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1 (* ========================================================================= *) |
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2 (* SOME SAMPLE PROBLEMS TO TEST PROOF PROCEDURES *) |
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3 (* Copyright (c) 2001-2007 Joe Hurd, distributed under the GNU GPL version 2 *) |
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4 (* ========================================================================= *) |
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5 |
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6 (* ========================================================================= *) |
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7 (* A type of problems. *) |
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8 (* ========================================================================= *) |
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9 |
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10 type problem = |
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11 {name : string, |
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12 comments : string list, |
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13 goal : Formula.quotation}; |
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14 |
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15 (* ========================================================================= *) |
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16 (* Helper functions. *) |
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17 (* ========================================================================= *) |
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18 |
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19 fun mkProblem description (problem : problem) : problem = |
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20 let |
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21 val {name,comments,goal} = problem |
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22 val comments = if null comments then [] else "" :: comments |
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23 val comments = "Collection: " ^ description :: comments |
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24 in |
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25 {name = name, comments = comments, goal = goal} |
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26 end; |
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27 |
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28 fun mkProblems description problems = map (mkProblem description) problems; |
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29 |
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30 (* ========================================================================= *) |
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31 (* The collection of problems. *) |
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32 (* ========================================================================= *) |
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33 |
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34 val problems : problem list = |
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35 |
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36 (* ========================================================================= *) |
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37 (* Problems without equality. *) |
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38 (* ========================================================================= *) |
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39 |
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40 mkProblems "Problems without equality from various sources" [ |
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41 |
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42 (* ------------------------------------------------------------------------- *) |
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43 (* Trivia (some of which demonstrate ex-bugs in provers). *) |
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44 (* ------------------------------------------------------------------------- *) |
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45 |
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46 {name = "TRUE", |
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47 comments = [], |
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48 goal = ` |
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49 T`}, |
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50 |
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51 {name = "SIMPLE", |
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52 comments = [], |
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53 goal = ` |
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54 !x y. ?z. p x \/ p y ==> p z`}, |
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55 |
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56 {name = "CYCLIC", |
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57 comments = [], |
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58 goal = ` |
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59 (!x. p (g (c x))) ==> ?z. p (g z)`}, |
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60 |
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61 {name = "MICHAEL_NORRISH_BUG", |
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62 comments = [], |
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63 goal = ` |
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64 (!x. ?y. f y x x) ==> ?z. f z 0 0`}, |
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65 |
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66 {name = "RELATION_COMPOSITION", |
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67 comments = [], |
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68 goal = ` |
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69 (!x. ?y. p x y) /\ (!x. ?y. q x y) /\ |
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70 (!x y z. p x y /\ q y z ==> r x z) ==> !x. ?y. r x y`}, |
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71 |
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72 (* ------------------------------------------------------------------------- *) |
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73 (* Propositional Logic. *) |
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74 (* ------------------------------------------------------------------------- *) |
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75 |
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76 {name = "PROP_1", |
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77 comments = [], |
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78 goal = ` |
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79 p ==> q <=> ~q ==> ~p`}, |
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80 |
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81 {name = "PROP_2", |
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82 comments = [], |
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83 goal = ` |
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84 ~~p <=> p`}, |
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85 |
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86 {name = "PROP_3", |
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87 comments = [], |
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88 goal = ` |
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89 ~(p ==> q) ==> q ==> p`}, |
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90 |
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91 {name = "PROP_4", |
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92 comments = [], |
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93 goal = ` |
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94 ~p ==> q <=> ~q ==> p`}, |
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95 |
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96 {name = "PROP_5", |
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97 comments = [], |
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98 goal = ` |
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99 (p \/ q ==> p \/ r) ==> p \/ (q ==> r)`}, |
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100 |
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101 {name = "PROP_6", |
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102 comments = [], |
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103 goal = ` |
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104 p \/ ~p`}, |
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105 |
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106 {name = "PROP_7", |
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107 comments = [], |
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108 goal = ` |
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109 p \/ ~~~p`}, |
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110 |
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111 {name = "PROP_8", |
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112 comments = [], |
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113 goal = ` |
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114 ((p ==> q) ==> p) ==> p`}, |
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115 |
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116 {name = "PROP_9", |
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117 comments = [], |
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118 goal = ` |
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119 (p \/ q) /\ (~p \/ q) /\ (p \/ ~q) ==> ~(~q \/ ~q)`}, |
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120 |
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121 {name = "PROP_10", |
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122 comments = [], |
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123 goal = ` |
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124 (q ==> r) /\ (r ==> p /\ q) /\ (p ==> q /\ r) ==> (p <=> q)`}, |
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125 |
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126 {name = "PROP_11", |
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127 comments = [], |
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128 goal = ` |
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129 p <=> p`}, |
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130 |
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131 {name = "PROP_12", |
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132 comments = [], |
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133 goal = ` |
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134 ((p <=> q) <=> r) <=> p <=> q <=> r`}, |
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135 |
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136 {name = "PROP_13", |
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137 comments = [], |
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138 goal = ` |
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139 p \/ q /\ r <=> (p \/ q) /\ (p \/ r)`}, |
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140 |
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141 {name = "PROP_14", |
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142 comments = [], |
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143 goal = ` |
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144 (p <=> q) <=> (q \/ ~p) /\ (~q \/ p)`}, |
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145 |
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146 {name = "PROP_15", |
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147 comments = [], |
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148 goal = ` |
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149 p ==> q <=> ~p \/ q`}, |
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150 |
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151 {name = "PROP_16", |
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152 comments = [], |
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153 goal = ` |
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154 (p ==> q) \/ (q ==> p)`}, |
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155 |
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156 {name = "PROP_17", |
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157 comments = [], |
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158 goal = ` |
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159 p /\ (q ==> r) ==> s <=> (~p \/ q \/ s) /\ (~p \/ ~r \/ s)`}, |
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160 |
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161 {name = "MATHS4_EXAMPLE", |
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162 comments = [], |
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163 goal = ` |
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164 (a \/ ~k ==> g) /\ (g ==> q) /\ ~q ==> k`}, |
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165 |
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166 {name = "LOGICPROOF_1996", |
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167 comments = [], |
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168 goal = ` |
