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(* Title: HOL/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 1994 University of Cambridge
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Higher-Order Logic: predicate calculus problems
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Taken from FOL/cla.ML; beware of precedence of = vs <->
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*)
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writeln"File HOL/ex/cla.";
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goal HOL.thy "(P --> Q | R) --> (P-->Q) | (P-->R)";
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by (fast_tac HOL_cs 1);
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result();
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(*If and only if*)
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goal HOL.thy "(P=Q) = (Q=P::bool)";
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by (fast_tac HOL_cs 1);
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result();
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goal HOL.thy "~ (P = (~P))";
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by (fast_tac HOL_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 HOL.thy "(P-->Q) = (~Q --> ~P)";
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by (fast_tac HOL_cs 1);
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result();
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(*2*)
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goal HOL.thy "(~ ~ P) = P";
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by (fast_tac HOL_cs 1);
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result();
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(*3*)
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goal HOL.thy "~(P-->Q) --> (Q-->P)";
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by (fast_tac HOL_cs 1);
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result();
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(*4*)
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goal HOL.thy "(~P-->Q) = (~Q --> P)";
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by (fast_tac HOL_cs 1);
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result();
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(*5*)
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goal HOL.thy "((P|Q)-->(P|R)) --> (P|(Q-->R))";
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by (fast_tac HOL_cs 1);
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result();
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(*6*)
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goal HOL.thy "P | ~ P";
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by (fast_tac HOL_cs 1);
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result();
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(*7*)
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goal HOL.thy "P | ~ ~ ~ P";
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by (fast_tac HOL_cs 1);
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result();
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(*8. Peirce's law*)
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goal HOL.thy "((P-->Q) --> P) --> P";
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by (fast_tac HOL_cs 1);
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result();
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(*9*)
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goal HOL.thy "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
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by (fast_tac HOL_cs 1);
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result();
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(*10*)
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goal HOL.thy "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)";
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by (fast_tac HOL_cs 1);
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result();
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(*11. Proved in each direction (incorrectly, says Pelletier!!) *)
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goal HOL.thy "P=P::bool";
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by (fast_tac HOL_cs 1);
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result();
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(*12. "Dijkstra's law"*)
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goal HOL.thy "((P = Q) = R) = (P = (Q = R))";
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by (fast_tac HOL_cs 1);
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result();
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(*13. Distributive law*)
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goal HOL.thy "(P | (Q & R)) = ((P | Q) & (P | R))";
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by (fast_tac HOL_cs 1);
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result();
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(*14*)
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goal HOL.thy "(P = Q) = ((Q | ~P) & (~Q|P))";
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by (fast_tac HOL_cs 1);
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result();
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(*15*)
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goal HOL.thy "(P --> Q) = (~P | Q)";
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by (fast_tac HOL_cs 1);
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result();
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(*16*)
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goal HOL.thy "(P-->Q) | (Q-->P)";
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by (fast_tac HOL_cs 1);
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result();
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(*17*)
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goal HOL.thy "((P & (Q-->R))-->S) = ((~P | Q | S) & (~P | ~R | S))";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Classical Logic: examples with quantifiers";
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goal HOL.thy "(! x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
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by (fast_tac HOL_cs 1);
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result();
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goal HOL.thy "(? x. P-->Q(x)) = (P --> (? x.Q(x)))";
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by (fast_tac HOL_cs 1);
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result();
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goal HOL.thy "(? x.P(x)-->Q) = ((! x.P(x)) --> Q)";
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by (fast_tac HOL_cs 1);
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result();
