(* Title: HOL/Old_Number_Theory/Legacy_GCD.thy Author: Christophe Tabacznyj and Lawrence C Paulson Copyright 1996 University of Cambridge *) 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" 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 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)" 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 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" 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: ac_simps 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: ac_simps 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: ac_simps) 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: ac_simps) 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" 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 ¦i¦) (nat ¦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 ¦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¦" by (simp add: zgcd_def abs_if) lemma zgcd_0_left [simp, algebra]: "zgcd 0 m = ¦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: 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" 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) = ¦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 ¦i¦) (nat ¦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 ¦k¦ dvd nat ¦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 ¦k¦ dvd nat ¦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 ¦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