(* Title: FOL/ex/Classical
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
*)
header{*Classical Predicate Calculus Problems*}
theory Classical imports FOL begin
lemma "(P --> Q | R) --> (P-->Q) | (P-->R)"
by blast
text{*If and only if*}
lemma "(P<->Q) <-> (Q<->P)"
by blast
lemma "~ (P <-> ~P)"
by blast
text{*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.
*}
subsection{*Pelletier's examples*}
text{*1*}
lemma "(P-->Q) <-> (~Q --> ~P)"
by blast
text{*2*}
lemma "~ ~ P <-> P"
by blast
text{*3*}
lemma "~(P-->Q) --> (Q-->P)"
by blast
text{*4*}
lemma "(~P-->Q) <-> (~Q --> P)"
by blast
text{*5*}
lemma "((P|Q)-->(P|R)) --> (P|(Q-->R))"
by blast
text{*6*}
lemma "P | ~ P"
by blast
text{*7*}
lemma "P | ~ ~ ~ P"
by blast
text{*8. Peirce's law*}
lemma "((P-->Q) --> P) --> P"
by blast
text{*9*}
lemma "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
by blast
text{*10*}
lemma "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P<->Q)"
by blast
text{*11. Proved in each direction (incorrectly, says Pelletier!!) *}
lemma "P<->P"
by blast
text{*12. "Dijkstra's law"*}
lemma "((P <-> Q) <-> R) <-> (P <-> (Q <-> R))"
by blast
text{*13. Distributive law*}
lemma "P | (Q & R) <-> (P | Q) & (P | R)"
by blast
text{*14*}
lemma "(P <-> Q) <-> ((Q | ~P) & (~Q|P))"
by blast
text{*15*}
lemma "(P --> Q) <-> (~P | Q)"
by blast
text{*16*}
lemma "(P-->Q) | (Q-->P)"
by blast
text{*17*}
lemma "((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))"
by blast
subsection{*Classical Logic: examples with quantifiers*}
lemma "(\<forall>x. P(x) & Q(x)) <-> (\<forall>x. P(x)) & (\<forall>x. Q(x))"
by blast
lemma "(\<exists>x. P-->Q(x)) <-> (P --> (\<exists>x. Q(x)))"
by blast
lemma "(\<exists>x. P(x)-->Q) <-> (\<forall>x. P(x)) --> Q"
by blast
lemma "(\<forall>x. P(x)) | Q <-> (\<forall>x. P(x) | Q)"
by blast
text{*Discussed in Avron, Gentzen-Type Systems, Resolution and Tableaux,
JAR 10 (265-281), 1993. Proof is trivial!*}
lemma "~((\<exists>x.~P(x)) & ((\<exists>x. P(x)) | (\<exists>x. P(x) & Q(x))) & ~ (\<exists>x. P(x)))"
by blast
subsection{*Problems requiring quantifier duplication*}
text{*Theorem B of Peter Andrews, Theorem Proving via General Matings,
JACM 28 (1981).*}
lemma "(\<exists>x. \<forall>y. P(x) <-> P(y)) --> ((\<exists>x. P(x)) <-> (\<forall>y. P(y)))"
by blast
text{*Needs multiple instantiation of ALL.*}
lemma "(\<forall>x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))"
by blast
text{*Needs double instantiation of the quantifier*}
lemma "\<exists>x. P(x) --> P(a) & P(b)"
by blast
lemma "\<exists>z. P(z) --> (\<forall>x. P(x))"
by blast
lemma "\<exists>x. (\<exists>y. P(y)) --> P(x)"
by blast
text{*V. Lifschitz, What Is the Inverse Method?, JAR 5 (1989), 1--23. NOT PROVED*}
lemma "\<exists>x x'. \<forall>y. \<exists>z z'.
