diff --git a/src/HOL/Divides.thy b/src/HOL/Divides.thy
--- a/src/HOL/Divides.thy
+++ b/src/HOL/Divides.thy
@@ -1,1338 +1,1280 @@
 (*  Title:      HOL/Divides.thy
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1999  University of Cambridge
 *)
 
 section \<open>More on quotient and remainder\<close>
 
 theory Divides
 imports Parity
 begin
 
 subsection \<open>More on division\<close>
 
 inductive eucl_rel_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool"
   where eucl_rel_int_by0: "eucl_rel_int k 0 (0, k)"
   | eucl_rel_int_dividesI: "l \<noteq> 0 \<Longrightarrow> k = q * l \<Longrightarrow> eucl_rel_int k l (q, 0)"
   | eucl_rel_int_remainderI: "sgn r = sgn l \<Longrightarrow> \<bar>r\<bar> < \<bar>l\<bar>
       \<Longrightarrow> k = q * l + r \<Longrightarrow> eucl_rel_int k l (q, r)"
 
 lemma eucl_rel_int_iff:    
   "eucl_rel_int k l (q, r) \<longleftrightarrow> 
     k = l * q + r \<and>
      (if 0 < l then 0 \<le> r \<and> r < l else if l < 0 then l < r \<and> r \<le> 0 else q = 0)"
   by (cases "r = 0")
     (auto elim!: eucl_rel_int.cases intro: eucl_rel_int_by0 eucl_rel_int_dividesI eucl_rel_int_remainderI
     simp add: ac_simps sgn_1_pos sgn_1_neg)
 
 lemma unique_quotient_lemma:
   assumes "b * q' + r' \<le> b * q + r" "0 \<le> r'" "r' < b" "r < b" shows "q' \<le> (q::int)"
 proof -
   have "r' + b * (q'-q) \<le> r"
     using assms by (simp add: right_diff_distrib)
   moreover have "0 < b * (1 + q - q') "
     using assms by (simp add: right_diff_distrib distrib_left)
   moreover have "b * q' < b * (1 + q)"
     using assms by (simp add: right_diff_distrib distrib_left)
   ultimately show ?thesis
     using assms by (simp add: mult_less_cancel_left)
 qed
 
 lemma unique_quotient_lemma_neg:
   "b * q' + r' \<le> b*q + r \<Longrightarrow> r \<le> 0 \<Longrightarrow> b < r \<Longrightarrow> b < r' \<Longrightarrow> q \<le> (q'::int)"
   by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma) auto
 
 lemma unique_quotient:
   "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> q = q'"
   apply (rule order_antisym)
    apply (simp_all add: eucl_rel_int_iff linorder_neq_iff split: if_split_asm)
      apply (blast intro: order_eq_refl [THEN unique_quotient_lemma] order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
   done
 
 lemma unique_remainder:
   "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> r = r'"
 apply (subgoal_tac "q = q'")
  apply (simp add: eucl_rel_int_iff)
 apply (blast intro: unique_quotient)
 done
 
 lemma eucl_rel_int:
   "eucl_rel_int k l (k div l, k mod l)"
 proof (cases k rule: int_cases3)
   case zero
   then show ?thesis
     by (simp add: eucl_rel_int_iff divide_int_def modulo_int_def)
 next
   case (pos n)
   then show ?thesis
     using div_mult_mod_eq [of n]
     by (cases l rule: int_cases3)
       (auto simp del: of_nat_mult of_nat_add
         simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
         eucl_rel_int_iff divide_int_def modulo_int_def)
 next
   case (neg n)
   then show ?thesis
     using div_mult_mod_eq [of n]
     by (cases l rule: int_cases3)
       (auto simp del: of_nat_mult of_nat_add
         simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
         eucl_rel_int_iff divide_int_def modulo_int_def)
 qed
 
 lemma divmod_int_unique:
   assumes "eucl_rel_int k l (q, r)"
   shows div_int_unique: "k div l = q" and mod_int_unique: "k mod l = r"
   using assms eucl_rel_int [of k l]
   using unique_quotient [of k l] unique_remainder [of k l]
   by auto
 
 lemma div_abs_eq_div_nat:
   "\<bar>k\<bar> div \<bar>l\<bar> = int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)"
   by (simp add: divide_int_def)
 
 lemma mod_abs_eq_div_nat:
   "\<bar>k\<bar> mod \<bar>l\<bar> = int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)"
   by (simp add: modulo_int_def)
 
 lemma zdiv_int:
   "int (a div b) = int a div int b"
   by (simp add: divide_int_def)
 
 lemma zmod_int:
   "int (a mod b) = int a mod int b"
   by (simp add: modulo_int_def)
 
 lemma div_sgn_abs_cancel:
   fixes k l v :: int
   assumes "v \<noteq> 0"
   shows "(sgn v * \<bar>k\<bar>) div (sgn v * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
 proof -
   from assms have "sgn v = - 1 \<or> sgn v = 1"
     by (cases "v \<ge> 0") auto
   then show ?thesis
     using assms unfolding divide_int_def [of "sgn v * \<bar>k\<bar>" "sgn v * \<bar>l\<bar>"]
     by (fastforce simp add: not_less div_abs_eq_div_nat)
 qed
 
 lemma div_eq_sgn_abs:
   fixes k l v :: int
   assumes "sgn k = sgn l"
   shows "k div l = \<bar>k\<bar> div \<bar>l\<bar>"
 proof (cases "l = 0")
   case True
   then show ?thesis
     by simp
 next
   case False
   with assms have "(sgn k * \<bar>k\<bar>) div (sgn l * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
     using div_sgn_abs_cancel [of l k l] by simp
   then show ?thesis
     by (simp add: sgn_mult_abs)
 qed
 
 lemma div_dvd_sgn_abs:
   fixes k l :: int
   assumes "l dvd k"
   shows "k div l = (sgn k * sgn l) * (\<bar>k\<bar> div \<bar>l\<bar>)"
 proof (cases "k = 0 \<or> l = 0")
   case True
   then show ?thesis
     by auto
 next
   case False
   then have "k \<noteq> 0" and "l \<noteq> 0"
     by auto
   show ?thesis
   proof (cases "sgn l = sgn k")
     case True
     then show ?thesis
       by (simp add: div_eq_sgn_abs)
   next
     case False
     with \<open>k \<noteq> 0\<close> \<open>l \<noteq> 0\<close>
     have "sgn l * sgn k = - 1"
       by (simp add: sgn_if split: if_splits)
     with assms show ?thesis
       unfolding divide_int_def [of k l]
       by (auto simp add: zdiv_int ac_simps)
   qed
 qed
 
 lemma div_noneq_sgn_abs:
   fixes k l :: int
   assumes "l \<noteq> 0"
   assumes "sgn k \<noteq> sgn l"
   shows "k div l = - (\<bar>k\<bar> div \<bar>l\<bar>) - of_bool (\<not> l dvd k)"
   using assms
   by (simp only: divide_int_def [of k l], auto simp add: not_less zdiv_int)
   
 
-subsubsection \<open>General Properties of div and mod\<close>
-
-lemma div_pos_pos_trivial [simp]:
-  "k div l = 0" if "k \<ge> 0" and "k < l" for k l :: int
-  using that by (simp add: unique_euclidean_semiring_class.div_eq_0_iff division_segment_int_def)
-
-lemma mod_pos_pos_trivial [simp]:
-  "k mod l = k" if "k \<ge> 0" and "k < l" for k l :: int
-  using that by (simp add: mod_eq_self_iff_div_eq_0)
-
-lemma div_neg_neg_trivial [simp]:
-  "k div l = 0" if "k \<le> 0" and "l < k" for k l :: int
-  using that by (cases "k = 0") (simp, simp add: unique_euclidean_semiring_class.div_eq_0_iff division_segment_int_def)
-
-lemma mod_neg_neg_trivial [simp]:
-  "k mod l = k" if "k \<le> 0" and "l < k" for k l :: int
-  using that by (simp add: mod_eq_self_iff_div_eq_0)
-
-lemma div_pos_neg_trivial:
-  "k div l = - 1" if "0 < k" and "k + l \<le> 0" for k l :: int
-  apply (rule div_int_unique [of _ _ _ "k + l"])
-  apply (use that in \<open>auto simp add: eucl_rel_int_iff\<close>)
-  done
-
-lemma mod_pos_neg_trivial:
-  "k mod l = k + l" if "0 < k" and "k + l \<le> 0" for k l :: int
-  apply (rule mod_int_unique [of _ _ "- 1"])
-  apply (use that in \<open>auto simp add: eucl_rel_int_iff\<close>)
-  done
-
-text \<open>There is neither \<open>div_neg_pos_trivial\<close> nor \<open>mod_neg_pos_trivial\<close>
-  because \<^term>\<open>0 div l = 0\<close> would supersede it.\<close>
-
-
-
 subsubsection \<open>Laws for div and mod with Unary Minus\<close>
 
 lemma zminus1_lemma:
      "eucl_rel_int a b (q, r) ==> b \<noteq> 0
       ==> eucl_rel_int (-a) b (if r=0 then -q else -q - 1,
                           if r=0 then 0 else b-r)"
 by (force simp add: eucl_rel_int_iff right_diff_distrib)
 
 
 lemma zdiv_zminus1_eq_if:
      "b \<noteq> (0::int)
       \<Longrightarrow> (-a) div b = (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
 by (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN div_int_unique])
 
 lemma zmod_zminus1_eq_if:
      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
 proof (cases "b = 0")
   case False
   then show ?thesis
     by (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN mod_int_unique])
 qed auto
 
 lemma zmod_zminus1_not_zero:
   fixes k l :: int
   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
   by (simp add: mod_eq_0_iff_dvd)
 
 lemma zmod_zminus2_not_zero:
   fixes k l :: int
   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
   by (simp add: mod_eq_0_iff_dvd)
 
 lemma zdiv_zminus2_eq_if:
   "b \<noteq> (0::int)
       ==> a div (-b) =
           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
   by (auto simp add: zdiv_zminus1_eq_if div_minus_right)
 
 lemma zmod_zminus2_eq_if:
   "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
   by (auto simp add: zmod_zminus1_eq_if mod_minus_right)
 
 
 subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
 
 lemma zdiv_mono1:
   fixes b::int
   assumes "a \<le> a'" "0 < b" shows "a div b \<le> a' div b"
 proof (rule unique_quotient_lemma)
   show "b * (a div b) + a mod b \<le> b * (a' div b) + a' mod b"
     using assms(1) by auto
 qed (use assms in auto)
 
 lemma zdiv_mono1_neg:
   fixes b::int
   assumes "a \<le> a'" "b < 0" shows "a' div b \<le> a div b"
 proof (rule unique_quotient_lemma_neg)
   show "b * (a div b) + a mod b \<le> b * (a' div b) + a' mod b"
     using assms(1) by auto
 qed (use assms in auto)
 
 
 subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
 
 lemma q_pos_lemma:
   fixes q'::int
   assumes "0 \<le> b'*q' + r'" "r' < b'" "0 < b'"
   shows "0 \<le> q'"
 proof -
   have "0 < b'* (q' + 1)"
     using assms by (simp add: distrib_left)
   with assms show ?thesis
     by (simp add: zero_less_mult_iff)
 qed
 
 lemma zdiv_mono2_lemma:
   fixes q'::int
   assumes eq: "b*q + r = b'*q' + r'" and le: "0 \<le> b'*q' + r'" and "r' < b'" "0 \<le> r" "0 < b'" "b' \<le> b"
   shows "q \<le> q'"
 proof -
   have "0 \<le> q'"
     using q_pos_lemma le \<open>r' < b'\<close> \<open>0 < b'\<close> by blast
   moreover have "b*q = r' - r + b'*q'"
     using eq by linarith
   ultimately have "b*q < b* (q' + 1)"
     using mult_right_mono assms unfolding distrib_left by fastforce
   with assms show ?thesis
     by (simp add: mult_less_cancel_left_pos)
 qed
 
 lemma zdiv_mono2:
   fixes a::int
   assumes "0 \<le> a" "0 < b'" "b' \<le> b" shows "a div b \<le> a div b'"
 proof (rule zdiv_mono2_lemma)
   have "b \<noteq> 0"
     using assms by linarith
   show "b * (a div b) + a mod b = b' * (a div b') + a mod b'"
     by simp
 qed (use assms in auto)
 
 lemma zdiv_mono2_neg_lemma:
     fixes q'::int
     assumes "b*q + r = b'*q' + r'" "b'*q' + r' < 0" "r < b" "0 \<le> r'" "0 < b'" "b' \<le> b"
     shows "q' \<le> q"
 proof -
   have "b'*q' < 0"
     using assms by linarith
   with assms have "q' \<le> 0"
     by (simp add: mult_less_0_iff)
   have "b*q' \<le> b'*q'"
     by (simp add: \<open>q' \<le> 0\<close> assms(6) mult_right_mono_neg)
   then have "b*q' < b* (q + 1)"
     using assms by (simp add: distrib_left)
   then show ?thesis
     using assms by (simp add: mult_less_cancel_left)
 qed
 
 lemma zdiv_mono2_neg:
   fixes a::int
   assumes "a < 0" "0 < b'" "b' \<le> b" shows "a div b' \<le> a div b"
 proof (rule zdiv_mono2_neg_lemma)
   have "b \<noteq> 0"
     using assms by linarith
   show "b * (a div b) + a mod b = b' * (a div b') + a mod b'"
     by simp
 qed (use assms in auto)
 
 lemma div_pos_geq:
   fixes k l :: int
   assumes "0 < l" and "l \<le> k"
   shows "k div l = (k - l) div l + 1"
 proof -
   have "k = (k - l) + l" by simp
   then obtain j where k: "k = j + l" ..
   with assms show ?thesis by (simp add: div_add_self2)
 qed
 
 lemma mod_pos_geq:
   fixes k l :: int
   assumes "0 < l" and "l \<le> k"
   shows "k mod l = (k - l) mod l"
 proof -
   have "k = (k - l) + l" by simp
   then obtain j where k: "k = j + l" ..
   with assms show ?thesis by simp
 qed
 
 
 subsubsection \<open>Splitting Rules for div and mod\<close>
 
 text\<open>The proofs of the two lemmas below are essentially identical\<close>
 
 lemma split_pos_lemma:
  "0<k \<Longrightarrow>
     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j \<and> j<k \<and> n = k*i + j \<longrightarrow> P i j)"
   by auto
 
 lemma split_neg_lemma:
  "k<0 \<Longrightarrow>
     P(n div k :: int)(n mod k) = (\<forall>i j. k<j \<and> j\<le>0 \<and> n = k*i + j \<longrightarrow> P i j)"
   by auto
 
 lemma split_zdiv:
  "P(n div k :: int) =
   ((k = 0 \<longrightarrow> P 0) \<and>
    (0<k \<longrightarrow> (\<forall>i j. 0\<le>j \<and> j<k \<and> n = k*i + j \<longrightarrow> P i)) \<and>
    (k<0 \<longrightarrow> (\<forall>i j. k<j \<and> j\<le>0 \<and> n = k*i + j \<longrightarrow> P i)))"
 proof (cases "k = 0")
   case False
   then show ?thesis
     unfolding linorder_neq_iff
     by (auto simp add: split_pos_lemma [of concl: "\<lambda>x y. P x"] split_neg_lemma [of concl: "\<lambda>x y. P x"])
 qed auto
 
 lemma split_zmod:
  "P(n mod k :: int) =
   ((k = 0 \<longrightarrow> P n) \<and>
    (0<k \<longrightarrow> (\<forall>i j. 0\<le>j \<and> j<k \<and> n = k*i + j \<longrightarrow> P j)) \<and>
    (k<0 \<longrightarrow> (\<forall>i j. k<j \<and> j\<le>0 \<and> n = k*i + j \<longrightarrow> P j)))"
 proof (cases "k = 0")
   case False
   then show ?thesis
     unfolding linorder_neq_iff
     by (auto simp add: split_pos_lemma [of concl: "\<lambda>x y. P y"] split_neg_lemma [of concl: "\<lambda>x y. P y"])
 qed auto
 
 text \<open>Enable (lin)arith to deal with \<^const>\<open>divide\<close> and \<^const>\<open>modulo\<close>
   when these are applied to some constant that is of the form
   \<^term>\<open>numeral k\<close>:\<close>
 declare split_zdiv [of _ _ "numeral k", arith_split] for k
 declare split_zmod [of _ _ "numeral k", arith_split] for k
 
 
 subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
 
 lemma pos_eucl_rel_int_mult_2:
   assumes "0 \<le> b"
   assumes "eucl_rel_int a b (q, r)"
   shows "eucl_rel_int (1 + 2*a) (2*b) (q, 1 + 2*r)"
   using assms unfolding eucl_rel_int_iff by auto
 
 lemma neg_eucl_rel_int_mult_2:
   assumes "b \<le> 0"
   assumes "eucl_rel_int (a + 1) b (q, r)"
   shows "eucl_rel_int (1 + 2*a) (2*b) (q, 2*r - 1)"
   using assms unfolding eucl_rel_int_iff by auto
 
 text\<open>computing div by shifting\<close>
 
 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
   using pos_eucl_rel_int_mult_2 [OF _ eucl_rel_int]
   by (rule div_int_unique)
 
 lemma neg_zdiv_mult_2:
   assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
   using neg_eucl_rel_int_mult_2 [OF A eucl_rel_int]
   by (rule div_int_unique)
 
 lemma zdiv_numeral_Bit0 [simp]:
   "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
     numeral v div (numeral w :: int)"
   unfolding numeral.simps unfolding mult_2 [symmetric]
   by (rule div_mult_mult1, simp)
 
 lemma zdiv_numeral_Bit1 [simp]:
   "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
     (numeral v div (numeral w :: int))"
   unfolding numeral.simps
   unfolding mult_2 [symmetric] add.commute [of _ 1]
   by (rule pos_zdiv_mult_2, simp)
 
 lemma pos_zmod_mult_2:
   fixes a b :: int
   assumes "0 \<le> a"
   shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
   using pos_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
   by (rule mod_int_unique)
 
 lemma neg_zmod_mult_2:
   fixes a b :: int
   assumes "a \<le> 0"
   shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
   using neg_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
   by (rule mod_int_unique)
 
 lemma zmod_numeral_Bit0 [simp]:
   "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
     (2::int) * (numeral v mod numeral w)"
   unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
   unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
 
 lemma zmod_numeral_Bit1 [simp]:
   "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
     2 * (numeral v mod numeral w) + (1::int)"
   unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
   unfolding mult_2 [symmetric] add.commute [of _ 1]
   by (rule pos_zmod_mult_2, simp)
 
 lemma zdiv_eq_0_iff:
   "i div k = 0 \<longleftrightarrow> k = 0 \<or> 0 \<le> i \<and> i < k \<or> i \<le> 0 \<and> k < i" (is "?L = ?R")
   for i k :: int
 proof
   assume ?L
   moreover have "?L \<longrightarrow> ?R"
     by (rule split_zdiv [THEN iffD2]) simp
   ultimately show ?R
     by blast
 next
   assume ?R then show ?L
     by auto
 qed
 
 lemma zmod_trival_iff:
   fixes i k :: int
   shows "i mod k = i \<longleftrightarrow> k = 0 \<or> 0 \<le> i \<and> i < k \<or> i \<le> 0 \<and> k < i"
 proof -
   have "i mod k = i \<longleftrightarrow> i div k = 0"
     by safe (insert div_mult_mod_eq [of i k], auto)
   with zdiv_eq_0_iff
   show ?thesis
     by simp
 qed
 
   
 subsubsection \<open>Quotients of Signs\<close>
 
 lemma div_eq_minus1: "0 < b \<Longrightarrow> - 1 div b = - 1" for b :: int
   by (simp add: divide_int_def)
 
 lemma zmod_minus1: "0 < b \<Longrightarrow> - 1 mod b = b - 1" for b :: int
   by (auto simp add: modulo_int_def)
 
 lemma minus_mod_int_eq:
   \<open>- k mod l = l - 1 - (k - 1) mod l\<close> if \<open>l \<ge> 0\<close> for k l :: int
 proof (cases \<open>l = 0\<close>)
   case True
   then show ?thesis
     by simp
 next
   case False
   with that have \<open>l > 0\<close>
     by simp
   then show ?thesis
   proof (cases \<open>l dvd k\<close>)
     case True
     then obtain j where \<open>k = l * j\<close> ..
     moreover have \<open>(l * j mod l - 1) mod l = l - 1\<close>
       using \<open>l > 0\<close> by (simp add: zmod_minus1)
     then have \<open>(l * j - 1) mod l = l - 1\<close>
       by (simp only: mod_simps)
     ultimately show ?thesis
       by simp
   next
     case False
     moreover have \<open>0 < k mod l\<close> \<open>k mod l < 1 + l\<close>
       using \<open>0 < l\<close> le_imp_0_less False apply auto
       using le_less apply fastforce
       using pos_mod_bound [of l k] apply linarith 
       done
     with \<open>l > 0\<close> have \<open>(k mod l - 1) mod l = k mod l - 1\<close>
       by (simp add: zmod_trival_iff)
     ultimately show ?thesis
       apply (simp only: zmod_zminus1_eq_if)
       apply (simp add: mod_eq_0_iff_dvd algebra_simps mod_simps)
       done
   qed
 qed
 
 lemma div_neg_pos_less0:
   fixes a::int
   assumes "a < 0" "0 < b" 
   shows "a div b < 0"
 proof -
   have "a div b \<le> - 1 div b"
     using zdiv_mono1 assms by auto
   also have "... \<le> -1"
     by (simp add: assms(2) div_eq_minus1)
   finally show ?thesis 
     by force
 qed
 
 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
   by (drule zdiv_mono1_neg, auto)
 
 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
   by (drule zdiv_mono1, auto)
 
 text\<open>Now for some equivalences of the form \<open>a div b >=< 0 \<longleftrightarrow> \<dots>\<close>
 conditional upon the sign of \<open>a\<close> or \<open>b\<close>. There are many more.
 They should all be simp rules unless that causes too much search.\<close>
 
 lemma pos_imp_zdiv_nonneg_iff:
       fixes a::int
       assumes "0 < b" 
       shows "(0 \<le> a div b) = (0 \<le> a)"
 proof
   show "0 \<le> a div b \<Longrightarrow> 0 \<le> a"
     using assms
     by (simp add: linorder_not_less [symmetric]) (blast intro: div_neg_pos_less0)
 next
   assume "0 \<le> a"
   then have "0 div b \<le> a div b"
     using zdiv_mono1 assms by blast
   then show "0 \<le> a div b"
     by auto
 qed
 
 lemma pos_imp_zdiv_pos_iff:
   "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
   using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k] by arith
 
 
 lemma neg_imp_zdiv_nonneg_iff:
   fixes a::int
   assumes "b < 0" 
   shows "(0 \<le> a div b) = (a \<le> 0)"
   using assms by (simp add: div_minus_minus [of a, symmetric] pos_imp_zdiv_nonneg_iff del: div_minus_minus)
 
 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
   by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
 
