Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 48,266 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 |
(* ========================================================================= *)
(* Parsing and printing for bitmatch expressions. *)
(* *)
(* This supports term constructions like *)
(* *)
(* bitmatch (word 17) with *)
(* | [hi3:3; 0b00:2; lo3:3] -> foo hi3 lo3 *)
(* | [hi3:3; 0b1:1; lo4:4] -> bar hi3 lo4 *)
(* | [0b1010101:7; b] -> baz b *)
(* *)
(* which decomposes the number 17 = 0b00010001 into the 3 high bits *)
(* (hi3 = 0), a 1 bit in the middle, and the 4 low bits (lo4 = 1) and calls *)
(* `bar`. The patterns are written with the low bits on the right. *)
(* A pattern variable without a digit matches one bit as a bool. *)
(* *)
(* (c) Copyright, Mario Carneiro 2020 *)
(* ========================================================================= *)
needs "Library/words.ml";;
prioritize_num();;
(* Like CHOOSE, but the variable is the same as the one in the existential *)
let TRIV_CHOOSE =
let P = `P:A->bool` and Q = `Q:bool` and an = `(/\)` in
let pth = (* `(\x:A. Q /\ P x) = P, (?) P |- Q` *)
let th1 = AP_THM (ASSUME `(\x:A. Q /\ P x) = P`) `x:A` in
let th1 = TRANS (SYM th1) (BETA `(\x:A. Q /\ P x) x`) in
let th1 = CONJUNCT1 (EQ_MP th1 (ASSUME `(P:A->bool) x`)) in
let th1 = GEN `x:A` (DISCH `(P:A->bool) x` th1) in
let th2 = CONV_RULE (RAND_CONV BETA_CONV) (AP_THM EXISTS_DEF P) in
MP (SPEC `Q:bool` (EQ_MP th2 (ASSUME `(?) (P:A->bool)`))) th1 in
fun th1 th2 ->
try let P' = rand (concl th1) in
let v,bod = dest_abs P' in
let Q' = concl th2 in
let anQ = mk_comb (an, Q') in
let th3 = DEDUCT_ANTISYM_RULE (CONJ th2 (ASSUME bod))
(CONJUNCT2 (ASSUME (mk_comb (anQ, bod)))) in
let th4 = AP_TERM anQ (BETA (mk_comb (P', v))) in
let th5 = PINST [snd(dest_var v),aty] [P',P; Q',Q] pth in
PROVE_HYP th1 (PROVE_HYP (ABS v (TRANS th4 th3)) th5)
with Failure _ -> failwith "TRIV_CHOOSE";;
let (WITH_GOAL:(term->tactic)->tactic) = fun tac (_,w as st) -> tac w st;;
let _BITMATCH = new_definition `_BITMATCH (s:N word) c = _MATCH (val s) c`;;
let _ELSEPATTERN = new_definition `_ELSEPATTERN (e:B) (a:A) = \x. e = x`;;
let bitpat =
let th = prove
(`?x:num#(num->bool). !a. SND x a ==> a < 2 EXP FST x`,
EXISTS_TAC `0, \i:num. F` THEN REWRITE_TAC[SND]) in
new_type_definition "bitpat" ("mk_bitpat","dest_bitpat") th;;
let pat_size = new_definition `pat_size p = FST (dest_bitpat p)`;;
let pat_set = new_definition `pat_set p = SND (dest_bitpat p)`;;
let pat_set_lt = prove (`!p s. pat_set p s ==> s < 2 EXP pat_size p`,
REWRITE_TAC [pat_set; pat_size; bitpat]);;
let pat_thm = prove (`p = mk_bitpat (n, f) /\ (!a. f a ==> a < 2 EXP n) ==>
pat_size p = n /\ !s. pat_set p s = f s`,
REPEAT DISCH_TAC THEN
SUBGOAL_THEN `dest_bitpat (mk_bitpat (n,f)) = (n,f)`
(fun th -> ASM_REWRITE_TAC [pat_size; pat_set; th]) THEN
ASM_REWRITE_TAC [GSYM (CONJUNCT2 bitpat)] THEN
FIRST_X_ASSUM (ASSUME_TAC o MATCH_MP pat_set_lt));;
let NILPAT = new_definition `NILPAT = mk_bitpat(0,\s:num. s = 0)`;;
let NILPAT_pat_size,NILPAT_pat_set = (CONJ_PAIR o prove)
(`pat_size NILPAT = 0 /\ !s. pat_set NILPAT s <=> s = 0`,
MATCH_MP_TAC pat_thm THEN REWRITE_TAC [NILPAT] THEN
STRIP_TAC THEN DISCH_THEN SUBST1_TAC THEN ARITH_TAC);;
let CONSPAT = new_definition `CONSPAT p (a:N word) =
let n = dimindex(:N) in
mk_bitpat(n + pat_size p,
\s:num. (?t. pat_set p t /\ s = val a + 2 EXP n * t))`;;
let CONSPAT_pat_size,CONSPAT_pat_set = (CONJ_PAIR o prove)
(`pat_size (CONSPAT p (a:N word)) = dimindex(:N) + pat_size p /\
!s. pat_set (CONSPAT p (a:N word)) s <=>
?t. pat_set p t /\ s = val a + 2 EXP dimindex(:N) * t`,
MATCH_MP_TAC pat_thm THEN
REWRITE_TAC [CONV_RULE (ONCE_DEPTH_CONV let_CONV) CONSPAT] THEN
REPEAT STRIP_TAC THEN POP_ASSUM SUBST1_TAC THEN
MATCH_MP_TAC LTE_TRANS THEN EXISTS_TAC `2 EXP dimindex (:N) * SUC t` THEN
POP_ASSUM (fun th -> REWRITE_TAC [EXP_ADD; LE_MULT_LCANCEL; LE_SUC_LT;
MATCH_MP pat_set_lt th]) THEN
REWRITE_TAC [MULT_SUC; LT_ADD_RCANCEL; VAL_BOUND]);;
reserve_words["BITPAT"; "bitmatch"];;
let word1 = new_definition `word1 p = (word (bitval p)):1 word`;;
let BIT0_WORD1 = prove (`bit 0 (word1 b) = b`,
REWRITE_TAC [word1; BIT_LSB; bitval; VAL_WORD] THEN DIMINDEX_TAC THEN
BOOL_CASES_TAC `b:bool` THEN REWRITE_TAC [] THEN ARITH_TAC);;
let VAL_WORD1 = prove (`val (word1 b) = bitval b`,
REWRITE_TAC [word1; VAL_WORD_BITVAL]);;
let WORD1_ODD = prove (`word1 (ODD n) = word n`,
CONV_TAC SYM_CONV THEN REWRITE_TAC [word1; WORD_EQ; DIMINDEX_CLAUSES] THEN
ASM_CASES_TAC `ODD n` THEN
ASM_REWRITE_TAC[ODD_MOD; BITVAL_CLAUSES; EXP_1; CONG] THEN
CONV_TAC NUM_REDUCE_CONV THEN
ASM_REWRITE_TAC[GSYM EVEN_MOD; GSYM ODD_MOD; GSYM NOT_ODD]);;
let preparse_bitpat,preparse_bitmatch =
let bitmatch_ptm = Varp("_BITMATCH",dpty)
and NILPAT_ptm = Varp("NILPAT",dpty)
and CONSPAT_ptm = Varp("CONSPAT",dpty)
and word_ptm = Varp("word",dpty)
and word1_ptm = Varp("word1",dpty)
and arb_ptm = Varp("ARB",dpty)
and ung_ptm = Varp("_UNGUARDED_PATTERN",dpty)
and seqp_ptm = Varp("_SEQPATTERN",dpty)
and else_ptm = Varp("_ELSEPATTERN",dpty) in
let rec pfrees ptm = match ptm with
| Varp(v,pty) ->
if v = "" && pty = dpty then []
else if can get_const_type v || can num_of_string v
|| exists (fun (w,_) -> v = w) (!the_interface) then []
else [ptm]
| Constp(_,_) -> []
| Combp(p1,p2) -> union (pfrees p1) (pfrees p2)
| Absp(p1,p2) -> subtract (pfrees p2) (pfrees p1)
| Typing(p,_) -> pfrees p in
let pgenvar =
let gcounter = ref 0 in
fun () -> let count = !gcounter in
(gcounter := count + 1;
Varp("BM%PVAR%"^(string_of_int count),dpty)) in
let pmk_exists v ptm = Combp(Varp("?",dpty),Absp(v,ptm)) in
let pmk_bitmatch (((_,e),_),cs) =
let x = pgenvar() and y = pgenvar() in
let rec pmk_clauses cs = match pmk_clauses_opt cs with
