aboutsummaryrefslogtreecommitdiffhomepage
path: root/test-suite/interactive/ParalITP_smallproofs.v
blob: 0d75d52a31f2d117f9b434d99ae5a4af4e867aec (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
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
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
2552
2553
2554
2555
2556
2557
2558
2559
2560
2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625
2626
2627
2628
2629
2630
2631
2632
2633
2634
2635
2636
2637
2638
2639
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
2653
2654
2655
2656
2657
2658
2659
2660
2661
2662
2663
2664
2665
2666
2667
2668
2669
2670
2671
2672
2673
2674
2675
2676
2677
2678
2679
2680
2681
2682
2683
2684
2685
2686
2687
2688
2689
2690
2691
2692
2693
2694
2695
2696
2697
2698
2699
2700
2701
2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
2732
2733
2734
2735
2736
2737
2738
2739
2740
2741
2742
2743
2744
2745
2746
2747
2748
2749
2750
2751
2752
2753
2754
2755
2756
2757
2758
2759
2760
2761
2762
2763
2764
2765
2766
2767
2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782
2783
2784
2785
2786
2787
2788
2789
2790
2791
2792
2793
2794
2795
2796
2797
2798
2799
2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852
2853
2854
2855
2856
2857
2858
2859
2860
2861
2862
2863
2864
2865
2866
2867
2868
2869
2870
2871
2872
2873
2874
2875
2876
2877
2878
2879
2880
2881
2882
2883
2884
2885
2886
2887
2888
2889
2890
2891
2892
2893
2894
2895
2896
2897
2898
2899
2900
2901
2902
2903
2904
2905
2906
2907
2908
2909
2910
2911
2912
2913
2914
2915
2916
2917
2918
2919
2920
2921
2922
2923
2924
2925
2926
2927
2928
2929
2930
2931
2932
2933
2934
2935
2936
2937
2938
2939
2940
2941
2942
2943
2944
2945
2946
2947
2948
2949
2950
2951
2952
2953
2954
2955
2956
2957
2958
2959
2960
2961
2962
2963
2964
2965
2966
2967
2968
2969
2970
2971
2972
2973
2974
2975
2976
2977
2978
2979
2980
2981
2982
2983
2984
2985
2986
2987
2988
2989
2990
2991
2992
2993
2994
2995
2996
2997
2998
2999
3000
3001
3002
3003
3004
3005
3006
3007
3008
3009
3010
3011
3012
3013
3014
3015
3016
3017
3018
3019
3020
3021
3022
3023
3024
3025
3026
3027
3028
3029
3030
3031
3032
3033
3034
3035
3036
3037
3038
3039
3040
3041
(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)


(** This file includes random facts about Integers (and natural numbers) which are not found in the standard library. Some of the lemma here are not used in the QArith developement but are rather useful.
*)

Require Export ZArith.
Require Export ZArithRing.

Tactic Notation "ElimCompare" constr(c) constr(d) := elim_compare c d.

Ltac Flip :=
  apply Zgt_lt || apply Zlt_gt || apply Zle_ge || apply Zge_le; assumption.

Ltac Falsum :=
  try intro; apply False_ind;
   repeat
    match goal with
    | id1:(~ ?X1) |- ?X2 =>
        (apply id1; assumption || reflexivity) || clear id1
    end.


Ltac Step_l a :=
  match goal with
  |  |- (?X1 < ?X2)%Z => replace X1 with a; [ idtac | try ring ]
  end. 

Ltac Step_r a :=
  match goal with
  |  |- (?X1 < ?X2)%Z => replace X2 with a; [ idtac | try ring ]
  end. 

Ltac CaseEq formula :=
  generalize (refl_equal formula); pattern formula at -1 in |- *;
   case formula.


Lemma pair_1 : forall (A B : Set) (H : A * B), H = pair (fst H) (snd H).
Proof.
 intros.
 case H.
 intros. 
 simpl in |- *.
 reflexivity.
Qed.

Lemma pair_2 :
 forall (A B : Set) (H1 H2 : A * B),
 fst H1 = fst H2 -> snd H1 = snd H2 -> H1 = H2.
Proof.
 intros A B H1 H2.
 case H1.
 case H2.
 simpl in |- *.
 intros.
 rewrite H.
 rewrite H0.
 reflexivity.
Qed.


Section projection.   
 Variable A : Set.
 Variable P : A -> Prop.
 
 Definition projP1 (H : sig P) := let (x, h) := H in x.
 Definition projP2 (H : sig P) :=
   let (x, h) as H return (P (projP1 H)) := H in h.
End projection.


(*###########################################################################*)
(* Declaring some realtions on natural numbers for stepl and stepr tactics.  *)
(*###########################################################################*)

Lemma le_stepl: forall x y z, le x y -> x=z -> le z y.
Proof.
 intros x y z H_le H_eq; subst z; trivial.
Qed.

Lemma le_stepr: forall x y z, le x y -> y=z -> le x z.
Proof.
 intros x y z H_le H_eq; subst z; trivial.
Qed.

Lemma lt_stepl: forall x y z, lt x y -> x=z -> lt z y.
Proof.
 intros x y z H_lt H_eq; subst z; trivial.
Qed.

Lemma lt_stepr: forall x y z, lt x y -> y=z -> lt x z.
Proof.
 intros x y z H_lt H_eq; subst z; trivial.
Qed.

Lemma neq_stepl:forall (x y z:nat), x<>y -> x=z -> z<>y.
Proof.
 intros x y z H_lt H_eq; subst; assumption.
Qed.

Lemma neq_stepr:forall (x y z:nat), x<>y -> y=z -> x<>z.
Proof.
 intros x y z H_lt H_eq; subst; assumption.
Qed.


Declare Left Step le_stepl.
Declare Right Step le_stepr.
Declare Left Step lt_stepl.
Declare Right Step lt_stepr.
Declare Left Step neq_stepl.
Declare Right Step neq_stepr.

(*###########################################################################*)
(** Some random facts about natural numbers, positive numbers and integers   *)
(*###########################################################################*)


Lemma not_O_S : forall n : nat, n <> 0 -> {p : nat | n = S p}.
Proof. 
 intros [| np] Hn; [ exists 0; apply False_ind; apply Hn | exists np ];
  reflexivity. 
Qed.
  

Lemma lt_minus_neq : forall m n : nat, m < n -> n - m <> 0.
Proof.
 intros.
 omega.
Qed.

Lemma lt_minus_eq_0 : forall m n : nat, m < n -> m - n = 0.
Proof.
 intros.
 omega.
Qed.

Lemma le_plus_Sn_1_SSn : forall n : nat, S n + 1 <= S (S n).
Proof.
 intros.
 omega.
Qed.

Lemma le_plus_O_l : forall p q : nat, p + q <= 0 -> p = 0.
Proof. 
 intros; omega.
Qed.

Lemma le_plus_O_r : forall p q : nat, p + q <= 0 -> q = 0.
Proof. 
 intros; omega.
Qed.

Lemma minus_pred : forall m n : nat, 0 < n -> pred m - pred n = m - n.
Proof.
 intros.
 omega.
Qed.


(*###########################################################################*)
(* Declaring some realtions on integers for stepl and stepr tactics.         *)
(*###########################################################################*)

Lemma Zle_stepl: forall x y z, (x<=y)%Z -> x=z -> (z<=y)%Z.
Proof.
 intros x y z H_le H_eq; subst z; trivial.
Qed.

Lemma Zle_stepr: forall x y z, (x<=y)%Z -> y=z -> (x<=z)%Z.
Proof.
 intros x y z H_le H_eq; subst z; trivial.
Qed.

Lemma Zlt_stepl: forall x y z, (x<y)%Z -> x=z -> (z<y)%Z.
Proof.
 intros x y z H_lt H_eq; subst z; trivial.
Qed.

Lemma Zlt_stepr: forall x y z, (x<y)%Z -> y=z -> (x<z)%Z.
Proof.
 intros x y z H_lt H_eq; subst z; trivial.
Qed.

Lemma Zneq_stepl:forall (x y z:Z), (x<>y)%Z -> x=z -> (z<>y)%Z.
Proof.
 intros x y z H_lt H_eq; subst; assumption.
Qed.

Lemma Zneq_stepr:forall (x y z:Z), (x<>y)%Z -> y=z -> (x<>z)%Z.
Proof.
 intros x y z H_lt H_eq; subst; assumption.
Qed.

Declare Left Step Zle_stepl.
Declare Right Step Zle_stepr.
Declare Left Step Zlt_stepl.
Declare Right Step Zlt_stepr.
Declare Left Step Zneq_stepl.
Declare Right Step Zneq_stepr.


(*###########################################################################*)
(** Informative case analysis                                                *)
(*###########################################################################*)

Lemma Zlt_cotrans :
 forall x y : Z, (x < y)%Z -> forall z : Z, {(x < z)%Z} + {(z < y)%Z}.
Proof.
 intros.
 case (Z_lt_ge_dec x z).
 intro.
 left.
 assumption.
 intro.
 right.
 apply Zle_lt_trans with (m := x).
 apply Zge_le.
 assumption.
 assumption.
Qed.

Lemma Zlt_cotrans_pos :
 forall x y : Z, (0 < x + y)%Z -> {(0 < x)%Z} + {(0 < y)%Z}.
Proof.
 intros.
 case (Zlt_cotrans 0 (x + y) H x).
 intro.
 left.
 assumption.
 intro.
 right.
 apply Zplus_lt_reg_l with (p := x).
 rewrite Zplus_0_r.
 assumption.
Qed.


Lemma Zlt_cotrans_neg :
 forall x y : Z, (x + y < 0)%Z -> {(x < 0)%Z} + {(y < 0)%Z}.
Proof.
 intros x y H; case (Zlt_cotrans (x + y) 0 H x); intro Hxy;
  [ right; apply Zplus_lt_reg_l with (p := x); rewrite Zplus_0_r | left ];
  assumption.
Qed.



Lemma not_Zeq_inf : forall x y : Z, x <> y -> {(x < y)%Z} + {(y < x)%Z}.
Proof.
 intros.
 case Z_lt_ge_dec with x y.
 intro.
 left.
 assumption.
 intro H0.
 generalize (Zge_le _ _ H0).
 intro.
 case (Z_le_lt_eq_dec _ _ H1).
 intro.
 right.
 assumption.
 intro.
 apply False_rec.
 apply H.
 symmetry  in |- *.
 assumption.
Qed.

Lemma Z_dec : forall x y : Z, {(x < y)%Z} + {(x > y)%Z} + {x = y}.
Proof.
 intros.
 case (Z_lt_ge_dec x y).
 intro H.
 left.
 left.
 assumption.
 intro H.
 generalize (Zge_le _ _ H).
 intro H0.
 case (Z_le_lt_eq_dec y x H0).
 intro H1.
 left.
 right.
 apply Zlt_gt.
 assumption.
 intro.
 right.
 symmetry  in |- *.
 assumption.
Qed.
 

Lemma Z_dec' : forall x y : Z, {(x < y)%Z} + {(y < x)%Z} + {x = y}.
Proof.
 intros x y.
 case (Z_eq_dec x y); intro H;
  [ right; assumption | left; apply (not_Zeq_inf _ _ H) ].
Qed.

Lemma Z_lt_le_dec : forall x y : Z, {(x < y)%Z} + {(y <= x)%Z}.
Proof.
 intros.
 case (Z_lt_ge_dec x y).
 intro.
 left.
 assumption.
 intro.
 right.
 apply Zge_le.
 assumption.
Qed.

Lemma Z_le_lt_dec : forall x y : Z, {(x <= y)%Z} + {(y < x)%Z}.
Proof.
 intros; case (Z_lt_le_dec y x); [ right | left ]; assumption.
Qed.

Lemma Z_lt_lt_S_eq_dec :
 forall x y : Z, (x < y)%Z -> {(x + 1 < y)%Z} + {(x + 1)%Z = y}.
Proof.
 intros.
 generalize (Zlt_le_succ _ _ H).
 unfold Zsucc in |- *.
 apply Z_le_lt_eq_dec.
Qed.

