1 (* Title: HOL/set |
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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 1991 University of Cambridge |
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5 |
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6 For set.thy. Set theory for higher-order logic. A set is simply a predicate. |
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7 *) |
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8 |
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9 open Set; |
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10 |
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11 val [prem] = goal Set.thy "[| P(a) |] ==> a : {x.P(x)}"; |
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12 by (rtac (mem_Collect_eq RS ssubst) 1); |
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13 by (rtac prem 1); |
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14 qed "CollectI"; |
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15 |
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16 val prems = goal Set.thy "[| a : {x.P(x)} |] ==> P(a)"; |
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17 by (resolve_tac (prems RL [mem_Collect_eq RS subst]) 1); |
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18 qed "CollectD"; |
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19 |
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20 val [prem] = goal Set.thy "[| !!x. (x:A) = (x:B) |] ==> A = B"; |
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21 by (rtac (prem RS ext RS arg_cong RS box_equals) 1); |
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22 by (rtac Collect_mem_eq 1); |
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23 by (rtac Collect_mem_eq 1); |
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24 qed "set_ext"; |
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25 |
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26 val [prem] = goal Set.thy "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}"; |
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27 by (rtac (prem RS ext RS arg_cong) 1); |
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28 qed "Collect_cong"; |
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29 |
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30 val CollectE = make_elim CollectD; |
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31 |
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32 (*** Bounded quantifiers ***) |
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33 |
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34 val prems = goalw Set.thy [Ball_def] |
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35 "[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)"; |
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36 by (REPEAT (ares_tac (prems @ [allI,impI]) 1)); |
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37 qed "ballI"; |
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38 |
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39 val [major,minor] = goalw Set.thy [Ball_def] |
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40 "[| ! x:A. P(x); x:A |] ==> P(x)"; |
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41 by (rtac (minor RS (major RS spec RS mp)) 1); |
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42 qed "bspec"; |
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43 |
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44 val major::prems = goalw Set.thy [Ball_def] |
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45 "[| ! x:A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q"; |
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46 by (rtac (major RS spec RS impCE) 1); |
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47 by (REPEAT (eresolve_tac prems 1)); |
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48 qed "ballE"; |
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49 |
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50 (*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*) |
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51 fun ball_tac i = etac ballE i THEN contr_tac (i+1); |
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52 |
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53 val prems = goalw Set.thy [Bex_def] |
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54 "[| P(x); x:A |] ==> ? x:A. P(x)"; |
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55 by (REPEAT (ares_tac (prems @ [exI,conjI]) 1)); |
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56 qed "bexI"; |
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57 |
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58 qed_goal "bexCI" Set.thy |
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59 "[| ! x:A. ~P(x) ==> P(a); a:A |] ==> ? x:A.P(x)" |
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60 (fn prems=> |
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61 [ (rtac classical 1), |
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62 (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1)) ]); |
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63 |
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64 val major::prems = goalw Set.thy [Bex_def] |
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65 "[| ? x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q"; |
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66 by (rtac (major RS exE) 1); |
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67 by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1)); |
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68 qed "bexE"; |
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69 |
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70 (*Trival rewrite rule; (! x:A.P)=P holds only if A is nonempty!*) |
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71 val prems = goal Set.thy |
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72 "(! x:A. True) = True"; |
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73 by (REPEAT (ares_tac [TrueI,ballI,iffI] 1)); |
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74 qed "ball_rew"; |
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75 |
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76 (** Congruence rules **) |
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77 |
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78 val prems = goal Set.thy |
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79 "[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ |
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80 \ (! x:A. P(x)) = (! x:B. Q(x))"; |
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81 by (resolve_tac (prems RL [ssubst]) 1); |
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82 by (REPEAT (ares_tac [ballI,iffI] 1 |
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83 ORELSE eresolve_tac ([make_elim bspec, mp] @ (prems RL [iffE])) 1)); |
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84 qed "ball_cong"; |
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85 |
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86 val prems = goal Set.thy |
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87 "[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ |
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88 \ (? x:A. P(x)) = (? x:B. Q(x))"; |
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89 by (resolve_tac (prems RL [ssubst]) 1); |
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90 by (REPEAT (etac bexE 1 |
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91 ORELSE ares_tac ([bexI,iffI] @ (prems RL [iffD1,iffD2])) 1)); |
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92 qed "bex_cong"; |
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93 |
