-
Notifications
You must be signed in to change notification settings - Fork 1
/
latex.html
5709 lines (5417 loc) · 280 KB
/
latex.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
<!DOCTYPE html>
<html lang="en">
<head>
<link rel="stylesheet" href="https://pygae.github.io/lean-ga-docs/style.css">
<link rel="stylesheet" href="https://pygae.github.io/lean-ga-docs/pygments.css">
<link rel="stylesheet" href="https://pygae.github.io/lean-ga-docs/pygments-dark.css">
<link rel="shortcut icon" href="https://pygae.github.io/lean-ga-docs/favicon.ico">
<title>latex tests - mathlib3 docs</title>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="description" content="The Lean 3 mathematical library, mathlib, is a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant." />
<link rel="canonical" href="" />
<meta property="og:title" content="latex tests - mathlib3 docs">
<meta property="og:site_name" content="mathlib for Lean 3 - API documentation">
<meta property="og:description" content="The Lean 3 mathematical library, mathlib, is a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant.">
<meta property="og:image" content="https://pygae.github.io/lean-ga-docs/meta-og.png">
<meta name="twitter:card" content="summary">
<script src="https://pygae.github.io/lean-ga-docs/color_scheme.js"></script>
</head>
<body>
<input id="nav_toggle" type="checkbox">
<header>
<h1><label for="nav_toggle"></label><a href="https://leanprover-community.github.io/lean3">mathlib3</a>
<span>documentation</span></h1>
<p class="header_filename break_within">latex tests</p>
<form action="https://google.com/search" method="get" id="search_form">
<input type="hidden" name="sitesearch" value="https://leanprover-community.github.io/mathlib_docs">
<input type="text" name="q" autocomplete="off">
<button>Google site search</button>
</form>
</header>
<nav class="internal_nav">
<h3><a href="#top">latex tests</a></h3>
</nav>
<main>
<a id="top"></a>
<h1 id="latex-examples" class="markdown-heading">LaTeX examples <a class="hover-link" href="#latex-examples">#</a></h1>
<p>This test page collects several examples of LaTeX in markdown.</p>
<p>Compare: <a href="https://math.meta.stackexchange.com/revisions/9386/164">https://math.meta.stackexchange.com/revisions/9386/164</a></p>
<h2 id="examples-from-doc-gen-issues" class="markdown-heading">Examples from doc-gen issues <a class="hover-link" href="#examples-from-doc-gen-issues">#</a></h2>
<p>From <a href="https://github.com/leanprover-community/doc-gen/issues/10">doc-gen#10</a>, what if I put $f[*a,*b](cd)$ in tex? $g_{x_0}(y)$ ?</p>
<p>From <a href="https://github.com/leanprover-community/doc-gen/issues/62">doc-gen#62</a>:</p>
<ul>
<li>might denote $R[X_i : i \in \sigma]$.</li>
<li>Example: $x \in \sigma$ and separately <code>y</code></li>
<li>Example: $x \in \sigma$ and separately <code>s</code></li>
<li>Example: $x \in \sigma$ and separately <code>sigm</code></li>
<li>Example: $x \in \sigma$ and separately <code>simg</code></li>
<li>Example: separately <code>s</code> and $x \in \sigma$</li>
</ul>
<p>Example from <a href="https://github.com/leanprover-community/mathlib/blob/c35672bbe581370c345d0862c078fbbbe1258fb3/src/dynamics/circle/rotation_number/translation_number.lean#L641">dynamics.circle.rotation_number.translation_number</a>:</p>
<p>For any <code>x : ℝ</code> the sequence $\frac{f^n(x)-x}{n}$ tends to the translation number of <code>f</code>.
In particular, this limit does not depend on <code>x</code>.</p>
<h2 id="a-hrefhttpsgithubcomleanprover-communitymathlibpull3776mathlib3776a" class="markdown-heading"><a href="https://github.com/leanprover-community/mathlib/pull/3776">mathlib#3776</a> <a class="hover-link" href="#a-hrefhttpsgithubcomleanprover-communitymathlibpull3776mathlib3776a">#</a></h2>
<p><a href="https://github.com/leanprover-community/mathlib/blob/d61bd4ae29222280e1b6dec421c840fb83c30438/src/algebra/classical_lie_algebras.lean#L45">Before PR</a> (has a stray <code>cc</code>), line spacing too large:</p>
<p>$$
J = \left[\begin{align}{cc}
0_l & 1_l\\\\
1_l & 0_l
\end{align}\right]
$$</p>
<p><a href="https://github.com/leanprover-community/mathlib/blob/0166d0baa856ca4c3d516025105cfe8f912f48dc/src/algebra/classical_lie_algebras.lean#L46">After PR</a>, line spacing too large:</p>
<p>$$
J = \left[\begin{array}{cc}
0_l & 1_l\\\\
1_l & 0_l
\end{array}\right]
$$</p>
<p>Actually fixed:</p>
<p>$$
J = \left[\begin{array}{cc}
0_l & 1_l\\
1_l & 0_l
\end{array}\right]
$$</p>
<h2 id="a-hrefhttpsgithubcomleanprover-communitymathlibpull6175mathlib6175a" class="markdown-heading"><a href="https://github.com/leanprover-community/mathlib/pull/6175">mathlib#6175</a> <a class="hover-link" href="#a-hrefhttpsgithubcomleanprover-communitymathlibpull6175mathlib6175a">#</a></h2>
<p><a href="https://github.com/leanprover-community/mathlib/blob/c70feebd43e143d81a695f7d7e5b21e5892286e8/src/analysis/analytic/basic.lean#L626">Before PR</a> (renders correctly):</p>
<p>If a function is analytic in a disk <code>D(x, R)</code>, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a <code>k</code>-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where <code>pₙ</code> is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets <code>s</code> of <code>fin n</code> of cardinal <code>k</code>, and attribute <code>z</code> to the indices in <code>s</code> and
<code>y</code> to the indices outside of <code>s</code>.
