author  haftmann 
Thu, 25 Jun 2015 15:01:42 +0200  
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parent 60569  f2f1f6860959 
child 60572  718b1ba06429 
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 "~~/src/HOL/Library/Polynomial" 
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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: 
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"b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<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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lemma normed_dvd [iff]: 
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"a div normalization_factor a dvd a" 
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proof (cases "a = 0") 
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case True then show ?thesis by simp 
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next 
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case False 
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then have "a = a div normalization_factor a * normalization_factor a" 
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by (auto intro: unit_div_mult_self) 
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then show ?thesis .. 
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qed 
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lemma dvd_normed [iff]: 
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"a dvd a div normalization_factor a" 
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proof (cases "a = 0") 
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case True then show ?thesis by simp 
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next 
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case False 
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then have "a div normalization_factor a = a * (1 div normalization_factor a)" 
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by (auto intro: unit_mult_div_div) 
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then show ?thesis .. 
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qed 
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lemma associated_normed: 
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"associated (a div normalization_factor a) a" 
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by (rule associatedI) simp_all 
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lemma normalization_factor_dvd' [simp]: 
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"normalization_factor a dvd a" 
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by (cases "a = 0", 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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144 
lemma euclidean_division: 

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

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assumes "b \<noteq> 0" and "\<not> b dvd a" 
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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 (auto simp add: mod_size_less) 
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qed 
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lemma dvd_euclidean_size_eq_imp_dvd: 

156 
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 (rule ccontr) 
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assume "\<not> a dvd b" 
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then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd) 
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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>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close> 
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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 

170 
qed 

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172 
function gcd_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" 

173 
where 

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"gcd_eucl a b = (if b = 0 then a div normalization_factor a 
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else if b dvd a then b div normalization_factor b 
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else gcd_eucl b (a mod b))" 
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by (pat_completeness, simp) 
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termination 
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by (relation "measure (euclidean_size \<circ> snd)") (simp_all add: mod_size_less) 
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declare gcd_eucl.simps [simp del] 

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lemma gcd_eucl_induct [case_names zero mod]: 
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assumes H1: "\<And>b. P b 0" 
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and H2: "\<And>a b. b \<noteq> 0 \<Longrightarrow> P b (a mod b) \<Longrightarrow> P a b" 
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shows "P 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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case True then show "P a b" by simp (rule H1) 
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next 
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case False 
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have "P b (a mod b)" 
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proof (cases "b dvd a") 
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case False with \<open>b \<noteq> 0\<close> show "P b (a mod b)" 
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by (rule "1.hyps") 
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next 
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case True then have "a mod b = 0" 
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by (simp add: mod_eq_0_iff_dvd) 
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then show "P b (a mod b)" by simp (rule H1) 
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qed 
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with \<open>b \<noteq> 0\<close> show "P a b" 
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by (blast intro: H2) 
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qed 
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qed 
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definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" 

209 
where 

60438  210 
"lcm_eucl a b = a * b div (gcd_eucl a b * normalization_factor (a * b))" 
58023  211 

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(* Somewhat complicated definition of Lcm that has the advantage of working 

213 
for infinite sets as well *) 

214 

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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 
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else 0)" 
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223 
definition Gcd_eucl :: "'a set \<Rightarrow> 'a" 

224 
where 

225 
"Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}" 

226 

227 
end 

228 

229 
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 

233 

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lemma gcd_red: 

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"gcd a b = gcd b (a mod b)" 
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by (cases "b dvd a") 
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(auto simp add: gcd_gcd_eucl gcd_eucl.simps [of a b] gcd_eucl.simps [of 0 a] gcd_eucl.simps [of b 0]) 
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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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lemma gcd_0_left: 

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"gcd 0 a = a div normalization_factor a" 
58023  245 
by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, subst gcd_eucl.simps, simp add: Let_def) 
246 

247 
lemma gcd_0: 

60438  248 
"gcd a 0 = a div normalization_factor a" 
58023  249 
by (simp only: gcd_gcd_eucl, subst gcd_eucl.simps, simp add: Let_def) 
250 

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251 
lemma gcd_dvd1 [iff]: "gcd a b dvd a" 
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252 
and gcd_dvd2 [iff]: "gcd a b dvd b" 
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253 
by (induct a b rule: gcd_eucl_induct) 
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254 
(simp_all add: gcd_0 gcd_non_0 dvd_mod_iff) 
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255 

58023  256 
lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m" 
257 
by (rule dvd_trans, assumption, rule gcd_dvd1) 

258 

259 
lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n" 

260 
by (rule dvd_trans, assumption, rule gcd_dvd2) 

261 

262 
lemma gcd_greatest: 

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263 
fixes k a b :: 'a 
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264 
shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b" 
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265 
proof (induct a b rule: gcd_eucl_induct) 
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266 
case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_0) 
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267 
next 
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268 
case (mod a b) 
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269 
then show ?case 
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270 
by (simp add: gcd_non_0 dvd_mod_iff) 
58023  271 
qed 
272 

273 
lemma dvd_gcd_iff: 

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274 
"k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b" 
58023  275 
by (blast intro!: gcd_greatest intro: dvd_trans) 
276 

277 
lemmas gcd_greatest_iff = dvd_gcd_iff 

278 

279 
lemma gcd_zero [simp]: 

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280 
"gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" 
58023  281 
by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+ 
282 

60438  283 
lemma normalization_factor_gcd [simp]: 
284 
"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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285 
by (induct a b rule: gcd_eucl_induct) 
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286 
(auto simp add: gcd_0 gcd_non_0) 
58023  287 

288 
lemma gcdI: 

60430
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289 
"k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> (\<And>l. l dvd a \<Longrightarrow> l dvd b \<Longrightarrow> l dvd k) 
60438  290 
\<Longrightarrow> normalization_factor k = (if k = 0 then 0 else 1) \<Longrightarrow> k = gcd a b" 
58023  291 
by (intro normed_associated_imp_eq) (auto simp: associated_def intro: gcd_greatest) 
292 