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94 (*** Subsets ***) |
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95 |
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96 val prems = goalw Set.thy [subset_def] "(!!x.x:A ==> x:B) ==> A <= B"; |
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97 by (REPEAT (ares_tac (prems @ [ballI]) 1)); |
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98 qed "subsetI"; |
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99 |
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100 (*Rule in Modus Ponens style*) |
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101 val major::prems = goalw Set.thy [subset_def] "[| A <= B; c:A |] ==> c:B"; |
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102 by (rtac (major RS bspec) 1); |
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103 by (resolve_tac prems 1); |
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104 qed "subsetD"; |
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105 |
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106 (*The same, with reversed premises for use with etac -- cf rev_mp*) |
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107 qed_goal "rev_subsetD" Set.thy "[| c:A; A <= B |] ==> c:B" |
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108 (fn prems=> [ (REPEAT (resolve_tac (prems@[subsetD]) 1)) ]); |
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109 |
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110 (*Classical elimination rule*) |
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111 val major::prems = goalw Set.thy [subset_def] |
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112 "[| A <= B; c~:A ==> P; c:B ==> P |] ==> P"; |
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113 by (rtac (major RS ballE) 1); |
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114 by (REPEAT (eresolve_tac prems 1)); |
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115 qed "subsetCE"; |
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116 |
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117 (*Takes assumptions A<=B; c:A and creates the assumption c:B *) |
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118 fun set_mp_tac i = etac subsetCE i THEN mp_tac i; |
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119 |
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120 qed_goal "subset_refl" Set.thy "A <= (A::'a set)" |
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121 (fn _=> [ (REPEAT (ares_tac [subsetI] 1)) ]); |
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122 |
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123 val prems = goal Set.thy "[| A<=B; B<=C |] ==> A<=(C::'a set)"; |
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124 by (cut_facts_tac prems 1); |
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125 by (REPEAT (ares_tac [subsetI] 1 ORELSE set_mp_tac 1)); |
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126 qed "subset_trans"; |
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127 |
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128 |
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129 (*** Equality ***) |
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130 |
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131 (*Anti-symmetry of the subset relation*) |
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132 val prems = goal Set.thy "[| A <= B; B <= A |] ==> A = (B::'a set)"; |
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133 by (rtac (iffI RS set_ext) 1); |
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134 by (REPEAT (ares_tac (prems RL [subsetD]) 1)); |
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135 qed "subset_antisym"; |
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136 val equalityI = subset_antisym; |
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137 |
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138 (* Equality rules from ZF set theory -- are they appropriate here? *) |
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139 val prems = goal Set.thy "A = B ==> A<=(B::'a set)"; |
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140 by (resolve_tac (prems RL [subst]) 1); |
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141 by (rtac subset_refl 1); |
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142 qed "equalityD1"; |
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143 |
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144 val prems = goal Set.thy "A = B ==> B<=(A::'a set)"; |
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145 by (resolve_tac (prems RL [subst]) 1); |
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146 by (rtac subset_refl 1); |
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147 qed "equalityD2"; |
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148 |
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149 val prems = goal Set.thy |
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150 "[| A = B; [| A<=B; B<=(A::'a set) |] ==> P |] ==> P"; |
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151 by (resolve_tac prems 1); |
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152 by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1)); |
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153 qed "equalityE"; |
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154 |
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155 val major::prems = goal Set.thy |
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156 "[| A = B; [| c:A; c:B |] ==> P; [| c~:A; c~:B |] ==> P |] ==> P"; |
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157 by (rtac (major RS equalityE) 1); |
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158 by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1)); |
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159 qed "equalityCE"; |
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160 |
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161 (*Lemma for creating induction formulae -- for "pattern matching" on p |
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162 To make the induction hypotheses usable, apply "spec" or "bspec" to |
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163 put universal quantifiers over the free variables in p. *) |
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164 val prems = goal Set.thy |
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165 "[| p:A; !!z. z:A ==> p=z --> R |] ==> R"; |
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166 by (rtac mp 1); |
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167 by (REPEAT (resolve_tac (refl::prems) 1)); |
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168 qed "setup_induction"; |
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169 |
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170 |
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171 (*** Set complement -- Compl ***) |
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172 |
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173 val prems = goalw Set.thy [Compl_def] |
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174 "[| c:A ==> False |] ==> c : Compl(A)"; |
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175 by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1)); |
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176 qed "ComplI"; |
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177 |
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178 (*This form, with negated conclusion, works well with the Classical prover. |
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179 Negated assumptions behave like formulae on the right side of the notional |
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180 turnstile...*) |
