diff --git a/src/HOL/Parity.thy b/src/HOL/Parity.thy
--- a/src/HOL/Parity.thy
+++ b/src/HOL/Parity.thy
@@ -1,1485 +1,1500 @@
 (*  Title:      HOL/Parity.thy
     Author:     Jeremy Avigad
     Author:     Jacques D. Fleuriot
 *)
 
 section \<open>Parity in rings and semirings\<close>
 
 theory Parity
   imports Euclidean_Division
 begin
 
 subsection \<open>Ring structures with parity and \<open>even\<close>/\<open>odd\<close> predicates\<close>
 
 class semiring_parity = comm_semiring_1 + semiring_modulo +
   assumes even_iff_mod_2_eq_zero: "2 dvd a \<longleftrightarrow> a mod 2 = 0"
     and odd_iff_mod_2_eq_one: "\<not> 2 dvd a \<longleftrightarrow> a mod 2 = 1"
     and odd_one [simp]: "\<not> 2 dvd 1"
 begin
 
 abbreviation even :: "'a \<Rightarrow> bool"
   where "even a \<equiv> 2 dvd a"
 
 abbreviation odd :: "'a \<Rightarrow> bool"
   where "odd a \<equiv> \<not> 2 dvd a"
 
 lemma parity_cases [case_names even odd]:
   assumes "even a \<Longrightarrow> a mod 2 = 0 \<Longrightarrow> P"
   assumes "odd a \<Longrightarrow> a mod 2 = 1 \<Longrightarrow> P"
   shows P
   using assms by (cases "even a")
     (simp_all add: even_iff_mod_2_eq_zero [symmetric] odd_iff_mod_2_eq_one [symmetric])
 
 lemma odd_of_bool_self [simp]:
   \<open>odd (of_bool p) \<longleftrightarrow> p\<close>
   by (cases p) simp_all
 
 lemma not_mod_2_eq_0_eq_1 [simp]:
   "a mod 2 \<noteq> 0 \<longleftrightarrow> a mod 2 = 1"
   by (cases a rule: parity_cases) simp_all
 
 lemma not_mod_2_eq_1_eq_0 [simp]:
   "a mod 2 \<noteq> 1 \<longleftrightarrow> a mod 2 = 0"
   by (cases a rule: parity_cases) simp_all
 
 lemma evenE [elim?]:
   assumes "even a"
   obtains b where "a = 2 * b"
   using assms by (rule dvdE)
 
 lemma oddE [elim?]:
   assumes "odd a"
   obtains b where "a = 2 * b + 1"
 proof -
   have "a = 2 * (a div 2) + a mod 2"
     by (simp add: mult_div_mod_eq)
   with assms have "a = 2 * (a div 2) + 1"
     by (simp add: odd_iff_mod_2_eq_one)
   then show ?thesis ..
 qed
 
 lemma mod_2_eq_odd:
   "a mod 2 = of_bool (odd a)"
   by (auto elim: oddE simp add: even_iff_mod_2_eq_zero)
 
 lemma of_bool_odd_eq_mod_2:
   "of_bool (odd a) = a mod 2"
   by (simp add: mod_2_eq_odd)
 
 lemma even_mod_2_iff [simp]:
   \<open>even (a mod 2) \<longleftrightarrow> even a\<close>
   by (simp add: mod_2_eq_odd)
 
 lemma mod2_eq_if:
   "a mod 2 = (if even a then 0 else 1)"
   by (simp add: mod_2_eq_odd)
 
 lemma even_zero [simp]:
   "even 0"
   by (fact dvd_0_right)
 
 lemma odd_even_add:
   "even (a + b)" if "odd a" and "odd b"
 proof -
   from that obtain c d where "a = 2 * c + 1" and "b = 2 * d + 1"
     by (blast elim: oddE)
   then have "a + b = 2 * c + 2 * d + (1 + 1)"
     by (simp only: ac_simps)
   also have "\<dots> = 2 * (c + d + 1)"
     by (simp add: algebra_simps)
   finally show ?thesis ..
 qed
 
 lemma even_add [simp]:
   "even (a + b) \<longleftrightarrow> (even a \<longleftrightarrow> even b)"
   by (auto simp add: dvd_add_right_iff dvd_add_left_iff odd_even_add)
 
 lemma odd_add [simp]:
   "odd (a + b) \<longleftrightarrow> \<not> (odd a \<longleftrightarrow> odd b)"
   by simp
 
 lemma even_plus_one_iff [simp]:
   "even (a + 1) \<longleftrightarrow> odd a"
   by (auto simp add: dvd_add_right_iff intro: odd_even_add)
 
 lemma even_mult_iff [simp]:
   "even (a * b) \<longleftrightarrow> even a \<or> even b" (is "?P \<longleftrightarrow> ?Q")
 proof
   assume ?Q
   then show ?P
     by auto
 next
   assume ?P
   show ?Q
   proof (rule ccontr)
     assume "\<not> (even a \<or> even b)"
     then have "odd a" and "odd b"
       by auto
     then obtain r s where "a = 2 * r + 1" and "b = 2 * s + 1"
       by (blast elim: oddE)
     then have "a * b = (2 * r + 1) * (2 * s + 1)"
       by simp
     also have "\<dots> = 2 * (2 * r * s + r + s) + 1"
       by (simp add: algebra_simps)
     finally have "odd (a * b)"
       by simp
     with \<open>?P\<close> show False
       by auto
   qed
 qed
 
 lemma even_numeral [simp]: "even (numeral (Num.Bit0 n))"
 proof -
   have "even (2 * numeral n)"
     unfolding even_mult_iff by simp
   then have "even (numeral n + numeral n)"
     unfolding mult_2 .
   then show ?thesis
     unfolding numeral.simps .
 qed
 
 lemma odd_numeral [simp]: "odd (numeral (Num.Bit1 n))"
 proof
   assume "even (numeral (num.Bit1 n))"
   then have "even (numeral n + numeral n + 1)"
     unfolding numeral.simps .
   then have "even (2 * numeral n + 1)"
     unfolding mult_2 .
   then have "2 dvd numeral n * 2 + 1"
     by (simp add: ac_simps)
   then have "2 dvd 1"
     using dvd_add_times_triv_left_iff [of 2 "numeral n" 1] by simp
   then show False by simp
 qed
 
 lemma even_power [simp]: "even (a ^ n) \<longleftrightarrow> even a \<and> n > 0"
   by (induct n) auto
 
 lemma mask_eq_sum_exp:
   \<open>2 ^ n - 1 = (\<Sum>m\<in>{q. q < n}. 2 ^ m)\<close>
 proof -
   have *: \<open>{q. q < Suc m} = insert m {q. q < m}\<close> for m
     by auto
   have \<open>2 ^ n = (\<Sum>m\<in>{q. q < n}. 2 ^ m) + 1\<close>
     by (induction n) (simp_all add: ac_simps mult_2 *)
   then have \<open>2 ^ n - 1 = (\<Sum>m\<in>{q. q < n}. 2 ^ m) + 1 - 1\<close>
     by simp
   then show ?thesis
     by simp
 qed
 
 end
 
 class ring_parity = ring + semiring_parity
 begin
 
 subclass comm_ring_1 ..
 
 lemma even_minus:
   "even (- a) \<longleftrightarrow> even a"
   by (fact dvd_minus_iff)
 
 lemma even_diff [simp]:
   "even (a - b) \<longleftrightarrow> even (a + b)"
   using even_add [of a "- b"] by simp
 
 end
 
 
 subsection \<open>Special case: euclidean rings containing the natural numbers\<close>
 
 context unique_euclidean_semiring_with_nat
 begin
 
 subclass semiring_parity
 proof
   show "2 dvd a \<longleftrightarrow> a mod 2 = 0" for a
     by (fact dvd_eq_mod_eq_0)
   show "\<not> 2 dvd a \<longleftrightarrow> a mod 2 = 1" for a
   proof
     assume "a mod 2 = 1"
     then show "\<not> 2 dvd a"
       by auto
   next
     assume "\<not> 2 dvd a"
     have eucl: "euclidean_size (a mod 2) = 1"
     proof (rule order_antisym)
       show "euclidean_size (a mod 2) \<le> 1"
         using mod_size_less [of 2 a] by simp
       show "1 \<le> euclidean_size (a mod 2)"
         using \<open>\<not> 2 dvd a\<close> by (simp add: Suc_le_eq dvd_eq_mod_eq_0)
     qed 
     from \<open>\<not> 2 dvd a\<close> have "\<not> of_nat 2 dvd division_segment a * of_nat (euclidean_size a)"
       by simp
     then have "\<not> of_nat 2 dvd of_nat (euclidean_size a)"
       by (auto simp only: dvd_mult_unit_iff' is_unit_division_segment)
     then have "\<not> 2 dvd euclidean_size a"
       using of_nat_dvd_iff [of 2] by simp
     then have "euclidean_size a mod 2 = 1"
       by (simp add: semidom_modulo_class.dvd_eq_mod_eq_0)
     then have "of_nat (euclidean_size a mod 2) = of_nat 1"
       by simp
     then have "of_nat (euclidean_size a) mod 2 = 1"
       by (simp add: of_nat_mod)
     from \<open>\<not> 2 dvd a\<close> eucl
     show "a mod 2 = 1"
       by (auto intro: division_segment_eq_iff simp add: division_segment_mod)
   qed
   show "\<not> is_unit 2"
   proof (rule notI)
     assume "is_unit 2"
     then have "of_nat 2 dvd of_nat 1"
       by simp
     then have "is_unit (2::nat)"
       by (simp only: of_nat_dvd_iff)
     then show False
       by simp
   qed
 qed
 
