src/HOL/Import/HOL/HOL4Word32.thy
author obua
Mon Aug 29 16:51:39 2005 +0200 (2005-08-29)
changeset 17188 a26a4fc323ed
parent 16417 9bc16273c2d4
child 17566 484ff733f29c
permissions -rw-r--r--
Updated import.
     1 (* AUTOMATICALLY GENERATED, DO NOT EDIT! *)
     2 
     3 theory HOL4Word32 = HOL4Base:
     4 
     5 ;setup_theory bits
     6 
     7 consts
     8   DIV2 :: "nat => nat" 
     9 
    10 defs
    11   DIV2_primdef: "DIV2 == %n. n div 2"
    12 
    13 lemma DIV2_def: "ALL n. DIV2 n = n div 2"
    14   by (import bits DIV2_def)
    15 
    16 consts
    17   TIMES_2EXP :: "nat => nat => nat" 
    18 
    19 defs
    20   TIMES_2EXP_primdef: "TIMES_2EXP == %x n. n * 2 ^ x"
    21 
    22 lemma TIMES_2EXP_def: "ALL x n. TIMES_2EXP x n = n * 2 ^ x"
    23   by (import bits TIMES_2EXP_def)
    24 
    25 consts
    26   DIV_2EXP :: "nat => nat => nat" 
    27 
    28 defs
    29   DIV_2EXP_primdef: "DIV_2EXP == %x n. n div 2 ^ x"
    30 
    31 lemma DIV_2EXP_def: "ALL x n. DIV_2EXP x n = n div 2 ^ x"
    32   by (import bits DIV_2EXP_def)
    33 
    34 consts
    35   MOD_2EXP :: "nat => nat => nat" 
    36 
    37 defs
    38   MOD_2EXP_primdef: "MOD_2EXP == %x n. n mod 2 ^ x"
    39 
    40 lemma MOD_2EXP_def: "ALL x n. MOD_2EXP x n = n mod 2 ^ x"
    41   by (import bits MOD_2EXP_def)
    42 
    43 consts
    44   DIVMOD_2EXP :: "nat => nat => nat * nat" 
    45 
    46 defs
    47   DIVMOD_2EXP_primdef: "DIVMOD_2EXP == %x n. (n div 2 ^ x, n mod 2 ^ x)"
    48 
    49 lemma DIVMOD_2EXP_def: "ALL x n. DIVMOD_2EXP x n = (n div 2 ^ x, n mod 2 ^ x)"
    50   by (import bits DIVMOD_2EXP_def)
    51 
    52 consts
    53   SBIT :: "bool => nat => nat" 
    54 
    55 defs
    56   SBIT_primdef: "SBIT == %b n. if b then 2 ^ n else 0"
    57 
    58 lemma SBIT_def: "ALL b n. SBIT b n = (if b then 2 ^ n else 0)"
    59   by (import bits SBIT_def)
    60 
    61 consts
    62   BITS :: "nat => nat => nat => nat" 
    63 
    64 defs
    65   BITS_primdef: "BITS == %h l n. MOD_2EXP (Suc h - l) (DIV_2EXP l n)"
    66 
    67 lemma BITS_def: "ALL h l n. BITS h l n = MOD_2EXP (Suc h - l) (DIV_2EXP l n)"
    68   by (import bits BITS_def)
    69 
    70 constdefs
    71   bit :: "nat => nat => bool" 
    72   "bit == %b n. BITS b b n = 1"
    73 
    74 lemma BIT_def: "ALL b n. bit b n = (BITS b b n = 1)"
    75   by (import bits BIT_def)
    76 
    77 consts
    78   SLICE :: "nat => nat => nat => nat" 
    79 
    80 defs
    81   SLICE_primdef: "SLICE == %h l n. MOD_2EXP (Suc h) n - MOD_2EXP l n"
    82 
    83 lemma SLICE_def: "ALL h l n. SLICE h l n = MOD_2EXP (Suc h) n - MOD_2EXP l n"
    84   by (import bits SLICE_def)
    85 
    86 consts
    87   LSBn :: "nat => bool" 
    88 
    89 defs
    90   LSBn_primdef: "LSBn == bit 0"
    91 
    92 lemma LSBn_def: "LSBn = bit 0"
    93   by (import bits LSBn_def)
    94 
    95 consts
    96   BITWISE :: "nat => (bool => bool => bool) => nat => nat => nat" 
    97 
    98 specification (BITWISE_primdef: BITWISE) BITWISE_def: "(ALL oper x y. BITWISE 0 oper x y = 0) &
    99 (ALL n oper x y.
   100     BITWISE (Suc n) oper x y =
   101     BITWISE n oper x y + SBIT (oper (bit n x) (bit n y)) n)"
   102   by (import bits BITWISE_def)
   103 
   104 lemma DIV1: "ALL x::nat. x div (1::nat) = x"
   105   by (import bits DIV1)
   106 
   107 lemma SUC_SUB: "Suc a - a = 1"
   108   by (import bits SUC_SUB)
   109 
   110 lemma DIV_MULT_1: "ALL (r::nat) n::nat. r < n --> (n + r) div n = (1::nat)"
   111   by (import bits DIV_MULT_1)
   112 
   113 lemma ZERO_LT_TWOEXP: "ALL n::nat. (0::nat) < (2::nat) ^ n"
   114   by (import bits ZERO_LT_TWOEXP)
   115 
   116 lemma MOD_2EXP_LT: "ALL (n::nat) k::nat. k mod (2::nat) ^ n < (2::nat) ^ n"
   117   by (import bits MOD_2EXP_LT)
   118 
   119 lemma TWOEXP_DIVISION: "ALL (n::nat) k::nat.
   120    k = k div (2::nat) ^ n * (2::nat) ^ n + k mod (2::nat) ^ n"
   121   by (import bits TWOEXP_DIVISION)
   122 
   123 lemma TWOEXP_MONO: "ALL (a::nat) b::nat. a < b --> (2::nat) ^ a < (2::nat) ^ b"
   124   by (import bits TWOEXP_MONO)
   125 
   126 lemma TWOEXP_MONO2: "ALL (a::nat) b::nat. a <= b --> (2::nat) ^ a <= (2::nat) ^ b"
   127   by (import bits TWOEXP_MONO2)
   128 
   129 lemma EXP_SUB_LESS_EQ: "ALL (a::nat) b::nat. (2::nat) ^ (a - b) <= (2::nat) ^ a"
   130   by (import bits EXP_SUB_LESS_EQ)
   131 
   132 lemma BITS_THM: "ALL x xa xb. BITS x xa xb = xb div 2 ^ xa mod 2 ^ (Suc x - xa)"
   133   by (import bits BITS_THM)
   134 
   135 lemma BITSLT_THM: "ALL h l n. BITS h l n < 2 ^ (Suc h - l)"
   136   by (import bits BITSLT_THM)
   137 
   138 lemma DIV_MULT_LEM: "ALL (m::nat) n::nat. (0::nat) < n --> m div n * n <= m"
   139   by (import bits DIV_MULT_LEM)
   140 
   141 lemma MOD_2EXP_LEM: "ALL (n::nat) x::nat.
   142    n mod (2::nat) ^ x = n - n div (2::nat) ^ x * (2::nat) ^ x"
   143   by (import bits MOD_2EXP_LEM)
   144 
   145 lemma BITS2_THM: "ALL h l n. BITS h l n = n mod 2 ^ Suc h div 2 ^ l"
   146   by (import bits BITS2_THM)
   147 
   148 lemma BITS_COMP_THM: "ALL h1 l1 h2 l2 n.
   149    h2 + l1 <= h1 --> BITS h2 l2 (BITS h1 l1 n) = BITS (h2 + l1) (l2 + l1) n"
   150   by (import bits BITS_COMP_THM)
   151 
   152 lemma BITS_DIV_THM: "ALL h l x n. BITS h l x div 2 ^ n = BITS h (l + n) x"
   153   by (import bits BITS_DIV_THM)
   154 
   155 lemma BITS_LT_HIGH: "ALL h l n. n < 2 ^ Suc h --> BITS h l n = n div 2 ^ l"
   156   by (import bits BITS_LT_HIGH)
   157 
   158 lemma BITS_ZERO: "ALL h l n. h < l --> BITS h l n = 0"
   159   by (import bits BITS_ZERO)
   160 
   161 lemma BITS_ZERO2: "ALL h l. BITS h l 0 = 0"
   162   by (import bits BITS_ZERO2)
   163 
   164 lemma BITS_ZERO3: "ALL h x. BITS h 0 x = x mod 2 ^ Suc h"
   165   by (import bits BITS_ZERO3)
   166 
   167 lemma BITS_COMP_THM2: "ALL h1 l1 h2 l2 n.
   168    BITS h2 l2 (BITS h1 l1 n) = BITS (min h1 (h2 + l1)) (l2 + l1) n"
   169   by (import bits BITS_COMP_THM2)
   170 
   171 lemma NOT_MOD2_LEM: "ALL n::nat. (n mod (2::nat) ~= (0::nat)) = (n mod (2::nat) = (1::nat))"
   172   by (import bits NOT_MOD2_LEM)
   173 
   174 lemma NOT_MOD2_LEM2: "ALL (n::nat) a::'a.
