--- a/src/HOL/Word/WordDefinition.thy Wed Aug 22 20:59:19 2007 +0200
+++ b/src/HOL/Word/WordDefinition.thy Wed Aug 22 21:09:21 2007 +0200
@@ -8,12 +8,13 @@
header {* Definition of Word Type *}
-theory WordDefinition imports Size BinOperations TdThs begin
+theory WordDefinition
+imports Numeral_Type BinOperations TdThs begin
typedef (open word) 'a word
- = "{(0::int) ..< 2^len_of TYPE('a::len0)}" by auto
+ = "{(0::int) ..< 2^CARD('a)}" by auto
-instance word :: (len0) number ..
+instance word :: (type) number ..
instance word :: (type) minus ..
instance word :: (type) plus ..
instance word :: (type) one ..
@@ -30,17 +31,17 @@
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)"
+ word_of_int :: "int => 'a word"
+ "word_of_int w == Abs_word (bintrunc CARD('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 :: "'a 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"
+ sint :: "'a :: finite word => int"
+ sint_uint: "sint w == sbintrunc (CARD('a) - 1) (uint w)"
+ unat :: "'a word => nat"
"unat w == nat (uint w)"
-- "the sets of integers representing the words"
@@ -54,11 +55,11 @@
"norm_sint n w == (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)"
defs (overloaded)
- word_size: "size (w :: 'a :: len0 word) == len_of TYPE('a)"
+ word_size: "size (w :: 'a word) == CARD('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 :: "(int => 'b) => ('a word) => 'b"
"word_int_case f w == f (uint w)"
syntax
@@ -70,18 +71,18 @@
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_1_wi: "(1 :: ('a) word) == word_of_int 1"
+ word_0_wi: "(0 :: ('a) word) == word_of_int 0"
constdefs
- word_succ :: "'a :: len0 word => 'a word"
+ word_succ :: "'a word => 'a word"
"word_succ a == word_of_int (Numeral.succ (uint a))"
- word_pred :: "'a :: len0 word => 'a word"
+ word_pred :: "'a word => 'a word"
"word_pred a == word_of_int (Numeral.pred (uint a))"
consts
- word_power :: "'a :: len0 word => nat => 'a word"
+ word_power :: "'a word => nat => 'a word"
primrec
"word_power a 0 = 1"
"word_power a (Suc n) = a * word_power a n"
@@ -98,46 +99,46 @@
defs (overloaded)
word_and_def:
- "(a::'a::len0 word) AND b == word_of_int (uint a AND uint b)"
+ "(a::'a 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)"
+ "(a::'a 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)"
+ "(a::'a 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))"
+ "NOT (a::'a word) == word_of_int (NOT (uint a))"
word_test_bit_def:
- "test_bit (a::'a::len0 word) == bin_nth (uint a)"
+ "test_bit (a::'a word) == bin_nth (uint a)"
word_set_bit_def:
- "set_bit (a::'a::len0 word) n x ==
+ "set_bit (a::'a word) n x ==
word_of_int (bin_sc n (If x bit.B1 bit.B0) (uint a))"
word_lsb_def:
- "lsb (a::'a::len0 word) == bin_last (uint a) = bit.B1"
+ "lsb (a::'a word) == bin_last (uint a) = bit.B1"
word_msb_def:
- "msb (a::'a::len word) == bin_sign (sint a) = Numeral.Min"
+ "msb (a::'a::finite word) == bin_sign (sint a) = Numeral.Min"
constdefs
- setBit :: "'a :: len0 word => nat => 'a word"
+ setBit :: "'a word => nat => 'a word"
"setBit w n == set_bit w n True"
- clearBit :: "'a :: len0 word => nat => 'a word"
+ clearBit :: "'a word => nat => 'a word"
"clearBit w n == set_bit w n False"
constdefs
-- "Largest representable machine integer."
