1 (* Title: HOL/Library/Abstract_Rat.thy |
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2 Author: Amine Chaieb |
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3 *) |
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4 |
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5 header {* Abstract rational numbers *} |
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6 |
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7 theory Abstract_Rat |
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8 imports Complex_Main |
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9 begin |
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10 |
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11 type_synonym Num = "int \<times> int" |
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12 |
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13 abbreviation Num0_syn :: Num ("0\<^sub>N") |
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14 where "0\<^sub>N \<equiv> (0, 0)" |
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15 |
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16 abbreviation Numi_syn :: "int \<Rightarrow> Num" ("'((_)')\<^sub>N") |
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17 where "(i)\<^sub>N \<equiv> (i, 1)" |
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18 |
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19 definition isnormNum :: "Num \<Rightarrow> bool" where |
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20 "isnormNum = (\<lambda>(a,b). (if a = 0 then b = 0 else b > 0 \<and> gcd a b = 1))" |
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21 |
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22 definition normNum :: "Num \<Rightarrow> Num" where |
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23 "normNum = (\<lambda>(a,b). |
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24 (if a=0 \<or> b = 0 then (0,0) else |
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25 (let g = gcd a b |
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26 in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))" |
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27 |
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28 declare gcd_dvd1_int[presburger] gcd_dvd2_int[presburger] |
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29 |
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30 lemma normNum_isnormNum [simp]: "isnormNum (normNum x)" |
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31 proof - |
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32 obtain a b where x: "x = (a, b)" by (cases x) |
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33 { assume "a=0 \<or> b = 0" hence ?thesis by (simp add: x normNum_def isnormNum_def) } |
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34 moreover |
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35 { assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0" |
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36 let ?g = "gcd a b" |
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37 let ?a' = "a div ?g" |
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38 let ?b' = "b div ?g" |
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39 let ?g' = "gcd ?a' ?b'" |
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40 from anz bnz have "?g \<noteq> 0" by simp with gcd_ge_0_int[of a b] |
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41 have gpos: "?g > 0" by arith |
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42 have gdvd: "?g dvd a" "?g dvd b" by arith+ |
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43 from dvd_mult_div_cancel[OF gdvd(1)] dvd_mult_div_cancel[OF gdvd(2)] anz bnz |
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44 have nz': "?a' \<noteq> 0" "?b' \<noteq> 0" by - (rule notI, simp)+ |
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45 from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith |
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46 from div_gcd_coprime_int[OF stupid] have gp1: "?g' = 1" . |
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47 from bnz have "b < 0 \<or> b > 0" by arith |
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48 moreover |
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49 { assume b: "b > 0" |
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50 from b have "?b' \<ge> 0" |
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51 by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos]) |
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52 with nz' have b': "?b' > 0" by arith |
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53 from b b' anz bnz nz' gp1 have ?thesis |
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54 by (simp add: x isnormNum_def normNum_def Let_def split_def) } |
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55 moreover { |
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56 assume b: "b < 0" |
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57 { assume b': "?b' \<ge> 0" |
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58 from gpos have th: "?g \<ge> 0" by arith |
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59 from mult_nonneg_nonneg[OF th b'] dvd_mult_div_cancel[OF gdvd(2)] |
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60 have False using b by arith } |
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61 hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric]) |
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62 from anz bnz nz' b b' gp1 have ?thesis |
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63 by (simp add: x isnormNum_def normNum_def Let_def split_def) } |
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64 ultimately have ?thesis by blast |
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65 } |
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66 ultimately show ?thesis by blast |
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67 qed |
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68 |
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69 text {* Arithmetic over Num *} |
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70 |
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71 definition Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "+\<^sub>N" 60) where |
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72 "Nadd = (\<lambda>(a,b) (a',b'). if a = 0 \<or> b = 0 then normNum(a',b') |
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73 else if a'=0 \<or> b' = 0 then normNum(a,b) |
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74 else normNum(a*b' + b*a', b*b'))" |
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75 |
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76 definition Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "*\<^sub>N" 60) where |
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77 "Nmul = (\<lambda>(a,b) (a',b'). let g = gcd (a*a') (b*b') |
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78 in (a*a' div g, b*b' div g))" |
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79 |
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80 definition Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N") |
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81 where "Nneg \<equiv> (\<lambda>(a,b). (-a,b))" |
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82 |
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83 definition Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "-\<^sub>N" 60) |
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84 where "Nsub = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)" |
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85 |
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86 definition Ninv :: "Num \<Rightarrow> Num" |
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87 where "Ninv = (\<lambda>(a,b). if a < 0 then (-b, \<bar>a\<bar>) else (b,a))" |
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88 |
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89 definition Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "\<div>\<^sub>N" 60) |
