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1 (* Title: FOL/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 1991 University of Cambridge |
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5 |
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6 First-Order Logic: quantifier examples (intuitionistic and classical) |
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7 Needs declarations of the theory "thy" and the tactic "tac" |
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8 *) |
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9 |
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10 writeln"File FOL/ex/quant."; |
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11 |
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12 goal thy "?p : (ALL x y.P(x,y)) --> (ALL y x.P(x,y))"; |
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13 by tac; |
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14 result(); |
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15 |
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16 |
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17 goal thy "?p : (EX x y.P(x,y)) --> (EX y x.P(x,y))"; |
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18 by tac; |
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19 result(); |
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20 |
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21 |
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22 (*Converse is false*) |
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23 goal thy "?p : (ALL x.P(x)) | (ALL x.Q(x)) --> (ALL x. P(x) | Q(x))"; |
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24 by tac; |
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25 result(); |
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26 |
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27 goal thy "?p : (ALL x. P-->Q(x)) <-> (P--> (ALL x.Q(x)))"; |
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28 by tac; |
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29 result(); |
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30 |
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31 |
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32 goal thy "?p : (ALL x.P(x)-->Q) <-> ((EX x.P(x)) --> Q)"; |
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33 by tac; |
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34 result(); |
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35 |
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36 |
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37 writeln"Some harder ones"; |
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38 |
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39 goal thy "?p : (EX x. P(x) | Q(x)) <-> (EX x.P(x)) | (EX x.Q(x))"; |
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40 by tac; |
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41 result(); |
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42 (*6 secs*) |
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43 |
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44 (*Converse is false*) |
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45 goal thy "?p : (EX x. P(x)&Q(x)) --> (EX x.P(x)) & (EX x.Q(x))"; |
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46 by tac; |
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47 result(); |
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48 |
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49 |
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50 writeln"Basic test of quantifier reasoning"; |
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51 (*TRUE*) |
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52 goal thy "?p : (EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))"; |
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53 by tac; |
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54 result(); |
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55 |
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56 |
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57 goal thy "?p : (ALL x. Q(x)) --> (EX x. Q(x))"; |
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58 by tac; |
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59 result(); |
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60 |
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61 |
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62 writeln"The following should fail, as they are false!"; |
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63 |
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64 goal thy "?p : (ALL x. EX y. Q(x,y)) --> (EX y. ALL x. Q(x,y))"; |
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65 by tac handle ERROR => writeln"Failed, as expected"; |
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66 (*Check that subgoals remain: proof failed.*) |
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67 getgoal 1; |
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68 |
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69 goal thy "?p : (EX x. Q(x)) --> (ALL x. Q(x))"; |
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70 by tac handle ERROR => writeln"Failed, as expected"; |
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71 getgoal 1; |
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72 |
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73 goal thy "?p : P(?a) --> (ALL x.P(x))"; |
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74 by tac handle ERROR => writeln"Failed, as expected"; |
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75 (*Check that subgoals remain: proof failed.*) |
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76 getgoal 1; |
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77 |
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78 goal thy |
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79 "?p : (P(?a) --> (ALL x.Q(x))) --> (ALL x. P(x) --> Q(x))"; |
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80 by tac handle ERROR => writeln"Failed, as expected"; |
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81 getgoal 1; |
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82 |
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83 |
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84 writeln"Back to things that are provable..."; |
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85 |
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86 goal thy "?p : (ALL x.P(x)-->Q(x)) & (EX x.P(x)) --> (EX x.Q(x))"; |
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87 by tac; |
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88 result(); |
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89 |
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90 |
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91 (*An example of why exI should be delayed as long as possible*) |
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92 goal thy "?p : (P --> (EX x.Q(x))) & P --> (EX x.Q(x))"; |
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93 by tac; |
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94 result(); |
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95 |
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96 goal thy "?p : (ALL x. P(x)-->Q(f(x))) & (ALL x. Q(x)-->R(g(x))) & P(d) --> R(?a)"; |
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97 by tac; |
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98 (*Verify that no subgoals remain.*) |
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99 uresult(); |
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100 |
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101 |
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102 goal thy "?p : (ALL x. Q(x)) --> (EX x. Q(x))"; |
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103 by tac; |
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104 result(); |
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105 |
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106 |
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107 writeln"Some slow ones"; |
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108 |
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109 |
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110 (*Principia Mathematica *11.53 *) |
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111 goal thy "?p : (ALL x y. P(x) --> Q(y)) <-> ((EX x. P(x)) --> (ALL y. Q(y)))"; |
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112 by tac; |
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113 result(); |
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114 (*6 secs*) |
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115 |
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116 (*Principia Mathematica *11.55 *) |
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117 goal thy "?p : (EX x y. P(x) & Q(x,y)) <-> (EX x. P(x) & (EX y. Q(x,y)))"; |
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118 by tac; |
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119 result(); |
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120 (*9 secs*) |
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121 |
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122 (*Principia Mathematica *11.61 *) |
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123 goal thy "?p : (EX y. ALL x. P(x) --> Q(x,y)) --> (ALL x. P(x) --> (EX y. Q(x,y)))"; |
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124 by tac; |
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125 result(); |
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126 (*3 secs*) |
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127 |
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128 writeln"Reached end of file."; |
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129 |