summaryrefslogtreecommitdiff
path: root/macros/latex/contrib/ebproof/ebproof.dtx
blob: df4f11f0cabb1dea474bff16e1ce2357a18fa00b (plain)
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
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
% \iffalse meta-comment
%
% Copyright (C) 2015-2021 by Emmanuel Beffara
%
% This file may be distributed and/or modified under the conditions of the
% LaTeX Project Public License, either version 1.3 of this license or (at
% your option) any later version. The latest version of this license is in:
%
%   http://www.latex-project.org/lppl.txt
%
% and version 1.3 or later is part of all distributions of LaTeX version
% 2005/12/01 or later.
%
%<*driver>
\documentclass[full]{l3doc}
\usepackage{ebproof}
\usepackage{multicol}
\usepackage{tikz}
\usetikzlibrary{decorations.pathmorphing}
\EnableCrossrefs
\newenvironment{example}{%
  \VerbatimEnvironment
  \begin{VerbatimOut}{example.tex}}{%
  \end{VerbatimOut}
  \begin{center}
  \begin{minipage}{.4\textwidth}
    \input{example.tex}
  \end{minipage}%
  \begin{minipage}{.6\textwidth}
    \small\VerbatimInput[gobble=0]{example.tex}
  \end{minipage}%
  \end{center}
}
\begin{document}
  \DocInput{\jobname.dtx}
\end{document}
%</driver>
% \fi
%
% \title{^^A
%   The \pkg{ebproof} package \\
%   Formal proofs in the style of sequent calculus^^A
% }
%
% \author{^^A
%   Emmanuel Beffara\thanks
%     {^^A
%       E-mail:  \href{mailto:manu@beffara.org}{manu@beffara.org}^^A
%     }^^A
% }
%
% \date{Version 2.1.1 -- Released 2021-01-28}
%
% \maketitle
%
% \setcounter{tocdepth}{2}
% \begin{multicols}{2}
% \tableofcontents
% \end{multicols}
%
% \begin{documentation}
%
% \section{Introduction}
%
% The \pkg{ebproof} package provides commands to typeset proof trees, in the
% style of sequent calculus and related systems:
%
% \begin{example}
%   \begin{prooftree}
%     \hypo{ \Gamma, A &\vdash B }
%     \infer1[abs]{ \Gamma &\vdash A\to B }
%     \hypo{ \Gamma \vdash A }
%     \infer2[app]{ \Gamma \vdash B }
%   \end{prooftree}
% \end{example}
%
% The structure is very much inspired by the
% \href{http://math.ucsd.edu/~sbuss/ResearchWeb/bussproofs/}{\pkg{bussproofs}}
% package, in particular for the postfix notation.
% I actually wrote \pkg{ebproof} because there were some limitations in
% \pkg{bussproofs} that I did not know how to lift, and also because I did
% not like some choices in that package (and also because it was fun to write).
%
% Any feedback is welcome, in the form of bug reports, feature requests or
% suggestions, through the web page of the project at \url{https://framagit.org/manu/ebproof}.
%
% \section{Environments}
%
% \begin{function}{prooftree,prooftree*}
%   \begin{syntax}
%     \verb|\begin{prooftree}|\oarg{options}
%     ~ \meta{statements}
%     \verb|\end{prooftree}|
%   \end{syntax}
%   The package provides the \env{prooftree} environment, in standard and
%   starred variants.
%   This typesets the proof tree described by the \meta{statements}, as
%   described in section~\ref{sec:statements}.
%   The \meta{options} provide default formatting options for the proof tree,
%   available options are described in section~\ref{sec:options}.
%
%   Following the conventions of \pkg{amsmath} for alignment environments, the
%   non-starred version produces a proof tree at the current position in the
%   text flow (it can be used in math mode or text mode) while the starred
%   version typesets the proof on a line of its own, like a displayed formula.
% \end{function}
%
% \begin{example}
%   \[
%     \begin{prooftree}
%       \infer0{ \vdash A }
%       \hypo{ \vdash B } \infer1{ \vdash B, C }
%       \infer2{ \vdash A\wedge B, C }
%     \end{prooftree}
%     \quad \rightsquigarrow \quad
%     \begin{prooftree}
%       \infer0{ \vdash A } \hypo{ \vdash B }
%       \infer2{ \vdash A\wedge B }
%       \infer1{ \vdash A\wedge B, C }
%     \end{prooftree}
%   \]
% \end{example}
%
% \section{Statements}
% \label{sec:statements}
%
% Statements describe proofs in postfix notation: when typesetting a proof tree
% whose last rule has, say, two premisses, you will first write statements for
% the subtree of the first premiss, then statements for the subtree of the
% second premiss, then a statement like \cs{infer2}\{\meta{conclusion}\} to
% build an inference with these two subtrees as premisses and the given text as
% conclusion.
%
% Hence statements operate on a stack of proof trees.
% At the beginning of a \env{prooftree} environment, the stack is empty.
% At the end, it must contain exactly one tree, which is the one that will be
% printed.
%
% Note that the commands defined in this section only exist right inside
% \env{prooftree} environments.
% If you have a macro with the same name as one of the statements, for instance
% \cs{hypo}, then this macro will keep its meaning outside \env{prooftree}
% environments as well as inside the arguments of a statement.
% If you really need to access the statements in another context, you can can
% always call them by prefixing their names with \texttt{ebproof}, for instance as
% \cs{ebproofhypo}.
%
% \subsection{Basic statements}
%
% The basic statements for building proofs are the following, where
% \meta{options} stands for arbitrary options as described in
% section~\ref{sec:options}.
%
% \begin{function}{\hypo}
%   \begin{syntax}
%     \cs{hypo}\oarg{options}\marg{text}
%   \end{syntax}
%   The statement \cs{hypo} pushes a new proof tree consisting only in one
%   conclusion line, with no premiss and no line above, in other words a tree
%   with only a leaf (\cs{hypo} stands for \emph{hypothesis}).
% \end{function}
%
% \begin{function}{\infer}
%   \begin{syntax}
%     \cs{infer}\oarg{options}\meta{arity}\oarg{label}\marg{text}
%   \end{syntax}
%   The statement \cs{infer} builds an inference step by taking some proof
%   trees from the top of the stack, assembling them with a rule joining their
%   conclusions and putting a new conclusion below.
%   The \meta{arity} is the number of sub-proofs, it may be any number
%   including 0 (in this case there will be a line above the conclusion but no
%   sub-proof).
%   If \meta{label} is present, it is used as the label on the right of the
%   inference line; it is equivalent to using the \cmd{right label} option.
% \end{function}
%
% \medskip
%
% The \meta{text} in these statements is the contents of the conclusion at the
% root of the tree that the statements create.
% It is typeset in math mode by default but any kind of formatting can be used
% instead, using the \cmd{template} option.
% The \meta{label} text is formatted in horizontal text mode by default.
%
% Each proof tree has a vertical axis, used for alignment of successive steps.
% The position of the axis is deduced from the text of the conclusion at the
% root of the tree: if \meta{text} contains the alignment character \verb|&|
% then the axis is set at that position, otherwise the axis is set at the center
% of the conclusion text.
% The \cs{infer} statement makes sure that the axis of the premiss is at the
% same position as the axis of the conclusion.
% If there are several premisses, it places the axis at the center between the
% left of the leftmost conclusion and the right of the rightmost conclusion:
%
% \begin{example}
%   \begin{prooftree}
%     \hypo{ &\vdash A, B, C }
%     \infer1{ A &\vdash B, C }
%     \infer1{ A, B &\vdash C }
%     \hypo{ D &\vdash E }
%     \infer2{ A, B, D &\vdash C, E }
%     \infer1{ A, B &\vdash C, D, E }
%     \infer1{ A &\vdash B, C, D, E }
%   \end{prooftree}
% \end{example}
%
% \begin{function}{\ellipsis}
%   \begin{syntax}
%     \cs{ellipsis}\marg{label}\marg{text}
%   \end{syntax}
%   The statement \cs{ellipsis} typesets vertical dots, with a label on the right,
%   and a new conclusion. No inference lines are inserted.