In this paragraph, we implement this. The new power series is called <code>p.change_origin y</code>. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.</p>
<p><a href="https://github.com/leanprover-community/mathlib/blob/676836509e16e6b6d3baf1354594658257f687bd/src/analysis/analytic/basic.lean#L626">After PR</a> (with two backslashes before <code>sum</code> as a workaround; should be broken):</p>
<p>If a function is analytic in a disk <code>D(x, R)</code>, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \\sum_{n} p_n (y + z)^n = \\sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \\sum_{k} \Bigl(\\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a <code>k</code>-th coefficient equal to
$\\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where <code>pₙ</code> is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets <code>s</code> of <code>fin n</code> of cardinal <code>k</code>, and attribute <code>z</code> to the indices in <code>s</code> and
<code>y</code> to the indices outside of <code>s</code>.</p>
<h2 id="line-spacing-tests" class="markdown-heading">Line spacing tests <a class="hover-link" href="#line-spacing-tests">#</a></h2>
<p>(math.stackexchange.com results given above each example)</p>
<hr />
<p>Renders OK ✅:
$\alpha
\alpha$</p>
<hr />
<p>Should not render ❌:
$\alpha</p>
<p>\alpha$</p>
<hr />
<p>Should not render ❌:
$$\alpha</p>
<p>\alpha$$</p>
<hr />
<p>Should not render ❌:
$$</p>
<p>\alpha
\alpha</p>
<p>$$</p>
<hr />
<p>Renders OK ✅:
$$ \alpha\alpha
$$</p>
<hr />
<p>Renders OK ✅:
$$
\alpha\alpha
[hi](345)
b
$$</p>
<hr />
<p>Should not render ❌:
$$ \alpha\alpha <a href="https://github.com">hi</a> \beta</p>
<p>$$</p>
<hr />
<p>Renders OK ✅:
\begin{align}
\begin{matrix}
a & b & c
\end{matrix}
\end{align}</p>
<hr />
<p>Only the inner matrix environment should render:
\begin{align}</p>
<p><a href="https://github.com">hi</a>
\begin{matrix}
a & b & c
\end{matrix}
\end{align}</p>
<hr />
<p>Should not render ❌:
\begin{align}</p>
<p>\begin{matrix}</p>
<p>a & b & c
\end{matrix}
\end{align}</p>
<hr />
<p>Renders OK ✅:</p>
<p>hi there \begin{align} 3 \\ 3 \end{align}</p>
<hr />
<p>Should not render ❌:
\begin{align}
\end{blah}</p>
<hr />
<p>Should not render ❌:
\begin{align}
[link](https://github.com) { {\alpha}
\end{align}</p>
<h2 id="nested-environments" class="markdown-heading">Nested environments <a class="hover-link" href="#nested-environments">#</a></h2>
<p>From <a href="http://web.archive.org/web/20120617014306/http://www.st.fmph.uniba.sk/~kiselak1/pdfka/tex/latexMath_align.pdf">http://web.archive.org/web/20120617014306/http://www.st.fmph.uniba.sk/~kiselak1/pdfka/tex/latexMath_align.pdf</a></p>
<p>\begin{align}T(n) & \leq 2(c\lfloor n/2 \rfloor \lg( \lfloor n/2 \rfloor )) + n \\T(n) & \leq 2(cn/2) \lg(n/2) + n \\T(n) & = cn (\lg n - 1) + n \\T(n) & \leq cn \lg n\end{align}</p>
<p>\begin{align}
T(n) & \leq 2(c\lfloor n/2 \rfloor \lg( \lfloor n/2 \rfloor )) + n \\
T(n) & \leq 2(cn/2) \lg(n/2) + n \\
T(n) & = cn (\lg n - 1) + n \\
T(n) & \leq cn \lg n
\end{align}</p>
<p>\begin{gather}
\begin{aligned}T(n) & \leq 2(c\lfloor n/2 \rfloor \lg( \lfloor n/2 \rfloor )) + n \\T(n) & \leq 2(cn/2) \lg(n/2) + n \\T(n) & = cn (\lg n - 1) + n \\T(n) & \leq cn \lg n\end{aligned}\end{gather}</p>
<p>\begin{gather}
\begin{aligned}
T(n) & \leq 2(c\lfloor n/2 \rfloor \lg( \lfloor n/2 \rfloor )) + n \\
T(n) & \leq 2(cn/2) \lg(n/2) + n \\
T(n) & = cn (\lg n - 1) + n \\
T(n) & \leq cn \lg n
\end{aligned}
\end{gather}</p>
<p>\begin{align}\begin{aligned}T(n) & \leq 2(c\lfloor n/2 \rfloor \lg( \lfloor n/2 \rfloor )) + n \\T(n) & \leq 2(cn/2) \lg(n/2) + n \\T(n) & = cn (\lg n - 1) + n \\T(n) & \leq cn \lg n\end{aligned}\end{align}</p>
<h2 id="mathjax-basic-tutorial-and-quick-reference" class="markdown-heading">MathJax basic tutorial and quick reference <a class="hover-link" href="#mathjax-basic-tutorial-and-quick-reference">#</a></h2>
<p>From <a href="https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference">https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference</a></p>
<p>(Deutsch: <a href="https://www.mathelounge.de/509545/mathjax-latex-basic-tutorial-und-referenz-deutsch">MathJax: LaTeX Basic Tutorial und Referenz</a>)</p>
<ol>
<li>
<p>To see how any formula was written in any question or answer, including this one, right-click on the expression it and choose "Show Math As > TeX Commands". (When you do this, the '$' will not display. Make sure you add these. See the next point. There are also <a href="https://math.meta.stackexchange.com/q/659">other possibilities</a> how to view the code for the formula or the whole post.)</p>
</li>
<li>
<p><strong>For inline formulas, enclose the formula in <code>$...$</code>. For displayed formulas, use <code>$$...$$</code>.</strong>
These render differently. For example,
type
<code>$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$</code>
to show $\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$ (which is inline mode) or type
<code>$$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$</code>
to show
$$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$
(which is display mode).</p>
</li>
<li>
<p>For <strong>Greek letters</strong>, use <code>\alpha</code>, <code>\beta</code>, …, <code>\omega</code>: $\alpha, \beta, … \omega$. For uppercase, use <code>\Gamma</code>, <code>\Delta</code>, …, <code>\Omega</code>: $\Gamma, \Delta, …, \Omega$. Some Greek letters have variant forms:
<code>\epsilon \varepsilon</code> $\epsilon$, $\varepsilon$, <code>\phi \varphi</code> $\phi$, $\varphi$, and others.</p>
</li>
<li>
<p>For <strong>superscripts and subscripts</strong>, use <code>^</code> and <code>_</code>. For example, <code>x_i^2</code>: $x_i^2$, <code>\log_2 x</code>: $\log_2 x$.</p>
</li>
<li>
<p><strong>Groups</strong>. Superscripts, subscripts, and other operations apply only to the next “group”. A “group” is either a single symbol, or any formula surrounded by curly braces <code>{</code>…<code>}</code>. If you do <code>10^10</code>, you will get a surprise: $10^10$. But <code>10^{10}</code> gives what you probably wanted: $10^{10}$. Use curly braces to delimit a formula to which a superscript or subscript applies: <code>x^5^6</code> is an error; <code>{x^y}^z</code> is ${x^y}^z$, and <code>x^{y^z}</code> is $x^{y^z}$. Observe the difference between <code>x_i^2</code> $x_i^2$ and <code>x_{i^2}</code> $x_{i^2}$.</p>
</li>
<li>
<p><strong>Parentheses</strong> Ordinary symbols <code>()[]</code> make parentheses and brackets $(2+3)[4+4]$. Use <code>\{</code> and <code>\}</code> for curly braces $\{\}$.</p>
