1 (* Title: HOL/GCD.thy |
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2 Author: Christophe Tabacznyj and Lawrence C Paulson |
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3 Copyright 1996 University of Cambridge |
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4 *) |
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5 |
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6 header {* The Greatest Common Divisor *} |
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7 |
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8 theory Legacy_GCD |
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9 imports Main |
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10 begin |
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11 |
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12 text {* |
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13 See \cite{davenport92}. \bigskip |
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14 *} |
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15 |
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16 subsection {* Specification of GCD on nats *} |
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17 |
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18 definition |
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19 is_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where -- {* @{term gcd} as a relation *} |
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20 [code del]: "is_gcd m n p \<longleftrightarrow> p dvd m \<and> p dvd n \<and> |
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21 (\<forall>d. d dvd m \<longrightarrow> d dvd n \<longrightarrow> d dvd p)" |
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22 |
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23 text {* Uniqueness *} |
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24 |
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25 lemma is_gcd_unique: "is_gcd a b m \<Longrightarrow> is_gcd a b n \<Longrightarrow> m = n" |
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26 by (simp add: is_gcd_def) (blast intro: dvd_anti_sym) |
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27 |
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28 text {* Connection to divides relation *} |
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29 |
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30 lemma is_gcd_dvd: "is_gcd a b m \<Longrightarrow> k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd m" |
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31 by (auto simp add: is_gcd_def) |
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32 |
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33 text {* Commutativity *} |
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34 |
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35 lemma is_gcd_commute: "is_gcd m n k = is_gcd n m k" |
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36 by (auto simp add: is_gcd_def) |
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37 |
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38 |
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39 subsection {* GCD on nat by Euclid's algorithm *} |
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40 |
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41 fun |
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42 gcd :: "nat => nat => nat" |
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43 where |
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44 "gcd m n = (if n = 0 then m else gcd n (m mod n))" |
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45 lemma gcd_induct [case_names "0" rec]: |
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46 fixes m n :: nat |
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47 assumes "\<And>m. P m 0" |
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48 and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n" |
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49 shows "P m n" |
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50 proof (induct m n rule: gcd.induct) |
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51 case (1 m n) with assms show ?case by (cases "n = 0") simp_all |
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52 qed |
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53 |
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54 lemma gcd_0 [simp, algebra]: "gcd m 0 = m" |
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55 by simp |
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56 |
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57 lemma gcd_0_left [simp,algebra]: "gcd 0 m = m" |
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58 by simp |
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59 |
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60 lemma gcd_non_0: "n > 0 \<Longrightarrow> gcd m n = gcd n (m mod n)" |
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61 by simp |
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62 |
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63 lemma gcd_1 [simp, algebra]: "gcd m (Suc 0) = Suc 0" |
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64 by simp |
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65 |
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66 lemma nat_gcd_1_right [simp, algebra]: "gcd m 1 = 1" |
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67 unfolding One_nat_def by (rule gcd_1) |
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68 |
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69 declare gcd.simps [simp del] |
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70 |
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71 text {* |
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72 \medskip @{term "gcd m n"} divides @{text m} and @{text n}. The |
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73 conjunctions don't seem provable separately. |
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74 *} |
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75 |
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76 lemma gcd_dvd1 [iff, algebra]: "gcd m n dvd m" |
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77 and gcd_dvd2 [iff, algebra]: "gcd m n dvd n" |
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78 apply (induct m n rule: gcd_induct) |
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79 apply (simp_all add: gcd_non_0) |
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80 apply (blast dest: dvd_mod_imp_dvd) |
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81 done |
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82 |
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83 text {* |
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84 \medskip Maximality: for all @{term m}, @{term n}, @{term k} |
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85 naturals, if @{term k} divides @{term m} and @{term k} divides |
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86 @{term n} then @{term k} divides @{term "gcd m n"}. |
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87 *} |
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88 |
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89 lemma gcd_greatest: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n" |
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90 by (induct m n rule: gcd_induct) (simp_all add: gcd_non_0 dvd_mod) |
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91 |
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92 text {* |
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93 \medskip Function gcd yields the Greatest Common Divisor. |
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94 *} |
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95 |
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96 lemma is_gcd: "is_gcd m n (gcd m n) " |
