author  wenzelm 
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permissions  rwrr 
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(* Author: Manuel Eberl *) 
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section \<open>Abstract euclidean algorithm\<close> 
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theory Euclidean_Algorithm 

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imports Complex_Main 

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begin 

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text \<open> 
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A Euclidean semiring is a semiring upon which the Euclidean algorithm can be 
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implemented. It must provide: 

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\begin{itemize} 

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\item division with remainder 

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\item a size function such that @{term "size (a mod b) < size b"} 

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for any @{term "b \<noteq> 0"} 

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\item a normalization factor such that two associated numbers are equal iff 
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they are the same when divd by their normalization factors. 

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\end{itemize} 
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The existence of these functions makes it possible to derive gcd and lcm functions 

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for any Euclidean semiring. 

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\<close> 
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class euclidean_semiring = semiring_div + 
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fixes euclidean_size :: "'a \<Rightarrow> nat" 

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fixes normalization_factor :: "'a \<Rightarrow> 'a" 
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assumes mod_size_less [simp]: 
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"b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b" 

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assumes size_mult_mono: 

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"b \<noteq> 0 \<Longrightarrow> euclidean_size (a * b) \<ge> euclidean_size a" 

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assumes normalization_factor_is_unit [intro,simp]: 
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"a \<noteq> 0 \<Longrightarrow> is_unit (normalization_factor a)" 

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assumes normalization_factor_mult: "normalization_factor (a * b) = 

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normalization_factor a * normalization_factor b" 

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assumes normalization_factor_unit: "is_unit a \<Longrightarrow> normalization_factor a = a" 

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assumes normalization_factor_0 [simp]: "normalization_factor 0 = 0" 

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begin 
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lemma normalization_factor_dvd [simp]: 
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"a \<noteq> 0 \<Longrightarrow> normalization_factor a dvd b" 

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by (rule unit_imp_dvd, simp) 
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lemma normalization_factor_1 [simp]: 
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"normalization_factor 1 = 1" 

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by (simp add: normalization_factor_unit) 

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lemma normalization_factor_0_iff [simp]: 
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"normalization_factor a = 0 \<longleftrightarrow> a = 0" 

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proof 
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assume "normalization_factor a = 0" 
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hence "\<not> is_unit (normalization_factor a)" 

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by simp 
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then show "a = 0" by auto 

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qed simp 

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lemma normalization_factor_pow: 
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"normalization_factor (a ^ n) = normalization_factor a ^ n" 

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by (induct n) (simp_all add: normalization_factor_mult power_Suc2) 

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lemma normalization_correct [simp]: 
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"normalization_factor (a div normalization_factor a) = (if a = 0 then 0 else 1)" 

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proof (cases "a = 0", simp) 
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assume "a \<noteq> 0" 
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let ?nf = "normalization_factor" 
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from normalization_factor_is_unit[OF \<open>a \<noteq> 0\<close>] have "?nf a \<noteq> 0" 
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by auto 
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have "?nf (a div ?nf a) * ?nf (?nf a) = ?nf (a div ?nf a * ?nf a)" 
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by (simp add: normalization_factor_mult) 
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also have "a div ?nf a * ?nf a = a" using \<open>a \<noteq> 0\<close> 
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by simp 
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also have "?nf (?nf a) = ?nf a" using \<open>a \<noteq> 0\<close> 
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normalization_factor_is_unit normalization_factor_unit by simp 
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finally have "normalization_factor (a div normalization_factor a) = 1" 

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using \<open>?nf a \<noteq> 0\<close> by (metis div_mult_self2_is_id div_self) 
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with \<open>a \<noteq> 0\<close> show ?thesis by simp 

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qed 
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lemma normalization_0_iff [simp]: 
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"a div normalization_factor a = 0 \<longleftrightarrow> a = 0" 

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by (cases "a = 0", simp, subst unit_eq_div1, blast, simp) 
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lemma mult_div_normalization [simp]: 
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"b * (1 div normalization_factor a) = b div normalization_factor a" 

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by (cases "a = 0") simp_all 
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lemma associated_iff_normed_eq: 
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"associated a b \<longleftrightarrow> a div normalization_factor a = b div normalization_factor b" 
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proof (cases "b = 0", simp, cases "a = 0", metis associated_0(1) normalization_0_iff, rule iffI) 

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let ?nf = normalization_factor 

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assume "a \<noteq> 0" "b \<noteq> 0" "a div ?nf a = b div ?nf b" 
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hence "a = b * (?nf a div ?nf b)" 

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apply (subst (asm) unit_eq_div1, blast, subst (asm) unit_div_commute, blast) 

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apply (subst div_mult_swap, simp, simp) 

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done 

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with \<open>a \<noteq> 0\<close> \<open>b \<noteq> 0\<close> have "\<exists>c. is_unit c \<and> a = c * b" 
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by (intro exI[of _ "?nf a div ?nf b"], force simp: mult_ac) 
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then obtain c where "is_unit c" and "a = c * b" by blast 
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then show "associated a b" by (rule is_unit_associatedI) 
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next 
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let ?nf = normalization_factor 
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assume "a \<noteq> 0" "b \<noteq> 0" "associated a b" 
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then obtain c where "is_unit c" and "a = c * b" by (blast elim: associated_is_unitE) 
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then show "a div ?nf a = b div ?nf b" 
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apply (simp only: \<open>a = c * b\<close> normalization_factor_mult normalization_factor_unit) 
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apply (rule div_mult_mult1, force) 
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done 

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qed 

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lemma normed_associated_imp_eq: 

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"associated a b \<Longrightarrow> normalization_factor a \<in> {0, 1} \<Longrightarrow> normalization_factor b \<in> {0, 1} \<Longrightarrow> a = b" 
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by (simp add: associated_iff_normed_eq, elim disjE, simp_all) 
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lemmas normalization_factor_dvd_iff [simp] = 
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unit_dvd_iff [OF normalization_factor_is_unit] 

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lemma euclidean_division: 

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fixes a :: 'a and b :: 'a 

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assumes "b \<noteq> 0" 

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obtains s and t where "a = s * b + t" 

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and "euclidean_size t < euclidean_size b" 

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proof  

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from div_mod_equality[of a b 0] 

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have "a = a div b * b + a mod b" by simp 

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with that and assms show ?thesis by force 

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qed 

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lemma dvd_euclidean_size_eq_imp_dvd: 

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assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b" 

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shows "a dvd b" 

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proof (subst dvd_eq_mod_eq_0, rule ccontr) 

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assume "b mod a \<noteq> 0" 

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from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff) 

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from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast 

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with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto 
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with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b" 

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using size_mult_mono by force 
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moreover from \<open>a \<noteq> 0\<close> have "euclidean_size (b mod a) < euclidean_size a" 
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using mod_size_less by blast 
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ultimately show False using size_eq by simp 

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qed 

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function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" 

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where 

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"gcd_eucl a b = (if b = 0 then a div normalization_factor a else gcd_eucl b (a mod b))" 
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by (pat_completeness, simp) 
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termination by (relation "measure (euclidean_size \<circ> snd)", simp_all) 

