Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 54,393 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 |
(* ========================================================================= *)
(* Formal semantics of HOL inside itself. *)
(* ========================================================================= *)
(* ------------------------------------------------------------------------- *)
(* Semantics of types. *)
(* ------------------------------------------------------------------------- *)
let typeset = new_recursive_definition type_RECURSION
`(typeset tau (Tyvar s) = tau(s)) /\
(typeset tau Bool = boolset) /\
(typeset tau Ind = indset) /\
(typeset tau (Fun a b) = funspace (typeset tau a) (typeset tau b))`;;
(* ------------------------------------------------------------------------- *)
(* Semantics of terms. *)
(* ------------------------------------------------------------------------- *)
let semantics = new_recursive_definition term_RECURSION
`(semantics sigma tau (Var n ty) = sigma(n,ty)) /\
(semantics sigma tau (Equal ty) =
abstract (typeset tau ty) (typeset tau (Fun ty Bool))
(\x. abstract (typeset tau ty) (typeset tau Bool)
(\y. boolean(x = y)))) /\
(semantics sigma tau (Select ty) =
abstract (typeset tau (Fun ty Bool)) (typeset tau ty)
(\s. if ?x. x <: ((typeset tau ty) suchthat (holds s))
then ch ((typeset tau ty) suchthat (holds s))
else ch (typeset tau ty))) /\
(semantics sigma tau (Comb s t) =
apply (semantics sigma tau s) (semantics sigma tau t)) /\
(semantics sigma tau (Abs n ty t) =
abstract (typeset tau ty) (typeset tau (typeof t))
(\x. semantics (((n,ty) |-> x) sigma) tau t))`;;
(* ------------------------------------------------------------------------- *)
(* Valid type and term valuations. *)
(* ------------------------------------------------------------------------- *)
let type_valuation = new_definition
`type_valuation tau <=> !x. (?y. y <: tau x)`;;
let term_valuation = new_definition
`term_valuation tau sigma <=> !n ty. sigma(n,ty) <: typeset tau ty`;;
let TERM_VALUATION_VALMOD = prove
(`!sigma taut n ty x.
term_valuation tau sigma /\ x <: typeset tau ty
==> term_valuation tau (((n,ty) |-> x) sigma)`,
REWRITE_TAC[term_valuation; valmod; PAIR_EQ] THEN MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* All the typesets are nonempty. *)
(* ------------------------------------------------------------------------- *)
let TYPESET_INHABITED = prove
(`!tau ty. type_valuation tau ==> ?x. x <: typeset tau ty`,
REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN GEN_TAC THEN DISCH_TAC THEN
MATCH_MP_TAC type_INDUCT THEN REWRITE_TAC[typeset] THEN
CONJ_TAC THENL
[ASM_MESON_TAC[type_valuation];
ASM_MESON_TAC[BOOLEAN_IN_BOOLSET; INDSET_INHABITED; FUNSPACE_INHABITED]]);;
(* ------------------------------------------------------------------------- *)
(* Semantics maps into the right place. *)
(* ------------------------------------------------------------------------- *)
let SEMANTICS_TYPESET_INDUCT = prove
(`!tm ty. tm has_type ty
==> tm has_type ty /\
!sigma tau. type_valuation tau /\ term_valuation tau sigma
==> (semantics sigma tau tm) <: (typeset tau ty)`,
MATCH_MP_TAC has_type_INDUCT THEN
ASM_SIMP_TAC[semantics; typeset; has_type_RULES] THEN
CONJ_TAC THENL [MESON_TAC[term_valuation]; ALL_TAC] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THENL
[MATCH_MP_TAC ABSTRACT_IN_FUNSPACE THEN REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC ABSTRACT_IN_FUNSPACE THEN
REWRITE_TAC[BOOLEAN_IN_BOOLSET];
MATCH_MP_TAC ABSTRACT_IN_FUNSPACE THEN
ASM_MESON_TAC[ch; SUCHTHAT; TYPESET_INHABITED];
ASM_MESON_TAC[has_type_RULES];
MATCH_MP_TAC APPLY_IN_RANSPACE THEN ASM_MESON_TAC[];
FIRST_ASSUM(SUBST1_TAC o MATCH_MP WELLTYPED_LEMMA) THEN
MATCH_MP_TAC ABSTRACT_IN_FUNSPACE THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[] THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD]]);;
let SEMANTICS_TYPESET = prove
(`!sigma tau tm ty.
type_valuation tau /\ term_valuation tau sigma /\ tm has_type ty
==> (semantics sigma tau tm) <: (typeset tau ty)`,
MESON_TAC[SEMANTICS_TYPESET_INDUCT]);;
(* ------------------------------------------------------------------------- *)
(* Semantics of equations. *)
(* ------------------------------------------------------------------------- *)
let SEMANTICS_EQUATION = prove
(`!sigma tau s t.
s has_type (typeof s) /\ t has_type (typeof s) /\
type_valuation tau /\ term_valuation tau sigma
==> (semantics sigma tau (s === t) =
boolean(semantics sigma tau s = semantics sigma tau t))`,
REPEAT STRIP_TAC THEN REWRITE_TAC[equation; semantics] THEN
ASM_SIMP_TAC[APPLY_ABSTRACT; typeset; SEMANTICS_TYPESET;
ABSTRACT_IN_FUNSPACE; BOOLEAN_IN_BOOLSET]);;
let SEMANTICS_EQUATION_ALT = prove
(`!sigma tau s t.
(s === t) has_type Bool /\
type_valuation tau /\ term_valuation tau sigma
==> (semantics sigma tau (s === t) =
boolean(semantics sigma tau s = semantics sigma tau t))`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC SEMANTICS_EQUATION THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `welltyped(s === t)` MP_TAC THENL
[ASM_MESON_TAC[welltyped]; ALL_TAC] THEN
REWRITE_TAC[equation; WELLTYPED_CLAUSES; typeof; codomain] THEN
MESON_TAC[welltyped; type_INJ; WELLTYPED; WELLTYPED_CLAUSES]);;
(* ------------------------------------------------------------------------- *)
(* Quick sanity check. *)
(* ------------------------------------------------------------------------- *)
let SEMANTICS_SELECT = prove
(`p has_type (Fun ty Bool) /\
type_valuation tau /\ term_valuation tau sigma
==> (semantics sigma tau (Comb (Select ty) p) =
if ?x. x <: (typeset tau ty) suchthat (holds (semantics sigma tau p))
then ch((typeset tau ty) suchthat (holds (semantics sigma tau p)))
else ch(typeset tau ty))`,
REPEAT STRIP_TAC THEN REWRITE_TAC[semantics] THEN
W(fun (asl,w) ->
let t = find_term (fun t ->
can (PART_MATCH (lhs o rand) APPLY_ABSTRACT) t) w in
MP_TAC(PART_MATCH (lhs o rand) APPLY_ABSTRACT t)) THEN
ANTS_TAC THENL
[CONJ_TAC THENL
[ASM_MESON_TAC[SEMANTICS_TYPESET; typeset];
REWRITE_TAC[SUCHTHAT] THEN
ASM_MESON_TAC[ch; SUCHTHAT; TYPESET_INHABITED]];
ALL_TAC] THEN
DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Semantics of a sequent. *)
(* ------------------------------------------------------------------------- *)
parse_as_infix("|=",(11,"right"));;
let sequent = new_definition
`asms |= p <=> ALL (\a. a has_type Bool) (CONS p asms) /\
!sigma tau. type_valuation tau /\
term_valuation tau sigma /\
ALL (\a. semantics sigma tau a = true) asms
==> (semantics sigma tau p = true)`;;
(* ------------------------------------------------------------------------- *)
(* Invariance of semantics under alpha-conversion. *)
(* ------------------------------------------------------------------------- *)
let SEMANTICS_RACONV = prove
(`!env tp.
RACONV env tp
==> !sigma1 sigma2 tau.
type_valuation tau /\
term_valuation tau sigma1 /\ term_valuation tau sigma2 /\
(!x1 ty1 x2 ty2.
