# Theory Legacy_GCD

theory Legacy_GCD
imports Main
```(*  Title:      HOL/Old_Number_Theory/Legacy_GCD.thy
Author:     Christophe Tabacznyj and Lawrence C Paulson
*)

section ‹The Greatest Common Divisor›

theory Legacy_GCD
imports Main
begin

text ‹
See @{cite davenport92}. \bigskip
›

subsection ‹Specification of GCD on nats›

definition
is_gcd :: "nat ⇒ nat ⇒ nat ⇒ bool" where -- ‹@{term gcd} as a relation›
"is_gcd m n p ⟷ p dvd m ∧ p dvd n ∧
(∀d. d dvd m ⟶ d dvd n ⟶ d dvd p)"

text ‹Uniqueness›

lemma is_gcd_unique: "is_gcd a b m ⟹ is_gcd a b n ⟹ m = n"
by (simp add: is_gcd_def) (blast intro: dvd_antisym)

text ‹Connection to divides relation›

lemma is_gcd_dvd: "is_gcd a b m ⟹ k dvd a ⟹ k dvd b ⟹ k dvd m"

text ‹Commutativity›

lemma is_gcd_commute: "is_gcd m n k = is_gcd n m k"

subsection ‹GCD on nat by Euclid's algorithm›

fun gcd :: "nat => nat => nat"
where "gcd m n = (if n = 0 then m else gcd n (m mod n))"

lemma gcd_induct [case_names "0" rec]:
fixes m n :: nat
assumes "⋀m. P m 0"
and "⋀m n. 0 < n ⟹ P n (m mod n) ⟹ P m n"
shows "P m n"
proof (induct m n rule: gcd.induct)
case (1 m n)
with assms show ?case by (cases "n = 0") simp_all
qed

lemma gcd_0 [simp, algebra]: "gcd m 0 = m"
by simp

lemma gcd_0_left [simp,algebra]: "gcd 0 m = m"
by simp

lemma gcd_non_0: "n > 0 ⟹ gcd m n = gcd n (m mod n)"
by simp

lemma gcd_1 [simp, algebra]: "gcd m (Suc 0) = Suc 0"
by simp

lemma nat_gcd_1_right [simp, algebra]: "gcd m 1 = 1"
unfolding One_nat_def by (rule gcd_1)

declare gcd.simps [simp del]

text ‹
\medskip @{term "gcd m n"} divides @{text m} and @{text n}.  The
conjunctions don't seem provable separately.
›

lemma gcd_dvd1 [iff, algebra]: "gcd m n dvd m"
and gcd_dvd2 [iff, algebra]: "gcd m n dvd n"
apply (induct m n rule: gcd_induct)
apply (blast dest: dvd_mod_imp_dvd)
done

text ‹
\medskip Maximality: for all @{term m}, @{term n}, @{term k}
naturals, if @{term k} divides @{term m} and @{term k} divides
@{term n} then @{term k} divides @{term "gcd m n"}.
›

lemma gcd_greatest: "k dvd m ⟹ k dvd n ⟹ k dvd gcd m n"
by (induct m n rule: gcd_induct) (simp_all add: gcd_non_0 dvd_mod)

text ‹
\medskip Function gcd yields the Greatest Common Divisor.
