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(* Title: FOL/ex/cla
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1992 University of Cambridge
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Classical First-Order Logic
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*)
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writeln"File FOL/ex/cla.";
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open Cla; (*in case structure Int is open!*)
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goal FOL.thy "(P --> Q | R) --> (P-->Q) | (P-->R)";
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by (fast_tac FOL_cs 1);
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result();
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(*If and only if*)
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goal FOL.thy "(P<->Q) <-> (Q<->P)";
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by (fast_tac FOL_cs 1);
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result();
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goal FOL.thy "~ (P <-> ~P)";
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by (fast_tac FOL_cs 1);
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result();
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(*Sample problems from
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F. J. Pelletier,
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Seventy-Five Problems for Testing Automatic Theorem Provers,
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J. Automated Reasoning 2 (1986), 191-216.
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Errata, JAR 4 (1988), 236-236.
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The hardest problems -- judging by experience with several theorem provers,
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including matrix ones -- are 34 and 43.
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*)
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writeln"Pelletier's examples";
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(*1*)
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goal FOL.thy "(P-->Q) <-> (~Q --> ~P)";
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by (fast_tac FOL_cs 1);
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result();
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(*2*)
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goal FOL.thy "~ ~ P <-> P";
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by (fast_tac FOL_cs 1);
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result();
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(*3*)
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goal FOL.thy "~(P-->Q) --> (Q-->P)";
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by (fast_tac FOL_cs 1);
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result();
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(*4*)
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goal FOL.thy "(~P-->Q) <-> (~Q --> P)";
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by (fast_tac FOL_cs 1);
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result();
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(*5*)
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goal FOL.thy "((P|Q)-->(P|R)) --> (P|(Q-->R))";
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by (fast_tac FOL_cs 1);
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result();
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(*6*)
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goal FOL.thy "P | ~ P";
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by (fast_tac FOL_cs 1);
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result();
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(*7*)
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goal FOL.thy "P | ~ ~ ~ P";
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by (fast_tac FOL_cs 1);
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result();
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(*8. Peirce's law*)
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goal FOL.thy "((P-->Q) --> P) --> P";
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by (fast_tac FOL_cs 1);
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result();
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(*9*)
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goal FOL.thy "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
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by (fast_tac FOL_cs 1);
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result();
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(*10*)
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goal FOL.thy "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P<->Q)";
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by (fast_tac FOL_cs 1);
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result();
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(*11. Proved in each direction (incorrectly, says Pelletier!!) *)
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goal FOL.thy "P<->P";
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by (fast_tac FOL_cs 1);
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result();
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(*12. "Dijkstra's law"*)
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goal FOL.thy "((P <-> Q) <-> R) <-> (P <-> (Q <-> R))";
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by (fast_tac FOL_cs 1);
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result();
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(*13. Distributive law*)
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goal FOL.thy "P | (Q & R) <-> (P | Q) & (P | R)";
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by (fast_tac FOL_cs 1);
