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1 (* Title: HOL/Finite_Set.thy |
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2 ID: $Id$ |
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3 Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel |
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4 License: GPL (GNU GENERAL PUBLIC LICENSE) |
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5 *) |
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6 |
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7 header {* Finite sets *} |
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8 |
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9 theory Finite_Set = Divides + Power + Inductive + SetInterval: |
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10 |
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11 subsection {* Collection of finite sets *} |
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12 |
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13 consts Finites :: "'a set set" |
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14 |
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15 inductive Finites |
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16 intros |
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17 emptyI [simp, intro!]: "{} : Finites" |
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18 insertI [simp, intro!]: "A : Finites ==> insert a A : Finites" |
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19 |
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20 syntax |
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21 finite :: "'a set => bool" |
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22 translations |
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23 "finite A" == "A : Finites" |
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24 |
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25 axclass finite \<subseteq> type |
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26 finite: "finite UNIV" |
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27 |
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28 |
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29 lemma finite_induct [case_names empty insert, induct set: Finites]: |
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30 "finite F ==> |
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31 P {} ==> (!!F x. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F" |
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32 -- {* Discharging @{text "x \<notin> F"} entails extra work. *} |
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33 proof - |
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34 assume "P {}" and insert: "!!F x. finite F ==> x \<notin> F ==> P F ==> P (insert x F)" |
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35 assume "finite F" |
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36 thus "P F" |
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37 proof induct |
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38 show "P {}" . |
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39 fix F x assume F: "finite F" and P: "P F" |
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40 show "P (insert x F)" |
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41 proof cases |
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42 assume "x \<in> F" |
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43 hence "insert x F = F" by (rule insert_absorb) |
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44 with P show ?thesis by (simp only:) |
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45 next |
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46 assume "x \<notin> F" |
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47 from F this P show ?thesis by (rule insert) |
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48 qed |
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49 qed |
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50 qed |
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51 |
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52 lemma finite_subset_induct [consumes 2, case_names empty insert]: |
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53 "finite F ==> F \<subseteq> A ==> |
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54 P {} ==> (!!F a. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==> |
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55 P F" |
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56 proof - |
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57 assume "P {}" and insert: "!!F a. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)" |
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58 assume "finite F" |
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59 thus "F \<subseteq> A ==> P F" |
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60 proof induct |
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61 show "P {}" . |
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62 fix F x assume "finite F" and "x \<notin> F" |
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63 and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A" |
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64 show "P (insert x F)" |
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65 proof (rule insert) |
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66 from i show "x \<in> A" by blast |
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67 from i have "F \<subseteq> A" by blast |
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68 with P show "P F" . |
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69 qed |
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70 qed |
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71 qed |
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72 |
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73 lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)" |
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74 -- {* The union of two finite sets is finite. *} |
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75 by (induct set: Finites) simp_all |
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76 |
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77 lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A" |
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78 -- {* Every subset of a finite set is finite. *} |
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79 proof - |
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80 assume "finite B" |
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81 thus "!!A. A \<subseteq> B ==> finite A" |
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82 proof induct |
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83 case empty |
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84 thus ?case by simp |
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85 next |
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86 case (insert F x A) |
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87 have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" . |
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88 show "finite A" |
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89 proof cases |
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90 assume x: "x \<in> A" |
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91 with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff) |
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92 with r have "finite (A - {x})" . |
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93 hence "finite (insert x (A - {x}))" .. |
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94 also have "insert x (A - {x}) = A" by (rule insert_Diff) |
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95 finally show ?thesis . |
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96 next |
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97 show "A \<subseteq> F ==> ?thesis" . |