| Some t -> t
| None -> Combp(else_ptm, arb_ptm)
and pmk_clauses_opt cs = match cs with
| [] -> None
| (None,res)::cs -> Some (Combp(else_ptm, res))
| (Some pat,res)::cs ->
let tx = Combp(Combp(Varp("pat_set",dpty),pat),x)
and ty = Combp(Combp(Varp("=",dpty),res),y) in
let t = itlist pmk_exists (pfrees pat) (Combp(Combp(ung_ptm,tx),ty)) in
Some(Combp(Combp(seqp_ptm, Absp(x, Absp(y, t))), pmk_clauses cs)) in
Combp(Combp(bitmatch_ptm,e), pmk_clauses cs) in
let rec to_bitpat p e = match e with
| Varp("NIL",_) -> p
| Combp(Combp(Varp("CONS",_),a),e) ->
let e2 = match a with
| Typing(a,i) ->
let a = match a with
| Varp(s,_) when can num_of_string s -> Combp(word_ptm,a)
| _ -> a in
Typing(a,Ptycon("word",[i]))
| a -> Combp(word1_ptm,a) in
to_bitpat (Combp(Combp(CONSPAT_ptm,p),e2)) e
| _ -> raise Noparse in
let bitpat = parse_preterm >> to_bitpat NILPAT_ptm in
let pattern = (a (Ident "_") >> (fun _ -> None))
||| (bitpat >> fun x -> Some(x)) in
let clause = pattern ++ (a (Resword "->") ++ parse_preterm >> snd) in
let clauses =
possibly (a (Resword "|")) ++
listof clause (a (Resword "|")) "pattern-match clause" >> snd in
(a (Resword "BITPAT") ++ bitpat >> snd),
(a (Resword "bitmatch") ++ parse_preterm ++ a (Resword "with") ++ clauses
>> pmk_bitmatch);;
install_parser("bitpat",preparse_bitpat);;
install_parser("bitmatch",preparse_bitmatch);;
let pp_print_bitpat,pp_print_bitmatch =
let rec dest_pat_rev tm =
match tm with
| Comb(Comb(Const("CONSPAT",_),p),a) -> a::dest_pat_rev p
| Const("NILPAT",_) -> []
| _ -> failwith "dest_pat" in
let dest_pat = rev o dest_pat_rev in
let dest_clause tm =
match snd(strip_exists(body(body tm))) with
| Comb(Comb(Const("_UNGUARDED_PATTERN",_),lhs),
Comb(Comb(Const("=",_),res),_)) ->
(match lhs with
| Comb(Comb(Const("pat_set",_),pat),_) -> dest_pat pat, res
| _ -> failwith "dest_clause")
| _ -> failwith "dest_clause" in
let rec dest_clauses tm =
match tm with
| Comb(Comb(Const("_SEQPATTERN",_),c),cs) ->
let c = dest_clause c and l,r = dest_clauses cs in c::l,r
| Comb(Const("_ELSEPATTERN",_),res) -> [],res
| _ -> failwith "dest_clauses" in
let f fmt =
let unword i = function
| Comb(Const("word",_),a) when is_numeral a ->
if dest_numeral a < power_num (Int 2) (dest_finty i)
then a
else failwith "numeral out of range"
| a -> a in
let print_pat = function
| Comb(Const("word1",_),a) -> pp_print_term fmt a
| a -> match type_of a with
| Tyapp("word",[i]) ->
(pp_print_term fmt (unword i a);
pp_print_string fmt ":";
pp_print_type fmt i)
| _ -> failwith "print_pat" in
let print_opat = function
| None -> pp_print_string fmt "_"
| Some [] -> pp_print_string fmt "[]"
| Some (x::xs) ->
(pp_print_string fmt "[";
print_pat x;
List.iter (fun x ->
pp_print_string fmt "; ";
print_pat x) xs;
pp_print_string fmt "]") in
let print_clause p r =
(print_opat p;
pp_print_string fmt " -> ";
pp_print_term fmt r) in
let rec print_clauses cls r = match cls,r with
| [p,res],Const("ARB",_) -> print_clause (Some p) res
| (p,res)::cs,_ -> (
print_clause (Some p) res;
pp_print_break fmt 1 0;
pp_print_string fmt "| ";
print_clauses cs r)
| [],r -> print_clause None r in
let pp_print_bitpat tm =
let pat = dest_pat tm in
(pp_open_hvbox fmt 0;
pp_print_string fmt "(BITPAT";
pp_print_space fmt ();
print_opat (Some pat);
pp_print_string fmt ")";
pp_close_box fmt ()) in
let pp_print_bitmatch = function
| Comb(Comb(Const("_BITMATCH",_),e),c) ->
let cls,r = dest_clauses c in
(pp_open_hvbox fmt 0;
pp_print_string fmt "(bitmatch ";
pp_print_term fmt e;
pp_print_string fmt " with";
pp_print_break fmt 1 2;
print_clauses cls r;
pp_close_box fmt ();
pp_print_string fmt ")")
| _ -> failwith "print_bitmatch" in
pp_print_bitpat,pp_print_bitmatch in
fst o f, snd o f;;
let print_bitpat = pp_print_bitpat std_formatter
and string_of_bitpat = print_to_string pp_print_bitpat;;
let print_bitmatch = pp_print_bitmatch std_formatter
and string_of_bitmatch = print_to_string pp_print_bitmatch;;
install_user_printer("bitpat",pp_print_bitpat);;
install_user_printer("bitmatch",pp_print_bitmatch);;
(* ------------------------------------------------------------------------- *)
(* Some tactics for dealing with bitmatch *)
(* ------------------------------------------------------------------------- *)
let dest_word_ty = function
| Tyapp("word",[i]) -> i
| _ -> failwith "dest_word_ty";;
let SPLIT_IF =
let th = (UNDISCH_ALL o TAUT)
`(p ==> a:A = t1) ==> (~p ==> a = t2) ==> a = if p then t1 else t2` in
let [Var(_,A) as a; p; t1; t2] = frees (concl th) and tnot = `~` in
fun p' th1 th2 ->
let a',t1' = dest_eq (concl th1) and
t2' = snd (dest_eq (concl th2)) in
(PROVE_HYP (DISCH p' th1) o PROVE_HYP (DISCH (mk_comb(tnot, p')) th2))
(PINST [type_of a',A] [p',p; a',a; t1',t1; t2',t2] th);;
let MATCH_SEQPATTERN = prove
(`_MATCH (e:A) (_SEQPATTERN c cs) =
if ?y:B. c e y then _MATCH e c else _MATCH e cs`,
COND_CASES_TAC THEN ASM_REWRITE_TAC [_MATCH; _SEQPATTERN]);;
let BITMATCH_SEQPATTERN = prove
(`_BITMATCH (e:N word) (_SEQPATTERN c cs) =
if ?y:B. c (val e) y then _BITMATCH e c else _BITMATCH e cs`,
REWRITE_TAC [_BITMATCH; MATCH_SEQPATTERN]);;
let BITMATCH_SEQPATTERN2 = prove
(`_BITMATCH (e:N word) (_SEQPATTERN c cs) =
if ?y:B. c (val e) y then
_BITMATCH e (_SEQPATTERN c (_ELSEPATTERN ARB))
else _BITMATCH e cs`,
REWRITE_TAC [BITMATCH_SEQPATTERN] THEN COND_CASES_TAC THEN REWRITE_TAC []);;
let MATCH_ELSEPATTERN = prove
(`_MATCH (e:B) (_ELSEPATTERN (a:A)) = a`,
REWRITE_TAC [_MATCH; _ELSEPATTERN] THEN METIS_TAC[]);;
let BITMATCH_ELSEPATTERN = prove
(`_BITMATCH (e:N word) (_ELSEPATTERN (a:A)) = a`,
REWRITE_TAC [_BITMATCH; MATCH_ELSEPATTERN]);;
let pat_extract = new_definition `pat_extract p i b <=>
(!n. pat_set p n ==> numbit i n = b)`;;
let PAT_EXTRACT_THM =
