--- a/src/HOL/Word/WordShift.thy Wed Aug 22 20:59:19 2007 +0200
+++ b/src/HOL/Word/WordShift.thy Wed Aug 22 21:09:21 2007 +0200
@@ -11,22 +11,22 @@
subsection "Bit shifting"
constdefs
- shiftl1 :: "'a :: len0 word => 'a word"
+ shiftl1 :: "'a 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 :: "'a word => 'a word"
"shiftr1 w == word_of_int (bin_rest (uint w))"
- sshiftr1 :: "'a :: len word => 'a word"
+ sshiftr1 :: "'a :: finite word => 'a word"
"sshiftr1 w == word_of_int (bin_rest (sint w))"
- sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55)
+ sshiftr :: "'a :: finite word => nat => 'a word" (infixl ">>>" 55)
"w >>> n == (sshiftr1 ^ n) w"
defs (overloaded)
- shiftl_def: "(w::'a::len0 word) << n == (shiftl1 ^ n) w"
- shiftr_def: "(w::'a::len0 word) >> n == (shiftr1 ^ n) w"
+ shiftl_def: "(w::'a word) << n == (shiftl1 ^ n) w"
+ shiftr_def: "(w::'a word) >> n == (shiftr1 ^ n) w"
lemma shiftl1_number [simp] :
"shiftl1 (number_of w) = number_of (w BIT bit.B0)"
@@ -58,10 +58,10 @@
lemma sshiftr1_n1 [simp] : "sshiftr1 -1 = -1"
unfolding sshiftr1_def by auto
-lemma shiftl_0 [simp] : "(0::'a::len0 word) << n = 0"
+lemma shiftl_0 [simp] : "(0::'a word) << n = 0"
unfolding shiftl_def by (induct n) auto
-lemma shiftr_0 [simp] : "(0::'a::len0 word) >> n = 0"
+lemma shiftr_0 [simp] : "(0::'a word) >> n = 0"
unfolding shiftr_def by (induct n) auto
lemma sshiftr_0 [simp] : "0 >>> n = 0"
@@ -78,7 +78,7 @@
done
lemma nth_shiftl' [rule_format]:
- "ALL n. ((w::'a::len0 word) << m) !! n = (n < size w & n >= m & w !! (n - m))"
+ "ALL n. ((w::'a word) << m) !! n = (n < size w & n >= m & w !! (n - m))"
apply (unfold shiftl_def)
apply (induct "m")
apply (force elim!: test_bit_size)
@@ -97,7 +97,7 @@
done
lemma nth_shiftr:
- "\<And>n. ((w::'a::len0 word) >> m) !! n = w !! (n + m)"
+ "\<And>n. ((w::'a word) >> m) !! n = w !! (n + m)"
apply (unfold shiftr_def)
apply (induct "m")
apply (auto simp add : nth_shiftr1)
@@ -188,7 +188,7 @@
subsubsection "shift functions in terms of lists of bools"
definition
- bshiftr1 :: "bool => 'a :: len word => 'a word" where
+ bshiftr1 :: "bool => 'a :: finite word => 'a word" where
"bshiftr1 b w == of_bl (b # butlast (to_bl w))"
lemmas bshiftr1_no_bin [simp] =
@@ -202,13 +202,11 @@
by (simp add: bl_to_bin_aux_append bl_to_bin_def)
lemmas shiftl1_bl = shiftl1_of_bl
- [where bl = "to_bl (?w :: ?'a :: len0 word)", simplified]
+ [where bl = "to_bl (?w :: ?'a word)", simplified]
lemma bl_shiftl1:
- "to_bl (shiftl1 (w :: 'a :: len word)) = tl (to_bl w) @ [False]"
- apply (simp add: shiftl1_bl word_rep_drop drop_Suc drop_Cons')
- apply (fast intro!: Suc_leI)
- done
+ "to_bl (shiftl1 (w :: 'a :: finite word)) = tl (to_bl w) @ [False]"
+ by (simp add: shiftl1_bl word_rep_drop drop_Suc drop_Cons')
lemma shiftr1_bl: "shiftr1 w = of_bl (butlast (to_bl w))"
apply (unfold shiftr1_def uint_bl of_bl_def)
@@ -217,15 +215,15 @@
done
lemma bl_shiftr1:
- "to_bl (shiftr1 (w :: 'a :: len word)) = False # butlast (to_bl w)"
