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1 (* Author: Andreas Lochbihler, KIT *) |
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2 |
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3 header {* A type class for computing the cardinality of a type's universe *} |
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4 |
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5 theory Card_Univ imports Main begin |
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6 |
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7 subsection {* A type class for computing the cardinality of a type's universe *} |
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8 |
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9 class card_UNIV = |
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10 fixes card_UNIV :: "'a itself \<Rightarrow> nat" |
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11 assumes card_UNIV: "card_UNIV x = card (UNIV :: 'a set)" |
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12 begin |
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13 |
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14 lemma card_UNIV_neq_0_finite_UNIV: |
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15 "card_UNIV x \<noteq> 0 \<longleftrightarrow> finite (UNIV :: 'a set)" |
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16 by(simp add: card_UNIV card_eq_0_iff) |
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17 |
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18 lemma card_UNIV_ge_0_finite_UNIV: |
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19 "card_UNIV x > 0 \<longleftrightarrow> finite (UNIV :: 'a set)" |
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20 by(auto simp add: card_UNIV intro: card_ge_0_finite finite_UNIV_card_ge_0) |
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21 |
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22 lemma card_UNIV_eq_0_infinite_UNIV: |
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23 "card_UNIV x = 0 \<longleftrightarrow> \<not> finite (UNIV :: 'a set)" |
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24 by(simp add: card_UNIV card_eq_0_iff) |
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25 |
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26 definition is_list_UNIV :: "'a list \<Rightarrow> bool" |
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27 where "is_list_UNIV xs = (let c = card_UNIV (TYPE('a)) in if c = 0 then False else size (remdups xs) = c)" |
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28 |
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29 lemma is_list_UNIV_iff: |
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30 fixes xs :: "'a list" |
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31 shows "is_list_UNIV xs \<longleftrightarrow> set xs = UNIV" |
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32 proof |
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33 assume "is_list_UNIV xs" |
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34 hence c: "card_UNIV (TYPE('a)) > 0" and xs: "size (remdups xs) = card_UNIV (TYPE('a))" |
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35 unfolding is_list_UNIV_def by(simp_all add: Let_def split: split_if_asm) |
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36 from c have fin: "finite (UNIV :: 'a set)" by(auto simp add: card_UNIV_ge_0_finite_UNIV) |
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37 have "card (set (remdups xs)) = size (remdups xs)" by(subst distinct_card) auto |
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38 also note set_remdups |
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39 finally show "set xs = UNIV" using fin unfolding xs card_UNIV by-(rule card_eq_UNIV_imp_eq_UNIV) |
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40 next |
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41 assume xs: "set xs = UNIV" |
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42 from finite_set[of xs] have fin: "finite (UNIV :: 'a set)" unfolding xs . |
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43 hence "card_UNIV (TYPE ('a)) \<noteq> 0" unfolding card_UNIV_neq_0_finite_UNIV . |
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44 moreover have "size (remdups xs) = card (set (remdups xs))" |
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45 by(subst distinct_card) auto |
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46 ultimately show "is_list_UNIV xs" using xs by(simp add: is_list_UNIV_def Let_def card_UNIV) |
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47 qed |
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48 |
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49 lemma card_UNIV_eq_0_is_list_UNIV_False: |
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50 assumes cU0: "card_UNIV x = 0" |
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51 shows "is_list_UNIV = (\<lambda>xs. False)" |
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52 proof(rule ext) |
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53 fix xs :: "'a list" |
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54 from cU0 have "\<not> finite (UNIV :: 'a set)" |
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55 by(auto simp only: card_UNIV_eq_0_infinite_UNIV) |
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56 moreover have "finite (set xs)" by(rule finite_set) |
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57 ultimately have "(UNIV :: 'a set) \<noteq> set xs" by(auto simp del: finite_set) |
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58 thus "is_list_UNIV xs = False" unfolding is_list_UNIV_iff by simp |
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59 qed |
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60 |
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61 end |
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62 |
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63 subsection {* Instantiations for @{text "card_UNIV"} *} |
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64 |
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65 subsubsection {* @{typ "nat"} *} |
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66 |
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67 instantiation nat :: card_UNIV begin |
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68 |
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69 definition card_UNIV_nat_def: |
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70 "card_UNIV_class.card_UNIV = (\<lambda>a :: nat itself. 0)" |