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declare gcd_eucl.simps [simp del] 

147 

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lemma gcd_induct: "\<lbrakk>\<And>b. P b 0; \<And>a b. 0 \<noteq> b \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b\<rbrakk> \<Longrightarrow> P a b" 

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proof (induct a b rule: gcd_eucl.induct) 

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case ("1" m n) 

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then show ?case by (cases "n = 0") auto 

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qed 

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definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" 

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where 

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"lcm_eucl a b = a * b div (gcd_eucl a b * normalization_factor (a * b))" 
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(* Somewhat complicated definition of Lcm that has the advantage of working 

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for infinite sets as well *) 

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definition Lcm_eucl :: "'a set \<Rightarrow> 'a" 

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where 

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"Lcm_eucl A = (if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) then 
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let l = SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = 
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(LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n) 
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in l div normalization_factor l 
58023  167 
else 0)" 
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definition Gcd_eucl :: "'a set \<Rightarrow> 'a" 

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where 

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"Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}" 

172 

173 
end 

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175 
class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd + 

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assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl" 

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assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl" 

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begin 

179 

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lemma gcd_red: 

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"gcd a b = gcd b (a mod b)" 
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by (metis gcd_eucl.simps mod_0 mod_by_0 gcd_gcd_eucl) 
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lemma gcd_non_0: 

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"b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)" 
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by (rule gcd_red) 
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188 
lemma gcd_0_left: 

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"gcd 0 a = a div normalization_factor a" 
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by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def) 
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lemma gcd_0: 

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"gcd a 0 = a div normalization_factor a" 
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by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def) 
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lemma gcd_dvd1 [iff]: "gcd a b dvd a" 
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and gcd_dvd2 [iff]: "gcd a b dvd b" 
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proof (induct a b rule: gcd_eucl.induct) 
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fix a b :: 'a 
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assume IH1: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd b" 
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assume IH2: "b \<noteq> 0 \<Longrightarrow> gcd b (a mod b) dvd (a mod b)" 
58023  202 

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have "gcd a b dvd a \<and> gcd a b dvd b" 
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proof (cases "b = 0") 
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case True 
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then show ?thesis by (cases "a = 0", simp_all add: gcd_0) 
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next 
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case False 

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with IH1 and IH2 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff) 

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qed 

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then show "gcd a b dvd a" "gcd a b dvd b" by simp_all 
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qed 
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lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m" 

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by (rule dvd_trans, assumption, rule gcd_dvd1) 

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lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n" 

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by (rule dvd_trans, assumption, rule gcd_dvd2) 

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lemma gcd_greatest: 

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fixes k a b :: 'a 
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shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b" 
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proof (induct a b rule: gcd_eucl.induct) 
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case (1 a b) 
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show ?case 
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proof (cases "b = 0") 
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assume "b = 0" 
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with 1 show ?thesis by (cases "a = 0", simp_all add: gcd_0) 
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next 
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assume "b \<noteq> 0" 
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with 1 show ?thesis by (simp add: gcd_non_0 dvd_mod_iff) 
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qed 

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qed 

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lemma dvd_gcd_iff: 

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"k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b" 
58023  237 
by (blast intro!: gcd_greatest intro: dvd_trans) 
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lemmas gcd_greatest_iff = dvd_gcd_iff 

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lemma gcd_zero [simp]: 

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"gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" 
58023  243 
by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+ 
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60438  245 
lemma normalization_factor_gcd [simp]: 
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"normalization_factor (gcd a b) = (if a = 0 \<and> b = 0 then 0 else 1)" (is "?f a b = ?g a b") 

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proof (induct a b rule: gcd_eucl.induct) 
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fix a b :: 'a 
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assume IH: "b \<noteq> 0 \<Longrightarrow> ?f b (a mod b) = ?g b (a mod b)" 
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then show "?f a b = ?g a b" by (cases "b = 0", auto simp: gcd_non_0 gcd_0) 
58023  251 
qed 
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lemma gcdI: 

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"k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k) 
60438  255 
\<Longrightarrow> normalization_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b" 
58023  256 
by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest) 
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258 
sublocale gcd!: abel_semigroup gcd 

259 
proof 

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fix a b c 
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show "gcd (gcd a b) c = gcd a (gcd b c)" 
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proof (rule gcdI) 
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have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all 
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then show "gcd (gcd a b) c dvd a" by (rule dvd_trans) 
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have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all 
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hence "gcd (gcd a b) c dvd b" by (rule dvd_trans) 
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moreover have "gcd (gcd a b) c dvd c" by simp 
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ultimately show "gcd (gcd a b) c dvd gcd b c" 
58023  269 
by (rule gcd_greatest) 
60438  270 
show "normalization_factor (gcd (gcd a b) c) = (if gcd (gcd a b) c = 0 then 0 else 1)" 
58023  271 
by auto 
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fix l assume "l dvd a" and "l dvd gcd b c" 
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with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2] 
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have "l dvd b" and "l dvd c" by blast+ 
60526  275 
with \<open>l dvd a\<close> show "l dvd gcd (gcd a b) c" 
58023  276 
by (intro gcd_greatest) 
277 
qed 

278 
next 

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fix a b 
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show "gcd a b = gcd b a" 
58023  281 
by (rule gcdI) (simp_all add: gcd_greatest) 
282 
qed 

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284 
lemma gcd_unique: "d dvd a \<and> d dvd b \<and> 

60438  285 
normalization_factor d = (if d = 0 then 0 else 1) \<and> 
58023  286 
(\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b" 
287 
by (rule, auto intro: gcdI simp: gcd_greatest) 

288 

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lemma gcd_dvd_prod: "gcd a b dvd k * b" 

290 
using mult_dvd_mono [of 1] by auto 

291 

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292 
lemma gcd_1_left [simp]: "gcd 1 a = 1" 
58023  293 
by (rule sym, rule gcdI, simp_all) 
294 

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295 
lemma gcd_1 [simp]: "gcd a 1 = 1" 
58023  296 
by (rule sym, rule gcdI, simp_all) 
297 

298 
lemma gcd_proj2_if_dvd: 

60438  299 
"b dvd a \<Longrightarrow> gcd a b = b div normalization_factor b" 
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300 
by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0) 
58023  301 

302 
lemma gcd_proj1_if_dvd: 

60438  303 
"a dvd b \<Longrightarrow> gcd a b = a div normalization_factor a" 
58023  304 
by (subst gcd.commute, simp add: gcd_proj2_if_dvd) 
305 

60438  306 
lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \<longleftrightarrow> m dvd n" 
58023  307 
proof 
60438  308 
assume A: "gcd m n = m div normalization_factor m" 
58023  309 
show "m dvd n" 
310 
proof (cases "m = 0") 

311 
assume [simp]: "m \<noteq> 0" 

60438  312 
from A have B: "m = gcd m n * normalization_factor m" 
58023  313 
by (simp add: unit_eq_div2) 
314 
show ?thesis by (subst B, simp add: mult_unit_dvd_iff) 

315 
qed (insert A, simp) 

316 
next 

317 
assume "m dvd n" 

60438  318 
then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd) 
58023  319 
qed 
320 