293 
sublocale gcd!: abel_semigroup gcd 

294 
proof 

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295 
fix a b c 
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296 
show "gcd (gcd a b) c = gcd a (gcd b c)" 
58023  297 
proof (rule gcdI) 
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298 
have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all 
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299 
then show "gcd (gcd a b) c dvd a" by (rule dvd_trans) 
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300 
have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all 
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301 
hence "gcd (gcd a b) c dvd b" by (rule dvd_trans) 
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302 
moreover have "gcd (gcd a b) c dvd c" by simp 
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303 
ultimately show "gcd (gcd a b) c dvd gcd b c" 
58023  304 
by (rule gcd_greatest) 
60438  305 
show "normalization_factor (gcd (gcd a b) c) = (if gcd (gcd a b) c = 0 then 0 else 1)" 
58023  306 
by auto 
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307 
fix l assume "l dvd a" and "l dvd gcd b c" 
58023  308 
with dvd_trans[OF _ gcd_dvd1] and dvd_trans[OF _ gcd_dvd2] 
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309 
have "l dvd b" and "l dvd c" by blast+ 
60526  310 
with \<open>l dvd a\<close> show "l dvd gcd (gcd a b) c" 
58023  311 
by (intro gcd_greatest) 
312 
qed 

313 
next 

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314 
fix a b 
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315 
show "gcd a b = gcd b a" 
58023  316 
by (rule gcdI) (simp_all add: gcd_greatest) 
317 
qed 

318 

319 
lemma gcd_unique: "d dvd a \<and> d dvd b \<and> 

60438  320 
normalization_factor d = (if d = 0 then 0 else 1) \<and> 
58023  321 
(\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b" 
322 
by (rule, auto intro: gcdI simp: gcd_greatest) 

323 

324 
lemma gcd_dvd_prod: "gcd a b dvd k * b" 

325 
using mult_dvd_mono [of 1] by auto 

326 

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lemma gcd_1_left [simp]: "gcd 1 a = 1" 
58023  328 
by (rule sym, rule gcdI, simp_all) 
329 

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330 
lemma gcd_1 [simp]: "gcd a 1 = 1" 
58023  331 
by (rule sym, rule gcdI, simp_all) 
332 

333 
lemma gcd_proj2_if_dvd: 

60438  334 
"b dvd a \<Longrightarrow> gcd a b = b div normalization_factor b" 
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335 
by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0) 
58023  336 

337 
lemma gcd_proj1_if_dvd: 

60438  338 
"a dvd b \<Longrightarrow> gcd a b = a div normalization_factor a" 
58023  339 
by (subst gcd.commute, simp add: gcd_proj2_if_dvd) 
340 

60438  341 
lemma gcd_proj1_iff: "gcd m n = m div normalization_factor m \<longleftrightarrow> m dvd n" 
58023  342 
proof 
60438  343 
assume A: "gcd m n = m div normalization_factor m" 
58023  344 
show "m dvd n" 
345 
proof (cases "m = 0") 

346 
assume [simp]: "m \<noteq> 0" 

60438  347 
from A have B: "m = gcd m n * normalization_factor m" 
58023  348 
by (simp add: unit_eq_div2) 
349 
show ?thesis by (subst B, simp add: mult_unit_dvd_iff) 

350 
qed (insert A, simp) 

351 
next 

352 
assume "m dvd n" 

60438  353 
then show "gcd m n = m div normalization_factor m" by (rule gcd_proj1_if_dvd) 
58023  354 
qed 
355 

60438  356 
lemma gcd_proj2_iff: "gcd m n = n div normalization_factor n \<longleftrightarrow> n dvd m" 
58023  357 
by (subst gcd.commute, simp add: gcd_proj1_iff) 
358 

359 
lemma gcd_mod1 [simp]: 

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360 
"gcd (a mod b) b = gcd a b" 
58023  361 
by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff) 
362 

363 
lemma gcd_mod2 [simp]: 

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364 
"gcd a (b mod a) = gcd a b" 
58023  365 
by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff) 
366 

367 
lemma gcd_mult_distrib': 

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368 
"c div normalization_factor c * gcd a b = gcd (c * a) (c * b)" 
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369 
proof (cases "c = 0") 
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370 
case True then show ?thesis by (simp_all add: gcd_0) 
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371 
next 
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372 
case False then have [simp]: "is_unit (normalization_factor c)" by simp 
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373 
show ?thesis 
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374 
proof (induct a b rule: gcd_eucl_induct) 
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375 
case (zero a) show ?case 
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376 
proof (cases "a = 0") 
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377 
case True then show ?thesis by (simp add: gcd_0) 
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378 
next 
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379 
case False then have "is_unit (normalization_factor a)" by simp 
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380 
then show ?thesis 
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381 
by (simp add: gcd_0 unit_div_commute unit_div_mult_swap normalization_factor_mult is_unit_div_mult2_eq) 
f2f1f6860959
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382 
qed 
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383 
case (mod a b) 
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384 
then show ?case by (simp add: mult_mod_right gcd.commute) 
58023  385 
qed 
386 
qed 

387 

388 
lemma gcd_mult_distrib: 

60438  389 
"k * gcd a b = gcd (k*a) (k*b) * normalization_factor k" 
58023  390 
proof 
60438  391 
let ?nf = "normalization_factor" 
58023  392 
from gcd_mult_distrib' 
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393 
have "gcd (k*a) (k*b) = k div ?nf k * gcd a b" .. 
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394 
also have "... = k * gcd a b div ?nf k" 
60438  395 
by (metis dvd_div_mult dvd_eq_mod_eq_0 mod_0 normalization_factor_dvd) 
58023  396 
finally show ?thesis 
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397 
by simp 
58023  398 
qed 
399 

400 
lemma euclidean_size_gcd_le1 [simp]: 

401 
assumes "a \<noteq> 0" 

402 
shows "euclidean_size (gcd a b) \<le> euclidean_size a" 

403 
proof  

404 
have "gcd a b dvd a" by (rule gcd_dvd1) 

405 
then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast 

60526  406 
with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto 
58023  407 
qed 
408 

409 
lemma euclidean_size_gcd_le2 [simp]: 

410 
"b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b" 

411 
by (subst gcd.commute, rule euclidean_size_gcd_le1) 

412 

413 
lemma euclidean_size_gcd_less1: 

414 
assumes "a \<noteq> 0" and "\<not>a dvd b" 

415 
shows "euclidean_size (gcd a b) < euclidean_size a" 

416 
proof (rule ccontr) 

417 
assume "\<not>euclidean_size (gcd a b) < euclidean_size a" 

60526  418 
with \<open>a \<noteq> 0\<close> have "euclidean_size (gcd a b) = euclidean_size a" 
58023  419 
by (intro le_antisym, simp_all) 
420 
with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd) 

421 
hence "a dvd b" using dvd_gcd_D2 by blast 

60526  422 
with \<open>\<not>a dvd b\<close> show False by contradiction 
58023  423 
qed 
424 

425 
lemma euclidean_size_gcd_less2: 

426 
assumes "b \<noteq> 0" and "\<not>b dvd a" 