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98 assume "x \<notin> A" |
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99 with A show "A \<subseteq> F" by (simp add: subset_insert_iff) |
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100 qed |
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101 qed |
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102 qed |
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103 |
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104 lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)" |
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105 by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI) |
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106 |
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107 lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)" |
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108 -- {* The converse obviously fails. *} |
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109 by (blast intro: finite_subset) |
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110 |
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111 lemma finite_insert [simp]: "finite (insert a A) = finite A" |
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112 apply (subst insert_is_Un) |
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113 apply (simp only: finite_Un) |
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114 apply blast |
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115 done |
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116 |
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117 lemma finite_imageI: "finite F ==> finite (h ` F)" |
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118 -- {* The image of a finite set is finite. *} |
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119 by (induct set: Finites) simp_all |
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120 |
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121 lemma finite_range_imageI: |
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122 "finite (range g) ==> finite (range (%x. f (g x)))" |
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123 apply (drule finite_imageI) |
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124 apply simp |
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125 done |
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126 |
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127 lemma finite_empty_induct: |
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128 "finite A ==> |
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129 P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}" |
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130 proof - |
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131 assume "finite A" |
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132 and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})" |
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133 have "P (A - A)" |
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134 proof - |
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135 fix c b :: "'a set" |
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136 presume c: "finite c" and b: "finite b" |
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137 and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})" |
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138 from c show "c \<subseteq> b ==> P (b - c)" |
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139 proof induct |
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140 case empty |
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141 from P1 show ?case by simp |
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142 next |
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143 case (insert F x) |
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144 have "P (b - F - {x})" |
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145 proof (rule P2) |
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146 from _ b show "finite (b - F)" by (rule finite_subset) blast |
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147 from insert show "x \<in> b - F" by simp |
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148 from insert show "P (b - F)" by simp |
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149 qed |
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150 also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric]) |
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151 finally show ?case . |
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152 qed |
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153 next |
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154 show "A \<subseteq> A" .. |
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155 qed |
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156 thus "P {}" by simp |
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157 qed |
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158 |
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159 lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)" |
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160 by (rule Diff_subset [THEN finite_subset]) |
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161 |
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162 lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)" |
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163 apply (subst Diff_insert) |
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164 apply (case_tac "a : A - B") |
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165 apply (rule finite_insert [symmetric, THEN trans]) |
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166 apply (subst insert_Diff) |
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167 apply simp_all |
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168 done |
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169 |
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170 |
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171 lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A" |
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172 proof - |
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173 have aux: "!!A. finite (A - {}) = finite A" by simp |
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174 fix B :: "'a set" |
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175 assume "finite B" |
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176 thus "!!A. f`A = B ==> inj_on f A ==> finite A" |
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177 apply induct |
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178 apply simp |
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179 apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})") |
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180 apply clarify |
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181 apply (simp (no_asm_use) add: inj_on_def) |
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182 apply (blast dest!: aux [THEN iffD1]) |
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183 apply atomize |
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184 apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl) |
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185 apply (frule subsetD [OF equalityD2 insertI1]) |
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186 apply clarify |
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187 apply (rule_tac x = xa in bexI) |
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188 apply (simp_all add: inj_on_image_set_diff) |
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189 done |
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190 qed (rule refl) |
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191 |
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192 |
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193 subsubsection {* The finite UNION of finite sets *} |
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194 |