(~P(y,y) | P(x,x) | ~S(z,x)) &
(S(x,y) | ~S(y,z) | Q(z',z')) &
(Q(x',y) | ~Q(y,z') | S(x',x'))"
oops
subsection{*Hard examples with quantifiers*}
text{*18*}
lemma "\<exists>y. \<forall>x. P(y)-->P(x)"
by blast
text{*19*}
lemma "\<exists>x. \<forall>y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"
by blast
text{*20*}
lemma "(\<forall>x y. \<exists>z. \<forall>w. (P(x)&Q(y)-->R(z)&S(w)))
--> (\<exists>x y. P(x) & Q(y)) --> (\<exists>z. R(z))"
by blast
text{*21*}
lemma "(\<exists>x. P-->Q(x)) & (\<exists>x. Q(x)-->P) --> (\<exists>x. P<->Q(x))"
by blast
text{*22*}
lemma "(\<forall>x. P <-> Q(x)) --> (P <-> (\<forall>x. Q(x)))"
by blast
text{*23*}
lemma "(\<forall>x. P | Q(x)) <-> (P | (\<forall>x. Q(x)))"
by blast
text{*24*}
lemma "~(\<exists>x. S(x)&Q(x)) & (\<forall>x. P(x) --> Q(x)|R(x)) &
(~(\<exists>x. P(x)) --> (\<exists>x. Q(x))) & (\<forall>x. Q(x)|R(x) --> S(x))
--> (\<exists>x. P(x)&R(x))"
by blast
text{*25*}
lemma "(\<exists>x. P(x)) &
(\<forall>x. L(x) --> ~ (M(x) & R(x))) &
(\<forall>x. P(x) --> (M(x) & L(x))) &
((\<forall>x. P(x)-->Q(x)) | (\<exists>x. P(x)&R(x)))
--> (\<exists>x. Q(x)&P(x))"
by blast
text{*26*}
lemma "((\<exists>x. p(x)) <-> (\<exists>x. q(x))) &
(\<forall>x. \<forall>y. p(x) & q(y) --> (r(x) <-> s(y)))
--> ((\<forall>x. p(x)-->r(x)) <-> (\<forall>x. q(x)-->s(x)))"
by blast
text{*27*}
lemma "(\<exists>x. P(x) & ~Q(x)) &
(\<forall>x. P(x) --> R(x)) &
(\<forall>x. M(x) & L(x) --> P(x)) &
((\<exists>x. R(x) & ~ Q(x)) --> (\<forall>x. L(x) --> ~ R(x)))
--> (\<forall>x. M(x) --> ~L(x))"
by blast
text{*28. AMENDED*}
lemma "(\<forall>x. P(x) --> (\<forall>x. Q(x))) &
((\<forall>x. Q(x)|R(x)) --> (\<exists>x. Q(x)&S(x))) &
((\<exists>x. S(x)) --> (\<forall>x. L(x) --> M(x)))
--> (\<forall>x. P(x) & L(x) --> M(x))"
by blast
text{*29. Essentially the same as Principia Mathematica *11.71*}
lemma "(\<exists>x. P(x)) & (\<exists>y. Q(y))
--> ((\<forall>x. P(x)-->R(x)) & (\<forall>y. Q(y)-->S(y)) <->
(\<forall>x y. P(x) & Q(y) --> R(x) & S(y)))"
by blast
text{*30*}
lemma "(\<forall>x. P(x) | Q(x) --> ~ R(x)) &
(\<forall>x. (Q(x) --> ~ S(x)) --> P(x) & R(x))
--> (\<forall>x. S(x))"
by blast
text{*31*}
lemma "~(\<exists>x. P(x) & (Q(x) | R(x))) &
(\<exists>x. L(x) & P(x)) &
(\<forall>x. ~ R(x) --> M(x))
--> (\<exists>x. L(x) & M(x))"
by blast
text{*32*}
lemma "(\<forall>x. P(x) & (Q(x)|R(x))-->S(x)) &
(\<forall>x. S(x) & R(x) --> L(x)) &
(\<forall>x. M(x) --> R(x))
--> (\<forall>x. P(x) & M(x) --> L(x))"
by blast
text{*33*}
lemma "(\<forall>x. P(a) & (P(x)-->P(b))-->P(c)) <->
(\<forall>x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))"
by blast
text{*34 AMENDED (TWICE!!). Andrews's challenge*}
lemma "((\<exists>x. \<forall>y. p(x) <-> p(y)) <->
((\<exists>x. q(x)) <-> (\<forall>y. p(y)))) <->
((\<exists>x. \<forall>y. q(x) <-> q(y)) <->
((\<exists>x. p(x)) <-> (\<forall>y. q(y))))"
by blast
text{*35*}
lemma "\<exists>x y. P(x,y) --> (\<forall>u v. P(u,v))"
by blast
text{*36*}
lemma "(\<forall>x. \<exists>y. J(x,y)) &
(\<forall>x. \<exists>y. G(x,y)) &
(\<forall>x y. J(x,y) | G(x,y) --> (\<forall>z. J(y,z) | G(y,z) --> H(x,z)))
--> (\<forall>x. \<exists>y. H(x,y))"
by blast
text{*37*}
lemma "(\<forall>z. \<exists>w. \<forall>x. \<exists>y.
