clasohm@1465: (*  Title:      HOL/ex/cla
clasohm@969:     ID:         $Id$
clasohm@1465:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
clasohm@969:     Copyright   1994  University of Cambridge
clasohm@969: 
clasohm@969: Higher-Order Logic: predicate calculus problems
clasohm@969: 
clasohm@969: Taken from FOL/cla.ML; beware of precedence of = vs <->
clasohm@969: *)
clasohm@969: 
clasohm@969: writeln"File HOL/ex/cla.";
clasohm@969: 
wenzelm@4089: context HOL.thy;  (*Boosts efficiency by omitting redundant rules*)
paulson@1912: 
clasohm@969: goal HOL.thy "(P --> Q | R) --> (P-->Q) | (P-->R)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*If and only if*)
clasohm@969: 
paulson@2997: goal HOL.thy "(P=Q) = (Q = (P::bool))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: goal HOL.thy "~ (P = (~P))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: 
clasohm@969: (*Sample problems from 
clasohm@969:   F. J. Pelletier, 
clasohm@969:   Seventy-Five Problems for Testing Automatic Theorem Provers,
clasohm@969:   J. Automated Reasoning 2 (1986), 191-216.
clasohm@969:   Errata, JAR 4 (1988), 236-236.
clasohm@969: 
clasohm@969: The hardest problems -- judging by experience with several theorem provers,
clasohm@969: including matrix ones -- are 34 and 43.
clasohm@969: *)
clasohm@969: 
clasohm@969: writeln"Pelletier's examples";
clasohm@969: (*1*)
clasohm@969: goal HOL.thy "(P-->Q)  =  (~Q --> ~P)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*2*)
clasohm@969: goal HOL.thy "(~ ~ P) =  P";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*3*)
clasohm@969: goal HOL.thy "~(P-->Q) --> (Q-->P)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*4*)
clasohm@969: goal HOL.thy "(~P-->Q)  =  (~Q --> P)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*5*)
clasohm@969: goal HOL.thy "((P|Q)-->(P|R)) --> (P|(Q-->R))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*6*)
clasohm@969: goal HOL.thy "P | ~ P";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*7*)
clasohm@969: goal HOL.thy "P | ~ ~ ~ P";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*8.  Peirce's law*)
clasohm@969: goal HOL.thy "((P-->Q) --> P)  -->  P";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*9*)
clasohm@969: goal HOL.thy "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*10*)
clasohm@969: goal HOL.thy "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*11.  Proved in each direction (incorrectly, says Pelletier!!)  *)
paulson@4061: goal HOL.thy "P=(P::bool)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*12.  "Dijkstra's law"*)
clasohm@969: goal HOL.thy "((P = Q) = R) = (P = (Q = R))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*13.  Distributive law*)
clasohm@969: goal HOL.thy "(P | (Q & R)) = ((P | Q) & (P | R))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*14*)
clasohm@969: goal HOL.thy "(P = Q) = ((Q | ~P) & (~Q|P))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*15*)
clasohm@969: goal HOL.thy "(P --> Q) = (~P | Q)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*16*)
clasohm@969: goal HOL.thy "(P-->Q) | (Q-->P)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*17*)
clasohm@969: goal HOL.thy "((P & (Q-->R))-->S)  =  ((~P | Q | S) & (~P | ~R | S))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Classical Logic: examples with quantifiers";
clasohm@969: 