   175    (n mod (2::nat) ~= (1::nat)) = (n mod (2::nat) = (0::nat))"
   176   by (import bits NOT_MOD2_LEM2)
   177 
   178 lemma EVEN_MOD2_LEM: "ALL n. EVEN n = (n mod 2 = 0)"
   179   by (import bits EVEN_MOD2_LEM)
   180 
   181 lemma ODD_MOD2_LEM: "ALL n. ODD n = (n mod 2 = 1)"
   182   by (import bits ODD_MOD2_LEM)
   183 
   184 lemma LSB_ODD: "LSBn = ODD"
   185   by (import bits LSB_ODD)
   186 
   187 lemma DIV_MULT_THM: "ALL (x::nat) n::nat.
   188    n div (2::nat) ^ x * (2::nat) ^ x = n - n mod (2::nat) ^ x"
   189   by (import bits DIV_MULT_THM)
   190 
   191 lemma DIV_MULT_THM2: "ALL x::nat. (2::nat) * (x div (2::nat)) = x - x mod (2::nat)"
   192   by (import bits DIV_MULT_THM2)
   193 
   194 lemma LESS_EQ_EXP_MULT: "ALL (a::nat) b::nat. a <= b --> (EX x::nat. (2::nat) ^ b = x * (2::nat) ^ a)"
   195   by (import bits LESS_EQ_EXP_MULT)
   196 
   197 lemma SLICE_LEM1: "ALL (a::nat) (x::nat) y::nat.
   198    a div (2::nat) ^ (x + y) * (2::nat) ^ (x + y) =
   199    a div (2::nat) ^ x * (2::nat) ^ x -
   200    a div (2::nat) ^ x mod (2::nat) ^ y * (2::nat) ^ x"
   201   by (import bits SLICE_LEM1)
   202 
   203 lemma SLICE_LEM2: "ALL (a::'a) (x::nat) y::nat.
   204    (n::nat) mod (2::nat) ^ (x + y) =
   205    n mod (2::nat) ^ x + n div (2::nat) ^ x mod (2::nat) ^ y * (2::nat) ^ x"
   206   by (import bits SLICE_LEM2)
   207 
   208 lemma SLICE_LEM3: "ALL (n::nat) (h::nat) l::nat.
   209    l < h --> n mod (2::nat) ^ Suc l <= n mod (2::nat) ^ h"
   210   by (import bits SLICE_LEM3)
   211 
   212 lemma SLICE_THM: "ALL n h l. SLICE h l n = BITS h l n * 2 ^ l"
   213   by (import bits SLICE_THM)
   214 
   215 lemma SLICELT_THM: "ALL h l n. SLICE h l n < 2 ^ Suc h"
   216   by (import bits SLICELT_THM)
   217 
   218 lemma BITS_SLICE_THM: "ALL h l n. BITS h l (SLICE h l n) = BITS h l n"
   219   by (import bits BITS_SLICE_THM)
   220 
   221 lemma BITS_SLICE_THM2: "ALL h l n. h <= h2 --> BITS h2 l (SLICE h l n) = BITS h l n"
   222   by (import bits BITS_SLICE_THM2)
   223 
   224 lemma MOD_2EXP_MONO: "ALL (n::nat) (h::nat) l::nat.
   225    l <= h --> n mod (2::nat) ^ l <= n mod (2::nat) ^ Suc h"
   226   by (import bits MOD_2EXP_MONO)
   227 
   228 lemma SLICE_COMP_THM: "ALL h m l n.
   229    Suc m <= h & l <= m --> SLICE h (Suc m) n + SLICE m l n = SLICE h l n"
   230   by (import bits SLICE_COMP_THM)
   231 
   232 lemma SLICE_ZERO: "ALL h l n. h < l --> SLICE h l n = 0"
   233   by (import bits SLICE_ZERO)
   234 
   235 lemma BIT_COMP_THM3: "ALL h m l n.
   236    Suc m <= h & l <= m -->
   237    BITS h (Suc m) n * 2 ^ (Suc m - l) + BITS m l n = BITS h l n"
   238   by (import bits BIT_COMP_THM3)
   239 
   240 lemma NOT_BIT: "ALL n a. (~ bit n a) = (BITS n n a = 0)"
   241   by (import bits NOT_BIT)
   242 
   243 lemma NOT_BITS: "ALL n a. (BITS n n a ~= 0) = (BITS n n a = 1)"
   244   by (import bits NOT_BITS)
   245 
   246 lemma NOT_BITS2: "ALL n a. (BITS n n a ~= 1) = (BITS n n a = 0)"
   247   by (import bits NOT_BITS2)
   248 
   249 lemma BIT_SLICE: "ALL n a b. (bit n a = bit n b) = (SLICE n n a = SLICE n n b)"
   250   by (import bits BIT_SLICE)
   251 
   252 lemma BIT_SLICE_LEM: "ALL y x n. SBIT (bit x n) (x + y) = SLICE x x n * 2 ^ y"
   253   by (import bits BIT_SLICE_LEM)
   254 
   255 lemma BIT_SLICE_THM: "ALL x xa. SBIT (bit x xa) x = SLICE x x xa"
   256   by (import bits BIT_SLICE_THM)
   257 
   258 lemma SBIT_DIV: "ALL b m n. n < m --> SBIT b (m - n) = SBIT b m div 2 ^ n"
   259   by (import bits SBIT_DIV)
   260 
   261 lemma BITS_SUC: "ALL h l n.
   262    l <= Suc h -->
   263    SBIT (bit (Suc h) n) (Suc h - l) + BITS h l n = BITS (Suc h) l n"
   264   by (import bits BITS_SUC)
   265 
   266 lemma BITS_SUC_THM: "ALL h l n.
   267    BITS (Suc h) l n =
   268    (if Suc h < l then 0 else SBIT (bit (Suc h) n) (Suc h - l) + BITS h l n)"
   269   by (import bits BITS_SUC_THM)
   270 
   271 lemma BIT_BITS_THM: "ALL h l a b.
   272    (ALL x. l <= x & x <= h --> bit x a = bit x b) =
   273    (BITS h l a = BITS h l b)"
   274   by (import bits BIT_BITS_THM)
   275 
   276 lemma BITWISE_LT_2EXP: "ALL n oper a b. BITWISE n oper a b < 2 ^ n"
   277   by (import bits BITWISE_LT_2EXP)
   278 
   279 lemma LESS_EXP_MULT2: "ALL (a::nat) b::nat.
   280    a < b -->
   281    (EX x::nat. (2::nat) ^ b = (2::nat) ^ (x + (1::nat)) * (2::nat) ^ a)"
   282   by (import bits LESS_EXP_MULT2)
   283 
   284 lemma BITWISE_THM: "ALL x n oper a b.
   285    x < n --> bit x (BITWISE n oper a b) = oper (bit x a) (bit x b)"
   286   by (import bits BITWISE_THM)
   287 
   288 lemma BITWISE_COR: "ALL x n oper a b.
   289    x < n -->
   290    oper (bit x a) (bit x b) --> BITWISE n oper a b div 2 ^ x mod 2 = 1"
   291   by (import bits BITWISE_COR)
   292 
   293 lemma BITWISE_NOT_COR: "ALL x n oper a b.
   294    x < n -->
   295    ~ oper (bit x a) (bit x b) --> BITWISE n oper a b div 2 ^ x mod 2 = 0"
   296   by (import bits BITWISE_NOT_COR)
   297 
   298 lemma MOD_PLUS_RIGHT: "ALL n>0::nat. ALL (j::nat) k::nat. (j + k mod n) mod n = (j + k) mod n"
   299   by (import bits MOD_PLUS_RIGHT)
   300 
   301 lemma MOD_PLUS_1: "ALL n>0::nat.
   302    ALL x::nat. ((x + (1::nat)) mod n = (0::nat)) = (x mod n + (1::nat) = n)"
   303   by (import bits MOD_PLUS_1)
   304 
   305 lemma MOD_ADD_1: "ALL n>0::nat.