Lemma quadro_leq_inf :
 forall a b c d : Z,
 {(c <= a)%Z /\ (d <= b)%Z} + {~ ((c <= a)%Z /\ (d <= b)%Z)}.
Proof.
 intros.
 case (Z_lt_le_dec a c).
 intro z.
 right.
 intro. 
 elim H.
 intros.
 generalize z.
 apply Zle_not_lt.
 assumption.
 intro.
 case (Z_lt_le_dec b d).
 intro z0.
 right. 
 intro. 
 elim H.
 intros.
 generalize z0.
 apply Zle_not_lt.
 assumption.
 intro.
 left.
 split.
 assumption.
 assumption. 
Qed.

(*###########################################################################*)
(** General auxiliary lemmata                                                *)
(*###########################################################################*)

Lemma Zminus_eq : forall x y : Z, (x - y)%Z = 0%Z -> x = y.
Proof.
 intros.
 apply Zplus_reg_l with (- y)%Z.
 rewrite Zplus_opp_l.
 unfold Zminus in H.
 rewrite Zplus_comm.
 assumption.
Qed.

Lemma Zlt_minus : forall a b : Z, (b < a)%Z -> (0 < a - b)%Z.
Proof.
 intros a b. 
 intros.
 apply Zplus_lt_reg_l with b.
 unfold Zminus in |- *. 
 rewrite (Zplus_comm a).
 rewrite (Zplus_assoc b (- b)). 
 rewrite Zplus_opp_r.
 simpl in |- *. 
 rewrite <- Zplus_0_r_reverse. 
 assumption.
Qed.


Lemma Zle_minus : forall a b : Z, (b <= a)%Z -> (0 <= a - b)%Z.
Proof.
 intros a b. 
 intros.
 apply Zplus_le_reg_l with b.
 unfold Zminus in |- *. 
 rewrite (Zplus_comm a).
 rewrite (Zplus_assoc b (- b)). 
 rewrite Zplus_opp_r.
 simpl in |- *. 
 rewrite <- Zplus_0_r_reverse. 
 assumption.
Qed.

Lemma Zlt_plus_plus :
 forall m n p q : Z, (m < n)%Z -> (p < q)%Z -> (m + p < n + q)%Z.
Proof.
 intros.
 apply Zlt_trans with (m := (n + p)%Z).
 rewrite Zplus_comm.
 rewrite Zplus_comm with (n := n).
 apply Zplus_lt_compat_l.
 assumption.
 apply Zplus_lt_compat_l.
 assumption.
Qed.

Lemma Zgt_plus_plus :
 forall m n p q : Z, (m > n)%Z -> (p > q)%Z -> (m + p > n + q)%Z.
 intros.
 apply Zgt_trans with (m := (n + p)%Z).
 rewrite Zplus_comm.
 rewrite Zplus_comm with (n := n).
 apply Zplus_gt_compat_l.
 assumption.
 apply Zplus_gt_compat_l.
 assumption.
Qed.

Lemma Zle_lt_plus_plus :
 forall m n p q : Z, (m <= n)%Z -> (p < q)%Z -> (m + p < n + q)%Z.
Proof.
 intros.
 case (Zle_lt_or_eq m n).
 assumption.
 intro.
 apply Zlt_plus_plus.
 assumption.
 assumption.
 intro.
 rewrite H1.
 apply Zplus_lt_compat_l.
 assumption.
Qed.

Lemma Zge_gt_plus_plus :
 forall m n p q : Z, (m >= n)%Z -> (p > q)%Z -> (m + p > n + q)%Z.
Proof.
 intros.
 case (Zle_lt_or_eq n m).
 apply Zge_le.
 assumption.
 intro.
 apply Zgt_plus_plus.
 apply Zlt_gt.
 assumption.
 assumption.
 intro.
 rewrite H1.
 apply Zplus_gt_compat_l.
 assumption.
Qed.

Lemma Zgt_ge_plus_plus :
 forall m n p q : Z, (m > n)%Z -> (p >= q)%Z -> (m + p > n + q)%Z.
Proof.
 intros.
 rewrite Zplus_comm.
 replace (n + q)%Z with (q + n)%Z.
 apply Zge_gt_plus_plus.
 assumption.
 assumption.
 apply Zplus_comm.
Qed.

Lemma Zlt_resp_pos : forall x y : Z, (0 < x)%Z -> (0 < y)%Z -> (0 < x + y)%Z.
Proof.
 intros.
 rewrite <- Zplus_0_r with 0%Z.
 apply Zlt_plus_plus; assumption.
Qed.


Lemma Zle_resp_neg :
 forall x y : Z, (x <= 0)%Z -> (y <= 0)%Z -> (x + y <= 0)%Z.
Proof.
 intros.
 rewrite <- Zplus_0_r with 0%Z.
 apply Zplus_le_compat; assumption.
Qed.


Lemma Zlt_pos_opp : forall x : Z, (0 < x)%Z -> (- x < 0)%Z.
Proof.
 intros.
 apply Zplus_lt_reg_l with x.
 rewrite Zplus_opp_r.
 rewrite Zplus_0_r.
 assumption.
Qed.

Lemma Zlt_neg_opp : forall x : Z, (x < 0)%Z -> (0 < - x)%Z.
Proof.
 intros.
 apply Zplus_lt_reg_l with x.
 rewrite Zplus_opp_r.
 rewrite Zplus_0_r.
 assumption.
Qed.


Lemma Zle_neg_opp : forall x : Z, (x <= 0)%Z -> (0 <= - x)%Z.
Proof. 
 intros.
 apply Zplus_le_reg_l with x.
 rewrite Zplus_opp_r.
 rewrite Zplus_0_r.
 assumption.
Qed.

Lemma Zle_pos_opp : forall x : Z, (0 <= x)%Z -> (- x <= 0)%Z.
Proof. 
 intros.
 apply Zplus_le_reg_l with x.
 rewrite Zplus_opp_r.
 rewrite Zplus_0_r.
 assumption.
Qed.


Lemma Zge_opp : forall x y : Z, (x <= y)%Z -> (- x >= - y)%Z.
Proof.
 intros.
 apply Zle_ge.
 apply Zplus_le_reg_l with (p := (x + y)%Z).
 ring_simplify (x + y + - y)%Z (x + y + - x)%Z.
 assumption.
Qed.



(* Omega can't solve this *)
Lemma Zmult_pos_pos : forall x y : Z, (0 < x)%Z -> (0 < y)%Z -> (0 < x * y)%Z.
Proof.
 intros [| px| px] [| py| py] Hx Hy; trivial || constructor.
Qed.

Lemma Zmult_neg_neg : forall x y : Z, (x < 0)%Z -> (y < 0)%Z -> (0 < x * y)%Z.
Proof.
 intros [| px| px] [| py| py] Hx Hy; trivial || constructor.
Qed.

Lemma Zmult_neg_pos : forall x y : Z, (x < 0)%Z -> (0 < y)%Z -> (x * y < 0)%Z.
Proof.
 intros [| px| px] [| py| py] Hx Hy; trivial || constructor.
Qed.

Lemma Zmult_pos_neg : forall x y : Z, (0 < x)%Z -> (y < 0)%Z -> (x * y < 0)%Z.
Proof.
 intros [| px| px] [| py| py] Hx Hy; trivial || constructor.
Qed.



Hint Resolve Zmult_pos_pos Zmult_neg_neg Zmult_neg_pos Zmult_pos_neg: zarith.


Lemma Zle_reg_mult_l :
 forall x y a : Z, (0 < a)%Z -> (x <= y)%Z -> (a * x <= a * y)%Z.
Proof.
 intros.
 apply Zplus_le_reg_l with (p := (- a * x)%Z).
 ring_simplify (- a * x + a * x)%Z.
 replace (- a * x + a * y)%Z with ((y - x) * a)%Z.
 apply Zmult_gt_0_le_0_compat.
 apply Zlt_gt.
 assumption. 
 unfold Zminus in |- *.
 apply Zle_left.
 assumption.
 ring.
Qed.

Lemma Zsimpl_plus_l_dep :
 forall x y m n : Z, (x + m)%Z = (y + n)%Z -> x = y -> m = n.
Proof.
 intros.
 apply Zplus_reg_l with x.
 rewrite <- H0 in H.
 assumption.
Qed.


Lemma Zsimpl_plus_r_dep :
 forall x y m n : Z, (m + x)%Z = (n + y)%Z -> x = y -> m = n.
Proof.
 intros.
 apply Zplus_reg_l with x.
 rewrite Zplus_comm.
 rewrite Zplus_comm with x n.
 rewrite <- H0 in H.
 assumption.
Qed.

Lemma Zmult_simpl :
 forall n m p q : Z, n = m -> p = q -> (n * p)%Z = (m * q)%Z.
Proof.
 intros.
 rewrite H.
 rewrite H0.
 reflexivity.
Qed.
 
Lemma Zsimpl_mult_l :
 forall n m p : Z, n <> 0%Z -> (n * m)%Z = (n * p)%Z -> m = p.
Proof.
 intros.
 apply Zplus_reg_l with (n := (- p)%Z).
 replace (- p + p)%Z with 0%Z.
 apply Zmult_integral_l with (n := n).
 assumption.
 replace ((- p + m) * n)%Z with (n * m + - (n * p))%Z.
 apply Zegal_left.
 assumption.
 ring.
 ring.
Qed.

Lemma Zlt_reg_mult_l :
 forall x y z : Z, (x > 0)%Z -> (y < z)%Z -> (x * y < x * z)%Z. (*QA*)
Proof.
 intros.
 case (Zcompare_Gt_spec x 0).
 unfold Zgt in H.
 assumption.
 intros.
 cut (x = Zpos x0). 
 intro.
 rewrite H2.
 unfold Zlt in H0.
 unfold Zlt in |- *.
 cut ((Zpos x0 * y ?= Zpos x0 * z)%Z = (y ?= z)%Z).
 intro.
 exact (trans_eq H3 H0).
 apply Zcompare_mult_compat.
 cut (x = (x + - (0))%Z).
 intro.
 exact (trans_eq H2 H1).
 simpl in |- *.
 apply (sym_eq (A:=Z)).
 exact (Zplus_0_r x).
Qed.


Lemma Zlt_opp : forall x y : Z, (x < y)%Z -> (- x > - y)%Z. (*QA*)
Proof.
 intros.
 red in |- *.
 apply sym_eq.
 cut (Datatypes.Gt = (y ?= x)%Z).
 intro.
 cut ((y ?= x)%Z = (- x ?= - y)%Z).
 intro.
 exact (trans_eq H0 H1). 
 exact (Zcompare_opp y x).
 apply sym_eq.
 exact (Zlt_gt x y H).
Qed.


Lemma Zlt_conv_mult_l :
 forall x y z : Z, (x < 0)%Z -> (y < z)%Z -> (x * y > x * z)%Z. (*QA*)
Proof.
 intros.
 cut (- x > 0)%Z.
 intro.
 cut (- x * y < - x * z)%Z.
 intro.
 cut (- (- x * y) > - (- x * z))%Z.
 intro.
 cut (- - (x * y) > - - (x * z))%Z.
 intro.
 cut ((- - (x * y))%Z = (x * y)%Z).
 intro.
 rewrite H5 in H4.
 cut ((- - (x * z))%Z = (x * z)%Z).
 intro.
 rewrite H6 in H4.
 assumption.
 exact (Zopp_involutive (x * z)).
 exact (Zopp_involutive (x * y)).
 cut ((- (- x * y))%Z = (- - (x * y))%Z).
 intro.
 rewrite H4 in H3.
 cut ((- (- x * z))%Z = (- - (x * z))%Z). 
 intro.
 rewrite H5 in H3.
 assumption.
 cut ((- x * z)%Z = (- (x * z))%Z).
 intro.
 exact (f_equal Zopp H5).
 exact (Zopp_mult_distr_l_reverse x z).
 cut ((- x * y)%Z = (- (x * y))%Z).
 intro.
 exact (f_equal Zopp H4).
 exact (Zopp_mult_distr_l_reverse x y).
 exact (Zlt_opp (- x * y) (- x * z) H2).
 exact (Zlt_reg_mult_l (- x) y z H1 H0).
 exact (Zlt_opp x 0 H).
Qed.

