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theory WordDefinition(* ID: $Id: WordDefinition.thy,v 1.14 2007/11/08 19:08:00 wenzelm Exp $ Author: Jeremy Dawson and Gerwin Klein, NICTA Basic definition of word type and basic theorems following from the definition of the word type *) header {* Definition of Word Type *} theory WordDefinition imports Size BinBoolList TdThs begin typedef (open word) 'a word = "{(0::int) ..< 2^len_of TYPE('a::len0)}" by auto instance word :: (len0) number .. instance word :: (type) minus .. instance word :: (type) plus .. instance word :: (type) one .. instance word :: (type) zero .. instance word :: (type) times .. instance word :: (type) Divides.div .. instance word :: (type) power .. instance word :: (type) ord .. instance word :: (type) size .. instance word :: (type) inverse .. instance word :: (type) bit .. subsection "Type conversions and casting" constdefs -- {* representation of words using unsigned or signed bins, only difference in these is the type class *} word_of_int :: "int => 'a :: len0 word" "word_of_int w == Abs_word (bintrunc (len_of TYPE ('a)) w)" -- {* uint and sint cast a word to an integer, uint treats the word as unsigned, sint treats the most-significant-bit as a sign bit *} uint :: "'a :: len0 word => int" "uint w == Rep_word w" sint :: "'a :: len word => int" sint_uint: "sint w == sbintrunc (len_of TYPE ('a) - 1) (uint w)" unat :: "'a :: len0 word => nat" "unat w == nat (uint w)" -- "the sets of integers representing the words" uints :: "nat => int set" "uints n == range (bintrunc n)" sints :: "nat => int set" "sints n == range (sbintrunc (n - 1))" unats :: "nat => nat set" "unats n == {i. i < 2 ^ n}" norm_sint :: "nat => int => int" "norm_sint n w == (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)" -- "cast a word to a different length" scast :: "'a :: len word => 'b :: len word" "scast w == word_of_int (sint w)" ucast :: "'a :: len0 word => 'b :: len0 word" "ucast w == word_of_int (uint w)" -- "whether a cast (or other) function is to a longer or shorter length" source_size :: "('a :: len0 word => 'b) => nat" "source_size c == let arb = arbitrary ; x = c arb in size arb" target_size :: "('a => 'b :: len0 word) => nat" "target_size c == size (c arbitrary)" is_up :: "('a :: len0 word => 'b :: len0 word) => bool" "is_up c == source_size c <= target_size c" is_down :: "('a :: len0 word => 'b :: len0 word) => bool" "is_down c == target_size c <= source_size c" constdefs of_bl :: "bool list => 'a :: len0 word" "of_bl bl == word_of_int (bl_to_bin bl)" to_bl :: "'a :: len0 word => bool list" "to_bl w == bin_to_bl (len_of TYPE ('a)) (uint w)" word_reverse :: "'a :: len0 word => 'a word" "word_reverse w == of_bl (rev (to_bl w))" defs (overloaded) word_size: "size (w :: 'a :: len0 word) == len_of TYPE('a)" word_number_of_def: "number_of w == word_of_int w" constdefs word_int_case :: "(int => 'b) => ('a :: len0 word) => 'b" "word_int_case f w == f (uint w)" syntax of_int :: "int => 'a" translations "case x of of_int y => b" == "word_int_case (%y. b) x" subsection "Arithmetic operations" defs (overloaded) word_1_wi: "(1 :: ('a :: len0) word) == word_of_int 1" word_0_wi: "(0 :: ('a :: len0) word) == word_of_int 0" word_le_def: "a <= b == uint a <= uint b" word_less_def: "x < y == x <= y & x ~= (y :: 'a :: len0 word)" constdefs word_succ :: "'a :: len0 word => 'a word" "word_succ a == word_of_int (Numeral.succ (uint a))" word_pred :: "'a :: len0 word => 'a word" "word_pred a == word_of_int (Numeral.pred (uint a))" udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) "a udvd b == EX n>=0. uint b = n * uint a" word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) "a <=s b == sint a <= sint b" word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) "(x <s y) == (x <=s y & x ~= y)" consts word_power :: "'a :: len0 word => nat => 'a word" primrec "word_power a 0 = 1" "word_power a (Suc n) = a * word_power a n" defs (overloaded) word_pow: "power == word_power" word_add_def: "a + b == word_of_int (uint a + uint b)" word_sub_wi: "a - b == word_of_int (uint a - uint b)" word_minus_def: "- a == word_of_int (- uint a)" word_mult_def: "a * b == word_of_int (uint a * uint b)" word_div_def: "a div b == word_of_int (uint a div uint b)" word_mod_def: "a mod b == word_of_int (uint a mod uint b)" subsection "Bit-wise operations" defs (overloaded) word_and_def: "(a::'a::len0 word) AND b == word_of_int (uint a AND uint b)" word_or_def: "(a::'a::len0 word) OR b == word_of_int (uint a OR uint b)" word_xor_def: "(a::'a::len0 word) XOR b == word_of_int (uint a XOR uint b)" word_not_def: "NOT (a::'a::len0 word) == word_of_int (NOT (uint a))" word_test_bit_def: "test_bit (a::'a::len0 word) == bin_nth (uint a)" word_set_bit_def: "set_bit (a::'a::len0 word) n x == word_of_int (bin_sc n (If x bit.B1 bit.B0) (uint