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195 lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)" |
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196 by (induct set: Finites) simp_all |
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197 |
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198 text {* |
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199 Strengthen RHS to |
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200 @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x ~= {}})"}? |
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201 |
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202 We'd need to prove |
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203 @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x ~= {}}"} |
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204 by induction. *} |
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205 |
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206 lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))" |
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207 by (blast intro: finite_UN_I finite_subset) |
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208 |
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209 |
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210 subsubsection {* Sigma of finite sets *} |
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211 |
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212 lemma finite_SigmaI [simp]: |
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213 "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)" |
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214 by (unfold Sigma_def) (blast intro!: finite_UN_I) |
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215 |
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216 lemma finite_Prod_UNIV: |
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217 "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)" |
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218 apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)") |
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219 apply (erule ssubst) |
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220 apply (erule finite_SigmaI) |
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221 apply auto |
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222 done |
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223 |
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224 instance unit :: finite |
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225 proof |
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226 have "finite {()}" by simp |
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227 also have "{()} = UNIV" by auto |
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228 finally show "finite (UNIV :: unit set)" . |
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229 qed |
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230 |
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231 instance * :: (finite, finite) finite |
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232 proof |
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233 show "finite (UNIV :: ('a \<times> 'b) set)" |
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234 proof (rule finite_Prod_UNIV) |
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235 show "finite (UNIV :: 'a set)" by (rule finite) |
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236 show "finite (UNIV :: 'b set)" by (rule finite) |
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237 qed |
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238 qed |
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239 |
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240 |
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241 subsubsection {* The powerset of a finite set *} |
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242 |
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243 lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A" |
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244 proof |
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245 assume "finite (Pow A)" |
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246 with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast |
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247 thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp |
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248 next |
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249 assume "finite A" |
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250 thus "finite (Pow A)" |
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251 by induct (simp_all add: finite_UnI finite_imageI Pow_insert) |
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252 qed |
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253 |
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254 lemma finite_converse [iff]: "finite (r^-1) = finite r" |
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255 apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r") |
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256 apply simp |
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257 apply (rule iffI) |
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258 apply (erule finite_imageD [unfolded inj_on_def]) |
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259 apply (simp split add: split_split) |
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260 apply (erule finite_imageI) |
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261 apply (simp add: converse_def image_def) |
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262 apply auto |
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263 apply (rule bexI) |
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264 prefer 2 apply assumption |
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265 apply simp |
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266 done |
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267 |
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268 lemma finite_lessThan [iff]: (fixes k :: nat) "finite {..k(}" |
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269 by (induct k) (simp_all add: lessThan_Suc) |
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270 |
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271 lemma finite_atMost [iff]: (fixes k :: nat) "finite {..k}" |
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272 by (induct k) (simp_all add: atMost_Suc) |
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273 |
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274 lemma bounded_nat_set_is_finite: |
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275 "(ALL i:N. i < (n::nat)) ==> finite N" |
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276 -- {* A bounded set of natural numbers is finite. *} |
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277 apply (rule finite_subset) |
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278 apply (rule_tac [2] finite_lessThan) |
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279 apply auto |
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280 done |
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281 |
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282 |
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283 subsubsection {* Finiteness of transitive closure *} |
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284 |
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285 text {* (Thanks to Sidi Ehmety) *} |
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286 |
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287 lemma finite_Field: "finite r ==> finite (Field r)" |
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288 -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *} |
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289 apply (induct set: Finites) |
