--- a/src/FOL/ex/cla.ML Wed Jul 15 18:26:15 1998 +0200
+++ b/src/FOL/ex/cla.ML Thu Jul 16 10:35:31 1998 +0200
@@ -8,19 +8,21 @@
writeln"File FOL/ex/cla.ML";
+context FOL.thy;
+
open Cla; (*in case structure IntPr is open!*)
-goal FOL.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 FOL.thy "(P<->Q) <-> (Q<->P)";
+Goal "(P<->Q) <-> (Q<->P)";
by (Blast_tac 1);
result();
-goal FOL.thy "~ (P <-> ~P)";
+Goal "~ (P <-> ~P)";
by (Blast_tac 1);
result();
@@ -37,183 +39,191 @@
writeln"Pelletier's examples";
(*1*)
-goal FOL.thy "(P-->Q) <-> (~Q --> ~P)";
+Goal "(P-->Q) <-> (~Q --> ~P)";
by (Blast_tac 1);
result();
(*2*)
-goal FOL.thy "~ ~ P <-> P";
+Goal "~ ~ P <-> P";
by (Blast_tac 1);
result();
(*3*)
-goal FOL.thy "~(P-->Q) --> (Q-->P)";
+Goal "~(P-->Q) --> (Q-->P)";
by (Blast_tac 1);
result();
(*4*)
-goal FOL.thy "(~P-->Q) <-> (~Q --> P)";
+Goal "(~P-->Q) <-> (~Q --> P)";
by (Blast_tac 1);
result();
(*5*)
-goal FOL.thy "((P|Q)-->(P|R)) --> (P|(Q-->R))";
+Goal "((P|Q)-->(P|R)) --> (P|(Q-->R))";
by (Blast_tac 1);
result();
(*6*)
-goal FOL.thy "P | ~ P";
+Goal "P | ~ P";
by (Blast_tac 1);
result();
(*7*)
-goal FOL.thy "P | ~ ~ ~ P";
+Goal "P | ~ ~ ~ P";
by (Blast_tac 1);
result();
(*8. Peirce's law*)
-goal FOL.thy "((P-->Q) --> P) --> P";
+Goal "((P-->Q) --> P) --> P";
by (Blast_tac 1);
result();
(*9*)
-goal FOL.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 FOL.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 FOL.thy "P<->P";
+Goal "P<->P";
by (Blast_tac 1);
result();
(*12. "Dijkstra's law"*)
-goal FOL.thy "((P <-> Q) <-> R) <-> (P <-> (Q <-> R))";
+Goal "((P <-> Q) <-> R) <-> (P <-> (Q <-> R))";
by (Blast_tac 1);
result();
(*13. Distributive law*)
-goal FOL.thy "P | (Q & R) <-> (P | Q) & (P | R)";
+Goal "P | (Q & R) <-> (P | Q) & (P | R)";
by (Blast_tac 1);
result();
(*14*)
-goal FOL.thy "(P <-> Q) <-> ((Q | ~P) & (~Q|P))";
+Goal "(P <-> Q) <-> ((Q | ~P) & (~Q|P))";
by (Blast_tac 1);
result();
(*15*)
-goal FOL.thy "(P --> Q) <-> (~P | Q)";
+Goal "(P --> Q) <-> (~P | Q)";
by (Blast_tac 1);
result();
(*16*)
-goal FOL.thy "(P-->Q) | (Q-->P)";
+Goal "(P-->Q) | (Q-->P)";
by (Blast_tac 1);
result();
(*17*)
-goal FOL.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 FOL.thy "(ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))";
+Goal "(ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))";
by (Blast_tac 1);
result();
-goal FOL.thy "(EX x. P-->Q(x)) <-> (P --> (EX x. Q(x)))";
+Goal "(EX x. P-->Q(x)) <-> (P --> (EX x. Q(x)))";
by (Blast_tac 1);
result();
-goal FOL.thy "(EX x. P(x)-->Q) <-> (ALL x. P(x)) --> Q";
+Goal "(EX x. P(x)-->Q) <-> (ALL x. P(x)) --> Q";
by (Blast_tac 1);
result();
-goal FOL.thy "(ALL x. P(x)) | Q <-> (ALL x. P(x) | Q)";
+Goal "(ALL x. P(x)) | Q <-> (ALL x. P(x) | Q)";
by (Blast_tac 1);
result();
(*Discussed in Avron, Gentzen-Type Systems, Resolution and Tableaux,
JAR 10 (265-281), 1993. Proof is trivial!*)
-goal FOL.thy
+Goal
