src/HOL/Divides.thy

Fixed splitting of div and mod on integers (split theorem differed from implementation).

(*  Title:      HOL/Divides.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1999  University of Cambridge
*)

header {* The division operators div and mod *}

theory Divides
imports Nat_Numeral Nat_Transfer
uses "~~/src/Provers/Arith/cancel_div_mod.ML"
begin

subsection {* Syntactic division operations *}

class div = dvd +
  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
    and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)


subsection {* Abstract division in commutative semirings. *}

class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +
  assumes mod_div_equality: "a div b * b + a mod b = a"
    and div_by_0 [simp]: "a div 0 = 0"
    and div_0 [simp]: "0 div a = 0"
    and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
    and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
begin

text {* @{const div} and @{const mod} *}

lemma mod_div_equality2: "b * (a div b) + a mod b = a"
  unfolding mult_commute [of b]
  by (rule mod_div_equality)

lemma mod_div_equality': "a mod b + a div b * b = a"
  using mod_div_equality [of a b]
  by (simp only: add_ac)

lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
  by (simp add: mod_div_equality)

lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
  by (simp add: mod_div_equality2)

lemma mod_by_0 [simp]: "a mod 0 = a"
  using mod_div_equality [of a zero] by simp

lemma mod_0 [simp]: "0 mod a = 0"
  using mod_div_equality [of zero a] div_0 by simp

lemma div_mult_self2 [simp]:
  assumes "b \<noteq> 0"
  shows "(a + b * c) div b = c + a div b"
  using assms div_mult_self1 [of b a c] by (simp add: mult_commute)

lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
proof (cases "b = 0")
  case True then show ?thesis by simp
next
  case False
  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
    by (simp add: mod_div_equality)
  also from False div_mult_self1 [of b a c] have
    "\<dots> = (c + a div b) * b + (a + c * b) mod b"
      by (simp add: algebra_simps)
  finally have "a = a div b * b + (a + c * b) mod b"
    by (simp add: add_commute [of a] add_assoc left_distrib)
  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
    by (simp add: mod_div_equality)
  then show ?thesis by simp
qed

lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"
  by (simp add: mult_commute [of b])

lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
  using div_mult_self2 [of b 0 a] by simp

lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
  using div_mult_self1 [of b 0 a] by simp

lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
  using mod_mult_self2 [of 0 b a] by simp

lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
  using mod_mult_self1 [of 0 a b] by simp

lemma div_by_1 [simp]: "a div 1 = a"
  using div_mult_self2_is_id [of 1 a] zero_neq_one by simp

lemma mod_by_1 [simp]: "a mod 1 = 0"
proof -
  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
  then have "a + a mod 1 = a + 0" by simp
  then show ?thesis by (rule add_left_imp_eq)
qed

lemma mod_self [simp]: "a mod a = 0"
  using mod_mult_self2_is_0 [of 1] by simp

lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
  using div_mult_self2_is_id [of _ 1] by simp

lemma div_add_self1 [simp]:
  assumes "b \<noteq> 0"
  shows "(b + a) div b = a div b + 1"
  using assms div_mult_self1 [of b a 1] by (simp add: add_commute)

lemma div_add_self2 [simp]:
  assumes "b \<noteq> 0"
  shows "(a + b) div b = a div b + 1"
  using assms div_add_self1 [of b a] by (simp add: add_commute)

lemma mod_add_self1 [simp]:
  "(b + a) mod b = a mod b"
  using mod_mult_self1 [of a 1 b] by (simp add: add_commute)

lemma mod_add_self2 [simp]:
  "(a + b) mod b = a mod b"
  using mod_mult_self1 [of a 1 b] by simp

lemma mod_div_decomp:
  fixes a b
  obtains q r where "q = a div b" and "r = a mod b"
    and "a = q * b + r"
proof -
  from mod_div_equality have "a = a div b * b + a mod b" by simp
  moreover have "a div b = a div b" ..
  moreover have "a mod b = a mod b" ..
  note that ultimately show thesis by blast
qed

lemma dvd_eq_mod_eq_0 [code, code_unfold, code_inline del]: "a dvd b \<longleftrightarrow> b mod a = 0"
proof
  assume "b mod a = 0"
  with mod_div_equality [of b a] have "b div a * a = b" by simp
  then have "b = a * (b div a)" unfolding mult_commute ..
  then have "\<exists>c. b = a * c" ..
  then show "a dvd b" unfolding dvd_def .
next
  assume "a dvd b"
  then have "\<exists>c. b = a * c" unfolding dvd_def .
  then obtain c where "b = a * c" ..
  then have "b mod a = a * c mod a" by simp
  then have "b mod a = c * a mod a" by (simp add: mult_commute)
  then show "b mod a = 0" by simp
qed

lemma mod_div_trivial [simp]: "a mod b div b = 0"
proof (cases "b = 0")
  assume "b = 0"
  thus ?thesis by simp
next
  assume "b \<noteq> 0"
  hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
    by (rule div_mult_self1 [symmetric])
  also have "\<dots> = a div b"
    by (simp only: mod_div_equality')
  also have "\<dots> = a div b + 0"
    by simp
  finally show ?thesis
    by (rule add_left_imp_eq)
qed

lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"
proof -
  have "a mod b mod b = (a mod b + a div b * b) mod b"
    by (simp only: mod_mult_self1)
  also have "\<dots> = a mod b"
    by (simp only: mod_div_equality')
  finally show ?thesis .
qed

lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"
by (rule dvd_eq_mod_eq_0[THEN iffD1])

lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)

lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b"
by (drule dvd_div_mult_self) (simp add: mult_commute)

lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"
apply (cases "a = 0")
 apply simp
apply (auto simp: dvd_def mult_assoc)
done

lemma div_dvd_div[simp]:
  "a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)"
apply (cases "a = 0")
 apply simp
apply (unfold dvd_def)
apply auto
 apply(blast intro:mult_assoc[symmetric])
apply(fastsimp simp add: mult_assoc)
done

lemma dvd_mod_imp_dvd: "[| k dvd m mod n;  k dvd n |] ==> k dvd m"
  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
   apply (simp add: mod_div_equality)
  apply (simp only: dvd_add dvd_mult)
  done

text {* Addition respects modular equivalence. *}

lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"
proof -
  have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
    by (simp only: mod_div_equality)
  also have "\<dots> = (a mod c + b + a div c * c) mod c"
    by (simp only: add_ac)
  also have "\<dots> = (a mod c + b) mod c"
    by (rule mod_mult_self1)
  finally show ?thesis .
qed

lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"
proof -
  have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"
    by (simp only: mod_div_equality)
  also have "\<dots> = (a + b mod c + b div c * c) mod c"
    by (simp only: add_ac)
  also have "\<dots> = (a + b mod c) mod c"
    by (rule mod_mult_self1)
  finally show ?thesis .
qed

lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"
by (rule trans [OF mod_add_left_eq mod_add_right_eq])

lemma mod_add_cong:
  assumes "a mod c = a' mod c"
  assumes "b mod c = b' mod c"
  shows "(a + b) mod c = (a' + b') mod c"
proof -
  have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
    unfolding assms ..
  thus ?thesis
    by (simp only: mod_add_eq [symmetric])
qed

lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y
  \<Longrightarrow> (x + y) div z = x div z + y div z"
by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)

text {* Multiplication respects modular equivalence. *}

lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"
proof -
  have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
    by (simp only: mod_div_equality)
  also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
    by (simp only: algebra_simps)
  also have "\<dots> = (a mod c * b) mod c"
    by (rule mod_mult_self1)
  finally show ?thesis .
qed

lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"
proof -
  have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"
    by (simp only: mod_div_equality)
  also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"
    by (simp only: algebra_simps)
  also have "\<dots> = (a * (b mod c)) mod c"
    by (rule mod_mult_self1)
  finally show ?thesis .
qed

lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"
by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])

lemma mod_mult_cong:
  assumes "a mod c = a' mod c"
  assumes "b mod c = b' mod c"
  shows "(a * b) mod c = (a' * b') mod c"
proof -
  have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
    unfolding assms ..
  thus ?thesis
    by (simp only: mod_mult_eq [symmetric])
qed

