author  eberlm <eberlm@in.tum.de> 
Wed, 13 Jul 2016 15:46:52 +0200  
changeset 63498  a3fe3250d05d 
parent 63167  0909deb8059b 
child 63633  2accfb71e33b 
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 "~~/src/HOL/GCD" Factorial_Ring 
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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 size_0 [simp]: "euclidean_size 0 = 0" 
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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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29 
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 

55 
qed 

56 

63498  57 
lemma size_mult_mono': "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (b * a)" 
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by (subst mult.commute) (rule size_mult_mono) 

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lemma euclidean_size_times_unit: 

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assumes "is_unit a" 

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shows "euclidean_size (a * b) = euclidean_size b" 

63 
proof (rule antisym) 

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from assms have [simp]: "a \<noteq> 0" by auto 

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thus "euclidean_size (a * b) \<ge> euclidean_size b" by (rule size_mult_mono') 

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from assms have "is_unit (1 div a)" by simp 

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hence "1 div a \<noteq> 0" by (intro notI) simp_all 

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hence "euclidean_size (a * b) \<le> euclidean_size ((1 div a) * (a * b))" 

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by (rule size_mult_mono') 

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also from assms have "(1 div a) * (a * b) = b" 

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by (simp add: algebra_simps unit_div_mult_swap) 

72 
finally show "euclidean_size (a * b) \<le> euclidean_size b" . 

73 
qed 

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lemma euclidean_size_unit: "is_unit x \<Longrightarrow> euclidean_size x = euclidean_size 1" 

76 
using euclidean_size_times_unit[of x 1] by simp 

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lemma unit_iff_euclidean_size: 

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"is_unit x \<longleftrightarrow> euclidean_size x = euclidean_size 1 \<and> x \<noteq> 0" 

80 
proof safe 

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assume A: "x \<noteq> 0" and B: "euclidean_size x = euclidean_size 1" 

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show "is_unit x" by (rule dvd_euclidean_size_eq_imp_dvd[OF A _ B]) simp_all 

83 
qed (auto intro: euclidean_size_unit) 

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lemma euclidean_size_times_nonunit: 

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assumes "a \<noteq> 0" "b \<noteq> 0" "\<not>is_unit a" 

87 
shows "euclidean_size b < euclidean_size (a * b)" 

88 
proof (rule ccontr) 

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assume "\<not>euclidean_size b < euclidean_size (a * b)" 

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with size_mult_mono'[OF assms(1), of b] 

91 
have eq: "euclidean_size (a * b) = euclidean_size b" by simp 

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have "a * b dvd b" 

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by (rule dvd_euclidean_size_eq_imp_dvd[OF _ _ eq]) (insert assms, simp_all) 

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hence "a * b dvd 1 * b" by simp 

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with \<open>b \<noteq> 0\<close> have "is_unit a" by (subst (asm) dvd_times_right_cancel_iff) 

96 
with assms(3) show False by contradiction 

97 
qed 

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lemma dvd_imp_size_le: 

100 
assumes "x dvd y" "y \<noteq> 0" 

101 
shows "euclidean_size x \<le> euclidean_size y" 

102 
using assms by (auto elim!: dvdE simp: size_mult_mono) 

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104 
lemma dvd_proper_imp_size_less: 

105 
assumes "x dvd y" "\<not>y dvd x" "y \<noteq> 0" 

106 
shows "euclidean_size x < euclidean_size y" 

107 
proof  

108 
from assms(1) obtain z where "y = x * z" by (erule dvdE) 

109 
hence z: "y = z * x" by (simp add: mult.commute) 

110 
from z assms have "\<not>is_unit z" by (auto simp: mult.commute mult_unit_dvd_iff) 

111 
with z assms show ?thesis 

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by (auto intro!: euclidean_size_times_nonunit simp: ) 

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qed 

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lemma irreducible_normalized_divisors: 

116 
assumes "irreducible x" "y dvd x" "normalize y = y" 

117 
shows "y = 1 \<or> y = normalize x" 

118 
proof  

119 
from assms have "is_unit y \<or> x dvd y" by (auto simp: irreducible_altdef) 

120 
thus ?thesis 

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proof (elim disjE) 

122 
assume "is_unit y" 

123 
hence "normalize y = 1" by (simp add: is_unit_normalize) 

124 
with assms show ?thesis by simp 

125 
next 

126 
assume "x dvd y" 

127 
with \<open>y dvd x\<close> have "normalize y = normalize x" by (rule associatedI) 

128 
with assms show ?thesis by simp 

129 
qed 

130 
qed 

131 

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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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159 
definition lcm_eucl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" 

160 
where 

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"lcm_eucl a b = normalize (a * b) div gcd_eucl a b" 
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63167  163 
definition Lcm_eucl :: "'a set \<Rightarrow> 'a" \<comment> \<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 
58023  171 
else 0)" 
172 

173 
definition Gcd_eucl :: "'a set \<Rightarrow> 'a" 

174 
where 

175 
"Gcd_eucl A = Lcm_eucl {d. \<forall>a\<in>A. d dvd a}" 

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declare Lcm_eucl_def Gcd_eucl_def [code del] 
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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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62422  191 
lemma gcd_eucl_dvd1 [iff]: "gcd_eucl a b dvd a" 
192 
and gcd_eucl_dvd2 [iff]: "gcd_eucl a b dvd b" 

193 
by (induct a b rule: gcd_eucl_induct) 

