diff --git a/src/HOL/Library/Bit_Operations.thy b/src/HOL/Library/Bit_Operations.thy
--- a/src/HOL/Library/Bit_Operations.thy
+++ b/src/HOL/Library/Bit_Operations.thy
@@ -1,1468 +1,1536 @@
 (*  Author:  Florian Haftmann, TUM
 *)
 
 section \<open>Bit operations in suitable algebraic structures\<close>
 
 theory Bit_Operations
   imports
     "HOL-Library.Boolean_Algebra"
     Main
 begin
 
 subsection \<open>Bit operations\<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)
     and mask :: \<open>nat \<Rightarrow> 'a\<close>
   assumes bit_and_iff: \<open>\<And>n. bit (a AND b) n \<longleftrightarrow> bit a n \<and> bit b n\<close>
     and bit_or_iff: \<open>\<And>n. bit (a OR b) n \<longleftrightarrow> bit a n \<or> bit b n\<close>
     and bit_xor_iff: \<open>\<And>n. bit (a XOR b) n \<longleftrightarrow> bit a n \<noteq> bit b n\<close>
     and mask_eq_exp_minus_1: \<open>mask n = 2 ^ n - 1\<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)
 
 lemma push_bit_and [simp]:
   \<open>push_bit n (a AND b) = push_bit n a AND push_bit n b\<close>
   by (rule bit_eqI) (auto simp add: bit_push_bit_iff bit_and_iff)
 
 lemma push_bit_or [simp]:
   \<open>push_bit n (a OR b) = push_bit n a OR push_bit n b\<close>
   by (rule bit_eqI) (auto simp add: bit_push_bit_iff bit_or_iff)
 
 lemma push_bit_xor [simp]:
   \<open>push_bit n (a XOR b) = push_bit n a XOR push_bit n b\<close>
   by (rule bit_eqI) (auto simp add: bit_push_bit_iff bit_xor_iff)
 
 lemma drop_bit_and [simp]:
   \<open>drop_bit n (a AND b) = drop_bit n a AND drop_bit n b\<close>
   by (rule bit_eqI) (auto simp add: bit_drop_bit_eq bit_and_iff)
 
 lemma drop_bit_or [simp]:
   \<open>drop_bit n (a OR b) = drop_bit n a OR drop_bit n b\<close>
   by (rule bit_eqI) (auto simp add: bit_drop_bit_eq bit_or_iff)
 
 lemma drop_bit_xor [simp]:
   \<open>drop_bit n (a XOR b) = drop_bit n a XOR drop_bit n b\<close>
   by (rule bit_eqI) (auto simp add: bit_drop_bit_eq bit_xor_iff)
 
 lemma bit_mask_iff:
   \<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]:
   \<open>mask 0 = 0\<close>
   by (simp add: mask_eq_exp_minus_1)
 
 lemma mask_Suc_0 [simp]:
   \<open>mask (Suc 0) = 1\<close>
   by (simp add: mask_eq_exp_minus_1 add_implies_diff sym)
 
 lemma mask_Suc_exp:
   \<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) = 1 OR 2 * mask n\<close>
 proof (rule bit_eqI)
   fix q
   assume \<open>2 ^ q \<noteq> 0\<close>
   show \<open>bit (mask (Suc n)) q \<longleftrightarrow> bit (1 OR 2 * mask n) 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 mask_numeral:
   \<open>mask (numeral n) = 1 + 2 * mask (pred_numeral n)\<close>
   by (simp add: numeral_eq_Suc mask_Suc_double one_or_eq ac_simps)
 
 lemma take_bit_eq_mask:
   \<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)
 
 lemma or_eq_0_iff:
   \<open>a OR b = 0 \<longleftrightarrow> a = 0 \<and> b = 0\<close>
 	by (auto simp add: bit_eq_iff bit_or_iff)
 
 lemma disjunctive_add:
   \<open>a + b = a OR b\<close> if \<open>\<And>n. \<not> bit a n \<or> \<not> bit b n\<close>
   by (rule bit_eqI) (use that in \<open>simp add: bit_disjunctive_add_iff bit_or_iff\<close>)
 
 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>
   by standard (rule bit_eqI, simp add: bit_and_iff)
 
 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 standard (auto simp add: bit_and_iff bit_or_iff bit_not_iff intro: bit_eqI)
   show \<open>boolean_algebra (AND) (OR) NOT 0 (- 1)\<close>
     by standard
   show \<open>boolean_algebra.xor (AND) (OR) NOT = (XOR)\<close>
     by (rule ext, rule ext, rule bit_eqI)
       (auto simp add: bit.xor_def bit_and_iff bit_or_iff bit_xor_iff bit_not_iff)
 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 not_add_distrib:
   \<open>NOT (a + b) = NOT a - b\<close>
   by (simp add: not_eq_complement algebra_simps)
 
 lemma not_diff_distrib:
   \<open>NOT (a - b) = NOT a + b\<close>
   using not_add_distrib [of a \<open>- b\<close>] by simp
 
 lemma (in ring_bit_operations) and_eq_minus_1_iff:
   \<open>a AND b = - 1 \<longleftrightarrow> a = - 1 \<and> b = - 1\<close>
 proof
   assume \<open>a = - 1 \<and> b = - 1\<close>
   then show \<open>a AND b = - 1\<close>
 	by simp
 next
   assume \<open>a AND b = - 1\<close>
   have *: \<open>bit a n\<close> \<open>bit b n\<close> if \<open>2 ^ n \<noteq> 0\<close> for n
   proof -
     from \<open>a AND b = - 1\<close>
     have \<open>bit (a AND b) n = bit (- 1) n\<close>
       by (simp add: bit_eq_iff)
     then show \<open>bit a n\<close> \<open>bit b n\<close>
 	    using that by (simp_all add: bit_and_iff)
   qed
   have \<open>a = - 1\<close>
     by (rule bit_eqI) (simp add: *)
   moreover have \<open>b = - 1\<close>
     by (rule bit_eqI) (simp add: *)
   ultimately show \<open>a = - 1 \<and> b = - 1\<close>
     by simp
 qed
 
 lemma disjunctive_diff:
   \<open>a - b = a AND NOT b\<close> if \<open>\<And>n. bit b n \<Longrightarrow> bit a n\<close>
 proof -
   have \<open>NOT a + b = NOT a OR b\<close>
     by (rule disjunctive_add) (auto simp add: bit_not_iff dest: that)
   then have \<open>NOT (NOT a + b) = NOT (NOT a OR b)\<close>
     by simp
   then show ?thesis
     by (simp add: not_add_distrib)
 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)
   apply (simp add: bit_not_iff bit_take_bit_iff bit_exp_iff)
   apply (use exp_eq_0_imp_not_bit in blast)
   done
 
 lemma take_bit_not_eq_mask_diff:
   \<open>take_bit n (NOT a) = mask n - take_bit n a\<close>
 proof -
   have \<open>take_bit n (NOT a) = take_bit n (NOT (take_bit n a))\<close>
     by (simp add: take_bit_not_take_bit)
   also have \<open>\<dots> = mask n AND NOT (take_bit n a)\<close>
     by (simp add: take_bit_eq_mask ac_simps)
   also have \<open>\<dots> = mask n - take_bit n a\<close>
     by (subst disjunctive_diff)
       (auto simp add: bit_take_bit_iff bit_mask_iff exp_eq_0_imp_not_bit)
   finally show ?thesis
     by simp
 qed
 
 lemma mask_eq_take_bit_minus_one:
   \<open>mask n = take_bit n (- 1)\<close>
   by (simp add: bit_eq_iff bit_mask_iff bit_take_bit_iff conj_commute)
 
 lemma take_bit_minus_one_eq_mask:
   \<open>take_bit n (- 1) = mask n\<close>
   by (simp add: mask_eq_take_bit_minus_one)
 
 lemma minus_exp_eq_not_mask:
   \<open>- (2 ^ n) = NOT (mask n)\<close>
   by (rule bit_eqI) (simp add: bit_minus_iff bit_not_iff flip: mask_eq_exp_minus_1)
 
 lemma push_bit_minus_one_eq_not_mask:
   \<open>push_bit n (- 1) = NOT (mask n)\<close>
   by (simp add: push_bit_eq_mult minus_exp_eq_not_mask)
 
 lemma take_bit_not_mask_eq_0:
   \<open>take_bit m (NOT (mask n)) = 0\<close> if \<open>n \<ge> m\<close>
   by (rule bit_eqI) (use that in \<open>simp add: bit_take_bit_iff bit_not_iff bit_mask_iff\<close>)
 
 lemma take_bit_mask [simp]:
   \<open>take_bit m (mask n) = mask (min m n)\<close>
   by (simp add: mask_eq_take_bit_minus_one)
 
 definition set_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   where \<open>set_bit n a = a OR push_bit n 1\<close>
 
 definition unset_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   where \<open>unset_bit n a = a AND NOT (push_bit n 1)\<close>
 
 definition flip_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   where \<open>flip_bit n a = a XOR push_bit n 1\<close>
 
 lemma bit_set_bit_iff:
   \<open>bit (set_bit m a) n \<longleftrightarrow> bit a n \<or> (m = n \<and> 2 ^ n \<noteq> 0)\<close>
   by (auto simp add: set_bit_def push_bit_of_1 bit_or_iff bit_exp_iff)
 
 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 push_bit_of_1 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 push_bit_of_1 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
 
 lemma flip_bit_eq_if:
   \<open>flip_bit n a = (if bit a n then unset_bit else set_bit) n a\<close>
   by (rule bit_eqI) (auto simp add: bit_set_bit_iff bit_unset_bit_iff bit_flip_bit_iff)
 
 lemma take_bit_set_bit_eq:
   \<open>take_bit n (set_bit m a) = (if n \<le> m then take_bit n a else set_bit m (take_bit n a))\<close>
   by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_set_bit_iff)
 
 lemma take_bit_unset_bit_eq:
   \<open>take_bit n (unset_bit m a) = (if n \<le> m then take_bit n a else unset_bit m (take_bit n a))\<close>
   by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_unset_bit_iff)
 
 lemma take_bit_flip_bit_eq:
   \<open>take_bit n (flip_bit m a) = (if n \<le> m then take_bit n a else flip_bit m (take_bit n a))\<close>
   by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_flip_bit_iff)
 
 end
 
 
 subsection \<open>Instance \<^typ>\<open>int\<close>\<close>
 
+lemma int_bit_bound:
+  fixes k :: int
+  obtains n where \<open>\<And>m. n \<le> m \<Longrightarrow> bit k m \<longleftrightarrow> bit k n\<close>
+    and \<open>n > 0 \<Longrightarrow> bit k (n - 1) \<noteq> bit k n\<close>
+proof -
+  obtain q where *: \<open>\<And>m. q \<le> m \<Longrightarrow> bit k m \<longleftrightarrow> bit k q\<close>
+  proof (cases \<open>k \<ge> 0\<close>)
+    case True
+    moreover from power_gt_expt [of 2 \<open>nat k\<close>]
+    have \<open>k < 2 ^ nat k\<close> by simp
+    ultimately have *: \<open>k div 2 ^ nat k = 0\<close>
+      by simp
+    show thesis
+    proof (rule that [of \<open>nat k\<close>])
+      fix m
+      assume \<open>nat k \<le> m\<close>
+      then show \<open>bit k m \<longleftrightarrow> bit k (nat k)\<close>
+        by (auto simp add: * bit_iff_odd power_add zdiv_zmult2_eq dest!: le_Suc_ex)
+    qed
+  next
+    case False
+    moreover from power_gt_expt [of 2 \<open>nat (- k)\<close>]
+    have \<open>- k \<le> 2 ^ nat (- k)\<close>
+      by simp
+    ultimately have \<open>- k div - (2 ^ nat (- k)) = - 1\<close>
+      by (subst div_pos_neg_trivial) simp_all
+    then have *: \<open>k div 2 ^ nat (- k) = - 1\<close>
+      by simp
+    show thesis
+    proof (rule that [of \<open>nat (- k)\<close>])
+      fix m
+      assume \<open>nat (- k) \<le> m\<close>
+      then show \<open>bit k m \<longleftrightarrow> bit k (nat (- k))\<close>
+        by (auto simp add: * bit_iff_odd power_add zdiv_zmult2_eq minus_1_div_exp_eq_int dest!: le_Suc_ex)
+    qed
+  qed
+  show thesis
+  proof (cases \<open>\<forall>m. bit k m \<longleftrightarrow> bit k q\<close>)
+    case True
+    then have \<open>bit k 0 \<longleftrightarrow> bit k q\<close>
+      by blast
+    with True that [of 0] show thesis
+      by simp
+  next
+    case False
+    then obtain r where **: \<open>bit k r \<noteq> bit k q\<close>
+      by blast
+    have \<open>r < q\<close>
+      by (rule ccontr) (use * [of r] ** in simp)
+    define N where \<open>N = {n. n < q \<and> bit k n \<noteq> bit k q}\<close>
+    moreover have \<open>finite N\<close> \<open>r \<in> N\<close>
+      using ** N_def \<open>r < q\<close> by auto
+    moreover define n where \<open>n = Suc (Max N)\<close>
+    ultimately have \<open>\<And>m. n \<le> m \<Longrightarrow> bit k m \<longleftrightarrow> bit k n\<close>
+      apply auto
+         apply (metis (full_types, lifting) "*" Max_ge_iff Suc_n_not_le_n \<open>finite N\<close> all_not_in_conv mem_Collect_eq not_le)
+        apply (metis "*" Max_ge Suc_n_not_le_n \<open>finite N\<close> linorder_not_less mem_Collect_eq)
+        apply (metis "*" Max_ge Suc_n_not_le_n \<open>finite N\<close> linorder_not_less mem_Collect_eq)
+      apply (metis (full_types, lifting) "*" Max_ge_iff Suc_n_not_le_n \<open>finite N\<close> all_not_in_conv mem_Collect_eq not_le)
+      done
+    have \<open>bit k (Max N) \<noteq> bit k n\<close>
+      by (metis (mono_tags, lifting) "*" Max_in N_def \<open>\<And>m. n \<le> m \<Longrightarrow> bit k m = bit k n\<close> \<open>finite N\<close> \<open>r \<in> N\<close> empty_iff le_cases mem_Collect_eq)
+    show thesis apply (rule that [of n])
+      using \<open>\<And>m. n \<le> m \<Longrightarrow> bit k m = bit k n\<close> apply blast
+      using \<open>bit k (Max N) \<noteq> bit k n\<close> n_def by auto
+  qed
+qed
+
 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)
 
 definition mask_int :: \<open>nat \<Rightarrow> int\<close>
   where \<open>mask n = (2 :: int) ^ n - 1\<close>
 
 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 mask_int_def)
 
 end
 
 
 lemma mask_half_int:
   \<open>mask n div 2 = (mask (n - 1) :: int)\<close>
   by (cases n) (simp_all add: mask_eq_exp_minus_1 algebra_simps)
 
 lemma mask_nonnegative_int [simp]:
   \<open>mask n \<ge> (0::int)\<close>
   by (simp add: mask_eq_exp_minus_1)
 
 lemma not_mask_negative_int [simp]:
   \<open>\<not> mask n < (0::int)\<close>
   by (simp add: not_less)
 
 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 and_less_eq:
   \<open>k AND l \<le> k\<close> if \<open>l < 0\<close> for k l :: int
 using that 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)
   from even.IH [of \<open>l div 2\<close>] even.hyps even.prems
   show ?case
     by (simp add: and_int_rec [of _ l])
 next
   case (odd k)
   from odd.IH [of \<open>l div 2\<close>] odd.hyps odd.prems
   show ?case
     by (simp add: and_int_rec [of _ l])
 qed
 
 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 or_greater_eq:
   \<open>k OR l \<ge> k\<close> if \<open>l \<ge> 0\<close> for k l :: int
 using that 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)
   from even.IH [of \<open>l div 2\<close>] even.hyps even.prems
   show ?case
     by (simp add: or_int_rec [of _ l])
 next
   case (odd k)
   from odd.IH [of \<open>l div 2\<close>] odd.hyps odd.prems
   show ?case
     by (simp add: or_int_rec [of _ l])
 qed
 
 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)
 
 lemma set_bit_greater_eq:
   \<open>set_bit n k \<ge> k\<close> for k :: int
   by (simp add: set_bit_def or_greater_eq)
 
 lemma unset_bit_less_eq:
   \<open>unset_bit n k \<le> k\<close> for k :: int
   by (simp add: unset_bit_def and_less_eq)
 
 lemma set_bit_eq:
   \<open>set_bit n k = k + of_bool (\<not> bit k n) * 2 ^ n\<close> for k :: int
 proof (rule bit_eqI)
   fix m
   show \<open>bit (set_bit n k) m \<longleftrightarrow> bit (k + of_bool (\<not> bit k n) * 2 ^ n) m\<close>
   proof (cases \<open>m = n\<close>)
     case True
     then show ?thesis
       apply (simp add: bit_set_bit_iff)
       apply (simp add: bit_iff_odd div_plus_div_distrib_dvd_right)
       done
   next
     case False
     then show ?thesis
       apply (clarsimp simp add: bit_set_bit_iff)
       apply (subst disjunctive_add)
       apply (clarsimp simp add: bit_exp_iff)
       apply (clarsimp simp add: bit_or_iff bit_exp_iff)
       done
   qed
 qed
 
 lemma unset_bit_eq:
   \<open>unset_bit n k = k - of_bool (bit k n) * 2 ^ n\<close> for k :: int
 proof (rule bit_eqI)
   fix m
   show \<open>bit (unset_bit n k) m \<longleftrightarrow> bit (k - of_bool (bit k n) * 2 ^ n) m\<close>
   proof (cases \<open>m = n\<close>)
     case True
     then show ?thesis
       apply (simp add: bit_unset_bit_iff)
       apply (simp add: bit_iff_odd)
       using div_plus_div_distrib_dvd_right [of \<open>2 ^ n\<close> \<open>- (2 ^ n)\<close> k]
       apply (simp add: dvd_neg_div)
       done
   next
     case False
     then show ?thesis
       apply (clarsimp simp add: bit_unset_bit_iff)
       apply (subst disjunctive_diff)
       apply (clarsimp simp add: bit_exp_iff)
       apply (clarsimp simp add: bit_and_iff bit_not_iff bit_exp_iff)
       done
   qed
 qed
 
 context ring_bit_operations
 begin
 
 lemma even_of_int_iff:
   \<open>even (of_int k) \<longleftrightarrow> even k\<close>
   by (induction k rule: int_bit_induct) simp_all
 
 lemma bit_of_int_iff:
   \<open>bit (of_int k) n \<longleftrightarrow> (2::'a) ^ n \<noteq> 0 \<and> bit k n\<close>
 proof (cases \<open>(2::'a) ^ n = 0\<close>)
   case True
   then show ?thesis
     by (simp add: exp_eq_0_imp_not_bit)
 next
   case False
   then have \<open>bit (of_int k) n \<longleftrightarrow> bit k n\<close>
   proof (induction k arbitrary: n 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 bit_double_iff [of \<open>of_int k\<close> n] Parity.bit_double_iff [of k n]
       by (cases n) (auto simp add: ac_simps dest: mult_not_zero)
   next
     case (odd k)
     then show ?case
       using bit_double_iff [of \<open>of_int k\<close> n]
       by (cases n) (auto simp add: ac_simps bit_double_iff even_bit_succ_iff Parity.bit_Suc dest: mult_not_zero)
   qed
   with False show ?thesis
     by simp
 qed
 
 lemma push_bit_of_int:
   \<open>push_bit n (of_int k) = of_int (push_bit n k)\<close>
   by (simp add: push_bit_eq_mult semiring_bit_shifts_class.push_bit_eq_mult)
 
 lemma of_int_push_bit:
   \<open>of_int (push_bit n k) = push_bit n (of_int k)\<close>
   by (simp add: push_bit_eq_mult semiring_bit_shifts_class.push_bit_eq_mult)
 
 lemma take_bit_of_int:
   \<open>take_bit n (of_int k) = of_int (take_bit n k)\<close>
   by (rule bit_eqI) (simp add: bit_take_bit_iff Parity.bit_take_bit_iff bit_of_int_iff)
 
 lemma of_int_take_bit:
   \<open>of_int (take_bit n k) = take_bit n (of_int k)\<close>
   by (rule bit_eqI) (simp add: bit_take_bit_iff Parity.bit_take_bit_iff bit_of_int_iff)
 
 lemma of_int_not_eq:
   \<open>of_int (NOT k) = NOT (of_int k)\<close>
   by (rule bit_eqI) (simp add: bit_not_iff Bit_Operations.bit_not_iff bit_of_int_iff)
 
 lemma of_int_and_eq:
   \<open>of_int (k AND l) = of_int k AND of_int l\<close>
   by (rule bit_eqI) (simp add: bit_of_int_iff bit_and_iff Bit_Operations.bit_and_iff)
 
 lemma of_int_or_eq:
   \<open>of_int (k OR l) = of_int k OR of_int l\<close>
   by (rule bit_eqI) (simp add: bit_of_int_iff bit_or_iff Bit_Operations.bit_or_iff)
 
 lemma of_int_xor_eq:
   \<open>of_int (k XOR l) = of_int k XOR of_int l\<close>
   by (rule bit_eqI) (simp add: bit_of_int_iff bit_xor_iff Bit_Operations.bit_xor_iff)
 
 lemma of_int_mask_eq:
   \<open>of_int (mask n) = mask n\<close>
   by (induction n) (simp_all add: mask_Suc_double Bit_Operations.mask_Suc_double of_int_or_eq)
 
 end
 
 
 subsection \<open>Bit concatenation\<close>
 
 definition concat_bit :: \<open>nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>concat_bit n k l = take_bit n k OR push_bit n l\<close>
 
 lemma bit_concat_bit_iff:
   \<open>bit (concat_bit m k l) n \<longleftrightarrow> n < m \<and> bit k n \<or> m \<le> n \<and> bit l (n - m)\<close>
   by (simp add: concat_bit_def bit_or_iff bit_and_iff bit_take_bit_iff bit_push_bit_iff ac_simps)
 
 lemma concat_bit_eq:
   \<open>concat_bit n k l = take_bit n k + push_bit n l\<close>
   by (simp add: concat_bit_def take_bit_eq_mask
     bit_and_iff bit_mask_iff bit_push_bit_iff disjunctive_add)
 
 lemma concat_bit_0 [simp]:
   \<open>concat_bit 0 k l = l\<close>
   by (simp add: concat_bit_def)
 
 lemma concat_bit_Suc:
   \<open>concat_bit (Suc n) k l = k mod 2 + 2 * concat_bit n (k div 2) l\<close>
   by (simp add: concat_bit_eq take_bit_Suc push_bit_double)
 
 lemma concat_bit_of_zero_1 [simp]:
   \<open>concat_bit n 0 l = push_bit n l\<close>
   by (simp add: concat_bit_def)
 
 lemma concat_bit_of_zero_2 [simp]:
   \<open>concat_bit n k 0 = take_bit n k\<close>
   by (simp add: concat_bit_def take_bit_eq_mask)
 
 lemma concat_bit_nonnegative_iff [simp]:
   \<open>concat_bit n k l \<ge> 0 \<longleftrightarrow> l \<ge> 0\<close>
   by (simp add: concat_bit_def)
 
 lemma concat_bit_negative_iff [simp]:
   \<open>concat_bit n k l < 0 \<longleftrightarrow> l < 0\<close>
   by (simp add: concat_bit_def)
 
 lemma concat_bit_assoc:
   \<open>concat_bit n k (concat_bit m l r) = concat_bit (m + n) (concat_bit n k l) r\<close>
   by (rule bit_eqI) (auto simp add: bit_concat_bit_iff ac_simps)
 
 lemma concat_bit_assoc_sym:
   \<open>concat_bit m (concat_bit n k l) r = concat_bit (min m n) k (concat_bit (m - n) l r)\<close>
   by (rule bit_eqI) (auto simp add: bit_concat_bit_iff ac_simps min_def)
 
 lemma concat_bit_eq_iff:
   \<open>concat_bit n k l = concat_bit n r s
     \<longleftrightarrow> take_bit n k = take_bit n r \<and> l = s\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
 proof
   assume ?Q
   then show ?P
     by (simp add: concat_bit_def)
 next
   assume ?P
   then have *: \<open>bit (concat_bit n k l) m = bit (concat_bit n r s) m\<close> for m
     by (simp add: bit_eq_iff)
   have \<open>take_bit n k = take_bit n r\<close>
   proof (rule bit_eqI)
     fix m
     from * [of m]
     show \<open>bit (take_bit n k) m \<longleftrightarrow> bit (take_bit n r) m\<close>
       by (auto simp add: bit_take_bit_iff bit_concat_bit_iff)
   qed
   moreover have \<open>push_bit n l = push_bit n s\<close>
   proof (rule bit_eqI)
     fix m
     from * [of m]
     show \<open>bit (push_bit n l) m \<longleftrightarrow> bit (push_bit n s) m\<close>
       by (auto simp add: bit_push_bit_iff bit_concat_bit_iff)
   qed
   then have \<open>l = s\<close>
     by (simp add: push_bit_eq_mult)
   ultimately show ?Q
     by (simp add: concat_bit_def)
 qed
 
 lemma take_bit_concat_bit_eq:
   \<open>take_bit m (concat_bit n k l) = concat_bit (min m n) k (take_bit (m - n) l)\<close>
   by (rule bit_eqI)
     (auto simp add: bit_take_bit_iff bit_concat_bit_iff min_def)  
 
 
 subsection \<open>Taking bits with sign propagation\<close>
 
 context ring_bit_operations
 begin
 
 definition signed_take_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   where \<open>signed_take_bit n a = take_bit n a OR (of_bool (bit a n) * NOT (mask n))\<close>
 
 lemma signed_take_bit_eq_if_positive:
   \<open>signed_take_bit n a = take_bit n a\<close> if \<open>\<not> bit a n\<close>
   using that by (simp add: signed_take_bit_def)
 
 lemma signed_take_bit_eq_if_negative:
   \<open>signed_take_bit n a = take_bit n a OR NOT (mask n)\<close> if \<open>bit a n\<close>
   using that by (simp add: signed_take_bit_def)
 
 lemma even_signed_take_bit_iff:
   \<open>even (signed_take_bit m a) \<longleftrightarrow> even a\<close>
   by (auto simp add: signed_take_bit_def even_or_iff even_mask_iff bit_double_iff)
 
 lemma bit_signed_take_bit_iff:
   \<open>bit (signed_take_bit m a) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> bit a (min m n)\<close>
   by (simp add: signed_take_bit_def bit_take_bit_iff bit_or_iff bit_not_iff bit_mask_iff min_def not_le)
     (use exp_eq_0_imp_not_bit in blast)
 
 lemma signed_take_bit_0 [simp]:
   \<open>signed_take_bit 0 a = - (a mod 2)\<close>
   by (simp add: signed_take_bit_def odd_iff_mod_2_eq_one)
 
 lemma signed_take_bit_Suc:
   \<open>signed_take_bit (Suc n) a = a mod 2 + 2 * signed_take_bit n (a div 2)\<close>
 proof (rule bit_eqI)
   fix m
   assume *: \<open>2 ^ m \<noteq> 0\<close>
   show \<open>bit (signed_take_bit (Suc n) a) m \<longleftrightarrow>
     bit (a mod 2 + 2 * signed_take_bit n (a div 2)) m\<close>
   proof (cases m)
     case 0
     then show ?thesis
       by (simp add: even_signed_take_bit_iff)
   next
     case (Suc m)
     with * have \<open>2 ^ m \<noteq> 0\<close>
       by (metis mult_not_zero power_Suc)
     with Suc show ?thesis
       by (simp add: bit_signed_take_bit_iff mod2_eq_if bit_double_iff even_bit_succ_iff
         ac_simps flip: bit_Suc)
   qed
 qed
 
 lemma signed_take_bit_of_0 [simp]:
   \<open>signed_take_bit n 0 = 0\<close>
   by (simp add: signed_take_bit_def)
 
 lemma signed_take_bit_of_minus_1 [simp]:
   \<open>signed_take_bit n (- 1) = - 1\<close>
   by (simp add: signed_take_bit_def take_bit_minus_one_eq_mask mask_eq_exp_minus_1)
 
 lemma signed_take_bit_Suc_1 [simp]:
   \<open>signed_take_bit (Suc n) 1 = 1\<close>
   by (simp add: signed_take_bit_Suc)
 
 lemma signed_take_bit_rec:
   \<open>signed_take_bit n a = (if n = 0 then - (a mod 2) else a mod 2 + 2 * signed_take_bit (n - 1) (a div 2))\<close>
   by (cases n) (simp_all add: signed_take_bit_Suc)
 
 lemma signed_take_bit_eq_iff_take_bit_eq:
   \<open>signed_take_bit n a = signed_take_bit n b \<longleftrightarrow> take_bit (Suc n) a = take_bit (Suc n) b\<close>
 proof -
   have \<open>bit (signed_take_bit n a) = bit (signed_take_bit n b) \<longleftrightarrow> bit (take_bit (Suc n) a) = bit (take_bit (Suc n) b)\<close>
     by (simp add: fun_eq_iff bit_signed_take_bit_iff bit_take_bit_iff not_le less_Suc_eq_le min_def)
       (use exp_eq_0_imp_not_bit in fastforce)
   then show ?thesis
     by (simp add: bit_eq_iff fun_eq_iff)
 qed
 
 lemma signed_take_bit_signed_take_bit [simp]:
   \<open>signed_take_bit m (signed_take_bit n a) = signed_take_bit (min m n) a\<close>
 proof (rule bit_eqI)
   fix q
   show \<open>bit (signed_take_bit m (signed_take_bit n a)) q \<longleftrightarrow>
     bit (signed_take_bit (min m n) a) q\<close>
     by (simp add: bit_signed_take_bit_iff min_def bit_or_iff bit_not_iff bit_mask_iff bit_take_bit_iff)
       (use le_Suc_ex exp_add_not_zero_imp in blast)
 qed
 
 lemma signed_take_bit_take_bit:
   \<open>signed_take_bit m (take_bit n a) = (if n \<le> m then take_bit n else signed_take_bit m) a\<close>
   by (rule bit_eqI) (auto simp add: bit_signed_take_bit_iff min_def bit_take_bit_iff)
 
 lemma take_bit_signed_take_bit:
   \<open>take_bit m (signed_take_bit n a) = take_bit m a\<close> if \<open>m \<le> Suc n\<close>
   using that by (rule le_SucE; intro bit_eqI)
    (auto simp add: bit_take_bit_iff bit_signed_take_bit_iff min_def less_Suc_eq)
 
 end
 
 text \<open>Modulus centered around 0\<close>
 
 lemma signed_take_bit_eq_concat_bit:
   \<open>signed_take_bit n k = concat_bit n k (- of_bool (bit k n))\<close>
   by (simp add: concat_bit_def signed_take_bit_def push_bit_minus_one_eq_not_mask)
 
 lemma signed_take_bit_add:
   \<open>signed_take_bit n (signed_take_bit n k + signed_take_bit n l) = signed_take_bit n (k + l)\<close>
   for k l :: int
 proof -
   have \<open>take_bit (Suc n)
      (take_bit (Suc n) (signed_take_bit n k) +
       take_bit (Suc n) (signed_take_bit n l)) =
     take_bit (Suc n) (k + l)\<close>
     by (simp add: take_bit_signed_take_bit take_bit_add)
   then show ?thesis
     by (simp only: signed_take_bit_eq_iff_take_bit_eq take_bit_add)
 qed
 
 lemma signed_take_bit_diff:
   \<open>signed_take_bit n (signed_take_bit n k - signed_take_bit n l) = signed_take_bit n (k - l)\<close>
   for k l :: int
 proof -
   have \<open>take_bit (Suc n)
      (take_bit (Suc n) (signed_take_bit n k) -
       take_bit (Suc n) (signed_take_bit n l)) =
     take_bit (Suc n) (k - l)\<close>
     by (simp add: take_bit_signed_take_bit take_bit_diff)
   then show ?thesis
     by (simp only: signed_take_bit_eq_iff_take_bit_eq take_bit_diff)
 qed
 
 lemma signed_take_bit_minus:
   \<open>signed_take_bit n (- signed_take_bit n k) = signed_take_bit n (- k)\<close>
   for k :: int
 proof -
   have \<open>take_bit (Suc n)
      (- take_bit (Suc n) (signed_take_bit n k)) =
     take_bit (Suc n) (- k)\<close>
     by (simp add: take_bit_signed_take_bit take_bit_minus)
   then show ?thesis
     by (simp only: signed_take_bit_eq_iff_take_bit_eq take_bit_minus)
 qed
 
 lemma signed_take_bit_mult:
   \<open>signed_take_bit n (signed_take_bit n k * signed_take_bit n l) = signed_take_bit n (k * l)\<close>
   for k l :: int
 proof -
   have \<open>take_bit (Suc n)
      (take_bit (Suc n) (signed_take_bit n k) *
       take_bit (Suc n) (signed_take_bit n l)) =
     take_bit (Suc n) (k * l)\<close>
     by (simp add: take_bit_signed_take_bit take_bit_mult)
   then show ?thesis
     by (simp only: signed_take_bit_eq_iff_take_bit_eq take_bit_mult)
 qed
 
 lemma signed_take_bit_eq_take_bit_minus:
   \<open>signed_take_bit n k = take_bit (Suc n) k - 2 ^ Suc n * of_bool (bit k n)\<close>
   for k :: int
 proof (cases \<open>bit k n\<close>)
   case True
   have \<open>signed_take_bit n k = take_bit (Suc n) k OR NOT (mask (Suc n))\<close>
     by (rule bit_eqI) (auto simp add: bit_signed_take_bit_iff min_def bit_take_bit_iff bit_or_iff bit_not_iff bit_mask_iff less_Suc_eq True)
   then have \<open>signed_take_bit n k = take_bit (Suc n) k + NOT (mask (Suc n))\<close>
     by (simp add: disjunctive_add bit_take_bit_iff bit_not_iff bit_mask_iff)
   with True show ?thesis
     by (simp flip: minus_exp_eq_not_mask)
 next
   case False
   show ?thesis
     by (rule bit_eqI) (simp add: False bit_signed_take_bit_iff bit_take_bit_iff min_def less_Suc_eq)
 qed
 
 lemma signed_take_bit_eq_take_bit_shift:
   \<open>signed_take_bit n k = take_bit (Suc n) (k + 2 ^ n) - 2 ^ n\<close>
   for k :: int
 proof -
   have *: \<open>take_bit n k OR 2 ^ n = take_bit n k + 2 ^ n\<close>
     by (simp add: disjunctive_add bit_exp_iff bit_take_bit_iff)
   have \<open>take_bit n k - 2 ^ n = take_bit n k + NOT (mask n)\<close>
     by (simp add: minus_exp_eq_not_mask)
   also have \<open>\<dots> = take_bit n k OR NOT (mask n)\<close>
     by (rule disjunctive_add)
       (simp add: bit_exp_iff bit_take_bit_iff bit_not_iff bit_mask_iff)
   finally have **: \<open>take_bit n k - 2 ^ n = take_bit n k OR NOT (mask n)\<close> .
   have \<open>take_bit (Suc n) (k + 2 ^ n) = take_bit (Suc n) (take_bit (Suc n) k + take_bit (Suc n) (2 ^ n))\<close>
     by (simp only: take_bit_add)
   also have \<open>take_bit (Suc n) k = 2 ^ n * of_bool (bit k n) + take_bit n k\<close>
     by (simp add: take_bit_Suc_from_most)
   finally have \<open>take_bit (Suc n) (k + 2 ^ n) = take_bit (Suc n) (2 ^ (n + of_bool (bit k n)) + take_bit n k)\<close>
     by (simp add: ac_simps)
   also have \<open>2 ^ (n + of_bool (bit k n)) + take_bit n k = 2 ^ (n + of_bool (bit k n)) OR take_bit n k\<close>
     by (rule disjunctive_add)
       (auto simp add: disjunctive_add bit_take_bit_iff bit_double_iff bit_exp_iff)
   finally show ?thesis
     using * ** by (simp add: signed_take_bit_def concat_bit_Suc min_def ac_simps)
 qed
 
 lemma signed_take_bit_nonnegative_iff [simp]:
   \<open>0 \<le> signed_take_bit n k \<longleftrightarrow> \<not> bit k n\<close>
   for k :: int
   by (simp add: signed_take_bit_def not_less concat_bit_def)
 
 lemma signed_take_bit_negative_iff [simp]:
   \<open>signed_take_bit n k < 0 \<longleftrightarrow> bit k n\<close>
   for k :: int
   by (simp add: signed_take_bit_def not_less concat_bit_def)
 
 lemma signed_take_bit_int_eq_self_iff:
   \<open>signed_take_bit n k = k \<longleftrightarrow> - (2 ^ n) \<le> k \<and> k < 2 ^ n\<close>
   for k :: int
   by (auto simp add: signed_take_bit_eq_take_bit_shift take_bit_int_eq_self_iff algebra_simps)
 
 lemma signed_take_bit_int_eq_self:
   \<open>signed_take_bit n k = k\<close> if \<open>- (2 ^ n) \<le> k\<close> \<open>k < 2 ^ n\<close>
   for k :: int
   using that by (simp add: signed_take_bit_int_eq_self_iff)
 
 lemma signed_take_bit_int_less_eq_self_iff:
   \<open>signed_take_bit n k \<le> k \<longleftrightarrow> - (2 ^ n) \<le> k\<close>
   for k :: int
   by (simp add: signed_take_bit_eq_take_bit_shift take_bit_int_less_eq_self_iff algebra_simps)
     linarith
 
 lemma signed_take_bit_int_less_self_iff:
   \<open>signed_take_bit n k < k \<longleftrightarrow> 2 ^ n \<le> k\<close>
   for k :: int
   by (simp add: signed_take_bit_eq_take_bit_shift take_bit_int_less_self_iff algebra_simps)
 
 lemma signed_take_bit_int_greater_self_iff:
   \<open>k < signed_take_bit n k \<longleftrightarrow> k < - (2 ^ n)\<close>
   for k :: int
   by (simp add: signed_take_bit_eq_take_bit_shift take_bit_int_greater_self_iff algebra_simps)
     linarith
 
 lemma signed_take_bit_int_greater_eq_self_iff:
   \<open>k \<le> signed_take_bit n k \<longleftrightarrow> k < 2 ^ n\<close>
   for k :: int
   by (simp add: signed_take_bit_eq_take_bit_shift take_bit_int_greater_eq_self_iff algebra_simps)
 
 lemma signed_take_bit_int_greater_eq:
   \<open>k + 2 ^ Suc n \<le> signed_take_bit n k\<close> if \<open>k < - (2 ^ n)\<close>
   for k :: int
   using that take_bit_int_greater_eq [of \<open>k + 2 ^ n\<close> \<open>Suc n\<close>]
   by (simp add: signed_take_bit_eq_take_bit_shift)
 
 lemma signed_take_bit_int_less_eq:
   \<open>signed_take_bit n k \<le> k - 2 ^ Suc n\<close> if \<open>k \<ge> 2 ^ n\<close>
   for k :: int
   using that take_bit_int_less_eq [of \<open>Suc n\<close> \<open>k + 2 ^ n\<close>]
   by (simp add: signed_take_bit_eq_take_bit_shift)
 
 lemma signed_take_bit_Suc_bit0 [simp]:
   \<open>signed_take_bit (Suc n) (numeral (Num.Bit0 k)) = signed_take_bit n (numeral k) * (2 :: int)\<close>
   by (simp add: signed_take_bit_Suc)
 
 lemma signed_take_bit_Suc_bit1 [simp]:
   \<open>signed_take_bit (Suc n) (numeral (Num.Bit1 k)) = signed_take_bit n (numeral k) * 2 + (1 :: int)\<close>
   by (simp add: signed_take_bit_Suc)
 
 lemma signed_take_bit_Suc_minus_bit0 [simp]:
   \<open>signed_take_bit (Suc n) (- numeral (Num.Bit0 k)) = signed_take_bit n (- numeral k) * (2 :: int)\<close>
   by (simp add: signed_take_bit_Suc)
 
 lemma signed_take_bit_Suc_minus_bit1 [simp]:
   \<open>signed_take_bit (Suc n) (- numeral (Num.Bit1 k)) = signed_take_bit n (- numeral k - 1) * 2 + (1 :: int)\<close>
   by (simp add: signed_take_bit_Suc)
 
 lemma signed_take_bit_numeral_bit0 [simp]:
   \<open>signed_take_bit (numeral l) (numeral (Num.Bit0 k)) = signed_take_bit (pred_numeral l) (numeral k) * (2 :: int)\<close>
   by (simp add: signed_take_bit_rec)
 
 lemma signed_take_bit_numeral_bit1 [simp]:
   \<open>signed_take_bit (numeral l) (numeral (Num.Bit1 k)) = signed_take_bit (pred_numeral l) (numeral k) * 2 + (1 :: int)\<close>
   by (simp add: signed_take_bit_rec)
 
 lemma signed_take_bit_numeral_minus_bit0 [simp]:
   \<open>signed_take_bit (numeral l) (- numeral (Num.Bit0 k)) = signed_take_bit (pred_numeral l) (- numeral k) * (2 :: int)\<close>
   by (simp add: signed_take_bit_rec)
 
 lemma signed_take_bit_numeral_minus_bit1 [simp]:
   \<open>signed_take_bit (numeral l) (- numeral (Num.Bit1 k)) = signed_take_bit (pred_numeral l) (- numeral k - 1) * 2 + (1 :: int)\<close>
   by (simp add: signed_take_bit_rec)
 
 lemma signed_take_bit_code [code]:
   \<open>signed_take_bit n a =
   (let l = take_bit (Suc n) a
    in if bit l n then l + push_bit (Suc n) (- 1) else l)\<close>
 proof -
   have *: \<open>take_bit (Suc n) a + push_bit n (- 2) =
     take_bit (Suc n) a OR NOT (mask (Suc n))\<close>
     by (auto simp add: bit_take_bit_iff bit_push_bit_iff bit_not_iff bit_mask_iff disjunctive_add
        simp flip: push_bit_minus_one_eq_not_mask)
   show ?thesis
     by (rule bit_eqI)
       (auto simp add: Let_def * bit_signed_take_bit_iff bit_take_bit_iff min_def less_Suc_eq bit_not_iff bit_mask_iff bit_or_iff)
 qed
 
 
 subsection \<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
 
 definition mask_nat :: \<open>nat \<Rightarrow> nat\<close>
   where \<open>mask n = (2 :: nat) ^ n - 1\<close>
 
 instance proof
   fix m n q :: nat
   show \<open>bit (m AND n) q \<longleftrightarrow> bit m q \<and> bit n q\<close>
     by (auto simp add: bit_nat_iff and_nat_def bit_and_iff less_le bit_eq_iff)
   show \<open>bit (m OR n) q \<longleftrightarrow> bit m q \<or> bit n q\<close>
     by (auto simp add: bit_nat_iff or_nat_def bit_or_iff less_le bit_eq_iff)
   show \<open>bit (m XOR n) q \<longleftrightarrow> bit m q \<noteq> bit n q\<close>
     by (auto simp add: bit_nat_iff xor_nat_def bit_xor_iff less_le bit_eq_iff)
 qed (simp add: mask_nat_def)
 