<p>These do <em>not</em> scale with the formula in between, so if you write <code>(\frac{\sqrt x}{y^3})</code> the parentheses will be too small: $(\frac{\sqrt x}{y^3})$. Using <code>\left(</code>…<code>\right)</code> will make the sizes adjust automatically to the formula they enclose: <code>\left(\frac{\sqrt x}{y^3}\right)</code> is $\left(\frac{\sqrt x}{y^3}\right)$.</p>
<p><code>\left</code> and<code>\right</code> apply to all the following sorts of parentheses: <code>(</code> and <code>)</code> $(x)$, <code>[</code> and <code>]</code> $[x]$, <code>\{</code> and <code>\}</code> $\{ x \}$, <code>|</code> $|x|$, <code>\vert</code> $\vert x \vert$, <code>\Vert</code> $\Vert x \Vert$, <code>\langle</code> and <code>\rangle</code> $\langle x \rangle$, <code>\lceil</code> and <code>\rceil</code> $\lceil x \rceil$, and <code>\lfloor</code> and <code>\rfloor</code> $\lfloor x \rfloor$. <code>\middle</code> can be used to add additional dividers. There are also invisible parentheses, denoted by <code>.</code>: <code>\left.\frac12\right\rbrace</code> is $\left.\frac12\right\rbrace$.</p>
<p>If manual size adjustments are required:
<code>\Biggl(\biggl(\Bigl(\bigl((x)\bigr)\Bigr)\biggr)\Biggr)</code> gives
$\Biggl(\biggl(\Bigl(\bigl((x)\bigr)\Bigr)\biggr)\Biggr)$.</p>
</li>
<li>
<p><strong>Sums and integrals</strong> <code>\sum</code> and <code>\int</code>; the subscript is the lower limit and the superscript is the upper limit, so for example <code>\sum_1^n</code> $\sum_1^n$. Don't forget <code>{</code>…<code>}</code> if the limits are more than a single symbol. For example, <code>\sum_{i=0}^\infty i^2</code> is $\sum_{i=0}^\infty i^2$. Similarly, <code>\prod</code> $\prod$, <code>\int</code> $\int$, <code>\bigcup</code> $\bigcup$, <code>\bigcap</code> $\bigcap$, <code>\iint</code> $\iint$, <code>\iiint</code> $\iiint$, <code>\idotsint</code> $\idotsint$.</p>
</li>
<li>
<p><strong>Fractions</strong> There are <a href="https://math.meta.stackexchange.com/q/12978/3111">three ways to make these</a>. <code>\frac ab</code> applies to the next two groups, and produces $\frac ab$; for more complicated numerators and denominators use <code>{</code>…<code>}</code>: <code>\frac{a+1}{b+1}</code> is $\frac{a+1}{b+1}$. If the numerator and denominator are complicated, you may prefer <code>\over</code>, which splits up the group that it is in: <code>{a+1\over b+1}</code> is ${a+1\over b+1}$.
Using <code>\cfrac{a}{b}</code> command is useful for continued fractions $\cfrac{a}{b}$, more details for which <a href="https://math.meta.stackexchange.com/a/5058/3111">are given in this sub-article</a>.</p>
</li>
<li>
<p><strong>Fonts</strong></p>
</li>
</ol>
<ul>
<li>Use <code>\mathbb</code> or <code>\Bbb</code> for "blackboard bold": $\mathbb{CHNQRZ}$.</li>
<li>Use <code>\mathbf</code> for boldface: $\mathbf{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ $\mathbf{abcdefghijklmnopqrstuvwxyz}$.
<ul>
<li>For expression based characters, use <code>\boldsymbol</code> instead: $\boldsymbol{\alpha}$</li>
</ul>
</li>
<li>Use <code>\mathit</code> for italics: $\mathit{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ $\mathit{abcdefghijklmnopqrstuvwxyz}$.</li>
<li>Use <code>\pmb</code> for boldfaced italics: $\pmb{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ $\pmb{abcdefghijklmnopqrstuvwxyz}$.</li>
<li>Use <code>\mathtt</code> for "typewriter" font: $\mathtt{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ $\mathtt{abcdefghijklmnopqrstuvwxyz}$.</li>
<li>Use <code>\mathrm</code> for roman font: $\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ $\mathrm{abcdefghijklmnopqrstuvwxyz}$.</li>
<li>Use <code>\mathsf</code> for sans-serif font: $\mathsf{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ $\mathsf{abcdefghijklmnopqrstuvwxyz}$.</li>
<li>Use <code>\mathcal</code> for "calligraphic" letters: $\mathcal{ ABCDEFGHIJKLMNOPQRSTUVWXYZ}$</li>
<li>Use <code>\mathscr</code> for script letters: $\mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$</li>
<li>Use <code>\mathfrak</code> for "Fraktur" (old German style) letters: $\mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ} \mathfrak{abcdefghijklmnopqrstuvwxyz}$.</li>
</ul>
<ol start="10">
<li>
<p><strong>Radical signs / roots</strong> Use <code>sqrt</code>, which adjusts to the size of its argument: <code>\sqrt{x^3}</code> $\sqrt{x^3}$; <code>\sqrt[3]{\frac xy}</code> $\sqrt[3]{\frac xy}$. For complicated expressions, consider using <code>{...}^{1/2}</code> instead.</p>
</li>
<li>
<p>Some <strong>special functions</strong> such as "lim", "sin", "max", "ln", and so on are normally set in roman font instead of italic font. Use <code>\lim</code>, <code>\sin</code>, etc. to make these: <code>\sin x</code> $\sin x$, not <code>sin x</code> $sin x$. Use subscripts to attach a notation to <code>\lim</code>: <code>\lim_{x\to 0}</code> $$\lim_{x\to 0}$$ Nonstandard function names can be set with <code>\operatorname{foo}(x)</code> $\operatorname{foo}(x)$.</p>
</li>
<li>
<p>There are a very large number of <strong>special symbols and notations</strong>, too many to list here; see <a href="http://pic.plover.com/MISC/symbols.pdf">this shorter listing</a>, or <a href="https://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf">this exhaustive listing</a>. Some of the most common include:</p>
</li>
</ol>
<ul>
<li><code>\lt \gt \le \leq \leqq \leqslant \ge \geq \geqq \geqslant \neq</code> $\lt$, $\gt$, $\le$, $\leq$, $\leqq$, $\leqslant$, $\ge$, $\geq$, $\geqq$, $\geqslant$, $\neq$. You can use <code>\not</code> to put a slash through almost anything: <code>\not\lt</code> $\not\lt$ but it often looks bad.</li>
<li><code>\times \div \pm \mp</code> $\times$, $\div$, $\pm$, $\mp$. <code>\cdot</code> is a centered dot: $x\cdot y$</li>
<li><code>\cup \cap \setminus \subset \subseteq \subsetneq \supset \in \notin \emptyset \varnothing</code> $\cup$, $\cap$, $\setminus$, $\subset$, $\subseteq$, $\subsetneq$, $\supset$, $\in$, $\notin$, $\emptyset$, $\varnothing$</li>
<li><code>{n+1 \choose 2k}</code> or <code>\binom{n+1}{2k}</code> ${n+1 \choose 2k}$</li>
<li><code>\to \rightarrow \leftarrow \Rightarrow \Leftarrow \mapsto</code> $\to$, $\rightarrow$, $\leftarrow$, $\Rightarrow$, $\Leftarrow$, $\mapsto$</li>
<li><code>\land \lor \lnot \forall \exists \top \bot \vdash \vDash</code> $\land$, $\lor$, $\lnot$, $\forall$, $\exists$, $\top$, $\bot$, $\vdash$, $\vDash$</li>
<li><code>\star \ast \oplus \circ \bullet</code> $\star$, $\ast$, $\oplus$, $\circ$, $\bullet$</li>
<li><code>\approx \sim \simeq \cong \equiv \prec \lhd \therefore</code> $\approx$, $\sim $, $\simeq$, $\cong$, $\equiv$, $\prec$, $\lhd$, $\therefore$</li>
<li><code>\infty \aleph_0</code> $\infty\, \aleph_0$ <code>\nabla \partial</code> $\nabla$, $\partial$ <code>\Im \Re</code> $\Im$, $\Re$</li>
