--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/FOL/ex/int.ML Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,421 @@
+(* Title: FOL/ex/int
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1991 University of Cambridge
+
+Intuitionistic First-Order Logic
+
+Single-step commands:
+by (Int.step_tac 1);
+by (biresolve_tac safe_brls 1);
+by (biresolve_tac haz_brls 1);
+by (assume_tac 1);
+by (Int.safe_tac 1);
+by (Int.mp_tac 1);
+by (Int.fast_tac 1);
+*)
+
+writeln"File FOL/ex/int.";
+
+(*Note: for PROPOSITIONAL formulae...
+ ~A is classically provable iff it is intuitionistically provable.
+ Therefore A is classically provable iff ~~A is intuitionistically provable.
+
+Let Q be the conjuction of the propositions A|~A, one for each atom A in
+P. If P is provable classically, then clearly P&Q is provable
+intuitionistically, so ~~(P&Q) is also provable intuitionistically.
+The latter is intuitionistically equivalent to ~~P&~~Q, hence to ~~P,
+since ~~Q is intuitionistically provable. Finally, if P is a negation then
+~~P is intuitionstically equivalent to P. [Andy Pitts]
+*)
+
+goal IFOL.thy "~~(P&Q) <-> ~~P & ~~Q";
+by (Int.fast_tac 1);
+result();
+
+goal IFOL.thy "~~~P <-> ~P";
+by (Int.fast_tac 1);
+result();
+
+goal IFOL.thy "~~((P --> Q | R) --> (P-->Q) | (P-->R))";
+by (Int.fast_tac 1);
+result();
+
+goal IFOL.thy "(P<->Q) <-> (Q<->P)";
+by (Int.fast_tac 1);
+result();
+
+
+writeln"Lemmas for the propositional double-negation translation";
+
+goal IFOL.thy "P --> ~~P";
+by (Int.fast_tac 1);
+result();
+
+goal IFOL.thy "~~(~~P --> P)";
+by (Int.fast_tac 1);
+result();
+
+goal IFOL.thy "~~P & ~~(P --> Q) --> ~~Q";
+by (Int.fast_tac 1);
+result();
+
+
+writeln"The following are classically but not constructively valid.";
+
+(*The attempt to prove them terminates quickly!*)
+goal IFOL.thy "((P-->Q) --> P) --> P";
+by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
+(*Check that subgoals remain: proof failed.*)
+getgoal 1;
+
+goal IFOL.thy "(P&Q-->R) --> (P-->R) | (Q-->R)";
+by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
+getgoal 1;
+
+
+writeln"Intuitionistic FOL: propositional problems based on Pelletier.";
+
+writeln"Problem ~~1";
+goal IFOL.thy "~~((P-->Q) <-> (~Q --> ~P))";
+by (Int.fast_tac 1);
+result();
+(*5 secs*)
+
+
+writeln"Problem ~~2";
+goal IFOL.thy "~~(~~P <-> P)";
+by (Int.fast_tac 1);
+result();
+(*1 secs*)
+
+
+writeln"Problem 3";
+goal IFOL.thy "~(P-->Q) --> (Q-->P)";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem ~~4";
+goal IFOL.thy "~~((~P-->Q) <-> (~Q --> P))";
+by (Int.fast_tac 1);
+result();
+(*9 secs*)
+
+writeln"Problem ~~5";
+goal IFOL.thy "~~((P|Q-->P|R) --> P|(Q-->R))";
+by (Int.fast_tac 1);
+result();
+(*10 secs*)
+
+
+writeln"Problem ~~6";
+goal IFOL.thy "~~(P | ~P)";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem ~~7";
+goal IFOL.thy "~~(P | ~~~P)";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem ~~8. Peirce's law";
+goal IFOL.thy "~~(((P-->Q) --> P) --> P)";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem 9";
+goal IFOL.thy "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
+by (Int.fast_tac 1);
+result();
+(*9 secs*)
+
+
+writeln"Problem 10";
+goal IFOL.thy "(Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)";
+by (Int.fast_tac 1);
+result();
+
+writeln"11. Proved in each direction (incorrectly, says Pelletier!!) ";
+goal IFOL.thy "P<->P";
+by (Int.fast_tac 1);
+
+writeln"Problem ~~12. Dijkstra's law ";
+goal IFOL.thy "~~(((P <-> Q) <-> R) <-> (P <-> (Q <-> R)))";
+by (Int.fast_tac 1);
+result();
+
+goal IFOL.thy "((P <-> Q) <-> R) --> ~~(P <-> (Q <-> R))";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem 13. Distributive law";
+goal IFOL.thy "P | (Q & R) <-> (P | Q) & (P | R)";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem ~~14";
+goal IFOL.thy "~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem ~~15";
+goal IFOL.thy "~~((P --> Q) <-> (~P | Q))";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem ~~16";
+goal IFOL.thy "~~((P-->Q) | (Q-->P))";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem ~~17";
+goal IFOL.thy
+ "~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))";
+by (Int.fast_tac 1);
+result();
+
+(*Dijkstra's "Golden Rule"*)
+goal IFOL.thy "(P&Q) <-> P <-> Q <-> (P|Q)";
+by (Int.fast_tac 1);
+result();
+
+
+writeln"U****Examples with quantifiers****";
+
+
+writeln"The converse is classical in the following implications...";
+
+goal IFOL.thy "(EX x.P(x)-->Q) --> (ALL x.P(x)) --> Q";
+by (Int.fast_tac 1);
+result();
+
+goal IFOL.thy "((ALL x.P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)";
+by (Int.fast_tac 1);
+result();
+
+goal IFOL.thy "((ALL x. ~P(x))-->Q) --> ~ (ALL x. ~ (P(x)|Q))";
+by (Int.fast_tac 1);
+result();
+
+goal IFOL.thy "(ALL x.P(x)) | Q --> (ALL x. P(x) | Q)";
