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src/HOL/Library/GCD.thy

author | haftmann |

Tue, 24 Jul 2007 15:20:49 +0200 | |

changeset 23951 | b188cac107ad |

parent 23687 | 06884f7ffb18 |

child 23983 | 79dc793bec43 |

permissions | -rw-r--r-- |

renamed lcm_lowest to lcm_least

(* 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 Main begin text {* 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 p m n \<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 m a b \<Longrightarrow> is_gcd n a b \<Longrightarrow> m = n" by (simp add: is_gcd_def) (blast intro: dvd_anti_sym) text {* Connection to divides relation *} lemma is_gcd_dvd: "is_gcd m a b \<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 k m n = is_gcd k n m" by (auto simp add: is_gcd_def) subsection {* GCD on nat by Euclid's algorithm *} fun gcd :: "nat \<times> nat => nat" where "gcd (m, n) = (if n = 0 then m else gcd (n, m mod n))" lemma gcd_induct: 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" apply (rule gcd.induct [of "split P" "(m, n)", unfolded Product_Type.split]) apply (case_tac "n = 0") apply simp_all using assms apply simp_all done lemma gcd_0 [simp]: "gcd (m, 0) = m" by simp lemma gcd_0_left [simp]: "gcd (0, m) = m" by simp lemma gcd_non_0: "n > 0 \<Longrightarrow> gcd (m, n) = gcd (n, m mod n)" by simp lemma gcd_1 [simp]: "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]: "gcd (m, n) dvd m" and gcd_dvd2 [iff]: "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 \<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 (gcd (m, n)) m n" by (simp add: is_gcd_def gcd_greatest) subsection {* Derived laws for GCD *} lemma gcd_greatest_iff [iff]: "k dvd gcd (m, n) \<longleftrightarrow> k dvd m \<and> k dvd n" by (blast intro!: gcd_greatest intro: dvd_trans) lemma gcd_zero: "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) 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]: "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]: "gcd (k, k * n) = k" apply (rule gcd_mult_distrib2 [of k 1 n, simplified, symmetric]) done lemma gcd_self [simp]: "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)" apply (blast intro: relprime_dvd_mult dvd_trans) done 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_trans) done text {* \medskip Addition laws *} lemma gcd_add1 [simp]: "gcd (m + n, n) = gcd (m, n)" apply (case_tac "n = 0") apply (simp_all add: gcd_non_0) done lemma gcd_add2 [simp]: "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]: "gcd (m, n + m) = gcd (m, n)" apply (subst add_commute) apply (rule gcd_add2) done lemma gcd_add_mult: "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 \<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 simp qed subsection {* LCM defined by GCD *} definition lcm :: "nat \<times> nat \<Rightarrow> nat" where "lcm = (\<lambda>(m, n). m * n div gcd (m, n))" lemma lcm_def: "lcm (m, n) = m * n div gcd (m, n)" unfolding lcm_def by simp 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 "\<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 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 "\<dots> = 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 "\<dots> = n * k" using mpos npos gcd_zero by simp finally show ?thesis by (simp add: lcm_def) qed qed subsection {* GCD and LCM on integers *} definition igcd :: "int \<Rightarrow> int \<Rightarrow> int" where "igcd i j = int (gcd (nat (abs i), nat (abs j)))" lemma igcd_dvd1 [simp]: "igcd i j dvd i" by (simp add: igcd_def int_dvd_iff) lemma igcd_dvd2 [simp]: "igcd i j dvd j" by (simp add: igcd_def int_dvd_iff) lemma igcd_pos: "igcd i j \<ge> 0" by (simp add: igcd_def) lemma igcd0 [simp]: "(igcd i j = 0) = (i = 0 \<and> j = 0)" by (simp add: igcd_def gcd_zero) arith lemma igcd_commute: "igcd i j = igcd j i" unfolding igcd_def by (simp add: gcd_commute) lemma igcd_neg1 [simp]: "igcd (- i) j = igcd i j" unfolding igcd_def by simp lemma igcd_neg2 [simp]: "igcd i (- j) = igcd i j" unfolding igcd_def by simp lemma zrelprime_dvd_mult: "igcd i j = 1 \<Longrightarrow> i dvd k * j \<Longrightarrow> i dvd k" unfolding igcd_def proof - 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 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 igcd_greatest: assumes "k dvd m" and "k dvd n" shows "k dvd igcd 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 zdvd_abs1 zdvd_abs2) from gcd_greatest [OF dvd'] have "int (nat \<bar>k\<bar>) dvd igcd m n" unfolding igcd_def by (simp only: zdvd_int) then have "\<bar>k\<bar> dvd igcd m n" by (simp only: int_nat_abs) then show "k dvd igcd m n" by (simp add: zdvd_abs1) qed lemma div_igcd_relprime: assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0" shows "igcd (a div (igcd a b)) (b div (igcd a b)) = 1" proof - from nz have nz': "nat \<bar>a\<bar> \<noteq> 0 \<or> nat \<bar>b\<bar> \<noteq> 0" by simp let ?g = "igcd a b" let ?a' = "a div ?g" let ?b' = "b div ?g" let ?g' = "igcd ?a' ?b'" have dvdg: "?g dvd a" "?g dvd b" by (simp_all add: igcd_dvd1 igcd_dvd2) have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by (simp_all add: igcd_dvd1 igcd_dvd2) 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 \<noteq> 0" using nz by simp then have gp: "?g \<noteq> 0" using igcd_pos[where i="a" and j="b"] by arith from igcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" . with zdvd_mult_cancel1 [OF gp] have "\<bar>?g'\<bar> = 1" by simp with igcd_pos show "?g' = 1" by simp qed definition "ilcm = (\<lambda>i j. int (lcm(nat(abs i),nat(abs j))))" (* ilcm_dvd12 are needed later *) lemma ilcm_dvd1: assumes anz: "a \<noteq> 0" and bnz: "b \<noteq> 0" shows "a dvd (ilcm 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 from nap nbp have "?na dvd lcm(?na,?nb)" using lcm_dvd1 by simp thus ?thesis by (simp add: ilcm_def dvd_int_iff) qed lemma ilcm_dvd2: assumes anz: "a \<noteq> 0" and bnz: "b \<noteq> 0" shows "b dvd (ilcm 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 from nap nbp have "?nb dvd lcm(?na,?nb)" using lcm_dvd2 by simp thus ?thesis by (simp add: ilcm_def dvd_int_iff) 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) lemma zdvd_abs1: "((d::int) dvd t) = ((abs d) dvd t)" by (cases "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 simp qed lemma ilcm_pos: assumes apos: " 0 < a" and bpos: "0 < b" shows "0 < ilcm a b" proof- let ?na = "nat (abs a)" let ?nb = "nat (abs b)" have nap: "?na >0" using apos by simp have nbp: "?nb >0" using bpos by simp have "0 < lcm (?na,?nb)" by (rule lcm_pos[OF nap nbp]) thus ?thesis by (simp add: ilcm_def) qed end