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
@@ -1,758 +1,758 @@
 (*  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
 
 context semiring_bit_shifts
 begin
 
 (*lemma bit_push_bit_iff:
   \<open>bit (push_bit m a) n \<longleftrightarrow> n \<ge> m \<and> 2 ^ n \<noteq> 0 \<and> bit a (n - m)\<close>*)
 
 end
 
 
 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 "AND" 64)
     and or :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close>  (infixr "OR"  59)
     and xor :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close>  (infixr "XOR" 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>
 
 definition map_bit :: \<open>nat \<Rightarrow> (bool \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   where \<open>map_bit n f a = take_bit n a + push_bit n (of_bool (f (bit a n)) + 2 * drop_bit (Suc n) a)\<close>
 
 definition set_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   where \<open>set_bit n = map_bit n top\<close>
 
 definition unset_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   where \<open>unset_bit n = map_bit n bot\<close>
 
 definition flip_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   where \<open>flip_bit n = map_bit n Not\<close>
 
 text \<open>
   Having 
   \<^const>\<open>set_bit\<close>, \<^const>\<open>unset_bit\<close> and \<^const>\<open>flip_bit\<close> as separate
   operations allows to implement them using bit masks later.
 \<close>
 
 lemma stable_imp_drop_eq:
   \<open>drop_bit n a = a\<close> if \<open>a div 2 = a\<close>
   by (induction n) (simp_all add: that)
 
 lemma map_bit_0 [simp]:
   \<open>map_bit 0 f a = of_bool (f (odd a)) + 2 * (a div 2)\<close>
   by (simp add: map_bit_def)
 
 lemma map_bit_Suc [simp]:
   \<open>map_bit (Suc n) f a = a mod 2 + 2 * map_bit n f (a div 2)\<close>
   by (auto simp add: map_bit_def algebra_simps mod2_eq_if push_bit_add mult_2
     elim: evenE oddE)
 
 lemma set_bit_0 [simp]:
   \<open>set_bit 0 a = 1 + 2 * (a div 2)\<close>
   by (simp add: set_bit_def)
 
 lemma set_bit_Suc [simp]:
   \<open>set_bit (Suc n) a = a mod 2 + 2 * set_bit n (a div 2)\<close>
   by (simp add: set_bit_def)
 
 lemma unset_bit_0 [simp]:
   \<open>unset_bit 0 a = 2 * (a div 2)\<close>
   by (simp add: unset_bit_def)
 
 lemma unset_bit_Suc [simp]:
   \<open>unset_bit (Suc n) a = a mod 2 + 2 * unset_bit n (a div 2)\<close>
   by (simp add: unset_bit_def)
 
 lemma flip_bit_0 [simp]:
   \<open>flip_bit 0 a = of_bool (even a) + 2 * (a div 2)\<close>
   by (simp add: flip_bit_def)
 
 lemma flip_bit_Suc [simp]:
   \<open>flip_bit (Suc n) a = a mod 2 + 2 * flip_bit n (a div 2)\<close>
   by (simp add: flip_bit_def)
 
 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 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 [simp]:
   "1 AND a = of_bool (odd a)"
   by (simp add: bit_eq_iff bit_and_iff) (auto simp add: bit_1_iff)
 
 lemma and_one_eq [simp]:
   "a AND 1 = of_bool (odd a)"
   using one_and_eq [of a] by (simp add: ac_simps)
 
 lemma one_or_eq [simp]:
   "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 [simp]:
   "a OR 1 = a + of_bool (even a)"
   using one_or_eq [of a] by (simp add: ac_simps)
 
 lemma one_xor_eq [simp]:
   "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_int_eq [simp]:
+lemma xor_one_eq [simp]:
   "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)
 
