n){n===u&&t!=e&&(this[t]=o)}),o},DFS:function(e,t,r,a){a=a||{};for(var l in e)e.hasOwnProperty(l)&&(t.call(e,l,e[l],r||l),"Object"!==n.util.type(e[l])||a[n.util.objId(e[l])]?"Array"!==n.util.type(e[l])||a[n.util.objId(e[l])]||(a[n.util.objId(e[l])]=!0,n.languages.DFS(e[l],t,l,a)):(a[n.util.objId(e[l])]=!0,n.languages.DFS(e[l],t,null,a)))}},plugins:{},highlightAll:function(e,t){n.highlightAllUnder(document,e,t)},highlightAllUnder:function(e,t,r){var a={callback:r,selector:'code[class*="language-"], [class*="language-"] code, code[class*="lang-"], [class*="lang-"] code'};n.hooks.run("before-highlightall",a);for(var l,i=a.elements||e.querySelectorAll(a.selector),o=0;l=i[o++];)n.highlightElement(l,t===!0,a.callback)},highlightElement:function(t,r,a){for(var l,i,o=t;o&&!e.test(o.className);)o=o.parentNode;o&&(l=(o.className.match(e)||[,""])[1].toLowerCase(),i=n.languages[l]),t.className=t.className.replace(e,"").replace(/\s+/g," ")+" language-"+l,t.parentNode&&(o=t.parentNode,/pre/i.test(o.nodeName)&&(o.className=o.className.replace(e,"").replace(/\s+/g," ")+" language-"+l));var s=t.textContent,u={element:t,language:l,grammar:i,code:s};if(n.hooks.run("before-sanity-check",u),!u.code||!u.grammar)return u.code&&(n.hooks.run("before-highlight",u),u.element.textContent=u.code,n.hooks.run("after-highlight",u)),n.hooks.run("complete",u),void 0;if(n.hooks.run("before-highlight",u),r&&_self.Worker){var g=new Worker(n.filename);g.onmessage=function(e){u.highlightedCode=e.data,n.hooks.run("before-insert",u),u.element.innerHTML=u.highlightedCode,a&&a.call(u.element),n.hooks.run("after-highlight",u),n.hooks.run("complete",u)},g.postMessage(JSON.stringify({language:u.language,code:u.code,immediateClose:!0}))}else u.highlightedCode=n.highlight(u.code,u.grammar,u.language),n.hooks.run("before-insert",u),u.element.innerHTML=u.highlightedCode,a&&a.call(t),n.hooks.run("after-highlight",u),n.hooks.run("complete",u)},highlight:function(e,t,a){var l={code:e,grammar:t,language:a};return n.hooks.run("before-tokenize",l),l.tokens=n.tokenize(l.code,l.grammar),n.hooks.run("after-tokenize",l),r.stringify(n.util.encode(l.tokens),l.language)},matchGrammar:function(e,t,r,a,l,i,o){var s=n.Token;for(var u in r)if(r.hasOwnProperty(u)&&r[u]){if(u==o)return;var g=r[u];g="Array"===n.util.type(g)?g:[g];for(var c=0;c<g.length;++c){var h=g[c],f=h.inside,d=!!h.lookbehind,m=!!h.greedy,p=0,y=h.alias;if(m&&!h.pattern.global){var v=h.pattern.toString().match(/[imuy]*$/)[0];h.pattern=RegExp(h.pattern.source,v+"g")}h=h.pattern||h;for(var b=a,k=l;b<t.length;k+=t[b].length,++b){var w=t[b];if(t.length>e.length)return;if(!(w instanceof s)){if(m&&b!=t.length-1){h.lastIndex=k;var _=h.exec(e);if(!_)break;for(var j=_.index+(d?_[1].length:0),P=_.index+_[0].length,A=b,x=k,O=t.length;O>A&&(P>x||!t[A].type&&!t[A-1].greedy);++A)x+=t[A].length,j>=x&&(++b,k=x);if(t[b]instanceof s)continue;I=A-b,w=e.slice(k,x),_.index-=k}else{h.lastIndex=0;var _=h.exec(w),I=1}if(_){d&&(p=_[1]?_[1].length:0);var j=_.index+p,_=_[0].slice(p),P=j+_.length,N=w.slice(0,j),S=w.slice(P),C=[b,I];N&&(++b,k+=N.length,C.push(N));var E=new s(u,f?n.tokenize(_,f):_,y,_,m);if(C.push(E),S&&C.push(S),Array.prototype.splice.apply(t,C),1!=I&&n.matchGrammar(e,t,r,b,k,!0,u),i)break}else if(i)break}}}}},tokenize:function(e,t){var r=[e],a=t.rest;if(a){for(var l in a)t[l]=a[l];delete t.rest}return n.matchGrammar(e,r,t,0,0,!1),r},hooks:{all:{},add:function(e,t){var r=n.hooks.all;r[e]=r[e]||[],r[e].push(t)},run:function(e,t){var r=n.hooks.all[e];if(r&&r.length)for(var a,l=0;a=r[l++];)a(t)}}},r=n.Token=function(e,t,n,r,a){this.type=e,this.content=t,this.alias=n,this.length=0|(r||"").length,this.greedy=!!a};if(r.stringify=function(e,t,a){if("string"==typeof e)return e;if("Array"===n.util.type(e))return e.map(function(n){return r.stringify(n,t,e)}).join("");var l={type:e.type,content:r.stringify(e.content,t,a),tag:"span",classes:["token",e.type],attributes:{},language:t,parent:a};if(e.alias){var i="Array"===n.util.type(e.alias)?e.alias:[e.alias];Array.prototype.push.apply(l.classes,i)}n.hooks.run("wrap",l);var o=Object.keys(l.attributes).map(function(e){return e+'="'+(l.attributes[e]||"").replace(/"/g,""")+'"'}).join(" ");return"<"+l.tag+' class="'+l.classes.join(" ")+'"'+(o?" "+o:"")+">"+l.content+"</"+l.tag+">"},!_self.document)return _self.addEventListener?(n.disableWorkerMessageHandler||_self.addEventListener("message",function(e){var t=JSON.parse(e.data),r=t.language,a=t.code,l=t.immediateClose;_self.postMessage(n.highlight(a,n.languages[r],r)),l&&_self.close()},!1),_self.Prism):_self.Prism;var a=document.currentScript||[].slice.call(document.getElementsByTagName("script")).pop();return a&&(n.filename=a.src,n.manual||a.hasAttribute("data-manual")||("loading"!==document.readyState?window.requestAnimationFrame?window.requestAnimationFrame(n.highlightAll):window.setTimeout(n.highlightAll,16):document.addEventListener("DOMContentLoaded",n.highlightAll))),_self.Prism}();"undefined"!=typeof module&&module.exports&&(module.exports=Prism),"undefined"!=typeof global&&(global.Prism=Prism);
