--- a/src/HOL/ex/cla.ML Wed Jul 15 18:26:15 1998 +0200
+++ b/src/HOL/ex/cla.ML Thu Jul 16 10:35:31 1998 +0200
@@ -10,19 +10,19 @@
writeln"File HOL/ex/cla.";
-context HOL.thy; (*Boosts efficiency by omitting redundant rules*)
+context HOL.thy;
-goal HOL.thy "(P --> Q | R) --> (P-->Q) | (P-->R)";
+Goal "(P --> Q | R) --> (P-->Q) | (P-->R)";
by (Blast_tac 1);
result();
(*If and only if*)
-goal HOL.thy "(P=Q) = (Q = (P::bool))";
+Goal "(P=Q) = (Q = (P::bool))";
by (Blast_tac 1);
result();
-goal HOL.thy "~ (P = (~P))";
+Goal "~ (P = (~P))";
by (Blast_tac 1);
result();
@@ -39,110 +39,110 @@
writeln"Pelletier's examples";
(*1*)
-goal HOL.thy "(P-->Q) = (~Q --> ~P)";
+Goal "(P-->Q) = (~Q --> ~P)";
by (Blast_tac 1);
result();
(*2*)
-goal HOL.thy "(~ ~ P) = P";
+Goal "(~ ~ P) = P";
by (Blast_tac 1);
result();
(*3*)
-goal HOL.thy "~(P-->Q) --> (Q-->P)";
+Goal "~(P-->Q) --> (Q-->P)";
by (Blast_tac 1);
result();
(*4*)
-goal HOL.thy "(~P-->Q) = (~Q --> P)";
+Goal "(~P-->Q) = (~Q --> P)";
by (Blast_tac 1);
result();
(*5*)
-goal HOL.thy "((P|Q)-->(P|R)) --> (P|(Q-->R))";
+Goal "((P|Q)-->(P|R)) --> (P|(Q-->R))";
by (Blast_tac 1);
result();
(*6*)
-goal HOL.thy "P | ~ P";
+Goal "P | ~ P";
by (Blast_tac 1);
result();
(*7*)
-goal HOL.thy "P | ~ ~ ~ P";
+Goal "P | ~ ~ ~ P";
by (Blast_tac 1);
result();
(*8. Peirce's law*)
-goal HOL.thy "((P-->Q) --> P) --> P";
+Goal "((P-->Q) --> P) --> P";
by (Blast_tac 1);
result();
(*9*)
-goal HOL.thy "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
+Goal "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
by (Blast_tac 1);
result();
(*10*)
-goal HOL.thy "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)";
+Goal "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P=Q)";
by (Blast_tac 1);
result();
(*11. Proved in each direction (incorrectly, says Pelletier!!) *)
-goal HOL.thy "P=(P::bool)";
+Goal "P=(P::bool)";
by (Blast_tac 1);
result();
(*12. "Dijkstra's law"*)
-goal HOL.thy "((P = Q) = R) = (P = (Q = R))";
+Goal "((P = Q) = R) = (P = (Q = R))";
by (Blast_tac 1);
result();
(*13. Distributive law*)
-goal HOL.thy "(P | (Q & R)) = ((P | Q) & (P | R))";
+Goal "(P | (Q & R)) = ((P | Q) & (P | R))";
by (Blast_tac 1);
result();
(*14*)
-goal HOL.thy "(P = Q) = ((Q | ~P) & (~Q|P))";
+Goal "(P = Q) = ((Q | ~P) & (~Q|P))";
by (Blast_tac 1);
result();
(*15*)
-goal HOL.thy "(P --> Q) = (~P | Q)";
+Goal "(P --> Q) = (~P | Q)";
by (Blast_tac 1);
result();
(*16*)
-goal HOL.thy "(P-->Q) | (Q-->P)";
+Goal "(P-->Q) | (Q-->P)";
by (Blast_tac 1);
result();
(*17*)
-goal HOL.thy "((P & (Q-->R))-->S) = ((~P | Q | S) & (~P | ~R | S))";