 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 (auto simp add: bit_eq_iff bit_and_iff)
   apply (simp add: bit_exp_iff)
   apply (metis local.bit_def local.bit_exp_iff local.bits_div_by_0)
   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
        apply (auto simp add: bit_eq_iff bit_and_iff bit_or_iff bit_not_iff)
     apply (simp add: bit_exp_iff)
     apply (metis local.bit_def local.bit_exp_iff local.bits_div_by_0)
     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 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
 
 end
 
 
 subsubsection \<open>Instance \<^typ>\<open>nat\<close>\<close>
 
 locale zip_nat = single: abel_semigroup f
     for f :: "bool \<Rightarrow> bool \<Rightarrow> bool"  (infixl "\<^bold>*" 70) +
   assumes end_of_bits: "\<not> False \<^bold>* False"
 begin
 
 lemma False_P_imp:
   "False \<^bold>* True \<and> P" if "False \<^bold>* P"
   using that end_of_bits by (cases P) simp_all
 
 function F :: "nat \<Rightarrow> nat \<Rightarrow> nat"  (infixl "\<^bold>\<times>" 70)
   where "m \<^bold>\<times> n = (if m = 0 \<and> n = 0 then 0
     else of_bool (odd m \<^bold>* odd n) + (m div 2) \<^bold>\<times> (n div 2) * 2)"
   by auto
 
 termination
   by (relation "measure (case_prod (+))") auto
 
 lemma zero_left_eq:
   "0 \<^bold>\<times> n = of_bool (False \<^bold>* True) * n"
   by (induction n rule: nat_bit_induct) (simp_all add: end_of_bits)
 
 lemma zero_right_eq:
   "m \<^bold>\<times> 0 = of_bool (True \<^bold>* False) * m"
   by (induction m rule: nat_bit_induct) (simp_all add: end_of_bits)
 
 lemma simps [simp]:
   "0 \<^bold>\<times> 0 = 0"
   "0 \<^bold>\<times> n = of_bool (False \<^bold>* True) * n"
   "m \<^bold>\<times> 0 = of_bool (True \<^bold>* False) * m"
   "m > 0 \<Longrightarrow> n > 0 \<Longrightarrow> m \<^bold>\<times> n = of_bool (odd m \<^bold>* odd n) + (m div 2) \<^bold>\<times> (n div 2) * 2"
   by (simp_all only: zero_left_eq zero_right_eq) simp
 
 lemma rec:
   "m \<^bold>\<times> n = of_bool (odd m \<^bold>* odd n) + (m div 2) \<^bold>\<times> (n div 2) * 2"
   by (cases "m = 0 \<and> n = 0") (auto simp add: end_of_bits)
 
 declare F.simps [simp del]
 
 sublocale abel_semigroup F
 proof
   show "m \<^bold>\<times> n \<^bold>\<times> q = m \<^bold>\<times> (n \<^bold>\<times> q)" for m n q :: nat
   proof (induction m arbitrary: n q rule: nat_bit_induct)
     case zero
     show ?case
       by simp
   next
     case (even m)
     with rec [of "2 * m"] rec [of _ q] show ?case
       by (cases "even n") (auto simp add: ac_simps dest: False_P_imp)
   next
     case (odd m)
     with rec [of "Suc (2 * m)"] rec [of _ q] show ?case
       by (cases "even n"; cases "even q")
         (auto dest: False_P_imp simp add: ac_simps)
   qed
   show "m \<^bold>\<times> n = n \<^bold>\<times> m" for m n :: nat
   proof (induction m arbitrary: n rule: nat_bit_induct)
     case zero
     show ?case
       by (simp add: ac_simps)
   next
     case (even m)
     with rec [of "2 * m" n] rec [of n "2 * m"] show ?case
       by (simp add: ac_simps)
   next
     case (odd m)
     with rec [of "Suc (2 * m)" n] rec [of n "Suc (2 * m)"] show ?case
       by (simp add: ac_simps)
   qed
 qed
 
 lemma self [simp]:
   "n \<^bold>\<times> n = of_bool (True \<^bold>* True) * n"
   by (induction n rule: nat_bit_induct) (simp_all add: end_of_bits)
 
 lemma even_iff [simp]:
   "even (m \<^bold>\<times> n) \<longleftrightarrow> \<not> (odd m \<^bold>* odd n)"
 proof (induction m arbitrary: n rule: nat_bit_induct)
   case zero
   show ?case
     by (cases "even n") (simp_all add: end_of_bits)
 next
   case (even m)
   then show ?case
     by (simp add: rec [of "2 * m"])
 next
   case (odd m)
   then show ?case
     by (simp add: rec [of "Suc (2 * m)"])
 qed
 
 end
 
 instantiation nat :: semiring_bit_operations
 begin
 
 global_interpretation and_nat: zip_nat "(\<and>)"
   defines and_nat = and_nat.F
   by standard auto
 
