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1 (* Title: ZF/zf.ML |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson and Martin D Coen, CU Computer Laboratory |
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4 Copyright 1992 University of Cambridge |
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5 |
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6 Basic introduction and elimination rules for Zermelo-Fraenkel Set Theory |
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7 *) |
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8 |
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9 open ZF; |
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10 |
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11 signature ZF_LEMMAS = |
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12 sig |
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13 val ballE : thm |
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14 val ballI : thm |
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15 val ball_cong : thm |
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16 val ball_rew : thm |
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17 val ball_tac : int -> tactic |
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18 val basic_ZF_congs : thm list |
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19 val bexCI : thm |
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20 val bexE : thm |
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21 val bexI : thm |
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22 val bex_cong : thm |
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23 val bspec : thm |
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24 val CollectD1 : thm |
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25 val CollectD2 : thm |
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26 val CollectE : thm |
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27 val CollectI : thm |
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28 val Collect_cong : thm |
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29 val emptyE : thm |
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30 val empty_subsetI : thm |
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31 val equalityCE : thm |
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32 val equalityD1 : thm |
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33 val equalityD2 : thm |
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34 val equalityE : thm |
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35 val equalityI : thm |
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36 val equality_iffI : thm |
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37 val equals0D : thm |
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38 val equals0I : thm |
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39 val ex1_functional : thm |
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40 val InterD : thm |
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41 val InterE : thm |
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42 val InterI : thm |
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43 val INT_E : thm |
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44 val INT_I : thm |
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45 val lemmas_cs : claset |
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46 val PowD : thm |
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47 val PowI : thm |
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48 val prove_cong_tac : thm list -> int -> tactic |
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49 val RepFunE : thm |
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50 val RepFunI : thm |
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51 val RepFun_eqI : thm |
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52 val RepFun_cong : thm |
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53 val ReplaceE : thm |
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54 val ReplaceI : thm |
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55 val Replace_iff : thm |
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56 val Replace_cong : thm |
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57 val rev_ballE : thm |
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58 val rev_bspec : thm |
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59 val rev_subsetD : thm |
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60 val separation : thm |
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61 val setup_induction : thm |
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62 val set_mp_tac : int -> tactic |
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63 val subsetCE : thm |
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64 val subsetD : thm |
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65 val subsetI : thm |
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66 val subset_refl : thm |
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67 val subset_trans : thm |
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68 val UnionE : thm |
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69 val UnionI : thm |
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70 val UN_E : thm |
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71 val UN_I : thm |
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72 end; |
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73 |
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74 |
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75 structure ZF_Lemmas : ZF_LEMMAS = |
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76 struct |
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77 |
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78 val basic_ZF_congs = mk_congs ZF.thy |
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79 ["op `", "op ``", "op Int", "op Un", "op -", "op <=", "op :", |
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80 "Pow", "Union", "Inter", "fst", "snd", "succ", "Pair", "Upair", "cons", |
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81 "domain", "range", "restrict"]; |
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82 |
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83 fun prove_cong_tac prems i = |
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84 REPEAT (ares_tac (prems@[refl]@FOL_congs@basic_ZF_congs) i); |
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85 |
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86 (*** Bounded universal quantifier ***) |
