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src/Sequents/LK/quant.ML

author | wenzelm |

Sun, 18 Sep 2005 15:20:08 +0200 | |

changeset 17481 | 75166ebb619b |

parent 6252 | 935f183bf406 |

child 18678 | dd0c569fa43d |

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

converted to Isar theory format;

(* Title: LK/ex/quant.ML ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge Classical sequent calculus: examples with quantifiers. *) goal (theory "LK") "|- (ALL x. P) <-> P"; by (fast_tac LK_pack 1); result(); goal (theory "LK") "|- (ALL x y. P(x,y)) <-> (ALL y x. P(x,y))"; by (fast_tac LK_pack 1); result(); goal (theory "LK") "ALL u. P(u), ALL v. Q(v) |- ALL u v. P(u) & Q(v)"; by (fast_tac LK_pack 1); result(); writeln"Permutation of existential quantifier."; goal (theory "LK") "|- (EX x y. P(x,y)) <-> (EX y x. P(x,y))"; by (fast_tac LK_pack 1); result(); goal (theory "LK") "|- (ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))"; by (fast_tac LK_pack 1); result(); (*Converse is invalid*) goal (theory "LK") "|- (ALL x. P(x)) | (ALL x. Q(x)) --> (ALL x. P(x)|Q(x))"; by (fast_tac LK_pack 1); result(); writeln"Pushing ALL into an implication."; goal (theory "LK") "|- (ALL x. P --> Q(x)) <-> (P --> (ALL x. Q(x)))"; by (fast_tac LK_pack 1); result(); goal (theory "LK") "|- (ALL x. P(x)-->Q) <-> ((EX x. P(x)) --> Q)"; by (fast_tac LK_pack 1); result(); goal (theory "LK") "|- (EX x. P) <-> P"; by (fast_tac LK_pack 1); result(); writeln"Distribution of EX over disjunction."; goal (theory "LK") "|- (EX x. P(x) | Q(x)) <-> (EX x. P(x)) | (EX x. Q(x))"; by (fast_tac LK_pack 1); result(); (*5 secs*) (*Converse is invalid*) goal (theory "LK") "|- (EX x. P(x) & Q(x)) --> (EX x. P(x)) & (EX x. Q(x))"; by (fast_tac LK_pack 1); result(); writeln"Harder examples: classical theorems."; goal (theory "LK") "|- (EX x. P-->Q(x)) <-> (P --> (EX x. Q(x)))"; by (fast_tac LK_pack 1); result(); (*3 secs*) goal (theory "LK") "|- (EX x. P(x)-->Q) <-> (ALL x. P(x)) --> Q"; by (fast_tac LK_pack 1); result(); (*5 secs*) goal (theory "LK") "|- (ALL x. P(x)) | Q <-> (ALL x. P(x) | Q)"; by (fast_tac LK_pack 1); result(); writeln"Basic test of quantifier reasoning"; goal (theory "LK") "|- (EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))"; by (fast_tac LK_pack 1); result(); goal (theory "LK") "|- (ALL x. Q(x)) --> (EX x. Q(x))"; by (fast_tac LK_pack 1); result(); writeln"The following are invalid!"; (*INVALID*) goal (theory "LK") "|- (ALL x. EX y. Q(x,y)) --> (EX y. ALL x. Q(x,y))"; by (fast_tac LK_pack 1) handle ERROR => writeln"Failed, as expected"; (*Check that subgoals remain: proof failed.*) getgoal 1; (*INVALID*) goal (theory "LK") "|- (EX x. Q(x)) --> (ALL x. Q(x))"; by (fast_tac LK_pack 1) handle ERROR => writeln"Failed, as expected"; getgoal 1; goal (theory "LK") "|- P(?a) --> (ALL x. P(x))"; by (fast_tac LK_pack 1) handle ERROR => writeln"Failed, as expected"; (*Check that subgoals remain: proof failed.*) getgoal 1; goal (theory "LK") "|- (P(?a) --> (ALL x. Q(x))) --> (ALL x. P(x) --> Q(x))"; by (fast_tac LK_pack 1) handle ERROR => writeln"Failed, as expected"; getgoal 1; writeln"Back to things that are provable..."; goal (theory "LK") "|- (ALL x. P(x)-->Q(x)) & (EX x. P(x)) --> (EX x. Q(x))"; by (fast_tac LK_pack 1); result(); (*An example of why exR should be delayed as long as possible*) goal (theory "LK") "|- (P--> (EX x. Q(x))) & P--> (EX x. Q(x))"; by (fast_tac LK_pack 1); result(); writeln"Solving for a Var"; goal (theory "LK") "|- (ALL x. P(x)-->Q(f(x))) & (ALL x. Q(x)-->R(g(x))) & P(d) --> R(?a)"; by (fast_tac LK_pack 1); result(); writeln"Principia Mathematica *11.53"; goal (theory "LK") "|- (ALL x y. P(x) --> Q(y)) <-> ((EX x. P(x)) --> (ALL y. Q(y)))"; by (fast_tac LK_pack 1); result(); writeln"Principia Mathematica *11.55"; goal (theory "LK") "|- (EX x y. P(x) & Q(x,y)) <-> (EX x. P(x) & (EX y. Q(x,y)))"; by (fast_tac LK_pack 1); result(); writeln"Principia Mathematica *11.61"; goal (theory "LK") "|- (EX y. ALL x. P(x) --> Q(x,y)) --> (ALL x. P(x) --> (EX y. Q(x,y)))"; by (fast_tac LK_pack 1); result(); (*21 August 88: loaded in 45.7 secs*) (*18 September 2005: loaded in 0.114 secs*)