--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Import/HOL/HOL4Word32.thy Fri Apr 02 17:37:45 2004 +0200
@@ -0,0 +1,1451 @@
+theory HOL4Word32 = HOL4Base:
+
+;setup_theory bits
+
+consts
+ DIV2 :: "nat => nat"
+
+defs
+ DIV2_primdef: "DIV2 == %n. n div 2"
+
+lemma DIV2_def: "ALL n. DIV2 n = n div 2"
+ by (import bits DIV2_def)
+
+consts
+ TIMES_2EXP :: "nat => nat => nat"
+
+defs
+ TIMES_2EXP_primdef: "TIMES_2EXP == %x n. n * 2 ^ x"
+
+lemma TIMES_2EXP_def: "ALL x n. TIMES_2EXP x n = n * 2 ^ x"
+ by (import bits TIMES_2EXP_def)
+
+consts
+ DIV_2EXP :: "nat => nat => nat"
+
+defs
+ DIV_2EXP_primdef: "DIV_2EXP == %x n. n div 2 ^ x"
+
+lemma DIV_2EXP_def: "ALL x n. DIV_2EXP x n = n div 2 ^ x"
+ by (import bits DIV_2EXP_def)
+
+consts
+ MOD_2EXP :: "nat => nat => nat"
+
+defs
+ MOD_2EXP_primdef: "MOD_2EXP == %x n. n mod 2 ^ x"
+
+lemma MOD_2EXP_def: "ALL x n. MOD_2EXP x n = n mod 2 ^ x"
+ by (import bits MOD_2EXP_def)
+
+consts
+ DIVMOD_2EXP :: "nat => nat => nat * nat"
+
+defs
+ DIVMOD_2EXP_primdef: "DIVMOD_2EXP == %x n. (n div 2 ^ x, n mod 2 ^ x)"
+
+lemma DIVMOD_2EXP_def: "ALL x n. DIVMOD_2EXP x n = (n div 2 ^ x, n mod 2 ^ x)"
+ by (import bits DIVMOD_2EXP_def)
+
+consts
+ SBIT :: "bool => nat => nat"
+
+defs
+ SBIT_primdef: "SBIT == %b n. if b then 2 ^ n else 0"
+
+lemma SBIT_def: "ALL b n. SBIT b n = (if b then 2 ^ n else 0)"
+ by (import bits SBIT_def)
+
+consts
+ BITS :: "nat => nat => nat => nat"
+
+defs
+ BITS_primdef: "BITS == %h l n. MOD_2EXP (Suc h - l) (DIV_2EXP l n)"
+
+lemma BITS_def: "ALL h l n. BITS h l n = MOD_2EXP (Suc h - l) (DIV_2EXP l n)"
+ by (import bits BITS_def)
+
+constdefs
+ bit :: "nat => nat => bool"
+ "bit == %b n. BITS b b n = 1"
+
+lemma BIT_def: "ALL b n. bit b n = (BITS b b n = 1)"
+ by (import bits BIT_def)
+
+consts
+ SLICE :: "nat => nat => nat => nat"
+
+defs
+ SLICE_primdef: "SLICE == %h l n. MOD_2EXP (Suc h) n - MOD_2EXP l n"
+
+lemma SLICE_def: "ALL h l n. SLICE h l n = MOD_2EXP (Suc h) n - MOD_2EXP l n"
+ by (import bits SLICE_def)
+
+consts
+ LSBn :: "nat => bool"
+
+defs
+ LSBn_primdef: "LSBn == bit 0"
+
+lemma LSBn_def: "LSBn = bit 0"
+ by (import bits LSBn_def)
+
+consts
+ BITWISE :: "nat => (bool => bool => bool) => nat => nat => nat"
+
+specification (BITWISE_primdef: BITWISE) BITWISE_def: "(ALL oper x y. BITWISE 0 oper x y = 0) &
+(ALL n oper x y.
+ BITWISE (Suc n) oper x y =
+ BITWISE n oper x y + SBIT (oper (bit n x) (bit n y)) n)"
+ by (import bits BITWISE_def)
+
+lemma DIV1: "ALL x::nat. x div (1::nat) = x"
+ by (import bits DIV1)
+
+lemma SUC_SUB: "Suc a - a = 1"
+ by (import bits SUC_SUB)
+
+lemma DIV_MULT_1: "ALL (r::nat) n::nat. r < n --> (n + r) div n = (1::nat)"
+ by (import bits DIV_MULT_1)
+
+lemma ZERO_LT_TWOEXP: "ALL n::nat. (0::nat) < (2::nat) ^ n"
+ by (import bits ZERO_LT_TWOEXP)
+
+lemma MOD_2EXP_LT: "ALL (n::nat) k::nat. k mod (2::nat) ^ n < (2::nat) ^ n"
+ by (import bits MOD_2EXP_LT)
+
+lemma TWOEXP_DIVISION: "ALL (n::nat) k::nat.
+ k = k div (2::nat) ^ n * (2::nat) ^ n + k mod (2::nat) ^ n"
+ by (import bits TWOEXP_DIVISION)
+
+lemma TWOEXP_MONO: "ALL (a::nat) b::nat. a < b --> (2::nat) ^ a < (2::nat) ^ b"
+ by (import bits TWOEXP_MONO)
+
+lemma TWOEXP_MONO2: "ALL (a::nat) b::nat. a <= b --> (2::nat) ^ a <= (2::nat) ^ b"
+ by (import bits TWOEXP_MONO2)
+
+lemma EXP_SUB_LESS_EQ: "ALL (a::nat) b::nat. (2::nat) ^ (a - b) <= (2::nat) ^ a"
+ by (import bits EXP_SUB_LESS_EQ)
+
+lemma BITS_THM: "ALL x xa xb. BITS x xa xb = xb div 2 ^ xa mod 2 ^ (Suc x - xa)"
+ by (import bits BITS_THM)
+
+lemma BITSLT_THM: "ALL h l n. BITS h l n < 2 ^ (Suc h - l)"
+ by (import bits BITSLT_THM)
+
+lemma DIV_MULT_LEM: "ALL (m::nat) n::nat. (0::nat) < n --> m div n * n <= m"
+ by (import bits DIV_MULT_LEM)
+
+lemma MOD_2EXP_LEM: "ALL (n::nat) x::nat.
+ n mod (2::nat) ^ x = n - n div (2::nat) ^ x * (2::nat) ^ x"
+ by (import bits MOD_2EXP_LEM)
+
+lemma BITS2_THM: "ALL h l n. BITS h l n = n mod 2 ^ Suc h div 2 ^ l"
+ by (import bits BITS2_THM)
+
+lemma BITS_COMP_THM: "ALL h1 l1 h2 l2 n.
+ h2 + l1 <= h1 --> BITS h2 l2 (BITS h1 l1 n) = BITS (h2 + l1) (l2 + l1) n"
+ by (import bits BITS_COMP_THM)
+
+lemma BITS_DIV_THM: "ALL h l x n. BITS h l x div 2 ^ n = BITS h (l + n) x"
+ by (import bits BITS_DIV_THM)
+
+lemma BITS_LT_HIGH: "ALL h l n. n < 2 ^ Suc h --> BITS h l n = n div 2 ^ l"
+ by (import bits BITS_LT_HIGH)
+
+lemma BITS_ZERO: "ALL h l n. h < l --> BITS h l n = 0"
+ by (import bits BITS_ZERO)
+
+lemma BITS_ZERO2: "ALL h l. BITS h l 0 = 0"
+ by (import bits BITS_ZERO2)
+
+lemma BITS_ZERO3: "ALL h x. BITS h 0 x = x mod 2 ^ Suc h"
+ by (import bits BITS_ZERO3)
+
+lemma BITS_COMP_THM2: "ALL h1 l1 h2 l2 n.
+ BITS h2 l2 (BITS h1 l1 n) = BITS (min h1 (h2 + l1)) (l2 + l1) n"
+ by (import bits BITS_COMP_THM2)
+
+lemma NOT_MOD2_LEM: "ALL n::nat. (n mod (2::nat) ~= (0::nat)) = (n mod (2::nat) = (1::nat))"
+ by (import bits NOT_MOD2_LEM)
+
+lemma NOT_MOD2_LEM2: "ALL (n::nat) a::'a.