let N,n,m,i,p,b = `:N`,`n:num`,`m:num`,`i:num`,`p:bitpat`,`b:bool`
and a,lt,pl,nb,di = `a:N word`,`(<)`,`(+)`,`numbit`,`dimindex(:N)` in
let word1_eq = prove
(`pat_extract (CONSPAT p (word1 b)) 0 b`,
REWRITE_TAC [pat_extract; numbit; CONSPAT_pat_set; word1; VAL_WORD] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM SUBST1_TAC THEN
DIMINDEX_TAC THEN NUM_REDUCE_TAC THEN
REWRITE_TAC [DIV_1; EVEN_ADD; GSYM NOT_EVEN; EVEN_DOUBLE] THEN
BOOL_CASES_TAC `b:bool` THEN REWRITE_TAC [bitval] THEN ARITH_TAC) in
let word_lt = (UNDISCH_ALL o prove)
(`dimindex(:N) = n ==> numbit i m = b ==> (i < n <=> T) ==>
pat_extract (CONSPAT p (word m:N word)) i b`,
REWRITE_TAC[] THEN REPEAT (DISCH_THEN (SUBST1_TAC o SYM)) THEN
REWRITE_TAC [pat_extract; CONSPAT_pat_set; numbit] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM SUBST1_TAC THEN
REWRITE_TAC [VAL_WORD; ODD_MOD; DIV_MOD] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
ONCE_REWRITE_TAC [MULT_SYM] THEN
REWRITE_TAC [GSYM (CONJUNCT2 EXP)] THEN
ONCE_REWRITE_TAC [GSYM MOD_ADD_MOD] THEN
FIRST_X_ASSUM (SUBST1_TAC o GSYM o
MATCH_MP (ARITH_RULE `m <= n ==> (m + (n - m) = n:num)`) o
REWRITE_RULE [GSYM LE_SUC_LT]) THEN
REWRITE_TAC [EXP_ADD; MOD_MOD] THEN ONCE_REWRITE_TAC [MULT_AC] THEN
REWRITE_TAC [MOD_MULT; ADD_0; MOD_MOD_REFL]) in
let word_skip = (UNDISCH_ALL o prove)
(`dimindex(:N) = n ==> m + n = i ==> pat_extract p m b ==>
pat_extract (CONSPAT p (a:N word)) i b`,
REPEAT (DISCH_THEN (SUBST1_TAC o SYM)) THEN
REWRITE_TAC [pat_extract; CONSPAT_pat_set; numbit] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM SUBST1_TAC THEN
POP_ASSUM (fun th -> POP_ASSUM (fun f ->
REWRITE_TAC [GSYM (MATCH_MP f th)])) THEN
REWRITE_TAC [SPECL [m;`dimindex(:N)`] ADD_SYM] THEN
REWRITE_TAC [EXP_ADD; GSYM DIV_DIV] THEN
AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
IMP_REWRITE_TAC [DIV_MULT_ADD; DIV_LT; ADD; VAL_BOUND; EXP_2_NE_0]) in
let rec go = function
| Comb(Comb(Const("CONSPAT",_),p'),a'),i' ->
let i'' = dest_numeral i' in
let aty = type_of a' in
let Tyapp(_,[N']) = aty in
let n'' = dest_finty N' in
if i'' < n'' then
match a' with
| Comb(Const("word1",_),b') -> INST [p',p; b',b] word1_eq
| Comb(Const("word",_),m') when is_numeral m' ->
let th = INST_TYPE [N',N] word_lt in
let th1 = DIMINDEX_CONV (inst [N',N] di) in
let n' = rhs (concl th1) in
let th2 = NUMBIT_CONV (mk_comb (mk_comb (nb,i'),m')) in
let b' = rhs (concl th2) in
let th3 = NUM_LT_CONV (mk_comb (mk_comb (lt,i'),n')) in
(PROVE_HYP th3 o PROVE_HYP th2 o PROVE_HYP th1)
(INST [i',i; m',m; n',n; p',p; b',b] th)
| _ -> failwith "PAT_EXTRACT_THM: not a constant at this position"
else
let th = INST_TYPE [N',N] word_skip in
let th1 = DIMINDEX_CONV (inst [N',N] di) in
let n' = rhs (concl th1) in
let m' = mk_numeral (sub_num i'' n'') in
let th2 = NUM_ADD_CONV (mk_comb (mk_comb (pl,m'),n')) in
let th3 = go (p', m') in
let b' = rand (concl th3) in
(PROVE_HYP th3 o PROVE_HYP th2 o PROVE_HYP th1)
(INST [i',i; m',m; n',n; p',p; b',b; a',mk_var("a",aty)] th)
| _ -> failwith "PAT_EXTRACT_THM: out of range"
in go;;
(* (pat_to_bit false `i` `pat_set p (val e)`) proves
`bit i e |- ~pat_set p (val e)` or
`~bit i e |- ~pat_set p (val e)`
(pat_to_bit true `i` `pat_set p (val e)`) proves
`pat_set p (val e) |- ~bit i e` or
`pat_set p (val e) |- bit i e` *)
let pat_to_bit =
(* thT := ~bit i e, pat_extract p i T |- ~pat_set p (val e)
thF := bit i e, pat_extract p i F |- ~pat_set p (val e) *)
let thT_pos,thF_pos =
(* TODO: I got frustrated with tactics so this is just a direct proof. *)
let a = PURE_REWRITE_RULE [pat_extract] (ASSUME `pat_extract p i b`) in
let a = UNDISCH (SPEC `val (e:N word)` a) in
let th = TRANS (SYM NUMBIT_VAL) a in
EQT_ELIM (INST [`T`,`b:bool`] th), EQF_ELIM (INST [`F`,`b:bool`] th) in
let thT_neg,thF_neg =
let f x y = NOT_INTRO (DISCH `pat_set p (val (e:N word))`
(MP (NOT_ELIM x) y)) in
f (ASSUME `~bit i (e:N word)`) thT_pos,
f thF_pos (ASSUME `bit i (e:N word)`) in
let N,i,e,p = `:N`,`i:num`,`e:N word`,`p:bitpat` in
fun pos i' -> function
| Comb(Comb(Const("pat_set",_),p'),
Comb(Const("val",Tyapp(_,[Tyapp(_, [N']); _])),e')) ->
let thp = PAT_EXTRACT_THM (p', i') in
let th = match rand (concl thp) with
| Const("T",_) -> if pos then thT_pos else thT_neg
| Const("F",_) -> if pos then thF_pos else thF_neg
| _ -> failwith "pat_to_bit" in
PROVE_HYP thp (PINST [N',N] [i',i; e',e; p',p] th)
| _ -> failwith "pat_to_bit";;
let bm_analyze_pat sz =
let A = Array.make sz None in
let rec go i = function
| Comb(Comb(Const("CONSPAT",_),p),a) ->
let n = Num.int_of_num (dest_finty (dest_word_ty (type_of a))) in
if i + n > sz then
raise (Invalid_argument "incorrect bit length") else
let () = match a with
| Comb(Const("word1",_),a) ->
A.(i) <- (match a with
| Const("T",_) -> Some true
| Const("F",_) -> Some false
| Var(_,_) -> None
| _ -> failwith "bm_analyze_pat")
| Comb(Const("word",_),Comb(Const("NUMERAL",_),a)) ->
let rec analyze_num n a = match a with
| Comb(Const("BIT0",_),a) ->
(A.(n) <- Some false; analyze_num (n+1) a)
| Comb(Const("BIT1",_),a) ->
(A.(n) <- Some true; analyze_num (n+1) a)
| Const("_0",_) -> Array.fill A n (sz - n) (Some false)
| _ -> failwith "bm_analyze_pat" in
analyze_num i a
| Var(_,_) -> Array.fill A i n None
| _ -> failwith "bm_analyze_pat" in
go (i + n) p
| Const("NILPAT",_) ->
if i = sz then () else
raise (Invalid_argument "incorrect bit length")
| Abs(_,c) -> go i c
| Comb(Const("?",_),c) -> go i c
| Comb(Comb(Const("_UNGUARDED_PATTERN",_),c),_) -> go i c
| Comb(Comb(Const("pat_set",_),c),_) -> go i c
| _ -> failwith "bm_analyze_pat" in
fun pat ->
try go 0 pat; A
with Invalid_argument _ -> failwith (sprintf
"bm_analyze_pat: pattern %s has incorrect bit length"
(string_of_term pat));;
let rec bm_analyze_clauses sz = function
