(* AUTOMATICALLY GENERATED, DO NOT EDIT! *)
theory HOL4Word32 imports HOL4Base begin
;setup_theory bits
consts
DIV2 :: "nat => nat"
defs
DIV2_primdef: "DIV2 == %n::nat. n div (2::nat)"
lemma DIV2_def: "ALL n::nat. DIV2 n = n div (2::nat)"
by (import bits DIV2_def)
consts
TIMES_2EXP :: "nat => nat => nat"
defs
TIMES_2EXP_primdef: "TIMES_2EXP == %(x::nat) n::nat. n * (2::nat) ^ x"
lemma TIMES_2EXP_def: "ALL (x::nat) n::nat. TIMES_2EXP x n = n * (2::nat) ^ x"
by (import bits TIMES_2EXP_def)
consts
DIV_2EXP :: "nat => nat => nat"
defs
DIV_2EXP_primdef: "DIV_2EXP == %(x::nat) n::nat. n div (2::nat) ^ x"
lemma DIV_2EXP_def: "ALL (x::nat) n::nat. DIV_2EXP x n = n div (2::nat) ^ x"
by (import bits DIV_2EXP_def)
consts
MOD_2EXP :: "nat => nat => nat"
defs
MOD_2EXP_primdef: "MOD_2EXP == %(x::nat) n::nat. n mod (2::nat) ^ x"
lemma MOD_2EXP_def: "ALL (x::nat) n::nat. MOD_2EXP x n = n mod (2::nat) ^ x"
by (import bits MOD_2EXP_def)
consts
DIVMOD_2EXP :: "nat => nat => nat * nat"
defs
DIVMOD_2EXP_primdef: "DIVMOD_2EXP == %(x::nat) n::nat. (n div (2::nat) ^ x, n mod (2::nat) ^ x)"
lemma DIVMOD_2EXP_def: "ALL (x::nat) n::nat.
DIVMOD_2EXP x n = (n div (2::nat) ^ x, n mod (2::nat) ^ x)"
by (import bits DIVMOD_2EXP_def)
consts
SBIT :: "bool => nat => nat"
defs
SBIT_primdef: "(op ==::(bool => nat => nat) => (bool => nat => nat) => prop)
(SBIT::bool => nat => nat)
(%(b::bool) n::nat.
(If::bool => nat => nat => nat) b
((op ^::nat => nat => nat)
((number_of::bin => nat)
((op BIT::bin => bit => bin)
((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
(bit.B0::bit)))
n)
(0::nat))"
lemma SBIT_def: "(All::(bool => bool) => bool)
(%b::bool.
(All::(nat => bool) => bool)
(%n::nat.
(op =::nat => nat => bool) ((SBIT::bool => nat => nat) b n)
((If::bool => nat => nat => nat) b
((op ^::nat => nat => nat)
((number_of::bin => nat)
((op BIT::bin => bit => bin)
((op BIT::bin => bit => bin) (Numeral.Pls::bin)
(bit.B1::bit))
(bit.B0::bit)))
n)
(0::nat))))"
by (import bits SBIT_def)
consts
BITS :: "nat => nat => nat => nat"
defs
BITS_primdef: "BITS == %(h::nat) (l::nat) n::nat. MOD_2EXP (Suc h - l) (DIV_2EXP l n)"
lemma BITS_def: "ALL (h::nat) (l::nat) n::nat.
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::nat) n::nat. BITS b b n = (1::nat)"
lemma BIT_def: "ALL (b::nat) n::nat. bit b n = (BITS b b n = (1::nat))"
by (import bits BIT_def)
consts
SLICE :: "nat => nat => nat => nat"
defs
SLICE_primdef: "SLICE == %(h::nat) (l::nat) n::nat. MOD_2EXP (Suc h) n - MOD_2EXP l n"
lemma SLICE_def: "ALL (h::nat) (l::nat) n::nat.
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::nat)"
lemma LSBn_def: "LSBn = bit (0::nat)"
by (import bits LSBn_def)
consts
BITWISE :: "nat => (bool => bool => bool) => nat => nat => nat"
specification (BITWISE_primdef: BITWISE) BITWISE_def: "(ALL (oper::bool => bool => bool) (x::nat) y::nat.
BITWISE (0::nat) oper x y = (0::nat)) &
(ALL (n::nat) (oper::bool => bool => bool) (x::nat) y::nat.
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::nat) - a = (1::nat)"
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::nat) (xa::nat) xb::nat.
BITS x xa xb = xb div (2::nat) ^ xa mod (2::nat) ^ (Suc x - xa)"
by (import bits BITS_THM)
lemma BITSLT_THM: "ALL (h::nat) (l::nat) n::nat. BITS h l n < (2::nat) ^ (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::nat) (l::nat) n::nat.
BITS h l n = n mod (2::nat) ^ Suc h div (2::nat) ^ l"
by (import bits BITS2_THM)
lemma BITS_COMP_THM: "ALL (h1::nat) (l1::nat) (h2::nat) (l2::nat) n::nat.
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::nat) (l::nat) (x::nat) n::nat.
BITS h l x div (2::nat) ^ n = BITS h (l + n) x"
by (import bits BITS_DIV_THM)
lemma BITS_LT_HIGH: "ALL (h::nat) (l::nat) n::nat.
n < (2::nat) ^ Suc h --> BITS h l n = n div (2::nat) ^ l"
by (import bits BITS_LT_HIGH)
lemma BITS_ZERO: "ALL (h::nat) (l::nat) n::nat. h < l --> BITS h l n = (0::nat)"
by (import bits BITS_ZERO)
lemma BITS_ZERO2: "ALL (h::nat) l::nat. BITS h l (0::nat) = (0::nat)"
by (import bits BITS_ZERO2)
lemma BITS_ZERO3: "ALL (h::nat) x::nat. BITS h (0::nat) x = x mod (2::nat) ^ Suc h"
by (import bits BITS_ZERO3)
lemma BITS_COMP_THM2: "ALL (h1::nat) (l1::nat) (h2::nat) (l2::nat) n::nat.
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::type.