lemma mod_mod_cancel:
  assumes "c dvd b"
  shows "a mod b mod c = a mod c"
proof -
  from `c dvd b` obtain k where "b = c * k"
    by (rule dvdE)
  have "a mod b mod c = a mod (c * k) mod c"
    by (simp only: `b = c * k`)
  also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
    by (simp only: mod_mult_self1)
  also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
    by (simp only: add_ac mult_ac)
  also have "\<dots> = a mod c"
    by (simp only: mod_div_equality)
  finally show ?thesis .
qed

lemma div_mult_div_if_dvd:
  "y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
  apply (cases "y = 0", simp)
  apply (cases "z = 0", simp)
  apply (auto elim!: dvdE simp add: algebra_simps)
  apply (subst mult_assoc [symmetric])
  apply (simp add: no_zero_divisors)
  done

lemma div_mult_mult2 [simp]:
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
  by (drule div_mult_mult1) (simp add: mult_commute)

lemma div_mult_mult1_if [simp]:
  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
  by simp_all

lemma mod_mult_mult1:
  "(c * a) mod (c * b) = c * (a mod b)"
proof (cases "c = 0")
  case True then show ?thesis by simp
next
  case False
  from mod_div_equality
  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
  with mod_div_equality show ?thesis by simp 
qed
  
lemma mod_mult_mult2:
  "(a * c) mod (b * c) = (a mod b) * c"
  using mod_mult_mult1 [of c a b] by (simp add: mult_commute)

lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
  unfolding dvd_def by (auto simp add: mod_mult_mult1)

lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
by (blast intro: dvd_mod_imp_dvd dvd_mod)

lemma div_power:
  "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
apply (induct n)
 apply simp
apply(simp add: div_mult_div_if_dvd dvd_power_same)
done

end

class ring_div = semiring_div + idom
begin

text {* Negation respects modular equivalence. *}

lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
proof -
  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
    by (simp only: mod_div_equality)
  also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
    by (simp only: minus_add_distrib minus_mult_left add_ac)
  also have "\<dots> = (- (a mod b)) mod b"
    by (rule mod_mult_self1)
  finally show ?thesis .
qed

lemma mod_minus_cong:
  assumes "a mod b = a' mod b"
  shows "(- a) mod b = (- a') mod b"
proof -
  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
    unfolding assms ..
  thus ?thesis
    by (simp only: mod_minus_eq [symmetric])
qed

text {* Subtraction respects modular equivalence. *}

lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"
  unfolding diff_minus
  by (intro mod_add_cong mod_minus_cong) simp_all

lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"
  unfolding diff_minus
  by (intro mod_add_cong mod_minus_cong) simp_all

lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"
  unfolding diff_minus
  by (intro mod_add_cong mod_minus_cong) simp_all

lemma mod_diff_cong:
  assumes "a mod c = a' mod c"
  assumes "b mod c = b' mod c"
  shows "(a - b) mod c = (a' - b') mod c"
  unfolding diff_minus using assms
  by (intro mod_add_cong mod_minus_cong)

lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
apply (case_tac "y = 0") apply simp
apply (auto simp add: dvd_def)
apply (subgoal_tac "-(y * k) = y * - k")
 apply (erule ssubst)
 apply (erule div_mult_self1_is_id)
apply simp
done

lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
apply (case_tac "y = 0") apply simp
apply (auto simp add: dvd_def)
apply (subgoal_tac "y * k = -y * -k")
 apply (erule ssubst)
 apply (rule div_mult_self1_is_id)
 apply simp
apply simp
done

end


subsection {* Division on @{typ nat} *}

text {*
  We define @{const div} and @{const mod} on @{typ nat} by means
  of a characteristic relation with two input arguments
  @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
  @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
*}

definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
  "divmod_nat_rel m n qr \<longleftrightarrow>
    m = fst qr * n + snd qr \<and>
      (if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)"

text {* @{const divmod_nat_rel} is total: *}

lemma divmod_nat_rel_ex:
  obtains q r where "divmod_nat_rel m n (q, r)"
proof (cases "n = 0")
  case True  with that show thesis
    by (auto simp add: divmod_nat_rel_def)
next
  case False
  have "\<exists>q r. m = q * n + r \<and> r < n"
  proof (induct m)
    case 0 with `n \<noteq> 0`
    have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
    then show ?case by blast
  next
    case (Suc m) then obtain q' r'
      where m: "m = q' * n + r'" and n: "r' < n" by auto
    then show ?case proof (cases "Suc r' < n")
      case True
      from m n have "Suc m = q' * n + Suc r'" by simp
      with True show ?thesis by blast
    next
      case False then have "n \<le> Suc r'" by auto
      moreover from n have "Suc r' \<le> n" by auto
      ultimately have "n = Suc r'" by auto
      with m have "Suc m = Suc q' * n + 0" by simp
      with `n \<noteq> 0` show ?thesis by blast
    qed
  qed
  with that show thesis
    using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def)
qed

text {* @{const divmod_nat_rel} is injective: *}

lemma divmod_nat_rel_unique:
  assumes "divmod_nat_rel m n qr"
    and "divmod_nat_rel m n qr'"
  shows "qr = qr'"
proof (cases "n = 0")
  case True with assms show ?thesis
    by (cases qr, cases qr')
      (simp add: divmod_nat_rel_def)
next
  case False
  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
  apply (rule leI)
  apply (subst less_iff_Suc_add)
  apply (auto simp add: add_mult_distrib)
  done
  from `n \<noteq> 0` assms have "fst qr = fst qr'"
    by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
  moreover from this assms have "snd qr = snd qr'"
    by (simp add: divmod_nat_rel_def)
  ultimately show ?thesis by (cases qr, cases qr') simp
qed

text {*
  We instantiate divisibility on the natural numbers by
  means of @{const divmod_nat_rel}:
*}

instantiation nat :: semiring_div
begin

definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
  [code del]: "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"

lemma divmod_nat_rel_divmod_nat:
  "divmod_nat_rel m n (divmod_nat m n)"
proof -
  from divmod_nat_rel_ex
    obtain qr where rel: "divmod_nat_rel m n qr" .
  then show ?thesis
  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
qed

lemma divmod_nat_eq:
  assumes "divmod_nat_rel m n qr" 
  shows "divmod_nat m n = qr"
  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)

definition div_nat where
  "m div n = fst (divmod_nat m n)"

definition mod_nat where
  "m mod n = snd (divmod_nat m n)"

lemma divmod_nat_div_mod:
  "divmod_nat m n = (m div n, m mod n)"
  unfolding div_nat_def mod_nat_def by simp

lemma div_eq:
  assumes "divmod_nat_rel m n (q, r)" 
  shows "m div n = q"
  using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)

lemma mod_eq:
  assumes "divmod_nat_rel m n (q, r)" 
  shows "m mod n = r"
  using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)

lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
  by (simp add: div_nat_def mod_nat_def divmod_nat_rel_divmod_nat)

lemma divmod_nat_zero:
  "divmod_nat m 0 = (0, m)"
proof -
  from divmod_nat_rel [of m 0] show ?thesis
    unfolding divmod_nat_div_mod divmod_nat_rel_def by simp
qed

lemma divmod_nat_base:
  assumes "m < n"
  shows "divmod_nat m n = (0, m)"
proof -
  from divmod_nat_rel [of m n] show ?thesis
    unfolding divmod_nat_div_mod divmod_nat_rel_def
    using assms by (cases "m div n = 0")
      (auto simp add: gr0_conv_Suc [of "m div n"])
qed

lemma divmod_nat_step:
  assumes "0 < n" and "n \<le> m"
  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"
proof -
  from divmod_nat_rel have divmod_nat_m_n: "divmod_nat_rel m n (m div n, m mod n)" .
  with assms have m_div_n: "m div n \<ge> 1"
    by (cases "m div n") (auto simp add: divmod_nat_rel_def)
  from assms divmod_nat_m_n have "divmod_nat_rel (m - n) n (m div n - Suc 0, m mod n)"
    by (cases "m div n") (auto simp add: divmod_nat_rel_def)
  with divmod_nat_eq have "divmod_nat (m - n) n = (m div n - Suc 0, m mod n)" by simp
  moreover from divmod_nat_div_mod have "divmod_nat (m - n) n = ((m - n) div n, (m - n) mod n)" .
  ultimately have "m div n = Suc ((m - n) div n)"
    and "m mod n = (m - n) mod n" using m_div_n by simp_all
  then show ?thesis using divmod_nat_div_mod by simp
qed