Prism.languages.markup={comment:/<!--[\s\S]*?-->/,prolog:/<\?[\s\S]+?\?>/,doctype:/<!DOCTYPE[\s\S]+?>/i,cdata:/<!\[CDATA\[[\s\S]*?]]>/i,tag:{pattern:/<\/?(?!\d)[^\s>\/=$<%]+(?:\s+[^\s>\/=]+(?:=(?:("|')(?:\\[\s\S]|(?!\1)[^\\])*\1|[^\s'">=]+))?)*\s*\/?>/i,greedy:!0,inside:{tag:{pattern:/^<\/?[^\s>\/]+/i,inside:{punctuation:/^<\/?/,namespace:/^[^\s>\/:]+:/}},"attr-value":{pattern:/=(?:("|')(?:\\[\s\S]|(?!\1)[^\\])*\1|[^\s'">=]+)/i,inside:{punctuation:[/^=/,{pattern:/(^|[^\\])["']/,lookbehind:!0}]}},punctuation:/\/?>/,"attr-name":{pattern:/[^\s>\/]+/,inside:{namespace:/^[^\s>\/:]+:/}}}},entity:/&#?[\da-z]{1,8};/i},Prism.languages.markup.tag.inside["attr-value"].inside.entity=Prism.languages.markup.entity,Prism.hooks.add("wrap",function(a){"entity"===a.type&&(a.attributes.title=a.content.replace(/&/,"&"))}),Prism.languages.xml=Prism.languages.markup,Prism.languages.html=Prism.languages.markup,Prism.languages.mathml=Prism.languages.markup,Prism.languages.svg=Prism.languages.markup;
Prism.languages.css={comment:/\/\*[\s\S]*?\*\//,atrule:{pattern:/@[\w-]+?.*?(?:;|(?=\s*\{))/i,inside:{rule:/@[\w-]+/}},url:/url\((?:(["'])(?:\\(?:\r\n|[\s\S])|(?!\1)[^\\\r\n])*\1|.*?)\)/i,selector:/[^{}\s][^{};]*?(?=\s*\{)/,string:{pattern:/("|')(?:\\(?:\r\n|[\s\S])|(?!\1)[^\\\r\n])*\1/,greedy:!0},property:/[-_a-z\xA0-\uFFFF][-\w\xA0-\uFFFF]*(?=\s*:)/i,important:/\B!important\b/i,"function":/[-a-z0-9]+(?=\()/i,punctuation:/[(){};:]/},Prism.languages.css.atrule.inside.rest=Prism.languages.css,Prism.languages.markup&&(Prism.languages.insertBefore("markup","tag",{style:{pattern:/(<style[\s\S]*?>)[\s\S]*?(?=<\/style>)/i,lookbehind:!0,inside:Prism.languages.css,alias:"language-css",greedy:!0}}),Prism.languages.insertBefore("inside","attr-value",{"style-attr":{pattern:/\s*style=("|')(?:\\[\s\S]|(?!\1)[^\\])*\1/i,inside:{"attr-name":{pattern:/^\s*style/i,inside:Prism.languages.markup.tag.inside},punctuation:/^\s*=\s*['"]|['"]\s*$/,"attr-value":{pattern:/.+/i,inside:Prism.languages.css}},alias:"language-css"}},Prism.languages.markup.tag));
Prism.languages.clike={comment:[{pattern:/(^|[^\\])\/\*[\s\S]*?(?:\*\/|$)/,lookbehind:!0},{pattern:/(^|[^\\:])\/\/.*/,lookbehind:!0,greedy:!0}],string:{pattern:/(["'])(?:\\(?:\r\n|[\s\S])|(?!\1)[^\\\r\n])*\1/,greedy:!0},"class-name":{pattern:/((?:\b(?:class|interface|extends|implements|trait|instanceof|new)\s+)|(?:catch\s+\())[\w.\\]+/i,lookbehind:!0,inside:{punctuation:/[.\\]/}},keyword:/\b(?:if|else|while|do|for|return|in|instanceof|function|new|try|throw|catch|finally|null|break|continue)\b/,"boolean":/\b(?:true|false)\b/,"function":/\w+(?=\()/,number:/\b0x[\da-f]+\b|(?:\b\d+\.?\d*|\B\.\d+)(?:e[+-]?\d+)?/i,operator:/--?|\+\+?|!=?=?|<=?|>=?|==?=?|&&?|\|\|?|\?|\*|\/|~|\^|%/,punctuation:/[{}[\];(),.:]/};
Prism.languages.c=Prism.languages.extend("clike",{keyword:/\b(?:_Alignas|_Alignof|_Atomic|_Bool|_Complex|_Generic|_Imaginary|_Noreturn|_Static_assert|_Thread_local|asm|typeof|inline|auto|break|case|char|const|continue|default|do|double|else|enum|extern|float|for|goto|if|int|long|register|return|short|signed|sizeof|static|struct|switch|typedef|union|unsigned|void|volatile|while)\b/,operator:/>>=?|<<=?|->|([-+&|:])\1|[?:~]|[-+*\/%&|^!=<>]=?/,number:/(?:\b0x[\da-f]+|(?:\b\d+\.?\d*|\B\.\d+)(?:e[+-]?\d+)?)[ful]*/i}),Prism.languages.insertBefore("c","string",{macro:{pattern:/(^\s*)#\s*[a-z]+(?:[^\r\n\\]|\\(?:\r\n|[\s\S]))*/im,lookbehind:!0,alias:"property",inside:{string:{pattern:/(#\s*include\s*)(?:<.+?>|("|')(?:\\?.)+?\2)/,lookbehind:!0},directive:{pattern:/(#\s*)\b(?:define|defined|elif|else|endif|error|ifdef|ifndef|if|import|include|line|pragma|undef|using)\b/,lookbehind:!0,alias:"keyword"}}},constant:/\b(?:__FILE__|__LINE__|__DATE__|__TIME__|__TIMESTAMP__|__func__|EOF|NULL|SEEK_CUR|SEEK_END|SEEK_SET|stdin|stdout|stderr)\b/}),delete Prism.languages.c["class-name"],delete Prism.languages.c["boolean"];
Prism.languages.cpp=Prism.languages.extend("c",{keyword:/\b(?:alignas|alignof|asm|auto|bool|break|case|catch|char|char16_t|char32_t|class|compl|concept|const|constexpr|const_cast|continue|decltype|default|delete|do|double|dynamic_cast|else|enum|explicit|export|extern|float|for|friend|goto|if|inline|int|int8_t|int16_t|int32_t|int64_t|uint8_t|uint16_t|uint32_t|uint64_t|long|mutable|namespace|new|noexcept|nullptr|operator|private|protected|public|register|reinterpret_cast|requires|return|short|signed|sizeof|static|static_assert|static_cast|struct|switch|template|this|thread_local|throw|try|typedef|typeid|typename|union|unsigned|using|virtual|void|volatile|wchar_t|while)\b/,"boolean":/\b(?:true|false)\b/,operator:/>>=?|<<=?|->|([-+&|:])\1|[?:~]|[-+*\/%&|^!=<>]=?|\b(?:and|and_eq|bitand|bitor|not|not_eq|or|or_eq|xor|xor_eq)\b/}),Prism.languages.insertBefore("cpp","keyword",{"class-name":{pattern:/(class\s+)\w+/i,lookbehind:!0}}),Prism.languages.insertBefore("cpp","string",{"raw-string":{pattern:/R"([^()\\ ]{0,16})\([\s\S]*?\)\1"/,alias:"string",greedy:!0}});