(n mod (2::nat) ~= (1::nat)) = (n mod (2::nat) = (0::nat))"
by (import bits NOT_MOD2_LEM2)
lemma EVEN_MOD2_LEM: "ALL n::nat. EVEN n = (n mod (2::nat) = (0::nat))"
by (import bits EVEN_MOD2_LEM)
lemma ODD_MOD2_LEM: "ALL n::nat. ODD n = (n mod (2::nat) = (1::nat))"
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::type) (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::nat) (h::nat) l::nat. SLICE h l n = BITS h l n * (2::nat) ^ l"
by (import bits SLICE_THM)
lemma SLICELT_THM: "ALL (h::nat) (l::nat) n::nat. SLICE h l n < (2::nat) ^ Suc h"
by (import bits SLICELT_THM)
lemma BITS_SLICE_THM: "ALL (h::nat) (l::nat) n::nat. BITS h l (SLICE h l n) = BITS h l n"
by (import bits BITS_SLICE_THM)
lemma BITS_SLICE_THM2: "ALL (h::nat) (l::nat) n::nat.
h <= (h2::nat) --> 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::nat) (m::nat) (l::nat) n::nat.
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::nat) (l::nat) n::nat. h < l --> SLICE h l n = (0::nat)"
by (import bits SLICE_ZERO)
lemma BIT_COMP_THM3: "ALL (h::nat) (m::nat) (l::nat) n::nat.
Suc m <= h & l <= m -->
BITS h (Suc m) n * (2::nat) ^ (Suc m - l) + BITS m l n = BITS h l n"
by (import bits BIT_COMP_THM3)
lemma NOT_BIT: "ALL (n::nat) a::nat. (~ bit n a) = (BITS n n a = (0::nat))"
by (import bits NOT_BIT)
lemma NOT_BITS: "ALL (n::nat) a::nat. (BITS n n a ~= (0::nat)) = (BITS n n a = (1::nat))"
by (import bits NOT_BITS)
lemma NOT_BITS2: "ALL (n::nat) a::nat. (BITS n n a ~= (1::nat)) = (BITS n n a = (0::nat))"
by (import bits NOT_BITS2)
lemma BIT_SLICE: "ALL (n::nat) (a::nat) b::nat.
(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::nat) (x::nat) n::nat.
SBIT (bit x n) (x + y) = SLICE x x n * (2::nat) ^ y"
by (import bits BIT_SLICE_LEM)
lemma BIT_SLICE_THM: "ALL (x::nat) xa::nat. SBIT (bit x xa) x = SLICE x x xa"
by (import bits BIT_SLICE_THM)
lemma SBIT_DIV: "ALL (b::bool) (m::nat) n::nat.
n < m --> SBIT b (m - n) = SBIT b m div (2::nat) ^ n"
by (import bits SBIT_DIV)
lemma BITS_SUC: "ALL (h::nat) (l::nat) n::nat.
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::nat) (l::nat) n::nat.
BITS (Suc h) l n =
(if Suc h < l then 0::nat
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::nat) (l::nat) (a::nat) b::nat.
(ALL x::nat. 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::nat) (oper::bool => bool => bool) (a::nat) b::nat.
BITWISE n oper a b < (2::nat) ^ 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::nat) (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
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::nat) (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
x < n -->
oper (bit x a) (bit x b) -->
BITWISE n oper a b div (2::nat) ^ x mod (2::nat) = (1::nat)"
by (import bits BITWISE_COR)
lemma BITWISE_NOT_COR: "ALL (x::nat) (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
x < n -->
~ oper (bit x a) (bit x b) -->
BITWISE n oper a b div (2::nat) ^ x mod (2::nat) = (0::nat)"
by (import bits BITWISE_NOT_COR)
lemma MOD_PLUS_RIGHT: "ALL n>0::nat. 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>0::nat.
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>0::nat.
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::nat. n mod (2::nat) ^ WL"
lemma MODw_def: "ALL n::nat. MODw n = n mod (2::nat) ^ WL"
by (import word32 MODw_def)
consts
INw :: "nat => bool"
defs
INw_primdef: "INw == %n::nat. n < (2::nat) ^ WL"
lemma INw_def: "ALL n::nat. INw n = (n < (2::nat) ^ WL)"
by (import word32 INw_def)
consts
EQUIV :: "nat => nat => bool"
defs
EQUIV_primdef: "EQUIV == %(x::nat) y::nat. MODw x = MODw y"
lemma EQUIV_def: "ALL (x::nat) y::nat. EQUIV x y = (MODw x = MODw y)"
by (import word32 EQUIV_def)
lemma EQUIV_QT: "ALL (x::nat) y::nat. EQUIV x y = (EQUIV x = EQUIV y)"
by (import word32 EQUIV_QT)
lemma FUNPOW_THM: "ALL (f::'a::type => 'a::type) (n::nat) x::'a::type.
(f ^ n) (f x) = f ((f ^ n) x)"
by (import word32 FUNPOW_THM)
lemma FUNPOW_THM2: "ALL (f::'a::type => 'a::type) (n::nat) x::'a::type.
(f ^ Suc n) x = f ((f ^ n) x)"
by (import word32 FUNPOW_THM2)
lemma FUNPOW_COMP: "ALL (f::'a::type => 'a::type) (m::nat) (n::nat) a::'a::type.