 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
 
 context semiring_bit_operations
 begin
 
 lemma of_nat_and_eq:
   \<open>of_nat (m AND n) = of_nat m AND of_nat n\<close>
   by (rule bit_eqI) (simp add: bit_of_nat_iff bit_and_iff Bit_Operations.bit_and_iff)
 
 lemma of_nat_or_eq:
   \<open>of_nat (m OR n) = of_nat m OR of_nat n\<close>
   by (rule bit_eqI) (simp add: bit_of_nat_iff bit_or_iff Bit_Operations.bit_or_iff)
 
 lemma of_nat_xor_eq:
   \<open>of_nat (m XOR n) = of_nat m XOR of_nat n\<close>
   by (rule bit_eqI) (simp add: bit_of_nat_iff bit_xor_iff Bit_Operations.bit_xor_iff)
 
 end
 
 context ring_bit_operations
 begin
 
 lemma of_nat_mask_eq:
   \<open>of_nat (mask n) = mask n\<close>
   by (induction n) (simp_all add: mask_Suc_double Bit_Operations.mask_Suc_double of_nat_or_eq)
 
 end
 
 
 subsection \<open>Instances for \<^typ>\<open>integer\<close> and \<^typ>\<open>natural\<close>\<close>
 
 unbundle integer.lifting natural.lifting
 
 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 .
 
 lift_definition mask_integer :: \<open>nat \<Rightarrow> integer\<close>
   is mask .
 
 instance by (standard; transfer)
   (simp_all add: minus_eq_not_minus_1 mask_eq_exp_minus_1
     bit_not_iff bit_and_iff bit_or_iff bit_xor_iff)
 
 end
 
 lemma [code]:
   \<open>mask n = 2 ^ n - (1::integer)\<close>
   by (simp add: mask_eq_exp_minus_1)
 
 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 .
 
 lift_definition mask_natural :: \<open>nat \<Rightarrow> natural\<close>
   is mask .
 
 instance by (standard; transfer)
   (simp_all add: mask_eq_exp_minus_1 bit_and_iff bit_or_iff bit_xor_iff)
 
 end
 
 lemma [code]:
   \<open>integer_of_natural (mask n) = mask n\<close>
   by transfer (simp add: mask_eq_exp_minus_1 of_nat_diff)
 
 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_iff_odd [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]}
 
       \<^item> Signed truncation, or modulus centered around \<^term>\<open>0::int\<close>: @{thm signed_take_bit_def [no_vars]}
 
       \<^item> Bit concatenation: @{thm concat_bit_def [no_vars]}
 
       \<^item> (Bounded) conversion from and to a list of bits: @{thm horner_sum_bit_eq_take_bit [where ?'a = int, no_vars]}
 \<close>
 
 end
diff --git a/src/HOL/Word/Bit_Comprehension.thy b/src/HOL/Word/Bit_Comprehension.thy
--- a/src/HOL/Word/Bit_Comprehension.thy
+++ b/src/HOL/Word/Bit_Comprehension.thy
@@ -1,37 +1,247 @@
 (*  Title:      HOL/Word/Bit_Comprehension.thy
     Author:     Brian Huffman, PSU; Jeremy Dawson and Gerwin Klein, NICTA
 *)
 
 section \<open>Comprehension syntax for bit expressions\<close>
 
 theory Bit_Comprehension
-  imports "HOL-Library.Bit_Operations"
+  imports Word
 begin
 
-class bit_comprehension = semiring_bits +
-  fixes set_bits :: "(nat \<Rightarrow> bool) \<Rightarrow> 'a"  (binder "BITS " 10)
+class bit_comprehension = ring_bit_operations +
+  fixes set_bits :: \<open>(nat \<Rightarrow> bool) \<Rightarrow> 'a\<close>  (binder \<open>BITS \<close> 10)
+  assumes set_bits_bit_eq: \<open>set_bits (bit a) = a\<close>
+begin
+
+lemma set_bits_False_eq [simp]:
+  \<open>(BITS _. False) = 0\<close>
+  using set_bits_bit_eq [of 0] by (simp add: bot_fun_def)
+
+end
 
 instantiation int :: bit_comprehension
 begin
 
 definition
-  "set_bits f =
-    (if \<exists>n. \<forall>n'\<ge>n. \<not> f n' then
-      let n = LEAST n. \<forall>n'\<ge>n. \<not> f n'
-      in horner_sum of_bool 2 (map f [0..<n])
-     else if \<exists>n. \<forall>n'\<ge>n. f n' then
-      let n = LEAST n. \<forall>n'\<ge>n. f n'
-      in signed_take_bit n (horner_sum of_bool 2 (map f [0..<n] @ [True]))
-     else 0 :: int)"
+  \<open>set_bits f = (
+      if \<exists>n. \<forall>m\<ge>n. f m = f n then
+      let n = LEAST n. \<forall>m\<ge>n. f m = f n
+      in signed_take_bit n (horner_sum of_bool 2 (map f [0..<Suc n]))
+     else 0 :: int)\<close>
 
-instance ..
+instance proof
+  fix k :: int
+  from int_bit_bound [of k] 
+  obtain n where *: \<open>\<And>m. n \<le> m \<Longrightarrow> bit k m \<longleftrightarrow> bit k n\<close>
+    and **: \<open>n > 0 \<Longrightarrow> bit k (n - 1) \<noteq> bit k n\<close>
+    by blast
+  then have ***: \<open>\<exists>n. \<forall>n'\<ge>n. bit k n' \<longleftrightarrow> bit k n\<close>
+    by meson
+  have l: \<open>(LEAST q. \<forall>m\<ge>q. bit k m \<longleftrightarrow> bit k q) = n\<close>
+    apply (rule Least_equality)
+    using * apply blast
+    apply (metis "**" One_nat_def Suc_pred le_cases le0 neq0_conv not_less_eq_eq)
+    done
+  show \<open>set_bits (bit k) = k\<close>
+    apply (simp only: *** set_bits_int_def horner_sum_bit_eq_take_bit l)
+    apply simp
+    apply (rule bit_eqI)
+    apply (simp add: bit_signed_take_bit_iff min_def)
+    using "*" by blast
+qed
 
 end
 
 lemma int_set_bits_K_False [simp]: "(BITS _. False) = (0 :: int)"
   by (simp add: set_bits_int_def)
 
 lemma int_set_bits_K_True [simp]: "(BITS _. True) = (-1 :: int)"
-  by (auto simp add: set_bits_int_def)
+  by (simp add: set_bits_int_def)
+
+instantiation word :: (len) bit_comprehension
+begin
+
+definition word_set_bits_def:
+  \<open>(BITS n. P n) = (horner_sum of_bool 2 (map P [0..<LENGTH('a)]) :: 'a word)\<close>
+
+instance by standard
+  (simp add: word_set_bits_def horner_sum_bit_eq_take_bit)
 
 end
+
+lemma bit_set_bits_word_iff:
+  \<open>bit (set_bits P :: 'a::len word) n \<longleftrightarrow> n < LENGTH('a) \<and> P n\<close>
+  by (auto simp add: word_set_bits_def bit_horner_sum_bit_word_iff)
+
+lemma set_bits_K_False [simp]:
+  \<open>set_bits (\<lambda>_. False) = (0 :: 'a :: len word)\<close>
+  by (rule bit_word_eqI) (simp add: bit_set_bits_word_iff)
+
+lemma set_bits_int_unfold':
+  \<open>set_bits f =
+    (if \<exists>n. \<forall>n'\<ge>n. \<not> f n' then
+      let n = LEAST n. \<forall>n'\<ge>n. \<not> f n'
+      in horner_sum of_bool 2 (map f [0..<n])
+     else if \<exists>n. \<forall>n'\<ge>n. f n' then
+      let n = LEAST n. \<forall>n'\<ge>n. f n'
+      in signed_take_bit n (horner_sum of_bool 2 (map f [0..<n] @ [True]))
+     else 0 :: int)\<close>
+proof (cases \<open>\<exists>n. \<forall>m\<ge>n. f m \<longleftrightarrow> f n\<close>)
+  case True
+  then obtain q where q: \<open>\<forall>m\<ge>q. f m \<longleftrightarrow> f q\<close>
+    by blast
+  define n where \<open>n = (LEAST n. \<forall>m\<ge>n. f m \<longleftrightarrow> f n)\<close>
+  have \<open>\<forall>m\<ge>n. f m \<longleftrightarrow> f n\<close> 
+    unfolding n_def
+    using q by (rule LeastI [of _ q])
+  then have n: \<open>\<And>m. n \<le> m \<Longrightarrow> f m \<longleftrightarrow> f n\<close>
+    by blast
+  from n_def have n_eq: \<open>(LEAST q. \<forall>m\<ge>q. f m \<longleftrightarrow> f n) = n\<close>
+    by (smt Least_equality Least_le \<open>\<forall>m\<ge>n. f m = f n\<close> dual_order.refl le_refl n order_refl)
+  show ?thesis
+  proof (cases \<open>f n\<close>)
+    case False
+    with n have *: \<open>\<exists>n. \<forall>n'\<ge>n. \<not> f n'\<close> 
+      by blast
+    have **: \<open>(LEAST n. \<forall>n'\<ge>n. \<not> f n') = n\<close>
+      using False n_eq by simp
+    from * False show ?thesis
+    apply (simp add: set_bits_int_def n_def [symmetric] ** del: upt.upt_Suc)
+    apply (auto simp add: take_bit_horner_sum_bit_eq
+      bit_horner_sum_bit_iff take_map
+      signed_take_bit_def set_bits_int_def
+      horner_sum_bit_eq_take_bit simp del: upt.upt_Suc)
+    done
+  next
+    case True
+    with n have *: \<open>\<exists>n. \<forall>n'\<ge>n. f n'\<close>
+      by blast
+    have ***: \<open>\<not> (\<exists>n. \<forall>n'\<ge>n. \<not> f n')\<close>
+      apply (rule ccontr)
+      using * nat_le_linear by auto 
+    have **: \<open>(LEAST n. \<forall>n'\<ge>n. f n') = n\<close>
+      using True n_eq by simp
+    from * *** True show ?thesis
+    apply (simp add: set_bits_int_def n_def [symmetric] ** del: upt.upt_Suc)
+    apply (auto simp add: take_bit_horner_sum_bit_eq
+      bit_horner_sum_bit_iff take_map
+      signed_take_bit_def set_bits_int_def
+      horner_sum_bit_eq_take_bit nth_append simp del: upt.upt_Suc)
+    done
+  qed
+next
+  case False
+  then show ?thesis
+    by (auto simp add: set_bits_int_def)
+qed
+
+inductive wf_set_bits_int :: "(nat \<Rightarrow> bool) \<Rightarrow> bool" 
+  for f :: "nat \<Rightarrow> bool"
+where
+  zeros: "\<forall>n' \<ge> n. \<not> f n' \<Longrightarrow> wf_set_bits_int f"
+| ones: "\<forall>n' \<ge> n. f n' \<Longrightarrow> wf_set_bits_int f"
+
+lemma wf_set_bits_int_simps: "wf_set_bits_int f \<longleftrightarrow> (\<exists>n. (\<forall>n'\<ge>n. \<not> f n') \<or> (\<forall>n'\<ge>n. f n'))"
+by(auto simp add: wf_set_bits_int.simps)
+
+lemma wf_set_bits_int_const [simp]: "wf_set_bits_int (\<lambda>_. b)"
+by(cases b)(auto intro: wf_set_bits_int.intros)
+
+lemma wf_set_bits_int_fun_upd [simp]: 
+  "wf_set_bits_int (f(n := b)) \<longleftrightarrow> wf_set_bits_int f" (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+  assume ?lhs
+  then obtain n'
+    where "(\<forall>n''\<ge>n'. \<not> (f(n := b)) n'') \<or> (\<forall>n''\<ge>n'. (f(n := b)) n'')"
+    by(auto simp add: wf_set_bits_int_simps)
+  hence "(\<forall>n''\<ge>max (Suc n) n'. \<not> f n'') \<or> (\<forall>n''\<ge>max (Suc n) n'. f n'')" by auto
+  thus ?rhs by(auto simp only: wf_set_bits_int_simps)
+next
+  assume ?rhs
+  then obtain n' where "(\<forall>n''\<ge>n'. \<not> f n'') \<or> (\<forall>n''\<ge>n'. f n'')" (is "?wf f n'")
+    by(auto simp add: wf_set_bits_int_simps)
+  hence "?wf (f(n := b)) (max (Suc n) n')" by auto
+  thus ?lhs by(auto simp only: wf_set_bits_int_simps)
+qed
+
+lemma wf_set_bits_int_Suc [simp]:
+  "wf_set_bits_int (\<lambda>n. f (Suc n)) \<longleftrightarrow> wf_set_bits_int f" (is "?lhs \<longleftrightarrow> ?rhs")
+by(auto simp add: wf_set_bits_int_simps intro: le_SucI dest: Suc_le_D)
+
+context
+  fixes f
+  assumes wff: "wf_set_bits_int f"
+begin
+
+lemma int_set_bits_unfold_BIT:
+  "set_bits f = of_bool (f 0) + (2 :: int) * set_bits (f \<circ> Suc)"
+using wff proof cases
+  case (zeros n)
+  show ?thesis
+  proof(cases "\<forall>n. \<not> f n")
+    case True
+    hence "f = (\<lambda>_. False)" by auto
+    thus ?thesis using True by(simp add: o_def)
+  next
+    case False
+    then obtain n' where "f n'" by blast
+    with zeros have "(LEAST n. \<forall>n'\<ge>n. \<not> f n') = Suc (LEAST n. \<forall>n'\<ge>Suc n. \<not> f n')"
+      by(auto intro: Least_Suc)
+    also have "(\<lambda>n. \<forall>n'\<ge>Suc n. \<not> f n') = (\<lambda>n. \<forall>n'\<ge>n. \<not> f (Suc n'))" by(auto dest: Suc_le_D)
+    also from zeros have "\<forall>n'\<ge>n. \<not> f (Suc n')" by auto
+    ultimately show ?thesis using zeros
+      apply (simp (no_asm_simp) add: set_bits_int_unfold' exI
+        del: upt.upt_Suc flip: map_map split del: if_split)
+      apply (simp only: map_Suc_upt upt_conv_Cons)
+      apply simp
+      done
+  qed
+next
+  case (ones n)
+  show ?thesis
+  proof(cases "\<forall>n. f n")
+    case True
+    hence "f = (\<lambda>_. True)" by auto
+    thus ?thesis using True by(simp add: o_def)
+  next
+    case False
+    then obtain n' where "\<not> f n'" by blast
+    with ones have "(LEAST n. \<forall>n'\<ge>n. f n') = Suc (LEAST n. \<forall>n'\<ge>Suc n. f n')"
+      by(auto intro: Least_Suc)
+    also have "(\<lambda>n. \<forall>n'\<ge>Suc n. f n') = (\<lambda>n. \<forall>n'\<ge>n. f (Suc n'))" by(auto dest: Suc_le_D)
+    also from ones have "\<forall>n'\<ge>n. f (Suc n')" by auto
+    moreover from ones have "(\<exists>n. \<forall>n'\<ge>n. \<not> f n') = False"
+      by(auto intro!: exI[where x="max n m" for n m] simp add: max_def split: if_split_asm)
+    moreover hence "(\<exists>n. \<forall>n'\<ge>n. \<not> f (Suc n')) = False"
+      by(auto elim: allE[where x="Suc n" for n] dest: Suc_le_D)
+    ultimately show ?thesis using ones
+      apply (simp (no_asm_simp) add: set_bits_int_unfold' exI split del: if_split)
+      apply (auto simp add: Let_def hd_map map_tl[symmetric] map_map[symmetric] map_Suc_upt upt_conv_Cons signed_take_bit_Suc
+        not_le simp del: map_map)
+      done
+  qed
+qed
+
+lemma bin_last_set_bits [simp]:
+  "bin_last (set_bits f) = f 0"
+  by (subst int_set_bits_unfold_BIT) simp_all
+
+lemma bin_rest_set_bits [simp]:
+  "bin_rest (set_bits f) = set_bits (f \<circ> Suc)"
+  by (subst int_set_bits_unfold_BIT) simp_all
+
+lemma bin_nth_set_bits [simp]:
+  "bin_nth (set_bits f) m = f m"
+using wff proof (induction m arbitrary: f)
+  case 0 
+  then show ?case
+    by (simp add: Bit_Comprehension.bin_last_set_bits)
+next
+  case Suc
+  from Suc.IH [of "f \<circ> Suc"] Suc.prems show ?case
+    by (simp add: Bit_Comprehension.bin_rest_set_bits comp_def bit_Suc)
+qed
+
+end
+
+end
diff --git a/src/HOL/Word/Misc_Typedef.thy b/src/HOL/Word/Misc_Typedef.thy
--- a/src/HOL/Word/Misc_Typedef.thy
+++ b/src/HOL/Word/Misc_Typedef.thy
@@ -1,357 +1,356 @@
 (*
   Author:     Jeremy Dawson and Gerwin Klein, NICTA
 
   Consequences of type definition theorems, and of extended type definition.
 *)
 
 section \<open>Type Definition Theorems\<close>
 
 theory Misc_Typedef
-  imports Main Word
+  imports Main Word Bit_Comprehension
 begin
 
 subsection "More lemmas about normal type definitions"
 
 lemma tdD1: "type_definition Rep Abs A \<Longrightarrow> \<forall>x. Rep x \<in> A"
   and tdD2: "type_definition Rep Abs A \<Longrightarrow> \<forall>x. Abs (Rep x) = x"
   and tdD3: "type_definition Rep Abs A \<Longrightarrow> \<forall>y. y \<in> A \<longrightarrow> Rep (Abs y) = y"
   by (auto simp: type_definition_def)
 
 lemma td_nat_int: "type_definition int nat (Collect ((\<le>) 0))"
   unfolding type_definition_def by auto
 
 context type_definition
 begin
 
 declare Rep [iff] Rep_inverse [simp] Rep_inject [simp]
 
 lemma Abs_eqD: "Abs x = Abs y \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x = y"
   by (simp add: Abs_inject)
 
 lemma Abs_inverse': "r \<in> A \<Longrightarrow> Abs r = a \<Longrightarrow> Rep a = r"
   by (safe elim!: Abs_inverse)
 
 lemma Rep_comp_inverse: "Rep \<circ> f = g \<Longrightarrow> Abs \<circ> g = f"
   using Rep_inverse by auto
 
 lemma Rep_eqD [elim!]: "Rep x = Rep y \<Longrightarrow> x = y"
   by simp
 
 lemma Rep_inverse': "Rep a = r \<Longrightarrow> Abs r = a"
   by (safe intro!: Rep_inverse)
 
 lemma comp_Abs_inverse: "f \<circ> Abs = g \<Longrightarrow> g \<circ> Rep = f"
   using Rep_inverse by auto
 
 lemma set_Rep: "A = range Rep"
 proof (rule set_eqI)
   show "x \<in> A \<longleftrightarrow> x \<in> range Rep" for x
     by (auto dest: Abs_inverse [of x, symmetric])
 qed
 
 lemma set_Rep_Abs: "A = range (Rep \<circ> Abs)"
 proof (rule set_eqI)
   show "x \<in> A \<longleftrightarrow> x \<in> range (Rep \<circ> Abs)" for x
     by (auto dest: Abs_inverse [of x, symmetric])
 qed
 
 lemma Abs_inj_on: "inj_on Abs A"
   unfolding inj_on_def
   by (auto dest: Abs_inject [THEN iffD1])
 
 lemma image: "Abs ` A = UNIV"
   by (fact Abs_image)
 
 lemmas td_thm = type_definition_axioms
 
 lemma fns1: "Rep \<circ> fa = fr \<circ> Rep \<or> fa \<circ> Abs = Abs \<circ> fr \<Longrightarrow> Abs \<circ> fr \<circ> Rep = fa"
   by (auto dest: Rep_comp_inverse elim: comp_Abs_inverse simp: o_assoc)
 
 lemmas fns1a = disjI1 [THEN fns1]
 lemmas fns1b = disjI2 [THEN fns1]
 
 lemma fns4: "Rep \<circ> fa \<circ> Abs = fr \<Longrightarrow> Rep \<circ> fa = fr \<circ> Rep \<and> fa \<circ> Abs = Abs \<circ> fr"
   by auto
 
 end
 
 interpretation nat_int: type_definition int nat "Collect ((\<le>) 0)"
   by (rule td_nat_int)
 
 declare
   nat_int.Rep_cases [cases del]
   nat_int.Abs_cases [cases del]
   nat_int.Rep_induct [induct del]
   nat_int.Abs_induct [induct del]
 
 
 subsection "Extended form of type definition predicate"
 
 lemma td_conds:
   "norm \<circ> norm = norm \<Longrightarrow>
     fr \<circ> norm = norm \<circ> fr \<longleftrightarrow> norm \<circ> fr \<circ> norm = fr \<circ> norm \<and> norm \<circ> fr \<circ> norm = norm \<circ> fr"
   apply safe
     apply (simp_all add: comp_assoc)
    apply (simp_all add: o_assoc)
   done
 
 lemma fn_comm_power: "fa \<circ> tr = tr \<circ> fr \<Longrightarrow> fa ^^ n \<circ> tr = tr \<circ> fr ^^ n"
   apply (rule ext)
   apply (induct n)
    apply (auto dest: fun_cong)
   done
 
 lemmas fn_comm_power' =
   ext [THEN fn_comm_power, THEN fun_cong, unfolded o_def]
 
 
 locale td_ext = type_definition +
   fixes norm
   assumes eq_norm: "\<And>x. Rep (Abs x) = norm x"
 begin
 
 lemma Abs_norm [simp]: "Abs (norm x) = Abs x"
   using eq_norm [of x] by (auto elim: Rep_inverse')
 
 lemma td_th: "g \<circ> Abs = f \<Longrightarrow> f (Rep x) = g x"
   by (drule comp_Abs_inverse [symmetric]) simp
 
 lemma eq_norm': "Rep \<circ> Abs = norm"
   by (auto simp: eq_norm)
 
 lemma norm_Rep [simp]: "norm (Rep x) = Rep x"
   by (auto simp: eq_norm' intro: td_th)
 
 lemmas td = td_thm
 
 lemma set_iff_norm: "w \<in> A \<longleftrightarrow> w = norm w"
   by (auto simp: set_Rep_Abs eq_norm' eq_norm [symmetric])
 
 lemma inverse_norm: "Abs n = w \<longleftrightarrow> Rep w = norm n"
   apply (rule iffI)
    apply (clarsimp simp add: eq_norm)
   apply (simp add: eq_norm' [symmetric])
   done
 
 lemma norm_eq_iff: "norm x = norm y \<longleftrightarrow> Abs x = Abs y"
   by (simp add: eq_norm' [symmetric])
 
 lemma norm_comps:
   "Abs \<circ> norm = Abs"
   "norm \<circ> Rep = Rep"
   "norm \<circ> norm = norm"
   by (auto simp: eq_norm' [symmetric] o_def)
 
 lemmas norm_norm [simp] = norm_comps
 
 lemma fns5: "Rep \<circ> fa \<circ> Abs = fr \<Longrightarrow> fr \<circ> norm = fr \<and> norm \<circ> fr = fr"
   by (fold eq_norm') auto
 
 text \<open>
   following give conditions for converses to \<open>td_fns1\<close>
   \<^item> the condition \<open>norm \<circ> fr \<circ> norm = fr \<circ> norm\<close> says that
     \<open>fr\<close> takes normalised arguments to normalised results
   \<^item> \<open>norm \<circ> fr \<circ> norm = norm \<circ> fr\<close> says that \<open>fr\<close>
     takes norm-equivalent arguments to norm-equivalent results
   \<^item> \<open>fr \<circ> norm = fr\<close> says that \<open>fr\<close>
     takes norm-equivalent arguments to the same result
   \<^item> \<open>norm \<circ> fr = fr\<close> says that \<open>fr\<close> takes any argument to a normalised result
 \<close>
 lemma fns2: "Abs \<circ> fr \<circ> Rep = fa \<Longrightarrow> norm \<circ> fr \<circ> norm = fr \<circ> norm \<longleftrightarrow> Rep \<circ> fa = fr \<circ> Rep"
   apply (fold eq_norm')
   apply safe
    prefer 2
    apply (simp add: o_assoc)
   apply (rule ext)
   apply (drule_tac x="Rep x" in fun_cong)
   apply auto
   done
 
 lemma fns3: "Abs \<circ> fr \<circ> Rep = fa \<Longrightarrow> norm \<circ> fr \<circ> norm = norm \<circ> fr \<longleftrightarrow> fa \<circ> Abs = Abs \<circ> fr"
   apply (fold eq_norm')
   apply safe
    prefer 2
    apply (simp add: comp_assoc)
   apply (rule ext)
   apply (drule_tac f="a \<circ> b" for a b in fun_cong)
   apply simp
   done
 
 lemma fns: "fr \<circ> norm = norm \<circ> fr \<Longrightarrow> fa \<circ> Abs = Abs \<circ> fr \<longleftrightarrow> Rep \<circ> fa = fr \<circ> Rep"
   apply safe
    apply (frule fns1b)
    prefer 2
    apply (frule fns1a)
    apply (rule fns3 [THEN iffD1])
      prefer 3
      apply (rule fns2 [THEN iffD1])
        apply (simp_all add: comp_assoc)
    apply (simp_all add: o_assoc)
   done
 
 lemma range_norm: "range (Rep \<circ> Abs) = A"
   by (simp add: set_Rep_Abs)
 
 end
 
 lemmas td_ext_def' =
   td_ext_def [unfolded type_definition_def td_ext_axioms_def]
 
 
 subsection \<open>Type-definition locale instantiations\<close>
 
 definition uints :: "nat \<Rightarrow> int set"
   \<comment> \<open>the sets of integers representing the words\<close>
   where "uints n = range (take_bit n)"
 
 definition sints :: "nat \<Rightarrow> int set"
   where "sints n = range (signed_take_bit (n - 1))"
 
 lemma uints_num: "uints n = {i. 0 \<le> i \<and> i < 2 ^ n}"
   by (simp add: uints_def range_bintrunc)
 
 lemma sints_num: "sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> i < 2 ^ (n - 1)}"
   by (simp add: sints_def range_sbintrunc)
 
 definition unats :: "nat \<Rightarrow> nat set"
   where "unats n = {i. i < 2 ^ n}"
 
 \<comment> \<open>naturals\<close>
 lemma uints_unats: "uints n = int ` unats n"
   apply (unfold unats_def uints_num)
   apply safe
     apply (rule_tac image_eqI)
      apply (erule_tac nat_0_le [symmetric])
   by auto
 
 lemma unats_uints: "unats n = nat ` uints n"
   by (auto simp: uints_unats image_iff)
 
 lemma td_ext_uint:
   "td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (LENGTH('a::len)))
     (\<lambda>w::int. w mod 2 ^ LENGTH('a))"
   apply (unfold td_ext_def')
   apply transfer
   apply (simp add: uints_num take_bit_eq_mod)
   done
 
 interpretation word_uint:
   td_ext
     "uint::'a::len word \<Rightarrow> int"
     word_of_int
     "uints (LENGTH('a::len))"
     "\<lambda>w. w mod 2 ^ LENGTH('a::len)"
   by (fact td_ext_uint)
 
 lemmas td_uint = word_uint.td_thm
 lemmas int_word_uint = word_uint.eq_norm
 
 lemma td_ext_ubin:
   "td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (LENGTH('a::len)))
     (take_bit (LENGTH('a)))"
   apply standard
   apply transfer
   apply simp
   done
 
 interpretation word_ubin:
   td_ext
     "uint::'a::len word \<Rightarrow> int"
     word_of_int
     "uints (LENGTH('a::len))"
     "take_bit (LENGTH('a::len))"
   by (fact td_ext_ubin)
 
 lemma td_ext_unat [OF refl]:
   "n = LENGTH('a::len) \<Longrightarrow>
     td_ext (unat :: 'a word \<Rightarrow> nat) of_nat (unats n) (\<lambda>i. i mod 2 ^ n)"
   apply (standard; transfer)
      apply (simp_all add: unats_def take_bit_of_nat take_bit_nat_eq_self_iff
       flip: take_bit_eq_mod)
   done
 
 lemmas unat_of_nat = td_ext_unat [THEN td_ext.eq_norm]
 
 interpretation word_unat:
   td_ext
     "unat::'a::len word \<Rightarrow> nat"
     of_nat
     "unats (LENGTH('a::len))"
     "\<lambda>i. i mod 2 ^ LENGTH('a::len)"
   by (rule td_ext_unat)
 
 lemmas td_unat = word_unat.td_thm
 
 lemmas unat_lt2p [iff] = word_unat.Rep [unfolded unats_def mem_Collect_eq]
 
 lemma unat_le: "y \<le> unat z \<Longrightarrow> y \<in> unats (LENGTH('a))"
   for z :: "'a::len word"
   apply (unfold unats_def)
   apply clarsimp
   apply (rule xtrans, rule unat_lt2p, assumption)
   done
 
 lemma td_ext_sbin:
   "td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (LENGTH('a::len)))
     (signed_take_bit (LENGTH('a) - 1))"
   by (standard; transfer) (auto simp add: sints_def)
 
 lemma td_ext_sint:
   "td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (LENGTH('a::len)))
      (\<lambda>w. (w + 2 ^ (LENGTH('a) - 1)) mod 2 ^ LENGTH('a) -
          2 ^ (LENGTH('a) - 1))"
   using td_ext_sbin [where ?'a = 'a] by (simp add: no_sbintr_alt2)
 
 text \<open>
   We do \<open>sint\<close> before \<open>sbin\<close>, before \<open>sint\<close> is the user version
   and interpretations do not produce thm duplicates. I.e.
   we get the name \<open>word_sint.Rep_eqD\<close>, but not \<open>word_sbin.Req_eqD\<close>,
   because the latter is the same thm as the former.
 \<close>
 interpretation word_sint:
   td_ext
     "sint ::'a::len word \<Rightarrow> int"
     word_of_int
     "sints (LENGTH('a::len))"
     "\<lambda>w. (w + 2^(LENGTH('a::len) - 1)) mod 2^LENGTH('a::len) -
       2 ^ (LENGTH('a::len) - 1)"
   by (rule td_ext_sint)
 
 interpretation word_sbin:
   td_ext
     "sint ::'a::len word \<Rightarrow> int"
     word_of_int
     "sints (LENGTH('a::len))"
     "signed_take_bit (LENGTH('a::len) - 1)"
   by (rule td_ext_sbin)
 
 lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]
 
 lemmas td_sint = word_sint.td
 
 lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"
   by (fact uints_def [unfolded no_bintr_alt1])
 
 lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq]
 lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq]
 
 lemmas bintr_num =
   word_ubin.norm_eq_iff [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
 lemmas sbintr_num =
   word_sbin.norm_eq_iff [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
 
 lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]]
 lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]]
 
 interpretation test_bit:
   td_ext
     "(!!) :: 'a::len word \<Rightarrow> nat \<Rightarrow> bool"
     set_bits
     "{f. \<forall>i. f i \<longrightarrow> i < LENGTH('a::len)}"
     "(\<lambda>h i. h i \<and> i < LENGTH('a::len))"
-  by standard
-    (auto simp add: test_bit_word_eq bit_imp_le_length bit_set_bits_word_iff set_bits_bit_eq)
+  by standard (auto simp add: test_bit_word_eq bit_imp_le_length bit_set_bits_word_iff set_bits_bit_eq)
 
 lemmas td_nth = test_bit.td_thm
 
 end
diff --git a/src/HOL/Word/More_Word.thy b/src/HOL/Word/More_Word.thy
--- a/src/HOL/Word/More_Word.thy
+++ b/src/HOL/Word/More_Word.thy
@@ -1,56 +1,57 @@
 (*  Title:      HOL/Word/More_thy
 *)
 
 section \<open>Ancient comprehensive Word Library\<close>
 
 theory More_Word
 imports
   Word
   Ancient_Numeral
   Reversed_Bit_Lists
   Bits_Int
+  Bit_Comprehension
   Misc_Auxiliary
   Misc_Arithmetic
   Misc_set_bit
   Misc_lsb
   Misc_Typedef
 begin
 
 declare signed_take_bit_Suc [simp]
 
 lemmas bshiftr1_def = bshiftr1_eq
 lemmas is_down_def = is_down_eq
 lemmas is_up_def = is_up_eq
 lemmas mask_def = mask_eq_decr_exp
 lemmas scast_def = scast_eq
 lemmas shiftl1_def = shiftl1_eq
 lemmas shiftr1_def = shiftr1_eq
 lemmas sshiftr1_def = sshiftr1_eq
 lemmas sshiftr_def = sshiftr_eq_funpow_sshiftr1
 lemmas to_bl_def = to_bl_eq
 lemmas ucast_def = ucast_eq
 lemmas unat_def = unat_eq_nat_uint
 lemmas word_cat_def = word_cat_eq
 lemmas word_reverse_def = word_reverse_eq_of_bl_rev_to_bl
 lemmas word_roti_def = word_roti_eq_word_rotr_word_rotl
 lemmas word_rotl_def = word_rotl_eq
 lemmas word_rotr_def = word_rotr_eq
 lemmas word_sle_def = word_sle_eq
 lemmas word_sless_def = word_sless_eq
 
 lemmas uint_0 = uint_nonnegative
 lemmas uint_lt = uint_bounded
 lemmas uint_mod_same = uint_idem
 lemmas of_nth_def = word_set_bits_def
 
 lemmas of_nat_word_eq_iff = word_of_nat_eq_iff
 lemmas of_nat_word_eq_0_iff = word_of_nat_eq_0_iff
 lemmas of_int_word_eq_iff = word_of_int_eq_iff
 lemmas of_int_word_eq_0_iff = word_of_int_eq_0_iff
 
 lemma shiftl_transfer [transfer_rule]:
   includes lifting_syntax
   shows "(pcr_word ===> (=) ===> pcr_word) (<<) (<<)"
   by (unfold shiftl_eq_push_bit) transfer_prover
 
 end
diff --git a/src/HOL/Word/Word.thy b/src/HOL/Word/Word.thy
--- a/src/HOL/Word/Word.thy
+++ b/src/HOL/Word/Word.thy
@@ -1,5339 +1,5310 @@
 (*  Title:      HOL/Word/Word.thy
     Author:     Jeremy Dawson and Gerwin Klein, NICTA
 *)
 
 section \<open>A type of finite bit strings\<close>
 
 theory Word
 imports
   "HOL-Library.Type_Length"
   "HOL-Library.Boolean_Algebra"
   "HOL-Library.Bit_Operations"
   Bits_Int
   Traditional_Syntax
-  Bit_Comprehension
 begin
 
 subsection \<open>Preliminaries\<close>
 
 lemma signed_take_bit_decr_length_iff:
   \<open>signed_take_bit (LENGTH('a::len) - Suc 0) k = signed_take_bit (LENGTH('a) - Suc 0) l
     \<longleftrightarrow> take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
   by (cases \<open>LENGTH('a)\<close>)
     (simp_all add: signed_take_bit_eq_iff_take_bit_eq)
 
 
 subsection \<open>Fundamentals\<close>
 
 subsubsection \<open>Type definition\<close>
 
 quotient_type (overloaded) 'a word = int / \<open>\<lambda>k l. take_bit LENGTH('a) k = take_bit LENGTH('a::len) l\<close>
   morphisms rep Word by (auto intro!: equivpI reflpI sympI transpI)
 
 hide_const (open) rep \<comment> \<open>only for foundational purpose\<close>
 hide_const (open) Word \<comment> \<open>only for code generation\<close>
 
 
 subsubsection \<open>Basic arithmetic\<close>
 
 instantiation word :: (len) comm_ring_1
 begin
 
 lift_definition zero_word :: \<open>'a word\<close>
   is 0 .
 