- max_word :: "'a::len word"
- "max_word \<equiv> word_of_int (2^len_of TYPE('a) - 1)"
+ max_word :: "'a::finite word"
+ "max_word \<equiv> word_of_int (2^CARD('a) - 1)"
consts
- of_bool :: "bool \<Rightarrow> 'a::len word"
+ of_bool :: "bool \<Rightarrow> 'a::finite word"
primrec
"of_bool False = 0"
"of_bool True = 1"
@@ -145,8 +146,8 @@
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
+ xtr1 [OF word_size [THEN meta_eq_to_obj_eq] zero_less_card_finite, standard]
+lemmas lens_gt_0 = word_size_gt_0 zero_less_card_finite
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}"
@@ -163,16 +164,16 @@
lemma
Rep_word_0:"0 <= Rep_word x" and
- Rep_word_lt: "Rep_word (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
+ Rep_word_lt: "Rep_word (x::'a word) < 2 ^ CARD('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)"
+ "Rep_word x mod 2 ^ CARD('a) = Rep_word (x::'a 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))"
+ "td_ext (uint :: 'a word => int) word_of_int (uints CARD('a))
+ (%w::int. w mod 2 ^ CARD('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
@@ -182,33 +183,34 @@
lemmas int_word_uint = td_ext_uint [THEN td_ext.eq_norm, standard]
interpretation word_uint:
- td_ext ["uint::'a::len0 word \<Rightarrow> int"
+ td_ext ["uint::'a word \<Rightarrow> int"
word_of_int
- "uints (len_of TYPE('a::len0))"
- "\<lambda>w. w mod 2 ^ len_of TYPE('a::len0)"]
+ "uints CARD('a)"
+ "\<lambda>w. w mod 2 ^ CARD('a)"]
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]]
+ [simplified zero_less_card_finite no_bintr_alt1 [symmetric]]
interpretation word_ubin:
- td_ext ["uint::'a::len0 word \<Rightarrow> int"
+ td_ext ["uint::'a word \<Rightarrow> int"
word_of_int
- "uints (len_of TYPE('a::len0))"
- "bintrunc (len_of TYPE('a::len0))"]
+ "uints CARD('a)"
+ "bintrunc CARD('a)"]
by (rule td_ext_ubin)
lemma sint_sbintrunc':
"sint (word_of_int bin :: 'a word) =
- (sbintrunc (len_of TYPE ('a :: len) - 1) bin)"
+ (sbintrunc (CARD('a :: finite) - 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))"
+ "uint w = bintrunc CARD('a) (sint (w :: 'a :: finite word))"
unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le)
+
lemma bintr_uint':
"n >= size w ==> bintrunc n (uint w) = uint w"
@@ -228,11 +230,11 @@
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))"
+ "td_ext (sint :: 'a word => int) word_of_int (sints CARD('a::finite))
+ (sbintrunc (CARD('a) - 1))"
apply (unfold td_ext_def' sint_uint)
apply (simp add : word_ubin.eq_norm)
- apply (cases "len_of TYPE('a)")
+ apply (cases "CARD('a)")
apply (auto simp add : sints_def)
apply (rule sym [THEN trans])
apply (rule word_ubin.Abs_norm)
@@ -242,25 +244,25 @@
done
lemmas td_ext_sint = td_ext_sbin
- [simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]]
+ [simplified zero_less_card_finite 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"
+ td_ext ["sint ::'a::finite 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)"]
+ "sints CARD('a::finite)"
+ "%w. (w + 2^(CARD('a::finite) - 1)) mod 2^CARD('a::finite) -
+ 2 ^ (CARD('a::finite) - 1)"]
by (rule td_ext_sint)
interpretation word_sbin:
- td_ext ["sint ::'a::len word => int"
+ td_ext ["sint ::'a::finite word => int"
word_of_int
- "sints (len_of TYPE('a::len))"
- "sbintrunc (len_of TYPE('a::len) - 1)"]
+ "sints CARD('a::finite)"
+ "sbintrunc (CARD('a::finite) - 1)"]
by (rule td_ext_sbin)
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm, standard]
@@ -276,18 +278,18 @@
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)"
+ number_of (bintrunc CARD('a) 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)"
+ number_of (sbintrunc (CARD('a :: finite) - 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)"
+ "unat (number_of bin :: 'a word) =
+ number_of (bintrunc CARD('a) bin)"
unfolding unat_def nat_number_of_def
by (simp only: uint_bintrunc)
@@ -297,7 +299,7 @@
sint_sbintrunc [simp]
unat_bintrunc [simp]
-lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 ==> v = w"
+lemma size_0_eq: "size (w :: 'a word) = 0 ==> v = w"
apply (unfold word_size)
apply (rule word_uint.Rep_eqD)
apply (rule box_equals)
@@ -322,7 +324,7 @@
iffD2 [OF linorder_not_le uint_m2p_neg, standard]
lemma lt2p_lem:
- "len_of TYPE('a) <= n ==> uint (w :: 'a :: len0 word) < 2 ^ n"
