summaryrefslogtreecommitdiff
path: root/plugins/micromega/sos.ml
blob: 6ddc48e733a071e2c0940c803118c0367ff5c3a8 (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
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
(* ========================================================================= *)
(* - This code originates from John Harrison's HOL LIGHT 2.30                *)
(*   (see file LICENSE.sos for license, copyright and disclaimer)            *)
(* - Laurent Théry (thery@sophia.inria.fr) has isolated the HOL              *)
(*   independent bits                                                        *)
(* - Frédéric Besson (fbesson@irisa.fr) is using it to feed  micromega       *)
(* ========================================================================= *)

(* ========================================================================= *)
(* Nonlinear universal reals procedure using SOS decomposition.              *)
(* ========================================================================= *)
open Num;;
open List;;
open Sos_types;;
open Sos_lib;;

(*
prioritize_real();;
*)

let debugging = ref false;;

exception Sanity;;

exception Unsolvable;;

(* ------------------------------------------------------------------------- *)
(* Turn a rational into a decimal string with d sig digits.                  *)
(* ------------------------------------------------------------------------- *)

let decimalize =
  let rec normalize y =
    if abs_num y </ Int 1 // Int 10 then normalize (Int 10 */ y) - 1
    else if abs_num y >=/ Int 1 then normalize (y // Int 10) + 1
    else 0 in
  fun d x ->
    if x =/ Int 0 then "0.0" else
    let y = abs_num x in
    let e = normalize y in
    let z = pow10(-e) */ y +/ Int 1 in
    let k = round_num(pow10 d */ z) in
    (if x </ Int 0 then "-0." else "0.") ^
    implode(tl(explode(string_of_num k))) ^
    (if e = 0 then "" else "e"^string_of_int e);;

(* ------------------------------------------------------------------------- *)
(* Iterations over numbers, and lists indexed by numbers.                    *)
(* ------------------------------------------------------------------------- *)

let rec itern k l f a =
  match l with
    [] -> a
  | h::t -> itern (k + 1) t f (f h k a);;

let rec iter (m,n) f a =
  if n < m then a
  else iter (m+1,n) f (f m a);;

(* ------------------------------------------------------------------------- *)
(* The main types.                                                           *)
(* ------------------------------------------------------------------------- *)

type vector = int*(int,num)func;;

type matrix = (int*int)*(int*int,num)func;;

type monomial = (vname,int)func;;

type poly = (monomial,num)func;;

(* ------------------------------------------------------------------------- *)
(* Assignment avoiding zeros.                                                *)
(* ------------------------------------------------------------------------- *)

let (|-->) x y a = if y =/ Int 0 then a else (x |-> y) a;;

(* ------------------------------------------------------------------------- *)
(* This can be generic.                                                      *)
(* ------------------------------------------------------------------------- *)

let element (d,v) i = tryapplyd v i (Int 0);;

let mapa f (d,v) =
  d,foldl (fun a i c -> (i |--> f(c)) a) undefined v;;

let is_zero (d,v) =
  match v with
    Empty -> true
  | _ -> false;;

(* ------------------------------------------------------------------------- *)
(* Vectors. Conventionally indexed 1..n.                                     *)
(* ------------------------------------------------------------------------- *)

let vector_0 n = (n,undefined:vector);;

let dim (v:vector) = fst v;;

let vector_const c n =
  if c =/ Int 0 then vector_0 n
  else (n,itlist (fun k -> k |-> c) (1--n) undefined :vector);;

let vector_1 = vector_const (Int 1);;

let vector_cmul c (v:vector) =
  let n = dim v in
  if c =/ Int 0 then vector_0 n
  else n,mapf (fun x -> c */ x) (snd v)

let vector_neg (v:vector) = (fst v,mapf minus_num (snd v) :vector);;

let vector_add (v1:vector) (v2:vector) =
  let m = dim v1 and n = dim v2 in
  if m <> n then failwith "vector_add: incompatible dimensions" else
  (n,combine (+/) (fun x -> x =/ Int 0) (snd v1) (snd v2) :vector);;

let vector_sub v1 v2 = vector_add v1 (vector_neg v2);;

let vector_dot (v1:vector) (v2:vector) =
  let m = dim v1 and n = dim v2 in
  if m <> n then failwith "vector_add: incompatible dimensions" else
  foldl (fun a i x -> x +/ a) (Int 0)
        (combine ( */ ) (fun x -> x =/ Int 0) (snd v1) (snd v2));;

let vector_of_list l =
  let n = length l in
  (n,itlist2 (|->) (1--n) l undefined :vector);;

(* ------------------------------------------------------------------------- *)
(* Matrices; again rows and columns indexed from 1.                          *)
(* ------------------------------------------------------------------------- *)

let matrix_0 (m,n) = ((m,n),undefined:matrix);;

let dimensions (m:matrix) = fst m;;

let matrix_const c (m,n as mn) =
  if m <> n then failwith "matrix_const: needs to be square"
  else if c =/ Int 0 then matrix_0 mn
  else (mn,itlist (fun k -> (k,k) |-> c) (1--n) undefined :matrix);;

let matrix_1 = matrix_const (Int 1);;

let matrix_cmul c (m:matrix) =
  let (i,j) = dimensions m in
  if c =/ Int 0 then matrix_0 (i,j)
  else (i,j),mapf (fun x -> c */ x) (snd m);;

let matrix_neg (m:matrix) = (dimensions m,mapf minus_num (snd m) :matrix);;

let matrix_add (m1:matrix) (m2:matrix) =
  let d1 = dimensions m1 and d2 = dimensions m2 in
  if d1 <> d2 then failwith "matrix_add: incompatible dimensions"
  else (d1,combine (+/) (fun x -> x =/ Int 0) (snd m1) (snd m2) :matrix);;

let matrix_sub m1 m2 = matrix_add m1 (matrix_neg m2);;

let row k (m:matrix) =
  let i,j = dimensions m in
  (j,
   foldl (fun a (i,j) c -> if i = k then (j |-> c) a else a) undefined (snd m)
   : vector);;

let column k (m:matrix) =
  let i,j = dimensions m in
  (i,
   foldl (fun a (i,j) c -> if j = k then (i |-> c) a else a) undefined (snd m)
   : vector);;

let transp (m:matrix) =
  let i,j = dimensions m in
  ((j,i),foldl (fun a (i,j) c -> ((j,i) |-> c) a) undefined (snd m) :matrix);;

let diagonal (v:vector) =
  let n = dim v in
  ((n,n),foldl (fun a i c -> ((i,i) |-> c) a) undefined (snd v) : matrix);;

let matrix_of_list l =
  let m = length l in
  if m = 0 then matrix_0 (0,0) else
  let n = length (hd l) in
  (m,n),itern 1 l (fun v i -> itern 1 v (fun c j -> (i,j) |-> c)) undefined;;

(* ------------------------------------------------------------------------- *)
(* Monomials.                                                                *)
(* ------------------------------------------------------------------------- *)

let monomial_eval assig (m:monomial) =
  foldl (fun a x k -> a */ power_num (apply assig x) (Int k))
        (Int 1) m;;

let monomial_1 = (undefined:monomial);;

let monomial_var x = (x |=> 1 :monomial);;

let (monomial_mul:monomial->monomial->monomial) =
  combine (+) (fun x -> false);;

let monomial_pow (m:monomial) k =
  if k = 0 then monomial_1
  else mapf (fun x -> k * x) m;;

let monomial_divides (m1:monomial) (m2:monomial) =
  foldl (fun a x k -> tryapplyd m2 x 0 >= k & a) true m1;;

let monomial_div (m1:monomial) (m2:monomial) =
  let m = combine (+) (fun x -> x = 0) m1 (mapf (fun x -> -x) m2) in
  if foldl (fun a x k -> k >= 0 & a) true m then m
  else failwith "monomial_div: non-divisible";;

let monomial_degree x (m:monomial) = tryapplyd m x 0;;

let monomial_lcm (m1:monomial) (m2:monomial) =
  (itlist (fun x -> x |-> max (monomial_degree x m1) (monomial_degree x m2))
          (union (dom m1) (dom m2)) undefined :monomial);;

let monomial_multidegree (m:monomial) = foldl (fun a x k -> k + a) 0 m;;

let monomial_variables m = dom m;;

(* ------------------------------------------------------------------------- *)
(* Polynomials.                                                              *)
(* ------------------------------------------------------------------------- *)

let eval assig (p:poly) =
  foldl (fun a m c -> a +/ c */ monomial_eval assig m) (Int 0) p;;

let poly_0 = (undefined:poly);;

let poly_isconst (p:poly) = foldl (fun a m c -> m = monomial_1 & a) true p;;

let poly_var x = ((monomial_var x) |=> Int 1 :poly);;

let poly_const c =
  if c =/ Int 0 then poly_0 else (monomial_1 |=> c);;

let poly_cmul c (p:poly) =
  if c =/ Int 0 then poly_0
  else mapf (fun x -> c */ x) p;;

let poly_neg (p:poly) = (mapf minus_num p :poly);;

let poly_add (p1:poly) (p2:poly) =
  (combine (+/) (fun x -> x =/ Int 0) p1 p2 :poly);;

let poly_sub p1 p2 = poly_add p1 (poly_neg p2);;

let poly_cmmul (c,m) (p:poly) =
  if c =/ Int 0 then poly_0
  else if m = monomial_1 then mapf (fun d -> c */ d) p
  else foldl (fun a m' d -> (monomial_mul m m' |-> c */ d) a) poly_0 p;;

let poly_mul (p1:poly) (p2:poly) =
  foldl (fun a m c -> poly_add (poly_cmmul (c,m) p2) a) poly_0 p1;;

let poly_div (p1:poly) (p2:poly) =
  if not(poly_isconst p2) then failwith "poly_div: non-constant" else
  let c = eval undefined p2 in
  if c =/ Int 0 then failwith "poly_div: division by zero"
  else poly_cmul (Int 1 // c) p1;;

let poly_square p = poly_mul p p;;

let rec poly_pow p k =
  if k = 0 then poly_const (Int 1)
  else if k = 1 then p
  else let q = poly_square(poly_pow p (k / 2)) in
       if k mod 2 = 1 then poly_mul p q else q;;

let poly_exp p1 p2 =
  if not(poly_isconst p2) then failwith "poly_exp: not a constant" else
  poly_pow p1 (Num.int_of_num (eval undefined p2));;

let degree x (p:poly) = foldl (fun a m c -> max (monomial_degree x m) a) 0 p;;

let multidegree (p:poly) =
  foldl (fun a m c -> max (monomial_multidegree m) a) 0 p;;

let poly_variables (p:poly) =
  foldr (fun m c -> union (monomial_variables m)) p [];;