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87 |
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88 val ballI = prove_goalw ZF.thy [Ball_def] |
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89 "[| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)" |
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90 (fn prems=> [ (REPEAT (ares_tac (prems @ [allI,impI]) 1)) ]); |
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91 |
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92 val bspec = prove_goalw ZF.thy [Ball_def] |
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93 "[| ALL x:A. P(x); x: A |] ==> P(x)" |
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94 (fn major::prems=> |
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95 [ (rtac (major RS spec RS mp) 1), |
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96 (resolve_tac prems 1) ]); |
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97 |
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98 val ballE = prove_goalw ZF.thy [Ball_def] |
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99 "[| ALL x:A. P(x); P(x) ==> Q; ~ x:A ==> Q |] ==> Q" |
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100 (fn major::prems=> |
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101 [ (rtac (major RS allE) 1), |
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102 (REPEAT (eresolve_tac (prems@[asm_rl,impCE]) 1)) ]); |
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103 |
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104 (*Used in the datatype package*) |
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105 val rev_bspec = prove_goal ZF.thy |
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106 "!!x A P. [| x: A; ALL x:A. P(x) |] ==> P(x)" |
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107 (fn _ => |
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108 [ REPEAT (ares_tac [bspec] 1) ]); |
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109 |
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110 (*Instantiates x first: better for automatic theorem proving?*) |
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111 val rev_ballE = prove_goal ZF.thy |
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112 "[| ALL x:A. P(x); ~ x:A ==> Q; P(x) ==> Q |] ==> Q" |
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113 (fn major::prems=> |
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114 [ (rtac (major RS ballE) 1), |
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115 (REPEAT (eresolve_tac prems 1)) ]); |
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116 |
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117 (*Takes assumptions ALL x:A.P(x) and a:A; creates assumption P(a)*) |
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118 val ball_tac = dtac bspec THEN' assume_tac; |
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119 |
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120 (*Trival rewrite rule; (ALL x:A.P)<->P holds only if A is nonempty!*) |
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121 val ball_rew = prove_goal ZF.thy "(ALL x:A. True) <-> True" |
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122 (fn prems=> [ (REPEAT (ares_tac [TrueI,ballI,iffI] 1)) ]); |
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123 |
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124 (*Congruence rule for rewriting*) |
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125 val ball_cong = prove_goalw ZF.thy [Ball_def] |
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126 "[| A=A'; !!x. x:A' ==> P(x) <-> P'(x) \ |
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127 \ |] ==> (ALL x:A. P(x)) <-> (ALL x:A'. P'(x))" |
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128 (fn prems=> [ (prove_cong_tac prems 1) ]); |
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129 |
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130 (*** Bounded existential quantifier ***) |
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131 |
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132 val bexI = prove_goalw ZF.thy [Bex_def] |
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133 "[| P(x); x: A |] ==> EX x:A. P(x)" |
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134 (fn prems=> [ (REPEAT (ares_tac (prems @ [exI,conjI]) 1)) ]); |
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135 |
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136 (*Not of the general form for such rules; ~EX has become ALL~ *) |
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137 val bexCI = prove_goal ZF.thy |
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138 "[| ALL x:A. ~P(x) ==> P(a); a: A |] ==> EX x:A.P(x)" |
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139 (fn prems=> |
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140 [ (rtac classical 1), |
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141 (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1)) ]); |
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142 |
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143 val bexE = prove_goalw ZF.thy [Bex_def] |
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144 "[| EX x:A. P(x); !!x. [| x:A; P(x) |] ==> Q \ |
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145 \ |] ==> Q" |
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146 (fn major::prems=> |
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147 [ (rtac (major RS exE) 1), |
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148 (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1)) ]); |
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149 |
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150 (*We do not even have (EX x:A. True) <-> True unless A is nonempty!!*) |
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151 |
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152 val bex_cong = prove_goalw ZF.thy [Bex_def] |
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153 "[| A=A'; !!x. x:A' ==> P(x) <-> P'(x) \ |
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154 \ |] ==> (EX x:A. P(x)) <-> (EX x:A'. P'(x))" |
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155 (fn prems=> [ (prove_cong_tac prems 1) ]); |
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156 |
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157 (*** Rules for subsets ***) |
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158 |
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159 val subsetI = prove_goalw ZF.thy [subset_def] |
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160 "(!!x.x:A ==> x:B) ==> A <= B" |