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290 apply (auto simp add: Field_def Domain_insert Range_insert) |
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291 done |
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292 |
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293 lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r" |
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294 apply clarify |
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295 apply (erule trancl_induct) |
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296 apply (auto simp add: Field_def) |
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297 done |
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298 |
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299 lemma finite_trancl: "finite (r^+) = finite r" |
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300 apply auto |
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301 prefer 2 |
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302 apply (rule trancl_subset_Field2 [THEN finite_subset]) |
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303 apply (rule finite_SigmaI) |
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304 prefer 3 |
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305 apply (blast intro: r_into_trancl finite_subset) |
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306 apply (auto simp add: finite_Field) |
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307 done |
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308 |
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309 |
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310 subsection {* Finite cardinality *} |
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311 |
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312 text {* |
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313 This definition, although traditional, is ugly to work with: @{text |
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314 "card A == LEAST n. EX f. A = {f i | i. i < n}"}. Therefore we have |
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315 switched to an inductive one: |
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316 *} |
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317 |
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318 consts cardR :: "('a set \<times> nat) set" |
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319 |
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320 inductive cardR |
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321 intros |
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322 EmptyI: "({}, 0) : cardR" |
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323 InsertI: "(A, n) : cardR ==> a \<notin> A ==> (insert a A, Suc n) : cardR" |
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324 |
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325 constdefs |
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326 card :: "'a set => nat" |
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327 "card A == THE n. (A, n) : cardR" |
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328 |
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329 inductive_cases cardR_emptyE: "({}, n) : cardR" |
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330 inductive_cases cardR_insertE: "(insert a A,n) : cardR" |
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331 |
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332 lemma cardR_SucD: "(A, n) : cardR ==> a : A --> (EX m. n = Suc m)" |
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333 by (induct set: cardR) simp_all |
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334 |
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335 lemma cardR_determ_aux1: |
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336 "(A, m): cardR ==> (!!n a. m = Suc n ==> a:A ==> (A - {a}, n) : cardR)" |
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337 apply (induct set: cardR) |
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338 apply auto |
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339 apply (simp add: insert_Diff_if) |
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340 apply auto |
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341 apply (drule cardR_SucD) |
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342 apply (blast intro!: cardR.intros) |
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343 done |
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344 |
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345 lemma cardR_determ_aux2: "(insert a A, Suc m) : cardR ==> a \<notin> A ==> (A, m) : cardR" |
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346 by (drule cardR_determ_aux1) auto |
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347 |
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348 lemma cardR_determ: "(A, m): cardR ==> (!!n. (A, n) : cardR ==> n = m)" |
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349 apply (induct set: cardR) |
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350 apply (safe elim!: cardR_emptyE cardR_insertE) |
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351 apply (rename_tac B b m) |
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352 apply (case_tac "a = b") |
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353 apply (subgoal_tac "A = B") |
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354 prefer 2 apply (blast elim: equalityE) |
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355 apply blast |
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356 apply (subgoal_tac "EX C. A = insert b C & B = insert a C") |
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357 prefer 2 |
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358 apply (rule_tac x = "A Int B" in exI) |
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359 apply (blast elim: equalityE) |
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360 apply (frule_tac A = B in cardR_SucD) |
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361 apply (blast intro!: cardR.intros dest!: cardR_determ_aux2) |
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362 done |
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363 |
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364 lemma cardR_imp_finite: "(A, n) : cardR ==> finite A" |
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365 by (induct set: cardR) simp_all |
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366 |
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367 lemma finite_imp_cardR: "finite A ==> EX n. (A, n) : cardR" |
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368 by (induct set: Finites) (auto intro!: cardR.intros) |
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369 |
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370 lemma card_equality: "(A,n) : cardR ==> card A = n" |
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371 by (unfold card_def) (blast intro: cardR_determ) |
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372 |
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373 lemma card_empty [simp]: "card {} = 0" |
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374 by (unfold card_def) (blast intro!: cardR.intros elim!: cardR_emptyE) |
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375 |
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376 lemma card_insert_disjoint [simp]: |
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377 "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)" |
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378 proof - |
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379 assume x: "x \<notin> A" |
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380 hence aux: "!!n. ((insert x A, n) : cardR) = (EX m. (A, m) : cardR & n = Suc m)" |
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381 apply (auto intro!: cardR.intros) |
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382 apply (rule_tac A1 = A in finite_imp_cardR [THEN exE]) |