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169 (p ==> r) /\ (~p ==> ~q) /\ (p \/ q) ==> p /\ r`}, |
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170 |
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171 {name = "XOR_ASSOC", |
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172 comments = [], |
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173 goal = ` |
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174 ~(~(p <=> q) <=> r) <=> ~(p <=> ~(q <=> r))`}, |
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175 |
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176 {name = "ALL_3_CLAUSES", |
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177 comments = [], |
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178 goal = ` |
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179 (p \/ q \/ r) /\ (p \/ q \/ ~r) /\ (p \/ ~q \/ r) /\ (p \/ ~q \/ ~r) /\ |
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180 (~p \/ q \/ r) /\ (~p \/ q \/ ~r) /\ (~p \/ ~q \/ r) /\ |
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181 (~p \/ ~q \/ ~r) ==> F`}, |
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182 |
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183 {name = "CLAUSE_SIMP", |
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184 comments = [], |
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185 goal = ` |
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186 (lit ==> clause) ==> (lit \/ clause <=> clause)`}, |
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187 |
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188 (* ------------------------------------------------------------------------- *) |
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189 (* Monadic Predicate Logic. *) |
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190 (* ------------------------------------------------------------------------- *) |
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191 |
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192 {name = "P18", |
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193 comments = ["The drinker's principle."], |
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194 goal = ` |
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195 ?very_popular_guy. !whole_pub. drinks very_popular_guy ==> drinks whole_pub`}, |
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196 |
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197 {name = "P19", |
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198 comments = [], |
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199 goal = ` |
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200 ?x. !y z. (p y ==> q z) ==> p x ==> q x`}, |
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201 |
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202 {name = "P20", |
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203 comments = [], |
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204 goal = ` |
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205 (!x y. ?z. !w. p x /\ q y ==> r z /\ u w) /\ (!x y. p x /\ q y) ==> ?z. r z`}, |
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206 |
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207 {name = "P21", |
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208 comments = [], |
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209 goal = ` |
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210 (?x. p ==> q x) /\ (?x. q x ==> p) ==> ?x. p <=> q x`}, |
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211 |
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212 {name = "P22", |
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213 comments = [], |
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214 goal = ` |
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215 (!x. p <=> q x) ==> (p <=> !x. q x)`}, |
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216 |
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217 {name = "P23", |
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218 comments = [], |
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219 goal = ` |
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220 (!x. p \/ q x) <=> p \/ !x. q x`}, |
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221 |
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222 {name = "P24", |
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223 comments = [], |
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224 goal = ` |
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225 ~(?x. u x /\ q x) /\ (!x. p x ==> q x \/ r x) /\ ~(?x. p x ==> ?x. q x) /\ |
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226 (!x. q x /\ r x ==> u x) ==> ?x. p x /\ r x`}, |
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227 |
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228 {name = "P25", |
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229 comments = [], |
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230 goal = ` |
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231 (?x. p x) /\ (!x. u x ==> ~g x /\ r x) /\ (!x. p x ==> g x /\ u x) /\ |
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232 ((!x. p x ==> q x) \/ ?x. q x /\ p x) ==> ?x. q x /\ p x`}, |
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233 |
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234 {name = "P26", |
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235 comments = [], |
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236 goal = ` |
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237 ((?x. p x) <=> ?x. q x) /\ (!x y. p x /\ q y ==> (r x <=> u y)) ==> |
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238 ((!x. p x ==> r x) <=> !x. q x ==> u x)`}, |
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239 |
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240 {name = "P27", |
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241 comments = [], |
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242 goal = ` |
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243 (?x. p x /\ ~q x) /\ (!x. p x ==> r x) /\ (!x. u x /\ s x ==> p x) /\ |
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244 (?x. r x /\ ~q x) ==> (!x. u x ==> ~r x) ==> !x. u x ==> ~s x`}, |
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245 |
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246 {name = "P28", |
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247 comments = [], |
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248 goal = ` |
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249 (!x. p x ==> !x. q x) /\ ((!x. q x \/ r x) ==> ?x. q x /\ r x) /\ |
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250 ((?x. r x) ==> !x. l x ==> m x) ==> !x. p x /\ l x ==> m x`}, |
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251 |
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252 {name = "P29", |
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253 comments = [], |
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254 goal = ` |
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255 (?x. p x) /\ (?x. g x) ==> |
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256 ((!x. p x ==> h x) /\ (!x. g x ==> j x) <=> |
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257 !x y. p x /\ g y ==> h x /\ j y)`}, |
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258 |
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259 {name = "P30", |
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260 comments = [], |
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261 goal = ` |
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262 (!x. p x \/ g x ==> ~h x) /\ (!x. (g x ==> ~u x) ==> p x /\ h x) ==> |
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263 !x. u x`}, |
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264 |
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265 {name = "P31", |
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266 comments = [], |
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267 goal = ` |
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268 ~(?x. p x /\ (g x \/ h x)) /\ (?x. q x /\ p x) /\ (!x. ~h x ==> j x) ==> |
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269 ?x. q x /\ j x`}, |
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270 |
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271 {name = "P32", |
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272 comments = [], |
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273 goal = ` |
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274 (!x. p x /\ (g x \/ h x) ==> q x) /\ (!x. q x /\ h x ==> j x) /\ |
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275 (!x. r x ==> h x) ==> !x. p x /\ r x ==> j x`}, |
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276 |
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277 {name = "P33", |
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278 comments = [], |
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279 goal = ` |
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280 (!x. p a /\ (p x ==> p b) ==> p c) <=> |
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281 (!x. p a ==> p x \/ p c) /\ (p a ==> p b ==> p c)`}, |
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282 |
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283 {name = "P34", |
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284 comments = |
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285 ["This gives rise to 5184 clauses when naively converted to CNF!"], |
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286 goal = ` |
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287 ((?x. !y. p x <=> p y) <=> (?x. q x) <=> !y. q y) <=> |
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288 (?x. !y. q x <=> q y) <=> (?x. p x) <=> !y. p y`}, |
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289 |
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290 {name = "P35", |
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291 comments = [], |
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292 goal = ` |
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293 ?x y. p x y ==> !x y. p x y`}, |
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294 |
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295 (* ------------------------------------------------------------------------- *) |
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296 (* Predicate logic without functions. *) |
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297 (* ------------------------------------------------------------------------- *) |
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298 |
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299 {name = "P36", |
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300 comments = [], |
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301 goal = ` |
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302 (!x. ?y. p x y) /\ (!x. ?y. g x y) /\ |
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303 (!x y. p x y \/ g x y ==> !z. p y z \/ g y z ==> h x z) ==> !x. ?y. h x y`}, |
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304 |
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305 {name = "P37", |
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306 comments = [], |
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307 goal = ` |
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308 (!z. ?w. !x. ?y. (p x z ==> p y w) /\ p y z /\ (p y w ==> ?v. q v w)) /\ |
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309 (!x z. ~p x z ==> ?y. q y z) /\ ((?x y. q x y) ==> !x. r x x) ==> |
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310 !x. ?y. r x y`}, |
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311 |
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312 {name = "P38", |
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313 comments = [], |
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314 goal = ` |
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315 (!x. p a /\ (p x ==> ?y. p y /\ r x y) ==> ?z w. p z /\ r x w /\ r w z) <=> |
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316 !x. |
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317 (~p a \/ p x \/ ?z w. p z /\ r x w /\ r w z) /\ |
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318 (~p a \/ ~(?y. p y /\ r x y) \/ ?z w. p z /\ r x w /\ r w z)`}, |
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319 |
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320 {name = "P39", |
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321 comments = [], |
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322 goal = ` |
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323 ~?x. !y. p y x <=> ~p y y`}, |
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324 |
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325 {name = "P40", |
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326 comments = [], |
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327 goal = ` |
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328 (?y. !x. p x y <=> p x x) ==> ~!x. ?y. !z. p z y <=> ~p z x`}, |
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329 |
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330 {name = "P41", |
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331 comments = [], |