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181 val major::prems = goalw Set.thy [Compl_def] |
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182 "[| c : Compl(A) |] ==> c~:A"; |
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183 by (rtac (major RS CollectD) 1); |
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184 qed "ComplD"; |
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185 |
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186 val ComplE = make_elim ComplD; |
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187 |
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188 |
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189 (*** Binary union -- Un ***) |
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190 |
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191 val prems = goalw Set.thy [Un_def] "c:A ==> c : A Un B"; |
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192 by (REPEAT (resolve_tac (prems @ [CollectI,disjI1]) 1)); |
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193 qed "UnI1"; |
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194 |
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195 val prems = goalw Set.thy [Un_def] "c:B ==> c : A Un B"; |
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196 by (REPEAT (resolve_tac (prems @ [CollectI,disjI2]) 1)); |
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197 qed "UnI2"; |
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198 |
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199 (*Classical introduction rule: no commitment to A vs B*) |
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200 qed_goal "UnCI" Set.thy "(c~:B ==> c:A) ==> c : A Un B" |
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201 (fn prems=> |
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202 [ (rtac classical 1), |
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203 (REPEAT (ares_tac (prems@[UnI1,notI]) 1)), |
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204 (REPEAT (ares_tac (prems@[UnI2,notE]) 1)) ]); |
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205 |
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206 val major::prems = goalw Set.thy [Un_def] |
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207 "[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P"; |
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208 by (rtac (major RS CollectD RS disjE) 1); |
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209 by (REPEAT (eresolve_tac prems 1)); |
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210 qed "UnE"; |
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211 |
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212 |
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213 (*** Binary intersection -- Int ***) |
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214 |
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215 val prems = goalw Set.thy [Int_def] |
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216 "[| c:A; c:B |] ==> c : A Int B"; |
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217 by (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)); |
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218 qed "IntI"; |
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219 |
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220 val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:A"; |
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221 by (rtac (major RS CollectD RS conjunct1) 1); |
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222 qed "IntD1"; |
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223 |
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224 val [major] = goalw Set.thy [Int_def] "c : A Int B ==> c:B"; |
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225 by (rtac (major RS CollectD RS conjunct2) 1); |
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226 qed "IntD2"; |
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227 |
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228 val [major,minor] = goal Set.thy |
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229 "[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P"; |
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230 by (rtac minor 1); |
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231 by (rtac (major RS IntD1) 1); |
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232 by (rtac (major RS IntD2) 1); |
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233 qed "IntE"; |
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234 |
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235 |
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236 (*** Set difference ***) |
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237 |
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238 qed_goalw "DiffI" Set.thy [set_diff_def] |
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239 "[| c : A; c ~: B |] ==> c : A - B" |
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240 (fn prems=> [ (REPEAT (resolve_tac (prems @ [CollectI,conjI]) 1)) ]); |
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241 |
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242 qed_goalw "DiffD1" Set.thy [set_diff_def] |
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243 "c : A - B ==> c : A" |
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244 (fn [major]=> [ (rtac (major RS CollectD RS conjunct1) 1) ]); |
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245 |
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246 qed_goalw "DiffD2" Set.thy [set_diff_def] |
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247 "[| c : A - B; c : B |] ==> P" |
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248 (fn [major,minor]=> |
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249 [rtac (minor RS (major RS CollectD RS conjunct2 RS notE)) 1]); |
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250 |
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251 qed_goal "DiffE" Set.thy |
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252 "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P" |
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253 (fn prems=> |
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254 [ (resolve_tac prems 1), |
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255 (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ]); |
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256 |
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257 qed_goal "Diff_iff" Set.thy "(c : A-B) = (c:A & c~:B)" |
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258 (fn _ => [ (fast_tac (HOL_cs addSIs [DiffI] addSEs [DiffE]) 1) ]); |
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259 |
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260 |
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261 (*** The empty set -- {} ***) |
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262 |
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263 qed_goalw "emptyE" Set.thy [empty_def] "a:{} ==> P" |
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264 (fn [prem] => [rtac (prem RS CollectD RS FalseE) 1]); |
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265 |
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266 qed_goal "empty_subsetI" Set.thy "{} <= A" |
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267 (fn _ => [ (REPEAT (ares_tac [equalityI,subsetI,emptyE] 1)) ]); |
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268 |
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269 qed_goal "equals0I" Set.thy "[| !!y. y:A ==> False |] ==> A={}" |
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270 (fn prems=> |
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271 [ (REPEAT (ares_tac (prems@[empty_subsetI,subsetI,equalityI]) 1 |
