1 (* Title: FOLP/ex/cla |
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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 Classical First-Order Logic. |
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7 *) |
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8 |
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9 goal (theory "FOLP") "?p : (P --> Q | R) --> (P-->Q) | (P-->R)"; |
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10 by (fast_tac FOLP_cs 1); |
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11 result(); |
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12 |
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13 (*If and only if*) |
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14 |
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15 goal (theory "FOLP") "?p : (P<->Q) <-> (Q<->P)"; |
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16 by (fast_tac FOLP_cs 1); |
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17 result(); |
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18 |
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19 goal (theory "FOLP") "?p : ~ (P <-> ~P)"; |
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20 by (fast_tac FOLP_cs 1); |
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21 result(); |
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22 |
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23 |
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24 (*Sample problems from |
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25 F. J. Pelletier, |
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26 Seventy-Five Problems for Testing Automatic Theorem Provers, |
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27 J. Automated Reasoning 2 (1986), 191-216. |
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28 Errata, JAR 4 (1988), 236-236. |
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29 |
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30 The hardest problems -- judging by experience with several theorem provers, |
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31 including matrix ones -- are 34 and 43. |
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32 *) |
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33 |
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34 writeln"Pelletier's examples"; |
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35 (*1*) |
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36 goal (theory "FOLP") "?p : (P-->Q) <-> (~Q --> ~P)"; |
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37 by (fast_tac FOLP_cs 1); |
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38 result(); |
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39 |
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40 (*2*) |
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41 goal (theory "FOLP") "?p : ~ ~ P <-> P"; |
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42 by (fast_tac FOLP_cs 1); |
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43 result(); |
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44 |
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45 (*3*) |
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46 goal (theory "FOLP") "?p : ~(P-->Q) --> (Q-->P)"; |
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47 by (fast_tac FOLP_cs 1); |
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48 result(); |
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49 |
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50 (*4*) |
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51 goal (theory "FOLP") "?p : (~P-->Q) <-> (~Q --> P)"; |
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52 by (fast_tac FOLP_cs 1); |
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53 result(); |
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54 |
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55 (*5*) |
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56 goal (theory "FOLP") "?p : ((P|Q)-->(P|R)) --> (P|(Q-->R))"; |
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57 by (fast_tac FOLP_cs 1); |
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58 result(); |
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59 |
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60 (*6*) |
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61 goal (theory "FOLP") "?p : P | ~ P"; |
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62 by (fast_tac FOLP_cs 1); |
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63 result(); |
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64 |
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65 (*7*) |
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66 goal (theory "FOLP") "?p : P | ~ ~ ~ P"; |
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67 by (fast_tac FOLP_cs 1); |
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68 result(); |
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69 |
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70 (*8. Peirce's law*) |
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71 goal (theory "FOLP") "?p : ((P-->Q) --> P) --> P"; |
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72 by (fast_tac FOLP_cs 1); |
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73 result(); |
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74 |
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75 (*9*) |
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76 goal (theory "FOLP") "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"; |
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77 by (fast_tac FOLP_cs 1); |
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78 result(); |
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79 |
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80 (*10*) |
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81 goal (theory "FOLP") "?p : (Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P<->Q)"; |
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82 by (fast_tac FOLP_cs 1); |
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83 result(); |
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84 |
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85 (*11. Proved in each direction (incorrectly, says Pelletier!!) *) |
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86 goal (theory "FOLP") "?p : P<->P"; |
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87 by (fast_tac FOLP_cs 1); |
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88 result(); |
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89 |
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90 (*12. "Dijkstra's law"*) |
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91 goal (theory "FOLP") "?p : ((P <-> Q) <-> R) <-> (P <-> (Q <-> R))"; |
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92 by (fast_tac FOLP_cs 1); |
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93 result(); |
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94 |
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95 (*13. Distributive law*) |
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96 goal (theory "FOLP") "?p : P | (Q & R) <-> (P | Q) & (P | R)"; |
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97 by (fast_tac FOLP_cs 1); |
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98 result(); |