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97 by (simp add: is_gcd_def gcd_greatest) |
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98 |
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99 |
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100 subsection {* Derived laws for GCD *} |
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101 |
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102 lemma gcd_greatest_iff [iff, algebra]: "k dvd gcd m n \<longleftrightarrow> k dvd m \<and> k dvd n" |
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103 by (blast intro!: gcd_greatest intro: dvd_trans) |
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104 |
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105 lemma gcd_zero[algebra]: "gcd m n = 0 \<longleftrightarrow> m = 0 \<and> n = 0" |
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106 by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff) |
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107 |
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108 lemma gcd_commute: "gcd m n = gcd n m" |
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109 apply (rule is_gcd_unique) |
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110 apply (rule is_gcd) |
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111 apply (subst is_gcd_commute) |
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112 apply (simp add: is_gcd) |
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113 done |
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114 |
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115 lemma gcd_assoc: "gcd (gcd k m) n = gcd k (gcd m n)" |
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116 apply (rule is_gcd_unique) |
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117 apply (rule is_gcd) |
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118 apply (simp add: is_gcd_def) |
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119 apply (blast intro: dvd_trans) |
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120 done |
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121 |
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122 lemma gcd_1_left [simp, algebra]: "gcd (Suc 0) m = Suc 0" |
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123 by (simp add: gcd_commute) |
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124 |
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125 lemma nat_gcd_1_left [simp, algebra]: "gcd 1 m = 1" |
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126 unfolding One_nat_def by (rule gcd_1_left) |
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127 |
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128 text {* |
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129 \medskip Multiplication laws |
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130 *} |
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131 |
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132 lemma gcd_mult_distrib2: "k * gcd m n = gcd (k * m) (k * n)" |
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133 -- {* \cite[page 27]{davenport92} *} |
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134 apply (induct m n rule: gcd_induct) |
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135 apply simp |
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136 apply (case_tac "k = 0") |
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137 apply (simp_all add: mod_geq gcd_non_0 mod_mult_distrib2) |
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138 done |
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139 |
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140 lemma gcd_mult [simp, algebra]: "gcd k (k * n) = k" |
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141 apply (rule gcd_mult_distrib2 [of k 1 n, simplified, symmetric]) |
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142 done |
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143 |
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144 lemma gcd_self [simp, algebra]: "gcd k k = k" |
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145 apply (rule gcd_mult [of k 1, simplified]) |
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146 done |
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147 |
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148 lemma relprime_dvd_mult: "gcd k n = 1 ==> k dvd m * n ==> k dvd m" |
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149 apply (insert gcd_mult_distrib2 [of m k n]) |
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150 apply simp |
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151 apply (erule_tac t = m in ssubst) |
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152 apply simp |
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153 done |
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154 |
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155 lemma relprime_dvd_mult_iff: "gcd k n = 1 ==> (k dvd m * n) = (k dvd m)" |
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156 by (auto intro: relprime_dvd_mult dvd_mult2) |
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157 |
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158 lemma gcd_mult_cancel: "gcd k n = 1 ==> gcd (k * m) n = gcd m n" |
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159 apply (rule dvd_anti_sym) |
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160 apply (rule gcd_greatest) |
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161 apply (rule_tac n = k in relprime_dvd_mult) |
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162 apply (simp add: gcd_assoc) |
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163 apply (simp add: gcd_commute) |
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164 apply (simp_all add: mult_commute) |
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165 done |
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166 |
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167 |
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168 text {* \medskip Addition laws *} |
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169 |
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170 lemma gcd_add1 [simp, algebra]: "gcd (m + n) n = gcd m n" |
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171 by (cases "n = 0") (auto simp add: gcd_non_0) |
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172 |
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173 lemma gcd_add2 [simp, algebra]: "gcd m (m + n) = gcd m n" |
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174 proof - |
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175 have "gcd m (m + n) = gcd (m + n) m" by (rule gcd_commute) |
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176 also have "... = gcd (n + m) m" by (simp add: add_commute) |
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177 also have "... = gcd n m" by simp |
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178 also have "... = gcd m n" by (rule gcd_commute) |
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179 finally show ?thesis . |
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180 qed |
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181 |
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182 lemma gcd_add2' [simp, algebra]: "gcd m (n + m) = gcd m n" |
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183 apply (subst add_commute) |
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184 apply (rule gcd_add2) |
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185 done |
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186 |
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187 lemma gcd_add_mult[algebra]: "gcd m (k * m + n) = gcd m n" |
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188 by (induct k) (simp_all add: add_assoc) |