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161 (fn prems=> [ (REPEAT (ares_tac (prems @ [ballI]) 1)) ]); |
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162 |
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163 (*Rule in Modus Ponens style [was called subsetE] *) |
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164 val subsetD = prove_goalw ZF.thy [subset_def] "[| A <= B; c:A |] ==> c:B" |
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165 (fn major::prems=> |
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166 [ (rtac (major RS bspec) 1), |
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167 (resolve_tac prems 1) ]); |
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168 |
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169 (*Classical elimination rule*) |
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170 val subsetCE = prove_goalw ZF.thy [subset_def] |
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171 "[| A <= B; ~(c:A) ==> P; c:B ==> P |] ==> P" |
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172 (fn major::prems=> |
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173 [ (rtac (major RS ballE) 1), |
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174 (REPEAT (eresolve_tac prems 1)) ]); |
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175 |
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176 (*Takes assumptions A<=B; c:A and creates the assumption c:B *) |
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177 val set_mp_tac = dtac subsetD THEN' assume_tac; |
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178 |
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179 (*Sometimes useful with premises in this order*) |
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180 val rev_subsetD = prove_goal ZF.thy "!!A B c. [| c:A; A<=B |] ==> c:B" |
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181 (fn _=> [REPEAT (ares_tac [subsetD] 1)]); |
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182 |
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183 val subset_refl = prove_goal ZF.thy "A <= A" |
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184 (fn _=> [ (rtac subsetI 1), atac 1 ]); |
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185 |
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186 val subset_trans = prove_goal ZF.thy "[| A<=B; B<=C |] ==> A<=C" |
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187 (fn prems=> [ (REPEAT (ares_tac ([subsetI]@(prems RL [subsetD])) 1)) ]); |
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188 |
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189 |
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190 (*** Rules for equality ***) |
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191 |
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192 (*Anti-symmetry of the subset relation*) |
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193 val equalityI = prove_goal ZF.thy "[| A <= B; B <= A |] ==> A = B" |
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194 (fn prems=> [ (REPEAT (resolve_tac (prems@[conjI, extension RS iffD2]) 1)) ]); |
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195 |
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196 val equality_iffI = prove_goal ZF.thy "(!!x. x:A <-> x:B) ==> A = B" |
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197 (fn [prem] => |
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198 [ (rtac equalityI 1), |
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199 (REPEAT (ares_tac [subsetI, prem RS iffD1, prem RS iffD2] 1)) ]); |
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200 |
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201 val equalityD1 = prove_goal ZF.thy "A = B ==> A<=B" |
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202 (fn prems=> |
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203 [ (rtac (extension RS iffD1 RS conjunct1) 1), |
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204 (resolve_tac prems 1) ]); |
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205 |
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206 val equalityD2 = prove_goal ZF.thy "A = B ==> B<=A" |
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207 (fn prems=> |
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208 [ (rtac (extension RS iffD1 RS conjunct2) 1), |
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209 (resolve_tac prems 1) ]); |
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210 |
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211 val equalityE = prove_goal ZF.thy |
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212 "[| A = B; [| A<=B; B<=A |] ==> P |] ==> P" |
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213 (fn prems=> |
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214 [ (DEPTH_SOLVE (resolve_tac (prems@[equalityD1,equalityD2]) 1)) ]); |
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215 |
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216 val equalityCE = prove_goal ZF.thy |
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217 "[| A = B; [| c:A; c:B |] ==> P; [| ~ c:A; ~ c:B |] ==> P |] ==> P" |
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218 (fn major::prems=> |
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219 [ (rtac (major RS equalityE) 1), |
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220 (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1)) ]); |
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221 |
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222 (*Lemma for creating induction formulae -- for "pattern matching" on p |
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223 To make the induction hypotheses usable, apply "spec" or "bspec" to |
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224 put universal quantifiers over the free variables in p. |
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225 Would it be better to do subgoal_tac "ALL z. p = f(z) --> R(z)" ??*) |
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226 val setup_induction = prove_goal ZF.thy |
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227 "[| p: A; !!z. z: A ==> p=z --> R |] ==> R" |
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228 (fn prems=> |
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229 [ (rtac mp 1), |
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230 (REPEAT (resolve_tac (refl::prems) 1)) ]); |
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231 |
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232 |
