1 (* Title: LK/ex/hard-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 Hard examples with quantifiers. Can be read to test the LK system. |
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7 From F. J. Pelletier, |
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8 Seventy-Five Problems for Testing Automatic Theorem Provers, |
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9 J. Automated Reasoning 2 (1986), 191-216. |
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10 Errata, JAR 4 (1988), 236-236. |
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11 |
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12 Uses pc_tac rather than fast_tac when the former is significantly faster. |
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13 *) |
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14 |
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15 writeln"File LK/ex/hard-quant."; |
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16 |
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17 goal LK.thy "|- (ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))"; |
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18 by (fast_tac LK_pack 1); |
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19 result(); |
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20 |
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21 goal LK.thy "|- (EX x. P-->Q(x)) <-> (P --> (EX x.Q(x)))"; |
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22 by (fast_tac LK_pack 1); |
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23 result(); |
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24 |
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25 goal LK.thy "|- (EX x.P(x)-->Q) <-> (ALL x.P(x)) --> Q"; |
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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 <-> (ALL x. P(x) | Q)"; |
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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 writeln"Problems requiring quantifier duplication"; |
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34 |
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35 (*Not provable by fast_tac LK_pack: needs multiple instantiation of ALL*) |
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36 goal LK.thy "|- (ALL x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))"; |
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37 by (best_tac LK_dup_pack 1); |
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38 result(); |
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39 |
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40 (*Needs double instantiation of the quantifier*) |
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41 goal LK.thy "|- EX x. P(x) --> P(a) & P(b)"; |
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42 by (fast_tac LK_dup_pack 1); |
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43 result(); |
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44 |
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45 goal LK.thy "|- EX z. P(z) --> (ALL x. P(x))"; |
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46 by (best_tac LK_dup_pack 1); |
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47 result(); |
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48 |
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49 writeln"Hard examples with quantifiers"; |
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50 |
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51 writeln"Problem 18"; |
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52 goal LK.thy "|- EX y. ALL x. P(y)-->P(x)"; |
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53 by (best_tac LK_dup_pack 1); |
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54 result(); |
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55 |
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56 writeln"Problem 19"; |
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57 goal LK.thy "|- EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"; |
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58 by (best_tac LK_dup_pack 1); |
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59 result(); |
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60 |
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61 writeln"Problem 20"; |
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62 goal LK.thy "|- (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w))) \ |
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63 \ --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))"; |
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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 writeln"Problem 21"; |
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68 goal LK.thy "|- (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))"; |
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69 by (best_tac LK_dup_pack 1); |
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70 result(); |
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71 |
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72 writeln"Problem 22"; |
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73 goal LK.thy "|- (ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))"; |
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74 by (fast_tac LK_pack 1); |
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75 result(); |
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76 |
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77 writeln"Problem 23"; |
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78 goal LK.thy "|- (ALL x. P | Q(x)) <-> (P | (ALL x. Q(x)))"; |
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79 by (best_tac LK_pack 1); |
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80 result(); |
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81 |
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82 writeln"Problem 24"; |
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83 goal LK.thy "|- ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) & \ |
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84 \ ~(EX x.P(x)) --> (EX x.Q(x)) & (ALL x. Q(x)|R(x) --> S(x)) \ |
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85 \ --> (EX x. P(x)&R(x))"; |
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86 by (pc_tac LK_pack 1); |
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87 result(); |
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88 |
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89 writeln"Problem 25"; |
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90 goal LK.thy "|- (EX x. P(x)) & \ |
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91 \ (ALL x. L(x) --> ~ (M(x) & R(x))) & \ |
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92 \ (ALL x. P(x) --> (M(x) & L(x))) & \ |
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93 \ ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x))) \ |
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94 \ --> (EX x. Q(x)&P(x))"; |
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95 by (best_tac LK_pack 1); |
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96 result(); |
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97 |
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98 writeln"Problem 26"; |
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99 goal LK.thy "|- ((EX x. p(x)) <-> (EX x. q(x))) & \ |
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100 \ (ALL x. ALL y. p(x) & q(y) --> (r(x) <-> s(y))) \ |
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101 \ --> ((ALL x. p(x)-->r(x)) <-> (ALL x. q(x)-->s(x)))"; |
