# HG changeset patch
# User haftmann
# Date 1666938866 0
# Node ID 8cb1413846826e2463129955b6dc992350000739
# Parent  6bc3bb9d0e3eb2fc5f946cee0b7f9fee954948d0
restructured

diff -r 6bc3bb9d0e3e -r 8cb141384682 src/HOL/Euclidean_Division.thy
--- a/src/HOL/Euclidean_Division.thy	Fri Oct 28 06:34:25 2022 +0000
+++ b/src/HOL/Euclidean_Division.thy	Fri Oct 28 06:34:26 2022 +0000
@@ -844,7 +844,7 @@
 end
 
 
-subsection \<open>Euclidean division on \<^typ>\<open>nat\<close>\<close>
+subsection \<open>Division on \<^typ>\<open>nat\<close>\<close>
 
 instantiation nat :: normalization_semidom
 begin
@@ -1652,7 +1652,7 @@
 
 
 
-subsection \<open>Elementary euclidean division on \<^typ>\<open>int\<close>\<close>
+subsection \<open>Division on \<^typ>\<open>int\<close>\<close>
 
 subsubsection \<open>Basic instantiation\<close>
 
@@ -1956,286 +1956,6 @@
 qed
 
 
-subsection \<open>Special case: euclidean rings containing the natural numbers\<close>
-
-class unique_euclidean_semiring_with_nat = semidom + semiring_char_0 + unique_euclidean_semiring +
-  assumes of_nat_div: "of_nat (m div n) = of_nat m div of_nat n"
-    and division_segment_of_nat [simp]: "division_segment (of_nat n) = 1"
-    and division_segment_euclidean_size [simp]: "division_segment a * of_nat (euclidean_size a) = a"
-begin
-
-lemma division_segment_eq_iff:
-  "a = b" if "division_segment a = division_segment b"
-    and "euclidean_size a = euclidean_size b"
-  using that division_segment_euclidean_size [of a] by simp
-
-lemma euclidean_size_of_nat [simp]:
-  "euclidean_size (of_nat n) = n"
-proof -
-  have "division_segment (of_nat n) * of_nat (euclidean_size (of_nat n)) = of_nat n"
-    by (fact division_segment_euclidean_size)
-  then show ?thesis by simp
-qed
-
-lemma of_nat_euclidean_size:
-  "of_nat (euclidean_size a) = a div division_segment a"
-proof -
-  have "of_nat (euclidean_size a) = division_segment a * of_nat (euclidean_size a) div division_segment a"
-    by (subst nonzero_mult_div_cancel_left) simp_all
-  also have "\<dots> = a div division_segment a"
-    by simp
-  finally show ?thesis .
-qed
-
-lemma division_segment_1 [simp]:
-  "division_segment 1 = 1"
-  using division_segment_of_nat [of 1] by simp
-
-lemma division_segment_numeral [simp]:
-  "division_segment (numeral k) = 1"
-  using division_segment_of_nat [of "numeral k"] by simp
-
-lemma euclidean_size_1 [simp]:
-  "euclidean_size 1 = 1"
-  using euclidean_size_of_nat [of 1] by simp
-
-lemma euclidean_size_numeral [simp]:
-  "euclidean_size (numeral k) = numeral k"
-  using euclidean_size_of_nat [of "numeral k"] by simp
-
-lemma of_nat_dvd_iff:
-  "of_nat m dvd of_nat n \<longleftrightarrow> m dvd n" (is "?P \<longleftrightarrow> ?Q")
-proof (cases "m = 0")
-  case True
-  then show ?thesis
-    by simp
-next
-  case False
-  show ?thesis
-  proof
-    assume ?Q
-    then show ?P
-      by auto
-  next
-    assume ?P
-    with False have "of_nat n = of_nat n div of_nat m * of_nat m"
-      by simp
-    then have "of_nat n = of_nat (n div m * m)"
-      by (simp add: of_nat_div)
-    then have "n = n div m * m"
-      by (simp only: of_nat_eq_iff)
-    then have "n = m * (n div m)"
-      by (simp add: ac_simps)
-    then show ?Q ..
-  qed
-qed
-
-lemma of_nat_mod:
-  "of_nat (m mod n) = of_nat m mod of_nat n"
-proof -
-  have "of_nat m div of_nat n * of_nat n + of_nat m mod of_nat n = of_nat m"
-    by (simp add: div_mult_mod_eq)
-  also have "of_nat m = of_nat (m div n * n + m mod n)"
-    by simp
-  finally show ?thesis
-    by (simp only: of_nat_div of_nat_mult of_nat_add) simp
-qed
-
-lemma one_div_two_eq_zero [simp]:
-  "1 div 2 = 0"
-proof -
-  from of_nat_div [symmetric] have "of_nat 1 div of_nat 2 = of_nat 0"
-    by (simp only:) simp
-  then show ?thesis
-    by simp
-qed
-
-lemma one_mod_two_eq_one [simp]:
-  "1 mod 2 = 1"
-proof -
-  from of_nat_mod [symmetric] have "of_nat 1 mod of_nat 2 = of_nat 1"
-    by (simp only:) simp
-  then show ?thesis
-    by simp
-qed
-
-lemma one_mod_2_pow_eq [simp]:
-  "1 mod (2 ^ n) = of_bool (n > 0)"
-proof -
-  have "1 mod (2 ^ n) = of_nat (1 mod (2 ^ n))"
-    using of_nat_mod [of 1 "2 ^ n"] by simp
-  also have "\<dots> = of_bool (n > 0)"
-    by simp
-  finally show ?thesis .
-qed
-
-lemma one_div_2_pow_eq [simp]:
-  "1 div (2 ^ n) = of_bool (n = 0)"
-  using div_mult_mod_eq [of 1 "2 ^ n"] by auto
-
-lemma div_mult2_eq':
-  \<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':
-  "a mod (of_nat m * of_nat n) = of_nat m * (a div of_nat m mod of_nat n) + a mod of_nat m"
-proof -
-  have "a div (of_nat m * of_nat n) * (of_nat m * of_nat n) + a mod (of_nat m * of_nat n) = a div of_nat m div of_nat n * of_nat n * of_nat m + (a div of_nat m mod of_nat n * of_nat m + a mod of_nat m)"
-    by (simp add: combine_common_factor div_mult_mod_eq)
-  moreover have "a div of_nat m div of_nat n * of_nat n * of_nat m = of_nat n * of_nat m * (a div of_nat m div of_nat n)"
-    by (simp add: ac_simps)
-  ultimately show ?thesis
-    by (simp add: div_mult2_eq' mult_commute)
-qed
-
-lemma div_mult2_numeral_eq:
-  "a div numeral k div numeral l = a div numeral (k * l)" (is "?A = ?B")
-proof -
-  have "?A = a div of_nat (numeral k) div of_nat (numeral l)"
-    by simp
-  also have "\<dots> = a div (of_nat (numeral k) * of_nat (numeral l))"
-    by (fact div_mult2_eq' [symmetric])
-  also have "\<dots> = ?B"
-    by simp
-  finally show ?thesis .
-qed
-
-lemma numeral_Bit0_div_2:
-  "numeral (num.Bit0 n) div 2 = numeral n"
-proof -
-  have "numeral (num.Bit0 n) = numeral n + numeral n"
-    by (simp only: numeral.simps)
-  also have "\<dots> = numeral n * 2"
-    by (simp add: mult_2_right)
-  finally have "numeral (num.Bit0 n) div 2 = numeral n * 2 div 2"
-    by simp
-  also have "\<dots> = numeral n"
-    by (rule nonzero_mult_div_cancel_right) simp
-  finally show ?thesis .
-qed
-
-lemma numeral_Bit1_div_2:
-  "numeral (num.Bit1 n) div 2 = numeral n"
-proof -
-  have "numeral (num.Bit1 n) = numeral n + numeral n + 1"
-    by (simp only: numeral.simps)
-  also have "\<dots> = numeral n * 2 + 1"
-    by (simp add: mult_2_right)
-  finally have "numeral (num.Bit1 n) div 2 = (numeral n * 2 + 1) div 2"
-    by simp
-  also have "\<dots> = numeral n * 2 div 2 + 1 div 2"
-    using dvd_triv_right by (rule div_plus_div_distrib_dvd_left)
-  also have "\<dots> = numeral n * 2 div 2"
-    by simp
-  also have "\<dots> = numeral n"
-    by (rule nonzero_mult_div_cancel_right) simp
-  finally show ?thesis .
-qed
-
-lemma exp_mod_exp:
-  \<open>2 ^ m mod 2 ^ n = of_bool (m < n) * 2 ^ m\<close>
-proof -
-  have \<open>(2::nat) ^ m mod 2 ^ n = of_bool (m < n) * 2 ^ m\<close> (is \<open>?lhs = ?rhs\<close>)
-    by (auto simp add: not_less monoid_mult_class.power_add dest!: le_Suc_ex)
-  then have \<open>of_nat ?lhs = of_nat ?rhs\<close>
-    by simp
-  then show ?thesis
-    by (simp add: of_nat_mod)
-qed
-
-lemma mask_mod_exp:
-  \<open>(2 ^ n - 1) mod 2 ^ m = 2 ^ min m n - 1\<close>
-proof -
-  have \<open>(2 ^ n - 1) mod 2 ^ m = 2 ^ min m n - (1::nat)\<close> (is \<open>?lhs = ?rhs\<close>)
-  proof (cases \<open>n \<le> m\<close>)
-    case True
-    then show ?thesis
-      by (simp add: Suc_le_lessD)
-  next
-    case False
-    then have \<open>m < n\<close>
-      by simp
-    then obtain q where n: \<open>n = Suc q + m\<close>
-      by (auto dest: less_imp_Suc_add)
-    then have \<open>min m n = m\<close>
-      by simp
-    moreover have \<open>(2::nat) ^ m \<le> 2 * 2 ^ q * 2 ^ m\<close>
-      using mult_le_mono1 [of 1 \<open>2 * 2 ^ q\<close> \<open>2 ^ m\<close>] by simp
-    with n have \<open>2 ^ n - 1 = (2 ^ Suc q - 1) * 2 ^ m + (2 ^ m - (1::nat))\<close>
-      by (simp add: monoid_mult_class.power_add algebra_simps)
-    ultimately show ?thesis
-      by (simp only: euclidean_semiring_cancel_class.mod_mult_self3) simp
-  qed
-  then have \<open>of_nat ?lhs = of_nat ?rhs\<close>
-    by simp
-  then show ?thesis
-    by (simp add: of_nat_mod of_nat_diff)
-qed
-
-lemma of_bool_half_eq_0 [simp]:
-  \<open>of_bool b div 2 = 0\<close>
-  by simp
-
-end
-
-class unique_euclidean_ring_with_nat = ring + unique_euclidean_semiring_with_nat
-
-instance nat :: unique_euclidean_semiring_with_nat
-  by standard (simp_all add: dvd_eq_mod_eq_0)
-
-instance int :: unique_euclidean_ring_with_nat
-  by standard (auto simp add: divide_int_def division_segment_int_def elim: contrapos_np)
-
-
-subsection \<open>More on euclidean division on \<^typ>\<open>int\<close>\<close>
-
 subsubsection \<open>Trivial reduction steps\<close>
 
