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src/FOLP/ex/int.ML

author | haftmann |

Tue, 10 Jul 2007 17:30:50 +0200 | |

changeset 23709 | fd31da8f752a |

parent 18678 | dd0c569fa43d |

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

moved lfp_induct2 here

(* Title: FOLP/ex/int.ML ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1991 University of Cambridge Intuitionistic First-Order Logic Single-step commands: by (IntPr.step_tac 1); by (biresolve_tac safe_brls 1); by (biresolve_tac haz_brls 1); by (assume_tac 1); by (IntPr.safe_tac 1); by (IntPr.mp_tac 1); by (IntPr.fast_tac 1); *) (*Note: for PROPOSITIONAL formulae... ~A is classically provable iff it is intuitionistically provable. Therefore A is classically provable iff ~~A is intuitionistically provable. Let Q be the conjuction of the propositions A|~A, one for each atom A in P. If P is provable classically, then clearly P&Q is provable intuitionistically, so ~~(P&Q) is also provable intuitionistically. The latter is intuitionistically equivalent to ~~P&~~Q, hence to ~~P, since ~~Q is intuitionistically provable. Finally, if P is a negation then ~~P is intuitionstically equivalent to P. [Andy Pitts] *) goal (theory "IFOLP") "?p : ~~(P&Q) <-> ~~P & ~~Q"; by (IntPr.fast_tac 1); result(); goal (theory "IFOLP") "?p : ~~~P <-> ~P"; by (IntPr.fast_tac 1); result(); goal (theory "IFOLP") "?p : ~~((P --> Q | R) --> (P-->Q) | (P-->R))"; by (IntPr.fast_tac 1); result(); goal (theory "IFOLP") "?p : (P<->Q) <-> (Q<->P)"; by (IntPr.fast_tac 1); result(); writeln"Lemmas for the propositional double-negation translation"; goal (theory "IFOLP") "?p : P --> ~~P"; by (IntPr.fast_tac 1); result(); goal (theory "IFOLP") "?p : ~~(~~P --> P)"; by (IntPr.fast_tac 1); result(); goal (theory "IFOLP") "?p : ~~P & ~~(P --> Q) --> ~~Q"; by (IntPr.fast_tac 1); result(); writeln"The following are classically but not constructively valid."; (*The attempt to prove them terminates quickly!*) goal (theory "IFOLP") "?p : ((P-->Q) --> P) --> P"; by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected"; (*Check that subgoals remain: proof failed.*) getgoal 1; goal (theory "IFOLP") "?p : (P&Q-->R) --> (P-->R) | (Q-->R)"; by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected"; getgoal 1; writeln"Intuitionistic FOL: propositional problems based on Pelletier."; writeln"Problem ~~1"; goal (theory "IFOLP") "?p : ~~((P-->Q) <-> (~Q --> ~P))"; by (IntPr.fast_tac 1); result(); (*5 secs*) writeln"Problem ~~2"; goal (theory "IFOLP") "?p : ~~(~~P <-> P)"; by (IntPr.fast_tac 1); result(); (*1 secs*) writeln"Problem 3"; goal (theory "IFOLP") "?p : ~(P-->Q) --> (Q-->P)"; by (IntPr.fast_tac 1); result(); writeln"Problem ~~4"; goal (theory "IFOLP") "?p : ~~((~P-->Q) <-> (~Q --> P))"; by (IntPr.fast_tac 1); result(); (*9 secs*) writeln"Problem ~~5"; goal (theory "IFOLP") "?p : ~~((P|Q-->P|R) --> P|(Q-->R))"; by (IntPr.fast_tac 1); result(); (*10 secs*) writeln"Problem ~~6"; goal (theory "IFOLP") "?p : ~~(P | ~P)"; by (IntPr.fast_tac 1); result(); writeln"Problem ~~7"; goal (theory "IFOLP") "?p : ~~(P | ~~~P)"; by (IntPr.fast_tac 1); result(); writeln"Problem ~~8. Peirce's law"; goal (theory "IFOLP") "?p : ~~(((P-->Q) --> P) --> P)"; by (IntPr.fast_tac 1); result(); writeln"Problem 9"; goal (theory "IFOLP") "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"; by (IntPr.fast_tac 