 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
   by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
 
 lemma nonneg1_imp_zdiv_pos_iff:
   fixes a::int
   assumes "0 \<le> a" 
   shows "a div b > 0 \<longleftrightarrow> a \<ge> b \<and> b>0"
 proof -
   have "0 < a div b \<Longrightarrow> b \<le> a"
     using div_pos_pos_trivial[of a b] assms by arith
   moreover have "0 < a div b \<Longrightarrow> b > 0"
     using assms div_nonneg_neg_le0[of a b]  by(cases "b=0"; force)
   moreover have "b \<le> a \<and> 0 < b \<Longrightarrow> 0 < a div b"
     using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b] by simp
   ultimately show ?thesis
     by blast
 qed
 
 lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 \<Longrightarrow> m mod k \<le> m"
   by (rule split_zmod[THEN iffD2]) (fastforce dest: q_pos_lemma intro: split_mult_pos_le)
 
 
 subsubsection \<open>Further properties\<close>
 
 lemma div_int_pos_iff:
   "k div l \<ge> 0 \<longleftrightarrow> k = 0 \<or> l = 0 \<or> k \<ge> 0 \<and> l \<ge> 0
     \<or> k < 0 \<and> l < 0"
   for k l :: int
 proof (cases "k = 0 \<or> l = 0")
   case False
   then show ?thesis
    apply (auto simp add: pos_imp_zdiv_nonneg_iff neg_imp_zdiv_nonneg_iff)
     by (meson neg_imp_zdiv_neg_iff not_le not_less_iff_gr_or_eq)
 qed auto
 
 lemma mod_int_pos_iff:
   "k mod l \<ge> 0 \<longleftrightarrow> l dvd k \<or> l = 0 \<and> k \<ge> 0 \<or> l > 0"
   for k l :: int
 proof (cases "l > 0")
   case False
   then show ?thesis 
     by (simp add: dvd_eq_mod_eq_0) (use neg_mod_sign [of l k] in \<open>auto simp add: le_less not_less\<close>)
 qed auto
 
 text \<open>Simplify expressions in which div and mod combine numerical constants\<close>
 
 lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
   by (rule div_int_unique [of a b q r]) (simp add: eucl_rel_int_iff)
 
 lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
   by (rule div_int_unique [of a b q r],
     simp add: eucl_rel_int_iff)
 
 lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
   by (rule mod_int_unique [of a b q r],
     simp add: eucl_rel_int_iff)
 
 lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
   by (rule mod_int_unique [of a b q r],
     simp add: eucl_rel_int_iff)
 
 lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
   unfolding dvd_def by (cases "y=0") (auto simp add: abs_mult)
 
 text\<open>Suggested by Matthias Daum\<close>
 lemma int_power_div_base:
   fixes k :: int
   assumes "0 < m" "0 < k"
   shows "k ^ m div k = (k::int) ^ (m - Suc 0)"
 proof -
   have eq: "k ^ m = k ^ ((m - Suc 0) + Suc 0)"
     by (simp add: assms)
   show ?thesis
     using assms by (simp only: power_add eq) auto
 qed
 
-text \<open>Distributive laws for function \<open>nat\<close>.\<close>
-
-lemma nat_div_distrib:
-  assumes "0 \<le> x"
-  shows "nat (x div y) = nat x div nat y"
-proof (cases y "0::int" rule: linorder_cases)
-  case less
-  with assms show ?thesis
-    using div_nonneg_neg_le0 by auto
-next
-  case greater
-  then show ?thesis
-    by (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
-qed auto
-
-(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
-lemma nat_mod_distrib:
-  assumes "0 \<le> x" "0 \<le> y"
-  shows "nat (x mod y) = nat x mod nat y"
-proof (cases "y = 0")
-  case False
-  with assms show ?thesis
-    by (simp add: nat_eq_iff zmod_int)
-qed auto
-
 text\<open>Suggested by Matthias Daum\<close>
 lemma int_div_less_self:
   fixes x::int
   assumes "0 < x" "1 < k"
   shows  "x div k < x"
 proof -
   have "nat x div nat k < nat x"
     by (simp add: assms)
   with assms show ?thesis
     by (simp add: nat_div_distrib [symmetric])
 qed
 
 lemma mod_eq_dvd_iff_nat:
   "m mod q = n mod q \<longleftrightarrow> q dvd m - n" if "m \<ge> n" for m n q :: nat
 proof -
   have "int m mod int q = int n mod int q \<longleftrightarrow> int q dvd int m - int n"
     by (simp add: mod_eq_dvd_iff)
   with that have "int (m mod q) = int (n mod q) \<longleftrightarrow> int q dvd int (m - n)"
     by (simp only: of_nat_mod of_nat_diff)
   then show ?thesis
     by simp
 qed
 
 lemma mod_eq_nat1E:
   fixes m n q :: nat
   assumes "m mod q = n mod q" and "m \<ge> n"
   obtains s where "m = n + q * s"
 proof -
   from assms have "q dvd m - n"
     by (simp add: mod_eq_dvd_iff_nat)
   then obtain s where "m - n = q * s" ..
   with \<open>m \<ge> n\<close> have "m = n + q * s"
     by simp
   with that show thesis .
 qed
 
 lemma mod_eq_nat2E:
   fixes m n q :: nat
   assumes "m mod q = n mod q" and "n \<ge> m"
   obtains s where "n = m + q * s"
   using assms mod_eq_nat1E [of n q m] by (auto simp add: ac_simps)
 
 lemma nat_mod_eq_lemma:
   assumes "(x::nat) mod n = y mod n" and "y \<le> x"
   shows "\<exists>q. x = y + n * q"
   using assms by (rule mod_eq_nat1E) rule
 
 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
   (is "?lhs = ?rhs")
 proof
   assume H: "x mod n = y mod n"
   {assume xy: "x \<le> y"
     from H have th: "y mod n = x mod n" by simp
     from nat_mod_eq_lemma[OF th xy] have ?rhs
       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
   moreover
   {assume xy: "y \<le> x"
     from nat_mod_eq_lemma[OF H xy] have ?rhs
       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
   ultimately  show ?rhs using linear[of x y] by blast
 next
   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
   thus  ?lhs by simp
 qed
 
 lemma take_bit_greater_eq:
   \<open>k + 2 ^ n \<le> take_bit n k\<close> if \<open>k < 0\<close> for k :: int
 proof -
   have \<open>k + 2 ^ n \<le> take_bit n (k + 2 ^ n)\<close>
   proof (cases \<open>k > - (2 ^ n)\<close>)
     case False
     then have \<open>k + 2 ^ n \<le> 0\<close>
       by simp
     also note take_bit_nonnegative
     finally show ?thesis .
   next
     case True
     with that have \<open>0 \<le> k + 2 ^ n\<close> and \<open>k + 2 ^ n < 2 ^ n\<close>
       by simp_all
     then show ?thesis
       by (simp only: take_bit_eq_mod mod_pos_pos_trivial)
   qed
   then show ?thesis
     by (simp add: take_bit_eq_mod)
 qed
 
 lemma take_bit_less_eq:
   \<open>take_bit n k \<le> k - 2 ^ n\<close> if \<open>2 ^ n \<le> k\<close> and \<open>n > 0\<close> for k :: int
   using that zmod_le_nonneg_dividend [of \<open>k - 2 ^ n\<close> \<open>2 ^ n\<close>]
   by (simp add: take_bit_eq_mod)
 
 
 subsection \<open>Numeral division with a pragmatic type class\<close>
 
 text \<open>
   The following type class contains everything necessary to formulate
   a division algorithm in ring structures with numerals, restricted
   to its positive segments.  This is its primary motivation, and it
   could surely be formulated using a more fine-grained, more algebraic
   and less technical class hierarchy.
 \<close>
 
 class unique_euclidean_semiring_numeral = unique_euclidean_semiring_with_nat + linordered_semidom +
   assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
     and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
     and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
     and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
     and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
     and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
     and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
     and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
   assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
   fixes divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
     and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
   assumes divmod_def: "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
     and divmod_step_def: "divmod_step l qr = (let (q, r) = qr
     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
     else (2 * q, r))"
     \<comment> \<open>These are conceptually definitions but force generated code
     to be monomorphic wrt. particular instances of this class which
     yields a significant speedup.\<close>
 begin
 
 lemma divmod_digit_1:
   assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
   shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
     and "a mod (2 * b) - b = a mod b" (is "?Q")
 proof -
   from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
     by (auto intro: trans)
   with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
   then have [simp]: "1 \<le> a div b" by (simp add: discrete)
   with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
   define w where "w = a div b mod 2"
   then have w_exhaust: "w = 0 \<or> w = 1" by auto
   have mod_w: "a mod (2 * b) = a mod b + b * w"
     by (simp add: w_def mod_mult2_eq ac_simps)
   from assms w_exhaust have "w = 1"
     by (auto simp add: mod_w) (insert mod_less, auto)
   with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
   have "2 * (a div (2 * b)) = a div b - w"
     by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
   with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
   then show ?P and ?Q
     by (simp_all add: div mod add_implies_diff [symmetric])
 qed
 
 lemma divmod_digit_0:
   assumes "0 < b" and "a mod (2 * b) < b"
   shows "2 * (a div (2 * b)) = a div b" (is "?P")
     and "a mod (2 * b) = a mod b" (is "?Q")
 proof -
   define w where "w = a div b mod 2"
   then have w_exhaust: "w = 0 \<or> w = 1" by auto
   have mod_w: "a mod (2 * b) = a mod b + b * w"
     by (simp add: w_def mod_mult2_eq ac_simps)
   moreover have "b \<le> a mod b + b"
   proof -
     from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
     then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
     then show ?thesis by simp
   qed
   moreover note assms w_exhaust
   ultimately have "w = 0" by auto
   with mod_w have mod: "a mod (2 * b) = a mod b" by simp
   have "2 * (a div (2 * b)) = a div b - w"
     by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
   with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
   then show ?P and ?Q
     by (simp_all add: div mod)
 qed
 
 lemma mod_double_modulus:
   assumes "m > 0" "x \<ge> 0"
   shows   "x mod (2 * m) = x mod m \<or> x mod (2 * m) = x mod m + m"
 proof (cases "x mod (2 * m) < m")
   case True
   thus ?thesis using assms using divmod_digit_0(2)[of m x] by auto
 next
   case False
   hence *: "x mod (2 * m) - m = x mod m"
     using assms by (intro divmod_digit_1) auto
   hence "x mod (2 * m) = x mod m + m"
     by (subst * [symmetric], subst le_add_diff_inverse2) (use False in auto)
   thus ?thesis by simp
 qed
 
 lemma fst_divmod:
   "fst (divmod m n) = numeral m div numeral n"
   by (simp add: divmod_def)
 
 lemma snd_divmod:
   "snd (divmod m n) = numeral m mod numeral n"
   by (simp add: divmod_def)
 
 text \<open>
   This is a formulation of one step (referring to one digit position)
   in school-method division: compare the dividend at the current
   digit position with the remainder from previous division steps
   and evaluate accordingly.
 \<close>
 
 lemma divmod_step_eq [simp]:
   "divmod_step l (q, r) = (if numeral l \<le> r
     then (2 * q + 1, r - numeral l) else (2 * q, r))"
   by (simp add: divmod_step_def)
 
 text \<open>
   This is a formulation of school-method division.
   If the divisor is smaller than the dividend, terminate.
   If not, shift the dividend to the right until termination
   occurs and then reiterate single division steps in the
   opposite direction.
 \<close>
 
 lemma divmod_divmod_step:
   "divmod m n = (if m < n then (0, numeral m)
     else divmod_step n (divmod m (Num.Bit0 n)))"
 proof (cases "m < n")
   case True then have "numeral m < numeral n" by simp
   then show ?thesis
     by (simp add: prod_eq_iff div_less mod_less fst_divmod snd_divmod)
 next
   case False
   have "divmod m n =
     divmod_step n (numeral m div (2 * numeral n),
       numeral m mod (2 * numeral n))"
   proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
     case True
     with divmod_step_eq
       have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
         (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
         by simp
     moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
       have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
       and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
       by simp_all
     ultimately show ?thesis by (simp only: divmod_def)
   next
     case False then have *: "numeral m mod (2 * numeral n) < numeral n"
       by (simp add: not_le)
     with divmod_step_eq
       have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
         (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
         by auto
     moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
       have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
       and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
       by (simp_all only: zero_less_numeral)
     ultimately show ?thesis by (simp only: divmod_def)
   qed
   then have "divmod m n =
     divmod_step n (numeral m div numeral (Num.Bit0 n),
       numeral m mod numeral (Num.Bit0 n))"
     by (simp only: numeral.simps distrib mult_1)
   then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
     by (simp add: divmod_def)
   with False show ?thesis by simp
 qed
 
 text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
 
 lemma divmod_trivial [simp]:
   "divmod m Num.One = (numeral m, 0)"
   "divmod num.One (num.Bit0 n) = (0, Numeral1)"
   "divmod num.One (num.Bit1 n) = (0, Numeral1)"
   using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
 
 text \<open>Division by an even number is a right-shift\<close>
 
 lemma divmod_cancel [simp]:
   "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
   "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
 proof -
   have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
     "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
     by (simp_all only: numeral_mult numeral.simps distrib) simp_all
   have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
   then show ?P and ?Q
     by (simp_all add: fst_divmod snd_divmod prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
       div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2]
       add.commute del: numeral_times_numeral)
 qed
 
 text \<open>The really hard work\<close>
 
 lemma divmod_steps [simp]:
   "divmod (num.Bit0 m) (num.Bit1 n) =
       (if m \<le> n then (0, numeral (num.Bit0 m))
        else divmod_step (num.Bit1 n)
              (divmod (num.Bit0 m)
                (num.Bit0 (num.Bit1 n))))"
   "divmod (num.Bit1 m) (num.Bit1 n) =
       (if m < n then (0, numeral (num.Bit1 m))
        else divmod_step (num.Bit1 n)
              (divmod (num.Bit1 m)
                (num.Bit0 (num.Bit1 n))))"
   by (simp_all add: divmod_divmod_step)
 
 lemmas divmod_algorithm_code = divmod_step_eq divmod_trivial divmod_cancel divmod_steps  
 
 text \<open>Special case: divisibility\<close>
 
 definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
 where
   "divides_aux qr \<longleftrightarrow> snd qr = 0"
 
 lemma divides_aux_eq [simp]:
   "divides_aux (q, r) \<longleftrightarrow> r = 0"
   by (simp add: divides_aux_def)
 
 lemma dvd_numeral_simp [simp]:
   "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
   by (simp add: divmod_def mod_eq_0_iff_dvd)
 
 text \<open>Generic computation of quotient and remainder\<close>  
 
 lemma numeral_div_numeral [simp]: 
   "numeral k div numeral l = fst (divmod k l)"
   by (simp add: fst_divmod)
 
 lemma numeral_mod_numeral [simp]: 
   "numeral k mod numeral l = snd (divmod k l)"
   by (simp add: snd_divmod)
 
 lemma one_div_numeral [simp]:
   "1 div numeral n = fst (divmod num.One n)"
   by (simp add: fst_divmod)
 
 lemma one_mod_numeral [simp]:
   "1 mod numeral n = snd (divmod num.One n)"
   by (simp add: snd_divmod)
 
 text \<open>Computing congruences modulo \<open>2 ^ q\<close>\<close>
 
 lemma cong_exp_iff_simps:
   "numeral n mod numeral Num.One = 0
     \<longleftrightarrow> True"
   "numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) = 0
     \<longleftrightarrow> numeral n mod numeral q = 0"
   "numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) = 0
     \<longleftrightarrow> False"
   "numeral m mod numeral Num.One = (numeral n mod numeral Num.One)
     \<longleftrightarrow> True"
   "numeral Num.One mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
     \<longleftrightarrow> True"
   "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
     \<longleftrightarrow> False"
   "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
     \<longleftrightarrow> (numeral n mod numeral q) = 0"
   "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
     \<longleftrightarrow> False"
   "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
     \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
   "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
     \<longleftrightarrow> False"
   "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
     \<longleftrightarrow> (numeral m mod numeral q) = 0"
   "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
     \<longleftrightarrow> False"
   "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
     \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
   by (auto simp add: case_prod_beta dest: arg_cong [of _ _ even])
 
 end
 
 hide_fact (open) div_less mod_less mod_less_eq_dividend mod_mult2_eq div_mult2_eq
 
 instantiation nat :: unique_euclidean_semiring_numeral
 begin
 
 definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
 where
   divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
 
 definition divmod_step_nat :: "num \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
 where
   "divmod_step_nat l qr = (let (q, r) = qr
     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
     else (2 * q, r))"
 
 instance by standard
   (auto simp add: divmod'_nat_def divmod_step_nat_def div_greater_zero_iff div_mult2_eq mod_mult2_eq)
 
 end
 
 declare divmod_algorithm_code [where ?'a = nat, code]
 
 lemma Suc_0_div_numeral [simp]:
   fixes k l :: num
   shows "Suc 0 div numeral k = fst (divmod Num.One k)"
   by (simp_all add: fst_divmod)
 
 lemma Suc_0_mod_numeral [simp]:
   fixes k l :: num
   shows "Suc 0 mod numeral k = snd (divmod Num.One k)"
   by (simp_all add: snd_divmod)
 
 instantiation int :: unique_euclidean_semiring_numeral
 begin
 
 definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
 where
   "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
 
 definition divmod_step_int :: "num \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
 where
   "divmod_step_int l qr = (let (q, r) = qr
     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
     else (2 * q, r))"
 
 instance
   by standard (auto intro: zmod_le_nonneg_dividend simp add: divmod_int_def divmod_step_int_def
     pos_imp_zdiv_pos_iff zmod_zmult2_eq zdiv_zmult2_eq)
 
 end
 
 declare divmod_algorithm_code [where ?'a = int, code]
 
 context
 begin
   
 qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
 where
   "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
 
 qualified lemma adjust_div_eq [simp, code]:
   "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
   by (simp add: adjust_div_def)
 
 qualified definition adjust_mod :: "int \<Rightarrow> int \<Rightarrow> int"
 where
   [simp]: "adjust_mod l r = (if r = 0 then 0 else l - r)"
 
 lemma minus_numeral_div_numeral [simp]:
   "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
 proof -
   have "int (fst (divmod m n)) = fst (divmod m n)"
     by (simp only: fst_divmod divide_int_def) auto
   then show ?thesis
     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
 qed
 
 lemma minus_numeral_mod_numeral [simp]:
   "- numeral m mod numeral n = adjust_mod (numeral n) (snd (divmod m n) :: int)"
 proof (cases "snd (divmod m n) = (0::int)")
   case True
   then show ?thesis
     by (simp add: mod_eq_0_iff_dvd divides_aux_def)
 next
   case False
   then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
     by (simp only: snd_divmod modulo_int_def) auto
   then show ?thesis
     by (simp add: divides_aux_def adjust_div_def)
       (simp add: divides_aux_def modulo_int_def)
 qed
 
 lemma numeral_div_minus_numeral [simp]:
   "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
 proof -
   have "int (fst (divmod m n)) = fst (divmod m n)"
     by (simp only: fst_divmod divide_int_def) auto
   then show ?thesis
     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
 qed
   
 lemma numeral_mod_minus_numeral [simp]:
   "numeral m mod - numeral n = - adjust_mod (numeral n) (snd (divmod m n) :: int)"
 proof (cases "snd (divmod m n) = (0::int)")
   case True
   then show ?thesis
     by (simp add: mod_eq_0_iff_dvd divides_aux_def)
 next
   case False
   then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
     by (simp only: snd_divmod modulo_int_def) auto
   then show ?thesis
     by (simp add: divides_aux_def adjust_div_def)
       (simp add: divides_aux_def modulo_int_def)
 qed
 
 lemma minus_one_div_numeral [simp]:
   "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
   using minus_numeral_div_numeral [of Num.One n] by simp  
 
 lemma minus_one_mod_numeral [simp]:
   "- 1 mod numeral n = adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
   using minus_numeral_mod_numeral [of Num.One n] by simp
 
 lemma one_div_minus_numeral [simp]:
   "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
   using numeral_div_minus_numeral [of Num.One n] by simp
   
 lemma one_mod_minus_numeral [simp]:
   "1 mod - numeral n = - adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
   using numeral_mod_minus_numeral [of Num.One n] by simp
 
 end
 
 lemma divmod_BitM_2_eq [simp]:
   \<open>divmod (Num.BitM m) (Num.Bit0 Num.One) = (numeral m - 1, (1 :: int))\<close>
   by (cases m) simp_all
 
 lemma bit_numeral_Bit0_Suc_iff [simp]:
   \<open>bit (numeral (Num.Bit0 m) :: int) (Suc n) \<longleftrightarrow> bit (numeral m :: int) n\<close>
   by (simp add: bit_Suc)
 
 lemma bit_numeral_Bit1_Suc_iff [simp]:
   \<open>bit (numeral (Num.Bit1 m) :: int) (Suc n) \<longleftrightarrow> bit (numeral m :: int) n\<close>
   by (simp add: bit_Suc)
 
 lemma div_positive_int:
   "k div l > 0" if "k \<ge> l" and "l > 0" for k l :: int
   using that div_positive [of l k] by blast
 
 
 subsubsection \<open>Dedicated simproc for calculation\<close>
 
 text \<open>
   There is space for improvement here: the calculation itself
   could be carried out outside the logic, and a generic simproc
   (simplifier setup) for generic calculation would be helpful. 
 \<close>
 
 simproc_setup numeral_divmod
   ("0 div 0 :: 'a :: unique_euclidean_semiring_numeral" | "0 mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
    "0 div 1 :: 'a :: unique_euclidean_semiring_numeral" | "0 mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
    "0 div - 1 :: int" | "0 mod - 1 :: int" |
    "0 div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "0 mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
    "0 div - numeral b :: int" | "0 mod - numeral b :: int" |
    "1 div 0 :: 'a :: unique_euclidean_semiring_numeral" | "1 mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
    "1 div 1 :: 'a :: unique_euclidean_semiring_numeral" | "1 mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
    "1 div - 1 :: int" | "1 mod - 1 :: int" |
    "1 div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "1 mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
    "1 div - numeral b :: int" |"1 mod - numeral b :: int" |
    "- 1 div 0 :: int" | "- 1 mod 0 :: int" | "- 1 div 1 :: int" | "- 1 mod 1 :: int" |
    "- 1 div - 1 :: int" | "- 1 mod - 1 :: int" | "- 1 div numeral b :: int" | "- 1 mod numeral b :: int" |
    "- 1 div - numeral b :: int" | "- 1 mod - numeral b :: int" |
    "numeral a div 0 :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
    "numeral a div 1 :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
    "numeral a div - 1 :: int" | "numeral a mod - 1 :: int" |
    "numeral a div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
    "numeral a div - numeral b :: int" | "numeral a mod - numeral b :: int" |
    "- numeral a div 0 :: int" | "- numeral a mod 0 :: int" |
    "- numeral a div 1 :: int" | "- numeral a mod 1 :: int" |
    "- numeral a div - 1 :: int" | "- numeral a mod - 1 :: int" |
    "- numeral a div numeral b :: int" | "- numeral a mod numeral b :: int" |
    "- numeral a div - numeral b :: int" | "- numeral a mod - numeral b :: int") =
 \<open> let
     val if_cong = the (Code.get_case_cong \<^theory> \<^const_name>\<open>If\<close>);
     fun successful_rewrite ctxt ct =
       let
         val thm = Simplifier.rewrite ctxt ct
       in if Thm.is_reflexive thm then NONE else SOME thm end;
   in fn phi =>
     let
       val simps = Morphism.fact phi (@{thms div_0 mod_0 div_by_0 mod_by_0 div_by_1 mod_by_1
         one_div_numeral one_mod_numeral minus_one_div_numeral minus_one_mod_numeral
         one_div_minus_numeral one_mod_minus_numeral
         numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
         numeral_div_minus_numeral numeral_mod_minus_numeral
         div_minus_minus mod_minus_minus Divides.adjust_div_eq of_bool_eq one_neq_zero
         numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
         divmod_cancel divmod_steps divmod_step_eq fst_conv snd_conv numeral_One
         case_prod_beta rel_simps Divides.adjust_mod_def div_minus1_right mod_minus1_right
         minus_minus numeral_times_numeral mult_zero_right mult_1_right}
         @ [@{lemma "0 = 0 \<longleftrightarrow> True" by simp}]);
       fun prepare_simpset ctxt = HOL_ss |> Simplifier.simpset_map ctxt
         (Simplifier.add_cong if_cong #> fold Simplifier.add_simp simps)
     in fn ctxt => successful_rewrite (Simplifier.put_simpset (prepare_simpset ctxt) ctxt) end
   end
 \<close>
 
 
 subsubsection \<open>Code generation\<close>
 
 definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat"
   where "divmod_nat m n = (m div n, m mod n)"
 
 lemma fst_divmod_nat [simp]:
   "fst (divmod_nat m n) = m div n"
   by (simp add: divmod_nat_def)
 
 lemma snd_divmod_nat [simp]:
   "snd (divmod_nat m n) = m mod n"
   by (simp add: divmod_nat_def)
 
 lemma divmod_nat_if [code]:
   "Divides.divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
     let (q, r) = Divides.divmod_nat (m - n) n in (Suc q, r))"
   by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
 
 lemma [code]:
   "m div n = fst (divmod_nat m n)"
   "m mod n = snd (divmod_nat m n)"
   by simp_all
 
 lemma [code]:
   fixes k :: int
   shows 
     "k div 0 = 0"
     "k mod 0 = k"
     "0 div k = 0"
     "0 mod k = 0"
     "k div Int.Pos Num.One = k"
     "k mod Int.Pos Num.One = 0"
     "k div Int.Neg Num.One = - k"
     "k mod Int.Neg Num.One = 0"
     "Int.Pos m div Int.Pos n = (fst (divmod m n) :: int)"
     "Int.Pos m mod Int.Pos n = (snd (divmod m n) :: int)"
     "Int.Neg m div Int.Pos n = - (Divides.adjust_div (divmod m n) :: int)"
     "Int.Neg m mod Int.Pos n = Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
     "Int.Pos m div Int.Neg n = - (Divides.adjust_div (divmod m n) :: int)"
     "Int.Pos m mod Int.Neg n = - Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
     "Int.Neg m div Int.Neg n = (fst (divmod m n) :: int)"
     "Int.Neg m mod Int.Neg n = - (snd (divmod m n) :: int)"
   by simp_all
 
 code_identifier
   code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
 
 
 subsection \<open>Lemmas of doubtful value\<close>
 
 lemma div_geq: "m div n = Suc ((m - n) div n)" if "0 < n" and " \<not> m < n" for m n :: nat
   by (rule le_div_geq) (use that in \<open>simp_all add: not_less\<close>)
 
 lemma mod_geq: "m mod n = (m - n) mod n" if "\<not> m < n" for m n :: nat
   by (rule le_mod_geq) (use that in \<open>simp add: not_less\<close>)
 
 lemma mod_eq_0D: "\<exists>q. m = d * q" if "m mod d = 0" for m d :: nat
   using that by (auto simp add: mod_eq_0_iff_dvd)
 
 lemma pos_mod_conj: "0 < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b" for a b :: int
   by simp
 
 lemma neg_mod_conj: "b < 0 \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b" for a b :: int
   by simp
 
 lemma zmod_eq_0_iff: "m mod d = 0 \<longleftrightarrow> (\<exists>q. m = d * q)" for m d :: int
   by (auto simp add: mod_eq_0_iff_dvd)
 