+ (n mod (2::nat) ~= (1::nat)) = (n mod (2::nat) = (0::nat))"
+ by (import bits NOT_MOD2_LEM2)
+
+lemma EVEN_MOD2_LEM: "ALL n. EVEN n = (n mod 2 = 0)"
+ by (import bits EVEN_MOD2_LEM)
+
+lemma ODD_MOD2_LEM: "ALL n. ODD n = (n mod 2 = 1)"
+ by (import bits ODD_MOD2_LEM)
+
+lemma LSB_ODD: "LSBn = ODD"
+ by (import bits LSB_ODD)
+
+lemma DIV_MULT_THM: "ALL (x::nat) n::nat.
+ n div (2::nat) ^ x * (2::nat) ^ x = n - n mod (2::nat) ^ x"
+ by (import bits DIV_MULT_THM)
+
+lemma DIV_MULT_THM2: "ALL x::nat. (2::nat) * (x div (2::nat)) = x - x mod (2::nat)"
+ by (import bits DIV_MULT_THM2)
+
+lemma LESS_EQ_EXP_MULT: "ALL (a::nat) b::nat. a <= b --> (EX x::nat. (2::nat) ^ b = x * (2::nat) ^ a)"
+ by (import bits LESS_EQ_EXP_MULT)
+
+lemma SLICE_LEM1: "ALL (a::nat) (x::nat) y::nat.
+ a div (2::nat) ^ (x + y) * (2::nat) ^ (x + y) =
+ a div (2::nat) ^ x * (2::nat) ^ x -
+ a div (2::nat) ^ x mod (2::nat) ^ y * (2::nat) ^ x"
+ by (import bits SLICE_LEM1)
+
+lemma SLICE_LEM2: "ALL (a::'a) (x::nat) y::nat.
+ (n::nat) mod (2::nat) ^ (x + y) =
+ n mod (2::nat) ^ x + n div (2::nat) ^ x mod (2::nat) ^ y * (2::nat) ^ x"
+ by (import bits SLICE_LEM2)
+
+lemma SLICE_LEM3: "ALL (n::nat) (h::nat) l::nat.
+ l < h --> n mod (2::nat) ^ Suc l <= n mod (2::nat) ^ h"
+ by (import bits SLICE_LEM3)
+
+lemma SLICE_THM: "ALL n h l. SLICE h l n = BITS h l n * 2 ^ l"
+ by (import bits SLICE_THM)
+
+lemma SLICELT_THM: "ALL h l n. SLICE h l n < 2 ^ Suc h"
+ by (import bits SLICELT_THM)
+
+lemma BITS_SLICE_THM: "ALL h l n. BITS h l (SLICE h l n) = BITS h l n"
+ by (import bits BITS_SLICE_THM)
+
+lemma BITS_SLICE_THM2: "ALL h l n. h <= h2 --> BITS h2 l (SLICE h l n) = BITS h l n"
+ by (import bits BITS_SLICE_THM2)
+
+lemma MOD_2EXP_MONO: "ALL (n::nat) (h::nat) l::nat.
+ l <= h --> n mod (2::nat) ^ l <= n mod (2::nat) ^ Suc h"
+ by (import bits MOD_2EXP_MONO)
+
+lemma SLICE_COMP_THM: "ALL h m l n.
+ Suc m <= h & l <= m --> SLICE h (Suc m) n + SLICE m l n = SLICE h l n"
+ by (import bits SLICE_COMP_THM)
+
+lemma SLICE_ZERO: "ALL h l n. h < l --> SLICE h l n = 0"
+ by (import bits SLICE_ZERO)
+
+lemma BIT_COMP_THM3: "ALL h m l n.
+ Suc m <= h & l <= m -->
+ BITS h (Suc m) n * 2 ^ (Suc m - l) + BITS m l n = BITS h l n"
+ by (import bits BIT_COMP_THM3)
+
+lemma NOT_BIT: "ALL n a. (~ bit n a) = (BITS n n a = 0)"
+ by (import bits NOT_BIT)
+
+lemma NOT_BITS: "ALL n a. (BITS n n a ~= 0) = (BITS n n a = 1)"
+ by (import bits NOT_BITS)
+
+lemma NOT_BITS2: "ALL n a. (BITS n n a ~= 1) = (BITS n n a = 0)"
+ by (import bits NOT_BITS2)
+
+lemma BIT_SLICE: "ALL n a b. (bit n a = bit n b) = (SLICE n n a = SLICE n n b)"
+ by (import bits BIT_SLICE)
+
+lemma BIT_SLICE_LEM: "ALL y x n. SBIT (bit x n) (x + y) = SLICE x x n * 2 ^ y"
+ by (import bits BIT_SLICE_LEM)
+
+lemma BIT_SLICE_THM: "ALL x xa. SBIT (bit x xa) x = SLICE x x xa"
+ by (import bits BIT_SLICE_THM)
+
+lemma SBIT_DIV: "ALL b m n. n < m --> SBIT b (m - n) = SBIT b m div 2 ^ n"
+ by (import bits SBIT_DIV)
+
+lemma BITS_SUC: "ALL h l n.
+ l <= Suc h -->
+ SBIT (bit (Suc h) n) (Suc h - l) + BITS h l n = BITS (Suc h) l n"
+ by (import bits BITS_SUC)
+
+lemma BITS_SUC_THM: "ALL h l n.
+ BITS (Suc h) l n =
+ (if Suc h < l then 0 else SBIT (bit (Suc h) n) (Suc h - l) + BITS h l n)"
+ by (import bits BITS_SUC_THM)
+
+lemma BIT_BITS_THM: "ALL h l a b.
+ (ALL x. l <= x & x <= h --> bit x a = bit x b) =
+ (BITS h l a = BITS h l b)"
+ by (import bits BIT_BITS_THM)
+
+lemma BITWISE_LT_2EXP: "ALL n oper a b. BITWISE n oper a b < 2 ^ n"
+ by (import bits BITWISE_LT_2EXP)
+
+lemma LESS_EXP_MULT2: "ALL (a::nat) b::nat.
+ a < b -->
+ (EX x::nat. (2::nat) ^ b = (2::nat) ^ (x + (1::nat)) * (2::nat) ^ a)"
+ by (import bits LESS_EXP_MULT2)
+
+lemma BITWISE_THM: "ALL x n oper a b.
+ x < n --> bit x (BITWISE n oper a b) = oper (bit x a) (bit x b)"
+ by (import bits BITWISE_THM)
+
+lemma BITWISE_COR: "ALL x n oper a b.
+ x < n -->
+ oper (bit x a) (bit x b) --> BITWISE n oper a b div 2 ^ x mod 2 = 1"
+ by (import bits BITWISE_COR)
+
+lemma BITWISE_NOT_COR: "ALL x n oper a b.
+ x < n -->
+ ~ oper (bit x a) (bit x b) --> BITWISE n oper a b div 2 ^ x mod 2 = 0"
+ by (import bits BITWISE_NOT_COR)
+
+lemma MOD_PLUS_RIGHT: "ALL n::nat.
+ (0::nat) < n -->
+ (ALL (j::nat) k::nat. (j + k mod n) mod n = (j + k) mod n)"
+ by (import bits MOD_PLUS_RIGHT)
+
+lemma MOD_PLUS_1: "ALL n::nat.
+ (0::nat) < n -->
+ (ALL x::nat.
+ ((x + (1::nat)) mod n = (0::nat)) = (x mod n + (1::nat) = n))"
+ by (import bits MOD_PLUS_1)
+
+lemma MOD_ADD_1: "ALL n::nat.
+ (0::nat) < n -->
+ (ALL x::nat.