›

lemma is_gcd: "is_gcd m n (gcd m n) "

subsection ‹Derived laws for GCD›

lemma gcd_greatest_iff [iff, algebra]: "k dvd gcd m n ⟷ k dvd m ∧ k dvd n"
by (blast intro!: gcd_greatest intro: dvd_trans)

lemma gcd_zero[algebra]: "gcd m n = 0 ⟷ m = 0 ∧ n = 0"
by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff)

lemma gcd_commute: "gcd m n = gcd n m"
apply (rule is_gcd_unique)
apply (rule is_gcd)
apply (subst is_gcd_commute)
done

lemma gcd_assoc: "gcd (gcd k m) n = gcd k (gcd m n)"
apply (rule is_gcd_unique)
apply (rule is_gcd)
apply (blast intro: dvd_trans)
done

lemma gcd_1_left [simp, algebra]: "gcd (Suc 0) m = Suc 0"

lemma nat_gcd_1_left [simp, algebra]: "gcd 1 m = 1"
unfolding One_nat_def by (rule gcd_1_left)

text ‹
\medskip Multiplication laws
›

lemma gcd_mult_distrib2: "k * gcd m n = gcd (k * m) (k * n)"
-- ‹@{cite ‹page 27› davenport92}›
apply (induct m n rule: gcd_induct)
apply simp
apply (case_tac "k = 0")
done

lemma gcd_mult [simp, algebra]: "gcd k (k * n) = k"
apply (rule gcd_mult_distrib2 [of k 1 n, simplified, symmetric])
done

lemma gcd_self [simp, algebra]: "gcd k k = k"
apply (rule gcd_mult [of k 1, simplified])
done

lemma relprime_dvd_mult: "gcd k n = 1 ==> k dvd m * n ==> k dvd m"
apply (insert gcd_mult_distrib2 [of m k n])
apply simp
apply (erule_tac t = m in ssubst)
apply simp
done

lemma relprime_dvd_mult_iff: "gcd k n = 1 ==> (k dvd m * n) = (k dvd m)"
by (auto intro: relprime_dvd_mult dvd_mult2)

lemma gcd_mult_cancel: "gcd k n = 1 ==> gcd (k * m) n = gcd m n"
apply (rule dvd_antisym)
apply (rule gcd_greatest)
apply (rule_tac n = k in relprime_dvd_mult)
done

lemma gcd_add1 [simp, algebra]: "gcd (m + n) n = gcd m n"
by (cases "n = 0") (auto simp add: gcd_non_0)

lemma gcd_add2 [simp, algebra]: "gcd m (m + n) = gcd m n"
proof -
have "gcd m (m + n) = gcd (m + n) m" by (rule gcd_commute)
also have "... = gcd (n + m) m" by (simp add: add.commute)
also have "... = gcd n m" by simp
also have  "... = gcd m n" by (rule gcd_commute)
finally show ?thesis .
qed

lemma gcd_add2' [simp, algebra]: "gcd m (n + m) = gcd m n"
done

lemma gcd_add_mult[algebra]: "gcd m (k * m + n) = gcd m n"

lemma gcd_dvd_prod: "gcd m n dvd m * n"
using mult_dvd_mono [of 1] by auto

text ‹
\medskip Division by gcd yields rrelatively primes.
›

lemma div_gcd_relprime:
assumes nz: "a ≠ 0 ∨ b ≠ 0"
shows "gcd (a div gcd a b) (b div gcd a b) = 1"
proof -
let ?g = "gcd a b"
let ?a' = "a div ?g"
let ?b' = "b div ?g"
let ?g' = "gcd ?a' ?b'"
have dvdg: "?g dvd a" "?g dvd b" by simp_all
have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
from dvdg dvdg' obtain ka kb ka' kb' where
kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
unfolding dvd_def by blast
from this(3-4) [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
by (simp_all only: ac_simps mult.left_commute [of _ "gcd a b"])
then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
have "?g ≠ 0" using nz by (simp add: gcd_zero)
then have gp: "?g > 0" by simp
from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
qed

lemma gcd_unique: "d dvd a∧d dvd b ∧ (∀e. e dvd a ∧ e dvd b ⟶ e dvd d) ⟷ d = gcd a b"
proof(auto)
assume H: "d dvd a" "d dvd b" "∀e. e dvd a ∧ e dvd b ⟶ e dvd d"
from H(3)[rule_format] gcd_dvd1[of a b] gcd_dvd2[of a b]
have th: "gcd a b dvd d" by blast
from dvd_antisym[OF th gcd_greatest[OF H(1,2)]]  show "d = gcd a b" by blast
qed

lemma gcd_eq: assumes H: "∀d. d dvd x ∧ d dvd y ⟷ d dvd u ∧ d dvd v"
shows "gcd x y = gcd u v"
proof-
from H have "∀d. d dvd x ∧ d dvd y ⟷ d dvd gcd u v" by simp
with gcd_unique[of "gcd u v" x y]  show ?thesis by auto
qed

lemma ind_euclid:
assumes c: " ∀a b. P (a::nat) b ⟷ P b a" and z: "∀a. P a 0"
and add: "∀a b. P a b ⟶ P a (a + b)"
shows "P a b"
proof(induct "a + b" arbitrary: a b rule: less_induct)
case less
have "a = b ∨ a < b ∨ b < a" by arith
moreover {assume eq: "a= b"
from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
by simp}
moreover
{assume lt: "a < b"
hence "a + b - a < a + b ∨ a = 0" by arith
moreover
{assume "a =0" with z c have "P a b" by blast }
moreover
{assume "a + b - a < a + b"
also have th0: "a + b - a = a + (b - a)" using lt by arith
finally have "a + (b - a) < a + b" .