 (* REVISIT: should this be generalized to all semiring_div types? *)
 lemma zmod_eq_0D [dest!]: "\<exists>q. m = d * q" if "m mod d = 0" for m d :: int
   using that by auto
 
+find_theorems \<open>(?k::int) mod _ = ?k\<close>
+
 end
diff --git a/src/HOL/Euclidean_Division.thy b/src/HOL/Euclidean_Division.thy
--- a/src/HOL/Euclidean_Division.thy
+++ b/src/HOL/Euclidean_Division.thy
@@ -1,2025 +1,2086 @@
 (*  Title:      HOL/Euclidean_Division.thy
     Author:     Manuel Eberl, TU Muenchen
     Author:     Florian Haftmann, TU Muenchen
 *)
 
 section \<open>Division in euclidean (semi)rings\<close>
 
 theory Euclidean_Division
   imports Int Lattices_Big
 begin
 
 subsection \<open>Euclidean (semi)rings with explicit division and remainder\<close>
   
 class euclidean_semiring = semidom_modulo + 
   fixes euclidean_size :: "'a \<Rightarrow> nat"
   assumes size_0 [simp]: "euclidean_size 0 = 0"
   assumes mod_size_less: 
     "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
   assumes size_mult_mono:
     "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)"
 begin
 
 lemma euclidean_size_eq_0_iff [simp]:
   "euclidean_size b = 0 \<longleftrightarrow> b = 0"
 proof
   assume "b = 0"
   then show "euclidean_size b = 0"
     by simp
 next
   assume "euclidean_size b = 0"
   show "b = 0"
   proof (rule ccontr)
     assume "b \<noteq> 0"
     with mod_size_less have "euclidean_size (b mod b) < euclidean_size b" .
     with \<open>euclidean_size b = 0\<close> show False
       by simp
   qed
 qed
 
 lemma euclidean_size_greater_0_iff [simp]:
   "euclidean_size b > 0 \<longleftrightarrow> b \<noteq> 0"
   using euclidean_size_eq_0_iff [symmetric, of b] by safe simp
 
 lemma size_mult_mono': "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (b * a)"
   by (subst mult.commute) (rule size_mult_mono)
 
 lemma dvd_euclidean_size_eq_imp_dvd:
   assumes "a \<noteq> 0" and "euclidean_size a = euclidean_size b"
     and "b dvd a" 
   shows "a dvd b"
 proof (rule ccontr)
   assume "\<not> a dvd b"
   hence "b mod a \<noteq> 0" using mod_0_imp_dvd [of b a] by blast
   then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
   from \<open>b dvd a\<close> have "b dvd b mod a" by (simp add: dvd_mod_iff)
   then obtain c where "b mod a = b * c" unfolding dvd_def by blast
     with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
   with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
     using size_mult_mono by force
   moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
   have "euclidean_size (b mod a) < euclidean_size a"
     using mod_size_less by blast
   ultimately show False using \<open>euclidean_size a = euclidean_size b\<close>
     by simp
 qed
 
 lemma euclidean_size_times_unit:
   assumes "is_unit a"
   shows   "euclidean_size (a * b) = euclidean_size b"
 proof (rule antisym)
   from assms have [simp]: "a \<noteq> 0" by auto
   thus "euclidean_size (a * b) \<ge> euclidean_size b" by (rule size_mult_mono')
   from assms have "is_unit (1 div a)" by simp
   hence "1 div a \<noteq> 0" by (intro notI) simp_all
   hence "euclidean_size (a * b) \<le> euclidean_size ((1 div a) * (a * b))"
     by (rule size_mult_mono')
   also from assms have "(1 div a) * (a * b) = b"
     by (simp add: algebra_simps unit_div_mult_swap)
   finally show "euclidean_size (a * b) \<le> euclidean_size b" .
 qed
 
 lemma euclidean_size_unit:
   "is_unit a \<Longrightarrow> euclidean_size a = euclidean_size 1"
   using euclidean_size_times_unit [of a 1] by simp
 
 lemma unit_iff_euclidean_size: 
   "is_unit a \<longleftrightarrow> euclidean_size a = euclidean_size 1 \<and> a \<noteq> 0"
 proof safe
   assume A: "a \<noteq> 0" and B: "euclidean_size a = euclidean_size 1"
   show "is_unit a"
     by (rule dvd_euclidean_size_eq_imp_dvd [OF A B]) simp_all
 qed (auto intro: euclidean_size_unit)
 
 lemma euclidean_size_times_nonunit:
   assumes "a \<noteq> 0" "b \<noteq> 0" "\<not> is_unit a"
   shows   "euclidean_size b < euclidean_size (a * b)"
 proof (rule ccontr)
   assume "\<not>euclidean_size b < euclidean_size (a * b)"
   with size_mult_mono'[OF assms(1), of b] 
     have eq: "euclidean_size (a * b) = euclidean_size b" by simp
   have "a * b dvd b"
     by (rule dvd_euclidean_size_eq_imp_dvd [OF _ eq]) (insert assms, simp_all)
   hence "a * b dvd 1 * b" by simp
   with \<open>b \<noteq> 0\<close> have "is_unit a" by (subst (asm) dvd_times_right_cancel_iff)
   with assms(3) show False by contradiction
 qed
 
 lemma dvd_imp_size_le:
   assumes "a dvd b" "b \<noteq> 0" 
   shows   "euclidean_size a \<le> euclidean_size b"
   using assms by (auto elim!: dvdE simp: size_mult_mono)
 
 lemma dvd_proper_imp_size_less:
   assumes "a dvd b" "\<not> b dvd a" "b \<noteq> 0" 
   shows   "euclidean_size a < euclidean_size b"
 proof -
   from assms(1) obtain c where "b = a * c" by (erule dvdE)
   hence z: "b = c * a" by (simp add: mult.commute)
   from z assms have "\<not>is_unit c" by (auto simp: mult.commute mult_unit_dvd_iff)
   with z assms show ?thesis
     by (auto intro!: euclidean_size_times_nonunit)
 qed
 
 lemma unit_imp_mod_eq_0:
   "a mod b = 0" if "is_unit b"
   using that by (simp add: mod_eq_0_iff_dvd unit_imp_dvd)
 
 lemma mod_eq_self_iff_div_eq_0:
   "a mod b = a \<longleftrightarrow> a div b = 0" (is "?P \<longleftrightarrow> ?Q")
 proof
   assume ?P
   with div_mult_mod_eq [of a b] show ?Q
     by auto
 next
   assume ?Q
   with div_mult_mod_eq [of a b] show ?P
     by simp
 qed
 
 lemma coprime_mod_left_iff [simp]:
   "coprime (a mod b) b \<longleftrightarrow> coprime a b" if "b \<noteq> 0"
   by (rule; rule coprimeI)
     (use that in \<open>auto dest!: dvd_mod_imp_dvd coprime_common_divisor simp add: dvd_mod_iff\<close>)
 
 lemma coprime_mod_right_iff [simp]:
   "coprime a (b mod a) \<longleftrightarrow> coprime a b" if "a \<noteq> 0"
   using that coprime_mod_left_iff [of a b] by (simp add: ac_simps)
 
 end
 
 class euclidean_ring = idom_modulo + euclidean_semiring
 begin
 
 lemma dvd_diff_commute [ac_simps]:
   "a dvd c - b \<longleftrightarrow> a dvd b - c"
 proof -
   have "a dvd c - b \<longleftrightarrow> a dvd (c - b) * - 1"
     by (subst dvd_mult_unit_iff) simp_all
   then show ?thesis
     by simp
 qed
  
 end
 
 
 subsection \<open>Euclidean (semi)rings with cancel rules\<close>
 
 class euclidean_semiring_cancel = euclidean_semiring +
   assumes div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
   and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
 begin
 
 lemma div_mult_self2 [simp]:
   assumes "b \<noteq> 0"
   shows "(a + b * c) div b = c + a div b"
   using assms div_mult_self1 [of b a c] by (simp add: mult.commute)
 
 lemma div_mult_self3 [simp]:
   assumes "b \<noteq> 0"
   shows "(c * b + a) div b = c + a div b"
   using assms by (simp add: add.commute)
 
 lemma div_mult_self4 [simp]:
   assumes "b \<noteq> 0"
   shows "(b * c + a) div b = c + a div b"
   using assms by (simp add: add.commute)
 
 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
 proof (cases "b = 0")
   case True then show ?thesis by simp
 next
   case False
   have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
     by (simp add: div_mult_mod_eq)
   also from False div_mult_self1 [of b a c] have
     "\<dots> = (c + a div b) * b + (a + c * b) mod b"
       by (simp add: algebra_simps)
   finally have "a = a div b * b + (a + c * b) mod b"
     by (simp add: add.commute [of a] add.assoc distrib_right)
   then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
     by (simp add: div_mult_mod_eq)
   then show ?thesis by simp
 qed
 
 lemma mod_mult_self2 [simp]:
   "(a + b * c) mod b = a mod b"
   by (simp add: mult.commute [of b])
 
 lemma mod_mult_self3 [simp]:
   "(c * b + a) mod b = a mod b"
   by (simp add: add.commute)
 
 lemma mod_mult_self4 [simp]:
   "(b * c + a) mod b = a mod b"
   by (simp add: add.commute)
 
 lemma mod_mult_self1_is_0 [simp]:
   "b * a mod b = 0"
   using mod_mult_self2 [of 0 b a] by simp
 
 lemma mod_mult_self2_is_0 [simp]:
   "a * b mod b = 0"
   using mod_mult_self1 [of 0 a b] by simp
 
 lemma div_add_self1:
   assumes "b \<noteq> 0"
   shows "(b + a) div b = a div b + 1"
   using assms div_mult_self1 [of b a 1] by (simp add: add.commute)
 
 lemma div_add_self2:
   assumes "b \<noteq> 0"
   shows "(a + b) div b = a div b + 1"
   using assms div_add_self1 [of b a] by (simp add: add.commute)
 
 lemma mod_add_self1 [simp]:
   "(b + a) mod b = a mod b"
   using mod_mult_self1 [of a 1 b] by (simp add: add.commute)
 
 lemma mod_add_self2 [simp]:
   "(a + b) mod b = a mod b"
   using mod_mult_self1 [of a 1 b] by simp
 
 lemma mod_div_trivial [simp]:
   "a mod b div b = 0"
 proof (cases "b = 0")
   assume "b = 0"
   thus ?thesis by simp
 next
   assume "b \<noteq> 0"
   hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
     by (rule div_mult_self1 [symmetric])
   also have "\<dots> = a div b"
     by (simp only: mod_div_mult_eq)
   also have "\<dots> = a div b + 0"
     by simp
   finally show ?thesis
     by (rule add_left_imp_eq)
 qed
 
 lemma mod_mod_trivial [simp]:
   "a mod b mod b = a mod b"
 proof -
   have "a mod b mod b = (a mod b + a div b * b) mod b"
     by (simp only: mod_mult_self1)
   also have "\<dots> = a mod b"
     by (simp only: mod_div_mult_eq)
   finally show ?thesis .
 qed
 
 lemma mod_mod_cancel:
   assumes "c dvd b"
   shows "a mod b mod c = a mod c"
 proof -
   from \<open>c dvd b\<close> obtain k where "b = c * k"
     by (rule dvdE)
   have "a mod b mod c = a mod (c * k) mod c"
     by (simp only: \<open>b = c * k\<close>)
   also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
     by (simp only: mod_mult_self1)
   also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
     by (simp only: ac_simps)
   also have "\<dots> = a mod c"
     by (simp only: div_mult_mod_eq)
   finally show ?thesis .
 qed
 
 lemma div_mult_mult2 [simp]:
   "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
   by (drule div_mult_mult1) (simp add: mult.commute)
 
 lemma div_mult_mult1_if [simp]:
   "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
   by simp_all
 
 lemma mod_mult_mult1:
   "(c * a) mod (c * b) = c * (a mod b)"
 proof (cases "c = 0")
   case True then show ?thesis by simp
 next
   case False
   from div_mult_mod_eq
   have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
   with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
     = c * a + c * (a mod b)" by (simp add: algebra_simps)
   with div_mult_mod_eq show ?thesis by simp
 qed
 
 lemma mod_mult_mult2:
   "(a * c) mod (b * c) = (a mod b) * c"
   using mod_mult_mult1 [of c a b] by (simp add: mult.commute)
 
 lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
   by (fact mod_mult_mult2 [symmetric])
 
 lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
   by (fact mod_mult_mult1 [symmetric])
 
 lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
   unfolding dvd_def by (auto simp add: mod_mult_mult1)
 
 lemma div_plus_div_distrib_dvd_left:
   "c dvd a \<Longrightarrow> (a + b) div c = a div c + b div c"
   by (cases "c = 0") (auto elim: dvdE)
 
 lemma div_plus_div_distrib_dvd_right:
   "c dvd b \<Longrightarrow> (a + b) div c = a div c + b div c"
   using div_plus_div_distrib_dvd_left [of c b a]
   by (simp add: ac_simps)
 
 lemma sum_div_partition:
   \<open>(\<Sum>a\<in>A. f a) div b = (\<Sum>a\<in>A \<inter> {a. b dvd f a}. f a div b) + (\<Sum>a\<in>A \<inter> {a. \<not> b dvd f a}. f a) div b\<close>
     if \<open>finite A\<close>
 proof -
   have \<open>A = A \<inter> {a. b dvd f a} \<union> A \<inter> {a. \<not> b dvd f a}\<close>
     by auto
   then have \<open>(\<Sum>a\<in>A. f a) = (\<Sum>a\<in>A \<inter> {a. b dvd f a} \<union> A \<inter> {a. \<not> b dvd f a}. f a)\<close>
     by simp
   also have \<open>\<dots> = (\<Sum>a\<in>A \<inter> {a. b dvd f a}. f a) + (\<Sum>a\<in>A \<inter> {a. \<not> b dvd f a}. f a)\<close>
     using \<open>finite A\<close> by (auto intro: sum.union_inter_neutral)
   finally have *: \<open>sum f A = sum f (A \<inter> {a. b dvd f a}) + sum f (A \<inter> {a. \<not> b dvd f a})\<close> .
   define B where B: \<open>B = A \<inter> {a. b dvd f a}\<close>
   with \<open>finite A\<close> have \<open>finite B\<close> and \<open>a \<in> B \<Longrightarrow> b dvd f a\<close> for a
     by simp_all
   then have \<open>(\<Sum>a\<in>B. f a) div b = (\<Sum>a\<in>B. f a div b)\<close> and \<open>b dvd (\<Sum>a\<in>B. f a)\<close>
     by induction (simp_all add: div_plus_div_distrib_dvd_left)
   then show ?thesis using *
     by (simp add: B div_plus_div_distrib_dvd_left)
 qed
 
 named_theorems mod_simps
 
 text \<open>Addition respects modular equivalence.\<close>
 
 lemma mod_add_left_eq [mod_simps]:
   "(a mod c + b) mod c = (a + b) mod c"
 proof -
   have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
     by (simp only: div_mult_mod_eq)
   also have "\<dots> = (a mod c + b + a div c * c) mod c"
     by (simp only: ac_simps)
   also have "\<dots> = (a mod c + b) mod c"
     by (rule mod_mult_self1)
   finally show ?thesis
     by (rule sym)
 qed
 
 lemma mod_add_right_eq [mod_simps]:
   "(a + b mod c) mod c = (a + b) mod c"
   using mod_add_left_eq [of b c a] by (simp add: ac_simps)
 
 lemma mod_add_eq:
   "(a mod c + b mod c) mod c = (a + b) mod c"
   by (simp add: mod_add_left_eq mod_add_right_eq)
 
 lemma mod_sum_eq [mod_simps]:
   "(\<Sum>i\<in>A. f i mod a) mod a = sum f A mod a"
 proof (induct A rule: infinite_finite_induct)
   case (insert i A)
   then have "(\<Sum>i\<in>insert i A. f i mod a) mod a
     = (f i mod a + (\<Sum>i\<in>A. f i mod a)) mod a"
     by simp
   also have "\<dots> = (f i + (\<Sum>i\<in>A. f i mod a) mod a) mod a"
     by (simp add: mod_simps)
   also have "\<dots> = (f i + (\<Sum>i\<in>A. f i) mod a) mod a"
     by (simp add: insert.hyps)
   finally show ?case
     by (simp add: insert.hyps mod_simps)
 qed simp_all
 
 lemma mod_add_cong:
   assumes "a mod c = a' mod c"
   assumes "b mod c = b' mod c"
   shows "(a + b) mod c = (a' + b') mod c"
 proof -
   have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
     unfolding assms ..
   then show ?thesis
     by (simp add: mod_add_eq)
 qed
 
 text \<open>Multiplication respects modular equivalence.\<close>
 
 lemma mod_mult_left_eq [mod_simps]:
   "((a mod c) * b) mod c = (a * b) mod c"
 proof -
   have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
     by (simp only: div_mult_mod_eq)
   also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
     by (simp only: algebra_simps)
   also have "\<dots> = (a mod c * b) mod c"
     by (rule mod_mult_self1)
   finally show ?thesis
     by (rule sym)
 qed
 
 lemma mod_mult_right_eq [mod_simps]:
   "(a * (b mod c)) mod c = (a * b) mod c"
   using mod_mult_left_eq [of b c a] by (simp add: ac_simps)
 
 lemma mod_mult_eq:
   "((a mod c) * (b mod c)) mod c = (a * b) mod c"
   by (simp add: mod_mult_left_eq mod_mult_right_eq)
 
 lemma mod_prod_eq [mod_simps]:
   "(\<Prod>i\<in>A. f i mod a) mod a = prod f A mod a"
 proof (induct A rule: infinite_finite_induct)
   case (insert i A)
   then have "(\<Prod>i\<in>insert i A. f i mod a) mod a
     = (f i mod a * (\<Prod>i\<in>A. f i mod a)) mod a"
     by simp
   also have "\<dots> = (f i * ((\<Prod>i\<in>A. f i mod a) mod a)) mod a"
     by (simp add: mod_simps)
   also have "\<dots> = (f i * ((\<Prod>i\<in>A. f i) mod a)) mod a"
     by (simp add: insert.hyps)
   finally show ?case
     by (simp add: insert.hyps mod_simps)
 qed simp_all
 
 lemma mod_mult_cong:
   assumes "a mod c = a' mod c"
   assumes "b mod c = b' mod c"
   shows "(a * b) mod c = (a' * b') mod c"
 proof -
   have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
     unfolding assms ..
   then show ?thesis
     by (simp add: mod_mult_eq)
 qed
 
 text \<open>Exponentiation respects modular equivalence.\<close>
 
 lemma power_mod [mod_simps]: 
   "((a mod b) ^ n) mod b = (a ^ n) mod b"
 proof (induct n)
   case 0
   then show ?case by simp
 next
   case (Suc n)
   have "(a mod b) ^ Suc n mod b = (a mod b) * ((a mod b) ^ n mod b) mod b"
     by (simp add: mod_mult_right_eq)
   with Suc show ?case
     by (simp add: mod_mult_left_eq mod_mult_right_eq)
 qed
 
 lemma power_diff_power_eq:
   \<open>a ^ m div a ^ n = (if n \<le> m then a ^ (m - n) else 1 div a ^ (n - m))\<close>
     if \<open>a \<noteq> 0\<close>
 proof (cases \<open>n \<le> m\<close>)
   case True
   with that power_diff [symmetric, of a n m] show ?thesis by simp
 next
   case False
   then obtain q where n: \<open>n = m + Suc q\<close>
     by (auto simp add: not_le dest: less_imp_Suc_add)
   then have \<open>a ^ m div a ^ n = (a ^ m * 1) div (a ^ m * a ^ Suc q)\<close>
     by (simp add: power_add ac_simps)
   moreover from that have \<open>a ^ m \<noteq> 0\<close>
     by simp
   ultimately have \<open>a ^ m div a ^ n = 1 div a ^ Suc q\<close>
     by (subst (asm) div_mult_mult1) simp
   with False n show ?thesis
     by simp
 qed
 
 end
 
 
 class euclidean_ring_cancel = euclidean_ring + euclidean_semiring_cancel
 begin
 
 subclass idom_divide ..
 
