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1 (* Title: ZF/Ordinal.thy |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1993 University of Cambridge |
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5 |
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6 For Ordinal.thy. Ordinals in Zermelo-Fraenkel Set Theory |
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7 *) |
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8 |
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9 open Ordinal; |
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10 |
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11 (*** Rules for Transset ***) |
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12 |
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13 (** Two neat characterisations of Transset **) |
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14 |
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15 goalw Ordinal.thy [Transset_def] "Transset(A) <-> A<=Pow(A)"; |
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16 by (fast_tac ZF_cs 1); |
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17 val Transset_iff_Pow = result(); |
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18 |
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19 goalw Ordinal.thy [Transset_def] "Transset(A) <-> Union(succ(A)) = A"; |
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20 by (fast_tac (eq_cs addSEs [equalityE]) 1); |
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21 val Transset_iff_Union_succ = result(); |
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22 |
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23 (** Consequences of downwards closure **) |
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24 |
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25 goalw Ordinal.thy [Transset_def] |
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26 "!!C a b. [| Transset(C); {a,b}: C |] ==> a:C & b: C"; |
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27 by (fast_tac ZF_cs 1); |
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28 val Transset_doubleton_D = result(); |
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29 |
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30 val [prem1,prem2] = goalw Ordinal.thy [Pair_def] |
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31 "[| Transset(C); <a,b>: C |] ==> a:C & b: C"; |
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32 by (cut_facts_tac [prem2] 1); |
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33 by (fast_tac (ZF_cs addSDs [prem1 RS Transset_doubleton_D]) 1); |
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34 val Transset_Pair_D = result(); |
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35 |
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36 val prem1::prems = goal Ordinal.thy |
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37 "[| Transset(C); A*B <= C; b: B |] ==> A <= C"; |
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38 by (cut_facts_tac prems 1); |
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39 by (fast_tac (ZF_cs addSDs [prem1 RS Transset_Pair_D]) 1); |
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40 val Transset_includes_domain = result(); |
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41 |
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42 val prem1::prems = goal Ordinal.thy |
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43 "[| Transset(C); A*B <= C; a: A |] ==> B <= C"; |
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44 by (cut_facts_tac prems 1); |
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45 by (fast_tac (ZF_cs addSDs [prem1 RS Transset_Pair_D]) 1); |
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46 val Transset_includes_range = result(); |
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47 |
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48 val [prem1,prem2] = goalw (merge_theories(Ordinal.thy,Sum.thy)) [sum_def] |
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49 "[| Transset(C); A+B <= C |] ==> A <= C & B <= C"; |
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50 by (rtac (prem2 RS (Un_subset_iff RS iffD1) RS conjE) 1); |
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51 by (REPEAT (etac (prem1 RS Transset_includes_range) 1 |
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52 ORELSE resolve_tac [conjI, singletonI] 1)); |
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53 val Transset_includes_summands = result(); |
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54 |
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55 val [prem] = goalw (merge_theories(Ordinal.thy,Sum.thy)) [sum_def] |
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56 "Transset(C) ==> (A+B) Int C <= (A Int C) + (B Int C)"; |
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57 by (rtac (Int_Un_distrib RS ssubst) 1); |
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58 by (fast_tac (ZF_cs addSDs [prem RS Transset_Pair_D]) 1); |
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59 val Transset_sum_Int_subset = result(); |
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60 |
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61 (** Closure properties **) |
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62 |
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63 goalw Ordinal.thy [Transset_def] "Transset(0)"; |
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64 by (fast_tac ZF_cs 1); |
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65 val Transset_0 = result(); |
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66 |
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67 goalw Ordinal.thy [Transset_def] |
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68 "!!i j. [| Transset(i); Transset(j) |] ==> Transset(i Un j)"; |
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69 by (fast_tac ZF_cs 1); |
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70 val Transset_Un = result(); |
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71 |
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72 goalw Ordinal.thy [Transset_def] |
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73 "!!i j. [| Transset(i); Transset(j) |] ==> Transset(i Int j)"; |
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74 by (fast_tac ZF_cs 1); |
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75 val Transset_Int = result(); |
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76 |
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77 goalw Ordinal.thy [Transset_def] "!!i. Transset(i) ==> Transset(succ(i))"; |
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78 by (fast_tac ZF_cs 1); |
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79 val Transset_succ = result(); |
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80 |
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81 goalw Ordinal.thy [Transset_def] "!!i. Transset(i) ==> Transset(Pow(i))"; |
