diff --git a/src/HOL/ex/Bit_Operations.thy b/src/HOL/ex/Bit_Operations.thy
--- a/src/HOL/ex/Bit_Operations.thy
+++ b/src/HOL/ex/Bit_Operations.thy
@@ -1,752 +1,780 @@
 (*  Author:  Florian Haftmann, TUM
 *)
 
 section \<open>Proof of concept for purely algebraically founded lists of bits\<close>
 
 theory Bit_Operations
   imports
     "HOL-Library.Boolean_Algebra"
     Main
 begin
 
 subsection \<open>Bit operations in suitable algebraic structures\<close>
 
 class semiring_bit_operations = semiring_bit_shifts +
   fixes "and" :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close>  (infixr \<open>AND\<close> 64)
     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)
   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>
 begin
 
 text \<open>
   We want the bitwise operations to bind slightly weaker
   than \<open>+\<close> and \<open>-\<close>.
   For the sake of code generation
   the operations \<^const>\<open>and\<close>, \<^const>\<open>or\<close> and \<^const>\<open>xor\<close>
   are specified as definitional class operations.
 \<close>
 
 sublocale "and": semilattice \<open>(AND)\<close>
   by standard (auto simp add: bit_eq_iff bit_and_iff)
 
 sublocale or: semilattice_neutr \<open>(OR)\<close> 0
   by standard (auto simp add: bit_eq_iff bit_or_iff)
 
 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)
 
 lemma and_zero_eq [simp]:
   "a AND 0 = 0"
   by (simp add: bit_eq_iff bit_and_iff)
 
 lemma one_and_eq:
   "1 AND a = a mod 2"
   by (simp add: bit_eq_iff bit_and_iff) (auto simp add: bit_1_iff)
 
 lemma and_one_eq:
   "a AND 1 = a mod 2"
   using one_and_eq [of a] by (simp add: ac_simps)
 
 lemma one_or_eq:
   "1 OR a = a + of_bool (even a)"
   by (simp add: bit_eq_iff bit_or_iff add.commute [of _ 1] even_bit_succ_iff) (auto simp add: bit_1_iff)
 
 lemma or_one_eq:
   "a OR 1 = a + of_bool (even a)"
   using one_or_eq [of a] by (simp add: ac_simps)
 
 lemma one_xor_eq:
   "1 XOR a = a + of_bool (even a) - of_bool (odd a)"
   by (simp add: bit_eq_iff bit_xor_iff add.commute [of _ 1] even_bit_succ_iff) (auto simp add: bit_1_iff odd_bit_iff_bit_pred elim: oddE)
 
 lemma xor_one_eq:
   "a XOR 1 = a + of_bool (even a) - of_bool (odd a)"
   using one_xor_eq [of a] by (simp add: ac_simps)
 
 lemma take_bit_and [simp]:
   \<open>take_bit n (a AND b) = take_bit n a AND take_bit n b\<close>
   by (auto simp add: bit_eq_iff bit_take_bit_iff bit_and_iff)
 
 lemma take_bit_or [simp]:
   \<open>take_bit n (a OR b) = take_bit n a OR take_bit n b\<close>
   by (auto simp add: bit_eq_iff bit_take_bit_iff bit_or_iff)
 
 lemma take_bit_xor [simp]:
   \<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 +
   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 minus_eq_not_minus_1: \<open>- a = NOT (a - 1)\<close>
 begin
 
 text \<open>
   For the sake of code generation \<^const>\<open>not\<close> is specified as
   definitional class operation.  Note that \<^const>\<open>not\<close> has no
   sensible definition for unlimited but only positive bit strings
   (type \<^typ>\<open>nat\<close>).
 \<close>
 
 lemma bits_minus_1_mod_2_eq [simp]:
   \<open>(- 1) mod 2 = 1\<close>
   by (simp add: mod_2_eq_odd)
 
 lemma not_eq_complement:
   \<open>NOT a = - a - 1\<close>
   using minus_eq_not_minus_1 [of \<open>a + 1\<close>] by simp
 
 lemma minus_eq_not_plus_1:
   \<open>- a = NOT a + 1\<close>
   using not_eq_complement [of a] by simp
 