<li>For modular equivalence, use <code>\pmod</code> like this: <code>a\equiv b\pmod n</code> $a\equiv b\pmod n$.</li>
<li>For the binary mod operator, use <code>\bmod</code> like this: <code>a\bmod 17</code> $a\bmod 17$.</li>
<li>Avoid using <code>\mod</code>, as it produces extra space: compare the above with <code>a\mod 17</code> $a\mod 17$.</li>
<li><code>\ldots</code> is the dots in $a_1, a_2, \ldots ,a_n$ <code>\cdots</code> is the dots in $a_1+a_2+\cdots+a_n$</li>
<li>Script lowercase l is <code>\ell</code> $\ell$.</li>
</ul>
<p><a href="http://detexify.kirelabs.org/classify.html">Detexify</a> lets you draw a symbol on a web page and then lists the $\TeX$ symbols that seem to resemble it. These are not guaranteed to work in MathJax but are a good place to start. To check that a command is supported, note that MathJax.org maintains a <a href="http://docs.mathjax.org/en/latest/tex.html#supported-latex-commands">list of currently supported $\LaTeX$ commands</a>, and one can also check Dr. Carol JVF Burns's page of <a href="http://www.onemathematicalcat.org/MathJaxDocumentation/TeXSyntax.htm">$\TeX$ Commands Available in MathJax</a>.</p>
<ol start="13">
<li><strong>Spaces</strong> MathJax usually decides for itself how to space formulas, using a complex set of rules. Putting extra literal spaces into formulas will not change the amount of space MathJax puts in: <code>a␣b</code> and <code>a␣␣␣␣b</code> are both $a b$. To add more space, use <code>\,</code> for a thin space $a\,b$; <code>\;</code> for a wider space $a\;b$. <code>\quad</code> and <code>\qquad</code> are large spaces: $a\quad b$, $a\qquad b$.</li>
</ol>
<p>To set plain text, use <code>\text{…}</code>: $\{x\in s\mid x\text{ is extra large}\}$. You can nest <code>$…$</code> inside of <code>\text{…}</code>, for example to access spaces.</p>
<ol start="14">
<li>
<p><strong>Accents and diacritical marks</strong> Use <code>\hat</code> for a single symbol $\hat x$, <code>\widehat</code> for a larger formula $\widehat{xy}$. If you make it too wide, it will look silly. Similarly, there are <code>\bar</code> $\bar x$ and <code>\overline</code> $\overline{xyz}$, and <code>\vec</code> $\vec x$ and <code>\overrightarrow</code> $\overrightarrow{xy}$ and <code>\overleftrightarrow</code> $\overleftrightarrow{xy}$. For dots, as in $\frac d{dx}x\dot x = \dot x^2 + x\ddot x$, use <code>\dot</code> and <code>\ddot</code>.</p>
</li>
<li>
<p>Special characters used for MathJax interpreting can be escaped using the <code>\</code> character: <code>\$</code> $\$$, <code>\{</code> $\{$, <code>\_</code> $\_$, etc. If you want <code>\</code> itself, you should use <code>\backslash</code> (symbol) or <code>\setminus</code> (<a href="https://tex.stackexchange.com/a/511332">binary operation</a>) for $\backslash$, because <code>\\</code> is for a new line.</p>
</li>
</ol>
<p>(Tutorial ends here.)</p>
<hr />
<p>It is important that this note be reasonably short and not suffer from too much bloat. To include more topics, please create short addenda and post them as answers instead of inserting them into this post.</p>
<h2 id="contents" class="markdown-heading">Contents <a class="hover-link" href="#contents">#</a></h2>
<p>Alphabetical list of links to To MathJax Topics, by title:</p>
<ul>
<li><a href="https://math.meta.stackexchange.com/a/15078/161490">Absolute values and norms</a> • <a href="https://math.meta.stackexchange.com/a/13081/161490">Additional symbolic decorations</a> • <a href="https://math.meta.stackexchange.com/a/5024/676335">Aligning Equations</a></li>
<li><a href="https://math.meta.stackexchange.com/a/27910/161490">Alternative Ways of Writing in LaTeX</a> • <a href="https://math.meta.stackexchange.com/a/21258/161490">Annotations of reasoning</a> • <a href="https://math.meta.stackexchange.com/a/15077/161490">Arbitrary operators</a></li>
<li><a href="https://math.meta.stackexchange.com/a/5044/161490">Arrays</a> • <a href="https://math.meta.stackexchange.com/a/11423/161490">Big braces</a> • <a href="https://math.meta.stackexchange.com/a/10116/161490">Colors</a></li>
<li><a href="https://math.meta.stackexchange.com/a/16888/161490">Commutative diagrams</a> • <a href="https://math.meta.stackexchange.com/a/5058/161490">Continued fractions</a> • <a href="https://math.meta.stackexchange.com/a/13183/161490">Crossing things out</a></li>
<li><a href="https://math.meta.stackexchange.com/a/5025/161490">Definitions by cases (piecewise functions)</a> • <a href="https://math.meta.stackexchange.com/a/19678/161490">Degree symbol</a> • <a href="https://math.meta.stackexchange.com/a/25054/161490">Display style</a></li>
<li><a href="https://math.meta.stackexchange.com/a/27793/161490">Equation numbering</a> • <a href="https://math.meta.stackexchange.com/a/5057/161490">Fussy spacing issues</a> • <a href="https://math.meta.stackexchange.com/a/22395/161490">Highlighting expressions</a></li>
<li><a href="https://math.meta.stackexchange.com/a/13310/161490">Left and right arrows</a> • <a href="https://math.meta.stackexchange.com/a/12850/161490">Limits</a> • <a href="https://math.meta.stackexchange.com/a/27756/161490">Linear programming</a></li>
<li><a href="https://math.meta.stackexchange.com/a/21096/161490">Long division</a> • <a href="https://math.meta.stackexchange.com/a/27756/676335">Math Programming</a> • <a href="https://math.meta.stackexchange.com/a/5023/676335">Matrices</a></li>
<li><a href="https://math.meta.stackexchange.com/a/31141/161490">Markov Chains</a> • <a href="https://math.meta.stackexchange.com/a/25251/161490">Mixing code and MathJax formatting on lines</a> • <a href="https://math.meta.stackexchange.com/a/11638/161490">The \newcommand function</a></li>
<li><a href="https://math.meta.stackexchange.com/a/11491/676335">Numbering Equations</a> • <a href="https://math.meta.stackexchange.com/a/32210/736802">Overlaying Symbols</a> • <a href="https://math.meta.stackexchange.com/a/22516/161490">Packs of cards</a></li>
<li><a href="https://math.meta.stackexchange.com/a/11284/161490">Symbols</a>
• <a href="https://math.meta.stackexchange.com/a/6267/161490">System of equations</a> • <a href="https://math.meta.stackexchange.com/a/29979/161490">Tables</a></li>
<li><a href="https://math.meta.stackexchange.com/a/11491/161490">Tags and references</a> • <a href="https://math.meta.stackexchange.com/a/30661/161490">Tensor indices</a> • <a href="https://math.meta.stackexchange.com/a/27212/161490">Units</a></li>
<li><a href="https://math.meta.stackexchange.com/a/25048/161490">Vertical spacing</a></li>
</ul>
<h2 id="aligned-equations" class="markdown-heading">Aligned equations <a class="hover-link" href="#aligned-equations">#</a></h2>