 lemma div_minus_minus [simp]: "(- a) div (- b) = a div b"
   using div_mult_mult1 [of "- 1" a b] by simp
 
 lemma mod_minus_minus [simp]: "(- a) mod (- b) = - (a mod b)"
   using mod_mult_mult1 [of "- 1" a b] by simp
 
 lemma div_minus_right: "a div (- b) = (- a) div b"
   using div_minus_minus [of "- a" b] by simp
 
 lemma mod_minus_right: "a mod (- b) = - ((- a) mod b)"
   using mod_minus_minus [of "- a" b] by simp
 
 lemma div_minus1_right [simp]: "a div (- 1) = - a"
   using div_minus_right [of a 1] by simp
 
 lemma mod_minus1_right [simp]: "a mod (- 1) = 0"
   using mod_minus_right [of a 1] by simp
 
 text \<open>Negation respects modular equivalence.\<close>
 
 lemma mod_minus_eq [mod_simps]:
   "(- (a mod b)) mod b = (- a) mod b"
 proof -
   have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
     by (simp only: div_mult_mod_eq)
   also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
     by (simp add: ac_simps)
   also have "\<dots> = (- (a mod b)) mod b"
     by (rule mod_mult_self1)
   finally show ?thesis
     by (rule sym)
 qed
 
 lemma mod_minus_cong:
   assumes "a mod b = a' mod b"
   shows "(- a) mod b = (- a') mod b"
 proof -
   have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
     unfolding assms ..
   then show ?thesis
     by (simp add: mod_minus_eq)
 qed
 
 text \<open>Subtraction respects modular equivalence.\<close>
 
 lemma mod_diff_left_eq [mod_simps]:
   "(a mod c - b) mod c = (a - b) mod c"
   using mod_add_cong [of a c "a mod c" "- b" "- b"]
   by simp
 
 lemma mod_diff_right_eq [mod_simps]:
   "(a - b mod c) mod c = (a - b) mod c"
   using mod_add_cong [of a c a "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b]
   by simp
 
 lemma mod_diff_eq:
   "(a mod c - b mod c) mod c = (a - b) mod c"
   using mod_add_cong [of a c "a mod c" "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b]
   by simp
 
 lemma mod_diff_cong:
   assumes "a mod c = a' mod c"
   assumes "b mod c = b' mod c"
   shows "(a - b) mod c = (a' - b') mod c"
   using assms mod_add_cong [of a c a' "- b" "- b'"] mod_minus_cong [of b c "b'"]
   by simp
 
 lemma minus_mod_self2 [simp]:
   "(a - b) mod b = a mod b"
   using mod_diff_right_eq [of a b b]
   by (simp add: mod_diff_right_eq)
 
 lemma minus_mod_self1 [simp]:
   "(b - a) mod b = - a mod b"
   using mod_add_self2 [of "- a" b] by simp
 
 lemma mod_eq_dvd_iff:
   "a mod c = b mod c \<longleftrightarrow> c dvd a - b" (is "?P \<longleftrightarrow> ?Q")
 proof
   assume ?P
   then have "(a mod c - b mod c) mod c = 0"
     by simp
   then show ?Q
     by (simp add: dvd_eq_mod_eq_0 mod_simps)
 next
   assume ?Q
   then obtain d where d: "a - b = c * d" ..
   then have "a = c * d + b"
     by (simp add: algebra_simps)
   then show ?P by simp
 qed
 
 lemma mod_eqE:
   assumes "a mod c = b mod c"
   obtains d where "b = a + c * d"
 proof -
   from assms have "c dvd a - b"
     by (simp add: mod_eq_dvd_iff)
   then obtain d where "a - b = c * d" ..
   then have "b = a + c * - d"
     by (simp add: algebra_simps)
   with that show thesis .
 qed
 
 lemma invertible_coprime:
   "coprime a c" if "a * b mod c = 1"
   by (rule coprimeI) (use that dvd_mod_iff [of _ c "a * b"] in auto)
 
 end
 
   
 subsection \<open>Uniquely determined division\<close>
   
 class unique_euclidean_semiring = euclidean_semiring + 
   assumes euclidean_size_mult: "euclidean_size (a * b) = euclidean_size a * euclidean_size b"
   fixes division_segment :: "'a \<Rightarrow> 'a"
   assumes is_unit_division_segment [simp]: "is_unit (division_segment a)"
     and division_segment_mult:
     "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> division_segment (a * b) = division_segment a * division_segment b"
     and division_segment_mod:
     "b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<Longrightarrow> division_segment (a mod b) = division_segment b"
   assumes div_bounded:
     "b \<noteq> 0 \<Longrightarrow> division_segment r = division_segment b
     \<Longrightarrow> euclidean_size r < euclidean_size b
     \<Longrightarrow> (q * b + r) div b = q"
 begin
 
 lemma division_segment_not_0 [simp]:
   "division_segment a \<noteq> 0"
   using is_unit_division_segment [of a] is_unitE [of "division_segment a"] by blast
 
 lemma divmod_cases [case_names divides remainder by0]:
   obtains 
     (divides) q where "b \<noteq> 0"
       and "a div b = q"
       and "a mod b = 0"
       and "a = q * b"
   | (remainder) q r where "b \<noteq> 0"
       and "division_segment r = division_segment b"
       and "euclidean_size r < euclidean_size b"
       and "r \<noteq> 0"
       and "a div b = q"
       and "a mod b = r"
       and "a = q * b + r"
   | (by0) "b = 0"
 proof (cases "b = 0")
   case True
   then show thesis
   by (rule by0)
 next
   case False
   show thesis
   proof (cases "b dvd a")
     case True
     then obtain q where "a = b * q" ..
     with \<open>b \<noteq> 0\<close> divides
     show thesis
       by (simp add: ac_simps)
   next
     case False
     then have "a mod b \<noteq> 0"
       by (simp add: mod_eq_0_iff_dvd)
     moreover from \<open>b \<noteq> 0\<close> \<open>\<not> b dvd a\<close> have "division_segment (a mod b) = division_segment b"
       by (rule division_segment_mod)
     moreover have "euclidean_size (a mod b) < euclidean_size b"
       using \<open>b \<noteq> 0\<close> by (rule mod_size_less)
     moreover have "a = a div b * b + a mod b"
       by (simp add: div_mult_mod_eq)
     ultimately show thesis
       using \<open>b \<noteq> 0\<close> by (blast intro!: remainder)
   qed
 qed
 
 lemma div_eqI:
   "a div b = q" if "b \<noteq> 0" "division_segment r = division_segment b"
     "euclidean_size r < euclidean_size b" "q * b + r = a"
 proof -
   from that have "(q * b + r) div b = q"
     by (auto intro: div_bounded)
   with that show ?thesis
     by simp
 qed
 
 lemma mod_eqI:
   "a mod b = r" if "b \<noteq> 0" "division_segment r = division_segment b"
     "euclidean_size r < euclidean_size b" "q * b + r = a" 
 proof -
   from that have "a div b = q"
     by (rule div_eqI)
   moreover have "a div b * b + a mod b = a"
     by (fact div_mult_mod_eq)
   ultimately have "a div b * b + a mod b = a div b * b + r"
     using \<open>q * b + r = a\<close> by simp
   then show ?thesis
     by simp
 qed
 
 subclass euclidean_semiring_cancel
 proof
   show "(a + c * b) div b = c + a div b" if "b \<noteq> 0" for a b c
   proof (cases a b rule: divmod_cases)
     case by0
     with \<open>b \<noteq> 0\<close> show ?thesis
       by simp
   next
     case (divides q)
     then show ?thesis
       by (simp add: ac_simps)
   next
     case (remainder q r)
     then show ?thesis
       by (auto intro: div_eqI simp add: algebra_simps)
   qed
 next
   show"(c * a) div (c * b) = a div b" if "c \<noteq> 0" for a b c
   proof (cases a b rule: divmod_cases)
     case by0
     then show ?thesis
       by simp
   next
     case (divides q)
     with \<open>c \<noteq> 0\<close> show ?thesis
       by (simp add: mult.left_commute [of c])
   next
     case (remainder q r)
     from \<open>b \<noteq> 0\<close> \<open>c \<noteq> 0\<close> have "b * c \<noteq> 0"
       by simp
     from remainder \<open>c \<noteq> 0\<close>
     have "division_segment (r * c) = division_segment (b * c)"
       and "euclidean_size (r * c) < euclidean_size (b * c)"
       by (simp_all add: division_segment_mult division_segment_mod euclidean_size_mult)
     with remainder show ?thesis
       by (auto intro!: div_eqI [of _ "c * (a mod b)"] simp add: algebra_simps)
         (use \<open>b * c \<noteq> 0\<close> in simp)
   qed
 qed
 
 lemma div_mult1_eq:
   "(a * b) div c = a * (b div c) + a * (b mod c) div c"
 proof (cases "a * (b mod c)" c rule: divmod_cases)
   case (divides q)
   have "a * b = a * (b div c * c + b mod c)"
     by (simp add: div_mult_mod_eq)
   also have "\<dots> = (a * (b div c) + q) * c"
     using divides by (simp add: algebra_simps)
   finally have "(a * b) div c = \<dots> div c"
     by simp
   with divides show ?thesis
     by simp
 next
   case (remainder q r)
   from remainder(1-3) show ?thesis
   proof (rule div_eqI)
     have "a * b = a * (b div c * c + b mod c)"
       by (simp add: div_mult_mod_eq)
     also have "\<dots> = a * c * (b div c) + q * c + r"
       using remainder by (simp add: algebra_simps)
     finally show "(a * (b div c) + a * (b mod c) div c) * c + r = a * b"
       using remainder(5-7) by (simp add: algebra_simps)
   qed
 next
   case by0
   then show ?thesis
     by simp
 qed
 
 lemma div_add1_eq:
   "(a + b) div c = a div c + b div c + (a mod c + b mod c) div c"
 proof (cases "a mod c + b mod c" c rule: divmod_cases)
   case (divides q)
   have "a + b = (a div c * c + a mod c) + (b div c * c + b mod c)"
     using mod_mult_div_eq [of a c] mod_mult_div_eq [of b c] by (simp add: ac_simps)
   also have "\<dots> = (a div c + b div c) * c + (a mod c + b mod c)"
     by (simp add: algebra_simps)
   also have "\<dots> = (a div c + b div c + q) * c"
     using divides by (simp add: algebra_simps)
   finally have "(a + b) div c = (a div c + b div c + q) * c div c"
     by simp
   with divides show ?thesis
     by simp
 next
   case (remainder q r)
   from remainder(1-3) show ?thesis
   proof (rule div_eqI)
     have "(a div c + b div c + q) * c + r + (a mod c + b mod c) =
         (a div c * c + a mod c) + (b div c * c + b mod c) + q * c + r"
       by (simp add: algebra_simps)
     also have "\<dots> = a + b + (a mod c + b mod c)"
       by (simp add: div_mult_mod_eq remainder) (simp add: ac_simps)
     finally show "(a div c + b div c + (a mod c + b mod c) div c) * c + r = a + b"
       using remainder by simp
   qed
 next
   case by0
   then show ?thesis
     by simp
 qed
 
 lemma div_eq_0_iff:
   "a div b = 0 \<longleftrightarrow> euclidean_size a < euclidean_size b \<or> b = 0" (is "_ \<longleftrightarrow> ?P")
   if "division_segment a = division_segment b"
 proof
   assume ?P
   with that show "a div b = 0"
     by (cases "b = 0") (auto intro: div_eqI)
 next
   assume "a div b = 0"
   then have "a mod b = a"
     using div_mult_mod_eq [of a b] by simp
   with mod_size_less [of b a] show ?P
     by auto
 qed
 
 end
 
 class unique_euclidean_ring = euclidean_ring + unique_euclidean_semiring
 begin
   
 subclass euclidean_ring_cancel ..
 
 end
 
 
 subsection \<open>Euclidean division on \<^typ>\<open>nat\<close>\<close>
 
 instantiation nat :: normalization_semidom
 begin
 
 definition normalize_nat :: "nat \<Rightarrow> nat"
   where [simp]: "normalize = (id :: nat \<Rightarrow> nat)"
 
 definition unit_factor_nat :: "nat \<Rightarrow> nat"
   where "unit_factor n = (if n = 0 then 0 else 1 :: nat)"
 
 lemma unit_factor_simps [simp]:
   "unit_factor 0 = (0::nat)"
   "unit_factor (Suc n) = 1"
   by (simp_all add: unit_factor_nat_def)
 
 definition divide_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
   where "m div n = (if n = 0 then 0 else Max {k::nat. k * n \<le> m})"
 
 instance
   by standard (auto simp add: divide_nat_def ac_simps unit_factor_nat_def intro: Max_eqI)
 
 end
 
 lemma coprime_Suc_0_left [simp]:
   "coprime (Suc 0) n"
   using coprime_1_left [of n] by simp
 
 lemma coprime_Suc_0_right [simp]:
   "coprime n (Suc 0)"
   using coprime_1_right [of n] by simp
 
 lemma coprime_common_divisor_nat: "coprime a b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> x = 1"
   for a b :: nat
   by (drule coprime_common_divisor [of _ _ x]) simp_all
 
 instantiation nat :: unique_euclidean_semiring
 begin
 
 definition euclidean_size_nat :: "nat \<Rightarrow> nat"
   where [simp]: "euclidean_size_nat = id"
 
 definition division_segment_nat :: "nat \<Rightarrow> nat"
   where [simp]: "division_segment_nat n = 1"
 
 definition modulo_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
   where "m mod n = m - (m div n * (n::nat))"
 
 instance proof
   fix m n :: nat
   have ex: "\<exists>k. k * n \<le> l" for l :: nat
     by (rule exI [of _ 0]) simp
   have fin: "finite {k. k * n \<le> l}" if "n > 0" for l
   proof -
     from that have "{k. k * n \<le> l} \<subseteq> {k. k \<le> l}"
       by (cases n) auto
     then show ?thesis
       by (rule finite_subset) simp
   qed
   have mult_div_unfold: "n * (m div n) = Max {l. l \<le> m \<and> n dvd l}"
   proof (cases "n = 0")
     case True
     moreover have "{l. l = 0 \<and> l \<le> m} = {0::nat}"
       by auto
     ultimately show ?thesis
       by simp
   next
     case False
     with ex [of m] fin have "n * Max {k. k * n \<le> m} = Max (times n ` {k. k * n \<le> m})"
       by (auto simp add: nat_mult_max_right intro: hom_Max_commute)
     also have "times n ` {k. k * n \<le> m} = {l. l \<le> m \<and> n dvd l}"
       by (auto simp add: ac_simps elim!: dvdE)
     finally show ?thesis
       using False by (simp add: divide_nat_def ac_simps)
   qed
   have less_eq: "m div n * n \<le> m"
     by (auto simp add: mult_div_unfold ac_simps intro: Max.boundedI)
   then show "m div n * n + m mod n = m"
     by (simp add: modulo_nat_def)
   assume "n \<noteq> 0" 
   show "euclidean_size (m mod n) < euclidean_size n"
   proof -
     have "m < Suc (m div n) * n"
     proof (rule ccontr)
       assume "\<not> m < Suc (m div n) * n"
       then have "Suc (m div n) * n \<le> m"
         by (simp add: not_less)
       moreover from \<open>n \<noteq> 0\<close> have "Max {k. k * n \<le> m} < Suc (m div n)"
         by (simp add: divide_nat_def)
       with \<open>n \<noteq> 0\<close> ex fin have "\<And>k. k * n \<le> m \<Longrightarrow> k < Suc (m div n)"
         by auto
       ultimately have "Suc (m div n) < Suc (m div n)"
         by blast
       then show False
         by simp
     qed
     with \<open>n \<noteq> 0\<close> show ?thesis
       by (simp add: modulo_nat_def)
   qed
   show "euclidean_size m \<le> euclidean_size (m * n)"
     using \<open>n \<noteq> 0\<close> by (cases n) simp_all
   fix q r :: nat
   show "(q * n + r) div n = q" if "euclidean_size r < euclidean_size n"
   proof -
     from that have "r < n"
       by simp
     have "k \<le> q" if "k * n \<le> q * n + r" for k
     proof (rule ccontr)
       assume "\<not> k \<le> q"
       then have "q < k"
         by simp
       then obtain l where "k = Suc (q + l)"
         by (auto simp add: less_iff_Suc_add)
       with \<open>r < n\<close> that show False
         by (simp add: algebra_simps)
     qed
     with \<open>n \<noteq> 0\<close> ex fin show ?thesis
       by (auto simp add: divide_nat_def Max_eq_iff)
   qed
 qed simp_all
 
 end
 
 text \<open>Tool support\<close>
 
 ML \<open>
 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
 (
   val div_name = \<^const_name>\<open>divide\<close>;
   val mod_name = \<^const_name>\<open>modulo\<close>;
   val mk_binop = HOLogic.mk_binop;
   val dest_plus = HOLogic.dest_bin \<^const_name>\<open>Groups.plus\<close> HOLogic.natT;
   val mk_sum = Arith_Data.mk_sum;
   fun dest_sum tm =
     if HOLogic.is_zero tm then []
     else
       (case try HOLogic.dest_Suc tm of
         SOME t => HOLogic.Suc_zero :: dest_sum t
       | NONE =>
           (case try dest_plus tm of
             SOME (t, u) => dest_sum t @ dest_sum u
           | NONE => [tm]));
 
   val div_mod_eqs = map mk_meta_eq @{thms cancel_div_mod_rules};
 
   val prove_eq_sums = Arith_Data.prove_conv2 all_tac
     (Arith_Data.simp_all_tac @{thms add_0_left add_0_right ac_simps})
 )
 \<close>
 
 simproc_setup cancel_div_mod_nat ("(m::nat) + n") =
   \<open>K Cancel_Div_Mod_Nat.proc\<close>
 
 lemma div_nat_eqI:
   "m div n = q" if "n * q \<le> m" and "m < n * Suc q" for m n q :: nat
   by (rule div_eqI [of _ "m - n * q"]) (use that in \<open>simp_all add: algebra_simps\<close>)
 
 lemma mod_nat_eqI:
   "m mod n = r" if "r < n" and "r \<le> m" and "n dvd m - r" for m n r :: nat
   by (rule mod_eqI [of _ _ "(m - r) div n"]) (use that in \<open>simp_all add: algebra_simps\<close>)
 
 lemma div_mult_self_is_m [simp]:
   "m * n div n = m" if "n > 0" for m n :: nat
   using that by simp
 
 lemma div_mult_self1_is_m [simp]:
   "n * m div n = m" if "n > 0" for m n :: nat
   using that by simp
 
 lemma mod_less_divisor [simp]:
   "m mod n < n" if "n > 0" for m n :: nat
   using mod_size_less [of n m] that by simp
 
 lemma mod_le_divisor [simp]:
   "m mod n \<le> n" if "n > 0" for m n :: nat
   using that by (auto simp add: le_less)
 
 lemma div_times_less_eq_dividend [simp]:
   "m div n * n \<le> m" for m n :: nat
   by (simp add: minus_mod_eq_div_mult [symmetric])
 
 lemma times_div_less_eq_dividend [simp]:
   "n * (m div n) \<le> m" for m n :: nat
   using div_times_less_eq_dividend [of m n]
   by (simp add: ac_simps)
 
 lemma dividend_less_div_times:
   "m < n + (m div n) * n" if "0 < n" for m n :: nat
 proof -
   from that have "m mod n < n"
     by simp
   then show ?thesis
     by (simp add: minus_mod_eq_div_mult [symmetric])
 qed
 
 lemma dividend_less_times_div:
   "m < n + n * (m div n)" if "0 < n" for m n :: nat
   using dividend_less_div_times [of n m] that
   by (simp add: ac_simps)
 
 lemma mod_Suc_le_divisor [simp]:
   "m mod Suc n \<le> n"
   using mod_less_divisor [of "Suc n" m] by arith
 
 lemma mod_less_eq_dividend [simp]:
   "m mod n \<le> m" for m n :: nat
 proof (rule add_leD2)
   from div_mult_mod_eq have "m div n * n + m mod n = m" .
   then show "m div n * n + m mod n \<le> m" by auto
 qed
 
 lemma
   div_less [simp]: "m div n = 0"
   and mod_less [simp]: "m mod n = m"
   if "m < n" for m n :: nat
   using that by (auto intro: div_eqI mod_eqI) 
 
 lemma le_div_geq:
   "m div n = Suc ((m - n) div n)" if "0 < n" and "n \<le> m" for m n :: nat
 proof -
   from \<open>n \<le> m\<close> obtain q where "m = n + q"
     by (auto simp add: le_iff_add)
   with \<open>0 < n\<close> show ?thesis
     by (simp add: div_add_self1)
 qed
 
 lemma le_mod_geq:
   "m mod n = (m - n) mod n" if "n \<le> m" for m n :: nat
 proof -
   from \<open>n \<le> m\<close> obtain q where "m = n + q"
     by (auto simp add: le_iff_add)
   then show ?thesis
     by simp
 qed
 
 lemma div_if:
   "m div n = (if m < n \<or> n = 0 then 0 else Suc ((m - n) div n))"
   by (simp add: le_div_geq)
 
 lemma mod_if:
   "m mod n = (if m < n then m else (m - n) mod n)" for m n :: nat
   by (simp add: le_mod_geq)
 
 lemma div_eq_0_iff:
   "m div n = 0 \<longleftrightarrow> m < n \<or> n = 0" for m n :: nat
   by (simp add: div_eq_0_iff)
 
 lemma div_greater_zero_iff:
   "m div n > 0 \<longleftrightarrow> n \<le> m \<and> n > 0" for m n :: nat
   using div_eq_0_iff [of m n] by auto
 
 lemma mod_greater_zero_iff_not_dvd:
   "m mod n > 0 \<longleftrightarrow> \<not> n dvd m" for m n :: nat
   by (simp add: dvd_eq_mod_eq_0)
 
 lemma div_by_Suc_0 [simp]:
   "m div Suc 0 = m"
   using div_by_1 [of m] by simp
 
 lemma mod_by_Suc_0 [simp]:
   "m mod Suc 0 = 0"
   using mod_by_1 [of m] by simp
 
 lemma div2_Suc_Suc [simp]:
   "Suc (Suc m) div 2 = Suc (m div 2)"
   by (simp add: numeral_2_eq_2 le_div_geq)
 
 lemma Suc_n_div_2_gt_zero [simp]:
   "0 < Suc n div 2" if "n > 0" for n :: nat
   using that by (cases n) simp_all
 
 lemma div_2_gt_zero [simp]:
   "0 < n div 2" if "Suc 0 < n" for n :: nat
   using that Suc_n_div_2_gt_zero [of "n - 1"] by simp
 
 lemma mod2_Suc_Suc [simp]:
   "Suc (Suc m) mod 2 = m mod 2"
   by (simp add: numeral_2_eq_2 le_mod_geq)
 
 lemma add_self_div_2 [simp]:
   "(m + m) div 2 = m" for m :: nat
   by (simp add: mult_2 [symmetric])
 
 lemma add_self_mod_2 [simp]:
   "(m + m) mod 2 = 0" for m :: nat
   by (simp add: mult_2 [symmetric])
 
 lemma mod2_gr_0 [simp]:
   "0 < m mod 2 \<longleftrightarrow> m mod 2 = 1" for m :: nat
 proof -
   have "m mod 2 < 2"
     by (rule mod_less_divisor) simp
   then have "m mod 2 = 0 \<or> m mod 2 = 1"
     by arith
   then show ?thesis
     by auto     
 qed
 
 lemma mod_Suc_eq [mod_simps]:
   "Suc (m mod n) mod n = Suc m mod n"
 proof -
   have "(m mod n + 1) mod n = (m + 1) mod n"
     by (simp only: mod_simps)
   then show ?thesis
     by simp
 qed
 
 lemma mod_Suc_Suc_eq [mod_simps]:
   "Suc (Suc (m mod n)) mod n = Suc (Suc m) mod n"
 proof -
   have "(m mod n + 2) mod n = (m + 2) mod n"
     by (simp only: mod_simps)
   then show ?thesis
     by simp
 qed
 
 lemma
   Suc_mod_mult_self1 [simp]: "Suc (m + k * n) mod n = Suc m mod n"
   and Suc_mod_mult_self2 [simp]: "Suc (m + n * k) mod n = Suc m mod n"
   and Suc_mod_mult_self3 [simp]: "Suc (k * n + m) mod n = Suc m mod n"
   and Suc_mod_mult_self4 [simp]: "Suc (n * k + m) mod n = Suc m mod n"
   by (subst mod_Suc_eq [symmetric], simp add: mod_simps)+
 
 lemma Suc_0_mod_eq [simp]:
   "Suc 0 mod n = of_bool (n \<noteq> Suc 0)"
   by (cases n) simp_all
 