 lemma bit_minus_iff:
   \<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)
 
 lemma even_not_iff [simp]:
   "even (NOT a) \<longleftrightarrow> odd a"
   using bit_not_iff [of a 0] by auto
 
 lemma bit_not_exp_iff:
   \<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)
 
 lemma bit_minus_1_iff [simp]:
   \<open>bit (- 1) n \<longleftrightarrow> 2 ^ n \<noteq> 0\<close>
   by (simp add: bit_minus_iff)
 
 lemma bit_minus_exp_iff:
   \<open>bit (- (2 ^ m)) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n \<ge> m\<close>
   oops
 
 lemma bit_minus_2_iff [simp]:
   \<open>bit (- 2) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n > 0\<close>
   by (simp add: bit_minus_iff bit_1_iff)
 
 lemma not_one [simp]:
   "NOT 1 = - 2"
   by (simp add: bit_eq_iff bit_not_iff) (simp add: bit_1_iff)
 
 sublocale "and": semilattice_neutr \<open>(AND)\<close> \<open>- 1\<close>
   apply standard
   apply (simp add: bit_eq_iff bit_and_iff)
   apply (auto simp add: exp_eq_0_imp_not_bit bit_exp_iff)
   done
 
 sublocale bit: boolean_algebra \<open>(AND)\<close> \<open>(OR)\<close> NOT 0 \<open>- 1\<close>
   rewrites \<open>bit.xor = (XOR)\<close>
 proof -
   interpret bit: boolean_algebra \<open>(AND)\<close> \<open>(OR)\<close> NOT 0 \<open>- 1\<close>
     apply standard
          apply (simp_all add: bit_eq_iff bit_and_iff bit_or_iff bit_not_iff)
       apply (auto simp add: exp_eq_0_imp_not_bit bit_exp_iff)
     done
   show \<open>boolean_algebra (AND) (OR) NOT 0 (- 1)\<close>
     by standard
   show \<open>boolean_algebra.xor (AND) (OR) NOT = (XOR)\<close>
     apply (auto simp add: fun_eq_iff bit.xor_def bit_eq_iff bit_and_iff bit_or_iff bit_not_iff bit_xor_iff)
          apply (simp_all add: bit_exp_iff, simp_all add: bit_def)
         apply (metis local.bit_exp_iff local.bits_div_by_0)
        apply (metis local.bit_exp_iff local.bits_div_by_0)
     done
 qed
 
 lemma and_eq_not_not_or:
   \<open>a AND b = NOT (NOT a OR NOT b)\<close>
   by simp
 
 lemma or_eq_not_not_and:
   \<open>a OR b = NOT (NOT a AND NOT b)\<close>
   by simp
 
 lemma push_bit_minus:
   \<open>push_bit n (- a) = - push_bit n a\<close>
   by (simp add: push_bit_eq_mult)
 
 lemma take_bit_not_take_bit:
   \<open>take_bit n (NOT (take_bit n a)) = take_bit n (NOT a)\<close>
   by (auto simp add: bit_eq_iff bit_take_bit_iff bit_not_iff)
 
 lemma take_bit_not_iff:
   "take_bit n (NOT a) = take_bit n (NOT b) \<longleftrightarrow> take_bit n a = take_bit n b"
   apply (simp add: bit_eq_iff bit_not_iff bit_take_bit_iff)
   apply (simp add: bit_exp_iff)
   apply (use local.exp_eq_0_imp_not_bit in blast)
   done
 
