(*  Title:      HOL/NumberTheory/IntPrimes.thy

     ID:         $Id$

     Author:     Thomas M. Rasmussen

     Copyright   2000  University of Cambridge

 *)

 header {* Divisibility and prime numbers (on integers) *}

 theory IntPrimes = Primes:

 text {*

   The @{text dvd} relation, GCD, Euclid's extended algorithm, primes,

   congruences (all on the Integers).  Comparable to theory @{text

   Primes}, but @{text dvd} is included here as it is not present in

   main HOL.  Also includes extended GCD and congruences not present in

   @{text Primes}.

 *}

 subsection {* Definitions *}

 consts

   xzgcda :: "int * int * int * int * int * int * int * int => int * int * int"

   xzgcd :: "int => int => int * int * int"

   zprime :: "int set"

   zcong :: "int => int => int => bool"    ("(1[_ = _] '(mod _'))")

 recdef xzgcda

   "measure ((\<lambda>(m, n, r', r, s', s, t', t). nat r)

     :: int * int * int * int *int * int * int * int => nat)"

   "xzgcda (m, n, r', r, s', s, t', t) =

     (if r \<le> 0 then (r', s', t')

      else xzgcda (m, n, r, r' mod r, s, s' - (r' div r) * s, t, t' - (r' div r) * t))"

   (hints simp: pos_mod_bound)

 constdefs

   zgcd :: "int * int => int"

   "zgcd == \<lambda>(x,y). int (gcd (nat (abs x), nat (abs y)))"

 defs

   xzgcd_def: "xzgcd m n == xzgcda (m, n, m, n, 1, 0, 0, 1)"

   zprime_def: "zprime == {p. 1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p)}"

   zcong_def: "[a = b] (mod m) == m dvd (a - b)"

 lemma zabs_eq_iff:

     "(abs (z::int) = w) = (z = w \<and> 0 <= z \<or> z = -w \<and> z < 0)"

   apply (auto simp add: zabs_def)

   done

 text {* \medskip @{term gcd} lemmas *}

 lemma gcd_add1_eq: "gcd (m + k, k) = gcd (m + k, m)"

   apply (simp add: gcd_commute)

   done

 lemma gcd_diff2: "m \<le> n ==> gcd (n, n - m) = gcd (n, m)"

   apply (subgoal_tac "n = m + (n - m)")

    apply (erule ssubst, rule gcd_add1_eq)

   apply simp

   done

 subsection {* Divides relation *}

 lemma zdvd_0_right [iff]: "(m::int) dvd 0"

   apply (unfold dvd_def)

   apply (blast intro: zmult_0_right [symmetric])

   done

 lemma zdvd_0_left [iff]: "(0 dvd (m::int)) = (m = 0)"

   apply (unfold dvd_def)

   apply auto

   done

 lemma zdvd_1_left [iff]: "1 dvd (m::int)"

   apply (unfold dvd_def)

   apply simp

   done

 lemma zdvd_refl [simp]: "m dvd (m::int)"

   apply (unfold dvd_def)

   apply (blast intro: zmult_1_right [symmetric])

   done

 lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"

   apply (unfold dvd_def)

   apply (blast intro: zmult_assoc)

   done

 lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))"

   apply (unfold dvd_def)

   apply auto

    apply (rule_tac [!] x = "-k" in exI)

   apply auto

   done

 lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))"

   apply (unfold dvd_def)

   apply auto

    apply (rule_tac [!] x = "-k" in exI)

   apply auto

   done

 lemma zdvd_anti_sym:

     "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"

   apply (unfold dvd_def)

   apply auto

   apply (simp add: zmult_assoc zmult_eq_self_iff int_0_less_mult_iff zmult_eq_1_iff)

   done

 lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"

   apply (unfold dvd_def)

   apply (blast intro: zadd_zmult_distrib2 [symmetric])

   done

 lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"

   apply (unfold dvd_def)

   apply (blast intro: zdiff_zmult_distrib2 [symmetric])

   done

 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"

   apply (subgoal_tac "m = n + (m - n)")

    apply (erule ssubst)

    apply (blast intro: zdvd_zadd)

   apply simp

   done

 lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"

   apply (unfold dvd_def)

   apply (blast intro: zmult_left_commute)

   done

 lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"

   apply (subst zmult_commute)

   apply (erule zdvd_zmult)

   done

 lemma [iff]: "(k::int) dvd m * k"

   apply (rule zdvd_zmult)

   apply (rule zdvd_refl)

   done

 lemma [iff]: "(k::int) dvd k * m"

   apply (rule zdvd_zmult2)

   apply (rule zdvd_refl)

   done

 lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"

   apply (unfold dvd_def)

   apply (simp add: zmult_assoc)

   apply blast

   done

 lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"

   apply (rule zdvd_zmultD2)

   apply (subst zmult_commute)

   apply assumption

   done

 lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"

   apply (unfold dvd_def)

   apply clarify

   apply (rule_tac x = "k * ka" in exI)

   apply (simp add: zmult_ac)

   done

 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"

   apply (rule iffI)

    apply (erule_tac [2] zdvd_zadd)

    apply (subgoal_tac "n = (n + k * m) - k * m")

     apply (erule ssubst)

     apply (erule zdvd_zdiff)

     apply simp_all

   done

 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"

   apply (unfold dvd_def)

   apply (auto simp add: zmod_zmult_zmult1)

   done

 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"

   apply (subgoal_tac "k dvd n * (m div n) + m mod n")

    apply (simp add: zmod_zdiv_equality [symmetric])

   apply (simp only: zdvd_zadd zdvd_zmult2)

   done

 lemma zdvd_iff_zmod_eq_0: "(k dvd n) = (n mod (k::int) = 0)"

   apply (unfold dvd_def)

   apply auto

   done

 lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"

   apply (unfold dvd_def)

   apply auto

   apply (subgoal_tac "0 < n")

    prefer 2

    apply (blast intro: zless_trans)

   apply (simp add: int_0_less_mult_iff)

   apply (subgoal_tac "n * k < n * 1")

    apply (drule zmult_zless_cancel1 [THEN iffD1])

    apply auto

   done

 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"

   apply (auto simp add: dvd_def nat_abs_mult_distrib)

   apply (auto simp add: nat_eq_iff zabs_eq_iff)

    apply (rule_tac [2] x = "-(int k)" in exI)

   apply (auto simp add: zmult_int [symmetric])

   done

 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"

   apply (auto simp add: dvd_def zabs_def zmult_int [symmetric])

     apply (rule_tac [3] x = "nat k" in exI)

     apply (rule_tac [2] x = "-(int k)" in exI)

     apply (rule_tac x = "nat (-k)" in exI)

     apply (cut_tac [3] k = m in int_less_0_conv)

     apply (cut_tac k = m in int_less_0_conv)

     apply (auto simp add: int_0_le_mult_iff zmult_less_0_iff

       nat_mult_distrib [symmetric] nat_eq_iff2)

   done

 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"

   apply (auto simp add: dvd_def zmult_int [symmetric])

   apply (rule_tac x = "nat k" in exI)

   apply (cut_tac k = m in int_less_0_conv)

   apply (auto simp add: int_0_le_mult_iff zmult_less_0_iff

     nat_mult_distrib [symmetric] nat_eq_iff2)

   done

 lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"

   apply (auto simp add: dvd_def)

    apply (rule_tac [!] x = "-k" in exI)

    apply auto

   done

 lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"

   apply (auto simp add: dvd_def)

    apply (drule zminus_equation [THEN iffD1])

    apply (rule_tac [!] x = "-k" in exI)

    apply auto

   done

 subsection {* Euclid's Algorithm and GCD *}

 lemma zgcd_0 [simp]: "zgcd (m, 0) = abs m"

   apply (simp add: zgcd_def zabs_def)

   done

 lemma zgcd_0_left [simp]: "zgcd (0, m) = abs m"

   apply (simp add: zgcd_def zabs_def)

   done

 lemma zgcd_zminus [simp]: "zgcd (-m, n) = zgcd (m, n)"

   apply (simp add: zgcd_def)

   done

 lemma zgcd_zminus2 [simp]: "zgcd (m, -n) = zgcd (m, n)"