 lemma even_of_nat [simp]:
   "even (of_nat a) \<longleftrightarrow> even a"
 proof -
   have "even (of_nat a) \<longleftrightarrow> of_nat 2 dvd of_nat a"
     by simp
   also have "\<dots> \<longleftrightarrow> even a"
     by (simp only: of_nat_dvd_iff)
   finally show ?thesis .
 qed
 
 lemma even_succ_div_two [simp]:
   "even a \<Longrightarrow> (a + 1) div 2 = a div 2"
   by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)
 
 lemma odd_succ_div_two [simp]:
   "odd a \<Longrightarrow> (a + 1) div 2 = a div 2 + 1"
   by (auto elim!: oddE simp add: add.assoc)
 
 lemma even_two_times_div_two:
   "even a \<Longrightarrow> 2 * (a div 2) = a"
   by (fact dvd_mult_div_cancel)
 
 lemma odd_two_times_div_two_succ [simp]:
   "odd a \<Longrightarrow> 2 * (a div 2) + 1 = a"
   using mult_div_mod_eq [of 2 a]
   by (simp add: even_iff_mod_2_eq_zero)
 
 lemma coprime_left_2_iff_odd [simp]:
   "coprime 2 a \<longleftrightarrow> odd a"
 proof
   assume "odd a"
   show "coprime 2 a"
   proof (rule coprimeI)
     fix b
     assume "b dvd 2" "b dvd a"
     then have "b dvd a mod 2"
       by (auto intro: dvd_mod)
     with \<open>odd a\<close> show "is_unit b"
       by (simp add: mod_2_eq_odd)
   qed
 next
   assume "coprime 2 a"
   show "odd a"
   proof (rule notI)
     assume "even a"
     then obtain b where "a = 2 * b" ..
     with \<open>coprime 2 a\<close> have "coprime 2 (2 * b)"
       by simp
     moreover have "\<not> coprime 2 (2 * b)"
       by (rule not_coprimeI [of 2]) simp_all
     ultimately show False
       by blast
   qed
 qed
 
 lemma coprime_right_2_iff_odd [simp]:
   "coprime a 2 \<longleftrightarrow> odd a"
   using coprime_left_2_iff_odd [of a] by (simp add: ac_simps)
 
 end
 
 context unique_euclidean_ring_with_nat
 begin
 
 subclass ring_parity ..
 
 lemma minus_1_mod_2_eq [simp]:
   "- 1 mod 2 = 1"
   by (simp add: mod_2_eq_odd)
 
 lemma minus_1_div_2_eq [simp]:
   "- 1 div 2 = - 1"
 proof -
   from div_mult_mod_eq [of "- 1" 2]
   have "- 1 div 2 * 2 = - 1 * 2"
     using add_implies_diff by fastforce
   then show ?thesis
     using mult_right_cancel [of 2 "- 1 div 2" "- 1"] by simp
 qed
 
 end
 
 
 subsection \<open>Instance for \<^typ>\<open>nat\<close>\<close>
 
 instance nat :: unique_euclidean_semiring_with_nat
   by standard (simp_all add: dvd_eq_mod_eq_0)
 
 lemma even_Suc_Suc_iff [simp]:
   "even (Suc (Suc n)) \<longleftrightarrow> even n"
   using dvd_add_triv_right_iff [of 2 n] by simp
 
 lemma even_Suc [simp]: "even (Suc n) \<longleftrightarrow> odd n"
   using even_plus_one_iff [of n] by simp
 
 lemma even_diff_nat [simp]:
   "even (m - n) \<longleftrightarrow> m < n \<or> even (m + n)" for m n :: nat
 proof (cases "n \<le> m")
   case True
   then have "m - n + n * 2 = m + n" by (simp add: mult_2_right)
   moreover have "even (m - n) \<longleftrightarrow> even (m - n + n * 2)" by simp
   ultimately have "even (m - n) \<longleftrightarrow> even (m + n)" by (simp only:)
   then show ?thesis by auto
 next
   case False
   then show ?thesis by simp
 qed
 
 lemma odd_pos:
   "odd n \<Longrightarrow> 0 < n" for n :: nat
   by (auto elim: oddE)
 
 lemma Suc_double_not_eq_double:
   "Suc (2 * m) \<noteq> 2 * n"
 proof
   assume "Suc (2 * m) = 2 * n"
   moreover have "odd (Suc (2 * m))" and "even (2 * n)"
     by simp_all
   ultimately show False by simp
 qed
 
 lemma double_not_eq_Suc_double:
   "2 * m \<noteq> Suc (2 * n)"
   using Suc_double_not_eq_double [of n m] by simp
 
 lemma odd_Suc_minus_one [simp]: "odd n \<Longrightarrow> Suc (n - Suc 0) = n"
   by (auto elim: oddE)
 
 lemma even_Suc_div_two [simp]:
   "even n \<Longrightarrow> Suc n div 2 = n div 2"
   using even_succ_div_two [of n] by simp
 
 lemma odd_Suc_div_two [simp]:
   "odd n \<Longrightarrow> Suc n div 2 = Suc (n div 2)"
   using odd_succ_div_two [of n] by simp
 
 lemma odd_two_times_div_two_nat [simp]:
   assumes "odd n"
   shows "2 * (n div 2) = n - (1 :: nat)"
 proof -
   from assms have "2 * (n div 2) + 1 = n"
     by (rule odd_two_times_div_two_succ)
   then have "Suc (2 * (n div 2)) - 1 = n - 1"
     by simp
   then show ?thesis
     by simp
 qed
 
 lemma not_mod2_eq_Suc_0_eq_0 [simp]:
   "n mod 2 \<noteq> Suc 0 \<longleftrightarrow> n mod 2 = 0"
   using not_mod_2_eq_1_eq_0 [of n] by simp
 
 lemma odd_card_imp_not_empty:
   \<open>A \<noteq> {}\<close> if \<open>odd (card A)\<close>
   using that by auto
 
 lemma nat_induct2 [case_names 0 1 step]:
   assumes "P 0" "P 1" and step: "\<And>n::nat. P n \<Longrightarrow> P (n + 2)"
   shows "P n"
 proof (induct n rule: less_induct)
   case (less n)
   show ?case
   proof (cases "n < Suc (Suc 0)")
     case True
     then show ?thesis
       using assms by (auto simp: less_Suc_eq)
   next
     case False
     then obtain k where k: "n = Suc (Suc k)"
       by (force simp: not_less nat_le_iff_add)
     then have "k<n"
       by simp
     with less assms have "P (k+2)"
       by blast
     then show ?thesis
       by (simp add: k)
   qed
 qed
 
 lemma mask_eq_sum_exp_nat:
   \<open>2 ^ n - Suc 0 = (\<Sum>m\<in>{q. q < n}. 2 ^ m)\<close>
   using mask_eq_sum_exp [where ?'a = nat] by simp
 
 context semiring_parity
 begin
 
 lemma even_sum_iff:
   \<open>even (sum f A) \<longleftrightarrow> even (card {a\<in>A. odd (f a)})\<close> if \<open>finite A\<close>
 using that proof (induction A)
   case empty
   then show ?case
     by simp
 next
   case (insert a A)
   moreover have \<open>{b \<in> insert a A. odd (f b)} = (if odd (f a) then {a} else {}) \<union> {b \<in> A. odd (f b)}\<close>
     by auto
   ultimately show ?case
     by simp
 qed
 
 lemma even_prod_iff:
   \<open>even (prod f A) \<longleftrightarrow> (\<exists>a\<in>A. even (f a))\<close> if \<open>finite A\<close>
   using that by (induction A) simp_all
 
