Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 55,792 Bytes
4365a98 |
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 |
(* ========================================================================= *)
(* Permutations, both general and specifically on finite sets. *)
(* ========================================================================= *)
parse_as_infix("permutes",(12,"right"));;
let permutes = new_definition
`p permutes s <=> (!x. ~(x IN s) ==> p(x) = x) /\ (!y. ?!x. p x = y)`;;
(* ------------------------------------------------------------------------- *)
(* Inverse function (on whole universe). *)
(* ------------------------------------------------------------------------- *)
let inverse = new_definition
`inverse(f) = \y. @x. f x = y`;;
let SURJECTIVE_INVERSE = prove
(`!f. (!y. ?x. f x = y) <=> !y. f(inverse f y) = y`,
REWRITE_TAC[SURJECTIVE_RIGHT_INVERSE; inverse] THEN MESON_TAC[]);;
let SURJECTIVE_INVERSE_o = prove
(`!f. (!y. ?x. f x = y) <=> (f o inverse f = I)`,
REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM; SURJECTIVE_INVERSE]);;
let INJECTIVE_INVERSE = prove
(`!f. (!x x'. f x = f x' ==> x = x') <=> !x. inverse f (f x) = x`,
MESON_TAC[inverse]);;
let INJECTIVE_INVERSE_o = prove
(`!f. (!x x'. f x = f x' ==> x = x') <=> (inverse f o f = I)`,
REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM; INJECTIVE_INVERSE]);;
let INVERSE_UNIQUE_o = prove
(`!f g. f o g = I /\ g o f = I ==> inverse f = g`,
REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN
MESON_TAC[INJECTIVE_INVERSE; SURJECTIVE_INVERSE]);;
let INVERSE_I = prove
(`inverse I = I`,
MATCH_MP_TAC INVERSE_UNIQUE_o THEN REWRITE_TAC[I_O_ID]);;
(* ------------------------------------------------------------------------- *)
(* Transpositions. *)
(* ------------------------------------------------------------------------- *)
let swap = new_definition
`swap(i,j) k = if k = i then j else if k = j then i else k`;;
let SWAP_REFL = prove
(`!a. swap(a,a) = I`,
REWRITE_TAC[FUN_EQ_THM; swap; I_THM] THEN MESON_TAC[]);;
let SWAP_SYM = prove
(`!a b. swap(a,b) = swap(b,a)`,
REWRITE_TAC[FUN_EQ_THM; swap; I_THM] THEN MESON_TAC[]);;
let SWAP_IDEMPOTENT = prove
(`!a b. swap(a,b) o swap(a,b) = I`,
REWRITE_TAC[FUN_EQ_THM; swap; o_THM; I_THM] THEN MESON_TAC[]);;
let INVERSE_SWAP = prove
(`!a b. inverse(swap(a,b)) = swap(a,b)`,
REPEAT GEN_TAC THEN MATCH_MP_TAC INVERSE_UNIQUE_o THEN
REWRITE_TAC[SWAP_SYM; SWAP_IDEMPOTENT]);;
let SWAP_GALOIS = prove
(`!a b x y. x = swap(a,b) y <=> y = swap(a,b) x`,
REWRITE_TAC[swap] THEN MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Basic consequences of the definition. *)
(* ------------------------------------------------------------------------- *)
let PERMUTES_IN_IMAGE = prove
(`!p s x. p permutes s ==> (p(x) IN s <=> x IN s)`,
REWRITE_TAC[permutes] THEN MESON_TAC[]);;
let PERMUTES_IMAGE = prove
(`!p s. p permutes s ==> IMAGE p s = s`,
REWRITE_TAC[permutes; EXTENSION; IN_IMAGE] THEN MESON_TAC[]);;
let PERMUTES_INJECTIVE = prove
(`!p s. p permutes s ==> !x y. p(x) = p(y) <=> x = y`,
REWRITE_TAC[permutes] THEN MESON_TAC[]);;
let PERMUTES_SURJECTIVE = prove
(`!p s. p permutes s ==> !y. ?x. p(x) = y`,
REWRITE_TAC[permutes] THEN MESON_TAC[]);;
let PERMUTES_INVERSES_o = prove
(`!p s. p permutes s ==> p o inverse(p) = I /\ inverse(p) o p = I`,
REWRITE_TAC[GSYM INJECTIVE_INVERSE_o; GSYM SURJECTIVE_INVERSE_o] THEN
REWRITE_TAC[permutes] THEN MESON_TAC[]);;
let PERMUTES_INVERSES = prove
(`!p s. p permutes s
==> (!x. p(inverse p x) = x) /\ (!x. inverse p (p x) = x)`,
REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP PERMUTES_INVERSES_o) THEN
REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM]);;
let PERMUTES_SUBSET = prove
(`!p s t. p permutes s /\ s SUBSET t ==> p permutes t`,
REWRITE_TAC[permutes; SUBSET] THEN MESON_TAC[]);;
let PERMUTES_EMPTY = prove
(`!p. p permutes {} <=> p = I`,
REWRITE_TAC[FUN_EQ_THM; I_THM; permutes; NOT_IN_EMPTY] THEN MESON_TAC[]);;
let PERMUTES_SING = prove
(`!p a. p permutes {a} <=> p = I`,
REWRITE_TAC[FUN_EQ_THM; I_THM; permutes; IN_SING] THEN MESON_TAC[]);;
let PERMUTES_UNIV = prove
(`!p. p permutes UNIV <=> !y:A. ?!x. p x = y`,
REWRITE_TAC[permutes; IN_UNIV] THEN MESON_TAC[]);;
let PERMUTES_INVERSE_EQ = prove
(`!p s. p permutes s ==> !x y. inverse p y = x <=> p x = y`,
REWRITE_TAC[permutes; inverse] THEN MESON_TAC[]);;
let PERMUTES_SWAP = prove
(`!a b s. a IN s /\ b IN s ==> swap(a,b) permutes s`,
REWRITE_TAC[permutes; swap] THEN MESON_TAC[]);;
let PERMUTES_SUPERSET = prove
(`!p s t. p permutes s /\ (!x. x IN (s DIFF t) ==> p(x) = x)
==> p permutes t`,
REWRITE_TAC[permutes; IN_DIFF] THEN MESON_TAC[]);;
let PERMUTES_BIJECTIONS = prove
(`!p q. (!x. x IN s ==> p x IN s) /\ (!x. ~(x IN s) ==> p x = x) /\
(!x. x IN s ==> q x IN s) /\ (!x. ~(x IN s) ==> q x = x) /\
(!x. p(q x) = x) /\ (!x. q(p x) = x)
==> p permutes s`,
REWRITE_TAC[permutes] THEN MESON_TAC[]);;
let PERMUTES_INVERSE_FUNCTION = prove
(`!s p:A->A.
p permutes s <=>
?q. (!x. ~(x IN s) ==> p x = x) /\
(!x. x IN s ==> p x IN s) /\
(!x. x IN s ==> p(q x) = x /\ q(p x) = x)`,
REPEAT GEN_TAC THEN EQ_TAC THENL
[DISCH_TAC THEN EXISTS_TAC `inverse p:A->A` THEN
ASM_MESON_TAC[permutes; PERMUTES_INVERSES];
STRIP_TAC THEN ASM_SIMP_TAC[permutes; SUBSET; FORALL_IN_IMAGE] THEN
X_GEN_TAC `x:A` THEN REWRITE_TAC[EXISTS_UNIQUE] THEN
ASM_CASES_TAC `(x:A) IN s` THENL
[EXISTS_TAC `(q:A->A) x`; EXISTS_TAC `x:A`] THEN
ASM_MESON_TAC[]]);;
let PERMUTES_ALT = prove
(`!(p:A->A) s.
p permutes s <=>
(!x. x IN s ==> p x IN s) /\
(!x. ~(x IN s) ==> p x = x) /\
(!y. y IN s ==> ?!x. x IN s /\ p x = y)`,
REPEAT GEN_TAC THEN REWRITE_TAC[permutes] THEN
EQ_TAC THEN SIMP_TAC[] THENL [MESON_TAC[]; DISCH_TAC] THEN
MATCH_MP_TAC(MESON[]
`!s. (!y. ~(y IN s) ==> P y) /\ (!y. y IN s ==> P y) ==> !y. P y`) THEN
EXISTS_TAC `IMAGE (p:A->A) s` THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN
ASM SET_TAC[]);;
let PERMUTES_RESTRICT_SET = prove
(`!Q p s:A->bool.
p permutes s /\ (!x. x IN s ==> (Q(p x) <=> Q x))
==> (\i. if Q i then p i else i) permutes {x | x IN s /\ Q x}`,
REWRITE_TAC[PERMUTES_ALT] THEN SET_TAC[]);;
let PERMUTES_RESTRICT = prove
(`!Q p s:A->bool.
p permutes s /\ (!x. x IN s ==> (Q(p x) <=> Q x))
==> (\i. if Q i then p i else i) permutes s`,
REWRITE_TAC[PERMUTES_ALT] THEN SET_TAC[]);;
let PERMUTES_CARTESIAN_PRODUCT = prove
(`!(p:A->A) (q:B->B) s t.
p permutes s /\ q permutes t
==> (\(i,j). if i IN s /\ j IN t then p i,q j else i,j)
permutes (s CROSS t)`,
REWRITE_TAC[permutes; EXISTS_UNIQUE_THM] THEN
REWRITE_TAC[FORALL_PAIR_THM; EXISTS_PAIR_THM; PAIR_EQ; IN_CROSS] THEN
REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[COND_RATOR] THEN
REWRITE_TAC[PAIR_EQ] THEN MESON_TAC[]);;
let PERMUTES_TRANSFER_BIJECTIONS = prove
(`!(f:A->B) f' p s t.