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result();
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(*14*)
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goal FOL.thy "(P <-> Q) <-> ((Q | ~P) & (~Q|P))";
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by (fast_tac FOL_cs 1);
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result();
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(*15*)
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goal FOL.thy "(P --> Q) <-> (~P | Q)";
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by (fast_tac FOL_cs 1);
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result();
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(*16*)
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goal FOL.thy "(P-->Q) | (Q-->P)";
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by (fast_tac FOL_cs 1);
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result();
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(*17*)
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goal FOL.thy "((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))";
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by (fast_tac FOL_cs 1);
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result();
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writeln"Classical Logic: examples with quantifiers";
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goal FOL.thy "(ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))";
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by (fast_tac FOL_cs 1);
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result();
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goal FOL.thy "(EX x. P-->Q(x)) <-> (P --> (EX x.Q(x)))";
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by (fast_tac FOL_cs 1);
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result();
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goal FOL.thy "(EX x.P(x)-->Q) <-> (ALL x.P(x)) --> Q";
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by (fast_tac FOL_cs 1);
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result();
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goal FOL.thy "(ALL x.P(x)) | Q <-> (ALL x. P(x) | Q)";
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by (fast_tac FOL_cs 1);
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result();
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writeln"Problems requiring quantifier duplication";
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(*Needs multiple instantiation of ALL.*)
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goal FOL.thy "(ALL x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))";
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by (best_tac FOL_dup_cs 1);
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result();
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(*Needs double instantiation of the quantifier*)
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goal FOL.thy "EX x. P(x) --> P(a) & P(b)";
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by (best_tac FOL_dup_cs 1);
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result();
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goal FOL.thy "EX z. P(z) --> (ALL x. P(x))";
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by (best_tac FOL_dup_cs 1);
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result();
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(*from Vladimir Lifschitz, What Is the Inverse Method?, JAR 5 (1989), 1--23*)
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goal FOL.thy "EX x x'. ALL y. EX z z'. \
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\ (~P(y,y) | P(x,x) | ~S(z,x)) & \
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\ (S(x,y) | ~S(y,z) | Q(z',z')) & \
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\ (Q(x',y) | ~Q(y,z') | S(x',x'))";
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writeln"Hard examples with quantifiers";
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writeln"Problem 18";
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goal FOL.thy "EX y. ALL x. P(y)-->P(x)";
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by (best_tac FOL_dup_cs 1);
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result();
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writeln"Problem 19";
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goal FOL.thy "EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";
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by (best_tac FOL_dup_cs 1);
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result();
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writeln"Problem 20";
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goal FOL.thy "(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 (fast_tac FOL_cs 1);
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goal FOL.thy "(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 (fast_tac FOL_cs 1);
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result();
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writeln"Problem 21";
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goal FOL.thy "(EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))";
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by (best_tac FOL_dup_cs 1);
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result();
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writeln"Problem 22";
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goal FOL.thy "(ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))";
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by (fast_tac FOL_cs 1);
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result();
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writeln"Problem 23";