1); result(); (*9 secs*) writeln"Problem 10"; goal (theory "IFOLP") "?p : (Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)"; by (IntPr.fast_tac 1); result(); writeln"11. Proved in each direction (incorrectly, says Pelletier!!) "; goal (theory "IFOLP") "?p : P<->P"; by (IntPr.fast_tac 1); writeln"Problem ~~12. Dijkstra's law "; goal (theory "IFOLP") "?p : ~~(((P <-> Q) <-> R) <-> (P <-> (Q <-> R)))"; by (IntPr.fast_tac 1); result(); goal (theory "IFOLP") "?p : ((P <-> Q) <-> R) --> ~~(P <-> (Q <-> R))"; by (IntPr.fast_tac 1); result(); writeln"Problem 13. Distributive law"; goal (theory "IFOLP") "?p : P | (Q & R) <-> (P | Q) & (P | R)"; by (IntPr.fast_tac 1); result(); writeln"Problem ~~14"; goal (theory "IFOLP") "?p : ~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))"; by (IntPr.fast_tac 1); result(); writeln"Problem ~~15"; goal (theory "IFOLP") "?p : ~~((P --> Q) <-> (~P | Q))"; by (IntPr.fast_tac 1); result(); writeln"Problem ~~16"; goal (theory "IFOLP") "?p : ~~((P-->Q) | (Q-->P))"; by (IntPr.fast_tac 1); result(); writeln"Problem ~~17"; goal (theory "IFOLP") "?p : ~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))"; by (IntPr.fast_tac 1); (*over 5 minutes?? -- printing the proof term takes 40 secs!!*) result(); writeln"** Examples with quantifiers **"; writeln"The converse is classical in the following implications..."; goal (theory "IFOLP") "?p : (EX x. P(x)-->Q) --> (ALL x. P(x)) --> Q"; by (IntPr.fast_tac 1); result(); goal (theory "IFOLP") "?p : ((ALL x. P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)"; by (IntPr.fast_tac 1); result(); goal (theory "IFOLP") "?p : ((ALL x. ~P(x))-->Q) --> ~ (ALL x. ~ (P(x)|Q))"; by (IntPr.fast_tac 1); result(); goal (theory "IFOLP") "?p : (ALL x. P(x)) | Q --> (ALL x. P(x) | Q)"; by (IntPr.fast_tac 1); result(); goal (theory "IFOLP") "?p : (EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))"; by (IntPr.fast_tac 1); result(); writeln"The following are not constructively valid!"; (*The attempt to prove them terminates quickly!*) goal (theory "IFOLP") "?p : ((ALL x. P(x))-->Q) --> (EX x. P(x)-->Q)"; by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected"; getgoal 1; goal (theory "IFOLP") "?p : (P --> (EX x. Q(x))) --> (EX x. P-->Q(x))"; by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected"; getgoal 1; goal (theory "IFOLP") "?p : (ALL x. P(x) | Q) --> ((ALL x. P(x)) | Q)"; by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected"; getgoal 1; goal (theory "IFOLP") "?p : (ALL x. ~~P(x)) --> ~~(ALL x. P(x))"; by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected"; getgoal 1; (*Classically but not intuitionistically valid. Proved by a bug in 1986!*) goal (theory "IFOLP") "?p : EX x. Q(x) --> (ALL x. Q(x))"; by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected"; getgoal 1; writeln"Hard examples with quantifiers"; (*The ones that have not been proved are not known to be valid! Some will require quantifier duplication -- not currently available*) writeln"Problem ~~18"; goal (theory "IFOLP") "?p : ~~(EX y. ALL x. P(y)-->P(x))"; (*NOT PROVED*) writeln"Problem ~~19"; goal (theory "IFOLP") "?p : ~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))"; (*NOT PROVED*) writeln"Problem 20"; goal (theory "IFOLP") "?p : (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w))) \ \ --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))"; by (IntPr.fast_tac 1); result(); writeln"Problem 21"; goal (theory "IFOLP") "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))"; (*NOT PROVED*) writeln"Problem 22"; goal (theory "IFOLP") "?p : (ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))"; by (IntPr.fast_tac 1); result(); writeln"Problem ~~23"; goal (theory "IFOLP") "?p : ~~ ((ALL x. P | Q(x)) <-> (P | (ALL x. Q(x))))"; by (IntPr.best_tac 1); result(); writeln"Problem 24"; goal (theory "IFOLP") "?p : ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) & \ \ (~(EX x. P(x)) --> (EX x. Q(x))) & (ALL x. Q(x)|R(x) --> S(x)) \ \ --> ~~(EX x. P(x)&R(x))"; (*Not clear why fast_tac, best_tac, ASTAR and ITER_DEEPEN all take forever*) by IntPr.safe_tac; by (etac impE 1); by (IntPr.fast_tac 1); by (IntPr.fast_tac 1); result(); writeln"Problem 25"; goal (theory "IFOLP") "?p : (EX x. P(x)) & \ \ (ALL x. L(x) --> ~ (M(x) & R(x))) & \ \ (ALL x. P(x) --> (M(x) & L(x))) & \ \ ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x))) \ \ --> (EX x. Q(x)&P(x))"; by (IntPr.best_tac 1); result(); writeln"Problem 29. Essentially the same as Principia Mathematica *11.71"; goal (theory "IFOLP") "?p : (EX x. P(x)) & (EX y. Q(y)) \ \ --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y)) <-> \ \ (ALL x y. P(x) & Q(y) --> R(x) & S(y)))"; by (IntPr.fast_tac 1); result(); writeln"Problem ~~30"; goal (theory "IFOLP") "?p : (ALL x. (P(x) | Q(x)) --> ~ R(x)) & \ \ (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \ \ --> (ALL x. ~~S(x))"; by (IntPr.fast_tac 1); result(); writeln"Problem 31"; goal (theory "IFOLP") "?p : ~(EX x. P(x) & (Q(x) | R(x))) & \ \ (EX x. L(x) & P(x)) & \ \ (ALL x. ~ R(x) --> M(x)) \ \ --> (EX x. L(x) & M(x))"; by (IntPr.fast_tac 1); result(); writeln"Problem 32"; goal (theory "IFOLP") "?p : (ALL x. P(x) & (Q(x)|R(x))-->S(x)) & \ \ (ALL x. S(x) & R(x) --> L(x)) & \ \ (ALL x. M(x) --> R(x)) \ \ --> (ALL x. P(x) & M(x) --> L(x))"; by (IntPr.best_tac 1); (*SLOW*) result(); writeln"Problem 39"; goal (theory "IFOLP") "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"; by (IntPr.fast_tac 1); result(); writeln"Problem 40. AMENDED"; goal (theory "IFOLP") "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) --> \ \ ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))"; by (IntPr.fast_tac 1); result(); writeln"Problem 44"; goal (theory "IFOLP") "?p : (ALL x. f(x) --> \ \ (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y)))) & \ \ (EX x. j(x) & (ALL y. g(y) --> h(x,y))) \ \ --> (EX x. j(x) & ~f(x))"; by (IntPr.fast_tac 1); result(); writeln"Problem 48"; goal (theory "IFOLP") "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c"; by (IntPr.fast_tac 1); result(); writeln"Problem 51"; goal (theory "IFOLP") "?p : (EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \ \ (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)"; by (IntPr.best_tac 1); (*60 seconds*) result(); writeln"Problem 56"; goal (theory "IFOLP") "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"; by (IntPr.fast_tac 1); result(); writeln"Problem 57"; goal (theory "IFOLP") "?p : P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \ \ (ALL x y z. P(x,y) & P(y,z) --> P(x,z)) --> P(f(a,b), f(a,c))"; by (IntPr.fast_tac 1); result(); writeln"Problem 60"; goal (theory "IFOLP") "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"; by (IntPr.fast_tac 1); result();