 context
   fixes m n q :: nat
 begin
 
 private lemma eucl_rel_mult2:
   "m mod n + n * (m div n mod q) < n * q"
   if "n > 0" and "q > 0"
 proof -
   from \<open>n > 0\<close> have "m mod n < n"
     by (rule mod_less_divisor)
   from \<open>q > 0\<close> have "m div n mod q < q"
     by (rule mod_less_divisor)
   then obtain s where "q = Suc (m div n mod q + s)"
     by (blast dest: less_imp_Suc_add)
   moreover have "m mod n + n * (m div n mod q) < n * Suc (m div n mod q + s)"
     using \<open>m mod n < n\<close> by (simp add: add_mult_distrib2)
   ultimately show ?thesis
     by simp
 qed
 
 lemma div_mult2_eq:
   "m div (n * q) = (m div n) div q"
 proof (cases "n = 0 \<or> q = 0")
   case True
   then show ?thesis
     by auto
 next
   case False
   with eucl_rel_mult2 show ?thesis
     by (auto intro: div_eqI [of _ "n * (m div n mod q) + m mod n"]
       simp add: algebra_simps add_mult_distrib2 [symmetric])
 qed
 
 lemma mod_mult2_eq:
   "m mod (n * q) = n * (m div n mod q) + m mod n"
 proof (cases "n = 0 \<or> q = 0")
   case True
   then show ?thesis
     by auto
 next
   case False
   with eucl_rel_mult2 show ?thesis
     by (auto intro: mod_eqI [of _ _ "(m div n) div q"]
       simp add: algebra_simps add_mult_distrib2 [symmetric])
 qed
 
 end
 
 lemma div_le_mono:
   "m div k \<le> n div k" if "m \<le> n" for m n k :: nat
 proof -
   from that obtain q where "n = m + q"
     by (auto simp add: le_iff_add)
   then show ?thesis
     by (simp add: div_add1_eq [of m q k])
 qed
 
 text \<open>Antimonotonicity of \<^const>\<open>divide\<close> in second argument\<close>
 
 lemma div_le_mono2:
   "k div n \<le> k div m" if "0 < m" and "m \<le> n" for m n k :: nat
 using that proof (induct k arbitrary: m rule: less_induct)
   case (less k)
   show ?case
   proof (cases "n \<le> k")
     case False
     then show ?thesis
       by simp
   next
     case True
     have "(k - n) div n \<le> (k - m) div n"
       using less.prems
       by (blast intro: div_le_mono diff_le_mono2)
     also have "\<dots> \<le> (k - m) div m"
       using \<open>n \<le> k\<close> less.prems less.hyps [of "k - m" m]
       by simp
     finally show ?thesis
       using \<open>n \<le> k\<close> less.prems
       by (simp add: le_div_geq)
   qed
 qed
 
 lemma div_le_dividend [simp]:
   "m div n \<le> m" for m n :: nat
   using div_le_mono2 [of 1 n m] by (cases "n = 0") simp_all
 
 lemma div_less_dividend [simp]:
   "m div n < m" if "1 < n" and "0 < m" for m n :: nat
 using that proof (induct m rule: less_induct)
   case (less m)
   show ?case
   proof (cases "n < m")
     case False
     with less show ?thesis
       by (cases "n = m") simp_all
   next
     case True
     then show ?thesis
       using less.hyps [of "m - n"] less.prems
       by (simp add: le_div_geq)
   qed
 qed
 
 lemma div_eq_dividend_iff:
   "m div n = m \<longleftrightarrow> n = 1" if "m > 0" for m n :: nat
 proof
   assume "n = 1"
   then show "m div n = m"
     by simp
 next
   assume P: "m div n = m"
   show "n = 1"
   proof (rule ccontr)
     have "n \<noteq> 0"
       by (rule ccontr) (use that P in auto)
     moreover assume "n \<noteq> 1"
     ultimately have "n > 1"
       by simp
     with that have "m div n < m"
       by simp
     with P show False
       by simp
   qed
 qed
 
 lemma less_mult_imp_div_less:
   "m div n < i" if "m < i * n" for m n i :: nat
 proof -
   from that have "i * n > 0"
     by (cases "i * n = 0") simp_all
   then have "i > 0" and "n > 0"
     by simp_all
   have "m div n * n \<le> m"
     by simp
   then have "m div n * n < i * n"
     using that by (rule le_less_trans)
   with \<open>n > 0\<close> show ?thesis
     by simp
 qed
 
 text \<open>A fact for the mutilated chess board\<close>
 
 lemma mod_Suc:
   "Suc m mod n = (if Suc (m mod n) = n then 0 else Suc (m mod n))" (is "_ = ?rhs")
 proof (cases "n = 0")
   case True
   then show ?thesis
     by simp
 next
   case False
   have "Suc m mod n = Suc (m mod n) mod n"
     by (simp add: mod_simps)
   also have "\<dots> = ?rhs"
     using False by (auto intro!: mod_nat_eqI intro: neq_le_trans simp add: Suc_le_eq)
   finally show ?thesis .
 qed
 
 lemma Suc_times_mod_eq:
   "Suc (m * n) mod m = 1" if "Suc 0 < m"
   using that by (simp add: mod_Suc)
 
 lemma Suc_times_numeral_mod_eq [simp]:
   "Suc (numeral k * n) mod numeral k = 1" if "numeral k \<noteq> (1::nat)"
   by (rule Suc_times_mod_eq) (use that in simp)
 
 lemma Suc_div_le_mono [simp]:
   "m div n \<le> Suc m div n"
   by (simp add: div_le_mono)
 
 text \<open>These lemmas collapse some needless occurrences of Suc:
   at least three Sucs, since two and fewer are rewritten back to Suc again!
   We already have some rules to simplify operands smaller than 3.\<close>
 
 lemma div_Suc_eq_div_add3 [simp]:
   "m div Suc (Suc (Suc n)) = m div (3 + n)"
   by (simp add: Suc3_eq_add_3)
 
 lemma mod_Suc_eq_mod_add3 [simp]:
   "m mod Suc (Suc (Suc n)) = m mod (3 + n)"
   by (simp add: Suc3_eq_add_3)
 
 lemma Suc_div_eq_add3_div:
   "Suc (Suc (Suc m)) div n = (3 + m) div n"
   by (simp add: Suc3_eq_add_3)
 
 lemma Suc_mod_eq_add3_mod:
   "Suc (Suc (Suc m)) mod n = (3 + m) mod n"
   by (simp add: Suc3_eq_add_3)
 
 lemmas Suc_div_eq_add3_div_numeral [simp] =
   Suc_div_eq_add3_div [of _ "numeral v"] for v
 
 lemmas Suc_mod_eq_add3_mod_numeral [simp] =
   Suc_mod_eq_add3_mod [of _ "numeral v"] for v
 
 lemma (in field_char_0) of_nat_div:
   "of_nat (m div n) = ((of_nat m - of_nat (m mod n)) / of_nat n)"
 proof -
   have "of_nat (m div n) = ((of_nat (m div n * n + m mod n) - of_nat (m mod n)) / of_nat n :: 'a)"
     unfolding of_nat_add by (cases "n = 0") simp_all
   then show ?thesis
     by simp
 qed
 
 text \<open>An ``induction'' law for modulus arithmetic.\<close>
 
 lemma mod_induct [consumes 3, case_names step]:
   "P m" if "P n" and "n < p" and "m < p"
     and step: "\<And>n. n < p \<Longrightarrow> P n \<Longrightarrow> P (Suc n mod p)"
 using \<open>m < p\<close> proof (induct m)
   case 0
   show ?case
   proof (rule ccontr)
     assume "\<not> P 0"
     from \<open>n < p\<close> have "0 < p"
       by simp
     from \<open>n < p\<close> obtain m where "0 < m" and "p = n + m"
       by (blast dest: less_imp_add_positive)
     with \<open>P n\<close> have "P (p - m)"
       by simp
     moreover have "\<not> P (p - m)"
     using \<open>0 < m\<close> proof (induct m)
       case 0
       then show ?case
         by simp
     next
       case (Suc m)
       show ?case
       proof
         assume P: "P (p - Suc m)"
         with \<open>\<not> P 0\<close> have "Suc m < p"
           by (auto intro: ccontr) 
         then have "Suc (p - Suc m) = p - m"
           by arith
         moreover from \<open>0 < p\<close> have "p - Suc m < p"
           by arith
         with P step have "P ((Suc (p - Suc m)) mod p)"
           by blast
         ultimately show False
           using \<open>\<not> P 0\<close> Suc.hyps by (cases "m = 0") simp_all
       qed
     qed
     ultimately show False
       by blast
   qed
 next
   case (Suc m)
   then have "m < p" and mod: "Suc m mod p = Suc m"
     by simp_all
   from \<open>m < p\<close> have "P m"
     by (rule Suc.hyps)
   with \<open>m < p\<close> have "P (Suc m mod p)"
     by (rule step)
   with mod show ?case
     by simp
 qed
 
 lemma split_div:
   "P (m div n) \<longleftrightarrow> (n = 0 \<longrightarrow> P 0) \<and> (n \<noteq> 0 \<longrightarrow>
      (\<forall>i j. j < n \<longrightarrow> m = n * i + j \<longrightarrow> P i))"
      (is "?P = ?Q") for m n :: nat
 proof (cases "n = 0")
   case True
   then show ?thesis
     by simp
 next
   case False
   show ?thesis
   proof
     assume ?P
     with False show ?Q
       by auto
   next
     assume ?Q
     with False have *: "\<And>i j. j < n \<Longrightarrow> m = n * i + j \<Longrightarrow> P i"
       by simp
     with False show ?P
       by (auto intro: * [of "m mod n"])
   qed
 qed
 
 lemma split_div':
   "P (m div n) \<longleftrightarrow> n = 0 \<and> P 0 \<or> (\<exists>q. (n * q \<le> m \<and> m < n * Suc q) \<and> P q)"
 proof (cases "n = 0")
   case True
   then show ?thesis
     by simp
 next
   case False
   then have "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> m div n = q" for q
     by (auto intro: div_nat_eqI dividend_less_times_div)
   then show ?thesis
     by auto
 qed
 
 lemma split_mod:
   "P (m mod n) \<longleftrightarrow> (n = 0 \<longrightarrow> P m) \<and> (n \<noteq> 0 \<longrightarrow>
      (\<forall>i j. j < n \<longrightarrow> m = n * i + j \<longrightarrow> P j))"
      (is "?P \<longleftrightarrow> ?Q") for m n :: nat
 proof (cases "n = 0")
   case True
   then show ?thesis
     by simp
 next
   case False
   show ?thesis
   proof
     assume ?P
     with False show ?Q
       by auto
   next
     assume ?Q
     with False have *: "\<And>i j. j < n \<Longrightarrow> m = n * i + j \<Longrightarrow> P j"
       by simp
     with False show ?P
       by (auto intro: * [of _ "m div n"])
   qed
 qed
 
 
 subsection \<open>Euclidean division on \<^typ>\<open>int\<close>\<close>
 
 instantiation int :: normalization_semidom
 begin
 
 definition normalize_int :: "int \<Rightarrow> int"
   where [simp]: "normalize = (abs :: int \<Rightarrow> int)"
 
 definition unit_factor_int :: "int \<Rightarrow> int"
   where [simp]: "unit_factor = (sgn :: int \<Rightarrow> int)"
 
 definition divide_int :: "int \<Rightarrow> int \<Rightarrow> int"
   where "k div l = (if l = 0 then 0
     else if sgn k = sgn l
       then int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
       else - int (nat \<bar>k\<bar> div nat \<bar>l\<bar> + of_bool (\<not> l dvd k)))"
 
 lemma divide_int_unfold:
   "(sgn k * int m) div (sgn l * int n) =
    (if sgn l = 0 \<or> sgn k = 0 \<or> n = 0 then 0
     else if sgn k = sgn l
       then int (m div n)
       else - int (m div n + of_bool (\<not> n dvd m)))"
   by (auto simp add: divide_int_def sgn_0_0 sgn_1_pos sgn_mult abs_mult
     nat_mult_distrib)
 
 instance proof
   fix k :: int show "k div 0 = 0"
   by (simp add: divide_int_def)
 next
   fix k l :: int
   assume "l \<noteq> 0"
   obtain n m and s t where k: "k = sgn s * int n" and l: "l = sgn t * int m" 
     by (blast intro: int_sgnE elim: that)
   then have "k * l = sgn (s * t) * int (n * m)"
     by (simp add: ac_simps sgn_mult)
   with k l \<open>l \<noteq> 0\<close> show "k * l div l = k"
     by (simp only: divide_int_unfold)
       (auto simp add: algebra_simps sgn_mult sgn_1_pos sgn_0_0)
 qed (auto simp add: sgn_mult mult_sgn_abs abs_eq_iff')
 
 end
 
 lemma coprime_int_iff [simp]:
   "coprime (int m) (int n) \<longleftrightarrow> coprime m n" (is "?P \<longleftrightarrow> ?Q")
 proof
   assume ?P
   show ?Q
   proof (rule coprimeI)
     fix q
     assume "q dvd m" "q dvd n"
     then have "int q dvd int m" "int q dvd int n"
       by simp_all
     with \<open>?P\<close> have "is_unit (int q)"
       by (rule coprime_common_divisor)
     then show "is_unit q"
       by simp
   qed
 next
   assume ?Q
   show ?P
   proof (rule coprimeI)
     fix k
     assume "k dvd int m" "k dvd int n"
     then have "nat \<bar>k\<bar> dvd m" "nat \<bar>k\<bar> dvd n"
       by simp_all
     with \<open>?Q\<close> have "is_unit (nat \<bar>k\<bar>)"
       by (rule coprime_common_divisor)
     then show "is_unit k"
       by simp
   qed
 qed
 
 lemma coprime_abs_left_iff [simp]:
   "coprime \<bar>k\<bar> l \<longleftrightarrow> coprime k l" for k l :: int
   using coprime_normalize_left_iff [of k l] by simp
 
 lemma coprime_abs_right_iff [simp]:
   "coprime k \<bar>l\<bar> \<longleftrightarrow> coprime k l" for k l :: int
   using coprime_abs_left_iff [of l k] by (simp add: ac_simps)
 
 lemma coprime_nat_abs_left_iff [simp]:
   "coprime (nat \<bar>k\<bar>) n \<longleftrightarrow> coprime k (int n)"
 proof -
   define m where "m = nat \<bar>k\<bar>"
   then have "\<bar>k\<bar> = int m"
     by simp
   moreover have "coprime k (int n) \<longleftrightarrow> coprime \<bar>k\<bar> (int n)"
     by simp
   ultimately show ?thesis
     by simp
 qed
 
 lemma coprime_nat_abs_right_iff [simp]:
   "coprime n (nat \<bar>k\<bar>) \<longleftrightarrow> coprime (int n) k"
   using coprime_nat_abs_left_iff [of k n] by (simp add: ac_simps)
 
 lemma coprime_common_divisor_int: "coprime a b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> \<bar>x\<bar> = 1"
   for a b :: int
   by (drule coprime_common_divisor [of _ _ x]) simp_all
 
 instantiation int :: idom_modulo
 begin
 
 definition modulo_int :: "int \<Rightarrow> int \<Rightarrow> int"
   where "k mod l = (if l = 0 then k
     else if sgn k = sgn l
       then sgn l * int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)
       else sgn l * (\<bar>l\<bar> * of_bool (\<not> l dvd k) - int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)))"
 
 lemma modulo_int_unfold:
   "(sgn k * int m) mod (sgn l * int n) =
    (if sgn l = 0 \<or> sgn k = 0 \<or> n = 0 then sgn k * int m
     else if sgn k = sgn l
       then sgn l * int (m mod n)
       else sgn l * (int (n * of_bool (\<not> n dvd m)) - int (m mod n)))"
   by (auto simp add: modulo_int_def sgn_0_0 sgn_1_pos sgn_mult abs_mult
     nat_mult_distrib)
 
 instance proof
   fix k l :: int
   obtain n m and s t where "k = sgn s * int n" and "l = sgn t * int m" 
     by (blast intro: int_sgnE elim: that)
   then show "k div l * l + k mod l = k"
     by (auto simp add: divide_int_unfold modulo_int_unfold algebra_simps dest!: sgn_not_eq_imp)
        (simp_all add: of_nat_mult [symmetric] of_nat_add [symmetric]
          distrib_left [symmetric] minus_mult_right
          del: of_nat_mult minus_mult_right [symmetric])
 qed
 
 end
 
 instantiation int :: unique_euclidean_ring
 begin
 
 definition euclidean_size_int :: "int \<Rightarrow> nat"
   where [simp]: "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
 
 definition division_segment_int :: "int \<Rightarrow> int"
   where "division_segment_int k = (if k \<ge> 0 then 1 else - 1)"
 
 lemma division_segment_eq_sgn:
   "division_segment k = sgn k" if "k \<noteq> 0" for k :: int
   using that by (simp add: division_segment_int_def)
 
 lemma abs_division_segment [simp]:
   "\<bar>division_segment k\<bar> = 1" for k :: int
   by (simp add: division_segment_int_def)
 
 lemma abs_mod_less:
   "\<bar>k mod l\<bar> < \<bar>l\<bar>" if "l \<noteq> 0" for k l :: int
 proof -
   obtain n m and s t where "k = sgn s * int n" and "l = sgn t * int m" 
     by (blast intro: int_sgnE elim: that)
   with that show ?thesis
     by (simp add: modulo_int_unfold sgn_0_0 sgn_1_pos sgn_1_neg
       abs_mult mod_greater_zero_iff_not_dvd)
 qed
 
 lemma sgn_mod:
   "sgn (k mod l) = sgn l" if "l \<noteq> 0" "\<not> l dvd k" for k l :: int
 proof -
   obtain n m and s t where "k = sgn s * int n" and "l = sgn t * int m" 
     by (blast intro: int_sgnE elim: that)
   with that show ?thesis
     by (simp add: modulo_int_unfold sgn_0_0 sgn_1_pos sgn_1_neg
       sgn_mult mod_eq_0_iff_dvd)
 qed
 
 instance proof
   fix k l :: int
   show "division_segment (k mod l) = division_segment l" if
     "l \<noteq> 0" and "\<not> l dvd k"
     using that by (simp add: division_segment_eq_sgn dvd_eq_mod_eq_0 sgn_mod)
 next
   fix l q r :: int
   obtain n m and s t
      where l: "l = sgn s * int n" and q: "q = sgn t * int m"
     by (blast intro: int_sgnE elim: that)
   assume \<open>l \<noteq> 0\<close>
   with l have "s \<noteq> 0" and "n > 0"
     by (simp_all add: sgn_0_0)
   assume "division_segment r = division_segment l"
   moreover have "r = sgn r * \<bar>r\<bar>"
     by (simp add: sgn_mult_abs)
   moreover define u where "u = nat \<bar>r\<bar>"
   ultimately have "r = sgn l * int u"
     using division_segment_eq_sgn \<open>l \<noteq> 0\<close> by (cases "r = 0") simp_all
   with l \<open>n > 0\<close> have r: "r = sgn s * int u"
     by (simp add: sgn_mult)
   assume "euclidean_size r < euclidean_size l"
   with l r \<open>s \<noteq> 0\<close> have "u < n"
     by (simp add: abs_mult)
   show "(q * l + r) div l = q"
   proof (cases "q = 0 \<or> r = 0")
     case True
     then show ?thesis
     proof
       assume "q = 0"
       then show ?thesis
         using l r \<open>u < n\<close> by (simp add: divide_int_unfold)
     next
       assume "r = 0"
       from \<open>r = 0\<close> have *: "q * l + r = sgn (t * s) * int (n * m)"
         using q l by (simp add: ac_simps sgn_mult)
       from \<open>s \<noteq> 0\<close> \<open>n > 0\<close> show ?thesis
         by (simp only: *, simp only: q l divide_int_unfold)
           (auto simp add: sgn_mult sgn_0_0 sgn_1_pos)
     qed
   next
     case False
     with q r have "t \<noteq> 0" and "m > 0" and "s \<noteq> 0" and "u > 0"
       by (simp_all add: sgn_0_0)
     moreover from \<open>0 < m\<close> \<open>u < n\<close> have "u \<le> m * n"
       using mult_le_less_imp_less [of 1 m u n] by simp
     ultimately have *: "q * l + r = sgn (s * t)
       * int (if t < 0 then m * n - u else m * n + u)"
       using l q r
       by (simp add: sgn_mult algebra_simps of_nat_diff)
     have "(m * n - u) div n = m - 1" if "u > 0"
       using \<open>0 < m\<close> \<open>u < n\<close> that
       by (auto intro: div_nat_eqI simp add: algebra_simps)
     moreover have "n dvd m * n - u \<longleftrightarrow> n dvd u"
       using \<open>u \<le> m * n\<close> dvd_diffD1 [of n "m * n" u]
       by auto
     ultimately show ?thesis
       using \<open>s \<noteq> 0\<close> \<open>m > 0\<close> \<open>u > 0\<close> \<open>u < n\<close> \<open>u \<le> m * n\<close>
       by (simp only: *, simp only: l q divide_int_unfold)
         (auto simp add: sgn_mult sgn_0_0 sgn_1_pos algebra_simps dest: dvd_imp_le)
   qed
 qed (use mult_le_mono2 [of 1] in \<open>auto simp add: division_segment_int_def not_le zero_less_mult_iff mult_less_0_iff abs_mult sgn_mult abs_mod_less sgn_mod nat_mult_distrib\<close>)
 
 end
 
 lemma pos_mod_bound [simp]:
   "k mod l < l" if "l > 0" for k l :: int
 proof -
   obtain m and s where "k = sgn s * int m"
     by (rule int_sgnE)
   moreover from that obtain n where "l = sgn 1 * int n"
     by (cases l) simp_all
   moreover from this that have "n > 0"
     by simp
   ultimately show ?thesis
     by (simp only: modulo_int_unfold)
       (simp add: mod_greater_zero_iff_not_dvd)
 qed
 
 lemma neg_mod_bound [simp]:
   "l < k mod l" if "l < 0" for k l :: int
 proof -
   obtain m and s where "k = sgn s * int m"
     by (rule int_sgnE)
   moreover from that obtain q where "l = sgn (- 1) * int (Suc q)"
     by (cases l) simp_all
   moreover define n where "n = Suc q"
   then have "Suc q = n"
     by simp
   ultimately show ?thesis
     by (simp only: modulo_int_unfold)
       (simp add: mod_greater_zero_iff_not_dvd)
 qed
 
 lemma pos_mod_sign [simp]:
   "0 \<le> k mod l" if "l > 0" for k l :: int
 proof -
   obtain m and s where "k = sgn s * int m"
     by (rule int_sgnE)
   moreover from that obtain n where "l = sgn 1 * int n"
     by (cases l) auto
   moreover from this that have "n > 0"
     by simp
   ultimately show ?thesis
     by (simp only: modulo_int_unfold) simp
 qed
 
 lemma neg_mod_sign [simp]:
   "k mod l \<le> 0" if "l < 0" for k l :: int
 proof -
   obtain m and s where "k = sgn s * int m"
     by (rule int_sgnE)
   moreover from that obtain q where "l = sgn (- 1) * int (Suc q)"
     by (cases l) simp_all
   moreover define n where "n = Suc q"
   then have "Suc q = n"
     by simp
   ultimately show ?thesis
     by (simp only: modulo_int_unfold) simp
 qed
 