| Comb(Comb(Const("_SEQPATTERN",_),c),cs) ->
let pat = (lhand o lhand o snd o strip_exists o body o body) c in
bm_analyze_pat sz pat :: bm_analyze_clauses sz cs
| _ -> [];;
(* (bm_skip_clause f `_BITMATCH e (_SEQPATTERN r rs)`) returns
`A |- _BITMATCH e (_SEQPATTERN r rs) = _BITMATCH e rs` if
(f `pat_set p (val e)`) proves `A |- ~pat_set p (val e)`
(probably via pat_to_bit false) *)
let bm_skip_clause =
let th = (UNDISCH o prove)
(`(?) (r (val e)) = F ==>
(_BITMATCH e (_SEQPATTERN r rs):B) = _BITMATCH (e:N word) rs`,
REPEAT DISCH_TAC THEN REWRITE_TAC [BITMATCH_SEQPATTERN] THEN
CONV_TAC (ONCE_DEPTH_CONV ETA_CONV) THEN ASM_REWRITE_TAC []) in
let N,B,e,r,rs =
`:N`,`:B`,`e:N word`,`r:num->B->bool`,`rs:num->B->bool` in
(* (strip_ex `x` `A |- P[x] = F`) proves `A |- (?x. P[x]) = F` *)
let strip_ex =
let th1 = (UNDISCH o prove)
(`(P = \x:A. F) ==> (?) P = F`, DISCH_TAC THEN ASM_REWRITE_TAC[]) in
let A = `:A` and P = `P:A->bool` in
fun x th ->
let th' = ABS x th in
PROVE_HYP th' (PINST [type_of x,A] [lhs (concl th'),P] th1) in
(* (skip_guard `A` `B`) proves `~A |- _UNGUARDED_PATTERN A B = F` *)
let skip_guard =
let th = (UNDISCH o prove)
(`~A ==> _UNGUARDED_PATTERN A B = F`,
DISCH_TAC THEN ASM_REWRITE_TAC [_UNGUARDED_PATTERN]) in
let A = `A:bool` and B = `B:bool` in
fun A' B' -> INST [A',A; B',B] th in
fun f -> function
| Comb(Comb(Const("_BITMATCH",ty),e'),
Comb(Comb(Const("_SEQPATTERN",_),c),rs')) ->
let Tyapp(_, [N']),Tyapp(_, [_; B']) = dest_fun_ty ty in
let th = PINST [N',N;B',B] [e',e; c,r; rs',rs] th in
let ex,h = dest_comb (lhs (hd (hyp th))) in
let th2 = BETA_CONV h in
let th3 = AP_TERM ex th2 in
let rec skip_exs = function
| Comb(Const("?",_),Abs(y,c)) -> strip_ex y (skip_exs c)
| Comb(Comb(Const("_UNGUARDED_PATTERN",_),p),res) ->
PROVE_HYP (f p) (skip_guard p res)
| _ -> failwith "skip_exs" in
PROVE_HYP (TRANS th3 (skip_exs (rhs (concl th3)))) th
| _ -> failwith "bm_skip_clause";;
type 'a discrim_tree =
Leaf_dt of 'a
| Split_dt of int * term * 'a discrim_tree * 'a discrim_tree;;
let rec map_dt f = function
| Leaf_dt a -> Leaf_dt (f a)
| Split_dt(i, bit, tr1, tr2) ->
Split_dt(i, bit, map_dt f tr1, map_dt f tr2);;
let bm_build_tree' =
let bit_tm =
let N = `:N` in
fun i e ->
let Tyapp("word",[N']) = type_of e in
let bit = mk_const("bit",[N',N]) in
mk_comb(mk_comb(bit,i),e) in
(* (seqp_rand `r` `A |- _BITMATCH e rs1 = _BITMATCH e rs2`) proves
`A |- _BITMATCH e (_SEQPATTERN r rs1) =
_BITMATCH e (_SEQPATTERN r rs2)` *)
let seqp_rand =
let th = (UNDISCH o prove)
(`_BITMATCH (e:N word) (rs1:num->B->bool) = _BITMATCH e rs2 ==>
_BITMATCH e (_SEQPATTERN r rs1) = _BITMATCH e (_SEQPATTERN r rs2)`,
DISCH_TAC THEN ASM_REWRITE_TAC [BITMATCH_SEQPATTERN]) in
let [rs1; e; r; rs2] = frees (concl th) and N,B = `:N`,`:B` in
fun r' th' ->
match dest_eq (concl th') with
| Comb(Comb(Const(_,ty),e'),rs1'),Comb(_,rs2') ->
let Tyapp(_, [N']),Tyapp(_, [_; B']) = dest_fun_ty ty in
PROVE_HYP th' (PINST [N',N; B',B] [e',e; r',r; rs1',rs1; rs2',rs2] th)
| _ -> failwith "seqp_rand" in
let aMerge a b = match a,b with
| None,_ -> a
| _,None -> None
| Some(a0,a1),Some false -> Some(a0+1, a1)
| Some(a0,a1),Some true -> Some(a0, a1+1) in
fun sz cls -> function
| Comb(Comb(Const("_BITMATCH",_),e) as m, cs) as tm ->
let rec build eqth cls cs =
let analysis = Array.make sz (Some(0,0)) in
let _ = List.iter (Array.iteri
(fun i a -> analysis.(i) <- aMerge analysis.(i) a)) cls in
let r =
let r = ref None in
let f i a = match a with
| Some(n1,n2) when n1 != 0 && n2 != 0 ->
let v = abs (n1 - n2) in
(match !r with
| Some(v',_) when v' <= v -> ()
| _ -> r := Some(v, i))
| _ -> () in
(Array.iteri f analysis; !r) in
match r with
| None -> Leaf_dt eqth
| Some(_,i) ->
let ii = mk_numeral (Int i) in
let bit = bit_tm ii e in
let skip_th sc th =
let sm, rs' = dest_comb (lhs (concl th)) in
let tm = mk_comb (sm, mk_comb (sc, rs')) in
TRANS (bm_skip_clause (pat_to_bit false ii) tm) th in
let rec split_ths = function
| [], cs -> let th = REFL (mk_comb(m,cs)) in [],[],th,th
| cl::cls, Comb(Comb(Const("_SEQPATTERN",_),c) as sc,cs) ->
let cls1,cls2,th1,th2 = split_ths(cls, cs) in
if let Some(b) = cl.(i) in b then
cls1, cl::cls2, skip_th sc th1, seqp_rand c th2
else
cl::cls1, cls2, seqp_rand c th1, skip_th sc th2
| cl::cls, cs ->
let cls1,cls2,th1,th2 = split_ths(cls, cs) in
if let Some(b) = cl.(i) in b then
cls1, cl::cls2, th1, th2
else
cl::cls1, cls2, th1, th2
| _ -> failwith "split_ths" in
let cls1,cls2,th1,th2 = split_ths(cls, cs) in
let tr1 = build (TRANS eqth th1) cls1 (rand (rhs (concl th1)))
and tr2 = build (TRANS eqth th2) cls2 (rand (rhs (concl th2))) in
Split_dt(i, bit, tr1, tr2) in
build (REFL tm) cls cs
| _ -> failwith "bm_build_tree'";;
let bm_build_tree = function
| Comb(Comb(Const("_BITMATCH",_),e),cs) as tm ->
let sz = Num.int_of_num (dest_finty (dest_word_ty (type_of e))) in
let cls = bm_analyze_clauses sz cs in
cls, bm_build_tree' sz cls tm
| _ -> failwith "bm_build_tree";;
let BM_IF_CONV =
let rec of_tree = function
| Leaf_dt th -> th
| Split_dt(_, bit, tr1, tr2) -> SPLIT_IF bit (of_tree tr2) (of_tree tr1) in
of_tree o snd o bm_build_tree;;
let MATCH_EQ = prove
(`(r:A->B->bool) e = (\y. x = y) ==> _MATCH e r = x`,
REWRITE_TAC [_MATCH] THEN DISCH_THEN SUBST1_TAC THEN METIS_TAC[]);;
let bitpat_inverts = new_definition
`bitpat_inverts p f (x:A) <=> (!y. pat_set p y ==> f y = x)`;;
let bitpat_down = new_definition
`bitpat_down(:N) (f:num->A) (n:num) = f (n DIV 2 EXP dimindex(:N))`;;
let CONSPAT_down_inverts = prove
(`bitpat_inverts p f (x:A) ==>
bitpat_inverts (CONSPAT p (a:N word)) (bitpat_down(:N) f) x`,
REWRITE_TAC [bitpat_inverts; bitpat_down; CONSPAT_pat_set] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
FIRST_X_ASSUM SUBST1_TAC THEN
