# HG changeset patch
# User haftmann
# Date 1674556256 0
# Node ID 5de3772609ea322face002d9875534ea55499092
# Parent  a5d3f3c07de87380cb66c915964c483822a5de05
generalized theory name: euclidean division denotes one particular division definition on integers

diff -r a5d3f3c07de8 -r 5de3772609ea NEWS
--- a/NEWS	Mon Jan 23 22:33:25 2023 +0100
+++ b/NEWS	Tue Jan 24 10:30:56 2023 +0000
@@ -43,6 +43,10 @@
 
 *** HOL ***
 
+* Theory "HOL.Euclidean_Division" renamed to "HOL.Euclidean_Rings";
+  "euclidean division" typically denotes a particular division on
+  integers.  Minor INCOMPATIBILITY.
+
 * Theory "HOL.Fun":
   - Renamed lemma inj_on_strict_subset to image_strict_mono.
     Minor INCOMPATIBILITY.
diff -r a5d3f3c07de8 -r 5de3772609ea src/HOL/Algebra/Multiplicative_Group.thy
--- a/src/HOL/Algebra/Multiplicative_Group.thy	Mon Jan 23 22:33:25 2023 +0100
+++ b/src/HOL/Algebra/Multiplicative_Group.thy	Tue Jan 24 10:30:56 2023 +0000
@@ -533,8 +533,7 @@
         using n
         by simp (metis (no_types, lifting) assms dvd_minus_mod dvd_trans int_pow_eq int_pow_eq_id int_pow_int)
       show "nat (a mod int n) \<in> {0..<n}"
-        using n  apply (simp add:  split: split_nat)
-        using Euclidean_Division.pos_mod_bound by presburger
+        using n by (simp add: nat_less_iff)
     qed
     then have "carrier (subgroup_generated G {x}) \<subseteq> ([^]) x ` {0..<n}"
       using carrier_subgroup_generated_by_singleton [OF assms] by auto
diff -r a5d3f3c07de8 -r 5de3772609ea src/HOL/Euclidean_Division.thy
--- a/src/HOL/Euclidean_Division.thy	Mon Jan 23 22:33:25 2023 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2733 +0,0 @@
-(*  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])
-       (use assms in 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 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 iffI; 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
-
-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: \<open>euclidean_size (a * b) = euclidean_size a * euclidean_size b\<close>
-  fixes division_segment :: \<open>'a \<Rightarrow> 'a\<close>
-  assumes is_unit_division_segment [simp]: \<open>is_unit (division_segment a)\<close>
-    and division_segment_mult:
-    \<open>a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> division_segment (a * b) = division_segment a * division_segment b\<close>
-    and division_segment_mod:
-    \<open>b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<Longrightarrow> division_segment (a mod b) = division_segment b\<close>
-  assumes div_bounded:
-    \<open>b \<noteq> 0 \<Longrightarrow> division_segment r = division_segment b
-    \<Longrightarrow> euclidean_size r < euclidean_size b
-    \<Longrightarrow> (q * b + r) div b = q\<close>
-begin
-
-lemma division_segment_not_0 [simp]:
-  \<open>division_segment a \<noteq> 0\<close>
-  using is_unit_division_segment [of a] is_unitE [of \<open>division_segment a\<close>] by blast
-
-lemma euclidean_relationI [case_names by0 divides euclidean_relation]:
-  \<open>(a div b, a mod b) = (q, r)\<close>
-    if by0: \<open>b = 0 \<Longrightarrow> q = 0 \<and> r = a\<close>
-    and divides: \<open>b \<noteq> 0 \<Longrightarrow> b dvd a \<Longrightarrow> r = 0 \<and> a = q * b\<close>
-    and euclidean_relation: \<open>b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<Longrightarrow> division_segment r = division_segment b
-      \<and> euclidean_size r < euclidean_size b \<and> a = q * b + r\<close>
-proof (cases \<open>b = 0\<close>)
-  case True
-  with by0 show ?thesis
-    by simp
-next
-  case False
-  show ?thesis
-  proof (cases \<open>b dvd a\<close>)
-    case True
-    with \<open>b \<noteq> 0\<close> divides
-    show ?thesis
-      by simp
-  next
-    case False
-    with \<open>b \<noteq> 0\<close> euclidean_relation
-    have \<open>division_segment r = division_segment b\<close>
-      \<open>euclidean_size r < euclidean_size b\<close> \<open>a = q * b + r\<close>
-      by simp_all
-    from \<open>b \<noteq> 0\<close> \<open>division_segment r = division_segment b\<close>
-      \<open>euclidean_size r < euclidean_size b\<close>
-    have \<open>(q * b + r) div b = q\<close>
-      by (rule div_bounded)
-    with \<open>a = q * b + r\<close>
-    have \<open>q = a div b\<close>
-      by simp
-    from \<open>a = q * b + r\<close>
-    have \<open>a div b * b + a mod b = q * b + r\<close>
-      by (simp add: div_mult_mod_eq)
-    with \<open>q = a div b\<close>
-    have \<open>q * b + a mod b = q * b + r\<close>
-      by simp
-    then have \<open>r = a mod b\<close>
-      by simp
-    with \<open>q = a div b\<close>
-    show ?thesis
-      by simp
-  qed
-qed
-
-subclass euclidean_semiring_cancel
-proof
-  fix a b c
-  assume \<open>b \<noteq> 0\<close>
-  have \<open>((a + c * b) div b, (a + c * b) mod b) = (c + a div b, a mod b)\<close>
-  proof (induction rule: euclidean_relationI)
-    case by0
-    with \<open>b \<noteq> 0\<close>
-    show ?case
-      by simp
-  next
-    case divides
-    then show ?case
-      by (simp add: algebra_simps dvd_add_left_iff)
-  next
-    case euclidean_relation
-    then have \<open>\<not> b dvd a\<close>
-      by (simp add: dvd_add_left_iff)
-    have \<open>a mod b + (b * c + b * (a div b)) = b * c + ((a div b) * b + a mod b)\<close>
-      by (simp add: ac_simps)
-    with \<open>b \<noteq> 0\<close> have *: \<open>a mod b + (b * c + b * (a div b)) = b * c + a\<close>
-      by (simp add: div_mult_mod_eq)
-    from \<open>\<not> b dvd a\<close> euclidean_relation show ?case
-      by (simp_all add: algebra_simps division_segment_mod mod_size_less *)
-  qed
-  then show \<open>(a + c * b) div b = c + a div b\<close>
-    by simp
-next
-  fix a b c
-  assume \<open>c \<noteq> 0\<close>
-  have \<open>((c * a) div (c * b), (c * a) mod (c * b)) = (a div b, c * (a mod b))\<close>
-  proof (induction rule: euclidean_relationI)
-    case by0
-    with \<open>c \<noteq> 0\<close> show ?case
-      by simp
-  next
-    case divides
-    then show ?case
-      by (auto simp add: algebra_simps)
-  next
-    case euclidean_relation
-    then have \<open>b \<noteq> 0\<close> \<open>a mod b \<noteq> 0\<close>
-      by (simp_all add: mod_eq_0_iff_dvd)
-    have \<open>c * (a mod b) + b * (c * (a div b)) = c * ((a div b) * b + a mod b)\<close>
-      by (simp add: algebra_simps)
-    with \<open>b \<noteq> 0\<close> have *: \<open>c * (a mod b) + b * (c * (a div b)) = c * a\<close>
-      by (simp add: div_mult_mod_eq)
-    from \<open>b \<noteq> 0\<close> \<open>c \<noteq> 0\<close> have \<open>euclidean_size c * euclidean_size (a mod b)
-      < euclidean_size c * euclidean_size b\<close>
-      using mod_size_less [of b a] by simp
-    with euclidean_relation \<open>b \<noteq> 0\<close> \<open>a mod b \<noteq> 0\<close> show ?case
-      by (simp add: algebra_simps division_segment_mult division_segment_mod euclidean_size_mult *)
-  qed
-  then show \<open>(c * a) div (c * b) = a div b\<close>
-    by simp
-qed
-
-lemma div_eq_0_iff:
-  \<open>a div b = 0 \<longleftrightarrow> euclidean_size a < euclidean_size b \<or> b = 0\<close> (is "_ \<longleftrightarrow> ?P")
-  if \<open>division_segment a = division_segment b\<close>
-proof (cases \<open>a = 0 \<or> b = 0\<close>)
-  case True
-  then show ?thesis by auto
-next
-  case False
-  then have \<open>a \<noteq> 0\<close> \<open>b \<noteq> 0\<close>
-    by simp_all
-  have \<open>a div b = 0 \<longleftrightarrow> euclidean_size a < euclidean_size b\<close>
-  proof
-    assume \<open>a div b = 0\<close>
-    then have \<open>a mod b = a\<close>
-      using div_mult_mod_eq [of a b] by simp
-    with \<open>b \<noteq> 0\<close> mod_size_less [of b a]
-    show \<open>euclidean_size a < euclidean_size b\<close>
-      by simp
-  next
-    assume \<open>euclidean_size a < euclidean_size b\<close>
-    have \<open>(a div b, a mod b) = (0, a)\<close>
-    proof (induction rule: euclidean_relationI)
-      case by0
-      show ?case
-        by simp
-    next
-      case divides
-      with \<open>euclidean_size a < euclidean_size b\<close> show ?case
-        using dvd_imp_size_le [of b a] \<open>a \<noteq> 0\<close> by simp
-    next
-      case euclidean_relation
-      with \<open>euclidean_size a < euclidean_size b\<close> that
-      show ?case
-        by simp
-    qed
-    then show \<open>a div b = 0\<close>
-      by simp
-  qed
-  with \<open>b \<noteq> 0\<close> show ?thesis
-    by simp
-qed
-
-lemma div_mult1_eq:
-  \<open>(a * b) div c = a * (b div c) + a * (b mod c) div c\<close>
-proof -
-  have *: \<open>(a * b) mod c + (a * (c * (b div c)) + c * (a * (b mod c) div c)) = a * b\<close> (is \<open>?A + (?B + ?C) = _\<close>)
-  proof -
-    have \<open>?A = a * (b mod c) mod c\<close>
-      by (simp add: mod_mult_right_eq)
-    then have \<open>?C + ?A = a * (b mod c)\<close>
-      by (simp add: mult_div_mod_eq)
-    then have \<open>?B + (?C + ?A) = a * (c * (b div c) + (b mod c))\<close>
-      by (simp add: algebra_simps)
-    also have \<open>\<dots> = a * b\<close>
-      by (simp add: mult_div_mod_eq)
-    finally show ?thesis
-      by (simp add: algebra_simps)
-  qed
-  have \<open>((a * b) div c, (a * b) mod c) = (a * (b div c) + a * (b mod c) div c, (a * b) mod c)\<close>
-  proof (induction rule: euclidean_relationI)
-    case by0
-    then show ?case by simp
-  next
-    case divides
-    with * show ?case
-      by (simp add: algebra_simps)
-  next
-    case euclidean_relation
-    with * show ?case
-      by (simp add: division_segment_mod mod_size_less algebra_simps)
-  qed
-  then show ?thesis
-    by simp
-qed
-
-lemma div_add1_eq:
-  \<open>(a + b) div c = a div c + b div c + (a mod c + b mod c) div c\<close>
-proof -
-  have *: \<open>(a + b) mod c + (c * (a div c) + (c * (b div c) + c * ((a mod c + b mod c) div c))) = a + b\<close>
-    (is \<open>?A + (?B + (?C + ?D)) = _\<close>)
-  proof -
-    have \<open>?A + (?B + (?C + ?D)) = ?A + ?D + (?B + ?C)\<close>
-      by (simp add: ac_simps)
-    also have \<open>?A + ?D = (a mod c + b mod c) mod c + ?D\<close>
-      by (simp add: mod_add_eq)
-    also have \<open>\<dots> = a mod c + b mod c\<close>
-      by (simp add: mod_mult_div_eq)
-    finally have \<open>?A + (?B + (?C + ?D)) = (a mod c + ?B) + (b mod c + ?C)\<close>
-      by (simp add: ac_simps)
-    then show ?thesis
-      by (simp add: mod_mult_div_eq)
-  qed
-  have \<open>((a + b) div c, (a + b) mod c) = (a div c + b div c + (a mod c + b mod c) div c, (a + b) mod c)\<close>
-  proof (induction rule: euclidean_relationI)
-    case by0
-    then show ?case
-      by simp
-  next
-    case divides
-    with * show ?case
-      by (simp add: algebra_simps)
-  next
-    case euclidean_relation
-    with * show ?case
-      by (simp add: division_segment_mod mod_size_less algebra_simps)
-  qed
-  then show ?thesis
-    by simp
-qed
-
-end
-
-class unique_euclidean_ring = euclidean_ring + unique_euclidean_semiring
-begin
-
-subclass euclidean_ring_cancel ..
-
-end
-
-
-subsection \<open>Division on \<^typ>\<open>nat\<close>\<close>
-
-instantiation nat :: normalization_semidom
-begin
-
-definition normalize_nat :: \<open>nat \<Rightarrow> nat\<close>
-  where [simp]: \<open>normalize = (id :: nat \<Rightarrow> nat)\<close>
-
-definition unit_factor_nat :: \<open>nat \<Rightarrow> nat\<close>
-  where \<open>unit_factor n = of_bool (n > 0)\<close> for n :: nat
-
-lemma unit_factor_simps [simp]:
-  \<open>unit_factor 0 = (0::nat)\<close>
-  \<open>unit_factor (Suc n) = 1\<close>
-  by (simp_all add: unit_factor_nat_def)
-
-definition divide_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close>
-  where \<open>m div n = (if n = 0 then 0 else Max {k. k * n \<le> m})\<close> for m n :: nat
-
-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 :: \<open>nat \<Rightarrow> nat\<close>
-  where [simp]: \<open>euclidean_size_nat = id\<close>
-
-definition division_segment_nat :: \<open>nat \<Rightarrow> nat\<close>
-  where [simp]: \<open>division_segment n = 1\<close> for n :: nat
-
-definition modulo_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close>
-  where \<open>m mod n = m - (m div n * n)\<close> for m 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
-
-lemma euclidean_relation_natI [case_names by0 divides euclidean_relation]:
-  \<open>(m div n, m mod n) = (q, r)\<close>
-    if by0: \<open>n = 0 \<Longrightarrow> q = 0 \<and> r = m\<close>
-    and divides: \<open>n > 0 \<Longrightarrow> n dvd m \<Longrightarrow> r = 0 \<and> m = q * n\<close>
-    and euclidean_relation: \<open>n > 0 \<Longrightarrow> \<not> n dvd m \<Longrightarrow> r < n \<and> m = q * n + r\<close> for m n q r :: nat
-  by (rule euclidean_relationI) (use that in simp_all)
-
-lemma div_nat_eqI:
-  \<open>m div n = q\<close> if \<open>n * q \<le> m\<close> and \<open>m < n * Suc q\<close> for m n q :: nat
-proof -
-  have \<open>(m div n, m mod n) = (q, m - n * q)\<close>
-  proof (induction rule: euclidean_relation_natI)
-    case by0
-    with that show ?case
-      by simp
-  next
-    case divides
-    from \<open>n dvd m\<close> obtain s where \<open>m = n * s\<close> ..
-    with \<open>n > 0\<close> that have \<open>s < Suc q\<close>
-      by (simp only: mult_less_cancel1)
-    with \<open>m = n * s\<close> \<open>n > 0\<close> that have \<open>q = s\<close>
-      by simp
-    with \<open>m = n * s\<close> show ?case
-      by (simp add: ac_simps)
-  next
-    case euclidean_relation
-    with that show ?case
-      by (simp add: ac_simps)
-  qed
-  then show ?thesis
-    by simp
-qed
-
-lemma mod_nat_eqI:
-  \<open>m mod n = r\<close> if \<open>r < n\<close> and \<open>r \<le> m\<close> and \<open>n dvd m - r\<close> for m n r :: nat
-proof -
-  have \<open>(m div n, m mod n) = ((m - r) div n, r)\<close>
-  proof (induction rule: euclidean_relation_natI)
-    case by0
-    with that show ?case
-      by simp
-  next
-    case divides
-    from that dvd_minus_add [of r \<open>m\<close> 1 n]
-    have \<open>n dvd m + (n - r)\<close>
-      by simp
-    with divides have \<open>n dvd n - r\<close>
-      by (simp add: dvd_add_right_iff)
-    then have \<open>n \<le> n - r\<close>
-      by (rule dvd_imp_le) (use \<open>r < n\<close> in simp)
-    with \<open>n > 0\<close> have \<open>r = 0\<close>
-      by simp
-    with \<open>n > 0\<close> that show ?case
-      by simp
-  next
-    case euclidean_relation
-    with that show ?case
-      by (simp add: ac_simps)
-  qed
-  then show ?thesis
-    by simp
-qed
-
-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_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_nat_eqI mod_nat_eqI)
-
-lemma split_div:
-  \<open>P (m div n) \<longleftrightarrow>
-    (n = 0 \<longrightarrow> P 0) \<and>
-    (n \<noteq> 0 \<longrightarrow> (\<forall>i j. j < n \<and> m = n * i + j \<longrightarrow> P i))\<close> (is ?div)
