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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 Main "~~/src/HOL/GCD" "~~/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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\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 + normalization_semidom + 
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fixes euclidean_size :: "'a \<Rightarrow> nat" 
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assumes mod_size_less: 
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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 \<le> euclidean_size (a * b)" 
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begin 
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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 (auto simp add: mod_size_less) 
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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 (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 

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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 normalize a 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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then have "P b (a mod b)" 
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by (rule "1.hyps") 
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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" 

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where 

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"lcm_eucl a b = normalize (a * b) div gcd_eucl a b" 
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definition Lcm_eucl :: "'a set \<Rightarrow> 'a"  \<open> 
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Somewhat complicated definition of Lcm that has the advantage of working 
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for infinite sets as well\<close> 
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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 normalize l 
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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}" 

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lemma gcd_eucl_0: 
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"gcd_eucl a 0 = normalize a" 
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by (simp add: gcd_eucl.simps [of a 0]) 
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lemma gcd_eucl_0_left: 
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"gcd_eucl 0 a = normalize a" 
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by (simp_all add: gcd_eucl_0 gcd_eucl.simps [of 0 a]) 
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lemma gcd_eucl_non_0: 
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"b \<noteq> 0 \<Longrightarrow> gcd_eucl a b = gcd_eucl b (a mod b)" 
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by (simp add: gcd_eucl.simps [of a b] gcd_eucl.simps [of b 0]) 
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end 
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class euclidean_ring = euclidean_semiring + idom 
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begin 
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function euclid_ext :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" where 
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"euclid_ext a b = 
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(if b = 0 then 
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(1 div unit_factor a, 0, normalize a) 
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else 
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case euclid_ext b (a mod b) of 
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(s, t, c) \<Rightarrow> (t, s  t * (a div b), c))" 
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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 euclid_ext.simps [simp del] 
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lemma euclid_ext_0: 
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"euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)" 
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by (simp add: euclid_ext.simps [of a 0]) 
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lemma euclid_ext_left_0: 
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"euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)" 
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by (simp add: euclid_ext_0 euclid_ext.simps [of 0 a]) 
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lemma euclid_ext_non_0: 
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"b \<noteq> 0 \<Longrightarrow> euclid_ext a b = (case euclid_ext b (a mod b) of 
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(s, t, c) \<Rightarrow> (t, s  t * (a div b), c))" 
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by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0]) 
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lemma euclid_ext_code [code]: 
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"euclid_ext a b = (if b = 0 then (1 div unit_factor a, 0, normalize a) 
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else let (s, t, c) = euclid_ext b (a mod b) in (t, s  t * (a div b), c))" 
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by (simp add: euclid_ext.simps [of a b] euclid_ext.simps [of b 0]) 
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lemma euclid_ext_correct: 
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"case euclid_ext a b of (s, t, c) \<Rightarrow> s * a + t * b = c" 
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proof (induct a b rule: gcd_eucl_induct) 
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case (zero a) then show ?case 
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by (simp add: euclid_ext_0 ac_simps) 
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next 
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case (mod a b) 
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obtain s t c where stc: "euclid_ext b (a mod b) = (s,t,c)" 
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by (cases "euclid_ext b (a mod b)") blast 
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with mod have "c = s * b + t * (a mod b)" by simp 
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also have "... = t * ((a div b) * b + a mod b) + (s  t * (a div b)) * b" 
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by (simp add: algebra_simps) 
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also have "(a div b) * b + a mod b = a" using mod_div_equality . 
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finally show ?case 
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by (subst euclid_ext.simps) (simp add: stc mod ac_simps) 
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qed 
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definition euclid_ext' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a" 
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where 
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"euclid_ext' a b = (case euclid_ext a b of (s, t, _) \<Rightarrow> (s, t))" 
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lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)" 
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by (simp add: euclid_ext'_def euclid_ext_0) 
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lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)" 
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by (simp add: euclid_ext'_def euclid_ext_left_0) 
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lemma euclid_ext'_non_0: "b \<noteq> 0 \<Longrightarrow> euclid_ext' a b = (snd (euclid_ext' b (a mod b)), 
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fst (euclid_ext' b (a mod b))  snd (euclid_ext' b (a mod b)) * (a div b))" 
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by (simp add: euclid_ext'_def euclid_ext_non_0 split_def) 
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end 
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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 

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lemma gcd_0_left: 

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"gcd 0 a = normalize a" 
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unfolding gcd_gcd_eucl by (fact gcd_eucl_0_left) 
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lemma gcd_0: 

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"gcd a 0 = normalize a" 
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193 
unfolding gcd_gcd_eucl by (fact gcd_eucl_0) 
58023  194 

195 
lemma gcd_non_0: 

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"b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)" 
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197 
unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0) 
58023  198 

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199 
lemma gcd_dvd1 [iff]: "gcd a b dvd a" 
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200 
and gcd_dvd2 [iff]: "gcd a b dvd b" 
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201 
by (induct a b rule: gcd_eucl_induct) 
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202 
(simp_all add: gcd_0 gcd_non_0 dvd_mod_iff) 
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203 

58023  204 
lemma dvd_gcd_D1: "k dvd gcd m n \<Longrightarrow> k dvd m" 
205 
by (rule dvd_trans, assumption, rule gcd_dvd1) 

206 

207 
lemma dvd_gcd_D2: "k dvd gcd m n \<Longrightarrow> k dvd n" 

208 
by (rule dvd_trans, assumption, rule gcd_dvd2) 

209 

210 
lemma gcd_greatest: 

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211 
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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214 
case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_0) 
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215 
next 
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216 
case (mod a b) 
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217 
then show ?case 
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by (simp add: gcd_non_0 dvd_mod_iff) 
58023  219 
qed 
220 

221 
lemma dvd_gcd_iff: 

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"k dvd gcd a b \<longleftrightarrow> k dvd a \<and> k dvd b" 
58023  223 
by (blast intro!: gcd_greatest intro: dvd_trans) 
224 

225 
lemmas gcd_greatest_iff = dvd_gcd_iff 

226 

227 
lemma gcd_zero [simp]: 

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228 
"gcd a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" 
58023  229 
by (metis dvd_0_left dvd_refl gcd_dvd1 gcd_dvd2 gcd_greatest)+ 
230 

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231 
lemma normalize_gcd [simp]: 
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"normalize (gcd a b) = gcd a b" 
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233 
by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_0 gcd_non_0) 
58023  234 

235 
lemma gcdI: 

60634  236 
assumes "c dvd a" and "c dvd b" and greatest: "\<And>d. d dvd a \<Longrightarrow> d dvd b \<Longrightarrow> d dvd c" 
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and "normalize c = c" 
60634  238 
shows "c = gcd a b" 
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239 
by (rule associated_eqI) (auto simp: assms intro: gcd_greatest) 
58023  240 

241 
sublocale gcd!: abel_semigroup gcd 

242 
proof 

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243 
fix a b c 
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244 
show "gcd (gcd a b) c = gcd a (gcd b c)" 
58023  245 
proof (rule gcdI) 
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246 
have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd a" by simp_all 
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247 
then show "gcd (gcd a b) c dvd a" by (rule dvd_trans) 
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248 
have "gcd (gcd a b) c dvd gcd a b" "gcd a b dvd b" by simp_all 
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249 
hence "gcd (gcd a b) c dvd b" by (rule dvd_trans) 
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250 
moreover have "gcd (gcd a b) c dvd c" by simp 
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251 
ultimately show "gcd (gcd a b) c dvd gcd b c" 
58023  252 
by (rule gcd_greatest) 
60688
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253 
show "normalize (gcd (gcd a b) c) = gcd (gcd a b) c" 
58023  254 
by auto 
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255 
fix l assume "l dvd a" and "l dvd gcd b c" 
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256 
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  258 
with \<open>l dvd a\<close> show "l dvd gcd (gcd a b) c" 
58023  259 
by (intro gcd_greatest) 
260 
qed 

