diff -r 1860f016886d -r 15a4b2cf8c34 src/HOL/GCD.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/GCD.thy Wed Dec 03 15:58:44 2008 +0100 @@ -0,0 +1,783 @@ +(* Title: HOL/GCD.thy + ID: \$Id\$ + Author: Christophe Tabacznyj and Lawrence C Paulson + Copyright 1996 University of Cambridge +*) + +header {* The Greatest Common Divisor *} + +theory GCD +imports Plain Presburger +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 *} + [code del]: "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_anti_sym) + +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) = 1" + by simp + +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 = 1" + by (simp add: gcd_commute) + +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: mod_geq gcd_non_0 mod_mult_distrib2) + 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_anti_sym) + 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) + apply (blast intro: dvd_mult) + 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_anti_sym[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 n\"a+b" arbitrary: a b rule: nat_less_induct) + fix n a b + assume H: "\m < n. \a b. m = a + b \ P a b" "n = a + b" + 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 < n \ a = 0" using H(2) by arith + moreover + {assume "a =0" with z c have "P a b" by blast } + moreover + {assume ab: "a + b - a < n" + have th0: "a + b - a = a + (b - a)" using lt by arith + from add[rule_format, OF H(1)[rule_format, OF ab th0]] + 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 < n \ b = 0" using H(2) by arith + moreover + {assume "b =0" with z c have "P a b" by blast } + moreover + {assume ab: "b + a - b < n" + have th0: "b + a - b = b + (a - b)" using lt by arith + from add[rule_format, OF H(1)[rule_format, OF ab th0]] + 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 add: dvd_add) +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 add: dvd_add) +apply clarsimp +apply (rule_tac x="d" in exI) +apply (case_tac "a * x = b * y + d", simp_all add: dvd_add) +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 prems 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 prems 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[OF dv(1,2)] dvd_diff[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: gcd_mult 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" +apply (simp_all add: gcd_add1) +by (simp add: gcd_commute gcd_add1) + +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,simp, algebra]: "zgcd i j dvd i" + by (simp add: zgcd_def int_dvd_iff) + +lemma zgcd_zdvd2 [iff,simp, 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) arith + +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 zdvd_abs1 [symmetric]) + apply (subst zdvd_abs2 [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 zdvd_abs1 zdvd_abs2) + 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 add: zdvd_abs1) +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 add: zgcd_zdvd1 zgcd_zdvd2) + have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by (simp_all add: zgcd_zdvd1 zgcd_zdvd2) + 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: zdvd_mult_div_cancel [OF dvdg(1)] + zdvd_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 (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, 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 gcd_1_left) + +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] zmult_int [symmetric]) + +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] zdvd_abs1) + 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] zdvd_abs1) + 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