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
@@ -1,778 +1,801 @@
 (*  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
 
+lemma minus_1_div_exp_eq_int [simp]:
+  \<open>- 1 div (2 :: int) ^ n = - 1\<close>
+  for n :: nat
+  by (induction n) (use div_exp_eq [symmetric, of \<open>- 1 :: int\<close> 1] in \<open>simp_all add: ac_simps\<close>)
+
+context semiring_bits
+begin
+
+lemma bits_div_by_0 [simp]:
+  \<open>a div 0 = 0\<close>
+  by (metis local.add_cancel_right_right local.bit_mod_div_trivial local.mod_mult_div_eq local.mult_not_zero)
+
+lemma bit_0_eq [simp]:
+  \<open>bit 0 = bot\<close>
+  by (simp add: fun_eq_iff bit_def)
+
+end
+
+context semiring_bit_shifts
+begin
+
+end
+
+
 subsection \<open>Bit operations in suitable algebraic structures\<close>
 
 class semiring_bit_operations = semiring_bit_shifts +
-  fixes "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixr "AND" 64)
-    and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixr "OR"  59)
-    and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixr "XOR" 59)
+  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
+  \<^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)
 
 end
 
 class ring_bit_operations = semiring_bit_operations + ring_parity +
   fixes not :: \<open>'a \<Rightarrow> 'a\<close>  (\<open>NOT\<close>)
-  assumes boolean_algebra: \<open>boolean_algebra (AND) (OR) NOT 0 (- 1)\<close>
-    and boolean_algebra_xor_eq: \<open>boolean_algebra.xor (AND) (OR) NOT = (XOR)\<close>
+  assumes bits_even_minus_1_div_exp_iff [simp]: \<open>even (- 1 div 2 ^ n) \<longleftrightarrow> 2 ^ n = 0\<close>
+  assumes bit_not_iff: \<open>\<And>n. bit (NOT a) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> \<not> bit a n\<close>
 begin
 
+lemma bits_minus_1_mod_2_eq [simp]:
+  \<open>(- 1) mod 2 = 1\<close>
+  by (simp add: mod_2_eq_odd)
+
+lemma bit_minus_1_iff [simp]:
+  \<open>bit (- 1) n \<longleftrightarrow> 2 ^ n \<noteq> 0\<close>
+  by (simp add: bit_def)
+
 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>
-    by (fact boolean_algebra)
+    apply standard
+             apply (auto simp add: bit_eq_iff bit_and_iff bit_or_iff bit_not_iff)
+     apply (metis local.bit_def local.bit_exp_iff local.bits_div_by_0)
+    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>  
-    by (fact boolean_algebra_xor_eq)
+  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 (metis local.bit_def local.bit_exp_iff local.bits_div_by_0)
+        apply (metis local.bit_def local.bit_exp_iff local.bits_div_by_0)
+       apply (metis local.bit_def local.bit_exp_iff local.bits_div_by_0)
+      apply (metis local.bit_def local.bit_exp_iff local.bits_div_by_0)
+     apply (metis local.bit_def local.bit_exp_iff local.bits_div_by_0)
+    apply (metis local.bit_def local.bit_exp_iff local.bits_div_by_0)
+    done
 qed
 
 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>
 
 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>
 
 lemma zero_nat_and_eq [simp]:
   "0 AND n = 0" for n :: nat
   by simp
 
 lemma and_zero_nat_eq [simp]:
   "n AND 0 = 0" for n :: nat
   by simp
 
 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>
 
 lemma zero_nat_or_eq [simp]:
   "0 OR n = n" for n :: nat
   by simp
 
 lemma or_zero_nat_eq [simp]:
   "n OR 0 = n" for n :: nat
   by simp
 
 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>
 
 lemma zero_nat_xor_eq [simp]:
   "0 XOR n = n" for n :: nat
   by simp
 
 lemma xor_zero_nat_eq [simp]:
   "n XOR 0 = n" for n :: nat
   by simp
 
-instance ..
+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
 
 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)
 
 lemma Suc_0_or_eq [simp]:
   "Suc 0 OR n = n + of_bool (even n)"
   by (cases n) (simp_all add: ac_simps)
 
 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)
 
 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)
 
 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)
 
 
 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
 
 lemma zero_int_and_eq [simp]:
   "0 AND k = 0" for k :: int
   by simp
 
 lemma and_zero_int_eq [simp]:
   "k AND 0 = 0" for k :: int
   by simp
 
 lemma minus_int_and_eq [simp]:
   "- 1 AND k = k" for k :: int
   by simp
 
 lemma and_minus_int_eq [simp]:
   "k AND - 1 = k" for k :: int
   by simp
 
 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
 
 lemma zero_int_or_eq [simp]:
   "0 OR k = k" for k :: int
   by simp
 
 lemma and_zero_or_eq [simp]:
   "k OR 0 = k" for k :: int
   by simp
 
 lemma minus_int_or_eq [simp]:
   "- 1 OR k = - 1" for k :: int
   by simp
 
 lemma or_minus_int_eq [simp]:
   "k OR - 1 = - 1" for k :: int
   by simp
 
 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 zero_int_xor_eq [simp]:
   "0 XOR k = k" for k :: int
   by simp
 
 lemma and_zero_xor_eq [simp]:
   "k XOR 0 = k" for k :: int
   by simp
 
 lemma minus_int_xor_eq [simp]:
   "- 1 XOR k = complement k" for k :: int
   by simp
 
 lemma xor_minus_int_eq [simp]:
   "k XOR - 1 = complement k" for k :: int
   by simp
 
 lemma not_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_simps [simp]:
   "NOT 0 = (- 1 :: int)"
   "NOT (- 1) = (0 :: int)"
   "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)
 
 lemma not_one_int [simp]:
   "NOT 1 = (- 2 :: int)"
   by simp
 
 lemma even_not_iff [simp]:
   "even (NOT k) \<longleftrightarrow> odd k"
-  for k :: int
+    for k :: int
   by (simp add: not_int_def)
 