 global_interpretation and_nat: semilattice "(AND) :: nat \<Rightarrow> nat \<Rightarrow> nat"
 proof (rule semilattice.intro, fact and_nat.abel_semigroup_axioms, standard)
   show "n AND n = n" for n :: nat
     by (simp add: and_nat.self)
 qed
 
 declare and_nat.simps [simp] \<comment> \<open>inconsistent declaration handling by
   \<open>global_interpretation\<close> in \<open>instantiation\<close>\<close>
 
 global_interpretation or_nat: zip_nat "(\<or>)"
   defines or_nat = or_nat.F
   by standard auto
 
 global_interpretation or_nat: semilattice "(OR) :: nat \<Rightarrow> nat \<Rightarrow> nat"
 proof (rule semilattice.intro, fact or_nat.abel_semigroup_axioms, standard)
   show "n OR n = n" for n :: nat
     by (simp add: or_nat.self)
 qed
 
 declare or_nat.simps [simp] \<comment> \<open>inconsistent declaration handling by
   \<open>global_interpretation\<close> in \<open>instantiation\<close>\<close>
 
 global_interpretation xor_nat: zip_nat "(\<noteq>)"
   defines xor_nat = xor_nat.F
   by standard auto
 
 declare xor_nat.simps [simp] \<comment> \<open>inconsistent declaration handling by
   \<open>global_interpretation\<close> in \<open>instantiation\<close>\<close>
 
 instance proof
   fix m n q :: nat
   show \<open>bit (m AND n) q \<longleftrightarrow> bit m q \<and> bit n q\<close>
   proof (rule sym, induction q arbitrary: m n)
     case 0
     then show ?case
       by (simp add: and_nat.even_iff)
   next
     case (Suc q)
     with and_nat.rec [of m n] show ?case
       by simp
   qed
   show \<open>bit (m OR n) q \<longleftrightarrow> bit m q \<or> bit n q\<close>
   proof (rule sym, induction q arbitrary: m n)
     case 0
     then show ?case
       by (simp add: or_nat.even_iff)
   next
     case (Suc q)
     with or_nat.rec [of m n] show ?case
       by simp
   qed
   show \<open>bit (m XOR n) q \<longleftrightarrow> bit m q \<noteq> bit n q\<close>
   proof (rule sym, induction q arbitrary: m n)
     case 0
     then show ?case
       by (simp add: xor_nat.even_iff)
   next
     case (Suc q)
     with xor_nat.rec [of m n] show ?case
       by simp
   qed
 qed
 
 end
 
 global_interpretation or_nat: semilattice_neutr "(OR)" "0 :: nat"
   by standard simp
 
 global_interpretation xor_nat: comm_monoid "(XOR)" "0 :: nat"
   by standard simp
 
 lemma Suc_0_and_eq [simp]:
-  "Suc 0 AND n = n mod 2"
-  by (cases n) auto
+  \<open>Suc 0 AND n = of_bool (odd n)\<close>
+  using one_and_eq [of n] by simp
 
 lemma and_Suc_0_eq [simp]:
-  "n AND Suc 0 = n mod 2"
-  using Suc_0_and_eq [of n] by (simp add: ac_simps)
+  \<open>n AND Suc 0 = of_bool (odd n)\<close>
+  using and_one_eq [of n] by simp
 
 lemma Suc_0_or_eq [simp]:
-  "Suc 0 OR n = n + of_bool (even n)"
-  by (cases n) (simp_all add: ac_simps)
+  \<open>Suc 0 OR n = n + of_bool (even n)\<close>
+  using one_or_eq [of n] by simp
 
 lemma or_Suc_0_eq [simp]:
-  "n OR Suc 0 = n + of_bool (even n)"
-  using Suc_0_or_eq [of n] by (simp add: ac_simps)
+  \<open>n OR Suc 0 = n + of_bool (even n)\<close>
+  using or_one_eq [of n] by simp
 
 lemma Suc_0_xor_eq [simp]:
-  "Suc 0 XOR n = n + of_bool (even n) - of_bool (odd n)"
-  by (cases n) (simp_all add: ac_simps)
+  \<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 [simp]:
-  "n XOR Suc 0 = n + of_bool (even n) - of_bool (odd n)"
-  using Suc_0_xor_eq [of n] by (simp add: ac_simps)
+  \<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>Instance \<^typ>\<open>int\<close>\<close>
 