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102 by (pc_tac LK_pack 1); |
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103 result(); |
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104 |
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105 writeln"Problem 27"; |
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106 goal LK.thy "|- (EX x. P(x) & ~Q(x)) & \ |
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107 \ (ALL x. P(x) --> R(x)) & \ |
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108 \ (ALL x. M(x) & L(x) --> P(x)) & \ |
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109 \ ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x))) \ |
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110 \ --> (ALL x. M(x) --> ~L(x))"; |
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111 by (pc_tac LK_pack 1); |
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112 result(); |
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113 |
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114 writeln"Problem 28. AMENDED"; |
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115 goal LK.thy "|- (ALL x. P(x) --> (ALL x. Q(x))) & \ |
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116 \ ((ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) & \ |
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117 \ ((EX x.S(x)) --> (ALL x. L(x) --> M(x))) \ |
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118 \ --> (ALL x. P(x) & L(x) --> M(x))"; |
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119 by (pc_tac LK_pack 1); |
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120 result(); |
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121 |
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122 writeln"Problem 29. Essentially the same as Principia Mathematica *11.71"; |
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123 goal LK.thy "|- (EX x. P(x)) & (EX y. Q(y)) \ |
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124 \ --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y)) <-> \ |
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125 \ (ALL x y. P(x) & Q(y) --> R(x) & S(y)))"; |
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126 by (pc_tac LK_pack 1); |
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127 result(); |
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128 |
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129 writeln"Problem 30"; |
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130 goal LK.thy "|- (ALL x. P(x) | Q(x) --> ~ R(x)) & \ |
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131 \ (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \ |
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132 \ --> (ALL x. S(x))"; |
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133 by (fast_tac LK_pack 1); |
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134 result(); |
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135 |
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136 writeln"Problem 31"; |
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137 goal LK.thy "|- ~(EX x.P(x) & (Q(x) | R(x))) & \ |
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138 \ (EX x. L(x) & P(x)) & \ |
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139 \ (ALL x. ~ R(x) --> M(x)) \ |
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140 \ --> (EX x. L(x) & M(x))"; |
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141 by (fast_tac LK_pack 1); |
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142 result(); |
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143 |
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144 writeln"Problem 32"; |
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145 goal LK.thy "|- (ALL x. P(x) & (Q(x)|R(x))-->S(x)) & \ |
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146 \ (ALL x. S(x) & R(x) --> L(x)) & \ |
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147 \ (ALL x. M(x) --> R(x)) \ |
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148 \ --> (ALL x. P(x) & M(x) --> L(x))"; |
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149 by (best_tac LK_pack 1); |
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150 result(); |
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151 |
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152 writeln"Problem 33"; |
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153 goal LK.thy "|- (ALL x. P(a) & (P(x)-->P(b))-->P(c)) <-> \ |
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154 \ (ALL x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))"; |
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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"Problem 34 AMENDED (TWICE!!) NOT PROVED AUTOMATICALLY"; |
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159 (*Andrews's challenge*) |
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160 goal LK.thy "|- ((EX x. ALL y. p(x) <-> p(y)) <-> \ |
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161 \ ((EX x. q(x)) <-> (ALL y. p(y)))) <-> \ |
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162 \ ((EX x. ALL y. q(x) <-> q(y)) <-> \ |
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163 \ ((EX x. p(x)) <-> (ALL y. q(y))))"; |
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164 by (safe_goal_tac LK_pack 1); (*53 secs*) (*13 secs*) |
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165 by (TRYALL (fast_tac LK_pack)); (*165 secs*) (*117 secs*) (*138 secs*) |
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166 (*for some reason, pc_tac leaves 14 subgoals instead of 6*) |
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167 by (TRYALL (best_tac LK_dup_pack)); (*55 secs*) (*29 secs*) (*54 secs*) |
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168 result(); |
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169 |
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170 |
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171 writeln"Problem 35"; |
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172 goal LK.thy "|- EX x y. P(x,y) --> (ALL u v. P(u,v))"; |
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173 by (best_tac LK_dup_pack 1); (*27 secs??*) |
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174 result(); |
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175 |
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176 writeln"Problem 36"; |
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177 goal LK.thy "|- (ALL x. EX y. J(x,y)) & \ |
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178 \ (ALL x. EX y. G(x,y)) & \ |
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179 \ (ALL x y. J(x,y) | G(x,y) --> \ |
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180 \ (ALL z. J(y,z) | G(y,z) --> H(x,z))) \ |
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181 \ --> (ALL x. EX y. H(x,y))"; |
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182 by (fast_tac LK_pack 1); |
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183 result(); |
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184 |
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185 writeln"Problem 37"; |
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186 goal LK.thy "|- (ALL z. EX w. ALL x. EX y. \ |
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187 \ (P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (EX u.Q(u,w)))) & \ |
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188 \ (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) & \ |
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189 \ ((EX x y. Q(x,y)) --> (ALL x. R(x,x))) \ |
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190 \ --> (ALL x. EX y. R(x,y))"; |