Prism.languages.nasm={comment:/;.*$/m,string:/(["'`])(?:\\.|(?!\1)[^\\\r\n])*\1/,label:{pattern:/(^\s*)[A-Za-z._?$][\w.?$@~#]*:/m,lookbehind:!0,alias:"function"},keyword:[/\[?BITS (?:16|32|64)\]?/,{pattern:/(^\s*)section\s*[a-zA-Z.]+:?/im,lookbehind:!0},/(?:extern|global)[^;\r\n]*/i,/(?:CPU|FLOAT|DEFAULT).*$/m],register:{pattern:/\b(?:st\d|[xyz]mm\d\d?|[cdt]r\d|r\d\d?[bwd]?|[er]?[abcd]x|[abcd][hl]|[er]?(?:bp|sp|si|di)|[cdefgs]s)\b/i,alias:"variable"},number:/(?:\b|(?=\$))(?:0[hx][\da-f]*\.?[\da-f]+(?:p[+-]?\d+)?|\d[\da-f]+[hx]|\$\d[\da-f]*|0[oq][0-7]+|[0-7]+[oq]|0[by][01]+|[01]+[by]|0[dt]\d+|\d*\.?\d+(?:\.?e[+-]?\d+)?[dt]?)\b/i,operator:/[\[\]*+\-\/%<>=&|$!]/};
Prism.languages.rust={comment:[{pattern:/(^|[^\\])\/\*[\s\S]*?\*\//,lookbehind:!0},{pattern:/(^|[^\\:])\/\/.*/,lookbehind:!0}],string:[{pattern:/b?r(#*)"(?:\\.|(?!"\1)[^\\\r\n])*"\1/,greedy:!0},{pattern:/b?"(?:\\.|[^\\\r\n"])*"/,greedy:!0}],"char":{pattern:/b?'(?:\\(?:x[0-7][\da-fA-F]|u{(?:[\da-fA-F]_*){1,6}|.)|[^\\\r\n\t'])'/,alias:"string"},"lifetime-annotation":{pattern:/'[^\s>']+/,alias:"symbol"},keyword:/\b(?:abstract|alignof|as|be|box|break|const|continue|crate|do|else|enum|extern|false|final|fn|for|if|impl|in|let|loop|match|mod|move|mut|offsetof|once|override|priv|pub|pure|ref|return|sizeof|static|self|struct|super|true|trait|type|typeof|unsafe|unsized|use|virtual|where|while|yield)\b/,attribute:{pattern:/#!?\[.+?\]/,greedy:!0,alias:"attr-name"},"function":[/\w+(?=\s*\()/,/\w+!(?=\s*\(|\[)/],"macro-rules":{pattern:/\w+!/,alias:"function"},number:/\b(?:0x[\dA-Fa-f](?:_?[\dA-Fa-f])*|0o[0-7](?:_?[0-7])*|0b[01](?:_?[01])*|(\d(?:_?\d)*)?\.?\d(?:_?\d)*(?:[Ee][+-]?\d+)?)(?:_?(?:[iu](?:8|16|32|64)?|f32|f64))?\b/,"closure-params":{pattern:/\|[^|]*\|(?=\s*[{-])/,inside:{punctuation:/[|:,]/,operator:/[&*]/}},punctuation:/[{}[\];(),:]|\.+|->/,operator:/[-+*\/%!^]=?|=[=>]?|@|&[&=]?|\|[|=]?|<<?=?|>>?=?/};
!function(){function e(e,t){return Array.prototype.slice.call((t||document).querySelectorAll(e))}function t(e,t){return t=" "+t+" ",(" "+e.className+" ").replace(/[\n\t]/g," ").indexOf(t)>-1}function n(e,n,i){n="string"==typeof n?n:e.getAttribute("data-line");for(var o,l=n.replace(/\s+/g,"").split(","),a=+e.getAttribute("data-line-offset")||0,s=r()?parseInt:parseFloat,d=s(getComputedStyle(e).lineHeight),u=t(e,"line-numbers"),c=0;o=l[c++];){var p=o.split("-"),m=+p[0],f=+p[1]||m,h=e.querySelector('.line-highlight[data-range="'+o+'"]')||document.createElement("div");if(h.setAttribute("aria-hidden","true"),h.setAttribute("data-range",o),h.className=(i||"")+" line-highlight",u&&Prism.plugins.lineNumbers){var g=Prism.plugins.lineNumbers.getLine(e,m),y=Prism.plugins.lineNumbers.getLine(e,f);g&&(h.style.top=g.offsetTop+"px"),y&&(h.style.height=y.offsetTop-g.offsetTop+y.offsetHeight+"px")}else h.setAttribute("data-start",m),f>m&&h.setAttribute("data-end",f),h.style.top=(m-a-1)*d+"px",h.textContent=new Array(f-m+2).join(" \n");u?e.appendChild(h):(e.querySelector("code")||e).appendChild(h)}}function i(){var t=location.hash.slice(1);e(".temporary.line-highlight").forEach(function(e){e.parentNode.removeChild(e)});var i=(t.match(/\.([\d,-]+)$/)||[,""])[1];if(i&&!document.getElementById(t)){var r=t.slice(0,t.lastIndexOf(".")),o=document.getElementById(r);o&&(o.hasAttribute("data-line")||o.setAttribute("data-line",""),n(o,i,"temporary "),document.querySelector(".temporary.line-highlight").scrollIntoView())}}if("undefined"!=typeof self&&self.Prism&&self.document&&document.querySelector){var r=function(){var e;return function(){if("undefined"==typeof e){var t=document.createElement("div");t.style.fontSize="13px",t.style.lineHeight="1.5",t.style.padding=0,t.style.border=0,t.innerHTML=" <br /> ",document.body.appendChild(t),e=38===t.offsetHeight,document.body.removeChild(t)}return e}}(),o=0;Prism.hooks.add("before-sanity-check",function(t){var n=t.element.parentNode,i=n&&n.getAttribute("data-line");if(n&&i&&/pre/i.test(n.nodeName)){var r=0;e(".line-highlight",n).forEach(function(e){r+=e.textContent.length,e.parentNode.removeChild(e)}),r&&/^( \n)+$/.test(t.code.slice(-r))&&(t.code=t.code.slice(0,-r))}}),Prism.hooks.add("complete",function l(e){var r=e.element.parentNode,a=r&&r.getAttribute("data-line");if(r&&a&&/pre/i.test(r.nodeName)){clearTimeout(o);var s=Prism.plugins.lineNumbers,d=e.plugins&&e.plugins.lineNumbers;t(r,"line-numbers")&&s&&!d?Prism.hooks.add("line-numbers",l):(n(r,a),o=setTimeout(i,1))}}),window.addEventListener("hashchange",i),window.addEventListener("resize",function(){var e=document.querySelectorAll("pre[data-line]");Array.prototype.forEach.call(e,function(e){n(e)})})}}();
</script>
</head>
<body>
<address align=right>
Document Number: P1189R0 <br />
Date: 2019-03-10 <br />
Audience: LEWG <br />
Reply-To: Barry Revzin, barry dot revzin at gmail dot com <br />
</address>
<hr /><h1 align=center><p>Adding <=> to library</p></h1>
<h2>Contents</h2>
<div class="toc">
<ol>
<li><a href="#revision-history">Revision History</a></li>
<li><a href="#introduction">Introduction</a><ol>
<li><a href="#motivation">Motivation</a></li>
<li><a href="#non-member-hidden-friend-or-member">Non-member, hidden friend, or member?</a></li>
</ol>
</li>
<li><a href="#dealing-with-operator-and-operator">Dealing with operator==() and operator!=()</a><ol>
<li><a href="#most-operators-are-obsolete">Most operator!=()s are obsolete</a></li>
<li><a href="#reversed-operators-too">Reversed operator==s too</a></li>