<p><a href="https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference/5024#5024">https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference/5024#5024</a></p>
<p>Often people want a series of equations where the equals signs are aligned. To get this, use <code>\begin{align}…\end{align}</code>. Each line should end with <code>\\</code>, and should contain an ampersand at the point to align at, typically immediately before the equals sign.</p>
<p>For example,</p>
<p>\begin{align}
\sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\
& = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\
& = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\
& = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\
& \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right)
\end{align}</p>
<p>is produced by</p>
<div class="codehilite"><pre><span></span><code><span class="bp">\</span><span class="kd">begin</span><span class="o">{</span><span class="n">align</span><span class="o">}</span>
<span class="bp">\</span><span class="n">sqrt</span><span class="o">{</span><span class="mi">37</span><span class="o">}</span> <span class="bp">&</span> <span class="bp">=</span> <span class="bp">\</span><span class="n">sqrt</span><span class="o">{</span><span class="bp">\</span><span class="n">frac</span><span class="o">{</span><span class="mi">73</span><span class="bp">^</span><span class="mi">2</span><span class="bp">-</span><span class="mi">1</span><span class="o">}{</span><span class="mi">12</span><span class="bp">^</span><span class="mi">2</span><span class="o">}}</span> <span class="bp">\\</span>
<span class="bp">&</span> <span class="bp">=</span> <span class="bp">\</span><span class="n">sqrt</span><span class="o">{</span><span class="bp">\</span><span class="n">frac</span><span class="o">{</span><span class="mi">73</span><span class="bp">^</span><span class="mi">2</span><span class="o">}{</span><span class="mi">12</span><span class="bp">^</span><span class="mi">2</span><span class="o">}</span><span class="bp">\</span><span class="n">cdot</span><span class="bp">\</span><span class="n">frac</span><span class="o">{</span><span class="mi">73</span><span class="bp">^</span><span class="mi">2</span><span class="bp">-</span><span class="mi">1</span><span class="o">}{</span><span class="mi">73</span><span class="bp">^</span><span class="mi">2</span><span class="o">}}</span> <span class="bp">\\</span>
<span class="bp">&</span> <span class="bp">=</span> <span class="bp">\</span><span class="n">sqrt</span><span class="o">{</span><span class="bp">\</span><span class="n">frac</span><span class="o">{</span><span class="mi">73</span><span class="bp">^</span><span class="mi">2</span><span class="o">}{</span><span class="mi">12</span><span class="bp">^</span><span class="mi">2</span><span class="o">}}</span><span class="bp">\</span><span class="n">sqrt</span><span class="o">{</span><span class="bp">\</span><span class="n">frac</span><span class="o">{</span><span class="mi">73</span><span class="bp">^</span><span class="mi">2</span><span class="bp">-</span><span class="mi">1</span><span class="o">}{</span><span class="mi">73</span><span class="bp">^</span><span class="mi">2</span><span class="o">}}</span> <span class="bp">\\</span>
<span class="bp">&</span> <span class="bp">=</span> <span class="bp">\</span><span class="n">frac</span><span class="o">{</span><span class="mi">73</span><span class="o">}{</span><span class="mi">12</span><span class="o">}</span><span class="bp">\</span><span class="n">sqrt</span><span class="o">{</span><span class="mi">1</span> <span class="bp">-</span> <span class="bp">\</span><span class="n">frac</span><span class="o">{</span><span class="mi">1</span><span class="o">}{</span><span class="mi">73</span><span class="bp">^</span><span class="mi">2</span><span class="o">}}</span> <span class="bp">\\</span>
<span class="bp">&</span> <span class="bp">\</span><span class="n">approx</span> <span class="bp">\</span><span class="n">frac</span><span class="o">{</span><span class="mi">73</span><span class="o">}{</span><span class="mi">12</span><span class="o">}</span><span class="bp">\</span><span class="n">left</span><span class="o">(</span><span class="mi">1</span> <span class="bp">-</span> <span class="bp">\</span><span class="n">frac</span><span class="o">{</span><span class="mi">1</span><span class="o">}{</span><span class="mi">2</span><span class="bp">\</span><span class="n">cdot73</span><span class="bp">^</span><span class="mi">2</span><span class="o">}</span><span class="bp">\</span><span class="n">right</span><span class="o">)</span>
<span class="bp">\</span><span class="kd">end</span><span class="o">{</span><span class="n">align</span><span class="o">}</span>
</code></pre></div>
<p>The usual <code>$$</code> marks that delimit the display may be omitted here.</p>
<h2 id="definitions-by-cases-piecewise-functions" class="markdown-heading">Definitions by cases (piecewise functions) <a class="hover-link" href="#definitions-by-cases-piecewise-functions">#</a></h2>
<p><a href="https://math.meta.stackexchange.com/a/5025">https://math.meta.stackexchange.com/a/5025</a></p>
<p>Use <code>\begin{cases}…\end{cases}</code>. End each case with a <code>\\</code>, and use <code>&</code> before parts that should be aligned.</p>
<p>For example, you get this:</p>
<p>$$f(n) =
\begin{cases}
n/2, & \text{if $n$ is even} \\
3n+1, & \text{if $n$ is odd}
\end{cases}$$</p>
<p>by writing this:</p>
<div class="codehilite"><pre><span></span><code> <span class="n">f</span><span class="o">(</span><span class="n">n</span><span class="o">)</span> <span class="bp">=</span>
<span class="bp">\</span><span class="kd">begin</span><span class="o">{</span><span class="n">cases</span><span class="o">}</span>
<span class="n">n</span><span class="bp">/</span><span class="mi">2</span><span class="o">,</span> <span class="bp">&</span> <span class="bp">\</span><span class="n">text</span><span class="o">{</span><span class="k">if</span> <span class="bp">$</span><span class="n">n</span><span class="bp">$</span> <span class="n">is</span> <span class="n">even</span><span class="o">}</span> <span class="bp">\\</span>
<span class="mi">3</span><span class="n">n</span><span class="bp">+</span><span class="mi">1</span><span class="o">,</span> <span class="bp">&</span> <span class="bp">\</span><span class="n">text</span><span class="o">{</span><span class="k">if</span> <span class="bp">$</span><span class="n">n</span><span class="bp">$</span> <span class="n">is</span> <span class="n">odd</span><span class="o">}</span>
<span class="bp">\</span><span class="kd">end</span><span class="o">{</span><span class="n">cases</span><span class="o">}</span>
</code></pre></div>
<p>The brace can be moved to the right:
$$
\left.