text {* The ''recursion'' equations for @{const div} and @{const mod} *}

lemma div_less [simp]:
  fixes m n :: nat
  assumes "m < n"
  shows "m div n = 0"
  using assms divmod_nat_base divmod_nat_div_mod by simp

lemma le_div_geq:
  fixes m n :: nat
  assumes "0 < n" and "n \<le> m"
  shows "m div n = Suc ((m - n) div n)"
  using assms divmod_nat_step divmod_nat_div_mod by simp

lemma mod_less [simp]:
  fixes m n :: nat
  assumes "m < n"
  shows "m mod n = m"
  using assms divmod_nat_base divmod_nat_div_mod by simp

lemma le_mod_geq:
  fixes m n :: nat
  assumes "n \<le> m"
  shows "m mod n = (m - n) mod n"
  using assms divmod_nat_step divmod_nat_div_mod by (cases "n = 0") simp_all

instance proof -
  have [simp]: "\<And>n::nat. n div 0 = 0"
    by (simp add: div_nat_def divmod_nat_zero)
  have [simp]: "\<And>n::nat. 0 div n = 0"
  proof -
    fix n :: nat
    show "0 div n = 0"
      by (cases "n = 0") simp_all
  qed
  show "OFCLASS(nat, semiring_div_class)" proof
    fix m n :: nat
    show "m div n * n + m mod n = m"
      using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
  next
    fix m n q :: nat
    assume "n \<noteq> 0"
    then show "(q + m * n) div n = m + q div n"
      by (induct m) (simp_all add: le_div_geq)
  next
    fix m n q :: nat
    assume "m \<noteq> 0"
    then show "(m * n) div (m * q) = n div q"
    proof (cases "n \<noteq> 0 \<and> q \<noteq> 0")
      case False then show ?thesis by auto
    next
      case True with `m \<noteq> 0`
        have "m > 0" and "n > 0" and "q > 0" by auto
      then have "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
        by (auto simp add: divmod_nat_rel_def) (simp_all add: algebra_simps)
      moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
      ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
      then show ?thesis by (simp add: div_eq)
    qed
  qed simp_all
qed

end

lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
by (simp add: divmod_nat_zero divmod_nat_base divmod_nat_step)
    (simp add: divmod_nat_div_mod)

text {* Simproc for cancelling @{const div} and @{const mod} *}

ML {*
local

structure CancelDivMod = CancelDivModFun(struct

  val div_name = @{const_name div};
  val mod_name = @{const_name mod};
  val mk_binop = HOLogic.mk_binop;
  val mk_sum = Nat_Arith.mk_sum;
  val dest_sum = Nat_Arith.dest_sum;

  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];

  val trans = trans;

  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
    (@{thm monoid_add_class.add_0_left} :: @{thm monoid_add_class.add_0_right} :: @{thms add_ac}))

end)

in

val cancel_div_mod_nat_proc = Simplifier.simproc @{theory}
  "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);

val _ = Addsimprocs [cancel_div_mod_nat_proc];

end
*}


subsubsection {* Quotient *}

lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
by (simp add: le_div_geq linorder_not_less)

lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
by (simp add: div_geq)

lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
by simp

lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
by simp


subsubsection {* Remainder *}

lemma mod_less_divisor [simp]:
  fixes m n :: nat
  assumes "n > 0"
  shows "m mod n < (n::nat)"
  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto

lemma mod_less_eq_dividend [simp]:
  fixes m n :: nat
  shows "m mod n \<le> m"
proof (rule add_leD2)
  from mod_div_equality have "m div n * n + m mod n = m" .
  then show "m div n * n + m mod n \<le> m" by auto
qed

lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
by (simp add: le_mod_geq linorder_not_less)

lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
by (simp add: le_mod_geq)

lemma mod_1 [simp]: "m mod Suc 0 = 0"
by (induct m) (simp_all add: mod_geq)

lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
  apply (cases "n = 0", simp)
  apply (cases "k = 0", simp)
  apply (induct m rule: nat_less_induct)
  apply (subst mod_if, simp)
  apply (simp add: mod_geq diff_mult_distrib)
  done

lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
by (simp add: mult_commute [of k] mod_mult_distrib)

(* a simple rearrangement of mod_div_equality: *)
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
by (cut_tac a = m and b = n in mod_div_equality2, arith)

lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
  apply (drule mod_less_divisor [where m = m])
  apply simp
  done

subsubsection {* Quotient and Remainder *}

lemma divmod_nat_rel_mult1_eq:
  "divmod_nat_rel b c (q, r) \<Longrightarrow> c > 0
   \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)

lemma div_mult1_eq:
  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
apply (cases "c = 0", simp)
apply (blast intro: divmod_nat_rel [THEN divmod_nat_rel_mult1_eq, THEN div_eq])
done

lemma divmod_nat_rel_add1_eq:
  "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br) \<Longrightarrow>  c > 0
   \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)

(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
lemma div_add1_eq:
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
apply (cases "c = 0", simp)
apply (blast intro: divmod_nat_rel_add1_eq [THEN div_eq] divmod_nat_rel)
done

lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
  apply (cut_tac m = q and n = c in mod_less_divisor)
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
  apply (simp add: add_mult_distrib2)
  done

lemma divmod_nat_rel_mult2_eq:
  "divmod_nat_rel a b (q, r) \<Longrightarrow> 0 < b \<Longrightarrow> 0 < c
   \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)

lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
  apply (cases "b = 0", simp)
  apply (cases "c = 0", simp)
  apply (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_eq])
  done

lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
  apply (cases "b = 0", simp)
  apply (cases "c = 0", simp)
  apply (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_eq])
  done


subsubsection{*Further Facts about Quotient and Remainder*}

lemma div_1 [simp]: "m div Suc 0 = m"
by (induct m) (simp_all add: div_geq)


(* Monotonicity of div in first argument *)
lemma div_le_mono [rule_format (no_asm)]:
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
apply (case_tac "k=0", simp)
apply (induct "n" rule: nat_less_induct, clarify)
apply (case_tac "n<k")
(* 1  case n<k *)
apply simp
(* 2  case n >= k *)
apply (case_tac "m<k")
(* 2.1  case m<k *)
apply simp
(* 2.2  case m>=k *)
apply (simp add: div_geq diff_le_mono)
done

(* Antimonotonicity of div in second argument *)
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
apply (subgoal_tac "0<n")
 prefer 2 apply simp
apply (induct_tac k rule: nat_less_induct)
apply (rename_tac "k")
apply (case_tac "k<n", simp)
apply (subgoal_tac "~ (k<m) ")
 prefer 2 apply simp
apply (simp add: div_geq)
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
 prefer 2
 apply (blast intro: div_le_mono diff_le_mono2)
apply (rule le_trans, simp)
apply (simp)
done

lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
apply (case_tac "n=0", simp)
apply (subgoal_tac "m div n \<le> m div 1", simp)
apply (rule div_le_mono2)
apply (simp_all (no_asm_simp))
done

(* Similar for "less than" *)
lemma div_less_dividend [rule_format]:
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
apply (induct_tac m rule: nat_less_induct)
apply (rename_tac "m")
apply (case_tac "m<n", simp)
apply (subgoal_tac "0<n")
 prefer 2 apply simp
apply (simp add: div_geq)
apply (case_tac "n<m")
 apply (subgoal_tac "(m-n) div n < (m-n) ")
  apply (rule impI less_trans_Suc)+
apply assumption
  apply (simp_all)
done

declare div_less_dividend [simp]

text{*A fact for the mutilated chess board*}
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
apply (case_tac "n=0", simp)
apply (induct "m" rule: nat_less_induct)
apply (case_tac "Suc (na) <n")
(* case Suc(na) < n *)
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
(* case n \<le> Suc(na) *)
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
apply (auto simp add: Suc_diff_le le_mod_geq)
done

lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)

lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]