a))" word_set_bits_def: "(BITS n. f n)::'a::len0 word == of_bl (bl_of_nth (len_of TYPE ('a)) f)" word_lsb_def: "lsb (a::'a::len0 word) == bin_last (uint a) = bit.B1" word_msb_def: "msb (a::'a::len word) == bin_sign (sint a) = Numeral.Min" constdefs setBit :: "'a :: len0 word => nat => 'a word" "setBit w n == set_bit w n True" clearBit :: "'a :: len0 word => nat => 'a word" "clearBit w n == set_bit w n False" subsection "Shift operations" constdefs shiftl1 :: "'a :: len0 word => 'a word" "shiftl1 w == word_of_int (uint w BIT bit.B0)" -- "shift right as unsigned or as signed, ie logical or arithmetic" shiftr1 :: "'a :: len0 word => 'a word" "shiftr1 w == word_of_int (bin_rest (uint w))" sshiftr1 :: "'a :: len word => 'a word" "sshiftr1 w == word_of_int (bin_rest (sint w))" bshiftr1 :: "bool => 'a :: len word => 'a word" "bshiftr1 b w == of_bl (b # butlast (to_bl w))" sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) "w >>> n == (sshiftr1 ^ n) w" mask :: "nat => 'a::len word" "mask n == (1 << n) - 1" revcast :: "'a :: len0 word => 'b :: len0 word" "revcast w == of_bl (takefill False (len_of TYPE('b)) (to_bl w))" slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" "slice1 n w == of_bl (takefill False n (to_bl w))" slice :: "nat => 'a :: len0 word => 'b :: len0 word" "slice n w == slice1 (size w - n) w" defs (overloaded) shiftl_def: "(w::'a::len0 word) << n == (shiftl1 ^ n) w" shiftr_def: "(w::'a::len0 word) >> n == (shiftr1 ^ n) w" subsection "Rotation" constdefs rotater1 :: "'a list => 'a list" "rotater1 ys == case ys of [] => [] | x # xs => last ys # butlast ys" rotater :: "nat => 'a list => 'a list" "rotater n == rotater1 ^ n" word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" "word_rotr n w == of_bl (rotater n (to_bl w))" word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" "word_rotl n w == of_bl (rotate n (to_bl w))" word_roti :: "int => 'a :: len0 word => 'a :: len0 word" "word_roti i w == if i >= 0 then word_rotr (nat i) w else word_rotl (nat (- i)) w" subsection "Split and cat operations" constdefs word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" "word_cat a b == word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))" word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" "word_split a == case bin_split (len_of TYPE ('c)) (uint a) of (u, v) => (word_of_int u, word_of_int v)" word_rcat :: "'a :: len0 word list => 'b :: len0 word" "word_rcat ws == word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))" word_rsplit :: "'a :: len0 word => 'b :: len word list" "word_rsplit w == map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))" constdefs -- "Largest representable machine integer." max_word :: "'a::len word" "max_word ≡ word_of_int (2^len_of TYPE('a) - 1)" consts of_bool :: "bool => 'a::len word" primrec "of_bool False = 0" "of_bool True = 1" lemmas of_nth_def = word_set_bits_def lemmas word_size_gt_0 [iff] = xtr1 [OF word_size [THEN meta_eq_to_obj_eq] len_gt_0, standard] lemmas lens_gt_0 = word_size_gt_0 len_gt_0 lemmas lens_not_0 [iff] = lens_gt_0 [THEN gr_implies_not0, standard] lemma uints_num: "uints n = {i. 0 ≤ i ∧ i < 2 ^ n}" by (simp add: uints_def range_bintrunc) lemma sints_num: "sints n = {i. - (2 ^ (n - 1)) ≤ i ∧ i < 2 ^ (n - 1)}" by (simp add: sints_def range_sbintrunc) lemmas atLeastLessThan_alt = atLeastLessThan_def [unfolded atLeast_def lessThan_def Collect_conj_eq [symmetric]] lemma mod_in_reps: "m > 0 ==> y mod m : {0::int ..< m}" unfolding atLeastLessThan_alt by auto lemma Rep_word_0:"0 <= Rep_word x" and Rep_word_lt: "Rep_word (x::'a::len0 word) < 2 ^ len_of TYPE('a)" by (auto simp: Rep_word [simplified]) lemma Rep_word_mod_same: "Rep_word x mod 2 ^ len_of TYPE('a) = Rep_word (x::'a::len0 word)" by (simp add: int_mod_eq Rep_word_lt Rep_word_0) lemma td_ext_uint: "td_ext (uint :: 'a word => int) word_of_int (uints (len_of TYPE('a::len0))) (%w::int. w mod 2 ^ len_of TYPE('a))" apply (unfold td_ext_def') apply (simp add: uints_num uint_def word_of_int_def bintrunc_mod2p) apply (simp add: Rep_word_mod_same Rep_word_0 Rep_word_lt word.Rep_word_inverse word.Abs_word_inverse int_mod_lem) done lemmas int_word_uint = td_ext_uint [THEN td_ext.eq_norm, standard] interpretation word_uint: td_ext ["uint::'a::len0 word => int" word_of_int "uints (len_of TYPE('a::len0))" "λw. w mod 2 ^ len_of TYPE('a::len0)"] by (rule td_ext_uint) lemmas td_uint = word_uint.td_thm lemmas td_ext_ubin = td_ext_uint [simplified len_gt_0 no_bintr_alt1 [symmetric]] interpretation word_ubin: td_ext ["uint::'a::len0 word => int" word_of_int "uints (len_of TYPE('a::len0))" "bintrunc (len_of TYPE('a::len0))"] by (rule td_ext_ubin) lemma sint_sbintrunc': "sint (word_of_int bin :: 'a word) = (sbintrunc (len_of TYPE ('a :: len) - 1) bin)" unfolding sint_uint by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt) lemma uint_sint: "uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))" unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le) lemma bintr_uint': "n >= size w ==> bintrunc n (uint w) = uint w" apply (unfold word_size) apply (subst word_ubin.norm_Rep [symmetric]) apply (simp only: bintrunc_bintrunc_min word_size min_def) apply simp done lemma wi_bintr': "wb = word_of_int bin ==> n >= size wb ==> word_of_int (bintrunc n bin) = wb" unfolding word_size by (clarsimp simp add : word_ubin.norm_eq_iff [symmetric] min_def) lemmas bintr_uint = bintr_uint' [unfolded word_size] lemmas wi_bintr = wi_bintr' [unfolded word_size] lemma td_ext_sbin: "td_ext (sint :: 'a word => int) word_of_int (sints (len_of TYPE('a::len))) (sbintrunc (len_of TYPE('a) - 1))" apply (unfold td_ext_def' sint_uint) apply (simp add : word_ubin.eq_norm) apply (cases "len_of TYPE('a)") apply (auto simp add : sints_def) apply (rule sym [THEN trans]) apply (rule word_ubin.Abs_norm) apply (simp only: bintrunc_sbintrunc) apply (drule sym) apply simp done lemmas td_ext_sint = td_ext_sbin [simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]] (* We do sint before sbin, before sint is the user version and interpretations do not produce thm duplicates. I.e. we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD, because the latter is the same thm as the former *) interpretation word_sint: td_ext ["sint ::'a::len word => int" word_of_int "sints (len_of TYPE('a::len))" "%w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) - 2 ^ (len_of TYPE('a::len) - 1)"] by (rule td_ext_sint) interpretation word_sbin: td_ext ["sint ::'a::len word => int" word_of_int "sints (len_of TYPE('a::len))" "sbintrunc (len_of TYPE('a::len) - 1)"] by (rule td_ext_sbin) lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm, standard] lemmas td_sint = word_sint.td lemma word_number_of_alt: "number_of b == word_of_int (number_of b)" unfolding word_number_of_def by (simp add: number_of_eq) lemma word_no_wi: "number_of = word_of_int" by (auto simp: word_number_of_def intro: ext) lemma to_bl_def': "(to_bl :: 'a :: len0 word => bool list) = bin_to_bl (len_of TYPE('a)) o uint" by (auto simp: to_bl_def intro: ext) lemmas word_reverse_no_def [simp] = word_reverse_def [of "number_of w", standard] lemmas uints_mod = uints_def [unfolded no_bintr_alt1] lemma uint_bintrunc: "uint (number_of bin :: 'a word) = number_of (bintrunc (len_of TYPE ('a :: len0)) bin)" unfolding word_number_of_def number_of_eq by (auto intro: word_ubin.eq_norm) lemma sint_sbintrunc: "sint (number_of bin :: 'a word) = number_of (sbintrunc (len_of TYPE ('a :: len) - 1) bin)" unfolding word_number_of_def number_of_eq by (subst word_sbin.eq_norm) simp lemma unat_bintrunc: "unat (number_of bin :: 'a :: len0 word) = number_of (bintrunc (len_of TYPE('a)) bin)" unfolding unat_def nat_number_of_def by (simp only: uint_bintrunc) (* WARNING - these may not always be helpful *) declare uint_bintrunc [simp] sint_sbintrunc [simp] unat_bintrunc [simp] lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 ==> v = w" apply (unfold word_size) apply (rule word_uint.Rep_eqD) apply (rule box_equals) defer apply (rule word_ubin.norm_Rep)+ apply simp done lemmas uint_lem = word_uint.Rep [unfolded uints_num mem_Collect_eq] lemmas sint_lem = word_sint.Rep [unfolded sints_num mem_Collect_eq] lemmas uint_ge_0 [iff] = uint_lem [THEN conjunct1, standard] lemmas uint_lt2p [iff] = uint_lem [THEN conjunct2, standard] lemmas sint_ge = sint_lem [THEN conjunct1, standard] lemmas sint_lt = sint_lem [THEN conjunct2, standard] lemma sign_uint_Pls [simp]: "bin_sign (uint x) = Numeral.Pls" by (simp add: sign_Pls_ge_0 number_of_eq) lemmas uint_m2p_neg = iffD2 [OF diff_less_0_iff_less uint_lt2p, standard] lemmas uint_m2p_not_non_neg = iffD2 [OF linorder_not_le uint_m2p_neg, standard] lemma lt2p_lem: "len_of TYPE('a) <= n ==> uint (w :: 'a :: len0 word) < 2 ^ n" by (rule xtr8 [OF _ uint_lt2p]) simp lemmas uint_le_0_iff [simp] = uint_ge_0 [THEN leD, THEN linorder_antisym_conv1, standard] lemma uint_nat: "uint w == int (unat w)" unfolding unat_def by auto lemma uint_number_of: "uint (number_of b :: 'a :: len0 word) = number_of b mod 2 ^ len_of TYPE('a)" unfolding word_number_of_alt by (simp only: int_word_uint) lemma unat_number_of: "bin_sign b = Numeral.Pls ==> unat (number_of b::'a::len0 word) = number_of b mod 2 ^ len_of TYPE ('a)" apply (unfold unat_def) apply (clarsimp simp only: uint_number_of) apply (rule nat_mod_distrib [THEN trans]) apply (erule sign_Pls_ge_0 [THEN iffD1]) apply (simp_all add: nat_power_eq) done lemma sint_number_of: "sint (number_of b :: 'a :: len word) = (number_of b + 2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) - 2 ^ (len_of TYPE('a) - 1)" unfolding word_number_of_alt by (rule int_word_sint) lemma word_of_int_bin [simp] : "(word_of_int (number_of bin) :: 'a :: len0 word) = (number_of