--- a/src/FOL/ex/cla.ML Thu Apr 03 19:32:03 1997 +0200
+++ b/src/FOL/ex/cla.ML Fri Apr 04 11:16:44 1997 +0200
@@ -11,17 +11,17 @@
open Cla; (*in case structure IntPr is open!*)
goal FOL.thy "(P --> Q | R) --> (P-->Q) | (P-->R)";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*If and only if*)
goal FOL.thy "(P<->Q) <-> (Q<->P)";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
goal FOL.thy "~ (P <-> ~P)";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
@@ -38,132 +38,132 @@
writeln"Pelletier's examples";
(*1*)
goal FOL.thy "(P-->Q) <-> (~Q --> ~P)";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*2*)
goal FOL.thy "~ ~ P <-> P";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*3*)
goal FOL.thy "~(P-->Q) --> (Q-->P)";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*4*)
goal FOL.thy "(~P-->Q) <-> (~Q --> P)";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*5*)
goal FOL.thy "((P|Q)-->(P|R)) --> (P|(Q-->R))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*6*)
goal FOL.thy "P | ~ P";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*7*)
goal FOL.thy "P | ~ ~ ~ P";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*8. Peirce's law*)
goal FOL.thy "((P-->Q) --> P) --> P";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*9*)
goal FOL.thy "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*10*)
goal FOL.thy "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P<->Q)";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*11. Proved in each direction (incorrectly, says Pelletier!!) *)
goal FOL.thy "P<->P";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*12. "Dijkstra's law"*)
goal FOL.thy "((P <-> Q) <-> R) <-> (P <-> (Q <-> R))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*13. Distributive law*)
goal FOL.thy "P | (Q & R) <-> (P | Q) & (P | R)";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*14*)
goal FOL.thy "(P <-> Q) <-> ((Q | ~P) & (~Q|P))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*15*)
goal FOL.thy "(P --> Q) <-> (~P | Q)";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*16*)
goal FOL.thy "(P-->Q) | (Q-->P)";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
(*17*)
goal FOL.thy "((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))";
-by (Fast_tac 1);
+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))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
goal FOL.thy "(EX x. P-->Q(x)) <-> (P --> (EX x.Q(x)))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
goal FOL.thy "(EX x.P(x)-->Q) <-> (ALL x.P(x)) --> Q";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
goal FOL.thy "(ALL x.P(x)) | Q <-> (ALL x. P(x) | Q)";
-by (Fast_tac 1);
+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
"~ ((EX x.~P(x)) & ((EX x.P(x)) | (EX x.P(x) & Q(x))) & ~ (EX x.P(x)))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problems requiring quantifier duplication";
(*Needs multiple instantiation of ALL.*)
goal FOL.thy "(ALL x. P(x)-->P(f(x))) & P(d)-->P(f(f(f(d))))";
-by (Deepen_tac 0 1);
+by (Blast_tac 1);
result();
(*Needs double instantiation of the quantifier*)
goal FOL.thy "EX x. P(x) --> P(a) & P(b)";
-by (Deepen_tac 0 1);
+by (Blast_tac 1);
result();
goal FOL.thy "EX z. P(z) --> (ALL x. P(x))";
-by (Deepen_tac 0 1);
+by (Blast_tac 1);
result();
goal FOL.thy "EX x. (EX y. P(y)) --> P(x)";
-by (Deepen_tac 0 1);
+by (Blast_tac 1);
result();
(*V. Lifschitz, What Is the Inverse Method?, JAR 5 (1989), 1--23. NOT PROVED*)
@@ -176,40 +176,40 @@
writeln"Problem 18";
goal FOL.thy "EX y. ALL x. P(y)-->P(x)";
-by (Deepen_tac 0 1);
+by (Blast_tac 1);
result();
writeln"Problem 19";
goal FOL.thy "EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";
-by (Deepen_tac 0 1);
+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))) \
\ --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))";
-by (Fast_tac 1);
+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))";
-by (Deepen_tac 0 1);
+by (Blast_tac 1);
result();
writeln"Problem 22";
goal FOL.thy "(ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 23";
goal FOL.thy "(ALL x. P | Q(x)) <-> (P | (ALL x. Q(x)))";
-by (Best_tac 1);
+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)) & \
\ (~(EX x.P(x)) --> (EX x.Q(x))) & (ALL x. Q(x)|R(x) --> S(x)) \
\ --> (EX x. P(x)&R(x))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 25";
@@ -218,14 +218,14 @@
\ (ALL x. P(x) --> (M(x) & L(x))) & \
\ ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x))) \
\ --> (EX x. Q(x)&P(x))";
-by (Best_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 26";
goal FOL.thy "((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 (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 27";
@@ -234,7 +234,7 @@
\ (ALL x. M(x) & L(x) --> P(x)) & \
\ ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x))) \
\ --> (ALL x. M(x) --> ~L(x))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 28. AMENDED";
@@ -242,21 +242,21 @@
\ ((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))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 29. Essentially the same as Principia Mathematica *11.71";