(P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (\<exists>u. Q(u,w)))) &
(\<forall>x z. ~P(x,z) --> (\<exists>y. Q(y,z))) &
((\<exists>x y. Q(x,y)) --> (\<forall>x. R(x,x)))
--> (\<forall>x. \<exists>y. R(x,y))"
by blast
text{*38*}
lemma "(\<forall>x. p(a) & (p(x) --> (\<exists>y. p(y) & r(x,y))) -->
(\<exists>z. \<exists>w. p(z) & r(x,w) & r(w,z))) <->
(\<forall>x. (~p(a) | p(x) | (\<exists>z. \<exists>w. p(z) & r(x,w) & r(w,z))) &
(~p(a) | ~(\<exists>y. p(y) & r(x,y)) |
(\<exists>z. \<exists>w. p(z) & r(x,w) & r(w,z))))"
by blast
text{*39*}
lemma "~ (\<exists>x. \<forall>y. F(y,x) <-> ~F(y,y))"
by blast
text{*40. AMENDED*}
lemma "(\<exists>y. \<forall>x. F(x,y) <-> F(x,x)) -->
~(\<forall>x. \<exists>y. \<forall>z. F(z,y) <-> ~ F(z,x))"
by blast
text{*41*}
lemma "(\<forall>z. \<exists>y. \<forall>x. f(x,y) <-> f(x,z) & ~ f(x,x))
--> ~ (\<exists>z. \<forall>x. f(x,z))"
by blast
text{*42*}
lemma "~ (\<exists>y. \<forall>x. p(x,y) <-> ~ (\<exists>z. p(x,z) & p(z,x)))"
by blast
text{*43*}
lemma "(\<forall>x. \<forall>y. q(x,y) <-> (\<forall>z. p(z,x) <-> p(z,y)))
--> (\<forall>x. \<forall>y. q(x,y) <-> q(y,x))"
by blast
(*Other proofs: Can use auto, which cheats by using rewriting!
Deepen_tac alone requires 253 secs. Or
by (mini_tac 1 THEN Deepen_tac 5 1) *)
text{*44*}
lemma "(\<forall>x. f(x) --> (\<exists>y. g(y) & h(x,y) & (\<exists>y. g(y) & ~ h(x,y)))) &
(\<exists>x. j(x) & (\<forall>y. g(y) --> h(x,y)))
--> (\<exists>x. j(x) & ~f(x))"
by blast
text{*45*}
lemma "(\<forall>x. f(x) & (\<forall>y. g(y) & h(x,y) --> j(x,y))
--> (\<forall>y. g(y) & h(x,y) --> k(y))) &
~ (\<exists>y. l(y) & k(y)) &
(\<exists>x. f(x) & (\<forall>y. h(x,y) --> l(y))
& (\<forall>y. g(y) & h(x,y) --> j(x,y)))
--> (\<exists>x. f(x) & ~ (\<exists>y. g(y) & h(x,y)))"
by blast
text{*46*}
lemma "(\<forall>x. f(x) & (\<forall>y. f(y) & h(y,x) --> g(y)) --> g(x)) &
((\<exists>x. f(x) & ~g(x)) -->
(\<exists>x. f(x) & ~g(x) & (\<forall>y. f(y) & ~g(y) --> j(x,y)))) &
(\<forall>x y. f(x) & f(y) & h(x,y) --> ~j(y,x))
--> (\<forall>x. f(x) --> g(x))"
by blast
subsection{*Problems (mainly) involving equality or functions*}
text{*48*}
lemma "(a=b | c=d) & (a=c | b=d) --> a=d | b=c"
by blast
text{*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. *}
lemma "(\<exists>x y::'a. \<forall>z. z=x | z=y) & P(a) & P(b) & a~=b
--> (\<forall>u::'a. P(u))"
apply safe
apply (rule_tac x = a in allE, assumption)
apply (rule_tac x = b in allE, assumption, fast)
--{*blast's treatment of equality can't do it*}
done
text{*50. (What has this to do with equality?) *}
lemma "(\<forall>x. P(a,x) | (\<forall>y. P(x,y))) --> (\<exists>x. \<forall>y. P(x,y))"
by blast
text{*51*}
lemma "(\<exists>z w. \<forall>x y. P(x,y) <-> (x=z & y=w)) -->
(\<exists>z. \<forall>x. \<exists>w. (\<forall>y. P(x,y) <-> y=w) <-> x=z)"
by blast
text{*52*}
text{*Almost the same as 51. *}
lemma "(\<exists>z w. \<forall>x y. P(x,y) <-> (x=z & y=w)) -->
(\<exists>w. \<forall>y. \<exists>z. (\<forall>x. P(x,y) <-> x=z) <-> y=w)"
by blast
text{*55*}
(*Original, equational version by Len Schubert, via Pelletier *** NOT PROVED
