--- a/src/HOL/Word/Word.thy Thu Oct 15 14:55:19 2020 +0200
+++ b/src/HOL/Word/Word.thy Sat Oct 17 18:56:36 2020 +0200
@@ -9,7 +9,6 @@
"HOL-Library.Type_Length"
"HOL-Library.Boolean_Algebra"
"HOL-Library.Bit_Operations"
- Bits_Int
Traditional_Syntax
begin
@@ -295,8 +294,7 @@
lemma signed_1 [simp]:
\<open>signed (1 :: 'b::len word) = (if LENGTH('b) = 1 then - 1 else 1)\<close>
- by (transfer fixing: uminus; cases \<open>LENGTH('b)\<close>)
- (simp_all add: sbintrunc_minus_simps)
+ by (transfer fixing: uminus; cases \<open>LENGTH('b)\<close>) (auto dest: gr0_implies_Suc)
lemma signed_minus_1 [simp]:
\<open>signed (- 1 :: 'b::len word) = - 1\<close>
@@ -476,7 +474,7 @@
lemma [code]:
\<open>Word.the_signed_int w = signed_take_bit (LENGTH('a) - Suc 0) (Word.the_int w)\<close>
for w :: \<open>'a::len word\<close>
- by transfer simp
+ by transfer (simp add: signed_take_bit_take_bit)
lemma [code_abbrev, simp]:
\<open>Word.the_signed_int = sint\<close>
@@ -1658,7 +1656,7 @@
by (fact word_coorder.extremum)
lemma word_m1_ge [simp] : "word_pred 0 \<ge> y" (* FIXME: delete *)
- by transfer (simp add: take_bit_minus_one_eq_mask mask_eq_exp_minus_1 bintr_lt2p)
+ by transfer (simp add: take_bit_minus_one_eq_mask mask_eq_exp_minus_1 )
lemma word_n1_ge [simp]: "y \<le> -1"
for y :: "'a::len word"
@@ -1766,7 +1764,8 @@
lift_definition shiftr1 :: \<open>'a::len word \<Rightarrow> 'a word\<close>
\<comment> \<open>shift right as unsigned or as signed, ie logical or arithmetic\<close>
- is \<open>\<lambda>k. take_bit LENGTH('a) k div 2\<close> by simp
+ is \<open>\<lambda>k. take_bit LENGTH('a) k div 2\<close>
+ by simp
lemma shiftr1_eq_div_2:
\<open>shiftr1 w = w div 2\<close>
@@ -1884,14 +1883,14 @@
lemma shiftr_def:
\<open>w >> n = (shiftr1 ^^ n) w\<close>
proof -
- have \<open>drop_bit n = (((\<lambda>k::int. k div 2) ^^ n))\<close> for n
- by (rule sym, induction n)
- (simp_all add: fun_eq_iff drop_bit_Suc flip: drop_bit_half)
+ have \<open>shiftr1 ^^ n = (drop_bit n :: 'a word \<Rightarrow> 'a word)\<close>
+ apply (induction n)
+ apply simp
+ apply (simp only: shiftr1_eq_div_2 [abs_def] drop_bit_eq_div [abs_def] funpow_Suc_right)
+ apply (use div_exp_eq [of _ 1, where ?'a = \<open>'a word\<close>] in simp)
+ done
then show ?thesis
- apply transfer
- apply simp
- apply (metis bintrunc_bintrunc rco_bintr)
- done
+ by (simp add: shiftr_word_eq)
qed
lemma bit_shiftl_word_iff:
@@ -2029,9 +2028,18 @@
apply (metis le_antisym less_eq_decr_length_iff)
done
+lemma signed_drop_bit_signed_drop_bit [simp]:
+ \<open>signed_drop_bit m (signed_drop_bit n w) = signed_drop_bit (m + n) w\<close>
+ for w :: \<open>'a::len word\<close>
+ apply (cases \<open>LENGTH('a)\<close>)
+ apply simp_all
+ apply (rule bit_word_eqI)
+ apply (auto simp add: bit_signed_drop_bit_iff not_le less_diff_conv ac_simps)
+ done
+
lemma signed_drop_bit_0 [simp]:
\<open>signed_drop_bit 0 w = w\<close>
- by transfer simp
+ by transfer (simp add: take_bit_signed_take_bit)
lemma sint_signed_drop_bit_eq:
\<open>sint (signed_drop_bit n w) = drop_bit n (sint w)\<close>
@@ -2041,42 +2049,37 @@
apply (auto simp add: bit_sint_iff bit_drop_bit_eq bit_signed_drop_bit_iff dest: bit_imp_le_length)
done
-lift_definition sshiftr1 :: \<open>'a::len word \<Rightarrow> 'a word\<close>
- is \<open>\<lambda>k. take_bit LENGTH('a) (signed_take_bit (LENGTH('a) - 1) k div 2)\<close>
+lift_definition sshiftr :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> 'a word\<close> (infixl \<open>>>>\<close> 55)