 lemma even_mask_iff [simp]:
   \<open>even (2 ^ n - 1) \<longleftrightarrow> n = 0\<close>
 proof (cases \<open>n = 0\<close>)
   case True
   then show ?thesis
     by simp
 next
   case False
   then have \<open>{a. a = 0 \<and> a < n} = {0}\<close>
     by auto
   then show ?thesis
     by (auto simp add: mask_eq_sum_exp even_sum_iff)
 qed
 
 end
 
 
 subsection \<open>Parity and powers\<close>
 
 context ring_1
 begin
 
 lemma power_minus_even [simp]: "even n \<Longrightarrow> (- a) ^ n = a ^ n"
   by (auto elim: evenE)
 
 lemma power_minus_odd [simp]: "odd n \<Longrightarrow> (- a) ^ n = - (a ^ n)"
   by (auto elim: oddE)
 
 lemma uminus_power_if:
   "(- a) ^ n = (if even n then a ^ n else - (a ^ n))"
   by auto
 
 lemma neg_one_even_power [simp]: "even n \<Longrightarrow> (- 1) ^ n = 1"
   by simp
 
 lemma neg_one_odd_power [simp]: "odd n \<Longrightarrow> (- 1) ^ n = - 1"
   by simp
 
 lemma neg_one_power_add_eq_neg_one_power_diff: "k \<le> n \<Longrightarrow> (- 1) ^ (n + k) = (- 1) ^ (n - k)"
   by (cases "even (n + k)") auto
 
 lemma minus_one_power_iff: "(- 1) ^ n = (if even n then 1 else - 1)"
   by (induct n) auto
 
 end
 
 context linordered_idom
 begin
 
 lemma zero_le_even_power: "even n \<Longrightarrow> 0 \<le> a ^ n"
   by (auto elim: evenE)
 
 lemma zero_le_odd_power: "odd n \<Longrightarrow> 0 \<le> a ^ n \<longleftrightarrow> 0 \<le> a"
   by (auto simp add: power_even_eq zero_le_mult_iff elim: oddE)
 
 lemma zero_le_power_eq: "0 \<le> a ^ n \<longleftrightarrow> even n \<or> odd n \<and> 0 \<le> a"
   by (auto simp add: zero_le_even_power zero_le_odd_power)
 
 lemma zero_less_power_eq: "0 < a ^ n \<longleftrightarrow> n = 0 \<or> even n \<and> a \<noteq> 0 \<or> odd n \<and> 0 < a"
 proof -
   have [simp]: "0 = a ^ n \<longleftrightarrow> a = 0 \<and> n > 0"
     unfolding power_eq_0_iff [of a n, symmetric] by blast
   show ?thesis
     unfolding less_le zero_le_power_eq by auto
 qed
 
 lemma power_less_zero_eq [simp]: "a ^ n < 0 \<longleftrightarrow> odd n \<and> a < 0"
   unfolding not_le [symmetric] zero_le_power_eq by auto
 
 lemma power_le_zero_eq: "a ^ n \<le> 0 \<longleftrightarrow> n > 0 \<and> (odd n \<and> a \<le> 0 \<or> even n \<and> a = 0)"
   unfolding not_less [symmetric] zero_less_power_eq by auto
 
 lemma power_even_abs: "even n \<Longrightarrow> \<bar>a\<bar> ^ n = a ^ n"
   using power_abs [of a n] by (simp add: zero_le_even_power)
 
 lemma power_mono_even:
   assumes "even n" and "\<bar>a\<bar> \<le> \<bar>b\<bar>"
   shows "a ^ n \<le> b ^ n"
 proof -
   have "0 \<le> \<bar>a\<bar>" by auto
   with \<open>\<bar>a\<bar> \<le> \<bar>b\<bar>\<close> have "\<bar>a\<bar> ^ n \<le> \<bar>b\<bar> ^ n"
     by (rule power_mono)
   with \<open>even n\<close> show ?thesis
     by (simp add: power_even_abs)
 qed
 
 lemma power_mono_odd:
   assumes "odd n" and "a \<le> b"
   shows "a ^ n \<le> b ^ n"
 proof (cases "b < 0")
   case True
   with \<open>a \<le> b\<close> have "- b \<le> - a" and "0 \<le> - b" by auto
   then have "(- b) ^ n \<le> (- a) ^ n" by (rule power_mono)
   with \<open>odd n\<close> show ?thesis by simp
 next
   case False
   then have "0 \<le> b" by auto
   show ?thesis
   proof (cases "a < 0")
     case True
     then have "n \<noteq> 0" and "a \<le> 0" using \<open>odd n\<close> [THEN odd_pos] by auto
     then have "a ^ n \<le> 0" unfolding power_le_zero_eq using \<open>odd n\<close> by auto
     moreover from \<open>0 \<le> b\<close> have "0 \<le> b ^ n" by auto
     ultimately show ?thesis by auto
   next
     case False
     then have "0 \<le> a" by auto
     with \<open>a \<le> b\<close> show ?thesis
       using power_mono by auto
   qed
 qed
 
 text \<open>Simplify, when the exponent is a numeral\<close>
 
 lemma zero_le_power_eq_numeral [simp]:
   "0 \<le> a ^ numeral w \<longleftrightarrow> even (numeral w :: nat) \<or> odd (numeral w :: nat) \<and> 0 \<le> a"
   by (fact zero_le_power_eq)
 
 lemma zero_less_power_eq_numeral [simp]:
   "0 < a ^ numeral w \<longleftrightarrow>
     numeral w = (0 :: nat) \<or>
     even (numeral w :: nat) \<and> a \<noteq> 0 \<or>
     odd (numeral w :: nat) \<and> 0 < a"
   by (fact zero_less_power_eq)
 
 lemma power_le_zero_eq_numeral [simp]:
   "a ^ numeral w \<le> 0 \<longleftrightarrow>
     (0 :: nat) < numeral w \<and>
     (odd (numeral w :: nat) \<and> a \<le> 0 \<or> even (numeral w :: nat) \<and> a = 0)"
   by (fact power_le_zero_eq)
 
 lemma power_less_zero_eq_numeral [simp]:
   "a ^ numeral w < 0 \<longleftrightarrow> odd (numeral w :: nat) \<and> a < 0"
   by (fact power_less_zero_eq)
 
 lemma power_even_abs_numeral [simp]:
   "even (numeral w :: nat) \<Longrightarrow> \<bar>a\<bar> ^ numeral w = a ^ numeral w"
   by (fact power_even_abs)
 
 end
 
 context unique_euclidean_semiring_with_nat
 begin
 
 lemma even_mask_div_iff':
   \<open>even ((2 ^ m - 1) div 2 ^ n) \<longleftrightarrow> m \<le> n\<close>
 proof -
   have \<open>even ((2 ^ m - 1) div 2 ^ n) \<longleftrightarrow> even (of_nat ((2 ^ m - Suc 0) div 2 ^ n))\<close>
     by (simp only: of_nat_div) (simp add: of_nat_diff)
   also have \<open>\<dots> \<longleftrightarrow> even ((2 ^ m - Suc 0) div 2 ^ n)\<close>
     by simp
   also have \<open>\<dots> \<longleftrightarrow> m \<le> n\<close>
   proof (cases \<open>m \<le> n\<close>)
     case True
     then show ?thesis
       by (simp add: Suc_le_lessD)
   next
     case False
     then obtain r where r: \<open>m = n + Suc r\<close>
       using less_imp_Suc_add by fastforce
     from r have \<open>{q. q < m} \<inter> {q. 2 ^ n dvd (2::nat) ^ q} = {q. n \<le> q \<and> q < m}\<close>
       by (auto simp add: dvd_power_iff_le)
     moreover from r have \<open>{q. q < m} \<inter> {q. \<not> 2 ^ n dvd (2::nat) ^ q} = {q. q < n}\<close>
       by (auto simp add: dvd_power_iff_le)
     moreover from False have \<open>{q. n \<le> q \<and> q < m \<and> q \<le> n} = {n}\<close>
       by auto
     then have \<open>odd ((\<Sum>a\<in>{q. n \<le> q \<and> q < m}. 2 ^ a div (2::nat) ^ n) + sum ((^) 2) {q. q < n} div 2 ^ n)\<close>
       by (simp_all add: euclidean_semiring_cancel_class.power_diff_power_eq semiring_parity_class.even_sum_iff not_less mask_eq_sum_exp_nat [symmetric])
     ultimately have \<open>odd (sum ((^) (2::nat)) {q. q < m} div 2 ^ n)\<close>
       by (subst euclidean_semiring_cancel_class.sum_div_partition) simp_all
     with False show ?thesis
       by (simp add: mask_eq_sum_exp_nat)
   qed
   finally show ?thesis .
 qed
 
