(* Title: HOL/Old_Number_Theory/Legacy_GCD.thy

Author: Christophe Tabacznyj and Lawrence C Paulson

Copyright 1996 University of Cambridge

*)

header {* 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"

by (auto simp add: is_gcd_def)

text {* Commutativity *}

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

by (auto simp add: is_gcd_def)

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 (simp_all add: gcd_non_0)

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) "

by (simp add: is_gcd_def gcd_greatest)

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)

apply (simp add: is_gcd)

done

lemma gcd_assoc: "gcd (gcd k m) n = gcd k (gcd m n)"

apply (rule is_gcd_unique)

apply (rule is_gcd)

apply (simp add: is_gcd_def)

apply (blast intro: dvd_trans)

done

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

by (simp add: gcd_commute)

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")

apply (simp_all add: gcd_non_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)

apply (simp add: gcd_assoc)

apply (simp add: gcd_commute)

apply (simp_all add: mult_commute)

done

text {* \medskip Addition laws *}

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"

apply (subst add_commute)

apply (rule gcd_add2)

done

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

by (induct k) (simp_all add: add_assoc)

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

then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'" by simp_all

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)"

using bezout_add[of a b]

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

from bezout_add[of a b]

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 add_commute mult_ac)

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"

by (algebra add: diff_mult_distrib)

hence "a * (x * k) - b * (y*k) = ?g ∨ b * (x * k) - a * (y*k) = ?g"

by (simp add: k mult_assoc)

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"

from bezout_add_strong[OF a, of 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"

by(simp add: gcd_mult_distrib2 mult_commute)

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"

by (simp add: k[symmetric] mult_assoc)

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

from dvd_add[OF dv] H

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"

by (simp_all add: gcd_commute)

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"

by (simp add: gcd_commute)}

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)"

by (simp add: mult_ac gcd_mult_distrib2)

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"

by (simp add: mult_ac gcd_mult_distrib2)

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"

by (simp add: mult_ac)

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"

by (simp add: mult_ac)

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: mult_ac)

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: mult_ac)

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"

by (simp add: gcd_commute)

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 (abs i)) (nat (abs j)))"

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

by (simp add: zgcd_def int_dvd_iff)

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

by (simp add: zgcd_def int_dvd_iff)

lemma zgcd_pos: "zgcd i j ≥ 0"

by (simp add: zgcd_def)

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

by (simp add: zgcd_def gcd_zero)

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 (abs x)) = abs 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

then have "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'" by simp_all

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 = abs m"

by (simp add: zgcd_def abs_if)

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

by (simp add: zgcd_def abs_if)

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: nat_diff_distrib gcd_diff2 nat_le_eq_zle)

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"

by (simp add: zgcd_def abs_if)

lemma zgcd_0_1_iff [simp, algebra]: "zgcd 0 m = 1 <-> ¦m¦ = 1"

by (simp add: zgcd_def abs_if)

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"

by (simp add: zgcd_def)

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

by (simp add: zgcd_def gcd_assoc)

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 ≤ k ==> k * zgcd m n = zgcd (k * m) (k * n)"

by (simp del: minus_mult_right [symmetric]

add: minus_mult_right nat_mult_distrib zgcd_def abs_if

mult_less_0_iff gcd_mult_distrib2 [symmetric] of_nat_mult)

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

by (simp add: abs_if zgcd_zmult_distrib2)

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(abs i)) (nat(abs j)))"

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

by(simp add:zlcm_def dvd_int_iff)

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

by(simp add:zlcm_def dvd_int_iff)

lemma dvd_imp_dvd_zlcm1:

assumes "k dvd i" shows "k dvd (zlcm i j)"

proof -

have "nat(abs k) dvd nat(abs i)" using `k dvd i`

by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric])

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(abs k) dvd nat(abs j)" using `k dvd j`

by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric])

thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)

qed

lemma zdvd_self_abs1: "(d::int) dvd (abs d)"

by (case_tac "d <0", simp_all)

lemma zdvd_self_abs2: "(abs (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 (abs a)"

let ?nb = "nat (abs 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