author haftmann Wed Jul 08 14:01:40 2015 +0200 (2015-07-08) changeset 60687 33dbbcb6a8a3 parent 60686 ea5bc46c11e6 child 60688 01488b559910
eliminated some duplication
 src/HOL/GCD.thy file | annotate | diff | revisions src/HOL/Number_Theory/Euclidean_Algorithm.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/GCD.thy	Wed Jul 08 14:01:39 2015 +0200
1.2 +++ b/src/HOL/GCD.thy	Wed Jul 08 14:01:40 2015 +0200
1.3 @@ -399,7 +399,7 @@
1.4    by (rule dvd_0_left, rule Gcd_greatest) simp
1.5
1.6  lemma Gcd_0_iff [simp]:
1.7 -  "Gcd A = 0 \<longleftrightarrow> (\<forall>a\<in>A. a = 0)" (is "?P \<longleftrightarrow> ?Q")
1.8 +  "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}" (is "?P \<longleftrightarrow> ?Q")
1.9  proof
1.10    assume ?P
1.11    show ?Q
1.12 @@ -407,7 +407,8 @@
1.13      fix a
1.14      assume "a \<in> A"
1.15      then have "Gcd A dvd a" by (rule Gcd_dvd)
1.16 -    with \<open>?P\<close> show "a = 0" by simp
1.17 +    with \<open>?P\<close> have "a = 0" by simp
1.18 +    then show "a \<in> {0}" by simp
1.19    qed
1.20  next
1.21    assume ?Q
1.22 @@ -415,7 +416,7 @@
1.23    proof (rule Gcd_greatest)
1.24      fix a
1.25      assume "a \<in> A"
1.26 -    with \<open>?Q\<close> have "a = 0" by simp
1.27 +    with \<open>?Q\<close> have "a = 0" by auto
1.28      then show "0 dvd a" by simp
1.29    qed
1.30    then show ?P by simp
1.31 @@ -424,7 +425,7 @@
1.32  lemma unit_factor_Gcd:
1.33    "unit_factor (Gcd A) = (if \<forall>a\<in>A. a = 0 then 0 else 1)"
1.34  proof (cases "Gcd A = 0")
1.35 -  case True then show ?thesis by simp
1.36 +  case True then show ?thesis by auto
1.37  next
1.38    case False
1.39    from unit_factor_mult_normalize
1.40 @@ -432,7 +433,7 @@
1.41    then have "unit_factor (Gcd A) * Gcd A = Gcd A" by simp
1.42    then have "unit_factor (Gcd A) * Gcd A div Gcd A = Gcd A div Gcd A" by simp
1.43    with False have "unit_factor (Gcd A) = 1" by simp
1.44 -  with False show ?thesis by simp
1.45 +  with False show ?thesis by auto
1.46  qed
1.47
1.48  lemma Gcd_UNIV [simp]:
1.49 @@ -473,7 +474,7 @@
1.50    ultimately have "associated (Gcd (insert a A)) (gcd a (Gcd A))"
1.51      by (rule associatedI)
1.52    then show ?thesis
1.53 -    by (rule associated_eqI) (simp_all add: unit_factor_gcd unit_factor_Gcd)
1.54 +    by (rule associated_eqI) (auto simp add: unit_factor_gcd unit_factor_Gcd)
1.55  qed
1.56
1.57  lemma dvd_Gcd: -- \<open>FIXME remove\<close>
1.58 @@ -509,7 +510,7 @@
1.59    then have "associated (Gcd (normalize ` A)) (Gcd A)"
1.60      by (auto intro!: associatedI Gcd_greatest intro: Gcd_dvd)
1.61    then show ?thesis
1.62 -    by (rule associated_eqI) (simp_all add: unit_factor_Gcd)
1.63 +    by (rule associated_eqI) (simp_all add: unit_factor_Gcd, auto dest!: bspec)
1.64  qed
1.65
1.66  lemma Lcm_least:
```
```     2.1 --- a/src/HOL/Number_Theory/Euclidean_Algorithm.thy	Wed Jul 08 14:01:39 2015 +0200
2.2 +++ b/src/HOL/Number_Theory/Euclidean_Algorithm.thy	Wed Jul 08 14:01:40 2015 +0200
2.3 @@ -753,84 +753,19 @@
2.4    "lcm a b = normalize (a * b) div gcd a b"
2.5    by (simp add: lcm_lcm_eucl gcd_gcd_eucl lcm_eucl_def)
2.6
2.7 +subclass semiring_gcd
2.8 +  apply standard
2.9 +  using gcd_right_idem