then have "P a (a + (b - a))" by (rule add[rule_format, OF less])
then have "P a b" by (simp add: th0[symmetric])}
ultimately have "P a b" by blast}
moreover
{assume lt: "a > b"
hence "b + a - b < a + b ∨ b = 0" by arith
moreover
{assume "b =0" with z c have "P a b" by blast }
moreover
{assume "b + a - b < a + b"
also have th0: "b + a - b = b + (a - b)" using lt by arith
finally have "b + (a - b) < a + b" .
then have "P b (b + (a - b))" by (rule add[rule_format, OF less])
then have "P b a" by (simp add: th0[symmetric])
hence "P a b" using c by blast }
ultimately have "P a b" by blast}
ultimately  show "P a b" by blast
qed

lemma bezout_lemma:
assumes ex: "∃(d::nat) x y. d dvd a ∧ d dvd b ∧ (a * x = b * y + d ∨ b * x = a * y + d)"
shows "∃d x y. d dvd a ∧ d dvd a + b ∧ (a * x = (a + b) * y + d ∨ (a + b) * x = a * y + d)"
using ex
apply clarsimp
apply (rule_tac x="d" in exI, simp)
apply (case_tac "a * x = b * y + d" , simp_all)
apply (rule_tac x="x + y" in exI)
apply (rule_tac x="y" in exI)
apply algebra
apply (rule_tac x="x" in exI)
apply (rule_tac x="x + y" in exI)
apply algebra
done

lemma bezout_add: "∃(d::nat) x y. d dvd a ∧ d dvd b ∧ (a * x = b * y + d ∨ b * x = a * y + d)"
apply(induct a b rule: ind_euclid)
apply blast
apply clarify
apply (rule_tac x="a" in exI, simp)
apply clarsimp
apply (rule_tac x="d" in exI)
apply (case_tac "a * x = b * y + d", simp_all)
apply (rule_tac x="x+y" in exI)
apply (rule_tac x="y" in exI)
apply algebra
apply (rule_tac x="x" in exI)
apply (rule_tac x="x+y" in exI)
apply algebra
done

lemma bezout: "∃(d::nat) x y. d dvd a ∧ d dvd b ∧ (a * x - b * y = d ∨ b * x - a * y = d)"
apply clarsimp
apply (rule_tac x="d" in exI, simp)
apply (rule_tac x="x" in exI)
apply (rule_tac x="y" in exI)
apply auto
done

text ‹We can get a stronger version with a nonzeroness assumption.›
lemma divides_le: "m dvd n ==> m <= n ∨ n = (0::nat)" by (auto simp add: dvd_def)

lemma bezout_add_strong: assumes nz: "a ≠ (0::nat)"
shows "∃d x y. d dvd a ∧ d dvd b ∧ a * x = b * y + d"
proof-
from nz have ap: "a > 0" by simp
have "(∃d x y. d dvd a ∧ d dvd b ∧ a * x = b * y + d) ∨ (∃d x y. d dvd a ∧ d dvd b ∧ b * x = a * y + d)" by blast
moreover
{fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
from H have ?thesis by blast }
moreover
{fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
{assume b0: "b = 0" with H  have ?thesis by simp}
moreover
{assume b: "b ≠ 0" hence bp: "b > 0" by simp
from divides_le[OF H(2)] b have "d < b ∨ d = b" using le_less by blast
moreover
{assume db: "d=b"
from nz H db have ?thesis apply simp
apply (rule exI[where x = b], simp)
apply (rule exI[where x = b])
by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
moreover
{assume db: "d < b"
{assume "x=0" hence ?thesis using nz H by simp }
moreover
{assume x0: "x ≠ 0" hence xp: "x > 0" by simp

from db have "d ≤ b - 1" by simp
hence "d*b ≤ b*(b - 1)" by simp
with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
have dble: "d*b ≤ x*b*(b - 1)" using bp by simp
from H (3) have "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)" by algebra
hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
by (simp only: diff_add_assoc[OF dble, of d, symmetric])
hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
by (simp only: diff_mult_distrib2 ac_simps)
hence ?thesis using H(1,2)
apply -
apply (rule exI[where x=d], simp)
apply (rule exI[where x="(b - 1) * y"])
by (rule exI[where x="x*(b - 1) - d"], simp)}
ultimately have ?thesis by blast}
ultimately have ?thesis by blast}
ultimately have ?thesis by blast}
ultimately show ?thesis by blast
qed

lemma bezout_gcd: "∃x y. a * x - b * y = gcd a b ∨ b * x - a * y = gcd a b"
proof-
let ?g = "gcd a b"
from bezout[of a b] obtain d x y where d: "d dvd a" "d dvd b" "a * x - b * y = d ∨ b * x - a * y = d" by blast
from d(1,2) have "d dvd ?g" by simp
then obtain k where k: "?g = d*k" unfolding dvd_def by blast
from d(3) have "(a * x - b * y)*k = d*k ∨ (b * x - a * y)*k = d*k" by blast
hence "a * x * k - b * y*k = d*k ∨ b * x * k - a * y*k = d*k"
hence "a * (x * k) - b * (y*k) = ?g ∨ b * (x * k) - a * (y*k) = ?g"
thus ?thesis by blast
qed

lemma bezout_gcd_strong: assumes a: "a ≠ 0"
shows "∃x y. a * x = b * y + gcd a b"
proof-
let ?g = "gcd a b"
obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
from d(1,2) have "d dvd ?g" by simp
then obtain k where k: "?g = d*k" unfolding dvd_def by blast
from d(3) have "a * x * k = (b * y + d) *k " by algebra
hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
thus ?thesis by blast
qed

lemma gcd_mult_distrib: "gcd(a * c) (b * c) = c * gcd a b"

lemma gcd_bezout: "(∃x y. a * x - b * y = d ∨ b * x - a * y = d) ⟷ gcd a b dvd d"
(is "?lhs ⟷ ?rhs")
proof-
let ?g = "gcd a b"
{assume H: ?rhs then obtain k where k: "d = ?g*k" unfolding dvd_def by blast
from bezout_gcd[of a b] obtain x y where xy: "a * x - b * y = ?g ∨ b * x - a * y = ?g"
by blast
hence "(a * x - b * y)*k = ?g*k ∨ (b * x - a * y)*k = ?g*k" by auto
hence "a * x*k - b * y*k = ?g*k ∨ b * x * k - a * y*k = ?g*k"
by (simp only: diff_mult_distrib)
hence "a * (x*k) - b * (y*k) = d ∨ b * (x * k) - a * (y*k) = d"
hence ?lhs by blast}
moreover
{fix x y assume H: "a * x - b * y = d ∨ b * x - a * y = d"
have dv: "?g dvd a*x" "?g dvd b * y" "?g dvd b*x" "?g dvd a * y"
using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
from dvd_diff_nat[OF dv(1,2)] dvd_diff_nat[OF dv(3,4)] H
have ?rhs by auto}
ultimately show ?thesis by blast
qed

lemma gcd_bezout_sum: assumes H:"a * x + b * y = d" shows "gcd a b dvd d"
proof-
let ?g = "gcd a b"
have dv: "?g dvd a*x" "?g dvd b * y"
using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
show ?thesis by auto
qed

lemma gcd_mult': "gcd b (a * b) = b"
by (simp add: mult.commute[of a b])

lemma gcd_add: "gcd(a + b) b = gcd a b"
"gcd(b + a) b = gcd a b" "gcd a (a + b) = gcd a b" "gcd a (b + a) = gcd a b"

lemma gcd_sub: "b <= a ==> gcd(a - b) b = gcd a b" "a <= b ==> gcd a (b - a) = gcd a b"
proof-
{fix a b assume H: "b ≤ (a::nat)"
hence th: "a - b + b = a" by arith
from gcd_add(1)[of "a - b" b] th  have "gcd(a - b) b = gcd a b" by simp}
note th = this
{
assume ab: "b ≤ a"
from th[OF ab] show "gcd (a - b)  b = gcd a b" by blast
next
assume ab: "a ≤ b"