<li><a href="#defaulting-operator">Defaulting operator==()</a></li>
</ol>
</li>
<li><a href="#adding-to-stdbasic_string">Adding <=> to std::basic_string</a><ol>
<li><a href="#add-a-type-alias">Add a type alias</a></li>
<li><a href="#add-a-static-member-function">Add a static member function</a></li>
<li><a href="#proposal">Proposal</a></li>
</ol>
</li>
<li><a href="#adding-to-stdvector">Adding <=> to std::vector</a><ol>
<li><a href="#proposal_1">Proposal</a></li>
</ol>
</li>
<li><a href="#adding-to-stdoptional">Adding <=> to std::optional</a><ol>
<li><a href="#proposal_2">Proposal</a></li>
</ol>
</li>
<li><a href="#adding-to-stdunique_ptr">Adding <=> to std::unique_ptr</a><ol>
<li><a href="#proposal_3">Proposal</a></li>
</ol>
</li>
<li><a href="#acknowledgements">Acknowledgements</a></li>
<li><a href="#references">References</a></li>
</ol>
</div>
<h2 id="revision-history">1. Revision History<a class="self-link" href="#revision-history"></a></h2>
<p>A draft of this paper was presented to LEWG in Kona, and it was approved with the following details:</p>
<ul>
<li>comparison operators should be hidden friends</li>
<li><code class="language-cpp">basic_string</code>'s comparison category should be driven by a <a href="#add-a-type-alias">type alias</a>, not a <a href="#add-a-static-member-function">static member function</a>.</li>
<li><code class="language-cpp">tuple</code>'s <code class="language-cpp">operator==</code> <em>cannot</em> be defaulted, because subobject order is currently unspecified. Defaulted <code class="language-cpp">==</code> compares in subobject order, but <code class="language-cpp">tuple</code> is specified to compare in template parameter order - so this would either force an ABI break on those implementations that don't order the subobjects from to back or break the actual equality semantics. <code class="language-cpp">pair</code> and <code class="language-cpp">array</code> have no such problems with defaulting <code class="language-cpp">operator==</code>.</li>
<li><code class="language-cpp">unique_ptr</code> cannot have its relational operators removed due to the <code class="language-cpp">less</code> specification - it ends up fitting in the <code class="language-cpp">optional</code> category. <code class="language-cpp">shared_ptr</code> is fine as proposed here.</li>
</ul>
<h2 id="introduction">2. Introduction<a class="self-link" href="#introduction"></a></h2>
<p>This paper builds on all of David Stone's effort in <a href="https://wg21.link/p0790r2" title="Effect of operator<=> on the C++ Standard Library">P0790R2</a> and exists just to fill in the missing holes. I believe it's very important to ship C++20 with <code class="language-cpp"><=></code> for all of these types - and this is my attempt to help get there. Of the types which P0790R2 does not provide a solution, they can be divided into four neat categories, each with a canonical representative. This paper will go through these four in turn:</p>
<ul>
<li><code class="language-cpp">std::basic_string</code></li>
<li><code class="language-cpp">std::vector</code></li>
<li><code class="language-cpp">std::optional</code></li>
<li><code class="language-cpp">std::unique_ptr</code></li>
</ul>
<p>Critically, no change proposed in this paper will change the semantics of any comparison. The answer that a particular comparison gave in C++17 will be the same as what that code will give in C++20. But the path taken to get to that answer may change.</p>
<h3 id="motivation">2.1. Motivation<a class="self-link" href="#motivation"></a></h3>
<p>But why should we bother? What's the motivation for doing all this work?</p>
<p>Major benefits of <code class="language-cpp">operator<=></code> are that it is easier to implement than the status quo comparisons implemented via <code class="language-cpp">operator<</code>, that it can provide better performance than the status quo, and that it can be more correct than the status quo. But of course, none of that is true if we don't actually use it in our own library. There is the inevitable user frustration than this doesn't work:</p>
<pre class="codehilite"><code class="language-cpp">struct Person {
std::string last;
std::string first;
auto operator<=>(Person const&) const = default;
};</code></pre>
<p>Here is a type ready-made for a three-way comparison (indeed, it already has one), but defaulting doesn't work? So much for ease of use! </p>
<p>A user might then, if <a href="https://wg21.link/p1186r1" title="When do you actually use <=>?">P1186R1</a> is adopted, try to do this:</p>
<pre class="codehilite"><code class="language-cpp">struct Person {
std::string last;
std::string first;
strong_ordering operator<=>(Person const&) const = default;
};</code></pre>
<p>This will, potentially, walk each string twice - when we could have performed this comparison by walking each string only once (at most). So much for performance!</p>
<p>We don't even need to go into user types to see this issue. Consider:</p>
<pre class="codehilite"><code class="language-cpp">void func(std::vector<std::string> const& names, std::vector<std::string> const& other_names)
{
if (names < other_names) {
// ...