427 
shows "euclidean_size (gcd a b) < euclidean_size b" 

428 
using assms by (subst gcd.commute, rule euclidean_size_gcd_less1) 

429 

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430 
lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c" 
58023  431 
apply (rule gcdI) 
432 
apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps) 

433 
apply (rule gcd_dvd2) 

434 
apply (rule gcd_greatest, simp add: unit_simps, assumption) 

60438  435 
apply (subst normalization_factor_gcd, simp add: gcd_0) 
58023  436 
done 
437 

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438 
lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c" 
58023  439 
by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute) 
440 

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441 
lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c" 
60433  442 
by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1) 
58023  443 

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444 
lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c" 
60433  445 
by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2) 
58023  446 

60438  447 
lemma gcd_idem: "gcd a a = a div normalization_factor a" 
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448 
by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all) 
58023  449 

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450 
lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b" 
58023  451 
apply (rule gcdI) 
452 
apply (simp add: ac_simps) 

453 
apply (rule gcd_dvd2) 

454 
apply (rule gcd_greatest, erule (1) gcd_greatest, assumption) 

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455 
apply simp 
58023  456 
done 
457 

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458 
lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b" 
58023  459 
apply (rule gcdI) 
460 
apply simp 

461 
apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2) 

462 
apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption) 

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463 
apply simp 
58023  464 
done 
465 

466 
lemma comp_fun_idem_gcd: "comp_fun_idem gcd" 

467 
proof 

468 
fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a" 

469 
by (simp add: fun_eq_iff ac_simps) 

470 
next 

471 
fix a show "gcd a \<circ> gcd a = gcd a" 

472 
by (simp add: fun_eq_iff gcd_left_idem) 

473 
qed 

474 

475 
lemma coprime_dvd_mult: 

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476 
assumes "gcd c b = 1" and "c dvd a * b" 
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477 
shows "c dvd a" 
58023  478 
proof  
60438  479 
let ?nf = "normalization_factor" 
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480 
from assms gcd_mult_distrib [of a c b] 
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481 
have A: "a = gcd (a * c) (a * b) * ?nf a" by simp 
60526  482 
from \<open>c dvd a * b\<close> show ?thesis by (subst A, simp_all add: gcd_greatest) 
58023  483 
qed 
484 

485 
lemma coprime_dvd_mult_iff: 

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486 
"gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)" 
58023  487 
by (rule, rule coprime_dvd_mult, simp_all) 
488 

489 
lemma gcd_dvd_antisym: 

490 
"gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d" 

491 
proof (rule gcdI) 

492 
assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b" 

493 
have "gcd c d dvd c" by simp 

494 
with A show "gcd a b dvd c" by (rule dvd_trans) 

495 
have "gcd c d dvd d" by simp 

496 
with A show "gcd a b dvd d" by (rule dvd_trans) 

60438  497 
show "normalization_factor (gcd a b) = (if gcd a b = 0 then 0 else 1)" 
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498 
by simp 
58023  499 
fix l assume "l dvd c" and "l dvd d" 
500 
hence "l dvd gcd c d" by (rule gcd_greatest) 

501 
from this and B show "l dvd gcd a b" by (rule dvd_trans) 

502 
qed 

503 

504 
lemma gcd_mult_cancel: 

505 
assumes "gcd k n = 1" 

506 
shows "gcd (k * m) n = gcd m n" 

507 
proof (rule gcd_dvd_antisym) 

508 
have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps) 

60526  509 
also note \<open>gcd k n = 1\<close> 
58023  510 
finally have "gcd (gcd (k * m) n) k = 1" by simp 
511 
hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps) 

512 
moreover have "gcd (k * m) n dvd n" by simp 

513 
ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest) 

514 
have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all 

515 
then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest) 

516 
qed 

517 

518 
lemma coprime_crossproduct: 

519 
assumes [simp]: "gcd a d = 1" "gcd b c = 1" 

520 
shows "associated (a * c) (b * d) \<longleftrightarrow> associated a b \<and> associated c d" (is "?lhs \<longleftrightarrow> ?rhs") 

521 
proof 

522 
assume ?rhs then show ?lhs unfolding associated_def by (fast intro: mult_dvd_mono) 

523 
next 

524 
assume ?lhs 

60526  525 
from \<open>?lhs\<close> have "a dvd b * d" unfolding associated_def by (metis dvd_mult_left) 
58023  526 
hence "a dvd b" by (simp add: coprime_dvd_mult_iff) 
60526  527 
moreover from \<open>?lhs\<close> have "b dvd a * c" unfolding associated_def by (metis dvd_mult_left) 
58023  528 
hence "b dvd a" by (simp add: coprime_dvd_mult_iff) 
60526  529 
moreover from \<open>?lhs\<close> have "c dvd d * b" 
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530 
unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps) 
58023  531 
hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute) 
60526  532 
moreover from \<open>?lhs\<close> have "d dvd c * a" 
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533 
unfolding associated_def by (auto dest: dvd_mult_right simp add: ac_simps) 
58023  534 
hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute) 
535 
ultimately show ?rhs unfolding associated_def by simp 

536 
qed 

537 

538 
lemma gcd_add1 [simp]: 

539 
"gcd (m + n) n = gcd m n" 

540 
by (cases "n = 0", simp_all add: gcd_non_0) 

541 

542 
lemma gcd_add2 [simp]: 

543 
"gcd m (m + n) = gcd m n" 

544 
using gcd_add1 [of n m] by (simp add: ac_simps) 

545 

546 
lemma gcd_add_mult: "gcd m (k * m + n) = gcd m n" 

547 
by (subst gcd.commute, subst gcd_red, simp) 

548 

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549 
lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1" 
58023  550 
by (rule sym, rule gcdI, simp_all) 
551 

552 
lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)" 

59061  553 
by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2) 
58023  554 

555 
lemma div_gcd_coprime: 

556 
assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0" 

557 
defines [simp]: "d \<equiv> gcd a b" 

558 
defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d" 

559 
shows "gcd a' b' = 1" 

560 
proof (rule coprimeI) 

561 
fix l assume "l dvd a'" "l dvd b'" 

562 
then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast 

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563 
moreover have "a = a' * d" "b = b' * d" by simp_all 
58023  564 
ultimately have "a = (l * d) * s" "b = (l * d) * t" 
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565 
by (simp_all only: ac_simps) 
58023  566 
hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left) 
567 
hence "l*d dvd d" by (simp add: gcd_greatest) 