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332 goal = ` |
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333 (!z. ?y. !x. p x y <=> p x z /\ ~p x x) ==> ~?z. !x. p x z`}, |
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334 |
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335 {name = "P42", |
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336 comments = [], |
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337 goal = ` |
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338 ~?y. !x. p x y <=> ~?z. p x z /\ p z x`}, |
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339 |
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340 {name = "P43", |
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341 comments = [], |
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342 goal = ` |
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343 (!x y. q x y <=> !z. p z x <=> p z y) ==> !x y. q x y <=> q y x`}, |
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344 |
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345 {name = "P44", |
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346 comments = [], |
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347 goal = ` |
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348 (!x. p x ==> (?y. g y /\ h x y) /\ ?y. g y /\ ~h x y) /\ |
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349 (?x. j x /\ !y. g y ==> h x y) ==> ?x. j x /\ ~p x`}, |
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350 |
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351 {name = "P45", |
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352 comments = [], |
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353 goal = ` |
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354 (!x. p x /\ (!y. g y /\ h x y ==> j x y) ==> !y. g y /\ h x y ==> r y) /\ |
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355 ~(?y. l y /\ r y) /\ |
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356 (?x. p x /\ (!y. h x y ==> l y) /\ !y. g y /\ h x y ==> j x y) ==> |
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357 ?x. p x /\ ~?y. g y /\ h x y`}, |
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358 |
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359 {name = "P46", |
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360 comments = [], |
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361 goal = ` |
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362 (!x. p x /\ (!y. p y /\ h y x ==> g y) ==> g x) /\ |
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363 ((?x. p x /\ ~g x) ==> ?x. p x /\ ~g x /\ !y. p y /\ ~g y ==> j x y) /\ |
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364 (!x y. p x /\ p y /\ h x y ==> ~j y x) ==> !x. p x ==> g x`}, |
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365 |
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366 {name = "P50", |
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367 comments = [], |
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368 goal = ` |
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369 (!x. f0 a x \/ !y. f0 x y) ==> ?x. !y. f0 x y`}, |
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370 |
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371 {name = "LOGICPROOF_L10", |
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372 comments = [], |
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373 goal = ` |
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374 !x. ?y. ~(P y x <=> ~P y y)`}, |
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375 |
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376 {name = "BARBER", |
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377 comments = ["One resolution of the barber paradox"], |
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378 goal = ` |
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379 (!x. man x ==> (shaves barber x <=> ~shaves x x)) ==> ~man barber`}, |
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380 |
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381 (* ------------------------------------------------------------------------- *) |
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382 (* Full predicate logic. *) |
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383 (* ------------------------------------------------------------------------- *) |
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384 |
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385 {name = "LOGICPROOF_1999", |
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386 comments = ["A non-theorem."], |
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387 goal = ` |
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388 (?x. p x /\ q x) ==> ?x. p (f x x) \/ !y. q y`}, |
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389 |
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390 {name = "P55", |
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391 comments = ["Example from Manthey and Bry, CADE-9. [JRH]"], |
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392 goal = ` |
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393 lives agatha /\ lives butler /\ lives charles /\ |
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394 (killed agatha agatha \/ killed butler agatha \/ killed charles agatha) /\ |
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395 (!x y. killed x y ==> hates x y /\ ~richer x y) /\ |
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396 (!x. hates agatha x ==> ~hates charles x) /\ |
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397 (hates agatha agatha /\ hates agatha charles) /\ |
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398 (!x. lives x /\ ~richer x agatha ==> hates butler x) /\ |
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399 (!x. hates agatha x ==> hates butler x) /\ |
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400 (!x. ~hates x agatha \/ ~hates x butler \/ ~hates x charles) ==> |
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401 killed agatha agatha /\ ~killed butler agatha /\ ~killed charles agatha`}, |
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402 |
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403 {name = "P57", |
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404 comments = [], |
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405 goal = ` |
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406 p (f a b) (f b c) /\ p (f b c) (f a c) /\ |
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407 (!x y z. p x y /\ p y z ==> p x z) ==> p (f a b) (f a c)`}, |
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408 |
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409 {name = "P58", |
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410 comments = ["See info-hol 1498 [JRH]"], |
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411 goal = ` |
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412 !x. ?v w. !y z. p x /\ q y ==> (p v \/ r w) /\ (r z ==> q v)`}, |
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413 |
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414 {name = "P59", |
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415 comments = [], |
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416 goal = ` |
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417 (!x. p x <=> ~p (f x)) ==> ?x. p x /\ ~p (f x)`}, |
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418 |
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419 {name = "P60", |
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420 comments = [], |
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421 goal = ` |
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422 !x. p x (f x) <=> ?y. (!z. p z y ==> p z (f x)) /\ p x y`}, |
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423 |
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424 (* ------------------------------------------------------------------------- *) |
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425 (* From Gilmore's classic paper. *) |
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426 (* ------------------------------------------------------------------------- *) |
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427 |
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428 {name = "GILMORE_1", |
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429 comments = |
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430 ["Amazingly, this still seems non-trivial... in HOL [MESON_TAC] it", |
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431 "works at depth 45! [JRH]", |
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432 "Confirmed (depth=45, inferences=263702, time=148s), though if", |
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433 "lemmaizing is used then a lemma is discovered at depth 29 that allows", |
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434 "a much quicker proof (depth=30, inferences=10039, time=5.5s)."], |
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435 goal = ` |
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436 ?x. !y z. |
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437 (f y ==> g y <=> f x) /\ (f y ==> h y <=> g x) /\ |
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438 ((f y ==> g y) ==> h y <=> h x) ==> f z /\ g z /\ h z`}, |
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439 |
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440 {name = "GILMORE_2", |
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441 comments = |
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442 ["This is not valid, according to Gilmore. [JRH]", |
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443 "Confirmed: ordered resolution quickly saturates."], |
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444 goal = ` |
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445 ?x y. !z. |
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446 (f x z <=> f z y) /\ (f z y <=> f z z) /\ (f x y <=> f y x) ==> |
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447 (f x y <=> f x z)`}, |
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448 |
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449 {name = "GILMORE_3", |
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450 comments = [], |
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451 goal = ` |
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452 ?x. !y z. |
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453 ((f y z ==> g y ==> h x) ==> f x x) /\ ((f z x ==> g x) ==> h z) /\ |
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454 f x y ==> f z z`}, |
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455 |
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456 {name = "GILMORE_4", |
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457 comments = [], |
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458 goal = ` |
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459 ?x y. !z. (f x y ==> f y z /\ f z z) /\ (f x y /\ g x y ==> g x z /\ g z z)`}, |
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460 |
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461 {name = "GILMORE_5", |
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462 comments = [], |
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463 goal = ` |
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464 (!x. ?y. f x y \/ f y x) /\ (!x y. f y x ==> f y y) ==> ?z. f z z`}, |
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465 |
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466 {name = "GILMORE_6", |
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467 comments = [], |
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468 goal = ` |
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469 !x. ?y. |
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470 (?w. !v. f w x ==> g v w /\ g w x) ==> |
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471 (?w. !v. f w y ==> g v w /\ g w y) \/ |
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472 !z v. ?w. g v z \/ h w y z ==> g z w`}, |
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473 |
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474 {name = "GILMORE_7", |
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475 comments = [], |
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476 goal = ` |
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477 (!x. k x ==> ?y. l y /\ (f x y ==> g x y)) /\ |
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478 (?z. k z /\ !w. l w ==> f z w) ==> ?v w. k v /\ l w /\ g v w`}, |
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479 |
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480 {name = "GILMORE_8", |
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481 comments = [], |
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482 goal = ` |
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483 ?x. !y z. |
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484 ((f y z ==> g y ==> !w. ?v. h w v x) ==> f x x) /\ |
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485 ((f z x ==> g x) ==> !w. ?v. h w v z) /\ f x y ==> f z z`}, |
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486 |
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487 {name = "GILMORE_9", |
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488 comments = |