clasohm@969: goal HOL.thy "(! x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result(); 
clasohm@969: 
wenzelm@3842: goal HOL.thy "(? x. P-->Q(x))  =  (P --> (? x. Q(x)))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result(); 
clasohm@969: 
wenzelm@3842: goal HOL.thy "(? x. P(x)-->Q) = ((! x. P(x)) --> Q)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result(); 
clasohm@969: 
wenzelm@3842: goal HOL.thy "((! x. P(x)) | Q)  =  (! x. P(x) | Q)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result(); 
clasohm@969: 
clasohm@969: (*From Wishnu Prasetya*)
clasohm@969: goal HOL.thy
clasohm@969:    "(!s. q(s) --> r(s)) & ~r(s) & (!s. ~r(s) & ~q(s) --> p(t) | q(t)) \
clasohm@969: \   --> p(t) | r(t)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result(); 
clasohm@969: 
clasohm@969: 
clasohm@969: writeln"Problems requiring quantifier duplication";
clasohm@969: 
clasohm@969: (*Needs multiple instantiation of the quantifier.*)
clasohm@969: goal HOL.thy "(! x. P(x)-->P(f(x)))  &  P(d)-->P(f(f(f(d))))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*Needs double instantiation of the quantifier*)
clasohm@969: goal HOL.thy "? x. P(x) --> P(a) & P(b)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: goal HOL.thy "? z. P(z) --> (! x. P(x))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: goal HOL.thy "? x. (? y. P(y)) --> P(x)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Hard examples with quantifiers";
clasohm@969: 
clasohm@969: writeln"Problem 18";
clasohm@969: goal HOL.thy "? y. ! x. P(y)-->P(x)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result(); 
clasohm@969: 
clasohm@969: writeln"Problem 19";
clasohm@969: goal HOL.thy "? x. ! y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 20";
clasohm@969: goal HOL.thy "(! x y. ? z. ! w. (P(x)&Q(y)-->R(z)&S(w)))     \
clasohm@969: \   --> (? x y. P(x) & Q(y)) --> (? z. R(z))";
paulson@2891: by (Blast_tac 1); 
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 21";
clasohm@969: goal HOL.thy "(? x. P-->Q(x)) & (? x. Q(x)-->P) --> (? x. P=Q(x))";
paulson@2891: by (Blast_tac 1); 
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 22";
clasohm@969: goal HOL.thy "(! x. P = Q(x))  -->  (P = (! x. Q(x)))";
paulson@2891: by (Blast_tac 1); 
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 23";
clasohm@969: goal HOL.thy "(! x. P | Q(x))  =  (P | (! x. Q(x)))";
paulson@2891: by (Blast_tac 1);  
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 24";
clasohm@969: goal HOL.thy "~(? x. S(x)&Q(x)) & (! x. P(x) --> Q(x)|R(x)) &  \
wenzelm@3842: \    (~(? x. P(x)) --> (? x. Q(x))) & (! x. Q(x)|R(x) --> S(x))  \
clasohm@969: \   --> (? x. P(x)&R(x))";
paulson@2891: by (Blast_tac 1); 
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 25";
clasohm@969: goal HOL.thy "(? x. P(x)) &  \
clasohm@969: \       (! x. L(x) --> ~ (M(x) & R(x))) &  \
clasohm@969: \       (! x. P(x) --> (M(x) & L(x))) &   \
clasohm@969: \       ((! x. P(x)-->Q(x)) | (? x. P(x)&R(x)))  \
clasohm@969: \   --> (? x. Q(x)&P(x))";
paulson@2891: by (Blast_tac 1); 
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 26";
clasohm@1465: goal HOL.thy "((? x. p(x)) = (? x. q(x))) &     \
clasohm@1465: \     (! x. ! y. p(x) & q(y) --> (r(x) = s(y))) \
clasohm@969: \ --> ((! x. p(x)-->r(x)) = (! x. q(x)-->s(x)))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 27";
