author | paulson |
Tue, 04 Mar 1997 10:21:16 +0100 | |
changeset 2715 | 79c35a051196 |
parent 2614 | 0b1364481214 |
child 2729 | 44cbfeebd0fe |
permissions | -rw-r--r-- |
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(* Title: FOL/ex/cla.ML |
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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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Classical First-Order Logic |
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*) |
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writeln"File FOL/ex/cla.ML"; |
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Renamed structure Int (intuitionistic prover) to IntPr to prevent clash
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open Cla; (*in case structure IntPr is open!*) |
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goal FOL.thy "(P --> Q | R) --> (P-->Q) | (P-->R)"; |
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by (Fast_tac 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 1); |
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result(); |
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goal FOL.thy "~ (P <-> ~P)"; |
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by (Fast_tac 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 1); |
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result(); |
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(*Discussed in Avron, Gentzen-Type Systems, Resolution and Tableaux, |
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JAR 10 (265-281), 1993. Proof is trivial!*) |
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goal FOL.thy |
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"~ ((EX x.~P(x)) & ((EX x.P(x)) | (EX x.P(x) & Q(x))) & ~ (EX x.P(x)))"; |
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by (Fast_tac 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 (Deepen_tac 0 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 (Deepen_tac 0 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 (Deepen_tac 0 1); |
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result(); |
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goal FOL.thy "EX x. (EX y. P(y)) --> P(x)"; |
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by (Deepen_tac 0 1); |
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result(); |
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(*V. Lifschitz, What Is the Inverse Method?, JAR 5 (1989), 1--23. NOT PROVED*) |
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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 (Deepen_tac 0 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 (Deepen_tac 0 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 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 (Deepen_tac 0 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 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 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 1); |
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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 1); |
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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 1); |
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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 1); |
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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 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 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 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 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 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 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 1); |
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result(); |
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writeln"Problem 34 AMENDED (TWICE!!)"; |
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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 (Deepen_tac 0 1); |
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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 (mini_tac 1); |
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by (Fast_tac 1); |
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(*Without miniscope, would have to use deepen_tac; would be slower*) |
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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 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 1); |
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result(); |
317 |
||