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71 |
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72 instance proof |
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73 fix x :: "nat itself" |
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74 show "card_UNIV x = card (UNIV :: nat set)" |
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75 unfolding card_UNIV_nat_def by simp |
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76 qed |
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77 |
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78 end |
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79 |
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80 subsubsection {* @{typ "int"} *} |
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81 |
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82 instantiation int :: card_UNIV begin |
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83 |
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84 definition card_UNIV_int_def: |
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85 "card_UNIV_class.card_UNIV = (\<lambda>a :: int itself. 0)" |
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86 |
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87 instance proof |
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88 fix x :: "int itself" |
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89 show "card_UNIV x = card (UNIV :: int set)" |
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90 unfolding card_UNIV_int_def by(simp add: infinite_UNIV_int) |
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91 qed |
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92 |
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93 end |
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94 |
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95 subsubsection {* @{typ "'a list"} *} |
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96 |
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97 instantiation list :: (type) card_UNIV begin |
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98 |
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99 definition card_UNIV_list_def: |
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100 "card_UNIV_class.card_UNIV = (\<lambda>a :: 'a list itself. 0)" |
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101 |
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102 instance proof |
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103 fix x :: "'a list itself" |
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104 show "card_UNIV x = card (UNIV :: 'a list set)" |
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105 unfolding card_UNIV_list_def by(simp add: infinite_UNIV_listI) |
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106 qed |
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107 |
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108 end |
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109 |
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110 subsubsection {* @{typ "unit"} *} |
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111 |
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112 lemma card_UNIV_unit: "card (UNIV :: unit set) = 1" |
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113 unfolding UNIV_unit by simp |
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114 |
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115 instantiation unit :: card_UNIV begin |
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116 |
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117 definition card_UNIV_unit_def: |
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118 "card_UNIV_class.card_UNIV = (\<lambda>a :: unit itself. 1)" |
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119 |
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120 instance proof |
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121 fix x :: "unit itself" |
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122 show "card_UNIV x = card (UNIV :: unit set)" |
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123 by(simp add: card_UNIV_unit_def card_UNIV_unit) |
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124 qed |
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125 |
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126 end |
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127 |
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128 subsubsection {* @{typ "bool"} *} |
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129 |
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130 lemma card_UNIV_bool: "card (UNIV :: bool set) = 2" |
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131 unfolding UNIV_bool by simp |
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132 |
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133 instantiation bool :: card_UNIV begin |
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134 |
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135 definition card_UNIV_bool_def: |
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136 "card_UNIV_class.card_UNIV = (\<lambda>a :: bool itself. 2)" |
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137 |
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138 instance proof |
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139 fix x :: "bool itself" |
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140 show "card_UNIV x = card (UNIV :: bool set)" |
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141 by(simp add: card_UNIV_bool_def card_UNIV_bool) |
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142 qed |
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143 |
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144 end |
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145 |
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146 subsubsection {* @{typ "char"} *} |
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147 |
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148 lemma card_UNIV_char: "card (UNIV :: char set) = 256" |
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149 proof - |
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150 from enum_distinct |
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151 have "card (set (Enum.enum :: char list)) = length (Enum.enum :: char list)" |
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152 by (rule distinct_card) |
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153 also have "set Enum.enum = (UNIV :: char set)" by (auto intro: in_enum) |