(* ------------------------------------------------------------------------- *)
(* Order monomials for human presentation.                                   *)
(* ------------------------------------------------------------------------- *)

let humanorder_varpow (x1,k1) (x2,k2) = x1 < x2 or x1 = x2 & k1 > k2;;

let humanorder_monomial =
  let rec ord l1 l2 = match (l1,l2) with
    _,[] -> true
  | [],_ -> false
  | h1::t1,h2::t2 -> humanorder_varpow h1 h2 or h1 = h2 & ord t1 t2 in
  fun m1 m2 -> m1 = m2 or
               ord (sort humanorder_varpow (graph m1))
                   (sort humanorder_varpow (graph m2));;

(* ------------------------------------------------------------------------- *)
(* Conversions to strings.                                                   *)
(* ------------------------------------------------------------------------- *)

let string_of_vector min_size max_size (v:vector) =
  let n_raw = dim v in
  if n_raw = 0 then "[]" else
  let n = max min_size (min n_raw max_size) in
  let xs = map ((o) string_of_num (element v)) (1--n) in
  "[" ^ end_itlist (fun s t -> s ^ ", " ^ t) xs ^
  (if n_raw > max_size then ", ...]" else "]");;

let string_of_matrix max_size (m:matrix) =
  let i_raw,j_raw = dimensions m in
  let i = min max_size i_raw and j = min max_size j_raw in
  let rstr = map (fun k -> string_of_vector j j (row k m)) (1--i) in
  "["^end_itlist(fun s t -> s^";\n "^t) rstr ^
  (if j > max_size then "\n ...]" else "]");;

let string_of_vname (v:vname): string = (v: string);;

let rec string_of_term t =
  match t with
  Opp t1 -> "(- " ^ string_of_term t1 ^ ")"
| Add (t1, t2) ->
   "(" ^ (string_of_term t1) ^ " + " ^ (string_of_term t2) ^ ")"
| Sub (t1, t2) ->
   "(" ^ (string_of_term t1) ^ " - " ^ (string_of_term t2) ^ ")"
| Mul (t1, t2) ->
   "(" ^ (string_of_term t1) ^ " * " ^ (string_of_term t2) ^ ")"
| Inv t1 -> "(/ " ^ string_of_term t1 ^ ")"
| Div (t1, t2) ->
   "(" ^ (string_of_term t1) ^ " / " ^ (string_of_term t2) ^ ")"
| Pow (t1, n1) ->
   "(" ^ (string_of_term t1) ^ " ^ " ^ (string_of_int n1) ^ ")"
| Zero -> "0"
| Var v -> "x" ^ (string_of_vname v)
| Const x -> string_of_num x;;


let string_of_varpow x k =
  if k = 1 then string_of_vname x else string_of_vname x^"^"^string_of_int k;;

let string_of_monomial m =
  if m = monomial_1 then "1" else
  let vps = List.fold_right (fun (x,k) a -> string_of_varpow x k :: a)
                            (sort humanorder_varpow (graph m)) [] in
  end_itlist (fun s t -> s^"*"^t) vps;;

let string_of_cmonomial (c,m) =
  if m = monomial_1 then string_of_num c
  else if c =/ Int 1 then string_of_monomial m
  else string_of_num c ^ "*" ^ string_of_monomial m;;

let string_of_poly (p:poly) =
  if p = poly_0 then "<<0>>" else
  let cms = sort (fun (m1,_) (m2,_) -> humanorder_monomial m1 m2) (graph p) in
  let s =
    List.fold_left (fun a (m,c) ->
             if c </ Int 0 then a ^ " - " ^ string_of_cmonomial(minus_num c,m)
             else a ^ " + " ^ string_of_cmonomial(c,m))
          "" cms in
  let s1 = String.sub s 0 3
  and s2 = String.sub s 3 (String.length s - 3) in
  "<<" ^(if s1 = " + " then s2 else "-"^s2)^">>";;

(* ------------------------------------------------------------------------- *)
(* Printers.                                                                 *)
(* ------------------------------------------------------------------------- *)

let print_vector v = Format.print_string(string_of_vector 0 20 v);;

let print_matrix m = Format.print_string(string_of_matrix 20 m);;

let print_monomial m = Format.print_string(string_of_monomial m);;

let print_poly m = Format.print_string(string_of_poly m);;

(*
#install_printer print_vector;;
#install_printer print_matrix;;
#install_printer print_monomial;;
#install_printer print_poly;;
*)

(* ------------------------------------------------------------------------- *)
(* Conversion from  term.                                                 *)
(* ------------------------------------------------------------------------- *)

let rec poly_of_term t = match t with
  Zero -> poly_0
| Const n -> poly_const n
| Var x -> poly_var x
| Opp t1 -> poly_neg (poly_of_term t1)
| Inv t1 ->
      let p = poly_of_term t1 in
      if poly_isconst p then poly_const(Int 1 // eval undefined p)
      else failwith "poly_of_term: inverse of non-constant polyomial"
| Add (l, r) -> poly_add (poly_of_term l) (poly_of_term r)
| Sub (l, r) -> poly_sub (poly_of_term l) (poly_of_term r)
| Mul (l, r) -> poly_mul (poly_of_term l) (poly_of_term r)
| Div (l, r) ->
      let p = poly_of_term l and q = poly_of_term r in
      if poly_isconst q then poly_cmul (Int 1 // eval undefined q) p
      else failwith "poly_of_term: division by non-constant polynomial"
| Pow (t, n) ->
      poly_pow (poly_of_term t) n;;

(* ------------------------------------------------------------------------- *)
(* String of vector (just a list of space-separated numbers).                *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_vector (v:vector) =
  let n = dim v in
  let strs = map (o (decimalize 20)  (element v)) (1--n) in
  end_itlist (fun x y -> x ^ " " ^ y) strs ^ "\n";;

(* ------------------------------------------------------------------------- *)
(* String for block diagonal matrix numbered k.                              *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_blockdiagonal k m =
  let pfx = string_of_int k ^" " in
  let ents =
    foldl (fun a (b,i,j) c -> if i > j then a else ((b,i,j),c)::a) [] m in
  let entss = sort (increasing fst) ents in
  itlist (fun ((b,i,j),c) a ->
     pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
     " " ^ decimalize 20 c ^ "\n" ^ a) entss "";;

(* ------------------------------------------------------------------------- *)
(* String for a matrix numbered k, in SDPA sparse format.                    *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_matrix k (m:matrix) =
  let pfx = string_of_int k ^ " 1 " in
  let ms = foldr (fun (i,j) c a -> if i > j then a else ((i,j),c)::a)
                 (snd m) [] in
  let mss = sort (increasing fst) ms in
  itlist (fun ((i,j),c) a ->
     pfx ^ string_of_int i ^ " " ^ string_of_int j ^
     " " ^ decimalize 20 c ^ "\n" ^ a) mss "";;

(* ------------------------------------------------------------------------- *)
(* String in SDPA sparse format for standard SDP problem:                    *)
(*                                                                           *)
(*    X = v_1 * [M_1] + ... + v_m * [M_m] - [M_0] must be PSD                *)
(*    Minimize obj_1 * v_1 + ... obj_m * v_m                                 *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_problem comment obj mats =
  let m = length mats - 1
  and n,_ = dimensions (hd mats) in
  "\"" ^ comment ^ "\"\n" ^
  string_of_int m ^ "\n" ^
  "1\n" ^
  string_of_int n ^ "\n" ^
  sdpa_of_vector obj ^
  itlist2 (fun k m a -> sdpa_of_matrix (k - 1) m ^ a)
          (1--length mats) mats "";;

(* ------------------------------------------------------------------------- *)
(* More parser basics.                                                       *)
(* ------------------------------------------------------------------------- *)

let word s =
   end_itlist (fun p1 p2 -> (p1 ++ p2) >> (fun (s,t) -> s^t))
              (map a (explode s));;
let token s =
  many (some isspace) ++ word s ++ many (some isspace)
  >> (fun ((_,t),_) -> t);;

let decimal =
  let numeral = some isnum in
  let decimalint = atleast 1 numeral >> ((o) Num.num_of_string  implode) in
  let decimalfrac = atleast 1 numeral
    >> (fun s -> Num.num_of_string(implode s) // pow10 (length s)) in
  let decimalsig =
    decimalint ++ possibly (a "." ++ decimalfrac >> snd)
    >> (function (h,[x]) -> h +/ x | (h,_) -> h) in
  let signed prs =
       a "-" ++ prs >> ((o) minus_num snd)
    || a "+" ++ prs >> snd
    || prs in
  let exponent = (a "e" || a "E") ++ signed decimalint >> snd in
    signed decimalsig ++ possibly exponent
    >> (function (h,[x]) -> h */ power_num (Int 10) x | (h,_) -> h);;

let mkparser p s =
  let x,rst = p(explode s) in
  if rst = [] then x else failwith "mkparser: unparsed input";;

let parse_decimal = mkparser decimal;;

(* ------------------------------------------------------------------------- *)
(* Parse back a vector.                                                      *)
(* ------------------------------------------------------------------------- *)

let parse_sdpaoutput,parse_csdpoutput =
  let vector =
    token "{" ++ listof decimal (token ",") "decimal" ++ token "}"
               >> (fun ((_,v),_) -> vector_of_list v) in
  let rec skipupto dscr prs inp =
      (dscr ++ prs >> snd
    || some (fun c -> true) ++ skipupto dscr prs >> snd) inp in
  let ignore inp = (),[] in
  let sdpaoutput =
    skipupto (word "xVec" ++ token "=")
             (vector ++ ignore >> fst) in
  let csdpoutput =
    (decimal ++ many(a " " ++ decimal >> snd) >> (fun (h,t) -> h::t)) ++
    (a " " ++ a "\n" ++ ignore) >> ((o) vector_of_list fst) in
  mkparser sdpaoutput,mkparser csdpoutput;;