(!x. f'(f x) = x) /\ (!y. f(f' y) = y) /\
(!x. x IN s ==> f x IN t) /\
(!y. y IN t ==> f' y IN s)
==> ((f' o p o f) permutes s <=> p permutes t)`,
REWRITE_TAC[permutes; o_THM] THEN MESON_TAC[]);;
let PERMUTES_TRANSFER = prove
(`!(f:A->B) p q s.
p permutes s /\
(!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\
(!x. x IN s ==> q(f x) = f(p x)) /\
(!y. ~(y IN IMAGE f s) ==> q y = y)
==> q permutes (IMAGE f s)`,
SIMP_TAC[PERMUTES_ALT] THEN SET_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Group properties. *)
(* ------------------------------------------------------------------------- *)
let PERMUTES_ID = prove
(`!s:A->bool. (\x. x) permutes s`,
REWRITE_TAC[permutes] THEN MESON_TAC[]);;
let PERMUTES_I = prove
(`!s. I permutes s`,
REWRITE_TAC[permutes; I_THM] THEN MESON_TAC[]);;
let PERMUTES_COMPOSE = prove
(`!p q s. p permutes s /\ q permutes s ==> (q o p) permutes s`,
REWRITE_TAC[permutes; o_THM] THEN METIS_TAC[]);;
let PERMUTES_INVERSE = prove
(`!p s. p permutes s ==> inverse(p) permutes s`,
REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP PERMUTES_INVERSE_EQ) THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[permutes] THEN MESON_TAC[]);;
let PERMUTES_INVERSE_INVERSE = prove
(`!p. p permutes s ==> inverse(inverse p) = p`,
SIMP_TAC[FUN_EQ_THM] THEN MESON_TAC[PERMUTES_INVERSE_EQ; PERMUTES_INVERSE]);;
(* ------------------------------------------------------------------------- *)
(* The number of permutations on a finite set. *)
(* ------------------------------------------------------------------------- *)
let PERMUTES_INSERT_LEMMA = prove
(`!p a:A s. p permutes (a INSERT s) ==> (swap(a,p(a)) o p) permutes s`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC PERMUTES_SUPERSET THEN
EXISTS_TAC `(a:A) INSERT s` THEN CONJ_TAC THENL
[ASM_MESON_TAC[PERMUTES_SWAP; PERMUTES_IN_IMAGE;
IN_INSERT; PERMUTES_COMPOSE];
REWRITE_TAC[o_THM; swap; IN_INSERT; IN_DIFF] THEN ASM_MESON_TAC[]]);;
let PERMUTES_INSERT = prove
(`{p:A->A | p permutes (a INSERT s)} =
IMAGE (\(b,p). swap(a,b) o p)
{(b,p) | b IN a INSERT s /\ p IN {p | p permutes s}}`,
REWRITE_TAC[EXTENSION; IN_ELIM_PAIR_THM; IN_IMAGE; EXISTS_PAIR_THM] THEN
X_GEN_TAC `p:A->A` THEN REWRITE_TAC[IN_ELIM_THM] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN EQ_TAC THENL
[DISCH_TAC THEN MAP_EVERY EXISTS_TAC
[`(p:A->A) a`; `swap(a,p a) o (p:A->A)`] THEN
ASM_SIMP_TAC[SWAP_IDEMPOTENT; o_ASSOC; I_O_ID; PERMUTES_INSERT_LEMMA] THEN
ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_INSERT];
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`b:A`; `q:A->A`] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC PERMUTES_COMPOSE THEN
CONJ_TAC THENL
[ASM_MESON_TAC[PERMUTES_SUBSET; SUBSET; IN_INSERT];
MATCH_MP_TAC PERMUTES_SWAP THEN
ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_INSERT]]]);;
let HAS_SIZE_PERMUTATIONS = prove
(`!s:A->bool n. s HAS_SIZE n ==> {p | p permutes s} HAS_SIZE (FACT n)`,
REWRITE_TAC[HAS_SIZE; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
SIMP_TAC[PERMUTES_EMPTY; CARD_CLAUSES; SET_RULE `{x | x = a} = {a}`] THEN
SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
REWRITE_TAC[NOT_IN_EMPTY] THEN CONJ_TAC THENL
[GEN_TAC THEN DISCH_THEN(SUBST1_TAC o SYM) THEN CONV_TAC NUM_REDUCE_CONV;
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`a:A`; `s:A->bool`] THEN REWRITE_TAC[GSYM HAS_SIZE] THEN
STRIP_TAC THEN X_GEN_TAC `k:num` THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
REWRITE_TAC[FACT; PERMUTES_INSERT] THEN MATCH_MP_TAC HAS_SIZE_IMAGE_INJ THEN
ASM_SIMP_TAC[HAS_SIZE_PRODUCT; HAS_SIZE; FINITE_INSERT; CARD_CLAUSES] THEN
REWRITE_TAC[FORALL_PAIR_THM; IN_ELIM_PAIR_THM; IN_ELIM_THM; PAIR_EQ] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
MAP_EVERY X_GEN_TAC [`b:A`; `q:A->A`; `c:A`; `r:A->A`] THEN
STRIP_TAC THEN SUBGOAL_THEN `c:A = b` SUBST_ALL_TAC THENL
[FIRST_X_ASSUM(MP_TAC o C AP_THM `a:A`) THEN REWRITE_TAC[o_THM; swap] THEN
SUBGOAL_THEN `(q:A->A) a = a /\ (r:A->A) a = a` (fun t -> SIMP_TAC[t]) THEN
ASM_MESON_TAC[permutes];
FIRST_X_ASSUM(MP_TAC o AP_TERM `(\q:A->A. swap(a:A,b) o q)`) THEN
ASM_SIMP_TAC[SWAP_IDEMPOTENT; o_ASSOC; I_O_ID]]);;
let FINITE_PERMUTATIONS = prove
(`!s. FINITE s ==> FINITE {p | p permutes s}`,
MESON_TAC[HAS_SIZE_PERMUTATIONS; HAS_SIZE]);;
let CARD_PERMUTATIONS = prove
(`!s. FINITE s ==> CARD {p | p permutes s} = FACT(CARD s)`,
MESON_TAC[HAS_SIZE; HAS_SIZE_PERMUTATIONS]);;
(* ------------------------------------------------------------------------- *)
(* Alternative characterizations of permutation of finite set. *)
(* ------------------------------------------------------------------------- *)
let PERMUTES_FINITE_INJECTIVE = prove
(`!s:A->bool p.
FINITE s
==> (p permutes s <=>
(!x. ~(x IN s) ==> p x = x) /\
(!x. x IN s ==> p x IN s) /\
(!x y. x IN s /\ y IN s /\ p x = p y ==> x = y))`,
REWRITE_TAC[permutes] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC(TAUT `(p ==> (q <=> r)) ==> (p /\ q <=> p /\ r)`) THEN
DISCH_TAC THEN EQ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `p:A->A` o MATCH_MP
(REWRITE_RULE[IMP_CONJ] SURJECTIVE_IFF_INJECTIVE)) THEN
ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IMP_IMP; GSYM CONJ_ASSOC] THEN
STRIP_TAC THEN X_GEN_TAC `y:A` THEN
ASM_CASES_TAC `(y:A) IN s` THEN ASM_MESON_TAC[]);;
let PERMUTES_FINITE_SURJECTIVE = prove
(`!s:A->bool p.