(f ^ m) ((f ^ n) a) = (f ^ (m + n)) a"
by (import word32 FUNPOW_COMP)
lemma INw_MODw: "ALL n::nat. INw (MODw n)"
by (import word32 INw_MODw)
lemma TOw_IDEM: "ALL a::nat. INw a --> MODw a = a"
by (import word32 TOw_IDEM)
lemma MODw_IDEM2: "ALL a::nat. MODw (MODw a) = MODw a"
by (import word32 MODw_IDEM2)
lemma TOw_QT: "ALL a::nat. EQUIV (MODw a) a"
by (import word32 TOw_QT)
lemma MODw_THM: "MODw = BITS HB (0::nat)"
by (import word32 MODw_THM)
lemma MOD_ADD: "ALL (a::nat) b::nat. MODw (a + b) = MODw (MODw a + MODw b)"
by (import word32 MOD_ADD)
lemma MODw_MULT: "ALL (a::nat) b::nat. MODw (a * b) = MODw (MODw a * MODw b)"
by (import word32 MODw_MULT)
consts
AONE :: "nat"
defs
AONE_primdef: "AONE == 1::nat"
lemma AONE_def: "AONE = (1::nat)"
by (import word32 AONE_def)
lemma ADD_QT: "(ALL n::nat. EQUIV ((0::nat) + n) n) &
(ALL (m::nat) n::nat. EQUIV (Suc m + n) (Suc (m + n)))"
by (import word32 ADD_QT)
lemma ADD_0_QT: "ALL a::nat. EQUIV (a + (0::nat)) a"
by (import word32 ADD_0_QT)
lemma ADD_COMM_QT: "ALL (a::nat) b::nat. EQUIV (a + b) (b + a)"
by (import word32 ADD_COMM_QT)
lemma ADD_ASSOC_QT: "ALL (a::nat) (b::nat) c::nat. EQUIV (a + (b + c)) (a + b + c)"
by (import word32 ADD_ASSOC_QT)
lemma MULT_QT: "(ALL n::nat. EQUIV ((0::nat) * n) (0::nat)) &
(ALL (m::nat) n::nat. EQUIV (Suc m * n) (m * n + n))"
by (import word32 MULT_QT)
lemma ADD1_QT: "ALL m::nat. EQUIV (Suc m) (m + AONE)"
by (import word32 ADD1_QT)
lemma ADD_CLAUSES_QT: "(ALL m::nat. EQUIV ((0::nat) + m) m) &
(ALL m::nat. EQUIV (m + (0::nat)) m) &
(ALL (m::nat) n::nat. EQUIV (Suc m + n) (Suc (m + n))) &
(ALL (m::nat) n::nat. EQUIV (m + Suc n) (Suc (m + n)))"
by (import word32 ADD_CLAUSES_QT)
lemma SUC_EQUIV_COMP: "ALL (a::nat) b::nat.
EQUIV (Suc a) b --> EQUIV a (b + ((2::nat) ^ WL - (1::nat)))"
by (import word32 SUC_EQUIV_COMP)
lemma INV_SUC_EQ_QT: "ALL (m::nat) n::nat. EQUIV (Suc m) (Suc n) = EQUIV m n"
by (import word32 INV_SUC_EQ_QT)
lemma ADD_INV_0_QT: "ALL (m::nat) n::nat. EQUIV (m + n) m --> EQUIV n (0::nat)"
by (import word32 ADD_INV_0_QT)
lemma ADD_INV_0_EQ_QT: "ALL (m::nat) n::nat. EQUIV (m + n) m = EQUIV n (0::nat)"
by (import word32 ADD_INV_0_EQ_QT)
lemma EQ_ADD_LCANCEL_QT: "ALL (m::nat) (n::nat) p::nat. EQUIV (m + n) (m + p) = EQUIV n p"
by (import word32 EQ_ADD_LCANCEL_QT)
lemma EQ_ADD_RCANCEL_QT: "ALL (x::nat) (xa::nat) xb::nat. EQUIV (x + xb) (xa + xb) = EQUIV x xa"
by (import word32 EQ_ADD_RCANCEL_QT)
lemma LEFT_ADD_DISTRIB_QT: "ALL (m::nat) (n::nat) p::nat. EQUIV (p * (m + n)) (p * m + p * n)"
by (import word32 LEFT_ADD_DISTRIB_QT)
lemma MULT_ASSOC_QT: "ALL (m::nat) (n::nat) p::nat. EQUIV (m * (n * p)) (m * n * p)"
by (import word32 MULT_ASSOC_QT)
lemma MULT_COMM_QT: "ALL (m::nat) n::nat. EQUIV (m * n) (n * m)"
by (import word32 MULT_COMM_QT)
lemma MULT_CLAUSES_QT: "ALL (m::nat) n::nat.
EQUIV ((0::nat) * m) (0::nat) &
EQUIV (m * (0::nat)) (0::nat) &
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::nat. (2::nat) ^ WL - (1::nat) - MODw x"
lemma ONE_COMP_def: "ALL x::nat. ONE_COMP x = (2::nat) ^ WL - (1::nat) - MODw x"
by (import word32 ONE_COMP_def)
consts
TWO_COMP :: "nat => nat"
defs
TWO_COMP_primdef: "TWO_COMP == %x::nat. (2::nat) ^ WL - MODw x"
lemma TWO_COMP_def: "ALL x::nat. TWO_COMP x = (2::nat) ^ WL - MODw x"
by (import word32 TWO_COMP_def)
lemma ADD_TWO_COMP_QT: "ALL a::nat. EQUIV (MODw a + TWO_COMP a) (0::nat)"
by (import word32 ADD_TWO_COMP_QT)
lemma TWO_COMP_ONE_COMP_QT: "ALL a::nat. EQUIV (TWO_COMP a) (ONE_COMP a + AONE)"
by (import word32 TWO_COMP_ONE_COMP_QT)
lemma BIT_EQUIV_THM: "(All::(nat => bool) => bool)
(%x::nat.
(All::(nat => bool) => bool)
(%xa::nat.
(op =::bool => bool => bool)
((All::(nat => bool) => bool)
(%xb::nat.
(op -->::bool => bool => bool)
((op <::nat => nat => bool) xb (WL::nat))
((op =::bool => bool => bool)
((bit::nat => nat => bool) xb x)
((bit::nat => nat => bool) xb xa))))
((EQUIV::nat => nat => bool) x xa)))"
by (import word32 BIT_EQUIV_THM)
lemma BITS_SUC2: "ALL (n::nat) a::nat.
BITS (Suc n) (0::nat) a = SLICE (Suc n) (Suc n) a + BITS n (0::nat) a"
by (import word32 BITS_SUC2)
lemma BITWISE_ONE_COMP_THM: "ALL (a::nat) b::nat. BITWISE WL (%(x::bool) y::bool. ~ x) a b = ONE_COMP a"
by (import word32 BITWISE_ONE_COMP_THM)
lemma ONE_COMP_THM: "ALL (x::nat) xa::nat. 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::bool) y::bool. x ~= y)"
lemma EOR_def: "EOR = BITWISE WL (%(x::bool) y::bool. x ~= y)"
by (import word32 EOR_def)
consts
COMP0 :: "nat"
defs
COMP0_primdef: "COMP0 == ONE_COMP (0::nat)"
lemma COMP0_def: "COMP0 = ONE_COMP (0::nat)"
by (import word32 COMP0_def)
lemma BITWISE_THM2: "(All::(nat => bool) => bool)
(%y::nat.