(*Loses information, namely we also have r<d provided d is nonzero*)
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
  apply (cut_tac a = m in mod_div_equality)
  apply (simp only: add_ac)
  apply (blast intro: sym)
  done

lemma split_div:
 "P(n div k :: nat) =
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
proof
  assume P: ?P
  show ?Q
  proof (cases)
    assume "k = 0"
    with P show ?Q by simp
  next
    assume not0: "k \<noteq> 0"
    thus ?Q
    proof (simp, intro allI impI)
      fix i j
      assume n: "n = k*i + j" and j: "j < k"
      show "P i"
      proof (cases)
        assume "i = 0"
        with n j P show "P i" by simp
      next
        assume "i \<noteq> 0"
        with not0 n j P show "P i" by(simp add:add_ac)
      qed
    qed
  qed
next
  assume Q: ?Q
  show ?P
  proof (cases)
    assume "k = 0"
    with Q show ?P by simp
  next
    assume not0: "k \<noteq> 0"
    with Q have R: ?R by simp
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
    show ?P by simp
  qed
qed

lemma split_div_lemma:
  assumes "0 < n"
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
proof
  assume ?rhs
  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
  then have A: "n * q \<le> m" by simp
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
  then have "m < m + (n - (m mod n))" by simp
  then have "m < n + (m - (m mod n))" by simp
  with nq have "m < n + n * q" by simp
  then have B: "m < n * Suc q" by simp
  from A B show ?lhs ..
next
  assume P: ?lhs
  then have "divmod_nat_rel m n (q, m - n * q)"
    unfolding divmod_nat_rel_def by (auto simp add: mult_ac)
  with divmod_nat_rel_unique divmod_nat_rel [of m n]
  have "(q, m - n * q) = (m div n, m mod n)" by auto
  then show ?rhs by simp
qed

theorem split_div':
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
  apply (case_tac "0 < n")
  apply (simp only: add: split_div_lemma)
  apply simp_all
  done

lemma split_mod:
 "P(n mod k :: nat) =
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
proof
  assume P: ?P
  show ?Q
  proof (cases)
    assume "k = 0"
    with P show ?Q by simp
  next
    assume not0: "k \<noteq> 0"
    thus ?Q
    proof (simp, intro allI impI)
      fix i j
      assume "n = k*i + j" "j < k"
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
    qed
  qed
next
  assume Q: ?Q
  show ?P
  proof (cases)
    assume "k = 0"
    with Q show ?P by simp
  next
    assume not0: "k \<noteq> 0"
    with Q have R: ?R by simp
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
    show ?P by simp
  qed
qed

theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
    subst [OF mod_div_equality [of _ n]])
  apply arith
  done

lemma div_mod_equality':
  fixes m n :: nat
  shows "m div n * n = m - m mod n"
proof -
  have "m mod n \<le> m mod n" ..
  from div_mod_equality have 
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
  with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
    "m div n * n + (m mod n - m mod n) = m - m mod n"
    by simp
  then show ?thesis by simp
qed


subsubsection {*An ``induction'' law for modulus arithmetic.*}

lemma mod_induct_0:
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
  and base: "P i" and i: "i<p"
  shows "P 0"
proof (rule ccontr)
  assume contra: "\<not>(P 0)"
  from i have p: "0<p" by simp
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
  proof
    fix k
    show "?A k"
    proof (induct k)
      show "?A 0" by simp  -- "by contradiction"
    next
      fix n
      assume ih: "?A n"
      show "?A (Suc n)"
      proof (clarsimp)
        assume y: "P (p - Suc n)"
        have n: "Suc n < p"
        proof (rule ccontr)
          assume "\<not>(Suc n < p)"
          hence "p - Suc n = 0"
            by simp
          with y contra show "False"
            by simp
        qed
        hence n2: "Suc (p - Suc n) = p-n" by arith
        from p have "p - Suc n < p" by arith
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
          by blast
        show "False"
        proof (cases "n=0")
          case True
          with z n2 contra show ?thesis by simp
        next
          case False
          with p have "p-n < p" by arith
          with z n2 False ih show ?thesis by simp
        qed
      qed
    qed
  qed
  moreover
  from i obtain k where "0<k \<and> i+k=p"
    by (blast dest: less_imp_add_positive)
  hence "0<k \<and> i=p-k" by auto
  moreover
  note base
  ultimately
  show "False" by blast
qed

lemma mod_induct:
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
  and base: "P i" and i: "i<p" and j: "j<p"
  shows "P j"
proof -
  have "\<forall>j<p. P j"
  proof
    fix j
    show "j<p \<longrightarrow> P j" (is "?A j")
    proof (induct j)
      from step base i show "?A 0"
        by (auto elim: mod_induct_0)
    next
      fix k
      assume ih: "?A k"
      show "?A (Suc k)"
      proof
        assume suc: "Suc k < p"
        hence k: "k<p" by simp
        with ih have "P k" ..
        with step k have "P (Suc k mod p)"
          by blast
        moreover
        from suc have "Suc k mod p = Suc k"
          by simp
        ultimately
        show "P (Suc k)" by simp
      qed
    qed
  qed
  with j show ?thesis by blast
qed

lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
by (auto simp add: numeral_2_eq_2 le_div_geq)

lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
by (simp add: nat_mult_2 [symmetric])

lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
apply (subgoal_tac "m mod 2 < 2")
apply (erule less_2_cases [THEN disjE])
apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
done

lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
proof -
  { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (induct n) simp_all }
  moreover have "m mod 2 < 2" by simp
  ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
  then show ?thesis by auto
qed

text{*These lemmas collapse some needless occurrences of Suc:
    at least three Sucs, since two and fewer are rewritten back to Suc again!
    We already have some rules to simplify operands smaller than 3.*}

lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
by (simp add: Suc3_eq_add_3)

lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
by (simp add: Suc3_eq_add_3)

lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
by (simp add: Suc3_eq_add_3)

lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
by (simp add: Suc3_eq_add_3)

lemmas Suc_div_eq_add3_div_number_of =
    Suc_div_eq_add3_div [of _ "number_of v", standard]
declare Suc_div_eq_add3_div_number_of [simp]

lemmas Suc_mod_eq_add3_mod_number_of =
    Suc_mod_eq_add3_mod [of _ "number_of v", standard]
declare Suc_mod_eq_add3_mod_number_of [simp]


lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 
apply (induct "m")
apply (simp_all add: mod_Suc)
done

declare Suc_times_mod_eq [of "number_of w", standard, simp]

lemma [simp]: "n div k \<le> (Suc n) div k"
by (simp add: div_le_mono) 

lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
by (cases n) simp_all

lemma div_2_gt_zero [simp]: "(1::nat) < n ==> 0 < n div 2" 
using Suc_n_div_2_gt_zero [of "n - 1"] by simp

  (* Potential use of algebra : Equality modulo n*)
lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
by (simp add: mult_ac add_ac)

lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
proof -
  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
  also have "... = Suc m mod n" by (rule mod_mult_self3) 
  finally show ?thesis .
qed

lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
apply (subst mod_Suc [of m]) 
apply (subst mod_Suc [of "m mod n"], simp) 
done


subsection {* Division on @{typ int} *}

definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
    --{*definition of quotient and remainder*}
    [code]: "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
               (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"

definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
    --{*for the division algorithm*}
    [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
                         else (2 * q, r))"

text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
  "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
     else adjust b (posDivAlg a (2 * b)))"
by auto
termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
  (auto simp add: mult_2)

text{*algorithm for the case @{text "a<0, b>0"}*}
function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
  "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
     else adjust b (negDivAlg a (2 * b)))"
by auto
termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
  (auto simp add: mult_2)

text{*algorithm for the general case @{term "b\<noteq>0"}*}
definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where
  [code_unfold]: "negateSnd = apsnd uminus"

definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
    --{*The full division algorithm considers all possible signs for a, b
       including the special case @{text "a=0, b<0"} because 
       @{term negDivAlg} requires @{term "a<0"}.*}
  "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
                  else if a = 0 then (0, 0)
                       else negateSnd (negDivAlg (-a) (-b))
               else 
                  if 0 < b then negDivAlg a b
                  else negateSnd (posDivAlg (-a) (-b)))"

instantiation int :: Divides.div
begin

definition
  "a div b = fst (divmod_int a b)"

definition
 "a mod b = snd (divmod_int a b)"

instance ..

