--- a/src/HOL/Library/Bit_Operations.thy Sat Sep 05 08:32:34 2020 +0000
+++ b/src/HOL/Library/Bit_Operations.thy Sat Sep 05 16:21:16 2020 +0000
@@ -552,6 +552,10 @@
end
+lemma mask_half_int:
+ \<open>mask n div 2 = (mask (n - 1) :: int)\<close>
+ by (cases n) (simp_all add: mask_eq_exp_minus_1 algebra_simps)
+
lemma mask_nonnegative_int [simp]:
\<open>mask n \<ge> (0::int)\<close>
by (simp add: mask_eq_exp_minus_1)
@@ -896,43 +900,55 @@
(auto simp add: bit_take_bit_iff bit_concat_bit_iff min_def)
-subsection \<open>Taking bit with sign propagation\<close>
+subsection \<open>Taking bits with sign propagation\<close>
-definition signed_take_bit :: \<open>nat \<Rightarrow> int \<Rightarrow> int\<close>
- where \<open>signed_take_bit n k = concat_bit n k (- of_bool (bit k n))\<close>
+context ring_bit_operations
+begin
-lemma signed_take_bit_unfold:
- \<open>signed_take_bit n k = take_bit n k OR (of_bool (bit k n) * NOT (mask n))\<close>
- by (simp add: signed_take_bit_def concat_bit_def push_bit_minus_one_eq_not_mask)
+definition signed_take_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
+ where \<open>signed_take_bit n a = take_bit n a OR (of_bool (bit a n) * NOT (mask n))\<close>
-lemma signed_take_bit_eq:
- \<open>signed_take_bit n k = take_bit n k\<close> if \<open>\<not> bit k n\<close>
+lemma signed_take_bit_eq_if_positive:
+ \<open>signed_take_bit n a = take_bit n a\<close> if \<open>\<not> bit a n\<close>
using that by (simp add: signed_take_bit_def)
-lemma signed_take_bit_eq_or:
- \<open>signed_take_bit n k = take_bit n k OR NOT (mask n)\<close> if \<open>bit k n\<close>
- using that by (simp add: signed_take_bit_def concat_bit_def take_bit_eq_mask push_bit_minus_one_eq_not_mask)
+lemma signed_take_bit_eq_if_negative:
+ \<open>signed_take_bit n a = take_bit n a OR NOT (mask n)\<close> if \<open>bit a n\<close>
+ using that by (simp add: signed_take_bit_def)
+
+lemma even_signed_take_bit_iff:
+ \<open>even (signed_take_bit m a) \<longleftrightarrow> even a\<close>
+ by (auto simp add: signed_take_bit_def even_or_iff even_mask_iff bit_double_iff)
+
+lemma bit_signed_take_bit_iff:
+ \<open>bit (signed_take_bit m a) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> bit a (min m n)\<close>
+ by (simp add: signed_take_bit_def bit_take_bit_iff bit_or_iff bit_not_iff bit_mask_iff min_def not_le)
+ (use exp_eq_0_imp_not_bit in blast)
lemma signed_take_bit_0 [simp]:
- \<open>signed_take_bit 0 k = - (k mod 2)\<close>
+ \<open>signed_take_bit 0 a = - (a mod 2)\<close>
by (simp add: signed_take_bit_def odd_iff_mod_2_eq_one)
-lemma mask_half_int:
- \<open>mask n div 2 = (mask (n - 1) :: int)\<close>
- by (cases n) (simp_all add: mask_eq_exp_minus_1 algebra_simps)
-
lemma signed_take_bit_Suc:
- \<open>signed_take_bit (Suc n) k = k mod 2 + 2 * signed_take_bit n (k div 2)\<close>
- by (unfold signed_take_bit_def or_int_rec [of \<open>take_bit (Suc n) k\<close>])
- (simp add: bit_Suc concat_bit_Suc even_or_iff even_mask_iff odd_iff_mod_2_eq_one not_int_div_2 mask_half_int)
-
-lemma signed_take_bit_rec:
- \<open>signed_take_bit n k = (if n = 0 then - (k mod 2) else k mod 2 + 2 * signed_take_bit (n - 1) (k div 2))\<close>