+Goal "((P & (Q-->R))-->S) = ((~P | Q | S) & (~P | ~R | S))";
by (Blast_tac 1);
result();
writeln"Classical Logic: examples with quantifiers";
-goal HOL.thy "(! x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
+Goal "(! x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
by (Blast_tac 1);
result();
-goal HOL.thy "(? x. P-->Q(x)) = (P --> (? x. Q(x)))";
+Goal "(? x. P-->Q(x)) = (P --> (? x. Q(x)))";
by (Blast_tac 1);
result();
-goal HOL.thy "(? x. P(x)-->Q) = ((! x. P(x)) --> Q)";
+Goal "(? x. P(x)-->Q) = ((! x. P(x)) --> Q)";
by (Blast_tac 1);
result();
-goal HOL.thy "((! x. P(x)) | Q) = (! x. P(x) | Q)";
+Goal "((! x. P(x)) | Q) = (! x. P(x) | Q)";
by (Blast_tac 1);
result();
(*From Wishnu Prasetya*)
-goal HOL.thy
+Goal
"(!s. q(s) --> r(s)) & ~r(s) & (!s. ~r(s) & ~q(s) --> p(t) | q(t)) \
\ --> p(t) | r(t)";
by (Blast_tac 1);
@@ -151,66 +151,72 @@
writeln"Problems requiring quantifier duplication";
+(*Theorem B of Peter Andrews, Theorem Proving via General Matings,
+ JACM 28 (1981).*)
+Goal "(EX x. ALL y. P(x) = P(y)) --> ((EX x. P(x)) = (ALL y. P(y)))";
+by (Blast_tac 1);
+result();
+
(*Needs multiple instantiation of the quantifier.*)
-goal HOL.thy "(! x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))";
+Goal "(! x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))";
by (Blast_tac 1);
result();
(*Needs double instantiation of the quantifier*)
-goal HOL.thy "? x. P(x) --> P(a) & P(b)";
+Goal "? x. P(x) --> P(a) & P(b)";
by (Blast_tac 1);
result();
-goal HOL.thy "? z. P(z) --> (! x. P(x))";
+Goal "? z. P(z) --> (! x. P(x))";
by (Blast_tac 1);
result();
-goal HOL.thy "? x. (? y. P(y)) --> P(x)";
+Goal "? x. (? y. P(y)) --> P(x)";
by (Blast_tac 1);
result();
writeln"Hard examples with quantifiers";
writeln"Problem 18";
-goal HOL.thy "? y. ! x. P(y)-->P(x)";
+Goal "? y. ! x. P(y)-->P(x)";
by (Blast_tac 1);
result();
writeln"Problem 19";
-goal HOL.thy "? x. ! y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";
+Goal "? x. ! y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";
by (Blast_tac 1);
result();
writeln"Problem 20";
-goal HOL.thy "(! x y. ? z. ! w. (P(x)&Q(y)-->R(z)&S(w))) \
+Goal "(! x y. ? z. ! w. (P(x)&Q(y)-->R(z)&S(w))) \
\ --> (? x y. P(x) & Q(y)) --> (? z. R(z))";
by (Blast_tac 1);
result();
writeln"Problem 21";
-goal HOL.thy "(? x. P-->Q(x)) & (? x. Q(x)-->P) --> (? x. P=Q(x))";
+Goal "(? x. P-->Q(x)) & (? x. Q(x)-->P) --> (? x. P=Q(x))";
by (Blast_tac 1);
result();
writeln"Problem 22";
-goal HOL.thy "(! x. P = Q(x)) --> (P = (! x. Q(x)))";
+Goal "(! x. P = Q(x)) --> (P = (! x. Q(x)))";