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writeln"Problem 38"; |
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goal FOL.thy |
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"(ALL x. p(a) & (p(x) --> (EX y. p(y) & r(x,y))) --> \ |
321 |
\ (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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by (Deepen_tac 0 1); (*beats fast_tac!*) |
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result(); |
0 | 327 |
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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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2469 | 330 |
by (Fast_tac 1); |
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result(); |
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writeln"Problem 40. AMENDED"; |
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334 |
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 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 (Fast_tac 1); |
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result(); |
344 |
||
428 | 345 |
writeln"Problem 42"; |
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goal FOL.thy "~ (EX y. ALL x. p(x,y) <-> ~ (EX z. p(x,z) & p(z,x)))"; |
2469 | 347 |
by (Deepen_tac 0 1); |
428 | 348 |
result(); |
0 | 349 |
|
732 | 350 |
writeln"Problem 43"; |
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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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by (Auto_tac()); |
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(*The proof above cheats by using rewriting! A purely logical proof is |
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by (mini_tac 1 THEN Deepen_tac 5 1); |
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Can use just deepen_tac but it requires 253 secs?!? |
2469 | 357 |
by (Deepen_tac 0 1); |
732 | 358 |
*) |
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result(); |
0 | 360 |
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writeln"Problem 44"; |
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goal FOL.thy "(ALL x. f(x) --> \ |
363 |
\ (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y)))) & \ |
|
364 |
\ (EX x. j(x) & (ALL y. g(y) --> h(x,y))) \ |
|
0 | 365 |
\ --> (EX x. j(x) & ~f(x))"; |
2469 | 366 |
by (Fast_tac 1); |
0 | 367 |
result(); |
368 |
||
369 |
writeln"Problem 45"; |
|
1459 | 370 |
goal FOL.thy "(ALL x. f(x) & (ALL y. g(y) & h(x,y) --> j(x,y)) \ |
371 |
\ --> (ALL y. g(y) & h(x,y) --> k(y))) & \ |
|
372 |
\ ~ (EX y. l(y) & k(y)) & \ |
|
373 |
\ (EX x. f(x) & (ALL y. h(x,y) --> l(y)) \ |
|
374 |
\ & (ALL y. g(y) & h(x,y) --> j(x,y))) \ |
|
0 | 375 |
\ --> (EX x. f(x) & ~ (EX y. g(y) & h(x,y)))"; |
2469 | 376 |
by (Best_tac 1); |
0 | 377 |
result(); |
378 |
||
379 |
||
380 |
writeln"Problems (mainly) involving equality or functions"; |
|
381 |
||
382 |
writeln"Problem 48"; |
|
383 |
goal FOL.thy "(a=b | c=d) & (a=c | b=d) --> a=d | b=c"; |
|
2469 | 384 |
by (Fast_tac 1); |
0 | 385 |
result(); |
386 |
||
387 |
writeln"Problem 49 NOT PROVED AUTOMATICALLY"; |
|
388 |
(*Hard because it involves substitution for Vars; |
|
389 |
the type constraint ensures that x,y,z have the same type as a,b,u. *) |
|
36 | 390 |
goal FOL.thy "(EX x y::'a. ALL z. z=x | z=y) & P(a) & P(b) & a~=b \ |
1459 | 391 |
\ --> (ALL u::'a.P(u))"; |
2469 | 392 |
by (Step_tac 1); |
0 | 393 |
by (res_inst_tac [("x","a")] allE 1); |
1459 | 394 |
by (assume_tac 1); |
0 | 395 |
by (res_inst_tac [("x","b")] allE 1); |
1459 | 396 |
by (assume_tac 1); |
2469 | 397 |
by (Fast_tac 1); |
0 | 398 |
result(); |
399 |
||
400 |
writeln"Problem 50"; |
|
401 |
(*What has this to do with equality?*) |
|
402 |
goal FOL.thy "(ALL x. P(a,x) | (ALL y.P(x,y))) --> (EX x. ALL y.P(x,y))"; |
|
732 | 403 |
by (mini_tac 1); |
2469 | 404 |
by (Deepen_tac 0 1); |
0 | 405 |
result(); |
406 |
||
407 |
writeln"Problem 51"; |
|
408 |
goal FOL.thy |
|
409 |
"(EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \ |
|
410 |
\ (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)"; |
|
2469 | 411 |
by (Fast_tac 1); |
0 | 412 |
result(); |
413 |
||
414 |
writeln"Problem 52"; |
|
415 |
(*Almost the same as 51. *) |
|
416 |
goal FOL.thy |
|
417 |
"(EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \ |
|
418 |
\ (EX w. ALL y. EX z. (ALL x. P(x,y) <-> x=z) <-> y=w)"; |
|
2469 | 419 |
by (Best_tac 1); |
0 | 420 |
result(); |
421 |
||
422 |
writeln"Problem 55"; |
|
423 |
||
424 |
(*Original, equational version by Len Schubert, via Pelletier *** NOT PROVED |
|
425 |
goal FOL.thy |
|
426 |
"(EX x. lives(x) & killed(x,agatha)) & \ |
|
427 |
