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theory HOL4Word32(* 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" lemma DIV2_def: "ALL n::nat. DIV2 n = n div 2" by (import bits DIV2_def) consts TIMES_2EXP :: "nat => nat => nat" defs TIMES_2EXP_primdef: "TIMES_2EXP == %(x::nat) n::nat. n * 2 ^ x" lemma TIMES_2EXP_def: "ALL (x::nat) n::nat. 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::nat) n::nat. n div 2 ^ x" lemma DIV_2EXP_def: "ALL (x::nat) n::nat. 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::nat) n::nat. n mod 2 ^ x" lemma MOD_2EXP_def: "ALL (x::nat) n::nat. 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::nat) n::nat. (n div 2 ^ x, n mod 2 ^ x)" lemma DIVMOD_2EXP_def: "ALL (x::nat) n::nat. 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::bool) n::nat. if b then 2 ^ n else 0" lemma SBIT_def: "ALL (b::bool) n::nat. 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::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" lemma BIT_def: "ALL (b::nat) n::nat. bit b n = (BITS b b n = 1)" 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" 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::bool => bool => bool) (x::nat) y::nat. BITWISE 0 oper x y = 0) & (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 = x" by (import bits DIV1) lemma SUC_SUB: "Suc (a::nat) - a = 1" by (import bits SUC_SUB) lemma DIV_MULT_1: "ALL (r::nat) n::nat. r < n --> (n + r) div n = 1" by (import bits DIV_MULT_1) lemma ZERO_LT_TWOEXP: "(All::(nat => bool) => bool) (%n::nat. (op <::nat => nat => bool) (0::nat) ((op ^::nat => nat => nat) ((number_of :: int => nat) ((Int.Bit0 :: int => int) ((Int.Bit1 :: int => int) (Int.Pls :: int)))) n))" by (import bits ZERO_LT_TWOEXP) lemma MOD_2EXP_LT: "ALL (n::nat) k::nat. k mod 2 ^ n < 2 ^ n" by (import bits MOD_2EXP_LT) lemma TWOEXP_DIVISION: "ALL (n::nat) k::nat. k = k div 2 ^ n * 2 ^ n + k mod 2 ^ n" by (import bits TWOEXP_DIVISION) lemma TWOEXP_MONO: "(All::(nat => bool) => bool) (%a::nat. (All::(nat => bool) => bool) (%b::nat. (op -->::bool => bool => bool) ((op <::nat => nat => bool) a b) ((op <::nat => nat => bool) ((op ^::nat => nat => nat) ((number_of :: int => nat) ((Int.Bit0 :: int => int) ((Int.Bit1 :: int => int) (Int.Pls :: int)))) a) ((op ^::nat => nat => nat) ((number_of :: int => nat) ((Int.Bit0 :: int => int) ((Int.Bit1 :: int => int) (Int.Pls :: int)))) b))))" by (import bits TWOEXP_MONO) lemma TWOEXP_MONO2: "(All::(nat => bool) => bool) (%a::nat. (All::(nat => bool) => bool) (%b::nat. (op -->::bool => bool => bool) ((op <=::nat => nat => bool) a b) ((op <=::nat => nat => bool) ((op ^::nat => nat => nat) ((number_of :: int => nat) ((Int.Bit0 :: int => int) ((Int.Bit1 :: int => int) (Int.Pls :: int)))) a) ((op ^::nat => nat => nat) ((number_of :: int => nat) ((Int.Bit0 :: int => int) ((Int.Bit1 :: int => int) (Int.Pls :: int)))) b))))" by (import bits TWOEXP_MONO2) lemma EXP_SUB_LESS_EQ: "(All::(nat => bool) => bool) (%a::nat. (All::(nat => bool) => bool) (%b::nat. (op <=::nat => nat => bool) ((op ^::nat => nat => nat) ((number_of :: int => nat) ((Int.Bit0 :: int => int) ((Int.Bit1 :: int => int) (Int.Pls :: int)))) ((op -::nat => nat => nat) a b)) ((op ^::nat => nat => nat) ((number_of :: int => nat) ((Int.Bit0 :: int => int) ((Int.Bit1 :: int => int) (Int.Pls :: int)))) 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 ^ xa mod 2 ^ (Suc x - xa)" by (import bits BITS_THM) lemma BITSLT_THM: "ALL (h::nat) (l::nat) n::nat. BITS h l n < 2 ^ (Suc h - l)" by (import bits BITSLT_THM) lemma DIV_MULT_LEM: "ALL (m::nat) n::nat. 0 < n --> m div n * n <= m" by (import bits DIV_MULT_LEM) lemma MOD_2EXP_LEM: "ALL (n::nat) x::nat. n mod 2 ^ x = n - n div 2 ^ x * 2 ^ x" by (import bits MOD_2EXP_LEM) lemma BITS2_THM: "ALL (h::nat) (l::nat) n::nat. BITS h l n = n mod 2 ^ Suc h div 2 ^ 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 ^ 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 ^ Suc h --> BITS h l n = n div 2 ^ l" by (import bits BITS_LT_HIGH) lemma BITS_ZERO: "ALL (h::nat) (l::nat) n::nat. h < l --> BITS h l n = 0" by (import bits BITS_ZERO) lemma BITS_ZERO2: "ALL (h::nat) l::nat. BITS h l 0 = 0" by (import bits BITS_ZERO2) lemma BITS_ZERO3: "ALL (h::nat) x::nat. BITS h 0 x = x mod 2 ^ 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 ~= 0) = (n mod 2 = 1)" by (import bits NOT_MOD2_LEM) lemma NOT_MOD2_LEM2: "ALL (n::nat) a::'a::type. (n mod 2 ~= 1) = (n mod 2 = 0)" by (import bits NOT_MOD2_LEM2) lemma EVEN_MOD2_LEM: "ALL n::nat. EVEN n = (n mod 2 = 0)" by (import bits EVEN_MOD2_LEM) lemma ODD_MOD2_LEM: "ALL n::nat. 