ALPHAVARS env (Var x1 ty1,Var x2 ty2)
==> (semantics sigma1 tau (Var x1 ty1) =
semantics sigma2 tau (Var x2 ty2)))
==> welltyped(FST tp) /\ welltyped(SND tp)
==> (semantics sigma1 tau (FST tp) =
semantics sigma2 tau (SND tp))`,
MATCH_MP_TAC RACONV_INDUCT THEN REWRITE_TAC[FORALL_PAIR_THM] THEN
REWRITE_TAC[semantics; WELLTYPED_CLAUSES] THEN REPEAT STRIP_TAC THENL
[ASM_MESON_TAC[];
BINOP_TAC THEN ASM_MESON_TAC[];
ALL_TAC] THEN
FIRST_X_ASSUM SUBST_ALL_TAC THEN MATCH_MP_TAC ABSTRACT_EQ THEN
ASM_SIMP_TAC[TYPESET_INHABITED] THEN
X_GEN_TAC `x:V` THEN DISCH_TAC THEN REPEAT CONJ_TAC THENL
[MATCH_MP_TAC SEMANTICS_TYPESET THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD; GSYM WELLTYPED];
MATCH_MP_TAC SEMANTICS_TYPESET THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD; GSYM WELLTYPED];
ALL_TAC] THEN
RULE_ASSUM_TAC(REWRITE_RULE[IMP_IMP]) THEN
FIRST_X_ASSUM(fun th -> MATCH_MP_TAC th THEN CONJ_TAC) THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD] THEN
REWRITE_TAC[ALPHAVARS; PAIR_EQ; term_INJ] THEN
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[VALMOD; PAIR_EQ] THEN
ASM_MESON_TAC[]);;
let SEMANTICS_ACONV = prove
(`!sigma tau s t.
type_valuation tau /\ term_valuation tau sigma /\
welltyped s /\ welltyped t /\ ACONV s t
==> (semantics sigma tau s = semantics sigma tau t)`,
REWRITE_TAC[ACONV] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_IMP; RIGHT_IMP_FORALL_THM; FORALL_PAIR_THM]
SEMANTICS_RACONV) THEN
EXISTS_TAC `[]:(term#term)list` THEN
ASM_SIMP_TAC[ALPHAVARS; term_INJ; PAIR_EQ]);;
(* ------------------------------------------------------------------------- *)
(* General semantic lemma about binary inference rules. *)
(* ------------------------------------------------------------------------- *)
let BINARY_INFERENCE_RULE = prove
(`(p1 has_type Bool /\ p2 has_type Bool
==> q has_type Bool /\
!sigma tau. type_valuation tau /\ term_valuation tau sigma /\
(semantics sigma tau p1 = true) /\
(semantics sigma tau p2 = true)
==> (semantics sigma tau q = true))
==> (asl1 |= p1 /\ asl2 |= p2 ==> TERM_UNION asl1 asl2 |= q)`,
REWRITE_TAC[sequent; ALL] THEN STRIP_TAC THEN STRIP_TAC THEN
ASM_SIMP_TAC[ALL_BOOL_TERM_UNION] THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MATCH_MP_TAC) THEN
ASM_REWRITE_TAC[] THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN UNDISCH_TAC
`ALL (\a. semantics sigma tau a = true) (TERM_UNION asl1 asl2)` THEN
REWRITE_TAC[GSYM ALL_MEM] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM ALL_MEM])) THEN
REWRITE_TAC[] THEN STRIP_TAC THEN STRIP_TAC THEN
DISCH_THEN(fun th -> X_GEN_TAC `r:term` THEN DISCH_TAC THEN MP_TAC th) THEN
MP_TAC(SPECL [`asl1:term list`; `asl2:term list`; `r:term`]
TERM_UNION_THM) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_TAC `s:term`) THEN
DISCH_THEN(MP_TAC o SPEC `s:term`) THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[SEMANTICS_ACONV; welltyped; TERM_UNION_NONEW]);;
(* ------------------------------------------------------------------------- *)
(* Semantics only depends on valuations of free variables. *)
(* ------------------------------------------------------------------------- *)
let TERM_VALUATION_VFREE_IN = prove
(`!tau sigma1 sigma2 t.
type_valuation tau /\
term_valuation tau sigma1 /\ term_valuation tau sigma2 /\
welltyped t /\
(!x ty. VFREE_IN (Var x ty) t ==> (sigma1(x,ty) = sigma2(x,ty)))
==> (semantics sigma1 tau t = semantics sigma2 tau t)`,
GEN_TAC THEN GEN_REWRITE_TAC BINDER_CONV [SWAP_FORALL_THM] THEN
GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN MATCH_MP_TAC term_INDUCT THEN
REWRITE_TAC[semantics; VFREE_IN; term_DISTINCT; term_INJ] THEN
REPEAT STRIP_TAC THENL
[ASM_MESON_TAC[];
BINOP_TAC THEN ASM_MESON_TAC[WELLTYPED_CLAUSES];
ALL_TAC] THEN
MATCH_MP_TAC ABSTRACT_EQ THEN ASM_SIMP_TAC[TYPESET_INHABITED] THEN
X_GEN_TAC `x:V` THEN DISCH_TAC THEN REPEAT(CONJ_TAC THENL
[ASM_MESON_TAC[TERM_VALUATION_VALMOD; WELLTYPED; WELLTYPED_CLAUSES;
SEMANTICS_TYPESET];
ALL_TAC]) THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[TERM_VALUATION_VALMOD] THEN
CONJ_TAC THENL [ASM_MESON_TAC[WELLTYPED_CLAUSES]; ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`x:string`; `ty:type`] THEN DISCH_TAC THEN
REWRITE_TAC[VALMOD; PAIR_EQ] THEN ASM_MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Prove some inference rules correct. *)
(* ------------------------------------------------------------------------- *)
let ASSUME_correct = prove
(`!p. p has_type Bool ==> [p] |= p`,
SIMP_TAC[sequent; ALL]);;
let REFL_correct = prove
(`!t. welltyped t ==> [] |= t === t`,
SIMP_TAC[sequent; SEMANTICS_EQUATION; ALL; WELLTYPED] THEN
REWRITE_TAC[boolean; equation] THEN MESON_TAC[has_type_RULES]);;
let TRANS_correct = prove
(`!asl1 asl2 l m1 m2 r.
asl1 |= l === m1 /\ asl2 |= m2 === r /\ ACONV m1 m2
==> TERM_UNION asl1 asl2 |= l === r`,
REPEAT GEN_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
MATCH_MP_TAC BINARY_INFERENCE_RULE THEN STRIP_TAC THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[ASM_MESON_TAC[EQUATION_HAS_TYPE_BOOL; ACONV_TYPE];
ASM_SIMP_TAC[SEMANTICS_EQUATION_ALT; IMP_CONJ; boolean] THEN
ASM_MESON_TAC[SEMANTICS_ACONV; TRUE_NE_FALSE; EQUATION_HAS_TYPE_BOOL]]);;
let MK_COMB_correct = prove
(`!asl1 l1 r1 asl2 l2 r2.
asl1 |= l1 === r1 /\ asl2 |= l2 === r2 /\
(?rty. typeof l1 = Fun (typeof l2) rty)
==> TERM_UNION asl1 asl2 |= Comb l1 l2 === Comb r1 r2`,
REPEAT GEN_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
MATCH_MP_TAC BINARY_INFERENCE_RULE THEN STRIP_TAC THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
REWRITE_TAC[EQUATION_HAS_TYPE_BOOL; WELLTYPED_CLAUSES; typeof] THEN
MESON_TAC[codomain];
ASM_SIMP_TAC[SEMANTICS_EQUATION_ALT; IMP_CONJ; boolean] THEN
REWRITE_TAC[semantics] THEN
ASM_MESON_TAC[SEMANTICS_ACONV; TRUE_NE_FALSE; EQUATION_HAS_TYPE_BOOL]]);;
let EQ_MP_correct = prove
(`!asl1 asl2 p q p'.