 lift_definition one_word :: \<open>'a word\<close>
   is 1 .
 
 lift_definition plus_word :: \<open>'a word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
   is \<open>(+)\<close>
   by (auto simp add: take_bit_eq_mod intro: mod_add_cong)
 
 lift_definition minus_word :: \<open>'a word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
   is \<open>(-)\<close>
   by (auto simp add: take_bit_eq_mod intro: mod_diff_cong)
 
 lift_definition uminus_word :: \<open>'a word \<Rightarrow> 'a word\<close>
   is uminus
   by (auto simp add: take_bit_eq_mod intro: mod_minus_cong)
 
 lift_definition times_word :: \<open>'a word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
   is \<open>(*)\<close>
   by (auto simp add: take_bit_eq_mod intro: mod_mult_cong)
 
 instance
   by (standard; transfer) (simp_all add: algebra_simps)
 
 end
 
 context
   includes lifting_syntax
   notes
     power_transfer [transfer_rule]
     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 power_transfer_word [transfer_rule]:
   \<open>(pcr_word ===> (=) ===> pcr_word) (^) (^)\<close>
   by transfer_prover
 
 lemma [transfer_rule]:
   \<open>((=) ===> pcr_word) of_bool of_bool\<close>
   by transfer_prover
 
 lemma [transfer_rule]:
   \<open>((=) ===> pcr_word) numeral numeral\<close>
   by transfer_prover
 
 lemma [transfer_rule]:
   \<open>((=) ===> pcr_word) int of_nat\<close>
   by transfer_prover
 
 lemma [transfer_rule]:
   \<open>((=) ===> pcr_word) (\<lambda>k. k) of_int\<close>
 proof -
   have \<open>((=) ===> pcr_word) of_int of_int\<close>
     by transfer_prover
   then show ?thesis by (simp add: id_def)
 qed
 
 lemma [transfer_rule]:
   \<open>(pcr_word ===> (\<longleftrightarrow>)) even ((dvd) 2 :: 'a::len word \<Rightarrow> bool)\<close>
 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
 
 lemma word_exp_length_eq_0 [simp]:
   \<open>(2 :: 'a::len word) ^ LENGTH('a) = 0\<close>
   by transfer simp
 
 lemma exp_eq_zero_iff:
   \<open>2 ^ n = (0 :: 'a::len word) \<longleftrightarrow> n \<ge> LENGTH('a)\<close>
   by transfer simp
 
 
 subsubsection \<open>Basic code generation setup\<close>
 
 context
 begin
 
 qualified lift_definition the_int :: \<open>'a::len word \<Rightarrow> int\<close>
   is \<open>take_bit LENGTH('a)\<close> .
 
 end
 
 lemma [code abstype]:
   \<open>Word.Word (Word.the_int w) = w\<close>
   by transfer simp
 
 lemma Word_eq_word_of_int [code_post, simp]:
   \<open>Word.Word = of_int\<close>
   by (rule; transfer) simp
 
 quickcheck_generator word
   constructors:
     \<open>0 :: 'a::len word\<close>,
     \<open>numeral :: num \<Rightarrow> 'a::len word\<close>
 
 instantiation word :: (len) equal
 begin
 
 lift_definition equal_word :: \<open>'a word \<Rightarrow> 'a word \<Rightarrow> bool\<close>
   is \<open>\<lambda>k l. take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
   by simp
 
 instance
   by (standard; transfer) rule
 
 end
 
 lemma [code]:
   \<open>HOL.equal v w \<longleftrightarrow> HOL.equal (Word.the_int v) (Word.the_int w)\<close>
   by transfer (simp add: equal)
 
 lemma [code]:
   \<open>Word.the_int 0 = 0\<close>
   by transfer simp
 
 lemma [code]:
   \<open>Word.the_int 1 = 1\<close>
   by transfer simp
 
 lemma [code]:
   \<open>Word.the_int (v + w) = take_bit LENGTH('a) (Word.the_int v + Word.the_int w)\<close>
   for v w :: \<open>'a::len word\<close>
   by transfer (simp add: take_bit_add)
 
 lemma [code]:
   \<open>Word.the_int (- w) = (let k = Word.the_int w in if w = 0 then 0 else 2 ^ LENGTH('a) - k)\<close>
   for w :: \<open>'a::len word\<close>
   by transfer (auto simp add: take_bit_eq_mod zmod_zminus1_eq_if)
 
 lemma [code]:
   \<open>Word.the_int (v - w) = take_bit LENGTH('a) (Word.the_int v - Word.the_int w)\<close>
   for v w :: \<open>'a::len word\<close>
   by transfer (simp add: take_bit_diff)
 
 lemma [code]:
   \<open>Word.the_int (v * w) = take_bit LENGTH('a) (Word.the_int v * Word.the_int w)\<close>
   for v w :: \<open>'a::len word\<close>
   by transfer (simp add: take_bit_mult)
 
 
 subsubsection \<open>Basic conversions\<close>
 
 abbreviation word_of_nat :: \<open>nat \<Rightarrow> 'a::len word\<close>
   where \<open>word_of_nat \<equiv> of_nat\<close>
 
 abbreviation word_of_int :: \<open>int \<Rightarrow> 'a::len word\<close>
   where \<open>word_of_int \<equiv> of_int\<close>
 
 lemma word_of_nat_eq_iff:
   \<open>word_of_nat m = (word_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 word_of_int_eq_iff:
   \<open>word_of_int k = (word_of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
   by transfer rule
 
 lemma word_of_nat_eq_0_iff [simp]:
   \<open>word_of_nat n = (0 :: 'a::len word) \<longleftrightarrow> 2 ^ LENGTH('a) dvd n\<close>
   using word_of_nat_eq_iff [where ?'a = 'a, of n 0] by (simp add: take_bit_eq_0_iff)
 
 lemma word_of_int_eq_0_iff [simp]:
   \<open>word_of_int k = (0 :: 'a::len word) \<longleftrightarrow> 2 ^ LENGTH('a) dvd k\<close>
   using word_of_int_eq_iff [where ?'a = 'a, of k 0] by (simp add: take_bit_eq_0_iff)
 
 context semiring_1
 begin
 
 lift_definition unsigned :: \<open>'b::len word \<Rightarrow> 'a\<close>
   is \<open>of_nat \<circ> nat \<circ> take_bit LENGTH('b)\<close>
   by simp
 
 lemma unsigned_0 [simp]:
   \<open>unsigned 0 = 0\<close>
   by transfer simp
 
 lemma unsigned_1 [simp]:
   \<open>unsigned 1 = 1\<close>
   by transfer simp
 
 lemma unsigned_numeral [simp]:
   \<open>unsigned (numeral n :: 'b::len word) = of_nat (take_bit LENGTH('b) (numeral n))\<close>
   by transfer (simp add: nat_take_bit_eq)
 
 lemma unsigned_neg_numeral [simp]:
   \<open>unsigned (- numeral n :: 'b::len word) = of_nat (nat (take_bit LENGTH('b) (- numeral n)))\<close>
   by transfer simp
 
 end
 
 context semiring_1
 begin
 
 lemma unsigned_of_nat [simp]:
   \<open>unsigned (word_of_nat n :: 'b::len word) = of_nat (take_bit LENGTH('b) n)\<close>
   by transfer (simp add: nat_eq_iff take_bit_of_nat)
 
 lemma unsigned_of_int [simp]:
   \<open>unsigned (word_of_int k :: 'b::len word) = of_nat (nat (take_bit LENGTH('b) k))\<close>
   by transfer simp
 
 end
 
 context semiring_char_0
 begin
 
 lemma unsigned_word_eqI:
   \<open>v = w\<close> if \<open>unsigned v = unsigned w\<close>
   using that by transfer (simp add: eq_nat_nat_iff)
 
 lemma word_eq_iff_unsigned:
   \<open>v = w \<longleftrightarrow> unsigned v = unsigned w\<close>
   by (auto intro: unsigned_word_eqI)
 
 lemma inj_unsigned [simp]:
   \<open>inj unsigned\<close>
   by (rule injI) (simp add: unsigned_word_eqI)
 
 lemma unsigned_eq_0_iff:
   \<open>unsigned w = 0 \<longleftrightarrow> w = 0\<close>
   using word_eq_iff_unsigned [of w 0] by simp
 
 end
 
 context ring_1
 begin
 
 lift_definition signed :: \<open>'b::len word \<Rightarrow> 'a\<close>
   is \<open>of_int \<circ> signed_take_bit (LENGTH('b) - Suc 0)\<close>
   by (simp flip: signed_take_bit_decr_length_iff)
 
 lemma signed_0 [simp]:
   \<open>signed 0 = 0\<close>
   by transfer simp
 
 lemma signed_1 [simp]:
   \<open>signed (1 :: 'b::len word) = (if LENGTH('b) = 1 then - 1 else 1)\<close>
   by (transfer fixing: uminus; cases \<open>LENGTH('b)\<close>)
     (simp_all add: sbintrunc_minus_simps)
 
 lemma signed_minus_1 [simp]:
   \<open>signed (- 1 :: 'b::len word) = - 1\<close>
   by (transfer fixing: uminus) simp
 
 lemma signed_numeral [simp]:
   \<open>signed (numeral n :: 'b::len word) = of_int (signed_take_bit (LENGTH('b) - 1) (numeral n))\<close>
   by transfer simp
 
 lemma signed_neg_numeral [simp]:
   \<open>signed (- numeral n :: 'b::len word) = of_int (signed_take_bit (LENGTH('b) - 1) (- numeral n))\<close>
   by transfer simp
 
 lemma signed_of_nat [simp]:
   \<open>signed (word_of_nat n :: 'b::len word) = of_int (signed_take_bit (LENGTH('b) - Suc 0) (int n))\<close>
   by transfer simp
 
 lemma signed_of_int [simp]:
   \<open>signed (word_of_int n :: 'b::len word) = of_int (signed_take_bit (LENGTH('b) - Suc 0) n)\<close>
   by transfer simp
 
 end
 
 context ring_char_0
 begin
 
 lemma signed_word_eqI:
   \<open>v = w\<close> if \<open>signed v = signed w\<close>
   using that by transfer (simp flip: signed_take_bit_decr_length_iff)
 
 lemma word_eq_iff_signed:
   \<open>v = w \<longleftrightarrow> signed v = signed w\<close>
   by (auto intro: signed_word_eqI)
 
 lemma inj_signed [simp]:
   \<open>inj signed\<close>
   by (rule injI) (simp add: signed_word_eqI)
 
 lemma signed_eq_0_iff:
   \<open>signed w = 0 \<longleftrightarrow> w = 0\<close>
   using word_eq_iff_signed [of w 0] by simp
 
 end
 
 abbreviation unat :: \<open>'a::len word \<Rightarrow> nat\<close>
   where \<open>unat \<equiv> unsigned\<close>
 
 abbreviation uint :: \<open>'a::len word \<Rightarrow> int\<close>
   where \<open>uint \<equiv> unsigned\<close>
 
 abbreviation sint :: \<open>'a::len word \<Rightarrow> int\<close>
   where \<open>sint \<equiv> signed\<close>
 
 abbreviation ucast :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
   where \<open>ucast \<equiv> unsigned\<close>
 
 abbreviation scast :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
   where \<open>scast \<equiv> signed\<close>
 
 context
   includes lifting_syntax
 begin
 
 lemma [transfer_rule]:
   \<open>(pcr_word ===> (=)) (nat \<circ> take_bit LENGTH('a)) (unat :: 'a::len word \<Rightarrow> nat)\<close>
   using unsigned.transfer [where ?'a = nat] by simp
 
 lemma [transfer_rule]:
   \<open>(pcr_word ===> (=)) (take_bit LENGTH('a)) (uint :: 'a::len word \<Rightarrow> int)\<close>
   using unsigned.transfer [where ?'a = int] by (simp add: comp_def)
 
 lemma [transfer_rule]:
   \<open>(pcr_word ===> (=)) (signed_take_bit (LENGTH('a) - Suc 0)) (sint :: 'a::len word \<Rightarrow> int)\<close>
   using signed.transfer [where ?'a = int] by simp
 
 lemma [transfer_rule]:
   \<open>(pcr_word ===> pcr_word) (take_bit LENGTH('a)) (ucast :: 'a::len word \<Rightarrow> 'b::len word)\<close>
 proof (rule rel_funI)
   fix k :: int and w :: \<open>'a word\<close>
   assume \<open>pcr_word k w\<close>
   then have \<open>w = word_of_int k\<close>
     by (simp add: pcr_word_def cr_word_def relcompp_apply)
   moreover have \<open>pcr_word (take_bit LENGTH('a) k) (ucast (word_of_int k :: 'a word))\<close>
     by transfer (simp add: pcr_word_def cr_word_def relcompp_apply)
   ultimately show \<open>pcr_word (take_bit LENGTH('a) k) (ucast w)\<close>
     by simp
 qed
 
 lemma [transfer_rule]:
   \<open>(pcr_word ===> pcr_word) (signed_take_bit (LENGTH('a) - Suc 0)) (scast :: 'a::len word \<Rightarrow> 'b::len word)\<close>
 proof (rule rel_funI)
   fix k :: int and w :: \<open>'a word\<close>
   assume \<open>pcr_word k w\<close>
   then have \<open>w = word_of_int k\<close>
     by (simp add: pcr_word_def cr_word_def relcompp_apply)
   moreover have \<open>pcr_word (signed_take_bit (LENGTH('a) - Suc 0) k) (scast (word_of_int k :: 'a word))\<close>
     by transfer (simp add: pcr_word_def cr_word_def relcompp_apply)
   ultimately show \<open>pcr_word (signed_take_bit (LENGTH('a) - Suc 0) k) (scast w)\<close>
     by simp
 qed
 
 end
 
 lemma of_nat_unat [simp]:
   \<open>of_nat (unat w) = unsigned w\<close>
   by transfer simp
 
 lemma of_int_uint [simp]:
   \<open>of_int (uint w) = unsigned w\<close>
   by transfer simp
 
 lemma of_int_sint [simp]:
   \<open>of_int (sint a) = signed a\<close>
   by transfer (simp_all add: take_bit_signed_take_bit)
 
 lemma nat_uint_eq [simp]:
   \<open>nat (uint w) = unat w\<close>
   by transfer simp
 
 lemma sgn_uint_eq [simp]:
   \<open>sgn (uint w) = of_bool (w \<noteq> 0)\<close>
   by transfer (simp add: less_le)
 
 text \<open>Aliasses only for code generation\<close>
 
 context
 begin
 
 qualified lift_definition of_int :: \<open>int \<Rightarrow> 'a::len word\<close>
   is \<open>take_bit LENGTH('a)\<close> .
 
 qualified lift_definition of_nat :: \<open>nat \<Rightarrow> 'a::len word\<close>
   is \<open>int \<circ> take_bit LENGTH('a)\<close> .
 
 qualified lift_definition the_nat :: \<open>'a::len word \<Rightarrow> nat\<close>
   is \<open>nat \<circ> take_bit LENGTH('a)\<close> by simp
 
 qualified lift_definition the_signed_int :: \<open>'a::len word \<Rightarrow> int\<close>
   is \<open>signed_take_bit (LENGTH('a) - Suc 0)\<close> by (simp add: signed_take_bit_decr_length_iff)
 
 qualified lift_definition cast :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
   is \<open>take_bit LENGTH('a)\<close> by simp
 
 qualified lift_definition signed_cast :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
   is \<open>signed_take_bit (LENGTH('a) - Suc 0)\<close> by (metis signed_take_bit_decr_length_iff)
 
 end
 
 lemma [code_abbrev, simp]:
   \<open>Word.the_int = uint\<close>
   by transfer rule
 
 lemma [code]:
   \<open>Word.the_int (Word.of_int k :: 'a::len word) = take_bit LENGTH('a) k\<close>
   by transfer simp
 
 lemma [code_abbrev, simp]:
   \<open>Word.of_int = word_of_int\<close>
   by (rule; transfer) simp
 
 lemma [code]:
   \<open>Word.the_int (Word.of_nat n :: 'a::len word) = take_bit LENGTH('a) (int n)\<close>
   by transfer (simp add: take_bit_of_nat)
 
 lemma [code_abbrev, simp]:
   \<open>Word.of_nat = word_of_nat\<close>
   by (rule; transfer) (simp add: take_bit_of_nat)
 
 lemma [code]:
   \<open>Word.the_nat w = nat (Word.the_int w)\<close>
   by transfer simp
 
 lemma [code_abbrev, simp]:
   \<open>Word.the_nat = unat\<close>
   by (rule; transfer) simp
 
 lemma [code]:
   \<open>Word.the_signed_int w = signed_take_bit (LENGTH('a) - Suc 0) (Word.the_int w)\<close>
   for w :: \<open>'a::len word\<close>
   by transfer simp
 
 lemma [code_abbrev, simp]:
   \<open>Word.the_signed_int = sint\<close>
   by (rule; transfer) simp
 
 lemma [code]:
   \<open>Word.the_int (Word.cast w :: 'b::len word) = take_bit LENGTH('b) (Word.the_int w)\<close>
   for w :: \<open>'a::len word\<close>
   by transfer simp
 
 lemma [code_abbrev, simp]:
   \<open>Word.cast = ucast\<close>
   by (rule; transfer) simp
 
 lemma [code]:
   \<open>Word.the_int (Word.signed_cast w :: 'b::len word) = take_bit LENGTH('b) (Word.the_signed_int w)\<close>
   for w :: \<open>'a::len word\<close>
   by transfer simp
 
 lemma [code_abbrev, simp]:
   \<open>Word.signed_cast = scast\<close>
   by (rule; transfer) simp
 
 lemma [code]:
   \<open>unsigned w = of_nat (nat (Word.the_int w))\<close>
   by transfer simp
 
 lemma [code]:
   \<open>signed w = of_int (Word.the_signed_int w)\<close>
   by transfer simp
 
 
 subsubsection \<open>Basic ordering\<close>
 
 instantiation word :: (len) 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
 
 interpretation word_order: ordering_top \<open>(\<le>)\<close> \<open>(<)\<close> \<open>- 1 :: 'a::len word\<close>
   by (standard; transfer) (simp add: take_bit_eq_mod zmod_minus1)
 
 interpretation word_coorder: ordering_top \<open>(\<ge>)\<close> \<open>(>)\<close> \<open>0 :: 'a::len word\<close>
   by (standard; transfer) simp
 
 lemma word_of_nat_less_eq_iff:
   \<open>word_of_nat m \<le> (word_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 word_of_int_less_eq_iff:
   \<open>word_of_int k \<le> (word_of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k \<le> take_bit LENGTH('a) l\<close>
   by transfer rule
 
 lemma word_of_nat_less_iff:
   \<open>word_of_nat m < (word_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 word_of_int_less_iff:
   \<open>word_of_int k < (word_of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k < take_bit LENGTH('a) l\<close>
   by transfer rule
 
 lemma word_le_def [code]:
   "a \<le> b \<longleftrightarrow> uint a \<le> uint b"
   by transfer rule
 
 lemma word_less_def [code]:
   "a < b \<longleftrightarrow> uint a < uint b"
   by transfer rule
 
 lemma word_greater_zero_iff:
   \<open>a > 0 \<longleftrightarrow> a \<noteq> 0\<close> for a :: \<open>'a::len word\<close>
   by transfer (simp add: less_le)
 
 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_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
 
 
 
 subsection \<open>Enumeration\<close>
 
 lemma inj_on_word_of_nat:
   \<open>inj_on (word_of_nat :: nat \<Rightarrow> 'a::len word) {0..<2 ^ LENGTH('a)}\<close>
   by (rule inj_onI; transfer) (simp_all add: take_bit_int_eq_self)
 
 lemma UNIV_word_eq_word_of_nat:
   \<open>(UNIV :: 'a::len word set) = word_of_nat ` {0..<2 ^ LENGTH('a)}\<close> (is \<open>_ = ?A\<close>)
 proof
   show \<open>word_of_nat ` {0..<2 ^ LENGTH('a)} \<subseteq> UNIV\<close>
     by simp
   show \<open>UNIV \<subseteq> ?A\<close>
   proof
     fix w :: \<open>'a word\<close>
     show \<open>w \<in> (word_of_nat ` {0..<2 ^ LENGTH('a)} :: 'a word set)\<close>
       by (rule image_eqI [of _ _ \<open>unat w\<close>]; transfer) simp_all
   qed
 qed
 
 instantiation word :: (len) enum
 begin
 
 definition enum_word :: \<open>'a word list\<close>
   where \<open>enum_word = map word_of_nat [0..<2 ^ LENGTH('a)]\<close>
 
 definition enum_all_word :: \<open>('a word \<Rightarrow> bool) \<Rightarrow> bool\<close>
   where \<open>enum_all_word = Ball UNIV\<close>
 
 definition enum_ex_word :: \<open>('a word \<Rightarrow> bool) \<Rightarrow> bool\<close>
   where \<open>enum_ex_word = Bex UNIV\<close>
 
 lemma [code]:
   \<open>Enum.enum_all P \<longleftrightarrow> Ball UNIV P\<close>
   \<open>Enum.enum_ex P \<longleftrightarrow> Bex UNIV P\<close> for P :: \<open>'a word \<Rightarrow> bool\<close>
   by (simp_all add: enum_all_word_def enum_ex_word_def)
 
 instance
   by standard (simp_all add: UNIV_word_eq_word_of_nat inj_on_word_of_nat enum_word_def enum_all_word_def enum_ex_word_def distinct_map)
 
 end
 
 
 subsection \<open>Bit-wise operations\<close>
 
 instantiation word :: (len) semiring_modulo
 begin
 
 lift_definition divide_word :: \<open>'a word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>a b. take_bit LENGTH('a) a div take_bit LENGTH('a) b\<close>
   by simp
 
 lift_definition modulo_word :: \<open>'a word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>a b. take_bit LENGTH('a) a mod take_bit LENGTH('a) b\<close>
   by simp
 
 instance 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
 
 end
 
 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 (simp_all add: mod_2_eq_odd take_bit_Suc)
   show "\<not> 2 dvd a \<longleftrightarrow> a mod 2 = 1"
     for a :: "'a word"
     by transfer (simp_all add: mod_2_eq_odd take_bit_Suc)
 qed
 
 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) - Suc 0) \<Longrightarrow> P (2 * a)\<close>
     and word_odd: \<open>\<And>a. P a \<Longrightarrow> a < 2 ^ (LENGTH('a) - Suc 0) \<Longrightarrow> P (1 + 2 * a)\<close>
   for P and a :: \<open>'a::len word\<close>
 proof -
   define m :: nat where \<open>m = LENGTH('a) - Suc 0\<close>
   then have l: \<open>LENGTH('a) = Suc m\<close>
     by simp
   define n :: nat where \<open>n = unat a\<close>
   then have \<open>n < 2 ^ LENGTH('a)\<close>
     by transfer (simp add: take_bit_eq_mod)
   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 l)
     moreover from \<open>n < 2 ^ m\<close> have \<open>(of_nat n :: 'a word) < 2 ^ (LENGTH('a) - Suc 0)\<close>
       using of_nat_word_less_iff [where ?'a = 'a, of n \<open>2 ^ m\<close>]
       by (simp add: l take_bit_eq_mod)
     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) - Suc 0)\<close>
       using of_nat_word_less_iff [where ?'a = 'a, of n \<open>2 ^ m\<close>]
       by (simp add: l take_bit_eq_mod)
     ultimately have \<open>P (1 + 2 * of_nat n)\<close>
       by (rule word_odd)
     then show ?case
       by simp
   qed
   moreover have \<open>of_nat (nat (uint a)) = a\<close>
     by transfer simp
   ultimately 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 \<open>2 \<le> LENGTH('a::len)\<close>)
   case False
   have \<open>of_bool (odd k) < (1 :: int) \<longleftrightarrow> even k\<close> for k :: int
     by auto
   with False that show ?thesis
     by transfer (simp add: eq_iff)
 next
   case True
   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 True 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 True 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 (auto simp add: take_bit_Suc)
     qed
     ultimately show ?thesis
       by simp
   qed
 qed
 
 lemma even_mult_exp_div_word_iff:
   \<open>even (a * 2 ^ m div 2 ^ n) \<longleftrightarrow> \<not> (
     m \<le> n \<and>
     n < LENGTH('a) \<and> odd (a div 2 ^ (n - m)))\<close> for a :: \<open>'a::len word\<close>
   by transfer
     (auto simp flip: drop_bit_eq_div simp add: even_drop_bit_iff_not_bit bit_take_bit_iff,
       simp_all flip: push_bit_eq_mult add: bit_push_bit_iff_int)
 
 instantiation word :: (len) semiring_bits
 begin
 
 lift_definition bit_word :: \<open>'a word \<Rightarrow> nat \<Rightarrow> bool\<close>
   is \<open>\<lambda>k n. n < LENGTH('a) \<and> bit k n\<close>
 proof
   fix k l :: int and n :: nat
   assume *: \<open>take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
   show \<open>n < LENGTH('a) \<and> bit k n \<longleftrightarrow> n < LENGTH('a) \<and> bit l n\<close>
   proof (cases \<open>n < LENGTH('a)\<close>)
     case True
     from * have \<open>bit (take_bit LENGTH('a) k) n \<longleftrightarrow> bit (take_bit LENGTH('a) l) n\<close>
       by simp
     then show ?thesis
       by (simp add: bit_take_bit_iff)
   next
     case False
     then show ?thesis
       by simp
   qed
 qed
 
 instance 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
     have \<open>0 div 2 = (0::'a word)\<close>
       by transfer simp
     with 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>bit a n \<longleftrightarrow> odd (a div 2 ^ n)\<close> for a :: \<open>'a word\<close> and n
     by transfer (simp flip: drop_bit_eq_div add: drop_bit_take_bit bit_iff_odd_drop_bit)
   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 simp add: mod_2_eq_odd take_bit_Suc elim!: evenE)
   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
     by transfer (auto simp flip: take_bit_eq_mod simp add: ac_simps)
   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
     by transfer (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)
   show \<open>even ((2 ^ m - 1) div (2::'a word) ^ n) \<longleftrightarrow> 2 ^ n = (0::'a word) \<or> m \<le> n\<close>
     for m n :: nat
     by transfer (auto simp add: take_bit_of_mask even_mask_div_iff)
   show \<open>even (a * 2 ^ m div 2 ^ n) \<longleftrightarrow> n < m \<or> (2::'a word) ^ n = 0 \<or> m \<le> n \<and> even (a div 2 ^ (n - m))\<close>
     for a :: \<open>'a word\<close> and m n :: nat
   proof transfer
     show \<open>even (take_bit LENGTH('a) (k * 2 ^ m) div take_bit LENGTH('a) (2 ^ n)) \<longleftrightarrow>
       n < m
       \<or> take_bit LENGTH('a) ((2::int) ^ n) = take_bit LENGTH('a) 0
       \<or> (m \<le> n \<and> even (take_bit LENGTH('a) k div take_bit LENGTH('a) (2 ^ (n - m))))\<close>
     for m n :: nat and k l :: int
       by (auto simp flip: take_bit_eq_mod drop_bit_eq_div push_bit_eq_mult
         simp add: div_push_bit_of_1_eq_drop_bit drop_bit_take_bit drop_bit_push_bit_int [of n m])
   qed
 qed
 
 end
 
 lemma bit_word_eqI:
   \<open>a = b\<close> if \<open>\<And>n. n < LENGTH('a) \<Longrightarrow> bit a n \<longleftrightarrow> bit b n\<close>
   for a b :: \<open>'a::len word\<close>
   using that by transfer (auto simp add: nat_less_le bit_eq_iff bit_take_bit_iff)
 
 lemma bit_imp_le_length:
   \<open>n < LENGTH('a)\<close> if \<open>bit w n\<close>
     for w :: \<open>'a::len word\<close>
   using that by transfer simp
 
 lemma not_bit_length [simp]:
   \<open>\<not> bit w LENGTH('a)\<close> for w :: \<open>'a::len word\<close>
   by transfer simp
 
 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>take_bit LENGTH('a) (push_bit n k) = take_bit LENGTH('a) (push_bit n l)\<close>
     if \<open>take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close> for k l :: int and n :: nat
   proof -
     from that
     have \<open>take_bit (LENGTH('a) - n) (take_bit LENGTH('a) k)
       = take_bit (LENGTH('a) - n) (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)
 
 lift_definition take_bit_word :: \<open>nat \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>n. take_bit (min LENGTH('a) n)\<close>
   by (simp add: ac_simps) (simp only: flip: take_bit_take_bit)
 
 instance proof
   show \<open>push_bit n a = a * 2 ^ n\<close> for n :: nat and a :: \<open>'a word\<close>
     by transfer (simp add: push_bit_eq_mult)
   show \<open>drop_bit n a = a div 2 ^ n\<close> for n :: nat and a :: \<open>'a word\<close>
     by transfer (simp flip: drop_bit_eq_div add: drop_bit_take_bit)
   show \<open>take_bit n a = a mod 2 ^ n\<close> for n :: nat and a :: \<open>'a word\<close>
     by transfer (auto simp flip: take_bit_eq_mod)
 qed
 
 end
 
 lemma [code]:
   \<open>Word.the_int (take_bit n w) = (if n < LENGTH('a::len) then take_bit n (Word.the_int w) else Word.the_int w)\<close>
   for w :: \<open>'a::len word\<close>
   by transfer (simp add: not_le not_less ac_simps min_absorb2)
 
 
 instantiation word :: (len) ring_bit_operations
 begin
 
 lift_definition not_word :: \<open>'a word \<Rightarrow> 'a word\<close>
   is not
   by (simp add: take_bit_not_iff)
 
 lift_definition and_word :: \<open>'a word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
   is \<open>and\<close>
   by simp
 
 lift_definition or_word :: \<open>'a word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
   is or
   by simp
 
 lift_definition xor_word ::  \<open>'a word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
   is xor
   by simp
 
 lift_definition mask_word :: \<open>nat \<Rightarrow> 'a word\<close>
   is mask
   .
 
 instance by (standard; transfer)
   (auto simp add: minus_eq_not_minus_1 mask_eq_exp_minus_1
     bit_not_iff bit_and_iff bit_or_iff bit_xor_iff)
 
 end
 
 lemma [code_abbrev]:
   \<open>push_bit n 1 = (2 :: 'a::len word) ^ n\<close>
   by (fact push_bit_of_1)
 
 lemma [code]:
   \<open>NOT w = Word.of_int (NOT (Word.the_int w))\<close>
   for w :: \<open>'a::len word\<close>
   by transfer (simp add: take_bit_not_take_bit) 
 
 lemma [code]:
   \<open>Word.the_int (v AND w) = Word.the_int v AND Word.the_int w\<close>
   by transfer simp
 
 lemma [code]:
   \<open>Word.the_int (v OR w) = Word.the_int v OR Word.the_int w\<close>
   by transfer simp
 
 lemma [code]:
   \<open>Word.the_int (v XOR w) = Word.the_int v XOR Word.the_int w\<close>
   by transfer simp
 
 lemma [code]:
   \<open>Word.the_int (mask n :: 'a::len word) = mask (min LENGTH('a) n)\<close>
   by transfer simp
 
 context
   includes lifting_syntax
 begin
 
 lemma set_bit_word_transfer [transfer_rule]:
   \<open>((=) ===> pcr_word ===> pcr_word) set_bit set_bit\<close>
   by (unfold set_bit_def) transfer_prover
 
 lemma unset_bit_word_transfer [transfer_rule]:
   \<open>((=) ===> pcr_word ===> pcr_word) unset_bit unset_bit\<close>
   by (unfold unset_bit_def) transfer_prover
 
 lemma flip_bit_word_transfer [transfer_rule]:
   \<open>((=) ===> pcr_word ===> pcr_word) flip_bit flip_bit\<close>
   by (unfold flip_bit_def) transfer_prover
 
 lemma signed_take_bit_word_transfer [transfer_rule]:
   \<open>((=) ===> pcr_word ===> pcr_word)
     (\<lambda>n k. signed_take_bit n (take_bit LENGTH('a::len) k))
     (signed_take_bit :: nat \<Rightarrow> 'a word \<Rightarrow> 'a word)\<close>
 proof -
   let ?K = \<open>\<lambda>n (k :: int). take_bit (min LENGTH('a) n) k OR of_bool (n < LENGTH('a) \<and> bit k n) * NOT (mask n)\<close>
   let ?W = \<open>\<lambda>n (w :: 'a word). take_bit n w OR of_bool (bit w n) * NOT (mask n)\<close>
   have \<open>((=) ===> pcr_word ===> pcr_word) ?K ?W\<close>
     by transfer_prover
   also have \<open>?K = (\<lambda>n k. signed_take_bit n (take_bit LENGTH('a::len) k))\<close>
     by (simp add: fun_eq_iff signed_take_bit_def bit_take_bit_iff ac_simps)
   also have \<open>?W = signed_take_bit\<close>
     by (simp add: fun_eq_iff signed_take_bit_def)
   finally show ?thesis .
 qed
 
 end
 
 
 subsection \<open>Conversions including casts\<close>
 
 subsubsection \<open>Generic unsigned conversion\<close>
 
 context semiring_bits
 begin
 
 lemma bit_unsigned_iff:
   \<open>bit (unsigned w) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> bit w n\<close>
   for w :: \<open>'b::len word\<close>
   by (transfer fixing: bit) (simp add: bit_of_nat_iff bit_nat_iff bit_take_bit_iff)
 
 end
 
 context semiring_bit_shifts
 begin
 
 lemma unsigned_push_bit_eq:
   \<open>unsigned (push_bit n w) = take_bit LENGTH('b) (push_bit n (unsigned w))\<close>
   for w :: \<open>'b::len word\<close>
 proof (rule bit_eqI)
   fix m
   assume \<open>2 ^ m \<noteq> 0\<close>
   show \<open>bit (unsigned (push_bit n w)) m = bit (take_bit LENGTH('b) (push_bit n (unsigned w))) m\<close>
   proof (cases \<open>n \<le> m\<close>)
     case True
     with \<open>2 ^ m \<noteq> 0\<close> have \<open>2 ^ (m - n) \<noteq> 0\<close>
       by (metis (full_types) diff_add exp_add_not_zero_imp)
     with True show ?thesis
       by (simp add: bit_unsigned_iff bit_push_bit_iff Parity.bit_push_bit_iff bit_take_bit_iff not_le exp_eq_zero_iff ac_simps)
   next
     case False
     then show ?thesis
       by (simp add: not_le bit_unsigned_iff bit_push_bit_iff Parity.bit_push_bit_iff bit_take_bit_iff)
   qed
 qed
 
 lemma unsigned_take_bit_eq:
   \<open>unsigned (take_bit n w) = take_bit n (unsigned w)\<close>
   for w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_unsigned_iff bit_take_bit_iff Parity.bit_take_bit_iff)
 
 end
 
 context semiring_bit_operations
 begin
 
 lemma unsigned_and_eq:
   \<open>unsigned (v AND w) = unsigned v AND unsigned w\<close>
   for v w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_unsigned_iff bit_and_iff Bit_Operations.bit_and_iff)
 
 lemma unsigned_or_eq:
   \<open>unsigned (v OR w) = unsigned v OR unsigned w\<close>
   for v w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_unsigned_iff bit_or_iff Bit_Operations.bit_or_iff)
 
 lemma unsigned_xor_eq:
   \<open>unsigned (v XOR w) = unsigned v XOR unsigned w\<close>
   for v w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_unsigned_iff bit_xor_iff Bit_Operations.bit_xor_iff)
 
 end
 
 context ring_bit_operations
 begin
 
 lemma unsigned_not_eq:
   \<open>unsigned (NOT w) = take_bit LENGTH('b) (NOT (unsigned w))\<close>
   for w :: \<open>'b::len word\<close>
   by (rule bit_eqI)
     (simp add: bit_unsigned_iff bit_take_bit_iff bit_not_iff Bit_Operations.bit_not_iff exp_eq_zero_iff not_le)
 
 end
 
 context unique_euclidean_semiring_numeral
 begin
 
 lemma unsigned_greater_eq [simp]:
   \<open>0 \<le> unsigned w\<close> for w :: \<open>'b::len word\<close>
   by (transfer fixing: less_eq) simp
 
 lemma unsigned_less [simp]:
   \<open>unsigned w < 2 ^ LENGTH('b)\<close> for w :: \<open>'b::len word\<close>
   by (transfer fixing: less) simp
 