(* ------------------------------------------------------------------------- *)
(* Also parse the SDPA output to test success (CSDP yields a return code).   *)
(* ------------------------------------------------------------------------- *)

let sdpa_run_succeeded =
   let rec skipupto dscr prs inp =
      (dscr ++ prs >> snd
    || some (fun c -> true) ++ skipupto dscr prs >> snd) inp in
  let prs = skipupto (word "phase.value" ++ token "=")
                     (possibly (a "p") ++ possibly (a "d") ++
                      (word "OPT" || word "FEAS")) in
  fun s -> try ignore (prs (explode s)); true with Noparse -> false;;

(* ------------------------------------------------------------------------- *)
(* The default parameters. Unfortunately this goes to a fixed file.          *)
(* ------------------------------------------------------------------------- *)

let sdpa_default_parameters =
"100     unsigned int maxIteration;\
\n1.0E-7  double 0.0 < epsilonStar;\
\n1.0E2   double 0.0 < lambdaStar;\
\n2.0     double 1.0 < omegaStar;\
\n-1.0E5  double lowerBound;\
\n1.0E5   double upperBound;\
\n0.1     double 0.0 <= betaStar <  1.0;\
\n0.2     double 0.0 <= betaBar  <  1.0, betaStar <= betaBar;\
\n0.9     double 0.0 < gammaStar  <  1.0;\
\n1.0E-7  double 0.0 < epsilonDash;\
\n";;

(* ------------------------------------------------------------------------- *)
(* These were suggested by Makoto Yamashita for problems where we are        *)
(* right at the edge of the semidefinite cone, as sometimes happens.         *)
(* ------------------------------------------------------------------------- *)

let sdpa_alt_parameters =
"1000    unsigned int maxIteration;\
\n1.0E-7  double 0.0 < epsilonStar;\
\n1.0E4   double 0.0 < lambdaStar;\
\n2.0     double 1.0 < omegaStar;\
\n-1.0E5  double lowerBound;\
\n1.0E5   double upperBound;\
\n0.1     double 0.0 <= betaStar <  1.0;\
\n0.2     double 0.0 <= betaBar  <  1.0, betaStar <= betaBar;\
\n0.9     double 0.0 < gammaStar  <  1.0;\
\n1.0E-7  double 0.0 < epsilonDash;\
\n";;

let sdpa_params = sdpa_alt_parameters;;

(* ------------------------------------------------------------------------- *)
(* CSDP parameters; so far I'm sticking with the defaults.                   *)
(* ------------------------------------------------------------------------- *)

let csdp_default_parameters =
"axtol=1.0e-8\
\natytol=1.0e-8\
\nobjtol=1.0e-8\
\npinftol=1.0e8\
\ndinftol=1.0e8\
\nmaxiter=100\
\nminstepfrac=0.9\
\nmaxstepfrac=0.97\
\nminstepp=1.0e-8\
\nminstepd=1.0e-8\
\nusexzgap=1\
\ntweakgap=0\
\naffine=0\
\nprintlevel=1\
\n";;

let csdp_params = csdp_default_parameters;;

(* ------------------------------------------------------------------------- *)
(* Now call CSDP on a problem and parse back the output.                     *)
(* ------------------------------------------------------------------------- *)

let run_csdp dbg obj mats =
  let input_file = Filename.temp_file "sos" ".dat-s" in
  let output_file =
    String.sub input_file 0 (String.length input_file - 6) ^ ".out"
  and params_file = Filename.concat (!temp_path) "param.csdp" in
  file_of_string input_file (sdpa_of_problem "" obj mats);
  file_of_string params_file csdp_params;
  let rv = Sys.command("cd "^(!temp_path)^"; csdp "^input_file ^
                        " " ^ output_file ^
                       (if dbg then "" else "> /dev/null")) in
  let op = string_of_file output_file in
  let res = parse_csdpoutput op in
  ((if dbg then ()
    else (Sys.remove input_file; Sys.remove output_file));
    rv,res);;

let csdp obj mats =
  let rv,res = run_csdp (!debugging) obj mats in
  (if rv = 1 or rv = 2 then failwith "csdp: Problem is infeasible"
   else if rv = 3 then ()
    (* Format.print_string "csdp warning: Reduced accuracy";
     Format.print_newline() *)
   else if rv <> 0 then failwith("csdp: error "^string_of_int rv)
   else ());
  res;;

(* ------------------------------------------------------------------------- *)
(* Try some apparently sensible scaling first. Note that this is purely to   *)
(* get a cleaner translation to floating-point, and doesn't affect any of    *)
(* the results, in principle. In practice it seems a lot better when there   *)
(* are extreme numbers in the original problem.                              *)
(* ------------------------------------------------------------------------- *)

let scale_then =
  let common_denominator amat acc =
    foldl (fun a m c -> lcm_num (denominator c) a) acc amat
  and maximal_element amat acc =
    foldl (fun maxa m c -> max_num maxa (abs_num c)) acc amat in
  fun solver obj mats ->
    let cd1 = itlist common_denominator mats (Int 1)
    and cd2 = common_denominator (snd obj)  (Int 1) in
    let mats' = map (mapf (fun x -> cd1 */ x)) mats
    and obj' = vector_cmul cd2 obj in
    let max1 = itlist maximal_element mats' (Int 0)
    and max2 = maximal_element (snd obj') (Int 0) in
    let scal1 = pow2 (20-int_of_float(log(float_of_num max1) /. log 2.0))
    and scal2 = pow2 (20-int_of_float(log(float_of_num max2) /. log 2.0)) in
    let mats'' = map (mapf (fun x -> x */ scal1)) mats'
    and obj'' = vector_cmul scal2 obj' in
    solver obj'' mats'';;

(* ------------------------------------------------------------------------- *)
(* Round a vector to "nice" rationals.                                       *)
(* ------------------------------------------------------------------------- *)

let nice_rational n x = round_num (n */ x) // n;;

let nice_vector n = mapa (nice_rational n);;

(* ------------------------------------------------------------------------- *)
(* Reduce linear program to SDP (diagonal matrices) and test with CSDP. This *)
(* one tests A [-1;x1;..;xn] >= 0 (i.e. left column is negated constants).   *)
(* ------------------------------------------------------------------------- *)

let linear_program_basic a =
  let m,n = dimensions a in
  let mats =  map (fun j -> diagonal (column j a)) (1--n)
  and obj = vector_const (Int 1) m in
  let rv,res = run_csdp false obj mats in
  if rv = 1 or rv = 2 then false
  else if rv = 0 then true
  else failwith "linear_program: An error occurred in the SDP solver";;

(* ------------------------------------------------------------------------- *)
(* Alternative interface testing A x >= b for matrix A, vector b.            *)
(* ------------------------------------------------------------------------- *)

let linear_program a b =
  let m,n = dimensions a in
  if dim b <> m then failwith "linear_program: incompatible dimensions" else
  let mats = diagonal b :: map (fun j -> diagonal (column j a)) (1--n)
  and obj = vector_const (Int 1) m in
  let rv,res = run_csdp false obj mats in
  if rv = 1 or rv = 2 then false
  else if rv = 0 then true
  else failwith "linear_program: An error occurred in the SDP solver";;

(* ------------------------------------------------------------------------- *)
(* Test whether a point is in the convex hull of others. Rather than use     *)
(* computational geometry, express as linear inequalities and call CSDP.     *)
(* This is a bit lazy of me, but it's easy and not such a bottleneck so far. *)
(* ------------------------------------------------------------------------- *)

let in_convex_hull pts pt =
  let pts1 = (1::pt) :: map (fun x -> 1::x) pts in
  let pts2 = map (fun p -> map (fun x -> -x) p @ p) pts1 in
  let n = length pts + 1
  and v = 2 * (length pt + 1) in
  let m = v + n - 1 in
  let mat =
    (m,n),
    itern 1 pts2 (fun pts j -> itern 1 pts (fun x i -> (i,j) |-> Int x))
                 (iter (1,n) (fun i -> (v + i,i+1) |-> Int 1) undefined) in
  linear_program_basic mat;;

(* ------------------------------------------------------------------------- *)
(* Filter down a set of points to a minimal set with the same convex hull.   *)
(* ------------------------------------------------------------------------- *)

let minimal_convex_hull =
  let augment1 = function
    | [] -> assert false
    | (m::ms) -> if in_convex_hull ms m then ms else ms@[m] in
  let augment m ms = funpow 3 augment1 (m::ms) in
  fun mons ->
    let mons' = itlist augment (tl mons) [hd mons] in
    funpow (length mons') augment1 mons';;

(* ------------------------------------------------------------------------- *)
(* Stuff for "equations" (generic A->num functions).                         *)
(* ------------------------------------------------------------------------- *)

let equation_cmul c eq =
  if c =/ Int 0 then Empty else mapf (fun d -> c */ d) eq;;

let equation_add eq1 eq2 = combine (+/) (fun x -> x =/ Int 0) eq1 eq2;;

let equation_eval assig eq =
  let value v = apply assig v in
  foldl (fun a v c -> a +/ value(v) */ c) (Int 0) eq;;

(* ------------------------------------------------------------------------- *)
(* Eliminate among linear equations: return unconstrained variables and      *)
(* assignments for the others in terms of them. We give one pseudo-variable  *)
(* "one" that's used for a constant term.                                    *)
(* ------------------------------------------------------------------------- *)

let failstore = ref [];;

let eliminate_equations =
  let rec extract_first p l =
    match l with
      [] -> failwith "extract_first"
    | h::t -> if p(h) then h,t else
              let k,s = extract_first p t in
              k,h::s in
  let rec eliminate vars dun eqs =
    match vars with
      [] -> if forall is_undefined eqs then dun
            else (failstore := [vars,dun,eqs]; raise Unsolvable)
    | v::vs ->
            try let eq,oeqs = extract_first (fun e -> defined e v) eqs in
                let a = apply eq v in
                let eq' = equation_cmul (Int(-1) // a) (undefine v eq) in
                let elim e =
                  let b = tryapplyd e v (Int 0) in
                  if b =/ Int 0 then e else
                  equation_add e (equation_cmul (minus_num b // a) eq) in
                eliminate vs ((v |-> eq') (mapf elim dun)) (map elim oeqs)
            with Failure _ -> eliminate vs dun eqs in
  fun one vars eqs ->
    let assig = eliminate vars undefined eqs in
    let vs = foldl (fun a x f -> subtract (dom f) [one] @ a) [] assig in
    setify vs,assig;;