+lemma div_pos_pos_trivial [simp]:
+  "k div l = 0" if "k \<ge> 0" and "k < l" for k l :: int
+  using that by (simp add: unique_euclidean_semiring_class.div_eq_0_iff division_segment_int_def)
+
+lemma mod_pos_pos_trivial [simp]:
+  "k mod l = k" if "k \<ge> 0" and "k < l" for k l :: int
+  using that by (simp add: mod_eq_self_iff_div_eq_0)
+
+lemma div_neg_neg_trivial [simp]:
+  "k div l = 0" if "k \<le> 0" and "l < k" for k l :: int
+  using that by (cases "k = 0") (simp, simp add: unique_euclidean_semiring_class.div_eq_0_iff division_segment_int_def)
+
+lemma mod_neg_neg_trivial [simp]:
+  "k mod l = k" if "k \<le> 0" and "l < k" for k l :: int
+  using that by (simp add: mod_eq_self_iff_div_eq_0)
+
+lemma div_pos_neg_trivial:
+  "k div l = - 1" if "0 < k" and "k + l \<le> 0" for k l :: int
+proof (cases \<open>l = - k\<close>)
+  case True
+  with that show ?thesis
+    by (simp add: divide_int_def)
+next
+  case False
+  show ?thesis
+    apply (rule div_eqI [of _ "k + l"])
+    using False that apply (simp_all add: division_segment_int_def)
+    done
+qed
+
+lemma mod_pos_neg_trivial:
+  "k mod l = k + l" if "0 < k" and "k + l \<le> 0" for k l :: int
+proof (cases \<open>l = - k\<close>)
+  case True
+  with that show ?thesis
+    by (simp add: divide_int_def)
+next
+  case False
+  show ?thesis
+    apply (rule mod_eqI [of _ _ \<open>- 1\<close>])
+    using False that apply (simp_all add: division_segment_int_def)
+    done
+qed
+
+text \<open>There is neither \<open>div_neg_pos_trivial\<close> nor \<open>mod_neg_pos_trivial\<close>
+  because \<^term>\<open>0 div l = 0\<close> would supersede it.\<close>
+
+text \<open>Distributive laws for function \<open>nat\<close>.\<close>
+
+lemma nat_div_distrib:
+  \<open>nat (x div y) = nat x div nat y\<close> if \<open>0 \<le> x\<close>
+  using that by (simp add: divide_int_def sgn_if)
+
+lemma nat_div_distrib':
+  \<open>nat (x div y) = nat x div nat y\<close> if \<open>0 \<le> y\<close>
+  using that by (simp add: divide_int_def sgn_if)
+
+lemma nat_mod_distrib: \<comment> \<open>Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't\<close>
+  \<open>nat (x mod y) = nat x mod nat y\<close> if \<open>0 \<le> x\<close> \<open>0 \<le> y\<close>
+  using that by (simp add: modulo_int_def sgn_if)
+
 
 subsection \<open>Special case: euclidean rings containing the natural numbers\<close>
 
 class unique_euclidean_semiring_with_nat = semidom + semiring_char_0 + unique_euclidean_semiring +
   assumes of_nat_div: "of_nat (m div n) = of_nat m div of_nat n"
     and division_segment_of_nat [simp]: "division_segment (of_nat n) = 1"
     and division_segment_euclidean_size [simp]: "division_segment a * of_nat (euclidean_size a) = a"
 begin
 
 lemma division_segment_eq_iff:
   "a = b" if "division_segment a = division_segment b"
     and "euclidean_size a = euclidean_size b"
   using that division_segment_euclidean_size [of a] by simp
 
 lemma euclidean_size_of_nat [simp]:
   "euclidean_size (of_nat n) = n"
 proof -
   have "division_segment (of_nat n) * of_nat (euclidean_size (of_nat n)) = of_nat n"
     by (fact division_segment_euclidean_size)
   then show ?thesis by simp
 qed
 
 lemma of_nat_euclidean_size:
   "of_nat (euclidean_size a) = a div division_segment a"
 proof -
   have "of_nat (euclidean_size a) = division_segment a * of_nat (euclidean_size a) div division_segment a"
     by (subst nonzero_mult_div_cancel_left) simp_all
   also have "\<dots> = a div division_segment a"
     by simp
   finally show ?thesis .
 qed
 
 lemma division_segment_1 [simp]:
   "division_segment 1 = 1"
   using division_segment_of_nat [of 1] by simp
 
 lemma division_segment_numeral [simp]:
   "division_segment (numeral k) = 1"
   using division_segment_of_nat [of "numeral k"] by simp
 
 lemma euclidean_size_1 [simp]:
   "euclidean_size 1 = 1"
   using euclidean_size_of_nat [of 1] by simp
 
 lemma euclidean_size_numeral [simp]:
   "euclidean_size (numeral k) = numeral k"
   using euclidean_size_of_nat [of "numeral k"] by simp
 
 lemma of_nat_dvd_iff:
   "of_nat m dvd of_nat n \<longleftrightarrow> m dvd n" (is "?P \<longleftrightarrow> ?Q")
 proof (cases "m = 0")
   case True
   then show ?thesis
     by simp
 next
   case False
   show ?thesis
   proof
     assume ?Q
     then show ?P
       by auto
   next
     assume ?P
     with False have "of_nat n = of_nat n div of_nat m * of_nat m"
       by simp
     then have "of_nat n = of_nat (n div m * m)"
       by (simp add: of_nat_div)
     then have "n = n div m * m"
       by (simp only: of_nat_eq_iff)
     then have "n = m * (n div m)"
       by (simp add: ac_simps)
     then show ?Q ..
   qed
 qed
 
 lemma of_nat_mod:
   "of_nat (m mod n) = of_nat m mod of_nat n"
 proof -
   have "of_nat m div of_nat n * of_nat n + of_nat m mod of_nat n = of_nat m"
     by (simp add: div_mult_mod_eq)
   also have "of_nat m = of_nat (m div n * n + m mod n)"
     by simp
   finally show ?thesis
     by (simp only: of_nat_div of_nat_mult of_nat_add) simp
 qed
 
 lemma one_div_two_eq_zero [simp]:
   "1 div 2 = 0"
 proof -
   from of_nat_div [symmetric] have "of_nat 1 div of_nat 2 = of_nat 0"
     by (simp only:) simp
   then show ?thesis
     by simp
 qed
 
 lemma one_mod_two_eq_one [simp]:
   "1 mod 2 = 1"
 proof -
   from of_nat_mod [symmetric] have "of_nat 1 mod of_nat 2 = of_nat 1"
     by (simp only:) simp
   then show ?thesis
     by simp
 qed
 
 lemma one_mod_2_pow_eq [simp]:
   "1 mod (2 ^ n) = of_bool (n > 0)"
 proof -
   have "1 mod (2 ^ n) = of_nat (1 mod (2 ^ n))"
     using of_nat_mod [of 1 "2 ^ n"] by simp
   also have "\<dots> = of_bool (n > 0)"
     by simp
   finally show ?thesis .
 qed
 
 lemma one_div_2_pow_eq [simp]:
   "1 div (2 ^ n) = of_bool (n = 0)"
   using div_mult_mod_eq [of 1 "2 ^ n"] by auto
 
 lemma div_mult2_eq':
   "a div (of_nat m * of_nat n) = a div of_nat m div of_nat n"
 proof (cases a "of_nat m * of_nat n" rule: divmod_cases)
   case (divides q)
   then show ?thesis
     using nonzero_mult_div_cancel_right [of "of_nat m" "q * of_nat n"]
     by (simp add: ac_simps)
 next
   case (remainder q r)
   then have "division_segment r = 1"
     using division_segment_of_nat [of "m * n"] by simp
   with division_segment_euclidean_size [of r]
   have "of_nat (euclidean_size r) = r"
     by simp
   have "a mod (of_nat m * of_nat n) div (of_nat m * of_nat n) = 0"
     by simp
   with remainder(6) have "r div (of_nat m * of_nat n) = 0"
     by simp
   with \<open>of_nat (euclidean_size r) = r\<close>
   have "of_nat (euclidean_size r) div (of_nat m * of_nat n) = 0"
     by simp
   then have "of_nat (euclidean_size r div (m * n)) = 0"
     by (simp add: of_nat_div)
   then have "of_nat (euclidean_size r div m div n) = 0"
     by (simp add: div_mult2_eq)
   with \<open>of_nat (euclidean_size r) = r\<close> have "r div of_nat m div of_nat n = 0"
     by (simp add: of_nat_div)
   with remainder(1)
   have "q = (r div of_nat m + q * of_nat n * of_nat m div of_nat m) div of_nat n"
     by simp
   with remainder(5) remainder(7) show ?thesis
     using div_plus_div_distrib_dvd_right [of "of_nat m" "q * (of_nat m * of_nat n)" r]
     by (simp add: ac_simps)
 next
   case by0
   then show ?thesis
     by auto
 qed
 
 lemma mod_mult2_eq':
   "a mod (of_nat m * of_nat n) = of_nat m * (a div of_nat m mod of_nat n) + a mod of_nat m"
 proof -
   have "a div (of_nat m * of_nat n) * (of_nat m * of_nat n) + a mod (of_nat m * of_nat n) = a div of_nat m div of_nat n * of_nat n * of_nat m + (a div of_nat m mod of_nat n * of_nat m + a mod of_nat m)"
     by (simp add: combine_common_factor div_mult_mod_eq)
   moreover have "a div of_nat m div of_nat n * of_nat n * of_nat m = of_nat n * of_nat m * (a div of_nat m div of_nat n)"
     by (simp add: ac_simps)
   ultimately show ?thesis
     by (simp add: div_mult2_eq' mult_commute)
 qed
 
 lemma div_mult2_numeral_eq:
   "a div numeral k div numeral l = a div numeral (k * l)" (is "?A = ?B")
 proof -
   have "?A = a div of_nat (numeral k) div of_nat (numeral l)"
     by simp
   also have "\<dots> = a div (of_nat (numeral k) * of_nat (numeral l))"
     by (fact div_mult2_eq' [symmetric])
   also have "\<dots> = ?B"
     by simp
   finally show ?thesis .
 qed
 
 lemma numeral_Bit0_div_2:
   "numeral (num.Bit0 n) div 2 = numeral n"
 proof -
   have "numeral (num.Bit0 n) = numeral n + numeral n"
     by (simp only: numeral.simps)
   also have "\<dots> = numeral n * 2"
     by (simp add: mult_2_right)
   finally have "numeral (num.Bit0 n) div 2 = numeral n * 2 div 2"
     by simp
   also have "\<dots> = numeral n"
     by (rule nonzero_mult_div_cancel_right) simp
   finally show ?thesis .
 qed
 
 lemma numeral_Bit1_div_2:
   "numeral (num.Bit1 n) div 2 = numeral n"
 proof -
   have "numeral (num.Bit1 n) = numeral n + numeral n + 1"
     by (simp only: numeral.simps)
   also have "\<dots> = numeral n * 2 + 1"
     by (simp add: mult_2_right)
   finally have "numeral (num.Bit1 n) div 2 = (numeral n * 2 + 1) div 2"
     by simp
   also have "\<dots> = numeral n * 2 div 2 + 1 div 2"
     using dvd_triv_right by (rule div_plus_div_distrib_dvd_left)
   also have "\<dots> = numeral n * 2 div 2"
     by simp
   also have "\<dots> = numeral n"
     by (rule nonzero_mult_div_cancel_right) simp
   finally show ?thesis .
 qed
 
 lemma exp_mod_exp:
   \<open>2 ^ m mod 2 ^ n = of_bool (m < n) * 2 ^ m\<close>
 proof -
   have \<open>(2::nat) ^ m mod 2 ^ n = of_bool (m < n) * 2 ^ m\<close> (is \<open>?lhs = ?rhs\<close>)
     by (auto simp add: not_less monoid_mult_class.power_add dest!: le_Suc_ex)
   then have \<open>of_nat ?lhs = of_nat ?rhs\<close>
     by simp
   then show ?thesis
     by (simp add: of_nat_mod)
 qed
 
 lemma mask_mod_exp:
   \<open>(2 ^ n - 1) mod 2 ^ m = 2 ^ min m n - 1\<close>
 proof -
   have \<open>(2 ^ n - 1) mod 2 ^ m = 2 ^ min m n - (1::nat)\<close> (is \<open>?lhs = ?rhs\<close>)
   proof (cases \<open>n \<le> m\<close>)
     case True
     then show ?thesis
       by (simp add: Suc_le_lessD min.absorb2)
   next
     case False
     then have \<open>m < n\<close>
       by simp
     then obtain q where n: \<open>n = Suc q + m\<close>
       by (auto dest: less_imp_Suc_add)
     then have \<open>min m n = m\<close>
       by simp
     moreover have \<open>(2::nat) ^ m \<le> 2 * 2 ^ q * 2 ^ m\<close>
       using mult_le_mono1 [of 1 \<open>2 * 2 ^ q\<close> \<open>2 ^ m\<close>] by simp
     with n have \<open>2 ^ n - 1 = (2 ^ Suc q - 1) * 2 ^ m + (2 ^ m - (1::nat))\<close>
       by (simp add: monoid_mult_class.power_add algebra_simps)
     ultimately show ?thesis
       by (simp only: euclidean_semiring_cancel_class.mod_mult_self3) simp
   qed
   then have \<open>of_nat ?lhs = of_nat ?rhs\<close>
     by simp
   then show ?thesis
     by (simp add: of_nat_mod of_nat_diff)
 qed
 
 lemma of_bool_half_eq_0 [simp]:
   \<open>of_bool b div 2 = 0\<close>
   by simp
 
 end
 
 class unique_euclidean_ring_with_nat = ring + unique_euclidean_semiring_with_nat
 
 instance nat :: unique_euclidean_semiring_with_nat
   by standard (simp_all add: dvd_eq_mod_eq_0)
 
 instance int :: unique_euclidean_ring_with_nat
   by standard (simp_all add: dvd_eq_mod_eq_0 divide_int_def division_segment_int_def)
 
 
 subsection \<open>Code generation\<close>
 
 code_identifier
   code_module Euclidean_Division \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
 
 end
diff --git a/src/HOL/Groups_List.thy b/src/HOL/Groups_List.thy
--- a/src/HOL/Groups_List.thy
+++ b/src/HOL/Groups_List.thy
@@ -1,561 +1,572 @@
 (* Author: Tobias Nipkow, TU Muenchen *)
 
 section \<open>Sum and product over lists\<close>
 
 theory Groups_List
 imports List
 begin
 
 locale monoid_list = monoid
 begin
 
 definition F :: "'a list \<Rightarrow> 'a"
 where
   eq_foldr [code]: "F xs = foldr f xs \<^bold>1"
 
 lemma Nil [simp]:
   "F [] = \<^bold>1"
   by (simp add: eq_foldr)
 
 lemma Cons [simp]:
   "F (x # xs) = x \<^bold>* F xs"
   by (simp add: eq_foldr)
 
 lemma append [simp]:
   "F (xs @ ys) = F xs \<^bold>* F ys"
   by (induct xs) (simp_all add: assoc)
 
 end
 
 locale comm_monoid_list = comm_monoid + monoid_list
 begin
 
 lemma rev [simp]:
   "F (rev xs) = F xs"
   by (simp add: eq_foldr foldr_fold  fold_rev fun_eq_iff assoc left_commute)
 
 end
 
 locale comm_monoid_list_set = list: comm_monoid_list + set: comm_monoid_set
 begin
 
 lemma distinct_set_conv_list:
   "distinct xs \<Longrightarrow> set.F g (set xs) = list.F (map g xs)"
   by (induct xs) simp_all
 
 lemma set_conv_list [code]:
   "set.F g (set xs) = list.F (map g (remdups xs))"
   by (simp add: distinct_set_conv_list [symmetric])
 
 end
 
 
 subsection \<open>List summation\<close>
 
 context monoid_add
 begin
 
 sublocale sum_list: monoid_list plus 0
 defines
   sum_list = sum_list.F ..
 
 end
 
 context comm_monoid_add
 begin
 
 sublocale sum_list: comm_monoid_list plus 0
 rewrites
   "monoid_list.F plus 0 = sum_list"
 proof -
   show "comm_monoid_list plus 0" ..
   then interpret sum_list: comm_monoid_list plus 0 .
   from sum_list_def show "monoid_list.F plus 0 = sum_list" by simp
 qed
 
 sublocale sum: comm_monoid_list_set plus 0
 rewrites
   "monoid_list.F plus 0 = sum_list"
   and "comm_monoid_set.F plus 0 = sum"
 proof -
   show "comm_monoid_list_set plus 0" ..
   then interpret sum: comm_monoid_list_set plus 0 .
   from sum_list_def show "monoid_list.F plus 0 = sum_list" by simp
   from sum_def show "comm_monoid_set.F plus 0 = sum" by (auto intro: sym)
 qed
 
 end
 
 text \<open>Some syntactic sugar for summing a function over a list:\<close>
 syntax (ASCII)
   "_sum_list" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
 syntax
   "_sum_list" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
 translations \<comment> \<open>Beware of argument permutation!\<close>
   "\<Sum>x\<leftarrow>xs. b" == "CONST sum_list (CONST map (\<lambda>x. b) xs)"
 
 context
   includes lifting_syntax
 begin
 
 lemma sum_list_transfer [transfer_rule]:
   "(list_all2 A ===> A) sum_list sum_list"
     if [transfer_rule]: "A 0 0" "(A ===> A ===> A) (+) (+)"
   unfolding sum_list.eq_foldr [abs_def]
   by transfer_prover
 
 end
 
 text \<open>TODO duplicates\<close>
 lemmas sum_list_simps = sum_list.Nil sum_list.Cons
 lemmas sum_list_append = sum_list.append
 lemmas sum_list_rev = sum_list.rev
 
 lemma (in monoid_add) fold_plus_sum_list_rev:
   "fold plus xs = plus (sum_list (rev xs))"
 proof
   fix x
   have "fold plus xs x = sum_list (rev xs @ [x])"
     by (simp add: foldr_conv_fold sum_list.eq_foldr)
   also have "\<dots> = sum_list (rev xs) + x"
     by simp
   finally show "fold plus xs x = sum_list (rev xs) + x"
     .
 qed
 
 lemma (in comm_monoid_add) sum_list_map_remove1:
   "x \<in> set xs \<Longrightarrow> sum_list (map f xs) = f x + sum_list (map f (remove1 x xs))"
   by (induct xs) (auto simp add: ac_simps)
 
 lemma (in monoid_add) size_list_conv_sum_list:
   "size_list f xs = sum_list (map f xs) + size xs"
   by (induct xs) auto
 
 lemma (in monoid_add) length_concat:
   "length (concat xss) = sum_list (map length xss)"
   by (induct xss) simp_all
 
 lemma (in monoid_add) length_product_lists:
   "length (product_lists xss) = foldr (*) (map length xss) 1"
 proof (induct xss)
   case (Cons xs xss) then show ?case by (induct xs) (auto simp: length_concat o_def)
 qed simp
 
 lemma (in monoid_add) sum_list_map_filter:
   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
   shows "sum_list (map f (filter P xs)) = sum_list (map f xs)"
   using assms by (induct xs) auto
 
 lemma sum_list_filter_le_nat:
   fixes f :: "'a \<Rightarrow> nat"
   shows "sum_list (map f (filter P xs)) \<le> sum_list (map f xs)"
 by(induction xs; simp)
 
 lemma (in comm_monoid_add) distinct_sum_list_conv_Sum:
   "distinct xs \<Longrightarrow> sum_list xs = Sum (set xs)"
   by (induct xs) simp_all
 
 lemma sum_list_upt[simp]:
   "m \<le> n \<Longrightarrow> sum_list [m..<n] = \<Sum> {m..<n}"
 by(simp add: distinct_sum_list_conv_Sum)
 
 context ordered_comm_monoid_add
 begin
 
 lemma sum_list_nonneg: "(\<And>x. x \<in> set xs \<Longrightarrow> 0 \<le> x) \<Longrightarrow> 0 \<le> sum_list xs"
 by (induction xs) auto
 
 lemma sum_list_nonpos: "(\<And>x. x \<in> set xs \<Longrightarrow> x \<le> 0) \<Longrightarrow> sum_list xs \<le> 0"
 by (induction xs) (auto simp: add_nonpos_nonpos)
 
 lemma sum_list_nonneg_eq_0_iff:
   "(\<And>x. x \<in> set xs \<Longrightarrow> 0 \<le> x) \<Longrightarrow> sum_list xs = 0 \<longleftrightarrow> (\<forall>x\<in> set xs. x = 0)"
 by (induction xs) (simp_all add: add_nonneg_eq_0_iff sum_list_nonneg)
 
 end
 
 context canonically_ordered_monoid_add
 begin
 
 lemma sum_list_eq_0_iff [simp]:
   "sum_list ns = 0 \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
 by (simp add: sum_list_nonneg_eq_0_iff)
 
 lemma member_le_sum_list:
   "x \<in> set xs \<Longrightarrow> x \<le> sum_list xs"
 by (induction xs) (auto simp: add_increasing add_increasing2)
 
 lemma elem_le_sum_list:
   "k < size ns \<Longrightarrow> ns ! k \<le> sum_list (ns)"
 by (rule member_le_sum_list) simp
 
 end
 
 lemma (in ordered_cancel_comm_monoid_diff) sum_list_update:
   "k < size xs \<Longrightarrow> sum_list (xs[k := x]) = sum_list xs + x - xs ! k"
 apply(induction xs arbitrary:k)
  apply (auto simp: add_ac split: nat.split)
 apply(drule elem_le_sum_list)
 by (simp add: local.add_diff_assoc local.add_increasing)
 
 lemma (in monoid_add) sum_list_triv:
   "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
   by (induct xs) (simp_all add: distrib_right)
 
 lemma (in monoid_add) sum_list_0 [simp]:
   "(\<Sum>x\<leftarrow>xs. 0) = 0"
   by (induct xs) (simp_all add: distrib_right)
 
 text\<open>For non-Abelian groups \<open>xs\<close> needs to be reversed on one side:\<close>
 lemma (in ab_group_add) uminus_sum_list_map:
   "- sum_list (map f xs) = sum_list (map (uminus \<circ> f) xs)"
   by (induct xs) simp_all
 
 lemma (in comm_monoid_add) sum_list_addf:
   "(\<Sum>x\<leftarrow>xs. f x + g x) = sum_list (map f xs) + sum_list (map g xs)"
   by (induct xs) (simp_all add: algebra_simps)
 
 lemma (in ab_group_add) sum_list_subtractf:
   "(\<Sum>x\<leftarrow>xs. f x - g x) = sum_list (map f xs) - sum_list (map g xs)"
   by (induct xs) (simp_all add: algebra_simps)
 
 lemma (in semiring_0) sum_list_const_mult:
   "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
   by (induct xs) (simp_all add: algebra_simps)
 
 lemma (in semiring_0) sum_list_mult_const:
   "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
   by (induct xs) (simp_all add: algebra_simps)
 
 lemma (in ordered_ab_group_add_abs) sum_list_abs:
   "\<bar>sum_list xs\<bar> \<le> sum_list (map abs xs)"
   by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
 
 lemma sum_list_mono:
   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
   shows "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sum>x\<leftarrow>xs. f x) \<le> (\<Sum>x\<leftarrow>xs. g x)"
 by (induct xs) (simp, simp add: add_mono)
 
 lemma sum_list_strict_mono:
   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, strict_ordered_ab_semigroup_add}"
   shows "\<lbrakk> xs \<noteq> [];  \<And>x. x \<in> set xs \<Longrightarrow> f x < g x \<rbrakk>
     \<Longrightarrow> sum_list (map f xs) < sum_list (map g xs)"
 proof (induction xs)
   case Nil thus ?case by simp
 next
   case C: (Cons _ xs)
   show ?case
   proof (cases xs)
     case Nil thus ?thesis using C.prems by simp
   next
     case Cons thus ?thesis using C by(simp add: add_strict_mono)
   qed
 qed
 
 lemma (in monoid_add) sum_list_distinct_conv_sum_set:
   "distinct xs \<Longrightarrow> sum_list (map f xs) = sum f (set xs)"
   by (induct xs) simp_all
 
 lemma (in monoid_add) interv_sum_list_conv_sum_set_nat:
   "sum_list (map f [m..<n]) = sum f (set [m..<n])"
   by (simp add: sum_list_distinct_conv_sum_set)
 
