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1 (* Title : HOL/Hyperreal/StarType.thy |
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2 ID : $Id$ |
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3 Author : Jacques D. Fleuriot and Brian Huffman |
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4 *) |
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5 |
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6 header {* Construction of Star Types Using Ultrafilters *} |
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7 |
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8 theory StarType |
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9 imports Filter |
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10 begin |
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11 |
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12 subsection {* A Free Ultrafilter over the Naturals *} |
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13 |
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14 constdefs |
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15 FreeUltrafilterNat :: "nat set set" ("\<U>") |
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16 "\<U> \<equiv> SOME U. freeultrafilter U" |
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17 |
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18 lemma freeultrafilter_FUFNat: "freeultrafilter \<U>" |
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19 apply (unfold FreeUltrafilterNat_def) |
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20 apply (rule someI_ex) |
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21 apply (rule freeultrafilter_Ex) |
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22 apply (rule nat_infinite) |
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23 done |
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24 |
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25 lemmas ultrafilter_FUFNat = |
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26 freeultrafilter_FUFNat [THEN freeultrafilter.ultrafilter] |
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27 |
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28 lemmas filter_FUFNat = |
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29 freeultrafilter_FUFNat [THEN freeultrafilter.filter] |
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30 |
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31 lemmas FUFNat_empty [iff] = |
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32 filter_FUFNat [THEN filter.empty] |
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33 |
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34 lemmas FUFNat_UNIV [iff] = |
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35 filter_FUFNat [THEN filter.UNIV] |
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36 |
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37 text {* This rule takes the place of the old ultra tactic *} |
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38 |
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39 lemma ultra: |
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40 "\<lbrakk>{n. P n} \<in> \<U>; {n. P n \<longrightarrow> Q n} \<in> \<U>\<rbrakk> \<Longrightarrow> {n. Q n} \<in> \<U>" |
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41 by (simp add: Collect_imp_eq |
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42 ultrafilter_FUFNat [THEN ultrafilter.Un_iff] |
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43 ultrafilter_FUFNat [THEN ultrafilter.Compl_iff]) |
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44 |
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45 |
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46 subsection {* Definition of @{text star} type constructor *} |
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47 |
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48 constdefs |
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49 starrel :: "((nat \<Rightarrow> 'a) \<times> (nat \<Rightarrow> 'a)) set" |
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50 "starrel \<equiv> {(X,Y). {n. X n = Y n} \<in> \<U>}" |
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51 |
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52 typedef 'a star = "(UNIV :: (nat \<Rightarrow> 'a) set) // starrel" |
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53 by (auto intro: quotientI) |
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54 |
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55 text {* Proving that @{term starrel} is an equivalence relation *} |
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56 |
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57 lemma starrel_iff [iff]: "((X,Y) \<in> starrel) = ({n. X n = Y n} \<in> \<U>)" |
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58 by (simp add: starrel_def) |
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59 |
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60 lemma equiv_starrel: "equiv UNIV starrel" |
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61 proof (rule equiv.intro) |
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62 show "reflexive starrel" by (simp add: refl_def) |
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63 show "sym starrel" by (simp add: sym_def eq_commute) |
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64 show "trans starrel" by (auto intro: transI elim!: ultra) |
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65 qed |
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66 |
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67 lemmas equiv_starrel_iff = |
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68 eq_equiv_class_iff [OF equiv_starrel UNIV_I UNIV_I] |
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69 -- {* @{term "(starrel `` {x} = starrel `` {y}) = ((x,y) \<in> starrel)"} *} |
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70 |
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71 lemma starrel_in_star: "starrel``{x} \<in> star" |
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72 by (simp add: star_def starrel_def quotient_def, fast) |
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73 |
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74 lemma eq_Abs_star: |
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75 "(\<And>x. z = Abs_star (starrel``{x}) \<Longrightarrow> P) \<Longrightarrow> P" |
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76 apply (rule_tac x=z in Abs_star_cases) |
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77 apply (unfold star_def) |
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78 apply (erule quotientE) |
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79 apply simp |
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80 done |
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81 |
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82 |
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83 subsection {* Constructors for type @{typ "'a star"} *} |
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84 |
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85 constdefs |
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86 star_n :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a star" |
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87 "star_n X \<equiv> Abs_star (starrel `` {X})" |