Lemma Zgt_not_eq : forall x y : Z, (x > y)%Z -> x <> y.   (*QA*)
Proof.
 intros.
 cut (y < x)%Z.
 intro.
 cut (y <> x).
 intro.
 red in |- *.
 intros.
 cut (y = x).
 intros.
 apply H1.
 assumption.
 exact (sym_eq H2).
 exact (Zorder.Zlt_not_eq y x H0).
 exact (Zgt_lt x y H).
Qed.

Lemma Zmult_resp_nonzero :
 forall x y : Z, x <> 0%Z -> y <> 0%Z -> (x * y)%Z <> 0%Z.
Proof.
 intros x y Hx Hy Hxy.
 apply Hx. 
 apply Zmult_integral_l with y; assumption.
Qed.


Lemma Zopp_app : forall y : Z, y <> 0%Z -> (- y)%Z <> 0%Z.
Proof.
 intros.
 intro.
 apply H.
 apply Zplus_reg_l with (- y)%Z.
 rewrite Zplus_opp_l.
 rewrite H0.
 simpl in |- *.
 reflexivity.
Qed.


Lemma Zle_neq_Zlt : forall a b : Z, (a <= b)%Z -> b <> a -> (a < b)%Z.
Proof.
 intros a b H H0.
 case (Z_le_lt_eq_dec _ _ H); trivial.
 intro; apply False_ind; apply H0; symmetry  in |- *; assumption.
Qed.

Lemma not_Zle_lt : forall x y : Z, ~ (y <= x)%Z -> (x < y)%Z.
Proof.
 intros; apply Zgt_lt; apply Znot_le_gt; assumption.
Qed.

Lemma not_Zlt : forall x y : Z, ~ (y < x)%Z -> (x <= y)%Z.
Proof.
 intros x y H1 H2; apply H1; apply Zgt_lt; assumption.
Qed.


Lemma Zmult_absorb :
 forall x y z : Z, x <> 0%Z -> (x * y)%Z = (x * z)%Z -> y = z.  (*QA*)
Proof.
 intros.
 case (dec_eq y z).
 intro.
 assumption.
 intro.
 case (not_Zeq y z).
 assumption.
 intro.
 case (not_Zeq x 0).
 assumption.
 intro.
 apply False_ind.
 cut (x * y > x * z)%Z.
 intro.
 cut ((x * y)%Z <> (x * z)%Z).
 intro.
 apply H5.
 assumption.
 exact (Zgt_not_eq (x * y) (x * z) H4).
 exact (Zlt_conv_mult_l x y z H3 H2).
 intro.
 apply False_ind.
 cut (x * y < x * z)%Z.
 intro.
 cut ((x * y)%Z <> (x * z)%Z).
 intro.
 apply H5.
 assumption.
 exact (Zorder.Zlt_not_eq (x * y) (x * z) H4).
 cut (x > 0)%Z.
 intro.
 exact (Zlt_reg_mult_l x y z H4 H2).
 exact (Zlt_gt 0 x H3).
 intro.
 apply False_ind.
 cut (x * z < x * y)%Z.
 intro.
 cut ((x * z)%Z <> (x * y)%Z).
 intro.
 apply H4.
 apply (sym_eq (A:=Z)).
 assumption.
 exact (Zorder.Zlt_not_eq (x * z) (x * y) H3).
 apply False_ind.
 case (not_Zeq x 0).
 assumption.
 intro.
 cut (x * z > x * y)%Z.
 intro.
 cut ((x * z)%Z <> (x * y)%Z).
 intro.
 apply H5.
 apply (sym_eq (A:=Z)).
 assumption.
 exact (Zgt_not_eq (x * z) (x * y) H4).
 exact (Zlt_conv_mult_l x z y H3 H2).
 intro.
 cut (x * z < x * y)%Z.
 intro.
 cut ((x * z)%Z <> (x * y)%Z).
 intro.
 apply H5.
 apply (sym_eq (A:=Z)).
 assumption.
 exact (Zorder.Zlt_not_eq (x * z) (x * y) H4).
 cut (x > 0)%Z.
 intro.
 exact (Zlt_reg_mult_l x z y H4 H2).
 exact (Zlt_gt 0 x H3).
Qed.

Lemma Zlt_mult_mult :
 forall a b c d : Z,
 (0 < a)%Z -> (0 < d)%Z -> (a < b)%Z -> (c < d)%Z -> (a * c < b * d)%Z.
Proof.
 intros.
 apply Zlt_trans with (a * d)%Z.
 apply Zlt_reg_mult_l.
 Flip. 
 assumption.
 rewrite Zmult_comm.
 rewrite Zmult_comm with b d.
 apply Zlt_reg_mult_l.
 Flip.
 assumption.
Qed.

Lemma Zgt_mult_conv_absorb_l :
 forall a x y : Z, (a < 0)%Z -> (a * x > a * y)%Z -> (x < y)%Z. (*QC*)
Proof.
 intros.
 case (dec_eq x y).
 intro.
 apply False_ind.
 rewrite H1 in H0.
 cut ((a * y)%Z = (a * y)%Z).
 change ((a * y)%Z <> (a * y)%Z) in |- *.
 apply Zgt_not_eq.
 assumption.
 trivial.
 
 intro.
 case (not_Zeq x y H1).
 trivial.
 
 intro.
 apply False_ind.
 cut (a * y > a * x)%Z.
 apply Zgt_asym with (m := (a * y)%Z) (n := (a * x)%Z).
 assumption.
 apply Zlt_conv_mult_l.
 assumption.
 assumption.
Qed.

Lemma Zgt_mult_reg_absorb_l :
 forall a x y : Z, (a > 0)%Z -> (a * x > a * y)%Z -> (x > y)%Z. (*QC*)
Proof.
 intros.
 cut (- - a > - - (0))%Z.
 intro.
 cut (- a < - (0))%Z.
 simpl in |- *.
 intro.
 replace x with (- - x)%Z.
 replace y with (- - y)%Z.
 apply Zlt_opp.
 apply Zgt_mult_conv_absorb_l with (a := (- a)%Z) (x := (- x)%Z).
 assumption.
 rewrite Zmult_opp_opp.
 rewrite Zmult_opp_opp.
 assumption.
 apply Zopp_involutive.
 apply Zopp_involutive.
 apply Zgt_lt.
 apply Zlt_opp.
 apply Zgt_lt.
 assumption.
 simpl in |- *.
 rewrite Zopp_involutive.
 assumption.
Qed.

Lemma Zopp_Zlt : forall x y : Z, (y < x)%Z -> (- x < - y)%Z.
Proof.
 intros x y Hyx.
 apply Zgt_mult_conv_absorb_l with (a := (-1)%Z).
 constructor.
 replace (-1 * - y)%Z with y.
 replace (-1 * - x)%Z with x.
 Flip.
 ring.
 ring.
Qed.


Lemma Zmin_cancel_Zlt : forall x y : Z, (- x < - y)%Z -> (y < x)%Z.
Proof.
 intros.
 apply Zgt_mult_conv_absorb_l with (a := (-1)%Z).
 constructor.
 replace (-1 * y)%Z with (- y)%Z.
 replace (-1 * x)%Z with (- x)%Z.
 apply Zlt_gt.
 assumption.
 ring.
 ring.
Qed.


Lemma Zmult_cancel_Zle :
 forall a x y : Z, (a < 0)%Z -> (a * x <= a * y)%Z -> (y <= x)%Z.
Proof.
 intros.
 case (Z_le_gt_dec y x).
 trivial.
 intro.
 apply False_ind.
 apply (Zlt_irrefl (a * x)).
 apply Zle_lt_trans with (m := (a * y)%Z).
 assumption.
 apply Zgt_lt.
 apply Zlt_conv_mult_l.
 assumption.
 apply Zgt_lt.
 assumption.
Qed.

Lemma Zlt_mult_cancel_l :
 forall x y z : Z, (0 < x)%Z -> (x * y < x * z)%Z -> (y < z)%Z.
Proof.
 intros.
 apply Zgt_lt.
 apply Zgt_mult_reg_absorb_l with x.
 apply Zlt_gt.
 assumption. 
 apply Zlt_gt.
 assumption.
Qed.
 

Lemma Zmin_cancel_Zle : forall x y : Z, (- x <= - y)%Z -> (y <= x)%Z.
Proof. 
 intros.
 apply Zmult_cancel_Zle with (a := (-1)%Z).
 constructor.
 replace (-1 * y)%Z with (- y)%Z.
 replace (-1 * x)%Z with (- x)%Z.
 assumption.
 ring.
 ring.
Qed.



Lemma Zmult_resp_Zle :
 forall a x y : Z, (0 < a)%Z -> (a * y <= a * x)%Z -> (y <= x)%Z.
Proof.
 intros.
 case (Z_le_gt_dec y x).
 trivial.
 intro.
 apply False_ind.
 apply (Zlt_irrefl (a * y)).
 apply Zle_lt_trans with (m := (a * x)%Z).
 assumption.
 apply Zlt_reg_mult_l.
 apply Zlt_gt.
 assumption.
 apply Zgt_lt.
 assumption.
Qed.

Lemma Zopp_Zle : forall x y : Z, (y <= x)%Z -> (- x <= - y)%Z.
Proof. 
 intros.
 apply Zmult_cancel_Zle with (a := (-1)%Z).
 constructor.
 replace (-1 * - y)%Z with y.
 replace (-1 * - x)%Z with x.
 assumption.
 clear y H; ring.
 clear x H; ring.
Qed.

 
Lemma Zle_lt_eq_S : forall x y : Z, (x <= y)%Z -> (y < x + 1)%Z -> y = x.
Proof.
 intros.
 case (Z_le_lt_eq_dec x y H).
 intro H1.
 apply False_ind.
 generalize (Zlt_le_succ x y H1).
 intro.
 apply (Zlt_not_le y (x + 1) H0).  
 replace (x + 1)%Z with (Zsucc x).
 assumption.
 reflexivity.
 intro H1.
 symmetry  in |- *.
 assumption.
Qed.

Lemma Zlt_le_eq_S :
 forall x y : Z, (x < y)%Z -> (y <= x + 1)%Z -> y = (x + 1)%Z.
Proof.
 intros.
 case (Z_le_lt_eq_dec y (x + 1) H0).
 intro H1.
 apply False_ind.
 generalize (Zlt_le_succ x y H).
 intro.
 apply (Zlt_not_le y (x + 1) H1).  
 replace (x + 1)%Z with (Zsucc x).
 assumption.
 reflexivity.
 trivial.
Qed.


Lemma double_not_equal_zero :
 forall c d : Z, ~ (c = 0%Z /\ d = 0%Z) -> c <> d \/ c <> 0%Z.
Proof.
 intros.
 case (Z_zerop c).
 intro.
 rewrite e.  
 left.
 apply sym_not_eq. 
 intro.
 apply H; repeat split; assumption.
 intro; right; assumption.
Qed.

Lemma triple_not_equal_zero :
 forall a b c : Z,
 ~ (a = 0%Z /\ b = 0%Z /\ c = 0%Z) -> a <> 0%Z \/ b <> 0%Z \/ c <> 0%Z.
Proof.
 intros a b c H; case (Z_zerop a); intro Ha;
            [ case (Z_zerop b); intro Hb;
               [ case (Z_zerop c); intro Hc;
                  [ apply False_ind; apply H; repeat split | right; right ]
               | right; left ]
            | left ]; assumption.
Qed. 

Lemma mediant_1 :
 forall m n m' n' : Z, (m' * n < m * n')%Z -> ((m + m') * n < m * (n + n'))%Z.
Proof.
 intros. 
 rewrite Zmult_plus_distr_r. 
 rewrite Zmult_plus_distr_l.
 apply Zplus_lt_compat_l.
 assumption.
Qed.
 
