File: entries.tex

package info (click to toggle)
hol88 2.02.19940316-35
  • links: PTS
  • area: main
  • in suites: buster
  • size: 65,988 kB
  • ctags: 21,623
  • sloc: ml: 199,939; ansic: 9,666; sh: 7,118; makefile: 6,095; lisp: 2,747; yacc: 894; sed: 201; cpp: 87; awk: 5
file content (1911 lines) | stat: -rw-r--r-- 65,440 bytes parent folder | download | duplicates (7)
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
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
\chapter{ML Functions in the {\tt res\_quan} Library}\label{entries}\input{entries-intro}\DOC{COND\_REWR\_CANON}

\TYPE {\small\verb%COND_REWR_CANON : thm -> thm%}\egroup

\SYNOPSIS
Transform a theorem into a form accepted by {\small\verb%COND_REWR_TAC%}.

\DESCRIBE
{\small\verb%COND_REWR_CANON%} transforms a theorem into a form accepted by {\small\verb%COND_REWR_TAC%}.
The input theorem should be an implication of the following form
{\par\samepage\setseps\small
\begin{verbatim}
   !x1 ... xn. P1[xi] ==> ... ==> !y1 ... ym. Pr[xi,yi] ==>
     (!z1 ... zk. u[xi,yi,zi] = v[xi,yi,zi])
\end{verbatim}
}
\noindent where each antecedent {\small\verb%Pi%} itself may be a conjunction or
disjunction. The output theorem will have all universal quantifications
moved to the outer most level with possible renaming to prevent
variable capture, and have all antecedents which are a conjunction
transformed to implications. The output theorem will be in the
following form
{\par\samepage\setseps\small
\begin{verbatim}
   !x1 ... xn y1 ... ym z1 ... zk. 
    P11[xi] ==> ... ==> P1p[xi] ==> ... ==>
     Pr1[xi,yi] ==> ... ==> Prq[x1,yi] ==> (u[xi,yi,zi] = v[xi,yi,zi])
\end{verbatim}
}

\FAILURE
This function fails if the input theorem is not in the correct form.

\EXAMPLE
{\small\verb%COND_REWR_CANON%} transforms the built-in theorem {\small\verb%CANCL_SUB%} into the
form for conditional rewriting:
{\par\samepage\setseps\small
\begin{verbatim}
   #COND_REWR_CANON CANCEL_SUB;;
   Theorem CANCEL_SUB autoloading from theory `arithmetic` ...
   CANCEL_SUB = |- !p n m. p <= n /\ p <= m ==> ((n - p = m - p) = (n = m))

   |- !p n m. p <= n ==> p <= m ==> ((n - p = m - p) = (n = m))
\end{verbatim}
}

\SEEALSO
COND_REWRITE1_TAC, COND_REWR_TAC, COND_REWRITE1_CONV, COND_REWR_CONV, 
search_top_down.

\ENDDOC

\DOC{COND\_REWR\_CONV}

{\small
\begin{verbatim}
COND_REWR_CONV : ((term -> term ->
 ((term # term) list # (type # type) list) list) -> thm -> conv)
\end{verbatim}
}\egroup

\SYNOPSIS
A lower level conversion implementing simple conditional rewriting.

\DESCRIBE
{\small\verb%COND_REWR_CONV%} is one of the basic building blocks for the
implementation of the simple conditional rewriting conversions in the
HOL system. In particular, the conditional term replacement or
rewriting done by all the conditional 
rewriting conversions in this library is ultimately done by applications of
{\small\verb%COND_REWR_CONV%}.  The description given here for {\small\verb%COND_REWR_CONV%} may
therefore be taken as a specification of the atomic action of
replacing equals by equals in a term under certain conditions that
are used in all these higher level conditional rewriting conversions.

The first argument to {\small\verb%COND_REWR_CONV%} is expected to be a function
which returns a list of matches. Each of these matches is in the form
of the value returned by the built-in function {\small\verb%match%}. It is used to
search the input term for instances which may be rewritten.

The second argument to {\small\verb%COND_REWR_CONV%} is expected to be an
implicative theorem in the following form:
{\par\samepage\setseps\small
\begin{verbatim}
   A |- !x1 ... xn. P1 ==> ... Pm ==> (Q[x1,...,xn] = R[x1,...,xn])
\end{verbatim}
}
\noindent where {\small\verb%x1%}, ..., {\small\verb%xn%} are all the variables that occur free in the
left hand side of the conclusion of the theorem but do not occur free in the
assumptions. 

The last argument to {\small\verb%COND_REWR_CONV%} is the term to be rewritten.

If {\small\verb%fn%} is a function and {\small\verb%th%} is an implicative theorem of the kind
shown above, then {\small\verb%COND_REWR_CONV fn th%} will be a conversion.
When applying to a term {\small\verb%tm%}, it will return a theorem
{\par\samepage\setseps\small
\begin{verbatim}
   P1', ..., Pm' |- tm = tm[R'/Q']
\end{verbatim}
}
\noindent if evaluating {\small\verb%fn Q[x1,...,xn] tm%} returns a
non-empty list of matches.
The assumptions of the resulting theorem are instances of the
antecedents of the input theorem {\small\verb%th%}. The right hand side of the
equation is obtained by rewriting the input term {\small\verb%tm%} with instances of
the conclusion of the input theorem.

\FAILURE
{\small\verb%COND_REWR_CONV fn th%} fails if {\small\verb%th%} is not an implication of the form
described above.  If {\small\verb%th%} is such an equation, but the function {\small\verb%fn%}
returns a null list of matches, or the function {\small\verb%fn%} returns a
non-empty list of matches, but the term or type instantiation fails.

\EXAMPLE
The following example illustrates a straightforward use of {\small\verb%COND_REWR_CONV%}.
We use the built-in theorem {\small\verb%LESS_MOD%} as the input theorem, and the function
{\small\verb%search_top_down%} as the search function.
{\par\samepage\setseps\small
\begin{verbatim}
   #LESS_MOD;;
   Theorem LESS_MOD autoloading from theory `arithmetic` ...
   LESS_MOD = |- !n k. k < n ==> (k MOD n = k)

   |- !n k. k < n ==> (k MOD n = k)

   #search_top_down;;
   - : (term -> term -> ((term # term) list # (type # type) list) list)

   #COND_REWR_CONV search_top_down LESS_MOD "2 MOD 3";;
   2 < 3 |- 2 MOD 3 = 2
\end{verbatim}
}

\SEEALSO
COND_REWR_TAC, COND_REWRITE1_TAC, COND_REWRITE1_CONV, COND_REWR_CANON,
search_top_down.

\ENDDOC
\DOC{COND\_REWRITE1\_CONV}

\TYPE {\small\verb%COND_REWRITE1_CONV : (thm list -> thm -> conv)%}\egroup

\SYNOPSIS
A simple conditional rewriting conversion.

\DESCRIBE
{\small\verb%COND_REWRITE1_CONV%} is a front end of the conditional rewriting
conversion {\small\verb%COND_REWR_CONV%}. The input theorem should be in the following form
{\par\samepage\setseps\small
\begin{verbatim}
   A |- !x11 ... . P1 ==> ... !xm1 ... . Pm ==> (!x ... . Q = R)
\end{verbatim}
}
\noindent where each antecedent {\small\verb%Pi%} itself may be a conjunction or disjunction.
This theorem is transformed to a standard form expected by
{\small\verb%COND_REWR_CONV%} which carries out the actual rewriting.
The transformation is performed by {\small\verb%COND_REWR_CANON%}. The search function
passed to {\small\verb%COND_REWR_CONV%} is {\small\verb%search_top_down%}. The effect of applying
the conversion {\small\verb%COND_REWRITE1_CONV ths th%} to a term {\small\verb%tm%} is to derive a theorem
{\par\samepage\setseps\small
\begin{verbatim}
  A' |- tm = tm[R'/Q']
\end{verbatim}
}
\noindent where the right hand side of the equation is obtained by rewriting
the input term {\small\verb%tm%} with an instance of the conclusion of the input theorem. 
The theorems in the list {\small\verb%ths%} are used to discharge the assumptions
generated from the antecedents of the input theorem. 

\FAILURE
{\small\verb%COND_REWRITE1_CONV ths th%}  fails if {\small\verb%th%} cannot be transformed into the
required form by {\small\verb%COND_REWR_CANON%}. Otherwise, it fails if no match
is found or the theorem cannot be instantiated.

\EXAMPLE
The following example illustrates a straightforward use of {\small\verb%COND_REWRITE1_CONV%}.
We use the built-in theorem {\small\verb%LESS_MOD%} as the input theorem.
{\par\samepage\setseps\small
\begin{verbatim}
   #LESS_MOD;;
   Theorem LESS_MOD autoloading from theory `arithmetic` ...
   LESS_MOD = |- !n k. k < n ==> (k MOD n = k)

   |- !n k. k < n ==> (k MOD n = k)

   #COND_REWRITE1_CONV [] LESS_MOD "2 MOD 3";;
   2 < 3 |- 2 MOD 3 = 2

   #let less_2_3 = REWRITE_RULE[LESS_MONO_EQ;LESS_0]
   #(REDEPTH_CONV num_CONV "2 < 3");;
   less_2_3 = |- 2 < 3

   #COND_REWRITE1_CONV [less_2_3] LESS_MOD "2 MOD 3";;
   |- 2 MOD 3 = 2

\end{verbatim}
}
\noindent In the first example, an empty theorem list is supplied to
{\small\verb%COND_REWRITE1_CONV%} so the resulting theorem has an assumption
{\small\verb%2 < 3%}. In the second example, a list containing a theorem {\small\verb%|- 2 < 3%}
is supplied, the resulting theorem has no assumptions.

\SEEALSO
COND_REWR_TAC, COND_REWRITE1_TAC, COND_REWR_CONV, 
COND_REWR_CANON, search_top_down.

