# Theory Divides

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theory Divides
imports Nat_Transfer
(*  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 Dividesimports Nat_Transferbeginsubsection {* Syntactic division operations *}class div = dvd +  fixes div :: "'a => 'a => 'a" (infixl "div" 70)    and mod :: "'a => 'a => '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 ≠ 0 ==> (a + c * b) div b = c + a div b"    and div_mult_mult1 [simp]: "c ≠ 0 ==> (c * a) div (c * b) = a div b"begintext {* @{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 simplemma mod_0 [simp]: "0 mod a = 0"  using mod_div_equality [of zero a] div_0 by simplemma div_mult_self2 [simp]:  assumes "b ≠ 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 simpnext  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    "… = (c + a div b) * b + (a + c * b) mod b"      by (simp add: algebra_simps)  finally have "a = a div b * b + (a + c * b) mod b"    by (simp add: add_commute [of a] add_assoc distrib_right)  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"    by (simp add: mod_div_equality)  then show ?thesis by simpqedlemma 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 ≠ 0 ==> b * a div b = a"  using div_mult_self2 [of b 0 a] by simplemma div_mult_self2_is_id [simp]: "b ≠ 0 ==> a * b div b = a"  using div_mult_self1 [of b 0 a] by simplemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"  using mod_mult_self2 [of 0 b a] by simplemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"  using mod_mult_self1 [of 0 a b] by simplemma div_by_1 [simp]: "a div 1 = a"  using div_mult_self2_is_id [of 1 a] zero_neq_one by simplemma 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)qedlemma mod_self [simp]: "a mod a = 0"  using mod_mult_self2_is_0 [of 1] by simplemma div_self [simp]: "a ≠ 0 ==> a div a = 1"  using div_mult_self2_is_id [of _ 1] by simplemma div_add_self1 [simp]:  assumes "b ≠ 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 ≠ 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 simplemma 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 blastqedlemma dvd_eq_mod_eq_0 [code]: "a dvd b <-> 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 "∃c. b = a * c" ..  then show "a dvd b" unfolding dvd_def .next  assume "a dvd b"  then have "∃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 simpqedlemma mod_div_trivial [simp]: "a mod b div b = 0"proof (cases "b = 0")  assume "b = 0"  thus ?thesis by simpnext  assume "b ≠ 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 "… = a div b"    by (simp only: mod_div_equality')  also have "… = a div b + 0"    by simp  finally show ?thesis    by (rule add_left_imp_eq)qedlemma 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 "… = a mod b"    by (simp only: mod_div_equality')  finally show ?thesis .qedlemma dvd_imp_mod_0: "a dvd b ==> b mod a = 0"by (rule dvd_eq_mod_eq_0[THEN iffD1])lemma dvd_div_mult_self: "a dvd b ==> (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 ==> a * (b div a) = b"by (drule dvd_div_mult_self) (simp add: mult_commute)lemma dvd_div_mult: "a dvd b ==> (b div a) * c = b * c div a"apply (cases "a = 0") apply simpapply (auto simp: dvd_def mult_assoc)donelemma div_dvd_div[simp]:  "a dvd b ==> a dvd c ==> (b div a dvd c div a) = (b dvd c)"apply (cases "a = 0") apply simpapply (unfold dvd_def)apply auto apply(blast intro:mult_assoc[symmetric])apply(fastforce simp add: mult_assoc)donelemma 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)  donetext {* 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 "… = (a mod c + b + a div c * c) mod c"    by (simp only: add_ac)  also have "… = (a mod c + b) mod c"    by (rule mod_mult_self1)  finally show ?thesis .qedlemma 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 "… = (a + b mod c + b div c * c) mod c"    by (simp only: add_ac)  also have "… = (a + b mod c) mod c"    by (rule mod_mult_self1)  finally show ?thesis .qedlemma 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])qedlemma div_add [simp]: "z dvd x ==> z dvd y  ==> (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 "… = (a mod c * b + a div c * b * c) mod c"    by (simp only: algebra_simps)  also have "… = (a mod c * b) mod c"    by (rule mod_mult_self1)  finally show ?thesis .qedlemma 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 "… = (a * (b mod c) + a * (b div c) * c) mod c"    by (simp only: algebra_simps)  also have "… = (a * (b mod c)) mod c"    by (rule mod_mult_self1)  finally show ?thesis .qedlemma 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])qedtext {* Exponentiation respects modular equivalence. *}lemma power_mod: "(a mod b)^n mod b = a^n mod b"apply (induct n, simp_all)apply (rule mod_mult_right_eq [THEN trans])apply (simp (no_asm_simp))apply (rule mod_mult_eq [symmetric])donelemma 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 "… = (a mod (c * k) + a div (c * k) * k * c) mod c"    by (simp only: mod_mult_self1)  also have "… = (a div (c * k) * (c * k) + a mod (c * k)) mod c"    by (simp only: add_ac mult_ac)  also have "… = a mod c"    by (simp only: mod_div_equality)  finally show ?thesis .qedlemma div_mult_div_if_dvd:  "y dvd x ==> z dvd w ==> (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)  donelemma div_mult_swap:  assumes "c dvd b"  shows "a * (b div c) = (a * b) div c"proof -  from assms have "b div c * (a div 1) = b * a div (c * 1)"    by (simp only: div_mult_div_if_dvd one_dvd)  then show ?thesis by (simp add: mult_commute)qed   lemma div_mult_mult2 [simp]:  "c ≠ 0 ==> (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_alllemma mod_mult_mult1:  "(c * a) mod (c * b) = c * (a mod b)"proof (cases "c = 0")  case True then show ?thesis by simpnext  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 mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"  by (fact mod_mult_mult2 [symmetric])lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"  