end

lemma divmod_int_mod_div:
  "divmod_int p q = (p div q, p mod q)"
  by (auto simp add: div_int_def mod_int_def)

text{*
Here is the division algorithm in ML:

\begin{verbatim}
    fun posDivAlg (a,b) =
      if a<b then (0,a)
      else let val (q,r) = posDivAlg(a, 2*b)
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
           end

    fun negDivAlg (a,b) =
      if 0\<le>a+b then (~1,a+b)
      else let val (q,r) = negDivAlg(a, 2*b)
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
           end;

    fun negateSnd (q,r:int) = (q,~r);

    fun divmod (a,b) = if 0\<le>a then 
                          if b>0 then posDivAlg (a,b) 
                           else if a=0 then (0,0)
                                else negateSnd (negDivAlg (~a,~b))
                       else 
                          if 0<b then negDivAlg (a,b)
                          else        negateSnd (posDivAlg (~a,~b));
\end{verbatim}
*}



subsubsection{*Uniqueness and Monotonicity of Quotients and Remainders*}

lemma unique_quotient_lemma:
     "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
      ==> q' \<le> (q::int)"
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
 prefer 2 apply (simp add: right_diff_distrib)
apply (subgoal_tac "0 < b * (1 + q - q') ")
apply (erule_tac [2] order_le_less_trans)
 prefer 2 apply (simp add: right_diff_distrib right_distrib)
apply (subgoal_tac "b * q' < b * (1 + q) ")
 prefer 2 apply (simp add: right_diff_distrib right_distrib)
apply (simp add: mult_less_cancel_left)
done

lemma unique_quotient_lemma_neg:
     "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
      ==> q \<le> (q'::int)"
by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
    auto)

lemma unique_quotient:
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
      ==> q = q'"
apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
apply (blast intro: order_antisym
             dest: order_eq_refl [THEN unique_quotient_lemma] 
             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
done


lemma unique_remainder:
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
      ==> r = r'"
apply (subgoal_tac "q = q'")
 apply (simp add: divmod_int_rel_def)
apply (blast intro: unique_quotient)
done


subsubsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}

text{*And positive divisors*}

lemma adjust_eq [simp]:
     "adjust b (q,r) = 
      (let diff = r-b in  
        if 0 \<le> diff then (2*q + 1, diff)   
                     else (2*q, r))"
by (simp add: Let_def adjust_def)

declare posDivAlg.simps [simp del]

text{*use with a simproc to avoid repeatedly proving the premise*}
lemma posDivAlg_eqn:
     "0 < b ==>  
      posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
by (rule posDivAlg.simps [THEN trans], simp)

text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
theorem posDivAlg_correct:
  assumes "0 \<le> a" and "0 < b"
  shows "divmod_int_rel a b (posDivAlg a b)"
using prems apply (induct a b rule: posDivAlg.induct)
apply auto
apply (simp add: divmod_int_rel_def)
apply (subst posDivAlg_eqn, simp add: right_distrib)
apply (case_tac "a < b")
apply simp_all
apply (erule splitE)
apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
done


subsubsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}

text{*And positive divisors*}

declare negDivAlg.simps [simp del]

text{*use with a simproc to avoid repeatedly proving the premise*}
lemma negDivAlg_eqn:
     "0 < b ==>  
      negDivAlg a b =       
       (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
by (rule negDivAlg.simps [THEN trans], simp)

(*Correctness of negDivAlg: it computes quotients correctly
  It doesn't work if a=0 because the 0/b equals 0, not -1*)
lemma negDivAlg_correct:
  assumes "a < 0" and "b > 0"
  shows "divmod_int_rel a b (negDivAlg a b)"
using prems apply (induct a b rule: negDivAlg.induct)
apply (auto simp add: linorder_not_le)
apply (simp add: divmod_int_rel_def)
apply (subst negDivAlg_eqn, assumption)
apply (case_tac "a + b < (0\<Colon>int)")
apply simp_all
apply (erule splitE)
apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
done


subsubsection{*Existence Shown by Proving the Division Algorithm to be Correct*}

(*the case a=0*)
lemma divmod_int_rel_0: "b \<noteq> 0 ==> divmod_int_rel 0 b (0, 0)"
by (auto simp add: divmod_int_rel_def linorder_neq_iff)

lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
by (subst posDivAlg.simps, auto)

lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
by (subst negDivAlg.simps, auto)

lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
by (simp add: negateSnd_def)

lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (negateSnd qr)"
by (auto simp add: split_ifs divmod_int_rel_def)

lemma divmod_int_correct: "b \<noteq> 0 ==> divmod_int_rel a b (divmod_int a b)"
by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
                    posDivAlg_correct negDivAlg_correct)

text{*Arbitrary definitions for division by zero.  Useful to simplify 
    certain equations.*}

lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
by (simp add: div_int_def mod_int_def divmod_int_def posDivAlg.simps)  


text{*Basic laws about division and remainder*}

lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
apply (case_tac "b = 0", simp)
apply (cut_tac a = a and b = b in divmod_int_correct)
apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
done

lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
by(simp add: zmod_zdiv_equality[symmetric])

lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
by(simp add: mult_commute zmod_zdiv_equality[symmetric])

text {* Tool setup *}

ML {*
local

structure CancelDivMod = CancelDivModFun(struct

  val div_name = @{const_name div};
  val mod_name = @{const_name mod};
  val mk_binop = HOLogic.mk_binop;
  val mk_sum = Arith_Data.mk_sum HOLogic.intT;
  val dest_sum = Arith_Data.dest_sum;

  val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];

  val trans = trans;

  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
    (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))

end)

in

val cancel_div_mod_int_proc = Simplifier.simproc @{theory}
  "cancel_zdiv_zmod" ["(k::int) + l"] (K CancelDivMod.proc);

val _ = Addsimprocs [cancel_div_mod_int_proc];

end
*}

lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
apply (cut_tac a = a and b = b in divmod_int_correct)
apply (auto simp add: divmod_int_rel_def mod_int_def)
done

lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]
   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]

lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
apply (cut_tac a = a and b = b in divmod_int_correct)
apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
done

lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]
   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]



subsubsection{*General Properties of div and mod*}

lemma divmod_int_rel_div_mod: "b \<noteq> 0 ==> divmod_int_rel a b (a div b, a mod b)"
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
apply (force simp add: divmod_int_rel_def linorder_neq_iff)
done

lemma divmod_int_rel_div: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"
by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])

lemma divmod_int_rel_mod: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"
by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])

lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
apply (rule divmod_int_rel_div)
apply (auto simp add: divmod_int_rel_def)
done

lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
apply (rule divmod_int_rel_div)
apply (auto simp add: divmod_int_rel_def)
done

lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
apply (rule divmod_int_rel_div)
apply (auto simp add: divmod_int_rel_def)
done

(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)

lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
apply (rule_tac q = 0 in divmod_int_rel_mod)
apply (auto simp add: divmod_int_rel_def)
done

lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
apply (rule_tac q = 0 in divmod_int_rel_mod)
apply (auto simp add: divmod_int_rel_def)
done

lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
apply (rule_tac q = "-1" in divmod_int_rel_mod)
apply (auto simp add: divmod_int_rel_def)
done

text{*There is no @{text mod_neg_pos_trivial}.*}


(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
apply (case_tac "b = 0", simp)
apply (simp add: divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, 
                                 THEN divmod_int_rel_div, THEN sym])

done

(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
apply (case_tac "b = 0", simp)
apply (subst divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, THEN divmod_int_rel_mod],
       auto)
done


subsubsection{*Laws for div and mod with Unary Minus*}

lemma zminus1_lemma:
     "divmod_int_rel a b (q, r)
      ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  
                          if r=0 then 0 else b-r)"
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)


lemma zdiv_zminus1_eq_if:
     "b \<noteq> (0::int)  
      ==> (-a) div b =  
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_div])

lemma zmod_zminus1_eq_if:
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
apply (case_tac "b = 0", simp)
apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_mod])
done

lemma zmod_zminus1_not_zero:
  fixes k l :: int
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
  unfolding zmod_zminus1_eq_if by auto

lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)

lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)

lemma zdiv_zminus2_eq_if:
     "b \<noteq> (0::int)  
      ==> a div (-b) =  
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)

lemma zmod_zminus2_eq_if:
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
by (simp add: zmod_zminus1_eq_if zmod_zminus2)

lemma zmod_zminus2_not_zero:
  fixes k l :: int
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
  unfolding zmod_zminus2_eq_if by auto 


subsubsection{*Division of a Number by Itself*}

lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
apply (subgoal_tac "0 < a*q")
 apply (simp add: zero_less_mult_iff, arith)
done

lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
apply (subgoal_tac "0 \<le> a* (1-q) ")
 apply (simp add: zero_le_mult_iff)
apply (simp add: right_diff_distrib)
done

lemma self_quotient: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"
apply (simp add: split_ifs divmod_int_rel_def linorder_neq_iff)
apply (rule order_antisym, safe, simp_all)
apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
done

lemma self_remainder: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"
apply (frule self_quotient, assumption)
apply (simp add: divmod_int_rel_def)
done

lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
by (simp add: divmod_int_rel_div_mod [THEN self_quotient])