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191 by (pc_tac LK_pack 1); (*slow*) |
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192 by flexflex_tac; |
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193 result(); |
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194 |
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195 writeln"Problem 38. NOT PROVED"; |
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196 goal LK.thy |
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197 "|- (ALL x. p(a) & (p(x) --> (EX y. p(y) & r(x,y))) --> \ |
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198 \ (EX z. EX w. p(z) & r(x,w) & r(w,z))) <-> \ |
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199 \ (ALL x. (~p(a) | p(x) | (EX z. EX w. p(z) & r(x,w) & r(w,z))) & \ |
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200 \ (~p(a) | ~(EX y. p(y) & r(x,y)) | \ |
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201 \ (EX z. EX w. p(z) & r(x,w) & r(w,z))))"; |
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202 |
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203 writeln"Problem 39"; |
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204 goal LK.thy "|- ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"; |
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205 by (fast_tac LK_pack 1); |
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206 result(); |
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207 |
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208 writeln"Problem 40. AMENDED"; |
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209 goal LK.thy "|- (EX y. ALL x. F(x,y) <-> F(x,x)) --> \ |
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210 \ ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))"; |
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211 by (fast_tac LK_pack 1); |
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212 result(); |
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213 |
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214 writeln"Problem 41"; |
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215 goal LK.thy "|- (ALL z. EX y. ALL x. f(x,y) <-> f(x,z) & ~ f(x,x)) \ |
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216 \ --> ~ (EX z. ALL x. f(x,z))"; |
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217 by (fast_tac LK_pack 1); |
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218 result(); |
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219 |
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220 writeln"Problem 42"; |
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221 goal LK.thy "|- ~ (EX y. ALL x. p(x,y) <-> ~ (EX z. p(x,z) & p(z,x)))"; |
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222 |
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223 writeln"Problem 43 NOT PROVED AUTOMATICALLY"; |
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224 goal LK.thy "|- (ALL x. ALL y. q(x,y) <-> (ALL z. p(z,x) <-> p(z,y))) \ |
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225 \ --> (ALL x. (ALL y. q(x,y) <-> q(y,x)))"; |
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226 |
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227 writeln"Problem 44"; |
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228 goal LK.thy "|- (ALL x. f(x) --> \ |
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229 \ (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y)))) & \ |
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230 \ (EX x. j(x) & (ALL y. g(y) --> h(x,y))) \ |
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231 \ --> (EX x. j(x) & ~f(x))"; |
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232 by (fast_tac LK_pack 1); |
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233 result(); |
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234 |
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235 writeln"Problem 45"; |
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236 goal LK.thy "|- (ALL x. f(x) & (ALL y. g(y) & h(x,y) --> j(x,y)) \ |
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237 \ --> (ALL y. g(y) & h(x,y) --> k(y))) & \ |
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238 \ ~ (EX y. l(y) & k(y)) & \ |
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239 \ (EX x. f(x) & (ALL y. h(x,y) --> l(y)) \ |
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240 \ & (ALL y. g(y) & h(x,y) --> j(x,y))) \ |
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241 \ --> (EX x. f(x) & ~ (EX y. g(y) & h(x,y)))"; |
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242 by (best_tac LK_pack 1); |
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243 result(); |
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244 |
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245 |
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246 writeln"Problem 50"; |
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247 goal LK.thy |
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248 "|- (ALL x. P(a,x) | (ALL y.P(x,y))) --> (EX x. ALL y.P(x,y))"; |
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249 by (best_tac LK_dup_pack 1); |
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250 result(); |
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251 |
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252 writeln"Problem 57"; |
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253 goal LK.thy |
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254 "|- P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \ |
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255 \ (ALL x y z. P(x,y) & P(y,z) --> P(x,z)) --> P(f(a,b), f(a,c))"; |
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256 by (fast_tac LK_pack 1); |
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257 result(); |
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258 |
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259 writeln"Problem 59"; |
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260 (*Unification works poorly here -- the abstraction %sobj prevents efficient |
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261 operation of the occurs check*) |
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262 Unify.trace_bound := !Unify.search_bound - 1; |
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263 goal LK.thy "|- (ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))"; |
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264 by (best_tac LK_dup_pack 1); |
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265 result(); |
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266 |
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267 writeln"Problem 60"; |
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268 goal LK.thy |
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269 "|- ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"; |
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270 by (fast_tac LK_pack 1); |
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271 result(); |
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272 |
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273 writeln"Reached end of file."; |
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274 |
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275 (*18 June 92: loaded in 372 secs*) |
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276 (*19 June 92: loaded in 166 secs except #34, using repeat_goal_tac*) |
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277 (*29 June 92: loaded in 370 secs*) |
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