 abbreviation (input) complement :: "int \<Rightarrow> int"
   where "complement k \<equiv> - k - 1"
 
 lemma complement_half:
   "complement (k * 2) div 2 = complement k"
   by simp
 
 lemma complement_div_2:
   "complement (k div 2) = complement k div 2"
   by linarith
 
 locale zip_int = single: abel_semigroup f
   for f :: "bool \<Rightarrow> bool \<Rightarrow> bool"  (infixl "\<^bold>*" 70)
 begin
  
 lemma False_False_imp_True_True:
   "True \<^bold>* True" if "False \<^bold>* False"
 proof (rule ccontr)
   assume "\<not> True \<^bold>* True"
   with that show False
     using single.assoc [of False True True]
     by (cases "False \<^bold>* True") simp_all
 qed
 
 function F :: "int \<Rightarrow> int \<Rightarrow> int"  (infixl "\<^bold>\<times>" 70)
   where "k \<^bold>\<times> l = (if k \<in> {0, - 1} \<and> l \<in> {0, - 1}
     then - of_bool (odd k \<^bold>* odd l)
     else of_bool (odd k \<^bold>* odd l) + (k div 2) \<^bold>\<times> (l div 2) * 2)"
   by auto
 
 termination
   by (relation "measure (\<lambda>(k, l). nat (\<bar>k\<bar> + \<bar>l\<bar>))") auto
 
 lemma zero_left_eq:
   "0 \<^bold>\<times> l = (case (False \<^bold>* False, False \<^bold>* True)
     of (False, False) \<Rightarrow> 0
      | (False, True) \<Rightarrow> l
      | (True, False) \<Rightarrow> complement l
      | (True, True) \<Rightarrow> - 1)"
   by (induction l rule: int_bit_induct)
    (simp_all split: bool.split) 
 
 lemma minus_left_eq:
   "- 1 \<^bold>\<times> l = (case (True \<^bold>* False, True \<^bold>* True)
     of (False, False) \<Rightarrow> 0
      | (False, True) \<Rightarrow> l
      | (True, False) \<Rightarrow> complement l
      | (True, True) \<Rightarrow> - 1)"
   by (induction l rule: int_bit_induct)
    (simp_all split: bool.split) 
 
 lemma zero_right_eq:
   "k \<^bold>\<times> 0 = (case (False \<^bold>* False, False \<^bold>* True)
     of (False, False) \<Rightarrow> 0
      | (False, True) \<Rightarrow> k
      | (True, False) \<Rightarrow> complement k
      | (True, True) \<Rightarrow> - 1)"
   by (induction k rule: int_bit_induct)
     (simp_all add: ac_simps split: bool.split)
 
 lemma minus_right_eq:
   "k \<^bold>\<times> - 1 = (case (True \<^bold>* False, True \<^bold>* True)
     of (False, False) \<Rightarrow> 0
      | (False, True) \<Rightarrow> k
      | (True, False) \<Rightarrow> complement k
      | (True, True) \<Rightarrow> - 1)"
   by (induction k rule: int_bit_induct)
     (simp_all add: ac_simps split: bool.split)
 