 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
 
 
 subsubsection \<open>Generic signed conversion\<close>
 
 context ring_bit_operations
 begin
 
 lemma bit_signed_iff:
   \<open>bit (signed w) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> bit w (min (LENGTH('b) - Suc 0) n)\<close>
   for w :: \<open>'b::len word\<close>
   by (transfer fixing: bit)
     (auto simp add: bit_of_int_iff Bit_Operations.bit_signed_take_bit_iff min_def)
 
 lemma signed_push_bit_eq:
   \<open>signed (push_bit n w) = signed_take_bit (LENGTH('b) - Suc 0) (push_bit n (signed w :: 'a))\<close>
   for w :: \<open>'b::len word\<close>
 proof (rule bit_eqI)
   fix m
   assume \<open>2 ^ m \<noteq> 0\<close>
   define q where \<open>q = LENGTH('b) - Suc 0\<close>
   then have *: \<open>LENGTH('b) = Suc q\<close>
     by simp
   show \<open>bit (signed (push_bit n w)) m \<longleftrightarrow>
     bit (signed_take_bit (LENGTH('b) - Suc 0) (push_bit n (signed w :: 'a))) m\<close>
   proof (cases \<open>q \<le> m\<close>)
     case True
     moreover define r where \<open>r = m - q\<close>
     ultimately have \<open>m = q + r\<close>
       by simp
     moreover from \<open>m = q + r\<close> \<open>2 ^ m \<noteq> 0\<close> have \<open>2 ^ q \<noteq> 0\<close> \<open>2 ^ r \<noteq> 0\<close>
       using exp_add_not_zero_imp_left [of q r] exp_add_not_zero_imp_right [of q r]
       by simp_all
     moreover from \<open>2 ^ q \<noteq> 0\<close> have \<open>2 ^ (q - n) \<noteq> 0\<close>
       by (rule exp_not_zero_imp_exp_diff_not_zero)
     ultimately show ?thesis
       by (auto simp add: bit_signed_iff bit_signed_take_bit_iff bit_push_bit_iff Parity.bit_push_bit_iff
       min_def * exp_eq_zero_iff le_diff_conv2)
   next
     case False
     then show ?thesis
       using exp_not_zero_imp_exp_diff_not_zero [of m n]
       by (auto simp add: bit_signed_iff bit_signed_take_bit_iff bit_push_bit_iff Parity.bit_push_bit_iff
       min_def not_le not_less * le_diff_conv2 less_diff_conv2 Parity.exp_eq_0_imp_not_bit exp_eq_0_imp_not_bit
       exp_eq_zero_iff)
   qed
 qed
 
 lemma signed_take_bit_eq:
   \<open>signed (take_bit n w) = (if n < LENGTH('b) then take_bit n (signed w) else signed w)\<close>
   for w :: \<open>'b::len word\<close>
   by (transfer fixing: take_bit; cases \<open>LENGTH('b)\<close>)
     (auto simp add: Bit_Operations.signed_take_bit_take_bit Bit_Operations.take_bit_signed_take_bit take_bit_of_int min_def less_Suc_eq)
 
 lemma signed_not_eq:
   \<open>signed (NOT w) = signed_take_bit LENGTH('b) (NOT (signed w))\<close>
   for w :: \<open>'b::len word\<close>
 proof (rule bit_eqI)
   fix n
   assume \<open>2 ^ n \<noteq> 0\<close>
   define q where \<open>q = LENGTH('b) - Suc 0\<close>
   then have *: \<open>LENGTH('b) = Suc q\<close>
     by simp
   show \<open>bit (signed (NOT w)) n \<longleftrightarrow>
     bit (signed_take_bit LENGTH('b) (NOT (signed w))) n\<close>
   proof (cases \<open>q < n\<close>)
     case True
     moreover define r where \<open>r = n - Suc q\<close>
     ultimately have \<open>n = r + Suc q\<close>
       by simp
     moreover from \<open>2 ^ n \<noteq> 0\<close> \<open>n = r + Suc q\<close>
       have \<open>2 ^ Suc q \<noteq> 0\<close>
       using exp_add_not_zero_imp_right by blast 
     ultimately show ?thesis
       by (simp add: * bit_signed_iff bit_not_iff bit_signed_take_bit_iff Bit_Operations.bit_not_iff min_def
         exp_eq_zero_iff)
   next
     case False
     then show ?thesis
       by (auto simp add: * bit_signed_iff bit_not_iff bit_signed_take_bit_iff Bit_Operations.bit_not_iff min_def
         exp_eq_zero_iff)
   qed
 qed
 
 lemma signed_and_eq:
   \<open>signed (v AND w) = signed v AND signed w\<close>
   for v w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_signed_iff bit_and_iff Bit_Operations.bit_and_iff)
 
 lemma signed_or_eq:
   \<open>signed (v OR w) = signed v OR signed w\<close>
   for v w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_signed_iff bit_or_iff Bit_Operations.bit_or_iff)
 
 lemma signed_xor_eq:
   \<open>signed (v XOR w) = signed v XOR signed w\<close>
   for v w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_signed_iff bit_xor_iff Bit_Operations.bit_xor_iff)
 
 end
 
 
 subsubsection \<open>More\<close>
 
 lemma sint_greater_eq:
   \<open>- (2 ^ (LENGTH('a) - Suc 0)) \<le> sint w\<close> for w :: \<open>'a::len word\<close>
 proof (cases \<open>bit w (LENGTH('a) - Suc 0)\<close>)
   case True
   then show ?thesis
     by transfer (simp add: signed_take_bit_eq_if_negative minus_exp_eq_not_mask or_greater_eq ac_simps)
 next
   have *: \<open>- (2 ^ (LENGTH('a) - Suc 0)) \<le> (0::int)\<close>
     by simp
   case False
   then show ?thesis
     by transfer (auto simp add: signed_take_bit_eq intro: order_trans *)
 qed
 
 lemma sint_less:
   \<open>sint w < 2 ^ (LENGTH('a) - Suc 0)\<close> for w :: \<open>'a::len word\<close>
   by (cases \<open>bit w (LENGTH('a) - Suc 0)\<close>; transfer)
     (simp_all add: signed_take_bit_eq signed_take_bit_def not_eq_complement mask_eq_exp_minus_1 OR_upper)
 
 lemma unat_div_distrib:
   \<open>unat (v div w) = unat v div unat w\<close>
 proof transfer
   fix k l
   have \<open>nat (take_bit LENGTH('a) k) div nat (take_bit LENGTH('a) l) \<le> nat (take_bit LENGTH('a) k)\<close>
     by (rule div_le_dividend)
   also have \<open>nat (take_bit LENGTH('a) k) < 2 ^ LENGTH('a)\<close>
     by (simp add: nat_less_iff)
   finally show \<open>(nat \<circ> take_bit LENGTH('a)) (take_bit LENGTH('a) k div take_bit LENGTH('a) l) =
     (nat \<circ> take_bit LENGTH('a)) k div (nat \<circ> take_bit LENGTH('a)) l\<close>
     by (simp add: nat_take_bit_eq div_int_pos_iff nat_div_distrib take_bit_nat_eq_self_iff)
 qed
 
 lemma unat_mod_distrib:
   \<open>unat (v mod w) = unat v mod unat w\<close>
 proof transfer
   fix k l
   have \<open>nat (take_bit LENGTH('a) k) mod nat (take_bit LENGTH('a) l) \<le> nat (take_bit LENGTH('a) k)\<close>
     by (rule mod_less_eq_dividend)
   also have \<open>nat (take_bit LENGTH('a) k) < 2 ^ LENGTH('a)\<close>
     by (simp add: nat_less_iff)
   finally show \<open>(nat \<circ> take_bit LENGTH('a)) (take_bit LENGTH('a) k mod take_bit LENGTH('a) l) =
     (nat \<circ> take_bit LENGTH('a)) k mod (nat \<circ> take_bit LENGTH('a)) l\<close>
     by (simp add: nat_take_bit_eq mod_int_pos_iff less_le nat_mod_distrib take_bit_nat_eq_self_iff)
 qed
 
 lemma uint_div_distrib:
   \<open>uint (v div w) = uint v div uint w\<close>
 proof -
   have \<open>int (unat (v div w)) = int (unat v div unat w)\<close>
     by (simp add: unat_div_distrib)
   then show ?thesis
     by (simp add: of_nat_div)
 qed
 
 lemma unat_drop_bit_eq:
   \<open>unat (drop_bit n w) = drop_bit n (unat w)\<close>
   by (rule bit_eqI) (simp add: bit_unsigned_iff bit_drop_bit_eq)
 
 lemma uint_mod_distrib:
   \<open>uint (v mod w) = uint v mod uint w\<close>
 proof -
   have \<open>int (unat (v mod w)) = int (unat v mod unat w)\<close>
     by (simp add: unat_mod_distrib)
   then show ?thesis
     by (simp add: of_nat_mod)
 qed
 
 context semiring_bit_shifts
 begin
 
 lemma unsigned_ucast_eq:
   \<open>unsigned (ucast w :: 'c::len word) = take_bit LENGTH('c) (unsigned w)\<close>
   for w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_unsigned_iff Word.bit_unsigned_iff bit_take_bit_iff exp_eq_zero_iff not_le)
 
 end
 
 context ring_bit_operations
 begin
 
 lemma signed_ucast_eq:
   \<open>signed (ucast w :: 'c::len word) = signed_take_bit (LENGTH('c) - Suc 0) (unsigned w)\<close>
   for w :: \<open>'b::len word\<close>
 proof (rule bit_eqI)
   fix n
   assume \<open>2 ^ n \<noteq> 0\<close>
   then have \<open>2 ^ (min (LENGTH('c) - Suc 0) n) \<noteq> 0\<close>
     by (simp add: min_def)
       (metis (mono_tags) diff_diff_cancel exp_not_zero_imp_exp_diff_not_zero)
   then show \<open>bit (signed (ucast w :: 'c::len word)) n \<longleftrightarrow> bit (signed_take_bit (LENGTH('c) - Suc 0) (unsigned w)) n\<close>
     by (simp add: bit_signed_iff bit_unsigned_iff Word.bit_unsigned_iff bit_signed_take_bit_iff exp_eq_zero_iff not_le)
 qed
 
 lemma signed_scast_eq:
   \<open>signed (scast w :: 'c::len word) = signed_take_bit (LENGTH('c) - Suc 0) (signed w)\<close>
   for w :: \<open>'b::len word\<close>
 proof (rule bit_eqI)
   fix n
   assume \<open>2 ^ n \<noteq> 0\<close>
   then have \<open>2 ^ (min (LENGTH('c) - Suc 0) n) \<noteq> 0\<close>
     by (simp add: min_def)
       (metis (mono_tags) diff_diff_cancel exp_not_zero_imp_exp_diff_not_zero)
   then show \<open>bit (signed (scast w :: 'c::len word)) n \<longleftrightarrow> bit (signed_take_bit (LENGTH('c) - Suc 0) (signed w)) n\<close>
     by (simp add: bit_signed_iff bit_unsigned_iff Word.bit_signed_iff bit_signed_take_bit_iff exp_eq_zero_iff not_le)
 qed
 
 end
 
 lemma uint_nonnegative: "0 \<le> uint w"
   by (fact unsigned_greater_eq)
 
 lemma uint_bounded: "uint w < 2 ^ LENGTH('a)"
   for w :: "'a::len word"
   by (fact unsigned_less)
 
 lemma uint_idem: "uint w mod 2 ^ LENGTH('a) = uint w"
   for w :: "'a::len word"
   by transfer (simp add: take_bit_eq_mod)
 
 lemma word_uint_eqI: "uint a = uint b \<Longrightarrow> a = b"
   by (fact unsigned_word_eqI)
 
 lemma word_uint_eq_iff: "a = b \<longleftrightarrow> uint a = uint b"
   by (fact word_eq_iff_unsigned)
 
 lemma uint_word_of_int_eq:
   \<open>uint (word_of_int k :: 'a::len word) = take_bit LENGTH('a) k\<close>
   by transfer rule
 
 lemma uint_word_of_int: "uint (word_of_int k :: 'a::len word) = k mod 2 ^ LENGTH('a)"
   by (simp add: uint_word_of_int_eq take_bit_eq_mod)
   
 lemma word_of_int_uint: "word_of_int (uint w) = w"
   by transfer simp
 
 lemma word_div_def [code]:
   "a div b = word_of_int (uint a div uint b)"
   by transfer rule
 
 lemma word_mod_def [code]:
   "a mod b = word_of_int (uint a mod uint b)"
   by transfer rule
 
 lemma split_word_all: "(\<And>x::'a::len word. PROP P x) \<equiv> (\<And>x. PROP P (word_of_int x))"
 proof
   fix x :: "'a word"
   assume "\<And>x. PROP P (word_of_int x)"
   then have "PROP P (word_of_int (uint x))" .
   then show "PROP P x"
     by (simp only: word_of_int_uint)
 qed
 
 lemma sint_uint:
   \<open>sint w = signed_take_bit (LENGTH('a) - Suc 0) (uint w)\<close>
   for w :: \<open>'a::len word\<close>
   by (cases \<open>LENGTH('a)\<close>; transfer) (simp_all add: signed_take_bit_take_bit)
 
 lemma unat_eq_nat_uint:
   \<open>unat w = nat (uint w)\<close>
   by simp
 
 lemma ucast_eq:
   \<open>ucast w = word_of_int (uint w)\<close>
   by transfer simp
 
 lemma scast_eq:
   \<open>scast w = word_of_int (sint w)\<close>
   by transfer simp
 
 lemma uint_0_eq:
   \<open>uint 0 = 0\<close>
   by (fact unsigned_0)
 
 lemma uint_1_eq:
   \<open>uint 1 = 1\<close>
   by (fact unsigned_1)
 
 lemma word_m1_wi: "- 1 = word_of_int (- 1)"
   by simp
 
 lemma uint_0_iff: "uint x = 0 \<longleftrightarrow> x = 0"
   by (auto simp add: unsigned_word_eqI)
 
 lemma unat_0_iff: "unat x = 0 \<longleftrightarrow> x = 0"
   by (auto simp add: unsigned_word_eqI)
 
 lemma unat_0: "unat 0 = 0"
   by (fact unsigned_0)
 
 lemma unat_gt_0: "0 < unat x \<longleftrightarrow> x \<noteq> 0"
   by (auto simp: unat_0_iff [symmetric])
 
 lemma ucast_0: "ucast 0 = 0"
   by (fact unsigned_0)
 
 lemma sint_0: "sint 0 = 0"
   by (fact signed_0)
 
 lemma scast_0: "scast 0 = 0"
   by (fact signed_0)
 
 lemma sint_n1: "sint (- 1) = - 1"
   by (fact signed_minus_1)
 
 lemma scast_n1: "scast (- 1) = - 1"
   by (fact signed_minus_1)
 
 lemma uint_1: "uint (1::'a::len word) = 1"
   by (fact uint_1_eq)
 
 lemma unat_1: "unat (1::'a::len word) = 1"
   by (fact unsigned_1)
 
 lemma ucast_1: "ucast (1::'a::len word) = 1"
   by (fact unsigned_1)
 
 instantiation word :: (len) size
 begin
 
 lift_definition size_word :: \<open>'a word \<Rightarrow> nat\<close>
   is \<open>\<lambda>_. LENGTH('a)\<close> ..
 
 instance ..
 
 end
 
 lemma word_size [code]:
   \<open>size w = LENGTH('a)\<close> for w :: \<open>'a::len word\<close>
   by (fact size_word.rep_eq)
 
 lemma word_size_gt_0 [iff]: "0 < size w"
   for w :: "'a::len word"
   by (simp add: word_size)
 
 lemmas lens_gt_0 = word_size_gt_0 len_gt_0
 
 lemma lens_not_0 [iff]:
   \<open>size w \<noteq> 0\<close> for  w :: \<open>'a::len word\<close>
   by auto
 
 lift_definition source_size :: \<open>('a::len word \<Rightarrow> 'b) \<Rightarrow> nat\<close>
   is \<open>\<lambda>_. LENGTH('a)\<close> .
 
 lift_definition target_size :: \<open>('a \<Rightarrow> 'b::len word) \<Rightarrow> nat\<close>
   is \<open>\<lambda>_. LENGTH('b)\<close> ..
 
 lift_definition is_up :: \<open>('a::len word \<Rightarrow> 'b::len word) \<Rightarrow> bool\<close>
   is \<open>\<lambda>_. LENGTH('a) \<le> LENGTH('b)\<close> ..
 
 lift_definition is_down :: \<open>('a::len word \<Rightarrow> 'b::len word) \<Rightarrow> bool\<close>
   is \<open>\<lambda>_. LENGTH('a) \<ge> LENGTH('b)\<close> ..
 
 lemma is_up_eq:
   \<open>is_up f \<longleftrightarrow> source_size f \<le> target_size f\<close>
   for f :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
   by (simp add: source_size.rep_eq target_size.rep_eq is_up.rep_eq)
 
 lemma is_down_eq:
   \<open>is_down f \<longleftrightarrow> target_size f \<le> source_size f\<close>
   for f :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
   by (simp add: source_size.rep_eq target_size.rep_eq is_down.rep_eq)
 
 lift_definition word_int_case :: \<open>(int \<Rightarrow> 'b) \<Rightarrow> 'a::len word \<Rightarrow> 'b\<close>
   is \<open>\<lambda>f. f \<circ> take_bit LENGTH('a)\<close> by simp
 
 lemma word_int_case_eq_uint [code]:
   \<open>word_int_case f w = f (uint w)\<close>
   by transfer simp
 
 translations
   "case x of XCONST of_int y \<Rightarrow> b" \<rightleftharpoons> "CONST word_int_case (\<lambda>y. b) x"
   "case x of (XCONST of_int :: 'a) y \<Rightarrow> b" \<rightharpoonup> "CONST word_int_case (\<lambda>y. b) x"
 
 
 subsection \<open>Arithmetic operations\<close>
 
 text \<open>Legacy theorems:\<close>
 
 lemma word_add_def [code]:
   "a + b = word_of_int (uint a + uint b)"
   by transfer (simp add: take_bit_add)
 
 lemma word_sub_wi [code]:
   "a - b = word_of_int (uint a - uint b)"
   by transfer (simp add: take_bit_diff)
 
 lemma word_mult_def [code]:
   "a * b = word_of_int (uint a * uint b)"
   by transfer (simp add: take_bit_eq_mod mod_simps)
 
 lemma word_minus_def [code]:
   "- a = word_of_int (- uint a)"
   by transfer (simp add: take_bit_minus)
 
 lemma word_0_wi:
   "0 = word_of_int 0"
   by transfer simp
 
 lemma word_1_wi:
   "1 = word_of_int 1"
   by transfer simp
 
 lift_definition word_succ :: "'a::len word \<Rightarrow> 'a word" is "\<lambda>x. x + 1"
   by (auto simp add: take_bit_eq_mod intro: mod_add_cong)
 
 lift_definition word_pred :: "'a::len word \<Rightarrow> 'a word" is "\<lambda>x. x - 1"
   by (auto simp add: take_bit_eq_mod intro: mod_diff_cong)
 
 lemma word_succ_alt [code]:
   "word_succ a = word_of_int (uint a + 1)"
   by transfer (simp add: take_bit_eq_mod mod_simps)
 
 lemma word_pred_alt [code]:
   "word_pred a = word_of_int (uint a - 1)"
   by transfer (simp add: take_bit_eq_mod mod_simps)
 
 lemmas word_arith_wis = 
   word_add_def word_sub_wi word_mult_def
   word_minus_def word_succ_alt word_pred_alt
   word_0_wi word_1_wi
 
 lemma wi_homs:
   shows wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)"
     and wi_hom_sub: "word_of_int a - word_of_int b = word_of_int (a - b)"
     and wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)"
     and wi_hom_neg: "- word_of_int a = word_of_int (- a)"
     and wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)"
     and wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a - 1)"
   by (transfer, simp)+
 
 lemmas wi_hom_syms = wi_homs [symmetric]
 
 lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi
 
 lemmas word_of_int_hom_syms = word_of_int_homs [symmetric]
 
 lemma double_eq_zero_iff:
   \<open>2 * a = 0 \<longleftrightarrow> a = 0 \<or> a = 2 ^ (LENGTH('a) - Suc 0)\<close>
   for a :: \<open>'a::len word\<close>
 proof -
   define n where \<open>n = LENGTH('a) - Suc 0\<close>
   then have *: \<open>LENGTH('a) = Suc n\<close>
     by simp
   have \<open>a = 0\<close> if \<open>2 * a = 0\<close> and \<open>a \<noteq> 2 ^ (LENGTH('a) - Suc 0)\<close>
     using that by transfer
       (auto simp add: take_bit_eq_0_iff take_bit_eq_mod *)
   moreover have \<open>2 ^ LENGTH('a) = (0 :: 'a word)\<close>
     by transfer simp
   then have \<open>2 * 2 ^ (LENGTH('a) - Suc 0) = (0 :: 'a word)\<close>
     by (simp add: *)
   ultimately show ?thesis
     by auto
 qed
 
 
 subsection \<open>Ordering\<close>
 
 lift_definition word_sle :: \<open>'a::len word \<Rightarrow> 'a word \<Rightarrow> bool\<close>
   is \<open>\<lambda>k l. signed_take_bit (LENGTH('a) - Suc 0) k \<le> signed_take_bit (LENGTH('a) - Suc 0) l\<close>
   by (simp flip: signed_take_bit_decr_length_iff)
 
 lift_definition word_sless :: \<open>'a::len word \<Rightarrow> 'a word \<Rightarrow> bool\<close>
   is \<open>\<lambda>k l. signed_take_bit (LENGTH('a) - Suc 0) k < signed_take_bit (LENGTH('a) - Suc 0) l\<close>
   by (simp flip: signed_take_bit_decr_length_iff)
 
 notation
   word_sle    ("'(\<le>s')") and
   word_sle    ("(_/ \<le>s _)"  [51, 51] 50) and
   word_sless  ("'(<s')") and
   word_sless  ("(_/ <s _)"  [51, 51] 50)
 
 notation (input)
   word_sle    ("(_/ <=s _)"  [51, 51] 50)
 
 lemma word_sle_eq [code]:
   \<open>a <=s b \<longleftrightarrow> sint a \<le> sint b\<close>
   by transfer simp
 
 lemma [code]:
   \<open>a <s b \<longleftrightarrow> sint a < sint b\<close>
   by transfer simp
 
 lemma signed_ordering: \<open>ordering word_sle word_sless\<close>
   apply (standard; transfer)
   apply simp_all
   using signed_take_bit_decr_length_iff apply force
   using signed_take_bit_decr_length_iff apply force
   done
 
 lemma signed_linorder: \<open>class.linorder word_sle word_sless\<close>
   by (standard; transfer) (auto simp add: signed_take_bit_decr_length_iff)
 
 interpretation signed: linorder word_sle word_sless
   by (fact signed_linorder)
 
 lemma word_sless_eq:
   \<open>x <s y \<longleftrightarrow> x <=s y \<and> x \<noteq> y\<close>
   by (fact signed.less_le)
 
 lemma word_less_alt: "a < b \<longleftrightarrow> uint a < uint b"
   by (fact word_less_def)
 
 lemma word_zero_le [simp]: "0 \<le> y"
   for y :: "'a::len word"
   by (fact word_coorder.extremum)
 
 lemma word_m1_ge [simp] : "word_pred 0 \<ge> y" (* FIXME: delete *)
   by transfer (simp add: take_bit_minus_one_eq_mask mask_eq_exp_minus_1 bintr_lt2p)
 
 lemma word_n1_ge [simp]: "y \<le> -1"
   for y :: "'a::len word"
   by (fact word_order.extremum)
 
 lemmas word_not_simps [simp] =
   word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD]
 
 lemma word_gt_0: "0 < y \<longleftrightarrow> 0 \<noteq> y"
   for y :: "'a::len word"
   by (simp add: less_le)
 
 lemmas word_gt_0_no [simp] = word_gt_0 [of "numeral y"] for y
 
 lemma word_sless_alt: "a <s b \<longleftrightarrow> sint a < sint b"
   by transfer simp
 
 lemma word_le_nat_alt: "a \<le> b \<longleftrightarrow> unat a \<le> unat b"
   by transfer (simp add: nat_le_eq_zle)
 
 lemma word_less_nat_alt: "a < b \<longleftrightarrow> unat a < unat b"
   by transfer (auto simp add: less_le [of 0])
 
 lemmas unat_mono = word_less_nat_alt [THEN iffD1]
 
 instance word :: (len) wellorder
 proof
   fix P :: "'a word \<Rightarrow> bool" and a
   assume *: "(\<And>b. (\<And>a. a < b \<Longrightarrow> P a) \<Longrightarrow> P b)"
   have "wf (measure unat)" ..
   moreover have "{(a, b :: ('a::len) word). a < b} \<subseteq> measure unat"
     by (auto simp add: word_less_nat_alt)
   ultimately have "wf {(a, b :: ('a::len) word). a < b}"
     by (rule wf_subset)
   then show "P a" using *
     by induction blast
 qed
 
 lemma wi_less:
   "(word_of_int n < (word_of_int m :: 'a::len word)) =
     (n mod 2 ^ LENGTH('a) < m mod 2 ^ LENGTH('a))"
   by transfer (simp add: take_bit_eq_mod)
 
 lemma wi_le:
   "(word_of_int n \<le> (word_of_int m :: 'a::len word)) =
     (n mod 2 ^ LENGTH('a) \<le> m mod 2 ^ LENGTH('a))"
   by transfer (simp add: take_bit_eq_mod)
 
 
 subsection \<open>Bit-wise operations\<close>
 
 lemma uint_take_bit_eq:
   \<open>uint (take_bit n w) = take_bit n (uint w)\<close>
   by transfer (simp add: ac_simps)
 
 lemma take_bit_word_eq_self:
   \<open>take_bit n w = w\<close> if \<open>LENGTH('a) \<le> n\<close> for w :: \<open>'a::len word\<close>
   using that by transfer simp
 
 lemma take_bit_length_eq [simp]:
   \<open>take_bit LENGTH('a) w = w\<close> for w :: \<open>'a::len word\<close>
   by (rule take_bit_word_eq_self) simp
 
 lemma bit_word_of_int_iff:
   \<open>bit (word_of_int k :: 'a::len word) n \<longleftrightarrow> n < LENGTH('a) \<and> bit k n\<close>
   by transfer rule
 
 lemma bit_uint_iff:
   \<open>bit (uint w) n \<longleftrightarrow> n < LENGTH('a) \<and> bit w n\<close>
     for w :: \<open>'a::len word\<close>
   by transfer (simp add: bit_take_bit_iff)
 
 lemma bit_sint_iff:
   \<open>bit (sint w) n \<longleftrightarrow> n \<ge> LENGTH('a) \<and> bit w (LENGTH('a) - 1) \<or> bit w n\<close>
   for w :: \<open>'a::len word\<close>
   by transfer (auto simp add: bit_signed_take_bit_iff min_def le_less not_less)
 
 lemma bit_word_ucast_iff:
   \<open>bit (ucast w :: 'b::len word) n \<longleftrightarrow> n < LENGTH('a) \<and> n < LENGTH('b) \<and> bit w n\<close>
   for w :: \<open>'a::len word\<close>
   by transfer (simp add: bit_take_bit_iff ac_simps)
 
 lemma bit_word_scast_iff:
   \<open>bit (scast w :: 'b::len word) n \<longleftrightarrow>
     n < LENGTH('b) \<and> (bit w n \<or> LENGTH('a) \<le> n \<and> bit w (LENGTH('a) - Suc 0))\<close>
   for w :: \<open>'a::len word\<close>
   by transfer (auto simp add: bit_signed_take_bit_iff le_less min_def)
 
 lift_definition shiftl1 :: \<open>'a::len word \<Rightarrow> 'a word\<close>
   is \<open>(*) 2\<close>
   by (auto simp add: take_bit_eq_mod intro: mod_mult_cong)
 
 lemma shiftl1_eq:
   \<open>shiftl1 w = word_of_int (2 * uint w)\<close>
   by transfer (simp add: take_bit_eq_mod mod_simps)
 
 lemma shiftl1_eq_mult_2:
   \<open>shiftl1 = (*) 2\<close>
   by (rule ext, transfer) simp
 
 lemma bit_shiftl1_iff:
   \<open>bit (shiftl1 w) n \<longleftrightarrow> 0 < n \<and> n < LENGTH('a) \<and> bit w (n - 1)\<close>
     for w :: \<open>'a::len word\<close>
   by (simp add: shiftl1_eq_mult_2 bit_double_iff exp_eq_zero_iff not_le) (simp add: ac_simps)
 
 lift_definition shiftr1 :: \<open>'a::len word \<Rightarrow> 'a word\<close>
   \<comment> \<open>shift right as unsigned or as signed, ie logical or arithmetic\<close>
   is \<open>\<lambda>k. take_bit LENGTH('a) k div 2\<close> by simp
 
 lemma shiftr1_eq_div_2:
   \<open>shiftr1 w = w div 2\<close>
   by transfer simp
 
 lemma bit_shiftr1_iff:
   \<open>bit (shiftr1 w) n \<longleftrightarrow> bit w (Suc n)\<close>
   by transfer (auto simp flip: bit_Suc simp add: bit_take_bit_iff)
 
 lemma shiftr1_eq:
   \<open>shiftr1 w = word_of_int (uint w div 2)\<close>
   by transfer simp
 
 instantiation word :: (len) semiring_bit_syntax
 begin
 
 lift_definition test_bit_word :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> bool\<close>
   is \<open>\<lambda>k n. n < LENGTH('a) \<and> bit k n\<close>
 proof
   fix k l :: int and n :: nat
   assume *: \<open>take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
   show \<open>n < LENGTH('a) \<and> bit k n \<longleftrightarrow> n < LENGTH('a) \<and> bit l n\<close>
   proof (cases \<open>n < LENGTH('a)\<close>)
     case True
     from * have \<open>bit (take_bit LENGTH('a) k) n \<longleftrightarrow> bit (take_bit LENGTH('a) l) n\<close>
       by simp
     then show ?thesis
       by (simp add: bit_take_bit_iff)
   next
     case False
     then show ?thesis
       by simp
   qed
 qed
 
 lemma test_bit_word_eq:
   \<open>test_bit = (bit :: 'a word \<Rightarrow> _)\<close>
   by transfer simp
 
 lemma bit_word_iff_drop_bit_and [code]:
   \<open>bit a n \<longleftrightarrow> drop_bit n a AND 1 = 1\<close> for a :: \<open>'a::len word\<close>
   by (simp add: bit_iff_odd_drop_bit odd_iff_mod_2_eq_one and_one_eq)
 
 lemma [code]:
   \<open>test_bit a n \<longleftrightarrow> drop_bit n a AND 1 = 1\<close> for a :: \<open>'a::len word\<close>
   by (simp add: test_bit_word_eq bit_word_iff_drop_bit_and)
 
 lift_definition shiftl_word :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>k n. push_bit n k\<close>
 proof -
   show \<open>take_bit LENGTH('a) (push_bit n k) = take_bit LENGTH('a) (push_bit n l)\<close>
     if \<open>take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close> for k l :: int and n :: nat
   proof -
     from that
     have \<open>take_bit (LENGTH('a) - n) (take_bit LENGTH('a) k)
       = take_bit (LENGTH('a) - n) (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
 
 lemma shiftl_word_eq:
   \<open>w << n = push_bit n w\<close> for w :: \<open>'a::len word\<close>
   by transfer rule
 
 lift_definition shiftr_word :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>k n. drop_bit n (take_bit LENGTH('a) k)\<close> by simp
   
 lemma shiftr_word_eq:
   \<open>w >> n = drop_bit n w\<close> for w :: \<open>'a::len word\<close>
   by transfer simp
 
 instance
   by (standard; transfer) simp_all
 
 end
 
 lemma shiftl_code [code]:
   \<open>w << n = w * 2 ^ n\<close>
   for w :: \<open>'a::len word\<close>
   by transfer (simp add: push_bit_eq_mult)
 
 lemma shiftl1_code [code]:
   \<open>shiftl1 w = w << 1\<close>
   by transfer (simp add: push_bit_eq_mult ac_simps)
 
 lemma uint_shiftr_eq:
   \<open>uint (w >> n) = uint w div 2 ^ n\<close>
   by transfer (simp flip: drop_bit_eq_div add: drop_bit_take_bit min_def le_less less_diff_conv)
 
 lemma [code]:
   \<open>Word.the_int (w >> n) = drop_bit n (Word.the_int w)\<close>
   by transfer (simp add: drop_bit_take_bit min_def le_less less_diff_conv)
 
 lemma shiftr1_code [code]:
   \<open>shiftr1 w = w >> 1\<close>
   by transfer (simp add: drop_bit_Suc)
 
 lemma word_test_bit_def: 
   \<open>test_bit a = bit (uint a)\<close>
   by transfer (simp add: fun_eq_iff bit_take_bit_iff)
 
 lemma shiftl_def:
   \<open>w << n = (shiftl1 ^^ n) w\<close>
 proof -
   have \<open>push_bit n = (((*) 2 ^^ n) :: int \<Rightarrow> int)\<close> for n
     by (induction n) (simp_all add: fun_eq_iff funpow_swap1, simp add: ac_simps)
   then show ?thesis
     by transfer simp
 qed
 
 lemma shiftr_def:
   \<open>w >> n = (shiftr1 ^^ n) w\<close>
 proof -
   have \<open>drop_bit n = (((\<lambda>k::int. k div 2) ^^ n))\<close> for n
     by (rule sym, induction n)
        (simp_all add: fun_eq_iff drop_bit_Suc flip: drop_bit_half)
   then show ?thesis
     apply transfer
     apply simp
     apply (metis bintrunc_bintrunc rco_bintr)
     done
 qed
 
 lemma bit_shiftl_word_iff:
   \<open>bit (w << m) n \<longleftrightarrow> m \<le> n \<and> n < LENGTH('a) \<and> bit w (n - m)\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: shiftl_word_eq bit_push_bit_iff exp_eq_zero_iff not_le)
 
 lemma [code]:
   \<open>push_bit n w = w << n\<close> for w :: \<open>'a::len word\<close>
   by (simp add: shiftl_word_eq)
 
 lemma [code]:
   \<open>drop_bit n w = w >> n\<close> for w :: \<open>'a::len word\<close>
   by (simp add: shiftr_word_eq)
 
 lemma bit_shiftr_word_iff:
   \<open>bit (w >> m) n \<longleftrightarrow> bit w (m + n)\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: shiftr_word_eq bit_drop_bit_eq)
 
 lemma
   word_not_def: "NOT (a::'a::len word) = word_of_int (NOT (uint a))"
     and word_and_def: "(a::'a word) AND b = word_of_int (uint a AND uint b)"
     and word_or_def: "(a::'a word) OR b = word_of_int (uint a OR uint b)"
     and word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"
   by (transfer, simp add: take_bit_not_take_bit)+
 
 lift_definition setBit :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>k n. set_bit n k\<close>
   by (simp add: take_bit_set_bit_eq)
 
 lemma set_Bit_eq:
   \<open>setBit w n = set_bit n w\<close>
   by transfer simp
 
 lemma bit_setBit_iff:
   \<open>bit (setBit w m) n \<longleftrightarrow> (m = n \<and> n < LENGTH('a) \<or> bit w n)\<close>
   for w :: \<open>'a::len word\<close>
   by transfer (auto simp add: bit_set_bit_iff)
 
 lift_definition clearBit :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>k n. unset_bit n k\<close>
   by (simp add: take_bit_unset_bit_eq)
 
 lemma clear_Bit_eq:
   \<open>clearBit w n = unset_bit n w\<close>
   by transfer simp
 
 lemma bit_clearBit_iff:
   \<open>bit (clearBit w m) n \<longleftrightarrow> m \<noteq> n \<and> bit w n\<close>
   for w :: \<open>'a::len word\<close>
   by transfer (auto simp add: bit_unset_bit_iff)
 
 definition even_word :: \<open>'a::len word \<Rightarrow> bool\<close>
   where [code_abbrev]: \<open>even_word = even\<close>
 
 lemma even_word_iff [code]:
   \<open>even_word a \<longleftrightarrow> a AND 1 = 0\<close>
   by (simp add: and_one_eq even_iff_mod_2_eq_zero even_word_def)
 
 lemma map_bit_range_eq_if_take_bit_eq:
   \<open>map (bit k) [0..<n] = map (bit l) [0..<n]\<close>
   if \<open>take_bit n k = take_bit n l\<close> for k l :: int
 using that proof (induction n arbitrary: k l)
   case 0
   then show ?case
     by simp
 next
   case (Suc n)
   from Suc.prems have \<open>take_bit n (k div 2) = take_bit n (l div 2)\<close>
     by (simp add: take_bit_Suc)
   then have \<open>map (bit (k div 2)) [0..<n] = map (bit (l div 2)) [0..<n]\<close>
     by (rule Suc.IH)
   moreover have \<open>bit (r div 2) = bit r \<circ> Suc\<close> for r :: int
     by (simp add: fun_eq_iff bit_Suc)
   moreover from Suc.prems have \<open>even k \<longleftrightarrow> even l\<close>
     by (auto simp add: take_bit_Suc elim!: evenE oddE) arith+
   ultimately show ?case
     by (simp only: map_Suc_upt upt_conv_Cons flip: list.map_comp) simp
 qed
 
 lemma
   take_bit_word_Bit0_eq [simp]: \<open>take_bit (numeral n) (numeral (num.Bit0 m) :: 'a::len word)
     = 2 * take_bit (pred_numeral n) (numeral m)\<close> (is ?P)
   and take_bit_word_Bit1_eq [simp]: \<open>take_bit (numeral n) (numeral (num.Bit1 m) :: 'a::len word)
     = 1 + 2 * take_bit (pred_numeral n) (numeral m)\<close> (is ?Q)
   and take_bit_word_minus_Bit0_eq [simp]: \<open>take_bit (numeral n) (- numeral (num.Bit0 m) :: 'a::len word)
     = 2 * take_bit (pred_numeral n) (- numeral m)\<close> (is ?R)
   and take_bit_word_minus_Bit1_eq [simp]: \<open>take_bit (numeral n) (- numeral (num.Bit1 m) :: 'a::len word)
     = 1 + 2 * take_bit (pred_numeral n) (- numeral (Num.inc m))\<close> (is ?S)
 proof -
   define w :: \<open>'a::len word\<close>
     where \<open>w = numeral m\<close>
   moreover define q :: nat
     where \<open>q = pred_numeral n\<close>
   ultimately have num:
     \<open>numeral m = w\<close>
     \<open>numeral (num.Bit0 m) = 2 * w\<close>
     \<open>numeral (num.Bit1 m) = 1 + 2 * w\<close>
     \<open>numeral (Num.inc m) = 1 + w\<close>
     \<open>pred_numeral n = q\<close>
     \<open>numeral n = Suc q\<close>
     by (simp_all only: w_def q_def numeral_Bit0 [of m] numeral_Bit1 [of m] ac_simps
       numeral_inc numeral_eq_Suc flip: mult_2)
   have even: \<open>take_bit (Suc q) (2 * w) = 2 * take_bit q w\<close> for w :: \<open>'a::len word\<close>
     by (rule bit_word_eqI)
       (auto simp add: bit_take_bit_iff bit_double_iff)
   have odd: \<open>take_bit (Suc q) (1 + 2 * w) = 1 + 2 * take_bit q w\<close> for w :: \<open>'a::len word\<close>
     by (rule bit_eqI)
       (auto simp add: bit_take_bit_iff bit_double_iff even_bit_succ_iff)
   show ?P
     using even [of w] by (simp add: num)
   show ?Q
     using odd [of w] by (simp add: num)
   show ?R
     using even [of \<open>- w\<close>] by (simp add: num)
   show ?S
     using odd [of \<open>- (1 + w)\<close>] by (simp add: num)
 qed
 
 
 subsection \<open>More shift operations\<close>
 
 lift_definition signed_drop_bit :: \<open>nat \<Rightarrow> 'a word \<Rightarrow> 'a::len word\<close>
   is \<open>\<lambda>n. drop_bit n \<circ> signed_take_bit (LENGTH('a) - Suc 0)\<close>
   using signed_take_bit_decr_length_iff
   by (simp add: take_bit_drop_bit) force
 
 lemma bit_signed_drop_bit_iff:
   \<open>bit (signed_drop_bit m w) n \<longleftrightarrow> bit w (if LENGTH('a) - m \<le> n \<and> n < LENGTH('a) then LENGTH('a) - 1 else m + n)\<close>
   for w :: \<open>'a::len word\<close>
   apply transfer
   apply (auto simp add: bit_drop_bit_eq bit_signed_take_bit_iff not_le min_def)
    apply (metis add.commute le_antisym less_diff_conv less_eq_decr_length_iff)
   apply (metis le_antisym less_eq_decr_length_iff)
   done
 
 lemma signed_drop_bit_0 [simp]:
   \<open>signed_drop_bit 0 w = w\<close>
   by transfer simp
 
 lemma sint_signed_drop_bit_eq:
   \<open>sint (signed_drop_bit n w) = drop_bit n (sint w)\<close>
   apply (cases \<open>LENGTH('a)\<close>; cases n)
      apply simp_all
   apply (rule bit_eqI)
   apply (auto simp add: bit_sint_iff bit_drop_bit_eq bit_signed_drop_bit_iff dest: bit_imp_le_length)
   done
 
 lift_definition sshiftr1 :: \<open>'a::len word \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>k. take_bit LENGTH('a) (signed_take_bit (LENGTH('a) - 1) k div 2)\<close>
   by (simp flip: signed_take_bit_decr_length_iff)
  
 lift_definition sshiftr :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> 'a word\<close>  (infixl \<open>>>>\<close> 55)
   is \<open>\<lambda>k n. take_bit LENGTH('a) (drop_bit n (signed_take_bit (LENGTH('a) - 1) k))\<close>
   by (simp flip: signed_take_bit_decr_length_iff)
 
 lift_definition bshiftr1 :: \<open>bool \<Rightarrow> 'a::len word \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>b k. take_bit LENGTH('a) k div 2 + of_bool b * 2 ^ (LENGTH('a) - Suc 0)\<close>
   by (fact arg_cong)
 
 lemma sshiftr1_eq:
   \<open>sshiftr1 w = word_of_int (sint w div 2)\<close>
   by transfer simp
 