FINITE s
==> (p permutes s <=>
(!x. ~(x IN s) ==> p x = x) /\ (!x. x IN s ==> p x IN s) /\
(!y. y IN s ==> ?x. x IN s /\ p x = y))`,
REWRITE_TAC[permutes] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC(TAUT `(p ==> (q <=> r)) ==> (p /\ q <=> p /\ r)`) THEN
DISCH_TAC THEN EQ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `p:A->A` o MATCH_MP
(REWRITE_RULE[IMP_CONJ] SURJECTIVE_IFF_INJECTIVE)) THEN
ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IMP_IMP; GSYM CONJ_ASSOC] THEN
STRIP_TAC THEN X_GEN_TAC `y:A` THEN
ASM_CASES_TAC `(y:A) IN s` THEN ASM_MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Permutations of index set for iterated operations. *)
(* ------------------------------------------------------------------------- *)
let ITERATE_PERMUTE = prove
(`!op. monoidal op
==> !f p s. p permutes s ==> iterate op s f = iterate op s (f o p)`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP ITERATE_BIJECTION) THEN
ASM_MESON_TAC[permutes]);;
let NSUM_PERMUTE = prove
(`!f p s. p permutes s ==> nsum s f = nsum s (f o p)`,
REWRITE_TAC[nsum] THEN MATCH_MP_TAC ITERATE_PERMUTE THEN
REWRITE_TAC[MONOIDAL_ADD]);;
let NSUM_PERMUTE_NUMSEG = prove
(`!f p m n. p permutes m..n ==> nsum(m..n) f = nsum(m..n) (f o p)`,
MESON_TAC[NSUM_PERMUTE; FINITE_NUMSEG]);;
let SUM_PERMUTE = prove
(`!f p s. p permutes s ==> sum s f = sum s (f o p)`,
REWRITE_TAC[sum] THEN MATCH_MP_TAC ITERATE_PERMUTE THEN
REWRITE_TAC[MONOIDAL_REAL_ADD]);;
let SUM_PERMUTE_NUMSEG = prove
(`!f p m n. p permutes m..n ==> sum(m..n) f = sum(m..n) (f o p)`,
MESON_TAC[SUM_PERMUTE; FINITE_NUMSEG]);;
(* ------------------------------------------------------------------------- *)
(* Various combinations of transpositions with 2, 1 and 0 common elements. *)
(* ------------------------------------------------------------------------- *)
let SWAP_COMMON = prove
(`!a b c:A. ~(a = c) /\ ~(b = c)
==> swap(a,b) o swap(a,c) = swap(b,c) o swap(a,b)`,
REPEAT GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM; swap; o_THM; I_THM] THEN
DISCH_TAC THEN X_GEN_TAC `x:A` THEN
MAP_EVERY ASM_CASES_TAC [`x:A = a`; `x:A = b`; `x:A = c`] THEN
REPEAT(FIRST_X_ASSUM SUBST_ALL_TAC) THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[]);;
let SWAP_COMMON' = prove
(`!a b c:A. ~(a = b) /\ ~(a = c)
==> swap(a,c) o swap(b,c) = swap(b,c) o swap(a,b)`,
REPEAT STRIP_TAC THEN
GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [SWAP_SYM] THEN
ASM_SIMP_TAC[GSYM SWAP_COMMON] THEN REWRITE_TAC[SWAP_SYM]);;
let SWAP_INDEPENDENT = prove
(`!a b c d:A. ~(a = c) /\ ~(a = d) /\ ~(b = c) /\ ~(b = d)
==> swap(a,b) o swap(c,d) = swap(c,d) o swap(a,b)`,
REPEAT GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM; swap; o_THM; I_THM] THEN
DISCH_TAC THEN X_GEN_TAC `x:A` THEN
MAP_EVERY ASM_CASES_TAC [`x:A = a`; `x:A = b`; `x:A = c`] THEN
REPEAT(FIRST_X_ASSUM SUBST_ALL_TAC) THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Permutations as transposition sequences. *)
(* ------------------------------------------------------------------------- *)
let swapseq_RULES,swapseq_INDUCT,swapseq_CASES = new_inductive_definition
`(swapseq 0 I) /\
(!a b p n. swapseq n p /\ ~(a = b) ==> swapseq (SUC n) (swap(a,b) o p))`;;
let permutation = new_definition
`permutation p <=> ?n. swapseq n p`;;
(* ------------------------------------------------------------------------- *)
(* Some closure properties of the set of permutations, with lengths. *)
(* ------------------------------------------------------------------------- *)
let SWAPSEQ_I = CONJUNCT1 swapseq_RULES;;
let PERMUTATION_I = prove
(`permutation I`,
REWRITE_TAC[permutation] THEN MESON_TAC[SWAPSEQ_I]);;
let SWAPSEQ_SWAP = prove
(`!a b. swapseq (if a = b then 0 else 1) (swap(a,b))`,
REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[num_CONV `1`] THEN
ASM_MESON_TAC[swapseq_RULES; I_O_ID; SWAPSEQ_I; SWAP_REFL]);;
let PERMUTATION_SWAP = prove
(`!a b. permutation(swap(a,b))`,
REWRITE_TAC[permutation] THEN MESON_TAC[SWAPSEQ_SWAP]);;
let SWAPSEQ_COMPOSE = prove
(`!n p m q. swapseq n p /\ swapseq m q ==> swapseq (n + m) (p o q)`,
REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN
MATCH_MP_TAC swapseq_INDUCT THEN
REWRITE_TAC[ADD_CLAUSES; I_O_ID; GSYM o_ASSOC] THEN
MESON_TAC[swapseq_RULES]);;
let PERMUTATION_COMPOSE = prove
(`!p q. permutation p /\ permutation q ==> permutation(p o q)`,
REWRITE_TAC[permutation] THEN MESON_TAC[SWAPSEQ_COMPOSE]);;
let SWAPSEQ_ENDSWAP = prove
(`!n p a b:A. swapseq n p /\ ~(a = b) ==> swapseq (SUC n) (p o swap(a,b))`,
REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN
MATCH_MP_TAC swapseq_INDUCT THEN REWRITE_TAC[I_O_ID; GSYM o_ASSOC] THEN
MESON_TAC[o_ASSOC; swapseq_RULES; I_O_ID]);;
let SWAPSEQ_INVERSE_EXISTS = prove
(`!n p:A->A. swapseq n p ==> ?q. swapseq n q /\ p o q = I /\ q o p = I`,
MATCH_MP_TAC swapseq_INDUCT THEN CONJ_TAC THENL
[MESON_TAC[I_O_ID; swapseq_RULES]; ALL_TAC] THEN
REPEAT STRIP_TAC THEN
MP_TAC(SPECL [`n:num`; `q:A->A`; `a:A`; `b:A`] SWAPSEQ_ENDSWAP) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
EXISTS_TAC `(q:A->A) o swap (a,b)` THEN
ASM_REWRITE_TAC[GSYM o_ASSOC] THEN
GEN_REWRITE_TAC (BINOP_CONV o LAND_CONV o RAND_CONV) [o_ASSOC] THEN
ASM_REWRITE_TAC[SWAP_IDEMPOTENT; I_O_ID]);;
let SWAPSEQ_INVERSE = prove
(`!n p. swapseq n p ==> swapseq n (inverse p)`,
MESON_TAC[SWAPSEQ_INVERSE_EXISTS; INVERSE_UNIQUE_o]);;
let PERMUTATION_INVERSE = prove
(`!p. permutation p ==> permutation(inverse p)`,
REWRITE_TAC[permutation] THEN MESON_TAC[SWAPSEQ_INVERSE]);;
(* ------------------------------------------------------------------------- *)
(* The identity map only has even transposition sequences. *)
(* ------------------------------------------------------------------------- *)
let SYMMETRY_LEMMA = prove
(`(!a b c d. P a b c d ==> P a b d c) /\
(!a b c d. ~(a = b) /\ ~(c = d) /\
(a = c /\ b = d \/ a = c /\ ~(b = d) \/ ~(a = c) /\ b = d \/
~(a = c) /\ ~(a = d) /\ ~(b = c) /\ ~(b = d))
==> P a b c d)
==> (!a b c d:A. ~(a = b) /\ ~(c = d) ==> P a b c d)`,
REPEAT STRIP_TAC THEN MAP_EVERY ASM_CASES_TAC
[`a:A = c`; `a:A = d`; `b:A = c`; `b:A = d`] THEN
ASM_MESON_TAC[]);;
let SWAP_GENERAL = prove
(`!a b c d:A.
~(a = b) /\ ~(c = d)
==> swap(a,b) o swap(c,d) = I \/
?x y z. ~(x = a) /\ ~(y = a) /\ ~(z = a) /\ ~(x = y) /\
swap(a,b) o swap(c,d) = swap(x,y) o swap(a,z)`,
MATCH_MP_TAC SYMMETRY_LEMMA THEN CONJ_TAC THENL
[REWRITE_TAC[SWAP_SYM] THEN SIMP_TAC[]; ALL_TAC] THEN
REPEAT STRIP_TAC THEN REPEAT(FIRST_X_ASSUM SUBST_ALL_TAC) THENL
[MESON_TAC[SWAP_IDEMPOTENT];
DISJ2_TAC THEN MAP_EVERY EXISTS_TAC [`b:A`; `d:A`; `b:A`] THEN
ASM_MESON_TAC[SWAP_COMMON];
DISJ2_TAC THEN MAP_EVERY EXISTS_TAC [`c:A`; `d:A`; `c:A`] THEN
ASM_MESON_TAC[SWAP_COMMON'];
DISJ2_TAC THEN MAP_EVERY EXISTS_TAC [`c:A`; `d:A`; `b:A`] THEN
ASM_MESON_TAC[SWAP_INDEPENDENT]]);;
let FIXING_SWAPSEQ_DECREASE = prove
(`!n p a b:A.