194 
(simp_all add: gcd_eucl_0 gcd_eucl_non_0 dvd_mod_iff) 

195 

196 
lemma normalize_gcd_eucl [simp]: 

197 
"normalize (gcd_eucl a b) = gcd_eucl a b" 

198 
by (induct a b rule: gcd_eucl_induct) (simp_all add: gcd_eucl_0 gcd_eucl_non_0) 

199 

200 
lemma gcd_eucl_greatest: 

201 
fixes k a b :: 'a 

202 
shows "k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd_eucl a b" 

203 
proof (induct a b rule: gcd_eucl_induct) 

204 
case (zero a) from zero(1) show ?case by (rule dvd_trans) (simp add: gcd_eucl_0) 

205 
next 

206 
case (mod a b) 

207 
then show ?case 

208 
by (simp add: gcd_eucl_non_0 dvd_mod_iff) 

209 
qed 

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lemma gcd_euclI: 
212 
fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" 

213 
assumes "d dvd a" "d dvd b" "normalize d = d" 

214 
"\<And>k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd d" 

215 
shows "gcd_eucl a b = d" 

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by (rule associated_eqI) (simp_all add: gcd_eucl_greatest assms) 

217 

62422  218 
lemma eq_gcd_euclI: 
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fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" 

220 
assumes "\<And>a b. gcd a b dvd a" "\<And>a b. gcd a b dvd b" "\<And>a b. normalize (gcd a b) = gcd a b" 

221 
"\<And>a b k. k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd gcd a b" 

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shows "gcd = gcd_eucl" 

223 
by (intro ext, rule associated_eqI) (simp_all add: gcd_eucl_greatest assms) 

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225 
lemma gcd_eucl_zero [simp]: 

226 
"gcd_eucl a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" 

227 
by (metis dvd_0_left dvd_refl gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest)+ 

228 

229 

230 
lemma dvd_Lcm_eucl [simp]: "a \<in> A \<Longrightarrow> a dvd Lcm_eucl A" 

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and Lcm_eucl_least: "(\<And>a. a \<in> A \<Longrightarrow> a dvd b) \<Longrightarrow> Lcm_eucl A dvd b" 

232 
and unit_factor_Lcm_eucl [simp]: 

233 
"unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" 

234 
proof  

235 
have "(\<forall>a\<in>A. a dvd Lcm_eucl A) \<and> (\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> Lcm_eucl A dvd l') \<and> 

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unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" (is ?thesis) 

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proof (cases "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l)") 

238 
case False 

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hence "Lcm_eucl A = 0" by (auto simp: Lcm_eucl_def) 

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with False show ?thesis by auto 

241 
next 

242 
case True 

243 
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 

63040  244 
define n where "n = (LEAST n. \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)" 
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define l where "l = (SOME l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)" 

62422  246 
have "\<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n" 
247 
apply (subst n_def) 

248 
apply (rule LeastI[of _ "euclidean_size l\<^sub>0"]) 

249 
apply (rule exI[of _ l\<^sub>0]) 

250 
apply (simp add: l\<^sub>0_props) 

251 
done 

252 
from someI_ex[OF this] have "l \<noteq> 0" and "\<forall>a\<in>A. a dvd l" and "euclidean_size l = n" 

253 
unfolding l_def by simp_all 

254 
{ 

255 
fix l' assume "\<forall>a\<in>A. a dvd l'" 

256 
with \<open>\<forall>a\<in>A. a dvd l\<close> have "\<forall>a\<in>A. a dvd gcd_eucl l l'" by (auto intro: gcd_eucl_greatest) 

257 
moreover from \<open>l \<noteq> 0\<close> have "gcd_eucl l l' \<noteq> 0" by simp 

258 
ultimately have "\<exists>b. b \<noteq> 0 \<and> (\<forall>a\<in>A. a dvd b) \<and> 

259 
euclidean_size b = euclidean_size (gcd_eucl l l')" 

260 
by (intro exI[of _ "gcd_eucl l l'"], auto) 

261 
hence "euclidean_size (gcd_eucl l l') \<ge> n" by (subst n_def) (rule Least_le) 

262 
moreover have "euclidean_size (gcd_eucl l l') \<le> n" 

263 
proof  

264 
have "gcd_eucl l l' dvd l" by simp 

265 
then obtain a where "l = gcd_eucl l l' * a" unfolding dvd_def by blast 

266 
with \<open>l \<noteq> 0\<close> have "a \<noteq> 0" by auto 

267 
hence "euclidean_size (gcd_eucl l l') \<le> euclidean_size (gcd_eucl l l' * a)" 

268 
by (rule size_mult_mono) 

269 
also have "gcd_eucl l l' * a = l" using \<open>l = gcd_eucl l l' * a\<close> .. 

270 
also note \<open>euclidean_size l = n\<close> 

271 
finally show "euclidean_size (gcd_eucl l l') \<le> n" . 

272 
qed 

273 
ultimately have *: "euclidean_size l = euclidean_size (gcd_eucl l l')" 

274 
by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>) 

275 
from \<open>l \<noteq> 0\<close> have "l dvd gcd_eucl l l'" 

276 
by (rule dvd_euclidean_size_eq_imp_dvd) (auto simp add: *) 

277 
hence "l dvd l'" by (rule dvd_trans[OF _ gcd_eucl_dvd2]) 

278 
} 

279 

280 
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> 

281 
have "(\<forall>a\<in>A. a dvd normalize l) \<and> 

282 
(\<forall>l'. (\<forall>a\<in>A. a dvd l') \<longrightarrow> normalize l dvd l') \<and> 

283 
unit_factor (normalize l) = 

284 
(if normalize l = 0 then 0 else 1)" 

285 
by (auto simp: unit_simps) 

286 
also from True have "normalize l = Lcm_eucl A" 

287 
by (simp add: Lcm_eucl_def Let_def n_def l_def) 

288 
finally show ?thesis . 