 lemma sshiftr_eq_funpow_sshiftr1:
   \<open>w >>> n = (sshiftr1 ^^ n) w\<close>
 proof -
   have *: \<open>(\<lambda>k::int. take_bit LENGTH('a) (signed_take_bit (LENGTH('a) - Suc 0) k div 2)) ^^ Suc n =
     take_bit LENGTH('a) \<circ> drop_bit (Suc n) \<circ> signed_take_bit (LENGTH('a) - Suc 0)\<close>
     for n
     apply (induction n)
      apply (auto simp add: fun_eq_iff drop_bit_Suc)
     apply (metis (no_types, lifting) Suc_pred funpow_swap1 len_gt_0 sbintrunc_bintrunc sbintrunc_rest)
     done
   show ?thesis
     apply transfer
     apply simp
     subgoal for k n
       apply (cases n)
        apply (simp_all only: *)
        apply simp_all
       done
     done
 qed
 
 lemma mask_eq:
   \<open>mask n = (1 << n) - (1 :: 'a::len word)\<close>
   by transfer (simp add: mask_eq_exp_minus_1 push_bit_of_1) 
 
 lemma uint_sshiftr_eq:
   \<open>uint (w >>> n) = take_bit LENGTH('a) (sint w div 2 ^  n)\<close>
   for w :: \<open>'a::len word\<close>
   by transfer (simp flip: drop_bit_eq_div)
 
 lemma [code]:
   \<open>Word.the_int (w >>> n) = take_bit LENGTH('a) (drop_bit n (Word.the_signed_int w))\<close>
   for w :: \<open>'a::len word\<close>
   by transfer simp
 
 lemma sshift1_code [code]:
   \<open>sshiftr1 w = w >>> 1\<close>
   by transfer (simp add: drop_bit_Suc)
 
 
 subsection \<open>Rotation\<close>
 
 lift_definition word_rotr :: \<open>nat \<Rightarrow> 'a::len word \<Rightarrow> 'a::len word\<close>
   is \<open>\<lambda>n k. concat_bit (LENGTH('a) - n mod LENGTH('a))
     (drop_bit (n mod LENGTH('a)) (take_bit LENGTH('a) k))
     (take_bit (n mod LENGTH('a)) k)\<close>
   subgoal for n k l
     apply (simp add: concat_bit_def nat_le_iff less_imp_le
       take_bit_tightened [of \<open>LENGTH('a)\<close> k l \<open>n mod LENGTH('a::len)\<close>])
     done
   done
 
 lift_definition word_rotl :: \<open>nat \<Rightarrow> 'a::len word \<Rightarrow> 'a::len word\<close>
   is \<open>\<lambda>n k. concat_bit (n mod LENGTH('a))
     (drop_bit (LENGTH('a) - n mod LENGTH('a)) (take_bit LENGTH('a) k))
     (take_bit (LENGTH('a) - n mod LENGTH('a)) k)\<close>
   subgoal for n k l
     apply (simp add: concat_bit_def nat_le_iff less_imp_le
       take_bit_tightened [of \<open>LENGTH('a)\<close> k l \<open>LENGTH('a) - n mod LENGTH('a::len)\<close>])
     done
   done
 
 lift_definition word_roti :: \<open>int \<Rightarrow> 'a::len word \<Rightarrow> 'a::len word\<close>
   is \<open>\<lambda>r k. concat_bit (LENGTH('a) - nat (r mod int LENGTH('a)))
     (drop_bit (nat (r mod int LENGTH('a))) (take_bit LENGTH('a) k))
     (take_bit (nat (r mod int LENGTH('a))) k)\<close>
   subgoal for r k l
     apply (simp add: concat_bit_def nat_le_iff less_imp_le
       take_bit_tightened [of \<open>LENGTH('a)\<close> k l \<open>nat (r mod int LENGTH('a::len))\<close>])
     done
   done
 
 lemma word_rotl_eq_word_rotr [code]:
   \<open>word_rotl n = (word_rotr (LENGTH('a) - n mod LENGTH('a)) :: 'a::len word \<Rightarrow> 'a word)\<close>
   by (rule ext, cases \<open>n mod LENGTH('a) = 0\<close>; transfer) simp_all
 
 lemma word_roti_eq_word_rotr_word_rotl [code]:
   \<open>word_roti i w =
     (if i \<ge> 0 then word_rotr (nat i) w else word_rotl (nat (- i)) w)\<close>
 proof (cases \<open>i \<ge> 0\<close>)
   case True
   moreover define n where \<open>n = nat i\<close>
   ultimately have \<open>i = int n\<close>
     by simp
   moreover have \<open>word_roti (int n) = (word_rotr n :: _ \<Rightarrow> 'a word)\<close>
     by (rule ext, transfer) (simp add: nat_mod_distrib)
   ultimately show ?thesis
     by simp
 next
   case False
   moreover define n where \<open>n = nat (- i)\<close>
   ultimately have \<open>i = - int n\<close> \<open>n > 0\<close>
     by simp_all
   moreover have \<open>word_roti (- int n) = (word_rotl n :: _ \<Rightarrow> 'a word)\<close>
     by (rule ext, transfer)
       (simp add: zmod_zminus1_eq_if flip: of_nat_mod of_nat_diff)
   ultimately show ?thesis
     by simp
 qed
 
 lemma bit_word_rotr_iff:
   \<open>bit (word_rotr m w) n \<longleftrightarrow>
     n < LENGTH('a) \<and> bit w ((n + m) mod LENGTH('a))\<close>
   for w :: \<open>'a::len word\<close>
 proof transfer
   fix k :: int and m n :: nat
   define q where \<open>q = m mod LENGTH('a)\<close>
   have \<open>q < LENGTH('a)\<close> 
     by (simp add: q_def)
   then have \<open>q \<le> LENGTH('a)\<close>
     by simp
   have \<open>m mod LENGTH('a) = q\<close>
     by (simp add: q_def)
   moreover have \<open>(n + m) mod LENGTH('a) = (n + q) mod LENGTH('a)\<close>
     by (subst mod_add_right_eq [symmetric]) (simp add: \<open>m mod LENGTH('a) = q\<close>)
   moreover have \<open>n < LENGTH('a) \<and>
     bit (concat_bit (LENGTH('a) - q) (drop_bit q (take_bit LENGTH('a) k)) (take_bit q k)) n \<longleftrightarrow>
     n < LENGTH('a) \<and> bit k ((n + q) mod LENGTH('a))\<close>
     using \<open>q < LENGTH('a)\<close>
     by (cases \<open>q + n \<ge> LENGTH('a)\<close>)
      (auto simp add: bit_concat_bit_iff bit_drop_bit_eq
         bit_take_bit_iff le_mod_geq ac_simps)
   ultimately show \<open>n < LENGTH('a) \<and>
     bit (concat_bit (LENGTH('a) - m mod LENGTH('a))
       (drop_bit (m mod LENGTH('a)) (take_bit LENGTH('a) k))
       (take_bit (m mod LENGTH('a)) k)) n
     \<longleftrightarrow> n < LENGTH('a) \<and>
       (n + m) mod LENGTH('a) < LENGTH('a) \<and>
       bit k ((n + m) mod LENGTH('a))\<close>
     by simp
 qed
 
 lemma bit_word_rotl_iff:
   \<open>bit (word_rotl m w) n \<longleftrightarrow>
     n < LENGTH('a) \<and> bit w ((n + (LENGTH('a) - m mod LENGTH('a))) mod LENGTH('a))\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: word_rotl_eq_word_rotr bit_word_rotr_iff)
 
 lemma bit_word_roti_iff:
   \<open>bit (word_roti k w) n \<longleftrightarrow>
     n < LENGTH('a) \<and> bit w (nat ((int n + k) mod int LENGTH('a)))\<close>
   for w :: \<open>'a::len word\<close>
 proof transfer
   fix k l :: int and n :: nat
   define m where \<open>m = nat (k mod int LENGTH('a))\<close>
   have \<open>m < LENGTH('a)\<close> 
     by (simp add: nat_less_iff m_def)
   then have \<open>m \<le> LENGTH('a)\<close>
     by simp
   have \<open>k mod int LENGTH('a) = int m\<close>
     by (simp add: nat_less_iff m_def)
   moreover have \<open>(int n + k) mod int LENGTH('a) = int ((n + m) mod LENGTH('a))\<close>
     by (subst mod_add_right_eq [symmetric]) (simp add: of_nat_mod \<open>k mod int LENGTH('a) = int m\<close>)
   moreover have \<open>n < LENGTH('a) \<and>
     bit (concat_bit (LENGTH('a) - m) (drop_bit m (take_bit LENGTH('a) l)) (take_bit m l)) n \<longleftrightarrow>
     n < LENGTH('a) \<and> bit l ((n + m) mod LENGTH('a))\<close>
     using \<open>m < LENGTH('a)\<close>
     by (cases \<open>m + n \<ge> LENGTH('a)\<close>)
      (auto simp add: bit_concat_bit_iff bit_drop_bit_eq
         bit_take_bit_iff nat_less_iff not_le not_less ac_simps
         le_diff_conv le_mod_geq)
   ultimately show \<open>n < LENGTH('a)
     \<and> bit (concat_bit (LENGTH('a) - nat (k mod int LENGTH('a)))
              (drop_bit (nat (k mod int LENGTH('a))) (take_bit LENGTH('a) l))
              (take_bit (nat (k mod int LENGTH('a))) l)) n \<longleftrightarrow>
        n < LENGTH('a) 
     \<and> nat ((int n + k) mod int LENGTH('a)) < LENGTH('a)
     \<and> bit l (nat ((int n + k) mod int LENGTH('a)))\<close>
     by simp
 qed
 
 lemma uint_word_rotr_eq:
   \<open>uint (word_rotr n w) = concat_bit (LENGTH('a) - n mod LENGTH('a))
     (drop_bit (n mod LENGTH('a)) (uint w))
     (uint (take_bit (n mod LENGTH('a)) w))\<close>
   for w :: \<open>'a::len word\<close>
   apply transfer
   apply (simp add: concat_bit_def take_bit_drop_bit push_bit_take_bit min_def)
   using mod_less_divisor not_less apply blast
   done
 
 lemma [code]:
   \<open>Word.the_int (word_rotr n w) = concat_bit (LENGTH('a) - n mod LENGTH('a))
     (drop_bit (n mod LENGTH('a)) (Word.the_int w))
     (Word.the_int (take_bit (n mod LENGTH('a)) w))\<close>
   for w :: \<open>'a::len word\<close>
   using uint_word_rotr_eq [of n w] by simp
 
     
 subsection \<open>Split and cat operations\<close>
 
 lift_definition word_cat :: \<open>'a::len word \<Rightarrow> 'b::len word \<Rightarrow> 'c::len word\<close>
   is \<open>\<lambda>k l. concat_bit LENGTH('b) l (take_bit LENGTH('a) k)\<close>
   by (simp add: bit_eq_iff bit_concat_bit_iff bit_take_bit_iff)
 
 lemma word_cat_eq:
   \<open>(word_cat v w :: 'c::len word) = push_bit LENGTH('b) (ucast v) + ucast w\<close>
   for v :: \<open>'a::len word\<close> and w :: \<open>'b::len word\<close>
   by transfer (simp add: concat_bit_eq ac_simps)
 
 lemma word_cat_eq' [code]:
   \<open>word_cat a b = word_of_int (concat_bit LENGTH('b) (uint b) (uint a))\<close>
   for a :: \<open>'a::len word\<close> and b :: \<open>'b::len word\<close>
   by transfer simp
 
 lemma bit_word_cat_iff:
   \<open>bit (word_cat v w :: 'c::len word) n \<longleftrightarrow> n < LENGTH('c) \<and> (if n < LENGTH('b) then bit w n else bit v (n - LENGTH('b)))\<close> 
   for v :: \<open>'a::len word\<close> and w :: \<open>'b::len word\<close>
   by transfer (simp add: bit_concat_bit_iff bit_take_bit_iff)
 
 definition word_split :: "'a::len word \<Rightarrow> 'b::len word \<times> 'c::len word"
   where "word_split a =
     (case bin_split (LENGTH('c)) (uint a) of
       (u, v) \<Rightarrow> (word_of_int u, word_of_int v))"
 
 definition word_rcat :: \<open>'a::len word list \<Rightarrow> 'b::len word\<close>
   where \<open>word_rcat = word_of_int \<circ> horner_sum uint (2 ^ LENGTH('a)) \<circ> rev\<close>
 
 lemma word_rcat_eq:
   \<open>word_rcat ws = word_of_int (bin_rcat (LENGTH('a::len)) (map uint ws))\<close>
   for ws :: \<open>'a::len word list\<close>
   apply (simp add: word_rcat_def bin_rcat_def rev_map)
   apply transfer
   apply (simp add: horner_sum_foldr foldr_map comp_def)
   done
 
 definition word_rsplit :: "'a::len word \<Rightarrow> 'b::len word list"
   where "word_rsplit w = map word_of_int (bin_rsplit (LENGTH('b)) (LENGTH('a), uint w))"
 
 abbreviation (input) max_word :: \<open>'a::len word\<close>
   \<comment> \<open>Largest representable machine integer.\<close>
   where "max_word \<equiv> - 1"
 
 
 subsection \<open>More on conversions\<close>
 
 lemma int_word_sint:
   \<open>sint (word_of_int x :: 'a::len word) = (x + 2 ^ (LENGTH('a) - 1)) mod 2 ^ LENGTH('a) - 2 ^ (LENGTH('a) - 1)\<close>
   by transfer (simp add: no_sbintr_alt2)
 
 lemma sint_sbintrunc': "sint (word_of_int bin :: 'a word) = signed_take_bit (LENGTH('a::len) - 1) bin"
   by simp
 
 lemma uint_sint: "uint w = take_bit (LENGTH('a)) (sint w)"
   for w :: "'a::len word"
   by transfer simp
 
 lemma bintr_uint: "LENGTH('a) \<le> n \<Longrightarrow> take_bit n (uint w) = uint w"
   for w :: "'a::len word"
   by transfer (simp add: min_def)
 
 lemma wi_bintr:
   "LENGTH('a::len) \<le> n \<Longrightarrow>
     word_of_int (take_bit n w) = (word_of_int w :: 'a word)"
   by transfer simp
 
 lemma word_numeral_alt: "numeral b = word_of_int (numeral b)"
   by (induct b, simp_all only: numeral.simps word_of_int_homs)
 
 declare word_numeral_alt [symmetric, code_abbrev]
 
 lemma word_neg_numeral_alt: "- numeral b = word_of_int (- numeral b)"
   by (simp only: word_numeral_alt wi_hom_neg)
 
 declare word_neg_numeral_alt [symmetric, code_abbrev]
 
 lemma uint_bintrunc [simp]:
   "uint (numeral bin :: 'a word) =
     take_bit (LENGTH('a::len)) (numeral bin)"
   by transfer rule
 
 lemma uint_bintrunc_neg [simp]:
   "uint (- numeral bin :: 'a word) = take_bit (LENGTH('a::len)) (- numeral bin)"
   by transfer rule
 
 lemma sint_sbintrunc [simp]:
   "sint (numeral bin :: 'a word) = signed_take_bit (LENGTH('a::len) - 1) (numeral bin)"
   by transfer simp
 
 lemma sint_sbintrunc_neg [simp]:
   "sint (- numeral bin :: 'a word) = signed_take_bit (LENGTH('a::len) - 1) (- numeral bin)"
   by transfer simp
 
 lemma unat_bintrunc [simp]:
   "unat (numeral bin :: 'a::len word) = nat (take_bit (LENGTH('a)) (numeral bin))"
   by transfer simp
 
 lemma unat_bintrunc_neg [simp]:
   "unat (- numeral bin :: 'a::len word) = nat (take_bit (LENGTH('a)) (- numeral bin))"
   by transfer simp
 
 lemma size_0_eq: "size w = 0 \<Longrightarrow> v = w"
   for v w :: "'a::len word"
   by transfer simp
 
 lemma uint_ge_0 [iff]: "0 \<le> uint x"
   by (fact unsigned_greater_eq)
 
 lemma uint_lt2p [iff]: "uint x < 2 ^ LENGTH('a)"
   for x :: "'a::len word"
   by (fact unsigned_less)
 
 lemma sint_ge: "- (2 ^ (LENGTH('a) - 1)) \<le> sint x"
   for x :: "'a::len word"
   using sint_greater_eq [of x] by simp
 
 lemma sint_lt: "sint x < 2 ^ (LENGTH('a) - 1)"
   for x :: "'a::len word"
   using sint_less [of x] by simp
 
 lemma sign_uint_Pls [simp]: "bin_sign (uint x) = 0"
   by (simp add: sign_Pls_ge_0)
 
 lemma uint_m2p_neg: "uint x - 2 ^ LENGTH('a) < 0"
   for x :: "'a::len word"
   by (simp only: diff_less_0_iff_less uint_lt2p)
 
 lemma uint_m2p_not_non_neg: "\<not> 0 \<le> uint x - 2 ^ LENGTH('a)"
   for x :: "'a::len word"
   by (simp only: not_le uint_m2p_neg)
 
 lemma lt2p_lem: "LENGTH('a) \<le> n \<Longrightarrow> uint w < 2 ^ n"
   for w :: "'a::len word"
   by (metis bintr_lt2p bintr_uint)
 
 lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0"
   by (fact uint_ge_0 [THEN leD, THEN antisym_conv1])
 
 lemma uint_nat: "uint w = int (unat w)"
   by transfer simp
 
 lemma uint_numeral: "uint (numeral b :: 'a::len word) = numeral b mod 2 ^ LENGTH('a)"
   by (simp flip: take_bit_eq_mod add: of_nat_take_bit)
 
 lemma uint_neg_numeral: "uint (- numeral b :: 'a::len word) = - numeral b mod 2 ^ LENGTH('a)"
   by (simp flip: take_bit_eq_mod add: of_nat_take_bit)
 
 lemma unat_numeral: "unat (numeral b :: 'a::len word) = numeral b mod 2 ^ LENGTH('a)"
   by transfer (simp add: take_bit_eq_mod nat_mod_distrib nat_power_eq)
 
 lemma sint_numeral:
   "sint (numeral b :: 'a::len word) =
     (numeral b + 2 ^ (LENGTH('a) - 1)) mod 2 ^ LENGTH('a) - 2 ^ (LENGTH('a) - 1)"
   apply (transfer fixing: b)
   using int_word_sint [of \<open>numeral b\<close>]
   apply simp
   done
 
 lemma word_of_int_0 [simp, code_post]: "word_of_int 0 = 0"
   by (fact of_int_0)
 
 lemma word_of_int_1 [simp, code_post]: "word_of_int 1 = 1"
   by (fact of_int_1)
 
 lemma word_of_int_neg_1 [simp]: "word_of_int (- 1) = - 1"
   by (simp add: wi_hom_syms)
 
 lemma word_of_int_numeral [simp] : "(word_of_int (numeral bin) :: 'a::len word) = numeral bin"
   by (fact of_int_numeral)
 
 lemma word_of_int_neg_numeral [simp]:
   "(word_of_int (- numeral bin) :: 'a::len word) = - numeral bin"
   by (fact of_int_neg_numeral)
 
 lemma word_int_case_wi:
   "word_int_case f (word_of_int i :: 'b word) = f (i mod 2 ^ LENGTH('b::len))"
   by transfer (simp add: take_bit_eq_mod)
 
 lemma word_int_split:
   "P (word_int_case f x) =
     (\<forall>i. x = (word_of_int i :: 'b::len word) \<and> 0 \<le> i \<and> i < 2 ^ LENGTH('b) \<longrightarrow> P (f i))"
   by transfer (auto simp add: take_bit_eq_mod)
 
 lemma word_int_split_asm:
   "P (word_int_case f x) =
     (\<nexists>n. x = (word_of_int n :: 'b::len word) \<and> 0 \<le> n \<and> n < 2 ^ LENGTH('b::len) \<and> \<not> P (f n))"
   by transfer (auto simp add: take_bit_eq_mod)
 
 lemma uint_range_size: "0 \<le> uint w \<and> uint w < 2 ^ size w"
   by transfer simp
 
 lemma sint_range_size: "- (2 ^ (size w - Suc 0)) \<le> sint w \<and> sint w < 2 ^ (size w - Suc 0)"
   by transfer (use sbintr_ge sbintr_lt in blast)
 
 lemma sint_above_size: "2 ^ (size w - 1) \<le> x \<Longrightarrow> sint w < x"
   for w :: "'a::len word"
   unfolding word_size by (rule less_le_trans [OF sint_lt])
 
 lemma sint_below_size: "x \<le> - (2 ^ (size w - 1)) \<Longrightarrow> x \<le> sint w"
   for w :: "'a::len word"
   unfolding word_size by (rule order_trans [OF _ sint_ge])
 
 
 subsection \<open>Testing bits\<close>
 
 lemma test_bit_eq_iff: "test_bit u = test_bit v \<longleftrightarrow> u = v"
   for u v :: "'a::len word"
   by (simp add: bit_eq_iff test_bit_eq_bit fun_eq_iff)
 
 lemma test_bit_size: "w !! n \<Longrightarrow> n < size w"
   for w :: "'a::len word"
   by transfer simp
 
 lemma word_eq_iff: "x = y \<longleftrightarrow> (\<forall>n<LENGTH('a). x !! n = y !! n)" (is \<open>?P \<longleftrightarrow> ?Q\<close>)
   for x y :: "'a::len word"
   by transfer (auto simp add: bit_eq_iff bit_take_bit_iff)
 
 lemma word_eqI: "(\<And>n. n < size u \<longrightarrow> u !! n = v !! n) \<Longrightarrow> u = v"
   for u :: "'a::len word"
   by (simp add: word_size word_eq_iff)
 
 lemma word_eqD: "u = v \<Longrightarrow> u !! x = v !! x"
   for u v :: "'a::len word"
   by simp
 
 lemma test_bit_bin': "w !! n \<longleftrightarrow> n < size w \<and> bin_nth (uint w) n"
   by transfer (simp add: bit_take_bit_iff)
 
 lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
 
 lemma bin_nth_uint_imp: "bin_nth (uint w) n \<Longrightarrow> n < LENGTH('a)"
   for w :: "'a::len word"
   by transfer (simp add: bit_take_bit_iff)
 
 lemma bin_nth_sint:
   "LENGTH('a) \<le> n \<Longrightarrow>
     bin_nth (sint w) n = bin_nth (sint w) (LENGTH('a) - 1)"
   for w :: "'a::len word"
   by (transfer fixing: n) (simp add: bit_signed_take_bit_iff le_diff_conv min_def)
 
 lemma num_of_bintr':
   "take_bit (LENGTH('a::len)) (numeral a :: int) = (numeral b) \<Longrightarrow>
     numeral a = (numeral b :: 'a word)"
 proof (transfer fixing: a b)
   assume \<open>take_bit LENGTH('a) (numeral a :: int) = numeral b\<close>
   then have \<open>take_bit LENGTH('a) (take_bit LENGTH('a) (numeral a :: int)) = take_bit LENGTH('a) (numeral b)\<close>
     by simp
   then show \<open>take_bit LENGTH('a) (numeral a :: int) = take_bit LENGTH('a) (numeral b)\<close>
     by simp
 qed
 
 lemma num_of_sbintr':
   "signed_take_bit (LENGTH('a::len) - 1) (numeral a :: int) = (numeral b) \<Longrightarrow>
     numeral a = (numeral b :: 'a word)"
 proof (transfer fixing: a b)
   assume \<open>signed_take_bit (LENGTH('a) - 1) (numeral a :: int) = numeral b\<close>
   then have \<open>take_bit LENGTH('a) (signed_take_bit (LENGTH('a) - 1) (numeral a :: int)) = take_bit LENGTH('a) (numeral b)\<close>
     by simp
   then show \<open>take_bit LENGTH('a) (numeral a :: int) = take_bit LENGTH('a) (numeral b)\<close>
     by simp
 qed
  
 lemma num_abs_bintr:
   "(numeral x :: 'a word) =
     word_of_int (take_bit (LENGTH('a::len)) (numeral x))"
   by transfer simp
 
 lemma num_abs_sbintr:
   "(numeral x :: 'a word) =
     word_of_int (signed_take_bit (LENGTH('a::len) - 1) (numeral x))"
   by transfer simp
 
 text \<open>
   \<open>cast\<close> -- note, no arg for new length, as it's determined by type of result,
   thus in \<open>cast w = w\<close>, the type means cast to length of \<open>w\<close>!
 \<close>
 
 lemma bit_ucast_iff:
   \<open>bit (ucast a :: 'a::len word) n \<longleftrightarrow> n < LENGTH('a::len) \<and> Parity.bit a n\<close>
   by transfer (simp add: bit_take_bit_iff)
 
 lemma ucast_id [simp]: "ucast w = w"
   by transfer simp
 
 lemma scast_id [simp]: "scast w = w"
   by transfer simp
 
 lemma nth_ucast: "(ucast w::'a::len word) !! n = (w !! n \<and> n < LENGTH('a))"
   by transfer (simp add: bit_take_bit_iff ac_simps)
 
 lemma ucast_mask_eq:
   \<open>ucast (mask n :: 'b word) = mask (min LENGTH('b::len) n)\<close>
   by (simp add: bit_eq_iff) (auto simp add: bit_mask_iff bit_ucast_iff exp_eq_zero_iff)
 
 \<comment> \<open>literal u(s)cast\<close>
 lemma ucast_bintr [simp]:
   "ucast (numeral w :: 'a::len word) =
     word_of_int (take_bit (LENGTH('a)) (numeral w))"
   by transfer simp
 
 (* TODO: neg_numeral *)
 
 lemma scast_sbintr [simp]:
   "scast (numeral w ::'a::len word) =
     word_of_int (signed_take_bit (LENGTH('a) - Suc 0) (numeral w))"
   by transfer simp
 
 lemma source_size: "source_size (c::'a::len word \<Rightarrow> _) = LENGTH('a)"
   by transfer simp
 
 lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len word) = LENGTH('b)"
   by transfer simp
 
 lemma is_down: "is_down c \<longleftrightarrow> LENGTH('b) \<le> LENGTH('a)"
   for c :: "'a::len word \<Rightarrow> 'b::len word"
   by transfer simp
 
 lemma is_up: "is_up c \<longleftrightarrow> LENGTH('a) \<le> LENGTH('b)"
   for c :: "'a::len word \<Rightarrow> 'b::len word"
   by transfer simp
 
 lemma is_up_down:
   \<open>is_up c \<longleftrightarrow> is_down d\<close>
   for c :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
   and d :: \<open>'b::len word \<Rightarrow> 'a::len word\<close>
   by transfer simp
 
 context
   fixes dummy_types :: \<open>'a::len \<times> 'b::len\<close>
 begin
 
 private abbreviation (input) UCAST :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
   where \<open>UCAST == ucast\<close>
 
 private abbreviation (input) SCAST :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
   where \<open>SCAST == scast\<close>
 
 lemma down_cast_same:
   \<open>UCAST = scast\<close> if \<open>is_down UCAST\<close>
   by (rule ext, use that in transfer) (simp add: take_bit_signed_take_bit)
 
 lemma sint_up_scast:
   \<open>sint (SCAST w) = sint w\<close> if \<open>is_up SCAST\<close>
   using that by transfer (simp add: min_def Suc_leI le_diff_iff)
 
 lemma uint_up_ucast:
   \<open>uint (UCAST w) = uint w\<close> if \<open>is_up UCAST\<close>
   using that by transfer (simp add: min_def)
 
 lemma ucast_up_ucast:
   \<open>ucast (UCAST w) = ucast w\<close> if \<open>is_up UCAST\<close>
   using that by transfer (simp add: ac_simps)
 
 lemma ucast_up_ucast_id:
   \<open>ucast (UCAST w) = w\<close> if \<open>is_up UCAST\<close>
   using that by (simp add: ucast_up_ucast)
 
 lemma scast_up_scast:
   \<open>scast (SCAST w) = scast w\<close> if \<open>is_up SCAST\<close>
   using that by transfer (simp add: ac_simps)
 
 lemma scast_up_scast_id:
   \<open>scast (SCAST w) = w\<close> if \<open>is_up SCAST\<close>
   using that by (simp add: scast_up_scast)
 
 lemma isduu:
   \<open>is_up UCAST\<close> if \<open>is_down d\<close>
     for d :: \<open>'b word \<Rightarrow> 'a word\<close>
   using that is_up_down [of UCAST d] by simp
 
 lemma isdus:
   \<open>is_up SCAST\<close> if \<open>is_down d\<close>
     for d :: \<open>'b word \<Rightarrow> 'a word\<close>
   using that is_up_down [of SCAST d] by simp
 
 lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id]
 lemmas scast_down_scast_id = isdus [THEN scast_up_scast_id]
 
 lemma up_ucast_surj:
   \<open>surj (ucast :: 'b word \<Rightarrow> 'a word)\<close> if \<open>is_up UCAST\<close>
   by (rule surjI) (use that in \<open>rule ucast_up_ucast_id\<close>)
 
 lemma up_scast_surj:
   \<open>surj (scast :: 'b word \<Rightarrow> 'a word)\<close> if \<open>is_up SCAST\<close>
   by (rule surjI) (use that in \<open>rule scast_up_scast_id\<close>)
 
 lemma down_ucast_inj:
   \<open>inj_on UCAST A\<close> if \<open>is_down (ucast :: 'b word \<Rightarrow> 'a word)\<close>
   by (rule inj_on_inverseI) (use that in \<open>rule ucast_down_ucast_id\<close>)
 
 lemma down_scast_inj:
   \<open>inj_on SCAST A\<close> if \<open>is_down (scast :: 'b word \<Rightarrow> 'a word)\<close>
   by (rule inj_on_inverseI) (use that in \<open>rule scast_down_scast_id\<close>)
   
 lemma ucast_down_wi:
   \<open>UCAST (word_of_int x) = word_of_int x\<close> if \<open>is_down UCAST\<close>
   using that by transfer simp
 
 lemma ucast_down_no:
   \<open>UCAST (numeral bin) = numeral bin\<close> if \<open>is_down UCAST\<close>
   using that by transfer simp
 
 end
 
 lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]
 
 lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def
 
 lemma bit_last_iff:
   \<open>bit w (LENGTH('a) - Suc 0) \<longleftrightarrow> sint w < 0\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
   for w :: \<open>'a::len word\<close>
 proof -
   have \<open>?P \<longleftrightarrow> bit (uint w) (LENGTH('a) - Suc 0)\<close>
     by (simp add: bit_uint_iff)
   also have \<open>\<dots> \<longleftrightarrow> ?Q\<close>
     by (simp add: sint_uint)
   finally show ?thesis .
 qed
 
 lemma drop_bit_eq_zero_iff_not_bit_last:
   \<open>drop_bit (LENGTH('a) - Suc 0) w = 0 \<longleftrightarrow> \<not> bit w (LENGTH('a) - Suc 0)\<close>
   for w :: "'a::len word"
     apply (cases \<open>LENGTH('a)\<close>)
     apply simp_all
     apply (simp add: bit_iff_odd_drop_bit)
     apply transfer
     apply (simp add: take_bit_drop_bit)
     apply (auto simp add: drop_bit_eq_div take_bit_eq_mod min_def)
     apply (auto elim!: evenE)
     apply (metis div_exp_eq mod_div_trivial mult.commute nonzero_mult_div_cancel_left power_Suc0_right power_add zero_neq_numeral)
     done
 
 
 subsection \<open>Word Arithmetic\<close>
 
 lemmas word_div_no [simp] = word_div_def [of "numeral a" "numeral b"] for a b
 lemmas word_mod_no [simp] = word_mod_def [of "numeral a" "numeral b"] for a b
 lemmas word_less_no [simp] = word_less_def [of "numeral a" "numeral b"] for a b
 lemmas word_le_no [simp] = word_le_def [of "numeral a" "numeral b"] for a b
 lemmas word_sless_no [simp] = word_sless_eq [of "numeral a" "numeral b"] for a b
 lemmas word_sle_no [simp] = word_sle_eq [of "numeral a" "numeral b"] for a b
 
 lemma size_0_same': "size w = 0 \<Longrightarrow> w = v"
   for v w :: "'a::len word"
   by (unfold word_size) simp
 
 lemmas size_0_same = size_0_same' [unfolded word_size]
 
 lemmas unat_eq_0 = unat_0_iff
 lemmas unat_eq_zero = unat_0_iff
 
 
 subsection \<open>Transferring goals from words to ints\<close>
 
 lemma word_ths:
   shows word_succ_p1: "word_succ a = a + 1"
     and word_pred_m1: "word_pred a = a - 1"
     and word_pred_succ: "word_pred (word_succ a) = a"
     and word_succ_pred: "word_succ (word_pred a) = a"
     and word_mult_succ: "word_succ a * b = b + a * b"
   by (transfer, simp add: algebra_simps)+
 
 lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y"
   by simp
 
 lemma uint_word_ariths:
   fixes a b :: "'a::len word"
   shows "uint (a + b) = (uint a + uint b) mod 2 ^ LENGTH('a::len)"
     and "uint (a - b) = (uint a - uint b) mod 2 ^ LENGTH('a)"
     and "uint (a * b) = uint a * uint b mod 2 ^ LENGTH('a)"
     and "uint (- a) = - uint a mod 2 ^ LENGTH('a)"
     and "uint (word_succ a) = (uint a + 1) mod 2 ^ LENGTH('a)"
     and "uint (word_pred a) = (uint a - 1) mod 2 ^ LENGTH('a)"
     and "uint (0 :: 'a word) = 0 mod 2 ^ LENGTH('a)"
     and "uint (1 :: 'a word) = 1 mod 2 ^ LENGTH('a)"
   by (simp_all only: word_arith_wis uint_word_of_int_eq flip: take_bit_eq_mod)
 
 lemma uint_word_arith_bintrs:
   fixes a b :: "'a::len word"
   shows "uint (a + b) = take_bit (LENGTH('a)) (uint a + uint b)"
     and "uint (a - b) = take_bit (LENGTH('a)) (uint a - uint b)"
     and "uint (a * b) = take_bit (LENGTH('a)) (uint a * uint b)"
     and "uint (- a) = take_bit (LENGTH('a)) (- uint a)"
     and "uint (word_succ a) = take_bit (LENGTH('a)) (uint a + 1)"
     and "uint (word_pred a) = take_bit (LENGTH('a)) (uint a - 1)"
     and "uint (0 :: 'a word) = take_bit (LENGTH('a)) 0"
     and "uint (1 :: 'a word) = take_bit (LENGTH('a)) 1"
   by (simp_all add: uint_word_ariths take_bit_eq_mod)
 
 lemma sint_word_ariths:
   fixes a b :: "'a::len word"
   shows "sint (a + b) = signed_take_bit (LENGTH('a) - 1) (sint a + sint b)"
     and "sint (a - b) = signed_take_bit (LENGTH('a) - 1) (sint a - sint b)"
     and "sint (a * b) = signed_take_bit (LENGTH('a) - 1) (sint a * sint b)"
     and "sint (- a) = signed_take_bit (LENGTH('a) - 1) (- sint a)"
     and "sint (word_succ a) = signed_take_bit (LENGTH('a) - 1) (sint a + 1)"
     and "sint (word_pred a) = signed_take_bit (LENGTH('a) - 1) (sint a - 1)"
     and "sint (0 :: 'a word) = signed_take_bit (LENGTH('a) - 1) 0"
     and "sint (1 :: 'a word) = signed_take_bit (LENGTH('a) - 1) 1"
          prefer 8
          apply (simp add: Suc_lessI sbintrunc_minus_simps(3))
         prefer 7
         apply simp
        apply transfer apply (simp add: signed_take_bit_add)
       apply transfer apply (simp add: signed_take_bit_diff)
      apply transfer apply (simp add: signed_take_bit_mult)
     apply transfer apply (simp add: signed_take_bit_minus)
   apply (metis One_nat_def id_apply of_int_eq_id sbintrunc_sbintrunc signed.rep_eq signed_word_eqI sint_sbintrunc' wi_hom_succ)
   apply (metis (no_types, lifting) One_nat_def signed_take_bit_decr_length_iff sint_uint uint_sint uint_word_of_int_eq wi_hom_pred word_of_int_uint)
   done
 
 lemma word_pred_0_n1: "word_pred 0 = word_of_int (- 1)"
   unfolding word_pred_m1 by simp
 
 lemma succ_pred_no [simp]:
     "word_succ (numeral w) = numeral w + 1"
     "word_pred (numeral w) = numeral w - 1"
     "word_succ (- numeral w) = - numeral w + 1"
     "word_pred (- numeral w) = - numeral w - 1"
   by (simp_all add: word_succ_p1 word_pred_m1)
 
 lemma word_sp_01 [simp]:
   "word_succ (- 1) = 0 \<and> word_succ 0 = 1 \<and> word_pred 0 = - 1 \<and> word_pred 1 = 0"
   by (simp_all add: word_succ_p1 word_pred_m1)
 