(* ------------------------------------------------------------------------- *)
(* Eliminate all variables, in an essentially arbitrary order.               *)
(* ------------------------------------------------------------------------- *)

let eliminate_all_equations one =
  let choose_variable eq =
    let (v,_) = choose eq in
    if v = one then
      let eq' = undefine v eq in
      if is_undefined eq' then failwith "choose_variable" else
      let (w,_) = choose eq' in w
    else v in
  let rec eliminate dun eqs =
    match eqs with
      [] -> dun
    | eq::oeqs ->
        if is_undefined eq then eliminate dun oeqs else
        let v = choose_variable eq in
        let a = apply eq v in
        let eq' = equation_cmul (Int(-1) // a) (undefine v eq) in
        let elim e =
          let b = tryapplyd e v (Int 0) in
          if b =/ Int 0 then e else
          equation_add e (equation_cmul (minus_num b // a) eq) in
        eliminate ((v |-> eq') (mapf elim dun)) (map elim oeqs) in
  fun eqs ->
    let assig = eliminate undefined eqs in
    let vs = foldl (fun a x f -> subtract (dom f) [one] @ a) [] assig in
    setify vs,assig;;

(* ------------------------------------------------------------------------- *)
(* Solve equations by assigning arbitrary numbers.                           *)
(* ------------------------------------------------------------------------- *)

let solve_equations one eqs =
  let vars,assigs = eliminate_all_equations one eqs in
  let vfn = itlist (fun v -> (v |-> Int 0)) vars (one |=> Int(-1)) in
  let ass =
    combine (+/) (fun c -> false) (mapf (equation_eval vfn) assigs) vfn in
  if forall (fun e -> equation_eval ass e =/ Int 0) eqs
  then undefine one ass else raise Sanity;;

(* ------------------------------------------------------------------------- *)
(* Hence produce the "relevant" monomials: those whose squares lie in the    *)
(* Newton polytope of the monomials in the input. (This is enough according  *)
(* to Reznik: "Extremal PSD forms with few terms", Duke Math. Journal,       *)
(* vol 45, pp. 363--374, 1978.                                               *)
(*                                                                           *)
(* These are ordered in sort of decreasing degree. In particular the         *)
(* constant monomial is last; this gives an order in diagonalization of the  *)
(* quadratic form that will tend to display constants.                       *)
(* ------------------------------------------------------------------------- *)

let newton_polytope pol =
  let vars = poly_variables pol in
  let mons = map (fun m -> map (fun x -> monomial_degree x m) vars) (dom pol)
  and ds = map (fun x -> (degree x pol + 1) / 2) vars in
  let all = itlist (fun n -> allpairs (fun h t -> h::t) (0--n)) ds [[]]
  and mons' = minimal_convex_hull mons in
  let all' =
    filter (fun m -> in_convex_hull mons' (map (fun x -> 2 * x) m)) all in
  map (fun m -> itlist2 (fun v i a -> if i = 0 then a else (v |-> i) a)
                        vars m monomial_1) (rev all');;

(* ------------------------------------------------------------------------- *)
(* Diagonalize (Cholesky/LDU) the matrix corresponding to a quadratic form.  *)
(* ------------------------------------------------------------------------- *)

let diag m =
  let nn = dimensions m in
  let n = fst nn in
  if snd nn <> n then failwith "diagonalize: non-square matrix" else
  let rec diagonalize i m =
    if is_zero m then [] else
    let a11 = element m (i,i) in
    if a11 </ Int 0 then failwith "diagonalize: not PSD"
    else if a11 =/ Int 0 then
      if is_zero(row i m) then diagonalize (i + 1) m
      else failwith "diagonalize: not PSD"
    else
      let v = row i m in
      let v' = mapa (fun a1k -> a1k // a11) v in
      let m' =
      (n,n),
      iter (i+1,n) (fun j ->
          iter (i+1,n) (fun k ->
              ((j,k) |--> (element m (j,k) -/ element v j */ element v' k))))
          undefined in
      (a11,v')::diagonalize (i + 1) m' in
  diagonalize 1 m;;

(* ------------------------------------------------------------------------- *)
(* Adjust a diagonalization to collect rationals at the start.               *)
(* ------------------------------------------------------------------------- *)

let deration d =
  if d = [] then Int 0,d else
  let adj(c,l) =
    let a = foldl (fun a i c -> lcm_num a (denominator c)) (Int 1) (snd l) //
            foldl (fun a i c -> gcd_num a (numerator c)) (Int 0) (snd l) in
    (c // (a */ a)),mapa (fun x -> a */ x) l in
  let d' = map adj d in
  let a = itlist ((o) lcm_num ( (o) denominator fst)) d' (Int 1) //
          itlist ((o) gcd_num ( (o) numerator fst)) d' (Int 0)  in
  (Int 1 // a),map (fun (c,l) -> (a */ c,l)) d';;

(* ------------------------------------------------------------------------- *)
(* Enumeration of monomials with given multidegree bound.                    *)
(* ------------------------------------------------------------------------- *)

let rec enumerate_monomials d vars =
  if d < 0 then []
  else if d = 0 then [undefined]
  else if vars = [] then [monomial_1] else
  let alts =
    map (fun k -> let oths = enumerate_monomials (d - k) (tl vars) in
                  map (fun ks -> if k = 0 then ks else (hd vars |-> k) ks) oths)
        (0--d) in
  end_itlist (@) alts;;

(* ------------------------------------------------------------------------- *)
(* Enumerate products of distinct input polys with degree <= d.              *)
(* We ignore any constant input polynomials.                                 *)
(* Give the output polynomial and a record of how it was derived.            *)
(* ------------------------------------------------------------------------- *)

let rec enumerate_products d pols =
  if d = 0 then [poly_const num_1,Rational_lt num_1] else if d < 0 then [] else
  match pols with
    [] -> [poly_const num_1,Rational_lt num_1]
  | (p,b)::ps -> let e = multidegree p in
                 if e = 0 then enumerate_products d ps else
                 enumerate_products d ps @
                 map (fun (q,c) -> poly_mul p q,Product(b,c))
                     (enumerate_products (d - e) ps);;

(* ------------------------------------------------------------------------- *)
(* Multiply equation-parametrized poly by regular poly and add accumulator.  *)
(* ------------------------------------------------------------------------- *)

let epoly_pmul p q acc =
  foldl (fun a m1 c ->
           foldl (fun b m2 e ->
                        let m =  monomial_mul m1 m2 in
                        let es = tryapplyd b m undefined in
                        (m |-> equation_add (equation_cmul c e) es) b)
                 a q) acc p;;

(* ------------------------------------------------------------------------- *)
(* Usual operations on equation-parametrized poly.                           *)
(* ------------------------------------------------------------------------- *)

let epoly_cmul c l =
  if c =/ Int 0 then undefined else mapf (equation_cmul c) l;;

let epoly_neg = epoly_cmul (Int(-1));;

let epoly_add = combine equation_add is_undefined;;

let epoly_sub p q = epoly_add p (epoly_neg q);;

(* ------------------------------------------------------------------------- *)
(* Convert regular polynomial. Note that we treat (0,0,0) as -1.             *)
(* ------------------------------------------------------------------------- *)

let epoly_of_poly p =
  foldl (fun a m c -> (m |-> ((0,0,0) |=> minus_num c)) a) undefined p;;

(* ------------------------------------------------------------------------- *)
(* String for block diagonal matrix numbered k.                              *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_blockdiagonal k m =
  let pfx = string_of_int k ^" " in
  let ents =
    foldl (fun a (b,i,j) c -> if i > j then a else ((b,i,j),c)::a) [] m in
  let entss = sort (increasing fst) ents in
  itlist (fun ((b,i,j),c) a ->
     pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
     " " ^ decimalize 20 c ^ "\n" ^ a) entss "";;

(* ------------------------------------------------------------------------- *)
(* SDPA for problem using block diagonal (i.e. multiple SDPs)                *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_blockproblem comment nblocks blocksizes obj mats =
  let m = length mats - 1 in
  "\"" ^ comment ^ "\"\n" ^
  string_of_int m ^ "\n" ^
  string_of_int nblocks ^ "\n" ^
  (end_itlist (fun s t -> s^" "^t) (map string_of_int blocksizes)) ^
  "\n" ^
  sdpa_of_vector obj ^
  itlist2 (fun k m a -> sdpa_of_blockdiagonal (k - 1) m ^ a)
          (1--length mats) mats "";;

(* ------------------------------------------------------------------------- *)
(* Hence run CSDP on a problem in block diagonal form.                       *)
(* ------------------------------------------------------------------------- *)

let run_csdp dbg nblocks blocksizes obj mats =
  let input_file = Filename.temp_file "sos" ".dat-s" in
  let output_file =
    String.sub input_file 0 (String.length input_file - 6) ^ ".out"
  and params_file = Filename.concat (!temp_path) "param.csdp" in
  file_of_string input_file
   (sdpa_of_blockproblem "" nblocks blocksizes obj mats);
  file_of_string params_file csdp_params;
  let rv = Sys.command("cd "^(!temp_path)^"; csdp "^input_file ^
                        " " ^ output_file ^
                       (if dbg then "" else "> /dev/null")) in
  let op = string_of_file output_file in
  let res = parse_csdpoutput op in
  ((if dbg then ()
    else (Sys.remove input_file; Sys.remove output_file));
    rv,res);;

let csdp nblocks blocksizes obj mats =
  let rv,res = run_csdp (!debugging) nblocks blocksizes obj mats in
  (if rv = 1 or rv = 2 then failwith "csdp: Problem is infeasible"
   else if rv = 3 then ()
    (*Format.print_string "csdp warning: Reduced accuracy";
     Format.print_newline() *)
   else if rv <> 0 then failwith("csdp: error "^string_of_int rv)
   else ());
  res;;

(* ------------------------------------------------------------------------- *)
(* 3D versions of matrix operations to consider blocks separately.           *)
(* ------------------------------------------------------------------------- *)

let bmatrix_add = combine (+/) (fun x -> x =/ Int 0);;

let bmatrix_cmul c bm =
  if c =/ Int 0 then undefined
  else mapf (fun x -> c */ x) bm;;

let bmatrix_neg = bmatrix_cmul (Int(-1));;

let bmatrix_sub m1 m2 = bmatrix_add m1 (bmatrix_neg m2);;

(* ------------------------------------------------------------------------- *)
(* Smash a block matrix into components.                                     *)
(* ------------------------------------------------------------------------- *)

let blocks blocksizes bm =
  map (fun (bs,b0) ->
        let m = foldl
          (fun a (b,i,j) c -> if b = b0 then ((i,j) |-> c) a else a)
          undefined bm in
        (((bs,bs),m):matrix))
      (zip blocksizes (1--length blocksizes));;