from th[OF ab] show "gcd a (b - a) = gcd a b"
qed

subsection ‹LCM defined by GCD›

definition
lcm :: "nat ⇒ nat ⇒ nat"
where
lcm_def: "lcm m n = m * n div gcd m n"

lemma prod_gcd_lcm:
"m * n = gcd m n * lcm m n"
unfolding lcm_def by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod])

lemma lcm_0 [simp]: "lcm m 0 = 0"
unfolding lcm_def by simp

lemma lcm_1 [simp]: "lcm m 1 = m"
unfolding lcm_def by simp

lemma lcm_0_left [simp]: "lcm 0 n = 0"
unfolding lcm_def by simp

lemma lcm_1_left [simp]: "lcm 1 m = m"
unfolding lcm_def by simp

lemma dvd_pos:
fixes n m :: nat
assumes "n > 0" and "m dvd n"
shows "m > 0"
using assms by (cases m) auto

lemma lcm_least:
assumes "m dvd k" and "n dvd k"
shows "lcm m n dvd k"
proof (cases k)
case 0 then show ?thesis by auto
next
case (Suc _) then have pos_k: "k > 0" by auto
from assms dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto
with gcd_zero [of m n] have pos_gcd: "gcd m n > 0" by simp
from assms obtain p where k_m: "k = m * p" using dvd_def by blast
from assms obtain q where k_n: "k = n * q" using dvd_def by blast
from pos_k k_m have pos_p: "p > 0" by auto
from pos_k k_n have pos_q: "q > 0" by auto
have "k * k * gcd q p = k * gcd (k * q) (k * p)"
also have "… = k * gcd (m * p * q) (n * q * p)"
by (simp add: k_m [symmetric] k_n [symmetric])
also have "… = k * p * q * gcd m n"
finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
by (simp only: k_m [symmetric] k_n [symmetric])
then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
by simp
with prod_gcd_lcm [of m n]
have "lcm m n * gcd q p * gcd m n = k * gcd m n"
with pos_gcd have "lcm m n * gcd q p = k" by simp
then show ?thesis using dvd_def by auto
qed

lemma lcm_dvd1 [iff]:
"m dvd lcm m n"
proof (cases m)
case 0 then show ?thesis by simp
next
case (Suc _)
then have mpos: "m > 0" by simp
show ?thesis
proof (cases n)
case 0 then show ?thesis by simp
next
case (Suc _)
then have npos: "n > 0" by simp
have "gcd m n dvd n" by simp
then obtain k where "n = gcd m n * k" using dvd_def by auto
then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n" by (simp add: ac_simps)
also have "… = m * k" using mpos npos gcd_zero by simp
finally show ?thesis by (simp add: lcm_def)
qed
qed

lemma lcm_dvd2 [iff]:
"n dvd lcm m n"
proof (cases n)
case 0 then show ?thesis by simp
next
case (Suc _)
then have npos: "n > 0" by simp
show ?thesis
proof (cases m)
case 0 then show ?thesis by simp
next
case (Suc _)
then have mpos: "m > 0" by simp
have "gcd m n dvd m" by simp
then obtain k where "m = gcd m n * k" using dvd_def by auto
then have "m * n div gcd m n = (gcd m n * k) * n div gcd m n" by (simp add: ac_simps)
also have "… = n * k" using mpos npos gcd_zero by simp
finally show ?thesis by (simp add: lcm_def)
qed
qed

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

lemma gcd_diff2: "m ≤ n ==> gcd n (n - m) = gcd n m"
apply (subgoal_tac "n = m + (n - m)")
apply (erule ssubst, rule gcd_add1_eq, simp)
done

subsection ‹GCD and LCM on integers›

definition
zgcd :: "int ⇒ int ⇒ int" where
"zgcd i j = int (gcd (nat ¦i¦) (nat ¦j¦))"

lemma zgcd_zdvd1 [iff, algebra]: "zgcd i j dvd i"