swapseq n p /\ ~(a = b) /\ (swap(a,b) o p) a = a
==> ~(n = 0) /\ swapseq (n - 1) (swap(a,b) o p)`,
INDUCT_TAC THEN REPEAT GEN_TAC THEN
GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [swapseq_CASES] THEN
REWRITE_TAC[NOT_SUC] THENL
[DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
ASM_REWRITE_TAC[I_THM; o_THM; swap] THEN MESON_TAC[];
ALL_TAC] THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM; LEFT_AND_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`c:A`; `d:A`; `q:A->A`; `m:num`] THEN
REWRITE_TAC[SUC_INJ; GSYM CONJ_ASSOC] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
FIRST_X_ASSUM SUBST_ALL_TAC THEN REWRITE_TAC[o_ASSOC] THEN STRIP_TAC THEN
MP_TAC(SPECL [`a:A`; `b:A`; `c:A`; `d:A`] SWAP_GENERAL) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(DISJ_CASES_THEN2 SUBST_ALL_TAC MP_TAC) THEN
ASM_REWRITE_TAC[I_O_ID; SUC_SUB1; LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`x:A`; `y:A`; `z:A`] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN SUBST_ALL_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL
[`q:A->A`; `a:A`; `z:A`]) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o check(is_eq o concl)) THEN
REWRITE_TAC[GSYM o_ASSOC] THEN
ABBREV_TAC `r:A->A = swap(a:A,z) o q` THEN
ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; swap] THEN ASM_MESON_TAC[];
SPEC_TAC(`n:num`,`n:num`) THEN INDUCT_TAC THEN
REWRITE_TAC[NOT_SUC; SUC_SUB1; GSYM o_ASSOC] THEN
ASM_MESON_TAC[swapseq_RULES]]);;
let SWAPSEQ_IDENTITY_EVEN = prove
(`!n. swapseq n (I:A->A) ==> EVEN n`,
MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
GEN_REWRITE_TAC LAND_CONV [swapseq_CASES] THEN
DISCH_THEN(DISJ_CASES_THEN2 (SUBST_ALL_TAC o CONJUNCT1) MP_TAC) THEN
REWRITE_TAC[EVEN; LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:A`; `b:A`; `p:A->A`; `m:num`] THEN
DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN
MP_TAC(SPECL [`m:num`; `p:A->A`; `a:A`; `b:A`] FIXING_SWAPSEQ_DECREASE) THEN
ASM_REWRITE_TAC[I_THM] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `m - 1`) THEN
UNDISCH_THEN `SUC m = n` (SUBST_ALL_TAC o SYM) THEN
ASM_REWRITE_TAC[ARITH_RULE `m - 1 < SUC m`] THEN UNDISCH_TAC `~(m = 0)` THEN
SPEC_TAC(`m:num`,`m:num`) THEN INDUCT_TAC THEN
REWRITE_TAC[SUC_SUB1; EVEN]);;
(* ------------------------------------------------------------------------- *)
(* Therefore we have a welldefined notion of parity. *)
(* ------------------------------------------------------------------------- *)
let evenperm = new_definition `evenperm(p) = EVEN(@n. swapseq n p)`;;
let SWAPSEQ_EVEN_EVEN = prove
(`!m n p:A->A. swapseq m p /\ swapseq n p ==> (EVEN m <=> EVEN n)`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP SWAPSEQ_INVERSE_EXISTS) THEN
STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o AP_TERM `swapseq (n + m) :(A->A)->bool`) THEN
ASM_SIMP_TAC[SWAPSEQ_COMPOSE] THEN
DISCH_THEN(MP_TAC o MATCH_MP SWAPSEQ_IDENTITY_EVEN) THEN
SIMP_TAC[EVEN_ADD]);;
let EVENPERM_UNIQUE = prove
(`!n p b. swapseq n p /\ EVEN n = b ==> evenperm p = b`,
REWRITE_TAC[evenperm] THEN MESON_TAC[SWAPSEQ_EVEN_EVEN]);;
(* ------------------------------------------------------------------------- *)
(* And it has the expected composition properties. *)
(* ------------------------------------------------------------------------- *)
let EVENPERM_I = prove
(`evenperm I = T`,
MATCH_MP_TAC EVENPERM_UNIQUE THEN MESON_TAC[swapseq_RULES; EVEN]);;
let EVENPERM_ID = prove
(`evenperm(\x:A. x)`,
REWRITE_TAC[GSYM I_DEF; EVENPERM_I]);;
let EVENPERM_SWAP = prove
(`!a b:A. evenperm(swap(a,b)) = (a = b)`,
REPEAT GEN_TAC THEN MATCH_MP_TAC EVENPERM_UNIQUE THEN
MESON_TAC[SWAPSEQ_SWAP; NUM_RED_CONV `EVEN 0`; NUM_RED_CONV `EVEN 1`]);;
let EVENPERM_COMPOSE = prove
(`!p q. permutation p /\ permutation q
==> evenperm (p o q) = (evenperm p = evenperm q)`,
REWRITE_TAC[permutation; LEFT_AND_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN REPEAT GEN_TAC THEN
DISCH_THEN(fun th -> ASSUME_TAC th THEN
ASSUME_TAC(MATCH_MP SWAPSEQ_COMPOSE th)) THEN
ASM_MESON_TAC[EVENPERM_UNIQUE; SWAPSEQ_COMPOSE; EVEN_ADD]);;
let EVENPERM_INVERSE = prove
(`!p. permutation p ==> evenperm(inverse p) = evenperm p`,
REWRITE_TAC[permutation] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC EVENPERM_UNIQUE THEN
ASM_MESON_TAC[SWAPSEQ_INVERSE; EVENPERM_UNIQUE]);;
(* ------------------------------------------------------------------------- *)
(* A more abstract characterization of permutations. *)
(* ------------------------------------------------------------------------- *)
let PERMUTATION_BIJECTIVE = prove
(`!p. permutation p ==> !y. ?!x. p(x) = y`,
REWRITE_TAC[permutation] THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP SWAPSEQ_INVERSE_EXISTS) THEN
REWRITE_TAC[FUN_EQ_THM; I_THM; o_THM; LEFT_IMP_EXISTS_THM] THEN
MESON_TAC[]);;
let PERMUTATION_FINITE_SUPPORT = prove
(`!p. permutation p ==> FINITE {x:A | ~(p x = x)}`,
REWRITE_TAC[permutation; LEFT_IMP_EXISTS_THM] THEN
ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN MATCH_MP_TAC swapseq_INDUCT THEN
REWRITE_TAC[I_THM; FINITE_RULES; SET_RULE `{x | F} = {}`] THEN
MAP_EVERY X_GEN_TAC [`a:A`; `b:A`; `p:A->A`] THEN
STRIP_TAC THEN MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `(a:A) INSERT b INSERT {x | ~(p x = x)}` THEN
ASM_REWRITE_TAC[FINITE_INSERT; SUBSET; IN_INSERT; IN_ELIM_THM] THEN
REWRITE_TAC[o_THM; swap] THEN MESON_TAC[]);;
let PERMUTATION_LEMMA = prove
(`!s p:A->A.
FINITE s /\
(!y. ?!x. p(x) = y) /\ (!x. ~(x IN s) ==> p x = x)
==> permutation p`,
ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN CONJ_TAC THENL
[REWRITE_TAC[NOT_IN_EMPTY] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `p:A->A = I` (fun th -> REWRITE_TAC[th; PERMUTATION_I]) THEN
ASM_REWRITE_TAC[FUN_EQ_THM; I_THM];
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`a:A`; `s:A->bool`] THEN STRIP_TAC THEN
REWRITE_TAC[IN_INSERT] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `permutation((swap(a,p(a)) o swap(a,p(a))) o (p:A->A))`
MP_TAC THENL [ALL_TAC; REWRITE_TAC[SWAP_IDEMPOTENT; I_O_ID]] THEN
REWRITE_TAC[GSYM o_ASSOC] THEN MATCH_MP_TAC PERMUTATION_COMPOSE THEN
REWRITE_TAC[PERMUTATION_SWAP] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
CONJ_TAC THENL
[UNDISCH_TAC `!y. ?!x. (p:A->A) x = y` THEN
REWRITE_TAC[EXISTS_UNIQUE_THM; swap; o_THM] THEN
ASM_CASES_TAC `(p:A->A) a = a` THEN ASM_REWRITE_TAC[] THENL
[ASM_MESON_TAC[]; ALL_TAC] THEN
REWRITE_TAC[TAUT
`(if p then x else y) = a <=> if p then x = a else y = a`] THEN
REWRITE_TAC[TAUT `(if p then x else y) <=> p /\ x \/ ~p /\ y`] THEN
ASM_MESON_TAC[];
REWRITE_TAC[swap; o_THM] THEN
ASM_CASES_TAC `(p:A->A) a = a` THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[]]);;
let PERMUTATION = prove
(`!p. permutation p <=> (!y. ?!x. p(x) = y) /\ FINITE {x:A | ~(p(x) = x)}`,
GEN_TAC THEN EQ_TAC THEN
SIMP_TAC[PERMUTATION_BIJECTIVE; PERMUTATION_FINITE_SUPPORT] THEN
STRIP_TAC THEN MATCH_MP_TAC PERMUTATION_LEMMA THEN
EXISTS_TAC `{x:A | ~(p x = x)}` THEN
ASM_SIMP_TAC[IN_ELIM_THM]);;
let PERMUTATION_INVERSE_WORKS = prove
(`!p. permutation p ==> inverse p o p = I /\ p o inverse p = I`,
MESON_TAC[PERMUTATION_BIJECTIVE; SURJECTIVE_INVERSE_o;
INJECTIVE_INVERSE_o]);;
let PERMUTATION_INVERSE_COMPOSE = prove
(`!p q. permutation p /\ permutation q
==> inverse(p o q) = inverse q o inverse p`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC INVERSE_UNIQUE_o THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP PERMUTATION_INVERSE_WORKS)) THEN
REWRITE_TAC[GSYM o_ASSOC] THEN REPEAT STRIP_TAC THEN
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [o_ASSOC] THEN
ASM_REWRITE_TAC[I_O_ID]);;
let PERMUTATION_COMPOSE_EQ = prove
(`(!p q:A->A. permutation(p) ==> (permutation(p o q) <=> permutation q)) /\
(!p q:A->A. permutation(q) ==> (permutation(p o q) <=> permutation p))`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP PERMUTATION_INVERSE) THEN
EQ_TAC THEN ASM_SIMP_TAC[PERMUTATION_COMPOSE] THENL
[DISCH_THEN(MP_TAC o SPEC `inverse(p:A->A)` o MATCH_MP
(REWRITE_RULE[IMP_CONJ_ALT] PERMUTATION_COMPOSE));
DISCH_THEN(MP_TAC o SPEC `inverse(q:A->A)` o MATCH_MP
(REWRITE_RULE[IMP_CONJ] PERMUTATION_COMPOSE))] THEN
ASM_SIMP_TAC[GSYM o_ASSOC; PERMUTATION_INVERSE_WORKS] THEN
ASM_SIMP_TAC[o_ASSOC; PERMUTATION_INVERSE_WORKS] THEN
REWRITE_TAC[I_O_ID]);;
let PERMUTATION_COMPOSE_SWAP = prove
(`(!p a b:A. permutation(swap(a,b) o p) <=> permutation p) /\
(!p a b:A. permutation(p o swap(a,b)) <=> permutation p)`,
SIMP_TAC[PERMUTATION_COMPOSE_EQ; PERMUTATION_SWAP]);;
(* ------------------------------------------------------------------------- *)
(* Relation to "permutes". *)
(* ------------------------------------------------------------------------- *)
let PERMUTATION_PERMUTES = prove
(`!p:A->A. permutation p <=> ?s. FINITE s /\ p permutes s`,
GEN_TAC THEN REWRITE_TAC[PERMUTATION; permutes] THEN
EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THENL
[EXISTS_TAC `{x:A | ~(p x = x)}` THEN ASM_SIMP_TAC[IN_ELIM_THM];
MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `s:A->bool` THEN
ASM_SIMP_TAC[IN_ELIM_THM; SUBSET] THEN ASM_MESON_TAC[]]);;
let PERMUTATION_RESTRICT = prove
(`!Q (p:A->A).