2.10 +  apply (simp_all add: lcm_gcd gcd_greatest_iff gcd_proj1_if_dvd)
2.11 +  done
2.12 +
2.13  lemma lcm_gcd_prod:
2.14    "lcm a b * gcd a b = normalize (a * b)"
2.16
2.17 -lemma lcm_dvd1 [iff]:
2.18 -  "a dvd lcm a b"
2.19 -proof (cases "a*b = 0")
2.20 -  assume "a * b \<noteq> 0"
2.21 -  hence "gcd a b \<noteq> 0" by simp
2.22 -  let ?c = "1 div unit_factor (a * b)"
2.23 -  from \<open>a * b \<noteq> 0\<close> have [simp]: "is_unit (unit_factor (a * b))" by simp
2.24 -  from lcm_gcd_prod[of a b] have "lcm a b * gcd a b = a * ?c * b"
2.25 -    by (simp add: div_mult_swap unit_div_commute)
2.26 -  hence "lcm a b * gcd a b div gcd a b = a * ?c * b div gcd a b" by simp
2.27 -  with \<open>gcd a b \<noteq> 0\<close> have "lcm a b = a * ?c * b div gcd a b"
2.28 -    by (subst (asm) div_mult_self2_is_id, simp_all)
2.29 -  also have "... = a * (?c * b div gcd a b)"
2.30 -    by (metis div_mult_swap gcd_dvd2 mult_assoc)
2.31 -  finally show ?thesis by (rule dvdI)
2.32 -qed (auto simp add: lcm_gcd)
2.33 -
2.34 -lemma lcm_least:
2.35 -  "\<lbrakk>a dvd k; b dvd k\<rbrakk> \<Longrightarrow> lcm a b dvd k"
2.36 -proof (cases "k = 0")
2.37 -  let ?nf = unit_factor
2.38 -  assume "k \<noteq> 0"
2.39 -  hence "is_unit (?nf k)" by simp
2.40 -  hence "?nf k \<noteq> 0" by (metis not_is_unit_0)
2.41 -  assume A: "a dvd k" "b dvd k"
2.42 -  hence "gcd a b \<noteq> 0" using \<open>k \<noteq> 0\<close> by auto
2.43 -  from A obtain r s where ar: "k = a * r" and bs: "k = b * s"
2.44 -    unfolding dvd_def by blast
2.45 -  with \<open>k \<noteq> 0\<close> have "r * s \<noteq> 0"
2.46 -    by auto (drule sym [of 0], simp)
2.47 -  hence "is_unit (?nf (r * s))" by simp
2.48 -  let ?c = "?nf k div ?nf (r*s)"
2.49 -  from \<open>is_unit (?nf k)\<close> and \<open>is_unit (?nf (r * s))\<close> have "is_unit ?c" by (rule unit_div)
2.50 -  hence "?c \<noteq> 0" using not_is_unit_0 by fast
2.51 -  from ar bs have "k * k * gcd s r = ?nf k * k * gcd (k * s) (k * r)"
2.52 -    by (subst mult_assoc, subst gcd_mult_distrib[of k s r], simp only: ac_simps)
2.53 -  also have "... = ?nf k * k * gcd ((r*s) * a) ((r*s) * b)"
2.54 -    by (subst (3) \<open>k = a * r\<close>, subst (3) \<open>k = b * s\<close>, simp add: algebra_simps)
2.55 -  also have "... = ?c * r*s * k * gcd a b" using \<open>r * s \<noteq> 0\<close>
2.56 -    by (subst gcd_mult_distrib'[symmetric], simp add: algebra_simps unit_simps)
2.57 -  finally have "(a*r) * (b*s) * gcd s r = ?c * k * r * s * gcd a b"
2.58 -    by (subst ar[symmetric], subst bs[symmetric], simp add: mult_ac)
2.59 -  hence "a * b * gcd s r * (r * s) = ?c * k * gcd a b * (r * s)"
2.60 -    by (simp add: algebra_simps)
2.61 -  hence "?c * k * gcd a b = a * b * gcd s r" using \<open>r * s \<noteq> 0\<close>
2.62 -    by (metis div_mult_self2_is_id)
2.63 -  also have "... = lcm a b * gcd a b * gcd s r * ?nf (a*b)"
2.64 -    by (subst lcm_gcd_prod[of a b], metis gcd_mult_distrib gcd_mult_distrib')
2.65 -  also have "... = lcm a b * gcd s r * ?nf (a*b) * gcd a b"
2.66 -    by (simp add: algebra_simps)
2.67 -  finally have "k * ?c = lcm a b * gcd s r * ?nf (a*b)" using \<open>gcd a b \<noteq> 0\<close>
2.68 -    by (metis mult.commute div_mult_self2_is_id)
