--- a/src/HOL/Library/Bit_Operations.thy Sun Nov 15 13:08:13 2020 +0000
+++ b/src/HOL/Library/Bit_Operations.thy Sun Nov 15 10:13:03 2020 +0000
@@ -16,9 +16,9 @@
and or :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixr \<open>OR\<close> 59)
and xor :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixr \<open>XOR\<close> 59)
and mask :: \<open>nat \<Rightarrow> 'a\<close>
- assumes bit_and_iff: \<open>\<And>n. bit (a AND b) n \<longleftrightarrow> bit a n \<and> bit b n\<close>
- and bit_or_iff: \<open>\<And>n. bit (a OR b) n \<longleftrightarrow> bit a n \<or> bit b n\<close>
- and bit_xor_iff: \<open>\<And>n. bit (a XOR b) n \<longleftrightarrow> bit a n \<noteq> bit b n\<close>
+ assumes bit_and_iff [bit_simps]: \<open>\<And>n. bit (a AND b) n \<longleftrightarrow> bit a n \<and> bit b n\<close>
+ and bit_or_iff [bit_simps]: \<open>\<And>n. bit (a OR b) n \<longleftrightarrow> bit a n \<or> bit b n\<close>
+ and bit_xor_iff [bit_simps]: \<open>\<And>n. bit (a XOR b) n \<longleftrightarrow> bit a n \<noteq> bit b n\<close>
and mask_eq_exp_minus_1: \<open>mask n = 2 ^ n - 1\<close>
begin
@@ -119,7 +119,7 @@
\<open>drop_bit n (a XOR b) = drop_bit n a XOR drop_bit n b\<close>
by (rule bit_eqI) (auto simp add: bit_drop_bit_eq bit_xor_iff)
-lemma bit_mask_iff:
+lemma bit_mask_iff [bit_simps]:
\<open>bit (mask m) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n < m\<close>
by (simp add: mask_eq_exp_minus_1 bit_mask_iff)
@@ -182,7 +182,7 @@
class ring_bit_operations = semiring_bit_operations + ring_parity +
fixes not :: \<open>'a \<Rightarrow> 'a\<close> (\<open>NOT\<close>)
- assumes bit_not_iff: \<open>\<And>n. bit (NOT a) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> \<not> bit a n\<close>
+ assumes bit_not_iff [bit_simps]: \<open>\<And>n. bit (NOT a) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> \<not> bit a n\<close>
assumes minus_eq_not_minus_1: \<open>- a = NOT (a - 1)\<close>
begin
@@ -205,7 +205,7 @@
\<open>- a = NOT a + 1\<close>
using not_eq_complement [of a] by simp
-lemma bit_minus_iff:
+lemma bit_minus_iff [bit_simps]:
\<open>bit (- a) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> \<not> bit (a - 1) n\<close>
by (simp add: minus_eq_not_minus_1 bit_not_iff)
@@ -213,7 +213,7 @@
"even (NOT a) \<longleftrightarrow> odd a"
using bit_not_iff [of a 0] by auto
-lemma bit_not_exp_iff:
+lemma bit_not_exp_iff [bit_simps]:
\<open>bit (NOT (2 ^ m)) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n \<noteq> m\<close>
by (auto simp add: bit_not_iff bit_exp_iff)
@@ -221,9 +221,9 @@
\<open>bit (- 1) n \<longleftrightarrow> 2 ^ n \<noteq> 0\<close>
by (simp add: bit_minus_iff)
-lemma bit_minus_exp_iff:
+lemma bit_minus_exp_iff [bit_simps]:
\<open>bit (- (2 ^ m)) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n \<ge> m\<close>
- oops
+ by (auto simp add: bit_simps simp flip: mask_eq_exp_minus_1)
lemma bit_minus_2_iff [simp]:
\<open>bit (- 2) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n > 0\<close>
@@ -361,7 +361,7 @@
definition flip_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
