author | paulson |
Wed, 09 Oct 1996 13:32:33 +0200 | |
changeset 2073 | fb0655539d05 |
permissions | -rw-r--r-- |
2073
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New unified treatment of sequent calculi by Sara Kalvala
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(* Title: LK/ex/prop |
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New unified treatment of sequent calculi by Sara Kalvala
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ID: $Id$ |
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New unified treatment of sequent calculi by Sara Kalvala
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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New unified treatment of sequent calculi by Sara Kalvala
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Copyright 1992 University of Cambridge |
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|
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New unified treatment of sequent calculi by Sara Kalvala
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Classical sequent calculus: examples with propositional connectives |
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Can be read to test the LK system. |
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*) |
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|
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writeln"LK/ex/prop: propositional examples"; |
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|
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New unified treatment of sequent calculi by Sara Kalvala
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writeln"absorptive laws of & and | "; |
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New unified treatment of sequent calculi by Sara Kalvala
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|
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New unified treatment of sequent calculi by Sara Kalvala
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goal LK.thy "|- P & P <-> P"; |
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New unified treatment of sequent calculi by Sara Kalvala
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by (fast_tac prop_pack 1); |
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New unified treatment of sequent calculi by Sara Kalvala
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result(); |
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New unified treatment of sequent calculi by Sara Kalvala
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|
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New unified treatment of sequent calculi by Sara Kalvala
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goal LK.thy "|- P | P <-> P"; |
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by (fast_tac prop_pack 1); |
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New unified treatment of sequent calculi by Sara Kalvala
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result(); |
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|
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writeln"commutative laws of & and | "; |
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|
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goal LK.thy "|- P & Q <-> Q & P"; |
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by (fast_tac prop_pack 1); |
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result(); |
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|
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goal LK.thy "|- P | Q <-> Q | P"; |
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by (fast_tac prop_pack 1); |
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result(); |
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|
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writeln"associative laws of & and | "; |
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|
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goal LK.thy "|- (P & Q) & R <-> P & (Q & R)"; |
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by (fast_tac prop_pack 1); |
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result(); |
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|
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goal LK.thy "|- (P | Q) | R <-> P | (Q | R)"; |
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by (fast_tac prop_pack 1); |
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result(); |
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writeln"distributive laws of & and | "; |
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goal LK.thy "|- (P & Q) | R <-> (P | R) & (Q | R)"; |
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by (fast_tac prop_pack 1); |
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result(); |
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|
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goal LK.thy "|- (P | Q) & R <-> (P & R) | (Q & R)"; |
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by (fast_tac prop_pack 1); |
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result(); |
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|
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writeln"Laws involving implication"; |
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|
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goal LK.thy "|- (P|Q --> R) <-> (P-->R) & (Q-->R)"; |
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by (fast_tac prop_pack 1); |
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New unified treatment of sequent calculi by Sara Kalvala
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result(); |
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|
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goal LK.thy "|- (P & Q --> R) <-> (P--> (Q-->R))"; |
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by (fast_tac prop_pack 1); |
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result(); |
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goal LK.thy "|- (P --> Q & R) <-> (P-->Q) & (P-->R)"; |
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by (fast_tac prop_pack 1); |
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result(); |
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writeln"Classical theorems"; |
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|
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goal LK.thy "|- P|Q --> P| ~P&Q"; |
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by (fast_tac prop_pack 1); |
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result(); |
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goal LK.thy "|- (P-->Q)&(~P-->R) --> (P&Q | R)"; |
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by (fast_tac prop_pack 1); |
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result(); |
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goal LK.thy "|- P&Q | ~P&R <-> (P-->Q)&(~P-->R)"; |
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by (fast_tac prop_pack 1); |
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result(); |
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goal LK.thy "|- (P-->Q) | (P-->R) <-> (P --> Q | R)"; |
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by (fast_tac prop_pack 1); |
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result(); |
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(*If and only if*) |
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|
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goal LK.thy "|- (P<->Q) <-> (Q<->P)"; |
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by (fast_tac prop_pack 1); |
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result(); |
