src/HOL/Number_Theory/Euclidean_Algorithm.thy
 changeset 60517 f16e4fb20652 parent 60516 0826b7025d07 child 60526 fad653acf58f
1.1 --- a/src/HOL/Number_Theory/Euclidean_Algorithm.thy	Fri Jun 19 07:53:33 2015 +0200
1.2 +++ b/src/HOL/Number_Theory/Euclidean_Algorithm.thy	Fri Jun 19 07:53:35 2015 +0200
1.3 @@ -5,276 +5,6 @@
1.4  theory Euclidean_Algorithm
1.5  imports Complex_Main
1.6  begin
1.7 -
1.8 -context semidom_divide
1.9 -begin
1.10 -
1.11 -lemma dvd_div_mult_self [simp]:
1.12 -  "a dvd b \<Longrightarrow> b div a * a = b"
1.13 -  by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
1.14 -
1.15 -lemma dvd_mult_div_cancel [simp]:
1.16 -  "a dvd b \<Longrightarrow> a * (b div a) = b"
1.17 -  using dvd_div_mult_self [of a b] by (simp add: ac_simps)
1.18 -
1.19 -lemma div_mult_swap:
1.20 -  assumes "c dvd b"
1.21 -  shows "a * (b div c) = (a * b) div c"
1.22 -proof (cases "c = 0")
1.23 -  case True then show ?thesis by simp
1.24 -next
1.25 -  case False from assms obtain d where "b = c * d" ..
1.26 -  moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
1.27 -    by simp
1.28 -  ultimately show ?thesis by (simp add: ac_simps)
1.29 -qed
1.30 -
1.31 -lemma dvd_div_mult:
1.32 -  assumes "c dvd b"
1.33 -  shows "b div c * a = (b * a) div c"
1.34 -  using assms div_mult_swap [of c b a] by (simp add: ac_simps)
1.35 -
1.36 -
1.37 -text \<open>Units: invertible elements in a ring\<close>
1.38 -
1.39 -abbreviation is_unit :: "'a \<Rightarrow> bool"
1.40 -where
1.41 -  "is_unit a \<equiv> a dvd 1"
1.42 -
1.43 -lemma not_is_unit_0 [simp]:
1.44 -  "\<not> is_unit 0"
1.45 -  by simp
1.46 -
1.47 -lemma unit_imp_dvd [dest]:
1.48 -  "is_unit b \<Longrightarrow> b dvd a"
1.49 -  by (rule dvd_trans [of _ 1]) simp_all
1.50 -
1.51 -lemma unit_dvdE:
1.52 -  assumes "is_unit a"
1.53 -  obtains c where "a \<noteq> 0" and "b = a * c"
1.54 -proof -
1.55 -  from assms have "a dvd b" by auto
1.56 -  then obtain c where "b = a * c" ..
1.57 -  moreover from assms have "a \<noteq> 0" by auto
1.58 -  ultimately show thesis using that by blast
1.59 -qed
1.60 -
1.61 -lemma dvd_unit_imp_unit:
1.62 -  "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
1.63 -  by (rule dvd_trans)
1.64 -
1.65 -lemma unit_div_1_unit [simp, intro]:
1.66 -  assumes "is_unit a"
1.67 -  shows "is_unit (1 div a)"
1.68 -proof -
1.69 -  from assms have "1 = 1 div a * a" by simp
1.70 -  then show "is_unit (1 div a)" by (rule dvdI)
1.71 -qed
1.72 -
1.73 -lemma is_unitE [elim?]:
1.74 -  assumes "is_unit a"
1.75 -  obtains b where "a \<noteq> 0" and "b \<noteq> 0"
1.76 -    and "is_unit b" and "1 div a = b" and "1 div b = a"
1.77 -    and "a * b = 1" and "c div a = c * b"
1.78 -proof (rule that)
1.79 -  def b \<equiv> "1 div a"
1.80 -  then show "1 div a = b" by simp
1.81 -  from b_def `is_unit a` show "is_unit b" by simp
1.82 -  from `is_unit a` and `is_unit b` show "a \<noteq> 0" and "b \<noteq> 0" by auto
1.83 -  from b_def `is_unit a` show "a * b = 1" by simp
1.84 -  then have "1 = a * b" ..