261 
next 

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262 
fix a b 
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263 
show "gcd a b = gcd b a" 
58023  264 
by (rule gcdI) (simp_all add: gcd_greatest) 
265 
qed 

266 

267 
lemma gcd_unique: "d dvd a \<and> d dvd b \<and> 

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normalize d = d \<and> 
58023  269 
(\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b" 
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by rule (auto intro: gcdI simp: gcd_greatest) 
58023  271 

272 
lemma gcd_dvd_prod: "gcd a b dvd k * b" 

273 
using mult_dvd_mono [of 1] by auto 

274 

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lemma gcd_1_left [simp]: "gcd 1 a = 1" 
58023  276 
by (rule sym, rule gcdI, simp_all) 
277 

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278 
lemma gcd_1 [simp]: "gcd a 1 = 1" 
58023  279 
by (rule sym, rule gcdI, simp_all) 
280 

281 
lemma gcd_proj2_if_dvd: 

60634  282 
"b dvd a \<Longrightarrow> gcd a b = normalize b" 
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283 
by (cases "b = 0", simp_all add: dvd_eq_mod_eq_0 gcd_non_0 gcd_0) 
58023  284 

285 
lemma gcd_proj1_if_dvd: 

60634  286 
"a dvd b \<Longrightarrow> gcd a b = normalize a" 
58023  287 
by (subst gcd.commute, simp add: gcd_proj2_if_dvd) 
288 

60634  289 
lemma gcd_proj1_iff: "gcd m n = normalize m \<longleftrightarrow> m dvd n" 
58023  290 
proof 
60634  291 
assume A: "gcd m n = normalize m" 
58023  292 
show "m dvd n" 
293 
proof (cases "m = 0") 

294 
assume [simp]: "m \<noteq> 0" 

60634  295 
from A have B: "m = gcd m n * unit_factor m" 
58023  296 
by (simp add: unit_eq_div2) 
297 
show ?thesis by (subst B, simp add: mult_unit_dvd_iff) 

298 
qed (insert A, simp) 

299 
next 

300 
assume "m dvd n" 

60634  301 
then show "gcd m n = normalize m" by (rule gcd_proj1_if_dvd) 
58023  302 
qed 
303 

60634  304 
lemma gcd_proj2_iff: "gcd m n = normalize n \<longleftrightarrow> n dvd m" 
305 
using gcd_proj1_iff [of n m] by (simp add: ac_simps) 

58023  306 

307 
lemma gcd_mod1 [simp]: 

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308 
"gcd (a mod b) b = gcd a b" 
58023  309 
by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff) 
310 

311 
lemma gcd_mod2 [simp]: 

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312 
"gcd a (b mod a) = gcd a b" 
58023  313 
by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff) 
314 

315 
lemma gcd_mult_distrib': 

60634  316 
"normalize c * gcd a b = gcd (c * a) (c * b)" 
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317 
proof (cases "c = 0") 
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318 
case True then show ?thesis by (simp_all add: gcd_0) 
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319 
next 
60634  320 
case False then have [simp]: "is_unit (unit_factor c)" by simp 
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321 
show ?thesis 
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322 
proof (induct a b rule: gcd_eucl_induct) 
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323 
case (zero a) show ?case 
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324 
proof (cases "a = 0") 
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325 
case True then show ?thesis by (simp add: gcd_0) 
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326 
next 
60634  327 
case False 
328 
then show ?thesis by (simp add: gcd_0 normalize_mult) 

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329 
qed 
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330 
case (mod a b) 
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331 
then show ?case by (simp add: mult_mod_right gcd.commute) 
58023  332 
qed 
333 
qed 

334 

335 
lemma gcd_mult_distrib: 

60634  336 
"k * gcd a b = gcd (k * a) (k * b) * unit_factor k" 
58023  337 
proof 
60634  338 
have "normalize k * gcd a b = gcd (k * a) (k * b)" 
339 
by (simp add: gcd_mult_distrib') 

340 
then have "normalize k * gcd a b * unit_factor k = gcd (k * a) (k * b) * unit_factor k" 

341 
by simp 

342 
then have "normalize k * unit_factor k * gcd a b = gcd (k * a) (k * b) * unit_factor k" 

343 
by (simp only: ac_simps) 

344 
then show ?thesis 

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345 
by simp 
58023  346 
qed 
347 

348 
lemma euclidean_size_gcd_le1 [simp]: 

349 
assumes "a \<noteq> 0" 

350 
shows "euclidean_size (gcd a b) \<le> euclidean_size a" 

351 
proof  

352 
have "gcd a b dvd a" by (rule gcd_dvd1) 

353 
then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast 

60526  354 
with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto 
58023  355 
qed 
356 

357 
lemma euclidean_size_gcd_le2 [simp]: 

358 
"b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b" 

359 
by (subst gcd.commute, rule euclidean_size_gcd_le1) 

360 

361 
lemma euclidean_size_gcd_less1: 

362 
assumes "a \<noteq> 0" and "\<not>a dvd b" 

363 
shows "euclidean_size (gcd a b) < euclidean_size a" 

364 
proof (rule ccontr) 

365 
assume "\<not>euclidean_size (gcd a b) < euclidean_size a" 

60526  366 
with \<open>a \<noteq> 0\<close> have "euclidean_size (gcd a b) = euclidean_size a" 
58023  367 
by (intro le_antisym, simp_all) 
368 
with assms have "a dvd gcd a b" by (auto intro: dvd_euclidean_size_eq_imp_dvd) 

369 
hence "a dvd b" using dvd_gcd_D2 by blast 

60526  370 
with \<open>\<not>a dvd b\<close> show False by contradiction 
58023  371 
qed 
372 

373 
lemma euclidean_size_gcd_less2: 

374 
assumes "b \<noteq> 0" and "\<not>b dvd a" 

375 
shows "euclidean_size (gcd a b) < euclidean_size b" 

376 
using assms by (subst gcd.commute, rule euclidean_size_gcd_less1) 

377 

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lemma gcd_mult_unit1: "is_unit a \<Longrightarrow> gcd (b * a) c = gcd b c" 
58023  379 
apply (rule gcdI) 
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380 
apply simp_all 
58023  381 
apply (rule dvd_trans, rule gcd_dvd1, simp add: unit_simps) 
382 
apply (rule gcd_greatest, simp add: unit_simps, assumption) 

383 
done 

384 

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385 
lemma gcd_mult_unit2: "is_unit a \<Longrightarrow> gcd b (c * a) = gcd b c" 
58023  386 
by (subst gcd.commute, subst gcd_mult_unit1, assumption, rule gcd.commute) 
387 

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388 
lemma gcd_div_unit1: "is_unit a \<Longrightarrow> gcd (b div a) c = gcd b c" 
60433  389 
by (erule is_unitE [of _ b]) (simp add: gcd_mult_unit1) 
58023  390 

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391 
lemma gcd_div_unit2: "is_unit a \<Longrightarrow> gcd b (c div a) = gcd b c" 
60433  392 
by (erule is_unitE [of _ c]) (simp add: gcd_mult_unit2) 
58023  393 

60634  394 
lemma normalize_gcd_left [simp]: 
395 
"gcd (normalize a) b = gcd a b" 

396 
proof (cases "a = 0") 

397 
case True then show ?thesis 

398 
by simp 

399 
next 

400 
case False then have "is_unit (unit_factor a)" 