60438  321 
lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \<longleftrightarrow> n dvd m" 
58023  322 
by (subst gcd.commute, simp add: gcd_proj1_iff) 
323 

324 
lemma gcd_mod1 [simp]: 

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325 
"gcd (a mod b) b = gcd a b" 
58023  326 
by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff) 
327 

328 
lemma gcd_mod2 [simp]: 

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329 
"gcd a (b mod a) = gcd a b" 
58023  330 
by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff) 
331 

60438  332 
lemma normalization_factor_dvd' [simp]: 
333 
"normalization_factor a dvd a" 

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334 
by (cases "a = 0", simp_all) 
58023  335 

336 
lemma gcd_mult_distrib': 

60438  337 
"k div normalization_factor k * gcd a b = gcd (k*a) (k*b)" 
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338 
proof (induct a b rule: gcd_eucl.induct) 
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339 
case (1 a b) 
58023  340 
show ?case 
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341 
proof (cases "b = 0") 
58023  342 
case True 
60438  343 
then show ?thesis by (simp add: normalization_factor_mult gcd_0 algebra_simps div_mult_div_if_dvd) 
58023  344 
next 
345 
case False 

60438  346 
hence "k div normalization_factor k * gcd a b = gcd (k * b) (k * (a mod b))" 
58023  347 
using 1 by (subst gcd_red, simp) 
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348 
also have "... = gcd (k * a) (k * b)" 
58023  349 
by (simp add: mult_mod_right gcd.commute) 
350 
finally show ?thesis . 

351 
qed 

352 
qed 

353 

354 
lemma gcd_mult_distrib: 

60438  355 
"k * gcd a b = gcd (k*a) (k*b) * normalization_factor k" 
58023  356 
proof 
60438  357 
let ?nf = "normalization_factor" 
58023  358 
from gcd_mult_distrib' 
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359 
have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" .. 
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360 
also have "... = k * gcd a b div ?nf k" 
60438  361 
by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd) 
58023  362 
finally show ?thesis 
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363 
by simp 
58023  364 
qed 
365 

366 
lemma euclidean_size_gcd_le1 [simp]: 

367 
assumes "a \<noteq> 0" 

368 
shows "euclidean_size (gcd a b) \<le> euclidean_size a" 

369 
proof  

370 
have "gcd a b dvd a" by (rule gcd_dvd1) 

371 
then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast 

60526  372 
with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto 
58023  373 
qed 
374 

375 
lemma euclidean_size_gcd_le2 [simp]: 

376 
"b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b" 

377 
by (subst gcd.commute, rule euclidean_size_gcd_le1) 

378 

379 
lemma euclidean_size_gcd_less1: 

380 
assumes "a \<noteq> 0" and "\<not>a dvd b" 

381 
shows "euclidean_size (gcd a b) < euclidean_size a" 

382 
proof (rule ccontr) 

383 
assume "\<not>euclidean_size (gcd a b) < euclidean_size a" 

60526  384 
with \<open>a \<noteq> 0\<close> have "euclidean_size (gcd a b) = euclidean_size a" 
58023  385 
by (intro le_antisym, simp_all) 
386 
with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd) 

387 
hence "a dvd b" using dvd_gcd_D2 by blast 

60526  388 
with \<open>\<not>a dvd b\<close> show False by contradiction 
58023  389 
qed 
390 

391 
lemma euclidean_size_gcd_less2: 

392 
assumes "b \<noteq> 0" and "\<not>b dvd a" 

393 
shows "euclidean_size (gcd a b) < euclidean_size b" 

394 
using assms by (subst gcd.commute, rule euclidean_size_gcd_less1) 

395 

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396 
lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c" 
58023  397 
apply (rule gcdI) 
398 
apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps) 

399 
apply (rule gcd_dvd2) 

400 
apply (rule gcd_greatest, simp add: unit_simps, assumption) 

60438  401 
apply (subst normalization_factor_gcd, simp add: gcd_0) 
58023  402 
done 
403 

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404 
lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c" 
58023  405 
by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute) 
406 

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407 
lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c" 
60433  408 
by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1) 
58023  409 

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410 
lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c" 
60433  411 
by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2) 
58023  412 

60438  413 
lemma gcd_idem: "gcd a a = a div normalization_factor a" 
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414 
by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all) 
58023  415 

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416 
lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b" 
58023  417 
apply (rule gcdI) 
418 
apply (simp add: ac_simps) 

419 
apply (rule gcd_dvd2) 

420 
apply (rule gcd_greatest, erule (1) gcd_greatest, assumption) 

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421 
apply simp 
58023  422 
done 
423 

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424 
lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b" 
58023  425 
apply (rule gcdI) 
426 
apply simp 

427 
apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2) 

428 
apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption) 

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429 
apply simp 
58023  430 
done 
431 

432 
lemma comp_fun_idem_gcd: "comp_fun_idem gcd" 

433 
proof 

434 
fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a" 

435 
by (simp add: fun_eq_iff ac_simps) 

436 
next 

437 
fix a show "gcd a \<circ> gcd a = gcd a" 

438 
by (simp add: fun_eq_iff gcd_left_idem) 

439 
qed 

440 

441 
lemma coprime_dvd_mult: 

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442 
assumes "gcd c b = 1" and "c dvd a * b" 
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443 
shows "c dvd a" 
58023  444 
proof  
60438  445 
let ?nf = "normalization_factor" 
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446 
from assms gcd_mult_distrib [of a c b] 
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447 
have A: "a = gcd (a * c) (a * b) * ?nf a" by simp 
60526  448 
from \<open>c dvd a * b\<close> show ?thesis by (subst A, simp_all add: gcd_greatest) 
58023  449 
qed 
450 

451 
lemma coprime_dvd_mult_iff: 

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452 
"gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)" 
58023  453 
by (rule, rule coprime_dvd_mult, simp_all) 
454 

455 
lemma gcd_dvd_antisym: 

456 
"gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d" 

457 
proof (rule gcdI) 

458 
assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b" 

459 
have "gcd c d dvd c" by simp 

460 
with A show "gcd a b dvd c" by (rule dvd_trans) 

461 
have "gcd c d dvd d" by simp 

462 
with A show "gcd a b dvd d" by (rule dvd_trans) 

60438  463 
show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)" 
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464 
by simp 
58023  465 
fix l assume "l dvd c" and "l dvd d" 
466 
hence "l dvd gcd c d" by (rule gcd_greatest) 

467 
from this and B show "l dvd gcd a b" by (rule dvd_trans) 

468 
qed 

469 

470 
lemma gcd_mult_cancel: 

471 
assumes "gcd k n = 1" 

472 
shows "gcd (k * m) n = gcd m n" 

473 
proof (rule gcd_dvd_antisym) 

474 
have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps) 

60526  475 
also note \<open>gcd k n = 1\<close> 
58023  476 
finally have "gcd (gcd (k * m) n) k = 1" by simp 
477 
hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps) 

478 
moreover have "gcd (k * m) n dvd n" by simp 

479 
ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest) 

480 
have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all 

481 
then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest) 

482 
qed 

483 

484 
lemma coprime_crossproduct: 

