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(* Title: FOL/ex/int


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ID: $Id$


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Author: Lawrence C Paulson, Cambridge University Computer Laboratory


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Copyright 1991 University of Cambridge


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Intuitionistic FirstOrder Logic


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Singlestep commands:


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by (Int.step_tac 1);


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by (biresolve_tac safe_brls 1);


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by (biresolve_tac haz_brls 1);


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by (assume_tac 1);


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by (Int.safe_tac 1);


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by (Int.mp_tac 1);


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by (Int.fast_tac 1);


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*)


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writeln"File FOL/ex/int.";


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(*Note: for PROPOSITIONAL formulae...


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~A is classically provable iff it is intuitionistically provable.


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Therefore A is classically provable iff ~~A is intuitionistically provable.


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Let Q be the conjuction of the propositions A~A, one for each atom A in


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P. If P is provable classically, then clearly P&Q is provable


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intuitionistically, so ~~(P&Q) is also provable intuitionistically.


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The latter is intuitionistically equivalent to ~~P&~~Q, hence to ~~P,


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since ~~Q is intuitionistically provable. Finally, if P is a negation then


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~~P is intuitionstically equivalent to P. [Andy Pitts]


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*)


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goal IFOLP.thy "?p : ~~(P&Q) <> ~~P & ~~Q";


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by (Int.fast_tac 1);


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result();


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goal IFOLP.thy "?p : ~~~P <> ~P";


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by (Int.fast_tac 1);


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result();


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goal IFOLP.thy "?p : ~~((P > Q  R) > (P>Q)  (P>R))";


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by (Int.fast_tac 1);


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result();


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goal IFOLP.thy "?p : (P<>Q) <> (Q<>P)";


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by (Int.fast_tac 1);


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result();


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writeln"Lemmas for the propositional doublenegation translation";


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goal IFOLP.thy "?p : P > ~~P";


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by (Int.fast_tac 1);


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result();


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goal IFOLP.thy "?p : ~~(~~P > P)";


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by (Int.fast_tac 1);


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result();


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goal IFOLP.thy "?p : ~~P & ~~(P > Q) > ~~Q";


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by (Int.fast_tac 1);


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result();


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writeln"The following are classically but not constructively valid.";


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(*The attempt to prove them terminates quickly!*)


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goal IFOLP.thy "?p : ((P>Q) > P) > P";


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by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";


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(*Check that subgoals remain: proof failed.*)


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getgoal 1;


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goal IFOLP.thy "?p : (P&Q>R) > (P>R)  (Q>R)";


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by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";


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getgoal 1;


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writeln"Intuitionistic FOL: propositional problems based on Pelletier.";


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writeln"Problem ~~1";


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goal IFOLP.thy "?p : ~~((P>Q) <> (~Q > ~P))";


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by (Int.fast_tac 1);


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result();


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(*5 secs*)


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writeln"Problem ~~2";


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goal IFOLP.thy "?p : ~~(~~P <> P)";


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by (Int.fast_tac 1);


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result();


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(*1 secs*)


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writeln"Problem 3";


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goal IFOLP.thy "?p : ~(P>Q) > (Q>P)";


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by (Int.fast_tac 1);


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result();


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writeln"Problem ~~4";


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goal IFOLP.thy "?p : ~~((~P>Q) <> (~Q > P))";


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by (Int.fast_tac 1);


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result();


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(*9 secs*)


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writeln"Problem ~~5";


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goal IFOLP.thy "?p : ~~((PQ>PR) > P(Q>R))";


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by (Int.fast_tac 1);


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result();


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(*10 secs*)


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writeln"Problem ~~6";


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goal IFOLP.thy "?p : ~~(P  ~P)";


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by (Int.fast_tac 1);


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result();


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writeln"Problem ~~7";


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goal IFOLP.thy "?p : ~~(P  ~~~P)";


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by (Int.fast_tac 1);


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result();


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writeln"Problem ~~8. Peirce's law";


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goal IFOLP.thy "?p : ~~(((P>Q) > P) > P)";


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by (Int.fast_tac 1);


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result();


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writeln"Problem 9";


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goal IFOLP.thy "?p : ((PQ) & (~PQ) & (P ~Q)) > ~ (~P  ~Q)";


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by (Int.fast_tac 1);


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result();


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(*9 secs*)