permutation p /\ (!x. Q(p x) <=> Q x)
==> permutation (\i. if Q i then p i else i)`,
REPEAT GEN_TAC THEN REWRITE_TAC[IMP_CONJ_ALT] THEN DISCH_TAC THEN
REWRITE_TAC[PERMUTATION_PERMUTES] THEN MATCH_MP_TAC MONO_EXISTS THEN
ASM_MESON_TAC[PERMUTES_RESTRICT]);;
(* ------------------------------------------------------------------------- *)
(* Hence a sort of induction principle composing by swaps. *)
(* ------------------------------------------------------------------------- *)
let PERMUTES_INDUCT = prove
(`!P s. FINITE s /\
P I /\
(!a b:A p. a IN s /\ b IN s /\ P p /\ permutation p
==> P (swap(a,b) o p))
==> (!p. p permutes s ==> P p)`,
ONCE_REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> b ==> a ==> c ==> d`] THEN
REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN GEN_TAC THEN DISCH_TAC THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
ASM_REWRITE_TAC[PERMUTES_EMPTY; IN_INSERT] THEN REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `p = swap(x,p x) o swap(x,p x) o (p:A->A)` SUBST1_TAC THENL
[REWRITE_TAC[o_ASSOC; SWAP_IDEMPOTENT; I_O_ID]; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN
ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_THEN(fun th -> FIRST_X_ASSUM MATCH_MP_TAC THEN ASSUME_TAC th) THEN
ASM_MESON_TAC[PERMUTES_IN_IMAGE; IN_INSERT; PERMUTES_INSERT_LEMMA;
PERMUTATION_PERMUTES; FINITE_INSERT; PERMUTATION_COMPOSE;
PERMUTATION_SWAP]);;
let PERMUTES_INDUCT_STRONG = prove
(`!P s:A->bool.
FINITE s /\
P I /\
(!a b p. a IN s /\ b IN s /\ ~(a = b) /\ P p /\ p permutes s
==> P (swap(a,b) o p))
==> !p. p permutes s ==> P p`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
ONCE_REWRITE_TAC[TAUT `p ==> q <=> p ==> p /\ q`] THEN
MATCH_MP_TAC PERMUTES_INDUCT THEN
ASM_SIMP_TAC[PERMUTES_I; PERMUTES_COMPOSE; PERMUTES_SWAP] THEN
MAP_EVERY X_GEN_TAC [`a:A`; `b:A`] THEN
ASM_CASES_TAC `a:A = b` THEN ASM_SIMP_TAC[SWAP_REFL; I_O_ID]);;
(* ------------------------------------------------------------------------- *)
(* Sign of a permutation as a real number. *)
(* ------------------------------------------------------------------------- *)
let sign = new_definition
`(sign p):real = if evenperm p then &1 else -- &1`;;
let SIGN_NZ = prove
(`!p. ~(sign p = &0)`,
REWRITE_TAC[sign] THEN REAL_ARITH_TAC);;
let SIGN_I = prove
(`sign I = &1`,
REWRITE_TAC[sign; EVENPERM_I]);;
let SIGN_ID = prove
(`sign(\x:A. x) = &1`,
REWRITE_TAC[sign; EVENPERM_ID]);;
let SIGN_INVERSE = prove
(`!p. permutation p ==> sign(inverse p) = sign p`,
SIMP_TAC[sign; EVENPERM_INVERSE] THEN REAL_ARITH_TAC);;
let SIGN_COMPOSE = prove
(`!p q. permutation p /\ permutation q ==> sign(p o q) = sign(p) * sign(q)`,
SIMP_TAC[sign; EVENPERM_COMPOSE] THEN REAL_ARITH_TAC);;
let SIGN_SWAP = prove
(`!a b. sign(swap(a,b)) = if a = b then &1 else -- &1`,
REWRITE_TAC[sign; EVENPERM_SWAP]);;
let SIGN_IDEMPOTENT = prove
(`!p. sign(p) * sign(p) = &1`,
GEN_TAC THEN REWRITE_TAC[sign] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV);;
let REAL_ABS_SIGN = prove
(`!p. abs(sign p) = &1`,
REWRITE_TAC[sign] THEN REAL_ARITH_TAC);;
let REAL_SGN_SIGN = prove
(`!p:A->A. real_sgn(sign p) = sign p`,
GEN_TAC THEN REWRITE_TAC[sign] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_SGN_EQ] THEN
CONV_TAC REAL_RAT_REDUCE_CONV);;
let EVENPERM_TRANSFER = prove
(`!(f:A->B) s p q.
FINITE s /\
(!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\
p permutes s /\
(!x. x IN s ==> q(f x) = f(p x)) /\
(!y. ~(y IN IMAGE f s) ==> q y = y)
==> (evenperm q <=> evenperm p)`,
GEN_TAC THEN GEN_TAC THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN
ONCE_REWRITE_TAC[IMP_CONJ] THEN
REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INJECTIVE_ON_LEFT_INVERSE]) THEN
DISCH_THEN(X_CHOOSE_TAC `g:B->A`) THEN
SUBGOAL_THEN
`!p q. (!x. x IN s ==> q (f x) = f (p x)) /\
(!y. ~(y IN IMAGE f s) ==> q y = y) <=>
q = \x. if x IN IMAGE f s then (f:A->B) (p(g x)) else x`
(fun th -> REWRITE_TAC[th])
THENL
[REWRITE_TAC[FUN_EQ_THM] THEN ASM SET_TAC[];
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_UNWIND_THM2]] THEN
MATCH_MP_TAC PERMUTES_INDUCT_STRONG THEN
ASM_REWRITE_TAC[I_THM; o_THM] THEN
SUBGOAL_THEN
`(\x. if x IN IMAGE (f:A->B) s then f (g x) else x) = I`
SUBST1_TAC THENL
[REWRITE_TAC[FUN_EQ_THM; I_THM] THEN ASM SET_TAC[];
REWRITE_TAC[EVENPERM_I]] THEN
MAP_EVERY X_GEN_TAC [`a:A`; `b:A`; `p:A->A`] THEN STRIP_TAC THEN
W(MP_TAC o PART_MATCH (lhand o rand) EVENPERM_COMPOSE o rand o snd) THEN
REWRITE_TAC[PERMUTATION_SWAP] THEN REWRITE_TAC[PERMUTATION_PERMUTES] THEN
ANTS_TAC THENL [ASM_MESON_TAC[]; DISCH_THEN SUBST1_TAC] THEN
FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
SUBGOAL_THEN `evenperm(swap(a,b)) = evenperm(swap((f:A->B) a,f b))`
SUBST1_TAC THENL
[REWRITE_TAC[EVENPERM_SWAP] THEN ASM_MESON_TAC[]; ALL_TAC] THEN
W(MP_TAC o PART_MATCH (rand o rand) EVENPERM_COMPOSE o rand o snd) THEN
REWRITE_TAC[PERMUTATION_SWAP] THEN ANTS_TAC THENL
[REWRITE_TAC[PERMUTATION_PERMUTES] THEN
EXISTS_TAC `IMAGE (f:A->B) s` THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN
MATCH_MP_TAC PERMUTES_TRANSFER THEN EXISTS_TAC `p:A->A` THEN
ASM SET_TAC[];
DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC] THEN
REWRITE_TAC[FUN_EQ_THM; o_DEF; swap] THEN
RULE_ASSUM_TAC(REWRITE_RULE[PERMUTES_ALT]) THEN
X_GEN_TAC `y:B` THEN REWRITE_TAC[IN_IMAGE] THEN
ASM_CASES_TAC `?x. y = (f:A->B) x /\ x IN s` THEN
ASM_REWRITE_TAC[] THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `x:A`
(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC)) THEN
ASM_SIMP_TAC[] THEN ASM_MESON_TAC[]);;
let SIGN_TRANSFER = prove
(`!(f:A->B) s p q.
FINITE s /\
(!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\
p permutes s /\
(!x. x IN s ==> q(f x) = f(p x)) /\
(!y. ~(y IN IMAGE f s) ==> q y = y)
==> sign q = sign p`,
REPEAT GEN_TAC THEN REWRITE_TAC[sign] THEN
DISCH_THEN(SUBST1_TAC o MATCH_MP EVENPERM_TRANSFER) THEN
REWRITE_TAC[]);;
let SIGN_CARTESIAN_PRODUCT = prove
(`!(p:A->A) (q:B->B) s t.