Goal "(\<exists>x. lives(x) & killed(x,agatha)) &
lives(agatha) & lives(butler) & lives(charles) &
(\<forall>x. lives(x) --> x=agatha | x=butler | x=charles) &
(\<forall>x y. killed(x,y) --> hates(x,y)) &
(\<forall>x y. killed(x,y) --> ~richer(x,y)) &
(\<forall>x. hates(agatha,x) --> ~hates(charles,x)) &
(\<forall>x. ~ x=butler --> hates(agatha,x)) &
(\<forall>x. ~richer(x,agatha) --> hates(butler,x)) &
(\<forall>x. hates(agatha,x) --> hates(butler,x)) &
(\<forall>x. \<exists>y. ~hates(x,y)) &
~ agatha=butler -->
killed(?who,agatha)"
by Safe_tac;
by (dres_inst_tac [("x1","x")] (spec RS mp) 1);
by (assume_tac 1);
by (etac (spec RS exE) 1);
by (REPEAT (etac allE 1));
by (Blast_tac 1);
result();
****)
text{*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
fast DISCOVERS who killed Agatha. *}
lemma "lives(agatha) & lives(butler) & lives(charles) &
(killed(agatha,agatha) | killed(butler,agatha) | killed(charles,agatha)) &
(\<forall>x y. killed(x,y) --> hates(x,y) & ~richer(x,y)) &
(\<forall>x. hates(agatha,x) --> ~hates(charles,x)) &
(hates(agatha,agatha) & hates(agatha,charles)) &
(\<forall>x. lives(x) & ~richer(x,agatha) --> hates(butler,x)) &
(\<forall>x. hates(agatha,x) --> hates(butler,x)) &
(\<forall>x. ~hates(x,agatha) | ~hates(x,butler) | ~hates(x,charles)) -->
killed(?who,agatha)"
by fast --{*MUCH faster than blast*}
text{*56*}
lemma "(\<forall>x. (\<exists>y. P(y) & x=f(y)) --> P(x)) <-> (\<forall>x. P(x) --> P(f(x)))"
by blast
text{*57*}
lemma "P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) &
(\<forall>x y z. P(x,y) & P(y,z) --> P(x,z)) --> P(f(a,b), f(a,c))"
by blast
text{*58 NOT PROVED AUTOMATICALLY*}
lemma "(\<forall>x y. f(x)=g(y)) --> (\<forall>x y. f(f(x))=f(g(y)))"
by (slow elim: subst_context)
text{*59*}
lemma "(\<forall>x. P(x) <-> ~P(f(x))) --> (\<exists>x. P(x) & ~P(f(x)))"
by blast
text{*60*}
lemma "\<forall>x. P(x,f(x)) <-> (\<exists>y. (\<forall>z. P(z,y) --> P(z,f(x))) & P(x,y))"
by blast
text{*62 as corrected in JAR 18 (1997), page 135*}
lemma "(\<forall>x. p(a) & (p(x) --> p(f(x))) --> p(f(f(x)))) <->
(\<forall>x. (~p(a) | p(x) | p(f(f(x)))) &
(~p(a) | ~p(f(x)) | p(f(f(x)))))"
by blast
text{*From Davis, Obvious Logical Inferences, IJCAI-81, 530-531
fast indeed copes!*}
lemma "(\<forall>x. F(x) & ~G(x) --> (\<exists>y. H(x,y) & J(y))) &
(\<exists>x. K(x) & F(x) & (\<forall>y. H(x,y) --> K(y))) &
(\<forall>x. K(x) --> ~G(x)) --> (\<exists>x. K(x) & J(x))"
by fast
text{*From Rudnicki, Obvious Inferences, JAR 3 (1987), 383-393.
It does seem obvious!*}
lemma "(\<forall>x. F(x) & ~G(x) --> (\<exists>y. H(x,y) & J(y))) &
(\<exists>x. K(x) & F(x) & (\<forall>y. H(x,y) --> K(y))) &
(\<forall>x. K(x) --> ~G(x)) --> (\<exists>x. K(x) --> ~G(x))"
by fast
text{*Halting problem: Formulation of Li Dafa (AAR Newsletter 27, Oct 1994.)
author U. Egly*}
lemma "((\<exists>x. A(x) & (\<forall>y. C(y) --> (\<forall>z. D(x,y,z)))) -->
(\<exists>w. C(w) & (\<forall>y. C(y) --> (\<forall>z. D(w,y,z)))))
&
(\<forall>w. C(w) & (\<forall>u. C(u) --> (\<forall>v. D(w,u,v))) -->
(\<forall>y z.
