ex/cla.ML

changed "." to "$" and Cons to infix "#" to eliminate ambiguity

(*  Title: 	HOL/ex/cla
    ID:         $Id$
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1992  University of Cambridge

Higher-Order Logic: predicate calculus problems

Taken from FOL/cla.ML; beware of precedence of = vs <->
*)

writeln"File HOL/ex/cla.";

goal HOL.thy "(P --> Q | R) --> (P-->Q) | (P-->R)";
by (fast_tac HOL_cs 1);
result();

(*If and only if*)

goal HOL.thy "(P=Q) = (Q=P::bool)";
by (fast_tac HOL_cs 1);
result();

goal HOL.thy "~ (P = (~P))";
by (fast_tac HOL_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 HOL.thy "(P-->Q)  =  (~Q --> ~P)";
by (fast_tac HOL_cs 1);
result();

(*2*)
goal HOL.thy "(~ ~ P) =  P";
by (fast_tac HOL_cs 1);
result();

(*3*)
goal HOL.thy "~(P-->Q) --> (Q-->P)";
by (fast_tac HOL_cs 1);
result();

(*4*)
goal HOL.thy "(~P-->Q)  =  (~Q --> P)";
by (fast_tac HOL_cs 1);
result();

(*5*)
goal HOL.thy "((P|Q)-->(P|R)) --> (P|(Q-->R))";
by (fast_tac HOL_cs 1);
result();

(*6*)
goal HOL.thy "P | ~ P";
by (fast_tac HOL_cs 1);
result();

(*7*)
goal HOL.thy "P | ~ ~ ~ P";
by (fast_tac HOL_cs 1);
result();

(*8.  Peirce's law*)
goal HOL.thy "((P-->Q) --> P)  -->  P";
by (fast_tac HOL_cs 1);
result();

(*9*)
goal HOL.thy "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
by (fast_tac HOL_cs 1);
result();

(*10*)
goal HOL.thy "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)";
by (fast_tac HOL_cs 1);
result();

(*11.  Proved in each direction (incorrectly, says Pelletier!!)  *)
goal HOL.thy "P=P::bool";
by (fast_tac HOL_cs 1);
result();

(*12.  "Dijkstra's law"*)
goal HOL.thy "((P = Q) = R) = (P = (Q = R))";
by (fast_tac HOL_cs 1);
result();

(*13.  Distributive law*)
goal HOL.thy "(P | (Q & R)) = ((P | Q) & (P | R))";
by (fast_tac HOL_cs 1);
result();

(*14*)
goal HOL.thy "(P = Q) = ((Q | ~P) & (~Q|P))";
by (fast_tac HOL_cs 1);
result();

(*15*)
goal HOL.thy "(P --> Q) = (~P | Q)";
by (fast_tac HOL_cs 1);
result();

(*16*)
goal HOL.thy "(P-->Q) | (Q-->P)";
by (fast_tac HOL_cs 1);
result();

(*17*)
goal HOL.thy "((P & (Q-->R))-->S)  =  ((~P | Q | S) & (~P | ~R | S))";
by (fast_tac HOL_cs 1);
result();

writeln"Classical Logic: examples with quantifiers";

goal HOL.thy "(! x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
by (fast_tac HOL_cs 1);
result(); 

goal HOL.thy "(? x. P-->Q(x))  =  (P --> (? x.Q(x)))";
by (fast_tac HOL_cs 1);
result(); 

goal HOL.thy "(? x.P(x)-->Q) = ((! x.P(x)) --> Q)";
by (fast_tac HOL_cs 1);
result(); 

goal HOL.thy "((! x.P(x)) | Q)  =  (! x. P(x) | Q)";
by (fast_tac HOL_cs 1);
result(); 

(*From Wishnu Prasetya*)
goal HOL.thy
   "(!s. q(s) --> r(s)) & ~r(s) & (!s. ~r(s) & ~q(s) --> p(t) | q(t)) \
\   --> p(t) | r(t)";
by (fast_tac HOL_cs 1);
result(); 


writeln"Problems requiring quantifier duplication";

(*Needs multiple instantiation of the quantifier.*)
goal HOL.thy "(! x. P(x)-->P(f(x)))  &  P(d)-->P(f(f(f(d))))";
by (best_tac HOL_dup_cs 1);
result();

