--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Classical.thy Wed Oct 08 15:57:41 2003 +0200
@@ -0,0 +1,786 @@
+(* Title: HOL/ex/Classical
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1994 University of Cambridge
+*)
+
+header{*Classical Predicate Calculus Problems*}
+
+theory Classical = Main:
+
+subsection{*Traditional Classical Reasoner*}
+
+text{*Taken from @{text "FOL/cla.ML"}. When porting examples from first-order
+logic, beware of the precedence of @{text "="} versus @{text "\<leftrightarrow>"}.*}
+
+lemma "(P --> Q | R) --> (P-->Q) | (P-->R)"
+by blast
+
+text{*If and only if*}
+
+lemma "(P=Q) = (Q = (P::bool))"
+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.
+*}
+
+subsubsection{*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::bool)"
+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
+
+subsubsection{*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{*From Wishnu Prasetya*}
+lemma "(\<forall>s. q(s) --> r(s)) & ~r(s) & (\<forall>s. ~r(s) & ~q(s) --> p(t) | q(t))
+ --> p(t) | r(t)"
+by blast
+
+
+subsubsection{*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 the quantifier.*}
+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
+
+subsubsection{*Hard examples with quantifiers*}
+
+text{*Problem 18*}
+lemma "\<exists>y. \<forall>x. P(y)-->P(x)"
+by blast
+
+text{*Problem 19*}
+lemma "\<exists>x. \<forall>y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))"
+by blast
+
+text{*Problem 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{*Problem 21*}
+lemma "(\<exists>x. P-->Q(x)) & (\<exists>x. Q(x)-->P) --> (\<exists>x. P=Q(x))"
+by blast
+
+text{*Problem 22*}
+lemma "(\<forall>x. P = Q(x)) --> (P = (\<forall>x. Q(x)))"
+by blast
+
+text{*Problem 23*}
+lemma "(\<forall>x. P | Q(x)) = (P | (\<forall>x. Q(x)))"
+by blast
+
+text{*Problem 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{*Problem 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{*Problem 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{*Problem 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{*Problem 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{*Problem 29. Essentially the same as Principia Mathematica *11.71*}
+lemma "(\<exists>x. F(x)) & (\<exists>y. G(y))
+ --> ( ((\<forall>x. F(x)-->H(x)) & (\<forall>y. G(y)-->J(y))) =
+ (\<forall>x y. F(x) & G(y) --> H(x) & J(y)))"
+by blast
+
+text{*Problem 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{*Problem 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{*Problem 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{*Problem 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{*Problem 34 AMENDED (TWICE!!)*}
+text{*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{*Problem 35*}
+lemma "\<exists>x y. P x y --> (\<forall>u v. P u v)"
+by blast
+
+text{*Problem 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{*Problem 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{*Problem 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 (*beats fast!*)
+
+text{*Problem 39*}
+lemma "~ (\<exists>x. \<forall>y. F y x = (~ F y y))"
+by blast
+
+text{*Problem 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{*Problem 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{*Problem 42*}
+lemma "~ (\<exists>y. \<forall>x. p x y = (~ (\<exists>z. p x z & p z x)))"
+by blast
+
+text{*Problem 43!!*}
+lemma "(\<forall>x::'a. \<forall>y::'a. q x y = (\<forall>z. p z x = (p z y::bool)))
+ --> (\<forall>x. (\<forall>y. q x y = (q y x::bool)))"
+by blast
+
+text{*Problem 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{*Problem 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
+
+
+subsubsection{*Problems (mainly) involving equality or functions*}
+
+text{*Problem 48*}
+lemma "(a=b | c=d) & (a=c | b=d) --> a=d | b=c"
+by blast
+
+text{*Problem 49 NOT PROVED AUTOMATICALLY*}
+text{*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{*Problem 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{*Problem 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{*Problem 52. 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{*Problem 55*}
+
+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
+
+text{*Problem 56*}
+lemma "(\<forall>x. (\<exists>y. P(y) & x=f(y)) --> P(x)) = (\<forall>x. P(x) --> P(f(x)))"
+by blast
+