289 
qed 

290 
note A = this 

291 

292 
{fix a assume "a \<in> A" then show "a dvd Lcm_eucl A" using A by blast} 

293 
{fix b assume "\<And>a. a \<in> A \<Longrightarrow> a dvd b" then show "Lcm_eucl A dvd b" using A by blast} 

294 
from A show "unit_factor (Lcm_eucl A) = (if Lcm_eucl A = 0 then 0 else 1)" by blast 

295 
qed 

63498  296 

62422  297 
lemma normalize_Lcm_eucl [simp]: 
298 
"normalize (Lcm_eucl A) = Lcm_eucl A" 

299 
proof (cases "Lcm_eucl A = 0") 

300 
case True then show ?thesis by simp 

301 
next 

302 
case False 

303 
have "unit_factor (Lcm_eucl A) * normalize (Lcm_eucl A) = Lcm_eucl A" 

304 
by (fact unit_factor_mult_normalize) 

305 
with False show ?thesis by simp 

306 
qed 

307 

308 
lemma eq_Lcm_euclI: 

309 
fixes lcm :: "'a set \<Rightarrow> 'a" 

310 
assumes "\<And>A a. a \<in> A \<Longrightarrow> a dvd lcm A" and "\<And>A c. (\<And>a. a \<in> A \<Longrightarrow> a dvd c) \<Longrightarrow> lcm A dvd c" 

311 
"\<And>A. normalize (lcm A) = lcm A" shows "lcm = Lcm_eucl" 

312 
by (intro ext, rule associated_eqI) (auto simp: assms intro: Lcm_eucl_least) 

313 

63498  314 
lemma Gcd_eucl_dvd: "x \<in> A \<Longrightarrow> Gcd_eucl A dvd x" 
315 
unfolding Gcd_eucl_def by (auto intro: Lcm_eucl_least) 

316 

317 
lemma Gcd_eucl_greatest: "(\<And>x. x \<in> A \<Longrightarrow> d dvd x) \<Longrightarrow> d dvd Gcd_eucl A" 

318 
unfolding Gcd_eucl_def by auto 

319 

320 
lemma normalize_Gcd_eucl [simp]: "normalize (Gcd_eucl A) = Gcd_eucl A" 

321 
by (simp add: Gcd_eucl_def) 

322 

323 
lemma Lcm_euclI: 

324 
assumes "\<And>x. x \<in> A \<Longrightarrow> x dvd d" "\<And>d'. (\<And>x. x \<in> A \<Longrightarrow> x dvd d') \<Longrightarrow> d dvd d'" "normalize d = d" 

325 
shows "Lcm_eucl A = d" 

326 
proof  

327 
have "normalize (Lcm_eucl A) = normalize d" 

328 
by (intro associatedI) (auto intro: dvd_Lcm_eucl Lcm_eucl_least assms) 

329 
thus ?thesis by (simp add: assms) 

330 
qed 

331 

332 
lemma Gcd_euclI: 

333 
assumes "\<And>x. x \<in> A \<Longrightarrow> d dvd x" "\<And>d'. (\<And>x. x \<in> A \<Longrightarrow> d' dvd x) \<Longrightarrow> d' dvd d" "normalize d = d" 

334 
shows "Gcd_eucl A = d" 

335 
proof  

336 
have "normalize (Gcd_eucl A) = normalize d" 

337 
by (intro associatedI) (auto intro: Gcd_eucl_dvd Gcd_eucl_greatest assms) 

338 
thus ?thesis by (simp add: assms) 

339 
qed 

340 

341 
lemmas lcm_gcd_eucl_facts = 

342 
gcd_eucl_dvd1 gcd_eucl_dvd2 gcd_eucl_greatest normalize_gcd_eucl lcm_eucl_def 

343 
Gcd_eucl_def Gcd_eucl_dvd Gcd_eucl_greatest normalize_Gcd_eucl 

344 
dvd_Lcm_eucl Lcm_eucl_least normalize_Lcm_eucl 

345 

346 
lemma normalized_factors_product: 

347 
"{p. p dvd a * b \<and> normalize p = p} = 

348 
(\<lambda>(x,y). x * y) ` ({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})" 

349 
proof safe 

350 
fix p assume p: "p dvd a * b" "normalize p = p" 

351 
interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op " normalize unit_factor 

352 
by standard (rule lcm_gcd_eucl_facts; assumption)+ 

353 
from dvd_productE[OF p(1)] guess x y . note xy = this 

354 
define x' y' where "x' = normalize x" and "y' = normalize y" 

355 
have "p = x' * y'" 

356 
by (subst p(2) [symmetric]) (simp add: xy x'_def y'_def normalize_mult) 

357 
moreover from xy have "normalize x' = x'" "normalize y' = y'" "x' dvd a" "y' dvd b" 

358 
by (simp_all add: x'_def y'_def) 

359 
ultimately show "p \<in> (\<lambda>(x, y). x * y) ` 

360 
({p. p dvd a \<and> normalize p = p} \<times> {p. p dvd b \<and> normalize p = p})" 

