(* Title: LK/ex/hard-quant ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge Hard examples with quantifiers. Can be read to test the LK system. From F. J. Pelletier, Seventy-Five Problems for Testing Automatic Theorem Provers, J. Automated Reasoning 2 (1986), 191-216. Errata, JAR 4 (1988), 236-236. Uses pc_tac rather than fast_tac when the former is significantly faster. *) context (theory "LK"); Goal "|- (ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))"; by (Fast_tac 1); result(); Goal "|- (EX x. P-->Q(x)) <-> (P --> (EX x. Q(x)))"; by (Fast_tac 1); result(); Goal "|- (EX x. P(x)-->Q) <-> (ALL x. P(x)) --> Q"; by (Fast_tac 1); result(); Goal "|- (ALL x. P(x)) | Q <-> (ALL x. P(x) | Q)"; by (Fast_tac 1); result(); writeln"Problems requiring quantifier duplication"; (*Not provable by Fast_tac: needs multiple instantiation of ALL*) Goal "|- (ALL x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))"; by (best_tac LK_dup_pack 1); result(); (*Needs double instantiation of the quantifier*) Goal "|- EX x. P(x) --> P(a) & P(b)"; by (fast_tac LK_dup_pack 1); result(); Goal "|- EX z. P(z) --> (ALL x. P(x))"; by (best_tac LK_dup_pack 1); result(); writeln"Hard examples with quantifiers"; writeln"Problem 18"; Goal "|- EX y. ALL x. P(y)-->P(x)"; by (best_tac LK_dup_pack 1); result(); writeln"Problem 19"; Goal "|- EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"; by (best_tac LK_dup_pack 1); result(); writeln"Problem 20"; Goal "|- (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w))) \ \ --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))"; by (Fast_tac 1); result(); writeln"Problem 21"; Goal "|- (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))"; by (best_tac LK_dup_pack 1); result(); writeln"Problem 22"; Goal "|- (ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))"; by (Fast_tac 1); result(); writeln"Problem 23"; Goal "|- (ALL x. P | Q(x)) <-> (P | (ALL x. Q(x)))"; by (best_tac LK_pack 1); result(); writeln"Problem 24"; Goal "|- ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) & \ \ ~(EX x. P(x)) --> (EX x. Q(x)) & (ALL x. Q(x)|R(x) --> S(x)) \ \ --> (EX x. P(x)&R(x))"; by (pc_tac LK_pack 1); result(); writeln"Problem 25"; Goal "|- (EX x. P(x)) & \ \ (ALL x. L(x) --> ~ (M(x) & R(x))) & \ \ (ALL x. P(x) --> (M(x) & L(x))) & \ \ ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x))) \ \ --> (EX x. Q(x)&P(x))"; by (best_tac LK_pack 1); result(); writeln"Problem 26"; Goal "|- ((EX x. p(x)) <-> (EX x. q(x))) & \ \ (ALL x. ALL y. p(x) & q(y) --> (r(x) <-> s(y))) \ \ --> ((ALL x. p(x)-->r(x)) <-> (ALL x. q(x)-->s(x)))"; by (pc_tac LK_pack 1); result(); writeln"Problem 27"; Goal "|- (EX x. P(x) & ~Q(x)) & \ \ (ALL x. P(x) --> R(x)) & \ \ (ALL x. M(x) & L(x) --> P(x)) & \ \ ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x))) \ \ --> (ALL x. M(x) --> ~L(x))"; by (pc_tac LK_pack 1); result(); writeln"Problem 28. AMENDED"; Goal "|- (ALL x. P(x) --> (ALL x. Q(x))) & \ \ ((ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) & \ \ ((EX x. S(x)) --> (ALL x. L(x) --> M(x))) \ \ --> (ALL x. P(x) & L(x) --> M(x))"; by (pc_tac LK_pack 1); result(); writeln"Problem 29. Essentially the same as Principia Mathematica *11.71"; Goal "|- (EX x. P(x)) & (EX y. Q(y)) \ \ --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y)) <-> \ \ (ALL x y. P(x) & Q(y) --> R(x) & S(y)))"; by (pc_tac LK_pack 1); result(); writeln"Problem 30"; Goal "|- (ALL x. P(x) | Q(x) --> ~ R(x)) & \ \ (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \ \ --> (ALL x. S(x))"; by (Fast_tac 1); result(); writeln"Problem 31"; Goal "|- ~(EX x. P(x) & (Q(x) | R(x))) & \ \ (EX x. L(x) & P(x)) & \ \ (ALL x. ~ R(x) --> M(x)) \ \ --> (EX x. L(x) & M(x))"; by (Fast_tac 1); result(); writeln"Problem 32"; Goal "|- (ALL x. P(x) & (Q(x)|R(x))-->S(x)) & \ \ (ALL x. S(x) & R(x) --> L(x)) & \ \ (ALL x. M(x) --> R(x)) \ \ --> (ALL x. P(x) & M(x) --> L(x))"; by (best_tac LK_pack 1); result(); writeln"Problem 33"; Goal "|- (ALL x. P(a) & (P(x)-->P(b))-->P(c)) <-> \ \ (ALL x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))"; by (Fast_tac 1); result(); writeln"Problem 34 AMENDED (TWICE!!)"; (*Andrews's challenge*) Goal "|- ((EX x. ALL y. p(x) <-> p(y)) <-> \ \ ((EX x. q(x)) <-> (ALL y. p(y)))) <-> \ \ ((EX x. ALL y. q(x) <-> q(y)) <-> \ \ ((EX x. p(x)) <-> (ALL y. q(y))))"; by (best_tac LK_dup_pack 1); (*10 secs!