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489 ["Model elimination (in HOL):", |
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490 "- With lemmaizing: (depth=18, inferences=15632, time=21s)", |
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491 "- Without: gave up waiting after (depth=25, inferences=2125072, ...)"], |
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492 goal = ` |
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493 !x. ?y. !z. |
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494 ((!w. ?v. f y w v /\ g y w /\ ~h y x) ==> |
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495 (!w. ?v. f x w v /\ g z u /\ ~h x z) ==> |
|
496 !w. ?v. f x w v /\ g y w /\ ~h x y) /\ |
|
497 ((!w. ?v. f x w v /\ g y w /\ ~h x y) ==> |
|
498 ~(!w. ?v. f x w v /\ g z w /\ ~h x z) ==> |
|
499 (!w. ?v. f y w v /\ g y w /\ ~h y x) /\ |
|
500 !w. ?v. f z w v /\ g y w /\ ~h z y)`}, |
|
501 |
|
502 {name = "GILMORE_9a", |
|
503 comments = |
|
504 ["Translation of Gilmore procedure using separate definitions. [JRH]"], |
|
505 goal = ` |
|
506 (!x y. p x y <=> !w. ?v. f x w v /\ g y w /\ ~h x y) ==> |
|
507 !x. ?y. !z. |
|
508 (p y x ==> p x z ==> p x y) /\ (p x y ==> ~p x z ==> p y x /\ p z y)`}, |
|
509 |
|
510 {name = "BAD_CONNECTIONS", |
|
511 comments = |
|
512 ["The interesting example where connections make the proof longer. [JRH]"], |
|
513 goal = ` |
|
514 ~a /\ (a \/ b) /\ (c \/ d) /\ (~b \/ e \/ f) /\ (~c \/ ~e) /\ (~c \/ ~f) /\ |
|
515 (~b \/ g \/ h) /\ (~d \/ ~g) /\ (~d \/ ~h) ==> F`}, |
|
516 |
|
517 {name = "LOS", |
|
518 comments = |
|
519 ["The classic Los puzzle. (Clausal version MSC006-1 in the TPTP library.)", |
|
520 "Note: this is actually in the decidable AE subset, though that doesn't", |
|
521 "yield a very efficient proof. [JRH]"], |
|
522 goal = ` |
|
523 (!x y z. p x y ==> p y z ==> p x z) /\ |
|
524 (!x y z. q x y ==> q y z ==> q x z) /\ (!x y. q x y ==> q y x) /\ |
|
525 (!x y. p x y \/ q x y) ==> (!x y. p x y) \/ !x y. q x y`}, |
|
526 |
|
527 {name = "STEAM_ROLLER", |
|
528 comments = [], |
|
529 goal = ` |
|
530 ((!x. p1 x ==> p0 x) /\ ?x. p1 x) /\ ((!x. p2 x ==> p0 x) /\ ?x. p2 x) /\ |
|
531 ((!x. p3 x ==> p0 x) /\ ?x. p3 x) /\ ((!x. p4 x ==> p0 x) /\ ?x. p4 x) /\ |
|
532 ((!x. p5 x ==> p0 x) /\ ?x. p5 x) /\ ((?x. q1 x) /\ !x. q1 x ==> q0 x) /\ |
|
533 (!x. |
|
534 p0 x ==> |
|
535 (!y. q0 y ==> r x y) \/ |
|
536 !y. p0 y /\ s0 y x /\ (?z. q0 z /\ r y z) ==> r x y) /\ |
|
537 (!x y. p3 y /\ (p5 x \/ p4 x) ==> s0 x y) /\ |
|
538 (!x y. p3 x /\ p2 y ==> s0 x y) /\ (!x y. p2 x /\ p1 y ==> s0 x y) /\ |
|
539 (!x y. p1 x /\ (p2 y \/ q1 y) ==> ~r x y) /\ |
|
540 (!x y. p3 x /\ p4 y ==> r x y) /\ (!x y. p3 x /\ p5 y ==> ~r x y) /\ |
|
541 (!x. p4 x \/ p5 x ==> ?y. q0 y /\ r x y) ==> |
|
542 ?x y. p0 x /\ p0 y /\ ?z. q1 z /\ r y z /\ r x y`}, |
|
543 |
|
544 {name = "MODEL_COMPLETENESS", |
|
545 comments = |
|
546 ["An incestuous example used to establish completeness", |
|
547 "characterization. [JRH]"], |
|
548 goal = ` |
|
549 (!w x. sentence x ==> holds w x \/ holds w (not x)) /\ |
|
550 (!w x. ~(holds w x /\ holds w (not x))) ==> |
|
551 ((!x. |
|
552 sentence x ==> |
|
553 (!w. models w s ==> holds w x) \/ |
|
554 !w. models w s ==> holds w (not x)) <=> |
|
555 !w v. |
|
556 models w s /\ models v s ==> |
|
557 !x. sentence x ==> (holds w x <=> holds v x))`} |
|
558 |
|
559 ] @ |
|
560 |
|
561 (* ========================================================================= *) |
|
562 (* Problems with equality. *) |
|
563 (* ========================================================================= *) |
|
564 |
|
565 mkProblems "Equality problems from various sources" [ |
|
566 |
|
567 (* ------------------------------------------------------------------------- *) |
|
568 (* Trivia (some of which demonstrate ex-bugs in the prover). *) |
|
569 (* ------------------------------------------------------------------------- *) |
|
570 |
|
571 {name = "REFLEXIVITY", |
|
572 comments = [], |
|
573 goal = ` |
|
574 c = c`}, |
|
575 |
|
576 {name = "SYMMETRY", |
|
577 comments = [], |
|
578 goal = ` |
|
579 !x y. x = y ==> y = x`}, |
|
580 |
|
581 {name = "TRANSITIVITY", |
|
582 comments = [], |
|
583 goal = ` |
|
584 !x y z. x = y /\ y = z ==> x = z`}, |
|
585 |
|
586 {name = "TRANS_SYMM", |
|
587 comments = [], |
|
588 goal = ` |
|
589 !x y z. x = y /\ y = z ==> z = x`}, |
|
590 |
|
591 {name = "SUBSTITUTIVITY", |
|
592 comments = [], |
|
593 goal = ` |
|
594 !x y. f x /\ x = y ==> f y`}, |
|
595 |
|
596 {name = "CYCLIC_SUBSTITUTION_BUG", |
|
597 comments = [], |
|
598 goal = ` |
|
599 !y. (!x. y = g (c x)) ==> ?z. y = g z`}, |
|
600 |
|
601 {name = "JUDITA_1", |
|
602 comments = [], |
|
603 goal = ` |
|
604 ~(a = b) /\ (!x. x = c) ==> F`}, |
|
605 |
|
606 {name = "JUDITA_2", |
|
607 comments = [], |
|
608 goal = ` |
|
609 ~(a = b) /\ (!x y. x = y) ==> F`}, |
|
610 |
|
611 {name = "JUDITA_3", |
|
612 comments = [], |
|
613 goal = ` |
|
614 p a /\ ~p b /\ (!x. x = c) ==> F`}, |
|
615 |
|
616 {name = "JUDITA_4", |
|
617 comments = [], |
|
618 goal = ` |
|
619 p a /\ ~p b /\ (!x y. x = y) ==> F`}, |
|
620 |
|
621 {name = "JUDITA_5", |
|
622 comments = [], |
|
623 goal = ` |
|
624 p a /\ p b /\ ~(a = b) /\ ~p c /\ (!x. x = a \/ x = d) ==> F`}, |
|
625 |
|
626 {name = "INJECTIVITY_1", |
|
627 comments = [], |
|
628 goal = ` |
|
629 (!x y. f x = f y ==> x = y) /\ f a = f b ==> a = b`}, |
|
630 |
|
631 {name = "INJECTIVITY_2", |
|
632 comments = [], |
|
633 goal = ` |
|
634 (!x y. g (f x) = g (f y) ==> x = y) /\ f a = f b ==> a = b`}, |
|
635 |
|
636 {name = "SELF_REWRITE", |
|
637 comments = [], |
|
638 goal = ` |
|
639 f (g (h c)) = h c /\ g (h c) = b /\ f b = a /\ (!x. ~(a = h x)) ==> F`}, |
|
640 |
|
641 (* ------------------------------------------------------------------------- *) |
|
642 (* Simple equality problems. *) |
|
643 (* ------------------------------------------------------------------------- *) |
|
644 |
|
645 {name = "P48", |
|
646 comments = [], |
|
647 goal = ` |
|
648 (a = b \/ c = d) /\ (a = c \/ b = d) ==> a = d \/ b = c`}, |
|
649 |
|
650 {name = "P49", |
|
651 comments = [], |
|
652 goal = ` |
|
653 (?x y. !z. z = x \/ z = y) /\ p a /\ p b /\ ~(a = b) ==> !x. p x`}, |
|
654 |
|
655 {name = "P51", |
|
656 comments = [], |
|
657 goal = ` |
|
658 (?z w. !x y. f0 x y <=> x = z /\ y = w) ==> |
|
659 ?z. !x. (?w. !y. f0 x y <=> y = w) <=> x = z`}, |
|
660 |
|
661 {name = "P52", |
|
662 comments = [], |
|
663 goal = ` |
|
664 (?z w. !x y. f0 x y <=> x = z /\ y = w) ==> |
|
665 ?w. !y. (?z. !x. f0 x y <=> x = z) <=> y = w`}, |
|
666 |
|
667 {name = "UNSKOLEMIZED_MELHAM", |
|
668 comments = ["The Melham problem after an inverse skolemization step."], |
|
669 goal = ` |
|
670 (!x y. g x = g y ==> f x = f y) ==> !y. ?w. !x. y = g x ==> w = f x`}, |
|
671 |
|
672 {name = "CONGRUENCE_CLOSURE_EXAMPLE", |
|
673 comments = ["The example always given for congruence closure."], |
|
674 goal = ` |
|
675 !x. f (f (f (f (f x)))) = x /\ f (f (f x)) = x ==> f x = x`}, |
|
676 |
|
677 {name = "EWD_1", |
|
678 comments = |
|
679 ["A simple example (see EWD1266a and the application to Morley's", |
|
680 "theorem). [JRH]"], |
|
681 goal = ` |
|
682 (!x. f x ==> g x) /\ (?x. f x) /\ (!x y. g x /\ g y ==> x = y) ==> |
|
683 !y. g y ==> f y`}, |
|
684 |
|
685 {name = "EWD_2", |
|
686 comments = [], |
|
687 goal = ` |
|
688 (!x. f (f x) = f x) /\ (!x. ?y. f y = x) ==> !x. f x = x`}, |
|
689 |
|
690 {name = "JIA", |
|
691 comments = ["Needs only the K combinator"], |
|
692 goal = ` |
|
693 (!x y. k . x . y = x) /\ (!v. P (v . a) a) /\ (!w. Q (w . b) b) ==> |
|
694 !z. ?x y. P (z . x . y) x /\ Q (z . x . y) y`}, |
|
695 |
|
696 {name = "WISHNU", |
|
697 comments = ["Wishnu Prasetya's example. [JRH]"], |
|
698 goal = ` |
|
699 (?x. x = f (g x) /\ !x'. x' = f (g x') ==> x = x') <=> |
|
700 ?y. y = g (f y) /\ !y'. y' = g (f y') ==> y = y'`}, |
|
701 |
|
702 {name = "AGATHA", |
|
703 comments = ["An equality version of the Agatha puzzle. [JRH]"], |
|
704 goal = ` |
|
705 (?x. lives x /\ killed x agatha) /\ |
|
706 (lives agatha /\ lives butler /\ lives charles) /\ |
|
707 (!x. lives x ==> x = agatha \/ x = butler \/ x = charles) /\ |
|
708 (!x y. killed x y ==> hates x y) /\ (!x y. killed x y ==> ~richer x y) /\ |
|
709 (!x. hates agatha x ==> ~hates charles x) /\ |
|
710 (!x. ~(x = butler) ==> hates agatha x) /\ |
|
711 (!x. ~richer x agatha ==> hates butler x) /\ |
|
712 (!x. hates agatha x ==> hates butler x) /\ (!x. ?y. ~hates x y) /\ |
|
713 ~(agatha = butler) ==> |
|
714 killed agatha agatha /\ ~killed butler agatha /\ ~killed charles agatha`}, |
|
715 |
|
716 {name = "DIVIDES_9_1", |
|
717 comments = [], |
|
718 goal = ` |
|
719 (!x y. x * y = y * x) /\ (!x y z. x * y * z = x * (y * z)) /\ |
|
720 (!x y. divides x y <=> ?z. y = z * x) ==> |
|
721 !x y z. divides x y ==> divides x (z * y)`}, |
|
722 |
|
723 {name = "DIVIDES_9_2", |
|
724 comments = [], |
|
725 goal = ` |
|
726 (!x y. x * y = y * x) /\ (!x y z. x * y * z = x * (y * z)) /\ |
|
727 (!x y. divides x y <=> ?z. z * x = y) ==> |
|
728 !x y z. divides x y ==> divides x (z * y)`}, |
|
729 |
|
730 (* ------------------------------------------------------------------------- *) |
|
731 (* Group theory examples. *) |
|
732 (* ------------------------------------------------------------------------- *) |
|
733 |
|
734 {name = "GROUP_INVERSE_INVERSE", |
|
735 comments = [], |
|
736 goal = ` |
|
737 (!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\ |
|
738 (!x. i x * x = e) ==> !x. i (i x) = x`}, |
|
739 |
|
740 {name = "GROUP_RIGHT_INVERSE", |
|
741 comments = ["Size 18, 61814 seconds. [JRH]"], |
|
742 goal = ` |
|
743 (!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\ |
|
744 (!x. i x * x = e) ==> !x. x * i x = e`}, |
|
745 |
|
746 {name = "GROUP_RIGHT_IDENTITY", |
|
747 comments = [], |
|
748 goal = ` |
|
749 (!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\ |
|
750 (!x. i x * x = e) ==> !x. x * e = x`}, |
|
751 |
|
752 {name = "GROUP_IDENTITY_EQUIV", |
|
753 comments = [], |
|
754 goal = ` |
|
755 (!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\ |
|
756 (!x. i x * x = e) ==> !x. x * x = x <=> x = e`}, |
|
757 |
|
758 {name = "KLEIN_GROUP_COMMUTATIVE", |
|
759 comments = [], |
|
760 goal = ` |
|
761 (!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\ (!x. x * e = x) /\ |
|
762 (!x. x * x = e) ==> !x y. x * y = y * x`}, |
|
763 |
|
764 (* ------------------------------------------------------------------------- *) |
|
765 (* Ring theory examples. *) |
|
766 (* ------------------------------------------------------------------------- *) |
|
767 |
|
768 {name = "JACOBSON_2", |
|
769 comments = [], |
|
770 goal = ` |
|
771 (!x. 0 + x = x) /\ (!x. x + 0 = x) /\ (!x. n x + x = 0) /\ |
|
772 (!x. x + n x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\ |
|
773 (!x y. x + y = y + x) /\ (!x y z. x * (y * z) = x * y * z) /\ |
|
774 (!x y z. x * (y + z) = x * y + x * z) /\ |
|
775 (!x y z. (x + y) * z = x * z + y * z) /\ (!x. x * x = x) ==> |
|
776 !x y. x * y = y * x`}, |
|
777 |
|
778 {name = "JACOBSON_3", |
|
779 comments = [], |
|
780 goal = ` |
|
781 (!x. 0 + x = x) /\ (!x. x + 0 = x) /\ (!x. n x + x = 0) /\ |
|
782 (!x. x + n x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\ |
|
783 (!x y. x + y = y + x) /\ (!x y z. x * (y * z) = x * y * z) /\ |
|
784 (!x y z. x * (y + z) = x * y + x * z) /\ |
|
785 (!x y z. (x + y) * z = x * z + y * z) /\ (!x. x * (x * x) = x) ==> |
|
786 !x y. x * y = y * x`} |
|
787 |
|
788 ] @ |
|
789 |
|
790 (* ========================================================================= *) |
|
791 (* Some sample problems from the TPTP archive. *) |
|
792 (* Note: for brevity some relation/function names have been shortened. *) |
|
793 (* ========================================================================= *) |
|
794 |
|
795 mkProblems "Sample problems from the TPTP collection" |
|
796 |
|
797 [ |
|
798 |
|
799 {name = "ALG005-1", |
|
800 comments = [], |
|
801 goal = ` |
|
802 (!x y. x + (y + x) = x) /\ (!x y. x + (x + y) = y + (y + x)) /\ |
|
803 (!x y z. x + y + z = x + z + (y + z)) /\ (!x y. x * y = x + (x + y)) ==> |
|
804 ~(a * b * c = a * (b * c)) ==> F`}, |
|
805 |
|
806 {name = "ALG006-1", |
|
807 comments = [], |
|
808 goal = ` |
|
809 (!x y. x + (y + x) = x) /\ (!x y. x + (x + y) = y + (y + x)) /\ |
|
810 (!x y z. x + y + z = x + z + (y + z)) ==> ~(a + c + b = a + b + c) ==> F`}, |
|
811 |
|
812 {name = "BOO021-1", |
|
813 comments = [], |
|
814 goal = ` |
|
815 (!x y. (x + y) * y = y) /\ (!x y z. x * (y + z) = y * x + z * x) /\ |
|
816 (!x. x + i x = 1) /\ (!x y. x * y + y = y) /\ |
|
817 (!x y z. x + y * z = (y + x) * (z + x)) /\ (!x. x * i x = 0) ==> |
|
818 ~(b * a = a * b) ==> F`}, |
|
819 |
|
820 {name = "COL058-2", |
|
821 comments = [], |
|
822 goal = ` |
|
823 (!x y. r (r 0 x) y = r x (r y y)) ==> |
|
824 ~(r (r (r 0 (r (r 0 (r 0 0)) (r 0 (r 0 0)))) (r 0 (r 0 0))) |
|
825 (r (r 0 (r (r 0 (r 0 0)) (r 0 (r 0 0)))) (r 0 (r 0 0))) = |
|
826 r (r 0 (r (r 0 (r 0 0)) (r 0 (r 0 0)))) (r 0 (r 0 0))) ==> F`}, |
|
827 |
|
828 {name = "COL060-3", |
|
829 comments = [], |
|
830 goal = ` |
|
831 (!x y z. b . x . y . z = x . (y . z)) /\ (!x y. t . x . y = y . x) ==> |
|
832 ~(b . (b . (t . b) . b) . t . x . y . z = y . (x . z)) ==> F`}, |
|
833 |
|
834 {name = "GEO002-4", |
|
835 comments = [], |
|
836 goal = ` |
|
837 (!x y z v. ~between x y z \/ ~between y v z \/ between x y v) /\ |
|
838 (!x y z. ~equidistant x y z z \/ x == y) /\ |
|
839 (!x y z v w. |
|
840 ~between x y z \/ ~between v z w \/ |
|