+ (x + (1::nat)) mod n ~= (0::nat) -->
+ (x + (1::nat)) mod n = x mod n + (1::nat))"
+ by (import bits MOD_ADD_1)
+
+;end_setup
+
+;setup_theory word32
+
+consts
+ HB :: "nat"
+
+defs
+ HB_primdef: "HB ==
+NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))"
+
+lemma HB_def: "HB =
+NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))"
+ by (import word32 HB_def)
+
+consts
+ WL :: "nat"
+
+defs
+ WL_primdef: "WL == Suc HB"
+
+lemma WL_def: "WL = Suc HB"
+ by (import word32 WL_def)
+
+consts
+ MODw :: "nat => nat"
+
+defs
+ MODw_primdef: "MODw == %n. n mod 2 ^ WL"
+
+lemma MODw_def: "ALL n. MODw n = n mod 2 ^ WL"
+ by (import word32 MODw_def)
+
+consts
+ INw :: "nat => bool"
+
+defs
+ INw_primdef: "INw == %n. n < 2 ^ WL"
+
+lemma INw_def: "ALL n. INw n = (n < 2 ^ WL)"
+ by (import word32 INw_def)
+
+consts
+ EQUIV :: "nat => nat => bool"
+
+defs
+ EQUIV_primdef: "EQUIV == %x y. MODw x = MODw y"
+
+lemma EQUIV_def: "ALL x y. EQUIV x y = (MODw x = MODw y)"
+ by (import word32 EQUIV_def)
+
+lemma EQUIV_QT: "ALL x y. EQUIV x y = (EQUIV x = EQUIV y)"
+ by (import word32 EQUIV_QT)
+
+lemma FUNPOW_THM: "ALL f n x. (f ^ n) (f x) = f ((f ^ n) x)"
+ by (import word32 FUNPOW_THM)
+
+lemma FUNPOW_THM2: "ALL f n x. (f ^ Suc n) x = f ((f ^ n) x)"
+ by (import word32 FUNPOW_THM2)
+
+lemma FUNPOW_COMP: "ALL f m n a. (f ^ m) ((f ^ n) a) = (f ^ (m + n)) a"
+ by (import word32 FUNPOW_COMP)
+
+lemma INw_MODw: "ALL n. INw (MODw n)"
+ by (import word32 INw_MODw)
+
+lemma TOw_IDEM: "ALL a. INw a --> MODw a = a"
+ by (import word32 TOw_IDEM)
+
+lemma MODw_IDEM2: "ALL a. MODw (MODw a) = MODw a"
+ by (import word32 MODw_IDEM2)
+
+lemma TOw_QT: "ALL a. EQUIV (MODw a) a"
+ by (import word32 TOw_QT)
+
+lemma MODw_THM: "MODw = BITS HB 0"
+ by (import word32 MODw_THM)
+
+lemma MOD_ADD: "ALL a b. MODw (a + b) = MODw (MODw a + MODw b)"
+ by (import word32 MOD_ADD)
+
+lemma MODw_MULT: "ALL a b. MODw (a * b) = MODw (MODw a * MODw b)"
+ by (import word32 MODw_MULT)
+
+consts
+ AONE :: "nat"
+
+defs
+ AONE_primdef: "AONE == 1"
+
+lemma AONE_def: "AONE = 1"
+ by (import word32 AONE_def)
+
+lemma ADD_QT: "(ALL n. EQUIV (0 + n) n) & (ALL m n. EQUIV (Suc m + n) (Suc (m + n)))"
+ by (import word32 ADD_QT)
+
+lemma ADD_0_QT: "ALL a. EQUIV (a + 0) a"
+ by (import word32 ADD_0_QT)
+
+lemma ADD_COMM_QT: "ALL a b. EQUIV (a + b) (b + a)"
+ by (import word32 ADD_COMM_QT)
+
+lemma ADD_ASSOC_QT: "ALL a b c. EQUIV (a + (b + c)) (a + b + c)"
+ by (import word32 ADD_ASSOC_QT)
+
+lemma MULT_QT: "(ALL n. EQUIV (0 * n) 0) & (ALL m n. EQUIV (Suc m * n) (m * n + n))"
+ by (import word32 MULT_QT)
+
+lemma ADD1_QT: "ALL m. EQUIV (Suc m) (m + AONE)"
+ by (import word32 ADD1_QT)
+
+lemma ADD_CLAUSES_QT: "(ALL m. EQUIV (0 + m) m) &
+(ALL m. EQUIV (m + 0) m) &
+(ALL m n. EQUIV (Suc m + n) (Suc (m + n))) &
+(ALL m n. EQUIV (m + Suc n) (Suc (m + n)))"
+ by (import word32 ADD_CLAUSES_QT)
+
+lemma SUC_EQUIV_COMP: "ALL a b. EQUIV (Suc a) b --> EQUIV a (b + (2 ^ WL - 1))"
+ by (import word32 SUC_EQUIV_COMP)
+
+lemma INV_SUC_EQ_QT: "ALL m n. EQUIV (Suc m) (Suc n) = EQUIV m n"
+ by (import word32 INV_SUC_EQ_QT)
+
+lemma ADD_INV_0_QT: "ALL m n. EQUIV (m + n) m --> EQUIV n 0"
+ by (import word32 ADD_INV_0_QT)
+
+lemma ADD_INV_0_EQ_QT: "ALL m n. EQUIV (m + n) m = EQUIV n 0"
+ by (import word32 ADD_INV_0_EQ_QT)
+
+lemma EQ_ADD_LCANCEL_QT: "ALL m n p. EQUIV (m + n) (m + p) = EQUIV n p"
+ by (import word32 EQ_ADD_LCANCEL_QT)
+
+lemma EQ_ADD_RCANCEL_QT: "ALL x xa xb. EQUIV (x + xb) (xa + xb) = EQUIV x xa"
+ by (import word32 EQ_ADD_RCANCEL_QT)
+
+lemma LEFT_ADD_DISTRIB_QT: "ALL m n p. EQUIV (p * (m + n)) (p * m + p * n)"
+ by (import word32 LEFT_ADD_DISTRIB_QT)
+
+lemma MULT_ASSOC_QT: "ALL m n p. EQUIV (m * (n * p)) (m * n * p)"
+ by (import word32 MULT_ASSOC_QT)
+
+lemma MULT_COMM_QT: "ALL m n. EQUIV (m * n) (n * m)"
+ by (import word32 MULT_COMM_QT)
+
+lemma MULT_CLAUSES_QT: "ALL m n.