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90 where "Ndiv = (\<lambda>a b. a *\<^sub>N Ninv b)" |
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91 |
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92 lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)" |
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93 by (simp add: isnormNum_def Nneg_def split_def) |
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94 |
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95 lemma Nadd_normN[simp]: "isnormNum (x +\<^sub>N y)" |
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96 by (simp add: Nadd_def split_def) |
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97 |
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98 lemma Nsub_normN[simp]: "\<lbrakk> isnormNum y\<rbrakk> \<Longrightarrow> isnormNum (x -\<^sub>N y)" |
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99 by (simp add: Nsub_def split_def) |
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100 |
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101 lemma Nmul_normN[simp]: |
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102 assumes xn: "isnormNum x" and yn: "isnormNum y" |
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103 shows "isnormNum (x *\<^sub>N y)" |
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104 proof - |
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105 obtain a b where x: "x = (a, b)" by (cases x) |
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106 obtain a' b' where y: "y = (a', b')" by (cases y) |
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107 { assume "a = 0" |
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108 hence ?thesis using xn x y |
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109 by (simp add: isnormNum_def Let_def Nmul_def split_def) } |
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110 moreover |
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111 { assume "a' = 0" |
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112 hence ?thesis using yn x y |
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113 by (simp add: isnormNum_def Let_def Nmul_def split_def) } |
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114 moreover |
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115 { assume a: "a \<noteq>0" and a': "a'\<noteq>0" |
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116 hence bp: "b > 0" "b' > 0" using xn yn x y by (simp_all add: isnormNum_def) |
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117 from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a * a', b * b')" |
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118 using x y a a' bp by (simp add: Nmul_def Let_def split_def normNum_def) |
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119 hence ?thesis by simp } |
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120 ultimately show ?thesis by blast |
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121 qed |
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122 |
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123 lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)" |
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124 by (simp add: Ninv_def isnormNum_def split_def) |
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125 (cases "fst x = 0", auto simp add: gcd_commute_int) |
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126 |
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127 lemma isnormNum_int[simp]: |
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128 "isnormNum 0\<^sub>N" "isnormNum ((1::int)\<^sub>N)" "i \<noteq> 0 \<Longrightarrow> isnormNum (i)\<^sub>N" |
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129 by (simp_all add: isnormNum_def) |
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130 |
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131 |
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132 text {* Relations over Num *} |
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133 |
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134 definition Nlt0:: "Num \<Rightarrow> bool" ("0>\<^sub>N") |
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135 where "Nlt0 = (\<lambda>(a,b). a < 0)" |
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136 |
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137 definition Nle0:: "Num \<Rightarrow> bool" ("0\<ge>\<^sub>N") |
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138 where "Nle0 = (\<lambda>(a,b). a \<le> 0)" |
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139 |
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140 definition Ngt0:: "Num \<Rightarrow> bool" ("0<\<^sub>N") |
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141 where "Ngt0 = (\<lambda>(a,b). a > 0)" |
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142 |
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143 definition Nge0:: "Num \<Rightarrow> bool" ("0\<le>\<^sub>N") |
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144 where "Nge0 = (\<lambda>(a,b). a \<ge> 0)" |
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145 |
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146 definition Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "<\<^sub>N" 55) |
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147 where "Nlt = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))" |
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148 |
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149 definition Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "\<le>\<^sub>N" 55) |
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150 where "Nle = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))" |
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151 |
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152 definition "INum = (\<lambda>(a,b). of_int a / of_int b)" |
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153 |
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154 lemma INum_int [simp]: "INum (i)\<^sub>N = ((of_int i) ::'a::field)" "INum 0\<^sub>N = (0::'a::field)" |
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155 by (simp_all add: INum_def) |
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156 |
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157 lemma isnormNum_unique[simp]: |
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158 assumes na: "isnormNum x" and nb: "isnormNum y" |
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159 shows "((INum x ::'a::{field_char_0, field_inverse_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs") |
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160 proof |
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161 obtain a b where x: "x = (a, b)" by (cases x) |
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162 obtain a' b' where y: "y = (a', b')" by (cases y) |
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163 assume H: ?lhs |
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164 { assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0" |
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165 hence ?rhs using na nb H |
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166 by (simp add: x y INum_def split_def isnormNum_def split: split_if_asm) } |
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167 moreover |
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168 { assume az: "a \<noteq> 0" and bz: "b \<noteq> 0" and a'z: "a'\<noteq>0" and b'z: "b'\<noteq>0" |
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169 from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: x y isnormNum_def) |
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170 from H bz b'z have eq: "a * b' = a'*b" |
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171 by (simp add: x y INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult) |
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172 from az a'z na nb have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1" |
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173 by (simp_all add: x y isnormNum_def add: gcd_commute_int) |