lemma zgcd_zdvd2 [iff, algebra]: "zgcd i j dvd j"

lemma zgcd_pos: "zgcd i j ≥ 0"

lemma zgcd0 [simp,algebra]: "(zgcd i j = 0) = (i = 0 ∧ j = 0)"

lemma zgcd_commute: "zgcd i j = zgcd j i"
unfolding zgcd_def by (simp add: gcd_commute)

lemma zgcd_zminus [simp, algebra]: "zgcd (- i) j = zgcd i j"
unfolding zgcd_def by simp

lemma zgcd_zminus2 [simp, algebra]: "zgcd i (- j) = zgcd i j"
unfolding zgcd_def by simp

(* should be solved by algebra*)
lemma zrelprime_dvd_mult: "zgcd i j = 1 ⟹ i dvd k * j ⟹ i dvd k"
unfolding zgcd_def
proof -
assume "int (gcd (nat ¦i¦) (nat ¦j¦)) = 1" "i dvd k * j"
then have g: "gcd (nat ¦i¦) (nat ¦j¦) = 1" by simp
from ‹i dvd k * j› obtain h where h: "k*j = i*h" unfolding dvd_def by blast
have th: "nat ¦i¦ dvd nat ¦k¦ * nat ¦j¦"
unfolding dvd_def
by (rule_tac x= "nat ¦h¦" in exI, simp add: h nat_abs_mult_distrib [symmetric])
from relprime_dvd_mult [OF g th] obtain h' where h': "nat ¦k¦ = nat ¦i¦ * h'"
unfolding dvd_def by blast
from h' have "int (nat ¦k¦) = int (nat ¦i¦ * h')" by simp
then have "¦k¦ = ¦i¦ * int h'" by (simp add: int_mult)
then show ?thesis
apply (subst abs_dvd_iff [symmetric])
apply (subst dvd_abs_iff [symmetric])
apply (unfold dvd_def)
apply (rule_tac x = "int h'" in exI, simp)
done
qed

lemma int_nat_abs: "int (nat ¦x¦) = ¦x¦" by arith

lemma zgcd_greatest:
assumes "k dvd m" and "k dvd n"
shows "k dvd zgcd m n"
proof -
let ?k' = "nat ¦k¦"
let ?m' = "nat ¦m¦"
let ?n' = "nat ¦n¦"
from ‹k dvd m› and ‹k dvd n› have dvd': "?k' dvd ?m'" "?k' dvd ?n'"
unfolding zdvd_int by (simp_all only: int_nat_abs abs_dvd_iff dvd_abs_iff)
from gcd_greatest [OF dvd'] have "int (nat ¦k¦) dvd zgcd m n"
unfolding zgcd_def by (simp only: zdvd_int)
then have "¦k¦ dvd zgcd m n" by (simp only: int_nat_abs)
then show "k dvd zgcd m n" by simp
qed

lemma div_zgcd_relprime:
assumes nz: "a ≠ 0 ∨ b ≠ 0"
shows "zgcd (a div (zgcd a b)) (b div (zgcd a b)) = 1"
proof -
from nz have nz': "nat ¦a¦ ≠ 0 ∨ nat ¦b¦ ≠ 0" by arith
let ?g = "zgcd a b"
let ?a' = "a div ?g"
let ?b' = "b div ?g"
let ?g' = "zgcd ?a' ?b'"
have dvdg: "?g dvd a" "?g dvd b" by simp_all
have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
from dvdg dvdg' obtain ka kb ka' kb' where
kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'"
unfolding dvd_def by blast
from this(3-4) [symmetric] have "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'"
by (simp_all only: ac_simps mult.left_commute [of _ "zgcd a b"])
then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
have "?g ≠ 0" using nz by simp
then have gp: "?g ≠ 0" using zgcd_pos[where i="a" and j="b"] by arith
from zgcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
with zdvd_mult_cancel1 [OF gp] have "¦?g'¦ = 1" by simp
with zgcd_pos show "?g' = 1" by simp
qed

lemma zgcd_0 [simp, algebra]: "zgcd m 0 = ¦m¦"

lemma zgcd_0_left [simp, algebra]: "zgcd 0 m = ¦m¦"

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 del: pos_mod_sign add: zgcd_def abs_if 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 del: pos_mod_bound add: algebra_simps nat_diff_distrib gcd_diff2 nat_le_eq_zle)