   apply (simp add: zgcd_def)

   done

 lemma zgcd_non_0: "0 < n ==> zgcd (m, n) = zgcd (n, m mod n)"

   apply (frule_tac b = n and a = m in pos_mod_sign)

   apply (simp add: zgcd_def zabs_def nat_mod_distrib)

   apply (auto simp add: gcd_non_0 nat_mod_distrib [symmetric] zmod_zminus1_eq_if)

   apply (frule_tac a = m in pos_mod_bound)

   apply (simp add: nat_diff_distrib gcd_diff2 nat_le_eq_zle)

   apply (simp add: gcd_non_0 nat_mod_distrib [symmetric])

   done

 lemma zgcd_eq: "zgcd (m, n) = zgcd (n, m mod n)"

   apply (case_tac "n = 0", simp add: DIVISION_BY_ZERO)

   apply (auto simp add: linorder_neq_iff zgcd_non_0)

   apply (cut_tac m = "-m" and n = "-n" in zgcd_non_0)

    apply auto

   done

 lemma zgcd_1 [simp]: "zgcd (m, 1) = 1"

   apply (simp add: zgcd_def zabs_def)

   done

 lemma zgcd_0_1_iff [simp]: "(zgcd (0, m) = 1) = (abs m = 1)"

   apply (simp add: zgcd_def zabs_def)

   done

 lemma zgcd_zdvd1 [iff]: "zgcd (m, n) dvd m"

   apply (simp add: zgcd_def zabs_def int_dvd_iff)

   done

 lemma zgcd_zdvd2 [iff]: "zgcd (m, n) dvd n"

   apply (simp add: zgcd_def zabs_def int_dvd_iff)

   done

 lemma zgcd_greatest_iff: "k dvd zgcd (m, n) = (k dvd m \<and> k dvd n)"

   apply (simp add: zgcd_def zabs_def int_dvd_iff dvd_int_iff nat_dvd_iff)

   done

 lemma zgcd_commute: "zgcd (m, n) = zgcd (n, m)"

   apply (simp add: zgcd_def gcd_commute)

   done

 lemma zgcd_1_left [simp]: "zgcd (1, m) = 1"

   apply (simp add: zgcd_def gcd_1_left)

   done

 lemma zgcd_assoc: "zgcd (zgcd (k, m), n) = zgcd (k, zgcd (m, n))"

   apply (simp add: zgcd_def gcd_assoc)

   done

 lemma zgcd_left_commute: "zgcd (k, zgcd (m, n)) = zgcd (m, zgcd (k, n))"

   apply (rule zgcd_commute [THEN trans])

   apply (rule zgcd_assoc [THEN trans])

   apply (rule zgcd_commute [THEN arg_cong])

   done

 lemmas zgcd_ac = zgcd_assoc zgcd_commute zgcd_left_commute

   -- {* addition is an AC-operator *}

 lemma zgcd_zmult_distrib2: "0 \<le> k ==> k * zgcd (m, n) = zgcd (k * m, k * n)"

   apply (simp del: zmult_zminus_right

     add: zmult_zminus_right [symmetric] nat_mult_distrib zgcd_def zabs_def

     zmult_less_0_iff gcd_mult_distrib2 [symmetric] zmult_int [symmetric])

   done

 lemma zgcd_zmult_distrib2_abs: "zgcd (k * m, k * n) = abs k * zgcd (m, n)"

   apply (simp add: zabs_def zgcd_zmult_distrib2)

   done

 lemma zgcd_self [simp]: "0 \<le> m ==> zgcd (m, m) = m"

   apply (cut_tac k = m and m = "1" and n = "1" in zgcd_zmult_distrib2)

    apply simp_all

   done

 lemma zgcd_zmult_eq_self [simp]: "0 \<le> k ==> zgcd (k, k * n) = k"

   apply (cut_tac k = k and m = "1" and n = n in zgcd_zmult_distrib2)

    apply simp_all

   done

 lemma zgcd_zmult_eq_self2 [simp]: "0 \<le> k ==> zgcd (k * n, k) = k"

   apply (cut_tac k = k and m = n and n = "1" in zgcd_zmult_distrib2)