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88 |
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89 star_of :: "'a \<Rightarrow> 'a star" |
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90 "star_of x \<equiv> star_n (\<lambda>n. x)" |
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91 |
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92 theorem star_cases: |
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93 "(\<And>X. x = star_n X \<Longrightarrow> P) \<Longrightarrow> P" |
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94 by (unfold star_n_def, rule eq_Abs_star[of x], blast) |
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95 |
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96 lemma all_star_eq: "(\<forall>x. P x) = (\<forall>X. P (star_n X))" |
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97 by (auto, rule_tac x=x in star_cases, simp) |
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98 |
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99 lemma ex_star_eq: "(\<exists>x. P x) = (\<exists>X. P (star_n X))" |
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100 by (auto, rule_tac x=x in star_cases, auto) |
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101 |
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102 lemma star_n_eq_iff: "(star_n X = star_n Y) = ({n. X n = Y n} \<in> \<U>)" |
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103 apply (unfold star_n_def) |
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104 apply (simp add: Abs_star_inject starrel_in_star equiv_starrel_iff) |
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105 done |
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106 |
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107 lemma star_of_inject: "(star_of x = star_of y) = (x = y)" |
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108 by (simp add: star_of_def star_n_eq_iff) |
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109 |
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110 |
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111 subsection {* Internal functions *} |
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112 |
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113 constdefs |
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114 Ifun :: "('a \<Rightarrow> 'b) star \<Rightarrow> 'a star \<Rightarrow> 'b star" ("_ \<star> _" [300,301] 300) |
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115 "Ifun f \<equiv> \<lambda>x. Abs_star |
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116 (\<Union>F\<in>Rep_star f. \<Union>X\<in>Rep_star x. starrel``{\<lambda>n. F n (X n)})" |
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117 |
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118 lemma Ifun_star_n: "star_n F \<star> star_n X = star_n (\<lambda>n. F n (X n))" |
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119 apply (unfold Ifun_def star_n_def) |
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120 apply (simp add: Abs_star_inverse starrel_in_star) |
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121 apply (rule_tac f=Abs_star in arg_cong) |
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122 apply safe |
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123 apply (erule ultra)+ |
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124 apply simp |
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125 apply force |
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126 done |
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127 |
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128 lemma Ifun [simp]: "star_of f \<star> star_of x = star_of (f x)" |
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129 by (simp only: star_of_def Ifun_star_n) |
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130 |
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131 |
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132 subsection {* Testing lifted booleans *} |
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133 |
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134 constdefs |
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135 unstar :: "bool star \<Rightarrow> bool" |
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136 "unstar b \<equiv> b = star_of True" |
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137 |
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138 lemma unstar_star_n: "unstar (star_n P) = ({n. P n} \<in> \<U>)" |
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139 by (simp add: unstar_def star_of_def star_n_eq_iff) |
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140 |
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141 lemma unstar [simp]: "unstar (star_of p) = p" |
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142 by (simp add: unstar_def star_of_inject) |
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143 |
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144 |
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145 subsection {* Internal functions and predicates *} |
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146 |
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147 constdefs |
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148 Ifun_of :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a star \<Rightarrow> 'b star)" |
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149 "Ifun_of f \<equiv> Ifun (star_of f)" |
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150 |
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151 Ifun2 :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) star \<Rightarrow> ('a star \<Rightarrow> 'b star \<Rightarrow> 'c star)" |
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152 "Ifun2 f \<equiv> \<lambda>x y. f \<star> x \<star> y" |
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153 |
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154 Ifun2_of :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a star \<Rightarrow> 'b star \<Rightarrow> 'c star)" |
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155 "Ifun2_of f \<equiv> \<lambda>x y. star_of f \<star> x \<star> y" |
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156 |
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157 Ipred :: "('a \<Rightarrow> bool) star \<Rightarrow> ('a star \<Rightarrow> bool)" |
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158 "Ipred P \<equiv> \<lambda>x. unstar (P \<star> x)" |
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159 |
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160 Ipred_of :: "('a \<Rightarrow> bool) \<Rightarrow> ('a star \<Rightarrow> bool)" |
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161 "Ipred_of P \<equiv> \<lambda>x. unstar (star_of P \<star> x)" |
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162 |
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163 Ipred2 :: "('a \<Rightarrow> 'b \<Rightarrow> bool) star \<Rightarrow> ('a star \<Rightarrow> 'b star \<Rightarrow> bool)" |
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164 "Ipred2 P \<equiv> \<lambda>x y. unstar (P \<star> x \<star> y)" |
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165 |