Lemma mediant_2 :
 forall m n m' n' : Z,
 (m' * n < m * n')%Z -> (m' * (n + n') < (m + m') * n')%Z. 
Proof.
 intros.
 rewrite Zmult_plus_distr_l.
 rewrite Zmult_plus_distr_r.
 apply Zplus_lt_compat_r.
 assumption.
Qed.


Lemma mediant_3 :
 forall a b m n m' n' : Z,
 (0 <= a * m + b * n)%Z ->
 (0 <= a * m' + b * n')%Z -> (0 <= a * (m + m') + b * (n + n'))%Z.
Proof.
 intros.
 replace (a * (m + m') + b * (n + n'))%Z with
  (a * m + b * n + (a * m' + b * n'))%Z.
 apply Zplus_le_0_compat.
 assumption.
 assumption.
 ring.
Qed. 

Lemma fraction_lt_trans :
 forall a b c d e f : Z,
 (0 < b)%Z ->
 (0 < d)%Z ->
 (0 < f)%Z -> (a * d < c * b)%Z -> (c * f < e * d)%Z -> (a * f < e * b)%Z.
Proof.
 intros.
 apply Zgt_lt.
 apply Zgt_mult_reg_absorb_l with d.
 Flip.
 apply Zgt_trans with (c * b * f)%Z.
 replace (d * (e * b))%Z with (b * (e * d))%Z.
 replace (c * b * f)%Z with (b * (c * f))%Z.
 apply Zlt_gt.
 apply Zlt_reg_mult_l.
 Flip.
 assumption.
 ring.
 ring.
 replace (c * b * f)%Z with (f * (c * b))%Z.  
 replace (d * (a * f))%Z with (f * (a * d))%Z.
 apply Zlt_gt.
 apply Zlt_reg_mult_l.
 Flip.
 assumption.
 ring.
 ring.
Qed.


Lemma square_pos : forall a : Z, a <> 0%Z -> (0 < a * a)%Z.
Proof.
 intros [| p| p]; intros; [ Falsum | constructor | constructor ].
Qed.
 
Hint Resolve square_pos: zarith.

(*###########################################################################*)
(** Properties of positive numbers, mapping between Z and nat                *)
(*###########################################################################*)


Definition Z2positive (z : Z) :=
  match z with
  | Zpos p => p
  | Zneg p => p
  | Z0 => 1%positive
  end.


Lemma ZL9 : forall p : positive, Z_of_nat (nat_of_P p) = Zpos p. (*QF*)
Proof.
 intro.
 cut (exists h : nat, nat_of_P p = S h).
 intro.
 case H.
 intros.
 unfold Z_of_nat in |- *.
 rewrite H0.
 
 apply f_equal with (A := positive) (B := Z) (f := Zpos).
 cut (P_of_succ_nat (nat_of_P p) = P_of_succ_nat (S x)).
 intro.
 rewrite P_of_succ_nat_o_nat_of_P_eq_succ in H1.
 cut (Ppred (Psucc p) = Ppred (P_of_succ_nat (S x))).
 intro.
 rewrite Ppred_succ in H2.
 simpl in H2.
 rewrite Ppred_succ in H2.
 apply sym_eq.
 assumption.
 apply f_equal with (A := positive) (B := positive) (f := Ppred).
 assumption.
 apply f_equal with (f := P_of_succ_nat).
 assumption.
 apply ZL4.
Qed.

Coercion Z_of_nat : nat >-> Z.

Lemma ZERO_lt_POS : forall p : positive, (0 < Zpos p)%Z.
Proof.
 intros.
 constructor.
Qed.


Lemma POS_neq_ZERO : forall p : positive, Zpos p <> 0%Z.
Proof.
 intros.
 apply sym_not_eq.
 apply Zorder.Zlt_not_eq.
 apply ZERO_lt_POS.
Qed.

Lemma NEG_neq_ZERO : forall p : positive, Zneg p <> 0%Z.
Proof.
 intros.
 apply Zorder.Zlt_not_eq.
 unfold Zlt in |- *.
 constructor.
Qed.


Lemma POS_resp_eq : forall p0 p1 : positive, Zpos p0 = Zpos p1 -> p0 = p1.
Proof.
 intros.
 injection H.
 trivial.
Qed.

Lemma nat_nat_pos : forall m n : nat, ((m + 1) * (n + 1) > 0)%Z. (*QF*)
Proof.
 intros.
 apply Zlt_gt.
 cut (Z_of_nat m + 1 > 0)%Z.
 intro.
 cut (0 < Z_of_nat n + 1)%Z.
 intro.
 cut ((Z_of_nat m + 1) * 0 < (Z_of_nat m + 1) * (Z_of_nat n + 1))%Z.
 rewrite Zmult_0_r.
 intro.
 assumption.
 
 apply Zlt_reg_mult_l.
 assumption.
 assumption.
 change (0 < Zsucc (Z_of_nat n))%Z in |- *.
 apply Zle_lt_succ.
 change (Z_of_nat 0 <= Z_of_nat n)%Z in |- *.
 apply Znat.inj_le.
 apply le_O_n.
 apply Zlt_gt.
 change (0 < Zsucc (Z_of_nat m))%Z in |- *.
 apply Zle_lt_succ.
 change (Z_of_nat 0 <= Z_of_nat m)%Z in |- *.
 apply Znat.inj_le.
 apply le_O_n.
Qed.
 
 
Theorem S_predn : forall m : nat, m <> 0 -> S (pred m) = m. (*QF*)
Proof.
 intros.
 case (O_or_S m).
 intro.
 case s.
 intros.
 rewrite <- e. 
 rewrite <- pred_Sn with (n := x). 
 trivial.
 intro.
 apply False_ind.
 apply H.
 apply sym_eq.
 assumption.
Qed.

Lemma absolu_1 : forall x : Z, Zabs_nat x = 0 -> x = 0%Z. (*QF*)
Proof.
 intros.
 case (dec_eq x 0).
 intro.
 assumption.
 intro.
 apply False_ind.
 cut ((x < 0)%Z \/ (x > 0)%Z).
 intro.
 ElimCompare x 0%Z.
 intro.
 cut (x = 0%Z).
 assumption.
 cut ((x ?= 0)%Z = Datatypes.Eq -> x = 0%Z).
 intro.
 apply H3.
 assumption.
 apply proj1 with (B := x = 0%Z -> (x ?= 0)%Z = Datatypes.Eq).
 change ((x ?= 0)%Z = Datatypes.Eq <-> x = 0%Z) in |- *.
 apply Zcompare_Eq_iff_eq.
 
 (***)
 intro.
 cut (exists h : nat, Zabs_nat x = S h).
 intro.
 case H3.
 rewrite H.
 exact O_S.
 
 change (x < 0)%Z in H2.
 cut (0 > x)%Z.
 intro.
 cut (exists p : positive, (0 + - x)%Z = Zpos p).
 simpl in |- *.
 intro.
 case H4.
 intros.
 cut (exists q : positive, x = Zneg q).
 intro.
 case H6.
 intros.
 rewrite H7.
 unfold Zabs_nat in |- *.
 generalize x1.
 exact ZL4.
 cut (x = (- Zpos x0)%Z).
 simpl in |- *.
 intro.
 exists x0.
 assumption.
 cut ((- - x)%Z = x).
 intro.
 rewrite <- H6.
 exact (f_equal Zopp H5). 
 apply Zopp_involutive.
 apply Zcompare_Gt_spec.
 assumption.
 apply Zlt_gt.
 assumption.
 
 (***)
 intro.
 cut (exists h : nat, Zabs_nat x = S h).
 intro.
 case H3.
 rewrite H.
 exact O_S.
 
 cut (exists p : positive, (x + - (0))%Z = Zpos p).
 simpl in |- *.
 rewrite Zplus_0_r.
 intro.
 case H3.
 intros.
 rewrite H4.
 unfold Zabs_nat in |- *.
 generalize x0.
 exact ZL4.
 apply Zcompare_Gt_spec.
 assumption.
 
 (***)
 cut ((x < 0)%Z \/ (0 < x)%Z).
 intro.
 apply
  or_ind with (A := (x < 0)%Z) (B := (0 < x)%Z) (P := (x < 0)%Z \/ (x > 0)%Z).
 intro.
 left.
 assumption.
 intro.
 right.
 apply Zlt_gt.
 assumption.
 assumption.
 apply not_Zeq.
 assumption.
Qed.

Lemma absolu_2 : forall x : Z, x <> 0%Z -> Zabs_nat x <> 0. (*QF*)
Proof.
 intros.
 intro.
 apply H.
 apply absolu_1.
 assumption.
Qed.




Lemma absolu_inject_nat : forall n : nat, Zabs_nat (Z_of_nat n) = n.
Proof.
 simple induction n; simpl in |- *.
 reflexivity.
 intros.
 apply nat_of_P_o_P_of_succ_nat_eq_succ.
Qed.


Lemma eq_inj : forall m n : nat, m = n :>Z -> m = n.
Proof.
 intros.
 generalize (f_equal Zabs_nat H). 
 intro.
 rewrite (absolu_inject_nat m) in H0.
 rewrite (absolu_inject_nat n) in H0.
 assumption.
Qed.

Lemma lt_inj : forall m n : nat, (m < n)%Z -> m < n.
Proof.
 intros.
 omega.
Qed.

Lemma le_inj : forall m n : nat, (m <= n)%Z -> m <= n.
Proof.
 intros.
 omega.
Qed.


Lemma inject_nat_S_inf : forall x : Z, (0 < x)%Z -> {n : nat | x = S n}.
Proof.
 intros [| p| p] Hp; try discriminate Hp.
 exists (pred (nat_of_P p)).
 rewrite S_predn.
 symmetry  in |- *; apply ZL9.
 clear Hp;
  apply sym_not_equal; apply lt_O_neq; apply lt_O_nat_of_P.
Qed.



Lemma le_absolu :
 forall x y : Z,
 (0 <= x)%Z -> (0 <= y)%Z -> (x <= y)%Z -> Zabs_nat x <= Zabs_nat y.
Proof.
 intros [| x| x] [| y| y] Hx Hy Hxy;
  apply le_O_n ||
    (try
      match goal with
      | id1:(0 <= Zneg _)%Z |- _ =>
          apply False_ind; apply id1; constructor
      | id1:(Zpos _ <= 0)%Z |- _ =>
          apply False_ind; apply id1; constructor
      | id1:(Zpos _ <= Zneg _)%Z |- _ =>
          apply False_ind; apply id1; constructor
      end).
 simpl in |- *. 
 apply le_inj.
 do 2 rewrite ZL9.
 assumption.
Qed.

Lemma lt_absolu :
 forall x y : Z,
 (0 <= x)%Z -> (0 <= y)%Z -> (x < y)%Z -> Zabs_nat x < Zabs_nat y.
Proof.
 intros [| x| x] [| y| y] Hx Hy Hxy; inversion Hxy;
  try
   match goal with
   | id1:(0 <= Zneg _)%Z |- _ =>
       apply False_ind; apply id1; constructor
   | id1:(Zpos _ <= 0)%Z |- _ =>
       apply False_ind; apply id1; constructor
   | id1:(Zpos _ <= Zneg _)%Z |- _ =>
       apply False_ind; apply id1; constructor
   end; simpl in |- *; apply lt_inj; repeat rewrite ZL9; 
  assumption.
Qed.

Lemma absolu_plus :
 forall x y : Z,
 (0 <= x)%Z -> (0 <= y)%Z -> Zabs_nat (x + y) = Zabs_nat x + Zabs_nat y.
Proof.
 intros [| x| x] [| y| y] Hx Hy; trivial;
  try
   match goal with
   | id1:(0 <= Zneg _)%Z |- _ =>
       apply False_ind; apply id1; constructor
   | id1:(Zpos _ <= 0)%Z |- _ =>
       apply False_ind; apply id1; constructor
   | id1:(Zpos _ <= Zneg _)%Z |- _ =>
       apply False_ind; apply id1; constructor
   end.
 rewrite <- BinInt.Zpos_plus_distr.
 unfold Zabs_nat in |- *.
 apply nat_of_P_plus_morphism.
Qed.