by (fact mod_mult_mult1 [symmetric])lemma dvd_mod: "k dvd m ==> k dvd n ==> k dvd (m mod n)"  unfolding dvd_def by (auto simp add: mod_mult_mult1)lemma dvd_mod_iff: "k dvd n ==> k dvd (m mod n) <-> k dvd m"by (blast intro: dvd_mod_imp_dvd dvd_mod)lemma div_power:  "y dvd x ==> (x div y) ^ n = x ^ n div y ^ n"apply (induct n) apply simpapply(simp add: div_mult_div_if_dvd dvd_power_same)donelemma dvd_div_eq_mult:  assumes "a ≠ 0" and "a dvd b"    shows "b div a = c <-> b = c * a"proof  assume "b = c * a"  then show "b div a = c" by (simp add: assms)next  assume "b div a = c"  then have "b div a * a = c * a" by simp  moreover from a dvd b have "b div a * a = b" by (simp add: dvd_div_mult_self)  ultimately show "b = c * a" by simpqed   lemma dvd_div_div_eq_mult:  assumes "a ≠ 0" "c ≠ 0" and "a dvd b" "c dvd d"  shows "b div a = d div c <-> b * c = a * d"  using assms by (auto simp add: mult_commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym)endclass ring_div = semiring_div + comm_ring_1beginsubclass ring_1_no_zero_divisors ..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 "… = (- (a mod b) + - (a div b) * b) mod b"    by (simp only: minus_add_distrib minus_mult_left add_ac)  also have "… = (- (a mod b)) mod b"    by (rule mod_mult_self1)  finally show ?thesis .qedlemma 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])qedtext {* 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_alllemma 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_alllemma 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_alllemma 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 ==> -x div y = - (x div y)"apply (case_tac "y = 0") apply simpapply (auto simp add: dvd_def)apply (subgoal_tac "-(y * k) = y * - k") apply (erule ssubst) apply (erule div_mult_self1_is_id)apply simpdonelemma dvd_div_neg: "y dvd x ==> x div -y = - (x div y)"apply (case_tac "y = 0") apply simpapply (auto simp add: dvd_def)apply (subgoal_tac "y * k = -y * -k") apply (erule ssubst) apply (rule div_mult_self1_is_id) apply simpapply simpdonelemma div_minus_minus [simp]: "(-a) div (-b) = a div b"  using div_mult_mult1 [of "- 1" a b]  unfolding neg_equal_0_iff_equal by simplemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"  using mod_mult_mult1 [of "- 1" a b] by simplemma div_minus_right: "a div (-b) = (-a) div b"  using div_minus_minus [of "-a" b] by simplemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"  using mod_minus_minus [of "-a" b] by simplemma div_minus1_right [simp]: "a div (-1) = -a"  using div_minus_right [of a 1] by simplemma mod_minus1_right [simp]: "a mod (-1) = 0"  using mod_minus_right [of a 1] by simpendsubsection {* 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::nat"}, @{term "n::nat"} and two output arguments  @{term "q::nat"}(uotient) and @{term "r::nat"}(emainder).*}definition divmod_nat_rel :: "nat => nat => nat × nat => bool" where  "divmod_nat_rel m n qr <->    m = fst qr * n + snd qr ∧      (if n = 0 then fst qr = 0 else if n > 0 then 0 ≤ snd qr ∧ snd qr < n else n < snd qr ∧ snd qr ≤ 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 "∃q r. m = q * n + r ∧ r < n"  proof (induct m)    case 0 with n ≠ 0    have "(0::nat) = 0 * n + 0 ∧ 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 ≤ Suc r'" by auto      moreover from n have "Suc r' ≤ n" by auto      ultimately have "n = Suc r'" by auto      with m have "Suc m = Suc q' * n + 0" by simp      with n ≠ 0 show ?thesis by blast    qed  qed  with that show thesis    using n ≠ 0 by (auto simp add: divmod_nat_rel_def)qedtext {* @{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: "!!q r q' r'. q' * n + r' = q * n + r ==> r < n ==> q' ≤ (q::nat)"  apply (rule leI)  apply (subst less_iff_Suc_add)  apply (auto simp add: add_mult_distrib)  done  from n ≠ 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') simpqedtext {*  We instantiate divisibility on the natural numbers by  means of @{const divmod_nat_rel}:*}definition divmod_nat :: "nat => nat => nat × nat" where  "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)qedlemma divmod_nat_unique:  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)instantiation nat :: semiring_divbegindefinition div_nat where  "m div n = fst (divmod_nat m n)"lemma fst_divmod_nat [simp]:  "fst (divmod_nat m n) = m div n"  by (simp add: div_nat_def)definition mod_nat where  "m mod n = snd (divmod_nat m n)"lemma snd_divmod_nat [simp]:  "snd (divmod_nat m n) = m mod n"  by (simp add: mod_nat_def)lemma divmod_nat_div_mod:  "divmod_nat m n = (m div n, m mod n)"  by (simp add: prod_eq_iff)lemma div_nat_unique:  assumes "divmod_nat_rel m n (q, r)"   shows "m div n = q"  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)lemma mod_nat_unique:  assumes "divmod_nat_rel m n (q, r)"   shows "m mod n = r"  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"  using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"  by (simp add: divmod_nat_unique divmod_nat_rel_def)lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"  by (simp add: divmod_nat_unique divmod_nat_rel_def)lemma divmod_nat_base: "m < n ==> divmod_nat m n = (0, m)"  by (simp add: divmod_nat_unique divmod_nat_rel_def)lemma divmod_nat_step:  assumes "0 < n" and "n ≤ m"  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"proof (rule divmod_nat_unique)  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"    by (rule divmod_nat_rel)  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"    unfolding divmod_nat_rel_def using assms by autoqedtext {* 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 by (simp add: prod_eq_iff)lemma le_div_geq:  fixes m n :: nat  assumes "0 < n" and "n ≤ m"  shows "m div n = Suc ((m - n) div n)"  using assms divmod_nat_step by (simp add: prod_eq_iff)lemma mod_less [simp]:  fixes m n :: nat  assumes "m < n"  shows "m mod n = m"  using assms divmod_nat_base by (simp add: prod_eq_iff)lemma le_mod_geq:  fixes m n :: nat  assumes "n ≤ m"  shows "m mod n = (m - n) mod n"  using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)instance 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 ≠ 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 ≠ 0"  hence "!!a b. divmod_nat_rel n q (a, b) ==> divmod_nat_rel (m * n) (m * q) (a, m * b)"    unfolding divmod_nat_rel_def    by (auto split: split_if_asm, 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))" .  