+ "to_bl (shiftr1 (w :: 'a :: finite word)) = False # butlast (to_bl w)"
unfolding shiftr1_bl
- by (simp add : word_rep_drop len_gt_0 [THEN Suc_leI])
+ by (simp add : word_rep_drop zero_less_card_finite [THEN Suc_leI])
-(* relate the two above : TODO - remove the :: len restriction on
+(* relate the two above : TODO - remove the :: finite restriction on
this theorem and others depending on it *)
lemma shiftl1_rev:
- "shiftl1 (w :: 'a :: len word) = word_reverse (shiftr1 (word_reverse w))"
+ "shiftl1 (w :: 'a :: finite word) = word_reverse (shiftr1 (word_reverse w))"
apply (unfold word_reverse_def)
apply (rule word_bl.Rep_inverse' [symmetric])
apply (simp add: bl_shiftl1 bl_shiftr1 word_bl.Abs_inverse)
@@ -234,7 +232,7 @@
done
lemma shiftl_rev:
- "shiftl (w :: 'a :: len word) n = word_reverse (shiftr (word_reverse w) n)"
+ "shiftl (w :: 'a :: finite word) n = word_reverse (shiftr (word_reverse w) n)"
apply (unfold shiftl_def shiftr_def)
apply (induct "n")
apply (auto simp add : shiftl1_rev)
@@ -247,7 +245,7 @@
lemmas rev_shiftr = shiftl_rev [THEN word_rev_gal', standard]
lemma bl_sshiftr1:
- "to_bl (sshiftr1 (w :: 'a :: len word)) = hd (to_bl w) # butlast (to_bl w)"
+ "to_bl (sshiftr1 (w :: 'a :: finite word)) = hd (to_bl w) # butlast (to_bl w)"
apply (unfold sshiftr1_def uint_bl word_size)
apply (simp add: butlast_rest_bin word_ubin.eq_norm)
apply (simp add: sint_uint)
@@ -259,14 +257,13 @@
nth_bin_to_bl bin_nth.Suc [symmetric]
nth_sbintr
del: bin_nth.Suc)
- apply force
apply (rule impI)
apply (rule_tac f = "bin_nth (uint w)" in arg_cong)
apply simp
done
lemma drop_shiftr:
- "drop n (to_bl ((w :: 'a :: len word) >> n)) = take (size w - n) (to_bl w)"
+ "drop n (to_bl ((w :: 'a :: finite word) >> n)) = take (size w - n) (to_bl w)"
apply (unfold shiftr_def)
apply (induct n)
prefer 2
@@ -276,7 +273,7 @@
done
lemma drop_sshiftr:
- "drop n (to_bl ((w :: 'a :: len word) >>> n)) = take (size w - n) (to_bl w)"
+ "drop n (to_bl ((w :: 'a :: finite word) >>> n)) = take (size w - n) (to_bl w)"
apply (unfold sshiftr_def)
apply (induct n)
prefer 2
@@ -286,7 +283,7 @@
done
lemma take_shiftr [rule_format] :
- "n <= size (w :: 'a :: len word) --> take n (to_bl (w >> n)) =
+ "n <= size (w :: 'a :: finite word) --> take n (to_bl (w >> n)) =
replicate n False"
apply (unfold shiftr_def)
apply (induct n)
@@ -298,7 +295,7 @@
done
lemma take_sshiftr' [rule_format] :
- "n <= size (w :: 'a :: len word) --> hd (to_bl (w >>> n)) = hd (to_bl w) &
+ "n <= size (w :: 'a :: finite word) --> hd (to_bl (w >>> n)) = hd (to_bl w) &
take n (to_bl (w >>> n)) = replicate n (hd (to_bl w))"
apply (unfold sshiftr_def)
apply (induct n)
@@ -323,7 +320,7 @@
by (induct n) (auto simp: shiftl1_of_bl replicate_app_Cons_same)
lemmas shiftl_bl =
- shiftl_of_bl [where bl = "to_bl (?w :: ?'a :: len0 word)", simplified]
+ shiftl_of_bl [where bl = "to_bl (?w :: ?'a word)", simplified]
lemmas shiftl_number [simp] = shiftl_def [where w="number_of ?w"]
@@ -332,46 +329,46 @@