 lemma simps [simp]:
   "0 \<^bold>\<times> 0 = - of_bool (False \<^bold>* False)"
   "- 1 \<^bold>\<times> 0 = - of_bool (True \<^bold>* False)"
   "0 \<^bold>\<times> - 1 = - of_bool (False \<^bold>* True)"
   "- 1 \<^bold>\<times> - 1 = - of_bool (True \<^bold>* True)"
   "0 \<^bold>\<times> l = (case (False \<^bold>* False, False \<^bold>* True)
     of (False, False) \<Rightarrow> 0
      | (False, True) \<Rightarrow> l
      | (True, False) \<Rightarrow> complement l
      | (True, True) \<Rightarrow> - 1)"
   "- 1 \<^bold>\<times> l = (case (True \<^bold>* False, True \<^bold>* True)
     of (False, False) \<Rightarrow> 0
      | (False, True) \<Rightarrow> l
      | (True, False) \<Rightarrow> complement l
      | (True, True) \<Rightarrow> - 1)"
   "k \<^bold>\<times> 0 = (case (False \<^bold>* False, False \<^bold>* True)
     of (False, False) \<Rightarrow> 0
      | (False, True) \<Rightarrow> k
      | (True, False) \<Rightarrow> complement k
      | (True, True) \<Rightarrow> - 1)"
   "k \<^bold>\<times> - 1 = (case (True \<^bold>* False, True \<^bold>* True)
     of (False, False) \<Rightarrow> 0
      | (False, True) \<Rightarrow> k
      | (True, False) \<Rightarrow> complement k
      | (True, True) \<Rightarrow> - 1)"
   "k \<noteq> 0 \<Longrightarrow> k \<noteq> - 1 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> l \<noteq> - 1
     \<Longrightarrow> k \<^bold>\<times> l = of_bool (odd k \<^bold>* odd l) + (k div 2) \<^bold>\<times> (l div 2) * 2"
   by simp_all[4] (simp_all only: zero_left_eq minus_left_eq zero_right_eq minus_right_eq, simp)
 
 declare F.simps [simp del]
 
 lemma rec:
   "k \<^bold>\<times> l = of_bool (odd k \<^bold>* odd l) + (k div 2) \<^bold>\<times> (l div 2) * 2"
   by (cases "k \<in> {0, - 1} \<and> l \<in> {0, - 1}") (auto simp add: ac_simps F.simps [of k l] split: bool.split)
 
 sublocale abel_semigroup F
 proof
   show "k \<^bold>\<times> l \<^bold>\<times> r = k \<^bold>\<times> (l \<^bold>\<times> r)" for k l r :: int
   proof (induction k arbitrary: l r rule: int_bit_induct)
     case zero
     have "complement l \<^bold>\<times> r = complement (l \<^bold>\<times> r)" if "False \<^bold>* False" "\<not> False \<^bold>* True"
     proof (induction l arbitrary: r rule: int_bit_induct)
       case zero
       from that show ?case
         by (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
     next
       case minus
       from that show ?case
         by (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
     next
       case (even l)
       with that rec [of _ r] show ?case
         by (cases "even r")
           (auto simp add: complement_half ac_simps False_False_imp_True_True split: bool.splits)
     next
       case (odd l)
       moreover have "- l - 1 = - 1 - l"
         by simp
       ultimately show ?case
         using that rec [of _ r]
         by (cases "even r")
           (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
     qed
     then show ?case
       by (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
   next
     case minus
     have "complement l \<^bold>\<times> r = complement (l \<^bold>\<times> r)" if "\<not> True \<^bold>* True" "False \<^bold>* True"
     proof (induction l arbitrary: r rule: int_bit_induct)
       case zero
       from that show ?case
         by (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
     next
       case minus
       from that show ?case
         by (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
     next
       case (even l)
       with that rec [of _ r] show ?case
         by (cases "even r")
           (auto simp add: complement_half ac_simps False_False_imp_True_True split: bool.splits)
     next
       case (odd l)
       moreover have "- l - 1 = - 1 - l"
         by simp
       ultimately show ?case
         using that rec [of _ r]
         by (cases "even r")
           (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
     qed
     then show ?case
       by (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
   next
     case (even k)
     with rec [of "k * 2"] rec [of _ r] show ?case
       by (cases "even r"; cases "even l") (auto simp add: ac_simps False_False_imp_True_True)
   next
     case (odd k)
     with rec [of "1 + k * 2"] rec [of _ r] show ?case
       by (cases "even r"; cases "even l") (auto simp add: ac_simps False_False_imp_True_True)
   qed
   show "k \<^bold>\<times> l = l \<^bold>\<times> k" for k l :: int
   proof (induction k arbitrary: l rule: int_bit_induct)
     case zero
     show ?case
       by simp
   next
     case minus
     show ?case
       by simp
   next
     case (even k)
     with rec [of "k * 2" l] rec [of l "k * 2"] show ?case
       by (simp add: ac_simps)
   next
     case (odd k)
     with rec [of "k * 2 + 1" l] rec [of l "k * 2 + 1"] show ?case
       by (simp add: ac_simps)
   qed
 qed
 