361 
by blast 

362 
qed (auto simp: normalize_mult mult_dvd_mono) 

363 

364 

365 
subclass factorial_semiring 

366 
proof (standard, rule factorial_semiring_altI_aux) 

367 
fix x assume "x \<noteq> 0" 

368 
thus "finite {p. p dvd x \<and> normalize p = p}" 

369 
proof (induction "euclidean_size x" arbitrary: x rule: less_induct) 

370 
case (less x) 

371 
show ?case 

372 
proof (cases "\<exists>y. y dvd x \<and> \<not>x dvd y \<and> \<not>is_unit y") 

373 
case False 

374 
have "{p. p dvd x \<and> normalize p = p} \<subseteq> {1, normalize x}" 

375 
proof 

376 
fix p assume p: "p \<in> {p. p dvd x \<and> normalize p = p}" 

377 
with False have "is_unit p \<or> x dvd p" by blast 

378 
thus "p \<in> {1, normalize x}" 

379 
proof (elim disjE) 

380 
assume "is_unit p" 

381 
hence "normalize p = 1" by (simp add: is_unit_normalize) 

382 
with p show ?thesis by simp 

383 
next 

384 
assume "x dvd p" 

385 
with p have "normalize p = normalize x" by (intro associatedI) simp_all 

386 
with p show ?thesis by simp 

387 
qed 

388 
qed 

389 
moreover have "finite \<dots>" by simp 

390 
ultimately show ?thesis by (rule finite_subset) 

391 

392 
next 

393 
case True 

394 
then obtain y where y: "y dvd x" "\<not>x dvd y" "\<not>is_unit y" by blast 

395 
define z where "z = x div y" 

396 
let ?fctrs = "\<lambda>x. {p. p dvd x \<and> normalize p = p}" 

397 
from y have x: "x = y * z" by (simp add: z_def) 

398 
with less.prems have "y \<noteq> 0" "z \<noteq> 0" by auto 

399 
from x y have "\<not>is_unit z" by (auto simp: mult_unit_dvd_iff) 

400 
have "?fctrs x = (\<lambda>(p,p'). p * p') ` (?fctrs y \<times> ?fctrs z)" 

401 
by (subst x) (rule normalized_factors_product) 

402 
also have "\<not>y * z dvd y * 1" "\<not>y * z dvd 1 * z" 

403 
by (subst dvd_times_left_cancel_iff dvd_times_right_cancel_iff; fact)+ 

404 
hence "finite ((\<lambda>(p,p'). p * p') ` (?fctrs y \<times> ?fctrs z))" 

405 
by (intro finite_imageI finite_cartesian_product less dvd_proper_imp_size_less) 

406 
(auto simp: x) 

407 
finally show ?thesis . 

408 
qed 

409 
qed 

410 
next 

411 
interpret semiring_gcd 1 0 "op *" gcd_eucl lcm_eucl "op div" "op +" "op " normalize unit_factor 

412 
by standard (rule lcm_gcd_eucl_facts; assumption)+ 

413 
fix p assume p: "irreducible p" 

414 
thus "is_prime_elem p" by (rule irreducible_imp_prime_gcd) 

415 
qed 

416 

417 
lemma gcd_eucl_eq_gcd_factorial: "gcd_eucl = gcd_factorial" 

418 
by (intro ext gcd_euclI gcd_lcm_factorial) 

419 

420 
lemma lcm_eucl_eq_lcm_factorial: "lcm_eucl = lcm_factorial" 

421 
by (intro ext) (simp add: lcm_eucl_def lcm_factorial_gcd_factorial gcd_eucl_eq_gcd_factorial) 

422 

423 
lemma Gcd_eucl_eq_Gcd_factorial: "Gcd_eucl = Gcd_factorial" 

424 
by (intro ext Gcd_euclI gcd_lcm_factorial) 

425 

426 
lemma Lcm_eucl_eq_Lcm_factorial: "Lcm_eucl = Lcm_factorial" 

427 
by (intro ext Lcm_euclI gcd_lcm_factorial) 

428 

429 
lemmas eucl_eq_factorial = 

430 
gcd_eucl_eq_gcd_factorial lcm_eucl_eq_lcm_factorial 

431 
Gcd_eucl_eq_Gcd_factorial Lcm_eucl_eq_Lcm_factorial 

432 

58023  433 
end 
434 

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435 
class euclidean_ring = euclidean_semiring + idom 
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436 
begin 
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437 

62457
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438 
subclass ring_div .. 
a3c7bd201da7
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439 

62442
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440 
function euclid_ext_aux :: "'a \<Rightarrow> _" where 
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441 
"euclid_ext_aux r' r s' s t' t = ( 
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442 
if r = 0 then let c = 1 div unit_factor r' in (s' * c, t' * c, normalize r') 
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443 
else let q = r' div r 
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444 
in euclid_ext_aux r (r' mod r) s (s'  q * s) t (t'  q * t))" 
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445 
by auto 
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446 
termination by (relation "measure (\<lambda>(_,b,_,_,_,_). euclidean_size b)") (simp_all add: mod_size_less) 
26e4be6a680f
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parents:
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447 

26e4be6a680f
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448 
declare euclid_ext_aux.simps [simp del] 
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449 