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383 apply (force dest: cardR_imp_finite) |
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384 apply (blast intro!: cardR.intros intro: cardR_determ) |
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385 done |
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386 assume "finite A" |
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387 thus ?thesis |
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388 apply (simp add: card_def aux) |
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389 apply (rule the_equality) |
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390 apply (auto intro: finite_imp_cardR |
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391 cong: conj_cong simp: card_def [symmetric] card_equality) |
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392 done |
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393 qed |
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394 |
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395 lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})" |
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396 apply auto |
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397 apply (drule_tac a = x in mk_disjoint_insert) |
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398 apply clarify |
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399 apply (rotate_tac -1) |
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400 apply auto |
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401 done |
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402 |
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403 lemma card_insert_if: |
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404 "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))" |
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405 by (simp add: insert_absorb) |
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406 |
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407 lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A" |
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408 apply (rule_tac t = A in insert_Diff [THEN subst]) |
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409 apply assumption |
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410 apply simp |
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411 done |
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412 |
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413 lemma card_Diff_singleton: |
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414 "finite A ==> x: A ==> card (A - {x}) = card A - 1" |
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415 by (simp add: card_Suc_Diff1 [symmetric]) |
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416 |
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417 lemma card_Diff_singleton_if: |
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418 "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)" |
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419 by (simp add: card_Diff_singleton) |
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420 |
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421 lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))" |
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422 by (simp add: card_insert_if card_Suc_Diff1) |
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423 |
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424 lemma card_insert_le: "finite A ==> card A <= card (insert x A)" |
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425 by (simp add: card_insert_if) |
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426 |
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427 lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)" |
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428 apply (induct set: Finites) |
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429 apply simp |
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430 apply clarify |
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431 apply (subgoal_tac "finite A & A - {x} <= F") |
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432 prefer 2 apply (blast intro: finite_subset) |
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433 apply atomize |
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434 apply (drule_tac x = "A - {x}" in spec) |
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435 apply (simp add: card_Diff_singleton_if split add: split_if_asm) |
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436 apply (case_tac "card A") |
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437 apply auto |
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438 done |
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439 |
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440 lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B" |
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441 apply (simp add: psubset_def linorder_not_le [symmetric]) |
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442 apply (blast dest: card_seteq) |
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443 done |
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444 |
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445 lemma card_mono: "finite B ==> A <= B ==> card A <= card B" |
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446 apply (case_tac "A = B") |
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447 apply simp |
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448 apply (simp add: linorder_not_less [symmetric]) |
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449 apply (blast dest: card_seteq intro: order_less_imp_le) |
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450 done |
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451 |
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452 lemma card_Un_Int: "finite A ==> finite B |
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453 ==> card A + card B = card (A Un B) + card (A Int B)" |
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454 apply (induct set: Finites) |
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455 apply simp |
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456 apply (simp add: insert_absorb Int_insert_left) |
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457 done |
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458 |
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459 lemma card_Un_disjoint: "finite A ==> finite B |
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460 ==> A Int B = {} ==> card (A Un B) = card A + card B" |
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461 by (simp add: card_Un_Int) |
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462 |
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463 lemma card_Diff_subset: |
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464 "finite A ==> B <= A ==> card A - card B = card (A - B)" |
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465 apply (subgoal_tac "(A - B) Un B = A") |
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466 prefer 2 apply blast |
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467 apply (rule add_right_cancel [THEN iffD1]) |
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468 apply (rule card_Un_disjoint [THEN subst]) |
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469 apply (erule_tac [4] ssubst) |
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470 prefer 3 apply blast |
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471 apply (simp_all add: add_commute not_less_iff_le |
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472 add_diff_inverse card_mono finite_subset) |
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473 done |
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474 |
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475 lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A" |
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476 apply (rule Suc_less_SucD) |