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189 |
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190 lemma gcd_dvd_prod: "gcd m n dvd m * n" |
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191 using mult_dvd_mono [of 1] by auto |
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192 |
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193 text {* |
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194 \medskip Division by gcd yields rrelatively primes. |
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195 *} |
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196 |
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197 lemma div_gcd_relprime: |
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198 assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0" |
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199 shows "gcd (a div gcd a b) (b div gcd a b) = 1" |
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200 proof - |
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201 let ?g = "gcd a b" |
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202 let ?a' = "a div ?g" |
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203 let ?b' = "b div ?g" |
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204 let ?g' = "gcd ?a' ?b'" |
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205 have dvdg: "?g dvd a" "?g dvd b" by simp_all |
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206 have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all |
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207 from dvdg dvdg' obtain ka kb ka' kb' where |
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208 kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'" |
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209 unfolding dvd_def by blast |
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210 then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'" by simp_all |
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211 then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b" |
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212 by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)] |
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213 dvd_mult_div_cancel [OF dvdg(2)] dvd_def) |
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214 have "?g \<noteq> 0" using nz by (simp add: gcd_zero) |
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215 then have gp: "?g > 0" by simp |
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216 from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" . |
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217 with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp |
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218 qed |
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219 |
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220 |
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221 lemma gcd_unique: "d dvd a\<and>d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b" |
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222 proof(auto) |
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223 assume H: "d dvd a" "d dvd b" "\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d" |
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224 from H(3)[rule_format] gcd_dvd1[of a b] gcd_dvd2[of a b] |
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225 have th: "gcd a b dvd d" by blast |
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226 from dvd_anti_sym[OF th gcd_greatest[OF H(1,2)]] show "d = gcd a b" by blast |
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227 qed |
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228 |
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229 lemma gcd_eq: assumes H: "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd u \<and> d dvd v" |
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230 shows "gcd x y = gcd u v" |
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231 proof- |
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232 from H have "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd gcd u v" by simp |
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233 with gcd_unique[of "gcd u v" x y] show ?thesis by auto |
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234 qed |
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235 |
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236 lemma ind_euclid: |
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237 assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0" |
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238 and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)" |
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239 shows "P a b" |
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240 proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct) |
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241 fix n a b |
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242 assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b" |
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243 have "a = b \<or> a < b \<or> b < a" by arith |
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244 moreover {assume eq: "a= b" |
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245 from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq by simp} |
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246 moreover |
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247 {assume lt: "a < b" |
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248 hence "a + b - a < n \<or> a = 0" using H(2) by arith |
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249 moreover |
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250 {assume "a =0" with z c have "P a b" by blast } |
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251 moreover |
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252 {assume ab: "a + b - a < n" |
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253 have th0: "a + b - a = a + (b - a)" using lt by arith |
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254 from add[rule_format, OF H(1)[rule_format, OF ab th0]] |
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255 have "P a b" by (simp add: th0[symmetric])} |
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256 ultimately have "P a b" by blast} |
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257 moreover |
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258 {assume lt: "a > b" |
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259 hence "b + a - b < n \<or> b = 0" using H(2) by arith |
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260 moreover |
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261 {assume "b =0" with z c have "P a b" by blast } |
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262 moreover |
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263 {assume ab: "b + a - b < n" |
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264 have th0: "b + a - b = b + (a - b)" using lt by arith |
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265 from add[rule_format, OF H(1)[rule_format, OF ab th0]] |
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266 have "P b a" by (simp add: th0[symmetric]) |
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267 hence "P a b" using c by blast } |
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268 ultimately have "P a b" by blast} |
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269 ultimately show "P a b" by blast |
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270 qed |
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271 |
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272 lemma bezout_lemma: |
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273 assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)" |