+lemma take_bit_minus_one_eq_mask:
+  \<open>take_bit n (- 1) = mask n\<close>
+  by (simp add: bit_eq_iff bit_mask_iff bit_take_bit_iff conj_commute)
+
+lemma push_bit_minus_one_eq_not_mask:
+  \<open>push_bit n (- 1) = NOT (mask n)\<close>
+proof (rule bit_eqI)
+  fix m
+  assume \<open>2 ^ m \<noteq> 0\<close>
+  show \<open>bit (push_bit n (- 1)) m \<longleftrightarrow> bit (NOT (mask n)) m\<close>
+  proof (cases \<open>n \<le> m\<close>)
+    case True
+    moreover define q where \<open>q = m - n\<close>
+    ultimately have \<open>m = n + q\<close> \<open>m - n = q\<close>
+      by simp_all
+    with \<open>2 ^ m \<noteq> 0\<close> have \<open>2 ^ n * 2 ^ q \<noteq> 0\<close>
+      by (simp add: power_add)
+    then have \<open>2 ^ q \<noteq> 0\<close>
+      using mult_not_zero by blast
+    with \<open>m - n = q\<close> show ?thesis
+      by (auto simp add: bit_not_iff bit_mask_iff bit_push_bit_iff not_less)
+  next
+    case False
+    then show ?thesis
+      by (simp add: bit_not_iff bit_mask_iff bit_push_bit_iff not_le)
+  qed
+qed
+
 definition set_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   where \<open>set_bit n a = a OR 2 ^ n\<close>
 
 definition unset_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   where \<open>unset_bit n a = a AND NOT (2 ^ n)\<close>
 
 definition flip_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   where \<open>flip_bit n a = a XOR 2 ^ n\<close>
 
 lemma bit_set_bit_iff:
   \<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 bit_or_iff bit_exp_iff)
 
 lemma even_set_bit_iff:
   \<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:
   \<open>bit (unset_bit m a) n \<longleftrightarrow> bit a n \<and> m \<noteq> n\<close>
   by (auto simp add: unset_bit_def bit_and_iff bit_not_iff bit_exp_iff exp_eq_0_imp_not_bit)
 
 lemma even_unset_bit_iff:
   \<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:
   \<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 bit_xor_iff bit_exp_iff exp_eq_0_imp_not_bit)
 
 lemma even_flip_bit_iff:
   \<open>even (flip_bit m a) \<longleftrightarrow> \<not> (even a \<longleftrightarrow> m = 0)\<close>
   using bit_flip_bit_iff [of m a 0] by auto
 
 lemma set_bit_0 [simp]:
   \<open>set_bit 0 a = 1 + 2 * (a div 2)\<close>
 proof (rule bit_eqI)
   fix m
   assume *: \<open>2 ^ m \<noteq> 0\<close>
   then show \<open>bit (set_bit 0 a) m = bit (1 + 2 * (a div 2)) m\<close>
     by (simp add: bit_set_bit_iff bit_double_iff even_bit_succ_iff)
       (cases m, simp_all add: bit_Suc)
 qed
 
 lemma set_bit_Suc:
   \<open>set_bit (Suc n) a = a mod 2 + 2 * set_bit n (a div 2)\<close>
 proof (rule bit_eqI)
   fix m
   assume *: \<open>2 ^ m \<noteq> 0\<close>
   show \<open>bit (set_bit (Suc n) a) m \<longleftrightarrow> bit (a mod 2 + 2 * set_bit n (a div 2)) m\<close>
   proof (cases m)
     case 0
     then show ?thesis
       by (simp add: even_set_bit_iff)
   next
     case (Suc m)
     with * have \<open>2 ^ m \<noteq> 0\<close>
       using mult_2 by auto
     show ?thesis
       by (cases a rule: parity_cases)
         (simp_all add: bit_set_bit_iff bit_double_iff even_bit_succ_iff *,
         simp_all add: Suc \<open>2 ^ m \<noteq> 0\<close> bit_Suc)
   qed
 qed
 
 lemma unset_bit_0 [simp]:
   \<open>unset_bit 0 a = 2 * (a div 2)\<close>
 proof (rule bit_eqI)
   fix m
   assume *: \<open>2 ^ m \<noteq> 0\<close>
   then show \<open>bit (unset_bit 0 a) m = bit (2 * (a div 2)) m\<close>
     by (simp add: bit_unset_bit_iff bit_double_iff)
       (cases m, simp_all add: bit_Suc)
 qed
 