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568 
then obtain u where "d = l * d * u" .. 
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569 
then have "d * (l * u) = d" by (simp add: ac_simps) 
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570 
moreover from nz have "d \<noteq> 0" by simp 
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571 
with div_mult_self1_is_id have "d * (l * u) div d = l * u" . 
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572 
ultimately have "1 = l * u" 
60526  573 
using \<open>d \<noteq> 0\<close> by simp 
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574 
then show "l dvd 1" .. 
58023  575 
qed 
576 

577 
lemma coprime_mult: 

578 
assumes da: "gcd d a = 1" and db: "gcd d b = 1" 

579 
shows "gcd d (a * b) = 1" 

580 
apply (subst gcd.commute) 

581 
using da apply (subst gcd_mult_cancel) 

582 
apply (subst gcd.commute, assumption) 

583 
apply (subst gcd.commute, rule db) 

584 
done 

585 

586 
lemma coprime_lmult: 

587 
assumes dab: "gcd d (a * b) = 1" 

588 
shows "gcd d a = 1" 

589 
proof (rule coprimeI) 

590 
fix l assume "l dvd d" and "l dvd a" 

591 
hence "l dvd a * b" by simp 

60526  592 
with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest) 
58023  593 
qed 
594 

595 
lemma coprime_rmult: 

596 
assumes dab: "gcd d (a * b) = 1" 

597 
shows "gcd d b = 1" 

598 
proof (rule coprimeI) 

599 
fix l assume "l dvd d" and "l dvd b" 

600 
hence "l dvd a * b" by simp 

60526  601 
with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest) 
58023  602 
qed 
603 

604 
lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1" 

605 
using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast 

606 

607 
lemma gcd_coprime: 

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608 
assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b" 
58023  609 
shows "gcd a' b' = 1" 
610 
proof  

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611 
from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp 
58023  612 
with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" . 
613 
also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+ 

614 
also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+ 

615 
finally show ?thesis . 

616 
qed 

617 

618 
lemma coprime_power: 

619 
assumes "0 < n" 

620 
shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1" 

621 
using assms proof (induct n) 

622 
case (Suc n) then show ?case 

623 
by (cases n) (simp_all add: coprime_mul_eq) 

624 
qed simp 

625 

626 
lemma gcd_coprime_exists: 

627 
assumes nz: "gcd a b \<noteq> 0" 

628 
shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1" 

629 
apply (rule_tac x = "a div gcd a b" in exI) 

630 
apply (rule_tac x = "b div gcd a b" in exI) 

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631 
apply (insert nz, auto intro: div_gcd_coprime) 
58023  632 
done 
633 

634 
lemma coprime_exp: 

635 
"gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1" 

636 
by (induct n, simp_all add: coprime_mult) 

637 

638 
lemma coprime_exp2 [intro]: 

639 
"gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1" 

640 
apply (rule coprime_exp) 

641 
apply (subst gcd.commute) 

642 
apply (rule coprime_exp) 

643 
apply (subst gcd.commute) 

644 
apply assumption 

645 
done 

646 

647 
lemma gcd_exp: 

648 
"gcd (a^n) (b^n) = (gcd a b) ^ n" 

649 
proof (cases "a = 0 \<and> b = 0") 

650 
assume "a = 0 \<and> b = 0" 

651 
then show ?thesis by (cases n, simp_all add: gcd_0_left) 

652 
next 

653 
assume A: "\<not>(a = 0 \<and> b = 0)" 

654 
hence "1 = gcd ((a div gcd a b)^n) ((b div gcd a b)^n)" 

655 
using div_gcd_coprime by (subst sym, auto simp: div_gcd_coprime) 

656 
hence "(gcd a b) ^ n = (gcd a b) ^ n * ..." by simp 

657 
also note gcd_mult_distrib 

60438  658 
also have "normalization_factor ((gcd a b)^n) = 1" 
659 
by (simp add: normalization_factor_pow A) 

58023  660 
also have "(gcd a b)^n * (a div gcd a b)^n = a^n" 
661 
by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp) 

662 
also have "(gcd a b)^n * (b div gcd a b)^n = b^n" 

663 
by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp) 

664 
finally show ?thesis by simp 

665 
qed 

666 

667 
lemma coprime_common_divisor: 

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668 
"gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a" 
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669 
apply (subgoal_tac "a dvd gcd a b") 
59061  670 
apply simp 
58023  671 
apply (erule (1) gcd_greatest) 
672 
done 

673 

674 
lemma division_decomp: 

675 
assumes dc: "a dvd b * c" 

676 
shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c" 

677 
proof (cases "gcd a b = 0") 

678 
assume "gcd a b = 0" 

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679 
hence "a = 0 \<and> b = 0" by simp 
58023  680 
hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp 
681 
then show ?thesis by blast 

682 
next 

683 
let ?d = "gcd a b" 

684 
assume "?d \<noteq> 0" 

685 
from gcd_coprime_exists[OF this] 

686 
obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1" 

687 
by blast 

688 
from ab'(1) have "a' dvd a" unfolding dvd_def by blast 

689 
with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp 

690 
from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp 

691 
hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac) 

60526  692 
with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp 
58023  693 
with coprime_dvd_mult[OF ab'(3)] 
694 
have "a' dvd c" by (subst (asm) ac_simps, blast) 

695 
with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac) 

696 
then show ?thesis by blast 

697 
qed 

698 

60433  699 
lemma pow_divs_pow: 
58023  700 
assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0" 
701 
shows "a dvd b" 

702 
proof (cases "gcd a b = 0") 

703 
assume "gcd a b = 0" 

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704 
then show ?thesis by simp 
58023  705 
next 
706 
let ?d = "gcd a b" 

707 
assume "?d \<noteq> 0" 

708 
from n obtain m where m: "n = Suc m" by (cases n, simp_all) 

60526  709 
from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero) 
710 
from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>] 

58023  711 
obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1" 
712 
by blast 

713 
from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n" 

714 
by (simp add: ab'(1,2)[symmetric]) 

715 
hence "?d^n * a'^n dvd ?d^n * b'^n" 

716 
by (simp only: power_mult_distrib ac_simps) 

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717 
with zn have "a'^n dvd b'^n" by simp 
58023  718 
hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m) 
719 
hence "a' dvd b'^m * b'" by (simp add: m ac_simps) 

720 
with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]] 

721 
have "a' dvd b'" by (subst (asm) ac_simps, blast) 

722 
hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp) 

723 
with ab'(1,2) show ?thesis by simp 

724 
qed 

725 

60433  726 
lemma pow_divs_eq [simp]: 
58023  727 
"n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b" 
60433  728 
by (auto intro: pow_divs_pow dvd_power_same) 
58023  729 