by (Blast_tac 1);
result();
writeln"Problem 23";
-goal HOL.thy "(! x. P | Q(x)) = (P | (! x. Q(x)))";
+Goal "(! x. P | Q(x)) = (P | (! x. Q(x)))";
by (Blast_tac 1);
result();
writeln"Problem 24";
-goal HOL.thy "~(? x. S(x)&Q(x)) & (! x. P(x) --> Q(x)|R(x)) & \
+Goal "~(? x. S(x)&Q(x)) & (! x. P(x) --> Q(x)|R(x)) & \
\ (~(? x. P(x)) --> (? x. Q(x))) & (! x. Q(x)|R(x) --> S(x)) \
\ --> (? x. P(x)&R(x))";
by (Blast_tac 1);
result();
writeln"Problem 25";
-goal HOL.thy "(? x. P(x)) & \
+Goal "(? x. P(x)) & \
\ (! x. L(x) --> ~ (M(x) & R(x))) & \
\ (! x. P(x) --> (M(x) & L(x))) & \
\ ((! x. P(x)-->Q(x)) | (? x. P(x)&R(x))) \
@@ -219,14 +225,14 @@
result();
writeln"Problem 26";
-goal HOL.thy "((? x. p(x)) = (? x. q(x))) & \
+Goal "((? x. p(x)) = (? x. q(x))) & \
\ (! x. ! y. p(x) & q(y) --> (r(x) = s(y))) \
\ --> ((! x. p(x)-->r(x)) = (! x. q(x)-->s(x)))";
by (Blast_tac 1);
result();
writeln"Problem 27";
-goal HOL.thy "(? x. P(x) & ~Q(x)) & \
+Goal "(? x. P(x) & ~Q(x)) & \
\ (! x. P(x) --> R(x)) & \
\ (! x. M(x) & L(x) --> P(x)) & \
\ ((? x. R(x) & ~ Q(x)) --> (! x. L(x) --> ~ R(x))) \
@@ -235,7 +241,7 @@
result();
writeln"Problem 28. AMENDED";
-goal HOL.thy "(! x. P(x) --> (! x. Q(x))) & \
+Goal "(! x. P(x) --> (! x. Q(x))) & \
\ ((! x. Q(x)|R(x)) --> (? x. Q(x)&S(x))) & \
\ ((? x. S(x)) --> (! x. L(x) --> M(x))) \
\ --> (! x. P(x) & L(x) --> M(x))";
@@ -243,21 +249,21 @@
result();
writeln"Problem 29. Essentially the same as Principia Mathematica *11.71";
-goal HOL.thy "(? x. F(x)) & (? y. G(y)) \
+Goal "(? x. F(x)) & (? y. G(y)) \
\ --> ( ((! x. F(x)-->H(x)) & (! y. G(y)-->J(y))) = \
\ (! x y. F(x) & G(y) --> H(x) & J(y)))";
by (Blast_tac 1);
result();
writeln"Problem 30";
-goal HOL.thy "(! x. P(x) | Q(x) --> ~ R(x)) & \
+Goal "(! x. P(x) | Q(x) --> ~ R(x)) & \
\ (! x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \
\ --> (! x. S(x))";
by (Blast_tac 1);
result();
writeln"Problem 31";
-goal HOL.thy "~(? x. P(x) & (Q(x) | R(x))) & \
+Goal "~(? x. P(x) & (Q(x) | R(x))) & \
\ (? x. L(x) & P(x)) & \
\ (! x. ~ R(x) --> M(x)) \
\ --> (? x. L(x) & M(x))";
@@ -265,7 +271,7 @@
result();
writeln"Problem 32";
-goal HOL.thy "(! x. P(x) & (Q(x)|R(x))-->S(x)) & \
+Goal "(! x. P(x) & (Q(x)|R(x))-->S(x)) & \
\ (! x. S(x) & R(x) --> L(x)) & \
\ (! x. M(x) --> R(x)) \
\ --> (! x. P(x) & M(x) --> L(x))";
@@ -273,14 +279,14 @@
result();
writeln"Problem 33";
-goal HOL.thy "(! x. P(a) & (P(x)-->P(b))-->P(c)) = \
+Goal "(! x. P(a) & (P(x)-->P(b))-->P(c)) = \
\ (! x. (~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c)))";
by (Blast_tac 1);
result();