\ENDDOC

\DOC{COND\_REWRITE1\_TAC}

\TYPE {\small\verb%COND_REWRITE1_TAC : thm_tactic%}\egroup

\SYNOPSIS
A simple conditional rewriting tactic.

\DESCRIBE
{\small\verb%COND_REWRITE1_TAC%} is a front end of the conditional rewriting
tactic {\small\verb%COND_REWR_TAC%}. The input theorem should be in the following form
{\par\samepage\setseps\small
\begin{verbatim}
   A |- !x11 ... . P1 ==> ... !xm1 ... . Pm ==> (!x ... . Q = R)
\end{verbatim}
}
\noindent where each antecedent {\small\verb%Pi%} itself may be a conjunction or disjunction.
This theorem is transformed to a standard form expected by
{\small\verb%COND_REWR_TAC%} which carries out the actual rewriting.
The transformation is performed by {\small\verb%COND_REWR_CANON%}. The search function
passed to {\small\verb%COND_REWR_TAC%} is {\small\verb%search_top_down%}. The effect of applying
this tactic is to substitute into the goal instances of the right hand
side of the conclusion of the input theorem {\small\verb%Ri'%} for the
corresponding instances of the left hand side. The search is top-down
left-to-right. All matches found by the search function are
substituted. New subgoals corresponding to the instances of the
antecedents which do not appear in the assumption of the original goal
are created. See manual page of {\small\verb%COND_REWR_TAC%} for details of how the
instantiation and substitution are done.

\FAILURE
{\small\verb%COND_REWRITE1_TAC th%}  fails if {\small\verb%th%} cannot be transformed into the
required form by the function {\small\verb%COND_REWR_CANON%}. Otherwise, it fails if no match
is found or the theorem cannot be instantiated.

\EXAMPLE
The following example illustrates a straightforward use of {\small\verb%COND_REWRITE1_TAC%}.
We use the built-in theorem {\small\verb%LESS_MOD%} as the input theorem.
{\par\samepage\setseps\small
\begin{verbatim}
   #LESS_MOD;;
   Theorem LESS_MOD autoloading from theory `arithmetic` ...
   LESS_MOD = |- !n k. k < n ==> (k MOD n = k)

   |- !n k. k < n ==> (k MOD n = k)
\end{verbatim}
}
\noindent We set up a goal
{\par\samepage\setseps\small
\begin{verbatim}
   #g"2 MOD 3 = 2";;
   "2 MOD 3 = 2"

   () : void
\end{verbatim}
}
\noindent and then apply the tactic
{\par\samepage\setseps\small
\begin{verbatim}
   #e(COND_REWRITE1_TAC LESS_MOD);;
   OK..
   2 subgoals
   "2 = 2"
       [ "2 < 3" ]

   "2 < 3"

   () : void
\end{verbatim}
}

\SEEALSO
COND_REWR_TAC, COND_REWRITE1_CONV, COND_REWR_CONV, 
COND_REWR_CANON, search_top_down.

\ENDDOC

\DOC{COND\_REWR\_TAC}

{\small
\begin{verbatim}
COND_REWR_TAC : ((term -> term ->
 ((term # term) list # (type # type) list) list) -> thm_tactic)
\end{verbatim}
}\egroup

\SYNOPSIS
A lower level tactic used to implement simple conditional rewriting tactic.

\DESCRIBE
{\small\verb%COND_REWR_TAC%} is one of the basic building blocks for the
implementation of conditional rewriting in the HOL system. In
particular, the conditional term replacement or rewriting done by all
the built-in  conditional
rewriting tactics is ultimately done by applications of
{\small\verb%COND_REWR_TAC%}.  The description given here for {\small\verb%COND_REWR_TAC%} may
therefore be taken as a specification of the atomic action of
replacing equals by equals in the goal under certain conditions that
aare used in all these higher level conditional rewriting tactics.

The first argument to {\small\verb%COND_REWR_TAC%} is expected to be a function
which returns a list of matches. Each of these matches is in the form
of the value returned by the built-in function {\small\verb%match%}. It is used to
search the goal for instances which may be rewritten.

The second argument to {\small\verb%COND_REWR_TAC%} is expected to be an implicative theorem
in the following form:
{\par\samepage\setseps\small
\begin{verbatim}
   A |- !x1 ... xn. P1 ==> ... Pm ==> (Q[x1,...,xn] = R[x1,...,xn])
\end{verbatim}
}
\noindent where {\small\verb%x1%}, ..., {\small\verb%xn%} are all the variables that occur free in the
left-hand side of the conclusion of the theorem but do not occur free in the
assumptions. 

If {\small\verb%fn%} is a function and {\small\verb%th%} is an implicative theorem of the kind
shown above, then {\small\verb%COND_REWR_TAC fn th%} will be a tactic which returns
a list of subgoals if evaluating 
{\par\samepage\setseps\small
\begin{verbatim}
   fn Q[x1,...,xn] gl
\end{verbatim}
}
\noindent returns a non-empty list of matches when applied to a goal {\small\verb%(asm,gl)%}.

Let {\small\verb%ml%} be the match list returned by evaluating {\small\verb%fn Q[x1,...,xn]
gl%}. Each element in this list is in the form of
{\par\samepage\setseps\small
\begin{verbatim}
   ([(e1,x1);...;(ep,xp)], [(ty1,vty1);...;(tyq,vtyq)])
\end{verbatim}
}
\noindent which specifies the term and type instantiations of the input theorem
{\small\verb%th%}. Either the term pair list or the type pair list may be empty. In
the case that both lists are empty, an exact match is found, i.e., no
instantiation is required. If {\small\verb%ml%} is an empty list, no match has been
found and the tactic will fail.

For each match in {\small\verb%ml%}, {\small\verb%COND_REWR_TAC%} will perform the following:
1) instantiate the input theorem {\small\verb%th%} to get
{\par\samepage\setseps\small
\begin{verbatim}
   th' = A |- P1' ==> ... ==> Pm' ==> (Q' = R')
\end{verbatim}
}
\noindent where the primed subterms are instances of the corresponding
unprimed subterms obtained by applying {\small\verb%INST_TYPE%} with
{\small\verb%[(ty1,vty1);...;(tyq,vtyq)]%} and then {\small\verb%INST%} with {\small\verb%[(e1,x1);...;(ep,xp)]%};
2) search the assumption list {\small\verb%asm%} for occurrences of any antecedents
{\small\verb%P1'%}, ..., {\small\verb%Pm'%};
3) if all antecedents appear in {\small\verb%asm%}, the goal {\small\verb%gl%} is reduced to
{\small\verb%gl'%} by substituting {\small\verb%R'%} for each free occurrence of {\small\verb%Q'%}, otherwise,
in addition to the substitution, all antecedents which do not appear
in {\small\verb%asm%} are added to it and new 
subgoals corresponding to these antecedents are created. For example,
if {\small\verb%Pk'%}, ..., {\small\verb%Pm'%} do not appear in {\small\verb%asm%}, the following subgoals
are returned:
{\par\samepage\setseps\small
\begin{verbatim}
   asm ?- Pk'  ...  asm ?- Pm'   {asm,Pk',...,Pm'} ?- gl'
\end{verbatim}
}

If {\small\verb%COND_REWR_TAC%} is given a theorem {\small\verb%th%}:
{\par\samepage\setseps\small
\begin{verbatim}
   A |- !x1 ... xn y1 ... yk. P1 ==> ... ==> Pm ==> (Q = R)
\end{verbatim}
}
\noindent where the variables {\small\verb%y1%}, ..., {\small\verb%ym%} do not occur free in the
left-hand side of the conclusion {\small\verb%Q%} but they do occur free in the
antecedents, then, when carrying out Step 2 described 
above, {\small\verb%COND_REWR_TAC%} will attempt to find instantiations for these
variables from the assumption {\small\verb%asm%}. For example, if {\small\verb%x1%} and {\small\verb%y1%}
occur free in {\small\verb%P1%}, and a match is found in which {\small\verb%e1%} is an
instantiation of {\small\verb%x1%}, then {\small\verb%P1'%} will become {\small\verb%P1[e1/x1, y1]%}. If a term
{\small\verb%P1'' = P1[e1,e1'/x1,y1]%} appears in  {\small\verb%asm%}, {\small\verb%th'%} is instantiated  with
{\small\verb%(e1', y1)%} to get
{\par\samepage\setseps\small
\begin{verbatim}
   th'' = A |- P1'' ==> ... ==> Pm'' ==> (Q' = R'')
\end{verbatim}
}
\noindent then {\small\verb%R''%} is substituted into {\small\verb%gl%} for all free occurrences
of {\small\verb%Q'%}. If no consistent instantiation is found, then {\small\verb%P1'%} which
contains the uninstantiated variable {\small\verb%y1%} will become one of the new
subgoals. In such a case, the user has no control over the
choice of the variable {\small\verb%yi%}.

\FAILURE
{\small\verb%COND_REWR_TAC fn th%} fails if {\small\verb%th%} is not an implication of the form
described above.  If {\small\verb%th%} is such an equation, but the function {\small\verb%fn%}
returns a null list of matches, or the function {\small\verb%fn%} returns a
non-empty list of matches, but the term or type instantiation fails.