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477 apply (simp add: card_Suc_Diff1) |
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478 done |
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479 |
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480 lemma card_Diff2_less: |
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481 "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A" |
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482 apply (case_tac "x = y") |
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483 apply (simp add: card_Diff1_less) |
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484 apply (rule less_trans) |
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485 prefer 2 apply (auto intro!: card_Diff1_less) |
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486 done |
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487 |
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488 lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A" |
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489 apply (case_tac "x : A") |
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490 apply (simp_all add: card_Diff1_less less_imp_le) |
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491 done |
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492 |
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493 lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B" |
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494 apply (erule psubsetI) |
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495 apply blast |
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496 done |
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497 |
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498 |
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499 subsubsection {* Cardinality of image *} |
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500 |
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501 lemma card_image_le: "finite A ==> card (f ` A) <= card A" |
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502 apply (induct set: Finites) |
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503 apply simp |
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504 apply (simp add: le_SucI finite_imageI card_insert_if) |
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505 done |
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506 |
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507 lemma card_image: "finite A ==> inj_on f A ==> card (f ` A) = card A" |
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508 apply (induct set: Finites) |
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509 apply simp_all |
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510 apply atomize |
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511 apply safe |
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512 apply (unfold inj_on_def) |
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513 apply blast |
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514 apply (subst card_insert_disjoint) |
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515 apply (erule finite_imageI) |
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516 apply blast |
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517 apply blast |
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518 done |
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519 |
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520 lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A" |
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521 by (simp add: card_seteq card_image) |
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522 |
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523 |
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524 subsubsection {* Cardinality of the Powerset *} |
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525 |
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526 lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *) |
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527 apply (induct set: Finites) |
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528 apply (simp_all add: Pow_insert) |
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529 apply (subst card_Un_disjoint) |
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530 apply blast |
|
531 apply (blast intro: finite_imageI) |
|
532 apply blast |
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533 apply (subgoal_tac "inj_on (insert x) (Pow F)") |
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534 apply (simp add: card_image Pow_insert) |
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535 apply (unfold inj_on_def) |
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536 apply (blast elim!: equalityE) |
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537 done |
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538 |
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539 text {* |
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540 \medskip Relates to equivalence classes. Based on a theorem of |
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541 F. Kammüller's. The @{prop "finite C"} premise is redundant. |
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542 *} |
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543 |
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544 lemma dvd_partition: |
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545 "finite C ==> finite (Union C) ==> |
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546 ALL c : C. k dvd card c ==> |
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547 (ALL c1: C. ALL c2: C. c1 ~= c2 --> c1 Int c2 = {}) ==> |
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548 k dvd card (Union C)" |
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549 apply (induct set: Finites) |
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550 apply simp_all |
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551 apply clarify |
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552 apply (subst card_Un_disjoint) |
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553 apply (auto simp add: dvd_add disjoint_eq_subset_Compl) |
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554 done |
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555 |
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556 |
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557 subsection {* A fold functional for finite sets *} |
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558 |
|
559 text {* |
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560 For @{text n} non-negative we have @{text "fold f e {x1, ..., xn} = |
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561 f x1 (... (f xn e))"} where @{text f} is at least left-commutative. |
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562 *} |
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563 |
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564 consts |
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565 foldSet :: "('b => 'a => 'a) => 'a => ('b set \<times> 'a) set" |
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566 |
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567 inductive "foldSet f e" |
|
568 intros |
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569 emptyI [intro]: "({}, e) : foldSet f e" |
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570 insertI [intro]: "x \<notin> A ==> (A, y) : foldSet f e ==> (insert x A, f x y) : foldSet f e" |
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571 |
|
572 inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f e" |
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573 |
|
574 constdefs |
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575 fold :: "('b => 'a => 'a) => 'a => 'b set => 'a" |
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576 "fold f e A == THE x. (A, x) : foldSet f e" |
|
577 |
|
578 lemma Diff1_foldSet: "(A - {x}, y) : foldSet f e ==> x: A ==> (A, f x y) : foldSet f e" |