writeln"Problem 34 AMENDED (TWICE!!)";
(*Andrews's challenge*)
-goal HOL.thy "((? x. ! y. p(x) = p(y)) = \
+Goal "((? x. ! y. p(x) = p(y)) = \
\ ((? x. q(x)) = (! y. p(y)))) = \
\ ((? x. ! y. q(x) = q(y)) = \
\ ((? x. p(x)) = (! y. q(y))))";
@@ -288,12 +294,12 @@
result();
writeln"Problem 35";
-goal HOL.thy "? x y. P x y --> (! u v. P u v)";
+Goal "? x y. P x y --> (! u v. P u v)";
by (Blast_tac 1);
result();
writeln"Problem 36";
-goal HOL.thy "(! x. ? y. J x y) & \
+Goal "(! x. ? y. J x y) & \
\ (! x. ? y. G x y) & \
\ (! x y. J x y | G x y --> \
\ (! z. J y z | G y z --> H x z)) \
@@ -302,7 +308,7 @@
result();
writeln"Problem 37";
-goal HOL.thy "(! z. ? w. ! x. ? y. \
+Goal "(! z. ? w. ! x. ? y. \
\ (P x z -->P y w) & P y z & (P y w --> (? u. Q u w))) & \
\ (! x z. ~(P x z) --> (? y. Q y z)) & \
\ ((? x y. Q x y) --> (! x. R x x)) \
@@ -311,7 +317,7 @@
result();
writeln"Problem 38";
-goal HOL.thy
+Goal
"(! x. p(a) & (p(x) --> (? y. p(y) & r x y)) --> \
\ (? z. ? w. p(z) & r x w & r w z)) = \
\ (! x. (~p(a) | p(x) | (? z. ? w. p(z) & r x w & r w z)) & \
@@ -321,36 +327,36 @@
result();
writeln"Problem 39";
-goal HOL.thy "~ (? x. ! y. F y x = (~ F y y))";
+Goal "~ (? x. ! y. F y x = (~ F y y))";
by (Blast_tac 1);
result();
writeln"Problem 40. AMENDED";
-goal HOL.thy "(? y. ! x. F x y = F x x) \
+Goal "(? y. ! x. F x y = F x x) \
\ --> ~ (! x. ? y. ! z. F z y = (~ F z x))";
by (Blast_tac 1);
result();
writeln"Problem 41";
-goal HOL.thy "(! z. ? y. ! x. f x y = (f x z & ~ f x x)) \
+Goal "(! z. ? y. ! x. f x y = (f x z & ~ f x x)) \
\ --> ~ (? z. ! x. f x z)";
by (Blast_tac 1);
result();
writeln"Problem 42";
-goal HOL.thy "~ (? y. ! x. p x y = (~ (? z. p x z & p z x)))";
+Goal "~ (? y. ! x. p x y = (~ (? z. p x z & p z x)))";
by (Blast_tac 1);
result();
writeln"Problem 43!!";
-goal HOL.thy
+Goal
"(! x::'a. ! y::'a. q x y = (! z. p z x = (p z y::bool))) \
\ --> (! x. (! y. q x y = (q y x::bool)))";
by (Blast_tac 1);
result();
writeln"Problem 44";
-goal HOL.thy "(! x. f(x) --> \
+Goal "(! x. f(x) --> \
\ (? y. g(y) & h x y & (? y. g(y) & ~ h x y))) & \
\ (? x. j(x) & (! y. g(y) --> h x y)) \
\ --> (? x. j(x) & ~f(x))";
@@ -358,7 +364,7 @@
result();
writeln"Problem 45";
-goal HOL.thy
+Goal
"(! x. f(x) & (! y. g(y) & h x y --> j x y) \
\ --> (! y. g(y) & h x y --> k(y))) & \
\ ~ (? y. l(y) & k(y)) & \
@@ -372,14 +378,14 @@
writeln"Problems (mainly) involving equality or functions";
writeln"Problem 48";
-goal HOL.thy "(a=b | c=d) & (a=c | b=d) --> a=d | b=c";
+Goal "(a=b | c=d) & (a=c | b=d) --> a=d | b=c";
by (Blast_tac 1);
result();