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goal FOL.thy "(ALL x. P | Q(x)) <-> (P | (ALL x. Q(x)))";
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by (best_tac FOL_cs 1);
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result();
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writeln"Problem 24";
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goal FOL.thy "~(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 (fast_tac FOL_cs 1);
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(*slow*)
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result();
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writeln"Problem 25";
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goal FOL.thy "(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 (best_tac FOL_cs 1);
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(*slow*)
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result();
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writeln"Problem 26";
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goal FOL.thy "((EX x. p(x)) <-> (EX x. q(x))) & \
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\ (ALL x. ALL y. p(x) & q(y) --> (r(x) <-> s(y))) \
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\ --> ((ALL x. p(x)-->r(x)) <-> (ALL x. q(x)-->s(x)))";
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by (fast_tac FOL_cs 1);
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(*slow*)
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result();
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writeln"Problem 27";
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goal FOL.thy "(EX x. P(x) & ~Q(x)) & \
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\ (ALL x. P(x) --> R(x)) & \
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\ (ALL x. M(x) & L(x) --> P(x)) & \
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\ ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x))) \
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\ --> (ALL x. M(x) --> ~L(x))";
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by (fast_tac FOL_cs 1);
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(*slow*)
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result();
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writeln"Problem 28. AMENDED";
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goal FOL.thy "(ALL x. P(x) --> (ALL x. Q(x))) & \
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\ ((ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) & \
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\ ((EX x.S(x)) --> (ALL x. L(x) --> M(x))) \
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\ --> (ALL x. P(x) & L(x) --> M(x))";
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by (fast_tac FOL_cs 1);
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(*slow*)
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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 FOL.thy "(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 (fast_tac FOL_cs 1);
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result();
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writeln"Problem 30";
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goal FOL.thy "(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 (fast_tac FOL_cs 1);
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result();
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writeln"Problem 31";
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goal FOL.thy "~(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 (fast_tac FOL_cs 1);
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result();
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writeln"Problem 32";
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goal FOL.thy "(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 (best_tac FOL_cs 1);
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result();
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writeln"Problem 33";
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goal FOL.thy "(ALL x. P(a) & (P(x)-->P(b))-->P(c)) <-> \
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\ (ALL x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))";
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by (best_tac FOL_cs 1);
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result();
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writeln"Problem 34 AMENDED (TWICE!!) NOT PROVED AUTOMATICALLY";
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(*Andrews's challenge*)
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goal FOL.thy "((EX x. ALL y. p(x) <-> p(y)) <-> \
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\ ((EX x. q(x)) <-> (ALL y. p(y)))) <-> \
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\ ((EX x. ALL y. q(x) <-> q(y)) <-> \
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\ ((EX x. p(x)) <-> (ALL y. q(y))))";
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by (safe_tac FOL_cs); (*22 secs*)
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by (TRYALL (fast_tac FOL_cs)); (*128 secs*)
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by (TRYALL (best_tac FOL_dup_cs)); (*77 secs*)
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result();
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writeln"Problem 35";
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goal FOL.thy "EX x y. P(x,y) --> (ALL u v. P(u,v))";
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by (best_tac FOL_dup_cs 1);
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(*6.1 secs*)
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result();
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writeln"Problem 36";
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goal FOL.thy