+ is \<open>\<lambda>k n. take_bit LENGTH('a) (drop_bit n (signed_take_bit (LENGTH('a) - Suc 0) k))\<close>
by (simp flip: signed_take_bit_decr_length_iff)
-
-lift_definition sshiftr :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> 'a word\<close> (infixl \<open>>>>\<close> 55)
- is \<open>\<lambda>k n. take_bit LENGTH('a) (drop_bit n (signed_take_bit (LENGTH('a) - 1) k))\<close>
+
+lift_definition sshiftr1 :: \<open>'a::len word \<Rightarrow> 'a word\<close>
+ is \<open>\<lambda>k. take_bit LENGTH('a) (signed_take_bit (LENGTH('a) - Suc 0) k div 2)\<close>
by (simp flip: signed_take_bit_decr_length_iff)
lift_definition bshiftr1 :: \<open>bool \<Rightarrow> 'a::len word \<Rightarrow> 'a word\<close>
is \<open>\<lambda>b k. take_bit LENGTH('a) k div 2 + of_bool b * 2 ^ (LENGTH('a) - Suc 0)\<close>
by (fact arg_cong)
+lemma sshiftr_eq:
+ \<open>w >>> n = signed_drop_bit n w\<close>
+ by transfer simp
+
+lemma sshiftr1_eq_signed_drop_bit_Suc_0:
+ \<open>sshiftr1 = signed_drop_bit (Suc 0)\<close>
+ by (rule ext) (transfer, simp add: drop_bit_Suc)
+
lemma sshiftr1_eq:
\<open>sshiftr1 w = word_of_int (sint w div 2)\<close>
by transfer simp
lemma sshiftr_eq_funpow_sshiftr1:
\<open>w >>> n = (sshiftr1 ^^ n) w\<close>
-proof -
- have *: \<open>(\<lambda>k::int. take_bit LENGTH('a) (signed_take_bit (LENGTH('a) - Suc 0) k div 2)) ^^ Suc n =
- take_bit LENGTH('a) \<circ> drop_bit (Suc n) \<circ> signed_take_bit (LENGTH('a) - Suc 0)\<close>
- for n
- apply (induction n)
- apply (auto simp add: fun_eq_iff drop_bit_Suc)
- apply (metis (no_types, lifting) Suc_pred funpow_swap1 len_gt_0 sbintrunc_bintrunc sbintrunc_rest)
- done
- show ?thesis
- apply transfer
- apply simp
- subgoal for k n
- apply (cases n)
- apply (simp_all only: *)
- apply simp_all
- done
- done
-qed
+ apply (rule sym)
+ apply (simp add: sshiftr1_eq_signed_drop_bit_Suc_0 sshiftr_eq)
+ apply (induction n)
+ apply simp_all
+ done
lemma mask_eq:
\<open>mask n = (1 << n) - (1 :: 'a::len word)\<close>
@@ -2260,29 +2263,20 @@
lemma word_cat_eq' [code]:
\<open>word_cat a b = word_of_int (concat_bit LENGTH('b) (uint b) (uint a))\<close>
for a :: \<open>'a::len word\<close> and b :: \<open>'b::len word\<close>
- by transfer simp
+ by transfer (simp add: concat_bit_take_bit_eq)
lemma bit_word_cat_iff:
\<open>bit (word_cat v w :: 'c::len word) n \<longleftrightarrow> n < LENGTH('c) \<and> (if n < LENGTH('b) then bit w n else bit v (n - LENGTH('b)))\<close>
for v :: \<open>'a::len word\<close> and w :: \<open>'b::len word\<close>
by transfer (simp add: bit_concat_bit_iff bit_take_bit_iff)
-definition word_split :: "'a::len word \<Rightarrow> 'b::len word \<times> 'c::len word"
- where "word_split a =
- (case bin_split (LENGTH('c)) (uint a) of
- (u, v) \<Rightarrow> (word_of_int u, word_of_int v))"
+definition word_split :: \<open>'a::len word \<Rightarrow> 'b::len word \<times> 'c::len word\<close>
+ where \<open>word_split w =
+ (ucast (drop_bit LENGTH('c) w) :: 'b::len word, ucast w :: 'c::len word)\<close>
definition word_rcat :: \<open>'a::len word list \<Rightarrow> 'b::len word\<close>
where \<open>word_rcat = word_of_int \<circ> horner_sum uint (2 ^ LENGTH('a)) \<circ> rev\<close>
-lemma word_rcat_eq:
- \<open>word_rcat ws = word_of_int (bin_rcat (LENGTH('a::len)) (map uint ws))\<close>
- for ws :: \<open>'a::len word list\<close>
- apply (simp add: word_rcat_def bin_rcat_def rev_map)