writeln"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. *)
-goal HOL.thy "(? x y::'a. ! z. z=x | z=y) & P(a) & P(b) & (~a=b) \
+Goal "(? x y::'a. ! z. z=x | z=y) & P(a) & P(b) & (~a=b) \
\ --> (! u::'a. P(u))";
by (Classical.Safe_tac);
by (res_inst_tac [("x","a")] allE 1);
@@ -391,12 +397,12 @@
writeln"Problem 50";
(*What has this to do with equality?*)
-goal HOL.thy "(! x. P a x | (! y. P x y)) --> (? x. ! y. P x y)";
+Goal "(! x. P a x | (! y. P x y)) --> (? x. ! y. P x y)";
by (Blast_tac 1);
result();
writeln"Problem 51";
-goal HOL.thy
+Goal
"(? z w. ! x y. P x y = (x=z & y=w)) --> \
\ (? z. ! x. ? w. (! y. P x y = (y=w)) = (x=z))";
by (Blast_tac 1);
@@ -404,7 +410,7 @@
writeln"Problem 52";
(*Almost the same as 51. *)
-goal HOL.thy
+Goal
"(? z w. ! x y. P x y = (x=z & y=w)) --> \
\ (? w. ! y. ? z. (! x. P x y = (x=z)) = (y=w))";
by (Blast_tac 1);
@@ -414,7 +420,7 @@
(*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
fast_tac DISCOVERS who killed Agatha. *)
-goal HOL.thy "lives(agatha) & lives(butler) & lives(charles) & \
+Goal "lives(agatha) & lives(butler) & lives(charles) & \
\ (killed agatha agatha | killed butler agatha | killed charles agatha) & \
\ (!x y. killed x y --> hates x y & ~richer x y) & \
\ (!x. hates agatha x --> ~hates charles x) & \
@@ -427,40 +433,39 @@
result();
writeln"Problem 56";
-goal HOL.thy
+Goal
"(! x. (? y. P(y) & x=f(y)) --> P(x)) = (! x. P(x) --> P(f(x)))";
by (Blast_tac 1);
result();
writeln"Problem 57";
-goal HOL.thy
+Goal
"P (f a b) (f b c) & P (f b c) (f a c) & \
\ (! x y z. P x y & P y z --> P x z) --> P (f a b) (f a c)";
by (Blast_tac 1);
result();
writeln"Problem 58 NOT PROVED AUTOMATICALLY";
-goal HOL.thy "(! x y. f(x)=g(y)) --> (! x y. f(f(x))=f(g(y)))";
+Goal "(! x y. f(x)=g(y)) --> (! x y. f(f(x))=f(g(y)))";
val f_cong = read_instantiate [("f","f")] arg_cong;
by (fast_tac (claset() addIs [f_cong]) 1);
result();
writeln"Problem 59";
-goal HOL.thy "(! x. P(x) = (~P(f(x)))) --> (? x. P(x) & ~P(f(x)))";
+Goal "(! x. P(x) = (~P(f(x)))) --> (? x. P(x) & ~P(f(x)))";
by (Blast_tac 1);
result();
writeln"Problem 60";
-goal HOL.thy
+Goal
"! x. P x (f x) = (? y. (! z. P z y --> P z (f x)) & P x y)";
by (Blast_tac 1);
result();
writeln"Problem 62 as corrected in JAR 18 (1997), page 135";
-goal HOL.thy
- "(ALL x. p a & (p x --> p(f x)) --> p(f(f x))) = \
-\ (ALL x. (~ p a | p x | p(f(f x))) & \
-\ (~ p a | ~ p(f x) | p(f(f x))))";
+Goal "(ALL x. p a & (p x --> p(f x)) --> p(f(f x))) = \
+\ (ALL x. (~ p a | p x | p(f(f x))) & \
+\ (~ p a | ~ p(f x) | p(f(f x))))";
by (Blast_tac 1);
result();