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goal HOL.thy "((! x.P(x)) | Q) = (! x. P(x) | Q)";
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by (fast_tac HOL_cs 1);
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result();
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(*From Wishnu Prasetya*)
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goal HOL.thy
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"(!s. q(s) --> r(s)) & ~r(s) & (!s. ~r(s) & ~q(s) --> p(t) | q(t)) \
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\ --> p(t) | r(t)";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problems requiring quantifier duplication";
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(*Needs multiple instantiation of the quantifier.*)
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goal HOL.thy "(! x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))";
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by (deepen_tac HOL_cs 1 1);
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result();
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(*Needs double instantiation of the quantifier*)
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goal HOL.thy "? x. P(x) --> P(a) & P(b)";
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by (deepen_tac HOL_cs 1 1);
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result();
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goal HOL.thy "? z. P(z) --> (! x. P(x))";
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by (deepen_tac HOL_cs 1 1);
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result();
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goal HOL.thy "? x. (? y. P(y)) --> P(x)";
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by (deepen_tac HOL_cs 1 1);
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result();
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writeln"Hard examples with quantifiers";
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writeln"Problem 18";
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goal HOL.thy "? y. ! x. P(y)-->P(x)";
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by (deepen_tac HOL_cs 1 1);
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result();
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writeln"Problem 19";
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goal HOL.thy "? x. ! y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";
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by (deepen_tac HOL_cs 1 1);
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result();
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writeln"Problem 20";
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goal HOL.thy "(! x y. ? z. ! w. (P(x)&Q(y)-->R(z)&S(w))) \
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\ --> (? x y. P(x) & Q(y)) --> (? z. R(z))";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problem 21";
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goal HOL.thy "(? x. P-->Q(x)) & (? x. Q(x)-->P) --> (? x. P=Q(x))";
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by (deepen_tac HOL_cs 1 1);
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result();
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writeln"Problem 22";
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goal HOL.thy "(! x. P = Q(x)) --> (P = (! x. Q(x)))";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problem 23";
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goal HOL.thy "(! x. P | Q(x)) = (P | (! x. Q(x)))";
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by (best_tac HOL_cs 1);
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result();
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writeln"Problem 24";
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goal HOL.thy "~(? x. S(x)&Q(x)) & (! x. P(x) --> Q(x)|R(x)) & \
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\ ~(? x.P(x)) --> (? x.Q(x)) & (! x. Q(x)|R(x) --> S(x)) \
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\ --> (? x. P(x)&R(x))";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problem 25";
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goal HOL.thy "(? x. P(x)) & \
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\ (! x. L(x) --> ~ (M(x) & R(x))) & \
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\ (! x. P(x) --> (M(x) & L(x))) & \
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\ ((! x. P(x)-->Q(x)) | (? x. P(x)&R(x))) \
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\ --> (? x. Q(x)&P(x))";
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by (best_tac HOL_cs 1);
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result();
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writeln"Problem 26";
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goal HOL.thy "((? x. p(x)) = (? x. q(x))) & \
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\ (! x. ! y. p(x) & q(y) --> (r(x) = s(y))) \
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\ --> ((! x. p(x)-->r(x)) = (! x. q(x)-->s(x)))";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problem 27";
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goal HOL.thy "(? x. P(x) & ~Q(x)) & \
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\ (! x. P(x) --> R(x)) & \
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\ (! x. M(x) & L(x) --> P(x)) & \
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\ ((? x. R(x) & ~ Q(x)) --> (! x. L(x) --> ~ R(x))) \
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\ --> (! x. M(x) --> ~L(x))";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problem 28. AMENDED";
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goal HOL.thy "(! x. P(x) --> (! x. Q(x))) & \
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\ ((! x. Q(x)|R(x)) --> (? x. Q(x)&S(x))) & \
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\ ((? x.S(x)) --> (! x. L(x) --> M(x))) \
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\ --> (! x. P(x) & L(x) --> M(x))";
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by (fast_tac HOL_cs 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 HOL.thy "(? x. F(x)) & (? y. G(y)) \
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\ --> ( ((! x. F(x)-->H(x)) & (! y. G(y)-->J(y))) = \
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\ (! x y. F(x) & G(y) --> H(x) & J(y)))";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problem 30";
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goal HOL.thy "(! x. P(x) | Q(x) --> ~ R(x)) & \
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\ (! x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \
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\ --> (! x. S(x))";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problem 31";
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goal HOL.thy "~(? x.P(x) & (Q(x) | R(x))) & \