(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
lemma zmod_self [simp]: "a mod a = (0::int)"
apply (case_tac "a = 0", simp)
apply (simp add: divmod_int_rel_div_mod [THEN self_remainder])
done


subsubsection{*Computation of Division and Remainder*}

lemma zdiv_zero [simp]: "(0::int) div b = 0"
by (simp add: div_int_def divmod_int_def)

lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
by (simp add: div_int_def divmod_int_def)

lemma zmod_zero [simp]: "(0::int) mod b = 0"
by (simp add: mod_int_def divmod_int_def)

lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
by (simp add: mod_int_def divmod_int_def)

text{*a positive, b positive *}

lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
by (simp add: div_int_def divmod_int_def)

lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
by (simp add: mod_int_def divmod_int_def)

text{*a negative, b positive *}

lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
by (simp add: div_int_def divmod_int_def)

lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
by (simp add: mod_int_def divmod_int_def)

text{*a positive, b negative *}

lemma div_pos_neg:
     "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
by (simp add: div_int_def divmod_int_def)

lemma mod_pos_neg:
     "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
by (simp add: mod_int_def divmod_int_def)

text{*a negative, b negative *}

lemma div_neg_neg:
     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
by (simp add: div_int_def divmod_int_def)

lemma mod_neg_neg:
     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
by (simp add: mod_int_def divmod_int_def)

text {*Simplify expresions in which div and mod combine numerical constants*}

lemma divmod_int_relI:
  "\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>
    \<Longrightarrow> divmod_int_rel a b (q, r)"
  unfolding divmod_int_rel_def by simp

lemmas divmod_int_rel_div_eq = divmod_int_relI [THEN divmod_int_rel_div, THEN eq_reflection]
lemmas divmod_int_rel_mod_eq = divmod_int_relI [THEN divmod_int_rel_mod, THEN eq_reflection]
lemmas arithmetic_simps =
  arith_simps
  add_special
  OrderedGroup.add_0_left
  OrderedGroup.add_0_right
  mult_zero_left
  mult_zero_right
  mult_1_left
  mult_1_right

(* simprocs adapted from HOL/ex/Binary.thy *)
ML {*
local
  val mk_number = HOLogic.mk_number HOLogic.intT;
  fun mk_cert u k l = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $
    (@{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $ u $ mk_number k) $
      mk_number l;
  fun prove ctxt prop = Goal.prove ctxt [] [] prop
    (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps @{thms arithmetic_simps}))));
  fun binary_proc proc ss ct =
    (case Thm.term_of ct of
      _ $ t $ u =>
      (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
        SOME args => proc (Simplifier.the_context ss) args
      | NONE => NONE)
    | _ => NONE);
in
  fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
    if n = 0 then NONE
    else let val (k, l) = Integer.div_mod m n;
    in SOME (rule OF [prove ctxt (Logic.mk_equals (t, mk_cert u k l))]) end);
end
*}

simproc_setup binary_int_div ("number_of m div number_of n :: int") =
  {* K (divmod_proc (@{thm divmod_int_rel_div_eq})) *}

simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
  {* K (divmod_proc (@{thm divmod_int_rel_mod_eq})) *}

lemmas posDivAlg_eqn_number_of [simp] =
    posDivAlg_eqn [of "number_of v" "number_of w", standard]

lemmas negDivAlg_eqn_number_of [simp] =
    negDivAlg_eqn [of "number_of v" "number_of w", standard]


text{*Special-case simplification *}

lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
apply (cut_tac a = a and b = "-1" in neg_mod_sign)
apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
apply (auto simp del: neg_mod_sign neg_mod_bound)
done

lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)

(** The last remaining special cases for constant arithmetic:
    1 div z and 1 mod z **)

lemmas div_pos_pos_1_number_of [simp] =
    div_pos_pos [OF int_0_less_1, of "number_of w", standard]

lemmas div_pos_neg_1_number_of [simp] =
    div_pos_neg [OF int_0_less_1, of "number_of w", standard]

lemmas mod_pos_pos_1_number_of [simp] =
    mod_pos_pos [OF int_0_less_1, of "number_of w", standard]

lemmas mod_pos_neg_1_number_of [simp] =
    mod_pos_neg [OF int_0_less_1, of "number_of w", standard]


lemmas posDivAlg_eqn_1_number_of [simp] =
    posDivAlg_eqn [of concl: 1 "number_of w", standard]

lemmas negDivAlg_eqn_1_number_of [simp] =
    negDivAlg_eqn [of concl: 1 "number_of w", standard]



subsubsection{*Monotonicity in the First Argument (Dividend)*}

lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
apply (rule unique_quotient_lemma)
apply (erule subst)
apply (erule subst, simp_all)
done

lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
apply (rule unique_quotient_lemma_neg)
apply (erule subst)
apply (erule subst, simp_all)
done


subsubsection{*Monotonicity in the Second Argument (Divisor)*}

lemma q_pos_lemma:
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
apply (subgoal_tac "0 < b'* (q' + 1) ")
 apply (simp add: zero_less_mult_iff)
apply (simp add: right_distrib)
done

lemma zdiv_mono2_lemma:
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
      ==> q \<le> (q'::int)"
apply (frule q_pos_lemma, assumption+) 
apply (subgoal_tac "b*q < b* (q' + 1) ")
 apply (simp add: mult_less_cancel_left)
apply (subgoal_tac "b*q = r' - r + b'*q'")
 prefer 2 apply simp
apply (simp (no_asm_simp) add: right_distrib)
apply (subst add_commute, rule zadd_zless_mono, arith)
apply (rule mult_right_mono, auto)
done

lemma zdiv_mono2:
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
apply (subgoal_tac "b \<noteq> 0")
 prefer 2 apply arith
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
apply (rule zdiv_mono2_lemma)
apply (erule subst)
apply (erule subst, simp_all)
done

lemma q_neg_lemma:
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
apply (subgoal_tac "b'*q' < 0")
 apply (simp add: mult_less_0_iff, arith)
done

lemma zdiv_mono2_neg_lemma:
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
      ==> q' \<le> (q::int)"
apply (frule q_neg_lemma, assumption+) 
apply (subgoal_tac "b*q' < b* (q + 1) ")
 apply (simp add: mult_less_cancel_left)
apply (simp add: right_distrib)
apply (subgoal_tac "b*q' \<le> b'*q'")
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
done

lemma zdiv_mono2_neg:
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
apply (rule zdiv_mono2_neg_lemma)
apply (erule subst)
apply (erule subst, simp_all)
done


subsubsection{*More Algebraic Laws for div and mod*}

text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}

lemma zmult1_lemma:
     "[| divmod_int_rel b c (q, r);  c \<noteq> 0 |]  
      ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)

lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
apply (case_tac "c = 0", simp)
apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_div])
done

lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
apply (case_tac "c = 0", simp)
apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_mod])
done

lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
apply (case_tac "b = 0", simp)
apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
done

text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}

lemma zadd1_lemma:
     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br);  c \<noteq> 0 |]  
      ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)