IMP_REWRITE_TAC [DIV_MULT_ADD; DIV_LT; VAL_BOUND; EXP_2_NE_0] THEN
ASM_REWRITE_TAC [ADD]);;
let CONSPAT_word_inverts = prove
(`bitpat_inverts (CONSPAT p (a:N word)) word a`,
REWRITE_TAC [bitpat_inverts; CONSPAT_pat_set] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM SUBST1_TAC THEN
REWRITE_TAC [WORD_VAL_GALOIS; MOD_MULT_ADD] THEN
REWRITE_TAC [GSYM WORD_VAL_GALOIS; WORD_VAL]);;
let bitpat_inverts_comp = MESON [bitpat_inverts; o_DEF]
`bitpat_inverts p f (x:A) ==> bitpat_inverts p (g o f) (g x:B)`;;
let CONSPAT_word1_inverts =
(* `bitpat_inverts (CONSPAT p (word1 b)) (bit 0 o word) b` *)
REWRITE_RULE [BIT0_WORD1]
(PART_MATCH rand (MATCH_MP bitpat_inverts_comp CONSPAT_word_inverts)
(lhs (concl BIT0_WORD1)));;
(* Given a pattern `p` and a target variable `z`,
(build_inverts `p`) produces an association list mapping `z`
to `|- bitpat_inverts p f z` for some term `f` not containing `z`. *)
let build_inverts =
let N,A,p,a,f,x,b =
`:N`,`:A`,`p:bitpat`,`a:N word`,`f:num->A`,`x:A`,`b:bool`
and th0 = ASSUME `bitpat_inverts p f (x:A)`
and th1 = UNDISCH CONSPAT_down_inverts in
let rec go = function
| Const("NILPAT",_),_ -> []
| (Comb(Comb(Const("CONSPAT",_),p'),a') as cp),thunk ->
let Tyapp(_, [N']) = type_of a' in
let next th =
let th' = PINST [N',N] [a',a] th1 in
let Comb(Comb(_,p'),f') = rator (concl th') in
PROVE_HYP th' (INST [p',p; f',f] th) in
let var v thv =
let th = thunk() in
let f' = rand (rator (concl thv)) in
let th' = PROVE_HYP thv (PINST [type_of v,A] [cp,p; f',f; v,x] th) in
(v,th') :: go (p', fun () -> next th) in
(match a' with
| Var(_,_) -> var a' (PINST [N',N] [p',p; a',a] CONSPAT_word_inverts)
| Comb(Const("word1",_), (Var(_,_) as b')) ->
var b' (INST [p',p; b',b] CONSPAT_word1_inverts)
| _ -> go (p', next o thunk))
| _ -> failwith "build_inverts" in
fun p' -> go (p', fun () -> th0);;
(* Given `bitmatch e with p -> res | ...` proves
`bit_set p (val e) |- (bitmatch e with p -> res | ...) = res`,
and given `bitmatch e with _ -> res` proves
`(bitmatch e with _ -> res) = res`. *)
let BM_FIRST_CASE_CONV =
let th1 = (UNDISCH o prove)
(`r (val (e:N word)) = (\y:A. x = y) ==> _BITMATCH e (_SEQPATTERN r s) = x`,
DISCH_TAC THEN
ASM_REWRITE_TAC [BITMATCH_SEQPATTERN; _BITMATCH; MESON[] `?y:A. x=y`] THEN
POP_ASSUM (ACCEPT_TAC o MATCH_MP MATCH_EQ)) in
let th2 = (UNDISCH o METIS[_UNGUARDED_PATTERN])
`A ==> (_UNGUARDED_PATTERN A B <=> B)` in
let th3 = EQ_MP (SYM th2) (ASSUME `B:bool`) in
let th4 = (UNDISCH_ALL o prove)
(`bitpat_inverts p f (x:A) ==> pat_set p e ==> f e = x`,
DISCH_THEN (MATCH_ACCEPT_TAC o REWRITE_RULE [bitpat_inverts])) in
let thg1,thg2 = (CONJ_PAIR o UNDISCH o METIS[_UNGUARDED_PATTERN])
`_UNGUARDED_PATTERN A B ==> A /\ B` in
let nty,eA,eB,ep,ef,ex = `:N`,`A:bool`,`B:bool`,`p:bitpat`,`f:num->A`,`x:A`
and ee,ea,er,es = `e:N word`,`a:A`,`r:num->A->bool`,`s:num->A->bool` in
function
| Comb(Comb(Const("_BITMATCH",_),e),
Comb(Comb(Const("_SEQPATTERN",_), (Abs(x,Abs(y,c')) as c)),cs)) as tm ->
let A' = type_of tm in
let N = dest_word_ty (type_of e) in
let val_e = mk_comb (mk_const("val", [N,nty]), e) in
let zs, Comb(Comb(Const("_UNGUARDED_PATTERN",_),
(Comb((Comb(_,p) as mp),_) as ps)),restm) = strip_exists c' in
let res = lhand restm in
let instAB = INST [ps,eA; restm,eB] in
let ps' = mk_comb (mp, val_e) in
let th' = if zs = [] then instAB th2 else
let inverts = build_inverts p in
let rec prove_ex c1 pr eqth = match c1 with
| Comb(Const("?",_),Abs(z,c')) ->
let inv = assoc z inverts in
let f = lhand (concl inv) in
let inv = PROVE_HYP inv (PINST [type_of z,aty]
[p,ep; f,ef; z,ex; x,`e:num`] th4) in
let abr = mk_abs (z, lhs (concl eqth)) in
let eqth1 = TRANS (AP_TERM abr (ASSUME (concl inv)))
(BETA (mk_comb (abr, z))) in
let pr',th = prove_ex c' (PROVE_HYP inv o pr) (TRANS eqth1 eqth) in
pr', TRANS (SYM eqth1) (TRIV_CHOOSE (ASSUME c1) th)
| _ -> pr, TRANS (PROVE_HYP (instAB thg1) (pr eqth)) (instAB thg2) in
let pr,thR = prove_ex c' I (REFL res) in
DEDUCT_ANTISYM_RULE (itlist SIMPLE_EXISTS zs (instAB th3)) (pr thR) in
let th' = INST [val_e,x] (TRANS (BETA (mk_comb (c, x))) (ABS y th')) in
PROVE_HYP th' (PINST [N,nty; A',aty]
[e,ee; res,ex; c,er; cs,es] th1)
| Comb(Comb(Const("_BITMATCH",_),e), Comb(Const("_ELSEPATTERN",_), a)) ->
let A' = type_of a in
let N = dest_word_ty (type_of e) in
PINST [N,nty; A',aty] [e,ee; a,ea] BITMATCH_ELSEPATTERN
| _ -> failwith "BM_FIRST_CASE_CONV";;
let bm_add_pos tr = function
| Comb(Comb(Const("_BITMATCH",_),e),cs) ->
let N = dest_word_ty (type_of e) in
let val_e = mk_comb (mk_const("val", [N,`:N`]), e) in
let rec build_cases stk mth = function
| Comb(Comb(Const("_SEQPATTERN",_), Abs(x,Abs(y,c'))),cs) ->
let ps' = mk_comb (rator (lhand (snd (strip_exists c'))), val_e) in
let th = itlist (fun n ->
try PROVE_HYP (pat_to_bit true n ps') with Failure _ -> I) stk mth in
TRANS th (BM_FIRST_CASE_CONV (rhs (concl th))) :: build_cases stk mth cs
| _ -> [] in
let rec build stk = function
| Leaf_dt mth ->
Leaf_dt (mth, build_cases stk mth (rand (rhs (concl mth))))
| Split_dt (i, bit, tr1, tr2) ->
let stk' = lhand bit :: stk in
Split_dt (i, bit, build stk' tr1, build stk' tr2) in
build [] tr
| _ -> failwith "bm_build_pos_tree";;
let bm_build_pos_tree tm =
let A, tr = bm_build_tree tm in A, bm_add_pos tr tm;;
let rec get_dt A = function
| Leaf_dt a -> [], a
| Split_dt(i, bit, tr1, tr2) ->
match A.(i) with
| None -> failwith ("get_dt splitting on " ^ string_of_int i)
| Some b ->
let stk,r = get_dt A (if b then tr2 else tr1) in
(b,bit)::stk, r;;
let BM_CASES tm =
let A, tr = bm_build_pos_tree tm in
map (fun cl -> hd (snd (snd (get_dt cl tr)))) A;;
(* (bitpat_matches `p` n) returns None if the pattern `p` would match
`word n`, and Some(i) where i is the smallest differing bit otherwise.