asl1 |= p === q /\ asl2 |= p' /\ ACONV p p'
==> TERM_UNION asl1 asl2 |= q`,
REPEAT GEN_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
MATCH_MP_TAC BINARY_INFERENCE_RULE THEN STRIP_TAC THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[ASM_MESON_TAC[EQUATION_HAS_TYPE_BOOL; WELLTYPED_LEMMA; WELLTYPED;
ACONV_TYPE];
ASM_SIMP_TAC[SEMANTICS_EQUATION_ALT; IMP_CONJ; boolean] THEN
ASM_MESON_TAC[EQUATION_HAS_TYPE_BOOL; TRUE_NE_FALSE; SEMANTICS_ACONV;
welltyped]]);;
let BETA_correct = prove
(`!x ty t. welltyped t ==> [] |= Comb (Abs x ty t) (Var x ty) === t`,
REPEAT STRIP_TAC THEN REWRITE_TAC[sequent; ALL] THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[REWRITE_TAC[EQUATION_HAS_TYPE_BOOL; typeof; WELLTYPED_CLAUSES] THEN
REWRITE_TAC[codomain; type_INJ] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
SIMP_TAC[SEMANTICS_EQUATION_ALT] THEN
DISCH_TAC THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
REWRITE_TAC[BOOLEAN_EQ_TRUE; semantics] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `semantics (((x,ty) |-> sigma(x,ty)) sigma) tau t` THEN
CONJ_TAC THENL [MATCH_MP_TAC APPLY_ABSTRACT; ALL_TAC] THEN
REWRITE_TAC[VALMOD_REPEAT] THEN
ASM_MESON_TAC[term_valuation; SEMANTICS_TYPESET; WELLTYPED]);;
let ABS_correct = prove
(`!asl x ty l r.
~(EX (VFREE_IN (Var x ty)) asl) /\ asl |= l === r
==> asl |= (Abs x ty l) === (Abs x ty r)`,
REPEAT GEN_TAC THEN REWRITE_TAC[sequent; ALL] THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN
CONJ_TAC THENL
[UNDISCH_TAC `(l === r) has_type Bool` THEN
SIMP_TAC[EQUATION_HAS_TYPE_BOOL; WELLTYPED_CLAUSES; typeof];
ALL_TAC] THEN
DISCH_TAC THEN ASM_SIMP_TAC[SEMANTICS_EQUATION_ALT; BOOLEAN_EQ_TRUE] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[semantics] THEN
SUBGOAL_THEN `typeof r = typeof l` SUBST1_TAC THENL
[ASM_MESON_TAC[EQUATION_HAS_TYPE_BOOL]; ALL_TAC] THEN
MATCH_MP_TAC ABSTRACT_EQ THEN ASM_SIMP_TAC[TYPESET_INHABITED] THEN
X_GEN_TAC `x:V` THEN DISCH_TAC THEN
REPEAT(CONJ_TAC THENL
[ASM_MESON_TAC[SEMANTICS_TYPESET; TERM_VALUATION_VALMOD;
WELLTYPED; EQUATION_HAS_TYPE_BOOL];
ALL_TAC]) THEN
FIRST_X_ASSUM(MP_TAC o check (is_forall o concl)) THEN
ASM_SIMP_TAC[SEMANTICS_EQUATION_ALT; BOOLEAN_EQ_TRUE] THEN
DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[TERM_VALUATION_VALMOD] THEN
SUBGOAL_THEN `ALL (\a. a has_type Bool) asl /\
ALL (\a. ~(VFREE_IN (Var x ty) a)) asl /\
ALL (\a. semantics sigma tau a = true) asl`
MP_TAC THENL [ASM_REWRITE_TAC[GSYM NOT_EX; ETA_AX]; ALL_TAC] THEN
REWRITE_TAC[AND_ALL] THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] ALL_IMP) THEN
X_GEN_TAC `p:term` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN
MATCH_MP_TAC TERM_VALUATION_VFREE_IN THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD] THEN
CONJ_TAC THENL [ASM_MESON_TAC[welltyped]; ALL_TAC] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[VALMOD; PAIR_EQ] THEN ASM_MESON_TAC[]);;
let DEDUCT_ANTISYM_RULE_correct = prove
(`!asl1 asl2 p q.
asl1 |= c1 /\ asl2 |= c2
==> let asl1' = FILTER((~) o ACONV c2) asl1
and asl2' = FILTER((~) o ACONV c1) asl2 in
(TERM_UNION asl1' asl2') |= c1 === c2`,
REPEAT GEN_TAC THEN
REWRITE_TAC[sequent; o_DEF; LET_DEF; LET_END_DEF; GSYM CONJ_ASSOC] THEN
MATCH_MP_TAC(TAUT `
(a1 /\ b1 ==> c1) /\ (a1 /\ b1 /\ c1 ==> a2 /\ b2 ==> c2)
==> a1 /\ a2 /\ b1 /\ b2 ==> c1 /\ c2`) THEN
CONJ_TAC THENL
[REWRITE_TAC[GSYM ALL_MEM; MEM] THEN REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[EQUATION_HAS_TYPE_BOOL] THEN
ASM_MESON_TAC[MEM_FILTER; TERM_UNION_NONEW; welltyped; WELLTYPED_LEMMA];
ALL_TAC] THEN
REWRITE_TAC[ALL; AND_FORALL_THM] THEN REWRITE_TAC[GSYM ALL_MEM] THEN
STRIP_TAC THEN
REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
ASM_SIMP_TAC[SEMANTICS_EQUATION_ALT; BOOLEAN_EQ_TRUE] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC BOOLEAN_EQ THEN
REPEAT(CONJ_TAC THENL
[ASM_MESON_TAC[typeset; SEMANTICS_TYPESET]; ALL_TAC]) THEN
EQ_TAC THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
X_GEN_TAC `a:term` THEN DISCH_TAC THENL
[ASM_CASES_TAC `ACONV c1 a` THENL
[ASM_MESON_TAC[SEMANTICS_ACONV; welltyped]; ALL_TAC];
ASM_CASES_TAC `ACONV c2 a` THENL
[ASM_MESON_TAC[SEMANTICS_ACONV; welltyped]; ALL_TAC]] THEN
(SUBGOAL_THEN
`MEM a (FILTER (\x. ~ACONV c2 x) asl1) \/
MEM a (FILTER (\x. ~ACONV c1 x) asl2)`
MP_TAC THENL
[REWRITE_TAC[MEM_FILTER] THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN
DISCH_THEN(MP_TAC o MATCH_MP TERM_UNION_THM) THEN
ASM_MESON_TAC[SEMANTICS_ACONV; welltyped]));;
(* ------------------------------------------------------------------------- *)
(* Correct semantics for term substitution. *)
(* ------------------------------------------------------------------------- *)
let DEST_VAR = new_recursive_definition term_RECURSION
`DEST_VAR (Var x ty) = (x,ty)`;;
let TERM_VALUATION_ITLIST = prove
(`!ilist sigma tau.
type_valuation tau /\ term_valuation tau sigma /\
(!s s'. MEM (s',s) ilist ==> ?x ty. (s = Var x ty) /\ s' has_type ty)
==> term_valuation tau
(ITLIST (\(t,x). DEST_VAR x |-> semantics sigma tau t) ilist sigma)`,
MATCH_MP_TAC list_INDUCT THEN SIMP_TAC[ITLIST] THEN
REWRITE_TAC[FORALL_PAIR_THM; MEM; PAIR_EQ] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[TAUT `a \/ b ==> c <=> (a ==> c) /\ (b ==> c)`] THEN
SIMP_TAC[LEFT_FORALL_IMP_THM; FORALL_AND_THM] THEN
REWRITE_TAC[LEFT_EXISTS_AND_THM; EXISTS_REFL] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[DEST_VAR] THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD; SEMANTICS_TYPESET]);;
let ITLIST_VALMOD_FILTER = prove
(`!ilist sigma sem x ty y yty.