62442
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450 
lemma euclid_ext_aux_correct: 
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451 
assumes "gcd_eucl r' r = gcd_eucl x y" 
26e4be6a680f
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parents:
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452 
assumes "s' * x + t' * y = r'" 
26e4be6a680f
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parents:
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453 
assumes "s * x + t * y = r" 
26e4be6a680f
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454 
shows "case euclid_ext_aux r' r s' s t' t of (a,b,c) \<Rightarrow> 
26e4be6a680f
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parents:
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455 
a * x + b * y = c \<and> c = gcd_eucl x y" (is "?P (euclid_ext_aux r' r s' s t' t)") 
26e4be6a680f
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parents:
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diff
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456 
using assms 
26e4be6a680f
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457 
proof (induction r' r s' s t' t rule: euclid_ext_aux.induct) 
26e4be6a680f
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changeset

458 
case (1 r' r s' s t' t) 
26e4be6a680f
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459 
show ?case 
26e4be6a680f
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460 
proof (cases "r = 0") 
26e4be6a680f
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parents:
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diff
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461 
case True 
26e4be6a680f
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462 
hence "euclid_ext_aux r' r s' s t' t = 
26e4be6a680f
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diff
changeset

463 
(s' div unit_factor r', t' div unit_factor r', normalize r')" 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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diff
changeset

464 
by (subst euclid_ext_aux.simps) (simp add: Let_def) 
26e4be6a680f
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465 
also have "?P \<dots>" 
26e4be6a680f
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466 
proof safe 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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parents:
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diff
changeset

467 
have "s' div unit_factor r' * x + t' div unit_factor r' * y = 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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parents:
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changeset

468 
(s' * x + t' * y) div unit_factor r'" 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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diff
changeset

469 
by (cases "r' = 0") (simp_all add: unit_div_commute) 
26e4be6a680f
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parents:
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470 
also have "s' * x + t' * y = r'" by fact 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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parents:
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diff
changeset

471 
also have "\<dots> div unit_factor r' = normalize r'" by simp 
26e4be6a680f
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parents:
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472 
finally show "s' div unit_factor r' * x + t' div unit_factor r' * y = normalize r'" . 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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diff
changeset

473 
next 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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474 
from "1.prems" True show "normalize r' = gcd_eucl x y" by (simp add: gcd_eucl_0) 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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475 
qed 
26e4be6a680f
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476 
finally show ?thesis . 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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parents:
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diff
changeset

477 
next 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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478 
case False 
26e4be6a680f
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479 
hence "euclid_ext_aux r' r s' s t' t = 
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parents:
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diff
changeset

480 
euclid_ext_aux r (r' mod r) s (s'  r' div r * s) t (t'  r' div r * t)" 
26e4be6a680f
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parents:
62429
diff
changeset

481 
by (subst euclid_ext_aux.simps) (simp add: Let_def) 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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parents:
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diff
changeset

482 
also from "1.prems" False have "?P \<dots>" 
26e4be6a680f
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parents:
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diff
changeset

483 
proof (intro "1.IH") 
26e4be6a680f
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parents:
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diff
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484 
have "(s'  r' div r * s) * x + (t'  r' div r * t) * y = 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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parents:
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diff
changeset

485 
(s' * x + t' * y)  r' div r * (s * x + t * y)" by (simp add: algebra_simps) 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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parents:
62429
diff
changeset

486 
also have "s' * x + t' * y = r'" by fact 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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parents:
62429
diff
changeset

487 
also have "s * x + t * y = r" by fact 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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parents:
62429
diff
changeset

488 
also have "r'  r' div r * r = r' mod r" using mod_div_equality[of r' r] 
26e4be6a680f
More efficient Extended Euclidean Algorithm
Manuel Eberl <eberlm@in.tum.de>
parents:
62429
diff
changeset

489 
by (simp add: algebra_simps) 
26e4be6a680f
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parents:
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diff
changeset

490 
finally show "(s'  r' div r * s) * x + (t'  r' div r * t) * y = r' mod r" . 
26e4be6a680f
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diff
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491 
qed (auto simp: gcd_eucl_non_0 algebra_simps div_mod_equality') 
26e4be6a680f
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parents:
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diff
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492 
finally show ?thesis . 
26e4be6a680f
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diff
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493 
qed 
26e4be6a680f
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diff
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494 
qed 
26e4be6a680f
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Manuel Eberl <eberlm@in.tum.de>
parents:
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diff
changeset

495 

26e4be6a680f
More efficient Extended Euclidean Algorithm
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diff
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496 
definition euclid_ext where 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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497 
"euclid_ext a b = euclid_ext_aux a b 1 0 0 1" 
60598
78ca5674c66a
rings follow immediately their corresponding semirings
haftmann
parents:
60582
diff
changeset

498 

78ca5674c66a
rings follow immediately their corresponding semirings
haftmann
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diff
changeset

499 
lemma euclid_ext_0: 
60634  500 
"euclid_ext a 0 = (1 div unit_factor a, 0, normalize a)" 
62442
26e4be6a680f
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parents:
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diff
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501 
by (simp add: euclid_ext_def euclid_ext_aux.simps) 
60598
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rings follow immediately their corresponding semirings
haftmann
parents:
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diff
changeset

502 

78ca5674c66a
rings follow immediately their corresponding semirings
haftmann
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changeset

503 
lemma euclid_ext_left_0: 
60634  504 
"euclid_ext 0 a = (0, 1 div unit_factor a, normalize a)" 
62442
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diff
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505 
by (simp add: euclid_ext_def euclid_ext_aux.simps) 
60598
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rings follow immediately their corresponding semirings
haftmann
parents:
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diff
changeset