bin)" unfolding word_number_of_alt by auto lemma word_int_case_wi: "word_int_case f (word_of_int i :: 'b word) = f (i mod 2 ^ len_of TYPE('b::len0))" unfolding word_int_case_def by (simp add: word_uint.eq_norm) lemma word_int_split: "P (word_int_case f x) = (ALL i. x = (word_of_int i :: 'b :: len0 word) & 0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))" unfolding word_int_case_def by (auto simp: word_uint.eq_norm int_mod_eq') lemma word_int_split_asm: "P (word_int_case f x) = (~ (EX n. x = (word_of_int n :: 'b::len0 word) & 0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))" unfolding word_int_case_def by (auto simp: word_uint.eq_norm int_mod_eq') lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq, standard] lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq, standard] lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w" unfolding word_size by (rule uint_range') lemma sint_range_size: "- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)" unfolding word_size by (rule sint_range') lemmas sint_above_size = sint_range_size [THEN conjunct2, THEN [2] xtr8, folded One_nat_def, standard] lemmas sint_below_size = sint_range_size [THEN conjunct1, THEN [2] order_trans, folded One_nat_def, standard] lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)" unfolding word_test_bit_def by (simp add: bin_nth_eq_iff) lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w" apply (unfold word_test_bit_def) apply (subst word_ubin.norm_Rep [symmetric]) apply (simp only: nth_bintr word_size) apply fast done lemma word_eqI [rule_format] : fixes u :: "'a::len0 word" shows "(ALL n. n < size u --> u !! n = v !! n) ==> u = v" apply (rule test_bit_eq_iff [THEN iffD1]) apply (rule ext) apply (erule allE) apply (erule impCE) prefer 2 apply assumption apply (auto dest!: test_bit_size simp add: word_size) done lemmas word_eqD = test_bit_eq_iff [THEN iffD2, THEN fun_cong, standard] lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)" unfolding word_test_bit_def word_size by (simp add: nth_bintr [symmetric]) lemmas test_bit_bin = test_bit_bin' [unfolded word_size] lemma bin_nth_uint_imp': "bin_nth (uint w) n --> n < size w" apply (unfold word_size) apply (rule impI) apply (rule nth_bintr [THEN iffD1, THEN conjunct1]) apply (subst word_ubin.norm_Rep) apply assumption done lemma bin_nth_sint': "n >= size w --> bin_nth (sint w) n = bin_nth (sint w) (size w - 1)" apply (rule impI) apply (subst word_sbin.norm_Rep [symmetric]) apply (simp add : nth_sbintr word_size) apply auto done lemmas bin_nth_uint_imp = bin_nth_uint_imp' [rule_format, unfolded word_size] lemmas bin_nth_sint = bin_nth_sint' [rule_format, unfolded word_size] (* type definitions theorem for in terms of equivalent bool list *) lemma td_bl: "type_definition (to_bl :: 'a::len0 word => bool list) of_bl {bl. length bl = len_of TYPE('a)}" apply (unfold type_definition_def of_bl_def to_bl_def) apply (simp add: word_ubin.eq_norm) apply safe apply (drule sym) apply simp done interpretation word_bl: type_definition ["to_bl :: 'a::len0 word => bool list" of_bl "{bl. length bl = len_of TYPE('a::len0)}"] by (rule td_bl) lemma word_size_bl: "size w == size (to_bl w)" unfolding word_size by auto lemma to_bl_use_of_bl: "(to_bl w = bl) = (w = of_bl bl ∧ length bl = length (to_bl w))" by (fastsimp elim!: word_bl.Abs_inverse [simplified]) lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)" unfolding word_reverse_def by (simp add: word_bl.Abs_inverse) lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w" unfolding word_reverse_def by (simp add : word_bl.Abs_inverse) lemma word_rev_gal: "word_reverse w = u ==> word_reverse u = w" by auto lemmas word_rev_gal' = sym [THEN word_rev_gal, symmetric, standard] lemmas length_bl_gt_0 [iff] = xtr1 [OF word_bl.Rep' len_gt_0, standard] lemmas bl_not_Nil [iff] = length_bl_gt_0 [THEN length_greater_0_conv [THEN iffD1], standard] lemmas length_bl_neq_0 [iff] = length_bl_gt_0 [THEN gr_implies_not0] lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = Numeral.Min)" apply (unfold to_bl_def sint_uint) apply (rule trans [OF _ bl_sbin_sign]) apply simp done lemma of_bl_drop': "lend = length bl - len_of TYPE ('a :: len0) ==> of_bl (drop lend bl) = (of_bl bl :: 'a word)" apply (unfold of_bl_def) apply (clarsimp simp add : trunc_bl2bin [symmetric]) done lemmas of_bl_no = of_bl_def [folded word_number_of_def] lemma test_bit_of_bl: "(of_bl bl::'a::len0 word) !! n = (rev bl ! n ∧ n < len_of TYPE('a) ∧ n < length bl)" apply (unfold of_bl_def word_test_bit_def) apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl) done lemma no_of_bl: "(number_of bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) bin)" unfolding word_size of_bl_no by (simp add : word_number_of_def) lemma uint_bl: "to_bl w == bin_to_bl (size w) (uint w)" unfolding word_size to_bl_def by auto lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w" unfolding uint_bl by (simp add : word_size) lemma to_bl_of_bin: "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin" unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size) lemmas to_bl_no_bin [simp] = to_bl_of_bin [folded word_number_of_def] lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w" unfolding uint_bl by (simp add : word_size) lemmas uint_bl_bin [simp] = trans [OF bin_bl_bin word_ubin.norm_Rep, standard] lemmas num_AB_u [simp] = word_uint.Rep_inverse [unfolded o_def word_number_of_def [symmetric], standard] lemmas num_AB_s [simp] = word_sint.Rep_inverse [unfolded o_def word_number_of_def [symmetric], standard] (* naturals *) lemma uints_unats: "uints n = int ` unats n" apply (unfold unats_def uints_num) apply safe apply (rule_tac image_eqI) apply (erule_tac nat_0_le [symmetric]) apply auto apply (erule_tac nat_less_iff [THEN iffD2]) apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1]) apply (auto simp add : nat_power_eq int_power) done lemma unats_uints: "unats n = nat ` uints n" by (auto simp add : uints_unats image_iff) lemmas bintr_num = word_ubin.norm_eq_iff [symmetric, folded word_number_of_def, standard] lemmas sbintr_num = word_sbin.norm_eq_iff [symmetric, folded word_number_of_def, standard] lemmas num_of_bintr = word_ubin.Abs_norm [folded word_number_of_def, standard] lemmas num_of_sbintr = word_sbin.Abs_norm [folded word_number_of_def, standard]; (* don't add these to simpset, since may want bintrunc n w to be simplified; may want these in reverse, but loop as simp rules, so use following *) lemma num_of_bintr': "bintrunc (len_of TYPE('a :: len0)) a = b ==> number_of a = (number_of b :: 'a word)" apply safe apply (rule_tac num_of_bintr [symmetric]) done lemma num_of_sbintr': "sbintrunc (len_of TYPE('a :: len) - 1) a = b ==> number_of a = (number_of b :: 'a word)" apply safe apply (rule_tac num_of_sbintr [symmetric]) done lemmas num_abs_bintr = sym [THEN trans, OF num_of_bintr word_number_of_def [THEN meta_eq_to_obj_eq], standard] lemmas num_abs_sbintr = sym [THEN trans, OF num_of_sbintr word_number_of_def [THEN meta_eq_to_obj_eq], standard] (** cast - note, no arg for new length, as it's determined by type of result, thus in "cast w = w, the type means cast to length of w! **) lemma ucast_id: "ucast w = w" unfolding ucast_def by auto lemma scast_id: "scast w = w" unfolding scast_def by auto lemma ucast_bl: "ucast w == of_bl (to_bl w)" unfolding ucast_def of_bl_def uint_bl by (auto simp add : word_size) lemma nth_ucast: "(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))" apply (unfold ucast_def test_bit_bin) apply (simp add: word_ubin.eq_norm nth_bintr word_size) apply (fast elim!: bin_nth_uint_imp) done (* for literal u(s)cast *) lemma ucast_bintr [simp]: "ucast (number_of w ::'a::len0 word) = number_of (bintrunc (len_of TYPE('a)) w)" unfolding ucast_def by simp lemma scast_sbintr [simp]: "scast (number_of w ::'a::len word) = number_of (sbintrunc (len_of TYPE('a) - Suc 0) w)" unfolding scast_def by simp lemmas source_size = source_size_def [unfolded Let_def word_size] lemmas target_size = target_size_def [unfolded Let_def word_size] lemmas is_down = is_down_def [unfolded source_size target_size] lemmas is_up = is_up_def [unfolded source_size target_size] lemmas is_up_down = trans [OF is_up [THEN meta_eq_to_obj_eq] is_down [THEN meta_eq_to_obj_eq, symmetric], standard] lemma down_cast_same': "uc = ucast ==> is_down uc ==> uc = scast" apply (unfold is_down) apply safe apply (rule ext) apply (unfold ucast_def scast_def uint_sint) apply (rule word_ubin.norm_eq_iff [THEN iffD1]) apply simp done lemma word_rev_tf': "r = to_bl (of_bl bl) ==> r = rev (takefill False (length r) (rev bl))" unfolding of_bl_def uint_bl by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size) lemmas word_rev_tf = refl [THEN word_rev_tf', unfolded word_bl.Rep', standard] lemmas word_rep_drop = word_rev_tf [simplified takefill_alt, simplified, simplified rev_take, simplified] lemma to_bl_ucast: "to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = replicate (len_of TYPE('a) - len_of TYPE('b)) False @ drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)" apply (unfold ucast_bl) apply (rule trans) apply (rule word_rep_drop) apply simp done lemma ucast_up_app': "uc = ucast ==> source_size uc + n = target_size uc ==> to_bl (uc w) = replicate n False @ (to_bl w)" apply (auto simp add : source_size target_size to_bl_ucast) apply (rule_tac f = "%n. replicate n False" in arg_cong) apply simp done lemma ucast_down_drop': "uc = ucast ==> source_size uc = target_size uc + n ==> to_bl (uc w) = drop n (to_bl w)" by (auto simp add : source_size target_size to_bl_ucast) lemma scast_down_drop': "sc = scast ==> source_size sc = target_size sc + n ==> to_bl (sc w) = drop n (to_bl w)" apply (subgoal_tac "sc = ucast") apply safe apply simp apply (erule refl [THEN ucast_down_drop']) apply (rule refl [THEN down_cast_same', symmetric]) apply (simp add : source_size target_size is_down) done lemma sint_up_scast': "sc = scast ==> is_up sc ==> sint (sc w) = sint w" apply (unfold is_up) apply safe apply (simp add: scast_def word_sbin.eq_norm) apply (rule box_equals) prefer 3 apply (rule word_sbin.norm_Rep) apply (rule sbintrunc_sbintrunc_l) defer apply (subst word_sbin.norm_Rep) apply (rule refl) apply simp done lemma uint_up_ucast': "uc = ucast ==> is_up uc ==> uint (uc w) = uint w" apply (unfold is_up) apply safe apply (rule bin_eqI) apply (fold word_test_bit_def) apply (auto simp add: nth_ucast) apply (auto simp add: test_bit_bin) done lemmas down_cast_same = refl [THEN down_cast_same'] lemmas ucast_up_app = refl [THEN ucast_up_app'] lemmas ucast_down_drop = refl [THEN ucast_down_drop'] lemmas scast_down_drop = refl [THEN scast_down_drop'] lemmas uint_up_ucast = refl [THEN uint_up_ucast'] lemmas sint_up_scast = refl [THEN sint_up_scast'] lemma ucast_up_ucast': "uc = ucast ==> is_up uc ==> ucast (uc w) = ucast w" apply (simp (no_asm) add: ucast_def) apply (clarsimp simp add: uint_up_ucast) done lemma scast_up_scast': "sc = scast ==> is_up sc ==> scast (sc w) = scast w" apply (simp (no_asm) add: scast_def) apply (clarsimp simp add: sint_up_scast) done lemma ucast_of_bl_up': "w = of_bl bl ==> size bl <= size w ==> ucast w = of_bl bl" by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI) lemmas ucast_up_ucast = refl [THEN ucast_up_ucast'] lemmas scast_up_scast = refl [THEN scast_up_scast'] lemmas ucast_of_bl_up = refl [THEN ucast_of_bl_up'] lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id] lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id] lemmas isduu = is_up_down [where c = "ucast", THEN iffD2] lemmas isdus = is_up_down [where c = "scast", THEN iffD2] lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id] lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id] lemma up_ucast_surj: "is_up (ucast :: 'b::len0 word => 'a::len0 word) ==> surj (ucast :: 'a word => 'b word)" by (rule surjI, erule ucast_up_ucast_id) lemma up_scast_surj: "is_up (scast :: 'b::len word => 'a::len word) ==> surj (scast :: 'a word => 'b word)" by (rule surjI, erule scast_up_scast_id) lemma down_scast_inj: "is_down (scast :: 'b::len word => 'a::len word) ==> inj_on (ucast :: 'a word => 'b word) A" by (rule inj_on_inverseI, erule scast_down_scast_id) lemma down_ucast_inj: "is_down (ucast :: 'b::len0 word => 'a::len0 word) ==> inj_on (ucast :: 'a word => 'b word) A" by (rule inj_on_inverseI, erule ucast_down_ucast_id) lemma of_bl_append_same: "of_bl (X @ to_bl w) = w" by (rule word_bl.Rep_eqD) (simp add: word_rep_drop) lemma ucast_down_no': "uc = ucast ==> is_down uc ==> uc (number_of bin) = number_of bin" apply (unfold word_number_of_def is_down) apply (clarsimp simp add: ucast_def word_ubin.eq_norm) apply (rule word_ubin.norm_eq_iff [THEN iffD1]) apply (erule bintrunc_bintrunc_ge) done lemmas ucast_down_no = ucast_down_no' [OF refl] lemma ucast_down_bl': "uc = ucast ==> is_down uc ==> uc (of_bl bl) = of_bl bl" unfolding of_bl_no by clarify (erule ucast_down_no) lemmas ucast_down_bl = ucast_down_bl' [OF refl] lemmas slice_def' = slice_def [unfolded word_size] lemmas test_bit_def' = word_test_bit_def [THEN meta_eq_to_obj_eq, THEN fun_cong] lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def lemmas word_log_bin_defs = word_log_defs end
lemma of_nth_def:
BITS n. f n == of_bl (bl_of_nth (len_of TYPE('a)) f)
lemma word_size_gt_0:
0 < size w
lemma lens_gt_0:
0 < size w
0 < len_of TYPE('a)
lemma lens_not_0:
size w ≠ 0
len_of TYPE('a) ≠ 0
lemma uints_num:
uints n = {i. 0 ≤ i ∧ i < 2 ^ n}
lemma sints_num:
sints n = {i. - (2 ^ (n - 1)) ≤ i ∧ i < 2 ^ (n - 1)}
lemma atLeastLessThan_alt:
{l..<u} == {x. l ≤ x ∧ x < u}
lemma mod_in_reps:
0 < m ==> y mod m ∈ {0..<m}
lemma Rep_word_0:
0 ≤ Rep_word x
and Rep_word_lt:
Rep_word x < 2 ^ len_of TYPE('a)
lemma Rep_word_mod_same:
Rep_word x mod 2 ^ len_of TYPE('a) = Rep_word x
lemma td_ext_uint:
td_ext uint word_of_int (uints (len_of TYPE('a)))
(λw. w mod 2 ^ len_of TYPE('a))
lemma int_word_uint:
uint (word_of_int x) = x mod 2 ^ len_of TYPE('a)
lemma td_uint:
type_definition uint word_of_int (uints (len_of TYPE('a)))
lemma td_ext_ubin:
td_ext uint word_of_int (uints (len_of TYPE('a))) (bintrunc (len_of TYPE('a)))
lemma sint_sbintrunc':
sint (word_of_int bin) = sbintrunc (len_of TYPE('a) - 1) bin
lemma uint_sint:
uint w = bintrunc (len_of TYPE('a)) (sint w)
lemma bintr_uint':
size w ≤ n ==> bintrunc n (uint w) = uint w
lemma wi_bintr':
[| wb = word_of_int bin; size wb ≤ n |] ==> word_of_int (bintrunc n bin) = wb
lemma bintr_uint:
len_of TYPE('a) ≤ n ==> bintrunc n (uint w) = uint w
lemma wi_bintr:
[| wb = word_of_int bin; len_of TYPE('a) ≤ n |]