   306    ALL x::nat.
   307       (x + (1::nat)) mod n ~= (0::nat) -->
   308       (x + (1::nat)) mod n = x mod n + (1::nat)"
   309   by (import bits MOD_ADD_1)
   310 
   311 ;end_setup
   312 
   313 ;setup_theory word32
   314 
   315 consts
   316   HB :: "nat" 
   317 
   318 defs
   319   HB_primdef: "HB ==
   320 NUMERAL
   321  (NUMERAL_BIT1
   322    (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))"
   323 
   324 lemma HB_def: "HB =
   325 NUMERAL
   326  (NUMERAL_BIT1
   327    (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))"
   328   by (import word32 HB_def)
   329 
   330 consts
   331   WL :: "nat" 
   332 
   333 defs
   334   WL_primdef: "WL == Suc HB"
   335 
   336 lemma WL_def: "WL = Suc HB"
   337   by (import word32 WL_def)
   338 
   339 consts
   340   MODw :: "nat => nat" 
   341 
   342 defs
   343   MODw_primdef: "MODw == %n. n mod 2 ^ WL"
   344 
   345 lemma MODw_def: "ALL n. MODw n = n mod 2 ^ WL"
   346   by (import word32 MODw_def)
   347 
   348 consts
   349   INw :: "nat => bool" 
   350 
   351 defs
   352   INw_primdef: "INw == %n. n < 2 ^ WL"
   353 
   354 lemma INw_def: "ALL n. INw n = (n < 2 ^ WL)"
   355   by (import word32 INw_def)
   356 
   357 consts
   358   EQUIV :: "nat => nat => bool" 
   359 
   360 defs
   361   EQUIV_primdef: "EQUIV == %x y. MODw x = MODw y"
   362 
   363 lemma EQUIV_def: "ALL x y. EQUIV x y = (MODw x = MODw y)"
   364   by (import word32 EQUIV_def)
   365 
   366 lemma EQUIV_QT: "ALL x y. EQUIV x y = (EQUIV x = EQUIV y)"
   367   by (import word32 EQUIV_QT)
   368 
   369 lemma FUNPOW_THM: "ALL f n x. (f ^ n) (f x) = f ((f ^ n) x)"
   370   by (import word32 FUNPOW_THM)
   371 
   372 lemma FUNPOW_THM2: "ALL f n x. (f ^ Suc n) x = f ((f ^ n) x)"
   373   by (import word32 FUNPOW_THM2)
   374 
   375 lemma FUNPOW_COMP: "ALL f m n a. (f ^ m) ((f ^ n) a) = (f ^ (m + n)) a"
   376   by (import word32 FUNPOW_COMP)
   377 
   378 lemma INw_MODw: "ALL n. INw (MODw n)"
   379   by (import word32 INw_MODw)
   380 
   381 lemma TOw_IDEM: "ALL a. INw a --> MODw a = a"
   382   by (import word32 TOw_IDEM)
   383 
   384 lemma MODw_IDEM2: "ALL a. MODw (MODw a) = MODw a"
   385   by (import word32 MODw_IDEM2)
   386 
   387 lemma TOw_QT: "ALL a. EQUIV (MODw a) a"
   388   by (import word32 TOw_QT)
   389 
   390 lemma MODw_THM: "MODw = BITS HB 0"
   391   by (import word32 MODw_THM)
   392 
   393 lemma MOD_ADD: "ALL a b. MODw (a + b) = MODw (MODw a + MODw b)"
   394   by (import word32 MOD_ADD)
   395 
   396 lemma MODw_MULT: "ALL a b. MODw (a * b) = MODw (MODw a * MODw b)"
   397   by (import word32 MODw_MULT)
   398 
   399 consts
   400   AONE :: "nat" 
   401 
   402 defs
   403   AONE_primdef: "AONE == 1"
   404 
   405 lemma AONE_def: "AONE = 1"
   406   by (import word32 AONE_def)
   407 
   408 lemma ADD_QT: "(ALL n. EQUIV (0 + n) n) & (ALL m n. EQUIV (Suc m + n) (Suc (m + n)))"
   409   by (import word32 ADD_QT)
   410 
   411 lemma ADD_0_QT: "ALL a. EQUIV (a + 0) a"
   412   by (import word32 ADD_0_QT)
   413 
   414 lemma ADD_COMM_QT: "ALL a b. EQUIV (a + b) (b + a)"
   415   by (import word32 ADD_COMM_QT)
   416 
   417 lemma ADD_ASSOC_QT: "ALL a b c. EQUIV (a + (b + c)) (a + b + c)"
   418   by (import word32 ADD_ASSOC_QT)
   419 
   420 lemma MULT_QT: "(ALL n. EQUIV (0 * n) 0) & (ALL m n. EQUIV (Suc m * n) (m * n + n))"
   421   by (import word32 MULT_QT)
   422 
   423 lemma ADD1_QT: "ALL m. EQUIV (Suc m) (m + AONE)"
   424   by (import word32 ADD1_QT)
   425 
   426 lemma ADD_CLAUSES_QT: "(ALL m. EQUIV (0 + m) m) &
   427 (ALL m. EQUIV (m + 0) m) &
   428 (ALL m n. EQUIV (Suc m + n) (Suc (m + n))) &
   429 (ALL m n. EQUIV (m + Suc n) (Suc (m + n)))"
   430   by (import word32 ADD_CLAUSES_QT)
   431 
   432 lemma SUC_EQUIV_COMP: "ALL a b. EQUIV (Suc a) b --> EQUIV a (b + (2 ^ WL - 1))"
   433   by (import word32 SUC_EQUIV_COMP)
   434 
   435 lemma INV_SUC_EQ_QT: "ALL m n. EQUIV (Suc m) (Suc n) = EQUIV m n"
   436   by (import word32 INV_SUC_EQ_QT)
   437 
   438 lemma ADD_INV_0_QT: "ALL m n. EQUIV (m + n) m --> EQUIV n 0"
   439   by (import word32 ADD_INV_0_QT)
   440 
   441 lemma ADD_INV_0_EQ_QT: "ALL m n. EQUIV (m + n) m = EQUIV n 0"
   442   by (import word32 ADD_INV_0_EQ_QT)
   443 
   444 lemma EQ_ADD_LCANCEL_QT: "ALL m n p. EQUIV (m + n) (m + p) = EQUIV n p"
   445   by (import word32 EQ_ADD_LCANCEL_QT)
   446 
   447 lemma EQ_ADD_RCANCEL_QT: "ALL x xa xb. EQUIV (x + xb) (xa + xb) = EQUIV x xa"
   448   by (import word32 EQ_ADD_RCANCEL_QT)
   449 
   450 lemma LEFT_ADD_DISTRIB_QT: "ALL m n p. EQUIV (p * (m + n)) (p * m + p * n)"
   451   by (import word32 LEFT_ADD_DISTRIB_QT)
   452 
   453 lemma MULT_ASSOC_QT: "ALL m n p. EQUIV (m * (n * p)) (m * n * p)"
   454   by (import word32 MULT_ASSOC_QT)
   455 
   456 lemma MULT_COMM_QT: "ALL m n. EQUIV (m * n) (n * m)"
   457   by (import word32 MULT_COMM_QT)
   458 
   459 lemma MULT_CLAUSES_QT: "ALL m n.
   460    EQUIV (0 * m) 0 &
   461    EQUIV (m * 0) 0 &
   462    EQUIV (AONE * m) m &
   463    EQUIV (m * AONE) m &
   464    EQUIV (Suc m * n) (m * n + n) & EQUIV (m * Suc n) (m + m * n)"
   465   by (import word32 MULT_CLAUSES_QT)
   466 
   467 consts
   468   MSBn :: "nat => bool" 
   469 
   470 defs
   471   MSBn_primdef: "MSBn == bit HB"
   472 
   473 lemma MSBn_def: "MSBn = bit HB"
   474   by (import word32 MSBn_def)
   475 
   476 consts
   477   ONE_COMP :: "nat => nat" 
   478 
   479 defs
   480   ONE_COMP_primdef: "ONE_COMP == %x. 2 ^ WL - 1 - MODw x"
   481 
   482 lemma ONE_COMP_def: "ALL x. ONE_COMP x = 2 ^ WL - 1 - MODw x"
   483   by (import word32 ONE_COMP_def)
   484 
   485 consts
   486   TWO_COMP :: "nat => nat" 
   487 
   488 defs
   489   TWO_COMP_primdef: "TWO_COMP == %x. 2 ^ WL - MODw x"
   490 
   491 lemma TWO_COMP_def: "ALL x. TWO_COMP x = 2 ^ WL - MODw x"
   492   by (import word32 TWO_COMP_def)
   493 
   494 lemma ADD_TWO_COMP_QT: "ALL a. EQUIV (MODw a + TWO_COMP a) 0"
   495   by (import word32 ADD_TWO_COMP_QT)
   496 
   497 lemma TWO_COMP_ONE_COMP_QT: "ALL a. EQUIV (TWO_COMP a) (ONE_COMP a + AONE)"
   498   by (import word32 TWO_COMP_ONE_COMP_QT)
   499 
   500 lemma BIT_EQUIV_THM: "(All::(nat => bool) => bool)
   501  (%x::nat.
   502      (All::(nat => bool) => bool)
   503       (%xa::nat.
   504           (op =::bool => bool => bool)
   505            ((All::(nat => bool) => bool)
   506              (%xb::nat.
   507                  (op -->::bool => bool => bool)
   508                   ((op <::nat => nat => bool) xb (WL::nat))
   509                   ((op =::bool => bool => bool)
   510                     ((bit::nat => nat => bool) xb x)
   511                     ((bit::nat => nat => bool) xb xa))))
   512            ((EQUIV::nat => nat => bool) x xa)))"
   513   by (import word32 BIT_EQUIV_THM)
   514 
   515 lemma BITS_SUC2: "ALL n a. BITS (Suc n) 0 a = SLICE (Suc n) (Suc n) a + BITS n 0 a"
   516   by (import word32 BITS_SUC2)
   517 
   518 lemma BITWISE_ONE_COMP_THM: "ALL a b. BITWISE WL (%x y. ~ x) a b = ONE_COMP a"
   519   by (import word32 BITWISE_ONE_COMP_THM)
   520 
   521 lemma ONE_COMP_THM: "ALL x xa. xa < WL --> bit xa (ONE_COMP x) = (~ bit xa x)"
   522   by (import word32 ONE_COMP_THM)
   523 
   524 consts
   525   OR :: "nat => nat => nat" 
   526 
   527 defs
   528   OR_primdef: "OR == BITWISE WL op |"
   529 
   530 lemma OR_def: "OR = BITWISE WL op |"
   531   by (import word32 OR_def)
   532 
   533 consts
   534   AND :: "nat => nat => nat" 
   535 
   536 defs
   537   AND_primdef: "AND == BITWISE WL op &"
   538 
   539 lemma AND_def: "AND = BITWISE WL op &"
   540   by (import word32 AND_def)
   541 
   542 consts
   543   EOR :: "nat => nat => nat" 
   544 
   545 defs
   546   EOR_primdef: "EOR == BITWISE WL (%x y. x ~= y)"
   547 
   548 lemma EOR_def: "EOR = BITWISE WL (%x y. x ~= y)"
   549   by (import word32 EOR_def)
   550 
   551 consts
   552   COMP0 :: "nat" 
   553 
   554 defs
   555   COMP0_primdef: "COMP0 == ONE_COMP 0"
   556 
   557 lemma COMP0_def: "COMP0 = ONE_COMP 0"
   558   by (import word32 COMP0_def)
   559 
   560 lemma BITWISE_THM2: "(All::(nat => bool) => bool)
   561  (%y::nat.