60433  730 
lemma divs_mult: 
58023  731 
assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1" 
732 
shows "m * n dvd r" 

733 
proof  

734 
from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'" 

735 
unfolding dvd_def by blast 

736 
from mr n' have "m dvd n'*n" by (simp add: ac_simps) 

737 
hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp 

738 
then obtain k where k: "n' = m*k" unfolding dvd_def by blast 

739 
with n' have "r = m * n * k" by (simp add: mult_ac) 

740 
then show ?thesis unfolding dvd_def by blast 

741 
qed 

742 

743 
lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1" 

744 
by (subst add_commute, simp) 

745 

746 
lemma setprod_coprime [rule_format]: 

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747 
"(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1" 
58023  748 
apply (cases "finite A") 
749 
apply (induct set: finite) 

750 
apply (auto simp add: gcd_mult_cancel) 

751 
done 

752 

753 
lemma coprime_divisors: 

754 
assumes "d dvd a" "e dvd b" "gcd a b = 1" 

755 
shows "gcd d e = 1" 

756 
proof  

757 
from assms obtain k l where "a = d * k" "b = e * l" 

758 
unfolding dvd_def by blast 

759 
with assms have "gcd (d * k) (e * l) = 1" by simp 

760 
hence "gcd (d * k) e = 1" by (rule coprime_lmult) 

761 
also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps) 

762 
finally have "gcd e d = 1" by (rule coprime_lmult) 

763 
then show ?thesis by (simp add: ac_simps) 

764 
qed 

765 

766 
lemma invertible_coprime: 

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767 
assumes "a * b mod m = 1" 
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768 
shows "coprime a m" 
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769 
proof  
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770 
from assms have "coprime m (a * b mod m)" 
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771 
by simp 
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772 
then have "coprime m (a * b)" 
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773 
by simp 
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774 
then have "coprime m a" 
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775 
by (rule coprime_lmult) 
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776 
then show ?thesis 
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777 
by (simp add: ac_simps) 
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778 
qed 
58023  779 

780 
lemma lcm_gcd: 

60438  781 
"lcm a b = a * b div (gcd a b * normalization_factor (a*b))" 
58023  782 
by (simp only: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def) 
783 

784 
lemma lcm_gcd_prod: 

60438  785 
"lcm a b * gcd a b = a * b div normalization_factor (a*b)" 
58023  786 
proof (cases "a * b = 0") 
60438  787 
let ?nf = normalization_factor 
58023  788 
assume "a * b \<noteq> 0" 
58953  789 
hence "gcd a b \<noteq> 0" by simp 
58023  790 
from lcm_gcd have "lcm a b * gcd a b = gcd a b * (a * b div (?nf (a*b) * gcd a b))" 
791 
by (simp add: mult_ac) 

60526  792 
also from \<open>a * b \<noteq> 0\<close> have "... = a * b div ?nf (a*b)" 
60432  793 
by (simp add: div_mult_swap mult.commute) 
58023  794 
finally show ?thesis . 
58953  795 
qed (auto simp add: lcm_gcd) 
58023  796 

797 
lemma lcm_dvd1 [iff]: 

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798 
"a dvd lcm a b" 
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799 
proof (cases "a*b = 0") 
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800 
assume "a * b \<noteq> 0" 
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801 
hence "gcd a b \<noteq> 0" by simp 
60438  802 
let ?c = "1 div normalization_factor (a * b)" 
60526  803 
from \<open>a * b \<noteq> 0\<close> have [simp]: "is_unit (normalization_factor (a * b))" by simp 
60430
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changeset

804 
from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b" 
60432  805 
by (simp add: div_mult_swap unit_div_commute) 
60430
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diff
changeset

806 
hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp 
60526  807 
with \<open>gcd a b \<noteq> 0\<close> have "lcm a b = a * ?c * b div gcd a b" 
58023  808 
by (subst (asm) div_mult_self2_is_id, simp_all) 
60430
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809 
also have "... = a * (?c * b div gcd a b)" 
58023  810 
by (metis div_mult_swap gcd_dvd2 mult_assoc) 
811 
finally show ?thesis by (rule dvdI) 

58953  812 
qed (auto simp add: lcm_gcd) 
58023  813 

814 
lemma lcm_least: 

815 
"\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k" 

816 
proof (cases "k = 0") 

60438  817 
let ?nf = normalization_factor 
58023  818 
assume "k \<noteq> 0" 
819 
hence "is_unit (?nf k)" by simp 

820 
hence "?nf k \<noteq> 0" by (metis not_is_unit_0) 

821 
assume A: "a dvd k" "b dvd k" 

60526  822 
hence "gcd a b \<noteq> 0" using \<open>k \<noteq> 0\<close> by auto 
58023  823 
from A obtain r s where ar: "k = a * r" and bs: "k = b * s" 
824 
unfolding dvd_def by blast 

60526  825 
with \<open>k \<noteq> 0\<close> have "r * s \<noteq> 0" 
58953  826 
by auto (drule sym [of 0], simp) 
58023  827 
hence "is_unit (?nf (r * s))" by simp 
828 
let ?c = "?nf k div ?nf (r*s)" 

60526  829 
from \<open>is_unit (?nf k)\<close> and \<open>is_unit (?nf (r * s))\<close> have "is_unit ?c" by (rule unit_div) 
58023  830 
hence "?c \<noteq> 0" using not_is_unit_0 by fast 
831 
from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)" 

58953  832 
by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps) 
58023  833 
also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)" 
60526  834 
by (subst (3) \<open>k = a * r\<close>, subst (3) \<open>k = b * s\<close>, simp add: algebra_simps) 
835 
also have "... = ?c * r*s * k * gcd a b" using \<open>r * s \<noteq> 0\<close> 

58023  836 
by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps) 
837 
finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b" 

838 
by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac) 

839 
hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)" 

840 
by (simp add: algebra_simps) 

60526  841 
hence "?c * k * gcd a b = a * b * gcd s r" using \<open>r * s \<noteq> 0\<close> 
58023  842 
by (metis div_mult_self2_is_id) 
843 
also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)" 

844 
by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib') 

845 
also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b" 

846 
by (simp add: algebra_simps) 

60526  847 
finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \<open>gcd a b \<noteq> 0\<close> 
58023  848 
by (metis mult.commute div_mult_self2_is_id) 
60526  849 
hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \<open>?c \<noteq> 0\<close> 
58023  850 
by (metis div_mult_self2_is_id mult_assoc) 
60526  851 
also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \<open>is_unit ?c\<close> 
58023  852 
by (simp add: unit_simps) 
853 
finally show ?thesis by (rule dvdI) 