==> word_of_int (bintrunc n bin) = wb
lemma td_ext_sbin:
td_ext sint word_of_int (sints (len_of TYPE('a)))
(sbintrunc (len_of TYPE('a) - 1))
lemma td_ext_sint:
td_ext sint word_of_int (sints (len_of TYPE('a)))
(λw. (w + 2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
2 ^ (len_of TYPE('a) - 1))
lemma int_word_sint:
sint (word_of_int x) =
(x + 2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
2 ^ (len_of TYPE('a) - 1)
lemma td_sint:
type_definition sint word_of_int (sints (len_of TYPE('a)))
lemma word_number_of_alt:
number_of b == word_of_int (number_of b)
lemma word_no_wi:
number_of = word_of_int
lemma to_bl_def':
to_bl = bin_to_bl (len_of TYPE('a)) o uint
lemma word_reverse_no_def:
word_reverse (number_of w) == of_bl (rev (to_bl (number_of w)))
lemma uints_mod:
uints n == range (λw. w mod 2 ^ n)
lemma uint_bintrunc:
uint (number_of bin) = number_of (bintrunc (len_of TYPE('a)) bin)
lemma sint_sbintrunc:
sint (number_of bin) = number_of (sbintrunc (len_of TYPE('a) - 1) bin)
lemma unat_bintrunc:
unat (number_of bin) = number_of (bintrunc (len_of TYPE('a)) bin)
lemma size_0_eq:
size w = 0 ==> v = w
lemma uint_lem:
0 ≤ uint x ∧ uint x < 2 ^ len_of TYPE('a)
lemma sint_lem:
- (2 ^ (len_of TYPE('a) - 1)) ≤ sint x ∧ sint x < 2 ^ (len_of TYPE('a) - 1)
lemma uint_ge_0:
0 ≤ uint x
lemma uint_lt2p:
uint x < 2 ^ len_of TYPE('a)
lemma sint_ge:
- (2 ^ (len_of TYPE('a) - 1)) ≤ sint x
lemma sint_lt:
sint x < 2 ^ (len_of TYPE('a) - 1)
lemma sign_uint_Pls:
bin_sign (uint x) = Numeral.Pls
lemma uint_m2p_neg:
uint x - 2 ^ len_of TYPE('a) < 0
lemma uint_m2p_not_non_neg:
¬ 0 ≤ uint x - 2 ^ len_of TYPE('a)
lemma lt2p_lem:
len_of TYPE('a) ≤ n ==> uint w < 2 ^ n
lemma uint_le_0_iff:
(uint x ≤ 0) = (uint x = 0)
lemma uint_nat:
uint w == int (unat w)
lemma uint_number_of:
uint (number_of b) = number_of b mod 2 ^ len_of TYPE('a)
lemma unat_number_of:
bin_sign b = Numeral.Pls
==> unat (number_of b) = number_of b mod 2 ^ len_of TYPE('a)
lemma sint_number_of:
sint (number_of b) =
(number_of b + 2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
2 ^ (len_of TYPE('a) - 1)
lemma word_of_int_bin:
word_of_int (number_of bin) = number_of bin
lemma word_int_case_wi:
word_int_case f (word_of_int i) = f (i mod 2 ^ len_of TYPE('b))
lemma word_int_split:
P (word_int_case f x) =
(∀i. x = word_of_int i ∧ 0 ≤ i ∧ i < 2 ^ len_of TYPE('b) --> P (f i))
lemma word_int_split_asm:
P (word_int_case f x) =
(¬ (∃n. x = word_of_int n ∧ 0 ≤ n ∧ n < 2 ^ len_of TYPE('b) ∧ ¬ P (f n)))
lemma uint_range':
0 ≤ uint x ∧ uint x < 2 ^ len_of TYPE('a)
lemma sint_range':
- (2 ^ (len_of TYPE('a) - Suc 0)) ≤ sint x ∧
sint x < 2 ^ (len_of TYPE('a) - Suc 0)
lemma uint_range_size:
0 ≤ uint w ∧ uint w < 2 ^ size w
lemma sint_range_size:
- (2 ^ (size w - Suc 0)) ≤ sint w ∧ sint w < 2 ^ (size w - Suc 0)
lemma sint_above_size:
2 ^ (size w - 1) ≤ x ==> sint w < x
lemma sint_below_size:
x ≤ - (2 ^ (size w - 1)) ==> x ≤ sint w
lemma test_bit_eq_iff:
(op !! u = op !! v) = (u = v)
lemma test_bit_size:
w !! n ==> n < size w
lemma word_eqI:
∀n<size u. u !! n = v !! n ==> u = v
lemma word_eqD:
u = v ==> u !! x = v !! x
lemma test_bit_bin':
w !! n = (n < size w ∧ bin_nth (uint w) n)
lemma test_bit_bin:
w !! n = (n < len_of TYPE('a) ∧ bin_nth (uint w) n)
lemma bin_nth_uint_imp':
bin_nth (uint w) n --> n < size w
lemma bin_nth_sint':
size w ≤ n --> bin_nth (sint w) n = bin_nth (sint w) (size w - 1)
lemma bin_nth_uint_imp:
bin_nth (uint w) n ==> n < len_of TYPE('a)
lemma bin_nth_sint:
len_of TYPE('a) ≤ n
==> bin_nth (sint w) n = bin_nth (sint w) (len_of TYPE('a) - 1)
lemma td_bl:
type_definition to_bl of_bl {bl. length bl = len_of TYPE('a)}
lemma word_size_bl:
size w == length (to_bl w)
lemma to_bl_use_of_bl:
(to_bl w = bl) = (w = of_bl bl ∧ length bl = length (to_bl w))
lemma to_bl_word_rev:
to_bl (word_reverse w) = rev (to_bl w)
lemma word_rev_rev:
word_reverse (word_reverse w) = w
lemma word_rev_gal:
word_reverse w = u ==> word_reverse u = w
lemma word_rev_gal':
u = word_reverse t ==> t = word_reverse u
lemma length_bl_gt_0:
0 < length (to_bl x)
lemma bl_not_Nil:
to_bl x ≠ []
lemma length_bl_neq_0:
length (to_bl x1) ≠ 0
lemma hd_bl_sign_sint:
hd (to_bl w) = (bin_sign (sint w) = Numeral.Min)
lemma of_bl_drop':
lend = length bl - len_of TYPE('a) ==> of_bl (drop lend bl) = of_bl bl
lemma of_bl_no:
of_bl bl == number_of (bl_to_bin bl)
lemma test_bit_of_bl:
of_bl bl !! n = (rev bl ! n ∧ n < len_of TYPE('a) ∧ n < length bl)
lemma no_of_bl:
number_of bin = of_bl (bin_to_bl (len_of TYPE('a)) bin)
lemma uint_bl:
to_bl w == bin_to_bl (size w) (uint w)
lemma to_bl_bin:
bl_to_bin (to_bl w) = uint w
lemma to_bl_of_bin:
to_bl (word_of_int bin) = bin_to_bl (len_of TYPE('a)) bin
lemma to_bl_no_bin:
to_bl (number_of bin) = bin_to_bl (len_of TYPE('a)) bin
lemma to_bl_to_bin:
bl_to_bin (to_bl w) = uint w
lemma uint_bl_bin:
bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x