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233 (*** Rules for Replace -- the derived form of replacement ***) |
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234 |
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235 val ex1_functional = prove_goal ZF.thy |
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236 "[| EX! z. P(a,z); P(a,b); P(a,c) |] ==> b = c" |
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237 (fn prems=> |
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238 [ (cut_facts_tac prems 1), |
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239 (best_tac FOL_dup_cs 1) ]); |
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240 |
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241 val Replace_iff = prove_goalw ZF.thy [Replace_def] |
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242 "b : {y. x:A, P(x,y)} <-> (EX x:A. P(x,b) & (ALL y. P(x,y) --> y=b))" |
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243 (fn _=> |
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244 [ (rtac (replacement RS iff_trans) 1), |
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245 (REPEAT (ares_tac [refl,bex_cong,iffI,ballI,allI,conjI,impI,ex1I] 1 |
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246 ORELSE eresolve_tac [conjE, spec RS mp, ex1_functional] 1)) ]); |
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247 |
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248 (*Introduction; there must be a unique y such that P(x,y), namely y=b. *) |
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249 val ReplaceI = prove_goal ZF.thy |
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250 "[| x: A; P(x,b); !!y. P(x,y) ==> y=b |] ==> \ |
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251 \ b : {y. x:A, P(x,y)}" |
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252 (fn prems=> |
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253 [ (rtac (Replace_iff RS iffD2) 1), |
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254 (REPEAT (ares_tac (prems@[bexI,conjI,allI,impI]) 1)) ]); |
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255 |
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256 (*Elimination; may asssume there is a unique y such that P(x,y), namely y=b. *) |
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257 val ReplaceE = prove_goal ZF.thy |
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258 "[| b : {y. x:A, P(x,y)}; \ |
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259 \ !!x. [| x: A; P(x,b); ALL y. P(x,y)-->y=b |] ==> R \ |
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260 \ |] ==> R" |
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261 (fn prems=> |
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262 [ (rtac (Replace_iff RS iffD1 RS bexE) 1), |
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263 (etac conjE 2), |
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264 (REPEAT (ares_tac prems 1)) ]); |
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265 |
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266 val Replace_cong = prove_goal ZF.thy |
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267 "[| A=B; !!x y. x:B ==> P(x,y) <-> Q(x,y) |] ==> \ |
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268 \ {y. x:A, P(x,y)} = {y. x:B, Q(x,y)}" |
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269 (fn prems=> |
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270 let val substprems = prems RL [subst, ssubst] |
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271 and iffprems = prems RL [iffD1,iffD2] |
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272 in [ (rtac equalityI 1), |
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273 (REPEAT (eresolve_tac (substprems@[asm_rl, ReplaceE, spec RS mp]) 1 |
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274 ORELSE resolve_tac [subsetI, ReplaceI] 1 |
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275 ORELSE (resolve_tac iffprems 1 THEN assume_tac 2))) ] |
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276 end); |
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277 |
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278 |
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279 (*** Rules for RepFun ***) |
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280 |
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281 val RepFunI = prove_goalw ZF.thy [RepFun_def] |
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282 "!!a A. a : A ==> f(a) : {f(x). x:A}" |
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283 (fn _ => [ (REPEAT (ares_tac [ReplaceI,refl] 1)) ]); |
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284 |
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285 (*Useful for co-induction proofs*) |
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286 val RepFun_eqI = prove_goal ZF.thy |
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287 "!!b a f. [| b=f(a); a : A |] ==> b : {f(x). x:A}" |
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288 (fn _ => [ etac ssubst 1, etac RepFunI 1 ]); |
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289 |
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290 val RepFunE = prove_goalw ZF.thy [RepFun_def] |
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291 "[| b : {f(x). x:A}; \ |
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292 \ !!x.[| x:A; b=f(x) |] ==> P |] ==> \ |
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293 \ P" |
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294 (fn major::prems=> |
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295 [ (rtac (major RS ReplaceE) 1), |
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296 (REPEAT (ares_tac prems 1)) ]); |
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297 |
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298 val RepFun_cong = prove_goalw ZF.thy [RepFun_def] |
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299 "[| A=B; !!x. x:B ==> f(x)=g(x) |] ==> \ |
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300 \ {f(x). x:A} = {g(x). x:B}" |
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301 (fn prems=> [ (prove_cong_tac (prems@[Replace_cong]) 1) ]); |
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302 |
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303 |
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304 (*** Rules for Collect -- forming a subset by separation ***) |
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305 |
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306 (*Separation is derivable from Replacement*) |
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307 val separation = prove_goalw ZF.thy [Collect_def] |