 end
 
 
 subsection \<open>Instance for \<^typ>\<open>int\<close>\<close>
 
 lemma even_diff_iff:
   "even (k - l) \<longleftrightarrow> even (k + l)" for k l :: int
   by (fact even_diff)
 
 lemma even_abs_add_iff:
   "even (\<bar>k\<bar> + l) \<longleftrightarrow> even (k + l)" for k l :: int
   by simp
 
 lemma even_add_abs_iff:
   "even (k + \<bar>l\<bar>) \<longleftrightarrow> even (k + l)" for k l :: int
   by simp
 
 lemma even_nat_iff: "0 \<le> k \<Longrightarrow> even (nat k) \<longleftrightarrow> even k"
   by (simp add: even_of_nat [of "nat k", where ?'a = int, symmetric])
 
 lemma zdiv_zmult2_eq:
   \<open>a div (b * c) = (a div b) div c\<close> if \<open>c \<ge> 0\<close> for a b c :: int
 proof (cases \<open>b \<ge> 0\<close>)
   case True
   with that show ?thesis
     using div_mult2_eq' [of a \<open>nat b\<close> \<open>nat c\<close>] by simp
 next
   case False
   with that show ?thesis
     using div_mult2_eq' [of \<open>- a\<close> \<open>nat (- b)\<close> \<open>nat c\<close>] by simp
 qed
 
 lemma zmod_zmult2_eq:
   \<open>a mod (b * c) = b * (a div b mod c) + a mod b\<close> if \<open>c \<ge> 0\<close> for a b c :: int
 proof (cases \<open>b \<ge> 0\<close>)
   case True
   with that show ?thesis
     using mod_mult2_eq' [of a \<open>nat b\<close> \<open>nat c\<close>] by simp
 next
   case False
   with that show ?thesis
     using mod_mult2_eq' [of \<open>- a\<close> \<open>nat (- b)\<close> \<open>nat c\<close>] by simp
 qed
 
 
 subsection \<open>Abstract bit structures\<close>
 
 class semiring_bits = semiring_parity +
   assumes bits_induct [case_names stable rec]:
     \<open>(\<And>a. a div 2 = a \<Longrightarrow> P a)
      \<Longrightarrow> (\<And>a b. P a \<Longrightarrow> (of_bool b + 2 * a) div 2 = a \<Longrightarrow> P (of_bool b + 2 * a))
         \<Longrightarrow> P a\<close>
   assumes bits_div_0 [simp]: \<open>0 div a = 0\<close>
     and bits_div_by_1 [simp]: \<open>a div 1 = a\<close>
     and bits_mod_div_trivial [simp]: \<open>a mod b div b = 0\<close>
     and even_succ_div_2 [simp]: \<open>even a \<Longrightarrow> (1 + a) div 2 = a div 2\<close>
     and even_mask_div_iff: \<open>even ((2 ^ m - 1) div 2 ^ n) \<longleftrightarrow> 2 ^ n = 0 \<or> m \<le> n\<close>
     and exp_div_exp_eq: \<open>2 ^ m div 2 ^ n = of_bool (2 ^ m \<noteq> 0 \<and> m \<ge> n) * 2 ^ (m - n)\<close>
     and div_exp_eq: \<open>a div 2 ^ m div 2 ^ n = a div 2 ^ (m + n)\<close>
     and mod_exp_eq: \<open>a mod 2 ^ m mod 2 ^ n = a mod 2 ^ min m n\<close>
     and mult_exp_mod_exp_eq: \<open>m \<le> n \<Longrightarrow> (a * 2 ^ m) mod (2 ^ n) = (a mod 2 ^ (n - m)) * 2 ^ m\<close>
     and div_exp_mod_exp_eq: \<open>a div 2 ^ n mod 2 ^ m = a mod (2 ^ (n + m)) div 2 ^ n\<close>
     and even_mult_exp_div_exp_iff: \<open>even (a * 2 ^ m div 2 ^ n) \<longleftrightarrow> m > n \<or> 2 ^ n = 0 \<or> (m \<le> n \<and> even (a div 2 ^ (n - m)))\<close>
 begin
 
 lemma bits_div_by_0 [simp]:
   \<open>a div 0 = 0\<close>
   by (metis add_cancel_right_right bits_mod_div_trivial mod_mult_div_eq mult_not_zero)
 
 lemma bits_1_div_2 [simp]:
   \<open>1 div 2 = 0\<close>
   using even_succ_div_2 [of 0] by simp
 
 lemma bits_1_div_exp [simp]:
   \<open>1 div 2 ^ n = of_bool (n = 0)\<close>
   using div_exp_eq [of 1 1] by (cases n) simp_all
 
 lemma even_succ_div_exp [simp]:
   \<open>(1 + a) div 2 ^ n = a div 2 ^ n\<close> if \<open>even a\<close> and \<open>n > 0\<close>
 proof (cases n)
   case 0
   with that show ?thesis
     by simp
 next
   case (Suc n)
   with \<open>even a\<close> have \<open>(1 + a) div 2 ^ Suc n = a div 2 ^ Suc n\<close>
   proof (induction n)
     case 0
     then show ?case
       by simp
   next
     case (Suc n)
     then show ?case
       using div_exp_eq [of _ 1 \<open>Suc n\<close>, symmetric]
       by simp
   qed
   with Suc show ?thesis
     by simp
 qed
 
 lemma even_succ_mod_exp [simp]:
   \<open>(1 + a) mod 2 ^ n = 1 + (a mod 2 ^ n)\<close> if \<open>even a\<close> and \<open>n > 0\<close>
   using div_mult_mod_eq [of \<open>1 + a\<close> \<open>2 ^ n\<close>] that
   apply simp
   by (metis local.add.left_commute local.add_left_cancel local.div_mult_mod_eq)
 
 lemma bits_mod_by_1 [simp]:
   \<open>a mod 1 = 0\<close>
   using div_mult_mod_eq [of a 1] by simp
 
 lemma bits_mod_0 [simp]:
   \<open>0 mod a = 0\<close>
   using div_mult_mod_eq [of 0 a] by simp
 
 lemma bits_one_mod_two_eq_one [simp]:
   \<open>1 mod 2 = 1\<close>
   by (simp add: mod2_eq_if)
 
 definition bit :: \<open>'a \<Rightarrow> nat \<Rightarrow> bool\<close>
   where \<open>bit a n \<longleftrightarrow> odd (a div 2 ^ n)\<close>
 
 lemma bit_0 [simp]:
   \<open>bit a 0 \<longleftrightarrow> odd a\<close>
   by (simp add: bit_def)
 
 lemma bit_Suc [simp]:
   \<open>bit a (Suc n) \<longleftrightarrow> bit (a div 2) n\<close>
   using div_exp_eq [of a 1 n] by (simp add: bit_def)
 
 lemma bit_0_eq [simp]:
   \<open>bit 0 = bot\<close>
   by (simp add: fun_eq_iff bit_def)
 
 context
   fixes a
   assumes stable: \<open>a div 2 = a\<close>
 begin
 
 lemma bits_stable_imp_add_self:
   \<open>a + a mod 2 = 0\<close>
 proof -
   have \<open>a div 2 * 2 + a mod 2 = a\<close>
     by (fact div_mult_mod_eq)
   then have \<open>a * 2 + a mod 2 = a\<close>
     by (simp add: stable)
   then show ?thesis
     by (simp add: mult_2_right ac_simps)
 qed
 
 lemma stable_imp_bit_iff_odd:
   \<open>bit a n \<longleftrightarrow> odd a\<close>
   by (induction n) (simp_all add: stable)
 
 end
 
 lemma bit_iff_idd_imp_stable:
   \<open>a div 2 = a\<close> if \<open>\<And>n. bit a n \<longleftrightarrow> odd a\<close>
 using that proof (induction a rule: bits_induct)
   case (stable a)
   then show ?case
     by simp
 next
   case (rec a b)
   from rec.prems [of 1] have [simp]: \<open>b = odd a\<close>
     by (simp add: rec.hyps)
   from rec.hyps have hyp: \<open>(of_bool (odd a) + 2 * a) div 2 = a\<close>
     by simp
   have \<open>bit a n \<longleftrightarrow> odd a\<close> for n
     using rec.prems [of \<open>Suc n\<close>] by (simp add: hyp)
   then have \<open>a div 2 = a\<close>
     by (rule rec.IH)
   then have \<open>of_bool (odd a) + 2 * a = 2 * (a div 2) + of_bool (odd a)\<close>
     by (simp add: ac_simps)
   also have \<open>\<dots> = a\<close>
     using mult_div_mod_eq [of 2 a]
     by (simp add: of_bool_odd_eq_mod_2)
   finally show ?case
     using \<open>a div 2 = a\<close> by (simp add: hyp)
 qed
 