goal FOL.thy "(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 (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 30";
goal FOL.thy "(ALL x. P(x) | Q(x) --> ~ R(x)) & \
\ (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \
\ --> (ALL x. S(x))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 31";
@@ -264,7 +264,7 @@
\ (EX x. L(x) & P(x)) & \
\ (ALL x. ~ R(x) --> M(x)) \
\ --> (EX x. L(x) & M(x))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 32";
@@ -272,13 +272,13 @@
\ (ALL x. S(x) & R(x) --> L(x)) & \
\ (ALL x. M(x) --> R(x)) \
\ --> (ALL x. P(x) & M(x) --> L(x))";
-by (Best_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 33";
goal FOL.thy "(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 (Best_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 34 AMENDED (TWICE!!)";
@@ -287,14 +287,12 @@
\ ((EX x. q(x)) <-> (ALL y. p(y)))) <-> \
\ ((EX x. ALL y. q(x) <-> q(y)) <-> \
\ ((EX x. p(x)) <-> (ALL y. q(y))))";
-by (Deepen_tac 0 1);
+by (Blast_tac 1);
result();
writeln"Problem 35";
goal FOL.thy "EX x y. P(x,y) --> (ALL u v. P(u,v))";
-by (mini_tac 1);
-by (Fast_tac 1);
-(*Without miniscope, would have to use deepen_tac; would be slower*)
+by (Blast_tac 1);
result();
writeln"Problem 36";
@@ -303,7 +301,7 @@
\ (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))) \
\ --> (ALL x. EX y. H(x,y))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 37";
@@ -312,7 +310,7 @@
\ (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) & \
\ ((EX x y. Q(x,y)) --> (ALL x. R(x,x))) \
\ --> (ALL x. EX y. R(x,y))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 38";
@@ -322,39 +320,38 @@
\ (ALL x. (~p(a) | p(x) | (EX z. EX w. p(z) & r(x,w) & r(w,z))) & \
\ (~p(a) | ~(EX y. p(y) & r(x,y)) | \
\ (EX z. EX w. p(z) & r(x,w) & r(w,z))))";
-by (Deepen_tac 0 1); (*beats fast_tac!*)
+by (Blast_tac 1); (*beats fast_tac!*)
result();
writeln"Problem 39";
goal FOL.thy "~ (EX x. ALL y. F(y,x) <-> ~F(y,y))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 40. AMENDED";
goal FOL.thy "(EX y. ALL x. F(x,y) <-> F(x,x)) --> \
\ ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))";
-by (Fast_tac 1);
+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)) \
\ --> ~ (EX z. ALL x. f(x,z))";
-by (Fast_tac 1);
+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)))";
-by (Deepen_tac 0 1);
+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))) \
\ --> (ALL x. ALL y. q(x,y) <-> q(y,x))";
-by (Auto_tac());
-(*The proof above cheats by using rewriting! A purely logical proof is
+by (Blast_tac 1);
+(*Other proofs: Can use Auto_tac(), whidh cheats by using rewriting!
+ Deepen_tac alone it requires 253 secs. Or
by (mini_tac 1 THEN Deepen_tac 5 1);
-Can use just deepen_tac but it requires 253 secs?!?
- by (Deepen_tac 0 1);
*)
result();
@@ -363,7 +360,7 @@
\ (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))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 45";
@@ -373,7 +370,7 @@
\ (EX x. f(x) & (ALL y. h(x,y) --> l(y)) \
\ & (ALL y. g(y) & h(x,y) --> j(x,y))) \
\ --> (EX x. f(x) & ~ (EX y. g(y) & h(x,y)))";
-by (Best_tac 1);
+by (Blast_tac 1);
result();
@@ -381,7 +378,7 @@
writeln"Problem 48";
goal FOL.thy "(a=b | c=d) & (a=c | b=d) --> a=d | b=c";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 49 NOT PROVED AUTOMATICALLY";
@@ -394,21 +391,20 @@
by (assume_tac 1);
by (res_inst_tac [("x","b")] allE 1);
by (assume_tac 1);
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
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))";
-by (mini_tac 1);
-by (Deepen_tac 0 1);
+by (Blast_tac 1);
result();
writeln"Problem 51";
goal FOL.thy
"(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 (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 52";
@@ -416,7 +412,7 @@
goal FOL.thy
"(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 (Best_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 55";
@@ -440,7 +436,7 @@
by (assume_tac 1);
by (etac (spec RS exE) 1);
by (REPEAT (etac allE 1));
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
****)
@@ -455,21 +451,21 @@
\ (ALL x. hates(agatha,x) --> hates(butler,x)) & \
\ (ALL x. ~hates(x,agatha) | ~hates(x,butler) | ~hates(x,charles)) --> \
\ killed(?who,agatha)";
-by (Fast_tac 1);
+by (Fast_tac 1); (*MUCH faster than Blast_tac: 8s against 29s*)
result();
writeln"Problem 56";
goal FOL.thy
"(ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 57";
goal FOL.thy
"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 (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 58 NOT PROVED AUTOMATICALLY";
@@ -479,13 +475,13 @@
writeln"Problem 59";
goal FOL.thy "(ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))";
-by (Deepen_tac 0 1);
+by (Blast_tac 1);
result();
writeln"Problem 60";
goal FOL.thy
"ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))";
-by (Fast_tac 1);
+by (Blast_tac 1);
result();
writeln"Problem 62 as corrected in JAR 18 (1997), page 135";
@@ -493,35 +489,63 @@
"(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 (Fast_tac 1);
+by (Blast_tac 1);
result();
(*Halting problem: Formulation of Li Dafa (AAR Newsletter 27, Oct 1994.)
