--- a/src/HOL/ex/Classical.thy Wed Oct 29 11:50:26 2003 +0100
+++ b/src/HOL/ex/Classical.thy Wed Oct 29 16:16:20 2003 +0100
@@ -10,8 +10,9 @@
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>"}.*}
+text{*Taken from @{text "FOL/Classical.thy"}. 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
@@ -25,8 +26,8 @@
by blast
-text{*Sample problems from
- F. J. Pelletier,
+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.
@@ -120,14 +121,14 @@
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))
+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,
+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
@@ -157,7 +158,7 @@
by blast
text{*Problem 20*}
-lemma "(\<forall>x y. \<exists>z. \<forall>w. (P(x)&Q(y)-->R(z)&S(w)))
+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
@@ -174,76 +175,76 @@
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))
+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)))
+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)))
+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)))
+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)))
+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))) =
+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))
+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))
+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))
+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)) =
+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)) =
+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
@@ -252,26 +253,26 @@
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))
+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))
+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) |
+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!*)
@@ -280,12 +281,12 @@
by blast
text{*Problem 40. AMENDED*}
-lemma "(\<exists>y. \<forall>x. F x y = F x x)
+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))
+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
@@ -294,23 +295,23 @@
by blast
text{*Problem 43!!*}
-lemma "(\<forall>x::'a. \<forall>y::'a. q x y = (\<forall>z. p z x = (p z y::bool)))
+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))
+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))
+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
@@ -321,10 +322,10 @@
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
+text{*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. *}
-lemma "(\<exists>x y::'a. \<forall>z. z=x | z=y) & P(a) & P(b) & (~a=b)
+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)
@@ -336,12 +337,12 @@
by blast
text{*Problem 51*}
-lemma "(\<exists>z w. \<forall>x y. P x y = (x=z & y=w)) -->
+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)) -->
+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
@@ -349,14 +350,14 @@
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) -->
+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
@@ -365,7 +366,7 @@
by blast
text{*Problem 57*}
-lemma "P (f a b) (f b c) & P (f b c) (f a c) &
+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
@@ -382,37 +383,37 @@
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))) &
+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))) &
+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.
+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))) &
+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
+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))
+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) &
+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
@@ -522,6 +523,10 @@
lemma "\<exists>z. P z --> (\<forall>x. P x)"
by meson
+text{*From a paper by Claire Quigley*}
+lemma "\<exists>y. ((P c & Q y) | (\<exists>z. ~ Q z)) | (\<exists>x. ~ P x & Q d)"
+by fast
+
subsubsection{*Hard examples with quantifiers*}
text{*Problem 18*}
@@ -533,7 +538,7 @@
by meson
text{*Problem 20*}
-lemma "(\<forall>x y. \<exists>z. \<forall>w. (P x & Q y --> R z & S w))
+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
@@ -550,78 +555,75 @@
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)
+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))
+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))
+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))
+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))
+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)) =
+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)
+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)
+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)
+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) =
+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)))"
+text{*Problem 34: Andrews's challenge has 924 Horn clauses*}
+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*}
@@ -629,27 +631,25 @@
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)"
+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))
+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) |
+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
@@ -658,12 +658,12 @@
by meson
text{*Problem 40. AMENDED*}
-lemma "(\<exists>y. \<forall>x. F x y = F x x)
+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))))
+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
@@ -672,72 +672,71 @@
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)))
+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)"
+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))
+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)"
+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))"
+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},
+ --{*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)"
+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)"
+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?*}
@@ -748,24 +747,23 @@
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)"
+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) &
+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 *}
+text{*Problem 58: 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
@@ -778,9 +776,9 @@
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))))"
+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