by (simp add: shiftl_bl word_rep_drop word_size min_def)
lemma shiftl_zero_size:
- fixes x :: "'a::len0 word"
+ fixes x :: "'a word"
shows "size x <= n ==> x << n = 0"
apply (unfold word_size)
apply (rule word_eqI)
apply (clarsimp simp add: shiftl_bl word_size test_bit_of_bl nth_append)
done
-(* note - the following results use 'a :: len word < number_ring *)
+(* note - the following results use 'a :: finite word < number_ring *)
-lemma shiftl1_2t: "shiftl1 (w :: 'a :: len word) = 2 * w"
+lemma shiftl1_2t: "shiftl1 (w :: 'a :: finite word) = 2 * w"
apply (simp add: shiftl1_def_u)
apply (simp only: double_number_of_BIT [symmetric])
apply simp
done
-lemma shiftl1_p: "shiftl1 (w :: 'a :: len word) = w + w"
+lemma shiftl1_p: "shiftl1 (w :: 'a :: finite word) = w + w"
apply (simp add: shiftl1_def_u)
apply (simp only: double_number_of_BIT [symmetric])
apply simp
done
-lemma shiftl_t2n: "shiftl (w :: 'a :: len word) n = 2 ^ n * w"
+lemma shiftl_t2n: "shiftl (w :: 'a :: finite word) n = 2 ^ n * w"
unfolding shiftl_def
by (induct n) (auto simp: shiftl1_2t power_Suc)
lemma shiftr1_bintr [simp]:
- "(shiftr1 (number_of w) :: 'a :: len0 word) =
- number_of (bin_rest (bintrunc (len_of TYPE ('a)) w))"
+ "(shiftr1 (number_of w) :: 'a word) =
+ number_of (bin_rest (bintrunc CARD('a) w))"
unfolding shiftr1_def word_number_of_def
by (simp add : word_ubin.eq_norm)
lemma sshiftr1_sbintr [simp] :
- "(sshiftr1 (number_of w) :: 'a :: len word) =
- number_of (bin_rest (sbintrunc (len_of TYPE ('a) - 1) w))"
+ "(sshiftr1 (number_of w) :: 'a :: finite word) =
+ number_of (bin_rest (sbintrunc (CARD('a) - 1) w))"
unfolding sshiftr1_def word_number_of_def
by (simp add : word_sbin.eq_norm)
lemma shiftr_no':
"w = number_of bin ==>
- (w::'a::len0 word) >> n = number_of ((bin_rest ^ n) (bintrunc (size w) bin))"
+ (w::'a word) >> n = number_of ((bin_rest ^ n) (bintrunc (size w) bin))"
apply clarsimp
apply (rule word_eqI)
apply (auto simp: nth_shiftr nth_rest_power_bin nth_bintr word_size)
@@ -383,7 +380,7 @@
apply clarsimp
apply (rule word_eqI)
apply (auto simp: nth_sshiftr nth_rest_power_bin nth_sbintr word_size)
- apply (subgoal_tac "na + n = len_of TYPE('a) - Suc 0", simp, simp)+
+ apply (subgoal_tac "na + n = CARD('a) - Suc 0", simp, simp)+
done
lemmas sshiftr_no [simp] =
@@ -419,7 +416,7 @@
lemmas shiftr_bl = word_bl.Rep' [THEN eq_imp_le, THEN shiftr_bl_of,
simplified word_size, simplified, THEN eq_reflection, standard]
-lemma msb_shift': "msb (w::'a::len word) <-> (w >> (size w - 1)) ~= 0"
+lemma msb_shift': "msb (w::'a::finite word) <-> (w >> (size w - 1)) ~= 0"
apply (unfold shiftr_bl word_msb_alt)
apply (simp add: word_size Suc_le_eq take_Suc)
apply (cases "hd (to_bl w)")
@@ -480,7 +477,7 @@
subsubsection "Mask"
definition
- mask :: "nat => 'a::len word" where
+ mask :: "nat => 'a::finite word" where
"mask n == (1 << n) - 1"
lemma nth_mask': "m = mask n ==> test_bit m i = (i < n & i < size m)"
@@ -514,9 +511,9 @@
lemmas and_mask_wi = and_mask_no [unfolded word_number_of_def]