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174 from eq have raw_dvd: "a dvd a' * b" "b dvd b' * a" "a' dvd a * b'" "b' dvd b * a'" |
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175 apply - |
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176 apply algebra |
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177 apply algebra |
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178 apply simp |
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179 apply algebra |
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180 done |
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181 from zdvd_antisym_abs[OF coprime_dvd_mult_int[OF gcd1(2) raw_dvd(2)] |
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182 coprime_dvd_mult_int[OF gcd1(4) raw_dvd(4)]] |
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183 have eq1: "b = b'" using pos by arith |
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184 with eq have "a = a'" using pos by simp |
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185 with eq1 have ?rhs by (simp add: x y) } |
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186 ultimately show ?rhs by blast |
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187 next |
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188 assume ?rhs thus ?lhs by simp |
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189 qed |
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190 |
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191 |
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192 lemma isnormNum0[simp]: |
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193 "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field_inverse_zero})) = (x = 0\<^sub>N)" |
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194 unfolding INum_int(2)[symmetric] |
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195 by (rule isnormNum_unique) simp_all |
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196 |
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197 lemma of_int_div_aux: "d ~= 0 ==> ((of_int x)::'a::field_char_0) / (of_int d) = |
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198 of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)" |
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199 proof - |
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200 assume "d ~= 0" |
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201 let ?t = "of_int (x div d) * ((of_int d)::'a) + of_int(x mod d)" |
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202 let ?f = "\<lambda>x. x / of_int d" |
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203 have "x = (x div d) * d + x mod d" |
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204 by auto |
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205 then have eq: "of_int x = ?t" |
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206 by (simp only: of_int_mult[symmetric] of_int_add [symmetric]) |
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207 then have "of_int x / of_int d = ?t / of_int d" |
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208 using cong[OF refl[of ?f] eq] by simp |
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209 then show ?thesis by (simp add: add_divide_distrib algebra_simps `d ~= 0`) |
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210 qed |
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211 |
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212 lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==> |
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213 (of_int(n div d)::'a::field_char_0) = of_int n / of_int d" |
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214 apply (frule of_int_div_aux [of d n, where ?'a = 'a]) |
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215 apply simp |
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216 apply (simp add: dvd_eq_mod_eq_0) |
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217 done |
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218 |
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219 |
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220 lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field_inverse_zero})" |
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221 proof - |
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222 obtain a b where x: "x = (a, b)" by (cases x) |
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223 { assume "a = 0 \<or> b = 0" |
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224 hence ?thesis by (simp add: x INum_def normNum_def split_def Let_def) } |
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225 moreover |
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226 { assume a: "a \<noteq> 0" and b: "b \<noteq> 0" |
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227 let ?g = "gcd a b" |
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228 from a b have g: "?g \<noteq> 0"by simp |
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229 from of_int_div[OF g, where ?'a = 'a] |
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230 have ?thesis by (auto simp add: x INum_def normNum_def split_def Let_def) } |
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231 ultimately show ?thesis by blast |
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232 qed |
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233 |
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234 lemma INum_normNum_iff: |
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235 "(INum x ::'a::{field_char_0, field_inverse_zero}) = INum y \<longleftrightarrow> normNum x = normNum y" |
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236 (is "?lhs = ?rhs") |
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237 proof - |
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238 have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)" |
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239 by (simp del: normNum) |
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240 also have "\<dots> = ?lhs" by simp |
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241 finally show ?thesis by simp |
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242 qed |
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243 |
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244 lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {field_char_0, field_inverse_zero})" |
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245 proof - |
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246 let ?z = "0:: 'a" |
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247 obtain a b where x: "x = (a, b)" by (cases x) |
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248 obtain a' b' where y: "y = (a', b')" by (cases y) |
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249 { assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0" |
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250 hence ?thesis |
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251 apply (cases "a=0", simp_all add: x y Nadd_def) |
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252 apply (cases "b= 0", simp_all add: INum_def) |
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253 apply (cases "a'= 0", simp_all) |
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254 apply (cases "b'= 0", simp_all) |
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255 done } |
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256 moreover |
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257 { assume aa': "a \<noteq> 0" "a'\<noteq> 0" and bb': "b \<noteq> 0" "b' \<noteq> 0" |
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258 { assume z: "a * b' + b * a' = 0" |
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259 hence "of_int (a*b' + b*a') / (of_int b* of_int b') = ?z" by simp |
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260 hence "of_int b' * of_int a / (of_int b * of_int b') + |
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261 of_int b * of_int a' / (of_int b * of_int b') = ?z" |
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262 by (simp add:add_divide_distrib) |