% \end{function}
%
% \begin{example}
%   \begin{prooftree}
%     \hypo{ \Gamma &\vdash A }
%     \ellipsis{foo}{ \Gamma &\vdash A, B }
%   \end{prooftree}
% \end{example}
%
%
% \subsection{Modifying proof trees}
%
% The following additional statements may be used to affect the format of the
% last proof tree on the stack.
%
% \begin{function}{\rewrite}
%   \begin{syntax}
%     \cs{rewrite}\marg{code}
%   \end{syntax}
%   The statement \cs{rewrite} is used to modify the proof of the stack while
%   preserving its size and alignment.
%   The \meta{code} is typeset in horizontal mode, with the following control
%   sequences defined:
%   \begin{itemize}
%   \item \cs{treebox} is a box register that contains the original material,
%   \item \cs{treemark}\marg{name} expands as the position of a given mark with
%     respect to the left of the box.
%   \end{itemize}
% \end{function}
%
% A simple use of this statement is to change the color of a proof tree:
% \begin{example}
%   \begin{prooftree}
%     \hypo{ \Gamma, A &\vdash B }
%     \infer1[abs]{ \Gamma &\vdash A\to B }
%     \rewrite{\color{red}\box\treebox}
%     \hypo{ \Gamma \vdash A }
%     \infer2[app]{ \Gamma \vdash B }
%   \end{prooftree}
% \end{example}
% Note the absence of spaces inside the call to \cs{rewrite}, because spaces
% would affect the position of the tree box.
% Note also that explicit use of \cs{treebox} is required to actually draw the
% subtree.
% Not using it will effectively not render the subtree, while still reserving
% its space in the enclosing tree:
% \begin{example}
%   \begin{prooftree}
%     \hypo{ \Gamma, A &\vdash B }
%     \infer1[abs]{ \Gamma &\vdash A\to B }
%     \rewrite{}
%     \hypo{ \Gamma \vdash A }
%     \infer2[app]{ \Gamma \vdash B }
%   \end{prooftree}
% \end{example}
% This kind of manipulation is useful for instance in conjunction with the
% \pkg{beamer} package to allow revealing subtrees of a proof tree
% progressively in successive slides of a given frame.
%
% \begin{function}{\delims}
%   \begin{syntax}
%     \cs{delims}\marg{left}\marg{right}
%   \end{syntax}
%   The statement \cs{delims} puts left and right delimiters around the whole
%   sub-proof, without changing the alignment (the spacing is affected by the
%   delimiters, however).
%   The \meta{left} text must contain an opening occurrence of \cs{left} and
%   the \meta{right} text must contain a matching occurrence of \cs{right}.
%   For instance, \verb|\delims{\left(}{\right)}| will put the sub-proof
%   between parentheses.
% \end{function}
%
% \begin{example}
%   \begin{prooftree}
%     \hypo{ A_1 \vee \cdots \vee A_n }
%     \hypo{ [A_i] }
%     \ellipsis{}{ B }
%     \delims{ \left( }{ \right)_{1\leq i\leq n} }
%     \infer2{ B }
%   \end{prooftree}
% \end{example}
%
% \begin{function}{\overlay}
%   \begin{syntax}
%     \cs{overlay}
%   \end{syntax}
%   The statement \cs{overlay} combines the last two proofs on the stack into
%   a single one, so that their conclusions are placed at the same point.
% \end{function}
%
% \begin{example}
%   \begin{prooftree}
%     \hypo{Z}
%     \hypo{A} \hypo{B} \infer2{C} \hypo{D} \infer2{D}
%       \rewrite{\color{blue}\box\treebox}
%     \hypo{E} \hypo{F} \hypo{G} \infer2{H} \infer2{I}
%       \rewrite{\color{red}\box\treebox}
%     \overlay \hypo{J} \infer3{K}
%   \end{prooftree}
% \end{example}
%
% The primary use of this feature is for building animated presentations where
% a subtree in a proof has to be modified without affecting the general
% alignment of the surrounding proof. For instance, the example above could be
% used in Beamer to build successive slides in a given frame with two
% different subtrees:
%
% \begin{verbatim}
%   \begin{prooftree}
%     \hypo{Z}
%     \hypo{A} \hypo{B} \infer2{C} \hypo{D} \infer2{D}
%       \only<2>{\rewrite{}} % erases this version on slide 2
%     \hypo{E} \hypo{F} \hypo{G} \infer2{H} \infer2{I}
%       \only<1>{\rewrite{}} % erases this version on slide 1
%     \overlay \hypo{J} \infer3{K}
%   \end{prooftree}
% \end{verbatim}
%
% \section{Options}
% \label{sec:options}
%
% The formatting of trees, conclusion texts and inference rules is affected by
% options, specified using the \LaTeX3 key-value system.
% All options are in the \texttt{ebproof} module in the key tree.
% They can be set locally for a proof tree or for a single statement using
% optional arguments in the associated commands.
%
% \begin{function}{\ebproofset,\set}
%   \begin{syntax}
%     \cs{ebproofset}\marg{options}
%   \end{syntax}
%   The statement \cs{ebproofset} is used to set some options. When used
%   inside a \env{prooftree} environment, it can written \cs{set}.
%   The options will apply in the current scope; using this in preamble will
%   effectively set options globally.
%   Specific options may also be specified for each proof tree and for each
%   statement in a proof tree, using optional arguments.
% \end{function}
%
% \subsection{General shape}
%
% The options in this section only make sense at the global level and at the
% proof level.
% Changing the proof style inside a \env{proof} environment has undefined
% behaviour.
%
% \begin{variable}{proof style}
%   The option \cmd{proof style} sets the general shape for representing proofs.
%   The following styles are provided:
%   \begin{description}
%   \item[upwards] 
%     This is the default style.
%     Proof trees grow upwards, with conclusions below and premisses above.
%   \item[downwards]
%     Proof trees grow downwards, with conclusions above and premisses below.
%     \fvset{gobble=6}
%     \begin{example}
%       \begin{prooftree}[proof style=downwards]
%         \hypo{ \Gamma, A &\vdash B }
%         \infer1[abs]{ \Gamma &\vdash A\to B }
%         \hypo{ \Gamma \vdash A }
%         \infer2[app]{ \Gamma \vdash B }
%       \end{prooftree}
%     \end{example}
%   \end{description}
% \end{variable}
%
% In the optional argument of \env{prooftree} environments, proof styles can
% be specified directly, without prefixing the name by ``\texttt{proof style=}''.
% For instance, the first line of the example above could be written
% \verb|\begin{prooftree}| equivalently.
%
% \begin{variable}{center}
%   The option \cmd{center} toggles vertical centering of typeset proofs.
%   If set to \texttt{true}, the tree produced by the \env{prooftree}
%   environment will be vertically centered around the text line.
%   If set to \texttt{false}, the base line of the tree will be the base line
%   of the conclusion.
%   The default value is \texttt{true}.
% \end{variable}
%
% \begin{example}
%   \begin{prooftree}[center=false]
%     \infer0{ A \vdash A }
%   \end{prooftree}
%   \qquad
%   \begin{prooftree}[center=false]
%     \hypo{ \Gamma, A \vdash B }
%     \infer1{ \Gamma \vdash A \to B }
%   \end{prooftree}
% \end{example}
%
% \subsection{Spacing}
%
% \begin{variable}{separation}
%   Horizontal separation between sub-proofs in an inference is defined by the
%   option \cmd{separation}.
%   The default value is \texttt{1.5em}.
% \end{variable}
%
% \begin{example}
%   \begin{prooftree}[separation=0.5em]
%     \hypo{ A } \hypo{ B } \infer2{ C }
%     \hypo{ D } \hypo{ E } \hypo{ F } \infer3{ G }
%     \hypo{ H } \infer[separation=3em]3{ K }
%   \end{prooftree}
% \end{example}
%
% \begin{variable}{rule margin}
%   The spacing above and below inference lines is defined by the option
%   \cmd{rule margin}.
%   The default value is \texttt{0.7ex}.