+ EQUIV (0 * m) 0 &
+ EQUIV (m * 0) 0 &
+ EQUIV (AONE * m) m &
+ EQUIV (m * AONE) m &
+ EQUIV (Suc m * n) (m * n + n) & EQUIV (m * Suc n) (m + m * n)"
+ by (import word32 MULT_CLAUSES_QT)
+
+consts
+ MSBn :: "nat => bool"
+
+defs
+ MSBn_primdef: "MSBn == bit HB"
+
+lemma MSBn_def: "MSBn = bit HB"
+ by (import word32 MSBn_def)
+
+consts
+ ONE_COMP :: "nat => nat"
+
+defs
+ ONE_COMP_primdef: "ONE_COMP == %x. 2 ^ WL - 1 - MODw x"
+
+lemma ONE_COMP_def: "ALL x. ONE_COMP x = 2 ^ WL - 1 - MODw x"
+ by (import word32 ONE_COMP_def)
+
+consts
+ TWO_COMP :: "nat => nat"
+
+defs
+ TWO_COMP_primdef: "TWO_COMP == %x. 2 ^ WL - MODw x"
+
+lemma TWO_COMP_def: "ALL x. TWO_COMP x = 2 ^ WL - MODw x"
+ by (import word32 TWO_COMP_def)
+
+lemma ADD_TWO_COMP_QT: "ALL a. EQUIV (MODw a + TWO_COMP a) 0"
+ by (import word32 ADD_TWO_COMP_QT)
+
+lemma TWO_COMP_ONE_COMP_QT: "ALL a. EQUIV (TWO_COMP a) (ONE_COMP a + AONE)"
+ by (import word32 TWO_COMP_ONE_COMP_QT)
+
+lemma BIT_EQUIV_THM: "ALL x xa. (ALL xb<WL. bit xb x = bit xb xa) = EQUIV x xa"
+ by (import word32 BIT_EQUIV_THM)
+
+lemma BITS_SUC2: "ALL n a. BITS (Suc n) 0 a = SLICE (Suc n) (Suc n) a + BITS n 0 a"
+ by (import word32 BITS_SUC2)
+
+lemma BITWISE_ONE_COMP_THM: "ALL a b. BITWISE WL (%x y. ~ x) a b = ONE_COMP a"
+ by (import word32 BITWISE_ONE_COMP_THM)
+
+lemma ONE_COMP_THM: "ALL x xa. xa < WL --> bit xa (ONE_COMP x) = (~ bit xa x)"
+ by (import word32 ONE_COMP_THM)
+
+consts
+ OR :: "nat => nat => nat"
+
+defs
+ OR_primdef: "OR == BITWISE WL op |"
+
+lemma OR_def: "OR = BITWISE WL op |"
+ by (import word32 OR_def)
+
+consts
+ AND :: "nat => nat => nat"
+
+defs
+ AND_primdef: "AND == BITWISE WL op &"
+
+lemma AND_def: "AND = BITWISE WL op &"
+ by (import word32 AND_def)
+
+consts
+ EOR :: "nat => nat => nat"
+
+defs
+ EOR_primdef: "EOR == BITWISE WL (%x y. x ~= y)"
+
+lemma EOR_def: "EOR = BITWISE WL (%x y. x ~= y)"
+ by (import word32 EOR_def)
+
+consts
+ COMP0 :: "nat"
+
+defs
+ COMP0_primdef: "COMP0 == ONE_COMP 0"
+
+lemma COMP0_def: "COMP0 = ONE_COMP 0"
+ by (import word32 COMP0_def)
+
+lemma BITWISE_THM2: "ALL y oper a b.
+ (ALL x<WL. oper (bit x a) (bit x b) = bit x y) =
+ EQUIV (BITWISE WL oper a b) y"
+ by (import word32 BITWISE_THM2)
+
+lemma OR_ASSOC_QT: "ALL a b c. EQUIV (OR a (OR b c)) (OR (OR a b) c)"
+ by (import word32 OR_ASSOC_QT)
+
+lemma OR_COMM_QT: "ALL a b. EQUIV (OR a b) (OR b a)"
+ by (import word32 OR_COMM_QT)
+
+lemma OR_ABSORB_QT: "ALL a b. EQUIV (AND a (OR a b)) a"
+ by (import word32 OR_ABSORB_QT)
+
+lemma OR_IDEM_QT: "ALL a. EQUIV (OR a a) a"
+ by (import word32 OR_IDEM_QT)
+
+lemma AND_ASSOC_QT: "ALL a b c. EQUIV (AND a (AND b c)) (AND (AND a b) c)"
+ by (import word32 AND_ASSOC_QT)
+
+lemma AND_COMM_QT: "ALL a b. EQUIV (AND a b) (AND b a)"
+ by (import word32 AND_COMM_QT)
+
+lemma AND_ABSORB_QT: "ALL a b. EQUIV (OR a (AND a b)) a"
+ by (import word32 AND_ABSORB_QT)
+
+lemma AND_IDEM_QT: "ALL a. EQUIV (AND a a) a"
+ by (import word32 AND_IDEM_QT)
+
+lemma OR_COMP_QT: "ALL a. EQUIV (OR a (ONE_COMP a)) COMP0"
+ by (import word32 OR_COMP_QT)
+
+lemma AND_COMP_QT: "ALL a. EQUIV (AND a (ONE_COMP a)) 0"
+ by (import word32 AND_COMP_QT)
+
+lemma ONE_COMP_QT: "ALL a. EQUIV (ONE_COMP (ONE_COMP a)) a"
+ by (import word32 ONE_COMP_QT)
+
+lemma RIGHT_AND_OVER_OR_QT: "ALL a b c. EQUIV (AND (OR a b) c) (OR (AND a c) (AND b c))"
+ by (import word32 RIGHT_AND_OVER_OR_QT)
+
+lemma RIGHT_OR_OVER_AND_QT: "ALL a b c. EQUIV (OR (AND a b) c) (AND (OR a c) (OR b c))"
+ by (import word32 RIGHT_OR_OVER_AND_QT)
+
+lemma DE_MORGAN_THM_QT: "ALL a b.
+ EQUIV (ONE_COMP (AND a b)) (OR (ONE_COMP a) (ONE_COMP b)) &
+ EQUIV (ONE_COMP (OR a b)) (AND (ONE_COMP a) (ONE_COMP b))"
+ by (import word32 DE_MORGAN_THM_QT)
+
+lemma BIT_EQUIV: "ALL n a b. n < WL --> EQUIV a b --> bit n a = bit n b"
+ by (import word32 BIT_EQUIV)
+
+lemma LSB_WELLDEF: "ALL a b. EQUIV a b --> LSBn a = LSBn b"
+ by (import word32 LSB_WELLDEF)
+
+lemma MSB_WELLDEF: "ALL a b. EQUIV a b --> MSBn a = MSBn b"
+ by (import word32 MSB_WELLDEF)
+
+lemma BITWISE_ISTEP: "ALL n oper a b.
+ 0 < n -->
+ BITWISE n oper (a div 2) (b div 2) =
+ BITWISE n oper a b div 2 + SBIT (oper (bit n a) (bit n b)) (n - 1)"
+ by (import word32 BITWISE_ISTEP)
+
+lemma BITWISE_EVAL: "ALL n oper a b.
+ BITWISE (Suc n) oper a b =
+ 2 * BITWISE n oper (a div 2) (b div 2) + SBIT (oper (LSBn a) (LSBn b)) 0"
+ by (import word32 BITWISE_EVAL)
+
+lemma BITWISE_WELLDEF: "ALL n oper a b c d.
+ EQUIV a b & EQUIV c d --> EQUIV (BITWISE n oper a c) (BITWISE n oper b d)"
+ by (import word32 BITWISE_WELLDEF)
+
+lemma BITWISEw_WELLDEF: "ALL oper a b c d.