(* ------------------------------------------------------------------------- *)
(* Positiv- and Nullstellensatz. Flag "linf" forces a linear representation. *)
(* ------------------------------------------------------------------------- *)

let real_positivnullstellensatz_general linf d eqs leqs pol =
  let vars = itlist ((o) union poly_variables) (pol::eqs @ map fst leqs) [] in
  let monoid =
    if linf then
      (poly_const num_1,Rational_lt num_1)::
      (filter (fun (p,c) -> multidegree p <= d) leqs)
    else enumerate_products d leqs in
  let nblocks = length monoid in
  let mk_idmultiplier k p =
    let e = d - multidegree p in
    let mons = enumerate_monomials e vars in
    let nons = zip mons (1--length mons) in
    mons,
    itlist (fun (m,n) -> (m |-> ((-k,-n,n) |=> Int 1))) nons undefined in
  let mk_sqmultiplier k (p,c) =
    let e = (d - multidegree p) / 2 in
    let mons = enumerate_monomials e vars in
    let nons = zip mons (1--length mons) in
    mons,
    itlist (fun (m1,n1) ->
      itlist (fun (m2,n2) a ->
          let m = monomial_mul m1 m2 in
          if n1 > n2 then a else
          let c = if n1 = n2 then Int 1 else Int 2 in
          let e = tryapplyd a m undefined in
          (m |-> equation_add ((k,n1,n2) |=> c) e) a)
         nons)
       nons undefined in
  let sqmonlist,sqs = unzip(map2 mk_sqmultiplier (1--length monoid) monoid)
  and idmonlist,ids =  unzip(map2 mk_idmultiplier (1--length eqs) eqs) in
  let blocksizes = map length sqmonlist in
  let bigsum =
    itlist2 (fun p q a -> epoly_pmul p q a) eqs ids
            (itlist2 (fun (p,c) s a -> epoly_pmul p s a) monoid sqs
                     (epoly_of_poly(poly_neg pol))) in
  let eqns = foldl (fun a m e -> e::a) [] bigsum in
  let pvs,assig = eliminate_all_equations (0,0,0) eqns in
  let qvars = (0,0,0)::pvs in
  let allassig = itlist (fun v -> (v |-> (v |=> Int 1))) pvs assig in
  let mk_matrix v =
    foldl (fun m (b,i,j) ass -> if b < 0 then m else
                                let c = tryapplyd ass v (Int 0) in
                                if c =/ Int 0 then m else
                                ((b,j,i) |-> c) (((b,i,j) |-> c) m))
          undefined allassig in
  let diagents = foldl
    (fun a (b,i,j) e -> if b > 0 & i = j then equation_add e a else a)
    undefined allassig in
  let mats = map mk_matrix qvars
  and obj = length pvs,
            itern 1 pvs (fun v i -> (i |--> tryapplyd diagents v (Int 0)))
                        undefined in
  let raw_vec = if pvs = [] then vector_0 0
                else scale_then (csdp nblocks blocksizes) obj mats in
  let find_rounding d =
   (if !debugging then
     (Format.print_string("Trying rounding with limit "^string_of_num d);
      Format.print_newline())
    else ());
    let vec = nice_vector d raw_vec in
    let blockmat = iter (1,dim vec)
     (fun i a -> bmatrix_add (bmatrix_cmul (element vec i) (el i mats)) a)
     (bmatrix_neg (el 0 mats)) in
    let allmats = blocks blocksizes blockmat in
    vec,map diag allmats in
  let vec,ratdias =
    if pvs = [] then find_rounding num_1
    else tryfind find_rounding (map Num.num_of_int (1--31) @
                                map pow2 (5--66)) in
  let newassigs =
    itlist (fun k -> el (k - 1) pvs |-> element vec k)
           (1--dim vec) ((0,0,0) |=> Int(-1)) in
  let finalassigs =
    foldl (fun a v e -> (v |-> equation_eval newassigs e) a) newassigs
          allassig in
  let poly_of_epoly p =
    foldl (fun a v e -> (v |--> equation_eval finalassigs e) a)
          undefined p in
  let mk_sos mons =
    let mk_sq (c,m) =
        c,itlist (fun k a -> (el (k - 1) mons |--> element m k) a)
                 (1--length mons) undefined in
    map mk_sq in
  let sqs = map2 mk_sos sqmonlist ratdias
  and cfs = map poly_of_epoly ids in
  let msq = filter (fun (a,b) -> b <> []) (map2 (fun a b -> a,b) monoid sqs) in
  let eval_sq sqs = itlist
   (fun (c,q) -> poly_add (poly_cmul c (poly_mul q q))) sqs poly_0 in
  let sanity =
    itlist (fun ((p,c),s) -> poly_add (poly_mul p (eval_sq s))) msq
           (itlist2 (fun p q -> poly_add (poly_mul p q)) cfs eqs
                    (poly_neg pol)) in
  if not(is_undefined sanity) then raise Sanity else
  cfs,map (fun (a,b) -> snd a,b) msq;;

(* ------------------------------------------------------------------------- *)
(* Iterative deepening.                                                      *)
(* ------------------------------------------------------------------------- *)

let rec deepen f n =
  try print_string "Searching with depth limit ";
      print_int n; print_newline(); f n
  with Failure _ -> deepen f (n + 1);;

(* ------------------------------------------------------------------------- *)
(* The ordering so we can create canonical HOL polynomials.                  *)
(* ------------------------------------------------------------------------- *)

let dest_monomial mon = sort (increasing fst) (graph mon);;

let monomial_order =
  let rec lexorder l1 l2 =
    match (l1,l2) with
      [],[] -> true
    | vps,[] -> false
    | [],vps -> true
    | ((x1,n1)::vs1),((x2,n2)::vs2) ->
          if x1 < x2 then true
          else if x2 < x1 then false
          else if n1 < n2 then false
          else if n2 < n1 then true
          else lexorder vs1 vs2 in
  fun m1 m2 ->
    if m2 = monomial_1 then true else if m1 = monomial_1 then false else
    let mon1 = dest_monomial m1 and mon2 = dest_monomial m2 in
    let deg1 = itlist ((o) (+) snd) mon1 0
    and deg2 = itlist ((o) (+) snd) mon2 0 in
    if deg1 < deg2 then false else if deg1 > deg2 then true
    else lexorder mon1 mon2;;

let dest_poly p =
  map (fun (m,c) -> c,dest_monomial m)
      (sort (fun (m1,_) (m2,_) -> monomial_order m1 m2) (graph p));;

(* ------------------------------------------------------------------------- *)
(* Map back polynomials and their composites to HOL.                         *)
(* ------------------------------------------------------------------------- *)

let term_of_varpow =
  fun x k ->
    if k = 1 then Var x else Pow (Var x, k);;

let term_of_monomial =
  fun m -> if m = monomial_1 then Const num_1 else
           let m' = dest_monomial m in
           let vps = itlist (fun (x,k) a -> term_of_varpow x k :: a) m' [] in
           end_itlist (fun s t -> Mul (s,t)) vps;;

let term_of_cmonomial =
  fun (m,c) ->
    if m = monomial_1 then Const c
    else if c =/ num_1 then term_of_monomial m
    else Mul (Const c,term_of_monomial m);;

let term_of_poly =
  fun p ->
    if p = poly_0 then Zero else
    let cms = map term_of_cmonomial
     (sort (fun (m1,_) (m2,_) -> monomial_order m1 m2) (graph p)) in
    end_itlist (fun t1 t2 -> Add (t1,t2)) cms;;

let term_of_sqterm (c,p) =
  Product(Rational_lt c,Square(term_of_poly p));;

let term_of_sos (pr,sqs) =
  if sqs = [] then pr
  else Product(pr,end_itlist (fun a b -> Sum(a,b)) (map term_of_sqterm sqs));;