 lemma exp_eq_0_imp_not_bit:
   \<open>\<not> bit a n\<close> if \<open>2 ^ n = 0\<close>
   using that by (simp add: bit_def)
 
 lemma bit_eqI:
   \<open>a = b\<close> if \<open>\<And>n. 2 ^ n \<noteq> 0 \<Longrightarrow> bit a n \<longleftrightarrow> bit b n\<close>
 proof -
   have \<open>bit a n \<longleftrightarrow> bit b n\<close> for n
   proof (cases \<open>2 ^ n = 0\<close>)
     case True
     then show ?thesis
       by (simp add: exp_eq_0_imp_not_bit)
   next
     case False
     then show ?thesis
       by (rule that)
   qed
   then show ?thesis proof (induction a arbitrary: b rule: bits_induct)
     case (stable a)
     from stable(2) [of 0] have **: \<open>even b \<longleftrightarrow> even a\<close>
       by simp
     have \<open>b div 2 = b\<close>
     proof (rule bit_iff_idd_imp_stable)
       fix n
       from stable have *: \<open>bit b n \<longleftrightarrow> bit a n\<close>
         by simp
       also have \<open>bit a n \<longleftrightarrow> odd a\<close>
         using stable by (simp add: stable_imp_bit_iff_odd)
       finally show \<open>bit b n \<longleftrightarrow> odd b\<close>
         by (simp add: **)
     qed
     from ** have \<open>a mod 2 = b mod 2\<close>
       by (simp add: mod2_eq_if)
     then have \<open>a mod 2 + (a + b) = b mod 2 + (a + b)\<close>
       by simp
     then have \<open>a + a mod 2 + b = b + b mod 2 + a\<close>
       by (simp add: ac_simps)
     with \<open>a div 2 = a\<close> \<open>b div 2 = b\<close> show ?case
       by (simp add: bits_stable_imp_add_self)
   next
     case (rec a p)
     from rec.prems [of 0] have [simp]: \<open>p = odd b\<close>
       by simp
     from rec.hyps have \<open>bit a n \<longleftrightarrow> bit (b div 2) n\<close> for n
       using rec.prems [of \<open>Suc n\<close>] by simp
     then have \<open>a = b div 2\<close>
       by (rule rec.IH)
     then have \<open>2 * a = 2 * (b div 2)\<close>
       by simp
     then have \<open>b mod 2 + 2 * a = b mod 2 + 2 * (b div 2)\<close>
       by simp
     also have \<open>\<dots> = b\<close>
       by (fact mod_mult_div_eq)
     finally show ?case
       by (auto simp add: mod2_eq_if)
   qed
 qed
 
 lemma bit_eq_iff:
   \<open>a = b \<longleftrightarrow> (\<forall>n. bit a n \<longleftrightarrow> bit b n)\<close>
   by (auto intro: bit_eqI)
 
 lemma bit_exp_iff:
   \<open>bit (2 ^ m) n \<longleftrightarrow> 2 ^ m \<noteq> 0 \<and> m = n\<close>
   by (auto simp add: bit_def exp_div_exp_eq)
 
 lemma bit_1_iff:
   \<open>bit 1 n \<longleftrightarrow> 1 \<noteq> 0 \<and> n = 0\<close>
   using bit_exp_iff [of 0 n] by simp
 
 lemma bit_2_iff:
   \<open>bit 2 n \<longleftrightarrow> 2 \<noteq> 0 \<and> n = 1\<close>
   using bit_exp_iff [of 1 n] by auto
 
 lemma even_bit_succ_iff:
   \<open>bit (1 + a) n \<longleftrightarrow> bit a n \<or> n = 0\<close> if \<open>even a\<close>
   using that by (cases \<open>n = 0\<close>) (simp_all add: bit_def)
 
 lemma odd_bit_iff_bit_pred:
   \<open>bit a n \<longleftrightarrow> bit (a - 1) n \<or> n = 0\<close> if \<open>odd a\<close>
 proof -
   from \<open>odd a\<close> obtain b where \<open>a = 2 * b + 1\<close> ..
   moreover have \<open>bit (2 * b) n \<or> n = 0 \<longleftrightarrow> bit (1 + 2 * b) n\<close>
     using even_bit_succ_iff by simp
   ultimately show ?thesis by (simp add: ac_simps)
 qed
 
 lemma bit_double_iff:
   \<open>bit (2 * a) n \<longleftrightarrow> bit a (n - 1) \<and> n \<noteq> 0 \<and> 2 ^ n \<noteq> 0\<close>
   using even_mult_exp_div_exp_iff [of a 1 n]
   by (cases n, auto simp add: bit_def ac_simps)
 
 lemma bit_eq_rec:
-  \<open>a = b \<longleftrightarrow> (even a \<longleftrightarrow> even b) \<and> a div 2 = b div 2\<close>
-  apply (auto simp add: bit_eq_iff)
-  using bit_0 apply blast
-  using bit_0 apply blast
-  using bit_Suc apply blast
-  using bit_Suc apply blast
-     apply (metis bit_eq_iff even_iff_mod_2_eq_zero mod_div_mult_eq)
-    apply (metis bit_eq_iff even_iff_mod_2_eq_zero mod_div_mult_eq)
-   apply (metis bit_eq_iff mod2_eq_if mod_div_mult_eq)
-  apply (metis bit_eq_iff mod2_eq_if mod_div_mult_eq)
-  done
+  \<open>a = b \<longleftrightarrow> (even a \<longleftrightarrow> even b) \<and> a div 2 = b div 2\<close> (is \<open>?P = ?Q\<close>)
+proof
+  assume ?P
+  then show ?Q
+    by simp
+next
+  assume ?Q
+  then have \<open>even a \<longleftrightarrow> even b\<close> and \<open>a div 2 = b div 2\<close>
+    by simp_all
+  show ?P
+  proof (rule bit_eqI)
+    fix n
+    show \<open>bit a n \<longleftrightarrow> bit b n\<close>
+    proof (cases n)
+      case 0
+      with \<open>even a \<longleftrightarrow> even b\<close> show ?thesis
+        by simp
+    next
+      case (Suc n)
+      moreover from \<open>a div 2 = b div 2\<close> have \<open>bit (a div 2) n = bit (b div 2) n\<close>
+        by simp
+      ultimately show ?thesis
+        by simp
+    qed
+  qed
+qed
 
 lemma bit_mask_iff:
   \<open>bit (2 ^ m - 1) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n < m\<close>
   by (simp add: bit_def even_mask_div_iff not_le)
 
 end
 
 lemma nat_bit_induct [case_names zero even odd]:
   "P n" if zero: "P 0"
     and even: "\<And>n. P n \<Longrightarrow> n > 0 \<Longrightarrow> P (2 * n)"
     and odd: "\<And>n. P n \<Longrightarrow> P (Suc (2 * n))"
 proof (induction n rule: less_induct)
   case (less n)
   show "P n"
   proof (cases "n = 0")
     case True with zero show ?thesis by simp
   next
     case False
     with less have hyp: "P (n div 2)" by simp
     show ?thesis
     proof (cases "even n")
       case True
       then have "n \<noteq> 1"
         by auto
       with \<open>n \<noteq> 0\<close> have "n div 2 > 0"
         by simp
       with \<open>even n\<close> hyp even [of "n div 2"] show ?thesis
         by simp
     next
       case False
       with hyp odd [of "n div 2"] show ?thesis
         by simp
     qed
   qed
 qed
 
