--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/FOL/ex/cla.ML Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,484 @@
+(* Title: FOL/ex/cla
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1992 University of Cambridge
+
+Classical First-Order Logic
+*)
+
+writeln"File FOL/ex/cla.";
+
+open Cla; (*in case structure Int is open!*)
+
+goal FOL.thy "(P --> Q | R) --> (P-->Q) | (P-->R)";
+by (fast_tac FOL_cs 1);
+result();
+
+(*If and only if*)
+
+goal FOL.thy "(P<->Q) <-> (Q<->P)";
+by (fast_tac FOL_cs 1);
+result();
+
+goal FOL.thy "~ (P <-> ~P)";
+by (fast_tac FOL_cs 1);
+result();
+
+
+(*Sample problems from
+ F. J. Pelletier,
+ Seventy-Five Problems for Testing Automatic Theorem Provers,
+ J. Automated Reasoning 2 (1986), 191-216.
+ Errata, JAR 4 (1988), 236-236.
+
+The hardest problems -- judging by experience with several theorem provers,
+including matrix ones -- are 34 and 43.
+*)
+
+writeln"Pelletier's examples";
+(*1*)
+goal FOL.thy "(P-->Q) <-> (~Q --> ~P)";
+by (fast_tac FOL_cs 1);
+result();
+
+(*2*)
+goal FOL.thy "~ ~ P <-> P";
+by (fast_tac FOL_cs 1);
+result();
+
+(*3*)
+goal FOL.thy "~(P-->Q) --> (Q-->P)";
+by (fast_tac FOL_cs 1);
+result();
+
+(*4*)
+goal FOL.thy "(~P-->Q) <-> (~Q --> P)";
+by (fast_tac FOL_cs 1);
+result();
+
+(*5*)
+goal FOL.thy "((P|Q)-->(P|R)) --> (P|(Q-->R))";
+by (fast_tac FOL_cs 1);
+result();
+
+(*6*)
+goal FOL.thy "P | ~ P";
+by (fast_tac FOL_cs 1);
+result();
+
+(*7*)
+goal FOL.thy "P | ~ ~ ~ P";
+by (fast_tac FOL_cs 1);
+result();
+
+(*8. Peirce's law*)
+goal FOL.thy "((P-->Q) --> P) --> P";
+by (fast_tac FOL_cs 1);
+result();
+
+(*9*)
+goal FOL.thy "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
+by (fast_tac FOL_cs 1);
+result();
+
+(*10*)
+goal FOL.thy "(Q-->R) & (R-->P&Q) & (P-->Q|R) --> (P<->Q)";
+by (fast_tac FOL_cs 1);
+result();
+
+(*11. Proved in each direction (incorrectly, says Pelletier!!) *)
+goal FOL.thy "P<->P";
+by (fast_tac FOL_cs 1);
+result();
+
+(*12. "Dijkstra's law"*)
+goal FOL.thy "((P <-> Q) <-> R) <-> (P <-> (Q <-> R))";
+by (fast_tac FOL_cs 1);
+result();
+
+(*13. Distributive law*)
+goal FOL.thy "P | (Q & R) <-> (P | Q) & (P | R)";
+by (fast_tac FOL_cs 1);
+result();
+
+(*14*)
+goal FOL.thy "(P <-> Q) <-> ((Q | ~P) & (~Q|P))";
+by (fast_tac FOL_cs 1);
+result();
+
+(*15*)
+goal FOL.thy "(P --> Q) <-> (~P | Q)";
+by (fast_tac FOL_cs 1);
+result();
+
+(*16*)
+goal FOL.thy "(P-->Q) | (Q-->P)";
+by (fast_tac FOL_cs 1);
+result();
+
+(*17*)
+goal FOL.thy "((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))";
+by (fast_tac FOL_cs 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 FOL_cs 1);
+result();
+
+goal FOL.thy "(EX x. P-->Q(x)) <-> (P --> (EX x.Q(x)))";
+by (fast_tac FOL_cs 1);
+result();
+
+goal FOL.thy "(EX x.P(x)-->Q) <-> (ALL x.P(x)) --> Q";
+by (fast_tac FOL_cs 1);
+result();
+
+goal FOL.thy "(ALL x.P(x)) | Q <-> (ALL x. P(x) | Q)";
+by (fast_tac FOL_cs 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 (best_tac FOL_dup_cs 1);
+result();
+
+(*Needs double instantiation of the quantifier*)
+goal FOL.thy "EX x. P(x) --> P(a) & P(b)";
+by (best_tac FOL_dup_cs 1);
+result();
+
+goal FOL.thy "EX z. P(z) --> (ALL x. P(x))";
+by (best_tac FOL_dup_cs 1);
+result();
+
+(*from Vladimir Lifschitz, What Is the Inverse Method?, JAR 5 (1989), 1--23*)
+goal FOL.thy "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)";
+by (best_tac FOL_dup_cs 1);
+result();
+
+writeln"Problem 19";
+goal FOL.thy "EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x))";
+by (best_tac FOL_dup_cs 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 FOL_cs 1);
+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 FOL_cs 1);
+result();
+
+writeln"Problem 21";
+goal FOL.thy "(EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> (EX x. P<->Q(x))";
+by (best_tac FOL_dup_cs 1);
+result();
+
+writeln"Problem 22";
+goal FOL.thy "(ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))";
+by (fast_tac FOL_cs 1);
+result();
+
+writeln"Problem 23";
+goal FOL.thy "(ALL x. P | Q(x)) <-> (P | (ALL x. Q(x)))";
+by (best_tac FOL_cs 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 FOL_cs 1);
+(*slow*)
+result();
+
+writeln"Problem 25";
+goal FOL.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 (best_tac FOL_cs 1);
+(*slow*)
+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 FOL_cs 1);
+(*slow*)
+result();
+
+writeln"Problem 27";
+goal FOL.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 (fast_tac FOL_cs 1);
+(*slow*)
+result();
+
+writeln"Problem 28. AMENDED";