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579 apply (erule insert_Diff [THEN subst], rule foldSet.intros) |
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580 apply auto |
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581 done |
|
582 |
|
583 lemma foldSet_imp_finite [simp]: "(A, x) : foldSet f e ==> finite A" |
|
584 by (induct set: foldSet) auto |
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585 |
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586 lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f e" |
|
587 by (induct set: Finites) auto |
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588 |
|
589 |
|
590 subsubsection {* Left-commutative operations *} |
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591 |
|
592 locale LC = |
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593 fixes f :: "'b => 'a => 'a" (infixl "\<cdot>" 70) |
|
594 assumes left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)" |
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595 |
|
596 lemma (in LC) foldSet_determ_aux: |
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597 "ALL A x. card A < n --> (A, x) : foldSet f e --> |
|
598 (ALL y. (A, y) : foldSet f e --> y = x)" |
|
599 apply (induct n) |
|
600 apply (auto simp add: less_Suc_eq) |
|
601 apply (erule foldSet.cases) |
|
602 apply blast |
|
603 apply (erule foldSet.cases) |
|
604 apply blast |
|
605 apply clarify |
|
606 txt {* force simplification of @{text "card A < card (insert ...)"}. *} |
|
607 apply (erule rev_mp) |
|
608 apply (simp add: less_Suc_eq_le) |
|
609 apply (rule impI) |
|
610 apply (rename_tac Aa xa ya Ab xb yb, case_tac "xa = xb") |
|
611 apply (subgoal_tac "Aa = Ab") |
|
612 prefer 2 apply (blast elim!: equalityE) |
|
613 apply blast |
|
614 txt {* case @{prop "xa \<notin> xb"}. *} |
|
615 apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb : Aa & xa : Ab") |
|
616 prefer 2 apply (blast elim!: equalityE) |
|
617 apply clarify |
|
618 apply (subgoal_tac "Aa = insert xb Ab - {xa}") |
|
619 prefer 2 apply blast |
|
620 apply (subgoal_tac "card Aa <= card Ab") |
|
621 prefer 2 |
|
622 apply (rule Suc_le_mono [THEN subst]) |
|
623 apply (simp add: card_Suc_Diff1) |
|
624 apply (rule_tac A1 = "Aa - {xb}" and f1 = f in finite_imp_foldSet [THEN exE]) |
|
625 apply (blast intro: foldSet_imp_finite finite_Diff) |
|
626 apply (frule (1) Diff1_foldSet) |
|
627 apply (subgoal_tac "ya = f xb x") |
|
628 prefer 2 apply (blast del: equalityCE) |
|
629 apply (subgoal_tac "(Ab - {xa}, x) : foldSet f e") |
|
630 prefer 2 apply simp |
|
631 apply (subgoal_tac "yb = f xa x") |
|
632 prefer 2 apply (blast del: equalityCE dest: Diff1_foldSet) |
|
633 apply (simp (no_asm_simp) add: left_commute) |
|
634 done |
|
635 |
|
636 lemma (in LC) foldSet_determ: "(A, x) : foldSet f e ==> (A, y) : foldSet f e ==> y = x" |
|
637 by (blast intro: foldSet_determ_aux [rule_format]) |
|
638 |
|
639 lemma (in LC) fold_equality: "(A, y) : foldSet f e ==> fold f e A = y" |
|
640 by (unfold fold_def) (blast intro: foldSet_determ) |
|
641 |
|
642 lemma fold_empty [simp]: "fold f e {} = e" |
|
643 by (unfold fold_def) blast |
|
644 |
|
645 lemma (in LC) fold_insert_aux: "x \<notin> A ==> |
|
646 ((insert x A, v) : foldSet f e) = |
|
647 (EX y. (A, y) : foldSet f e & v = f x y)" |
|
648 apply auto |
|
649 apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE]) |
|
650 apply (fastsimp dest: foldSet_imp_finite) |
|
651 apply (blast intro: foldSet_determ) |
|
652 done |
|
653 |
|
654 lemma (in LC) fold_insert: |
|
655 "finite A ==> x \<notin> A ==> fold f e (insert x A) = f x (fold f e A)" |
|
656 apply (unfold fold_def) |
|
657 apply (simp add: fold_insert_aux) |
|
658 apply (rule the_equality) |
|
659 apply (auto intro: finite_imp_foldSet |
|
660 cong add: conj_cong simp add: fold_def [symmetric] fold_equality) |
|
661 done |
|
662 |
|
663 lemma (in LC) fold_commute: "finite A ==> (!!e. f x (fold f e A) = fold f (f x e) A)" |
|
664 apply (induct set: Finites) |
|
665 apply simp |
|
666 apply (simp add: left_commute fold_insert) |
|
667 done |
|
668 |
|
669 lemma (in LC) fold_nest_Un_Int: |
|
670 "finite A ==> finite B |
|
671 ==> fold f (fold f e B) A = fold f (fold f e (A Int B)) (A Un B)" |
|
672 apply (induct set: Finites) |
|
673 apply simp |
|
674 apply (simp add: fold_insert fold_commute Int_insert_left insert_absorb) |
|
675 done |
|
676 |
|
677 lemma (in LC) fold_nest_Un_disjoint: |
|
678 "finite A ==> finite B ==> A Int B = {} |
|
679 ==> fold f e (A Un B) = fold f (fold f e B) A" |
|
680 by (simp add: fold_nest_Un_Int) |
|
681 |
|
682 declare foldSet_imp_finite [simp del] |
|
683 empty_foldSetE [rule del] foldSet.intros [rule del] |
|
684 -- {* Delete rules to do with @{text foldSet} relation. *} |
|
685 |
|
686 |
|
687 |
|
688 subsubsection {* Commutative monoids *} |
|
689 |
|
690 text {* |
|
691 We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"} |
|
692 instead of @{text "'b => 'a => 'a"}. |
|
693 *} |
|
694 |
|
695 locale ACe = |
|
696 fixes f :: "'a => 'a => 'a" (infixl "\<cdot>" 70) |
|
697 and e :: 'a |
|
698 assumes ident [simp]: "x \<cdot> e = x" |
|
699 and commute: "x \<cdot> y = y \<cdot> x" |
|
700 and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" |
|
701 |
|
702 lemma (in ACe) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)" |
|
703 proof - |
|
704 have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute) |
|
705 also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc) |
|
706 also have "z \<cdot> x = x \<cdot> z" by (simp only: commute) |
|
707 finally show ?thesis . |
|
708 qed |
|
709 |
|
710 lemma (in ACe) |
|
711 AC: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" "x \<cdot> y = y \<cdot> x" "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)" |
|
712 by (rule assoc, rule commute, rule left_commute) (* FIXME localize "lemmas" (!??) *) |
|
713 |
|
714 lemma (in ACe [simp]) left_ident: "e \<cdot> x = x" |
|
715 proof - |
|
716 have "x \<cdot> e = x" by (rule ident) |
|
717 thus ?thesis by (subst commute) |
|
718 qed |
|
719 |
|
720 lemma (in ACe) fold_Un_Int: |
|
721 "finite A ==> finite B ==> |
|
722 fold f e A \<cdot> fold f e B = fold f e (A Un B) \<cdot> fold f e (A Int B)" |
|
723 apply (induct set: Finites) |
|
724 apply simp |
|
725 apply (simp add: AC fold_insert insert_absorb Int_insert_left) |
|
726 done |
|
727 |
|
728 lemma (in ACe) fold_Un_disjoint: |
|
729 "finite A ==> finite B ==> A Int B = {} ==> |
|
730 fold f e (A Un B) = fold f e A \<cdot> fold f e B" |
|
731 by (simp add: fold_Un_Int) |
|
732 |
|
733 lemma (in ACe) fold_Un_disjoint2: |
|
734 "finite A ==> finite B ==> A Int B = {} ==> |
|
735 fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B" |
|
736 proof - |
|
737 assume b: "finite B" |
|
738 assume "finite A" |
|
739 thus "A Int B = {} ==> |
|
740 fold (f o g) e (A Un B) = fold (f o g) e A \<cdot> fold (f o g) e B" |
|
741 proof induct |
|
742 case empty |
|
743 thus ?case by simp |
|
744 next |
|
745 case (insert F x) |
|
746 have "fold (f \<circ> g) e (insert x F \<union> B) = fold (f \<circ> g) e (insert x (F \<union> B))" |
|
747 by simp |
|
748 also have "... = (f \<circ> g) x (fold (f \<circ> g) e (F \<union> B))" |
|
749 by (rule fold_insert) (insert b insert, auto simp add: left_commute) (* FIXME import of fold_insert (!?) *) |
|
750 also from insert have "fold (f \<circ> g) e (F \<union> B) = |
|
751 fold (f \<circ> g) e F \<cdot> fold (f \<circ> g) e B" by blast |
|
752 also have "(f \<circ> g) x ... = (f \<circ> g) x (fold (f \<circ> g) e F) \<cdot> fold (f \<circ> g) e B" |
|
753 by (simp add: AC) |
|
754 also have "(f \<circ> g) x (fold (f \<circ> g) e F) = fold (f \<circ> g) e (insert x F)" |
|
755 by (rule fold_insert [symmetric]) (insert b insert, auto simp add: left_commute) |
|
756 finally show ?case . |
|
757 qed |
|
758 qed |
|
759 |
|
760 |
|
761 subsection {* Generalized summation over a set *} |
|
762 |
|
763 constdefs |
|
764 setsum :: "('a => 'b) => 'a set => 'b::plus_ac0" |
|
765 "setsum f A == if finite A then fold (op + o f) 0 A else 0" |
|
766 |
|
767 syntax |
|
768 "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0" ("\<Sum>_:_. _" [0, 51, 10] 10) |