    apply simp_all

   done

 lemma zrelprime_zdvd_zmult_aux: "zgcd (n, k) = 1 ==> k dvd m * n ==> 0 \<le> m ==> k dvd m"

   apply (subgoal_tac "m = zgcd (m * n, m * k)")

    apply (erule ssubst, rule zgcd_greatest_iff [THEN iffD2])

    apply (simp_all add: zgcd_zmult_distrib2 [symmetric] int_0_le_mult_iff)

   done

 lemma zrelprime_zdvd_zmult: "zgcd (n, k) = 1 ==> k dvd m * n ==> k dvd m"

   apply (case_tac "0 \<le> m")

    apply (blast intro: zrelprime_zdvd_zmult_aux)

   apply (subgoal_tac "k dvd -m")

    apply (rule_tac [2] zrelprime_zdvd_zmult_aux)

      apply auto

   done

 lemma zprime_imp_zrelprime:

     "p \<in> zprime ==> \<not> p dvd n ==> zgcd (n, p) = 1"

   apply (unfold zprime_def)

   apply auto

   done

 lemma zless_zprime_imp_zrelprime:

     "p \<in> zprime ==> 0 < n ==> n < p ==> zgcd (n, p) = 1"

   apply (erule zprime_imp_zrelprime)

   apply (erule zdvd_not_zless)

   apply assumption

   done

 lemma zprime_zdvd_zmult:

     "0 \<le> (m::int) ==> p \<in> zprime ==> p dvd m * n ==> p dvd m \<or> p dvd n"

   apply safe

   apply (rule zrelprime_zdvd_zmult)

    apply (rule zprime_imp_zrelprime)

     apply auto

   done

 lemma zgcd_zadd_zmult [simp]: "zgcd (m + n * k, n) = zgcd (m, n)"

   apply (rule zgcd_eq [THEN trans])

   apply (simp add: zmod_zadd1_eq)

   apply (rule zgcd_eq [symmetric])

   done

 lemma zgcd_zdvd_zgcd_zmult: "zgcd (m, n) dvd zgcd (k * m, n)"

   apply (simp add: zgcd_greatest_iff)

   apply (blast intro: zdvd_trans)

   done

 lemma zgcd_zmult_zdvd_zgcd:

     "zgcd (k, n) = 1 ==> zgcd (k * m, n) dvd zgcd (m, n)"

   apply (simp add: zgcd_greatest_iff)

   apply (rule_tac n = k in zrelprime_zdvd_zmult)

    prefer 2

    apply (simp add: zmult_commute)

   apply (subgoal_tac "zgcd (k, zgcd (k * m, n)) = zgcd (k * m, zgcd (k, n))")

    apply simp

   apply (simp (no_asm) add: zgcd_ac)

   done

 lemma zgcd_zmult_cancel: "zgcd (k, n) = 1 ==> zgcd (k * m, n) = zgcd (m, n)"

   apply (simp add: zgcd_def nat_abs_mult_distrib gcd_mult_cancel)

   done

 lemma zgcd_zgcd_zmult:

     "zgcd (k, m) = 1 ==> zgcd (n, m) = 1 ==> zgcd (k * n, m) = 1"

   apply (simp (no_asm_simp) add: zgcd_zmult_cancel)

   done

 lemma zdvd_iff_zgcd: "0 < m ==> (m dvd n) = (zgcd (n, m) = m)"

   apply safe

    apply (rule_tac [2] n = "zgcd (n, m)" in zdvd_trans)

     apply (rule_tac [3] zgcd_zdvd1)

    apply simp_all

   apply (unfold dvd_def)

   apply auto

   done

 subsection {* Congruences *}

 lemma zcong_1 [simp]: "[a = b] (mod 1)"

   apply (unfold zcong_def)

   apply auto

   done

 lemma zcong_refl [simp]: "[k = k] (mod m)"

   apply (unfold zcong_def)

   apply auto

   done

 lemma zcong_sym: "[a = b] (mod m) = [b = a] (mod m)"

   apply (unfold zcong_def dvd_def)

   apply auto

    apply (rule_tac [!] x = "-k" in exI)

    apply auto

   done

 lemma zcong_zadd:

     "[a = b] (mod m) ==> [c = d] (mod m) ==> [a + c = b + d] (mod m)"

   apply (unfold zcong_def)

   apply (rule_tac s = "(a - b) + (c - d)" in subst)

    apply (rule_tac [2] zdvd_zadd)

     apply auto

   done

 lemma zcong_zdiff:

     "[a = b] (mod m) ==> [c = d] (mod m) ==> [a - c = b - d] (mod m)"

   apply (unfold zcong_def)

   apply (rule_tac s = "(a - b) - (c - d)" in subst)

    apply (rule_tac [2] zdvd_zdiff)

     apply auto

   done

 lemma zcong_trans:

     "[a = b] (mod m) ==> [b = c] (mod m) ==> [a = c] (mod m)"

   apply (unfold zcong_def dvd_def)

   apply auto

   apply (rule_tac x = "k + ka" in exI)

   apply (simp add: zadd_ac zadd_zmult_distrib2)

   done

 lemma zcong_zmult:

     "[a = b] (mod m) ==> [c = d] (mod m) ==> [a * c = b * d] (mod m)"

   apply (rule_tac b = "b * c" in zcong_trans)

    apply (unfold zcong_def)

    apply (rule_tac s = "c * (a - b)" in subst)

     apply (rule_tac [3] s = "b * (c - d)" in subst)

      prefer 4

      apply (blast intro: zdvd_zmult)

     prefer 2

     apply (blast intro: zdvd_zmult)

    apply (simp_all add: zdiff_zmult_distrib2 zmult_commute)

   done

 lemma zcong_scalar: "[a = b] (mod m) ==> [a * k = b * k] (mod m)"

   apply (rule zcong_zmult)

   apply simp_all

   done

 lemma zcong_scalar2: "[a = b] (mod m) ==> [k * a = k * b] (mod m)"

   apply (rule zcong_zmult)

   apply simp_all

   done

 lemma zcong_zmult_self: "[a * m = b * m] (mod m)"

   apply (unfold zcong_def)

   apply (rule zdvd_zdiff)

    apply simp_all

   done

 lemma zcong_square:

   "p \<in> zprime ==> 0 < a ==> [a * a = 1] (mod p)

     ==> [a = 1] (mod p) \<or> [a = p - 1] (mod p)"

   apply (unfold zcong_def)

   apply (rule zprime_zdvd_zmult)

     apply (rule_tac [3] s = "a * a - 1 + p * (1 - a)" in subst)

      prefer 4

      apply (simp add: zdvd_reduce)

     apply (simp_all add: zdiff_zmult_distrib zmult_commute zdiff_zmult_distrib2)

   done

 lemma zcong_cancel:

   "0 \<le> m ==>

     zgcd (k, m) = 1 ==> [a * k = b * k] (mod m) = [a = b] (mod m)"

   apply safe

    prefer 2

    apply (blast intro: zcong_scalar)

   apply (case_tac "b < a")

    prefer 2

    apply (subst zcong_sym)

    apply (unfold zcong_def)

    apply (rule_tac [!] zrelprime_zdvd_zmult)

      apply (simp_all add: zdiff_zmult_distrib)

   apply (subgoal_tac "m dvd (-(a * k - b * k))")

    apply (simp add: zminus_zdiff_eq)

   apply (subst zdvd_zminus_iff)

   apply assumption

   done

 lemma zcong_cancel2:

   "0 \<le> m ==>

     zgcd (k, m) = 1 ==> [k * a = k * b] (mod m) = [a = b] (mod m)"

   apply (simp add: zmult_commute zcong_cancel)

   done

 lemma zcong_zgcd_zmult_zmod:

   "[a = b] (mod m) ==> [a = b] (mod n) ==> zgcd (m, n) = 1

     ==> [a = b] (mod m * n)"

   apply (unfold zcong_def dvd_def)

   apply auto

   apply (subgoal_tac "m dvd n * ka")

    apply (subgoal_tac "m dvd ka")

     apply (case_tac [2] "0 \<le> ka")

      prefer 3

      apply (subst zdvd_zminus_iff [symmetric])

      apply (rule_tac n = n in zrelprime_zdvd_zmult)

       apply (simp add: zgcd_commute)

      apply (simp add: zmult_commute zdvd_zminus_iff)

     prefer 2

     apply (rule_tac n = n in zrelprime_zdvd_zmult)

      apply (simp add: zgcd_commute)

     apply (simp add: zmult_commute)

    apply (auto simp add: dvd_def)

   apply (blast intro: sym)

   done

 lemma zcong_zless_imp_eq:

   "0 \<le> a ==>

     a < m ==> 0 \<le> b ==> b < m ==> [a = b] (mod m) ==> a = b"

   apply (unfold zcong_def dvd_def)

   apply auto

   apply (drule_tac f = "\<lambda>z. z mod m" in arg_cong)

   apply (cut_tac z = a and w = b in zless_linear)

   apply auto

    apply (subgoal_tac [2] "(a - b) mod m = a - b")

     apply (rule_tac [3] mod_pos_pos_trivial)

      apply auto

   apply (subgoal_tac "(m + (a - b)) mod m = m + (a - b)")

    apply (rule_tac [2] mod_pos_pos_trivial)

     apply auto

   done

 lemma zcong_square_zless:

   "p \<in> zprime ==> 0 < a ==> a < p ==>

     [a * a = 1] (mod p) ==> a = 1 \<or> a = p - 1"

   apply (cut_tac p = p and a = a in zcong_square)

      apply (simp add: zprime_def)

     apply (auto intro: zcong_zless_imp_eq)

   done

 lemma zcong_not:

     "0 < a ==> a < m ==> 0 < b ==> b < a ==> \<not> [a = b] (mod m)"

   apply (unfold zcong_def)

   apply (rule zdvd_not_zless)

    apply auto

   done

 lemma zcong_zless_0:

     "0 \<le> a ==> a < m ==> [a = 0] (mod m) ==> a = 0"

   apply (unfold zcong_def dvd_def)

   apply auto

   apply (subgoal_tac "0 < m")

    apply (rotate_tac -1)

    apply (simp add: int_0_le_mult_iff)

    apply (subgoal_tac "m * k < m * 1")

     apply (drule zmult_zless_cancel1 [THEN iffD1])

     apply (auto simp add: linorder_neq_iff)

   done

 lemma zcong_zless_unique:

     "0 < m ==> (\<exists>!b. 0 \<le> b \<and> b < m \<and> [a = b] (mod m))"

   apply auto

    apply (subgoal_tac [2] "[b = y] (mod m)")

     apply (case_tac [2] "b = 0")

      apply (case_tac [3] "y = 0")

       apply (auto intro: zcong_trans zcong_zless_0 zcong_zless_imp_eq order_less_le

         simp add: zcong_sym)

   apply (unfold zcong_def dvd_def)

   apply (rule_tac x = "a mod m" in exI)

   apply (auto simp add: pos_mod_sign pos_mod_bound)

   apply (rule_tac x = "-(a div m)" in exI)

   apply (simp add:zdiff_eq_eq eq_zdiff_eq zadd_commute)

   done

 lemma zcong_iff_lin: "([a = b] (mod m)) = (\<exists>k. b = a + m * k)"

   apply (unfold zcong_def dvd_def)

   apply auto

    apply (rule_tac [!] x = "-k" in exI)

    apply auto

   done

 lemma zgcd_zcong_zgcd:

   "0 < m ==>

     zgcd (a, m) = 1 ==> [a = b] (mod m) ==> zgcd (b, m) = 1"

   apply (auto simp add: zcong_iff_lin)

   done

 lemma zcong_zmod_aux: "a - b = (m::int) * (a div m - b div m) + (a mod m - b mod m)"

 by(simp add: zdiff_zmult_distrib2 zadd_zdiff_eq eq_zdiff_eq zadd_ac)

 lemma zcong_zmod: "[a = b] (mod m) = [a mod m = b mod m] (mod m)"

   apply (unfold zcong_def)

   apply (rule_tac t = "a - b" in ssubst)

   apply (rule_tac "m" = "m" in zcong_zmod_aux)