lemma num_AB_u:
number_of (uint x) = x
lemma num_AB_s:
number_of (sint x) = x
lemma uints_unats:
uints n = int ` unats n
lemma unats_uints:
unats n = nat ` uints n
lemma bintr_num:
(number_of x = number_of y) =
(bintrunc (len_of TYPE('a)) x = bintrunc (len_of TYPE('a)) y)
lemma sbintr_num:
(number_of x = number_of y) =
(sbintrunc (len_of TYPE('a) - 1) x = sbintrunc (len_of TYPE('a) - 1) y)
lemma num_of_bintr:
number_of (bintrunc (len_of TYPE('a)) x) = number_of x
lemma num_of_sbintr:
number_of (sbintrunc (len_of TYPE('a) - 1) x) = number_of x
lemma num_of_bintr':
bintrunc (len_of TYPE('a)) a = b ==> number_of a = number_of b
lemma num_of_sbintr':
sbintrunc (len_of TYPE('a) - 1) a = b ==> number_of a = number_of b
lemma num_abs_bintr:
number_of x = word_of_int (bintrunc (len_of TYPE('a)) x)
lemma num_abs_sbintr:
number_of x = word_of_int (sbintrunc (len_of TYPE('a) - 1) x)
lemma ucast_id:
ucast w = w
lemma scast_id:
scast w = w
lemma ucast_bl:
ucast w == of_bl (to_bl w)
lemma nth_ucast:
ucast w !! n = (w !! n ∧ n < len_of TYPE('a))
lemma ucast_bintr:
ucast (number_of w) = number_of (bintrunc (len_of TYPE('a)) w)
lemma scast_sbintr:
scast (number_of w) = number_of (sbintrunc (len_of TYPE('a) - Suc 0) w)
lemma source_size:
source_size c == len_of TYPE('a)
lemma target_size:
target_size c == len_of TYPE('b)
lemma is_down:
is_down c == len_of TYPE('b) ≤ len_of TYPE('a)
lemma is_up:
is_up c == len_of TYPE('a) ≤ len_of TYPE('b)
lemma is_up_down:
is_up ca = is_down c
lemma down_cast_same':
[| uc = ucast; is_down uc |] ==> uc = scast
lemma word_rev_tf':
r = to_bl (of_bl bl) ==> r = rev (takefill False (length r) (rev bl))
lemma word_rev_tf:
to_bl (of_bl bl) = rev (takefill False (len_of TYPE('a)) (rev bl))
lemma word_rep_drop:
to_bl (of_bl bl) =
replicate (len_of TYPE('a) - length bl) False @
drop (length bl - len_of TYPE('a)) bl
lemma to_bl_ucast:
to_bl (ucast w) =
replicate (len_of TYPE('a) - len_of TYPE('b)) False @
drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)
lemma ucast_up_app':
[| uc = ucast; source_size uc + n = target_size uc |]
==> to_bl (uc w) = replicate n False @ to_bl w
lemma ucast_down_drop':
[| uc = ucast; source_size uc = target_size uc + n |]
==> to_bl (uc w) = drop n (to_bl w)
lemma scast_down_drop':
[| sc = scast; source_size sc = target_size sc + n |]
==> to_bl (sc w) = drop n (to_bl w)
lemma sint_up_scast':
[| sc = scast; is_up sc |] ==> sint (sc w) = sint w
lemma uint_up_ucast':
[| uc = ucast; is_up uc |] ==> uint (uc w) = uint w
lemma down_cast_same:
is_down ucast ==> ucast = scast
lemma ucast_up_app:
source_size ucast + n = target_size ucast
==> to_bl (ucast w) = replicate n False @ to_bl w
lemma ucast_down_drop:
source_size ucast = target_size ucast + n ==> to_bl (ucast w) = drop n (to_bl w)
lemma scast_down_drop:
source_size scast = target_size scast + n ==> to_bl (scast w) = drop n (to_bl w)
lemma uint_up_ucast:
is_up ucast ==> uint (ucast w) = uint w
lemma sint_up_scast:
is_up scast ==> sint (scast w) = sint w
lemma ucast_up_ucast':
[| uc = ucast; is_up uc |] ==> ucast (uc w) = ucast w
lemma scast_up_scast':
[| sc = scast; is_up sc |] ==> scast (sc w) = scast w
lemma ucast_of_bl_up':
[| w = of_bl bl; length bl ≤ size w |] ==> ucast w = of_bl bl
lemma ucast_up_ucast:
is_up ucast ==> ucast (ucast w) = ucast w
lemma scast_up_scast:
is_up scast ==> scast (scast w) = scast w
lemma ucast_of_bl_up:
length bl ≤ size (of_bl bl) ==> ucast (of_bl bl) = of_bl bl
lemma ucast_up_ucast_id:
is_up ucast ==> ucast (ucast t) = t
lemma scast_up_scast_id:
is_up scast ==> scast (scast t) = t
lemma isduu:
is_down ucast ==> is_up ca1
lemma isdus:
is_down scast ==> is_up ca1
lemma ucast_down_ucast_id:
is_down ucast ==> ucast (ucast t) = t
lemma scast_down_scast_id:
is_down scast ==> ucast (ucast t) = t
lemma up_ucast_surj:
is_up ucast ==> surj ucast
lemma up_scast_surj:
is_up scast ==> surj scast
lemma down_scast_inj:
is_down scast ==> inj_on ucast A
lemma down_ucast_inj:
is_down ucast ==> inj_on ucast A
lemma of_bl_append_same:
of_bl (X @ to_bl w) = w
lemma ucast_down_no':
[| uc = ucast; is_down uc |] ==> uc (number_of bin) = number_of bin
lemma ucast_down_no:
is_down ucast ==> ucast (number_of bin) = number_of bin
lemma ucast_down_bl':
[| uc = ucast; is_down uc |] ==> uc (of_bl bl) = of_bl bl
lemma ucast_down_bl:
is_down ucast ==> ucast (of_bl bl) = of_bl bl
lemma slice_def':
slice n w == slice1 (len_of TYPE('a) - n) w
lemma test_bit_def':
a2 !! x = bin_nth (uint a2) x
lemma word_log_defs:
a AND b == word_of_int (uint a AND uint b)
a OR b == word_of_int (uint a OR uint b)
a XOR b == word_of_int (uint a XOR uint b)
NOT a == word_of_int (NOT uint a)
lemma word_log_bin_defs:
a AND b == word_of_int (uint a AND uint b)
a OR b == word_of_int (uint a OR uint b)
a XOR b == word_of_int (uint a XOR uint b)
NOT a == word_of_int (NOT uint a)