src/HOL/ex/cla.ML
author nipkow
Thu Oct 12 18:38:23 2000 +0200 (2000-10-12)
changeset 10212 33fe2d701ddd
parent 6799 95abcc002a21
child 11025 a70b796d9af8
permissions -rw-r--r--
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(*  Title:      HOL/ex/cla
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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Higher-Order Logic: predicate calculus problems
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Taken from FOL/cla.ML; beware of precedence of = vs <->
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*)
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writeln"File HOL/ex/cla.";
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context HOL.thy; 
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Goal "(P --> Q | R) --> (P-->Q) | (P-->R)";
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by (Blast_tac 1);
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result();
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(*If and only if*)
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Goal "(P=Q) = (Q = (P::bool))";
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by (Blast_tac 1);
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result();
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Goal "~ (P = (~P))";
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by (Blast_tac 1);
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result();
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(*Sample problems from 
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  F. J. Pelletier, 
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  Seventy-Five Problems for Testing Automatic Theorem Provers,
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  J. Automated Reasoning 2 (1986), 191-216.
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  Errata, JAR 4 (1988), 236-236.
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The hardest problems -- judging by experience with several theorem provers,
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including matrix ones -- are 34 and 43.
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*)
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writeln"Pelletier's examples";
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(*1*)
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Goal "(P-->Q)  =  (~Q --> ~P)";
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by (Blast_tac 1);
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result();
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(*2*)
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Goal "(~ ~ P) =  P";
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by (Blast_tac 1);
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result();
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(*3*)
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Goal "~(P-->Q) --> (Q-->P)";
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by (Blast_tac 1);
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result();
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(*4*)
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Goal "(~P-->Q)  =  (~Q --> P)";
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by (Blast_tac 1);
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result();
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(*5*)
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Goal "((P|Q)-->(P|R)) --> (P|(Q-->R))";
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by (Blast_tac 1);
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result();
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(*6*)
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Goal "P | ~ P";
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by (Blast_tac 1);
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result();
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(*7*)
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Goal "P | ~ ~ ~ P";
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by (Blast_tac 1);
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result();
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(*8.  Peirce's law*)
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Goal "((P-->Q) --> P)  -->  P";
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by (Blast_tac 1);
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result();
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(*9*)
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Goal "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
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by (Blast_tac 1);
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result();
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(*10*)
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Goal "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)";
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by (Blast_tac 1);
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result();
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(*11.  Proved in each direction (incorrectly, says Pelletier!!)  *)
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Goal "P=(P::bool)";
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by (Blast_tac 1);
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result();
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(*12.  "Dijkstra's law"*)
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Goal "((P = Q) = R) = (P = (Q = R))";
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by (Blast_tac 1);
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result();
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(*13.  Distributive law*)
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Goal "(P | (Q & R)) = ((P | Q) & (P | R))";
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by (Blast_tac 1);
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result();
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(*14*)