1.85 -  with b_def `b \<noteq> 0` show "1 div b = a" by simp
1.86 -  from `is_unit a` have "a dvd c" ..
1.87 -  then obtain d where "c = a * d" ..
1.88 -  with `a \<noteq> 0` `a * b = 1` show "c div a = c * b"
1.89 -    by (simp add: mult.assoc mult.left_commute [of a])
1.90 -qed
1.91 -
1.92 -lemma unit_prod [intro]:
1.93 -  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
1.94 -  by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
1.95 -
1.96 -lemma unit_div [intro]:
1.97 -  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
1.98 -  by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
1.99 -
1.100 -lemma mult_unit_dvd_iff:
1.101 -  assumes "is_unit b"
1.102 -  shows "a * b dvd c \<longleftrightarrow> a dvd c"
1.103 -proof
1.104 -  assume "a * b dvd c"
1.105 -  with assms show "a dvd c"
1.106 -    by (simp add: dvd_mult_left)
1.107 -next
1.108 -  assume "a dvd c"
1.109 -  then obtain k where "c = a * k" ..
1.110 -  with assms have "c = (a * b) * (1 div b * k)"
1.111 -    by (simp add: mult_ac)
1.112 -  then show "a * b dvd c" by (rule dvdI)
1.113 -qed
1.115 -lemma dvd_mult_unit_iff:
1.116 -  assumes "is_unit b"
1.117 -  shows "a dvd c * b \<longleftrightarrow> a dvd c"
1.118 -proof
1.119 -  assume "a dvd c * b"
1.120 -  with assms have "c * b dvd c * (b * (1 div b))"
1.121 -    by (subst mult_assoc [symmetric]) simp
1.122 -  also from `is_unit b` have "b * (1 div b) = 1" by (rule is_unitE) simp
1.123 -  finally have "c * b dvd c" by simp
1.124 -  with `a dvd c * b` show "a dvd c" by (rule dvd_trans)
1.125 -next
1.126 -  assume "a dvd c"
1.127 -  then show "a dvd c * b" by simp
1.128 -qed
1.130 -lemma div_unit_dvd_iff:
1.131 -  "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
1.132 -  by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
1.134 -lemma dvd_div_unit_iff:
1.135 -  "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
1.136 -  by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
1.138 -lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
1.139 -  dvd_mult_unit_iff dvd_div_unit_iff -- \<open>FIXME consider fact collection\<close>
1.141 -lemma unit_mult_div_div [simp]:
1.142 -  "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
1.143 -  by (erule is_unitE [of _ b]) simp
1.145 -lemma unit_div_mult_self [simp]:
1.146 -  "is_unit a \<Longrightarrow> b div a * a = b"
1.147 -  by (rule dvd_div_mult_self) auto
1.149 -lemma unit_div_1_div_1 [simp]:
1.150 -  "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
1.151 -  by (erule is_unitE) simp
1.153 -lemma unit_div_mult_swap:
1.154 -  "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
1.155 -  by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
1.157 -lemma unit_div_commute:
1.158 -  "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
1.159 -  using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
1.161 -lemma unit_eq_div1:
1.162 -  "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
1.163 -  by (auto elim: is_unitE)
1.165 -lemma unit_eq_div2:
1.166 -  "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
1.167 -  using unit_eq_div1 [of b c a] by auto
1.169 -lemma unit_mult_left_cancel:
1.170 -  assumes "is_unit a"
1.171 -  shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
1.172 -  using assms mult_cancel_left [of a b c] by auto
1.174 -lemma unit_mult_right_cancel:
1.175 -  "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
1.176 -  using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
1.178 -lemma unit_div_cancel:
1.179 -  assumes "is_unit a"
1.180 -  shows "b div a = c div a \<longleftrightarrow> b = c"
1.181 -proof -
1.182 -  from assms have "is_unit (1 div a)" by simp