lemma bl_and_mask:
- "to_bl (w AND mask n :: 'a :: len word) =
- replicate (len_of TYPE('a) - n) False @
- drop (len_of TYPE('a) - n) (to_bl w)"
+ "to_bl (w AND mask n :: 'a :: finite word) =
+ replicate (CARD('a) - n) False @
+ drop (CARD('a) - n) (to_bl w)"
apply (rule nth_equalityI)
apply simp
apply (clarsimp simp add: to_bl_nth word_size)
@@ -563,14 +560,14 @@
done
lemma word_2p_lem:
- "n < size w ==> w < 2 ^ n = (uint (w :: 'a :: len word) < 2 ^ n)"
+ "n < size w ==> w < 2 ^ n = (uint (w :: 'a :: finite word) < 2 ^ n)"
apply (unfold word_size word_less_alt word_number_of_alt)
apply (clarsimp simp add: word_of_int_power_hom word_uint.eq_norm
int_mod_eq'
simp del: word_of_int_bin)
done
-lemma less_mask_eq: "x < 2 ^ n ==> x AND mask n = (x :: 'a :: len word)"
+lemma less_mask_eq: "x < 2 ^ n ==> x AND mask n = (x :: 'a :: finite word)"
apply (unfold word_less_alt word_number_of_alt)
apply (clarsimp simp add: and_mask_mod_2p word_of_int_power_hom
word_uint.eq_norm
@@ -590,7 +587,7 @@
unfolding word_size by (erule and_mask_less')
lemma word_mod_2p_is_mask':
- "c = 2 ^ n ==> c > 0 ==> x mod c = (x :: 'a :: len word) AND mask n"
+ "c = 2 ^ n ==> c > 0 ==> x mod c = (x :: 'a :: finite word) AND mask n"
by (clarsimp simp add: word_mod_def uint_2p and_mask_mod_2p)
lemmas word_mod_2p_is_mask = refl [THEN word_mod_2p_is_mask']
@@ -620,8 +617,8 @@
subsubsection "Revcast"
definition
- revcast :: "'a :: len0 word => 'b :: len0 word" where
- "revcast w == of_bl (takefill False (len_of TYPE('b)) (to_bl w))"
+ revcast :: "'a word => 'b word" where
+ "revcast w == of_bl (takefill False CARD('b) (to_bl w))"
lemmas revcast_def' = revcast_def [simplified]
lemmas revcast_def'' = revcast_def' [simplified word_size]
@@ -629,8 +626,8 @@
revcast_def' [where w="number_of ?w", unfolded word_size]
lemma to_bl_revcast:
- "to_bl (revcast w :: 'a :: len0 word) =
- takefill False (len_of TYPE ('a)) (to_bl w)"
+ "to_bl (revcast w :: 'a word) =
+ takefill False CARD('a) (to_bl w)"
apply (unfold revcast_def' word_size)
apply (rule word_bl.Abs_inverse)
apply simp
@@ -659,7 +656,7 @@
lemma revcast_down_uu':
"rc = revcast ==> source_size rc = target_size rc + n ==>
- rc (w :: 'a :: len word) = ucast (w >> n)"
+ rc (w :: 'a :: finite word) = ucast (w >> n)"
apply (simp add: revcast_def')
apply (rule word_bl.Rep_inverse')
apply (rule trans, rule ucast_down_drop)
@@ -670,7 +667,7 @@
lemma revcast_down_us':
"rc = revcast ==> source_size rc = target_size rc + n ==>
- rc (w :: 'a :: len word) = ucast (w >>> n)"
+ rc (w :: 'a :: finite word) = ucast (w >>> n)"
apply (simp add: revcast_def')
apply (rule word_bl.Rep_inverse')
apply (rule trans, rule ucast_down_drop)
@@ -681,7 +678,7 @@
lemma revcast_down_su':
"rc = revcast ==> source_size rc = target_size rc + n ==>
- rc (w :: 'a :: len word) = scast (w >> n)"
+ rc (w :: 'a :: finite word) = scast (w >> n)"
apply (simp add: revcast_def')
apply (rule word_bl.Rep_inverse')