It throws if n >= 2^pat_size p. *)
let rec bitpat_matches p i = match p with
| Comb(Comb(Const("CONSPAT",_),p),a) ->
let N = dest_word_ty (type_of a) in
let n = Num.int_of_num (dest_finty N) in
let m = power_num (Int 2) (Int n) in
let i' = quo_num i m and a' = mod_num i m in
let r = match a with
| Comb(Const("word1",_),Const("T",_)) -> if a' = Int 1 then None else Some 0
| Comb(Const("word1",_),Const("F",_)) -> if a' = Int 0 then None else Some 0
| Comb(Const("word1",_),Var(_,_)) -> None
| Comb(Const("word",_),n) ->
let n' = dest_numeral n in
if a' = n' then None else
let rec f i r = if i land 1 != 0 then r else f (i lsr 1) (r+1) in
Some (f ((Num.int_of_num a') lxor (Num.int_of_num n')) 0)
| Var(_,_) -> None
| _ -> failwith "bitpat_matches" in
(match r with
| Some j -> Some j
| None ->
match bitpat_matches p i' with
| Some j -> Some (j + n)
| None -> None)
| Const("NILPAT",_) -> if i = Int 0 then None else
failwith "bitpat_matches: out of range"
| Abs(_,c) -> bitpat_matches c i
| Comb(Const("?",_),c) -> bitpat_matches c i
| Comb(Comb(Const("_UNGUARDED_PATTERN",_),c),_) -> bitpat_matches c i
| Comb(Comb(Const("pat_set",_),c),_) -> bitpat_matches c i
| _ -> failwith "bitpat_matches";;
(* (inst_bitpat_numeral `pat_set p (val e)` n) will produce an instantiation
theta for p and e such that e[theta] = word n, and a proof of
`|- (pat_set p (val e))[theta]`.
(inst_bitpat_numeral `pat_set p e` n) will produce an instantiation
theta for p and e such that e[theta] = n, and a proof of
`|- (pat_set p e)[theta]`. *)
let inst_bitpat_numeral =
let en,ep,ex,ei,ea = `n:num`,`p:bitpat`,`x:num`,`i:num`,`a:num`
and eN,T,F = `:N`,`T`,`F` in
let dim =
let dN = `dimindex(:N)` in
fun N -> DIMINDEX_CONV (inst [N,eN] dN) in
let w0 = prove (`pat_set NILPAT _0`,
REWRITE_TAC [NILPAT_pat_set; NUMERAL])
and wS,(w1T,w1F) =
let pth = prove
(`pat_set p x ==>
pat_set (CONSPAT p (word a:N word)) (num_shift_add a x (dimindex(:N)))`,
REWRITE_TAC [CONSPAT_pat_set; num_shift_add] THEN
DISCH_THEN (fun th ->
EXISTS_TAC ex THEN REWRITE_TAC [th; VAL_WORD; MULT_SYM])) in
(UNDISCH_ALL o prove)
(`dimindex(:N) = NUMERAL n ==> num_shift_add a x n = i ==> pat_set p x ==>
pat_set (CONSPAT p (word (NUMERAL a):N word)) i`,
REWRITE_TAC [NUMERAL] THEN
REPEAT (DISCH_THEN (SUBST1_TAC o SYM)) THEN ACCEPT_TAC pth),
(CONJ_PAIR o UNDISCH o prove)
(`pat_set p x ==>
pat_set (CONSPAT p (word1 T)) (BIT1 x) /\
pat_set (CONSPAT p (word1 F)) (BIT0 x)`,
REWRITE_TAC [word1; bitval] THEN
DISCH_THEN (fun th -> CONJ_TAC THEN ASSUME_TAC th) THENL [
POP_ASSUM (MP_TAC o MP (PINST [`:1`,eN] [`1`,ea] pth)) THEN
SUBGOAL_THEN
`num_shift_add 1 x (dimindex(:1)) = num_shift_add (BIT1 0) x (SUC 0)`
(fun th -> REWRITE_TAC [th; num_shift_add_SUC; num_shift_add_0]) THEN
CONV_TAC (ONCE_DEPTH_CONV DIMINDEX_CONV) THEN
REWRITE_TAC [BIT1_0; ONE];
POP_ASSUM (MP_TAC o MP (PINST [`:1`,eN] [`0`,ea] pth)) THEN
SUBGOAL_THEN
`num_shift_add 0 x (dimindex(:1)) = num_shift_add (BIT0 0) x (SUC 0)`
(fun th -> REWRITE_TAC [th; num_shift_add_SUC; num_shift_add_0]) THEN
CONV_TAC (ONCE_DEPTH_CONV DIMINDEX_CONV) THEN
REWRITE_TAC [BIT0_0; ONE]]) in
let w1F0 = REWRITE_RULE [ARITH_ZERO] (INST [`_0`,ex] w1F) in
let rec go i = function
| Comb(Comb(Const("CONSPAT",_),p),a) ->
let N = dest_word_ty (type_of a) in
let n = Num.int_of_num (dest_finty N) in
let m = power_num (Int 2) (Int n) in
let i' = quo_num i m and a' = mod_num i m in
let ls, th' = go i' p in
let p',x = dest_comb (concl th') in let p' = rand p' in
(match a with
| Comb(Const("word1",_),a) ->
let ls, b = match a with
| Const("T",_) -> ls,true
| Const("F",_) -> ls,false
| Var(_,_) -> let b = a' = Int 1 in ((if b then T else F),a)::ls, b
| _ -> failwith "inst_bitpat_numeral" in
ls, PROVE_HYP th' (
if b then INST [x,ex; p',ep] w1T
else if i = Int 0 then INST [p',ep] w1F0
else INST [x,ex; p',ep] w1F)
| _ ->
let thd = dim N in
let n' = rand (rhs (concl thd)) in
let ls, a = match a with
| Comb(Const("word",_),Comb(Const("NUMERAL",_),a)) -> ls, a
| Var(_,_) ->
let n = mk_numeral a' in
(mk_comb (mk_const ("word", [N,eN]), n), a) :: ls, rand n
| _ -> failwith "inst_bitpat_numeral" in
let thn = NUM_SHIFT_ADD_CORE a x n' in
let e = rhs (concl thn) in
ls, PROVE_HYP th' (PROVE_HYP thn (PROVE_HYP thd
(INST [n',en; a,ea; x,ex; e,ei; p',ep] (INST_TYPE [N,eN] wS)))))
| Const("NILPAT",_) -> [], w0
| _ -> failwith "inst_bitpat_numeral" in
let pth1 = SYM (SPEC en NUMERAL)
and pth2 = (UNDISCH_ALL o prove)
(`pat_size p = NUMERAL n ==> dimindex(:N) = NUMERAL n ==>
pat_set p x ==> pat_set p (val (word (NUMERAL x):N word))`,
DISCH_THEN (SUBST1_TAC o SYM) THEN REPEAT STRIP_TAC THEN
REWRITE_TAC [NUMERAL] THEN IMP_REWRITE_TAC [VAL_WORD_EQ] THEN
POP_ASSUM (ACCEPT_TAC o MATCH_MP pat_set_lt))
and conv =
let ps = `pat_size` in
REWRITE_CONV [CONSPAT_pat_size; NILPAT_pat_size] THENC
ONCE_DEPTH_CONV DIMINDEX_CONV THENC REDEPTH_CONV NUM_ADD_CONV o
mk_comb o (fun tm -> (ps, tm)) in
