(* Title: LK/ex/prop
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Classical sequent calculus: examples with propositional connectives
Can be read to test the LK system.
*)
writeln"absorptive laws of & and | ";
goal (theory "LK") "|- P & P <-> P";
by (fast_tac prop_pack 1);
result();
goal (theory "LK") "|- P | P <-> P";
by (fast_tac prop_pack 1);
result();
writeln"commutative laws of & and | ";
goal (theory "LK") "|- P & Q <-> Q & P";
by (fast_tac prop_pack 1);
result();
goal (theory "LK") "|- P | Q <-> Q | P";
by (fast_tac prop_pack 1);
result();
writeln"associative laws of & and | ";
goal (theory "LK") "|- (P & Q) & R <-> P & (Q & R)";
by (fast_tac prop_pack 1);
result();
goal (theory "LK") "|- (P | Q) | R <-> P | (Q | R)";
by (fast_tac prop_pack 1);
result();
writeln"distributive laws of & and | ";
goal (theory "LK") "|- (P & Q) | R <-> (P | R) & (Q | R)";
by (fast_tac prop_pack 1);
result();
goal (theory "LK") "|- (P | Q) & R <-> (P & R) | (Q & R)";
by (fast_tac prop_pack 1);
result();
writeln"Laws involving implication";
goal (theory "LK") "|- (P|Q --> R) <-> (P-->R) & (Q-->R)";
by (fast_tac prop_pack 1);
result();
goal (theory "LK") "|- (P & Q --> R) <-> (P--> (Q-->R))";
by (fast_tac prop_pack 1);
result();
goal (theory "LK") "|- (P --> Q & R) <-> (P-->Q) & (P-->R)";
by (fast_tac prop_pack 1);
result();
writeln"Classical theorems";
goal (theory "LK") "|- P|Q --> P| ~P&Q";
by (fast_tac prop_pack 1);
result();
goal (theory "LK") "|- (P-->Q)&(~P-->R) --> (P&Q | R)";
by (fast_tac prop_pack 1);
result();
goal (theory "LK") "|- P&Q | ~P&R <-> (P-->Q)&(~P-->R)";
by (fast_tac prop_pack 1);
result();
goal (theory "LK") "|- (P-->Q) | (P-->R) <-> (P --> Q | R)";
by (fast_tac prop_pack 1);
result();
(*If and only if*)
goal (theory "LK") "|- (P<->Q) <-> (Q<->P)";
by (fast_tac prop_pack 1);
result();
goal (theory "LK") "|- ~ (P <-> ~P)";
by (fast_tac prop_pack 1);
result();
(*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*)
goal (theory "LK") "|- (P-->Q) <-> (~Q --> ~P)";
by (fast_tac prop_pack 1);
result();
(*2*)
goal (theory "LK") "|- ~ ~ P <-> P";
by (fast_tac prop_pack 1);
result();
(*3*)
goal (theory "LK") "|- ~(P-->Q) --> (Q-->P)";
by (fast_tac prop_pack 1);
result();
(*4*)
goal (theory "LK") "|- (~P-->Q) <-> (~Q --> P)";
by (fast_tac prop_pack 1);
result();
(*5*)
goal (theory "LK") "|- ((P|Q)-->(P|R)) --> (P|(Q-->R))";
by (fast_tac prop_pack 1);
result();
(*6*)
goal (theory "LK") "|- P | ~ P";
by (fast_tac prop_pack 1);
result();
(*7*)
goal (theory "LK") "|- P | ~ ~ ~ P";
by (fast_tac prop_pack 1);
result();
(*8. Peirce's law*)
goal (theory "LK") "|- ((P-->Q) --> P) --> P";
by (fast_tac prop_pack 1);
result();
(*9*)
goal (theory "LK") "|- ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
by (fast_tac prop_pack 1);
result();
(*10*)
goal (theory "LK") "Q-->R, R-->P&Q, P-->(Q|R) |- P<->Q";
by (fast_tac prop_pack 1);
result();
(*11. Proved in each direction (incorrectly, says Pelletier!!) *)
goal (theory "LK") "|- P<->P";
by (fast_tac prop_pack 1);
result();
(*12. "Dijkstra's law"*)
goal (theory "LK") "|- ((P <-> Q) <-> R) <-> (P <-> (Q <-> R))";
by (fast_tac prop_pack 1);
result();
(*13. Distributive law*)
goal (theory "LK") "|- P | (Q & R) <-> (P | Q) & (P | R)";
by (fast_tac prop_pack 1);
result();
(*14*)
goal (theory "LK") "|- (P <-> Q) <-> ((Q | ~P) & (~Q|P))";
by (fast_tac prop_pack 1);
result();
(*15*)
goal (theory "LK") "|- (P --> Q) <-> (~P | Q)";
by (fast_tac prop_pack 1);
result();
(*16*)
goal (theory "LK") "|- (P-->Q) | (Q-->P)";
by (fast_tac prop_pack 1);
result();
(*17*)
goal (theory "LK") "|- ((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))";
by (fast_tac prop_pack 1);
result();