-  and split_mod:
-  \<open>Q (m mod n) \<longleftrightarrow>
-    (n = 0 \<longrightarrow> Q m) \<and>
-    (n \<noteq> 0 \<longrightarrow> (\<forall>i j. j < n \<and> m = n * i + j \<longrightarrow> Q j))\<close> (is ?mod)
-  for m n :: nat
-proof -
-  have *: \<open>R (m div n) (m mod n) \<longleftrightarrow>
-    (n = 0 \<longrightarrow> R 0 m) \<and>
-    (n \<noteq> 0 \<longrightarrow> (\<forall>i j. j < n \<and> m = n * i + j \<longrightarrow> R i j))\<close> for R
-    by (cases \<open>n = 0\<close>) auto
-  from * [of \<open>\<lambda>q _. P q\<close>] show ?div .
-  from * [of \<open>\<lambda>_ r. Q r\<close>] show ?mod .
-qed
-
-declare split_div [of _ _ \<open>numeral n\<close>, linarith_split] for n
-declare split_mod [of _ _ \<open>numeral n\<close>, linarith_split] for n
-
-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 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
-
-lemma div_mult2_eq:
-    \<open>m div (n * q) = (m div n) div q\<close> (is ?Q)
-  and mod_mult2_eq:
-    \<open>m mod (n * q) = n * (m div n mod q) + m mod n\<close> (is ?R)
-  for m n q :: nat
-proof -
-  have \<open>(m div (n * q), m mod (n * q)) = ((m div n) div q, n * (m div n mod q) + m mod n)\<close>
-  proof (induction rule: euclidean_relation_natI)
-    case by0
-    then show ?case
-      by auto
-  next
-    case divides
-    from \<open>n * q dvd m\<close> obtain t where \<open>m = n * q * t\<close> ..
-    with \<open>n * q > 0\<close> show ?case
-      by (simp add: algebra_simps)
-  next
-    case euclidean_relation
-    then have \<open>n > 0\<close> \<open>q > 0\<close>
-      by simp_all
-    from \<open>n > 0\<close> have \<open>m mod n < n\<close>
-      by (rule mod_less_divisor)
-    from \<open>q > 0\<close> have \<open>m div n mod q < q\<close>
-      by (rule mod_less_divisor)
-    then obtain s where \<open>q = Suc (m div n mod q + s)\<close>
-      by (blast dest: less_imp_Suc_add)
-    moreover have \<open>m mod n + n * (m div n mod q) < n * Suc (m div n mod q + s)\<close>
-      using \<open>m mod n < n\<close> by (simp add: add_mult_distrib2)
-    ultimately have \<open>m mod n + n * (m div n mod q) < n * q\<close>
-      by simp
-    then show ?case
-      by (simp add: algebra_simps flip: add_mult_distrib2)
-  qed
-  then show ?Q and ?R
-    by simp_all
-qed
-
-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
-
-lemma div_less_iff_less_mult:
-  \<open>m div q < n \<longleftrightarrow> m < n * q\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
-  if \<open>q > 0\<close> for m n q :: nat
-proof
-  assume ?Q then show ?P
-    by (rule less_mult_imp_div_less)
-next
-  assume ?P
-  then obtain h where \<open>n = Suc (m div q + h)\<close>
-    using less_natE by blast
-  moreover have \<open>m < m + (Suc h * q - m mod q)\<close>
-    using that by (simp add: trans_less_add1)
-  ultimately show ?Q
-    by (simp add: algebra_simps flip: minus_mod_eq_mult_div)
-qed
-
-lemma less_eq_div_iff_mult_less_eq:
-  \<open>m \<le> n div q \<longleftrightarrow> m * q \<le> n\<close> if \<open>q > 0\<close> for m n q :: nat
-  using div_less_iff_less_mult [of q n m] that by auto
-
-lemma div_Suc:
-  \<open>Suc m div n = (if Suc m mod n = 0 then Suc (m div n) else m div n)\<close>
-proof (cases \<open>n = 0 \<or> n = 1\<close>)
-  case True
-  then show ?thesis by auto
-next
-  case False
-  then have \<open>n > 1\<close>
-    by simp
-  then have \<open>Suc m div n = m div n + Suc (m mod n) div n\<close>
-    using div_add1_eq [of m 1 n] by simp
-  also have \<open>Suc (m mod n) div n = of_bool (n dvd Suc m)\<close>
-  proof (cases \<open>n dvd Suc m\<close>)
-    case False
-    moreover have \<open>Suc (m mod n) \<noteq> n\<close>
-    proof (rule ccontr)
-      assume \<open>\<not> Suc (m mod n) \<noteq> n\<close>
-      then have \<open>m mod n = n - Suc 0\<close>
-        by simp
-      with \<open>n > 1\<close> have \<open>(m + 1) mod n = 0\<close>
-        by (subst mod_add_left_eq [symmetric]) simp
-      then have \<open>n dvd Suc m\<close>
-        by auto
-      with False show False ..
-    qed
-    moreover have \<open>Suc (m mod n) \<le> n\<close>
-      using \<open>n > 1\<close> by (simp add: Suc_le_eq)
-    ultimately show ?thesis
-      by (simp add: div_eq_0_iff)
-  next
-    case True
-    then obtain q where q: \<open>Suc m = n * q\<close> ..
-    moreover have \<open>q > 0\<close> by (rule ccontr)
-      (use q in simp)
-    ultimately have \<open>m mod n = n - Suc 0\<close>
-      using \<open>n > 1\<close> mult_le_cancel1 [of n \<open>Suc 0\<close> q]
-      by (auto intro: mod_nat_eqI)
-    with True \<open>n > 1\<close> show ?thesis
-      by simp
-  qed
-  finally show ?thesis
-    by (simp add: mod_greater_zero_iff_not_dvd)
-qed
-
-lemma mod_Suc:
-  \<open>Suc m mod n = (if Suc (m mod n) = n then 0 else Suc (m mod n))\<close>
-proof (cases \<open>n = 0\<close>)
-  case True
-  then show ?thesis
-    by simp
-next
-  case False
-  moreover have \<open>Suc m mod n = Suc (m mod n) mod n\<close>
-    by (simp add: mod_simps)
-  ultimately show ?thesis
-    by (auto intro!: mod_nat_eqI intro: neq_le_trans simp add: Suc_le_eq)
-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 funpow_mod_eq: \<^marker>\<open>contributor \<open>Lars Noschinski\<close>\<close>
-  \<open>(f ^^ (m mod n)) x = (f ^^ m) x\<close> if \<open>(f ^^ n) x = x\<close>
-proof -
-  have \<open>(f ^^ m) x = (f ^^ (m mod n + m div n * n)) x\<close>
-    by simp
-  also have \<open>\<dots> = (f ^^ (m mod n)) (((f ^^ n) ^^ (m div n)) x)\<close>
-    by (simp only: funpow_add funpow_mult ac_simps) simp
-  also have \<open>((f ^^ n) ^^ q) x = x\<close> for q
-    by (induction q) (use \<open>(f ^^ n) x = x\<close> in simp_all)
-  finally show ?thesis
-    by simp
-qed
-
-lemma mod_eq_dvd_iff_nat:
-  \<open>m mod q = n mod q \<longleftrightarrow> q dvd m - n\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
-    if \<open>m \<ge> n\<close> for m n q :: nat
-proof
-  assume ?Q
-  then obtain s where \<open>m - n = q * s\<close> ..
-  with that have \<open>m = q * s + n\<close>
-    by simp
-  then show ?P
-    by simp
-next
-  assume ?P
-  have \<open>m - n = m div q * q + m mod q - (n div q * q + n mod q)\<close>
-    by simp
-  also have \<open>\<dots> = q * (m div q - n div q)\<close>
-    by (simp only: algebra_simps \<open>?P\<close>)
-  finally show ?Q ..
-qed
-
-lemma mod_eq_iff_dvd_symdiff_nat:
-  \<open>m mod q = n mod q \<longleftrightarrow> q dvd nat \<bar>int m - int n\<bar>\<close>
-  by (auto simp add: abs_if mod_eq_dvd_iff_nat nat_diff_distrib dest: sym intro: sym)
-
-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_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 mod_eq_nat1E [OF th xy] obtain q where "y = x + n * q" .
-    then have "x + n * q = y + n * 0"
-      by simp
-    then have "\<exists>q1 q2. x + n * q1 = y + n * q2"
-      by blast
-  }
-  moreover
-  { assume xy: "y \<le> x"
-    from mod_eq_nat1E [OF H xy] obtain q where "x = y + n * q" .
-    then have "x + n * 0 = y + n * q"
-      by simp
-    then have "\<exists>q1 q2. x + n * q1 = y + n * q2"
-      by blast
-  }
-  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
-
-
-
-subsection \<open>Division on \<^typ>\<open>int\<close>\<close>
-
-subsubsection \<open>Basic instantiation\<close>
-
-instantiation int :: "{normalization_semidom, idom_modulo}"
-begin
-
-definition normalize_int :: \<open>int \<Rightarrow> int\<close>
-  where [simp]: \<open>normalize = (abs :: int \<Rightarrow> int)\<close>
-
-definition unit_factor_int :: \<open>int \<Rightarrow> int\<close>
-  where [simp]: \<open>unit_factor = (sgn :: int \<Rightarrow> int)\<close>
-
-definition divide_int :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close>
-  where \<open>k div l = (sgn k * sgn l * int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
-    - of_bool (l \<noteq> 0 \<and> sgn k \<noteq> sgn l \<and> \<not> l dvd k))\<close>
-
-lemma divide_int_unfold:
-  \<open>(sgn k * int m) div (sgn l * int n) = (sgn k * sgn l * int (m div n)
-    - of_bool ((k = 0 \<longleftrightarrow> m = 0) \<and> l \<noteq> 0 \<and> n \<noteq> 0 \<and> sgn k \<noteq> sgn l \<and> \<not> n dvd m))\<close>
-  by (simp add: divide_int_def sgn_mult nat_mult_distrib abs_mult sgn_eq_0_iff ac_simps)
-
-definition modulo_int :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close>
-  where \<open>k mod l = sgn k * int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>) + l * of_bool (sgn k \<noteq> sgn l \<and> \<not> l dvd k)\<close>
-
-lemma modulo_int_unfold:
-  \<open>(sgn k * int m) mod (sgn l * int n) =
-    sgn k * int (m mod (of_bool (l \<noteq> 0) * n)) + (sgn l * int n) * of_bool ((k = 0 \<longleftrightarrow> m = 0) \<and> sgn k \<noteq> sgn l \<and> \<not> n dvd m)\<close>
-  by (auto simp add: modulo_int_def sgn_mult abs_mult)
-
-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)
-next
-  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 (simp add: divide_int_unfold modulo_int_unfold algebra_simps modulo_nat_def of_nat_diff)
-qed (auto simp add: sgn_mult mult_sgn_abs abs_eq_iff')
-
-end
-
-
-subsubsection \<open>Algebraic foundations\<close>
-
-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
-
-
-subsubsection \<open>Basic conversions\<close>
-
-lemma div_abs_eq_div_nat:
-  "\<bar>k\<bar> div \<bar>l\<bar> = int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)"
-  by (auto simp add: divide_int_def)
-
-lemma div_eq_div_abs:
-  \<open>k div l = sgn k * sgn l * (\<bar>k\<bar> div \<bar>l\<bar>)
-    - of_bool (l \<noteq> 0 \<and> sgn k \<noteq> sgn l \<and> \<not> l dvd k)\<close>
-  for k l :: int
-  by (simp add: divide_int_def [of k l] div_abs_eq_div_nat)
-
-lemma div_abs_eq:
-  \<open>\<bar>k\<bar> div \<bar>l\<bar> = sgn k * sgn l * (k div l + of_bool (sgn k \<noteq> sgn l \<and> \<not> l dvd k))\<close>
-  for k l :: int
-  by (simp add: div_eq_div_abs [of k l] ac_simps)
-
-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 mod_eq_mod_abs:
-  \<open>k mod l = sgn k * (\<bar>k\<bar> mod \<bar>l\<bar>) + l * of_bool (sgn k \<noteq> sgn l \<and> \<not> l dvd k)\<close>
-  for k l :: int
-  by (simp add: modulo_int_def [of k l] mod_abs_eq_div_nat)
-
-lemma mod_abs_eq:
-  \<open>\<bar>k\<bar> mod \<bar>l\<bar> = sgn k * (k mod l - l * of_bool (sgn k \<noteq> sgn l \<and> \<not> l dvd k))\<close>
-  for k l :: int
-  by (auto simp: mod_eq_mod_abs [of k l])
-
-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>"
-  using assms by (simp add: sgn_mult abs_mult sgn_0_0
-    divide_int_def [of "sgn v * \<bar>k\<bar>" "sgn v * \<bar>l\<bar>"] flip: div_abs_eq_div_nat)
-
-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>"
-  using assms by (auto simp add: div_abs_eq)
-
-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>)"
-  using assms by (auto simp add: div_abs_eq ac_simps)
-
-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 (auto simp add: div_abs_eq ac_simps sgn_0_0 dest!: sgn_not_eq_imp)
-
-
-subsubsection \<open>Euclidean division\<close>
-
-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 (auto simp add: modulo_int_unfold abs_mult mod_greater_zero_iff_not_dvd
-        simp flip: right_diff_distrib dest!: sgn_not_eq_imp)
-      (simp add: sgn_0_0)
-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 (auto simp add: modulo_int_unfold sgn_mult mod_greater_zero_iff_not_dvd
-      simp flip: right_diff_distrib dest!: sgn_not_eq_imp)
-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 ac_simps)
-    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 euclidean_relation_intI [case_names by0 divides euclidean_relation]:
-  \<open>(k div l, k mod l) = (q, r)\<close>
-    if by0': \<open>l = 0 \<Longrightarrow> q = 0 \<and> r = k\<close>
-    and divides': \<open>l \<noteq> 0 \<Longrightarrow> l dvd k \<Longrightarrow> r = 0 \<and> k = q * l\<close>
-    and euclidean_relation': \<open>l \<noteq> 0 \<Longrightarrow> \<not> l dvd k \<Longrightarrow> sgn r = sgn l
-      \<and> \<bar>r\<bar> < \<bar>l\<bar> \<and> k = q * l + r\<close> for k l :: int
-proof (induction rule: euclidean_relationI)
-  case by0
-  then show ?case
-    by (rule by0')
-next
-  case divides
-  then show ?case
-    by (rule divides')
-next
-  case euclidean_relation
-  with euclidean_relation' have \<open>sgn r = sgn l\<close> \<open>\<bar>r\<bar> < \<bar>l\<bar>\<close> \<open>k = q * l + r\<close>
-    by simp_all
-  from \<open>sgn r = sgn l\<close> \<open>l \<noteq> 0\<close> have \<open>division_segment r = division_segment l\<close>
-    by (simp add: division_segment_int_def sgn_if split: if_splits)
-  with \<open>\<bar>r\<bar> < \<bar>l\<bar>\<close> \<open>k = q * l + r\<close>
-  show ?case
-    by simp
-qed
-
-
-subsubsection \<open>Trivial reduction steps\<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: \<open>k div l = - 1\<close>  (is ?Q)
-  and mod_pos_neg_trivial: \<open>k mod l = k + l\<close>  (is ?R)
-    if \<open>0 < k\<close> and \<open>k + l \<le> 0\<close> for k l :: int
-proof -
-  from that have \<open>l < 0\<close>
-    by simp
-  have \<open>(k div l, k mod l) = (- 1, k + l)\<close>
-  proof (induction rule: euclidean_relation_intI)
-    case by0
-    with \<open>l < 0\<close> show ?case
-      by simp
-  next
-    case divides
-    from \<open>l dvd k\<close> obtain j where \<open>k = l * j\<close> ..
-    with \<open>l < 0\<close> \<open>0 < k\<close> have \<open>j < 0\<close>
-      by (simp add: zero_less_mult_iff)
-    moreover from \<open>k + l \<le> 0\<close> \<open>k = l * j\<close> have \<open>l * (j + 1) \<le> 0\<close>
-      by (simp add: algebra_simps)
-    with \<open>l < 0\<close> have \<open>j + 1 \<ge> 0\<close>
-      by (simp add: mult_le_0_iff)
-    with \<open>j < 0\<close> have \<open>j = - 1\<close>
-      by simp
-    with \<open>k = l * j\<close> show ?case
-      by simp
-  next
-    case euclidean_relation
-    with \<open>k + l \<le> 0\<close> have \<open>k + l < 0\<close>
-      by (auto simp add: less_le add_eq_0_iff)
-    with \<open>0 < k\<close> show ?case
-      by simp
-  qed
-  then show ?Q and ?R
-    by simp_all
-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>
-
-
-subsubsection \<open>More uniqueness rules\<close>
-
-lemma
-  fixes a b q r :: int
-  assumes \<open>a = b * q + r\<close> \<open>0 \<le> r\<close> \<open>r < b\<close>
-  shows int_div_pos_eq:
-      \<open>a div b = q\<close> (is ?Q)
-    and int_mod_pos_eq:
-      \<open>a mod b = r\<close> (is ?R)
-proof -
-  have \<open>(a div b, a mod b) = (q, r)\<close>
-    by (induction rule: euclidean_relation_intI)
-      (use assms in \<open>auto simp add: ac_simps dvd_add_left_iff sgn_1_pos le_less dest: zdvd_imp_le\<close>)
-  then show ?Q and ?R
-    by simp_all
-qed
-
-lemma int_div_neg_eq:
-  \<open>a div b = q\<close> if \<open>a = b * q + r\<close> \<open>r \<le> 0\<close> \<open>b < r\<close> for a b q r :: int
-  using that int_div_pos_eq [of a \<open>- b\<close> \<open>- q\<close> \<open>- r\<close>] by simp_all
-
-lemma int_mod_neg_eq:
-  \<open>a mod b = r\<close> if \<open>a = b * q + r\<close> \<open>r \<le> 0\<close> \<open>b < r\<close> for a b q r :: int
-  using that int_div_neg_eq [of a b q r] by simp
-
-
-subsubsection \<open>Laws for unary minus\<close>
-
-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_zminus1_eq_if:
-  \<open>(- a) div b = (if a mod b = 0 then - (a div b) else - (a div b) - 1)\<close>
-  if \<open>b \<noteq> 0\<close> for a b :: int
-  using that sgn_not_eq_imp [of b \<open>- a\<close>]
-  by (cases \<open>a = 0\<close>) (auto simp add: div_eq_div_abs [of \<open>- a\<close> b] div_eq_div_abs [of a b] sgn_eq_0_iff)
-
-lemma zdiv_zminus2_eq_if:
-  \<open>a div (- b) = (if a mod b = 0 then - (a div b) else - (a div b) - 1)\<close>
-  if \<open>b \<noteq> 0\<close> for a b :: int
-  using that by (auto simp add: zdiv_zminus1_eq_if div_minus_right)
-
-lemma zmod_zminus1_eq_if:
-  \<open>(- a) mod b = (if a mod b = 0 then 0 else b - (a mod b))\<close>
-  for a b :: int
-  by (cases \<open>b = 0\<close>)
-    (auto simp flip: minus_div_mult_eq_mod simp add: zdiv_zminus1_eq_if algebra_simps)
-
-lemma zmod_zminus2_eq_if:
-  \<open>a mod (- b) = (if a mod b = 0 then 0 else (a mod b) - b)\<close>
-  for a b :: int
-  by (auto simp add: zmod_zminus1_eq_if mod_minus_right)
-
-
-subsubsection \<open>Borders\<close>
-
-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)
-      (auto simp add: mod_greater_zero_iff_not_dvd sgn_1_pos)
-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)
-      (auto simp add: mod_greater_zero_iff_not_dvd sgn_1_neg)
-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) (auto simp add: sgn_1_pos)
-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
-  moreover have \<open>int (m mod n) \<le> int n\<close>
-    using \<open>Suc q = n\<close> by simp
-  then have \<open>sgn s * int (m mod n) \<le> int n\<close>
-    by (cases s \<open>0::int\<close> rule: linorder_cases) simp_all
-  ultimately show ?thesis
-    by (simp only: modulo_int_unfold) auto
-qed
-
-
-subsubsection \<open>Splitting Rules for div and mod\<close>
-
-lemma split_zdiv:
-  \<open>P (n div k) \<longleftrightarrow>
-    (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))\<close> (is ?div)
-  and split_zmod:
-  \<open>Q (n mod k) \<longleftrightarrow>
-    (k = 0 \<longrightarrow> Q n) \<and>
-    (0 < k \<longrightarrow> (\<forall>i j. 0 \<le> j \<and> j < k \<and> n = k * i + j \<longrightarrow> Q j)) \<and>
-    (k < 0 \<longrightarrow> (\<forall>i j. k < j \<and> j \<le> 0 \<and> n = k * i + j \<longrightarrow> Q j))\<close> (is ?mod)
-  for n k :: int
-proof -
-  have *: \<open>R (n div k) (n mod k) \<longleftrightarrow>
-    (k = 0 \<longrightarrow> R 0 n) \<and>
-    (0 < k \<longrightarrow> (\<forall>i j. 0 \<le> j \<and> j < k \<and> n = k * i + j \<longrightarrow> R i j)) \<and>
-    (k < 0 \<longrightarrow> (\<forall>i j. k < j \<and> j \<le> 0 \<and> n = k * i + j \<longrightarrow> R i j))\<close> for R
-    by (cases \<open>k = 0\<close>)
-      (auto simp add: linorder_class.neq_iff)
-  from * [of \<open>\<lambda>q _. P q\<close>] show ?div .
-  from * [of \<open>\<lambda>_ r. Q r\<close>] show ?mod .
-qed
-
-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 _ _ \<open>numeral n\<close>, linarith_split] for n
-declare split_zdiv [of _ _ \<open>- numeral n\<close>, linarith_split] for n
-declare split_zmod [of _ _ \<open>numeral n\<close>, linarith_split] for n
-declare split_zmod [of _ _ \<open>- numeral n\<close>, linarith_split] for n
-
-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_trivial_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"
-    using div_mult_mod_eq [of i k] by safe auto
-  with zdiv_eq_0_iff
-  show ?thesis
-    by simp
-qed
-
-
-subsubsection \<open>Algebraic rewrites\<close>
-
-lemma zdiv_zmult2_eq: \<open>a div (b * c) = (a div b) div c\<close>  (is ?Q)
-  and zmod_zmult2_eq: \<open>a mod (b * c) = b * (a div b mod c) + a mod b\<close>  (is ?P)
-  if \<open>c \<ge> 0\<close> for a b c :: int
-proof -
-  have *: \<open>(a div (b * c), a mod (b * c)) = ((a div b) div c, b * (a div b mod c) + a mod b)\<close>
-    if \<open>b > 0\<close> for a b
-  proof (induction rule: euclidean_relationI)
-    case by0
-    then show ?case by auto
-  next
-    case divides
-    then obtain d where \<open>a = b * c * d\<close>
-      by blast
-    with divides that show ?case
-      by (simp add: ac_simps)
-  next
-    case euclidean_relation
-    with \<open>b > 0\<close> \<open>c \<ge> 0\<close> have \<open>0 < c\<close> \<open>b > 0\<close>
-      by simp_all
-    then have \<open>a mod b < b\<close>
-      by simp
-    moreover have \<open>1 \<le> c - a div b mod c\<close>
-      using \<open>c > 0\<close> by (simp add: int_one_le_iff_zero_less)
-    ultimately have \<open>a mod b * 1 < b * (c - a div b mod c)\<close>
-      by (rule mult_less_le_imp_less) (use \<open>b > 0\<close> in simp_all)
-    with \<open>0 < b\<close> \<open>0 < c\<close> show ?case
-      by (simp add: division_segment_int_def algebra_simps flip: minus_mod_eq_mult_div)
-  qed
-  show ?Q
-  proof (cases \<open>b \<ge> 0\<close>)
-    case True
-    with * [of b a] show ?thesis
-      by (cases \<open>b = 0\<close>) simp_all
-  next
-    case False
-    with * [of \<open>- b\<close> \<open>- a\<close>] show ?thesis
-      by simp
-  qed
-  show ?P
-  proof (cases \<open>b \<ge> 0\<close>)
-    case True
-    with * [of b a] show ?thesis
-      by (cases \<open>b = 0\<close>) simp_all
-  next
-    case False
-    with * [of \<open>- b\<close> \<open>- a\<close>] show ?thesis
-      by simp
-  qed
-qed
-
-lemma zdiv_zmult2_eq':
-  \<open>k div (l * j) = ((sgn j * k) div l) div \<bar>j\<bar>\<close> for k l j :: int
-proof -
-  have \<open>k div (l * j) = (sgn j * k) div (sgn j * (l * j))\<close>
-    by (simp add: sgn_0_0)
-  also have \<open>sgn j * (l * j) = l * \<bar>j\<bar>\<close>
-    by (simp add: mult.left_commute [of _ l] abs_sgn) (simp add: ac_simps)
-  also have \<open>(sgn j * k) div (l * \<bar>j\<bar>) = ((sgn j * k) div l) div \<bar>j\<bar>\<close>
-    by (simp add: zdiv_zmult2_eq)
-  finally show ?thesis .
-qed
-
-lemma half_nonnegative_int_iff [simp]:
-  \<open>k div 2 \<ge> 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int
-  by auto
-
-lemma half_negative_int_iff [simp]:
-  \<open>k div 2 < 0 \<longleftrightarrow> k < 0\<close> for k :: int
-  by auto
-
-
-subsubsection \<open>Distributive laws for conversions.\<close>
-
-lemma zdiv_int:
-  \<open>int (m div n) = int m div int n\<close>
-  by (cases \<open>m = 0\<close>) (auto simp add: divide_int_def)
-
-lemma zmod_int:
-  \<open>int (m mod n) = int m mod int n\<close>
-  by (cases \<open>m = 0\<close>) (auto simp add: modulo_int_def)
-
-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)
-
-
-subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
-
-lemma zdiv_mono1:
-  \<open>a div b \<le> a' div b\<close>
-    if \<open>a \<le> a'\<close> \<open>0 < b\<close>
-    for a b b' :: int
-proof -
-  from \<open>a \<le> a'\<close> have \<open>b * (a div b) + a mod b \<le> b * (a' div b) + a' mod b\<close>
-    by simp
-  then have \<open>b * (a div b) \<le> (a' mod b - a mod b) + b * (a' div b)\<close>
-    by (simp add: algebra_simps)
-  moreover have \<open>a' mod b < b + a mod b\<close>
-    by (rule less_le_trans [of _ b]) (use \<open>0 < b\<close> in simp_all)
-  ultimately have \<open>b * (a div b) < b * (1 + a' div b)\<close>
-    by (simp add: distrib_left)
-  with \<open>0 < b\<close> have \<open>a div b < 1 + a' div b\<close>
-    by (simp add: mult_less_cancel_left)
-  then show ?thesis
-    by simp
-qed
-
-lemma zdiv_mono1_neg:
-  \<open>a' div b \<le> a div b\<close>
-    if \<open>a \<le> a'\<close> \<open>b < 0\<close>
-    for a a' b :: int
-  using that zdiv_mono1 [of \<open>- a'\<close> \<open>- a\<close> \<open>- b\<close>] by simp
-
-
-subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
-
-lemma zdiv_mono2:
-  \<open>a div b \<le> a div b'\<close> if \<open>0 \<le> a\<close> \<open>0 < b'\<close> \<open>b' \<le> b\<close> for a b b' :: int
-proof -
-  define q q' r r' where **: \<open>q = a div b\<close> \<open>q' = a div b'\<close> \<open>r = a mod b\<close> \<open>r' = a mod b'\<close>
-  then have *: \<open>b * q + r = b' * q' + r'\<close> \<open>0 \<le> b' * q' + r'\<close>
-    \<open>r' < b'\<close> \<open>0 \<le> r\<close> \<open>0 < b'\<close> \<open>b' \<le> b\<close>
-    using that by simp_all
-  have \<open>0 < b' * (q' + 1)\<close>
-    using * by (simp add: distrib_left)
-  with * have \<open>0 \<le> q'\<close>
-    by (simp add: zero_less_mult_iff)
-  moreover have \<open>b * q = r' - r + b' * q'\<close>
-    using * by linarith
-  ultimately have \<open>b * q < b * (q' + 1)\<close>
-    using mult_right_mono * unfolding distrib_left by fastforce
-  with * have \<open>q \<le> q'\<close>
-    by (simp add: mult_less_cancel_left_pos)
-  with ** show ?thesis
-    by simp
-qed
-
-lemma zdiv_mono2_neg:
-  \<open>a div b' \<le> a div b\<close> if \<open>a < 0\<close> \<open>0 < b'\<close> \<open>b' \<le> b\<close> for a b b' :: int
-proof -
-  define q q' r r' where **: \<open>q = a div b\<close> \<open>q' = a div b'\<close> \<open>r = a mod b\<close> \<open>r' = a mod b'\<close>
-  then have *: \<open>b * q + r = b' * q' + r'\<close> \<open>b' * q' + r' < 0\<close>
-    \<open>r < b\<close> \<open>0 \<le> r'\<close> \<open>0 < b'\<close> \<open>b' \<le> b\<close>
-    using that by simp_all
-  have \<open>b' * q' < 0\<close>
-    using * by linarith
-  with * have \<open>q' \<le> 0\<close>
-    by (simp add: mult_less_0_iff)
-  have \<open>b * q' \<le> b' * q'\<close>
-    by (simp add: \<open>q' \<le> 0\<close> * mult_right_mono_neg)
-  then have "b * q' < b * (q + 1)"
-    using * by (simp add: distrib_left)
-  then have \<open>q' \<le> q\<close>
-    using * by (simp add: mult_less_cancel_left)
-  then show ?thesis
-    by (simp add: **)
-qed
-
-
-subsubsection \<open>Quotients of Signs\<close>
-
-lemma div_eq_minus1:
-  \<open>0 < b \<Longrightarrow> - 1 div b = - 1\<close> for b :: int
-  by (simp add: divide_int_def)
-
-lemma zmod_minus1:
-  \<open>0 < b \<Longrightarrow> - 1 mod b = b - 1\<close> 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 1: \<open>0 < k mod l\<close>
-      using \<open>0 < l\<close> False le_less by fastforce
-    moreover have 2: \<open>k mod l < 1 + l\<close>
-      using \<open>0 < l\<close> pos_mod_bound[of l k] by linarith
-    from 1 2 \<open>l > 0\<close> have \<open>(k mod l - 1) mod l = k mod l - 1\<close>
-      by (simp add: zmod_trivial_iff)
-    ultimately show ?thesis
-      by (simp only: zmod_zminus1_eq_if)
-         (simp add: mod_eq_0_iff_dvd algebra_simps mod_simps)
-  qed
-qed
-
-lemma div_neg_pos_less0:
-  \<open>a div b < 0\<close> if \<open>a < 0\<close> \<open>0 < b\<close> for a b :: int
-proof -
-  have "a div b \<le> - 1 div b"
-    using zdiv_mono1 that by auto
-  also have "... \<le> -1"
-    by (simp add: that(2) div_eq_minus1)
-  finally show ?thesis
-    by force
-qed
-
-lemma div_nonneg_neg_le0:
-  \<open>a div b \<le> 0\<close> if \<open>0 \<le> a\<close> \<open>b < 0\<close> for a b :: int
-  using that by (auto dest: zdiv_mono1_neg)
-
-lemma div_nonpos_pos_le0:
-  \<open>a div b \<le> 0\<close> if \<open>a \<le> 0\<close> \<open>0 < b\<close> for a b :: int
-  using that by (auto dest: zdiv_mono1)
-
-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:
-  \<open>0 \<le> a div b \<longleftrightarrow> 0 \<le> a\<close>
-  if \<open>0 < b\<close> for a b :: int
-proof
-  assume \<open>0 \<le> a div b\<close>
-  show \<open>0 \<le> a\<close>
-  proof (rule ccontr)
-    assume \<open>\<not> 0 \<le> a\<close>
-    then have \<open>a < 0\<close>
-      by simp
-    then have \<open>a div b < 0\<close>
-      using that by (rule div_neg_pos_less0)
-    with \<open>0 \<le> a div b\<close> show False
-      by simp
-  qed
-next
-  assume "0 \<le> a"
-  then have "0 div b \<le> a div b"
-    using zdiv_mono1 that by blast
-  then show "0 \<le> a div b"
-    by auto
-qed
-
-lemma neg_imp_zdiv_nonneg_iff:
-  \<open>0 \<le> a div b \<longleftrightarrow> a \<le> 0\<close> if \<open>b < 0\<close> for a b :: int
-  using that pos_imp_zdiv_nonneg_iff [of \<open>- b\<close> \<open>- a\<close>] by simp
-
-lemma pos_imp_zdiv_pos_iff:
-  \<open>0 < (i::int) div k \<longleftrightarrow> k \<le> i\<close> if \<open>0 < k\<close> for i k :: int
-  using that pos_imp_zdiv_nonneg_iff [of k i] zdiv_eq_0_iff [of i k] by arith
-
-lemma pos_imp_zdiv_neg_iff:
-  \<open>a div b < 0 \<longleftrightarrow> a < 0\<close> if \<open>0 < b\<close> for a b :: int
-    \<comment> \<open>But not \<^prop>\<open>a div b \<le> 0 \<longleftrightarrow> a \<le> 0\<close>; consider \<^prop>\<open>a = 1\<close>, \<^prop>\<open>b = 2\<close> when \<^prop>\<open>a div b = 0\<close>.\<close>
-  using that by (simp add: pos_imp_zdiv_nonneg_iff flip: linorder_not_le)
-
-lemma neg_imp_zdiv_neg_iff:
-    \<comment> \<open>But not \<^prop>\<open>a div b \<le> 0 \<longleftrightarrow> 0 \<le> a\<close>; consider \<^prop>\<open>a = - 1\<close>, \<^prop>\<open>b = - 2\<close> when \<^prop>\<open>a div b = 0\<close>.\<close>
-  \<open>a div b < 0 \<longleftrightarrow> 0 < a\<close> if \<open>b < 0\<close> for a b :: int
-  using that by (simp add: neg_imp_zdiv_nonneg_iff flip: linorder_not_le)
-
-lemma nonneg1_imp_zdiv_pos_iff:
-  \<open>a div b > 0 \<longleftrightarrow> a \<ge> b \<and> b > 0\<close> if \<open>0 \<le> a\<close> for a b :: int
-proof -
-  have "0 < a div b \<Longrightarrow> b \<le> a"
-    using div_pos_pos_trivial[of a b] that by arith
-  moreover have "0 < a div b \<Longrightarrow> b > 0"
-    using that 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:
-  \<open>m mod k \<le> m\<close> if \<open>(m::int) \<ge> 0\<close> for m k :: int
-proof -
-  from that have \<open>m > 0 \<or> m = 0\<close>
-    by auto
-  then show ?thesis proof
-    assume \<open>m = 0\<close> then show ?thesis
-      by simp
-  next
-    assume \<open>m > 0\<close> then show ?thesis
-    proof (cases k \<open>0::int\<close> rule: linorder_cases)
-      case less
-      moreover define l where \<open>l = - k\<close>
-      ultimately have \<open>l > 0\<close>
-        by simp
-      with \<open>m > 0\<close> have \<open>int (nat m mod nat l) \<le> m\<close>
-        by (simp flip: le_nat_iff)
-      then have \<open>int (nat m mod nat l) - l \<le> m\<close>
-        using \<open>l > 0\<close> by simp
-      with \<open>m > 0\<close> \<open>l > 0\<close> show ?thesis
-        by (simp add: modulo_int_def l_def flip: le_nat_iff)
-    qed (simp_all add: modulo_int_def flip: le_nat_iff)