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274 shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and> (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)" |
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275 using ex |
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276 apply clarsimp |
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277 apply (rule_tac x="d" in exI, simp add: dvd_add) |
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278 apply (case_tac "a * x = b * y + d" , simp_all) |
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279 apply (rule_tac x="x + y" in exI) |
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280 apply (rule_tac x="y" in exI) |
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281 apply algebra |
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282 apply (rule_tac x="x" in exI) |
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283 apply (rule_tac x="x + y" in exI) |
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284 apply algebra |
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285 done |
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286 |
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287 lemma bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)" |
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288 apply(induct a b rule: ind_euclid) |
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289 apply blast |
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290 apply clarify |
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291 apply (rule_tac x="a" in exI, simp add: dvd_add) |
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292 apply clarsimp |
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293 apply (rule_tac x="d" in exI) |
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294 apply (case_tac "a * x = b * y + d", simp_all add: dvd_add) |
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295 apply (rule_tac x="x+y" in exI) |
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296 apply (rule_tac x="y" in exI) |
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297 apply algebra |
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298 apply (rule_tac x="x" in exI) |
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299 apply (rule_tac x="x+y" in exI) |
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300 apply algebra |
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301 done |
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302 |
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303 lemma bezout: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x - b * y = d \<or> b * x - a * y = d)" |
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304 using bezout_add[of a b] |
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305 apply clarsimp |
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306 apply (rule_tac x="d" in exI, simp) |
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307 apply (rule_tac x="x" in exI) |
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308 apply (rule_tac x="y" in exI) |
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309 apply auto |
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310 done |
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311 |
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312 |
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313 text {* We can get a stronger version with a nonzeroness assumption. *} |
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314 lemma divides_le: "m dvd n ==> m <= n \<or> n = (0::nat)" by (auto simp add: dvd_def) |
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315 |
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316 lemma bezout_add_strong: assumes nz: "a \<noteq> (0::nat)" |
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317 shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d" |
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318 proof- |
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319 from nz have ap: "a > 0" by simp |
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320 from bezout_add[of a b] |
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321 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or> (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast |
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322 moreover |
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323 {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d" |
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324 from H have ?thesis by blast } |
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325 moreover |
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326 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d" |
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327 {assume b0: "b = 0" with H have ?thesis by simp} |
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328 moreover |
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329 {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp |
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330 from divides_le[OF H(2)] b have "d < b \<or> d = b" using le_less by blast |
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331 moreover |
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332 {assume db: "d=b" |
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333 from prems have ?thesis apply simp |
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334 apply (rule exI[where x = b], simp) |
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335 apply (rule exI[where x = b]) |
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336 by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)} |
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337 moreover |
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338 {assume db: "d < b" |
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339 {assume "x=0" hence ?thesis using prems by simp } |
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340 moreover |
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341 {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp |
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342 |
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343 from db have "d \<le> b - 1" by simp |
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344 hence "d*b \<le> b*(b - 1)" by simp |
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345 with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"] |
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346 have dble: "d*b \<le> x*b*(b - 1)" using bp by simp |
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347 from H (3) have "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)" by algebra |
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348 hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp |
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349 hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)" |
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350 by (simp only: diff_add_assoc[OF dble, of d, symmetric]) |
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351 hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d" |
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352 by (simp only: diff_mult_distrib2 add_commute mult_ac) |
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353 hence ?thesis using H(1,2) |
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354 apply - |
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355 apply (rule exI[where x=d], simp) |
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356 apply (rule exI[where x="(b - 1) * y"]) |
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357 by (rule exI[where x="x*(b - 1) - d"], simp)} |
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358 ultimately have ?thesis by blast} |