\EXAMPLE
The following example illustrates a straightforward use of {\small\verb%COND_REWR_TAC%}.
We use the built-in theorem {\small\verb%LESS_MOD%} as the input theorem, and the function
{\small\verb%search_top_down%} as the search function.
{\par\samepage\setseps\small
\begin{verbatim}
   #LESS_MOD;;
   Theorem LESS_MOD autoloading from theory `arithmetic` ...
   LESS_MOD = |- !n k. k < n ==> (k MOD n = k)

   |- !n k. k < n ==> (k MOD n = k)

   #search_top_down;;
   - : (term -> term -> ((term # term) list # (type # type) list) list)
\end{verbatim}
}
\noindent We set up a goal
{\par\samepage\setseps\small
\begin{verbatim}
   #g"2 MOD 3 = 2";;
   "2 MOD 3 = 2"

   () : void
\end{verbatim}
}
\noindent and then apply the tactic
{\par\samepage\setseps\small
\begin{verbatim}
   #e(COND_REWR_TAC search_top_down LESS_MOD);;
   OK..
   2 subgoals
   "2 = 2"
       [ "2 < 3" ]

   "2 < 3"

    () : void
\end{verbatim}
}

\SEEALSO
COND_REWRITE1_TAC, COND_REWRITE1_CONV, COND_REWR_CONV, COND_REWR_CANON,
search_top_down.

\ENDDOC
\DOC{dest\_resq\_abstract}

\TYPE {\small\verb%dest_resq_abstract : (term -> (term # term # term))%}\egroup

\SYNOPSIS
Breaks apart a restricted abstract term into
the quantified variable, predicate and body.

\DESCRIBE
{\small\verb%dest_resq_abstract%} is a term destructor for restricted abstraction:
{\par\samepage\setseps\small
\begin{verbatim}
   dest_resq_abstract "\var::P. t"
\end{verbatim}
}
\noindent returns {\small\verb%("var","P","t")%}.

\FAILURE
Fails with {\small\verb%dest_resq_abstract%} if the term is not a restricted
abstraction.

\SEEALSO
mk_resq_abstract, is_resq_abstract, strip_resq_abstract.

\ENDDOC
\DOC{dest\_resq\_exists}

\TYPE {\small\verb%dest_resq_exists : (term -> (term # term # term))%}\egroup

\SYNOPSIS
Breaks apart a restricted existentially quantified term into
the quantified variable, predicate and body.

\DESCRIBE
{\small\verb%dest_resq_exists%} is a term destructor for restricted existential
quantification: 
{\par\samepage\setseps\small
\begin{verbatim}
   dest_resq_exists "?var::P. t"
\end{verbatim}
}
\noindent returns {\small\verb%("var","P","t")%}.

\FAILURE
Fails with {\small\verb%dest_resq_exists%} if the term is not a restricted
existential quantification.

\SEEALSO
mk_resq_exists, is_resq_exists, strip_resq_exists.

\ENDDOC
\DOC{dest\_resq\_forall}

\TYPE {\small\verb%dest_resq_forall : (term -> (term # term # term))%}\egroup

\SYNOPSIS
Breaks apart a restricted universally quantified term into
the quantified variable, predicate and body.

\DESCRIBE
{\small\verb%dest_resq_forall%} is a term destructor for restricted universal
quantification: 
{\par\samepage\setseps\small
\begin{verbatim}
   dest_resq_forall "!var::P. t"
\end{verbatim}
}
\noindent returns {\small\verb%("var","P","t")%}.

\FAILURE
Fails with {\small\verb%dest_resq_forall%} if the term is not a restricted
universal quantification.

\SEEALSO
mk_resq_forall, is_resq_forall, strip_resq_forall.

\ENDDOC
\DOC{dest\_resq\_select}

\TYPE {\small\verb%dest_resq_select : (term -> (term # term # term))%}\egroup

\SYNOPSIS
Breaks apart a restricted choice quantified term into
the quantified variable, predicate and body.

\DESCRIBE
{\small\verb%dest_resq_select%} is a term destructor for restricted choice
quantification: 
{\par\samepage\setseps\small
\begin{verbatim}
   dest_resq_select "@var::P. t"
\end{verbatim}
}
\noindent returns {\small\verb%("var","P","t")%}.

\FAILURE
Fails with {\small\verb%dest_resq_select%} if the term is not a restricted
choice quantification.

\SEEALSO
mk_resq_select, is_resq_select, strip_resq_select.

\ENDDOC
\DOC{GQSPEC\_ALL}

\TYPE {\small\verb%GQSPEC_ALL : (thm -> thm)%}\egroup

\SYNOPSIS
Specializes the conclusion of a theorem with its own quantified variables.

\DESCRIBE
When applied to a theorem whose conclusion has zero or more 
ordinary or restricted universal quantifications,
the inference rule {\small\verb%GQSPEC_ALL%} returns a theorem which is the result
of specializing the quantified variables with its own variables. If this
will cause name clashes, a variant of the variable is used instead.
Normally {\small\verb%xi'%} is just {\small\verb%xi%}, in which case {\small\verb%GQSPEC_ALL%} simply removes
all universal quantifiers.
{\par\samepage\setseps\small
\begin{verbatim}
       A |- !x1::P1. ...!xk. ... !xn::Pn. t
   ------------------------------------------------------  GQSPEC_ALL
    A,P1 x1,...,Pn xn |- t[x1'/x1]...[xk'/xk]...[xn'/xn]
\end{verbatim}
}
\FAILURE
Never fails.

\SEEALSO
GQSPEC, GQSPECL, SPEC, SPECL, SPEC_ALL, RESQ_GEN, RESQ_GENL,
RESQ_GEN_ALL, RESQ_GEN_TAC, RESQ_SPEC, RESQ_SPECL, RESQ_SPEC_ALL.

\ENDDOC
\DOC{GQSPECL}

\TYPE {\small\verb%GQSPECL : (term list -> thm -> thm)%}\egroup

\SYNOPSIS
Specializes zero or more variables in the conclusion of a 
universally quantified theorem.

\DESCRIBE
When applied to a term list {\small\verb%[u1;...;un]%} and a theorem whose conclusion has
zero or more ordinary or restricted universal quantifications,
the inference rule {\small\verb%GQSPECL%} returns a theorem which is the result of
specializing the quantified variables. The substitutions are made
sequentially left-to-right in the same way as for {\small\verb%GQSPEC%}, with the same
sort of alpha-conversions applied to the body of the conclusion.
The two kinds of universal quantification can be mixed.
{\par\samepage\setseps\small
\begin{verbatim}
       A |- !x1::P1. ... !xk. ... !xn::Pn. t
   --------------------------------------------------  GQSPECL "[u1;...;un]"
    A,P1 u1,...,Pn un |- t[u1/x1]...[uk/xk]...[un/xn]
\end{verbatim}
}
\noindent It is permissible for the term-list to be empty, in which case
the application of {\small\verb%GQSPECL%} has no effect.

\FAILURE
Fails if one of the specialization of the 
quantified variable in the original theorem fails.

\SEEALSO
GQSPEC, GQSPEC_ALL, SPECL, GENL, RESQ_GEN, RESQ_GENL, RESQ_GEN_ALL,
RESQ_GEN_TAC, RESQ_SPEC, RESQ_SPECL, RESQ_SPEC_ALL.

\ENDDOC
\DOC{IMP\_RESQ\_FORALL\_CONV}

\TYPE {\small\verb%IMP_RESQ_FORALL_CONV : conv%}\egroup

\SYNOPSIS
Converts an implication to a restricted universal quantification.

\DESCRIBE
When applied to a term of the form {\small\verb%!x.P x ==> Q%}, the conversion
{\small\verb%IMP_RESQ_FORALL_CONV%} returns the theorem:
{\par\samepage\setseps\small
\begin{verbatim}
   |- (!x. P x ==> Q) = !x::P. Q 
\end{verbatim}
}

\FAILURE
Fails if applied to a term not of the form {\small\verb%!x.P x ==> Q%}.

\SEEALSO
RESQ_FORALL_CONV, LIST_RESQ_FORALL_CONV.

\ENDDOC
\DOC{is\_resq\_abstract}

\TYPE {\small\verb%is_resq_abstract : (term -> bool)%}\egroup

\SYNOPSIS
Tests a term to see if it is a restricted abstraction.

\DESCRIBE
{\small\verb%is_resq_abstract "\var::P. t"%} returns {\small\verb%true%}. If the term is not a
restricted abstraction the result is {\small\verb%false%}.

\FAILURE
Never fails.

\SEEALSO
mk_resq_abstract, dest_resq_abstract.

\ENDDOC
\DOC{is\_resq\_exists}

\TYPE {\small\verb%is_resq_exists : (term -> bool)%}\egroup

\SYNOPSIS
Tests a term to see if it is a restricted existential quantification.

\DESCRIBE
{\small\verb%is_resq_exists "?var::P. t"%} returns {\small\verb%true%}. If the term is not a 
restricted existential quantification the result is {\small\verb%false%}.

\FAILURE
Never fails.

\SEEALSO
mk_resq_exists, dest_resq_exists.

\ENDDOC
\DOC{is\_resq\_forall}

\TYPE {\small\verb%is_resq_forall : (term -> bool)%}\egroup

\SYNOPSIS
Tests a term to see if it is a restricted universal quantification.

\DESCRIBE
{\small\verb%is_resq_forall "!var::P. t"%} returns {\small\verb%true%}. If the term is not a 
restricted universal quantification the result is {\small\verb%false%}.

\FAILURE
Never fails.

\SEEALSO
mk_resq_forall, dest_resq_forall.

\ENDDOC
\DOC{is\_resq\_select}

\TYPE {\small\verb%is_resq_select : (term -> bool)%}\egroup

\SYNOPSIS
Tests a term to see if it is a restricted choice quantification.

\DESCRIBE
{\small\verb%is_resq_select "@var::P. t"%} returns {\small\verb%true%}. If the term is not a
restricted choice quantification the result is {\small\verb%false%}.

\FAILURE
Never fails.

\SEEALSO
mk_resq_select, dest_resq_select.

\ENDDOC
\DOC{list\_mk\_resq\_exists}

\TYPE {\small\verb%list_mk_resq_exists : ((term # term) list # term) -> term)%}\egroup

\SYNOPSIS
Iteratively constructs a restricted existential quantification.