 lemma unset_bit_Suc:
   \<open>unset_bit (Suc n) a = a mod 2 + 2 * unset_bit n (a div 2)\<close>
 proof (rule bit_eqI)
   fix m
   assume *: \<open>2 ^ m \<noteq> 0\<close>
   then show \<open>bit (unset_bit (Suc n) a) m \<longleftrightarrow> bit (a mod 2 + 2 * unset_bit n (a div 2)) m\<close>
   proof (cases m)
     case 0
     then show ?thesis
       by (simp add: even_unset_bit_iff)
   next
     case (Suc m)
     show ?thesis
       by (cases a rule: parity_cases)
         (simp_all add: bit_unset_bit_iff bit_double_iff even_bit_succ_iff *,
          simp_all add: Suc bit_Suc)
   qed
 qed
 
 lemma flip_bit_0 [simp]:
   \<open>flip_bit 0 a = of_bool (even a) + 2 * (a div 2)\<close>
 proof (rule bit_eqI)
   fix m
   assume *: \<open>2 ^ m \<noteq> 0\<close>
   then show \<open>bit (flip_bit 0 a) m = bit (of_bool (even a) + 2 * (a div 2)) m\<close>
     by (simp add: bit_flip_bit_iff bit_double_iff even_bit_succ_iff)
       (cases m, simp_all add: bit_Suc)
 qed
 
 lemma flip_bit_Suc:
   \<open>flip_bit (Suc n) a = a mod 2 + 2 * flip_bit n (a div 2)\<close>
 proof (rule bit_eqI)
   fix m
   assume *: \<open>2 ^ m \<noteq> 0\<close>
   show \<open>bit (flip_bit (Suc n) a) m \<longleftrightarrow> bit (a mod 2 + 2 * flip_bit n (a div 2)) m\<close>
   proof (cases m)
     case 0
     then show ?thesis
       by (simp add: even_flip_bit_iff)
   next
     case (Suc m)
     with * have \<open>2 ^ m \<noteq> 0\<close>
       using mult_2 by auto
     show ?thesis
       by (cases a rule: parity_cases)
         (simp_all add: bit_flip_bit_iff bit_double_iff even_bit_succ_iff,
         simp_all add: Suc \<open>2 ^ m \<noteq> 0\<close> bit_Suc)
   qed
 qed
 
 end
 
 
 subsubsection \<open>Instance \<^typ>\<open>int\<close>\<close>
 
 instantiation int :: ring_bit_operations
 begin
 
 definition not_int :: \<open>int \<Rightarrow> int\<close>
   where \<open>not_int k = - k - 1\<close>
 
 lemma not_int_rec:
   "NOT k = of_bool (even k) + 2 * NOT (k div 2)" for k :: int
   by (auto simp add: not_int_def elim: oddE)
 
 lemma even_not_iff_int:
   \<open>even (NOT k) \<longleftrightarrow> odd k\<close> for k :: int
   by (simp add: not_int_def)
 
 lemma not_int_div_2:
   \<open>NOT k div 2 = NOT (k div 2)\<close> for k :: int
   by (simp add: not_int_def)
 
 lemma bit_not_int_iff:
   \<open>bit (NOT k) n \<longleftrightarrow> \<not> bit k n\<close>
     for k :: int
   by (induction n arbitrary: k) (simp_all add: not_int_div_2 even_not_iff_int bit_Suc)
 
 function and_int :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>(k::int) AND l = (if k \<in> {0, - 1} \<and> l \<in> {0, - 1}
     then - of_bool (odd k \<and> odd l)
     else of_bool (odd k \<and> odd l) + 2 * ((k div 2) AND (l div 2)))\<close>
   by auto
 
 termination
   by (relation \<open>measure (\<lambda>(k, l). nat (\<bar>k\<bar> + \<bar>l\<bar>))\<close>) auto
 
 declare and_int.simps [simp del]
 
 lemma and_int_rec:
   \<open>k AND l = of_bool (odd k \<and> odd l) + 2 * ((k div 2) AND (l div 2))\<close>
     for k l :: int
 proof (cases \<open>k \<in> {0, - 1} \<and> l \<in> {0, - 1}\<close>)
   case True
   then show ?thesis
     by auto (simp_all add: and_int.simps)
 next
   case False
   then show ?thesis
     by (auto simp add: ac_simps and_int.simps [of k l])
 qed
 