 lemma (in monoid_add) interv_sum_list_conv_sum_set_int:
   "sum_list (map f [k..l]) = sum f (set [k..l])"
   by (simp add: sum_list_distinct_conv_sum_set)
 
 text \<open>General equivalence between \<^const>\<open>sum_list\<close> and \<^const>\<open>sum\<close>\<close>
 lemma (in monoid_add) sum_list_sum_nth:
   "sum_list xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
   using interv_sum_list_conv_sum_set_nat [of "(!) xs" 0 "length xs"] by (simp add: map_nth)
 
 lemma sum_list_map_eq_sum_count:
   "sum_list (map f xs) = sum (\<lambda>x. count_list xs x * f x) (set xs)"
 proof(induction xs)
   case (Cons x xs)
   show ?case (is "?l = ?r")
   proof cases
     assume "x \<in> set xs"
     have "?l = f x + (\<Sum>x\<in>set xs. count_list xs x * f x)" by (simp add: Cons.IH)
     also have "set xs = insert x (set xs - {x})" using \<open>x \<in> set xs\<close>by blast
     also have "f x + (\<Sum>x\<in>insert x (set xs - {x}). count_list xs x * f x) = ?r"
       by (simp add: sum.insert_remove eq_commute)
     finally show ?thesis .
   next
     assume "x \<notin> set xs"
     hence "\<And>xa. xa \<in> set xs \<Longrightarrow> x \<noteq> xa" by blast
     thus ?thesis by (simp add: Cons.IH \<open>x \<notin> set xs\<close>)
   qed
 qed simp
 
 lemma sum_list_map_eq_sum_count2:
 assumes "set xs \<subseteq> X" "finite X"
 shows "sum_list (map f xs) = sum (\<lambda>x. count_list xs x * f x) X"
 proof-
   let ?F = "\<lambda>x. count_list xs x * f x"
   have "sum ?F X = sum ?F (set xs \<union> (X - set xs))"
     using Un_absorb1[OF assms(1)] by(simp)
   also have "\<dots> = sum ?F (set xs)"
     using assms(2)
     by(simp add: sum.union_disjoint[OF _ _ Diff_disjoint] del: Un_Diff_cancel)
   finally show ?thesis by(simp add:sum_list_map_eq_sum_count)
 qed
 
 lemma sum_list_nonneg:
     "(\<And>x. x \<in> set xs \<Longrightarrow> (x :: 'a :: ordered_comm_monoid_add) \<ge> 0) \<Longrightarrow> sum_list xs \<ge> 0"
   by (induction xs) simp_all
 
 lemma sum_list_Suc:
   "sum_list (map (\<lambda>x. Suc(f x)) xs) = sum_list (map f xs) + length xs"
 by(induction xs; simp)
 
 lemma (in monoid_add) sum_list_map_filter':
   "sum_list (map f (filter P xs)) = sum_list (map (\<lambda>x. if P x then f x else 0) xs)"
   by (induction xs) simp_all
 
 text \<open>Summation of a strictly ascending sequence with length \<open>n\<close>
   can be upper-bounded by summation over \<open>{0..<n}\<close>.\<close>
 
 lemma sorted_wrt_less_sum_mono_lowerbound:
   fixes f :: "nat \<Rightarrow> ('b::ordered_comm_monoid_add)"
   assumes mono: "\<And>x y. x\<le>y \<Longrightarrow> f x \<le> f y"
   shows "sorted_wrt (<) ns \<Longrightarrow>
     (\<Sum>i\<in>{0..<length ns}. f i) \<le> (\<Sum>i\<leftarrow>ns. f i)"
 proof (induction ns rule: rev_induct)
   case Nil
   then show ?case by simp
 next
   case (snoc n ns)
   have "sum f {0..<length (ns @ [n])}
       = sum f {0..<length ns} + f (length ns)"
     by simp
   also have "sum f {0..<length ns} \<le> sum_list (map f ns)"
     using snoc by (auto simp: sorted_wrt_append)
   also have "length ns \<le> n"
     using sorted_wrt_less_idx[OF snoc.prems(1), of "length ns"] by auto
   finally have "sum f {0..<length (ns @ [n])} \<le> sum_list (map f ns) + f n"
     using mono add_mono by blast
   thus ?case by simp
 qed
 
 
 subsection \<open>Horner sums\<close>
 
 context comm_semiring_0
 begin
 
 definition horner_sum :: \<open>('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'a\<close>
   where horner_sum_foldr: \<open>horner_sum f a xs = foldr (\<lambda>x b. f x + a * b) xs 0\<close>
 
 lemma horner_sum_simps [simp]:
   \<open>horner_sum f a [] = 0\<close>
   \<open>horner_sum f a (x # xs) = f x + a * horner_sum f a xs\<close>
   by (simp_all add: horner_sum_foldr)
 
 lemma horner_sum_eq_sum_funpow:
   \<open>horner_sum f a xs = (\<Sum>n = 0..<length xs. ((*) a ^^ n) (f (xs ! n)))\<close>
 proof (induction xs)
   case Nil
   then show ?case
     by simp
 next
   case (Cons x xs)
   then show ?case
     by (simp add: sum.atLeast0_lessThan_Suc_shift sum_distrib_left del: sum.op_ivl_Suc)
 qed
 
 end
 
 context
   includes lifting_syntax
 begin
 
 lemma horner_sum_transfer [transfer_rule]:
   \<open>((B ===> A) ===> A ===> list_all2 B ===> A) horner_sum horner_sum\<close>
   if [transfer_rule]: \<open>A 0 0\<close>
     and [transfer_rule]: \<open>(A ===> A ===> A) (+) (+)\<close>
     and [transfer_rule]: \<open>(A ===> A ===> A) (*) (*)\<close>
   by (unfold horner_sum_foldr) transfer_prover
 
 end
 
 context comm_semiring_1
 begin
 
 lemma horner_sum_eq_sum:
   \<open>horner_sum f a xs = (\<Sum>n = 0..<length xs. f (xs ! n) * a ^ n)\<close>
 proof -
   have \<open>(*) a ^^ n = (*) (a ^ n)\<close> for n
     by (induction n) (simp_all add: ac_simps)
   then show ?thesis
     by (simp add: horner_sum_eq_sum_funpow ac_simps)
 qed
 
 end
 
 context semiring_bit_shifts
 begin
 
 lemma horner_sum_bit_eq_take_bit:
   \<open>horner_sum of_bool 2 (map (bit a) [0..<n]) = take_bit n a\<close>
 proof (induction a arbitrary: n rule: bits_induct)
   case (stable a)
   moreover have \<open>bit a = (\<lambda>_. odd a)\<close>
     using stable by (simp add: stable_imp_bit_iff_odd fun_eq_iff)
   moreover have \<open>{q. q < n} = {0..<n}\<close>
     by auto
   ultimately show ?case
     by (simp add: stable_imp_take_bit_eq horner_sum_eq_sum mask_eq_sum_exp)
 next
   case (rec a b)
   show ?case
   proof (cases n)
     case 0
     then show ?thesis
       by simp
   next
     case (Suc m)
     have \<open>map (bit (of_bool b + 2 * a)) [0..<Suc m] = b # map (bit (of_bool b + 2 * a)) [Suc 0..<Suc m]\<close>
       by (simp only: upt_conv_Cons) simp
     also have \<open>\<dots> = b # map (bit a) [0..<m]\<close>
       by (simp only: flip: map_Suc_upt) (simp add: bit_Suc rec.hyps)
     finally show ?thesis
       using Suc rec.IH [of m] by (simp add: take_bit_Suc rec.hyps, simp add: ac_simps mod_2_eq_odd)
   qed
 qed
 
 end
 
 context unique_euclidean_semiring_with_bit_shifts
 begin
 
 lemma bit_horner_sum_bit_iff:
   \<open>bit (horner_sum of_bool 2 bs) n \<longleftrightarrow> n < length bs \<and> bs ! n\<close>
 proof (induction bs arbitrary: n)
   case Nil
   then show ?case
     by simp
 next
   case (Cons b bs)
   show ?case
   proof (cases n)
     case 0
     then show ?thesis
       by simp
   next
     case (Suc m)
     with bit_rec [of _ n] Cons.prems Cons.IH [of m]
     show ?thesis by simp
   qed
 qed
 
 lemma take_bit_horner_sum_bit_eq:
   \<open>take_bit n (horner_sum of_bool 2 bs) = horner_sum of_bool 2 (take n bs)\<close>
   by (auto simp add: bit_eq_iff bit_take_bit_iff bit_horner_sum_bit_iff)
 
 end
 
+lemma horner_sum_of_bool_2_less:
+  \<open>(horner_sum of_bool 2 bs :: int) < 2 ^ length bs\<close>
+proof -
+  have \<open>(\<Sum>n = 0..<length bs. of_bool (bs ! n) * (2::int) ^ n) \<le> (\<Sum>n = 0..<length bs. 2 ^ n)\<close>
+    by (rule sum_mono) simp
+  also have \<open>\<dots> = 2 ^ length bs - 1\<close>
+    by (induction bs) simp_all
+  finally show ?thesis
+    by (simp add: horner_sum_eq_sum)
+qed
+
 
 subsection \<open>Further facts about \<^const>\<open>List.n_lists\<close>\<close>
 
 lemma length_n_lists: "length (List.n_lists n xs) = length xs ^ n"
   by (induct n) (auto simp add: comp_def length_concat sum_list_triv)
 
 lemma distinct_n_lists:
   assumes "distinct xs"
   shows "distinct (List.n_lists n xs)"
 proof (rule card_distinct)
   from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
   have "card (set (List.n_lists n xs)) = card (set xs) ^ n"
   proof (induct n)
     case 0 then show ?case by simp
   next
     case (Suc n)
     moreover have "card (\<Union>ys\<in>set (List.n_lists n xs). (\<lambda>y. y # ys) ` set xs)
       = (\<Sum>ys\<in>set (List.n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
       by (rule card_UN_disjoint) auto
     moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
       by (rule card_image) (simp add: inj_on_def)
     ultimately show ?case by auto
   qed
   also have "\<dots> = length xs ^ n" by (simp add: card_length)
   finally show "card (set (List.n_lists n xs)) = length (List.n_lists n xs)"
     by (simp add: length_n_lists)
 qed
 
 
 subsection \<open>Tools setup\<close>
 
 lemmas sum_code = sum.set_conv_list
 
 lemma sum_set_upto_conv_sum_list_int [code_unfold]:
   "sum f (set [i..j::int]) = sum_list (map f [i..j])"
   by (simp add: interv_sum_list_conv_sum_set_int)
 
 lemma sum_set_upt_conv_sum_list_nat [code_unfold]:
   "sum f (set [m..<n]) = sum_list (map f [m..<n])"
   by (simp add: interv_sum_list_conv_sum_set_nat)
 
 
 subsection \<open>List product\<close>
 
 context monoid_mult
 begin
 
 sublocale prod_list: monoid_list times 1
 defines
   prod_list = prod_list.F ..
 
 end
 
 context comm_monoid_mult
 begin
 
 sublocale prod_list: comm_monoid_list times 1
 rewrites
   "monoid_list.F times 1 = prod_list"
 proof -
   show "comm_monoid_list times 1" ..
   then interpret prod_list: comm_monoid_list times 1 .
   from prod_list_def show "monoid_list.F times 1 = prod_list" by simp
 qed
 
 sublocale prod: comm_monoid_list_set times 1
 rewrites
   "monoid_list.F times 1 = prod_list"
   and "comm_monoid_set.F times 1 = prod"
 proof -
   show "comm_monoid_list_set times 1" ..
   then interpret prod: comm_monoid_list_set times 1 .
   from prod_list_def show "monoid_list.F times 1 = prod_list" by simp
   from prod_def show "comm_monoid_set.F times 1 = prod" by (auto intro: sym)
 qed
 
 end
 
 text \<open>Some syntactic sugar:\<close>
 
 syntax (ASCII)
   "_prod_list" :: "pttrn => 'a list => 'b => 'b"    ("(3PROD _<-_. _)" [0, 51, 10] 10)
 syntax
   "_prod_list" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Prod>_\<leftarrow>_. _)" [0, 51, 10] 10)
 translations \<comment> \<open>Beware of argument permutation!\<close>
   "\<Prod>x\<leftarrow>xs. b" \<rightleftharpoons> "CONST prod_list (CONST map (\<lambda>x. b) xs)"
 
 context
   includes lifting_syntax
 begin
 
 lemma prod_list_transfer [transfer_rule]:
   "(list_all2 A ===> A) prod_list prod_list"
     if [transfer_rule]: "A 1 1" "(A ===> A ===> A) (*) (*)"
   unfolding prod_list.eq_foldr [abs_def]
   by transfer_prover
 
 end
 
 lemma prod_list_zero_iff:
   "prod_list xs = 0 \<longleftrightarrow> (0 :: 'a :: {semiring_no_zero_divisors, semiring_1}) \<in> set xs"
   by (induction xs) simp_all
 
 end
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
@@ -1,1197 +1,1227 @@
 (*  Author:  Florian Haftmann, TUM
 *)
 
 section \<open>Bit operations in suitable algebraic structures\<close>
 
 theory Bit_Operations
   imports
     "HOL-Library.Boolean_Algebra"
     Main
 begin
 
-lemma nat_take_bit_eq:
-  \<open>nat (take_bit n k) = take_bit n (nat k)\<close>
-  if \<open>k \<ge> 0\<close>
-  using that by (simp add: take_bit_eq_mod nat_mod_distrib nat_power_eq)
-
-lemma take_bit_eq_self:
-  \<open>take_bit m n = n\<close> if \<open>n < 2 ^ m\<close> for n :: nat
-  using that by (simp add: take_bit_eq_mod)
-
-lemma horner_sum_of_bool_2_less:
-  \<open>(horner_sum of_bool 2 bs :: int) < 2 ^ length bs\<close>
-proof -
-  have \<open>(\<Sum>n = 0..<length bs. of_bool (bs ! n) * (2::int) ^ n) \<le> (\<Sum>n = 0..<length bs. 2 ^ n)\<close>
-    by (rule sum_mono) simp
-  also have \<open>\<dots> = 2 ^ length bs - 1\<close>
-    by (induction bs) simp_all
-  finally show ?thesis
-    by (simp add: horner_sum_eq_sum)
-qed
-
-
 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 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)
 
 end
 
 class ring_bit_operations = semiring_bit_operations + ring_parity +
   fixes not :: \<open>'a \<Rightarrow> 'a\<close>  (\<open>NOT\<close>)
   assumes bit_not_iff: \<open>\<And>n. bit (NOT a) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> \<not> bit a n\<close>
   assumes minus_eq_not_minus_1: \<open>- a = NOT (a - 1)\<close>
 begin
 
 text \<open>
   For the sake of code generation \<^const>\<open>not\<close> is specified as
   definitional class operation.  Note that \<^const>\<open>not\<close> has no
   sensible definition for unlimited but only positive bit strings
   (type \<^typ>\<open>nat\<close>).
 \<close>
 
 lemma bits_minus_1_mod_2_eq [simp]:
   \<open>(- 1) mod 2 = 1\<close>
   by (simp add: mod_2_eq_odd)
 
 lemma not_eq_complement:
   \<open>NOT a = - a - 1\<close>
   using minus_eq_not_minus_1 [of \<open>a + 1\<close>] by simp
 
 lemma minus_eq_not_plus_1:
   \<open>- a = NOT a + 1\<close>
   using not_eq_complement [of a] by simp
 
 lemma bit_minus_iff:
   \<open>bit (- a) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> \<not> bit (a - 1) n\<close>
   by (simp add: minus_eq_not_minus_1 bit_not_iff)
 
 lemma even_not_iff [simp]:
   "even (NOT a) \<longleftrightarrow> odd a"
   using bit_not_iff [of a 0] by auto
 
 lemma bit_not_exp_iff:
   \<open>bit (NOT (2 ^ m)) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n \<noteq> m\<close>
   by (auto simp add: bit_not_iff bit_exp_iff)
 
 lemma bit_minus_1_iff [simp]:
   \<open>bit (- 1) n \<longleftrightarrow> 2 ^ n \<noteq> 0\<close>
   by (simp add: bit_minus_iff)
 
 lemma bit_minus_exp_iff:
   \<open>bit (- (2 ^ m)) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n \<ge> m\<close>
   oops
 
 lemma bit_minus_2_iff [simp]:
   \<open>bit (- 2) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n > 0\<close>
   by (simp add: bit_minus_iff bit_1_iff)
 
 lemma not_one [simp]:
   "NOT 1 = - 2"
   by (simp add: bit_eq_iff bit_not_iff) (simp add: bit_1_iff)
 
 sublocale "and": semilattice_neutr \<open>(AND)\<close> \<open>- 1\<close>
   apply standard
   apply (simp add: bit_eq_iff bit_and_iff)
   apply (auto simp add: exp_eq_0_imp_not_bit bit_exp_iff)
   done
 
 sublocale bit: boolean_algebra \<open>(AND)\<close> \<open>(OR)\<close> NOT 0 \<open>- 1\<close>
   rewrites \<open>bit.xor = (XOR)\<close>
 proof -
   interpret bit: boolean_algebra \<open>(AND)\<close> \<open>(OR)\<close> NOT 0 \<open>- 1\<close>
     apply standard
          apply (simp_all add: bit_eq_iff)
        apply (auto simp add: bit_and_iff bit_or_iff bit_not_iff bit_exp_iff exp_eq_0_imp_not_bit)
     done
   show \<open>boolean_algebra (AND) (OR) NOT 0 (- 1)\<close>
     by standard
   show \<open>boolean_algebra.xor (AND) (OR) NOT = (XOR)\<close>
     apply (simp add: fun_eq_iff bit_eq_iff bit.xor_def)
     apply (auto simp add: bit_and_iff bit_or_iff bit_not_iff bit_xor_iff exp_eq_0_imp_not_bit)
     done
 qed
 
 lemma and_eq_not_not_or:
   \<open>a AND b = NOT (NOT a OR NOT b)\<close>
   by simp
 
 lemma or_eq_not_not_and:
   \<open>a OR b = NOT (NOT a AND NOT b)\<close>
   by simp
 
 lemma 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 push_bit_minus:
   \<open>push_bit n (- a) = - push_bit n a\<close>
   by (simp add: push_bit_eq_mult)
 
 lemma take_bit_not_take_bit:
   \<open>take_bit n (NOT (take_bit n a)) = take_bit n (NOT a)\<close>
   by (auto simp add: bit_eq_iff bit_take_bit_iff bit_not_iff)
 
 lemma take_bit_not_iff:
   "take_bit n (NOT a) = take_bit n (NOT b) \<longleftrightarrow> take_bit n a = take_bit n b"
   apply (simp add: bit_eq_iff bit_not_iff bit_take_bit_iff)
   apply (simp add: bit_exp_iff)
   apply (use local.exp_eq_0_imp_not_bit in blast)
   done
 
 lemma 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>
 
 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 disjunctive_add:
   \<open>k + l = k OR l\<close> if \<open>\<And>n. \<not> bit k n \<or> \<not> bit l n\<close> for k l :: int
   \<comment> \<open>TODO: may integrate (indirectly) into \<^class>\<open>semiring_bits\<close> premises\<close>
 proof (rule bit_eqI)
   fix n
   from that have \<open>bit (k + l) n \<longleftrightarrow> bit k n \<or> bit l n\<close>
   proof (induction n arbitrary: k l)
     case 0
     from this [of 0] show ?case
       by auto
   next
     case (Suc n)
     have \<open>bit ((k + l) div 2) n \<longleftrightarrow> bit (k div 2 + l div 2) n\<close>
       using Suc.prems [of 0] div_add1_eq [of k l] by auto
     also have \<open>bit (k div 2 + l div 2) n \<longleftrightarrow> bit (k div 2) n \<or> bit (l div 2) n\<close>
       by (rule Suc.IH) (use Suc.prems in \<open>simp flip: bit_Suc\<close>)
     finally show ?case
       by (simp add: bit_Suc)
   qed
   also have \<open>\<dots> \<longleftrightarrow> bit (k OR l) n\<close>
     by (simp add: bit_or_iff)
   finally show \<open>bit (k + l) n \<longleftrightarrow> bit (k OR l) n\<close> .
 qed
 
 lemma disjunctive_diff:
   \<open>k - l = k AND NOT l\<close> if \<open>\<And>n. bit l n \<Longrightarrow> bit k n\<close> for k l :: int
 proof -
   have \<open>NOT k + l = NOT k OR l\<close>
     by (rule disjunctive_add) (auto simp add: bit_not_iff dest: that)
   then have \<open>NOT (NOT k + l) = NOT (NOT k OR l)\<close>
     by simp
   then show ?thesis
     by (simp add: not_add_distrib)
 qed
 
 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
 
 
 subsection \<open>Bit concatenation\<close>
 
 definition concat_bit :: \<open>nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>concat_bit n k l = (k AND mask n) 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_mask_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)
 
 
 subsection \<open>Taking bit with sign propagation\<close>
 
 definition signed_take_bit :: \<open>nat \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>signed_take_bit n k = concat_bit n k (- of_bool (bit k n))\<close>
 
 lemma signed_take_bit_eq:
   \<open>signed_take_bit n k = take_bit n k\<close> if \<open>\<not> bit k n\<close>
   using that by (simp add: signed_take_bit_def)
 
 lemma signed_take_bit_eq_or:
   \<open>signed_take_bit n k = take_bit n k OR NOT (mask n)\<close> if \<open>bit k n\<close>
   using that by (simp add: signed_take_bit_def concat_bit_def take_bit_eq_mask push_bit_minus_one_eq_not_mask)
 
 lemma signed_take_bit_0 [simp]:
   \<open>signed_take_bit 0 k = - (k mod 2)\<close>
   by (simp add: signed_take_bit_def odd_iff_mod_2_eq_one)
 
 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 signed_take_bit_Suc:
   \<open>signed_take_bit (Suc n) k = k mod 2 + 2 * signed_take_bit n (k div 2)\<close>
   by (unfold signed_take_bit_def or_int_rec [of \<open>take_bit (Suc n) k\<close>])
     (simp add: bit_Suc concat_bit_Suc even_or_iff even_mask_iff odd_iff_mod_2_eq_one not_int_div_2 mask_half_int)
 
 lemma signed_take_bit_rec:
   \<open>signed_take_bit n k = (if n = 0 then - (k mod 2) else k mod 2 + 2 * signed_take_bit (n - 1) (k div 2))\<close>
   by (cases n) (simp_all add: signed_take_bit_Suc)
 
 lemma bit_signed_take_bit_iff:
   \<open>bit (signed_take_bit m k) n = bit k (min m n)\<close>
   by (simp add: signed_take_bit_def bit_or_iff bit_concat_bit_iff bit_not_iff bit_mask_iff min_def)
 
+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 concat_bit_def push_bit_minus_one_eq_not_mask take_bit_minus_one_eq_mask)
+
+lemma signed_take_bit_signed_take_bit [simp]:
+  \<open>signed_take_bit m (signed_take_bit n k) = signed_take_bit (min m n) k\<close>
+  by (rule bit_eqI) (auto simp add: bit_signed_take_bit_iff min_def bit_or_iff bit_not_iff bit_mask_iff bit_take_bit_iff)
+
+lemma signed_take_bit_eq_iff_take_bit_eq:
+  \<open>signed_take_bit n k = signed_take_bit n l \<longleftrightarrow> take_bit (Suc n) k = take_bit (Suc n) l\<close>
+proof (cases \<open>bit k n \<longleftrightarrow> bit l n\<close>)
+  case True
+  moreover have \<open>take_bit n k OR NOT (mask n) = take_bit n k - 2 ^ n\<close>
+    for k :: int
+    by (auto simp add: disjunctive_add [symmetric] bit_not_iff bit_mask_iff bit_take_bit_iff minus_exp_eq_not_mask)
+  ultimately show ?thesis
+    by (simp add: signed_take_bit_def take_bit_Suc_from_most concat_bit_eq)
+next
+  case False
+  then have \<open>signed_take_bit n k \<noteq> signed_take_bit n l\<close> and \<open>take_bit (Suc n) k \<noteq> take_bit (Suc n) l\<close>
+    by (auto simp add: bit_eq_iff bit_take_bit_iff bit_signed_take_bit_iff min_def)
+  then show ?thesis
+    by simp
+qed
+
+lemma take_bit_signed_take_bit:
+  \<open>take_bit m (signed_take_bit n k) = take_bit m k\<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)
+
+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>
+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>
+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>
+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>
+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
+
 text \<open>Modulus centered around 0\<close>
 
