# HG changeset patch
# User haftmann
# Date 1662388763 -7200
# Node ID 3310317cc4840eeea7de3cb9115fbde225e0d3e7
# Parent  6a20d0ebd5b3c55be2b355ede95c8ea395c04f80
clarified generic euclidean relation

diff -r 6a20d0ebd5b3 -r 3310317cc484 src/HOL/Code_Numeral.thy
--- a/src/HOL/Code_Numeral.thy	Mon Sep 05 12:54:05 2022 +0200
+++ b/src/HOL/Code_Numeral.thy	Mon Sep 05 16:39:23 2022 +0200
@@ -276,9 +276,11 @@
 declare division_segment_integer.rep_eq [simp]
 
 instance
-  by (standard; transfer)
-    (use mult_le_mono2 [of 1] in \<open>auto simp add: sgn_mult_abs abs_mult sgn_mult abs_mod_less sgn_mod nat_mult_distrib
-     division_segment_mult division_segment_mod intro: div_eqI\<close>)
+  apply (standard; transfer)
+  apply (use mult_le_mono2 [of 1] in \<open>auto simp add: sgn_mult_abs abs_mult sgn_mult abs_mod_less sgn_mod nat_mult_distrib
+     division_segment_mult division_segment_mod\<close>)
+  apply (simp add: division_segment_int_def split: if_splits)
+  done
 
 end
 
diff -r 6a20d0ebd5b3 -r 3310317cc484 src/HOL/Divides.thy
--- a/src/HOL/Divides.thy	Mon Sep 05 12:54:05 2022 +0200
+++ b/src/HOL/Divides.thy	Mon Sep 05 16:39:23 2022 +0200
@@ -114,155 +114,6 @@
 qed (use assms in auto)
 