   562      (All::((bool => bool => bool) => bool) => bool)
   563       (%oper::bool => bool => bool.
   564           (All::(nat => bool) => bool)
   565            (%a::nat.
   566                (All::(nat => bool) => bool)
   567                 (%b::nat.
   568                     (op =::bool => bool => bool)
   569                      ((All::(nat => bool) => bool)
   570                        (%x::nat.
   571                            (op -->::bool => bool => bool)
   572                             ((op <::nat => nat => bool) x (WL::nat))
   573                             ((op =::bool => bool => bool)
   574                               (oper ((bit::nat => nat => bool) x a)
   575                                 ((bit::nat => nat => bool) x b))
   576                               ((bit::nat => nat => bool) x y))))
   577                      ((EQUIV::nat => nat => bool)
   578                        ((BITWISE::nat
   579                                   => (bool => bool => bool)
   580                                      => nat => nat => nat)
   581                          (WL::nat) oper a b)
   582                        y)))))"
   583   by (import word32 BITWISE_THM2)
   584 
   585 lemma OR_ASSOC_QT: "ALL a b c. EQUIV (OR a (OR b c)) (OR (OR a b) c)"
   586   by (import word32 OR_ASSOC_QT)
   587 
   588 lemma OR_COMM_QT: "ALL a b. EQUIV (OR a b) (OR b a)"
   589   by (import word32 OR_COMM_QT)
   590 
   591 lemma OR_ABSORB_QT: "ALL a b. EQUIV (AND a (OR a b)) a"
   592   by (import word32 OR_ABSORB_QT)
   593 
   594 lemma OR_IDEM_QT: "ALL a. EQUIV (OR a a) a"
   595   by (import word32 OR_IDEM_QT)
   596 
   597 lemma AND_ASSOC_QT: "ALL a b c. EQUIV (AND a (AND b c)) (AND (AND a b) c)"
   598   by (import word32 AND_ASSOC_QT)
   599 
   600 lemma AND_COMM_QT: "ALL a b. EQUIV (AND a b) (AND b a)"
   601   by (import word32 AND_COMM_QT)
   602 
   603 lemma AND_ABSORB_QT: "ALL a b. EQUIV (OR a (AND a b)) a"
   604   by (import word32 AND_ABSORB_QT)
   605 
   606 lemma AND_IDEM_QT: "ALL a. EQUIV (AND a a) a"
   607   by (import word32 AND_IDEM_QT)
   608 
   609 lemma OR_COMP_QT: "ALL a. EQUIV (OR a (ONE_COMP a)) COMP0"
   610   by (import word32 OR_COMP_QT)
   611 
   612 lemma AND_COMP_QT: "ALL a. EQUIV (AND a (ONE_COMP a)) 0"
   613   by (import word32 AND_COMP_QT)
   614 
   615 lemma ONE_COMP_QT: "ALL a. EQUIV (ONE_COMP (ONE_COMP a)) a"
   616   by (import word32 ONE_COMP_QT)
   617 
   618 lemma RIGHT_AND_OVER_OR_QT: "ALL a b c. EQUIV (AND (OR a b) c) (OR (AND a c) (AND b c))"
   619   by (import word32 RIGHT_AND_OVER_OR_QT)
   620 
   621 lemma RIGHT_OR_OVER_AND_QT: "ALL a b c. EQUIV (OR (AND a b) c) (AND (OR a c) (OR b c))"
   622   by (import word32 RIGHT_OR_OVER_AND_QT)
   623 
   624 lemma DE_MORGAN_THM_QT: "ALL a b.
   625    EQUIV (ONE_COMP (AND a b)) (OR (ONE_COMP a) (ONE_COMP b)) &
   626    EQUIV (ONE_COMP (OR a b)) (AND (ONE_COMP a) (ONE_COMP b))"
   627   by (import word32 DE_MORGAN_THM_QT)
   628 
   629 lemma BIT_EQUIV: "ALL n a b. n < WL --> EQUIV a b --> bit n a = bit n b"
   630   by (import word32 BIT_EQUIV)
   631 
   632 lemma LSB_WELLDEF: "ALL a b. EQUIV a b --> LSBn a = LSBn b"
   633   by (import word32 LSB_WELLDEF)
   634 
   635 lemma MSB_WELLDEF: "ALL a b. EQUIV a b --> MSBn a = MSBn b"
   636   by (import word32 MSB_WELLDEF)
   637 
   638 lemma BITWISE_ISTEP: "ALL n oper a b.
   639    0 < n -->
   640    BITWISE n oper (a div 2) (b div 2) =
   641    BITWISE n oper a b div 2 + SBIT (oper (bit n a) (bit n b)) (n - 1)"
   642   by (import word32 BITWISE_ISTEP)
   643 
   644 lemma BITWISE_EVAL: "ALL n oper a b.
   645    BITWISE (Suc n) oper a b =
   646    2 * BITWISE n oper (a div 2) (b div 2) + SBIT (oper (LSBn a) (LSBn b)) 0"
   647   by (import word32 BITWISE_EVAL)
   648 
   649 lemma BITWISE_WELLDEF: "ALL n oper a b c d.
   650    EQUIV a b & EQUIV c d --> EQUIV (BITWISE n oper a c) (BITWISE n oper b d)"
   651   by (import word32 BITWISE_WELLDEF)
   652 
   653 lemma BITWISEw_WELLDEF: "ALL oper a b c d.
   654    EQUIV a b & EQUIV c d -->
   655    EQUIV (BITWISE WL oper a c) (BITWISE WL oper b d)"
   656   by (import word32 BITWISEw_WELLDEF)
   657 
   658 lemma SUC_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (Suc a) (Suc b)"
   659   by (import word32 SUC_WELLDEF)
   660 
   661 lemma ADD_WELLDEF: "ALL a b c d. EQUIV a b & EQUIV c d --> EQUIV (a + c) (b + d)"
   662   by (import word32 ADD_WELLDEF)
   663 
   664 lemma MUL_WELLDEF: "ALL a b c d. EQUIV a b & EQUIV c d --> EQUIV (a * c) (b * d)"
   665   by (import word32 MUL_WELLDEF)
   666 
   667 lemma ONE_COMP_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (ONE_COMP a) (ONE_COMP b)"
   668   by (import word32 ONE_COMP_WELLDEF)
   669 
   670 lemma TWO_COMP_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (TWO_COMP a) (TWO_COMP b)"
   671   by (import word32 TWO_COMP_WELLDEF)
   672 
   673 lemma TOw_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (MODw a) (MODw b)"
   674   by (import word32 TOw_WELLDEF)
   675 
   676 consts
   677   LSR_ONE :: "nat => nat" 
   678 
   679 defs
   680   LSR_ONE_primdef: "LSR_ONE == %a. MODw a div 2"
   681 
   682 lemma LSR_ONE_def: "ALL a. LSR_ONE a = MODw a div 2"
   683   by (import word32 LSR_ONE_def)
   684 
   685 consts
   686   ASR_ONE :: "nat => nat" 
   687 
   688 defs
   689   ASR_ONE_primdef: "ASR_ONE == %a. LSR_ONE a + SBIT (MSBn a) HB"
   690 
   691 lemma ASR_ONE_def: "ALL a. ASR_ONE a = LSR_ONE a + SBIT (MSBn a) HB"
   692   by (import word32 ASR_ONE_def)
   693 
   694 consts
   695   ROR_ONE :: "nat => nat" 
   696 
   697 defs
   698   ROR_ONE_primdef: "ROR_ONE == %a. LSR_ONE a + SBIT (LSBn a) HB"
   699 
   700 lemma ROR_ONE_def: "ALL a. ROR_ONE a = LSR_ONE a + SBIT (LSBn a) HB"
   701   by (import word32 ROR_ONE_def)
   702 
   703 consts
   704   RRXn :: "bool => nat => nat" 
   705 
   706 defs
   707   RRXn_primdef: "RRXn == %c a. LSR_ONE a + SBIT c HB"
   708 
   709 lemma RRXn_def: "ALL c a. RRXn c a = LSR_ONE a + SBIT c HB"
   710   by (import word32 RRXn_def)
   711 
   712 lemma LSR_ONE_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (LSR_ONE a) (LSR_ONE b)"
   713   by (import word32 LSR_ONE_WELLDEF)
   714 
   715 lemma ASR_ONE_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (ASR_ONE a) (ASR_ONE b)"
   716   by (import word32 ASR_ONE_WELLDEF)
   717 
   718 lemma ROR_ONE_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (ROR_ONE a) (ROR_ONE b)"
   719   by (import word32 ROR_ONE_WELLDEF)
   720 
   721 lemma RRX_WELLDEF: "ALL a b c. EQUIV a b --> EQUIV (RRXn c a) (RRXn c b)"
   722   by (import word32 RRX_WELLDEF)
   723 
   724 lemma LSR_ONE: "LSR_ONE = BITS HB 1"
   725   by (import word32 LSR_ONE)
   726 
   727 typedef (open) word32 = "{x. EX xa. x = EQUIV xa}" 
   728   by (rule typedef_helper,import word32 word32_TY_DEF)
   729 
   730 lemmas word32_TY_DEF = typedef_hol2hol4 [OF type_definition_word32]
   731 
   732 consts
   733   mk_word32 :: "(nat => bool) => word32" 
   734   dest_word32 :: "word32 => nat => bool" 
   735 
   736 specification (dest_word32 mk_word32) word32_tybij: "(ALL a. mk_word32 (dest_word32 a) = a) &
   737 (ALL r. (EX x. r = EQUIV x) = (dest_word32 (mk_word32 r) = r))"
   738   by (import word32 word32_tybij)
   739 
   740 consts
   741   w_0 :: "word32" 
   742 
   743 defs
   744   w_0_primdef: "w_0 == mk_word32 (EQUIV 0)"
   745 
   746 lemma w_0_def: "w_0 = mk_word32 (EQUIV 0)"
   747   by (import word32 w_0_def)
   748 
   749 consts
   750   w_1 :: "word32" 
   751 
   752 defs
   753   w_1_primdef: "w_1 == mk_word32 (EQUIV AONE)"
   754 
   755 lemma w_1_def: "w_1 = mk_word32 (EQUIV AONE)"
   756   by (import word32 w_1_def)
   757 
   758 consts
   759   w_T :: "word32" 
   760 
   761 defs
   762   w_T_primdef: "w_T == mk_word32 (EQUIV COMP0)"
   763 
   764 lemma w_T_def: "w_T = mk_word32 (EQUIV COMP0)"
   765   by (import word32 w_T_def)
   766 
   767 constdefs
   768   word_suc :: "word32 => word32" 
   769   "word_suc == %T1. mk_word32 (EQUIV (Suc (Eps (dest_word32 T1))))"
   770 
   771 lemma word_suc: "ALL T1. word_suc T1 = mk_word32 (EQUIV (Suc (Eps (dest_word32 T1))))"
   772   by (import word32 word_suc)
   773 
   774 constdefs
   775   word_add :: "word32 => word32 => word32" 
   776   "word_add ==
   777 %T1 T2. mk_word32 (EQUIV (Eps (dest_word32 T1) + Eps (dest_word32 T2)))"
   778 
   779 lemma word_add: "ALL T1 T2.