- apply transfer
- apply (simp add: horner_sum_foldr foldr_map comp_def)
- done
-
abbreviation (input) max_word :: \<open>'a::len word\<close>
\<comment> \<open>Largest representable machine integer.\<close>
where "max_word \<equiv> - 1"
@@ -2292,14 +2286,14 @@
lemma int_word_sint:
\<open>sint (word_of_int x :: 'a::len word) = (x + 2 ^ (LENGTH('a) - 1)) mod 2 ^ LENGTH('a) - 2 ^ (LENGTH('a) - 1)\<close>
- by transfer (simp add: no_sbintr_alt2)
+ by transfer (simp flip: take_bit_eq_mod add: signed_take_bit_eq_take_bit_shift)
lemma sint_sbintrunc': "sint (word_of_int bin :: 'a word) = signed_take_bit (LENGTH('a::len) - 1) bin"
by simp
-lemma uint_sint: "uint w = take_bit (LENGTH('a)) (sint w)"
+lemma uint_sint: "uint w = take_bit LENGTH('a) (sint w)"
for w :: "'a::len word"
- by transfer simp
+ by transfer (simp add: take_bit_signed_take_bit)
lemma bintr_uint: "LENGTH('a) \<le> n \<Longrightarrow> take_bit n (uint w) = uint w"
for w :: "'a::len word"
@@ -2364,9 +2358,6 @@
for x :: "'a::len word"
using sint_less [of x] by simp
-lemma sign_uint_Pls [simp]: "bin_sign (uint x) = 0"
- by (simp add: sign_Pls_ge_0)
-
lemma uint_m2p_neg: "uint x - 2 ^ LENGTH('a) < 0"
for x :: "'a::len word"
by (simp only: diff_less_0_iff_less uint_lt2p)
@@ -2377,7 +2368,7 @@
lemma lt2p_lem: "LENGTH('a) \<le> n \<Longrightarrow> uint w < 2 ^ n"
for w :: "'a::len word"
- by (metis bintr_lt2p bintr_uint)
+ using uint_bounded [of w] by (rule less_le_trans) simp
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0"
by (fact uint_ge_0 [THEN leD, THEN antisym_conv1])
@@ -2436,7 +2427,7 @@
by transfer simp
lemma sint_range_size: "- (2 ^ (size w - Suc 0)) \<le> sint w \<and> sint w < 2 ^ (size w - Suc 0)"
- by transfer (use sbintr_ge sbintr_lt in blast)
+ by (simp add: word_size sint_greater_eq sint_less)
lemma sint_above_size: "2 ^ (size w - 1) \<le> x \<Longrightarrow> sint w < x"
for w :: "'a::len word"
@@ -2469,18 +2460,18 @@
for u v :: "'a::len word"
by simp
-lemma test_bit_bin': "w !! n \<longleftrightarrow> n < size w \<and> bin_nth (uint w) n"
+lemma test_bit_bin': "w !! n \<longleftrightarrow> n < size w \<and> bit (uint w) n"
by transfer (simp add: bit_take_bit_iff)
lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
-lemma bin_nth_uint_imp: "bin_nth (uint w) n \<Longrightarrow> n < LENGTH('a)"
+lemma bin_nth_uint_imp: "bit (uint w) n \<Longrightarrow> n < LENGTH('a)"
for w :: "'a::len word"
by transfer (simp add: bit_take_bit_iff)
lemma bin_nth_sint:
"LENGTH('a) \<le> n \<Longrightarrow>
- bin_nth (sint w) n = bin_nth (sint w) (LENGTH('a) - 1)"
+ bit (sint w) n = bit (sint w) (LENGTH('a) - 1)"
for w :: "'a::len word"
by (transfer fixing: n) (simp add: bit_signed_take_bit_iff le_diff_conv min_def)
@@ -2503,7 +2494,7 @@
then have \<open>take_bit LENGTH('a) (signed_take_bit (LENGTH('a) - 1) (numeral a :: int)) = take_bit LENGTH('a) (numeral b)\<close>
by simp
then show \<open>take_bit LENGTH('a) (numeral a :: int) = take_bit LENGTH('a) (numeral b)\<close>
- by simp
+ by (simp add: take_bit_signed_take_bit)
qed
lemma num_abs_bintr:
@@ -2514,7 +2505,7 @@
lemma num_abs_sbintr:
"(numeral x :: 'a word) =
word_of_int (signed_take_bit (LENGTH('a::len) - 1) (numeral x))"
- by transfer simp
+ by transfer (simp add: take_bit_signed_take_bit)
text \<open>
\<open>cast\<close> -- note, no arg for new length, as it's determined by type of result,
@@ -2529,7 +2520,7 @@
by transfer simp
lemma scast_id [simp]: "scast w = w"