+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
-  interpret bit_int: boolean_algebra "(AND)" "(OR)" NOT 0 "- 1 :: int"
-  proof
-    show "k AND (l OR r) = k AND l OR k AND r"
-      for k l r :: int
-    proof (induction k arbitrary: l r rule: int_bit_induct)
-      case zero
-      show ?case
-        by simp
-    next
-      case minus
-      show ?case
-        by simp
-    next
-      case (even k)
-      then show ?case by (simp add: and_int.rec [of "k * 2"]
-        or_int.rec [of "(k AND l div 2) * 2"] or_int.rec [of l])
-    next
-      case (odd k)
-      then show ?case by (simp add: and_int.rec [of "1 + k * 2"]
-        or_int.rec [of "(k AND l div 2) * 2"] or_int.rec [of "1 + (k AND l div 2) * 2"] or_int.rec [of l])
-    qed
-    show "k OR l AND r = (k OR l) AND (k OR r)"
-      for k l r :: int
-    proof (induction k arbitrary: l r 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 by (simp add: or_int.rec [of "k * 2"]
-        and_int.rec [of "(k OR l div 2) * 2"] and_int.rec [of "1 + (k OR l div 2) * 2"] and_int.rec [of l])
-    next
-      case (odd k)
-      then show ?case by (simp add: or_int.rec [of "1 + k * 2"]
-        and_int.rec [of "1 + (k OR l div 2) * 2"] and_int.rec [of l])
-    qed
-    show "k AND NOT k = 0" for k :: int
-      by (induction k rule: int_bit_induct)
-        (simp_all add: not_int_def complement_half minus_diff_commute [of 1])
-    show "k OR NOT k = - 1" for k :: int
-      by (induction k rule: int_bit_induct)
-        (simp_all add: not_int_def complement_half minus_diff_commute [of 1])
-  qed (simp_all add: and_int.assoc or_int.assoc,
-    simp_all add: and_int.commute or_int.commute)
-  show "boolean_algebra (AND) (OR) NOT 0 (- 1 :: int)"
-    by (fact bit_int.boolean_algebra_axioms)
-  show "bit_int.xor = ((XOR) :: int \<Rightarrow> _)"
-  proof (rule ext)+
-    fix k l :: int
-    have "k XOR l = k AND NOT l OR NOT k AND l"
-    proof (induction k arbitrary: l rule: int_bit_induct)
-      case zero
-      show ?case
-        by simp
-    next
-      case minus
-      show ?case
-        by (simp add: not_int_def)
-    next
-      case (even k)
-      then show ?case
-        by (simp add: xor_int.rec [of "k * 2"] and_int.rec [of "k * 2"]
-          or_int.rec [of _ "1 + 2 * NOT k AND l"] not_div_2)
-          (simp add: and_int.rec [of _ l])
-    next
-      case (odd k)
-      then show ?case
-        by (simp add: xor_int.rec [of "1 + k * 2"] and_int.rec [of "1 + k * 2"]
-          or_int.rec [of _ "2 * NOT k AND l"] not_div_2)
-          (simp add: and_int.rec [of _ l])
-    qed
-    then show "bit_int.xor k l = k XOR l"
-      by (simp add: bit_int.xor_def)
+  fix k l :: int and n :: nat
+  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
-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: bit_not_iff_int)
 
 end
 
 lemma one_and_int_eq [simp]:
   "1 AND k = k mod 2" for k :: int
   using and_int.rec [of 1] by (simp add: mod2_eq_if)
 
 lemma and_one_int_eq [simp]:
   "k AND 1 = k mod 2" for k :: int
   using one_and_int_eq [of 1] by (simp add: ac_simps)
 
 lemma one_or_int_eq [simp]:
   "1 OR k = k + of_bool (even k)" for k :: int
   using or_int.rec [of 1] by (auto elim: oddE)
 
 lemma or_one_int_eq [simp]:
   "k OR 1 = k + of_bool (even k)" for k :: int
   using one_or_int_eq [of k] by (simp add: ac_simps)
 
 lemma one_xor_int_eq [simp]:
   "1 XOR k = k + of_bool (even k) - of_bool (odd k)" for k :: int
   using xor_int.rec [of 1] by (auto elim: oddE)
 
 lemma xor_one_int_eq [simp]:
   "k XOR 1 = k + of_bool (even k) - of_bool (odd k)" for k :: int
   using one_xor_int_eq [of k] by (simp add: ac_simps)
 
 lemma take_bit_complement_iff:
   "Parity.take_bit n (complement k) = Parity.take_bit n (complement l) \<longleftrightarrow> Parity.take_bit n k = Parity.take_bit n l"
   for k l :: int
   by (simp add: Parity.take_bit_eq_mod mod_eq_dvd_iff dvd_diff_commute)
 
 lemma take_bit_not_iff:
   "Parity.take_bit n (NOT k) = Parity.take_bit n (NOT l) \<longleftrightarrow> Parity.take_bit n k = Parity.take_bit n l"
   for k l :: int
-  by (simp add: not_int_def take_bit_complement_iff)
+  by (auto simp add: bit_eq_iff bit_take_bit_iff bit_not_iff_int)
 
 lemma take_bit_and [simp]:
   "Parity.take_bit n (k AND l) = Parity.take_bit n k AND Parity.take_bit n l"
   for k l :: int
-  apply (induction n arbitrary: k l)
-   apply simp
-  apply (subst and_int.rec)
-  apply (subst (2) and_int.rec)
-  apply simp
-  done
+  by (auto simp add: bit_eq_iff bit_take_bit_iff bit_and_iff)
 
 lemma take_bit_or [simp]:
   "Parity.take_bit n (k OR l) = Parity.take_bit n k OR Parity.take_bit n l"
   for k l :: int
-  apply (induction n arbitrary: k l)
-   apply simp
-  apply (subst or_int.rec)
-  apply (subst (2) or_int.rec)
-  apply simp
-  done
+  by (auto simp add: bit_eq_iff bit_take_bit_iff bit_or_iff)
 