485 
assumes [simp]: "gcd a d = 1" "gcd b c = 1" 

486 
shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs") 

487 
proof 

488 
assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono) 

489 
next 

490 
assume ?lhs 

60526  491 
from \<open>?lhs\<close> have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left) 
58023  492 
hence "a dvd b" by (simp add: coprime_dvd_mult_iff) 
60526  493 
moreover from \<open>?lhs\<close> have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left) 
58023  494 
hence "b dvd a" by (simp add: coprime_dvd_mult_iff) 
60526  495 
moreover from \<open>?lhs\<close> have "c dvd d * b" 
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496 
unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps) 
58023  497 
hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute) 
60526  498 
moreover from \<open>?lhs\<close> have "d dvd c * a" 
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499 
unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps) 
58023  500 
hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute) 
501 
ultimately show ?rhs unfolding associated_def by simp 

502 
qed 

503 

504 
lemma gcd_add1 [simp]: 

505 
"gcd (m + n) n = gcd m n" 

506 
by (cases "n = 0", simp_all add: gcd_non_0) 

507 

508 
lemma gcd_add2 [simp]: 

509 
"gcd m (m + n) = gcd m n" 

510 
using gcd_add1 [of n m] by (simp add: ac_simps) 

511 

512 
lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n" 

513 
by (subst gcd.commute, subst gcd_red, simp) 

514 

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515 
lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1" 
58023  516 
by (rule sym, rule gcdI, simp_all) 
517 

518 
lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)" 

59061  519 
by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2) 
58023  520 

521 
lemma div_gcd_coprime: 

522 
assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0" 

523 
defines [simp]: "d \<equiv> gcd a b" 

524 
defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d" 

525 
shows "gcd a' b' = 1" 

526 
proof (rule coprimeI) 

527 
fix l assume "l dvd a'" "l dvd b'" 

528 
then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast 

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529 
moreover have "a = a' * d" "b = b' * d" by simp_all 
58023  530 
ultimately have "a = (l * d) * s" "b = (l * d) * t" 
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531 
by (simp_all only: ac_simps) 
58023  532 
hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left) 
533 
hence "l*d dvd d" by (simp add: gcd_greatest) 

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534 
then obtain u where "d = l * d * u" .. 
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535 
then have "d * (l * u) = d" by (simp add: ac_simps) 
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536 
moreover from nz have "d \<noteq> 0" by simp 
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537 
with div_mult_self1_is_id have "d * (l * u) div d = l * u" . 
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538 
ultimately have "1 = l * u" 
60526  539 
using \<open>d \<noteq> 0\<close> by simp 
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540 
then show "l dvd 1" .. 
58023  541 
qed 
542 

543 
lemma coprime_mult: 

544 
assumes da: "gcd d a = 1" and db: "gcd d b = 1" 

545 
shows "gcd d (a * b) = 1" 

546 
apply (subst gcd.commute) 

547 
using da apply (subst gcd_mult_cancel) 

548 
apply (subst gcd.commute, assumption) 

549 
apply (subst gcd.commute, rule db) 

550 
done 

551 

552 
lemma coprime_lmult: 

553 
assumes dab: "gcd d (a * b) = 1" 

554 
shows "gcd d a = 1" 

555 
proof (rule coprimeI) 

556 
fix l assume "l dvd d" and "l dvd a" 

557 
hence "l dvd a * b" by simp 

60526  558 
with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest) 
58023  559 
qed 
560 

561 
lemma coprime_rmult: 

562 
assumes dab: "gcd d (a * b) = 1" 

563 
shows "gcd d b = 1" 

564 
proof (rule coprimeI) 

565 
fix l assume "l dvd d" and "l dvd b" 

566 
hence "l dvd a * b" by simp 

60526  567 
with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest) 
58023  568 
qed 
569 

570 
lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1" 

571 
using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast 

572 

573 
lemma gcd_coprime: 

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574 
assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b" 
58023  575 
shows "gcd a' b' = 1" 
576 
proof  

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577 
from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp 
58023  578 
with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" . 
579 
also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+ 

580 
also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+ 

581 
finally show ?thesis . 

582 
qed 

583 

584 
lemma coprime_power: 

585 
assumes "0 < n" 

586 
shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1" 

587 
using assms proof (induct n) 

588 
case (Suc n) then show ?case 

589 
by (cases n) (simp_all add: coprime_mul_eq) 

590 
qed simp 

591 

592 
lemma gcd_coprime_exists: 

593 
assumes nz: "gcd a b \<noteq> 0" 

594 
shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1" 

595 
apply (rule_tac x = "a div gcd a b" in exI) 

596 
apply (rule_tac x = "b div gcd a b" in exI) 

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597 
apply (insert nz, auto intro: div_gcd_coprime) 
58023  598 
done 
599 

600 
lemma coprime_exp: 

601 
"gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1" 

602 
by (induct n, simp_all add: coprime_mult) 

603 

604 
lemma coprime_exp2 [intro]: 

605 
"gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1" 

606 
apply (rule coprime_exp) 

607 
apply (subst gcd.commute) 

608 
apply (rule coprime_exp) 

609 
apply (subst gcd.commute) 

610 
apply assumption 

611 
done 

612 

613 
lemma gcd_exp: 

614 
"gcd (a^n) (b^n) = (gcd a b) ^ n" 

615 
proof (cases "a = 0 \<and> b = 0") 

616 
assume "a = 0 \<and> b = 0" 

617 
then show ?thesis by (cases n, simp_all add: gcd_0_left) 

618 
next 

619 
assume A: "\<not>(a = 0 \<and> b = 0)" 

620 
hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)" 

621 
using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime) 

622 
hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp 

623 
also note gcd_mult_distrib 

60438  624 
also have "normalization_factor ((gcd a b)^n) = 1" 
625 
by (simp add: normalization_factor_pow A) 

58023  626 
also have "(gcd a b)^n * (a div gcd a b)^n = a^n" 
627 
by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp) 

628 
also have "(gcd a b)^n * (b div gcd a b)^n = b^n" 

629 
by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp) 

630 
finally show ?thesis by simp 

631 
qed 

632 

633 
lemma coprime_common_divisor: 

60430
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634 
"gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a" 
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635 
apply (subgoal_tac "a dvd gcd a b") 
59061  636 
apply simp 
58023  637 
apply (erule (1) gcd_greatest) 
638 
done 

639 

640 
lemma division_decomp: 

641 
assumes dc: "a dvd b * c" 

642 
shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c" 

643 
proof (cases "gcd a b = 0") 

644 
assume "gcd a b = 0" 

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645 
hence "a = 0 \<and> b = 0" by simp 
58023  646 
hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp 
647 
then show ?thesis by blast 

648 
next 

649 
let ?d = "gcd a b" 

650 
assume "?d \<noteq> 0" 

651 
from gcd_coprime_exists[OF this] 

652 
obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1" 

653 
by blast 

654 
from ab'(1) have "a' dvd a" unfolding dvd_def by blast 

655 
with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp 

656 
from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp 

657 
hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac) 