 instance nat :: semiring_bits
 proof
   show \<open>P n\<close> if stable: \<open>\<And>n. n div 2 = n \<Longrightarrow> P n\<close>
     and rec: \<open>\<And>n b. P n \<Longrightarrow> (of_bool b + 2 * n) div 2 = n \<Longrightarrow> P (of_bool b + 2 * n)\<close>
     for P and n :: nat
   proof (induction n rule: nat_bit_induct)
     case zero
     from stable [of 0] show ?case
       by simp
   next
     case (even n)
     with rec [of n False] show ?case
       by simp
   next
     case (odd n)
     with rec [of n True] show ?case
       by simp
   qed
   show \<open>q mod 2 ^ m mod 2 ^ n = q mod 2 ^ min m n\<close>
     for q m n :: nat
     apply (auto simp add: less_iff_Suc_add power_add mod_mod_cancel split: split_min_lin)
     apply (metis div_mult2_eq mod_div_trivial mod_eq_self_iff_div_eq_0 mod_mult_self2_is_0 power_commutes)
     done
   show \<open>(q * 2 ^ m) mod (2 ^ n) = (q mod 2 ^ (n - m)) * 2 ^ m\<close> if \<open>m \<le> n\<close>
     for q m n :: nat
     using that
     apply (auto simp add: mod_mod_cancel div_mult2_eq power_add mod_mult2_eq le_iff_add split: split_min_lin)
     apply (simp add: mult.commute)
     done
   show \<open>even ((2 ^ m - (1::nat)) div 2 ^ n) \<longleftrightarrow> 2 ^ n = (0::nat) \<or> m \<le> n\<close>
     for m n :: nat
     using even_mask_div_iff' [where ?'a = nat, of m n] by simp
   show \<open>even (q * 2 ^ m div 2 ^ n) \<longleftrightarrow> n < m \<or> (2::nat) ^ n = 0 \<or> m \<le> n \<and> even (q div 2 ^ (n - m))\<close>
     for m n q r :: nat
     apply (auto simp add: not_less power_add ac_simps dest!: le_Suc_ex)
     apply (metis (full_types) dvd_mult dvd_mult_imp_div dvd_power_iff_le not_less not_less_eq order_refl power_Suc)
     done
 qed (auto simp add: div_mult2_eq mod_mult2_eq power_add power_diff)
 
 lemma int_bit_induct [case_names zero minus even odd]:
   "P k" if zero_int: "P 0"
     and minus_int: "P (- 1)"
     and even_int: "\<And>k. P k \<Longrightarrow> k \<noteq> 0 \<Longrightarrow> P (k * 2)"
     and odd_int: "\<And>k. P k \<Longrightarrow> k \<noteq> - 1 \<Longrightarrow> P (1 + (k * 2))" for k :: int
 proof (cases "k \<ge> 0")
   case True
   define n where "n = nat k"
   with True have "k = int n"
     by simp
   then show "P k"
   proof (induction n arbitrary: k rule: nat_bit_induct)
     case zero
     then show ?case
       by (simp add: zero_int)
   next
     case (even n)
     have "P (int n * 2)"
       by (rule even_int) (use even in simp_all)
     with even show ?case
       by (simp add: ac_simps)
   next
     case (odd n)
     have "P (1 + (int n * 2))"
       by (rule odd_int) (use odd in simp_all)
     with odd show ?case
       by (simp add: ac_simps)
   qed
 next
   case False
   define n where "n = nat (- k - 1)"
   with False have "k = - int n - 1"
     by simp
   then show "P k"
   proof (induction n arbitrary: k rule: nat_bit_induct)
     case zero
     then show ?case
       by (simp add: minus_int)
   next
     case (even n)
     have "P (1 + (- int (Suc n) * 2))"
       by (rule odd_int) (use even in \<open>simp_all add: algebra_simps\<close>)
     also have "\<dots> = - int (2 * n) - 1"
       by (simp add: algebra_simps)
     finally show ?case
       using even by simp
   next
     case (odd n)
     have "P (- int (Suc n) * 2)"
       by (rule even_int) (use odd in \<open>simp_all add: algebra_simps\<close>)
     also have "\<dots> = - int (Suc (2 * n)) - 1"
       by (simp add: algebra_simps)
     finally show ?case
       using odd by simp
   qed
 qed
 
 instance int :: semiring_bits
 proof
   show \<open>P k\<close> if stable: \<open>\<And>k. k div 2 = k \<Longrightarrow> P k\<close>
     and rec: \<open>\<And>k b. P k \<Longrightarrow> (of_bool b + 2 * k) div 2 = k \<Longrightarrow> P (of_bool b + 2 * k)\<close>
     for P and k :: int
   proof (induction k rule: int_bit_induct)
     case zero
     from stable [of 0] show ?case
       by simp
   next
     case minus
     from stable [of \<open>- 1\<close>] show ?case
       by simp
   next
     case (even k)
     with rec [of k False] show ?case
       by (simp add: ac_simps)
   next
     case (odd k)
     with rec [of k True] show ?case
       by (simp add: ac_simps)
   qed
   show \<open>(2::int) ^ m div 2 ^ n = of_bool ((2::int) ^ m \<noteq> 0 \<and> n \<le> m) * 2 ^ (m - n)\<close>
     for m n :: nat
   proof (cases \<open>m < n\<close>)
     case True
     then have \<open>n = m + (n - m)\<close>
       by simp
     then have \<open>(2::int) ^ m div 2 ^ n = (2::int) ^ m div 2 ^ (m + (n - m))\<close>
       by simp
     also have \<open>\<dots> = (2::int) ^ m div (2 ^ m * 2 ^ (n - m))\<close>
       by (simp add: power_add)
     also have \<open>\<dots> = (2::int) ^ m div 2 ^ m div 2 ^ (n - m)\<close>
       by (simp add: zdiv_zmult2_eq)
     finally show ?thesis using \<open>m < n\<close> by simp
   next
     case False
     then show ?thesis
       by (simp add: power_diff)
   qed
   show \<open>k mod 2 ^ m mod 2 ^ n = k mod 2 ^ min m n\<close>
     for m n :: nat and k :: int
     using mod_exp_eq [of \<open>nat k\<close> m n]
     apply (auto simp add: mod_mod_cancel zdiv_zmult2_eq power_add zmod_zmult2_eq le_iff_add split: split_min_lin)
      apply (auto simp add: less_iff_Suc_add mod_mod_cancel power_add)
     apply (simp only: flip: mult.left_commute [of \<open>2 ^ m\<close>])
     apply (subst zmod_zmult2_eq) apply simp_all
     done
   show \<open>(k * 2 ^ m) mod (2 ^ n) = (k mod 2 ^ (n - m)) * 2 ^ m\<close>
     if \<open>m \<le> n\<close> for m n :: nat and k :: int
     using that
     apply (auto simp add: power_add zmod_zmult2_eq le_iff_add split: split_min_lin)
     apply (simp add: ac_simps)
     done
   show \<open>even ((2 ^ m - (1::int)) div 2 ^ n) \<longleftrightarrow> 2 ^ n = (0::int) \<or> m \<le> n\<close>
     for m n :: nat
     using even_mask_div_iff' [where ?'a = int, of m n] by simp
   show \<open>even (k * 2 ^ m div 2 ^ n) \<longleftrightarrow> n < m \<or> (2::int) ^ n = 0 \<or> m \<le> n \<and> even (k div 2 ^ (n - m))\<close>
     for m n :: nat and k l :: int
     apply (auto simp add: not_less power_add ac_simps dest!: le_Suc_ex)
     apply (metis Suc_leI dvd_mult dvd_mult_imp_div dvd_power_le dvd_refl power.simps(2))
     done
 qed (auto simp add: zdiv_zmult2_eq zmod_zmult2_eq power_add power_diff not_le)
 
 class semiring_bit_shifts = semiring_bits +
   fixes push_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   assumes push_bit_eq_mult: \<open>push_bit n a = a * 2 ^ n\<close>
   fixes drop_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   assumes drop_bit_eq_div: \<open>drop_bit n a = a div 2 ^ n\<close>
 begin
 
 definition take_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   where take_bit_eq_mod: \<open>take_bit n a = a mod 2 ^ n\<close>
 
 text \<open>
   Logically, \<^const>\<open>push_bit\<close>,
   \<^const>\<open>drop_bit\<close> and \<^const>\<open>take_bit\<close> are just aliases; having them
   as separate operations makes proofs easier, otherwise proof automation
   would fiddle with concrete expressions \<^term>\<open>2 ^ n\<close> in a way obfuscating the basic
   algebraic relationships between those operations.
   Having
   \<^const>\<open>push_bit\<close> and \<^const>\<open>drop_bit\<close> as definitional class operations
   takes into account that specific instances of these can be implemented
   differently wrt. code generation.
 \<close>
 
 lemma bit_iff_odd_drop_bit:
   \<open>bit a n \<longleftrightarrow> odd (drop_bit n a)\<close>
   by (simp add: bit_def drop_bit_eq_div)
 
 lemma even_drop_bit_iff_not_bit:
   \<open>even (drop_bit n a) \<longleftrightarrow> \<not> bit a n\<close>
   by (simp add: bit_iff_odd_drop_bit)
 
 lemma div_push_bit_of_1_eq_drop_bit:
   \<open>a div push_bit n 1 = drop_bit n a\<close>
   by (simp add: push_bit_eq_mult drop_bit_eq_div)
 
 lemma bits_ident:
   "push_bit n (drop_bit n a) + take_bit n a = a"
   using div_mult_mod_eq by (simp add: push_bit_eq_mult take_bit_eq_mod drop_bit_eq_div)
 