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82 by (fast_tac ZF_cs 1); |
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83 val Transset_Pow = result(); |
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84 |
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85 goalw Ordinal.thy [Transset_def] "!!A. Transset(A) ==> Transset(Union(A))"; |
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86 by (fast_tac ZF_cs 1); |
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87 val Transset_Union = result(); |
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88 |
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89 val [Transprem] = goalw Ordinal.thy [Transset_def] |
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90 "[| !!i. i:A ==> Transset(i) |] ==> Transset(Union(A))"; |
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91 by (fast_tac (ZF_cs addEs [Transprem RS bspec RS subsetD]) 1); |
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92 val Transset_Union_family = result(); |
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93 |
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94 val [prem,Transprem] = goalw Ordinal.thy [Transset_def] |
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95 "[| j:A; !!i. i:A ==> Transset(i) |] ==> Transset(Inter(A))"; |
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96 by (cut_facts_tac [prem] 1); |
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97 by (fast_tac (ZF_cs addEs [Transprem RS bspec RS subsetD]) 1); |
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98 val Transset_Inter_family = result(); |
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99 |
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100 (*** Natural Deduction rules for Ord ***) |
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101 |
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102 val prems = goalw Ordinal.thy [Ord_def] |
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103 "[| Transset(i); !!x. x:i ==> Transset(x) |] ==> Ord(i) "; |
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104 by (REPEAT (ares_tac (prems@[ballI,conjI]) 1)); |
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105 val OrdI = result(); |
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106 |
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107 val [major] = goalw Ordinal.thy [Ord_def] |
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108 "Ord(i) ==> Transset(i)"; |
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109 by (rtac (major RS conjunct1) 1); |
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110 val Ord_is_Transset = result(); |
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111 |
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112 val [major,minor] = goalw Ordinal.thy [Ord_def] |
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113 "[| Ord(i); j:i |] ==> Transset(j) "; |
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114 by (rtac (minor RS (major RS conjunct2 RS bspec)) 1); |
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115 val Ord_contains_Transset = result(); |
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116 |
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117 (*** Lemmas for ordinals ***) |
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118 |
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119 goalw Ordinal.thy [Ord_def,Transset_def] "!!i j.[| Ord(i); j:i |] ==> Ord(j)"; |
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120 by (fast_tac ZF_cs 1); |
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121 val Ord_in_Ord = result(); |
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122 |
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123 (* Ord(succ(j)) ==> Ord(j) *) |
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124 val Ord_succD = succI1 RSN (2, Ord_in_Ord); |
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125 |
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126 goal Ordinal.thy "!!i j. [| Ord(i); Transset(j); j<=i |] ==> Ord(j)"; |
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127 by (REPEAT (ares_tac [OrdI] 1 |
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128 ORELSE eresolve_tac [Ord_contains_Transset, subsetD] 1)); |
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129 val Ord_subset_Ord = result(); |
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130 |
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131 goalw Ordinal.thy [Ord_def,Transset_def] "!!i j. [| j:i; Ord(i) |] ==> j<=i"; |
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132 by (fast_tac ZF_cs 1); |
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133 val OrdmemD = result(); |
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134 |
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135 goal Ordinal.thy "!!i j k. [| i:j; j:k; Ord(k) |] ==> i:k"; |
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136 by (REPEAT (ares_tac [OrdmemD RS subsetD] 1)); |
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137 val Ord_trans = result(); |
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138 |
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139 goal Ordinal.thy "!!i j. [| i:j; Ord(j) |] ==> succ(i) <= j"; |
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140 by (REPEAT (ares_tac [OrdmemD RSN (2,succ_subsetI)] 1)); |
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141 val Ord_succ_subsetI = result(); |
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142 |
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143 |
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144 (*** The construction of ordinals: 0, succ, Union ***) |
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145 |
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146 goal Ordinal.thy "Ord(0)"; |
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147 by (REPEAT (ares_tac [OrdI,Transset_0] 1 ORELSE etac emptyE 1)); |
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148 val Ord_0 = result(); |
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149 |
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150 goal Ordinal.thy "!!i. Ord(i) ==> Ord(succ(i))"; |
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151 by (REPEAT (ares_tac [OrdI,Transset_succ] 1 |
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152 ORELSE eresolve_tac [succE,ssubst,Ord_is_Transset, |
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153 Ord_contains_Transset] 1)); |
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154 val Ord_succ = result(); |
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155 |
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156 goal Ordinal.thy "Ord(succ(i)) <-> Ord(i)"; |
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157 by (fast_tac (ZF_cs addIs [Ord_succ] addDs [Ord_succD]) 1); |
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158 val Ord_succ_iff = result(); |
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159 |