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Goal "(P = Q) = ((Q | ~P) & (~Q|P))";
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by (Blast_tac 1);
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result();
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(*15*)
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Goal "(P --> Q) = (~P | Q)";
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by (Blast_tac 1);
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result();
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(*16*)
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Goal "(P-->Q) | (Q-->P)";
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by (Blast_tac 1);
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result();
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(*17*)
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Goal "((P & (Q-->R))-->S)  =  ((~P | Q | S) & (~P | ~R | S))";
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by (Blast_tac 1);
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result();
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writeln"Classical Logic: examples with quantifiers";
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Goal "(! x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
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by (Blast_tac 1);
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result(); 
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Goal "(? x. P-->Q(x))  =  (P --> (? x. Q(x)))";
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by (Blast_tac 1);
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result(); 
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Goal "(? x. P(x)-->Q) = ((! x. P(x)) --> Q)";
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by (Blast_tac 1);
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result(); 
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Goal "((! x. P(x)) | Q)  =  (! x. P(x) | Q)";
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by (Blast_tac 1);
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result(); 
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(*From Wishnu Prasetya*)
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Goal "(!s. q(s) --> r(s)) & ~r(s) & (!s. ~r(s) & ~q(s) --> p(t) | q(t)) \
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\   --> p(t) | r(t)";
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by (Blast_tac 1);
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result(); 
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writeln"Problems requiring quantifier duplication";
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(*Theorem B of Peter Andrews, Theorem Proving via General Matings, 
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  JACM 28 (1981).*)
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Goal "(EX x. ALL y. P(x) = P(y)) --> ((EX x. P(x)) = (ALL y. P(y)))";
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by (Blast_tac 1);
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result();
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(*Needs multiple instantiation of the quantifier.*)
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Goal "(! x. P(x)-->P(f(x)))  &  P(d)-->P(f(f(f(d))))";
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by (Blast_tac 1);
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result();
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(*Needs double instantiation of the quantifier*)
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Goal "? x. P(x) --> P(a) & P(b)";
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by (Blast_tac 1);
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result();
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Goal "? z. P(z) --> (! x. P(x))";
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by (Blast_tac 1);
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result();
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Goal "? x. (? y. P(y)) --> P(x)";
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by (Blast_tac 1);
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result();
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writeln"Hard examples with quantifiers";
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writeln"Problem 18";
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Goal "? y. ! x. P(y)-->P(x)";
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by (Blast_tac 1);
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result(); 
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writeln"Problem 19";
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Goal "? x. ! y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";
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by (Blast_tac 1);
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result();
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writeln"Problem 20";
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Goal "(! x y. ? z. ! w. (P(x)&Q(y)-->R(z)&S(w)))     \
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\   --> (? x y. P(x) & Q(y)) --> (? z. R(z))";
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by (Blast_tac 1); 
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result();
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writeln"Problem 21";
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Goal "(? x. P-->Q(x)) & (? x. Q(x)-->P) --> (? x. P=Q(x))";
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by (Blast_tac 1); 
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result();
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writeln"Problem 22";
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Goal "(! x. P = Q(x))  -->  (P = (! x. Q(x)))";
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by (Blast_tac 1); 
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result();
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writeln"Problem 23";