(* ------------------------------------------------------------------------- *)
(* Interface to HOL.                                                         *)
(* ------------------------------------------------------------------------- *)
(*
let REAL_NONLINEAR_PROVER translator (eqs,les,lts) =
  let eq0 = map (poly_of_term o lhand o concl) eqs
  and le0 = map (poly_of_term o lhand o concl) les
  and lt0 = map (poly_of_term o lhand o concl) lts in
  let eqp0 = map (fun (t,i) -> t,Axiom_eq i) (zip eq0 (0--(length eq0 - 1)))
  and lep0 = map (fun (t,i) -> t,Axiom_le i) (zip le0 (0--(length le0 - 1)))
  and ltp0 = map (fun (t,i) -> t,Axiom_lt i) (zip lt0 (0--(length lt0 - 1))) in
  let keq,eq = partition (fun (p,_) -> multidegree p = 0) eqp0
  and klep,lep = partition (fun (p,_) -> multidegree p = 0) lep0
  and kltp,ltp = partition (fun (p,_) -> multidegree p = 0) ltp0 in
  let trivial_axiom (p,ax) =
    match ax with
      Axiom_eq n when eval undefined p <>/ num_0 -> el n eqs
    | Axiom_le n when eval undefined p </ num_0 -> el n les
    | Axiom_lt n when eval undefined p <=/ num_0 -> el n lts
    | _ -> failwith "not a trivial axiom" in
  try let th = tryfind trivial_axiom (keq @ klep @ kltp) in
      CONV_RULE (LAND_CONV REAL_POLY_CONV THENC REAL_RAT_RED_CONV) th
  with Failure _ ->
  let pol = itlist poly_mul (map fst ltp) (poly_const num_1) in
  let leq = lep @ ltp in
  let tryall d =
    let e = multidegree pol in
    let k = if e = 0 then 0 else d / e in
    let eq' = map fst eq in
    tryfind (fun i -> d,i,real_positivnullstellensatz_general false d eq' leq
                          (poly_neg(poly_pow pol i)))
            (0--k) in
  let d,i,(cert_ideal,cert_cone) = deepen tryall 0 in
  let proofs_ideal =
    map2 (fun q (p,ax) -> Eqmul(term_of_poly q,ax)) cert_ideal eq
  and proofs_cone = map term_of_sos cert_cone
  and proof_ne =
    if ltp = [] then Rational_lt num_1 else
    let p = end_itlist (fun s t -> Product(s,t)) (map snd ltp) in
    funpow i (fun q -> Product(p,q)) (Rational_lt num_1) in
  let proof = end_itlist (fun s t -> Sum(s,t))
                         (proof_ne :: proofs_ideal @ proofs_cone) in
  print_string("Translating proof certificate to HOL");
  print_newline();
  translator (eqs,les,lts) proof;;
*)
(* ------------------------------------------------------------------------- *)
(* A wrapper that tries to substitute away variables first.                  *)
(* ------------------------------------------------------------------------- *)
(*
let REAL_NONLINEAR_SUBST_PROVER =
  let zero = `&0:real`
  and mul_tm = `( * ):real->real->real`
  and shuffle1 =
    CONV_RULE(REWR_CONV(REAL_ARITH `a + x = (y:real) <=> x = y - a`))
  and shuffle2 =
    CONV_RULE(REWR_CONV(REAL_ARITH `x + a = (y:real) <=> x = y - a`)) in
  let rec substitutable_monomial fvs tm =
    match tm with
      Var(_,Tyapp("real",[])) when not (mem tm fvs) -> Int 1,tm
    | Comb(Comb(Const("real_mul",_),c),(Var(_,_) as t))
         when is_ratconst c & not (mem t fvs)
          -> rat_of_term c,t
    | Comb(Comb(Const("real_add",_),s),t) ->
         (try substitutable_monomial (union (frees t) fvs) s
          with Failure _ -> substitutable_monomial (union (frees s) fvs) t)
    | _ -> failwith "substitutable_monomial"
  and isolate_variable v th =
    match lhs(concl th) with
      x when x = v -> th
    | Comb(Comb(Const("real_add",_),(Var(_,Tyapp("real",[])) as x)),t)
        when x = v -> shuffle2 th
    | Comb(Comb(Const("real_add",_),s),t) ->
        isolate_variable v(shuffle1 th) in
  let make_substitution th =
    let (c,v) = substitutable_monomial [] (lhs(concl th)) in
    let th1 = AP_TERM (mk_comb(mul_tm,term_of_rat(Int 1 // c))) th in
    let th2 = CONV_RULE(BINOP_CONV REAL_POLY_MUL_CONV) th1 in
    CONV_RULE (RAND_CONV REAL_POLY_CONV) (isolate_variable v th2) in
  fun translator ->
    let rec substfirst(eqs,les,lts) =
      try let eth = tryfind make_substitution eqs in
          let modify =
            CONV_RULE(LAND_CONV(SUBS_CONV[eth] THENC REAL_POLY_CONV)) in
          substfirst(filter (fun t -> lhand(concl t) <> zero) (map modify eqs),
                     map modify les,map modify lts)
      with Failure _ -> REAL_NONLINEAR_PROVER translator (eqs,les,lts) in
    substfirst;;
*)
(* ------------------------------------------------------------------------- *)
(* Overall function.                                                         *)
(* ------------------------------------------------------------------------- *)
(*
let REAL_SOS =
  let init = GEN_REWRITE_CONV ONCE_DEPTH_CONV [DECIMAL]
  and pure = GEN_REAL_ARITH REAL_NONLINEAR_SUBST_PROVER in
  fun tm -> let th = init tm in EQ_MP (SYM th) (pure(rand(concl th)));;
*)
(* ------------------------------------------------------------------------- *)
(* Add hacks for division.                                                   *)
(* ------------------------------------------------------------------------- *)
(*
let REAL_SOSFIELD =
  let inv_tm = `inv:real->real` in
  let prenex_conv =
    TOP_DEPTH_CONV BETA_CONV THENC
    PURE_REWRITE_CONV[FORALL_SIMP; EXISTS_SIMP; real_div;
                      REAL_INV_INV; REAL_INV_MUL; GSYM REAL_POW_INV] THENC
    NNFC_CONV THENC DEPTH_BINOP_CONV `(/\)` CONDS_CELIM_CONV THENC
    PRENEX_CONV
  and setup_conv = NNF_CONV THENC WEAK_CNF_CONV THENC CONJ_CANON_CONV
  and core_rule t =
    try REAL_ARITH t
    with Failure _ -> try REAL_RING t
    with Failure _ -> REAL_SOS t
  and is_inv =
    let is_div = is_binop `(/):real->real->real` in
    fun tm -> (is_div tm or (is_comb tm & rator tm = inv_tm)) &
              not(is_ratconst(rand tm)) in
  let BASIC_REAL_FIELD tm =
    let is_freeinv t = is_inv t & free_in t tm in
    let itms = setify(map rand (find_terms is_freeinv tm)) in
    let hyps = map (fun t -> SPEC t REAL_MUL_RINV) itms in
    let tm' = itlist (fun th t -> mk_imp(concl th,t)) hyps tm in
    let itms' = map (curry mk_comb inv_tm) itms in
    let gvs = map (genvar o type_of) itms' in
    let tm'' = subst (zip gvs itms') tm' in
    let th1 = setup_conv tm'' in
    let cjs = conjuncts(rand(concl th1)) in
    let ths = map core_rule cjs in
    let th2 = EQ_MP (SYM th1) (end_itlist CONJ ths) in
    rev_itlist (C MP) hyps (INST (zip itms' gvs) th2) in
  fun tm ->
    let th0 = prenex_conv tm in
    let tm0 = rand(concl th0) in
    let avs,bod = strip_forall tm0 in
    let th1 = setup_conv bod in
    let ths = map BASIC_REAL_FIELD (conjuncts(rand(concl th1))) in
    EQ_MP (SYM th0) (GENL avs (EQ_MP (SYM th1) (end_itlist CONJ ths)));;
*)
(* ------------------------------------------------------------------------- *)
(* Integer version.                                                          *)
(* ------------------------------------------------------------------------- *)
(*
let INT_SOS =
  let atom_CONV =
    let pth = prove
      (`(~(x <= y) <=> y + &1 <= x:int) /\
        (~(x < y) <=> y <= x) /\
        (~(x = y) <=> x + &1 <= y \/ y + &1 <= x) /\
        (x < y <=> x + &1 <= y)`,
       REWRITE_TAC[INT_NOT_LE; INT_NOT_LT; INT_NOT_EQ; INT_LT_DISCRETE]) in
    GEN_REWRITE_CONV I [pth]
  and bub_CONV = GEN_REWRITE_CONV TOP_SWEEP_CONV
   [int_eq; int_le; int_lt; int_ge; int_gt;
    int_of_num_th; int_neg_th; int_add_th; int_mul_th;
    int_sub_th; int_pow_th; int_abs_th; int_max_th; int_min_th] in
  let base_CONV = TRY_CONV atom_CONV THENC bub_CONV in
  let NNF_NORM_CONV = GEN_NNF_CONV false
   (base_CONV,fun t -> base_CONV t,base_CONV(mk_neg t)) in
  let init_CONV =
    GEN_REWRITE_CONV DEPTH_CONV [FORALL_SIMP; EXISTS_SIMP] THENC
    GEN_REWRITE_CONV DEPTH_CONV [INT_GT; INT_GE] THENC
    CONDS_ELIM_CONV THENC NNF_NORM_CONV in
  let p_tm = `p:bool`
  and not_tm = `(~)` in
  let pth = TAUT(mk_eq(mk_neg(mk_neg p_tm),p_tm)) in
  fun tm ->
    let th0 = INST [tm,p_tm] pth
    and th1 = NNF_NORM_CONV(mk_neg tm) in
    let th2 = REAL_SOS(mk_neg(rand(concl th1))) in
    EQ_MP th0 (EQ_MP (AP_TERM not_tm (SYM th1)) th2);;
*)
(* ------------------------------------------------------------------------- *)
(* Natural number version.                                                   *)
(* ------------------------------------------------------------------------- *)
(*
let SOS_RULE tm =
  let avs = frees tm in
  let tm' = list_mk_forall(avs,tm) in
  let th1 = NUM_TO_INT_CONV tm' in
  let th2 = INT_SOS (rand(concl th1)) in
  SPECL avs (EQ_MP (SYM th1) th2);;
*)
(* ------------------------------------------------------------------------- *)
(* Now pure SOS stuff.                                                       *)
(* ------------------------------------------------------------------------- *)

(*prioritize_real();;*)

(* ------------------------------------------------------------------------- *)
(* Some combinatorial helper functions.                                      *)
(* ------------------------------------------------------------------------- *)

let rec allpermutations l =
  if l = [] then [[]] else
  itlist (fun h acc -> map (fun t -> h::t)
                (allpermutations (subtract l [h])) @ acc) l [];;

let allvarorders l =
  map (fun vlis x -> index x vlis) (allpermutations l);;

let changevariables_monomial zoln (m:monomial) =
  foldl (fun a x k -> (assoc x zoln |-> k) a) monomial_1 m;;

let changevariables zoln pol =
  foldl (fun a m c -> (changevariables_monomial zoln m |-> c) a)
        poly_0 pol;;

(* ------------------------------------------------------------------------- *)
(* Return to original non-block matrices.                                    *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_vector (v:vector) =
  let n = dim v in
  let strs = map (o (decimalize 20) (element v)) (1--n) in
  end_itlist (fun x y -> x ^ " " ^ y) strs ^ "\n";;

let sdpa_of_blockdiagonal k m =
  let pfx = string_of_int k ^" " in
  let ents =
    foldl (fun a (b,i,j) c -> if i > j then a else ((b,i,j),c)::a) [] m in
  let entss = sort (increasing fst) ents in
  itlist (fun ((b,i,j),c) a ->
     pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
     " " ^ decimalize 20 c ^ "\n" ^ a) entss "";;

let sdpa_of_matrix k (m:matrix) =
  let pfx = string_of_int k ^ " 1 " in
  let ms = foldr (fun (i,j) c a -> if i > j then a else ((i,j),c)::a)
                 (snd m) [] in
  let mss = sort (increasing fst) ms in
  itlist (fun ((i,j),c) a ->
     pfx ^ string_of_int i ^ " " ^ string_of_int j ^
     " " ^ decimalize 20 c ^ "\n" ^ a) mss "";;

let sdpa_of_problem comment obj mats =
  let m = length mats - 1
  and n,_ = dimensions (hd mats) in
  "\"" ^ comment ^ "\"\n" ^
  string_of_int m ^ "\n" ^
  "1\n" ^
  string_of_int n ^ "\n" ^
  sdpa_of_vector obj ^
  itlist2 (fun k m a -> sdpa_of_matrix (k - 1) m ^ a)
          (1--length mats) mats "";;

let run_csdp dbg obj mats =
  let input_file = Filename.temp_file "sos" ".dat-s" in
  let output_file =
    String.sub input_file 0 (String.length input_file - 6) ^ ".out"
  and params_file = Filename.concat (!temp_path) "param.csdp" in
  file_of_string input_file (sdpa_of_problem "" obj mats);
  file_of_string params_file csdp_params;
  let rv = Sys.command("cd "^(!temp_path)^"; csdp "^input_file ^
                       " " ^ output_file ^
                       (if dbg then "" else "> /dev/null")) in
  let op = string_of_file output_file in
  let res = parse_csdpoutput op in
  ((if dbg then ()
    else (Sys.remove input_file; Sys.remove output_file));
    rv,res);;

let csdp obj mats =
  let rv,res = run_csdp (!debugging) obj mats in
  (if rv = 1 or rv = 2 then failwith "csdp: Problem is infeasible"
    else if rv = 3 then ()
(*    (Format.print_string "csdp warning: Reduced accuracy";
     Format.print_newline()) *)
   else if rv <> 0 then failwith("csdp: error "^string_of_int rv)
   else ());
  res;;