apply (metis dual_order.strict_implies_order gcd.simps gcd_0_left gcd_diff2 mod_by_0 nat_mono)
done

lemma zgcd_eq: "zgcd m n = zgcd n (m mod n)"
apply (cases "n = 0", simp)
apply (auto simp add: linorder_neq_iff zgcd_non_0)
apply (cut_tac m = "-m" and n = "-n" in zgcd_non_0, auto)
done

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

lemma zgcd_0_1_iff [simp, algebra]: "zgcd 0 m = 1 ⟷ ¦m¦ = 1"

lemma zgcd_greatest_iff[algebra]: "k dvd zgcd m n = (k dvd m ∧ k dvd n)"
by (simp add: zgcd_def abs_if int_dvd_iff dvd_int_iff nat_dvd_iff)

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

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

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

lemma zgcd_zmult_distrib2: "0 ≤ k ==> k * zgcd m n = zgcd (k * m) (k * n)"
by (simp del: minus_mult_right [symmetric]
mult_less_0_iff gcd_mult_distrib2 [symmetric] of_nat_mult)

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

lemma zgcd_self [simp]: "0 ≤ m ==> zgcd m m = m"
by (cut_tac k = m and m = 1 and n = 1 in zgcd_zmult_distrib2, simp_all)

lemma zgcd_zmult_eq_self [simp]: "0 ≤ k ==> zgcd k (k * n) = k"
by (cut_tac k = k and m = 1 and n = n in zgcd_zmult_distrib2, simp_all)

lemma zgcd_zmult_eq_self2 [simp]: "0 ≤ k ==> zgcd (k * n) k = k"
by (cut_tac k = k and m = n and n = 1 in zgcd_zmult_distrib2, simp_all)

definition "zlcm i j = int (lcm (nat ¦i¦) (nat ¦j¦))"

lemma dvd_zlcm_self1[simp, algebra]: "i dvd zlcm i j"

lemma dvd_zlcm_self2[simp, algebra]: "j dvd zlcm i j"

lemma dvd_imp_dvd_zlcm1:
assumes "k dvd i" shows "k dvd (zlcm i j)"
proof -
have "nat ¦k¦ dvd nat ¦i¦" using ‹k dvd i›
thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
qed

lemma dvd_imp_dvd_zlcm2:
assumes "k dvd j" shows "k dvd (zlcm i j)"
proof -
have "nat ¦k¦ dvd nat ¦j¦" using ‹k dvd j›
thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
qed

lemma zdvd_self_abs1: "(d::int) dvd ¦d¦"
by (case_tac "d <0", simp_all)

lemma zdvd_self_abs2: "¦d::int¦ dvd d"
by (case_tac "d<0", simp_all)

(* lcm a b is positive for positive a and b *)

lemma lcm_pos:
assumes mpos: "m > 0"
and npos: "n>0"
shows "lcm m n > 0"
proof (rule ccontr, simp add: lcm_def gcd_zero)
assume h:"m*n div gcd m n = 0"
from mpos npos have "gcd m n ≠ 0" using gcd_zero by simp
hence gcdp: "gcd m n > 0" by simp
with h
have "m*n < gcd m n"
by (cases "m * n < gcd m n") (auto simp add: div_if[OF gcdp, where m="m*n"])
moreover
have "gcd m n dvd m" by simp
with mpos dvd_imp_le have t1:"gcd m n ≤ m" by simp
with npos have t1:"gcd m n *n ≤ m*n" by simp
have "gcd m n ≤ gcd m n*n" using npos by simp
with t1 have "gcd m n ≤ m*n" by arith
ultimately show "False" by simp
qed

lemma zlcm_pos:
assumes anz: "a ≠ 0"
and bnz: "b ≠ 0"
shows "0 < zlcm a b"
proof-
let ?na = "nat ¦a¦"
let ?nb = "nat ¦b¦"
have nap: "?na >0" using anz by simp
have nbp: "?nb >0" using bnz by simp
have "0 < lcm ?na ?nb" by (rule lcm_pos[OF nap nbp])
thus ?thesis by (simp add: zlcm_def)
qed

lemma zgcd_code [code]:
"zgcd k l = ¦if l = 0 then k else zgcd l (¦k¦ mod ¦l¦)¦"
by (simp add: zgcd_def gcd.simps [of "nat ¦k¦"] nat_mod_distrib)

end
```