FINITE s /\ FINITE t /\ p permutes s /\ q permutes t
==> sign (\(i,j). if i IN s /\ j IN t then p i,q j else i,j) =
sign p pow CARD t * sign q pow CARD s`,
let lemma1 = prove
(`!p (s:A->bool) (t:B->bool).
p permutes s /\ FINITE s /\ FINITE t
==> sign (\(i,j). if i IN s /\ j IN t then p i,j else i,j) =
sign p pow CARD t`,
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
REPEAT GEN_TAC THEN REPEAT DISCH_TAC THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN CONJ_TAC THENL
[REWRITE_TAC[CARD_CLAUSES; NOT_IN_EMPTY] THEN
REWRITE_TAC[GSYM LAMBDA_PAIR_THM; SIGN_ID; real_pow];
MAP_EVERY X_GEN_TAC [`b:B`; `t:B->bool`] THEN STRIP_TAC] THEN
SUBGOAL_THEN
`(\(i:A,j:B). if i IN s /\ j IN b INSERT t then p i,j else i,j) =
(\(i,j). if i IN s /\ j IN {b} then p i,j else i,j) o
(\(i,j). if i IN s /\ j IN t then p i,j else i,j)`
SUBST1_TAC THENL
[REWRITE_TAC[FUN_EQ_THM; FORALL_PAIR_THM; o_THM] THEN
MAP_EVERY X_GEN_TAC [`i:A`; `j:B`] THEN
ASM_CASES_TAC `(i:A) IN s` THEN ASM_REWRITE_TAC[IN_INSERT] THEN
ASM_CASES_TAC `j:B = b` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
ASM_CASES_TAC `(j:B) IN t` THEN ASM_REWRITE_TAC[];
ALL_TAC] THEN
W(MP_TAC o PART_MATCH (lhand o rand) SIGN_COMPOSE o lhand o snd) THEN
ANTS_TAC THENL
[REWRITE_TAC[PERMUTATION_PERMUTES] THEN CONJ_TAC THENL
[EXISTS_TAC `(s:A->bool) CROSS {b:B}`;
EXISTS_TAC `(s:A->bool) CROSS (t:B->bool)`] THEN
ASM_REWRITE_TAC[FINITE_CROSS_EQ; FINITE_SING] THEN
MATCH_MP_TAC PERMUTES_CARTESIAN_PRODUCT THEN
ASM_REWRITE_TAC[PERMUTES_ID];
DISCH_THEN SUBST1_TAC] THEN
ASM_SIMP_TAC[CARD_CLAUSES; real_pow; IN_SING] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
MATCH_MP_TAC SIGN_TRANSFER THEN
EXISTS_TAC `\x:A. x,(b:B)` THEN EXISTS_TAC `s:A->bool` THEN
ASM_SIMP_TAC[PAIR_EQ; IN_IMAGE; FORALL_PAIR_THM] THEN
ASM_MESON_TAC[]) in
let lemma2 = prove
(`!p (s:A->bool) (t:B->bool).
FINITE s /\ p permutes t /\ FINITE t
==> sign (\(i:A,j:B). if i IN s /\ j IN t then i,p j else i,j) =
sign p pow CARD s`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`p:B->B`; `t:B->bool`; `s:A->bool`]
lemma1) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
MATCH_MP_TAC SIGN_TRANSFER THEN
EXISTS_TAC `\(i:B,j:A). j,i` THEN
EXISTS_TAC `(t:B->bool) CROSS (s:A->bool)` THEN
ASM_REWRITE_TAC[FINITE_CROSS_EQ; FORALL_PAIR_THM; IN_CROSS] THEN
SIMP_TAC[PAIR_EQ; IN_IMAGE; IN_CROSS; EXISTS_PAIR_THM] THEN
CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
MATCH_MP_TAC PERMUTES_CARTESIAN_PRODUCT THEN
ASM_REWRITE_TAC[PERMUTES_ID]) in
REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`(\(i,j). if i IN s /\ j IN t then (p:A->A) i,(q:B->B) j else i,j) =
(\(i,j). if i IN s /\ j IN t then p i,j else i,j) o
(\(i,j). if i IN s /\ j IN t then i,q j else i,j)`
SUBST1_TAC THENL
[REWRITE_TAC[FUN_EQ_THM; FORALL_PAIR_THM; o_THM] THEN REPEAT GEN_TAC THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[PAIR_EQ]) THEN
RULE_ASSUM_TAC(REWRITE_RULE[PERMUTES_ALT]) THEN ASM SET_TAC[];
W(MP_TAC o PART_MATCH (lhand o rand) SIGN_COMPOSE o lhand o snd)] THEN
ANTS_TAC THENL
[REWRITE_TAC[PERMUTATION_PERMUTES] THEN CONJ_TAC THEN
EXISTS_TAC `(s:A->bool) CROSS (t:B->bool)` THEN
ASM_SIMP_TAC[FINITE_CROSS_EQ; PERMUTES_CARTESIAN_PRODUCT; PERMUTES_ID];
DISCH_THEN SUBST1_TAC] THEN
ASM_SIMP_TAC[lemma1; lemma2]);;
(* ------------------------------------------------------------------------- *)
(* More lemmas about permutations. *)
(* ------------------------------------------------------------------------- *)
let PERMUTES_NUMSET_LE = prove
(`!p s:num->bool. p permutes s /\ (!i. i IN s ==> p(i) <= i) ==> p = I`,
REPEAT GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM; I_THM] THEN STRIP_TAC THEN
MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
ASM_CASES_TAC `(n:num) IN s` THENL [ALL_TAC; ASM_MESON_TAC[permutes]] THEN
ASM_SIMP_TAC[GSYM LE_ANTISYM] THEN REWRITE_TAC[GSYM NOT_LT] THEN
ASM_MESON_TAC[PERMUTES_INJECTIVE; LT_REFL]);;
let PERMUTES_NUMSET_GE = prove
(`!p s:num->bool. p permutes s /\ (!i. i IN s ==> i <= p(i)) ==> p = I`,
REPEAT STRIP_TAC THEN
MP_TAC(SPECL [`inverse(p:num->num)`; `s:num->bool`] PERMUTES_NUMSET_LE) THEN
ANTS_TAC THENL
[ASM_MESON_TAC[PERMUTES_INVERSE; PERMUTES_INVERSES; PERMUTES_IN_IMAGE];
ASM_MESON_TAC[PERMUTES_INVERSE_INVERSE; INVERSE_I]]);;
let IMAGE_INVERSE_PERMUTATIONS = prove
(`!s:A->bool. {inverse p | p permutes s} = {p | p permutes s}`,
REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
MESON_TAC[PERMUTES_INVERSE_INVERSE; PERMUTES_INVERSE]);;
let IMAGE_COMPOSE_PERMUTATIONS_L = prove
(`!s q:A->A. q permutes s ==> {q o p | p permutes s} = {p | p permutes s}`,
REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
X_GEN_TAC `p:A->A` THEN EQ_TAC THENL
[ASM_MESON_TAC[PERMUTES_COMPOSE];
DISCH_TAC THEN EXISTS_TAC `inverse(q:A->A) o (p:A->A)` THEN
ASM_SIMP_TAC[o_ASSOC; PERMUTES_INVERSE; PERMUTES_COMPOSE] THEN
ASM_MESON_TAC[PERMUTES_INVERSES_o; I_O_ID]]);;
let IMAGE_COMPOSE_PERMUTATIONS_R = prove
(`!s q:A->A. q permutes s ==> {p o q | p permutes s} = {p | p permutes s}`,
REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
X_GEN_TAC `p:A->A` THEN EQ_TAC THENL
[ASM_MESON_TAC[PERMUTES_COMPOSE];
DISCH_TAC THEN EXISTS_TAC `(p:A->A) o inverse(q:A->A)` THEN
ASM_SIMP_TAC[GSYM o_ASSOC; PERMUTES_INVERSE; PERMUTES_COMPOSE] THEN
ASM_MESON_TAC[PERMUTES_INVERSES_o; I_O_ID]]);;
let PERMUTES_IN_NUMSEG = prove
(`!p n i. p permutes 1..n /\ i IN 1..n ==> 1 <= p(i) /\ p(i) <= n`,
REWRITE_TAC[permutes; IN_NUMSEG] THEN MESON_TAC[]);;
let SUM_PERMUTATIONS_INVERSE = prove
(`!f m n. sum {p | p permutes m..n} f =
sum {p | p permutes m..n} (\p. f(inverse p))`,
REPEAT GEN_TAC THEN
GEN_REWRITE_TAC (funpow 2 LAND_CONV) [GSYM IMAGE_INVERSE_PERMUTATIONS] THEN
GEN_REWRITE_TAC (funpow 2 LAND_CONV)
[SET_RULE `{f x | p x} = IMAGE f {x | p x}`] THEN
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM o_DEF] THEN
MATCH_MP_TAC SUM_IMAGE THEN
SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG; IN_ELIM_THM] THEN
MESON_TAC[PERMUTES_INVERSE_INVERSE]);;
let SUM_PERMUTATIONS_COMPOSE_L = prove
(`!f m n q.