506 

62442
26e4be6a680f
More efficient Extended Euclidean Algorithm
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diff
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507 
lemma euclid_ext_correct': 
26e4be6a680f
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508 
"case euclid_ext x y of (a,b,c) \<Rightarrow> a * x + b * y = c \<and> c = gcd_eucl x y" 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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parents:
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diff
changeset

509 
unfolding euclid_ext_def by (rule euclid_ext_aux_correct) simp_all 
60598
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rings follow immediately their corresponding semirings
haftmann
parents:
60582
diff
changeset

510 

62457
a3c7bd201da7
Minor adjustments to euclidean rings
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parents:
62442
diff
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511 
lemma euclid_ext_gcd_eucl: 
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512 
"(case euclid_ext x y of (a,b,c) \<Rightarrow> c) = gcd_eucl x y" 
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Minor adjustments to euclidean rings
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513 
using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold) 
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Minor adjustments to euclidean rings
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changeset

514 

62442
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515 
definition euclid_ext' where 
26e4be6a680f
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516 
"euclid_ext' x y = (case euclid_ext x y of (a, b, _) \<Rightarrow> (a, b))" 
60598
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rings follow immediately their corresponding semirings
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517 

62442
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518 
lemma euclid_ext'_correct': 
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519 
"case euclid_ext' x y of (a,b) \<Rightarrow> a * x + b * y = gcd_eucl x y" 
26e4be6a680f
More efficient Extended Euclidean Algorithm
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parents:
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changeset

520 
using euclid_ext_correct'[of x y] by (simp add: case_prod_unfold euclid_ext'_def) 
60598
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521 

60634  522 
lemma euclid_ext'_0: "euclid_ext' a 0 = (1 div unit_factor a, 0)" 
60598
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523 
by (simp add: euclid_ext'_def euclid_ext_0) 
78ca5674c66a
rings follow immediately their corresponding semirings
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524 

60634  525 
lemma euclid_ext'_left_0: "euclid_ext' 0 a = (0, 1 div unit_factor a)" 
60598
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526 
by (simp add: euclid_ext'_def euclid_ext_left_0) 
78ca5674c66a
rings follow immediately their corresponding semirings
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527 

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528 
end 
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529 

58023  530 
class euclidean_semiring_gcd = euclidean_semiring + gcd + Gcd + 
531 
assumes gcd_gcd_eucl: "gcd = gcd_eucl" and lcm_lcm_eucl: "lcm = lcm_eucl" 

532 
assumes Gcd_Gcd_eucl: "Gcd = Gcd_eucl" and Lcm_Lcm_eucl: "Lcm = Lcm_eucl" 

533 
begin 

534 

62422  535 
subclass semiring_gcd 
536 
by standard (simp_all add: gcd_gcd_eucl gcd_eucl_greatest lcm_lcm_eucl lcm_eucl_def) 

58023  537 

62422  538 
subclass semiring_Gcd 
539 
by standard (auto simp: Gcd_Gcd_eucl Lcm_Lcm_eucl Gcd_eucl_def intro: Lcm_eucl_least) 

63498  540 

541 
subclass factorial_semiring_gcd 

542 
proof 

543 
fix a b 

544 
show "gcd a b = gcd_factorial a b" 

545 
by (rule sym, rule gcdI) (rule gcd_lcm_factorial; assumption)+ 

546 
thus "lcm a b = lcm_factorial a b" 

547 
by (simp add: lcm_factorial_gcd_factorial lcm_gcd) 

548 
next 

549 
fix A 

550 
show "Gcd A = Gcd_factorial A" 

551 
by (rule sym, rule GcdI) (rule gcd_lcm_factorial; assumption)+ 

552 
show "Lcm A = Lcm_factorial A" 

553 
by (rule sym, rule LcmI) (rule gcd_lcm_factorial; assumption)+ 

554 
qed 

555 

58023  556 
lemma gcd_non_0: 
60430
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557 
"b \<noteq> 0 \<Longrightarrow> gcd a b = gcd b (a mod b)" 
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558 
unfolding gcd_gcd_eucl by (fact gcd_eucl_non_0) 
58023  559 

62422  560 
lemmas gcd_0 = gcd_0_right 
561 
lemmas dvd_gcd_iff = gcd_greatest_iff 

58023  562 
lemmas gcd_greatest_iff = dvd_gcd_iff 
563 

564 
lemma gcd_mod1 [simp]: 

60430
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565 
"gcd (a mod b) b = gcd a b" 
58023  566 
by (rule gcdI, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff) 
567 

568 
lemma gcd_mod2 [simp]: 

60430
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changeset

569 
"gcd a (b mod a) = gcd a b" 
58023  570 
by (rule gcdI, simp, metis dvd_mod_iff gcd_dvd1 gcd_dvd2, simp_all add: gcd_greatest dvd_mod_iff) 
571 

572 
lemma euclidean_size_gcd_le1 [simp]: 

573 
assumes "a \<noteq> 0" 

574 
shows "euclidean_size (gcd a b) \<le> euclidean_size a" 

575 
proof  

576 
have "gcd a b dvd a" by (rule gcd_dvd1) 

577 
then obtain c where A: "a = gcd a b * c" unfolding dvd_def by blast 

60526  578 
with \<open>a \<noteq> 0\<close> show ?thesis by (subst (2) A, intro size_mult_mono) auto 
58023  579 
qed 
580 