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160 goalw Ordinal.thy [Ord_def] "!!i j. [| Ord(i); Ord(j) |] ==> Ord(i Un j)"; |
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161 by (fast_tac (ZF_cs addSIs [Transset_Un]) 1); |
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162 val Ord_Un = result(); |
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163 |
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164 goalw Ordinal.thy [Ord_def] "!!i j. [| Ord(i); Ord(j) |] ==> Ord(i Int j)"; |
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165 by (fast_tac (ZF_cs addSIs [Transset_Int]) 1); |
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166 val Ord_Int = result(); |
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167 |
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168 val nonempty::prems = goal Ordinal.thy |
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169 "[| j:A; !!i. i:A ==> Ord(i) |] ==> Ord(Inter(A))"; |
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170 by (rtac (nonempty RS Transset_Inter_family RS OrdI) 1); |
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171 by (rtac Ord_is_Transset 1); |
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172 by (REPEAT (ares_tac ([Ord_contains_Transset,nonempty]@prems) 1 |
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173 ORELSE etac InterD 1)); |
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174 val Ord_Inter = result(); |
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175 |
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176 val jmemA::prems = goal Ordinal.thy |
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177 "[| j:A; !!x. x:A ==> Ord(B(x)) |] ==> Ord(INT x:A. B(x))"; |
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178 by (rtac (jmemA RS RepFunI RS Ord_Inter) 1); |
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179 by (etac RepFunE 1); |
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180 by (etac ssubst 1); |
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181 by (eresolve_tac prems 1); |
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182 val Ord_INT = result(); |
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183 |
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184 (*There is no set of all ordinals, for then it would contain itself*) |
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185 goal Ordinal.thy "~ (ALL i. i:X <-> Ord(i))"; |
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186 by (rtac notI 1); |
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187 by (forw_inst_tac [("x", "X")] spec 1); |
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188 by (safe_tac (ZF_cs addSEs [mem_anti_refl])); |
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189 by (swap_res_tac [Ord_is_Transset RSN (2,OrdI)] 1); |
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190 by (fast_tac ZF_cs 2); |
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191 bw Transset_def; |
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192 by (safe_tac ZF_cs); |
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193 by (asm_full_simp_tac ZF_ss 1); |
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194 by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 1)); |
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195 val ON_class = result(); |
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196 |
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197 (*** < is 'less than' for ordinals ***) |
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198 |
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199 goalw Ordinal.thy [lt_def] "!!i j. [| i:j; Ord(j) |] ==> i<j"; |
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200 by (REPEAT (ares_tac [conjI] 1)); |
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201 val ltI = result(); |
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202 |
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203 val major::prems = goalw Ordinal.thy [lt_def] |
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204 "[| i<j; [| i:j; Ord(i); Ord(j) |] ==> P |] ==> P"; |
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205 by (rtac (major RS conjE) 1); |
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206 by (REPEAT (ares_tac (prems@[Ord_in_Ord]) 1)); |
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207 val ltE = result(); |
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208 |
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209 goal Ordinal.thy "!!i j. i<j ==> i:j"; |
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210 by (etac ltE 1); |
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211 by (assume_tac 1); |
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212 val ltD = result(); |
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213 |
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214 goalw Ordinal.thy [lt_def] "~ i<0"; |
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215 by (fast_tac ZF_cs 1); |
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216 val not_lt0 = result(); |
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217 |
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218 (* i<0 ==> R *) |
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219 val lt0E = standard (not_lt0 RS notE); |
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220 |
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221 goal Ordinal.thy "!!i j k. [| i<j; j<k |] ==> i<k"; |
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222 by (fast_tac (ZF_cs addSIs [ltI] addSEs [ltE, Ord_trans]) 1); |
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223 val lt_trans = result(); |
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224 |
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225 goalw Ordinal.thy [lt_def] "!!i j. [| i<j; j<i |] ==> P"; |
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226 by (REPEAT (eresolve_tac [asm_rl, conjE, mem_anti_sym] 1)); |
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227 val lt_anti_sym = result(); |
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228 |
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229 val lt_anti_refl = prove_goal Ordinal.thy "i<i ==> P" |
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230 (fn [major]=> [ (rtac (major RS (major RS lt_anti_sym)) 1) ]); |
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231 |
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232 val lt_not_refl = prove_goal Ordinal.thy "~ i<i" |
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233 (fn _=> [ (rtac notI 1), (etac lt_anti_refl 1) ]); |
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234 |
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235 (** le is less than or equals; recall i le j abbrevs i<succ(j) !! **) |
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236 |
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237 goalw Ordinal.thy [lt_def] "i le j <-> i<j | (i=j & Ord(j))"; |
