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1 (* Title: ZF/qpair.ML |
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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 qpair.thy. |
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7 |
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8 Quine-inspired ordered pairs and disjoint sums, for non-well-founded data |
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9 structures in ZF. Does not precisely follow Quine's construction. Thanks |
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10 to Thomas Forster for suggesting this approach! |
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11 |
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12 W. V. Quine, On Ordered Pairs and Relations, in Selected Logic Papers, |
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13 1966. |
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14 |
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15 Many proofs are borrowed from pair.ML and sum.ML |
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16 |
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17 Do we EVER have rank(a) < rank(<a;b>) ? Perhaps if the latter rank |
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18 is not a limit ordinal? |
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19 *) |
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20 |
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21 |
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22 open QPair; |
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23 |
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24 (**** Quine ordered pairing ****) |
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25 |
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26 (** Lemmas for showing that <a;b> uniquely determines a and b **) |
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27 |
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28 val QPair_iff = prove_goalw QPair.thy [QPair_def] |
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29 "<a;b> = <c;d> <-> a=c & b=d" |
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30 (fn _=> [rtac sum_equal_iff 1]); |
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31 |
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32 val QPair_inject = standard (QPair_iff RS iffD1 RS conjE); |
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33 |
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34 val QPair_inject1 = prove_goal QPair.thy "<a;b> = <c;d> ==> a=c" |
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35 (fn [major]=> |
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36 [ (rtac (major RS QPair_inject) 1), (assume_tac 1) ]); |
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37 |
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38 val QPair_inject2 = prove_goal QPair.thy "<a;b> = <c;d> ==> b=d" |
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39 (fn [major]=> |
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40 [ (rtac (major RS QPair_inject) 1), (assume_tac 1) ]); |
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41 |
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42 |
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43 (*** QSigma: Disjoint union of a family of sets |
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44 Generalizes Cartesian product ***) |
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45 |
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46 val QSigmaI = prove_goalw QPair.thy [QSigma_def] |
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47 "[| a:A; b:B(a) |] ==> <a;b> : QSigma(A,B)" |
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48 (fn prems=> [ (REPEAT (resolve_tac (prems@[singletonI,UN_I]) 1)) ]); |
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49 |
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50 (*The general elimination rule*) |
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51 val QSigmaE = prove_goalw QPair.thy [QSigma_def] |
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52 "[| c: QSigma(A,B); \ |
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53 \ !!x y.[| x:A; y:B(x); c=<x;y> |] ==> P \ |
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54 \ |] ==> P" |
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55 (fn major::prems=> |
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56 [ (cut_facts_tac [major] 1), |
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57 (REPEAT (eresolve_tac [UN_E, singletonE] 1 ORELSE ares_tac prems 1)) ]); |
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58 |
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59 (** Elimination rules for <a;b>:A*B -- introducing no eigenvariables **) |
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60 |
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61 val QSigmaE2 = |
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62 rule_by_tactic (REPEAT_FIRST (etac QPair_inject ORELSE' bound_hyp_subst_tac) |
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63 THEN prune_params_tac) |
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64 (read_instantiate [("c","<a;b>")] QSigmaE); |
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65 |
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66 val QSigmaD1 = prove_goal QPair.thy "<a;b> : QSigma(A,B) ==> a : A" |
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67 (fn [major]=> |
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68 [ (rtac (major RS QSigmaE2) 1), (assume_tac 1) ]); |
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69 |
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70 val QSigmaD2 = prove_goal QPair.thy "<a;b> : QSigma(A,B) ==> b : B(a)" |
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71 (fn [major]=> |
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72 [ (rtac (major RS QSigmaE2) 1), (assume_tac 1) ]); |
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73 |
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74 val QSigma_cong = prove_goalw QPair.thy [QSigma_def] |
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75 "[| A=A'; !!x. x:A' ==> B(x)=B'(x) |] ==> \ |
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76 \ QSigma(A,B) = QSigma(A',B')" |
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77 (fn prems=> [ (prove_cong_tac (prems@[RepFun_cong]) 1) ]); |
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78 |
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79 val QSigma_empty1 = prove_goal QPair.thy "QSigma(0,B) = 0" |
