src/LK/ex/hard-quant.ML

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(*  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.
*)

writeln"File LK/ex/hard-quant.";

goal LK.thy "|- (ALL x. P(x) & Q(x)) <-> (ALL x. P(x))  &  (ALL x. Q(x))";
by (fast_tac LK_pack 1);
result(); 

goal LK.thy "|- (EX x. P-->Q(x))  <->  (P --> (EX x.Q(x)))";
by (fast_tac LK_pack 1);
result(); 

goal LK.thy "|- (EX x.P(x)-->Q)  <->  (ALL x.P(x)) --> Q";
by (fast_tac LK_pack 1);
result(); 

goal LK.thy "|- (ALL x.P(x)) | Q  <->  (ALL x. P(x) | Q)";
by (fast_tac LK_pack 1);
result(); 

writeln"Problems requiring quantifier duplication";

(*Not provable by fast_tac LK_pack: needs multiple instantiation of ALL*)
goal LK.thy "|- (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 LK.thy "|- EX x. P(x) --> P(a) & P(b)";
by (fast_tac LK_dup_pack 1);
result();

goal LK.thy "|- 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 LK.thy "|- EX y. ALL x. P(y)-->P(x)";
by (best_tac LK_dup_pack 1);
result(); 

writeln"Problem 19";
goal LK.thy "|- 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 LK.thy "|- (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 LK_pack 1); 
result();

writeln"Problem 21";
goal LK.thy "|- (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 LK.thy "|- (ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))";
by (fast_tac LK_pack 1); 
result();

writeln"Problem 23";
goal LK.thy "|- (ALL x. P | Q(x))  <->  (P | (ALL x. Q(x)))";
by (best_tac LK_pack 1);  
result();

writeln"Problem 24";
goal LK.thy "|- ~(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 LK.thy "|- (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 LK.thy "|- ((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 LK.thy "|- (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 LK.thy "|- (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 LK.thy "|- (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 LK.thy "|- (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 LK_pack 1);  
result();

writeln"Problem 31";
goal LK.thy "|- ~(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 LK_pack 1);
result();

writeln"Problem 32";
goal LK.thy "|- (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 LK.thy "|- (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 LK_pack 1);
result();

writeln"Problem 34  AMENDED (TWICE!!)  NOT PROVED AUTOMATICALLY";
(*Andrews's challenge*)
goal LK.thy "|- ((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 (safe_goal_tac LK_pack 1);   (*53 secs*) (*13 secs*)
by (TRYALL (fast_tac LK_pack));	(*165 secs*)  (*117 secs*)  (*138 secs*)
(*for some reason, pc_tac leaves 14 subgoals instead of 6*)
by (TRYALL (best_tac LK_dup_pack)); (*55 secs*) (*29 secs*) (*54 secs*)
result();


writeln"Problem 35";
goal LK.thy "|- EX x y. P(x,y) -->  (ALL u v. P(u,v))";
by (best_tac LK_dup_pack 1);  (*27 secs??*)
result();

writeln"Problem 36";
goal LK.thy "|- (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 LK_pack 1);
result();

writeln"Problem 37";
goal LK.thy "|- (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);  (*slow*)
by flexflex_tac;
result();

writeln"Problem 38. NOT PROVED";
goal LK.thy
 "|- (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))))";

writeln"Problem 39";
goal LK.thy "|- ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))";
by (fast_tac LK_pack 1);
result();

writeln"Problem 40.  AMENDED";
goal LK.thy "|- (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 LK_pack 1);
result();

writeln"Problem 41";
goal LK.thy "|- (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 LK_pack 1);
result();

writeln"Problem 42";
goal LK.thy "|- ~ (EX y. ALL x. p(x,y) <-> ~ (EX z. p(x,z) & p(z,x)))";

writeln"Problem 43  NOT PROVED AUTOMATICALLY";
goal LK.thy "|- (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 LK.thy "|- (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 LK_pack 1);
result();

writeln"Problem 45";
goal LK.thy "|- (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"Problem 50";  
goal LK.thy
    "|- (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 57";
goal LK.thy
    "|- 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 LK_pack 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 LK.thy "|- (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 LK.thy
    "|- ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))";
by (fast_tac LK_pack 1);
result();

writeln"Reached end of file.";

(*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*)