- by (cases n) (simp_all add: signed_take_bit_Suc)
-
-lemma bit_signed_take_bit_iff:
- \<open>bit (signed_take_bit m k) n = bit k (min m n)\<close>
- by (simp add: signed_take_bit_def bit_or_iff bit_concat_bit_iff bit_not_iff bit_mask_iff min_def)
+ \<open>signed_take_bit (Suc n) a = a mod 2 + 2 * signed_take_bit n (a div 2)\<close>
+proof (rule bit_eqI)
+ fix m
+ assume *: \<open>2 ^ m \<noteq> 0\<close>
+ show \<open>bit (signed_take_bit (Suc n) a) m \<longleftrightarrow>
+ bit (a mod 2 + 2 * signed_take_bit n (a div 2)) m\<close>
+ proof (cases m)
+ case 0
+ then show ?thesis
+ by (simp add: even_signed_take_bit_iff)
+ next
+ case (Suc m)
+ with * have \<open>2 ^ m \<noteq> 0\<close>
+ by (metis mult_not_zero power_Suc)
+ with Suc show ?thesis
+ by (simp add: bit_signed_take_bit_iff mod2_eq_if bit_double_iff even_bit_succ_iff
+ ac_simps flip: bit_Suc)
+ qed
+qed
lemma signed_take_bit_of_0 [simp]:
\<open>signed_take_bit n 0 = 0\<close>
@@ -940,36 +956,56 @@
lemma signed_take_bit_of_minus_1 [simp]:
\<open>signed_take_bit n (- 1) = - 1\<close>
- by (simp add: signed_take_bit_def concat_bit_def push_bit_minus_one_eq_not_mask take_bit_minus_one_eq_mask)
+ by (simp add: signed_take_bit_def take_bit_minus_one_eq_mask mask_eq_exp_minus_1)
-lemma signed_take_bit_signed_take_bit [simp]:
- \<open>signed_take_bit m (signed_take_bit n k) = signed_take_bit (min m n) k\<close>
- by (rule bit_eqI) (auto simp add: bit_signed_take_bit_iff min_def bit_or_iff bit_not_iff bit_mask_iff bit_take_bit_iff)
+lemma signed_take_bit_Suc_1 [simp]:
+ \<open>signed_take_bit (Suc n) 1 = 1\<close>
+ by (simp add: signed_take_bit_Suc)
+
+lemma signed_take_bit_rec:
+ \<open>signed_take_bit n a = (if n = 0 then - (a mod 2) else a mod 2 + 2 * signed_take_bit (n - 1) (a div 2))\<close>
+ by (cases n) (simp_all add: signed_take_bit_Suc)
lemma signed_take_bit_eq_iff_take_bit_eq:
- \<open>signed_take_bit n k = signed_take_bit n l \<longleftrightarrow> take_bit (Suc n) k = take_bit (Suc n) l\<close>
-proof (cases \<open>bit k n \<longleftrightarrow> bit l n\<close>)
- case True
- moreover have \<open>take_bit n k OR NOT (mask n) = take_bit n k - 2 ^ n\<close>
- for k :: int
- by (auto simp add: disjunctive_add [symmetric] bit_not_iff bit_mask_iff bit_take_bit_iff minus_exp_eq_not_mask)
- ultimately show ?thesis
- by (simp add: signed_take_bit_def take_bit_Suc_from_most concat_bit_eq)
-next
- case False
- then have \<open>signed_take_bit n k \<noteq> signed_take_bit n l\<close> and \<open>take_bit (Suc n) k \<noteq> take_bit (Suc n) l\<close>
- by (auto simp add: bit_eq_iff bit_take_bit_iff bit_signed_take_bit_iff min_def)
+ \<open>signed_take_bit n a = signed_take_bit n b \<longleftrightarrow> take_bit (Suc n) a = take_bit (Suc n) b\<close>
+proof -
+ have \<open>bit (signed_take_bit n a) = bit (signed_take_bit n b) \<longleftrightarrow> bit (take_bit (Suc n) a) = bit (take_bit (Suc n) b)\<close>
+ by (simp add: fun_eq_iff bit_signed_take_bit_iff bit_take_bit_iff not_le less_Suc_eq_le min_def)
+ (use exp_eq_0_imp_not_bit in fastforce)
then show ?thesis
- by simp
+ by (simp add: bit_eq_iff fun_eq_iff)