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154 also note enum_chars |
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155 finally show ?thesis by (simp add: chars_def) |
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156 qed |
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157 |
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158 instantiation char :: card_UNIV begin |
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159 |
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160 definition card_UNIV_char_def: |
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161 "card_UNIV_class.card_UNIV = (\<lambda>a :: char itself. 256)" |
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162 |
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163 instance proof |
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164 fix x :: "char itself" |
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165 show "card_UNIV x = card (UNIV :: char set)" |
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166 by(simp add: card_UNIV_char_def card_UNIV_char) |
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167 qed |
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168 |
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169 end |
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170 |
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171 subsubsection {* @{typ "'a \<times> 'b"} *} |
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172 |
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173 instantiation prod :: (card_UNIV, card_UNIV) card_UNIV begin |
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174 |
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175 definition card_UNIV_product_def: |
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176 "card_UNIV_class.card_UNIV = (\<lambda>a :: ('a \<times> 'b) itself. card_UNIV (TYPE('a)) * card_UNIV (TYPE('b)))" |
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177 |
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178 instance proof |
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179 fix x :: "('a \<times> 'b) itself" |
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180 show "card_UNIV x = card (UNIV :: ('a \<times> 'b) set)" |
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181 by(simp add: card_UNIV_product_def card_UNIV UNIV_Times_UNIV[symmetric] card_cartesian_product del: UNIV_Times_UNIV) |
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182 qed |
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183 |
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184 end |
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185 |
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186 subsubsection {* @{typ "'a + 'b"} *} |
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187 |
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188 instantiation sum :: (card_UNIV, card_UNIV) card_UNIV begin |
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189 |
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190 definition card_UNIV_sum_def: |
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191 "card_UNIV_class.card_UNIV = (\<lambda>a :: ('a + 'b) itself. let ca = card_UNIV (TYPE('a)); cb = card_UNIV (TYPE('b)) |
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192 in if ca \<noteq> 0 \<and> cb \<noteq> 0 then ca + cb else 0)" |
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193 |
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194 instance proof |
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195 fix x :: "('a + 'b) itself" |
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196 show "card_UNIV x = card (UNIV :: ('a + 'b) set)" |
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197 by (auto simp add: card_UNIV_sum_def card_UNIV card_eq_0_iff UNIV_Plus_UNIV[symmetric] finite_Plus_iff Let_def card_Plus simp del: UNIV_Plus_UNIV dest!: card_ge_0_finite) |
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198 qed |
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199 |
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200 end |
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201 |
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202 subsubsection {* @{typ "'a \<Rightarrow> 'b"} *} |
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203 |
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204 instantiation "fun" :: (card_UNIV, card_UNIV) card_UNIV begin |
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205 |
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206 definition card_UNIV_fun_def: |
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207 "card_UNIV_class.card_UNIV = (\<lambda>a :: ('a \<Rightarrow> 'b) itself. let ca = card_UNIV (TYPE('a)); cb = card_UNIV (TYPE('b)) |
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208 in if ca \<noteq> 0 \<and> cb \<noteq> 0 \<or> cb = 1 then cb ^ ca else 0)" |
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209 |
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210 instance proof |
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211 fix x :: "('a \<Rightarrow> 'b) itself" |
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212 |
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213 { assume "0 < card (UNIV :: 'a set)" |
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214 and "0 < card (UNIV :: 'b set)" |
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215 hence fina: "finite (UNIV :: 'a set)" and finb: "finite (UNIV :: 'b set)" |
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216 by(simp_all only: card_ge_0_finite) |
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217 from finite_distinct_list[OF finb] obtain bs |
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218 where bs: "set bs = (UNIV :: 'b set)" and distb: "distinct bs" by blast |
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219 from finite_distinct_list[OF fina] obtain as |
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220 where as: "set as = (UNIV :: 'a set)" and dista: "distinct as" by blast |
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221 have cb: "card (UNIV :: 'b set) = length bs" |
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222 unfolding bs[symmetric] distinct_card[OF distb] .. |
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223 have ca: "card (UNIV :: 'a set) = length as" |
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224 unfolding as[symmetric] distinct_card[OF dista] .. |