\DESCRIBE
{\par\samepage\setseps\small
\begin{verbatim}
   list_mk_resq_exists([("x1","P1");...;("xn","Pn")],"t")
\end{verbatim}
}
\noindent returns {\small\verb%"?x1::P1. ... ?xn::Pn. t"%}.

\FAILURE
Fails with {\small\verb%list_mk_resq_exists%} if the first terms {\small\verb%xi%} in the pairs are
not a variable or if the second terms {\small\verb%Pi%} in the pairs and {\small\verb%t%} 
are not of type {\small\verb%":bool"%} if the list is non-empty. If the list is
empty the type of {\small\verb%t%} can be anything.

\SEEALSO
strip_resq_exists, mk_resq_exists.

\ENDDOC
\DOC{list\_mk\_resq\_forall}

\TYPE {\small\verb%list_mk_resq_forall : ((term # term) list # term) -> term)%}\egroup

\SYNOPSIS
Iteratively constructs a restricted universal quantification.

\DESCRIBE
{\par\samepage\setseps\small
\begin{verbatim}
   list_mk_resq_forall([("x1","P1");...;("xn","Pn")],"t")
\end{verbatim}
}
\noindent returns {\small\verb%"!x1::P1. ... !xn::Pn. t"%}.

\FAILURE
Fails with {\small\verb%list_mk_resq_forall%} if the first terms {\small\verb%xi%} in the pairs are
not a variable or if the second terms {\small\verb%Pi%} in the pairs and {\small\verb%t%} 
are not of type {\small\verb%":bool"%} if the list is non-empty. If the list is
empty the type of {\small\verb%t%} can be anything.

\SEEALSO
strip_resq_forall, mk_resq_forall.

\ENDDOC
\DOC{LIST\_RESQ\_FORALL\_CONV}

\TYPE {\small\verb%LIST_RESQ_FORALL_CONV : conv%}\egroup

\SYNOPSIS
Converts restricted universal quantifications iteratively to implications.

\DESCRIBE
When applied to a term whose outer level is a series of restricted universal
quantifications, the conversion
{\small\verb%LIST_RESQ_FORALL_CONV%} returns the theorem:
{\par\samepage\setseps\small
\begin{verbatim}
   |- !x1::P1. ... !xn::Pn. Q = (!x1...xn. P1 x1 ==> ... ==> Pn xn ==> Q)
\end{verbatim}
}

\FAILURE
Never fails.

\SEEALSO
IMP_RESQ_FORALL_CONV, RESQ_FORALL_CONV.

\ENDDOC
\DOC{mk\_resq\_abstract}

\TYPE {\small\verb%mk_resq_abstract : ((term # term # term) -> term)%}\egroup

\SYNOPSIS
Term constructor for restricted abstraction.

\DESCRIBE
{\small\verb%mk_resq_abstract("var","P","t")%} returns {\small\verb%"\var :: P . t"%}.

\FAILURE
Fails with {\small\verb%mk_resq_abstract%} if the first term is not a variable or if {\small\verb%P%} and {\small\verb%t%}
are not of type {\small\verb%":bool"%}.

\SEEALSO
dest_resq_abstract, is_resq_abstract, list_mk_resq_abstract.

\ENDDOC
\DOC{mk\_resq\_exists}

\TYPE {\small\verb%mk_resq_exists : ((term # term # term) -> term)%}\egroup

\SYNOPSIS
Term constructor for restricted existential quantification.

\DESCRIBE
{\small\verb%mk_resq_exists("var","P","t")%} returns {\small\verb%"?var :: P . t"%}.

\FAILURE
Fails with {\small\verb%mk_resq_exists%} if the first term is not a variable or if {\small\verb%P%} and {\small\verb%t%}
are not of type {\small\verb%":bool"%}.

\SEEALSO
dest_resq_exists, is_resq_exists, list_mk_resq_exists.

\ENDDOC
\DOC{mk\_resq\_forall}

\TYPE {\small\verb%mk_resq_forall : ((term # term # term) -> term)%}\egroup

\SYNOPSIS
Term constructor for restricted universal quantification.

\DESCRIBE
{\small\verb%mk_resq_forall("var","P","t")%} returns {\small\verb%"!var :: P . t"%}.

\FAILURE
Fails with {\small\verb%mk_resq_forall%} if the first term is not a variable or if {\small\verb%P%} and {\small\verb%t%}
are not of type {\small\verb%":bool"%}.

\SEEALSO
dest_resq_forall, is_resq_forall, list_mk_resq_forall.

\ENDDOC
\DOC{mk\_resq\_select}

\TYPE {\small\verb%mk_resq_select : ((term # term # term) -> term)%}\egroup

\SYNOPSIS
Term constructor for restricted choice quantification.

\DESCRIBE
{\small\verb%mk_resq_select("var","P","t")%} returns {\small\verb%"@var :: P . t"%}.

\FAILURE
Fails with {\small\verb%mk_resq_select%} if the first term is not a variable or if {\small\verb%P%} and {\small\verb%t%}
are not of type {\small\verb%":bool"%}.

\SEEALSO
dest_resq_select, is_resq_select, list_mk_resq_select.

\ENDDOC
\DOC{new\_binder\_resq\_definition}

\TYPE {\small\verb%new_binder_resq_definition : ((string # term) -> thm)%}\egroup

\SYNOPSIS
Declare a new binder and install a definitional axiom in the current theory.

\DESCRIBE
The function {\small\verb%new_binder_resq_definition%} provides a facility for definitional
extensions to the current theory. The new constant defined using this
function may take arguments which are restricted quantified.  The
function {\small\verb%new_binder_resq_definition%} takes a pair argument consisting 
of the name under which the resulting definition will be saved
in the current theory segment, and a term giving the desired definition.  The
value returned by {\small\verb%new_binder_resq_definition%} is a theorem which states the
definition requested by the user.

Let {\small\verb%x_1,...,x_n%} be distinct variables.  Evaluating
{\par\samepage\setseps\small
\begin{verbatim}
   new_binder_resq_definition (`name`,
    "!x_i::P_i. ... !x_j::P_j. B x_1 ... x_n = t")
\end{verbatim}
}
where {\small\verb%B%} is not already a constant, {\small\verb%i%} is greater or equal to 1 and
{\small\verb%i <= j <= n%}, declares {\small\verb%B%} to be a new constant in the current theory
with this definition as its specification. 
This constant specification is returned as a theorem with the form
{\par\samepage\setseps\small
\begin{verbatim}
   |- !x_i::P_i. ... !x_j::P_j. !x_k .... B x_1 ... x_n = t
\end{verbatim}
}
\noindent where the variables {\small\verb%x_k%} are the free variables occurring on
the left hand side of the definition and are not restricted
quantified. This theorem is saved in the current theory under
(the name) {\small\verb%name%}. 

The constant {\small\verb%B%} defined by this function will have the binder status
only after the definition has been processed. It is therefore necessary to use
the constant in normal prefix position when making the definition.
 
If the restricting predicates {\small\verb%P_l%} contains free occurrence of
variable(s) of the left hand side, the constant {\small\verb%B%} will stand for a
family of functions.

\FAILURE
{\small\verb%new_binder_resq_definition%} fails if called when HOL is not in draft mode.  It also
fails if there is already an axiom, definition or specification of the given
name in the current theory segment; if {\small\verb%`B`%} is already a constant in the
current theory or is not an allowed name for a constant; if {\small\verb%t%} contains free
variables that do not occur in the left hand side, or if
any variable occurs more than once in {\small\verb%x_1, ..., x_n%}.  Finally, failure occurs
if there is a type variable in {\small\verb%x_1%}, ..., {\small\verb%x_n%} or {\small\verb%t%} that does not occur in
the type of {\small\verb%B%}.

\SEEALSO
new_infix_resq_definition, new_resq_definition,
new_definition, new_specification.


\ENDDOC
\DOC{new\_infix\_resq\_definition}

\TYPE {\small\verb%new_infix_resq_definition : ((string # term) -> thm)%}\egroup

\SYNOPSIS
Declare a new infix constant and install a definitional axiom in the current theory.

\DESCRIBE
The function {\small\verb%new_infix_resq_definition%} provides a facility for definitional
extensions to the current theory. The new constant defined using this
function may take arguments which are restricted quantified.  The
function {\small\verb%new_infix_resq_definition%} takes a pair argument consisting 
of the name under which the resulting definition will be saved
in the current theory segment, and a term giving the desired definition.  The
value returned by {\small\verb%new_infix_resq_definition%} is a theorem which states the
definition requested by the user.

Let {\small\verb%x_1,...,x_n%} be distinct variables.  Evaluating
{\par\samepage\setseps\small
\begin{verbatim}
   new_infix_resq_definition (`name`,
    "!x_i::P_i. ... !x_j::P_j. IX x_1 ... x_n = t")
\end{verbatim}
}
where {\small\verb%IX%} is not already a constant, {\small\verb%i%} is greater or equal to 1 and
{\small\verb%i <= j <= n%}, declares {\small\verb%IX%} to be a new constant in the current theory
with this definition as its specification. 
This constant specification is returned as a theorem with the form
{\par\samepage\setseps\small
\begin{verbatim}
   |- !x_i::P_i. ... !x_j::P_j. !x_k .... IX x_1 ... x_n = t
\end{verbatim}
}
\noindent where the variables {\small\verb%x_k%} are the free variables occurring on
the left hand side of the definition and are not restricted
quantified. This theorem is saved in the current theory under
(the name) {\small\verb%name%}. 

The constant {\small\verb%IX%} defined by this function will have the infix status
only after the definition has been processed. It is therefore necessary to use
the constant in normal prefix position when making the definition.
 
If the restricting predicates {\small\verb%P_l%} contains free occurrence of
variable(s) of the left hand side, the constant {\small\verb%IX%} will stand for a
family of functions.