- by transfer simp
+ by transfer (simp add: take_bit_signed_take_bit)
lemma nth_ucast: "(ucast w::'a::len word) !! n = (w !! n \<and> n < LENGTH('a))"
by transfer (simp add: bit_take_bit_iff ac_simps)
@@ -2743,16 +2734,13 @@
and "sint (word_pred a) = signed_take_bit (LENGTH('a) - 1) (sint a - 1)"
and "sint (0 :: 'a word) = signed_take_bit (LENGTH('a) - 1) 0"
and "sint (1 :: 'a word) = signed_take_bit (LENGTH('a) - 1) 1"
- prefer 8
- apply (simp add: Suc_lessI sbintrunc_minus_simps(3))
- prefer 7
- apply simp
- apply transfer apply (simp add: signed_take_bit_add)
- apply transfer apply (simp add: signed_take_bit_diff)
- apply transfer apply (simp add: signed_take_bit_mult)
- apply transfer apply (simp add: signed_take_bit_minus)
- apply (metis One_nat_def id_apply of_int_eq_id sbintrunc_sbintrunc signed.rep_eq signed_word_eqI sint_sbintrunc' wi_hom_succ)
- apply (metis (no_types, lifting) One_nat_def signed_take_bit_decr_length_iff sint_uint uint_sint uint_word_of_int_eq wi_hom_pred word_of_int_uint)
+ apply transfer apply (simp add: signed_take_bit_add)
+ apply transfer apply (simp add: signed_take_bit_diff)
+ apply transfer apply (simp add: signed_take_bit_mult)
+ apply transfer apply (simp add: signed_take_bit_minus)
+ apply (metis of_int_sint scast_id sint_sbintrunc' wi_hom_succ)
+ apply (metis of_int_sint scast_id sint_sbintrunc' wi_hom_pred)
+ apply (simp_all add: sint_uint)
done
lemma word_pred_0_n1: "word_pred 0 = word_of_int (- 1)"
@@ -2937,18 +2925,31 @@
unfolding uint_word_ariths by (simp add: zmod_le_nonneg_dividend)
lemma uint_sub_ge: "uint (x - y) \<ge> uint x - uint y"
- unfolding uint_word_ariths by (simp add: int_mod_ge)
-
+ unfolding uint_word_ariths
+ by (simp flip: take_bit_eq_mod add: take_bit_int_greater_eq_self_iff)
+
+lemma int_mod_ge: \<open>a \<le> a mod n\<close> if \<open>a < n\<close> \<open>0 < n\<close>
+ for a n :: int
+proof (cases \<open>a < 0\<close>)
+ case True
+ with \<open>0 < n\<close> show ?thesis
+ by (metis less_trans not_less pos_mod_conj)
+
+next
+ case False
+ with \<open>a < n\<close> show ?thesis
+ by simp
+qed
+
lemma mod_add_if_z:
"x < z \<Longrightarrow> y < z \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> z \<Longrightarrow>
(x + y) mod z = (if x + y < z then x + y else x + y - z)"
for x y z :: int
apply (auto simp add: not_less)
apply (rule antisym)
- apply (metis diff_ge_0_iff_ge minus_mod_self2 zmod_le_nonneg_dividend)
- apply (simp only: flip: minus_mod_self2 [of \<open>x + y\<close> z])
- apply (rule int_mod_ge)
- apply simp_all
+ apply (metis diff_ge_0_iff_ge minus_mod_self2 zmod_le_nonneg_dividend)
+ apply (simp only: flip: minus_mod_self2 [of \<open>x + y\<close> z])
+ apply (metis add.commute add_less_cancel_left add_mono_thms_linordered_field(5) diff_add_cancel diff_ge_0_iff_ge mod_pos_pos_trivial order_refl)
done
lemma uint_plus_if':
@@ -3593,16 +3594,13 @@
by (fact inj_unsigned)
lemma range_uint: \<open>range (uint :: 'a word \<Rightarrow> int) = {0..<2 ^ LENGTH('a::len)}\<close>
- by transfer (auto simp add: bintr_lt2p range_bintrunc)
+ apply transfer
+ apply (auto simp add: image_iff)
+ apply (metis take_bit_int_eq_self_iff)
+ done
lemma UNIV_eq: \<open>(UNIV :: 'a word set) = word_of_int ` {0..<2 ^ LENGTH('a::len)}\<close>
-proof -
- have \<open>uint ` (UNIV :: 'a word set) = uint ` (word_of_int :: int \<Rightarrow> 'a word) ` {0..<2 ^ LENGTH('a::len)}\<close>