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225 let ?xs = "map (\<lambda>ys. the o map_of (zip as ys)) (Enum.n_lists (length as) bs)" |
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226 have "UNIV = set ?xs" |
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227 proof(rule UNIV_eq_I) |
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228 fix f :: "'a \<Rightarrow> 'b" |
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229 from as have "f = the \<circ> map_of (zip as (map f as))" |
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230 by(auto simp add: map_of_zip_map intro: ext) |
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231 thus "f \<in> set ?xs" using bs by(auto simp add: set_n_lists) |
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232 qed |
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233 moreover have "distinct ?xs" unfolding distinct_map |
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234 proof(intro conjI distinct_n_lists distb inj_onI) |
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235 fix xs ys :: "'b list" |
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236 assume xs: "xs \<in> set (Enum.n_lists (length as) bs)" |
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237 and ys: "ys \<in> set (Enum.n_lists (length as) bs)" |
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238 and eq: "the \<circ> map_of (zip as xs) = the \<circ> map_of (zip as ys)" |
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239 from xs ys have [simp]: "length xs = length as" "length ys = length as" |
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240 by(simp_all add: length_n_lists_elem) |
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241 have "map_of (zip as xs) = map_of (zip as ys)" |
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242 proof |
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243 fix x |
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244 from as bs have "\<exists>y. map_of (zip as xs) x = Some y" "\<exists>y. map_of (zip as ys) x = Some y" |
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245 by(simp_all add: map_of_zip_is_Some[symmetric]) |
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246 with eq show "map_of (zip as xs) x = map_of (zip as ys) x" |
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247 by(auto dest: fun_cong[where x=x]) |
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248 qed |
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249 with dista show "xs = ys" by(simp add: map_of_zip_inject) |
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250 qed |
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251 hence "card (set ?xs) = length ?xs" by(simp only: distinct_card) |
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252 moreover have "length ?xs = length bs ^ length as" by(simp add: length_n_lists) |
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253 ultimately have "card (UNIV :: ('a \<Rightarrow> 'b) set) = card (UNIV :: 'b set) ^ card (UNIV :: 'a set)" |
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254 using cb ca by simp } |
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255 moreover { |
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256 assume cb: "card (UNIV :: 'b set) = Suc 0" |
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257 then obtain b where b: "UNIV = {b :: 'b}" by(auto simp add: card_Suc_eq) |
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258 have eq: "UNIV = {\<lambda>x :: 'a. b ::'b}" |
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259 proof(rule UNIV_eq_I) |
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260 fix x :: "'a \<Rightarrow> 'b" |
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261 { fix y |
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262 have "x y \<in> UNIV" .. |
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263 hence "x y = b" unfolding b by simp } |
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264 thus "x \<in> {\<lambda>x. b}" by(auto intro: ext) |
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265 qed |
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266 have "card (UNIV :: ('a \<Rightarrow> 'b) set) = Suc 0" unfolding eq by simp } |
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267 ultimately show "card_UNIV x = card (UNIV :: ('a \<Rightarrow> 'b) set)" |
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268 unfolding card_UNIV_fun_def card_UNIV Let_def |
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269 by(auto simp del: One_nat_def)(auto simp add: card_eq_0_iff dest: finite_fun_UNIVD2 finite_fun_UNIVD1) |
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270 qed |
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271 |
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272 end |
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273 |
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274 subsubsection {* @{typ "'a option"} *} |
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275 |
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276 instantiation option :: (card_UNIV) card_UNIV |
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277 begin |
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278 |
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279 definition card_UNIV_option_def: |
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280 "card_UNIV_class.card_UNIV = (\<lambda>a :: 'a option itself. let c = card_UNIV (TYPE('a)) |
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281 in if c \<noteq> 0 then Suc c else 0)" |
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282 |
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283 instance proof |
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284 fix x :: "'a option itself" |
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285 show "card_UNIV x = card (UNIV :: 'a option set)" |
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286 unfolding UNIV_option_conv |
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287 by(auto simp add: card_UNIV_option_def card_UNIV card_eq_0_iff Let_def intro: inj_Some dest: finite_imageD) |
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288 (subst card_insert_disjoint, auto simp add: card_eq_0_iff card_image inj_Some intro: finite_imageI card_ge_0_finite) |
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289 qed |
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290 |
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291 end |
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292 |
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293 end |