   780    word_add T1 T2 =
   781    mk_word32 (EQUIV (Eps (dest_word32 T1) + Eps (dest_word32 T2)))"
   782   by (import word32 word_add)
   783 
   784 constdefs
   785   word_mul :: "word32 => word32 => word32" 
   786   "word_mul ==
   787 %T1 T2. mk_word32 (EQUIV (Eps (dest_word32 T1) * Eps (dest_word32 T2)))"
   788 
   789 lemma word_mul: "ALL T1 T2.
   790    word_mul T1 T2 =
   791    mk_word32 (EQUIV (Eps (dest_word32 T1) * Eps (dest_word32 T2)))"
   792   by (import word32 word_mul)
   793 
   794 constdefs
   795   word_1comp :: "word32 => word32" 
   796   "word_1comp == %T1. mk_word32 (EQUIV (ONE_COMP (Eps (dest_word32 T1))))"
   797 
   798 lemma word_1comp: "ALL T1. word_1comp T1 = mk_word32 (EQUIV (ONE_COMP (Eps (dest_word32 T1))))"
   799   by (import word32 word_1comp)
   800 
   801 constdefs
   802   word_2comp :: "word32 => word32" 
   803   "word_2comp == %T1. mk_word32 (EQUIV (TWO_COMP (Eps (dest_word32 T1))))"
   804 
   805 lemma word_2comp: "ALL T1. word_2comp T1 = mk_word32 (EQUIV (TWO_COMP (Eps (dest_word32 T1))))"
   806   by (import word32 word_2comp)
   807 
   808 constdefs
   809   word_lsr1 :: "word32 => word32" 
   810   "word_lsr1 == %T1. mk_word32 (EQUIV (LSR_ONE (Eps (dest_word32 T1))))"
   811 
   812 lemma word_lsr1: "ALL T1. word_lsr1 T1 = mk_word32 (EQUIV (LSR_ONE (Eps (dest_word32 T1))))"
   813   by (import word32 word_lsr1)
   814 
   815 constdefs
   816   word_asr1 :: "word32 => word32" 
   817   "word_asr1 == %T1. mk_word32 (EQUIV (ASR_ONE (Eps (dest_word32 T1))))"
   818 
   819 lemma word_asr1: "ALL T1. word_asr1 T1 = mk_word32 (EQUIV (ASR_ONE (Eps (dest_word32 T1))))"
   820   by (import word32 word_asr1)
   821 
   822 constdefs
   823   word_ror1 :: "word32 => word32" 
   824   "word_ror1 == %T1. mk_word32 (EQUIV (ROR_ONE (Eps (dest_word32 T1))))"
   825 
   826 lemma word_ror1: "ALL T1. word_ror1 T1 = mk_word32 (EQUIV (ROR_ONE (Eps (dest_word32 T1))))"
   827   by (import word32 word_ror1)
   828 
   829 consts
   830   RRX :: "bool => word32 => word32" 
   831 
   832 defs
   833   RRX_primdef: "RRX == %T1 T2. mk_word32 (EQUIV (RRXn T1 (Eps (dest_word32 T2))))"
   834 
   835 lemma RRX_def: "ALL T1 T2. RRX T1 T2 = mk_word32 (EQUIV (RRXn T1 (Eps (dest_word32 T2))))"
   836   by (import word32 RRX_def)
   837 
   838 consts
   839   LSB :: "word32 => bool" 
   840 
   841 defs
   842   LSB_primdef: "LSB == %T1. LSBn (Eps (dest_word32 T1))"
   843 
   844 lemma LSB_def: "ALL T1. LSB T1 = LSBn (Eps (dest_word32 T1))"
   845   by (import word32 LSB_def)
   846 
   847 consts
   848   MSB :: "word32 => bool" 
   849 
   850 defs
   851   MSB_primdef: "MSB == %T1. MSBn (Eps (dest_word32 T1))"
   852 
   853 lemma MSB_def: "ALL T1. MSB T1 = MSBn (Eps (dest_word32 T1))"
   854   by (import word32 MSB_def)
   855 
   856 constdefs
   857   bitwise_or :: "word32 => word32 => word32" 
   858   "bitwise_or ==
   859 %T1 T2. mk_word32 (EQUIV (OR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
   860 
   861 lemma bitwise_or: "ALL T1 T2.
   862    bitwise_or T1 T2 =
   863    mk_word32 (EQUIV (OR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
   864   by (import word32 bitwise_or)
   865 
   866 constdefs
   867   bitwise_eor :: "word32 => word32 => word32" 
   868   "bitwise_eor ==
   869 %T1 T2.
   870    mk_word32 (EQUIV (EOR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
   871 
   872 lemma bitwise_eor: "ALL T1 T2.
   873    bitwise_eor T1 T2 =
   874    mk_word32 (EQUIV (EOR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
   875   by (import word32 bitwise_eor)
   876 
   877 constdefs
   878   bitwise_and :: "word32 => word32 => word32" 
   879   "bitwise_and ==
   880 %T1 T2.
   881    mk_word32 (EQUIV (AND (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
   882 
   883 lemma bitwise_and: "ALL T1 T2.
   884    bitwise_and T1 T2 =
   885    mk_word32 (EQUIV (AND (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
   886   by (import word32 bitwise_and)
   887 
   888 consts
   889   TOw :: "word32 => word32" 
   890 
   891 defs
   892   TOw_primdef: "TOw == %T1. mk_word32 (EQUIV (MODw (Eps (dest_word32 T1))))"
   893 
   894 lemma TOw_def: "ALL T1. TOw T1 = mk_word32 (EQUIV (MODw (Eps (dest_word32 T1))))"
   895   by (import word32 TOw_def)
   896 
   897 consts
   898   n2w :: "nat => word32" 
   899 
   900 defs
   901   n2w_primdef: "n2w == %n. mk_word32 (EQUIV n)"
   902 
   903 lemma n2w_def: "ALL n. n2w n = mk_word32 (EQUIV n)"
   904   by (import word32 n2w_def)
   905 
   906 consts
   907   w2n :: "word32 => nat" 
   908 
   909 defs
   910   w2n_primdef: "w2n == %w. MODw (Eps (dest_word32 w))"
   911 
   912 lemma w2n_def: "ALL w. w2n w = MODw (Eps (dest_word32 w))"
   913   by (import word32 w2n_def)
   914 
   915 lemma ADDw: "(ALL x. word_add w_0 x = x) &
   916 (ALL x xa. word_add (word_suc x) xa = word_suc (word_add x xa))"
   917   by (import word32 ADDw)
   918 
   919 lemma ADD_0w: "ALL x. word_add x w_0 = x"
   920   by (import word32 ADD_0w)
   921 
   922 lemma ADD1w: "ALL x. word_suc x = word_add x w_1"
   923   by (import word32 ADD1w)
   924 
   925 lemma ADD_ASSOCw: "ALL x xa xb. word_add x (word_add xa xb) = word_add (word_add x xa) xb"
   926   by (import word32 ADD_ASSOCw)
   927 
   928 lemma ADD_CLAUSESw: "(ALL x. word_add w_0 x = x) &
   929 (ALL x. word_add x w_0 = x) &
   930 (ALL x xa. word_add (word_suc x) xa = word_suc (word_add x xa)) &
   931 (ALL x xa. word_add x (word_suc xa) = word_suc (word_add x xa))"
   932   by (import word32 ADD_CLAUSESw)
   933 
   934 lemma ADD_COMMw: "ALL x xa. word_add x xa = word_add xa x"
   935   by (import word32 ADD_COMMw)
   936 
   937 lemma ADD_INV_0_EQw: "ALL x xa. (word_add x xa = x) = (xa = w_0)"
   938   by (import word32 ADD_INV_0_EQw)
   939 
   940 lemma EQ_ADD_LCANCELw: "ALL x xa xb. (word_add x xa = word_add x xb) = (xa = xb)"
   941   by (import word32 EQ_ADD_LCANCELw)
   942 
   943 lemma EQ_ADD_RCANCELw: "ALL x xa xb. (word_add x xb = word_add xa xb) = (x = xa)"
   944   by (import word32 EQ_ADD_RCANCELw)
   945 
   946 lemma LEFT_ADD_DISTRIBw: "ALL x xa xb.