   apply (rule trans)

    apply (rule_tac [2] k = m and m = "a div m - b div m" in zdvd_reduce)

   apply (simp add: zadd_commute)

   done

 lemma zcong_zmod_eq: "0 < m ==> [a = b] (mod m) = (a mod m = b mod m)"

   apply auto

    apply (rule_tac m = m in zcong_zless_imp_eq)

        prefer 5

        apply (subst zcong_zmod [symmetric])

        apply (simp_all add: pos_mod_bound pos_mod_sign)

   apply (unfold zcong_def dvd_def)

   apply (rule_tac x = "a div m - b div m" in exI)

   apply (rule_tac m1 = m in zcong_zmod_aux [THEN trans])

   apply auto

   done

 lemma zcong_zminus [iff]: "[a = b] (mod -m) = [a = b] (mod m)"

   apply (auto simp add: zcong_def)

   done

 lemma zcong_zero [iff]: "[a = b] (mod 0) = (a = b)"

   apply (auto simp add: zcong_def)

   done

 lemma "[a = b] (mod m) = (a mod m = b mod m)"

   apply (case_tac "m = 0", simp add: DIVISION_BY_ZERO)

   apply (simp add: linorder_neq_iff)

   apply (erule disjE)  

    prefer 2 apply (simp add: zcong_zmod_eq)

   txt{*Remainding case: @{term "m<0"}*}

   apply (rule_tac t = m in zminus_zminus [THEN subst])

   apply (subst zcong_zminus)

   apply (subst zcong_zmod_eq)

    apply arith

   apply (frule neg_mod_bound [of _ a], frule neg_mod_bound [of _ b]) 

   apply (simp add: zmod_zminus2_eq_if)

   done

 subsection {* Modulo *}

 lemma zmod_zdvd_zmod:

     "0 < (m::int) ==> m dvd b ==> (a mod b mod m) = (a mod m)"

   apply (unfold dvd_def)

   apply auto

   apply (subst zcong_zmod_eq [symmetric])

    prefer 2

    apply (subst zcong_iff_lin)

    apply (rule_tac x = "k * (a div (m * k))" in exI)

    apply(simp add:zmult_assoc [symmetric])

   apply assumption

   done

 subsection {* Extended GCD *}

 declare xzgcda.simps [simp del]

 lemma xzgcd_correct_aux1:

   "zgcd (r', r) = k --> 0 < r -->

     (\<exists>sn tn. xzgcda (m, n, r', r, s', s, t', t) = (k, sn, tn))"

   apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and

     z = s and aa = t' and ab = t in xzgcda.induct)

   apply (subst zgcd_eq)

   apply (subst xzgcda.simps)

   apply auto

   apply (case_tac "r' mod r = 0")

    prefer 2

    apply (frule_tac a = "r'" in pos_mod_sign)

    apply auto

   apply (rule exI)

   apply (rule exI)

   apply (subst xzgcda.simps)

   apply auto

   apply (simp add: zabs_def)

   done

 lemma xzgcd_correct_aux2:

   "(\<exists>sn tn. xzgcda (m, n, r', r, s', s, t', t) = (k, sn, tn)) --> 0 < r -->

     zgcd (r', r) = k"

   apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and

     z = s and aa = t' and ab = t in xzgcda.induct)

   apply (subst zgcd_eq)

   apply (subst xzgcda.simps)

   apply (auto simp add: linorder_not_le)

   apply (case_tac "r' mod r = 0")

    prefer 2

    apply (frule_tac a = "r'" in pos_mod_sign)

    apply auto

   apply (erule_tac P = "xzgcda ?u = ?v" in rev_mp)

   apply (subst xzgcda.simps)

   apply auto

   apply (simp add: zabs_def)

   done

 lemma xzgcd_correct:

     "0 < n ==> (zgcd (m, n) = k) = (\<exists>s t. xzgcd m n = (k, s, t))"

   apply (unfold xzgcd_def)

   apply (rule iffI)

    apply (rule_tac [2] xzgcd_correct_aux2 [THEN mp, THEN mp])

     apply (rule xzgcd_correct_aux1 [THEN mp, THEN mp])

      apply auto

   done

 text {* \medskip @{term xzgcd} linear *}

 lemma xzgcda_linear_aux1:

   "(a - r * b) * m + (c - r * d) * (n::int) =

     (a * m + c * n) - r * (b * m + d * n)"

   apply (simp add: zdiff_zmult_distrib zadd_zmult_distrib2 zmult_assoc)

   done

 lemma xzgcda_linear_aux2:

   "r' = s' * m + t' * n ==> r = s * m + t * n

     ==> (r' mod r) = (s' - (r' div r) * s) * m + (t' - (r' div r) * t) * (n::int)"

   apply (rule trans)

    apply (rule_tac [2] xzgcda_linear_aux1 [symmetric])

   apply (simp add: eq_zdiff_eq zmult_commute)

   done

 lemma order_le_neq_implies_less: "(x::'a::order) \<le> y ==> x \<noteq> y ==> x < y"

   by (rule iffD2 [OF order_less_le conjI])

 lemma xzgcda_linear [rule_format]:

   "0 < r --> xzgcda (m, n, r', r, s', s, t', t) = (rn, sn, tn) -->

     r' = s' * m + t' * n -->  r = s * m + t * n --> rn = sn * m + tn * n"

   apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and

     z = s and aa = t' and ab = t in xzgcda.induct)

   apply (subst xzgcda.simps)

   apply (simp (no_asm))

   apply (rule impI)+

   apply (case_tac "r' mod r = 0")

    apply (simp add: xzgcda.simps)

    apply clarify

   apply (subgoal_tac "0 < r' mod r")

    apply (rule_tac [2] order_le_neq_implies_less)

    apply (rule_tac [2] pos_mod_sign)

     apply (cut_tac m = m and n = n and r' = r' and r = r and s' = s' and

       s = s and t' = t' and t = t in xzgcda_linear_aux2)

       apply auto

   done

 lemma xzgcd_linear:

     "0 < n ==> xzgcd m n = (r, s, t) ==> r = s * m + t * n"

   apply (unfold xzgcd_def)

   apply (erule xzgcda_linear)

     apply assumption

    apply auto

   done

 lemma zgcd_ex_linear:

     "0 < n ==> zgcd (m, n) = k ==> (\<exists>s t. k = s * m + t * n)"

   apply (simp add: xzgcd_correct)

   apply safe

   apply (rule exI)+

   apply (erule xzgcd_linear)

   apply auto

   done

 lemma zcong_lineq_ex:

     "0 < n ==> zgcd (a, n) = 1 ==> \<exists>x. [a * x = 1] (mod n)"

   apply (cut_tac m = a and n = n and k = "1" in zgcd_ex_linear)

     apply safe

   apply (rule_tac x = s in exI)

   apply (rule_tac b = "s * a + t * n" in zcong_trans)

    prefer 2

    apply simp

   apply (unfold zcong_def)

   apply (simp (no_asm) add: zmult_commute zdvd_zminus_iff)

   done

 lemma zcong_lineq_unique:

   "0 < n ==>

     zgcd (a, n) = 1 ==> \<exists>!x. 0 \<le> x \<and> x < n \<and> [a * x = b] (mod n)"

   apply auto

    apply (rule_tac [2] zcong_zless_imp_eq)

        apply (tactic {* stac (thm "zcong_cancel2" RS sym) 6 *})

          apply (rule_tac [8] zcong_trans)

           apply (simp_all (no_asm_simp))

    prefer 2

    apply (simp add: zcong_sym)

   apply (cut_tac a = a and n = n in zcong_lineq_ex)

     apply auto

   apply (rule_tac x = "x * b mod n" in exI)

   apply safe

     apply (simp_all (no_asm_simp) add: pos_mod_bound pos_mod_sign)

   apply (subst zcong_zmod)

   apply (subst zmod_zmult1_eq [symmetric])

   apply (subst zcong_zmod [symmetric])

   apply (subgoal_tac "[a * x * b = 1 * b] (mod n)")

    apply (rule_tac [2] zcong_zmult)

     apply (simp_all add: zmult_assoc)

   done

 end