 
-subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
-
-inductive eucl_rel_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool"
-  where eucl_rel_int_by0: "eucl_rel_int k 0 (0, k)"
-  | eucl_rel_int_dividesI: "l \<noteq> 0 \<Longrightarrow> k = q * l \<Longrightarrow> eucl_rel_int k l (q, 0)"
-  | eucl_rel_int_remainderI: "sgn r = sgn l \<Longrightarrow> \<bar>r\<bar> < \<bar>l\<bar>
-      \<Longrightarrow> k = q * l + r \<Longrightarrow> eucl_rel_int k l (q, r)"
-
-lemma eucl_rel_int_iff:    
-  "eucl_rel_int k l (q, r) \<longleftrightarrow> 
-    k = l * q + r \<and>
-     (if 0 < l then 0 \<le> r \<and> r < l else if l < 0 then l < r \<and> r \<le> 0 else q = 0)"
-  by (cases "r = 0")
-    (auto elim!: eucl_rel_int.cases intro: eucl_rel_int_by0 eucl_rel_int_dividesI eucl_rel_int_remainderI
-    simp add: ac_simps sgn_1_pos sgn_1_neg)
-
-lemma unique_quotient:
-  "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> q = q'"
-  apply (rule order_antisym)
-   apply (simp_all add: eucl_rel_int_iff linorder_neq_iff split: if_split_asm)
-     apply (blast intro: order_eq_refl [THEN unique_quotient_lemma] order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
-  done
-
-lemma unique_remainder:
-  assumes "eucl_rel_int a b (q, r)"
-    and "eucl_rel_int a b (q', r')"
-  shows "r = r'"
-proof -
-  have "q = q'"
-    using assms by (blast intro: unique_quotient)
-  then show "r = r'"
-    using assms by (simp add: eucl_rel_int_iff)
-qed
-
-lemma eucl_rel_int:
-  "eucl_rel_int k l (k div l, k mod l)"
-proof (cases k rule: int_cases3)
-  case zero
-  then show ?thesis
-    by (simp add: eucl_rel_int_iff divide_int_def modulo_int_def)
-next
-  case (pos n)
-  then show ?thesis
-    using div_mult_mod_eq [of n]
-    by (cases l rule: int_cases3)
-      (auto simp del: of_nat_mult of_nat_add
-        simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
-        eucl_rel_int_iff divide_int_def modulo_int_def)
-next
-  case (neg n)
-  then show ?thesis
-    using div_mult_mod_eq [of n]
-    by (cases l rule: int_cases3)
-      (auto simp del: of_nat_mult of_nat_add
-        simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
-        eucl_rel_int_iff divide_int_def modulo_int_def)
-qed
-
-lemma divmod_int_unique:
-  assumes "eucl_rel_int k l (q, r)"
-  shows div_int_unique: "k div l = q" and mod_int_unique: "k mod l = r"
-  using assms eucl_rel_int [of k l]
-  using unique_quotient [of k l] unique_remainder [of k l]
-  by auto
-
-lemma div_pos_geq:
-  fixes k l :: int
-  assumes "0 < l" and "l \<le> k"
-  shows "k div l = (k - l) div l + 1"
-proof -
-  have "k = (k - l) + l" by simp
-  then obtain j where k: "k = j + l" ..
-  with assms show ?thesis by (simp add: div_add_self2)
-qed
-
-lemma mod_pos_geq:
-  fixes k l :: int
-  assumes "0 < l" and "l \<le> k"
-  shows "k mod l = (k - l) mod l"
-proof -
-  have "k = (k - l) + l" by simp
-  then obtain j where k: "k = j + l" ..
-  with assms show ?thesis by simp
-qed
-
-lemma pos_eucl_rel_int_mult_2:
-  assumes "0 \<le> b"
-  assumes "eucl_rel_int a b (q, r)"
-  shows "eucl_rel_int (1 + 2*a) (2*b) (q, 1 + 2*r)"
-  using assms unfolding eucl_rel_int_iff by auto
-
-lemma neg_eucl_rel_int_mult_2:
-  assumes "b \<le> 0"
-  assumes "eucl_rel_int (a + 1) b (q, r)"
-  shows "eucl_rel_int (1 + 2*a) (2*b) (q, 2*r - 1)"
-  using assms unfolding eucl_rel_int_iff by auto
-
-text\<open>computing div by shifting\<close>
-
-lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
-  using pos_eucl_rel_int_mult_2 [OF _ eucl_rel_int]
-  by (rule div_int_unique)
-
-lemma neg_zdiv_mult_2:
-  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
-  using neg_eucl_rel_int_mult_2 [OF A eucl_rel_int]
-  by (rule div_int_unique)
-
-lemma zdiv_numeral_Bit0 [simp]:
-  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
-    numeral v div (numeral w :: int)"
-  unfolding numeral.simps unfolding mult_2 [symmetric]
-  by (rule div_mult_mult1, simp)
-
-lemma zdiv_numeral_Bit1 [simp]:
-  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
-    (numeral v div (numeral w :: int))"
-  unfolding numeral.simps
-  unfolding mult_2 [symmetric] add.commute [of _ 1]
-  by (rule pos_zdiv_mult_2, simp)
-
-lemma pos_zmod_mult_2:
-  fixes a b :: int
-  assumes "0 \<le> a"
-  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
-  using pos_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
-  by (rule mod_int_unique)
-
-lemma neg_zmod_mult_2:
-  fixes a b :: int
-  assumes "a \<le> 0"
-  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
-  using neg_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
-  by (rule mod_int_unique)
-
-lemma zmod_numeral_Bit0 [simp]:
-  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
-    (2::int) * (numeral v mod numeral w)"
-  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
-  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
-
-lemma zmod_numeral_Bit1 [simp]:
-  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
-    2 * (numeral v mod numeral w) + (1::int)"
-  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
-  unfolding mult_2 [symmetric] add.commute [of _ 1]
-  by (rule pos_zmod_mult_2, simp)
-
-  
 subsubsection \<open>Quotients of Signs\<close>
 
 lemma div_eq_minus1: "0 < b \<Longrightarrow> - 1 div b = - 1" for b :: int
@@ -402,6 +253,29 @@
     sgn_mult sgn_1_pos sgn_1_neg sgn_eq_0_iff nonneg1_imp_zdiv_pos_iff * dest: sgn_not_eq_imp)
 qed
 
+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 -
+  from assms have \<open>(a div b, a mod b) = (q, r)\<close>
+    by (cases b q r a rule: euclidean_relationI)
+      (auto simp add: division_segment_int_def ac_simps dvd_add_left_iff dest: zdvd_imp_le)
+  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>Further properties\<close>
 
@@ -430,21 +304,6 @@
 
 text \<open>Simplify expressions in which div and mod combine numerical constants\<close>
 
-lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
-  by (rule div_int_unique [of a b q r]) (simp add: eucl_rel_int_iff)
-
-lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
-  by (rule div_int_unique [of a b q r],
-    simp add: eucl_rel_int_iff)
-
-lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
-  by (rule mod_int_unique [of a b q r],
-    simp add: eucl_rel_int_iff)
-
-lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
-  by (rule mod_int_unique [of a b q r],
-    simp add: eucl_rel_int_iff)
-
 lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
   unfolding dvd_def by (cases "y=0") (auto simp add: abs_mult)
 