"~ ((EX x.~P(x)) & ((EX x. P(x)) | (EX x. P(x) & Q(x))) & ~ (EX x. P(x)))";
by (Blast_tac 1);
result();
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 ALL.*)
-goal FOL.thy "(ALL x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))";
+Goal "(ALL 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 FOL.thy "EX x. P(x) --> P(a) & P(b)";
+Goal "EX x. P(x) --> P(a) & P(b)";
by (Blast_tac 1);
result();
-goal FOL.thy "EX z. P(z) --> (ALL x. P(x))";
+Goal "EX z. P(z) --> (ALL x. P(x))";
by (Blast_tac 1);
result();
-goal FOL.thy "EX x. (EX y. P(y)) --> P(x)";
+Goal "EX x. (EX y. P(y)) --> P(x)";
by (Blast_tac 1);
result();
(*V. Lifschitz, What Is the Inverse Method?, JAR 5 (1989), 1--23. NOT PROVED*)
-goal FOL.thy "EX x x'. ALL y. EX z z'. \
+Goal "EX x x'. ALL y. EX 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'))";
+
+
writeln"Hard examples with quantifiers";
writeln"Problem 18";
-goal FOL.thy "EX y. ALL x. P(y)-->P(x)";
+Goal "EX y. ALL x. P(y)-->P(x)";
by (Blast_tac 1);
result();
writeln"Problem 19";
-goal FOL.thy "EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";
+Goal "EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";
by (Blast_tac 1);
result();
writeln"Problem 20";
-goal FOL.thy "(ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w))) \
+Goal "(ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w))) \
\ --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))";
by (Blast_tac 1);
result();
writeln"Problem 21";
-goal FOL.thy "(EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))";
+Goal "(EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))";
by (Blast_tac 1);
result();
writeln"Problem 22";
-goal FOL.thy "(ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))";
+Goal "(ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))";
by (Blast_tac 1);
result();
writeln"Problem 23";
-goal FOL.thy "(ALL x. P | Q(x)) <-> (P | (ALL x. Q(x)))";
+Goal "(ALL x. P | Q(x)) <-> (P | (ALL x. Q(x)))";
by (Blast_tac 1);
result();
writeln"Problem 24";
-goal FOL.thy "~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) & \
+Goal "~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) & \
\ (~(EX x. P(x)) --> (EX x. Q(x))) & (ALL x. Q(x)|R(x) --> S(x)) \
\ --> (EX x. P(x)&R(x))";
by (Blast_tac 1);
result();
writeln"Problem 25";
-goal FOL.thy "(EX x. P(x)) & \
+Goal "(EX x. P(x)) & \
\ (ALL x. L(x) --> ~ (M(x) & R(x))) & \
\ (ALL x. P(x) --> (M(x) & L(x))) & \
\ ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x))) \
@@ -222,14 +232,14 @@
result();
writeln"Problem 26";
-goal FOL.thy "((EX x. p(x)) <-> (EX x. q(x))) & \
+Goal "((EX x. p(x)) <-> (EX x. q(x))) & \
\ (ALL x. ALL y. p(x) & q(y) --> (r(x) <-> s(y))) \
\ --> ((ALL x. p(x)-->r(x)) <-> (ALL x. q(x)-->s(x)))";
by (Blast_tac 1);
result();
writeln"Problem 27";
-goal FOL.thy "(EX x. P(x) & ~Q(x)) & \
+Goal "(EX x. P(x) & ~Q(x)) & \
\ (ALL x. P(x) --> R(x)) & \
\ (ALL x. M(x) & L(x) --> P(x)) & \
\ ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x))) \