\FAILURE
{\small\verb%new_infix_resq_definition%} fails if called when HOL is not in draft mode.  It also
fails if there is already an axiom, definition or specification of the given
name in the current theory segment; if {\small\verb%`IX`%} is already a constant in the
current theory or is not an allowed name for a constant; if {\small\verb%t%} contains free
variables that do not occur in the left hand side, or if
any variable occurs more than once in {\small\verb%x_1, ..., x_n%}.  Finally, failure occurs
if there is a type variable in {\small\verb%x_1%}, ..., {\small\verb%x_n%} or {\small\verb%t%} that does not occur in
the type of {\small\verb%IX%}.

\EXAMPLE
A function for indexing list element starting from 1 can be defined as follows:
{\par\samepage\setseps\small
\begin{verbatim}
   #let IXEL1_DEF = new_infix_resq_definition (`IXEL1_DEF`,
   # "!n:: (\k. 0 < k). IXEL1 n (l:* list) = EL (n -1) l");;
   IXEL1_DEF = |- !n :: \k. 0 < k. !l. IXEL1 n l = EL(n - 1)l
\end{verbatim}
}
One can then use {\small\verb%IXEL1%} as an infix and do the following proof:
{\par\samepage\setseps\small
\begin{verbatim}
   #g"2 IXEL1 [1;2;3] = 2";;
   "2 IXEL1 [1;2;3] = 2"

   #e(RESQ_REWRITE1_TAC IXEL1_DEF THENL[
   #   CONV_TAC(ONCE_DEPTH_CONV num_CONV) THEN MATCH_ACCEPT_TAC LESS_0;
   #   CONV_TAC((LHS_CONV o LHS_CONV)(REDEPTH_CONV num_CONV))
   #   THEN REWRITE_TAC[SUB_MONO_EQ;SUB_0;EL;HD;TL]]);;
   OK..
   goal proved
   |- 2 IXEL1 [1;2;3] = 2

   Previous subproof:
   goal proved
   () : void
\end{verbatim}
}
\SEEALSO
new_binder_resq_definition, new_resq_definition,
new_definition, new_specification.


\ENDDOC
\DOC{new\_resq\_definition}

\TYPE {\small\verb%new_resq_definition : ((string # term) -> thm)%}\egroup

\SYNOPSIS
Declare a new constant and install a definitional axiom in the current theory.

\DESCRIBE
The function {\small\verb%new_resq_definition%} provides a facility for definitional
extensions to the current theory. The new constant defined using this
function may take arguments which are restricted quantified.  The
function {\small\verb%new_resq_definition%} takes a pair argument consisting 
of the name under which the resulting definition will be saved
in the current theory segment, and a term giving the desired definition.  The
value returned by {\small\verb%new_resq_definition%} is a theorem which states the
definition requested by the user.

Let {\small\verb%x_1,...,x_n%} be distinct variables.  Evaluating
{\par\samepage\setseps\small
\begin{verbatim}
   new_resq_definition (`name`,
    "!x_i::P_i. ... !x_j::P_j. C x_1 ... x_n = t")
\end{verbatim}
}
\noindent where {\small\verb%C%} is not already a constant, {\small\verb%i%} is greater or equal to 1 and
{\small\verb%i <= j <= n%}, declares {\small\verb%C%} to be a new constant in the current theory
with this definition as its specification. 
This constant specification is returned as a theorem with the form
{\par\samepage\setseps\small
\begin{verbatim}
   |- !x_i::P_i. ... !x_j::P_j. !x_k .... C x_1 ... x_n = t
\end{verbatim}
}
\noindent where the variables {\small\verb%x_k%} are the free variables occurring on
the left hand side of the definition and are not restricted
quantified. This theorem is saved in the current theory under
(the name) {\small\verb%name%}. 

If the restricting predicates {\small\verb%P_l%} contains free occurrence of
variable(s) of the left hand side, the constant {\small\verb%C%} will stand for a
family of functions.


\FAILURE
{\small\verb%new_resq_definition%} fails if called when HOL is not in draft mode.  It also
fails if there is already an axiom, definition or specification of the given
name in the current theory segment; if {\small\verb%`C`%} is already a constant in the
current theory or is not an allowed name for a constant; if {\small\verb%t%} contains free
variables that do not occur in the left hand side, or if
any variable occurs more than once in {\small\verb%x_1, ..., x_n%}.  Finally, failure occurs
if there is a type variable in {\small\verb%x_1%}, ..., {\small\verb%x_n%} or {\small\verb%t%} that does not occur in
the type of {\small\verb%C%}.

\EXAMPLE
A function for indexing list elements starting from 1 can be defined as follows:
{\par\samepage\setseps\small
\begin{verbatim}
   #new_resq_definition (`EL1_DEF`,
   # "!n:: (\k. 0 < k). EL1 n (l:* list) = EL (n - 1) l");;
   |- !n :: \k. 0 < k. !l. EL1 n l = EL(n - 1)l
\end{verbatim}
}
The following example shows how a family of constants may be defined
if the restricting predicate involves free variable on the left hand
side of the definition.
{\par\samepage\setseps\small
\begin{verbatim}
   #new_resq_definition (`ELL_DEF`,
   # "!n:: (\k. k < (LENGTH l)). ELL n (l:* list) = EL n  l");;
   |- !l. !n :: \k. k < (LENGTH l). !l'. ELL l n l' = EL n l'
\end{verbatim}
}

\SEEALSO
new_resq_binder_definition, new_resq_infix_definition,
new_definition, new_specification.


\ENDDOC
\DOC{RESQ\_EXISTS\_CONV}

\TYPE {\small\verb%RESQ_EXISTS_CONV : conv%}\egroup

\SYNOPSIS
Converts a restricted existential quantification to a conjunction.

\DESCRIBE
When applied to a term of the form {\small\verb%?x::P. Q[x]%}, the conversion
{\small\verb%RESQ_EXISTS_CONV%} returns the theorem:
{\par\samepage\setseps\small
\begin{verbatim}
   |- ?x::P. Q[x] = (?x. P x /\ Q[x])
\end{verbatim}
}
\noindent which is the underlying semantic representation of the restricted
existential quantification.

\FAILURE
Fails if applied to a term not of the form {\small\verb%?x::P. Q%}.

\SEEALSO
RESQ_FORALL_CONV, RESQ_EXISTS_TAC.

\ENDDOC
\DOC{RESQ\_EXISTS\_TAC}

\TYPE {\small\verb%RESQ_EXISTS_TAC : term -> tactic%}\egroup

\SYNOPSIS
Strips the outermost restricted existential quantifier from
the conclusion of a goal.

\DESCRIBE
When applied to a goal {\small\verb%A ?- ?x::P. t%}, the tactic {\small\verb%RESQ_EXISTS_TAC%}
reduces it to a new subgoal {\small\verb%A ?- P x' /\ t[x'/x]%} where {\small\verb%x'%} is a variant
of {\small\verb%x%} chosen to avoid clashing with any variables free in the goal's
assumption list. Normally {\small\verb%x'%} is just {\small\verb%x%}.
{\par\samepage\setseps\small
\begin{verbatim}
     A ?- ?x::P. t
   ======================  RESQ_EXISTS_TAC
    A ?- P x' /\ t[x'/x]
\end{verbatim}
}
\FAILURE
Fails unless the goal's conclusion is a restricted extistential quantification.

\SEEALSO
RESQ_HALF_EXISTS.

\ENDDOC
\DOC{RESQ\_FORALL\_AND\_CONV}

\TYPE {\small\verb%RESQ_FORALL_AND_CONV : conv%}\egroup

\SYNOPSIS
Splits a restricted universal quantification across a conjunction.

\DESCRIBE
When applied to a term of the form {\small\verb%!x::P. Q /\ R%}, the conversion
{\small\verb%RESQ_FORALL_AND_CONV%} returns the theorem:
{\par\samepage\setseps\small
\begin{verbatim}
   |- (!x::P. Q /\ R)  = ((!x::P. Q) /\ (!x::P. R))
\end{verbatim}
}

\FAILURE
Fails if applied to a term not of the form {\small\verb%!x::P. Q /\ R%}.

\SEEALSO
AND_RESQ_FORALL_CONV.

\ENDDOC
\DOC{RESQ\_FORALL\_CONV}

\TYPE {\small\verb%RESQ_FORALL_CONV : conv%}\egroup

\SYNOPSIS
Converts a restricted universal quantification to an implication.

\DESCRIBE
When applied to a term of the form {\small\verb%!x::P. Q%}, the conversion
{\small\verb%RESQ_FORALL_CONV%} returns the theorem:
{\par\samepage\setseps\small
\begin{verbatim}
   |- !x::P. Q = (!x. P x ==> Q)
\end{verbatim}
}
\noindent which is the underlying semantic representation of the restricted
universal quantification.

\FAILURE
Fails if applied to a term not of the form {\small\verb%!x::P. Q%}.

\SEEALSO
IMP_RESQ_FORALL_CONV, LIST_RESQ_FORALL_CONV.

\ENDDOC
\DOC{RESQ\_FORALL\_SWAP\_CONV}

\TYPE {\small\verb%RESQ_FORALL_SWAP_CONV : conv%}\egroup

\SYNOPSIS
Changes the order of two restricted universal quantifications.

\DESCRIBE
When applied to a term of the form {\small\verb%!x::P. !y::Q. R%}, the conversion
{\small\verb%RESQ_FORALL_SWAP_CONV%} returns the theorem:
{\par\samepage\setseps\small
\begin{verbatim}
   |- (!x::P. !y::Q. R) =  !y::Q. !x::P. R
\end{verbatim}
}
\noindent providing that {\small\verb%x%} does not occur free in {\small\verb%Q%} and {\small\verb%y%} does not
occur free in {\small\verb%P%}.

\FAILURE
Fails if applied to a term not of the correct form.

\SEEALSO
RESQ_FORALL_CONV.