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359 ultimately have ?thesis by blast} |
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360 ultimately have ?thesis by blast} |
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361 ultimately show ?thesis by blast |
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362 qed |
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363 |
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364 |
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365 lemma bezout_gcd: "\<exists>x y. a * x - b * y = gcd a b \<or> b * x - a * y = gcd a b" |
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366 proof- |
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367 let ?g = "gcd a b" |
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368 from bezout[of a b] obtain d x y where d: "d dvd a" "d dvd b" "a * x - b * y = d \<or> b * x - a * y = d" by blast |
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369 from d(1,2) have "d dvd ?g" by simp |
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370 then obtain k where k: "?g = d*k" unfolding dvd_def by blast |
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371 from d(3) have "(a * x - b * y)*k = d*k \<or> (b * x - a * y)*k = d*k" by blast |
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372 hence "a * x * k - b * y*k = d*k \<or> b * x * k - a * y*k = d*k" |
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373 by (algebra add: diff_mult_distrib) |
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374 hence "a * (x * k) - b * (y*k) = ?g \<or> b * (x * k) - a * (y*k) = ?g" |
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375 by (simp add: k mult_assoc) |
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376 thus ?thesis by blast |
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377 qed |
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378 |
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379 lemma bezout_gcd_strong: assumes a: "a \<noteq> 0" |
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380 shows "\<exists>x y. a * x = b * y + gcd a b" |
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381 proof- |
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382 let ?g = "gcd a b" |
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383 from bezout_add_strong[OF a, of b] |
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384 obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast |
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385 from d(1,2) have "d dvd ?g" by simp |
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386 then obtain k where k: "?g = d*k" unfolding dvd_def by blast |
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387 from d(3) have "a * x * k = (b * y + d) *k " by algebra |
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388 hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k) |
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389 thus ?thesis by blast |
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390 qed |
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391 |
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392 lemma gcd_mult_distrib: "gcd(a * c) (b * c) = c * gcd a b" |
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393 by(simp add: gcd_mult_distrib2 mult_commute) |
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394 |
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395 lemma gcd_bezout: "(\<exists>x y. a * x - b * y = d \<or> b * x - a * y = d) \<longleftrightarrow> gcd a b dvd d" |
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396 (is "?lhs \<longleftrightarrow> ?rhs") |
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397 proof- |
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398 let ?g = "gcd a b" |
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399 {assume H: ?rhs then obtain k where k: "d = ?g*k" unfolding dvd_def by blast |
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400 from bezout_gcd[of a b] obtain x y where xy: "a * x - b * y = ?g \<or> b * x - a * y = ?g" |
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401 by blast |
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402 hence "(a * x - b * y)*k = ?g*k \<or> (b * x - a * y)*k = ?g*k" by auto |
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403 hence "a * x*k - b * y*k = ?g*k \<or> b * x * k - a * y*k = ?g*k" |
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404 by (simp only: diff_mult_distrib) |
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405 hence "a * (x*k) - b * (y*k) = d \<or> b * (x * k) - a * (y*k) = d" |
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406 by (simp add: k[symmetric] mult_assoc) |
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407 hence ?lhs by blast} |
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408 moreover |
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409 {fix x y assume H: "a * x - b * y = d \<or> b * x - a * y = d" |
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410 have dv: "?g dvd a*x" "?g dvd b * y" "?g dvd b*x" "?g dvd a * y" |
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411 using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all |
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412 from dvd_diff_nat[OF dv(1,2)] dvd_diff_nat[OF dv(3,4)] H |
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413 have ?rhs by auto} |
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414 ultimately show ?thesis by blast |
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415 qed |
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416 |
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417 lemma gcd_bezout_sum: assumes H:"a * x + b * y = d" shows "gcd a b dvd d" |
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418 proof- |
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419 let ?g = "gcd a b" |
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420 have dv: "?g dvd a*x" "?g dvd b * y" |
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421 using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all |
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422 from dvd_add[OF dv] H |
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423 show ?thesis by auto |
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424 qed |
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425 |
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426 lemma gcd_mult': "gcd b (a * b) = b" |
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427 by (simp add: gcd_mult mult_commute[of a b]) |
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428 |
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429 lemma gcd_add: "gcd(a + b) b = gcd a b" |
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430 "gcd(b + a) b = gcd a b" "gcd a (a + b) = gcd a b" "gcd a (b + a) = gcd a b" |
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431 apply (simp_all add: gcd_add1) |
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432 by (simp add: gcd_commute gcd_add1) |
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433 |
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434 lemma gcd_sub: "b <= a ==> gcd(a - b) b = gcd a b" "a <= b ==> gcd a (b - a) = gcd a b" |
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435 proof- |
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436 {fix a b assume H: "b \<le> (a::nat)" |
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437 hence th: "a - b + b = a" by arith |
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438 from gcd_add(1)[of "a - b" b] th have "gcd(a - b) b = gcd a b" by simp} |
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439 note th = this |
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440 { |
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441 assume ab: "b \<le> a" |