(* ------------------------------------------------------------------------- *)
(* Sum-of-squares function with some lowbrow symmetry reductions.            *)
(* ------------------------------------------------------------------------- *)

let sumofsquares_general_symmetry tool pol =
  let vars = poly_variables pol
  and lpps = newton_polytope pol in
  let n = length lpps in
  let sym_eqs =
    let invariants = filter
     (fun vars' ->
        is_undefined(poly_sub pol (changevariables (zip vars vars') pol)))
     (allpermutations vars) in
    let lpns = zip lpps (1--length lpps) in
    let lppcs =
      filter (fun (m,(n1,n2)) -> n1 <= n2)
             (allpairs
               (fun (m1,n1) (m2,n2) -> (m1,m2),(n1,n2)) lpns lpns) in
    let clppcs = end_itlist (@)
       (map (fun ((m1,m2),(n1,n2)) ->
                map (fun vars' ->
                        (changevariables_monomial (zip vars vars') m1,
                         changevariables_monomial (zip vars vars') m2),(n1,n2))
                    invariants)
            lppcs) in
    let clppcs_dom = setify(map fst clppcs) in
    let clppcs_cls = map (fun d -> filter (fun (e,_) -> e = d) clppcs)
                         clppcs_dom in
    let eqvcls = map (o setify (map snd)) clppcs_cls in
    let mk_eq cls acc =
      match cls with
        [] -> raise Sanity
      | [h] -> acc
      | h::t -> map (fun k -> (k |-> Int(-1)) (h |=> Int 1)) t @ acc in
    itlist mk_eq eqvcls [] in
  let eqs = foldl (fun a x y -> y::a) []
   (itern 1 lpps (fun m1 n1 ->
        itern 1 lpps (fun m2 n2 f ->
                let m = monomial_mul m1 m2 in
                if n1 > n2 then f else
                let c = if n1 = n2 then Int 1 else Int 2 in
                (m |-> ((n1,n2) |-> c) (tryapplyd f m undefined)) f))
       (foldl (fun a m c -> (m |-> ((0,0)|=>c)) a)
              undefined pol)) @
    sym_eqs in
  let pvs,assig = eliminate_all_equations (0,0) eqs in
  let allassig = itlist (fun v -> (v |-> (v |=> Int 1))) pvs assig in
  let qvars = (0,0)::pvs in
  let diagents =
    end_itlist equation_add (map (fun i -> apply allassig (i,i)) (1--n)) in
  let mk_matrix v =
   ((n,n),
    foldl (fun m (i,j) ass -> let c = tryapplyd ass v (Int 0) in
                              if c =/ Int 0 then m else
                              ((j,i) |-> c) (((i,j) |-> c) m))
          undefined allassig :matrix) in
  let mats = map mk_matrix qvars
  and obj = length pvs,
            itern 1 pvs (fun v i -> (i |--> tryapplyd diagents v (Int 0)))
                undefined in
  let raw_vec = if pvs = [] then vector_0 0 else tool obj mats in
  let find_rounding d =
   (if !debugging then
     (Format.print_string("Trying rounding with limit "^string_of_num d);
      Format.print_newline())
    else ());
    let vec = nice_vector d raw_vec in
    let mat = iter (1,dim vec)
     (fun i a -> matrix_add (matrix_cmul (element vec i) (el i mats)) a)
     (matrix_neg (el 0 mats)) in
    deration(diag mat) in
  let rat,dia =
    if pvs = [] then
       let mat = matrix_neg (el 0 mats) in
       deration(diag mat)
    else
       tryfind find_rounding (map Num.num_of_int (1--31) @
                              map pow2 (5--66)) in
  let poly_of_lin(d,v) =
    d,foldl(fun a i c -> (el (i - 1) lpps |-> c) a) undefined (snd v) in
  let lins = map poly_of_lin dia in
  let sqs = map (fun (d,l) -> poly_mul (poly_const d) (poly_pow l 2)) lins in
  let sos = poly_cmul rat (end_itlist poly_add sqs) in
  if is_undefined(poly_sub sos pol) then rat,lins else raise Sanity;;

let sumofsquares = sumofsquares_general_symmetry csdp;;

(* ------------------------------------------------------------------------- *)
(* Pure HOL SOS conversion.                                                  *)
(* ------------------------------------------------------------------------- *)
(*
let SOS_CONV =
  let mk_square =
    let pow_tm = `(pow)` and two_tm = `2` in
    fun tm -> mk_comb(mk_comb(pow_tm,tm),two_tm)
  and mk_prod = mk_binop `( * )`
  and mk_sum = mk_binop `(+)` in
  fun tm ->
    let k,sos = sumofsquares(poly_of_term tm) in
    let mk_sqtm(c,p) =
      mk_prod (term_of_rat(k */ c)) (mk_square(term_of_poly p)) in
    let tm' = end_itlist mk_sum (map mk_sqtm sos) in
    let th = REAL_POLY_CONV tm and th' = REAL_POLY_CONV tm' in
    TRANS th (SYM th');;
*)
(* ------------------------------------------------------------------------- *)
(* Attempt to prove &0 <= x by direct SOS decomposition.                     *)
(* ------------------------------------------------------------------------- *)
(*
let PURE_SOS_TAC =
  let tac =
    MATCH_ACCEPT_TAC(REWRITE_RULE[GSYM REAL_POW_2] REAL_LE_SQUARE) ORELSE
    MATCH_ACCEPT_TAC REAL_LE_SQUARE ORELSE
    (MATCH_MP_TAC REAL_LE_ADD THEN CONJ_TAC) ORELSE
    (MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC) ORELSE
    CONV_TAC(RAND_CONV REAL_RAT_REDUCE_CONV THENC REAL_RAT_LE_CONV) in
  REPEAT GEN_TAC THEN REWRITE_TAC[real_ge] THEN
  GEN_REWRITE_TAC I [GSYM REAL_SUB_LE] THEN
  CONV_TAC(RAND_CONV SOS_CONV) THEN
  REPEAT tac THEN NO_TAC;;

let PURE_SOS tm = prove(tm,PURE_SOS_TAC);;
*)
(* ------------------------------------------------------------------------- *)
(* Examples.                                                                 *)
(* ------------------------------------------------------------------------- *)

(*****

time REAL_SOS
  `a1 >= &0 /\ a2 >= &0 /\
   (a1 * a1 + a2 * a2 = b1 * b1 + b2 * b2 + &2) /\
   (a1 * b1 + a2 * b2 = &0)
   ==> a1 * a2 - b1 * b2 >= &0`;;

time REAL_SOS `&3 * x + &7 * a < &4 /\ &3 < &2 * x ==> a < &0`;;

time REAL_SOS
  `b pow 2 < &4 * a * c ==> ~(a * x pow 2 + b * x + c = &0)`;;

time REAL_SOS
  `(a * x pow 2 + b * x + c = &0) ==> b pow 2 >= &4 * a * c`;;

time REAL_SOS
 `&0 <= x /\ x <= &1 /\ &0 <= y /\ y <= &1
  ==> x pow 2 + y pow 2 < &1 \/
      (x - &1) pow 2 + y pow 2 < &1 \/
      x pow 2 + (y - &1) pow 2 < &1 \/
      (x - &1) pow 2 + (y - &1) pow 2 < &1`;;

time REAL_SOS
 `&0 <= b /\ &0 <= c /\ &0 <= x /\ &0 <= y /\
  (x pow 2 = c) /\ (y pow 2 = a pow 2 * c + b)
  ==> a * c <= y * x`;;

time REAL_SOS
 `&0 <= x /\ &0 <= y /\ &0 <= z /\ x + y + z <= &3
  ==> x * y + x * z + y * z >= &3 * x * y * z`;;

time REAL_SOS
 `(x pow 2 + y pow 2 + z pow 2 = &1) ==> (x + y + z) pow 2 <= &3`;;

time REAL_SOS
 `(w pow 2 + x pow 2 + y pow 2 + z pow 2 = &1)
  ==> (w + x + y + z) pow 2 <= &4`;;

time REAL_SOS
 `x >= &1 /\ y >= &1 ==> x * y >= x + y - &1`;;

time REAL_SOS
 `x > &1 /\ y > &1 ==> x * y > x + y - &1`;;

time REAL_SOS
 `abs(x) <= &1
  ==> abs(&64 * x pow 7 - &112 * x pow 5 + &56 * x pow 3 - &7 * x) <= &1`;;

time REAL_SOS
 `abs(x - z) <= e /\ abs(y - z) <= e /\ &0 <= u /\ &0 <= v /\ (u + v = &1)
  ==> abs((u * x + v * y) - z) <= e`;;

(* ------------------------------------------------------------------------- *)
(* One component of denominator in dodecahedral example.                     *)
(* ------------------------------------------------------------------------- *)

time REAL_SOS
  `&2 <= x /\ x <= &125841 / &50000 /\
   &2 <= y /\ y <= &125841 / &50000 /\
   &2 <= z /\ z <= &125841 / &50000
   ==> &2 * (x * z + x * y + y * z) - (x * x + y * y + z * z) >= &0`;;

(* ------------------------------------------------------------------------- *)
(* Over a larger but simpler interval.                                       *)
(* ------------------------------------------------------------------------- *)

time REAL_SOS
 `&2 <= x /\ x <= &4 /\ &2 <= y /\ y <= &4 /\ &2 <= z /\ z <= &4
  ==> &0 <= &2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)`;;