apply (rule trans, rule scast_down_drop)
@@ -692,7 +689,7 @@
lemma revcast_down_ss':
"rc = revcast ==> source_size rc = target_size rc + n ==>
- rc (w :: 'a :: len word) = scast (w >>> n)"
+ rc (w :: 'a :: finite word) = scast (w >>> n)"
apply (simp add: revcast_def')
apply (rule word_bl.Rep_inverse')
apply (rule trans, rule scast_down_drop)
@@ -708,7 +705,7 @@
lemma cast_down_rev:
"uc = ucast ==> source_size uc = target_size uc + n ==>
- uc w = revcast ((w :: 'a :: len word) << n)"
+ uc w = revcast ((w :: 'a :: finite word) << n)"
apply (unfold shiftl_rev)
apply clarify
apply (simp add: revcast_rev_ucast)
@@ -720,7 +717,7 @@
lemma revcast_up':
"rc = revcast ==> source_size rc + n = target_size rc ==>
- rc w = (ucast w :: 'a :: len word) << n"
+ rc w = (ucast w :: 'a :: finite word) << n"
apply (simp add: revcast_def')
apply (rule word_bl.Rep_inverse')
apply (simp add: takefill_alt)
@@ -747,11 +744,11 @@
subsubsection "Slices"
definition
- slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" where
+ slice1 :: "nat => 'a word => 'b word" where
"slice1 n w == of_bl (takefill False n (to_bl w))"
definition
- slice :: "nat => 'a :: len0 word => 'b :: len0 word" where
+ slice :: "nat => 'a word => 'b word" where
"slice n w == slice1 (size w - n) w"
lemmas slice_def' = slice_def [unfolded word_size]
@@ -788,8 +785,8 @@
done
lemma nth_slice:
- "(slice n w :: 'a :: len0 word) !! m =
- (w !! (m + n) & m < len_of TYPE ('a))"
+ "(slice n w :: 'a word) !! m =
+ (w !! (m + n) & m < CARD('a))"
unfolding slice_shiftr
by (simp add : nth_ucast nth_shiftr)
@@ -805,8 +802,8 @@
apply (unfold slice1_def word_size of_bl_def uint_bl)
apply (clarsimp simp: word_ubin.eq_norm bl_bin_bl_rep_drop
takefill_append [symmetric])
- apply (rule_tac f = "%k. takefill False (len_of TYPE('a))
- (replicate k False @ bin_to_bl (len_of TYPE('b)) (uint w))" in arg_cong)
+ apply (rule_tac f = "%k. takefill False CARD('a)
+ (replicate k False @ bin_to_bl CARD('b) (uint w))" in arg_cong)
apply arith
done
@@ -833,17 +830,17 @@
lemmas revcast_slice1 = refl [THEN revcast_slice1']
lemma slice1_tf_tf':
- "to_bl (slice1 n w :: 'a :: len0 word) =
- rev (takefill False (len_of TYPE('a)) (rev (takefill False n (to_bl w))))"
+ "to_bl (slice1 n w :: 'a word) =
+ rev (takefill False CARD('a) (rev (takefill False n (to_bl w))))"
unfolding slice1_def by (rule word_rev_tf)
lemmas slice1_tf_tf = slice1_tf_tf'
[THEN word_bl.Rep_inverse', symmetric, standard]
lemma rev_slice1:
- "n + k = len_of TYPE('a) + len_of TYPE('b) \<Longrightarrow>
- slice1 n (word_reverse w :: 'b :: len0 word) =
- word_reverse (slice1 k w :: 'a :: len0 word)"
+ "n + k = CARD('a) + CARD('b) \<Longrightarrow>
+ slice1 n (word_reverse w :: 'b word) =
+ word_reverse (slice1 k w :: 'a word)"
apply (unfold word_reverse_def slice1_tf_tf)
apply (rule word_bl.Rep_inverse')
apply (rule rev_swap [THEN iffD1])
@@ -871,10 +868,10 @@
criterion for overflow of addition of signed integers *}
lemma sofl_test:
- "(sint (x :: 'a :: len word) + sint y = sint (x + y)) =
+ "(sint (x :: 'a :: finite word) + sint y = sint (x + y)) =
((((x+y) XOR x) AND ((x+y) XOR y)) >> (size x - 1) = 0)"
apply (unfold word_size)
- apply (cases "len_of TYPE('a)", simp)
+ apply (cases "CARD('a)", simp)
apply (subst msb_shift [THEN sym_notr])
apply (simp add: word_ops_msb)
apply (simp add: word_msb_sint)
@@ -902,29 +899,29 @@
subsection "Split and cat"
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_cat :: "'a word => 'b word => 'c word"
+ "word_cat a b == word_of_int (bin_cat (uint a) CARD('b) (uint b))"
- word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)"
+ word_split :: "'a word => ('b word) * ('c word)"
"word_split a ==
- case bin_split (len_of TYPE ('c)) (uint a) of
+ case bin_split CARD('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 :: "'a word list => 'b word"
"word_rcat ws ==
- word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))"
+ word_of_int (bin_rcat CARD('a) (map uint ws))"
- word_rsplit :: "'a :: len0 word => 'b :: len word list"
+ word_rsplit :: "'a word => 'b :: finite word list"
"word_rsplit w ==
- map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))"
+ map word_of_int (bin_rsplit CARD('b) (CARD('a), uint w))"
lemmas word_split_bin' = word_split_def [THEN meta_eq_to_obj_eq, standard]
lemmas word_cat_bin' = word_cat_def [THEN meta_eq_to_obj_eq, standard]
lemma word_rsplit_no:
- "(word_rsplit (number_of bin :: 'b :: len0 word) :: 'a word list) =
- map number_of (bin_rsplit (len_of TYPE('a :: len))
- (len_of TYPE('b), bintrunc (len_of TYPE('b)) bin))"
+ "(word_rsplit (number_of bin :: 'b word) :: 'a word list) =
+ map number_of (bin_rsplit CARD('a :: finite)
+ (CARD('b), bintrunc CARD('b) bin))"
apply (unfold word_rsplit_def word_no_wi)
apply (simp add: word_ubin.eq_norm)
done
@@ -946,7 +943,7 @@
done
lemma of_bl_append:
- "(of_bl (xs @ ys) :: 'a :: len word) = of_bl xs * 2^(length ys) + of_bl ys"
+ "(of_bl (xs @ ys) :: 'a :: finite word) = of_bl xs * 2^(length ys) + of_bl ys"
apply (unfold of_bl_def)
apply (simp add: bl_to_bin_app_cat bin_cat_num)
apply (simp add: word_of_int_power_hom [symmetric] new_word_of_int_hom_syms)
@@ -958,7 +955,7 @@
(auto simp add: test_bit_of_bl nth_append)
lemma of_bl_True:
- "(of_bl (True#xs)::'a::len word) = 2^length xs + of_bl xs"
+ "(of_bl (True#xs)::'a::finite word) = 2^length xs + of_bl xs"
by (subst of_bl_append [where xs="[True]", simplified])
(simp add: word_1_bl)
@@ -966,8 +963,8 @@
"of_bl (x#xs) = of_bool x * 2^length xs + of_bl xs"
by (cases x) (simp_all add: of_bl_True)
-lemma split_uint_lem: "bin_split n (uint (w :: 'a :: len0 word)) = (a, b) ==>
- a = bintrunc (len_of TYPE('a) - n) a & b = bintrunc (len_of TYPE('a)) b"
+lemma split_uint_lem: "bin_split n (uint (w :: 'a word)) = (a, b) ==>
+ a = bintrunc (CARD('a) - n) a & b = bintrunc CARD('a) b"
apply (frule word_ubin.norm_Rep [THEN ssubst])
apply (drule bin_split_trunc1)
apply (drule sym [THEN trans])
@@ -989,7 +986,7 @@