60526  658 
with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp 
58023  659 
with coprime_dvd_mult[OF ab'(3)] 
660 
have "a' dvd c" by (subst (asm) ac_simps, blast) 

661 
with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac) 

662 
then show ?thesis by blast 

663 
qed 

664 

60433  665 
lemma pow_divs_pow: 
58023  666 
assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0" 
667 
shows "a dvd b" 

668 
proof (cases "gcd a b = 0") 

669 
assume "gcd a b = 0" 

59009
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670 
then show ?thesis by simp 
58023  671 
next 
672 
let ?d = "gcd a b" 

673 
assume "?d \<noteq> 0" 

674 
from n obtain m where m: "n = Suc m" by (cases n, simp_all) 

60526  675 
from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero) 
676 
from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>] 

58023  677 
obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1" 
678 
by blast 

679 
from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n" 

680 
by (simp add: ab'(1,2)[symmetric]) 

681 
hence "?d^n * a'^n dvd ?d^n * b'^n" 

682 
by (simp only: power_mult_distrib ac_simps) 

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683 
with zn have "a'^n dvd b'^n" by simp 
58023  684 
hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m) 
685 
hence "a' dvd b'^m * b'" by (simp add: m ac_simps) 

686 
with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]] 

687 
have "a' dvd b'" by (subst (asm) ac_simps, blast) 

688 
hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp) 

689 
with ab'(1,2) show ?thesis by simp 

690 
qed 

691 

60433  692 
lemma pow_divs_eq [simp]: 
58023  693 
"n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b" 
60433  694 
by (auto intro: pow_divs_pow dvd_power_same) 
58023  695 

60433  696 
lemma divs_mult: 
58023  697 
assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1" 
698 
shows "m * n dvd r" 

699 
proof  

700 
from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'" 

701 
unfolding dvd_def by blast 

702 
from mr n' have "m dvd n'*n" by (simp add: ac_simps) 

703 
hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp 

704 
then obtain k where k: "n' = m*k" unfolding dvd_def by blast 

705 
with n' have "r = m * n * k" by (simp add: mult_ac) 

706 
then show ?thesis unfolding dvd_def by blast 

707 
qed 

708 

709 
lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1" 

710 
by (subst add_commute, simp) 

711 

712 
lemma setprod_coprime [rule_format]: 

60430
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713 
"(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1" 
58023  714 
apply (cases "finite A") 
715 
apply (induct set: finite) 

716 
apply (auto simp add: gcd_mult_cancel) 

717 
done 

718 

719 
lemma coprime_divisors: 

720 
assumes "d dvd a" "e dvd b" "gcd a b = 1" 

721 
shows "gcd d e = 1" 

722 
proof  

723 
from assms obtain k l where "a = d * k" "b = e * l" 

724 
unfolding dvd_def by blast 

725 
with assms have "gcd (d * k) (e * l) = 1" by simp 

726 
hence "gcd (d * k) e = 1" by (rule coprime_lmult) 

727 
also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps) 

728 
finally have "gcd e d = 1" by (rule coprime_lmult) 

729 
then show ?thesis by (simp add: ac_simps) 

730 
qed 

731 

732 
lemma invertible_coprime: 

60430
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733 
assumes "a * b mod m = 1" 
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changeset

734 
shows "coprime a m" 
59009
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735 
proof  
60430
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736 
from assms have "coprime m (a * b mod m)" 
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737 
by simp 
60430
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738 
then have "coprime m (a * b)" 
59009
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changeset

739 
by simp 
60430
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740 
then have "coprime m a" 
59009
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741 
by (rule coprime_lmult) 
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changeset

742 
then show ?thesis 
348561aa3869
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changeset

743 
by (simp add: ac_simps) 
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744 
qed 
58023  745 

746 
lemma lcm_gcd: 

60438  747 
"lcm a b = a * b div (gcd a b * normalization_factor (a*b))" 
58023  748 
by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def) 
749 

750 
lemma lcm_gcd_prod: 

60438  751 
"lcm a b * gcd a b = a * b div normalization_factor (a*b)" 
58023  752 
proof (cases "a * b = 0") 
60438  753 
let ?nf = normalization_factor 
58023  754 
assume "a * b \<noteq> 0" 
58953  755 
hence "gcd a b \<noteq> 0" by simp 
58023  756 
from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))" 
757 
by (simp add: mult_ac) 

60526  758 
also from \<open>a * b \<noteq> 0\<close> have "... = a * b div ?nf (a*b)" 
60432  759 
by (simp add: div_mult_swap mult.commute) 
58023  760 
finally show ?thesis . 
58953  761 
qed (auto simp add: lcm_gcd) 
58023  762 

763 
lemma lcm_dvd1 [iff]: 

60430
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changeset

764 
"a dvd lcm a b" 
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changeset

765 
proof (cases "a*b = 0") 
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766 
assume "a * b \<noteq> 0" 
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changeset

767 
hence "gcd a b \<noteq> 0" by simp 
60438  768 
let ?c = "1 div normalization_factor (a * b)" 
60526  769 
from \<open>a * b \<noteq> 0\<close> have [simp]: "is_unit (normalization_factor (a * b))" by simp 
60430
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770 
from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b" 
60432  771 
by (simp add: div_mult_swap unit_div_commute) 
60430
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772 
hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp 
60526  773 
with \<open>gcd a b \<noteq> 0\<close> have "lcm a b = a * ?c * b div gcd a b" 
58023  774 
by (subst (asm) div_mult_self2_is_id, simp_all) 
60430
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775 
also have "... = a * (?c * b div gcd a b)" 
58023  776 
by (metis div_mult_swap gcd_dvd2 mult_assoc) 
777 
finally show ?thesis by (rule dvdI) 

58953  778 
qed (auto simp add: lcm_gcd) 
58023  779 

780 
lemma lcm_least: 

781 
"\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k" 

782 
proof (cases "k = 0") 

60438  783 
let ?nf = normalization_factor 
58023  784 
assume "k \<noteq> 0" 
785 
hence "is_unit (?nf k)" by simp 

786 
hence "?nf k \<noteq> 0" by (metis not_is_unit_0) 

787 
assume A: "a dvd k" "b dvd k" 

60526  788 
hence "gcd a b \<noteq> 0" using \<open>k \<noteq> 0\<close> by auto 
58023  789 
from A obtain r s where ar: "k = a * r" and bs: "k = b * s" 
790 
unfolding dvd_def by blast 

60526  791 
with \<open>k \<noteq> 0\<close> have "r * s \<noteq> 0" 
58953  792 
by auto (drule sym [of 0], simp) 
58023  793 
hence "is_unit (?nf (r * s))" by simp 
794 
let ?c = "?nf k div ?nf (r*s)" 

60526  795 
from \<open>is_unit (?nf k)\<close> and \<open>is_unit (?nf (r * s))\<close> have "is_unit ?c" by (rule unit_div) 
58023  796 
hence "?c \<noteq> 0" using not_is_unit_0 by fast 
797 
from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)" 