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263 hence th: "of_int a / of_int b + of_int a' / of_int b' = ?z" using bb' aa' |
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264 by simp |
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265 from z aa' bb' have ?thesis |
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266 by (simp add: x y th Nadd_def normNum_def INum_def split_def) } |
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267 moreover { |
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268 assume z: "a * b' + b * a' \<noteq> 0" |
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269 let ?g = "gcd (a * b' + b * a') (b*b')" |
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270 have gz: "?g \<noteq> 0" using z by simp |
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271 have ?thesis using aa' bb' z gz |
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272 of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a * b' + b * a'" and y="b*b'"]] |
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273 of_int_div[where ?'a = 'a, OF gz gcd_dvd2_int[where x="a * b' + b * a'" and y="b*b'"]] |
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274 by (simp add: x y Nadd_def INum_def normNum_def Let_def add_divide_distrib) } |
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275 ultimately have ?thesis using aa' bb' |
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276 by (simp add: x y Nadd_def INum_def normNum_def Let_def) } |
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277 ultimately show ?thesis by blast |
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278 qed |
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279 |
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280 lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {field_char_0, field_inverse_zero})" |
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281 proof - |
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282 let ?z = "0::'a" |
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283 obtain a b where x: "x = (a, b)" by (cases x) |
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284 obtain a' b' where y: "y = (a', b')" by (cases y) |
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285 { assume "a=0 \<or> a'= 0 \<or> b = 0 \<or> b' = 0" |
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286 hence ?thesis |
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287 apply (cases "a=0", simp_all add: x y Nmul_def INum_def Let_def) |
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288 apply (cases "b=0", simp_all) |
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289 apply (cases "a'=0", simp_all) |
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290 done } |
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291 moreover |
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292 { assume z: "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0" |
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293 let ?g="gcd (a*a') (b*b')" |
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294 have gz: "?g \<noteq> 0" using z by simp |
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295 from z of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a*a'" and y="b*b'"]] |
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296 of_int_div[where ?'a = 'a , OF gz gcd_dvd2_int[where x="a*a'" and y="b*b'"]] |
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297 have ?thesis by (simp add: Nmul_def x y Let_def INum_def) } |
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298 ultimately show ?thesis by blast |
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299 qed |
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300 |
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301 lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x ::'a:: field)" |
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302 by (simp add: Nneg_def split_def INum_def) |
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303 |
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304 lemma Nsub[simp]: "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0, field_inverse_zero})" |
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305 by (simp add: Nsub_def split_def) |
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306 |
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307 lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field_inverse_zero) / (INum x)" |
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308 by (simp add: Ninv_def INum_def split_def) |
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309 |
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310 lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {field_char_0, field_inverse_zero})" |
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311 by (simp add: Ndiv_def) |
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312 |
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313 lemma Nlt0_iff[simp]: |
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314 assumes nx: "isnormNum x" |
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315 shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})< 0) = 0>\<^sub>N x" |
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316 proof - |
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317 obtain a b where x: "x = (a, b)" by (cases x) |
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318 { assume "a = 0" hence ?thesis by (simp add: x Nlt0_def INum_def) } |
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319 moreover |
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320 { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0" |
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321 using nx by (simp add: x isnormNum_def) |
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322 from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"] |
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323 have ?thesis by (simp add: x Nlt0_def INum_def) } |
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324 ultimately show ?thesis by blast |
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325 qed |
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326 |
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327 lemma Nle0_iff[simp]: |
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328 assumes nx: "isnormNum x" |
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329 shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<le> 0) = 0\<ge>\<^sub>N x" |
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330 proof - |
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331 obtain a b where x: "x = (a, b)" by (cases x) |
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332 { assume "a = 0" hence ?thesis by (simp add: x Nle0_def INum_def) } |
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333 moreover |
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334 { assume a: "a \<noteq> 0" hence b: "(of_int b :: 'a) > 0" |
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335 using nx by (simp add: x isnormNum_def) |
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336 from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"] |
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337 have ?thesis by (simp add: x Nle0_def INum_def) } |
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338 ultimately show ?thesis by blast |
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339 qed |
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340 |
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341 lemma Ngt0_iff[simp]: |
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342 assumes nx: "isnormNum x" |
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343 shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<\<^sub>N x" |
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344 proof - |
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345 obtain a b where x: "x = (a, b)" by (cases x) |
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346 { assume "a = 0" hence ?thesis by (simp add: x Ngt0_def INum_def) } |