 lemma div_pos_pos_trivial [simp]:
@@ -2478,16 +2198,54 @@
 
 subsubsection \<open>Algebraic rewrites\<close>
 
-lemma zdiv_zmult2_eq:
-  \<open>a div (b * c) = (a div b) div c\<close> if \<open>c \<ge> 0\<close> for a b c :: int
-proof (cases \<open>b \<ge> 0\<close>)
-  case True
-  with that show ?thesis
-    using div_mult2_eq' [of a \<open>nat b\<close> \<open>nat c\<close>] by simp
-next
-  case False
-  with that show ?thesis
-    using div_mult2_eq' [of \<open>- a\<close> \<open>nat (- b)\<close> \<open>nat c\<close>] by simp
+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':
@@ -2502,18 +2260,6 @@
   finally show ?thesis .
 qed
 
-lemma zmod_zmult2_eq:
-  \<open>a mod (b * c) = b * (a div b mod c) + a mod b\<close> if \<open>c \<ge> 0\<close> for a b c :: int
-proof (cases \<open>b \<ge> 0\<close>)
-  case True
-  with that show ?thesis
-    using mod_mult2_eq' [of a \<open>nat b\<close> \<open>nat c\<close>] by simp
-next
-  case False
-  with that show ?thesis
-    using mod_mult2_eq' [of \<open>- a\<close> \<open>nat (- b)\<close> \<open>nat c\<close>] by simp
-qed
-
 lemma half_nonnegative_int_iff [simp]:
   \<open>k div 2 \<ge> 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int
   by auto
@@ -2526,12 +2272,12 @@
 subsubsection \<open>Distributive laws for conversions.\<close>
 
 lemma zdiv_int:
-  "int (a div b) = int a div int b"
-  by (fact of_nat_div)
+  \<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:
-  "int (a mod b) = int a mod int b"
-  by (fact of_nat_mod)
+  \<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>
@@ -2961,384 +2707,7 @@
   by (rule pos_zmod_mult_2) simp
 