- by transfer (simp add: range_bintrunc, auto simp add: take_bit_eq_mod)
- then show ?thesis
- using inj_image_eq_iff [of \<open>uint :: 'a word \<Rightarrow> int\<close> \<open>UNIV :: 'a word set\<close> \<open>word_of_int ` {0..<2 ^ LENGTH('a)} :: 'a word set\<close>, OF inj_uint]
- by simp
-qed
+ by (auto simp add: image_iff) (metis atLeastLessThan_iff linorder_not_le uint_split)
lemma card_word: "CARD('a word) = 2 ^ LENGTH('a::len)"
by (simp add: UNIV_eq card_image inj_on_word_of_int)
@@ -3617,18 +3615,6 @@
subsection \<open>Bitwise Operations on Words\<close>
-lemmas bin_log_bintrs = bin_trunc_not bin_trunc_xor bin_trunc_and bin_trunc_or
-
-\<comment> \<open>following definitions require both arithmetic and bit-wise word operations\<close>
-
-\<comment> \<open>to get \<open>word_no_log_defs\<close> from \<open>word_log_defs\<close>, using \<open>bin_log_bintrs\<close>\<close>
-lemmas wils1 = bin_log_bintrs [THEN word_of_int_eq_iff [THEN iffD2],
- folded uint_word_of_int_eq, THEN eq_reflection]
-
-\<comment> \<open>the binary operations only\<close> (* BH: why is this needed? *)
-lemmas word_log_binary_defs =
- word_and_def word_or_def word_xor_def
-
lemma word_wi_log_defs:
"NOT (word_of_int a) = word_of_int (NOT a)"
"word_of_int a AND word_of_int b = word_of_int (a AND b)"
@@ -3701,7 +3687,7 @@
by (simp only: test_bit_eq_bit) transfer_prover
lemma test_bit_wi [simp]:
- "(word_of_int x :: 'a::len word) !! n \<longleftrightarrow> n < LENGTH('a) \<and> bin_nth x n"
+ "(word_of_int x :: 'a::len word) !! n \<longleftrightarrow> n < LENGTH('a) \<and> bit x n"
by transfer simp
lemma word_ops_nth_size:
@@ -3711,29 +3697,29 @@
(x XOR y) !! n = (x !! n \<noteq> y !! n) \<and>
(NOT x) !! n = (\<not> x !! n)"
for x :: "'a::len word"
- unfolding word_size by transfer (simp add: bin_nth_ops)
+ by transfer (simp add: bit_or_iff bit_and_iff bit_xor_iff bit_not_iff)
lemma word_ao_nth:
"(x OR y) !! n = (x !! n | y !! n) \<and>
(x AND y) !! n = (x !! n \<and> y !! n)"
for x :: "'a::len word"
- by transfer (auto simp add: bin_nth_ops)
+ by transfer (auto simp add: bit_or_iff bit_and_iff)
lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN word_ops_nth_size [unfolded word_size]]
lemmas msb1 = msb0 [where i = 0]
lemma test_bit_numeral [simp]:
"(numeral w :: 'a::len word) !! n \<longleftrightarrow>
- n < LENGTH('a) \<and> bin_nth (numeral w) n"
+ n < LENGTH('a) \<and> bit (numeral w :: int) n"
by transfer (rule refl)
lemma test_bit_neg_numeral [simp]:
"(- numeral w :: 'a::len word) !! n \<longleftrightarrow>
- n < LENGTH('a) \<and> bin_nth (- numeral w) n"
+ n < LENGTH('a) \<and> bit (- numeral w :: int) n"
by transfer (rule refl)
lemma test_bit_1 [simp]: "(1 :: 'a::len word) !! n \<longleftrightarrow> n = 0"
- by transfer auto
+ by transfer (auto simp add: bit_1_iff)
lemma nth_0 [simp]: "\<not> (0 :: 'a::len word) !! n"
by transfer simp
@@ -3822,7 +3808,7 @@
lemma word_add_not [simp]: "x + NOT x = -1"
for x :: "'a::len word"
- by transfer (simp add: bin_add_not)
+ by transfer (simp add: not_int_def)
lemma word_plus_and_or [simp]: "(x AND y) + (x OR y) = x + y"
for x :: "'a::len word"
@@ -3842,7 +3828,7 @@
lemma le_word_or2: "x \<le> x OR y"
for x y :: "'a::len word"
- by (auto simp: word_le_def uint_or intro: le_int_or)
+ by (simp add: or_greater_eq uint_or word_le_def)
lemmas le_word_or1 = xtrans(3) [OF word_bw_comms (2) le_word_or2]
lemmas word_and_le1 = xtrans(3) [OF word_ao_absorbs (4) [symmetric] le_word_or2]