(!s s'. MEM (s',s) ilist ==> ?x ty. (s = Var x ty) /\ s' has_type ty)
==> (ITLIST (\(t,x). DEST_VAR x |-> sem x t)
(FILTER (\(s',s). ~(s = Var x ty)) ilist) sigma (y,yty) =
if (y = x) /\ (yty = ty) then sigma(y,yty)
else ITLIST (\(t,x). DEST_VAR x |-> sem x t) ilist sigma (y,yty))`,
MATCH_MP_TAC list_INDUCT THEN REWRITE_TAC[FILTER; ITLIST; COND_ID] THEN
REWRITE_TAC[FORALL_PAIR_THM] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[MEM; TAUT `a \/ b ==> c <=> (a ==> c) /\ (b ==> c)`] THEN
SIMP_TAC[FORALL_AND_THM; LEFT_FORALL_IMP_THM; PAIR_EQ] THEN
REWRITE_TAC[WELLTYPED_CLAUSES; LEFT_EXISTS_AND_THM; EXISTS_REFL] THEN
MAP_EVERY X_GEN_TAC [`t:term`; `pp:term`; `ilist:(term#term)list`] THEN
DISCH_TAC THEN REPEAT GEN_TAC THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `s:string` MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `sty:type` MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [COND_RAND] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [COND_RATOR] THEN
ASM_REWRITE_TAC[ITLIST] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[DEST_VAR] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [COND_RATOR] THEN
REWRITE_TAC[VALMOD] THEN REWRITE_TAC[term_INJ] THEN
ASM_CASES_TAC `(s:string = x) /\ (sty:type = ty)` THEN
ASM_SIMP_TAC[PAIR_EQ] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[]);;
let ITLIST_VALMOD_EQ = prove
(`!l. (!t x. MEM (t,x) l /\ (f x = a) ==> (g x t = h x t)) /\ (i a = j a)
==> (ITLIST (\(t,x). f(x) |-> g x t) l i a =
ITLIST (\(t,x). f(x) |-> h x t) l j a)`,
MATCH_MP_TAC list_INDUCT THEN SIMP_TAC[MEM; ITLIST; FORALL_PAIR_THM] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[PAIR_EQ; VALMOD] THEN MESON_TAC[]);;
let SEMANTICS_VSUBST = prove
(`!tm sigma tau ilist.
welltyped tm /\
(!s s'. MEM (s',s) ilist ==> ?x ty. (s = Var x ty) /\ s' has_type ty)
==> !sigma tau. type_valuation tau /\ term_valuation tau sigma
==> (semantics sigma tau (VSUBST ilist tm) =
semantics
(ITLIST
(\(t,x). DEST_VAR x |-> semantics sigma tau t)
ilist sigma)
tau tm)`,
MATCH_MP_TAC term_INDUCT THEN REWRITE_TAC[VSUBST; semantics] THEN
CONJ_TAC THENL
[MAP_EVERY X_GEN_TAC [`x:string`; `ty:type`] THEN
MATCH_MP_TAC list_INDUCT THEN
REWRITE_TAC[MEM; REV_ASSOCD; ITLIST; semantics; FORALL_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`t:term`; `s:term`; `ilist:(term#term)list`] THEN
REWRITE_TAC[TAUT `a \/ b ==> c <=> (a ==> c) /\ (b ==> c)`] THEN
SIMP_TAC[FORALL_AND_THM; LEFT_FORALL_IMP_THM; PAIR_EQ] THEN
REWRITE_TAC[WELLTYPED_CLAUSES; LEFT_EXISTS_AND_THM; EXISTS_REFL] THEN
DISCH_THEN(fun th ->
DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN MP_TAC th) THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `y:string` MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `tty:type` MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[DEST_VAR; VALMOD; term_INJ; PAIR_EQ] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
CONJ_TAC THENL
[REWRITE_TAC[WELLTYPED_CLAUSES] THEN REPEAT STRIP_TAC THEN
BINOP_TAC THEN ASM_MESON_TAC[];
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`x:string`; `ty:type`; `t:term`] THEN
REWRITE_TAC[WELLTYPED_CLAUSES] THEN
ASM_CASES_TAC `welltyped t` THEN ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN LET_TAC THEN LET_TAC THEN
SUBGOAL_THEN
`!s s'. MEM (s',s) ilist' ==> (?x ty. (s = Var x ty) /\ s' has_type ty)`
ASSUME_TAC THENL
[EXPAND_TAC "ilist'" THEN ASM_SIMP_TAC[MEM_FILTER]; ALL_TAC] THEN
COND_CASES_TAC THENL
[REPEAT LET_TAC THEN
SUBGOAL_THEN
`!s s'. MEM (s',s) ilist'' ==> (?x ty. (s = Var x ty) /\ s' has_type ty)`
ASSUME_TAC THENL
[EXPAND_TAC "ilist''" THEN REWRITE_TAC[MEM; PAIR_EQ] THEN
ASM_MESON_TAC[has_type_RULES];
ALL_TAC];
ALL_TAC] THEN
REWRITE_TAC[semantics] THEN
MATCH_MP_TAC ABSTRACT_EQ THEN ASM_SIMP_TAC[TYPESET_INHABITED] THEN
X_GEN_TAC `a:V` THEN DISCH_TAC THEN
REPEAT CONJ_TAC THEN TRY(MATCH_MP_TAC SEMANTICS_TYPESET) THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD; TERM_VALUATION_ITLIST] THEN
EXPAND_TAC "t'" THEN
ASM_SIMP_TAC[VSUBST_WELLTYPED; GSYM WELLTYPED; TERM_VALUATION_VALMOD] THEN
MATCH_MP_TAC TERM_VALUATION_VFREE_IN THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD; TERM_VALUATION_ITLIST] THEN
MAP_EVERY X_GEN_TAC [`u:string`; `uty:type`] THEN DISCH_TAC THENL
[EXPAND_TAC "ilist''" THEN REWRITE_TAC[ITLIST] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[DEST_VAR; VALMOD; PAIR_EQ] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[semantics; VALMOD];
ALL_TAC] THEN
EXPAND_TAC "ilist'" THEN ASM_SIMP_TAC[ITLIST_VALMOD_FILTER] THEN
REWRITE_TAC[VALMOD] THENL
[ALL_TAC;
REWRITE_TAC[PAIR_EQ] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC ITLIST_VALMOD_EQ THEN ASM_REWRITE_TAC[VALMOD; PAIR_EQ] THEN
MAP_EVERY X_GEN_TAC [`s':term`; `s:term`] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o C MATCH_MP
(ASSUME `MEM (s':term,s:term) ilist`)) THEN
DISCH_THEN(X_CHOOSE_THEN `w:string` MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `wty:type` MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
UNDISCH_TAC `DEST_VAR (Var w wty) = u,uty` THEN
REWRITE_TAC[DEST_VAR; PAIR_EQ] THEN
DISCH_THEN(CONJUNCTS_THEN SUBST_ALL_TAC) THEN
MATCH_MP_TAC TERM_VALUATION_VFREE_IN THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD] THEN
CONJ_TAC THENL [ASM_MESON_TAC[welltyped]; ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`v:string`; `vty:type`] THEN
DISCH_TAC THEN REWRITE_TAC[VALMOD; PAIR_EQ] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(CONJUNCTS_THEN SUBST_ALL_TAC) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EX]) THEN
REWRITE_TAC[GSYM ALL_MEM] THEN
DISCH_THEN(MP_TAC o SPEC `(s':term,Var u uty)`) THEN
ASM_REWRITE_TAC[] THEN CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
ASM_REWRITE_TAC[] THEN EXPAND_TAC "ilist'" THEN
REWRITE_TAC[MEM_FILTER] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
ASM_REWRITE_TAC[term_INJ]] THEN
MP_TAC(ISPECL [`t':term`; `x:string`; `ty:type`] VARIANT) THEN
ASM_REWRITE_TAC[] THEN EXPAND_TAC "t'" THEN
REWRITE_TAC[VFREE_IN_VSUBST] THEN
REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(a /\ b) = b ==> ~a`] THEN
DISCH_THEN(MP_TAC o SPECL [`u:string`; `uty:type`]) THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN
`REV_ASSOCD (Var u uty) ilist' (Var u uty) =
REV_ASSOCD (Var u uty) ilist (Var u uty)`
SUBST1_TAC THENL
[EXPAND_TAC "ilist'" THEN REWRITE_TAC[REV_ASSOCD_FILTER] THEN
ASM_REWRITE_TAC[term_INJ];
ALL_TAC] THEN
UNDISCH_TAC
`!s s'. MEM (s',s) ilist ==> ?x ty. (s = Var x ty) /\ s' has_type ty` THEN
SPEC_TAC(`ilist:(term#term)list`,`l:(term#term)list`) THEN
MATCH_MP_TAC list_INDUCT THEN
REWRITE_TAC[REV_ASSOCD; ITLIST; VFREE_IN; VALMOD; term_INJ] THEN
SIMP_TAC[PAIR_EQ] THEN REWRITE_TAC[FORALL_PAIR_THM] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[VALMOD; REV_ASSOCD; MEM] THEN
REWRITE_TAC[TAUT `a \/ b ==> c <=> (a ==> c) /\ (b ==> c)`] THEN
SIMP_TAC[FORALL_AND_THM; LEFT_FORALL_IMP_THM; PAIR_EQ] THEN
REWRITE_TAC[WELLTYPED_CLAUSES; LEFT_EXISTS_AND_THM; EXISTS_REFL] THEN
MAP_EVERY X_GEN_TAC [`t1:term`; `t2:term`; `i:(term#term)list`] THEN
DISCH_THEN(fun th ->
DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN MP_TAC th) THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `v:string` MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `vty:type` MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
ASM_REWRITE_TAC[DEST_VAR; term_INJ; PAIR_EQ] THEN
SUBGOAL_THEN `(v:string = u) /\ (vty:type = uty) <=> (u = v) /\ (uty = vty)`
SUBST1_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC TERM_VALUATION_VFREE_IN THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD; VALMOD] THEN
REWRITE_TAC[PAIR_EQ] THEN ASM_MESON_TAC[welltyped; term_INJ]);;
(* ------------------------------------------------------------------------- *)
(* Hence correctness of INST. *)
(* ------------------------------------------------------------------------- *)
let INST_correct = prove
(`!ilist asl p.