841 between x (outer_pasch y x v w z) v) /\ |
|
842 (!x y z v w. |
|
843 ~between x y z \/ ~between v z w \/ |
|
844 between w y (outer_pasch y x v w z)) /\ |
|
845 (!x y z v. between x y (extension x y z v)) /\ |
|
846 (!x y z v. equidistant x (extension y x z v) z v) /\ |
|
847 (!x y z v. ~(x == y) \/ ~between z v x \/ between z v y) ==> |
|
848 ~between a a b ==> F`}, |
|
849 |
|
850 {name = "GEO036-2", |
|
851 comments = [], |
|
852 goal = ` |
|
853 (!x y. equidistant x y y x) /\ |
|
854 (!x y z x' y' z'. |
|
855 ~equidistant x y z x' \/ ~equidistant x y y' z' \/ |
|
856 equidistant z x' y' z') /\ (!x y z. ~equidistant x y z z \/ x = y) /\ |
|
857 (!x y z v. between x y (extension x y z v)) /\ |
|
858 (!x y z v. equidistant x (extension y x z v) z v) /\ |
|
859 (!x y z v w x' y' z'. |
|
860 ~equidistant x y z v \/ ~equidistant y w v x' \/ |
|
861 ~equidistant x y' z z' \/ ~equidistant y y' v z' \/ ~between x y w \/ |
|
862 ~between z v x' \/ x = y \/ equidistant w y' x' z') /\ |
|
863 (!x y. ~between x y x \/ x = y) /\ |
|
864 (!x y z v w. |
|
865 ~between x y z \/ ~between v w z \/ |
|
866 between y (inner_pasch x y z w v) v) /\ |
|
867 (!x y z v w. |
|
868 ~between x y z \/ ~between v w z \/ |
|
869 between w (inner_pasch x y z w v) x) /\ |
|
870 ~between lower_dimension_point_1 lower_dimension_point_2 |
|
871 lower_dimension_point_3 /\ |
|
872 ~between lower_dimension_point_2 lower_dimension_point_3 |
|
873 lower_dimension_point_1 /\ |
|
874 ~between lower_dimension_point_3 lower_dimension_point_1 |
|
875 lower_dimension_point_2 /\ |
|
876 (!x y z v w. |
|
877 ~equidistant x y x z \/ ~equidistant v y v z \/ ~equidistant w y w z \/ |
|
878 between x v w \/ between v w x \/ between w x v \/ y = z) /\ |
|
879 (!x y z v w. |
|
880 ~between x y z \/ ~between v y w \/ x = y \/ |
|
881 between x v (euclid1 x v y w z)) /\ |
|
882 (!x y z v w. |
|
883 ~between x y z \/ ~between v y w \/ x = y \/ |
|
884 between x w (euclid2 x v y w z)) /\ |
|
885 (!x y z v w. |
|
886 ~between x y z \/ ~between v y w \/ x = y \/ |
|
887 between (euclid1 x v y w z) z (euclid2 x v y w z)) /\ |
|
888 (!x y z x' y' z'. |
|
889 ~equidistant x y x z \/ ~equidistant x x' x y' \/ ~between x y x' \/ |
|
890 ~between y z' x' \/ between z (continuous x y z z' x' y') y') /\ |
|
891 (!x y z x' y' z'. |
|
892 ~equidistant x y x z \/ ~equidistant x x' x y' \/ ~between x y x' \/ |
|
893 ~between y z' x' \/ equidistant x z' x (continuous x y z z' x' y')) /\ |
|
894 (!x y z v. ~(x = y) \/ ~between x z v \/ between y z v) /\ |
|
895 (!x y z v. ~(x = y) \/ ~between z x v \/ between z y v) /\ |
|
896 (!x y z v. ~(x = y) \/ ~between z v x \/ between z v y) /\ |
|
897 (!x y z v w. ~(x = y) \/ ~equidistant x z v w \/ equidistant y z v w) /\ |
|
898 (!x y z v w. ~(x = y) \/ ~equidistant z x v w \/ equidistant z y v w) /\ |
|
899 (!x y z v w. ~(x = y) \/ ~equidistant z v x w \/ equidistant z v y w) /\ |
|
900 (!x y z v w. ~(x = y) \/ ~equidistant z v w x \/ equidistant z v w y) /\ |
|
901 (!x y z x' y' z'. |
|
902 ~(x = y) \/ inner_pasch x z x' y' z' = inner_pasch y z x' y' z') /\ |
|
903 (!x y z x' y' z'. |
|
904 ~(x = y) \/ inner_pasch z x x' y' z' = inner_pasch z y x' y' z') /\ |
|
905 (!x y z x' y' z'. |
|
906 ~(x = y) \/ inner_pasch z x' x y' z' = inner_pasch z x' y y' z') /\ |
|
907 (!x y z x' y' z'. |
|
908 ~(x = y) \/ inner_pasch z x' y' x z' = inner_pasch z x' y' y z') /\ |
|
909 (!x y z x' y' z'. |
|
910 ~(x = y) \/ inner_pasch z x' y' z' x = inner_pasch z x' y' z' y) /\ |
|
911 (!x y z x' y' z'. |
|
912 ~(x = y) \/ euclid1 x z x' y' z' = euclid1 y z x' y' z') /\ |
|
913 (!x y z x' y' z'. |
|
914 ~(x = y) \/ euclid1 z x x' y' z' = euclid1 z y x' y' z') /\ |
|
915 (!x y z x' y' z'. |
|
916 ~(x = y) \/ euclid1 z x' x y' z' = euclid1 z x' y y' z') /\ |
|
917 (!x y z x' y' z'. |
|
918 ~(x = y) \/ euclid1 z x' y' x z' = euclid1 z x' y' y z') /\ |
|
919 (!x y z x' y' z'. |
|
920 ~(x = y) \/ euclid1 z x' y' z' x = euclid1 z x' y' z' y) /\ |
|
921 (!x y z x' y' z'. |
|
922 ~(x = y) \/ euclid2 x z x' y' z' = euclid2 y z x' y' z') /\ |
|
923 (!x y z x' y' z'. |
|
924 ~(x = y) \/ euclid2 z x x' y' z' = euclid2 z y x' y' z') /\ |
|
925 (!x y z x' y' z'. |
|
926 ~(x = y) \/ euclid2 z x' x y' z' = euclid2 z x' y y' z') /\ |
|
927 (!x y z x' y' z'. |
|
928 ~(x = y) \/ euclid2 z x' y' x z' = euclid2 z x' y' y z') /\ |
|
929 (!x y z x' y' z'. |
|
930 ~(x = y) \/ euclid2 z x' y' z' x = euclid2 z x' y' z' y) /\ |
|
931 (!x y z v w. ~(x = y) \/ extension x z v w = extension y z v w) /\ |
|
932 (!x y z v w. ~(x = y) \/ extension z x v w = extension z y v w) /\ |
|
933 (!x y z v w. ~(x = y) \/ extension z v x w = extension z v y w) /\ |
|
934 (!x y z v w. ~(x = y) \/ extension z v w x = extension z v w y) /\ |
|
935 (!x y z v w x' y'. |
|
936 ~(x = y) \/ continuous x z v w x' y' = continuous y z v w x' y') /\ |
|
937 (!x y z v w x' y'. |
|
938 ~(x = y) \/ continuous z x v w x' y' = continuous z y v w x' y') /\ |
|
939 (!x y z v w x' y'. |
|
940 ~(x = y) \/ continuous z v x w x' y' = continuous z v y w x' y') /\ |
|
941 (!x y z v w x' y'. |
|
942 ~(x = y) \/ continuous z v w x x' y' = continuous z v w y x' y') /\ |
|
943 (!x y z v w x' y'. |
|
944 ~(x = y) \/ continuous z v w x' x y' = continuous z v w x' y y') /\ |
|
945 (!x y z v w x' y'. |
|
946 ~(x = y) \/ continuous z v w x' y' x = continuous z v w x' y' y) ==> |
|
947 lower_dimension_point_1 = lower_dimension_point_2 \/ |
|
948 lower_dimension_point_2 = lower_dimension_point_3 \/ |
|
949 lower_dimension_point_1 = lower_dimension_point_3 ==> F`}, |
|
950 |
|
951 {name = "GRP010-4", |
|
952 comments = [], |
|
953 goal = ` |
|
954 (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ |
|
955 (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x y. ~(x = y) \/ i x = i y) /\ |
|
956 (!x y z. ~(x = y) \/ x * z = y * z) /\ |
|
957 (!x y z. ~(x = y) \/ z * x = z * y) /\ (!x y z. x * y * z = x * (y * z)) /\ |
|
958 (!x. 1 * x = x) /\ (!x. i x * x = 1) /\ c * b = 1 ==> ~(b * c = 1) ==> F`}, |
|
959 |
|
960 {name = "GRP057-1", |
|
961 comments = [], |
|
962 goal = ` |
|
963 (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ |
|
964 (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ |
|
965 (!x y z v. x * i (i (i y * (i x * z)) * v * i (y * v)) = z) /\ |
|
966 (!x y. ~(x = y) \/ i x = i y) /\ (!x y z. ~(x = y) \/ x * z = y * z) /\ |
|
967 (!x y z. ~(x = y) \/ z * x = z * y) ==> |
|
968 ~(i a1 * a1 = i b1 * b1) \/ ~(i b2 * b2 * a2 = a2) \/ |
|
969 ~(a3 * b3 * c3 = a3 * (b3 * c3)) ==> F`}, |
|
970 |
|
971 {name = "GRP086-1", |
|
972 comments = ["Bug check: used to be unsolvable without sticky constraints"], |
|
973 goal = ` |
|
974 (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ |
|
975 (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ |
|
976 (!x y z. x * (y * z * i (x * z)) = y) /\ (!x y. ~(x = y) \/ i x = i y) /\ |
|
977 (!x y z. ~(x = y) \/ x * z = y * z) /\ |
|
978 (!x y z. ~(x = y) \/ z * x = z * y) ==> |
|
979 (!x y. |
|
980 ~(i a1 * a1 = i b1 * b1) \/ ~(i b2 * b2 * a2 = a2) \/ |
|
981 ~(a3 * b3 * c3 = a3 * (b3 * c3)) \/ ~(x * y = y * x)) ==> F`}, |
|
982 |
|
983 {name = "GRP104-1", |
|
984 comments = [], |
|
985 goal = ` |
|
986 (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ |
|
987 (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ |
|
988 (!x y z. |
|
989 double_divide x |
|
990 (inverse |
|
991 (double_divide |
|
992 (inverse (double_divide (double_divide x y) (inverse z))) y)) = |
|
993 z) /\ (!x y. x * y = inverse (double_divide y x)) /\ |
|
994 (!x y. ~(x = y) \/ inverse x = inverse y) /\ |
|
995 (!x y z. ~(x = y) \/ x * z = y * z) /\ |
|
996 (!x y z. ~(x = y) \/ z * x = z * y) /\ |
|
997 (!x y z. ~(x = y) \/ double_divide x z = double_divide y z) /\ |
|
998 (!x y z. ~(x = y) \/ double_divide z x = double_divide z y) ==> |
|
999 (!x y. |
|
1000 ~(inverse a1 * a1 = inverse b1 * b1) \/ ~(inverse b2 * b2 * a2 = a2) \/ |
|
1001 ~(a3 * b3 * c3 = a3 * (b3 * c3)) \/ ~(x * y = y * x)) ==> F`}, |
|
1002 |
|
1003 {name = "GRP128-4.003", |
|
1004 comments = [], |
|
1005 goal = ` |
|
1006 (!x y. |
|
1007 ~elt x \/ ~elt y \/ product e_1 x y \/ product e_2 x y \/ |
|
1008 product e_3 x y) /\ |
|
1009 (!x y. |
|
1010 ~elt x \/ ~elt y \/ product x e_1 y \/ product x e_2 y \/ |
|
1011 product x e_3 y) /\ elt e_1 /\ elt e_2 /\ elt e_3 /\ ~(e_1 == e_2) /\ |
|
1012 ~(e_1 == e_3) /\ ~(e_2 == e_1) /\ ~(e_2 == e_3) /\ ~(e_3 == e_1) /\ |
|
1013 ~(e_3 == e_2) /\ |
|
1014 (!x y. |
|
1015 ~elt x \/ ~elt y \/ product x y e_1 \/ product x y e_2 \/ |
|
1016 product x y e_3) /\ |
|
1017 (!x y z v. ~product x y z \/ ~product x y v \/ z == v) /\ |
|
1018 (!x y z v. ~product x y z \/ ~product x v z \/ y == v) /\ |
|
1019 (!x y z v. ~product x y z \/ ~product v y z \/ x == v) ==> |
|
1020 (!x y z v. product x y z \/ ~product x z v \/ ~product z y v) /\ |
|
1021 (!x y z v. product x y z \/ ~product v x z \/ ~product v y x) /\ |
|
1022 (!x y z v. ~product x y z \/ ~product z y v \/ product x z v) ==> F`}, |
|
1023 |
|
1024 {name = "HEN006-3", |
|
1025 comments = [], |
|
1026 goal = ` |
|
1027 (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ |
|
1028 (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ |
|
1029 (!x y. ~(x <= y) \/ x / y = 0) /\ (!x y. ~(x / y = 0) \/ x <= y) /\ |
|
1030 (!x y. x / y <= x) /\ (!x y z. x / y / (z / y) <= x / z / y) /\ |
|
1031 (!x. 0 <= x) /\ (!x y. ~(x <= y) \/ ~(y <= x) \/ x = y) /\ (!x. x <= 1) /\ |
|
1032 (!x y z. ~(x = y) \/ x / z = y / z) /\ |
|
1033 (!x y z. ~(x = y) \/ z / x = z / y) /\ |
|
1034 (!x y z. ~(x = y) \/ ~(x <= z) \/ y <= z) /\ |
|
1035 (!x y z. ~(x = y) \/ ~(z <= x) \/ z <= y) /\ a / b <= d ==> |
|
1036 ~(a / d <= b) ==> F`}, |
|
1037 |
|
1038 {name = "LAT005-3", |
|
1039 comments = ["SAM's lemma"], |
|
1040 goal = ` |
|
1041 (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ |
|
1042 (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x. meet x x = x) /\ |
|
1043 (!x. join x x = x) /\ (!x y. meet x (join x y) = x) /\ |
|
1044 (!x y. join x (meet x y) = x) /\ (!x y. meet x y = meet y x) /\ |
|
1045 (!x y. join x y = join y x) /\ |
|
1046 (!x y z. meet (meet x y) z = meet x (meet y z)) /\ |
|
1047 (!x y z. join (join x y) z = join x (join y z)) /\ |
|
1048 (!x y z. ~(x = y) \/ join x z = join y z) /\ |
|
1049 (!x y z. ~(x = y) \/ join z x = join z y) /\ |
|
1050 (!x y z. ~(x = y) \/ meet x z = meet y z) /\ |
|
1051 (!x y z. ~(x = y) \/ meet z x = meet z y) /\ (!x. meet x 0 = 0) /\ |
|
1052 (!x. join x 0 = x) /\ (!x. meet x 1 = x) /\ (!x. join x 1 = 1) /\ |
|
1053 (!x y z. ~(meet x y = x) \/ meet y (join x z) = join x (meet z y)) /\ |
|
1054 (!x y. ~complement x y \/ meet x y = 0) /\ |
|
1055 (!x y. ~complement x y \/ join x y = 1) /\ |
|
1056 (!x y. ~(meet x y = 0) \/ ~(join x y = 1) \/ complement x y) /\ |
|
1057 (!x y z. ~(x = y) \/ ~complement x z \/ complement y z) /\ |
|
1058 (!x y z. ~(x = y) \/ ~complement z x \/ complement z y) /\ |
|
1059 complement r1 (join a b) /\ complement r2 (meet a b) ==> |
|
1060 ~(r1 = meet (join r1 (meet r2 b)) (join r1 (meet r2 a))) ==> F`}, |
|
1061 |
|
1062 {name = "LCL009-1", |
|
1063 comments = [], |
|
1064 goal = ` |
|
1065 (!x y. ~p (x - y) \/ ~p x \/ p y) /\ |
|
1066 (!x y z. p (x - y - (z - y - (x - z)))) ==> |
|
1067 ~p (a - b - c - (a - (b - c))) ==> F`}, |
|
1068 |
|
1069 {name = "LCL087-1", |
|
1070 comments = |
|
1071 ["Solved quickly by resolution when NOT tracking term-ordering constraints."], |
|
1072 goal = ` |
|
1073 (!x y. ~p (implies x y) \/ ~p x \/ p y) /\ |
|
1074 (!x y z v w. |
|
1075 p |
|
1076 (implies (implies (implies x y) (implies z v)) |
|
1077 (implies w (implies (implies v x) (implies z x))))) ==> |
|
1078 ~p (implies a (implies b a)) ==> F`}, |
|
1079 |
|
1080 {name = "LCL107-1", |
|
1081 comments = ["A very small problem that's tricky to prove."], |
|
1082 goal = ` |
|
1083 (!x y. ~p (x - y) \/ ~p x \/ p y) /\ |
|
1084 (!x y z v w x' y'. |
|
1085 p |
|
1086 (x - y - z - (v - w - (x' - w - (x' - v) - y')) - |
|
1087 (z - (y - x - y')))) ==> ~p (a - b - c - (e - b - (a - e - c))) ==> F`}, |
|
1088 |
|
1089 {name = "LCL187-1", |
|
1090 comments = [], |
|
1091 goal = ` |
|
1092 (!x. axiom (or (not (or x x)) x)) /\ (!x y. axiom (or (not x) (or y x))) /\ |
|
1093 (!x y. axiom (or (not (or x y)) (or y x))) /\ |
|
1094 (!x y z. axiom (or (not (or x (or y z))) (or y (or x z)))) /\ |
|
1095 (!x y z. axiom (or (not (or (not x) y)) (or (not (or z x)) (or z y)))) /\ |
|
1096 (!x. theorem x \/ ~axiom x) /\ |
|
1097 (!x y. theorem x \/ ~axiom (or (not y) x) \/ ~theorem y) /\ |
|
1098 (!x y z. |
|
1099 theorem (or (not x) y) \/ ~axiom (or (not x) z) \/ |
|
1100 ~theorem (or (not z) y)) ==> |
|
1101 ~theorem (or (not p) (or (not (not p)) q)) ==> F`}, |
|
1102 |
|
1103 {name = "LDA007-3", |
|
1104 comments = [], |
|
1105 goal = ` |
|
1106 (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ |
|
1107 (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ |
|
1108 (!x y z. f x (f y z) = f (f x y) (f x z)) /\ |
|
1109 (!x y z. ~(x = y) \/ f x z = f y z) /\ |
|
1110 (!x y z. ~(x = y) \/ f z x = f z y) /\ tt = f t t /\ ts = f t s /\ |
|
1111 tt_ts = f tt ts /\ tk = f t k /\ tsk = f ts k ==> |
|
1112 ~(f t tsk = f tt_ts tk) ==> F`}, |
|
1113 |
|
1114 {name = "NUM001-1", |
|
1115 comments = [], |
|
1116 goal = ` |
|
1117 (!x. x == x) /\ (!x y z. ~(x == y) \/ ~(y == z) \/ x == z) /\ |
|
1118 (!x y. x + y == y + x) /\ (!x y z. x + (y + z) == x + y + z) /\ |
|
1119 (!x y. x + y - y == x) /\ (!x y. x == x + y - y) /\ |
|
1120 (!x y z. x - y + z == x + z - y) /\ (!x y z. x + y - z == x - z + y) /\ |
|
1121 (!x y z v. ~(x == y) \/ ~(z == x + v) \/ z == y + v) /\ |
|
