author | lcp |
Tue, 25 Apr 1995 11:14:03 +0200 | |
changeset 1072 | 0140ff702b23 |
parent 650 | ab49d4f96a09 |
child 1461 | 6bcb44e4d6e5 |
permissions | -rw-r--r-- |
0 | 1 |
(* Title: LK/ex/hard-quant |
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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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Hard examples with quantifiers. Can be read to test the LK system. |
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From 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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Uses pc_tac rather than fast_tac when the former is significantly faster. |
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*) |
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writeln"File LK/ex/hard-quant."; |
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goal LK.thy "|- (ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))"; |
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by (fast_tac LK_pack 1); |
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result(); |
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goal LK.thy "|- (EX x. P-->Q(x)) <-> (P --> (EX x.Q(x)))"; |
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by (fast_tac LK_pack 1); |
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result(); |
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goal LK.thy "|- (EX x.P(x)-->Q) <-> (ALL x.P(x)) --> Q"; |
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by (fast_tac LK_pack 1); |
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result(); |
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goal LK.thy "|- (ALL x.P(x)) | Q <-> (ALL x. P(x) | Q)"; |
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by (fast_tac LK_pack 1); |
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result(); |
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writeln"Problems requiring quantifier duplication"; |
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(*Not provable by fast_tac LK_pack: needs multiple instantiation of ALL*) |
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goal LK.thy "|- (ALL x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))"; |
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by (best_tac LK_dup_pack 1); |
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result(); |
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(*Needs double instantiation of the quantifier*) |
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goal LK.thy "|- EX x. P(x) --> P(a) & P(b)"; |
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by (fast_tac LK_dup_pack 1); |
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result(); |
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goal LK.thy "|- EX z. P(z) --> (ALL x. P(x))"; |
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by (best_tac LK_dup_pack 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 LK.thy "|- EX y. ALL x. P(y)-->P(x)"; |
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by (best_tac LK_dup_pack 1); |
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result(); |
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writeln"Problem 19"; |
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goal LK.thy "|- EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"; |
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by (best_tac LK_dup_pack 1); |
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result(); |
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writeln"Problem 20"; |
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goal LK.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 LK_pack 1); |
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result(); |
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writeln"Problem 21"; |
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goal LK.thy "|- (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))"; |
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by (best_tac LK_dup_pack 1); |
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result(); |
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writeln"Problem 22"; |
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goal LK.thy "|- (ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))"; |
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by (fast_tac LK_pack 1); |
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result(); |
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writeln"Problem 23"; |
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goal LK.thy "|- (ALL x. P | Q(x)) <-> (P | (ALL x. Q(x)))"; |
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by (best_tac LK_pack 1); |
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result(); |
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writeln"Problem 24"; |
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goal LK.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 (pc_tac LK_pack 1); |
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result(); |
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writeln"Problem 25"; |
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goal LK.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 LK_pack 1); |
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result(); |
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writeln"Problem 26"; |
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goal LK.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 (pc_tac LK_pack 1); |
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result(); |
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writeln"Problem 27"; |
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goal LK.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 (pc_tac LK_pack 1); |
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result(); |
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writeln"Problem 28. AMENDED"; |
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goal LK.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 (pc_tac LK_pack 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 LK.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 (pc_tac LK_pack 1); |
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result(); |
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writeln"Problem 30"; |
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goal LK.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 LK_pack 1); |
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result(); |
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writeln"Problem 31"; |
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goal LK.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 LK_pack 1); |
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result(); |
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writeln"Problem 32"; |
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goal LK.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 LK_pack 1); |
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result(); |
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writeln"Problem 33"; |
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goal LK.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 (fast_tac LK_pack 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 LK.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_goal_tac LK_pack 1); (*53 secs*) (*13 secs*) |
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by (TRYALL (fast_tac LK_pack)); (*165 secs*) (*117 secs*) (*138 secs*) |
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(*for some reason, pc_tac leaves 14 subgoals instead of 6*) |
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by (TRYALL (best_tac LK_dup_pack)); (*55 secs*) (*29 secs*) (*54 secs*) |
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result(); |
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writeln"Problem 35"; |
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goal LK.thy "|- EX x y. P(x,y) --> (ALL u v. P(u,v))"; |
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by (best_tac LK_dup_pack 1); (*27 secs??*) |
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result(); |
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writeln"Problem 36"; |
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goal LK.thy "|- (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) --> \ |
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\ (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 LK_pack 1); |
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result(); |
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writeln"Problem 37"; |
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goal LK.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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ab49d4f96a09
LK/hardquant/37: deleted call to flexflex_tac: no longer needed
lcp
parents:
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diff
changeset
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by (pc_tac LK_pack 1); |
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result(); |
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ab49d4f96a09
LK/hardquant/37: deleted call to flexflex_tac: no longer needed
lcp
parents:
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changeset
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writeln"Problem 38"; |
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goal LK.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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ab49d4f96a09
LK/hardquant/37: deleted call to flexflex_tac: no longer needed
lcp
parents:
0
diff
changeset
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by (pc_tac LK_pack 1); |
ab49d4f96a09
LK/hardquant/37: deleted call to flexflex_tac: no longer needed
lcp
parents:
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diff
changeset
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result(); |
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writeln"Problem 39"; |
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goal LK.thy "|- ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"; |
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by (fast_tac LK_pack 1); |
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result(); |
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writeln"Problem 40. AMENDED"; |
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goal LK.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 LK_pack 1); |
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result(); |
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writeln"Problem 41"; |
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goal LK.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 LK_pack 1); |
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result(); |
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writeln"Problem 42"; |
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goal LK.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 LK.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 LK.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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by (fast_tac LK_pack 1); |
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result(); |
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writeln"Problem 45"; |
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goal LK.thy "|- (ALL x. f(x) & (ALL y. g(y) & h(x,y) --> j(x,y)) \ |
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\ --> (ALL y. g(y) & h(x,y) --> k(y))) & \ |
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\ ~ (EX y. l(y) & k(y)) & \ |
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\ (EX x. f(x) & (ALL y. h(x,y) --> l(y)) \ |
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\ & (ALL y. g(y) & h(x,y) --> j(x,y))) \ |
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\ --> (EX x. f(x) & ~ (EX y. g(y) & h(x,y)))"; |
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by (best_tac LK_pack 1); |
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result(); |
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writeln"Problem 50"; |
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goal LK.thy |
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"|- (ALL x. P(a,x) | (ALL y.P(x,y))) --> (EX x. ALL y.P(x,y))"; |
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by (best_tac LK_dup_pack 1); |
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result(); |
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writeln"Problem 57"; |
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goal LK.thy |
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"|- P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \ |
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\ (ALL x y z. P(x,y) & P(y,z) --> P(x,z)) --> P(f(a,b), f(a,c))"; |
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by (fast_tac LK_pack 1); |
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result(); |
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writeln"Problem 59"; |
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(*Unification works poorly here -- the abstraction %sobj prevents efficient |
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operation of the occurs check*) |
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Unify.trace_bound := !Unify.search_bound - 1; |
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goal LK.thy "|- (ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))"; |
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by (best_tac LK_dup_pack 1); |
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result(); |
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writeln"Problem 60"; |
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goal LK.thy |
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"|- ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"; |
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by (fast_tac LK_pack 1); |
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result(); |
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writeln"Reached end of file."; |
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(*18 June 92: loaded in 372 secs*) |
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(*19 June 92: loaded in 166 secs except #34, using repeat_goal_tac*) |
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(*29 June 92: loaded in 370 secs*) |