\begin{array}{l}
\text{if $n$ is even:}&n/2\\
\text{if $n$ is odd:}&3n+1
\end{array}
\right\}
=f(n)
$$
by writing this:</p>
<div class="codehilite"><pre><span></span><code><span class="bp">\</span><span class="n">left.</span>
<span class="bp">\</span><span class="kd">begin</span><span class="o">{</span><span class="n">array</span><span class="o">}{</span><span class="n">l</span><span class="o">}</span>
<span class="bp">\</span><span class="n">text</span><span class="o">{</span><span class="k">if</span> <span class="bp">$</span><span class="n">n</span><span class="bp">$</span> <span class="n">is</span> <span class="n">even</span><span class="o">:}</span><span class="bp">&</span><span class="n">n</span><span class="bp">/</span><span class="mi">2</span><span class="bp">\\</span>
<span class="bp">\</span><span class="n">text</span><span class="o">{</span><span class="k">if</span> <span class="bp">$</span><span class="n">n</span><span class="bp">$</span> <span class="n">is</span> <span class="n">odd</span><span class="o">:}</span><span class="bp">&</span><span class="mi">3</span><span class="n">n</span><span class="bp">+</span><span class="mi">1</span>
<span class="bp">\</span><span class="kd">end</span><span class="o">{</span><span class="n">array</span><span class="o">}</span>
<span class="bp">\</span><span class="n">right</span><span class="bp">\</span><span class="o">}</span>
<span class="bp">=</span><span class="n">f</span><span class="o">(</span><span class="n">n</span><span class="o">)</span>
</code></pre></div>
<p>To get a larger vertical space between cases we can use <code>\\[2ex]</code> instead of <code>\\</code>. For example, you get this:</p>
<p>$$f(n) =
\begin{cases}
\frac{n}{2}, & \text{if $n$ is even} \\[2ex]
3n+1, & \text{if $n$ is odd}
\end{cases}$$</p>
<p>by writing this:</p>
<div class="codehilite"><pre><span></span><code><span class="n">f</span><span class="o">(</span><span class="n">n</span><span class="o">)</span> <span class="bp">=</span>
<span class="bp">\</span><span class="kd">begin</span><span class="o">{</span><span class="n">cases</span><span class="o">}</span>
<span class="bp">\</span><span class="n">frac</span><span class="o">{</span><span class="n">n</span><span class="o">}{</span><span class="mi">2</span><span class="o">},</span> <span class="bp">&</span> <span class="bp">\</span><span class="n">text</span><span class="o">{</span><span class="k">if</span> <span class="bp">$</span><span class="n">n</span><span class="bp">$</span> <span class="n">is</span> <span class="n">even</span><span class="o">}</span> <span class="bp">\\</span><span class="o">[</span><span class="mi">2</span><span class="n">ex</span><span class="o">]</span>
<span class="mi">3</span><span class="n">n</span><span class="bp">+</span><span class="mi">1</span><span class="o">,</span> <span class="bp">&</span> <span class="bp">\</span><span class="n">text</span><span class="o">{</span><span class="k">if</span> <span class="bp">$</span><span class="n">n</span><span class="bp">$</span> <span class="n">is</span> <span class="n">odd</span><span class="o">}</span>
<span class="bp">\</span><span class="kd">end</span><span class="o">{</span><span class="n">cases</span><span class="o">}</span>
</code></pre></div>
<p>(An ‘ex’ is a length equal to the height of the letter <code>x</code>; <code>2ex</code> here means the space should be two exes high.)</p>
<h2 id="matrices" class="markdown-heading">Matrices <a class="hover-link" href="#matrices">#</a></h2>
<p><a href="https://math.meta.stackexchange.com/a/5023/">https://math.meta.stackexchange.com/a/5023/</a></p>
<ol>
<li>
<p>Use <code>$$\begin{matrix}…\end{matrix}$$</code> In between the <code>\begin</code> and <code>\end</code>, put the matrix elements. End each matrix row with <code>\\</code>, and separate matrix elements with <code>&</code>. For example,</p>
<div class="codehilite"><pre><span></span><code> <span class="bp">$$</span>
<span class="bp">\</span><span class="kd">begin</span><span class="o">{</span><span class="n">matrix</span><span class="o">}</span>
<span class="mi">1</span> <span class="bp">&</span> <span class="n">x</span> <span class="bp">&</span> <span class="n">x</span><span class="bp">^</span><span class="mi">2</span> <span class="bp">\\</span>
<span class="mi">1</span> <span class="bp">&</span> <span class="n">y</span> <span class="bp">&</span> <span class="n">y</span><span class="bp">^</span><span class="mi">2</span> <span class="bp">\\</span>
<span class="mi">1</span> <span class="bp">&</span> <span class="n">z</span> <span class="bp">&</span> <span class="n">z</span><span class="bp">^</span><span class="mi">2</span> <span class="bp">\\</span>
<span class="bp">\</span><span class="kd">end</span><span class="o">{</span><span class="n">matrix</span><span class="o">}</span>
<span class="bp">$$</span>
</code></pre></div>
<p>produces:</p>
</li>
</ol>
<p>$$
\begin{matrix}
1 & x & x^2 \\
1 & y & y^2 \\
1 & z & z^2 \\
\end{matrix}
$$</p>
<p>MathJax will adjust the sizes of the rows and columns so that everything fits.</p>
<ol start="2">
<li>
<p>To add brackets, either use <code>\left…\right</code> as in section 6 of the tutorial, or replace <code>matrix</code> with <code>pmatrix</code> $\begin{pmatrix}1&2\\3&4\\ \end{pmatrix}$, <code>bmatrix</code> $\begin{bmatrix}1&2\\3&4\\ \end{bmatrix}$, <code>Bmatrix</code> $\begin{Bmatrix}1&2\\3&4\\ \end{Bmatrix}$, <code>vmatrix</code> $\begin{vmatrix}1&2\\3&4\\ \end{vmatrix}$, <code>Vmatrix</code> $\begin{Vmatrix}1&2\\3&4\\ \end{Vmatrix}$.</p>
</li>
<li>
<p>Use <code>\cdots</code> $\cdots$ <code>\ddots</code> $\ddots$ <code>vdots</code> $\vdots$ when you want to omit some of the entries:</p>
<p>$$\begin{pmatrix}
1 & a_1 & a_1^2 & \cdots & a_1^n \\
1 & a_2 & a_2^2 & \cdots & a_2^n \\
\vdots & \vdots& \vdots & \ddots & \vdots \\
1 & a_m & a_m^2 & \cdots & a_m^n
\end{pmatrix}$$</p>
</li>
<li>
<p>For horizontally "augmented" matrices, put parentheses or brackets around a suitably-formatted table; see <a href="http://meta.math.stackexchange.com/a/5044/">arrays</a> below for details. Here is an example:</p>
</li>
</ol>
<p>$$ \left[\begin{array}{cc|c}
1&2&3\\
4&5&6
\end{array}\right] $$</p>
<p>is produced by:</p>
<div class="codehilite"><pre><span></span><code> <span class="bp">$$</span> <span class="bp">\</span><span class="n">left</span><span class="o">[</span>
<span class="bp">\</span><span class="kd">begin</span><span class="o">{</span><span class="n">array</span><span class="o">}{</span><span class="n">cc</span><span class="bp">|</span><span class="n">c</span><span class="o">}</span>
<span class="mi">1</span><span class="bp">&</span><span class="mi">2</span><span class="bp">&</span><span class="mi">3</span><span class="bp">\\</span>
<span class="mi">4</span><span class="bp">&</span><span class="mi">5</span><span class="bp">&</span><span class="mi">6</span>
<span class="bp">\</span><span class="kd">end</span><span class="o">{</span><span class="n">array</span><span class="o">}</span>
<span class="bp">\</span><span class="n">right</span><span class="o">]</span> <span class="bp">$$</span>
</code></pre></div>
<p>The <code>cc|c</code> is the crucial part here; it says that there are three centered columns with a vertical bar between the second and third.</p>
<ol start="5">
<li>For vertically "augmented" matrices, use <code>\hline</code>. For example</li>
</ol>
<p>$$
\begin{pmatrix}
a & b \\
c & d\\
\hline
1 & 0\\
0 & 1
\end{pmatrix}
$$
is produced by</p>
<div class="codehilite"><pre><span></span><code><span class="bp">$$</span>
<span class="bp">\</span><span class="kd">begin</span><span class="o">{</span><span class="n">pmatrix</span><span class="o">}</span>
<span class="n">a</span> <span class="bp">&</span> <span class="n">b</span><span class="bp">\\</span>
<span class="n">c</span> <span class="bp">&</span> <span class="n">d</span><span class="bp">\\</span>
<span class="bp">\</span><span class="n">hline</span>