   947    word_mul xb (word_add x xa) = word_add (word_mul xb x) (word_mul xb xa)"
   948   by (import word32 LEFT_ADD_DISTRIBw)
   949 
   950 lemma MULT_ASSOCw: "ALL x xa xb. word_mul x (word_mul xa xb) = word_mul (word_mul x xa) xb"
   951   by (import word32 MULT_ASSOCw)
   952 
   953 lemma MULT_COMMw: "ALL x xa. word_mul x xa = word_mul xa x"
   954   by (import word32 MULT_COMMw)
   955 
   956 lemma MULT_CLAUSESw: "ALL x xa.
   957    word_mul w_0 x = w_0 &
   958    word_mul x w_0 = w_0 &
   959    word_mul w_1 x = x &
   960    word_mul x w_1 = x &
   961    word_mul (word_suc x) xa = word_add (word_mul x xa) xa &
   962    word_mul x (word_suc xa) = word_add x (word_mul x xa)"
   963   by (import word32 MULT_CLAUSESw)
   964 
   965 lemma TWO_COMP_ONE_COMP: "ALL x. word_2comp x = word_add (word_1comp x) w_1"
   966   by (import word32 TWO_COMP_ONE_COMP)
   967 
   968 lemma OR_ASSOCw: "ALL x xa xb.
   969    bitwise_or x (bitwise_or xa xb) = bitwise_or (bitwise_or x xa) xb"
   970   by (import word32 OR_ASSOCw)
   971 
   972 lemma OR_COMMw: "ALL x xa. bitwise_or x xa = bitwise_or xa x"
   973   by (import word32 OR_COMMw)
   974 
   975 lemma OR_IDEMw: "ALL x. bitwise_or x x = x"
   976   by (import word32 OR_IDEMw)
   977 
   978 lemma OR_ABSORBw: "ALL x xa. bitwise_and x (bitwise_or x xa) = x"
   979   by (import word32 OR_ABSORBw)
   980 
   981 lemma AND_ASSOCw: "ALL x xa xb.
   982    bitwise_and x (bitwise_and xa xb) = bitwise_and (bitwise_and x xa) xb"
   983   by (import word32 AND_ASSOCw)
   984 
   985 lemma AND_COMMw: "ALL x xa. bitwise_and x xa = bitwise_and xa x"
   986   by (import word32 AND_COMMw)
   987 
   988 lemma AND_IDEMw: "ALL x. bitwise_and x x = x"
   989   by (import word32 AND_IDEMw)
   990 
   991 lemma AND_ABSORBw: "ALL x xa. bitwise_or x (bitwise_and x xa) = x"
   992   by (import word32 AND_ABSORBw)
   993 
   994 lemma ONE_COMPw: "ALL x. word_1comp (word_1comp x) = x"
   995   by (import word32 ONE_COMPw)
   996 
   997 lemma RIGHT_AND_OVER_ORw: "ALL x xa xb.
   998    bitwise_and (bitwise_or x xa) xb =
   999    bitwise_or (bitwise_and x xb) (bitwise_and xa xb)"
  1000   by (import word32 RIGHT_AND_OVER_ORw)
  1001 
  1002 lemma RIGHT_OR_OVER_ANDw: "ALL x xa xb.
  1003    bitwise_or (bitwise_and x xa) xb =
  1004    bitwise_and (bitwise_or x xb) (bitwise_or xa xb)"
  1005   by (import word32 RIGHT_OR_OVER_ANDw)
  1006 
  1007 lemma DE_MORGAN_THMw: "ALL x xa.
  1008    word_1comp (bitwise_and x xa) =
  1009    bitwise_or (word_1comp x) (word_1comp xa) &
  1010    word_1comp (bitwise_or x xa) = bitwise_and (word_1comp x) (word_1comp xa)"
  1011   by (import word32 DE_MORGAN_THMw)
  1012 
  1013 lemma w_0: "w_0 = n2w 0"
  1014   by (import word32 w_0)
  1015 
  1016 lemma w_1: "w_1 = n2w 1"
  1017   by (import word32 w_1)
  1018 
  1019 lemma w_T: "w_T =
  1020 n2w (NUMERAL
  1021       (NUMERAL_BIT1
  1022         (NUMERAL_BIT1
  1023           (NUMERAL_BIT1
  1024             (NUMERAL_BIT1
  1025               (NUMERAL_BIT1
  1026                 (NUMERAL_BIT1
  1027                   (NUMERAL_BIT1
  1028                     (NUMERAL_BIT1
  1029                       (NUMERAL_BIT1
  1030                         (NUMERAL_BIT1
  1031                           (NUMERAL_BIT1
  1032                             (NUMERAL_BIT1
  1033                               (NUMERAL_BIT1
  1034                                 (NUMERAL_BIT1
  1035                                   (NUMERAL_BIT1
  1036                                     (NUMERAL_BIT1
  1037 (NUMERAL_BIT1
  1038   (NUMERAL_BIT1
  1039     (NUMERAL_BIT1
  1040       (NUMERAL_BIT1
  1041         (NUMERAL_BIT1
  1042           (NUMERAL_BIT1
  1043             (NUMERAL_BIT1
  1044               (NUMERAL_BIT1
  1045                 (NUMERAL_BIT1
  1046                   (NUMERAL_BIT1
  1047                     (NUMERAL_BIT1
  1048                       (NUMERAL_BIT1
  1049                         (NUMERAL_BIT1
  1050                           (NUMERAL_BIT1
  1051                             (NUMERAL_BIT1
  1052                               (NUMERAL_BIT1
  1053                                 ALT_ZERO)))))))))))))))))))))))))))))))))"
  1054   by (import word32 w_T)
  1055 
  1056 lemma ADD_TWO_COMP: "ALL x. word_add x (word_2comp x) = w_0"
  1057   by (import word32 ADD_TWO_COMP)
  1058 
  1059 lemma ADD_TWO_COMP2: "ALL x. word_add (word_2comp x) x = w_0"
  1060   by (import word32 ADD_TWO_COMP2)
  1061 
  1062 constdefs
  1063   word_sub :: "word32 => word32 => word32" 
  1064   "word_sub == %a b. word_add a (word_2comp b)"
  1065 
  1066 lemma word_sub: "ALL a b. word_sub a b = word_add a (word_2comp b)"
  1067   by (import word32 word_sub)
  1068 
  1069 constdefs
  1070   word_lsl :: "word32 => nat => word32" 
  1071   "word_lsl == %a n. word_mul a (n2w (2 ^ n))"
  1072 
  1073 lemma word_lsl: "ALL a n. word_lsl a n = word_mul a (n2w (2 ^ n))"
  1074   by (import word32 word_lsl)
  1075 
  1076 constdefs
  1077   word_lsr :: "word32 => nat => word32" 
  1078   "word_lsr == %a n. (word_lsr1 ^ n) a"
  1079 
  1080 lemma word_lsr: "ALL a n. word_lsr a n = (word_lsr1 ^ n) a"
  1081   by (import word32 word_lsr)
  1082 
  1083 constdefs
  1084   word_asr :: "word32 => nat => word32" 
  1085   "word_asr == %a n. (word_asr1 ^ n) a"
  1086 
  1087 lemma word_asr: "ALL a n. word_asr a n = (word_asr1 ^ n) a"
  1088   by (import word32 word_asr)
  1089 
  1090 constdefs
  1091   word_ror :: "word32 => nat => word32" 
  1092   "word_ror == %a n. (word_ror1 ^ n) a"
  1093 
  1094 lemma word_ror: "ALL a n. word_ror a n = (word_ror1 ^ n) a"
  1095   by (import word32 word_ror)
  1096 
  1097 consts
  1098   BITw :: "nat => word32 => bool" 
  1099 
  1100 defs
  1101   BITw_primdef: "BITw == %b n. bit b (w2n n)"
  1102 
  1103 lemma BITw_def: "ALL b n. BITw b n = bit b (w2n n)"
  1104   by (import word32 BITw_def)
  1105 
  1106 consts
  1107   BITSw :: "nat => nat => word32 => nat" 
  1108 
  1109 defs
  1110   BITSw_primdef: "BITSw == %h l n. BITS h l (w2n n)"
  1111 
  1112 lemma BITSw_def: "ALL h l n. BITSw h l n = BITS h l (w2n n)"