% \end{variable}
%
% \begin{example}
%   \begin{prooftree}[rule margin=2ex]
%     \hypo{ \Gamma, A &\vdash B }
%     \infer1[abs]{ \Gamma &\vdash A\to B }
%     \hypo{ \Gamma \vdash A }
%     \infer2[app]{ \Gamma \vdash B }
%   \end{prooftree}
% \end{example}
%
% \subsection{Shape of inference lines}
%
% \begin{variable}{rule style}
%   The shape of inference lines is set by the option \cmd{rule style}.
%   The following values are provided:
%   \begin{center}
%     \begin{tabular}{@{}l@{\qquad}l@{}}
%       \toprule
%       \cmd{simple}  & a simple line (this is the default style) \\
%       \cmd{no rule} & no rule, only a single space of length \texttt{rule margin} \\
%       \cmd{double}  & a double line \\
%       \cmd{dashed}  & a single dashed line \\
%       \bottomrule
%     \end{tabular}
%   \end{center}
% \end{variable}
%
% The precise rendering is influenced by parameters specified below.
% Arbitrary new shapes can defined using the \cs{ebproofnewrulestyle} command
% described in section~\ref{sec:styles}, using \texttt{rule code} option
% described below.
%
% In the optional argument of the \cs{infer} statement, rule styles can be
% specified directly, without prefixing the style name by ``\texttt{rule style=}''.
% For instance, \verb|\infer[dashed]| is equivalent to
% \verb|\infer[rule style=dashed]|.
%
% \begin{example}
%   \begin{prooftree}
%     \hypo{ \Gamma &\vdash A \to B }
%     \infer[no rule]1{ \Gamma &\vdash {!A} \multimap B }
%     \hypo{ \Delta &\vdash A }
%     \infer[rule thickness=2pt]1{ \Delta &\vdash {!A} }
%     \infer0{ B \vdash B }
%     \infer[dashed]2{ \Delta, {!A}\multimap B \vdash B }
%     \infer2{ \Gamma, \Delta &\vdash B }
%     \infer[double]1{ \Gamma \cup \Delta &\vdash B }
%   \end{prooftree}
% \end{example}
%
% \begin{variable}{rule thickness,rule separation}
%   The thickness of inference lines is defined by option \cmd{rule
%   thickness}, it is \texttt{0.4pt} by default.
%   The distance between the two lines in the \texttt{double} rule style is
%   defined by the \cmd{rule separation} option.
%   It is \texttt{2pt} by default.
% \end{variable}
% \begin{variable}{rule dash length,rule dash space}
%   For dashed rules, the length of dashes is defined by the option
%   \cmd{rule dash length} and the space between dashes is defined by the
%   option \cmd{rule dash space}.
%   The default values are \texttt{0.2em} and \texttt{0.3em} respectively.
% \end{variable}
%
% \begin{variable}{rule code}
%   Arbitrary rule shapes can be optained using the \cmd{rule code} option.
%   The argument is code is used to render the rule, it is executed in
%   vertical mode in a \cs{vbox} whose \cs{hsize} is set to the width of the
%   rule.
%   Margins above and below are inserted automatically (they can be removed by
%   setting \texttt{rule margin} to \texttt{0pt}).
% \end{variable}
%
% \begin{example}
%   \begin{prooftree}[rule code={\hbox{\tikz
%       \draw[decorate,decoration={snake,amplitude=.3ex}]
%       (0,0) -- (\hsize,0);}}]
%     \hypo{ \Gamma &\vdash A }
%     \infer1{ \Gamma &\vdash A, \ldots, A }
%     \hypo{ \Delta, A, \ldots, A \vdash \Theta }
%     \infer2{ \Gamma, \Delta \vdash \Theta }
%   \end{prooftree}
% \end{example}
% Note that this example requires the \pkg{tikz} package, with the
% \pkg{decorations.pathmorphing} library for the \texttt{snake} decoration.
%
% \subsection{Format of conclusions and labels}
%
% \begin{variable}{template,left template,right template}
%   The format of text in inferences is defined by templates.
%   The option \cmd{template} is used for text with no alignment mark, the
%   options \cmd{left template} and \cmd{right template} are used for the left
%   and right side of the alignment mark when it is present.
%   The value of these options is arbitrary \TeX\ code, composed in horizontal mode.
%   The macro \cs{inserttext} is used to insert the actual text passed to the
%   \cs{hypo} and \cs{infer} statements.
%   The default value for \cmd{template} is simply \verb|$\inserttext$|, so
%   that conclusions are set in math mode.
%   The default values for \cmd{left template} and \cmd{right template} are
%   similar, with spacing assuming that a relation symbol is put near the
%   alignment mark, so that \verb|\infer1{A &\vdash B}| is spaced correctly.
% \end{variable}
%
% \begin{example}
%   \begin{prooftree}[template=(\textbf\inserttext)]
%     \hypo{ foo }
%     \hypo{ bar }
%     \infer1{ baz }
%     \infer2{ quux }
%   \end{prooftree}
% \end{example}
%
% \begin{variable}{left label,right label}
%   The text to use as the labels of the rules, on the left and on the right
%   of the inference line, is defined by the options \cmd{left label} and
%   \cmd{right label}.
%   Using the second optional argument in \cs{infer} is equivalent to setting
%   the \env{right label} option with the value of that argument.
% \end{variable}
%
% \begin{example}
%   \begin{prooftree}
%     \hypo{ \Gamma, A &\vdash B }
%     \infer[left label=$\lambda$]1[abs]
%       { \Gamma &\vdash A\to B }
%     \hypo{ \Gamma \vdash A }
%     \infer[left label=@]2[app]{ \Gamma \vdash B }
%   \end{prooftree}
% \end{example}
%
% \begin{variable}{left label template,right label template}
%   Similarly to conclusions, labels are formatted according to templates.
%   The code is arbitrary \TeX\ code, composed in horizontal mode, where the
%   macro \cs{inserttext} can be used to insert the actual label text.
%   The default values are simply \cs{inserttext} so that labels are set in
%   plain text mode.
% \end{variable}
% \begin{variable}{label separation}
%   The spacing between an inference line and its labels is defined by the
%   option \cmd{label separation}, the default value is \texttt{0.5em}.
%   The height of the horizontal axis used for aligning the labels with the
%   rules is defined by the option \cmd{label axis}, the default value is
%   \texttt{0.5ex}.
% \end{variable}
%
%
% \subsection{Style macros}
% \label{sec:styles}
%
% The following commands allow for the definition of custom styles using the
% basic style options, in a way similar to PGF's ``styles'' and \LaTeX3's
% ``meta-keys''.
% This allows setting a bunch of options with the same values in many proofs
% using a single definition.
%
% \begin{function}{\ebproofnewstyle}
%   \begin{syntax}
%     \cs{ebproofnewstyle}\marg{name}\marg{options}
%   \end{syntax}
%   The statement \cs{ebproofnewstyle} defines a new style option with some
%   \meta{name} that sets a given set of \meta{options}.
% \end{function}
% For instance, the following code defines a new option \cmd{small} that sets
% various parameters so that proofs are rendered smaller.
% \begin{example}
%   \ebproofnewstyle{small}{
%     separation = 1em, rule margin = .5ex,
%     template = \footnotesize$\inserttext$ }
%   \begin{prooftree}[small]
%     \hypo{ \Gamma, A \vdash B }
%     \infer1{ \Gamma \vdash A\to B }
%     \hypo{ \Gamma \vdash A } \infer2{ \Gamma \vdash B }
%   \end{prooftree}
% \end{example}
%
% \begin{function}{\ebproofnewrulestyle}
%   \begin{syntax}
%     \cs{ebproofnewrulestyle}\marg{name}\marg{options}
%   \end{syntax}
%   The statement \cs{ebproofnewrulestyle} does the same for rule styles.
%   The \meta{options} part includes options used to set how to draw rules in
%   the new style.
% \end{function}
%
% The option \cmd{rule code} is useful in this command as it allows to
% define arbitrary rule styles.