(!s s'. MEM (s',s) ilist ==> ?x ty. (s = Var x ty) /\ s' has_type ty)
==> asl |= p ==> MAP (VSUBST ilist) asl |= VSUBST ilist p`,
REWRITE_TAC[sequent] THEN REPEAT STRIP_TAC THENL
[UNDISCH_TAC `ALL (\a. a has_type Bool) (CONS p asl)` THEN
REWRITE_TAC[ALL; ALL_MAP] THEN MATCH_MP_TAC MONO_AND THEN
CONJ_TAC THENL
[ALL_TAC;
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] ALL_IMP) THEN
GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[o_THM]] THEN
DISCH_TAC THEN MATCH_MP_TAC VSUBST_HAS_TYPE THEN ASM_REWRITE_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `welltyped p` ASSUME_TAC THENL
[ASM_MESON_TAC[welltyped; ALL]; ALL_TAC] THEN
ASM_SIMP_TAC[SEMANTICS_VSUBST] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_SIMP_TAC[TERM_VALUATION_ITLIST] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ALL_MAP]) THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] ALL_IMP) THEN
X_GEN_TAC `a:term` THEN DISCH_TAC THEN
SUBGOAL_THEN `welltyped a` MP_TAC THENL
[ASM_MESON_TAC[ALL_MEM; MEM; welltyped]; ALL_TAC] THEN
ASM_SIMP_TAC[SEMANTICS_VSUBST; o_THM]);;
(* ------------------------------------------------------------------------- *)
(* Lemma about typesets to simplify some later goals. *)
(* ------------------------------------------------------------------------- *)
let TYPESET_LEMMA = prove
(`!ty tau tyin.
typeset (\s. typeset tau (REV_ASSOCD (Tyvar s) tyin (Tyvar s))) ty =
typeset tau (TYPE_SUBST tyin ty)`,
MATCH_MP_TAC type_INDUCT THEN SIMP_TAC[typeset; TYPE_SUBST]);;
(* ------------------------------------------------------------------------- *)
(* Semantics of type instantiation core. *)
(* ------------------------------------------------------------------------- *)
let SEMANTICS_INST_CORE = prove
(`!n tm env tyin.
welltyped tm /\ (sizeof tm = n) /\
(!s s'. MEM (s,s') env
==> ?x ty. (s = Var x ty) /\
(s' = Var x (TYPE_SUBST tyin ty)))
==> (?x ty. (INST_CORE env tyin tm =
Clash(Var x (TYPE_SUBST tyin ty))) /\
VFREE_IN (Var x ty) tm /\
~(REV_ASSOCD (Var x (TYPE_SUBST tyin ty))
env (Var x ty) = Var x ty)) \/
(!x ty. VFREE_IN (Var x ty) tm
==> (REV_ASSOCD (Var x (TYPE_SUBST tyin ty))
env (Var x ty) = Var x ty)) /\
(?tm'. (INST_CORE env tyin tm = Result tm') /\
tm' has_type (TYPE_SUBST tyin (typeof tm)) /\
(!u uty. VFREE_IN (Var u uty) tm' <=>
?oty. VFREE_IN (Var u oty) tm /\
(uty = TYPE_SUBST tyin oty)) /\
!sigma tau.
type_valuation tau /\ term_valuation tau sigma
==> (semantics sigma tau tm' =
semantics
(\(x,ty). sigma(x,TYPE_SUBST tyin ty))
(\s. typeset tau (TYPE_SUBST tyin (Tyvar s)))
tm))`,
MATCH_MP_TAC num_WF THEN GEN_TAC THEN DISCH_TAC THEN
MATCH_MP_TAC term_INDUCT THEN
ONCE_REWRITE_TAC[INST_CORE] THEN REWRITE_TAC[semantics] THEN
REPEAT CONJ_TAC THENL
[POP_ASSUM(K ALL_TAC) THEN
REWRITE_TAC[REV_ASSOCD; LET_DEF; LET_END_DEF] THEN
REPEAT GEN_TAC THEN COND_CASES_TAC THEN
ASM_REWRITE_TAC[result_DISTINCT; result_INJ; UNWIND_THM1] THEN
REWRITE_TAC[typeof; has_type_RULES] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[RESULT; semantics; VFREE_IN; term_INJ] THEN ASM_MESON_TAC[];
POP_ASSUM(K ALL_TAC) THEN
REWRITE_TAC[TYPE_SUBST; RESULT; VFREE_IN; term_DISTINCT] THEN
ASM_REWRITE_TAC[result_DISTINCT; result_INJ; UNWIND_THM1] THEN
REWRITE_TAC[typeof; has_type_RULES; TYPE_SUBST; VFREE_IN] THEN
REWRITE_TAC[semantics; typeset; TYPESET_LEMMA; TYPE_SUBST; term_DISTINCT];
POP_ASSUM(K ALL_TAC) THEN
REWRITE_TAC[TYPE_SUBST; RESULT; VFREE_IN; term_DISTINCT] THEN
ASM_REWRITE_TAC[result_DISTINCT; result_INJ; UNWIND_THM1] THEN
REWRITE_TAC[typeof; has_type_RULES; TYPE_SUBST; VFREE_IN] THEN
REWRITE_TAC[semantics; typeset; TYPESET_LEMMA; TYPE_SUBST; term_DISTINCT];
MAP_EVERY X_GEN_TAC [`s:term`; `t:term`] THEN DISCH_THEN(K ALL_TAC) THEN
POP_ASSUM MP_TAC THEN ASM_CASES_TAC `n = sizeof(Comb s t)` THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(fun th -> MP_TAC(SPEC `sizeof t` th) THEN
MP_TAC(SPEC `sizeof s` th)) THEN
REWRITE_TAC[sizeof; ARITH_RULE `s < 1 + s + t /\ t < 1 + s + t`] THEN
DISCH_THEN(fun th -> DISCH_THEN(MP_TAC o SPEC `t:term`) THEN
MP_TAC(SPEC `s:term` th)) THEN
REWRITE_TAC[IMP_IMP; AND_FORALL_THM; WELLTYPED_CLAUSES] THEN
REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN
GEN_REWRITE_TAC I [IMP_CONJ] THEN
DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL
[DISCH_THEN(fun th -> DISCH_THEN(K ALL_TAC) THEN MP_TAC th) THEN
DISCH_THEN(fun th -> DISJ1_TAC THEN MP_TAC th) THEN
REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
STRIP_TAC THEN ASM_REWRITE_TAC[LET_DEF; LET_END_DEF; IS_CLASH; VFREE_IN];
ALL_TAC] THEN
REWRITE_TAC[TYPE_SUBST] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `s':term` STRIP_ASSUME_TAC) THEN
DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL
[DISCH_THEN(fun th -> DISJ1_TAC THEN MP_TAC th) THEN
REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
STRIP_TAC THEN ASM_REWRITE_TAC[LET_DEF; LET_END_DEF; IS_CLASH; VFREE_IN];
ALL_TAC] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `t':term` STRIP_ASSUME_TAC) THEN
DISJ2_TAC THEN CONJ_TAC THENL
[REWRITE_TAC[VFREE_IN] THEN ASM_MESON_TAC[]; ALL_TAC] THEN
EXISTS_TAC `Comb s' t'` THEN
ASM_SIMP_TAC[LET_DEF; LET_END_DEF; IS_CLASH; semantics; RESULT] THEN
ASM_REWRITE_TAC[VFREE_IN] THEN
ASM_REWRITE_TAC[typeof] THEN ONCE_REWRITE_TAC[has_type_CASES] THEN
REWRITE_TAC[term_DISTINCT; term_INJ; codomain] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`x:string`; `ty:type`; `t:term`] THEN
DISCH_THEN(K ALL_TAC) THEN POP_ASSUM MP_TAC THEN
ASM_CASES_TAC `n = sizeof (Abs x ty t)` THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM(K ALL_TAC) THEN DISCH_TAC THEN REPEAT GEN_TAC THEN
REWRITE_TAC[WELLTYPED_CLAUSES] THEN STRIP_TAC THEN REPEAT LET_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `sizeof t`) THEN
REWRITE_TAC[sizeof; ARITH_RULE `t < 2 + t`] THEN
DISCH_THEN(MP_TAC o SPECL
[`t:term`; `env':(term#term)list`; `tyin:(type#type)list`]) THEN
ASM_REWRITE_TAC[] THEN
ANTS_TAC THENL
[EXPAND_TAC "env'" THEN REWRITE_TAC[MEM; PAIR_EQ] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL
[ALL_TAC;
FIRST_X_ASSUM(K ALL_TAC o SPEC `0`) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `t':term` STRIP_ASSUME_TAC) THEN
DISJ2_TAC THEN ASM_REWRITE_TAC[IS_RESULT] THEN CONJ_TAC THENL
[FIRST_X_ASSUM(fun th ->
MP_TAC th THEN MATCH_MP_TAC MONO_FORALL THEN
GEN_TAC THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN
DISCH_THEN(MP_TAC o check (is_imp o concl))) THEN
EXPAND_TAC "env'" THEN
REWRITE_TAC[VFREE_IN; REV_ASSOCD; term_INJ] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[term_INJ] THEN MESON_TAC[];
ALL_TAC] THEN
REWRITE_TAC[result_INJ; UNWIND_THM1; RESULT] THEN
MATCH_MP_TAC(TAUT `a /\ b /\ (b ==> c) ==> b /\ a /\ c`) THEN
CONJ_TAC THENL
[ASM_REWRITE_TAC[VFREE_IN; term_INJ] THEN
MAP_EVERY X_GEN_TAC [`u:string`; `uty:type`] THEN
ASM_CASES_TAC `u:string = x` THEN ASM_REWRITE_TAC[] THEN
UNDISCH_THEN `u:string = x` SUBST_ALL_TAC THEN
REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
X_GEN_TAC `oty:type` THEN REWRITE_TAC[] THEN
ASM_CASES_TAC `uty = TYPE_SUBST tyin oty` THEN ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `VFREE_IN (Var x oty) t` THEN ASM_REWRITE_TAC[] THEN
EQ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
REWRITE_TAC[CONTRAPOS_THM] THEN DISCH_THEN(ASSUME_TAC o SYM) THEN
FIRST_X_ASSUM(fun th ->
MP_TAC(SPECL [`x:string`; `oty:type`] th) THEN
ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN NO_TAC; ALL_TAC]) THEN
EXPAND_TAC "env'" THEN REWRITE_TAC[REV_ASSOCD] THEN
ASM_MESON_TAC[term_INJ];
ALL_TAC] THEN
CONJ_TAC THENL
[REWRITE_TAC[typeof; TYPE_SUBST] THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[has_type_RULES];
ALL_TAC] THEN
DISCH_TAC THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
REWRITE_TAC[semantics] THEN
ASM_REWRITE_TAC[TYPE_SUBST; TYPESET_LEMMA] THEN
MATCH_MP_TAC ABSTRACT_EQ THEN
CONJ_TAC THENL [ASM_SIMP_TAC[TYPESET_INHABITED]; ALL_TAC] THEN
X_GEN_TAC `a:V` THEN REWRITE_TAC[] THEN DISCH_TAC THEN CONJ_TAC THENL
[MATCH_MP_TAC SEMANTICS_TYPESET THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD] THEN
ASM_MESON_TAC[welltyped; WELLTYPED];
ALL_TAC] THEN
MATCH_MP_TAC(TAUT `(b ==> a) /\ b ==> a /\ b`) THEN CONJ_TAC THENL
[DISCH_THEN(SUBST1_TAC o SYM) THEN
MATCH_MP_TAC SEMANTICS_TYPESET THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o SPECL
[`(x,ty' |-> a) (sigma:(string#type)->V)`; `tau:string->V`]) THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD] THEN DISCH_TAC THEN
REWRITE_TAC[GSYM(CONJUNCT1 TYPE_SUBST)] THEN
MATCH_MP_TAC TERM_VALUATION_VFREE_IN THEN CONJ_TAC THENL
[REWRITE_TAC[type_valuation] THEN ASM_SIMP_TAC[TYPESET_INHABITED];
ALL_TAC] THEN
CONJ_TAC THENL
[REWRITE_TAC[term_valuation] THEN
MAP_EVERY X_GEN_TAC [`y:string`; `yty:type`] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[VALMOD; PAIR_EQ] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[TYPE_SUBST; TYPESET_LEMMA] THEN
ASM_MESON_TAC[term_valuation];
ALL_TAC] THEN
CONJ_TAC THENL
[REWRITE_TAC[term_valuation] THEN
MAP_EVERY X_GEN_TAC [`y:string`; `yty:type`] THEN
REWRITE_TAC[VALMOD] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[VALMOD; PAIR_EQ] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[TYPE_SUBST; TYPESET_LEMMA] THEN
ASM_MESON_TAC[term_valuation];
ALL_TAC] THEN
UNDISCH_THEN
`!u uty.