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308 "a : {x:A. P(x)} <-> a:A & P(a)" |
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309 (fn _=> [ (fast_tac (FOL_cs addIs [bexI,ReplaceI] |
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310 addSEs [bexE,ReplaceE]) 1) ]); |
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311 |
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312 val CollectI = prove_goal ZF.thy |
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313 "[| a:A; P(a) |] ==> a : {x:A. P(x)}" |
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314 (fn prems=> |
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315 [ (rtac (separation RS iffD2) 1), |
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316 (REPEAT (resolve_tac (prems@[conjI]) 1)) ]); |
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317 |
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318 val CollectE = prove_goal ZF.thy |
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319 "[| a : {x:A. P(x)}; [| a:A; P(a) |] ==> R |] ==> R" |
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320 (fn prems=> |
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321 [ (rtac (separation RS iffD1 RS conjE) 1), |
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322 (REPEAT (ares_tac prems 1)) ]); |
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323 |
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324 val CollectD1 = prove_goal ZF.thy "a : {x:A. P(x)} ==> a:A" |
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325 (fn [major]=> |
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326 [ (rtac (major RS CollectE) 1), |
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327 (assume_tac 1) ]); |
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328 |
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329 val CollectD2 = prove_goal ZF.thy "a : {x:A. P(x)} ==> P(a)" |
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330 (fn [major]=> |
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331 [ (rtac (major RS CollectE) 1), |
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332 (assume_tac 1) ]); |
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333 |
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334 val Collect_cong = prove_goalw ZF.thy [Collect_def] |
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335 "[| A=B; !!x. x:B ==> P(x) <-> Q(x) |] ==> \ |
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336 \ {x:A. P(x)} = {x:B. Q(x)}" |
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337 (fn prems=> [ (prove_cong_tac (prems@[Replace_cong]) 1) ]); |
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338 |
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339 (*** Rules for Unions ***) |
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340 |
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341 (*The order of the premises presupposes that C is rigid; A may be flexible*) |
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342 val UnionI = prove_goal ZF.thy "[| B: C; A: B |] ==> A: Union(C)" |
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343 (fn prems=> |
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344 [ (resolve_tac [union_iff RS iffD2] 1), |
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345 (REPEAT (resolve_tac (prems @ [bexI]) 1)) ]); |
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346 |
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347 val UnionE = prove_goal ZF.thy |
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348 "[| A : Union(C); !!B.[| A: B; B: C |] ==> R |] ==> R" |
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349 (fn prems=> |
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350 [ (resolve_tac [union_iff RS iffD1 RS bexE] 1), |
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351 (REPEAT (ares_tac prems 1)) ]); |
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352 |
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353 (*** Rules for Inter ***) |
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354 |
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355 (*Not obviously useful towards proving InterI, InterD, InterE*) |
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356 val Inter_iff = prove_goalw ZF.thy [Inter_def,Ball_def] |
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357 "A : Inter(C) <-> (ALL x:C. A: x) & (EX x. x:C)" |
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358 (fn _=> [ (rtac (separation RS iff_trans) 1), |
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359 (fast_tac (FOL_cs addIs [UnionI] addSEs [UnionE]) 1) ]); |
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360 |
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361 (* Intersection is well-behaved only if the family is non-empty! *) |
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362 val InterI = prove_goalw ZF.thy [Inter_def] |
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363 "[| !!x. x: C ==> A: x; c:C |] ==> A : Inter(C)" |
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364 (fn prems=> |
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365 [ (DEPTH_SOLVE (ares_tac ([CollectI,UnionI,ballI] @ prems) 1)) ]); |
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366 |
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367 (*A "destruct" rule -- every B in C contains A as an element, but |
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368 A:B can hold when B:C does not! This rule is analogous to "spec". *) |
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369 val InterD = prove_goalw ZF.thy [Inter_def] |
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370 "[| A : Inter(C); B : C |] ==> A : B" |
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371 (fn [major,minor]=> |
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372 [ (rtac (major RS CollectD2 RS bspec) 1), |
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373 (rtac minor 1) ]); |
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374 |
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375 (*"Classical" elimination rule -- does not require exhibiting B:C *) |
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376 val InterE = prove_goalw ZF.thy [Inter_def] |
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377 "[| A : Inter(C); A:B ==> R; ~ B:C ==> R |] ==> R" |
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378 (fn major::prems=> |
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379 [ (rtac (major RS CollectD2 RS ballE) 1), |
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380 (REPEAT (eresolve_tac prems 1)) ]); |