  1113   by (import word32 BITSw_def)
  1114 
  1115 consts
  1116   SLICEw :: "nat => nat => word32 => nat" 
  1117 
  1118 defs
  1119   SLICEw_primdef: "SLICEw == %h l n. SLICE h l (w2n n)"
  1120 
  1121 lemma SLICEw_def: "ALL h l n. SLICEw h l n = SLICE h l (w2n n)"
  1122   by (import word32 SLICEw_def)
  1123 
  1124 lemma TWO_COMP_ADD: "ALL a b. word_2comp (word_add a b) = word_add (word_2comp a) (word_2comp b)"
  1125   by (import word32 TWO_COMP_ADD)
  1126 
  1127 lemma TWO_COMP_ELIM: "ALL a. word_2comp (word_2comp a) = a"
  1128   by (import word32 TWO_COMP_ELIM)
  1129 
  1130 lemma ADD_SUB_ASSOC: "ALL a b c. word_sub (word_add a b) c = word_add a (word_sub b c)"
  1131   by (import word32 ADD_SUB_ASSOC)
  1132 
  1133 lemma ADD_SUB_SYM: "ALL a b c. word_sub (word_add a b) c = word_add (word_sub a c) b"
  1134   by (import word32 ADD_SUB_SYM)
  1135 
  1136 lemma SUB_EQUALw: "ALL a. word_sub a a = w_0"
  1137   by (import word32 SUB_EQUALw)
  1138 
  1139 lemma ADD_SUBw: "ALL a b. word_sub (word_add a b) b = a"
  1140   by (import word32 ADD_SUBw)
  1141 
  1142 lemma SUB_SUBw: "ALL a b c. word_sub a (word_sub b c) = word_sub (word_add a c) b"
  1143   by (import word32 SUB_SUBw)
  1144 
  1145 lemma ONE_COMP_TWO_COMP: "ALL a. word_1comp a = word_sub (word_2comp a) w_1"
  1146   by (import word32 ONE_COMP_TWO_COMP)
  1147 
  1148 lemma SUBw: "ALL m n. word_sub (word_suc m) n = word_suc (word_sub m n)"
  1149   by (import word32 SUBw)
  1150 
  1151 lemma ADD_EQ_SUBw: "ALL m n p. (word_add m n = p) = (m = word_sub p n)"
  1152   by (import word32 ADD_EQ_SUBw)
  1153 
  1154 lemma CANCEL_SUBw: "ALL m n p. (word_sub n p = word_sub m p) = (n = m)"
  1155   by (import word32 CANCEL_SUBw)
  1156 
  1157 lemma SUB_PLUSw: "ALL a b c. word_sub a (word_add b c) = word_sub (word_sub a b) c"
  1158   by (import word32 SUB_PLUSw)
  1159 
  1160 lemma word_nchotomy: "ALL w. EX n. w = n2w n"
  1161   by (import word32 word_nchotomy)
  1162 
  1163 lemma dest_word_mk_word_eq3: "ALL a. dest_word32 (mk_word32 (EQUIV a)) = EQUIV a"
  1164   by (import word32 dest_word_mk_word_eq3)
  1165 
  1166 lemma MODw_ELIM: "ALL n. n2w (MODw n) = n2w n"
  1167   by (import word32 MODw_ELIM)
  1168 
  1169 lemma w2n_EVAL: "ALL n. w2n (n2w n) = MODw n"
  1170   by (import word32 w2n_EVAL)
  1171 
  1172 lemma w2n_ELIM: "ALL a. n2w (w2n a) = a"
  1173   by (import word32 w2n_ELIM)
  1174 
  1175 lemma n2w_11: "ALL a b. (n2w a = n2w b) = (MODw a = MODw b)"
  1176   by (import word32 n2w_11)
  1177 
  1178 lemma ADD_EVAL: "word_add (n2w a) (n2w b) = n2w (a + b)"
  1179   by (import word32 ADD_EVAL)
  1180 
  1181 lemma MUL_EVAL: "word_mul (n2w a) (n2w b) = n2w (a * b)"
  1182   by (import word32 MUL_EVAL)
  1183 
  1184 lemma ONE_COMP_EVAL: "word_1comp (n2w a) = n2w (ONE_COMP a)"
  1185   by (import word32 ONE_COMP_EVAL)
  1186 
  1187 lemma TWO_COMP_EVAL: "word_2comp (n2w a) = n2w (TWO_COMP a)"
  1188   by (import word32 TWO_COMP_EVAL)
  1189 
  1190 lemma LSR_ONE_EVAL: "word_lsr1 (n2w a) = n2w (LSR_ONE a)"
  1191   by (import word32 LSR_ONE_EVAL)
  1192 
  1193 lemma ASR_ONE_EVAL: "word_asr1 (n2w a) = n2w (ASR_ONE a)"
  1194   by (import word32 ASR_ONE_EVAL)
  1195 
  1196 lemma ROR_ONE_EVAL: "word_ror1 (n2w a) = n2w (ROR_ONE a)"
  1197   by (import word32 ROR_ONE_EVAL)
  1198 
  1199 lemma RRX_EVAL: "RRX c (n2w a) = n2w (RRXn c a)"
  1200   by (import word32 RRX_EVAL)
  1201 
  1202 lemma LSB_EVAL: "LSB (n2w a) = LSBn a"
  1203   by (import word32 LSB_EVAL)
  1204 
  1205 lemma MSB_EVAL: "MSB (n2w a) = MSBn a"
  1206   by (import word32 MSB_EVAL)
  1207 
  1208 lemma OR_EVAL: "bitwise_or (n2w a) (n2w b) = n2w (OR a b)"
  1209   by (import word32 OR_EVAL)
  1210 
  1211 lemma EOR_EVAL: "bitwise_eor (n2w a) (n2w b) = n2w (EOR a b)"
  1212   by (import word32 EOR_EVAL)
  1213 
  1214 lemma AND_EVAL: "bitwise_and (n2w a) (n2w b) = n2w (AND a b)"
  1215   by (import word32 AND_EVAL)
  1216 
  1217 lemma BITS_EVAL: "ALL h l a. BITSw h l (n2w a) = BITS h l (MODw a)"
  1218   by (import word32 BITS_EVAL)
  1219 
  1220 lemma BIT_EVAL: "ALL b a. BITw b (n2w a) = bit b (MODw a)"
  1221   by (import word32 BIT_EVAL)
  1222 
  1223 lemma SLICE_EVAL: "ALL h l a. SLICEw h l (n2w a) = SLICE h l (MODw a)"
  1224   by (import word32 SLICE_EVAL)
  1225 
  1226 lemma LSL_ADD: "ALL a m n. word_lsl (word_lsl a m) n = word_lsl a (m + n)"
  1227   by (import word32 LSL_ADD)
  1228 
  1229 lemma LSR_ADD: "ALL x xa xb. word_lsr (word_lsr x xa) xb = word_lsr x (xa + xb)"
  1230   by (import word32 LSR_ADD)
  1231 
  1232 lemma ASR_ADD: "ALL x xa xb. word_asr (word_asr x xa) xb = word_asr x (xa + xb)"
  1233   by (import word32 ASR_ADD)
  1234 
  1235 lemma ROR_ADD: "ALL x xa xb. word_ror (word_ror x xa) xb = word_ror x (xa + xb)"
  1236   by (import word32 ROR_ADD)
  1237 
  1238 lemma LSL_LIMIT: "ALL w n. HB < n --> word_lsl w n = w_0"
  1239   by (import word32 LSL_LIMIT)
  1240 
  1241 lemma MOD_MOD_DIV: "ALL a b. INw (MODw a div 2 ^ b)"
  1242   by (import word32 MOD_MOD_DIV)
  1243 
  1244 lemma MOD_MOD_DIV_2EXP: "ALL a n. MODw (MODw a div 2 ^ n) div 2 = MODw a div 2 ^ Suc n"
  1245   by (import word32 MOD_MOD_DIV_2EXP)
  1246 
  1247 lemma LSR_EVAL: "ALL n. word_lsr (n2w a) n = n2w (MODw a div 2 ^ n)"
  1248   by (import word32 LSR_EVAL)
  1249 
  1250 lemma LSR_THM: "ALL x n. word_lsr (n2w n) x = n2w (BITS HB (min WL x) n)"
  1251   by (import word32 LSR_THM)
  1252 
  1253 lemma LSR_LIMIT: "ALL x w. HB < x --> word_lsr w x = w_0"
  1254   by (import word32 LSR_LIMIT)
  1255 
  1256 lemma LEFT_SHIFT_LESS: "ALL (n::nat) (m::nat) a::nat.
  1257    a < (2::nat) ^ m -->
  1258    (2::nat) ^ n + a * (2::nat) ^ n <= (2::nat) ^ (m + n)"
  1259   by (import word32 LEFT_SHIFT_LESS)
  1260 
  1261 lemma ROR_THM: "ALL x n.
  1262    word_ror (n2w n) x =
  1263    (let x' = x mod WL
  1264     in n2w (BITS HB x' n + BITS (x' - 1) 0 n * 2 ^ (WL - x')))"
  1265   by (import word32 ROR_THM)
  1266 
  1267 lemma ROR_CYCLE: "ALL x w. word_ror w (x * WL) = w"
  1268   by (import word32 ROR_CYCLE)
  1269 
  1270 lemma ASR_THM: "ALL x n.