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80 (fn _ => [ (fast_tac (ZF_cs addIs [equalityI] addSEs [QSigmaE]) 1) ]); |
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81 |
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82 val QSigma_empty2 = prove_goal QPair.thy "A <*> 0 = 0" |
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83 (fn _ => [ (fast_tac (ZF_cs addIs [equalityI] addSEs [QSigmaE]) 1) ]); |
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84 |
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85 |
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86 (*** Eliminator - qsplit ***) |
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87 |
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88 val qsplit = prove_goalw QPair.thy [qsplit_def] |
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89 "qsplit(%x y.c(x,y), <a;b>) = c(a,b)" |
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90 (fn _ => [ (fast_tac (ZF_cs addIs [the_equality] addEs [QPair_inject]) 1) ]); |
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91 |
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92 val qsplit_type = prove_goal QPair.thy |
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93 "[| p:QSigma(A,B); \ |
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94 \ !!x y.[| x:A; y:B(x) |] ==> c(x,y):C(<x;y>) \ |
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95 \ |] ==> qsplit(%x y.c(x,y), p) : C(p)" |
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96 (fn major::prems=> |
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97 [ (rtac (major RS QSigmaE) 1), |
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98 (etac ssubst 1), |
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99 (REPEAT (ares_tac (prems @ [qsplit RS ssubst]) 1)) ]); |
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100 |
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101 (*This congruence rule uses NO typing information...*) |
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102 val qsplit_cong = prove_goalw QPair.thy [qsplit_def] |
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103 "[| p=p'; !!x y.c(x,y) = c'(x,y) |] ==> \ |
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104 \ qsplit(%x y.c(x,y), p) = qsplit(%x y.c'(x,y), p')" |
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105 (fn prems=> [ (prove_cong_tac (prems@[the_cong]) 1) ]); |
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106 |
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107 |
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108 val qpair_cs = ZF_cs addSIs [QSigmaI] addSEs [QSigmaE2, QSigmaE, QPair_inject]; |
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109 |
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110 (*** qconverse ***) |
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111 |
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112 val qconverseI = prove_goalw QPair.thy [qconverse_def] |
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113 "!!a b r. <a;b>:r ==> <b;a>:qconverse(r)" |
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114 (fn _ => [ (fast_tac qpair_cs 1) ]); |
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115 |
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116 val qconverseD = prove_goalw QPair.thy [qconverse_def] |
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117 "!!a b r. <a;b> : qconverse(r) ==> <b;a> : r" |
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118 (fn _ => [ (fast_tac qpair_cs 1) ]); |
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119 |
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120 val qconverseE = prove_goalw QPair.thy [qconverse_def] |
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121 "[| yx : qconverse(r); \ |
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122 \ !!x y. [| yx=<y;x>; <x;y>:r |] ==> P \ |
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123 \ |] ==> P" |
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124 (fn [major,minor]=> |
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125 [ (rtac (major RS ReplaceE) 1), |
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126 (REPEAT (eresolve_tac [exE, conjE, minor] 1)), |
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127 (hyp_subst_tac 1), |
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128 (assume_tac 1) ]); |
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129 |
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130 val qconverse_cs = qpair_cs addSIs [qconverseI] |
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131 addSEs [qconverseD,qconverseE]; |
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132 |
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133 val qconverse_of_qconverse = prove_goal QPair.thy |
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134 "!!A B r. r<=QSigma(A,B) ==> qconverse(qconverse(r)) = r" |
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135 (fn _ => [ (fast_tac (qconverse_cs addSIs [equalityI]) 1) ]); |
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136 |
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137 val qconverse_type = prove_goal QPair.thy |
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138 "!!A B r. r <= A <*> B ==> qconverse(r) <= B <*> A" |
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139 (fn _ => [ (fast_tac qconverse_cs 1) ]); |
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140 |
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141 val qconverse_of_prod = prove_goal QPair.thy "qconverse(A <*> B) = B <*> A" |
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142 (fn _ => [ (fast_tac (qconverse_cs addSIs [equalityI]) 1) ]); |
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143 |
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144 val qconverse_empty = prove_goal QPair.thy "qconverse(0) = 0" |
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145 (fn _ => [ (fast_tac (qconverse_cs addSIs [equalityI]) 1) ]); |
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146 |
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147 |
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148 (*** qsplit for predicates: result type o ***) |
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149 |
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150 goalw QPair.thy [qfsplit_def] "!!R a b. R(a,b) ==> qfsplit(R, <a;b>)"; |
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151 by (REPEAT (ares_tac [refl,exI,conjI] 1)); |