@@ -541,6 +400,157 @@
 qed
 
 
+subsubsection \<open>Uniqueness rules\<close>
+
+inductive eucl_rel_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool"
+  where eucl_rel_int_by0: "eucl_rel_int k 0 (0, k)"
+  | eucl_rel_int_dividesI: "l \<noteq> 0 \<Longrightarrow> k = q * l \<Longrightarrow> eucl_rel_int k l (q, 0)"
+  | eucl_rel_int_remainderI: "sgn r = sgn l \<Longrightarrow> \<bar>r\<bar> < \<bar>l\<bar>
+      \<Longrightarrow> k = q * l + r \<Longrightarrow> eucl_rel_int k l (q, r)"
+
+lemma eucl_rel_int_iff:    
+  "eucl_rel_int k l (q, r) \<longleftrightarrow> 
+    k = l * q + r \<and>
+     (if 0 < l then 0 \<le> r \<and> r < l else if l < 0 then l < r \<and> r \<le> 0 else q = 0)"
+  by (cases "r = 0")
+    (auto elim!: eucl_rel_int.cases intro: eucl_rel_int_by0 eucl_rel_int_dividesI eucl_rel_int_remainderI
+    simp add: ac_simps sgn_1_pos sgn_1_neg)
+
+lemma eucl_rel_int:
+  "eucl_rel_int k l (k div l, k mod l)"
+proof (cases k rule: int_cases3)
+  case zero
+  then show ?thesis
+    by (simp add: eucl_rel_int_iff divide_int_def modulo_int_def)
+next
+  case (pos n)
+  then show ?thesis
+    using div_mult_mod_eq [of n]
+    by (cases l rule: int_cases3)
+      (auto simp del: of_nat_mult of_nat_add
+        simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
+        eucl_rel_int_iff divide_int_def modulo_int_def)
+next
+  case (neg n)
+  then show ?thesis
+    using div_mult_mod_eq [of n]
+    by (cases l rule: int_cases3)
+      (auto simp del: of_nat_mult of_nat_add
+        simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
+        eucl_rel_int_iff divide_int_def modulo_int_def)
+qed
+
+lemma unique_quotient:
+  \<open>q = q'\<close> if \<open>eucl_rel_int a b (q, r)\<close> \<open>eucl_rel_int a b (q', r')\<close>
+  apply (rule order_antisym)
+  using that
+   apply (simp_all add: eucl_rel_int_iff linorder_neq_iff split: if_split_asm)
+     apply (blast intro: order_eq_refl [THEN unique_quotient_lemma] order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
+  done
+
+lemma unique_remainder:
+  \<open>r = r'\<close> if \<open>eucl_rel_int a b (q, r)\<close> \<open>eucl_rel_int a b (q', r')\<close>
+proof -
+  have \<open>q = q'\<close>
+    using that by (blast intro: unique_quotient)
+  then show ?thesis
+    using that by (simp add: eucl_rel_int_iff)
+qed
+
+lemma divmod_int_unique:
+  assumes \<open>eucl_rel_int k l (q, r)\<close>
+  shows div_int_unique: \<open>k div l = q\<close> and mod_int_unique: \<open>k mod l = r\<close>
+  using assms eucl_rel_int [of k l]
+  using unique_quotient [of k l] unique_remainder [of k l]
+  by auto
+
+
+subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
+
+lemma div_pos_geq:
+  fixes k l :: int
+  assumes "0 < l" and "l \<le> k"
+  shows "k div l = (k - l) div l + 1"
+proof -
+  have "k = (k - l) + l" by simp
+  then obtain j where k: "k = j + l" ..
+  with assms show ?thesis by (simp add: div_add_self2)
+qed
+
+lemma mod_pos_geq:
+  fixes k l :: int
+  assumes "0 < l" and "l \<le> k"
+  shows "k mod l = (k - l) mod l"
+proof -
+  have "k = (k - l) + l" by simp
+  then obtain j where k: "k = j + l" ..
+  with assms show ?thesis by simp
+qed
+
+lemma pos_eucl_rel_int_mult_2:
+  assumes "0 \<le> b"
+  assumes "eucl_rel_int a b (q, r)"
+  shows "eucl_rel_int (1 + 2*a) (2*b) (q, 1 + 2*r)"
+  using assms unfolding eucl_rel_int_iff by auto
+
+lemma neg_eucl_rel_int_mult_2:
+  assumes "b \<le> 0"
+  assumes "eucl_rel_int (a + 1) b (q, r)"
+  shows "eucl_rel_int (1 + 2*a) (2*b) (q, 2*r - 1)"
+  using assms unfolding eucl_rel_int_iff by auto
+
+text\<open>computing div by shifting\<close>
+
+lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
+  using pos_eucl_rel_int_mult_2 [OF _ eucl_rel_int]
+  by (rule div_int_unique)
+
+lemma neg_zdiv_mult_2:
+  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
+  using neg_eucl_rel_int_mult_2 [OF A eucl_rel_int]
+  by (rule div_int_unique)
+
+lemma zdiv_numeral_Bit0 [simp]:
+  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
+    numeral v div (numeral w :: int)"
+  unfolding numeral.simps unfolding mult_2 [symmetric]
+  by (rule div_mult_mult1, simp)
+
+lemma zdiv_numeral_Bit1 [simp]:
+  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
+    (numeral v div (numeral w :: int))"
+  unfolding numeral.simps
+  unfolding mult_2 [symmetric] add.commute [of _ 1]
+  by (rule pos_zdiv_mult_2, simp)
+
+lemma pos_zmod_mult_2:
+  fixes a b :: int
+  assumes "0 \<le> a"
+  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
+  using pos_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
+  by (rule mod_int_unique)
+
+lemma neg_zmod_mult_2:
+  fixes a b :: int
+  assumes "a \<le> 0"
+  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
+  using neg_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
+  by (rule mod_int_unique)
+
+lemma zmod_numeral_Bit0 [simp]:
+  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
+    (2::int) * (numeral v mod numeral w)"
+  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
+  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
+
+lemma zmod_numeral_Bit1 [simp]:
+  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
+    2 * (numeral v mod numeral w) + (1::int)"
+  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
+  unfolding mult_2 [symmetric] add.commute [of _ 1]
+  by (rule pos_zmod_mult_2, simp)
+
+  
 code_identifier
   code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
 