% For instance, the squiggly rule example above could be turned into a new
% rule style \texttt{zigzag} with the following code:
% \begin{example}
%   \ebproofnewrulestyle{zigzag}{
%     rule code = {\hbox{\tikz
%       \draw[decorate,decoration={snake,amplitude=.3ex}]
%       (0,0) -- (\hsize,0);}}}
%   \begin{prooftree}
%     \hypo{ \Gamma &\vdash A }
%     \infer1{ \Gamma &\vdash A, \ldots, A }
%     \hypo{ \Delta, A, \ldots, A \vdash \Theta }
%     \infer[zigzag]2{ \Gamma, \Delta \vdash \Theta }
%   \end{prooftree}
% \end{example}
%
%
% \section{License}
%
% This work may be distributed and/or modified under the
% conditions of the \LaTeX\ Project Public License, either version 1.3
% of this license or (at your option) any later version.
% The latest version of this license is in
% \begin{center}
%   \url{http://www.latex-project.org/lppl.txt}
% \end{center}
% and version 1.3 or later is part of all distributions of \LaTeX\
% version 2005/12/01 or later.
%
% This work has the LPPL maintenance status `maintained'.
%
% The Current Maintainer of this work is Emmanuel Beffara.
%
% This work consists of the files \texttt{ebproof.sty} and \texttt{ebproof.tex}.
%
% \section{History}
%
% This section lists the principal evolutions of the package, in reverse
% chronological order.
% \begin{description}
% \item[Version 2.1.1 (2021-01-28)]
%   Bugfix release, no changes in the user interface.
%   \begin{itemize}
%   \item Fixes a deprecation issue with \LaTeX3 release 2021-01-09 and
%     various warnings that appear in \LaTeX3 debugging mode.
%   \item Fixes \cmd{proof style=downwards}.
%   \end{itemize}
% \item[Version 2.1 (2020-08-19)]
%   Mostly a bugfix release.
%   \begin{itemize}
%   \item Makes the \env{prooftree} environment robust to use in tabular
%     contexts.
%   \item Adds the \cs{overlay} statement.
%   \item Fixes a compatibility issue with \LaTeX\ release 2020-10-01.
%   \end{itemize}
% \item[Version 2.0 (2017-05-17)]
%   A complete rewrite of the code using the \LaTeX3 programming environment. 
%   The incompatible changes from the user's point of view are the following:
%   \begin{itemize}
%   \item Proof statements are now writtten in lowercase ({i.e.} \cs{Infer} is
%     now written \cs{infer} etc.) but the syntax is otherwise unchanged.
%     The old uppercase commands still work but produce a deprecation warning,
%     they will be removed in a future version.
%   \item New styles are now defined using \cs{ebproofnewstyle} and
%     \cs{ebproofnewrulestyle}. The previous method using PGF styles does not
%     work anymore (because PGF is not used anymore).
%   \end{itemize}
%   The new commands and options are the following:
%   \begin{itemize}
%   \item The statement \cs{rewrite} generalizes \cs{Alter},
%   \item The option \cmd{label axis} controls vertical alignment of labels.
%   \end{itemize}
% \item[Version 1.1 (2015-03-13)]
%   A bugfix release.
%   In \cmd{template} options, one now uses \cs{inserttext} instead of \verb|#1|
%   for the text arguments, which improves robustness.
% \item[Version 1.0 (2015-02-04)]
%   The first public release.
% \end{description}
%
% \end{documentation}
%
% \clearpage \appendix
%
% \begin{implementation}
%
% \section{Implementation}
%
%    \begin{macrocode}
%<*package>
\NeedsTeXFormat{LaTeX2e}
\RequirePackage{expl3}
\RequirePackage{xparse}
\ProvidesExplPackage{ebproof}{2021/01/28}{2.1.1}{EB's proof trees}
%<@@=ebproof>
%    \end{macrocode}
%
% \subsection{Parameters}
%
% We first declare all options. For the meaning of options, see
% section~\ref{sec:options}.
%
%    \begin{macrocode}
\bool_new:N \l_@@_updown_bool
\keys_define:nn { ebproof } {
center .bool_set:N = \l_@@_center_bool,
proof~style .choice: ,
proof~style / upwards .code:n = \bool_set_false:N \l_@@_updown_bool,
proof~style / downwards .code:n = \bool_set_true:N \l_@@_updown_bool,
separation .dim_set:N = \l_@@_separation_dim,
rule~margin .dim_set:N = \l_@@_rule_margin_dim,
rule~thickness .dim_set:N = \l_@@_rule_thickness_dim,
rule~separation .dim_set:N = \l_@@_rule_separation_dim,
rule~dash~length .dim_set:N = \l_@@_rule_dash_length_dim,
rule~dash~space .dim_set:N = \l_@@_rule_dash_space_dim,
rule~code .tl_set:N = \l_@@_rule_code_tl,
rule~style .choice:,
template .tl_set:N = \l_@@_template_tl,
left~template .tl_set:N = \l_@@_left_template_tl,
right~template .tl_set:N = \l_@@_right_template_tl,
left~label .tl_set:N = \l_@@_left_label_tl,
right~label .tl_set:N = \l_@@_right_label_tl,
left~label~template .tl_set:N = \l_@@_left_label_template_tl,
right~label~template .tl_set:N = \l_@@_right_label_template_tl,
label~separation .dim_set:N = \l_@@_label_separation_dim,
label~axis .dim_set:N = \l_@@_label_axis_dim,
}
%    \end{macrocode}
%
% \begin{macro}{\ebproofnewrulestyle}
%   We then define the document-level macro \cs{ebproofnewrulestyle} and use
%   it to define the default styles. This simply consists in defining a
%   meta-key.
%    \begin{macrocode}
\NewDocumentCommand \ebproofnewrulestyle { mm } {
  \keys_define:nn { ebproof } {
    rule~style / #1 .meta:nn = { ebproof } { #2 }
  }
}
%    \end{macrocode}
% \end{macro}
%
% The styles |simple|, |no rule| and |double| are defined in a straightforward
% way.
%
%    \begin{macrocode}
\ebproofnewrulestyle { simple } {
  rule~code = { \tex_hrule:D height \l_@@_rule_thickness_dim }
}
\ebproofnewrulestyle { no~rule } {
  rule~code =
}
\ebproofnewrulestyle { double } {
  rule~code = {
    \tex_hrule:D height \l_@@_rule_thickness_dim
    \skip_vertical:N \l_@@_rule_separation_dim
    \tex_hrule:D height \l_@@_rule_thickness_dim
  }
}
%    \end{macrocode}
%
% The |dashed| style uses leaders and filling for repeating a single dash. We
% use \TeX\ primitives that have no \LaTeX3 counterpart for this.
%
%    \begin{macrocode}
\ebproofnewrulestyle { dashed } {
  rule~code = {
    \hbox_to_wd:nn { \tex_hsize:D } {
      \dim_set:Nn \l_tmpa_dim { \l_@@_rule_dash_space_dim / 2 }
      \skip_horizontal:n { -\l_tmpa_dim }
      \tex_cleaders:D \hbox:n {
        \skip_horizontal:N \l_tmpa_dim
        \tex_vrule:D
          height \l_@@_rule_thickness_dim
          width \l_@@_rule_dash_length_dim
        \skip_horizontal:N \l_tmpa_dim
      } \tex_hfill:D
      \skip_horizontal:n { -\l_tmpa_dim }
    }
  }
}
%    \end{macrocode}
%
% Now we can define the default values, including the default rule style.
%
%    \begin{macrocode}
\keys_set:nn { ebproof } {
  center = true,
  proof~style = upwards,
  separation = 1.5em,
  rule~margin = .7ex,
  rule~thickness = .4pt,
  rule~separation = 2pt,
  rule~dash~length = .2em,
  rule~dash~space = .3em,
  rule~style = simple,
  template = $\inserttext$,
  left~template = $\inserttext\mathrel{}$,
  right~template = $\mathrel{}\inserttext$,
  left~label = ,
  right~label = ,
  left~label~template = \inserttext,
  right~label~template = \inserttext,
  label~separation = 0.5em,
  label~axis = 0.5ex
}
%    \end{macrocode}
%
% \begin{macro}{\ebproofnewstyle}
%   Defining a style simply means defining a meta-key.