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|
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goal LK.thy "|- ~ (P <-> ~P)"; |
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by (fast_tac prop_pack 1); |
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result(); |
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(*Sample problems from |
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F. J. Pelletier, |
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Seventy-Five Problems for Testing Automatic Theorem Provers, |
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104 |
J. Automated Reasoning 2 (1986), 191-216. |
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Errata, JAR 4 (1988), 236-236. |
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*) |
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107 |
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108 |
(*1*) |
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goal LK.thy "|- (P-->Q) <-> (~Q --> ~P)"; |
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by (fast_tac prop_pack 1); |
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111 |
result(); |
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112 |
|
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113 |
(*2*) |
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goal LK.thy "|- ~ ~ P <-> P"; |
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by (fast_tac prop_pack 1); |
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result(); |
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|
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118 |
(*3*) |
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goal LK.thy "|- ~(P-->Q) --> (Q-->P)"; |
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120 |
by (fast_tac prop_pack 1); |
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result(); |
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122 |
|
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123 |
(*4*) |
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124 |
goal LK.thy "|- (~P-->Q) <-> (~Q --> P)"; |
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by (fast_tac prop_pack 1); |
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result(); |
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127 |
|
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128 |
(*5*) |
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129 |
goal LK.thy "|- ((P|Q)-->(P|R)) --> (P|(Q-->R))"; |
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130 |
by (fast_tac prop_pack 1); |
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131 |
result(); |
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|
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133 |
(*6*) |
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134 |
goal LK.thy "|- P | ~ P"; |
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by (fast_tac prop_pack 1); |
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result(); |
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137 |
|
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138 |
(*7*) |
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139 |
goal LK.thy "|- P | ~ ~ ~ P"; |
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140 |
by (fast_tac prop_pack 1); |
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141 |
result(); |
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142 |
|
fb0655539d05
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143 |
(*8. Peirce's law*) |
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144 |
goal LK.thy "|- ((P-->Q) --> P) --> P"; |
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145 |
by (fast_tac prop_pack 1); |
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146 |
result(); |
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147 |
|
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148 |
(*9*) |
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149 |
goal LK.thy "|- ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"; |
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150 |
by (fast_tac prop_pack 1); |
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151 |
result(); |
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152 |
|
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153 |
(*10*) |
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154 |
goal LK.thy "Q-->R, R-->P&Q, P-->(Q|R) |- P<->Q"; |
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155 |
by (fast_tac prop_pack 1); |
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156 |
result(); |
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157 |
|
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158 |
(*11. Proved in each direction (incorrectly, says Pelletier!!) *) |
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159 |
goal LK.thy "|- P<->P"; |
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160 |
by (fast_tac prop_pack 1); |
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161 |
result(); |
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162 |
|
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163 |
(*12. "Dijkstra's law"*) |
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164 |
goal LK.thy "|- ((P <-> Q) <-> R) <-> (P <-> (Q <-> R))"; |
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165 |
by (fast_tac prop_pack 1); |
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166 |
result(); |
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167 |
|
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168 |
(*13. Distributive law*) |
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169 |
goal LK.thy "|- P | (Q & R) <-> (P | Q) & (P | R)"; |
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170 |
by (fast_tac prop_pack 1); |
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171 |
result(); |
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172 |
|
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173 |
(*14*) |
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174 |
goal LK.thy "|- (P <-> Q) <-> ((Q | ~P) & (~Q|P))"; |
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175 |
by (fast_tac prop_pack 1); |
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176 |
result(); |
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|
177 |
|
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178 |
|
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179 |
(*15*) |
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180 |
goal LK.thy "|- (P --> Q) <-> (~P | Q)"; |
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181 |
by (fast_tac prop_pack 1); |
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182 |
result(); |
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|
183 |
|
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184 |
(*16*) |
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185 |
goal LK.thy "|- (P-->Q) | (Q-->P)"; |
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|
186 |
by (fast_tac prop_pack 1); |
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|
187 |
result(); |
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|
188 |
|
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189 |
(*17*) |
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190 |
goal LK.thy "|- ((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))"; |
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|
191 |
by (fast_tac prop_pack 1); |
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|
192 |
result(); |
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|
193 |
|
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|
194 |
writeln"Reached end of file."; |