*) result(); writeln"Problem 35"; Goal "|- EX x y. P(x,y) --> (ALL u v. P(u,v))"; by (best_tac LK_dup_pack 1); result(); writeln"Problem 36"; Goal "|- (ALL x. EX y. J(x,y)) & \ \ (ALL x. EX y. G(x,y)) & \ \ (ALL x y. J(x,y) | G(x,y) --> \ \ (ALL z. J(y,z) | G(y,z) --> H(x,z))) \ \ --> (ALL x. EX y. H(x,y))"; by (Fast_tac 1); result(); writeln"Problem 37"; Goal "|- (ALL z. EX w. ALL x. EX y. \ \ (P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (EX u. Q(u,w)))) & \ \ (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) & \ \ ((EX x y. Q(x,y)) --> (ALL x. R(x,x))) \ \ --> (ALL x. EX y. R(x,y))"; by (pc_tac LK_pack 1); result(); writeln"Problem 38"; Goal "|- (ALL x. p(a) & (p(x) --> (EX y. p(y) & r(x,y))) --> \ \ (EX z. EX w. p(z) & r(x,w) & r(w,z))) <-> \ \ (ALL x. (~p(a) | p(x) | (EX z. EX w. p(z) & r(x,w) & r(w,z))) & \ \ (~p(a) | ~(EX y. p(y) & r(x,y)) | \ \ (EX z. EX w. p(z) & r(x,w) & r(w,z))))"; by (pc_tac LK_pack 1); result(); writeln"Problem 39"; Goal "|- ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"; by (Fast_tac 1); result(); writeln"Problem 40. AMENDED"; Goal "|- (EX y. ALL x. F(x,y) <-> F(x,x)) --> \ \ ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))"; by (Fast_tac 1); result(); writeln"Problem 41"; Goal "|- (ALL z. EX y. ALL x. f(x,y) <-> f(x,z) & ~ f(x,x)) \ \ --> ~ (EX z. ALL x. f(x,z))"; by (Fast_tac 1); result(); writeln"Problem 42"; Goal "|- ~ (EX y. ALL x. p(x,y) <-> ~ (EX z. p(x,z) & p(z,x)))"; writeln"Problem 43 NOT PROVED AUTOMATICALLY"; Goal "|- (ALL x. ALL y. q(x,y) <-> (ALL z. p(z,x) <-> p(z,y))) \ \ --> (ALL x. (ALL y. q(x,y) <-> q(y,x)))"; writeln"Problem 44"; Goal "|- (ALL x. f(x) --> \ \ (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y)))) & \ \ (EX x. j(x) & (ALL y. g(y) --> h(x,y))) \ \ --> (EX x. j(x) & ~f(x))"; by (Fast_tac 1); result(); writeln"Problem 45"; Goal "|- (ALL x. f(x) & (ALL y. g(y) & h(x,y) --> j(x,y)) \ \ --> (ALL y. g(y) & h(x,y) --> k(y))) & \ \ ~ (EX y. l(y) & k(y)) & \ \ (EX x. f(x) & (ALL y. h(x,y) --> l(y)) \ \ & (ALL y. g(y) & h(x,y) --> j(x,y))) \ \ --> (EX x. f(x) & ~ (EX y. g(y) & h(x,y)))"; by (best_tac LK_pack 1); result(); writeln"Problems (mainly) involving equality or functions"; writeln"Problem 48"; Goal "|- (a=b | c=d) & (a=c | b=d) --> a=d | b=c"; by (fast_tac (pack() add_safes [subst]) 1); result(); writeln"Problem 50"; Goal "|- (ALL x. P(a,x) | (ALL y. P(x,y))) --> (EX x. ALL y. P(x,y))"; by (best_tac LK_dup_pack 1); result(); writeln"Problem 51"; Goal "|- (EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \ \ (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)"; by (fast_tac (pack() add_safes [subst]) 1); result(); writeln"Problem 52"; (*Almost the same as 51. *) Goal "|- (EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \ \ (EX w. ALL y. EX z. (ALL x. P(x,y) <-> x=z) <-> y=w)"; by (fast_tac (pack() add_safes [subst]) 1); result(); writeln"Problem 56"; Goal "|- (ALL x.(EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"; by (best_tac (pack() add_unsafes [symL, subst]) 1); (*requires tricker to orient the equality properly*) result(); writeln"Problem 57"; Goal "|- P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \ \ (ALL x y z. P(x,y) & P(y,z) --> P(x,z)) --> P(f(a,b), f(a,c))"; by (Fast_tac 1); result(); writeln"Problem 58!"; Goal "|- (ALL x y. f(x)=g(y)) --> (ALL x y. f(f(x))=f(g(y)))"; by (fast_tac (pack() add_safes [subst]) 1); result(); writeln"Problem 59"; (*Unification works poorly here -- the abstraction %sobj prevents efficient operation of the occurs check*) Unify.trace_bound := !Unify.search_bound - 1; Goal "|- (ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))"; by (best_tac LK_dup_pack 1); result(); writeln"Problem 60"; Goal "|- ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"; by (Fast_tac 1); result(); writeln"Problem 62 as corrected in JAR 18 (1997), page 135"; Goal "|- (ALL x. p(a) & (p(x) --> p(f(x))) --> p(f(f(x)))) <-> \ \ (ALL x. (~p(a) | p(x) | p(f(f(x)))) & \ \ (~p(a) | ~p(f(x)) | p(f(f(x)))))"; by (Fast_tac 1); result(); (*18 June 92: loaded in 372 secs*) (*19 June 92: loaded in 166 secs except #34, using repeat_goal_tac*) (*29 June 92: loaded in 370 secs*) (*18 September 2005: loaded in 1.809 secs*)