 
-subsection \<open>Generic symbolic computations\<close>
-
-text \<open>
-  The following type class contains everything necessary to formulate
-  a division algorithm in ring structures with numerals, restricted
-  to its positive segments.
-\<close>
-
-class unique_euclidean_semiring_with_nat_division = unique_euclidean_semiring_with_nat +
-  fixes divmod :: \<open>num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a\<close>
-    and divmod_step :: \<open>'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a\<close> \<comment> \<open>
-      These are conceptually definitions but force generated code
-      to be monomorphic wrt. particular instances of this class which
-      yields a significant speedup.\<close>
-  assumes divmod_def: \<open>divmod m n = (numeral m div numeral n, numeral m mod numeral n)\<close>
-    and divmod_step_def [simp]: \<open>divmod_step l (q, r) =
-      (if euclidean_size l \<le> euclidean_size r then (2 * q + 1, r - l)
-       else (2 * q, r))\<close> \<comment> \<open>
-         This is a formulation of one step (referring to one digit position)
-         in school-method division: compare the dividend at the current
-         digit position with the remainder from previous division steps
-         and evaluate accordingly.\<close>
-begin
-
-lemma fst_divmod:
-  \<open>fst (divmod m n) = numeral m div numeral n\<close>
-  by (simp add: divmod_def)
-
-lemma snd_divmod:
-  \<open>snd (divmod m n) = numeral m mod numeral n\<close>
-  by (simp add: divmod_def)
-
-text \<open>
-  Following a formulation of school-method division.
-  If the divisor is smaller than the dividend, terminate.
-  If not, shift the dividend to the right until termination
-  occurs and then reiterate single division steps in the
-  opposite direction.
-\<close>
-
-lemma divmod_divmod_step:
-  \<open>divmod m n = (if m < n then (0, numeral m)
-    else divmod_step (numeral n) (divmod m (Num.Bit0 n)))\<close>
-proof (cases \<open>m < n\<close>)
-  case True
-  then show ?thesis
-    by (simp add: prod_eq_iff fst_divmod snd_divmod flip: of_nat_numeral of_nat_div of_nat_mod)
-next
-  case False
-  define r s t where \<open>r = (numeral m :: nat)\<close> \<open>s = (numeral n :: nat)\<close> \<open>t = 2 * s\<close>
-  then have *: \<open>numeral m = of_nat r\<close> \<open>numeral n = of_nat s\<close> \<open>numeral (num.Bit0 n) = of_nat t\<close>
-    and \<open>\<not> s \<le> r mod s\<close>
-    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>)
-      (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
-  from \<open>\<not> s \<le> r mod s\<close> have \<open>s \<le> r mod t \<Longrightarrow>
-     r div s = Suc (2 * (r div t)) \<and>
-     r mod s = r mod t - s\<close>
-    using rs
-    by (auto simp add: t)
-  moreover have \<open>r mod t < s \<Longrightarrow>
-     r div s = 2 * (r div t) \<and>
-     r mod s = r mod t\<close>
-    using rs
-    by (auto simp add: t)
-  ultimately show ?thesis
-    by (simp add: divmod_def prod_eq_iff split_def Let_def
-        not_less mod_eq_0_iff_dvd Rings.mod_eq_0_iff_dvd False not_le *)
-    (simp add: flip: of_nat_numeral of_nat_mult add.commute [of 1] of_nat_div of_nat_mod of_nat_Suc of_nat_diff)
-qed
-
-text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
-
-lemma divmod_trivial [simp]:
-  "divmod m Num.One = (numeral m, 0)"
-  "divmod num.One (num.Bit0 n) = (0, Numeral1)"
-  "divmod num.One (num.Bit1 n) = (0, Numeral1)"
-  using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
-
-text \<open>Division by an even number is a right-shift\<close>
-
-lemma divmod_cancel [simp]:
-  \<open>divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))\<close> (is ?P)
-  \<open>divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))\<close> (is ?Q)
-proof -
-  define r s where \<open>r = (numeral m :: nat)\<close> \<open>s = (numeral n :: nat)\<close>
-  then have *: \<open>numeral m = of_nat r\<close> \<open>numeral n = of_nat s\<close>
-    \<open>numeral (num.Bit0 m) = of_nat (2 * r)\<close> \<open>numeral (num.Bit0 n) = of_nat (2 * s)\<close>
-    \<open>numeral (num.Bit1 m) = of_nat (Suc (2 * r))\<close>
-    by simp_all
-  have **: \<open>Suc (2 * r) div 2 = r\<close>
-    by simp
-  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] **)
-qed
-
-text \<open>The really hard work\<close>
-
-lemma divmod_steps [simp]:
-  "divmod (num.Bit0 m) (num.Bit1 n) =
-      (if m \<le> n then (0, numeral (num.Bit0 m))
-       else divmod_step (numeral (num.Bit1 n))
-             (divmod (num.Bit0 m)
-               (num.Bit0 (num.Bit1 n))))"
-  "divmod (num.Bit1 m) (num.Bit1 n) =
-      (if m < n then (0, numeral (num.Bit1 m))
-       else divmod_step (numeral (num.Bit1 n))
-             (divmod (num.Bit1 m)
-               (num.Bit0 (num.Bit1 n))))"
-  by (simp_all add: divmod_divmod_step)
-
-lemmas divmod_algorithm_code = divmod_trivial divmod_cancel divmod_steps
-
-text \<open>Special case: divisibility\<close>
-
-definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
-where
-  "divides_aux qr \<longleftrightarrow> snd qr = 0"
-
-lemma divides_aux_eq [simp]:
-  "divides_aux (q, r) \<longleftrightarrow> r = 0"
-  by (simp add: divides_aux_def)
-
-lemma dvd_numeral_simp [simp]:
-  "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
-  by (simp add: divmod_def mod_eq_0_iff_dvd)
-
-text \<open>Generic computation of quotient and remainder\<close>
-
-lemma numeral_div_numeral [simp]:
-  "numeral k div numeral l = fst (divmod k l)"
-  by (simp add: fst_divmod)
-
-lemma numeral_mod_numeral [simp]:
-  "numeral k mod numeral l = snd (divmod k l)"
-  by (simp add: snd_divmod)
-
-lemma one_div_numeral [simp]:
-  "1 div numeral n = fst (divmod num.One n)"
-  by (simp add: fst_divmod)
-
-lemma one_mod_numeral [simp]:
-  "1 mod numeral n = snd (divmod num.One n)"
-  by (simp add: snd_divmod)
-
-end
-
-instantiation nat :: unique_euclidean_semiring_with_nat_division
-begin
-
-definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
-where
-  divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
-
-definition divmod_step_nat :: "nat \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
-where
-  "divmod_step_nat l qr = (let (q, r) = qr
-    in if r \<ge> l then (2 * q + 1, r - l)
-    else (2 * q, r))"
-
-instance
-  by standard (simp_all add: divmod'_nat_def divmod_step_nat_def)
-
-end
-
-declare divmod_algorithm_code [where ?'a = nat, code]
-
-lemma Suc_0_div_numeral [simp]:
-  \<open>Suc 0 div numeral Num.One = 1\<close>
-  \<open>Suc 0 div numeral (Num.Bit0 n) = 0\<close>
-  \<open>Suc 0 div numeral (Num.Bit1 n) = 0\<close>
-  by simp_all
-
-lemma Suc_0_mod_numeral [simp]:
-  \<open>Suc 0 mod numeral Num.One = 0\<close>
-  \<open>Suc 0 mod numeral (Num.Bit0 n) = 1\<close>
-  \<open>Suc 0 mod numeral (Num.Bit1 n) = 1\<close>
-  by simp_all
-
-instantiation int :: unique_euclidean_semiring_with_nat_division
-begin
-
-definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
-where
-  "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
-
-definition divmod_step_int :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
-where
-  "divmod_step_int l qr = (let (q, r) = qr
-    in if \<bar>l\<bar> \<le> \<bar>r\<bar> then (2 * q + 1, r - l)
-    else (2 * q, r))"
-
-instance
-  by standard (auto simp add: divmod_int_def divmod_step_int_def)
-
-end
-
-declare divmod_algorithm_code [where ?'a = int, code]
-
-context
-begin
-
-qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
-where
-  "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
-
-qualified lemma adjust_div_eq [simp, code]:
-  "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
-  by (simp add: adjust_div_def)
-
-qualified definition adjust_mod :: "num \<Rightarrow> int \<Rightarrow> int"
-where
-  [simp]: "adjust_mod l r = (if r = 0 then 0 else numeral l - r)"
-
-lemma minus_numeral_div_numeral [simp]:
-  "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
-proof -
-  have "int (fst (divmod m n)) = fst (divmod m n)"
-    by (simp only: fst_divmod divide_int_def) auto
-  then show ?thesis
-    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
-qed
-
-lemma minus_numeral_mod_numeral [simp]:
-  "- numeral m mod numeral n = adjust_mod n (snd (divmod m n) :: int)"
-proof (cases "snd (divmod m n) = (0::int)")
-  case True
-  then show ?thesis
-    by (simp add: mod_eq_0_iff_dvd divides_aux_def)
-next
-  case False
-  then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
-    by (simp only: snd_divmod modulo_int_def) auto
-  then show ?thesis
-    by (simp add: divides_aux_def adjust_div_def)
-      (simp add: divides_aux_def modulo_int_def)
-qed
-
-lemma numeral_div_minus_numeral [simp]:
-  "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
-proof -
-  have "int (fst (divmod m n)) = fst (divmod m n)"
-    by (simp only: fst_divmod divide_int_def) auto
-  then show ?thesis
-    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
-qed
-
-lemma numeral_mod_minus_numeral [simp]:
-  "numeral m mod - numeral n = - adjust_mod n (snd (divmod m n) :: int)"
-proof (cases "snd (divmod m n) = (0::int)")
-  case True
-  then show ?thesis
-    by (simp add: mod_eq_0_iff_dvd divides_aux_def)
-next
-  case False
-  then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
-    by (simp only: snd_divmod modulo_int_def) auto
-  then show ?thesis
-    by (simp add: divides_aux_def adjust_div_def)
-      (simp add: divides_aux_def modulo_int_def)
-qed
-
-lemma minus_one_div_numeral [simp]:
-  "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
-  using minus_numeral_div_numeral [of Num.One n] by simp
-
-lemma minus_one_mod_numeral [simp]:
-  "- 1 mod numeral n = adjust_mod n (snd (divmod Num.One n) :: int)"
-  using minus_numeral_mod_numeral [of Num.One n] by simp
-
-lemma one_div_minus_numeral [simp]:
-  "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
-  using numeral_div_minus_numeral [of Num.One n] by simp
-
-lemma one_mod_minus_numeral [simp]:
-  "1 mod - numeral n = - adjust_mod n (snd (divmod Num.One n) :: int)"
-  using numeral_mod_minus_numeral [of Num.One n] by simp
-
-lemma [code]:
-  fixes k :: int
-  shows
-    "k div 0 = 0"
-    "k mod 0 = k"
-    "0 div k = 0"
-    "0 mod k = 0"
-    "k div Int.Pos Num.One = k"
-    "k mod Int.Pos Num.One = 0"
-    "k div Int.Neg Num.One = - k"
-    "k mod Int.Neg Num.One = 0"
-    "Int.Pos m div Int.Pos n = (fst (divmod m n) :: int)"
-    "Int.Pos m mod Int.Pos n = (snd (divmod m n) :: int)"
-    "Int.Neg m div Int.Pos n = - (adjust_div (divmod m n) :: int)"
-    "Int.Neg m mod Int.Pos n = adjust_mod n (snd (divmod m n) :: int)"
-    "Int.Pos m div Int.Neg n = - (adjust_div (divmod m n) :: int)"
-    "Int.Pos m mod Int.Neg n = - adjust_mod n (snd (divmod m n) :: int)"
-    "Int.Neg m div Int.Neg n = (fst (divmod m n) :: int)"
-    "Int.Neg m mod Int.Neg n = - (snd (divmod m n) :: int)"
-  by simp_all
-
-end
-
-lemma divmod_BitM_2_eq [simp]:
-  \<open>divmod (Num.BitM m) (Num.Bit0 Num.One) = (numeral m - 1, (1 :: int))\<close>
-  by (cases m) simp_all
-
-
-subsubsection \<open>Computation by simplification\<close>
-
-lemma euclidean_size_nat_less_eq_iff:
-  \<open>euclidean_size m \<le> euclidean_size n \<longleftrightarrow> m \<le> n\<close> for m n :: nat
-  by simp
-
-lemma euclidean_size_int_less_eq_iff:
-  \<open>euclidean_size k \<le> euclidean_size l \<longleftrightarrow> \<bar>k\<bar> \<le> \<bar>l\<bar>\<close> for k l :: int
-  by auto
-
-simproc_setup numeral_divmod
-  ("0 div 0 :: 'a :: unique_euclidean_semiring_with_nat_division" | "0 mod 0 :: 'a :: unique_euclidean_semiring_with_nat_division" |
-   "0 div 1 :: 'a :: unique_euclidean_semiring_with_nat_division" | "0 mod 1 :: 'a :: unique_euclidean_semiring_with_nat_division" |
-   "0 div - 1 :: int" | "0 mod - 1 :: int" |
-   "0 div numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" | "0 mod numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" |
-   "0 div - numeral b :: int" | "0 mod - numeral b :: int" |
-   "1 div 0 :: 'a :: unique_euclidean_semiring_with_nat_division" | "1 mod 0 :: 'a :: unique_euclidean_semiring_with_nat_division" |
-   "1 div 1 :: 'a :: unique_euclidean_semiring_with_nat_division" | "1 mod 1 :: 'a :: unique_euclidean_semiring_with_nat_division" |
-   "1 div - 1 :: int" | "1 mod - 1 :: int" |
-   "1 div numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" | "1 mod numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" |
-   "1 div - numeral b :: int" |"1 mod - numeral b :: int" |
-   "- 1 div 0 :: int" | "- 1 mod 0 :: int" | "- 1 div 1 :: int" | "- 1 mod 1 :: int" |
-   "- 1 div - 1 :: int" | "- 1 mod - 1 :: int" | "- 1 div numeral b :: int" | "- 1 mod numeral b :: int" |
-   "- 1 div - numeral b :: int" | "- 1 mod - numeral b :: int" |
-   "numeral a div 0 :: 'a :: unique_euclidean_semiring_with_nat_division" | "numeral a mod 0 :: 'a :: unique_euclidean_semiring_with_nat_division" |
-   "numeral a div 1 :: 'a :: unique_euclidean_semiring_with_nat_division" | "numeral a mod 1 :: 'a :: unique_euclidean_semiring_with_nat_division" |
-   "numeral a div - 1 :: int" | "numeral a mod - 1 :: int" |
-   "numeral a div numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" | "numeral a mod numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" |
-   "numeral a div - numeral b :: int" | "numeral a mod - numeral b :: int" |
-   "- numeral a div 0 :: int" | "- numeral a mod 0 :: int" |
-   "- numeral a div 1 :: int" | "- numeral a mod 1 :: int" |
-   "- numeral a div - 1 :: int" | "- numeral a mod - 1 :: int" |
-   "- numeral a div numeral b :: int" | "- numeral a mod numeral b :: int" |
-   "- numeral a div - numeral b :: int" | "- numeral a mod - numeral b :: int") = \<open>
-  let
-    val if_cong = the (Code.get_case_cong \<^theory> \<^const_name>\<open>If\<close>);
-    fun successful_rewrite ctxt ct =
-      let
-        val thm = Simplifier.rewrite ctxt ct
-      in if Thm.is_reflexive thm then NONE else SOME thm end;
-  in fn phi =>
-    let
-      val simps = Morphism.fact phi (@{thms div_0 mod_0 div_by_0 mod_by_0 div_by_1 mod_by_1
-        one_div_numeral one_mod_numeral minus_one_div_numeral minus_one_mod_numeral
-        one_div_minus_numeral one_mod_minus_numeral
-        numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
-        numeral_div_minus_numeral numeral_mod_minus_numeral
-        div_minus_minus mod_minus_minus Euclidean_Division.adjust_div_eq of_bool_eq one_neq_zero
-        numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
-        divmod_cancel divmod_steps divmod_step_def fst_conv snd_conv numeral_One
-        case_prod_beta rel_simps Euclidean_Division.adjust_mod_def div_minus1_right mod_minus1_right
-        minus_minus numeral_times_numeral mult_zero_right mult_1_right
-        euclidean_size_nat_less_eq_iff euclidean_size_int_less_eq_iff diff_nat_numeral nat_numeral}
-        @ [@{lemma "0 = 0 \<longleftrightarrow> True" by simp}]);
-      fun prepare_simpset ctxt = HOL_ss |> Simplifier.simpset_map ctxt
-        (Simplifier.add_cong if_cong #> fold Simplifier.add_simp simps)
-    in fn ctxt => successful_rewrite (Simplifier.put_simpset (prepare_simpset ctxt) ctxt) end
-  end
-\<close> \<comment> \<open>
-  There is space for improvement here: the calculation itself
-  could be carried out outside the logic, and a generic simproc
-  (simplifier setup) for generic calculation would be helpful.
-\<close>
-
-
-subsubsection \<open>Code generation\<close>
+subsection \<open>Code generation\<close>
 