 \<comment> \<open>alternative approach to lifting arithmetic equalities\<close>
 lemma word_of_int_Ex: "\<exists>y. x = word_of_int y"
   by (rule_tac x="uint x" in exI) simp
 
 
 subsection \<open>Order on fixed-length words\<close>
 
 lift_definition udvd :: \<open>'a::len word \<Rightarrow> 'a::len word \<Rightarrow> bool\<close> (infixl \<open>udvd\<close> 50)
   is \<open>\<lambda>k l. take_bit LENGTH('a) k dvd take_bit LENGTH('a) l\<close> by simp
 
 lemma udvd_iff_dvd:
   \<open>x udvd y \<longleftrightarrow> unat x dvd unat y\<close>
   by transfer (simp add: nat_dvd_iff)
 
 lemma udvd_iff_dvd_int:
   \<open>v udvd w \<longleftrightarrow> uint v dvd uint w\<close>
   by transfer rule
 
 lemma udvdI [intro]:
   \<open>v udvd w\<close> if \<open>unat w = unat v * unat u\<close>
 proof -
   from that have \<open>unat v dvd unat w\<close> ..
   then show ?thesis
     by (simp add: udvd_iff_dvd)
 qed
 
 lemma udvdE [elim]:
   fixes v w :: \<open>'a::len word\<close>
   assumes \<open>v udvd w\<close>
   obtains u :: \<open>'a word\<close> where \<open>unat w = unat v * unat u\<close>
 proof (cases \<open>v = 0\<close>)
   case True
   moreover from True \<open>v udvd w\<close> have \<open>w = 0\<close>
     by transfer simp
   ultimately show thesis
     using that by simp
 next
   case False
   then have \<open>unat v > 0\<close>
     by (simp add: unat_gt_0)
   from \<open>v udvd w\<close> have \<open>unat v dvd unat w\<close>
     by (simp add: udvd_iff_dvd)
   then obtain n where \<open>unat w = unat v * n\<close> ..
   moreover have \<open>n < 2 ^ LENGTH('a)\<close>
   proof (rule ccontr)
     assume \<open>\<not> n < 2 ^ LENGTH('a)\<close>
     then have \<open>n \<ge> 2 ^ LENGTH('a)\<close>
       by (simp add: not_le)
     then have \<open>unat v * n \<ge> 2 ^ LENGTH('a)\<close>
       using \<open>unat v > 0\<close> mult_le_mono [of 1 \<open>unat v\<close> \<open>2 ^ LENGTH('a)\<close> n]
       by simp
     with \<open>unat w = unat v * n\<close>
     have \<open>unat w \<ge> 2 ^ LENGTH('a)\<close>
       by simp
     with unsigned_less [of w, where ?'a = nat] show False
       by linarith
   qed
   ultimately have \<open>unat w = unat v * unat (word_of_nat n :: 'a word)\<close>
     by (auto simp add: take_bit_nat_eq_self_iff intro: sym)
   with that show thesis .
 qed
 
 lemma udvd_imp_mod_eq_0:
   \<open>w mod v = 0\<close> if \<open>v udvd w\<close>
   using that by transfer simp
 
 lemma mod_eq_0_imp_udvd [intro?]:
   \<open>v udvd w\<close> if \<open>w mod v = 0\<close>
 proof -
   from that have \<open>unat (w mod v) = unat 0\<close>
     by simp
   then have \<open>unat w mod unat v = 0\<close>
     by (simp add: unat_mod_distrib)
   then have \<open>unat v dvd unat w\<close> ..
   then show ?thesis
     by (simp add: udvd_iff_dvd)
 qed
 
 lemma udvd_imp_dvd:
   \<open>v dvd w\<close> if \<open>v udvd w\<close> for v w :: \<open>'a::len word\<close>
 proof -
   from that obtain u :: \<open>'a word\<close> where \<open>unat w = unat v * unat u\<close> ..
   then have \<open>(word_of_nat (unat w) :: 'a word) = word_of_nat (unat v * unat u)\<close>
     by simp
   then have \<open>w = v * u\<close>
     by simp
   then show \<open>v dvd w\<close> ..
 qed
 
 lemma exp_dvd_iff_exp_udvd:
   \<open>2 ^ n dvd w \<longleftrightarrow> 2 ^ n udvd w\<close> for v w :: \<open>'a::len word\<close>
 proof
   assume \<open>2 ^ n udvd w\<close> then show \<open>2 ^ n dvd w\<close>
     by (rule udvd_imp_dvd) 
 next
   assume \<open>2 ^ n dvd w\<close>
   then obtain u :: \<open>'a word\<close> where \<open>w = 2 ^ n * u\<close> ..
   then have \<open>w = push_bit n u\<close>
     by (simp add: push_bit_eq_mult)
   then show \<open>2 ^ n udvd w\<close>
     by transfer (simp add: take_bit_push_bit dvd_eq_mod_eq_0 flip: take_bit_eq_mod)
 qed
 
 lemma udvd_nat_alt:
   \<open>a udvd b \<longleftrightarrow> (\<exists>n. unat b = n * unat a)\<close>
   by (auto simp add: udvd_iff_dvd)
 
 lemma udvd_unfold_int:
   \<open>a udvd b \<longleftrightarrow> (\<exists>n\<ge>0. uint b = n * uint a)\<close>
   apply (auto elim!: dvdE simp add: udvd_iff_dvd)
    apply (simp only: uint_nat)
    apply auto
   using of_nat_0_le_iff apply blast
   apply (simp only: unat_eq_nat_uint)
   apply (simp add: nat_mult_distrib)
   done
 
 lemma unat_minus_one:
   \<open>unat (w - 1) = unat w - 1\<close> if \<open>w \<noteq> 0\<close>
 proof -
   have "0 \<le> uint w" by (fact uint_nonnegative)
   moreover from that have "0 \<noteq> uint w"
     by (simp add: uint_0_iff)
   ultimately have "1 \<le> uint w"
     by arith
   from uint_lt2p [of w] have "uint w - 1 < 2 ^ LENGTH('a)"
     by arith
   with \<open>1 \<le> uint w\<close> have "(uint w - 1) mod 2 ^ LENGTH('a) = uint w - 1"
     by (auto intro: mod_pos_pos_trivial)
   with \<open>1 \<le> uint w\<close> have "nat ((uint w - 1) mod 2 ^ LENGTH('a)) = nat (uint w) - 1"
     by (auto simp del: nat_uint_eq)
   then show ?thesis
     by (simp only: unat_eq_nat_uint word_arith_wis mod_diff_right_eq)
       (metis of_int_1 uint_word_of_int unsigned_1)
 qed
 
 lemma measure_unat: "p \<noteq> 0 \<Longrightarrow> unat (p - 1) < unat p"
   by (simp add: unat_minus_one) (simp add: unat_0_iff [symmetric])
 
 lemmas uint_add_ge0 [simp] = add_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
 lemmas uint_mult_ge0 [simp] = mult_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
 
 lemma uint_sub_lt2p [simp]: "uint x - uint y < 2 ^ LENGTH('a)"
   for x :: "'a::len word" and y :: "'b::len word"
   using uint_ge_0 [of y] uint_lt2p [of x] by arith
 
 
 subsection \<open>Conditions for the addition (etc) of two words to overflow\<close>
 
 lemma uint_add_lem:
   "(uint x + uint y < 2 ^ LENGTH('a)) =
     (uint (x + y) = uint x + uint y)"
   for x y :: "'a::len word"
   by (metis add.right_neutral add_mono_thms_linordered_semiring(1) mod_pos_pos_trivial of_nat_0_le_iff uint_lt2p uint_nat uint_word_ariths(1))
 
 lemma uint_mult_lem:
   "(uint x * uint y < 2 ^ LENGTH('a)) =
     (uint (x * y) = uint x * uint y)"
   for x y :: "'a::len word"
   by (metis mod_pos_pos_trivial uint_lt2p uint_mult_ge0 uint_word_ariths(3))
 
 lemma uint_sub_lem: "uint x \<ge> uint y \<longleftrightarrow> uint (x - y) = uint x - uint y"
   by (metis diff_ge_0_iff_ge of_nat_0_le_iff uint_nat uint_sub_lt2p uint_word_of_int unique_euclidean_semiring_numeral_class.mod_less word_sub_wi)
 
 lemma uint_add_le: "uint (x + y) \<le> uint x + uint y"
   unfolding uint_word_ariths by (simp add: zmod_le_nonneg_dividend) 
 
 lemma uint_sub_ge: "uint (x - y) \<ge> uint x - uint y"
   unfolding uint_word_ariths by (simp add: int_mod_ge)
   
 lemma mod_add_if_z:
   "x < z \<Longrightarrow> y < z \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> z \<Longrightarrow>
     (x + y) mod z = (if x + y < z then x + y else x + y - z)"
   for x y z :: int
   apply (auto simp add: not_less)
   apply (rule antisym)
   apply (metis diff_ge_0_iff_ge minus_mod_self2 zmod_le_nonneg_dividend)
    apply (simp only: flip: minus_mod_self2 [of \<open>x + y\<close> z])
   apply (rule int_mod_ge)
    apply simp_all
   done
 
 lemma uint_plus_if':
   "uint (a + b) =
     (if uint a + uint b < 2 ^ LENGTH('a) then uint a + uint b
      else uint a + uint b - 2 ^ LENGTH('a))"
   for a b :: "'a::len word"
   using mod_add_if_z [of "uint a" _ "uint b"] by (simp add: uint_word_ariths)
 
 lemma mod_sub_if_z:
   "x < z \<Longrightarrow> y < z \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> z \<Longrightarrow>
     (x - y) mod z = (if y \<le> x then x - y else x - y + z)"
   for x y z :: int
   apply (auto simp add: not_le)
   apply (rule antisym)
    apply (simp only: flip: mod_add_self2 [of \<open>x - y\<close> z])
    apply (rule zmod_le_nonneg_dividend)
    apply simp
   apply (metis add.commute add.right_neutral add_le_cancel_left diff_ge_0_iff_ge int_mod_ge le_less le_less_trans mod_add_self1 not_less)
   done
 
 lemma uint_sub_if':
   "uint (a - b) =
     (if uint b \<le> uint a then uint a - uint b
      else uint a - uint b + 2 ^ LENGTH('a))"
   for a b :: "'a::len word"
   using mod_sub_if_z [of "uint a" _ "uint b"] by (simp add: uint_word_ariths)
 
 
 subsection \<open>Definition of \<open>uint_arith\<close>\<close>
 
 lemma word_of_int_inverse:
   "word_of_int r = a \<Longrightarrow> 0 \<le> r \<Longrightarrow> r < 2 ^ LENGTH('a) \<Longrightarrow> uint a = r"
   for a :: "'a::len word"
   apply transfer
   apply (drule sym)
   apply (simp add: take_bit_int_eq_self)
   done
 
 lemma uint_split:
   "P (uint x) = (\<forall>i. word_of_int i = x \<and> 0 \<le> i \<and> i < 2^LENGTH('a) \<longrightarrow> P i)"
   for x :: "'a::len word"
   by transfer (auto simp add: take_bit_eq_mod)
 
 lemma uint_split_asm:
   "P (uint x) = (\<nexists>i. word_of_int i = x \<and> 0 \<le> i \<and> i < 2^LENGTH('a) \<and> \<not> P i)"
   for x :: "'a::len word"
   by auto (metis take_bit_int_eq_self_iff)
 
 lemmas uint_splits = uint_split uint_split_asm
 
 lemmas uint_arith_simps =
   word_le_def word_less_alt
   word_uint_eq_iff
   uint_sub_if' uint_plus_if'
 
 \<comment> \<open>use this to stop, eg. \<open>2 ^ LENGTH(32)\<close> being simplified\<close>
 lemma power_False_cong: "False \<Longrightarrow> a ^ b = c ^ d"
   by auto
 
 \<comment> \<open>\<open>uint_arith_tac\<close>: reduce to arithmetic on int, try to solve by arith\<close>
 ML \<open>
 val uint_arith_simpset =
   @{context}
   |> fold Simplifier.add_simp @{thms uint_arith_simps}
   |> fold Splitter.add_split @{thms if_split_asm}
   |> fold Simplifier.add_cong @{thms power_False_cong}
   |> simpset_of;
   
 fun uint_arith_tacs ctxt =
   let
     fun arith_tac' n t =
       Arith_Data.arith_tac ctxt n t
         handle Cooper.COOPER _ => Seq.empty;
   in
     [ clarify_tac ctxt 1,
       full_simp_tac (put_simpset uint_arith_simpset ctxt) 1,
       ALLGOALS (full_simp_tac
         (put_simpset HOL_ss ctxt
           |> fold Splitter.add_split @{thms uint_splits}
           |> fold Simplifier.add_cong @{thms power_False_cong})),
       rewrite_goals_tac ctxt @{thms word_size},
       ALLGOALS  (fn n => REPEAT (resolve_tac ctxt [allI, impI] n) THEN
                          REPEAT (eresolve_tac ctxt [conjE] n) THEN
                          REPEAT (dresolve_tac ctxt @{thms word_of_int_inverse} n
                                  THEN assume_tac ctxt n
                                  THEN assume_tac ctxt n)),
       TRYALL arith_tac' ]
   end
 
 fun uint_arith_tac ctxt = SELECT_GOAL (EVERY (uint_arith_tacs ctxt))
 \<close>
 
 method_setup uint_arith =
   \<open>Scan.succeed (SIMPLE_METHOD' o uint_arith_tac)\<close>
   "solving word arithmetic via integers and arith"
 
 
 subsection \<open>More on overflows and monotonicity\<close>
 
 lemma no_plus_overflow_uint_size: "x \<le> x + y \<longleftrightarrow> uint x + uint y < 2 ^ size x"
   for x y :: "'a::len word"
   unfolding word_size by uint_arith
 
 lemmas no_olen_add = no_plus_overflow_uint_size [unfolded word_size]
 
 lemma no_ulen_sub: "x \<ge> x - y \<longleftrightarrow> uint y \<le> uint x"
   for x y :: "'a::len word"
   by uint_arith
 
 lemma no_olen_add': "x \<le> y + x \<longleftrightarrow> uint y + uint x < 2 ^ LENGTH('a)"
   for x y :: "'a::len word"
   by (simp add: ac_simps no_olen_add)
 
 lemmas olen_add_eqv = trans [OF no_olen_add no_olen_add' [symmetric]]
 
 lemmas uint_plus_simple_iff = trans [OF no_olen_add uint_add_lem]
 lemmas uint_plus_simple = uint_plus_simple_iff [THEN iffD1]
 lemmas uint_minus_simple_iff = trans [OF no_ulen_sub uint_sub_lem]
 lemmas uint_minus_simple_alt = uint_sub_lem [folded word_le_def]
 lemmas word_sub_le_iff = no_ulen_sub [folded word_le_def]
 lemmas word_sub_le = word_sub_le_iff [THEN iffD2]
 
 lemma word_less_sub1: "x \<noteq> 0 \<Longrightarrow> 1 < x \<longleftrightarrow> 0 < x - 1"
   for x :: "'a::len word"
   by uint_arith
 
 lemma word_le_sub1: "x \<noteq> 0 \<Longrightarrow> 1 \<le> x \<longleftrightarrow> 0 \<le> x - 1"
   for x :: "'a::len word"
   by uint_arith
 
 lemma sub_wrap_lt: "x < x - z \<longleftrightarrow> x < z"
   for x z :: "'a::len word"
   by uint_arith
 
 lemma sub_wrap: "x \<le> x - z \<longleftrightarrow> z = 0 \<or> x < z"
   for x z :: "'a::len word"
   by uint_arith
 
 lemma plus_minus_not_NULL_ab: "x \<le> ab - c \<Longrightarrow> c \<le> ab \<Longrightarrow> c \<noteq> 0 \<Longrightarrow> x + c \<noteq> 0"
   for x ab c :: "'a::len word"
   by uint_arith
 
 lemma plus_minus_no_overflow_ab: "x \<le> ab - c \<Longrightarrow> c \<le> ab \<Longrightarrow> x \<le> x + c"
   for x ab c :: "'a::len word"
   by uint_arith
 
 lemma le_minus': "a + c \<le> b \<Longrightarrow> a \<le> a + c \<Longrightarrow> c \<le> b - a"
   for a b c :: "'a::len word"
   by uint_arith
 
 lemma le_plus': "a \<le> b \<Longrightarrow> c \<le> b - a \<Longrightarrow> a + c \<le> b"
   for a b c :: "'a::len word"
   by uint_arith
 
 lemmas le_plus = le_plus' [rotated]
 
 lemmas le_minus = leD [THEN thin_rl, THEN le_minus'] (* FIXME *)
 
 lemma word_plus_mono_right: "y \<le> z \<Longrightarrow> x \<le> x + z \<Longrightarrow> x + y \<le> x + z"
   for x y z :: "'a::len word"
   by uint_arith
 
 lemma word_less_minus_cancel: "y - x < z - x \<Longrightarrow> x \<le> z \<Longrightarrow> y < z"
   for x y z :: "'a::len word"
   by uint_arith
 
 lemma word_less_minus_mono_left: "y < z \<Longrightarrow> x \<le> y \<Longrightarrow> y - x < z - x"
   for x y z :: "'a::len word"
   by uint_arith
 
 lemma word_less_minus_mono: "a < c \<Longrightarrow> d < b \<Longrightarrow> a - b < a \<Longrightarrow> c - d < c \<Longrightarrow> a - b < c - d"
   for a b c d :: "'a::len word"
   by uint_arith
 
 lemma word_le_minus_cancel: "y - x \<le> z - x \<Longrightarrow> x \<le> z \<Longrightarrow> y \<le> z"
   for x y z :: "'a::len word"
   by uint_arith
 
 lemma word_le_minus_mono_left: "y \<le> z \<Longrightarrow> x \<le> y \<Longrightarrow> y - x \<le> z - x"
   for x y z :: "'a::len word"
   by uint_arith
 
 lemma word_le_minus_mono:
   "a \<le> c \<Longrightarrow> d \<le> b \<Longrightarrow> a - b \<le> a \<Longrightarrow> c - d \<le> c \<Longrightarrow> a - b \<le> c - d"
   for a b c d :: "'a::len word"
   by uint_arith
 
 lemma plus_le_left_cancel_wrap: "x + y' < x \<Longrightarrow> x + y < x \<Longrightarrow> x + y' < x + y \<longleftrightarrow> y' < y"
   for x y y' :: "'a::len word"
   by uint_arith
 
 lemma plus_le_left_cancel_nowrap: "x \<le> x + y' \<Longrightarrow> x \<le> x + y \<Longrightarrow> x + y' < x + y \<longleftrightarrow> y' < y"
   for x y y' :: "'a::len word"
   by uint_arith
 
 lemma word_plus_mono_right2: "a \<le> a + b \<Longrightarrow> c \<le> b \<Longrightarrow> a \<le> a + c"
   for a b c :: "'a::len word"
   by uint_arith
 
 lemma word_less_add_right: "x < y - z \<Longrightarrow> z \<le> y \<Longrightarrow> x + z < y"
   for x y z :: "'a::len word"
   by uint_arith
 
 lemma word_less_sub_right: "x < y + z \<Longrightarrow> y \<le> x \<Longrightarrow> x - y < z"
   for x y z :: "'a::len word"
   by uint_arith
 
 lemma word_le_plus_either: "x \<le> y \<or> x \<le> z \<Longrightarrow> y \<le> y + z \<Longrightarrow> x \<le> y + z"
   for x y z :: "'a::len word"
   by uint_arith
 
 lemma word_less_nowrapI: "x < z - k \<Longrightarrow> k \<le> z \<Longrightarrow> 0 < k \<Longrightarrow> x < x + k"
   for x z k :: "'a::len word"
   by uint_arith
 
 lemma inc_le: "i < m \<Longrightarrow> i + 1 \<le> m"
   for i m :: "'a::len word"
   by uint_arith
 
 lemma inc_i: "1 \<le> i \<Longrightarrow> i < m \<Longrightarrow> 1 \<le> i + 1 \<and> i + 1 \<le> m"
   for i m :: "'a::len word"
   by uint_arith
 
 lemma udvd_incr_lem:
   "up < uq \<Longrightarrow> up = ua + n * uint K \<Longrightarrow>
     uq = ua + n' * uint K \<Longrightarrow> up + uint K \<le> uq"
   by auto (metis int_distrib(1) linorder_not_less mult.left_neutral mult_right_mono uint_nonnegative zless_imp_add1_zle)
 
 lemma udvd_incr':
   "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow>
     uint q = ua + n' * uint K \<Longrightarrow> p + K \<le> q"
   apply (unfold word_less_alt word_le_def)
   apply (drule (2) udvd_incr_lem)
   apply (erule uint_add_le [THEN order_trans])
   done
 
 lemma udvd_decr':
   "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow>
     uint q = ua + n' * uint K \<Longrightarrow> p \<le> q - K"
   apply (unfold word_less_alt word_le_def)
   apply (drule (2) udvd_incr_lem)
   apply (drule le_diff_eq [THEN iffD2])
   apply (erule order_trans)
   apply (rule uint_sub_ge)
   done
 
 lemmas udvd_incr_lem0 = udvd_incr_lem [where ua=0, unfolded add_0_left]
 lemmas udvd_incr0 = udvd_incr' [where ua=0, unfolded add_0_left]
 lemmas udvd_decr0 = udvd_decr' [where ua=0, unfolded add_0_left]
 
 lemma udvd_minus_le': "xy < k \<Longrightarrow> z udvd xy \<Longrightarrow> z udvd k \<Longrightarrow> xy \<le> k - z"
   apply (unfold udvd_unfold_int)
   apply clarify
   apply (erule (2) udvd_decr0)
   done
 
 lemma udvd_incr2_K:
   "p < a + s \<Longrightarrow> a \<le> a + s \<Longrightarrow> K udvd s \<Longrightarrow> K udvd p - a \<Longrightarrow> a \<le> p \<Longrightarrow>
     0 < K \<Longrightarrow> p \<le> p + K \<and> p + K \<le> a + s"
   supply [[simproc del: linordered_ring_less_cancel_factor]]
   apply (unfold udvd_unfold_int)
   apply clarify
   apply (simp add: uint_arith_simps split: if_split_asm)
    prefer 2
   using uint_lt2p [of s] apply simp
   apply (drule add.commute [THEN xtrans(1)])
   apply (simp flip: diff_less_eq)
   apply (subst (asm) mult_less_cancel_right)
   apply simp
   apply (simp add: diff_eq_eq not_less)
   apply (subst (asm) (3) zless_iff_Suc_zadd)
   apply auto
     apply (auto simp add: algebra_simps)
   apply (drule less_le_trans [of _ \<open>2 ^ LENGTH('a)\<close>]) apply assumption
   apply (simp add: mult_less_0_iff)
   done
 
 
 subsection \<open>Arithmetic type class instantiations\<close>
 
 lemmas word_le_0_iff [simp] =
   word_zero_le [THEN leD, THEN antisym_conv1]
 
 lemma word_of_int_nat: "0 \<le> x \<Longrightarrow> word_of_int x = of_nat (nat x)"
   by simp
 
 text \<open>
   note that \<open>iszero_def\<close> is only for class \<open>comm_semiring_1_cancel\<close>,
   which requires word length \<open>\<ge> 1\<close>, ie \<open>'a::len word\<close>
 \<close>
 lemma iszero_word_no [simp]:
   "iszero (numeral bin :: 'a::len word) =
     iszero (take_bit LENGTH('a) (numeral bin :: int))"
   apply (simp add: iszero_def)
   apply transfer
   apply simp
   done
 
 text \<open>Use \<open>iszero\<close> to simplify equalities between word numerals.\<close>
 
 lemmas word_eq_numeral_iff_iszero [simp] =
   eq_numeral_iff_iszero [where 'a="'a::len word"]
 
 
 subsection \<open>Word and nat\<close>
 
 lemma word_nchotomy: "\<forall>w :: 'a::len word. \<exists>n. w = of_nat n \<and> n < 2 ^ LENGTH('a)"
   apply (rule allI)
   apply (rule exI [of _ \<open>unat w\<close> for w :: \<open>'a word\<close>])
   apply simp
   done
 
 lemma of_nat_eq: "of_nat n = w \<longleftrightarrow> (\<exists>q. n = unat w + q * 2 ^ LENGTH('a))"
   for w :: "'a::len word"
   using mod_div_mult_eq [of n "2 ^ LENGTH('a)", symmetric]
   by (auto simp flip: take_bit_eq_mod)
 
 lemma of_nat_eq_size: "of_nat n = w \<longleftrightarrow> (\<exists>q. n = unat w + q * 2 ^ size w)"
   unfolding word_size by (rule of_nat_eq)
 
 lemma of_nat_0: "of_nat m = (0::'a::len word) \<longleftrightarrow> (\<exists>q. m = q * 2 ^ LENGTH('a))"
   by (simp add: of_nat_eq)
 
 lemma of_nat_2p [simp]: "of_nat (2 ^ LENGTH('a)) = (0::'a::len word)"
   by (fact mult_1 [symmetric, THEN iffD2 [OF of_nat_0 exI]])
 
 lemma of_nat_gt_0: "of_nat k \<noteq> 0 \<Longrightarrow> 0 < k"
   by (cases k) auto
 
 lemma of_nat_neq_0: "0 < k \<Longrightarrow> k < 2 ^ LENGTH('a::len) \<Longrightarrow> of_nat k \<noteq> (0 :: 'a word)"
   by (auto simp add : of_nat_0)
 
 lemma Abs_fnat_hom_add: "of_nat a + of_nat b = of_nat (a + b)"
   by simp
 
 lemma Abs_fnat_hom_mult: "of_nat a * of_nat b = (of_nat (a * b) :: 'a::len word)"
   by (simp add: wi_hom_mult)
 
 lemma Abs_fnat_hom_Suc: "word_succ (of_nat a) = of_nat (Suc a)"
   by transfer (simp add: ac_simps)
 
 lemma Abs_fnat_hom_0: "(0::'a::len word) = of_nat 0"
   by simp
 
 lemma Abs_fnat_hom_1: "(1::'a::len word) = of_nat (Suc 0)"
   by simp
 
 lemmas Abs_fnat_homs =
   Abs_fnat_hom_add Abs_fnat_hom_mult Abs_fnat_hom_Suc
   Abs_fnat_hom_0 Abs_fnat_hom_1
 
 lemma word_arith_nat_add: "a + b = of_nat (unat a + unat b)"
   by simp
 
 lemma word_arith_nat_mult: "a * b = of_nat (unat a * unat b)"
   by simp
 
 lemma word_arith_nat_Suc: "word_succ a = of_nat (Suc (unat a))"
   by (subst Abs_fnat_hom_Suc [symmetric]) simp
 
 lemma word_arith_nat_div: "a div b = of_nat (unat a div unat b)"
   by (metis of_int_of_nat_eq of_nat_unat of_nat_div word_div_def)
   
 lemma word_arith_nat_mod: "a mod b = of_nat (unat a mod unat b)"
   by (metis of_int_of_nat_eq of_nat_mod of_nat_unat word_mod_def)
 
 lemmas word_arith_nat_defs =
   word_arith_nat_add word_arith_nat_mult
   word_arith_nat_Suc Abs_fnat_hom_0
   Abs_fnat_hom_1 word_arith_nat_div
   word_arith_nat_mod
 
 lemma unat_cong: "x = y \<Longrightarrow> unat x = unat y"
   by (fact arg_cong)
 
 lemma unat_of_nat:
   \<open>unat (word_of_nat x :: 'a::len word) = x mod 2 ^ LENGTH('a)\<close>
   by transfer (simp flip: take_bit_eq_mod add: nat_take_bit_eq)
 
 lemmas unat_word_ariths = word_arith_nat_defs
   [THEN trans [OF unat_cong unat_of_nat]]
 
 lemmas word_sub_less_iff = word_sub_le_iff
   [unfolded linorder_not_less [symmetric] Not_eq_iff]
 
 lemma unat_add_lem:
   "unat x + unat y < 2 ^ LENGTH('a) \<longleftrightarrow> unat (x + y) = unat x + unat y"
   for x y :: "'a::len word"
   apply (auto simp: unat_word_ariths)
   apply (drule sym)
   apply (metis unat_of_nat unsigned_less)
   done
 
 lemma unat_mult_lem:
   "unat x * unat y < 2 ^ LENGTH('a) \<longleftrightarrow> unat (x * y) = unat x * unat y"
   for x y :: "'a::len word"
   apply (auto simp: unat_word_ariths)
   apply (drule sym)
   apply (metis unat_of_nat unsigned_less)
   done
 
 lemma unat_plus_if':
   \<open>unat (a + b) =
     (if unat a + unat b < 2 ^ LENGTH('a)
     then unat a + unat b
     else unat a + unat b - 2 ^ LENGTH('a))\<close> for a b :: \<open>'a::len word\<close>
   apply (auto simp: unat_word_ariths not_less)
   apply (subst (asm) le_iff_add)
   apply auto
   apply (simp flip: take_bit_eq_mod add: take_bit_nat_eq_self_iff)
   apply (metis add.commute add_less_cancel_right le_less_trans less_imp_le unsigned_less)
   done
 
 lemma le_no_overflow: "x \<le> b \<Longrightarrow> a \<le> a + b \<Longrightarrow> x \<le> a + b"
   for a b x :: "'a::len word"
   apply (erule order_trans)
   apply (erule olen_add_eqv [THEN iffD1])
   done
 
 lemmas un_ui_le =
   trans [OF word_le_nat_alt [symmetric] word_le_def]
 
 lemma unat_sub_if_size:
   "unat (x - y) =
     (if unat y \<le> unat x
      then unat x - unat y
      else unat x + 2 ^ size x - unat y)"
   supply nat_uint_eq [simp del]
   apply (unfold word_size)
   apply (simp add: un_ui_le)
   apply (auto simp add: unat_eq_nat_uint uint_sub_if')
    apply (rule nat_diff_distrib)
     prefer 3
     apply (simp add: algebra_simps)
     apply (rule nat_diff_distrib [THEN trans])
       prefer 3
       apply (subst nat_add_distrib)
         prefer 3
         apply (simp add: nat_power_eq)
        apply auto
   apply uint_arith
   done
 
 lemmas unat_sub_if' = unat_sub_if_size [unfolded word_size]
 
 lemma uint_div:
   \<open>uint (x div y) = uint x div uint y\<close>
   by (fact uint_div_distrib)
 
 lemma unat_div:
   \<open>unat (x div y) = unat x div unat y\<close>
   by (fact unat_div_distrib)
 
 lemma uint_mod:
   \<open>uint (x mod y) = uint x mod uint y\<close>
   by (fact uint_mod_distrib)
   
 lemma unat_mod:
   \<open>unat (x mod y) = unat x mod unat y\<close>
   by (fact unat_mod_distrib)
 
 
 text \<open>Definition of \<open>unat_arith\<close> tactic\<close>
 
 lemma unat_split: "P (unat x) \<longleftrightarrow> (\<forall>n. of_nat n = x \<and> n < 2^LENGTH('a) \<longrightarrow> P n)"
   for x :: "'a::len word"
   by auto (metis take_bit_nat_eq_self_iff)
 
 lemma unat_split_asm: "P (unat x) \<longleftrightarrow> (\<nexists>n. of_nat n = x \<and> n < 2^LENGTH('a) \<and> \<not> P n)"
   for x :: "'a::len word"
   by auto (metis take_bit_nat_eq_self_iff)
 
 lemma of_nat_inverse:
   \<open>word_of_nat r = a \<Longrightarrow> r < 2 ^ LENGTH('a) \<Longrightarrow> unat a = r\<close>
   for a :: \<open>'a::len word\<close>
   apply (drule sym)
   apply transfer
   apply (simp add: take_bit_int_eq_self)
   done
 
 lemma word_unat_eq_iff:
   \<open>v = w \<longleftrightarrow> unat v = unat w\<close>
   for v w :: \<open>'a::len word\<close>
   by (fact word_eq_iff_unsigned)
 
 lemmas unat_splits = unat_split unat_split_asm
 
 lemmas unat_arith_simps =
   word_le_nat_alt word_less_nat_alt
   word_unat_eq_iff
   unat_sub_if' unat_plus_if' unat_div unat_mod
 
 \<comment> \<open>\<open>unat_arith_tac\<close>: tactic to reduce word arithmetic to \<open>nat\<close>, try to solve via \<open>arith\<close>\<close>
 ML \<open>
 val unat_arith_simpset =
   @{context}
   |> fold Simplifier.add_simp @{thms unat_arith_simps}
   |> fold Splitter.add_split @{thms if_split_asm}
   |> fold Simplifier.add_cong @{thms power_False_cong}
   |> simpset_of
 
 fun unat_arith_tacs ctxt =
   let
     fun arith_tac' n t =
       Arith_Data.arith_tac ctxt n t
         handle Cooper.COOPER _ => Seq.empty;
   in
     [ clarify_tac ctxt 1,
       full_simp_tac (put_simpset unat_arith_simpset ctxt) 1,
       ALLGOALS (full_simp_tac
         (put_simpset HOL_ss ctxt
           |> fold Splitter.add_split @{thms unat_splits}
           |> fold Simplifier.add_cong @{thms power_False_cong})),
       rewrite_goals_tac ctxt @{thms word_size},
       ALLGOALS (fn n => REPEAT (resolve_tac ctxt [allI, impI] n) THEN
                          REPEAT (eresolve_tac ctxt [conjE] n) THEN
                          REPEAT (dresolve_tac ctxt @{thms of_nat_inverse} n THEN assume_tac ctxt n)),
       TRYALL arith_tac' ]
   end
 
 fun unat_arith_tac ctxt = SELECT_GOAL (EVERY (unat_arith_tacs ctxt))
 \<close>
 
 method_setup unat_arith =
   \<open>Scan.succeed (SIMPLE_METHOD' o unat_arith_tac)\<close>
   "solving word arithmetic via natural numbers and arith"
 
 lemma no_plus_overflow_unat_size: "x \<le> x + y \<longleftrightarrow> unat x + unat y < 2 ^ size x"
   for x y :: "'a::len word"
   unfolding word_size by unat_arith
 
 lemmas no_olen_add_nat =
   no_plus_overflow_unat_size [unfolded word_size]
 
 lemmas unat_plus_simple =
   trans [OF no_olen_add_nat unat_add_lem]
 
 lemma word_div_mult: "0 < y \<Longrightarrow> unat x * unat y < 2 ^ LENGTH('a) \<Longrightarrow> x * y div y = x"
   for x y :: "'a::len word"
   apply unat_arith
   apply clarsimp
   apply (subst unat_mult_lem [THEN iffD1])
    apply auto
   done
 
 lemma div_lt': "i \<le> k div x \<Longrightarrow> unat i * unat x < 2 ^ LENGTH('a)"
   for i k x :: "'a::len word"
   apply unat_arith
   apply clarsimp
   apply (drule mult_le_mono1)
   apply (erule order_le_less_trans)
   apply (metis add_lessD1 div_mult_mod_eq unsigned_less)
   done
 
 lemmas div_lt'' = order_less_imp_le [THEN div_lt']
 
 lemma div_lt_mult: "i < k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x < k"
   for i k x :: "'a::len word"
   apply (frule div_lt'' [THEN unat_mult_lem [THEN iffD1]])
   apply (simp add: unat_arith_simps)
   apply (drule (1) mult_less_mono1)
   apply (erule order_less_le_trans)
   apply auto
   done
 
 lemma div_le_mult: "i \<le> k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x \<le> k"
   for i k x :: "'a::len word"
   apply (frule div_lt' [THEN unat_mult_lem [THEN iffD1]])
   apply (simp add: unat_arith_simps)
   apply (drule mult_le_mono1)
   apply (erule order_trans)
   apply auto
   done
 
 lemma div_lt_uint': "i \<le> k div x \<Longrightarrow> uint i * uint x < 2 ^ LENGTH('a)"
   for i k x :: "'a::len word"
   apply (unfold uint_nat)
   apply (drule div_lt')
   apply (metis of_nat_less_iff of_nat_mult of_nat_numeral of_nat_power)
   done
 
 lemmas div_lt_uint'' = order_less_imp_le [THEN div_lt_uint']
 
 lemma word_le_exists': "x \<le> y \<Longrightarrow> \<exists>z. y = x + z \<and> uint x + uint z < 2 ^ LENGTH('a)"
   for x y z :: "'a::len word"
   by (metis add_diff_cancel_left' add_diff_eq uint_add_lem uint_plus_simple)
   
 lemmas plus_minus_not_NULL = order_less_imp_le [THEN plus_minus_not_NULL_ab]
 
 lemmas plus_minus_no_overflow =
   order_less_imp_le [THEN plus_minus_no_overflow_ab]
 
 lemmas mcs = word_less_minus_cancel word_less_minus_mono_left
   word_le_minus_cancel word_le_minus_mono_left
 
 lemmas word_l_diffs = mcs [where y = "w + x", unfolded add_diff_cancel] for w x
 lemmas word_diff_ls = mcs [where z = "w + x", unfolded add_diff_cancel] for w x
 lemmas word_plus_mcs = word_diff_ls [where y = "v + x", unfolded add_diff_cancel] for v x
 
 lemma le_unat_uoi:
   \<open>y \<le> unat z \<Longrightarrow> unat (word_of_nat y :: 'a word) = y\<close>
   for z :: \<open>'a::len word\<close>
   by transfer (simp add: nat_take_bit_eq take_bit_nat_eq_self_iff le_less_trans)
 
 lemmas thd = times_div_less_eq_dividend
 
 lemmas uno_simps [THEN le_unat_uoi] = mod_le_divisor div_le_dividend
 
 lemma word_mod_div_equality: "(n div b) * b + (n mod b) = n"
   for n b :: "'a::len word"
   by (fact div_mult_mod_eq)
 
 lemma word_div_mult_le: "a div b * b \<le> a"
   for a b :: "'a::len word"
   by (metis div_le_mult mult_not_zero order.not_eq_order_implies_strict order_refl word_zero_le)
 
 lemma word_mod_less_divisor: "0 < n \<Longrightarrow> m mod n < n"
   for m n :: "'a::len word"
   by (simp add: unat_arith_simps)
   
 lemma word_of_int_power_hom: "word_of_int a ^ n = (word_of_int (a ^ n) :: 'a::len word)"
   by (induct n) (simp_all add: wi_hom_mult [symmetric])
 
 lemma word_arith_power_alt: "a ^ n = (word_of_int (uint a ^ n) :: 'a::len word)"
   by (simp add : word_of_int_power_hom [symmetric])
 
 lemma unatSuc: "1 + n \<noteq> 0 \<Longrightarrow> unat (1 + n) = Suc (unat n)"
   for n :: "'a::len word"
   by unat_arith
 
 
 subsection \<open>Cardinality, finiteness of set of words\<close>
 
 lemma inj_on_word_of_int: \<open>inj_on (word_of_int :: int \<Rightarrow> 'a word) {0..<2 ^ LENGTH('a::len)}\<close>
   apply (rule inj_onI)
   apply transfer
   apply (simp add: take_bit_eq_mod)
   done
 
 lemma inj_uint: \<open>inj uint\<close>
   by (fact inj_unsigned)
 
 lemma range_uint: \<open>range (uint :: 'a word \<Rightarrow> int) = {0..<2 ^ LENGTH('a::len)}\<close>
   by transfer (auto simp add: bintr_lt2p range_bintrunc)
 
 lemma UNIV_eq: \<open>(UNIV :: 'a word set) = word_of_int ` {0..<2 ^ LENGTH('a::len)}\<close>
 proof -
   have \<open>uint ` (UNIV :: 'a word set) = uint ` (word_of_int :: int \<Rightarrow> 'a word) ` {0..<2 ^ LENGTH('a::len)}\<close>
     by transfer (simp add: range_bintrunc, auto simp add: take_bit_eq_mod)
   then show ?thesis
     using inj_image_eq_iff [of \<open>uint :: 'a word \<Rightarrow> int\<close> \<open>UNIV :: 'a word set\<close> \<open>word_of_int ` {0..<2 ^ LENGTH('a)} :: 'a word set\<close>, OF inj_uint]
     by simp
 qed
 
 lemma card_word: "CARD('a word) = 2 ^ LENGTH('a::len)"
   by (simp add: UNIV_eq card_image inj_on_word_of_int)
 
 lemma card_word_size: "CARD('a word) = 2 ^ size x"
   for x :: "'a::len word"
   unfolding word_size by (rule card_word)
 
 instance word :: (len) finite
   by standard (simp add: UNIV_eq)
 
 
 subsection \<open>Bitwise Operations on Words\<close>
 
 lemmas bin_log_bintrs = bin_trunc_not bin_trunc_xor bin_trunc_and bin_trunc_or
 
 \<comment> \<open>following definitions require both arithmetic and bit-wise word operations\<close>
 
 \<comment> \<open>to get \<open>word_no_log_defs\<close> from \<open>word_log_defs\<close>, using \<open>bin_log_bintrs\<close>\<close>
 lemmas wils1 = bin_log_bintrs [THEN word_of_int_eq_iff [THEN iffD2],
   folded uint_word_of_int_eq, THEN eq_reflection]
 