%    \begin{macrocode}
\NewDocumentCommand \ebproofnewstyle { mm } {
  \keys_define:nn { ebproof } { #1 .meta:n = { #2 } }
}
%    \end{macrocode}
% \end{macro}
%
%
% \subsection{Proof boxes}
%
% \TeX\ does not actually provide data structures, so we have to encode things.
% We provide an allocator for ``registers'' holding boxes with attributes. Such
% a register consists in a box register and a property list for marks, which
% maps mark names to values as explicit dimensions with units.
%
% \begin{macro}{\@@_new:N}
%   Using only public interfaces forces a convoluted approach to allocation:
%   we use a global counter \cs{g_ebproof_register_int} to number registers,
%   then each allocation creates registers named \cs{S_ebproof_K_N} where
%   S is the scope of the register (local or global, deduced from the argument),
%   K is the kind of component (box or marks) and N is the identifier of the
%   register. The proof box register itself only contains the identifier used
%   for indirection.
%    \begin{macrocode}
\int_new:N \g_@@_register_int
\cs_new:Nn \@@_box:N {
  \str_item:nn { #1 } { 2 } __ebproof_ \tl_use:N #1 _box
}
\cs_new:Nn \@@_marks:N {
  \str_item:nn { #1 } { 2 } __ebproof_ \tl_use:N #1 _prop
}
\cs_new:Nn \@@_new:N {
  \tl_new:N #1
  \int_gincr:N \g_@@_register_int
  \str_if_eq:eeTF { \str_item:nn { #1 } { 2 } } { g }
    { \tl_gset:Nx #1 { \int_to_arabic:n { \g_@@_register_int } } }
    { \tl_set:Nx #1 { \int_to_arabic:n { \g_@@_register_int } } }
  \box_new:c { \@@_box:N #1 }
  \prop_new:c { \@@_marks:N #1 }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_clear:N}
%   The box is cleared by setting it to an empty hbox.
%   Using \cs{box_clear:N} instead would not work because trying to push this
%   box on the stack would not actually append any box.
%    \begin{macrocode}
\cs_new:Nn \@@_clear:N {
  \hbox_set:cn { \@@_box:N #1 } {}
  \prop_clear:c { \@@_marks:N #1 }
  \@@_set_mark:Nnn #1 { left } { 0pt }
  \@@_set_mark:Nnn #1 { right } { 0pt }
  \@@_set_mark:Nnn #1 { axis } { 0pt }
}
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Mark operations}
%
% \begin{macro}{\@@_set_mark:Nnn}
%   Setting the value of a mark uses a temporary register to evaluate the
%   dimension expression because values are stored textually in a property
%   list.
%    \begin{macrocode}
\dim_new:N \l_@@_transit_dim
\cs_new:Nn \@@_set_mark:Nnn {
  \dim_set:Nn \l_@@_transit_dim { #3 }
  \prop_put:cnV { \@@_marks:N #1 } { #2 }
    \l_@@_transit_dim
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_mark:Nn}
%   Getting the value of a mark simply consists in getting an item in a
%   property list.
%    \begin{macrocode}
\cs_new:Nn \@@_mark:Nn {
  \prop_item:cn { \@@_marks:N #1 } { #2 }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_shift_x:Nn}
%   This function shifts the marks by a specified amount, without modifying
%   the box.
%    \begin{macrocode}
\cs_new:Nn \@@_shift_x:Nn {
  \prop_map_inline:cn { \@@_marks:N #1 } {
    \@@_set_mark:Nnn #1 { ##1 } { ##2 + #2 }
  }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_enlarge_conclusion:NN}
%   This function moves the left and right marks of the first tree so that
%   they are at least as far from the axis as they are in the second tree. 
%   For instance we get the following:
%   \begin{center}
%     \begin{tikzpicture}[y=-3ex]
%       \node (L1) at (1,0) {L}; \node (A1) at (2,0) {A}; \node (R1) at (4,0) {R};
%       \node (L2) at (0,1) {L}; \node (A2) at (2,1) {A}; \node (R2) at (3,1) {R};
%       \node (L3) at (0,2) {L}; \node (A3) at (2,2) {A}; \node (R3) at (4,2) {R};
%       \draw (L1) -- (A1) -- (R1);
%       \draw (L2) -- (A2) -- (R2);
%       \draw (L3) -- (A3) -- (R3);
%       \node[anchor=west] at (5,0) {box 1 before};
%       \node[anchor=west] at (5,1) {box 2 before};
%       \node[anchor=west] at (5,2) {box 1 after};
%     \end{tikzpicture}
%   \end{center}
%   The contents of the trees are unchanged.
%    \begin{macrocode}
\cs_new:Nn \@@_enlarge_conclusion:NN {
  \dim_set:Nn \l_tmpa_dim { \@@_mark:Nn #1 {axis}
    + \@@_mark:Nn #2 {left} - \@@_mark:Nn #2 {axis} }
  \dim_compare:nNnT { \l_tmpa_dim } < { \@@_mark:Nn #1 {left} } {
    \@@_set_mark:Nnn #1 {left} { \l_tmpa_dim } }
  \dim_set:Nn \l_tmpa_dim { \@@_mark:Nn #1 {axis}
    + \@@_mark:Nn #2 {right} - \@@_mark:Nn #2 {axis} }
  \dim_compare:nNnT { \l_tmpa_dim } > { \@@_mark:Nn #1 {right} } {
    \@@_set_mark:Nnn #1 {right} { \l_tmpa_dim } }
}
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Building blocks}
%
% \begin{macro}{\@@_make_simple:Nn}
%   Make a tree with explicit material in horizontal mode. Set the left and
%   right marks to extremal positions and set the axis in the middle.
%    \begin{macrocode}
\cs_new:Nn \@@_make_simple:Nn {
  \hbox_set:cn { \@@_box:N #1 } { #2 }
  \@@_set_mark:Nnn #1 { left } { 0pt }
  \@@_set_mark:Nnn #1 { axis } { \box_wd:c { \@@_box:N #1 } / 2 }
  \@@_set_mark:Nnn #1 { right } { \box_wd:c { \@@_box:N #1 } }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_make_split:Nnn}
%   Make a tree with explicit material in horizontal mode, split in two parts.
%   Set the left and right marks to extremal positions and set the axis
%   between the two parts.
%    \begin{macrocode}
\cs_new:Nn \@@_make_split:Nnn {
  \@@_set_mark:Nnn #1 { left } { 0pt }
  \hbox_set:cn { \@@_box:N #1 } { #2 }
  \@@_set_mark:Nnn #1 { axis } { \box_wd:c { \@@_box:N #1 } }
  \hbox_set:cn { \@@_box:N #1 } { \hbox_unpack:c { \@@_box:N #1 } #3 }
  \@@_set_mark:Nnn #1 { right } { \box_wd:c { \@@_box:N #1 } }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_make_vertical:Nnnn}
%   Make a tree with explicit material in vertical mode, using an explicit
%   width and axis.
%    \begin{macrocode}
\cs_new:Nn \@@_make_vertical:Nnnn {
  \@@_set_mark:Nnn #1 { left } { 0pt }
  \@@_set_mark:Nnn #1 { axis } { #2 }
  \@@_set_mark:Nnn #1 { right } { #3 }
  \vbox_set:cn { \@@_box:N #1 } {
    \dim_set:Nn \tex_hsize:D { \@@_mark:Nn #1 {right} }
    #4
  }
  \box_set_wd:cn { \@@_box:N #1 } { \@@_mark:Nn #1 {right} }
}
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Assembling boxes}
%
% \begin{macro}{\@@_extend:Nnnnn}
%   Extend a tree box. The marks are shifted so that alignment is preserved.
%   The arguments are dimensions for the left, top, right and bottom sides
%   respectively.
%    \begin{macrocode}
\cs_new:Nn \@@_extend:Nnnnn {
  \dim_compare:nNnF { #2 } = { 0pt } {
    \hbox_set:cn { \@@_box:N #1 } {
      \skip_horizontal:n { #2 }
      \box_use:c { \@@_box:N #1 }
    }
    \@@_shift_x:Nn #1 { #2 }
  }
  \box_set_ht:Nn #1 { \box_ht:c { \@@_box:N #1 } + #3 }
  \box_set_wd:Nn #1 { \box_wd:c { \@@_box:N #1 } + #4 }
  \box_set_dp:Nn #1 { \box_dp:c { \@@_box:N #1 } + #5 }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_append_right:NnN}
%   Append the contents of the second tree to the first one on the right, with
%   matching baselines. The marks of both trees are preserved. The middle
%   argument specifies the space to insert between boxes.
