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src/Sequents/LK/Propositional.thy

author | wenzelm |

Sat, 13 Mar 2010 16:44:12 +0100 | |

changeset 35762 | af3ff2ba4c54 |

parent 21426 | 87ac12bed1ab |

child 41959 | b460124855b8 |

permissions | -rw-r--r-- |

removed old CVS Ids;
tuned headers;

(* Title: LK/Propositional.thy Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge *) header {* Classical sequent calculus: examples with propositional connectives *} theory Propositional imports LK begin text "absorptive laws of & and | " lemma "|- P & P <-> P" by fast_prop lemma "|- P | P <-> P" by fast_prop text "commutative laws of & and | " lemma "|- P & Q <-> Q & P" by fast_prop lemma "|- P | Q <-> Q | P" by fast_prop text "associative laws of & and | " lemma "|- (P & Q) & R <-> P & (Q & R)" by fast_prop lemma "|- (P | Q) | R <-> P | (Q | R)" by fast_prop text "distributive laws of & and | " lemma "|- (P & Q) | R <-> (P | R) & (Q | R)" by fast_prop lemma "|- (P | Q) & R <-> (P & R) | (Q & R)" by fast_prop text "Laws involving implication" lemma "|- (P|Q --> R) <-> (P-->R) & (Q-->R)" by fast_prop lemma "|- (P & Q --> R) <-> (P--> (Q-->R))" by fast_prop lemma "|- (P --> Q & R) <-> (P-->Q) & (P-->R)" by fast_prop text "Classical theorems" lemma "|- P|Q --> P| ~P&Q" by fast_prop lemma "|- (P-->Q)&(~P-->R) --> (P&Q | R)" by fast_prop lemma "|- P&Q | ~P&R <-> (P-->Q)&(~P-->R)" by fast_prop lemma "|- (P-->Q) | (P-->R) <-> (P --> Q | R)" by fast_prop (*If and only if*) lemma "|- (P<->Q) <-> (Q<->P)" by fast_prop lemma "|- ~ (P <-> ~P)" by fast_prop (*Sample problems from F. J. Pelletier, Seventy-Five Problems for Testing Automatic Theorem Provers, J. Automated Reasoning 2 (1986), 191-216. Errata, JAR 4 (1988), 236-236. *) (*1*) lemma "|- (P-->Q) <-> (~Q --> ~P)" by fast_prop (*2*) lemma "|- ~ ~ P <-> P" by fast_prop (*3*) lemma "|- ~(P-->Q) --> (Q-->P)" by fast_prop (*4*) lemma "|- (~P-->Q) <-> (~Q --> P)" by fast_prop (*5*) lemma "|- ((P|Q)-->(P|R)) --> (P|(Q-->R))" by fast_prop (*6*) lemma "|- P | ~ P" by fast_prop (*7*) lemma "|- P | ~ ~ ~ P" by fast_prop (*8. Peirce's law*) lemma "|- ((P-->Q) --> P) --> P" by fast_prop (*9*) lemma "|- ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)" by fast_prop (*10*) lemma "Q-->R, R-->P&Q, P-->(Q|R) |- P<->Q" by fast_prop (*11. Proved in each direction (incorrectly, says Pelletier!!) *) lemma "|- P<->P" by fast_prop (*12. "Dijkstra's law"*) lemma "|- ((P <-> Q) <-> R) <-> (P <-> (Q <-> R))" by fast_prop (*13. Distributive law*) lemma "|- P | (Q & R) <-> (P | Q) & (P | R)" by fast_prop (*14*) lemma "|- (P <-> Q) <-> ((Q | ~P) & (~Q|P))" by fast_prop (*15*) lemma "|- (P --> Q) <-> (~P | Q)" by fast_prop (*16*) lemma "|- (P-->Q) | (Q-->P)" by fast_prop (*17*) lemma "|- ((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))" by fast_prop end