(*Needs double instantiation of the quantifier*)
goal HOL.thy "? x. P(x) --> P(a) & P(b)";
by (best_tac HOL_dup_cs 1);
result();

goal HOL.thy "? z. P(z) --> (! x. P(x))";
by (best_tac HOL_dup_cs 1);
result();

writeln"Hard examples with quantifiers";

writeln"Problem 18";
goal HOL.thy "? y. ! x. P(y)-->P(x)";
by (best_tac HOL_dup_cs 1);
result(); 

writeln"Problem 19";
goal HOL.thy "? x. ! y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";
by (best_tac HOL_dup_cs 1);
result();

writeln"Problem 20";
goal HOL.thy "(! x y. ? z. ! w. (P(x)&Q(y)-->R(z)&S(w)))     \
\   --> (? x y. P(x) & Q(y)) --> (? z. R(z))";
by (fast_tac HOL_cs 1); 
result();

writeln"Problem 21";
goal HOL.thy "(? x. P-->Q(x)) & (? x. Q(x)-->P) --> (? x. P=Q(x))";
by (best_tac HOL_dup_cs 1); 
result();

writeln"Problem 22";
goal HOL.thy "(! x. P = Q(x))  -->  (P = (! x. Q(x)))";
by (fast_tac HOL_cs 1); 
result();

writeln"Problem 23";
goal HOL.thy "(! x. P | Q(x))  =  (P | (! x. Q(x)))";
by (best_tac HOL_cs 1);  
result();

writeln"Problem 24";
goal HOL.thy "~(? x. S(x)&Q(x)) & (! x. P(x) --> Q(x)|R(x)) &  \
\    ~(? x.P(x)) --> (? x.Q(x)) & (! x. Q(x)|R(x) --> S(x))  \
\   --> (? x. P(x)&R(x))";
by (fast_tac HOL_cs 1); 
result();

writeln"Problem 25";
goal HOL.thy "(? x. P(x)) &  \
\       (! x. L(x) --> ~ (M(x) & R(x))) &  \
\       (! x. P(x) --> (M(x) & L(x))) &   \
\       ((! x. P(x)-->Q(x)) | (? x. P(x)&R(x)))  \
\   --> (? x. Q(x)&P(x))";
by (best_tac HOL_cs 1); 
result();

writeln"Problem 26";
goal HOL.thy "((? x. p(x)) = (? x. q(x))) &	\
\     (! x. ! y. p(x) & q(y) --> (r(x) = s(y)))	\
\ --> ((! x. p(x)-->r(x)) = (! x. q(x)-->s(x)))";
by (fast_tac HOL_cs 1);
result();

writeln"Problem 27";
goal HOL.thy "(? x. P(x) & ~Q(x)) &   \
\             (! x. P(x) --> R(x)) &   \
\             (! x. M(x) & L(x) --> P(x)) &   \
\             ((? x. R(x) & ~ Q(x)) --> (! x. L(x) --> ~ R(x)))  \
\         --> (! x. M(x) --> ~L(x))";
by (fast_tac HOL_cs 1); 
result();

writeln"Problem 28.  AMENDED";
goal HOL.thy "(! x. P(x) --> (! x. Q(x))) &   \
\       ((! x. Q(x)|R(x)) --> (? x. Q(x)&S(x))) &  \
\       ((? x.S(x)) --> (! x. L(x) --> M(x)))  \
\   --> (! x. P(x) & L(x) --> M(x))";
by (fast_tac HOL_cs 1);  
result();

writeln"Problem 29.  Essentially the same as Principia Mathematica *11.71";
goal HOL.thy "(? x. F(x)) & (? y. G(y))  \
\   --> ( ((! x. F(x)-->H(x)) & (! y. G(y)-->J(y)))  =   \
\         (! x y. F(x) & G(y) --> H(x) & J(y)))";
by (fast_tac HOL_cs 1); 
result();

writeln"Problem 30";
goal HOL.thy "(! x. P(x) | Q(x) --> ~ R(x)) & \
\       (! x. (Q(x) --> ~ S(x)) --> P(x) & R(x))  \
\   --> (! x. S(x))";
by (fast_tac HOL_cs 1);  
result();

writeln"Problem 31";
goal HOL.thy "~(? x.P(x) & (Q(x) | R(x))) & \
\       (? x. L(x) & P(x)) & \
\       (! x. ~ R(x) --> M(x))  \
\   --> (? x. L(x) & M(x))";
by (fast_tac HOL_cs 1);
result();