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\ (? x. L(x) & P(x)) & \
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\ (! x. ~ R(x) --> M(x)) \
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\ --> (? x. L(x) & M(x))";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problem 32";
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goal HOL.thy "(! x. P(x) & (Q(x)|R(x))-->S(x)) & \
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\ (! x. S(x) & R(x) --> L(x)) & \
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\ (! x. M(x) --> R(x)) \
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\ --> (! x. P(x) & M(x) --> L(x))";
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by (best_tac HOL_cs 1);
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result();
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writeln"Problem 33";
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goal HOL.thy "(! x. P(a) & (P(x)-->P(b))-->P(c)) = \
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\ (! x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))";
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by (best_tac HOL_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 HOL.thy "((? x. ! y. p(x) = p(y)) = \
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\ ((? x. q(x)) = (! y. p(y)))) = \
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\ ((? x. ! y. q(x) = q(y)) = \
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\ ((? x. p(x)) = (! y. q(y))))";
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by (deepen_tac HOL_cs 3 1);
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(*slower with smaller bounds*)
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result();
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writeln"Problem 35";
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goal HOL.thy "? x y. P x y --> (! u v. P u v)";
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by (deepen_tac HOL_cs 1 1);
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result();
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writeln"Problem 36";
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goal HOL.thy "(! x. ? y. J x y) & \
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\ (! x. ? y. G x y) & \
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\ (! x y. J x y | G x y --> \
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\ (! z. J y z | G y z --> H x z)) \
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\ --> (! x. ? y. H x y)";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problem 37";
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goal HOL.thy "(! z. ? w. ! x. ? y. \
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\ (P x z -->P y w) & P y z & (P y w --> (? u.Q u w))) & \
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\ (! x z. ~(P x z) --> (? y. Q y z)) & \
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\ ((? x y. Q x y) --> (! x. R x x)) \
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\ --> (! x. ? y. R x y)";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problem 38";
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goal HOL.thy
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"(! x. p(a) & (p(x) --> (? y. p(y) & r x y)) --> \
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\ (? z. ? w. p(z) & r x w & r w z)) = \
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\ (! x. (~p(a) | p(x) | (? z. ? w. p(z) & r x w & r w z)) & \
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\ (~p(a) | ~(? y. p(y) & r x y) | \
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\ (? z. ? w. p(z) & r x w & r w z)))";
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writeln"Problem 39";
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goal HOL.thy "~ (? x. ! y. F y x = (~ F y y))";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problem 40. AMENDED";
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goal HOL.thy "(? y. ! x. F x y = F x x) \
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\ --> ~ (! x. ? y. ! z. F z y = (~ F z x))";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problem 41";
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goal HOL.thy "(! z. ? y. ! x. f x y = (f x z & ~ f x x)) \
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\ --> ~ (? z. ! x. f x z)";
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by (best_tac HOL_cs 1);
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result();
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writeln"Problem 42";
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goal HOL.thy "~ (? y. ! x. p x y = (~ (? z. p x z & p z x)))";
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by (deepen_tac HOL_cs 3 1);
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result();
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writeln"Problem 43 NOT PROVED AUTOMATICALLY";
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goal HOL.thy
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"(! x::'a. ! y::'a. q x y = (! z. p z x = (p z y::bool))) \
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\ --> (! x. (! y. q x y = (q y x::bool)))";
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writeln"Problem 44";
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goal HOL.thy "(! x. f(x) --> \
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\ (? y. g(y) & h x y & (? y. g(y) & ~ h x y))) & \
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\ (? x. j(x) & (! y. g(y) --> h x y)) \
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\ --> (? x. j(x) & ~f(x))";
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by (fast_tac HOL_cs 1);
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result();
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writeln"Problem 45";
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goal HOL.thy
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"(! x. f(x) & (! y. g(y) & h x y --> j x y) \
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\ --> (! y. g(y) & h x y --> k(y))) & \
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\ ~ (? y. l(y) & k(y)) & \
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\ (? x. f(x) & (! y. h x y --> l(y)) \
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\ & (! y. g(y) & h x y --> j x y)) \
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\ --> (? x. f(x) & ~ (? y. g(y) & h x y))";
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by (best_tac HOL_cs 1);
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result();
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367 |
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368 |
writeln"Problems (mainly) involving equality or functions";
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369 |
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370 |
writeln"Problem 48";
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371 |