\ENDDOC
\DOC{RESQ\_GEN\_ALL}

\TYPE {\small\verb%RESQ_GEN_ALL : (thm -> thm)%}\egroup

\SYNOPSIS
Generalizes the conclusion of a theorem over its own assumptions.

\DESCRIBE
When applied to a theorem {\small\verb%A |- t%}, the inference rule {\small\verb%RESQ_GEN_ALL%} returns
the theorem {\small\verb%A' |- !x1::P1. ...!xn::Pn. t%}, where the {\small\verb%Pi xi%} are in the 
assumptions.
{\par\samepage\setseps\small
\begin{verbatim}
         A |- t
   ------------------------------------------------  RESQ_GEN_ALL
   A - (P1 x1,...,Pn xn) |- !x1::P1. ... !xn::Pn. t
\end{verbatim}
}
\FAILURE
Never fails.

\SEEALSO
RESQ_GEN, RESQ_GENL, GEN_ALL, RESQ_SPEC, RESQ_SPECL, RESQ_SPEC_ALL.

\ENDDOC
\DOC{RESQ\_GEN}

\TYPE {\small\verb%RESQ_GEN : ((term # term) -> thm -> thm)%}\egroup

\SYNOPSIS
Generalizes the conclusion of a theorem to a restricted universal quantification.

\DESCRIBE
When applied to a pair of terms {\small\verb%x%}, {\small\verb%P%} and a theorem {\small\verb%A |- t%},
the inference rule {\small\verb%RESQ_GEN%} returns the theorem {\small\verb%A |- !x::P. t%},
provided that {\small\verb%P%} is a predicate taking an argument of the same type
as {\small\verb%x%} and that {\small\verb%x%} is a variable not free in any of the assumptions
except {\small\verb%P x%} if it occurs. There is no compulsion that {\small\verb%x%} should
be free in {\small\verb%t%} or {\small\verb%P x%} should be in the assumptions.
{\par\samepage\setseps\small
\begin{verbatim}
      A |- t
   --------------- RESQ_GEN ("x","P") [where x is not free in A except P x]
    A |- !x::P. t
\end{verbatim}
}
\FAILURE
Fails if {\small\verb%x%} is not a variable, or if it is free in any of the assumptions
other than {\small\verb%P x%}.

\SEEALSO
RESQ_GENL, RESQ_GEN_ALL, RESQ_GEN_TAC, RESQ_SPEC, RESQ_SPECL, RESQ_SPEC_ALL.

\ENDDOC
\DOC{RESQ\_GENL}

\TYPE {\small\verb%RESQ_GENL : ((term # term) list -> thm -> thm)%}\egroup

\SYNOPSIS
Generalizes zero or more variables to restricted universal quantification
in the conclusion of a theorem.

\DESCRIBE
When applied to a term-pair list {\small\verb%[(x1,P1);...;(xn,Pn)]%} and a theorem 
{\small\verb%A |- t%}, the inference rule {\small\verb%RESQ_GENL%} returns the theorem 
{\small\verb%A |- !x1::P1. ... !xn::Pn. t%}, provided none of the
variables {\small\verb%xi%} are free in any of the assumptions except in the corresponding
{\small\verb%Pi%}. It is not necessary that any or all of the {\small\verb%xi%} should be free in {\small\verb%t%}.
{\par\samepage\setseps\small
\begin{verbatim}
         A |- t
   ------------------------------  RESQ_GENL "[(x1,P1);...;(xn,Pn)]" 
    A |- !x1::P1. ... !xn::Pn. t   [where no xi is free in A except in Pi]
\end{verbatim}
}
\FAILURE
Fails unless all the terms {\small\verb%xi%} in the list are variables, none of which are
free in the assumption list except in {\small\verb%Pi%}.

\SEEALSO
RESQ_GEN, RESQ_GEN_ALL, RESQ_GEN_TAC, RESQ_SPEC, RESQ_SPECL, RESQ_SPEC_ALL.

\ENDDOC
\DOC{RESQ\_GEN\_TAC}

\TYPE {\small\verb%RESQ_GEN_TAC : tactic%}\egroup

\SYNOPSIS
Strips the outermost restricted universal quantifier from
the conclusion of a goal.

\DESCRIBE
When applied to a goal {\small\verb%A ?- !x::P. t%}, the tactic {\small\verb%RESQ_GEN_TAC%}
reduces it to a new goal {\small\verb%A,P x' ?- t[x'/x]%} where {\small\verb%x'%} is a variant of {\small\verb%x%}
chosen to avoid clashing with any variables free in the goal's
assumption list. Normally {\small\verb%x'%} is just {\small\verb%x%}.
{\par\samepage\setseps\small
\begin{verbatim}
     A ?- !x::P. t
   ===================  RESQ_GEN_TAC
    A,P x' ?- t[x'/x]
\end{verbatim}
}
\FAILURE
Fails unless the goal's conclusion is a restricted universal quantification.

\USES
The tactic {\small\verb%REPEAT RESQ_GEN_TAC%} strips away a series of restricted
universal quantifiers, and is commonly used before tactics relying on
the  underlying term structure. 

\SEEALSO
RESQ_HALF_GEN_TAC, RESQ_GEN, RESQ_GENL, RESQ_GEN_ALL, RESQ_SPEC, RESQ_SPECL,
RESQ_SPEC_ALL, GGEN_TAC, STRIP_TAC, GEN_TAC, X_GEN_TAC.

\ENDDOC
\DOC{RESQ\_HALF\_EXISTS}

\TYPE {\small\verb%RESQ_HALF_EXISTS : (thm -> thm)%}\egroup

\SYNOPSIS
Strip a restricted existential quantification from the conclusion of a theorem.

\DESCRIBE
When applied to a theorem {\small\verb%A |- ?x::P. t%}, {\small\verb%RESQ_HALF_EXISTS%} returns
the theorem
{\par\samepage\setseps\small
\begin{verbatim}
   A |- ?x. P x /\ t
\end{verbatim}
}
\noindent i.e., it transforms the restricted existential
quantification to its underlying semantic representation.
{\par\samepage\setseps\small
\begin{verbatim}
      A |- ?x::P. t
   --------------------  RESQ_HALF_EXISTS
    A |- ?x. P x ==> t
\end{verbatim}
}
\FAILURE
Fails if the theorem's conclusion is not a restricted existential quantification.

\SEEALSO
RESQ_EXISTS_TAC, EXISTS.

\ENDDOC
\DOC{RESQ\_HALF\_GEN\_TAC}

\TYPE {\small\verb%RESQ_HALF_GEN_TAC : tactic%}\egroup

\SYNOPSIS
Strips the outermost restricted universal quantifier from
the conclusion of a goal.

\DESCRIBE
When applied to a goal {\small\verb%A ?- !x::P. t%}, {\small\verb%RESQ_GEN_TAC%}
reduces it to {\small\verb%A ?- !x. P x ==> t%} which is the underlying semantic
representation of the restricted universal quantification.
{\par\samepage\setseps\small
\begin{verbatim}
     A ?- !x::P. t
   ====================  RESQ_HALF_GEN_TAC
    A ?- !x. P x ==> t
\end{verbatim}
}
\FAILURE
Fails unless the goal's conclusion is a restricted universal quantification.

\USES
The tactic {\small\verb%REPEAT RESQ_GEN_TAC%} strips away a series of restricted
universal quantifiers, and is commonly used before tactics relying on
the  underlying term structure. 

\SEEALSO
RESQ_GEN_TAC, RESQ_GEN, RESQ_GENL, RESQ_GEN_ALL, RESQ_SPEC, RESQ_SPECL,
RESQ_SPEC_ALL, GGEN_TAC, STRIP_TAC, GEN_TAC, X_GEN_TAC.

\ENDDOC
\DOC{RESQ\_HALF\_SPEC}

\TYPE {\small\verb%RESQ_HALF_SPEC : (thm -> thm)%}\egroup

\SYNOPSIS
Strip a restricted universal quantification in the conclusion of a theorem.

\DESCRIBE
When applied to a theorem {\small\verb%A |- !x::P. t%}, the derived inference rule
{\small\verb%RESQ_HALF_SPEC%} returns
the theorem {\small\verb%A |- !x. P x ==> t%}, i.e., it transforms the restricted universal
quantification to its underlying semantic representation.
{\par\samepage\setseps\small
\begin{verbatim}
      A |- !x::P. t
   --------------------  RESQ_HALF_SPEC
    A |- !x. P x ==> t
\end{verbatim}
}
\FAILURE
Fails if the theorem's conclusion is not a restricted universal quantification.

\SEEALSO
RESQ_SPEC, RESQ_SPECL, RESQ_SPEC_ALL, RESQ_GEN, RESQ_GENL, RESQ_GEN_ALL.

\ENDDOC
\DOC{RESQ\_IMP\_RES\_TAC}

\TYPE {\small\verb%RESQ_IMP_RES_TAC : thm_tactic%}\egroup

\SYNOPSIS
Repeatedly resolves a restricted universally quantified theorem with
the assumptions of a goal.

\DESCRIBE
The function {\small\verb%RESQ_IMP_RES_TAC%} performs repeatedly
resolution using a restricted quantified theorem.
It takes a restricted quantified theorem and transforms it into an
implication. This resulting theorem is used in the resolution.

Given a theorem {\small\verb%th%}, the theorem-tactic {\small\verb%RESQ_IMP_RES_TAC%}
applies {\small\verb%RESQ_IMP_RES_THEN%} repeatedly to resolve the theorem with the
assumptions. 

\FAILURE
Never fails

\SEEALSO
RESQ_IMP_RES_THEN, RESQ_RES_THEN, RESQ_RES_TAC,
IMP_RES_THEN, IMP_RES_TAC, MATCH_MP, RES_CANON, RES_TAC, RES_THEN.