 lemma take_bit_xor [simp]:
   "Parity.take_bit n (k XOR l) = Parity.take_bit n k XOR Parity.take_bit n l"
   for k l :: int
-  apply (induction n arbitrary: k l)
-   apply simp
-  apply (subst xor_int.rec)
-  apply (subst (2) xor_int.rec)
-  apply simp
-  done
+  by (auto simp add: bit_eq_iff bit_take_bit_iff bit_xor_iff)
 
 end
diff --git a/src/HOL/ex/Word.thy b/src/HOL/ex/Word.thy
--- a/src/HOL/ex/Word.thy
+++ b/src/HOL/ex/Word.thy
@@ -1,754 +1,769 @@
 (*  Author:  Florian Haftmann, TUM
 *)
 
 section \<open>Proof of concept for algebraically founded bit word types\<close>
 
 theory Word
   imports
     Main
     "HOL-Library.Type_Length"
     "HOL-ex.Bit_Operations"
 begin
 
+context
+  includes lifting_syntax
+begin
+
+lemma transfer_rule_of_bool:
+  \<open>((\<longleftrightarrow>) ===> (\<cong>)) of_bool of_bool\<close>
+    if [transfer_rule]: \<open>0 \<cong> 0\<close> \<open>1 \<cong> 1\<close>
+    for R :: \<open>'a::zero_neq_one \<Rightarrow> 'b::zero_neq_one \<Rightarrow> bool\<close>  (infix \<open>\<cong>\<close> 50)
+  by (unfold of_bool_def [abs_def]) transfer_prover
+
+end
+
+
 subsection \<open>Preliminaries\<close>
 
 lemma length_not_greater_eq_2_iff [simp]:
   \<open>\<not> 2 \<le> LENGTH('a::len) \<longleftrightarrow> LENGTH('a) = 1\<close>
   by (auto simp add: not_le dest: less_2_cases)
 
 lemma take_bit_uminus:
   "take_bit n (- (take_bit n k)) = take_bit n (- k)" for k :: int
   by (simp add: take_bit_eq_mod mod_minus_eq)
 
 lemma take_bit_minus:
   "take_bit n (take_bit n k - take_bit n l) = take_bit n (k - l)" for k l :: int
   by (simp add: take_bit_eq_mod mod_diff_eq)
 
 lemma take_bit_nonnegative [simp]:
   "take_bit n k \<ge> 0" for k :: int
   by (simp add: take_bit_eq_mod)
 
 definition signed_take_bit :: "nat \<Rightarrow> int \<Rightarrow> int"
   where signed_take_bit_eq_take_bit:
     "signed_take_bit n k = take_bit (Suc n) (k + 2 ^ n) - 2 ^ n"
 
 lemma signed_take_bit_eq_take_bit':
   "signed_take_bit (n - Suc 0) k = take_bit n (k + 2 ^ (n - 1)) - 2 ^ (n - 1)" if "n > 0"
   using that by (simp add: signed_take_bit_eq_take_bit)
 
 lemma signed_take_bit_0 [simp]:
   "signed_take_bit 0 k = - (k mod 2)"
 proof (cases "even k")
   case True
   then have "odd (k + 1)"
     by simp
   then have "(k + 1) mod 2 = 1"
     by (simp add: even_iff_mod_2_eq_zero)
   with True show ?thesis
     by (simp add: signed_take_bit_eq_take_bit)
 next
   case False
   then show ?thesis
     by (simp add: signed_take_bit_eq_take_bit odd_iff_mod_2_eq_one)
 qed
 
 lemma signed_take_bit_Suc [simp]:
   "signed_take_bit (Suc n) k = signed_take_bit n (k div 2) * 2 + k mod 2"
   by (simp add: odd_iff_mod_2_eq_one signed_take_bit_eq_take_bit algebra_simps)
 
 lemma signed_take_bit_of_0 [simp]:
   "signed_take_bit n 0 = 0"
   by (simp add: signed_take_bit_eq_take_bit take_bit_eq_mod)
 
 lemma signed_take_bit_of_minus_1 [simp]:
   "signed_take_bit n (- 1) = - 1"
   by (induct n) simp_all
 
 lemma signed_take_bit_eq_iff_take_bit_eq:
   "signed_take_bit (n - Suc 0) k = signed_take_bit (n - Suc 0) l \<longleftrightarrow> take_bit n k = take_bit n l" (is "?P \<longleftrightarrow> ?Q")
   if "n > 0"
 proof -
   from that obtain m where m: "n = Suc m"
     by (cases n) auto
   show ?thesis
   proof
     assume ?Q
     have "take_bit (Suc m) (k + 2 ^ m) =
       take_bit (Suc m) (take_bit (Suc m) k + take_bit (Suc m) (2 ^ m))"
       by (simp only: take_bit_add)
     also have "\<dots> =
       take_bit (Suc m) (take_bit (Suc m) l + take_bit (Suc m) (2 ^ m))"
       by (simp only: \<open>?Q\<close> m [symmetric])
     also have "\<dots> = take_bit (Suc m) (l + 2 ^ m)"
       by (simp only: take_bit_add)
     finally show ?P
       by (simp only: signed_take_bit_eq_take_bit m) simp
   next
     assume ?P
     with that have "(k + 2 ^ (n - Suc 0)) mod 2 ^ n = (l + 2 ^ (n - Suc 0)) mod 2 ^ n"
       by (simp add: signed_take_bit_eq_take_bit' take_bit_eq_mod)
     then have "(i + (k + 2 ^ (n - Suc 0))) mod 2 ^ n = (i + (l + 2 ^ (n - Suc 0))) mod 2 ^ n" for i
       by (metis mod_add_eq)
     then have "k mod 2 ^ n = l mod 2 ^ n"
       by (metis add_diff_cancel_right' uminus_add_conv_diff)
     then show ?Q
       by (simp add: take_bit_eq_mod)
   qed
 qed
 
 
 subsection \<open>Bit strings as quotient type\<close>
 
 subsubsection \<open>Basic properties\<close>
 
 quotient_type (overloaded) 'a word = int / "\<lambda>k l. take_bit LENGTH('a) k = take_bit LENGTH('a::len0) l"
   by (auto intro!: equivpI reflpI sympI transpI)
 
 instantiation word :: (len0) "{semiring_numeral, comm_semiring_0, comm_ring}"
 begin
 
 lift_definition zero_word :: "'a word"
   is 0
   .
 