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 ^ x * 2 ^ x = n - n mod 2 ^ x" by (import bits DIV_MULT_THM) lemma DIV_MULT_THM2: "ALL x::nat. 2 * (x div 2) = x - x mod 2" by (import bits DIV_MULT_THM2) lemma LESS_EQ_EXP_MULT: "ALL (a::nat) b::nat. a <= b --> (EX x::nat. 2 ^ b = x * 2 ^ a)" by (import bits LESS_EQ_EXP_MULT) lemma SLICE_LEM1: "ALL (a::nat) (x::nat) y::nat. a div 2 ^ (x + y) * 2 ^ (x + y) = a div 2 ^ x * 2 ^ x - a div 2 ^ x mod 2 ^ y * 2 ^ x" by (import bits SLICE_LEM1) lemma SLICE_LEM2: "ALL (a::'a::type) (x::nat) y::nat. (n::nat) mod 2 ^ (x + y) = n mod 2 ^ x + n div 2 ^ x mod 2 ^ y * 2 ^ x" by (import bits SLICE_LEM2) lemma SLICE_LEM3: "ALL (n::nat) (h::nat) l::nat. l < h --> n mod 2 ^ Suc l <= n mod 2 ^ 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 ^ l" by (import bits SLICE_THM) lemma SLICELT_THM: "ALL (h::nat) (l::nat) n::nat. SLICE h l n < 2 ^ 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 ^ l <= n mod 2 ^ 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" 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 ^ (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)" by (import bits NOT_BIT) lemma NOT_BITS: "ALL (n::nat) a::nat. (BITS n n a ~= 0) = (BITS n n a = 1)" by (import bits NOT_BITS) lemma NOT_BITS2: "ALL (n::nat) a::nat. (BITS n n a ~= 1) = (BITS n n a = 0)" 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 ^ 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 ^ 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 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 ^ n" by (import bits BITWISE_LT_2EXP) lemma LESS_EXP_MULT2: "(All::(nat => bool) => bool) (%a::nat. (All::(nat => bool) => bool) (%b::nat. (op -->::bool => bool => bool) ((op <::nat => nat => bool) a b) ((Ex::(nat => bool) => bool) (%x::nat. (op =::nat => nat => bool) ((op ^::nat => nat => nat) ((number_of :: int => nat) ((Int.Bit0 :: int => int) ((Int.Bit1 :: int => int) (Int.Pls :: int)))) b) ((op *::nat => nat => nat) ((op ^::nat => nat => nat) ((number_of :: int => nat) ((Int.Bit0 :: int => int) ((Int.Bit1 :: int => int) (Int.Pls :: int)))) ((op +::nat => nat => nat) x (1::nat))) ((op ^::nat => nat => nat) ((number_of :: int => nat) ((Int.Bit0 :: int => int) ((Int.Bit1 :: int => int) (Int.Pls :: int)))) 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 ^ x mod 2 = 1" 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 ^ x mod 2 = 0" by (import bits BITWISE_NOT_COR) lemma MOD_PLUS_RIGHT: "ALL n>0. 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. ALL x::nat. ((x + 1) mod n = 0) = (x mod n + 1 = n)" by (import bits MOD_PLUS_1) lemma MOD_ADD_1: "ALL n>0. ALL x::nat. (x + 1) mod n ~= 0 --> (x + 1) mod n = x mod n + 1" 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 ^ WL" lemma MODw_def: "ALL n::nat. MODw n = n mod 2 ^ WL" by (import word32 MODw_def) consts INw :: "nat => bool" defs INw_primdef: "INw == %n::nat. n < 2 ^ WL" lemma INw_def: "ALL n::nat. INw n = (n < 2 ^ 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" 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" lemma AONE_def: "AONE = 1" by (import word32 AONE_def) lemma ADD_QT: "(ALL n::nat. EQUIV (0 + 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) 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 * n) 0) & (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 + m) m) & (ALL m::nat. EQUIV (m + 0) 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 ^ WL - 1))" 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" 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" 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 * 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::nat. 2 ^ WL - 1 - MODw x" lemma ONE_COMP_def: "ALL x::nat. 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::nat. 2 ^ WL - MODw x" lemma TWO_COMP_def: "ALL x::nat. TWO_COMP x = 2 ^ WL - MODw x" by (import word32 TWO_COMP_def) lemma ADD_TWO_COMP_QT: "ALL a::nat. EQUIV (MODw a + TWO_COMP a) 0" 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 a = SLICE (Suc n) (Suc n) a + BITS n 0 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" lemma COMP0_def: "COMP0 = ONE_COMP 0" 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" 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 < 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::nat) (oper::bool => bool => bool) (a::nat) b::nat. 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::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" lemma LSR_ONE_def: "ALL a::nat. 