854 
qed simp 

855 

856 
lemma lcm_zero: 

857 
"lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" 

858 
proof  

60438  859 
let ?nf = normalization_factor 
58023  860 
{ 
861 
assume "a \<noteq> 0" "b \<noteq> 0" 

862 
hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors) 

60526  863 
moreover from \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "gcd a b \<noteq> 0" by simp 
58023  864 
ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp) 
865 
} moreover { 

866 
assume "a = 0 \<or> b = 0" 

867 
hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd) 

868 
} 

869 
ultimately show ?thesis by blast 

870 
qed 

871 

872 
lemmas lcm_0_iff = lcm_zero 

873 

874 
lemma gcd_lcm: 

875 
assumes "lcm a b \<noteq> 0" 

60438  876 
shows "gcd a b = a * b div (lcm a b * normalization_factor (a * b))" 
58023  877 
proof 
59009
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generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
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diff
changeset

878 
from assms have "gcd a b \<noteq> 0" by (simp add: lcm_zero) 
60438  879 
let ?c = "normalization_factor (a * b)" 
60526  880 
from \<open>lcm a b \<noteq> 0\<close> have "?c \<noteq> 0" by (intro notI, simp add: lcm_zero no_zero_divisors) 
58023  881 
hence "is_unit ?c" by simp 
882 
from lcm_gcd_prod [of a b] have "gcd a b = a * b div ?c div lcm a b" 

60526  883 
by (subst (2) div_mult_self2_is_id[OF \<open>lcm a b \<noteq> 0\<close>, symmetric], simp add: mult_ac) 
884 
also from \<open>is_unit ?c\<close> have "... = a * b div (lcm a b * ?c)" 

885 
by (metis \<open>?c \<noteq> 0\<close> div_mult_mult1 dvd_mult_div_cancel mult_commute normalization_factor_dvd') 

60433  886 
finally show ?thesis . 
58023  887 
qed 
888 

60438  889 
lemma normalization_factor_lcm [simp]: 
890 
"normalization_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)" 

58023  891 
proof (cases "a = 0 \<or> b = 0") 
892 
case True then show ?thesis 

58953  893 
by (auto simp add: lcm_gcd) 
58023  894 
next 
895 
case False 

60438  896 
let ?nf = normalization_factor 
58023  897 
from lcm_gcd_prod[of a b] 
898 
have "?nf (lcm a b) * ?nf (gcd a b) = ?nf (a*b) div ?nf (a*b)" 

60438  899 
by (metis div_by_0 div_self normalization_correct normalization_factor_0 normalization_factor_mult) 
58023  900 
also have "... = (if a*b = 0 then 0 else 1)" 
58953  901 
by simp 
902 
finally show ?thesis using False by simp 

58023  903 
qed 
904 

60430
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changeset

905 
lemma lcm_dvd2 [iff]: "b dvd lcm a b" 
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906 
using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps) 
58023  907 

908 
lemma lcmI: 

60430
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909 
"\<lbrakk>a dvd k; b dvd k; \<And>l. a dvd l \<Longrightarrow> b dvd l \<Longrightarrow> k dvd l; 
60438  910 
normalization_factor k = (if k = 0 then 0 else 1)\<rbrakk> \<Longrightarrow> k = lcm a b" 
58023  911 
by (intro normed_associated_imp_eq) (auto simp: associated_def intro: lcm_least) 
912 

913 
sublocale lcm!: abel_semigroup lcm 

914 
proof 

60430
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915 
fix a b c 
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diff
changeset

916 
show "lcm (lcm a b) c = lcm a (lcm b c)" 
58023  917 
proof (rule lcmI) 
60430
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918 
have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all 
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919 
then show "a dvd lcm (lcm a b) c" by (rule dvd_trans) 
58023  920 

60430
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changeset

921 
have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all 
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922 
hence "b dvd lcm (lcm a b) c" by (rule dvd_trans) 
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923 
moreover have "c dvd lcm (lcm a b) c" by simp 
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924 
ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least) 
58023  925 

60430
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926 
fix l assume "a dvd l" and "lcm b c dvd l" 
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diff
changeset

927 
have "b dvd lcm b c" by simp 
60526  928 
from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans) 
60430
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changeset

929 
have "c dvd lcm b c" by simp 
60526  930 
from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans) 
931 
from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least) 

932 
from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least) 

58023  933 
qed (simp add: lcm_zero) 
934 
next 

60430
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diff
changeset

935 
fix a b 
ce559c850a27
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parents:
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diff
changeset

936 
show "lcm a b = lcm b a" 
58023  937 
by (simp add: lcm_gcd ac_simps) 
938 
qed 

939 

940 
lemma dvd_lcm_D1: 

941 
"lcm m n dvd k \<Longrightarrow> m dvd k" 

942 
by (rule dvd_trans, rule lcm_dvd1, assumption) 

943 

944 
lemma dvd_lcm_D2: 

945 
"lcm m n dvd k \<Longrightarrow> n dvd k" 

946 
by (rule dvd_trans, rule lcm_dvd2, assumption) 

947 

948 
lemma gcd_dvd_lcm [simp]: 

949 
"gcd a b dvd lcm a b" 

950 
by (metis dvd_trans gcd_dvd2 lcm_dvd2) 

951 

952 
lemma lcm_1_iff: 

953 
"lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b" 

954 
proof 

955 
assume "lcm a b = 1" 

59061  956 
then show "is_unit a \<and> is_unit b" by auto 
58023  957 
next 
958 
assume "is_unit a \<and> is_unit b" 

59061  959 
hence "a dvd 1" and "b dvd 1" by simp_all 
960 
hence "is_unit (lcm a b)" by (rule lcm_least) 

60438  961 
hence "lcm a b = normalization_factor (lcm a b)" 
962 
by (subst normalization_factor_unit, simp_all) 

60526  963 
also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close> 
59061  964 
by auto 
58023  965 
finally show "lcm a b = 1" . 
966 
qed 

967 

968 
lemma lcm_0_left [simp]: 

60430
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diff
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969 
"lcm 0 a = 0" 
58023  970 
by (rule sym, rule lcmI, simp_all) 
971 

972 
lemma lcm_0 [simp]: 

60430
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haftmann
parents:
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diff
changeset

973 
"lcm a 0 = 0" 
58023  974 
by (rule sym, rule lcmI, simp_all) 
975 

976 
lemma lcm_unique: 