let check p i = match bitpat_matches p i with
| None -> go i p
| _ -> failwith "inst_bitpat_numeral: number does not match pattern" in
function
| Comb(Comb(Const("pat_set",_),p), Comb(Const("val",_), e)) ->
let N' = dest_word_ty (type_of e) in
let thd = dim N' in
let pth2 = (PROVE_HYP thd o PROVE_HYP (conv p) o
INST [p,ep; rand (rhs (concl thd)),en] o INST_TYPE [N',eN]) pth2 in
fun i ->
let ls, th = check p i in
let e' = rand (concl th) in
let th1 = PROVE_HYP th (INST ((e',ex)::ls) pth2) in
(match e with
| Var(_,_) -> (rand (rand (concl th1)),e)::ls, th1
| _ when aconv e (rand (rand (concl th1))) -> ls, th1
| _ -> failwith "inst_bitpat_numeral: pattern failed")
| Comb(Comb(Const("pat_set",_),p), e) ->
fun i ->
let ls, th = check p i in
let f, e' = dest_comb (concl th) in
let th1 = INST [e',en] pth1 in
let th2 = EQ_MP (AP_TERM f th1) th in
(match e with
| Var(_,_) -> (rhs (concl th1), e)::ls, th2
| _ when aconv e (rhs (concl th1)) -> ls, th2
| _ -> failwith "inst_bitpat_numeral: pattern failed")
| _ -> failwith "inst_bitpat_numeral";;
let BITMATCH_CONV =
fun tm -> match tm with
| Comb(Comb(Const("_BITMATCH",_),
Comb(Const("word",Tyapp(_,[_;Tyapp(_,[N])])),n)),_) when is_numeral n ->
let A, tr = bm_build_pos_tree tm in
let n = Num.int_of_num (dest_numeral n)
and sz = Num.int_of_num (dest_finty N) in
let a = Array.init sz (fun i -> Some (n land (1 lsl i) != 0)) in
(match snd (snd (get_dt a tr)) with
| th::_ ->
let ps = hd (hyp th) in
let ls, th' = inst_bitpat_numeral ps (Int n) in
PROVE_HYP th' (INST ls th)
| _ -> failwith "BITMATCH_CONV")
| _ -> failwith "BITMATCH_CONV";;
let BITMATCH_SIMP_CONV asl =
let pos,neg =
let rec go = function
| [] -> [],[]
| th::ths ->
let pos,neg = go ths in
match concl th with
| Comb(Const("~",_),c) when
(match snd (strip_exists c) with
| Comb(Comb(Const("pat_set",_),_),_) -> true
| _ -> false) -> pos,th::neg
| Comb(Comb(Const("pat_set",_),_),_) -> th::pos,neg
| _ -> pos,neg in
go asl in
let rec conv = function
| Comb(Comb(Const("_BITMATCH",_),_),Comb(Const("_ELSEPATTERN",_),_)) as tm ->
PART_MATCH lhs BITMATCH_ELSEPATTERN tm
| Comb(Comb(Const("_BITMATCH",_),
Comb(Const("word",_),n)),_) as tm when is_numeral n -> BITMATCH_CONV tm
| Comb(Comb(Const("_BITMATCH",_),e),
Comb(Comb(Const("_SEQPATTERN",_),c),cs)) as tm ->
let pat = mk_comb (rator (lhand (snd (strip_exists (body (body c))))),
mk_comb (mk_const ("val", [dest_word_ty (type_of e),`:N`]), e)) in
let vars = frees (lhand pat) in
let rec check_pos = function
| th::ths -> (try
let _,ls,_ = term_unify vars pat (concl th) in
PROVE_HYP th (INST ls (BM_FIRST_CASE_CONV tm))
with Failure _ -> check_pos ths)
| [] ->
let rec check_neg = function
| th::ths -> (try
let pat' = snd (strip_exists (rand (concl th))) in
let _,ls,_ = term_unify (frees (lhand pat')) pat' pat in
let ath = INST ls (SPEC_ALL
(PURE_REWRITE_RULE [NOT_EXISTS_THM] (ASSUME (concl th)))) in
let th' = PROVE_HYP th (bm_skip_clause (K ath) tm) in
TRANS th' (TRY_CONV conv (rhs (concl th')))
with Failure _ -> check_neg ths)
| [] ->
let sz = Num.int_of_num (dest_finty (dest_word_ty (type_of e))) in
let a = bm_analyze_pat sz pat in
let rec check_disj = function
| th::ths -> (try
let h = concl th in
let a' = bm_analyze_pat sz h in
let r = ref None in
Array.iteri (fun i x -> match x,a'.(i),!r with
| Some b, Some c, None when b != c -> r := Some i
| _ -> ()) a;
let i = match !r with
| Some i -> mk_numeral (Int i)
| _ -> fail () in
let th' = PROVE_HYP th (PROVE_HYP (pat_to_bit true i h)
(bm_skip_clause (pat_to_bit false i) tm)) in
TRANS th' (TRY_CONV conv (rhs (concl th')))
with Failure _ -> check_disj ths)
| [] -> failwith "BITMATCH_SIMP_CONV" in
check_disj pos in
check_neg neg in
check_pos pos
| _ -> failwith "BITMATCH_SIMP_CONV" in
conv;;
let rec bitpat_irrefutable = function
| Comb(Comb(Const("CONSPAT",_),p),a) ->
(match a with
| Comb(Const("word1",_),Var(_,_)) -> bitpat_irrefutable p
| Var(_,_) -> bitpat_irrefutable p
| _ -> false)
| Const("NILPAT",_) -> true
| Abs(_,c) -> bitpat_irrefutable c
| Comb(Const("?",_),c) -> bitpat_irrefutable c
| Comb(Comb(Const("_UNGUARDED_PATTERN",_),c),_) -> bitpat_irrefutable c
| Comb(Comb(Const("pat_set",_),c),_) -> bitpat_irrefutable c
| _ -> failwith "bitpat_irrefutable";;
let bitpat_irrefutable_thm =
let eN,ee,ee',en,em,ek = `:N`,`e:num`,`e:N word`,`n:num`,`m:num`,`k:num`
and ep,dN,pl,_1 = `p:bitpat`,`dimindex(:N)`,`(+)`,`1`
and e2n = `e DIV 2 EXP n`
and pth,pth1 =
let pth = prove
(`dimindex(:N) = n ==> pat_set p (e DIV 2 EXP n) ==>
pat_set (CONSPAT p (word e:N word)) e`,
REWRITE_TAC [CONSPAT_pat_set] THEN DISCH_THEN (SUBST1_TAC o SYM) THEN
DISCH_THEN (fun th -> EXISTS_TAC `e DIV 2 EXP dimindex(:N)` THEN
REWRITE_TAC [th; VAL_WORD; ADD_SYM; MULT_SYM;
GSYM (MATCH_MP DIVISION (SPEC `n:num` EXP_2_NE_0))])) in
UNDISCH_ALL pth,
(UNDISCH o prove)
(`pat_set p (e DIV 2 EXP 1) ==> pat_set (CONSPAT p (word1 (ODD e))) e`,