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"(ALL x. EX y. J(x,y)) & \
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\ (ALL x. EX y. G(x,y)) & \
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\ (ALL x y. J(x,y) | G(x,y) --> (ALL z. J(y,z) | G(y,z) --> H(x,z))) \
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\ --> (ALL x. EX y. H(x,y))";
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by (fast_tac FOL_cs 1);
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result();
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writeln"Problem 37";
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goal FOL.thy "(ALL z. EX w. ALL x. EX y. \
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\ (P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (EX u.Q(u,w)))) & \
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\ (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) & \
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\ ((EX x y. Q(x,y)) --> (ALL x. R(x,x))) \
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\ --> (ALL x. EX y. R(x,y))";
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by (fast_tac FOL_cs 1);
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(*slow...Poly/ML: 9.7 secs*)
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result();
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writeln"Problem 38. NOT PROVED";
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goal FOL.thy
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"(ALL x. p(a) & (p(x) --> (EX y. p(y) & r(x,y))) --> \
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\ (EX z. EX w. p(z) & r(x,w) & r(w,z))) <-> \
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\ (ALL x. (~p(a) | p(x) | (EX z. EX w. p(z) & r(x,w) & r(w,z))) & \
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\ (~p(a) | ~(EX y. p(y) & r(x,y)) | \
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\ (EX z. EX w. p(z) & r(x,w) & r(w,z))))";
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writeln"Problem 39";
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goal FOL.thy "~ (EX x. ALL y. F(y,x) <-> ~F(y,y))";
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by (fast_tac FOL_cs 1);
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result();
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writeln"Problem 40. AMENDED";
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goal FOL.thy "(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 (fast_tac FOL_cs 1);
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result();
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writeln"Problem 41";
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goal FOL.thy "(ALL z. EX y. ALL x. f(x,y) <-> f(x,z) & ~ f(x,x)) \
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\ --> ~ (EX z. ALL x. f(x,z))";
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by (best_tac FOL_cs 1);
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result();
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writeln"Problem 42 NOT PROVED";
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goal FOL.thy "~ (EX y. ALL x. p(x,y) <-> ~ (EX z. p(x,z) & p(z,x)))";
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writeln"Problem 43 NOT PROVED AUTOMATICALLY";
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goal FOL.thy "(ALL x. ALL y. q(x,y) <-> (ALL z. p(z,x) <-> p(z,y))) \
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\ --> (ALL x. (ALL y. q(x,y) <-> q(y,x)))";
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writeln"Problem 44";
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goal FOL.thy "(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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355 |
by (fast_tac FOL_cs 1);
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356 |
result();
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357 |
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358 |
writeln"Problem 45";
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359 |
goal FOL.thy "(ALL x. f(x) & (ALL y. g(y) & h(x,y) --> j(x,y)) \
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360 |
\ --> (ALL y. g(y) & h(x,y) --> k(y))) & \
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361 |
\ ~ (EX y. l(y) & k(y)) & \
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362 |
\ (EX x. f(x) & (ALL y. h(x,y) --> l(y)) \
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363 |
\ & (ALL y. g(y) & h(x,y) --> j(x,y))) \
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364 |
\ --> (EX x. f(x) & ~ (EX y. g(y) & h(x,y)))";
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|
365 |
by (best_tac FOL_cs 1);
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366 |
(*41.5 secs*)
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|
367 |
result();
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368 |
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369 |
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370 |
writeln"Problems (mainly) involving equality or functions";
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371 |
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372 |
writeln"Problem 48";
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373 |
goal FOL.thy "(a=b | c=d) & (a=c | b=d) --> a=d | b=c";
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374 |
by (fast_tac FOL_cs 1);
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|
375 |
result();
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376 |
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|
377 |
writeln"Problem 49 NOT PROVED AUTOMATICALLY";
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|
378 |
(*Hard because it involves substitution for Vars;
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|
379 |
the type constraint ensures that x,y,z have the same type as a,b,u. *)