 context
 begin
diff -r 6bc3bb9d0e3e -r 8cb141384682 src/HOL/Matrix_LP/ComputeNumeral.thy
--- a/src/HOL/Matrix_LP/ComputeNumeral.thy	Fri Oct 28 06:34:25 2022 +0000
+++ b/src/HOL/Matrix_LP/ComputeNumeral.thy	Fri Oct 28 06:34:26 2022 +0000
@@ -51,10 +51,10 @@
   one_div_minus_numeral one_mod_minus_numeral
   numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
   numeral_div_minus_numeral numeral_mod_minus_numeral
-  div_minus_minus mod_minus_minus Euclidean_Division.adjust_div_eq of_bool_eq one_neq_zero
+  div_minus_minus mod_minus_minus Parity.adjust_div_eq of_bool_eq one_neq_zero
   numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
   divmod_steps divmod_cancel divmod_step_def fst_conv snd_conv numeral_One
-  case_prod_beta rel_simps Euclidean_Division.adjust_mod_def div_minus1_right mod_minus1_right
+  case_prod_beta rel_simps Parity.adjust_mod_def div_minus1_right mod_minus1_right
   minus_minus numeral_times_numeral mult_zero_right mult_1_right
 
 
diff -r 6bc3bb9d0e3e -r 8cb141384682 src/HOL/Parity.thy
--- a/src/HOL/Parity.thy	Fri Oct 28 06:34:25 2022 +0000
+++ b/src/HOL/Parity.thy	Fri Oct 28 06:34:26 2022 +0000
@@ -23,6 +23,24 @@
 abbreviation odd :: "'a \<Rightarrow> bool"
   where "odd a \<equiv> \<not> 2 dvd a"
 
+end
+
+class ring_parity = ring + semiring_parity
+begin
+
+subclass comm_ring_1 ..
+
+end
+
+instance nat :: semiring_parity
+  by standard (simp_all add: dvd_eq_mod_eq_0)
+
+instance int :: ring_parity
+  by standard (auto simp add: dvd_eq_mod_eq_0)
+
+context semiring_parity
+begin
+
 lemma parity_cases [case_names even odd]:
   assumes "even a \<Longrightarrow> a mod 2 = 0 \<Longrightarrow> P"
   assumes "odd a \<Longrightarrow> a mod 2 = 1 \<Longrightarrow> P"
@@ -159,6 +177,10 @@
 lemma even_power [simp]: "even (a ^ n) \<longleftrightarrow> even a \<and> n > 0"
   by (induct n) auto
 
+lemma even_prod_iff:
+  \<open>even (prod f A) \<longleftrightarrow> (\<exists>a\<in>A. even (f a))\<close> if \<open>finite A\<close>
+  using that by (induction A) simp_all
+
 lemma mask_eq_sum_exp:
   \<open>2 ^ n - 1 = (\<Sum>m\<in>{q. q < n}. 2 ^ m)\<close>
 proof -
@@ -172,13 +194,15 @@
     by simp
 qed
 
+lemma (in -) mask_eq_sum_exp_nat:
+  \<open>2 ^ n - Suc 0 = (\<Sum>m\<in>{q. q < n}. 2 ^ m)\<close>
+  using mask_eq_sum_exp [where ?'a = nat] by simp
+
 end
 
-class ring_parity = ring + semiring_parity
+context ring_parity
 begin
 
-subclass comm_ring_1 ..
-
 lemma even_minus:
   "even (- a) \<longleftrightarrow> even a"
   by (fact dvd_minus_iff)
@@ -190,135 +214,8 @@
 end
 
 
-subsection \<open>Special case: euclidean rings containing the natural numbers\<close>
-
-context unique_euclidean_semiring_with_nat
-begin
-
-subclass semiring_parity
-proof
-  show "2 dvd a \<longleftrightarrow> a mod 2 = 0" for a
-    by (fact dvd_eq_mod_eq_0)
-  show "\<not> 2 dvd a \<longleftrightarrow> a mod 2 = 1" for a
-  proof
-    assume "a mod 2 = 1"
-    then show "\<not> 2 dvd a"
-      by auto
-  next
-    assume "\<not> 2 dvd a"
-    have eucl: "euclidean_size (a mod 2) = 1"
-    proof (rule order_antisym)
-      show "euclidean_size (a mod 2) \<le> 1"
-        using mod_size_less [of 2 a] by simp
-      show "1 \<le> euclidean_size (a mod 2)"
-        using \<open>\<not> 2 dvd a\<close> by (simp add: Suc_le_eq dvd_eq_mod_eq_0)
-    qed 
-    from \<open>\<not> 2 dvd a\<close> have "\<not> of_nat 2 dvd division_segment a * of_nat (euclidean_size a)"
-      by simp
-    then have "\<not> of_nat 2 dvd of_nat (euclidean_size a)"
-      by (auto simp only: dvd_mult_unit_iff' is_unit_division_segment)
-    then have "\<not> 2 dvd euclidean_size a"
-      using of_nat_dvd_iff [of 2] by simp
-    then have "euclidean_size a mod 2 = 1"
-      by (simp add: semidom_modulo_class.dvd_eq_mod_eq_0)
-    then have "of_nat (euclidean_size a mod 2) = of_nat 1"
-      by simp
-    then have "of_nat (euclidean_size a) mod 2 = 1"
-      by (simp add: of_nat_mod)
-    from \<open>\<not> 2 dvd a\<close> eucl
-    show "a mod 2 = 1"
-      by (auto intro: division_segment_eq_iff simp add: division_segment_mod)
-  qed
-  show "\<not> is_unit 2"
-  proof (rule notI)
-    assume "is_unit 2"
-    then have "of_nat 2 dvd of_nat 1"
-      by simp
-    then have "is_unit (2::nat)"
-      by (simp only: of_nat_dvd_iff)
-    then show False
-      by simp
-  qed
-qed
-
-lemma even_succ_div_two [simp]:
-  "even a \<Longrightarrow> (a + 1) div 2 = a div 2"
-  by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)
-
-lemma odd_succ_div_two [simp]:
-  "odd a \<Longrightarrow> (a + 1) div 2 = a div 2 + 1"
-  by (auto elim!: oddE simp add: add.assoc)
-
-lemma even_two_times_div_two:
-  "even a \<Longrightarrow> 2 * (a div 2) = a"
-  by (fact dvd_mult_div_cancel)
-
-lemma odd_two_times_div_two_succ [simp]:
-  "odd a \<Longrightarrow> 2 * (a div 2) + 1 = a"
-  using mult_div_mod_eq [of 2 a]
-  by (simp add: even_iff_mod_2_eq_zero)
-
-lemma coprime_left_2_iff_odd [simp]:
-  "coprime 2 a \<longleftrightarrow> odd a"
-proof
-  assume "odd a"
-  show "coprime 2 a"
-  proof (rule coprimeI)
-    fix b
-    assume "b dvd 2" "b dvd a"
-    then have "b dvd a mod 2"
-      by (auto intro: dvd_mod)
-    with \<open>odd a\<close> show "is_unit b"
-      by (simp add: mod_2_eq_odd)
-  qed
-next
-  assume "coprime 2 a"
-  show "odd a"
-  proof (rule notI)
-    assume "even a"
-    then obtain b where "a = 2 * b" ..
-    with \<open>coprime 2 a\<close> have "coprime 2 (2 * b)"
-      by simp
-    moreover have "\<not> coprime 2 (2 * b)"
-      by (rule not_coprimeI [of 2]) simp_all
-    ultimately show False
-      by blast
-  qed
-qed
-
-lemma coprime_right_2_iff_odd [simp]:
-  "coprime a 2 \<longleftrightarrow> odd a"
-  using coprime_left_2_iff_odd [of a] by (simp add: ac_simps)
-
-end
-
-context unique_euclidean_ring_with_nat
-begin
-
-subclass ring_parity ..
-
-lemma minus_1_mod_2_eq [simp]:
-  "- 1 mod 2 = 1"
-  by (simp add: mod_2_eq_odd)
-
-lemma minus_1_div_2_eq [simp]:
-  "- 1 div 2 = - 1"
-proof -
-  from div_mult_mod_eq [of "- 1" 2]
-  have "- 1 div 2 * 2 = - 1 * 2"
-    using add_implies_diff by fastforce
-  then show ?thesis
-    using mult_right_cancel [of 2 "- 1 div 2" "- 1"] by simp
-qed
-
-end
-
-
 subsection \<open>Instance for \<^typ>\<open>nat\<close>\<close>
 