<span class="mi">1</span> <span class="bp">&</span> <span class="mi">0</span><span class="bp">\\</span>
<span class="mi">0</span> <span class="bp">&</span> <span class="mi">1</span>
<span class="bp">\</span><span class="kd">end</span><span class="o">{</span><span class="n">pmatrix</span><span class="o">}</span>
<span class="bp">$$</span>
</code></pre></div>
<ol start="6">
<li>
<p>For small inline matrices use <code>\bigl(\begin{smallmatrix} ... \end{smallmatrix}\bigr)</code>, e.g. $\bigl( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \bigr)$ is produced by:</p>
<div class="codehilite"><pre><span></span><code> <span class="bp">$\</span><span class="n">bigl</span><span class="o">(</span> <span class="bp">\</span><span class="kd">begin</span><span class="o">{</span><span class="n">smallmatrix</span><span class="o">}</span> <span class="n">a</span> <span class="bp">&</span> <span class="n">b</span> <span class="bp">\\</span> <span class="n">c</span> <span class="bp">&</span> <span class="n">d</span> <span class="bp">\</span><span class="kd">end</span><span class="o">{</span><span class="n">smallmatrix</span><span class="o">}</span> <span class="bp">\</span><span class="n">bigr</span><span class="o">)</span><span class="bp">$</span>
</code></pre></div>
</li>
</ol>
<h2 id="from-codetactic_writingmdcode" class="markdown-heading">From <code>tactic_writing.md</code> <a class="hover-link" href="#from-codetactic_writingmdcode">#</a></h2>
<ul>
<li><code>return</code>: produce a value in the monad (type: <code>A → m A</code>)</li>
<li><code>ma >>= f</code>: get the value of type <code>A</code> from <code>ma : m A</code> and pass it to <code>f : A → m B</code>. Alternate
syntax: <code>do a ← ma, f a</code></li>
<li><code>f <$> ma</code>: apply the function <code>f : A → B</code> to the value in <code>ma : m A</code> to get a <code>m B</code>. Same as
<code>do a ← ma, return (f a)</code></li>
<li><code>ma >> mb</code>: same as <code>do a ← ma, mb</code>; here the return value of <code>ma</code> is ignored and then <code>mb</code> is
called. Alternate syntax: <code>do ma, mb</code></li>
<li><code>mf <*> ma</code>: same as <code>do f ← mf, f <$> ma</code>, or <code>do f ← mf, a ← ma, return (f a)</code></li>
<li><code>ma <* mb</code>: same as <code>do a ← ma, mb, return a</code></li>
<li><code>ma *> mb</code>: same as <code>do ma, mb</code>, or <code>ma >> mb</code>. Why two notations for the same thing? Historical
reasons.</li>
<li><code>pure</code>: same as <code>return</code>. Again, historical reasons.</li>
<li><code>failure</code>: failed value (specific monads usually have a more useful form of this, like <code>fail</code> and
<code>failed</code> for tactics).</li>
<li><code>ma <|> ma'</code> recover from failure: runs <code>ma</code> and if it fails then runs <code>ma'</code>.</li>
<li><code>a $> mb</code>: same as <code>do mb, return a</code></li>
<li><code>ma <$ b</code>: same as <code>do ma, return b</code></li>
</ul>
</main>
<nav class="nav">
<h3>General documentation</h3>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/">index</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/foundational_types.html">foundational types</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/tactics.html">tactics</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/commands.html">commands</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/hole_commands.html">hole commands</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/attributes.html">attributes</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/notes.html">notes</a></div>
<div class="nav_link"><a href="https://pygae.github.io/lean-ga-docs/references.html">references</a></div>
<h3>Additional documentation</h3>
<h3>Library</h3>
<h4>core</h4>
<details class="nav_sect" data-path="data" >
<summary>data</summary>
<details class="nav_sect" data-path="data/buffer" >
<summary>buffer</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/data/buffer/parser.html">parser</a>
</div>
</details>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/data/buffer.html">buffer</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/data/dlist.html">dlist</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/data/vector.html">vector</a>
</div>
</details>
<details class="nav_sect" data-path="init" >
<summary>init</summary>
<details class="nav_sect" data-path="init/algebra" >
<summary>algebra</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/algebra/classes.html">classes</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/algebra/functions.html">functions</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/algebra/order.html">order</a>
</div>
</details>
<details class="nav_sect" data-path="init/control" >
<summary>control</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/alternative.html">alternative</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/applicative.html">applicative</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/combinators.html">combinators</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/except.html">except</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/functor.html">functor</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/id.html">id</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/lawful.html">lawful</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/lift.html">lift</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/monad.html">monad</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/monad_fail.html">monad_fail</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/option.html">option</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/reader.html">reader</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/control/state.html">state</a>
</div>
</details>
<details class="nav_sect" data-path="init/data" >
<summary>data</summary>
<details class="nav_sect" data-path="init/data/array" >
<summary>array</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/array/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/array/slice.html">slice</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/bool" >
<summary>bool</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/bool/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/bool/lemmas.html">lemmas</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/char" >
<summary>char</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/char/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/char/classes.html">classes</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/char/lemmas.html">lemmas</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/fin" >
<summary>fin</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/fin/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/fin/ops.html">ops</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/int" >
<summary>int</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/int/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/int/bitwise.html">bitwise</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/int/comp_lemmas.html">comp_lemmas</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/int/order.html">order</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/list" >
<summary>list</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/list/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/list/instances.html">instances</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/list/lemmas.html">lemmas</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/list/qsort.html">qsort</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/nat" >
<summary>nat</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/nat/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/nat/bitwise.html">bitwise</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/nat/div.html">div</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/nat/gcd.html">gcd</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/nat/lemmas.html">lemmas</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/option" >