+ EQUIV a b & EQUIV c d -->
+ EQUIV (BITWISE WL oper a c) (BITWISE WL oper b d)"
+ by (import word32 BITWISEw_WELLDEF)
+
+lemma SUC_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (Suc a) (Suc b)"
+ by (import word32 SUC_WELLDEF)
+
+lemma ADD_WELLDEF: "ALL a b c d. EQUIV a b & EQUIV c d --> EQUIV (a + c) (b + d)"
+ by (import word32 ADD_WELLDEF)
+
+lemma MUL_WELLDEF: "ALL a b c d. EQUIV a b & EQUIV c d --> EQUIV (a * c) (b * d)"
+ by (import word32 MUL_WELLDEF)
+
+lemma ONE_COMP_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (ONE_COMP a) (ONE_COMP b)"
+ by (import word32 ONE_COMP_WELLDEF)
+
+lemma TWO_COMP_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (TWO_COMP a) (TWO_COMP b)"
+ by (import word32 TWO_COMP_WELLDEF)
+
+lemma TOw_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (MODw a) (MODw b)"
+ by (import word32 TOw_WELLDEF)
+
+consts
+ LSR_ONE :: "nat => nat"
+
+defs
+ LSR_ONE_primdef: "LSR_ONE == %a. MODw a div 2"
+
+lemma LSR_ONE_def: "ALL a. LSR_ONE a = MODw a div 2"
+ by (import word32 LSR_ONE_def)
+
+consts
+ ASR_ONE :: "nat => nat"
+
+defs
+ ASR_ONE_primdef: "ASR_ONE == %a. LSR_ONE a + SBIT (MSBn a) HB"
+
+lemma ASR_ONE_def: "ALL a. ASR_ONE a = LSR_ONE a + SBIT (MSBn a) HB"
+ by (import word32 ASR_ONE_def)
+
+consts
+ ROR_ONE :: "nat => nat"
+
+defs
+ ROR_ONE_primdef: "ROR_ONE == %a. LSR_ONE a + SBIT (LSBn a) HB"
+
+lemma ROR_ONE_def: "ALL a. ROR_ONE a = LSR_ONE a + SBIT (LSBn a) HB"
+ by (import word32 ROR_ONE_def)
+
+consts
+ RRXn :: "bool => nat => nat"
+
+defs
+ RRXn_primdef: "RRXn == %c a. LSR_ONE a + SBIT c HB"
+
+lemma RRXn_def: "ALL c a. RRXn c a = LSR_ONE a + SBIT c HB"
+ by (import word32 RRXn_def)
+
+lemma LSR_ONE_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (LSR_ONE a) (LSR_ONE b)"
+ by (import word32 LSR_ONE_WELLDEF)
+
+lemma ASR_ONE_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (ASR_ONE a) (ASR_ONE b)"
+ by (import word32 ASR_ONE_WELLDEF)
+
+lemma ROR_ONE_WELLDEF: "ALL a b. EQUIV a b --> EQUIV (ROR_ONE a) (ROR_ONE b)"
+ by (import word32 ROR_ONE_WELLDEF)
+
+lemma RRX_WELLDEF: "ALL a b c. EQUIV a b --> EQUIV (RRXn c a) (RRXn c b)"
+ by (import word32 RRX_WELLDEF)
+
+lemma LSR_ONE: "LSR_ONE = BITS HB 1"
+ by (import word32 LSR_ONE)
+
+typedef (open) word32 = "{x. EX xa. x = EQUIV xa}"
+ by (rule typedef_helper,import word32 word32_TY_DEF)
+
+lemmas word32_TY_DEF = typedef_hol2hol4 [OF type_definition_word32]
+
+consts
+ mk_word32 :: "(nat => bool) => word32"
+ dest_word32 :: "word32 => nat => bool"
+
+specification (dest_word32 mk_word32) word32_tybij: "(ALL a. mk_word32 (dest_word32 a) = a) &
+(ALL r. (EX x. r = EQUIV x) = (dest_word32 (mk_word32 r) = r))"
+ by (import word32 word32_tybij)
+
+consts
+ w_0 :: "word32"
+
+defs
+ w_0_primdef: "w_0 == mk_word32 (EQUIV 0)"
+
+lemma w_0_def: "w_0 = mk_word32 (EQUIV 0)"
+ by (import word32 w_0_def)
+
+consts
+ w_1 :: "word32"
+
+defs
+ w_1_primdef: "w_1 == mk_word32 (EQUIV AONE)"
+
+lemma w_1_def: "w_1 = mk_word32 (EQUIV AONE)"
+ by (import word32 w_1_def)
+
+consts
+ w_T :: "word32"
+
+defs
+ w_T_primdef: "w_T == mk_word32 (EQUIV COMP0)"
+
+lemma w_T_def: "w_T = mk_word32 (EQUIV COMP0)"
+ by (import word32 w_T_def)
+
+constdefs
+ word_suc :: "word32 => word32"
+ "word_suc == %T1. mk_word32 (EQUIV (Suc (Eps (dest_word32 T1))))"
+
+lemma word_suc: "ALL T1. word_suc T1 = mk_word32 (EQUIV (Suc (Eps (dest_word32 T1))))"
+ by (import word32 word_suc)
+
+constdefs
+ word_add :: "word32 => word32 => word32"
+ "word_add ==
+%T1 T2. mk_word32 (EQUIV (Eps (dest_word32 T1) + Eps (dest_word32 T2)))"
+
+lemma word_add: "ALL T1 T2.
+ word_add T1 T2 =
+ mk_word32 (EQUIV (Eps (dest_word32 T1) + Eps (dest_word32 T2)))"
+ by (import word32 word_add)
+
+constdefs
+ word_mul :: "word32 => word32 => word32"
+ "word_mul ==
+%T1 T2. mk_word32 (EQUIV (Eps (dest_word32 T1) * Eps (dest_word32 T2)))"
+
+lemma word_mul: "ALL T1 T2.
+ word_mul T1 T2 =
+ mk_word32 (EQUIV (Eps (dest_word32 T1) * Eps (dest_word32 T2)))"
+ by (import word32 word_mul)
+
+constdefs
+ word_1comp :: "word32 => word32"
+ "word_1comp == %T1. mk_word32 (EQUIV (ONE_COMP (Eps (dest_word32 T1))))"
+
+lemma word_1comp: "ALL T1. word_1comp T1 = mk_word32 (EQUIV (ONE_COMP (Eps (dest_word32 T1))))"
+ by (import word32 word_1comp)
+
+constdefs
+ word_2comp :: "word32 => word32"
+ "word_2comp == %T1. mk_word32 (EQUIV (TWO_COMP (Eps (dest_word32 T1))))"
+
+lemma word_2comp: "ALL T1. word_2comp T1 = mk_word32 (EQUIV (TWO_COMP (Eps (dest_word32 T1))))"
+ by (import word32 word_2comp)
+
+constdefs
+ word_lsr1 :: "word32 => word32"
+ "word_lsr1 == %T1. mk_word32 (EQUIV (LSR_ONE (Eps (dest_word32 T1))))"
+
+lemma word_lsr1: "ALL T1. word_lsr1 T1 = mk_word32 (EQUIV (LSR_ONE (Eps (dest_word32 T1))))"
+ by (import word32 word_lsr1)
+
+constdefs
+ word_asr1 :: "word32 => word32"
+ "word_asr1 == %T1. mk_word32 (EQUIV (ASR_ONE (Eps (dest_word32 T1))))"
+
+lemma word_asr1: "ALL T1. word_asr1 T1 = mk_word32 (EQUIV (ASR_ONE (Eps (dest_word32 T1))))"
+ by (import word32 word_asr1)
+
+constdefs
+ word_ror1 :: "word32 => word32"
+ "word_ror1 == %T1. mk_word32 (EQUIV (ROR_ONE (Eps (dest_word32 T1))))"
+
+lemma word_ror1: "ALL T1. word_ror1 T1 = mk_word32 (EQUIV (ROR_ONE (Eps (dest_word32 T1))))"
+ by (import word32 word_ror1)
+
+consts
+ RRX :: "bool => word32 => word32"
+
+defs
+ RRX_primdef: "RRX == %T1 T2. mk_word32 (EQUIV (RRXn T1 (Eps (dest_word32 T2))))"
+
+lemma RRX_def: "ALL T1 T2. RRX T1 T2 = mk_word32 (EQUIV (RRXn T1 (Eps (dest_word32 T2))))"
+ by (import word32 RRX_def)
+
+consts
+ LSB :: "word32 => bool"
+
+defs
+ LSB_primdef: "LSB == %T1. LSBn (Eps (dest_word32 T1))"
+
+lemma LSB_def: "ALL T1. LSB T1 = LSBn (Eps (dest_word32 T1))"
+ by (import word32 LSB_def)
+
+consts
+ MSB :: "word32 => bool"
+
+defs
+ MSB_primdef: "MSB == %T1. MSBn (Eps (dest_word32 T1))"
+
+lemma MSB_def: "ALL T1. MSB T1 = MSBn (Eps (dest_word32 T1))"
+ by (import word32 MSB_def)
+
+constdefs
+ bitwise_or :: "word32 => word32 => word32"
+ "bitwise_or ==
+%T1 T2. mk_word32 (EQUIV (OR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
+
+lemma bitwise_or: "ALL T1 T2.
+ bitwise_or T1 T2 =
+ mk_word32 (EQUIV (OR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
+ by (import word32 bitwise_or)
+
+constdefs
+ bitwise_eor :: "word32 => word32 => word32"
+ "bitwise_eor ==
+%T1 T2.
+ mk_word32 (EQUIV (EOR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
+
+lemma bitwise_eor: "ALL T1 T2.
+ bitwise_eor T1 T2 =
+ mk_word32 (EQUIV (EOR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
+ by (import word32 bitwise_eor)
+
+constdefs
+ bitwise_and :: "word32 => word32 => word32"
+ "bitwise_and ==
+%T1 T2.