 lift_definition one_word :: "'a word"
   is 1
   .
 
 lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
   is plus
   by (subst take_bit_add [symmetric]) (simp add: take_bit_add)
 
 lift_definition uminus_word :: "'a word \<Rightarrow> 'a word"
   is uminus
   by (subst take_bit_uminus [symmetric]) (simp add: take_bit_uminus)
 
 lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
   is minus
   by (subst take_bit_minus [symmetric]) (simp add: take_bit_minus)
 
 lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
   is times
   by (auto simp add: take_bit_eq_mod intro: mod_mult_cong)
 
 instance
   by standard (transfer; simp add: algebra_simps)+
 
 end
 
 instance word :: (len) comm_ring_1
   by standard (transfer; simp)+
 
 quickcheck_generator word
   constructors:
     "zero_class.zero :: ('a::len0) word",
     "numeral :: num \<Rightarrow> ('a::len0) word",
     "uminus :: ('a::len0) word \<Rightarrow> ('a::len0) word"
 
 context
   includes lifting_syntax
   notes power_transfer [transfer_rule]
 begin
 
 lemma power_transfer_word [transfer_rule]:
   \<open>(pcr_word ===> (=) ===> pcr_word) (^) (^)\<close>
   by transfer_prover
 
 end
 
 
 subsubsection \<open>Conversions\<close>
 
 context
   includes lifting_syntax
   notes 
     transfer_rule_of_bool [transfer_rule]
     transfer_rule_numeral [transfer_rule]
     transfer_rule_of_nat [transfer_rule]
     transfer_rule_of_int [transfer_rule]
 begin
 
 lemma [transfer_rule]:
   "((=) ===> (pcr_word :: int \<Rightarrow> 'a::len word \<Rightarrow> bool)) of_bool of_bool"
   by transfer_prover
 
 lemma [transfer_rule]:
   "((=) ===> (pcr_word :: int \<Rightarrow> 'a::len word \<Rightarrow> bool)) numeral numeral"
   by transfer_prover
 
 lemma [transfer_rule]:
   "((=) ===> pcr_word) int of_nat"
   by transfer_prover
 
 lemma [transfer_rule]:
   "((=) ===> pcr_word) (\<lambda>k. k) of_int"
 proof -
   have "((=) ===> pcr_word) of_int of_int"
     by transfer_prover
   then show ?thesis by (simp add: id_def)
 qed
 
 end
 
 lemma abs_word_eq:
   "abs_word = of_int"
   by (rule ext) (transfer, rule)
 
 context semiring_1
 begin
 
 lift_definition unsigned :: "'b::len0 word \<Rightarrow> 'a"
   is "of_nat \<circ> nat \<circ> take_bit LENGTH('b)"
   by simp
 
 lemma unsigned_0 [simp]:
   "unsigned 0 = 0"
   by transfer simp
 
 end
 
 context semiring_char_0
 begin
 
 lemma word_eq_iff_unsigned:
   "a = b \<longleftrightarrow> unsigned a = unsigned b"
   by safe (transfer; simp add: eq_nat_nat_iff)
 
 end
 
 instantiation word :: (len0) equal
 begin
 
 definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
   where "equal_word a b \<longleftrightarrow> (unsigned a :: int) = unsigned b"
 
 instance proof
   fix a b :: "'a word"
   show "HOL.equal a b \<longleftrightarrow> a = b"
     using word_eq_iff_unsigned [of a b] by (auto simp add: equal_word_def)
 qed
 
 end
 
 context ring_1
 begin
 
 lift_definition signed :: "'b::len word \<Rightarrow> 'a"
   is "of_int \<circ> signed_take_bit (LENGTH('b) - 1)"
   by (simp add: signed_take_bit_eq_iff_take_bit_eq [symmetric])
 
 lemma signed_0 [simp]:
   "signed 0 = 0"
   by transfer simp
 
 end
 
 lemma unsigned_of_nat [simp]:
   "unsigned (of_nat n :: 'a word) = take_bit LENGTH('a::len) n"
   by transfer (simp add: nat_eq_iff take_bit_eq_mod zmod_int)
 
 lemma of_nat_unsigned [simp]:
   "of_nat (unsigned a) = a"
   by transfer simp
 
 lemma of_int_unsigned [simp]:
   "of_int (unsigned a) = a"
   by transfer simp
 
 lemma unsigned_nat_less:
   \<open>unsigned a < (2 ^ LENGTH('a) :: nat)\<close> for a :: \<open>'a::len0 word\<close>
   by transfer (simp add: take_bit_eq_mod)
 
 lemma unsigned_int_less:
   \<open>unsigned a < (2 ^ LENGTH('a) :: int)\<close> for a :: \<open>'a::len0 word\<close>
   by transfer (simp add: take_bit_eq_mod)
 
 context ring_char_0
 begin
 
 lemma word_eq_iff_signed:
   "a = b \<longleftrightarrow> signed a = signed b"
   by safe (transfer; auto simp add: signed_take_bit_eq_iff_take_bit_eq)
 
 end
 
 lemma signed_of_int [simp]:
   "signed (of_int k :: 'a word) = signed_take_bit (LENGTH('a::len) - 1) k"
   by transfer simp
 
 lemma of_int_signed [simp]:
   "of_int (signed a) = a"
   by transfer (simp add: signed_take_bit_eq_take_bit take_bit_eq_mod mod_simps)
 