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)next  fix n :: nat show "n div 0 = 0"    by (simp add: div_nat_def divmod_nat_zero)next  fix n :: nat show "0 div n = 0"    by (simp add: div_nat_def divmod_nat_zero_left)qedendlemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 ∨ m < n then (0, m) else  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"  by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq)text {* Simproc for cancelling @{const div} and @{const mod} *}ML {*structure Cancel_Div_Mod_Nat = Cancel_Div_Mod(  val div_name = @{const_name div};  val mod_name = @{const_name mod};  val mk_binop = HOLogic.mk_binop;  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;  fun mk_sum [] = HOLogic.zero    | mk_sum [t] = t    | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);  fun dest_sum tm =    if HOLogic.is_zero tm then []    else      (case try HOLogic.dest_Suc tm of        SOME t => HOLogic.Suc_zero :: dest_sum t      | NONE =>          (case try dest_plus tm of            SOME (t, u) => dest_sum t @ dest_sum u          | NONE => [tm]));  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac    (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac})))*}simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}subsubsection {* Quotient *}lemma div_geq: "0 < n ==>  ¬ m < n ==> m div n = Suc ((m - n) div n)"by (simp add: le_div_geq linorder_not_less)lemma div_if: "0 < n ==> 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 simplemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"by simpsubsubsection {* 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 autolemma mod_less_eq_dividend [simp]:  fixes m n :: nat  shows "m mod n ≤ 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 ≤ m" by autoqedlemma mod_geq: "¬ m < (n::nat) ==> m mod n = (m - n) mod n"by (simp add: le_mod_geq linorder_not_less)lemma mod_if: "m mod (n::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)(* a simple rearrangement of mod_div_equality: *)lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"  using mod_div_equality2 [of n m] by arithlemma mod_le_divisor[simp]: "0 < n ==> m mod n ≤ (n::nat)"  apply (drule mod_less_divisor [where m = m])  apply simp  donesubsubsection {* Quotient and Remainder *}lemma divmod_nat_rel_mult1_eq:  "divmod_nat_rel b c (q, r)   ==> 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)"by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)lemma divmod_nat_rel_add1_eq:  "divmod_nat_rel a c (aq, ar) ==> divmod_nat_rel b c (bq, br)   ==> 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)"by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)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)  donelemma divmod_nat_rel_mult2_eq:  "divmod_nat_rel a b (q, r)   ==> 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)"by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])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)]:    "∀m::nat. m ≤ n --> (m div k) ≤ (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≤n |] ==> (k div n) ≤ (k div m)"apply (subgoal_tac "0<n") prefer 2 apply simpapply (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 simpapply (simp add: div_geq)apply (subgoal_tac "(k-n) div n ≤ (k-m) div n") prefer 2 apply (blast intro: div_le_mono diff_le_mono2)apply (rule le_trans, simp)apply (simp)donelemma div_le_dividend [simp]: "m div n ≤ (m::nat)"apply (case_tac "n=0", simp)apply (subgoal_tac "m div n ≤ m div 1", simp)apply (rule div_le_mono2)apply (simp_all (no_asm_simp))done(* Similar for "less than" *)lemma div_less_dividend [simp]:  "[|(1::nat) < n; 0 < m|] ==> m div n < m"apply (induct m rule: nat_less_induct)apply (rename_tac "m")apply (case_tac "m<n", simp)apply (subgoal_tac "0<n") prefer 2 apply simpapply (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)donetext{*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 ≤ Suc(na) *)apply (simp add: linorder_not_less le_Suc_eq mod_geq)apply (auto simp add: Suc_diff_le le_mod_geq)donelemma mod_eq_0_iff: "(m mod d = 0) = (∃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) ==> ∃q::nat. m = r + q*d"  apply (cut_tac a = m in mod_div_equality)  apply (simp only: add_ac)  apply (blast intro: sym)  donelemma split_div: "P(n div k :: nat) = ((k = 0 --> P 0) ∧ (k ≠ 0 --> (!i. !j<k. n = k*i + j --> P i)))" (is "?P = ?Q" is "_ = (_ ∧ (_ --> ?R))")proof  assume P: ?P  show ?Q  proof (cases)    assume "k = 0"    with P show ?Q by simp  next    assume not0: "k ≠ 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 ≠ 0"        with not0 n j P show "P i" by(simp add:add_ac)      qed    qed  qednext  assume Q: ?Q  show ?P  proof (cases)    assume "k = 0"    with Q show ?P by simp  next    assume not0: "k ≠ 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  qedqedlemma split_div_lemma:  assumes "0 < n"  shows "n * q ≤ m ∧ m < n * Suc q <-> q = ((m::nat) div n)" (is "?lhs <-> ?rhs")proof  assume ?rhs  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp  then have A: "n * q ≤ 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 simpqedtheorem split_div':  "P ((m::nat) div n) = ((n = 0 ∧ P 0) ∨   (∃q. (n * q ≤ m ∧ m < n * (Suc q)) ∧ P q))"  apply (case_tac "0 < n")  apply (simp only: add: split_div_lemma)  apply simp_all  donelemma split_mod: "P(n mod k :: nat) = ((k = 0 --> P n) ∧ (k ≠ 0 --> (!i. !j<k. n = k*i + j --> P j)))" (is "?P = ?Q" is "_ = (_ ∧ (_ --> ?R))")proof  assume P: ?P  show ?Q  proof (cases)    assume "k = 0"    with P show ?Q by simp  next    assume not0: "k ≠ 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  qednext  assume Q: ?Q  show ?P  proof (cases)    assume "k = 0"    with Q show ?P by simp  next    assume not0: "k ≠ 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  qedqedtheorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"  using mod_div_equality [of m n] by arithlemma div_mod_equality': "(m::nat) div n * n = m - m mod n"  using mod_div_equality [of m n] by arith(* FIXME: very similar to mult_div_cancel *)subsubsection {* An induction'' law for modulus arithmetic. *}lemma mod_induct_0:  assumes step: "∀i<p. P i --> P ((Suc i) mod p)"  and base: "P i" and i: "i<p"  shows "P 0"proof (rule ccontr)  assume contra: "¬(P 0)"  from i have p: "0<p" by simp  have "∀k. 0<k --> ¬ P (p-k)" (is "∀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 "¬(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 ∧ i+k=p"    by (blast dest: less_imp_add_positive)  hence "0<k ∧ i=p-k" by auto  moreover  note base  ultimately  show "False" by blastqedlemma mod_induct:  assumes step: "∀i<p. P i --> P ((Suc i) mod p)"  and base: "P i" and i: "i<p" and j: "j<p"  shows "P j"proof -  have "∀j<p. P j"  proof    fix j    show "j<p --> 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 blastqedlemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"  by (simp add: numeral_2_eq_2 le_div_geq)lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"  by (simp add: numeral_2_eq_2 le_mod_geq)lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"by (simp add: mult_2 [symmetric])lemma mod2_gr_0 [simp]: "0 < (m::nat) mod 2 <-> m mod 2 = 1"proof -  { fix n :: nat have  "(n::nat) < 2 ==> n = 0 ∨ n = 1" by (cases n) simp_all }  moreover have "m mod 2 < 2" by simp  ultimately have "m mod 2 = 0 ∨ m mod 2 = 1" .  then show ?thesis by autoqedtext{*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_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for vlemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for vlemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" apply (induct "m")apply (simp_all add: mod_Suc)donedeclare Suc_times_mod_eq [of "numeral w", simp] for wlemma Suc_div_le_mono [simp]: "n div k ≤ (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_alllemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"proof -  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp qed  (* 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 .qedlemma 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) donelemma mod_2_not_eq_zero_eq_one_nat:  fixes n :: nat  shows "n mod 2 ≠ 0 <-> n mod 2 = 1"  by simpsubsection {* Division on @{typ int} *}definition divmod_int_rel :: "int => int => int × int => bool" where    --{*definition of quotient and remainder*}  "divmod_int_rel a b = (λ(q, r). a = b * q + r ∧    (if 0 < b then 0 ≤ r ∧ r < b else if b < 0 then b < r ∧ r ≤ 0 else q = 0))"definition adjust :: "int => int × int => int × int" where    --{*for the division algorithm*}    "adjust b = (λ(q, r). if 0 ≤ r - b then (2 * q + 1, r - b)                         else (2 * q, r))"text{*algorithm for the case @{text "a≥0, b>0"}*}function posDivAlg :: "int => int => int × int" where  "posDivAlg a b = (if a < b ∨  b ≤ 0 then (0, a)     else adjust b (posDivAlg a (2 * b)))"by autotermination by (relation "measure (λ(a, b). nat (a - b + 1))")  (auto simp add: mult_2)text{*algorithm for the case @{text "a<0, b>0"}*}function negDivAlg :: "int => int => int × int" where  "negDivAlg a b = (if 0 ≤a + b ∨ b ≤ 0  then (-1, a + b)     else adjust b (negDivAlg a (2 * b)))"by autotermination by (relation "measure (λ(a, b). nat (- a - b))")  (auto simp add: mult_2)text{*algorithm for the general case @{term "b≠0"}*}definition divmod_int :: "int => int => int × 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 ≤ a then if 0 ≤ b then posDivAlg a b                  else if a = 0 then (0, 0)                       else apsnd uminus (negDivAlg (-a) (-b))               else                   if 0 < b then negDivAlg a b                  else apsnd uminus (posDivAlg (-a) (-b)))"instantiation int :: Divides.divbegindefinition div_int where  "a div b = fst (divmod_int a b)"lemma fst_divmod_int [simp]:  "fst (divmod_int a b) = a div b"  by (simp add: div_int_def)definition mod_int where  "a mod b = snd (divmod_int a b)"lemma snd_divmod_int [simp]:  "snd (divmod_int a b) = a mod b"  by (simp add: mod_int_def)instance ..endlemma divmod_int_mod_div:  "divmod_int p q = (p div q, p mod q)"  by (simp add: prod_eq_iff)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≤r-b then (2*q+1, r-b) else (2*q, r)           end    fun negDivAlg (a,b) =      if 0≤a+b then (~1,a+b)      else let val (q,r) = negDivAlg(a, 2*b)               in  if 0≤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≤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'  ≤ b*q + r;  0 ≤ r';  r' < b;  r < b |]        ==> q' ≤ (q::int)"apply (subgoal_tac "r' + b * (q'-q) ≤ 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 distrib_left)apply (subgoal_tac "b * q' < b * (1 + q) ") prefer 2 apply (simp add: right_diff_distrib distrib_left)apply (simp add: mult_less_cancel_left)donelemma unique_quotient_lemma_neg:     "[| b*q' + r' ≤ b*q + r;  r ≤ 0;  b < r;  b < r' |]        ==> q ≤ (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') |]        ==> 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)+donelemma unique_remainder:     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]        ==> r = r'"apply (subgoal_tac "q = q'") apply (simp add: divmod_int_rel_def)apply (blast intro: unique_quotient)donesubsubsection {* 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 ≤ 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 ≤ a" and "0 < b"  shows "divmod_int_rel a b (posDivAlg a b)"  using assms  apply (induct a b rule: posDivAlg.induct)  apply auto  apply (simp add: divmod_int_rel_def)  apply (subst posDivAlg_eqn, simp add: distrib_left)  apply (case_tac "a < b")  apply simp_all  apply (erule splitE)  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)  donesubsubsection {* 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≤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 assms  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::int)")  apply simp_all  apply (erule splitE)  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)  donesubsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}(*the case a=0*)lemma divmod_int_rel_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 posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"by (subst posDivAlg.simps, auto)lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"by (subst negDivAlg.simps, auto)lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"by (auto simp add: divmod_int_rel_def)lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg                    posDivAlg_correct negDivAlg_correct)lemma divmod_int_unique:  assumes "divmod_int_rel a b qr"   shows "divmod_int a b = qr"  using assms divmod_int_correct [of a b]  using unique_quotient [of a b] unique_remainder [of a b]  by (metis pair_collapse)lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"  using divmod_int_correct by (simp add: divmod_int_mod_div)lemma div_int_unique: "divmod_int_rel a b (q, r) ==> a div b = q"  by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])lemma mod_int_unique: "divmod_int_rel a b (q, r) ==> a mod b = r"  by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])instance int :: ring_divproof  fix a b :: int  show "a div b * b + a mod b = a"    using divmod_int_rel_div_mod [of a b]    unfolding divmod_int_rel_def by (simp add: mult_commute)next  fix a b c :: int  assume "b ≠ 0"  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"    using divmod_int_rel_div_mod [of a b]    unfolding divmod_int_rel_def by (auto simp: algebra_simps)  thus "(a + c * b) div b = c + a div b"    by (rule div_int_unique)next  fix a b c :: int  assume "c ≠ 0"  hence "!!q r. divmod_int_rel a b (q, r)    ==> divmod_int_rel (c * a) (c * b) (q, c * r)"    unfolding divmod_int_rel_def    by - (rule linorder_cases [of 0 b], auto simp: algebra_simps      mult_less_0_iff zero_less_mult_iff mult_strict_right_mono      mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"    using divmod_int_rel_div_mod [of a b] .  thus "(c * a) div (c * b) = a div b"    by (rule div_int_unique)next  fix a :: int show "a div 0 = 0"    by (rule div_int_unique, simp add: divmod_int_rel_def)next  fix a :: int show "0 div a = 0"    by (rule div_int_unique, auto simp add: divmod_int_rel_def)qedtext{*Basic laws about division and remainder*}lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"  by (fact mod_div_equality2 [symmetric])text {* Tool setup *}(* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)lemmas add_0s = add_0_left add_0_rightML {*structure Cancel_Div_Mod_Int = Cancel_Div_Mod(  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 div_mod_equality}, @{thm div_mod_equality2}];  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac     (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac})))*}simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}lemma pos_mod_conj: "(0::int) < b ==> 0 ≤ a mod b ∧ a mod b < b"  using divmod_int_correct [of a b]  by (auto simp add: divmod_int_rel_def prod_eq_iff)lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]lemma neg_mod_conj: "b < (0::int) ==> a mod b ≤ 0 ∧ b < a mod b"  using divmod_int_correct [of a b]  by (auto simp add: divmod_int_rel_def prod_eq_iff)lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]subsubsection {* General Properties of div and mod *}lemma div_pos_pos_trivial: "[| (0::int) ≤ a;  a < b |] ==> a div b = 0"apply (rule div_int_unique)apply (auto simp add: divmod_int_rel_def)donelemma div_neg_neg_trivial: "[| a ≤ (0::int);  b < a |] ==> a div b = 0"apply (rule div_int_unique)apply (auto simp add: divmod_int_rel_def)donelemma div_pos_neg_trivial: "[| (0::int) < a;  a+b ≤ 0 |] ==> a div b = -1"apply (rule div_int_unique)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) ≤ a;  a < b |] ==> a mod b = a"apply (rule_tac q = 0 in mod_int_unique)apply (auto simp add: divmod_int_rel_def)donelemma mod_neg_neg_trivial: "[| a ≤ (0::int);  b < a |] ==> a mod b = a"apply (rule_tac q = 0 in mod_int_unique)apply (auto simp add: divmod_int_rel_def)donelemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b ≤ 0 |] ==> a mod b = a+b"apply (rule_tac q = "-1" in mod_int_unique)apply (auto simp add: divmod_int_rel_def)donetext{*There is no @{text mod_neg_pos_trivial}.*}subsubsection {* Laws for div and mod with Unary Minus *}lemma zminus1_lemma:     "divmod_int_rel a b (q, r) ==> b ≠ 0      ==> 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 ≠ (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 div_int_unique])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 mod_int_unique])donelemma zmod_zminus1_not_zero:  fixes k l :: int  shows "- k mod l ≠ 0 ==> k mod l ≠ 0"  unfolding zmod_zminus1_eq_if by autolemma zdiv_zminus2_eq_if:     "b ≠ (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 div_minus_right)lemma zmod_zminus2_eq_if:     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"by (simp add: zmod_zminus1_eq_if mod_minus_right)lemma zmod_zminus2_not_zero:  fixes k l :: int  shows "k mod - l ≠ 0 ==> k mod l ≠ 0"  unfolding zmod_zminus2_eq_if by auto subsubsection {* Computation of Division and Remainder *}lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"by (simp add: div_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 ≤ b |] ==> a div b = fst (posDivAlg a b)"by (simp add: div_int_def divmod_int_def)lemma mod_pos_pos: "[| 0 < a;  0 ≤ 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 (apsnd uminus (negDivAlg (-a) (-b)))"by (simp add: div_int_def divmod_int_def)lemma mod_pos_neg:     "[| 0 < a;  b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"by (simp add: mod_int_def divmod_int_def)text{*a negative, b negative *}lemma div_neg_neg:     "[| a < 0;  b ≤ 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"by (simp add: div_int_def divmod_int_def)lemma mod_neg_neg:     "[| a < 0;  b ≤ 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"by (simp add: mod_int_def divmod_int_def)text {*Simplify expresions in which div and mod combine numerical constants*}lemma int_div_pos_eq: "[|(a::int) = b * q + r; 0 ≤ r; r < b|] ==> a div b = q"  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)lemma int_div_neg_eq: "[|(a::int) = b * q + r; r ≤ 0; b < r|] ==> a div b = q"  by (rule div_int_unique [of a b q r],    simp add: divmod_int_rel_def)lemma int_mod_pos_eq: "[|(a::int) = b * q + r; 0 ≤ r; r < b|] ==> a mod b = r"  by (rule mod_int_unique [of a b q r],    simp add: divmod_int_rel_def)lemma int_mod_neg_eq: "[|(a::int) = b * q + r; r ≤ 0; b < r|] ==> a mod b = r"  by (rule mod_int_unique [of a b q r],    simp add: divmod_int_rel_def)(* simprocs adapted from HOL/ex/Binary.thy *)ML {*local  val mk_number = HOLogic.mk_number HOLogic.intT  val plus = @{term "plus :: int => int => int"}  val times = @{term "times :: int => int => int"}  val zero = @{term "0 :: int"}  val less = @{term "op < :: int => int => bool"}  val le = @{term "op ≤ :: int => int => bool"}  val simps = @{thms arith_simps} @ @{thms rel_simps} @    map (fn th => th RS sym) [@{thm numeral_1_eq_1}]  fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)    (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps 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 posrule negrule =    binary_proc (fn ctxt => fn ((a, t), (b, u)) =>      if b = 0 then NONE else let        val (q, r) = pairself mk_number (Integer.div_mod a b)        val goal1 = HOLogic.mk_eq (t, plus $(times$ u $q)$ r)        val (goal2, goal3, rule) = if b > 0          then (le $zero$ r, less $r$ u, posrule RS eq_reflection)          else (le $r$ zero, less $u$ r, negrule RS eq_reflection)      in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)end*}simproc_setup binary_int_div  ("numeral m div numeral n :: int" |   "numeral m div neg_numeral n :: int" |   "neg_numeral m div numeral n :: int" |   "neg_numeral m div neg_numeral n :: int") =  {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}simproc_setup binary_int_mod  ("numeral m mod numeral n :: int" |   "numeral m mod neg_numeral n :: int" |   "neg_numeral m mod numeral n :: int" |   "neg_numeral m mod neg_numeral n :: int") =  {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}lemmas posDivAlg_eqn_numeral [simp] =    posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v wlemmas negDivAlg_eqn_numeral [simp] =    negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v wtext{*Special-case simplification *}(** The last remaining special cases for constant arithmetic:    1 div z and 1 mod z **)lemmas div_pos_pos_1_numeral [simp] =  div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for wlemmas div_pos_neg_1_numeral [simp] =  div_pos_neg [OF zero_less_one, of "neg_numeral w",  OF neg_numeral_less_zero] for wlemmas mod_pos_pos_1_numeral [simp] =  mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for wlemmas mod_pos_neg_1_numeral [simp] =  mod_pos_neg [OF zero_less_one, of "neg_numeral w",  OF neg_numeral_less_zero] for wlemmas posDivAlg_eqn_1_numeral [simp] =    posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for wlemmas negDivAlg_eqn_1_numeral [simp] =    negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for wsubsubsection {* Monotonicity in the First Argument (Dividend) *}lemma zdiv_mono1: "[| a ≤ a';  0 < (b::int) |] ==> a div b ≤ 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)donelemma zdiv_mono1_neg: "[| a ≤ a';  (b::int) < 0 |] ==> a' div b ≤ 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)donesubsubsection {* Monotonicity in the Second Argument (Divisor) *}lemma q_pos_lemma:     "[| 0 ≤ b'*q' + r'; r' < b';  0 < b' |] ==> 0 ≤ (q'::int)"apply (subgoal_tac "0 < b'* (q' + 1) ") apply (simp add: zero_less_mult_iff)apply (simp add: distrib_left)donelemma zdiv_mono2_lemma:     "[| b*q + r = b'*q' + r';  0 ≤ b'*q' + r';            r' < b';  0 ≤ r;  0 < b';  b' ≤ b |]         ==> q ≤ (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 simpapply (simp (no_asm_simp) add: distrib_left)apply (subst add_commute, rule add_less_le_mono, arith)apply (rule mult_right_mono, auto)donelemma zdiv_mono2:     "[| (0::int) ≤ a;  0 < b';  b' ≤ b |] ==> a div b ≤ a div b'"apply (subgoal_tac "b ≠ 0") prefer 2 apply arithapply (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)donelemma q_neg_lemma:     "[| b'*q' + r' < 0;  0 ≤ r';  0 < b' |] ==> q' ≤ (0::int)"apply (subgoal_tac "b'*q' < 0") apply (simp add: mult_less_0_iff, arith)donelemma zdiv_mono2_neg_lemma:     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;            r < b;  0 ≤ r';  0 < b';  b' ≤ b |]         ==> q' ≤ (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: distrib_left)apply (subgoal_tac "b*q' ≤ b'*q'") prefer 2 apply (simp add: mult_right_mono_neg, arith)donelemma zdiv_mono2_neg:     "[| a < (0::int);  0 < b';  b' ≤ b |] ==> a div b' ≤ 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)donesubsubsection {* 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) |]        ==> 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 distrib_left 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 div_int_unique])donetext{*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) |]        ==> 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 distrib_left)(*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] div_int_unique)donelemma posDivAlg_div_mod:  assumes "k ≥ 0"  and "l ≥ 