+ mk_word32 (EQUIV (AND (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
+
+lemma bitwise_and: "ALL T1 T2.
+ bitwise_and T1 T2 =
+ mk_word32 (EQUIV (AND (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
+ by (import word32 bitwise_and)
+
+consts
+ TOw :: "word32 => word32"
+
+defs
+ TOw_primdef: "TOw == %T1. mk_word32 (EQUIV (MODw (Eps (dest_word32 T1))))"
+
+lemma TOw_def: "ALL T1. TOw T1 = mk_word32 (EQUIV (MODw (Eps (dest_word32 T1))))"
+ by (import word32 TOw_def)
+
+consts
+ n2w :: "nat => word32"
+
+defs
+ n2w_primdef: "n2w == %n. mk_word32 (EQUIV n)"
+
+lemma n2w_def: "ALL n. n2w n = mk_word32 (EQUIV n)"
+ by (import word32 n2w_def)
+
+consts
+ w2n :: "word32 => nat"
+
+defs
+ w2n_primdef: "w2n == %w. MODw (Eps (dest_word32 w))"
+
+lemma w2n_def: "ALL w. w2n w = MODw (Eps (dest_word32 w))"
+ by (import word32 w2n_def)
+
+lemma ADDw: "(ALL x. word_add w_0 x = x) &
+(ALL x xa. word_add (word_suc x) xa = word_suc (word_add x xa))"
+ by (import word32 ADDw)
+
+lemma ADD_0w: "ALL x. word_add x w_0 = x"
+ by (import word32 ADD_0w)
+
+lemma ADD1w: "ALL x. word_suc x = word_add x w_1"
+ by (import word32 ADD1w)
+
+lemma ADD_ASSOCw: "ALL x xa xb. word_add x (word_add xa xb) = word_add (word_add x xa) xb"
+ by (import word32 ADD_ASSOCw)
+
+lemma ADD_CLAUSESw: "(ALL x. word_add w_0 x = x) &
+(ALL x. word_add x w_0 = x) &
+(ALL x xa. word_add (word_suc x) xa = word_suc (word_add x xa)) &
+(ALL x xa. word_add x (word_suc xa) = word_suc (word_add x xa))"
+ by (import word32 ADD_CLAUSESw)
+
+lemma ADD_COMMw: "ALL x xa. word_add x xa = word_add xa x"
+ by (import word32 ADD_COMMw)
+
+lemma ADD_INV_0_EQw: "ALL x xa. (word_add x xa = x) = (xa = w_0)"
+ by (import word32 ADD_INV_0_EQw)
+
+lemma EQ_ADD_LCANCELw: "ALL x xa xb. (word_add x xa = word_add x xb) = (xa = xb)"
+ by (import word32 EQ_ADD_LCANCELw)
+
+lemma EQ_ADD_RCANCELw: "ALL x xa xb. (word_add x xb = word_add xa xb) = (x = xa)"
+ by (import word32 EQ_ADD_RCANCELw)
+
+lemma LEFT_ADD_DISTRIBw: "ALL x xa xb.
+ word_mul xb (word_add x xa) = word_add (word_mul xb x) (word_mul xb xa)"
+ by (import word32 LEFT_ADD_DISTRIBw)
+
+lemma MULT_ASSOCw: "ALL x xa xb. word_mul x (word_mul xa xb) = word_mul (word_mul x xa) xb"
+ by (import word32 MULT_ASSOCw)
+
+lemma MULT_COMMw: "ALL x xa. word_mul x xa = word_mul xa x"
+ by (import word32 MULT_COMMw)
+
+lemma MULT_CLAUSESw: "ALL x xa.
+ word_mul w_0 x = w_0 &
+ word_mul x w_0 = w_0 &
+ word_mul w_1 x = x &
+ word_mul x w_1 = x &
+ word_mul (word_suc x) xa = word_add (word_mul x xa) xa &
+ word_mul x (word_suc xa) = word_add x (word_mul x xa)"
+ by (import word32 MULT_CLAUSESw)
+
+lemma TWO_COMP_ONE_COMP: "ALL x. word_2comp x = word_add (word_1comp x) w_1"
+ by (import word32 TWO_COMP_ONE_COMP)
+
+lemma OR_ASSOCw: "ALL x xa xb.
+ bitwise_or x (bitwise_or xa xb) = bitwise_or (bitwise_or x xa) xb"
+ by (import word32 OR_ASSOCw)
+
+lemma OR_COMMw: "ALL x xa. bitwise_or x xa = bitwise_or xa x"
+ by (import word32 OR_COMMw)
+
+lemma OR_IDEMw: "ALL x. bitwise_or x x = x"
+ by (import word32 OR_IDEMw)
+
+lemma OR_ABSORBw: "ALL x xa. bitwise_and x (bitwise_or x xa) = x"
+ by (import word32 OR_ABSORBw)
+
+lemma AND_ASSOCw: "ALL x xa xb.
+ bitwise_and x (bitwise_and xa xb) = bitwise_and (bitwise_and x xa) xb"
+ by (import word32 AND_ASSOCw)
+
+lemma AND_COMMw: "ALL x xa. bitwise_and x xa = bitwise_and xa x"
+ by (import word32 AND_COMMw)
+
+lemma AND_IDEMw: "ALL x. bitwise_and x x = x"
+ by (import word32 AND_IDEMw)
+
+lemma AND_ABSORBw: "ALL x xa. bitwise_or x (bitwise_and x xa) = x"
+ by (import word32 AND_ABSORBw)
+
+lemma ONE_COMPw: "ALL x. word_1comp (word_1comp x) = x"
+ by (import word32 ONE_COMPw)
+
+lemma RIGHT_AND_OVER_ORw: "ALL x xa xb.
+ bitwise_and (bitwise_or x xa) xb =
+ bitwise_or (bitwise_and x xb) (bitwise_and xa xb)"
+ by (import word32 RIGHT_AND_OVER_ORw)
+
+lemma RIGHT_OR_OVER_ANDw: "ALL x xa xb.
+ bitwise_or (bitwise_and x xa) xb =
+ bitwise_and (bitwise_or x xb) (bitwise_or xa xb)"
+ by (import word32 RIGHT_OR_OVER_ANDw)
+
+lemma DE_MORGAN_THMw: "ALL x xa.