(All::((bool => bool => bool) => bool) => bool)
(%oper::bool => bool => bool.
(All::(nat => bool) => bool)
(%a::nat.
(All::(nat => bool) => bool)
(%b::nat.
(op =::bool => bool => bool)
((All::(nat => bool) => bool)
(%x::nat.
(op -->::bool => bool => bool)
((op <::nat => nat => bool) x (WL::nat))
((op =::bool => bool => bool)
(oper ((bit::nat => nat => bool) x a)
((bit::nat => nat => bool) x b))
((bit::nat => nat => bool) x y))))
((EQUIV::nat => nat => bool)
((BITWISE::nat
=> (bool => bool => bool)
=> nat => nat => nat)
(WL::nat) oper a b)
y)))))"
by (import word32 BITWISE_THM2)
lemma OR_ASSOC_QT: "ALL (a::nat) (b::nat) c::nat. EQUIV (OR a (OR b c)) (OR (OR a b) c)"
by (import word32 OR_ASSOC_QT)
lemma OR_COMM_QT: "ALL (a::nat) b::nat. EQUIV (OR a b) (OR b a)"
by (import word32 OR_COMM_QT)
lemma OR_ABSORB_QT: "ALL (a::nat) b::nat. EQUIV (AND a (OR a b)) a"
by (import word32 OR_ABSORB_QT)
lemma OR_IDEM_QT: "ALL a::nat. EQUIV (OR a a) a"
by (import word32 OR_IDEM_QT)
lemma AND_ASSOC_QT: "ALL (a::nat) (b::nat) c::nat. EQUIV (AND a (AND b c)) (AND (AND a b) c)"
by (import word32 AND_ASSOC_QT)
lemma AND_COMM_QT: "ALL (a::nat) b::nat. EQUIV (AND a b) (AND b a)"
by (import word32 AND_COMM_QT)
lemma AND_ABSORB_QT: "ALL (a::nat) b::nat. EQUIV (OR a (AND a b)) a"
by (import word32 AND_ABSORB_QT)
lemma AND_IDEM_QT: "ALL a::nat. EQUIV (AND a a) a"
by (import word32 AND_IDEM_QT)
lemma OR_COMP_QT: "ALL a::nat. EQUIV (OR a (ONE_COMP a)) COMP0"
by (import word32 OR_COMP_QT)
lemma AND_COMP_QT: "ALL a::nat. EQUIV (AND a (ONE_COMP a)) (0::nat)"
by (import word32 AND_COMP_QT)
lemma ONE_COMP_QT: "ALL a::nat. EQUIV (ONE_COMP (ONE_COMP a)) a"
by (import word32 ONE_COMP_QT)
lemma RIGHT_AND_OVER_OR_QT: "ALL (a::nat) (b::nat) c::nat.
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::nat) (b::nat) c::nat. 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::nat) b::nat.
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::nat) (a::nat) b::nat. n < WL --> EQUIV a b --> bit n a = bit n b"
by (import word32 BIT_EQUIV)
lemma LSB_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> LSBn a = LSBn b"
by (import word32 LSB_WELLDEF)
lemma MSB_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> MSBn a = MSBn b"
by (import word32 MSB_WELLDEF)
lemma BITWISE_ISTEP: "ALL (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
(0::nat) < n -->
BITWISE n oper (a div (2::nat)) (b div (2::nat)) =
BITWISE n oper a b div (2::nat) +
SBIT (oper (bit n a) (bit n b)) (n - (1::nat))"
by (import word32 BITWISE_ISTEP)
lemma BITWISE_EVAL: "ALL (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
BITWISE (Suc n) oper a b =
(2::nat) * BITWISE n oper (a div (2::nat)) (b div (2::nat)) +
SBIT (oper (LSBn a) (LSBn b)) (0::nat)"
by (import word32 BITWISE_EVAL)
lemma BITWISE_WELLDEF: "ALL (n::nat) (oper::bool => bool => bool) (a::nat) (b::nat) (c::nat) d::nat.
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::bool => bool => bool) (a::nat) (b::nat) (c::nat) d::nat.
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::nat) b::nat. EQUIV a b --> EQUIV (Suc a) (Suc b)"
by (import word32 SUC_WELLDEF)
lemma ADD_WELLDEF: "ALL (a::nat) (b::nat) (c::nat) d::nat.
EQUIV a b & EQUIV c d --> EQUIV (a + c) (b + d)"
by (import word32 ADD_WELLDEF)
lemma MUL_WELLDEF: "ALL (a::nat) (b::nat) (c::nat) d::nat.
EQUIV a b & EQUIV c d --> EQUIV (a * c) (b * d)"
by (import word32 MUL_WELLDEF)
lemma ONE_COMP_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> EQUIV (ONE_COMP a) (ONE_COMP b)"
by (import word32 ONE_COMP_WELLDEF)
lemma TWO_COMP_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> EQUIV (TWO_COMP a) (TWO_COMP b)"
by (import word32 TWO_COMP_WELLDEF)
lemma TOw_WELLDEF: "ALL (a::nat) b::nat. 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::nat. MODw a div (2::nat)"
lemma LSR_ONE_def: "ALL a::nat. LSR_ONE a = MODw a div (2::nat)"
by (import word32 LSR_ONE_def)
consts
ASR_ONE :: "nat => nat"
defs
ASR_ONE_primdef: "ASR_ONE == %a::nat. LSR_ONE a + SBIT (MSBn a) HB"
lemma ASR_ONE_def: "ALL a::nat. 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::nat. LSR_ONE a + SBIT (LSBn a) HB"
lemma ROR_ONE_def: "ALL a::nat. 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::bool) a::nat. LSR_ONE a + SBIT c HB"
lemma RRXn_def: "ALL (c::bool) a::nat. RRXn c a = LSR_ONE a + SBIT c HB"
by (import word32 RRXn_def)
lemma LSR_ONE_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> EQUIV (LSR_ONE a) (LSR_ONE b)"
by (import word32 LSR_ONE_WELLDEF)
lemma ASR_ONE_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> EQUIV (ASR_ONE a) (ASR_ONE b)"
by (import word32 ASR_ONE_WELLDEF)
lemma ROR_ONE_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> EQUIV (ROR_ONE a) (ROR_ONE b)"
by (import word32 ROR_ONE_WELLDEF)
lemma RRX_WELLDEF: "ALL (a::nat) (b::nat) c::bool. EQUIV a b --> EQUIV (RRXn c a) (RRXn c b)"
by (import word32 RRX_WELLDEF)
lemma LSR_ONE: "LSR_ONE = BITS HB (1::nat)"
by (import word32 LSR_ONE)
typedef (open) word32 = "{x::nat => bool. EX xa::nat. 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::word32. mk_word32 (dest_word32 a) = a) &
(ALL r::nat => bool.