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99 |
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100 (*14*) |
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101 goal (theory "FOLP") "?p : (P <-> Q) <-> ((Q | ~P) & (~Q|P))"; |
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102 by (fast_tac FOLP_cs 1); |
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103 result(); |
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104 |
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105 (*15*) |
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106 goal (theory "FOLP") "?p : (P --> Q) <-> (~P | Q)"; |
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107 by (fast_tac FOLP_cs 1); |
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108 result(); |
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109 |
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110 (*16*) |
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111 goal (theory "FOLP") "?p : (P-->Q) | (Q-->P)"; |
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112 by (fast_tac FOLP_cs 1); |
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113 result(); |
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114 |
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115 (*17*) |
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116 goal (theory "FOLP") "?p : ((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))"; |
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117 by (fast_tac FOLP_cs 1); |
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118 result(); |
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119 |
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120 writeln"Classical Logic: examples with quantifiers"; |
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121 |
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122 goal (theory "FOLP") "?p : (ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))"; |
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123 by (fast_tac FOLP_cs 1); |
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124 result(); |
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125 |
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126 goal (theory "FOLP") "?p : (EX x. P-->Q(x)) <-> (P --> (EX x. Q(x)))"; |
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127 by (fast_tac FOLP_cs 1); |
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128 result(); |
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129 |
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130 goal (theory "FOLP") "?p : (EX x. P(x)-->Q) <-> (ALL x. P(x)) --> Q"; |
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131 by (fast_tac FOLP_cs 1); |
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132 result(); |
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133 |
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134 goal (theory "FOLP") "?p : (ALL x. P(x)) | Q <-> (ALL x. P(x) | Q)"; |
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135 by (fast_tac FOLP_cs 1); |
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136 result(); |
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137 |
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138 writeln"Problems requiring quantifier duplication"; |
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139 |
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140 (*Needs multiple instantiation of ALL.*) |
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141 (* |
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142 goal (theory "FOLP") "?p : (ALL x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))"; |
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143 by (best_tac FOLP_dup_cs 1); |
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144 result(); |
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145 *) |
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146 (*Needs double instantiation of the quantifier*) |
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147 goal (theory "FOLP") "?p : EX x. P(x) --> P(a) & P(b)"; |
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148 by (best_tac FOLP_dup_cs 1); |
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149 result(); |
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150 |
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151 goal (theory "FOLP") "?p : EX z. P(z) --> (ALL x. P(x))"; |
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152 by (best_tac FOLP_dup_cs 1); |
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153 result(); |
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154 |
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155 |
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156 writeln"Hard examples with quantifiers"; |
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157 |
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158 writeln"Problem 18"; |
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159 goal (theory "FOLP") "?p : EX y. ALL x. P(y)-->P(x)"; |
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160 by (best_tac FOLP_dup_cs 1); |
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161 result(); |
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162 |
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163 writeln"Problem 19"; |
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164 goal (theory "FOLP") "?p : EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"; |
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165 by (best_tac FOLP_dup_cs 1); |
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166 result(); |
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167 |
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168 writeln"Problem 20"; |
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169 goal (theory "FOLP") "?p : (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w))) \ |
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170 \ --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))"; |
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171 by (fast_tac FOLP_cs 1); |
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172 result(); |
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173 (* |
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174 writeln"Problem 21"; |
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175 goal (theory "FOLP") "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))"; |
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176 by (best_tac FOLP_dup_cs 1); |
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177 result(); |
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178 *) |
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179 writeln"Problem 22"; |
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180 goal (theory "FOLP") "?p : (ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))"; |
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181 by (fast_tac FOLP_cs 1); |
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182 result(); |
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183 |
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184 writeln"Problem 23"; |
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185 goal (theory "FOLP") "?p : (ALL x. P | Q(x)) <-> (P | (ALL x. Q(x)))"; |
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186 by (best_tac FOLP_cs 1); |
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187 result(); |