1122 (!x y z v. ~(x == y) \/ ~(z == v + x) \/ z == v + y) /\ |
|
1123 (!x y z v. ~(x == y) \/ ~(z == x - v) \/ z == y - v) /\ |
|
1124 (!x y z v. ~(x == y) \/ ~(z == v - x) \/ z == v - y) ==> |
|
1125 ~(a + b + c == a + (b + c)) ==> F`}, |
|
1126 |
|
1127 {name = "NUM014-1", |
|
1128 comments = [], |
|
1129 goal = ` |
|
1130 (!x. product x x (square x)) /\ |
|
1131 (!x y z. ~product x y z \/ product y x z) /\ |
|
1132 (!x y z. ~product x y z \/ divides x z) /\ |
|
1133 (!x y z v. |
|
1134 ~prime x \/ ~product y z v \/ ~divides x v \/ divides x y \/ |
|
1135 divides x z) /\ prime a /\ product a (square c) (square b) ==> |
|
1136 ~divides a b ==> F`}, |
|
1137 |
|
1138 {name = "PUZ001-1", |
|
1139 comments = [], |
|
1140 goal = ` |
|
1141 lives agatha /\ lives butler /\ lives charles /\ |
|
1142 (!x y. ~killed x y \/ ~richer x y) /\ |
|
1143 (!x. ~hates agatha x \/ ~hates charles x) /\ |
|
1144 (!x. ~hates x agatha \/ ~hates x butler \/ ~hates x charles) /\ |
|
1145 hates agatha agatha /\ hates agatha charles /\ |
|
1146 (!x y. ~killed x y \/ hates x y) /\ |
|
1147 (!x. ~hates agatha x \/ hates butler x) /\ |
|
1148 (!x. ~lives x \/ richer x agatha \/ hates butler x) ==> |
|
1149 killed butler agatha \/ killed charles agatha ==> F`}, |
|
1150 |
|
1151 {name = "PUZ011-1", |
|
1152 comments = |
|
1153 ["Curiosity: solved trivially by meson without cache cutting, but not with."], |
|
1154 goal = ` |
|
1155 ocean atlantic /\ ocean indian /\ borders atlantic brazil /\ |
|
1156 borders atlantic uruguay /\ borders atlantic venesuela /\ |
|
1157 borders atlantic zaire /\ borders atlantic nigeria /\ |
|
1158 borders atlantic angola /\ borders indian india /\ |
|
1159 borders indian pakistan /\ borders indian iran /\ borders indian somalia /\ |
|
1160 borders indian kenya /\ borders indian tanzania /\ south_american brazil /\ |
|
1161 south_american uruguay /\ south_american venesuela /\ african zaire /\ |
|
1162 african nigeria /\ african angola /\ african somalia /\ african kenya /\ |
|
1163 african tanzania /\ asian india /\ asian pakistan /\ asian iran ==> |
|
1164 (!x y z. |
|
1165 ~ocean x \/ ~borders x y \/ ~african y \/ ~borders x z \/ ~asian z) ==> |
|
1166 F`}, |
|
1167 |
|
1168 {name = "PUZ020-1", |
|
1169 comments = [], |
|
1170 goal = ` |
|
1171 (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ |
|
1172 (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ |
|
1173 (!x y. ~(x = y) \/ statement_by x = statement_by y) /\ |
|
1174 (!x. ~person x \/ knight x \/ knave x) /\ |
|
1175 (!x. ~person x \/ ~knight x \/ ~knave x) /\ |
|
1176 (!x y. ~says x y \/ a_truth y \/ ~a_truth y) /\ |
|
1177 (!x y. ~says x y \/ ~(x = y)) /\ (!x y. ~says x y \/ y = statement_by x) /\ |
|
1178 (!x y. ~person x \/ ~(x = statement_by y)) /\ |
|
1179 (!x. ~person x \/ ~a_truth (statement_by x) \/ knight x) /\ |
|
1180 (!x. ~person x \/ a_truth (statement_by x) \/ knave x) /\ |
|
1181 (!x y. ~(x = y) \/ ~knight x \/ knight y) /\ |
|
1182 (!x y. ~(x = y) \/ ~knave x \/ knave y) /\ |
|
1183 (!x y. ~(x = y) \/ ~person x \/ person y) /\ |
|
1184 (!x y z. ~(x = y) \/ ~says x z \/ says y z) /\ |
|
1185 (!x y z. ~(x = y) \/ ~says z x \/ says z y) /\ |
|
1186 (!x y. ~(x = y) \/ ~a_truth x \/ a_truth y) /\ |
|
1187 (!x y. ~knight x \/ ~says x y \/ a_truth y) /\ |
|
1188 (!x y. ~knave x \/ ~says x y \/ ~a_truth y) /\ person husband /\ |
|
1189 person wife /\ ~(husband = wife) /\ says husband (statement_by husband) /\ |
|
1190 (~a_truth (statement_by husband) \/ ~knight husband \/ knight wife) /\ |
|
1191 (a_truth (statement_by husband) \/ ~knight husband) /\ |
|
1192 (a_truth (statement_by husband) \/ knight wife) /\ |
|
1193 (~knight wife \/ a_truth (statement_by husband)) ==> ~knight husband ==> F`}, |
|
1194 |
|
1195 {name = "ROB002-1", |
|
1196 comments = [], |
|
1197 goal = ` |
|
1198 (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ |
|
1199 (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x y. x + y = y + x) /\ |
|
1200 (!x y z. x + y + z = x + (y + z)) /\ |
|
1201 (!x y. negate (negate (x + y) + negate (x + negate y)) = x) /\ |
|
1202 (!x y z. ~(x = y) \/ x + z = y + z) /\ |
|
1203 (!x y z. ~(x = y) \/ z + x = z + y) /\ |
|
1204 (!x y. ~(x = y) \/ negate x = negate y) /\ (!x. negate (negate x) = x) ==> |
|
1205 ~(negate (a + negate b) + negate (negate a + negate b) = b) ==> F`} |
|
1206 |
|
1207 ] @ |
|
1208 |
|
1209 (* ========================================================================= *) |
|
1210 (* Some problems from HOL. *) |
|
1211 (* ========================================================================= *) |
|
1212 |
|
1213 mkProblems "HOL subgoals sent to MESON_TAC" [ |
|
1214 |
|
1215 {name = "Coder_4_0", |
|
1216 comments = [], |
|
1217 goal = ` |
|
1218 (!x y. |
|
1219 x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\ |
|
1220 (!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\ |
|
1221 (!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\ |
|
1222 (!x y. ~{x} \/ ~(x = y) \/ {y}) /\ ~{existential . (K . falsity)} /\ |
|
1223 {existential . (K . truth)} /\ ~{universal . (K . falsity)} /\ |
|
1224 {universal . (K . truth)} /\ ~{falsity} /\ {truth} /\ |
|
1225 (!x y z. ~(APPEND . x . y = APPEND . z . y) \/ x = z) /\ |
|
1226 (!x y z. ~(APPEND . x . y = APPEND . x . z) \/ y = z) ==> |
|
1227 {wf_encoder . p . e} /\ (!x. e . x = f . x \/ ~{p . x}) /\ |
|
1228 {wf_encoder . p . f} /\ {p . q} /\ {p . q'} /\ |
|
1229 APPEND . (f . q) . r = APPEND . (f . q') . r' /\ q = q' /\ ~(r' = r) ==> F`}, |
|
1230 |
|
1231 {name = "DeepSyntax_47", |
|
1232 comments = [], |
|
1233 goal = ` |
|
1234 (!x y z v. ~(x = y) \/ ~(z = v) \/ int_add x z = int_add y v) /\ |
|
1235 (!x y. ~(x = y) \/ int_neg x = int_neg y) /\ |
|
1236 (!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\ |
|
1237 (!x y z v. ~(x = y) \/ ~(z = v) \/ int_lt y v \/ ~int_lt x z) /\ |
|
1238 (!x y z v. ~(x = y) \/ ~(z = v) \/ eval_form y v \/ ~eval_form x z) /\ |
|
1239 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1240 (!x y z v. |
|
1241 int_lt (int_add x y) (int_add z v) \/ ~int_lt x z \/ ~int_lt y v) /\ |
|
1242 (!x. int_add x (int_of_num 0) = x) /\ |
|
1243 (!x. int_add x (int_neg x) = int_of_num 0) /\ |
|
1244 (!x y z. int_add x (int_add y z) = int_add (int_add x y) z) ==> |
|
1245 int_lt (int_neg d) (int_of_num 0) /\ eval_form g x /\ |
|
1246 int_lt (int_add x d) i /\ ~int_lt x i ==> F`}, |
|
1247 |
|
1248 {name = "divides_9", |
|
1249 comments = [], |
|
1250 goal = ` |
|
1251 (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ |
|
1252 (!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\ |
|
1253 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1254 (!x y z. x * (y * z) = x * y * z) /\ (!x y. x * y = y * x) /\ |
|
1255 (!x y. ~divides x y \/ y = gv1556 x y * x) /\ |
|
1256 (!x y z. divides x y \/ ~(y = z * x)) ==> |
|
1257 divides gv1546 gv1547 /\ ~divides gv1546 (gv1547 * gv1548) ==> F`}, |
|
1258 |
|
1259 {name = "Encode_28", |
|
1260 comments = [], |
|
1261 goal = ` |
|
1262 (!x y z v. ~(x = y) \/ ~(z = v) \/ MOD x z = MOD y v) /\ |
|
1263 (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ |
|
1264 (!x y z v. ~(x = y) \/ ~(z = v) \/ x + z = y + v) /\ |
|
1265 (!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\ |
|
1266 (!x y. ~(x = y) \/ BIT2 x = BIT2 y) /\ |
|
1267 (!x y z v. ~(x = y) \/ ~(z = v) \/ EXP x z = EXP y v) /\ |
|
1268 (!x y z v. ~(x = y) \/ ~(z = v) \/ DIV x z = DIV y v) /\ |
|
1269 (!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\ |
|
1270 (!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\ |
|
1271 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1272 (!x y. x * y = y * x) /\ |
|
1273 (!x y z. MOD (MOD x (y * z)) y = MOD x y \/ ~(0 < y) \/ ~(0 < z)) ==> |
|
1274 (!x. |
|
1275 MOD x (NUMERAL (BIT2 ZERO)) = 0 \/ |
|
1276 MOD x (NUMERAL (BIT2 ZERO)) = NUMERAL (BIT1 ZERO)) /\ |
|
1277 MOD |
|
1278 (DIV x (NUMERAL (BIT2 ZERO)) * NUMERAL (BIT2 ZERO) + |
|
1279 MOD x (NUMERAL (BIT2 ZERO))) |
|
1280 (NUMERAL (BIT2 ZERO) * EXP (NUMERAL (BIT2 ZERO)) m) = |
|
1281 MOD |
|
1282 (DIV y (NUMERAL (BIT2 ZERO)) * NUMERAL (BIT2 ZERO) + |
|
1283 MOD y (NUMERAL (BIT2 ZERO))) |
|
1284 (NUMERAL (BIT2 ZERO) * EXP (NUMERAL (BIT2 ZERO)) m) /\ |
|
1285 0 < EXP (NUMERAL (BIT2 ZERO)) m /\ 0 < NUMERAL (BIT2 ZERO) /\ |
|
1286 (!x y. |
|
1287 ~(MOD (x * NUMERAL (BIT2 ZERO) + MOD (x) (NUMERAL (BIT2 ZERO))) |
|
1288 (NUMERAL (BIT2 ZERO)) = |
|
1289 MOD (y * NUMERAL (BIT2 ZERO) + MOD (y) (NUMERAL (BIT2 ZERO))) |
|
1290 (NUMERAL (BIT2 ZERO)))) ==> F`}, |
|
1291 |
|
1292 {name = "euclid_4", |
|
1293 comments = [], |
|
1294 goal = ` |
|
1295 (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ |
|
1296 (!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\ |
|
1297 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1298 (!x y z. x * (y * z) = x * y * z) /\ |
|
1299 (!x y. ~divides x y \/ y = x * gv5371 x y) /\ |
|
1300 (!x y z. divides x y \/ ~(y = x * z)) ==> |
|
1301 divides gv5316 gv5317 /\ divides gv5315 gv5316 /\ |
|
1302 ~divides gv5315 gv5317 ==> F`}, |
|
1303 |
|
1304 {name = "euclid_8", |
|
1305 comments = [], |
|
1306 goal = ` |
|
1307 (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ |
|
1308 (!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\ |
|
1309 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1310 (!x y. x * y = y * x) /\ (!x y z. x * (y * z) = x * y * z) /\ |
|
1311 (!x y. ~divides x y \/ y = x * gv7050 x y) /\ |
|
1312 (!x y z. divides x y \/ ~(y = x * z)) ==> |
|
1313 divides gv7000 gv7001 /\ ~divides gv7000 (gv7002 * gv7001) ==> F`}, |
|
1314 |
|
1315 {name = "extra_arith_6", |
|
1316 comments = [], |
|
1317 goal = ` |
|
1318 (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ |
|
1319 (!x y. ~(x = y) \/ SUC x = SUC y) /\ |
|
1320 (!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\ |
|
1321 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1322 (!x y z. ~(SUC x * y = SUC x * z) \/ y = z) /\ |
|
1323 (!x y z. SUC x * y = SUC x * z \/ ~(y = z)) /\ |
|
1324 (!x y z. x * (y * z) = x * y * z) /\ (!x y. x * y = y * x) ==> |
|
1325 SUC n * b = q * (SUC n * a) /\ 0 < SUC n /\ ~(b = q * a) ==> F`}, |
|
1326 |
|
1327 {name = "extra_real_5", |
|
1328 comments = [], |
|
1329 goal = ` |
|
1330 ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ |
|
1331 ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ |
|
1332 {truth} ==> |
|
1333 (!x y z v. |
|
1334 {real_lt . x . (sup . P)} \/ ~{P . y} \/ ~{real_lt . x . y} \/ |
|
1335 ~{P . z} \/ ~{real_lte . (gv6327 . v) . v}) /\ |
|
1336 (!x y z. |
|
1337 ~{real_lt . x . (sup . P)} \/ {P . (gv6327 . x)} \/ ~{P . y} \/ |
|
1338 ~{real_lte . (gv6327 . z) . z}) /\ |
|
1339 (!x y z. |
|
1340 ~{real_lt . x . (sup . P)} \/ {real_lt . x . (gv6327 . x)} \/ |
|
1341 ~{P . y} \/ ~{real_lte . (gv6327 . z) . z}) /\ |
|
1342 (!x y z. |
|
1343 ~{real_lt . x . (sup . P)} \/ {real_lt . x . (gv6327 . x)} \/ |
|
1344 ~{P . y} \/ {P . (gv6327 . z)}) /\ |
|
1345 (!x y z. |
|
1346 ~{real_lt . x . (sup . P)} \/ {P . (gv6327 . x)} \/ ~{P . y} \/ |
|
1347 {P . (gv6327 . z)}) /\ |
|
1348 (!x y z v. |
|
1349 {real_lt . x . (sup . P)} \/ ~{P . y} \/ ~{real_lt . x . y} \/ |
|
1350 ~{P . z} \/ {P . (gv6327 . v)}) /\ {P . x} /\ |
|
1351 (!x. {real_lte . x . z} \/ ~{P . x}) /\ |
|
1352 (!x. |
|
1353 {real_lt . (gv6328 . x) . (gv6329 . x)} \/ |
|
1354 {real_lt . (gv6328 . x) . x}) /\ |
|
1355 (!x. {P . (gv6329 . x)} \/ {real_lt . (gv6328 . x) . x}) /\ |
|
1356 (!x y. |
|
1357 ~{real_lt . (gv6328 . x) . y} \/ ~{P . y} \/ |
|
1358 ~{real_lt . (gv6328 . x) . x}) ==> F`}, |
|
1359 |
|
1360 {name = "gcd_19", |
|
1361 comments = [], |
|
1362 goal = ` |
|
1363 (!x y z v. ~(x = y) \/ ~(z = v) \/ x + z = y + v) /\ |
|
1364 (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ |
|
1365 (!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\ |
|
1366 (!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\ (!x y. ~(x = y) \/ SUC x = SUC y) /\ |
|
1367 (!x y z v. ~(x = y) \/ ~(z = v) \/ y <= v \/ ~(x <= z)) /\ |
|
1368 (!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\ |
|
1369 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1370 (!x y z. x * (y + z) = x * y + x * z) /\ (!x y. x + SUC y = SUC (x + y)) /\ |
|
1371 (!x y. SUC x + y = SUC (x + y)) /\ (!x. x + 0 = x) /\ (!x. 0 + x = x) /\ |
|
1372 (!x y. x * SUC y = x + x * y) /\ (!x y. SUC x * y = x * y + y) /\ |
|
1373 (!x. x * NUMERAL (BIT1 ZERO) = x) /\ (!x. NUMERAL (BIT1 ZERO) * x = x) /\ |
|
1374 (!x. x * 0 = 0) /\ (!x. 0 * x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\ |
|
1375 (!x y z. divides x y \/ ~divides x z \/ ~divides x (z + y)) ==> |
|
1376 ~(x + z <= x) /\ divides c (d * SUC x) /\ divides c (d * SUC (x + z)) /\ |
|
1377 ~divides c (d * z) ==> F`}, |
|
1378 |
|
1379 {name = "gcd_20", |
|
1380 comments = [], |
|
1381 goal = ` |
|
1382 (!x y z v. ~(x = y) \/ ~(z = v) \/ x + z = y + v) /\ |
|
1383 (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ |
|
1384 (!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\ |
|
1385 (!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\ (!x y. ~(x = y) \/ SUC x = SUC y) /\ |
|
1386 (!x y z v. ~(x = y) \/ ~(z = v) \/ y <= v \/ ~(x <= z)) /\ |
|
1387 (!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\ |
|
1388 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1389 (!x y z. x * (y + z) = x * y + x * z) /\ (!x y. x + SUC y = SUC (x + y)) /\ |
|
1390 (!x y. SUC x + y = SUC (x + y)) /\ (!x. x + 0 = x) /\ (!x. 0 + x = x) /\ |
|
1391 (!x y. x * SUC y = x + x * y) /\ (!x y. SUC x * y = x * y + y) /\ |
|
1392 (!x. x * NUMERAL (BIT1 ZERO) = x) /\ (!x. NUMERAL (BIT1 ZERO) * x = x) /\ |
|