-  qed
-qed
-
-lemma sgn_div_eq_sgn_mult:
-  \<open>sgn (k div l) = of_bool (k div l \<noteq> 0) * sgn (k * l)\<close>
-  for k l :: int
-proof (cases \<open>k div l = 0\<close>)
-  case True
-  then show ?thesis
-    by simp
-next
-  case False
-  have \<open>0 \<le> \<bar>k\<bar> div \<bar>l\<bar>\<close>
-    by (cases \<open>l = 0\<close>) (simp_all add: pos_imp_zdiv_nonneg_iff)
-  then have \<open>\<bar>k\<bar> div \<bar>l\<bar> \<noteq> 0 \<longleftrightarrow> 0 < \<bar>k\<bar> div \<bar>l\<bar>\<close>
-    by (simp add: less_le)
-  also have \<open>\<dots> \<longleftrightarrow> \<bar>k\<bar> \<ge> \<bar>l\<bar>\<close>
-    using False nonneg1_imp_zdiv_pos_iff by auto
-  finally have *: \<open>\<bar>k\<bar> div \<bar>l\<bar> \<noteq> 0 \<longleftrightarrow> \<bar>l\<bar> \<le> \<bar>k\<bar>\<close> .
-  show ?thesis
-    using \<open>0 \<le> \<bar>k\<bar> div \<bar>l\<bar>\<close> False
-  by (auto simp add: div_eq_div_abs [of k l] div_eq_sgn_abs [of k l]
-    sgn_mult sgn_1_pos sgn_1_neg sgn_eq_0_iff nonneg1_imp_zdiv_pos_iff * dest: sgn_not_eq_imp)
-qed
-
-
-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 have *: "k \<noteq> 0" "l \<noteq> 0"
-    by auto
-  then have "0 \<le> k div l \<Longrightarrow> \<not> k < 0 \<Longrightarrow> 0 \<le> l"
-    by (meson neg_imp_zdiv_neg_iff not_le not_less_iff_gr_or_eq)
-  then show ?thesis
-   using * by (auto simp add: pos_imp_zdiv_nonneg_iff neg_imp_zdiv_nonneg_iff)
-qed auto
-
-lemma mod_int_pos_iff:
-  \<open>k mod l \<ge> 0 \<longleftrightarrow> l dvd k \<or> l = 0 \<and> k \<ge> 0 \<or> l > 0\<close>
-  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
-
-lemma abs_div:
-  \<open>\<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>\<close> if \<open>y dvd x\<close> for x y :: int
-  using that by (cases \<open>y = 0\<close>) (auto simp add: abs_mult)
-
-lemma int_power_div_base: \<^marker>\<open>contributor \<open>Matthias Daum\<close>\<close>
-  \<open>k ^ m div k = k ^ (m - Suc 0)\<close> if \<open>0 < m\<close> \<open>0 < k\<close> for k :: int
-  using that by (cases m) simp_all
-
-lemma int_div_less_self: \<^marker>\<open>contributor \<open>Matthias Daum\<close>\<close>
-  \<open>x div k < x\<close> if \<open>0 < x\<close> \<open>1 < k\<close> for x k :: int
-proof -
-  from that have \<open>nat (x div k) = nat x div nat k\<close>
-    by (simp add: nat_div_distrib)
-  also from that have \<open>nat x div nat k < nat x\<close>
-    by simp
-  finally show ?thesis
-    by simp
-qed
-
-
-subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> by shifting\<close>
-
-lemma div_pos_geq:
-  \<open>k div l = (k - l) div l + 1\<close> if \<open>0 < l\<close> \<open>l \<le> k\<close> for k l :: int
-proof -
-  have "k = (k - l) + l" by simp
-  then obtain j where k: "k = j + l" ..
-  with that show ?thesis by (simp add: div_add_self2)
-qed
-
-lemma mod_pos_geq:
-  \<open>k mod l = (k - l) mod l\<close>  if \<open>0 < l\<close> \<open>l \<le> k\<close> for k l :: int
-proof -
-  have "k = (k - l) + l" by simp
-  then obtain j where k: "k = j + l" ..
-  with that show ?thesis by simp
-qed
-
-lemma pos_zdiv_mult_2: \<open>(1 + 2 * b) div (2 * a) = b div a\<close> (is ?Q)
-  and pos_zmod_mult_2: \<open>(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)\<close> (is ?R)
-  if \<open>0 \<le> a\<close> for a b :: int
-proof -
-  have \<open>((1 + 2 * b) div (2 * a), (1 + 2 * b) mod (2 * a)) = (b div a, 1 + 2 * (b mod a))\<close>
-  proof (induction rule: euclidean_relation_intI)
-    case by0
-    then show ?case
-      by simp
-  next
-    case divides
-    have \<open>2 dvd (2 * a)\<close>
-      by simp
-    then have \<open>2 dvd (1 + 2 * b)\<close>
-      using \<open>2 * a dvd 1 + 2 * b\<close> by (rule dvd_trans)
-    then have \<open>2 dvd (1 + b * 2)\<close>
-      by (simp add: ac_simps)
-    then have \<open>is_unit (2 :: int)\<close>
-      by simp
-    then show ?case
-      by simp
-  next
-    case euclidean_relation
-    with that have \<open>a > 0\<close>
-      by simp
-    moreover have \<open>b mod a < a\<close>
-      using \<open>a > 0\<close> by simp
-    then have \<open>1 + 2 * (b mod a) < 2 * a\<close>
-      by simp
-    moreover have \<open>2 * (b mod a) + a * (2 * (b div a)) = 2 * (b div a * a + b mod a)\<close>
-      by (simp only: algebra_simps)
-    moreover have \<open>0 \<le> 2 * (b mod a)\<close>
-      using \<open>a > 0\<close> by simp
-    ultimately show ?case
-      by (simp add: algebra_simps)
-  qed
-  then show ?Q and ?R
-    by simp_all
-qed
-
-lemma neg_zdiv_mult_2: \<open>(1 + 2 * b) div (2 * a) = (b + 1) div a\<close> (is ?Q)
-  and neg_zmod_mult_2: \<open>(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1\<close> (is ?R)
-  if \<open>a \<le> 0\<close> for a b :: int
-proof -
-  have \<open>((1 + 2 * b) div (2 * a), (1 + 2 * b) mod (2 * a)) = ((b + 1) div a, 2 * ((b + 1) mod a) - 1)\<close>
-  proof (induction rule: euclidean_relation_intI)
-    case by0
-    then show ?case
-      by simp
-  next
-    case divides
-    have \<open>2 dvd (2 * a)\<close>
-      by simp
-    then have \<open>2 dvd (1 + 2 * b)\<close>
-      using \<open>2 * a dvd 1 + 2 * b\<close> by (rule dvd_trans)
-    then have \<open>2 dvd (1 + b * 2)\<close>
-      by (simp add: ac_simps)
-    then have \<open>is_unit (2 :: int)\<close>
-      by simp
-    then show ?case
-      by simp
-  next
-    case euclidean_relation
-    with that have \<open>a < 0\<close>
-      by simp
-    moreover have \<open>(b + 1) mod a > a\<close>
-      using \<open>a < 0\<close> by simp
-    then have \<open>2 * ((b + 1) mod a) > 1 + 2 * a\<close>
-      by simp
-    moreover have \<open>((1 + b) mod a) \<le> 0\<close>
-      using \<open>a < 0\<close> by simp
-    then have \<open>2 * ((1 + b) mod a) \<le> 0\<close>
-      by simp
-    moreover have \<open>2 * ((1 + b) mod a) + a * (2 * ((1 + b) div a)) =
-      2 * ((1 + b) div a * a + (1 + b) mod a)\<close>
-      by (simp only: algebra_simps)
-    ultimately show ?case
-      by (simp add: algebra_simps sgn_mult abs_mult)
-  qed
-  then show ?Q and ?R
-    by simp_all
-qed
-
-lemma zdiv_numeral_Bit0 [simp]:
-  \<open>numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
-    numeral v div (numeral w :: int)\<close>
-  unfolding numeral.simps unfolding mult_2 [symmetric]
-  by (rule div_mult_mult1) simp
-
-lemma zdiv_numeral_Bit1 [simp]:
-  \<open>numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
-    (numeral v div (numeral w :: int))\<close>
-  unfolding numeral.simps
-  unfolding mult_2 [symmetric] add.commute [of _ 1]
-  by (rule pos_zdiv_mult_2) simp
-
-lemma zmod_numeral_Bit0 [simp]:
-  \<open>numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
-    (2::int) * (numeral v mod numeral w)\<close>
-  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
-  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
-
-lemma zmod_numeral_Bit1 [simp]:
-  \<open>numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
-    2 * (numeral v mod numeral w) + (1::int)\<close>
-  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
-  unfolding mult_2 [symmetric] add.commute [of _ 1]
-  by (rule pos_zmod_mult_2) simp
-
-
-subsection \<open>Code generation\<close>
-
-context
-begin
-
-qualified definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat"
-  where "divmod_nat m n = (m div n, m mod n)"
-
-qualified lemma divmod_nat_if [code]:
-  "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
-    let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
-  by (simp add: divmod_nat_def prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
-
-qualified lemma [code]:
-  "m div n = fst (divmod_nat m n)"
-  "m mod n = snd (divmod_nat m n)"
-  by (simp_all add: divmod_nat_def)
-
-end
-
-code_identifier
-  code_module Euclidean_Division \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
-
-end
diff -r a5d3f3c07de8 -r 5de3772609ea src/HOL/Euclidean_Rings.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Euclidean_Rings.thy	Tue Jan 24 10:30:56 2023 +0000
@@ -0,0 +1,2733 @@
+(*  Title:      HOL/Euclidean_Rgins.thy
+    Author:     Manuel Eberl, TU Muenchen
+    Author:     Florian Haftmann, TU Muenchen
+*)
+
+section \<open>Division in euclidean (semi)rings\<close>
+
+theory Euclidean_Rings
+  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])
+       (use assms in 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 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 iffI; 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
+
+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: \<open>euclidean_size (a * b) = euclidean_size a * euclidean_size b\<close>
+  fixes division_segment :: \<open>'a \<Rightarrow> 'a\<close>
+  assumes is_unit_division_segment [simp]: \<open>is_unit (division_segment a)\<close>
+    and division_segment_mult:
+    \<open>a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> division_segment (a * b) = division_segment a * division_segment b\<close>
+    and division_segment_mod:
+    \<open>b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<Longrightarrow> division_segment (a mod b) = division_segment b\<close>
+  assumes div_bounded:
+    \<open>b \<noteq> 0 \<Longrightarrow> division_segment r = division_segment b
+    \<Longrightarrow> euclidean_size r < euclidean_size b
+    \<Longrightarrow> (q * b + r) div b = q\<close>
+begin
+
+lemma division_segment_not_0 [simp]:
+  \<open>division_segment a \<noteq> 0\<close>
+  using is_unit_division_segment [of a] is_unitE [of \<open>division_segment a\<close>] by blast
+
+lemma euclidean_relationI [case_names by0 divides euclidean_relation]:
+  \<open>(a div b, a mod b) = (q, r)\<close>
+    if by0: \<open>b = 0 \<Longrightarrow> q = 0 \<and> r = a\<close>
+    and divides: \<open>b \<noteq> 0 \<Longrightarrow> b dvd a \<Longrightarrow> r = 0 \<and> a = q * b\<close>
+    and euclidean_relation: \<open>b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<Longrightarrow> division_segment r = division_segment b
+      \<and> euclidean_size r < euclidean_size b \<and> a = q * b + r\<close>
+proof (cases \<open>b = 0\<close>)
+  case True
+  with by0 show ?thesis
+    by simp
+next
+  case False
+  show ?thesis
+  proof (cases \<open>b dvd a\<close>)
+    case True
+    with \<open>b \<noteq> 0\<close> divides
+    show ?thesis
+      by simp
+  next
+    case False
+    with \<open>b \<noteq> 0\<close> euclidean_relation
+    have \<open>division_segment r = division_segment b\<close>
+      \<open>euclidean_size r < euclidean_size b\<close> \<open>a = q * b + r\<close>
+      by simp_all
+    from \<open>b \<noteq> 0\<close> \<open>division_segment r = division_segment b\<close>
+      \<open>euclidean_size r < euclidean_size b\<close>
+    have \<open>(q * b + r) div b = q\<close>
+      by (rule div_bounded)
+    with \<open>a = q * b + r\<close>
+    have \<open>q = a div b\<close>
+      by simp
+    from \<open>a = q * b + r\<close>
+    have \<open>a div b * b + a mod b = q * b + r\<close>
+      by (simp add: div_mult_mod_eq)
+    with \<open>q = a div b\<close>
+    have \<open>q * b + a mod b = q * b + r\<close>
+      by simp
+    then have \<open>r = a mod b\<close>
+      by simp
+    with \<open>q = a div b\<close>
+    show ?thesis
+      by simp
+  qed
+qed
+
+subclass euclidean_semiring_cancel
+proof
+  fix a b c
+  assume \<open>b \<noteq> 0\<close>
+  have \<open>((a + c * b) div b, (a + c * b) mod b) = (c + a div b, a mod b)\<close>
+  proof (induction rule: euclidean_relationI)
+    case by0
+    with \<open>b \<noteq> 0\<close>
+    show ?case
+      by simp
+  next
+    case divides
+    then show ?case
+      by (simp add: algebra_simps dvd_add_left_iff)
+  next
+    case euclidean_relation
+    then have \<open>\<not> b dvd a\<close>
+      by (simp add: dvd_add_left_iff)
+    have \<open>a mod b + (b * c + b * (a div b)) = b * c + ((a div b) * b + a mod b)\<close>
+      by (simp add: ac_simps)
+    with \<open>b \<noteq> 0\<close> have *: \<open>a mod b + (b * c + b * (a div b)) = b * c + a\<close>
+      by (simp add: div_mult_mod_eq)
+    from \<open>\<not> b dvd a\<close> euclidean_relation show ?case
+      by (simp_all add: algebra_simps division_segment_mod mod_size_less *)
+  qed
+  then show \<open>(a + c * b) div b = c + a div b\<close>
+    by simp
+next
+  fix a b c
+  assume \<open>c \<noteq> 0\<close>
+  have \<open>((c * a) div (c * b), (c * a) mod (c * b)) = (a div b, c * (a mod b))\<close>
+  proof (induction rule: euclidean_relationI)
+    case by0
+    with \<open>c \<noteq> 0\<close> show ?case
+      by simp
+  next
+    case divides
+    then show ?case
+      by (auto simp add: algebra_simps)
+  next
+    case euclidean_relation
+    then have \<open>b \<noteq> 0\<close> \<open>a mod b \<noteq> 0\<close>
+      by (simp_all add: mod_eq_0_iff_dvd)
+    have \<open>c * (a mod b) + b * (c * (a div b)) = c * ((a div b) * b + a mod b)\<close>
+      by (simp add: algebra_simps)
+    with \<open>b \<noteq> 0\<close> have *: \<open>c * (a mod b) + b * (c * (a div b)) = c * a\<close>
+      by (simp add: div_mult_mod_eq)
+    from \<open>b \<noteq> 0\<close> \<open>c \<noteq> 0\<close> have \<open>euclidean_size c * euclidean_size (a mod b)
+      < euclidean_size c * euclidean_size b\<close>
+      using mod_size_less [of b a] by simp
+    with euclidean_relation \<open>b \<noteq> 0\<close> \<open>a mod b \<noteq> 0\<close> show ?case
+      by (simp add: algebra_simps division_segment_mult division_segment_mod euclidean_size_mult *)
+  qed
+  then show \<open>(c * a) div (c * b) = a div b\<close>
+    by simp
+qed
+
+lemma div_eq_0_iff:
+  \<open>a div b = 0 \<longleftrightarrow> euclidean_size a < euclidean_size b \<or> b = 0\<close> (is "_ \<longleftrightarrow> ?P")
+  if \<open>division_segment a = division_segment b\<close>
+proof (cases \<open>a = 0 \<or> b = 0\<close>)
+  case True
+  then show ?thesis by auto
+next
+  case False
+  then have \<open>a \<noteq> 0\<close> \<open>b \<noteq> 0\<close>
+    by simp_all
+  have \<open>a div b = 0 \<longleftrightarrow> euclidean_size a < euclidean_size b\<close>
+  proof
+    assume \<open>a div b = 0\<close>
+    then have \<open>a mod b = a\<close>
+      using div_mult_mod_eq [of a b] by simp
+    with \<open>b \<noteq> 0\<close> mod_size_less [of b a]
+    show \<open>euclidean_size a < euclidean_size b\<close>
+      by simp
+  next
+    assume \<open>euclidean_size a < euclidean_size b\<close>
+    have \<open>(a div b, a mod b) = (0, a)\<close>
+    proof (induction rule: euclidean_relationI)
+      case by0
+      show ?case
+        by simp
+    next
+      case divides
+      with \<open>euclidean_size a < euclidean_size b\<close> show ?case
+        using dvd_imp_size_le [of b a] \<open>a \<noteq> 0\<close> by simp
+    next
+      case euclidean_relation