401 
by simp 

402 
moreover have "normalize a = a div unit_factor a" 

403 
by simp 

404 
ultimately show ?thesis 

405 
by (simp only: gcd_div_unit1) 

406 
qed 

407 

408 
lemma normalize_gcd_right [simp]: 

409 
"gcd a (normalize b) = gcd a b" 

410 
using normalize_gcd_left [of b a] by (simp add: ac_simps) 

411 

412 
lemma gcd_idem: "gcd a a = normalize a" 

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413 
by (cases "a = 0") (simp add: gcd_0_left, rule sym, rule gcdI, simp_all) 
58023  414 

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415 
lemma gcd_right_idem: "gcd (gcd a b) b = gcd a b" 
58023  416 
apply (rule gcdI) 
417 
apply (simp add: ac_simps) 

418 
apply (rule gcd_dvd2) 

419 
apply (rule gcd_greatest, erule (1) gcd_greatest, assumption) 

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420 
apply simp 
58023  421 
done 
422 

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423 
lemma gcd_left_idem: "gcd a (gcd a b) = gcd a b" 
58023  424 
apply (rule gcdI) 
425 
apply simp 

426 
apply (rule dvd_trans, rule gcd_dvd2, rule gcd_dvd2) 

427 
apply (rule gcd_greatest, assumption, erule gcd_greatest, assumption) 

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428 
apply simp 
58023  429 
done 
430 

431 
lemma comp_fun_idem_gcd: "comp_fun_idem gcd" 

432 
proof 

433 
fix a b show "gcd a \<circ> gcd b = gcd b \<circ> gcd a" 

434 
by (simp add: fun_eq_iff ac_simps) 

435 
next 

436 
fix a show "gcd a \<circ> gcd a = gcd a" 

437 
by (simp add: fun_eq_iff gcd_left_idem) 

438 
qed 

439 

440 
lemma coprime_dvd_mult: 

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441 
assumes "gcd c b = 1" and "c dvd a * b" 
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442 
shows "c dvd a" 
58023  443 
proof  
60634  444 
let ?nf = "unit_factor" 
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445 
from assms gcd_mult_distrib [of a c b] 
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446 
have A: "a = gcd (a * c) (a * b) * ?nf a" by simp 
60526  447 
from \<open>c dvd a * b\<close> show ?thesis by (subst A, simp_all add: gcd_greatest) 
58023  448 
qed 
449 

450 
lemma coprime_dvd_mult_iff: 

60430
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451 
"gcd c b = 1 \<Longrightarrow> (c dvd a * b) = (c dvd a)" 
58023  452 
by (rule, rule coprime_dvd_mult, simp_all) 
453 

454 
lemma gcd_dvd_antisym: 

455 
"gcd a b dvd gcd c d \<Longrightarrow> gcd c d dvd gcd a b \<Longrightarrow> gcd a b = gcd c d" 

456 
proof (rule gcdI) 

457 
assume A: "gcd a b dvd gcd c d" and B: "gcd c d dvd gcd a b" 

458 
have "gcd c d dvd c" by simp 

459 
with A show "gcd a b dvd c" by (rule dvd_trans) 

460 
have "gcd c d dvd d" by simp 

461 
with A show "gcd a b dvd d" by (rule dvd_trans) 

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462 
show "normalize (gcd a b) = gcd a b" 
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changeset

463 
by simp 
58023  464 
fix l assume "l dvd c" and "l dvd d" 
465 
hence "l dvd gcd c d" by (rule gcd_greatest) 

466 
from this and B show "l dvd gcd a b" by (rule dvd_trans) 

467 
qed 

468 

469 
lemma gcd_mult_cancel: 

470 
assumes "gcd k n = 1" 

471 
shows "gcd (k * m) n = gcd m n" 

472 
proof (rule gcd_dvd_antisym) 

473 
have "gcd (gcd (k * m) n) k = gcd (gcd k n) (k * m)" by (simp add: ac_simps) 

60526  474 
also note \<open>gcd k n = 1\<close> 
58023  475 
finally have "gcd (gcd (k * m) n) k = 1" by simp 
476 
hence "gcd (k * m) n dvd m" by (rule coprime_dvd_mult, simp add: ac_simps) 

477 
moreover have "gcd (k * m) n dvd n" by simp 

478 
ultimately show "gcd (k * m) n dvd gcd m n" by (rule gcd_greatest) 

479 
have "gcd m n dvd (k * m)" and "gcd m n dvd n" by simp_all 

480 
then show "gcd m n dvd gcd (k * m) n" by (rule gcd_greatest) 

481 
qed 

482 

483 
lemma coprime_crossproduct: 

484 
assumes [simp]: "gcd a d = 1" "gcd b c = 1" 

60688
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485 
shows "normalize (a * c) = normalize (b * d) \<longleftrightarrow> normalize a = normalize b \<and> normalize c = normalize d" 
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486 
(is "?lhs \<longleftrightarrow> ?rhs") 
58023  487 
proof 
60688
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488 
assume ?rhs 
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489 
then have "a dvd b" "b dvd a" "c dvd d" "d dvd c" by (simp_all add: associated_iff_dvd) 
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490 
then have "a * c dvd b * d" "b * d dvd a * c" by (fast intro: mult_dvd_mono)+ 
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491 
then show ?lhs by (simp add: associated_iff_dvd) 
58023  492 
next 
493 
assume ?lhs 

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494 
then have dvd: "a * c dvd b * d" "b * d dvd a * c" by (simp_all add: associated_iff_dvd) 
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495 
then have "a dvd b * d" by (metis dvd_mult_left) 
58023  496 
hence "a dvd b" by (simp add: coprime_dvd_mult_iff) 
60688
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497 
moreover from dvd have "b dvd a * c" by (metis dvd_mult_left) 
58023  498 
hence "b dvd a" by (simp add: coprime_dvd_mult_iff) 
60688
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499 
moreover from dvd have "c dvd d * b" 
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500 
by (auto dest: dvd_mult_right simp add: ac_simps) 
58023  501 
hence "c dvd d" by (simp add: coprime_dvd_mult_iff gcd.commute) 
60688
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502 
moreover from dvd have "d dvd c * a" 
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503 
by (auto dest: dvd_mult_right simp add: ac_simps) 
58023  504 
hence "d dvd c" by (simp add: coprime_dvd_mult_iff gcd.commute) 
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505 
ultimately show ?rhs by (simp add: associated_iff_dvd) 
58023  506 
qed 
507 

508 
lemma gcd_add1 [simp]: 

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

510 
by (cases "n = 0", simp_all add: gcd_non_0) 

511 

512 
lemma gcd_add2 [simp]: 

513 
"gcd m (m + n) = gcd m n" 

514 
using gcd_add1 [of n m] by (simp add: ac_simps) 

515 

60572
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516 
lemma gcd_add_mult: 
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517 
"gcd m (k * m + n) = gcd m n" 
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518 
proof  
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519 
have "gcd m ((k * m + n) mod m) = gcd m (k * m + n)" 
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520 
by (fact gcd_mod2) 
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521 
then show ?thesis by simp 
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522 
qed 
58023  523 

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524 
lemma coprimeI: "(\<And>l. \<lbrakk>l dvd a; l dvd b\<rbrakk> \<Longrightarrow> l dvd 1) \<Longrightarrow> gcd a b = 1" 
58023  525 
by (rule sym, rule gcdI, simp_all) 
526 

527 
lemma coprime: "gcd a b = 1 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> is_unit d)" 

59061  528 
by (auto intro: coprimeI gcd_greatest dvd_gcd_D1 dvd_gcd_D2) 
58023  529 

530 
lemma div_gcd_coprime: 

531 
assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0" 