 lemma bit_and_int_iff:
   \<open>bit (k AND l) n \<longleftrightarrow> bit k n \<and> bit l n\<close> for k l :: int
 proof (induction n arbitrary: k l)
   case 0
   then show ?case
     by (simp add: and_int_rec [of k l])
 next
   case (Suc n)
   then show ?case
     by (simp add: and_int_rec [of k l] bit_Suc)
 qed
 
 lemma even_and_iff_int:
   \<open>even (k AND l) \<longleftrightarrow> even k \<or> even l\<close> for k l :: int
   using bit_and_int_iff [of k l 0] by auto
 
 definition or_int :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>k OR l = NOT (NOT k AND NOT l)\<close> for k l :: int
 
 lemma or_int_rec:
   \<open>k OR l = of_bool (odd k \<or> odd l) + 2 * ((k div 2) OR (l div 2))\<close>
   for k l :: int
   using and_int_rec [of \<open>NOT k\<close> \<open>NOT l\<close>]
   by (simp add: or_int_def even_not_iff_int not_int_div_2)
     (simp add: not_int_def)
 
 lemma bit_or_int_iff:
   \<open>bit (k OR l) n \<longleftrightarrow> bit k n \<or> bit l n\<close> for k l :: int
   by (simp add: or_int_def bit_not_int_iff bit_and_int_iff)
 
 definition xor_int :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>k XOR l = k AND NOT l OR NOT k AND l\<close> for k l :: int
 
 lemma xor_int_rec:
   \<open>k XOR l = of_bool (odd k \<noteq> odd l) + 2 * ((k div 2) XOR (l div 2))\<close>
   for k l :: int
   by (simp add: xor_int_def or_int_rec [of \<open>k AND NOT l\<close> \<open>NOT k AND l\<close>] even_and_iff_int even_not_iff_int)
     (simp add: and_int_rec [of \<open>NOT k\<close> \<open>l\<close>] and_int_rec [of \<open>k\<close> \<open>NOT l\<close>] not_int_div_2)
 
 lemma bit_xor_int_iff:
   \<open>bit (k XOR l) n \<longleftrightarrow> bit k n \<noteq> bit l n\<close> for k l :: int
   by (auto simp add: xor_int_def bit_or_int_iff bit_and_int_iff bit_not_int_iff)
 
 instance proof
   fix k l :: int and n :: nat
   show \<open>- k = NOT (k - 1)\<close>
     by (simp add: not_int_def)
   show \<open>bit (k AND l) n \<longleftrightarrow> bit k n \<and> bit l n\<close>
     by (fact bit_and_int_iff)
   show \<open>bit (k OR l) n \<longleftrightarrow> bit k n \<or> bit l n\<close>
     by (fact bit_or_int_iff)
   show \<open>bit (k XOR l) n \<longleftrightarrow> bit k n \<noteq> bit l n\<close>
     by (fact bit_xor_int_iff)
 qed (simp_all add: bit_not_int_iff)
 
 end
 
 lemma not_nonnegative_int_iff [simp]:
   \<open>NOT k \<ge> 0 \<longleftrightarrow> k < 0\<close> for k :: int
   by (simp add: not_int_def)
 
 lemma not_negative_int_iff [simp]:
   \<open>NOT k < 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int
   by (subst Not_eq_iff [symmetric]) (simp add: not_less not_le)
 
 lemma and_nonnegative_int_iff [simp]:
   \<open>k AND l \<ge> 0 \<longleftrightarrow> k \<ge> 0 \<or> l \<ge> 0\<close> for k l :: int
 proof (induction k arbitrary: l rule: int_bit_induct)
   case zero
   then show ?case
     by simp
 next
   case minus
   then show ?case
     by simp
 next
   case (even k)
   then show ?case
     using and_int_rec [of \<open>k * 2\<close> l] by (simp add: pos_imp_zdiv_nonneg_iff)
 next
   case (odd k)
   from odd have \<open>0 \<le> k AND l div 2 \<longleftrightarrow> 0 \<le> k \<or> 0 \<le> l div 2\<close>
     by simp
   then have \<open>0 \<le> (1 + k * 2) div 2 AND l div 2 \<longleftrightarrow> 0 \<le> (1 + k * 2) div 2\<or> 0 \<le> l div 2\<close>
     by simp
   with and_int_rec [of \<open>1 + k * 2\<close> l]
   show ?case
     by auto
 qed
 