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238 by (fast_tac (ZF_cs addSIs [Ord_succ] addSDs [Ord_succD]) 1); |
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239 val le_iff = result(); |
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240 |
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241 goal Ordinal.thy "!!i j. i<j ==> i le j"; |
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242 by (asm_simp_tac (ZF_ss addsimps [le_iff]) 1); |
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243 val leI = result(); |
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244 |
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245 goal Ordinal.thy "!!i. [| i=j; Ord(j) |] ==> i le j"; |
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246 by (asm_simp_tac (ZF_ss addsimps [le_iff]) 1); |
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247 val le_eqI = result(); |
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248 |
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249 val le_refl = refl RS le_eqI; |
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250 |
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251 val [prem] = goal Ordinal.thy "(~ (i=j & Ord(j)) ==> i<j) ==> i le j"; |
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252 by (rtac (disjCI RS (le_iff RS iffD2)) 1); |
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253 by (etac prem 1); |
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254 val leCI = result(); |
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255 |
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256 val major::prems = goal Ordinal.thy |
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257 "[| i le j; i<j ==> P; [| i=j; Ord(j) |] ==> P |] ==> P"; |
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258 by (rtac (major RS (le_iff RS iffD1 RS disjE)) 1); |
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259 by (DEPTH_SOLVE (ares_tac prems 1 ORELSE etac conjE 1)); |
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260 val leE = result(); |
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261 |
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262 goal Ordinal.thy "!!i j. [| i le j; j le i |] ==> i=j"; |
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263 by (asm_full_simp_tac (ZF_ss addsimps [le_iff]) 1); |
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264 by (fast_tac (ZF_cs addEs [lt_anti_sym]) 1); |
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265 val le_asym = result(); |
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266 |
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267 goal Ordinal.thy "i le 0 <-> i=0"; |
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268 by (fast_tac (ZF_cs addSIs [Ord_0 RS le_refl] addSEs [leE, lt0E]) 1); |
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269 val le0_iff = result(); |
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270 |
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271 val le0D = standard (le0_iff RS iffD1); |
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272 |
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273 val lt_cs = |
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274 ZF_cs addSIs [le_refl, leCI] |
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275 addSDs [le0D] |
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276 addSEs [lt_anti_refl, lt0E, leE]; |
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277 |
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278 |
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279 (*** Natural Deduction rules for Memrel ***) |
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280 |
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281 goalw Ordinal.thy [Memrel_def] "<a,b> : Memrel(A) <-> a:b & a:A & b:A"; |
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282 by (fast_tac ZF_cs 1); |
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283 val Memrel_iff = result(); |
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284 |
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285 val prems = goal Ordinal.thy "[| a: b; a: A; b: A |] ==> <a,b> : Memrel(A)"; |
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286 by (REPEAT (resolve_tac (prems@[conjI, Memrel_iff RS iffD2]) 1)); |
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287 val MemrelI = result(); |
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288 |
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289 val [major,minor] = goal Ordinal.thy |
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290 "[| <a,b> : Memrel(A); \ |
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291 \ [| a: A; b: A; a:b |] ==> P \ |
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292 \ |] ==> P"; |
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293 by (rtac (major RS (Memrel_iff RS iffD1) RS conjE) 1); |
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294 by (etac conjE 1); |
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295 by (rtac minor 1); |
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296 by (REPEAT (assume_tac 1)); |
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297 val MemrelE = result(); |
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298 |
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299 (*The membership relation (as a set) is well-founded. |
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300 Proof idea: show A<=B by applying the foundation axiom to A-B *) |
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301 goalw Ordinal.thy [wf_def] "wf(Memrel(A))"; |
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302 by (EVERY1 [rtac (foundation RS disjE RS allI), |
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303 etac disjI1, |
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304 etac bexE, |
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305 rtac (impI RS allI RS bexI RS disjI2), |
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306 etac MemrelE, |
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307 etac bspec, |
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308 REPEAT o assume_tac]); |
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309 val wf_Memrel = result(); |
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310 |
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311 (*Transset(i) does not suffice, though ALL j:i.Transset(j) does*) |
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312 goalw Ordinal.thy [Ord_def, Transset_def, trans_def] |
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313 "!!i. Ord(i) ==> trans(Memrel(i))"; |
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314 by (fast_tac (ZF_cs addSIs [MemrelI] addSEs [MemrelE]) 1); |
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315 val trans_Memrel = result(); |
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316 |