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Goal "(! x. P | Q(x))  =  (P | (! x. Q(x)))";
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by (Blast_tac 1);  
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result();
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writeln"Problem 24";
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Goal "~(? x. S(x)&Q(x)) & (! x. P(x) --> Q(x)|R(x)) &  \
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\    (~(? x. P(x)) --> (? x. Q(x))) & (! x. Q(x)|R(x) --> S(x))  \
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\   --> (? x. P(x)&R(x))";
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by (Blast_tac 1); 
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result();
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writeln"Problem 25";
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Goal "(? x. P(x)) &  \
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\       (! x. L(x) --> ~ (M(x) & R(x))) &  \
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\       (! x. P(x) --> (M(x) & L(x))) &   \
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\       ((! x. P(x)-->Q(x)) | (? x. P(x)&R(x)))  \
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\   --> (? x. Q(x)&P(x))";
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by (Blast_tac 1); 
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result();
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writeln"Problem 26";
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Goal "((? x. p(x)) = (? x. q(x))) &     \
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\     (! x. ! y. p(x) & q(y) --> (r(x) = s(y))) \
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\ --> ((! x. p(x)-->r(x)) = (! x. q(x)-->s(x)))";
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by (Blast_tac 1);
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result();
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writeln"Problem 27";
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Goal "(? x. P(x) & ~Q(x)) &   \
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\             (! x. P(x) --> R(x)) &   \
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\             (! x. M(x) & L(x) --> P(x)) &   \
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\             ((? x. R(x) & ~ Q(x)) --> (! x. L(x) --> ~ R(x)))  \
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\         --> (! x. M(x) --> ~L(x))";
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by (Blast_tac 1); 
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result();
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writeln"Problem 28.  AMENDED";
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Goal "(! x. P(x) --> (! x. Q(x))) &   \
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\       ((! x. Q(x)|R(x)) --> (? x. Q(x)&S(x))) &  \
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\       ((? x. S(x)) --> (! x. L(x) --> M(x)))  \
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\   --> (! x. P(x) & L(x) --> M(x))";
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by (Blast_tac 1);  
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result();
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writeln"Problem 29.  Essentially the same as Principia Mathematica *11.71";
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Goal "(? x. F(x)) & (? y. G(y))  \
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\   --> ( ((! x. F(x)-->H(x)) & (! y. G(y)-->J(y)))  =   \
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\         (! x y. F(x) & G(y) --> H(x) & J(y)))";
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by (Blast_tac 1); 
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result();
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writeln"Problem 30";
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Goal "(! x. P(x) | Q(x) --> ~ R(x)) & \
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\       (! x. (Q(x) --> ~ S(x)) --> P(x) & R(x))  \
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\   --> (! x. S(x))";
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by (Blast_tac 1);  
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result();
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writeln"Problem 31";
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Goal "~(? x. P(x) & (Q(x) | R(x))) & \
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\       (? x. L(x) & P(x)) & \
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\       (! x. ~ R(x) --> M(x))  \
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\   --> (? x. L(x) & M(x))";
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by (Blast_tac 1);
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result();
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writeln"Problem 32";
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Goal "(! x. P(x) & (Q(x)|R(x))-->S(x)) & \
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\       (! x. S(x) & R(x) --> L(x)) & \
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\       (! x. M(x) --> R(x))  \
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\   --> (! x. P(x) & M(x) --> L(x))";
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by (Blast_tac 1);
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result();
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writeln"Problem 33";
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Goal "(! x. P(a) & (P(x)-->P(b))-->P(c))  =    \
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\    (! x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))";
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by (Blast_tac 1);
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result();
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writeln"Problem 34  AMENDED (TWICE!!)";
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(*Andrews's challenge*)
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Goal "((? x. ! y. p(x) = p(y))  =               \