@@ -3877,16 +3863,9 @@
lemma nth_sint:
fixes w :: "'a::len word"
defines "l \<equiv> LENGTH('a)"
- shows "bin_nth (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))"
+ shows "bit (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))"
unfolding sint_uint l_def
- by (auto simp: nth_sbintr word_test_bit_def [symmetric])
-
-lemma setBit_no [simp]: "setBit (numeral bin) n = word_of_int (bin_sc n True (numeral bin))"
- by transfer (simp add: bin_sc_eq)
-
-lemma clearBit_no [simp]:
- "clearBit (numeral bin) n = word_of_int (bin_sc n False (numeral bin))"
- by transfer (simp add: bin_sc_eq)
+ by (auto simp: bit_signed_take_bit_iff word_test_bit_def not_less min_def)
lemma test_bit_2p: "(word_of_int (2 ^ n)::'a::len word) !! m \<longleftrightarrow> m = n \<and> m < LENGTH('a)"
by transfer (auto simp add: bit_exp_iff)
@@ -3973,12 +3952,13 @@
text \<open>
see paper page 10, (1), (2), \<open>shiftr1_def\<close> is of the form of (1),
- where \<open>f\<close> (ie \<open>bin_rest\<close>) takes normal arguments to normal results,
+ where \<open>f\<close> (ie \<open>_ div 2\<close>) takes normal arguments to normal results,
thus we get (2) from (1)
\<close>
-lemma uint_shiftr1: "uint (shiftr1 w) = bin_rest (uint w)"
- by transfer simp
+lemma uint_shiftr1: "uint (shiftr1 w) = uint w div 2"
+ using uint_shiftr_eq [of w 1]
+ by (simp add: shiftr1_code)
lemma bit_sshiftr1_iff:
\<open>bit (sshiftr1 w) n \<longleftrightarrow> bit w (if n = LENGTH('a) - 1 then LENGTH('a) - 1 else Suc n)\<close>
@@ -3998,10 +3978,6 @@
using le_less_Suc_eq apply fastforce
done
-lemma sshiftr_eq:
- \<open>w >>> m = signed_drop_bit m w\<close>
- by (rule bit_eqI) (simp add: bit_signed_drop_bit_iff bit_sshiftr_word_iff)
-
lemma nth_sshiftr1: "sshiftr1 w !! n = (if n = size w - 1 then w !! n else w !! Suc n)"
apply transfer
apply (auto simp add: bit_take_bit_iff bit_signed_take_bit_iff min_def simp flip: bit_Suc)
@@ -4022,20 +3998,15 @@
by (fact uint_shiftr1)
lemma sshiftr1_div_2: "sint (sshiftr1 w) = sint w div 2"
- by transfer simp
+ using sint_signed_drop_bit_eq [of 1 w]
+ by (simp add: drop_bit_Suc sshiftr1_eq_signed_drop_bit_Suc_0)
lemma shiftr_div_2n: "uint (shiftr w n) = uint w div 2 ^ n"
- apply (unfold shiftr_def)
- apply (induct n)
- apply simp
- apply (simp add: shiftr1_div_2 mult.commute zdiv_zmult2_eq [symmetric])
- done
+ by (fact uint_shiftr_eq)
lemma sshiftr_div_2n: "sint (sshiftr w n) = sint w div 2 ^ n"
- apply transfer
- apply (rule bit_eqI)
- apply (simp add: bit_signed_take_bit_iff bit_drop_bit_eq min_def flip: drop_bit_eq_div)
- done
+ using sint_signed_drop_bit_eq [of n w]
+ by (simp add: drop_bit_eq_div sshiftr_eq)
lemma bit_bshiftr1_iff:
\<open>bit (bshiftr1 b w) n \<longleftrightarrow> b \<and> n = LENGTH('a) - 1 \<or> bit w (Suc n)\<close>
@@ -4092,12 +4063,12 @@
lemma shiftr1_bintr [simp]:
"(shiftr1 (numeral w) :: 'a::len word) =
- word_of_int (bin_rest (take_bit (LENGTH('a)) (numeral w)))"
+ word_of_int (take_bit LENGTH('a) (numeral w) div 2)"
by transfer simp
lemma sshiftr1_sbintr [simp]:
"(sshiftr1 (numeral w) :: 'a::len word) =
- word_of_int (bin_rest (signed_take_bit (LENGTH('a) - 1) (numeral w)))"
+ word_of_int (signed_take_bit (LENGTH('a) - 1) (numeral w) div 2)"
by transfer simp
text \<open>TODO: rules for \<^term>\<open>- (numeral n)\<close>\<close>
@@ -4116,7 +4087,7 @@
word_of_int (drop_bit (numeral n) (signed_take_bit (LENGTH('a) - 1) (numeral k)))\<close>
apply (rule word_eqI)