532 
defines [simp]: "d \<equiv> gcd a b" 

533 
defines [simp]: "a' \<equiv> a div d" and [simp]: "b' \<equiv> b div d" 

534 
shows "gcd a' b' = 1" 

535 
proof (rule coprimeI) 

536 
fix l assume "l dvd a'" "l dvd b'" 

537 
then obtain s t where "a' = l * s" "b' = l * t" unfolding dvd_def by blast 

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538 
moreover have "a = a' * d" "b = b' * d" by simp_all 
58023  539 
ultimately have "a = (l * d) * s" "b = (l * d) * t" 
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changeset

540 
by (simp_all only: ac_simps) 
58023  541 
hence "l*d dvd a" and "l*d dvd b" by (simp_all only: dvd_triv_left) 
542 
hence "l*d dvd d" by (simp add: gcd_greatest) 

59009
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changeset

543 
then obtain u where "d = l * d * u" .. 
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changeset

544 
then have "d * (l * u) = d" by (simp add: ac_simps) 
348561aa3869
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changeset

545 
moreover from nz have "d \<noteq> 0" by simp 
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changeset

546 
with div_mult_self1_is_id have "d * (l * u) div d = l * u" . 
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547 
ultimately have "1 = l * u" 
60526  548 
using \<open>d \<noteq> 0\<close> by simp 
59009
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549 
then show "l dvd 1" .. 
58023  550 
qed 
551 

552 
lemma coprime_mult: 

553 
assumes da: "gcd d a = 1" and db: "gcd d b = 1" 

554 
shows "gcd d (a * b) = 1" 

555 
apply (subst gcd.commute) 

556 
using da apply (subst gcd_mult_cancel) 

557 
apply (subst gcd.commute, assumption) 

558 
apply (subst gcd.commute, rule db) 

559 
done 

560 

561 
lemma coprime_lmult: 

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

563 
shows "gcd d a = 1" 

564 
proof (rule coprimeI) 

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

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_rmult: 

571 
assumes dab: "gcd d (a * b) = 1" 

572 
shows "gcd d b = 1" 

573 
proof (rule coprimeI) 

574 
fix l assume "l dvd d" and "l dvd b" 

575 
hence "l dvd a * b" by simp 

60526  576 
with \<open>l dvd d\<close> and dab show "l dvd 1" by (auto intro: gcd_greatest) 
58023  577 
qed 
578 

579 
lemma coprime_mul_eq: "gcd d (a * b) = 1 \<longleftrightarrow> gcd d a = 1 \<and> gcd d b = 1" 

580 
using coprime_rmult[of d a b] coprime_lmult[of d a b] coprime_mult[of d a b] by blast 

581 

582 
lemma gcd_coprime: 

60430
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haftmann
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diff
changeset

583 
assumes c: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b" 
58023  584 
shows "gcd a' b' = 1" 
585 
proof  

60430
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haftmann
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diff
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586 
from c have "a \<noteq> 0 \<or> b \<noteq> 0" by simp 
58023  587 
with div_gcd_coprime have "gcd (a div gcd a b) (b div gcd a b) = 1" . 
588 
also from assms have "a div gcd a b = a'" by (metis div_mult_self2_is_id)+ 

589 
also from assms have "b div gcd a b = b'" by (metis div_mult_self2_is_id)+ 

590 
finally show ?thesis . 

591 
qed 

592 

593 
lemma coprime_power: 

594 
assumes "0 < n" 

595 
shows "gcd a (b ^ n) = 1 \<longleftrightarrow> gcd a b = 1" 

596 
using assms proof (induct n) 

597 
case (Suc n) then show ?case 

598 
by (cases n) (simp_all add: coprime_mul_eq) 

599 
qed simp 

600 

601 
lemma gcd_coprime_exists: 

602 
assumes nz: "gcd a b \<noteq> 0" 

603 
shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> gcd a' b' = 1" 

604 
apply (rule_tac x = "a div gcd a b" in exI) 

605 
apply (rule_tac x = "b div gcd a b" in exI) 

59009
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generalized lemmas (particularly concerning dvd) as far as appropriate
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changeset

606 
apply (insert nz, auto intro: div_gcd_coprime) 
58023  607 
done 
608 

609 
lemma coprime_exp: 

610 
"gcd d a = 1 \<Longrightarrow> gcd d (a^n) = 1" 

611 
by (induct n, simp_all add: coprime_mult) 

612 

613 
lemma coprime_exp2 [intro]: 

614 
"gcd a b = 1 \<Longrightarrow> gcd (a^n) (b^m) = 1" 

615 
apply (rule coprime_exp) 

616 
apply (subst gcd.commute) 

617 
apply (rule coprime_exp) 

618 
apply (subst gcd.commute) 

619 
apply assumption 

620 
done 

621 

60688
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changeset

622 
lemma lcm_gcd: 
01488b559910
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changeset

623 
"lcm a b = normalize (a * b) div gcd a b" 
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diff
changeset

624 
by (simp add: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def) 
01488b559910
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changeset

625 

01488b559910
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changeset

626 
subclass semiring_gcd 
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haftmann
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diff
changeset

627 
apply standard 
01488b559910
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haftmann
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diff
changeset

628 
using gcd_right_idem 
01488b559910
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haftmann
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diff
changeset

629 
apply (simp_all add: lcm_gcd gcd_greatest_iff gcd_proj1_if_dvd) 
01488b559910
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630 
done 
01488b559910
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631 

58023  632 
lemma gcd_exp: 
60688
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changeset

633 
"gcd (a ^ n) (b ^ n) = gcd a b ^ n" 
58023  634 
proof (cases "a = 0 \<and> b = 0") 
60688
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635 
case True 
01488b559910
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haftmann
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diff
changeset

636 
then show ?thesis by (cases n) simp_all 
58023  637 
next 
60688
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diff
changeset

638 
case False 
01488b559910
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changeset

639 
then have "1 = gcd ((a div gcd a b) ^ n) ((b div gcd a b) ^ n)" 
01488b559910
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diff
changeset

640 
using div_gcd_coprime by (subst sym) (auto simp: div_gcd_coprime) 
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haftmann
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diff
changeset

641 
then have "gcd a b ^ n = gcd a b ^ n * ..." by simp 
58023  642 
also note gcd_mult_distrib 
60688
01488b559910
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changeset

643 
also have "unit_factor (gcd a b ^ n) = 1" 
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haftmann
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changeset

644 
using False by (auto simp add: unit_factor_power unit_factor_gcd) 
58023  645 
also have "(gcd a b)^n * (a div gcd a b)^n = a^n" 
646 
by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp) 

647 
also have "(gcd a b)^n * (b div gcd a b)^n = b^n" 

648 
by (subst ac_simps, subst div_power, simp, rule dvd_div_mult_self, rule dvd_power_same, simp) 

649 
finally show ?thesis by simp 

650 
qed 

651 

652 
lemma coprime_common_divisor: 

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

653 
"gcd a b = 1 \<Longrightarrow> a dvd a \<Longrightarrow> a dvd b \<Longrightarrow> is_unit a" 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

654 
apply (subgoal_tac "a dvd gcd a b") 
59061  655 
apply simp 
58023  656 
apply (erule (1) gcd_greatest) 
657 
done 

658 

659 
lemma division_decomp: 

660 
assumes dc: "a dvd b * c" 

661 
shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c" 

662 
proof (cases "gcd a b = 0") 

663 
assume "gcd a b = 0" 

59009
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generalized lemmas (particularly concerning dvd) as far as appropriate
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changeset

664 
hence "a = 0 \<and> b = 0" by simp 
58023  665 
hence "a = 0 * c \<and> 0 dvd b \<and> c dvd c" by simp 
666 
then show ?thesis by blast 