|
769 syntax (xsymbols) |
|
770 "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0" ("\<Sum>_\<in>_. _" [0, 51, 10] 10) |
|
771 translations |
|
772 "\<Sum>i:A. b" == "setsum (%i. b) A" -- {* Beware of argument permutation! *} |
|
773 |
|
774 |
|
775 lemma setsum_empty [simp]: "setsum f {} = 0" |
|
776 by (simp add: setsum_def) |
|
777 |
|
778 lemma setsum_insert [simp]: |
|
779 "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F" |
|
780 by (simp add: setsum_def fold_insert plus_ac0_left_commute) |
|
781 |
|
782 lemma setsum_0: "setsum (\<lambda>i. 0) A = 0" |
|
783 apply (case_tac "finite A") |
|
784 prefer 2 apply (simp add: setsum_def) |
|
785 apply (erule finite_induct) |
|
786 apply auto |
|
787 done |
|
788 |
|
789 lemma setsum_eq_0_iff [simp]: |
|
790 "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))" |
|
791 by (induct set: Finites) auto |
|
792 |
|
793 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a" |
|
794 apply (case_tac "finite A") |
|
795 prefer 2 apply (simp add: setsum_def) |
|
796 apply (erule rev_mp) |
|
797 apply (erule finite_induct) |
|
798 apply auto |
|
799 done |
|
800 |
|
801 lemma card_eq_setsum: "finite A ==> card A = setsum (\<lambda>x. 1) A" |
|
802 -- {* Could allow many @{text "card"} proofs to be simplified. *} |
|
803 by (induct set: Finites) auto |
|
804 |
|
805 lemma setsum_Un_Int: "finite A ==> finite B |
|
806 ==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B" |
|
807 -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *} |
|
808 apply (induct set: Finites) |
|
809 apply simp |
|
810 apply (simp add: plus_ac0 Int_insert_left insert_absorb) |
|
811 done |
|
812 |
|
813 lemma setsum_Un_disjoint: "finite A ==> finite B |
|
814 ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B" |
|
815 apply (subst setsum_Un_Int [symmetric]) |
|
816 apply auto |
|
817 done |
|
818 |
|
819 lemma setsum_UN_disjoint: (fixes f :: "'a => 'b::plus_ac0") |
|
820 "finite I ==> (ALL i:I. finite (A i)) ==> |
|
821 (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==> |
|
822 setsum f (UNION I A) = setsum (\<lambda>i. setsum f (A i)) I" |
|
823 apply (induct set: Finites) |
|
824 apply simp |
|
825 apply atomize |
|
826 apply (subgoal_tac "ALL i:F. x \<noteq> i") |
|
827 prefer 2 apply blast |
|
828 apply (subgoal_tac "A x Int UNION F A = {}") |
|
829 prefer 2 apply blast |
|
830 apply (simp add: setsum_Un_disjoint) |
|
831 done |
|
832 |
|
833 lemma setsum_addf: "setsum (\<lambda>x. f x + g x) A = (setsum f A + setsum g A)" |
|
834 apply (case_tac "finite A") |
|
835 prefer 2 apply (simp add: setsum_def) |
|
836 apply (erule finite_induct) |
|
837 apply auto |
|
838 apply (simp add: plus_ac0) |
|
839 done |
|
840 |
|
841 lemma setsum_Un: "finite A ==> finite B ==> |
|
842 (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)" |
|
843 -- {* For the natural numbers, we have subtraction. *} |
|
844 apply (subst setsum_Un_Int [symmetric]) |
|
845 apply auto |
|
846 done |
|
847 |
|
848 lemma setsum_diff1: "(setsum f (A - {a}) :: nat) = |
|
849 (if a:A then setsum f A - f a else setsum f A)" |
|
850 apply (case_tac "finite A") |
|
851 prefer 2 apply (simp add: setsum_def) |
|
852 apply (erule finite_induct) |
|
853 apply (auto simp add: insert_Diff_if) |
|
854 apply (drule_tac a = a in mk_disjoint_insert) |
|
855 apply auto |
|
856 done |
|
857 |
|
858 lemma setsum_cong: |
|
859 "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B" |
|
860 apply (case_tac "finite B") |
|
861 prefer 2 apply (simp add: setsum_def) |
|
862 apply simp |
|
863 apply (subgoal_tac "ALL C. C <= B --> (ALL x:C. f x = g x) --> setsum f C = setsum g C") |
|
864 apply simp |
|
865 apply (erule finite_induct) |
|
866 apply simp |
|
867 apply (simp add: subset_insert_iff) |
|
868 apply clarify |
|
869 apply (subgoal_tac "finite C") |
|
870 prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl]) |
|
871 apply (subgoal_tac "C = insert x (C - {x})") |
|
872 prefer 2 apply blast |
|
873 apply (erule ssubst) |
|
874 apply (drule spec) |
|
875 apply (erule (1) notE impE) |
|
876 apply (simp add: Ball_def) |
|
877 done |
|
878 |
|
879 |
|
880 text {* |
|
881 \medskip Basic theorem about @{text "choose"}. By Florian |
|
882 Kammüller, tidied by LCP. |
|
883 *} |
|
884 |
|
885 lemma card_s_0_eq_empty: |
|
886 "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1" |
|
887 apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq]) |
|
888 apply (simp cong add: rev_conj_cong) |
|
889 done |
|
890 |
|
891 lemma choose_deconstruct: "finite M ==> x \<notin> M |
|
892 ==> {s. s <= insert x M & card(s) = Suc k} |
|
893 = {s. s <= M & card(s) = Suc k} Un |
|
894 {s. EX t. t <= M & card(t) = k & s = insert x t}" |
|
895 apply safe |
|
896 apply (auto intro: finite_subset [THEN card_insert_disjoint]) |
|
897 apply (drule_tac x = "xa - {x}" in spec) |
|
898 apply (subgoal_tac "x ~: xa") |
|
899 apply auto |
|
900 apply (erule rev_mp, subst card_Diff_singleton) |
|
901 apply (auto intro: finite_subset) |
|
902 done |
|
903 |
|
904 lemma card_inj_on_le: |
|
905 "finite A ==> finite B ==> f ` A \<subseteq> B ==> inj_on f A ==> card A <= card B" |
|
906 by (auto intro: card_mono simp add: card_image [symmetric]) |
|
907 |
|
908 lemma card_bij_eq: "finite A ==> finite B ==> |
|
909 f ` A \<subseteq> B ==> inj_on f A ==> g ` B \<subseteq> A ==> inj_on g B ==> card A = card B" |
|
910 by (auto intro: le_anti_sym card_inj_on_le) |
|
911 |
|
912 lemma constr_bij: "finite A ==> x \<notin> A ==> |
|
913 card {B. EX C. C <= A & card(C) = k & B = insert x C} = |
|
914 card {B. B <= A & card(B) = k}" |
|
915 apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq) |
|
916 apply (rule_tac B = "Pow (insert x A) " in finite_subset) |
|
917 apply (rule_tac [3] B = "Pow (A) " in finite_subset) |
|
918 apply fast+ |
|
919 txt {* arity *} |
|
920 apply (auto elim!: equalityE simp add: inj_on_def) |
|
921 apply (subst Diff_insert0) |
|
922 apply auto |
|
923 done |
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924 |
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925 text {* |
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926 Main theorem: combinatorial statement about number of subsets of a set. |
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927 *} |
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928 |
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929 lemma n_sub_lemma: |
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930 "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)" |
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931 apply (induct k) |
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932 apply (simp add: card_s_0_eq_empty) |
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933 apply atomize |
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934 apply (rotate_tac -1, erule finite_induct) |
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935 apply (simp_all (no_asm_simp) cong add: conj_cong add: card_s_0_eq_empty choose_deconstruct) |
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936 apply (subst card_Un_disjoint) |
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937 prefer 4 apply (force simp add: constr_bij) |
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938 prefer 3 apply force |
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939 prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2] |
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940 finite_subset [of _ "Pow (insert x F)", standard]) |
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941 apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset]) |
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942 done |
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943 |
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944 theorem n_subsets: "finite A ==> card {B. B <= A & card(B) = k} = (card A choose k)" |
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945 by (simp add: n_sub_lemma) |
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946 |
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947 end |