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 div_int_unique [OF this] mod_int_unique [OF this]  show ?thesis by simpqedlemma negDivAlg_div_mod:  assumes "k < 0"  and "l > 0"  shows "negDivAlg k l = (k div l, k mod l)"proof -  from assms have "l ≠ 0" by simp  from assms negDivAlg_correct    have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"    by simp  from div_int_unique [OF this] mod_int_unique [OF this]  show ?thesis by simpqedlemma 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]lemma zmod_zdiv_equality':  "(m::int) mod n = m - (m div n) * n"  using mod_div_equality [of m n] by arithsubsubsection {* 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 ≤ 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) apply (simp)apply (simp)donelemma zmult2_lemma_aux2:     "[| (0::int) < c;   b < r;  r ≤ 0 |] ==> b * (q mod c) + r ≤ 0"apply (subgoal_tac "b * (q mod c) ≤ 0") apply arithapply (simp add: mult_le_0_iff)donelemma zmult2_lemma_aux3: "[| (0::int) < c;  0 ≤ r;  r < b |] ==> 0 ≤ b * (q mod c) + r"apply (subgoal_tac "0 ≤ b * (q mod c) ")apply arithapply (simp add: zero_le_mult_iff)donelemma zmult2_lemma_aux4: "[| (0::int) < c; 0 ≤ 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)apply simpdonelemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 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 distrib_left [symmetric]                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)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 div_int_unique])donelemma 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 mod_int_unique])donelemma div_pos_geq:  fixes k l :: int  assumes "0 < l" and "l ≤ k"  shows "k div l = (k - l) div l + 1"proof -  have "k = (k - l) + l" by simp  then obtain j where k: "k = j + l" ..  with assms show ?thesis by simpqedlemma mod_pos_geq:  fixes k l :: int  assumes "0 < l" and "l ≤ k"  shows "k mod l = (k - l) mod l"proof -  have "k = (k - l) + l" by simp  then obtain j where k: "k = j + l" ..  with assms show ?thesis by simpqedsubsubsection {* 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) = (∀i j. 0≤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)donelemma split_neg_lemma: "k<0 ==>    P(n div k :: int)(n mod k) = (∀i j. k<j & j≤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)donelemma split_zdiv: "P(n div k :: int) =  ((k = 0 --> P 0) &    (0<k --> (∀i j. 0≤j & j<k & n = k*i + j --> P i)) &    (k<0 --> (∀i j. k<j & j≤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"])donelemma split_zmod: "P(n mod k :: int) =  ((k = 0 --> P n) &    (0<k --> (∀i j. 0≤j & j<k & n = k*i + j --> P j)) &    (k<0 --> (∀i j. k<j & j≤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"])donetext {* Enable (lin)arith to deal with @{const div} and @{const mod}  when these are applied to some constant that is of the form  @{term "numeral k"}: *}declare split_zdiv [of _ _ "numeral k", arith_split] for kdeclare split_zmod [of _ _ "numeral k", arith_split] for ksubsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *}lemma pos_divmod_int_rel_mult_2:  assumes "0 ≤ b"  assumes "divmod_int_rel a b (q, r)"  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"  using assms unfolding divmod_int_rel_def by autolemma neg_divmod_int_rel_mult_2:  assumes "b ≤ 0"  assumes "divmod_int_rel (a + 1) b (q, r)"  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"  using assms unfolding divmod_int_rel_def by autotext{*computing div by shifting *}lemma pos_zdiv_mult_2: "(0::int) ≤ a ==> (1 + 2*b) div (2*a) = b div a"  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]  by (rule div_int_unique)lemma neg_zdiv_mult_2:   assumes A: "a ≤ (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]  by (rule div_int_unique)(* FIXME: add rules for negative numerals *)lemma zdiv_numeral_Bit0 [simp]:  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =    numeral v div (numeral w :: int)"  unfolding numeral.simps unfolding mult_2 [symmetric]  by (rule div_mult_mult1, simp)lemma zdiv_numeral_Bit1 [simp]:  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =      (numeral v div (numeral w :: int))"  unfolding numeral.simps  unfolding mult_2 [symmetric] add_commute [of _ 1]  by (rule pos_zdiv_mult_2, simp)lemma pos_zmod_mult_2:  fixes a b :: int  assumes "0 ≤ a"  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"  using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]  by (rule mod_int_unique)lemma neg_zmod_mult_2:  fixes a b :: int  assumes "a ≤ 0"  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"  using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]  by (rule mod_int_unique)(* FIXME: add rules for negative numerals *)lemma zmod_numeral_Bit0 [simp]:  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =      (2::int) * (numeral v mod numeral w)"  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)lemma zmod_numeral_Bit1 [simp]:  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =    2 * (numeral v mod numeral w) + (1::int)"  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]  unfolding mult_2 [symmetric] add_commute [of _ 1]  by (rule pos_zmod_mult_2, simp)lemma zdiv_eq_0_iff: "(i::int) div k = 0 <-> k=0 ∨ 0≤i ∧ i<k ∨ i≤0 ∧ k<i" (is "?L = ?R")proof  assume ?L  have "?L --> ?R" by (rule split_zdiv[THEN iffD2]) simp  with ?L show ?R by blastnext  assume ?R thus ?L    by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)qedsubsubsection {* Quotients of Signs *}lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"apply (subgoal_tac "a div b ≤ -1", force)apply (rule order_trans)apply (rule_tac a' = "-1" in zdiv_mono1)apply (auto simp add: div_eq_minus1)donelemma div_nonneg_neg_le0: "[| (0::int) ≤ a; b < 0 |] ==> a div b ≤ 0"by (drule zdiv_mono1_neg, auto)lemma div_nonpos_pos_le0: "[| (a::int) ≤ 0; b > 0 |] ==> a div b ≤ 0"by (drule zdiv_mono1, auto)text{* Now for some equivalences of the form @{text"a div b >=< 0 <-> …"}conditional upon