@@ -238,7 +248,7 @@
result();
writeln"Problem 28. AMENDED";
-goal FOL.thy "(ALL x. P(x) --> (ALL x. Q(x))) & \
+Goal "(ALL x. P(x) --> (ALL x. Q(x))) & \
\ ((ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) & \
\ ((EX x. S(x)) --> (ALL x. L(x) --> M(x))) \
\ --> (ALL x. P(x) & L(x) --> M(x))";
@@ -246,21 +256,21 @@
result();
writeln"Problem 29. Essentially the same as Principia Mathematica *11.71";
-goal FOL.thy "(EX x. P(x)) & (EX y. Q(y)) \
+Goal "(EX x. P(x)) & (EX y. Q(y)) \
\ --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y)) <-> \
\ (ALL x y. P(x) & Q(y) --> R(x) & S(y)))";
by (Blast_tac 1);
result();
writeln"Problem 30";
-goal FOL.thy "(ALL x. P(x) | Q(x) --> ~ R(x)) & \
+Goal "(ALL x. P(x) | Q(x) --> ~ R(x)) & \
\ (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \
\ --> (ALL x. S(x))";
by (Blast_tac 1);
result();
writeln"Problem 31";
-goal FOL.thy "~(EX x. P(x) & (Q(x) | R(x))) & \
+Goal "~(EX x. P(x) & (Q(x) | R(x))) & \
\ (EX x. L(x) & P(x)) & \
\ (ALL x. ~ R(x) --> M(x)) \
\ --> (EX x. L(x) & M(x))";
@@ -268,7 +278,7 @@
result();
writeln"Problem 32";
-goal FOL.thy "(ALL x. P(x) & (Q(x)|R(x))-->S(x)) & \
+Goal "(ALL x. P(x) & (Q(x)|R(x))-->S(x)) & \
\ (ALL x. S(x) & R(x) --> L(x)) & \
\ (ALL x. M(x) --> R(x)) \
\ --> (ALL x. P(x) & M(x) --> L(x))";
@@ -276,14 +286,14 @@
result();
writeln"Problem 33";
-goal FOL.thy "(ALL x. P(a) & (P(x)-->P(b))-->P(c)) <-> \
+Goal "(ALL x. P(a) & (P(x)-->P(b))-->P(c)) <-> \
\ (ALL 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 FOL.thy "((EX x. ALL y. p(x) <-> p(y)) <-> \
+Goal "((EX x. ALL y. p(x) <-> p(y)) <-> \
\ ((EX x. q(x)) <-> (ALL y. p(y)))) <-> \
\ ((EX x. ALL y. q(x) <-> q(y)) <-> \
\ ((EX x. p(x)) <-> (ALL y. q(y))))";
@@ -291,12 +301,12 @@
result();
writeln"Problem 35";
-goal FOL.thy "EX x y. P(x,y) --> (ALL u v. P(u,v))";
+Goal "EX x y. P(x,y) --> (ALL u v. P(u,v))";
by (Blast_tac 1);
result();
writeln"Problem 36";
-goal FOL.thy
+Goal
"(ALL x. EX y. J(x,y)) & \
\ (ALL x. EX y. G(x,y)) & \
\ (ALL x y. J(x,y) | G(x,y) --> (ALL z. J(y,z) | G(y,z) --> H(x,z))) \
@@ -305,7 +315,7 @@
result();
writeln"Problem 37";
-goal FOL.thy "(ALL z. EX w. ALL x. EX y. \
+Goal "(ALL z. EX w. ALL x. EX y. \
\ (P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (EX u. Q(u,w)))) & \
\ (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) & \
\ ((EX x y. Q(x,y)) --> (ALL x. R(x,x))) \
@@ -314,7 +324,7 @@
result();
writeln"Problem 38";
-goal FOL.thy
+Goal
"(ALL x. p(a) & (p(x) --> (EX y. p(y) & r(x,y))) --> \
\ (EX z. EX w. p(z) & r(x,w) & r(w,z))) <-> \
\ (ALL x. (~p(a) | p(x) | (EX z. EX w. p(z) & r(x,w) & r(w,z))) & \
@@ -324,29 +334,29 @@
result();
writeln"Problem 39";
-goal FOL.thy "~ (EX x. ALL y. F(y,x) <-> ~F(y,y))";
+Goal "~ (EX x. ALL y. F(y,x) <-> ~F(y,y))";
by (Blast_tac 1);
result();
writeln"Problem 40. AMENDED";
-goal FOL.thy "(EX y. ALL x. F(x,y) <-> F(x,x)) --> \
+Goal "(EX y. ALL x. F(x,y) <-> F(x,x)) --> \
\ ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))";
by (Blast_tac 1);
result();
writeln"Problem 41";
-goal FOL.thy "(ALL z. EX y. ALL x. f(x,y) <-> f(x,z) & ~ f(x,x)) \
+Goal "(ALL z. EX y. ALL x. f(x,y) <-> f(x,z) & ~ f(x,x)) \
\ --> ~ (EX z. ALL x. f(x,z))";
by (Blast_tac 1);
result();
writeln"Problem 42";
-goal FOL.thy "~ (EX y. ALL x. p(x,y) <-> ~ (EX z. p(x,z) & p(z,x)))";
+Goal "~ (EX y. ALL x. p(x,y) <-> ~ (EX z. p(x,z) & p(z,x)))";
by (Blast_tac 1);
result();
writeln"Problem 43";
-goal FOL.thy "(ALL x. ALL y. q(x,y) <-> (ALL z. p(z,x) <-> p(z,y))) \
+Goal "(ALL x. ALL y. q(x,y) <-> (ALL z. p(z,x) <-> p(z,y))) \
\ --> (ALL x. ALL y. q(x,y) <-> q(y,x))";
by (Blast_tac 1);
(*Other proofs: Can use Auto_tac(), which cheats by using rewriting!
@@ -356,7 +366,7 @@
result();
writeln"Problem 44";
-goal FOL.thy "(ALL x. f(x) --> \
+Goal "(ALL x. f(x) --> \
\ (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y)))) & \
\ (EX x. j(x) & (ALL y. g(y) --> h(x,y))) \
\ --> (EX x. j(x) & ~f(x))";
@@ -364,7 +374,7 @@
result();
writeln"Problem 45";
-goal FOL.thy "(ALL x. f(x) & (ALL y. g(y) & h(x,y) --> j(x,y)) \
+Goal "(ALL x. f(x) & (ALL y. g(y) & h(x,y) --> j(x,y)) \
\ --> (ALL y. g(y) & h(x,y) --> k(y))) & \
\ ~ (EX y. l(y) & k(y)) & \
\ (EX x. f(x) & (ALL y. h(x,y) --> l(y)) \
@@ -375,7 +385,7 @@
writeln"Problem 46";
-goal FOL.thy
+Goal
"(ALL x. f(x) & (ALL y. f(y) & h(y,x) --> g(y)) --> g(x)) & \
\ ((EX x. f(x) & ~g(x)) --> \
\ (EX x. f(x) & ~g(x) & (ALL y. f(y) & ~g(y) --> j(x,y)))) & \
@@ -388,14 +398,14 @@
writeln"Problems (mainly) involving equality or functions";
writeln"Problem 48";
-goal FOL.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 FOL.thy "(EX x y::'a. ALL z. z=x | z=y) & P(a) & P(b) & a~=b \
+Goal "(EX x y::'a. ALL z. z=x | z=y) & P(a) & P(b) & a~=b \
\ --> (ALL u::'a. P(u))";
by Safe_tac;
by (res_inst_tac [("x","a")] allE 1);
@@ -407,12 +417,12 @@
writeln"Problem 50";
(*What has this to do with equality?*)
-goal FOL.thy "(ALL x. P(a,x) | (ALL y. P(x,y))) --> (EX x. ALL y. P(x,y))";
+Goal "(ALL x. P(a,x) | (ALL y. P(x,y))) --> (EX x. ALL y. P(x,y))";
by (Blast_tac 1);
result();
writeln"Problem 51";
-goal FOL.thy
+Goal
"(EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \
\ (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)";
by (Blast_tac 1);
@@ -420,7 +430,7 @@
writeln"Problem 52";
(*Almost the same as 51. *)
-goal FOL.thy
+Goal
"(EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) --> \
\ (EX w. ALL y. EX z. (ALL x. P(x,y) <-> x=z) <-> y=w)";
by (Blast_tac 1);
@@ -429,7 +439,7 @@
writeln"Problem 55";
(*Original, equational version by Len Schubert, via Pelletier *** NOT PROVED
-goal FOL.thy
+Goal
"(EX x. lives(x) & killed(x,agatha)) & \
\ lives(agatha) & lives(butler) & lives(charles) & \
\ (ALL x. lives(x) --> x=agatha | x=butler | x=charles) & \
@@ -453,7 +463,7 @@
(*Non-equational version, from Manthey and Bry, CADE-9 (Springer, 1988).