 lemma and_negative_int_iff [simp]:
   \<open>k AND l < 0 \<longleftrightarrow> k < 0 \<and> l < 0\<close> for k l :: int
   by (subst Not_eq_iff [symmetric]) (simp add: not_less)
 
 lemma or_nonnegative_int_iff [simp]:
   \<open>k OR l \<ge> 0 \<longleftrightarrow> k \<ge> 0 \<and> l \<ge> 0\<close> for k l :: int
   by (simp only: or_eq_not_not_and not_nonnegative_int_iff) simp
 
 lemma or_negative_int_iff [simp]:
   \<open>k OR l < 0 \<longleftrightarrow> k < 0 \<or> l < 0\<close> for k l :: int
   by (subst Not_eq_iff [symmetric]) (simp add: not_less)
 
 lemma xor_nonnegative_int_iff [simp]:
   \<open>k XOR l \<ge> 0 \<longleftrightarrow> (k \<ge> 0 \<longleftrightarrow> l \<ge> 0)\<close> for k l :: int
   by (simp only: bit.xor_def or_nonnegative_int_iff) auto
 
 lemma xor_negative_int_iff [simp]:
   \<open>k XOR l < 0 \<longleftrightarrow> (k < 0) \<noteq> (l < 0)\<close> for k l :: int
   by (subst Not_eq_iff [symmetric]) (auto simp add: not_less)
 
 lemma set_bit_nonnegative_int_iff [simp]:
   \<open>set_bit n k \<ge> 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int
   by (simp add: set_bit_def)
 
 lemma set_bit_negative_int_iff [simp]:
   \<open>set_bit n k < 0 \<longleftrightarrow> k < 0\<close> for k :: int
   by (simp add: set_bit_def)
 
 lemma unset_bit_nonnegative_int_iff [simp]:
   \<open>unset_bit n k \<ge> 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int
   by (simp add: unset_bit_def)
 
 lemma unset_bit_negative_int_iff [simp]:
   \<open>unset_bit n k < 0 \<longleftrightarrow> k < 0\<close> for k :: int
   by (simp add: unset_bit_def)
 
 lemma flip_bit_nonnegative_int_iff [simp]:
   \<open>flip_bit n k \<ge> 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int
   by (simp add: flip_bit_def)
 
 lemma flip_bit_negative_int_iff [simp]:
   \<open>flip_bit n k < 0 \<longleftrightarrow> k < 0\<close> for k :: int
   by (simp add: flip_bit_def)
 
 
 subsubsection \<open>Instance \<^typ>\<open>nat\<close>\<close>
 
 instantiation nat :: semiring_bit_operations
 begin
 
 definition and_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close>
   where \<open>m AND n = nat (int m AND int n)\<close> for m n :: nat
 
 definition or_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close>
   where \<open>m OR n = nat (int m OR int n)\<close> for m n :: nat
 
 definition xor_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close>
   where \<open>m XOR n = nat (int m XOR int n)\<close> for m n :: nat
 
 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: and_nat_def bit_and_iff less_le bit_eq_iff)
   show \<open>bit (m OR n) q \<longleftrightarrow> bit m q \<or> bit n q\<close>
     by (auto simp add: or_nat_def bit_or_iff less_le bit_eq_iff)
   show \<open>bit (m XOR n) q \<longleftrightarrow> bit m q \<noteq> bit n q\<close>
     by (auto simp add: xor_nat_def bit_xor_iff less_le bit_eq_iff)
 qed
 
 end
 
 lemma and_nat_rec:
   \<open>m AND n = of_bool (odd m \<and> odd n) + 2 * ((m div 2) AND (n div 2))\<close> for m n :: nat
   by (simp add: and_nat_def and_int_rec [of \<open>int m\<close> \<open>int n\<close>] zdiv_int nat_add_distrib nat_mult_distrib)
 