 lemma push_bit_push_bit [simp]:
   "push_bit m (push_bit n a) = push_bit (m + n) a"
   by (simp add: push_bit_eq_mult power_add ac_simps)
 
 lemma push_bit_0_id [simp]:
   "push_bit 0 = id"
   by (simp add: fun_eq_iff push_bit_eq_mult)
 
 lemma push_bit_of_0 [simp]:
   "push_bit n 0 = 0"
   by (simp add: push_bit_eq_mult)
 
 lemma push_bit_of_1:
   "push_bit n 1 = 2 ^ n"
   by (simp add: push_bit_eq_mult)
 
 lemma push_bit_Suc [simp]:
   "push_bit (Suc n) a = push_bit n (a * 2)"
   by (simp add: push_bit_eq_mult ac_simps)
 
 lemma push_bit_double:
   "push_bit n (a * 2) = push_bit n a * 2"
   by (simp add: push_bit_eq_mult ac_simps)
 
 lemma push_bit_add:
   "push_bit n (a + b) = push_bit n a + push_bit n b"
   by (simp add: push_bit_eq_mult algebra_simps)
 
 lemma take_bit_0 [simp]:
   "take_bit 0 a = 0"
   by (simp add: take_bit_eq_mod)
 
 lemma take_bit_Suc [simp]:
   \<open>take_bit (Suc n) a = take_bit n (a div 2) * 2 + of_bool (odd a)\<close>
 proof -
   have \<open>take_bit (Suc n) (a div 2 * 2 + of_bool (odd a)) = take_bit n (a div 2) * 2 + of_bool (odd a)\<close>
     using even_succ_mod_exp [of \<open>2 * (a div 2)\<close> \<open>Suc n\<close>]
       mult_exp_mod_exp_eq [of 1 \<open>Suc n\<close> \<open>a div 2\<close>]
     by (auto simp add: take_bit_eq_mod ac_simps)
   then show ?thesis
     using div_mult_mod_eq [of a 2] by (simp add: mod_2_eq_odd)
 qed
 
 lemma take_bit_of_0 [simp]:
   "take_bit n 0 = 0"
   by (simp add: take_bit_eq_mod)
 
 lemma take_bit_of_1 [simp]:
   "take_bit n 1 = of_bool (n > 0)"
   by (cases n) simp_all
 
 lemma drop_bit_of_0 [simp]:
   "drop_bit n 0 = 0"
   by (simp add: drop_bit_eq_div)
 
 lemma drop_bit_of_1 [simp]:
   "drop_bit n 1 = of_bool (n = 0)"
   by (simp add: drop_bit_eq_div)
 
 lemma drop_bit_0 [simp]:
   "drop_bit 0 = id"
   by (simp add: fun_eq_iff drop_bit_eq_div)
 
 lemma drop_bit_Suc [simp]:
   "drop_bit (Suc n) a = drop_bit n (a div 2)"
   using div_exp_eq [of a 1] by (simp add: drop_bit_eq_div)
 
 lemma drop_bit_half:
   "drop_bit n (a div 2) = drop_bit n a div 2"
   by (induction n arbitrary: a) simp_all
 
 lemma drop_bit_of_bool [simp]:
   "drop_bit n (of_bool d) = of_bool (n = 0 \<and> d)"
   by (cases n) simp_all
 
 lemma take_bit_eq_0_imp_dvd:
   "take_bit n a = 0 \<Longrightarrow> 2 ^ n dvd a"
   by (simp add: take_bit_eq_mod mod_0_imp_dvd)
 
 lemma even_take_bit_eq [simp]:
   \<open>even (take_bit n a) \<longleftrightarrow> n = 0 \<or> even a\<close>
   by (cases n) simp_all
 
 lemma take_bit_take_bit [simp]:
   "take_bit m (take_bit n a) = take_bit (min m n) a"
   by (simp add: take_bit_eq_mod mod_exp_eq ac_simps)
 
 lemma drop_bit_drop_bit [simp]:
   "drop_bit m (drop_bit n a) = drop_bit (m + n) a"
   by (simp add: drop_bit_eq_div power_add div_exp_eq ac_simps)
 
 lemma push_bit_take_bit:
   "push_bit m (take_bit n a) = take_bit (m + n) (push_bit m a)"
   apply (simp add: push_bit_eq_mult take_bit_eq_mod power_add ac_simps)
   using mult_exp_mod_exp_eq [of m \<open>m + n\<close> a] apply (simp add: ac_simps power_add)
   done
 
 lemma take_bit_push_bit:
   "take_bit m (push_bit n a) = push_bit n (take_bit (m - n) a)"
 proof (cases "m \<le> n")
   case True
   then show ?thesis
     apply (simp add:)
     apply (simp_all add: push_bit_eq_mult take_bit_eq_mod)
     apply (auto dest!: le_Suc_ex simp add: power_add ac_simps)
     using mult_exp_mod_exp_eq [of m m \<open>a * 2 ^ n\<close> for n]
     apply (simp add: ac_simps)
     done
 next
   case False
   then show ?thesis
     using push_bit_take_bit [of n "m - n" a]
     by simp
 qed
 
 lemma take_bit_drop_bit:
   "take_bit m (drop_bit n a) = drop_bit n (take_bit (m + n) a)"
   by (simp add: drop_bit_eq_div take_bit_eq_mod ac_simps div_exp_mod_exp_eq)
 
 lemma drop_bit_take_bit:
   "drop_bit m (take_bit n a) = take_bit (n - m) (drop_bit m a)"
 proof (cases "m \<le> n")
   case True
   then show ?thesis
     using take_bit_drop_bit [of "n - m" m a] by simp
 next
   case False
   then obtain q where \<open>m = n + q\<close>
     by (auto simp add: not_le dest: less_imp_Suc_add)
   then have \<open>drop_bit m (take_bit n a) = 0\<close>
     using div_exp_eq [of \<open>a mod 2 ^ n\<close> n q]
     by (simp add: take_bit_eq_mod drop_bit_eq_div)
   with False show ?thesis
     by simp
 qed
 
 lemma even_push_bit_iff [simp]:
   \<open>even (push_bit n a) \<longleftrightarrow> n \<noteq> 0 \<or> even a\<close>
   by (simp add: push_bit_eq_mult) auto
 
 lemma bit_push_bit_iff:
   \<open>bit (push_bit m a) n \<longleftrightarrow> n \<ge> m \<and> 2 ^ n \<noteq> 0 \<and> (n < m \<or> bit a (n - m))\<close>
   by (auto simp add: bit_def push_bit_eq_mult even_mult_exp_div_exp_iff)
 
 lemma bit_drop_bit_eq:
   \<open>bit (drop_bit n a) = bit a \<circ> (+) n\<close>
   by (simp add: bit_def fun_eq_iff ac_simps flip: drop_bit_eq_div)
 
 lemma bit_take_bit_iff:
   \<open>bit (take_bit m a) n \<longleftrightarrow> n < m \<and> bit a n\<close>
   by (simp add: bit_def drop_bit_take_bit not_le flip: drop_bit_eq_div)
 
 end
 
 instantiation nat :: semiring_bit_shifts
 begin
 
 definition push_bit_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close>
   where \<open>push_bit_nat n m = m * 2 ^ n\<close>
 
 definition drop_bit_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close>
   where \<open>drop_bit_nat n m = m div 2 ^ n\<close>
 
 instance proof
   show \<open>push_bit n m = m * 2 ^ n\<close> for n m :: nat
     by (simp add: push_bit_nat_def)
   show \<open>drop_bit n m = m div 2 ^ n\<close> for n m :: nat
     by (simp add: drop_bit_nat_def)
 qed
 
 end
 
 instantiation int :: semiring_bit_shifts
 begin
 
 definition push_bit_int :: \<open>nat \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>push_bit_int n k = k * 2 ^ n\<close>
 
 definition drop_bit_int :: \<open>nat \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>drop_bit_int n k = k div 2 ^ n\<close>
 
 instance proof
   show \<open>push_bit n k = k * 2 ^ n\<close> for n :: nat and k :: int
     by (simp add: push_bit_int_def)
   show \<open>drop_bit n k = k div 2 ^ n\<close> for n :: nat and k :: int
     by (simp add: drop_bit_int_def)
 qed
 
 end
 
 lemma bit_push_bit_iff_nat:
   \<open>bit (push_bit m q) n \<longleftrightarrow> m \<le> n \<and> bit q (n - m)\<close> for q :: nat
   by (auto simp add: bit_push_bit_iff)
 