667 
next 

668 
let ?d = "gcd a b" 

669 
assume "?d \<noteq> 0" 

670 
from gcd_coprime_exists[OF this] 

671 
obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1" 

672 
by blast 

673 
from ab'(1) have "a' dvd a" unfolding dvd_def by blast 

674 
with dc have "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp 

675 
from dc ab'(1,2) have "a'*?d dvd (b'*?d) * c" by simp 

676 
hence "?d * a' dvd ?d * (b' * c)" by (simp add: mult_ac) 

60526  677 
with \<open>?d \<noteq> 0\<close> have "a' dvd b' * c" by simp 
58023  678 
with coprime_dvd_mult[OF ab'(3)] 
679 
have "a' dvd c" by (subst (asm) ac_simps, blast) 

680 
with ab'(1) have "a = ?d * a' \<and> ?d dvd b \<and> a' dvd c" by (simp add: mult_ac) 

681 
then show ?thesis by blast 

682 
qed 

683 

60433  684 
lemma pow_divs_pow: 
58023  685 
assumes ab: "a ^ n dvd b ^ n" and n: "n \<noteq> 0" 
686 
shows "a dvd b" 

687 
proof (cases "gcd a b = 0") 

688 
assume "gcd a b = 0" 

59009
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changeset

689 
then show ?thesis by simp 
58023  690 
next 
691 
let ?d = "gcd a b" 

692 
assume "?d \<noteq> 0" 

693 
from n obtain m where m: "n = Suc m" by (cases n, simp_all) 

60526  694 
from \<open>?d \<noteq> 0\<close> have zn: "?d ^ n \<noteq> 0" by (rule power_not_zero) 
695 
from gcd_coprime_exists[OF \<open>?d \<noteq> 0\<close>] 

58023  696 
obtain a' b' where ab': "a = a' * ?d" "b = b' * ?d" "gcd a' b' = 1" 
697 
by blast 

698 
from ab have "(a' * ?d) ^ n dvd (b' * ?d) ^ n" 

699 
by (simp add: ab'(1,2)[symmetric]) 

700 
hence "?d^n * a'^n dvd ?d^n * b'^n" 

701 
by (simp only: power_mult_distrib ac_simps) 

59009
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generalized lemmas (particularly concerning dvd) as far as appropriate
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diff
changeset

702 
with zn have "a'^n dvd b'^n" by simp 
58023  703 
hence "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by (simp add: m) 
704 
hence "a' dvd b'^m * b'" by (simp add: m ac_simps) 

705 
with coprime_dvd_mult[OF coprime_exp[OF ab'(3), of m]] 

706 
have "a' dvd b'" by (subst (asm) ac_simps, blast) 

707 
hence "a'*?d dvd b'*?d" by (rule mult_dvd_mono, simp) 

708 
with ab'(1,2) show ?thesis by simp 

709 
qed 

710 

60433  711 
lemma pow_divs_eq [simp]: 
58023  712 
"n \<noteq> 0 \<Longrightarrow> a ^ n dvd b ^ n \<longleftrightarrow> a dvd b" 
60433  713 
by (auto intro: pow_divs_pow dvd_power_same) 
58023  714 

60433  715 
lemma divs_mult: 
58023  716 
assumes mr: "m dvd r" and nr: "n dvd r" and mn: "gcd m n = 1" 
717 
shows "m * n dvd r" 

718 
proof  

719 
from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'" 

720 
unfolding dvd_def by blast 

721 
from mr n' have "m dvd n'*n" by (simp add: ac_simps) 

722 
hence "m dvd n'" using coprime_dvd_mult_iff[OF mn] by simp 

723 
then obtain k where k: "n' = m*k" unfolding dvd_def by blast 

724 
with n' have "r = m * n * k" by (simp add: mult_ac) 

725 
then show ?thesis unfolding dvd_def by blast 

726 
qed 

727 

728 
lemma coprime_plus_one [simp]: "gcd (n + 1) n = 1" 

729 
by (subst add_commute, simp) 

730 

731 
lemma setprod_coprime [rule_format]: 

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

732 
"(\<forall>i\<in>A. gcd (f i) a = 1) \<longrightarrow> gcd (\<Prod>i\<in>A. f i) a = 1" 
58023  733 
apply (cases "finite A") 
734 
apply (induct set: finite) 

735 
apply (auto simp add: gcd_mult_cancel) 

736 
done 

737 

738 
lemma coprime_divisors: 

739 
assumes "d dvd a" "e dvd b" "gcd a b = 1" 

740 
shows "gcd d e = 1" 

741 
proof  

742 
from assms obtain k l where "a = d * k" "b = e * l" 

743 
unfolding dvd_def by blast 

744 
with assms have "gcd (d * k) (e * l) = 1" by simp 

745 
hence "gcd (d * k) e = 1" by (rule coprime_lmult) 

746 
also have "gcd (d * k) e = gcd e (d * k)" by (simp add: ac_simps) 

747 
finally have "gcd e d = 1" by (rule coprime_lmult) 

748 
then show ?thesis by (simp add: ac_simps) 

749 
qed 

750 

751 
lemma invertible_coprime: 

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

752 
assumes "a * b mod m = 1" 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

753 
shows "coprime a m" 
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset

754 
proof  
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

755 
from assms have "coprime m (a * b mod m)" 
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset

756 
by simp 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

757 
then have "coprime m (a * b)" 
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset

758 
by simp 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

759 
then have "coprime m a" 
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset

760 
by (rule coprime_lmult) 
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset

761 
then show ?thesis 
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset

762 
by (simp add: ac_simps) 
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset

763 
qed 
58023  764 

765 
lemma lcm_gcd_prod: 

60634  766 
"lcm a b * gcd a b = normalize (a * b)" 
767 
by (simp add: lcm_gcd) 

58023  768 

769 
lemma lcm_zero: 

770 
"lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" 

60687  771 
by (fact lcm_eq_0_iff) 
58023  772 

773 
lemmas lcm_0_iff = lcm_zero 

774 

775 
lemma gcd_lcm: 

776 
assumes "lcm a b \<noteq> 0" 

60634  777 
shows "gcd a b = normalize (a * b) div lcm a b" 
778 
proof  

779 
have "lcm a b * gcd a b = normalize (a * b)" 

780 
by (fact lcm_gcd_prod) 

781 
with assms show ?thesis 

782 
by (metis nonzero_mult_divide_cancel_left) 

58023  783 
qed 
784 

60687  785 
declare unit_factor_lcm [simp] 
58023  786 

787 
lemma lcmI: 

60634  788 
assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d" 
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

789 
and "normalize c = c" 
60634  790 
shows "c = lcm a b" 
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

791 
by (rule associated_eqI) (auto simp: assms intro: lcm_least) 
58023  792 

60687  793 
sublocale lcm!: abel_semigroup lcm .. 
58023  794 

795 
lemma dvd_lcm_D1: 

796 
"lcm m n dvd k \<Longrightarrow> m dvd k" 

60690  797 
by (rule dvd_trans, rule dvd_lcm1, assumption) 
58023  798 

799 
lemma dvd_lcm_D2: 

800 
"lcm m n dvd k \<Longrightarrow> n dvd k" 

60690  801 
by (rule dvd_trans, rule dvd_lcm2, assumption) 
58023  802 

803 
lemma gcd_dvd_lcm [simp]: 

804 
"gcd a b dvd lcm a b" 

60690  805 
using gcd_dvd2 by (rule dvd_lcmI2) 
58023  806 

807 
lemma lcm_1_iff: 

808 
"lcm a b = 1 \<longleftrightarrow> is_unit a \<and> is_unit b" 