2.69 -  hence "k = lcm a b * (gcd s r * ?nf (a*b)) div ?c" using \<open>?c \<noteq> 0\<close>
2.70 -    by (metis div_mult_self2_is_id mult_assoc)
2.71 -  also have "... = lcm a b * (gcd s r * ?nf (a*b) div ?c)" using \<open>is_unit ?c\<close>
2.72 -    by (simp add: unit_simps)
2.73 -  finally show ?thesis by (rule dvdI)
2.74 -qed simp
2.75 -
2.76  lemma lcm_zero:
2.77    "lcm a b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
2.78 -proof -
2.79 -  let ?nf = unit_factor
2.80 -  {
2.81 -    assume "a \<noteq> 0" "b \<noteq> 0"
2.82 -    hence "a * b div ?nf (a * b) \<noteq> 0" by (simp add: no_zero_divisors)
2.83 -    moreover from \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> have "gcd a b \<noteq> 0" by simp
2.84 -    ultimately have "lcm a b \<noteq> 0" using lcm_gcd_prod[of a b] by (intro notI, simp)
2.85 -  } moreover {
2.86 -    assume "a = 0 \<or> b = 0"
2.87 -    hence "lcm a b = 0" by (elim disjE, simp_all add: lcm_gcd)
2.88 -  }
2.89 -  ultimately show ?thesis by blast
2.90 -qed
2.91 +  by (fact lcm_eq_0_iff)
2.92
2.93  lemmas lcm_0_iff = lcm_zero
2.94
2.95 @@ -844,46 +779,15 @@
2.96      by (metis nonzero_mult_divide_cancel_left)
2.97  qed
2.98
2.99 -lemma unit_factor_lcm [simp]:
2.100 -  "unit_factor (lcm a b) = (if a = 0 \<or> b = 0 then 0 else 1)"
2.101 -  by (simp add: dvd_unit_factor_div lcm_gcd)
2.102 -
2.103 -lemma lcm_dvd2 [iff]: "b dvd lcm a b"
2.104 -  using lcm_dvd1 [of b a] by (simp add: lcm_gcd ac_simps)
2.105 +declare unit_factor_lcm [simp]
2.106
2.107  lemma lcmI:
2.108    assumes "a dvd c" and "b dvd c" and "\<And>d. a dvd d \<Longrightarrow> b dvd d \<Longrightarrow> c dvd d"
2.109      and "unit_factor c = (if c = 0 then 0 else 1)"
2.110    shows "c = lcm a b"
2.111 -  by (rule associated_eqI)
2.112 -    (auto simp: assms associated_def intro: lcm_least, simp_all add: lcm_gcd)
2.113 +  by (rule associated_eqI) (auto simp: assms intro: associatedI lcm_least)
2.114
2.115 -sublocale lcm!: abel_semigroup lcm
2.116 -proof
2.117 -  fix a b c
2.118 -  show "lcm (lcm a b) c = lcm a (lcm b c)"
2.119 -  proof (rule lcmI)
2.120 -    have "a dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
2.121 -    then show "a dvd lcm (lcm a b) c" by (rule dvd_trans)
2.122 -
2.123 -    have "b dvd lcm a b" and "lcm a b dvd lcm (lcm a b) c" by simp_all
2.124 -    hence "b dvd lcm (lcm a b) c" by (rule dvd_trans)
2.125 -    moreover have "c dvd lcm (lcm a b) c" by simp
2.126 -    ultimately show "lcm b c dvd lcm (lcm a b) c" by (rule lcm_least)
2.127 -
2.128 -    fix l assume "a dvd l" and "lcm b c dvd l"
2.129 -    have "b dvd lcm b c" by simp
2.130 -    from this and \<open>lcm b c dvd l\<close> have "b dvd l" by (rule dvd_trans)
2.131 -    have "c dvd lcm b c" by simp
2.132 -    from this and \<open>lcm b c dvd l\<close> have "c dvd l" by (rule dvd_trans)
2.133 -    from \<open>a dvd l\<close> and \<open>b dvd l\<close> have "lcm a b dvd l" by (rule lcm_least)
2.134 -    from this and \<open>c dvd l\<close> show "lcm (lcm a b) c dvd l" by (rule lcm_least)
2.135 -  qed (simp add: lcm_zero)
2.136 -next
2.137 -  fix a b
2.138 -  show "lcm a b = lcm b a"
2.139 -    by (simp add: lcm_gcd ac_simps)
2.140 -qed
2.141 +sublocale lcm!: abel_semigroup lcm ..