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|
380 |
goal FOL.thy "(EX x y::'a. ALL z. z=x | z=y) & P(a) & P(b) & (~a=b) \
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|
381 |
\ --> (ALL u::'a.P(u))";
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|
382 |
by (safe_tac FOL_cs);
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|
383 |
by (res_inst_tac [("x","a")] allE 1);
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|
384 |
ba 1;
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|
385 |
by (res_inst_tac [("x","b")] allE 1);
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|
386 |
ba 1;
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|
387 |
by (fast_tac FOL_cs 1);
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|
388 |
result();
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|
389 |
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|
390 |
writeln"Problem 50";
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|
391 |
(*What has this to do with equality?*)
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|
392 |
goal FOL.thy "(ALL x. P(a,x) | (ALL y.P(x,y))) --> (EX x. ALL y.P(x,y))";
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|
393 |
by (best_tac FOL_dup_cs 1);
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|
394 |
result();
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|
395 |
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|
396 |
writeln"Problem 51";
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|
397 |
goal FOL.thy
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|
398 |
"(EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \
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|
399 |
\ (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)";
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|
400 |
by (fast_tac FOL_cs 1);
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|
401 |
result();
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|
402 |
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|
403 |
writeln"Problem 52";
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|
404 |
(*Almost the same as 51. *)
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|
405 |
goal FOL.thy
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|
406 |
"(EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \
|
|
407 |
\ (EX w. ALL y. EX z. (ALL x. P(x,y) <-> x=z) <-> y=w)";
|
|
408 |
by (best_tac FOL_cs 1);
|
|
409 |
result();
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|
410 |
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|
411 |
writeln"Problem 55";
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|
412 |
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|
413 |
(*Original, equational version by Len Schubert, via Pelletier *** NOT PROVED
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|
414 |
goal FOL.thy
|
|
415 |
"(EX x. lives(x) & killed(x,agatha)) & \
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|
416 |
\ lives(agatha) & lives(butler) & lives(charles) & \
|
|
417 |
\ (ALL x. lives(x) --> x=agatha | x=butler | x=charles) & \
|
|
418 |
\ (ALL x y. killed(x,y) --> hates(x,y)) & \
|
|
419 |
\ (ALL x y. killed(x,y) --> ~richer(x,y)) & \
|
|
420 |
\ (ALL x. hates(agatha,x) --> ~hates(charles,x)) & \
|
|
421 |
\ (ALL x. ~ x=butler --> hates(agatha,x)) & \
|
|
422 |
\ (ALL x. ~richer(x,agatha) --> hates(butler,x)) & \
|
|
423 |
\ (ALL x. hates(agatha,x) --> hates(butler,x)) & \
|
|
424 |
\ (ALL x. EX y. ~hates(x,y)) & \
|
|
425 |
\ ~ agatha=butler --> \
|
|
426 |
\ killed(?who,agatha)";
|
|
427 |
by (safe_tac FOL_cs);
|
|
428 |
by (dres_inst_tac [("x1","x")] (spec RS mp) 1);
|
|
429 |
ba 1;
|
|
430 |
be (spec RS exE) 1;
|
|
431 |
by (REPEAT (etac allE 1));
|
|
432 |
by (fast_tac FOL_cs 1);
|
|
433 |
result();
|
|
434 |
****)
|
|
435 |
|
|
436 |
(*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
|
|
437 |
fast_tac DISCOVERS who killed Agatha. *)
|
|
438 |
goal FOL.thy "lives(agatha) & lives(butler) & lives(charles) & \
|
|
439 |
\ (killed(agatha,agatha) | killed(butler,agatha) | killed(charles,agatha)) & \
|
|
440 |
\ (ALL x y. killed(x,y) --> hates(x,y) & ~richer(x,y)) & \
|
|
441 |
\ (ALL x. hates(agatha,x) --> ~hates(charles,x)) & \
|
|
442 |
\ (hates(agatha,agatha) & hates(agatha,charles)) & \
|
|
443 |
\ (ALL x. lives(x) & ~richer(x,agatha) --> hates(butler,x)) & \
|
|
444 |
\ (ALL x. hates(agatha,x) --> hates(butler,x)) & \
|
|
445 |
\ (ALL x. ~hates(x,agatha) | ~hates(x,butler) | ~hates(x,charles)) --> \
|
|
446 |
\ killed(?who,agatha)";
|
|
447 |
by (fast_tac FOL_cs 1);
|
|
448 |
result();
|
|
449 |
|
|
450 |
|
|
451 |
writeln"Problem 56";
|
|
452 |
goal FOL.thy
|
|
453 |
"(ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))";
|
|
454 |
by (fast_tac FOL_cs 1);
|
|
455 |
result();
|
|
456 |
|
|
457 |
writeln"Problem 57";
|
|
458 |
goal FOL.thy
|
|
459 |
"P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \
|
|
460 |
\ (ALL x y z. P(x,y) & P(y,z) --> P(x,z)) --> P(f(a,b), f(a,c))";
|
|
461 |
by (fast_tac FOL_cs 1);
|
|
462 |
result();
|
|
463 |
|
|
464 |
writeln"Problem 58 NOT PROVED AUTOMATICALLY";
|
|
465 |
goal FOL.thy "(ALL x y. f(x)=g(y)) --> (ALL x y. f(f(x))=f(g(y)))";
|
|
466 |
val f_cong = read_instantiate [("t","f")] subst_context;
|
|
467 |
by (fast_tac (FOL_cs addIs [f_cong]) 1);
|
|
468 |
result();
|
|
469 |
|
|
470 |
writeln"Problem 59";
|
|
471 |
goal FOL.thy "(ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))";
|
|
472 |
by (best_tac FOL_dup_cs 1);
|
|
473 |
result();
|
|
474 |
|
|
475 |
writeln"Problem 60";
|
|
476 |
goal FOL.thy
|
|
477 |
"ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))";
|
|
478 |
by (fast_tac FOL_cs 1);
|
|
479 |
result();
|
|
480 |
|
|
481 |
|
|
482 |
writeln"Reached end of file.";
|
|
483 |
|
|
484 |
(*Thu Jul 23 1992: loaded in 467.1s using iffE*)
|