809 
proof 

810 
assume "lcm a b = 1" 

59061  811 
then show "is_unit a \<and> is_unit b" by auto 
58023  812 
next 
813 
assume "is_unit a \<and> is_unit b" 

59061  814 
hence "a dvd 1" and "b dvd 1" by simp_all 
815 
hence "is_unit (lcm a b)" by (rule lcm_least) 

60634  816 
hence "lcm a b = unit_factor (lcm a b)" 
817 
by (blast intro: sym is_unit_unit_factor) 

60526  818 
also have "\<dots> = 1" using \<open>is_unit a \<and> is_unit b\<close> 
59061  819 
by auto 
58023  820 
finally show "lcm a b = 1" . 
821 
qed 

822 

60687  823 
lemma lcm_0: 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

824 
"lcm a 0 = 0" 
60687  825 
by (fact lcm_0_right) 
58023  826 

827 
lemma lcm_unique: 

828 
"a dvd d \<and> b dvd d \<and> 

60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

829 
normalize d = d \<and> 
58023  830 
(\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b" 
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

831 
by rule (auto intro: lcmI simp: lcm_least lcm_zero) 
58023  832 

833 
lemma lcm_coprime: 

60634  834 
"gcd a b = 1 \<Longrightarrow> lcm a b = normalize (a * b)" 
58023  835 
by (subst lcm_gcd) simp 
836 

837 
lemma lcm_proj1_if_dvd: 

60634  838 
"b dvd a \<Longrightarrow> lcm a b = normalize a" 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

839 
by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all) 
58023  840 

841 
lemma lcm_proj2_if_dvd: 

60634  842 
"a dvd b \<Longrightarrow> lcm a b = normalize b" 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

843 
using lcm_proj1_if_dvd [of a b] by (simp add: ac_simps) 
58023  844 

845 
lemma lcm_proj1_iff: 

60634  846 
"lcm m n = normalize m \<longleftrightarrow> n dvd m" 
58023  847 
proof 
60634  848 
assume A: "lcm m n = normalize m" 
58023  849 
show "n dvd m" 
850 
proof (cases "m = 0") 

851 
assume [simp]: "m \<noteq> 0" 

60634  852 
from A have B: "m = lcm m n * unit_factor m" 
58023  853 
by (simp add: unit_eq_div2) 
854 
show ?thesis by (subst B, simp) 

855 
qed simp 

856 
next 

857 
assume "n dvd m" 

60634  858 
then show "lcm m n = normalize m" by (rule lcm_proj1_if_dvd) 
58023  859 
qed 
860 

861 
lemma lcm_proj2_iff: 

60634  862 
"lcm m n = normalize n \<longleftrightarrow> m dvd n" 
58023  863 
using lcm_proj1_iff [of n m] by (simp add: ac_simps) 
864 

865 
lemma euclidean_size_lcm_le1: 

866 
assumes "a \<noteq> 0" and "b \<noteq> 0" 

867 
shows "euclidean_size a \<le> euclidean_size (lcm a b)" 

868 
proof  

60690  869 
have "a dvd lcm a b" by (rule dvd_lcm1) 
870 
then obtain c where A: "lcm a b = a * c" .. 

60526  871 
with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_zero) 
58023  872 
then show ?thesis by (subst A, intro size_mult_mono) 
873 
qed 

874 

875 
lemma euclidean_size_lcm_le2: 

876 
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)" 

877 
using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps) 

878 

879 
lemma euclidean_size_lcm_less1: 

880 
assumes "b \<noteq> 0" and "\<not>b dvd a" 

881 
shows "euclidean_size a < euclidean_size (lcm a b)" 

882 
proof (rule ccontr) 

883 
from assms have "a \<noteq> 0" by auto 

884 
assume "\<not>euclidean_size a < euclidean_size (lcm a b)" 

60526  885 
with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a" 
58023  886 
by (intro le_antisym, simp, intro euclidean_size_lcm_le1) 
887 
with assms have "lcm a b dvd a" 

888 
by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_zero) 

889 
hence "b dvd a" by (rule dvd_lcm_D2) 

60526  890 
with \<open>\<not>b dvd a\<close> show False by contradiction 
58023  891 
qed 
892 

893 
lemma euclidean_size_lcm_less2: 

894 
assumes "a \<noteq> 0" and "\<not>a dvd b" 

895 
shows "euclidean_size b < euclidean_size (lcm a b)" 

896 
using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps) 

897 

898 
lemma lcm_mult_unit1: 

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

899 
"is_unit a \<Longrightarrow> lcm (b * a) c = lcm b c" 
60690  900 
by (rule associated_eqI) (simp_all add: mult_unit_dvd_iff dvd_lcmI1) 
58023  901 

902 
lemma lcm_mult_unit2: 

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

903 
"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

904 
using lcm_mult_unit1 [of a c b] by (simp add: ac_simps) 
58023  905 

906 
lemma lcm_div_unit1: 

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

907 
"is_unit a \<Longrightarrow> lcm (b div a) c = lcm b c" 
60433  908 
by (erule is_unitE [of _ b]) (simp add: lcm_mult_unit1) 
58023  909 

910 
lemma lcm_div_unit2: 

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

911 
"is_unit a \<Longrightarrow> lcm b (c div a) = lcm b c" 
60433  912 
by (erule is_unitE [of _ c]) (simp add: lcm_mult_unit2) 
58023  913 

60634  914 
lemma normalize_lcm_left [simp]: 
915 
"lcm (normalize a) b = lcm a b" 

916 
proof (cases "a = 0") 

917 
case True then show ?thesis 

918 
by simp 

919 
next 

920 
case False then have "is_unit (unit_factor a)" 

921 
by simp 

922 
moreover have "normalize a = a div unit_factor a" 

923 
by simp 

924 
ultimately show ?thesis 

925 
by (simp only: lcm_div_unit1) 

926 
qed 

927 

928 
lemma normalize_lcm_right [simp]: 

929 
"lcm a (normalize b) = lcm a b" 

930 
using normalize_lcm_left [of b a] by (simp add: ac_simps) 

931 

58023  932 
lemma lcm_left_idem: 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

933 
"lcm a (lcm a b) = lcm a b" 
60690  934 
by (rule associated_eqI) simp_all 
58023  935 

936 
lemma lcm_right_idem: 

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

937 
"lcm (lcm a b) b = lcm a b" 
60690  938 
by (rule associated_eqI) simp_all 
58023  939 

940 
lemma comp_fun_idem_lcm: "comp_fun_idem lcm" 

941 
proof 

942 
fix a b show "lcm a \<circ> lcm b = lcm b \<circ> lcm a" 

943 
by (simp add: fun_eq_iff ac_simps) 

944 
next 

945 
fix a show "lcm a \<circ> lcm a = lcm a" unfolding o_def 

946 
by (intro ext, simp add: lcm_left_idem) 

947 
qed 

948 

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

949 
lemma dvd_Lcm [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm A" 
60634  950 
and Lcm_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm A dvd b" 
951 
and unit_factor_Lcm [simp]: 

952 
"unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" 

58023  953 
proof  
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

954 
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> 
60634  955 
unit_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

956 
proof (cases "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)") 
58023  957 
case False 
958 
hence "Lcm A = 0" by (auto simp: Lcm_Lcm_eucl Lcm_eucl_def) 

959 
with False show ?thesis by auto 

960 
next 

961 
case True 

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

962 
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

963 
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

964 
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

965 
have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" 
58023  966 
apply (subst n_def) 
967 
apply (rule LeastI[of _ "euclidean_size l\<^sub>0"]) 

968 
apply (rule exI[of _ l\<^sub>0]) 