1393 (!x. x * 0 = 0) /\ (!x. 0 * x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\ |
|
1394 (!x y z. divides x y \/ ~divides x z \/ ~divides x (z + y)) ==> |
|
1395 y <= y + z /\ divides c (d * SUC (y + z)) /\ divides c (d * SUC y) /\ |
|
1396 ~divides c (d * z) ==> F`}, |
|
1397 |
|
1398 {name = "gcd_21", |
|
1399 comments = [], |
|
1400 goal = ` |
|
1401 (!x y z v. ~(x = y) \/ ~(z = v) \/ x + z = y + v) /\ |
|
1402 (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ |
|
1403 (!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\ |
|
1404 (!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\ (!x y. ~(x = y) \/ SUC x = SUC y) /\ |
|
1405 (!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\ |
|
1406 (!x y z v. ~(x = y) \/ ~(z = v) \/ y <= v \/ ~(x <= z)) /\ |
|
1407 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1408 (!x y z. x * (y + z) = x * y + x * z) /\ (!x y. x + SUC y = SUC (x + y)) /\ |
|
1409 (!x y. SUC x + y = SUC (x + y)) /\ (!x. x + 0 = x) /\ (!x. 0 + x = x) /\ |
|
1410 (!x y. x * SUC y = x + x * y) /\ (!x y. SUC x * y = x * y + y) /\ |
|
1411 (!x. x * NUMERAL (BIT1 ZERO) = x) /\ (!x. NUMERAL (BIT1 ZERO) * x = x) /\ |
|
1412 (!x. x * 0 = 0) /\ (!x. 0 * x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\ |
|
1413 (!x y z. divides x y \/ ~divides x z \/ ~divides x (z + y)) ==> |
|
1414 divides c (d * SUC y) /\ y <= y + z /\ divides c (d * SUC (y + z)) /\ |
|
1415 ~divides c (d * z) ==> F`}, |
|
1416 |
|
1417 {name = "int_arith_6", |
|
1418 comments = [], |
|
1419 goal = ` |
|
1420 (!x y z v. ~(x = y) \/ ~(z = v) \/ int_mul x z = int_mul y v) /\ |
|
1421 (!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\ |
|
1422 (!x y z v. ~(x = y) \/ ~(z = v) \/ int_lt y v \/ ~int_lt x z) /\ |
|
1423 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1424 (!x y. int_mul x y = int_mul y x) /\ (!x. ~int_lt x x) /\ |
|
1425 (!x y z. ~(int_mul x y = int_mul z y) \/ y = int_of_num 0 \/ x = z) /\ |
|
1426 (!x y z. int_mul x y = int_mul z y \/ ~(y = int_of_num 0)) /\ |
|
1427 (!x y z. int_mul x y = int_mul z y \/ ~(x = z)) ==> |
|
1428 int_lt (int_of_num 0) gv1085 /\ |
|
1429 int_mul gv1085 gv1086 = int_mul gv1085 gv1087 /\ ~(gv1086 = gv1087) ==> F`}, |
|
1430 |
|
1431 {name = "int_arith_139", |
|
1432 comments = [], |
|
1433 goal = ` |
|
1434 (!x y z v. ~(x = y) \/ ~(z = v) \/ int_add x z = int_add y v) /\ |
|
1435 (!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\ |
|
1436 (!x y z v. ~(x = y) \/ ~(z = v) \/ int_le y v \/ ~int_le x z) /\ |
|
1437 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1438 (!x. int_add (int_of_num 0) x = x) /\ |
|
1439 (!x y z v. |
|
1440 int_le (int_add x y) (int_add z v) \/ ~int_le x z \/ ~int_le y v) /\ |
|
1441 (!x y z. int_add x (int_add y z) = int_add (int_add x y) z) /\ |
|
1442 (!x y. int_add x y = int_add y x) /\ |
|
1443 (!x y z. ~int_le (int_add x y) (int_add x z) \/ int_le y z) /\ |
|
1444 (!x y z. int_le (int_add x y) (int_add x z) \/ ~int_le y z) ==> |
|
1445 int_le x y /\ int_le (int_of_num 0) (int_add c x) /\ |
|
1446 ~int_le (int_of_num 0) (int_add c y) ==> F`}, |
|
1447 |
|
1448 {name = "llist_69", |
|
1449 comments = [], |
|
1450 goal = ` |
|
1451 (!x y. ~(x = y) \/ LTL x = LTL y) /\ (!x y. ~(x = y) \/ SOME x = SOME y) /\ |
|
1452 (!x y. ~(x = y) \/ LHD x = LHD y) /\ |
|
1453 (!x y z v. ~(x = y) \/ ~(z = v) \/ LCONS x z = LCONS y v) /\ |
|
1454 (!x y. ~(x = y) \/ g x = g y) /\ (!x y. ~(x = y) \/ THE x = THE y) /\ |
|
1455 (!x y z v. ~(x = y) \/ ~(z = v) \/ LNTH z x = LNTH v y) /\ |
|
1456 (!x y z v. ~(x = y) \/ ~(z = v) \/ LDROP z x = LDROP v y) /\ |
|
1457 (!x y. ~(x = y) \/ SUC x = SUC y) /\ |
|
1458 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1459 (!x y z. ~(x = LCONS y z) \/ LTL x = SOME z) /\ |
|
1460 (!x y z. ~(x = LCONS y z) \/ LHD x = SOME y) /\ |
|
1461 (!x y z. x = LCONS y z \/ ~(LHD x = SOME y) \/ ~(LTL x = SOME z)) ==> |
|
1462 LTL (g (LCONS LNIL t)) = |
|
1463 SOME (g (LCONS (THE (LTL (THE (LNTH n t)))) (THE (LDROP (SUC n) t)))) /\ |
|
1464 LHD (g (LCONS LNIL t)) = SOME (THE (LHD (THE (LNTH n t)))) /\ |
|
1465 LHD (g t) = SOME (THE (LHD (THE (LNTH n t)))) /\ |
|
1466 LTL (g t) = |
|
1467 SOME (g (LCONS (THE (LTL (THE (LNTH n t)))) (THE (LDROP (SUC n) t)))) /\ |
|
1468 ~(g (LCONS LNIL t) = g t) ==> F`}, |
|
1469 |
|
1470 {name = "MachineTransition_0", |
|
1471 comments = [], |
|
1472 goal = ` |
|
1473 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1474 (!x y. |
|
1475 x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\ |
|
1476 (!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\ |
|
1477 (!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\ |
|
1478 (!x y. ~{x} \/ ~(x = y) \/ {y}) /\ |
|
1479 (!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\ |
|
1480 ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ |
|
1481 ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ |
|
1482 {truth} /\ Eq = equality /\ |
|
1483 (!x y z. ~{Next . x . y . z} \/ {x . (pair . (gv940 . x . y . z) . z)}) /\ |
|
1484 (!x y z. ~{Next . x . y . z} \/ {y . (gv940 . x . y . z)}) /\ |
|
1485 (!x y z v. {Next . x . y . z} \/ ~{y . v} \/ ~{x . (pair . v . z)}) /\ |
|
1486 (!x y z. ~{Prev . x . y . z} \/ {y . (gv935 . x . y . z)}) /\ |
|
1487 (!x y z. ~{Prev . x . y . z} \/ {x . (pair . z . (gv935 . x . y . z))}) /\ |
|
1488 (!x y z v. {Prev . x . y . z} \/ ~{x . (pair . z . v)} \/ ~{y . v}) ==> |
|
1489 {Next . gv914 . (Eq . gv915) . gv916} /\ |
|
1490 ~{Prev . gv914 . (Eq . gv916) . gv915} ==> F`}, |
|
1491 |
|
1492 {name = "MachineTransition_2_1", |
|
1493 comments = [], |
|
1494 goal = ` |
|
1495 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1496 (!x y. |
|
1497 x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\ |
|
1498 (!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\ |
|
1499 (!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\ |
|
1500 (!x y. ~{x} \/ ~(x = y) \/ {y}) /\ |
|
1501 (!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\ |
|
1502 ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ |
|
1503 ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ |
|
1504 {truth} /\ |
|
1505 (!x y z. ReachIn . x . (Next . x . y) . z = ReachIn . x . y . (SUC . z)) /\ |
|
1506 (!x y z. ~{Next . x . y . z} \/ {x . (pair . (gv5488 . x . y . z) . z)}) /\ |
|
1507 (!x y z. ~{Next . x . y . z} \/ {y . (gv5488 . x . y . z)}) /\ |
|
1508 (!x y z v. {Next . x . y . z} \/ ~{y . v} \/ ~{x . (pair . v . z)}) /\ |
|
1509 (!x y z. ReachIn . x . y . (SUC . z) = Next . x . (ReachIn . x . y . z)) /\ |
|
1510 (!x y. ReachIn . x . y . 0 = y) ==> |
|
1511 {ReachIn . R . (Next . R . B) . gv5278 . state} /\ |
|
1512 (!x. ~{ReachIn . R . B . gv5278 . x} \/ ~{R . (pair . x . state)}) ==> F`}, |
|
1513 |
|
1514 {name = "MachineTransition_52", |
|
1515 comments = [], |
|
1516 goal = ` |
|
1517 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1518 (!x y. |
|
1519 x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\ |
|
1520 (!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\ |
|
1521 (!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\ |
|
1522 (!x y. ~{x} \/ ~(x = y) \/ {y}) /\ |
|
1523 (!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\ |
|
1524 ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ |
|
1525 ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ |
|
1526 {truth} /\ |
|
1527 (!x y z. |
|
1528 {(<=) . (gv7028 . x . y . z) . (add . x . (NUMERAL . (BIT1 . ZERO)))} \/ |
|
1529 z . (add . x . (NUMERAL . (BIT1 . ZERO))) = |
|
1530 y . (add . x . (NUMERAL . (BIT1 . ZERO)))) /\ |
|
1531 (!x y z. |
|
1532 ~(x . (gv7028 . y . z . x) = z . (gv7028 . y . z . x)) \/ |
|
1533 x . (add . y . (NUMERAL . (BIT1 . ZERO))) = |
|
1534 z . (add . y . (NUMERAL . (BIT1 . ZERO)))) /\ |
|
1535 (!x y z v. |
|
1536 ~(x . (gv7028 . y . z . x) = z . (gv7028 . y . z . x)) \/ |
|
1537 x . v = z . v \/ ~{(<=) . v . y}) /\ |
|
1538 (!x y z v. |
|
1539 {(<=) . (gv7028 . x . y . z) . (add . x . (NUMERAL . (BIT1 . ZERO)))} \/ |
|
1540 z . v = y . v \/ ~{(<=) . v . x}) /\ |
|
1541 (!x y z v. |
|
1542 ~{(<=) . x . (add . y . (NUMERAL . (BIT1 . ZERO)))} \/ z . x = v . x \/ |
|
1543 ~(z . (gv7027 . y . v . z) = v . (gv7027 . y . v . z)) \/ |
|
1544 ~(z . (add . y . (NUMERAL . (BIT1 . ZERO))) = |
|
1545 v . (add . y . (NUMERAL . (BIT1 . ZERO))))) /\ |
|
1546 (!x y z v. |
|
1547 ~{(<=) . x . (add . y . (NUMERAL . (BIT1 . ZERO)))} \/ z . x = v . x \/ |
|
1548 {(<=) . (gv7027 . y . v . z) . y} \/ |
|
1549 ~(z . (add . y . (NUMERAL . (BIT1 . ZERO))) = |
|
1550 v . (add . y . (NUMERAL . (BIT1 . ZERO))))) ==> |
|
1551 ({FinPath . (pair . R . s) . f2 . n} \/ |
|
1552 ~{FinPath . (pair . R . s) . f1 . n} \/ ~(f1 . gv7034 = f2 . gv7034)) /\ |
|
1553 (~{FinPath . (pair . R . s) . f2 . n} \/ |
|
1554 {FinPath . (pair . R . s) . f1 . n} \/ ~(f1 . gv7034 = f2 . gv7034)) /\ |
|
1555 (~{FinPath . (pair . R . s) . f2 . n} \/ |
|
1556 {FinPath . (pair . R . s) . f1 . n} \/ {(<=) . gv7034 . n}) /\ |
|
1557 ({FinPath . (pair . R . s) . f2 . n} \/ |
|
1558 ~{FinPath . (pair . R . s) . f1 . n} \/ {(<=) . gv7034 . n}) /\ |
|
1559 (!x. |
|
1560 f1 . x = f2 . x \/ |
|
1561 ~{(<=) . x . (add . n . (NUMERAL . (BIT1 . ZERO)))}) /\ |
|
1562 {FinPath . (pair . R . s) . f2 . n} /\ |
|
1563 {R . (pair . (f2 . n) . (f2 . (add . n . (NUMERAL . (BIT1 . ZERO)))))} /\ |
|
1564 ~{FinPath . (pair . R . s) . f1 . n} ==> F`}, |
|
1565 |
|
1566 {name = "measure_138", |
|
1567 comments = [], |
|
1568 goal = ` |
|
1569 (!x y z. ~SUBSET x y \/ IN z y \/ ~IN z x) /\ |
|
1570 (!x y. SUBSET x y \/ IN (gv122874 x y) x) /\ |
|
1571 (!x y. SUBSET x y \/ ~IN (gv122874 x y) y) /\ |
|
1572 (!x. sigma_algebra (sigma x)) /\ (!x y z. ~IN x (INTER y z) \/ IN x z) /\ |
|
1573 (!x y z. ~IN x (INTER y z) \/ IN x y) /\ |
|
1574 (!x y z. IN x (INTER y z) \/ ~IN x y \/ ~IN x z) /\ |
|
1575 (!x y. |
|
1576 ~sigma_algebra x \/ IN (BIGUNION y) x \/ ~countable y \/ ~SUBSET y x) /\ |
|
1577 (!x y. ~sigma_algebra x \/ IN (COMPL y) x \/ ~IN y x) /\ |
|
1578 (!x. ~sigma_algebra x \/ IN EMPTY x) /\ |
|
1579 (!x. |
|
1580 sigma_algebra x \/ ~IN EMPTY x \/ ~IN (COMPL (gv122851 x)) x \/ |
|
1581 SUBSET (gv122852 x) x) /\ |
|
1582 (!x. |
|
1583 sigma_algebra x \/ ~IN EMPTY x \/ IN (gv122851 x) x \/ |
|
1584 SUBSET (gv122852 x) x) /\ |
|
1585 (!x. |
|
1586 sigma_algebra x \/ ~IN EMPTY x \/ IN (gv122851 x) x \/ |
|
1587 countable (gv122852 x)) /\ |
|
1588 (!x. |
|
1589 sigma_algebra x \/ ~IN EMPTY x \/ ~IN (COMPL (gv122851 x)) x \/ |
|
1590 countable (gv122852 x)) /\ |
|
1591 (!x. |
|
1592 sigma_algebra x \/ ~IN EMPTY x \/ IN (gv122851 x) x \/ |
|
1593 ~IN (BIGUNION (gv122852 x)) x) /\ |
|
1594 (!x. |
|
1595 sigma_algebra x \/ ~IN EMPTY x \/ ~IN (COMPL (gv122851 x)) x \/ |
|
1596 ~IN (BIGUNION (gv122852 x)) x) ==> |
|
1597 SUBSET c (INTER p (sigma a)) /\ |
|
1598 (!x. IN (BIGUNION x) p \/ ~countable x \/ ~SUBSET x (INTER p (sigma a))) /\ |
|
1599 SUBSET a p /\ IN EMPTY p /\ |
|
1600 (!x. IN (COMPL x) p \/ ~IN x (INTER p (sigma a))) /\ countable c /\ |
|
1601 ~IN (BIGUNION c) (sigma a) ==> F`}, |
|
1602 |
|
1603 {name = "Omega_13", |
|
1604 comments = [], |
|
1605 goal = ` |
|
1606 (!x y z v. ~(x = y) \/ ~(z = v) \/ int_mul x z = int_mul y v) /\ |
|
1607 (!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\ |
|
1608 (!x y z v. ~(x = y) \/ ~(z = v) \/ evalupper y v \/ ~evalupper x z) /\ |
|
1609 (!x y z v. ~(x = y) \/ ~(z = v) \/ int_lt y v \/ ~int_lt x z) /\ |
|
1610 (!x y z v. ~(x = y) \/ ~(z = v) \/ int_le y v \/ ~int_le x z) /\ |
|
1611 (!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\ |
|
1612 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1613 (!x y. ~int_le x y \/ int_lt x y \/ x = y) /\ |
|
1614 (!x y. int_le x y \/ ~int_lt x y) /\ (!x y. int_le x y \/ ~(x = y)) /\ |
|
1615 (!x y z. |
|
1616 int_lt x y \/ |
|
1617 ~int_lt (int_mul (int_of_num z) x) (int_mul (int_of_num z) y) \/ |
|
1618 ~(0 < z)) /\ |
|
1619 (!x y z. |
|
1620 ~int_lt x y \/ |
|
1621 int_lt (int_mul (int_of_num z) x) (int_mul (int_of_num z) y) \/ |
|
1622 ~(0 < z)) /\ (!x y z. int_lt x y \/ ~int_le x z \/ ~int_lt z y) ==> |
|
1623 (!x y. evalupper x uppers \/ ~evalupper y uppers \/ ~int_lt x y) /\ |
|
1624 int_le (int_mul (int_of_num p_1) x) p_2 /\ evalupper x uppers /\ |
|
1625 int_lt y x /\ 0 < p_1 /\ ~int_le (int_mul (int_of_num p_1) y) p_2 ==> F`}, |
|
1626 |
|
1627 {name = "Omega_71", |
|
1628 comments = [], |
|
1629 goal = ` |
|
1630 (!x y z v. ~(x = y) \/ ~(z = v) \/ int_mul x z = int_mul y v) /\ |
|
1631 (!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\ |
|
1632 (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ |
|
1633 (!x y z v. ~(x = y) \/ ~(z = v) \/ int_add x z = int_add y v) /\ |
|
1634 (!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\ |
|
1635 (!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\ |
|
1636 (!x y z v. ~(x = y) \/ ~(z = v) \/ (x, z) = (y, v)) /\ |
|
1637 (!x y z v. ~(x = y) \/ ~(z = v) \/ evallower y v \/ ~evallower x z) /\ |
|
1638 (!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\ |
|
1639 (!x y z v. ~(x = y) \/ ~(z = v) \/ rshadow_row v y \/ ~rshadow_row z x) /\ |
|
1640 (!x y z v. |
|
1641 ~(x = y) \/ ~(z = v) \/ dark_shadow_cond_row v y \/ |
|
1642 ~dark_shadow_cond_row z x) /\ |
|
1643 (!x y z v. ~(x = y) \/ ~(z = v) \/ int_le y v \/ ~int_le x z) /\ |
|
1644 (!x y z v. ~(x = y) \/ ~(z = v) \/ EVERY y v \/ ~EVERY x z) /\ |
|
1645 (!x y z v. ~(x = y) \/ ~(z = v) \/ int_lt y v \/ ~int_lt x z) /\ |
|
1646 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1647 (!x y. int_mul x y = int_mul y x) /\ |
|
1648 (!x y z. int_mul x (int_mul y z) = int_mul (int_mul x y) z) /\ |
|
1649 (!x y z. |
|
1650 int_le x y \/ |
|