 lemma or_nat_rec:
   \<open>m OR n = of_bool (odd m \<or> odd n) + 2 * ((m div 2) OR (n div 2))\<close> for m n :: nat
   by (simp add: or_nat_def or_int_rec [of \<open>int m\<close> \<open>int n\<close>] zdiv_int nat_add_distrib nat_mult_distrib)
 
 lemma xor_nat_rec:
   \<open>m XOR n = of_bool (odd m \<noteq> odd n) + 2 * ((m div 2) XOR (n div 2))\<close> for m n :: nat
   by (simp add: xor_nat_def xor_int_rec [of \<open>int m\<close> \<open>int n\<close>] zdiv_int nat_add_distrib nat_mult_distrib)
 
 lemma Suc_0_and_eq [simp]:
   \<open>Suc 0 AND n = n mod 2\<close>
   using one_and_eq [of n] by simp
 
 lemma and_Suc_0_eq [simp]:
   \<open>n AND Suc 0 = n mod 2\<close>
   using and_one_eq [of n] by simp
 
 lemma Suc_0_or_eq:
   \<open>Suc 0 OR n = n + of_bool (even n)\<close>
   using one_or_eq [of n] by simp
 
 lemma or_Suc_0_eq:
   \<open>n OR Suc 0 = n + of_bool (even n)\<close>
   using or_one_eq [of n] by simp
 
 lemma Suc_0_xor_eq:
   \<open>Suc 0 XOR n = n + of_bool (even n) - of_bool (odd n)\<close>
   using one_xor_eq [of n] by simp
 
 lemma xor_Suc_0_eq:
   \<open>n XOR Suc 0 = n + of_bool (even n) - of_bool (odd n)\<close>
   using xor_one_eq [of n] by simp
 
 
 subsubsection \<open>Instances for \<^typ>\<open>integer\<close> and \<^typ>\<open>natural\<close>\<close>
 
 unbundle integer.lifting natural.lifting
 
 context
   includes lifting_syntax
 begin
 
 lemma transfer_rule_bit_integer [transfer_rule]:
   \<open>((pcr_integer :: int \<Rightarrow> integer \<Rightarrow> bool) ===> (=)) bit bit\<close>
   by (unfold bit_def) transfer_prover
 
 lemma transfer_rule_bit_natural [transfer_rule]:
   \<open>((pcr_natural :: nat \<Rightarrow> natural \<Rightarrow> bool) ===> (=)) bit bit\<close>
   by (unfold bit_def) transfer_prover
 
 end
 
 instantiation integer :: ring_bit_operations
 begin
 
 lift_definition not_integer :: \<open>integer \<Rightarrow> integer\<close>
   is not .
 
 lift_definition and_integer :: \<open>integer \<Rightarrow> integer \<Rightarrow> integer\<close>
   is \<open>and\<close> .
 
 lift_definition or_integer :: \<open>integer \<Rightarrow> integer \<Rightarrow> integer\<close>
   is or .
 
 lift_definition xor_integer ::  \<open>integer \<Rightarrow> integer \<Rightarrow> integer\<close>
   is xor .
 
 instance proof
   fix k l :: \<open>integer\<close> and n :: nat
   show \<open>- k = NOT (k - 1)\<close>
     by transfer (simp add: minus_eq_not_minus_1)
   show \<open>bit (NOT k) n \<longleftrightarrow> (2 :: integer) ^ n \<noteq> 0 \<and> \<not> bit k n\<close>
     by transfer (fact bit_not_iff)
   show \<open>bit (k AND l) n \<longleftrightarrow> bit k n \<and> bit l n\<close>
     by transfer (fact bit_and_iff)
   show \<open>bit (k OR l) n \<longleftrightarrow> bit k n \<or> bit l n\<close>
     by transfer (fact bit_or_iff)
   show \<open>bit (k XOR l) n \<longleftrightarrow> bit k n \<noteq> bit l n\<close>
     by transfer (fact bit_xor_iff)
 qed
 
 end
 
 instantiation natural :: semiring_bit_operations
 begin
 
 lift_definition and_natural :: \<open>natural \<Rightarrow> natural \<Rightarrow> natural\<close>
   is \<open>and\<close> .
 