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442 from th[OF ab] show "gcd (a - b) b = gcd a b" by blast |
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443 next |
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444 assume ab: "a \<le> b" |
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445 from th[OF ab] show "gcd a (b - a) = gcd a b" |
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446 by (simp add: gcd_commute)} |
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447 qed |
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448 |
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449 |
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450 subsection {* LCM defined by GCD *} |
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451 |
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452 |
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453 definition |
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454 lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" |
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455 where |
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456 lcm_def: "lcm m n = m * n div gcd m n" |
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457 |
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458 lemma prod_gcd_lcm: |
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459 "m * n = gcd m n * lcm m n" |
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460 unfolding lcm_def by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod]) |
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461 |
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462 lemma lcm_0 [simp]: "lcm m 0 = 0" |
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463 unfolding lcm_def by simp |
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464 |
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465 lemma lcm_1 [simp]: "lcm m 1 = m" |
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466 unfolding lcm_def by simp |
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467 |
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468 lemma lcm_0_left [simp]: "lcm 0 n = 0" |
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469 unfolding lcm_def by simp |
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470 |
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471 lemma lcm_1_left [simp]: "lcm 1 m = m" |
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472 unfolding lcm_def by simp |
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473 |
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474 lemma dvd_pos: |
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475 fixes n m :: nat |
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476 assumes "n > 0" and "m dvd n" |
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477 shows "m > 0" |
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478 using assms by (cases m) auto |
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479 |
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480 lemma lcm_least: |
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481 assumes "m dvd k" and "n dvd k" |
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482 shows "lcm m n dvd k" |
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483 proof (cases k) |
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484 case 0 then show ?thesis by auto |
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485 next |
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486 case (Suc _) then have pos_k: "k > 0" by auto |
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487 from assms dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto |
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488 with gcd_zero [of m n] have pos_gcd: "gcd m n > 0" by simp |
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489 from assms obtain p where k_m: "k = m * p" using dvd_def by blast |
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490 from assms obtain q where k_n: "k = n * q" using dvd_def by blast |
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491 from pos_k k_m have pos_p: "p > 0" by auto |
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492 from pos_k k_n have pos_q: "q > 0" by auto |
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493 have "k * k * gcd q p = k * gcd (k * q) (k * p)" |
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494 by (simp add: mult_ac gcd_mult_distrib2) |
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495 also have "\<dots> = k * gcd (m * p * q) (n * q * p)" |
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496 by (simp add: k_m [symmetric] k_n [symmetric]) |
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497 also have "\<dots> = k * p * q * gcd m n" |
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498 by (simp add: mult_ac gcd_mult_distrib2) |
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499 finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n" |
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500 by (simp only: k_m [symmetric] k_n [symmetric]) |
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501 then have "p * q * m * n * gcd q p = p * q * k * gcd m n" |
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502 by (simp add: mult_ac) |
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503 with pos_p pos_q have "m * n * gcd q p = k * gcd m n" |
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504 by simp |
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505 with prod_gcd_lcm [of m n] |
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506 have "lcm m n * gcd q p * gcd m n = k * gcd m n" |
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507 by (simp add: mult_ac) |
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508 with pos_gcd have "lcm m n * gcd q p = k" by simp |
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509 then show ?thesis using dvd_def by auto |
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510 qed |
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511 |
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512 lemma lcm_dvd1 [iff]: |
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513 "m dvd lcm m n" |
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514 proof (cases m) |
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515 case 0 then show ?thesis by simp |
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516 next |
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517 case (Suc _) |
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518 then have mpos: "m > 0" by simp |
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519 show ?thesis |
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520 proof (cases n) |
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521 case 0 then show ?thesis by simp |
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522 next |
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523 case (Suc _) |
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524 then have npos: "n > 0" by simp |
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525 have "gcd m n dvd n" by simp |
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526 then obtain k where "n = gcd m n * k" using dvd_def by auto |
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527 then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n" by (simp add: mult_ac) |
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528 also have "\<dots> = m * k" using mpos npos gcd_zero by simp |
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529 finally show ?thesis by (simp add: lcm_def) |
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530 qed |
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531 qed |
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532 |
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533 lemma lcm_dvd2 [iff]: |
