(*  Title:      HOL/ex/cla

    ID:         $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1994  University of Cambridge

Higher-Order Logic: predicate calculus problems

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

*)

writeln"File HOL/ex/cla.";

context HOL.thy;  (*Boosts efficiency by omitting redundant rules*)

goal HOL.thy "(P --> Q | R) --> (P-->Q) | (P-->R)";

by (Blast_tac 1);

result();

(*If and only if*)

goal HOL.thy "(P=Q) = (Q = (P::bool))";

by (Blast_tac 1);

result();

goal HOL.thy "~ (P = (~P))";

by (Blast_tac 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 (Blast_tac 1);

result();

(*2*)

goal HOL.thy "(~ ~ P) =  P";

by (Blast_tac 1);

result();

(*3*)

goal HOL.thy "~(P-->Q) --> (Q-->P)";

by (Blast_tac 1);

result();

(*4*)

goal HOL.thy "(~P-->Q)  =  (~Q --> P)";

by (Blast_tac 1);

result();

(*5*)

goal HOL.thy "((P|Q)-->(P|R)) --> (P|(Q-->R))";

by (Blast_tac 1);

result();

(*6*)

goal HOL.thy "P | ~ P";

by (Blast_tac 1);

result();

(*7*)

goal HOL.thy "P | ~ ~ ~ P";

by (Blast_tac 1);

result();

(*8.  Peirce's law*)

goal HOL.thy "((P-->Q) --> P)  -->  P";

by (Blast_tac 1);

result();

(*9*)

goal HOL.thy "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";

by (Blast_tac 1);

result();

(*10*)

goal HOL.thy "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)";

by (Blast_tac 1);

result();

(*11.  Proved in each direction (incorrectly, says Pelletier!!)  *)

goal HOL.thy "P=(P::bool)";

by (Blast_tac 1);

result();

(*12.  "Dijkstra's law"*)

goal HOL.thy "((P = Q) = R) = (P = (Q = R))";

by (Blast_tac 1);

result();

(*13.  Distributive law*)

goal HOL.thy "(P | (Q & R)) = ((P | Q) & (P | R))";

by (Blast_tac 1);

result();

(*14*)

goal HOL.thy "(P = Q) = ((Q | ~P) & (~Q|P))";

by (Blast_tac 1);

result();

(*15*)

goal HOL.thy "(P --> Q) = (~P | Q)";

by (Blast_tac 1);

result();

(*16*)

goal HOL.thy "(P-->Q) | (Q-->P)";

by (Blast_tac 1);

result();

(*17*)

goal HOL.thy "((P & (Q-->R))-->S)  =  ((~P | Q | S) & (~P | ~R | S))";

by (Blast_tac 1);

result();

writeln"Classical Logic: examples with quantifiers";

goal HOL.thy "(! x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";

by (Blast_tac 1);

result(); 

goal HOL.thy "(? x. P-->Q(x))  =  (P --> (? x. Q(x)))";

by (Blast_tac 1);

result(); 

goal HOL.thy "(? x. P(x)-->Q) = ((! x. P(x)) --> Q)";

by (Blast_tac 1);

result(); 

goal HOL.thy "((! x. P(x)) | Q)  =  (! x. P(x) | Q)";

by (Blast_tac 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 (Blast_tac 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 (Blast_tac 1);

result();

(*Needs double instantiation of the quantifier*)

goal HOL.thy "? x. P(x) --> P(a) & P(b)";

by (Blast_tac 1);

result();

goal HOL.thy "? z. P(z) --> (! x. P(x))";

by (Blast_tac 1);

result();

goal HOL.thy "? x. (? y. P(y)) --> P(x)";

by (Blast_tac 1);

result();

writeln"Hard examples with quantifiers";

writeln"Problem 18";

goal HOL.thy "? y. ! x. P(y)-->P(x)";

by (Blast_tac 1);

result(); 

writeln"Problem 19";

goal HOL.thy "? x. ! y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";

by (Blast_tac 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 (Blast_tac 1); 

result();

writeln"Problem 21";

goal HOL.thy "(? x. P-->Q(x)) & (? x. Q(x)-->P) --> (? x. P=Q(x))";

by (Blast_tac 1); 

result();

writeln"Problem 22";

goal HOL.thy "(! x. P = Q(x))  -->  (P = (! x. Q(x)))";

by (Blast_tac 1); 

result();

writeln"Problem 23";

goal HOL.thy "(! x. P | Q(x))  =  (P | (! x. Q(x)))";

by (Blast_tac 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 (Blast_tac 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 (Blast_tac 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 (Blast_tac 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 (Blast_tac 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 (Blast_tac 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 (Blast_tac 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 (Blast_tac 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 (Blast_tac 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 (Blast_tac 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 (Blast_tac 1);

result();

writeln"Problem 34  AMENDED (TWICE!!)";

(*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 (Blast_tac 1);

result();

writeln"Problem 35";

goal HOL.thy "? x y. P x y -->  (! u v. P u v)";

by (Blast_tac 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 (Blast_tac 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 (Blast_tac 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)))";

by (Blast_tac 1);  (*beats fast_tac!*)

result();

writeln"Problem 39";

goal HOL.thy "~ (? x. ! y. F y x = (~ F y y))";

by (Blast_tac 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 (Blast_tac 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 (Blast_tac 1);

result();

writeln"Problem 42";

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

by (Blast_tac 1);

result();

writeln"Problem 43!!";

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)))";

by (Blast_tac 1);

result();

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 (Blast_tac 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 (Blast_tac 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 (Blast_tac 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);

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 (Blast_tac 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 (Blast_tac 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 (Blast_tac 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 (Blast_tac 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 1);

result();

writeln"Problem 56";

goal HOL.thy

    "(! x. (? y. P(y) & x=f(y)) --> P(x)) = (! x. P(x) --> P(f(x)))";

by (Blast_tac 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 (Blast_tac 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 (claset() addIs [f_cong]) 1);

result();

writeln"Problem 59";

goal HOL.thy "(! x. P(x) = (~P(f(x)))) --> (? x. P(x) & ~P(f(x)))";

by (Blast_tac 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 (Blast_tac 1);

result();

writeln"Problem 62 as corrected in JAR 18 (1997), page 135";

goal HOL.thy

    "(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 (Blast_tac 1);

result();

(*From Rudnicki, Obvious Inferences, JAR 3 (1987), 383-393.  

  It does seem obvious!*)

goal Prod.thy

    "(ALL x. F(x) & ~G(x) --> (EX y. H(x,y) & J(y))) &        \

\    (EX x. K(x) & F(x) & (ALL y. H(x,y) --> K(y)))  &        \

\    (ALL x. K(x) --> ~G(x))   -->   (EX x. K(x) --> ~G(x))";

by (Blast_tac 1);

result();

goal Prod.thy

    "(ALL x y. R(x,y) | R(y,x)) &                \

\    (ALL x y. S(x,y) & S(y,x) --> x=y) &        \

\    (ALL x y. R(x,y) --> S(x,y))    -->   (ALL x y. S(x,y) --> R(x,y))";

by (Blast_tac 1);

result();

writeln"Reached end of file.";