+      with \<open>euclidean_size a < euclidean_size b\<close> that
+      show ?case
+        by simp
+    qed
+    then show \<open>a div b = 0\<close>
+      by simp
+  qed
+  with \<open>b \<noteq> 0\<close> show ?thesis
+    by simp
+qed
+
+lemma div_mult1_eq:
+  \<open>(a * b) div c = a * (b div c) + a * (b mod c) div c\<close>
+proof -
+  have *: \<open>(a * b) mod c + (a * (c * (b div c)) + c * (a * (b mod c) div c)) = a * b\<close> (is \<open>?A + (?B + ?C) = _\<close>)
+  proof -
+    have \<open>?A = a * (b mod c) mod c\<close>
+      by (simp add: mod_mult_right_eq)
+    then have \<open>?C + ?A = a * (b mod c)\<close>
+      by (simp add: mult_div_mod_eq)
+    then have \<open>?B + (?C + ?A) = a * (c * (b div c) + (b mod c))\<close>
+      by (simp add: algebra_simps)
+    also have \<open>\<dots> = a * b\<close>
+      by (simp add: mult_div_mod_eq)
+    finally show ?thesis
+      by (simp add: algebra_simps)
+  qed
+  have \<open>((a * b) div c, (a * b) mod c) = (a * (b div c) + a * (b mod c) div c, (a * b) mod c)\<close>
+  proof (induction rule: euclidean_relationI)
+    case by0
+    then show ?case by simp
+  next
+    case divides
+    with * show ?case
+      by (simp add: algebra_simps)
+  next
+    case euclidean_relation
+    with * show ?case
+      by (simp add: division_segment_mod mod_size_less algebra_simps)
+  qed
+  then show ?thesis
+    by simp
+qed
+
+lemma div_add1_eq:
+  \<open>(a + b) div c = a div c + b div c + (a mod c + b mod c) div c\<close>
+proof -
+  have *: \<open>(a + b) mod c + (c * (a div c) + (c * (b div c) + c * ((a mod c + b mod c) div c))) = a + b\<close>
+    (is \<open>?A + (?B + (?C + ?D)) = _\<close>)
+  proof -
+    have \<open>?A + (?B + (?C + ?D)) = ?A + ?D + (?B + ?C)\<close>
+      by (simp add: ac_simps)
+    also have \<open>?A + ?D = (a mod c + b mod c) mod c + ?D\<close>
+      by (simp add: mod_add_eq)
+    also have \<open>\<dots> = a mod c + b mod c\<close>
+      by (simp add: mod_mult_div_eq)
+    finally have \<open>?A + (?B + (?C + ?D)) = (a mod c + ?B) + (b mod c + ?C)\<close>
+      by (simp add: ac_simps)
+    then show ?thesis
+      by (simp add: mod_mult_div_eq)
+  qed
+  have \<open>((a + b) div c, (a + b) mod c) = (a div c + b div c + (a mod c + b mod c) div c, (a + b) mod c)\<close>
+  proof (induction rule: euclidean_relationI)
+    case by0
+    then show ?case
+      by simp
+  next
+    case divides
+    with * show ?case
+      by (simp add: algebra_simps)
+  next
+    case euclidean_relation
+    with * show ?case
+      by (simp add: division_segment_mod mod_size_less algebra_simps)
+  qed
+  then show ?thesis
+    by simp
+qed
+
+end
+
+class unique_euclidean_ring = euclidean_ring + unique_euclidean_semiring
+begin
+
+subclass euclidean_ring_cancel ..
+
+end
+
+
+subsection \<open>Division on \<^typ>\<open>nat\<close>\<close>
+
+instantiation nat :: normalization_semidom
+begin
+
+definition normalize_nat :: \<open>nat \<Rightarrow> nat\<close>
+  where [simp]: \<open>normalize = (id :: nat \<Rightarrow> nat)\<close>
+
+definition unit_factor_nat :: \<open>nat \<Rightarrow> nat\<close>
+  where \<open>unit_factor n = of_bool (n > 0)\<close> for n :: nat
+
+lemma unit_factor_simps [simp]:
+  \<open>unit_factor 0 = (0::nat)\<close>
+  \<open>unit_factor (Suc n) = 1\<close>
+  by (simp_all add: unit_factor_nat_def)
+
+definition divide_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close>
+  where \<open>m div n = (if n = 0 then 0 else Max {k. k * n \<le> m})\<close> for m n :: nat
+
+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 :: \<open>nat \<Rightarrow> nat\<close>
+  where [simp]: \<open>euclidean_size_nat = id\<close>
+
+definition division_segment_nat :: \<open>nat \<Rightarrow> nat\<close>
+  where [simp]: \<open>division_segment n = 1\<close> for n :: nat
+
+definition modulo_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close>
+  where \<open>m mod n = m - (m div n * n)\<close> for m 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
+
+lemma euclidean_relation_natI [case_names by0 divides euclidean_relation]:
+  \<open>(m div n, m mod n) = (q, r)\<close>
+    if by0: \<open>n = 0 \<Longrightarrow> q = 0 \<and> r = m\<close>
+    and divides: \<open>n > 0 \<Longrightarrow> n dvd m \<Longrightarrow> r = 0 \<and> m = q * n\<close>
+    and euclidean_relation: \<open>n > 0 \<Longrightarrow> \<not> n dvd m \<Longrightarrow> r < n \<and> m = q * n + r\<close> for m n q r :: nat
+  by (rule euclidean_relationI) (use that in simp_all)
+
+lemma div_nat_eqI:
+  \<open>m div n = q\<close> if \<open>n * q \<le> m\<close> and \<open>m < n * Suc q\<close> for m n q :: nat
+proof -
+  have \<open>(m div n, m mod n) = (q, m - n * q)\<close>
+  proof (induction rule: euclidean_relation_natI)
+    case by0
+    with that show ?case
+      by simp
+  next
+    case divides
+    from \<open>n dvd m\<close> obtain s where \<open>m = n * s\<close> ..
+    with \<open>n > 0\<close> that have \<open>s < Suc q\<close>
+      by (simp only: mult_less_cancel1)
+    with \<open>m = n * s\<close> \<open>n > 0\<close> that have \<open>q = s\<close>
+      by simp
+    with \<open>m = n * s\<close> show ?case
+      by (simp add: ac_simps)
+  next
+    case euclidean_relation
+    with that show ?case
+      by (simp add: ac_simps)
+  qed
+  then show ?thesis
+    by simp
+qed
+
+lemma mod_nat_eqI:
+  \<open>m mod n = r\<close> if \<open>r < n\<close> and \<open>r \<le> m\<close> and \<open>n dvd m - r\<close> for m n r :: nat
+proof -
+  have \<open>(m div n, m mod n) = ((m - r) div n, r)\<close>
+  proof (induction rule: euclidean_relation_natI)
+    case by0
+    with that show ?case
+      by simp
+  next
+    case divides
+    from that dvd_minus_add [of r \<open>m\<close> 1 n]
+    have \<open>n dvd m + (n - r)\<close>
+      by simp
+    with divides have \<open>n dvd n - r\<close>
+      by (simp add: dvd_add_right_iff)
+    then have \<open>n \<le> n - r\<close>
+      by (rule dvd_imp_le) (use \<open>r < n\<close> in simp)
+    with \<open>n > 0\<close> have \<open>r = 0\<close>
+      by simp
+    with \<open>n > 0\<close> that show ?case
+      by simp
+  next
+    case euclidean_relation
+    with that show ?case
+      by (simp add: ac_simps)
+  qed
+  then show ?thesis
+    by simp
+qed
+
+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_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_nat_eqI mod_nat_eqI)
+
+lemma split_div:
+  \<open>P (m div n) \<longleftrightarrow>
+    (n = 0 \<longrightarrow> P 0) \<and>
+    (n \<noteq> 0 \<longrightarrow> (\<forall>i j. j < n \<and> m = n * i + j \<longrightarrow> P i))\<close> (is ?div)
+  and split_mod:
+  \<open>Q (m mod n) \<longleftrightarrow>
+    (n = 0 \<longrightarrow> Q m) \<and>
+    (n \<noteq> 0 \<longrightarrow> (\<forall>i j. j < n \<and> m = n * i + j \<longrightarrow> Q j))\<close> (is ?mod)
+  for m n :: nat
+proof -
+  have *: \<open>R (m div n) (m mod n) \<longleftrightarrow>
+    (n = 0 \<longrightarrow> R 0 m) \<and>
+    (n \<noteq> 0 \<longrightarrow> (\<forall>i j. j < n \<and> m = n * i + j \<longrightarrow> R i j))\<close> for R
+    by (cases \<open>n = 0\<close>) auto
+  from * [of \<open>\<lambda>q _. P q\<close>] show ?div .
+  from * [of \<open>\<lambda>_ r. Q r\<close>] show ?mod .
+qed
+
+declare split_div [of _ _ \<open>numeral n\<close>, linarith_split] for n
+declare split_mod [of _ _ \<open>numeral n\<close>, linarith_split] for n
+
+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 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
+
+lemma div_mult2_eq:
+    \<open>m div (n * q) = (m div n) div q\<close> (is ?Q)
+  and mod_mult2_eq:
+    \<open>m mod (n * q) = n * (m div n mod q) + m mod n\<close> (is ?R)
+  for m n q :: nat
+proof -
+  have \<open>(m div (n * q), m mod (n * q)) = ((m div n) div q, n * (m div n mod q) + m mod n)\<close>
+  proof (induction rule: euclidean_relation_natI)
+    case by0
+    then show ?case
+      by auto
+  next
+    case divides
+    from \<open>n * q dvd m\<close> obtain t where \<open>m = n * q * t\<close> ..
+    with \<open>n * q > 0\<close> show ?case
+      by (simp add: algebra_simps)
+  next
+    case euclidean_relation
+    then have \<open>n > 0\<close> \<open>q > 0\<close>
+      by simp_all
+    from \<open>n > 0\<close> have \<open>m mod n < n\<close>
+      by (rule mod_less_divisor)
+    from \<open>q > 0\<close> have \<open>m div n mod q < q\<close>
+      by (rule mod_less_divisor)
+    then obtain s where \<open>q = Suc (m div n mod q + s)\<close>
+      by (blast dest: less_imp_Suc_add)
+    moreover have \<open>m mod n + n * (m div n mod q) < n * Suc (m div n mod q + s)\<close>
+      using \<open>m mod n < n\<close> by (simp add: add_mult_distrib2)
+    ultimately have \<open>m mod n + n * (m div n mod q) < n * q\<close>
+      by simp
+    then show ?case
+      by (simp add: algebra_simps flip: add_mult_distrib2)
+  qed
+  then show ?Q and ?R
+    by simp_all
+qed
+
+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
+
+lemma div_less_iff_less_mult:
+  \<open>m div q < n \<longleftrightarrow> m < n * q\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
+  if \<open>q > 0\<close> for m n q :: nat
+proof
+  assume ?Q then show ?P
+    by (rule less_mult_imp_div_less)
+next
+  assume ?P
+  then obtain h where \<open>n = Suc (m div q + h)\<close>
+    using less_natE by blast
+  moreover have \<open>m < m + (Suc h * q - m mod q)\<close>
+    using that by (simp add: trans_less_add1)
+  ultimately show ?Q
+    by (simp add: algebra_simps flip: minus_mod_eq_mult_div)
+qed
+
+lemma less_eq_div_iff_mult_less_eq:
+  \<open>m \<le> n div q \<longleftrightarrow> m * q \<le> n\<close> if \<open>q > 0\<close> for m n q :: nat
+  using div_less_iff_less_mult [of q n m] that by auto
+
+lemma div_Suc:
+  \<open>Suc m div n = (if Suc m mod n = 0 then Suc (m div n) else m div n)\<close>
+proof (cases \<open>n = 0 \<or> n = 1\<close>)
+  case True
+  then show ?thesis by auto
+next
+  case False
+  then have \<open>n > 1\<close>
+    by simp
+  then have \<open>Suc m div n = m div n + Suc (m mod n) div n\<close>
+    using div_add1_eq [of m 1 n] by simp
+  also have \<open>Suc (m mod n) div n = of_bool (n dvd Suc m)\<close>
+  proof (cases \<open>n dvd Suc m\<close>)
+    case False
+    moreover have \<open>Suc (m mod n) \<noteq> n\<close>
+    proof (rule ccontr)
+      assume \<open>\<not> Suc (m mod n) \<noteq> n\<close>
+      then have \<open>m mod n = n - Suc 0\<close>
+        by simp
+      with \<open>n > 1\<close> have \<open>(m + 1) mod n = 0\<close>
+        by (subst mod_add_left_eq [symmetric]) simp
+      then have \<open>n dvd Suc m\<close>
+        by auto
+      with False show False ..
+    qed
+    moreover have \<open>Suc (m mod n) \<le> n\<close>
+      using \<open>n > 1\<close> by (simp add: Suc_le_eq)
+    ultimately show ?thesis
+      by (simp add: div_eq_0_iff)
+  next
+    case True
+    then obtain q where q: \<open>Suc m = n * q\<close> ..
+    moreover have \<open>q > 0\<close> by (rule ccontr)
+      (use q in simp)
+    ultimately have \<open>m mod n = n - Suc 0\<close>
+      using \<open>n > 1\<close> mult_le_cancel1 [of n \<open>Suc 0\<close> q]
+      by (auto intro: mod_nat_eqI)
+    with True \<open>n > 1\<close> show ?thesis
+      by simp
+  qed
+  finally show ?thesis
+    by (simp add: mod_greater_zero_iff_not_dvd)
+qed
+
+lemma mod_Suc:
+  \<open>Suc m mod n = (if Suc (m mod n) = n then 0 else Suc (m mod n))\<close>
+proof (cases \<open>n = 0\<close>)
+  case True
+  then show ?thesis
+    by simp
+next
+  case False
+  moreover have \<open>Suc m mod n = Suc (m mod n) mod n\<close>
+    by (simp add: mod_simps)
+  ultimately show ?thesis
+    by (auto intro!: mod_nat_eqI intro: neq_le_trans simp add: Suc_le_eq)
+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 funpow_mod_eq: \<^marker>\<open>contributor \<open>Lars Noschinski\<close>\<close>
+  \<open>(f ^^ (m mod n)) x = (f ^^ m) x\<close> if \<open>(f ^^ n) x = x\<close>
+proof -
+  have \<open>(f ^^ m) x = (f ^^ (m mod n + m div n * n)) x\<close>
+    by simp
+  also have \<open>\<dots> = (f ^^ (m mod n)) (((f ^^ n) ^^ (m div n)) x)\<close>
+    by (simp only: funpow_add funpow_mult ac_simps) simp
+  also have \<open>((f ^^ n) ^^ q) x = x\<close> for q
+    by (induction q) (use \<open>(f ^^ n) x = x\<close> in simp_all)
+  finally show ?thesis
+    by simp
+qed
+
+lemma mod_eq_dvd_iff_nat:
+  \<open>m mod q = n mod q \<longleftrightarrow> q dvd m - n\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
+    if \<open>m \<ge> n\<close> for m n q :: nat
+proof
+  assume ?Q
+  then obtain s where \<open>m - n = q * s\<close> ..
+  with that have \<open>m = q * s + n\<close>
+    by simp
+  then show ?P
+    by simp
+next
+  assume ?P
+  have \<open>m - n = m div q * q + m mod q - (n div q * q + n mod q)\<close>
+    by simp
+  also have \<open>\<dots> = q * (m div q - n div q)\<close>
+    by (simp only: algebra_simps \<open>?P\<close>)
+  finally show ?Q ..
+qed
+
+lemma mod_eq_iff_dvd_symdiff_nat:
+  \<open>m mod q = n mod q \<longleftrightarrow> q dvd nat \<bar>int m - int n\<bar>\<close>
+  by (auto simp add: abs_if mod_eq_dvd_iff_nat nat_diff_distrib dest: sym intro: sym)
+
+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_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 mod_eq_nat1E [OF th xy] obtain q where "y = x + n * q" .
+    then have "x + n * q = y + n * 0"
+      by simp
+    then have "\<exists>q1 q2. x + n * q1 = y + n * q2"
+      by blast
+  }
+  moreover
+  { assume xy: "y \<le> x"
+    from mod_eq_nat1E [OF H xy] obtain q where "x = y + n * q" .
+    then have "x + n * 0 = y + n * q"
+      by simp
+    then have "\<exists>q1 q2. x + n * q1 = y + n * q2"
+      by blast
+  }
+  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
+
+
+
+subsection \<open>Division on \<^typ>\<open>int\<close>\<close>
+
+subsubsection \<open>Basic instantiation\<close>
+
+instantiation int :: "{normalization_semidom, idom_modulo}"
+begin
+
+definition normalize_int :: \<open>int \<Rightarrow> int\<close>
+  where [simp]: \<open>normalize = (abs :: int \<Rightarrow> int)\<close>
+
+definition unit_factor_int :: \<open>int \<Rightarrow> int\<close>
+  where [simp]: \<open>unit_factor = (sgn :: int \<Rightarrow> int)\<close>
+
+definition divide_int :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close>
+  where \<open>k div l = (sgn k * sgn l * int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
+    - of_bool (l \<noteq> 0 \<and> sgn k \<noteq> sgn l \<and> \<not> l dvd k))\<close>
+
+lemma divide_int_unfold:
+  \<open>(sgn k * int m) div (sgn l * int n) = (sgn k * sgn l * int (m div n)
+    - of_bool ((k = 0 \<longleftrightarrow> m = 0) \<and> l \<noteq> 0 \<and> n \<noteq> 0 \<and> sgn k \<noteq> sgn l \<and> \<not> n dvd m))\<close>