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534 "n dvd lcm m n" |
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535 proof (cases n) |
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536 case 0 then show ?thesis by simp |
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537 next |
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538 case (Suc _) |
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539 then have npos: "n > 0" by simp |
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540 show ?thesis |
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541 proof (cases m) |
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542 case 0 then show ?thesis by simp |
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543 next |
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544 case (Suc _) |
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545 then have mpos: "m > 0" by simp |
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546 have "gcd m n dvd m" by simp |
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547 then obtain k where "m = gcd m n * k" using dvd_def by auto |
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548 then have "m * n div gcd m n = (gcd m n * k) * n div gcd m n" by (simp add: mult_ac) |
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549 also have "\<dots> = n * k" using mpos npos gcd_zero by simp |
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550 finally show ?thesis by (simp add: lcm_def) |
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551 qed |
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552 qed |
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553 |
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554 lemma gcd_add1_eq: "gcd (m + k) k = gcd (m + k) m" |
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555 by (simp add: gcd_commute) |
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556 |
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557 lemma gcd_diff2: "m \<le> n ==> gcd n (n - m) = gcd n m" |
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558 apply (subgoal_tac "n = m + (n - m)") |
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559 apply (erule ssubst, rule gcd_add1_eq, simp) |
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560 done |
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561 |
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562 |
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563 subsection {* GCD and LCM on integers *} |
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564 |
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565 definition |
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566 zgcd :: "int \<Rightarrow> int \<Rightarrow> int" where |
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567 "zgcd i j = int (gcd (nat (abs i)) (nat (abs j)))" |
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568 |
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569 lemma zgcd_zdvd1 [iff,simp, algebra]: "zgcd i j dvd i" |
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570 by (simp add: zgcd_def int_dvd_iff) |
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571 |
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572 lemma zgcd_zdvd2 [iff,simp, algebra]: "zgcd i j dvd j" |
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573 by (simp add: zgcd_def int_dvd_iff) |
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574 |
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575 lemma zgcd_pos: "zgcd i j \<ge> 0" |
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576 by (simp add: zgcd_def) |
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577 |
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578 lemma zgcd0 [simp,algebra]: "(zgcd i j = 0) = (i = 0 \<and> j = 0)" |
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579 by (simp add: zgcd_def gcd_zero) |
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580 |
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581 lemma zgcd_commute: "zgcd i j = zgcd j i" |
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582 unfolding zgcd_def by (simp add: gcd_commute) |
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583 |
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584 lemma zgcd_zminus [simp, algebra]: "zgcd (- i) j = zgcd i j" |
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585 unfolding zgcd_def by simp |
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586 |
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587 lemma zgcd_zminus2 [simp, algebra]: "zgcd i (- j) = zgcd i j" |
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588 unfolding zgcd_def by simp |
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589 |
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590 (* should be solved by algebra*) |
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591 lemma zrelprime_dvd_mult: "zgcd i j = 1 \<Longrightarrow> i dvd k * j \<Longrightarrow> i dvd k" |
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592 unfolding zgcd_def |
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593 proof - |
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594 assume "int (gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>)) = 1" "i dvd k * j" |
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595 then have g: "gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>) = 1" by simp |
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596 from `i dvd k * j` obtain h where h: "k*j = i*h" unfolding dvd_def by blast |
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597 have th: "nat \<bar>i\<bar> dvd nat \<bar>k\<bar> * nat \<bar>j\<bar>" |
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598 unfolding dvd_def |
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599 by (rule_tac x= "nat \<bar>h\<bar>" in exI, simp add: h nat_abs_mult_distrib [symmetric]) |
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600 from relprime_dvd_mult [OF g th] obtain h' where h': "nat \<bar>k\<bar> = nat \<bar>i\<bar> * h'" |
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601 unfolding dvd_def by blast |
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602 from h' have "int (nat \<bar>k\<bar>) = int (nat \<bar>i\<bar> * h')" by simp |
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603 then have "\<bar>k\<bar> = \<bar>i\<bar> * int h'" by (simp add: int_mult) |
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604 then show ?thesis |
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605 apply (subst abs_dvd_iff [symmetric]) |
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606 apply (subst dvd_abs_iff [symmetric]) |
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607 apply (unfold dvd_def) |
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608 apply (rule_tac x = "int h'" in exI, simp) |
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609 done |
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610 qed |
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611 |
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612 lemma int_nat_abs: "int (nat (abs x)) = abs x" by arith |
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613 |
|
614 lemma zgcd_greatest: |
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615 assumes "k dvd m" and "k dvd n" |
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616 shows "k dvd zgcd m n" |
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617 proof - |
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618 let ?k' = "nat \<bar>k\<bar>" |
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619 let ?m' = "nat \<bar>m\<bar>" |
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620 let ?n' = "nat \<bar>n\<bar>" |
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621 from `k dvd m` and `k dvd n` have dvd': "?k' dvd ?m'" "?k' dvd ?n'" |