969 
apply (simp add: l\<^sub>0_props) 

970 
done 

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

971 
from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n" 
58023  972 
unfolding l_def by simp_all 
973 
{ 

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

974 
fix l' assume "\<forall>a\<in>A. a dvd l'" 
60526  975 
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) 
976 
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

977 
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  978 
by (intro exI[of _ "gcd l l'"], auto) 
979 
hence "euclidean_size (gcd l l') \<ge> n" by (subst n_def) (rule Least_le) 

980 
moreover have "euclidean_size (gcd l l') \<le> n" 

981 
proof  

982 
have "gcd l l' dvd l" by simp 

983 
then obtain a where "l = gcd l l' * a" unfolding dvd_def by blast 

60526  984 
with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto 
58023  985 
hence "euclidean_size (gcd l l') \<le> euclidean_size (gcd l l' * a)" 
986 
by (rule size_mult_mono) 

60526  987 
also have "gcd l l' * a = l" using \<open>l = gcd l l' * a\<close> .. 
988 
also note \<open>euclidean_size l = n\<close> 

58023  989 
finally show "euclidean_size (gcd l l') \<le> n" . 
990 
qed 

60690  991 
ultimately have *: "euclidean_size l = euclidean_size (gcd l l')" 
60526  992 
by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>) 
60690  993 
from \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" 
994 
by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *) 

58023  995 
hence "l dvd l'" by (blast dest: dvd_gcd_D2) 
996 
} 

997 

60634  998 
with \<open>(\<forall>a\<in>A. a dvd l)\<close> and unit_factor_is_unit[OF \<open>l \<noteq> 0\<close>] and \<open>l \<noteq> 0\<close> 
999 
have "(\<forall>a\<in>A. a dvd normalize l) \<and> 

1000 
(\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and> 

1001 
unit_factor (normalize l) = 

1002 
(if normalize l = 0 then 0 else 1)" 

58023  1003 
by (auto simp: unit_simps) 
60634  1004 
also from True have "normalize l = Lcm A" 
58023  1005 
by (simp add: Lcm_Lcm_eucl Lcm_eucl_def Let_def n_def l_def) 
1006 
finally show ?thesis . 

1007 
qed 

1008 
note A = this 

1009 

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

1010 
{fix a assume "a \<in> A" then show "a dvd Lcm A" using A by blast} 
60634  1011 
{fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm A dvd b" using A by blast} 
1012 
from A show "unit_factor (Lcm A) = (if Lcm A = 0 then 0 else 1)" by blast 

58023  1013 
qed 
60634  1014 

1015 
lemma normalize_Lcm [simp]: 

1016 
"normalize (Lcm A) = Lcm A" 

60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1017 
proof (cases "Lcm A = 0") 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1018 
case True then show ?thesis by simp 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1019 
next 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1020 
case False 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1021 
have "unit_factor (Lcm A) * normalize (Lcm A) = Lcm A" 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1022 
by (fact unit_factor_mult_normalize) 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1023 
with False show ?thesis by simp 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1024 
qed 
60634  1025 

58023  1026 
lemma LcmI: 
60634  1027 
assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd b" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> b dvd c" 
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1028 
and "normalize b = b" shows "b = Lcm A" 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1029 
by (rule associated_eqI) (auto simp: assms intro: Lcm_least) 
58023  1030 

1031 
lemma Lcm_subset: 

1032 
"A \<subseteq> B \<Longrightarrow> Lcm A dvd Lcm B" 

60634  1033 
by (blast intro: Lcm_least dvd_Lcm) 
58023  1034 

1035 
lemma Lcm_Un: 

1036 
"Lcm (A \<union> B) = lcm (Lcm A) (Lcm B)" 

1037 
apply (rule lcmI) 

1038 
apply (blast intro: Lcm_subset) 

1039 
apply (blast intro: Lcm_subset) 

60634  1040 
apply (intro Lcm_least ballI, elim UnE) 
58023  1041 
apply (rule dvd_trans, erule dvd_Lcm, assumption) 
1042 
apply (rule dvd_trans, erule dvd_Lcm, assumption) 

1043 
apply simp 

1044 
done 

1045 

1046 
lemma Lcm_1_iff: 

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

1047 
"Lcm A = 1 \<longleftrightarrow> (\<forall>a\<in>A. is_unit a)" 
58023  1048 
proof 
1049 
assume "Lcm A = 1" 

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

1050 
then show "\<forall>a\<in>A. is_unit a" by auto 
58023  1051 
qed (rule LcmI [symmetric], auto) 
1052 

1053 
lemma Lcm_no_units: 

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

1054 
"Lcm A = Lcm (A  {a. is_unit a})" 
58023  1055 
proof  
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1056 
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

1057 
hence "Lcm A = lcm (Lcm (A  {a. is_unit a})) (Lcm {a\<in>A. is_unit a})" 
60634  1058 
by (simp add: Lcm_Un [symmetric]) 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1059 
also have "Lcm {a\<in>A. is_unit a} = 1" by (simp add: Lcm_1_iff) 
58023  1060 
finally show ?thesis by simp 
1061 
qed 

1062 

1063 
lemma Lcm_empty [simp]: 

1064 
"Lcm {} = 1" 

1065 
by (simp add: Lcm_1_iff) 

1066 

1067 
lemma Lcm_eq_0 [simp]: 

1068 
"0 \<in> A \<Longrightarrow> Lcm A = 0" 

1069 
by (drule dvd_Lcm) simp 

1070 

1071 
lemma Lcm0_iff': 

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

1072 
"Lcm A = 0 \<longleftrightarrow> \<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" 
58023  1073 
proof 
1074 
assume "Lcm A = 0" 

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

1075 
show "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" 
58023  1076 
proof 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1077 
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

1078 
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

1079 
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

1080 
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

1081 
have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" 
58023  1082 
apply (subst n_def) 
1083 
apply (rule LeastI[of _ "euclidean_size l\<^sub>0"]) 

1084 
apply (rule exI[of _ l\<^sub>0]) 

1085 
apply (simp add: l\<^sub>0_props) 

1086 
done 

1087 
from someI_ex[OF this] have "l \<noteq> 0" unfolding l_def by simp_all 

60634  1088 
hence "normalize l \<noteq> 0" by simp 
1089 
also from ex have "normalize l = Lcm A" 

58023  1090 
by (simp only: Lcm_Lcm_eucl Lcm_eucl_def n_def l_def if_True Let_def) 
60526  1091 
finally show False using \<open>Lcm A = 0\<close> by contradiction 
58023  1092 
qed 
1093 
qed (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False) 

1094 

1095 
lemma Lcm0_iff [simp]: 

1096 
"finite A \<Longrightarrow> Lcm A = 0 \<longleftrightarrow> 0 \<in> A" 

1097 
proof  

1098 
assume "finite A" 

1099 
have "0 \<in> A \<Longrightarrow> Lcm A = 0" by (intro dvd_0_left dvd_Lcm) 

1100 
moreover { 

1101 
assume "0 \<notin> A" 

1102 
hence "\<Prod>A \<noteq> 0" 

60526  1103 
apply (induct rule: finite_induct[OF \<open>finite A\<close>]) 
58023  1104 
apply simp 
1105 
apply (subst setprod.insert, assumption, assumption) 

1106 
apply (rule no_zero_divisors) 

1107 
apply blast+ 

1108 
done 

60526  1109 
moreover from \<open>finite A\<close> have "\<forall>a\<in>A. a dvd \<Prod>A" by blast 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1110 
ultimately have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)" by blast 
58023  1111 
with Lcm0_iff' have "Lcm A \<noteq> 0" by simp 
1112 
} 