%    \begin{macrocode}
\cs_new:Nn \@@_append_right:NnN {
  \hbox_set:cn { \@@_box:N #1 } {
    \box_use:c { \@@_box:N #1 }
    \dim_compare:nNnF { #2 } = { 0pt } { \skip_horizontal:n { #2 } }
    \box_use:c { \@@_box:N #3 }
  }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_append_left:NnN}
%   Append the contents of the second tree to the first one on the left, with
%   matching baselines. The marks of the first tree are shifted accordingly.
%   The middle argument specifies the space to insert between boxes.
%    \begin{macrocode}
\cs_new:Nn \@@_append_left:NnN {
  \@@_shift_x:Nn #1 { \box_wd:c { \@@_box:N #3 } + #2 }
  \hbox_set:cn { \@@_box:N #1 } {
    \box_use:c { \@@_box:N #3 }
    \dim_compare:nNnF { #2 } = { 0pt } { \skip_horizontal:n { #2 } }
    \box_use:c { \@@_box:N #1 }
  }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_align:NN}
%   Shift one of two trees to the right so that their axes match. The marks of
%   the one that is shifted are updated accordingly.
%    \begin{macrocode}
\cs_new:Nn \@@_align:NN {
  \dim_set:Nn \l_tmpa_dim
    { \@@_mark:Nn #2 {axis} - \@@_mark:Nn #1 {axis} }
  \dim_compare:nNnTF \l_tmpa_dim < { 0pt } {
    \@@_extend:Nnnnn #2 { -\l_tmpa_dim } { 0pt } { 0pt } { 0pt }
  } {
    \@@_extend:Nnnnn #1 { \l_tmpa_dim } { 0pt } { 0pt } { 0pt }
  }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_append_above:NN}
%   Append the contents of the second tree above the first one, with matching
%   axes. The marks of the first tree are preserved.
%    \begin{macrocode}
\cs_new:Nn \@@_append_above:NN {
  \@@_align:NN #1 #2
  \vbox_set:cn  { \@@_box:N #1 } {
    \box_use:c { \@@_box:N #2 }
    \tex_prevdepth:D -1000pt
    \box_use:c { \@@_box:N #1 }
  }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_append_below:NN}
%   Append the contents of the second tree below the first one, with matching
%   axes. The marks of the first tree are preserved.
%    \begin{macrocode}
\cs_new:Nn \@@_append_below:NN {
  \@@_align:NN #1 #2
  \vbox_set_top:cn { \@@_box:N #1 } {
    \box_use:c { \@@_box:N #1 }
    \tex_prevdepth:D -1000pt
    \box_use:c { \@@_box:N #2 }
  }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_overlay:NN}
%   Append the second tree as an overlay over the first one, so that the
%   baselines and axes match. The bounding box of the result adjusts to
%   contain both trees.
%    \begin{macrocode}
\cs_new:Nn \@@_overlay:NN {
  \@@_align:NN #1 #2
  \hbox_set:cn { \@@_box:N #1 } {
    \hbox_overlap_right:n { \box_use:c { \@@_box:N #1 } }
    \box_use:c { \@@_box:N #2 }
    \dim_compare:nNnT
      { \box_wd:c { \@@_box:N #2 } } < { \box_wd:c { \@@_box:N #1 } }
      { \skip_horizontal:n
        { \box_wd:c { \@@_box:N #1 } - \box_wd:c { \@@_box:N #2 } } }
  }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_vcenter:N}
%   Shift the material in a tree vertically so that the height and depth are
%   equal (like \TeX's \cs{vcenter} but around the baseline).
%    \begin{macrocode}
\cs_new:Nn \@@_vcenter:N {
  \dim_set:Nn \l_tmpa_dim
    { ( \box_ht:c { \@@_box:N #1 } - \box_dp:c { \@@_box:N #1 } ) / 2 }
  \box_set_eq:Nc \l_tmpa_box { \@@_box:N #1 }
  \hbox_set:cn { \@@_box:N #1 }
    { \box_move_down:nn { \l_tmpa_dim } { \box_use:N \l_tmpa_box } }
}
%    \end{macrocode}
% \end{macro}
%
% \subsection{Making inferences}
%
% The following commands use the parameters defined at the beginning of the
% package for actually building proof trees using the commands defined above.
%
% \begin{macro}{\@@_append_vertical:NN}
%   Append the contents of the second tree above or below the first one,
%   depending on current settings. Axes are aligned and the marks of the first
%   tree are preserved.
%    \begin{macrocode}
\cs_new:Nn \@@_append_vertical:NN {
  \bool_if:NTF \l_@@_updown_bool
    { \@@_append_below:NN #1 #2 }
    { \@@_append_above:NN #1 #2 }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_make_rule_for:NNN}
%   Make a box containing an inference rule with labels, using the current
%   settings. The width and axis position are taken as those of the conclusion
%   of another tree box. The third argument is used as a temporary register
%   for building labels.
%    \begin{macrocode}
\cs_new:Nn \@@_make_rule_for:NNN {
%    \end{macrocode}
%   Build the rule.
%    \begin{macrocode}
  \@@_make_vertical:Nnnn #1
    { \@@_mark:Nn #2 {axis} - \@@_mark:Nn #2 {left} }
    { \@@_mark:Nn #2 {right} - \@@_mark:Nn #2 {left} }
    {
      \skip_vertical:N \l_@@_rule_margin_dim
      \tl_if_empty:NF { \l_@@_rule_code_tl } {
        \tl_use:N \l_@@_rule_code_tl
        \skip_vertical:N \l_@@_rule_margin_dim
      }
    }
  \@@_vcenter:N #1
%    \end{macrocode}
%   Append the left label.
%    \begin{macrocode}
  \tl_if_blank:VF \l_@@_left_label_tl {
    \@@_make_simple:Nn #3 {
      \box_move_down:nn { \l_@@_label_axis_dim } { \hbox:n {
        \cs_set_eq:NN \inserttext \l_@@_left_label_tl
        \tl_use:N \l_@@_left_label_template_tl
      } }
    }
    \box_set_ht:cn { \@@_box:N #3 } { 0pt }
    \box_set_dp:cn { \@@_box:N #3 } { 0pt }
    \@@_append_left:NnN
      \l_@@_c_box \l_@@_label_separation_dim \l_@@_d_box
  }
%    \end{macrocode}
%   Append the right label.
%    \begin{macrocode}
  \tl_if_blank:VF \l_@@_right_label_tl {
    \@@_make_simple:Nn #3 {
      \box_move_down:nn { \l_@@_label_axis_dim } { \hbox:n {
        \cs_set_eq:NN \inserttext \l_@@_right_label_tl
        \tl_use:N \l_@@_right_label_template_tl
      } }
    }
    \box_set_ht:cn { \@@_box:N #3 } { 0pt }
    \box_set_dp:cn { \@@_box:N #3 } { 0pt }
    \@@_append_right:NnN
      \l_@@_c_box \l_@@_label_separation_dim \l_@@_d_box
  }
}
%    \end{macrocode}
% \end{macro}
%
% \subsection{Stack-based interface}
%
% \subsubsection{The stack}
%
% Logically, box structures are stored on a stack. However, \TeX\ does not
% provide data structures for that and the grouping mechanism is not flexible
% enough, so we encode them using what we actually have. A stack for boxes is
% implemented using a global hbox \cs{g_@@_stack_box} that contains all the
% boxes successively. A sequence \cs{g_@@_stack_seq} is used to store the
% dimensions property lists textually. We maintain a counter
% \cs{g_@@_level_int} with the number of elements on the stack, for
% consistency checks.
%    \begin{macrocode}
\int_new:N \g_@@_level_int
\box_new:N \g_@@_stack_box
\seq_new:N \g_@@_stack_seq
%    \end{macrocode}
%
% \begin{macro}{\@@_clear_stack:}
%   Clear the stack.