writeln"Problem 32";
goal HOL.thy "(! x. P(x) & (Q(x)|R(x))-->S(x)) & \
\       (! x. S(x) & R(x) --> L(x)) & \
\       (! x. M(x) --> R(x))  \
\   --> (! x. P(x) & M(x) --> L(x))";
by (best_tac HOL_cs 1);
result();

writeln"Problem 33";
goal HOL.thy "(! x. P(a) & (P(x)-->P(b))-->P(c))  =    \
\    (! x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))";
by (best_tac HOL_cs 1);
result();

writeln"Problem 34  AMENDED (TWICE!!)  NOT PROVED AUTOMATICALLY";
(*Andrews's challenge*)
goal HOL.thy "((? x. ! y. p(x) = p(y))  =		\
\                   ((? x. q(x)) = (! y. p(y))))   =	\
\                  ((? x. ! y. q(x) = q(y))  =		\
\                   ((? x. p(x)) = (! y. q(y))))";
by (safe_tac HOL_cs);			(*24 secs*)
by (TRYALL (fast_tac HOL_cs));		(*161 secs*)
by (TRYALL (best_tac HOL_dup_cs));	(*86 secs -- much faster than FOL!*)
result();

writeln"Problem 35";
goal HOL.thy "? x y. P(x,y) -->  (! u v. P(u,v))";
by (best_tac HOL_dup_cs 1);
result();

writeln"Problem 36";
goal HOL.thy "(! x. ? y. J(x,y)) & \
\       (! x. ? y. G(x,y)) & \
\       (! x y. J(x,y) | G(x,y) -->	\
\       (! z. J(y,z) | G(y,z) --> H(x,z)))   \
\   --> (! x. ? y. H(x,y))";
by (fast_tac HOL_cs 1);
result();

writeln"Problem 37";
goal HOL.thy "(! z. ? w. ! x. ? y. \
\          (P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (? u.Q(u,w)))) & \
\       (! x z. ~P(x,z) --> (? y. Q(y,z))) & \
\       ((? x y. Q(x,y)) --> (! x. R(x,x)))  \
\   --> (! x. ? y. R(x,y))";
by (fast_tac HOL_cs 1);
result();

writeln"Problem 38";
goal HOL.thy
    "(! x. p(a) & (p(x) --> (? y. p(y) & r(x,y))) -->		\
\          (? z. ? w. p(z) & r(x,w) & r(w,z)))  =			\
\    (! x. (~p(a) | p(x) | (? z. ? w. p(z) & r(x,w) & r(w,z))) &	\
\          (~p(a) | ~(? y. p(y) & r(x,y)) |				\
\           (? z. ? w. p(z) & r(x,w) & r(w,z))))";

writeln"Problem 39";
goal HOL.thy "~ (? x. ! y. F(y,x) = (~F(y,y)))";
by (fast_tac HOL_cs 1);
result();

writeln"Problem 40.  AMENDED";
goal HOL.thy "(? y. ! x. F(x,y) = F(x,x))  \
\       -->  ~ (! x. ? y. ! z. F(z,y) = (~F(z,x)))";
by (fast_tac HOL_cs 1);
result();

writeln"Problem 41";
goal HOL.thy "(! z. ? y. ! x. f(x,y) = (f(x,z) & ~ f(x,x)))	\
\              --> ~ (? z. ! x. f(x,z))";
by (best_tac HOL_cs 1);
result();

writeln"Problem 42";
goal HOL.thy "~ (? y. ! x. p(x,y) = (~ (? z. p(x,z) & p(z,x))))";

writeln"Problem 43  NOT PROVED AUTOMATICALLY";
goal HOL.thy
    "(! x::'a. ! y::'a. q(x,y) = (! z. p(z,x) = p(z,y)::bool))	\
\ --> (! x. (! y. q(x,y) = q(y,x)::bool))";


writeln"Problem 44";
goal HOL.thy "(! x. f(x) -->					\
\             (? y. g(y) & h(x,y) & (? y. g(y) & ~ h(x,y))))  &   	\
\             (? x. j(x) & (! y. g(y) --> h(x,y)))			\
\             --> (? x. j(x) & ~f(x))";
by (fast_tac HOL_cs 1);
result();