+ "CARD('a) <= n ==> uint (w :: 'a word) < 2 ^ n"
by (rule xtr8 [OF _ uint_lt2p]) simp
lemmas uint_le_0_iff [simp] =
@@ -332,13 +334,13 @@
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)"
+ "uint (number_of b :: 'a word) = number_of b mod 2 ^ CARD('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)"
+ unat (number_of b::'a word) = number_of b mod 2 ^ CARD('a)"
apply (unfold unat_def)
apply (clarsimp simp only: uint_number_of)
apply (rule nat_mod_distrib [THEN trans])
@@ -346,31 +348,31 @@
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)"
+lemma sint_number_of: "sint (number_of b :: 'a :: finite word) = (number_of b +
+ 2 ^ (CARD('a) - 1)) mod 2 ^ CARD('a) -
+ 2 ^ (CARD('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)"
+ "(word_of_int (number_of bin) :: 'a 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))"
+ f (i mod 2 ^ CARD('b))"
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))"
+ (ALL i. x = (word_of_int i :: 'b word) &
+ 0 <= i & i < 2 ^ CARD('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)))"
+ (~ (EX n. x = (word_of_int n :: 'b word) &
+ 0 <= n & n < 2 ^ CARD('b) & ~ P (f n)))"
unfolding word_int_case_def
by (auto simp: word_uint.eq_norm int_mod_eq')
@@ -392,10 +394,10 @@
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)"
+lemma test_bit_eq_iff: "(test_bit (u::'a 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"
+lemma test_bit_size [rule_format] : "(w::'a 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)
@@ -403,7 +405,7 @@
done
lemma word_eqI [rule_format] :
- fixes u :: "'a::len0 word"
+ fixes u :: "'a word"
shows "(ALL n. n < size u --> u !! n = v !! n) ==> u = v"
apply (rule test_bit_eq_iff [THEN iffD1])
apply (rule ext)
@@ -475,14 +477,14 @@
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 ==>
+ "bintrunc CARD('a) 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 ==>
+ "sbintrunc (CARD('a :: finite) - 1) a = b ==>
number_of a = (number_of b :: 'a word)"
apply safe
apply (rule_tac num_of_sbintr [symmetric])
@@ -503,19 +505,19 @@
constdefs
-- "cast a word to a different length"
- scast :: "'a :: len word => 'b :: len word"
+ scast :: "'a :: finite word => 'b :: finite word"
"scast w == word_of_int (sint w)"
- ucast :: "'a :: len0 word => 'b :: len0 word"
+ ucast :: "'a word => 'b 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 :: "('a word => 'b) => nat"
"source_size c == let arb = arbitrary ; x = c arb in size arb"
- target_size :: "('a => 'b :: len0 word) => nat"
+ target_size :: "('a => 'b word) => nat"
"target_size c == size (c arbitrary)"
- is_up :: "('a :: len0 word => 'b :: len0 word) => bool"
+ is_up :: "('a word => 'b word) => bool"
"is_up c == source_size c <= target_size c"
- is_down :: "('a :: len0 word => 'b :: len0 word) => bool"
+ is_down :: "('a word => 'b word) => bool"
"is_down c == target_size c <= source_size c"
(** cast - note, no arg for new length, as it's determined by type of result,
@@ -528,7 +530,7 @@
unfolding scast_def by auto
lemma nth_ucast:
- "(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))"
+ "(ucast w::'a word) !! n = (w !! n & n < CARD('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)
@@ -537,13 +539,13 @@
(* for literal u(s)cast *)
lemma ucast_bintr [simp]:
- "ucast (number_of w ::'a::len0 word) =
- number_of (bintrunc (len_of TYPE('a)) w)"
+ "ucast (number_of w ::'a word) =
+ number_of (bintrunc CARD('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)"
+ "scast (number_of w ::'a::finite word) =
+ number_of (sbintrunc (CARD('a) - Suc 0) w)"
unfolding scast_def by simp
lemmas source_size = source_size_def [unfolded Let_def word_size]
@@ -616,22 +618,22 @@
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id]
lemma up_ucast_surj:
- "is_up (ucast :: 'b::len0 word => 'a::len0 word) ==>
+ "is_up (ucast :: 'b word => 'a 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) ==>
+ "is_up (scast :: 'b::finite word => 'a::finite 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) ==>
+ "is_down (scast :: 'b::finite word => 'a::finite 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) ==>
+ "is_down (ucast :: 'b word => 'a word) ==>
inj_on (ucast :: 'a word => 'b word) A"
by (rule inj_on_inverseI, erule ucast_down_ucast_id)