 lift_definition or_natural :: \<open>natural \<Rightarrow> natural \<Rightarrow> natural\<close>
   is or .
 
 lift_definition xor_natural ::  \<open>natural \<Rightarrow> natural \<Rightarrow> natural\<close>
   is xor .
 
 instance proof
   fix m n :: \<open>natural\<close> and q :: nat
   show \<open>bit (m AND n) q \<longleftrightarrow> bit m q \<and> bit n q\<close>
     by transfer (fact bit_and_iff)
   show \<open>bit (m OR n) q \<longleftrightarrow> bit m q \<or> bit n q\<close>
     by transfer (fact bit_or_iff)
   show \<open>bit (m XOR n) q \<longleftrightarrow> bit m q \<noteq> bit n q\<close>
     by transfer (fact bit_xor_iff)
 qed
 
 end
 
 lifting_update integer.lifting
 lifting_forget integer.lifting
 
 lifting_update natural.lifting
 lifting_forget natural.lifting
 
 
 subsection \<open>Key ideas of bit operations\<close>
 
 text \<open>
   When formalizing bit operations, it is tempting to represent
   bit values as explicit lists over a binary type. This however
   is a bad idea, mainly due to the inherent ambiguities in
   representation concerning repeating leading bits.
 
   Hence this approach avoids such explicit lists altogether
   following an algebraic path:
 
   \<^item> Bit values are represented by numeric types: idealized
     unbounded bit values can be represented by type \<^typ>\<open>int\<close>,
     bounded bit values by quotient types over \<^typ>\<open>int\<close>.
 
   \<^item> (A special case are idealized unbounded bit values ending
     in @{term [source] 0} which can be represented by type \<^typ>\<open>nat\<close> but
     only support a restricted set of operations).
 
   \<^item> From this idea follows that
 
       \<^item> multiplication by \<^term>\<open>2 :: int\<close> is a bit shift to the left and
 
       \<^item> division by \<^term>\<open>2 :: int\<close> is a bit shift to the right.
 
   \<^item> Concerning bounded bit values, iterated shifts to the left
     may result in eliminating all bits by shifting them all
     beyond the boundary.  The property \<^prop>\<open>(2 :: int) ^ n \<noteq> 0\<close>
     represents that \<^term>\<open>n\<close> is \<^emph>\<open>not\<close> beyond that boundary.
 
   \<^item> The projection on a single bit is then @{thm bit_def [where ?'a = int, no_vars]}.
 
   \<^item> This leads to the most fundamental properties of bit values:
 
       \<^item> Equality rule: @{thm bit_eqI [where ?'a = int, no_vars]}
 
       \<^item> Induction rule: @{thm bits_induct [where ?'a = int, no_vars]}
 
   \<^item> Typical operations are characterized as follows:
 
       \<^item> Singleton \<^term>\<open>n\<close>th bit: \<^term>\<open>(2 :: int) ^ n\<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]}
 
       \<^item> Right shift: @{thm drop_bit_eq_div [where ?'a = int, no_vars]}
 
       \<^item> Truncation: @{thm take_bit_eq_mod [where ?'a = int, no_vars]}
 
       \<^item> Negation: @{thm bit_not_iff [where ?'a = int, no_vars]}
 
       \<^item> And: @{thm bit_and_iff [where ?'a = int, no_vars]}
 
       \<^item> Or: @{thm bit_or_iff [where ?'a = int, no_vars]}
 
       \<^item> Xor: @{thm bit_xor_iff [where ?'a = int, no_vars]}
 
       \<^item> Set a single bit: @{thm set_bit_def [where ?'a = int, no_vars]}
 
       \<^item> Unset a single bit: @{thm unset_bit_def [where ?'a = int, no_vars]}
 
       \<^item> Flip a single bit: @{thm flip_bit_def [where ?'a = int, no_vars]}
 \<close>
 
 end