}
}</code></pre>
<p>Even though <code class="language-cpp">std::string</code> already has a three-way comparison function (by way of <code class="language-cpp">compare()</code>), the invocation of <code class="language-cpp"><</code> above won't use it - it'll instead invoke <code class="language-cpp"><</code> up to twice on each string. That's twice as much work as needs to be done to answer this question. If we adopt <code class="language-cpp"><=></code> for all of thse library types as we should, then the above comparison would go through <code class="language-cpp">std::vector</code>'s <code class="language-cpp"><=></code> which would go through <code class="language-cpp">std::string</code>'s <code class="language-cpp"><=></code> which ensures that we only compare each <code class="language-cpp">string</code> at most one time. We're not just talking about making comparisons for user-defined types easier to write and more performant, we're also talking about making already existing comparisons of standard library types more performant.</p>
<p>And these problems stack, as the standard library types are inevitably used together. A <code class="language-cpp">std::optional<std::vector<std::string>></code> should provide an <code class="language-cpp">operator<=></code> with a comparison category of <code class="language-cpp">strong_ordering</code>, which should be efficient, and it should really be there for C++20.</p>
<h3 id="non-member-hidden-friend-or-member">2.2. Non-member, hidden friend, or member?<a class="self-link" href="#non-member-hidden-friend-or-member"></a></h3>
<p>This paper proposes adding a whole lot of <code class="language-cpp">operator<=></code>s into the standard library. There are three ways we could do this:</p>
<ul>
<li>non-member functions</li>
<li>hidden friends</li>
<li>member functions</li>
</ul>
<p>Most (if not all?) of these operators are non-member functions today. This has the downside of polluting scope and increasing then number of candidates for each of the comparisons that overload resolution has to look through. It has the additional downside of allowing surprising conversions in certain cases simply because of ADL (e.g. <a href="https://wg21.link/lwg3065" title="LWG 2989 missed that all path's other operators should be hidden friends as well">LWG3065</a>).</p>
<p>This leads many people to rightly suggest to make operators hidden friends instead, which maintains the advantage of non-member functions by allowing conversions in both arguments but reduces the number of candidates for unrelated types and doesn't have surprise conversions in both arguments. However, it does introduce some new possibilities for conversions that did not exist with the non-member function approach. Consider (on <a href="https://godbolt.org/z/s7wxPE">godbolt</a>):</p>
<pre class="codehilite"><code class="language-cpp">template <typename T>
struct C {
#ifdef FRIEND
friend bool operator<(C, C);
#endif
};
#ifndef FRIEND
template <typename T>
bool operator<(C<T>, C<T>);
#endif
void ex(C<int> a, C<int> b) {
a < b; // #1
std::ref(a) < b; // #2
a < std::ref(b); // #3
std::ref(a) < std::ref(b); // #4
}</code></pre>
<p>With the non-member operator template (i.e. without <code class="language-cpp">FRIEND</code> defined), only <code class="language-cpp">#1</code> is a valid expression. But with the hidden friend, <em>all four</em> are valid. These kind of additional conversions may be surprising - and it is certainly possible that they could lead to ambiguities. </p>
<p>But with <code class="language-cpp">operator<=></code> we can actually take a third approach: member functions. Because <code class="language-cpp"><=></code> considers reversed candidates as well, a member function is sufficient:</p>
<pre class="codehilite"><code class="language-cpp">struct X {
X(int);
bool operator==(X const&) const;
strong_ordering operator<=>(X const&) const;
};
// all of these are ok
X{42} < X{57};
X{42} < 57;
42 < X{57};
42 == X{57};
X{42} != 57;</code></pre>
<p>Using a member function does not pollute the global scope and does not add to the work that overload resolution has to do needlessly. It does have unique behavior where conversions are concerned though. If we return to the earlier example with <code class="language-cpp">C<T></code>:</p>
<pre class="codehilite"><code class="language-cpp">template <typename T>
struct C {
std::strong_ordering operator<=>(C);
};
void ex(C<int> a, C<int> b) {
a < b; // #1
std::ref(a) < b; // #2
a < std::ref(b); // #3
std::ref(a) < std::ref(b); // #4
}</code></pre>
<p>With this implementation, <code class="language-cpp">#1</code> compiles straightforwardly. But <code class="language-cpp">#2</code> and <code class="language-cpp">#3</code> are <em>also</em> valid. <code class="language-cpp">#3</code> evaluates as <code class="language-cpp">a.operator<=>(std::ref(b)) < 0</code> while <code class="language-cpp">#2</code> evaluates as <code class="language-cpp">0 < b.operator<=>(std::ref(a))</code>. Only <code class="language-cpp">#4</code> is ill-formed in this case. Recall that in the non-member case, only <code class="language-cpp">#1</code> was valid and in the hidden friend case, all four were. </p>
<p>There, however, does exist a situation where moving from non-member comparison operator templates to a member spaceship function can change behavior. It's not completely free. This example courtesy of Tim Song:</p>
<table style="width:100%">
<tr>
<th style="width:50%">
<p>Non-member <code class="language-cpp"><</code> template</p>
</th>
<th style="width:50%">
<p>Member <code class="language-cpp"><=></code></p>
</th>
</tr>
<tr>
<td>
<pre style="background:transparent;border:0px" data-line="4,5,6,7,10,11,12,13"><code class="language-cpp">template <typename CharT, typename Traits, typename Alloc>
struct basic_string { /* ... */ };
// #1
template <typename CharT, typename Traits, typename Alloc>
bool operator<(basic_string<CharT, Traits, Alloc> const&,
basic_string<CharT, Traits, Alloc> const&);
struct B { };
// #2
template<typename CharT, typename Traits, typename Alloc>
bool operator<(basic_string<CharT, Traits, Alloc> const&,
B const&);
struct C {
operator string() const;
operator B() const;
};
""s < C();</code></pre>
</td>
<td>