(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
lemma zdiv_zadd1_eq:
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
apply (case_tac "c = 0", simp)
apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] divmod_int_rel_div)
done

instance int :: ring_div
proof
  fix a b c :: int
  assume not0: "b \<noteq> 0"
  show "(a + c * b) div b = c + a div b"
    unfolding zdiv_zadd1_eq [of a "c * b"] using not0 
      by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
next
  fix a b c :: int
  assume "a \<noteq> 0"
  then show "(a * b) div (a * c) = b div c"
  proof (cases "b \<noteq> 0 \<and> c \<noteq> 0")
    case False then show ?thesis by auto
  next
    case True then have "b \<noteq> 0" and "c \<noteq> 0" by auto
    with `a \<noteq> 0`
    have "\<And>q r. divmod_int_rel b c (q, r) \<Longrightarrow> divmod_int_rel (a * b) (a * c) (q, a * r)"
      apply (auto simp add: divmod_int_rel_def) 
      apply (auto simp add: algebra_simps)
      apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff mult_commute [of a] mult_less_cancel_right)
      done
    moreover with `c \<noteq> 0` divmod_int_rel_div_mod have "divmod_int_rel b c (b div c, b mod c)" by auto
    ultimately have "divmod_int_rel (a * b) (a * c) (b div c, a * (b mod c))" .
    moreover from  `a \<noteq> 0` `c \<noteq> 0` have "a * c \<noteq> 0" by simp
    ultimately show ?thesis by (rule divmod_int_rel_div)
  qed
qed auto

lemma posDivAlg_div_mod:
  assumes "k \<ge> 0"
  and "l \<ge> 0"
  shows "posDivAlg k l = (k div l, k mod l)"
proof (cases "l = 0")
  case True then show ?thesis by (simp add: posDivAlg.simps)
next
  case False with assms posDivAlg_correct
    have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
    by simp
  from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
  show ?thesis by simp
qed

lemma negDivAlg_div_mod:
  assumes "k < 0"
  and "l > 0"
  shows "negDivAlg k l = (k div l, k mod l)"
proof -
  from assms have "l \<noteq> 0" by simp
  from assms negDivAlg_correct
    have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
    by simp
  from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
  show ?thesis by simp
qed

lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)

(* REVISIT: should this be generalized to all semiring_div types? *)
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]


subsubsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}

(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
  to cause particular problems.*)

text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}

lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
apply (subgoal_tac "b * (c - q mod c) < r * 1")
 apply (simp add: algebra_simps)
apply (rule order_le_less_trans)
 apply (erule_tac [2] mult_strict_right_mono)
 apply (rule mult_left_mono_neg)
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)
 apply (simp)
apply (simp)
done

lemma zmult2_lemma_aux2:
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
apply (subgoal_tac "b * (q mod c) \<le> 0")
 apply arith
apply (simp add: mult_le_0_iff)
done

lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
apply (subgoal_tac "0 \<le> b * (q mod c) ")
apply arith
apply (simp add: zero_le_mult_iff)
done

lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
 apply (simp add: right_diff_distrib)
apply (rule order_less_le_trans)
 apply (erule mult_strict_right_mono)
 apply (rule_tac [2] mult_left_mono)
  apply simp
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)
apply simp
done

lemma zmult2_lemma: "[| divmod_int_rel a b (q, r);  b \<noteq> 0;  0 < c |]  
      ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
                   zero_less_mult_iff right_distrib [symmetric] 
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)

lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
apply (case_tac "b = 0", simp)
apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_div])
done

lemma zmod_zmult2_eq:
     "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
apply (case_tac "b = 0", simp)
apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_mod])
done


subsubsection {*Splitting Rules for div and mod*}

text{*The proofs of the two lemmas below are essentially identical*}

lemma split_pos_lemma:
 "0<k ==> 
    P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
apply (rule iffI, clarify)
 apply (erule_tac P="P ?x ?y" in rev_mp)  
 apply (subst mod_add_eq) 
 apply (subst zdiv_zadd1_eq) 
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
txt{*converse direction*}
apply (drule_tac x = "n div k" in spec) 
apply (drule_tac x = "n mod k" in spec, simp)
done

lemma split_neg_lemma:
 "k<0 ==>
    P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
apply (rule iffI, clarify)
 apply (erule_tac P="P ?x ?y" in rev_mp)  
 apply (subst mod_add_eq) 
 apply (subst zdiv_zadd1_eq) 
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
txt{*converse direction*}
apply (drule_tac x = "n div k" in spec) 
apply (drule_tac x = "n mod k" in spec, simp)
done

lemma split_zdiv:
 "P(n div k :: int) =
  ((k = 0 --> P 0) & 
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
apply (case_tac "k=0", simp)
apply (simp only: linorder_neq_iff)
apply (erule disjE) 
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
                      split_neg_lemma [of concl: "%x y. P x"])
done

lemma split_zmod:
 "P(n mod k :: int) =
  ((k = 0 --> P n) & 
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
apply (case_tac "k=0", simp)
apply (simp only: linorder_neq_iff)
apply (erule disjE) 
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
                      split_neg_lemma [of concl: "%x y. P y"])
done

text {* Enable (lin)arith to deal with @{const div} and @{const mod}
  when these are applied to some constant that is of the form
  @{term "number_of k"}: *}
declare split_zdiv [of _ _ "number_of k", standard, arith_split]
declare split_zmod [of _ _ "number_of k", standard, arith_split]


subsubsection{*Speeding up the Division Algorithm with Shifting*}

text{*computing div by shifting *}

lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
proof cases
  assume "a=0"
    thus ?thesis by simp
next
  assume "a\<noteq>0" and le_a: "0\<le>a"   
  hence a_pos: "1 \<le> a" by arith
  hence one_less_a2: "1 < 2 * a" by arith
  hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
    unfolding mult_le_cancel_left
    by (simp add: add1_zle_eq add_commute [of 1])
  with a_pos have "0 \<le> b mod a" by simp
  hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
    by (simp add: mod_pos_pos_trivial one_less_a2)
  with  le_2a
  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
    by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
                  right_distrib) 
  thus ?thesis
    by (subst zdiv_zadd1_eq,
        simp add: mod_mult_mult1 one_less_a2
                  div_pos_pos_trivial)
qed

lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
apply (rule_tac [2] pos_zdiv_mult_2)
apply (auto simp add: right_diff_distrib)
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric])
apply (simp_all add: algebra_simps)
apply (simp only: ab_diff_minus minus_add_distrib [symmetric] number_of_Min zdiv_zminus_zminus)
done

lemma zdiv_number_of_Bit0 [simp]:
     "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  
          number_of v div (number_of w :: int)"
by (simp only: number_of_eq numeral_simps) (simp add: mult_2 [symmetric])

lemma zdiv_number_of_Bit1 [simp]:
     "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  
          (if (0::int) \<le> number_of w                    
           then number_of v div (number_of w)     
           else (number_of v + (1::int)) div (number_of w))"
apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) 
apply (simp add: pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac mult_2 [symmetric])
done


subsubsection{*Computing mod by Shifting (proofs resemble those for div)*}

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)"
proof (cases "0 < a")
  case False with assms show ?thesis by simp
next
  case True
  then have "b mod a < a" by (rule pos_mod_bound)
  then have "1 + b mod a \<le> a" by simp
  then have A: "2 * (1 + b mod a) \<le> 2 * a" by simp
  from `0 < a` have "0 \<le> b mod a" by (rule pos_mod_sign)
  then have B: "0 \<le> 1 + 2 * (b mod a)" by simp
  have "((1\<Colon>int) mod ((2\<Colon>int) * a) + (2\<Colon>int) * b mod ((2\<Colon>int) * a)) mod ((2\<Colon>int) * a) = (1\<Colon>int) + (2\<Colon>int) * (b mod a)"
    using `0 < a` and A
    by (auto simp add: mod_mult_mult1 mod_pos_pos_trivial ring_distribs intro!: mod_pos_pos_trivial B)
  then show ?thesis by (subst mod_add_eq)
qed

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"
proof -
  from assms have "0 \<le> - a" by auto
  then have "(1 + 2 * (- b - 1)) mod (2 * (- a)) = 1 + 2 * ((- b - 1) mod (- a))"
    by (rule pos_zmod_mult_2)
  then show ?thesis by (simp add: zmod_zminus2 algebra_simps)
     (simp add: diff_minus add_ac)
qed

lemma zmod_number_of_Bit0 [simp]:
     "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  
      (2::int) * (number_of v mod number_of w)"
apply (simp only: number_of_eq numeral_simps) 
apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
                 neg_zmod_mult_2 add_ac mult_2 [symmetric])
done

lemma zmod_number_of_Bit1 [simp]:
     "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  
      (if (0::int) \<le> number_of w  
                then 2 * (number_of v mod number_of w) + 1     
                else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
apply (simp only: number_of_eq numeral_simps) 
apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
                 neg_zmod_mult_2 add_ac mult_2 [symmetric])
done


subsubsection{*Quotients of Signs*}

lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
apply (subgoal_tac "a div b \<le> -1", force)
apply (rule order_trans)
apply (rule_tac a' = "-1" in zdiv_mono1)
apply (auto simp add: div_eq_minus1)
done

lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
by (drule zdiv_mono1_neg, auto)

lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
by (drule zdiv_mono1, auto)

lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
apply auto
apply (drule_tac [2] zdiv_mono1)
apply (auto simp add: linorder_neq_iff)
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
apply (blast intro: div_neg_pos_less0)
done

lemma neg_imp_zdiv_nonneg_iff:
     "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
apply (subst zdiv_zminus_zminus [symmetric])
apply (subst pos_imp_zdiv_nonneg_iff, auto)
done