  1271    word_asr (n2w n) x =
  1272    (let x' = min HB x; s = BITS HB x' n
  1273     in n2w (if MSBn n then 2 ^ WL - 2 ^ (WL - x') + s else s))"
  1274   by (import word32 ASR_THM)
  1275 
  1276 lemma ASR_LIMIT: "ALL x w. HB <= x --> word_asr w x = (if MSB w then w_T else w_0)"
  1277   by (import word32 ASR_LIMIT)
  1278 
  1279 lemma ZERO_SHIFT: "(ALL n. word_lsl w_0 n = w_0) &
  1280 (ALL n. word_asr w_0 n = w_0) &
  1281 (ALL n. word_lsr w_0 n = w_0) & (ALL n. word_ror w_0 n = w_0)"
  1282   by (import word32 ZERO_SHIFT)
  1283 
  1284 lemma ZERO_SHIFT2: "(ALL a. word_lsl a 0 = a) &
  1285 (ALL a. word_asr a 0 = a) &
  1286 (ALL a. word_lsr a 0 = a) & (ALL a. word_ror a 0 = a)"
  1287   by (import word32 ZERO_SHIFT2)
  1288 
  1289 lemma ASR_w_T: "ALL n. word_asr w_T n = w_T"
  1290   by (import word32 ASR_w_T)
  1291 
  1292 lemma ROR_w_T: "ALL n. word_ror w_T n = w_T"
  1293   by (import word32 ROR_w_T)
  1294 
  1295 lemma MODw_EVAL: "ALL x.
  1296    MODw x =
  1297    x mod
  1298    NUMERAL
  1299     (NUMERAL_BIT2
  1300       (NUMERAL_BIT1
  1301         (NUMERAL_BIT1
  1302           (NUMERAL_BIT1
  1303             (NUMERAL_BIT1
  1304               (NUMERAL_BIT1
  1305                 (NUMERAL_BIT1
  1306                   (NUMERAL_BIT1
  1307                     (NUMERAL_BIT1
  1308                       (NUMERAL_BIT1
  1309                         (NUMERAL_BIT1
  1310                           (NUMERAL_BIT1
  1311                             (NUMERAL_BIT1
  1312                               (NUMERAL_BIT1
  1313                                 (NUMERAL_BIT1
  1314                                   (NUMERAL_BIT1
  1315                                     (NUMERAL_BIT1
  1316 (NUMERAL_BIT1
  1317   (NUMERAL_BIT1
  1318     (NUMERAL_BIT1
  1319       (NUMERAL_BIT1
  1320         (NUMERAL_BIT1
  1321           (NUMERAL_BIT1
  1322             (NUMERAL_BIT1
  1323               (NUMERAL_BIT1
  1324                 (NUMERAL_BIT1
  1325                   (NUMERAL_BIT1
  1326                     (NUMERAL_BIT1
  1327                       (NUMERAL_BIT1
  1328                         (NUMERAL_BIT1
  1329                           (NUMERAL_BIT1
  1330                             (NUMERAL_BIT1
  1331                               ALT_ZERO))))))))))))))))))))))))))))))))"
  1332   by (import word32 MODw_EVAL)
  1333 
  1334 lemma ADD_EVAL2: "ALL b a. word_add (n2w a) (n2w b) = n2w (MODw (a + b))"
  1335   by (import word32 ADD_EVAL2)
  1336 
  1337 lemma MUL_EVAL2: "ALL b a. word_mul (n2w a) (n2w b) = n2w (MODw (a * b))"
  1338   by (import word32 MUL_EVAL2)
  1339 
  1340 lemma ONE_COMP_EVAL2: "ALL a.
  1341    word_1comp (n2w a) =
  1342    n2w (2 ^
  1343         NUMERAL
  1344          (NUMERAL_BIT2
  1345            (NUMERAL_BIT1
  1346              (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))) -
  1347         1 -
  1348         MODw a)"
  1349   by (import word32 ONE_COMP_EVAL2)
  1350 
  1351 lemma TWO_COMP_EVAL2: "ALL a.
  1352    word_2comp (n2w a) =
  1353    n2w (MODw
  1354          (2 ^
  1355           NUMERAL
  1356            (NUMERAL_BIT2
  1357              (NUMERAL_BIT1
  1358                (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))) -
  1359           MODw a))"
  1360   by (import word32 TWO_COMP_EVAL2)
  1361 
  1362 lemma LSR_ONE_EVAL2: "ALL a. word_lsr1 (n2w a) = n2w (MODw a div 2)"
  1363   by (import word32 LSR_ONE_EVAL2)
  1364 
  1365 lemma ASR_ONE_EVAL2: "ALL a.
  1366    word_asr1 (n2w a) =
  1367    n2w (MODw a div 2 +
  1368         SBIT (MSBn a)
  1369          (NUMERAL
  1370            (NUMERAL_BIT1
  1371              (NUMERAL_BIT1
  1372                (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))))"
  1373   by (import word32 ASR_ONE_EVAL2)
  1374 
  1375 lemma ROR_ONE_EVAL2: "ALL a.
  1376    word_ror1 (n2w a) =
  1377    n2w (MODw a div 2 +
  1378         SBIT (LSBn a)
  1379          (NUMERAL
  1380            (NUMERAL_BIT1
  1381              (NUMERAL_BIT1
  1382                (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))))"
  1383   by (import word32 ROR_ONE_EVAL2)
  1384 
  1385 lemma RRX_EVAL2: "ALL c a.
  1386    RRX c (n2w a) =
  1387    n2w (MODw a div 2 +
  1388         SBIT c
  1389          (NUMERAL
  1390            (NUMERAL_BIT1
  1391              (NUMERAL_BIT1
  1392                (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))))"
  1393   by (import word32 RRX_EVAL2)
  1394 
  1395 lemma LSB_EVAL2: "ALL a. LSB (n2w a) = ODD a"
  1396   by (import word32 LSB_EVAL2)
  1397 
  1398 lemma MSB_EVAL2: "ALL a.
  1399    MSB (n2w a) =
  1400    bit (NUMERAL
  1401          (NUMERAL_BIT1
  1402            (NUMERAL_BIT1
  1403              (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
  1404     a"
  1405   by (import word32 MSB_EVAL2)
  1406 
  1407 lemma OR_EVAL2: "ALL b a.
  1408    bitwise_or (n2w a) (n2w b) =
  1409    n2w (BITWISE
  1410          (NUMERAL
  1411            (NUMERAL_BIT2
  1412              (NUMERAL_BIT1
  1413                (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
  1414          op | a b)"
  1415   by (import word32 OR_EVAL2)
  1416 
  1417 lemma AND_EVAL2: "ALL b a.
  1418    bitwise_and (n2w a) (n2w b) =
  1419    n2w (BITWISE
  1420          (NUMERAL
  1421            (NUMERAL_BIT2
  1422              (NUMERAL_BIT1
  1423                (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
  1424          op & a b)"
  1425   by (import word32 AND_EVAL2)
  1426 
  1427 lemma EOR_EVAL2: "ALL b a.
  1428    bitwise_eor (n2w a) (n2w b) =
  1429    n2w (BITWISE
  1430          (NUMERAL
  1431            (NUMERAL_BIT2
  1432              (NUMERAL_BIT1
  1433                (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
  1434          (%x y. x ~= y) a b)"
  1435   by (import word32 EOR_EVAL2)
  1436 
  1437 lemma BITWISE_EVAL2: "ALL n oper x y.
  1438    BITWISE n oper x y =
  1439    (if n = 0 then 0
  1440     else 2 * BITWISE (n - 1) oper (x div 2) (y div 2) +
  1441          (if oper (ODD x) (ODD y) then 1 else 0))"
  1442   by (import word32 BITWISE_EVAL2)
  1443 
  1444 lemma BITSwLT_THM: "ALL h l n. BITSw h l n < 2 ^ (Suc h - l)"
  1445   by (import word32 BITSwLT_THM)
  1446 
  1447 lemma BITSw_COMP_THM: "ALL h1 l1 h2 l2 n.
  1448    h2 + l1 <= h1 -->
  1449    BITS h2 l2 (BITSw h1 l1 n) = BITSw (h2 + l1) (l2 + l1) n"
  1450   by (import word32 BITSw_COMP_THM)
  1451 
  1452 lemma BITSw_DIV_THM: "ALL h l n x. BITSw h l x div 2 ^ n = BITSw h (l + n) x"
  1453   by (import word32 BITSw_DIV_THM)
  1454 
  1455 lemma BITw_THM: "ALL b n. BITw b n = (BITSw b b n = 1)"
  1456   by (import word32 BITw_THM)
  1457 
  1458 lemma SLICEw_THM: "ALL n h l. SLICEw h l n = BITSw h l n * 2 ^ l"
  1459   by (import word32 SLICEw_THM)
  1460 
  1461 lemma BITS_SLICEw_THM: "ALL h l n. BITS h l (SLICEw h l n) = BITSw h l n"
  1462   by (import word32 BITS_SLICEw_THM)
  1463 
  1464 lemma SLICEw_ZERO_THM: "ALL n h. SLICEw h 0 n = BITSw h 0 n"
  1465   by (import word32 SLICEw_ZERO_THM)
  1466 
  1467 lemma SLICEw_COMP_THM: "ALL h m l a.
  1468    Suc m <= h & l <= m --> SLICEw h (Suc m) a + SLICEw m l a = SLICEw h l a"
  1469   by (import word32 SLICEw_COMP_THM)
  1470 
  1471 lemma BITSw_ZERO: "ALL h l n. h < l --> BITSw h l n = 0"
  1472   by (import word32 BITSw_ZERO)
  1473 
  1474 lemma SLICEw_ZERO: "ALL h l n. h < l --> SLICEw h l n = 0"
  1475   by (import word32 SLICEw_ZERO)
  1476 
  1477 ;end_setup
  1478 
  1479 end
  1480