-instance nat :: unique_euclidean_semiring_with_nat
-  by standard (simp_all add: dvd_eq_mod_eq_0)
-
 lemma even_Suc_Suc_iff [simp]:
   "even (Suc (Suc n)) \<longleftrightarrow> even n"
   using dvd_add_triv_right_iff [of 2 n] by simp
@@ -361,18 +258,18 @@
 
 lemma even_Suc_div_two [simp]:
   "even n \<Longrightarrow> Suc n div 2 = n div 2"
-  using even_succ_div_two [of n] by simp
+  by auto
 
 lemma odd_Suc_div_two [simp]:
   "odd n \<Longrightarrow> Suc n div 2 = Suc (n div 2)"
-  using odd_succ_div_two [of n] by simp
+  by (auto elim: oddE)
 
 lemma odd_two_times_div_two_nat [simp]:
   assumes "odd n"
   shows "2 * (n div 2) = n - (1 :: nat)"
 proof -
   from assms have "2 * (n div 2) + 1 = n"
-    by (rule odd_two_times_div_two_succ)
+    by (auto elim: oddE)
   then have "Suc (2 * (n div 2)) - 1 = n - 1"
     by simp
   then show ?thesis
@@ -410,17 +307,9 @@
   qed
 qed
 
-lemma mask_eq_sum_exp_nat:
-  \<open>2 ^ n - Suc 0 = (\<Sum>m\<in>{q. q < n}. 2 ^ m)\<close>
-  using mask_eq_sum_exp [where ?'a = nat] by simp
-
 context semiring_parity
 begin
 
-lemma even_of_nat_iff [simp]:
-  "even (of_nat n) \<longleftrightarrow> even n"
-  by (induction n) simp_all
-
 lemma even_sum_iff:
   \<open>even (sum f A) \<longleftrightarrow> even (card {a\<in>A. odd (f a)})\<close> if \<open>finite A\<close>
 using that proof (induction A)
@@ -435,10 +324,6 @@
     by simp
 qed
 
-lemma even_prod_iff:
-  \<open>even (prod f A) \<longleftrightarrow> (\<exists>a\<in>A. even (f a))\<close> if \<open>finite A\<close>
-  using that by (induction A) simp_all
-
 lemma even_mask_iff [simp]:
   \<open>even (2 ^ n - 1) \<longleftrightarrow> n = 0\<close>
 proof (cases \<open>n = 0\<close>)
@@ -453,6 +338,10 @@
     by (auto simp add: mask_eq_sum_exp even_sum_iff)
 qed
 
+lemma even_of_nat_iff [simp]:
+  "even (of_nat n) \<longleftrightarrow> even n"
+  by (induction n) simp_all
+
 end
 
 
@@ -580,46 +469,9 @@
 
 end
 
-context unique_euclidean_semiring_with_nat
-begin
-
-lemma even_mask_div_iff':
-  \<open>even ((2 ^ m - 1) div 2 ^ n) \<longleftrightarrow> m \<le> n\<close>
-proof -
-  have \<open>even ((2 ^ m - 1) div 2 ^ n) \<longleftrightarrow> even (of_nat ((2 ^ m - Suc 0) div 2 ^ n))\<close>
-    by (simp only: of_nat_div) (simp add: of_nat_diff)
-  also have \<open>\<dots> \<longleftrightarrow> even ((2 ^ m - Suc 0) div 2 ^ n)\<close>
-    by simp
-  also have \<open>\<dots> \<longleftrightarrow> m \<le> n\<close>
-  proof (cases \<open>m \<le> n\<close>)
-    case True
-    then show ?thesis
-      by (simp add: Suc_le_lessD)
-  next
-    case False
-    then obtain r where r: \<open>m = n + Suc r\<close>
-      using less_imp_Suc_add by fastforce
-    from r have \<open>{q. q < m} \<inter> {q. 2 ^ n dvd (2::nat) ^ q} = {q. n \<le> q \<and> q < m}\<close>
-      by (auto simp add: dvd_power_iff_le)
-    moreover from r have \<open>{q. q < m} \<inter> {q. \<not> 2 ^ n dvd (2::nat) ^ q} = {q. q < n}\<close>
-      by (auto simp add: dvd_power_iff_le)
-    moreover from False have \<open>{q. n \<le> q \<and> q < m \<and> q \<le> n} = {n}\<close>
-      by auto
-    then have \<open>odd ((\<Sum>a\<in>{q. n \<le> q \<and> q < m}. 2 ^ a div (2::nat) ^ n) + sum ((^) 2) {q. q < n} div 2 ^ n)\<close>
-      by (simp_all add: euclidean_semiring_cancel_class.power_diff_power_eq semiring_parity_class.even_sum_iff not_less mask_eq_sum_exp_nat [symmetric])
-    ultimately have \<open>odd (sum ((^) (2::nat)) {q. q < m} div 2 ^ n)\<close>
-      by (subst euclidean_semiring_cancel_class.sum_div_partition) simp_all
-    with False show ?thesis
-      by (simp add: mask_eq_sum_exp_nat)
-  qed
-  finally show ?thesis .
-qed
-
-end
-
 
 subsection \<open>Instance for \<^typ>\<open>int\<close>\<close>
-
+  
 lemma even_diff_iff:
   "even (k - l) \<longleftrightarrow> even (k + l)" for k l :: int
   by (fact even_diff)
@@ -670,6 +522,820 @@
 end
 