qed
+lemma signed_take_bit_signed_take_bit [simp]:
+ \<open>signed_take_bit m (signed_take_bit n a) = signed_take_bit (min m n) a\<close>
+proof (rule bit_eqI)
+ fix q
+ show \<open>bit (signed_take_bit m (signed_take_bit n a)) q \<longleftrightarrow>
+ bit (signed_take_bit (min m n) a) q\<close>
+ by (simp add: bit_signed_take_bit_iff min_def bit_or_iff bit_not_iff bit_mask_iff bit_take_bit_iff)
+ (use le_Suc_ex exp_add_not_zero_imp in blast)
+qed
+
+lemma signed_take_bit_take_bit:
+ \<open>signed_take_bit m (take_bit n a) = (if n \<le> m then take_bit n else signed_take_bit m) a\<close>
+ by (rule bit_eqI) (auto simp add: bit_signed_take_bit_iff min_def bit_take_bit_iff)
+
lemma take_bit_signed_take_bit:
- \<open>take_bit m (signed_take_bit n k) = take_bit m k\<close> if \<open>m \<le> Suc n\<close>
+ \<open>take_bit m (signed_take_bit n a) = take_bit m a\<close> if \<open>m \<le> Suc n\<close>
using that by (rule le_SucE; intro bit_eqI)
(auto simp add: bit_take_bit_iff bit_signed_take_bit_iff min_def less_Suc_eq)
+end
+
+text \<open>Modulus centered around 0\<close>
+
+lemma signed_take_bit_eq_concat_bit:
+ \<open>signed_take_bit n k = concat_bit n k (- of_bool (bit k n))\<close>
+ by (simp add: concat_bit_def signed_take_bit_def push_bit_minus_one_eq_not_mask)
+
lemma signed_take_bit_add:
\<open>signed_take_bit n (signed_take_bit n k + signed_take_bit n l) = signed_take_bit n (k + l)\<close>
+ for k l :: int
proof -
have \<open>take_bit (Suc n)
(take_bit (Suc n) (signed_take_bit n k) +
@@ -982,6 +1018,7 @@
lemma signed_take_bit_diff:
\<open>signed_take_bit n (signed_take_bit n k - signed_take_bit n l) = signed_take_bit n (k - l)\<close>
+ for k l :: int
proof -
have \<open>take_bit (Suc n)
(take_bit (Suc n) (signed_take_bit n k) -
@@ -994,6 +1031,7 @@
lemma signed_take_bit_minus:
\<open>signed_take_bit n (- signed_take_bit n k) = signed_take_bit n (- k)\<close>
+ for k :: int
proof -
have \<open>take_bit (Suc n)
(- take_bit (Suc n) (signed_take_bit n k)) =
@@ -1005,6 +1043,7 @@
lemma signed_take_bit_mult:
\<open>signed_take_bit n (signed_take_bit n k * signed_take_bit n l) = signed_take_bit n (k * l)\<close>
+ for k l :: int
proof -
have \<open>take_bit (Suc n)
(take_bit (Suc n) (signed_take_bit n k) *
@@ -1015,10 +1054,9 @@
by (simp only: signed_take_bit_eq_iff_take_bit_eq take_bit_mult)
qed
-text \<open>Modulus centered around 0\<close>
-
lemma signed_take_bit_eq_take_bit_minus:
\<open>signed_take_bit n k = take_bit (Suc n) k - 2 ^ Suc n * of_bool (bit k n)\<close>
+ for k :: int
proof (cases \<open>bit k n\<close>)
case True
have \<open>signed_take_bit n k = take_bit (Suc n) k OR NOT (mask (Suc n))\<close>
@@ -1029,13 +1067,13 @@
by (simp flip: minus_exp_eq_not_mask)
next
case False
- then show ?thesis
- by (simp add: bit_eq_iff bit_take_bit_iff bit_signed_take_bit_iff min_def)
- (auto intro: bit_eqI simp add: less_Suc_eq)
+ show ?thesis
+ by (rule bit_eqI) (simp add: False bit_signed_take_bit_iff bit_take_bit_iff min_def less_Suc_eq)
qed
lemma signed_take_bit_eq_take_bit_shift:
\<open>signed_take_bit n k = take_bit (Suc n) (k + 2 ^ n) - 2 ^ n\<close>
+ for k :: int
proof -