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272 ORELSE eresolve_tac (prems RL [FalseE]) 1)) ]); |
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273 |
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274 qed_goal "equals0D" Set.thy "[| A={}; a:A |] ==> P" |
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275 (fn [major,minor]=> |
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276 [ (rtac (minor RS (major RS equalityD1 RS subsetD RS emptyE)) 1) ]); |
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277 |
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278 |
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279 (*** Augmenting a set -- insert ***) |
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280 |
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281 qed_goalw "insertI1" Set.thy [insert_def] "a : insert(a,B)" |
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282 (fn _ => [rtac (CollectI RS UnI1) 1, rtac refl 1]); |
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283 |
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284 qed_goalw "insertI2" Set.thy [insert_def] "a : B ==> a : insert(b,B)" |
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285 (fn [prem]=> [ (rtac (prem RS UnI2) 1) ]); |
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286 |
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287 qed_goalw "insertE" Set.thy [insert_def] |
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288 "[| a : insert(b,A); a=b ==> P; a:A ==> P |] ==> P" |
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289 (fn major::prems=> |
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290 [ (rtac (major RS UnE) 1), |
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291 (REPEAT (eresolve_tac (prems @ [CollectE]) 1)) ]); |
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292 |
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293 qed_goal "insert_iff" Set.thy "a : insert(b,A) = (a=b | a:A)" |
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294 (fn _ => [fast_tac (HOL_cs addIs [insertI1,insertI2] addSEs [insertE]) 1]); |
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295 |
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296 (*Classical introduction rule*) |
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297 qed_goal "insertCI" Set.thy "(a~:B ==> a=b) ==> a: insert(b,B)" |
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298 (fn [prem]=> |
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299 [ (rtac (disjCI RS (insert_iff RS iffD2)) 1), |
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300 (etac prem 1) ]); |
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301 |
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302 (*** Singletons, using insert ***) |
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303 |
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304 qed_goal "singletonI" Set.thy "a : {a}" |
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305 (fn _=> [ (rtac insertI1 1) ]); |
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306 |
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307 qed_goal "singletonE" Set.thy "[| a: {b}; a=b ==> P |] ==> P" |
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308 (fn major::prems=> |
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309 [ (rtac (major RS insertE) 1), |
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310 (REPEAT (eresolve_tac (prems @ [emptyE]) 1)) ]); |
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311 |
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312 goalw Set.thy [insert_def] "!!a. b : {a} ==> b=a"; |
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313 by(fast_tac (HOL_cs addSEs [emptyE,CollectE,UnE]) 1); |
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314 qed "singletonD"; |
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315 |
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316 val singletonE = make_elim singletonD; |
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317 |
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318 val [major] = goal Set.thy "{a}={b} ==> a=b"; |
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319 by (rtac (major RS equalityD1 RS subsetD RS singletonD) 1); |
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320 by (rtac singletonI 1); |
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321 qed "singleton_inject"; |
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322 |
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323 (*** Unions of families -- UNION x:A. B(x) is Union(B``A) ***) |
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324 |
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325 (*The order of the premises presupposes that A is rigid; b may be flexible*) |
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326 val prems = goalw Set.thy [UNION_def] |
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327 "[| a:A; b: B(a) |] ==> b: (UN x:A. B(x))"; |
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328 by (REPEAT (resolve_tac (prems @ [bexI,CollectI]) 1)); |
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329 qed "UN_I"; |
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330 |
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331 val major::prems = goalw Set.thy [UNION_def] |
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332 "[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R"; |
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333 by (rtac (major RS CollectD RS bexE) 1); |
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334 by (REPEAT (ares_tac prems 1)); |
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335 qed "UN_E"; |
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336 |
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337 val prems = goal Set.thy |
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338 "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
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339 \ (UN x:A. C(x)) = (UN x:B. D(x))"; |
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340 by (REPEAT (etac UN_E 1 |
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341 ORELSE ares_tac ([UN_I,equalityI,subsetI] @ |
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342 (prems RL [equalityD1,equalityD2] RL [subsetD])) 1)); |
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343 qed "UN_cong"; |
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344 |
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345 |
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346 (*** Intersections of families -- INTER x:A. B(x) is Inter(B``A) *) |
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347 |
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348 val prems = goalw Set.thy [INTER_def] |
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349 "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))"; |
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350 by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1)); |
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351 qed "INT_I"; |
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352 |
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353 val major::prems = goalw Set.thy [INTER_def] |
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354 "[| b : (INT x:A. B(x)); a:A |] ==> b: B(a)"; |
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355 by (rtac (major RS CollectD RS bspec) 1); |
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356 by (resolve_tac prems 1); |
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357 qed "INT_D"; |
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358 |
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359 (*"Classical" elimination -- by the Excluded Middle on a:A *) |