 
+subsection \<open>Special case: euclidean rings containing the natural numbers\<close>
+
+class unique_euclidean_semiring_with_nat = semidom + semiring_char_0 + unique_euclidean_semiring +
+  assumes of_nat_div: "of_nat (m div n) = of_nat m div of_nat n"
+    and division_segment_of_nat [simp]: "division_segment (of_nat n) = 1"
+    and division_segment_euclidean_size [simp]: "division_segment a * of_nat (euclidean_size a) = a"
+begin
+
+lemma division_segment_eq_iff:
+  "a = b" if "division_segment a = division_segment b"
+    and "euclidean_size a = euclidean_size b"
+  using that division_segment_euclidean_size [of a] by simp
+
+lemma euclidean_size_of_nat [simp]:
+  "euclidean_size (of_nat n) = n"
+proof -
+  have "division_segment (of_nat n) * of_nat (euclidean_size (of_nat n)) = of_nat n"
+    by (fact division_segment_euclidean_size)
+  then show ?thesis by simp
+qed
+
+lemma of_nat_euclidean_size:
+  "of_nat (euclidean_size a) = a div division_segment a"
+proof -
+  have "of_nat (euclidean_size a) = division_segment a * of_nat (euclidean_size a) div division_segment a"
+    by (subst nonzero_mult_div_cancel_left) simp_all
+  also have "\<dots> = a div division_segment a"
+    by simp
+  finally show ?thesis .
+qed
+
+lemma division_segment_1 [simp]:
+  "division_segment 1 = 1"
+  using division_segment_of_nat [of 1] by simp
+
+lemma division_segment_numeral [simp]:
+  "division_segment (numeral k) = 1"
+  using division_segment_of_nat [of "numeral k"] by simp
+
+lemma euclidean_size_1 [simp]:
+  "euclidean_size 1 = 1"
+  using euclidean_size_of_nat [of 1] by simp
+
+lemma euclidean_size_numeral [simp]:
+  "euclidean_size (numeral k) = numeral k"
+  using euclidean_size_of_nat [of "numeral k"] by simp
+
+lemma of_nat_dvd_iff:
+  "of_nat m dvd of_nat n \<longleftrightarrow> m dvd n" (is "?P \<longleftrightarrow> ?Q")
+proof (cases "m = 0")
+  case True
+  then show ?thesis
+    by simp
+next
+  case False
+  show ?thesis
+  proof
+    assume ?Q
+    then show ?P
+      by auto
+  next
+    assume ?P
+    with False have "of_nat n = of_nat n div of_nat m * of_nat m"
+      by simp
+    then have "of_nat n = of_nat (n div m * m)"
+      by (simp add: of_nat_div)
+    then have "n = n div m * m"
+      by (simp only: of_nat_eq_iff)
+    then have "n = m * (n div m)"
+      by (simp add: ac_simps)
+    then show ?Q ..
+  qed
+qed
+
+lemma of_nat_mod:
+  "of_nat (m mod n) = of_nat m mod of_nat n"
+proof -
+  have "of_nat m div of_nat n * of_nat n + of_nat m mod of_nat n = of_nat m"
+    by (simp add: div_mult_mod_eq)
+  also have "of_nat m = of_nat (m div n * n + m mod n)"
+    by simp
+  finally show ?thesis
+    by (simp only: of_nat_div of_nat_mult of_nat_add) simp
+qed
+
+lemma one_div_two_eq_zero [simp]:
+  "1 div 2 = 0"
+proof -
+  from of_nat_div [symmetric] have "of_nat 1 div of_nat 2 = of_nat 0"
+    by (simp only:) simp
+  then show ?thesis
+    by simp
+qed
+
+lemma one_mod_two_eq_one [simp]:
+  "1 mod 2 = 1"
+proof -
+  from of_nat_mod [symmetric] have "of_nat 1 mod of_nat 2 = of_nat 1"
+    by (simp only:) simp
+  then show ?thesis
+    by simp
+qed
+
+lemma one_mod_2_pow_eq [simp]:
+  "1 mod (2 ^ n) = of_bool (n > 0)"
+proof -
+  have "1 mod (2 ^ n) = of_nat (1 mod (2 ^ n))"
+    using of_nat_mod [of 1 "2 ^ n"] by simp
+  also have "\<dots> = of_bool (n > 0)"
+    by simp
+  finally show ?thesis .
+qed
+
+lemma one_div_2_pow_eq [simp]:
+  "1 div (2 ^ n) = of_bool (n = 0)"
+  using div_mult_mod_eq [of 1 "2 ^ n"] by auto
+
+lemma div_mult2_eq':
+  \<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':
+  "a mod (of_nat m * of_nat n) = of_nat m * (a div of_nat m mod of_nat n) + a mod of_nat m"
+proof -
+  have "a div (of_nat m * of_nat n) * (of_nat m * of_nat n) + a mod (of_nat m * of_nat n) = a div of_nat m div of_nat n * of_nat n * of_nat m + (a div of_nat m mod of_nat n * of_nat m + a mod of_nat m)"
+    by (simp add: combine_common_factor div_mult_mod_eq)
+  moreover have "a div of_nat m div of_nat n * of_nat n * of_nat m = of_nat n * of_nat m * (a div of_nat m div of_nat n)"
+    by (simp add: ac_simps)
+  ultimately show ?thesis
+    by (simp add: div_mult2_eq' mult_commute)
+qed
+
+lemma div_mult2_numeral_eq:
+  "a div numeral k div numeral l = a div numeral (k * l)" (is "?A = ?B")
+proof -
+  have "?A = a div of_nat (numeral k) div of_nat (numeral l)"
+    by simp
+  also have "\<dots> = a div (of_nat (numeral k) * of_nat (numeral l))"
+    by (fact div_mult2_eq' [symmetric])
+  also have "\<dots> = ?B"
+    by simp
+  finally show ?thesis .
+qed
+
+lemma numeral_Bit0_div_2:
+  "numeral (num.Bit0 n) div 2 = numeral n"
+proof -
+  have "numeral (num.Bit0 n) = numeral n + numeral n"
+    by (simp only: numeral.simps)
+  also have "\<dots> = numeral n * 2"
+    by (simp add: mult_2_right)
+  finally have "numeral (num.Bit0 n) div 2 = numeral n * 2 div 2"
+    by simp
+  also have "\<dots> = numeral n"
+    by (rule nonzero_mult_div_cancel_right) simp
+  finally show ?thesis .
+qed
+
+lemma numeral_Bit1_div_2:
+  "numeral (num.Bit1 n) div 2 = numeral n"
+proof -
+  have "numeral (num.Bit1 n) = numeral n + numeral n + 1"
+    by (simp only: numeral.simps)
+  also have "\<dots> = numeral n * 2 + 1"
+    by (simp add: mult_2_right)
+  finally have "numeral (num.Bit1 n) div 2 = (numeral n * 2 + 1) div 2"
+    by simp
+  also have "\<dots> = numeral n * 2 div 2 + 1 div 2"
+    using dvd_triv_right by (rule div_plus_div_distrib_dvd_left)
+  also have "\<dots> = numeral n * 2 div 2"
+    by simp
+  also have "\<dots> = numeral n"
+    by (rule nonzero_mult_div_cancel_right) simp
+  finally show ?thesis .
+qed
+
+lemma exp_mod_exp:
+  \<open>2 ^ m mod 2 ^ n = of_bool (m < n) * 2 ^ m\<close>
+proof -
+  have \<open>(2::nat) ^ m mod 2 ^ n = of_bool (m < n) * 2 ^ m\<close> (is \<open>?lhs = ?rhs\<close>)
+    by (auto simp add: not_less monoid_mult_class.power_add dest!: le_Suc_ex)
+  then have \<open>of_nat ?lhs = of_nat ?rhs\<close>
+    by simp
+  then show ?thesis
+    by (simp add: of_nat_mod)
+qed
+
+lemma mask_mod_exp:
+  \<open>(2 ^ n - 1) mod 2 ^ m = 2 ^ min m n - 1\<close>
+proof -
+  have \<open>(2 ^ n - 1) mod 2 ^ m = 2 ^ min m n - (1::nat)\<close> (is \<open>?lhs = ?rhs\<close>)
+  proof (cases \<open>n \<le> m\<close>)
+    case True
+    then show ?thesis
+      by (simp add: Suc_le_lessD)
+  next
+    case False
+    then have \<open>m < n\<close>
+      by simp
+    then obtain q where n: \<open>n = Suc q + m\<close>
+      by (auto dest: less_imp_Suc_add)
+    then have \<open>min m n = m\<close>
+      by simp
+    moreover have \<open>(2::nat) ^ m \<le> 2 * 2 ^ q * 2 ^ m\<close>
+      using mult_le_mono1 [of 1 \<open>2 * 2 ^ q\<close> \<open>2 ^ m\<close>] by simp
+    with n have \<open>2 ^ n - 1 = (2 ^ Suc q - 1) * 2 ^ m + (2 ^ m - (1::nat))\<close>
+      by (simp add: monoid_mult_class.power_add algebra_simps)
+    ultimately show ?thesis
+      by (simp only: euclidean_semiring_cancel_class.mod_mult_self3) simp
+  qed
+  then have \<open>of_nat ?lhs = of_nat ?rhs\<close>
+    by simp
+  then show ?thesis
+    by (simp add: of_nat_mod of_nat_diff)
+qed
+
+lemma of_bool_half_eq_0 [simp]:
+  \<open>of_bool b div 2 = 0\<close>
+  by simp
+
+end
+
+class unique_euclidean_ring_with_nat = ring + unique_euclidean_semiring_with_nat
+
+instance nat :: unique_euclidean_semiring_with_nat
+  by standard (simp_all add: dvd_eq_mod_eq_0)
+
+instance int :: unique_euclidean_ring_with_nat
+  by standard (auto simp add: divide_int_def division_segment_int_def elim: contrapos_np)
+
+
+context unique_euclidean_semiring_with_nat
+begin
+
+subclass semiring_parity
+proof
+  show "2 dvd a \<longleftrightarrow> a mod 2 = 0" for a
+    by (fact dvd_eq_mod_eq_0)
+  show "\<not> 2 dvd a \<longleftrightarrow> a mod 2 = 1" for a
+  proof
+    assume "a mod 2 = 1"
+    then show "\<not> 2 dvd a"
+      by auto
+  next
+    assume "\<not> 2 dvd a"
+    have eucl: "euclidean_size (a mod 2) = 1"
+    proof (rule order_antisym)
+      show "euclidean_size (a mod 2) \<le> 1"
+        using mod_size_less [of 2 a] by simp
+      show "1 \<le> euclidean_size (a mod 2)"
+        using \<open>\<not> 2 dvd a\<close> by (simp add: Suc_le_eq dvd_eq_mod_eq_0)
+    qed 
+    from \<open>\<not> 2 dvd a\<close> have "\<not> of_nat 2 dvd division_segment a * of_nat (euclidean_size a)"
+      by simp
+    then have "\<not> of_nat 2 dvd of_nat (euclidean_size a)"
+      by (auto simp only: dvd_mult_unit_iff' is_unit_division_segment)
+    then have "\<not> 2 dvd euclidean_size a"
+      using of_nat_dvd_iff [of 2] by simp
+    then have "euclidean_size a mod 2 = 1"
+      by (simp add: semidom_modulo_class.dvd_eq_mod_eq_0)
+    then have "of_nat (euclidean_size a mod 2) = of_nat 1"
+      by simp
+    then have "of_nat (euclidean_size a) mod 2 = 1"
+      by (simp add: of_nat_mod)
+    from \<open>\<not> 2 dvd a\<close> eucl
+    show "a mod 2 = 1"
+      by (auto intro: division_segment_eq_iff simp add: division_segment_mod)
+  qed
+  show "\<not> is_unit 2"
+  proof (rule notI)
+    assume "is_unit 2"
+    then have "of_nat 2 dvd of_nat 1"
+      by simp
+    then have "is_unit (2::nat)"
+      by (simp only: of_nat_dvd_iff)
+    then show False
+      by simp
+  qed
+qed
+
+lemma even_succ_div_two [simp]:
+  "even a \<Longrightarrow> (a + 1) div 2 = a div 2"
+  by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)
+
+lemma odd_succ_div_two [simp]:
+  "odd a \<Longrightarrow> (a + 1) div 2 = a div 2 + 1"
