(* 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_GCDimports Mainbegintext {* See \cite{davenport92}. \bigskip*}subsection {* Specification of GCD on nats *}definition is_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where -- {* @{term gcd} as a relation *} "is_gcd m n p \<longleftrightarrow> p dvd m \<and> p dvd n \<and> (\<forall>d. d dvd m \<longrightarrow> d dvd n \<longrightarrow> d dvd p)"text {* Uniqueness *}lemma is_gcd_unique: "is_gcd a b m \<Longrightarrow> is_gcd a b n \<Longrightarrow> 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 \<Longrightarrow> k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> 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 "\<And>m. P m 0" and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> 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_allqedlemma gcd_0 [simp, algebra]: "gcd m 0 = m" by simplemma gcd_0_left [simp,algebra]: "gcd 0 m = m" by simplemma gcd_non_0: "n > 0 \<Longrightarrow> gcd m n = gcd n (m mod n)" by simplemma gcd_1 [simp, algebra]: "gcd m (Suc 0) = Suc 0" by simplemma 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) donetext {* \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 \<Longrightarrow> k dvd n \<Longrightarrow> 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 \<longleftrightarrow> k dvd m \<and> k dvd n" by (blast intro!: gcd_greatest intro: dvd_trans)lemma gcd_zero[algebra]: "gcd m n = 0 \<longleftrightarrow> m = 0 \<and> 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) donelemma 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) donelemma 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) donelemma gcd_mult [simp, algebra]: "gcd k (k * n) = k" apply (rule gcd_mult_distrib2 [of k 1 n, simplified, symmetric]) donelemma gcd_self [simp, algebra]: "gcd k k = k" apply (rule gcd_mult [of k 1, simplified]) donelemma 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 donelemma 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) donetext {* \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 .qedlemma gcd_add2' [simp, algebra]: "gcd m (n + m) = gcd m n" apply (subst add_commute) apply (rule gcd_add2) donelemma 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 autotext {* \medskip Division by gcd yields rrelatively primes.*}lemma div_gcd_relprime: assumes nz: "a \<noteq> 0 \<or> b \<noteq> 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 \<noteq> 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 simpqedlemma gcd_unique: "d dvd a\<and>d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"proof(auto) assume H: "d dvd a" "d dvd b" "\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> 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 qedlemma gcd_eq: assumes H: "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd u \<and> d dvd v" shows "gcd x y = gcd u v"proof- from H have "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd gcd u v" by simp with gcd_unique[of "gcd u v" x y] show ?thesis by autoqedlemma ind_euclid: assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0" and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)" shows "P a b"proof(induct "a + b" arbitrary: a b rule: less_induct) case less have "a = b \<or> a < b \<or> 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 \<or> 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 \<or> 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 blastqedlemma bezout_lemma: assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)" shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and> (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"using exapply clarsimpapply (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 algebraapply (rule_tac x="x" in exI)apply (rule_tac x="x + y" in exI)apply algebradonelemma bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"apply(induct a b rule: ind_euclid)apply blastapply clarifyapply (rule_tac x="a" in exI, simp)apply clarsimpapply (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 algebraapply (rule_tac x="x" in exI)apply (rule_tac x="x+y" in exI)apply algebradonelemma bezout: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x - b * y = d \<or> b * x - a * y = d)"using bezout_add[of a b]apply clarsimpapply (rule_tac x="d" in exI, simp)apply (rule_tac x="x" in exI)apply (rule_tac x="y" in exI)apply autodonetext {* We can get a stronger version with a nonzeroness assumption. *}lemma divides_le: "m dvd n ==> m <= n \<or> n = (0::nat)" by (auto simp add: dvd_def)lemma bezout_add_strong: assumes nz: "a \<noteq> (0::nat)" shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"proof- from nz have ap: "a > 0" by simp from bezout_add[of a b] have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or> (\<exists>d x y. d dvd a \<and> d dvd b \<and> 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 \<noteq> 0" hence bp: "b > 0" by simp from divides_le[OF H(2)] b have "d < b \<or> 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 \<noteq> 0" hence xp: "x > 0" by simp from db have "d \<le> b - 1" by simp hence "d*b \<le> b*(b - 1)" by simp with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"] have dble: "d*b \<le> 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 blastqedlemma bezout_gcd: "\<exists>x y. a * x - b * y = gcd a b \<or> 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 \<or> 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 \<or> (b * x - a * y)*k = d*k" by blast hence "a * x * k - b * y*k = d*k \<or> b * x * k - a * y*k = d*k" by (algebra add: diff_mult_distrib) hence "a * (x * k) - b * (y*k) = ?g \<or> b * (x * k) - a * (y*k) = ?g" by (simp add: k mult_assoc) thus ?thesis by blastqedlemma bezout_gcd_strong: assumes a: "a \<noteq> 0" shows "\<exists>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 blastqedlemma gcd_mult_distrib: "gcd(a * c) (b * c) = c * gcd a b"by(simp add: gcd_mult_distrib2 mult_commute)lemma gcd_bezout: "(\<exists>x y. a * x - b * y = d \<or> b * x - a * y = d) \<longleftrightarrow> gcd a b dvd d" (is "?lhs \<longleftrightarrow> ?