<pre style="background:transparent;border:0px" data-line="3,4,10,11,12,13"><code class="language-cpp">template <typename CharT, typename Traits, typename Alloc>
struct basic_string {
// #3
auto operator<=>(basic_string const&) const;
};
struct B { };
// #2
template<typename CharT, typename Traits, typename Alloc>
bool operator<(basic_string<CharT, Traits, Alloc> const&,
B const&);
struct C {
operator string() const;
operator B() const;
};
""s < C();</code></pre>
</td>
</tr>
</table>
<p>Today, this goes through the global <code class="language-cpp">operator<</code> template (<code class="language-cpp">#2</code>). The <code class="language-cpp">operator<</code> taking two <code class="language-cpp">basic_string</code>s (<code class="language-cpp">#1</code>) is not a candidate because <code class="language-cpp">C</code> is not a <code class="language-cpp">basic_string</code>. However, if we change <code class="language-cpp">basic_string</code> to instead have a member spaceship operator... then this spaceship (<code class="language-cpp">#3</code>) suddenly not only becomes a candidate (since <code class="language-cpp">C</code> is convertible to <code class="language-cpp">string</code>) but actually becomes the best viable candidate because the tie-breaker preferring a non-template to a template is higher than the tie-breaker preferring a non-rewritten candidate to a rewritten one. The same thing would happen in the case of a hidden (non-template) friend.</p>
<p>This seems like a fairly contrived scenario. Though like all fairly contrived scenarios, it assuredly exists in some C++ code base somewhere. The most conservative approach would be to stay put and keep the proposed <code class="language-cpp">operator<=></code>s as non-member operator templates. But there are very clear benefits of making them member functions, so I think member <code class="language-cpp">operator<=></code> is still the way to go.</p>
<h2 id="dealing-with-operator-and-operator">3. Dealing with <code class="language-cpp">operator==()</code> and <code class="language-cpp">operator!=()</code><a class="self-link" href="#dealing-with-operator-and-operator"></a></h2>
<p>Before getting to how to handle <code class="language-cpp">operator<=>()</code> for all of these situations, let's first talk about the equality operators. There are several changes to how these operators behave as a result of <a href="https://wg21.link/p1185r1" title="<=> != ==">P1185R1</a> (R0 was approved by EWG in San Diego, R1 will likely be moved in Kona - there is still an open design question, but it is about a part of that proposal which is irrelevant to this paper). </p>
<h3 id="most-operators-are-obsolete">3.1. Most <code class="language-cpp">operator!=()</code>s are obsolete<a class="self-link" href="#most-operators-are-obsolete"></a></h3>
<p>The first part of P1185R1 changes the candidate set for inequality operators to include equality operators. In other words, for types in which <code class="language-cpp">a != b</code> is defined to mean <code class="language-cpp">!(a == b)</code>, we no longer need to define <code class="language-cpp">operator!=</code>. The language will simply do the right thing for us.
Just about every <code class="language-cpp">operator!=()</code> in the library does this. Indeed, we have blanket wording in [operators] which lets us avoid having to write the boilerplate every time. The only exceptions to this are:</p>
<ul>
<li>the types in the group represented by <code class="language-cpp">std::optional</code>, which will be discussed <a href="#adding-to-stdoptional">later</a></li>
<li><code class="language-cpp">std::valarray</code>, for which <code class="language-cpp">==</code> doesn't really mean equality anyway</li>
</ul>
<p>The other ~250 declarations of <code class="language-cpp">operator!=()</code> can just be removed entirely. The semantics for all callers will remain the same. </p>
<h3 id="reversed-operators-too">3.2. Reversed <code class="language-cpp">operator==</code>s too<a class="self-link" href="#reversed-operators-too"></a></h3>
<p>Some library types provided mixed-type equality operators. For example, the class template <code class="language-cpp">std::basic_string</code> provides the following equality and inequality operators today:</p>
<pre class="codehilite"><code class="language-cpp">// #1
template <typename CharT, typename Traits, typename Alloc>
bool operator==(basic_string<CharT, Traits, Alloc> const&, basic_string<CharT, Traits, Alloc> const&);
// #2
template <typename CharT, typename Traits, typename Alloc>
bool operator!=(basic_string<CharT, Traits, Alloc> const&, basic_string<CharT, Traits, Alloc> const&);
// #3
template <typename CharT, typename Traits, typename Alloc>
bool operator==(basic_string<CharT, Traits, Alloc> const&, CharT const*);
// #4
template <typename CharT, typename Traits, typename Alloc>
bool operator==(CharT const*, basic_string<CharT, Traits, Alloc> const&);
// #5
template <typename CharT, typename Traits, typename Alloc>
bool operator!=(basic_string<CharT, Traits, Alloc> const&, CharT const*);
// #6
template <typename CharT, typename Traits, typename Alloc>
bool operator!=(CharT const*, basic_string<CharT, Traits, Alloc> const&);</code></pre>
<p>The previous section suggests removing the inequality operators (<code class="language-cpp">#2</code>, <code class="language-cpp">#5</code>, and <code class="language-cpp">#6</code>) and just relying on their equality counterparts to be used as rewritten candidates. But P1185R1, in addition to allowing inequality to be able to be rewritten as equality, also allows equality to be reversed. That is, a source expression <code class="language-cpp">a == b</code> can find <code class="language-cpp">b == a</code> as a candidate.</p>
<p>This means that <code class="language-cpp">#4</code> in the above isn't necessary either, since we can rely on <code class="language-cpp">"hello" == "hello"s</code> to invoke <code class="language-cpp">#3</code>. We really only need <code class="language-cpp">#1</code> and <code class="language-cpp">#3</code> in the above declarations. In other words, in order to be able to provide full equality and inequality between a <code class="language-cpp">basic_string</code> and its corresponding <code class="language-cpp">CharT const*</code>, we just need to write a single operator (<code class="language-cpp">#3</code>) instead of today's four. </p>