 lemma bit_push_bit_iff_int:
   \<open>bit (push_bit m k) n \<longleftrightarrow> m \<le> n \<and> bit k (n - m)\<close> for k :: int
   by (auto simp add: bit_push_bit_iff)
 
 class unique_euclidean_semiring_with_bit_shifts =
   unique_euclidean_semiring_with_nat + semiring_bit_shifts
 begin
 
 lemma take_bit_of_exp [simp]:
   \<open>take_bit m (2 ^ n) = of_bool (n < m) * 2 ^ n\<close>
   by (simp add: take_bit_eq_mod exp_mod_exp)
 
 lemma take_bit_of_2 [simp]:
   \<open>take_bit n 2 = of_bool (2 \<le> n) * 2\<close>
   using take_bit_of_exp [of n 1] by simp
 
 lemma take_bit_of_mask:
   \<open>take_bit m (2 ^ n - 1) = 2 ^ min m n - 1\<close>
   by (simp add: take_bit_eq_mod mask_mod_exp)
 
 lemma push_bit_eq_0_iff [simp]:
   "push_bit n a = 0 \<longleftrightarrow> a = 0"
   by (simp add: push_bit_eq_mult)
 
 lemma push_bit_numeral [simp]:
   "push_bit (numeral l) (numeral k) = push_bit (pred_numeral l) (numeral (Num.Bit0 k))"
   by (simp only: numeral_eq_Suc power_Suc numeral_Bit0 [of k] mult_2 [symmetric]) (simp add: ac_simps)
 
 lemma push_bit_of_nat:
   "push_bit n (of_nat m) = of_nat (push_bit n m)"
   by (simp add: push_bit_eq_mult Parity.push_bit_eq_mult)
 
 lemma take_bit_add:
   "take_bit n (take_bit n a + take_bit n b) = take_bit n (a + b)"
   by (simp add: take_bit_eq_mod mod_simps)
 
 lemma take_bit_eq_0_iff:
   "take_bit n a = 0 \<longleftrightarrow> 2 ^ n dvd a"
   by (simp add: take_bit_eq_mod mod_eq_0_iff_dvd)
 
 lemma take_bit_of_1_eq_0_iff [simp]:
   "take_bit n 1 = 0 \<longleftrightarrow> n = 0"
   by (simp add: take_bit_eq_mod)
 
 lemma take_bit_numeral_bit0 [simp]:
   "take_bit (numeral l) (numeral (Num.Bit0 k)) = take_bit (pred_numeral l) (numeral k) * 2"
   by (simp only: numeral_eq_Suc power_Suc numeral_Bit0 [of k] mult_2 [symmetric] take_bit_Suc
     ac_simps even_mult_iff nonzero_mult_div_cancel_right [OF numeral_neq_zero]) simp
 
 lemma take_bit_numeral_bit1 [simp]:
   "take_bit (numeral l) (numeral (Num.Bit1 k)) = take_bit (pred_numeral l) (numeral k) * 2 + 1"
   by (simp only: numeral_eq_Suc power_Suc numeral_Bit1 [of k] mult_2 [symmetric] take_bit_Suc
     ac_simps even_add even_mult_iff div_mult_self1 [OF numeral_neq_zero]) (simp add: ac_simps)
 
 lemma take_bit_of_nat:
   "take_bit n (of_nat m) = of_nat (take_bit n m)"
   by (simp add: take_bit_eq_mod Parity.take_bit_eq_mod of_nat_mod [of m "2 ^ n"])
 
 lemma drop_bit_numeral_bit0 [simp]:
   "drop_bit (numeral l) (numeral (Num.Bit0 k)) = drop_bit (pred_numeral l) (numeral k)"
   by (simp only: numeral_eq_Suc power_Suc numeral_Bit0 [of k] mult_2 [symmetric] drop_bit_Suc
     nonzero_mult_div_cancel_left [OF numeral_neq_zero])
 
 lemma drop_bit_numeral_bit1 [simp]:
   "drop_bit (numeral l) (numeral (Num.Bit1 k)) = drop_bit (pred_numeral l) (numeral k)"
   by (simp only: numeral_eq_Suc power_Suc numeral_Bit1 [of k] mult_2 [symmetric] drop_bit_Suc
     div_mult_self4 [OF numeral_neq_zero]) simp
 
 lemma drop_bit_of_nat:
   "drop_bit n (of_nat m) = of_nat (drop_bit n m)"
   by (simp add: drop_bit_eq_div Parity.drop_bit_eq_div of_nat_div [of m "2 ^ n"])
 
 lemma bit_of_nat_iff_bit [simp]:
   \<open>bit (of_nat m) n \<longleftrightarrow> bit m n\<close>
 proof -
   have \<open>even (m div 2 ^ n) \<longleftrightarrow> even (of_nat (m div 2 ^ n))\<close>
     by simp
   also have \<open>of_nat (m div 2 ^ n) = of_nat m div of_nat (2 ^ n)\<close>
     by (simp add: of_nat_div)
   finally show ?thesis
     by (simp add: bit_def semiring_bits_class.bit_def)
 qed
 
 lemma of_nat_push_bit:
   \<open>of_nat (push_bit m n) = push_bit m (of_nat n)\<close>
   by (simp add: push_bit_eq_mult semiring_bit_shifts_class.push_bit_eq_mult)
 
 lemma of_nat_drop_bit:
   \<open>of_nat (drop_bit m n) = drop_bit m (of_nat n)\<close>
   by (simp add: drop_bit_eq_div semiring_bit_shifts_class.drop_bit_eq_div of_nat_div)
 
 lemma of_nat_take_bit:
   \<open>of_nat (take_bit m n) = take_bit m (of_nat n)\<close>
   by (simp add: take_bit_eq_mod semiring_bit_shifts_class.take_bit_eq_mod of_nat_mod)
 
 lemma bit_push_bit_iff_of_nat_iff:
   \<open>bit (push_bit m (of_nat r)) n \<longleftrightarrow> m \<le> n \<and> bit (of_nat r) (n - m)\<close>
   by (auto simp add: bit_push_bit_iff)
 
 end
 
 instance nat :: unique_euclidean_semiring_with_bit_shifts ..
 
 instance int :: unique_euclidean_semiring_with_bit_shifts ..
 
 lemma push_bit_of_Suc_0 [simp]:
   "push_bit n (Suc 0) = 2 ^ n"
   using push_bit_of_1 [where ?'a = nat] by simp
 
 lemma take_bit_of_Suc_0 [simp]:
   "take_bit n (Suc 0) = of_bool (0 < n)"
   using take_bit_of_1 [where ?'a = nat] by simp
 
 lemma drop_bit_of_Suc_0 [simp]:
   "drop_bit n (Suc 0) = of_bool (n = 0)"
   using drop_bit_of_1 [where ?'a = nat] by simp
 
 lemma take_bit_eq_self:
   \<open>take_bit n m = m\<close> if \<open>m < 2 ^ n\<close> for n m :: nat
   using that by (simp add: take_bit_eq_mod)
 
 lemma push_bit_minus_one:
   "push_bit n (- 1 :: int) = - (2 ^ n)"
   by (simp add: push_bit_eq_mult)
 
 lemma minus_1_div_exp_eq_int:
   \<open>- 1 div (2 :: int) ^ n = - 1\<close>
   by (induction n) (use div_exp_eq [symmetric, of \<open>- 1 :: int\<close> 1] in \<open>simp_all add: ac_simps\<close>)
 
 lemma drop_bit_minus_one [simp]:
   \<open>drop_bit n (- 1 :: int) = - 1\<close>
   by (simp add: drop_bit_eq_div minus_1_div_exp_eq_int)
 
 lemma take_bit_minus:
   "take_bit n (- (take_bit n k)) = take_bit n (- k)"
     for k :: int
   by (simp add: take_bit_eq_mod mod_minus_eq)
 
 lemma take_bit_diff:
   "take_bit n (take_bit n k - take_bit n l) = take_bit n (k - l)"
     for k l :: int
   by (simp add: take_bit_eq_mod mod_diff_eq)
 
 lemma take_bit_nonnegative [simp]:
   "take_bit n k \<ge> 0"
     for k :: int
   by (simp add: take_bit_eq_mod)
 
 lemma drop_bit_push_bit_int:
   \<open>drop_bit m (push_bit n k) = drop_bit (m - n) (push_bit (n - m) k)\<close> for k :: int
   by (cases \<open>m \<le> n\<close>) (auto simp add: mult.left_commute [of _ \<open>2 ^ n\<close>] mult.commute [of _ \<open>2 ^ n\<close>] mult.assoc
      mult.commute [of k] drop_bit_eq_div push_bit_eq_mult not_le power_add dest!: le_Suc_ex less_imp_Suc_add)
 
 end