 \<comment> \<open>the binary operations only\<close>  (* BH: why is this needed? *)
 lemmas word_log_binary_defs =
   word_and_def word_or_def word_xor_def
 
 lemma word_wi_log_defs:
   "NOT (word_of_int a) = word_of_int (NOT a)"
   "word_of_int a AND word_of_int b = word_of_int (a AND b)"
   "word_of_int a OR word_of_int b = word_of_int (a OR b)"
   "word_of_int a XOR word_of_int b = word_of_int (a XOR b)"
   by (transfer, rule refl)+
 
 lemma word_no_log_defs [simp]:
   "NOT (numeral a) = word_of_int (NOT (numeral a))"
   "NOT (- numeral a) = word_of_int (NOT (- numeral a))"
   "numeral a AND numeral b = word_of_int (numeral a AND numeral b)"
   "numeral a AND - numeral b = word_of_int (numeral a AND - numeral b)"
   "- numeral a AND numeral b = word_of_int (- numeral a AND numeral b)"
   "- numeral a AND - numeral b = word_of_int (- numeral a AND - numeral b)"
   "numeral a OR numeral b = word_of_int (numeral a OR numeral b)"
   "numeral a OR - numeral b = word_of_int (numeral a OR - numeral b)"
   "- numeral a OR numeral b = word_of_int (- numeral a OR numeral b)"
   "- numeral a OR - numeral b = word_of_int (- numeral a OR - numeral b)"
   "numeral a XOR numeral b = word_of_int (numeral a XOR numeral b)"
   "numeral a XOR - numeral b = word_of_int (numeral a XOR - numeral b)"
   "- numeral a XOR numeral b = word_of_int (- numeral a XOR numeral b)"
   "- numeral a XOR - numeral b = word_of_int (- numeral a XOR - numeral b)"
   by (transfer, rule refl)+
 
 text \<open>Special cases for when one of the arguments equals 1.\<close>
 
 lemma word_bitwise_1_simps [simp]:
   "NOT (1::'a::len word) = -2"
   "1 AND numeral b = word_of_int (1 AND numeral b)"
   "1 AND - numeral b = word_of_int (1 AND - numeral b)"
   "numeral a AND 1 = word_of_int (numeral a AND 1)"
   "- numeral a AND 1 = word_of_int (- numeral a AND 1)"
   "1 OR numeral b = word_of_int (1 OR numeral b)"
   "1 OR - numeral b = word_of_int (1 OR - numeral b)"
   "numeral a OR 1 = word_of_int (numeral a OR 1)"
   "- numeral a OR 1 = word_of_int (- numeral a OR 1)"
   "1 XOR numeral b = word_of_int (1 XOR numeral b)"
   "1 XOR - numeral b = word_of_int (1 XOR - numeral b)"
   "numeral a XOR 1 = word_of_int (numeral a XOR 1)"
   "- numeral a XOR 1 = word_of_int (- numeral a XOR 1)"
   by (transfer, simp)+
 
 text \<open>Special cases for when one of the arguments equals -1.\<close>
 
 lemma word_bitwise_m1_simps [simp]:
   "NOT (-1::'a::len word) = 0"
   "(-1::'a::len word) AND x = x"
   "x AND (-1::'a::len word) = x"
   "(-1::'a::len word) OR x = -1"
   "x OR (-1::'a::len word) = -1"
   " (-1::'a::len word) XOR x = NOT x"
   "x XOR (-1::'a::len word) = NOT x"
   by (transfer, simp)+
 
 lemma uint_and:
   \<open>uint (x AND y) = uint x AND uint y\<close>
   by transfer simp
 
 lemma uint_or:
   \<open>uint (x OR y) = uint x OR uint y\<close>
   by transfer simp
 
 lemma uint_xor:
   \<open>uint (x XOR y) = uint x XOR uint y\<close>
   by transfer simp
 
 lemma word_test_bit_transfer [transfer_rule]:
   "(rel_fun pcr_word (rel_fun (=) (=)))
     (\<lambda>x n. n < LENGTH('a) \<and> bit x n) (test_bit :: 'a::len word \<Rightarrow> _)"
   by (simp only: test_bit_eq_bit) transfer_prover
 
 lemma test_bit_wi [simp]:
   "(word_of_int x :: 'a::len word) !! n \<longleftrightarrow> n < LENGTH('a) \<and> bin_nth x n"
   by transfer simp
 
 lemma word_ops_nth_size:
   "n < size x \<Longrightarrow>
     (x OR y) !! n = (x !! n | y !! n) \<and>
     (x AND y) !! n = (x !! n \<and> y !! n) \<and>
     (x XOR y) !! n = (x !! n \<noteq> y !! n) \<and>
     (NOT x) !! n = (\<not> x !! n)"
   for x :: "'a::len word"
   unfolding word_size by transfer (simp add: bin_nth_ops)
 
 lemma word_ao_nth:
   "(x OR y) !! n = (x !! n | y !! n) \<and>
     (x AND y) !! n = (x !! n \<and> y !! n)"
   for x :: "'a::len word"
   by transfer (auto simp add: bin_nth_ops)
 
 lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN word_ops_nth_size [unfolded word_size]]
 lemmas msb1 = msb0 [where i = 0]
 
 lemma test_bit_numeral [simp]:
   "(numeral w :: 'a::len word) !! n \<longleftrightarrow>
     n < LENGTH('a) \<and> bin_nth (numeral w) n"
   by transfer (rule refl)
 
 lemma test_bit_neg_numeral [simp]:
   "(- numeral w :: 'a::len word) !! n \<longleftrightarrow>
     n < LENGTH('a) \<and> bin_nth (- numeral w) n"
   by transfer (rule refl)
 
 lemma test_bit_1 [simp]: "(1 :: 'a::len word) !! n \<longleftrightarrow> n = 0"
   by transfer auto
 
 lemma nth_0 [simp]: "\<not> (0 :: 'a::len word) !! n"
   by transfer simp
 
 lemma nth_minus1 [simp]: "(-1 :: 'a::len word) !! n \<longleftrightarrow> n < LENGTH('a)"
   by transfer simp
 
 \<comment> \<open>get from commutativity, associativity etc of \<open>int_and\<close> etc to same for \<open>word_and etc\<close>\<close>
 lemmas bwsimps =
   wi_hom_add
   word_wi_log_defs
 
 lemma word_bw_assocs:
   "(x AND y) AND z = x AND y AND z"
   "(x OR y) OR z = x OR y OR z"
   "(x XOR y) XOR z = x XOR y XOR z"
   for x :: "'a::len word"
   by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
 
 lemma word_bw_comms:
   "x AND y = y AND x"
   "x OR y = y OR x"
   "x XOR y = y XOR x"
   for x :: "'a::len word"
   by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
 
 lemma word_bw_lcs:
   "y AND x AND z = x AND y AND z"
   "y OR x OR z = x OR y OR z"
   "y XOR x XOR z = x XOR y XOR z"
   for x :: "'a::len word"
   by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
 
 lemma word_log_esimps:
   "x AND 0 = 0"
   "x AND -1 = x"
   "x OR 0 = x"
   "x OR -1 = -1"
   "x XOR 0 = x"
   "x XOR -1 = NOT x"
   "0 AND x = 0"
   "-1 AND x = x"
   "0 OR x = x"
   "-1 OR x = -1"
   "0 XOR x = x"
   "-1 XOR x = NOT x"
   for x :: "'a::len word"
   by simp_all
 
 lemma word_not_dist:
   "NOT (x OR y) = NOT x AND NOT y"
   "NOT (x AND y) = NOT x OR NOT y"
   for x :: "'a::len word"
   by simp_all
 
 lemma word_bw_same:
   "x AND x = x"
   "x OR x = x"
   "x XOR x = 0"
   for x :: "'a::len word"
   by simp_all
 
 lemma word_ao_absorbs [simp]:
   "x AND (y OR x) = x"
   "x OR y AND x = x"
   "x AND (x OR y) = x"
   "y AND x OR x = x"
   "(y OR x) AND x = x"
   "x OR x AND y = x"
   "(x OR y) AND x = x"
   "x AND y OR x = x"
   for x :: "'a::len word"
   by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
 
 lemma word_not_not [simp]: "NOT (NOT x) = x"
   for x :: "'a::len word"
   by simp
 
 lemma word_ao_dist: "(x OR y) AND z = x AND z OR y AND z"
   for x :: "'a::len word"
   by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
 
 lemma word_oa_dist: "x AND y OR z = (x OR z) AND (y OR z)"
   for x :: "'a::len word"
   by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
 
 lemma word_add_not [simp]: "x + NOT x = -1"
   for x :: "'a::len word"
   by transfer (simp add: bin_add_not)
 
 lemma word_plus_and_or [simp]: "(x AND y) + (x OR y) = x + y"
   for x :: "'a::len word"
   by transfer (simp add: plus_and_or)
 
 lemma leoa: "w = x OR y \<Longrightarrow> y = w AND y"
   for x :: "'a::len word"
   by auto
 
 lemma leao: "w' = x' AND y' \<Longrightarrow> x' = x' OR w'"
   for x' :: "'a::len word"
   by auto
 
 lemma word_ao_equiv: "w = w OR w' \<longleftrightarrow> w' = w AND w'"
   for w w' :: "'a::len word"
   by (auto intro: leoa leao)
 
 lemma le_word_or2: "x \<le> x OR y"
   for x y :: "'a::len word"
   by (auto simp: word_le_def uint_or intro: le_int_or)
 
 lemmas le_word_or1 = xtrans(3) [OF word_bw_comms (2) le_word_or2]
 lemmas word_and_le1 = xtrans(3) [OF word_ao_absorbs (4) [symmetric] le_word_or2]
 lemmas word_and_le2 = xtrans(3) [OF word_ao_absorbs (8) [symmetric] le_word_or2]
 
 lemma bit_horner_sum_bit_word_iff:
   \<open>bit (horner_sum of_bool (2 :: 'a::len word) bs) n
     \<longleftrightarrow> n < min LENGTH('a) (length bs) \<and> bs ! n\<close>
   by transfer (simp add: bit_horner_sum_bit_iff)
 
 definition word_reverse :: \<open>'a::len word \<Rightarrow> 'a word\<close>
   where \<open>word_reverse w = horner_sum of_bool 2 (rev (map (bit w) [0..<LENGTH('a)]))\<close>
 
 lemma bit_word_reverse_iff:
   \<open>bit (word_reverse w) n \<longleftrightarrow> n < LENGTH('a) \<and> bit w (LENGTH('a) - Suc n)\<close>
   for w :: \<open>'a::len word\<close>
   by (cases \<open>n < LENGTH('a)\<close>)
     (simp_all add: word_reverse_def bit_horner_sum_bit_word_iff rev_nth)
 
 lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w"
   by (rule bit_word_eqI)
     (auto simp add: bit_word_reverse_iff bit_imp_le_length Suc_diff_Suc)
 
 lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w"
   by (metis word_rev_rev)
 
 lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u"
   by simp
 
 lemmas lsb0 = len_gt_0 [THEN word_ops_nth_size [unfolded word_size]]
 
 lemma nth_sint:
   fixes w :: "'a::len word"
   defines "l \<equiv> LENGTH('a)"
   shows "bin_nth (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))"
   unfolding sint_uint l_def
   by (auto simp: nth_sbintr word_test_bit_def [symmetric])
 
 lemma setBit_no [simp]: "setBit (numeral bin) n = word_of_int (bin_sc n True (numeral bin))"
   by transfer (simp add: bin_sc_eq)
  
 lemma clearBit_no [simp]:
   "clearBit (numeral bin) n = word_of_int (bin_sc n False (numeral bin))"
   by transfer (simp add: bin_sc_eq)
 
 lemma test_bit_2p: "(word_of_int (2 ^ n)::'a::len word) !! m \<longleftrightarrow> m = n \<and> m < LENGTH('a)"
   by transfer (auto simp add: bit_exp_iff)
 
 lemma nth_w2p: "((2::'a::len word) ^ n) !! m \<longleftrightarrow> m = n \<and> m < LENGTH('a::len)"
   by transfer (auto simp add: bit_exp_iff)
 
 lemma uint_2p: "(0::'a::len word) < 2 ^ n \<Longrightarrow> uint (2 ^ n::'a::len word) = 2 ^ n"
   apply (cases \<open>n < LENGTH('a)\<close>; transfer)
   apply auto
   done
 
 lemma word_of_int_2p: "(word_of_int (2 ^ n) :: 'a::len word) = 2 ^ n"
   by (induct n) (simp_all add: wi_hom_syms)
 
 lemma bang_is_le: "x !! m \<Longrightarrow> 2 ^ m \<le> x"
   for x :: "'a::len word"
   apply (rule xtrans(3))
    apply (rule_tac [2] y = "x" in le_word_or2)
   apply (rule word_eqI)
   apply (auto simp add: word_ao_nth nth_w2p word_size)
   done
 
 
-subsection \<open>Bit comprehension\<close>
-
-instantiation word :: (len) bit_comprehension
-begin
-
-definition word_set_bits_def:
-  \<open>(BITS n. P n) = (horner_sum of_bool 2 (map P [0..<LENGTH('a)]) :: 'a word)\<close>
-
-instance ..
-
-end
-
-lemma bit_set_bits_word_iff:
-  \<open>bit (set_bits P :: 'a::len word) n \<longleftrightarrow> n < LENGTH('a) \<and> P n\<close>
-  by (auto simp add: word_set_bits_def bit_horner_sum_bit_word_iff)
-
-lemma set_bits_bit_eq:
-  \<open>set_bits (bit w) = w\<close> for w :: \<open>'a::len word\<close>
-  by (rule bit_word_eqI) (auto simp add: bit_set_bits_word_iff bit_imp_le_length)
-
-lemma set_bits_K_False [simp]:
-  \<open>set_bits (\<lambda>_. False) = (0 :: 'a :: len word)\<close>
-  by (rule bit_word_eqI) (simp add: bit_set_bits_word_iff)
-
-
 subsection \<open>Shifting, Rotating, and Splitting Words\<close>
 
 lemma shiftl1_wi [simp]: "shiftl1 (word_of_int w) = word_of_int (2 * w)"
   by transfer simp
 
 lemma shiftl1_numeral [simp]: "shiftl1 (numeral w) = numeral (Num.Bit0 w)"
   unfolding word_numeral_alt shiftl1_wi by simp
 
 lemma shiftl1_neg_numeral [simp]: "shiftl1 (- numeral w) = - numeral (Num.Bit0 w)"
   unfolding word_neg_numeral_alt shiftl1_wi by simp
 
 lemma shiftl1_0 [simp] : "shiftl1 0 = 0"
   by transfer simp
 
 lemma shiftl1_def_u: "shiftl1 w = word_of_int (2 * uint w)"
   by (fact shiftl1_eq)
 
 lemma shiftl1_def_s: "shiftl1 w = word_of_int (2 * sint w)"
   by (simp add: shiftl1_def_u wi_hom_syms)
 
 lemma shiftr1_0 [simp]: "shiftr1 0 = 0"
   by transfer simp
 
 lemma sshiftr1_0 [simp]: "sshiftr1 0 = 0"
   by transfer simp
 
 lemma sshiftr1_n1 [simp]: "sshiftr1 (- 1) = - 1"
   by transfer simp
 
 lemma shiftl_0 [simp]: "(0::'a::len word) << n = 0"
   by transfer simp
 
 lemma shiftr_0 [simp]: "(0::'a::len word) >> n = 0"
   by transfer simp
 
 lemma sshiftr_0 [simp]: "0 >>> n = 0"
   by transfer simp
 
 lemma sshiftr_n1 [simp]: "-1 >>> n = -1"
   by transfer simp
 
 lemma nth_shiftl1: "shiftl1 w !! n \<longleftrightarrow> n < size w \<and> n > 0 \<and> w !! (n - 1)"
   by transfer (auto simp add: bit_double_iff)
 
 lemma nth_shiftl': "(w << m) !! n \<longleftrightarrow> n < size w \<and> n >= m \<and> w !! (n - m)"
   for w :: "'a::len word"
   by transfer (auto simp add: bit_push_bit_iff)
 
 lemmas nth_shiftl = nth_shiftl' [unfolded word_size]
 
 lemma nth_shiftr1: "shiftr1 w !! n = w !! Suc n"
   by transfer (auto simp add: bit_take_bit_iff simp flip: bit_Suc)
 
 lemma nth_shiftr: "(w >> m) !! n = w !! (n + m)"
   for w :: "'a::len word"
   apply (unfold shiftr_def)
   apply (induct "m" arbitrary: n)
    apply (auto simp add: nth_shiftr1)
   done
 
 text \<open>
   see paper page 10, (1), (2), \<open>shiftr1_def\<close> is of the form of (1),
   where \<open>f\<close> (ie \<open>bin_rest\<close>) takes normal arguments to normal results,
   thus we get (2) from (1)
 \<close>
 
 lemma uint_shiftr1: "uint (shiftr1 w) = bin_rest (uint w)"
   by transfer simp
 
 lemma bit_sshiftr1_iff:
   \<open>bit (sshiftr1 w) n \<longleftrightarrow> bit w (if n = LENGTH('a) - 1 then LENGTH('a) - 1 else Suc n)\<close>
   for w :: \<open>'a::len word\<close>
   apply transfer
   apply (auto simp add: bit_take_bit_iff bit_signed_take_bit_iff min_def simp flip: bit_Suc)
   using le_less_Suc_eq apply fastforce
   using le_less_Suc_eq apply fastforce
   done
 
 lemma bit_sshiftr_word_iff:
   \<open>bit (w >>> m) n \<longleftrightarrow> bit w (if LENGTH('a) - m \<le> n \<and> n < LENGTH('a) then LENGTH('a) - 1 else (m + n))\<close>
   for w :: \<open>'a::len word\<close>
   apply transfer
   apply (auto simp add: bit_take_bit_iff bit_drop_bit_eq bit_signed_take_bit_iff min_def not_le simp flip: bit_Suc)
   using le_less_Suc_eq apply fastforce
   using le_less_Suc_eq apply fastforce
   done
 
 lemma sshiftr_eq:
   \<open>w >>> m = signed_drop_bit m w\<close>
   by (rule bit_eqI) (simp add: bit_signed_drop_bit_iff bit_sshiftr_word_iff)
 
 lemma nth_sshiftr1: "sshiftr1 w !! n = (if n = size w - 1 then w !! n else w !! Suc n)"
   apply transfer
   apply (auto simp add: bit_take_bit_iff bit_signed_take_bit_iff min_def simp flip: bit_Suc)
   using le_less_Suc_eq apply fastforce
   using le_less_Suc_eq apply fastforce
   done
 
 lemma nth_sshiftr :
   "sshiftr w m !! n =
     (n < size w \<and> (if n + m \<ge> size w then w !! (size w - 1) else w !! (n + m)))"
   apply transfer
   apply (auto simp add: bit_take_bit_iff bit_drop_bit_eq bit_signed_take_bit_iff min_def not_le ac_simps)
   using le_less_Suc_eq apply fastforce
   using le_less_Suc_eq apply fastforce
   done
 
 lemma shiftr1_div_2: "uint (shiftr1 w) = uint w div 2"
   by (fact uint_shiftr1)
 
 lemma sshiftr1_div_2: "sint (sshiftr1 w) = sint w div 2"
   by transfer simp
 
 lemma shiftr_div_2n: "uint (shiftr w n) = uint w div 2 ^ n"
   apply (unfold shiftr_def)
   apply (induct n)
    apply simp
   apply (simp add: shiftr1_div_2 mult.commute zdiv_zmult2_eq [symmetric])
   done
 
 lemma sshiftr_div_2n: "sint (sshiftr w n) = sint w div 2 ^ n"
   apply transfer
   apply (rule bit_eqI)
   apply (simp add: bit_signed_take_bit_iff bit_drop_bit_eq min_def flip: drop_bit_eq_div)
   done
 
 lemma bit_bshiftr1_iff:
   \<open>bit (bshiftr1 b w) n \<longleftrightarrow> b \<and> n = LENGTH('a) - 1 \<or> bit w (Suc n)\<close>
   for w :: \<open>'a::len word\<close>
   apply transfer
   apply (simp add: bit_take_bit_iff flip: bit_Suc)
     apply (subst disjunctive_add)
    apply (auto simp add: bit_take_bit_iff bit_or_iff bit_exp_iff simp flip: bit_Suc)
   done
 
 
 subsubsection \<open>shift functions in terms of lists of bools\<close>
 
 lemma shiftl1_rev: "shiftl1 w = word_reverse (shiftr1 (word_reverse w))"
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_shiftl1_iff bit_word_reverse_iff bit_shiftr1_iff Suc_diff_Suc)
   done
 
 lemma shiftl_rev: "shiftl w n = word_reverse (shiftr (word_reverse w) n)"
   by (induct n) (auto simp add: shiftl_def shiftr_def shiftl1_rev)
 
 lemma rev_shiftl: "word_reverse w << n = word_reverse (w >> n)"
   by (simp add: shiftl_rev)
 
 lemma shiftr_rev: "w >> n = word_reverse (word_reverse w << n)"
   by (simp add: rev_shiftl)
 
 lemma rev_shiftr: "word_reverse w >> n = word_reverse (w << n)"
   by (simp add: shiftr_rev)
 
 lemma shiftl_numeral [simp]:
   \<open>numeral k << numeral l = (push_bit (numeral l) (numeral k) :: 'a::len word)\<close>
   by (fact shiftl_word_eq)
 
 lemma shiftl_zero_size: "size x \<le> n \<Longrightarrow> x << n = 0"
   for x :: "'a::len word"
   apply transfer
   apply (simp add: take_bit_push_bit)
   done
 
 \<comment> \<open>note -- the following results use \<open>'a::len word < number_ring\<close>\<close>
 
 lemma shiftl1_2t: "shiftl1 w = 2 * w"
   for w :: "'a::len word"
   by (simp add: shiftl1_eq wi_hom_mult [symmetric])
 
 lemma shiftl1_p: "shiftl1 w = w + w"
   for w :: "'a::len word"
   by (simp add: shiftl1_2t)
 
 lemma shiftl_t2n: "shiftl w n = 2 ^ n * w"
   for w :: "'a::len word"
   by (induct n) (auto simp: shiftl_def shiftl1_2t)
 
 lemma shiftr1_bintr [simp]:
   "(shiftr1 (numeral w) :: 'a::len word) =
     word_of_int (bin_rest (take_bit (LENGTH('a)) (numeral w)))"
   by transfer simp
 
 lemma sshiftr1_sbintr [simp]:
   "(sshiftr1 (numeral w) :: 'a::len word) =
     word_of_int (bin_rest (signed_take_bit (LENGTH('a) - 1) (numeral w)))"
   by transfer simp
 
 text \<open>TODO: rules for \<^term>\<open>- (numeral n)\<close>\<close>
 
 lemma drop_bit_word_numeral [simp]:
   \<open>drop_bit (numeral n) (numeral k) =
     (word_of_int (drop_bit (numeral n) (take_bit LENGTH('a) (numeral k))) :: 'a::len word)\<close>
   by transfer simp
 
 lemma shiftr_numeral [simp]:
   \<open>(numeral k >> numeral n :: 'a::len word) = drop_bit (numeral n) (numeral k)\<close>
   by (fact shiftr_word_eq)
 
 lemma sshiftr_numeral [simp]:
   \<open>(numeral k >>> numeral n :: 'a::len word) =
     word_of_int (drop_bit (numeral n) (signed_take_bit (LENGTH('a) - 1) (numeral k)))\<close>
   apply (rule word_eqI)
   apply (cases \<open>LENGTH('a)\<close>)
    apply (simp_all add: word_size bit_drop_bit_eq nth_sshiftr nth_sbintr not_le not_less less_Suc_eq_le ac_simps)
   done
 
 lemma zip_replicate: "n \<ge> length ys \<Longrightarrow> zip (replicate n x) ys = map (\<lambda>y. (x, y)) ys"
   apply (induct ys arbitrary: n)
    apply simp_all
   apply (case_tac n)
    apply simp_all
   done
 
 lemma align_lem_or [rule_format] :
   "\<forall>x m. length x = n + m \<longrightarrow> length y = n + m \<longrightarrow>
     drop m x = replicate n False \<longrightarrow> take m y = replicate m False \<longrightarrow>
     map2 (|) x y = take m x @ drop m y"
   apply (induct y)
    apply force
   apply clarsimp
   apply (case_tac x)
    apply force
   apply (case_tac m)
    apply auto
   apply (drule_tac t="length xs" for xs in sym)
   apply (auto simp: zip_replicate o_def)
   done
 
 lemma align_lem_and [rule_format] :
   "\<forall>x m. length x = n + m \<longrightarrow> length y = n + m \<longrightarrow>
     drop m x = replicate n False \<longrightarrow> take m y = replicate m False \<longrightarrow>
     map2 (\<and>) x y = replicate (n + m) False"
   apply (induct y)
    apply force
   apply clarsimp
   apply (case_tac x)
    apply force
   apply (case_tac m)
   apply auto
   apply (drule_tac t="length xs" for xs in sym)
   apply (auto simp: zip_replicate o_def map_replicate_const)
   done
 
 
 subsubsection \<open>Mask\<close>
 
 lemma minus_1_eq_mask:
   \<open>- 1 = (mask LENGTH('a) :: 'a::len word)\<close>
   by (rule bit_eqI) (simp add: bit_exp_iff bit_mask_iff exp_eq_zero_iff)
 
 lemma mask_eq_decr_exp:
   \<open>mask n = 2 ^ n - (1 :: 'a::len word)\<close>
   by (fact mask_eq_exp_minus_1)
 
 lemma mask_Suc_rec:
   \<open>mask (Suc n) = 2 * mask n + (1 :: 'a::len word)\<close>
   by (simp add: mask_eq_exp_minus_1)
 
 context
 begin
 
 qualified lemma bit_mask_iff:
   \<open>bit (mask m :: 'a::len word) n \<longleftrightarrow> n < min LENGTH('a) m\<close>
   by (simp add: bit_mask_iff exp_eq_zero_iff not_le)
 
 end
 
 lemma nth_mask [simp]:
   \<open>(mask n :: 'a::len word) !! i \<longleftrightarrow> i < n \<and> i < size (mask n :: 'a word)\<close>
   by (auto simp add: test_bit_word_eq word_size Word.bit_mask_iff)
 
 lemma mask_bin: "mask n = word_of_int (take_bit n (- 1))"
   by (auto simp add: nth_bintr word_size intro: word_eqI)
 
 lemma and_mask_bintr: "w AND mask n = word_of_int (take_bit n (uint w))"
   apply (rule word_eqI)
   apply (simp add: nth_bintr word_size word_ops_nth_size)
   apply (auto simp add: test_bit_bin)
   done
 
 lemma and_mask_wi: "word_of_int i AND mask n = word_of_int (take_bit n i)"
   by (auto simp add: nth_bintr word_size word_ops_nth_size word_eq_iff)
 
 lemma and_mask_wi':
   "word_of_int i AND mask n = (word_of_int (take_bit (min LENGTH('a) n) i) :: 'a::len word)"
   by (auto simp add: nth_bintr word_size word_ops_nth_size word_eq_iff)
 
 lemma and_mask_no: "numeral i AND mask n = word_of_int (take_bit n (numeral i))"
   unfolding word_numeral_alt by (rule and_mask_wi)
 
 lemma and_mask_mod_2p: "w AND mask n = word_of_int (uint w mod 2 ^ n)"
   by (simp only: and_mask_bintr take_bit_eq_mod)
 
 lemma uint_mask_eq:
   \<open>uint (mask n :: 'a::len word) = mask (min LENGTH('a) n)\<close>
   by transfer simp
 
 lemma and_mask_lt_2p: "uint (w AND mask n) < 2 ^ n"
   apply (simp add: uint_and uint_mask_eq)
   apply (rule AND_upper2'')
   apply simp
   apply (auto simp add: mask_eq_exp_minus_1 min_def power_add algebra_simps dest!: le_Suc_ex)
   apply (metis add_minus_cancel le_add2 one_le_numeral power_add power_increasing uminus_add_conv_diff zle_diff1_eq)
   done
 
 lemma eq_mod_iff: "0 < n \<Longrightarrow> b = b mod n \<longleftrightarrow> 0 \<le> b \<and> b < n"
   for b n :: int
   by auto (metis pos_mod_conj)+
 
 lemma mask_eq_iff: "w AND mask n = w \<longleftrightarrow> uint w < 2 ^ n"
   apply (auto simp flip: take_bit_eq_mask)
    apply (metis take_bit_int_eq_self_iff uint_take_bit_eq)
   apply (simp add: take_bit_int_eq_self unsigned_take_bit_eq word_uint_eqI)
   done
 
 lemma and_mask_dvd: "2 ^ n dvd uint w \<longleftrightarrow> w AND mask n = 0"
   by (simp flip: take_bit_eq_mask take_bit_eq_mod unsigned_take_bit_eq add: dvd_eq_mod_eq_0 uint_0_iff)
 
 lemma and_mask_dvd_nat: "2 ^ n dvd unat w \<longleftrightarrow> w AND mask n = 0"
   by (simp flip: take_bit_eq_mask take_bit_eq_mod unsigned_take_bit_eq add: dvd_eq_mod_eq_0 unat_0_iff uint_0_iff)
 
 lemma word_2p_lem: "n < size w \<Longrightarrow> w < 2 ^ n = (uint w < 2 ^ n)"
   for w :: "'a::len word"
   by transfer simp
 
 lemma less_mask_eq: "x < 2 ^ n \<Longrightarrow> x AND mask n = x"
   for x :: "'a::len word"
   apply (simp add: and_mask_bintr)
   apply transfer
   apply (simp add: ac_simps)
   apply (auto simp add: min_def)
   apply (metis bintrunc_bintrunc_ge mod_pos_pos_trivial mult.commute mult.left_neutral mult_zero_left not_le of_bool_def take_bit_eq_mod take_bit_nonnegative)
   done
 
 lemmas mask_eq_iff_w2p = trans [OF mask_eq_iff word_2p_lem [symmetric]]
 
 lemmas and_mask_less' = iffD2 [OF word_2p_lem and_mask_lt_2p, simplified word_size]
 
 lemma and_mask_less_size: "n < size x \<Longrightarrow> x AND mask n < 2 ^ n"
   for x :: \<open>'a::len word\<close>
   unfolding word_size by (erule and_mask_less')
 
 lemma word_mod_2p_is_mask [OF refl]: "c = 2 ^ n \<Longrightarrow> c > 0 \<Longrightarrow> x mod c = x AND mask n"
   for c x :: "'a::len word"
   by (auto simp: word_mod_def uint_2p and_mask_mod_2p)
 
 lemma mask_eqs:
   "(a AND mask n) + b AND mask n = a + b AND mask n"
   "a + (b AND mask n) AND mask n = a + b AND mask n"
   "(a AND mask n) - b AND mask n = a - b AND mask n"
   "a - (b AND mask n) AND mask n = a - b AND mask n"
   "a * (b AND mask n) AND mask n = a * b AND mask n"
   "(b AND mask n) * a AND mask n = b * a AND mask n"
   "(a AND mask n) + (b AND mask n) AND mask n = a + b AND mask n"
   "(a AND mask n) - (b AND mask n) AND mask n = a - b AND mask n"
   "(a AND mask n) * (b AND mask n) AND mask n = a * b AND mask n"
   "- (a AND mask n) AND mask n = - a AND mask n"
   "word_succ (a AND mask n) AND mask n = word_succ a AND mask n"
   "word_pred (a AND mask n) AND mask n = word_pred a AND mask n"
   using word_of_int_Ex [where x=a] word_of_int_Ex [where x=b]
              apply (auto simp flip: take_bit_eq_mask)
              apply transfer
              apply (simp add: take_bit_eq_mod mod_simps)
             apply transfer
             apply (simp add: take_bit_eq_mod mod_simps)
            apply transfer
            apply (simp add: take_bit_eq_mod mod_simps)
           apply transfer
           apply (simp add: take_bit_eq_mod mod_simps)
          apply transfer
          apply (simp add: take_bit_eq_mod mod_simps)
         apply transfer
         apply (simp add: take_bit_eq_mod mod_simps)
        apply transfer
        apply (simp add: take_bit_eq_mod mod_simps)
       apply transfer
       apply (simp add: take_bit_eq_mod mod_simps)
      apply transfer
      apply (simp add: take_bit_eq_mod mod_simps)
     apply transfer
     apply (simp add: take_bit_eq_mod mod_simps)
    apply transfer
    apply (simp add: take_bit_eq_mod mod_simps)
   apply transfer
   apply (simp add: take_bit_eq_mod mod_simps)
   done
 
 lemma mask_power_eq: "(x AND mask n) ^ k AND mask n = x ^ k AND mask n"
   for x :: \<open>'a::len word\<close>
   using word_of_int_Ex [where x=x]
   apply (auto simp flip: take_bit_eq_mask)
   apply transfer
   apply (simp add: take_bit_eq_mod mod_simps)
   done
 
 lemma mask_full [simp]: "mask LENGTH('a) = (- 1 :: 'a::len word)"
   by transfer (simp add: take_bit_minus_one_eq_mask)
 
 
 subsubsection \<open>Slices\<close>
 
 definition slice1 :: \<open>nat \<Rightarrow> 'a::len word \<Rightarrow> 'b::len word\<close>
   where \<open>slice1 n w = (if n < LENGTH('a)
     then ucast (drop_bit (LENGTH('a) - n) w)
     else push_bit (n - LENGTH('a)) (ucast w))\<close>
 
 lemma bit_slice1_iff:
   \<open>bit (slice1 m w :: 'b::len word) n \<longleftrightarrow> m - LENGTH('a) \<le> n \<and> n < min LENGTH('b) m
     \<and> bit w (n + (LENGTH('a) - m) - (m - LENGTH('a)))\<close>
   for w :: \<open>'a::len word\<close>
   by (auto simp add: slice1_def bit_ucast_iff bit_drop_bit_eq bit_push_bit_iff exp_eq_zero_iff not_less not_le ac_simps
     dest: bit_imp_le_length)
 
 definition slice :: \<open>nat \<Rightarrow> 'a::len word \<Rightarrow> 'b::len word\<close>
   where \<open>slice n = slice1 (LENGTH('a) - n)\<close>
 
 lemma bit_slice_iff:
   \<open>bit (slice m w :: 'b::len word) n \<longleftrightarrow> n < min LENGTH('b) (LENGTH('a) - m) \<and> bit w (n + LENGTH('a) - (LENGTH('a) - m))\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: slice_def word_size bit_slice1_iff)
 
 lemma slice1_0 [simp] : "slice1 n 0 = 0"
   unfolding slice1_def by simp
 
 lemma slice_0 [simp] : "slice n 0 = 0"
   unfolding slice_def by auto
 
 lemma slice_shiftr: "slice n w = ucast (w >> n)"
   apply (rule bit_word_eqI)
   apply (cases \<open>n \<le> LENGTH('b)\<close>)
    apply (auto simp add: bit_slice_iff bit_ucast_iff bit_shiftr_word_iff ac_simps
     dest: bit_imp_le_length)
   done
 
 lemma nth_slice: "(slice n w :: 'a::len word) !! m = (w !! (m + n) \<and> m < LENGTH('a))"
   by (simp add: slice_shiftr nth_ucast nth_shiftr)
 
 lemma ucast_slice1: "ucast w = slice1 (size w) w"
   apply (simp add: slice1_def)
   apply transfer
   apply simp
   done
 
 lemma ucast_slice: "ucast w = slice 0 w"
   by (simp add: slice_def slice1_def)
 
 lemma slice_id: "slice 0 t = t"
   by (simp only: ucast_slice [symmetric] ucast_id)
 
 lemma rev_slice1:
   \<open>slice1 n (word_reverse w :: 'b::len word) = word_reverse (slice1 k w :: 'a::len word)\<close>
   if \<open>n + k = LENGTH('a) + LENGTH('b)\<close>
 proof (rule bit_word_eqI)
   fix m
   assume *: \<open>m < LENGTH('a)\<close>
   from that have **: \<open>LENGTH('b) = n + k - LENGTH('a)\<close>
     by simp
   show \<open>bit (slice1 n (word_reverse w :: 'b word) :: 'a word) m \<longleftrightarrow> bit (word_reverse (slice1 k w :: 'a word)) m\<close>
     apply (simp add: bit_slice1_iff bit_word_reverse_iff)
     using * **
     apply (cases \<open>n \<le> LENGTH('a)\<close>; cases \<open>k \<le> LENGTH('a)\<close>)
        apply auto
     done
 qed
 
 lemma rev_slice:
   "n + k + LENGTH('a::len) = LENGTH('b::len) \<Longrightarrow>
     slice n (word_reverse (w::'b word)) = word_reverse (slice k w :: 'a word)"
   apply (unfold slice_def word_size)
   apply (rule rev_slice1)
   apply arith
   done
 
 
 subsubsection \<open>Revcast\<close>
 
 definition revcast :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
   where \<open>revcast = slice1 LENGTH('b)\<close>
 
 lemma bit_revcast_iff:
   \<open>bit (revcast w :: 'b::len word) n \<longleftrightarrow> LENGTH('b) - LENGTH('a) \<le> n \<and> n < LENGTH('b)
     \<and> bit w (n + (LENGTH('a) - LENGTH('b)) - (LENGTH('b) - LENGTH('a)))\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: revcast_def bit_slice1_iff)
 
 lemma revcast_slice1 [OF refl]: "rc = revcast w \<Longrightarrow> slice1 (size rc) w = rc"
   by (simp add: revcast_def word_size)
 
 lemma revcast_rev_ucast [OF refl refl refl]:
   "cs = [rc, uc] \<Longrightarrow> rc = revcast (word_reverse w) \<Longrightarrow> uc = ucast w \<Longrightarrow>
     rc = word_reverse uc"
   apply auto
   apply (rule bit_word_eqI)
   apply (cases \<open>LENGTH('a) \<le> LENGTH('b)\<close>)
    apply (simp_all add: bit_revcast_iff bit_word_reverse_iff bit_ucast_iff not_le
      bit_imp_le_length)
   using bit_imp_le_length apply fastforce
   using bit_imp_le_length apply fastforce
   done
 
 lemma revcast_ucast: "revcast w = word_reverse (ucast (word_reverse w))"
   using revcast_rev_ucast [of "word_reverse w"] by simp
 
 lemma ucast_revcast: "ucast w = word_reverse (revcast (word_reverse w))"
   by (fact revcast_rev_ucast [THEN word_rev_gal'])
 
 lemma ucast_rev_revcast: "ucast (word_reverse w) = word_reverse (revcast w)"
   by (fact revcast_ucast [THEN word_rev_gal'])
 
 
 text "linking revcast and cast via shift"
 
 lemmas wsst_TYs = source_size target_size word_size
 
 lemma revcast_down_uu [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = ucast (w >> n)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_revcast_iff bit_ucast_iff bit_shiftr_word_iff ac_simps)
   done
 
 lemma revcast_down_us [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = ucast (w >>> n)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_revcast_iff bit_ucast_iff bit_sshiftr_word_iff ac_simps)
   done
 
 lemma revcast_down_su [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = scast (w >> n)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_revcast_iff bit_word_scast_iff bit_shiftr_word_iff ac_simps)
   done
 
 lemma revcast_down_ss [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = scast (w >>> n)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_revcast_iff bit_word_scast_iff bit_sshiftr_word_iff ac_simps)
   done
 
 lemma cast_down_rev [OF refl]:
   "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> uc w = revcast (w << n)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_revcast_iff bit_word_ucast_iff bit_shiftl_word_iff)
   done
 
 lemma revcast_up [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc + n = target_size rc \<Longrightarrow>
     rc w = (ucast w :: 'a::len word) << n"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_revcast_iff bit_word_ucast_iff bit_shiftl_word_iff)
   apply auto
   apply (metis add.commute add_diff_cancel_right)
   apply (metis diff_add_inverse2 diff_diff_add)
   done
 
 lemmas rc1 = revcast_up [THEN
   revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]]
 lemmas rc2 = revcast_down_uu [THEN
   revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]]
 
 lemmas ucast_up =
  rc1 [simplified rev_shiftr [symmetric] revcast_ucast [symmetric]]
 lemmas ucast_down =
   rc2 [simplified rev_shiftr revcast_ucast [symmetric]]
 
 lemmas sym_notr =
   not_iff [THEN iffD2, THEN not_sym, THEN not_iff [THEN iffD1]]
 