977 
"a dvd d \<and> b dvd d \<and> 

60438  978 
normalization_factor d = (if d = 0 then 0 else 1) \<and> 
58023  979 
(\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b" 
980 
by (rule, auto intro: lcmI simp: lcm_least lcm_zero) 

981 

982 
lemma dvd_lcm_I1 [simp]: 

983 
"k dvd m \<Longrightarrow> k dvd lcm m n" 

984 
by (metis lcm_dvd1 dvd_trans) 

985 

986 
lemma dvd_lcm_I2 [simp]: 

987 
"k dvd n \<Longrightarrow> k dvd lcm m n" 

988 
by (metis lcm_dvd2 dvd_trans) 

989 

990 
lemma lcm_1_left [simp]: 

60438  991 
"lcm 1 a = a div normalization_factor a" 
60430
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diff
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992 
by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all) 
58023  993 

994 
lemma lcm_1_right [simp]: 

60438  995 
"lcm a 1 = a div normalization_factor a" 
60430
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diff
changeset

996 
using lcm_1_left [of a] by (simp add: ac_simps) 
58023  997 

998 
lemma lcm_coprime: 

60438  999 
"gcd a b = 1 \<Longrightarrow> lcm a b = a * b div normalization_factor (a*b)" 
58023  1000 
by (subst lcm_gcd) simp 
1001 

1002 
lemma lcm_proj1_if_dvd: 

60438  1003 
"b dvd a \<Longrightarrow> lcm a b = a div normalization_factor a" 
60430
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1004 
by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all) 
58023  1005 

1006 
lemma lcm_proj2_if_dvd: 

60438  1007 
"a dvd b \<Longrightarrow> lcm a b = b div normalization_factor b" 
60430
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diff
changeset

1008 
using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps) 
58023  1009 

1010 
lemma lcm_proj1_iff: 

60438  1011 
"lcm m n = m div normalization_factor m \<longleftrightarrow> n dvd m" 
58023  1012 
proof 
60438  1013 
assume A: "lcm m n = m div normalization_factor m" 
58023  1014 
show "n dvd m" 
1015 
proof (cases "m = 0") 

1016 
assume [simp]: "m \<noteq> 0" 

60438  1017 
from A have B: "m = lcm m n * normalization_factor m" 
58023  1018 
by (simp add: unit_eq_div2) 
1019 
show ?thesis by (subst B, simp) 

1020 
qed simp 

1021 
next 

1022 
assume "n dvd m" 

60438  1023 
then show "lcm m n = m div normalization_factor m" by (rule lcm_proj1_if_dvd) 
58023  1024 
qed 
1025 

1026 
lemma lcm_proj2_iff: 

60438  1027 
"lcm m n = n div normalization_factor n \<longleftrightarrow> m dvd n" 
58023  1028 
using lcm_proj1_iff [of n m] by (simp add: ac_simps) 
1029 

1030 
lemma euclidean_size_lcm_le1: 

1031 
assumes "a \<noteq> 0" and "b \<noteq> 0" 

1032 
shows "euclidean_size a \<le> euclidean_size (lcm a b)" 

1033 
proof  

1034 
have "a dvd lcm a b" by (rule lcm_dvd1) 

1035 
then obtain c where A: "lcm a b = a * c" unfolding dvd_def by blast 

60526  1036 
with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero) 
58023  1037 
then show ?thesis by (subst A, intro size_mult_mono) 
1038 
qed 

1039 

1040 
lemma euclidean_size_lcm_le2: 

1041 
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)" 

1042 
using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps) 

1043 

1044 
lemma euclidean_size_lcm_less1: 

1045 
assumes "b \<noteq> 0" and "\<not>b dvd a" 

1046 
shows "euclidean_size a < euclidean_size (lcm a b)" 

1047 
proof (rule ccontr) 

1048 
from assms have "a \<noteq> 0" by auto 

1049 
assume "\<not>euclidean_size a < euclidean_size (lcm a b)" 

60526  1050 
with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a" 
58023  1051 
by (intro le_antisym, simp, intro euclidean_size_lcm_le1) 
1052 
with assms have "lcm a b dvd a" 

1053 
by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero) 

1054 
hence "b dvd a" by (rule dvd_lcm_D2) 

60526  1055 
with \<open>\<not>b dvd a\<close> show False by contradiction 
58023  1056 
qed 
1057 

1058 
lemma euclidean_size_lcm_less2: 

1059 
assumes "a \<noteq> 0" and "\<not>a dvd b" 

1060 
shows "euclidean_size b < euclidean_size (lcm a b)" 

1061 
using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps) 

1062 

1063 
lemma lcm_mult_unit1: 

60430
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diff
changeset

1064 
"is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c" 
58023  1065 
apply (rule lcmI) 
60430
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diff
changeset

1066 
apply (rule dvd_trans[of _ "b * a"], simp, rule lcm_dvd1) 
58023  1067 
apply (rule lcm_dvd2) 
1068 
apply (rule lcm_least, simp add: unit_simps, assumption) 

60438  1069 
apply (subst normalization_factor_lcm, simp add: lcm_zero) 
58023  1070 
done 
1071 

1072 
lemma lcm_mult_unit2: 

60430
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haftmann
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diff
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1073 
"is_unit a \<Longrightarrow> lcm b (c * a) = lcm b c" 
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diff
changeset

1074 
using lcm_mult_unit1 [of a c b] by (simp add: ac_simps) 
58023  1075 

1076 
lemma lcm_div_unit1: 

60430
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haftmann
parents:
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diff
changeset

1077 
"is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c" 
60433  1078 
by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1) 
58023  1079 

1080 
lemma lcm_div_unit2: 

60430
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haftmann
parents:
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diff
changeset

1081 
"is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c" 
60433  1082 
by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2) 
58023  1083 

1084 
lemma lcm_left_idem: 

60430
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haftmann
parents:
59061
diff
changeset

1085 
"lcm a (lcm a b) = lcm a b" 
58023  1086 
apply (rule lcmI) 
1087 
apply simp 

1088 
apply (subst lcm.assoc [symmetric], rule lcm_dvd2) 

1089 
apply (rule lcm_least, assumption) 

1090 
apply (erule (1) lcm_least) 

1091 
apply (auto simp: lcm_zero) 

1092 
done 

1093 

1094 
lemma lcm_right_idem: 

60430
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haftmann
parents:
59061
diff
changeset

1095 
"lcm (lcm a b) b = lcm a b" 
58023  1096 
apply (rule lcmI) 
1097 
apply (subst lcm.assoc, rule lcm_dvd1) 