 
 subsubsection \<open>Properties\<close>
 
 lemma length_cases: \<comment> \<open>TODO get rid of\<close>
   obtains (triv) "LENGTH('a::len) = 1" "take_bit LENGTH('a) 2 = (0 :: int)"
     | (take_bit_2) "take_bit LENGTH('a) 2 = (2 :: int)"
 proof (cases "LENGTH('a) \<ge> 2")
   case False
   then have "LENGTH('a) = 1"
     by (auto simp add: not_le dest: less_2_cases)
   then have "take_bit LENGTH('a) 2 = (0 :: int)"
     by simp
   with \<open>LENGTH('a) = 1\<close> triv show ?thesis
     by simp
 next
   case True
   then obtain n where "LENGTH('a) = Suc (Suc n)"
     by (auto dest: le_Suc_ex)
   then have "take_bit LENGTH('a) 2 = (2 :: int)"
     by simp
   with take_bit_2 show ?thesis
     by simp
 qed
 
 
 subsubsection \<open>Division\<close>
 
 instantiation word :: (len0) modulo
 begin
 
 lift_definition divide_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
   is "\<lambda>a b. take_bit LENGTH('a) a div take_bit LENGTH('a) b"
   by simp
 
 lift_definition modulo_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
   is "\<lambda>a b. take_bit LENGTH('a) a mod take_bit LENGTH('a) b"
   by simp
 
 instance ..
 
 end
 
 lemma zero_word_div_eq [simp]:
   \<open>0 div a = 0\<close> for a :: \<open>'a::len0 word\<close>
   by transfer simp
 
 lemma div_zero_word_eq [simp]:
   \<open>a div 0 = 0\<close> for a :: \<open>'a::len0 word\<close>
   by transfer simp
 
+(*lemma
+  \<open>a div a = of_bool (a \<noteq> 0)\<close> for a :: \<open>'a::len word\<close>
+  by transfer  simp*)
+
 context
   includes lifting_syntax
 begin
 
 lemma [transfer_rule]:
   "(pcr_word ===> (\<longleftrightarrow>)) even ((dvd) 2 :: 'a::len word \<Rightarrow> bool)"
 proof -
   have even_word_unfold: "even k \<longleftrightarrow> (\<exists>l. take_bit LENGTH('a) k = take_bit LENGTH('a) (2 * l))" (is "?P \<longleftrightarrow> ?Q")
     for k :: int
   proof
     assume ?P
     then show ?Q
       by auto
   next
     assume ?Q
     then obtain l where "take_bit LENGTH('a) k = take_bit LENGTH('a) (2 * l)" ..
     then have "even (take_bit LENGTH('a) k)"
       by simp
     then show ?P
       by simp
   qed
   show ?thesis by (simp only: even_word_unfold [abs_def] dvd_def [where ?'a = "'a word", abs_def])
     transfer_prover
 qed
 
 end
 
 instance word :: (len) semiring_modulo
 proof
   show "a div b * b + a mod b = a" for a b :: "'a word"
   proof transfer
     fix k l :: int
     define r :: int where "r = 2 ^ LENGTH('a)"
     then have r: "take_bit LENGTH('a) k = k mod r" for k
       by (simp add: take_bit_eq_mod)
     have "k mod r = ((k mod r) div (l mod r) * (l mod r)
       + (k mod r) mod (l mod r)) mod r"
       by (simp add: div_mult_mod_eq)
     also have "... = (((k mod r) div (l mod r) * (l mod r)) mod r
       + (k mod r) mod (l mod r)) mod r"
       by (simp add: mod_add_left_eq)
     also have "... = (((k mod r) div (l mod r) * l) mod r
       + (k mod r) mod (l mod r)) mod r"
       by (simp add: mod_mult_right_eq)
     finally have "k mod r = ((k mod r) div (l mod r) * l
       + (k mod r) mod (l mod r)) mod r"
       by (simp add: mod_simps)
     with r show "take_bit LENGTH('a) (take_bit LENGTH('a) k div take_bit LENGTH('a) l * l
       + take_bit LENGTH('a) k mod take_bit LENGTH('a) l) = take_bit LENGTH('a) k"
       by simp
   qed
 qed
 
 instance word :: (len) semiring_parity
 proof
   show "\<not> 2 dvd (1::'a word)"
     by transfer simp
   show even_iff_mod_2_eq_0: "2 dvd a \<longleftrightarrow> a mod 2 = 0"
     for a :: "'a word"
     by (transfer; cases rule: length_cases [where ?'a = 'a]) (simp_all add: mod_2_eq_odd)
   show "\<not> 2 dvd a \<longleftrightarrow> a mod 2 = 1"
     for a :: "'a word"
     by (transfer; cases rule: length_cases [where ?'a = 'a]) (simp_all add: mod_2_eq_odd)
 qed
 
+(*lemma
+  \<open>2 ^ n = (0 :: 'a word) \<longleftrightarrow> LENGTH('a::len) \<le> n\<close>
+  apply transfer*)
+  
+
 
 subsubsection \<open>Orderings\<close>
 
 instantiation word :: (len0) linorder
 begin
 
 lift_definition less_eq_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
   is "\<lambda>a b. take_bit LENGTH('a) a \<le> take_bit LENGTH('a) b"
   by simp
 
 lift_definition less_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
   is "\<lambda>a b. take_bit LENGTH('a) a < take_bit LENGTH('a) b"
   by simp
 
 instance
   by standard (transfer; auto)+
 
 end
 
 context linordered_semidom
 begin
 
 lemma word_less_eq_iff_unsigned:
   "a \<le> b \<longleftrightarrow> unsigned a \<le> unsigned b"
   by (transfer fixing: less_eq) (simp add: nat_le_eq_zle)
 
 lemma word_less_iff_unsigned:
   "a < b \<longleftrightarrow> unsigned a < unsigned b"
   by (transfer fixing: less) (auto dest: preorder_class.le_less_trans [OF take_bit_nonnegative])
 