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152 val qfsplitI = result(); |
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153 |
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154 val major::prems = goalw QPair.thy [qfsplit_def] |
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155 "[| qfsplit(R,z); !!x y. [| z = <x;y>; R(x,y) |] ==> P |] ==> P"; |
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156 by (cut_facts_tac [major] 1); |
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157 by (REPEAT (eresolve_tac (prems@[asm_rl,exE,conjE]) 1)); |
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158 val qfsplitE = result(); |
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159 |
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160 goal QPair.thy "!!R a b. qfsplit(R,<a;b>) ==> R(a,b)"; |
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161 by (REPEAT (eresolve_tac [asm_rl,qfsplitE,QPair_inject,ssubst] 1)); |
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162 val qfsplitD = result(); |
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163 |
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164 |
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165 (**** The Quine-inspired notion of disjoint sum ****) |
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166 |
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167 val qsum_defs = [qsum_def,QInl_def,QInr_def,qcase_def]; |
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168 |
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169 (** Introduction rules for the injections **) |
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170 |
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171 goalw QPair.thy qsum_defs "!!a A B. a : A ==> QInl(a) : A <+> B"; |
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172 by (REPEAT (ares_tac [UnI1,QSigmaI,singletonI] 1)); |
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173 val QInlI = result(); |
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174 |
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175 goalw QPair.thy qsum_defs "!!b A B. b : B ==> QInr(b) : A <+> B"; |
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176 by (REPEAT (ares_tac [UnI2,QSigmaI,singletonI] 1)); |
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177 val QInrI = result(); |
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178 |
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179 (** Elimination rules **) |
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180 |
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181 val major::prems = goalw QPair.thy qsum_defs |
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182 "[| u: A <+> B; \ |
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183 \ !!x. [| x:A; u=QInl(x) |] ==> P; \ |
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184 \ !!y. [| y:B; u=QInr(y) |] ==> P \ |
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185 \ |] ==> P"; |
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186 by (rtac (major RS UnE) 1); |
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187 by (REPEAT (rtac refl 1 |
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188 ORELSE eresolve_tac (prems@[QSigmaE,singletonE,ssubst]) 1)); |
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189 val qsumE = result(); |
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190 |
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191 (** QInjection and freeness rules **) |
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192 |
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193 val [major] = goalw QPair.thy qsum_defs "QInl(a)=QInl(b) ==> a=b"; |
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194 by (EVERY1 [rtac (major RS QPair_inject), assume_tac]); |
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195 val QInl_inject = result(); |
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196 |
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197 val [major] = goalw QPair.thy qsum_defs "QInr(a)=QInr(b) ==> a=b"; |
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198 by (EVERY1 [rtac (major RS QPair_inject), assume_tac]); |
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199 val QInr_inject = result(); |
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200 |
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201 val [major] = goalw QPair.thy qsum_defs "QInl(a)=QInr(b) ==> P"; |
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202 by (rtac (major RS QPair_inject) 1); |
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203 by (etac (sym RS one_neq_0) 1); |
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204 val QInl_neq_QInr = result(); |
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205 |
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206 val QInr_neq_QInl = sym RS QInl_neq_QInr; |
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207 |
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208 (** Injection and freeness equivalences, for rewriting **) |
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209 |
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210 goal QPair.thy "QInl(a)=QInl(b) <-> a=b"; |
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211 by (rtac iffI 1); |
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212 by (REPEAT (eresolve_tac [QInl_inject,subst_context] 1)); |
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213 val QInl_iff = result(); |
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214 |
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215 goal QPair.thy "QInr(a)=QInr(b) <-> a=b"; |
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216 by (rtac iffI 1); |
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217 by (REPEAT (eresolve_tac [QInr_inject,subst_context] 1)); |
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218 val QInr_iff = result(); |
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219 |
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220 goal QPair.thy "QInl(a)=QInr(b) <-> False"; |
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221 by (rtac iffI 1); |
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222 by (REPEAT (eresolve_tac [QInl_neq_QInr,FalseE] 1)); |
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223 val QInl_QInr_iff = result(); |
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224 |
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225 goal QPair.thy "QInr(b)=QInl(a) <-> False"; |