clasohm@969: goal HOL.thy "(? x. P(x) & ~Q(x)) &   \
clasohm@969: \             (! x. P(x) --> R(x)) &   \
clasohm@969: \             (! x. M(x) & L(x) --> P(x)) &   \
clasohm@969: \             ((? x. R(x) & ~ Q(x)) --> (! x. L(x) --> ~ R(x)))  \
clasohm@969: \         --> (! x. M(x) --> ~L(x))";
paulson@2891: by (Blast_tac 1); 
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 28.  AMENDED";
clasohm@969: goal HOL.thy "(! x. P(x) --> (! x. Q(x))) &   \
clasohm@969: \       ((! x. Q(x)|R(x)) --> (? x. Q(x)&S(x))) &  \
wenzelm@3842: \       ((? x. S(x)) --> (! x. L(x) --> M(x)))  \
clasohm@969: \   --> (! x. P(x) & L(x) --> M(x))";
paulson@2891: by (Blast_tac 1);  
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 29.  Essentially the same as Principia Mathematica *11.71";
clasohm@969: goal HOL.thy "(? x. F(x)) & (? y. G(y))  \
clasohm@969: \   --> ( ((! x. F(x)-->H(x)) & (! y. G(y)-->J(y)))  =   \
clasohm@969: \         (! x y. F(x) & G(y) --> H(x) & J(y)))";
paulson@2891: by (Blast_tac 1); 
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 30";
clasohm@969: goal HOL.thy "(! x. P(x) | Q(x) --> ~ R(x)) & \
clasohm@969: \       (! x. (Q(x) --> ~ S(x)) --> P(x) & R(x))  \
clasohm@969: \   --> (! x. S(x))";
paulson@2891: by (Blast_tac 1);  
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 31";
wenzelm@3842: goal HOL.thy "~(? x. P(x) & (Q(x) | R(x))) & \
clasohm@969: \       (? x. L(x) & P(x)) & \
clasohm@969: \       (! x. ~ R(x) --> M(x))  \
clasohm@969: \   --> (? x. L(x) & M(x))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 32";
clasohm@969: goal HOL.thy "(! x. P(x) & (Q(x)|R(x))-->S(x)) & \
clasohm@969: \       (! x. S(x) & R(x) --> L(x)) & \
clasohm@969: \       (! x. M(x) --> R(x))  \
clasohm@969: \   --> (! x. P(x) & M(x) --> L(x))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 33";
clasohm@969: goal HOL.thy "(! x. P(a) & (P(x)-->P(b))-->P(c))  =    \
clasohm@969: \    (! x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
paulson@1768: writeln"Problem 34  AMENDED (TWICE!!)";
clasohm@969: (*Andrews's challenge*)
clasohm@1465: goal HOL.thy "((? x. ! y. p(x) = p(y))  =               \
paulson@3019: \              ((? x. q(x)) = (! y. p(y))))   =    \
paulson@3019: \             ((? x. ! y. q(x) = q(y))  =          \
paulson@3019: \              ((? x. p(x)) = (! y. q(y))))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 35";
clasohm@969: goal HOL.thy "? x y. P x y -->  (! u v. P u v)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 36";
clasohm@969: goal HOL.thy "(! x. ? y. J x y) & \
clasohm@969: \       (! x. ? y. G x y) & \
clasohm@1465: \       (! x y. J x y | G x y -->       \
clasohm@969: \       (! z. J y z | G y z --> H x z))   \
clasohm@969: \   --> (! x. ? y. H x y)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 37";
clasohm@969: goal HOL.thy "(! z. ? w. ! x. ? y. \
wenzelm@3842: \          (P x z -->P y w) & P y z & (P y w --> (? u. Q u w))) & \
clasohm@969: \       (! x z. ~(P x z) --> (? y. Q y z)) & \
clasohm@969: \       ((? x y. Q x y) --> (! x. R x x))  \
clasohm@969: \   --> (! x. ? y. R x y)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 38";
clasohm@969: goal HOL.thy
clasohm@1465:     "(! x. p(a) & (p(x) --> (? y. p(y) & r x y)) -->            \
clasohm@1465: \          (? z. ? w. p(z) & r x w & r w z))  =                 \