%    \begin{macrocode}
\cs_new:Nn \@@_clear_stack: {
  \int_gset:Nn \g_@@_level_int { 0 }
  \hbox_gset:Nn \g_@@_stack_box { }
  \seq_gclear:N \g_@@_stack_seq
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_push:N}
%   Push the contents of a register on the stack.
%    \begin{macrocode}
\cs_new:Nn \@@_push:N {
  \int_gincr:N \g_@@_level_int
  \hbox_gset:Nn \g_@@_stack_box
    { \hbox_unpack:N \g_@@_stack_box \box_use:c { \@@_box:N #1 } }
  \seq_gput_left:Nv \g_@@_stack_seq
    { \@@_marks:N #1 }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_pop:N}
%   Pop the value from the top of the stack into a register.
%    \begin{macrocode}
\cs_new:Nn \@@_pop:N {
  \int_compare:nNnTF { \g_@@_level_int } > { 0 } {
    \int_gdecr:N \g_@@_level_int
    \hbox_gset:Nn \g_@@_stack_box {
      \hbox_unpack:N \g_@@_stack_box
      \box_gset_to_last:N \g_tmpa_box
    }
    \box_set_eq_drop:cN { \@@_box:N #1 } \g_tmpa_box
    \seq_gpop_left:NN \g_@@_stack_seq \l_tmpa_tl
    \tl_set_eq:cN { \@@_marks:N #1 } \l_tmpa_tl
  } {
    \PackageError{ebproof}{Missing~premiss~in~a~proof~tree}{}
    \@@_clear:N #1
  }
}
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Assembling trees}
%
%    \begin{macrocode}
\@@_new:N \l_@@_a_box
\@@_new:N \l_@@_b_box
\@@_new:N \l_@@_c_box
\@@_new:N \l_@@_d_box
%    \end{macrocode}
%
% \begin{macro}{\@@_join_horizontal:n}
%   Join horizontally a number of elements at the top of the stack. If several
%   trees are joined, use the left mark of the left tree, the right mark of
%   the right tree and set the axis in the middle of these marks.
%    \begin{macrocode}
\cs_new:Nn \@@_join_horizontal:n {
  \int_case:nnF { #1 } {
  { 0 } {
    \group_begin:
    \@@_clear:N \l_@@_a_box
    \@@_push:N \l_@@_a_box
    \group_end:
  }
  { 1 } { }
  } {
    \group_begin:
    \@@_pop:N \l_@@_a_box
    \prg_replicate:nn { #1 - 1 } {
      \@@_pop:N \l_@@_b_box
      \@@_append_left:NnN
        \l_@@_a_box \l_@@_separation_dim \l_@@_b_box
    }
    \@@_set_mark:Nnn \l_@@_a_box { left }
      { \@@_mark:Nn \l_@@_b_box { left } }
    \@@_set_mark:Nnn \l_@@_a_box { axis }
      { ( \@@_mark:Nn \l_@@_a_box { left }
        + \@@_mark:Nn \l_@@_a_box { right } ) / 2 }
    \@@_push:N \l_@@_a_box
    \group_end:
  }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_join_vertical:}
%   Join vertically the two elements at the top of the stack, with a
%   horizontal rule of the appropriate size.
%    \begin{macrocode}
\cs_new:Nn \@@_join_vertical: {
  \group_begin:
  \@@_pop:N \l_@@_a_box
  \@@_pop:N \l_@@_b_box
  \@@_enlarge_conclusion:NN \l_@@_b_box \l_@@_a_box
  \@@_make_rule_for:NNN \l_@@_c_box \l_@@_b_box
    \l_@@_d_box
  \@@_append_vertical:NN \l_@@_a_box \l_@@_c_box
  \@@_append_vertical:NN \l_@@_a_box \l_@@_b_box
  \@@_push:N \l_@@_a_box
  \group_end:
}
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{High-level commands}
%
% \begin{macro}{\@@_statement_parse:w}
%   An auxiliary function for parsing the argument in
%   \cs{@@_push_statement:n}.
%    \begin{macrocode}
\cs_new:Npn \@@_statement_parse:w #1 & #2 & #3 \q_stop {
  \tl_if_empty:nTF { #3 } {
    \@@_make_simple:Nn \l_@@_a_box
      { \cs_set:Npn \inserttext { #1 } \tl_use:N \l_@@_template_tl }
  } {
    \@@_make_split:Nnn \l_@@_a_box
      { \cs_set:Npn \inserttext { #1 } \tl_use:N \l_@@_left_template_tl }
      { \cs_set:Npn \inserttext { #2 } \tl_use:N \l_@@_right_template_tl }
  }
  \@@_push:N \l_@@_a_box
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_push_statement:n}
%   Push a box with default formatting, using explicit alignment if the code
%   contains a |&| character
%    \begin{macrocode}
\cs_new:Nn \@@_push_statement:n {
  \@@_statement_parse:w #1 & & \q_stop
}
%    \end{macrocode}
% \end{macro}
%
% \subsection{Document interface}
%
% \subsubsection{Functions to define statements}
%
% The \cs{g_@@_statements_seq} variable contains the list of all defined
% statements. For each statement |X|, there is a document command \cs{ebproofX}
% and the alias \cs{X} is defined when entering a |prooftree| environment.
%    \begin{macrocode}
\seq_new:N \g_@@_statements_seq
%    \end{macrocode}
%
% \begin{macro}{\@@_setup_statements:}
%   Install the aliases for statements, saving the original value of the control
%   sequences.
%    \begin{macrocode}
\cs_new:Nn \@@_setup_statements: {
  \seq_map_inline:Nn \g_@@_statements_seq {
    \cs_set_eq:cc { ebproof_saved_ ##1 } { ##1 }
    \cs_set_eq:cc { ##1 } { ebproof ##1 }
  }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_restore_statements:}
%   Restore the saved meanings of the control sequences. This is useful when
%   interpreting user-provided code in statement arguments. The meanings are
%   automatically restored when leaving a |prooftree| environment because of
%   grouping.
%    \begin{macrocode}
\cs_new:Nn \@@_restore_statements: {
  \seq_map_inline:Nn \g_@@_statements_seq {
    \cs_set_eq:cc { ##1 } { ebproof_saved_ ##1 }
  }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_new_statement:nnn}
%   Define a new statement. The first argument is the name, the second one is
%   an argument specifier as used by |xparse| and the third one is the body of
%   the command.
%    \begin{macrocode}
\cs_new:Nn \@@_new_statement:nnn {
  \exp_args:Nc \NewDocumentCommand { ebproof#1 }{ #2 } { #3 }
  \seq_gput_right:Nn \g_@@_statements_seq { #1 }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_new_deprecated_statement:nnnn}
%   Define a deprecated statement. The syntax is the same as above except that
%   an extra argument in third position indicates what should be used instead.
%   The effect is the same except that a warning message is issued the first
%   time the statement is used.
%    \begin{macrocode}
\cs_new:Nn \@@_new_deprecated_statement:nnnn {
  \cs_new:cpn { ebproof_#1_warning: } {
    \PackageWarning { ebproof } { \token_to_str:c{#1}~is~deprecated,~#3 }
    \cs_gset:cn { ebproof_#1_warning: } { }
  }
  \@@_new_statement:nnn { #1 } { #2 }
    { \use:c { ebproof_#1_warning: } #4 }
}
%    \end{macrocode}
% \end{macro}
%
%
% \subsubsection{Basic commands}
%
% \begin{macro}{\ebproofset,\set}
%   This is a simple wrapper around \cs{keys_set:nn}.
%    \begin{macrocode}
\@@_new_statement:nnn { set } { m } {
  \keys_set:nn { ebproof } { #1 }
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\hypo}
%   This is mostly a wrapper around \cs{ebproof_push_statement:n}, with
%   material to handle options and the statements macros.
%    \begin{macrocode}
\@@_new_statement:nnn { hypo } { O{} m } {
  \group_begin:
  \@@_restore_statements:
  \keys_set:nn { ebproof } { #1 }
  \@@_push_statement:n { #2 }
  \group_end:
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\infer}
%   This is a bit more involved than \cs{hypo} because we have to handle rule
%   style options and joining.