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\              ((? x. q(x)) = (! y. p(y))))   =    \
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\             ((? x. ! y. q(x) = q(y))  =          \
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\              ((? x. p(x)) = (! y. q(y))))";
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by (Blast_tac 1);
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result();
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writeln"Problem 35";
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Goal "? x y. P x y -->  (! u v. P u v)";
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by (Blast_tac 1);
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result();
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writeln"Problem 36";
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Goal "(! x. ? y. J x y) & \
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\       (! x. ? y. G x y) & \
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\       (! x y. J x y | G x y -->       \
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\       (! z. J y z | G y z --> H x z))   \
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\   --> (! x. ? y. H x y)";
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by (Blast_tac 1);
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result();
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writeln"Problem 37";
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Goal "(! z. ? w. ! x. ? y. \
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\          (P x z -->P y w) & P y z & (P y w --> (? u. Q u w))) & \
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\       (! x z. ~(P x z) --> (? y. Q y z)) & \
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\       ((? x y. Q x y) --> (! x. R x x))  \
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\   --> (! x. ? y. R x y)";
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by (Blast_tac 1);
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result();
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writeln"Problem 38";
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Goal "(! x. p(a) & (p(x) --> (? y. p(y) & r x y)) -->            \
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\          (? z. ? w. p(z) & r x w & r w z))  =                 \
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\    (! x. (~p(a) | p(x) | (? z. ? w. p(z) & r x w & r w z)) &  \
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\          (~p(a) | ~(? y. p(y) & r x y) |                      \
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\           (? z. ? w. p(z) & r x w & r w z)))";
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by (Blast_tac 1);  (*beats fast_tac!*)
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result();
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writeln"Problem 39";
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Goal "~ (? x. ! y. F y x = (~ F y y))";
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by (Blast_tac 1);
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result();
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writeln"Problem 40.  AMENDED";
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Goal "(? y. ! x. F x y = F x x)  \
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\       -->  ~ (! x. ? y. ! z. F z y = (~ F z x))";
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by (Blast_tac 1);
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result();
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writeln"Problem 41";
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Goal "(! z. ? y. ! x. f x y = (f x z & ~ f x x))        \
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\              --> ~ (? z. ! x. f x z)";
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by (Blast_tac 1);
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   342
result();
clasohm@969
   343
clasohm@969
   344
writeln"Problem 42";
paulson@5150
   345
Goal "~ (? y. ! x. p x y = (~ (? z. p x z & p z x)))";
paulson@2891
   346
by (Blast_tac 1);
clasohm@969
   347
result();
clasohm@969
   348
paulson@2891
   349
writeln"Problem 43!!";
paulson@5278
   350
Goal "(! x::'a. ! y::'a. q x y = (! z. p z x = (p z y::bool)))   \
clasohm@969
   351
\ --> (! x. (! y. q x y = (q y x::bool)))";
paulson@2891
   352
by (Blast_tac 1);
paulson@3347
   353
result();
clasohm@969
   354
clasohm@969
   355
writeln"Problem 44";
paulson@5150
   356
Goal "(! x. f(x) -->                                    \
clasohm@969
   357
\             (? y. g(y) & h x y & (? y. g(y) & ~ h x y)))  &   \
clasohm@1465
   358
\             (? x. j(x) & (! y. g(y) --> h x y))               \
clasohm@969
   359
\             --> (? x. j(x) & ~f(x))";
paulson@2891
   360
by (Blast_tac 1);
clasohm@969
   361
result();
clasohm@969
   362
clasohm@969
   363
writeln"Problem 45";
paulson@5278
   364
Goal "(! x. f(x) & (! y. g(y) & h x y --> j x y) \
clasohm@1465
   365
\                     --> (! y. g(y) & h x y --> k(y))) &       \
clasohm@1465
   366
\    ~ (? y. l(y) & k(y)) &                                     \
clasohm@1465
   367
\    (? x. f(x) & (! y. h x y --> l(y))                         \
clasohm@1465
   368
\               & (! y. g(y) & h x y --> j x y))                \
clasohm@969
   369
\     --> (? x. f(x) & ~ (? y. g(y) & h x y))";
paulson@2891
   370
by (Blast_tac 1); 
clasohm@969
   371
result();
clasohm@969
   372
clasohm@969
   373
clasohm@969
   374
writeln"Problems (mainly) involving equality or functions";
clasohm@969
   375
clasohm@969
   376
writeln"Problem 48";
paulson@5150
   377
Goal "(a=b | c=d) & (a=c | b=d) --> a=d | b=c";
paulson@2891
   378
by (Blast_tac 1);
clasohm@969
   379
result();
clasohm@969
   380
clasohm@969
   381
writeln"Problem 49  NOT PROVED AUTOMATICALLY";
clasohm@969
   382
(*Hard because it involves substitution for Vars;
clasohm@969
   383