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188 |
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189 writeln"Problem 24"; |
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190 goal (theory "FOLP") "?p : ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) & \ |
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191 \ (~(EX x. P(x)) --> (EX x. Q(x))) & (ALL x. Q(x)|R(x) --> S(x)) \ |
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192 \ --> (EX x. P(x)&R(x))"; |
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193 by (fast_tac FOLP_cs 1); |
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194 result(); |
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195 (* |
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196 writeln"Problem 25"; |
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197 goal (theory "FOLP") "?p : (EX x. P(x)) & \ |
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198 \ (ALL x. L(x) --> ~ (M(x) & R(x))) & \ |
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199 \ (ALL x. P(x) --> (M(x) & L(x))) & \ |
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200 \ ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x))) \ |
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201 \ --> (EX x. Q(x)&P(x))"; |
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202 by (best_tac FOLP_cs 1); |
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203 result(); |
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204 |
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205 writeln"Problem 26"; |
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206 goal (theory "FOLP") "?u : ((EX x. p(x)) <-> (EX x. q(x))) & \ |
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207 \ (ALL x. ALL y. p(x) & q(y) --> (r(x) <-> s(y))) \ |
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208 \ --> ((ALL x. p(x)-->r(x)) <-> (ALL x. q(x)-->s(x)))"; |
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209 by (fast_tac FOLP_cs 1); |
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210 result(); |
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211 *) |
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212 writeln"Problem 27"; |
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213 goal (theory "FOLP") "?p : (EX x. P(x) & ~Q(x)) & \ |
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214 \ (ALL x. P(x) --> R(x)) & \ |
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215 \ (ALL x. M(x) & L(x) --> P(x)) & \ |
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216 \ ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x))) \ |
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217 \ --> (ALL x. M(x) --> ~L(x))"; |
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218 by (fast_tac FOLP_cs 1); |
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219 result(); |
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220 |
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221 writeln"Problem 28. AMENDED"; |
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222 goal (theory "FOLP") "?p : (ALL x. P(x) --> (ALL x. Q(x))) & \ |
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223 \ ((ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) & \ |
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224 \ ((EX x. S(x)) --> (ALL x. L(x) --> M(x))) \ |
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225 \ --> (ALL x. P(x) & L(x) --> M(x))"; |
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226 by (fast_tac FOLP_cs 1); |
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227 result(); |
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228 |
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229 writeln"Problem 29. Essentially the same as Principia Mathematica *11.71"; |
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230 goal (theory "FOLP") "?p : (EX x. P(x)) & (EX y. Q(y)) \ |
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231 \ --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y)) <-> \ |
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232 \ (ALL x y. P(x) & Q(y) --> R(x) & S(y)))"; |
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233 by (fast_tac FOLP_cs 1); |
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234 result(); |
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235 |
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236 writeln"Problem 30"; |
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237 goal (theory "FOLP") "?p : (ALL x. P(x) | Q(x) --> ~ R(x)) & \ |
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238 \ (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \ |
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239 \ --> (ALL x. S(x))"; |
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240 by (fast_tac FOLP_cs 1); |
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241 result(); |
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242 |
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243 writeln"Problem 31"; |
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244 goal (theory "FOLP") "?p : ~(EX x. P(x) & (Q(x) | R(x))) & \ |
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245 \ (EX x. L(x) & P(x)) & \ |
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246 \ (ALL x. ~ R(x) --> M(x)) \ |
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247 \ --> (EX x. L(x) & M(x))"; |
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248 by (fast_tac FOLP_cs 1); |
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249 result(); |
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250 |
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251 writeln"Problem 32"; |
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252 goal (theory "FOLP") "?p : (ALL x. P(x) & (Q(x)|R(x))-->S(x)) & \ |
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253 \ (ALL x. S(x) & R(x) --> L(x)) & \ |
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254 \ (ALL x. M(x) --> R(x)) \ |
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255 \ --> (ALL x. P(x) & M(x) --> L(x))"; |
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256 by (best_tac FOLP_cs 1); |
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257 result(); |
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258 |
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259 writeln"Problem 33"; |
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260 goal (theory "FOLP") "?p : (ALL x. P(a) & (P(x)-->P(b))-->P(c)) <-> \ |
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261 \ (ALL x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))"; |
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262 by (best_tac FOLP_cs 1); |
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263 result(); |
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264 |
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265 writeln"Problem 35"; |
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266 goal (theory "FOLP") "?p : EX x y. P(x,y) --> (ALL u v. P(u,v))"; |
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267 by (best_tac FOLP_dup_cs 1); |
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268 result(); |
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269 |
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270 writeln"Problem 36"; |
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271 goal (theory "FOLP") |