Lemma pred_absolu :
 forall x : Z, (0 < x)%Z -> pred (Zabs_nat x) = Zabs_nat (x - 1).
Proof.
 intros x Hx.
 generalize (Z_lt_lt_S_eq_dec 0 x Hx); simpl in |- *; intros [H1| H1];
  [ replace (Zabs_nat x) with (Zabs_nat (x - 1 + 1));
     [ idtac | apply f_equal with Z; auto with zarith ];
     rewrite absolu_plus;
     [ unfold Zabs_nat at 2, nat_of_P, Piter_op in |- *; omega
     | auto with zarith
     | intro; discriminate ]
  | rewrite <- H1; reflexivity ].
Qed. 

Definition pred_nat : forall (x : Z) (Hx : (0 < x)%Z), nat.
intros [| px| px] Hx; try abstract (discriminate Hx).
exact (pred (nat_of_P px)).
Defined.

Lemma pred_nat_equal :
 forall (x : Z) (Hx1 Hx2 : (0 < x)%Z), pred_nat x Hx1 = pred_nat x Hx2.
Proof.
 intros [| px| px] Hx1 Hx2; try (discriminate Hx1); trivial.
Qed.

Let pred_nat_unfolded_subproof px :
  Pos.to_nat px <> 0.
Proof.
apply sym_not_equal; apply lt_O_neq; apply lt_O_nat_of_P.
Qed.

Lemma pred_nat_unfolded :
 forall (x : Z) (Hx : (0 < x)%Z), x = S (pred_nat x Hx).
Proof.
 intros [| px| px] Hx; try discriminate Hx.
 unfold pred_nat in |- *.
 rewrite S_predn.
 symmetry  in |- *; apply ZL9.
 clear Hx; apply pred_nat_unfolded_subproof.
Qed.

Lemma absolu_pred_nat :
 forall (m : Z) (Hm : (0 < m)%Z), S (pred_nat m Hm) = Zabs_nat m.
Proof.
 intros [| px| px] Hx; try discriminate Hx.
 unfold pred_nat in |- *.
 rewrite S_predn.
 reflexivity.
 apply pred_nat_unfolded_subproof.
Qed.

Lemma pred_nat_absolu :
 forall (m : Z) (Hm : (0 < m)%Z), pred_nat m Hm = Zabs_nat (m - 1).
Proof.
 intros [| px| px] Hx; try discriminate Hx.
 unfold pred_nat in |- *.
 rewrite <- pred_absolu; reflexivity || assumption.
Qed.

Lemma minus_pred_nat :
 forall (n m : Z) (Hn : (0 < n)%Z) (Hm : (0 < m)%Z) (Hnm : (0 < n - m)%Z),
 S (pred_nat n Hn) - S (pred_nat m Hm) = S (pred_nat (n - m) Hnm).
Proof.
 intros.
 simpl in |- *.  
 destruct n; try discriminate Hn.
 destruct m; try discriminate Hm.
 unfold pred_nat at 1 2 in |- *.
 rewrite minus_pred; try apply lt_O_nat_of_P.
 apply eq_inj.
 rewrite <- pred_nat_unfolded. 
 rewrite Znat.inj_minus1.
 repeat rewrite ZL9. 
 reflexivity.
 apply le_inj.
 apply Zlt_le_weak.
 repeat rewrite ZL9.
 apply Zlt_O_minus_lt.
 assumption.
Qed.


(*###########################################################################*)
(** Properties of Zsgn                                                       *)
(*###########################################################################*)


Lemma Zsgn_1 :
 forall x : Z, {Zsgn x = 0%Z} + {Zsgn x = 1%Z} + {Zsgn x = (-1)%Z}. (*QF*)
Proof.
 intros.
 case x.
 left.
 left.
 unfold Zsgn in |- *.
 reflexivity.
 intro.
 simpl in |- *.
 left.
 right.
 reflexivity.
 intro.
 right.
 simpl in |- *.
 reflexivity.
Qed.


Lemma Zsgn_2 : forall x : Z, Zsgn x = 0%Z -> x = 0%Z.   (*QF*)
Proof.
 intros [| p1| p1]; simpl in |- *; intro H; constructor || discriminate H.
Qed.


Lemma Zsgn_3 : forall x : Z, x <> 0%Z -> Zsgn x <> 0%Z.   (*QF*)
Proof.
 intro.
 case x.
 intros.
 apply False_ind.
 apply H.
 reflexivity.
 intros.
 simpl in |- *.
 discriminate.
 intros.
 simpl in |- *.
 discriminate.
Qed.




Theorem Zsgn_4 : forall a : Z, a = (Zsgn a * Zabs_nat a)%Z.  (*QF*)
Proof.
 intro.
 case a.
 simpl in |- *.
 reflexivity.
 intro.
 unfold Zsgn in |- *.
 unfold Zabs_nat in |- *.
 rewrite Zmult_1_l.
 symmetry  in |- *.
 apply ZL9.
 intros.
 unfold Zsgn in |- *.
 unfold Zabs_nat in |- *.
 rewrite ZL9.
 constructor.
Qed.


Theorem Zsgn_5 :
 forall a b x y : Z,
 x <> 0%Z ->
 y <> 0%Z ->
 (Zsgn a * x)%Z = (Zsgn b * y)%Z -> (Zsgn a * y)%Z = (Zsgn b * x)%Z.  (*QF*)
Proof.
 intros a b x y H H0.
 case a.

 case b.
 simpl in |- *.
 trivial.

 intro.
 unfold Zsgn in |- *.
 intro.
 rewrite Zmult_1_l in H1.
 simpl in H1.
 apply False_ind.
 apply H0.
 symmetry  in |- *.
 assumption.
 intro.
 unfold Zsgn in |- *.
 intro.
 apply False_ind.
 apply H0.
 apply Zopp_inj.
 simpl in |- *.
 transitivity (-1 * y)%Z.
 constructor.
 transitivity (0 * x)%Z.
 symmetry  in |- *.
 assumption.
 simpl in |- *.
 reflexivity.
 intro.
 unfold Zsgn at 1 in |- *.
 unfold Zsgn at 2 in |- *.
 intro.
 transitivity y.
 rewrite Zmult_1_l.
 reflexivity.
 transitivity (Zsgn b * (Zsgn b * y))%Z.
 case (Zsgn_1 b).
 intro.
 case s.
 intro.
 apply False_ind.
 apply H.
 rewrite e in H1.
 change ((1 * x)%Z = 0%Z) in H1.
 rewrite Zmult_1_l in H1.
 assumption.
 intro.
 rewrite e.
 rewrite Zmult_1_l.
 rewrite Zmult_1_l.
 reflexivity.
 intro.
 rewrite e.
 ring.
 rewrite Zmult_1_l in H1.
 rewrite H1.
 reflexivity.
 intro.
 unfold Zsgn at 1 in |- *.
 unfold Zsgn at 2 in |- *.
 intro.
 transitivity (Zsgn b * (-1 * (Zsgn b * y)))%Z.
 case (Zsgn_1 b).
 intros.
 case s.
 intro.
 apply False_ind.
 apply H.
 apply Zopp_inj.
 transitivity (-1 * x)%Z.
 ring.
 unfold Zopp in |- *.
 rewrite e in H1.
 transitivity (0 * y)%Z.
 assumption.
 simpl in |- *.
 reflexivity.
 intro.
 rewrite e.
 ring.
 intro.
 rewrite e.
 ring.
 rewrite <- H1.
 ring.
Qed.

Lemma Zsgn_6 : forall x : Z, x = 0%Z -> Zsgn x = 0%Z.
Proof.
 intros.
 rewrite H.
 simpl in |- *.
 reflexivity.
Qed.


Lemma Zsgn_7 : forall x : Z, (x > 0)%Z -> Zsgn x = 1%Z.
Proof.
 intro.
 case x.
 intro.
 apply False_ind.
 apply (Zlt_irrefl 0).
 Flip.
 intros.
 simpl in |- *.
 reflexivity.
 intros.
 apply False_ind.
 apply (Zlt_irrefl (Zneg p)).
 apply Zlt_trans with 0%Z.
 constructor.
 Flip.
Qed.


Lemma Zsgn_7' : forall x : Z, (0 < x)%Z -> Zsgn x = 1%Z.
Proof.
 intros; apply Zsgn_7; Flip.
Qed.


Lemma Zsgn_8 : forall x : Z, (x < 0)%Z -> Zsgn x = (-1)%Z.
Proof.
 intro.
 case x.
 intro.
 apply False_ind.
 apply (Zlt_irrefl 0).
 assumption.
 intros.
 apply False_ind.
 apply (Zlt_irrefl 0).
 apply Zlt_trans with (Zpos p).
 constructor.
 assumption.
 intros.
 simpl in |- *.
 reflexivity.
Qed.

Lemma Zsgn_9 : forall x : Z, Zsgn x = 1%Z -> (0 < x)%Z.
Proof.
 intro.
 case x.
 intro.
 apply False_ind.
 simpl in H.
 discriminate.
 intros.
 constructor.
 intros.
 apply False_ind.
 discriminate.
Qed.
 
Lemma Zsgn_10 : forall x : Z, Zsgn x = (-1)%Z -> (x < 0)%Z.
Proof.
 intro.
 case x.
 intro.
 apply False_ind.
 discriminate.
 intros.
 apply False_ind.
 discriminate.
 intros.
 constructor.
Qed. 

Lemma Zsgn_11 : forall x : Z, (Zsgn x < 0)%Z -> (x < 0)%Z.
Proof.
 intros.
 apply Zsgn_10.
 case (Zsgn_1 x).
 intro.
 apply False_ind.
 case s.
 intro.
 generalize (Zorder.Zlt_not_eq _ _ H). 
 intro.
 apply (H0 e).
 intro.
 rewrite e in H.
 generalize (Zorder.Zlt_not_eq _ _ H).
 intro.
 discriminate.
 trivial.
Qed. 

Lemma Zsgn_12 : forall x : Z, (0 < Zsgn x)%Z -> (0 < x)%Z.
Proof.
 intros.
 apply Zsgn_9.
 case (Zsgn_1 x).
 intro.
 case s.
 intro.
 generalize (Zorder.Zlt_not_eq _ _ H). 
 intro.
 generalize (sym_eq e).
 intro.
 apply False_ind.
 apply (H0 H1).
 trivial.
 intro.
 rewrite e in H.
 generalize (Zorder.Zlt_not_eq _ _ H).
 intro.
 apply False_ind.
 discriminate.
Qed.   

Lemma Zsgn_13 : forall x : Z, (0 <= Zsgn x)%Z -> (0 <= x)%Z.
Proof.
 intros. 
 case (Z_le_lt_eq_dec 0 (Zsgn x) H).
 intro.
 apply Zlt_le_weak.
 apply Zsgn_12.
 assumption. 
 intro.
 assert (x = 0%Z). 
 apply Zsgn_2.
 symmetry  in |- *.
 assumption.
 rewrite H0.
 apply Zle_refl.
Qed.

Lemma Zsgn_14 : forall x : Z, (Zsgn x <= 0)%Z -> (x <= 0)%Z.
Proof.
 intros. 
 case (Z_le_lt_eq_dec (Zsgn x) 0 H).
 intro.
 apply Zlt_le_weak.
 apply Zsgn_11.
 assumption. 
 intro.
 assert (x = 0%Z). 
 apply Zsgn_2.
 assumption.
 rewrite H0.
 apply Zle_refl.
Qed.

Lemma Zsgn_15 : forall x y : Z, Zsgn (x * y) = (Zsgn x * Zsgn y)%Z.
Proof.
 intros [y| p1 [| p2| p2]| p1 [| p2| p2]]; simpl in |- *; constructor.
Qed.

Lemma Zsgn_16 :
 forall x y : Z,
 Zsgn (x * y) = 1%Z -> {(0 < x)%Z /\ (0 < y)%Z} + {(x < 0)%Z /\ (y < 0)%Z}.
Proof.
 intros [y| p1 [| p2| p2]| p1 [| p2| p2]]; simpl in |- *; intro H;
  try discriminate H; [ left | right ]; repeat split.
Qed. 