have *: \<open>take_bit n k OR 2 ^ n = take_bit n k + 2 ^ n\<close>
by (simp add: disjunctive_add bit_exp_iff bit_take_bit_iff)
@@ -1055,87 +1093,80 @@
by (rule disjunctive_add)
(auto simp add: disjunctive_add bit_take_bit_iff bit_double_iff bit_exp_iff)
finally show ?thesis
- using * **
- by (simp add: signed_take_bit_def concat_bit_Suc min_def ac_simps)
- (simp add: concat_bit_def take_bit_eq_mask push_bit_minus_one_eq_not_mask ac_simps)
+ using * ** by (simp add: signed_take_bit_def concat_bit_Suc min_def ac_simps)
qed
-lemma signed_take_bit_take_bit:
- \<open>signed_take_bit m (take_bit n k) = (if n \<le> m then take_bit n else signed_take_bit m) k\<close>
- by (rule bit_eqI) (auto simp add: bit_signed_take_bit_iff min_def bit_take_bit_iff)
-
lemma signed_take_bit_nonnegative_iff [simp]:
\<open>0 \<le> signed_take_bit n k \<longleftrightarrow> \<not> bit k n\<close>
+ for k :: int
by (simp add: signed_take_bit_def not_less concat_bit_def)
lemma signed_take_bit_negative_iff [simp]:
\<open>signed_take_bit n k < 0 \<longleftrightarrow> bit k n\<close>
+ for k :: int
by (simp add: signed_take_bit_def not_less concat_bit_def)
lemma signed_take_bit_greater_eq:
\<open>k + 2 ^ Suc n \<le> signed_take_bit n k\<close> if \<open>k < - (2 ^ n)\<close>
+ for k :: int
using that take_bit_greater_eq [of \<open>k + 2 ^ n\<close> \<open>Suc n\<close>]
by (simp add: signed_take_bit_eq_take_bit_shift)
lemma signed_take_bit_less_eq:
\<open>signed_take_bit n k \<le> k - 2 ^ Suc n\<close> if \<open>k \<ge> 2 ^ n\<close>
+ for k :: int
using that take_bit_less_eq [of \<open>Suc n\<close> \<open>k + 2 ^ n\<close>]
by (simp add: signed_take_bit_eq_take_bit_shift)
lemma signed_take_bit_eq_self:
\<open>signed_take_bit n k = k\<close> if \<open>- (2 ^ n) \<le> k\<close> \<open>k < 2 ^ n\<close>
+ for k :: int
using that by (simp add: signed_take_bit_eq_take_bit_shift take_bit_int_eq_self)
-lemma signed_take_bit_Suc_1 [simp]:
- \<open>signed_take_bit (Suc n) 1 = 1\<close>
- by (simp add: signed_take_bit_Suc)
-
lemma signed_take_bit_Suc_bit0 [simp]:
- \<open>signed_take_bit (Suc n) (numeral (Num.Bit0 k)) = signed_take_bit n (numeral k) * 2\<close>
+ \<open>signed_take_bit (Suc n) (numeral (Num.Bit0 k)) = signed_take_bit n (numeral k) * (2 :: int)\<close>
by (simp add: signed_take_bit_Suc)
lemma signed_take_bit_Suc_bit1 [simp]:
- \<open>signed_take_bit (Suc n) (numeral (Num.Bit1 k)) = signed_take_bit n (numeral k) * 2 + 1\<close>
+ \<open>signed_take_bit (Suc n) (numeral (Num.Bit1 k)) = signed_take_bit n (numeral k) * 2 + (1 :: int)\<close>
by (simp add: signed_take_bit_Suc)
lemma signed_take_bit_Suc_minus_bit0 [simp]:
- \<open>signed_take_bit (Suc n) (- numeral (Num.Bit0 k)) = signed_take_bit n (- numeral k) * 2\<close>
+ \<open>signed_take_bit (Suc n) (- numeral (Num.Bit0 k)) = signed_take_bit n (- numeral k) * (2 :: int)\<close>
by (simp add: signed_take_bit_Suc)
lemma signed_take_bit_Suc_minus_bit1 [simp]:
- \<open>signed_take_bit (Suc n) (- numeral (Num.Bit1 k)) = signed_take_bit n (- numeral k - 1) * 2 + 1\<close>
+ \<open>signed_take_bit (Suc n) (- numeral (Num.Bit1 k)) = signed_take_bit n (- numeral k - 1) * 2 + (1 :: int)\<close>