+by (Int.fast_tac 1);
+result();
+
+goal IFOL.thy "(EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))";
+by (Int.fast_tac 1);
+result();
+
+
+
+
+writeln"The following are not constructively valid!";
+(*The attempt to prove them terminates quickly!*)
+
+goal IFOL.thy "((ALL x.P(x))-->Q) --> (EX x.P(x)-->Q)";
+by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
+getgoal 1;
+
+goal IFOL.thy "(P --> (EX x.Q(x))) --> (EX x. P-->Q(x))";
+by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
+getgoal 1;
+
+goal IFOL.thy "(ALL x. P(x) | Q) --> ((ALL x.P(x)) | Q)";
+by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
+getgoal 1;
+
+goal IFOL.thy "(ALL x. ~~P(x)) --> ~~(ALL x. P(x))";
+by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
+getgoal 1;
+
+(*Classically but not intuitionistically valid. Proved by a bug in 1986!*)
+goal IFOL.thy "EX x. Q(x) --> (ALL x. Q(x))";
+by (Int.fast_tac 1) handle ERROR => writeln"Failed, as expected";
+getgoal 1;
+
+
+writeln"Hard examples with quantifiers";
+
+(*The ones that have not been proved are not known to be valid!
+ Some will require quantifier duplication -- not currently available*)
+
+writeln"Problem ~~18";
+goal IFOL.thy "~~(EX y. ALL x. P(y)-->P(x))";
+(*NOT PROVED*)
+
+writeln"Problem ~~19";
+goal IFOL.thy "~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))";
+(*NOT PROVED*)
+
+writeln"Problem 20";
+goal IFOL.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 (Int.fast_tac 1);
+result();
+
+writeln"Problem 21";
+goal IFOL.thy "(EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))";
+(*NOT PROVED*)
+
+writeln"Problem 22";
+goal IFOL.thy "(ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem ~~23";
+goal IFOL.thy "~~ ((ALL x. P | Q(x)) <-> (P | (ALL x. Q(x))))";
+by (Int.best_tac 1);
+result();
+
+writeln"Problem 24";
+goal IFOL.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 (Int.fast_tac 1);
+result();
+
+writeln"Problem 25";
+goal IFOL.thy "(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))) \
+\ --> (EX x. Q(x)&P(x))";
+by (Int.best_tac 1);
+result();
+
+writeln"Problem ~~26";
+goal IFOL.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)))";
+(*NOT PROVED*)
+
+writeln"Problem 27";
+goal IFOL.thy "(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))) \
+\ --> (ALL x. M(x) --> ~L(x))";
+by (Int.fast_tac 1); (*44 secs*)
+result();
+
+writeln"Problem ~~28. AMENDED";
+goal IFOL.thy "(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))";
+by (Int.fast_tac 1); (*101 secs*)
+result();
+
+writeln"Problem 29. Essentially the same as Principia Mathematica *11.71";
+goal IFOL.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 (Int.fast_tac 1);
+result();
+
+writeln"Problem ~~30";
+goal IFOL.thy "(ALL x. (P(x) | Q(x)) --> ~ R(x)) & \
+\ (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \
+\ --> (ALL x. ~~S(x))";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem 31";
+goal IFOL.thy "~(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))";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem 32";
+goal IFOL.thy "(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))";
+by (Int.best_tac 1);
+result();
+
+writeln"Problem ~~33";
+goal IFOL.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 (Int.best_tac 1);
+result();
+
+
+writeln"Problem 36";
+goal IFOL.thy
+ "(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))) \
+\ --> (ALL x. EX y. H(x,y))";
+by (Int.fast_tac 1); (*35 secs*)
+result();
+
+writeln"Problem 37";
+goal IFOL.thy
+ "(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))) \
+\ --> ~~(ALL x. EX y. R(x,y))";
+(*NOT PROVED*)
+
+writeln"Problem 39";
+goal IFOL.thy "~ (EX x. ALL y. F(y,x) <-> ~F(y,y))";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem 40. AMENDED";
+goal IFOL.thy "(EX y. ALL x. F(x,y) <-> F(x,x)) --> \
+\ ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem 44";
+goal IFOL.thy "(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))";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem 48";
+goal IFOL.thy "(a=b | c=d) & (a=c | b=d) --> a=d | b=c";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem 51";
+goal IFOL.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 (Int.best_tac 1); (*60 seconds*)
+result();
+
+writeln"Problem 52";
+(*Almost the same as 51. *)
+goal IFOL.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 (Int.best_tac 1); (*60 seconds*)
+result();
+
+writeln"Problem 56";
+goal IFOL.thy
+ "(ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))";
+by (Int.fast_tac 1);
+result();
+
+writeln"Problem 57";
+goal IFOL.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 (Int.fast_tac 1);
+result();
+
+writeln"Problem 60";
+goal IFOL.thy
+ "ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))";
+by (Int.fast_tac 1);
+result();
+
+writeln"Reached end of file.";