(* ------------------------------------------------------------------------- *)
(* We can do 12. I think 12 is a sharp bound; see PP's certificate.          *)
(* ------------------------------------------------------------------------- *)

time REAL_SOS
 `&2 <= x /\ x <= &4 /\ &2 <= y /\ y <= &4 /\ &2 <= z /\ z <= &4
  ==> &12 <= &2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)`;;

(* ------------------------------------------------------------------------- *)
(* Gloptipoly example.                                                       *)
(* ------------------------------------------------------------------------- *)

(*** This works but normalization takes minutes

time REAL_SOS
 `(x - y - &2 * x pow 4 = &0) /\ &0 <= x /\ x <= &2 /\ &0 <= y /\ y <= &3
  ==> y pow 2 - &7 * y - &12 * x + &17 >= &0`;;

 ***)

(* ------------------------------------------------------------------------- *)
(* Inequality from sci.math (see "Leon-Sotelo, por favor").                  *)
(* ------------------------------------------------------------------------- *)

time REAL_SOS
  `&0 <= x /\ &0 <= y /\ (x * y = &1)
   ==> x + y <= x pow 2 + y pow 2`;;

time REAL_SOS
  `&0 <= x /\ &0 <= y /\ (x * y = &1)
   ==> x * y * (x + y) <= x pow 2 + y pow 2`;;

time REAL_SOS
  `&0 <= x /\ &0 <= y ==> x * y * (x + y) pow 2 <= (x pow 2 + y pow 2) pow 2`;;

(* ------------------------------------------------------------------------- *)
(* Some examples over integers and natural numbers.                          *)
(* ------------------------------------------------------------------------- *)

time SOS_RULE `!m n. 2 * m + n = (n + m) + m`;;
time SOS_RULE `!n. ~(n = 0) ==> (0 MOD n = 0)`;;
time SOS_RULE  `!m n. m < n ==> (m DIV n = 0)`;;
time SOS_RULE `!n:num. n <= n * n`;;
time SOS_RULE  `!m n. n * (m DIV n) <= m`;;
time SOS_RULE `!n. ~(n = 0) ==> (0 DIV n = 0)`;;
time SOS_RULE `!m n p. ~(p = 0) /\ m <= n ==> m DIV p <= n DIV p`;;
time SOS_RULE `!a b n. ~(a = 0) ==> (n <= b DIV a <=> a * n <= b)`;;

(* ------------------------------------------------------------------------- *)
(* This is particularly gratifying --- cf hideous manual proof in arith.ml   *)
(* ------------------------------------------------------------------------- *)

(*** This doesn't now seem to work as well as it did; what changed?

time SOS_RULE
 `!a b c d. ~(b = 0) /\ b * c < (a + 1) * d ==> c DIV d <= a DIV b`;;

 ***)

(* ------------------------------------------------------------------------- *)
(* Key lemma for injectivity of Cantor-type pairing functions.               *)
(* ------------------------------------------------------------------------- *)

time SOS_RULE
 `!x1 y1 x2 y2. ((x1 + y1) EXP 2 + x1 + 1 = (x2 + y2) EXP 2 + x2 + 1)
                ==> (x1 + y1 = x2 + y2)`;;

time SOS_RULE
 `!x1 y1 x2 y2. ((x1 + y1) EXP 2 + x1 + 1 = (x2 + y2) EXP 2 + x2 + 1) /\
                (x1 + y1 = x2 + y2)
                ==> (x1 = x2) /\ (y1 = y2)`;;

time SOS_RULE
 `!x1 y1 x2 y2.
      (((x1 + y1) EXP 2 + 3 * x1 + y1) DIV 2 =
       ((x2 + y2) EXP 2 + 3 * x2 + y2) DIV 2)
       ==> (x1 + y1 = x2 + y2)`;;

time SOS_RULE
 `!x1 y1 x2 y2.
      (((x1 + y1) EXP 2 + 3 * x1 + y1) DIV 2 =
       ((x2 + y2) EXP 2 + 3 * x2 + y2) DIV 2) /\
      (x1 + y1 = x2 + y2)
      ==> (x1 = x2) /\ (y1 = y2)`;;

(* ------------------------------------------------------------------------- *)
(* Reciprocal multiplication (actually just ARITH_RULE does these).          *)
(* ------------------------------------------------------------------------- *)

time SOS_RULE `x <= 127 ==> ((86 * x) DIV 256 = x DIV 3)`;;

time SOS_RULE `x < 2 EXP 16 ==> ((104858 * x) DIV (2 EXP 20) = x DIV 10)`;;

(* ------------------------------------------------------------------------- *)
(* This is more impressive since it's really nonlinear. See REMAINDER_DECODE *)
(* ------------------------------------------------------------------------- *)

time SOS_RULE `0 < m /\ m < n  ==> ((m * ((n * x) DIV m + 1)) DIV n = x)`;;

(* ------------------------------------------------------------------------- *)
(* Some conversion examples.                                                 *)
(* ------------------------------------------------------------------------- *)

time SOS_CONV
 `&2 * x pow 4 + &2 * x pow 3 * y - x pow 2 * y pow 2 + &5 * y pow 4`;;

time SOS_CONV
 `x pow 4 - (&2 * y * z + &1) * x pow 2 +
  (y pow 2 * z pow 2 + &2 * y * z + &2)`;;

time SOS_CONV `&4 * x pow 4 +
          &4 * x pow 3 * y - &7 * x pow 2 * y pow 2 - &2 * x * y pow 3 +
          &10 * y pow 4`;;

time SOS_CONV `&4 * x pow 4 * y pow 6 + x pow 2 - x * y pow 2 + y pow 2`;;

time SOS_CONV
  `&4096 * (x pow 4 + x pow 2 + z pow 6 - &3 * x pow 2 * z pow 2) + &729`;;

time SOS_CONV
 `&120 * x pow 2 - &63 * x pow 4 + &10 * x pow 6 +
  &30 * x * y - &120 * y pow 2 + &120 * y pow 4 + &31`;;

time SOS_CONV
 `&9 * x pow 2 * y pow 4 + &9 * x pow 2 * z pow 4 + &36 * x pow 2 * y pow 3 +
  &36 * x pow 2 * y pow 2 - &48 * x * y * z pow 2 + &4 * y pow 4 +
  &4 * z pow 4 - &16 * y pow 3 + &16 * y pow 2`;;

time SOS_CONV
 `(x pow 2 + y pow 2 + z pow 2) *
  (x pow 4 * y pow 2 + x pow 2 * y pow 4 +
   z pow 6 - &3 * x pow 2 * y pow 2 *   z pow 2)`;;

time SOS_CONV
 `x pow 4 + y pow 4 + z pow 4 - &4 * x * y * z + x + y + z + &3`;;

(*** I think this will work, but normalization is slow

time SOS_CONV
 `&100 * (x pow 4 + y pow 4 + z pow 4 - &4 * x * y * z + x + y + z) + &212`;;

 ***)

time SOS_CONV
 `&100 * ((&2 * x - &2) pow 2 + (x pow 3 - &8 * x - &2) pow 2) - &588`;;

time SOS_CONV
 `x pow 2 * (&120 - &63 * x pow 2 + &10 * x pow 4) + &30 * x * y +
    &30 * y pow 2 * (&4 * y pow 2 - &4) + &31`;;

(* ------------------------------------------------------------------------- *)
(* Example of basic rule.                                                    *)
(* ------------------------------------------------------------------------- *)

time PURE_SOS
  `!x. x pow 4 + y pow 4 + z pow 4 - &4 * x * y * z + x + y + z + &3
       >= &1 / &7`;;

time PURE_SOS
 `&0 <= &98 * x pow 12 +
        -- &980 * x pow 10 +
        &3038 * x pow 8 +
        -- &2968 * x pow 6 +
        &1022 * x pow 4 +
        -- &84 * x pow 2 +
        &2`;;

time PURE_SOS
 `!x. &0 <= &2 * x pow 14 +
            -- &84 * x pow 12 +
            &1022 * x pow 10 +
            -- &2968 * x pow 8 +
            &3038 * x pow 6 +
            -- &980 * x pow 4 +
            &98 * x pow 2`;;

(* ------------------------------------------------------------------------- *)
(* From Zeng et al, JSC vol 37 (2004), p83-99.                               *)
(* All of them work nicely with pure SOS_CONV, except (maybe) the one noted. *)
(* ------------------------------------------------------------------------- *)

PURE_SOS
  `x pow 6 + y pow 6 + z pow 6 - &3 * x pow 2 * y pow 2 * z pow 2 >= &0`;;

PURE_SOS `x pow 4 + y pow 4 + z pow 4 + &1 - &4*x*y*z >= &0`;;

PURE_SOS `x pow 4 + &2*x pow 2*z + x pow 2 - &2*x*y*z + &2*y pow 2*z pow 2 +
&2*y*z pow 2 + &2*z pow 2 - &2*x + &2* y*z + &1 >= &0`;;

(**** This is harder. Interestingly, this fails the pure SOS test, it seems.
      Yet only on rounding(!?) Poor Newton polytope optimization or something?
      But REAL_SOS does finally converge on the second run at level 12!

REAL_SOS
`x pow 4*y pow 4 - &2*x pow 5*y pow 3*z pow 2 + x pow 6*y pow 2*z pow 4 + &2*x
pow 2*y pow 3*z - &4* x pow 3*y pow 2*z pow 3 + &2*x pow 4*y*z pow 5 + z pow
2*y pow 2 - &2*z pow 4*y*x + z pow 6*x pow 2 >= &0`;;

 ****)

PURE_SOS
`x pow 4 + &4*x pow 2*y pow 2 + &2*x*y*z pow 2 + &2*x*y*w pow 2 + y pow 4 + z
pow 4 + w pow 4 + &2*z pow 2*w pow 2 + &2*x pow 2*w + &2*y pow 2*w + &2*x*y +
&3*w pow 2 + &2*z pow 2 + &1 >= &0`;;

PURE_SOS
`w pow 6 + &2*z pow 2*w pow 3 + x pow 4 + y pow 4 + z pow 4 + &2*x pow 2*w +
&2*x pow 2*z + &3*x pow 2 + w pow 2 + &2*z*w + z pow 2 + &2*z + &2*w + &1 >=
&0`;;

*****)