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::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" 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)" 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::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" 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::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 ^ n))" lemma word_lsl: "ALL (a::word32) n::nat. word_lsl a n = word_mul a (n2w (2 ^ 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 ^ b)" by (import word32 MOD_MOD_DIV) lemma MOD_MOD_DIV_2EXP: "ALL (a::nat) n::nat. 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::nat. word_lsr (n2w (a::nat)) n = n2w (MODw a div 2 ^ 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 ^ m --> 2 ^ n + a * 2 ^ n <= 2 ^ (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) 0 n * 2 ^ (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 ^ WL - 2 ^ (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 = a) & (ALL a::word32. word_asr a 0 = a) & (ALL a::word32. word_lsr a 0 = a) & (ALL a::word32. word_ror a 0 = 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 ^ 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::nat. 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::nat. word_lsr1 (n2w a) = n2w (MODw a div 2)" by (import word32 LSR_ONE_EVAL2) lemma ASR_ONE_EVAL2: "ALL a::nat. 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::nat. 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::bool) a::nat. 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::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 (n::nat) (oper::bool => bool => bool) (x::nat) y::nat. 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::nat) (l::nat) n::word32. BITSw h l n < 2 ^ (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 ^ 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)" by (import word32 BITw_THM) lemma SLICEw_THM: "ALL (n::word32) (h::nat) l::nat. SLICEw h l n = BITSw h l n * 2 ^ 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 n = BITSw h 0 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" by (import word32 BITSw_ZERO) lemma SLICEw_ZERO: "ALL (h::nat) (l::nat) n::word32. h < l --> SLICEw h l n = 0" by (import word32 SLICEw_ZERO) ;end_setup end
lemma DIV2_def:
∀n. DIV2 n = n div 2
lemma TIMES_2EXP_def:
∀x n. TIMES_2EXP x n = n * 2 ^ x
lemma DIV_2EXP_def:
∀x n. DIV_2EXP x n = n div 2 ^ x
lemma MOD_2EXP_def:
∀x n. MOD_2EXP x n = n mod 2 ^ x
lemma DIVMOD_2EXP_def:
∀x n. DIVMOD_2EXP x n = (n div 2 ^ x, n mod 2 ^ x)
lemma SBIT_def:
∀b n. SBIT b n = (if b then 2 ^ n else 0)
lemma BITS_def:
∀h l n. BITS h l n = MOD_2EXP (Suc h - l) (DIV_2EXP l n)
lemma BIT_def:
∀b n. bit b n = (BITS b b n = 1)
lemma SLICE_def:
∀h l n. SLICE h l n = MOD_2EXP (Suc h) n - MOD_2EXP l n
lemma LSBn_def:
LSBn = bit 0
lemma DIV1:
∀x. x div 1 = x
lemma SUC_SUB:
Suc a - a = 1
lemma DIV_MULT_1:
∀r n. r < n --> (n + r) div n = 1
lemma ZERO_LT_TWOEXP:
∀n. 0 < 2 ^ n
lemma MOD_2EXP_LT:
∀n k. k mod 2 ^ n < 2 ^ n
lemma TWOEXP_DIVISION:
∀n k. k = k div 2 ^ n * 2 ^ n + k mod 2 ^ n
lemma TWOEXP_MONO:
∀a b. a < b --> 2 ^ a < 2 ^ b
lemma TWOEXP_MONO2:
∀a b. a ≤ b --> 2 ^ a ≤ 2 ^ b
lemma EXP_SUB_LESS_EQ:
∀a b. 2 ^ (a - b) ≤ 2 ^ a
lemma BITS_THM:
∀x xa xb. BITS x xa xb = xb div 2 ^ xa mod 2 ^ (Suc x - xa)
lemma BITSLT_THM:
∀h l n. BITS h l n < 2 ^ (Suc h - l)
lemma DIV_MULT_LEM:
∀m n. 0 < n --> m div n * n ≤ m
lemma MOD_2EXP_LEM:
∀n x. n mod 2 ^ x = n - n div 2 ^ x * 2 ^ x
lemma BITS2_THM:
∀h l n. BITS h l n = n mod 2 ^ Suc h div 2 ^ l
lemma BITS_COMP_THM:
∀h1 l1 h2 l2 n.
h2 + l1 ≤ h1 --> BITS h2 l2 (BITS h1 l1 n) = BITS (h2 + l1) (l2 + l1) n
lemma BITS_DIV_THM:
∀h l x n. BITS h l x div 2 ^ n = BITS h (l + n) x
lemma BITS_LT_HIGH:
∀h l n. n < 2 ^ Suc h --> BITS h l n = n div 2 ^ l
lemma BITS_ZERO:
∀h l n. h < l --> BITS h l n = 0
lemma BITS_ZERO2:
∀h l. BITS h l 0 = 0
lemma BITS_ZERO3:
∀h x. BITS h 0 x = x mod 2 ^ Suc h
lemma BITS_COMP_THM2:
∀h1 l1 h2 l2 n. BITS h2 l2 (BITS h1 l1 n) = BITS (min h1 (h2 + l1)) (l2 + l1) n
lemma NOT_MOD2_LEM:
∀n. (n mod 2 ≠ 0) = (n mod 2 = 1)
lemma NOT_MOD2_LEM2:
∀n a. (n mod 2 ≠ 1) = (n mod 2 = 0)
lemma EVEN_MOD2_LEM:
∀n. EVEN n = (n mod 2 = 0)
lemma ODD_MOD2_LEM:
∀n. ODD n = (n mod 2 = 1)
lemma LSB_ODD:
LSBn = ODD
lemma DIV_MULT_THM:
∀x n. n div 2 ^ x * 2 ^ x = n - n mod 2 ^ x
lemma DIV_MULT_THM2:
∀x. 2 * (x div 2) = x - x mod 2
lemma LESS_EQ_EXP_MULT:
∀a b. a ≤ b --> (∃x. 2 ^ b = x * 2 ^ a)
lemma SLICE_LEM1:
∀a x y.