58953  798 
by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps) 
58023  799 
also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)" 
60526  800 
by (subst (3) \<open>k = a * r\<close>, subst (3) \<open>k = b * s\<close>, simp add: algebra_simps) 
801 
also have "... = ?c * r*s * k * gcd a b" using \<open>r * s \<noteq> 0\<close> 

58023  802 
by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps) 
803 
finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b" 

804 
by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac) 

805 
hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)" 

806 
by (simp add: algebra_simps) 

60526  807 
hence "?c * k * gcd a b = a * b * gcd s r" using \<open>r * s \<noteq> 0\<close> 
58023  808 
by (metis div_mult_self2_is_id) 
809 
also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)" 

810 
by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib') 

811 
also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b" 

812 
by (simp add: algebra_simps) 

60526  813 
finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \<open>gcd a b \<noteq> 0\<close> 
58023  814 
by (metis mult.commute div_mult_self2_is_id) 
60526  815 
hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \<open>?c \<noteq> 0\<close> 
58023  816 
by (metis div_mult_self2_is_id mult_assoc) 
60526  817 
also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \<open>is_unit ?c\<close> 
58023  818 
by (simp add: unit_simps) 
819 
finally show ?thesis by (rule dvdI) 

820 
qed simp 

821 

822 
lemma lcm_zero: 

823 
"lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" 

824 
proof  

60438  825 
let ?nf = normalization_factor 
58023  826 
{ 
827 
assume "a \<noteq> 0" "b \<noteq> 0" 

828 
hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors) 

60526  829 
moreover from \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "gcd a b \<noteq> 0" by simp 
58023  830 
ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp) 
831 
} moreover { 

832 
assume "a = 0 \<or> b = 0" 

833 
hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd) 

834 
} 

835 
ultimately show ?thesis by blast 

836 
qed 

837 

838 
lemmas lcm_0_iff = lcm_zero 

839 

840 
lemma gcd_lcm: 

841 
assumes "lcm a b \<noteq> 0" 

60438  842 
shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))" 
58023  843 
proof 
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changeset

844 
from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero) 
60438  845 
let ?c = "normalization_factor (a * b)" 
60526  846 
from \<open>lcm a b \<noteq> 0\<close> have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors) 
58023  847 
hence "is_unit ?c" by simp 
848 
from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b" 

60526  849 
by (subst (2) div_mult_self2_is_id[OF \<open>lcm a b \<noteq> 0\<close>, symmetric], simp add: mult_ac) 
850 
also from \<open>is_unit ?c\<close> have "... = a * b div (lcm a b * ?c)" 

851 
by (metis \<open>?c \<noteq> 0\<close> div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd') 

60433  852 
finally show ?thesis . 
58023  853 
qed 
854 

60438  855 
lemma normalization_factor_lcm [simp]: 
856 
"normalization_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)" 

58023  857 
proof (cases "a = 0 \<or> b = 0") 
858 
case True then show ?thesis 

58953  859 
by (auto simp add: lcm_gcd) 
58023  860 
next 
861 
case False 

60438  862 
let ?nf = normalization_factor 
58023  863 
from lcm_gcd_prod[of a b] 
864 
have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)" 

60438  865 
by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult) 
58023  866 
also have "... = (if a*b = 0 then 0 else 1)" 
58953  867 
by simp 
868 
finally show ?thesis using False by simp 

58023  869 
qed 
870 

60430
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871 
lemma lcm_dvd2 [iff]: "b dvd lcm a b" 
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changeset

872 
using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps) 
58023  873 

874 
lemma lcmI: 

60430
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875 
"\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l; 
60438  876 
normalization_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b" 
58023  877 
by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least) 
878 

879 
sublocale lcm!: abel_semigroup lcm 

880 
proof 

60430
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changeset

881 
fix a b c 
ce559c850a27
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882 
show "lcm (lcm a b) c = lcm a (lcm b c)" 
58023  883 
proof (rule lcmI) 
60430
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884 
have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all 
ce559c850a27
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changeset

885 
then show "a dvd lcm (lcm a b) c" by (rule dvd_trans) 
58023  886 

60430
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changeset

887 
have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

888 
hence "b dvd lcm (lcm a b) c" by (rule dvd_trans) 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

889 
moreover have "c dvd lcm (lcm a b) c" by simp 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

890 
ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least) 
58023  891 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

892 
fix l assume "a dvd l" and "lcm b c dvd l" 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

893 
have "b dvd lcm b c" by simp 
60526  894 
from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans) 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

895 
have "c dvd lcm b c" by simp 
60526  896 
from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans) 
897 
from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least) 

898 
from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least) 

58023  899 
qed (simp add: lcm_zero) 
900 
next 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

901 
fix a b 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

902 
show "lcm a b = lcm b a" 
58023  903 
by (simp add: lcm_gcd ac_simps) 
904 
qed 

905 

906 
lemma dvd_lcm_D1: 

907 
"lcm m n dvd k \<Longrightarrow> m dvd k" 

908 
by (rule dvd_trans, rule lcm_dvd1, assumption) 

909 

910 
lemma dvd_lcm_D2: 

911 
"lcm m n dvd k \<Longrightarrow> n dvd k" 

912 
by (rule dvd_trans, rule lcm_dvd2, assumption) 

913 

914 
lemma gcd_dvd_lcm [simp]: 

915 
"gcd a b dvd lcm a b" 

916 
by (metis dvd_trans gcd_dvd2 lcm_dvd2) 

917 

918 
lemma lcm_1_iff: 

919 
"lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b" 

920 
proof 

921 
assume "lcm a b = 1" 

59061  922 
then show "is_unit a \<and> is_unit b" by auto 
58023  923 
next 
924 
assume "is_unit a \<and> is_unit b" 

59061  925 
hence "a dvd 1" and "b dvd 1" by simp_all 
926 
hence "is_unit (lcm a b)" by (rule lcm_least) 

60438  927 
hence "lcm a b = normalization_factor (lcm a b)" 
928 
by (subst normalization_factor_unit, simp_all) 

60526  929 
also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close> 
59061  930 
by auto 
58023  931 
finally show "lcm a b = 1" . 
932 
qed 

933 

934 
lemma lcm_0_left [simp]: 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

935 
"lcm 0 a = 0" 
58023  936 
by (rule sym, rule lcmI, simp_all) 
937 

938 
lemma lcm_0 [simp]: 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

939 
"lcm a 0 = 0" 
58023  940 
by (rule sym, rule lcmI, simp_all) 
941 

942 
lemma lcm_unique: 

943 
"a dvd d \<and> b dvd d \<and> 

60438  944 
normalization_factor d = (if d = 0 then 0 else 1) \<and> 
58023  945 
(\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b" 
946 
by (rule, auto intro: lcmI simp: lcm_least lcm_zero) 

947 

948 
lemma dvd_lcm_I1 [simp]: 

949 
"k dvd m \<Longrightarrow> k dvd lcm m n" 

950 
by (metis lcm_dvd1 dvd_trans) 

951 

952 
lemma dvd_lcm_I2 [simp]: 

953 
"k dvd n \<Longrightarrow> k dvd lcm m n" 

954 
by (metis lcm_dvd2 dvd_trans) 