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317 (*If Transset(A) then Memrel(A) internalizes the membership relation below A*) |
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318 goalw Ordinal.thy [Transset_def] |
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319 "!!A. Transset(A) ==> <a,b> : Memrel(A) <-> a:b & b:A"; |
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320 by (fast_tac (ZF_cs addSIs [MemrelI] addSEs [MemrelE]) 1); |
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321 val Transset_Memrel_iff = result(); |
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322 |
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323 |
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324 (*** Transfinite induction ***) |
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325 |
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326 (*Epsilon induction over a transitive set*) |
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327 val major::prems = goalw Ordinal.thy [Transset_def] |
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328 "[| i: k; Transset(k); \ |
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329 \ !!x.[| x: k; ALL y:x. P(y) |] ==> P(x) \ |
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330 \ |] ==> P(i)"; |
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331 by (rtac (major RS (wf_Memrel RS wf_induct2)) 1); |
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332 by (fast_tac (ZF_cs addEs [MemrelE]) 1); |
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333 by (resolve_tac prems 1); |
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334 by (assume_tac 1); |
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335 by (cut_facts_tac prems 1); |
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336 by (fast_tac (ZF_cs addIs [MemrelI]) 1); |
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337 val Transset_induct = result(); |
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338 |
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339 (*Induction over an ordinal*) |
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340 val Ord_induct = Ord_is_Transset RSN (2, Transset_induct); |
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341 |
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342 (*Induction over the class of ordinals -- a useful corollary of Ord_induct*) |
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343 val [major,indhyp] = goal Ordinal.thy |
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344 "[| Ord(i); \ |
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345 \ !!x.[| Ord(x); ALL y:x. P(y) |] ==> P(x) \ |
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346 \ |] ==> P(i)"; |
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347 by (rtac (major RS Ord_succ RS (succI1 RS Ord_induct)) 1); |
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348 by (rtac indhyp 1); |
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349 by (rtac (major RS Ord_succ RS Ord_in_Ord) 1); |
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350 by (REPEAT (assume_tac 1)); |
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351 val trans_induct = result(); |
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352 |
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353 (*Perform induction on i, then prove the Ord(i) subgoal using prems. *) |
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354 fun trans_ind_tac a prems i = |
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355 EVERY [res_inst_tac [("i",a)] trans_induct i, |
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356 rename_last_tac a ["1"] (i+1), |
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357 ares_tac prems i]; |
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358 |
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359 |
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360 (*** Fundamental properties of the epsilon ordering (< on ordinals) ***) |
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361 |
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362 (*Finds contradictions for the following proof*) |
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363 val Ord_trans_tac = EVERY' [etac notE, etac Ord_trans, REPEAT o atac]; |
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364 |
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365 (** Proving that < is a linear ordering on the ordinals **) |
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366 |
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367 val prems = goal Ordinal.thy |
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368 "Ord(i) ==> (ALL j. Ord(j) --> i:j | i=j | j:i)"; |
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369 by (trans_ind_tac "i" prems 1); |
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370 by (rtac (impI RS allI) 1); |
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371 by (trans_ind_tac "j" [] 1); |
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372 by (DEPTH_SOLVE (step_tac eq_cs 1 ORELSE Ord_trans_tac 1)); |
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373 val Ord_linear_lemma = result(); |
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374 val Ord_linear = standard (Ord_linear_lemma RS spec RS mp); |
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375 |
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376 (*The trichotomy law for ordinals!*) |
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377 val ordi::ordj::prems = goalw Ordinal.thy [lt_def] |
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378 "[| Ord(i); Ord(j); i<j ==> P; i=j ==> P; j<i ==> P |] ==> P"; |
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379 by (rtac ([ordi,ordj] MRS Ord_linear RS disjE) 1); |
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380 by (etac disjE 2); |
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381 by (DEPTH_SOLVE (ares_tac ([ordi,ordj,conjI] @ prems) 1)); |
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382 val Ord_linear_lt = result(); |
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383 |
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384 val prems = goal Ordinal.thy |
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385 "[| Ord(i); Ord(j); i<j ==> P; j le i ==> P |] ==> P"; |
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386 by (res_inst_tac [("i","i"),("j","j")] Ord_linear_lt 1); |
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387 by (DEPTH_SOLVE (ares_tac ([leI, sym RS le_eqI] @ prems) 1)); |
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388 val Ord_linear2 = result(); |
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389 |
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390 val prems = goal Ordinal.thy |
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391 "[| Ord(i); Ord(j); i le j ==> P; j le i ==> P |] ==> P"; |
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392 by (res_inst_tac [("i","i"),("j","j")] Ord_linear_lt 1); |