where \<open>flip_bit n a = a XOR push_bit n 1\<close>
-lemma bit_set_bit_iff:
+lemma bit_set_bit_iff [bit_simps]:
\<open>bit (set_bit m a) n \<longleftrightarrow> bit a n \<or> (m = n \<and> 2 ^ n \<noteq> 0)\<close>
by (auto simp add: set_bit_def push_bit_of_1 bit_or_iff bit_exp_iff)
@@ -369,7 +369,7 @@
\<open>even (set_bit m a) \<longleftrightarrow> even a \<and> m \<noteq> 0\<close>
using bit_set_bit_iff [of m a 0] by auto
-lemma bit_unset_bit_iff:
+lemma bit_unset_bit_iff [bit_simps]:
\<open>bit (unset_bit m a) n \<longleftrightarrow> bit a n \<and> m \<noteq> n\<close>
by (auto simp add: unset_bit_def push_bit_of_1 bit_and_iff bit_not_iff bit_exp_iff exp_eq_0_imp_not_bit)
@@ -377,7 +377,7 @@
\<open>even (unset_bit m a) \<longleftrightarrow> even a \<or> m = 0\<close>
using bit_unset_bit_iff [of m a 0] by auto
-lemma bit_flip_bit_iff:
+lemma bit_flip_bit_iff [bit_simps]:
\<open>bit (flip_bit m a) n \<longleftrightarrow> (m = n \<longleftrightarrow> \<not> bit a n) \<and> 2 ^ n \<noteq> 0\<close>
by (auto simp add: flip_bit_def push_bit_of_1 bit_xor_iff bit_exp_iff exp_eq_0_imp_not_bit)
@@ -583,7 +583,7 @@
\<open>NOT k div 2 = NOT (k div 2)\<close> for k :: int
by (simp add: not_int_def)
-lemma bit_not_int_iff:
+lemma bit_not_int_iff [bit_simps]:
\<open>bit (NOT k) n \<longleftrightarrow> \<not> bit k n\<close>
for k :: int
by (simp add: bit_not_int_iff' not_int_def)
@@ -973,7 +973,7 @@
\<open>even (of_int k) \<longleftrightarrow> even k\<close>
by (induction k rule: int_bit_induct) simp_all
-lemma bit_of_int_iff:
+lemma bit_of_int_iff [bit_simps]:
\<open>bit (of_int k) n \<longleftrightarrow> (2::'a) ^ n \<noteq> 0 \<and> bit k n\<close>
proof (cases \<open>(2::'a) ^ n = 0\<close>)
case True
@@ -1157,7 +1157,7 @@
definition concat_bit :: \<open>nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> int\<close>
where \<open>concat_bit n k l = take_bit n k OR push_bit n l\<close>
-lemma bit_concat_bit_iff:
+lemma bit_concat_bit_iff [bit_simps]:
\<open>bit (concat_bit m k l) n \<longleftrightarrow> n < m \<and> bit k n \<or> m \<le> n \<and> bit l (n - m)\<close>
by (simp add: concat_bit_def bit_or_iff bit_and_iff bit_take_bit_iff bit_push_bit_iff ac_simps)
@@ -1259,7 +1259,7 @@
\<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:
+lemma bit_signed_take_bit_iff [bit_simps]:
\<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)
@@ -1560,11 +1560,11 @@
instance proof
fix m n q :: nat
show \<open>bit (m AND n) q \<longleftrightarrow> bit m q \<and> bit n q\<close>
- by (auto simp add: bit_nat_iff and_nat_def bit_and_iff less_le bit_eq_iff)
+ by (simp add: and_nat_def bit_simps)
show \<open>bit (m OR n) q \<longleftrightarrow> bit m q \<or> bit n q\<close>
- by (auto simp add: bit_nat_iff or_nat_def bit_or_iff less_le bit_eq_iff)
+ by (simp add: or_nat_def bit_simps)
show \<open>bit (m XOR n) q \<longleftrightarrow> bit m q \<noteq> bit n q\<close>
- by (auto simp add: bit_nat_iff xor_nat_def bit_xor_iff less_le bit_eq_iff)
+ by (simp add: xor_nat_def bit_simps)
qed (simp add: mask_nat_def)
end