2.142
2.143  lemma dvd_lcm_D1:
2.144    "lcm m n dvd k \<Longrightarrow> m dvd k"
2.145 @@ -913,13 +817,9 @@
2.146    finally show "lcm a b = 1" .
2.147  qed
2.148
2.149 -lemma lcm_0_left [simp]:
2.150 -  "lcm 0 a = 0"
2.151 -  by (rule sym, rule lcmI, simp_all)
2.152 -
2.153 -lemma lcm_0 [simp]:
2.154 +lemma lcm_0:
2.155    "lcm a 0 = 0"
2.156 -  by (rule sym, rule lcmI, simp_all)
2.157 +  by (fact lcm_0_right)
2.158
2.159  lemma lcm_unique:
2.160    "a dvd d \<and> b dvd d \<and>
2.161 @@ -935,14 +835,6 @@
2.162    "k dvd n \<Longrightarrow> k dvd lcm m n"
2.163    by (metis lcm_dvd2 dvd_trans)
2.164
2.165 -lemma lcm_1_left [simp]:
2.166 -  "lcm 1 a = normalize a"
2.167 -  by (cases "a = 0") (simp, rule sym, rule lcmI, simp_all)
2.168 -
2.169 -lemma lcm_1_right [simp]:
2.170 -  "lcm a 1 = normalize a"
2.171 -  using lcm_1_left [of a] by (simp add: ac_simps)
2.172 -
2.173  lemma lcm_coprime:
2.174    "gcd a b = 1 \<Longrightarrow> lcm a b = normalize (a * b)"
2.175    by (subst lcm_gcd) simp
2.176 @@ -1119,7 +1011,8 @@
2.177        qed
2.178        ultimately have "euclidean_size l = euclidean_size (gcd l l')"
2.179          by (intro le_antisym, simp_all add: \<open>euclidean_size l = n\<close>)
2.180 -      with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'" by (blast intro: dvd_euclidean_size_eq_imp_dvd)
2.181 +      with \<open>l \<noteq> 0\<close> have "l dvd gcd l l'"
2.182 +        using dvd_euclidean_size_eq_imp_dvd by auto
2.183        hence "l dvd l'" by (blast dest: dvd_gcd_D2)
2.184      }
2.185
2.186 @@ -1246,7 +1139,7 @@
2.187    "Lcm (insert a A) = lcm a (Lcm A)"
2.188  proof (rule lcmI)
2.189    fix l assume "a dvd l" and "Lcm A dvd l"
2.190 -  hence "\<forall>a\<in>A. a dvd l" by (blast intro: dvd_trans dvd_Lcm)
2.191 +  then have "\<forall>a\<in>A. a dvd l" by (auto intro: dvd_trans [of _ "Lcm A" l])
2.192    with \<open>a dvd l\<close> show "Lcm (insert a A) dvd l" by (force intro: Lcm_least)
2.193  qed (auto intro: Lcm_least dvd_Lcm)
2.194
2.195 @@ -1313,6 +1206,9 @@
2.196    "normalize (Gcd A) = Gcd A"
2.197    by (cases "Gcd A = 0") (auto intro: associated_eqI)
2.198
2.199 +subclass semiring_Gcd
2.200 +  by standard (simp_all add: Gcd_greatest)
2.201 +
2.202  lemma GcdI:
2.203    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"
2.204      and "unit_factor b = (if b = 0 then 0 else 1)"
2.205 @@ -1323,28 +1219,12 @@
2.206    "Lcm A = Gcd {m. \<forall>a\<in>A. a dvd m}"