apply (cases \<open>LENGTH('a)\<close>)
- apply (simp_all add: word_size bit_drop_bit_eq nth_sshiftr nth_sbintr not_le not_less less_Suc_eq_le ac_simps)
+ apply (simp_all add: word_size bit_drop_bit_eq nth_sshiftr bit_signed_take_bit_iff min_def not_le not_less less_Suc_eq_le ac_simps)
done
lemma zip_replicate: "n \<ge> length ys \<Longrightarrow> zip (replicate n x) ys = map (\<lambda>y. (x, y)) ys"
@@ -4185,20 +4156,17 @@
by (auto simp add: test_bit_word_eq word_size Word.bit_mask_iff)
lemma mask_bin: "mask n = word_of_int (take_bit n (- 1))"
- by (auto simp add: nth_bintr word_size intro: word_eqI)
+ by transfer (simp add: take_bit_minus_one_eq_mask)
lemma and_mask_bintr: "w AND mask n = word_of_int (take_bit n (uint w))"
- apply (rule word_eqI)
- apply (simp add: nth_bintr word_size word_ops_nth_size)
- apply (auto simp add: test_bit_bin)
- done
+ by transfer (simp add: ac_simps take_bit_eq_mask)
lemma and_mask_wi: "word_of_int i AND mask n = word_of_int (take_bit n i)"
- by (auto simp add: nth_bintr word_size word_ops_nth_size word_eq_iff)
+ by (auto simp add: and_mask_bintr min_def not_le wi_bintr)
lemma and_mask_wi':
"word_of_int i AND mask n = (word_of_int (take_bit (min LENGTH('a) n) i) :: 'a::len word)"
- by (auto simp add: nth_bintr word_size word_ops_nth_size word_eq_iff)
+ by (auto simp add: and_mask_wi min_def wi_bintr)
lemma and_mask_no: "numeral i AND mask n = word_of_int (take_bit n (numeral i))"
unfolding word_numeral_alt by (rule and_mask_wi)
@@ -4211,16 +4179,10 @@
by transfer simp
lemma and_mask_lt_2p: "uint (w AND mask n) < 2 ^ n"
- apply (simp add: uint_and uint_mask_eq)
- apply (rule AND_upper2'')
- apply simp
- apply (auto simp add: mask_eq_exp_minus_1 min_def power_add algebra_simps dest!: le_Suc_ex)
- apply (metis add_minus_cancel le_add2 one_le_numeral power_add power_increasing uminus_add_conv_diff zle_diff1_eq)
- done
-
-lemma eq_mod_iff: "0 < n \<Longrightarrow> b = b mod n \<longleftrightarrow> 0 \<le> b \<and> b < n"
- for b n :: int
- by auto (metis pos_mod_conj)+
+ apply (simp flip: take_bit_eq_mask)
+ apply transfer
+ apply (auto simp add: min_def)
+ using antisym_conv take_bit_int_eq_self_iff by fastforce
lemma mask_eq_iff: "w AND mask n = w \<longleftrightarrow> uint w < 2 ^ n"
apply (auto simp flip: take_bit_eq_mask)
@@ -4240,11 +4202,11 @@
lemma less_mask_eq: "x < 2 ^ n \<Longrightarrow> x AND mask n = x"
for x :: "'a::len word"
- apply (simp add: and_mask_bintr)
+ apply (cases \<open>n < LENGTH('a)\<close>)
+ apply (simp_all add: not_less flip: take_bit_eq_mask exp_eq_zero_iff)
apply transfer
- apply (simp add: ac_simps)
- apply (auto simp add: min_def)
- apply (metis bintrunc_bintrunc_ge mod_pos_pos_trivial mult.commute mult.left_neutral mult_zero_left not_le of_bool_def take_bit_eq_mod take_bit_nonnegative)
+ apply (simp add: min_def)
+ apply (metis min_def nat_less_le take_bit_int_eq_self_iff take_bit_take_bit)
done
lemmas mask_eq_iff_w2p = trans [OF mask_eq_iff word_2p_lem [symmetric]]
@@ -4544,27 +4506,15 @@
apply (auto simp add: bit_concat_bit_iff bit_take_bit_iff)
done
-lemma split_uint_lem: "bin_split n (uint w) = (a, b) \<Longrightarrow>
- a = take_bit (LENGTH('a) - n) a \<and> b = take_bit (LENGTH('a)) b"
- for w :: "'a::len word"
- by transfer (simp add: drop_bit_take_bit ac_simps)
-
\<comment> \<open>keep quantifiers for use in simplification\<close>
lemma test_bit_split':
"word_split c = (a, b) \<longrightarrow>
(\<forall>n m.