by (simp add: signed_take_bit_Suc)
lemma signed_take_bit_numeral_bit0 [simp]:
- \<open>signed_take_bit (numeral l) (numeral (Num.Bit0 k)) = signed_take_bit (pred_numeral l) (numeral k) * 2\<close>
+ \<open>signed_take_bit (numeral l) (numeral (Num.Bit0 k)) = signed_take_bit (pred_numeral l) (numeral k) * (2 :: int)\<close>
by (simp add: signed_take_bit_rec)
lemma signed_take_bit_numeral_bit1 [simp]:
- \<open>signed_take_bit (numeral l) (numeral (Num.Bit1 k)) = signed_take_bit (pred_numeral l) (numeral k) * 2 + 1\<close>
+ \<open>signed_take_bit (numeral l) (numeral (Num.Bit1 k)) = signed_take_bit (pred_numeral l) (numeral k) * 2 + (1 :: int)\<close>
by (simp add: signed_take_bit_rec)
lemma signed_take_bit_numeral_minus_bit0 [simp]:
- \<open>signed_take_bit (numeral l) (- numeral (Num.Bit0 k)) = signed_take_bit (pred_numeral l) (- numeral k) * 2\<close>
+ \<open>signed_take_bit (numeral l) (- numeral (Num.Bit0 k)) = signed_take_bit (pred_numeral l) (- numeral k) * (2 :: int)\<close>
by (simp add: signed_take_bit_rec)
lemma signed_take_bit_numeral_minus_bit1 [simp]:
- \<open>signed_take_bit (numeral l) (- numeral (Num.Bit1 k)) = signed_take_bit (pred_numeral l) (- numeral k - 1) * 2 + 1\<close>
+ \<open>signed_take_bit (numeral l) (- numeral (Num.Bit1 k)) = signed_take_bit (pred_numeral l) (- numeral k - 1) * 2 + (1 :: int)\<close>
by (simp add: signed_take_bit_rec)
lemma signed_take_bit_code [code]:
- \<open>signed_take_bit n k =
- (let l = take_bit (Suc n) k
- in if bit l n then l - (push_bit n 2) else l)\<close>
+ \<open>signed_take_bit n a =
+ (let l = take_bit (Suc n) a
+ in if bit l n then l + push_bit (Suc n) (- 1) else l)\<close>
proof -
- have *: \<open>take_bit (Suc n) k - 2 * 2 ^ n = take_bit (Suc n) k OR NOT (mask (Suc n))\<close>
- apply (subst disjunctive_add [symmetric])
- apply (simp_all add: bit_and_iff bit_mask_iff bit_not_iff bit_take_bit_iff)
- apply (simp flip: minus_exp_eq_not_mask)
- done
+ have *: \<open>take_bit (Suc n) a + push_bit n (- 2) =
+ take_bit (Suc n) a OR NOT (mask (Suc n))\<close>
+ by (auto simp add: bit_take_bit_iff bit_push_bit_iff bit_not_iff bit_mask_iff disjunctive_add
+ simp flip: push_bit_minus_one_eq_not_mask)
show ?thesis
by (rule bit_eqI)
- (auto simp add: Let_def bit_and_iff bit_signed_take_bit_iff push_bit_eq_mult min_def not_le
- bit_mask_iff bit_exp_iff less_Suc_eq * bit_or_iff bit_take_bit_iff bit_not_iff)
+ (auto simp add: Let_def * bit_signed_take_bit_iff bit_take_bit_iff min_def less_Suc_eq bit_not_iff bit_mask_iff bit_or_iff)
qed
@@ -1358,9 +1389,9 @@
\<^item> Flip a single bit: @{thm flip_bit_def [where ?'a = int, no_vars]}
- \<^item> Bit concatenation: @{thm concat_bit_def [no_vars]}
+ \<^item> Signed truncation, or modulus centered around \<^term>\<open>0::int\<close>: @{thm signed_take_bit_def [no_vars]}
- \<^item> Signed truncation, or modulus centered around \<^term>\<open>0::int\<close>: @{thm signed_take_bit_def [no_vars]}
+ \<^item> Bit concatenation: @{thm concat_bit_def [no_vars]}
\<^item> (Bounded) conversion from and to a list of bits: @{thm horner_sum_bit_eq_take_bit [where ?'a = int, no_vars]}
\<close>