+  by (auto elim!: oddE simp add: add.assoc)
+
+lemma even_two_times_div_two:
+  "even a \<Longrightarrow> 2 * (a div 2) = a"
+  by (fact dvd_mult_div_cancel)
+
+lemma odd_two_times_div_two_succ [simp]:
+  "odd a \<Longrightarrow> 2 * (a div 2) + 1 = a"
+  using mult_div_mod_eq [of 2 a]
+  by (simp add: even_iff_mod_2_eq_zero)
+
+lemma coprime_left_2_iff_odd [simp]:
+  "coprime 2 a \<longleftrightarrow> odd a"
+proof
+  assume "odd a"
+  show "coprime 2 a"
+  proof (rule coprimeI)
+    fix b
+    assume "b dvd 2" "b dvd a"
+    then have "b dvd a mod 2"
+      by (auto intro: dvd_mod)
+    with \<open>odd a\<close> show "is_unit b"
+      by (simp add: mod_2_eq_odd)
+  qed
+next
+  assume "coprime 2 a"
+  show "odd a"
+  proof (rule notI)
+    assume "even a"
+    then obtain b where "a = 2 * b" ..
+    with \<open>coprime 2 a\<close> have "coprime 2 (2 * b)"
+      by simp
+    moreover have "\<not> coprime 2 (2 * b)"
+      by (rule not_coprimeI [of 2]) simp_all
+    ultimately show False
+      by blast
+  qed
+qed
+
+lemma coprime_right_2_iff_odd [simp]:
+  "coprime a 2 \<longleftrightarrow> odd a"
+  using coprime_left_2_iff_odd [of a] by (simp add: ac_simps)
+
+end
+
+context unique_euclidean_ring_with_nat
+begin
+
+subclass ring_parity ..
+
+lemma minus_1_mod_2_eq [simp]:
+  "- 1 mod 2 = 1"
+  by (simp add: mod_2_eq_odd)
+
+lemma minus_1_div_2_eq [simp]:
+  "- 1 div 2 = - 1"
+proof -
+  from div_mult_mod_eq [of "- 1" 2]
+  have "- 1 div 2 * 2 = - 1 * 2"
+    using add_implies_diff by fastforce
+  then show ?thesis
+    using mult_right_cancel [of 2 "- 1 div 2" "- 1"] by simp
+qed
+
+end
+
+context unique_euclidean_semiring_with_nat
+begin
+
+lemma even_mask_div_iff':
+  \<open>even ((2 ^ m - 1) div 2 ^ n) \<longleftrightarrow> m \<le> n\<close>
+proof -
+  have \<open>even ((2 ^ m - 1) div 2 ^ n) \<longleftrightarrow> even (of_nat ((2 ^ m - Suc 0) div 2 ^ n))\<close>
+    by (simp only: of_nat_div) (simp add: of_nat_diff)
+  also have \<open>\<dots> \<longleftrightarrow> even ((2 ^ m - Suc 0) div 2 ^ n)\<close>
+    by simp
+  also have \<open>\<dots> \<longleftrightarrow> m \<le> n\<close>
+  proof (cases \<open>m \<le> n\<close>)
+    case True
+    then show ?thesis
+      by (simp add: Suc_le_lessD)
+  next
+    case False
+    then obtain r where r: \<open>m = n + Suc r\<close>
+      using less_imp_Suc_add by fastforce
+    from r have \<open>{q. q < m} \<inter> {q. 2 ^ n dvd (2::nat) ^ q} = {q. n \<le> q \<and> q < m}\<close>
+      by (auto simp add: dvd_power_iff_le)
+    moreover from r have \<open>{q. q < m} \<inter> {q. \<not> 2 ^ n dvd (2::nat) ^ q} = {q. q < n}\<close>
+      by (auto simp add: dvd_power_iff_le)
+    moreover from False have \<open>{q. n \<le> q \<and> q < m \<and> q \<le> n} = {n}\<close>
+      by auto
+    then have \<open>odd ((\<Sum>a\<in>{q. n \<le> q \<and> q < m}. 2 ^ a div (2::nat) ^ n) + sum ((^) 2) {q. q < n} div 2 ^ n)\<close>
+      by (simp_all add: euclidean_semiring_cancel_class.power_diff_power_eq semiring_parity_class.even_sum_iff not_less mask_eq_sum_exp_nat [symmetric])
+    ultimately have \<open>odd (sum ((^) (2::nat)) {q. q < m} div 2 ^ n)\<close>
+      by (subst euclidean_semiring_cancel_class.sum_div_partition) simp_all
+    with False show ?thesis
+      by (simp add: mask_eq_sum_exp_nat)
+  qed
+  finally show ?thesis .
+qed
+
+end
+
+
+subsection \<open>Generic symbolic computations\<close>
+
+text \<open>
+  The following type class contains everything necessary to formulate
+  a division algorithm in ring structures with numerals, restricted
+  to its positive segments.
+\<close>
+
+class unique_euclidean_semiring_with_nat_division = unique_euclidean_semiring_with_nat +
+  fixes divmod :: \<open>num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a\<close>
+    and divmod_step :: \<open>'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a\<close> \<comment> \<open>
+      These are conceptually definitions but force generated code
+      to be monomorphic wrt. particular instances of this class which
+      yields a significant speedup.\<close>
+  assumes divmod_def: \<open>divmod m n = (numeral m div numeral n, numeral m mod numeral n)\<close>
+    and divmod_step_def [simp]: \<open>divmod_step l (q, r) =
+      (if euclidean_size l \<le> euclidean_size r then (2 * q + 1, r - l)
+       else (2 * q, r))\<close> \<comment> \<open>
+         This is a formulation of one step (referring to one digit position)
+         in school-method division: compare the dividend at the current
+         digit position with the remainder from previous division steps
+         and evaluate accordingly.\<close>
+begin
+
+lemma fst_divmod:
+  \<open>fst (divmod m n) = numeral m div numeral n\<close>
+  by (simp add: divmod_def)
+
+lemma snd_divmod:
+  \<open>snd (divmod m n) = numeral m mod numeral n\<close>
+  by (simp add: divmod_def)
+
+text \<open>
+  Following a formulation of school-method division.
+  If the divisor is smaller than the dividend, terminate.
+  If not, shift the dividend to the right until termination
+  occurs and then reiterate single division steps in the
+  opposite direction.
+\<close>
+
+lemma divmod_divmod_step:
+  \<open>divmod m n = (if m < n then (0, numeral m)
+    else divmod_step (numeral n) (divmod m (Num.Bit0 n)))\<close>
+proof (cases \<open>m < n\<close>)
+  case True
+  then show ?thesis
+    by (simp add: prod_eq_iff fst_divmod snd_divmod flip: of_nat_numeral of_nat_div of_nat_mod)
+next
+  case False
+  define r s t where \<open>r = (numeral m :: nat)\<close> \<open>s = (numeral n :: nat)\<close> \<open>t = 2 * s\<close>
+  then have *: \<open>numeral m = of_nat r\<close> \<open>numeral n = of_nat s\<close> \<open>numeral (num.Bit0 n) = of_nat t\<close>
+    and \<open>\<not> s \<le> r mod s\<close>
+    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>)
+      (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
+  from \<open>\<not> s \<le> r mod s\<close> have \<open>s \<le> r mod t \<Longrightarrow>
+     r div s = Suc (2 * (r div t)) \<and>
+     r mod s = r mod t - s\<close>
+    using rs
+    by (auto simp add: t)
+  moreover have \<open>r mod t < s \<Longrightarrow>
+     r div s = 2 * (r div t) \<and>
+     r mod s = r mod t\<close>
+    using rs
+    by (auto simp add: t)
+  ultimately show ?thesis
+    by (simp add: divmod_def prod_eq_iff split_def Let_def
+        not_less mod_eq_0_iff_dvd Rings.mod_eq_0_iff_dvd False not_le *)
+    (simp add: flip: of_nat_numeral of_nat_mult add.commute [of 1] of_nat_div of_nat_mod of_nat_Suc of_nat_diff)
+qed
+
+text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
+
+lemma divmod_trivial [simp]:
+  "divmod m Num.One = (numeral m, 0)"
+  "divmod num.One (num.Bit0 n) = (0, Numeral1)"
+  "divmod num.One (num.Bit1 n) = (0, Numeral1)"
+  using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
+
+text \<open>Division by an even number is a right-shift\<close>
+
+lemma divmod_cancel [simp]:
+  \<open>divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))\<close> (is ?P)
+  \<open>divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))\<close> (is ?Q)
+proof -
+  define r s where \<open>r = (numeral m :: nat)\<close> \<open>s = (numeral n :: nat)\<close>
+  then have *: \<open>numeral m = of_nat r\<close> \<open>numeral n = of_nat s\<close>
+    \<open>numeral (num.Bit0 m) = of_nat (2 * r)\<close> \<open>numeral (num.Bit0 n) = of_nat (2 * s)\<close>
+    \<open>numeral (num.Bit1 m) = of_nat (Suc (2 * r))\<close>
+    by simp_all
+  have **: \<open>Suc (2 * r) div 2 = r\<close>
+    by simp
+  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] **)
+qed
+
+text \<open>The really hard work\<close>
+
+lemma divmod_steps [simp]:
+  "divmod (num.Bit0 m) (num.Bit1 n) =
+      (if m \<le> n then (0, numeral (num.Bit0 m))
+       else divmod_step (numeral (num.Bit1 n))
+             (divmod (num.Bit0 m)
+               (num.Bit0 (num.Bit1 n))))"
+  "divmod (num.Bit1 m) (num.Bit1 n) =
+      (if m < n then (0, numeral (num.Bit1 m))
+       else divmod_step (numeral (num.Bit1 n))
+             (divmod (num.Bit1 m)
+               (num.Bit0 (num.Bit1 n))))"
+  by (simp_all add: divmod_divmod_step)
+
+lemmas divmod_algorithm_code = divmod_trivial divmod_cancel divmod_steps
+
+text \<open>Special case: divisibility\<close>
+
+definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
+where
+  "divides_aux qr \<longleftrightarrow> snd qr = 0"
+
+lemma divides_aux_eq [simp]:
+  "divides_aux (q, r) \<longleftrightarrow> r = 0"
+  by (simp add: divides_aux_def)
+
+lemma dvd_numeral_simp [simp]:
+  "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
+  by (simp add: divmod_def mod_eq_0_iff_dvd)
+
+text \<open>Generic computation of quotient and remainder\<close>
+
+lemma numeral_div_numeral [simp]:
+  "numeral k div numeral l = fst (divmod k l)"
+  by (simp add: fst_divmod)
+
+lemma numeral_mod_numeral [simp]:
+  "numeral k mod numeral l = snd (divmod k l)"
+  by (simp add: snd_divmod)
+
+lemma one_div_numeral [simp]:
+  "1 div numeral n = fst (divmod num.One n)"
+  by (simp add: fst_divmod)
+
+lemma one_mod_numeral [simp]:
+  "1 mod numeral n = snd (divmod num.One n)"
+  by (simp add: snd_divmod)
+
+end
+
+instantiation nat :: unique_euclidean_semiring_with_nat_division
+begin
+
+definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
+where
+  divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
+