a div 2 ^ (x + y) * 2 ^ (x + y) =
a div 2 ^ x * 2 ^ x - a div 2 ^ x mod 2 ^ y * 2 ^ x
lemma SLICE_LEM2:
∀a x y. n mod 2 ^ (x + y) = n mod 2 ^ x + n div 2 ^ x mod 2 ^ y * 2 ^ x
lemma SLICE_LEM3:
∀n h l. l < h --> n mod 2 ^ Suc l ≤ n mod 2 ^ h
lemma SLICE_THM:
∀n h l. SLICE h l n = BITS h l n * 2 ^ l
lemma SLICELT_THM:
∀h l n. SLICE h l n < 2 ^ Suc h
lemma BITS_SLICE_THM:
∀h l n. BITS h l (SLICE h l n) = BITS h l n
lemma BITS_SLICE_THM2:
∀h l n. h ≤ h2.0 --> BITS h2.0 l (SLICE h l n) = BITS h l n
lemma MOD_2EXP_MONO:
∀n h l. l ≤ h --> n mod 2 ^ l ≤ n mod 2 ^ Suc h
lemma SLICE_COMP_THM:
∀h m l n. Suc m ≤ h ∧ l ≤ m --> SLICE h (Suc m) n + SLICE m l n = SLICE h l n
lemma SLICE_ZERO:
∀h l n. h < l --> SLICE h l n = 0
lemma BIT_COMP_THM3:
∀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
lemma NOT_BIT:
∀n a. (¬ bit n a) = (BITS n n a = 0)
lemma NOT_BITS:
∀n a. (BITS n n a ≠ 0) = (BITS n n a = 1)
lemma NOT_BITS2:
∀n a. (BITS n n a ≠ 1) = (BITS n n a = 0)
lemma BIT_SLICE:
∀n a b. (bit n a = bit n b) = (SLICE n n a = SLICE n n b)
lemma BIT_SLICE_LEM:
∀y x n. SBIT (bit x n) (x + y) = SLICE x x n * 2 ^ y
lemma BIT_SLICE_THM:
∀x xa. SBIT (bit x xa) x = SLICE x x xa
lemma SBIT_DIV:
∀b m n. n < m --> SBIT b (m - n) = SBIT b m div 2 ^ n
lemma BITS_SUC:
∀h l n.
l ≤ Suc h -->
SBIT (bit (Suc h) n) (Suc h - l) + BITS h l n = BITS (Suc h) l n
lemma BITS_SUC_THM:
∀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)
lemma BIT_BITS_THM:
∀h l a b. (∀x. l ≤ x ∧ x ≤ h --> bit x a = bit x b) = (BITS h l a = BITS h l b)
lemma BITWISE_LT_2EXP:
∀n oper a b. BITWISE n oper a b < 2 ^ n
lemma LESS_EXP_MULT2:
∀a b. a < b --> (∃x. 2 ^ b = 2 ^ (x + 1) * 2 ^ a)
lemma BITWISE_THM:
∀x n oper a b. x < n --> bit x (BITWISE n oper a b) = oper (bit x a) (bit x b)
lemma BITWISE_COR:
∀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
lemma BITWISE_NOT_COR:
∀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
lemma MOD_PLUS_RIGHT:
∀n>0. ∀j k. (j + k mod n) mod n = (j + k) mod n
lemma MOD_PLUS_1:
∀n>0. ∀x. ((x + 1) mod n = 0) = (x mod n + 1 = n)
lemma MOD_ADD_1:
∀n>0. ∀x. (x + 1) mod n ≠ 0 --> (x + 1) mod n = x mod n + 1
lemma HB_def:
HB =
NUMERAL
(NUMERAL_BIT1
(NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))
lemma WL_def:
WL = Suc HB
lemma MODw_def:
∀n. MODw n = n mod 2 ^ WL
lemma INw_def:
∀n. INw n = (n < 2 ^ WL)
lemma EQUIV_def:
∀x y. EQUIV x y = (MODw x = MODw y)
lemma EQUIV_QT:
∀x y. EQUIV x y = (EQUIV x = EQUIV y)
lemma FUNPOW_THM:
∀f n x. (f ^ n) (f x) = f ((f ^ n) x)
lemma FUNPOW_THM2:
∀f n x. (f ^ Suc n) x = f ((f ^ n) x)
lemma FUNPOW_COMP:
∀f m n a. (f ^ m) ((f ^ n) a) = (f ^ (m + n)) a
lemma INw_MODw:
∀n. INw (MODw n)
lemma TOw_IDEM:
∀a. INw a --> MODw a = a
lemma MODw_IDEM2:
∀a. MODw (MODw a) = MODw a
lemma TOw_QT:
∀a. EQUIV (MODw a) a
lemma MODw_THM:
MODw = BITS HB 0
lemma MOD_ADD:
∀a b. MODw (a + b) = MODw (MODw a + MODw b)
lemma MODw_MULT:
∀a b. MODw (a * b) = MODw (MODw a * MODw b)
lemma AONE_def:
AONE = 1
lemma ADD_QT:
(∀n. EQUIV (0 + n) n) ∧ (∀m n. EQUIV (Suc m + n) (Suc (m + n)))
lemma ADD_0_QT:
∀a. EQUIV (a + 0) a
lemma ADD_COMM_QT:
∀a b. EQUIV (a + b) (b + a)