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393 by (DEPTH_SOLVE (ares_tac ([leI,le_eqI] @ prems) 1)); |
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394 val Ord_linear_le = result(); |
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395 |
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396 goal Ordinal.thy "!!i j. j le i ==> ~ i<j"; |
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397 by (fast_tac (lt_cs addEs [lt_anti_sym]) 1); |
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398 val le_imp_not_lt = result(); |
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399 |
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400 goal Ordinal.thy "!!i j. [| ~ i<j; Ord(i); Ord(j) |] ==> j le i"; |
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401 by (res_inst_tac [("i","i"),("j","j")] Ord_linear2 1); |
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402 by (REPEAT (SOMEGOAL assume_tac)); |
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403 by (fast_tac (lt_cs addEs [lt_anti_sym]) 1); |
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404 val not_lt_imp_le = result(); |
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405 |
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406 goal Ordinal.thy "!!i j. [| Ord(i); Ord(j) |] ==> ~ i<j <-> j le i"; |
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407 by (REPEAT (ares_tac [iffI, le_imp_not_lt, not_lt_imp_le] 1)); |
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408 val not_lt_iff_le = result(); |
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409 |
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410 goal Ordinal.thy "!!i j. [| Ord(i); Ord(j) |] ==> ~ i le j <-> j<i"; |
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411 by (asm_simp_tac (ZF_ss addsimps [not_lt_iff_le RS iff_sym]) 1); |
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412 val not_le_iff_lt = result(); |
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413 |
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414 goal Ordinal.thy "!!i. Ord(i) ==> 0 le i"; |
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415 by (etac (not_lt_iff_le RS iffD1) 1); |
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416 by (REPEAT (resolve_tac [Ord_0, not_lt0] 1)); |
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417 val Ord_0_le = result(); |
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418 |
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419 goal Ordinal.thy "!!i. [| Ord(i); i~=0 |] ==> 0<i"; |
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420 by (etac (not_le_iff_lt RS iffD1) 1); |
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421 by (rtac Ord_0 1); |
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422 by (fast_tac lt_cs 1); |
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423 val Ord_0_lt = result(); |
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424 |
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425 (*** Results about less-than or equals ***) |
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426 |
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427 (** For ordinals, j<=i (subset) implies j le i (less-than or equals) **) |
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428 |
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429 goal Ordinal.thy "!!i j. [| j<=i; Ord(i); Ord(j) |] ==> j le i"; |
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430 by (rtac (not_lt_iff_le RS iffD1) 1); |
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431 by (assume_tac 1); |
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432 by (assume_tac 1); |
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433 by (fast_tac (ZF_cs addEs [ltE, mem_anti_refl]) 1); |
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434 val subset_imp_le = result(); |
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435 |
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436 goal Ordinal.thy "!!i j. i le j ==> i<=j"; |
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437 by (etac leE 1); |
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438 by (fast_tac ZF_cs 2); |
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439 by (fast_tac (subset_cs addIs [OrdmemD] addEs [ltE]) 1); |
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440 val le_imp_subset = result(); |
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441 |
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442 goal Ordinal.thy "j le i <-> j<=i & Ord(i) & Ord(j)"; |
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443 by (fast_tac (ZF_cs addSEs [subset_imp_le, le_imp_subset] |
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444 addEs [ltE, make_elim Ord_succD]) 1); |
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445 val le_subset_iff = result(); |
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446 |
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447 goal Ordinal.thy "i le succ(j) <-> i le j | i=succ(j) & Ord(i)"; |
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448 by (simp_tac (ZF_ss addsimps [le_iff]) 1); |
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449 by (fast_tac (ZF_cs addIs [Ord_succ] addDs [Ord_succD]) 1); |
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450 val le_succ_iff = result(); |
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451 |
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452 (*Just a variant of subset_imp_le*) |
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453 val [ordi,ordj,minor] = goal Ordinal.thy |
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454 "[| Ord(i); Ord(j); !!x. x<j ==> x<i |] ==> j le i"; |
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455 by (REPEAT_FIRST (ares_tac [notI RS not_lt_imp_le, ordi, ordj])); |
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456 be (minor RS lt_anti_refl) 1; |
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457 val all_lt_imp_le = result(); |
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458 |
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459 (** Transitive laws **) |
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460 |
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461 goal Ordinal.thy "!!i j. [| i le j; j<k |] ==> i<k"; |
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462 by (fast_tac (ZF_cs addEs [leE, lt_trans]) 1); |
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463 val lt_trans1 = result(); |
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464 |
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465 goal Ordinal.thy "!!i j. [| i<j; j le k |] ==> i<k"; |
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466 by (fast_tac (ZF_cs addEs [leE, lt_trans]) 1); |
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467 val lt_trans2 = result(); |
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468 |
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469 goal Ordinal.thy "!!i j. [| i le j; j le k |] ==> i le k"; |