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622 unfolding zdvd_int by (simp_all only: int_nat_abs abs_dvd_iff dvd_abs_iff) |
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623 from gcd_greatest [OF dvd'] have "int (nat \<bar>k\<bar>) dvd zgcd m n" |
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624 unfolding zgcd_def by (simp only: zdvd_int) |
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625 then have "\<bar>k\<bar> dvd zgcd m n" by (simp only: int_nat_abs) |
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626 then show "k dvd zgcd m n" by simp |
|
627 qed |
|
628 |
|
629 lemma div_zgcd_relprime: |
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630 assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0" |
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631 shows "zgcd (a div (zgcd a b)) (b div (zgcd a b)) = 1" |
|
632 proof - |
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633 from nz have nz': "nat \<bar>a\<bar> \<noteq> 0 \<or> nat \<bar>b\<bar> \<noteq> 0" by arith |
|
634 let ?g = "zgcd a b" |
|
635 let ?a' = "a div ?g" |
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636 let ?b' = "b div ?g" |
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637 let ?g' = "zgcd ?a' ?b'" |
|
638 have dvdg: "?g dvd a" "?g dvd b" by (simp_all add: zgcd_zdvd1 zgcd_zdvd2) |
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639 have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by (simp_all add: zgcd_zdvd1 zgcd_zdvd2) |
|
640 from dvdg dvdg' obtain ka kb ka' kb' where |
|
641 kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'" |
|
642 unfolding dvd_def by blast |
|
643 then have "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'" by simp_all |
|
644 then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b" |
|
645 by (auto simp add: zdvd_mult_div_cancel [OF dvdg(1)] |
|
646 zdvd_mult_div_cancel [OF dvdg(2)] dvd_def) |
|
647 have "?g \<noteq> 0" using nz by simp |
|
648 then have gp: "?g \<noteq> 0" using zgcd_pos[where i="a" and j="b"] by arith |
|
649 from zgcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" . |
|
650 with zdvd_mult_cancel1 [OF gp] have "\<bar>?g'\<bar> = 1" by simp |
|
651 with zgcd_pos show "?g' = 1" by simp |
|
652 qed |
|
653 |
|
654 lemma zgcd_0 [simp, algebra]: "zgcd m 0 = abs m" |
|
655 by (simp add: zgcd_def abs_if) |
|
656 |
|
657 lemma zgcd_0_left [simp, algebra]: "zgcd 0 m = abs m" |
|
658 by (simp add: zgcd_def abs_if) |
|
659 |
|
660 lemma zgcd_non_0: "0 < n ==> zgcd m n = zgcd n (m mod n)" |
|
661 apply (frule_tac b = n and a = m in pos_mod_sign) |
|
662 apply (simp del: pos_mod_sign add: zgcd_def abs_if nat_mod_distrib) |
|
663 apply (auto simp add: gcd_non_0 nat_mod_distrib [symmetric] zmod_zminus1_eq_if) |
|
664 apply (frule_tac a = m in pos_mod_bound) |
|
665 apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2 nat_le_eq_zle) |
|
666 done |
|
667 |
|
668 lemma zgcd_eq: "zgcd m n = zgcd n (m mod n)" |
|
669 apply (case_tac "n = 0", simp add: DIVISION_BY_ZERO) |
|
670 apply (auto simp add: linorder_neq_iff zgcd_non_0) |
|
671 apply (cut_tac m = "-m" and n = "-n" in zgcd_non_0, auto) |
|
672 done |
|
673 |
|
674 lemma zgcd_1 [simp, algebra]: "zgcd m 1 = 1" |
|
675 by (simp add: zgcd_def abs_if) |
|
676 |
|
677 lemma zgcd_0_1_iff [simp, algebra]: "zgcd 0 m = 1 \<longleftrightarrow> \<bar>m\<bar> = 1" |
|
678 by (simp add: zgcd_def abs_if) |
|
679 |
|
680 lemma zgcd_greatest_iff[algebra]: "k dvd zgcd m n = (k dvd m \<and> k dvd n)" |
|
681 by (simp add: zgcd_def abs_if int_dvd_iff dvd_int_iff nat_dvd_iff) |
|
682 |
|
683 lemma zgcd_1_left [simp, algebra]: "zgcd 1 m = 1" |
|
684 by (simp add: zgcd_def gcd_1_left) |
|
685 |
|
686 lemma zgcd_assoc: "zgcd (zgcd k m) n = zgcd k (zgcd m n)" |
|
687 by (simp add: zgcd_def gcd_assoc) |
|
688 |
|
689 lemma zgcd_left_commute: "zgcd k (zgcd m n) = zgcd m (zgcd k n)" |
|
690 apply (rule zgcd_commute [THEN trans]) |
|
691 apply (rule zgcd_assoc [THEN trans]) |
|
692 apply (rule zgcd_commute [THEN arg_cong]) |
|
693 done |
|
694 |
|
695 lemmas zgcd_ac = zgcd_assoc zgcd_commute zgcd_left_commute |
|
696 -- {* addition is an AC-operator *} |
|
697 |
|
698 lemma zgcd_zmult_distrib2: "0 \<le> k ==> k * zgcd m n = zgcd (k * m) (k * n)" |
|
699 by (simp del: minus_mult_right [symmetric] |
|
700 add: minus_mult_right nat_mult_distrib zgcd_def abs_if |
|
701 mult_less_0_iff gcd_mult_distrib2 [symmetric] zmult_int [symmetric]) |
|
702 |
|
703 lemma zgcd_zmult_distrib2_abs: "zgcd (k * m) (k * n) = abs k * zgcd m n" |
|
704 by (simp add: abs_if zgcd_zmult_distrib2) |
|
705 |
|
706 lemma zgcd_self [simp]: "0 \<le> m ==> zgcd m m = m" |
|
707 by (cut_tac k = m and m = 1 and n = 1 in zgcd_zmult_distrib2, simp_all) |
|
708 |
|
709 lemma zgcd_zmult_eq_self [simp]: "0 \<le> k ==> zgcd k (k * n) = k" |
|
710 by (cut_tac k = k and m = 1 and n = n in zgcd_zmult_distrib2, simp_all) |
|
711 |
|
712 lemma zgcd_zmult_eq_self2 [simp]: "0 \<le> k ==> zgcd (k * n) k = k" |
|
713 by (cut_tac k = k and m = n and n = 1 in zgcd_zmult_distrib2, simp_all) |
|
714 |
|
715 |
|
716 definition "zlcm i j = int (lcm(nat(abs i)) (nat(abs j)))" |
|
717 |
|
718 lemma dvd_zlcm_self1[simp, algebra]: "i dvd zlcm i j" |
|
719 by(simp add:zlcm_def dvd_int_iff) |
|
720 |
|
721 lemma dvd_zlcm_self2[simp, algebra]: "j dvd zlcm i j" |
|
722 by(simp add:zlcm_def dvd_int_iff) |
|
723 |
|
724 |
|
725 lemma dvd_imp_dvd_zlcm1: |
|
726 assumes "k dvd i" shows "k dvd (zlcm i j)" |
|
727 proof - |
|
728 have "nat(abs k) dvd nat(abs i)" using `k dvd i` |
|
729 by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric]) |
|
730 thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans) |
|
731 qed |
|
732 |
|
733 lemma dvd_imp_dvd_zlcm2: |
|
734 assumes "k dvd j" shows "k dvd (zlcm i j)" |
|
735 proof - |
|
736 have "nat(abs k) dvd nat(abs j)" using `k dvd j` |
|
737 by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric]) |
|
738 thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans) |
|
739 qed |
|
740 |
|
741 |
|
742 lemma zdvd_self_abs1: "(d::int) dvd (abs d)" |
|
743 by (case_tac "d <0", simp_all) |
|
744 |
|
745 lemma zdvd_self_abs2: "(abs (d::int)) dvd d" |
|
746 by (case_tac "d<0", simp_all) |
|
747 |
|
748 (* lcm a b is positive for positive a and b *) |
|
749 |
|
750 lemma lcm_pos: |
|
751 assumes mpos: "m > 0" |
|
752 and npos: "n>0" |
|
753 shows "lcm m n > 0" |
|
754 proof(rule ccontr, simp add: lcm_def gcd_zero) |
|
755 assume h:"m*n div gcd m n = 0" |
|
756 from mpos npos have "gcd m n \<noteq> 0" using gcd_zero by simp |
|
757 hence gcdp: "gcd m n > 0" by simp |
|
758 with h |
|
759 have "m*n < gcd m n" |
|
760 by (cases "m * n < gcd m n") (auto simp add: div_if[OF gcdp, where m="m*n"]) |
|
761 moreover |
|
762 have "gcd m n dvd m" by simp |
|
763 with mpos dvd_imp_le have t1:"gcd m n \<le> m" by simp |
|
764 with npos have t1:"gcd m n *n \<le> m*n" by simp |
|
765 have "gcd m n \<le> gcd m n*n" using npos by simp |
|
766 with t1 have "gcd m n \<le> m*n" by arith |
|
767 ultimately show "False" by simp |
|
768 qed |
|
769 |
|
770 lemma zlcm_pos: |
|
771 assumes anz: "a \<noteq> 0" |
|
772 and bnz: "b \<noteq> 0" |
|
773 shows "0 < zlcm a b" |
|
774 proof- |
|
775 let ?na = "nat (abs a)" |
|
776 let ?nb = "nat (abs b)" |
|
777 have nap: "?na >0" using anz by simp |
|
778 have nbp: "?nb >0" using bnz by simp |
|
779 have "0 < lcm ?na ?nb" by (rule lcm_pos[OF nap nbp]) |
|
780 thus ?thesis by (simp add: zlcm_def) |
|
781 qed |
|
782 |
|
783 lemma zgcd_code [code]: |
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784 "zgcd k l = \<bar>if l = 0 then k else zgcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>" |
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785 by (simp add: zgcd_def gcd.simps [of "nat \<bar>k\<bar>"] nat_mod_distrib) |
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786 |
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787 end |
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