diff -r 6a20d0ebd5b3 -r 3310317cc484 src/HOL/Euclidean_Division.thy
--- a/src/HOL/Euclidean_Division.thy	Mon Sep 05 12:54:05 2022 +0200
+++ b/src/HOL/Euclidean_Division.thy	Mon Sep 05 16:39:23 2022 +0200
@@ -605,205 +605,235 @@
 subsection \<open>Uniquely determined division\<close>
 
 class unique_euclidean_semiring = euclidean_semiring + 
-  assumes euclidean_size_mult: "euclidean_size (a * b) = euclidean_size a * euclidean_size b"
-  fixes division_segment :: "'a \<Rightarrow> 'a"
-  assumes is_unit_division_segment [simp]: "is_unit (division_segment a)"
+  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:
-    "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> division_segment (a * b) = division_segment a * division_segment b"
+    \<open>a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> division_segment (a * b) = division_segment a * division_segment b\<close>
     and division_segment_mod:
-    "b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<Longrightarrow> division_segment (a mod b) = division_segment b"
+    \<open>b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<Longrightarrow> division_segment (a mod b) = division_segment b\<close>
   assumes div_bounded:
-    "b \<noteq> 0 \<Longrightarrow> division_segment r = division_segment b
+    \<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"
+    \<Longrightarrow> (q * b + r) div b = q\<close>
 begin
 
 lemma division_segment_not_0 [simp]:
-  "division_segment a \<noteq> 0"
-  using is_unit_division_segment [of a] is_unitE [of "division_segment a"] by blast
-
-lemma divmod_cases [case_names divides remainder by0]:
-  obtains 
-    (divides) q where "b \<noteq> 0"
-      and "a div b = q"
-      and "a mod b = 0"
-      and "a = q * b"
-  | (remainder) q r where "b \<noteq> 0"
-      and "division_segment r = division_segment b"
-      and "euclidean_size r < euclidean_size b"
-      and "r \<noteq> 0"
-      and "a div b = q"
-      and "a mod b = r"
-      and "a = q * b + r"
-  | (by0) "b = 0"
-proof (cases "b = 0")
+  \<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
-  then show thesis
-  by (rule by0)
+  with by0 show ?thesis
+    by simp
 next
   case False
-  show thesis
-  proof (cases "b dvd a")
+  show ?thesis
+  proof (cases \<open>b dvd a\<close>)
     case True
-    then obtain q where "a = b * q" ..
     with \<open>b \<noteq> 0\<close> divides
-    show thesis
-      by (simp add: ac_simps)
+    show ?thesis
+      by simp
   next
     case False
-    then have "a mod b \<noteq> 0"
-      by (simp add: mod_eq_0_iff_dvd)
-    moreover from \<open>b \<noteq> 0\<close> \<open>\<not> b dvd a\<close> have "division_segment (a mod b) = division_segment b"
-      by (rule division_segment_mod)
-    moreover have "euclidean_size (a mod b) < euclidean_size b"
-      using \<open>b \<noteq> 0\<close> by (rule mod_size_less)
-    moreover have "a = a div b * b + a mod b"
+    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)
-    ultimately show thesis
-      using \<open>b \<noteq> 0\<close> by (blast intro!: remainder)
+    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
 
