--- a/src/FOL/ex/int.ML Wed Jun 07 23:34:37 2006 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,478 +0,0 @@
-(* Title: FOL/ex/int.ML
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1991 University of Cambridge
-
-Intuitionistic First-Order Logic.
-
-Single-step commands:
-by (IntPr.step_tac 1);
-by (biresolve_tac safe_brls 1);
-by (biresolve_tac haz_brls 1);
-by (assume_tac 1);
-by (IntPr.safe_tac 1);
-by (IntPr.mp_tac 1);
-by (IntPr.fast_tac 1);
-*)
-
-
-(*Metatheorem (for PROPOSITIONAL formulae...):
- P is classically provable iff ~~P is intuitionistically provable.
- Therefore ~P is classically provable iff it is intuitionistically provable.
-
-Proof: Let Q be the conjuction of the propositions A|~A, one for each atom A
-in P. Now ~~Q is intuitionistically provable because ~~(A|~A) is and because
-~~ distributes over &. If P is provable classically, then clearly Q-->P is
-provable intuitionistically, so ~~(Q-->P) is also provable intuitionistically.
-The latter is intuitionistically equivalent to ~~Q-->~~P, hence to ~~P, since
-~~Q is intuitionistically provable. Finally, if P is a negation then ~~P is
-intuitionstically equivalent to P. [Andy Pitts] *)
-
-Goal "~~(P&Q) <-> ~~P & ~~Q";
-by (IntPr.fast_tac 1);
-qed "";
-
-Goal "~~ ((~P --> Q) --> (~P --> ~Q) --> P)";
-by (IntPr.fast_tac 1);
-qed "";
-
-(* ~~ does NOT distribute over | *)
-
-Goal "~~(P-->Q) <-> (~~P --> ~~Q)";
-by (IntPr.fast_tac 1);
-qed "";
-
-Goal "~~~P <-> ~P";
-by (IntPr.fast_tac 1);
-qed "";
-
-Goal "~~((P --> Q | R) --> (P-->Q) | (P-->R))";
-by (IntPr.fast_tac 1);
-qed "";
-
-Goal "(P<->Q) <-> (Q<->P)";
-by (IntPr.fast_tac 1);
-qed "";
-
-Goal "((P --> (Q | (Q-->R))) --> R) --> R";
-by (IntPr.fast_tac 1);
-qed "";
-
-Goal "(((G-->A) --> J) --> D --> E) --> (((H-->B)-->I)-->C-->J) \
-\ --> (A-->H) --> F --> G --> (((C-->B)-->I)-->D)-->(A-->C) \
-\ --> (((F-->A)-->B) --> I) --> E";
-by (IntPr.fast_tac 1);
-qed "";
-
-
-writeln"Lemmas for the propositional double-negation translation";
-
-Goal "P --> ~~P";
-by (IntPr.fast_tac 1);
-qed "";
-
-Goal "~~(~~P --> P)";
-by (IntPr.fast_tac 1);
-qed "";
-
-Goal "~~P & ~~(P --> Q) --> ~~Q";
-by (IntPr.fast_tac 1);
-qed "";
-
-
-writeln"The following are classically but not constructively valid.";
-
-(*The attempt to prove them terminates quickly!*)
-Goal "((P-->Q) --> P) --> P";
-by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected";
-(*Check that subgoals remain: proof failed.*)
-getgoal 1;
-
-Goal "(P&Q-->R) --> (P-->R) | (Q-->R)";
-by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected";
-getgoal 1;
-
-
-writeln"de Bruijn formulae";
-
-(*de Bruijn formula with three predicates*)
-Goal "((P<->Q) --> P&Q&R) & \
-\ ((Q<->R) --> P&Q&R) & \
-\ ((R<->P) --> P&Q&R) --> P&Q&R";
-by (IntPr.fast_tac 1);
-qed "";
-
-
-(*de Bruijn formula with five predicates*)
-Goal "((P<->Q) --> P&Q&R&S&T) & \
-\ ((Q<->R) --> P&Q&R&S&T) & \
-\ ((R<->S) --> P&Q&R&S&T) & \
-\ ((S<->T) --> P&Q&R&S&T) & \
-\ ((T<->P) --> P&Q&R&S&T) --> P&Q&R&S&T";
-by (IntPr.fast_tac 1);
-qed "";
-
-
-(*** Problems from of Sahlin, Franzen and Haridi,
- An Intuitionistic Predicate Logic Theorem Prover.