\ENDDOC
\DOC{RESQ\_IMP\_RES\_THEN}

\TYPE {\small\verb%RESQ_IMP_RES_THEN : thm_tactical%}\egroup

\SYNOPSIS
Resolves a restricted universally quantified theorem with
the assumptions of a goal.

\DESCRIBE
The function {\small\verb%RESQ_IMP_RES_THEN%} is the basic building block for
resolution using a restricted quantified theorem.
It takes a restricted quantified theorem and transforms it into an
implication. This resulting theorem is used in the resolution.

Given a theorem-tactic {\small\verb%ttac%} and a theorem {\small\verb%th%}, the theorem-tactical
{\small\verb%RESQ_IMP_RES_THEN%} transforms the theorem into an implication {\small\verb%th'%}. It
then passes {\small\verb%th'%} together with {\small\verb%ttac%} to {\small\verb%IMP_RES_THEN%} to carry out
the resolution.

\FAILURE
Evaluating {\small\verb%RESQ_IMP_RES_THEN ttac th%} fails if the supplied
theorem {\small\verb%th%} is not restricted universally quantified, or if the call
to {\small\verb%IMP_RES_THEN%} fails.


\SEEALSO
RESQ_IMP_RES_TAC, RESQ_RES_THEN, RESQ_RES_TAC,
IMP_RES_THEN, IMP_RES_TAC, MATCH_MP, RES_CANON, RES_TAC, RES_THEN.

\ENDDOC
\DOC{RESQ\_MATCH\_MP}

\TYPE {\small\verb%RESQ_MATCH_MP : (thm -> thm -> thm)%}\egroup

\SYNOPSIS
Eliminating a restricted universal quantification with automatic matching.

\DESCRIBE
When applied to theorems {\small\verb%A1 |- !x::P. Q[x]%} and {\small\verb%A2 |- P x'%}, the
derived inference rule {\small\verb%RESQ_MATCH_MP%} matches {\small\verb%x'%} to {\small\verb%x%} by instantiating 
free or universally quantified variables in the first theorem (only),
and returns a theorem {\small\verb%A1 u A2 |- Q[x'/x]%}. Polymorphic types are also
instantiated if necessary.

{\par\samepage\setseps\small
\begin{verbatim}
    A1 |- !x::P.Q[x]   A2 |- P x'
   --------------------------------------  RESQ_MATCH_MP
          A1 u A2 |- Q[x'/x]
\end{verbatim}
}
\FAILURE
Fails unless the first theorem is a (possibly repeatedly) restricted
universal quantification whose quantified variable can be instantiated
to match the conclusion of the second theorem, without instantiating
any variables which are free in {\small\verb%A1%}, the first theorem's assumption list.

\SEEALSO
MATCH_MP, RESQ_HALF_SPEC.

\ENDDOC
\DOC{RESQ\_RES\_TAC}

\TYPE {\small\verb%RESQ_RES_TAC : tactic%}\egroup

\SYNOPSIS
Enriches assumptions by repeatedly resolving restricted universal
quantifications in them against the others.

\DESCRIBE
{\small\verb%RESQ_RES_TAC%} uses those assumptions which are restricted universal
quantifications in resolution in a way similar to {\small\verb%RES_TAC%}. It calls
{\small\verb%RESQ_RES_THEN%} repeatedly until there is no more resolution can be done.
The conclusions of all the new results are returned as additional
assumptions of the subgoal(s).  The effect of {\small\verb%RESQ_RES_TAC%} 
on a goal is to enrich the assumption set with some of its collective
consequences.


\FAILURE
{\small\verb%RESQ_RES_TAC%} cannot fail and so should not be unconditionally {\small\verb%REPEAT%}ed.

\SEEALSO
RESQ_IMP_RES_TAC, RESQ_IMP_RES_THEN, RESQ_RES_THEN,
IMP_RES_TAC, IMP_RES_THEN, RES_CANON, RES_THEN, RES_TAC.

\ENDDOC
\DOC{RESQ\_RES\_THEN}

\TYPE {\small\verb%RESQ_RES_THEN : thm_tactic -> tactic%}\egroup

\SYNOPSIS
Resolves all restricted universally quantified assumptions against
other assumptions of a goal.

\DESCRIBE
Like the function {\small\verb%RESQ_IMP_RES_THEN%}, the function {\small\verb%RESQ_RES_THEN%}
performs a single step resolution. The difference is that the
restricted universal quantification used in the resolution is taken
from the assumptions.

Given a theorem-tactic {\small\verb%ttac%}, applying the tactic {\small\verb%RESQ_RES_THEN
ttac%} to a goal {\small\verb%(asml,gl)%} has the effect of:
{\par\samepage\setseps\small
\begin{verbatim}
   MAP_EVERY (mapfilter ttac [... ; (ai,aj |- vi) ; ...]) (amsl ?- g)
\end{verbatim}
}
where the theorems {\small\verb%ai,aj |- vi%} are all the consequences that can be
drawn by a (single) matching modus-ponens inference from the
assumptions {\small\verb%amsl%} and the implications derived from the restricted
universal quantifications in the assumptions.

\FAILURE
Evaluating {\small\verb%RESQ_RES_TAC ttac th%} fails if there are no restricted
universal quantifications in the assumptions, or if the theorem-tactic
{\small\verb%ttac%} applied to all the consequences fails.


\SEEALSO
RESQ_IMP_RES_TAC, RESQ_IMP_RES_THEN, RESQ_RES_TAC,
IMP_RES_THEN, IMP_RES_TAC, MATCH_MP, RES_CANON, RES_TAC, RES_THEN.

\ENDDOC
\DOC{RESQ\_REWR\_CANON}

\TYPE {\small\verb%RESQ_REWR_CANON : thm -> thm%}\egroup

\SYNOPSIS
Transform a theorem into a form accepted for rewriting.

\DESCRIBE
{\small\verb%RESQ_REWR_CANON%} transforms a theorem into a form accepted by {\small\verb%COND_REWR_TAC%}.
The input theorem should be headed by a series of restricted universal
quantifications in the following form
{\par\samepage\setseps\small
\begin{verbatim}
   !x1::P1. ... !xn::Pn. u[xi] = v[xi])
\end{verbatim}
}
\noindent Other variables occurring in {\small\verb%u%} and {\small\verb%v%} may be universally quantified.
The output theorem will have all ordinary universal quantifications
moved to the outer most level with possible renaming to prevent
variable capture, and have all restricted universal quantifications
converted to implications. The output theorem will be in the
form accepted by {\small\verb%COND_REWR_TAC%}.

\FAILURE
This function fails is the input theorem is not in the correct form.


\SEEALSO
RESQ_REWRITE1_TAC, RESQ_REWRITE1_CONV,
COND_REWR_CANON, COND_REWR_TAC, COND_REWR_CONV,.


\ENDDOC

\DOC{RESQ\_REWRITE1\_CONV}

\TYPE {\small\verb%RESQ_REWRITE1_CONV : thm list -> thm -> conv%}\egroup

\SYNOPSIS
Rewriting conversion using a restricted universally quantified theorem.

\DESCRIBE
{\small\verb%RESQ_REWRITE1_CONV%} is a rewriting conversion similar to
{\small\verb%COND_REWRITE1_CONV%}. The only difference is the rewriting theorem it
takes. This should be an equation with restricted universal
quantification at the outer level. It is converted to a theorem in the
form accepted by the conditional rewriting conversion.

Suppose that {\small\verb%th%} is the following theorem
{\par\samepage\setseps\small
\begin{verbatim}
   A |- !x::P. Q[x] = R[x])
\end{verbatim}
}
\noindent evaluating {\small\verb%RESQ_REWRITE1_CONV thms th "t[x']"%}
will return a theorem
{\par\samepage\setseps\small
\begin{verbatim}
   A, P x' |- t[x'] = t'[x']
\end{verbatim}
}
\noindent where {\small\verb%t'%} is the result of substituting instances of
{\small\verb%R[x'/x]%} for corresponding instances of {\small\verb%Q[x'/x]%} in the original
term {\small\verb%t[x]%}. All instances of {\small\verb%P x'%} which 
do not appear in the original assumption {\small\verb%asml%} are added to the assumption.
The theorems in the list {\small\verb%thms%} are used to eliminate the instances {\small\verb%P
x'%} if it is possible.

\FAILURE
{\small\verb%RESQ_REWRITE1_CONV%}  fails if {\small\verb%th%} cannot be transformed into the
required form by the function {\small\verb%RESQ_REWR_CANON%}. Otherwise, it fails if no 
match is found or the theorem cannot be instantiated.


\SEEALSO
RESQ_REWRITE1_TAC, RESQ_REWR_CANON, COND_REWR_TAC,
COND_REWRITE1_CONV, COND_REWR_CONV, COND_REWR_CANON, search_top_down.

\ENDDOC

\DOC{RESQ\_REWRITE1\_TAC}

\TYPE {\small\verb%RESQ_REWRITE1_TAC : thm_tactic%}\egroup

\SYNOPSIS
Rewriting with a restricted universally quantified theorem.

\DESCRIBE
{\small\verb%RESQ_REWRITE1_TAC%} takes an equational theorem which is restricted universally
quantified at the outer level. It calls {\small\verb%RESQ_REWR_CANON%} to convert
the theorem to the form accepted by {\small\verb%COND_REWR_TAC%} and passes the
resulting theorem to this tactic which carries out conditional
rewriting.

Suppose that {\small\verb%th%} is the following theorem
{\par\samepage\setseps\small
\begin{verbatim}
   A |- !x::P. Q[x] = R[x])
\end{verbatim}
}
\noindent Applying the tactic {\small\verb%RESQ_REWRITE1_TAC th%} to a goal {\small\verb%(asml,gl)%}
will return a main subgoal {\small\verb%(asml',gl')%} where {\small\verb%gl'%} is obtained by
substituting instances of {\small\verb%R[x'/x]%} for corresponding instances of
{\small\verb%Q[x'/x]%} in the original goal {\small\verb%gl%}. All instances of {\small\verb%P x'%} which
do not appear in the original assumption {\small\verb%asml%} are added to it to
form {\small\verb%asml'%}, and they also become new subgoals {\small\verb%(asml,P x')%}.