 end
 
 lemma word_greater_zero_iff:
   \<open>a > 0 \<longleftrightarrow> a \<noteq> 0\<close> for a :: \<open>'a::len0 word\<close>
   by transfer (simp add: less_le)
 
 lemma of_nat_word_eq_iff:
   \<open>of_nat m = (of_nat n :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) m = take_bit LENGTH('a) n\<close>
   by transfer (simp add: take_bit_of_nat)
 
 lemma of_nat_word_less_eq_iff:
   \<open>of_nat m \<le> (of_nat n :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) m \<le> take_bit LENGTH('a) n\<close>
   by transfer (simp add: take_bit_of_nat)
 
 lemma of_nat_word_less_iff:
   \<open>of_nat m < (of_nat n :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) m < take_bit LENGTH('a) n\<close>
   by transfer (simp add: take_bit_of_nat)
 
 lemma of_nat_word_eq_0_iff:
   \<open>of_nat n = (0 :: 'a::len word) \<longleftrightarrow> 2 ^ LENGTH('a) dvd n\<close>
   using of_nat_word_eq_iff [where ?'a = 'a, of n 0] by (simp add: take_bit_eq_0_iff)
 
 lemma of_int_word_eq_iff:
   \<open>of_int k = (of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
   by transfer rule
 
 lemma of_int_word_less_eq_iff:
   \<open>of_int k \<le> (of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k \<le> take_bit LENGTH('a) l\<close>
   by transfer rule
 
 lemma of_int_word_less_iff:
   \<open>of_int k < (of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k < take_bit LENGTH('a) l\<close>
   by transfer rule
 
 lemma of_int_word_eq_0_iff:
   \<open>of_int k = (0 :: 'a::len word) \<longleftrightarrow> 2 ^ LENGTH('a) dvd k\<close>
   using of_int_word_eq_iff [where ?'a = 'a, of k 0] by (simp add: take_bit_eq_0_iff)
 
 
 subsection \<open>Bit structure on \<^typ>\<open>'a word\<close>\<close>
 
 lemma word_bit_induct [case_names zero even odd]:
   \<open>P a\<close> if word_zero: \<open>P 0\<close>
     and word_even: \<open>\<And>a. P a \<Longrightarrow> 0 < a \<Longrightarrow> a < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> P (2 * a)\<close>
     and word_odd: \<open>\<And>a. P a \<Longrightarrow> a < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> P (1 + 2 * a)\<close>
   for P and a :: \<open>'a::len word\<close>
 proof -
   define m :: nat where \<open>m = LENGTH('a) - 1\<close>
   then have l: \<open>LENGTH('a) = Suc m\<close>
     by simp
   define n :: nat where \<open>n = unsigned a\<close>
   then have \<open>n < 2 ^ LENGTH('a)\<close>
     by (simp add: unsigned_nat_less)
   then have \<open>n < 2 * 2 ^ m\<close>
     by (simp add: l)
   then have \<open>P (of_nat n)\<close>
   proof (induction n rule: nat_bit_induct)
     case zero
     show ?case
       by simp (rule word_zero)
   next
     case (even n)
     then have \<open>n < 2 ^ m\<close>
       by simp
     with even.IH have \<open>P (of_nat n)\<close>
       by simp
     moreover from \<open>n < 2 ^ m\<close> even.hyps have \<open>0 < (of_nat n :: 'a word)\<close>
       by (auto simp add: word_greater_zero_iff of_nat_word_eq_0_iff l)
     moreover from \<open>n < 2 ^ m\<close> have \<open>(of_nat n :: 'a word) < 2 ^ (LENGTH('a) - 1)\<close>
       using of_nat_word_less_iff [where ?'a = 'a, of n \<open>2 ^ m\<close>]
       by (cases \<open>m = 0\<close>) (simp_all add: not_less take_bit_eq_self ac_simps l)
     ultimately have \<open>P (2 * of_nat n)\<close>
       by (rule word_even)
     then show ?case
       by simp
   next
     case (odd n)
     then have \<open>Suc n \<le> 2 ^ m\<close>
       by simp
     with odd.IH have \<open>P (of_nat n)\<close>
       by simp
     moreover from \<open>Suc n \<le> 2 ^ m\<close> have \<open>(of_nat n :: 'a word) < 2 ^ (LENGTH('a) - 1)\<close>
       using of_nat_word_less_iff [where ?'a = 'a, of n \<open>2 ^ m\<close>]
       by (cases \<open>m = 0\<close>) (simp_all add: not_less take_bit_eq_self ac_simps l)
     ultimately have \<open>P (1 + 2 * of_nat n)\<close>
       by (rule word_odd)
     then show ?case
       by simp
   qed
   then show ?thesis
     by (simp add: n_def)
 qed
 