- J. Logic and Comp. 2 (5), October 1992, 619-656.
-***)
-
-(*Problem 1.1*)
-Goal "(ALL x. EX y. ALL z. p(x) & q(y) & r(z)) <-> \
-\ (ALL z. EX y. ALL x. p(x) & q(y) & r(z))";
-(*
-by (IntPr.best_dup_tac 1); (*65 seconds on a Pentium III! Is it worth it?*)
-*)
-
-(*Problem 3.1*)
-Goal "~ (EX x. ALL y. mem(y,x) <-> ~ mem(x,x))";
-by (IntPr.fast_tac 1);
-qed "";
-
-(*Problem 4.1: hopeless!*)
-Goal "(ALL x. p(x) --> p(h(x)) | p(g(x))) & (EX x. p(x)) & (ALL x. ~p(h(x))) \
-\ --> (EX x. p(g(g(g(g(g(x)))))))";
-
-
-writeln"Intuitionistic FOL: propositional problems based on Pelletier.";
-
-writeln"Problem ~~1";
-Goal "~~((P-->Q) <-> (~Q --> ~P))";
-by (IntPr.fast_tac 1);
-qed "";
-
-
-writeln"Problem ~~2";
-Goal "~~(~~P <-> P)";
-by (IntPr.fast_tac 1);
-qed "";
-(*1 secs*)
-
-
-writeln"Problem 3";
-Goal "~(P-->Q) --> (Q-->P)";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem ~~4";
-Goal "~~((~P-->Q) <-> (~Q --> P))";
-by (IntPr.fast_tac 1);
-qed "";
-(*9 secs*)
-
-writeln"Problem ~~5";
-Goal "~~((P|Q-->P|R) --> P|(Q-->R))";
-by (IntPr.fast_tac 1);
-qed "";
-(*10 secs*)
-
-
-writeln"Problem ~~6";
-Goal "~~(P | ~P)";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem ~~7";
-Goal "~~(P | ~~~P)";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem ~~8. Peirce's law";
-Goal "~~(((P-->Q) --> P) --> P)";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 9";
-Goal "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
-by (IntPr.fast_tac 1);
-qed "";
-(*9 secs*)
-
-
-writeln"Problem 10";
-Goal "(Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"11. Proved in each direction (incorrectly, says Pelletier!!) ";
-Goal "P<->P";
-by (IntPr.fast_tac 1);
-
-writeln"Problem ~~12. Dijkstra's law ";
-Goal "~~(((P <-> Q) <-> R) <-> (P <-> (Q <-> R)))";
-by (IntPr.fast_tac 1);
-qed "";
-
-Goal "((P <-> Q) <-> R) --> ~~(P <-> (Q <-> R))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 13. Distributive law";
-Goal "P | (Q & R) <-> (P | Q) & (P | R)";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem ~~14";
-Goal "~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem ~~15";
-Goal "~~((P --> Q) <-> (~P | Q))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem ~~16";
-Goal "~~((P-->Q) | (Q-->P))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem ~~17";
-Goal
- "~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))";
-by (IntPr.fast_tac 1);
-qed "";
-
-(*Dijkstra's "Golden Rule"*)
-Goal "(P&Q) <-> P <-> Q <-> (P|Q)";
-by (IntPr.fast_tac 1);
-qed "";
-
-
-writeln"****Examples with quantifiers****";
-
-
-writeln"The converse is classical in the following implications...";
-
-Goal "(EX x. P(x)-->Q) --> (ALL x. P(x)) --> Q";
-by (IntPr.fast_tac 1);
-qed "";
-
-Goal "((ALL x. P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)";
-by (IntPr.fast_tac 1);
-qed "";
-
-Goal "((ALL x. ~P(x))-->Q) --> ~ (ALL x. ~ (P(x)|Q))";
-by (IntPr.fast_tac 1);
-qed "";
-
-Goal "(ALL x. P(x)) | Q --> (ALL x. P(x) | Q)";
-by (IntPr.fast_tac 1);
-qed "";
-
-Goal "(EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))";
-by (IntPr.fast_tac 1);
-qed "";
-
-
-
-
-writeln"The following are not constructively valid!";
-(*The attempt to prove them terminates quickly!*)
-
-Goal "((ALL x. P(x))-->Q) --> (EX x. P(x)-->Q)";
-by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected";
-getgoal 1;
-
-Goal "(P --> (EX x. Q(x))) --> (EX x. P-->Q(x))";
-by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected";
-getgoal 1;
-
-Goal "(ALL x. P(x) | Q) --> ((ALL x. P(x)) | Q)";
-by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected";
-getgoal 1;
-
-Goal "(ALL x. ~~P(x)) --> ~~(ALL x. P(x))";
-by (IntPr.fast_tac 1) handle ERROR _ => writeln"Failed, as expected";