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 \<or> b * x - a * y = ?g" by blast hence "(a * x - b * y)*k = ?g*k \<or> (b * x - a * y)*k = ?g*k" by auto hence "a * x*k - b * y*k = ?g*k \<or> b * x * k - a * y*k = ?g*k" by (simp only: diff_mult_distrib) hence "a * (x*k) - b * (y*k) = d \<or> 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 \<or> 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 blastqedlemma 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 autoqedlemma 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 \<le> (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 \<le> a" from th[OF ab] show "gcd (a - b) b = gcd a b" by blastnext assume ab: "a \<le> b" from th[OF ab] show "gcd a (b - a) = gcd a b" by (simp add: gcd_commute)}qedsubsection {* LCM defined by GCD *}definition lcm :: "nat \<Rightarrow> nat \<Rightarrow> 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 simplemma lcm_1 [simp]: "lcm m 1 = m" unfolding lcm_def by simplemma lcm_0_left [simp]: "lcm 0 n = 0" unfolding lcm_def by simplemma lcm_1_left [simp]: "lcm 1 m = m" unfolding lcm_def by simplemma dvd_pos: fixes n m :: nat assumes "n > 0" and "m dvd n" shows "m > 0"using assms by (cases m) autolemma 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 autonext 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 "\<dots> = k * gcd (m * p * q) (n * q * p)" by (simp add: k_m [symmetric] k_n [symmetric]) also have "\<dots> = 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 autoqedlemma lcm_dvd1 [iff]: "m dvd lcm m n"proof (cases m) case 0 then show ?thesis by simpnext 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 "\<dots> = m * k" using mpos npos gcd_zero by simp finally show ?thesis by (simp add: lcm_def) qedqedlemma lcm_dvd2 [iff]: "n dvd lcm m n"proof (cases n) case 0 then show ?thesis by simpnext 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 "\<dots> = n * k" using mpos npos gcd_zero by simp finally show ?thesis by (simp add: lcm_def) qedqedlemma gcd_add1_eq: "gcd (m + k) k = gcd (m + k) m" by (simp add: gcd_commute)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, simp) donesubsection {* GCD and LCM on integers *}definition zgcd :: "int \<Rightarrow> int \<Rightarrow> 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 \<ge> 0"by (simp add: zgcd_def)lemma zgcd0 [simp,algebra]: "(zgcd i j = 0) = (i = 0 \<and> 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 simplemma 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 \<Longrightarrow> i dvd k * j \<Longrightarrow> i dvd k" unfolding zgcd_defproof - assume "int (gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>)) = 1" "i dvd k * j" then have g: "gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>) = 1" by simp from `i dvd k * j` obtain h where h: "k*j = i*h" unfolding dvd_def by blast have th: "nat \<bar>i\<bar> dvd nat \<bar>k\<bar> * nat \<bar>j\<bar>" unfolding dvd_def by (rule_tac x= "nat \<bar>h\<bar>" in exI, simp add: h nat_abs_mult_distrib [symmetric]) from relprime_dvd_mult [OF g th] obtain h' where h': "nat \<bar>k\<bar> = nat \<bar>i\<bar> * h'" unfolding dvd_def by blast from h' have "int (nat \<bar>k\<bar>) = int (nat \<bar>i\<bar> * h')" by simp then have "\<bar>k\<bar> = \<bar>i\<bar> * 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) doneqedlemma int_nat_abs: "int (nat (abs x)) = abs x" by arithlemma zgcd_greatest: assumes "k dvd m" and "k dvd n" shows "k dvd zgcd m n"proof - let ?k' = "nat \<bar>k\<bar>" let ?m' = "nat \<bar>m\<bar>" let ?n' = "nat \<bar>n\<bar>" 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 \<bar>k\<bar>) dvd zgcd m n" unfolding zgcd_def by (simp only: zdvd_int) then have "\<bar>k\<bar> dvd zgcd m n" by (simp only: int_nat_abs) then show "k dvd zgcd m n" by simpqedlemma div_zgcd_relprime: assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0" shows "zgcd (a div (zgcd a b)) (b div (zgcd a b)) = 1"proof - from nz have nz': "nat \<bar>a\<bar> \<noteq> 0 \<or> nat \<bar>b\<bar> \<noteq> 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 \<noteq> 0" using nz by simp then have gp: "?g \<noteq> 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 "\<bar>?g'\<bar> = 1" by simp with zgcd_pos show "?g' = 1" by simpqedlemma 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) donelemma 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) donelemma 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 \<longleftrightarrow> \<bar>m\<bar> = 1" by (simp add: zgcd_def abs_if)lemma zgcd_greatest_iff[algebra]: "k dvd zgcd m n = (k dvd m \<and> 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]) donelemmas 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)" 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 \<le> 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 \<le> 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 \<le> 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)qedlemma 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)qedlemma 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 \<noteq> 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 \<le> m" by simp with npos have t1:"gcd m n *n \<le> m*n" by simp have "gcd m n \<le> gcd m n*n" using npos by simp with t1 have "gcd m n \<le> m*n" by arith ultimately show "False" by simpqedlemma zlcm_pos: assumes anz: "a \<noteq> 0" and bnz: "b \<noteq> 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)qedlemma zgcd_code [code]: "zgcd k l = \<bar>if l = 0 then k else zgcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>" by (simp add: zgcd_def gcd.simps [of "nat \<bar>k\<bar>"] nat_mod_distrib)end