955 

956 
lemma lcm_1_left [simp]: 

60438  957 
"lcm 1 a = a div normalization_factor a" 
60430
ce559c850a27
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haftmann
parents:
59061
diff
changeset

958 
by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all) 
58023  959 

960 
lemma lcm_1_right [simp]: 

60438  961 
"lcm a 1 = a div normalization_factor a" 
60430
ce559c850a27
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haftmann
parents:
59061
diff
changeset

962 
using lcm_1_left [of a] by (simp add: ac_simps) 
58023  963 

964 
lemma lcm_coprime: 

60438  965 
"gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)" 
58023  966 
by (subst lcm_gcd) simp 
967 

968 
lemma lcm_proj1_if_dvd: 

60438  969 
"b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a" 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

970 
by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all) 
58023  971 

972 
lemma lcm_proj2_if_dvd: 

60438  973 
"a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b" 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

974 
using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps) 
58023  975 

976 
lemma lcm_proj1_iff: 

60438  977 
"lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m" 
58023  978 
proof 
60438  979 
assume A: "lcm m n = m div normalization_factor m" 
58023  980 
show "n dvd m" 
981 
proof (cases "m = 0") 

982 
assume [simp]: "m \<noteq> 0" 

60438  983 
from A have B: "m = lcm m n * normalization_factor m" 
58023  984 
by (simp add: unit_eq_div2) 
985 
show ?thesis by (subst B, simp) 

986 
qed simp 

987 
next 

988 
assume "n dvd m" 

60438  989 
then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd) 
58023  990 
qed 
991 

992 
lemma lcm_proj2_iff: 

60438  993 
"lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n" 
58023  994 
using lcm_proj1_iff [of n m] by (simp add: ac_simps) 
995 

996 
lemma euclidean_size_lcm_le1: 

997 
assumes "a \<noteq> 0" and "b \<noteq> 0" 

998 
shows "euclidean_size a \<le> euclidean_size (lcm a b)" 

999 
proof  

1000 
have "a dvd lcm a b" by (rule lcm_dvd1) 

1001 
then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast 

60526  1002 
with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero) 
58023  1003 
then show ?thesis by (subst A, intro size_mult_mono) 
1004 
qed 

1005 

1006 
lemma euclidean_size_lcm_le2: 

1007 
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)" 

1008 
using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps) 

1009 

1010 
lemma euclidean_size_lcm_less1: 

1011 
assumes "b \<noteq> 0" and "\<not>b dvd a" 

1012 
shows "euclidean_size a < euclidean_size (lcm a b)" 

1013 
proof (rule ccontr) 

1014 
from assms have "a \<noteq> 0" by auto 

1015 
assume "\<not>euclidean_size a < euclidean_size (lcm a b)" 

60526  1016 
with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a" 
58023  1017 
by (intro le_antisym, simp, intro euclidean_size_lcm_le1) 
1018 
with assms have "lcm a b dvd a" 

1019 
by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero) 

1020 
hence "b dvd a" by (rule dvd_lcm_D2) 

60526  1021 
with \<open>\<not>b dvd a\<close> show False by contradiction 
58023  1022 
qed 
1023 

1024 
lemma euclidean_size_lcm_less2: 

1025 
assumes "a \<noteq> 0" and "\<not>a dvd b" 

1026 
shows "euclidean_size b < euclidean_size (lcm a b)" 

1027 
using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps) 

1028 

1029 
lemma lcm_mult_unit1: 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1030 
"is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c" 
58023  1031 
apply (rule lcmI) 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1032 
apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1) 
58023  1033 
apply (rule lcm_dvd2) 
1034 
apply (rule lcm_least, simp add: unit_simps, assumption) 

60438  1035 
apply (subst normalization_factor_lcm, simp add: lcm_zero) 
58023  1036 
done 
1037 

1038 
lemma lcm_mult_unit2: 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1039 
"is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c" 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1040 
using lcm_mult_unit1 [of a c b] by (simp add: ac_simps) 
58023  1041 

1042 
lemma lcm_div_unit1: 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1043 
"is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c" 
60433  1044 
by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1) 
58023  1045 

1046 
lemma lcm_div_unit2: 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1047 
"is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c" 
60433  1048 
by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2) 
58023  1049 

1050 
lemma lcm_left_idem: 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1051 
"lcm a (lcm a b) = lcm a b" 
58023  1052 
apply (rule lcmI) 
1053 
apply simp 

1054 
apply (subst lcm.assoc [symmetric], rule lcm_dvd2) 

1055 
apply (rule lcm_least, assumption) 

1056 
apply (erule (1) lcm_least) 

1057 
apply (auto simp: lcm_zero) 

1058 
done 

1059 

1060 
lemma lcm_right_idem: 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1061 
"lcm (lcm a b) b = lcm a b" 
58023  1062 
apply (rule lcmI) 
1063 
apply (subst lcm.assoc, rule lcm_dvd1) 

1064 
apply (rule lcm_dvd2) 

1065 
apply (rule lcm_least, erule (1) lcm_least, assumption) 

1066 
apply (auto simp: lcm_zero) 

1067 
done 

1068 

1069 
lemma comp_fun_idem_lcm: "comp_fun_idem lcm" 

1070 
proof 

1071 
fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a" 

1072 
by (simp add: fun_eq_iff ac_simps) 

1073 
next 

1074 
fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def 

1075 
by (intro ext, simp add: lcm_left_idem) 

1076 
qed 

1077 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1078 
lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A" 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1079 
and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'" 
60438  1080 
and normalization_factor_Lcm [simp]: 
1081 
"normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" 

58023  1082 
proof  
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1083 
have "(\<forall>a\<in>A. a dvd Lcm A) \<and> (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> Lcm A dvd l') \<and> 
60438  1084 
normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" (is ?thesis) 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1085 
proof (cases "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)") 
58023  1086 
case False 
1087 
hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def) 

1088 
with False show ?thesis by auto 

1089 
next 

1090 
case True 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1091 
then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l\<^sub>0)" by blast 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1092 
def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1093 
def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1094 
have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" 
58023  1095 
apply (subst n_def) 
1096 
apply (rule LeastI[of _ "euclidean_size l\<^sub>0"]) 

1097 
apply (rule exI[of _ l\<^sub>0]) 

1098 
apply (simp add: l\<^sub>0_props) 

1099 
done 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1100 
from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n" 
58023  1101 
unfolding l_def by simp_all 
1102 
{ 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1103 
fix l' assume "\<forall>a\<in>A. a dvd l'" 
60526  1104 
with \<open>\<forall>a\<in>A. a dvd l\<close> have "\<forall>a\<in>A. a dvd gcd l l'" by (auto intro: gcd_greatest) 
1105 
moreover from \<open>l \<noteq> 0\<close> have "gcd l l' \<noteq> 0" by simp 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1106 
ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> euclidean_size b = euclidean_size (gcd l l')" 
58023  1107 
by (intro exI[of _ "gcd l l'"], auto) 
1108 
hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le) 

1109 
moreover have "euclidean_size (gcd l l') \<le> n" 