-getgoal 1;
-
-(*Classically but not intuitionistically valid. Proved by a bug in 1986!*)
-Goal "EX x. Q(x) --> (ALL x. Q(x))";
-by (IntPr.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 "~~(EX y. ALL x. P(y)-->P(x))";
-(*NOT PROVED*)
-
-writeln"Problem ~~19";
-Goal "~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))";
-(*NOT PROVED*)
-
-writeln"Problem 20";
-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 (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 21";
-Goal "(EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))";
-(*NOT PROVED; needs quantifier duplication*)
-
-writeln"Problem 22";
-Goal "(ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem ~~23";
-Goal "~~ ((ALL x. P | Q(x)) <-> (P | (ALL x. Q(x))))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 24";
-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))";
-(*Not clear why fast_tac, best_tac, ASTAR and ITER_DEEPEN all take forever*)
-by IntPr.safe_tac;
-by (etac impE 1);
-by (IntPr.fast_tac 1);
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 25";
-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))) \
-\ --> (EX x. Q(x)&P(x))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem ~~26";
-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)))";
-(*NOT PROVED*)
-
-writeln"Problem 27";
-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))) \
-\ --> (ALL x. M(x) --> ~L(x))";
-by (IntPr.fast_tac 1); (*21 secs*)
-qed "";
-
-writeln"Problem ~~28. AMENDED";
-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))";
-by (IntPr.fast_tac 1); (*48 secs*)
-qed "";
-
-writeln"Problem 29. Essentially the same as Principia Mathematica *11.71";
-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 (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem ~~30";
-Goal "(ALL x. (P(x) | Q(x)) --> ~ R(x)) & \
-\ (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) \
-\ --> (ALL x. ~~S(x))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 31";
-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))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 32";
-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))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem ~~33";
-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 (IntPr.best_tac 1); (*1.67s*)
-qed "";
-
-
-writeln"Problem 36";
-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))) \
-\ --> (ALL x. EX y. H(x,y))";
-by (IntPr.fast_tac 1); (*5 secs*)
-qed "";
-
-writeln"Problem 37";
-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))) \
-\ --> ~~(ALL x. EX y. R(x,y))";
-(*NOT PROVED*)
-
-writeln"Problem 39";
-Goal "~ (EX x. ALL y. F(y,x) <-> ~F(y,y))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 40. AMENDED";
-Goal "(EX y. ALL x. F(x,y) <-> F(x,x)) --> \
-\ ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 44";
-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))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 48";
-Goal "(a=b | c=d) & (a=c | b=d) --> a=d | b=c";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 51";
-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 (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 52";
-(*Almost the same as 51. *)
-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 (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 56";
-Goal "(ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))";
-by (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 57";
-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 (IntPr.fast_tac 1);
-qed "";
-
-writeln"Problem 60";
-Goal "ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))";
-by (IntPr.fast_tac 1);
-qed "";