 \<comment> \<open>problem posed by TPHOLs referee:
       criterion for overflow of addition of signed integers\<close>
 
 lemma sofl_test:
   \<open>sint x + sint y = sint (x + y) \<longleftrightarrow>
     (x + y XOR x) AND (x + y XOR y) >> (size x - 1) = 0\<close>
   for x y :: \<open>'a::len word\<close>
 proof -
   obtain n where n: \<open>LENGTH('a) = Suc n\<close>
     by (cases \<open>LENGTH('a)\<close>) simp_all
   have *: \<open>sint x + sint y + 2 ^ Suc n > signed_take_bit n (sint x + sint y) \<Longrightarrow> sint x + sint y \<ge> - (2 ^ n)\<close>
     \<open>signed_take_bit n (sint x + sint y) > sint x + sint y - 2 ^ Suc n \<Longrightarrow> 2 ^ n > sint x + sint y\<close>
     using signed_take_bit_int_greater_eq [of \<open>sint x + sint y\<close> n] signed_take_bit_int_less_eq [of n \<open>sint x + sint y\<close>]
     by (auto intro: ccontr)
   have \<open>sint x + sint y = sint (x + y) \<longleftrightarrow>
     (sint (x + y) < 0 \<longleftrightarrow> sint x < 0) \<or>
     (sint (x + y) < 0 \<longleftrightarrow> sint y < 0)\<close>
     using sint_less [of x] sint_greater_eq [of x] sint_less [of y] sint_greater_eq [of y]
     signed_take_bit_int_eq_self [of \<open>LENGTH('a) - 1\<close> \<open>sint x + sint y\<close>]
     apply (auto simp add: not_less)
        apply (unfold sint_word_ariths)
        apply (subst signed_take_bit_int_eq_self)
          prefer 4
        apply (subst signed_take_bit_int_eq_self)
          prefer 7
        apply (subst signed_take_bit_int_eq_self)
          prefer 10
              apply (subst signed_take_bit_int_eq_self)
        apply (auto simp add: signed_take_bit_int_eq_self signed_take_bit_eq_take_bit_minus take_bit_Suc_from_most n not_less intro!: *)
     done
   then show ?thesis
     apply (simp only: One_nat_def word_size shiftr_word_eq drop_bit_eq_zero_iff_not_bit_last bit_and_iff bit_xor_iff)
     apply (simp add: bit_last_iff)
     done
 qed
 
 lemma shiftr_zero_size: "size x \<le> n \<Longrightarrow> x >> n = 0"
   for x :: "'a :: len word"
   by (rule word_eqI) (auto simp add: nth_shiftr dest: test_bit_size)
 
 
 subsection \<open>Split and cat\<close>
 
 lemmas word_split_bin' = word_split_def
 lemmas word_cat_bin' = word_cat_eq
 
 lemma word_rsplit_no:
   "(word_rsplit (numeral bin :: 'b::len word) :: 'a word list) =
     map word_of_int (bin_rsplit (LENGTH('a::len))
       (LENGTH('b), take_bit (LENGTH('b)) (numeral bin)))"
   by (simp add: word_rsplit_def of_nat_take_bit)
 
 lemmas word_rsplit_no_cl [simp] = word_rsplit_no
   [unfolded bin_rsplitl_def bin_rsplit_l [symmetric]]
 
 lemma test_bit_cat [OF refl]:
   "wc = word_cat a b \<Longrightarrow> wc !! n = (n < size wc \<and>
     (if n < size b then b !! n else a !! (n - size b)))"
   apply (simp add: word_size not_less; transfer)
        apply (auto simp add: bit_concat_bit_iff bit_take_bit_iff)
   done
 
 lemma split_uint_lem: "bin_split n (uint w) = (a, b) \<Longrightarrow>
     a = take_bit (LENGTH('a) - n) a \<and> b = take_bit (LENGTH('a)) b"
   for w :: "'a::len word"
   by transfer (simp add: drop_bit_take_bit ac_simps)
 
 \<comment> \<open>keep quantifiers for use in simplification\<close>
 lemma test_bit_split':
   "word_split c = (a, b) \<longrightarrow>
     (\<forall>n m.
       b !! n = (n < size b \<and> c !! n) \<and>
       a !! m = (m < size a \<and> c !! (m + size b)))"
   apply (unfold word_split_bin' test_bit_bin)
   apply (clarify)
   apply simp
   apply (erule conjE)
   apply (drule sym)
   apply (drule sym)
   apply (simp add: bit_take_bit_iff bit_drop_bit_eq)
   apply transfer
   apply (simp add: bit_take_bit_iff ac_simps)
   done
 
 lemma test_bit_split:
   "word_split c = (a, b) \<Longrightarrow>
     (\<forall>n::nat. b !! n \<longleftrightarrow> n < size b \<and> c !! n) \<and>
     (\<forall>m::nat. a !! m \<longleftrightarrow> m < size a \<and> c !! (m + size b))"
   by (simp add: test_bit_split')
 
 lemma test_bit_split_eq:
   "word_split c = (a, b) \<longleftrightarrow>
     ((\<forall>n::nat. b !! n = (n < size b \<and> c !! n)) \<and>
      (\<forall>m::nat. a !! m = (m < size a \<and> c !! (m + size b))))"
   apply (rule_tac iffI)
    apply (rule_tac conjI)
     apply (erule test_bit_split [THEN conjunct1])
    apply (erule test_bit_split [THEN conjunct2])
   apply (case_tac "word_split c")
   apply (frule test_bit_split)
   apply (erule trans)
   apply (fastforce intro!: word_eqI simp add: word_size)
   done
 
 \<comment> \<open>this odd result is analogous to \<open>ucast_id\<close>,
       result to the length given by the result type\<close>
 
 lemma word_cat_id: "word_cat a b = b"
   by transfer simp
 
 \<comment> \<open>limited hom result\<close>
 lemma word_cat_hom:
   "LENGTH('a::len) \<le> LENGTH('b::len) + LENGTH('c::len) \<Longrightarrow>
     (word_cat (word_of_int w :: 'b word) (b :: 'c word) :: 'a word) =
     word_of_int (bin_cat w (size b) (uint b))"
   apply transfer
   using bintr_cat by auto
 
 lemma word_cat_split_alt: "size w \<le> size u + size v \<Longrightarrow> word_split w = (u, v) \<Longrightarrow> word_cat u v = w"
   apply (rule word_eqI)
   apply (drule test_bit_split)
   apply (clarsimp simp add : test_bit_cat word_size)
   apply safe
   apply arith
   done
 
 lemmas word_cat_split_size = sym [THEN [2] word_cat_split_alt [symmetric]]
 
 
 subsubsection \<open>Split and slice\<close>
 
 lemma split_slices: "word_split w = (u, v) \<Longrightarrow> u = slice (size v) w \<and> v = slice 0 w"
   apply (drule test_bit_split)
   apply (rule conjI)
    apply (rule word_eqI, clarsimp simp: nth_slice word_size)+
   done
 
 lemma slice_cat1 [OF refl]:
   "wc = word_cat a b \<Longrightarrow> size wc >= size a + size b \<Longrightarrow> slice (size b) wc = a"
   apply safe
   apply (rule word_eqI)
   apply (simp add: nth_slice test_bit_cat word_size)
   done
 
 lemmas slice_cat2 = trans [OF slice_id word_cat_id]
 
 lemma cat_slices:
   "a = slice n c \<Longrightarrow> b = slice 0 c \<Longrightarrow> n = size b \<Longrightarrow>
     size a + size b >= size c \<Longrightarrow> word_cat a b = c"
   apply safe
   apply (rule word_eqI)
   apply (simp add: nth_slice test_bit_cat word_size)
   apply safe
   apply arith
   done
 
 lemma word_split_cat_alt:
   "w = word_cat u v \<Longrightarrow> size u + size v \<le> size w \<Longrightarrow> word_split w = (u, v)"
   apply (case_tac "word_split w")
   apply (rule trans, assumption)
   apply (drule test_bit_split)
   apply safe
    apply (rule word_eqI, clarsimp simp: test_bit_cat word_size)+
   done
 
 text \<open>
   This odd result arises from the fact that the statement of the
   result implies that the decoded words are of the same type,
   and therefore of the same length, as the original word.\<close>
 
 lemma word_rsplit_same: "word_rsplit w = [w]"
   apply (simp add: word_rsplit_def bin_rsplit_all)
   apply transfer
   apply simp
   done
 
 lemma word_rsplit_empty_iff_size: "word_rsplit w = [] \<longleftrightarrow> size w = 0"
   by (simp add: word_rsplit_def bin_rsplit_def word_size bin_rsplit_aux_simp_alt Let_def
       split: prod.split)
 
 lemma test_bit_rsplit:
   "sw = word_rsplit w \<Longrightarrow> m < size (hd sw) \<Longrightarrow>
     k < length sw \<Longrightarrow> (rev sw ! k) !! m = w !! (k * size (hd sw) + m)"
   for sw :: "'a::len word list"
   apply (unfold word_rsplit_def word_test_bit_def)
   apply (rule trans)
    apply (rule_tac f = "\<lambda>x. bin_nth x m" in arg_cong)
    apply (rule nth_map [symmetric])
    apply simp
   apply (rule bin_nth_rsplit)
      apply simp_all
   apply (simp add : word_size rev_map)
   apply (rule trans)
    defer
    apply (rule map_ident [THEN fun_cong])
   apply (rule refl [THEN map_cong])
   apply simp
   using bin_rsplit_size_sign take_bit_int_eq_self_iff by blast
 
 lemma horner_sum_uint_exp_Cons_eq:
   \<open>horner_sum uint (2 ^ LENGTH('a)) (w # ws) =
     concat_bit LENGTH('a) (uint w) (horner_sum uint (2 ^ LENGTH('a)) ws)\<close>
   for ws :: \<open>'a::len word list\<close>
   apply (simp add: concat_bit_eq push_bit_eq_mult)
   apply transfer
   apply simp
   done
 
 lemma bit_horner_sum_uint_exp_iff:
   \<open>bit (horner_sum uint (2 ^ LENGTH('a)) ws) n \<longleftrightarrow>
     n div LENGTH('a) < length ws \<and> bit (ws ! (n div LENGTH('a))) (n mod LENGTH('a))\<close>
   for ws :: \<open>'a::len word list\<close>
 proof (induction ws arbitrary: n)
   case Nil
   then show ?case
     by simp
 next
   case (Cons w ws)
   then show ?case
     by (cases \<open>n \<ge> LENGTH('a)\<close>)
       (simp_all only: horner_sum_uint_exp_Cons_eq, simp_all add: bit_concat_bit_iff le_div_geq le_mod_geq bit_uint_iff Cons)
 qed
 
 lemma test_bit_rcat:
   "sw = size (hd wl) \<Longrightarrow> rc = word_rcat wl \<Longrightarrow> rc !! n =
     (n < size rc \<and> n div sw < size wl \<and> (rev wl) ! (n div sw) !! (n mod sw))"
   for wl :: "'a::len word list"
   by (simp add: word_size word_rcat_def bin_rcat_def foldl_map rev_map bit_horner_sum_uint_exp_iff)
     (simp add: test_bit_eq_bit)
 
 lemmas test_bit_cong = arg_cong [where f = "test_bit", THEN fun_cong]
 
 lemma test_bit_rsplit_alt:
   \<open>(word_rsplit w  :: 'b::len word list) ! i !! m \<longleftrightarrow>
     w !! ((length (word_rsplit w :: 'b::len word list) - Suc i) * size (hd (word_rsplit w :: 'b::len word list)) + m)\<close>
   if \<open>i < length (word_rsplit w :: 'b::len word list)\<close> \<open>m < size (hd (word_rsplit w :: 'b::len word list))\<close> \<open>0 < length (word_rsplit w :: 'b::len word list)\<close>
   for w :: \<open>'a::len word\<close>
   apply (rule trans)
    apply (rule test_bit_cong)
    apply (rule rev_nth [of _ \<open>rev (word_rsplit w)\<close>, simplified rev_rev_ident])
   apply simp
    apply (rule that(1))
   apply simp
   apply (rule test_bit_rsplit)
     apply (rule refl)
   apply (rule asm_rl)
    apply (rule that(2))
   apply (rule diff_Suc_less)
   apply (rule that(3))
   done
 
 lemma word_rsplit_len_indep [OF refl refl refl refl]:
   "[u,v] = p \<Longrightarrow> [su,sv] = q \<Longrightarrow> word_rsplit u = su \<Longrightarrow>
     word_rsplit v = sv \<Longrightarrow> length su = length sv"
   by (auto simp: word_rsplit_def bin_rsplit_len_indep)
 
 lemma length_word_rsplit_size:
   "n = LENGTH('a::len) \<Longrightarrow>
     length (word_rsplit w :: 'a word list) \<le> m \<longleftrightarrow> size w \<le> m * n"
   by (auto simp: word_rsplit_def word_size bin_rsplit_len_le)
 
 lemmas length_word_rsplit_lt_size =
   length_word_rsplit_size [unfolded Not_eq_iff linorder_not_less [symmetric]]
 
 lemma length_word_rsplit_exp_size:
   "n = LENGTH('a::len) \<Longrightarrow>
     length (word_rsplit w :: 'a word list) = (size w + n - 1) div n"
   by (auto simp: word_rsplit_def word_size bin_rsplit_len)
 
 lemma length_word_rsplit_even_size:
   "n = LENGTH('a::len) \<Longrightarrow> size w = m * n \<Longrightarrow>
     length (word_rsplit w :: 'a word list) = m"
   by (cases \<open>LENGTH('a)\<close>) (simp_all add: length_word_rsplit_exp_size div_nat_eqI)
 
 lemmas length_word_rsplit_exp_size' = refl [THEN length_word_rsplit_exp_size]
 
 \<comment> \<open>alternative proof of \<open>word_rcat_rsplit\<close>\<close>
 lemmas tdle = times_div_less_eq_dividend
 lemmas dtle = xtrans(4) [OF tdle mult.commute]
 
 lemma word_rcat_rsplit: "word_rcat (word_rsplit w) = w"
   apply (rule word_eqI)
   apply (clarsimp simp: test_bit_rcat word_size)
   apply (subst refl [THEN test_bit_rsplit])
     apply (simp_all add: word_size
       refl [THEN length_word_rsplit_size [simplified not_less [symmetric], simplified]])
    apply safe
    apply (erule xtrans(7), rule dtle)+
   done
 
 lemma size_word_rsplit_rcat_size:
   "word_rcat ws = frcw \<Longrightarrow> size frcw = length ws * LENGTH('a)
     \<Longrightarrow> length (word_rsplit frcw::'a word list) = length ws"
   for ws :: "'a::len word list" and frcw :: "'b::len word"
   by (cases \<open>LENGTH('a)\<close>) (simp_all add: word_size length_word_rsplit_exp_size' div_nat_eqI)
 
 lemma msrevs:
   "0 < n \<Longrightarrow> (k * n + m) div n = m div n + k"
   "(k * n + m) mod n = m mod n"
   for n :: nat
   by (auto simp: add.commute)
 
 lemma word_rsplit_rcat_size [OF refl]:
   "word_rcat ws = frcw \<Longrightarrow>
     size frcw = length ws * LENGTH('a) \<Longrightarrow> word_rsplit frcw = ws"
   for ws :: "'a::len word list"
   apply (frule size_word_rsplit_rcat_size, assumption)
   apply (clarsimp simp add : word_size)
   apply (rule nth_equalityI, assumption)
   apply clarsimp
   apply (rule word_eqI [rule_format])
   apply (rule trans)
    apply (rule test_bit_rsplit_alt)
      apply (clarsimp simp: word_size)+
   apply (rule trans)
    apply (rule test_bit_rcat [OF refl refl])
   apply (simp add: word_size)
   apply (subst rev_nth)
    apply arith
   apply (simp add: le0 [THEN [2] xtrans(7), THEN diff_Suc_less])
   apply safe
   apply (simp add: diff_mult_distrib)
    apply (cases "size ws")
     apply simp_all
   done
 
 
 subsection \<open>Rotation\<close>
 
 lemma word_rotr_word_rotr_eq:
   \<open>word_rotr m (word_rotr n w) = word_rotr (m + n) w\<close>
   by (rule bit_word_eqI) (simp add: bit_word_rotr_iff ac_simps mod_add_right_eq)
 
 lemma word_rot_rl [simp]:
   \<open>word_rotl k (word_rotr k v) = v\<close>
   apply (rule bit_word_eqI)
   apply (simp add: word_rotl_eq_word_rotr word_rotr_word_rotr_eq bit_word_rotr_iff algebra_simps)
   apply (auto dest: bit_imp_le_length)
    apply (metis (no_types, lifting) add.right_neutral add_diff_cancel_right' div_mult_mod_eq mod_add_right_eq mod_add_self2 mod_if mod_mult_self2_is_0)
   apply (metis (no_types, lifting) add.right_neutral add_diff_cancel_right' div_mult_mod_eq mod_add_right_eq mod_add_self2 mod_less mod_mult_self2_is_0)
   done
 
 lemma word_rot_lr [simp]:
   \<open>word_rotr k (word_rotl k v) = v\<close>
   apply (rule bit_word_eqI)
   apply (simp add: word_rotl_eq_word_rotr word_rotr_word_rotr_eq bit_word_rotr_iff algebra_simps)
   apply (auto dest: bit_imp_le_length)
    apply (metis (no_types, lifting) add.right_neutral add_diff_cancel_right' div_mult_mod_eq mod_add_right_eq mod_add_self2 mod_if mod_mult_self2_is_0)
   apply (metis (no_types, lifting) add.right_neutral add_diff_cancel_right' div_mult_mod_eq mod_add_right_eq mod_add_self2 mod_less mod_mult_self2_is_0)
   done
 
 lemma word_rot_gal:
   \<open>word_rotr n v = w \<longleftrightarrow> word_rotl n w = v\<close>
   by auto
 
 lemma word_rot_gal':
   \<open>w = word_rotr n v \<longleftrightarrow> v = word_rotl n w\<close>
   by auto
 
 lemma word_rotr_rev:
   \<open>word_rotr n w = word_reverse (word_rotl n (word_reverse w))\<close>
 proof (rule bit_word_eqI)
   fix m
   assume \<open>m < LENGTH('a)\<close>
   moreover have \<open>1 +
     ((int m + int n mod int LENGTH('a)) mod int LENGTH('a) +
      ((int LENGTH('a) * 2) mod int LENGTH('a) - (1 + (int m + int n mod int LENGTH('a)))) mod int LENGTH('a)) =
     int LENGTH('a)\<close>
     apply (cases \<open>(1 + (int m + int n mod int LENGTH('a))) mod
          int LENGTH('a) = 0\<close>)
     using zmod_zminus1_eq_if [of \<open>1 + (int m + int n mod int LENGTH('a))\<close> \<open>int LENGTH('a)\<close>]
     apply simp_all
      apply (auto simp add: algebra_simps)
      apply (simp add: minus_equation_iff [of \<open>int m\<close>])
      apply (drule sym [of _ \<open>int m\<close>])
     apply simp
     apply (metis add.commute add_minus_cancel diff_minus_eq_add len_gt_0 less_imp_of_nat_less less_nat_zero_code mod_mult_self3 of_nat_gt_0 zmod_minus1)
     apply (metis (no_types, hide_lams) Abs_fnat_hom_add less_not_refl mod_Suc of_nat_Suc of_nat_gt_0 of_nat_mod)
     done
   then have \<open>int ((m + n) mod LENGTH('a)) =
     int (LENGTH('a) - Suc ((LENGTH('a) - Suc m + LENGTH('a) - n mod LENGTH('a)) mod LENGTH('a)))\<close>
     using \<open>m < LENGTH('a)\<close>
     by (simp only: of_nat_mod mod_simps)
       (simp add: of_nat_diff of_nat_mod Suc_le_eq add_less_mono algebra_simps mod_simps)
   then have \<open>(m + n) mod LENGTH('a) =
     LENGTH('a) - Suc ((LENGTH('a) - Suc m + LENGTH('a) - n mod LENGTH('a)) mod LENGTH('a))\<close>
     by simp
   ultimately show \<open>bit (word_rotr n w) m \<longleftrightarrow> bit (word_reverse (word_rotl n (word_reverse w))) m\<close>
     by (simp add: word_rotl_eq_word_rotr bit_word_rotr_iff bit_word_reverse_iff)
 qed
 
 lemma word_roti_0 [simp]: "word_roti 0 w = w"
   by transfer simp
 
 lemma word_roti_add: "word_roti (m + n) w = word_roti m (word_roti n w)"
   by (rule bit_word_eqI)
     (simp add: bit_word_roti_iff nat_less_iff mod_simps ac_simps)
 
 lemma word_roti_conv_mod':
   "word_roti n w = word_roti (n mod int (size w)) w"
   by transfer simp
 
 lemmas word_roti_conv_mod = word_roti_conv_mod' [unfolded word_size]
 
 
 subsubsection \<open>"Word rotation commutes with bit-wise operations\<close>
 
 \<comment> \<open>using locale to not pollute lemma namespace\<close>
 locale word_rotate
 begin
 
 lemma word_rot_logs:
   "word_rotl n (NOT v) = NOT (word_rotl n v)"
   "word_rotr n (NOT v) = NOT (word_rotr n v)"
   "word_rotl n (x AND y) = word_rotl n x AND word_rotl n y"
   "word_rotr n (x AND y) = word_rotr n x AND word_rotr n y"
   "word_rotl n (x OR y) = word_rotl n x OR word_rotl n y"
   "word_rotr n (x OR y) = word_rotr n x OR word_rotr n y"
   "word_rotl n (x XOR y) = word_rotl n x XOR word_rotl n y"
   "word_rotr n (x XOR y) = word_rotr n x XOR word_rotr n y"
          apply (rule bit_word_eqI)
          apply (auto simp add: bit_word_rotl_iff bit_not_iff algebra_simps exp_eq_zero_iff not_le)
         apply (rule bit_word_eqI)
         apply (auto simp add: bit_word_rotr_iff bit_not_iff algebra_simps exp_eq_zero_iff not_le)
        apply (rule bit_word_eqI)
        apply (auto simp add: bit_word_rotl_iff bit_and_iff algebra_simps exp_eq_zero_iff not_le)
       apply (rule bit_word_eqI)
       apply (auto simp add: bit_word_rotr_iff bit_and_iff algebra_simps exp_eq_zero_iff not_le)
      apply (rule bit_word_eqI)
      apply (auto simp add: bit_word_rotl_iff bit_or_iff algebra_simps exp_eq_zero_iff not_le)
     apply (rule bit_word_eqI)
     apply (auto simp add: bit_word_rotr_iff bit_or_iff algebra_simps exp_eq_zero_iff not_le)
    apply (rule bit_word_eqI)
    apply (auto simp add: bit_word_rotl_iff bit_xor_iff algebra_simps exp_eq_zero_iff not_le)
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_word_rotr_iff bit_xor_iff algebra_simps exp_eq_zero_iff not_le)
   done
 
 end
 
 lemmas word_rot_logs = word_rotate.word_rot_logs
 
 lemma word_rotx_0 [simp] : "word_rotr i 0 = 0 \<and> word_rotl i 0 = 0"
   by transfer simp_all
 
 lemma word_roti_0' [simp] : "word_roti n 0 = 0"
   by transfer simp
 
 declare word_roti_eq_word_rotr_word_rotl [simp]
 
 
 subsection \<open>Maximum machine word\<close>
 
 lemma word_int_cases:
   fixes x :: "'a::len word"
   obtains n where "x = word_of_int n" and "0 \<le> n" and "n < 2^LENGTH('a)"
   by (rule that [of \<open>uint x\<close>]) simp_all
 
 lemma word_nat_cases [cases type: word]:
   fixes x :: "'a::len word"
   obtains n where "x = of_nat n" and "n < 2^LENGTH('a)"
   by (rule that [of \<open>unat x\<close>]) simp_all
 
 lemma max_word_max [intro!]: "n \<le> max_word"
   by (fact word_order.extremum)
 
 lemma word_of_int_2p_len: "word_of_int (2 ^ LENGTH('a)) = (0::'a::len word)"
   by simp
 
 lemma word_pow_0: "(2::'a::len word) ^ LENGTH('a) = 0"
   by (fact word_exp_length_eq_0)
 
 lemma max_word_wrap: "x + 1 = 0 \<Longrightarrow> x = max_word"
   by (simp add: eq_neg_iff_add_eq_0)
 
 lemma max_test_bit: "(max_word::'a::len word) !! n \<longleftrightarrow> n < LENGTH('a)"
   by (fact nth_minus1)
 
 lemma word_and_max: "x AND max_word = x"
   by (fact word_log_esimps)
 
 lemma word_or_max: "x OR max_word = max_word"
   by (fact word_log_esimps)
 
 lemma word_ao_dist2: "x AND (y OR z) = x AND y OR x AND z"
   for x y z :: "'a::len word"
   by (rule word_eqI) (auto simp add: word_ops_nth_size word_size)
 
 lemma word_oa_dist2: "x OR y AND z = (x OR y) AND (x OR z)"
   for x y z :: "'a::len word"
   by (rule word_eqI) (auto simp add: word_ops_nth_size word_size)
 
 lemma word_and_not [simp]: "x AND NOT x = 0"
   for x :: "'a::len word"
   by (rule word_eqI) (auto simp add: word_ops_nth_size word_size)
 
 lemma word_or_not [simp]: "x OR NOT x = max_word"
   by (rule word_eqI) (auto simp add: word_ops_nth_size word_size)
 
 lemma word_xor_and_or: "x XOR y = x AND NOT y OR NOT x AND y"
   for x y :: "'a::len word"
   by (rule word_eqI) (auto simp add: word_ops_nth_size word_size)
 
 lemma shiftr_x_0 [iff]: "x >> 0 = x"
   for x :: "'a::len word"
   by transfer simp
 
 lemma shiftl_x_0 [simp]: "x << 0 = x"
   for x :: "'a::len word"
   by (simp add: shiftl_t2n)
 
 lemma shiftl_1 [simp]: "(1::'a::len word) << n = 2^n"
   by (simp add: shiftl_t2n)
 
 lemma uint_lt_0 [simp]: "uint x < 0 = False"
   by (simp add: linorder_not_less)
 
 lemma shiftr1_1 [simp]: "shiftr1 (1::'a::len word) = 0"
   by transfer simp
 
 lemma shiftr_1[simp]: "(1::'a::len word) >> n = (if n = 0 then 1 else 0)"
   by (induct n) (auto simp: shiftr_def)
 
 lemma word_less_1 [simp]: "x < 1 \<longleftrightarrow> x = 0"
   for x :: "'a::len word"
   by (simp add: word_less_nat_alt unat_0_iff)
 
 lemma map_nth_0 [simp]: "map ((!!) (0::'a::len word)) xs = replicate (length xs) False"
   by (induct xs) auto
 
 lemma uint_plus_if_size:
   "uint (x + y) =
     (if uint x + uint y < 2^size x
      then uint x + uint y
      else uint x + uint y - 2^size x)"
   apply (simp only: word_arith_wis word_size uint_word_of_int_eq)
   apply (auto simp add: not_less take_bit_int_eq_self_iff)
   apply (metis not_less take_bit_eq_mod uint_plus_if' uint_word_ariths(1))
   done
 
 lemma unat_plus_if_size:
   "unat (x + y) =
     (if unat x + unat y < 2^size x
      then unat x + unat y
      else unat x + unat y - 2^size x)"
   for x y :: "'a::len word"
   apply (subst word_arith_nat_defs)
   apply (subst unat_of_nat)
   apply (auto simp add: not_less word_size)
   apply (metis not_le unat_plus_if' unat_word_ariths(1))
   done
 
 lemma word_neq_0_conv: "w \<noteq> 0 \<longleftrightarrow> 0 < w"
   for w :: "'a::len word"
   by (fact word_coorder.not_eq_extremum)
 
 lemma max_lt: "unat (max a b div c) = unat (max a b) div unat c"
   for c :: "'a::len word"
   by (fact unat_div)
 
 lemma uint_sub_if_size:
   "uint (x - y) =
     (if uint y \<le> uint x
      then uint x - uint y
      else uint x - uint y + 2^size x)"
   apply (simp only: word_arith_wis word_size uint_word_of_int_eq)
   apply (auto simp add: take_bit_int_eq_self_iff not_le)
   apply (metis not_less uint_sub_if' uint_word_arith_bintrs(2))
   done
 
 lemma unat_sub:
   \<open>unat (a - b) = unat a - unat b\<close>
   if \<open>b \<le> a\<close>
 proof -
   from that have \<open>unat b \<le> unat a\<close>
     by transfer simp
   with that show ?thesis
     apply transfer
   apply simp
     apply (subst take_bit_diff [symmetric])
     apply (subst nat_take_bit_eq)
     apply (simp add: nat_le_eq_zle)
     apply (simp add: nat_diff_distrib take_bit_nat_eq_self_iff less_imp_diff_less bintr_lt2p)
     done
 qed
 
 lemmas word_less_sub1_numberof [simp] = word_less_sub1 [of "numeral w"] for w
 lemmas word_le_sub1_numberof [simp] = word_le_sub1 [of "numeral w"] for w
 
 lemma word_of_int_minus: "word_of_int (2^LENGTH('a) - i) = (word_of_int (-i)::'a::len word)"
   apply transfer
   apply (subst take_bit_diff [symmetric])
   apply (simp add: take_bit_minus)
   done
 
 lemma word_of_int_inj:
   \<open>(word_of_int x :: 'a::len word) = word_of_int y \<longleftrightarrow> x = y\<close>
   if \<open>0 \<le> x \<and> x < 2 ^ LENGTH('a)\<close> \<open>0 \<le> y \<and> y < 2 ^ LENGTH('a)\<close>
   using that by (transfer fixing: x y) (simp add: take_bit_int_eq_self) 
 
 lemma word_le_less_eq: "x \<le> y \<longleftrightarrow> x = y \<or> x < y"
   for x y :: "'z::len word"
   by (auto simp add: order_class.le_less)
 
 lemma mod_plus_cong:
   fixes b b' :: int
   assumes 1: "b = b'"
     and 2: "x mod b' = x' mod b'"
     and 3: "y mod b' = y' mod b'"
     and 4: "x' + y' = z'"
   shows "(x + y) mod b = z' mod b'"
 proof -
   from 1 2[symmetric] 3[symmetric] have "(x + y) mod b = (x' mod b' + y' mod b') mod b'"
     by (simp add: mod_add_eq)
   also have "\<dots> = (x' + y') mod b'"
     by (simp add: mod_add_eq)
   finally show ?thesis
     by (simp add: 4)
 qed
 
 lemma mod_minus_cong:
   fixes b b' :: int
   assumes "b = b'"
     and "x mod b' = x' mod b'"
     and "y mod b' = y' mod b'"
     and "x' - y' = z'"
   shows "(x - y) mod b = z' mod b'"
   using assms [symmetric] by (auto intro: mod_diff_cong)
 
 lemma word_induct_less:
   \<open>P m\<close> if zero: \<open>P 0\<close> and less: \<open>\<And>n. n < m \<Longrightarrow> P n \<Longrightarrow> P (1 + n)\<close>
   for m :: \<open>'a::len word\<close>
 proof -
   define q where \<open>q = unat m\<close>
   with less have \<open>\<And>n. n < word_of_nat q \<Longrightarrow> P n \<Longrightarrow> P (1 + n)\<close>
     by simp
   then have \<open>P (word_of_nat q :: 'a word)\<close>
   proof (induction q)
     case 0
     show ?case
       by (simp add: zero)
   next
     case (Suc q)
     show ?case
     proof (cases \<open>1 + word_of_nat q = (0 :: 'a word)\<close>)
       case True
       then show ?thesis
         by (simp add: zero)
     next
       case False
       then have *: \<open>word_of_nat q < (word_of_nat (Suc q) :: 'a word)\<close>
         by (simp add: unatSuc word_less_nat_alt)
       then have **: \<open>n < (1 + word_of_nat q :: 'a word) \<longleftrightarrow> n \<le> (word_of_nat q :: 'a word)\<close> for n
         by (metis (no_types, lifting) add.commute inc_le le_less_trans not_less of_nat_Suc)
       have \<open>P (word_of_nat q)\<close>
         apply (rule Suc.IH)
         apply (rule Suc.prems)
          apply (erule less_trans)
          apply (rule *)
         apply assumption
         done
       with * have \<open>P (1 + word_of_nat q)\<close>
         by (rule Suc.prems)
       then show ?thesis
         by simp
     qed
   qed
   with \<open>q = unat m\<close> show ?thesis
     by simp
 qed
 
 lemma word_induct: "P 0 \<Longrightarrow> (\<And>n. P n \<Longrightarrow> P (1 + n)) \<Longrightarrow> P m"
   for P :: "'a::len word \<Rightarrow> bool"
   by (rule word_induct_less)
 
 lemma word_induct2 [induct type]: "P 0 \<Longrightarrow> (\<And>n. 1 + n \<noteq> 0 \<Longrightarrow> P n \<Longrightarrow> P (1 + n)) \<Longrightarrow> P n"
   for P :: "'b::len word \<Rightarrow> bool"
   apply (rule word_induct_less)
    apply simp_all
   apply (case_tac "1 + na = 0")
    apply auto
   done
 
 
 subsection \<open>Recursion combinator for words\<close>
 
 definition word_rec :: "'a \<Rightarrow> ('b::len word \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'b word \<Rightarrow> 'a"
   where "word_rec forZero forSuc n = rec_nat forZero (forSuc \<circ> of_nat) (unat n)"
 
 lemma word_rec_0: "word_rec z s 0 = z"
   by (simp add: word_rec_def)
 
 lemma word_rec_Suc: "1 + n \<noteq> 0 \<Longrightarrow> word_rec z s (1 + n) = s n (word_rec z s n)"
   for n :: "'a::len word"
   apply (auto simp add: word_rec_def unat_word_ariths)
   apply (metis (mono_tags, lifting) Abs_fnat_hom_add add_diff_cancel_left' o_def of_nat_1 old.nat.simps(7) plus_1_eq_Suc unatSuc unat_word_ariths(1) unsigned_1 word_arith_nat_add)
   done
 
 lemma word_rec_Pred: "n \<noteq> 0 \<Longrightarrow> word_rec z s n = s (n - 1) (word_rec z s (n - 1))"
   apply (rule subst[where t="n" and s="1 + (n - 1)"])
    apply simp
   apply (subst word_rec_Suc)
    apply simp
   apply simp
   done
 
 lemma word_rec_in: "f (word_rec z (\<lambda>_. f) n) = word_rec (f z) (\<lambda>_. f) n"
   by (induct n) (simp_all add: word_rec_0 word_rec_Suc)
 
 lemma word_rec_in2: "f n (word_rec z f n) = word_rec (f 0 z) (f \<circ> (+) 1) n"
   by (induct n) (simp_all add: word_rec_0 word_rec_Suc)
 
 lemma word_rec_twice:
   "m \<le> n \<Longrightarrow> word_rec z f n = word_rec (word_rec z f (n - m)) (f \<circ> (+) (n - m)) m"
   apply (erule rev_mp)
   apply (rule_tac x=z in spec)
   apply (rule_tac x=f in spec)
   apply (induct n)
    apply (simp add: word_rec_0)
   apply clarsimp
   apply (rule_tac t="1 + n - m" and s="1 + (n - m)" in subst)
    apply simp
   apply (case_tac "1 + (n - m) = 0")
    apply (simp add: word_rec_0)
    apply (rule_tac f = "word_rec a b" for a b in arg_cong)
    apply (rule_tac t="m" and s="m + (1 + (n - m))" in subst)
     apply simp
    apply (simp (no_asm_use))
   apply (simp add: word_rec_Suc word_rec_in2)
   apply (erule impE)
    apply uint_arith
   apply (drule_tac x="x \<circ> (+) 1" in spec)
   apply (drule_tac x="x 0 xa" in spec)
   apply simp
   apply (rule_tac t="\<lambda>a. x (1 + (n - m + a))" and s="\<lambda>a. x (1 + (n - m) + a)" in subst)
    apply (clarsimp simp add: fun_eq_iff)
    apply (rule_tac t="(1 + (n - m + xb))" and s="1 + (n - m) + xb" in subst)
     apply simp
    apply (rule refl)
   apply (rule refl)
   done
 
 lemma word_rec_id: "word_rec z (\<lambda>_. id) n = z"
   by (induct n) (auto simp add: word_rec_0 word_rec_Suc)
 
 lemma word_rec_id_eq: "\<forall>m < n. f m = id \<Longrightarrow> word_rec z f n = z"
   apply (erule rev_mp)
   apply (induct n)
    apply (auto simp add: word_rec_0 word_rec_Suc)
    apply (drule spec, erule mp)
    apply uint_arith
   apply (drule_tac x=n in spec, erule impE)
    apply uint_arith
   apply simp
   done
 
 lemma word_rec_max:
   "\<forall>m\<ge>n. m \<noteq> - 1 \<longrightarrow> f m = id \<Longrightarrow> word_rec z f (- 1) = word_rec z f n"
   apply (subst word_rec_twice[where n="-1" and m="-1 - n"])
    apply simp
   apply simp
   apply (rule word_rec_id_eq)
   apply clarsimp
   apply (drule spec, rule mp, erule mp)
    apply (rule word_plus_mono_right2[OF _ order_less_imp_le])
     prefer 2
     apply assumption
    apply simp
   apply (erule contrapos_pn)
   apply simp
   apply (drule arg_cong[where f="\<lambda>x. x - n"])
   apply simp
   done
 
 
 subsection \<open>More\<close>
 
 lemma test_bit_1' [simp]:
   "(1 :: 'a :: len word) !! n \<longleftrightarrow> 0 < LENGTH('a) \<and> n = 0"
   by simp
 
 lemma shiftl0:
   "x << 0 = (x :: 'a :: len word)"
   by (fact shiftl_x_0)
 
 lemma mask_1: "mask 1 = 1"
   by transfer (simp add: min_def mask_Suc_exp)
 
 lemma mask_Suc_0: "mask (Suc 0) = 1"
   using mask_1 by simp
 
-lemma mask_numeral: "mask (numeral n) = 2 * mask (pred_numeral n) + (1 :: 'a::len word)"
-  by (simp add: mask_Suc_rec numeral_eq_Suc)
-
 lemma bin_last_bintrunc: "bin_last (take_bit l n) = (l > 0 \<and> bin_last n)"
   by simp
 
 lemma word_and_1:
   "n AND 1 = (if n !! 0 then 1 else 0)" for n :: "_ word"
   by (rule bit_word_eqI) (auto simp add: bit_and_iff test_bit_eq_bit bit_1_iff intro: gr0I)
 
 lemma bintrunc_shiftl:
   "take_bit n (m << i) = take_bit (n - i) m << i"
   for m :: int
   by (rule bit_eqI) (auto simp add: bit_take_bit_iff)
 
 lemma uint_shiftl:
   "uint (n << i) = take_bit (size n) (uint n << i)"
   by transfer (simp add: push_bit_take_bit shiftl_eq_push_bit)
 
 
 subsection \<open>Misc\<close>
 
 ML_file \<open>Tools/word_lib.ML\<close>
 ML_file \<open>Tools/smt_word.ML\<close>
 
 end