clasohm@1465: \    (! x. (~p(a) | p(x) | (? z. ? w. p(z) & r x w & r w z)) &  \
paulson@1716: \          (~p(a) | ~(? y. p(y) & r x y) |                      \
clasohm@969: \           (? z. ? w. p(z) & r x w & r w z)))";
paulson@2891: by (Blast_tac 1);  (*beats fast_tac!*)
paulson@1716: result();
clasohm@969: 
clasohm@969: writeln"Problem 39";
clasohm@969: goal HOL.thy "~ (? x. ! y. F y x = (~ F y y))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 40.  AMENDED";
clasohm@969: goal HOL.thy "(? y. ! x. F x y = F x x)  \
clasohm@969: \       -->  ~ (! x. ? y. ! z. F z y = (~ F z x))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 41";
clasohm@1465: goal HOL.thy "(! z. ? y. ! x. f x y = (f x z & ~ f x x))        \
clasohm@969: \              --> ~ (? z. ! x. f x z)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 42";
clasohm@969: goal HOL.thy "~ (? y. ! x. p x y = (~ (? z. p x z & p z x)))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
paulson@2891: writeln"Problem 43!!";
clasohm@969: goal HOL.thy
clasohm@1465:     "(! x::'a. ! y::'a. q x y = (! z. p z x = (p z y::bool)))   \
clasohm@969: \ --> (! x. (! y. q x y = (q y x::bool)))";
paulson@2891: by (Blast_tac 1);
paulson@3347: result();
clasohm@969: 
clasohm@969: writeln"Problem 44";
clasohm@1465: goal HOL.thy "(! x. f(x) -->                                    \
clasohm@969: \             (? y. g(y) & h x y & (? y. g(y) & ~ h x y)))  &   \
clasohm@1465: \             (? x. j(x) & (! y. g(y) --> h x y))               \
clasohm@969: \             --> (? x. j(x) & ~f(x))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 45";
clasohm@969: goal HOL.thy
clasohm@1465:     "(! x. f(x) & (! y. g(y) & h x y --> j x y) \
clasohm@1465: \                     --> (! y. g(y) & h x y --> k(y))) &       \
clasohm@1465: \    ~ (? y. l(y) & k(y)) &                                     \
clasohm@1465: \    (? x. f(x) & (! y. h x y --> l(y))                         \
clasohm@1465: \               & (! y. g(y) & h x y --> j x y))                \
clasohm@969: \     --> (? x. f(x) & ~ (? y. g(y) & h x y))";
paulson@2891: by (Blast_tac 1); 
clasohm@969: result();
clasohm@969: 
clasohm@969: 
clasohm@969: writeln"Problems (mainly) involving equality or functions";
clasohm@969: 
clasohm@969: writeln"Problem 48";
clasohm@969: goal HOL.thy "(a=b | c=d) & (a=c | b=d) --> a=d | b=c";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 49  NOT PROVED AUTOMATICALLY";
clasohm@969: (*Hard because it involves substitution for Vars;
clasohm@969:   the type constraint ensures that x,y,z have the same type as a,b,u. *)
clasohm@969: goal HOL.thy "(? x y::'a. ! z. z=x | z=y) & P(a) & P(b) & (~a=b) \
wenzelm@3842: \               --> (! u::'a. P(u))";
paulson@4153: by (Classical.Safe_tac);
clasohm@969: by (res_inst_tac [("x","a")] allE 1);
clasohm@969: by (assume_tac 1);
clasohm@969: by (res_inst_tac [("x","b")] allE 1);
clasohm@969: by (assume_tac 1);
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 50";  
clasohm@969: (*What has this to do with equality?*)
wenzelm@3842: goal HOL.thy "(! x. P a x | (! y. P x y)) --> (? x. ! y. P x y)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 51";
clasohm@969: goal HOL.thy
clasohm@969:     "(? z w. ! x y. P x y = (x=z & y=w)) -->  \
clasohm@969: \    (? z. ! x. ? w. (! y. P x y = (y=w)) = (x=z))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 52";