VFREE_IN (Var u uty) t' <=>
(?oty. VFREE_IN (Var u oty) t /\ (uty = TYPE_SUBST tyin oty))`
(K ALL_TAC) THEN
ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC [`y:string`; `yty:type`] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[VALMOD] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
ASM_CASES_TAC `y:string = x` THEN ASM_REWRITE_TAC[PAIR_EQ] THEN
ASM_CASES_TAC `yty:type = ty` THEN ASM_REWRITE_TAC[] THEN
UNDISCH_THEN `y:string = x` SUBST_ALL_TAC THEN COND_CASES_TAC THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`x:string`; `yty:type`]) THEN
ASM_REWRITE_TAC[] THEN EXPAND_TAC "env'" THEN
ASM_REWRITE_TAC[REV_ASSOCD; term_INJ]] THEN
DISCH_THEN(X_CHOOSE_THEN `z:string` (X_CHOOSE_THEN `zty:type`
(CONJUNCTS_THEN2 SUBST_ALL_TAC MP_TAC))) THEN
EXPAND_TAC "w" THEN REWRITE_TAC[CLASH; IS_RESULT; term_INJ] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
[FIRST_X_ASSUM(K ALL_TAC o SPEC `0`) THEN
DISCH_THEN(fun th ->
DISJ1_TAC THEN REWRITE_TAC[result_INJ] THEN
MAP_EVERY EXISTS_TAC [`z:string`; `zty:type`] THEN
MP_TAC th) THEN
ASM_REWRITE_TAC[VFREE_IN; term_INJ] THEN
EXPAND_TAC "env'" THEN ASM_REWRITE_TAC[REV_ASSOCD; term_INJ] THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
FIRST_X_ASSUM(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN STRIP_TAC THEN
ONCE_REWRITE_TAC[INST_CORE] THEN ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[letlemma] THEN
ABBREV_TAC `env'' = CONS (Var x' ty,Var x' ty') env` THEN
ONCE_REWRITE_TAC[letlemma] THEN
ABBREV_TAC
`ures = INST_CORE env'' tyin (VSUBST[Var x' ty,Var x ty] t)` THEN
ONCE_REWRITE_TAC[letlemma] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `sizeof t`) THEN
REWRITE_TAC[sizeof; ARITH_RULE `t < 2 + t`] THEN
DISCH_THEN(fun th ->
MP_TAC(SPECL [`VSUBST [Var x' ty,Var x ty] t`;
`env'':(term#term)list`; `tyin:(type#type)list`] th) THEN
MP_TAC(SPECL [`t:term`; `[]:(term#term)list`; `tyin:(type#type)list`]
th)) THEN
REWRITE_TAC[MEM; REV_ASSOCD] THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `t':term` MP_TAC) THEN STRIP_TAC THEN
UNDISCH_TAC `VARIANT (RESULT (INST_CORE [] tyin t)) x ty' = x'` THEN
ASM_REWRITE_TAC[RESULT] THEN DISCH_TAC THEN
MP_TAC(SPECL [`t':term`; `x:string`; `ty':type`] VARIANT) THEN
ASM_REWRITE_TAC[] THEN
GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV)
[NOT_EXISTS_THM; TAUT `~(a /\ b) <=> a ==> ~b`] THEN DISCH_TAC THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[MATCH_MP_TAC VSUBST_WELLTYPED THEN ASM_REWRITE_TAC[MEM; PAIR_EQ] THEN
ASM_MESON_TAC[has_type_RULES];
ALL_TAC] THEN
CONJ_TAC THENL
[MATCH_MP_TAC SIZEOF_VSUBST THEN
ASM_REWRITE_TAC[MEM; PAIR_EQ] THEN ASM_MESON_TAC[has_type_RULES];
ALL_TAC] THEN
EXPAND_TAC "env''" THEN REWRITE_TAC[MEM; PAIR_EQ] THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL
[DISCH_THEN(fun th -> DISJ1_TAC THEN MP_TAC th) THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:string` THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `vty:type` THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC MP_TAC) THEN
ASM_REWRITE_TAC[IS_RESULT; CLASH] THEN
ONCE_REWRITE_TAC[letlemma] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
[REWRITE_TAC[VFREE_IN_VSUBST] THEN EXPAND_TAC "env''" THEN
REWRITE_TAC[REV_ASSOCD] THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[term_INJ] THEN
DISCH_THEN(REPEAT_TCL CHOOSE_THEN MP_TAC) THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[VFREE_IN; term_INJ] THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [term_INJ]) THEN
DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN
EXPAND_TAC "env''" THEN REWRITE_TAC[REV_ASSOCD] THEN
ASM_CASES_TAC `vty:type = ty` THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o CONJUNCT1) THEN
REWRITE_TAC[VFREE_IN_VSUBST; NOT_EXISTS_THM; REV_ASSOCD] THEN
ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[VFREE_IN; term_INJ] THEN
MAP_EVERY X_GEN_TAC [`k:string`; `kty:type`] THEN
MP_TAC(SPECL [`t':term`; `x:string`; `ty':type`] VARIANT) THEN
ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `t'':term` STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[IS_RESULT; result_INJ; UNWIND_THM1; result_DISTINCT] THEN
REWRITE_TAC[RESULT] THEN
MATCH_MP_TAC(TAUT `b /\ (b ==> c /\ a /\ d) ==> a /\ b /\ c /\ d`) THEN
CONJ_TAC THENL
[ASM_REWRITE_TAC[typeof; TYPE_SUBST] THEN
MATCH_MP_TAC(last(CONJUNCTS has_type_RULES)) THEN
SUBGOAL_THEN `(VSUBST [Var x' ty,Var x ty] t) has_type (typeof t)`
(fun th -> ASM_MESON_TAC[th; WELLTYPED_LEMMA]) THEN
MATCH_MP_TAC VSUBST_HAS_TYPE THEN ASM_REWRITE_TAC[GSYM WELLTYPED] THEN
REWRITE_TAC[MEM; PAIR_EQ] THEN MESON_TAC[has_type_RULES];
ALL_TAC] THEN
DISCH_TAC THEN MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN
CONJ_TAC THENL
[ASM_REWRITE_TAC[VFREE_IN] THEN
MAP_EVERY X_GEN_TAC [`k:string`; `kty:type`] THEN
ASM_REWRITE_TAC[VFREE_IN_VSUBST; REV_ASSOCD] THEN
ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[VFREE_IN; term_INJ] THEN
SIMP_TAC[] THEN
REWRITE_TAC[TAUT `x /\ (if p then a /\ b else c /\ b) <=>
b /\ x /\ (if p then a else c)`] THEN
REWRITE_TAC[UNWIND_THM2] THEN
REWRITE_TAC[TAUT `x /\ (if p /\ q then a else b) <=>
p /\ q /\ a /\ x \/ b /\ ~(p /\ q) /\ x`] THEN
REWRITE_TAC[EXISTS_OR_THM; UNWIND_THM1; UNWIND_THM2] THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
DISCH_TAC THEN CONJ_TAC THENL
[MAP_EVERY X_GEN_TAC [`k:string`; `kty:type`] THEN
REWRITE_TAC[VFREE_IN] THEN STRIP_TAC THEN
UNDISCH_TAC `!x'' ty'.
VFREE_IN (Var x'' ty') (VSUBST [Var x' ty,Var x ty] t)
==> (REV_ASSOCD (Var x'' (TYPE_SUBST tyin ty')) env''
(Var x'' ty') = Var x'' ty')` THEN
DISCH_THEN(MP_TAC o SPECL [`k:string`; `kty:type`]) THEN
REWRITE_TAC[VFREE_IN_VSUBST; REV_ASSOCD] THEN
ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[VFREE_IN] THEN
REWRITE_TAC[VFREE_IN; term_INJ] THEN
SIMP_TAC[] THEN
REWRITE_TAC[TAUT `x /\ (if p then a /\ b else c /\ b) <=>
b /\ x /\ (if p then a else c)`] THEN
REWRITE_TAC[UNWIND_THM2] THEN
REWRITE_TAC[TAUT `x /\ (if p /\ q then a else b) <=>
p /\ q /\ a /\ x \/ b /\ ~(p /\ q) /\ x`] THEN
REWRITE_TAC[EXISTS_OR_THM; UNWIND_THM1; UNWIND_THM2] THEN
UNDISCH_TAC `~(Var x ty = Var k kty)` THEN
REWRITE_TAC[term_INJ] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
EXPAND_TAC "env''" THEN REWRITE_TAC[REV_ASSOCD] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[semantics] THEN
REWRITE_TAC[TYPE_SUBST; TYPESET_LEMMA] THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC ABSTRACT_EQ THEN
CONJ_TAC THENL [ASM_SIMP_TAC[TYPESET_INHABITED]; ALL_TAC] THEN
X_GEN_TAC `a:V` THEN REWRITE_TAC[] THEN DISCH_TAC THEN CONJ_TAC THENL