+  by (simp add: divide_int_def sgn_mult nat_mult_distrib abs_mult sgn_eq_0_iff ac_simps)
+
+definition modulo_int :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close>
+  where \<open>k mod l = sgn k * int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>) + l * of_bool (sgn k \<noteq> sgn l \<and> \<not> l dvd k)\<close>
+
+lemma modulo_int_unfold:
+  \<open>(sgn k * int m) mod (sgn l * int n) =
+    sgn k * int (m mod (of_bool (l \<noteq> 0) * n)) + (sgn l * int n) * of_bool ((k = 0 \<longleftrightarrow> m = 0) \<and> sgn k \<noteq> sgn l \<and> \<not> n dvd m)\<close>
+  by (auto simp add: modulo_int_def sgn_mult abs_mult)
+
+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)
+next
+  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 (simp add: divide_int_unfold modulo_int_unfold algebra_simps modulo_nat_def of_nat_diff)
+qed (auto simp add: sgn_mult mult_sgn_abs abs_eq_iff')
+
+end
+
+
+subsubsection \<open>Algebraic foundations\<close>
+
+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
+
+
+subsubsection \<open>Basic conversions\<close>
+
+lemma div_abs_eq_div_nat:
+  "\<bar>k\<bar> div \<bar>l\<bar> = int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)"
+  by (auto simp add: divide_int_def)
+
+lemma div_eq_div_abs:
+  \<open>k div l = sgn k * sgn l * (\<bar>k\<bar> div \<bar>l\<bar>)
+    - of_bool (l \<noteq> 0 \<and> sgn k \<noteq> sgn l \<and> \<not> l dvd k)\<close>
+  for k l :: int
+  by (simp add: divide_int_def [of k l] div_abs_eq_div_nat)
+
+lemma div_abs_eq:
+  \<open>\<bar>k\<bar> div \<bar>l\<bar> = sgn k * sgn l * (k div l + of_bool (sgn k \<noteq> sgn l \<and> \<not> l dvd k))\<close>
+  for k l :: int
+  by (simp add: div_eq_div_abs [of k l] ac_simps)
+
+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 mod_eq_mod_abs:
+  \<open>k mod l = sgn k * (\<bar>k\<bar> mod \<bar>l\<bar>) + l * of_bool (sgn k \<noteq> sgn l \<and> \<not> l dvd k)\<close>
+  for k l :: int
+  by (simp add: modulo_int_def [of k l] mod_abs_eq_div_nat)
+
+lemma mod_abs_eq:
+  \<open>\<bar>k\<bar> mod \<bar>l\<bar> = sgn k * (k mod l - l * of_bool (sgn k \<noteq> sgn l \<and> \<not> l dvd k))\<close>
+  for k l :: int
+  by (auto simp: mod_eq_mod_abs [of k l])
+
+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>"
+  using assms by (simp add: sgn_mult abs_mult sgn_0_0
+    divide_int_def [of "sgn v * \<bar>k\<bar>" "sgn v * \<bar>l\<bar>"] flip: div_abs_eq_div_nat)
+
+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>"
+  using assms by (auto simp add: div_abs_eq)
+
+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>)"
+  using assms by (auto simp add: div_abs_eq ac_simps)
+
+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 (auto simp add: div_abs_eq ac_simps sgn_0_0 dest!: sgn_not_eq_imp)
+
+
+subsubsection \<open>Euclidean division\<close>
+
+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 (auto simp add: modulo_int_unfold abs_mult mod_greater_zero_iff_not_dvd
+        simp flip: right_diff_distrib dest!: sgn_not_eq_imp)
+      (simp add: sgn_0_0)
+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 (auto simp add: modulo_int_unfold sgn_mult mod_greater_zero_iff_not_dvd
+      simp flip: right_diff_distrib dest!: sgn_not_eq_imp)
+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 ac_simps)
+    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 euclidean_relation_intI [case_names by0 divides euclidean_relation]:
+  \<open>(k div l, k mod l) = (q, r)\<close>
+    if by0': \<open>l = 0 \<Longrightarrow> q = 0 \<and> r = k\<close>
+    and divides': \<open>l \<noteq> 0 \<Longrightarrow> l dvd k \<Longrightarrow> r = 0 \<and> k = q * l\<close>
+    and euclidean_relation': \<open>l \<noteq> 0 \<Longrightarrow> \<not> l dvd k \<Longrightarrow> sgn r = sgn l
+      \<and> \<bar>r\<bar> < \<bar>l\<bar> \<and> k = q * l + r\<close> for k l :: int
+proof (induction rule: euclidean_relationI)
+  case by0
+  then show ?case
+    by (rule by0')
+next
+  case divides
+  then show ?case
+    by (rule divides')
+next
+  case euclidean_relation
+  with euclidean_relation' have \<open>sgn r = sgn l\<close> \<open>\<bar>r\<bar> < \<bar>l\<bar>\<close> \<open>k = q * l + r\<close>
+    by simp_all
+  from \<open>sgn r = sgn l\<close> \<open>l \<noteq> 0\<close> have \<open>division_segment r = division_segment l\<close>
+    by (simp add: division_segment_int_def sgn_if split: if_splits)
+  with \<open>\<bar>r\<bar> < \<bar>l\<bar>\<close> \<open>k = q * l + r\<close>
+  show ?case
+    by simp
+qed
+
+
+subsubsection \<open>Trivial reduction steps\<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: \<open>k div l = - 1\<close>  (is ?Q)
+  and mod_pos_neg_trivial: \<open>k mod l = k + l\<close>  (is ?R)
+    if \<open>0 < k\<close> and \<open>k + l \<le> 0\<close> for k l :: int
+proof -
+  from that have \<open>l < 0\<close>
+    by simp
+  have \<open>(k div l, k mod l) = (- 1, k + l)\<close>
+  proof (induction rule: euclidean_relation_intI)
+    case by0
+    with \<open>l < 0\<close> show ?case
+      by simp
+  next
+    case divides
+    from \<open>l dvd k\<close> obtain j where \<open>k = l * j\<close> ..
+    with \<open>l < 0\<close> \<open>0 < k\<close> have \<open>j < 0\<close>
+      by (simp add: zero_less_mult_iff)
+    moreover from \<open>k + l \<le> 0\<close> \<open>k = l * j\<close> have \<open>l * (j + 1) \<le> 0\<close>
+      by (simp add: algebra_simps)
+    with \<open>l < 0\<close> have \<open>j + 1 \<ge> 0\<close>
+      by (simp add: mult_le_0_iff)
+    with \<open>j < 0\<close> have \<open>j = - 1\<close>
+      by simp
+    with \<open>k = l * j\<close> show ?case
+      by simp
+  next
+    case euclidean_relation
+    with \<open>k + l \<le> 0\<close> have \<open>k + l < 0\<close>
+      by (auto simp add: less_le add_eq_0_iff)
+    with \<open>0 < k\<close> show ?case
+      by simp
+  qed
+  then show ?Q and ?R
+    by simp_all
+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>
+
+
+subsubsection \<open>More uniqueness rules\<close>
+
+lemma
+  fixes a b q r :: int
+  assumes \<open>a = b * q + r\<close> \<open>0 \<le> r\<close> \<open>r < b\<close>
+  shows int_div_pos_eq:
+      \<open>a div b = q\<close> (is ?Q)
+    and int_mod_pos_eq:
+      \<open>a mod b = r\<close> (is ?R)
+proof -
+  have \<open>(a div b, a mod b) = (q, r)\<close>
+    by (induction rule: euclidean_relation_intI)
+      (use assms in \<open>auto simp add: ac_simps dvd_add_left_iff sgn_1_pos le_less dest: zdvd_imp_le\<close>)
+  then show ?Q and ?R
+    by simp_all
+qed
+
+lemma int_div_neg_eq:
+  \<open>a div b = q\<close> if \<open>a = b * q + r\<close> \<open>r \<le> 0\<close> \<open>b < r\<close> for a b q r :: int
+  using that int_div_pos_eq [of a \<open>- b\<close> \<open>- q\<close> \<open>- r\<close>] by simp_all
+
+lemma int_mod_neg_eq:
+  \<open>a mod b = r\<close> if \<open>a = b * q + r\<close> \<open>r \<le> 0\<close> \<open>b < r\<close> for a b q r :: int
+  using that int_div_neg_eq [of a b q r] by simp
+
+
+subsubsection \<open>Laws for unary minus\<close>
+
+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_zminus1_eq_if:
+  \<open>(- a) div b = (if a mod b = 0 then - (a div b) else - (a div b) - 1)\<close>
+  if \<open>b \<noteq> 0\<close> for a b :: int
+  using that sgn_not_eq_imp [of b \<open>- a\<close>]
+  by (cases \<open>a = 0\<close>) (auto simp add: div_eq_div_abs [of \<open>- a\<close> b] div_eq_div_abs [of a b] sgn_eq_0_iff)
+
+lemma zdiv_zminus2_eq_if:
+  \<open>a div (- b) = (if a mod b = 0 then - (a div b) else - (a div b) - 1)\<close>
+  if \<open>b \<noteq> 0\<close> for a b :: int
+  using that by (auto simp add: zdiv_zminus1_eq_if div_minus_right)
+
+lemma zmod_zminus1_eq_if:
+  \<open>(- a) mod b = (if a mod b = 0 then 0 else b - (a mod b))\<close>
+  for a b :: int
+  by (cases \<open>b = 0\<close>)
+    (auto simp flip: minus_div_mult_eq_mod simp add: zdiv_zminus1_eq_if algebra_simps)
+
+lemma zmod_zminus2_eq_if:
+  \<open>a mod (- b) = (if a mod b = 0 then 0 else (a mod b) - b)\<close>
+  for a b :: int
+  by (auto simp add: zmod_zminus1_eq_if mod_minus_right)
+
+
+subsubsection \<open>Borders\<close>
+
+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)
+      (auto simp add: mod_greater_zero_iff_not_dvd sgn_1_pos)
+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)
+      (auto simp add: mod_greater_zero_iff_not_dvd sgn_1_neg)
+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) (auto simp add: sgn_1_pos)
+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
+  moreover have \<open>int (m mod n) \<le> int n\<close>
+    using \<open>Suc q = n\<close> by simp
+  then have \<open>sgn s * int (m mod n) \<le> int n\<close>
+    by (cases s \<open>0::int\<close> rule: linorder_cases) simp_all
+  ultimately show ?thesis
+    by (simp only: modulo_int_unfold) auto
+qed
+
+
+subsubsection \<open>Splitting Rules for div and mod\<close>
+
+lemma split_zdiv:
+  \<open>P (n div k) \<longleftrightarrow>
+    (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))\<close> (is ?div)
+  and split_zmod:
+  \<open>Q (n mod k) \<longleftrightarrow>
+    (k = 0 \<longrightarrow> Q n) \<and>
+    (0 < k \<longrightarrow> (\<forall>i j. 0 \<le> j \<and> j < k \<and> n = k * i + j \<longrightarrow> Q j)) \<and>
+    (k < 0 \<longrightarrow> (\<forall>i j. k < j \<and> j \<le> 0 \<and> n = k * i + j \<longrightarrow> Q j))\<close> (is ?mod)
+  for n k :: int
+proof -
+  have *: \<open>R (n div k) (n mod k) \<longleftrightarrow>
+    (k = 0 \<longrightarrow> R 0 n) \<and>
+    (0 < k \<longrightarrow> (\<forall>i j. 0 \<le> j \<and> j < k \<and> n = k * i + j \<longrightarrow> R i j)) \<and>
+    (k < 0 \<longrightarrow> (\<forall>i j. k < j \<and> j \<le> 0 \<and> n = k * i + j \<longrightarrow> R i j))\<close> for R
+    by (cases \<open>k = 0\<close>)
+      (auto simp add: linorder_class.neq_iff)
+  from * [of \<open>\<lambda>q _. P q\<close>] show ?div .
+  from * [of \<open>\<lambda>_ r. Q r\<close>] show ?mod .
+qed
+
+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 _ _ \<open>numeral n\<close>, linarith_split] for n
+declare split_zdiv [of _ _ \<open>- numeral n\<close>, linarith_split] for n
+declare split_zmod [of _ _ \<open>numeral n\<close>, linarith_split] for n
+declare split_zmod [of _ _ \<open>- numeral n\<close>, linarith_split] for n
+
+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_trivial_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"
+    using div_mult_mod_eq [of i k] by safe auto
+  with zdiv_eq_0_iff
+  show ?thesis
+    by simp
+qed
+
+
+subsubsection \<open>Algebraic rewrites\<close>
+
+lemma zdiv_zmult2_eq: \<open>a div (b * c) = (a div b) div c\<close>  (is ?Q)
+  and zmod_zmult2_eq: \<open>a mod (b * c) = b * (a div b mod c) + a mod b\<close>  (is ?P)
+  if \<open>c \<ge> 0\<close> for a b c :: int
+proof -
+  have *: \<open>(a div (b * c), a mod (b * c)) = ((a div b) div c, b * (a div b mod c) + a mod b)\<close>
+    if \<open>b > 0\<close> for a b
+  proof (induction rule: euclidean_relationI)
+    case by0
+    then show ?case by auto
+  next
+    case divides
+    then obtain d where \<open>a = b * c * d\<close>
+      by blast
+    with divides that show ?case
+      by (simp add: ac_simps)
+  next
+    case euclidean_relation
+    with \<open>b > 0\<close> \<open>c \<ge> 0\<close> have \<open>0 < c\<close> \<open>b > 0\<close>
+      by simp_all
+    then have \<open>a mod b < b\<close>
+      by simp
+    moreover have \<open>1 \<le> c - a div b mod c\<close>
+      using \<open>c > 0\<close> by (simp add: int_one_le_iff_zero_less)
+    ultimately have \<open>a mod b * 1 < b * (c - a div b mod c)\<close>
+      by (rule mult_less_le_imp_less) (use \<open>b > 0\<close> in simp_all)
+    with \<open>0 < b\<close> \<open>0 < c\<close> show ?case
+      by (simp add: division_segment_int_def algebra_simps flip: minus_mod_eq_mult_div)
+  qed
+  show ?Q
+  proof (cases \<open>b \<ge> 0\<close>)
+    case True
+    with * [of b a] show ?thesis
+      by (cases \<open>b = 0\<close>) simp_all
+  next
+    case False
+    with * [of \<open>- b\<close> \<open>- a\<close>] show ?thesis
+      by simp
+  qed
+  show ?P
+  proof (cases \<open>b \<ge> 0\<close>)
+    case True
+    with * [of b a] show ?thesis
+      by (cases \<open>b = 0\<close>) simp_all
+  next
+    case False
+    with * [of \<open>- b\<close> \<open>- a\<close>] show ?thesis
+      by simp
+  qed
+qed
+
+lemma zdiv_zmult2_eq':
+  \<open>k div (l * j) = ((sgn j * k) div l) div \<bar>j\<bar>\<close> for k l j :: int
+proof -
+  have \<open>k div (l * j) = (sgn j * k) div (sgn j * (l * j))\<close>
+    by (simp add: sgn_0_0)
+  also have \<open>sgn j * (l * j) = l * \<bar>j\<bar>\<close>
+    by (simp add: mult.left_commute [of _ l] abs_sgn) (simp add: ac_simps)
+  also have \<open>(sgn j * k) div (l * \<bar>j\<bar>) = ((sgn j * k) div l) div \<bar>j\<bar>\<close>
+    by (simp add: zdiv_zmult2_eq)
+  finally show ?thesis .
+qed
+
+lemma half_nonnegative_int_iff [simp]:
+  \<open>k div 2 \<ge> 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int
+  by auto
+
+lemma half_negative_int_iff [simp]:
+  \<open>k div 2 < 0 \<longleftrightarrow> k < 0\<close> for k :: int
+  by auto
+
+
+subsubsection \<open>Distributive laws for conversions.\<close>
+
+lemma zdiv_int:
+  \<open>int (m div n) = int m div int n\<close>
+  by (cases \<open>m = 0\<close>) (auto simp add: divide_int_def)
+
+lemma zmod_int:
+  \<open>int (m mod n) = int m mod int n\<close>
+  by (cases \<open>m = 0\<close>) (auto simp add: modulo_int_def)
+
+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)
+
+
+subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
+
+lemma zdiv_mono1:
+  \<open>a div b \<le> a' div b\<close>
+    if \<open>a \<le> a'\<close> \<open>0 < b\<close>
+    for a b b' :: int
+proof -
+  from \<open>a \<le> a'\<close> have \<open>b * (a div b) + a mod b \<le> b * (a' div b) + a' mod b\<close>
+    by simp
+  then have \<open>b * (a div b) \<le> (a' mod b - a mod b) + b * (a' div b)\<close>
+    by (simp add: algebra_simps)
+  moreover have \<open>a' mod b < b + a mod b\<close>
+    by (rule less_le_trans [of _ b]) (use \<open>0 < b\<close> in simp_all)
+  ultimately have \<open>b * (a div b) < b * (1 + a' div b)\<close>
+    by (simp add: distrib_left)
+  with \<open>0 < b\<close> have \<open>a div b < 1 + a' div b\<close>
+    by (simp add: mult_less_cancel_left)
+  then show ?thesis
+    by simp
+qed
+
+lemma zdiv_mono1_neg:
+  \<open>a' div b \<le> a div b\<close>
+    if \<open>a \<le> a'\<close> \<open>b < 0\<close>
+    for a a' b :: int
+  using that zdiv_mono1 [of \<open>- a'\<close> \<open>- a\<close> \<open>- b\<close>] by simp
+
+
+subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
+
+lemma zdiv_mono2:
+  \<open>a div b \<le> a div b'\<close> if \<open>0 \<le> a\<close> \<open>0 < b'\<close> \<open>b' \<le> b\<close> for a b b' :: int
+proof -
+  define q q' r r' where **: \<open>q = a div b\<close> \<open>q' = a div b'\<close> \<open>r = a mod b\<close> \<open>r' = a mod b'\<close>
+  then have *: \<open>b * q + r = b' * q' + r'\<close> \<open>0 \<le> b' * q' + r'\<close>
+    \<open>r' < b'\<close> \<open>0 \<le> r\<close> \<open>0 < b'\<close> \<open>b' \<le> b\<close>
+    using that by simp_all
+  have \<open>0 < b' * (q' + 1)\<close>
+    using * by (simp add: distrib_left)
+  with * have \<open>0 \<le> q'\<close>
+    by (simp add: zero_less_mult_iff)
+  moreover have \<open>b * q = r' - r + b' * q'\<close>
+    using * by linarith
+  ultimately have \<open>b * q < b * (q' + 1)\<close>
+    using mult_right_mono * unfolding distrib_left by fastforce
+  with * have \<open>q \<le> q'\<close>
+    by (simp add: mult_less_cancel_left_pos)
+  with ** show ?thesis
+    by simp
+qed
+
+lemma zdiv_mono2_neg:
+  \<open>a div b' \<le> a div b\<close> if \<open>a < 0\<close> \<open>0 < b'\<close> \<open>b' \<le> b\<close> for a b b' :: int
+proof -
+  define q q' r r' where **: \<open>q = a div b\<close> \<open>q' = a div b'\<close> \<open>r = a mod b\<close> \<open>r' = a mod b'\<close>
+  then have *: \<open>b * q + r = b' * q' + r'\<close> \<open>b' * q' + r' < 0\<close>
+    \<open>r < b\<close> \<open>0 \<le> r'\<close> \<open>0 < b'\<close> \<open>b' \<le> b\<close>
+    using that by simp_all
+  have \<open>b' * q' < 0\<close>
+    using * by linarith
+  with * have \<open>q' \<le> 0\<close>
+    by (simp add: mult_less_0_iff)
+  have \<open>b * q' \<le> b' * q'\<close>
+    by (simp add: \<open>q' \<le> 0\<close> * mult_right_mono_neg)
+  then have "b * q' < b * (q + 1)"
+    using * by (simp add: distrib_left)
+  then have \<open>q' \<le> q\<close>
+    using * by (simp add: mult_less_cancel_left)
+  then show ?thesis
+    by (simp add: **)
+qed
+
+
+subsubsection \<open>Quotients of Signs\<close>
+
+lemma div_eq_minus1:
+  \<open>0 < b \<Longrightarrow> - 1 div b = - 1\<close> for b :: int
+  by (simp add: divide_int_def)
+
+lemma zmod_minus1:
+  \<open>0 < b \<Longrightarrow> - 1 mod b = b - 1\<close> 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 1: \<open>0 < k mod l\<close>
+      using \<open>0 < l\<close> False le_less by fastforce
+    moreover have 2: \<open>k mod l < 1 + l\<close>
+      using \<open>0 < l\<close> pos_mod_bound[of l k] by linarith
+    from 1 2 \<open>l > 0\<close> have \<open>(k mod l - 1) mod l = k mod l - 1\<close>
+      by (simp add: zmod_trivial_iff)
+    ultimately show ?thesis
+      by (simp only: zmod_zminus1_eq_if)
+         (simp add: mod_eq_0_iff_dvd algebra_simps mod_simps)
+  qed
+qed
+
+lemma div_neg_pos_less0:
+  \<open>a div b < 0\<close> if \<open>a < 0\<close> \<open>0 < b\<close> for a b :: int
+proof -
+  have "a div b \<le> - 1 div b"
+    using zdiv_mono1 that by auto
+  also have "... \<le> -1"
+    by (simp add: that(2) div_eq_minus1)
+  finally show ?thesis
+    by force
+qed
+
+lemma div_nonneg_neg_le0:
+  \<open>a div b \<le> 0\<close> if \<open>0 \<le> a\<close> \<open>b < 0\<close> for a b :: int
+  using that by (auto dest: zdiv_mono1_neg)
+
+lemma div_nonpos_pos_le0:
+  \<open>a div b \<le> 0\<close> if \<open>a \<le> 0\<close> \<open>0 < b\<close> for a b :: int
+  using that by (auto dest: zdiv_mono1)
+
+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:
+  \<open>0 \<le> a div b \<longleftrightarrow> 0 \<le> a\<close>
+  if \<open>0 < b\<close> for a b :: int
+proof
+  assume \<open>0 \<le> a div b\<close>
+  show \<open>0 \<le> a\<close>
+  proof (rule ccontr)
+    assume \<open>\<not> 0 \<le> a\<close>
+    then have \<open>a < 0\<close>
+      by simp
+    then have \<open>a div b < 0\<close>
+      using that by (rule div_neg_pos_less0)
+    with \<open>0 \<le> a div b\<close> show False
+      by simp
+  qed
+next
+  assume "0 \<le> a"
+  then have "0 div b \<le> a div b"
+    using zdiv_mono1 that by blast
+  then show "0 \<le> a div b"
+    by auto
+qed
+
+lemma neg_imp_zdiv_nonneg_iff:
+  \<open>0 \<le> a div b \<longleftrightarrow> a \<le> 0\<close> if \<open>b < 0\<close> for a b :: int
+  using that pos_imp_zdiv_nonneg_iff [of \<open>- b\<close> \<open>- a\<close>] by simp
+
+lemma pos_imp_zdiv_pos_iff:
+  \<open>0 < (i::int) div k \<longleftrightarrow> k \<le> i\<close> if \<open>0 < k\<close> for i k :: int
+  using that pos_imp_zdiv_nonneg_iff [of k i] zdiv_eq_0_iff [of i k] by arith
+
+lemma pos_imp_zdiv_neg_iff:
+  \<open>a div b < 0 \<longleftrightarrow> a < 0\<close> if \<open>0 < b\<close> for a b :: int
+    \<comment> \<open>But not \<^prop>\<open>a div b \<le> 0 \<longleftrightarrow> a \<le> 0\<close>; consider \<^prop>\<open>a = 1\<close>, \<^prop>\<open>b = 2\<close> when \<^prop>\<open>a div b = 0\<close>.\<close>
+  using that by (simp add: pos_imp_zdiv_nonneg_iff flip: linorder_not_le)
+
+lemma neg_imp_zdiv_neg_iff:
+    \<comment> \<open>But not \<^prop>\<open>a div b \<le> 0 \<longleftrightarrow> 0 \<le> a\<close>; consider \<^prop>\<open>a = - 1\<close>, \<^prop>\<open>b = - 2\<close> when \<^prop>\<open>a div b = 0\<close>.\<close>
+  \<open>a div b < 0 \<longleftrightarrow> 0 < a\<close> if \<open>b < 0\<close> for a b :: int
+  using that by (simp add: neg_imp_zdiv_nonneg_iff flip: linorder_not_le)
+
+lemma nonneg1_imp_zdiv_pos_iff:
+  \<open>a div b > 0 \<longleftrightarrow> a \<ge> b \<and> b > 0\<close> if \<open>0 \<le> a\<close> for a b :: int
+proof -
+  have "0 < a div b \<Longrightarrow> b \<le> a"
+    using div_pos_pos_trivial[of a b] that by arith
+  moreover have "0 < a div b \<Longrightarrow> b > 0"
+    using that 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:
+  \<open>m mod k \<le> m\<close> if \<open>(m::int) \<ge> 0\<close> for m k :: int
+proof -
+  from that have \<open>m > 0 \<or> m = 0\<close>
+    by auto
+  then show ?thesis proof
+    assume \<open>m = 0\<close> then show ?thesis
+      by simp
+  next
+    assume \<open>m > 0\<close> then show ?thesis
+    proof (cases k \<open>0::int\<close> rule: linorder_cases)
+      case less
+      moreover define l where \<open>l = - k\<close>
+      ultimately have \<open>l > 0\<close>
+        by simp
+      with \<open>m > 0\<close> have \<open>int (nat m mod nat l) \<le> m\<close>
+        by (simp flip: le_nat_iff)
+      then have \<open>int (nat m mod nat l) - l \<le> m\<close>
+        using \<open>l > 0\<close> by simp
+      with \<open>m > 0\<close> \<open>l > 0\<close> show ?thesis
+        by (simp add: modulo_int_def l_def flip: le_nat_iff)
+    qed (simp_all add: modulo_int_def flip: le_nat_iff)
+  qed
+qed
+
+lemma sgn_div_eq_sgn_mult:
+  \<open>sgn (k div l) = of_bool (k div l \<noteq> 0) * sgn (k * l)\<close>
+  for k l :: int
+proof (cases \<open>k div l = 0\<close>)
+  case True
+  then show ?thesis
+    by simp
+next
+  case False
+  have \<open>0 \<le> \<bar>k\<bar> div \<bar>l\<bar>\<close>
+    by (cases \<open>l = 0\<close>) (simp_all add: pos_imp_zdiv_nonneg_iff)
+  then have \<open>\<bar>k\<bar> div \<bar>l\<bar> \<noteq> 0 \<longleftrightarrow> 0 < \<bar>k\<bar> div \<bar>l\<bar>\<close>
+    by (simp add: less_le)
+  also have \<open>\<dots> \<longleftrightarrow> \<bar>k\<bar> \<ge> \<bar>l\<bar>\<close>
+    using False nonneg1_imp_zdiv_pos_iff by auto
+  finally have *: \<open>\<bar>k\<bar> div \<bar>l\<bar> \<noteq> 0 \<longleftrightarrow> \<bar>l\<bar> \<le> \<bar>k\<bar>\<close> .
+  show ?thesis
+    using \<open>0 \<le> \<bar>k\<bar> div \<bar>l\<bar>\<close> False
+  by (auto simp add: div_eq_div_abs [of k l] div_eq_sgn_abs [of k l]
+    sgn_mult sgn_1_pos sgn_1_neg sgn_eq_0_iff nonneg1_imp_zdiv_pos_iff * dest: sgn_not_eq_imp)
+qed
+
+
+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 have *: "k \<noteq> 0" "l \<noteq> 0"
+    by auto
+  then have "0 \<le> k div l \<Longrightarrow> \<not> k < 0 \<Longrightarrow> 0 \<le> l"
+    by (meson neg_imp_zdiv_neg_iff not_le not_less_iff_gr_or_eq)
+  then show ?thesis
+   using * by (auto simp add: pos_imp_zdiv_nonneg_iff neg_imp_zdiv_nonneg_iff)
+qed auto
+
+lemma mod_int_pos_iff:
+  \<open>k mod l \<ge> 0 \<longleftrightarrow> l dvd k \<or> l = 0 \<and> k \<ge> 0 \<or> l > 0\<close>
+  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
+
+lemma abs_div:
+  \<open>\<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>\<close> if \<open>y dvd x\<close> for x y :: int
+  using that by (cases \<open>y = 0\<close>) (auto simp add: abs_mult)
+
+lemma int_power_div_base: \<^marker>\<open>contributor \<open>Matthias Daum\<close>\<close>
+  \<open>k ^ m div k = k ^ (m - Suc 0)\<close> if \<open>0 < m\<close> \<open>0 < k\<close> for k :: int
+  using that by (cases m) simp_all
+
+lemma int_div_less_self: \<^marker>\<open>contributor \<open>Matthias Daum\<close>\<close>
+  \<open>x div k < x\<close> if \<open>0 < x\<close> \<open>1 < k\<close> for x k :: int
+proof -
+  from that have \<open>nat (x div k) = nat x div nat k\<close>
+    by (simp add: nat_div_distrib)
+  also from that have \<open>nat x div nat k < nat x\<close>
+    by simp
+  finally show ?thesis
+    by simp
+qed
+
+
+subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> by shifting\<close>
+
+lemma div_pos_geq:
+  \<open>k div l = (k - l) div l + 1\<close> if \<open>0 < l\<close> \<open>l \<le> k\<close> for k l :: int
+proof -
+  have "k = (k - l) + l" by simp
+  then obtain j where k: "k = j + l" ..
+  with that show ?thesis by (simp add: div_add_self2)
+qed
+
+lemma mod_pos_geq:
+  \<open>k mod l = (k - l) mod l\<close>  if \<open>0 < l\<close> \<open>l \<le> k\<close> for k l :: int
+proof -
+  have "k = (k - l) + l" by simp
+  then obtain j where k: "k = j + l" ..
+  with that show ?thesis by simp
+qed
+
+lemma pos_zdiv_mult_2: \<open>(1 + 2 * b) div (2 * a) = b div a\<close> (is ?Q)
+  and pos_zmod_mult_2: \<open>(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)\<close> (is ?R)
+  if \<open>0 \<le> a\<close> for a b :: int
+proof -
+  have \<open>((1 + 2 * b) div (2 * a), (1 + 2 * b) mod (2 * a)) = (b div a, 1 + 2 * (b mod a))\<close>
+  proof (induction rule: euclidean_relation_intI)
+    case by0
+    then show ?case
+      by simp
+  next
+    case divides
+    have \<open>2 dvd (2 * a)\<close>
+      by simp
+    then have \<open>2 dvd (1 + 2 * b)\<close>
+      using \<open>2 * a dvd 1 + 2 * b\<close> by (rule dvd_trans)
+    then have \<open>2 dvd (1 + b * 2)\<close>
+      by (simp add: ac_simps)
+    then have \<open>is_unit (2 :: int)\<close>
+      by simp
+    then show ?case
+      by simp
+  next
+    case euclidean_relation
+    with that have \<open>a > 0\<close>
+      by simp
+    moreover have \<open>b mod a < a\<close>
+      using \<open>a > 0\<close> by simp
+    then have \<open>1 + 2 * (b mod a) < 2 * a\<close>
+      by simp
+    moreover have \<open>2 * (b mod a) + a * (2 * (b div a)) = 2 * (b div a * a + b mod a)\<close>
+      by (simp only: algebra_simps)
+    moreover have \<open>0 \<le> 2 * (b mod a)\<close>
+      using \<open>a > 0\<close> by simp
+    ultimately show ?case
+      by (simp add: algebra_simps)
+  qed
+  then show ?Q and ?R
+    by simp_all
+qed
+
+lemma neg_zdiv_mult_2: \<open>(1 + 2 * b) div (2 * a) = (b + 1) div a\<close> (is ?Q)
+  and neg_zmod_mult_2: \<open>(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1\<close> (is ?R)
+  if \<open>a \<le> 0\<close> for a b :: int
+proof -
+  have \<open>((1 + 2 * b) div (2 * a), (1 + 2 * b) mod (2 * a)) = ((b + 1) div a, 2 * ((b + 1) mod a) - 1)\<close>
+  proof (induction rule: euclidean_relation_intI)
+    case by0
+    then show ?case
+      by simp
+  next
+    case divides
+    have \<open>2 dvd (2 * a)\<close>
+      by simp
+    then have \<open>2 dvd (1 + 2 * b)\<close>
+      using \<open>2 * a dvd 1 + 2 * b\<close> by (rule dvd_trans)
+    then have \<open>2 dvd (1 + b * 2)\<close>
+      by (simp add: ac_simps)
+    then have \<open>is_unit (2 :: int)\<close>
+      by simp
+    then show ?case
+      by simp
+  next
+    case euclidean_relation
+    with that have \<open>a < 0\<close>
+      by simp
+    moreover have \<open>(b + 1) mod a > a\<close>
+      using \<open>a < 0\<close> by simp
+    then have \<open>2 * ((b + 1) mod a) > 1 + 2 * a\<close>
+      by simp
+    moreover have \<open>((1 + b) mod a) \<le> 0\<close>
+      using \<open>a < 0\<close> by simp
+    then have \<open>2 * ((1 + b) mod a) \<le> 0\<close>
+      by simp
+    moreover have \<open>2 * ((1 + b) mod a) + a * (2 * ((1 + b) div a)) =
+      2 * ((1 + b) div a * a + (1 + b) mod a)\<close>
+      by (simp only: algebra_simps)
+    ultimately show ?case
+      by (simp add: algebra_simps sgn_mult abs_mult)
+  qed
+  then show ?Q and ?R
+    by simp_all
+qed
+
+lemma zdiv_numeral_Bit0 [simp]:
+  \<open>numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
+    numeral v div (numeral w :: int)\<close>
+  unfolding numeral.simps unfolding mult_2 [symmetric]
+  by (rule div_mult_mult1) simp
+
+lemma zdiv_numeral_Bit1 [simp]:
+  \<open>numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
+    (numeral v div (numeral w :: int))\<close>
+  unfolding numeral.simps
+  unfolding mult_2 [symmetric] add.commute [of _ 1]
+  by (rule pos_zdiv_mult_2) simp
+
+lemma zmod_numeral_Bit0 [simp]:
+  \<open>numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
+    (2::int) * (numeral v mod numeral w)\<close>
+  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
+  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
+
+lemma zmod_numeral_Bit1 [simp]:
+  \<open>numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
+    2 * (numeral v mod numeral w) + (1::int)\<close>
+  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
+  unfolding mult_2 [symmetric] add.commute [of _ 1]
+  by (rule pos_zmod_mult_2) simp
+
+
+subsection \<open>Code generation\<close>
+
+context
+begin
+
+qualified definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat"
+  where "divmod_nat m n = (m div n, m mod n)"
+
+qualified lemma divmod_nat_if [code]:
+  "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
+    let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
+  by (simp add: divmod_nat_def prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
+
+qualified lemma [code]:
+  "m div n = fst (divmod_nat m n)"
+  "m mod n = snd (divmod_nat m n)"
+  by (simp_all add: divmod_nat_def)
+
+end
+
+code_identifier
+  code_module Euclidean_Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
+
+end
diff -r a5d3f3c07de8 -r 5de3772609ea src/HOL/Library/Code_Binary_Nat.thy
--- a/src/HOL/Library/Code_Binary_Nat.thy	Mon Jan 23 22:33:25 2023 +0100
+++ b/src/HOL/Library/Code_Binary_Nat.thy	Tue Jan 24 10:30:56 2023 +0000
@@ -127,13 +127,13 @@
   "nat_of_num k < nat_of_num l \<longleftrightarrow> k < l"
   by (simp_all add: nat_of_num_numeral)
 