-lemma div_eqI:
-  "a div b = q" if "b \<noteq> 0" "division_segment r = division_segment b"
-    "euclidean_size r < euclidean_size b" "q * b + r = a"
-proof -
-  from that have "(q * b + r) div b = q"
-    by (auto intro: div_bounded)
-  with that show ?thesis
-    by simp
-qed
-
-lemma mod_eqI:
-  "a mod b = r" if "b \<noteq> 0" "division_segment r = division_segment b"
-    "euclidean_size r < euclidean_size b" "q * b + r = a" 
-proof -
-  from that have "a div b = q"
-    by (rule div_eqI)
-  moreover have "a div b * b + a mod b = a"
-    by (fact div_mult_mod_eq)
-  ultimately have "a div b * b + a mod b = a div b * b + r"
-    using \<open>q * b + r = a\<close> by simp
-  then show ?thesis
-    by simp
-qed
-
 subclass euclidean_semiring_cancel
 proof
-  show "(a + c * b) div b = c + a div b" if "b \<noteq> 0" for a b c
-  proof (cases a b rule: divmod_cases)
+  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 (cases b \<open>c + a div b\<close> \<open>a mod b\<close> \<open>a + c * b\<close> rule: euclidean_relationI)
     case by0
-    with \<open>b \<noteq> 0\<close> show ?thesis
+    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 (cases \<open>c * b\<close> \<open>a div b\<close> \<open>c * (a mod b)\<close> \<open>c * a\<close> rule: euclidean_relationI)
+    case by0
+    with \<open>c \<noteq> 0\<close> show ?case
       by simp
   next
-    case (divides q)
-    then show ?thesis
-      by (simp add: ac_simps)
+    case divides
+    then show ?case
+      by (auto simp add: algebra_simps)
   next
-    case (remainder q r)
-    then show ?thesis
-      by (auto intro: div_eqI simp add: algebra_simps)
+    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
-  show"(c * a) div (c * b) = a div b" if "c \<noteq> 0" for a b c
-  proof (cases a b rule: divmod_cases)
-    case by0
-    then show ?thesis
+  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
-    case (divides q)
-    with \<open>c \<noteq> 0\<close> show ?thesis
-      by (simp add: mult.left_commute [of c])
-  next
-    case (remainder q r)
-    from \<open>b \<noteq> 0\<close> \<open>c \<noteq> 0\<close> have "b * c \<noteq> 0"
+    assume \<open>euclidean_size a < euclidean_size b\<close>
+    have \<open>(a div b, a mod b) = (0, a)\<close>
+    proof (cases b 0 a a 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
-    from remainder \<open>c \<noteq> 0\<close>
-    have "division_segment (r * c) = division_segment (b * c)"
-      and "euclidean_size (r * c) < euclidean_size (b * c)"
-      by (simp_all add: division_segment_mult division_segment_mod euclidean_size_mult)
-    with remainder show ?thesis
-      by (auto intro!: div_eqI [of _ "c * (a mod b)"] simp add: algebra_simps)
-        (use \<open>b * c \<noteq> 0\<close> in simp)
   qed
+  with \<open>b \<noteq> 0\<close> show ?thesis
+    by simp
 qed
 
 lemma div_mult1_eq:
-  "(a * b) div c = a * (b div c) + a * (b mod c) div c"
-proof (cases "a * (b mod c)" c rule: divmod_cases)
-  case (divides q)
-  have "a * b = a * (b div c * c + b mod c)"
-    by (simp add: div_mult_mod_eq)
-  also have "\<dots> = (a * (b div c) + q) * c"
-    using divides by (simp add: algebra_simps)
-  finally have "(a * b) div c = \<dots> div c"
-    by simp
-  with divides show ?thesis
-    by simp
-next
-  case (remainder q r)
-  from remainder(1-3) show ?thesis
-  proof (rule div_eqI)
-    have "a * b = a * (b div c * c + b mod c)"
-      by (simp add: div_mult_mod_eq)
-    also have "\<dots> = a * c * (b div c) + q * c + r"
-      using remainder by (simp add: algebra_simps)
-    finally show "(a * (b div c) + a * (b mod c) div c) * c + r = a * b"
-      using remainder(5-7) by (simp add: algebra_simps)
+  \<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
-next
-  case by0
+  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 (cases c \<open>a * (b div c) + a * (b mod c) div c\<close> \<open>(a * b) mod c\<close> \<open>a * b\<close> 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:
-  "(a + b) div c = a div c + b div c + (a mod c + b mod c) div c"
-proof (cases "a mod c + b mod c" c rule: divmod_cases)
-  case (divides q)
-  have "a + b = (a div c * c + a mod c) + (b div c * c + b mod c)"
-    using mod_mult_div_eq [of a c] mod_mult_div_eq [of b c] by (simp add: ac_simps)
-  also have "\<dots> = (a div c + b div c) * c + (a mod c + b mod c)"
-    by (simp add: algebra_simps)
-  also have "\<dots> = (a div c + b div c + q) * c"
-    using divides by (simp add: algebra_simps)
-  finally have "(a + b) div c = (a div c + b div c + q) * c div c"
-    by simp
-  with divides show ?thesis
-    by simp
-next
-  case (remainder q r)
-  from remainder(1-3) show ?thesis
-  proof (rule div_eqI)
-    have "(a div c + b div c + q) * c + r + (a mod c + b mod c) =
-        (a div c * c + a mod c) + (b div c * c + b mod c) + q * c + r"
+  \<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 (cases c \<open>a div c + b div c + (a mod c + b mod c) div c\<close> \<open>(a + b) mod c\<close> \<open>a + b\<close> rule: euclidean_relationI)
+    case by0
+    then show ?case
+      by simp
+  next
+    case divides
+    with * show ?case
       by (simp add: algebra_simps)
-    also have "\<dots> = a + b + (a mod c + b mod c)"
-      by (simp add: div_mult_mod_eq remainder) (simp add: ac_simps)
-    finally show "(a div c + b div c + (a mod c + b mod c) div c) * c + r = a + b"
-      using remainder by simp
+  next
+    case euclidean_relation
+    with * show ?case
+      by (simp add: division_segment_mod mod_size_less algebra_simps)
   qed
-next
-  case by0
   then show ?thesis
     by simp
 qed
 