%    \begin{macrocode}
\@@_new_statement:nnn { infer } { O{} m O{} m } {
  \group_begin:
  \@@_restore_statements:
  \keys_set_known:nnN { ebproof / rule~style } { #1 } \l_tmpa_tl
  \keys_set:nV { ebproof } \l_tmpa_tl
  \tl_set:Nn \l_@@_right_label_tl { #3 }
  \@@_join_horizontal:n { #2 }
  \@@_push_statement:n { #4 }
  \@@_join_vertical:
  \group_end:
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\ellipsis}
%   An ellipsis is made by hand using vertical leaders to render the dots
%   after rendering the label.
%    \begin{macrocode}
\@@_new_statement:nnn { ellipsis } { m m } {
  \group_begin:
  \@@_restore_statements:
  \tl_clear:N \l_@@_rule_code_tl
  \@@_make_split:Nnn \l_@@_a_box { } {
    \vbox_set:Nn \l_tmpa_box {
      \skip_vertical:n { 1.2ex }
      \hbox:n { \tex_ignorespaces:D #1 }
      \skip_vertical:n { 1.2ex }
    }
    \vbox_to_ht:nn { \box_ht:N \l_tmpa_box } {
      \tex_xleaders:D \vbox_to_ht:nn { .8ex }
        { \tex_vss:D \hbox:n { . } \tex_vss:D }
      \tex_vfill:D
    }
    \hbox_overlap_right:n { ~ \box_use:N \l_tmpa_box }
  }
  \@@_push:N \l_@@_a_box
  \@@_join_vertical:
  \@@_push_statement:n {#2}
  \@@_join_vertical:
  \group_end:
}
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Modifying trees}
%
% \begin{macro}{\rewrite}
%   Rewrite the box at the top of the stack while preserving its dimensions an
%   marks. The code is typeset in horizontal mode, with control sequences to
%   access the original box and its marks.
%    \begin{macrocode}
\@@_new_statement:nnn { rewrite } { m } {
  \group_begin:
  \@@_restore_statements:
  \@@_pop:N \l_@@_a_box
  \box_set_eq:Nc \l_tmpa_box { \@@_box:N \l_@@_a_box }
  \hbox_set:Nn \l_tmpb_box {
    \cs_set_eq:NN \treebox \l_tmpa_box
    \cs_set:Npn \treemark { \@@_mark:Nn \l_@@_a_box }
    { #1 }
  }
  \box_set_wd:Nn \l_tmpb_box { \box_wd:c { \@@_box:N \l_@@_a_box } }
  \box_set_ht:Nn \l_tmpb_box { \box_ht:c { \@@_box:N \l_@@_a_box } }
  \box_set_dp:Nn \l_tmpb_box { \box_dp:c { \@@_box:N \l_@@_a_box } }
  \box_set_eq:cN { \@@_box:N \l_@@_a_box } \l_tmpb_box
  \@@_push:N \l_@@_a_box
  \group_end:
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\delims}
%   Insert \cs{left} and \cs{right} delimiters without changing the alignment.
%    \begin{macrocode}
\@@_new_statement:nnn { delims } { m m } {
  \group_begin:
  \@@_restore_statements:
  \@@_pop:N \l_@@_a_box
  \hbox_set:Nn \l_tmpa_box
    { $ \tex_vcenter:D { \box_use:c { \@@_box:N \l_@@_a_box } } $ }
  \dim_set:Nn \l_tmpa_dim
    { \box_ht:N \l_tmpa_box - \box_ht:c { \@@_box:N \l_@@_a_box } }
  \hbox_set:cn { \@@_box:N \l_@@_a_box } {
    $ #1 \tex_vrule:D
      height \box_ht:N \l_tmpa_box depth \box_dp:N \l_tmpa_box width 0pt
    \tex_right:D . $
  }
  \@@_shift_x:Nn \l_@@_a_box
    { \box_wd:c { \@@_box:N \l_@@_a_box } }
  \hbox_set:cn { \@@_box:N \l_@@_a_box } {
    \hbox_unpack:c { \@@_box:N \l_@@_a_box }
    $ \tex_left:D . \box_use:N \l_tmpa_box #2 $
  }
  \hbox_set:cn { \@@_box:N \l_@@_a_box }
    { \box_move_down:nn { \l_tmpa_dim }
        { \box_use:c { \@@_box:N \l_@@_a_box } } }
  \@@_push:N \l_@@_a_box
  \group_end:
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\overlay}
%   Pop two trees and append the second tree as an overlay over the first one,
%   so that the baselines and axes match. The bounding box of the result
%   adjusts to contain both trees.
%    \begin{macrocode}
\@@_new_statement:nnn { overlay } { } {
  \group_begin:
  \@@_pop:N \l_@@_a_box
  \@@_pop:N \l_@@_b_box
  \@@_overlay:NN \l_@@_a_box \l_@@_b_box
  \@@_push:N \l_@@_a_box
  \group_end:
}
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Deprecated statements}
%
% These statements were defined in versions 1.x of the package, they are
% preserved for temporary upwards compatibility and will be removed in a
% future version.
%    \begin{macrocode}
\@@_new_deprecated_statement:nnnn { Alter } { m }
  { use~\token_to_str:c{rewrite}~instead } { \ebproofrewrite{ #1 \box\treebox } }
\@@_new_deprecated_statement:nnnn { Delims } { }
  { use~\token_to_str:c{delims}~instead } { \ebproofdelims }
\@@_new_deprecated_statement:nnnn { Ellipsis } { }
  { use~\token_to_str:c{ellipsis}~instead } { \ebproofellipsis }
\@@_new_deprecated_statement:nnnn { Hypo } { }
  { use~\token_to_str:c{hypo}~instead } { \ebproofhypo }
\@@_new_deprecated_statement:nnnn { Infer } { }
  { use~\token_to_str:c{infer}~instead } { \ebproofinfer }
%    \end{macrocode}
%
%
% \subsubsection{Environment interface}
%
% The stack is initialised globally. The \env{prooftree} environment does not
% clear the stack, instead it saves the initial level in order to check that
% statements are properly balanced. This allows for nested uses of the
% environment, if it ever happens to be useful.
%
%    \begin{macrocode}
\@@_clear_stack:
\tl_new:N \l_@@_start_level_tl
%    \end{macrocode}
%
% \begin{macro}{prooftree,prooftree*}
%   The \env{prooftree} environment.
%    \begin{macrocode}
\NewDocumentEnvironment { prooftree } { s O{} } {
  \group_align_safe_begin:
  \keys_set_known:nnN { ebproof / proof~style } { #2 } \l_tmpa_tl
  \keys_set:nV { ebproof } \l_tmpa_tl
  \tl_set:Nx \l_@@_start_level_tl { \int_use:N \g_@@_level_int }
  \vbox_set:Nw \l_tmpa_box
  \@@_setup_statements:
} {
  \vbox_set_end:
  \@@_pop:N \l_@@_a_box
  \int_compare:nNnF { \g_@@_level_int } = { \tl_use:N \l_@@_start_level_tl } {
    \PackageError{ebproof}{Malformed~proof~tree}{
      Some~hypotheses~were~declared~but~not~used~in~this~tree.}
  }
  \IfBooleanTF { #1 } {
    \[ \box_use:c { \@@_box:N \l_@@_a_box } \]
    \ignorespacesafterend
  } {
    \hbox_unpack:N \c_empty_box
    \bool_if:NTF \l_@@_center_bool {
      \hbox:n { $ \tex_vcenter:D { \box_use:c { \@@_box:N \l_@@_a_box } } $ }
    } {
      \box_use:c { \@@_box:N \l_@@_a_box }
    }
  }
  \group_align_safe_end:
}
%    \end{macrocode}
% A trick for the starred version:
%    \begin{macrocode}
\cs_new:cpn { prooftree* } { \prooftree* }
\cs_new:cpn { endprooftree* } { \endprooftree }
%    \end{macrocode}
% \end{macro}
%
%    \begin{macrocode}
%</package>
%    \end{macrocode}
%
% \end{implementation}