1 (* Title: LK/ex/quant |
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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 1992 University of Cambridge |
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5 |
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6 Classical sequent calculus: examples with quantifiers. |
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7 *) |
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8 |
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9 |
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10 writeln"LK/ex/quant: Examples with quantifiers"; |
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11 |
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12 goal LK.thy "|- (ALL x. P) <-> P"; |
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13 by (fast_tac LK_pack 1); |
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14 result(); |
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15 |
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16 goal LK.thy "|- (ALL x y.P(x,y)) <-> (ALL y x.P(x,y))"; |
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17 by (fast_tac LK_pack 1); |
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18 result(); |
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19 |
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20 goal LK.thy "ALL u.P(u), ALL v.Q(v) |- ALL u v. P(u) & Q(v)"; |
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21 by (fast_tac LK_pack 1); |
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22 result(); |
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23 |
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24 writeln"Permutation of existential quantifier."; |
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25 goal LK.thy "|- (EX x y.P(x,y)) <-> (EX y x.P(x,y))"; |
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26 by (fast_tac LK_pack 1); |
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27 result(); |
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28 |
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29 goal LK.thy "|- (ALL x. P(x) & Q(x)) <-> (ALL x.P(x)) & (ALL x.Q(x))"; |
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30 by (fast_tac LK_pack 1); |
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31 result(); |
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32 |
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33 |
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34 (*Converse is invalid*) |
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35 goal LK.thy "|- (ALL x. P(x)) | (ALL x. Q(x)) --> (ALL x. P(x)|Q(x))"; |
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36 by (fast_tac LK_pack 1); |
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37 result(); |
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38 |
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39 |
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40 writeln"Pushing ALL into an implication."; |
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41 goal LK.thy "|- (ALL x. P --> Q(x)) <-> (P --> (ALL x. Q(x)))"; |
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42 by (fast_tac LK_pack 1); |
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43 result(); |
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44 |
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45 |
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46 goal LK.thy "|- (ALL x.P(x)-->Q) <-> ((EX x.P(x)) --> Q)"; |
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47 by (fast_tac LK_pack 1); |
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48 result(); |
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49 |
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50 |
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51 goal LK.thy "|- (EX x. P) <-> P"; |
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52 by (fast_tac LK_pack 1); |
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53 result(); |
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54 |
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55 |
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56 writeln"Distribution of EX over disjunction."; |
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57 goal LK.thy "|- (EX x. P(x) | Q(x)) <-> (EX x. P(x)) | (EX x. Q(x))"; |
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58 by (fast_tac LK_pack 1); |
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59 result(); |
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60 (*5 secs*) |
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61 |
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62 (*Converse is invalid*) |
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63 goal LK.thy "|- (EX x. P(x) & Q(x)) --> (EX x. P(x)) & (EX x. Q(x))"; |
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64 by (fast_tac LK_pack 1); |
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65 result(); |
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66 |
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67 |
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68 writeln"Harder examples: classical theorems."; |
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69 |
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70 goal LK.thy "|- (EX x. P-->Q(x)) <-> (P --> (EX x.Q(x)))"; |
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71 by (fast_tac LK_pack 1); |
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72 result(); |
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73 (*3 secs*) |
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74 |
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75 |
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76 goal LK.thy "|- (EX x.P(x)-->Q) <-> (ALL x.P(x)) --> Q"; |
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77 by (fast_tac LK_pack 1); |
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78 result(); |
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79 (*5 secs*) |
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80 |
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81 |
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82 goal LK.thy "|- (ALL x.P(x)) | Q <-> (ALL x. P(x) | Q)"; |
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83 by (fast_tac LK_pack 1); |
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84 result(); |
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85 |
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86 |
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87 writeln"Basic test of quantifier reasoning"; |
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88 goal LK.thy |
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89 "|- (EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))"; |
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90 by (fast_tac LK_pack 1); |
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91 result(); |
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92 |
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93 |
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94 goal LK.thy "|- (ALL x. Q(x)) --> (EX x. Q(x))"; |
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95 by (fast_tac LK_pack 1); |
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96 result(); |
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97 |
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98 |
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99 writeln"The following are invalid!"; |
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100 |
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101 (*INVALID*) |
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102 goal LK.thy "|- (ALL x. EX y. Q(x,y)) --> (EX y. ALL x. Q(x,y))"; |
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103 by (fast_tac LK_pack 1) handle ERROR => writeln"Failed, as expected"; |
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104 (*Check that subgoals remain: proof failed.*) |
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105 getgoal 1; |
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106 |
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107 (*INVALID*) |
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108 goal LK.thy "|- (EX x. Q(x)) --> (ALL x. Q(x))"; |
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109 by (fast_tac LK_pack 1) handle ERROR => writeln"Failed, as expected"; |
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110 getgoal 1; |
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111 |
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112 goal LK.thy "|- P(?a) --> (ALL x.P(x))"; |
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113 by (fast_tac LK_pack 1) handle ERROR => writeln"Failed, as expected"; |
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114 (*Check that subgoals remain: proof failed.*) |
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115 getgoal 1; |
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116 |
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117 goal LK.thy "|- (P(?a) --> (ALL x.Q(x))) --> (ALL x. P(x) --> Q(x))"; |
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118 by (fast_tac LK_pack 1) handle ERROR => writeln"Failed, as expected"; |
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119 getgoal 1; |
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120 |
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121 |
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122 writeln"Back to things that are provable..."; |
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123 |
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124 goal LK.thy "|- (ALL x. P(x)-->Q(x)) & (EX x.P(x)) --> (EX x.Q(x))"; |
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125 by (fast_tac LK_pack 1); |
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126 result(); |
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127 |
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128 (*An example of why exR should be delayed as long as possible*) |
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129 goal LK.thy "|- (P--> (EX x.Q(x))) & P--> (EX x.Q(x))"; |
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130 by (fast_tac LK_pack 1); |
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131 result(); |
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132 |
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133 writeln"Solving for a Var"; |
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134 goal LK.thy |
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135 "|- (ALL x. P(x)-->Q(f(x))) & (ALL x. Q(x)-->R(g(x))) & P(d) --> R(?a)"; |
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136 by (fast_tac LK_pack 1); |
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137 result(); |
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138 |
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139 |
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140 writeln"Principia Mathematica *11.53"; |
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141 goal LK.thy |
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142 "|- (ALL x y. P(x) --> Q(y)) <-> ((EX x. P(x)) --> (ALL y. Q(y)))"; |
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143 by (fast_tac LK_pack 1); |
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144 result(); |
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145 |
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146 |
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147 writeln"Principia Mathematica *11.55"; |
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148 goal LK.thy "|- (EX x y. P(x) & Q(x,y)) <-> (EX x. P(x) & (EX y.Q(x,y)))"; |
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149 by (fast_tac LK_pack 1); |
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150 result(); |
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151 |
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152 writeln"Principia Mathematica *11.61"; |
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153 goal LK.thy |
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154 "|- (EX y. ALL x. P(x) --> Q(x,y)) --> (ALL x. P(x) --> (EX y. Q(x,y)))"; |
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155 by (fast_tac LK_pack 1); |
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156 result(); |
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157 |
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158 writeln"Reached end of file."; |
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159 |
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160 (*21 August 88: loaded in 45.7 secs*) |
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