q permutes m..n
==> sum {p | p permutes m..n} f =
sum {p | p permutes m..n} (\p. f(q o p))`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (funpow 2 LAND_CONV)
[GSYM(MATCH_MP IMAGE_COMPOSE_PERMUTATIONS_L th)]) THEN
GEN_REWRITE_TAC (funpow 2 LAND_CONV)
[SET_RULE `{f x | p x} = IMAGE f {x | p x}`] THEN
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM o_DEF] THEN
MATCH_MP_TAC SUM_IMAGE THEN
SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG; IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o AP_TERM `\p:num->num. inverse(q:num->num) o p`) THEN
REWRITE_TAC[o_ASSOC] THEN
EVERY_ASSUM(CONJUNCTS_THEN SUBST1_TAC o MATCH_MP PERMUTES_INVERSES_o) THEN
REWRITE_TAC[I_O_ID]);;
let SUM_PERMUTATIONS_COMPOSE_R = prove
(`!f m n q.
q permutes m..n
==> sum {p | p permutes m..n} f =
sum {p | p permutes m..n} (\p. f(p o q))`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (funpow 2 LAND_CONV)
[GSYM(MATCH_MP IMAGE_COMPOSE_PERMUTATIONS_R th)]) THEN
GEN_REWRITE_TAC (funpow 2 LAND_CONV)
[SET_RULE `{f x | p x} = IMAGE f {x | p x}`] THEN
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM o_DEF] THEN
MATCH_MP_TAC SUM_IMAGE THEN
SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG; IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o AP_TERM `\p:num->num. p o inverse(q:num->num)`) THEN
REWRITE_TAC[GSYM o_ASSOC] THEN
EVERY_ASSUM(CONJUNCTS_THEN SUBST1_TAC o MATCH_MP PERMUTES_INVERSES_o) THEN
REWRITE_TAC[I_O_ID]);;
let CARD_EVEN_PERMUTATIONS = prove
(`!s:A->bool. FINITE s /\ 2 <= CARD s
==> 2 * CARD {p | p permutes s /\ evenperm p} = FACT(CARD s)`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `?a b:A. a IN s /\ b IN s /\ ~(a = b)` STRIP_ASSUME_TAC THENL
[MP_TAC(SPECL [`2`; `s:A->bool`] CHOOSE_SUBSET_STRONG) THEN
ASM_REWRITE_TAC[HAS_SIZE_CONV `s HAS_SIZE 2`] THEN SET_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `!p:A->A. p permutes s ==> permutation p` ASSUME_TAC THENL
[ASM_MESON_TAC[PERMUTATION_PERMUTES]; ALL_TAC] THEN
SUBGOAL_THEN `!Q. FINITE {p:A->A | p permutes s /\ Q p}` ASSUME_TAC THENL
[REWRITE_TAC[SET_RULE `{p | p permutes s /\ Q p} =
{p | p IN {p | p permutes s} /\ Q p}`] THEN
ASM_SIMP_TAC[FINITE_RESTRICT; FINITE_PERMUTATIONS];
ALL_TAC] THEN
SUBGOAL_THEN
`FACT(CARD s) = CARD ({p | p permutes s /\ evenperm p} UNION
IMAGE (\p. swap(a:A,b) o p)
{p | p permutes s /\ evenperm p})`
SUBST1_TAC THENL
[FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP CARD_PERMUTATIONS) THEN
AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE
`(!x. P x ==> P(f x)) /\ (!x. f(f x) = x) /\
(!x. P x ==> Q x \/ Q(f x))
==> {x | P x} = {x | P x /\ Q x} UNION IMAGE f {x | P x /\ Q x}`) THEN
ASM_SIMP_TAC[PERMUTES_COMPOSE; PERMUTES_SWAP; SWAP_IDEMPOTENT; o_ASSOC;
I_O_ID; EVENPERM_COMPOSE; PERMUTATION_SWAP; EVENPERM_SWAP] THEN
CONV_TAC TAUT;
W(MP_TAC o PART_MATCH (lhs o rand) CARD_UNION o rand o snd) THEN
ASM_SIMP_TAC[FINITE_IMAGE] THEN ANTS_TAC THENL
[MATCH_MP_TAC(SET_RULE
`(!x. P x ==> ~P(f x)) ==> {x | P x} INTER IMAGE f {x | P x} = {}`) THEN
ASM_SIMP_TAC[IN_ELIM_THM; EVENPERM_COMPOSE; PERMUTATION_SWAP] THEN
ASM_REWRITE_TAC[EVENPERM_SWAP];
DISCH_THEN SUBST1_TAC] THEN
MATCH_MP_TAC(ARITH_RULE `b = a ==> 2 * a = a + b`) THEN
MATCH_MP_TAC CARD_IMAGE_INJ THEN ASM_SIMP_TAC[IN_ELIM_THM] THEN
MATCH_MP_TAC(MESON[]
`(!x. f(f x) = x) ==> (!x y. P x /\ P y /\ f x = f y ==> x = y)`) THEN
ASM_SIMP_TAC[SWAP_IDEMPOTENT; o_ASSOC; I_O_ID]]);;
(* ------------------------------------------------------------------------- *)
(* The special case of involutions. *)
(* ------------------------------------------------------------------------- *)
let PERMUTES_INVOLUTION = prove
(`!p s:A->bool. (!x. p(p x) = x) /\ (!x. ~(x IN s) ==> p x = x)
==> p permutes s`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC PERMUTES_BIJECTIONS THEN
EXISTS_TAC `p:A->A` THEN ASM_MESON_TAC[]);;
let SIGN_INVOLUTION = prove
(`!p:A->A s. FINITE s /\ (!x. p(p x) = x) /\ (!x. ~(x IN s) ==> p x = x)
==> sign p = --(&1) pow (CARD {x | ~(p x = x)} DIV 2)`,
ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN GEN_TAC THEN
WF_INDUCT_TAC `CARD(s:A->bool)` THEN REPEAT STRIP_TAC THEN
ASM_CASES_TAC `p:A->A = I` THEN
ASM_SIMP_TAC[I_THM; EMPTY_GSPEC; CARD_CLAUSES; SIGN_I; DIV_0; real_pow;
ARITH_EQ] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [FUN_EQ_THM]) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM; I_THM; NOT_FORALL_THM] THEN
X_GEN_TAC `a:A` THEN DISCH_TAC THEN
SUBGOAL_THEN `(a:A) IN s /\ p a IN s` STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[]; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `s DELETE (a:A) DELETE (p a)`) THEN
ASM_SIMP_TAC[CARD_DELETE; FINITE_DELETE; IN_DELETE] THEN
ASM_SIMP_TAC[CARD_EQ_0; ARITH_RULE `n - 1 - 1 < n <=> ~(n = 0)`] THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `permutation(p:A->A)` ASSUME_TAC THENL
[ASM_MESON_TAC[PERMUTATION_PERMUTES; PERMUTES_INVOLUTION]; ALL_TAC] THEN
DISCH_THEN(MP_TAC o SPEC `p o swap(a:A,p a)`) THEN
ASM_SIMP_TAC[SIGN_COMPOSE; PERMUTATION_SWAP; SIGN_SWAP] THEN
REWRITE_TAC[o_THM; swap] THEN ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC(REAL_RING `-- &1 * a:real = b ==> s * -- &1 = a ==> s = b`) THEN
SUBGOAL_THEN
`{x | ~(p (if x = a then p a else if x = p a then a else x) = x)} =
{x:A | ~(p x = x)} DELETE a DELETE (p a)`
SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `FINITE {x:A | ~(p x = x)}` ASSUME_TAC THENL
[MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `s:A->bool` THEN
ASM SET_TAC[];
ASM_SIMP_TAC[CARD_DELETE; IN_ELIM_THM; FINITE_DELETE; IN_DELETE]] THEN
REWRITE_TAC[GSYM(CONJUNCT2 real_pow)] THEN AP_TERM_TAC THEN
MATCH_MP_TAC(ARITH_RULE `m + 2 = n ==> SUC(m DIV 2) = n DIV 2`) THEN
MATCH_MP_TAC(ARITH_RULE `2 <= n ==> n - 1 - 1 + 2 = n`) THEN
TRANS_TAC LE_TRANS `CARD {a:A,p a}` THEN CONJ_TAC THENL
[ASM_SIMP_TAC[CARD_CLAUSES; FINITE_RULES; IN_SING; NOT_IN_EMPTY] THEN
CONV_TAC NUM_REDUCE_CONV;
MATCH_MP_TAC CARD_SUBSET THEN ASM SET_TAC[]]);;
(* ------------------------------------------------------------------------- *)
(* Cyclic permutations realized via modular addition. *)
(* ------------------------------------------------------------------------- *)
let PERMUTES_CYCLIC = prove
(`!n. (\i. if i < n then (i + 1) MOD n else i) permutes {i | i < n}`,