<p>This paper proposes removing all of these duplicated <code class="language-cpp">operator==</code> declarations as well. The full list is:</p>
<ul>
<li><code class="language-cpp">error_code</code> / <code class="language-cpp">error_condition</code></li>
<li><code class="language-cpp">optional<T></code> / <code class="language-cpp">nullopt</code></li>
<li><code class="language-cpp">unique_ptr<T,D></code> / <code class="language-cpp">nullptr_t</code></li>
<li><code class="language-cpp">shared_ptr<T></code> / <code class="language-cpp">nullptr_t</code></li>
<li><code class="language-cpp">function<R(Args...)></code> / <code class="language-cpp">nullptr_t</code></li>
<li><code class="language-cpp">move_iterator</code> / <code class="language-cpp">move_sentinel<S></code></li>
<li><code class="language-cpp">counted_iterator</code> / <code class="language-cpp">default_sentinel_t</code></li>
<li><code class="language-cpp">unreachable_sentinel_t</code> / <code class="language-cpp">I</code></li>
<li><code class="language-cpp">istream_iterator</code> / <code class="language-cpp">default_sentinel_t</code></li>
<li><code class="language-cpp">istreambuf_iterator</code> / <code class="language-cpp">default_sentinel_t</code></li>
<li><code class="language-cpp">filter_view::iterator</code> / <code class="language-cpp">filter_view::sentinel</code></li>
<li><code class="language-cpp">transform_view::iterator</code> / <code class="language-cpp">transform_view::sentinel</code></li>
<li><code class="language-cpp">iota_view::iterator</code> / <code class="language-cpp">iota_view::sentinel</code></li>
<li><code class="language-cpp">take_view::iterator</code> / <code class="language-cpp">take_view::sentinel</code></li>
<li><code class="language-cpp">join_view::iterator</code> / <code class="language-cpp">join_view::sentinel</code></li>
<li><code class="language-cpp">split_view::outer_iterator</code> / <code class="language-cpp">default_sentinel_t</code></li>
<li><code class="language-cpp">split_view::inner_iterator</code> / <code class="language-cpp">default_sentinel_t</code></li>
<li><code class="language-cpp">complex<T></code> / <code class="language-cpp">T</code></li>
<li><code class="language-cpp">valarray<T></code> / <code class="language-cpp">valarray<T>::value_type</code></li>
<li><code class="language-cpp">leap</code> / <code class="language-cpp">sys_time<D></code></li>
<li><code class="language-cpp">sub_match</code> / <code class="language-cpp">basic_string</code></li>
<li><code class="language-cpp">sub_match</code> / <code class="language-cpp">value_type const*</code></li>
<li><code class="language-cpp">sub_match</code> / <code class="language-cpp">value_type const&</code></li>
</ul>
<p>Note that even though this paper is not removing <code class="language-cpp">optional<T></code>'s or <code class="language-cpp">valarray<T></code>s <code class="language-cpp">operator!=</code>s as a whole, it can still remove the reversed <code class="language-cpp">operator==</code>s and the corresponding reversed <code class="language-cpp">operator!=</code>s.</p>
<h3 id="defaulting-operator">3.3. Defaulting <code class="language-cpp">operator==()</code><a class="self-link" href="#defaulting-operator"></a></h3>
<p>Another important aspect of P1185 is that it redefines "strong structural equality" to be based on a defaulted <code class="language-cpp">operator==()</code> rather than a defaulted <code class="language-cpp">operator<=>()</code>, which is the requirement for a type being usable as a non-type template parameter - a new C++20 language feature introduced by way of <a href="https://wg21.link/p0732r2" title="Class Types in Non-Type Template Parameters">P0732R2</a></p>
<p>There are many library types which currently define their <code class="language-cpp">operator==()</code> as member-wise equality. Respecifying all of these <code class="language-cpp">operator==()</code>s as being <code class="language-cpp">default</code>ed instead will not change any semantics or ABI, but will simply directly allow these types to be usable as non-type template parameters. So this paper proposes to do that for:</p>
<ul>
<li><code class="language-cpp">pair</code></li>
<li><code class="language-cpp">tuple</code></li>
<li><code class="language-cpp">array</code></li>
</ul>
<p>... and any library type that meets this member-wise equality condition that is missing from the above list.</p>
<h2 id="adding-to-stdbasic_string">4. Adding <code class="language-cpp"><=></code> to <code class="language-cpp">std::basic_string</code><a class="self-link" href="#adding-to-stdbasic_string"></a></h2>
<p>This group is composed of templates which implement their comparisons via <code class="language-cpp">Traits::compare()</code>: <code class="language-cpp">std::basic_string</code>, <code class="language-cpp">std::basic_string_view</code>, and <code class="language-cpp">std::sub_match</code>. Because <code class="language-cpp">Traits::compare()</code> is already a three-way comparison, this seems like a trivial case. Just write this and ship it right?</p>
<pre class="codehilite"><code class="language-cpp">template <typename CharT, typename Traits, typename Alloc>
auto operator<=>(basic_string<CharT, Traits, Alloc> const& lhs,
basic_string<CharT, Traits, Alloc> const& rhs)
{
return lhs.compare(rhs) <=> 0;
}</code></pre>
<p>Unfortunately, since <code class="language-cpp">lhs.compare(rhs)</code> is an <code class="language-cpp">int</code>, this would unconditionally yield a <code class="language-cpp">std::strong_ordering</code>. The result of this will be correct as far as all the binary comparisons though, but comparing arbitrary <code class="language-cpp">basic_string</code>'s is not necessarily a <code class="language-cpp">strong_ordering</code>. Indeed, the <a href="http://www.gotw.ca/gotw/029.htm" title="GotW #29: Strings">canonical example</a> of providing a custom <code class="language-cpp">Traits</code> for <code class="language-cpp">basic_string</code> is to implement a case-insensitive comparison - which should be a <code class="language-cpp">weak_ordering</code>. So we cannot simply pick <code class="language-cpp">strong_ordering</code>. On the other hand, <code class="language-cpp">std::string</code>'s <code class="language-cpp">operator<=></code> should <em>certainly</em> yield <code class="language-cpp">strong_ordering</code> so we cannot simply pick <code class="language-cpp">weak_ordering</code> either. We'll have to do something slightly more complex.</p>
