(*
ID: $Id$
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 \<equiv> word_of_int (2^len_of TYPE('a) - 1)"
consts
of_bool :: "bool \<Rightarrow> '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 \<le> i \<and> i < 2 ^ n}"
by (simp add: uints_def range_bintrunc)
lemma sints_num: "sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> 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 \<Rightarrow> int"
word_of_int
"uints (len_of TYPE('a::len0))"
"\<lambda>w. 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 \<Rightarrow> 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"]
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 (auto intro!: word_sbin.eq_norm simp del: one_is_Suc_zero)
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 \<and> 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 \<and> n < len_of TYPE('a) \<and> 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"
apply (auto simp add : uints_unats image_iff)
done
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