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226 by (rtac iffI 1); |
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227 by (REPEAT (eresolve_tac [QInr_neq_QInl,FalseE] 1)); |
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228 val QInr_QInl_iff = result(); |
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229 |
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230 val qsum_cs = |
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231 ZF_cs addIs [QInlI,QInrI] addSEs [qsumE,QInl_neq_QInr,QInr_neq_QInl] |
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232 addSDs [QInl_inject,QInr_inject]; |
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233 |
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234 (** <+> is itself injective... who cares?? **) |
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235 |
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236 goal QPair.thy |
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237 "u: A <+> B <-> (EX x. x:A & u=QInl(x)) | (EX y. y:B & u=QInr(y))"; |
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238 by (fast_tac qsum_cs 1); |
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239 val qsum_iff = result(); |
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240 |
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241 goal QPair.thy "A <+> B <= C <+> D <-> A<=C & B<=D"; |
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242 by (fast_tac qsum_cs 1); |
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243 val qsum_subset_iff = result(); |
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244 |
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245 goal QPair.thy "A <+> B = C <+> D <-> A=C & B=D"; |
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246 by (SIMP_TAC (ZF_ss addrews [extension,qsum_subset_iff]) 1); |
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247 by (fast_tac ZF_cs 1); |
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248 val qsum_equal_iff = result(); |
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249 |
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250 (*** Eliminator -- qcase ***) |
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251 |
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252 goalw QPair.thy qsum_defs "qcase(c, d, QInl(a)) = c(a)"; |
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253 by (rtac (qsplit RS trans) 1); |
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254 by (rtac cond_0 1); |
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255 val qcase_QInl = result(); |
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256 |
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257 goalw QPair.thy qsum_defs "qcase(c, d, QInr(b)) = d(b)"; |
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258 by (rtac (qsplit RS trans) 1); |
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259 by (rtac cond_1 1); |
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260 val qcase_QInr = result(); |
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261 |
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262 val prems = goalw QPair.thy [qcase_def] |
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263 "[| u=u'; !!x. c(x)=c'(x); !!y. d(y)=d'(y) |] ==> \ |
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264 \ qcase(c,d,u)=qcase(c',d',u')"; |
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265 by (REPEAT (resolve_tac ([refl,qsplit_cong,cond_cong] @ prems) 1)); |
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266 val qcase_cong = result(); |
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267 |
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268 val major::prems = goal QPair.thy |
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269 "[| u: A <+> B; \ |
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270 \ !!x. x: A ==> c(x): C(QInl(x)); \ |
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271 \ !!y. y: B ==> d(y): C(QInr(y)) \ |
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272 \ |] ==> qcase(c,d,u) : C(u)"; |
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273 by (rtac (major RS qsumE) 1); |
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274 by (ALLGOALS (etac ssubst)); |
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275 by (ALLGOALS (ASM_SIMP_TAC (ZF_ss addrews |
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276 (prems@[qcase_QInl,qcase_QInr])))); |
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277 val qcase_type = result(); |
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278 |
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279 (** Rules for the Part primitive **) |
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280 |
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281 goal QPair.thy "Part(A <+> B,QInl) = {QInl(x). x: A}"; |
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282 by (fast_tac (qsum_cs addIs [PartI,equalityI] addSEs [PartE]) 1); |
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283 val Part_QInl = result(); |
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284 |
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285 goal QPair.thy "Part(A <+> B,QInr) = {QInr(y). y: B}"; |
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286 by (fast_tac (qsum_cs addIs [PartI,equalityI] addSEs [PartE]) 1); |
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287 val Part_QInr = result(); |
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288 |
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289 goal QPair.thy "Part(A <+> B, %x.QInr(h(x))) = {QInr(y). y: Part(B,h)}"; |
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290 by (fast_tac (qsum_cs addIs [PartI,equalityI] addSEs [PartE]) 1); |
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291 val Part_QInr2 = result(); |
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292 |
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293 goal QPair.thy "!!A B C. C <= A <+> B ==> Part(C,QInl) Un Part(C,QInr) = C"; |
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294 by (rtac equalityI 1); |
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295 by (rtac Un_least 1); |
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296 by (rtac Part_subset 1); |
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297 by (rtac Part_subset 1); |
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298 by (fast_tac (ZF_cs addIs [PartI] addSEs [qsumE]) 1); |
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299 val Part_qsum_equality = result(); |