<summary>option</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/option/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/option/instances.html">instances</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/ordering" >
<summary>ordering</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/ordering/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/ordering/lemmas.html">lemmas</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/sigma" >
<summary>sigma</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/sigma/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/sigma/lex.html">lex</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/string" >
<summary>string</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/string/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/string/ops.html">ops</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/subtype" >
<summary>subtype</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/subtype/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/subtype/instances.html">instances</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/sum" >
<summary>sum</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/sum/basic.html">basic</a>
</div>
</details>
<details class="nav_sect" data-path="init/data/unsigned" >
<summary>unsigned</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/unsigned/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/unsigned/ops.html">ops</a>
</div>
</details>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/prod.html">prod</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/punit.html">punit</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/quot.html">quot</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/repr.html">repr</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/set.html">set</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/setoid.html">setoid</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/data/to_string.html">to_string</a>
</div>
</details>
<details class="nav_sect" data-path="init/meta" >
<summary>meta</summary>
<details class="nav_sect" data-path="init/meta/converter" >
<summary>converter</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/converter/conv.html">conv</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/converter/interactive.html">interactive</a>
</div>
</details>
<details class="nav_sect" data-path="init/meta/lean" >
<summary>lean</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/lean/parser.html">parser</a>
</div>
</details>
<details class="nav_sect" data-path="init/meta/smt" >
<summary>smt</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/smt/congruence_closure.html">congruence_closure</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/smt/ematch.html">ematch</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/smt/interactive.html">interactive</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/smt/rsimp.html">rsimp</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/smt/smt_tactic.html">smt_tactic</a>
</div>
</details>
<details class="nav_sect" data-path="init/meta/widget" >
<summary>widget</summary>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/widget/basic.html">basic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/widget/html_cmd.html">html_cmd</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/widget/interactive_expr.html">interactive_expr</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/widget/replace_save_info.html">replace_save_info</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/widget/tactic_component.html">tactic_component</a>
</div>
</details>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/ac_tactics.html">ac_tactics</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/async_tactic.html">async_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/attribute.html">attribute</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/backward.html">backward</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/case_tag.html">case_tag</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/comp_value_tactics.html">comp_value_tactics</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/congr_lemma.html">congr_lemma</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/congr_tactic.html">congr_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/constructor_tactic.html">constructor_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/contradiction_tactic.html">contradiction_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/declaration.html">declaration</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/derive.html">derive</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/environment.html">environment</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/exceptional.html">exceptional</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/expr.html">expr</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/expr_address.html">expr_address</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/feature_search.html">feature_search</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/float.html">float</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/format.html">format</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/fun_info.html">fun_info</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/has_reflect.html">has_reflect</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/hole_command.html">hole_command</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/injection_tactic.html">injection_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/instance_cache.html">instance_cache</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/interaction_monad.html">interaction_monad</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/interactive.html">interactive</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/interactive_base.html">interactive_base</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/json.html">json</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/level.html">level</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/local_context.html">local_context</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/match_tactic.html">match_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/mk_dec_eq_instance.html">mk_dec_eq_instance</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/mk_has_reflect_instance.html">mk_has_reflect_instance</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/mk_has_sizeof_instance.html">mk_has_sizeof_instance</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/mk_inhabited_instance.html">mk_inhabited_instance</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/module_info.html">module_info</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/name.html">name</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/occurrences.html">occurrences</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/options.html">options</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/pexpr.html">pexpr</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/rb_map.html">rb_map</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/rec_util.html">rec_util</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/ref.html">ref</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/relation_tactics.html">relation_tactics</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/rewrite_tactic.html">rewrite_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/set_get_option_tactics.html">set_get_option_tactics</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/simp_tactic.html">simp_tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/tactic.html">tactic</a>
</div>
<div class="nav_link ">
<a href="https://pygae.github.io/lean-ga-docs/init/meta/tagged_format.html">tagged_format</a>
</div>
<div class="nav_link ">