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166 Ipred2_of :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a star \<Rightarrow> 'b star \<Rightarrow> bool)" |
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167 "Ipred2_of P \<equiv> \<lambda>x y. unstar (star_of P \<star> x \<star> y)" |
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168 |
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169 lemma Ifun_of [simp]: |
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170 "Ifun_of f (star_of x) = star_of (f x)" |
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171 by (simp only: Ifun_of_def Ifun) |
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172 |
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173 lemma Ifun2_of [simp]: |
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174 "Ifun2_of f (star_of x) (star_of y) = star_of (f x y)" |
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175 by (simp only: Ifun2_of_def Ifun) |
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176 |
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177 lemma Ipred_of [simp]: |
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178 "Ipred_of P (star_of x) = P x" |
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179 by (simp only: Ipred_of_def Ifun unstar) |
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180 |
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181 lemma Ipred2_of [simp]: |
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182 "Ipred2_of P (star_of x) (star_of y) = P x y" |
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183 by (simp only: Ipred2_of_def Ifun unstar) |
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184 |
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185 lemmas Ifun_defs = |
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186 star_of_def Ifun_of_def Ifun2_def Ifun2_of_def |
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187 Ipred_def Ipred_of_def Ipred2_def Ipred2_of_def |
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188 |
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189 |
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190 subsection {* Internal sets *} |
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191 |
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192 constdefs |
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193 Iset :: "'a set star \<Rightarrow> 'a star set" |
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194 "Iset A \<equiv> {x. Ipred2_of (op \<in>) x A}" |
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195 |
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196 Iset_of :: "'a set \<Rightarrow> 'a star set" |
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197 "Iset_of A \<equiv> Iset (star_of A)" |
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198 |
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199 lemma Iset_star_n: |
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200 "(star_n X \<in> Iset (star_n A)) = ({n. X n \<in> A n} \<in> \<U>)" |
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201 by (simp add: Iset_def Ipred2_of_def star_of_def Ifun_star_n unstar_star_n) |
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202 |
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203 |
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204 subsection {* Class constants *} |
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205 |
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206 instance star :: (ord) ord .. |
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207 instance star :: (zero) zero .. |
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208 instance star :: (one) one .. |
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209 instance star :: (plus) plus .. |
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210 instance star :: (times) times .. |
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211 instance star :: (minus) minus .. |
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212 instance star :: (inverse) inverse .. |
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213 instance star :: (number) number .. |
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214 instance star :: ("Divides.div") "Divides.div" .. |
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215 instance star :: (power) power .. |
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216 |
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217 defs (overloaded) |
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218 star_zero_def: "0 \<equiv> star_of 0" |
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219 star_one_def: "1 \<equiv> star_of 1" |
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220 star_number_def: "number_of b \<equiv> star_of (number_of b)" |
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221 star_add_def: "(op +) \<equiv> Ifun2_of (op +)" |
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222 star_diff_def: "(op -) \<equiv> Ifun2_of (op -)" |
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223 star_minus_def: "uminus \<equiv> Ifun_of uminus" |
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224 star_mult_def: "(op *) \<equiv> Ifun2_of (op *)" |
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225 star_divide_def: "(op /) \<equiv> Ifun2_of (op /)" |
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226 star_inverse_def: "inverse \<equiv> Ifun_of inverse" |
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227 star_le_def: "(op \<le>) \<equiv> Ipred2_of (op \<le>)" |
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228 star_less_def: "(op <) \<equiv> Ipred2_of (op <)" |
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229 star_abs_def: "abs \<equiv> Ifun_of abs" |
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230 star_div_def: "(op div) \<equiv> Ifun2_of (op div)" |
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231 star_mod_def: "(op mod) \<equiv> Ifun2_of (op mod)" |
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232 star_power_def: "(op ^) \<equiv> \<lambda>x n. Ifun_of (\<lambda>x. x ^ n) x" |
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233 |
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234 lemmas star_class_defs = |
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235 star_zero_def star_one_def star_number_def |
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236 star_add_def star_diff_def star_minus_def |
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237 star_mult_def star_divide_def star_inverse_def |
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238 star_le_def star_less_def star_abs_def |
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239 star_div_def star_mod_def star_power_def |
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240 |
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241 text {* @{term star_of} preserves class operations *} |
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242 |
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243 lemma star_of_add: "star_of (x + y) = star_of x + star_of y" |
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244 by (simp add: star_add_def) |
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245 |
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246 lemma star_of_diff: "star_of (x - y) = star_of x - star_of y" |