2.207    by (rule LcmI[symmetric]) (auto intro: dvd_Gcd Gcd_greatest)
2.208
2.209 -lemma Gcd_0_iff:
2.210 -  "Gcd A = 0 \<longleftrightarrow> A \<subseteq> {0}"
2.211 -  apply (rule iffI)
2.212 -  apply (rule subsetI, drule Gcd_dvd, simp)
2.213 -  apply (auto intro: GcdI[symmetric])
2.214 -  done
2.215 -
2.216 -lemma Gcd_empty [simp]:
2.217 -  "Gcd {} = 0"
2.218 -  by (simp add: Gcd_0_iff)
2.219 +subclass semiring_Lcm
2.220 +  by standard (simp add: Lcm_Gcd)
2.221
2.222  lemma Gcd_1:
2.223    "1 \<in> A \<Longrightarrow> Gcd A = 1"
2.224 -  by (intro GcdI[symmetric]) (auto intro: Gcd_dvd dvd_Gcd)
2.225 -
2.226 -lemma Gcd_insert [simp]:
2.227 -  "Gcd (insert a A) = gcd a (Gcd A)"
2.228 -proof (rule gcdI)
2.229 -  fix l assume "l dvd a" and "l dvd Gcd A"
2.230 -  hence "\<forall>a\<in>A. l dvd a" by (blast intro: dvd_trans Gcd_dvd)
2.231 -  with \<open>l dvd a\<close> show "l dvd Gcd (insert a A)" by (force intro: Gcd_dvd Gcd_greatest)
2.232 -qed (auto intro: Gcd_greatest)
2.233 +  by (auto intro!: Gcd_eq_1_I)
2.234
2.235  lemma Gcd_finite:
2.236    assumes "finite A"
2.237 @@ -1357,22 +1237,10 @@
2.238    using comp_fun_idem.fold_set_fold[OF comp_fun_idem_gcd] Gcd_finite by (simp add: ac_simps)
2.239
2.240  lemma Gcd_singleton [simp]: "Gcd {a} = normalize a"
2.241 -  by (simp add: gcd_0)
2.242 +  by simp
2.243
2.244  lemma Gcd_2 [simp]: "Gcd {a,b} = gcd a b"
2.245 -  by (simp add: gcd_0)
2.246 -
2.247 -subclass semiring_gcd
2.248 -  apply standard
2.249 -  using gcd_right_idem
2.250 -  apply (simp_all add: lcm_gcd gcd_greatest_iff gcd_proj1_if_dvd)
2.251 -  done
2.252 -
2.253 -subclass semiring_Gcd
2.254 -  by standard (simp_all add: Gcd_greatest)
2.255 -
2.256 -subclass semiring_Lcm
2.257 -  by standard (simp add: Lcm_Gcd)
2.258 +  by simp
2.259
2.260  end
2.261
2.262 @@ -1515,4 +1383,19 @@
2.263
2.264  end
2.265
2.266 +(*instance nat :: euclidean_semiring_gcd
2.267 +proof (standard, auto intro!: ext)
2.268 +  fix m n :: nat
2.269 +  show *: "gcd m n = gcd_eucl m n"
2.270 +  proof (induct m n rule: gcd_eucl_induct)
2.271 +    case zero then show ?case by (simp add: gcd_eucl_0)
2.272 +  next
2.273 +    case (mod m n)
2.274 +    with gcd_eucl_non_0 [of n m, symmetric]
2.275 +    show ?case by (simp add: gcd_non_0_nat)
2.276 +  qed
2.277 +  show "lcm m n = lcm_eucl m n"
2.278 +    by (simp add: lcm_eucl_def lcm_gcd * [symmetric] lcm_nat_def)
2.279 +qed*)
2.280 +
2.281  end
```