1 (* Title: LK/ex/prop |
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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 propositional connectives |
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7 Can be read to test the LK system. |
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8 *) |
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9 |
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10 writeln"LK/ex/prop: propositional examples"; |
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11 |
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12 writeln"absorptive laws of & and | "; |
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13 |
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14 goal LK.thy "|- P & P <-> P"; |
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15 by (fast_tac prop_pack 1); |
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16 result(); |
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17 |
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18 goal LK.thy "|- P | P <-> P"; |
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19 by (fast_tac prop_pack 1); |
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20 result(); |
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21 |
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22 writeln"commutative laws of & and | "; |
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23 |
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24 goal LK.thy "|- P & Q <-> Q & P"; |
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25 by (fast_tac prop_pack 1); |
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26 result(); |
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27 |
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28 goal LK.thy "|- P | Q <-> Q | P"; |
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29 by (fast_tac prop_pack 1); |
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30 result(); |
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31 |
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32 |
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33 writeln"associative laws of & and | "; |
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34 |
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35 goal LK.thy "|- (P & Q) & R <-> P & (Q & R)"; |
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36 by (fast_tac prop_pack 1); |
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37 result(); |
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38 |
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39 goal LK.thy "|- (P | Q) | R <-> P | (Q | R)"; |
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40 by (fast_tac prop_pack 1); |
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41 result(); |
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42 |
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43 writeln"distributive laws of & and | "; |
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44 goal LK.thy "|- (P & Q) | R <-> (P | R) & (Q | R)"; |
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45 by (fast_tac prop_pack 1); |
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46 result(); |
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47 |
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48 goal LK.thy "|- (P | Q) & R <-> (P & R) | (Q & R)"; |
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49 by (fast_tac prop_pack 1); |
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50 result(); |
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51 |
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52 writeln"Laws involving implication"; |
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53 |
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54 goal LK.thy "|- (P|Q --> R) <-> (P-->R) & (Q-->R)"; |
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55 by (fast_tac prop_pack 1); |
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56 result(); |
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57 |
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58 goal LK.thy "|- (P & Q --> R) <-> (P--> (Q-->R))"; |
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59 by (fast_tac prop_pack 1); |
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60 result(); |
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61 |
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62 |
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63 goal LK.thy "|- (P --> Q & R) <-> (P-->Q) & (P-->R)"; |
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64 by (fast_tac prop_pack 1); |
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65 result(); |
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66 |
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67 |
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68 writeln"Classical theorems"; |
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69 |
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70 goal LK.thy "|- P|Q --> P| ~P&Q"; |
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71 by (fast_tac prop_pack 1); |
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72 result(); |
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73 |
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74 |
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75 goal LK.thy "|- (P-->Q)&(~P-->R) --> (P&Q | R)"; |
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76 by (fast_tac prop_pack 1); |
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77 result(); |
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78 |
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79 |
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80 goal LK.thy "|- P&Q | ~P&R <-> (P-->Q)&(~P-->R)"; |
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81 by (fast_tac prop_pack 1); |
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82 result(); |
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83 |
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84 |
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85 goal LK.thy "|- (P-->Q) | (P-->R) <-> (P --> Q | R)"; |
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86 by (fast_tac prop_pack 1); |
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87 result(); |
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88 |
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89 |
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90 (*If and only if*) |
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91 |
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92 goal LK.thy "|- (P<->Q) <-> (Q<->P)"; |
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93 by (fast_tac prop_pack 1); |
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94 result(); |
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95 |
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96 goal LK.thy "|- ~ (P <-> ~P)"; |
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97 by (fast_tac prop_pack 1); |
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98 result(); |
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99 |
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100 |
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101 (*Sample problems from |
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102 F. J. Pelletier, |
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103 Seventy-Five Problems for Testing Automatic Theorem Provers, |
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104 J. Automated Reasoning 2 (1986), 191-216. |
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105 Errata, JAR 4 (1988), 236-236. |
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106 *) |
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107 |
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108 (*1*) |
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109 goal LK.thy "|- (P-->Q) <-> (~Q --> ~P)"; |
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110 by (fast_tac prop_pack 1); |
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111 result(); |
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112 |
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113 (*2*) |
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114 goal LK.thy "|- ~ ~ P <-> P"; |
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115 by (fast_tac prop_pack 1); |
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116 result(); |
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117 |
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118 (*3*) |
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119 goal LK.thy "|- ~(P-->Q) --> (Q-->P)"; |
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120 by (fast_tac prop_pack 1); |
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121 result(); |
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122 |
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123 (*4*) |
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124 goal LK.thy "|- (~P-->Q) <-> (~Q --> P)"; |
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125 by (fast_tac prop_pack 1); |
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126 result(); |
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127 |
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128 (*5*) |
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129 goal LK.thy "|- ((P|Q)-->(P|R)) --> (P|(Q-->R))"; |
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130 by (fast_tac prop_pack 1); |
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131 result(); |
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132 |
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133 (*6*) |
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134 goal LK.thy "|- P | ~ P"; |
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135 by (fast_tac prop_pack 1); |
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136 result(); |
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137 |
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138 (*7*) |
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139 goal LK.thy "|- P | ~ ~ ~ P"; |
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140 by (fast_tac prop_pack 1); |
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141 result(); |
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142 |
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143 (*8. Peirce's law*) |
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144 goal LK.thy "|- ((P-->Q) --> P) --> P"; |
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145 by (fast_tac prop_pack 1); |
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146 result(); |
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147 |
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148 (*9*) |
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149 goal LK.thy "|- ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"; |
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150 by (fast_tac prop_pack 1); |
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151 result(); |
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152 |
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153 (*10*) |
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154 goal LK.thy "Q-->R, R-->P&Q, P-->(Q|R) |- P<->Q"; |
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155 by (fast_tac prop_pack 1); |
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156 result(); |
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157 |
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158 (*11. Proved in each direction (incorrectly, says Pelletier!!) *) |
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159 goal LK.thy "|- P<->P"; |
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160 by (fast_tac prop_pack 1); |
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161 result(); |
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162 |
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163 (*12. "Dijkstra's law"*) |
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164 goal LK.thy "|- ((P <-> Q) <-> R) <-> (P <-> (Q <-> R))"; |
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165 by (fast_tac prop_pack 1); |
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166 result(); |
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167 |
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168 (*13. Distributive law*) |
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169 goal LK.thy "|- P | (Q & R) <-> (P | Q) & (P | R)"; |
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170 by (fast_tac prop_pack 1); |
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171 result(); |
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172 |
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173 (*14*) |
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174 goal LK.thy "|- (P <-> Q) <-> ((Q | ~P) & (~Q|P))"; |
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175 by (fast_tac prop_pack 1); |
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176 result(); |
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177 |
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178 |
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179 (*15*) |
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180 goal LK.thy "|- (P --> Q) <-> (~P | Q)"; |
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181 by (fast_tac prop_pack 1); |
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182 result(); |
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183 |
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184 (*16*) |
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185 goal LK.thy "|- (P-->Q) | (Q-->P)"; |
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186 by (fast_tac prop_pack 1); |
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187 result(); |
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188 |
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189 (*17*) |
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190 goal LK.thy "|- ((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))"; |
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191 by (fast_tac prop_pack 1); |
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192 result(); |
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193 |
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194 writeln"Reached end of file."; |
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