(EX x::nat. 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::nat))"
lemma w_0_def: "w_0 = mk_word32 (EQUIV (0::nat))"
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::word32. mk_word32 (EQUIV (Suc (Eps (dest_word32 T1))))"
lemma word_suc: "ALL T1::word32. 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::word32) T2::word32.
mk_word32 (EQUIV (Eps (dest_word32 T1) + Eps (dest_word32 T2)))"
lemma word_add: "ALL (T1::word32) T2::word32.
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::word32) T2::word32.
mk_word32 (EQUIV (Eps (dest_word32 T1) * Eps (dest_word32 T2)))"
lemma word_mul: "ALL (T1::word32) T2::word32.
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::word32. mk_word32 (EQUIV (ONE_COMP (Eps (dest_word32 T1))))"
lemma word_1comp: "ALL T1::word32.
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::word32. mk_word32 (EQUIV (TWO_COMP (Eps (dest_word32 T1))))"
lemma word_2comp: "ALL T1::word32.
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::word32. mk_word32 (EQUIV (LSR_ONE (Eps (dest_word32 T1))))"
lemma word_lsr1: "ALL T1::word32.
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::word32. mk_word32 (EQUIV (ASR_ONE (Eps (dest_word32 T1))))"
lemma word_asr1: "ALL T1::word32.
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::word32. mk_word32 (EQUIV (ROR_ONE (Eps (dest_word32 T1))))"
lemma word_ror1: "ALL T1::word32.
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::bool) T2::word32. mk_word32 (EQUIV (RRXn T1 (Eps (dest_word32 T2))))"
lemma RRX_def: "ALL (T1::bool) T2::word32.
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::word32. LSBn (Eps (dest_word32 T1))"
lemma LSB_def: "ALL T1::word32. LSB T1 = LSBn (Eps (dest_word32 T1))"
by (import word32 LSB_def)
consts
MSB :: "word32 => bool"
defs
MSB_primdef: "MSB == %T1::word32. MSBn (Eps (dest_word32 T1))"
lemma MSB_def: "ALL T1::word32. MSB T1 = MSBn (Eps (dest_word32 T1))"
by (import word32 MSB_def)
constdefs
bitwise_or :: "word32 => word32 => word32"
"bitwise_or ==
%(T1::word32) T2::word32.
mk_word32 (EQUIV (OR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
lemma bitwise_or: "ALL (T1::word32) T2::word32.
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::word32) T2::word32.
mk_word32 (EQUIV (EOR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
lemma bitwise_eor: "ALL (T1::word32) T2::word32.
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::word32) T2::word32.
mk_word32 (EQUIV (AND (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
lemma bitwise_and: "ALL (T1::word32) T2::word32.
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::word32. mk_word32 (EQUIV (MODw (Eps (dest_word32 T1))))"
lemma TOw_def: "ALL T1::word32. TOw T1 = mk_word32 (EQUIV (MODw (Eps (dest_word32 T1))))"
by (import word32 TOw_def)
consts
n2w :: "nat => word32"
defs
n2w_primdef: "n2w == %n::nat. mk_word32 (EQUIV n)"
lemma n2w_def: "ALL n::nat. n2w n = mk_word32 (EQUIV n)"
by (import word32 n2w_def)
consts
w2n :: "word32 => nat"
defs
w2n_primdef: "w2n == %w::word32. MODw (Eps (dest_word32 w))"
lemma w2n_def: "ALL w::word32. w2n w = MODw (Eps (dest_word32 w))"
by (import word32 w2n_def)
lemma ADDw: "(ALL x::word32. word_add w_0 x = x) &
(ALL (x::word32) xa::word32.
word_add (word_suc x) xa = word_suc (word_add x xa))"
by (import word32 ADDw)
lemma ADD_0w: "ALL x::word32. word_add x w_0 = x"
by (import word32 ADD_0w)
lemma ADD1w: "ALL x::word32. word_suc x = word_add x w_1"
by (import word32 ADD1w)
lemma ADD_ASSOCw: "ALL (x::word32) (xa::word32) xb::word32.
word_add x (word_add xa xb) = word_add (word_add x xa) xb"
by (import word32 ADD_ASSOCw)
lemma ADD_CLAUSESw: "(ALL x::word32. word_add w_0 x = x) &
(ALL x::word32. word_add x w_0 = x) &
(ALL (x::word32) xa::word32.
word_add (word_suc x) xa = word_suc (word_add x xa)) &
(ALL (x::word32) xa::word32.
word_add x (word_suc xa) = word_suc (word_add x xa))"
by (import word32 ADD_CLAUSESw)
lemma ADD_COMMw: "ALL (x::word32) xa::word32. word_add x xa = word_add xa x"
by (import word32 ADD_COMMw)
lemma ADD_INV_0_EQw: "ALL (x::word32) xa::word32. (word_add x xa = x) = (xa = w_0)"
by (import word32 ADD_INV_0_EQw)
lemma EQ_ADD_LCANCELw: "ALL (x::word32) (xa::word32) xb::word32.
(word_add x xa = word_add x xb) = (xa = xb)"
by (import word32 EQ_ADD_LCANCELw)
lemma EQ_ADD_RCANCELw: "ALL (x::word32) (xa::word32) xb::word32.
(word_add x xb = word_add xa xb) = (x = xa)"
by (import word32 EQ_ADD_RCANCELw)
lemma LEFT_ADD_DISTRIBw: "ALL (x::word32) (xa::word32) xb::word32.