+goal FOL.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 (fast_tac FOL_cs 1);
+(*slow*)
+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 FOL_cs 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 FOL_cs 1);
+result();
+
+writeln"Problem 31";
+goal FOL.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 (fast_tac FOL_cs 1);
+result();
+
+writeln"Problem 32";
+goal FOL.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 (best_tac FOL_cs 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 FOL_cs 1);
+result();
+
+writeln"Problem 34 AMENDED (TWICE!!) NOT PROVED AUTOMATICALLY";
+(*Andrews's challenge*)
+goal FOL.thy "((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))))";
+by (safe_tac FOL_cs); (*22 secs*)
+by (TRYALL (fast_tac FOL_cs)); (*128 secs*)
+by (TRYALL (best_tac FOL_dup_cs)); (*77 secs*)
+result();
+
+writeln"Problem 35";
+goal FOL.thy "EX x y. P(x,y) --> (ALL u v. P(u,v))";
+by (best_tac FOL_dup_cs 1);
+(*6.1 secs*)
+result();
+
+writeln"Problem 36";
+goal FOL.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 (fast_tac FOL_cs 1);
+result();
+
+writeln"Problem 37";
+goal FOL.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))";
+by (fast_tac FOL_cs 1);
+(*slow...Poly/ML: 9.7 secs*)
+result();
+
+writeln"Problem 38. NOT PROVED";
+goal FOL.thy
+ "(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))) & \
+\ (~p(a) | ~(EX y. p(y) & r(x,y)) | \
+\ (EX z. EX w. p(z) & r(x,w) & r(w,z))))";
+
+writeln"Problem 39";
+goal FOL.thy "~ (EX x. ALL y. F(y,x) <-> ~F(y,y))";
+by (fast_tac FOL_cs 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 FOL_cs 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 (best_tac FOL_cs 1);
+result();
+
+writeln"Problem 42 NOT PROVED";
+goal FOL.thy "~ (EX y. ALL x. p(x,y) <-> ~ (EX z. p(x,z) & p(z,x)))";
+
+writeln"Problem 43 NOT PROVED AUTOMATICALLY";
+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)))";
+
+
+writeln"Problem 44";
+goal FOL.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 (fast_tac FOL_cs 1);
+result();
+
+writeln"Problem 45";
+goal FOL.thy "(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)) \
+\ & (ALL y. g(y) & h(x,y) --> j(x,y))) \
+\ --> (EX x. f(x) & ~ (EX y. g(y) & h(x,y)))";
+by (best_tac FOL_cs 1);
+(*41.5 secs*)
+result();
+
+
+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";
+by (fast_tac FOL_cs 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) \
+\ --> (ALL u::'a.P(u))";
+by (safe_tac FOL_cs);
+by (res_inst_tac [("x","a")] allE 1);
+ba 1;
+by (res_inst_tac [("x","b")] allE 1);
+ba 1;
+by (fast_tac FOL_cs 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 (best_tac FOL_dup_cs 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 FOL_cs 1);
+result();
+
+writeln"Problem 52";
+(*Almost the same as 51. *)
+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 FOL_cs 1);
+result();
+
+writeln"Problem 55";
+
+(*Original, equational version by Len Schubert, via Pelletier *** NOT PROVED
+goal FOL.thy
+ "(EX x. lives(x) & killed(x,agatha)) & \
+\ lives(agatha) & lives(butler) & lives(charles) & \
+\ (ALL x. lives(x) --> x=agatha | x=butler | x=charles) & \
+\ (ALL x y. killed(x,y) --> hates(x,y)) & \
+\ (ALL x y. killed(x,y) --> ~richer(x,y)) & \
+\ (ALL x. hates(agatha,x) --> ~hates(charles,x)) & \
+\ (ALL x. ~ x=butler --> hates(agatha,x)) & \
+\ (ALL x. ~richer(x,agatha) --> hates(butler,x)) & \
+\ (ALL x. hates(agatha,x) --> hates(butler,x)) & \
+\ (ALL x. EX y. ~hates(x,y)) & \
+\ ~ agatha=butler --> \
+\ killed(?who,agatha)";
+by (safe_tac FOL_cs);
+by (dres_inst_tac [("x1","x")] (spec RS mp) 1);
+ba 1;
+be (spec RS exE) 1;
+by (REPEAT (etac allE 1));
+by (fast_tac FOL_cs 1);
+result();
+****)
+
+(*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) & \
+\ (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)) & \
+\ (hates(agatha,agatha) & hates(agatha,charles)) & \
+\ (ALL x. lives(x) & ~richer(x,agatha) --> hates(butler,x)) & \
+\ (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 FOL_cs 1);
+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 FOL_cs 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 FOL_cs 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)))";
+val f_cong = read_instantiate [("t","f")] subst_context;
+by (fast_tac (FOL_cs addIs [f_cong]) 1);
+result();
+
+writeln"Problem 59";
+goal FOL.thy "(ALL x. P(x) <-> ~P(f(x))) --> (EX x. P(x) & ~P(f(x)))";
+by (best_tac FOL_dup_cs 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 FOL_cs 1);
+result();
+
+
+writeln"Reached end of file.";
+
+(*Thu Jul 23 1992: loaded in 467.1s using iffE*)