1110 
proof  

1111 
have "gcd l l' dvd l" by simp 

1112 
then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast 

60526  1113 
with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto 
58023  1114 
hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)" 
1115 
by (rule size_mult_mono) 

60526  1116 
also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> .. 
1117 
also note \<open>euclidean_size l = n\<close> 

58023  1118 
finally show "euclidean_size (gcd l l') \<le> n" . 
1119 
qed 

1120 
ultimately have "euclidean_size l = euclidean_size (gcd l l')" 

60526  1121 
by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>) 
1122 
with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd) 

58023  1123 
hence "l dvd l'" by (blast dest: dvd_gcd_D2) 
1124 
} 

1125 

60526  1126 
with \<open>(\<forall>a\<in>A. a dvd l)\<close> and normalization_factor_is_unit[OF \<open>l \<noteq> 0\<close>] and \<open>l \<noteq> 0\<close> 
60438  1127 
have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and> 
1128 
(\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and> 

1129 
normalization_factor (l div normalization_factor l) = 

1130 
(if l div normalization_factor l = 0 then 0 else 1)" 

58023  1131 
by (auto simp: unit_simps) 
60438  1132 
also from True have "l div normalization_factor l = Lcm A" 
58023  1133 
by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def) 
1134 
finally show ?thesis . 

1135 
qed 

1136 
note A = this 

1137 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1138 
{fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast} 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1139 
{fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast} 
60438  1140 
from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast 
58023  1141 
qed 
1142 

1143 
lemma LcmI: 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1144 
"(\<And>a. a\<in>A \<Longrightarrow> a dvd l) \<Longrightarrow> (\<And>l'. (\<forall>a\<in>A. a dvd l') \<Longrightarrow> l dvd l') \<Longrightarrow> 
60438  1145 
normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A" 
58023  1146 
by (intro normed_associated_imp_eq) 
1147 
(auto intro: Lcm_dvd dvd_Lcm simp: associated_def) 

1148 

1149 
lemma Lcm_subset: 

1150 
"A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B" 

1151 
by (blast intro: Lcm_dvd dvd_Lcm) 

1152 

1153 
lemma Lcm_Un: 

1154 
"Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)" 

1155 
apply (rule lcmI) 

1156 
apply (blast intro: Lcm_subset) 

1157 
apply (blast intro: Lcm_subset) 

1158 
apply (intro Lcm_dvd ballI, elim UnE) 

1159 
apply (rule dvd_trans, erule dvd_Lcm, assumption) 

1160 
apply (rule dvd_trans, erule dvd_Lcm, assumption) 

1161 
apply simp 

1162 
done 

1163 

1164 
lemma Lcm_1_iff: 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1165 
"Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)" 
58023  1166 
proof 
1167 
assume "Lcm A = 1" 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1168 
then show "\<forall>a\<in>A. is_unit a" by auto 
58023  1169 
qed (rule LcmI [symmetric], auto) 
1170 

1171 
lemma Lcm_no_units: 

60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1172 
"Lcm A = Lcm (A  {a. is_unit a})" 
58023  1173 
proof  
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1174 
have "(A  {a. is_unit a}) \<union> {a\<in>A. is_unit a} = A" by blast 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1175 
hence "Lcm A = lcm (Lcm (A  {a. is_unit a})) (Lcm {a\<in>A. is_unit a})" 
58023  1176 
by (simp add: Lcm_Un[symmetric]) 
60430
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diff
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1177 
also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff) 
58023  1178 
finally show ?thesis by simp 
1179 
qed 

1180 

1181 
lemma Lcm_empty [simp]: 

1182 
"Lcm {} = 1" 

1183 
by (simp add: Lcm_1_iff) 

1184 

1185 
lemma Lcm_eq_0 [simp]: 

1186 
"0 \<in> A \<Longrightarrow> Lcm A = 0" 

1187 
by (drule dvd_Lcm) simp 

1188 

1189 
lemma Lcm0_iff': 

60430
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parents:
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diff
changeset

1190 
"Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" 
58023  1191 
proof 
1192 
assume "Lcm A = 0" 

60430
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changeset

1193 
show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" 
58023  1194 
proof 
60430
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diff
changeset

1195 
assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" 
ce559c850a27
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diff
changeset

1196 
then obtain l\<^sub>0 where l\<^sub>0_props: "l\<^sub>0 \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l\<^sub>0)" by blast 
ce559c850a27
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diff
changeset

1197 
def n \<equiv> "LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" 
ce559c850a27
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parents:
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diff
changeset

1198 
def l \<equiv> "SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" 
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parents:
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diff
changeset

1199 
have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" 
58023  1200 
apply (subst n_def) 
1201 
apply (rule LeastI[of _ "euclidean_size l\<^sub>0"]) 

1202 
apply (rule exI[of _ l\<^sub>0]) 

1203 
apply (simp add: l\<^sub>0_props) 

1204 
done 

1205 
from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all 

60438  1206 
hence "l div normalization_factor l \<noteq> 0" by simp 
1207 
also from ex have "l div normalization_factor l = Lcm A" 

58023  1208 
by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def) 
60526  1209 
finally show False using \<open>Lcm A = 0\<close> by contradiction 
58023  1210 
qed 
1211 
qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False) 

1212 

1213 
lemma Lcm0_iff [simp]: 

1214 
"finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A" 

1215 
proof  

1216 
assume "finite A" 

1217 
have "0 \<in> A \<Longrightarrow> Lcm A = 0" by (intro dvd_0_left dvd_Lcm) 

1218 
moreover { 

1219 
assume "0 \<notin> A" 

1220 
hence "\<Prod>A \<noteq> 0" 

60526  1221 
apply (induct rule: finite_induct[OF \<open>finite A\<close>]) 
58023  1222 
apply simp 
1223 
apply (subst setprod.insert, assumption, assumption) 

1224 
apply (rule no_zero_divisors) 

1225 
apply blast+ 

1226 
done 

60526  1227 
moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast 
60430
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diff
changeset

1228 
ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast 
58023  1229 
with Lcm0_iff' have "Lcm A \<noteq> 0" by simp 
1230 
} 

1231 
ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast 

1232 
qed 

1233 

1234 
lemma Lcm_no_multiple: 

60430
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parents:
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diff
changeset

1235 
"(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0" 
58023  1236 
proof  
60430
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diff
changeset

1237 
assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)" 
ce559c850a27
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parents:
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diff
changeset

1238 
hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast 
58023  1239 
then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False) 
1240 
qed 

1241 

1242 
lemma Lcm_insert [simp]: 

1243 
"Lcm (insert a A) = lcm a (Lcm A)" 

1244 
proof (rule lcmI) 

1245 
fix l assume "a dvd l" and "Lcm A dvd l" 

60430
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parents:
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diff
changeset

1246 
hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm) 
60526  1247 
with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_dvd) 
58023  1248 
qed (auto intro: Lcm_dvd dvd_Lcm) 
1249 

1250 
lemma Lcm_finite: 

1251 
assumes "finite A" 

1252 
shows "Lcm A = Finite_Set.fold lcm 1 A" 

60526  1253 
by (induct rule: finite.induct[OF \<open>finite A\<close>]) 