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1 (* Title: HOL/Hilbert_Choice_lemmas |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson |
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4 Copyright 2001 University of Cambridge |
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5 |
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6 Lemmas for Hilbert's epsilon-operator and the Axiom of Choice |
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7 *) |
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8 |
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9 |
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10 (* ML bindings *) |
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11 val someI = thm "someI"; |
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12 |
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13 section "SOME: Hilbert's Epsilon-operator"; |
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14 |
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15 (*Easier to apply than someI if witness ?a comes from an EX-formula*) |
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16 Goal "EX x. P x ==> P (SOME x. P x)"; |
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17 by (etac exE 1); |
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18 by (etac someI 1); |
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19 qed "someI_ex"; |
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20 |
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21 (*Easier to apply than someI: conclusion has only one occurrence of P*) |
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22 val prems = Goal "[| P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"; |
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23 by (resolve_tac prems 1); |
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24 by (rtac someI 1); |
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25 by (resolve_tac prems 1) ; |
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26 qed "someI2"; |
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27 |
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28 (*Easier to apply than someI2 if witness ?a comes from an EX-formula*) |
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29 val [major,minor] = Goal "[| EX a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"; |
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30 by (rtac (major RS exE) 1); |
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31 by (etac someI2 1 THEN etac minor 1); |
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32 qed "someI2_ex"; |
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33 |
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34 val prems = Goal "[| P a; !!x. P x ==> x=a |] ==> (SOME x. P x) = a"; |
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35 by (rtac someI2 1); |
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36 by (REPEAT (ares_tac prems 1)) ; |
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37 qed "some_equality"; |
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38 AddIs [some_equality]; |
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39 |
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40 Goal "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"; |
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41 by (rtac some_equality 1); |
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42 by (atac 1); |
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43 by (etac ex1E 1); |
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44 by (etac all_dupE 1); |
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45 by (dtac mp 1); |
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46 by (atac 1); |
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47 by (etac ssubst 1); |
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48 by (etac allE 1); |
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49 by (etac mp 1); |
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50 by (atac 1); |
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51 qed "some1_equality"; |
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52 |
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53 Goal "P (SOME x. P x) = (EX x. P x)"; |
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54 by (rtac iffI 1); |
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55 by (etac exI 1); |
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56 by (etac exE 1); |
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57 by (etac someI 1); |
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58 qed "some_eq_ex"; |
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59 |
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60 Goal "(SOME y. y=x) = x"; |
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61 by (rtac some_equality 1); |
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62 by (rtac refl 1); |
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63 by (atac 1); |
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64 qed "some_eq_trivial"; |
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65 |
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66 Goal "(SOME y. x=y) = x"; |
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67 by (rtac some_equality 1); |
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68 by (rtac refl 1); |
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69 by (etac sym 1); |
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70 qed "some_sym_eq_trivial"; |
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71 |
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72 |
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73 AddXEs [someI_ex]; |
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74 AddIs [some_equality]; |
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75 |
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76 Addsimps [some_eq_trivial, some_sym_eq_trivial]; |
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77 |
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78 |
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79 (** "Axiom" of Choice, proved using the description operator **) |
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80 |
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81 (*"choice" is now proved in Tools/meson.ML*) |
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82 |
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83 Goal "ALL x:S. EX y. Q x y ==> EX f. ALL x:S. Q x (f x)"; |
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84 by (fast_tac (claset() addEs [someI]) 1); |
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85 qed "bchoice"; |
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86 |
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87 |
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88 (**** Function Inverse ****) |
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89 |
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90 val inv_def = thm "inv_def"; |
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91 val Inv_def = thm "Inv_def"; |
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92 |
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93 |
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94 Goal "inv id = id"; |
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95 by (simp_tac (simpset() addsimps [inv_def,id_def]) 1); |