<p>We could simply bless all the standard-mandated specializations of <code class="language-cpp">std::char_traits</code> and say that those are <code class="language-cpp">strong_ordering</code>, and any other trait is a <code class="language-cpp">weak_ordering</code>. But that seems overly restrictive on users who might want to provide a custom traits which provides an ordering which is either strong or partial. </p>
<p>Instead, I propose one of two choices: add a member type alias to the traits expressing the comparison category, or add a new static member function performing a three-way comparison and yielding a comparison category.</p>
<h3 id="add-a-type-alias">4.1. Add a type alias<a class="self-link" href="#add-a-type-alias"></a></h3>
<p>This is the least intrusive option. Simply add a member alias to the standard-mandated specializations of <code class="language-cpp">char_traits</code>:</p>
<pre class="codehilite"><code class="language-cpp">template <> struct char_traits<char> {
using comparison_category = strong_ordering;
};</code></pre>
<p>And replace the four relational operators with an <code class="language-cpp">operator<=></code> whose return type is based on that category:</p>
<pre class="codehilite"><code class="language-cpp">// exposition-only
template <typename T>
struct traits_category {
using type = weak_ordering;
};
template <typename T>
requires requires { typename T::comparison_category; }
struct traits_category {
using type = typename T::comparison_category;
};
template <typename T>
using traits_category_t = typename traits_category<T>::type;
// actual implementation
template <typename CharT, typename Traits, typename Alloc>
class basic_string {
// ...
traits_category_t<Traits> operator<=>(basic_string const& rhs) const
{
return compare(rhs) <=> 0;
}
// ...
};</code></pre>
<p>This works because <code class="language-cpp">int</code>'s comparison category is <code class="language-cpp">strong_ordering</code>, which is implicitly convertible to any other comparison category.</p>
<p>This is a fairly small change. The downside is that any future custom traits will still have to write <code class="language-cpp">compare()</code> functions that return an <code class="language-cpp">int</code> instead of one of the standard comparison categories. To that end, we could:</p>
<h3 id="add-a-static-member-function">4.2. Add a static member function<a class="self-link" href="#add-a-static-member-function"></a></h3>
<p>Add a static member function returning a comparison category:</p>
<pre class="codehilite"><code class="language-cpp">template <> struct char_traits<char> {
static constexpr strong_ordering compare_3way(const char* s1, const char* s2, size_t count)
{
return compare(s1, s2, count) <=> 0;
}
};</code></pre>
<p>And then use that if it is present:</p>
<pre class="codehilite"><code class="language-cpp">template <typename CharT, typename Traits, typename Alloc>
class basic_string {
// ...
auto operator<=>(basic_string const& rhs) const
{
auto impl = [](CharT const* lhs, CharT const* rhs, size_t sz) {
if constexpr (requires { Traits::compare_3way(lhs, rhs, sz); }) {
return Traits::compare_3way(lhs, rhs, sz);
} else {
return static_cast<weak_ordering>(
Traits::compare(lhs, rhs, sz) <=> 0);
}
};
auto cmp = impl(data(), rhs.data(), min(size(), rhs.size()));
return cmp != 0 ? cmp : size() <=> rhs.size();
}
// ...
};</code></pre>
<p>This is somewhat more involved than the previous alternative, but not by much. Since we'd still need to provide a version of <code class="language-cpp">compare</code> that returns an <code class="language-cpp">int</code>, this may not actually be worth it - since while it's very easy to convert an <code class="language-cpp">int</code> to a comparison category (simply <code class="language-cpp"><=></code> it against <code class="language-cpp">0</code>), there's no actual easy way of going the other way without branching. But it <em>does</em> offer a way to make a <code class="language-cpp">basic_string</code> with a partial ordering, if that is desired. </p>
<h3 id="proposal">4.3. Proposal<a class="self-link" href="#proposal"></a></h3>
<p>I weakly favor the member type alias approach. Remove <code class="language-cpp">operator<</code>, <code class="language-cpp">operator></code>, <code class="language-cpp">operator<=</code>, and <code class="language-cpp">operator>=</code> for <code class="language-cpp">basic_string</code>, <code class="language-cpp">basic_string_view</code>, and <code class="language-cpp">sub_match</code> and replace them with <code class="language-cpp">operator<=></code>s which invoke <code class="language-cpp">x.compare(y) <=> 0</code> cast to the appropriate comparison category (with argument adjustments as appropriate). This <code class="language-cpp">operator<=></code> should be a member function. </p>
<h2 id="adding-to-stdvector">5. Adding <code class="language-cpp"><=></code> to <code class="language-cpp">std::vector</code><a class="self-link" href="#adding-to-stdvector"></a></h2>
<p>There are many types in the standard library that are, fundamentally, sequences of objects. Those sequences can be heterogeneous or homogeneous, fixed-length or variable-length, but they're all still sequences. The way we order these sequences, the way we write the comparison operators for them today, is lexicographically. Basically, this general algorithm:</p>
<pre class="codehilite"><code class="language-cpp">bool operator<(Sequence const& x, Sequence const& y) {
size_t const N = min(x.size(), y.size());
if (x[0] < y[0]) return true;
if (y[0] < x[0]) return false;
if (x[1] < y[1]) return true;
if (y[1] < x[1]) return false;
// ...
if (x[N-1] < y[N-1]) return true;
if (y[N-1] < x[N-1]) return false;
return x.size() < y.size();
}</code></pre>
<p>For variable-sized sequences, this means we perform up to <code class="language-cpp">2N</code> invocations of an underlying <code class="language-cpp">operator<</code>. For fixed-size sequences, we can optimize out the last two comparisons here so we only do up to <code class="language-cpp">2N-1</code> such invocations. </p>