+text{*Problem 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{*Problem 58 NOT PROVED AUTOMATICALLY*}
+lemma "(\<forall>x y. f(x)=g(y)) --> (\<forall>x y. f(f(x))=f(g(y)))"
+by (fast intro: arg_cong [of concl: f])
+
+text{*Problem 59*}
+lemma "(\<forall>x. P(x) = (~P(f(x)))) --> (\<exists>x. P(x) & ~P(f(x)))"
+by blast
+
+text{*Problem 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{*Problem 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{*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
+
+lemma "(\<forall>x y. R(x,y) | R(y,x)) &
+ (\<forall>x y. S(x,y) & S(y,x) --> x=y) &
+ (\<forall>x y. R(x,y) --> S(x,y)) --> (\<forall>x y. S(x,y) --> R(x,y))"
+by blast
+
+
+subsection{*Model Elimination Prover*}
+
+text{*The "small example" from Bezem, Hendriks and de Nivelle,
+Automatic Proof Construction in Type Theory Using Resolution,
+JAR 29: 3-4 (2002), pages 253-275 *}
+lemma "(\<forall>x y z. R(x,y) & R(y,z) --> R(x,z)) &
+ (\<forall>x. \<exists>y. R(x,y)) -->
+ ~ (\<forall>x. P x = (\<forall>y. R(x,y) --> ~ P y))"
+by (tactic{*safe_best_meson_tac 1*})
+ --{*In contrast, @{text meson} is SLOW: 15s on a 1.8GHz machine!*}
+
+
+subsubsection{*Pelletier's examples*}
+text{*1*}
+lemma "(P --> Q) = (~Q --> ~P)"
+by meson
+
+text{*2*}
+lemma "(~ ~ P) = P"
+by meson
+
+text{*3*}
+lemma "~(P-->Q) --> (Q-->P)"
+by meson
+
+text{*4*}
+lemma "(~P-->Q) = (~Q --> P)"
+by meson
+
+text{*5*}
+lemma "((P|Q)-->(P|R)) --> (P|(Q-->R))"
+by meson
+
+text{*6*}
+lemma "P | ~ P"
+by meson
+
+text{*7*}
+lemma "P | ~ ~ ~ P"
+by meson
+
+text{*8. Peirce's law*}
+lemma "((P-->Q) --> P) --> P"
+by meson
+
+text{*9*}
+lemma "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
+by meson
+
+text{*10*}
+lemma "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)"
+by meson
+
+text{*11. Proved in each direction (incorrectly, says Pelletier!!) *}
+lemma "P=(P::bool)"
+by meson
+
+text{*12. "Dijkstra's law"*}
+lemma "((P = Q) = R) = (P = (Q = R))"
+by meson
+
+text{*13. Distributive law*}
+lemma "(P | (Q & R)) = ((P | Q) & (P | R))"
+by meson
+
+text{*14*}
+lemma "(P = Q) = ((Q | ~P) & (~Q|P))"
+by meson
+
+text{*15*}
+lemma "(P --> Q) = (~P | Q)"
+by meson
+
+text{*16*}
+lemma "(P-->Q) | (Q-->P)"
+by meson
+
+text{*17*}
+lemma "((P & (Q-->R))-->S) = ((~P | Q | S) & (~P | ~R | S))"
+by meson
+
+subsubsection{*Classical Logic: examples with quantifiers*}
+
+lemma "(\<forall>x. P x & Q x) = ((\<forall>x. P x) & (\<forall>x. Q x))"
+by meson
+
+lemma "(\<exists>x. P --> Q x) = (P --> (\<exists>x. Q x))"
+by meson
+
+lemma "(\<exists>x. P x --> Q) = ((\<forall>x. P x) --> Q)"
+by meson
+
+lemma "((\<forall>x. P x) | Q) = (\<forall>x. P x | Q)"
+by meson
+
+lemma "(\<forall>x. P x --> P(f x)) & P d --> P(f(f(f d)))"
+by meson
+
+text{*Needs double instantiation of EXISTS*}
+lemma "\<exists>x. P x --> P a & P b"
+by meson
+
+lemma "\<exists>z. P z --> (\<forall>x. P x)"
+by meson
+
+subsubsection{*Hard examples with quantifiers*}
+
+text{*Problem 18*}
+lemma "\<exists>y. \<forall>x. P y --> P x"
+by meson
+
+text{*Problem 19*}
+lemma "\<exists>x. \<forall>y z. (P y --> Q z) --> (P x --> Q x)"
+by meson
+
+text{*Problem 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 meson
+
+text{*Problem 21*}
+lemma "(\<exists>x. P --> Q x) & (\<exists>x. Q x --> P) --> (\<exists>x. P=Q x)"
+by meson
+
+text{*Problem 22*}
+lemma "(\<forall>x. P = Q x) --> (P = (\<forall>x. Q x))"
+by meson
+
+text{*Problem 23*}
+lemma "(\<forall>x. P | Q x) = (P | (\<forall>x. Q x))"
+by meson
+
+text{*Problem 24*} (*The first goal clause is useless*)
+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 meson
+
+text{*Problem 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 meson
+
+text{*Problem 26; has 24 Horn clauses*}
+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 meson
+
+text{*Problem 27; has 13 Horn clauses*}
+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 meson
+
+text{*Problem 28. AMENDED; has 14 Horn clauses*}
+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 meson
+
+text{*Problem 29. Essentially the same as Principia Mathematica
+*11.71. 62 Horn clauses*}
+lemma "(\<exists>x. F x) & (\<exists>y. G y)
+ --> ( ((\<forall>x. F x --> H x) & (\<forall>y. G y --> J y)) =
+ (\<forall>x y. F x & G y --> H x & J y))"
+by meson
+
+
+text{*Problem 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 meson
+