+ word_1comp (bitwise_and x xa) =
+ bitwise_or (word_1comp x) (word_1comp xa) &
+ word_1comp (bitwise_or x xa) = bitwise_and (word_1comp x) (word_1comp xa)"
+ by (import word32 DE_MORGAN_THMw)
+
+lemma w_0: "w_0 = n2w 0"
+ by (import word32 w_0)
+
+lemma w_1: "w_1 = n2w 1"
+ by (import word32 w_1)
+
+lemma w_T: "w_T =
+n2w (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+(NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ ALT_ZERO)))))))))))))))))))))))))))))))))"
+ by (import word32 w_T)
+
+lemma ADD_TWO_COMP: "ALL x. word_add x (word_2comp x) = w_0"
+ by (import word32 ADD_TWO_COMP)
+
+lemma ADD_TWO_COMP2: "ALL x. word_add (word_2comp x) x = w_0"
+ by (import word32 ADD_TWO_COMP2)
+
+constdefs
+ word_sub :: "word32 => word32 => word32"
+ "word_sub == %a b. word_add a (word_2comp b)"
+
+lemma word_sub: "ALL a b. word_sub a b = word_add a (word_2comp b)"
+ by (import word32 word_sub)
+
+constdefs
+ word_lsl :: "word32 => nat => word32"
+ "word_lsl == %a n. word_mul a (n2w (2 ^ n))"
+
+lemma word_lsl: "ALL a n. word_lsl a n = word_mul a (n2w (2 ^ n))"
+ by (import word32 word_lsl)
+
+constdefs
+ word_lsr :: "word32 => nat => word32"
+ "word_lsr == %a n. (word_lsr1 ^ n) a"
+
+lemma word_lsr: "ALL a n. word_lsr a n = (word_lsr1 ^ n) a"
+ by (import word32 word_lsr)
+
+constdefs
+ word_asr :: "word32 => nat => word32"
+ "word_asr == %a n. (word_asr1 ^ n) a"
+
+lemma word_asr: "ALL a n. word_asr a n = (word_asr1 ^ n) a"
+ by (import word32 word_asr)
+
+constdefs
+ word_ror :: "word32 => nat => word32"
+ "word_ror == %a n. (word_ror1 ^ n) a"
+
+lemma word_ror: "ALL a n. word_ror a n = (word_ror1 ^ n) a"
+ by (import word32 word_ror)
+
+consts
+ BITw :: "nat => word32 => bool"
+
+defs
+ BITw_primdef: "BITw == %b n. bit b (w2n n)"
+
+lemma BITw_def: "ALL b n. BITw b n = bit b (w2n n)"
+ by (import word32 BITw_def)
+
+consts
+ BITSw :: "nat => nat => word32 => nat"
+
+defs
+ BITSw_primdef: "BITSw == %h l n. BITS h l (w2n n)"
+
+lemma BITSw_def: "ALL h l n. BITSw h l n = BITS h l (w2n n)"
+ by (import word32 BITSw_def)
+
+consts
+ SLICEw :: "nat => nat => word32 => nat"
+
+defs
+ SLICEw_primdef: "SLICEw == %h l n. SLICE h l (w2n n)"
+
+lemma SLICEw_def: "ALL h l n. SLICEw h l n = SLICE h l (w2n n)"
+ by (import word32 SLICEw_def)
+
+lemma TWO_COMP_ADD: "ALL a b. word_2comp (word_add a b) = word_add (word_2comp a) (word_2comp b)"
+ by (import word32 TWO_COMP_ADD)
+
+lemma TWO_COMP_ELIM: "ALL a. word_2comp (word_2comp a) = a"
+ by (import word32 TWO_COMP_ELIM)
+
+lemma ADD_SUB_ASSOC: "ALL a b c. word_sub (word_add a b) c = word_add a (word_sub b c)"
+ by (import word32 ADD_SUB_ASSOC)
+
+lemma ADD_SUB_SYM: "ALL a b c. word_sub (word_add a b) c = word_add (word_sub a c) b"
+ by (import word32 ADD_SUB_SYM)
+
+lemma SUB_EQUALw: "ALL a. word_sub a a = w_0"
+ by (import word32 SUB_EQUALw)
+
+lemma ADD_SUBw: "ALL a b. word_sub (word_add a b) b = a"
+ by (import word32 ADD_SUBw)
+
+lemma SUB_SUBw: "ALL a b c. word_sub a (word_sub b c) = word_sub (word_add a c) b"
+ by (import word32 SUB_SUBw)
+
+lemma ONE_COMP_TWO_COMP: "ALL a. word_1comp a = word_sub (word_2comp a) w_1"
+ by (import word32 ONE_COMP_TWO_COMP)
+
+lemma SUBw: "ALL m n. word_sub (word_suc m) n = word_suc (word_sub m n)"
+ by (import word32 SUBw)
+
+lemma ADD_EQ_SUBw: "ALL m n p. (word_add m n = p) = (m = word_sub p n)"
+ by (import word32 ADD_EQ_SUBw)
+
+lemma CANCEL_SUBw: "ALL m n p. (word_sub n p = word_sub m p) = (n = m)"
+ by (import word32 CANCEL_SUBw)
+
+lemma SUB_PLUSw: "ALL a b c. word_sub a (word_add b c) = word_sub (word_sub a b) c"
+ by (import word32 SUB_PLUSw)
+
+lemma word_nchotomy: "ALL w. EX n. w = n2w n"
+ by (import word32 word_nchotomy)
+
+lemma dest_word_mk_word_eq3: "ALL a. dest_word32 (mk_word32 (EQUIV a)) = EQUIV a"
+ by (import word32 dest_word_mk_word_eq3)
+
+lemma MODw_ELIM: "ALL n. n2w (MODw n) = n2w n"
+ by (import word32 MODw_ELIM)
+
+lemma w2n_EVAL: "ALL n. w2n (n2w n) = MODw n"
+ by (import word32 w2n_EVAL)
+
+lemma w2n_ELIM: "ALL a. n2w (w2n a) = a"
+ by (import word32 w2n_ELIM)
+
+lemma n2w_11: "ALL a b. (n2w a = n2w b) = (MODw a = MODw b)"
+ by (import word32 n2w_11)
+
+lemma ADD_EVAL: "word_add (n2w a) (n2w b) = n2w (a + b)"
+ by (import word32 ADD_EVAL)
+
+lemma MUL_EVAL: "word_mul (n2w a) (n2w b) = n2w (a * b)"
+ by (import word32 MUL_EVAL)
+
+lemma ONE_COMP_EVAL: "word_1comp (n2w a) = n2w (ONE_COMP a)"
+ by (import word32 ONE_COMP_EVAL)
+
+lemma TWO_COMP_EVAL: "word_2comp (n2w a) = n2w (TWO_COMP a)"
+ by (import word32 TWO_COMP_EVAL)
+
+lemma LSR_ONE_EVAL: "word_lsr1 (n2w a) = n2w (LSR_ONE a)"
+ by (import word32 LSR_ONE_EVAL)
+
+lemma ASR_ONE_EVAL: "word_asr1 (n2w a) = n2w (ASR_ONE a)"
+ by (import word32 ASR_ONE_EVAL)
+
+lemma ROR_ONE_EVAL: "word_ror1 (n2w a) = n2w (ROR_ONE a)"
+ by (import word32 ROR_ONE_EVAL)
+
+lemma RRX_EVAL: "RRX c (n2w a) = n2w (RRXn c a)"
+ by (import word32 RRX_EVAL)
+
+lemma LSB_EVAL: "LSB (n2w a) = LSBn a"
+ by (import word32 LSB_EVAL)
+
+lemma MSB_EVAL: "MSB (n2w a) = MSBn a"
+ by (import word32 MSB_EVAL)
+
+lemma OR_EVAL: "bitwise_or (n2w a) (n2w b) = n2w (OR a b)"
+ by (import word32 OR_EVAL)
+
+lemma EOR_EVAL: "bitwise_eor (n2w a) (n2w b) = n2w (EOR a b)"
+ by (import word32 EOR_EVAL)
+
+lemma AND_EVAL: "bitwise_and (n2w a) (n2w b) = n2w (AND a b)"
+ by (import word32 AND_EVAL)
+
+lemma BITS_EVAL: "ALL h l a. BITSw h l (n2w a) = BITS h l (MODw a)"
+ by (import word32 BITS_EVAL)
+
+lemma BIT_EVAL: "ALL b a. BITw b (n2w a) = bit b (MODw a)"
+ by (import word32 BIT_EVAL)
+
+lemma SLICE_EVAL: "ALL h l a. SLICEw h l (n2w a) = SLICE h l (MODw a)"
+ by (import word32 SLICE_EVAL)
+
+lemma LSL_ADD: "ALL a m n. word_lsl (word_lsl a m) n = word_lsl a (m + n)"
+ by (import word32 LSL_ADD)
+
+lemma LSR_ADD: "ALL x xa xb. word_lsr (word_lsr x xa) xb = word_lsr x (xa + xb)"
+ by (import word32 LSR_ADD)
+
+lemma ASR_ADD: "ALL x xa xb. word_asr (word_asr x xa) xb = word_asr x (xa + xb)"
+ by (import word32 ASR_ADD)
+
+lemma ROR_ADD: "ALL x xa xb. word_ror (word_ror x xa) xb = word_ror x (xa + xb)"
+ by (import word32 ROR_ADD)
+
+lemma LSL_LIMIT: "ALL w n. HB < n --> word_lsl w n = w_0"
+ by (import word32 LSL_LIMIT)
+
+lemma MOD_MOD_DIV: "ALL a b. INw (MODw a div 2 ^ b)"
+ by (import word32 MOD_MOD_DIV)
+
+lemma MOD_MOD_DIV_2EXP: "ALL a n. MODw (MODw a div 2 ^ n) div 2 = MODw a div 2 ^ Suc n"
+ by (import word32 MOD_MOD_DIV_2EXP)
+
+lemma LSR_EVAL: "ALL n. word_lsr (n2w a) n = n2w (MODw a div 2 ^ n)"
+ by (import word32 LSR_EVAL)
+
+lemma LSR_THM: "ALL x n. word_lsr (n2w n) x = n2w (BITS HB (min WL x) n)"
+ by (import word32 LSR_THM)
+
+lemma LSR_LIMIT: "ALL x w. HB < x --> word_lsr w x = w_0"
+ by (import word32 LSR_LIMIT)
+
+lemma LEFT_SHIFT_LESS: "ALL (n::nat) (m::nat) a::nat.