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::word32) (xa::word32) xb::word32.
word_mul x (word_mul xa xb) = word_mul (word_mul x xa) xb"
by (import word32 MULT_ASSOCw)
lemma MULT_COMMw: "ALL (x::word32) xa::word32. word_mul x xa = word_mul xa x"
by (import word32 MULT_COMMw)
lemma MULT_CLAUSESw: "ALL (x::word32) xa::word32.
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::word32. word_2comp x = word_add (word_1comp x) w_1"
by (import word32 TWO_COMP_ONE_COMP)
lemma OR_ASSOCw: "ALL (x::word32) (xa::word32) xb::word32.
bitwise_or x (bitwise_or xa xb) = bitwise_or (bitwise_or x xa) xb"
by (import word32 OR_ASSOCw)
lemma OR_COMMw: "ALL (x::word32) xa::word32. bitwise_or x xa = bitwise_or xa x"
by (import word32 OR_COMMw)
lemma OR_IDEMw: "ALL x::word32. bitwise_or x x = x"
by (import word32 OR_IDEMw)
lemma OR_ABSORBw: "ALL (x::word32) xa::word32. bitwise_and x (bitwise_or x xa) = x"
by (import word32 OR_ABSORBw)
lemma AND_ASSOCw: "ALL (x::word32) (xa::word32) xb::word32.
bitwise_and x (bitwise_and xa xb) = bitwise_and (bitwise_and x xa) xb"
by (import word32 AND_ASSOCw)
lemma AND_COMMw: "ALL (x::word32) xa::word32. bitwise_and x xa = bitwise_and xa x"
by (import word32 AND_COMMw)
lemma AND_IDEMw: "ALL x::word32. bitwise_and x x = x"
by (import word32 AND_IDEMw)
lemma AND_ABSORBw: "ALL (x::word32) xa::word32. bitwise_or x (bitwise_and x xa) = x"
by (import word32 AND_ABSORBw)
lemma ONE_COMPw: "ALL x::word32. word_1comp (word_1comp x) = x"
by (import word32 ONE_COMPw)
lemma RIGHT_AND_OVER_ORw: "ALL (x::word32) (xa::word32) xb::word32.
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::word32) (xa::word32) xb::word32.
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::word32) xa::word32.
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::nat)"
by (import word32 w_0)
lemma w_1: "w_1 = n2w (1::nat)"
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::word32. word_add x (word_2comp x) = w_0"
by (import word32 ADD_TWO_COMP)
lemma ADD_TWO_COMP2: "ALL x::word32. word_add (word_2comp x) x = w_0"
by (import word32 ADD_TWO_COMP2)
constdefs
word_sub :: "word32 => word32 => word32"
"word_sub == %(a::word32) b::word32. word_add a (word_2comp b)"
lemma word_sub: "ALL (a::word32) b::word32. word_sub a b = word_add a (word_2comp b)"
by (import word32 word_sub)
constdefs
word_lsl :: "word32 => nat => word32"
"word_lsl == %(a::word32) n::nat. word_mul a (n2w ((2::nat) ^ n))"
lemma word_lsl: "ALL (a::word32) n::nat. word_lsl a n = word_mul a (n2w ((2::nat) ^ n))"
by (import word32 word_lsl)
constdefs
word_lsr :: "word32 => nat => word32"
"word_lsr == %(a::word32) n::nat. (word_lsr1 ^ n) a"
lemma word_lsr: "ALL (a::word32) n::nat. word_lsr a n = (word_lsr1 ^ n) a"
by (import word32 word_lsr)
constdefs
word_asr :: "word32 => nat => word32"
"word_asr == %(a::word32) n::nat. (word_asr1 ^ n) a"
lemma word_asr: "ALL (a::word32) n::nat. word_asr a n = (word_asr1 ^ n) a"
by (import word32 word_asr)
constdefs
word_ror :: "word32 => nat => word32"
"word_ror == %(a::word32) n::nat. (word_ror1 ^ n) a"
lemma word_ror: "ALL (a::word32) n::nat. word_ror a n = (word_ror1 ^ n) a"
by (import word32 word_ror)
consts
BITw :: "nat => word32 => bool"
defs
BITw_primdef: "BITw == %(b::nat) n::word32. bit b (w2n n)"
lemma BITw_def: "ALL (b::nat) n::word32. BITw b n = bit b (w2n n)"
by (import word32 BITw_def)
consts
BITSw :: "nat => nat => word32 => nat"
defs
BITSw_primdef: "BITSw == %(h::nat) (l::nat) n::word32. BITS h l (w2n n)"
lemma BITSw_def: "ALL (h::nat) (l::nat) n::word32. 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::nat) (l::nat) n::word32. SLICE h l (w2n n)"
lemma SLICEw_def: "ALL (h::nat) (l::nat) n::word32. SLICEw h l n = SLICE h l (w2n n)"
by (import word32 SLICEw_def)
lemma TWO_COMP_ADD: "ALL (a::word32) b::word32.
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::word32. word_2comp (word_2comp a) = a"
by (import word32 TWO_COMP_ELIM)
lemma ADD_SUB_ASSOC: "ALL (a::word32) (b::word32) c::word32.
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::word32) (b::word32) c::word32.
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::word32. word_sub a a = w_0"
by (import word32 SUB_EQUALw)
lemma ADD_SUBw: "ALL (a::word32) b::word32. word_sub (word_add a b) b = a"
by (import word32 ADD_SUBw)
lemma SUB_SUBw: "ALL (a::word32) (b::word32) c::word32.
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::word32. word_1comp a = word_sub (word_2comp a) w_1"
by (import word32 ONE_COMP_TWO_COMP)
lemma SUBw: "ALL (m::word32) n::word32. word_sub (word_suc m) n = word_suc (word_sub m n)"
by (import word32 SUBw)
lemma ADD_EQ_SUBw: "ALL (m::word32) (n::word32) p::word32.
(word_add m n = p) = (m = word_sub p n)"
by (import word32 ADD_EQ_SUBw)
lemma CANCEL_SUBw: "ALL (m::word32) (n::word32) p::word32.