581 
lemma euclidean_size_gcd_le2 [simp]: 

582 
"b \<noteq> 0 \<Longrightarrow> euclidean_size (gcd a b) \<le> euclidean_size b" 

583 
by (subst gcd.commute, rule euclidean_size_gcd_le1) 

584 

585 
lemma euclidean_size_gcd_less1: 

586 
assumes "a \<noteq> 0" and "\<not>a dvd b" 

587 
shows "euclidean_size (gcd a b) < euclidean_size a" 

588 
proof (rule ccontr) 

589 
assume "\<not>euclidean_size (gcd a b) < euclidean_size a" 

62422  590 
with \<open>a \<noteq> 0\<close> have A: "euclidean_size (gcd a b) = euclidean_size a" 
58023  591 
by (intro le_antisym, simp_all) 
62422  592 
have "a dvd gcd a b" 
593 
by (rule dvd_euclidean_size_eq_imp_dvd) (simp_all add: assms A) 

594 
hence "a dvd b" using dvd_gcdD2 by blast 

60526  595 
with \<open>\<not>a dvd b\<close> show False by contradiction 
58023  596 
qed 
597 

598 
lemma euclidean_size_gcd_less2: 

599 
assumes "b \<noteq> 0" and "\<not>b dvd a" 

600 
shows "euclidean_size (gcd a b) < euclidean_size b" 

601 
using assms by (subst gcd.commute, rule euclidean_size_gcd_less1) 

602 

603 
lemma euclidean_size_lcm_le1: 

604 
assumes "a \<noteq> 0" and "b \<noteq> 0" 

605 
shows "euclidean_size a \<le> euclidean_size (lcm a b)" 

606 
proof  

60690  607 
have "a dvd lcm a b" by (rule dvd_lcm1) 
608 
then obtain c where A: "lcm a b = a * c" .. 

62429
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Tuned Euclidean Rings/GCD rings
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parents:
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changeset

609 
with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "c \<noteq> 0" by (auto simp: lcm_eq_0_iff) 
58023  610 
then show ?thesis by (subst A, intro size_mult_mono) 
611 
qed 

612 

613 
lemma euclidean_size_lcm_le2: 

614 
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> euclidean_size b \<le> euclidean_size (lcm a b)" 

615 
using euclidean_size_lcm_le1 [of b a] by (simp add: ac_simps) 

616 

617 
lemma euclidean_size_lcm_less1: 

618 
assumes "b \<noteq> 0" and "\<not>b dvd a" 

619 
shows "euclidean_size a < euclidean_size (lcm a b)" 

620 
proof (rule ccontr) 

621 
from assms have "a \<noteq> 0" by auto 

622 
assume "\<not>euclidean_size a < euclidean_size (lcm a b)" 

60526  623 
with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "euclidean_size (lcm a b) = euclidean_size a" 
58023  624 
by (intro le_antisym, simp, intro euclidean_size_lcm_le1) 
625 
with assms have "lcm a b dvd a" 

62429
25271ff79171
Tuned Euclidean Rings/GCD rings
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parents:
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changeset

626 
by (rule_tac dvd_euclidean_size_eq_imp_dvd) (auto simp: lcm_eq_0_iff) 
62422  627 
hence "b dvd a" by (rule lcm_dvdD2) 
60526  628 
with \<open>\<not>b dvd a\<close> show False by contradiction 
58023  629 
qed 
630 

631 
lemma euclidean_size_lcm_less2: 

632 
assumes "a \<noteq> 0" and "\<not>a dvd b" 

633 
shows "euclidean_size b < euclidean_size (lcm a b)" 

634 
using assms euclidean_size_lcm_less1 [of a b] by (simp add: ac_simps) 

635 

62428
4d5fbec92bb1
Fixed code equations for Gcd/Lcm
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parents:
62425
diff
changeset

636 
lemma Lcm_eucl_set [code]: 
4d5fbec92bb1
Fixed code equations for Gcd/Lcm
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parents:
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diff
changeset

637 
"Lcm_eucl (set xs) = foldl lcm_eucl 1 xs" 
4d5fbec92bb1
Fixed code equations for Gcd/Lcm
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parents:
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changeset

638 
by (simp add: Lcm_Lcm_eucl [symmetric] lcm_lcm_eucl Lcm_set) 
58023  639 

62428
4d5fbec92bb1
Fixed code equations for Gcd/Lcm
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parents:
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diff
changeset

640 
lemma Gcd_eucl_set [code]: 
4d5fbec92bb1
Fixed code equations for Gcd/Lcm
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parents:
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diff
changeset

641 
"Gcd_eucl (set xs) = foldl gcd_eucl 0 xs" 
4d5fbec92bb1
Fixed code equations for Gcd/Lcm
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parents:
62425
diff
changeset

642 
by (simp add: Gcd_Gcd_eucl [symmetric] gcd_gcd_eucl Gcd_set) 
58023  643 

644 
end 

645 

63498  646 

60526  647 
text \<open> 
58023  648 
A Euclidean ring is a Euclidean semiring with additive inverses. It provides a 
649 
few more lemmas; in particular, Bezout's lemma holds for any Euclidean ring. 

60526  650 
\<close> 
58023  651 

652 
class euclidean_ring_gcd = euclidean_semiring_gcd + idom 

653 
begin 

654 

655 
subclass euclidean_ring .. 