+text{*Problem 31; has 10 Horn clauses; first negative clauses is useless*}
+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 meson
+
+text{*Problem 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 meson
+
+text{*Problem 33; has 55 Horn clauses*}
+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 meson
+
+text{*Problem 34 AMENDED (TWICE!!); has 924 Horn clauses*}
+text{*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 meson
+
+text{*Problem 35*}
+lemma "\<exists>x y. P x y --> (\<forall>u v. P u v)"
+by meson
+
+text{*Problem 36; has 15 Horn clauses*}
+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 meson
+
+text{*Problem 37; has 10 Horn clauses*}
+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 meson --{*causes unification tracing messages*}
+
+
+text{*Problem 38*} text{*Quite hard: 422 Horn clauses!!*}
+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 meson
+
+text{*Problem 39*}
+lemma "~ (\<exists>x. \<forall>y. F y x = (~F y y))"
+by meson
+
+text{*Problem 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 meson
+
+text{*Problem 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 meson
+
+text{*Problem 42*}
+lemma "~ (\<exists>y. \<forall>x. p x y = (~ (\<exists>z. p x z & p z x)))"
+by meson
+
+text{*Problem 43 NOW PROVED AUTOMATICALLY!!*}
+lemma "(\<forall>x. \<forall>y. q x y = (\<forall>z. p z x = (p z y::bool)))
+ --> (\<forall>x. (\<forall>y. q x y = (q y x::bool)))"
+by meson
+
+text{*Problem 44: 13 Horn clauses; 7-step proof*}
+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 meson
+
+text{*Problem 45; has 27 Horn clauses; 54-step proof*}
+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 meson
+
+text{*Problem 46; has 26 Horn clauses; 21-step proof*}
+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 meson
+
+text{*Problem 47. Schubert's Steamroller*}
+ text{*26 clauses; 63 Horn clauses
+ 87094 inferences so far. Searching to depth 36*}
+lemma "(\<forall>x. P1 x --> P0 x) & (\<exists>x. P1 x) &
+ (\<forall>x. P2 x --> P0 x) & (\<exists>x. P2 x) &
+ (\<forall>x. P3 x --> P0 x) & (\<exists>x. P3 x) &
+ (\<forall>x. P4 x --> P0 x) & (\<exists>x. P4 x) &
+ (\<forall>x. P5 x --> P0 x) & (\<exists>x. P5 x) &
+ (\<forall>x. Q1 x --> Q0 x) & (\<exists>x. Q1 x) &
+ (\<forall>x. P0 x --> ((\<forall>y. Q0 y-->R x y) |
+ (\<forall>y. P0 y & S y x &
+ (\<exists>z. Q0 z&R y z) --> R x y))) &
+ (\<forall>x y. P3 y & (P5 x|P4 x) --> S x y) &
+ (\<forall>x y. P3 x & P2 y --> S x y) &
+ (\<forall>x y. P2 x & P1 y --> S x y) &
+ (\<forall>x y. P1 x & (P2 y|Q1 y) --> ~R x y) &
+ (\<forall>x y. P3 x & P4 y --> R x y) &
+ (\<forall>x y. P3 x & P5 y --> ~R x y) &
+ (\<forall>x. (P4 x|P5 x) --> (\<exists>y. Q0 y & R x y))
+ --> (\<exists>x y. P0 x & P0 y & (\<exists>z. Q1 z & R y z & R x y))"
+by (tactic{*safe_best_meson_tac 1*})
+ --{*Considerably faster than @{text meson},
+ which does iterative deepening rather than best-first search*}
+
+text{*The Los problem. Circulated by John Harrison*}
+lemma "(\<forall>x y z. P x y & P y z --> P x z) &
+ (\<forall>x y z. Q x y & Q y z --> Q x z) &
+ (\<forall>x y. P x y --> P y x) &
+ (\<forall>x y. P x y | Q x y)
+ --> (\<forall>x y. P x y) | (\<forall>x y. Q x y)"
+by meson
+
+text{*A similar example, suggested by Johannes Schumann and
+ credited to Pelletier*}
+lemma "(\<forall>x y z. P x y --> P y z --> P x z) -->
+ (\<forall>x y z. Q x y --> Q y z --> Q x z) -->
+ (\<forall>x y. Q x y --> Q y x) --> (\<forall>x y. P x y | Q x y) -->
+ (\<forall>x y. P x y) | (\<forall>x y. Q x y)"
+by meson
+
+text{*Problem 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 meson
+
+text{*Problem 55*}
+
+text{*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
+ @{text meson} cannot report 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) -->
+ (\<exists>x. killed x agatha)"
+by meson
+
+text{*Problem 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 meson
+
+text{*Problem 58*}
+text{* Challenge found on info-hol *}
+lemma "\<forall>P Q R x. \<exists>v w. \<forall>y z. P x & Q y --> (P v | R w) & (R z --> Q v)"
+by meson
+
+text{*Problem 59*}
+lemma "(\<forall>x. P x = (~P(f x))) --> (\<exists>x. P x & ~P(f x))"
+by meson
+
+text{*Problem 60*}
+lemma "\<forall>x. P x (f x) = (\<exists>y. (\<forall>z. P z y --> P z (f x)) & P x y)"
+by meson
+
+text{*Problem 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 meson
+
+end