clasohm@969: (*Almost the same as 51. *)
clasohm@969: goal HOL.thy
clasohm@969:     "(? z w. ! x y. P x y = (x=z & y=w)) -->  \
clasohm@969: \    (? w. ! y. ? z. (! x. P x y = (x=z)) = (y=w))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 55";
clasohm@969: 
clasohm@969: (*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
clasohm@969:   fast_tac DISCOVERS who killed Agatha. *)
clasohm@969: goal HOL.thy "lives(agatha) & lives(butler) & lives(charles) & \
clasohm@969: \  (killed agatha agatha | killed butler agatha | killed charles agatha) & \
clasohm@969: \  (!x y. killed x y --> hates x y & ~richer x y) & \
clasohm@969: \  (!x. hates agatha x --> ~hates charles x) & \
clasohm@969: \  (hates agatha agatha & hates agatha charles) & \
clasohm@969: \  (!x. lives(x) & ~richer x agatha --> hates butler x) & \
clasohm@969: \  (!x. hates agatha x --> hates butler x) & \
clasohm@969: \  (!x. ~hates x agatha | ~hates x butler | ~hates x charles) --> \
clasohm@969: \   killed ?who agatha";
paulson@2922: by (Fast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 56";
clasohm@969: goal HOL.thy
clasohm@969:     "(! x. (? y. P(y) & x=f(y)) --> P(x)) = (! x. P(x) --> P(f(x)))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 57";
clasohm@969: goal HOL.thy
clasohm@969:     "P (f a b) (f b c) & P (f b c) (f a c) & \
clasohm@969: \    (! x y z. P x y & P y z --> P x z)    -->   P (f a b) (f a c)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 58  NOT PROVED AUTOMATICALLY";
clasohm@969: goal HOL.thy "(! x y. f(x)=g(y)) --> (! x y. f(f(x))=f(g(y)))";
clasohm@969: val f_cong = read_instantiate [("f","f")] arg_cong;
wenzelm@4089: by (fast_tac (claset() addIs [f_cong]) 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 59";
clasohm@969: goal HOL.thy "(! x. P(x) = (~P(f(x)))) --> (? x. P(x) & ~P(f(x)))";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
clasohm@969: writeln"Problem 60";
clasohm@969: goal HOL.thy
clasohm@969:     "! x. P x (f x) = (? y. (! z. P z y --> P z (f x)) & P x y)";
paulson@2891: by (Blast_tac 1);
clasohm@969: result();
clasohm@969: 
paulson@2715: writeln"Problem 62 as corrected in JAR 18 (1997), page 135";
paulson@1404: goal HOL.thy
clasohm@1465:     "(ALL x. p a & (p x --> p(f x)) --> p(f(f x)))  =   \
clasohm@1465: \    (ALL x. (~ p a | p x | p(f(f x))) &                        \
paulson@1404: \            (~ p a | ~ p(f x) | p(f(f x))))";
paulson@2891: by (Blast_tac 1);
paulson@1404: result();
paulson@1404: 
paulson@1712: (*From Rudnicki, Obvious Inferences, JAR 3 (1987), 383-393.  
paulson@1712:   It does seem obvious!*)
paulson@1712: goal Prod.thy
paulson@1712:     "(ALL x. F(x) & ~G(x) --> (EX y. H(x,y) & J(y))) &        \
paulson@1712: \    (EX x. K(x) & F(x) & (ALL y. H(x,y) --> K(y)))  &        \
paulson@1712: \    (ALL x. K(x) --> ~G(x))   -->   (EX x. K(x) --> ~G(x))";
paulson@2891: by (Blast_tac 1);
paulson@1712: result();
paulson@1712: 
paulson@1712: goal Prod.thy
paulson@1712:     "(ALL x y. R(x,y) | R(y,x)) &                \
paulson@1712: \    (ALL x y. S(x,y) & S(y,x) --> x=y) &        \
paulson@1712: \    (ALL x y. R(x,y) --> S(x,y))    -->   (ALL x y. S(x,y) --> R(x,y))";
paulson@2891: by (Blast_tac 1);
paulson@1712: result();
paulson@1712: 
paulson@1712: 
paulson@1712: 
clasohm@969: writeln"Reached end of file.";