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360 val major::prems = goalw Set.thy [INTER_def] |
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361 "[| b : (INT x:A. B(x)); b: B(a) ==> R; a~:A ==> R |] ==> R"; |
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362 by (rtac (major RS CollectD RS ballE) 1); |
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363 by (REPEAT (eresolve_tac prems 1)); |
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364 qed "INT_E"; |
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365 |
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366 val prems = goal Set.thy |
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367 "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
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368 \ (INT x:A. C(x)) = (INT x:B. D(x))"; |
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369 by (REPEAT_FIRST (resolve_tac [INT_I,equalityI,subsetI])); |
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370 by (REPEAT (dtac INT_D 1 |
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371 ORELSE ares_tac (prems RL [equalityD1,equalityD2] RL [subsetD]) 1)); |
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372 qed "INT_cong"; |
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373 |
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374 |
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375 (*** Unions over a type; UNION1(B) = Union(range(B)) ***) |
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376 |
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377 (*The order of the premises presupposes that A is rigid; b may be flexible*) |
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378 val prems = goalw Set.thy [UNION1_def] |
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379 "b: B(x) ==> b: (UN x. B(x))"; |
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380 by (REPEAT (resolve_tac (prems @ [TrueI, CollectI RS UN_I]) 1)); |
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381 qed "UN1_I"; |
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382 |
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383 val major::prems = goalw Set.thy [UNION1_def] |
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384 "[| b : (UN x. B(x)); !!x. b: B(x) ==> R |] ==> R"; |
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385 by (rtac (major RS UN_E) 1); |
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386 by (REPEAT (ares_tac prems 1)); |
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387 qed "UN1_E"; |
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388 |
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389 |
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390 (*** Intersections over a type; INTER1(B) = Inter(range(B)) *) |
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391 |
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392 val prems = goalw Set.thy [INTER1_def] |
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393 "(!!x. b: B(x)) ==> b : (INT x. B(x))"; |
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394 by (REPEAT (ares_tac (INT_I::prems) 1)); |
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395 qed "INT1_I"; |
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396 |
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397 val [major] = goalw Set.thy [INTER1_def] |
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398 "b : (INT x. B(x)) ==> b: B(a)"; |
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399 by (rtac (TrueI RS (CollectI RS (major RS INT_D))) 1); |
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400 qed "INT1_D"; |
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401 |
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402 (*** Unions ***) |
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403 |
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404 (*The order of the premises presupposes that C is rigid; A may be flexible*) |
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405 val prems = goalw Set.thy [Union_def] |
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406 "[| X:C; A:X |] ==> A : Union(C)"; |
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407 by (REPEAT (resolve_tac (prems @ [UN_I]) 1)); |
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408 qed "UnionI"; |
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409 |
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410 val major::prems = goalw Set.thy [Union_def] |
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411 "[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R"; |
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412 by (rtac (major RS UN_E) 1); |
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413 by (REPEAT (ares_tac prems 1)); |
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414 qed "UnionE"; |
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415 |
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416 (*** Inter ***) |
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417 |
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418 val prems = goalw Set.thy [Inter_def] |
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419 "[| !!X. X:C ==> A:X |] ==> A : Inter(C)"; |
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420 by (REPEAT (ares_tac ([INT_I] @ prems) 1)); |
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421 qed "InterI"; |
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422 |
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423 (*A "destruct" rule -- every X in C contains A as an element, but |
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424 A:X can hold when X:C does not! This rule is analogous to "spec". *) |
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425 val major::prems = goalw Set.thy [Inter_def] |
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426 "[| A : Inter(C); X:C |] ==> A:X"; |
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427 by (rtac (major RS INT_D) 1); |
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428 by (resolve_tac prems 1); |
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429 qed "InterD"; |
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430 |
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431 (*"Classical" elimination rule -- does not require proving X:C *) |
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432 val major::prems = goalw Set.thy [Inter_def] |
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433 "[| A : Inter(C); A:X ==> R; X~:C ==> R |] ==> R"; |
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434 by (rtac (major RS INT_E) 1); |
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435 by (REPEAT (eresolve_tac prems 1)); |
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436 qed "InterE"; |
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437 |
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438 (*** Powerset ***) |
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439 |
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440 qed_goalw "PowI" Set.thy [Pow_def] "!!A B. A <= B ==> A : Pow(B)" |
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441 (fn _ => [ (etac CollectI 1) ]); |
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442 |
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443 qed_goalw "PowD" Set.thy [Pow_def] "!!A B. A : Pow(B) ==> A<=B" |
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444 (fn _=> [ (etac CollectD 1) ]); |
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445 |
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446 val Pow_bottom = empty_subsetI RS PowI; (* {}: Pow(B) *) |
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447 val Pow_top = subset_refl RS PowI; (* A : Pow(A) *) |
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