\FAILURE
{\small\verb%RESQ_REWRITE1_TAC th%}  fails if {\small\verb%th%} cannot be transformed into the
required form by the function {\small\verb%RESQ_REWR_CANON%}. Otherwise, it fails if no 
match is found or the theorem cannot be instantiated.


\SEEALSO
RESQ_REWRITE1_CONV, RESQ_REWR_CANON, COND_REWR_TAC,
COND_REWRITE1_CONV, COND_REWR_CONV, COND_REWR_CANON, search_top_down.

\ENDDOC

\DOC{RESQ\_SPEC\_ALL}

\TYPE {\small\verb%RESQ_SPEC_ALL : (thm -> thm)%}\egroup

\SYNOPSIS
Specializes the conclusion of a theorem with its own restricted
quantified variables.

\DESCRIBE
When applied to a theorem {\small\verb%A |- !x1::P1. ...!xn::Pn. t%},
the inference rule {\small\verb%RESQ_SPEC_ALL%}
returns the theorem {\small\verb%A,P1 x1',...,Pn xn' |- t[x1'/x1]...[xn'/xn]%}
 where the {\small\verb%xi'%} are distinct
variants of the corresponding {\small\verb%xi%}, chosen to avoid clashes with any variables
free in the assumption list and with the names of constants. Normally {\small\verb%xi'%} is
just {\small\verb%xi%}, in which case {\small\verb%RESQ_SPEC_ALL%} simply removes all restricted
 universal quantifiers.
{\par\samepage\setseps\small
\begin{verbatim}
       A |- !x1::P1. ... !xn::Pn. t
   -------------------------------------------  RESQ_SPEC_ALL
    A,P1 x1,...,Pn xn |- t[x1'/x1]...[xn'/xn]
\end{verbatim}
}

\FAILURE
Never fails.

\SEEALSO
RESQ_GEN, RESQ_GENL, RESQ_GEN_ALL, RESQ_GEN_TAC, RESQ_SPEC, RESQ_SPECL.

\ENDDOC
\DOC{RESQ\_SPEC}

\TYPE {\small\verb%RESQ_SPEC : (term -> thm -> thm)%}\egroup

\SYNOPSIS
Specializes the conclusion of a restricted universally quantified theorem.

\DESCRIBE
When applied to a term {\small\verb%u%} and a theorem {\small\verb%A |- !x::P. t%}, {\small\verb%RESQ_SPEC%} returns
the theorem {\small\verb%A, P u |- t[u/x]%}. If necessary, variables will be renamed prior
to the specialization to ensure that {\small\verb%u%} is free for {\small\verb%x%} in {\small\verb%t%}, that is,
no variables free in {\small\verb%u%} become bound after substitution.
{\par\samepage\setseps\small
\begin{verbatim}
      A |- !x::P. t
   ------------------  RESQ_SPEC "u"
    A, P u |- t[u/x]
\end{verbatim}
}
\FAILURE
Fails if the theorem's conclusion is not restricted universally quantified,
or if type instantiation fails.

\EXAMPLE
The following example shows how {\small\verb%RESQ_SPEC%} renames bound variables if necessary,
prior to substitution: a straightforward substitution would result in the
clearly invalid theorem {\small\verb%(\y. 0 < y)y |- y = y%}.
{\par\samepage\setseps\small
\begin{verbatim}
   #let th = RESQ_GEN "x:num" "\y.0<y" (REFL "x:num");;
   th = |- !x :: \y. 0 < y. x = x

   #RESQ_SPEC "y:num" th;;
   (\y'. 0 < y')y |- y = y
\end{verbatim}
}
\SEEALSO
RESQ_SPECL, RESQ_SPEC_ALL, RESQ_GEN, RESQ_GENL, RESQ_GEN_ALL.

\ENDDOC
\DOC{RESQ\_SPECL}

\TYPE {\small\verb%RESQ_SPECL : (term list -> thm -> thm)%}\egroup

\SYNOPSIS
Specializes zero or more variables in the conclusion of a restricted
universally quantified theorem.

\DESCRIBE
When applied to a term list {\small\verb%[u1;...;un]%} and a theorem
{\small\verb%A |- !x1::P1. ... !xn::Pn. t%}, the inference rule {\small\verb%RESQ_SPECL%} returns
the theorem
{\par\samepage\setseps\small
\begin{verbatim}
   A,P1 u1,...,Pn un |- t[u1/x1]...[un/xn]
\end{verbatim}
}
\noindent where the substitutions are made
sequentially left-to-right in the same way as for {\small\verb%RESQ_SPEC%}, with the same
sort of alpha-conversions applied to {\small\verb%t%} if necessary to ensure that no
variables which are free in {\small\verb%ui%} become bound after substitution.
{\par\samepage\setseps\small
\begin{verbatim}
           A |- !x1::P1. ... !xn::Pn. t
   --------------------------------------------  RESQ_SPECL "[u1;...;un]"
     A,P1 u1, ..., Pn un |- t[u1/x1]...[un/xn]
\end{verbatim}
}
\noindent It is permissible for the term-list to be empty, in which case
the application of {\small\verb%RESQ_SPECL%} has no effect.

\FAILURE
Fails if one of the specialization of the 
restricted universally quantified variable in the original theorem fails.

\SEEALSO
RESQ_GEN, RESQ_GENL, RESQ_GEN_ALL, RESQ_GEN_TAC, RESQ_SPEC, RESQ_SPEC_ALL.

\ENDDOC
\DOC{search\_top\_down}

{\small
\begin{verbatim}
search_top_down
 : (term -> term -> ((term # term) list # (type # type) list) list)
\end{verbatim}
}\egroup

\SYNOPSIS
Search a term in a top-down fashion to find matches to another term.

\DESCRIBE
{\small\verb%search_top_down tm1 tm2%} returns a list of instantiations which make
the whole  or part of {\small\verb%tm2%} match {\small\verb%tm1%}. The first term should not have
a quantifier at the outer most level. {\small\verb%search_top_down%} first
attempts to match the whole second term to {\small\verb%tm1%}. If this fails, it
recursively descend into the subterms of {\small\verb%tm2%} to find all matches.

The length of the returned list indicates the number of matches found.
An empty list means no match can be found between {\small\verb%tm1%} and {\small\verb%tm2%} or
any subterms of {\small\verb%tm2%}.
The instantiations returned in the list are in the same format as for  the
function {\small\verb%match%}. Each instantiation is a pair of lists: the first is
a list of term pairs and the second is a list of type pairs. Either of
these lists may be empty. The situation in which both lists are empty
indicates that there is an exact match between the two terms, i.e., no
instantiation is required to make the entire {\small\verb%tm2%} or a part of {\small\verb%tm2%}
the same as {\small\verb%tm1%}.

\FAILURE
Never fails.

\EXAMPLE
{\par\samepage\setseps\small
\begin{verbatim}
   #search_top_down "x = y:*" "3 = 5";;
   [([("5", "y"); ("3", "x")], [(":num", ":*")])]
   : ((term # term) list # (type # type) list) list

   #search_top_down "x = y:*" "x =y:*";;
   [([], [])] : ((term # term) list # (type # type) list) list

   #search_top_down "x = y:*" "0 < p ==> (x <= p = y <= p)";;
   [([("y <= p", "y"); ("x <= p", "x")], [(":bool", ":*")])]
   : ((term # term) list # (type # type) list) list
\end{verbatim}
}
\noindent The first example above shows the entire {\small\verb%tm2%} matching {\small\verb%tm1%}.
The second example shows the two terms match exactly. No
instantiation is required. The last example shows that a subterm of
{\small\verb%tm2%} can be instantiated to match {\small\verb%tm1%}.

\SEEALSO
match, COND_REWR_TAC, CONV_REWRITE_TAC, COND_REWR_CONV, CONV_REWRITE_CONV.

\ENDDOC


\DOC{strip\_resq\_exists}

\TYPE {\small\verb%strip_resq_exists : (term -> ((term # term) list # term))%}\egroup

\SYNOPSIS
Iteratively breaks apart a restricted existentially quantified term.

\DESCRIBE
{\small\verb%strip_resq_exists%} is an iterative term destructor for restricted existential
quantifications. It iteratively breaks apart a restricted existentially
quantified term into a list of pairs which are the restricted quantified
variables and predicates and the body.
{\par\samepage\setseps\small
\begin{verbatim}
   strip_resq_exists "?x1::P1. ... ?xn::Pn. t"
\end{verbatim}
}
\noindent returns {\small\verb%([("x1","P1");...;("xn","Pn")],"t")%}.

\FAILURE
Never fails.

\SEEALSO
list_mk_resq_exists, is_resq_exists, dest_resq_exists.

\ENDDOC
\DOC{strip\_resq\_forall}

\TYPE {\small\verb%strip_resq_forall : (term -> ((term # term) list # term))%}\egroup

\SYNOPSIS
Iteratively breaks apart a restricted universally quantified term.

\DESCRIBE
{\small\verb%strip_resq_forall%} is an iterative term destructor for restricted universal
quantifications. It iteratively breaks apart a restricted universally
quantified term into a list of pairs which are the restricted quantified
variables and predicates and the body.
{\par\samepage\setseps\small
\begin{verbatim}
   strip_resq_forall "!x1::P1. ... !xn::Pn. t"
\end{verbatim}
}
\noindent returns {\small\verb%([("x1","P1");...;("xn","Pn")],"t")%}.

\FAILURE
Never fails.

\SEEALSO
list_mk_resq_forall, is_resq_forall, dest_resq_forall.

\ENDDOC