  the type constraint ensures that x,y,z have the same type as a,b,u. *)
paulson@5150
   384
Goal "(? x y::'a. ! z. z=x | z=y) & P(a) & P(b) & (~a=b) \
wenzelm@3842
   385
\               --> (! u::'a. P(u))";
paulson@4153
   386
by (Classical.Safe_tac);
clasohm@969
   387
by (res_inst_tac [("x","a")] allE 1);
clasohm@969
   388
by (assume_tac 1);
clasohm@969
   389
by (res_inst_tac [("x","b")] allE 1);
clasohm@969
   390
by (assume_tac 1);
paulson@4353
   391
by (Fast_tac 1);    (*Blast_tac's treatment of equality can't do it*)
clasohm@969
   392
result();
clasohm@969
   393
clasohm@969
   394
writeln"Problem 50";  
clasohm@969
   395
(*What has this to do with equality?*)
paulson@5150
   396
Goal "(! x. P a x | (! y. P x y)) --> (? x. ! y. P x y)";
paulson@2891
   397
by (Blast_tac 1);
clasohm@969
   398
result();
clasohm@969
   399
clasohm@969
   400
writeln"Problem 51";
paulson@5278
   401
Goal "(? z w. ! x y. P x y = (x=z & y=w)) -->  \
clasohm@969
   402
\    (? z. ! x. ? w. (! y. P x y = (y=w)) = (x=z))";
paulson@2891
   403
by (Blast_tac 1);
clasohm@969
   404
result();
clasohm@969
   405
clasohm@969
   406
writeln"Problem 52";
clasohm@969
   407
(*Almost the same as 51. *)
paulson@5278
   408
Goal "(? z w. ! x y. P x y = (x=z & y=w)) -->  \
clasohm@969
   409
\    (? w. ! y. ? z. (! x. P x y = (x=z)) = (y=w))";
paulson@2891
   410
by (Blast_tac 1);
clasohm@969
   411
result();
clasohm@969
   412
clasohm@969
   413
writeln"Problem 55";
clasohm@969
   414
clasohm@969
   415
(*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
clasohm@969
   416
  fast_tac DISCOVERS who killed Agatha. *)
paulson@5150
   417
Goal "lives(agatha) & lives(butler) & lives(charles) & \
clasohm@969
   418
\  (killed agatha agatha | killed butler agatha | killed charles agatha) & \
clasohm@969
   419
\  (!x y. killed x y --> hates x y & ~richer x y) & \
clasohm@969
   420
\  (!x. hates agatha x --> ~hates charles x) & \
clasohm@969
   421
\  (hates agatha agatha & hates agatha charles) & \
clasohm@969
   422
\  (!x. lives(x) & ~richer x agatha --> hates butler x) & \
clasohm@969
   423
\  (!x. hates agatha x --> hates butler x) & \
clasohm@969
   424
\  (!x. ~hates x agatha | ~hates x butler | ~hates x charles) --> \
clasohm@969
   425
\   killed ?who agatha";
paulson@2922
   426
by (Fast_tac 1);
clasohm@969
   427
result();
clasohm@969
   428
clasohm@969
   429
writeln"Problem 56";
paulson@5278
   430
Goal "(! x. (? y. P(y) & x=f(y)) --> P(x)) = (! x. P(x) --> P(f(x)))";
paulson@2891
   431
by (Blast_tac 1);
clasohm@969
   432
result();
clasohm@969
   433
clasohm@969
   434
writeln"Problem 57";
paulson@5278
   435
Goal "P (f a b) (f b c) & P (f b c) (f a c) & \
clasohm@969
   436
\    (! x y z. P x y & P y z --> P x z)    -->   P (f a b) (f a c)";
paulson@2891
   437
by (Blast_tac 1);
clasohm@969
   438
result();
clasohm@969
   439
clasohm@969
   440
writeln"Problem 58  NOT PROVED AUTOMATICALLY";
paulson@5150
   441
Goal "(! x y. f(x)=g(y)) --> (! x y. f(f(x))=f(g(y)))";
clasohm@969
   442
val f_cong = read_instantiate [("f","f")] arg_cong;
wenzelm@4089
   443
by (fast_tac (claset() addIs [f_cong]) 1);
clasohm@969
   444
result();
clasohm@969
   445
clasohm@969
   446
writeln"Problem 59";
paulson@5150
   447
Goal "(! x. P(x) = (~P(f(x)))) --> (? x. P(x) & ~P(f(x)))";
paulson@2891
   448
by (Blast_tac 1);
clasohm@969
   449
result();
clasohm@969
   450
clasohm@969
   451
writeln"Problem 60";
paulson@5278
   452
Goal "! x. P x (f x) = (? y. (! z. P z y --> P z (f x)) & P x y)";
paulson@2891
   453
by (Blast_tac 1);
clasohm@969
   454
result();
clasohm@969
   455
paulson@2715
   456
writeln"Problem 62 as corrected in JAR 18 (1997), page 135";
paulson@5150
   457
Goal "(ALL x. p a & (p x --> p(f x)) --> p(f(f x)))  =   \
paulson@5150
   458
\     (ALL x. (~ p a | p x | p(f(f x))) &                        \
paulson@5150
   459
\             (~ p a | ~ p(f x) | p(f(f x))))";
paulson@2891
   460
by (Blast_tac 1);
paulson@1404
   461
result();
paulson@1404
   462
paulson@4463
   463
(*From Davis, Obvious Logical Inferences, IJCAI-81, 530-531
paulson@4463
   464
  Fast_tac indeed copes!*)
nipkow@10212
   465
goal Product_Type.thy "(ALL x. F(x) & ~G(x) --> (EX y. H(x,y) & J(y))) & \
paulson@4463
   466
\             (EX x. K(x) & F(x) & (ALL y. H(x,y) --> K(y))) &   \
paulson@4463
   467
\             (ALL x. K(x) --> ~G(x))  -->  (EX x. K(x) & J(x))";
paulson@4463
   468
by (Fast_tac 1);
paulson@4463
   469
result();
paulson@4463
   470
paulson@1712
   471
(*From Rudnicki, Obvious Inferences, JAR 3 (1987), 383-393.  
paulson@1712
   472
  It does seem obvious!*)
nipkow@10212
   473
goal Product_Type.thy
paulson@1712
   474
    "(ALL x. F(x) & ~G(x) --> (EX y. H(x,y) & J(y))) &        \
paulson@1712
   475
\    (EX x. K(x) & F(x) & (ALL y. H(x,y) --> K(y)))  &        \
paulson@1712
   476
\    (ALL x. K(x) --> ~G(x))   -->   (EX x. K(x) --> ~G(x))";
paulson@4463
   477
by (Fast_tac 1);
paulson@1712
   478
result();
paulson@1712
   479
paulson@6799
   480
(*Attributed to Lewis Carroll by S. G. Pulman.  The first or last assumption
paulson@6799
   481
can be deleted.*)
paulson@6799
   482
Goal "(ALL x. honest(x) & industrious(x) --> healthy(x)) & \
paulson@6799
   483
\     ~ (EX x. grocer(x) & healthy(x)) & \
paulson@6799
   484
\     (ALL x. industrious(x) & grocer(x) --> honest(x)) & \
paulson@6799
   485
\     (ALL x. cyclist(x) --> industrious(x)) & \
paulson@6799
   486
\     (ALL x. ~healthy(x) & cyclist(x) --> ~honest(x))  \
paulson@6799
   487
\     --> (ALL x. grocer(x) --> ~cyclist(x))";
paulson@6799
   488
by (Blast_tac 1);
paulson@6799
   489
result();
paulson@6799
   490
nipkow@10212
   491
goal Product_Type.thy
paulson@1712
   492
    "(ALL x y. R(x,y) | R(y,x)) &                \
paulson@1712
   493
\    (ALL x y. S(x,y) & S(y,x) --> x=y) &        \
paulson@1712
   494
\    (ALL x y. R(x,y) --> S(x,y))    -->   (ALL x y. S(x,y) --> R(x,y))";
paulson@2891
   495
by (Blast_tac 1);
paulson@1712
   496
result();
paulson@1712
   497
paulson@1712
   498
paulson@1712
   499
clasohm@969
   500
writeln"Reached end of file.";