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96 qed "inv_id"; |
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97 Addsimps [inv_id]; |
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98 |
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99 (*A one-to-one function has an inverse.*) |
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100 Goalw [inv_def] "inj(f) ==> inv f (f x) = x"; |
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101 by (asm_simp_tac (simpset() addsimps [inj_eq]) 1); |
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102 qed "inv_f_f"; |
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103 Addsimps [inv_f_f]; |
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104 |
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105 Goal "[| inj(f); f x = y |] ==> inv f y = x"; |
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106 by (etac subst 1); |
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107 by (etac inv_f_f 1); |
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108 qed "inv_f_eq"; |
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109 |
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110 Goal "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"; |
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111 by (blast_tac (claset() addIs [ext, inv_f_eq]) 1); |
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112 qed "inj_imp_inv_eq"; |
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113 |
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114 (* Useful??? *) |
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115 val [oneone,minor] = Goal |
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116 "[| inj(f); !!y. y: range(f) ==> P(inv f y) |] ==> P(x)"; |
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117 by (res_inst_tac [("t", "x")] (oneone RS (inv_f_f RS subst)) 1); |
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118 by (rtac (rangeI RS minor) 1); |
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119 qed "inj_transfer"; |
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120 |
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121 Goal "(inj f) = (inv f o f = id)"; |
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122 by (asm_simp_tac (simpset() addsimps [o_def, expand_fun_eq]) 1); |
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123 by (blast_tac (claset() addIs [inj_inverseI, inv_f_f]) 1); |
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124 qed "inj_iff"; |
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125 |
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126 Goal "inj f ==> surj (inv f)"; |
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127 by (blast_tac (claset() addIs [surjI, inv_f_f]) 1); |
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128 qed "inj_imp_surj_inv"; |
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129 |
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130 Goalw [inv_def] "y : range(f) ==> f(inv f y) = y"; |
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131 by (fast_tac (claset() addIs [someI]) 1); |
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132 qed "f_inv_f"; |
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133 |
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134 Goal "surj f ==> f(inv f y) = y"; |
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135 by (asm_simp_tac (simpset() addsimps [f_inv_f, surj_range]) 1); |
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136 qed "surj_f_inv_f"; |
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137 |
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138 Goal "[| inv f x = inv f y; x: range(f); y: range(f) |] ==> x=y"; |
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139 by (rtac (arg_cong RS box_equals) 1); |
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140 by (REPEAT (ares_tac [f_inv_f] 1)); |
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141 qed "inv_injective"; |
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142 |
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143 Goal "A <= range(f) ==> inj_on (inv f) A"; |
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144 by (fast_tac (claset() addIs [inj_onI] |
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145 addEs [inv_injective, injD]) 1); |
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146 qed "inj_on_inv"; |
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147 |
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148 Goal "surj f ==> inj (inv f)"; |
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149 by (asm_simp_tac (simpset() addsimps [inj_on_inv, surj_range]) 1); |
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150 qed "surj_imp_inj_inv"; |
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151 |
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152 Goal "(surj f) = (f o inv f = id)"; |
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153 by (asm_simp_tac (simpset() addsimps [o_def, expand_fun_eq]) 1); |
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154 by (blast_tac (claset() addIs [surjI, surj_f_inv_f]) 1); |
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155 qed "surj_iff"; |
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156 |
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157 Goal "[| surj f; ALL x. g(f x) = x |] ==> inv f = g"; |
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158 by (rtac ext 1); |
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159 by (dres_inst_tac [("x","inv f x")] spec 1); |
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160 by (asm_full_simp_tac (simpset() addsimps [surj_f_inv_f]) 1); |
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161 qed "surj_imp_inv_eq"; |
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162 |
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163 Goalw [bij_def] "bij f ==> bij (inv f)"; |
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164 by (asm_simp_tac (simpset() addsimps [inj_imp_surj_inv, surj_imp_inj_inv]) 1); |
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165 qed "bij_imp_bij_inv"; |
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166 |
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167 val prems = |
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168 Goalw [inv_def] "[| !! x. g (f x) = x; !! y. f (g y) = y |] ==> inv f = g"; |
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169 by (rtac ext 1); |
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170 by (auto_tac (claset(), simpset() addsimps prems)); |
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171 qed "inv_equality"; |
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172 |
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173 Goalw [bij_def] "bij f ==> inv (inv f) = f"; |
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174 by (rtac inv_equality 1); |
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175 by (auto_tac (claset(), simpset() addsimps [surj_f_inv_f])); |
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176 qed "inv_inv_eq"; |
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177 |
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178 (** bij(inv f) implies little about f. Consider f::bool=>bool such that |
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179 f(True)=f(False)=True. Then it's consistent with axiom someI that |
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180 inv(f) could be any function at all, including the identity function. |