apply (clarsimp split: prod.splits)
apply (frule split_uint_lem [THEN conjunct1])
apply (unfold word_size)
- apply (cases "len_of TYPE('a) >= len_of TYPE('b)")
+ apply (cases "CARD('a) >= CARD('b)")
defer
apply (simp add: word_0_bl word_0_wi_Pls)
apply (simp add : of_bl_def to_bl_def)
@@ -1015,9 +1012,9 @@
done
lemma word_split_bl_eq:
- "(word_split (c::'a::len word) :: ('c :: len0 word * 'd :: len0 word)) =
- (of_bl (take (len_of TYPE('a::len) - len_of TYPE('d::len0)) (to_bl c)),
- of_bl (drop (len_of TYPE('a) - len_of TYPE('d)) (to_bl c)))"
+ "(word_split (c::'a::finite word) :: ('c word * 'd word)) =
+ (of_bl (take (CARD('a::finite) - CARD('d)) (to_bl c)),
+ of_bl (drop (CARD('a) - CARD('d)) (to_bl c)))"
apply (rule word_split_bl [THEN iffD1])
apply (unfold word_size)
apply (rule refl conjI)+
@@ -1060,14 +1057,13 @@
-- "limited hom result"
lemma word_cat_hom:
- "len_of TYPE('a::len0) <= len_of TYPE('b::len0) + len_of TYPE ('c::len0)
+ "CARD('a) <= CARD('b) + CARD('c)
==>
(word_cat (word_of_int w :: 'b word) (b :: 'c word) :: 'a word) =
word_of_int (bin_cat w (size b) (uint b))"
apply (unfold word_cat_def word_size)
apply (clarsimp simp add : word_ubin.norm_eq_iff [symmetric]
word_ubin.eq_norm bintr_cat min_def)
- apply arith
done
lemma word_cat_split_alt:
@@ -1142,7 +1138,7 @@
by (simp add: bin_rsplit_aux_simp_alt Let_def split: split_split)
lemma test_bit_rsplit:
- "sw = word_rsplit w ==> m < size (hd sw :: 'a :: len word) ==>
+ "sw = word_rsplit w ==> m < size (hd sw :: 'a :: finite word) ==>
k < length sw ==> (rev sw ! k) !! m = (w !! (k * size (hd sw) + m))"
apply (unfold word_rsplit_def word_test_bit_def)
apply (rule trans)
@@ -1157,7 +1153,7 @@
apply (rule map_ident [THEN fun_cong])
apply (rule refl [THEN map_cong])
apply (simp add : word_ubin.eq_norm)
- apply (erule bin_rsplit_size_sign [OF len_gt_0 refl])
+ apply (erule bin_rsplit_size_sign [OF zero_less_card_finite refl])
done
lemma word_rcat_bl: "word_rcat wl == of_bl (concat (map to_bl wl))"
@@ -1170,10 +1166,10 @@
lemmas size_rcat_lem = size_rcat_lem' [unfolded word_size]
-lemmas td_gal_lt_len = len_gt_0 [THEN td_gal_lt, standard]
+lemmas td_gal_lt_len = zero_less_card_finite [THEN td_gal_lt, standard]
lemma nth_rcat_lem' [rule_format] :
- "sw = size (hd wl :: 'a :: len word) ==> (ALL n. n < size wl * sw -->
+ "sw = size (hd wl :: 'a :: finite word) ==> (ALL n. n < size wl * sw -->
rev (concat (map to_bl wl)) ! n =
rev (to_bl (rev wl ! (n div sw))) ! (n mod sw))"
apply (unfold word_size)
@@ -1188,7 +1184,7 @@
lemmas nth_rcat_lem = refl [THEN nth_rcat_lem', unfolded word_size]
lemma test_bit_rcat:
- "sw = size (hd wl :: 'a :: len word) ==> rc = word_rcat wl ==> rc !! n =
+ "sw = size (hd wl :: 'a :: finite word) ==> rc = word_rcat wl ==> rc !! n =
(n < size rc & n div sw < size wl & (rev wl) ! (n div sw) !! (n mod sw))"
apply (unfold word_rcat_bl word_size)
apply (clarsimp simp add :
@@ -1219,7 +1215,7 @@
lemmas word_rsplit_len_indep = word_rsplit_len_indep' [OF refl refl refl refl]