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272 "?p : (ALL x. EX y. J(x,y)) & \ |
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273 \ (ALL x. EX y. G(x,y)) & \ |
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274 \ (ALL x y. J(x,y) | G(x,y) --> (ALL z. J(y,z) | G(y,z) --> H(x,z))) \ |
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275 \ --> (ALL x. EX y. H(x,y))"; |
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276 by (fast_tac FOLP_cs 1); |
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277 result(); |
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278 |
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279 writeln"Problem 37"; |
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280 goal (theory "FOLP") "?p : (ALL z. EX w. ALL x. EX y. \ |
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281 \ (P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (EX u. Q(u,w)))) & \ |
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282 \ (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) & \ |
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283 \ ((EX x y. Q(x,y)) --> (ALL x. R(x,x))) \ |
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284 \ --> (ALL x. EX y. R(x,y))"; |
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285 by (fast_tac FOLP_cs 1); |
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286 result(); |
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287 |
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288 writeln"Problem 39"; |
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289 goal (theory "FOLP") "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"; |
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290 by (fast_tac FOLP_cs 1); |
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291 result(); |
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292 |
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293 writeln"Problem 40. AMENDED"; |
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294 goal (theory "FOLP") "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) --> \ |
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295 \ ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))"; |
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296 by (fast_tac FOLP_cs 1); |
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297 result(); |
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298 |
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299 writeln"Problem 41"; |
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300 goal (theory "FOLP") "?p : (ALL z. EX y. ALL x. f(x,y) <-> f(x,z) & ~ f(x,x)) \ |
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301 \ --> ~ (EX z. ALL x. f(x,z))"; |
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302 by (best_tac FOLP_cs 1); |
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303 result(); |
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304 |
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305 writeln"Problem 44"; |
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306 goal (theory "FOLP") "?p : (ALL x. f(x) --> \ |
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307 \ (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y)))) & \ |
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308 \ (EX x. j(x) & (ALL y. g(y) --> h(x,y))) \ |
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309 \ --> (EX x. j(x) & ~f(x))"; |
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310 by (fast_tac FOLP_cs 1); |
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311 result(); |
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312 |
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313 writeln"Problems (mainly) involving equality or functions"; |
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314 |
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315 writeln"Problem 48"; |
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316 goal (theory "FOLP") "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c"; |
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317 by (fast_tac FOLP_cs 1); |
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318 result(); |
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319 |
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320 writeln"Problem 50"; |
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321 (*What has this to do with equality?*) |
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322 goal (theory "FOLP") "?p : (ALL x. P(a,x) | (ALL y. P(x,y))) --> (EX x. ALL y. P(x,y))"; |
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323 by (best_tac FOLP_dup_cs 1); |
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324 result(); |
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325 |
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326 writeln"Problem 56"; |
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327 goal (theory "FOLP") |
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328 "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"; |
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329 by (fast_tac FOLP_cs 1); |
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330 result(); |
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331 |
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332 writeln"Problem 57"; |
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333 goal (theory "FOLP") |
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334 "?p : P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \ |
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335 \ (ALL x y z. P(x,y) & P(y,z) --> P(x,z)) --> P(f(a,b), f(a,c))"; |
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336 by (fast_tac FOLP_cs 1); |
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337 result(); |
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338 |
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339 writeln"Problem 58 NOT PROVED AUTOMATICALLY"; |
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340 goal (theory "FOLP") "?p : (ALL x y. f(x)=g(y)) --> (ALL x y. f(f(x))=f(g(y)))"; |
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341 val f_cong = read_instantiate [("t","f")] subst_context; |
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342 by (fast_tac (FOLP_cs addIs [f_cong]) 1); |
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343 result(); |
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344 |
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345 writeln"Problem 59"; |
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346 goal (theory "FOLP") "?p : (ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))"; |
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347 by (best_tac FOLP_dup_cs 1); |
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348 result(); |
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349 |
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350 writeln"Problem 60"; |
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351 goal (theory "FOLP") |
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352 "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"; |
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353 by (fast_tac FOLP_cs 1); |
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354 result(); |
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