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181 If inv(f)=id then inv(f) is a bijection, but inj(f), surj(f) and |
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182 inv(inv(f))=f all fail. |
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183 **) |
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184 |
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185 Goalw [bij_def] "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"; |
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186 by (rtac (inv_equality) 1); |
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187 by (auto_tac (claset(), simpset() addsimps [surj_f_inv_f])); |
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188 qed "o_inv_distrib"; |
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189 |
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190 |
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191 Goal "surj f ==> f ` (inv f ` A) = A"; |
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192 by (asm_simp_tac (simpset() addsimps [image_eq_UN, surj_f_inv_f]) 1); |
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193 qed "image_surj_f_inv_f"; |
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194 |
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195 Goal "inj f ==> (inv f) ` (f ` A) = A"; |
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196 by (asm_simp_tac (simpset() addsimps [image_eq_UN]) 1); |
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197 qed "image_inv_f_f"; |
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198 |
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199 Goalw [image_def] "inj(f) ==> inv(f)`(f`X) = X"; |
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200 by Auto_tac; |
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201 qed "inv_image_comp"; |
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202 |
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203 Goal "bij f ==> f ` Collect P = {y. P (inv f y)}"; |
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204 by Auto_tac; |
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205 by (force_tac (claset(), simpset() addsimps [bij_is_inj]) 1); |
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206 by (blast_tac (claset() addIs [bij_is_surj RS surj_f_inv_f RS sym]) 1); |
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207 qed "bij_image_Collect_eq"; |
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208 |
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209 Goal "bij f ==> f -` A = inv f ` A"; |
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210 by Safe_tac; |
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211 by (asm_simp_tac (simpset() addsimps [bij_is_surj RS surj_f_inv_f]) 2); |
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212 by (blast_tac (claset() addIs [bij_is_inj RS inv_f_f RS sym]) 1); |
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213 qed "bij_vimage_eq_inv_image"; |
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214 |
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215 (*** Inverse ***) |
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216 |
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217 Goalw [Inv_def] "f ` A = B ==> (lam x: B. (Inv A f) x) : B funcset A"; |
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218 by (fast_tac (claset() addIs [restrict_in_funcset, someI2]) 1); |
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219 qed "Inv_funcset"; |
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220 |
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221 Goal "[| inj_on f A; x : A |] ==> Inv A f (f x) = x"; |
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222 by (asm_full_simp_tac (simpset() addsimps [Inv_def, inj_on_def]) 1); |
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223 by (blast_tac (claset() addIs [someI2]) 1); |
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224 qed "Inv_f_f"; |
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225 |
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226 Goal "y : f`A ==> f (Inv A f y) = y"; |
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227 by (asm_simp_tac (simpset() addsimps [Inv_def]) 1); |
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228 by (fast_tac (claset() addIs [someI2]) 1); |
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229 qed "f_Inv_f"; |
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230 |
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231 Goal "[| Inv A f x = Inv A f y; x : f`A; y : f`A |] ==> x=y"; |
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232 by (rtac (arg_cong RS box_equals) 1); |
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233 by (REPEAT (ares_tac [f_Inv_f] 1)); |
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234 qed "Inv_injective"; |
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235 |
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236 Goal "B <= f`A ==> inj_on (Inv A f) B"; |
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237 by (rtac inj_onI 1); |
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238 by (blast_tac (claset() addIs [inj_onI] addDs [Inv_injective, injD]) 1); |
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239 qed "inj_on_Inv"; |
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240 |
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241 Goal "[| inj_on f A; f ` A = B |] \ |
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242 \ ==> compose A (lam y:B. (Inv A f) y) f = (lam x: A. x)"; |
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243 by (asm_simp_tac (simpset() addsimps [compose_def]) 1); |
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244 by (rtac restrict_ext 1); |
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245 by Auto_tac; |
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246 by (etac subst 1); |
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247 by (asm_full_simp_tac (simpset() addsimps [Inv_f_f]) 1); |
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248 qed "compose_Inv_id"; |
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249 |
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250 |
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251 (**** split ****) |
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252 |
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253 (*Can't be added to simpset: loops!*) |
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254 Goal "(SOME x. P x) = (SOME (a,b). P(a,b))"; |
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255 by (simp_tac (simpset() addsimps [split_Pair_apply]) 1); |
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256 qed "split_paired_Eps"; |
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257 |
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258 Goalw [split_def] "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"; |
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259 by (rtac refl 1); |
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260 qed "Eps_split"; |
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261 |
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262 Goal "(@(x',y'). x = x' & y = y') = (x,y)"; |
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263 by (Blast_tac 1); |
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264 qed "Eps_split_eq"; |
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265 Addsimps [Eps_split_eq]; |