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writeln"Problem 10";


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goal IFOLP.thy "?p : (Q>R) > (R>P&Q) > (P>(QR)) > (P<>Q)";


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by (Int.fast_tac 1);


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result();


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writeln"11. Proved in each direction (incorrectly, says Pelletier!!) ";


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goal IFOLP.thy "?p : P<>P";


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by (Int.fast_tac 1);


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writeln"Problem ~~12. Dijkstra's law ";


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goal IFOLP.thy "?p : ~~(((P <> Q) <> R) <> (P <> (Q <> R)))";


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by (Int.fast_tac 1);


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result();


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goal IFOLP.thy "?p : ((P <> Q) <> R) > ~~(P <> (Q <> R))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem 13. Distributive law";


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goal IFOLP.thy "?p : P  (Q & R) <> (P  Q) & (P  R)";


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by (Int.fast_tac 1);


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result();


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writeln"Problem ~~14";


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goal IFOLP.thy "?p : ~~((P <> Q) <> ((Q  ~P) & (~QP)))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem ~~15";


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goal IFOLP.thy "?p : ~~((P > Q) <> (~P  Q))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem ~~16";


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goal IFOLP.thy "?p : ~~((P>Q)  (Q>P))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem ~~17";


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goal IFOLP.thy


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"?p : ~~(((P & (Q>R))>S) <> ((~P  Q  S) & (~P  ~R  S)))";


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by (Int.fast_tac 1);


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(*over 5 minutes??  printing the proof term takes 40 secs!!*)


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result();


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writeln"** Examples with quantifiers **";


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writeln"The converse is classical in the following implications...";


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goal IFOLP.thy "?p : (EX x.P(x)>Q) > (ALL x.P(x)) > Q";


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by (Int.fast_tac 1);


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result();


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goal IFOLP.thy "?p : ((ALL x.P(x))>Q) > ~ (ALL x. P(x) & ~Q)";


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by (Int.fast_tac 1);


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result();


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goal IFOLP.thy "?p : ((ALL x. ~P(x))>Q) > ~ (ALL x. ~ (P(x)Q))";


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by (Int.fast_tac 1);


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result();


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goal IFOLP.thy "?p : (ALL x.P(x))  Q > (ALL x. P(x)  Q)";


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by (Int.fast_tac 1);


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result();


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goal IFOLP.thy "?p : (EX x. P > Q(x)) > (P > (EX x. Q(x)))";


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by (Int.fast_tac 1);


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result();


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writeln"The following are not constructively valid!";


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(*The attempt to prove them terminates quickly!*)


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goal IFOLP.thy "?p : ((ALL x.P(x))>Q) > (EX x.P(x)>Q)";


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by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";


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getgoal 1;


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goal IFOLP.thy "?p : (P > (EX x.Q(x))) > (EX x. P>Q(x))";


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by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";


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getgoal 1;


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goal IFOLP.thy "?p : (ALL x. P(x)  Q) > ((ALL x.P(x))  Q)";


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by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";


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getgoal 1;


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goal IFOLP.thy "?p : (ALL x. ~~P(x)) > ~~(ALL x. P(x))";


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by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";


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getgoal 1;


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(*Classically but not intuitionistically valid. Proved by a bug in 1986!*)


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goal IFOLP.thy "?p : EX x. Q(x) > (ALL x. Q(x))";


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by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";


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getgoal 1;


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writeln"Hard examples with quantifiers";


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(*The ones that have not been proved are not known to be valid!


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Some will require quantifier duplication  not currently available*)


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writeln"Problem ~~18";


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goal IFOLP.thy "?p : ~~(EX y. ALL x. P(y)>P(x))";


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(*NOT PROVED*)


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writeln"Problem ~~19";


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goal IFOLP.thy "?p : ~~(EX x. ALL y z. (P(y)>Q(z)) > (P(x)>Q(x)))";


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(*NOT PROVED*)


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writeln"Problem 20";


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goal IFOLP.thy "?p : (ALL x y. EX z. ALL w. (P(x)&Q(y)>R(z)&S(w))) \


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\ > (EX x y. P(x) & Q(y)) > (EX z. R(z))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem 21";


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goal IFOLP.thy


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"?p : (EX x. P>Q(x)) & (EX x. Q(x)>P) > ~~(EX x. P<>Q(x))";