(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)

(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)


subsubsection {* The Divides Relation *}

lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
  dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]

lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
  by (rule dvd_mod) (* TODO: remove *)

lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
  by (rule dvd_mod_imp_dvd) (* TODO: remove *)

lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
  using zmod_zdiv_equality[where a="m" and b="n"]
  by (simp add: algebra_simps)

lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
apply (induct "y", auto)
apply (rule zmod_zmult1_eq [THEN trans])
apply (simp (no_asm_simp))
apply (rule mod_mult_eq [symmetric])
done

lemma zdiv_int: "int (a div b) = (int a) div (int b)"
apply (subst split_div, auto)
apply (subst split_zdiv, auto)
apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)
apply (auto simp add: divmod_int_rel_def of_nat_mult)
done

lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
apply (subst split_mod, auto)
apply (subst split_zmod, auto)
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
       in unique_remainder)
apply (auto simp add: divmod_int_rel_def of_nat_mult)
done

lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)

lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
apply (subgoal_tac "m mod n = 0")
 apply (simp add: zmult_div_cancel)
apply (simp only: dvd_eq_mod_eq_0)
done

text{*Suggested by Matthias Daum*}
lemma int_power_div_base:
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
 apply (erule ssubst)
 apply (simp only: power_add)
 apply simp_all
done

text {* by Brian Huffman *}
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
by (rule mod_minus_eq [symmetric])

lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
by (rule mod_diff_left_eq [symmetric])

lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
by (rule mod_diff_right_eq [symmetric])

lemmas zmod_simps =
  mod_add_left_eq  [symmetric]
  mod_add_right_eq [symmetric]
  zmod_zmult1_eq   [symmetric]
  mod_mult_left_eq [symmetric]
  zpower_zmod
  zminus_zmod zdiff_zmod_left zdiff_zmod_right

text {* Distributive laws for function @{text nat}. *}

lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
apply (rule linorder_cases [of y 0])
apply (simp add: div_nonneg_neg_le0)
apply simp
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
done

(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
lemma nat_mod_distrib:
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
apply (case_tac "y = 0", simp)
apply (simp add: nat_eq_iff zmod_int)
done

text  {* transfer setup *}

lemma transfer_nat_int_functions:
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
  by (auto simp add: nat_div_distrib nat_mod_distrib)

lemma transfer_nat_int_function_closures:
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
  apply (cases "y = 0")
  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
  apply (cases "y = 0")
  apply auto
done

declare TransferMorphism_nat_int [transfer add return:
  transfer_nat_int_functions
  transfer_nat_int_function_closures
]

lemma transfer_int_nat_functions:
    "(int x) div (int y) = int (x div y)"
    "(int x) mod (int y) = int (x mod y)"
  by (auto simp add: zdiv_int zmod_int)

lemma transfer_int_nat_function_closures:
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
  by (simp_all only: is_nat_def transfer_nat_int_function_closures)

declare TransferMorphism_int_nat [transfer add return:
  transfer_int_nat_functions
  transfer_int_nat_function_closures
]

text{*Suggested by Matthias Daum*}
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
apply (subgoal_tac "nat x div nat k < nat x")
 apply (simp (asm_lr) add: nat_div_distrib [symmetric])
apply (rule Divides.div_less_dividend, simp_all)
done

text {* code generator setup *}

context ring_1
begin

lemma of_int_num [code]:
  "of_int k = (if k = 0 then 0 else if k < 0 then
     - of_int (- k) else let
       (l, m) = divmod_int k 2;
       l' = of_int l
     in if m = 0 then l' + l' else l' + l' + 1)"
proof -
  have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow> 
    of_int k = of_int (k div 2 * 2 + 1)"
  proof -
    have "k mod 2 < 2" by (auto intro: pos_mod_bound)
    moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)
    moreover assume "k mod 2 \<noteq> 0"
    ultimately have "k mod 2 = 1" by arith
    moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
    ultimately show ?thesis by auto
  qed
  have aux2: "\<And>x. of_int 2 * x = x + x"
  proof -
    fix x
    have int2: "(2::int) = 1 + 1" by arith
    show "of_int 2 * x = x + x"
    unfolding int2 of_int_add left_distrib by simp
  qed
  have aux3: "\<And>x. x * of_int 2 = x + x"
  proof -
    fix x
    have int2: "(2::int) = 1 + 1" by arith
    show "x * of_int 2 = x + x" 
    unfolding int2 of_int_add right_distrib by simp
  qed
  from aux1 show ?thesis by (auto simp add: divmod_int_mod_div Let_def aux2 aux3)
qed

end

lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
proof
  assume H: "x mod n = y mod n"
  hence "x mod n - y mod n = 0" by simp
  hence "(x mod n - y mod n) mod n = 0" by simp 
  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
next
  assume H: "n dvd x - y"
  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
  hence "x = n*k + y" by simp
  hence "x mod n = (n*k + y) mod n" by simp
  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
qed

lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
  shows "\<exists>q. x = y + n * q"
proof-
  from xy have th: "int x - int y = int (x - y)" by simp 
  from xyn have "int x mod int n = int y mod int n" 
    by (simp add: zmod_int[symmetric])
  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
  hence "n dvd x - y" by (simp add: th zdvd_int)
  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
qed

lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
  (is "?lhs = ?rhs")
proof
  assume H: "x mod n = y mod n"
  {assume xy: "x \<le> y"
    from H have th: "y mod n = x mod n" by simp
    from nat_mod_eq_lemma[OF th xy] have ?rhs 
      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
  moreover
  {assume xy: "y \<le> x"
    from nat_mod_eq_lemma[OF H xy] have ?rhs 
      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
  ultimately  show ?rhs using linear[of x y] by blast  
next
  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
  thus  ?lhs by simp
qed

lemma div_nat_number_of [simp]:
     "(number_of v :: nat)  div  number_of v' =  
          (if neg (number_of v :: int) then 0  
           else nat (number_of v div number_of v'))"
  unfolding nat_number_of_def number_of_is_id neg_def
  by (simp add: nat_div_distrib)

lemma one_div_nat_number_of [simp]:
     "Suc 0 div number_of v' = nat (1 div number_of v')" 
by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 

lemma mod_nat_number_of [simp]:
     "(number_of v :: nat)  mod  number_of v' =  
        (if neg (number_of v :: int) then 0  
         else if neg (number_of v' :: int) then number_of v  
         else nat (number_of v mod number_of v'))"
  unfolding nat_number_of_def number_of_is_id neg_def
  by (simp add: nat_mod_distrib)

lemma one_mod_nat_number_of [simp]:
     "Suc 0 mod number_of v' =  
        (if neg (number_of v' :: int) then Suc 0
         else nat (1 mod number_of v'))"
by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 

lemmas dvd_eq_mod_eq_0_number_of =
  dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]

declare dvd_eq_mod_eq_0_number_of [simp]


subsubsection {* Code generation *}

definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
  "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"

lemma pdivmod_posDivAlg [code]:
  "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)

lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
  apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
    then pdivmod k l
    else (let (r, s) = pdivmod k l in
      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
proof -
  have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
  show ?thesis
    by (simp add: divmod_int_mod_div pdivmod_def)
      (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
      zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
qed

lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
  apsnd ((op *) (sgn l)) (if sgn k = sgn l
    then pdivmod k l
    else (let (r, s) = pdivmod k l in
      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
proof -
  have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"
    by (auto simp add: not_less sgn_if)
  then show ?thesis by (simp add: divmod_int_pdivmod)
qed

code_modulename SML
  Divides Arith

code_modulename OCaml
  Divides Arith

code_modulename Haskell
  Divides Arith

end