1.183 -  then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
1.184 -    by (rule unit_mult_right_cancel)
1.185 -  with assms show ?thesis by simp
1.186 -qed
1.189 -text \<open>Associated elements in a ring – an equivalence relation induced by the quasi-order divisibility \<close>
1.191 -definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
1.192 -where
1.193 -  "associated a b \<longleftrightarrow> a dvd b \<and> b dvd a"
1.195 -lemma associatedI:
1.196 -  "a dvd b \<Longrightarrow> b dvd a \<Longrightarrow> associated a b"
1.197 -  by (simp add: associated_def)
1.199 -lemma associatedD1:
1.200 -  "associated a b \<Longrightarrow> a dvd b"
1.201 -  by (simp add: associated_def)
1.203 -lemma associatedD2:
1.204 -  "associated a b \<Longrightarrow> b dvd a"
1.205 -  by (simp add: associated_def)
1.207 -lemma associated_refl [simp]:
1.208 -  "associated a a"
1.209 -  by (auto intro: associatedI)
1.211 -lemma associated_sym:
1.212 -  "associated b a \<longleftrightarrow> associated a b"
1.213 -  by (auto intro: associatedI dest: associatedD1 associatedD2)
1.215 -lemma associated_trans:
1.216 -  "associated a b \<Longrightarrow> associated b c \<Longrightarrow> associated a c"
1.217 -  by (auto intro: associatedI dvd_trans dest: associatedD1 associatedD2)
1.219 -lemma equivp_associated:
1.220 -  "equivp associated"
1.221 -proof (rule equivpI)
1.222 -  show "reflp associated"
1.223 -    by (rule reflpI) simp
1.224 -  show "symp associated"
1.225 -    by (rule sympI) (simp add: associated_sym)
1.226 -  show "transp associated"
1.227 -    by (rule transpI) (fact associated_trans)
1.228 -qed
1.230 -lemma associated_0 [simp]:
1.231 -  "associated 0 b \<longleftrightarrow> b = 0"
1.232 -  "associated a 0 \<longleftrightarrow> a = 0"
1.233 -  by (auto dest: associatedD1 associatedD2)
1.235 -lemma associated_unit:
1.236 -  "associated a b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
1.237 -  using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
1.239 -lemma is_unit_associatedI:
1.240 -  assumes "is_unit c" and "a = c * b"
1.241 -  shows "associated a b"
1.242 -proof (rule associatedI)
1.243 -  from `a = c * b` show "b dvd a" by auto
1.244 -  from `is_unit c` obtain d where "c * d = 1" by (rule is_unitE)
1.245 -  moreover from `a = c * b` have "d * a = d * (c * b)" by simp
1.246 -  ultimately have "b = a * d" by (simp add: ac_simps)
1.247 -  then show "a dvd b" ..
1.248 -qed
1.250 -lemma associated_is_unitE:
1.251 -  assumes "associated a b"
1.252 -  obtains c where "is_unit c" and "a = c * b"
1.253 -proof (cases "b = 0")
1.254 -  case True with assms have "is_unit 1" and "a = 1 * b" by simp_all
1.255 -  with that show thesis .
1.256 -next
1.257 -  case False
1.258 -  from assms have "a dvd b" and "b dvd a" by (auto dest: associatedD1 associatedD2)
1.259 -  then obtain c d where "b = a * d" and "a = b * c" by (blast elim: dvdE)
1.260 -  then have "a = c * b" and "(c * d) * b = 1 * b" by (simp_all add: ac_simps)
1.261 -  with False have "c * d = 1" using mult_cancel_right [of "c * d" b 1] by simp
1.262 -  then have "is_unit c" by auto
1.263 -  with `a = c * b` that show thesis by blast
1.264 -qed
1.266 -lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
1.267 -  dvd_div_unit_iff unit_div_mult_swap unit_div_commute
1.268 -  unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
1.269 -  unit_eq_div1 unit_eq_div2
1.271 -end
1.273 -lemma is_unit_int:
1.274 -  "is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
1.275 -  by auto
1.278  text {*
1.279    A Euclidean semiring is a semiring upon which the Euclidean algorithm can be