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247 by (simp add: star_diff_def) |
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248 |
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249 lemma star_of_minus: "star_of (-x) = - star_of x" |
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250 by (simp add: star_minus_def) |
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251 |
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252 lemma star_of_mult: "star_of (x * y) = star_of x * star_of y" |
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253 by (simp add: star_mult_def) |
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254 |
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255 lemma star_of_divide: "star_of (x / y) = star_of x / star_of y" |
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256 by (simp add: star_divide_def) |
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257 |
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258 lemma star_of_inverse: "star_of (inverse x) = inverse (star_of x)" |
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259 by (simp add: star_inverse_def) |
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260 |
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261 lemma star_of_div: "star_of (x div y) = star_of x div star_of y" |
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262 by (simp add: star_div_def) |
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263 |
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264 lemma star_of_mod: "star_of (x mod y) = star_of x mod star_of y" |
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265 by (simp add: star_mod_def) |
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266 |
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267 lemma star_of_power: "star_of (x ^ n) = star_of x ^ n" |
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268 by (simp add: star_power_def) |
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269 |
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270 lemma star_of_abs: "star_of (abs x) = abs (star_of x)" |
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271 by (simp add: star_abs_def) |
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272 |
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273 text {* @{term star_of} preserves numerals *} |
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274 |
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275 lemma star_of_zero: "star_of 0 = 0" |
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276 by (simp add: star_zero_def) |
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277 |
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278 lemma star_of_one: "star_of 1 = 1" |
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279 by (simp add: star_one_def) |
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280 |
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281 lemma star_of_number_of: "star_of (number_of x) = number_of x" |
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282 by (simp add: star_number_def) |
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283 |
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284 text {* @{term star_of} preserves orderings *} |
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285 |
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286 lemma star_of_less: "(star_of x < star_of y) = (x < y)" |
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287 by (simp add: star_less_def) |
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288 |
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289 lemma star_of_le: "(star_of x \<le> star_of y) = (x \<le> y)" |
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290 by (simp add: star_le_def) |
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291 |
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292 lemma star_of_eq: "(star_of x = star_of y) = (x = y)" |
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293 by (rule star_of_inject) |
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294 |
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295 text{*As above, for 0*} |
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296 |
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297 lemmas star_of_0_less = star_of_less [of 0, simplified star_of_zero] |
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298 lemmas star_of_0_le = star_of_le [of 0, simplified star_of_zero] |
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299 lemmas star_of_0_eq = star_of_eq [of 0, simplified star_of_zero] |
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300 |
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301 lemmas star_of_less_0 = star_of_less [of _ 0, simplified star_of_zero] |
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302 lemmas star_of_le_0 = star_of_le [of _ 0, simplified star_of_zero] |
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303 lemmas star_of_eq_0 = star_of_eq [of _ 0, simplified star_of_zero] |
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304 |
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305 text{*As above, for 1*} |
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306 |
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307 lemmas star_of_1_less = star_of_less [of 1, simplified star_of_one] |
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308 lemmas star_of_1_le = star_of_le [of 1, simplified star_of_one] |
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309 lemmas star_of_1_eq = star_of_eq [of 1, simplified star_of_one] |
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310 |
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311 lemmas star_of_less_1 = star_of_less [of _ 1, simplified star_of_one] |
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312 lemmas star_of_le_1 = star_of_le [of _ 1, simplified star_of_one] |
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313 lemmas star_of_eq_1 = star_of_eq [of _ 1, simplified star_of_one] |
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314 |
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315 lemmas star_of_simps = |
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316 star_of_add star_of_diff star_of_minus |
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317 star_of_mult star_of_divide star_of_inverse |
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318 star_of_div star_of_mod |
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319 star_of_power star_of_abs |
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320 star_of_zero star_of_one star_of_number_of |
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321 star_of_less star_of_le star_of_eq |
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322 star_of_0_less star_of_0_le star_of_0_eq |
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323 star_of_less_0 star_of_le_0 star_of_eq_0 |
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324 star_of_1_less star_of_1_le star_of_1_eq |
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325 star_of_less_1 star_of_le_1 star_of_eq_1 |
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326 |
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327 declare star_of_simps [simp] |
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328 |
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329 end |