--- a/src/HOL/ex/Bit_Operations.thy Fri May 08 06:26:28 2020 +0000
+++ b/src/HOL/ex/Bit_Operations.thy Fri May 08 06:26:29 2020 +0000
@@ -37,6 +37,18 @@
sublocale xor: comm_monoid \<open>(XOR)\<close> 0
by standard (auto simp add: bit_eq_iff bit_xor_iff)
+lemma even_and_iff:
+ \<open>even (a AND b) \<longleftrightarrow> even a \<or> even b\<close>
+ using bit_and_iff [of a b 0] by auto
+
+lemma even_or_iff:
+ \<open>even (a OR b) \<longleftrightarrow> even a \<and> even b\<close>
+ using bit_or_iff [of a b 0] by auto
+
+lemma even_xor_iff:
+ \<open>even (a XOR b) \<longleftrightarrow> (even a \<longleftrightarrow> even b)\<close>
+ using bit_xor_iff [of a b 0] by auto
+
lemma zero_and_eq [simp]:
"0 AND a = 0"
by (simp add: bit_eq_iff bit_and_iff)
@@ -81,6 +93,41 @@
\<open>take_bit n (a XOR b) = take_bit n a XOR take_bit n b\<close>
by (auto simp add: bit_eq_iff bit_take_bit_iff bit_xor_iff)
+definition mask :: \<open>nat \<Rightarrow> 'a\<close>
+ where mask_eq_exp_minus_1: \<open>mask n = 2 ^ n - 1\<close>
+
+lemma bit_mask_iff:
+ \<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)
+
+lemma even_mask_iff:
+ \<open>even (mask n) \<longleftrightarrow> n = 0\<close>
+ using bit_mask_iff [of n 0] by auto
+
+lemma mask_0 [simp, code]:
+ \<open>mask 0 = 0\<close>
+ by (simp add: mask_eq_exp_minus_1)
+
+lemma mask_Suc_exp [code]:
+ \<open>mask (Suc n) = 2 ^ n OR mask n\<close>
+ by (rule bit_eqI)
+ (auto simp add: bit_or_iff bit_mask_iff bit_exp_iff not_less le_less_Suc_eq)
+
+lemma mask_Suc_double:
+ \<open>mask (Suc n) = 2 * mask n OR 1\<close>
+proof (rule bit_eqI)
+ fix q
+ assume \<open>2 ^ q \<noteq> 0\<close>
+ show \<open>bit (mask (Suc n)) q \<longleftrightarrow> bit (2 * mask n OR 1) q\<close>
+ by (cases q)
+ (simp_all add: even_mask_iff even_or_iff bit_or_iff bit_mask_iff bit_exp_iff bit_double_iff not_less le_less_Suc_eq bit_1_iff, auto simp add: mult_2)
+qed
+
+lemma take_bit_eq_mask [code]:
+ \<open>take_bit n a = a AND mask n\<close>
+ by (rule bit_eqI)
+ (auto simp add: bit_take_bit_iff bit_and_iff bit_mask_iff)
+
end
class ring_bit_operations = semiring_bit_operations + ring_parity +
@@ -679,7 +726,7 @@
\<^item> Singleton \<^term>\<open>n\<close>th bit: \<^term>\<open>(2 :: int) ^ n\<close>
- \<^item> Bit mask upto bit \<^term>\<open>n\<close>: \<^term>\<open>(2 :: int) ^ n - 1\<close>
+ \<^item> Bit mask upto bit \<^term>\<open>n\<close>: @{thm mask_eq_exp_minus_1 [where ?'a = int, no_vars]}}
\<^item> Left shift: @{thm push_bit_eq_mult [where ?'a = int, no_vars]}