1098 
apply (rule lcm_dvd2) 

1099 
apply (rule lcm_least, erule (1) lcm_least, assumption) 

1100 
apply (auto simp: lcm_zero) 

1101 
done 

1102 

1103 
lemma comp_fun_idem_lcm: "comp_fun_idem lcm" 

1104 
proof 

1105 
fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a" 

1106 
by (simp add: fun_eq_iff ac_simps) 

1107 
next 

1108 
fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def 

1109 
by (intro ext, simp add: lcm_left_idem) 

1110 
qed 

1111 

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

1112 
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

1113 
and Lcm_dvd [simp]: "(\<forall>a\<in>A. a dvd l') \<Longrightarrow> Lcm A dvd l'" 
60438  1114 
and normalization_factor_Lcm [simp]: 
1115 
"normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" 

58023  1116 
proof  
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1117 
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  1118 
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

1119 
proof (cases "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)") 
58023  1120 
case False 
1121 
hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def) 

1122 
with False show ?thesis by auto 

1123 
next 

1124 
case True 

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

1125 
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

1126 
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

1127 
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

1128 
have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" 
58023  1129 
apply (subst n_def) 
1130 
apply (rule LeastI[of _ "euclidean_size l\<^sub>0"]) 

1131 
apply (rule exI[of _ l\<^sub>0]) 

1132 
apply (simp add: l\<^sub>0_props) 

1133 
done 

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

1134 
from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n" 
58023  1135 
unfolding l_def by simp_all 
1136 
{ 

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

1137 
fix l' assume "\<forall>a\<in>A. a dvd l'" 
60526  1138 
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) 
1139 
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

1140 
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  1141 
by (intro exI[of _ "gcd l l'"], auto) 
1142 
hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le) 

1143 
moreover have "euclidean_size (gcd l l') \<le> n" 

1144 
proof  

1145 
have "gcd l l' dvd l" by simp 

1146 
then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast 

60526  1147 
with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto 
58023  1148 
hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)" 
1149 
by (rule size_mult_mono) 

60526  1150 
also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> .. 
1151 
also note \<open>euclidean_size l = n\<close> 

58023  1152 
finally show "euclidean_size (gcd l l') \<le> n" . 
1153 
qed 

1154 
ultimately have "euclidean_size l = euclidean_size (gcd l l')" 

60526  1155 
by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>) 
1156 
with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd) 

58023  1157 
hence "l dvd l'" by (blast dest: dvd_gcd_D2) 
1158 
} 

1159 

60526  1160 
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  1161 
have "(\<forall>a\<in>A. a dvd l div normalization_factor l) \<and> 
1162 
(\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> l div normalization_factor l dvd l') \<and> 

1163 
normalization_factor (l div normalization_factor l) = 

1164 
(if l div normalization_factor l = 0 then 0 else 1)" 

58023  1165 
by (auto simp: unit_simps) 
60438  1166 
also from True have "l div normalization_factor l = Lcm A" 
58023  1167 
by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def) 
1168 
finally show ?thesis . 

1169 
qed 

1170 
note A = this 

1171 

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

1172 
{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

1173 
{fix l' assume "\<forall>a\<in>A. a dvd l'" then show "Lcm A dvd l'" using A by blast} 
60438  1174 
from A show "normalization_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast 
58023  1175 
qed 
1176 

1177 
lemma LcmI: 

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

1178 
"(\<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  1179 
normalization_factor l = (if l = 0 then 0 else 1) \<Longrightarrow> l = Lcm A" 
58023  1180 
by (intro normed_associated_imp_eq) 
1181 
(auto intro: Lcm_dvd dvd_Lcm simp: associated_def) 

1182 

1183 
lemma Lcm_subset: 

1184 
"A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B" 

1185 
by (blast intro: Lcm_dvd dvd_Lcm) 

1186 

1187 
lemma Lcm_Un: 

1188 
"Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)" 

1189 
apply (rule lcmI) 

1190 
apply (blast intro: Lcm_subset) 

1191 
apply (blast intro: Lcm_subset) 

1192 
apply (intro Lcm_dvd ballI, elim UnE) 

1193 
apply (rule dvd_trans, erule dvd_Lcm, assumption) 

1194 
apply (rule dvd_trans, erule dvd_Lcm, assumption) 

1195 
apply simp 

1196 
done 

1197 

1198 
lemma Lcm_1_iff: 

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

1199 
"Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)" 
58023  1200 
proof 
1201 
assume "Lcm A = 1" 

60430
ce559c850a27
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haftmann
parents:
59061
diff
changeset

1202 
then show "\<forall>a\<in>A. is_unit a" by auto 
58023  1203 
qed (rule LcmI [symmetric], auto) 
1204 

1205 
lemma Lcm_no_units: 

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

1206 
"Lcm A = Lcm (A  {a. is_unit a})" 
58023  1207 
proof  
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1208 
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

1209 
hence "Lcm A = lcm (Lcm (A  {a. is_unit a})) (Lcm {a\<in>A. is_unit a})" 
58023  1210 
by (simp add: Lcm_Un[symmetric]) 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1211 
also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff) 
58023  1212 
finally show ?thesis by simp 
1213 
qed 

1214 

1215 
lemma Lcm_empty [simp]: 

1216 
"Lcm {} = 1" 

1217 
by (simp add: Lcm_1_iff) 

1218 

1219 
lemma Lcm_eq_0 [simp]: 

1220 
"0 \<in> A \<Longrightarrow> Lcm A = 0" 

1221 
by (drule dvd_Lcm) simp 

1222 

1223 
lemma Lcm0_iff': 

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

1224 
"Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" 
58023  1225 
proof 
1226 
assume "Lcm A = 0" 

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

1227 
show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" 
58023  1228 
proof 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1229 
assume ex: "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1230 
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

1231 
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

1232 
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

1233 
have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" 
58023  1234 
apply (subst n_def) 
1235 
apply (rule LeastI[of _ "euclidean_size l\<^sub>0"]) 

1236 
apply (rule exI[of _ l\<^sub>0]) 

1237 
apply (simp add: l\<^sub>0_props) 

1238 
done 

1239 
from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all 

60438  1240 
hence "l div normalization_factor l \<noteq> 0" by simp 
1241 
also from ex have "l div normalization_factor l = Lcm A" 

58023  1242 
by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def) 
60526  1243 
finally show False using \<open>Lcm A = 0\<close> by contradiction 
58023  1244 
qed 
1245 
qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False) 

1246 