GEN_TAC THEN SIMP_TAC[PERMUTES_FINITE_INJECTIVE; FINITE_NUMSEG_LT] THEN
REWRITE_TAC[IMP_CONJ; IN_ELIM_THM] THEN SIMP_TAC[] THEN
SIMP_TAC[MOD_CASES; ARITH_RULE `x < n ==> x + 1 < 2 * n`] THEN ARITH_TAC);;
let PERMUTES_CYCLIC_N = prove
(`!n k. (\i. if i < n then (i + k) MOD n else i) permutes {i | i < n}`,
GEN_TAC THEN INDUCT_TAC THEN
SIMP_TAC[ADD_CLAUSES; MOD_LT; COND_ID; PERMUTES_ID] THEN
SUBGOAL_THEN
`(\i. if i < n then SUC(i + k) MOD n else i) =
(\i. if i < n then (i + 1) MOD n else i) o
(\i. if i < n then (i + k) MOD n else i)`
SUBST1_TAC THENL
[REWRITE_TAC[FUN_EQ_THM; o_THM] THEN X_GEN_TAC `m:num` THEN
ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[CONJUNCT1 LT] THEN
ASM_CASES_TAC `m:num < n` THEN ASM_REWRITE_TAC[MOD_LT_EQ] THEN
REWRITE_TAC[ADD1] THEN MESON_TAC[MOD_ADD_MOD; MOD_MOD_REFL];
MATCH_MP_TAC PERMUTES_COMPOSE THEN
ASM_REWRITE_TAC[PERMUTES_CYCLIC]]);;
let PERMUTATION_CYCLIC = prove
(`!n. permutation (\i. if i < n then (i + 1) MOD n else i)`,
GEN_TAC THEN REWRITE_TAC[PERMUTATION_PERMUTES] THEN
EXISTS_TAC `{i:num | i < n}` THEN
ASM_REWRITE_TAC[PERMUTES_CYCLIC; FINITE_NUMSEG_LT]);;
let PERMUTATION_CYCLIC_N = prove
(`!n k. permutation (\i. if i < n then (i + k) MOD n else i)`,
REPEAT GEN_TAC THEN REWRITE_TAC[PERMUTATION_PERMUTES] THEN
EXISTS_TAC `{i:num | i < n}` THEN
ASM_REWRITE_TAC[PERMUTES_CYCLIC_N; FINITE_NUMSEG_LT]);;
let EVENPERM_CYCLIC = prove
(`!n. evenperm(\i. if i < n then (i + 1) MOD n else i) <=> n = 0 \/ ODD n`,
INDUCT_TAC THEN REWRITE_TAC[CONJUNCT1 LT; EVENPERM_ID; ODD] THEN
REWRITE_TAC[NOT_SUC] THEN
SUBGOAL_THEN
`(\i. if i < SUC n then (i + 1) MOD SUC n else i) =
(swap(0,n)) o (\i. if i < n then (i + 1) MOD n else i)`
SUBST1_TAC THENL
[SIMP_TAC[MOD_CASES; ARITH_RULE `x < n ==> x + 1 < 2 * n`] THEN
REWRITE_TAC[FUN_EQ_THM; o_THM; swap] THEN
POP_ASSUM_LIST(K ALL_TAC) THEN
REWRITE_TAC[ARITH_RULE `m + 1 < SUC n <=> m < n`] THEN
X_GEN_TAC `m:num` THEN
ASM_CASES_TAC `m + 1 < n` THEN
ASM_SIMP_TAC[ARITH_RULE `m + 1 < n ==> m < n /\ m < SUC n`] THENL
[ASM_ARITH_TAC; ALL_TAC] THEN
ASM_CASES_TAC `m:num < n` THEN
ASM_SIMP_TAC[ARITH_RULE `m < n ==> m < SUC n`] THEN
ASM_ARITH_TAC;
ASM_SIMP_TAC[EVENPERM_COMPOSE; PERMUTATION_SWAP; PERMUTATION_CYCLIC] THEN
REWRITE_TAC[EVENPERM_SWAP] THEN
MESON_TAC[ODD]]);;
let EVENPERM_CYCLIC_N = prove
(`!n k. evenperm(\i. if i < n then (i + k) MOD n else i) <=>
n = 0 \/ ODD n \/ EVEN k`,
GEN_TAC THEN INDUCT_TAC THEN
SIMP_TAC[ADD_CLAUSES; MOD_LT; COND_ID; EVENPERM_ID; ARITH] THEN
SUBGOAL_THEN
`(\i. if i < n then SUC(i + k) MOD n else i) =
(\i. if i < n then (i + 1) MOD n else i) o
(\i. if i < n then (i + k) MOD n else i)`
SUBST1_TAC THENL
[REWRITE_TAC[FUN_EQ_THM; o_THM] THEN X_GEN_TAC `m:num` THEN
ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[CONJUNCT1 LT] THEN
ASM_CASES_TAC `m:num < n` THEN ASM_REWRITE_TAC[MOD_LT_EQ] THEN
REWRITE_TAC[ADD1] THEN MESON_TAC[MOD_ADD_MOD; MOD_MOD_REFL];
ALL_TAC] THEN
ASM_SIMP_TAC[EVENPERM_COMPOSE; PERMUTATION_CYCLIC_N] THEN
REWRITE_TAC[EVENPERM_CYCLIC; EVEN; NOT_EVEN] THEN
ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[] THEN
MESON_TAC[NOT_EVEN]);;
let SIGN_CYCLIC = prove
(`!n. sign(\i. if i < n then (i + 1) MOD n else i) = --(&1) pow (n - 1)`,
GEN_TAC THEN REWRITE_TAC[sign; EVENPERM_CYCLIC] THEN
REWRITE_TAC[REAL_POW_NEG; REAL_POW_ONE; EVEN_SUB; ARITH; NOT_EVEN] THEN
ASM_CASES_TAC `n = 1` THEN ASM_REWRITE_TAC[LE_REFL; ARITH] THEN
ASM_REWRITE_TAC[ARITH_RULE `n <= 1 <=> n = 1 \/ n = 0`]);;
let SIGN_CYCLIC_N = prove
(`!n k. sign(\i. if i < n then (i + k) MOD n else i) =
--(&1) pow (k * (n - 1))`,
REPEAT GEN_TAC THEN REWRITE_TAC[sign; EVENPERM_CYCLIC_N] THEN
REWRITE_TAC[REAL_POW_NEG; REAL_POW_ONE; EVEN_MULT;
EVEN_SUB; ARITH; NOT_EVEN] THEN
ASM_CASES_TAC `n = 1` THEN ASM_REWRITE_TAC[LE_REFL; ARITH] THEN
ASM_REWRITE_TAC[ARITH_RULE `n <= 1 <=> n = 1 \/ n = 0`] THEN
MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Conversion for `{p | p permutes s}` where s is a set enumeration. *)
(* ------------------------------------------------------------------------- *)
let PERMSET_CONV =
let pth_empty = prove
(`{p | p permutes {}} = {I}`,
REWRITE_TAC[PERMUTES_EMPTY] THEN SET_TAC[])
and pth_cross = SET_RULE
`IMAGE f {x,y | x IN {} /\ y IN t} = {} /\
IMAGE f {x,y | x IN (a INSERT s) /\ y IN t} =
(IMAGE (\y. f(a,y)) t) UNION (IMAGE f {x,y | x IN s /\ y IN t})`
and pth_union = SET_RULE
`{} UNION t = t /\
(x INSERT s) UNION t = x INSERT (s UNION t)` in
let rec PERMSET_CONV tm =
(GEN_REWRITE_CONV I [pth_empty] ORELSEC
(GEN_REWRITE_CONV I [PERMUTES_INSERT] THENC
ONCE_DEPTH_CONV PERMSET_CONV THENC
REWRITE_CONV[pth_cross] THENC
REWRITE_CONV[IMAGE_CLAUSES] THENC
REWRITE_CONV[pth_union] THENC
REWRITE_CONV[SWAP_REFL; I_O_ID])) tm in
PERMSET_CONV;;
(* ------------------------------------------------------------------------- *)
(* Sum over a set of permutations (could generalize to iteration). *)
(* ------------------------------------------------------------------------- *)
let SUM_OVER_PERMUTATIONS_INSERT = prove
(`!f a s. FINITE s /\ ~(a IN s)
==> sum {p:A->A | p permutes (a INSERT s)} f =
sum (a INSERT s)
(\b. sum {p | p permutes s} (\q. f(swap(a,b) o q)))`,
let lemma = prove
(`(\(b,p). f (swap (a,b) o p)) = f o (\(b,p). swap(a,b) o p)`,
REWRITE_TAC[FUN_EQ_THM; FORALL_PAIR_THM; o_THM]) in
REPEAT STRIP_TAC THEN REWRITE_TAC[PERMUTES_INSERT] THEN
ASM_SIMP_TAC[FINITE_PERMUTATIONS; FINITE_INSERT; SUM_SUM_PRODUCT] THEN
REWRITE_TAC[lemma] THEN MATCH_MP_TAC SUM_IMAGE THEN
REWRITE_TAC[FORALL_PAIR_THM; IN_ELIM_PAIR_THM] THEN
REWRITE_TAC[IN_ELIM_THM] THEN
MAP_EVERY X_GEN_TAC [`b:A`; `p:A->A`; `c:A`; `q:A->A`] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[PAIR_EQ] THEN
MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL
[FIRST_X_ASSUM(MP_TAC o C AP_THM `a:A`) THEN
REWRITE_TAC[o_THM; swap] THEN ASM_MESON_TAC[permutes];
DISCH_THEN SUBST_ALL_TAC THEN
FIRST_X_ASSUM(MP_TAC o AP_TERM `(\p:A->A. swap(a:A,c) o p)`) THEN
REWRITE_TAC[o_ASSOC; SWAP_IDEMPOTENT; I_O_ID]]);;
let SUM_OVER_PERMUTATIONS_NUMSEG = prove
(`!f m n. m <= n
==> sum {p | p permutes (m..n)} f =
sum(m..n) (\i. sum {p | p permutes (m+1..n)}
(\q. f(swap(m,i) o q)))`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[GSYM NUMSEG_LREC] THEN
MATCH_MP_TAC SUM_OVER_PERMUTATIONS_INSERT THEN
REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN ARITH_TAC);;
|