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347 moreover |
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348 { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx |
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349 by (simp add: x isnormNum_def) |
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350 from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"] |
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351 have ?thesis by (simp add: x Ngt0_def INum_def) } |
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352 ultimately show ?thesis by blast |
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353 qed |
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354 |
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355 lemma Nge0_iff[simp]: |
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356 assumes nx: "isnormNum x" |
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357 shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<ge> 0) = 0\<le>\<^sub>N x" |
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358 proof - |
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359 obtain a b where x: "x = (a, b)" by (cases x) |
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360 { assume "a = 0" hence ?thesis by (simp add: x Nge0_def INum_def) } |
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361 moreover |
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362 { assume "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx |
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363 by (simp add: x isnormNum_def) |
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364 from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"] |
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365 have ?thesis by (simp add: x Nge0_def INum_def) } |
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366 ultimately show ?thesis by blast |
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367 qed |
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368 |
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369 lemma Nlt_iff[simp]: |
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370 assumes nx: "isnormNum x" and ny: "isnormNum y" |
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371 shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) < INum y) = (x <\<^sub>N y)" |
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372 proof - |
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373 let ?z = "0::'a" |
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374 have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)" |
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375 using nx ny by simp |
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376 also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))" |
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377 using Nlt0_iff[OF Nsub_normN[OF ny]] by simp |
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378 finally show ?thesis by (simp add: Nlt_def) |
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379 qed |
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380 |
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381 lemma Nle_iff[simp]: |
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382 assumes nx: "isnormNum x" and ny: "isnormNum y" |
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383 shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})\<le> INum y) = (x \<le>\<^sub>N y)" |
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384 proof - |
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385 have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))" |
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386 using nx ny by simp |
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387 also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))" |
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388 using Nle0_iff[OF Nsub_normN[OF ny]] by simp |
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389 finally show ?thesis by (simp add: Nle_def) |
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390 qed |
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391 |
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392 lemma Nadd_commute: |
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393 assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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394 shows "x +\<^sub>N y = y +\<^sub>N x" |
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395 proof - |
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396 have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)" by simp_all |
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397 have "(INum (x +\<^sub>N y)::'a) = INum (y +\<^sub>N x)" by simp |
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398 with isnormNum_unique[OF n] show ?thesis by simp |
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399 qed |
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400 |
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401 lemma [simp]: |
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402 assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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403 shows "(0, b) +\<^sub>N y = normNum y" |
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404 and "(a, 0) +\<^sub>N y = normNum y" |
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405 and "x +\<^sub>N (0, b) = normNum x" |
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406 and "x +\<^sub>N (a, 0) = normNum x" |
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407 apply (simp add: Nadd_def split_def) |
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408 apply (simp add: Nadd_def split_def) |
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409 apply (subst Nadd_commute, simp add: Nadd_def split_def) |
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410 apply (subst Nadd_commute, simp add: Nadd_def split_def) |
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411 done |
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412 |
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413 lemma normNum_nilpotent_aux[simp]: |
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414 assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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415 assumes nx: "isnormNum x" |
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416 shows "normNum x = x" |
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417 proof - |
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418 let ?a = "normNum x" |
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419 have n: "isnormNum ?a" by simp |
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420 have th: "INum ?a = (INum x ::'a)" by simp |
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421 with isnormNum_unique[OF n nx] show ?thesis by simp |
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422 qed |
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423 |
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424 lemma normNum_nilpotent[simp]: |
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425 assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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426 shows "normNum (normNum x) = normNum x" |
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427 by simp |
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428 |
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429 lemma normNum0[simp]: "normNum (0,b) = 0\<^sub>N" "normNum (a,0) = 0\<^sub>N" |
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430 by (simp_all add: normNum_def) |
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431 |
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432 lemma normNum_Nadd: |
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433 assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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434 shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp |
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435 |