[MATCH_MP_TAC SEMANTICS_TYPESET THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD] THEN
ASM_MESON_TAC[welltyped; WELLTYPED];
ALL_TAC] THEN
MATCH_MP_TAC(TAUT `(b ==> a) /\ b ==> a /\ b`) THEN CONJ_TAC THENL
[DISCH_THEN(SUBST1_TAC o SYM) THEN
MATCH_MP_TAC SEMANTICS_TYPESET THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD] THEN
SUBGOAL_THEN `(VSUBST [Var x' ty,Var x ty] t) has_type (typeof t)`
(fun th -> ASM_MESON_TAC[th; WELLTYPED_LEMMA]) THEN
MATCH_MP_TAC VSUBST_HAS_TYPE THEN ASM_REWRITE_TAC[GSYM WELLTYPED] THEN
REWRITE_TAC[MEM; PAIR_EQ] THEN MESON_TAC[has_type_RULES];
ALL_TAC] THEN
W(fun (asl,w) -> FIRST_X_ASSUM(fun th ->
MP_TAC(PART_MATCH (lhand o rand) th (lhand w)))) THEN
ASM_SIMP_TAC[TERM_VALUATION_VALMOD] THEN DISCH_TAC THEN
REWRITE_TAC[GSYM(CONJUNCT1 TYPE_SUBST)] THEN
MP_TAC SEMANTICS_VSUBST THEN
REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP] THEN
DISCH_THEN(fun th ->
W(fun (asl,w) -> MP_TAC(PART_MATCH (lhand o rand) th (lhand w)))) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[MEM; PAIR_EQ] THEN CONJ_TAC THENL
[MESON_TAC[has_type_RULES]; ALL_TAC] THEN
CONJ_TAC THENL
[REWRITE_TAC[type_valuation] THEN ASM_SIMP_TAC[TYPESET_INHABITED];
ALL_TAC] THEN
REWRITE_TAC[term_valuation] THEN
MAP_EVERY X_GEN_TAC [`y:string`; `yty:type`] THEN
REWRITE_TAC[VALMOD] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[VALMOD; PAIR_EQ] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[TYPE_SUBST; TYPESET_LEMMA] THEN
ASM_MESON_TAC[term_valuation];
ALL_TAC] THEN
DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[GSYM(CONJUNCT1 TYPE_SUBST)] THEN
MATCH_MP_TAC TERM_VALUATION_VFREE_IN THEN CONJ_TAC THENL
[REWRITE_TAC[type_valuation] THEN ASM_SIMP_TAC[TYPESET_INHABITED];
ALL_TAC] THEN
REWRITE_TAC[ITLIST] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[DEST_VAR] THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL
[ASM_REWRITE_TAC[] THEN CONJ_TAC THEN
REWRITE_TAC[term_valuation; semantics] THEN
MAP_EVERY X_GEN_TAC [`k:string`; `kty:type`] THEN
REWRITE_TAC[VALMOD] THEN CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[TYPESET_LEMMA; TYPE_SUBST] THEN
SIMP_TAC[PAIR_EQ] THEN ASM_REWRITE_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[term_valuation];
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`k:string`; `kty:type`] THEN DISCH_TAC THEN
REWRITE_TAC[VALMOD; semantics] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
SIMP_TAC[PAIR_EQ] THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* So in particular, we get key properties of INST itself. *)
(* ------------------------------------------------------------------------- *)
let SEMANTICS_INST = prove
(`!tyin tm.
welltyped tm
==> (INST tyin tm) has_type (TYPE_SUBST tyin (typeof tm)) /\
(!u uty. VFREE_IN (Var u uty) (INST tyin tm) <=>
?oty. VFREE_IN (Var u oty) tm /\
(uty = TYPE_SUBST tyin oty)) /\
!sigma tau.
type_valuation tau /\ term_valuation tau sigma
==> (semantics sigma tau (INST tyin tm) =
semantics
(\(x,ty). sigma(x,TYPE_SUBST tyin ty))
(\s. typeset tau (TYPE_SUBST tyin (Tyvar s))) tm)`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
MP_TAC(SPECL [`sizeof tm`; `tm:term`; `[]:(term#term)list`;
`tyin:(type#type)list`] SEMANTICS_INST_CORE) THEN
ASM_REWRITE_TAC[MEM; INST_DEF; REV_ASSOCD] THEN MESON_TAC[RESULT]);;
(* ------------------------------------------------------------------------- *)
(* Hence soundness of the INST_TYPE rule. *)
(* ------------------------------------------------------------------------- *)
let INST_TYPE_correct = prove
(`!tyin asl p. asl |= p ==> MAP (INST tyin) asl |= INST tyin p`,
REWRITE_TAC[sequent] THEN REPEAT STRIP_TAC THENL
[UNDISCH_TAC `ALL (\a. a has_type Bool) (CONS p asl)` THEN
REWRITE_TAC[ALL; ALL_MAP] THEN MATCH_MP_TAC MONO_AND THEN
CONJ_TAC THENL
[ALL_TAC;
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] ALL_IMP) THEN
GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[o_THM]] THEN
ASM_MESON_TAC[SEMANTICS_INST; TYPE_SUBST; welltyped; WELLTYPED;
WELLTYPED_LEMMA];
ALL_TAC] THEN
SUBGOAL_THEN `welltyped p` ASSUME_TAC THENL
[ASM_MESON_TAC[welltyped; ALL]; ALL_TAC] THEN
ASM_SIMP_TAC[SEMANTICS_INST] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL
[REWRITE_TAC[type_valuation] THEN ASM_MESON_TAC[TYPESET_INHABITED];
ALL_TAC] THEN
CONJ_TAC THENL
[REWRITE_TAC[term_valuation] THEN
REWRITE_TAC[TYPE_SUBST; TYPESET_LEMMA] THEN
ASM_MESON_TAC[term_valuation];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ALL_MAP]) THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] ALL_IMP) THEN
X_GEN_TAC `a:term` THEN DISCH_TAC THEN
SUBGOAL_THEN `welltyped a` MP_TAC THENL
[ASM_MESON_TAC[ALL_MEM; MEM; welltyped]; ALL_TAC] THEN
ASM_SIMP_TAC[SEMANTICS_INST; o_THM]);;
(* ------------------------------------------------------------------------- *)
(* Soundness. *)
(* ------------------------------------------------------------------------- *)
let HOL_IS_SOUND = prove
(`!asl p. asl |- p ==> asl |= p`,
MATCH_MP_TAC proves_INDUCT THEN
REWRITE_TAC[REFL_correct; TRANS_correct; ABS_correct;
BETA_correct; ASSUME_correct; EQ_MP_correct; INST_TYPE_correct;
REWRITE_RULE[LET_DEF; LET_END_DEF] DEDUCT_ANTISYM_RULE_correct;
REWRITE_RULE[IMP_IMP] INST_correct] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC MK_COMB_correct THEN
ASM_MESON_TAC[WELLTYPED_CLAUSES; MK_COMB_correct]);;
(* ------------------------------------------------------------------------- *)
(* Consistency. *)
(* ------------------------------------------------------------------------- *)
let HOL_IS_CONSISTENT = prove
(`?p. p has_type Bool /\ ~([] |- p)`,
SUBGOAL_THEN `?p. p has_type Bool /\ ~([] |= p)`
(fun th -> MESON_TAC[th; HOL_IS_SOUND]) THEN
EXISTS_TAC `Var x Bool === Var (VARIANT (Var x Bool) x Bool) Bool` THEN
SIMP_TAC[EQUATION_HAS_TYPE_BOOL; WELLTYPED_CLAUSES; typeof;
sequent; ALL; SEMANTICS_EQUATION; has_type_RULES; semantics;
BOOLEAN_EQ_TRUE] THEN
MP_TAC(SPECL [`Var x Bool`; `x:string`; `Bool`] VARIANT) THEN
ABBREV_TAC `y = VARIANT (Var x Bool) x Bool` THEN
REWRITE_TAC[VFREE_IN; term_INJ; NOT_FORALL_THM] THEN DISCH_TAC THEN
EXISTS_TAC `((x:string,Bool) |-> false) (((y,Bool) |-> true)
(\(x,ty). @a. a <: typeset (\x. boolset) ty))` THEN
EXISTS_TAC `\x:string. boolset` THEN
ASM_REWRITE_TAC[type_valuation; VALMOD; PAIR_EQ; TRUE_NE_FALSE] THEN
CONJ_TAC THENL [MESON_TAC[IN_BOOL]; ALL_TAC] THEN
REWRITE_TAC[term_valuation] THEN REPEAT GEN_TAC THEN
REWRITE_TAC[VALMOD; PAIR_EQ] THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[typeset; IN_BOOL]) THEN
CONV_TAC SELECT_CONV THEN MATCH_MP_TAC TYPESET_INHABITED THEN
REWRITE_TAC[type_valuation] THEN MESON_TAC[IN_BOOL]);;
|