-lemma div_eq_0_iff:
-  "a div b = 0 \<longleftrightarrow> euclidean_size a < euclidean_size b \<or> b = 0" (is "_ \<longleftrightarrow> ?P")
-  if "division_segment a = division_segment b"
-proof
-  assume ?P
-  with that show "a div b = 0"
-    by (cases "b = 0") (auto intro: div_eqI)
-next
-  assume "a div b = 0"
-  then have "a mod b = a"
-    using div_mult_mod_eq [of a b] by simp
-  with mod_size_less [of b a] show ?P
-    by auto
-qed
-
 end
 
 class unique_euclidean_ring = euclidean_ring + unique_euclidean_semiring
@@ -819,19 +849,19 @@
 instantiation nat :: normalization_semidom
 begin
 
-definition normalize_nat :: "nat \<Rightarrow> nat"
-  where [simp]: "normalize = (id :: nat \<Rightarrow> nat)"
-
-definition unit_factor_nat :: "nat \<Rightarrow> nat"
-  where "unit_factor n = (if n = 0 then 0 else 1 :: nat)"
+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]:
-  "unit_factor 0 = (0::nat)"
-  "unit_factor (Suc n) = 1"
+  \<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 :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-  where "m div n = (if n = 0 then 0 else Max {k::nat. k * n \<le> m})"
+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)
@@ -853,14 +883,14 @@
 instantiation nat :: unique_euclidean_semiring
 begin
 
-definition euclidean_size_nat :: "nat \<Rightarrow> nat"
-  where [simp]: "euclidean_size_nat = id"
-
-definition division_segment_nat :: "nat \<Rightarrow> nat"
-  where [simp]: "division_segment_nat n = 1"
-
-definition modulo_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-  where "m mod n = m - (m div n * (n::nat))"
+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
@@ -938,12 +968,60 @@
 end
 
 lemma div_nat_eqI:
-  "m div n = q" if "n * q \<le> m" and "m < n * Suc q" for m n q :: nat
-  by (rule div_eqI [of _ "m - n * q"]) (use that in \<open>simp_all add: algebra_simps\<close>)
+  \<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 (cases n q \<open>m - n * q\<close>  m rule: euclidean_relationI)
+    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 \<noteq> 0\<close> that have \<open>s < Suc q\<close>
+      by (simp only: mult_less_cancel1)
+    with \<open>m = n * s\<close> \<open>n \<noteq> 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:
-  "m mod n = r" if "r < n" and "r \<le> m" and "n dvd m - r" for m n r :: nat
-  by (rule mod_eqI [of _ _ "(m - r) div n"]) (use that in \<open>simp_all add: algebra_simps\<close>)
+  \<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 (cases n \<open>(m - r) div n\<close> r  m rule: euclidean_relationI)
+    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 \<noteq> 0\<close> have \<open>r = 0\<close>
+      by simp
+    with \<open>n \<noteq> 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>
 
@@ -1029,7 +1107,7 @@
   div_less [simp]: "m div n = 0"
   and mod_less [simp]: "m mod n = m"
   if "m < n" for m n :: nat
-  using that by (auto intro: div_eqI mod_eqI) 
+  using that by (auto intro: div_nat_eqI mod_nat_eqI)
  
 lemma split_div:
   \<open>P (m div n) \<longleftrightarrow>
@@ -1176,54 +1254,43 @@
   "Suc 0 mod n = of_bool (n \<noteq> Suc 0)"
   by (cases n) simp_all
 