 lemma bit_word_half_eq:
   \<open>(of_bool b + a * 2) div 2 = a\<close>
     if \<open>a < 2 ^ (LENGTH('a) - Suc 0)\<close>
     for a :: \<open>'a::len word\<close>
 proof (cases rule: length_cases [where ?'a = 'a])
   case triv
   have \<open>of_bool (odd k) < (1 :: int) \<longleftrightarrow> even k\<close> for k :: int
     by auto
   with triv that show ?thesis
     by (auto; transfer) simp_all
 next
   case take_bit_2
   obtain n where length: \<open>LENGTH('a) = Suc n\<close>
     by (cases \<open>LENGTH('a)\<close>) simp_all
   show ?thesis proof (cases b)
     case False
     moreover have \<open>a * 2 div 2 = a\<close>
     using that proof transfer
       fix k :: int
       from length have \<open>k * 2 mod 2 ^ LENGTH('a) = (k mod 2 ^ n) * 2\<close>
         by simp
       moreover assume \<open>take_bit LENGTH('a) k < take_bit LENGTH('a) (2 ^ (LENGTH('a) - Suc 0))\<close>
       with \<open>LENGTH('a) = Suc n\<close>
       have \<open>k mod 2 ^ LENGTH('a) = k mod 2 ^ n\<close>
         by (simp add: take_bit_eq_mod divmod_digit_0)
       ultimately have \<open>take_bit LENGTH('a) (k * 2) = take_bit LENGTH('a) k * 2\<close>
         by (simp add: take_bit_eq_mod)
       with take_bit_2 show \<open>take_bit LENGTH('a) (take_bit LENGTH('a) (k * 2) div take_bit LENGTH('a) 2)
         = take_bit LENGTH('a) k\<close>
         by simp
     qed
     ultimately show ?thesis
       by simp
   next
     case True
     moreover have \<open>(1 + a * 2) div 2 = a\<close>
     using that proof transfer
       fix k :: int
       from length have \<open>(1 + k * 2) mod 2 ^ LENGTH('a) = 1 + (k mod 2 ^ n) * 2\<close>
         using pos_zmod_mult_2 [of \<open>2 ^ n\<close> k] by (simp add: ac_simps)
       moreover assume \<open>take_bit LENGTH('a) k < take_bit LENGTH('a) (2 ^ (LENGTH('a) - Suc 0))\<close>
       with \<open>LENGTH('a) = Suc n\<close>
       have \<open>k mod 2 ^ LENGTH('a) = k mod 2 ^ n\<close>
         by (simp add: take_bit_eq_mod divmod_digit_0)
       ultimately have \<open>take_bit LENGTH('a) (1 + k * 2) = 1 + take_bit LENGTH('a) k * 2\<close>
         by (simp add: take_bit_eq_mod)
       with take_bit_2 show \<open>take_bit LENGTH('a) (take_bit LENGTH('a) (1 + k * 2) div take_bit LENGTH('a) 2)
         = take_bit LENGTH('a) k\<close>
         by simp
     qed
     ultimately show ?thesis
       by simp
   qed
 qed
 
 instance word :: (len) semiring_bits
 proof
   show \<open>P a\<close> if stable: \<open>\<And>a. a div 2 = a \<Longrightarrow> P a\<close>
     and rec: \<open>\<And>a b. P a \<Longrightarrow> (of_bool b + 2 * a) div 2 = a \<Longrightarrow> P (of_bool b + 2 * a)\<close>
   for P and a :: \<open>'a word\<close>
   proof (induction a rule: word_bit_induct)
     case zero
     from stable [of 0] show ?case
       by simp
   next
     case (even a)
     with rec [of a False] show ?case
       using bit_word_half_eq [of a False] by (simp add: ac_simps)
   next
     case (odd a)
     with rec [of a True] show ?case
       using bit_word_half_eq [of a True] by (simp add: ac_simps)
   qed
   show \<open>0 div a = 0\<close>
     for a :: \<open>'a word\<close>
     by transfer simp
   show \<open>a div 1 = a\<close>
     for a :: \<open>'a word\<close>
     by transfer simp
   show \<open>a mod b div b = 0\<close>
     for a b :: \<open>'a word\<close>
     apply transfer
     apply (simp add: take_bit_eq_mod)
     apply (subst (3) mod_pos_pos_trivial [of _ \<open>2 ^ LENGTH('a)\<close>])
       apply simp_all
      apply (metis le_less mod_by_0 pos_mod_conj zero_less_numeral zero_less_power)
     using pos_mod_bound [of \<open>2 ^ LENGTH('a)\<close>] apply simp
   proof -
     fix aa :: int and ba :: int
     have f1: "\<And>i n. (i::int) mod 2 ^ n = 0 \<or> 0 < i mod 2 ^ n"
       by (metis le_less take_bit_eq_mod take_bit_nonnegative)
     have "(0::int) < 2 ^ len_of (TYPE('a)::'a itself) \<and> ba mod 2 ^ len_of (TYPE('a)::'a itself) \<noteq> 0 \<or> aa mod 2 ^ len_of (TYPE('a)::'a itself) mod (ba mod 2 ^ len_of (TYPE('a)::'a itself)) < 2 ^ len_of (TYPE('a)::'a itself)"
       by (metis (no_types) mod_by_0 unique_euclidean_semiring_numeral_class.pos_mod_bound zero_less_numeral zero_less_power)
     then show "aa mod 2 ^ len_of (TYPE('a)::'a itself) mod (ba mod 2 ^ len_of (TYPE('a)::'a itself)) < 2 ^ len_of (TYPE('a)::'a itself)"
       using f1 by (meson le_less less_le_trans unique_euclidean_semiring_numeral_class.pos_mod_bound)
   qed
   show \<open>(1 + a) div 2 = a div 2\<close>
     if \<open>even a\<close>
     for a :: \<open>'a word\<close>
     using that by transfer (auto dest: le_Suc_ex)
   show \<open>(2 :: 'a word) ^ m div 2 ^ n = of_bool ((2 :: 'a word) ^ m \<noteq> 0 \<and> n \<le> m) * 2 ^ (m - n)\<close>
     for m n :: nat
     by transfer (simp, simp add: exp_div_exp_eq)
   show "a div 2 ^ m div 2 ^ n = a div 2 ^ (m + n)"
     for a :: "'a word" and m n :: nat
     apply transfer
     apply (auto simp add: not_less take_bit_drop_bit ac_simps simp flip: drop_bit_eq_div)
     apply (simp add: drop_bit_take_bit)
     done
   show "a mod 2 ^ m mod 2 ^ n = a mod 2 ^ min m n"
     for a :: "'a word" and m n :: nat
     apply transfer
     apply (auto simp flip: take_bit_eq_mod)
     apply (simp add: ac_simps)
     done
   show \<open>a * 2 ^ m mod 2 ^ n = a mod 2 ^ (n - m) * 2 ^ m\<close>
     if \<open>m \<le> n\<close> for a :: "'a word" and m n :: nat
     using that apply transfer
     apply (auto simp flip: take_bit_eq_mod)
            apply (auto simp flip: push_bit_eq_mult simp add: push_bit_take_bit split: split_min_lin)
     done
   show \<open>a div 2 ^ n mod 2 ^ m = a mod (2 ^ (n + m)) div 2 ^ n\<close>
     for a :: "'a word" and m n :: nat
     apply transfer
     apply (auto simp add: not_less take_bit_drop_bit ac_simps simp flip: take_bit_eq_mod drop_bit_eq_div split: split_min_lin)
     done
 qed
 