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(*NOT PROVED*)


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writeln"Problem 22";


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goal IFOLP.thy "?p : (ALL x. P <> Q(x)) > (P <> (ALL x. Q(x)))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem ~~23";


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goal IFOLP.thy "?p : ~~ ((ALL x. P  Q(x)) <> (P  (ALL x. Q(x))))";


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by (Int.best_tac 1);


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result();


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writeln"Problem 24";


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goal IFOLP.thy "?p : ~(EX x. S(x)&Q(x)) & (ALL x. P(x) > Q(x)R(x)) & \


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\ ~(EX x.P(x)) > (EX x.Q(x)) & (ALL x. Q(x)R(x) > S(x)) \


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\ > (EX x. P(x)&R(x))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem 25";


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goal IFOLP.thy "?p : (EX x. P(x)) & \


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\ (ALL x. L(x) > ~ (M(x) & R(x))) & \


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\ (ALL x. P(x) > (M(x) & L(x))) & \


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\ ((ALL x. P(x)>Q(x))  (EX x. P(x)&R(x))) \


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\ > (EX x. Q(x)&P(x))";


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by (Int.best_tac 1);


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result();


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writeln"Problem 29. Essentially the same as Principia Mathematica *11.71";


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goal IFOLP.thy "?p : (EX x. P(x)) & (EX y. Q(y)) \


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\ > ((ALL x. P(x)>R(x)) & (ALL y. Q(y)>S(y)) <> \


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\ (ALL x y. P(x) & Q(y) > R(x) & S(y)))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem ~~30";


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goal IFOLP.thy "?p : (ALL x. (P(x)  Q(x)) > ~ R(x)) & \


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\ (ALL x. (Q(x) > ~ S(x)) > P(x) & R(x)) \


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\ > (ALL x. ~~S(x))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem 31";


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goal IFOLP.thy "?p : ~(EX x.P(x) & (Q(x)  R(x))) & \


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\ (EX x. L(x) & P(x)) & \


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\ (ALL x. ~ R(x) > M(x)) \


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\ > (EX x. L(x) & M(x))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem 32";


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goal IFOLP.thy "?p : (ALL x. P(x) & (Q(x)R(x))>S(x)) & \


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\ (ALL x. S(x) & R(x) > L(x)) & \


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\ (ALL x. M(x) > R(x)) \


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\ > (ALL x. P(x) & M(x) > L(x))";


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by (Int.best_tac 1); (*SLOW*)


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result();


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writeln"Problem 39";


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goal IFOLP.thy "?p : ~ (EX x. ALL y. F(y,x) <> ~F(y,y))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem 40. AMENDED";


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goal IFOLP.thy "?p : (EX y. ALL x. F(x,y) <> F(x,x)) > \


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\ ~(ALL x. EX y. ALL z. F(z,y) <> ~ F(z,x))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem 44";


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goal IFOLP.thy "?p : (ALL x. f(x) > \


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\ (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y)))) & \


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\ (EX x. j(x) & (ALL y. g(y) > h(x,y))) \


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\ > (EX x. j(x) & ~f(x))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem 48";


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goal IFOLP.thy "?p : (a=b  c=d) & (a=c  b=d) > a=d  b=c";


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by (Int.fast_tac 1);


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result();


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writeln"Problem 51";


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goal IFOLP.thy


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"?p : (EX z w. ALL x y. P(x,y) <> (x=z & y=w)) > \


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\ (EX z. ALL x. EX w. (ALL y. P(x,y) <> y=w) <> x=z)";


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by (Int.best_tac 1); (*60 seconds*)


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result();


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writeln"Problem 56";


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goal IFOLP.thy


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"?p : (ALL x. (EX y. P(y) & x=f(y)) > P(x)) <> (ALL x. P(x) > P(f(x)))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem 57";


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goal IFOLP.thy


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"?p : P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \


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\ (ALL x y z. P(x,y) & P(y,z) > P(x,z)) > P(f(a,b), f(a,c))";


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by (Int.fast_tac 1);


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result();


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writeln"Problem 60";


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goal IFOLP.thy


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"?p : ALL x. P(x,f(x)) <> (EX y. (ALL z. P(z,y) > P(z,f(x))) & P(x,y))";


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by (Int.fast_tac 1);


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result();


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writeln"Reached end of file.";