Lemma Zsgn_17 :
 forall x y : Z,
 Zsgn (x * y) = (-1)%Z -> {(0 < x)%Z /\ (y < 0)%Z} + {(x < 0)%Z /\ (0 < y)%Z}.
Proof.
 intros [y| p1 [| p2| p2]| p1 [| p2| p2]]; simpl in |- *; intro H;
  try discriminate H; [ left | right ]; repeat split.
Qed. 

Lemma Zsgn_18 : forall x y : Z, Zsgn (x * y) = 0%Z -> {x = 0%Z} + {y = 0%Z}.
Proof.
 intros [y| p1 [| p2| p2]| p1 [| p2| p2]]; simpl in |- *; intro H;
  try discriminate H; [ left | right | right ]; constructor.
Qed. 



Lemma Zsgn_19 : forall x y : Z, (0 < Zsgn x + Zsgn y)%Z -> (0 < x + y)%Z.
Proof.
 Proof.
 intros [y| p1 [| p2| p2]| p1 [| p2| p2]]; simpl in |- *; intro H;
  discriminate H || (constructor || apply Zsgn_12; assumption).
Qed.

Lemma Zsgn_20 : forall x y : Z, (Zsgn x + Zsgn y < 0)%Z -> (x + y < 0)%Z.
Proof.
 Proof.
 intros [y| p1 [| p2| p2]| p1 [| p2| p2]]; simpl in |- *; intro H;
  discriminate H || (constructor || apply Zsgn_11; assumption).
Qed.


Lemma Zsgn_21 : forall x y : Z, (0 < Zsgn x + Zsgn y)%Z -> (0 <= x)%Z.
Proof.
 intros [y| p1 [| p2| p2]| p1 [| p2| p2]]; simpl in |- *; intros H H0;
  discriminate H || discriminate H0.
Qed.

Lemma Zsgn_22 : forall x y : Z, (Zsgn x + Zsgn y < 0)%Z -> (x <= 0)%Z.
Proof.
 Proof.
 intros [y| p1 [| p2| p2]| p1 [| p2| p2]]; simpl in |- *; intros H H0;
  discriminate H || discriminate H0.
Qed.

Lemma Zsgn_23 : forall x y : Z, (0 < Zsgn x + Zsgn y)%Z -> (0 <= y)%Z.
Proof.
 intros [[| p2| p2]| p1 [| p2| p2]| p1 [| p2| p2]]; simpl in |- *;
  intros H H0; discriminate H || discriminate H0.
Qed.

Lemma Zsgn_24 : forall x y : Z, (Zsgn x + Zsgn y < 0)%Z -> (y <= 0)%Z.
Proof.
 intros [[| p2| p2]| p1 [| p2| p2]| p1 [| p2| p2]]; simpl in |- *;
  intros H H0; discriminate H || discriminate H0.
Qed.

Lemma Zsgn_25 : forall x : Z, Zsgn (- x) = (- Zsgn x)%Z.
Proof.
 intros [| p1| p1]; simpl in |- *; reflexivity.
Qed.


Lemma Zsgn_26 : forall x : Z, (0 < x)%Z -> (0 < Zsgn x)%Z.
Proof.
 intros [| p| p] Hp; trivial.
Qed.

Lemma Zsgn_27 : forall x : Z, (x < 0)%Z -> (Zsgn x < 0)%Z.
Proof.
 intros [| p| p] Hp; trivial.
Qed.

Hint Resolve Zsgn_1 Zsgn_2 Zsgn_3 Zsgn_4 Zsgn_5 Zsgn_6 Zsgn_7 Zsgn_7' Zsgn_8
  Zsgn_9 Zsgn_10 Zsgn_11 Zsgn_12 Zsgn_13 Zsgn_14 Zsgn_15 Zsgn_16 Zsgn_17
  Zsgn_18 Zsgn_19 Zsgn_20 Zsgn_21 Zsgn_22 Zsgn_23 Zsgn_24 Zsgn_25 Zsgn_26
  Zsgn_27: zarith.

(*###########################################################################*)
(** Properties of Zabs                                                       *)
(*###########################################################################*)

Lemma Zabs_1 : forall z p : Z, (Zabs z < p)%Z -> (z < p)%Z /\ (- p < z)%Z.
Proof.
 intros z p.
 case z.
 intros.
 simpl in H.
 split.
 assumption.
 apply Zgt_mult_conv_absorb_l with (a := (-1)%Z).
 replace (-1)%Z with (Zpred 0).
 apply Zlt_pred.
 simpl; trivial.
 ring_simplify (-1 * - p)%Z (-1 * 0)%Z.
 apply Zlt_gt.
 assumption.

 intros.
 simpl in H.
 split.
 assumption.
 apply Zlt_trans with (m := 0%Z).
 apply Zgt_mult_conv_absorb_l with (a := (-1)%Z).
 replace (-1)%Z with (Zpred 0).
 apply Zlt_pred.
 simpl; trivial.
 ring_simplify (-1 * - p)%Z (-1 * 0)%Z.
 apply Zlt_gt.
 apply Zlt_trans with (m := Zpos p0).
 constructor.
 assumption.
 constructor.

 intros.
 simpl in H.
 split.
 apply Zlt_trans with (m := Zpos p0).
 constructor.
 assumption.
 
 apply Zgt_mult_conv_absorb_l with (a := (-1)%Z).
 replace (-1)%Z with (Zpred 0).
 apply Zlt_pred.
 simpl;trivial.
 ring_simplify (-1 * - p)%Z.
 replace (-1 * Zneg p0)%Z with (- Zneg p0)%Z.
 replace (- Zneg p0)%Z with (Zpos p0). 
 apply Zlt_gt.
 assumption.
 symmetry  in |- *.
 apply Zopp_neg.
 rewrite Zopp_mult_distr_l_reverse with (n := 1%Z). 
 simpl in |- *.
 constructor.
Qed.


Lemma Zabs_2 : forall z p : Z, (Zabs z > p)%Z -> (z > p)%Z \/ (- p > z)%Z.
Proof.
 intros z p.
 case z.
 intros.
 simpl in H.
 left.
 assumption.

 intros.
 simpl in H.
 left.
 assumption.

 intros.
 simpl in H.
 right.
 apply Zlt_gt.
 apply Zgt_mult_conv_absorb_l with (a := (-1)%Z).
 constructor.
 ring_simplify (-1 * - p)%Z.
 replace (-1 * Zneg p0)%Z with (Zpos p0).
 assumption.
 reflexivity.
Qed.

Lemma Zabs_3 : forall z p : Z, (z < p)%Z /\ (- p < z)%Z -> (Zabs z < p)%Z.
Proof.
 intros z p.
 case z.
  intro. 
  simpl in |- *.
  elim H.
  intros.
  assumption.
  
  intros.
  elim H.
  intros.
  simpl in |- *.
  assumption.
  
  intros.
  elim H.
  intros.
  simpl in |- *.
  apply Zgt_mult_conv_absorb_l with (a := (-1)%Z).
  constructor.
  replace (-1 * Zpos p0)%Z with (Zneg p0).
  replace (-1 * p)%Z with (- p)%Z.
  apply Zlt_gt.
  assumption.
  ring.  
  simpl in |- *.
  reflexivity.
Qed.

Lemma Zabs_4 : forall z p : Z, (Zabs z < p)%Z -> (- p < z < p)%Z.
Proof.
 intros.
 split.
 apply proj2 with (A := (z < p)%Z).
 apply Zabs_1.
 assumption.
 apply proj1 with (B := (- p < z)%Z).
 apply Zabs_1.
 assumption.
Qed.
 

Lemma Zabs_5 : forall z p : Z, (Zabs z <= p)%Z -> (- p <= z <= p)%Z.
Proof.
 intros.
 split.
 replace (- p)%Z with (Zsucc (- Zsucc p)).
 apply Zlt_le_succ.
 apply proj2 with (A := (z < Zsucc p)%Z).
 apply Zabs_1.
 apply Zle_lt_succ.
 assumption.
 unfold Zsucc in |- *.
 ring.
 apply Zlt_succ_le.
 apply proj1 with (B := (- Zsucc p < z)%Z).
 apply Zabs_1.
 apply Zle_lt_succ.
 assumption.
Qed.

Lemma Zabs_6 : forall z p : Z, (Zabs z <= p)%Z -> (z <= p)%Z.
Proof.
 intros.
 apply proj2 with (A := (- p <= z)%Z).
 apply Zabs_5.
 assumption.
Qed.

Lemma Zabs_7 : forall z p : Z, (Zabs z <= p)%Z -> (- p <= z)%Z.
Proof.
 intros.
 apply proj1 with (B := (z <= p)%Z).
 apply Zabs_5.
 assumption.
Qed.

Lemma Zabs_8 : forall z p : Z, (- p <= z <= p)%Z -> (Zabs z <= p)%Z.
Proof.
 intros.
 apply Zlt_succ_le.
 apply Zabs_3.
 elim H.
 intros.
 split.
 apply Zle_lt_succ.
 assumption.
 apply Zlt_le_trans with (m := (- p)%Z).
 apply Zgt_lt. 
 apply Zlt_opp.
 apply Zlt_succ.
 assumption.
Qed.

Lemma Zabs_min : forall z : Z, Zabs z = Zabs (- z).
Proof.
 intro.
 case z.
 simpl in |- *.
 reflexivity.
 intro.
 simpl in |- *.
 reflexivity.
 intro.
 simpl in |- *.
 reflexivity.
Qed.

Lemma Zabs_9 :
 forall z p : Z, (0 <= p)%Z -> (p < z)%Z \/ (z < - p)%Z -> (p < Zabs z)%Z.
Proof.
 intros.
 case H0.
 intro.
 replace (Zabs z) with z.
 assumption.
 symmetry  in |- *.
 apply Zabs_eq.
 apply Zlt_le_weak.
 apply Zle_lt_trans with (m := p).
 assumption.
 assumption.
 intro.
 cut (Zabs z = (- z)%Z).
 intro.
 rewrite H2.
 apply Zmin_cancel_Zlt.
 ring_simplify (- - z)%Z.
 assumption.
 rewrite Zabs_min.
 apply Zabs_eq.
 apply Zlt_le_weak.
 apply Zle_lt_trans with (m := p).
 assumption.
 apply Zmin_cancel_Zlt.
 ring_simplify (- - z)%Z.
 assumption.
Qed.

Lemma Zabs_10 : forall z : Z, (0 <= Zabs z)%Z.
Proof.
 intro.
 case (Z_zerop z).
 intro.
 rewrite e.
 simpl in |- *.
 apply Zle_refl.
 intro.
 case (not_Zeq z 0 n).
 intro.
 apply Zlt_le_weak.
 apply Zabs_9.
 apply Zle_refl.
 simpl in |- *.
 right.
 assumption.
 intro.
 apply Zlt_le_weak.
 apply Zabs_9.
 apply Zle_refl.
 simpl in |- *.
 left.
 assumption.
Qed.

Lemma Zabs_11 : forall z : Z, z <> 0%Z -> (0 < Zabs z)%Z.
Proof.
 intros.
 apply Zabs_9.
 apply Zle_refl.
 simpl in |- *.
 apply not_Zeq.
 intro.
 apply H.
 symmetry  in |- *.
 assumption.
Qed.

Lemma Zabs_12 : forall z m : Z, (m < Zabs z)%Z -> {(m < z)%Z} + {(z < - m)%Z}.
Proof.
 intros [| p| p] m; simpl in |- *; intros H;
  [ left | left | right; apply Zmin_cancel_Zlt; rewrite Zopp_involutive ];
  assumption.
Qed.

Lemma Zabs_mult : forall z p : Z, Zabs (z * p) = (Zabs z * Zabs p)%Z.
Proof.
 intros.
 case z.
 simpl in |- *.
 reflexivity.
 case p.
 simpl in |- *.
 reflexivity.
 intros.
 simpl in |- *.
 reflexivity.
 intros.
 simpl in |- *.
 reflexivity.
 case p.
 intro.
 simpl in |- *.
 reflexivity.
 intros.
 simpl in |- *.
 reflexivity.
 intros.
 simpl in |- *.
 reflexivity.
Qed.