-context
-  fixes m n q :: nat
-begin
-
-private lemma eucl_rel_mult2:
-  "m mod n + n * (m div n mod q) < n * q"
-  if "n > 0" and "q > 0"
+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 -
-  from \<open>n > 0\<close> have "m mod n < n"
-    by (rule mod_less_divisor)
-  from \<open>q > 0\<close> have "m div n mod q < q"
-    by (rule mod_less_divisor)
-  then obtain s where "q = Suc (m div n mod q + s)"
-    by (blast dest: less_imp_Suc_add)
-  moreover have "m mod n + n * (m div n mod q) < n * Suc (m div n mod q + s)"
-    using \<open>m mod n < n\<close> by (simp add: add_mult_distrib2)
-  ultimately show ?thesis
-    by simp
+  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 (cases \<open>n * q\<close> \<open>(m div n) div q\<close> \<open>n * (m div n mod q) + m mod n\<close> m rule: euclidean_relationI)
+    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 \<noteq> 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_mult2_eq:
-  "m div (n * q) = (m div n) div q"
-proof (cases "n = 0 \<or> q = 0")
-  case True
-  then show ?thesis
-    by auto
-next
-  case False
-  with eucl_rel_mult2 show ?thesis
-    by (auto intro: div_eqI [of _ "n * (m div n mod q) + m mod n"]
-      simp add: algebra_simps add_mult_distrib2 [symmetric])
-qed
-
-lemma mod_mult2_eq:
-  "m mod (n * q) = n * (m div n mod q) + m mod n"
-proof (cases "n = 0 \<or> q = 0")
-  case True
-  then show ?thesis
-    by auto
-next
-  case False
-  with eucl_rel_mult2 show ?thesis
-    by (auto intro: mod_eqI [of _ _ "(m div n) div q"]
-      simp add: algebra_simps add_mult_distrib2 [symmetric])
-qed
-
-end
-
 lemma div_le_mono:
   "m div k \<le> n div k" if "m \<le> n" for m n k :: nat
 proof -
@@ -1864,42 +1931,54 @@
   using div_mult_mod_eq [of 1 "2 ^ n"] by auto
 
 lemma div_mult2_eq':
-  "a div (of_nat m * of_nat n) = a div of_nat m div of_nat n"
-proof (cases a "of_nat m * of_nat n" rule: divmod_cases)
-  case (divides q)
-  then show ?thesis
-    using nonzero_mult_div_cancel_right [of "of_nat m" "q * of_nat n"]
-    by (simp add: ac_simps)
-next
-  case (remainder q r)
-  then have "division_segment r = 1"
-    using division_segment_of_nat [of "m * n"] by simp
-  with division_segment_euclidean_size [of r]
-  have "of_nat (euclidean_size r) = r"
-    by simp
-  have "a mod (of_nat m * of_nat n) div (of_nat m * of_nat n) = 0"
-    by simp
-  with remainder(6) have "r div (of_nat m * of_nat n) = 0"
-    by simp
-  with \<open>of_nat (euclidean_size r) = r\<close>
-  have "of_nat (euclidean_size r) div (of_nat m * of_nat n) = 0"
-    by simp
-  then have "of_nat (euclidean_size r div (m * n)) = 0"
-    by (simp add: of_nat_div)
-  then have "of_nat (euclidean_size r div m div n) = 0"
-    by (simp add: div_mult2_eq)
-  with \<open>of_nat (euclidean_size r) = r\<close> have "r div of_nat m div of_nat n = 0"
-    by (simp add: of_nat_div)
-  with remainder(1)
-  have "q = (r div of_nat m + q * of_nat n * of_nat m div of_nat m) div of_nat n"
-    by simp
-  with remainder(5) remainder(7) show ?thesis
-    using div_plus_div_distrib_dvd_right [of "of_nat m" "q * (of_nat m * of_nat n)" r]
-    by (simp add: ac_simps)
-next
-  case by0
+  \<open>a div (of_nat m * of_nat n) = a div of_nat m div of_nat n\<close>
+proof (cases \<open>m = 0 \<or> n = 0\<close>)
+  case True
   then show ?thesis
     by auto
+next
+  case False
+  then have \<open>m > 0\<close> \<open>n > 0\<close>
+    by simp_all
+  show ?thesis
+  proof (cases \<open>of_nat m * of_nat n dvd a\<close>)
+    case True
+    then obtain b where \<open>a = (of_nat m * of_nat n) * b\<close> ..
+    then have \<open>a = of_nat m * (of_nat n * b)\<close>
+      by (simp add: ac_simps)
+    then show ?thesis
+      by simp
+  next
+    case False
+    define q where \<open>q = a div (of_nat m * of_nat n)\<close>
+    define r where \<open>r = a mod (of_nat m * of_nat n)\<close>
+    from \<open>m > 0\<close> \<open>n > 0\<close> \<open>\<not> of_nat m * of_nat n dvd a\<close> r_def have "division_segment r = 1"
+      using division_segment_of_nat [of "m * n"] by (simp add: division_segment_mod)
+    with division_segment_euclidean_size [of r]
+    have "of_nat (euclidean_size r) = r"
+      by simp
+    have "a mod (of_nat m * of_nat n) div (of_nat m * of_nat n) = 0"
+      by simp
+    with \<open>m > 0\<close> \<open>n > 0\<close> r_def have "r div (of_nat m * of_nat n) = 0"
+      by simp
+    with \<open>of_nat (euclidean_size r) = r\<close>
+    have "of_nat (euclidean_size r) div (of_nat m * of_nat n) = 0"
+      by simp
+    then have "of_nat (euclidean_size r div (m * n)) = 0"
+      by (simp add: of_nat_div)
+    then have "of_nat (euclidean_size r div m div n) = 0"
+      by (simp add: div_mult2_eq)
+    with \<open>of_nat (euclidean_size r) = r\<close> have "r div of_nat m div of_nat n = 0"
+      by (simp add: of_nat_div)
+    with \<open>m > 0\<close> \<open>n > 0\<close> q_def
+    have "q = (r div of_nat m + q * of_nat n * of_nat m div of_nat m) div of_nat n"
+      by simp
+    moreover have \<open>a = q * (of_nat m * of_nat n) + r\<close>
+      by (simp add: q_def r_def div_mult_mod_eq)
+    ultimately show \<open>a div (of_nat m * of_nat n) = a div of_nat m div of_nat n\<close>
+      using q_def [symmetric] div_plus_div_distrib_dvd_right [of \<open>of_nat m\<close> \<open>q * (of_nat m * of_nat n)\<close> r]
+      by (simp add: ac_simps)
+  qed
 qed
 