(C(y) & P(y,z) --> Q(w,y,z) & OO(w,g)) &
(C(y) & ~P(y,z) --> Q(w,y,z) & OO(w,b))))
&
(\<forall>w. C(w) &
(\<forall>y z.
(C(y) & P(y,z) --> Q(w,y,z) & OO(w,g)) &
(C(y) & ~P(y,z) --> Q(w,y,z) & OO(w,b))) -->
(\<exists>v. C(v) &
(\<forall>y. ((C(y) & Q(w,y,y)) & OO(w,g) --> ~P(v,y)) &
((C(y) & Q(w,y,y)) & OO(w,b) --> P(v,y) & OO(v,b)))))
-->
~ (\<exists>x. A(x) & (\<forall>y. C(y) --> (\<forall>z. D(x,y,z))))"
by (tactic{*Blast.depth_tac (claset ()) 12 1*})
--{*Needed because the search for depths below 12 is very slow*}
text{*Halting problem II: credited to M. Bruschi by Li Dafa in JAR 18(1), p.105*}
lemma "((\<exists>x. A(x) & (\<forall>y. C(y) --> (\<forall>z. D(x,y,z)))) -->
(\<exists>w. C(w) & (\<forall>y. C(y) --> (\<forall>z. D(w,y,z)))))
&
(\<forall>w. C(w) & (\<forall>u. C(u) --> (\<forall>v. D(w,u,v))) -->
(\<forall>y z.
(C(y) & P(y,z) --> Q(w,y,z) & OO(w,g)) &
(C(y) & ~P(y,z) --> Q(w,y,z) & OO(w,b))))
&
((\<exists>w. C(w) & (\<forall>y. (C(y) & P(y,y) --> Q(w,y,y) & OO(w,g)) &
(C(y) & ~P(y,y) --> Q(w,y,y) & OO(w,b))))
-->
(\<exists>v. C(v) & (\<forall>y. (C(y) & P(y,y) --> P(v,y) & OO(v,g)) &
(C(y) & ~P(y,y) --> P(v,y) & OO(v,b)))))
-->
((\<exists>v. C(v) & (\<forall>y. (C(y) & P(y,y) --> P(v,y) & OO(v,g)) &
(C(y) & ~P(y,y) --> P(v,y) & OO(v,b))))
-->
(\<exists>u. C(u) & (\<forall>y. (C(y) & P(y,y) --> ~P(u,y)) &
(C(y) & ~P(y,y) --> P(u,y) & OO(u,b)))))
-->
~ (\<exists>x. A(x) & (\<forall>y. C(y) --> (\<forall>z. D(x,y,z))))"
by blast
text{* Challenge found on info-hol *}
lemma "\<forall>x. \<exists>v w. \<forall>y z. P(x) & Q(y) --> (P(v) | R(w)) & (R(z) --> Q(v))"
by blast
text{*Attributed to Lewis Carroll by S. G. Pulman. The first or last assumption
can be deleted.*}
lemma "(\<forall>x. honest(x) & industrious(x) --> healthy(x)) &
~ (\<exists>x. grocer(x) & healthy(x)) &
(\<forall>x. industrious(x) & grocer(x) --> honest(x)) &
(\<forall>x. cyclist(x) --> industrious(x)) &
(\<forall>x. ~healthy(x) & cyclist(x) --> ~honest(x))
--> (\<forall>x. grocer(x) --> ~cyclist(x))"
by blast
(*Runtimes for old versions of this file:
Thu Jul 23 1992: loaded in 467s using iffE [on SPARC2]
Mon Nov 14 1994: loaded in 144s [on SPARC10, with deepen_tac]
Wed Nov 16 1994: loaded in 138s [after addition of norm_term_skip]
Mon Nov 21 1994: loaded in 131s [DEPTH_FIRST suppressing repetitions]
Further runtimes on a Sun-4
Tue Mar 4 1997: loaded in 93s (version 94-7)
Tue Mar 4 1997: loaded in 89s
Thu Apr 3 1997: loaded in 44s--using mostly Blast_tac
Thu Apr 3 1997: loaded in 96s--addition of two Halting Probs
Thu Apr 3 1997: loaded in 98s--using lim-1 for all haz rules
Tue Dec 2 1997: loaded in 107s--added 46; new equalSubst
Fri Dec 12 1997: loaded in 91s--faster proof reconstruction
Thu Dec 18 1997: loaded in 94s--two new "obvious theorems" (??)
*)
end