lemma ADD_ASSOC_QT:
∀a b c. EQUIV (a + (b + c)) (a + b + c)
lemma MULT_QT:
(∀n. EQUIV (0 * n) 0) ∧ (∀m n. EQUIV (Suc m * n) (m * n + n))
lemma ADD1_QT:
∀m. EQUIV (Suc m) (m + AONE)
lemma ADD_CLAUSES_QT:
(∀m. EQUIV (0 + m) m) ∧
(∀m. EQUIV (m + 0) m) ∧
(∀m n. EQUIV (Suc m + n) (Suc (m + n))) ∧
(∀m n. EQUIV (m + Suc n) (Suc (m + n)))
lemma SUC_EQUIV_COMP:
∀a b. EQUIV (Suc a) b --> EQUIV a (b + (2 ^ WL - 1))
lemma INV_SUC_EQ_QT:
∀m n. EQUIV (Suc m) (Suc n) = EQUIV m n
lemma ADD_INV_0_QT:
∀m n. EQUIV (m + n) m --> EQUIV n 0
lemma ADD_INV_0_EQ_QT:
∀m n. EQUIV (m + n) m = EQUIV n 0
lemma EQ_ADD_LCANCEL_QT:
∀m n p. EQUIV (m + n) (m + p) = EQUIV n p
lemma EQ_ADD_RCANCEL_QT:
∀x xa xb. EQUIV (x + xb) (xa + xb) = EQUIV x xa
lemma LEFT_ADD_DISTRIB_QT:
∀m n p. EQUIV (p * (m + n)) (p * m + p * n)
lemma MULT_ASSOC_QT:
∀m n p. EQUIV (m * (n * p)) (m * n * p)
lemma MULT_COMM_QT:
∀m n. EQUIV (m * n) (n * m)
lemma MULT_CLAUSES_QT:
∀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)
lemma MSBn_def:
MSBn = bit HB
lemma ONE_COMP_def:
∀x. ONE_COMP x = 2 ^ WL - 1 - MODw x
lemma TWO_COMP_def:
∀x. TWO_COMP x = 2 ^ WL - MODw x
lemma ADD_TWO_COMP_QT:
∀a. EQUIV (MODw a + TWO_COMP a) 0
lemma TWO_COMP_ONE_COMP_QT:
∀a. EQUIV (TWO_COMP a) (ONE_COMP a + AONE)
lemma BIT_EQUIV_THM:
∀x xa. (∀xb<WL. bit xb x = bit xb xa) = EQUIV x xa
lemma BITS_SUC2:
∀n a. BITS (Suc n) 0 a = SLICE (Suc n) (Suc n) a + BITS n 0 a
lemma BITWISE_ONE_COMP_THM:
∀a b. BITWISE WL (λx y. ¬ x) a b = ONE_COMP a
lemma ONE_COMP_THM:
∀x xa. xa < WL --> bit xa (ONE_COMP x) = (¬ bit xa x)
lemma OR_def:
OR = BITWISE WL op ∨
lemma AND_def:
AND = BITWISE WL op ∧
lemma EOR_def:
EOR = BITWISE WL op ≠
lemma COMP0_def:
COMP0 = ONE_COMP 0
lemma BITWISE_THM2:
∀y oper a b.
(∀x<WL. oper (bit x a) (bit x b) = bit x y) = EQUIV (BITWISE WL oper a b) y
lemma OR_ASSOC_QT:
∀a b c. EQUIV (OR a (OR b c)) (OR (OR a b) c)
lemma OR_COMM_QT:
∀a b. EQUIV (OR a b) (OR b a)
lemma OR_ABSORB_QT:
∀a b. EQUIV (AND a (OR a b)) a
lemma OR_IDEM_QT:
∀a. EQUIV (OR a a) a
lemma AND_ASSOC_QT:
∀a b c. EQUIV (AND a (AND b c)) (AND (AND a b) c)
lemma AND_COMM_QT:
∀a b. EQUIV (AND a b) (AND b a)
lemma AND_ABSORB_QT:
∀a b. EQUIV (OR a (AND a b)) a
lemma AND_IDEM_QT:
∀a. EQUIV (AND a a) a
lemma OR_COMP_QT:
∀a. EQUIV (OR a (ONE_COMP a)) COMP0
lemma AND_COMP_QT:
∀a. EQUIV (AND a (ONE_COMP a)) 0
lemma ONE_COMP_QT:
∀a. EQUIV (ONE_COMP (ONE_COMP a)) a
lemma RIGHT_AND_OVER_OR_QT:
∀a b c. EQUIV (AND (OR a b) c) (OR (AND a c) (AND b c))
lemma RIGHT_OR_OVER_AND_QT:
∀a b c. EQUIV (OR (AND a b) c) (AND (OR a c) (OR b c))
lemma DE_MORGAN_THM_QT:
∀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))
lemma BIT_EQUIV:
∀n a b. n < WL --> EQUIV a b --> bit n a = bit n b
lemma LSB_WELLDEF:
∀a b. EQUIV a b --> LSBn a = LSBn b
lemma MSB_WELLDEF:
∀a b. EQUIV a b --> MSBn a = MSBn b
lemma BITWISE_ISTEP:
∀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)
lemma BITWISE_EVAL:
∀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
lemma BITWISE_WELLDEF:
∀n oper a b c d.
EQUIV a b ∧ EQUIV c d --> EQUIV (BITWISE n oper a c) (BITWISE n oper b d)
lemma BITWISEw_WELLDEF:
∀oper a b c d.