Lemma Zabs_plus : forall z p : Z, (Zabs (z + p) <= Zabs z + Zabs p)%Z.
Proof.
 intros.
 case z.
 simpl in |- *.
 apply Zle_refl.
 case p.
 intro.
 simpl in |- *.
 apply Zle_refl.
 intros.
 simpl in |- *.
 apply Zle_refl.
 intros.
 unfold Zabs at 2 in |- *.
 unfold Zabs at 2 in |- *.
 apply Zabs_8.
 split.
 apply Zplus_le_reg_l with (Zpos p1 - Zneg p0)%Z.
 replace (Zpos p1 - Zneg p0 + - (Zpos p1 + Zpos p0))%Z with
  (- (Zpos p0 + Zneg p0))%Z.
 replace (Zpos p1 - Zneg p0 + (Zpos p1 + Zneg p0))%Z with (2 * Zpos p1)%Z.
 replace (- (Zpos p0 + Zneg p0))%Z with 0%Z.
 apply Zmult_gt_0_le_0_compat.
 constructor.
 apply Zlt_le_weak.
 constructor.
 rewrite <- Zopp_neg with p0.
 ring.
 ring.
 ring.
 apply Zplus_le_compat.
 apply Zle_refl.
 apply Zlt_le_weak.
 constructor.
 
 case p.
 simpl in |- *.
 intro.
 apply Zle_refl.
 intros.
 unfold Zabs at 2 in |- *.
 unfold Zabs at 2 in |- *.
 apply Zabs_8.
 split.
 apply Zplus_le_reg_l with (Zpos p1 + Zneg p0)%Z.
 replace (Zpos p1 + Zneg p0 + - (Zpos p1 + Zpos p0))%Z with
  (Zneg p0 - Zpos p0)%Z.
 replace (Zpos p1 + Zneg p0 + (Zneg p1 + Zpos p0))%Z with 0%Z.
 apply Zplus_le_reg_l with (Zpos p0).
 replace (Zpos p0 + (Zneg p0 - Zpos p0))%Z with (Zneg p0).
 simpl in |- *.
 apply Zlt_le_weak.
 constructor.
 ring.
 replace (Zpos p1 + Zneg p0 + (Zneg p1 + Zpos p0))%Z with
  (Zpos p1 + Zneg p1 + (Zpos p0 + Zneg p0))%Z.
 replace 0%Z with (0 + 0)%Z.
 apply Zplus_eq_compat.
 rewrite <- Zopp_neg with p1.
 ring.
 rewrite <- Zopp_neg with p0.
 ring.
 simpl in |- *.
 constructor.
 ring.
 ring.
 apply Zplus_le_compat.
 apply Zlt_le_weak.
 constructor.
 apply Zle_refl.
 intros.
 simpl in |- *.
 apply Zle_refl.
Qed.

Lemma Zabs_neg : forall z : Z, (z <= 0)%Z -> Zabs z = (- z)%Z.
Proof.
 intro.
 case z.
 simpl in |- *.
 intro.
 reflexivity.
 intros.
 apply False_ind.
 apply H.
 simpl in |- *.
 reflexivity.
 intros.
 simpl in |- *.
 reflexivity.
Qed.

Lemma Zle_Zabs: forall z, (z <= Zabs z)%Z.
Proof.
 intros [|z|z]; simpl; auto with zarith; apply Zle_neg_pos. 
Qed.
 
Hint Resolve Zabs_1 Zabs_2 Zabs_3 Zabs_4 Zabs_5 Zabs_6 Zabs_7 Zabs_8 Zabs_9
  Zabs_10 Zabs_11 Zabs_12 Zabs_min Zabs_neg Zabs_mult Zabs_plus Zle_Zabs: zarith.


(*###########################################################################*)
(** Induction on Z                                                           *)
(*###########################################################################*)

Lemma Zind :
 forall (P : Z -> Prop) (p : Z),
 P p ->
 (forall q : Z, (p <= q)%Z -> P q -> P (q + 1)%Z) ->
 forall q : Z, (p <= q)%Z -> P q. 
Proof.
 intros P p.
 intro.
 intro.
 cut (forall q : Z, (p <= q)%Z -> exists k : nat, q = (p + k)%Z).
 intro.
 cut (forall k : nat, P (p + k)%Z).
 intro.
 intros.
 cut (exists k : nat, q = (p + Z_of_nat k)%Z).
 intro.
 case H4.
 intros.
 rewrite H5.
 apply H2.
 apply H1.
 assumption.
 intro.
 induction  k as [| k Hreck].
 simpl in |- *.
 ring_simplify (p + 0)%Z.
 assumption.
 replace (p + Z_of_nat (S k))%Z with (p + k + 1)%Z.
 apply H0.
 apply Zplus_le_reg_l with (p := (- p)%Z).
 replace (- p + p)%Z with (Z_of_nat 0). 
 ring_simplify (- p + (p + Z_of_nat k))%Z.
 apply Znat.inj_le.
 apply le_O_n.
 ring_simplify; auto with arith. 
 assumption.
 rewrite (Znat.inj_S k).
 unfold Zsucc in |- *.
 ring.
 intros.
 cut (exists k : nat, (q - p)%Z = Z_of_nat k).
 intro.
 case H2.
 intro k.
 intros.
 exists k.
 apply Zplus_reg_l with (n := (- p)%Z).
 replace (- p + q)%Z with (q - p)%Z.
 rewrite H3.
 ring.
 ring.
 apply Z_of_nat_complete.
 unfold Zminus in |- *.
 apply Zle_left.
 assumption.
Qed.

Lemma Zrec :
 forall (P : Z -> Set) (p : Z),
 P p ->
 (forall q : Z, (p <= q)%Z -> P q -> P (q + 1)%Z) ->
 forall q : Z, (p <= q)%Z -> P q. 
Proof.
 intros F p.
 intro.
 intro.
 cut (forall q : Z, (p <= q)%Z -> {k : nat | q = (p + k)%Z}).
 intro.
 cut (forall k : nat, F (p + k)%Z).
 intro.
 intros.
 cut {k : nat | q = (p + Z_of_nat k)%Z}.
 intro.
 case H4.
 intros.
 rewrite e.
 apply H2.
 apply H1.
 assumption.
 intro.
 induction  k as [| k Hreck].
 simpl in |- *.
 rewrite Zplus_0_r.
 assumption.
 replace (p + Z_of_nat (S k))%Z with (p + k + 1)%Z.
 apply H0.
 apply Zplus_le_reg_l with (p := (- p)%Z).
 replace (- p + p)%Z with (Z_of_nat 0). 
 replace (- p + (p + Z_of_nat k))%Z with (Z_of_nat k).
 apply Znat.inj_le.
 apply le_O_n.
 rewrite Zplus_assoc; rewrite Zplus_opp_l; reflexivity.
 rewrite Zplus_opp_l; reflexivity.
 assumption.
 rewrite (Znat.inj_S k).
 unfold Zsucc in |- *.
 apply Zplus_assoc_reverse.
 intros.
 cut {k : nat | (q - p)%Z = Z_of_nat k}.
 intro H2.
 case H2.
 intro k.
 intros.
 exists k.
 apply Zplus_reg_l with (n := (- p)%Z).
 replace (- p + q)%Z with (q - p)%Z.
 rewrite e.
 rewrite Zplus_assoc; rewrite Zplus_opp_l; reflexivity.
 unfold Zminus in |- *.
 apply Zplus_comm.
 apply Z_of_nat_complete_inf.
 unfold Zminus in |- *.
 apply Zle_left.
 assumption.
Qed.

Lemma Zrec_down :
 forall (P : Z -> Set) (p : Z),
 P p ->
 (forall q : Z, (q <= p)%Z -> P q -> P (q - 1)%Z) ->
 forall q : Z, (q <= p)%Z -> P q.
Proof.
 intros F p.
 intro.
 intro.
 cut (forall q : Z, (q <= p)%Z -> {k : nat | q = (p - k)%Z}).
 intro.
 cut (forall k : nat, F (p - k)%Z).
 intro.
 intros.
 cut {k : nat | q = (p - Z_of_nat k)%Z}.
 intro.
 case H4.
 intros.
 rewrite e.
 apply H2.
 apply H1.
 assumption.
 intro.
 induction  k as [| k Hreck].
 simpl in |- *.
 replace (p - 0)%Z with p.
 assumption.
 unfold Zminus in |- *.
 unfold Zopp in |- *.
 rewrite Zplus_0_r; reflexivity.
 replace (p - Z_of_nat (S k))%Z with (p - k - 1)%Z.
 apply H0.
 apply Zplus_le_reg_l with (p := (- p)%Z).
 replace (- p + p)%Z with (- Z_of_nat 0)%Z. 
 replace (- p + (p - Z_of_nat k))%Z with (- Z_of_nat k)%Z.
 apply Zge_le.
 apply Zge_opp.
 apply Znat.inj_le.
 apply le_O_n.
 unfold Zminus in |- *; rewrite Zplus_assoc; rewrite Zplus_opp_l; reflexivity.
 rewrite Zplus_opp_l; reflexivity.
 assumption.
 rewrite (Znat.inj_S k).
 unfold Zsucc in |- *.
 unfold Zminus at 1 2 in |- *.
 rewrite Zplus_assoc_reverse.
 rewrite <- Zopp_plus_distr.
 reflexivity.
 intros.
 cut {k : nat | (p - q)%Z = Z_of_nat k}.
 intro.
 case H2.
 intro k.
 intros.
 exists k.
 apply Zopp_inj. 
 apply Zplus_reg_l with (n := p).
 replace (p + - (p - Z_of_nat k))%Z with (Z_of_nat k). 
 rewrite <- e.
 reflexivity.
 unfold Zminus in |- *.
 rewrite Zopp_plus_distr.
 rewrite Zplus_assoc.
 rewrite Zplus_opp_r.
 rewrite Zopp_involutive.
 reflexivity.
 apply Z_of_nat_complete_inf.
 unfold Zminus in |- *.
 apply Zle_left.
 assumption.
Qed.

Lemma Zind_down :
 forall (P : Z -> Prop) (p : Z),
 P p ->
 (forall q : Z, (q <= p)%Z -> P q -> P (q - 1)%Z) ->
 forall q : Z, (q <= p)%Z -> P q.
Proof.
 intros F p.
 intro.
 intro.
 cut (forall q : Z, (q <= p)%Z -> exists k : nat, q = (p - k)%Z).
 intro.
 cut (forall k : nat, F (p - k)%Z).
 intro.
 intros.
 cut (exists k : nat, q = (p - Z_of_nat k)%Z).
 intro.
 case H4.
 intros x e.
 rewrite e.
 apply H2.
 apply H1.
 assumption.
 intro.
 induction  k as [| k Hreck].
 simpl in |- *.
 replace (p - 0)%Z with p.
 assumption.
 ring.
 replace (p - Z_of_nat (S k))%Z with (p - k - 1)%Z.
 apply H0.
 apply Zplus_le_reg_l with (p := (- p)%Z).
 replace (- p + p)%Z with (- Z_of_nat 0)%Z. 
 replace (- p + (p - Z_of_nat k))%Z with (- Z_of_nat k)%Z.
 apply Zge_le.
 apply Zge_opp.
 apply Znat.inj_le.
 apply le_O_n.
 ring. 
 ring_simplify; auto with arith.
 assumption.
 rewrite (Znat.inj_S k).
 unfold Zsucc in |- *.
 ring.
 intros.
 cut (exists k : nat, (p - q)%Z = Z_of_nat k).
 intro.
 case H2.
 intro k.
 intros.
 exists k.
 apply Zopp_inj. 
 apply Zplus_reg_l with (n := p).
 replace (p + - (p - Z_of_nat k))%Z with (Z_of_nat k). 
 rewrite <- H3.
 ring.
 ring.
 apply Z_of_nat_complete.
 unfold Zminus in |- *.
 apply Zle_left.
 assumption.
Qed.