 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>
 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
   then show ?thesis
     by (simp add: bit_eq_iff bit_take_bit_iff bit_signed_take_bit_iff min_def)
       (auto intro: bit_eqI simp add: 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>
 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)
       (simp add: concat_bit_def take_bit_eq_mask push_bit_minus_one_eq_not_mask ac_simps)
 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 concat_bit_def push_bit_minus_one_eq_not_mask take_bit_minus_one_eq_mask)
-
-lemma signed_take_bit_eq_iff_take_bit_eq:
-  \<open>signed_take_bit n k = signed_take_bit n l \<longleftrightarrow> take_bit (Suc n) k = take_bit (Suc n) l\<close>
-proof (cases \<open>bit k n \<longleftrightarrow> bit l n\<close>)
-  case True
-  moreover have \<open>take_bit n k OR NOT (mask n) = take_bit n k - 2 ^ n\<close>
-    for k :: int
-    by (auto simp add: disjunctive_add [symmetric] bit_not_iff bit_mask_iff bit_take_bit_iff minus_exp_eq_not_mask)
-  ultimately show ?thesis
-    by (simp add: signed_take_bit_def take_bit_Suc_from_most concat_bit_eq)
-next
-  case False
-  then have \<open>signed_take_bit n k \<noteq> signed_take_bit n l\<close> and \<open>take_bit (Suc n) k \<noteq> take_bit (Suc n) l\<close>
-    by (auto simp add: bit_eq_iff bit_take_bit_iff bit_signed_take_bit_iff min_def)
-  then show ?thesis
-    by simp
-qed
-
-lemma signed_take_bit_signed_take_bit [simp]:
-  \<open>signed_take_bit m (signed_take_bit n k) = signed_take_bit (min m n) k\<close>
-  by (rule bit_eqI) (auto simp add: bit_signed_take_bit_iff min_def bit_or_iff bit_not_iff bit_mask_iff bit_take_bit_iff)
-
-lemma take_bit_signed_take_bit:
-  \<open>take_bit m (signed_take_bit n k) = take_bit m k\<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)
-
 lemma signed_take_bit_take_bit:
   \<open>signed_take_bit m (take_bit n k) = (if n \<le> m then take_bit n else signed_take_bit m) k\<close>
   by (rule bit_eqI) (auto simp add: bit_signed_take_bit_iff min_def bit_take_bit_iff)
 
 lemma signed_take_bit_nonnegative_iff [simp]:
   \<open>0 \<le> signed_take_bit n k \<longleftrightarrow> \<not> bit k n\<close>
   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>
   by (simp add: signed_take_bit_def not_less concat_bit_def)
 
 lemma signed_take_bit_greater_eq:
   \<open>k + 2 ^ Suc n \<le> signed_take_bit n k\<close> if \<open>k < - (2 ^ n)\<close>
   using that take_bit_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_less_eq:
   \<open>signed_take_bit n k \<le> k - 2 ^ Suc n\<close> if \<open>k \<ge> 2 ^ n\<close>
   using that take_bit_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_eq_self:
+  \<open>signed_take_bit n k = k\<close> if \<open>- (2 ^ n) \<le> k\<close> \<open>k < 2 ^ n\<close>
+  using that by (simp add: signed_take_bit_eq_take_bit_shift take_bit_int_eq_self)
+
 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_Suc_bit0 [simp]:
   \<open>signed_take_bit (Suc n) (numeral (Num.Bit0 k)) = signed_take_bit n (numeral k) * 2\<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\<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\<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\<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\<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\<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\<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\<close>
   by (simp add: signed_take_bit_rec)
 
 lemma signed_take_bit_code [code]:
   \<open>signed_take_bit n k =
   (let l = take_bit (Suc n) k
    in if bit l n then l - (push_bit n 2) else l)\<close>
 proof -
   have *: \<open>(take_bit (Suc n) k) - 2 * 2 ^ n = take_bit (Suc n) k OR NOT (mask (Suc n))\<close>
     apply (subst disjunctive_add [symmetric])
     apply (simp_all add: bit_and_iff bit_mask_iff bit_not_iff bit_take_bit_iff)
     apply (simp flip: minus_exp_eq_not_mask)
     done
   show ?thesis
     by (rule bit_eqI)
      (auto simp add: Let_def bit_and_iff bit_signed_take_bit_iff push_bit_eq_mult min_def not_le
        bit_mask_iff bit_exp_iff less_Suc_eq * bit_or_iff bit_take_bit_iff bit_not_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: and_nat_def bit_and_iff less_le bit_eq_iff)
   show \<open>bit (m OR n) q \<longleftrightarrow> bit m q \<or> bit n q\<close>
     by (auto simp add: or_nat_def bit_or_iff less_le bit_eq_iff)
   show \<open>bit (m XOR n) q \<longleftrightarrow> bit m q \<noteq> bit n q\<close>
     by (auto simp add: xor_nat_def bit_xor_iff less_le bit_eq_iff)
 qed (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
 
 
 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> Bit concatenation: @{thm concat_bit_def [no_vars]}
 
       \<^item> Truncation centered towards \<^term>\<open>0::int\<close>: @{thm signed_take_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/Parity.thy b/src/HOL/Parity.thy
--- a/src/HOL/Parity.thy
+++ b/src/HOL/Parity.thy
@@ -1,1735 +1,1761 @@
 (*  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 odd_numeral_BitM [simp]:
   \<open>odd (numeral (Num.BitM w))\<close>
   by (cases w) simp_all
 
 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
 
 context
   assumes "SORT_CONSTRAINT('a::division_ring)"
 begin
 
 lemma power_int_minus_left:
   "power_int (-a :: 'a) n = (if even n then power_int a n else -power_int a n)"
   by (auto simp: power_int_def minus_one_power_iff even_nat_iff)
 
 lemma power_int_minus_left_even [simp]: "even n \<Longrightarrow> power_int (-a :: 'a) n = power_int a n"
   by (simp add: power_int_minus_left)
 
 lemma power_int_minus_left_odd [simp]: "odd n \<Longrightarrow> power_int (-a :: 'a) n = -power_int a n"
   by (simp add: power_int_minus_left)
 
 lemma power_int_minus_left_distrib:
   "NO_MATCH (-1) x \<Longrightarrow> power_int (-a :: 'a) n = power_int (-1) n * power_int a n"
   by (simp add: power_int_minus_left)
 
 lemma power_int_minus_one_minus: "power_int (-1 :: 'a) (-n) = power_int (-1) n"
   by (simp add: power_int_minus_left)
 
 lemma power_int_minus_one_diff_commute: "power_int (-1 :: 'a) (a - b) = power_int (-1) (b - a)"
   by (subst power_int_minus_one_minus [symmetric]) auto
 
 lemma power_int_minus_one_mult_self [simp]:
   "power_int (-1 :: 'a) m * power_int (-1) m = 1"
   by (simp add: power_int_minus_left)
 
 lemma power_int_minus_one_mult_self' [simp]:
   "power_int (-1 :: 'a) m * (power_int (-1) m * b) = b"
   by (simp add: power_int_minus_left)
 
 end
 
 
 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>
   fixes bit :: \<open>'a \<Rightarrow> nat \<Rightarrow> bool\<close>
   assumes bit_iff_odd: \<open>bit a n \<longleftrightarrow> odd (a div 2 ^ n)\<close>
 begin
 
 text \<open>
   Having \<^const>\<open>bit\<close> as definitional class operation
   takes into account that specific instances can be implemented
   differently wrt. code generation.
 \<close>
 
 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)
 
 lemma bit_0 [simp]:
   \<open>bit a 0 \<longleftrightarrow> odd a\<close>
   by (simp add: bit_iff_odd)
 
 lemma bit_Suc:
   \<open>bit a (Suc n) \<longleftrightarrow> bit (a div 2) n\<close>
   using div_exp_eq [of a 1 n] by (simp add: bit_iff_odd)
 
 lemma bit_rec:
   \<open>bit a n \<longleftrightarrow> (if n = 0 then odd a else bit (a div 2) (n - 1))\<close>
   by (cases n) (simp_all add: bit_Suc)
 
 lemma bit_0_eq [simp]:
   \<open>bit 0 = bot\<close>
   by (simp add: fun_eq_iff bit_iff_odd)
 
 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 bit_Suc)
 
 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 bit_Suc)
   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 bit_Suc)
   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_iff_odd)
 
 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 add: bit_Suc)
     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_iff_odd 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_iff_odd)
 
 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_iff_odd ac_simps)
 
 lemma bit_eq_rec:
   \<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 add: bit_Suc)
     qed
   qed
 qed
 
 lemma bit_mod_2_iff [simp]:
   \<open>bit (a mod 2) n \<longleftrightarrow> n = 0 \<and> odd a\<close>
   by (cases a rule: parity_cases) (simp_all add: bit_iff_odd)
 
 lemma bit_mask_iff:
   \<open>bit (2 ^ m - 1) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n < m\<close>
   by (simp add: bit_iff_odd even_mask_div_iff not_le)
 
 lemma bit_Numeral1_iff [simp]:
   \<open>bit (numeral Num.One) n \<longleftrightarrow> n = 0\<close>
   by (simp add: bit_rec)
 
 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
 
 instantiation nat :: semiring_bits
 begin
 
 definition bit_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> bool\<close>
   where \<open>bit_nat m n \<longleftrightarrow> odd (m div 2 ^ n)\<close>
 
 instance
 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 bit_nat_def)
 
 end
 
 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
 
 instantiation int :: semiring_bits
 begin
 
 definition bit_int :: \<open>int \<Rightarrow> nat \<Rightarrow> bool\<close>
   where \<open>bit_int k n \<longleftrightarrow> odd (k div 2 ^ n)\<close>
 
 instance
 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 bit_int_def)
 
 end
 
 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>
   fixes take_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
   assumes take_bit_eq_mod: \<open>take_bit n a = a mod 2 ^ n\<close>
 begin
 
 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
   them 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_iff_odd 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 push_bit_numeral [simp]:
   \<open>push_bit (numeral l) (numeral k) = push_bit (pred_numeral l) (numeral (Num.Bit0 k))\<close>
   by (simp add: numeral_eq_Suc mult_2_right) (simp add: numeral_Bit0)
 
 lemma take_bit_0 [simp]:
   "take_bit 0 a = 0"
   by (simp add: take_bit_eq_mod)
 
 lemma take_bit_Suc:
   \<open>take_bit (Suc n) a = take_bit n (a div 2) * 2 + a mod 2\<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_rec:
   \<open>take_bit n a = (if n = 0 then 0 else take_bit (n - 1) (a div 2) * 2 + a mod 2)\<close>
   by (cases n) (simp_all add: take_bit_Suc)
 
 lemma take_bit_Suc_0 [simp]:
   \<open>take_bit (Suc 0) a = a mod 2\<close>
   by (simp add: take_bit_eq_mod)
 
 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 add: take_bit_Suc)
 
 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:
   "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_rec:
   "drop_bit n a = (if n = 0 then a else drop_bit (n - 1) (a div 2))"
   by (cases n) (simp_all add: drop_bit_Suc)
 
 lemma drop_bit_half:
   "drop_bit n (a div 2) = drop_bit n a div 2"
   by (induction n arbitrary: a) (simp_all add: drop_bit_Suc)
 
 lemma drop_bit_of_bool [simp]:
   "drop_bit n (of_bool b) = of_bool (n = 0 \<and> b)"
   by (cases n) simp_all
 
 lemma even_take_bit_eq [simp]:
   \<open>even (take_bit n a) \<longleftrightarrow> n = 0 \<or> even a\<close>
   by (simp add: take_bit_rec [of n a])
 
 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> m \<le> n \<and> 2 ^ n \<noteq> 0 \<and> bit a (n - m)\<close>
   by (auto simp add: bit_iff_odd 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_iff_odd 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_iff_odd drop_bit_take_bit not_le flip: drop_bit_eq_div)
 
 lemma stable_imp_drop_bit_eq:
   \<open>drop_bit n a = a\<close>
   if \<open>a div 2 = a\<close>
   by (induction n) (simp_all add: that drop_bit_Suc)
 
 lemma stable_imp_take_bit_eq:
   \<open>take_bit n a = (if even a then 0 else 2 ^ n - 1)\<close>
     if \<open>a div 2 = a\<close>
 proof (rule bit_eqI)
   fix m
   assume \<open>2 ^ m \<noteq> 0\<close>
   with that show \<open>bit (take_bit n a) m \<longleftrightarrow> bit (if even a then 0 else 2 ^ n - 1) m\<close>
     by (simp add: bit_take_bit_iff bit_mask_iff stable_imp_bit_iff_odd)
 qed
 
 lemma exp_dvdE:
   assumes \<open>2 ^ n dvd a\<close>
   obtains b where \<open>a = push_bit n b\<close>
 proof -
   from assms obtain b where \<open>a = 2 ^ n * b\<close> ..
   then have \<open>a = push_bit n b\<close>
     by (simp add: push_bit_eq_mult ac_simps)
   with that show thesis .
 qed
 
 lemma take_bit_eq_0_iff:
   \<open>take_bit n a = 0 \<longleftrightarrow> 2 ^ n dvd a\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
 proof
   assume ?P
   then show ?Q
 	  by (simp add: take_bit_eq_mod mod_0_imp_dvd)
 next
   assume ?Q
   then obtain b where \<open>a = push_bit n b\<close>
     by (rule exp_dvdE)
   then show ?P
     by (simp add: take_bit_push_bit)
 qed
 
 lemma take_bit_tightened:
   \<open>take_bit m a = take_bit m b\<close> if \<open>take_bit n a = take_bit n b\<close> and \<open>m \<le> n\<close> 
 proof -
   from that have \<open>take_bit m (take_bit n a) = take_bit m (take_bit n b)\<close>
     by simp
   then have \<open>take_bit (min m n) a = take_bit (min m n) b\<close>
     by simp
   with that show ?thesis
     by (simp add: min_def)
 qed
 
 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>
 
 definition take_bit_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close>
   where \<open>take_bit_nat n m = m mod 2 ^ n\<close>
 
 instance
   by standard (simp_all add: push_bit_nat_def drop_bit_nat_def take_bit_nat_def)
 
 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>
 
 definition take_bit_int :: \<open>nat \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>take_bit_int n k = k mod 2 ^ n\<close>
 
 instance
   by standard (simp_all add: push_bit_int_def drop_bit_int_def take_bit_int_def)
 
 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_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_of_1_eq_0_iff [simp]:
   "take_bit n 1 = 0 \<longleftrightarrow> n = 0"
   by (simp add: take_bit_eq_mod)
 
 lemma take_bit_Suc_1 [simp]:
   \<open>take_bit (Suc n) 1 = 1\<close>
   by (simp add: take_bit_Suc)
 
 lemma take_bit_Suc_bit0 [simp]:
   \<open>take_bit (Suc n) (numeral (Num.Bit0 k)) = take_bit n (numeral k) * 2\<close>
   by (simp add: take_bit_Suc numeral_Bit0_div_2)
 
 lemma take_bit_Suc_bit1 [simp]:
   \<open>take_bit (Suc n) (numeral (Num.Bit1 k)) = take_bit n (numeral k) * 2 + 1\<close>
   by (simp add: take_bit_Suc numeral_Bit1_div_2 mod_2_eq_odd)
 
 lemma take_bit_numeral_1 [simp]:
   \<open>take_bit (numeral l) 1 = 1\<close>
   by (simp add: take_bit_rec [of \<open>numeral l\<close> 1])
 
 lemma take_bit_numeral_bit0 [simp]:
   \<open>take_bit (numeral l) (numeral (Num.Bit0 k)) = take_bit (pred_numeral l) (numeral k) * 2\<close>
   by (simp add: take_bit_rec numeral_Bit0_div_2)
 
 lemma take_bit_numeral_bit1 [simp]:
   \<open>take_bit (numeral l) (numeral (Num.Bit1 k)) = take_bit (pred_numeral l) (numeral k) * 2 + 1\<close>
   by (simp add: take_bit_rec numeral_Bit1_div_2 mod_2_eq_odd)
 
 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_Suc_bit0 [simp]:
   \<open>drop_bit (Suc n) (numeral (Num.Bit0 k)) = drop_bit n (numeral k)\<close>
   by (simp add: drop_bit_Suc numeral_Bit0_div_2)
 
 lemma drop_bit_Suc_bit1 [simp]:
   \<open>drop_bit (Suc n) (numeral (Num.Bit1 k)) = drop_bit n (numeral k)\<close>
   by (simp add: drop_bit_Suc numeral_Bit1_div_2)
 
 lemma drop_bit_numeral_bit0 [simp]:
   \<open>drop_bit (numeral l) (numeral (Num.Bit0 k)) = drop_bit (pred_numeral l) (numeral k)\<close>
   by (simp add: drop_bit_rec numeral_Bit0_div_2)
 
 lemma drop_bit_numeral_bit1 [simp]:
   \<open>drop_bit (numeral l) (numeral (Num.Bit1 k)) = drop_bit (pred_numeral l) (numeral k)\<close>
   by (simp add: drop_bit_rec numeral_Bit1_div_2)
 
 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_iff_odd semiring_bits_class.bit_iff_odd)
 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 bit_nat_iff [simp]:
   \<open>bit (nat k) n \<longleftrightarrow> k > 0 \<and> bit k n\<close>
 proof (cases \<open>k > 0\<close>)
   case True
   moreover define m where \<open>m = nat k\<close>
   ultimately have \<open>k = int m\<close>
     by simp
   then show ?thesis
     by (auto intro: ccontr)
 next
   case False
   then show ?thesis
     by simp
 qed
 
+lemma nat_take_bit_eq:
+  \<open>nat (take_bit n k) = take_bit n (nat k)\<close>
+  if \<open>k \<ge> 0\<close>
+  using that by (simp add: take_bit_eq_mod nat_mod_distrib nat_power_eq)
+
 lemma not_exp_less_eq_0_int [simp]:
   \<open>\<not> 2 ^ n \<le> (0::int)\<close>
   by (simp add: power_le_zero_eq)
 
 lemma half_nonnegative_int_iff [simp]:
   \<open>k div 2 \<ge> 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int
 proof (cases \<open>k \<ge> 0\<close>)
   case True
   then show ?thesis
     by (auto simp add: divide_int_def sgn_1_pos)
 next
   case False
   then show ?thesis
     apply (auto simp add: divide_int_def not_le elim!: evenE)
     apply (simp only: minus_mult_right)
     apply (subst nat_mult_distrib)
      apply simp_all
     done
 qed
 
 lemma half_negative_int_iff [simp]:
   \<open>k div 2 < 0 \<longleftrightarrow> k < 0\<close> for k :: int
   by (subst Not_eq_iff [symmetric]) (simp add: not_less)
 
 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_Suc_from_most:
   \<open>take_bit (Suc n) k = 2 ^ n * of_bool (bit k n) + take_bit n k\<close> for k :: int
   by (simp only: take_bit_eq_mod power_Suc2) (simp_all add: bit_iff_odd odd_iff_mod_2_eq_one zmod_zmult2_eq)
 
 lemma take_bit_minus:
   \<open>take_bit n (- take_bit n k) = take_bit n (- k)\<close>
     for k :: int
   by (simp add: take_bit_eq_mod mod_minus_eq)
 
 lemma take_bit_diff:
   \<open>take_bit n (take_bit n k - take_bit n l) = take_bit n (k - l)\<close>
     for k l :: int
   by (simp add: take_bit_eq_mod mod_diff_eq)
 
 lemma take_bit_nonnegative [simp]:
   \<open>take_bit n k \<ge> 0\<close>
     for k :: int
   by (simp add: take_bit_eq_mod)
 
 lemma not_take_bit_negative [simp]:
   \<open>\<not> take_bit n k < 0\<close>
     for k :: int
   by (simp add: not_less)
 
 lemma take_bit_int_less_exp:
   \<open>take_bit n k < 2 ^ n\<close> for k :: int
   by (simp add: take_bit_eq_mod)
 
+lemma take_bit_int_eq_self:
+  \<open>take_bit n k = k\<close> if \<open>0 \<le> k\<close> \<open>k < 2 ^ n\<close> for k :: int
+  using that by (simp add: take_bit_eq_mod)
+
+lemma bit_imp_take_bit_positive:
+  \<open>0 < take_bit m k\<close> if \<open>n < m\<close> and \<open>bit k n\<close> for k :: int
+proof (rule ccontr)
+  assume \<open>\<not> 0 < take_bit m k\<close>
+  then have \<open>take_bit m k = 0\<close>
+    by (auto simp add: not_less intro: order_antisym)
+  then have \<open>bit (take_bit m k) n = bit 0 n\<close>
+    by simp
+  with that show False
+    by (simp add: bit_take_bit_iff)
+qed
+
+lemma take_bit_mult:
+  \<open>take_bit n (take_bit n k * take_bit n l) = take_bit n (k * l)\<close>
+  for k l :: int
+  by (simp add: take_bit_eq_mod mod_mult_eq)
+
 lemma (in ring_1) of_nat_nat_take_bit_eq [simp]:
   \<open>of_nat (nat (take_bit n k)) = of_int (take_bit n k)\<close>
   by simp
 
 lemma take_bit_minus_small_eq:
   \<open>take_bit n (- k) = 2 ^ n - k\<close> if \<open>0 < k\<close> \<open>k \<le> 2 ^ n\<close> for k :: int
 proof -
   define m where \<open>m = nat k\<close>
   with that have \<open>k = int m\<close> and \<open>0 < m\<close> and \<open>m \<le> 2 ^ n\<close>
     by simp_all
   have \<open>(2 ^ n - m) mod 2 ^ n = 2 ^ n - m\<close>
     using \<open>0 < m\<close> by simp
   then have \<open>int ((2 ^ n - m) mod 2 ^ n) = int (2 ^ n - m)\<close>
     by simp
   then have \<open>(2 ^ n - int m) mod 2 ^ n = 2 ^ n - int m\<close>
     using \<open>m \<le> 2 ^ n\<close> by (simp only: of_nat_mod of_nat_diff) simp
   with \<open>k = int m\<close> have \<open>(2 ^ n - k) mod 2 ^ n = 2 ^ n - k\<close>
     by simp
   then show ?thesis
     by (simp add: take_bit_eq_mod)
 qed
 
 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)
 
 lemma push_bit_nonnegative_int_iff [simp]:
   \<open>push_bit n k \<ge> 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int
   by (simp add: push_bit_eq_mult zero_le_mult_iff)
 
 lemma push_bit_negative_int_iff [simp]:
   \<open>push_bit n k < 0 \<longleftrightarrow> k < 0\<close> for k :: int
   by (subst Not_eq_iff [symmetric]) (simp add: not_less)
 
 lemma drop_bit_nonnegative_int_iff [simp]:
   \<open>drop_bit n k \<ge> 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int
   by (induction n) (simp_all add: drop_bit_Suc drop_bit_half)
 
 lemma drop_bit_negative_int_iff [simp]:
   \<open>drop_bit n k < 0 \<longleftrightarrow> k < 0\<close> for k :: int
   by (subst Not_eq_iff [symmetric]) (simp add: not_less)
 
 code_identifier
   code_module Parity \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
 
 end