+definition divmod_step_nat :: "nat \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
+where
+  "divmod_step_nat l qr = (let (q, r) = qr
+    in if r \<ge> l then (2 * q + 1, r - l)
+    else (2 * q, r))"
+
+instance
+  by standard (simp_all add: divmod'_nat_def divmod_step_nat_def)
+
+end
+
+declare divmod_algorithm_code [where ?'a = nat, code]
+
+lemma Suc_0_div_numeral [simp]:
+  \<open>Suc 0 div numeral Num.One = 1\<close>
+  \<open>Suc 0 div numeral (Num.Bit0 n) = 0\<close>
+  \<open>Suc 0 div numeral (Num.Bit1 n) = 0\<close>
+  by simp_all
+
+lemma Suc_0_mod_numeral [simp]:
+  \<open>Suc 0 mod numeral Num.One = 0\<close>
+  \<open>Suc 0 mod numeral (Num.Bit0 n) = 1\<close>
+  \<open>Suc 0 mod numeral (Num.Bit1 n) = 1\<close>
+  by simp_all
+
+instantiation int :: unique_euclidean_semiring_with_nat_division
+begin
+
+definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
+where
+  "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
+
+definition divmod_step_int :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
+where
+  "divmod_step_int l qr = (let (q, r) = qr
+    in if \<bar>l\<bar> \<le> \<bar>r\<bar> then (2 * q + 1, r - l)
+    else (2 * q, r))"
+
+instance
+  by standard (auto simp add: divmod_int_def divmod_step_int_def)
+
+end
+
+declare divmod_algorithm_code [where ?'a = int, code]
+
+context
+begin
+
+qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
+where
+  "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
+
+qualified lemma adjust_div_eq [simp, code]:
+  "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
+  by (simp add: adjust_div_def)
+
+qualified definition adjust_mod :: "num \<Rightarrow> int \<Rightarrow> int"
+where
+  [simp]: "adjust_mod l r = (if r = 0 then 0 else numeral l - r)"
+
+lemma minus_numeral_div_numeral [simp]:
+  "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
+proof -
+  have "int (fst (divmod m n)) = fst (divmod m n)"
+    by (simp only: fst_divmod divide_int_def) auto
+  then show ?thesis
+    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
+qed
+
+lemma minus_numeral_mod_numeral [simp]:
+  "- numeral m mod numeral n = adjust_mod n (snd (divmod m n) :: int)"
+proof (cases "snd (divmod m n) = (0::int)")
+  case True
+  then show ?thesis
+    by (simp add: mod_eq_0_iff_dvd divides_aux_def)
+next
+  case False
+  then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
+    by (simp only: snd_divmod modulo_int_def) auto
+  then show ?thesis
+    by (simp add: divides_aux_def adjust_div_def)
+      (simp add: divides_aux_def modulo_int_def)
+qed
+
+lemma numeral_div_minus_numeral [simp]:
+  "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
+proof -
+  have "int (fst (divmod m n)) = fst (divmod m n)"
+    by (simp only: fst_divmod divide_int_def) auto
+  then show ?thesis
+    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
+qed
+
+lemma numeral_mod_minus_numeral [simp]:
+  "numeral m mod - numeral n = - adjust_mod n (snd (divmod m n) :: int)"
+proof (cases "snd (divmod m n) = (0::int)")
+  case True
+  then show ?thesis
+    by (simp add: mod_eq_0_iff_dvd divides_aux_def)
+next
+  case False
+  then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
+    by (simp only: snd_divmod modulo_int_def) auto
+  then show ?thesis
+    by (simp add: divides_aux_def adjust_div_def)
+      (simp add: divides_aux_def modulo_int_def)
+qed
+
+lemma minus_one_div_numeral [simp]:
+  "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
+  using minus_numeral_div_numeral [of Num.One n] by simp
+
+lemma minus_one_mod_numeral [simp]:
+  "- 1 mod numeral n = adjust_mod n (snd (divmod Num.One n) :: int)"
+  using minus_numeral_mod_numeral [of Num.One n] by simp
+
+lemma one_div_minus_numeral [simp]:
+  "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
+  using numeral_div_minus_numeral [of Num.One n] by simp
+
+lemma one_mod_minus_numeral [simp]:
+  "1 mod - numeral n = - adjust_mod n (snd (divmod Num.One n) :: int)"
+  using numeral_mod_minus_numeral [of Num.One n] by simp
+
+lemma [code]:
+  fixes k :: int
+  shows
+    "k div 0 = 0"
+    "k mod 0 = k"
+    "0 div k = 0"
+    "0 mod k = 0"
+    "k div Int.Pos Num.One = k"
+    "k mod Int.Pos Num.One = 0"
+    "k div Int.Neg Num.One = - k"
+    "k mod Int.Neg Num.One = 0"
+    "Int.Pos m div Int.Pos n = (fst (divmod m n) :: int)"
+    "Int.Pos m mod Int.Pos n = (snd (divmod m n) :: int)"
+    "Int.Neg m div Int.Pos n = - (adjust_div (divmod m n) :: int)"
+    "Int.Neg m mod Int.Pos n = adjust_mod n (snd (divmod m n) :: int)"
+    "Int.Pos m div Int.Neg n = - (adjust_div (divmod m n) :: int)"
+    "Int.Pos m mod Int.Neg n = - adjust_mod n (snd (divmod m n) :: int)"
+    "Int.Neg m div Int.Neg n = (fst (divmod m n) :: int)"
+    "Int.Neg m mod Int.Neg n = - (snd (divmod m n) :: int)"
+  by simp_all
+
+end
+
+lemma divmod_BitM_2_eq [simp]:
+  \<open>divmod (Num.BitM m) (Num.Bit0 Num.One) = (numeral m - 1, (1 :: int))\<close>
+  by (cases m) simp_all
+
+
+subsubsection \<open>Computation by simplification\<close>
+
+lemma euclidean_size_nat_less_eq_iff:
+  \<open>euclidean_size m \<le> euclidean_size n \<longleftrightarrow> m \<le> n\<close> for m n :: nat
+  by simp
+
+lemma euclidean_size_int_less_eq_iff:
+  \<open>euclidean_size k \<le> euclidean_size l \<longleftrightarrow> \<bar>k\<bar> \<le> \<bar>l\<bar>\<close> for k l :: int
+  by auto
+
+simproc_setup numeral_divmod
+  ("0 div 0 :: 'a :: unique_euclidean_semiring_with_nat_division" | "0 mod 0 :: 'a :: unique_euclidean_semiring_with_nat_division" |
+   "0 div 1 :: 'a :: unique_euclidean_semiring_with_nat_division" | "0 mod 1 :: 'a :: unique_euclidean_semiring_with_nat_division" |
+   "0 div - 1 :: int" | "0 mod - 1 :: int" |
+   "0 div numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" | "0 mod numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" |
+   "0 div - numeral b :: int" | "0 mod - numeral b :: int" |
+   "1 div 0 :: 'a :: unique_euclidean_semiring_with_nat_division" | "1 mod 0 :: 'a :: unique_euclidean_semiring_with_nat_division" |
+   "1 div 1 :: 'a :: unique_euclidean_semiring_with_nat_division" | "1 mod 1 :: 'a :: unique_euclidean_semiring_with_nat_division" |
+   "1 div - 1 :: int" | "1 mod - 1 :: int" |
+   "1 div numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" | "1 mod numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" |
+   "1 div - numeral b :: int" |"1 mod - numeral b :: int" |
+   "- 1 div 0 :: int" | "- 1 mod 0 :: int" | "- 1 div 1 :: int" | "- 1 mod 1 :: int" |
+   "- 1 div - 1 :: int" | "- 1 mod - 1 :: int" | "- 1 div numeral b :: int" | "- 1 mod numeral b :: int" |
+   "- 1 div - numeral b :: int" | "- 1 mod - numeral b :: int" |
+   "numeral a div 0 :: 'a :: unique_euclidean_semiring_with_nat_division" | "numeral a mod 0 :: 'a :: unique_euclidean_semiring_with_nat_division" |
+   "numeral a div 1 :: 'a :: unique_euclidean_semiring_with_nat_division" | "numeral a mod 1 :: 'a :: unique_euclidean_semiring_with_nat_division" |
+   "numeral a div - 1 :: int" | "numeral a mod - 1 :: int" |
+   "numeral a div numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" | "numeral a mod numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" |
+   "numeral a div - numeral b :: int" | "numeral a mod - numeral b :: int" |
+   "- numeral a div 0 :: int" | "- numeral a mod 0 :: int" |
+   "- numeral a div 1 :: int" | "- numeral a mod 1 :: int" |
+   "- numeral a div - 1 :: int" | "- numeral a mod - 1 :: int" |
+   "- numeral a div numeral b :: int" | "- numeral a mod numeral b :: int" |
+   "- numeral a div - numeral b :: int" | "- numeral a mod - numeral b :: int") = \<open>
+  let
+    val if_cong = the (Code.get_case_cong \<^theory> \<^const_name>\<open>If\<close>);
+    fun successful_rewrite ctxt ct =
+      let
+        val thm = Simplifier.rewrite ctxt ct
+      in if Thm.is_reflexive thm then NONE else SOME thm end;
+  in fn phi =>
+    let
+      val simps = Morphism.fact phi (@{thms div_0 mod_0 div_by_0 mod_by_0 div_by_1 mod_by_1
+        one_div_numeral one_mod_numeral minus_one_div_numeral minus_one_mod_numeral
+        one_div_minus_numeral one_mod_minus_numeral
+        numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
+        numeral_div_minus_numeral numeral_mod_minus_numeral
+        div_minus_minus mod_minus_minus Parity.adjust_div_eq of_bool_eq one_neq_zero
+        numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
+        divmod_cancel divmod_steps divmod_step_def fst_conv snd_conv numeral_One
+        case_prod_beta rel_simps Parity.adjust_mod_def div_minus1_right mod_minus1_right
+        minus_minus numeral_times_numeral mult_zero_right mult_1_right
+        euclidean_size_nat_less_eq_iff euclidean_size_int_less_eq_iff diff_nat_numeral nat_numeral}
+        @ [@{lemma "0 = 0 \<longleftrightarrow> True" by simp}]);
+      fun prepare_simpset ctxt = HOL_ss |> Simplifier.simpset_map ctxt
+        (Simplifier.add_cong if_cong #> fold Simplifier.add_simp simps)
+    in fn ctxt => successful_rewrite (Simplifier.put_simpset (prepare_simpset ctxt) ctxt) end
+  end
+\<close> \<comment> \<open>
+  There is space for improvement here: the calculation itself
+  could be carried out outside the logic, and a generic simproc
+  (simplifier setup) for generic calculation would be helpful.
+\<close>
+
+
 subsection \<open>Computing congruences modulo \<open>2 ^ q\<close>\<close>
 
 context unique_euclidean_semiring_with_nat_division