lemma length_word_rsplit_size:
- "n = len_of TYPE ('a :: len) ==>
+ "n = CARD('a :: finite) ==>
(length (word_rsplit w :: 'a word list) <= m) = (size w <= m * n)"
apply (unfold word_rsplit_def word_size)
apply (clarsimp simp add : bin_rsplit_len_le)
@@ -1229,12 +1225,12 @@
length_word_rsplit_size [unfolded Not_eq_iff linorder_not_less [symmetric]]
lemma length_word_rsplit_exp_size:
- "n = len_of TYPE ('a :: len) ==>
+ "n = CARD('a :: finite) ==>
length (word_rsplit w :: 'a word list) = (size w + n - 1) div n"
unfolding word_rsplit_def by (clarsimp simp add : word_size bin_rsplit_len)
lemma length_word_rsplit_even_size:
- "n = len_of TYPE ('a :: len) ==> size w = m * n ==>
+ "n = CARD('a :: finite) ==> size w = m * n ==>
length (word_rsplit w :: 'a word list) = m"
by (clarsimp simp add : length_word_rsplit_exp_size given_quot_alt)
@@ -1251,15 +1247,15 @@
apply (simp_all add: word_size
refl [THEN length_word_rsplit_size [simplified le_def, simplified]])
apply safe
- apply (erule xtr7, rule len_gt_0 [THEN dtle])+
+ apply (erule xtr7, rule zero_less_card_finite [THEN dtle])+
done
lemma size_word_rsplit_rcat_size':
- "word_rcat (ws :: 'a :: len word list) = frcw ==>
- size frcw = length ws * len_of TYPE ('a) ==>
+ "word_rcat (ws :: 'a :: finite word list) = frcw ==>
+ size frcw = length ws * CARD('a) ==>
size (hd [word_rsplit frcw, ws]) = size ws"
apply (clarsimp simp add : word_size length_word_rsplit_exp_size')
- apply (fast intro: given_quot_alt)
+ apply (fast intro: given_quot_alt zero_less_card_finite)
done
lemmas size_word_rsplit_rcat_size =
@@ -1272,8 +1268,8 @@
by (auto simp: add_commute)
lemma word_rsplit_rcat_size':
- "word_rcat (ws :: 'a :: len word list) = frcw ==>
- size frcw = length ws * len_of TYPE ('a) ==> word_rsplit frcw = ws"
+ "word_rcat (ws :: 'a :: finite word list) = frcw ==>
+ size frcw = length ws * CARD('a) ==> word_rsplit frcw = ws"
apply (frule size_word_rsplit_rcat_size, assumption)
apply (clarsimp simp add : word_size)
apply (rule nth_equalityI, assumption)
@@ -1308,13 +1304,13 @@
rotater :: "nat => 'a list => 'a list"
"rotater n == rotater1 ^ n"
- word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word"
+ word_rotr :: "nat => 'a word => 'a word"
"word_rotr n w == of_bl (rotater n (to_bl w))"
- word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word"
+ word_rotl :: "nat => 'a word => 'a word"
"word_rotl n w == of_bl (rotate n (to_bl w))"
- word_roti :: "int => 'a :: len0 word => 'a :: len0 word"
+ word_roti :: "int => 'a word => 'a word"
"word_roti i w == if i >= 0 then word_rotr (nat i) w
else word_rotl (nat (- i)) w"
@@ -1632,7 +1628,7 @@
simplified word_bl.Rep', standard]
lemma bl_word_roti_dt':
- "n = nat ((- i) mod int (size (w :: 'a :: len word))) ==>
+ "n = nat ((- i) mod int (size (w :: 'a :: finite word))) ==>
to_bl (word_roti i w) = drop n (to_bl w) @ take n (to_bl w)"
apply (unfold word_roti_def)
apply (simp add: bl_word_rotl_dt bl_word_rotr_dt word_size)