1113 
ultimately show "Lcm A = 0 \<longleftrightarrow> 0 \<in> A" by blast 

1114 
qed 

1115 

1116 
lemma Lcm_no_multiple: 

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

1117 
"(\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)) \<Longrightarrow> Lcm A = 0" 
58023  1118 
proof  
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1119 
assume "\<forall>m. m \<noteq> 0 \<longrightarrow> (\<exists>a\<in>A. \<not>a dvd m)" 
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1120 
hence "\<not>(\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l))" by blast 
58023  1121 
then show "Lcm A = 0" by (simp only: Lcm_Lcm_eucl Lcm_eucl_def if_False) 
1122 
qed 

1123 

1124 
lemma Lcm_insert [simp]: 

1125 
"Lcm (insert a A) = lcm a (Lcm A)" 

1126 
proof (rule lcmI) 

1127 
fix l assume "a dvd l" and "Lcm A dvd l" 

60687  1128 
then have "\<forall>a\<in>A. a dvd l" by (auto intro: dvd_trans [of _ "Lcm A" l]) 
60634  1129 
with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_least) 
1130 
qed (auto intro: Lcm_least dvd_Lcm) 

58023  1131 

1132 
lemma Lcm_finite: 

1133 
assumes "finite A" 

1134 
shows "Lcm A = Finite_Set.fold lcm 1 A" 

60526  1135 
by (induct rule: finite.induct[OF \<open>finite A\<close>]) 
58023  1136 
(simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_lcm]) 
1137 

60431
db9c67b760f1
dropped warnings by dropping ineffective code declarations
haftmann
parents:
60430
diff
changeset

1138 
lemma Lcm_set [code_unfold]: 
58023  1139 
"Lcm (set xs) = fold lcm xs 1" 
1140 
using comp_fun_idem.fold_set_fold[OF comp_fun_idem_lcm] Lcm_finite by (simp add: ac_simps) 

1141 

1142 
lemma Lcm_singleton [simp]: 

60634  1143 
"Lcm {a} = normalize a" 
58023  1144 
by simp 
1145 

1146 
lemma Lcm_2 [simp]: 

1147 
"Lcm {a,b} = lcm a b" 

60634  1148 
by simp 
58023  1149 

1150 
lemma Lcm_coprime: 

1151 
assumes "finite A" and "A \<noteq> {}" 

1152 
assumes "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1" 

60634  1153 
shows "Lcm A = normalize (\<Prod>A)" 
58023  1154 
using assms proof (induct rule: finite_ne_induct) 
1155 
case (insert a A) 

1156 
have "Lcm (insert a A) = lcm a (Lcm A)" by simp 

60634  1157 
also from insert have "Lcm A = normalize (\<Prod>A)" by blast 
58023  1158 
also have "lcm a \<dots> = lcm a (\<Prod>A)" by (cases "\<Prod>A = 0") (simp_all add: lcm_div_unit2) 
1159 
also from insert have "gcd a (\<Prod>A) = 1" by (subst gcd.commute, intro setprod_coprime) auto 

60634  1160 
with insert have "lcm a (\<Prod>A) = normalize (\<Prod>(insert a A))" 
58023  1161 
by (simp add: lcm_coprime) 
1162 
finally show ?case . 

1163 
qed simp 

1164 

1165 
lemma Lcm_coprime': 

1166 
"card A \<noteq> 0 \<Longrightarrow> (\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> gcd a b = 1) 

60634  1167 
\<Longrightarrow> Lcm A = normalize (\<Prod>A)" 
58023  1168 
by (rule Lcm_coprime) (simp_all add: card_eq_0_iff) 
1169 

1170 
lemma Gcd_Lcm: 

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

1171 
"Gcd A = Lcm {d. \<forall>a\<in>A. d dvd a}" 
58023  1172 
by (simp add: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def) 
1173 

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

1174 
lemma Gcd_dvd [simp]: "a \<in> A \<Longrightarrow> Gcd A dvd a" 
60634  1175 
and Gcd_greatest: "(\<And>a. a \<in> A \<Longrightarrow> b dvd a) \<Longrightarrow> b dvd Gcd A" 
1176 
and unit_factor_Gcd [simp]: 

1177 
"unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)" 

58023  1178 
proof  
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1179 
fix a assume "a \<in> A" 
60634  1180 
hence "Lcm {d. \<forall>a\<in>A. d dvd a} dvd a" by (intro Lcm_least) blast 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1181 
then show "Gcd A dvd a" by (simp add: Gcd_Lcm) 
58023  1182 
next 
60634  1183 
fix g' assume "\<And>a. a \<in> A \<Longrightarrow> g' dvd a" 
60430
ce559c850a27
standardized algebraic conventions: prefer a, b, c over x, y, z
haftmann
parents:
59061
diff
changeset

1184 
hence "g' dvd Lcm {d. \<forall>a\<in>A. d dvd a}" by (intro dvd_Lcm) blast 
58023  1185 
then show "g' dvd Gcd A" by (simp add: Gcd_Lcm) 
1186 
next 

60634  1187 
show "unit_factor (Gcd A) = (if Gcd A = 0 then 0 else 1)" 
59009
348561aa3869
generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents:
58953
diff
changeset

1188 
by (simp add: Gcd_Lcm) 
58023  1189 
qed 
1190 

60634  1191 
lemma normalize_Gcd [simp]: 
1192 
"normalize (Gcd A) = Gcd A" 

60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1193 
proof (cases "Gcd A = 0") 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1194 
case True then show ?thesis by simp 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1195 
next 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1196 
case False 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1197 
have "unit_factor (Gcd A) * normalize (Gcd A) = Gcd A" 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1198 
by (fact unit_factor_mult_normalize) 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1199 
with False show ?thesis by simp 
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1200 
qed 
60634  1201 

60687  1202 
subclass semiring_Gcd 
1203 
by standard (simp_all add: Gcd_greatest) 

1204 

58023  1205 
lemma GcdI: 
60634  1206 
assumes "\<And>a. a \<in> A \<Longrightarrow> b dvd a" and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> c dvd a) \<Longrightarrow> c dvd b" 
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1207 
and "normalize b = b" 
60634  1208 
shows "b = Gcd A" 
60688
01488b559910
avoid explicit definition of the relation of associated elements in a ring  prefer explicit normalization instead
haftmann
parents:
60687
diff
changeset

1209 
by (rule associated_eqI) (auto simp: assms intro: Gcd_greatest) 
58023  1210 

1211 
lemma Lcm_Gcd: 

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

1212 
"Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}" 
60634  1213 
by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_greatest) 
58023  1214 

60687  1215 
subclass semiring_Lcm 
1216 
by standard (simp add: Lcm_Gcd) 

58023  1217 

1218 
lemma Gcd_1: 

1219 
"1 \<in> A \<Longrightarrow> Gcd A = 1" 

60687  1220 
by (auto intro!: Gcd_eq_1_I) 
58023  1221 

1222 
lemma Gcd_finite: 

1223 
assumes "finite A" 

1224 
shows "Gcd A = Finite_Set.fold gcd 0 A" 

60526  1225 
by (induct rule: finite.induct[OF \<open>finite A\<close>]) 
58023  1226 
(simp_all add: comp_fun_idem.fold_insert_idem[OF comp_fun_idem_gcd]) 
1227 

60431
db9c67b760f1
dropped warnings by dropping ineffective code declarations
haftmann
parents:
60430
diff
changeset

1228 
lemma Gcd_set [code_unfold]: 
58023  1229 
"Gcd (set xs) = fold gcd xs 0" 
1230 
using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps) 

1231 

60634  1232 
lemma Gcd_singleton [simp]: "Gcd {a} = normalize a" 
60687  1233 
by simp 