1651 ~int_le (int_mul (int_of_num z) x) (int_mul (int_of_num z) y) \/ |
|
1652 ~(0 < z)) /\ |
|
1653 (!x y z. |
|
1654 ~int_le x y \/ |
|
1655 int_le (int_mul (int_of_num z) x) (int_mul (int_of_num z) y) \/ |
|
1656 ~(0 < z)) ==> |
|
1657 (!x y z. |
|
1658 evallower (gv6249 x y z) lowers \/ ~(0 < y) \/ ~evallower x lowers \/ |
|
1659 ~rshadow_row (y, z) lowers \/ ~dark_shadow_cond_row (y, z) lowers) /\ |
|
1660 (!x y z. |
|
1661 int_le (int_mul (int_of_num x) (gv6249 y x z)) z \/ ~(0 < x) \/ |
|
1662 ~evallower y lowers \/ ~rshadow_row (x, z) lowers \/ |
|
1663 ~dark_shadow_cond_row (x, z) lowers) /\ 0 < c /\ |
|
1664 int_le R (int_mul (int_of_num d) x) /\ evallower x lowers /\ 0 < d /\ |
|
1665 EVERY fst_nzero lowers /\ |
|
1666 int_le (int_mul (int_of_num c) R) (int_mul (int_of_num d) L) /\ |
|
1667 rshadow_row (c, L) lowers /\ dark_shadow_cond_row (c, L) lowers /\ |
|
1668 (!x. |
|
1669 ~int_lt (int_mul (int_of_num d) L) |
|
1670 (int_mul (int_of_num (c * d)) |
|
1671 (int_add x (int_of_num (NUMERAL (BIT1 ZERO))))) \/ |
|
1672 ~int_lt (int_mul (int_of_num (c * d)) x) (int_mul (int_of_num c) R)) /\ |
|
1673 int_le (int_mul (int_of_num c) y) L /\ evallower y lowers /\ |
|
1674 int_le (int_mul (int_of_num (c * d)) y) (int_mul (int_of_num d) L) /\ |
|
1675 int_le (int_mul (int_of_num c) R) (int_mul (int_of_num (c * d)) x) /\ |
|
1676 int_lt (int_mul (int_of_num (c * d)) y) (int_mul (int_of_num c) R) /\ |
|
1677 0 < c * d /\ |
|
1678 int_le (int_mul (int_of_num c) R) (int_mul (int_of_num (c * d)) j) /\ |
|
1679 int_le (int_mul (int_of_num (c * d)) j) (int_mul (int_of_num d) L) /\ |
|
1680 int_le (int_mul (int_mul (int_of_num c) (int_of_num d)) j) |
|
1681 (int_mul (int_of_num d) L) /\ ~int_le (int_mul (int_of_num c) j) L ==> F`}, |
|
1682 |
|
1683 {name = "pred_set_1", |
|
1684 comments = ["Small problem that's hard for ordered resolution"], |
|
1685 goal = ` |
|
1686 (!x y z. ~(x <= y) \/ p z y \/ ~p z x) /\ (!x y. x <= y \/ ~p (a x y) y) /\ |
|
1687 (!x y. x <= y \/ p (a x y) x) /\ (!x y z. ~p x (y * z) \/ p x z) /\ |
|
1688 (!x y z. ~p x (y * z) \/ p x y) /\ |
|
1689 (!x y z. p x (y * z) \/ ~p x y \/ ~p x z) ==> |
|
1690 b <= c /\ b <= d /\ ~(b <= c * d) ==> F`}, |
|
1691 |
|
1692 {name = "pred_set_54_1", |
|
1693 comments = [], |
|
1694 goal = ` |
|
1695 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1696 (!x y. |
|
1697 x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\ |
|
1698 (!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\ |
|
1699 (!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\ |
|
1700 (!x y. ~{x} \/ ~(x = y) \/ {y}) /\ |
|
1701 (!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\ |
|
1702 ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ |
|
1703 ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ |
|
1704 {truth} /\ |
|
1705 (!x y z. ~{IN . x . (INSERT . y . z)} \/ x = y \/ {IN . x . z}) /\ |
|
1706 (!x y z. {IN . x . (INSERT . y . z)} \/ ~(x = y)) /\ |
|
1707 (!x y z. {IN . x . (INSERT . y . z)} \/ ~{IN . x . z}) ==> |
|
1708 (!x y z. f . x . (f . y . z) = f . y . (f . x . z)) /\ |
|
1709 (!x y z. |
|
1710 ITSET . f . (INSERT . x . y) . z = |
|
1711 ITSET . f . (DELETE . y . x) . (f . x . z) \/ |
|
1712 ~{less_than . (CARD . y) . v} \/ ~{FINITE . y}) /\ v = CARD . s /\ |
|
1713 {FINITE . s} /\ REST . (INSERT . x . s) = t /\ |
|
1714 CHOICE . (INSERT . x . s) = y /\ ~{IN . y . t} /\ ~{IN . x . s} /\ |
|
1715 INSERT . x . s = INSERT . y . t /\ ~(x = y) /\ ~{IN . x . t} ==> F`}, |
|
1716 |
|
1717 {name = "prob_44", |
|
1718 comments = [], |
|
1719 goal = ` |
|
1720 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1721 (!x y. |
|
1722 x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\ |
|
1723 (!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\ |
|
1724 (!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\ |
|
1725 (!x y. ~{x} \/ ~(x = y) \/ {y}) /\ |
|
1726 (!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\ |
|
1727 ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ |
|
1728 ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ |
|
1729 {truth} ==> |
|
1730 (!x y z. |
|
1731 ~{IN . x . (prefix_set . x')} \/ ~{IN . x . (prefix_set . (x))} \/ |
|
1732 ~{IN . y . c} \/ ~{IN . (gv24939 . y) . (prefix_set . (x))} \/ |
|
1733 ~{IN . (gv24939 . y) . (prefix_set . y)} \/ |
|
1734 ~{IN . (gv24940 . z) . (prefix_set . z)} \/ |
|
1735 ~{IN . (gv24940 . z) . (prefix_set . x')} \/ ~{IN . z . c}) /\ |
|
1736 (!x y z. |
|
1737 ~{IN . x . (prefix_set . x')} \/ ~{IN . x . (prefix_set . (x))} \/ |
|
1738 ~{IN . y . c} \/ {IN . (gv24939 . y) . (prefix_set . (x))} \/ |
|
1739 {IN . (gv24939 . y) . (prefix_set . y)} \/ |
|
1740 ~{IN . (gv24940 . z) . (prefix_set . z)} \/ |
|
1741 ~{IN . (gv24940 . z) . (prefix_set . x')} \/ ~{IN . z . c}) /\ |
|
1742 (!x y z. |
|
1743 ~{IN . x . (prefix_set . x')} \/ ~{IN . x . (prefix_set . (x))} \/ |
|
1744 ~{IN . y . c} \/ {IN . (gv24939 . y) . (prefix_set . (x))} \/ |
|
1745 {IN . (gv24939 . y) . (prefix_set . y)} \/ |
|
1746 {IN . (gv24940 . z) . (prefix_set . z)} \/ |
|
1747 {IN . (gv24940 . z) . (prefix_set . x')} \/ ~{IN . z . c}) /\ |
|
1748 (!x y z. |
|
1749 ~{IN . x . (prefix_set . x')} \/ ~{IN . x . (prefix_set . (x))} \/ |
|
1750 ~{IN . y . c} \/ ~{IN . (gv24939 . y) . (prefix_set . (x))} \/ |
|
1751 ~{IN . (gv24939 . y) . (prefix_set . y)} \/ |
|
1752 {IN . (gv24940 . z) . (prefix_set . z)} \/ |
|
1753 {IN . (gv24940 . z) . (prefix_set . x')} \/ ~{IN . z . c}) /\ |
|
1754 {IN . x' . c} /\ |
|
1755 {IN . (PREIMAGE . ((o) . SND . f) . s) . (events . bern)} /\ |
|
1756 (!x y. |
|
1757 f . x = |
|
1758 pair . (FST . (f . (prefix_seq . y))) . (sdrop . (LENGTH . y) . x) \/ |
|
1759 ~{IN . y . c} \/ ~{IN . x . (prefix_set . y)}) /\ |
|
1760 {IN . ((o) . SND . f) . |
|
1761 (measurable . (events . bern) . (events . bern))} /\ |
|
1762 {countable . (range . ((o) . FST . f))} /\ |
|
1763 {IN . ((o) . FST . f) . (measurable . (events . bern) . UNIV)} /\ |
|
1764 {prefix_cover . c} /\ {IN . s . (events . bern)} /\ {IN . x . c} /\ |
|
1765 ({IN . x'' . (prefix_set . x)} \/ {IN . x'' . (prefix_set . x')}) /\ |
|
1766 (~{IN . x'' . (prefix_set . x)} \/ ~{IN . x'' . (prefix_set . x')}) /\ |
|
1767 {IN . x''' . (prefix_set . x)} /\ {IN . x''' . (prefix_set . x')} ==> F`}, |
|
1768 |
|
1769 {name = "prob_53", |
|
1770 comments = [], |
|
1771 goal = ` |
|
1772 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1773 (!x y. |
|
1774 x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\ |
|
1775 (!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\ |
|
1776 (!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\ |
|
1777 (!x y. ~{x} \/ ~(x = y) \/ {y}) /\ |
|
1778 (!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\ |
|
1779 ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ |
|
1780 ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ |
|
1781 {truth} ==> |
|
1782 (!x y z. |
|
1783 ~{IN . x . (prefix_set . x''')} \/ ~{IN . x . (prefix_set . x'')} \/ |
|
1784 ~{IN . y . c} \/ ~{IN . (gv39280 . y) . (prefix_set . x'')} \/ |
|
1785 ~{IN . (gv39280 . y) . (prefix_set . y)} \/ |
|
1786 ~{IN . (gv39280 . z) . (prefix_set . z)} \/ |
|
1787 ~{IN . (gv39280 . z) . (prefix_set . x''')} \/ ~{IN . z . c}) /\ |
|
1788 (!x y z. |
|
1789 ~{IN . x . (prefix_set . x''')} \/ ~{IN . x . (prefix_set . x'')} \/ |
|
1790 ~{IN . y . c} \/ {IN . (gv39280 . y) . (prefix_set . x'')} \/ |
|
1791 {IN . (gv39280 . y) . (prefix_set . y)} \/ |
|
1792 ~{IN . (gv39280 . z) . (prefix_set . z)} \/ |
|
1793 ~{IN . (gv39280 . z) . (prefix_set . x''')} \/ ~{IN . z . c}) /\ |
|
1794 (!x y z. |
|
1795 ~{IN . x . (prefix_set . x''')} \/ ~{IN . x . (prefix_set . x'')} \/ |
|
1796 ~{IN . y . c} \/ {IN . (gv39280 . y) . (prefix_set . x'')} \/ |
|
1797 {IN . (gv39280 . y) . (prefix_set . y)} \/ |
|
1798 {IN . (gv39280 . z) . (prefix_set . z)} \/ |
|
1799 {IN . (gv39280 . z) . (prefix_set . x''')} \/ ~{IN . z . c}) /\ |
|
1800 (!x y z. |
|
1801 ~{IN . x . (prefix_set . x''')} \/ ~{IN . x . (prefix_set . x'')} \/ |
|
1802 ~{IN . y . c} \/ ~{IN . (gv39280 . y) . (prefix_set . x'')} \/ |
|
1803 ~{IN . (gv39280 . y) . (prefix_set . y)} \/ |
|
1804 {IN . (gv39280 . z) . (prefix_set . z)} \/ |
|
1805 {IN . (gv39280 . z) . (prefix_set . x''')} \/ ~{IN . z . c}) /\ |
|
1806 {IN . x''' . c} /\ |
|
1807 {IN . (PREIMAGE . ((o) . SND . f) . x') . (events . bern)} /\ |
|
1808 {IN . x' . (events . bern)} /\ {prefix_cover . c} /\ |
|
1809 {IN . ((o) . FST . f) . (measurable . (events . bern) . UNIV)} /\ |
|
1810 {countable . (range . ((o) . FST . f))} /\ |
|
1811 {IN . ((o) . SND . f) . |
|
1812 (measurable . (events . bern) . (events . bern))} /\ |
|
1813 (!x y. |
|
1814 f . x = |
|
1815 pair . (FST . (f . (prefix_seq . y))) . (sdrop . (LENGTH . y) . x) \/ |
|
1816 ~{IN . y . c} \/ ~{IN . x . (prefix_set . y)}) /\ |
|
1817 {IN . (PREIMAGE . ((o) . FST . f) . x) . (events . bern)} /\ |
|
1818 {IN . x'' . c} /\ |
|
1819 ({IN . x'''' . (prefix_set . x'')} \/ |
|
1820 {IN . x'''' . (prefix_set . x''')}) /\ |
|
1821 (~{IN . x'''' . (prefix_set . x'')} \/ |
|
1822 ~{IN . x'''' . (prefix_set . x''')}) /\ |
|
1823 {IN . x''''' . (prefix_set . x'')} /\ |
|
1824 {IN . x''''' . (prefix_set . x''')} ==> F`}, |
|
1825 |
|
1826 {name = "prob_extra_22", |
|
1827 comments = [], |
|
1828 goal = ` |
|
1829 ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ |
|
1830 ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ |
|
1831 {truth} ==> |
|
1832 {P . x} /\ (!x. {real_lte . x . z} \/ ~{P . x}) /\ |
|
1833 (!x y z v. |
|
1834 {real_lt . x . (sup . P)} \/ ~{P . y} \/ ~{real_lt . x . y} \/ |
|
1835 ~{P . z} \/ ~{real_lte . (gv13960 . v) . v}) /\ |
|
1836 (!x y z. |
|
1837 ~{real_lt . x . (sup . P)} \/ {P . (gv13960 . x)} \/ ~{P . y} \/ |
|
1838 ~{real_lte . (gv13960 . z) . z}) /\ |
|
1839 (!x y z. |
|
1840 ~{real_lt . x . (sup . P)} \/ {real_lt . x . (gv13960 . x)} \/ |
|
1841 ~{P . y} \/ ~{real_lte . (gv13960 . z) . z}) /\ |
|
1842 (!x y z. |
|
1843 ~{real_lt . x . (sup . P)} \/ {real_lt . x . (gv13960 . x)} \/ |
|
1844 ~{P . y} \/ {P . (gv13960 . z)}) /\ |
|
1845 (!x y z. |
|
1846 ~{real_lt . x . (sup . P)} \/ {P . (gv13960 . x)} \/ ~{P . y} \/ |
|
1847 {P . (gv13960 . z)}) /\ |
|
1848 (!x y z v. |
|
1849 {real_lt . x . (sup . P)} \/ ~{P . y} \/ ~{real_lt . x . y} \/ |
|
1850 ~{P . z} \/ {P . (gv13960 . v)}) /\ |
|
1851 (!x. |
|
1852 {real_lt . (gv13925 . x) . (gv13926 . x)} \/ |
|
1853 {real_lt . (gv13925 . x) . x}) /\ |
|
1854 (!x. {P . (gv13926 . x)} \/ {real_lt . (gv13925 . x) . x}) /\ |
|
1855 (!x y. |
|
1856 ~{real_lt . (gv13925 . x) . y} \/ ~{P . y} \/ |
|
1857 ~{real_lt . (gv13925 . x) . x}) ==> F`}, |
|
1858 |
|
1859 {name = "root2_2", |
|
1860 comments = [], |
|
1861 goal = ` |
|
1862 (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ |
|
1863 (!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\ |
|
1864 (!x y. ~(x = y) \/ BIT2 x = BIT2 y) /\ |
|
1865 (!x y z v. ~(x = y) \/ ~(z = v) \/ EXP x z = EXP y v) /\ |
|
1866 (!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\ |
|
1867 (!x y. ~(x = y) \/ EVEN y \/ ~EVEN x) /\ |
|
1868 (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ |
|
1869 (!x y. |
|
1870 ~(EXP (NUMERAL (BIT2 ZERO) * x) (NUMERAL (BIT2 ZERO)) = |
|
1871 NUMERAL (BIT2 ZERO) * EXP y (NUMERAL (BIT2 ZERO))) \/ |
|
1872 NUMERAL (BIT2 ZERO) * EXP x (NUMERAL (BIT2 ZERO)) = |
|
1873 EXP y (NUMERAL (BIT2 ZERO))) /\ |
|
1874 (!x y. |
|
1875 EXP (NUMERAL (BIT2 ZERO) * x) (NUMERAL (BIT2 ZERO)) = |
|
1876 NUMERAL (BIT2 ZERO) * EXP y (NUMERAL (BIT2 ZERO)) \/ |
|
1877 ~(NUMERAL (BIT2 ZERO) * EXP x (NUMERAL (BIT2 ZERO)) = |
|
1878 EXP y (NUMERAL (BIT2 ZERO)))) /\ |
|
1879 (!x. ~EVEN x \/ x = NUMERAL (BIT2 ZERO) * gv4671 x) /\ |
|
1880 (!x y. EVEN x \/ ~(x = NUMERAL (BIT2 ZERO) * y)) /\ |
|
1881 (!x y. ~EVEN (x * y) \/ EVEN x \/ EVEN y) /\ |
|
1882 (!x y. EVEN (x * y) \/ ~EVEN x) /\ (!x y. EVEN (x * y) \/ ~EVEN y) /\ |
|
1883 (!x. EXP x (NUMERAL (BIT2 ZERO)) = x * x) /\ |
|
1884 (!x. EVEN (NUMERAL (BIT2 ZERO) * x)) ==> |
|
1885 EXP (NUMERAL (BIT2 ZERO) * k) (NUMERAL (BIT2 ZERO)) = |
|
1886 NUMERAL (BIT2 ZERO) * EXP n (NUMERAL (BIT2 ZERO)) /\ |
|
1887 (!x y. |
|
1888 ~(EXP x (NUMERAL (BIT2 ZERO)) = |
|
1889 NUMERAL (BIT2 ZERO) * EXP y (NUMERAL (BIT2 ZERO))) \/ x = 0 \/ |
|
1890 ~(x < NUMERAL (BIT2 ZERO) * k)) /\ |
|
1891 (!x y. |
|
1892 ~(EXP x (NUMERAL (BIT2 ZERO)) = |
|
1893 NUMERAL (BIT2 ZERO) * EXP y (NUMERAL (BIT2 ZERO))) \/ y = 0 \/ |
|
1894 ~(x < NUMERAL (BIT2 ZERO) * k)) /\ |
|
1895 (!x. ~(n = NUMERAL (BIT2 ZERO) * x)) ==> F`}, |
|
1896 |
|
1897 {name = "TermRewriting_13", |
|
1898 comments = [], |
|
1899 goal = ` |
|
1900 ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ |
|
1901 ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ |
|
1902 {truth} /\ |
|
1903 (!x y z v. |
|
1904 ~{RTC . x . y . z} \/ {RTC . x . v . z} \/ ~{RTC . x . v . y}) ==> |
|
1905 {WCR . R} /\ {SN . R} /\ |
|
1906 (!x y z. |
|
1907 ~{RTC . R . x . y} \/ ~{RTC . R . x . z} \/ |
|
1908 {RTC . R . y . (gv1300 . x . z . y)} \/ ~{TC . R . (x) . x}) /\ |
|
1909 (!x y z. |
|
1910 ~{RTC . R . x . y} \/ ~{RTC . R . x . z} \/ |
|
1911 {RTC . R . z . (gv1300 . x . z . y)} \/ ~{TC . R . (x) . x}) /\ |
|
1912 {RTC . R . x . y} /\ {RTC . R . x . z} /\ {R . x . y1} /\ |
|
1913 {RTC . R . y1 . y} /\ {R . x . z1} /\ {RTC . R . z1 . z} /\ |
|
1914 {RTC . R . y1 . x0} /\ {RTC . R . z1 . x0} /\ {TC . R . x . y1} /\ |
|
1915 {TC . R . x . z1} /\ {RTC . R . y . y2} /\ {RTC . R . x0 . y2} /\ |
|
1916 {RTC . R . z . z2} /\ {RTC . R . x0 . z2} /\ {TC . R . x . x0} /\ |
|
1917 (!x. ~{RTC . R . y . x} \/ ~{RTC . R . z . x}) ==> F`} |
|
1918 |
|
1919 ]; |