goal HOL.thy "(a=b | c=d) & (a=c | b=d) --> a=d | b=c";
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372 |
by (fast_tac HOL_cs 1);
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373 |
result();
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374 |
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375 |
writeln"Problem 49 NOT PROVED AUTOMATICALLY";
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376 |
(*Hard because it involves substitution for Vars;
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377 |
the type constraint ensures that x,y,z have the same type as a,b,u. *)
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378 |
goal HOL.thy "(? x y::'a. ! z. z=x | z=y) & P(a) & P(b) & (~a=b) \
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379 |
\ --> (! u::'a.P(u))";
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380 |
by (Classical.safe_tac HOL_cs);
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381 |
by (res_inst_tac [("x","a")] allE 1);
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382 |
by (assume_tac 1);
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383 |
by (res_inst_tac [("x","b")] allE 1);
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384 |
by (assume_tac 1);
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385 |
by (fast_tac HOL_cs 1);
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386 |
result();
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387 |
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388 |
writeln"Problem 50";
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389 |
(*What has this to do with equality?*)
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390 |
goal HOL.thy "(! x. P a x | (! y.P x y)) --> (? x. ! y.P x y)";
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391 |
by (deepen_tac HOL_cs 1 1);
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392 |
result();
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393 |
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394 |
writeln"Problem 51";
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395 |
goal HOL.thy
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396 |
"(? z w. ! x y. P x y = (x=z & y=w)) --> \
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397 |
\ (? z. ! x. ? w. (! y. P x y = (y=w)) = (x=z))";
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398 |
by (best_tac HOL_cs 1);
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399 |
result();
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400 |
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401 |
writeln"Problem 52";
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402 |
(*Almost the same as 51. *)
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403 |
goal HOL.thy
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|
404 |
"(? z w. ! x y. P x y = (x=z & y=w)) --> \
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|
405 |
\ (? w. ! y. ? z. (! x. P x y = (x=z)) = (y=w))";
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|
406 |
by (best_tac HOL_cs 1);
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|
407 |
result();
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408 |
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|
409 |
writeln"Problem 55";
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410 |
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411 |
(*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
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412 |
fast_tac DISCOVERS who killed Agatha. *)
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413 |
goal HOL.thy "lives(agatha) & lives(butler) & lives(charles) & \
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|
414 |
\ (killed agatha agatha | killed butler agatha | killed charles agatha) & \
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|
415 |
\ (!x y. killed x y --> hates x y & ~richer x y) & \
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|
416 |
\ (!x. hates agatha x --> ~hates charles x) & \
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|
417 |
\ (hates agatha agatha & hates agatha charles) & \
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418 |
\ (!x. lives(x) & ~richer x agatha --> hates butler x) & \
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|
419 |
\ (!x. hates agatha x --> hates butler x) & \
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|
420 |
\ (!x. ~hates x agatha | ~hates x butler | ~hates x charles) --> \
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|
421 |
\ killed ?who agatha";
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|
422 |
by (fast_tac HOL_cs 1);
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|
423 |
result();
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|
424 |
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|
425 |
writeln"Problem 56";
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|
426 |
goal HOL.thy
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|
427 |
"(! x. (? y. P(y) & x=f(y)) --> P(x)) = (! x. P(x) --> P(f(x)))";
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|
428 |
by (fast_tac HOL_cs 1);
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|
429 |
result();
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|
430 |
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|
431 |
writeln"Problem 57";
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|
432 |
goal HOL.thy
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|
433 |
"P (f a b) (f b c) & P (f b c) (f a c) & \
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|
434 |
\ (! x y z. P x y & P y z --> P x z) --> P (f a b) (f a c)";
|
|
435 |
by (fast_tac HOL_cs 1);
|
|
436 |
result();
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|
437 |
|
|
438 |
writeln"Problem 58 NOT PROVED AUTOMATICALLY";
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|
439 |
goal HOL.thy "(! x y. f(x)=g(y)) --> (! x y. f(f(x))=f(g(y)))";
|
|
440 |
val f_cong = read_instantiate [("f","f")] arg_cong;
|
|
441 |
by (fast_tac (HOL_cs addIs [f_cong]) 1);
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|
442 |
result();
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|
443 |
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|
444 |
writeln"Problem 59";
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|
445 |
goal HOL.thy "(! x. P(x) = (~P(f(x)))) --> (? x. P(x) & ~P(f(x)))";
|
|
446 |
by (deepen_tac HOL_cs 1 1);
|
|
447 |
result();
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|
448 |
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|
449 |
writeln"Problem 60";
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|
450 |
goal HOL.thy
|
|
451 |
"! x. P x (f x) = (? y. (! z. P z y --> P z (f x)) & P x y)";
|
|
452 |
by (fast_tac HOL_cs 1);
|
|
453 |
result();
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|
454 |
|
|
455 |
writeln"Reached end of file.";
|