(word_sub n p = word_sub m p) = (n = m)"
by (import word32 CANCEL_SUBw)
lemma SUB_PLUSw: "ALL (a::word32) (b::word32) c::word32.
word_sub a (word_add b c) = word_sub (word_sub a b) c"
by (import word32 SUB_PLUSw)
lemma word_nchotomy: "ALL w::word32. EX n::nat. w = n2w n"
by (import word32 word_nchotomy)
lemma dest_word_mk_word_eq3: "ALL a::nat. dest_word32 (mk_word32 (EQUIV a)) = EQUIV a"
by (import word32 dest_word_mk_word_eq3)
lemma MODw_ELIM: "ALL n::nat. n2w (MODw n) = n2w n"
by (import word32 MODw_ELIM)
lemma w2n_EVAL: "ALL n::nat. w2n (n2w n) = MODw n"
by (import word32 w2n_EVAL)
lemma w2n_ELIM: "ALL a::word32. n2w (w2n a) = a"
by (import word32 w2n_ELIM)
lemma n2w_11: "ALL (a::nat) b::nat. (n2w a = n2w b) = (MODw a = MODw b)"
by (import word32 n2w_11)
lemma ADD_EVAL: "word_add (n2w (a::nat)) (n2w (b::nat)) = n2w (a + b)"
by (import word32 ADD_EVAL)
lemma MUL_EVAL: "word_mul (n2w (a::nat)) (n2w (b::nat)) = n2w (a * b)"
by (import word32 MUL_EVAL)
lemma ONE_COMP_EVAL: "word_1comp (n2w (a::nat)) = n2w (ONE_COMP a)"
by (import word32 ONE_COMP_EVAL)
lemma TWO_COMP_EVAL: "word_2comp (n2w (a::nat)) = n2w (TWO_COMP a)"
by (import word32 TWO_COMP_EVAL)
lemma LSR_ONE_EVAL: "word_lsr1 (n2w (a::nat)) = n2w (LSR_ONE a)"
by (import word32 LSR_ONE_EVAL)
lemma ASR_ONE_EVAL: "word_asr1 (n2w (a::nat)) = n2w (ASR_ONE a)"
by (import word32 ASR_ONE_EVAL)
lemma ROR_ONE_EVAL: "word_ror1 (n2w (a::nat)) = n2w (ROR_ONE a)"
by (import word32 ROR_ONE_EVAL)
lemma RRX_EVAL: "RRX (c::bool) (n2w (a::nat)) = n2w (RRXn c a)"
by (import word32 RRX_EVAL)
lemma LSB_EVAL: "LSB (n2w (a::nat)) = LSBn a"
by (import word32 LSB_EVAL)
lemma MSB_EVAL: "MSB (n2w (a::nat)) = MSBn a"
by (import word32 MSB_EVAL)
lemma OR_EVAL: "bitwise_or (n2w (a::nat)) (n2w (b::nat)) = n2w (OR a b)"
by (import word32 OR_EVAL)
lemma EOR_EVAL: "bitwise_eor (n2w (a::nat)) (n2w (b::nat)) = n2w (EOR a b)"
by (import word32 EOR_EVAL)
lemma AND_EVAL: "bitwise_and (n2w (a::nat)) (n2w (b::nat)) = n2w (AND a b)"
by (import word32 AND_EVAL)
lemma BITS_EVAL: "ALL (h::nat) (l::nat) a::nat. BITSw h l (n2w a) = BITS h l (MODw a)"
by (import word32 BITS_EVAL)
lemma BIT_EVAL: "ALL (b::nat) a::nat. BITw b (n2w a) = bit b (MODw a)"
by (import word32 BIT_EVAL)
lemma SLICE_EVAL: "ALL (h::nat) (l::nat) a::nat. SLICEw h l (n2w a) = SLICE h l (MODw a)"
by (import word32 SLICE_EVAL)
lemma LSL_ADD: "ALL (a::word32) (m::nat) n::nat.
word_lsl (word_lsl a m) n = word_lsl a (m + n)"
by (import word32 LSL_ADD)
lemma LSR_ADD: "ALL (x::word32) (xa::nat) xb::nat.
word_lsr (word_lsr x xa) xb = word_lsr x (xa + xb)"
by (import word32 LSR_ADD)
lemma ASR_ADD: "ALL (x::word32) (xa::nat) xb::nat.
word_asr (word_asr x xa) xb = word_asr x (xa + xb)"
by (import word32 ASR_ADD)
lemma ROR_ADD: "ALL (x::word32) (xa::nat) xb::nat.
word_ror (word_ror x xa) xb = word_ror x (xa + xb)"
by (import word32 ROR_ADD)
lemma LSL_LIMIT: "ALL (w::word32) n::nat. HB < n --> word_lsl w n = w_0"
by (import word32 LSL_LIMIT)
lemma MOD_MOD_DIV: "ALL (a::nat) b::nat. INw (MODw a div (2::nat) ^ b)"
by (import word32 MOD_MOD_DIV)
lemma MOD_MOD_DIV_2EXP: "ALL (a::nat) n::nat.
MODw (MODw a div (2::nat) ^ n) div (2::nat) = MODw a div (2::nat) ^ Suc n"
by (import word32 MOD_MOD_DIV_2EXP)
lemma LSR_EVAL: "ALL n::nat. word_lsr (n2w (a::nat)) n = n2w (MODw a div (2::nat) ^ n)"
by (import word32 LSR_EVAL)
lemma LSR_THM: "ALL (x::nat) n::nat. word_lsr (n2w n) x = n2w (BITS HB (min WL x) n)"
by (import word32 LSR_THM)
lemma LSR_LIMIT: "ALL (x::nat) w::word32. 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::nat) n::nat.
word_ror (n2w n) x =
(let x'::nat = x mod WL
in n2w (BITS HB x' n +
BITS (x' - (1::nat)) (0::nat) n * (2::nat) ^ (WL - x')))"
by (import word32 ROR_THM)
lemma ROR_CYCLE: "ALL (x::nat) w::word32. word_ror w (x * WL) = w"
by (import word32 ROR_CYCLE)
lemma ASR_THM: "ALL (x::nat) n::nat.
word_asr (n2w n) x =
(let x'::nat = min HB x; s::nat = BITS HB x' n
in n2w (if MSBn n then (2::nat) ^ WL - (2::nat) ^ (WL - x') + s else s))"
by (import word32 ASR_THM)
lemma ASR_LIMIT: "ALL (x::nat) w::word32.