EQUIV a b ∧ EQUIV c d --> EQUIV (BITWISE WL oper a c) (BITWISE WL oper b d)
lemma SUC_WELLDEF:
∀a b. EQUIV a b --> EQUIV (Suc a) (Suc b)
lemma ADD_WELLDEF:
∀a b c d. EQUIV a b ∧ EQUIV c d --> EQUIV (a + c) (b + d)
lemma MUL_WELLDEF:
∀a b c d. EQUIV a b ∧ EQUIV c d --> EQUIV (a * c) (b * d)
lemma ONE_COMP_WELLDEF:
∀a b. EQUIV a b --> EQUIV (ONE_COMP a) (ONE_COMP b)
lemma TWO_COMP_WELLDEF:
∀a b. EQUIV a b --> EQUIV (TWO_COMP a) (TWO_COMP b)
lemma TOw_WELLDEF:
∀a b. EQUIV a b --> EQUIV (MODw a) (MODw b)
lemma LSR_ONE_def:
∀a. LSR_ONE a = MODw a div 2
lemma ASR_ONE_def:
∀a. ASR_ONE a = LSR_ONE a + SBIT (MSBn a) HB
lemma ROR_ONE_def:
∀a. ROR_ONE a = LSR_ONE a + SBIT (LSBn a) HB
lemma RRXn_def:
∀c a. RRXn c a = LSR_ONE a + SBIT c HB
lemma LSR_ONE_WELLDEF:
∀a b. EQUIV a b --> EQUIV (LSR_ONE a) (LSR_ONE b)
lemma ASR_ONE_WELLDEF:
∀a b. EQUIV a b --> EQUIV (ASR_ONE a) (ASR_ONE b)
lemma ROR_ONE_WELLDEF:
∀a b. EQUIV a b --> EQUIV (ROR_ONE a) (ROR_ONE b)
lemma RRX_WELLDEF:
∀a b c. EQUIV a b --> EQUIV (RRXn c a) (RRXn c b)
lemma LSR_ONE:
LSR_ONE = BITS HB 1
lemma word32_TY_DEF:
∃rep. TYPE_DEFINITION (λx. ∃xa. x = EQUIV xa) rep
lemma w_0_def:
w_0 = mk_word32 (EQUIV 0)
lemma w_1_def:
w_1 = mk_word32 (EQUIV AONE)
lemma w_T_def:
w_T = mk_word32 (EQUIV COMP0)
lemma word_suc:
∀T1. word_suc T1 = mk_word32 (EQUIV (Suc (Eps (dest_word32 T1))))
lemma word_add:
∀T1 T2.
word_add T1 T2 =
mk_word32 (EQUIV (Eps (dest_word32 T1) + Eps (dest_word32 T2)))
lemma word_mul:
∀T1 T2.
word_mul T1 T2 =
mk_word32 (EQUIV (Eps (dest_word32 T1) * Eps (dest_word32 T2)))
lemma word_1comp:
∀T1. word_1comp T1 = mk_word32 (EQUIV (ONE_COMP (Eps (dest_word32 T1))))
lemma word_2comp:
∀T1. word_2comp T1 = mk_word32 (EQUIV (TWO_COMP (Eps (dest_word32 T1))))
lemma word_lsr1:
∀T1. word_lsr1 T1 = mk_word32 (EQUIV (LSR_ONE (Eps (dest_word32 T1))))
lemma word_asr1:
∀T1. word_asr1 T1 = mk_word32 (EQUIV (ASR_ONE (Eps (dest_word32 T1))))
lemma word_ror1:
∀T1. word_ror1 T1 = mk_word32 (EQUIV (ROR_ONE (Eps (dest_word32 T1))))
lemma RRX_def:
∀T1 T2. RRX T1 T2 = mk_word32 (EQUIV (RRXn T1 (Eps (dest_word32 T2))))
lemma LSB_def:
∀T1. LSB T1 = LSBn (Eps (dest_word32 T1))
lemma MSB_def:
∀T1. MSB T1 = MSBn (Eps (dest_word32 T1))
lemma bitwise_or:
∀T1 T2.
bitwise_or T1 T2 =
mk_word32 (EQUIV (OR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))
lemma bitwise_eor:
∀T1 T2.
bitwise_eor T1 T2 =
mk_word32 (EQUIV (EOR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))
lemma bitwise_and:
∀T1 T2.
bitwise_and T1 T2 =
mk_word32 (EQUIV (AND (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))
lemma TOw_def:
∀T1. TOw T1 = mk_word32 (EQUIV (MODw (Eps (dest_word32 T1))))
lemma n2w_def:
∀n. n2w n = mk_word32 (EQUIV n)
lemma w2n_def:
∀w. w2n w = MODw (Eps (dest_word32 w))
lemma ADDw:
(∀x. word_add w_0 x = x) ∧
(∀x xa. word_add (word_suc x) xa = word_suc (word_add x xa))
lemma ADD_0w:
∀x. word_add x w_0 = x
lemma ADD1w:
∀x. word_suc x = word_add x w_1
lemma ADD_ASSOCw:
∀x xa xb. word_add x (word_add xa xb) = word_add (word_add x xa) xb
lemma ADD_CLAUSESw:
(∀x. word_add w_0 x = x) ∧
(∀x. word_add x w_0 = x) ∧
(∀x xa. word_add (word_suc x) xa = word_suc (word_add x xa)) ∧
(∀x xa. word_add x (word_suc xa) = word_suc (word_add x xa))
lemma ADD_COMMw:
∀x xa. word_add x xa = word_add xa x
lemma ADD_INV_0_EQw:
∀x xa. (word_add x xa = x) = (xa = w_0)
lemma EQ_ADD_LCANCELw:
∀x xa xb. (word_add x xa = word_add x xb) = (xa = xb)
lemma EQ_ADD_RCANCELw:
∀x xa xb. (word_add x xb = word_add xa xb) = (x = xa)
lemma LEFT_ADD_DISTRIBw:
∀x xa xb.