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470 by (REPEAT (ares_tac [lt_trans1] 1)); |
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471 val le_trans = result(); |
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472 |
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473 goal Ordinal.thy "!!i j. i<j ==> succ(i) le j"; |
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474 by (rtac (not_lt_iff_le RS iffD1) 1); |
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475 by (fast_tac (lt_cs addEs [lt_anti_sym]) 3); |
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476 by (ALLGOALS (fast_tac (ZF_cs addEs [ltE] addIs [Ord_succ]))); |
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477 val succ_leI = result(); |
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478 |
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479 goal Ordinal.thy "!!i j. succ(i) le j ==> i<j"; |
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480 by (rtac (not_le_iff_lt RS iffD1) 1); |
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481 by (fast_tac (lt_cs addEs [lt_anti_sym]) 3); |
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482 by (ALLGOALS (fast_tac (ZF_cs addEs [ltE, make_elim Ord_succD]))); |
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483 val succ_leE = result(); |
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484 |
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485 goal Ordinal.thy "succ(i) le j <-> i<j"; |
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486 by (REPEAT (ares_tac [iffI,succ_leI,succ_leE] 1)); |
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487 val succ_le_iff = result(); |
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488 |
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489 (** Union and Intersection **) |
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490 |
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491 goal Ordinal.thy "!!i j. [| Ord(i); Ord(j) |] ==> i le i Un j"; |
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492 by (rtac (Un_upper1 RS subset_imp_le) 1); |
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493 by (REPEAT (ares_tac [Ord_Un] 1)); |
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494 val Un_upper1_le = result(); |
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495 |
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496 goal Ordinal.thy "!!i j. [| Ord(i); Ord(j) |] ==> j le i Un j"; |
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497 by (rtac (Un_upper2 RS subset_imp_le) 1); |
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498 by (REPEAT (ares_tac [Ord_Un] 1)); |
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499 val Un_upper2_le = result(); |
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500 |
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501 (*Replacing k by succ(k') yields the similar rule for le!*) |
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502 goal Ordinal.thy "!!i j k. [| i<k; j<k |] ==> i Un j < k"; |
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503 by (res_inst_tac [("i","i"),("j","j")] Ord_linear_le 1); |
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504 by (rtac (Un_commute RS ssubst) 4); |
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505 by (asm_full_simp_tac (ZF_ss addsimps [le_subset_iff, subset_Un_iff]) 4); |
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506 by (asm_full_simp_tac (ZF_ss addsimps [le_subset_iff, subset_Un_iff]) 3); |
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507 by (REPEAT (eresolve_tac [asm_rl, ltE] 1)); |
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508 val Un_least_lt = result(); |
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509 |
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510 goal Ordinal.thy "!!i j. [| Ord(i); Ord(j) |] ==> i Un j < k <-> i<k & j<k"; |
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511 by (safe_tac (ZF_cs addSIs [Un_least_lt])); |
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512 br (Un_upper2_le RS lt_trans1) 2; |
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513 br (Un_upper1_le RS lt_trans1) 1; |
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514 by (REPEAT_SOME assume_tac); |
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515 val Un_least_lt_iff = result(); |
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516 |
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517 val [ordi,ordj,ordk] = goal Ordinal.thy |
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518 "[| Ord(i); Ord(j); Ord(k) |] ==> i Un j : k <-> i:k & j:k"; |
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519 by (cut_facts_tac [[ordi,ordj] MRS |
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520 read_instantiate [("k","k")] Un_least_lt_iff] 1); |
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521 by (asm_full_simp_tac (ZF_ss addsimps [lt_def,ordi,ordj,ordk]) 1); |
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522 val Un_least_mem_iff = result(); |
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523 |
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524 (*Replacing k by succ(k') yields the similar rule for le!*) |
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525 goal Ordinal.thy "!!i j k. [| i<k; j<k |] ==> i Int j < k"; |
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526 by (res_inst_tac [("i","i"),("j","j")] Ord_linear_le 1); |
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527 by (rtac (Int_commute RS ssubst) 4); |
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528 by (asm_full_simp_tac (ZF_ss addsimps [le_subset_iff, subset_Int_iff]) 4); |
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529 by (asm_full_simp_tac (ZF_ss addsimps [le_subset_iff, subset_Int_iff]) 3); |
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530 by (REPEAT (eresolve_tac [asm_rl, ltE] 1)); |
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531 val Int_greatest_lt = result(); |
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532 |
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533 (*FIXME: the Intersection duals are missing!*) |
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534 |
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535 |
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536 (*** Results about limits ***) |
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537 |
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538 val prems = goal Ordinal.thy "[| !!i. i:A ==> Ord(i) |] ==> Ord(Union(A))"; |
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539 by (rtac (Ord_is_Transset RS Transset_Union_family RS OrdI) 1); |
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540 by (REPEAT (etac UnionE 1 ORELSE ares_tac ([Ord_contains_Transset]@prems) 1)); |
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541 val Ord_Union = result(); |
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542 |