the sign of @{text a} or @{text b}. There are many more.They should all be simp rules unless that causes too much search. *}lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 ≤ a div b) = (0 ≤ a)"apply autoapply (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)donelemma neg_imp_zdiv_nonneg_iff:  "b < (0::int) ==> (0 ≤ a div b) = (a ≤ (0::int))"apply (subst div_minus_minus [symmetric])apply (subst pos_imp_zdiv_nonneg_iff, auto)done(*But not (a div b ≤ 0 iff a≤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)lemma pos_imp_zdiv_pos_iff:  "0<k ==> 0 < (i::int) div k <-> k ≤ i"using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]by arith(*Again the law fails for ≤: consider a = -1, b = -2 when a div b = 0*)lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)lemma nonneg1_imp_zdiv_pos_iff:  "(0::int) <= a ==> (a div b > 0) = (a >= b & b>0)"apply rule apply rule  using div_pos_pos_trivial[of a b]apply arith apply(cases "b=0")apply simp using div_nonneg_neg_le0[of a b]apply arithusing int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simpdonelemma zmod_le_nonneg_dividend: "(m::int) ≥ 0 ==> m mod k ≤ m"apply (rule split_zmod[THEN iffD2])apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)donesubsubsection {* The Divides Relation *}lemma dvd_neg_numeral_left [simp]:  fixes y :: "'a::comm_ring_1"  shows "(neg_numeral k) dvd y <-> (numeral k) dvd y"  unfolding neg_numeral_def minus_dvd_iff ..lemma dvd_neg_numeral_right [simp]:  fixes x :: "'a::comm_ring_1"  shows "x dvd (neg_numeral k) <-> x dvd (numeral k)"  unfolding neg_numeral_def dvd_minus_iff ..lemmas dvd_eq_mod_eq_0_numeral [simp] =  dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x ysubsubsection {* Further properties *}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) (* FIXME: generalize *)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)donelemma 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)donelemma abs_div: "(y::int) dvd x ==> abs (x div y) = abs x div abs y"by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)text{*Suggested by Matthias Daum*}lemma int_power_div_base:     "[|0 < m; 0 < k|] ==> 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_alldonetext {* 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]  mod_mult_right_eq[symmetric]  mod_mult_left_eq [symmetric]  power_mod  zminus_zmod zdiff_zmod_left zdiff_zmod_righttext {* Distributive laws for function @{text nat}. *}lemma nat_div_distrib: "0 ≤ x ==> nat (x div y) = nat x div nat y"apply (rule linorder_cases [of y 0])apply (simp add: div_nonneg_neg_le0)apply simpapply (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:  "[|0 ≤ x; 0 ≤ y|] ==> nat (x mod y) = nat x mod nat y"apply (case_tac "y = 0", simp)apply (simp add: nat_eq_iff zmod_int)donetext  {* transfer setup *}lemma transfer_nat_int_functions:    "(x::int) >= 0 ==> y >= 0 ==> (nat x) div (nat y) = nat (x div y)"    "(x::int) >= 0 ==> y >= 0 ==> (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 ==> y >= 0 ==> x div y >= 0"    "(x::int) >= 0 ==> y >= 0 ==> x mod y >= 0"  apply (cases "y = 0")  apply (auto simp add: pos_imp_zdiv_nonneg_iff)  apply (cases "y = 0")  apply autodonedeclare transfer_morphism_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 ==> is_nat y ==> is_nat (x div y)"    "is_nat x ==> is_nat y ==> is_nat (x mod y)"  by (simp_all only: is_nat_def transfer_nat_int_function_closures)declare transfer_morphism_int_nat [transfer add return:  transfer_int_nat_functions  transfer_int_nat_function_closures]text{*Suggested by Matthias Daum*}lemma int_div_less_self: "[|0 < x; 1 < k|] ==> x div k < (x::int)"apply (subgoal_tac "nat x div nat k < nat x") apply (simp add: nat_div_distrib [symmetric])apply (rule Divides.div_less_dividend, simp_all)donelemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n <-> 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)qedlemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y ≤ x"  shows "∃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 arithqedlemma nat_mod_eq_iff: "(x::nat) mod n = y mod n <-> (∃q1 q2. x + n * q1 = y + n * q2)"   (is "?lhs = ?rhs")proof  assume H: "x mod n = y mod n"  {assume xy: "x ≤ 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 ≤ 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 simpqedlemma div_nat_numeral [simp]:  "(numeral v :: nat) div numeral v' = nat (numeral v div numeral v')"  by (simp add: nat_div_distrib)lemma one_div_nat_numeral [simp]:  "Suc 0 div numeral v' = nat (1 div numeral v')"  by (subst nat_div_distrib, simp_all)lemma mod_nat_numeral [simp]:  "(numeral v :: nat) mod numeral v' = nat (numeral v mod numeral v')"  by (simp add: nat_mod_distrib)lemma one_mod_nat_numeral [simp]:  "Suc 0 mod numeral v' = nat (1 mod numeral v')"  by (subst nat_mod_distrib) simp_alllemma mod_2_not_eq_zero_eq_one_int:  fixes k :: int  shows "k mod 2 ≠ 0 <-> k mod 2 = 1"  by autosubsubsection {* Tools setup *}text {* Nitpick *}lemmas [nitpick_unfold] = dvd_eq_mod_eq_0 mod_div_equality' zmod_zdiv_equality'subsubsection {* Code generation *}definition pdivmod :: "int => int => int × int" where  "pdivmod k l = (¦k¦ div ¦l¦, ¦k¦ mod ¦l¦)"lemma pdivmod_posDivAlg [code]:  "pdivmod k l = (if l = 0 then (0, ¦k¦) else posDivAlg ¦k¦ ¦l¦)"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 ∧ 0 ≤ k ∨ l < 0 ∧ k < 0    then pdivmod k l    else (let (r, s) = pdivmod k l in       if s = 0 then (- r, 0) else (- r - 1, ¦l¦ - s))))"proof -  have aux: "!!q::int. - k = l * q <-> 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)qedlemma 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, ¦l¦ - s))))"proof -  have "k ≠ 0 ==> l ≠ 0 ==> 0 < l ∧ 0 ≤ k ∨ l < 0 ∧ k < 0 <-> sgn k = sgn l"    by (auto simp add: not_less sgn_if)  then show ?thesis by (simp add: divmod_int_pdivmod)qedcode_modulename SML  Divides Arithcode_modulename OCaml  Divides Arithcode_modulename Haskell  Divides Arithend`