writeln"Problem 45";
goal HOL.thy
    "(! x. f(x) & (! y. g(y) & h(x,y) --> j(x,y))	\
\                     --> (! y. g(y) & h(x,y) --> k(y))) &	\
\    ~ (? y. l(y) & k(y)) &					\
\    (? x. f(x) & (! y. h(x,y) --> l(y))			\
\               & (! y. g(y) & h(x,y) --> j(x,y)))		\
\     --> (? x. f(x) & ~ (? y. g(y) & h(x,y)))";
by (best_tac HOL_cs 1); 
result();


writeln"Problems (mainly) involving equality or functions";

writeln"Problem 48";
goal HOL.thy "(a=b | c=d) & (a=c | b=d) --> a=d | b=c";
by (fast_tac HOL_cs 1);
result();

writeln"Problem 49  NOT PROVED AUTOMATICALLY";
(*Hard because it involves substitution for Vars;
  the type constraint ensures that x,y,z have the same type as a,b,u. *)
goal HOL.thy "(? x y::'a. ! z. z=x | z=y) & P(a) & P(b) & (~a=b) \
\		--> (! u::'a.P(u))";
by (Classical.safe_tac HOL_cs);
by (res_inst_tac [("x","a")] allE 1);
by (assume_tac 1);
by (res_inst_tac [("x","b")] allE 1);
by (assume_tac 1);
by (fast_tac HOL_cs 1);
result();

writeln"Problem 50";  
(*What has this to do with equality?*)
goal HOL.thy "(! x. P(a,x) | (! y.P(x,y))) --> (? x. ! y.P(x,y))";
by (best_tac HOL_dup_cs 1);
result();

writeln"Problem 51";
goal HOL.thy
    "(? z w. ! x y. P(x,y) = (x=z & y=w)) -->  \
\    (? z. ! x. ? w. (! y. P(x,y) = (y=w)) = (x=z))";
by (best_tac HOL_cs 1);
result();

writeln"Problem 52";
(*Almost the same as 51. *)
goal HOL.thy
    "(? z w. ! x y. P(x,y) = (x=z & y=w)) -->  \
\    (? w. ! y. ? z. (! x. P(x,y) = (x=z)) = (y=w))";
by (best_tac HOL_cs 1);
result();

writeln"Problem 55";

(*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
  fast_tac DISCOVERS who killed Agatha. *)
goal HOL.thy "lives(agatha) & lives(butler) & lives(charles) & \
\  (killed(agatha,agatha) | killed(butler,agatha) | killed(charles,agatha)) & \
\  (!x y. killed(x,y) --> hates(x,y) & ~richer(x,y)) & \
\  (!x. hates(agatha,x) --> ~hates(charles,x)) & \
\  (hates(agatha,agatha) & hates(agatha,charles)) & \
\  (!x. lives(x) & ~richer(x,agatha) --> hates(butler,x)) & \
\  (!x. hates(agatha,x) --> hates(butler,x)) & \
\  (!x. ~hates(x,agatha) | ~hates(x,butler) | ~hates(x,charles)) --> \
\   killed(?who,agatha)";
by (fast_tac HOL_cs 1);
result();

writeln"Problem 56";
goal HOL.thy
    "(! x. (? y. P(y) & x=f(y)) --> P(x)) = (! x. P(x) --> P(f(x)))";
by (fast_tac HOL_cs 1);
result();

writeln"Problem 57";
goal HOL.thy
    "P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \
\    (! x y z. P(x,y) & P(y,z) --> P(x,z))    -->   P(f(a,b), f(a,c))";
by (fast_tac HOL_cs 1);
result();

writeln"Problem 58  NOT PROVED AUTOMATICALLY";
goal HOL.thy "(! x y. f(x)=g(y)) --> (! x y. f(f(x))=f(g(y)))";
val f_cong = read_instantiate [("f","f")] arg_cong;
by (fast_tac (HOL_cs addIs [f_cong]) 1);
result();

writeln"Problem 59";
goal HOL.thy "(! x. P(x) = (~P(f(x)))) --> (? x. P(x) & ~P(f(x)))";
by (best_tac HOL_dup_cs 1);
result();

writeln"Problem 60";
goal HOL.thy
    "! x. P(x,f(x)) = (? y. (! z. P(z,y) --> P(z,f(x))) & P(x,y))";
by (fast_tac HOL_cs 1);
result();

writeln"Reached end of file.";