author U. Egly*)
goal FOL.thy
-"((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))))) \
-\ & \
-\ (ALL W. c(W) & (ALL U. c(U) --> (ALL V. d(W, U, V))) --> \
-\ (ALL Y Z. \
-\ (c(Y) & h2(Y, Z) --> h3(W, Y, Z) & o(W, g)) & \
-\ (c(Y) & ~h2(Y, Z) --> h3(W, Y, Z) & o(W, b)))) \
-\ & \
-\ (ALL W. c(W) & \
-\ (ALL Y Z. \
-\ (c(Y) & h2(Y, Z) --> h3(W, Y, Z) & o(W, g)) & \
-\ (c(Y) & ~h2(Y, Z) --> h3(W, Y, Z) & o(W, b))) --> \
-\ (EX V. c(V) & \
-\ (ALL Y. ((c(Y) & h3(W, Y, Y)) & o(W, g) --> ~h2(V, Y)) & \
-\ ((c(Y) & h3(W, Y, Y)) & o(W, b) --> h2(V, Y) & o(V, b))))) \
+ "((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))))) \
+\ & \
+\ (ALL w. C(w) & (ALL u. C(u) --> (ALL v. D(w,u,v))) --> \
+\ (ALL 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)))) \
+\ & \
+\ (ALL w. C(w) & \
+\ (ALL 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))) --> \
+\ (EX v. C(v) & \
+\ (ALL 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))))) \
\ --> \
-\ ~ (EX X. a(X) & (ALL Y. c(Y) --> (ALL Z. d(X, Y, Z))))";
+\ ~ (EX x. A(x) & (ALL y. C(y) --> (ALL z. D(x,y,z))))";
+by (Blast_tac 1);
+result();
+
+(*Halting problem II: credited to M. Bruschi by Li Dafa in JAR 18(1), p.105*)
+goal FOL.thy
+ "((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))))) \
+\ & \
+\ (ALL w. C(w) & (ALL u. C(u) --> (ALL v. D(w,u,v))) --> \
+\ (ALL 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)))) \
+\ & \
+\ ((EX w. C(w) & (ALL 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)))) \
+\ --> \
+\ (EX v. C(v) & (ALL y. (C(y) & P(y,y) --> P(v,y) & OO(v,g)) & \
+\ (C(y) & ~P(y,y) --> P(v,y) & OO(v,b))))) \
+\ --> \
+\ ((EX v. C(v) & (ALL y. (C(y) & P(y,y) --> P(v,y) & OO(v,g)) & \
+\ (C(y) & ~P(y,y) --> P(v,y) & OO(v,b)))) \
+\ --> \
+\ (EX u. C(u) & (ALL y. (C(y) & P(y,y) --> ~P(u,y)) & \
+\ (C(y) & ~P(y,y) --> P(u,y) & OO(u,b))))) \
+\ --> \
+\ ~ (EX x. A(x) & (ALL y. C(y) --> (ALL z. D(x,y,z))))";
+by (Blast_tac 1);
+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))";
-by (Deepen_tac 0 1);
+by (Blast_tac 1);
result();
writeln"Reached end of file.";
@@ -533,4 +557,6 @@
(*Tue Mar 4 1997: loaded in 93s (on pochard, version 94-7) *)
(*Tue Mar 4 1997: loaded in 89s (on pochard) *)
+(*Thu Apr 3 1997: loaded in 44s (on pochard)--using mostly Blast_tac*)
+(*Thu Apr 3 1997: loaded in 96s (on pochard)--addition of two Halting Probs*)