 context
   includes lifting_syntax
 begin
 
-lemma transfer_rule_bit_word:
+lemma transfer_rule_bit_word [transfer_rule]:
   \<open>((pcr_word :: int \<Rightarrow> 'a::len word \<Rightarrow> bool) ===> (=)) (\<lambda>k n. n < LENGTH('a) \<and> bit k n) bit\<close>
 proof -
   let ?t = \<open>\<lambda>a n. odd (take_bit LENGTH('a) a div take_bit LENGTH('a) ((2::int) ^ n))\<close>
   have \<open>((pcr_word :: int \<Rightarrow> 'a word \<Rightarrow> bool) ===> (=)) ?t bit\<close>
     by (unfold bit_def) transfer_prover
   also have \<open>?t = (\<lambda>k n. n < LENGTH('a) \<and> bit k n)\<close>
     by (simp add: fun_eq_iff bit_take_bit_iff flip: bit_def)
   finally show ?thesis .
 qed
 
 end
 
 instantiation word :: (len) semiring_bit_shifts
 begin
 
 lift_definition push_bit_word :: \<open>nat \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
   is push_bit
 proof -
   show \<open>Parity.take_bit LENGTH('a) (push_bit n k) = Parity.take_bit LENGTH('a) (push_bit n l)\<close>
     if \<open>Parity.take_bit LENGTH('a) k = Parity.take_bit LENGTH('a) l\<close> for k l :: int and n :: nat
   proof -
     from that
     have \<open>Parity.take_bit (LENGTH('a) - n) (Parity.take_bit LENGTH('a) k)
       = Parity.take_bit (LENGTH('a) - n) (Parity.take_bit LENGTH('a) l)\<close>
       by simp
     moreover have \<open>min (LENGTH('a) - n) LENGTH('a) = LENGTH('a) - n\<close>
       by simp
     ultimately show ?thesis
       by (simp add: take_bit_push_bit)
   qed
 qed
 
 lift_definition drop_bit_word :: \<open>nat \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>n. drop_bit n \<circ> take_bit LENGTH('a)\<close>
   by (simp add: take_bit_eq_mod)
 
 instance proof
   show \<open>push_bit n a = a * 2 ^ n\<close> for n :: nat and a :: "'a word"
     by transfer (simp add: push_bit_eq_mult)
   show \<open>drop_bit n a = a div 2 ^ n\<close> for n :: nat and a :: "'a word"
   proof (cases \<open>n < LENGTH('a)\<close>)
     case True
     then show ?thesis
       by transfer
         (simp add: take_bit_eq_mod drop_bit_eq_div)
   next
     case False
     then obtain m where n: \<open>n = LENGTH('a) + m\<close>
       by (auto simp add: not_less dest: le_Suc_ex)
     then show ?thesis
       by transfer
         (simp add: take_bit_eq_mod drop_bit_eq_div power_add zdiv_zmult2_eq)
   qed
 qed
 
 end
 
 instantiation word :: (len) ring_bit_operations
 begin
 
 lift_definition not_word :: "'a word \<Rightarrow> 'a word"
   is not
   by (simp add: take_bit_not_iff)
 
 lift_definition and_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
   is "and"
   by simp
 
 lift_definition or_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
   is or
   by simp
 
 lift_definition xor_word ::  "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
   is xor
   by simp
 
 instance proof
-  interpret bit_word: boolean_algebra "(AND)" "(OR)" NOT 0 "- 1 :: 'a word"
-  proof
-    show "a AND (b OR c) = a AND b OR a AND c"
-      for a b c :: "'a word"
-      by transfer (simp add: bit.conj_disj_distrib)
-    show "a OR b AND c = (a OR b) AND (a OR c)"
-      for a b c :: "'a word"
-      by transfer (simp add: bit.disj_conj_distrib)
-  qed (transfer; simp add: ac_simps)+
-  show "boolean_algebra (AND) (OR) NOT 0 (- 1 :: 'a word)"
-    by (fact bit_word.boolean_algebra_axioms)
-  show "bit_word.xor = ((XOR) :: 'a word \<Rightarrow> _)"
-  proof (rule ext)+
-    fix a b :: "'a word"
-    have "a XOR b = a AND NOT b OR NOT a AND b"
-      by transfer (simp add: bit.xor_def)
-    then show "bit_word.xor a b = a XOR b"
-      by (simp add: bit_word.xor_def)
-  qed
+  fix a b :: \<open>'a word\<close> and n :: nat
+  show \<open>even (- 1 div (2 :: 'a word) ^ n) \<longleftrightarrow> (2 :: 'a word) ^ n = 0\<close>
+    by transfer
+      (simp flip: drop_bit_eq_div add: drop_bit_take_bit, simp add: drop_bit_eq_div)
+  show \<open>bit (NOT a) n \<longleftrightarrow> (2 :: 'a word) ^ n \<noteq> 0 \<and> \<not> bit a n\<close>
+    by transfer (simp add: bit_not_iff)
+  show \<open>bit (a AND b) n \<longleftrightarrow> bit a n \<and> bit b n\<close>
+    by transfer (auto simp add: bit_and_iff)
+  show \<open>bit (a OR b) n \<longleftrightarrow> bit a n \<or> bit b n\<close>
+    by transfer (auto simp add: bit_or_iff)
+  show \<open>bit (a XOR b) n \<longleftrightarrow> bit a n \<noteq> bit b n\<close>
+    by transfer (auto simp add: bit_xor_iff)
 qed
 
 end
 
 end