(* Title: FOLP/ex/cla ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1993 University of Cambridge Classical First-Order Logic *) writeln"File FOLP/ex/cla.ML"; open Cla; (*in case structure Int is open!*) goal FOLP.thy "?p : (P --> Q | R) --> (P-->Q) | (P-->R)"; by (fast_tac FOLP_cs 1); result(); (*If and only if*) goal FOLP.thy "?p : (P<->Q) <-> (Q<->P)"; by (fast_tac FOLP_cs 1); result(); goal FOLP.thy "?p : ~ (P <-> ~P)"; by (fast_tac FOLP_cs 1); result(); (*Sample problems 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. The hardest problems -- judging by experience with several theorem provers, including matrix ones -- are 34 and 43. *) writeln"Pelletier's examples"; (*1*) goal FOLP.thy "?p : (P-->Q) <-> (~Q --> ~P)"; by (fast_tac FOLP_cs 1); result(); (*2*) goal FOLP.thy "?p : ~ ~ P <-> P"; by (fast_tac FOLP_cs 1); result(); (*3*) goal FOLP.thy "?p : ~(P-->Q) --> (Q-->P)"; by (fast_tac FOLP_cs 1); result(); (*4*) goal FOLP.thy "?p : (~P-->Q) <-> (~Q --> P)"; by (fast_tac FOLP_cs 1); result(); (*5*) goal FOLP.thy "?p : ((P|Q)-->(P|R)) --> (P|(Q-->R))"; by (fast_tac FOLP_cs 1); result(); (*6*) goal FOLP.thy "?p : P | ~ P"; by (fast_tac FOLP_cs 1); result(); (*7*) goal FOLP.thy "?p : P | ~ ~ ~ P"; by (fast_tac FOLP_cs 1); result(); (*8. Peirce's law*) goal FOLP.thy "?p : ((P-->Q) --> P) --> P"; by (fast_tac FOLP_cs 1); result(); (*9*) goal FOLP.thy "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"; by (fast_tac FOLP_cs 1); result(); (*10*) goal FOLP.thy "?p : (Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P<->Q)"; by (fast_tac FOLP_cs 1); result(); (*11. Proved in each direction (incorrectly, says Pelletier!!) *) goal FOLP.thy "?p : P<->P"; by (fast_tac FOLP_cs 1); result(); (*12. "Dijkstra's law"*) goal FOLP.thy "?p : ((P <-> Q) <-> R) <-> (P <-> (Q <-> R))"; by (fast_tac FOLP_cs 1); result(); (*13. Distributive law*) goal FOLP.thy "?p : P | (Q & R) <-> (P | Q) & (P | R)"; by (fast_tac FOLP_cs 1); result(); (*14*) goal FOLP.thy "?p : (P <-> Q) <-> ((Q | ~P) & (~Q|P))"; by (fast_tac FOLP_cs 1); result(); (*15*) goal FOLP.thy "?p : (P --> Q) <-> (~P | Q)"; by (fast_tac FOLP_cs 1); result(); (*16*) goal FOLP.thy "?p : (P-->Q) | (Q-->P)"; by (fast_tac FOLP_cs 1); result(); (*17*) goal FOLP.thy "?p : ((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))"; by (fast_tac FOLP_cs 1); result(); writeln"Classical Logic: examples with quantifiers"; goal FOLP.thy "?p : (ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))"; by (fast_tac FOLP_cs 1); result(); goal FOLP.thy "?p : (EX x. P-->Q(x)) <-> (P --> (EX x.Q(x)))"; by (fast_tac FOLP_cs 1); result(); goal FOLP.thy "?p : (EX x.P(x)-->Q) <-> (ALL x.P(x)) --> Q"; by (fast_tac FOLP_cs 1); result(); goal FOLP.thy "?p : (ALL x.P(x)) | Q <-> (ALL x. P(x) | Q)"; by (fast_tac FOLP_cs 1); result(); writeln"Problems requiring quantifier duplication"; (*Needs multiple instantiation of ALL.*) (* goal FOLP.thy "?p : (ALL x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))"; by (best_tac FOLP_dup_cs 1); result(); *) (*Needs double instantiation of the quantifier*) goal FOLP.thy "?p : EX x. P(x) --> P(a) & P(b)"; by (best_tac FOLP_dup_cs 1); result(); goal FOLP.thy "?p : EX z. P(z) --> (ALL x. P(x))"; by (best_tac FOLP_dup_cs 1); result(); writeln"Hard examples with quantifiers"; writeln"Problem 18"; goal FOLP.thy "?p : EX y. ALL x. P(y)-->P(x)"; by (best_tac FOLP_dup_cs 1); result(); writeln"Problem 19"; goal FOLP.thy "?p : EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"; by (best_tac FOLP_dup_cs 1); result(); writeln"Problem 20"; goal FOLP.thy "?p : (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 FOLP_cs 1); result(); (* writeln"Problem 21"; goal FOLP.thy "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))"; by (best_tac FOLP_dup_cs 1); result(); *) writeln"Problem 22"; goal FOLP.thy "?p : (ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))"; by (fast_tac FOLP_cs 1); result(); writeln"Problem 23"; goal FOLP.thy "?p : (ALL x. P | Q(x)) <-> (P | (ALL x. Q(x)))"; by (best_tac FOLP_cs 1); result(); writeln"Problem 24"; goal FOLP.thy "?p : ~(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 (fast_tac FOLP_cs 1); result(); (* writeln"Problem 25"; goal FOLP.thy "?p : (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 FOLP_cs 1); result(); writeln"Problem 26"; goal FOLP.thy "?u : ((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 (fast_tac FOLP_cs 1); result(); *) writeln"Problem 27"; goal FOLP.thy "?p : (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 (fast_tac FOLP_cs 1); result(); writeln"Problem 28. AMENDED"; goal FOLP.thy "?p : (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 (fast_tac FOLP_cs 1); result(); writeln"Problem 29. Essentially the same as Principia Mathematica *11.71"; goal FOLP.thy "?p : (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 (fast_tac FOLP_cs 1); result(); writeln"Problem 30"; goal FOLP.thy "?p : (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 FOLP_cs 1); result(); writeln"Problem 31"; goal FOLP.thy "?p : ~(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 FOLP_cs 1); result(); writeln"Problem 32"; goal FOLP.thy "?p : (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 FOLP_cs 1); result(); writeln"Problem 33"; goal FOLP.thy "?p : (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 (best_tac FOLP_cs 1); result(); writeln"Problem 35"; goal FOLP.thy "?p : EX x y. P(x,y) --> (ALL u v. P(u,v))"; by (best_tac FOLP_dup_cs 1); result(); writeln"Problem 36"; goal FOLP.thy "?p : (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 FOLP_cs 1); result(); writeln"Problem 37"; goal FOLP.thy "?p : (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 (fast_tac FOLP_cs 1); result(); writeln"Problem 39"; goal FOLP.thy "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"; by (fast_tac FOLP_cs 1); result(); writeln"Problem 40. AMENDED"; goal FOLP.thy "?p : (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 FOLP_cs 1); result(); writeln"Problem 41"; goal FOLP.thy "?p : (ALL z. EX y. ALL x. f(x,y) <-> f(x,z) & ~ f(x,x)) \ \ --> ~ (EX z. ALL x. f(x,z))"; by (best_tac FOLP_cs 1); result(); writeln"Problem 44"; goal FOLP.thy "?p : (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 FOLP_cs 1); result(); writeln"Problems (mainly) involving equality or functions"; writeln"Problem 48"; goal FOLP.thy "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c"; by (fast_tac FOLP_cs 1); result(); writeln"Problem 50"; (*What has this to do with equality?*) goal FOLP.thy "?p : (ALL x. P(a,x) | (ALL y.P(x,y))) --> (EX x. ALL y.P(x,y))"; by (best_tac FOLP_dup_cs 1); result(); writeln"Problem 56"; goal FOLP.thy "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"; by (fast_tac FOLP_cs 1); result(); writeln"Problem 57"; goal FOLP.thy "?p : 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 FOLP_cs 1); result(); writeln"Problem 58 NOT PROVED AUTOMATICALLY"; goal FOLP.thy "?p : (ALL x y. f(x)=g(y)) --> (ALL x y. f(f(x))=f(g(y)))"; val f_cong = read_instantiate [("t","f")] subst_context; by (fast_tac (FOLP_cs addIs [f_cong]) 1); result(); writeln"Problem 59"; goal FOLP.thy "?p : (ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))"; by (best_tac FOLP_dup_cs 1); result(); writeln"Problem 60"; goal FOLP.thy "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"; by (fast_tac FOLP_cs 1); result(); writeln"Reached end of file.";