 lemma self [simp]:
   "k \<^bold>\<times> k = (case (False \<^bold>* False, True \<^bold>* True)
     of (False, False) \<Rightarrow> 0
      | (False, True) \<Rightarrow> k
      | (True, True) \<Rightarrow> - 1)"
   by (induction k rule: int_bit_induct) (auto simp add: False_False_imp_True_True split: bool.split)
 
 lemma even_iff [simp]:
   "even (k \<^bold>\<times> l) \<longleftrightarrow> \<not> (odd k \<^bold>* odd l)"
 proof (induction k arbitrary: l rule: int_bit_induct)
   case zero
   show ?case
     by (cases "even l") (simp_all split: bool.splits)
 next
   case minus
   show ?case
     by (cases "even l") (simp_all split: bool.splits)
 next
   case (even k)
   then show ?case
     by (simp add: rec [of "k * 2"])
 next
   case (odd k)
   then show ?case
     by (simp add: rec [of "1 + k * 2"])
 qed
 
 end
 
 instantiation int :: ring_bit_operations
 begin
 
 definition not_int :: "int \<Rightarrow> int"
   where "not_int = complement"
 
 global_interpretation and_int: zip_int "(\<and>)"
   defines and_int = and_int.F
   by standard
 
 declare and_int.simps [simp] \<comment> \<open>inconsistent declaration handling by
   \<open>global_interpretation\<close> in \<open>instantiation\<close>\<close>
 
 global_interpretation and_int: semilattice "(AND) :: int \<Rightarrow> int \<Rightarrow> int"
 proof (rule semilattice.intro, fact and_int.abel_semigroup_axioms, standard)
   show "k AND k = k" for k :: int
     by (simp add: and_int.self)
 qed
 
 global_interpretation or_int: zip_int "(\<or>)"
   defines or_int = or_int.F
   by standard
 
 declare or_int.simps [simp] \<comment> \<open>inconsistent declaration handling by
   \<open>global_interpretation\<close> in \<open>instantiation\<close>\<close>
 
 global_interpretation or_int: semilattice "(OR) :: int \<Rightarrow> int \<Rightarrow> int"
 proof (rule semilattice.intro, fact or_int.abel_semigroup_axioms, standard)
   show "k OR k = k" for k :: int
     by (simp add: or_int.self)
 qed
 
 global_interpretation xor_int: zip_int "(\<noteq>)"
   defines xor_int = xor_int.F
   by standard
 
 declare xor_int.simps [simp] \<comment> \<open>inconsistent declaration handling by
   \<open>global_interpretation\<close> in \<open>instantiation\<close>\<close>
 
 lemma bit_not_iff_int:
   \<open>bit (NOT k) n \<longleftrightarrow> \<not> bit k n\<close>
     for k :: int
     by (induction n arbitrary: k)
       (simp_all add: not_int_def flip: complement_div_2)
 
 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>
   proof (rule sym, induction n arbitrary: k l)
     case 0
     then show ?case
       by (simp add: and_int.even_iff)
   next
     case (Suc n)
     with and_int.rec [of k l] show ?case
       by simp
   qed
   show \<open>bit (k OR l) n \<longleftrightarrow> bit k n \<or> bit l n\<close>
   proof (rule sym, induction n arbitrary: k l)
     case 0
     then show ?case
       by (simp add: or_int.even_iff)
   next
     case (Suc n)
     with or_int.rec [of k l] show ?case
       by simp
   qed
   show \<open>bit (k XOR l) n \<longleftrightarrow> bit k n \<noteq> bit l n\<close>
   proof (rule sym, induction n arbitrary: k l)
     case 0
     then show ?case
       by (simp add: xor_int.even_iff)
   next
     case (Suc n)
     with xor_int.rec [of k l] show ?case
       by simp
   qed
 qed (simp_all add: minus_1_div_exp_eq_int bit_not_iff_int)
 
 end
 
 lemma not_int_div_2:
   "NOT k div 2 = NOT (k div 2)" for k :: int
   by (simp add: complement_div_2 not_int_def)
 
 lemma not_int_rec [simp]:
   "k \<noteq> 0 \<Longrightarrow> k \<noteq> - 1 \<Longrightarrow> NOT k = of_bool (even k) + 2 * NOT (k div 2)" for k :: int
   by (auto simp add: not_int_def elim: oddE)
 
 end