 lemma mod_mult2_eq':
@@ -2033,31 +2112,45 @@
   using that by (simp add: mod_eq_self_iff_div_eq_0)
 
 lemma div_pos_neg_trivial:
-  "k div l = - 1" if "0 < k" and "k + l \<le> 0" for k l :: int
-proof (cases \<open>l = - k\<close>)
+  \<open>k div l = - 1\<close> if \<open>0 < k\<close> and \<open>k + l \<le> 0\<close> for k l :: int
+proof (cases \<open>l dvd k\<close>)
   case True
-  with that show ?thesis
-    by (simp add: divide_int_def)
+  then obtain j where \<open>k = l * j\<close> ..
+  from that have \<open>l < 0\<close>
+    by simp
+  with \<open>k = l * j\<close> \<open>0 < k\<close> have \<open>j \<le> - 1\<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>- 1 \<le> j\<close>
+    by (simp add: mult_le_0_iff)
+  ultimately have \<open>j = - 1\<close>
+    by (rule order.antisym)
+  with \<open>k = l * j\<close> \<open>l < 0\<close> show ?thesis
+    by (simp add: dvd_neg_div)
 next
   case False
-  show ?thesis
-    apply (rule div_eqI [of _ "k + l"])
-    using False that apply (simp_all add: division_segment_int_def)
-    done
+  have \<open>k + l < 0\<close>
+  proof (rule ccontr)
+    assume \<open>\<not> k + l < 0\<close>
+    with \<open>k + l \<le> 0\<close> have \<open>k + l = 0\<close>
+      by simp
+    then have \<open>k = - l\<close>
+      by simp
+    then have \<open>l dvd k\<close>
+      by simp
+    with \<open>\<not> l dvd k\<close> show False ..
+  qed
+  with that \<open>\<not> l dvd k\<close> show ?thesis
+    by (simp add: div_eq_div_abs [of k l])
 qed
 
 lemma mod_pos_neg_trivial:
-  "k mod l = k + l" if "0 < k" and "k + l \<le> 0" for k l :: int
-proof (cases \<open>l = - k\<close>)
-  case True
-  with that show ?thesis
-    by (simp add: divide_int_def)
-next
-  case False
-  show ?thesis
-    apply (rule mod_eqI [of _ _ \<open>- 1\<close>])
-    using False that apply (simp_all add: division_segment_int_def)
-    done
+  \<open>k mod l = k + l\<close> if \<open>0 < k\<close> and \<open>k + l \<le> 0\<close> for k l :: int
+proof -
+  from that have \<open>k mod l = k div l * l + k mod l + l\<close>
+    by (simp add: div_pos_neg_trivial)
+  then show ?thesis by simp
 qed
 
 text \<open>There is neither \<open>div_neg_pos_trivial\<close> nor \<open>mod_neg_pos_trivial\<close>