+ a < (2::nat) ^ m -->
+ (2::nat) ^ n + a * (2::nat) ^ n <= (2::nat) ^ (m + n)"
+ by (import word32 LEFT_SHIFT_LESS)
+
+lemma ROR_THM: "ALL x n.
+ word_ror (n2w n) x =
+ (let x' = x mod WL
+ in n2w (BITS HB x' n + BITS (x' - 1) 0 n * 2 ^ (WL - x')))"
+ by (import word32 ROR_THM)
+
+lemma ROR_CYCLE: "ALL x w. word_ror w (x * WL) = w"
+ by (import word32 ROR_CYCLE)
+
+lemma ASR_THM: "ALL x n.
+ word_asr (n2w n) x =
+ (let x' = min HB x; s = BITS HB x' n
+ in n2w (if MSBn n then 2 ^ WL - 2 ^ (WL - x') + s else s))"
+ by (import word32 ASR_THM)
+
+lemma ASR_LIMIT: "ALL x w. HB <= x --> word_asr w x = (if MSB w then w_T else w_0)"
+ by (import word32 ASR_LIMIT)
+
+lemma ZERO_SHIFT: "(ALL n. word_lsl w_0 n = w_0) &
+(ALL n. word_asr w_0 n = w_0) &
+(ALL n. word_lsr w_0 n = w_0) & (ALL n. word_ror w_0 n = w_0)"
+ by (import word32 ZERO_SHIFT)
+
+lemma ZERO_SHIFT2: "(ALL a. word_lsl a 0 = a) &
+(ALL a. word_asr a 0 = a) &
+(ALL a. word_lsr a 0 = a) & (ALL a. word_ror a 0 = a)"
+ by (import word32 ZERO_SHIFT2)
+
+lemma ASR_w_T: "ALL n. word_asr w_T n = w_T"
+ by (import word32 ASR_w_T)
+
+lemma ROR_w_T: "ALL n. word_ror w_T n = w_T"
+ by (import word32 ROR_w_T)
+
+lemma MODw_EVAL: "ALL x.
+ MODw x =
+ x mod
+ NUMERAL
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+(NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ ALT_ZERO))))))))))))))))))))))))))))))))"
+ by (import word32 MODw_EVAL)
+
+lemma ADD_EVAL2: "ALL b a. word_add (n2w a) (n2w b) = n2w (MODw (a + b))"
+ by (import word32 ADD_EVAL2)
+
+lemma MUL_EVAL2: "ALL b a. word_mul (n2w a) (n2w b) = n2w (MODw (a * b))"
+ by (import word32 MUL_EVAL2)
+
+lemma ONE_COMP_EVAL2: "ALL a.
+ word_1comp (n2w a) =
+ n2w (2 ^
+ NUMERAL
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))) -
+ 1 -
+ MODw a)"
+ by (import word32 ONE_COMP_EVAL2)
+
+lemma TWO_COMP_EVAL2: "ALL a.
+ word_2comp (n2w a) =
+ n2w (MODw
+ (2 ^
+ NUMERAL
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))) -
+ MODw a))"
+ by (import word32 TWO_COMP_EVAL2)
+
+lemma LSR_ONE_EVAL2: "ALL a. word_lsr1 (n2w a) = n2w (MODw a div 2)"
+ by (import word32 LSR_ONE_EVAL2)
+
+lemma ASR_ONE_EVAL2: "ALL a.
+ word_asr1 (n2w a) =
+ n2w (MODw a div 2 +
+ SBIT (MSBn a)
+ (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))))"
+ by (import word32 ASR_ONE_EVAL2)
+
+lemma ROR_ONE_EVAL2: "ALL a.
+ word_ror1 (n2w a) =
+ n2w (MODw a div 2 +
+ SBIT (LSBn a)
+ (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))))"
+ by (import word32 ROR_ONE_EVAL2)
+
+lemma RRX_EVAL2: "ALL c a.
+ RRX c (n2w a) =
+ n2w (MODw a div 2 +
+ SBIT c
+ (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))))"
+ by (import word32 RRX_EVAL2)
+
+lemma LSB_EVAL2: "ALL a. LSB (n2w a) = ODD a"
+ by (import word32 LSB_EVAL2)
+
+lemma MSB_EVAL2: "ALL a.
+ MSB (n2w a) =
+ bit (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
+ a"
+ by (import word32 MSB_EVAL2)
+
+lemma OR_EVAL2: "ALL b a.
+ bitwise_or (n2w a) (n2w b) =
+ n2w (BITWISE
+ (NUMERAL
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
+ op | a b)"
+ by (import word32 OR_EVAL2)
+
+lemma AND_EVAL2: "ALL b a.
+ bitwise_and (n2w a) (n2w b) =
+ n2w (BITWISE
+ (NUMERAL
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
+ op & a b)"
+ by (import word32 AND_EVAL2)
+
+lemma EOR_EVAL2: "ALL b a.
+ bitwise_eor (n2w a) (n2w b) =
+ n2w (BITWISE
+ (NUMERAL
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
+ (%x y. x ~= y) a b)"
+ by (import word32 EOR_EVAL2)
+
+lemma BITWISE_EVAL2: "ALL n oper x y.
+ BITWISE n oper x y =
+ (if n = 0 then 0
+ else 2 * BITWISE (n - 1) oper (x div 2) (y div 2) +
+ (if oper (ODD x) (ODD y) then 1 else 0))"
+ by (import word32 BITWISE_EVAL2)
+
+lemma BITSwLT_THM: "ALL h l n. BITSw h l n < 2 ^ (Suc h - l)"
+ by (import word32 BITSwLT_THM)
+
+lemma BITSw_COMP_THM: "ALL h1 l1 h2 l2 n.
+ h2 + l1 <= h1 -->
+ BITS h2 l2 (BITSw h1 l1 n) = BITSw (h2 + l1) (l2 + l1) n"
+ by (import word32 BITSw_COMP_THM)
+
+lemma BITSw_DIV_THM: "ALL h l n x. BITSw h l x div 2 ^ n = BITSw h (l + n) x"
+ by (import word32 BITSw_DIV_THM)
+
+lemma BITw_THM: "ALL b n. BITw b n = (BITSw b b n = 1)"
+ by (import word32 BITw_THM)
+
+lemma SLICEw_THM: "ALL n h l. SLICEw h l n = BITSw h l n * 2 ^ l"
+ by (import word32 SLICEw_THM)
+
+lemma BITS_SLICEw_THM: "ALL h l n. BITS h l (SLICEw h l n) = BITSw h l n"
+ by (import word32 BITS_SLICEw_THM)
+
+lemma SLICEw_ZERO_THM: "ALL n h. SLICEw h 0 n = BITSw h 0 n"
+ by (import word32 SLICEw_ZERO_THM)
+
+lemma SLICEw_COMP_THM: "ALL h m l a.
+ Suc m <= h & l <= m --> SLICEw h (Suc m) a + SLICEw m l a = SLICEw h l a"
+ by (import word32 SLICEw_COMP_THM)
+
+lemma BITSw_ZERO: "ALL h l n. h < l --> BITSw h l n = 0"
+ by (import word32 BITSw_ZERO)
+
+lemma SLICEw_ZERO: "ALL h l n. h < l --> SLICEw h l n = 0"
+ by (import word32 SLICEw_ZERO)
+
+;end_setup
+
+end
+