b !! n = (n < size b \<and> c !! n) \<and>
a !! m = (m < size a \<and> c !! (m + size b)))"
- apply (unfold word_split_bin' test_bit_bin)
- apply (clarify)
- apply simp
- apply (erule conjE)
- apply (drule sym)
- apply (drule sym)
- apply (simp add: bit_take_bit_iff bit_drop_bit_eq)
- apply transfer
- apply (simp add: bit_take_bit_iff ac_simps)
- done
+ by (auto simp add: word_split_bin' test_bit_bin bit_unsigned_iff word_size
+ bit_drop_bit_eq ac_simps exp_eq_zero_iff
+ dest: bit_imp_le_length)
lemma test_bit_split:
"word_split c = (a, b) \<Longrightarrow>
@@ -4590,15 +4540,7 @@
result to the length given by the result type\<close>
lemma word_cat_id: "word_cat a b = b"
- by transfer simp
-
-\<comment> \<open>limited hom result\<close>
-lemma word_cat_hom:
- "LENGTH('a::len) \<le> LENGTH('b::len) + LENGTH('c::len) \<Longrightarrow>
- (word_cat (word_of_int w :: 'b word) (b :: 'c word) :: 'a word) =
- word_of_int (bin_cat w (size b) (uint b))"
- apply transfer
- using bintr_cat by auto
+ by transfer (simp add: take_bit_concat_bit_eq)
lemma word_cat_split_alt: "size w \<le> size u + size v \<Longrightarrow> word_split w = (u, v) \<Longrightarrow> word_cat u v = w"
apply (rule word_eqI)
@@ -4675,7 +4617,7 @@
"sw = size (hd wl) \<Longrightarrow> rc = word_rcat wl \<Longrightarrow> rc !! n =
(n < size rc \<and> n div sw < size wl \<and> (rev wl) ! (n div sw) !! (n mod sw))"
for wl :: "'a::len word list"
- by (simp add: word_size word_rcat_def bin_rcat_def foldl_map rev_map bit_horner_sum_uint_exp_iff)
+ by (simp add: word_size word_rcat_def foldl_map rev_map bit_horner_sum_uint_exp_iff)
(simp add: test_bit_eq_bit)
lemmas test_bit_cong = arg_cong [where f = "test_bit", THEN fun_cong]
@@ -4932,11 +4874,11 @@
by transfer simp
with that show ?thesis
apply transfer
- apply simp
+ apply simp
apply (subst take_bit_diff [symmetric])
apply (subst nat_take_bit_eq)
- apply (simp add: nat_le_eq_zle)
- apply (simp add: nat_diff_distrib take_bit_nat_eq_self_iff less_imp_diff_less bintr_lt2p)
+ apply (simp add: nat_le_eq_zle)
+ apply (simp add: nat_diff_distrib take_bit_nat_eq_self_iff less_imp_diff_less)
done
qed
@@ -5145,22 +5087,13 @@
lemma mask_Suc_0: "mask (Suc 0) = 1"
using mask_1 by simp
-lemma bin_last_bintrunc: "bin_last (take_bit l n) = (l > 0 \<and> bin_last n)"
+lemma bin_last_bintrunc: "odd (take_bit l n) \<longleftrightarrow> l > 0 \<and> odd n"
by simp
lemma word_and_1:
"n AND 1 = (if n !! 0 then 1 else 0)" for n :: "_ word"
by (rule bit_word_eqI) (auto simp add: bit_and_iff test_bit_eq_bit bit_1_iff intro: gr0I)
-lemma bintrunc_shiftl:
- "take_bit n (m << i) = take_bit (n - i) m << i"
- for m :: int
- by (rule bit_eqI) (auto simp add: bit_take_bit_iff)
-
-lemma uint_shiftl:
- "uint (n << i) = take_bit (size n) (uint n << i)"
- by transfer (simp add: push_bit_take_bit shiftl_eq_push_bit)
-
subsection \<open>Misc\<close>