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543 val prems = goal Ordinal.thy |
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544 "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(UN x:A. B(x))"; |
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545 by (rtac Ord_Union 1); |
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546 by (etac RepFunE 1); |
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547 by (etac ssubst 1); |
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548 by (eresolve_tac prems 1); |
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549 val Ord_UN = result(); |
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550 |
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551 (* No < version; consider (UN i:nat.i)=nat *) |
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552 val [ordi,limit] = goal Ordinal.thy |
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553 "[| Ord(i); !!x. x:A ==> b(x) le i |] ==> (UN x:A. b(x)) le i"; |
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554 by (rtac (le_imp_subset RS UN_least RS subset_imp_le) 1); |
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555 by (REPEAT (ares_tac [ordi, Ord_UN, limit] 1 ORELSE etac (limit RS ltE) 1)); |
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556 val UN_least_le = result(); |
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557 |
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558 val [jlti,limit] = goal Ordinal.thy |
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559 "[| j<i; !!x. x:A ==> b(x)<j |] ==> (UN x:A. succ(b(x))) < i"; |
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560 by (rtac (jlti RS ltE) 1); |
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561 by (rtac (UN_least_le RS lt_trans2) 1); |
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562 by (REPEAT (ares_tac [jlti, succ_leI, limit] 1)); |
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563 val UN_succ_least_lt = result(); |
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564 |
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565 val prems = goal Ordinal.thy |
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566 "[| a: A; i le b(a); !!x. x:A ==> Ord(b(x)) |] ==> i le (UN x:A. b(x))"; |
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567 by (resolve_tac (prems RL [ltE]) 1); |
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568 by (rtac (le_imp_subset RS subset_trans RS subset_imp_le) 1); |
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569 by (REPEAT (ares_tac (prems @ [UN_upper, Ord_UN]) 1)); |
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570 val UN_upper_le = result(); |
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571 |
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572 goal Ordinal.thy "!!i. Ord(i) ==> (UN y:i. succ(y)) = i"; |
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573 by (fast_tac (eq_cs addEs [Ord_trans]) 1); |
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574 val Ord_equality = result(); |
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575 |
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576 (*Holds for all transitive sets, not just ordinals*) |
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577 goal Ordinal.thy "!!i. Ord(i) ==> Union(i) <= i"; |
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578 by (fast_tac (ZF_cs addSEs [Ord_trans]) 1); |
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579 val Ord_Union_subset = result(); |
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580 |
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581 |
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582 (*** Limit ordinals -- general properties ***) |
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583 |
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584 goalw Ordinal.thy [Limit_def] "!!i. Limit(i) ==> Union(i) = i"; |
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585 by (fast_tac (eq_cs addSIs [ltI] addSEs [ltE] addEs [Ord_trans]) 1); |
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586 val Limit_Union_eq = result(); |
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587 |
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588 goalw Ordinal.thy [Limit_def] "!!i. Limit(i) ==> Ord(i)"; |
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589 by (etac conjunct1 1); |
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590 val Limit_is_Ord = result(); |
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591 |
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592 goalw Ordinal.thy [Limit_def] "!!i. Limit(i) ==> 0 < i"; |
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593 by (etac (conjunct2 RS conjunct1) 1); |
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594 val Limit_has_0 = result(); |
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595 |
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596 goalw Ordinal.thy [Limit_def] "!!i. [| Limit(i); j<i |] ==> succ(j) < i"; |
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597 by (fast_tac ZF_cs 1); |
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598 val Limit_has_succ = result(); |
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599 |
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600 goalw Ordinal.thy [Limit_def] |
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601 "!!i. [| 0<i; ALL y. succ(y) ~= i |] ==> Limit(i)"; |
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602 by (safe_tac subset_cs); |
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603 by (rtac (not_le_iff_lt RS iffD1) 2); |
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604 by (fast_tac (lt_cs addEs [lt_anti_sym]) 4); |
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605 by (REPEAT (eresolve_tac [asm_rl, ltE, Ord_succ] 1)); |
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606 val non_succ_LimitI = result(); |
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607 |
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608 goal Ordinal.thy "!!i. Limit(succ(i)) ==> P"; |
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609 br lt_anti_refl 1; |
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610 br Limit_has_succ 1; |
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611 ba 1; |
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612 be (Limit_is_Ord RS Ord_succD RS le_refl) 1; |
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613 val succ_LimitE = result(); |
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614 |
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615 goal Ordinal.thy "!!i. [| Limit(i); i le succ(j) |] ==> i le j"; |
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616 by (safe_tac (ZF_cs addSEs [succ_LimitE, leE])); |
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617 val Limit_le_succD = result(); |
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618 |
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619 |