-declare [[code drop: Euclidean_Division.divmod_nat]]
+declare [[code drop: Euclidean_Rings.divmod_nat]]
   
 lemma divmod_nat_code [code]:
-  "Euclidean_Division.divmod_nat (nat_of_num k) (nat_of_num l) = divmod k l"
-  "Euclidean_Division.divmod_nat m 0 = (0, m)"
-  "Euclidean_Division.divmod_nat 0 n = (0, 0)"
-  by (simp_all add: Euclidean_Division.divmod_nat_def nat_of_num_numeral)
+  "Euclidean_Rings.divmod_nat (nat_of_num k) (nat_of_num l) = divmod k l"
+  "Euclidean_Rings.divmod_nat m 0 = (0, m)"
+  "Euclidean_Rings.divmod_nat 0 n = (0, 0)"
+  by (simp_all add: Euclidean_Rings.divmod_nat_def nat_of_num_numeral)
 
 end
 
diff -r a5d3f3c07de8 -r 5de3772609ea src/HOL/Library/Code_Target_Nat.thy
--- a/src/HOL/Library/Code_Target_Nat.thy	Mon Jan 23 22:33:25 2023 +0100
+++ b/src/HOL/Library/Code_Target_Nat.thy	Tue Jan 24 10:30:56 2023 +0000
@@ -98,13 +98,13 @@
 begin
 
 lemma divmod_nat_code [code]: \<^marker>\<open>contributor \<open>René Thiemann\<close>\<close> \<^marker>\<open>contributor \<open>Akihisa Yamada\<close>\<close>
-  "Euclidean_Division.divmod_nat m n = (
+  "Euclidean_Rings.divmod_nat m n = (
      let k = integer_of_nat m; l = integer_of_nat n
      in map_prod nat_of_integer nat_of_integer
        (if k = 0 then (0, 0)
         else if l = 0 then (0, k) else
           Code_Numeral.divmod_abs k l))"
-  by (simp add: prod_eq_iff Let_def Euclidean_Division.divmod_nat_def; transfer)
+  by (simp add: prod_eq_iff Let_def Euclidean_Rings.divmod_nat_def; transfer)
     (simp add: nat_div_distrib nat_mod_distrib)
 
 end
@@ -136,11 +136,11 @@
 lemma (in semiring_1) of_nat_code_if:
   "of_nat n = (if n = 0 then 0
      else let
-       (m, q) = Euclidean_Division.divmod_nat n 2;
+       (m, q) = Euclidean_Rings.divmod_nat n 2;
        m' = 2 * of_nat m
      in if q = 0 then m' else m' + 1)"
   by (cases n)
-    (simp_all add: Let_def Euclidean_Division.divmod_nat_def ac_simps
+    (simp_all add: Let_def Euclidean_Rings.divmod_nat_def ac_simps
       flip: of_nat_numeral of_nat_mult minus_mod_eq_mult_div)
 
 declare of_nat_code_if [code]
diff -r a5d3f3c07de8 -r 5de3772609ea src/HOL/Library/RBT_Impl.thy
--- a/src/HOL/Library/RBT_Impl.thy	Mon Jan 23 22:33:25 2023 +0100
+++ b/src/HOL/Library/RBT_Impl.thy	Tue Jan 24 10:30:56 2023 +0000
@@ -1154,24 +1154,24 @@
     else if n = 1 then
       case kvs of (k, v) # kvs' \<Rightarrow> 
         (Branch R Empty k v Empty, kvs')
-    else let (n', r) = Euclidean_Division.divmod_nat n 2 in
+    else let (n', r) = Euclidean_Rings.divmod_nat n 2 in
       if r = 0 then
         case rbtreeify_f n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
           apfst (Branch B t1 k v) (rbtreeify_g n' kvs')
       else case rbtreeify_f n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
           apfst (Branch B t1 k v) (rbtreeify_f n' kvs'))"
-by (subst rbtreeify_f.simps) (simp only: Let_def Euclidean_Division.divmod_nat_def prod.case)
+by (subst rbtreeify_f.simps) (simp only: Let_def Euclidean_Rings.divmod_nat_def prod.case)
 
 lemma rbtreeify_g_code [code]:
   "rbtreeify_g n kvs =
    (if n = 0 \<or> n = 1 then (Empty, kvs)
-    else let (n', r) = Euclidean_Division.divmod_nat n 2 in
+    else let (n', r) = Euclidean_Rings.divmod_nat n 2 in
       if r = 0 then
         case rbtreeify_g n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
           apfst (Branch B t1 k v) (rbtreeify_g n' kvs')
       else case rbtreeify_f n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
           apfst (Branch B t1 k v) (rbtreeify_g n' kvs'))"
-by(subst rbtreeify_g.simps)(simp only: Let_def Euclidean_Division.divmod_nat_def prod.case)
+by(subst rbtreeify_g.simps)(simp only: Let_def Euclidean_Rings.divmod_nat_def prod.case)
 
 lemma Suc_double_half: "Suc (2 * n) div 2 = n"
 by simp
diff -r a5d3f3c07de8 -r 5de3772609ea src/HOL/Library/Word.thy
--- a/src/HOL/Library/Word.thy	Mon Jan 23 22:33:25 2023 +0100
+++ b/src/HOL/Library/Word.thy	Tue Jan 24 10:30:56 2023 +0000
@@ -837,15 +837,13 @@
   show \<open>a div 1 = a\<close>
     for a :: \<open>'a word\<close>
     by transfer simp
-  have \<section>: "\<And>i n. (i::int) mod 2 ^ n = 0 \<or> 0 < i mod 2 ^ n"
-    by (metis le_less take_bit_eq_mod take_bit_nonnegative)
-  have less_power: "\<And>n i p. (i::int) mod numeral p ^ n < numeral p ^ n"
-    by simp
   show \<open>a mod b div b = 0\<close>
     for a b :: \<open>'a word\<close>
     apply transfer
-    apply (simp add: take_bit_eq_mod mod_eq_0_iff_dvd dvd_def)
-    by (metis (no_types, opaque_lifting) "\<section>" Euclidean_Division.pos_mod_bound Euclidean_Division.pos_mod_sign le_less_trans mult_eq_0_iff take_bit_eq_mod take_bit_nonnegative zdiv_eq_0_iff zmod_le_nonneg_dividend)
+    apply (simp add: take_bit_eq_mod)
+    apply (smt (verit, best) Euclidean_Rings.pos_mod_bound Euclidean_Rings.pos_mod_sign div_int_pos_iff
+        nonneg1_imp_zdiv_pos_iff zero_less_power zmod_le_nonneg_dividend)
+    done
   show \<open>(1 + a) div 2 = a div 2\<close>
     if \<open>even a\<close>
     for a :: \<open>'a word\<close>
diff -r a5d3f3c07de8 -r 5de3772609ea src/HOL/Parity.thy
--- a/src/HOL/Parity.thy	Mon Jan 23 22:33:25 2023 +0100
+++ b/src/HOL/Parity.thy	Tue Jan 24 10:30:56 2023 +0000
@@ -6,7 +6,7 @@
 section \<open>Parity in rings and semirings\<close>
 
 theory Parity
-  imports Euclidean_Division
+  imports Euclidean_Rings
 begin
 
 subsection \<open>Ring structures with parity and \<open>even\<close>/\<open>odd\<close> predicates\<close>
@@ -1014,7 +1014,7 @@
     by (simp_all add: not_le)
   have t: \<open>2 * (r div t) = r div s - r div s mod 2\<close>
     \<open>r mod t = s * (r div s mod 2) + r mod s\<close>
-    by (simp add: Rings.minus_mod_eq_mult_div Groups.mult.commute [of 2] Euclidean_Division.div_mult2_eq \<open>t = 2 * s\<close>)
+    by (simp add: Rings.minus_mod_eq_mult_div Groups.mult.commute [of 2] Euclidean_Rings.div_mult2_eq \<open>t = 2 * s\<close>)
       (use mod_mult2_eq [of r s 2] in \<open>simp add: ac_simps \<open>t = 2 * s\<close>\<close>)
   have rs: \<open>r div s mod 2 = 0 \<or> r div s mod 2 = Suc 0\<close>
     by auto
@@ -1058,7 +1058,7 @@
   show ?P and ?Q
     by (simp_all add: divmod_def *)
       (simp_all flip: of_nat_numeral of_nat_div of_nat_mod of_nat_mult add.commute [of 1] of_nat_Suc
-       add: Euclidean_Division.mod_mult_mult1 div_mult2_eq [of _ 2] mod_mult2_eq [of _ 2] **)
+       add: Euclidean_Rings.mod_mult_mult1 div_mult2_eq [of _ 2] mod_mult2_eq [of _ 2] **)
 qed
 
 text \<open>The really hard work\<close>
diff -r a5d3f3c07de8 -r 5de3772609ea src/HOL/ex/Parallel_Example.thy
--- a/src/HOL/ex/Parallel_Example.thy	Mon Jan 23 22:33:25 2023 +0100
+++ b/src/HOL/ex/Parallel_Example.thy	Tue Jan 24 10:30:56 2023 +0000
@@ -61,7 +61,7 @@
 function factorise_from :: "nat \<Rightarrow> nat \<Rightarrow> nat list" where
   "factorise_from k n = (if 1 < k \<and> k \<le> n
     then
-      let (q, r) = Euclidean_Division.divmod_nat n k 
+      let (q, r) = Euclidean_Rings.divmod_nat n k
       in if r = 0 then k # factorise_from k q
         else factorise_from (Suc k) n
     else [])" 
@@ -69,7 +69,7 @@
 
 termination factorise_from \<comment> \<open>tuning of this proof is left as an exercise to the reader\<close>
   apply (relation "measure (\<lambda>(k, n). 2 * n - k)")
-  apply (auto simp add: Euclidean_Division.divmod_nat_def algebra_simps elim!: dvdE)
+  apply (auto simp add: Euclidean_Rings.divmod_nat_def algebra_simps elim!: dvdE)
   subgoal for m n
     apply (cases "m \<le> n * 2")
      apply (auto intro: diff_less_mono)