60439
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proper subclass instances for existing gcd (semi)rings
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656 
subclass ring_gcd .. 
63498  657 
subclass factorial_ring_gcd .. 
60439
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proper subclass instances for existing gcd (semi)rings
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658 

60572
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659 
lemma euclid_ext_gcd [simp]: 
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660 
"(case euclid_ext a b of (_, _ , t) \<Rightarrow> t) = gcd a b" 
62442
26e4be6a680f
More efficient Extended Euclidean Algorithm
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parents:
62429
diff
changeset

661 
using euclid_ext_correct'[of a b] by (simp add: case_prod_unfold Let_def gcd_gcd_eucl) 
60572
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streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
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changeset

662 

718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
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663 
lemma euclid_ext_gcd' [simp]: 
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streamlined definitions and primitive lemma of euclidean algorithm, including code generation
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664 
"euclid_ext a b = (r, s, t) \<Longrightarrow> t = gcd a b" 
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
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diff
changeset

665 
by (insert euclid_ext_gcd[of a b], drule (1) subst, simp) 
62442
26e4be6a680f
More efficient Extended Euclidean Algorithm
Manuel Eberl <eberlm@in.tum.de>
parents:
62429
diff
changeset

666 

26e4be6a680f
More efficient Extended Euclidean Algorithm
Manuel Eberl <eberlm@in.tum.de>
parents:
62429
diff
changeset

667 
lemma euclid_ext_correct: 
26e4be6a680f
More efficient Extended Euclidean Algorithm
Manuel Eberl <eberlm@in.tum.de>
parents:
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diff
changeset

668 
"case euclid_ext x y of (a,b,c) \<Rightarrow> a * x + b * y = c \<and> c = gcd x y" 
26e4be6a680f
More efficient Extended Euclidean Algorithm
Manuel Eberl <eberlm@in.tum.de>
parents:
62429
diff
changeset

669 
using euclid_ext_correct'[of x y] 
26e4be6a680f
More efficient Extended Euclidean Algorithm
Manuel Eberl <eberlm@in.tum.de>
parents:
62429
diff
changeset

670 
by (simp add: gcd_gcd_eucl case_prod_unfold) 
60572
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streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
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diff
changeset

671 

718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
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diff
changeset

672 
lemma euclid_ext'_correct: 
718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
parents:
60571
diff
changeset

673 
"fst (euclid_ext' a b) * a + snd (euclid_ext' a b) * b = gcd a b" 
62442
26e4be6a680f
More efficient Extended Euclidean Algorithm
Manuel Eberl <eberlm@in.tum.de>
parents:
62429
diff
changeset

674 
using euclid_ext_correct'[of a b] 
26e4be6a680f
More efficient Extended Euclidean Algorithm
Manuel Eberl <eberlm@in.tum.de>
parents:
62429
diff
changeset

675 
by (simp add: gcd_gcd_eucl case_prod_unfold euclid_ext'_def) 
60572
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streamlined definitions and primitive lemma of euclidean algorithm, including code generation
haftmann
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changeset

676 

718b1ba06429
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changeset

677 
lemma bezout: "\<exists>s t. s * a + t * b = gcd a b" 
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678 
using euclid_ext'_correct by blast 
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679 

718b1ba06429
streamlined definitions and primitive lemma of euclidean algorithm, including code generation
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680 
end 
58023  681 

682 

60572
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changeset

683 
subsection \<open>Typical instances\<close> 
58023  684 

685 
instantiation nat :: euclidean_semiring 

686 
begin 

687 

688 
definition [simp]: 

689 
"euclidean_size_nat = (id :: nat \<Rightarrow> nat)" 

690 

63498  691 
instance by standard simp_all 
58023  692 

693 
end 

694 

62422  695 

58023  696 
instantiation int :: euclidean_ring 
697 
begin 

698 

699 
definition [simp]: 

700 
"euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)" 

701 

63498  702 
instance by standard (auto simp add: abs_mult nat_mult_distrib split: abs_split) 
58023  703 

704 
end 

705 

62422  706 
instance nat :: euclidean_semiring_gcd 
707 
proof 

708 
show [simp]: "gcd = (gcd_eucl :: nat \<Rightarrow> _)" "Lcm = (Lcm_eucl :: nat set \<Rightarrow> _)" 

709 
by (simp_all add: eq_gcd_euclI eq_Lcm_euclI) 

710 
show "lcm = (lcm_eucl :: nat \<Rightarrow> _)" "Gcd = (Gcd_eucl :: nat set \<Rightarrow> _)" 

711 
by (intro ext, simp add: lcm_eucl_def lcm_nat_def Gcd_nat_def Gcd_eucl_def)+ 

712 
qed 

713 

714 
instance int :: euclidean_ring_gcd 

715 
proof 

716 
show [simp]: "gcd = (gcd_eucl :: int \<Rightarrow> _)" "Lcm = (Lcm_eucl :: int set \<Rightarrow> _)" 

717 
by (simp_all add: eq_gcd_euclI eq_Lcm_euclI) 

718 
show "lcm = (lcm_eucl :: int \<Rightarrow> _)" "Gcd = (Gcd_eucl :: int set \<Rightarrow> _)" 

719 
by (intro ext, simp add: lcm_eucl_def lcm_altdef_int 

720 
semiring_Gcd_class.Gcd_Lcm Gcd_eucl_def abs_mult)+ 

721 
qed 

722 

63498  723 
end 