Lemma Zrec_wf :
 forall (P : Z -> Set) (p : Z),
 (forall q : Z, (forall r : Z, (p <= r < q)%Z -> P r) -> P q) ->
 forall q : Z, (p <= q)%Z -> P q.
Proof.
 intros P p WF_ind_step q Hq.
 cut (forall x : Z, (p <= x)%Z -> forall y : Z, (p <= y < x)%Z -> P y).
 intro.
 apply (H (Zsucc q)).
 apply Zle_le_succ.
 assumption.
 
 split; [ assumption | exact (Zlt_succ q) ].

 intros x0 Hx0; generalize Hx0; pattern x0 in |- *. 
 apply Zrec with (p := p).
 intros.
 absurd (p <= p)%Z.
 apply Zgt_not_le.
 apply Zgt_le_trans with (m := y).
 apply Zlt_gt.
 elim H.
 intros.
 assumption.
 elim H.
 intros.
 assumption.
 apply Zle_refl.

 intros. 
 apply WF_ind_step. 
 intros.
 apply (H0 H).
 split. 
 elim H2.
 intros.
 assumption.
 apply Zlt_le_trans with y. 
 elim H2.
 intros.
 assumption.
 apply Zgt_succ_le. 
 apply Zlt_gt. 
 elim H1.
 intros.
 unfold Zsucc in |- *.
 assumption.
 assumption.
Qed.

Lemma Zrec_wf2 :
 forall (q : Z) (P : Z -> Set) (p : Z),
 (forall q : Z, (forall r : Z, (p <= r < q)%Z -> P r) -> P q) ->
 (p <= q)%Z -> P q.
Proof.
 intros.
 apply Zrec_wf with (p := p).
 assumption.
 assumption.
Qed.

Lemma Zrec_wf_double :
 forall (P : Z -> Z -> Set) (p0 q0 : Z),
 (forall n m : Z,
  (forall p q : Z, (q0 <= q)%Z -> (p0 <= p < n)%Z -> P p q) ->
  (forall p : Z, (q0 <= p < m)%Z -> P n p) -> P n m) ->
 forall p q : Z, (q0 <= q)%Z -> (p0 <= p)%Z -> P p q.
Proof.
 intros P p0 q0 Hrec p.
 intros.
 generalize q H.
 pattern p in |- *.
 apply Zrec_wf with (p := p0).
 intros p1 H1.
 intros.
 pattern q1 in |- *.
 apply Zrec_wf with (p := q0).
 intros q2 H3.
 apply Hrec.
 intros.
 apply H1.
 assumption.
 assumption.
 intros.
 apply H3.
 assumption.
 assumption.
 assumption.
Qed.

Lemma Zind_wf :
 forall (P : Z -> Prop) (p : Z),
 (forall q : Z, (forall r : Z, (p <= r < q)%Z -> P r) -> P q) ->
 forall q : Z, (p <= q)%Z -> P q.
Proof.
 intros P p WF_ind_step q Hq.
 cut (forall x : Z, (p <= x)%Z -> forall y : Z, (p <= y < x)%Z -> P y).
 intro.
 apply (H (Zsucc q)).
 apply Zle_le_succ.
 assumption.
 
 split; [ assumption | exact (Zlt_succ q) ].

 intros x0 Hx0; generalize Hx0; pattern x0 in |- *. 
 apply Zind with (p := p).
 intros.
 absurd (p <= p)%Z.
 apply Zgt_not_le.
 apply Zgt_le_trans with (m := y).
 apply Zlt_gt.
 elim H.
 intros.
 assumption.
 elim H.
 intros.
 assumption.
 apply Zle_refl.

 intros. 
 apply WF_ind_step. 
 intros.
 apply (H0 H).
 split. 
 elim H2.
 intros.
 assumption.
 apply Zlt_le_trans with y. 
 elim H2.
 intros.
 assumption.
 apply Zgt_succ_le. 
 apply Zlt_gt. 
 elim H1.
 intros.
 unfold Zsucc in |- *.
 assumption.
 assumption.
Qed.

Lemma Zind_wf2 :
 forall (q : Z) (P : Z -> Prop) (p : Z),
 (forall q : Z, (forall r : Z, (p <= r < q)%Z -> P r) -> P q) ->
 (p <= q)%Z -> P q.
Proof.
 intros.
 apply Zind_wf with (p := p).
 assumption.
 assumption.
Qed.

Lemma Zind_wf_double :
 forall (P : Z -> Z -> Prop) (p0 q0 : Z),
 (forall n m : Z,
  (forall p q : Z, (q0 <= q)%Z -> (p0 <= p < n)%Z -> P p q) ->
  (forall p : Z, (q0 <= p < m)%Z -> P n p) -> P n m) ->
 forall p q : Z, (q0 <= q)%Z -> (p0 <= p)%Z -> P p q.
Proof.
 intros P p0 q0 Hrec p.
 intros.
 generalize q H.
 pattern p in |- *.
 apply Zind_wf with (p := p0).
 intros p1 H1.
 intros.
 pattern q1 in |- *.
 apply Zind_wf with (p := q0).
 intros q2 H3.
 apply Hrec.
 intros.
 apply H1.
 assumption.
 assumption.
 intros.
 apply H3.
 assumption.
 assumption.
 assumption.
Qed.

(*###########################################################################*)
(** Properties of Zmax                                                       *)
(*###########################################################################*)

Definition Zmax (n m : Z) := (n + m - Zmin n m)%Z.

Lemma ZmaxSS : forall n m : Z, (Zmax n m + 1)%Z = Zmax (n + 1) (m + 1).
Proof.
 intros.
 unfold Zmax in |- *.
 replace (Zmin (n + 1) (m + 1)) with (Zmin n m + 1)%Z.
 ring.
 symmetry  in |- *.
 change (Zmin (Zsucc n) (Zsucc m) = Zsucc (Zmin n m)) in |- *.
 symmetry  in |- *.
 apply Zmin_SS.
Qed.

Lemma Zle_max_l : forall n m : Z, (n <= Zmax n m)%Z.
Proof.
 intros.
 unfold Zmax in |- *.
 apply Zplus_le_reg_l with (p := (- n + Zmin n m)%Z).
 ring_simplify (- n + Zmin n m + n)%Z.
 ring_simplify (- n + Zmin n m + (n + m - Zmin n m))%Z.
 apply Zle_min_r.
Qed.

Lemma Zle_max_r : forall n m : Z, (m <= Zmax n m)%Z.
Proof.
 intros.
 unfold Zmax in |- *.
 apply Zplus_le_reg_l with (p := (- m + Zmin n m)%Z).
 ring_simplify (- m + Zmin n m + m)%Z.
 ring_simplify (- m + Zmin n m + (n + m - Zmin n m))%Z.
 apply Zle_min_l.
Qed.


Lemma Zmin_or_informative : forall n m : Z, {Zmin n m = n} + {Zmin n m = m}.
Proof.
 intros.
 case (Z_lt_ge_dec n m).
 unfold Zmin in |- *.
 unfold Zlt in |- *.
 intro z.
 rewrite z.
 left.
 reflexivity.
 intro.
 cut ({(n > m)%Z} + {n = m :>Z}).
 intro.
 case H.
 intros z0.
 unfold Zmin in |- *.
 unfold Zgt in z0.
 rewrite z0.
 right.
 reflexivity.
 intro.
 rewrite e.
 right.
 apply Zmin_n_n.
 cut ({(m < n)%Z} + {m = n :>Z}).
 intro.
 elim H.
 intro.
 left.
 apply Zlt_gt.
 assumption.
 intro.
 right.
 symmetry  in |- *.
 assumption.
 apply Z_le_lt_eq_dec. 
 apply Zge_le.
 assumption.
Qed.

Lemma Zmax_case : forall (n m : Z) (P : Z -> Set), P n -> P m -> P (Zmax n m).
Proof.
 intros.
 unfold Zmax in |- *.
 case Zmin_or_informative with (n := n) (m := m).
 intro.
 rewrite e.
 cut ((n + m - n)%Z = m).
 intro.
 rewrite H1.
 assumption.
 ring.
 intro.
 rewrite e.
 cut ((n + m - m)%Z = n).
 intro.
 rewrite H1.
 assumption.
 ring.
Qed.
 
Lemma Zmax_or_informative : forall n m : Z, {Zmax n m = n} + {Zmax n m = m}. 
Proof.
 intros.
 unfold Zmax in |- *.
 case Zmin_or_informative with (n := n) (m := m).
 intro.
 rewrite e.
 right.
 ring.
 intro.
 rewrite e.
 left.
 ring.
Qed.

Lemma Zmax_n_n : forall n : Z, Zmax n n = n.
Proof.
 intros.
 unfold Zmax in |- *.
 rewrite (Zmin_n_n n).
 ring.
Qed.

Hint Resolve ZmaxSS Zle_max_r Zle_max_l Zmax_n_n: zarith.

(*###########################################################################*)
(** Properties of Arity                                                      *)
(*###########################################################################*)

Lemma Zeven_S : forall x : Z, Zeven.Zodd x -> Zeven.Zeven (x + 1).
Proof.
 exact Zeven.Zeven_Sn.
Qed.

Lemma Zeven_pred : forall x : Z, Zeven.Zodd x -> Zeven.Zeven (x - 1). 
Proof.
 exact Zeven.Zeven_pred.
Qed.

(* This lemma used to be useful since it was mentioned with an unnecessary premise 
   `x>=0` as Z_modulo_2 in ZArith, but the ZArith version has been fixed. *)

Definition Z_modulo_2_always :
  forall x : Z, {y : Z | x = (2 * y)%Z} + {y : Z | x = (2 * y + 1)%Z} :=
  Zeven.Z_modulo_2.

(*###########################################################################*)
(** Properties of Zdiv                                                       *)
(*###########################################################################*)

Lemma Z_div_mod_eq_2 :
 forall a b : Z, (0 < b)%Z -> (b * (a / b))%Z = (a - a mod b)%Z.
Proof.
 intros.
 apply Zplus_minus_eq.
 rewrite Zplus_comm.
 apply Z_div_mod_eq.
 Flip.
Qed.

Lemma Z_div_le :
 forall a b c : Z, (0 < c)%Z -> (b <= a)%Z -> (b / c <= a / c)%Z. 
Proof.
 intros.
 apply Zge_le.
 apply Z_div_ge; Flip; assumption.
Qed.

Lemma Z_div_nonneg :
 forall a b : Z, (0 < b)%Z -> (0 <= a)%Z -> (0 <= a / b)%Z.
Proof.
 intros.
 apply Zge_le.
 apply Z_div_ge0; Flip; assumption.
Qed.

Lemma Z_div_neg : forall a b : Z, (0 < b)%Z -> (a < 0)%Z -> (a / b < 0)%Z.
Proof.
 intros.
 rewrite (Z_div_mod_eq a b) in H0.
 elim (Z_mod_lt a b).
 intros H1 _.
 apply Znot_ge_lt.
 intro.
 apply (Zlt_not_le (b * (a / b) + a mod b) 0 H0).
 apply Zplus_le_0_compat.
 apply Zmult_le_0_compat. 
 apply Zlt_le_weak; assumption.
 Flip.
 assumption.
 Flip.
 Flip.
Qed.

Hint Resolve Z_div_mod_eq_2 Z_div_le Z_div_nonneg Z_div_neg: zarith.

(*###########################################################################*)
(** Properties of Zpower                                                       *)
(*###########################################################################*)

Lemma Zpower_1 : forall a : Z, (a ^ 1)%Z = a.
Proof.
 intros; unfold Zpower in |- *; unfold Zpower_pos in |- *; simpl in |- *;
  auto with zarith.
Qed.

Lemma Zpower_2 : forall a : Z, (a ^ 2)%Z = (a * a)%Z.
Proof.
 intros; unfold Zpower in |- *; unfold Zpower_pos in |- *; simpl in |- *;
  ring.
Qed.

Hint Resolve Zpower_1 Zpower_2: zarith.