fast_tac DISCOVERS who killed Agatha. *)
-goal FOL.thy "lives(agatha) & lives(butler) & lives(charles) & \
+Goal "lives(agatha) & lives(butler) & lives(charles) & \
\ (killed(agatha,agatha) | killed(butler,agatha) | killed(charles,agatha)) & \
\ (ALL x y. killed(x,y) --> hates(x,y) & ~richer(x,y)) & \
\ (ALL x. hates(agatha,x) --> ~hates(charles,x)) & \
@@ -467,36 +477,36 @@
writeln"Problem 56";
-goal FOL.thy
+Goal
"(ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))";
by (Blast_tac 1);
result();
writeln"Problem 57";
-goal FOL.thy
+Goal
"P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) & \
\ (ALL 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 FOL.thy "(ALL x y. f(x)=g(y)) --> (ALL x y. f(f(x))=f(g(y)))";
+Goal "(ALL x y. f(x)=g(y)) --> (ALL x y. f(f(x))=f(g(y)))";
by (slow_tac (claset() addEs [subst_context]) 1);
result();
writeln"Problem 59";
-goal FOL.thy "(ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))";
+Goal "(ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))";
by (Blast_tac 1);
result();
writeln"Problem 60";
-goal FOL.thy
+Goal
"ALL x. P(x,f(x)) <-> (EX y. (ALL 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 FOL.thy
+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)))))";
@@ -505,7 +515,7 @@
(*From Davis, Obvious Logical Inferences, IJCAI-81, 530-531
Fast_tac indeed copes!*)
-goal FOL.thy "(ALL x. F(x) & ~G(x) --> (EX y. H(x,y) & J(y))) & \
+Goal "(ALL x. F(x) & ~G(x) --> (EX y. H(x,y) & J(y))) & \
\ (EX x. K(x) & F(x) & (ALL y. H(x,y) --> K(y))) & \
\ (ALL x. K(x) --> ~G(x)) --> (EX x. K(x) & J(x))";
by (Fast_tac 1);
@@ -513,7 +523,7 @@
(*From Rudnicki, Obvious Inferences, JAR 3 (1987), 383-393.
It does seem obvious!*)
-goal FOL.thy
+Goal
"(ALL x. F(x) & ~G(x) --> (EX y. H(x,y) & J(y))) & \
\ (EX x. K(x) & F(x) & (ALL y. H(x,y) --> K(y))) & \
\ (ALL x. K(x) --> ~G(x)) --> (EX x. K(x) --> ~G(x))";
@@ -522,7 +532,7 @@
(*Halting problem: Formulation of Li Dafa (AAR Newsletter 27, Oct 1994.)
author U. Egly*)
-goal FOL.thy
+Goal
"((EX x. A(x) & (ALL y. C(y) --> (ALL z. D(x,y,z)))) --> \
\ (EX w. C(w) & (ALL y. C(y) --> (ALL z. D(w,y,z))))) \
\ & \
@@ -545,7 +555,7 @@
(*Halting problem II: credited to M. Bruschi by Li Dafa in JAR 18(1), p.105*)
-goal FOL.thy
+Goal
"((EX x. A(x) & (ALL y. C(y) --> (ALL z. D(x,y,z)))) --> \
\ (EX w. C(w) & (ALL y. C(y) --> (ALL z. D(w,y,z))))) \
\ & \
@@ -571,8 +581,7 @@
result();
(* Challenge found on info-hol *)
-goal FOL.thy
- "ALL x. EX v w. ALL y z. P(x) & Q(y) --> (P(v) | R(w)) & (R(z) --> Q(v))";
+Goal "ALL x. EX v w. ALL y z. P(x) & Q(y) --> (P(v) | R(w)) & (R(z) --> Q(v))";
by (Blast_tac 1);
result();
--- 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();