word_mul xb (word_add x xa) = word_add (word_mul xb x) (word_mul xb xa)
lemma MULT_ASSOCw:
∀x xa xb. word_mul x (word_mul xa xb) = word_mul (word_mul x xa) xb
lemma MULT_COMMw:
∀x xa. word_mul x xa = word_mul xa x
lemma MULT_CLAUSESw:
∀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)
lemma TWO_COMP_ONE_COMP:
∀x. word_2comp x = word_add (word_1comp x) w_1
lemma OR_ASSOCw:
∀x xa xb. bitwise_or x (bitwise_or xa xb) = bitwise_or (bitwise_or x xa) xb
lemma OR_COMMw:
∀x xa. bitwise_or x xa = bitwise_or xa x
lemma OR_IDEMw:
∀x. bitwise_or x x = x
lemma OR_ABSORBw:
∀x xa. bitwise_and x (bitwise_or x xa) = x
lemma AND_ASSOCw:
∀x xa xb. bitwise_and x (bitwise_and xa xb) = bitwise_and (bitwise_and x xa) xb
lemma AND_COMMw:
∀x xa. bitwise_and x xa = bitwise_and xa x
lemma AND_IDEMw:
∀x. bitwise_and x x = x
lemma AND_ABSORBw:
∀x xa. bitwise_or x (bitwise_and x xa) = x
lemma ONE_COMPw:
∀x. word_1comp (word_1comp x) = x
lemma RIGHT_AND_OVER_ORw:
∀x xa xb.
bitwise_and (bitwise_or x xa) xb =
bitwise_or (bitwise_and x xb) (bitwise_and xa xb)
lemma RIGHT_OR_OVER_ANDw:
∀x xa xb.
bitwise_or (bitwise_and x xa) xb =
bitwise_and (bitwise_or x xb) (bitwise_or xa xb)
lemma DE_MORGAN_THMw:
∀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)
lemma w_0:
w_0 = n2w 0
lemma w_1:
w_1 = n2w 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)))))))))))))))))))))))))))))))))
lemma ADD_TWO_COMP:
∀x. word_add x (word_2comp x) = w_0
lemma ADD_TWO_COMP2:
∀x. word_add (word_2comp x) x = w_0
lemma word_sub:
∀a b. word_sub a b = word_add a (word_2comp b)
lemma word_lsl:
∀a n. word_lsl a n = word_mul a (n2w (2 ^ n))
lemma word_lsr:
∀a n. word_lsr a n = (word_lsr1 ^ n) a
lemma word_asr:
∀a n. word_asr a n = (word_asr1 ^ n) a
lemma word_ror:
∀a n. word_ror a n = (word_ror1 ^ n) a
lemma BITw_def:
∀b n. BITw b n = bit b (w2n n)
lemma BITSw_def:
∀h l n. BITSw h l n = BITS h l (w2n n)
lemma SLICEw_def:
∀h l n. SLICEw h l n = SLICE h l (w2n n)
lemma TWO_COMP_ADD:
∀a b. word_2comp (word_add a b) = word_add (word_2comp a) (word_2comp b)
lemma TWO_COMP_ELIM:
∀a. word_2comp (word_2comp a) = a
lemma ADD_SUB_ASSOC:
∀a b c. word_sub (word_add a b) c = word_add a (word_sub b c)
lemma ADD_SUB_SYM:
∀a b c. word_sub (word_add a b) c = word_add (word_sub a c) b
lemma SUB_EQUALw:
∀a. word_sub a a = w_0
lemma ADD_SUBw:
∀a b. word_sub (word_add a b) b = a
lemma SUB_SUBw:
∀a b c. word_sub a (word_sub b c) = word_sub (word_add a c) b
lemma ONE_COMP_TWO_COMP:
∀a. word_1comp a = word_sub (word_2comp a) w_1
lemma SUBw:
∀m n. word_sub (word_suc m) n = word_suc (word_sub m n)
lemma ADD_EQ_SUBw:
∀m n p. (word_add m n = p) = (m = word_sub p n)
lemma CANCEL_SUBw:
∀m n p. (word_sub n p = word_sub m p) = (n = m)
lemma SUB_PLUSw:
∀a b c. word_sub a (word_add b c) = word_sub (word_sub a b) c
lemma word_nchotomy:
∀w. ∃n. w = n2w n
lemma dest_word_mk_word_eq3:
∀a. dest_word32 (mk_word32 (EQUIV a)) = EQUIV a
lemma MODw_ELIM:
∀n. n2w (MODw n) = n2w n
lemma w2n_EVAL:
∀n. w2n (n2w n) = MODw n
lemma w2n_ELIM:
∀a. n2w (w2n a) = a
lemma n2w_11:
∀a b. (n2w a = n2w b) = (MODw a = MODw b)
lemma ADD_EVAL:
word_add (n2w a) (n2w b) = n2w (a + b)
lemma MUL_EVAL:
word_mul (n2w a) (n2w b) = n2w (a * b)
lemma ONE_COMP_EVAL:
word_1comp (n2w a) = n2w (ONE_COMP a)
lemma TWO_COMP_EVAL:
word_2comp (n2w a) = n2w (TWO_COMP a)
lemma LSR_ONE_EVAL:
word_lsr1 (n2w a) = n2w (LSR_ONE a)
lemma ASR_ONE_EVAL:
word_asr1 (n2w a) = n2w (ASR_ONE a)
lemma ROR_ONE_EVAL:
word_ror1 (n2w a) = n2w (ROR_ONE a)
lemma RRX_EVAL:
RRX c (n2w a) = n2w (RRXn c a)
lemma LSB_EVAL:
LSB (n2w a) = LSBn a
lemma MSB_EVAL:
MSB (n2w a) = MSBn a
lemma OR_EVAL:
bitwise_or (n2w a) (n2w b) = n2w (OR a b)
lemma EOR_EVAL:
bitwise_eor (n2w a) (n2w b) = n2w (EOR a b)
lemma AND_EVAL:
bitwise_and (n2w a) (n2w b) = n2w (AND a b)
lemma BITS_EVAL:
∀h l a. BITSw h l (n2w a) = BITS h l (MODw a)
lemma BIT_EVAL:
∀b a. BITw b (n2w a) = bit b (MODw a)
lemma SLICE_EVAL:
∀h l a. SLICEw h l (n2w a) = SLICE h l (MODw a)
lemma LSL_ADD:
∀a m n. word_lsl (word_lsl a m) n = word_lsl a (m + n)
lemma LSR_ADD:
∀x xa xb. word_lsr (word_lsr x xa) xb = word_lsr x (xa + xb)
lemma ASR_ADD:
∀x xa xb. word_asr (word_asr x xa) xb = word_asr x (xa + xb)
lemma ROR_ADD:
∀x xa xb. word_ror (word_ror x xa) xb = word_ror x (xa + xb)
lemma LSL_LIMIT:
∀w n. HB < n --> word_lsl w n = w_0
lemma MOD_MOD_DIV:
∀a b. INw (MODw a div 2 ^ b)
lemma MOD_MOD_DIV_2EXP:
∀a n. MODw (MODw a div 2 ^ n) div 2 = MODw a div 2 ^ Suc n
lemma LSR_EVAL:
∀n. word_lsr (n2w a) n = n2w (MODw a div 2 ^ n)
lemma LSR_THM:
∀x n. word_lsr (n2w n) x = n2w (BITS HB (min WL x) n)
lemma LSR_LIMIT:
∀x w. HB < x --> word_lsr w x = w_0
lemma LEFT_SHIFT_LESS:
∀n m a. a < 2 ^ m --> 2 ^ n + a * 2 ^ n ≤ 2 ^ (m + n)
lemma ROR_THM:
∀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')))
lemma ROR_CYCLE:
∀x w. word_ror w (x * WL) = w
lemma ASR_THM:
∀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))
lemma ASR_LIMIT:
∀x w. HB ≤ x --> word_asr w x = (if MSB w then w_T else w_0)
lemma ZERO_SHIFT:
(∀n. word_lsl w_0 n = w_0) ∧
(∀n. word_asr w_0 n = w_0) ∧
(∀n. word_lsr w_0 n = w_0) ∧ (∀n. word_ror w_0 n = w_0)
lemma ZERO_SHIFT2:
(∀a. word_lsl a 0 = a) ∧
(∀a. word_asr a 0 = a) ∧ (∀a. word_lsr a 0 = a) ∧ (∀a. word_ror a 0 = a)
lemma ASR_w_T:
∀n. word_asr w_T n = w_T
lemma ROR_w_T:
∀n. word_ror w_T n = w_T
lemma MODw_EVAL:
∀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))))))))))))))))))))))))))))))))
lemma ADD_EVAL2:
∀b a. word_add (n2w a) (n2w b) = n2w (MODw (a + b))
lemma MUL_EVAL2:
∀b a. word_mul (n2w a) (n2w b) = n2w (MODw (a * b))
lemma ONE_COMP_EVAL2:
∀a. word_1comp (n2w a) =
n2w (2 ^ NUMERAL
(NUMERAL_BIT2
(NUMERAL_BIT1
(NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))) -
1 -
MODw a)
lemma TWO_COMP_EVAL2:
∀a. word_2comp (n2w a) =
n2w (MODw (2 ^ NUMERAL
(NUMERAL_BIT2
(NUMERAL_BIT1
(NUMERAL_BIT1
(NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))) -
MODw a))
lemma LSR_ONE_EVAL2:
∀a. word_lsr1 (n2w a) = n2w (MODw a div 2)
lemma ASR_ONE_EVAL2:
∀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)))))))
lemma ROR_ONE_EVAL2:
∀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)))))))
lemma RRX_EVAL2:
∀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)))))))
lemma LSB_EVAL2:
∀a. LSB (n2w a) = ODD a
lemma MSB_EVAL2:
∀a. MSB (n2w a) =
bit (NUMERAL
(NUMERAL_BIT1
(NUMERAL_BIT1
(NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
a
lemma OR_EVAL2:
∀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)
lemma AND_EVAL2:
∀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)
lemma EOR_EVAL2:
∀b a. bitwise_eor (n2w a) (n2w b) =
n2w (BITWISE
(NUMERAL
(NUMERAL_BIT2
(NUMERAL_BIT1
(NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
op ≠ a b)
lemma BITWISE_EVAL2:
∀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))
lemma BITSwLT_THM:
∀h l n. BITSw h l n < 2 ^ (Suc h - l)
lemma BITSw_COMP_THM:
∀h1 l1 h2 l2 n.
h2 + l1 ≤ h1 --> BITS h2 l2 (BITSw h1 l1 n) = BITSw (h2 + l1) (l2 + l1) n
lemma BITSw_DIV_THM:
∀h l n x. BITSw h l x div 2 ^ n = BITSw h (l + n) x
lemma BITw_THM:
∀b n. BITw b n = (BITSw b b n = 1)
lemma SLICEw_THM:
∀n h l. SLICEw h l n = BITSw h l n * 2 ^ l
lemma BITS_SLICEw_THM:
∀h l n. BITS h l (SLICEw h l n) = BITSw h l n
lemma SLICEw_ZERO_THM:
∀n h. SLICEw h 0 n = BITSw h 0 n
lemma SLICEw_COMP_THM:
∀h m l a. Suc m ≤ h ∧ l ≤ m --> SLICEw h (Suc m) a + SLICEw m l a = SLICEw h l a
lemma BITSw_ZERO:
∀h l n. h < l --> BITSw h l n = 0
lemma SLICEw_ZERO:
∀h l n. h < l --> SLICEw h l n = 0