\ lives(agatha) & lives(butler) & lives(charles) & \ |
|
428 |
\ (ALL x. lives(x) --> x=agatha | x=butler | x=charles) & \ |
|
429 |
\ (ALL x y. killed(x,y) --> hates(x,y)) & \ |
|
430 |
\ (ALL x y. killed(x,y) --> ~richer(x,y)) & \ |
|
431 |
\ (ALL x. hates(agatha,x) --> ~hates(charles,x)) & \ |
|
432 |
\ (ALL x. ~ x=butler --> hates(agatha,x)) & \ |
|
433 |
\ (ALL x. ~richer(x,agatha) --> hates(butler,x)) & \ |
|
434 |
\ (ALL x. hates(agatha,x) --> hates(butler,x)) & \ |
|
435 |
\ (ALL x. EX y. ~hates(x,y)) & \ |
|
436 |
\ ~ agatha=butler --> \ |
|
437 |
\ killed(?who,agatha)"; |
|
2469 | 438 |
by (Step_tac 1); |
0 | 439 |
by (dres_inst_tac [("x1","x")] (spec RS mp) 1); |
1459 | 440 |
by (assume_tac 1); |
441 |
by (etac (spec RS exE) 1); |
|
0 | 442 |
by (REPEAT (etac allE 1)); |
2469 | 443 |
by (Fast_tac 1); |
0 | 444 |
result(); |
445 |
****) |
|
446 |
||
447 |
(*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988). |
|
448 |
fast_tac DISCOVERS who killed Agatha. *) |
|
449 |
goal FOL.thy "lives(agatha) & lives(butler) & lives(charles) & \ |
|
450 |
\ (killed(agatha,agatha) | killed(butler,agatha) | killed(charles,agatha)) & \ |
|
451 |
\ (ALL x y. killed(x,y) --> hates(x,y) & ~richer(x,y)) & \ |
|
452 |
\ (ALL x. hates(agatha,x) --> ~hates(charles,x)) & \ |
|
453 |
\ (hates(agatha,agatha) & hates(agatha,charles)) & \ |
|
454 |
\ (ALL x. lives(x) & ~richer(x,agatha) --> hates(butler,x)) & \ |
|
455 |
\ (ALL x. hates(agatha,x) --> hates(butler,x)) & \ |
|
456 |
\ (ALL x. ~hates(x,agatha) | ~hates(x,butler) | ~hates(x,charles)) --> \ |
|
457 |
\ killed(?who,agatha)"; |
|
2469 | 458 |
by (Fast_tac 1); |
0 | 459 |
result(); |
460 |
||
461 |
||
462 |
writeln"Problem 56"; |
|
463 |
goal FOL.thy |
|
464 |
"(ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"; |
|
2469 | 465 |
by (Fast_tac 1); |
0 | 466 |
result(); |
467 |
||
468 |
writeln"Problem 57"; |
|
469 |
goal FOL.thy |
|
470 |
"P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \ |
|
471 |
\ (ALL x y z. P(x,y) & P(y,z) --> P(x,z)) --> P(f(a,b), f(a,c))"; |
|
2469 | 472 |
by (Fast_tac 1); |
0 | 473 |
result(); |
474 |
||
475 |
writeln"Problem 58 NOT PROVED AUTOMATICALLY"; |
|
476 |
goal FOL.thy "(ALL x y. f(x)=g(y)) --> (ALL x y. f(f(x))=f(g(y)))"; |
|
2469 | 477 |
by (slow_tac (!claset addEs [subst_context]) 1); |
0 | 478 |
result(); |
479 |
||
480 |
writeln"Problem 59"; |
|
481 |
goal FOL.thy "(ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))"; |
|
2469 | 482 |
by (Deepen_tac 0 1); |
0 | 483 |
result(); |
484 |
||
485 |
writeln"Problem 60"; |
|
486 |
goal FOL.thy |
|
487 |
"ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"; |
|
2469 | 488 |
by (Fast_tac 1); |
0 | 489 |
result(); |
490 |
||
2715 | 491 |
writeln"Problem 62 as corrected in JAR 18 (1997), page 135"; |
1404 | 492 |
goal FOL.thy |
1459 | 493 |
"(ALL x. p(a) & (p(x) --> p(f(x))) --> p(f(f(x)))) <-> \ |
494 |
\ (ALL x. (~p(a) | p(x) | p(f(f(x)))) & \ |
|
1404 | 495 |
\ (~p(a) | ~p(f(x)) | p(f(f(x)))))"; |
2469 | 496 |
by (Fast_tac 1); |
1404 | 497 |
result(); |
498 |
||
1560 | 499 |
(*Halting problem: Formulation of Li Dafa (AAR Newsletter 27, Oct 1994.) |
500 |
author U. Egly*) |
|
501 |
goal FOL.thy |
|
502 |
"((EX X. a(X) & (ALL Y. c(Y) --> (ALL Z. d(X, Y, Z)))) --> \ |
|
503 |
\ (EX W. c(W) & (ALL Y. c(Y) --> (ALL Z. d(W, Y, Z))))) \ |
|
504 |
\ & \ |
|
505 |
\ (ALL W. c(W) & (ALL U. c(U) --> (ALL V. d(W, U, V))) --> \ |
|
506 |
\ (ALL Y Z. \ |
|
507 |
\ (c(Y) & h2(Y, Z) --> h3(W, Y, Z) & o(W, g)) & \ |
|
508 |
\ (c(Y) & ~h2(Y, Z) --> h3(W, Y, Z) & o(W, b)))) \ |
|
509 |
\ & \ |
|
510 |
\ (ALL W. c(W) & \ |
|
511 |
\ (ALL Y Z. \ |
|
512 |
\ (c(Y) & h2(Y, Z) --> h3(W, Y, Z) & o(W, g)) & \ |
|
513 |
\ (c(Y) & ~h2(Y, Z) --> h3(W, Y, Z) & o(W, b))) --> \ |
|
514 |
\ (EX V. c(V) & \ |
|
515 |
\ (ALL Y. ((c(Y) & h3(W, Y, Y)) & o(W, g) --> ~h2(V, Y)) & \ |
|
516 |
\ ((c(Y) & h3(W, Y, Y)) & o(W, b) --> h2(V, Y) & o(V, b))))) \ |
|
517 |
\ --> \ |
|
518 |
\ ~ (EX X. a(X) & (ALL Y. c(Y) --> (ALL Z. d(X, Y, Z))))"; |
|
519 |
||
0 | 520 |
|
2614 | 521 |
(* Challenge found on info-hol *) |
522 |
goal FOL.thy |
|
523 |
"ALL x. EX v w. ALL y z. P(x) & Q(y) --> (P(v) | R(w)) & (R(z) --> Q(v))"; |
|
524 |
by (Deepen_tac 0 1); |
|
525 |
result(); |
|
526 |
||
0 | 527 |
writeln"Reached end of file."; |
528 |
||
732 | 529 |
(*Thu Jul 23 1992: loaded in 467s using iffE [on SPARC2] *) |
530 |
(*Mon Nov 14 1994: loaded in 144s [on SPARC10, with deepen_tac] *) |
|
531 |
(*Wed Nov 16 1994: loaded in 138s [after addition of norm_term_skip] *) |
|
532 |
(*Mon Nov 21 1994: loaded in 131s [DEPTH_FIRST suppressing repetitions] *) |
|
533 |