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381 |
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382 (*** Rules for Unions of families ***) |
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383 (* UN x:A. B(x) abbreviates Union({B(x). x:A}) *) |
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384 |
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385 (*The order of the premises presupposes that A is rigid; b may be flexible*) |
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386 val UN_I = prove_goal ZF.thy "[| a: A; b: B(a) |] ==> b: (UN x:A. B(x))" |
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387 (fn prems=> |
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388 [ (REPEAT (resolve_tac (prems@[UnionI,RepFunI]) 1)) ]); |
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389 |
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390 val UN_E = prove_goal ZF.thy |
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391 "[| b : (UN x:A. B(x)); !!x.[| x: A; b: B(x) |] ==> R |] ==> R" |
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392 (fn major::prems=> |
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393 [ (rtac (major RS UnionE) 1), |
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394 (REPEAT (eresolve_tac (prems@[asm_rl, RepFunE, subst]) 1)) ]); |
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395 |
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396 |
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397 (*** Rules for Intersections of families ***) |
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398 (* INT x:A. B(x) abbreviates Inter({B(x). x:A}) *) |
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399 |
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400 val INT_I = prove_goal ZF.thy |
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401 "[| !!x. x: A ==> b: B(x); a: A |] ==> b: (INT x:A. B(x))" |
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402 (fn prems=> |
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403 [ (REPEAT (ares_tac (prems@[InterI,RepFunI]) 1 |
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404 ORELSE eresolve_tac [RepFunE,ssubst] 1)) ]); |
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405 |
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406 val INT_E = prove_goal ZF.thy |
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407 "[| b : (INT x:A. B(x)); a: A |] ==> b : B(a)" |
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408 (fn [major,minor]=> |
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409 [ (rtac (major RS InterD) 1), |
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410 (rtac (minor RS RepFunI) 1) ]); |
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411 |
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412 |
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413 (*** Rules for Powersets ***) |
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414 |
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415 val PowI = prove_goal ZF.thy "A <= B ==> A : Pow(B)" |
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416 (fn [prem]=> [ (rtac (prem RS (power_set RS iffD2)) 1) ]); |
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417 |
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418 val PowD = prove_goal ZF.thy "A : Pow(B) ==> A<=B" |
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419 (fn [major]=> [ (rtac (major RS (power_set RS iffD1)) 1) ]); |
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420 |
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421 |
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422 (*** Rules for the empty set ***) |
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423 |
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424 (*The set {x:0.False} is empty; by foundation it equals 0 |
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425 See Suppes, page 21.*) |
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426 val emptyE = prove_goal ZF.thy "a:0 ==> P" |
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427 (fn [major]=> |
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428 [ (rtac (foundation RS disjE) 1), |
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429 (etac (equalityD2 RS subsetD RS CollectD2 RS FalseE) 1), |
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430 (rtac major 1), |
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431 (etac bexE 1), |
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432 (etac (CollectD2 RS FalseE) 1) ]); |
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433 |
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434 val empty_subsetI = prove_goal ZF.thy "0 <= A" |
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435 (fn _ => [ (REPEAT (ares_tac [equalityI,subsetI,emptyE] 1)) ]); |
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436 |
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437 val equals0I = prove_goal ZF.thy "[| !!y. y:A ==> False |] ==> A=0" |
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438 (fn prems=> |
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439 [ (REPEAT (ares_tac (prems@[empty_subsetI,subsetI,equalityI]) 1 |
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440 ORELSE eresolve_tac (prems RL [FalseE]) 1)) ]); |
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441 |
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442 val equals0D = prove_goal ZF.thy "[| A=0; a:A |] ==> P" |
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443 (fn [major,minor]=> |
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444 [ (rtac (minor RS (major RS equalityD1 RS subsetD RS emptyE)) 1) ]); |
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445 |
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446 val lemmas_cs = FOL_cs |
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447 addSIs [ballI, InterI, CollectI, PowI, subsetI] |
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448 addIs [bexI, UnionI, ReplaceI, RepFunI] |
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449 addSEs [bexE, make_elim PowD, UnionE, ReplaceE, RepFunE, |
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450 CollectE, emptyE] |
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451 addEs [rev_ballE, InterD, make_elim InterD, subsetD, subsetCE]; |
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452 |
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453 end; |
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454 |
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455 open ZF_Lemmas; |