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::nat. word_lsl w_0 n = w_0) &
(ALL n::nat. word_asr w_0 n = w_0) &
(ALL n::nat. word_lsr w_0 n = w_0) & (ALL n::nat. word_ror w_0 n = w_0)"
by (import word32 ZERO_SHIFT)
lemma ZERO_SHIFT2: "(ALL a::word32. word_lsl a (0::nat) = a) &
(ALL a::word32. word_asr a (0::nat) = a) &
(ALL a::word32. word_lsr a (0::nat) = a) &
(ALL a::word32. word_ror a (0::nat) = a)"
by (import word32 ZERO_SHIFT2)
lemma ASR_w_T: "ALL n::nat. word_asr w_T n = w_T"
by (import word32 ASR_w_T)
lemma ROR_w_T: "ALL n::nat. word_ror w_T n = w_T"
by (import word32 ROR_w_T)
lemma MODw_EVAL: "ALL x::nat.
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::nat) a::nat. word_add (n2w a) (n2w b) = n2w (MODw (a + b))"
by (import word32 ADD_EVAL2)
lemma MUL_EVAL2: "ALL (b::nat) a::nat. word_mul (n2w a) (n2w b) = n2w (MODw (a * b))"
by (import word32 MUL_EVAL2)
lemma ONE_COMP_EVAL2: "ALL a::nat.
word_1comp (n2w a) =
n2w ((2::nat) ^
NUMERAL
(NUMERAL_BIT2
(NUMERAL_BIT1
(NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))) -
(1::nat) -
MODw a)"
by (import word32 ONE_COMP_EVAL2)
lemma TWO_COMP_EVAL2: "ALL a::nat.
word_2comp (n2w a) =
n2w (MODw
((2::nat) ^
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::nat. word_lsr1 (n2w a) = n2w (MODw a div (2::nat))"
by (import word32 LSR_ONE_EVAL2)
lemma ASR_ONE_EVAL2: "ALL a::nat.
word_asr1 (n2w a) =
n2w (MODw a div (2::nat) +
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::nat.
word_ror1 (n2w a) =
n2w (MODw a div (2::nat) +
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::bool) a::nat.
RRX c (n2w a) =
n2w (MODw a div (2::nat) +
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::nat. LSB (n2w a) = ODD a"
by (import word32 LSB_EVAL2)
lemma MSB_EVAL2: "ALL a::nat.
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::nat) a::nat.
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::nat) a::nat.
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::nat) a::nat.
bitwise_eor (n2w a) (n2w b) =
n2w (BITWISE
(NUMERAL
(NUMERAL_BIT2
(NUMERAL_BIT1
(NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
(%(x::bool) y::bool. x ~= y) a b)"
by (import word32 EOR_EVAL2)
lemma BITWISE_EVAL2: "(All::(nat => bool) => bool)
(%n::nat.
(All::((bool => bool => bool) => bool) => bool)
(%oper::bool => bool => bool.
(All::(nat => bool) => bool)
(%x::nat.
(All::(nat => bool) => bool)
(%y::nat.
(op =::nat => nat => bool)
((BITWISE::nat
=> (bool => bool => bool)
=> nat => nat => nat)
n oper x y)
((If::bool => nat => nat => nat)
((op =::nat => nat => bool) n (0::nat)) (0::nat)
((op +::nat => nat => nat)
((op *::nat => nat => nat)
((number_of::bin => nat)
((op BIT::bin => bit => bin)
((op BIT::bin => bit => bin)
(Numeral.Pls::bin) (bit.B1::bit))
(bit.B0::bit)))
((BITWISE::nat
=> (bool => bool => bool) => nat => nat => nat)
((op -::nat => nat => nat) n (1::nat)) oper
((op div::nat => nat => nat) x
((number_of::bin => nat)
((op BIT::bin => bit => bin)
((op BIT::bin => bit => bin)
(Numeral.Pls::bin) (bit.B1::bit))
(bit.B0::bit))))
((op div::nat => nat => nat) y
((number_of::bin => nat)
((op BIT::bin => bit => bin)
((op BIT::bin => bit => bin)
(Numeral.Pls::bin) (bit.B1::bit))
(bit.B0::bit))))))
((If::bool => nat => nat => nat)
(oper ((ODD::nat => bool) x)
((ODD::nat => bool) y))
(1::nat) (0::nat))))))))"
by (import word32 BITWISE_EVAL2)
lemma BITSwLT_THM: "ALL (h::nat) (l::nat) n::word32. BITSw h l n < (2::nat) ^ (Suc h - l)"
by (import word32 BITSwLT_THM)
lemma BITSw_COMP_THM: "ALL (h1::nat) (l1::nat) (h2::nat) (l2::nat) n::word32.
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::nat) (l::nat) (n::nat) x::word32.
BITSw h l x div (2::nat) ^ n = BITSw h (l + n) x"
by (import word32 BITSw_DIV_THM)
lemma BITw_THM: "ALL (b::nat) n::word32. BITw b n = (BITSw b b n = (1::nat))"
by (import word32 BITw_THM)
lemma SLICEw_THM: "ALL (n::word32) (h::nat) l::nat. SLICEw h l n = BITSw h l n * (2::nat) ^ l"
by (import word32 SLICEw_THM)
lemma BITS_SLICEw_THM: "ALL (h::nat) (l::nat) n::word32. BITS h l (SLICEw h l n) = BITSw h l n"
by (import word32 BITS_SLICEw_THM)
lemma SLICEw_ZERO_THM: "ALL (n::word32) h::nat. SLICEw h (0::nat) n = BITSw h (0::nat) n"
by (import word32 SLICEw_ZERO_THM)
lemma SLICEw_COMP_THM: "ALL (h::nat) (m::nat) (l::nat) a::word32.
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::nat) (l::nat) n::word32. h < l --> BITSw h l n = (0::nat)"
by (import word32 BITSw_ZERO)
lemma SLICEw_ZERO: "ALL (h::nat) (l::nat) n::word32. h < l --> SLICEw h l n = (0::nat)"
by (import word32 SLICEw_ZERO)
;end_setup
end