REWRITE_TAC [WORD1_ODD] THEN
ACCEPT_TAC (MATCH_MP pth (DIMINDEX_CONV `dimindex(:1)`)))
and pth0 = (UNDISCH o prove) (`e < 2 EXP 0 ==> pat_set NILPAT e`,
REWRITE_TAC [EXP; ARITH_RULE `n < 1 <=> n = 0`; NILPAT_pat_set])
and pthS = (UNDISCH_ALL o prove) (`n + m = k ==>
e < 2 EXP k ==> e DIV 2 EXP n < 2 EXP m`,
DISCH_THEN (SUBST1_TAC o SYM) THEN
IMP_REWRITE_TAC [EXP_ADD; RDIV_LT_EQ; EXP_2_NE_0])
and pthW = (UNDISCH_ALL o prove)
(`dimindex(:N) = n ==> val (e:N word) < 2 EXP n`,
DISCH_THEN (SUBST1_TAC o SYM) THEN REWRITE_TAC [VAL_BOUND]) in
let rec build = function
| Const("NILPAT",_),e -> [], INST [e,ee] pth0
| Comb(Comb(Const("CONSPAT",_),p),v),e ->
let pthS n m =
let th = NUM_ADD_CONV (mk_comb (mk_comb (pl, n), m)) in
PROVE_HYP th (INST [n,en; m,em; rhs (concl th),ek; e,ee] pthS) in
(match v with
| Comb(Const("word1",_),(Var(_,_) as v)) ->
let ls, th = build (p, vsubst [e,ee; _1,en] e2n) in
let th' = PROVE_HYP th (INST [lhand (concl th),ep; e,ee] pth1) in
(rand (rand (lhand (concl th'))),v)::ls,
PROVE_HYP (pthS _1 (rand (rand (hd (hyp th))))) th'
| Var(_,ty) ->
let N = dest_word_ty ty in
let th1 = DIMINDEX_CONV (inst [N,eN] dN) in
let n = rhs (concl th1) in
let ls, th = build (p, vsubst [e,ee; n,en] e2n) in
let th' = PROVE_HYP th1
(INST [n,en; lhand (concl th),ep; e,ee] (INST_TYPE [N,eN] pth)) in
let th' = PROVE_HYP th th' in
(rand (lhand (concl th')),v)::ls,
PROVE_HYP (pthS n (rand (rand (hd (hyp th))))) th'
| _ -> failwith "bitpat_irrefutable_thm: not irrefutable")
| _ -> failwith "bitpat_irrefutable_thm" in
fun tm -> match snd (strip_exists tm) with
| Comb(Comb(Const("pat_set",_),p),(Comb(Const("val",_),e') as e)) ->
let ls,th = build (p, e) in
let rec build_ex = function
| Comb(Const("?",_),Abs(v,c)) as tm ->
let e = rev_assoc v ls in
EXISTS (tm, e) (build_ex (vsubst [e,v] c))
| tm when aconv (concl th) tm -> th
| _ -> failwith "bitpat_irrefutable_thm: nonlinear pattern" in
let th = build_ex tm in
let N = dest_word_ty (type_of e') in
let th1 = DIMINDEX_CONV (inst [N,eN] dN) in
let n = rhs (concl th1) in
if aconv n (rand (rand (hd (hyp th)))) then
PROVE_HYP (PROVE_HYP th1 (PINST [N,eN] [n,en; e',ee'] pthW)) th
else failwith "bitpat_irrefutable_thm: incorrect bit length"
| _ -> failwith "bitpat_irrefutable_thm";;
let ONLY_BITMATCH_CASES_THEN thltac = WITH_GOAL (fun w ->
let e,cs =
let f = function
| Comb(Comb(Const("_BITMATCH",_),_),_) -> true
| _ -> false in
(rand F_F I) (dest_comb (find_term f w)) in
let rec tac thl = function
| Comb(Comb(Const("_SEQPATTERN",_),Abs(_,Abs(_,c))),cs) ->
let rec go = function
| Comb((Const("?",_) as f),Abs(z,c)) ->
let tm, tac1 = go c in
mk_comb (f, mk_abs (z, tm)), POP_ASSUM CHOOSE_TAC THEN tac1
| tm ->
mk_comb (rator (lhand tm),
mk_comb (mk_const ("val", [dest_word_ty (type_of e),`:N`]), e)),
ALL_TAC in
let tm, tac1 = go c in
if bitpat_irrefutable c then
ASSUME_TAC (bitpat_irrefutable_thm tm) THEN
tac1 THEN POP_ASSUM (fun th -> thltac (th::thl))
else
ASM_CASES_TAC tm THENL [
tac1 THEN POP_ASSUM (fun th -> thltac (th::thl));
POP_ASSUM (fun th -> tac (th::thl) cs)]
| _ -> thltac thl in
tac [] cs);;
let BITMATCH_ASM_CASES_TAC =
ONLY_BITMATCH_CASES_THEN (fun thl ->
CONV_TAC (TOP_SWEEP_CONV (BITMATCH_SIMP_CONV thl)) THEN
MAP_EVERY ASSUME_TAC thl);;
let BITMATCH_CASES_TAC =
ONLY_BITMATCH_CASES_THEN (CONV_TAC o
TOP_SWEEP_CONV o BITMATCH_SIMP_CONV);;
(* (bm_seq_numeral `bitmatch e with ...` n) will
return `word n` and `(bitmatch word n with ...) = res` where `res` is the
appropriate match branch. Unlike BITMATCH_CONV this also works with matches
with non-disjoint cases. *)
let bm_seq_numeral = function
| Comb((Comb(Const("_BITMATCH",_),e) as me),cs) ->
let N = dest_word_ty (type_of e) in
let sz = Num.int_of_num (dest_finty N) in
let word = mk_const ("word", [N,`:N`]) in
let rec mk_fun cs =
let tm = mk_comb (me, cs) in
let th = BM_FIRST_CASE_CONV tm in
let inst e' th = if is_var e then INST [e',e] th else th in
match cs with
| Comb(Comb(Const("_SEQPATTERN",_),c),cs') ->
let ps = hd (hyp th) in
let pats = Array.init sz (fun i -> try
Some (bm_skip_clause (pat_to_bit false (mk_numeral (Int i))) tm)
with Failure _ -> None) in
let f = mk_fun cs' in
fun n e' ->
(match bitpat_matches c n with
| None ->
let ls, th' = inst_bitpat_numeral ps n in
PROVE_HYP th' (INST ls th)
| Some i ->
let Some th' = pats.(i) in
let th1 = inst e' th' in
let th2 = match hd (hyp th1) with
| Comb(Const("~",_),p) -> EQF_ELIM (WORD_RED_CONV p)
| p -> EQT_ELIM (WORD_RED_CONV p) in
TRANS (PROVE_HYP th2 th1) (f n e'))
| Comb(Const("_ELSEPATTERN",_),_) -> fun _ e' -> inst e' th
| _ -> failwith "bm_seq_numeral" in
let f = mk_fun cs in
fun n -> let e = mk_comb (word, mk_numeral n) in e, f n e
| _ -> failwith "bm_seq_numeral";;
let BITMATCH_SEQ_CONV = function
| Comb(Comb(Const("_BITMATCH",_), Comb(Const("word",_),n)),_) as tm ->
snd (bm_seq_numeral tm (dest_numeral n))
| _ -> failwith "BITMATCH_CONV";;
|