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436 lemma Nadd_normNum1[simp]: |
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437 assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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438 shows "normNum x +\<^sub>N y = x +\<^sub>N y" |
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439 proof - |
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440 have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)" by simp_all |
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441 have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)" by simp |
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442 also have "\<dots> = INum (x +\<^sub>N y)" by simp |
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443 finally show ?thesis using isnormNum_unique[OF n] by simp |
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444 qed |
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445 |
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446 lemma Nadd_normNum2[simp]: |
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447 assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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448 shows "x +\<^sub>N normNum y = x +\<^sub>N y" |
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449 proof - |
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450 have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)" by simp_all |
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451 have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)" by simp |
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452 also have "\<dots> = INum (x +\<^sub>N y)" by simp |
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453 finally show ?thesis using isnormNum_unique[OF n] by simp |
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454 qed |
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455 |
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456 lemma Nadd_assoc: |
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457 assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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458 shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)" |
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459 proof - |
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460 have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))" by simp_all |
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461 have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp |
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462 with isnormNum_unique[OF n] show ?thesis by simp |
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463 qed |
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464 |
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465 lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x" |
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466 by (simp add: Nmul_def split_def Let_def gcd_commute_int mult_commute) |
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467 |
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468 lemma Nmul_assoc: |
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469 assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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470 assumes nx: "isnormNum x" and ny: "isnormNum y" and nz: "isnormNum z" |
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471 shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)" |
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472 proof - |
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473 from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))" |
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474 by simp_all |
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475 have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp |
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476 with isnormNum_unique[OF n] show ?thesis by simp |
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477 qed |
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478 |
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479 lemma Nsub0: |
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480 assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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481 assumes x: "isnormNum x" and y: "isnormNum y" |
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482 shows "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = y" |
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483 proof - |
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484 fix h :: 'a |
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485 from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"] |
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486 have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp |
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487 also have "\<dots> = (INum x = (INum y :: 'a))" by simp |
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488 also have "\<dots> = (x = y)" using x y by simp |
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489 finally show ?thesis . |
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490 qed |
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491 |
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492 lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N" |
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493 by (simp_all add: Nmul_def Let_def split_def) |
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494 |
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495 lemma Nmul_eq0[simp]: |
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496 assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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497 assumes nx: "isnormNum x" and ny: "isnormNum y" |
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498 shows "x*\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = 0\<^sub>N \<or> y = 0\<^sub>N" |
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499 proof - |
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500 fix h :: 'a |
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501 obtain a b where x: "x = (a, b)" by (cases x) |
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502 obtain a' b' where y: "y = (a', b')" by (cases y) |
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503 have n0: "isnormNum 0\<^sub>N" by simp |
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504 show ?thesis using nx ny |
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505 apply (simp only: isnormNum_unique[where ?'a = 'a, OF Nmul_normN[OF nx ny] n0, symmetric] |
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506 Nmul[where ?'a = 'a]) |
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507 apply (simp add: x y INum_def split_def isnormNum_def split: split_if_asm) |
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508 done |
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509 qed |
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510 |
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511 lemma Nneg_Nneg[simp]: "~\<^sub>N (~\<^sub>N c) = c" |
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512 by (simp add: Nneg_def split_def) |
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513 |
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514 lemma Nmul1[simp]: |
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515 "isnormNum c \<Longrightarrow> (1)\<^sub>N *\<^sub>N c = c" |
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516 "isnormNum c \<Longrightarrow> c *\<^sub>N (1)\<^sub>N = c" |
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517 apply (simp_all add: Nmul_def Let_def split_def isnormNum_def) |
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518 apply (cases "fst c = 0", simp_all, cases c, simp_all)+ |
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519 done |
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520 |
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521 end |
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